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In mathematics , symmetrization is a process that converts any function in n {\displaystyle n} variables to a symmetric function in n {\displaystyle n} variables.
Similarly, antisymmetrization converts any function in n {\displaystyle n} variables into an antisymmetric function.
Let S {\displaystyle S} be a set and A {\displaystyle A} be an additive abelian group . A map α : S × S → A {\displaystyle \alpha :S\times S\to A} is called a symmetric map if α ( s , t ) = α ( t , s ) for all s , t ∈ S . {\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.} It is called an antisymmetric map if instead α ( s , t ) = − α ( t , s ) for all s , t ∈ S . {\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.}
The symmetrization of a map α : S × S → A {\displaystyle \alpha :S\times S\to A} is the map ( x , y ) ↦ α ( x , y ) + α ( y , x ) . {\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).} Similarly, the antisymmetrization or skew-symmetrization of a map α : S × S → A {\displaystyle \alpha :S\times S\to A} is the map ( x , y ) ↦ α ( x , y ) − α ( y , x ) . {\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).}
The sum of the symmetrization and the antisymmetrization of a map α {\displaystyle \alpha } is 2 α . {\displaystyle 2\alpha .} Thus, away from 2 , meaning if 2 is invertible , such as for the real numbers , one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the antisymmetrization of a symmetric map is zero, while the antisymmetrization of an anti-symmetric map is its double.
The symmetrization and antisymmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form.
At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form. For instance, over the integers , the associated symmetric form (over the rationals ) may take half-integer values, while over Z / 2 Z , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} a function is skew-symmetric if and only if it is symmetric (as 1 = − 1 {\displaystyle 1=-1} ).
This leads to the notion of ε-quadratic forms and ε-symmetric forms.
In terms of representation theory :
As the symmetric group of order two equals the cyclic group of order two ( S 2 = C 2 {\displaystyle \mathrm {S} _{2}=\mathrm {C} _{2}} ), this corresponds to the discrete Fourier transform of order two.
More generally, given a function in n {\displaystyle n} variables, one can symmetrize by taking the sum over all n ! {\displaystyle n!} permutations of the variables, [ 1 ] or antisymmetrize by taking the sum over all n ! / 2 {\displaystyle n!/2} even permutations and subtracting the sum over all n ! / 2 {\displaystyle n!/2} odd permutations (except that when n ≤ 1 , {\displaystyle n\leq 1,} the only permutation is even).
Here symmetrizing a symmetric function multiplies by n ! {\displaystyle n!} – thus if n ! {\displaystyle n!} is invertible, such as when working over a field of characteristic 0 {\displaystyle 0} or p > n , {\displaystyle p>n,} then these yield projections when divided by n ! . {\displaystyle n!.}
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for n > 2 {\displaystyle n>2} there are others – see representation theory of the symmetric group and symmetric polynomials .
Given a function in k {\displaystyle k} variables, one can obtain a symmetric function in n {\displaystyle n} variables by taking the sum over k {\displaystyle k} -element subsets of the variables. In statistics, this is referred to as bootstrapping , and the associated statistics are called U-statistics . | https://en.wikipedia.org/wiki/Symmetrization |
Symmetry (from Ancient Greek συμμετρία ( summetría ) ' agreement in dimensions, due proportion, arrangement ' ) [ 1 ] in everyday life refers to a sense of harmonious and beautiful proportion and balance. [ 2 ] [ 3 ] [ a ] In mathematics , the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations , such as translation , reflection , rotation , or scaling . Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time ; as a spatial relationship ; through geometric transformations ; through other kinds of functional transformations; and as an aspect of abstract objects , including theoretic models , language , and music . [ 4 ] [ b ]
This article describes symmetry from three perspectives: in mathematics , including geometry , the most familiar type of symmetry for many people; in science and nature ; and in the arts, covering architecture , art , and music.
The opposite of symmetry is asymmetry , which refers to the absence of symmetry.
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. [ 5 ] This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
A dyadic relation R = S × S is symmetric if for all elements a , b in S , whenever it is true that Rab , it is also true that Rba . [ 13 ] Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.
In propositional logic, symmetric binary logical connectives include and (∧, or &), or (∨, or |) and if and only if (↔), while the connective if (→) is not symmetric. [ 14 ] Other symmetric logical connectives include nand (not-and, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽).
Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is symmetric with respect to a given mathematical operation , if, when applied to the object, this operation preserves some property of the object. [ 15 ] The set of operations that preserve a given property of the object form a group .
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions in calculus , symmetric groups in abstract algebra , symmetric matrices in linear algebra , and Galois groups in Galois theory . In statistics , symmetry also manifests as symmetric probability distributions , and as skewness —the asymmetry of distributions. [ 16 ]
Symmetry in physics has been generalized to mean invariance —that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations . [ 17 ] This concept has become one of the most powerful tools of theoretical physics , as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that "it is only slightly overstating the case to say that physics is the study of symmetry." [ 18 ] See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language); [ 19 ] and also, Wigner's classification , which says that the symmetries of the laws of physics determine the properties of the particles found in nature. [ 20 ]
Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime ; internal symmetries of particles; and supersymmetry of physical theories.
In biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals , including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. [ 21 ] Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialized with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric. [ 22 ]
Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry , which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms , the group that includes starfish , sea urchins , and sea lilies . [ 23 ]
In biology, the notion of symmetry is also used as in physics, that is to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. [ 24 ] [ 25 ]
Symmetry is important to chemistry because it undergirds essentially all specific interactions between molecules in nature (i.e., via the interaction of natural and human-made chiral molecules with inherently chiral biological systems). The control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects . A rigorous understanding of symmetry explains fundamental observations in quantum chemistry , and in the applied areas of spectroscopy and crystallography . The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory . [ 26 ]
For a human observer, some symmetry types are more salient than others, in particular the most salient is a reflection with a vertical axis, like that present in the human face. Ernst Mach made this observation in his book "The analysis of sensations" (1897), [ 27 ] and this implies that perception of symmetry is not a general response to all types of regularities. Both behavioural and neurophysiological studies have confirmed the special sensitivity to reflection symmetry in humans and also in other animals. [ 28 ] Early studies within the Gestalt tradition suggested that bilateral symmetry was one of the key factors in perceptual grouping . This is known as the Law of Symmetry . The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry is faster when this is a property of a single object. [ 29 ] Studies of human perception and psychophysics have shown that detection of symmetry is fast, efficient and robust to perturbations. For example, symmetry can be detected with presentations between 100 and 150 milliseconds. [ 30 ]
More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al. [ 31 ] used functional magnetic resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots. A strong activity was present in extrastriate regions of the occipital cortex but not in the primary visual cortex. The extrastriate regions included V3A, V4, V7, and the lateral occipital complex (LOC). Electrophysiological studies have found a late posterior negativity that originates from the same areas. [ 32 ] In general, a large part of the visual system seems to be involved in processing visual symmetry, and these areas involve similar networks to those responsible for detecting and recognising objects. [ 33 ]
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity , empathy , sympathy , apology , dialogue , respect, justice , and revenge . Reflective equilibrium is the balance that may be attained through deliberative mutual adjustment among general principles and specific judgments . [ 34 ] Symmetrical interactions send the moral message "we are all the same" while asymmetrical interactions may send the message "I am special; better than you." Peer relationships, such as can be governed by the Golden Rule , are based on symmetry, whereas power relationships are based on asymmetry. [ 35 ] Symmetrical relationships can to some degree be maintained by simple ( game theory ) strategies seen in symmetric games such as tit for tat . [ 36 ]
There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts. [ 37 ]
Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic cathedrals and The White House , through the layout of the individual floor plans , and down to the design of individual building elements such as tile mosaics . Islamic buildings such as the Taj Mahal and the Lotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation. [ 38 ] [ 39 ] Moorish buildings like the Alhambra are ornamented with complex patterns made using translational and reflection symmetries as well as rotations. [ 40 ]
It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures"; [ 41 ] Modernist architecture , starting with International style , relies instead on "wings and balance of masses". [ 41 ]
Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives.
Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese , for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design. [ 42 ]
A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a rectangle —that is, motifs that are reflected across both the horizontal and vertical axes (see Klein four-group § Geometry ). [ 43 ] [ 44 ]
As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. [ 45 ]
Symmetries appear in the design of objects of all kinds. Examples include beadwork , furniture , sand paintings , knotwork , masks , and musical instruments . Symmetries are central to the art of M.C. Escher and the many applications of tessellation in art and craft forms such as wallpaper , ceramic tilework such as in Islamic geometric decoration , batik , ikat , carpet-making, and many kinds of textile and embroidery patterns. [ 46 ]
Symmetry is also used in designing logos. [ 47 ] By creating a logo on a grid and using the theory of symmetry, designers can organize their work, create a symmetric or asymmetrical design, determine the space between letters, determine how much negative space is required in the design, and how to accentuate parts of the logo to make it stand out.
Symmetry is not restricted to the visual arts. Its role in the history of music touches many aspects of the creation and perception of music.
Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve Reich , Béla Bartók , and James Tenney . In classical music, Johann Sebastian Bach used the symmetry concepts of permutation and invariance. [ 48 ]
Symmetry is also an important consideration in the formation of scales and chords , traditional or tonal music being made up of non-symmetrical groups of pitches , such as the diatonic scale or the major chord . Symmetrical scales or chords, such as the whole tone scale , augmented chord , or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality . However, composers such as Alban Berg , Béla Bartók , and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non- tonal tonal centers . [ 49 ] George Perle explains that "C–E, D–F♯, [and] Eb–G, are different instances of the same interval … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:" [ 49 ]
Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0). [ 49 ]
Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin , Edgard Varèse , and the Vienna school. At the same time, these progressions signal the end of tonality. [ 49 ] [ 50 ]
The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet , Op. 3 (1910). [ 50 ]
Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm .
The relationship of symmetry to aesthetics is complex. Humans find bilateral symmetry in faces physically attractive; [ 51 ] it indicates health and genetic fitness. [ 52 ] [ 53 ] Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Rudolf Arnheim suggested that people prefer shapes that have some symmetry, and enough complexity to make them interesting. [ 54 ]
Symmetry can be found in various forms in literature , a simple example being the palindrome where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern of Beowulf . [ 55 ] | https://en.wikipedia.org/wiki/Symmetry |
Symmetry-adapted perturbation theory or SAPT [ 1 ] [ 2 ] is a methodology in electronic structure theory developed to describe non-covalent interactions between atoms and/or molecules . SAPT is a member of the family of methods known as energy decomposition analysis (EDA) . Most EDA methods decompose a total interaction energy that is computed via a supermolecular approach, such that:
Δ E i n t = E A B − E A − E B {\displaystyle \Delta E_{\rm {int}}=E_{\rm {AB}}-E_{\rm {A}}-E_{\rm {B}}}
where Δ E i n t {\textstyle \Delta E_{\rm {int}}} is the total interaction energy obtained via subtracting isolated monomer energies E A {\textstyle E_{\rm {A}}} and E B {\displaystyle E_{\rm {B}}} from the dimer energy E A B {\textstyle E_{\rm {AB}}} . A key deficiency of the supermolecular interaction energy is that it is susceptible to basis set superposition error (BSSE).
The major difference between SAPT and supermolecular EDA methods is that, as the name suggests, SAPT computes the interaction energy directly via a perturbative approach. One consequence of capturing the total interaction energy as a perturbation to the total system energy rather than using the subtractive supermolecular method outlined above, is that the interaction energy is made free of BSSE in a natural way.
Being a perturbation expansion, SAPT also provides insight into the contributing components to the interaction energy. The lowest-order expansion at which all interaction energy components are obtained is second-order in the intermolecular perturbation. The simplest such SAPT approach is called SAPT0 because it neglects intramolecular correlation effects (i.e., it is based on Hartree–Fock densities). SAPT0 captures the classical electrostatic interaction of two charge densities and exchange (or Pauli repulsion ) at first-order, and at second-order the terms for electrostatic induction (the polarization of the molecular orbitals in the electric field of the interacting atom/molecule) and dispersion (see London dispersion ) appear, along with their exchange counterparts.
E i n t S A P T 0 = E e l s t ( 1 ) + E e x c h ( 1 ) + E i n d ( 2 ) + E exch-ind ( 2 ) + E d i s p ( 2 ) + E exch-disp ( 2 ) {\displaystyle E_{\rm {int}}^{\rm {SAPT0}}=E_{\rm {elst}}^{(1)}+E_{\rm {exch}}^{(1)}+E_{\rm {ind}}^{(2)}+E_{\text{exch-ind}}^{(2)}+E_{\rm {disp}}^{(2)}+E_{\text{exch-disp}}^{(2)}}
Higher terms in the perturbation series can be accounted for using many-body perturbation theory or coupled-cluster approaches. Alternatively, density functional theory variants of SAPT have been formulated. The higher-level SAPT methods approach supermolecular coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] in accuracy. [ 3 ]
This molecular physics –related article is a stub . You can help Wikipedia by expanding it .
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it .
This quantum mechanics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symmetry-adapted_perturbation_theory |
Symmetry-protected topological (SPT) order [ 1 ] [ 2 ] is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap.
To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points). [ 1 ] The SPT order has the following defining properties:
(a) distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry . (b) however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation .
The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.
Using the notion of quantum entanglement , we can say that SPT states are short-range entangled states with a symmetry (by contrast: for long-range entanglement see topological order , which is not related to the famous EPR paradox ). Since short-range entangled states have only trivial topological orders we may also refer the SPT order as Symmetry Protected "Trivial" order.
SPT states are short-range entangled while topologically ordered states are long-range entangled.
Both intrinsic topological order , and also SPT order, can sometimes have protected gapless boundary excitations . The difference is subtle: the gapless boundary excitations in intrinsic topological order can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations that do not break the symmetry . So the gapless boundary excitations in intrinsic topological order are topologically protected, while the gapless boundary excitations in SPT order are symmetry protected . [ 9 ]
We also know that an intrinsic topological order has emergent fractional charge , emergent fractional statistics , and emergent gauge theory . In contrast, a SPT order has no emergent fractional charge / fractional statistics for finite-energy excitations, nor emergent gauge theory (due to its short-range entanglement). Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by modifying the Hamiltonian.
The first example of SPT order is the Haldane phase of odd-integer spin chain. [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] It is a SPT phase protected by SO(3) spin rotation symmetry. [ 1 ] Note that Haldane phases of even-integer-spin chain do not have SPT order.
A more well known example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry .
On the other hand, fractional quantum Hall states are not SPT states. They are states with (intrinsic) topological order and long-range entanglements.
Using the notion of quantum entanglement , one obtains the following general picture of gapped
phases at zero temperature. All gapped zero-temperature phases can be divided into two classes: long-range entangled phases ( ie phases with intrinsic topological order ) and short-range entangled phases ( ie phases with no intrinsic topological order ). All short-range entangled phases can be further divided into three classes: symmetry-breaking phases, SPT phases, and their mix (symmetry breaking order and SPT order can appear together).
It is well known that symmetry-breaking orders are described by group theory . For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by group cohomology theory: [ 15 ] [ 16 ] those (d+1)D SPT states with symmetry G are labeled by the elements in group cohomology class H d + 1 [ G , U ( 1 ) ] {\displaystyle H^{d+1}[G,U(1)]} .
For other (d+1)D SPT states [ 17 ] [ 18 ] [ 19 ] [ 20 ] with mixed gauge-gravity anomalous boundary, they can be described by ⊕ k = 1 d H k [ G , i T O d + 1 − k ] {\displaystyle \oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]} , [ 21 ] where i T O d + 1 {\displaystyle iTO^{d+1}} is the Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).
From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected by U(1) and time-reversal symmetry) and bosonic topological superconductors (the SPT states protected by time-reversal symmetry), as well as many other new SPT states protected by other symmetries.
A list of bosonic SPT states from group cohomology H d + 1 [ G , U ( 1 ) ] ⊕ k = 1 d H k [ G , i T O d + 1 − k ] {\displaystyle H^{d+1}[G,U(1)]\oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]} ( Z 2 T {\displaystyle Z_{2}^{T}} = time-reversal-symmetry group)
The phases before "+" come from H d + 1 [ G , U ( 1 ) ] {\displaystyle H^{d+1}[G,U(1)]} . The phases after "+" come from ⊕ k = 1 d H k [ G , i T O d + 1 − k ] {\displaystyle \oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]} .
Just like group theory can give us 230 crystal structures in 3+1D, group cohomology theory can give us various SPT phases in any dimensions with any on-site symmetry groups.
On the other hand, the fermionic SPT orders are described by group super-cohomology theory. [ 22 ] So the group (super-)cohomology theory allows us to construct many
SPT orders even for interacting systems, which include interacting topological insulator/superconductor.
Using the notions of quantum entanglement and SPT order, one can obtain
a complete classification of all 1D gapped quantum phases.
First, it is shown that there is no (intrinsic) topological order in 1D ( ie all 1D gapped states
are short-range entangled). [ 23 ] Thus, if the Hamiltonians have no symmetry, all their 1D gapped quantum states belong to one phase—the phase of trivial product states.
On the other hand, if the Hamiltonians do have a symmetry, their 1D gapped quantum states
are either symmetry-breaking phases, SPT phases, and their mix.
Such an understanding allows one to classify all 1D gapped quantum phases: [ 15 ] [ 24 ] [ 25 ] [ 26 ] [ 27 ] All 1D gapped phases are classified by
the following three mathematical objects: ( G H , G Ψ , H 2 [ G Ψ , U ( 1 ) ] ) {\displaystyle (G_{H},G_{\Psi },H^{2}[G_{\Psi },U(1)])} , where G H {\displaystyle G_{H}} is the symmetry group of the Hamiltonian, G Ψ {\displaystyle G_{\Psi }} the symmetry group of the ground states, and H 2 [ G Ψ , U ( 1 ) ] {\displaystyle H^{2}[G_{\Psi },U(1)]} the second group cohomology class of G Ψ {\displaystyle G_{\Psi }} . (Note that H 2 [ G , U ( 1 ) ] {\displaystyle H^{2}[G,U(1)]} classifies the projective representations of G {\displaystyle G} .) If there is no symmetry breaking ( ie G Ψ = G H {\displaystyle G_{\Psi }=G_{H}} ), the 1D gapped phases are classified by the projective representations of symmetry group G H {\displaystyle G_{H}} . | https://en.wikipedia.org/wiki/Symmetry-protected_topological_order |
In geometry , an object has symmetry if there is an operation or transformation (such as translation , scaling , rotation or reflection ) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [ 1 ] Thus, a symmetry can be thought of as an immunity to change. [ 2 ] For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry . If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry ; [ 3 ] it is also possible for a figure/object to have more than one line of symmetry. [ 4 ]
The types of symmetries that are possible for a geometric object depend on the set of geometric transforms available, and on what object properties should remain unchanged after a transformation. Because the composition of two transforms is also a transform and every transform has, by definition, an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group , the symmetry group of the object. [ 5 ]
The most common group of transforms applied to objects are termed the Euclidean group of " isometries ", which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces ). These isometries consist of reflections , rotations , translations , and combinations of these basic operations. [ 6 ] Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation. [ 7 ] A geometric object is typically symmetric only under a subset or " subgroup " of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups, and by the types of object invariance that are possible in geometry.
By the Cartan–Dieudonné theorem , an orthogonal transformation in n -dimensional space can be represented by the composition of at most n reflections.
Reflectional symmetry, linear symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. [ 8 ]
In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions, there is an axis of symmetry (a.k.a., line of symmetry), and in three dimensions there is a plane of symmetry. [ 3 ] [ 9 ] An object or figure for which every point has a one-to-one mapping onto another, equidistant from and on opposite sides of a common plane is called mirror symmetric (for more, see mirror image ).
The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical as mirror images of each other. For example. a square has four axes of symmetry, because there are four different ways to fold it and have the edges match each other. Another example would be that of a circle , which has infinitely many axes of symmetry passing through its center for the same reason. [ 10 ]
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. Thus one can describe this phenomenon unambiguously by saying that "T has a vertical symmetry axis", or that "T has left-right symmetry".
The triangles with reflection symmetry are isosceles , the quadrilaterals with this symmetry are kites and isosceles trapezoids . [ 11 ]
For each line or plane of reflection, the symmetry group is isomorphic with C s (see point groups in three dimensions for more), one of the three types of order two ( involutions ), hence algebraically isomorphic to C 2 . The fundamental domain is a half-plane or half-space . [ 12 ]
Reflection symmetry can be generalized to other isometries of m -dimensional space which are involutions , such as
in a certain system of Cartesian coordinates . This reflects the space along an ( m − k ) -dimensional affine subspace . [ 13 ] If k = m , then such a transformation is known as a point reflection , or an inversion through a point . On the plane ( m = 2), a point reflection is the same as a half- turn (180°) rotation; see below. Antipodal symmetry is an alternative name for a point reflection symmetry through the origin. [ 14 ]
Such a "reflection" preserves orientation if and only if k is an even number. [ 15 ] This implies that for m = 3 (as well as for other odd m ), a point reflection changes the orientation of the space, like a mirror-image symmetry. That explains why in physics, the term P- symmetry (P stands for parity ) is used for both point reflection and mirror symmetry. Since a point reflection in three dimensions changes a left-handed coordinate system into a right-handed coordinate system , symmetry under a point reflection is also called a left-right symmetry. [ 16 ]
Rotational symmetry is symmetry with respect to some or all rotations in m -dimensional Euclidean space. Rotations are direct isometries , which are isometries that preserve orientation . [ 17 ] Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean group E + ( m ) .
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations (because translations are compositions of rotations about distinct points), [ 18 ] and the symmetry group is the whole E + ( m ). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
For symmetry with respect to rotations about a point, one can take that point as origin. These rotations form the special orthogonal group SO( m ), which can be represented by the group of m × m orthogonal matrices with determinant 1. For m = 3, this is the rotation group SO(3) . [ 19 ]
Phrased slightly differently, the rotation group of an object is the symmetry group within E + ( m ), the group of rigid motions; [ 20 ] that is, the intersection of the full symmetry group and the group of rigid motions. For chiral objects, it is the same as the full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem , rotational symmetry of a physical system is equivalent to the angular momentum conservation law . [ 21 ] For more, see rotational invariance .
Translational symmetry leaves an object invariant under a discrete or continuous group of translations T a ( p ) = p + a {\displaystyle \scriptstyle T_{a}(p)\;=\;p\,+\,a} . [ 22 ] The illustration on the right shows four congruent footprints generated by translations along the arrow. If the line of footprints were to extend to infinity in both directions, then they would have a discrete translational symmetry; any translation that mapped one footprint onto another would leave the whole line unchanged.
In 2D, a glide reflection symmetry (also called a glide plane symmetry in 3D, and a transflection in general) means that a reflection in a line or plane combined with a translation along the line or in the plane, results in the same object (such as in the case of footprints). [ 2 ] [ 23 ] The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the frieze group p11g , and is isomorphic with the infinite cyclic group Z .
In 3D, a rotary reflection , rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis. [ 24 ] The symmetry groups associated with rotoreflections include:
For more, see point groups in three dimensions .
In 3D geometry and higher, a screw axis (or rotary translation) is a combination of a rotation and a translation along the rotation axis. [ 25 ]
Helical symmetry is the kind of symmetry seen in everyday objects such as springs , Slinky toys, drill bits , and augers . The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant angular speed , while simultaneously translating at a constant linear speed along its axis of rotation. At any point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix. [ 26 ] When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the object rotates slowly and translates quickly, the coiling angle will approach 90°.
Three main classes of helical symmetry can be distinguished, based on the interplay of the angle of coiling and translation symmetries along the axis:
In 4D, a double rotation symmetry can be generated as the composite of two orthogonal rotations. [ 29 ] It is similar to 3D screw axis which is the composite of a rotation and an orthogonal translation.
A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
In Felix Klein 's Erlangen program , each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. [ 32 ] For example, the Euclidean group defines Euclidean geometry , whereas the group of Möbius transformations defines projective geometry .
Scale symmetry means that if an object is expanded or reduced in size, the new object has the same properties as the original. [ 33 ] This self-similarity is seen in many natural structures such as cumulus clouds, lightning, ferns and coastlines, over a wide range of scales. It is generally not found in gravitationally bound structures, for example the shape of the legs of an elephant and a mouse (so-called allometric scaling ). Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight.
A more subtle form of scale symmetry is demonstrated by fractals . As conceived by Benoît Mandelbrot , fractals are a mathematical concept in which the structure of a complex form looks similar at any degree of magnification , [ 34 ] well seen in the Mandelbrot set . A coast is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables small twigs to stand in for full trees in dioramas , is another example.
Because fractals can generate the appearance of patterns in nature , they have a beauty and familiarity not typically seen with mathematically generated functions. Fractals have also found a place in computer-generated movie effects , where their ability to create complex curves with fractal symmetries results in more realistic virtual worlds .
With every geometry, Felix Klein associated an underlying group of symmetries . The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups , and hierarchy of their invariants . For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations . A concept of parallelism , which is preserved in affine geometry , is not meaningful in projective geometry . Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
William Thurston introduced a similar version of symmetries in geometry. A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers. The Lie group can be thought of as the group of symmetries of the geometry.
A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers, i.e. if it is the maximal group of symmetries. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold M is a diffeomorphism from M to X /Γ for some model geometry X , where Γ is a discrete subgroup of G acting freely on X . If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X . Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries . (There are also uncountably many model geometries without compact quotients.) | https://en.wikipedia.org/wiki/Symmetry_(geometry) |
The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation .
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group ).
These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.
Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is described in special relativity by a group of transformations of the spacetime known as the Poincaré group . Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity .
Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, temperature may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is invariant under a shift in an observer's position within the room.
Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry . A rotation about any axis of the sphere will preserve the shape of its surface from any given vantage point.
The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well.
For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry , because the electric field strength at a given distance r from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius r . Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.
In Newton's theory of mechanics, given two bodies, each with mass m , starting at the origin and moving along the x -axis in opposite directions, one with speed v 1 and the other with speed v 2 the total kinetic energy of the system (as calculated from an observer at the origin) is 1 / 2 m ( v 1 2 + v 2 2 ) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y -axis.
The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v 1 and v 2 are interchanged.
Symmetries may be broadly classified as global or local . A global symmetry is one that keeps a property invariant for a transformation that is applied simultaneously at all points of spacetime , whereas a local symmetry is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of spacetime ; specifically a local symmetry transformation is parameterised by the spacetime coordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for gauge theories .
The two examples of rotational symmetry described above – spherical and cylindrical – are each instances of continuous symmetry . These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change continuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.
Continuous spacetime symmetries are symmetries involving transformations of space and time . These may be further classified as spatial symmetries , involving only the spatial geometry associated with a physical system; temporal symmetries , involving only changes in time; or spatio-temporal symmetries , involving changes in both space and time.
Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold . The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.
Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries .
A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges .
The Standard Model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.
These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry . CP violation , the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe. CP violation is a fruitful area of current research in particle physics .
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between bosons and fermions . Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.
Generalized symmetries encompass a number of recently recognized generalizations of the concept of a global symmetry. These include higher form symmetries, higher group symmetries, non-invertible symmetries, and subsystem symmetries. [ 1 ]
The transformations describing physical symmetries typically form a mathematical group . Group theory is an important area of mathematics for physicists.
Continuous symmetries are specified mathematically by continuous groups (called Lie groups ). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group ).
Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the symmetric group S 3 .
A type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries . Gauge symmetries in the Standard Model , used to describe three of the fundamental interactions , are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force , the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force .)
Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology ).
The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum , and temporal translation symmetry (i.e. homogeneity of time) gives rise to conservation of energy .
The following table summarizes some fundamental symmetries and the associated conserved quantity.
Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields . The commutator of two of these infinitesimal transformations is equivalent to a third infinitesimal transformation of the same kind hence they form a Lie algebra .
A general coordinate transformation described as the general field h ( x ) {\displaystyle h(x)} (also known as a diffeomorphism ) has the infinitesimal effect on a scalar ϕ ( x ) {\displaystyle \phi (x)} , spinor ψ ( x ) {\displaystyle \psi (x)} or vector field A ( x ) {\displaystyle A(x)} that can be expressed (using the Einstein summation convention ):
Without gravity only the Poincaré symmetries are preserved which restricts h ( x ) {\displaystyle h(x)} to be of the form:
where M is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:
where τ {\displaystyle \tau } are generators of a particular Lie group . So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of different types.
Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:
If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:
with D generating scale transformations and K generating special conformal transformations. For example, N = 4 supersymmetric Yang–Mills theory has this symmetry while general relativity does not although other theories of gravity such as conformal gravity do. The 'action' of a field theory is an invariant under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models.
In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields. | https://en.wikipedia.org/wiki/Symmetry_(physics) |
Symmetry aspects of M. C. Escher's periodic drawings is a book by crystallographer Caroline H. MacGillavry published for the International Union of Crystallography (IUCr) by Oosthoek in 1965. The book analyzes the symmetry of M. C. Escher's colored periodic drawings using the international crystallographic notation .
In 1959, MacGillavry met Escher. His work, the regular tiling of the plane, showed obvious links with the symmetry principles of crystallography. After seeking approval from the organisers (Joseph and Gabrielle Donnay ), MacGillavry asked Escher to exhibit his lithographic works at the IUCr Congress in Cambridge, U.K. in 1960. The exhibition was a success, and as a consequence the IUCr commissioned MacGillavry to write the book under its auspices. [ 1 ] [ 2 ]
The book has three chapters. In the first chapter, entitled Patterns with Classical Symmetry, the author introduces the concepts of motif , symmetry operations , lattice and unit cell , and uses these to analyze the symmetry of 13 of Escher's tiling designs.
In the second chapter, Patterns with Black-white Symmetry, the antisymmetry operation (indicated by a prime ') is introduced. The chapter analyzes 22 of Escher's design in terms of black-white symmetry and assigns each a symbol in the international notation describing its symmetries.
In the third chapter, Patterns with Polychromatic Symmetry , the analysis is extended to 7 of Escher's design possessing three or more colors. The book is printed in full color to facilitate the recognition of color symmetries in the images.
The publication of the book was sponsored by the IUCr and the original target audience was crystallography students learning the principles of symmetry, particularly color symmetry . In the introduction to the book the author states "Although the book is meant primarily for undergraduate students, I hope that many people who are simply amused and intrigued by Escher's designs will be interested to see how they illustrate the laws of symmetry". [ 3 ]
The reception of the book was positive. Robert M. Mengel in Scientific American wrote "[the author] has organized this unique and beautiful book from the corpus of marvelous spacefilling periodic drawings made over two decades by the artist Maurits C. Escher. Adding a few specially drawn for this work, Escher has here given us the classical crystal groups in the plane, and a good many more that exploit the latest extensions to color symmetry, foreseen by the artist before mathematicians had officially recognized and classified them." [ 4 ] [ 5 ]
F. I. G. Rawlins in Acta Crystallographica wrote "Under [the author's] sure guidance the reader is skilfully conducted through such regions of the theory of symmetry as are necessary for a tolerable grasp of the full significance of these patterns, several of them produced in full colour." [ 6 ]
J. Bohm reviewed the book in Kristall Und Technik. Bohm acknowledged the special value of Escher's art as crystallographic teaching material. He praised the author for preparing the material in a detailed, crystallographically valid and didactically appealing way. Overall he stated that the book was a successful collaboration between the artist, author, publisher and the IUCr. [ 7 ]
In 1976 an announcement in the IUCr's journals stated that the book was "extremely popular" and this had a necessitated a reprint in both the Netherlands and the U.S.A. [ 8 ] [ 9 ] [ 10 ] In an obituary of the author it is stated that the publication of the book helped to popularise M. C. Escher's work in the U.S.A. [ 1 ] MacGillavry's book inspired further work on the symmetry analysis of M. C. Escher's work, particularly by Doris Schattschneider in M. C. Escher: Visions of Symmetry . [ 11 ] [ 12 ] | https://en.wikipedia.org/wiki/Symmetry_aspects_of_M._C._Escher's_periodic_drawings |
In physics , symmetry breaking is a phenomenon where a disordered but symmetric state collapses into an ordered, but less symmetric state. [ 1 ] This collapse is often one of many possible bifurcations that a particle can take as it approaches a lower energy state. Due to the many possibilities, an observer may assume the result of the collapse to be arbitrary. This phenomenon is fundamental to quantum field theory (QFT), and further, contemporary understandings of physics . [ 2 ] Specifically, it plays a central role in the Glashow–Weinberg–Salam model which forms part of the Standard model modelling the electroweak sector.
In an infinite system ( Minkowski spacetime ) symmetry breaking occurs, however in a finite system (that is, any real super-condensed system), the system is less predictable, but in many cases quantum tunneling occurs. [ 2 ] [ 3 ] Symmetry breaking and tunneling relate through the collapse of a particle into non-symmetric state as it seeks a lower energy. [ 4 ]
Symmetry breaking can be distinguished into two types, explicit and spontaneous . They are characterized by whether the equations of motion fail to be invariant, or the ground state fails to be invariant.
This section describes spontaneous symmetry breaking. This is the idea that for a physical system, the lowest energy configuration (the vacuum state ) is not the most symmetric configuration of the system. Roughly speaking, there are three types of symmetry that can be broken: discrete, continuous and gauge, ordered in increasing technicality.
An example of a system with discrete symmetry is given by the figure with the red graph: consider a particle moving on this graph, subject to gravity . A similar graph could be given by the function f ( x ) = ( x 2 − a 2 ) 2 {\displaystyle f(x)=(x^{2}-a^{2})^{2}} . This system is symmetric under reflection in the y-axis. There are three possible stationary states for the particle: the top of the hill at x = 0 {\displaystyle x=0} , or the bottom, at x = ± a {\displaystyle x=\pm a} . When the particle is at the top, the configuration respects the reflection symmetry: the particle stays in the same place when reflected. However, the lowest energy configurations are those at x = ± a {\displaystyle x=\pm a} . When the particle is in either of these configurations, it is no longer fixed under reflection in the y-axis: reflection swaps the two vacuum states.
An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph f ( x , y ) = ( x 2 + y 2 − a 2 ) 2 {\displaystyle f(x,y)=(x^{2}+y^{2}-a^{2})^{2}} . This is essentially the graph of the Mexican hat potential . This has a continuous symmetry given by rotation about the axis through the top of the hill (as well as a discrete symmetry by reflection through any radial plane). Again, if the particle is at the top of the hill it is fixed under rotations, but it has higher gravitational energy at the top. At the bottom, it is no longer invariant under rotations but minimizes its gravitational potential energy. Furthermore rotations move the particle from one energy minimizing configuration to another. There is a novelty here not seen in the previous example: from any of the vacuum states it is possible to access any other vacuum state with only a small amount of energy, by moving around the trough at the bottom of the hill, whereas in the previous example, to access the other vacuum, the particle would have to cross the hill, requiring a large amount of energy.
Gauge symmetry breaking is the most subtle, but has important physical consequences. Roughly speaking, for the purposes of this section a gauge symmetry is an assignment of systems with continuous symmetry to every point in spacetime . Gauge symmetry forbids mass generation for gauge fields , yet massive gauge fields ( W and Z bosons ) have been observed. Spontaneous symmetry breaking was developed to resolve this inconsistency. The idea is that in an early stage of the universe it was in a high energy state, analogous to the particle being at the top of the hill, and so had full gauge symmetry and all the gauge fields were massless. As it cooled, it settled into a choice of vacuum, thus spontaneously breaking the symmetry, thus removing the gauge symmetry and allowing mass generation of those gauge fields. A full explanation is highly technical: see electroweak interaction .
In spontaneous symmetry breaking (SSB), the equations of motion of the system are invariant, but any vacuum state (lowest energy state) is not.
For an example with two-fold symmetry, if there is some atom that has two vacuum states, occupying either one of these states breaks the two-fold symmetry. This act of selecting one of the states as the system reaches a lower energy is SSB. When this happens, the atom is no longer Z 2 {\displaystyle \mathbb {Z} _{2}} symmetric (reflectively symmetric) and has collapsed into a lower energy state.
Such a symmetry breaking is parametrized by an order parameter . A special case of this type of symmetry breaking is dynamical symmetry breaking .
In the Lagrangian setting of quantum field theory (QFT), the Lagrangian L {\displaystyle L} is a functional of quantum fields which is invariant under the action of a symmetry group G {\displaystyle G} . However, the vacuum expectation value formed when the particle collapses to a lower energy may not be invariant under G {\displaystyle G} . In this instance, it will partially break the symmetry of G {\displaystyle G} , into a subgroup H {\displaystyle H} . This is spontaneous symmetry breaking.
Within the context of gauge symmetry however, SSB is the phenomenon by which gauge fields 'acquire mass' despite gauge-invariance enforcing that such fields be massless. This is because the SSB of gauge symmetry breaks gauge-invariance, and such a break allows for the existence of massive gauge fields. This is an important exemption from Goldstone's theorem , where a Nambu-Goldstone boson can gain mass, becoming a Higgs boson in the process. [ 5 ]
Further, in this context the usage of 'symmetry breaking' while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of trivialization , somewhat analogous to redundancy arising from a choice of basis.
Spontaneous symmetry breaking is also associated with phase transitions . For example in the Ising model , as the temperature of the system falls below the critical temperature the Z 2 {\displaystyle \mathbb {Z} _{2}} symmetry of the vacuum is broken, giving a phase transition of the system.
In explicit symmetry breaking (ESB), the equations of motion describing a system are variant under the broken symmetry. In Hamiltonian mechanics or Lagrangian mechanics , this happens when there is at least one term in the Hamiltonian (or Lagrangian) that explicitly breaks the given symmetry.
In the Hamiltonian setting, this is often studied when the Hamiltonian can be written H = H 0 + H int {\displaystyle H=H_{0}+H_{\text{int}}} .
Here H 0 {\displaystyle H_{0}} is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group G {\displaystyle G} . Often this is an integrable Hamiltonian.
The H int {\displaystyle H_{\text{int}}} is a perturbation or interaction Hamiltonian. This is not invariant under the action of G {\displaystyle G} . It is often proportional to a small, perturbative parameter.
This is essentially the paradigm for perturbation theory in quantum mechanics. An example of its use is in finding the fine structure of atomic spectra.
Symmetry breaking can cover any of the following scenarios:
One of the first cases of broken symmetry discussed in the physics literature is related to the form taken by a uniformly rotating body of incompressible fluid in gravitational and hydrostatic equilibrium . Jacobi [ 6 ] and soon later Liouville , [ 7 ] in 1834, discussed the fact that a tri-axial ellipsoid was an equilibrium solution for this problem when the kinetic energy compared to the gravitational energy of the rotating body exceeded a certain critical value. The axial symmetry presented by the McLaurin spheroids is broken at this bifurcation point. Furthermore, above this bifurcation point, and for constant angular momentum, the solutions that minimize the kinetic energy are the non -axially symmetric Jacobi ellipsoids instead of the Maclaurin spheroids . | https://en.wikipedia.org/wiki/Symmetry_breaking |
Symmetry breaking in biology is the process by which uniformity is broken, or the number of points to view invariance are reduced, to generate a more structured and improbable state. [ 1 ] Symmetry breaking is the event where symmetry along a particular axis is lost to establish a polarity. Polarity is a measure for a biological system to distinguish poles along an axis. This measure is important because it is the first step to building complexity. For example, during organismal development, one of the first steps for the embryo is to distinguish its dorsal-ventral axis . The symmetry-breaking event that occurs here will determine which end of this axis will be the ventral side, and which end will be the dorsal side. Once this distinction is made, then all the structures that are located along this axis can develop at the proper location. As an example, during human development, the embryo needs to establish where is ‘back’ and where is ‘front’ before complex structures, such as the spine and lungs, can develop in the right location (where the lungs are placed ‘in front’ of the spine). This relationship between symmetry breaking and complexity was articulated by P.W. Anderson . He speculated that increasing levels of broken symmetry in many-body systems correlates with increasing complexity and functional specialization. [ 2 ] In a biological perspective, the more complex an organism is, the higher number of symmetry-breaking events can be found.
The importance of symmetry breaking in biology is also reflected in the fact that it's found at all scales. Symmetry breaking can be found at the macromolecular level, [ 3 ] at the subcellular level [ 4 ] and even at the tissues and organ level. [ 5 ] It's also interesting to note that most asymmetry on a higher scale is a reflection of symmetry breaking on a lower scale. Cells first need to establish a polarity through a symmetry-breaking event before tissues and organs themselves can be polar. For example, one model proposes that left-right body axis asymmetry in vertebrates is determined by asymmetry of cilia rotation during early development, which will produce a constant, unidirectional flow. [ 6 ] [ 7 ] However, there is also evidence that earlier asymmetries in serotonin distribution and ion-channel mRNA and protein localization occur in zebrafish , chicken and Xenopus development, [ 8 ] [ 9 ] [ 10 ] and similar to observations of intrinsic chirality generated by the cytoskeleton [ 11 ] [ 12 ] leading to organ and whole organism asymmetries in Arabidopsis [ 13 ] [ 14 ] [ 15 ] [ 16 ] this itself seems to be controlled from the macromolecular level by the cytoskeleton. [ 10 ]
There are several examples of symmetry breaking that are currently being studied. One of the most studied examples is the cortical rotation during Xenopus development, where this rotation acts as the symmetry-breaking event that determines the dorsal-ventral axis of the developing embryo. This example is discussed in more detail below. Another example that involves symmetry breaking is the establishment of dendrites and axon during neuron development, and the PAR protein network in C. elegans . It is thought that a protein called shootin-1 determines which outgrowth in neurons eventually becomes the axon, at it does this by breaking symmetry and accumulating in only one outgrowth. [ 17 ] The PAR protein network works under similar mechanisms, where the certain PAR proteins, which are initially homogenous throughout the cell, break their symmetry and are segregated to different ends of the zygote to establish a polarity during development. [ 18 ]
Cortical rotation is a phenomenon that seems to be limited to Xenopus and few ancient teleosts , however the underlying mechanisms of cortical rotation have conserved elements that are found in other chordates .
A sperm can bind a Xenopus egg at any position of the pigmented animal hemisphere; however, once bound, this position then determines the dorsal side of the animal. The dorsal side of the egg is always directly opposite the sperm entry point. The sperm's centriole acts as an organizing center for the egg's microtubules , which transport the maternal dorsalizing factors, such as wnt11 mRNA, wnt5a mRNA, and Dishevelled protein. [ 19 ]
A series of experiments utilizing UV irradiation, cold temperature and pressure (all of which cause microtubule depolymerization) demonstrated that without polymerized microtubules, cortical rotation did not occur and resulted in a mutant ventral phenotype. [ 20 ] Another study also revealed that mutant phenotype could be rescued (returned to normal) by physically turning the embryo, thus mimicking cortical rotation and demonstrating that microtubules were not the determinant of dorsal development. [ 21 ] From this it was hypothesized that there were other elements within the embryo being moved during cortical rotation.
To identify these elements, researchers looked for mRNA and protein that demonstrated localization to either the vegetal pole or the dorsal side of the embryo to find candidates. The early candidates for the determinant were β-catenin and disheveled (Dsh). [ 22 ] [ 23 ] When maternal β-catenin mRNA was degraded in the oocyte, the resulting embryo developed into mutant ventral phenotype and this could be rescued by injecting the fertilized egg with β-catenin mRNA. β-catenin is observed to be enriched in the dorsal side of the embryo following cortical rotation. The Dsh protein was fused to a GFP and tracked during cortical rotation, it was observed to be in vesicles that were couriered along microtubules to the dorsal side. This led researchers to look into other candidates of the Wnt pathway. Wnt 11 was found to be located specifically at the vegetal pole prior to cortical rotation and is moved to the dorsal side where it activates the wnt signaling pathway . [ 24 ] VegT, a T-box transcription factor, is localized to the vegetal cortex and upon cortical rotation is released in a gradient fashion into the embryo to regulate mesoderm development. [ 25 ] VegT activates Wnt expression, so while not acted on or moved during cortical rotation, it is active in dorsal-ventral axis formation.
The question still remains, how are these molecules being moved to the dorsal side? This is still not completely known, however evidence suggests that microtubule bundles within the cortex are interacting with kinesin (plus-end directed) motors to become organized into parallel arrays within the cortex and this motion of the motors is the cause of the rotation of the cortex. [ 26 ] Also unclear is whether Wnt 11 is the main dorsal determinant or is β-catenin also required, as these two molecules have both been demonstrated to be necessary and sufficient for dorsal development. This along with all of the other factors are important for activating Nodal genes that propagate normal dorsoventral development. | https://en.wikipedia.org/wiki/Symmetry_breaking_and_cortical_rotation |
Symmetry breaking of escaping ants is a herd behavior phenomenon observed when ants are constrained to a cell with two equidistant exits and then sprayed with an insect repellent . The ants tend to crowd one door more while trying to escape (i.e., there is a symmetry breaking in their escape behavior), thereby decreasing evacuation efficiency.
This phenomenon arises in experiments where worker ants are enclosed in circular cells with a glass cover in such a way that they can only move in two dimensions (i.e., ants cannot pass over one other). The cell has two exits located symmetrically relative to its center. The experiments consisted of two different sets of trials. In the first set of trials, both exits were opened at the same time, letting the ants escape. After 30 repetitions, one door was used 13.666% more than the other. In the second set of trials, the configuration was identical, but a few seconds before opening the doors, a dose of 50 μL of insect repellent was injected into the cell at its center through a small hole in the glass cover. After 30 repetitions, one door was used 38.3% more than the other.
Inspired by earlier computer simulations that predicted a symmetry-breaking phenomenon when panicked humans escape from a room with two equivalent exits, a team of researchers led by E. Altschuler carried out the two experiments described above, which revealed the symmetry-breaking effect in the leafcutter ant Atta insular in the presence of insect repellent. [ 1 ]
Another team of researchers led by Geng Li investigated the influence of the ant group's density on the symmetry breaking. They used the red imported fire ant to repeat the experiment with different amounts of ants. The results show that symmetry breaking is high at low densities of ants, but decreases beyond a certain point in the density of ants. In other words, when density is low, the ant group produces a collective escaping behavior, while at high density, their behavior is more like random particles. [ 2 ]
The common idea is that the action of injecting the insect repellent induces herd behavior in the ants. When ants are in "panic", they experience a strong tendency to follow each other. As a result, if a random fluctuation in the system produces a locally large amount of ants trying to reach one of the two doors, the fluctuation can be amplified because ants tend to follow the direction of the majority of individuals, resulting in that door getting crowded.
Altshuler and coworkers were able to reproduce their symmetry-breaking experiments previously done in ants in humans, using a simplified version of the theoretical model proposed earlier by Helbing et al. [ 3 ] based on the fact that walkers tend to follow the general direction of motion of their neighbors (" Vicsek's rule " [ 4 ] ), and such herd behavior increases as the so-called "panic parameter" increases. In the case of ants, the panic parameter is supposed to be low when no repellent is used and high when the repellent is used.
A more "biologically sensible" model based on the deposition of an alarm pheromone by ants under stress also reproduces the symmetry-breaking phenomenon, with the advantage that it also predicts the experimental output for different concentrations of ants in the cell. [ 2 ] The pheromone mechanism shares the key elements of the previous models: stressed ants tend to "follow the crowd". | https://en.wikipedia.org/wiki/Symmetry_breaking_of_escaping_ants |
In chemistry and crystallography , a symmetry element is a point , line , or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane , an axis of rotation (either proper and improper), or a center of inversion . [ 1 ] [ 2 ] [ 3 ] For an object such as a molecule or a crystal , a symmetry element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the object unchanged. The set containing these operations form one of the symmetry groups of the object. The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.
The identity symmetry element is found in all objects and is denoted E . [ 4 ] It corresponds to an operation of doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. An object having no symmetry elements other than E is called asymmetric. Such an object is necessarily chiral. [ 5 ]
Mirror planes are denoted by σ . In a molecule that also has an axis of symmetry, a mirror plane that includes the axis is called a vertical mirror plane and is labeled σ v , while one perpendicular to the axis is called a horizontal mirror plane and is labeled σ h . A vertical mirror plane that bisects the angle between two C2 axes is called a dihedral mirror plane, σ d . [ 6 ]
Rotational symmetry, also known as radial symmetry, is represented by an axis about which the object rotates in its corresponding symmetry operation. A group of proper rotations is denoted as C n , where the degrees of rotation that restore the object is 360/n ( C 2 = 180º rotation, C 3 = 120º rotation, C 4 = 90º rotation, C 5 = 72º rotation). [ 4 ] The C n notation is also used for the related, more abstract, cyclic group .
An improper rotation is the composition of a rotation about an axis, and reflection in a plane perpendicular to that axis. [ 2 ] The order in which the rotation and reflection are performed does not matter (that is, these operations commute). Improper rotation is also defined as the composition of a rotation about an axis, and inversion about a point on the axis. [ 3 ] These definitions are equivalent because inversion about a point is equivalent to rotation by 180° about any axis, followed by mirroring about a plane perpendicular to that axis. The symmetry elements for improper rotation are the rotation axis, and either the mirror plane, the inversion point, or both. The improper rotation group of order 2 n is denoted S 2 n .
For inversion, denoted i , there must be a point in the center of an object that is the inversion center. Inversion consists of passing each point through the center of inversion and out to the same distance on the other side of the molecule. In the inversion operation for 3D coordinates, the inversion center is the origin (0,0,0). When an object is inverted, the position vector of a point in an object, ⟨x,y,z⟩, is inverted to ⟨-x,-y,-z⟩. | https://en.wikipedia.org/wiki/Symmetry_element |
In nuclear physics , the symmetry energy reflects the variation of the binding energy of the nucleons in the nuclear matter depending on its neutron to proton ratio as a function of baryon density. Symmetry energy is an important parameter in the equation of state describing the nuclear structure of heavy nuclei and neutron stars . [ 1 ] [ 2 ] [ 3 ] [ 4 ]
Let n p {\displaystyle n_{p}} and n n {\displaystyle n_{n}} be the number density of protons and neutrons in nuclear matter, and n = n p + n n {\displaystyle n=n_{p}+n_{n}} . Let E 0 ( n ) {\displaystyle E_{0}(n)} be the binding energy per nucleon in symmetric matter, with equally many protons as neutrons, as a function of density. The binding energy per nucleon E {\displaystyle E} of non-symmetric matter is then a function that also depends on the isospin asymmetry,
so to lowest order the energy per baryon is
where S {\displaystyle S} is the symmetry energy. [ 2 ] There are no odd powers of δ {\displaystyle \delta } in the expansion because the nuclear force acts the same between two protons as between two neutrons. [ 5 ] At saturation density n 0 {\displaystyle n_{0}} , the symmetry energy is 32.0 ± 1.1 MeV . [ 4 ]
This nuclear physics or atomic physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symmetry_energy |
In group theory , the symmetry group of a geometric object is the group of all transformations under which the object is invariant , endowed with the group operation of composition . Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym( X ).
For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry , but the concept may also be studied for more general types of geometric structure.
We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern . For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field , a function of position with values in a set of colors or substances; as a vector field ; or as a more general function on the object.) The group of isometries of space induces a group action on objects in it, and the symmetry group Sym( X ) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X .
The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations ), as long as those isometries map this particular X to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group . An object is chiral when it has no orientation -reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.
Any symmetry group whose elements have a common fixed point , which is true if the group is finite or the figure is bounded, can be represented as a subgroup of the orthogonal group O( n ) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO( n ), and is called the rotation group of the figure.
In a discrete symmetry group , the points symmetric to a given point do not accumulate toward a limit point . That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set . All finite symmetry groups are discrete.
Discrete symmetry groups come in three types: (1) finite point groups , which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O( n ); (2) infinite lattice groups , which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups ( Lie groups ), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3) , the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E( n ) (the isometry group of R n ).
Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H 1 , H 2 are related by H 1 = g −1 H 2 g for some g in E( n ). For example:
In the following sections, we only consider isometry groups whose orbits are topologically closed , including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a rational number ; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.
The isometry groups in one dimension are:
Up to conjugacy the discrete point groups in two-dimensional space are the following classes:
C 1 is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C 2 is the symmetry group of the letter "Z", C 3 that of a triskelion , C 4 of a swastika , and C 5 , C 6 , etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four.
D 1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry , for example the letter "A".
D 2 , which is isomorphic to the Klein four-group , is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes.
D 3 , D 4 etc. are the symmetry groups of the regular polygons .
Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.
The remaining isometry groups in two dimensions with a fixed point are:
Non-bounded figures may have isometry groups including translations; these are:
Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In crystallography , only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 individual groups from the 7 series, and 5 of the 7 other individuals).
The continuous symmetry groups with a fixed point include those of:
For objects with scalar field patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field patterns: for example, in cylindrical coordinates with respect to some axis, the vector field A = A ρ ρ ^ + A ϕ ϕ ^ + A z z ^ {\displaystyle \mathbf {A} =A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}} has cylindrical symmetry with respect to the axis whenever A ρ , A ϕ , {\displaystyle A_{\rho },A_{\phi },} and A z {\displaystyle A_{z}} have this symmetry (no dependence on ϕ {\displaystyle \phi } ); and it has reflectional symmetry only when A ϕ = 0 {\displaystyle A_{\phi }=0} .
For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.
The continuous symmetry groups without a fixed point include those with a screw axis , such as an infinite helix . See also subgroups of the Euclidean group .
In wider contexts, a symmetry group may be any kind of transformation group , or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the Erlangen programme .
For example, objects in a hyperbolic non-Euclidean geometry have Fuchsian symmetry groups , which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of Escher .) Similarly, automorphism groups of finite geometries preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space.
Another example of a symmetry group is that of a combinatorial graph : a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph ; the free group is the symmetry group of an infinite tree graph .
Cayley's theorem states that any abstract group is a subgroup of the permutations of some set X , and so can be considered as the symmetry group of X with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries.
For example, let G = Sym( X ) be the finite symmetry group of a figure X in a Euclidean space , and let H ⊂ G be a subgroup. Then H can be interpreted as the symmetry group of X + , a "decorated" version of X . Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to X so as to break all symmetry, obtaining a figure X # with Sym( X # ) = {1}, the trivial subgroup; that is, gX # ≠ X # for all non-trivial g ∈ G . Now we get:
Normal subgroups may also be characterized in this framework.
The symmetry group of the translation gX + is the conjugate subgroup gHg −1 . Thus H is normal whenever:
that is, whenever the decoration of X + may be drawn in any orientation, with respect to any side or feature of X , and still yield the same symmetry group gHg −1 = H .
As an example, consider the dihedral group G = D 3 = Sym( X ), where X is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure X # . Letting τ ∈ G be the reflection of the arrowed edge, the composite figure X + = X # ∪ τ X # has a bidirectional arrow on that edge, and its symmetry group is H = {1, τ}. This subgroup is not normal, since gX + may have the bi-arrow on a different edge, giving a different reflection symmetry group.
However, letting H = {1, ρ, ρ 2 } ⊂ D 3 be the cyclic subgroup generated by a rotation, the decorated figure X + consists of a 3-cycle of arrows with consistent orientation. Then H is normal, since drawing such a cycle with either orientation yields the same symmetry group H . | https://en.wikipedia.org/wiki/Symmetry_group |
Symmetry in Science and Art is a book by A.V. Shubnikov and V.A. Koptsik published by Plenum Press in 1974. The book is a translation of Simmetrija v nauke i iskusstve (Russian: Симметрия в науке и искусстве) published by Nauka in 1972. The book was notable because it gave English-language speakers access to Russian work in the fields of dichromatic and polychromatic symmetry .
The book is divided into two parts. The first part is an updated version of A.V. Shubnikov's 1940 book Symmetry: laws of symmetry and their application in science, technology and applied arts (Russian: Симметрия : законы симметрии и их применение в науке, технике и прикладном искусстве). [ 1 ] The following types of classical (one-color) and dichromatic (two-color) symmetries are covered in the first part of the book: one-sided rosettes , figures with a singular point , one-sided bands , two-sided bands , rods , network patterns , layers and space groups .
The second part consists of three new chapters written by V.A. Koptsik covering the following subjects: group theory , crystallographic groups , antisymmetry , colored symmetry , symmetry in science and art , and conservation laws .
The book is written for crystallographers, mathematicians and physicists who are interested in the application of color symmetry to crystal structure analysis and physics experiments involving magnetic or ferroelectric materials. Werner Nowacki in his review of the book for Science stated: "This is an extraordinary book, dealing with symmetry in all its aspects and written for the nonspecialist as well as the specialist (crystallographer and physicist) in this domain of natural sciences." [ 2 ]
The book had a mixed reception from contemporary reviewers. Marc H. Bornstein in a review for Leonardo praised the book: "Shubnikov and Koptsik, I find, should stand beside Weyl's classic treatise, Symmetry ". [ 3 ] Werner Nowacki wrote a positive review: "This clearly written, beautifully illustrated book will become a standard work for all who are interested in unifying branches of natural sciences and of art, and we must be grateful to the translator, the editor, and the publisher for having produced such a precious publication." [ 2 ]
However, Herbert Callen in American Scientist , criticised the book:"The book remains as it was in its original edition - an exhaustive classification of symmetry groups for systems with particular types of symmetry operations, now updated by Koptsik. The larger philosophical and aesthetic extensions, however, do not meet Western standards of critical accuracy, rigour, or precision of statement; they are not pursued in any depth, and they draw on no currents of thought outside the Soviet Union." [ 4 ]
Tony Crilly, when reviewing Jaswon and Rose's Crystal symmetry, theory of colour crystallography in The Mathematical Gazette in 1984 commented: "The beginning student would find Symmetry in Science and Art (by A. V. Shubnikov and V. A. Koptsick, 1974) a stimulating introduction to the ideas worked out in technical detail by Jaswon and Rose." [ 5 ] István and Magdolna Hargittai in the preface to their book Symmetry through the eyes of a chemist remarked: "We would like especially to note here two classics in the literature of symmetry which have strongly influenced us: Weyl 's Symmetry and Shubnikov and Koptsik's Symmetry in Science and Art ". [ 6 ]
In later reviews of the literature by R.L.E. Schwarzenberger [ 7 ] and by Branko Grünbaum and G.C. Shephard in their book Tilings and patterns [ 8 ] the work of the Russian color symmetry school led by A.V. Shubnikov and N.V. Belov was put into its proper historical context. Schwarzenberger, and Grünbaum and Shephard, give credit to Shubnikov and Belov for relaunching the field of color symmetry after the work of Heinrich Heesch and H.J. Woods in the 1930s was largely ignored. However, they criticise Shubnikov and Koptsik for taking a crystallographic rather than a group-theoretic approach, and for continuing to use their own confusing notation rather than adopting the international standard Hermann–Mauguin notation for crystallographic symmetry elements. | https://en.wikipedia.org/wiki/Symmetry_in_Science_and_Art |
Symmetry in biology refers to the symmetry observed in organisms , including plants, animals, fungi , and bacteria . External symmetry can be easily seen by just looking at an organism. For example, the face of a human being has a plane of symmetry down its centre, or a pine cone displays a clear symmetrical spiral pattern. Internal features can also show symmetry, for example the tubes in the human body (responsible for transporting gases , nutrients , and waste products) which are cylindrical and have several planes of symmetry.
Biological symmetry can be thought of as a balanced distribution of duplicate body parts or shapes within the body of an organism. Importantly, unlike in mathematics, symmetry in biology is always approximate. For example, plant leaves – while considered symmetrical – rarely match up exactly when folded in half. Symmetry is one class of patterns in nature whereby there is near-repetition of the pattern element, either by reflection or rotation .
While sponges and placozoans represent two groups of animals which do not show any symmetry (i.e. are asymmetrical), the body plans of most multicellular organisms exhibit, and are defined by, some form of symmetry. There are only a few types of symmetry which are possible in body plans. These are radial (cylindrical) symmetry, bilateral , biradial and spherical symmetry. [ 1 ] While the classification of viruses as an "organism" remains controversial, viruses also contain icosahedral symmetry .
The importance of symmetry is illustrated by the fact that groups of animals have traditionally been defined by this feature in taxonomic groupings. The Radiata , animals with radial symmetry, formed one of the four branches of Georges Cuvier 's classification of the animal kingdom . [ 2 ] [ 3 ] [ 4 ] Meanwhile, Bilateria is a taxonomic grouping still used today to represent organisms with embryonic bilateral symmetry.
Organisms with radial symmetry show a repeating pattern around a central axis such that they can be separated into several identical pieces when cut through the central point, much like pieces of a pie. Typically, this involves repeating a body part 4, 5, 6 or 8 times around the axis – referred to as tetramerism, pentamerism, hexamerism and octamerism, respectively. Such organisms exhibit no left or right sides but do have a top and a bottom surface, or a front and a back.
George Cuvier classified animals with radial symmetry in the taxon Radiata ( Zoophytes ), [ 5 ] [ 4 ] which is now generally accepted to be an assemblage of different animal phyla that do not share a single common ancestor (a polyphyletic group). [ 6 ] Most radially symmetric animals are symmetrical about an axis extending from the center of the oral surface, which contains the mouth, to the center of the opposite (aboral) end. Animals in the phyla Cnidaria and Echinodermata generally show radial symmetry, [ 7 ] although many sea anemones and some corals within the Cnidaria have bilateral symmetry defined by a single structure, the siphonoglyph . [ 8 ] Radial symmetry is especially suitable for sessile animals such as the sea anemone, floating animals such as jellyfish , and slow moving organisms such as starfish ; whereas bilateral symmetry favours locomotion by generating a streamlined body.
Many flowers are also radially symmetric, or " actinomorphic ". Roughly identical floral structures – petals , sepals , and stamens – occur at regular intervals around the axis of the flower, which is often the female reproductive organ containing the carpel , style and stigma . [ 9 ]
Three-fold triradial symmetry was present in Trilobozoa from the Late Ediacaran period.
Four-fold tetramerism appears in some jellyfish, such as Aurelia marginalis . This is immediately obvious when looking at the jellyfish due to the presence of four gonads , visible through its translucent body. This radial symmetry is ecologically important in allowing the jellyfish to detect and respond to stimuli (mainly food and danger) from all directions.
Flowering plants show five-fold pentamerism, in many of their flowers and fruits. This is easily seen through the arrangement of five carpels (seed pockets) in an apple when cut transversely . Among animals, only the echinoderms such as sea stars , sea urchins , and sea lilies are pentamerous as adults, with five arms arranged around the mouth. Being bilaterian animals, however, they initially develop with mirror symmetry as larvae, then gain pentaradial symmetry later. [ 10 ]
Hexamerism is found in the corals and sea anemones (class Anthozoa ), which are divided into two groups based on their symmetry. The most common corals in the subclass Hexacorallia have a hexameric body plan; their polyps have six-fold internal symmetry and a number of tentacles that is a multiple of six.
Octamerism is found in corals of the subclass Octocorallia . These have polyps with eight tentacles and octameric radial symmetry. The octopus , however, has bilateral symmetry, despite its eight arms.
Icosahedral symmetry occurs in an organism which contains 60 subunits generated by 20 faces, each an equilateral triangle , and 12 corners. Within the icosahedron there is 2-fold, 3-fold and 5-fold symmetry . Many viruses, including canine parvovirus , show this form of symmetry due to the presence of an icosahedral viral shell . Such symmetry has evolved because it allows the viral particle to be built up of repetitive subunits consisting of a limited number of structural proteins (encoded by viral genes ), thereby saving space in the viral genome . The icosahedral symmetry can still be maintained with more than 60 subunits, but only in multiples of 60. For example, the T=3 Tomato bushy stunt virus has 60x3 protein subunits (180 copies of the same structural protein). [ 11 ] [ 12 ] Although these viruses are often referred to as 'spherical', they do not show true mathematical spherical symmetry.
In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria , some of whose skeletons are shaped like various regular polyhedra. Examples include Circoporus octahedrus , Circogonia icosahedra , Lithocubus geometricus and Circorrhegma dodecahedra . The shapes of these creatures should be obvious from their names. Tetrahedral symmetry is not present in Callimitra agnesae .
Spherical symmetry is characterised by the ability to draw an endless, or great but finite, number of symmetry axes through the body. This means that spherical symmetry occurs in an organism if it is able to be cut into two identical halves through any cut that runs through the organism's center. True spherical symmetry is not found in animal body plans. [ 1 ] Organisms which show approximate spherical symmetry include the freshwater green alga Volvox . [ 7 ]
Bacteria are often referred to as having a 'spherical' shape. Bacteria are categorized based on their shapes into three classes: cocci (spherical-shaped), bacillus (rod-shaped) and spirochetes (spiral-shaped) cells. In reality, this is a severe over-simplification as bacterial cells can be curved, bent, flattened, oblong spheroids and many more shapes. [ 13 ] Due to the huge number of bacteria considered to be cocci (coccus if a single cell), it is unlikely that all of these show true spherical symmetry. It is important to distinguish between the generalized use of the word 'spherical' to describe organisms at ease, and the true meaning of spherical symmetry. The same situation is seen in the description of viruses – 'spherical' viruses do not necessarily show spherical symmetry, being usually icosahedral.
Organisms with bilateral symmetry contain a single plane of symmetry, the sagittal plane , which divides the organism into two roughly mirror image left and right halves – approximate reflectional symmetry.
Animals with bilateral symmetry are classified into a large group called the bilateria , which contains 99% of all animals (comprising over 32 phyla and 1 million described species). All bilaterians have some asymmetrical features; for example, the human heart and liver are positioned asymmetrically despite the body having external bilateral symmetry. [ 14 ]
The bilateral symmetry of bilaterians is a complex trait which develops due to the expression of many genes . The bilateria have two axes of polarity . The first is an anterior – posterior (AP) axis which can be visualised as an imaginary axis running from the head or mouth to the tail or other end of an organism. The second is the dorsal – ventral (DV) axis which runs perpendicular to the AP axis. [ 15 ] [ 1 ] During development the AP axis is always specified before the DV axis, [ 16 ] which is known as the second embryonic axis .
The AP axis is essential in defining the polarity of bilateria and allowing the development of a front and back to give the organism direction. The front end encounters the environment before the rest of the body so sensory organs such as eyes tend to be clustered there. This is also the site where a mouth develops since it is the first part of the body to encounter food. Therefore, a distinct head, with sense organs connected to a central nervous system, tends to develop. [ 17 ] This pattern of development (with a distinct head and tail) is called cephalization . It is also argued that the development of an AP axis is important in locomotion – bilateral symmetry gives the body an intrinsic direction and allows streamlining to reduce drag .
In addition to animals, the flowers of some plants also show bilateral symmetry. Such plants are referred to as zygomorphic and include the orchid ( Orchidaceae ) and pea ( Fabaceae ) families, and most of the figwort family ( Scrophulariaceae ). [ 18 ] [ 19 ] The leaves of plants also commonly show approximate bilateral symmetry.
Biradial symmetry is found in organisms which show morphological features (internal or external) of both bilateral and radial symmetry. Unlike radially symmetrical organisms which can be divided equally along many planes, biradial organisms can only be cut equally along two planes. This could represent an intermediate stage in the evolution of bilateral symmetry from a radially symmetric ancestor. [ 20 ]
The animal group with the most obvious biradial symmetry is the ctenophores . In ctenophores the two planes of symmetry are (1) the plane of the tentacles and (2) the plane of the pharynx. [ 1 ] In addition to this group, evidence for biradial symmetry has even been found in the 'perfectly radial' freshwater polyp Hydra (a cnidarian). Biradial symmetry, especially when considering both internal and external features, is more common than originally accounted for. [ 21 ]
Like all the traits of organisms, symmetry (or indeed asymmetry) evolves due to an advantage to the organism – a process of natural selection . This involves changes in the frequency of symmetry-related genes throughout time.
Early flowering plants had radially symmetric flowers but since then many plants have evolved bilaterally symmetrical flowers. The evolution of bilateral symmetry is due to the expression of CYCLOIDEA genes. Evidence for the role of the CYCLOIDEA gene family comes from mutations in these genes which cause a reversion to radial symmetry. The CYCLOIDEA genes encode transcription factors , proteins which control the expression of other genes. This allows their expression to influence developmental pathways relating to symmetry. [ 22 ] [ 23 ] For example, in Antirrhinum majus , CYCLOIDEA is expressed during early development in the dorsal domain of the flower meristem and continues to be expressed later on in the dorsal petals to control their size and shape. It is believed that the evolution of specialized pollinators may play a part in the transition of radially symmetrical flowers to bilaterally symmetrical flowers. [ 24 ]
Symmetry is often selected for in the evolution of animals. This is unsurprising since asymmetry is often an indication of unfitness – either defects during development or injuries throughout a lifetime. This is most apparent during mating during which females of some species select males with highly symmetrical features. Additionally, female barn swallows , a species where adults have long tail streamers, prefer to mate with males that have the most symmetrical tails. [ 26 ]
While symmetry is known to be under selection, the evolutionary history of different types of symmetry in animals is an area of extensive debate. Traditionally it has been suggested that bilateral animals evolved from a radial ancestor . Cnidarians , a phylum containing animals with radial symmetry, are the most closely related group to the bilaterians. Cnidarians are one of two groups of early animals considered to have defined structure, the second being the ctenophores . Ctenophores show biradial symmetry leading to the suggestion that they represent an intermediate step in the evolution of bilateral symmetry from radial symmetry. [ 27 ]
Interpretations based only on morphology are not sufficient to explain the evolution of symmetry. Two different explanations are proposed for the different symmetries in cnidarians and bilateria. The first suggestion is that an ancestral animal had no symmetry (was asymmetric) before cnidarians and bilaterians separated into different evolutionary lineages . Radial symmetry could have then evolved in cnidarians and bilateral symmetry in bilaterians. Alternatively, the second suggestion is that an ancestor of cnidarians and bilaterians had bilateral symmetry before the cnidarians evolved and became different by having radial symmetry. Both potential explanations are being explored and evidence continues to fuel the debate.
Although asymmetry is typically associated with being unfit, some species have evolved to be asymmetrical as an important adaptation . Many members of the phylum Porifera (sponges) have no symmetry, though some are radially symmetric. [ 28 ]
The presence of these asymmetrical features requires a process of symmetry breaking during development, both in plants and animals. Symmetry breaking occurs at several different levels in order to generate the anatomical asymmetry which we observe. These levels include asymmetric gene expression, protein expression, and activity of cells.
For example, left–right asymmetry in mammals has been investigated extensively in the embryos of mice. Such studies have led to support for the nodal flow hypothesis. In a region of the embryo referred to as the node there are small hair-like structures ( monocilia ) that all rotate together in a particular direction. This creates a unidirectional flow of signalling molecules causing these signals to accumulate on one side of the embryo and not the other. This results in the activation of different developmental pathways on each side, and subsequent asymmetry. [ 37 ] [ 38 ]
Much of the investigation of the genetic basis of symmetry breaking has been done on chick embryos. In chick embryos the left side expresses genes called NODAL and LEFTY2 that activate PITX2 to signal the development of left side structures. Whereas, the right side does not express PITX2 and consequently develops right side structures. [ 39 ] [ 40 ] A more complete pathway is shown in the image at the side of the page.
For more information about symmetry breaking in animals please refer to the left–right asymmetry page.
Plants also show asymmetry. For example the direction of helical growth in Arabidopsis , the most commonly studied model plant, shows left-handedness. Interestingly, the genes involved in this asymmetry are similar (closely related) to those in animal asymmetry – both LEFTY1 and LEFTY2 play a role. In the same way as animals, symmetry breaking in plants can occur at a molecular (genes/proteins), subcellular, cellular, tissue and organ level. [ 41 ]
Fluctuating asymmetry (FA), is a form of biological asymmetry , along with anti-symmetry and direction asymmetry. Fluctuating asymmetry refers to small, random deviations away from perfect bilateral symmetry. [ 42 ] [ 43 ] This deviation from perfection is thought to reflect the genetic and environmental pressures experienced throughout development, with greater pressures resulting in higher levels of asymmetry. [ 42 ] Examples of FA in the human body include unequal sizes (asymmetry) of bilateral features in the face and body, such as left and right eyes, ears, wrists, breasts , testicles , and thighs. | https://en.wikipedia.org/wiki/Symmetry_in_biology |
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics , relativistic quantum mechanics and quantum field theory , and with applications in the mathematical formulation of the standard model and condensed matter physics . In general, symmetry in physics , invariance , and conservation laws , are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.
This article outlines the connection between the classical form of continuous symmetries as well as their quantum operators , and relates them to the Lie groups , and relativistic transformations in the Lorentz group and Poincaré group .
The notational conventions used in this article are as follows. Boldface indicates vectors , four vectors , matrices , and vectorial operators , while quantum states use bra–ket notation . Wide hats are for operators , narrow hats are for unit vectors (including their components in tensor index notation ). The summation convention on the repeated tensor indices is used, unless stated otherwise. The Minkowski metric signature is (+−−−).
Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem .
The form of the fundamental quantum operators, for example the energy operator as a partial time derivative and momentum operator as a spatial gradient , becomes clear when one considers the initial state, then changes one parameter of it slightly. This can be done for displacements (lengths), durations (time), and angles (rotations). Additionally, the invariance of certain quantities can be seen by making such changes in lengths and angles, illustrating conservation of these quantities.
In what follows, transformations on only one-particle wavefunctions in the form:
Ω ^ ψ ( r , t ) = ψ ( r ′ , t ′ ) {\displaystyle {\widehat {\Omega }}\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t')}
are considered, where Ω ^ {\displaystyle {\widehat {\Omega }}} denotes a unitary operator . Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations. The inverse is the Hermitian conjugate Ω ^ − 1 = Ω ^ † {\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }} . The results can be extended to many-particle wavefunctions. Written in Dirac notation as standard, the transformations on quantum state vectors are:
Ω ^ | r ( t ) ⟩ = | r ′ ( t ′ ) ⟩ {\displaystyle {\widehat {\Omega }}\left|\mathbf {r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle }
Now, the action of Ω ^ {\displaystyle {\widehat {\Omega }}} changes ψ ( r , t ) to ψ ( r ′, t ′) , so the inverse Ω ^ − 1 = Ω ^ † {\displaystyle {\widehat {\Omega }}^{-1}={\widehat {\Omega }}^{\dagger }} changes ψ ( r ′, t ′) back to ψ ( r , t ) . Thus, an operator A ^ {\displaystyle {\widehat {A}}} invariant under Ω ^ {\displaystyle {\widehat {\Omega }}} satisfies [I am sorry, but this is non-sequitor. You have not laid a foundation for this proposition]:
A ^ ψ = Ω ^ † A ^ Ω ^ ψ ⇒ Ω ^ A ^ ψ = A ^ Ω ^ ψ . {\displaystyle {\widehat {A}}\psi ={\widehat {\Omega }}^{\dagger }{\widehat {A}}{\widehat {\Omega }}\psi \quad \Rightarrow \quad {\widehat {\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat {\Omega }}\psi .}
Concomitantly,
[ Ω ^ , A ^ ] ψ = 0 {\displaystyle [{\widehat {\Omega }},{\widehat {A}}]\psi =0}
for any state ψ . Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers , i.e. the operator equals its Hermitian conjugate , A ^ = A ^ † {\displaystyle {\widehat {A}}={\widehat {A}}^{\dagger }} .
Following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall [ 1 ] [ 2 ]
Let G be a Lie group , which is a group that locally is parameterized by a finite number N of real continuously varying parameters ξ 1 , ξ 2 , ..., ξ N . In more mathematical language, this means that G is a smooth manifold that is also a group, for which the group operations are smooth.
A representation which cannot be decomposed into a direct sum of other representations, is called irreducible . It is conventional to label irreducible representations by a superscripted number n in brackets, as in D ( n ) , or if there is more than one number, we write D ( n , m , ...) .
There is an additional subtlety that arises in quantum theory, where two vectors that differ by multiplication by a scalar represent the same physical state. Here, the pertinent notion of representation is a projective representation , one that only satisfies the composition law up to a scalar. In the context of quantum mechanical spin, such representations are called spinorial .
The space translation operator T ^ ( Δ r ) {\displaystyle {\widehat {T}}(\Delta \mathbf {r} )} acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δ r . The explicit expression T ^ {\displaystyle {\widehat {T}}} can be quickly determined by a Taylor expansion of ψ ( r + Δ r , t ) about r , then (keeping the first order term and neglecting second and higher order terms), replace the space derivatives by the momentum operator p ^ {\displaystyle {\widehat {\mathbf {p} }}} . Similarly for the time translation operator acting on the time parameter, the Taylor expansion of ψ ( r , t + Δ t ) is about t , and the time derivative replaced by the energy operator E ^ {\displaystyle {\widehat {E}}} .
The exponential functions arise by definition as those limits, due to Euler , and can be understood physically and mathematically as follows. A net translation can be composed of many small translations, so to obtain the translation operator for a finite increment, replace Δ r by Δ r / N and Δ t by Δ t / N , where N is a positive non-zero integer. Then as N increases, the magnitude of Δ r and Δ t become even smaller, while leaving the directions unchanged. Acting the infinitesimal operators on the wavefunction N times and taking the limit as N tends to infinity gives the finite operators.
Space and time translations commute, which means the operators and generators commute.
For a time-independent Hamiltonian, energy is conserved in time and quantum states are stationary states : the eigenstates of the Hamiltonian are the energy eigenvalues E :
U ^ ( t ) = exp ( − i Δ t E ℏ ) {\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac {i\Delta tE}{\hbar }}\right)}
and all stationary states have the form
ψ ( r , t + t 0 ) = U ^ ( t − t 0 ) ψ ( r , t 0 ) {\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat {U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}
where t 0 is the initial time, usually set to zero since there is no loss of continuity when the initial time is set.
An alternative notation is U ^ ( t − t 0 ) ≡ U ( t , t 0 ) {\displaystyle {\widehat {U}}(t-t_{0})\equiv U(t,t_{0})} .
The rotation operator, R ^ {\displaystyle {\widehat {R}}} , acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δ θ :
R ^ ( Δ θ , a ^ ) ψ ( r , t ) = ψ ( r ′ , t ) {\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ',t)}
where r′ are the rotated coordinates about an axis defined by a unit vector a ^ = ( a 1 , a 2 , a 3 ) {\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})} through an angular increment Δ θ , given by:
r ′ = R ^ ( Δ θ , a ^ ) r . {\displaystyle \mathbf {r} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})\mathbf {r} \,.}
where R ^ ( Δ θ , a ^ ) {\displaystyle {\widehat {R}}(\Delta \theta ,{\hat {\mathbf {a} }})} is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis Δ θ a ^ = Δ θ ( a 1 , a 2 , a 3 ) {\displaystyle \Delta \theta {\hat {\mathbf {a} }}=\Delta \theta (a_{1},a_{2},a_{3})} are the parameters, of the three-dimensional special orthogonal group , SO(3). The rotation matrices about the standard Cartesian basis vector e ^ x , e ^ y , e ^ z {\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}} through angle Δ θ , and the corresponding generators of rotations J = ( J x , J y , J z ) , are:
More generally for rotations about an axis defined by a ^ {\displaystyle {\hat {\mathbf {a} }}} , the rotation matrix elements are: [ 3 ]
[ R ^ ( θ , a ^ ) ] i j = ( δ i j − a i a j ) cos θ − ε i j k a k sin θ + a i a j {\displaystyle [{\widehat {R}}(\theta ,{\hat {\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos \theta -\varepsilon _{ijk}a_{k}\sin \theta +a_{i}a_{j}}
where δ ij is the Kronecker delta , and ε ijk is the Levi-Civita symbol .
It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x , y , or z -axis) then infer the general result, or use the general rotation matrix directly and tensor index notation with δ ij and ε ijk . To derive the infinitesimal rotation operator, which corresponds to small Δ θ , we use the small angle approximations sin(Δ θ ) ≈ Δ θ and cos(Δ θ ) ≈ 1 , then Taylor expand about r or r i , keep the first order term, and substitute the angular momentum operator components.
The z -component of angular momentum can be replaced by the component along the axis defined by a ^ {\displaystyle {\hat {\mathbf {a} }}} , using the dot product a ^ ⋅ L ^ {\displaystyle {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {L} }}} .
Again, a finite rotation can be made from many small rotations, replacing Δ θ by Δ θ / N and taking the limit as N tends to infinity gives the rotation operator for a finite rotation.
Rotations about the same axis do commute, for example a rotation through angles θ 1 and θ 2 about axis i can be written
R ( θ 1 + θ 2 , e i ) = R ( θ 1 e i ) R ( θ 2 e i ) , [ R ( θ 1 e i ) , R ( θ 2 e i ) ] = 0 . {\displaystyle R(\theta _{1}+\theta _{2},\mathbf {e} _{i})=R(\theta _{1}\mathbf {e} _{i})R(\theta _{2}\mathbf {e} _{i})\,,\quad [R(\theta _{1}\mathbf {e} _{i}),R(\theta _{2}\mathbf {e} _{i})]=0\,.}
However, rotations about different axes do not commute. The general commutation rules are summarized by
[ L i , L j ] = i ℏ ε i j k L k . {\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon _{ijk}L_{k}.}
In this sense, orbital angular momentum has the common sense properties of rotations. Each of the above commutators can be easily demonstrated by holding an everyday object and rotating it through the same angle about any two different axes in both possible orderings; the final configurations are different.
In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.
All previous quantities have classical definitions. Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum. The spin vector operator is denoted S ^ = ( S x ^ , S y ^ , S z ^ ) {\displaystyle {\widehat {\mathbf {S} }}=({\widehat {S_{x}}},{\widehat {S_{y}}},{\widehat {S_{z}}})} . The eigenvalues of its components are the possible outcomes (in units of ℏ {\displaystyle \hbar } ) of a measurement of the spin projected onto one of the basis directions.
Rotations (of ordinary space) about an axis a ^ {\displaystyle {\hat {\mathbf {a} }}} through angle θ about the unit vector a ^ {\displaystyle {\hat {a}}} in space acting on a multicomponent wave function (spinor) at a point in space is represented by:
S ^ ( θ , a ^ ) = exp ( − i ℏ θ a ^ ⋅ S ^ ) {\displaystyle {\widehat {S}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {S} }}\right)}
However, unlike orbital angular momentum in which the z -projection quantum number ℓ can only take positive or negative integer values (including zero), the z -projection spin quantum number s can take all positive and negative half-integer values. There are rotational matrices for each spin quantum number.
Evaluating the exponential for a given z -projection spin quantum number s gives a (2 s + 1)-dimensional spin matrix. This can be used to define a spinor as a column vector of 2 s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space.
For the simplest non-trivial case of s = 1/2, the spin operator is given by
S ^ = ℏ 2 σ {\displaystyle {\widehat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }}}
where the Pauli matrices in the standard representation are:
σ 1 = σ x = ( 0 1 1 0 ) , σ 2 = σ y = ( 0 − i i 0 ) , σ 3 = σ z = ( 1 0 0 − 1 ) {\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad \sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad \sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
The total angular momentum operator is the sum of the orbital and spin
J ^ = L ^ + S ^ {\displaystyle {\widehat {\mathbf {J} }}={\widehat {\mathbf {L} }}+{\widehat {\mathbf {S} }}}
and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules.
We have a similar rotation matrix:
J ^ ( θ , a ^ ) = exp ( − i ℏ θ a ^ ⋅ J ^ ) {\displaystyle {\widehat {J}}(\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar }}\theta {\hat {\mathbf {a} }}\cdot {\widehat {\mathbf {J} }}\right)}
The dynamical symmetry group of the n dimensional quantum harmonic oscillator is the special unitary group SU( n ). As an example, the number of infinitesimal generators of the corresponding Lie algebras of SU(2) and SU(3) are three and eight respectively. This leads to exactly three and eight independent conserved quantities (other than the Hamiltonian) in these systems.
The two dimensional quantum harmonic oscillator has the expected conserved quantities of the Hamiltonian and the angular momentum, but has additional hidden conserved quantities of energy level difference and another form of angular momentum.
Following is an overview of the Lorentz group; a treatment of boosts and rotations in spacetime. Throughout this section, see (for example) T. Ohlsson (2011) [ 4 ] and E. Abers (2004). [ 5 ]
Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector n ^ = ( n 1 , n 2 , n 3 ) {\displaystyle {\hat {\mathbf {n} }}=(n_{1},n_{2},n_{3})} , and a rotation angle θ about a three-dimensional unit vector a ^ = ( a 1 , a 2 , a 3 ) {\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})} defining an axis, so φ n ^ = φ ( n 1 , n 2 , n 3 ) {\displaystyle \varphi {\hat {\mathbf {n} }}=\varphi (n_{1},n_{2},n_{3})} and θ a ^ = θ ( a 1 , a 2 , a 3 ) {\displaystyle \theta {\hat {\mathbf {a} }}=\theta (a_{1},a_{2},a_{3})} are together six parameters of the Lorentz group (three for rotations and three for boosts). The Lorentz group is 6-dimensional.
The rotation matrices and rotation generators considered above form the spacelike part of a four-dimensional matrix, representing pure-rotation Lorentz transformations. Three of the Lorentz group elements R ^ x , R ^ y , R ^ z {\displaystyle {\widehat {R}}_{x},{\widehat {R}}_{y},{\widehat {R}}_{z}} and generators J = ( J 1 , J 2 , J 3 ) for pure rotations are:
The rotation matrices act on any four vector A = ( A 0 , A 1 , A 2 , A 3 ) and rotate the space-like components according to
A ′ = R ^ ( Δ θ , n ^ ) A {\displaystyle \mathbf {A} '={\widehat {R}}(\Delta \theta ,{\hat {\mathbf {n} }})\mathbf {A} }
leaving the time-like coordinate unchanged. In matrix expressions, A is treated as a column vector .
A boost with velocity c tanh φ in the x , y , or z directions given by the standard Cartesian basis vector e ^ x , e ^ y , e ^ z {\displaystyle {\hat {\mathbf {e} }}_{x},{\hat {\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}} , are the boost transformation matrices. These matrices B ^ x , B ^ y , B ^ z {\displaystyle {\widehat {B}}_{x},{\widehat {B}}_{y},{\widehat {B}}_{z}} and the corresponding generators K = ( K 1 , K 2 , K 3 ) are the remaining three group elements and generators of the Lorentz group:
The boost matrices act on any four vector A = ( A 0 , A 1 , A 2 , A 3 ) and mix the time-like and the space-like components, according to:
A ′ = B ^ ( φ , n ^ ) A {\displaystyle \mathbf {A} '={\widehat {B}}(\varphi ,{\hat {\mathbf {n} }})\mathbf {A} }
The term "boost" refers to the relative velocity between two frames, and is not to be conflated with momentum as the generator of translations , as explained below .
Products of rotations give another rotation (a frequent exemplification of a subgroup), while products of boosts and boosts or of rotations and boosts cannot be expressed as pure boosts or pure rotations. In general, any Lorentz transformation can be expressed as a product of a pure rotation and a pure boost. For more background see (for example) B.R. Durney (2011) [ 6 ] and H.L. Berk et al. [ 7 ] and references therein.
The boost and rotation generators have representations denoted D ( K ) and D ( J ) respectively, the capital D in this context indicates a group representation .
For the Lorentz group, the representations D ( K ) and D ( J ) of the generators K and J fulfill the following commutation rules.
In all commutators, the boost entities mixed with those for rotations, although rotations alone simply give another rotation. Exponentiating the generators gives the boost and rotation operators which combine into the general Lorentz transformation, under which the spacetime coordinates transform from one rest frame to another boosted and/or rotating frame. Likewise, exponentiating the representations of the generators gives the representations of the boost and rotation operators, under which a particle's spinor field transforms.
In the literature, the boost generators K and rotation generators J are sometimes combined into one generator for Lorentz transformations M , an antisymmetric four-dimensional matrix with entries:
M 0 a = − M a 0 = K a , M a b = ε a b c J c . {\displaystyle M^{0a}=-M^{a0}=K_{a}\,,\quad M^{ab}=\varepsilon _{abc}J_{c}\,.}
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω , with entries:
ω 0 a = − ω a 0 = φ n a , ω a b = θ ε a b c a c , {\displaystyle \omega _{0a}=-\omega _{a0}=\varphi n_{a}\,,\quad \omega _{ab}=\theta \varepsilon _{abc}a_{c}\,,}
The general Lorentz transformation is then:
Λ ( φ , n ^ , θ , a ^ ) = exp ( − i 2 ω α β M α β ) = exp [ − i 2 ( φ n ^ ⋅ K + θ a ^ ⋅ J ) ] {\displaystyle \Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})=\exp \left(-{\frac {i}{2}}\omega _{\alpha \beta }M^{\alpha \beta }\right)=\exp \left[-{\frac {i}{2}}\left(\varphi {\hat {\mathbf {n} }}\cdot \mathbf {K} +\theta {\hat {\mathbf {a} }}\cdot \mathbf {J} \right)\right]}
with summation over repeated matrix indices α and β . The Λ matrices act on any four vector A = ( A 0 , A 1 , A 2 , A 3 ) and mix the time-like and the space-like components, according to:
A ′ = Λ ( φ , n ^ , θ , a ^ ) A {\displaystyle \mathbf {A} '=\Lambda (\varphi ,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf {a} }})\mathbf {A} }
In relativistic quantum mechanics , wavefunctions are no longer single-component scalar fields, but now 2(2 s + 1) component spinor fields, where s is the spin of the particle. The transformations of these functions in spacetime are given below.
Under a proper orthochronous Lorentz transformation ( r , t ) → Λ( r , t ) in Minkowski space , all one-particle quantum states ψ σ locally transform under some representation D of the Lorentz group : [ 8 ] [ 9 ]
ψ σ ( r , t ) → D ( Λ ) ψ σ ( Λ − 1 ( r , t ) ) {\displaystyle \psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))}
where D (Λ) is a finite-dimensional representation, in other words a (2 s + 1)×(2 s + 1) dimensional square matrix , and ψ is thought of as a column vector containing components with the (2 s + 1) allowed values of σ :
ψ ( r , t ) = [ ψ σ = s ( r , t ) ψ σ = s − 1 ( r , t ) ⋮ ψ σ = − s + 1 ( r , t ) ψ σ = − s ( r , t ) ] ⇌ ψ ( r , t ) † = [ ψ σ = s ( r , t ) ⋆ ψ σ = s − 1 ( r , t ) ⋆ ⋯ ψ σ = − s + 1 ( r , t ) ⋆ ψ σ = − s ( r , t ) ⋆ ] {\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}}
The irreducible representations of D ( K ) and D ( J ) , in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new operators:
A = J + i K 2 , B = J − i K 2 , {\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,}
so A and B are simply complex conjugates of each other, it follows they satisfy the symmetrically formed commutators:
[ A i , A j ] = ε i j k A k , [ B i , B j ] = ε i j k B k , [ A i , B j ] = 0 , {\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}
and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore, A and B form operator algebras analogous to angular momentum; same ladder operators , z -projections, etc., independently of each other as each of their components mutually commute. By the analogy to the spin quantum number, we can introduce positive integers or half integers, a, b , with corresponding sets of values m = a , a − 1, ... − a + 1, − a and n = b , b − 1, ... − b + 1, − b . The matrices satisfying the above commutation relations are the same as for spins a and b have components given by multiplying Kronecker delta values with angular momentum matrix elements:
( A x ) m ′ n ′ , m n = δ n ′ n ( J x ( m ) ) m ′ m ( B x ) m ′ n ′ , m n = δ m ′ m ( J x ( n ) ) n ′ n {\displaystyle \left(A_{x}\right)_{m'n',mn}=\delta _{n'n}\left(J_{x}^{(m)}\right)_{m'm}\,\quad \left(B_{x}\right)_{m'n',mn}=\delta _{m'm}\left(J_{x}^{(n)}\right)_{n'n}} ( A y ) m ′ n ′ , m n = δ n ′ n ( J y ( m ) ) m ′ m ( B y ) m ′ n ′ , m n = δ m ′ m ( J y ( n ) ) n ′ n {\displaystyle \left(A_{y}\right)_{m'n',mn}=\delta _{n'n}\left(J_{y}^{(m)}\right)_{m'm}\,\quad \left(B_{y}\right)_{m'n',mn}=\delta _{m'm}\left(J_{y}^{(n)}\right)_{n'n}} ( A z ) m ′ n ′ , m n = δ n ′ n ( J z ( m ) ) m ′ m ( B z ) m ′ n ′ , m n = δ m ′ m ( J z ( n ) ) n ′ n {\displaystyle \left(A_{z}\right)_{m'n',mn}=\delta _{n'n}\left(J_{z}^{(m)}\right)_{m'm}\,\quad \left(B_{z}\right)_{m'n',mn}=\delta _{m'm}\left(J_{z}^{(n)}\right)_{n'n}}
where in each case the row number m′n′ and column number mn are separated by a comma, and in turn:
( J z ( m ) ) m ′ m = m δ m ′ m ( J x ( m ) ± i J y ( m ) ) m ′ m = m δ a ′ , a ± 1 ( a ∓ m ) ( a ± m + 1 ) {\displaystyle \left(J_{z}^{(m)}\right)_{m'm}=m\delta _{m'm}\,\quad \left(J_{x}^{(m)}\pm iJ_{y}^{(m)}\right)_{m'm}=m\delta _{a',a\pm 1}{\sqrt {(a\mp m)(a\pm m+1)}}}
and similarly for J ( n ) . [ note 1 ] The three J ( m ) matrices are each (2 m + 1)×(2 m + 1) square matrices, and the three J ( n ) are each (2 n + 1)×(2 n + 1) square matrices. The integers or half-integers m and n numerate all the irreducible representations by, in equivalent notations used by authors: D ( m , n ) ≡ ( m , n ) ≡ D ( m ) ⊗ D ( n ) , which are each [(2 m + 1)(2 n + 1)]×[(2 m + 1)(2 n + 1)] square matrices.
Applying this to particles with spin s ;
In these cases the D refers to any of D ( J ) , D ( K ) , or a full Lorentz transformation D (Λ) .
In the context of the Dirac equation and Weyl equation , the Weyl spinors satisfying the Weyl equation transform under the simplest irreducible spin representations of the Lorentz group, since the spin quantum number in this case is the smallest non-zero number allowed: 1/2. The 2-component left-handed Weyl spinor transforms under D (1/2, 0) and the 2-component right-handed Weyl spinor transforms under D (0, 1/2) . Dirac spinors satisfying the Dirac equation transform under the representation D (1/2, 0) ⊕ D (0, 1/2) , the direct sum of the irreps for the Weyl spinors.
Space translations , time translations , rotations , and boosts , all taken together, constitute the Poincaré group . The group elements are the three rotation matrices and three boost matrices (as in the Lorentz group), and one for time translations and three for space translations in spacetime. There is a generator for each. Therefore, the Poincaré group is 10-dimensional.
In special relativity , space and time can be collected into a four-position vector X = ( ct , − r ) , and in parallel so can energy and momentum which combine into a four-momentum vector P = ( E / c , − p ) . With relativistic quantum mechanics in mind, the time duration and spatial displacement parameters (four in total, one for time and three for space) combine into a spacetime displacement Δ X = ( c Δ t , −Δ r ) , and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,
P ^ = ( E ^ c , − p ^ ) = i ℏ ( 1 c ∂ ∂ t , ∇ ) , {\displaystyle {\widehat {\mathbf {P} }}=\left({\frac {\widehat {E}}{c}},-{\widehat {\mathbf {p} }}\right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)\,,}
which are the generators of spacetime translations (four in total, one time and three space):
X ^ ( Δ X ) = exp ( − i ℏ Δ X ⋅ P ^ ) = exp [ − i ℏ ( Δ t E ^ + Δ r ⋅ p ^ ) ] . {\displaystyle {\widehat {X}}(\Delta \mathbf {X} )=\exp \left(-{\frac {i}{\hbar }}\Delta \mathbf {X} \cdot {\widehat {\mathbf {P} }}\right)=\exp \left[-{\frac {i}{\hbar }}\left(\Delta t{\widehat {E}}+\Delta \mathbf {r} \cdot {\widehat {\mathbf {p} }}\right)\right]\,.}
There are commutation relations between the components four-momentum P (generators of spacetime translations), and angular momentum M (generators of Lorentz transformations), that define the Poincaré algebra: [ 10 ] [ 11 ]
where η is the Minkowski metric tensor. (It is common to drop any hats for the four-momentum operators in the commutation relations). These equations are an expression of the fundamental properties of space and time as far as they are known today. They have a classical counterpart where the commutators are replaced by Poisson brackets .
To describe spin in relativistic quantum mechanics, the Pauli–Lubanski pseudovector
W μ = 1 2 ε μ ν ρ σ J ν ρ P σ , {\displaystyle W_{\mu }={\frac {1}{2}}\varepsilon _{\mu \nu \rho \sigma }J^{\nu \rho }P^{\sigma },}
a Casimir operator , is the constant spin contribution to the total angular momentum, and there are commutation relations between P and W and between M and W :
[ P μ , W ν ] = 0 , {\displaystyle \left[P^{\mu },W^{\nu }\right]=0\,,} [ J μ ν , W ρ ] = i ( η ρ ν W μ − η ρ μ W ν ) , {\displaystyle \left[J^{\mu \nu },W^{\rho }\right]=i\left(\eta ^{\rho \nu }W^{\mu }-\eta ^{\rho \mu }W^{\nu }\right)\,,} [ W μ , W ν ] = − i ϵ μ ν ρ σ W ρ P σ . {\displaystyle \left[W_{\mu },W_{\nu }\right]=-i\epsilon _{\mu \nu \rho \sigma }W^{\rho }P^{\sigma }\,.}
Invariants constructed from W , instances of Casimir invariants can be used to classify irreducible representations of the Lorentz group.
Group theory is an abstract way of mathematically analyzing symmetries. Unitary operators are paramount to quantum theory, so unitary groups are important in particle physics. The group of N dimensional unitary square matrices is denoted U( N ). Unitary operators preserve inner products which means probabilities are also preserved, so the quantum mechanics of the system is invariant under unitary transformations. Let U ^ {\displaystyle {\widehat {U}}} be a unitary operator, so the inverse is the Hermitian adjoint U ^ − 1 = U ^ † {\displaystyle {\widehat {U}}^{-1}={\widehat {U}}^{\dagger }} , which commutes with the Hamiltonian:
[ U ^ , H ^ ] = 0 {\displaystyle \left[{\widehat {U}},{\widehat {H}}\right]=0}
then the observable corresponding to the operator U ^ {\displaystyle {\widehat {U}}} is conserved, and the Hamiltonian is invariant under the transformation U ^ {\displaystyle {\widehat {U}}} .
Since the predictions of quantum mechanics should be invariant under the action of a group, physicists look for unitary transformations to represent the group.
Important subgroups of each U( N ) are those unitary matrices which have unit determinant (or are "unimodular"): these are called the special unitary groups and are denoted SU( N ).
The simplest unitary group is U(1), which is just the complex numbers of modulus 1. This one-dimensional matrix entry is of the form:
U = e − i θ {\displaystyle U=e^{-i\theta }}
in which θ is the parameter of the group, and the group is Abelian since one-dimensional matrices always commute under matrix multiplication. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If there is a quantum number a associated with the U(1) symmetry, for example baryon and the three lepton numbers in electromagnetic interactions, we have:
U = e − i a θ {\displaystyle U=e^{-ia\theta }}
The general form of an element of a U(2) element is parametrized by two complex numbers a and b :
U = ( a b − b ⋆ a ⋆ ) {\displaystyle U={\begin{pmatrix}a&b\\-b^{\star }&a^{\star }\\\end{pmatrix}}}
and for SU(2), the determinant is restricted to 1:
det ( U ) = a a ⋆ + b b ⋆ = | a | 2 + | b | 2 = 1 {\displaystyle \det(U)=aa^{\star }+bb^{\star }={|a|}^{2}+{|b|}^{2}=1}
In group theoretic language, the Pauli matrices are the generators of the special unitary group in two dimensions, denoted SU(2). Their commutation relation is the same as for orbital angular momentum, aside from a factor of 2:
[ σ a , σ b ] = 2 i ℏ ε a b c σ c {\displaystyle [\sigma _{a},\sigma _{b}]=2i\hbar \varepsilon _{abc}\sigma _{c}}
A group element of SU(2) can be written:
U ( θ , e ^ j ) = e i θ σ j / 2 {\displaystyle U(\theta ,{\hat {\mathbf {e} }}_{j})=e^{i\theta \sigma _{j}/2}}
where σ j is a Pauli matrix, and the group parameters are the angles turned through about an axis.
The two-dimensional isotropic quantum harmonic oscillator has symmetry group SU(2), while the symmetry algebra of the rational anisotropic oscillator is a nonlinear extension of u(2). [ 12 ]
The eight Gell-Mann matrices λ n (see article for them and the structure constants) are important for quantum chromodynamics . They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics. They are the generators for the SU(3) group, so an element of SU(3) can be written analogously to an element of SU(2):
U ( θ , e ^ j ) = exp ( − i 2 ∑ n = 1 8 θ n λ n ) {\displaystyle U(\theta ,{\hat {\mathbf {e} }}_{j})=\exp \left(-{\frac {i}{2}}\sum _{n=1}^{8}\theta _{n}\lambda _{n}\right)}
where θ n are eight independent parameters. The λ n matrices satisfy the commutator:
[ λ a , λ b ] = 2 i f a b c λ c {\displaystyle \left[\lambda _{a},\lambda _{b}\right]=2if_{abc}\lambda _{c}}
where the indices a , b , c take the values 1, 2, 3, ..., 8. The structure constants f abc are totally antisymmetric in all indices analogous to those of SU(2). In the standard colour charge basis ( r for red, g for green, b for blue):
| r ⟩ = ( 1 0 0 ) , | g ⟩ = ( 0 1 0 ) , | b ⟩ = ( 0 0 1 ) {\displaystyle |r\rangle ={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad |g\rangle ={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad |b\rangle ={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
the colour states are eigenstates of the λ 3 and λ 8 matrices, while the other matrices mix colour states together.
The eight gluons states (8-dimensional column vectors) are simultaneous eigenstates of the adjoint representation of SU(3) , the 8-dimensional representation acting on its own Lie algebra su(3) , for the λ 3 and λ 8 matrices. By forming tensor products of representations (the standard representation and its dual) and taking appropriate quotients, protons and neutrons, and other hadrons are eigenstates of various representations of SU(3) of color. The representations of SU(3) can be described by a "theorem of the highest weight". [ 13 ]
In relativistic quantum mechanics, relativistic wave equations predict a remarkable symmetry of nature: that every particle has a corresponding antiparticle . This is mathematically contained in the spinor fields which are the solutions of the relativistic wave equations.
Charge conjugation switches particles and antiparticles. Physical laws and interactions unchanged by this operation have C symmetry .
In quantum electrodynamics , the local symmetry group is U(1) and is abelian . In quantum chromodynamics , the local symmetry group is SU(3) and is non-abelian .
The electromagnetic interaction is mediated by photons , which have no electric charge. The electromagnetic tensor has an electromagnetic four-potential field possessing gauge symmetry.
The strong (color) interaction is mediated by gluons , which can have eight color charges . There are eight gluon field strength tensors with corresponding gluon four potentials field, each possessing gauge symmetry.
Analogous to the spin operator, there are color charge operators in terms of the Gell-Mann matrices λ j :
F ^ j = 1 2 λ j {\displaystyle {\hat {F}}_{j}={\frac {1}{2}}\lambda _{j}}
and since color charge is a conserved charge, all color charge operators must commute with the Hamiltonian:
[ F ^ j , H ^ ] = 0 {\displaystyle \left[{\hat {F}}_{j},{\hat {H}}\right]=0}
Isospin is conserved in strong interactions.
Magnetic monopoles can be theoretically realized, although current observations and theory are consistent with them existing or not existing. Electric and magnetic charges can effectively be "rotated into one another" by a duality transformation .
A Lie superalgebra is an algebra in which (suitable) basis elements either have a commutation relation or have an anticommutation relation. Symmetries have been proposed to the effect that all fermionic particles have bosonic analogues, and vice versa. These symmetry have theoretical appeal in that no extra assumptions (such as existence of strings) barring symmetries are made. In addition, by assuming supersymmetry, a number of puzzling issues can be resolved. These symmetries, which are represented by Lie superalgebras, have not been confirmed experimentally. It is now believed that they are broken symmetries, if they exist. But it has been speculated that dark matter is constitutes gravitinos , a spin 3/2 particle with mass, its supersymmetric partner being the graviton .
The concept of exchange symmetry is derived from a fundamental postulate of quantum statistics , which states that no observable physical quantity should change after exchanging two identical particles . It states that because all observables are proportional to | ψ | 2 {\displaystyle \left|\psi \right|^{2}} for a system of identical particles, the wave function ψ {\displaystyle \psi } must either remain the same or change sign upon such an exchange. More generally, for a system of n identical particles the wave function ψ {\displaystyle \psi } must transform as an irreducible representation of the finite symmetric group S n . It turns out that, according to the spin-statistics theorem , fermion states transform as the antisymmetric irreducible representation of S n and boson states as the symmetric irreducible representation.
Because the exchange of two identical particles is mathematically equivalent to the rotation of each particle by 180 degrees (and so to the rotation of one particle's frame by 360 degrees), [ 14 ] the symmetric nature of the wave function depends on the particle's spin after the rotation operator is applied to it. Integer spin particles do not change the sign of their wave function upon a 360 degree rotation—therefore the sign of the wave function of the entire system does not change. Semi-integer spin particles change the sign of their wave function upon a 360 degree rotation (see more in spin–statistics theorem ).
Particles for which the wave function does not change sign upon exchange are called bosons , or particles with a symmetric wave function. The particles for which the wave function of the system changes sign are called fermions , or particles with an antisymmetric wave function.
Fermions therefore obey different statistics (called Fermi–Dirac statistics ) than bosons (which obey Bose–Einstein statistics ). One of the consequences of Fermi–Dirac statistics is the exclusion principle for fermions—no two identical fermions can share the same quantum state (in other words, the wave function of two identical fermions in the same state is zero). This in turn results in degeneracy pressure for fermions—the strong resistance of fermions to compression into smaller volume. This resistance gives rise to the “stiffness” or “rigidity” of ordinary atomic matter (as atoms contain electrons which are fermions). | https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics |
The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, that is, it is the order of its symmetry group . The object can be a molecule, crystal lattice, lattice, tiling, or in general any kind of mathematical object that admits symmetries. [ 1 ]
In statistical thermodynamics , the symmetry number corrects for any overcounting of equivalent molecular conformations in the partition function . In this sense, the symmetry number depends upon how the partition function is formulated. For example, if one writes the partition function of ethane so that the integral includes full rotation of a methyl , then the 3-fold rotational symmetry of the methyl group contributes a factor of 3 to the symmetry number; but if one writes the partition function so that the integral includes only one rotational energy well of the methyl, then the methyl rotation does not contribute to the symmetry number. [ 2 ] | https://en.wikipedia.org/wiki/Symmetry_number |
Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of quantum mechanics in physics and chemistry, for example, it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions (based on selection rules ), without doing the exact rigorous calculations (which, in some cases, may not even be possible). To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and are relatively easier to analyze.
The physical laws governing a system is generally written as a relation (equations, differential equations, integral equations etc.). An operation on the ingredients of this relation, which keeps the form of the relations invariant is called a symmetry transformation or a symmetry of the system.
Symmetry is a fundamentally important concept in quantum mechanics. It can predict conserved quantities and provide quantum numbers. It can predict degeneracies of eigenstates and gives insights about the matrix elements of the Hamiltonian without calculating them. Rather than looking into individual symmetries, it is sometimes more convenient to look into the general relations between the symmetries. It turns out that Group theory is the most efficient way of doing this.
A group is a mathematical structure (usually denoted in the form ( G ,*)) consisting of a set G and a binary operation ′ ∗ ′ {\displaystyle '*'} (sometimes loosely called 'multiplication'), satisfying the following properties:
The set of all symmetry transformations of a Hamiltonian has the structure of a group, with group multiplication equivalent to applying the transformations one after the other. The group elements can be represented as matrices, so that the group operation becomes the ordinary matrix multiplication. In quantum mechanics, the evolution of an arbitrary superposition of states are given by unitary operators, so each of the elements of the symmetry groups are unitary operators. Now any unitary operator can be expressed as the exponential of some Hermitian operator . So, the corresponding Hermitian operators are the ' generators ' of the symmetry group . These unitary transformations act on the Hamiltonian operator in some Hilbert space in a way that the Hamiltonian remains invariant under the transformations. In other words, the symmetry operators commute with the Hamiltonian. If U {\displaystyle U} represents the unitary symmetry operator and acts on the Hamiltonian H {\displaystyle H} , then;
These operators have the above-mentioned properties of a group:
So, by the symmetry of a system, we mean a set of operators, each of which commutes with the Hamiltonian, and they form a symmetry group . This group may be abelian or non-abelian. Depending upon which one it is, the properties of the system changes (for example, if the group is abelian, there would be no degeneracy ). Corresponding to every different kind of symmetry in a system, we can find a symmetry group associated with it.
It follows that the generator T {\displaystyle T} of the symmetry group also commutes with the Hamiltonian. Now, it follows that:
d ⟨ T ⟩ d t = d ⟨ Ψ | T | Ψ ⟩ d t = ⟨ ∂ Ψ ∂ t | T | Ψ ⟩ + ⟨ Ψ | ∂ T ∂ t | Ψ ⟩ + ⟨ Ψ | T | ∂ Ψ ∂ t ⟩ {\displaystyle {\frac {d\left\langle T\right\rangle }{dt}}={\frac {d\left\langle \Psi |T|\Psi \right\rangle }{dt}}=\left\langle {\frac {\partial \Psi }{\partial t}}|T|\Psi \right\rangle +\left\langle \Psi |{\frac {\partial T}{\partial t}}|\Psi \right\rangle +\left\langle \Psi |T|{\frac {\partial \Psi }{\partial t}}\right\rangle }
Now,
i ℏ ∂ | Ψ ⟩ ∂ t = H | Ψ ⟩ {\displaystyle i\hbar {\frac {\partial \left|\Psi \right\rangle }{\partial t}}=H\left|\Psi \right\rangle }
So,
d ⟨ T ⟩ d t = − 1 i ℏ ⟨ Ψ | H T | Ψ ⟩ + 1 i ℏ ⟨ Ψ | T H | Ψ ⟩ + ⟨ ∂ T ∂ t ⟩ {\displaystyle {\frac {d\left\langle T\right\rangle }{dt}}=-{\frac {1}{i\hbar }}\left\langle \Psi |HT|\Psi \right\rangle +{\frac {1}{i\hbar }}\left\langle \Psi |TH|\Psi \right\rangle +\left\langle {\frac {\partial T}{\partial t}}\right\rangle }
as H is also Hermitian. So we have,
d ⟨ T ⟩ d t = 1 i ℏ ⟨ [ H , T ] ⟩ + ⟨ ∂ T ∂ t ⟩ {\displaystyle {\frac {d\left\langle T\right\rangle }{dt}}={\frac {1}{i\hbar }}\left\langle [H,T]\right\rangle +\left\langle {\frac {\partial T}{\partial t}}\right\rangle }
Now, [ H , T ] = 0 {\displaystyle [H,T]=0} as stated above, and if the operator T does not have any explicit time-dependence;
d ⟨ T ⟩ d t = 0 {\displaystyle {\frac {d\left\langle T\right\rangle }{dt}}=0} ⇒ ⟨ T ⟩ {\displaystyle \Rightarrow \left\langle T\right\rangle } is a constant, independent of what the state | Ψ ⟩ {\displaystyle \left|\Psi \right\rangle } may be.
So the observable corresponding to the operator T, is conserved.
Some specific examples can be systems having rotational , translational invariance etc. For a rotationally invariant system, the symmetry group of the Hamiltonian is the general rotation group. Now, if (say) the system is invariant about any rotation about Z-axis (i.e., the system has axial symmetry ), then the symmetry group of the Hamiltonian is the group of rotation about the symmetry axis. Now, this group is generated by the Z-component of the orbital angular momentum, L z {\displaystyle {L}_{z}} (general group element R ( α ) = e − i α L z ℏ {\displaystyle R(\alpha )={{e}^{\frac {-i\alpha {{L}_{z}}}{\hbar }}}} ). Thus, L z {\displaystyle {L}_{z}} commutes with H {\displaystyle H} for this system and Z-component of the angular momentum is conserved. Similarly, translation symmetry gives rise to conservation of linear momentum, inversion symmetry gives rise to parity conservation and so on.
A molecule at equilibrium in a certain electronic state usually has some geometrical symmetry. This symmetry is described by a certain point group which consists of operations (called symmetry operations) that produce a spatial orientation of the molecule that is indistinguishable from the starting configuration. There are five types of point group symmetry operation: identity, rotation, reflection, inversion and improper rotation or rotation-reflection. Common to all symmetry operations is that the geometrical center-point of the molecule does not change its position; hence the name point group . One can determine the elements of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group, the elements are not to be interpreted in the same way. Instead the elements rotate and/or reflect the vibronic (vibration-electronic) coordinates and these elements commute with the vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates. The symmetry classification of the rotational levels, the eigenstates of the full (rovibronic nuclear spin) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins . [ 1 ] See the Section Inversion symmetry and nuclear permutation symmetry below. The elements of permutation-inversion groups commute with the full molecular Hamiltonian. In addition to point groups, there exists another kind of group important in crystallography , where translation in 3-D also needs to be taken care of. They are known as space groups .
The five basic symmetry operations mentioned above are: [ 2 ]
All other symmetry present in a specific molecule are a combination of these 5 operations.
The Schoenflies (or Schönflies ) notation , named after the German mathematician Arthur Moritz Schoenflies , is one of two conventions commonly used to describe point groups. This notation is used in spectroscopy and is used here to specify a molecular point group.
There are two point groups for diatomic molecules: C ∞ v {\displaystyle {{C}_{\infty v}}} for heteronuclear diatomics, and D ∞ h {\displaystyle {{D}_{\infty h}}} for homonuclear diatomics.
The group C ∞ v {\displaystyle {{C}_{\infty v}}} , contains rotations C ( ϕ ) {\displaystyle C(\phi )} through any angle ϕ {\displaystyle \phi } about the axis of symmetry and an infinite number of reflections σ v {\displaystyle {{\sigma }_{v}}} through the planes containing the inter-nuclear axis (or the vertical axis, that is reason of the subscript ' v ').In the group C ∞ v {\displaystyle {{C}_{\infty v}}} all planes of symmetry are equivalent, so that all reflections σ v {\displaystyle {{\sigma }_{v}}} form a single class with a continuous series of elements; the axis of symmetry is bilateral, so that there is a continuous series of classes, each containing two elements C ( ± ϕ ) {\displaystyle C(\pm \phi )} . Note that this group is non-abelian and there exists an infinite number of irreducible representations in the group. The character table of the group is as follows:
E
2c ∞ ϕ {\displaystyle {\phi }}
...
∞ σ v {\displaystyle \infty {{\sigma }_{v}}}
linear functions,
rotations
quadratic
In addition to axial reflection symmetry, homonuclear diatomic molecules are symmetric with respect to inversion or reflection through any axis in the plane passing through the point of symmetry and perpendicular to the inter-nuclear axis.
The classes of the group D ∞ h {\displaystyle {{D}_{\infty h}}} can be obtained from those of the group C ∞ v {\displaystyle {{C}_{\infty v}}} using the relation between the two groups: D ∞ h = C ∞ v × C i {\displaystyle {{D}_{\infty h}}={{C}_{\infty v}}\times {{C}_{i}}} . Like C ∞ v {\displaystyle {{C}_{\infty v}}} , D ∞ h {\displaystyle {{D}_{\infty h}}} is non-abelian and there are an infinite number of irreducible representations in the group. The character table of this group is as follows:
E
2c ∞ ϕ {\displaystyle {\phi }}
...
∞ σ h {\displaystyle \infty {{\sigma }_{h}}}
∞ {\displaystyle \infty } c 2 ′ {\displaystyle c_{2}^{'}}
linear functions,
rotations
quadratic
Point group
Symmetry operations
or group operations
or group elements
Simple description of typical geometry
and irreducible
representations (irreps)
Example
Unlike a single atom, the Hamiltonian of a diatomic molecule doesn't commute with L 2 {\displaystyle {{L}^{2}}} . So the quantum number l {\displaystyle l} is no longer a good quantum number . The internuclear axis chooses a specific direction in space and the potential is no longer spherically symmetric. Instead, L z {\displaystyle {{L}_{z}}} and J z {\displaystyle {{J}_{z}}} commutes with the Hamiltonian H {\displaystyle H} (taking the arbitrary internuclear axis as the Z axis). But L x , L y {\displaystyle {{L}_{x}},{{L}_{y}}} do not commute with H {\displaystyle H} due to the fact that the electronic Hamiltonian of a diatomic molecule is invariant under rotations about the internuclear line (the Z axis), but not under rotations about the X or Y axes. Again, S 2 {\displaystyle {{S}^{2}}} and S z {\displaystyle {{S}_{z}}} act on a different Hilbert space, so they commute with H {\displaystyle H} in this case also. The electronic Hamiltonian for a diatomic molecule is also invariant under reflections in all planes containing the internuclear line. The ( X-Z ) plane is such a plane, and reflection of the coordinates of the electrons in this plane corresponds to the operation y i → − y i {\displaystyle {{y}_{i}}\to -{{y}_{i}}} . If A y {\displaystyle {{A}_{y}}} is the operator that performs this reflection, then [ A y , H ] = 0 {\displaystyle [{{A}_{y}},H]=0} . So the Complete Set of Commuting Operators (CSCO) for a general heteronuclear diatomic molecule is { H , J z , L z , S 2 , S z , A } {\displaystyle \{H,{\text{ }}{{J}_{z}},{{L}_{z}},{{S}^{2}},{{S}_{z}},A\}} ; where A {\displaystyle A} is an operator that inverts only one of the two spatial co-ordinates ( x or y).
In the special case of a homonuclear diatomic molecule, there is an extra symmetry since in addition to the axis of symmetry provided by the internuclear axis, there is a centre of symmetry at the midpoint of the distance between the two nuclei (the symmetry discussed in this paragraph only depends on the two nuclear charges being the same. The two nuclei can therefore have different mass, that is they can be two isotopes of the same species such as the proton and the deuteron, or O 16 {\displaystyle {{O}^{16}}} and O 18 {\displaystyle {{O}^{18}}} , and so on). Choosing this point as the origin of the coordinates, the Hamiltonian is invariant under an inversion of the coordinates of all electrons with respect to that origin, namely in the operation r → i → − r → i {\displaystyle {{\vec {r}}_{i}}\to -{{\vec {r}}_{i}}} . Thus the parity operator Π {\displaystyle \Pi } . Thus the CSCO for a homonuclear diatomic molecule is { H , J z , L z , S 2 , S z , A , Π } {\displaystyle \left\{H,{\text{ }}{{J}_{z}},{{L}_{z}},{{S}^{2}},{{S}_{z}},A,{\text{ }}\Pi \right\}} .
Molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule. It is the equivalent of the term symbol for the atomic case. We already know the CSCO of the most general diatomic molecule. So, the good quantum numbers can sufficiently describe the state of the diatomic molecule. Here, the symmetry is explicitly stated in the nomenclature.
Here, the system is not spherically symmetric. So, [ H , L 2 ] ≠ 0 {\displaystyle [H,L^{2}]\neq 0} , and the state cannot be depicted in terms of l {\displaystyle l} as an eigenstate of the Hamiltonian is not an eigenstate of L 2 {\displaystyle L^{2}} anymore (in contrast to the atomic term symbol, where the states were written as 2 S + 1 L J {\displaystyle ^{2S+1}L_{J}} ). But, as [ H , L z ] = 0 {\displaystyle [H,L_{z}]=0} , the eigenvalues corresponding to L z {\displaystyle L_{z}} can still be used. If L z | Ψ ⟩ = M L ℏ | Ψ ⟩ ; M L = 0 , ± 1 , ± 2 , … , {\displaystyle L_{z}|\Psi \rangle =M_{L}\hbar |\Psi \rangle ;\quad M_{L}=0,\pm 1,\pm 2,\dots ,} then L z | Ψ ⟩ = ± Λ ℏ | Ψ ⟩ ; Λ = 0 , 1 , 2 , … , {\displaystyle L_{z}|\Psi \rangle =\pm \Lambda \hbar |\Psi \rangle ;\quad \Lambda =0,1,2,\dots ,} where Λ = | M L | {\displaystyle \Lambda =|M_{L}|} is the absolute value (in a.u.) of the projection of the total electronic angular momentum on the internuclear axis; Λ {\displaystyle \Lambda } can be used as a term symbol. By analogy with the spectroscopic notation S, P, D, F, ... used for atoms, it is customary to associate code letters with the values of Λ {\displaystyle \Lambda } according to the correspondence value of Λ : 0 1 2 3 … ↕ ↕ ↕ ↕ code letter: Σ Π Δ Φ … {\displaystyle {\begin{array}{rcccc}{\text{value of}}\ \Lambda \colon &0&1&2&3&\dots \\&\updownarrow &\updownarrow &\updownarrow &\updownarrow \\{\text{code letter:}}&\Sigma &\Pi &\Delta &\Phi &\dots \end{array}}}
For the individual electrons, the notation and the correspondence used are λ = | m l | {\displaystyle \lambda =|m_{l}|} and value of λ : 0 1 2 3 … ↕ ↕ ↕ ↕ code letter: σ π δ ϕ … {\displaystyle {\begin{array}{rcccc}{\text{value of}}\ \lambda \colon &0&1&2&3&\dots \\&\updownarrow &\updownarrow &\updownarrow &\updownarrow \\{\text{code letter:}}&\sigma &\pi &\delta &\phi &\dots \end{array}}}
Again, [ A y , H ] = 0 {\displaystyle [A_{y},H]=0} , and in addition A y L z = − L z A y {\displaystyle A_{y}L_{z}=-L_{z}A_{y}} , since L z = − i ℏ ( x ∂ ∂ y − y ∂ ∂ x ) . {\displaystyle L_{z}=-i\hbar \left(x{\frac {\partial }{\partial y}}-y{\frac {\partial }{\partial x}}\right).} It follows immediately that if Λ ≠ 0 , {\displaystyle \Lambda \neq 0,} the action of the operator A y {\displaystyle A_{y}} on an eigenstate corresponding to the eigenvalue Λ ℏ {\displaystyle \Lambda \hbar } of L z {\displaystyle L_{z}} converts this state into another one, corresponding to the eigenvalue − Λ ℏ {\displaystyle -\Lambda \hbar } , and that both eigenstates have the same energy. The electronic terms such that Λ ≠ 0 {\displaystyle \Lambda \neq 0} (that is, the terms Π , Δ , Φ , … {\displaystyle \Pi ,\Delta ,\Phi ,\dots } ) are thus doubly degenerate, each value of the energy corresponding to two states which differ by the direction of the projection of the orbital angular momentum along the molecular axis. This twofold degeneracy is actually only approximate, and it is possible to show that the interaction between the electronic and rotational motions leads to a splitting of the terms with Λ ≠ 0 {\displaystyle \Lambda \neq 0} into two nearby levels, which is called Λ {\displaystyle {\boldsymbol {\Lambda }}} -doubling . [ 3 ]
Λ = 0 {\displaystyle \Lambda =0} corresponds to the Σ {\displaystyle \Sigma } states. These states are non-degenerate, so that the states of a Σ {\displaystyle \Sigma } term can only be multiplied by a constant in a reflection through a plane containing the molecular axis. When Λ = 0 {\displaystyle \Lambda =0} , simultaneous eigenfunctions of H {\displaystyle H} , L z {\displaystyle L_{z}} and A y {\displaystyle A_{y}} can be constructed. Since A y 2 = 1 {\displaystyle A_{y}^{2}=1} , the eigenfunctions of A y {\displaystyle A_{y}} have eigenvalues ± 1 {\displaystyle \pm 1} . So to completely specify Σ {\displaystyle \Sigma } states of diatomic molecules, Σ + {\displaystyle \Sigma ^{+}} states, which are left unchanged upon reflection in a plane containing the nuclei, need to be distinguished from Σ − {\displaystyle \Sigma ^{-}} states, which change sign upon reflection.
Homonuclear diatomic molecules have a center of symmetry at their midpoint. Choosing this point (which is the nuclear center of mass) as the origin of the coordinates, the electronic Hamiltonian is invariant under the point group operation i of inversion of the coordinates of all electrons at that origin. This operation is not the parity operation P (or E*); the parity operation involves the inversion of nuclear and electronic spatial coordinates at the molecular center of mass. Electronic states either remain unchanged by the operation i , or they are changed in sign by i . The former are denoted by the subscript g and are called gerade, while the latter are denoted by the subscript u and are called ungerade. The subscripts g or u are therefore added to the term symbol, so that for homonuclear diatomic molecules electronic states can have the symmetries Σ g + , Σ g − , Σ u + , Σ u − , Π g , Π u {\displaystyle \Sigma _{g}^{+},\Sigma _{g}^{-},\Sigma _{u}^{+},\Sigma _{u}^{-},{{\Pi }_{g}},{{\Pi }_{u}}} ,......according to the irreducible representations of the D ∞ h {\displaystyle {{D}_{\infty h}}} point group.
The complete Hamiltonian of a diatomic molecule (as for all molecules) commutes with the parity operation P or E* and rovibronic (rotation-vibration-electronic) energy levels (often called rotational levels) can be given the parity symmetry label + or - . The complete Hamiltonian of a homonuclear diatomic molecule also commutes with the operation
of permuting (or exchanging) the coordinates of the two (identical) nuclei and rotational levels
gain the additional label s or a depending on whether the total wavefunction is
unchanged (symmetric) or changed in sign (antisymmetric) by the permutation operation. Thus, the rotational levels of heteronuclear diatomic molecules are labelled + or - , whereas those of homonuclear diatomic
molecules are labelled +s , +a , -s or -a . The rovibronic nuclear spin states are classified using the appropriate permutation-inversion group.
The complete Hamiltonian of a homonuclear diatomic molecule (as for all centro-symmetric molecules)
does not commute with the point group inversion operation i because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of g and u vibronic states (called ortho - para mixing) and give
rise to ortho - para transitions [ 4 ] [ 5 ]
If S denotes the resultant of the individual electron spins, s ( s + 1 ) ℏ 2 {\displaystyle s(s+1){{\hbar }^{2}}} are the eigenvalues of S and as in the case of atoms, each electronic term of the molecule is also characterised by the value of S . If spin-orbit coupling is neglected, there is a degeneracy of order 2 s + 1 {\displaystyle 2s+1} associated with each s {\displaystyle s} for a given Λ {\displaystyle \Lambda } . Just as for atoms, the quantity 2 s + 1 {\displaystyle 2s+1} is called the multiplicity of the term and.is written as a (left) superscript, so that the term symbol is written as 2 s + 1 Λ {\displaystyle {}^{2s+1}\Lambda } . For example, the symbol 3 Π {\displaystyle {}^{3}\Pi } denotes a term such that Λ = 1 {\displaystyle \Lambda =1} and s = 1 {\displaystyle s=1} . It is worth noting that the ground state (often labelled by the symbol X {\displaystyle X} ) of most diatomic molecules is such that s = 0 {\displaystyle s=0} and exhibits maximum symmetry. Thus, in most cases it is a 1 Σ + {\displaystyle {}^{1}{{\Sigma }^{+}}} state (written as X 1 Σ + {\displaystyle X{}^{1}{{\Sigma }^{+}}} , excited states are written with A , B , C , . . . {\displaystyle A,B,C,...} in front) for a heteronuclear molecule and a 1 Σ g + {\displaystyle {}^{1}\Sigma _{g}^{+}} state (written as X 1 Σ g + {\displaystyle X{}^{1}\Sigma _{g}^{+}} ) for a homonuclear molecule.
Spin–orbit coupling lifts the degeneracy of the electronic states. This is because the z -component of spin interacts with the z -component of the orbital angular momentum, generating a total electronic angular momentum along the molecule axis J z . This is characterized by the quantum number M J {\displaystyle {{M}_{J}}} , where M J = M S + M L {\displaystyle {{M}_{J}}={{M}_{S}}+{{M}_{L}}} . Again, positive and negative values of M J {\displaystyle {{M}_{J}}} are degenerate, so the pairs ( M L , M S ) and (− M L , − M S ) are degenerate. These pairs are grouped together with the quantum number Ω {\displaystyle \Omega } , which is defined as the sum of the pair of values ( M L , M S ) for which M L is positive: Ω = Λ + M S {\displaystyle \Omega =\Lambda +{{M}_{S}}}
So, the overall molecular term symbol for the most general diatomic molecule is given by 2 S + 1 Λ Ω , (g/u) ( + / − ) , {\displaystyle {}^{2S+1}\!\Lambda _{\Omega ,{\text{(g/u)}}}^{(+/-)},} where
The electronic terms or potential curves E S ( R ) {\displaystyle {{E}_{S}}(R)} of a diatomic molecule depend only on the internuclear distance R {\displaystyle R} , and it is important to investigate the behaviour of these potential curves as R varies. It is of considerable interest to examine the intersection of the curves representing the different terms.
Let E 1 ( R ) {\displaystyle {{E}_{1}}(R)} and E 2 ( R ) {\displaystyle {{E}_{2}}(R)} two different electronic potential curves. If they intersect at some point, then the functions E 1 ( R ) {\displaystyle {{E}_{1}}(R)} and E 2 ( R ) {\displaystyle {{E}_{2}}(R)} will have neighbouring values near this point. To decide whether such an intersection can occur, it is convenient to put the problem as follows. Suppose at some internuclear distance R c {\displaystyle {R}_{c}} the values E 1 ( R C ) {\displaystyle {{E}_{1}}({{R}_{C}})} and E 2 ( R C ) {\displaystyle {{E}_{2}}({{R}_{C}})} are close, but distinct (as shown in the figure). Then it is to be examined whether or E 1 ( R ) {\displaystyle {{E}_{1}}(R)} and E 2 ( R ) {\displaystyle {{E}_{2}}(R)} can be made to intersect by the modification R C → R C + Δ R {\displaystyle {{R}_{C}}\to {{R}_{C}}+\Delta R} . The energies E 1 ( 0 ) = E 1 ( R C ) {\displaystyle E_{1}^{(0)}={{E}_{1}}({{R}_{C}})} and E 2 ( 0 ) = E 2 ( R C ) {\displaystyle E_{2}^{(0)}={{E}_{2}}({{R}_{C}})} are eigenvalues of the Hamiltonian H 0 = H ( R C ) {\displaystyle {{H}_{0}}=H({{R}_{C}})} . The corresponding orthonormal electronic eigenstates will be denoted by | Φ 1 ( 0 ) ⟩ {\displaystyle \left|\Phi _{1}^{(0)}\right\rangle } and | Φ 2 ( 0 ) ⟩ {\displaystyle \left|\Phi _{2}^{(0)}\right\rangle } and are assumed to be real.
The Hamiltonian now becomes H ≡ H ( R C + Δ R ) = H 0 + H ′ {\displaystyle H\equiv H({{R}_{C}}+\Delta R)={{H}_{0}}+H'} , where H ′ = ∂ H 0 ∂ R C Δ R {\displaystyle H'={\frac {\partial {{H}_{0}}}{\partial {{R}_{C}}}}\Delta R} is the small perturbation operator (though it is a degenerate case, so ordinary method of perturbation won't work). setting H i j ′ = ⟨ Φ i ( 0 ) | H ′ | Φ j ( 0 ) ⟩ ; i , j = 1 , 2 {\displaystyle H_{ij}^{'}=\left\langle \Phi _{i}^{(0)}|H'|\Phi _{j}^{(0)}\right\rangle ;i,j=1,2} , it can be deduced that in order for E 1 ( R ) {\displaystyle {{E}_{1}}(R)} and E 2 ( R ) {\displaystyle {{E}_{2}}(R)} to be equal at the point R C + Δ R {\displaystyle {{R}_{C}}+\Delta R} the following two conditions are required to be fulfilled:
( H 0 + H ′ ) | Φ ( 0 ) ⟩ = E | Φ ( 0 ) ⟩ {\displaystyle ({{H}_{0}}+H')\left|{{\Phi }^{(0)}}\right\rangle =E\left|{{\Phi }^{(0)}}\right\rangle }
Expanding:
c 1 ( E 1 ( 0 ) + H ′ − E ) | Φ 1 ( 0 ) ⟩ + c 2 ( E 2 ( 0 ) + H ′ − E ) | Φ 2 ( 0 ) ⟩ = 0 {\displaystyle {{c}_{1}}(E_{1}^{(0)}+H'-E)\left|\Phi _{1}^{(0)}\right\rangle +{{c}_{2}}(E_{2}^{(0)}+H'-E)\left|\Phi _{2}^{(0)}\right\rangle =0}
Taking inner product with the respective bra's:
c 1 ( E 1 ( 0 ) + H ′ − E ) ⟨ Φ 1 ( 0 ) | Φ 1 ( 0 ) ⟩ + c 2 ( E 2 ( 0 ) + H ′ − E ) ⟨ Φ 1 ( 0 ) | Φ 2 ( 0 ) ⟩ = 0 {\displaystyle {{c}_{1}}(E_{1}^{(0)}+H'-E)\left\langle \Phi _{1}^{(0)}|\Phi _{1}^{(0)}\right\rangle +{{c}_{2}}(E_{2}^{(0)}+H'-E)\left\langle \Phi _{1}^{(0)}|\Phi _{2}^{(0)}\right\rangle =0} ; and
c 1 ( E 1 ( 0 ) + H ′ − E ) ⟨ Φ 2 ( 0 ) | Φ 1 ( 0 ) ⟩ + c 2 ( E 2 ( 0 ) + H ′ − E ) ⟨ Φ 2 ( 0 ) | Φ 2 ( 0 ) ⟩ = 0 {\displaystyle {{c}_{1}}(E_{1}^{(0)}+H'-E)\left\langle \Phi _{2}^{(0)}|\Phi _{1}^{(0)}\right\rangle +{{c}_{2}}(E_{2}^{(0)}+H'-E)\left\langle \Phi _{2}^{(0)}|\Phi _{2}^{(0)}\right\rangle =0}
Now, | Φ 1 ( 0 ) ⟩ {\displaystyle \left|\Phi _{1}^{(0)}\right\rangle } and | Φ 2 ( 0 ) ⟩ {\displaystyle \left|\Phi _{2}^{(0)}\right\rangle } are eigenstates of the Hamiltonian H 0 {\displaystyle {{H}_{0}}} corresponding to different eigenvalues and as H 0 {\displaystyle {{H}_{0}}} is itself Hermitian, they are orthonormal : ⟨ Φ i ( 0 ) | Φ j ( 0 ) ⟩ = δ i j {\displaystyle \left\langle \Phi _{i}^{(0)}|\Phi _{j}^{(0)}\right\rangle ={{\delta }_{ij}}}
Thus:
c 1 ( E 1 ( 0 ) + H 11 ′ − E ) + c 2 H 12 ′ = 0 {\displaystyle {{c}_{1}}(E_{1}^{(0)}+H_{11}^{'}-E)+{{c}_{2}}H_{12}^{'}=0} ; and
c 1 H 21 ′ + c 2 ( E 2 ( 0 ) + H 22 ′ − E ) = 0 {\displaystyle {{c}_{1}}H_{21}^{'}+{{c}_{2}}(E_{2}^{(0)}+H_{22}^{'}-E)=0}
Since the operator H ′ {\displaystyle H'} is Hermitian, the matrix elements H 11 ′ {\displaystyle H_{11}^{'}} and H 22 ′ {\displaystyle H_{22}^{'}} are real, while H 12 ′ = H 21 ′ ∗ {\displaystyle H_{12}^{'}=H_{21}^{'*}} . The compatibility condition for these equations is (such that both c 1 {\displaystyle {{c}_{1}}} and c 2 {\displaystyle {{c}_{2}}} are not simultaneously zero):
| E 1 ( 0 ) + H 11 ′ − E H 12 ′ H 21 ′ E 2 ( 0 ) + H 22 ′ − E | = 0 {\displaystyle \left|{\begin{matrix}E_{1}^{(0)}+H_{11}^{'}-E&H_{12}^{'}\\H_{21}^{'}&E_{2}^{(0)}+H_{22}^{'}-E\\\end{matrix}}\right|=0}
This gives: E = 1 2 ( E 1 ( 0 ) + E 2 ( 0 ) + H 11 ′ + H 22 ′ ) ± 1 4 ( E 1 ( 0 ) − E 2 ( 0 ) + H 11 ′ − H 22 ′ ) 2 + | H 12 ′ | 2 {\displaystyle E={\frac {1}{2}}(E_{1}^{(0)}+E_{2}^{(0)}+H_{11}^{'}+H_{22}^{'})\pm {\sqrt {{\frac {1}{4}}{{(E_{1}^{(0)}-E_{2}^{(0)}+H_{11}^{'}-H_{22}^{'})}^{2}}+{{\left|H_{12}^{'}\right|}^{2}}}}}
This formula gives the required eigenvalues of the energy in the first approximation.
If the energy values of the two terms become equal at the point ( R C + Δ R ) {\displaystyle ({{R}_{C}}+\Delta R)} (i.e. the terms intersect), this means that the two values of E {\displaystyle E} given by formula, are the same. For this to happen, the expression under the radical must vanish. Since it is the sum of two squares, both are simultaneously zero. So, it gives the conditions:
E 1 ( 0 ) − E 2 ( 0 ) + H 11 ′ − H 22 ′ = 0 {\displaystyle E_{1}^{(0)}-E_{2}^{(0)}+H_{11}^{'}-H_{22}^{'}=0}
and
H 12 ′ = 0 {\displaystyle H_{12}^{'}=0}
However, we have at our disposal only one arbitrary parameter Δ R {\displaystyle \Delta R} giving the perturbation H ′ {\displaystyle H'} . Hence the
two conditions involving more than one parameter cannot in general be simultaneously satisfied (the initial assumption that | Φ 1 ( 0 ) ⟩ {\displaystyle \left|\Phi _{1}^{(0)}\right\rangle } and | Φ 2 ( 0 ) ⟩ {\displaystyle \left|\Phi _{2}^{(0)}\right\rangle } real, implies that H 12 ′ {\displaystyle H_{12}^{'}} is also real). So, two case can arise:
Thus, in a diatomic molecule, only terms of different symmetry can intersect, while the intersection of terms of like symmetry is forbidden. This is, in general, true for any case in quantum mechanics where the Hamiltonian contains some parameter and its eigenvalues are consequently functions of that parameter. This general rule is known as von Neumann - Wigner non-crossing rule. [ notes 1 ]
This general symmetry principle has important consequences is molecular spectra.
In fact, in the applications of valence bond method in case of diatomic molecules, three main correspondence between the atomic and the molecular orbitals are taken care of:
Thus, von Neumann-Wigner non-crossing rule also acts as a starting point for valence bond theory.
Symmetry in diatomic molecules manifests itself directly by influencing the molecular spectra of the molecule. The effect of symmetry on different types of spectra in diatomic molecules are:
In the electric dipole approximation the transition amplitude for emission or absorption of radiation can be shown to be proportional to the vibronic matrix element of the component of the electric dipole operator D {\displaystyle D} along the molecular axis. This is the permanent electric dipole moment.
In homonuclear diatomic molecules, the permanent electric dipole moment vanishes and there is no pure rotation spectrum (but see N.B. below).
Heteronuclear diatomic molecules possess a permanent electric dipole moment and exhibit spectra corresponding to rotational transitions, without change in the vibronic state. For Λ = 0 {\displaystyle \Lambda =0} , the selection rules for a rotational transition are: Δ ℑ = ± 1 Δ M ℑ = 0 , ± 1 {\displaystyle {\begin{aligned}&\Delta \Im =\pm 1\\&\Delta {{M}_{\Im }}=0,\pm 1\\\end{aligned}}} . For Λ ≠ 0 {\displaystyle \Lambda \neq 0} , the selection rules become: Δ ℑ = 0 , ± 1 Δ M ℑ = 0 , ± 1 {\displaystyle {\begin{aligned}&\Delta \Im =0,\pm 1\\&\Delta {{M}_{\Im }}=0,\pm 1\\\end{aligned}}} .This is due to the fact that although the photon absorbed or emitted carries one unit of angular momentum, the nuclear rotation can change, with no change in ℑ {\displaystyle \Im } , if the electronic angular momentum makes an equal and opposite change. Symmetry considerations require that the electric dipole moment of a diatomic molecule is directed along the internuclear line, and this leads to the additional selection rule Δ Λ = 0 {\displaystyle \Delta \Lambda =0} .The pure rotational spectrum of a diatomic molecule consists of lines in the far infra-red or the microwave region, the frequencies of these lines given by:
ℏ ω ℑ + 1 , ℑ = E r ( ℑ + 1 ) − E r ( ℑ ) = 2 B ( ℑ + 1 ) {\displaystyle \hbar {{\omega }_{\Im +1,\Im }}={{E}_{r}}(\Im +1)-{{E}_{r}}(\Im )=2B(\Im +1)} ; where B = ℏ 2 2 μ R 0 2 {\displaystyle B={\frac {{\hbar }^{2}}{2\mu R_{0}^{2}}}} , and ℑ ≥ Λ {\displaystyle \Im \geq \Lambda }
The transition matrix elements for pure vibrational transition are μ v , v ′ = ⟨ v ′ | μ | v ⟩ {\displaystyle {{\mu }_{v,v'}}=\left\langle v'|\mu |v\right\rangle } , where μ {\displaystyle \mu } is the dipole moment of the diatomic molecule in the electronic state α {\displaystyle \alpha } . Because the dipole moment depends on the bond length R {\displaystyle R} , its variation with displacement of the nuclei from equilibrium can be expressed as: μ = μ 0 + ( d μ d x ) 0 x + 1 2 ( d 2 μ d x 2 ) 0 x 2 + . . . . . . . {\displaystyle \mu ={{\mu }_{0}}+{{({\frac {d\mu }{dx}})}_{0}}x+{\frac {1}{2}}{{({\frac {{{d}^{2}}\mu }{d{{x}^{2}}}})}_{0}}{{x}^{2}}+.......} ; where μ 0 {\displaystyle {{\mu }_{0}}} is the dipole moment when the displacement is zero. The transition matrix elements are, therefore: ⟨ v ′ | μ | v ⟩ = μ 0 ⟨ v ′ | v ⟩ + ( d μ d x ) 0 ⟨ v ′ | x | v ⟩ + 1 2 ( d 2 μ d x 2 ) 0 ⟨ v ′ | x 2 | v ⟩ + . . . . . . . = ( d μ d x ) 0 ⟨ v ′ | x | v ⟩ + 1 2 ( d 2 μ d x 2 ) 0 ⟨ v ′ | x 2 | v ⟩ + . . . . . . . {\displaystyle \left\langle v'|\mu |v\right\rangle ={{\mu }_{0}}\left\langle v'|v\right\rangle +{{({\frac {d\mu }{dx}})}_{0}}\left\langle v'|x|v\right\rangle +{\frac {1}{2}}{{({\frac {{{d}^{2}}\mu }{d{{x}^{2}}}})}_{0}}\left\langle v'|{{x}^{2}}|v\right\rangle +.......={{({\frac {d\mu }{dx}})}_{0}}\left\langle v'|x|v\right\rangle +{\frac {1}{2}}{{({\frac {{{d}^{2}}\mu }{d{{x}^{2}}}})}_{0}}\left\langle v'|{{x}^{2}}|v\right\rangle +.......} using orthogonality of the states. So, the transition matrix is non-zero only if the molecular dipole moment varies with displacement, for otherwise the derivatives of μ {\displaystyle \mu } would be zero. The gross selection rule for the vibrational transitions of diatomic molecules is then: To show a vibrational spectrum, a diatomic molecule must have a dipole moment that varies with extension. So, homonuclear diatomic molecules do not undergo electric-dipole vibrational transitions. So, a homonuclear diatomic molecule doesn't show purely vibrational spectra.
For small displacements, the electric dipole moment of a molecule can be expected to vary linearly with the extension of the bond. This would be the case for a heteronuclear molecule in which the partial charges on the two atoms were independent of the internuclear distance. In such cases (known as harmonic approximation), the quadratic and higher terms in the expansion can be ignored and μ v , v ′ = ⟨ v ′ | μ | v ⟩ = ( d μ d x ) 0 ⟨ v ′ | x | v ⟩ {\displaystyle {{\mu }_{v,v'}}=\left\langle v'|\mu |v\right\rangle ={{({\frac {d\mu }{dx}})}_{0}}\left\langle v'|x|v\right\rangle } . Now, the matrix elements can be expressed in position basis in terms of the harmonic oscillator wavefunctions: Hermite polynomials. Using the property of Hermite polynomials: 2 ( α x ) H v ( α x ) = 2 v H v − 1 ( α x ) + H v + 1 ( α x ) {\displaystyle 2(\alpha x){{H}_{v}}(\alpha x)=2v{{H}_{v-1}}(\alpha x)+{{H}_{v+1}}(\alpha x)} , it is evident that x | v ⟩ {\displaystyle x\left|v\right\rangle } which is proportional to x H v ( α x ) {\displaystyle x{{H}_{v}}(\alpha x)} , produces two terms, one proportional to | v + 1 ⟩ {\displaystyle \left|v+1\right\rangle } and the other to | v − 1 ⟩ {\displaystyle \left|v-1\right\rangle } . So, the only non-zero contributions to μ v , v ′ {\displaystyle {{\mu }_{v,v'}}} comes from v ′ = v ± 1 {\displaystyle v'=v\pm 1} . So, the selection rule for heteronuclear diatomic molecules is: Δ v = ± 1 {\displaystyle \Delta v=\pm 1}
Homonuclear diatomic molecules show neither pure vibrational nor pure rotational spectra. However, as the absorption of a photon requires the molecule to take up one unit of angular momentum , vibrational transitions are accompanied by a change in rotational state, which is subject to the same selection rules as for the pure rotational spectrum. For a molecule in a Σ {\displaystyle \Sigma } state, the transitions between two vibration-rotation (or rovibrational ) levels ( v , ℑ ) {\displaystyle (v,\Im )} and ( v ′ , ℑ ′ ) {\displaystyle (v',\Im ')} , with vibrational quantum numbers v {\displaystyle v} and v ′ = v + 1 {\displaystyle v'=v+1} , fall into two sets according to whether Δ ℑ = + 1 {\displaystyle \Delta \Im =+1} or Δ ℑ = − 1 {\displaystyle \Delta \Im =-1} . The set corresponding to Δ ℑ = + 1 {\displaystyle \Delta \Im =+1} is called the R branch . The corresponding frequencies are given by: ℏ ω R = E ( v + 1 , ℑ + 1 ) − E ( v , ℑ ) = 2 B ( ℑ + 1 ) + ℏ ω 0 ; ℑ = 0 , 1 , 2 , . . . . . . {\displaystyle \hbar {{\omega }^{R}}=E(v+1,\Im +1)-E(v,\Im )=2B(\Im +1)+\hbar {{\omega }_{0}};{\text{ }}\Im =0,1,2,......}
The set corresponding to Δ ℑ = − 1 {\displaystyle \Delta \Im =-1} is called the P branch . The corresponding frequencies are given by: ℏ ω P = E ( v + 1 , ℑ − 1 ) − E ( v , ℑ ) = − 2 B ℑ + ℏ ω 0 ; ℑ = 1 , 2 , 3 , . . . . . . {\displaystyle \hbar {{\omega }^{P}}=E(v+1,\Im -1)-E(v,\Im )=-2B\Im +\hbar {{\omega }_{0}};{\text{ }}\Im =1,2,3,......}
Both branches make up what is called a rotational-vibrational band or a rovibrational band . These bands are in the infra-red part of the spectrum.
If the molecule is not in a Σ {\displaystyle \Sigma } state, so that Λ ≠ 0 {\displaystyle \Lambda \neq 0} , transitions with Δ ℑ = 0 {\displaystyle \Delta \Im =0} are allowed. This gives rise to a further branch of the vibrational-rotational spectrum, called the Q branch . The frequencies ω Q {\displaystyle {{\omega }^{Q}}} corresponding to the lines in this branch are given by a quadratic function of ℑ {\displaystyle \Im } if B v {\displaystyle {{B}_{v}}} and B v + 1 {\displaystyle {{B}_{v+1}}} are unequal, and reduce to the single frequency: ℏ ω Q = E ( v + 1 , ℑ ) − E ( v , ℑ ) = ℏ ω 0 {\displaystyle \hbar {{\omega }^{Q}}=E(v+1,\Im )-E(v,\Im )=\hbar {{\omega }_{0}}} if B v + 1 = B v {\displaystyle {{B}_{v+1}}={{B}_{v}}} .
For a heteronuclear diatomic molecule, this selection rule has two consequences:
Homonuclear diatomic molecules also show this kind of spectra. The selection rules, however, are a bit different.
An explicit implication of symmetry on the molecular structure can be shown in case of the simplest bi-nuclear system: a hydrogen molecule ion or a di-hydrogen cation, H 2 + {\displaystyle {\text{H}}_{2}^{+}} . A natural trial wave function for the H 2 + {\displaystyle {\text{H}}_{2}^{+}} is determined by first considering the lowest-energy state of the system when the two protons are widely separated. Then there are clearly two possible states: the electron is attached either to one of the protons, forming a hydrogen atom in the ground state , or the electron is attached to the other proton, again in the ground state of a hydrogen atom (as depicted in the picture).
The trial states in the position basis (or the ' wave functions ') are then:
⟨ r | 1 ⟩ = 1 π a 0 3 e − | r − R 2 | a 0 {\displaystyle \left\langle \mathbf {r} |\mathbf {1} \right\rangle ={\frac {1}{\sqrt {\pi a_{0}^{3}}}}{{e}^{-{\frac {\left|\mathbf {r} -{\frac {\mathbf {R} }{2}}\right|}{{a}_{0}}}}}} and ⟨ r | 2 ⟩ = 1 π a 0 3 e − | r + R 2 | a 0 {\displaystyle \left\langle \mathbf {r} |\mathbf {2} \right\rangle ={\frac {1}{\sqrt {\pi a_{0}^{3}}}}{{e}^{-{\frac {\left|\mathbf {r} +{\frac {\mathbf {R} }{2}}\right|}{{a}_{0}}}}}}
The analysis of H 2 + {\displaystyle {\text{H}}_{2}^{+}} using variational method starts assuming these forms. Again, this is only one possible combination of states. There can be other combination of states also, for example, the electron is in an excited state of the hydrogen atom. The corresponding Hamiltonian of the system is:
H = p 2 2 m e − e 2 | r − R / 2 | − e 2 | r + R / 2 | + e 2 R {\displaystyle H={\frac {{\mathbf {p} }^{2}}{2{{m}_{e}}}}-{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}-{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}+{\frac {{e}^{2}}{R}}}
Clearly, using the states | 1 ⟩ {\displaystyle \left|1\right\rangle } and | 2 ⟩ {\displaystyle \left|2\right\rangle } as basis will introduce off-diagonal elements in the Hamiltonian. Here, because of the relative simplicity of the H 2 + {\displaystyle {\text{H}}_{2}^{+}} ion, the matrix elements can actually be calculated. The electronic Hamiltonian of H 2 + {\displaystyle {\text{H}}_{2}^{+}} commutes with the point group inversion symmetry operation i . Using its symmetry properties, we can relate the diagonal and off-diagonal elements of the Hamiltonian as:
H 11 = ⟨ 1 | p 2 2 m e − e 2 | r − R / 2 | | 1 ⟩ − ⟨ 1 | e 2 | r + R / 2 | | 1 ⟩ + e 2 R ⟨ 1 | 1 ⟩ {\displaystyle {{H}_{11}}=\left\langle 1|{\frac {{\mathbf {p} }^{2}}{2{{m}_{e}}}}-{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}|1\right\rangle -\left\langle 1|{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}|1\right\rangle +{\frac {{e}^{2}}{R}}\left\langle 1|1\right\rangle } ⇒ H 11 = E 1 − ∫ d 3 r e 2 | r + R / 2 | | ⟨ r | 1 ⟩ | 2 + e 2 R {\displaystyle \Rightarrow {{H}_{11}}={{E}_{1}}-\int {{{d}^{3}}r}{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}{{\left|\left\langle \mathbf {r} |1\right\rangle \right|}^{2}}+{\frac {{e}^{2}}{R}}} ⇒ H 11 = E 1 − ∫ d 3 r e 2 | r + R / 2 | | ⟨ r | 1 ⟩ | 2 + e 2 R {\displaystyle \Rightarrow {{H}_{11}}={{E}_{1}}-\int {{{d}^{3}}r}{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}{{\left|\left\langle \mathbf {r} |1\right\rangle \right|}^{2}}+{\frac {{e}^{2}}{R}}}
Where, E 1 {\displaystyle {{E}_{1}}} is the ground-state energy of the hydrogen atom.
Again, H 22 = ⟨ 2 | p 2 2 m e − e 2 | r + R / 2 | | 2 ⟩ − ⟨ 2 | e 2 | r − R / 2 | | 2 ⟩ + e 2 R ⟨ 2 | 2 ⟩ {\displaystyle {{H}_{22}}=\left\langle 2|{\frac {{\mathbf {p} }^{2}}{2{{m}_{e}}}}-{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}|2\right\rangle -\left\langle 2|{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}|2\right\rangle +{\frac {{e}^{2}}{R}}\left\langle 2|2\right\rangle }
⇒ H 22 = E 1 − ∫ d 3 r e 2 | r − R / 2 | | ⟨ r | 2 ⟩ | 2 + e 2 R = H 11 {\displaystyle \Rightarrow {{H}_{22}}={{E}_{1}}-\int {{{d}^{3}}r}{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}{{\left|\left\langle \mathbf {r} |2\right\rangle \right|}^{2}}+{\frac {{e}^{2}}{R}}={{H}_{11}}}
where the last step follows from the fact that | ⟨ r | 2 ⟩ | 2 = | ⟨ r | 1 ⟩ | 2 = 1 π a 0 3 {\displaystyle {{\left|\left\langle \mathbf {r} |2\right\rangle \right|}^{2}}={{\left|\left\langle \mathbf {r} |1\right\rangle \right|}^{2}}={\frac {1}{\pi a_{0}^{3}}}} and from the symmetry of the system, the value of the integrals are same.
Now the off-diagonal terms:
H 12 = ⟨ 1 | p 2 2 m e − e 2 | r + R / 2 | | 2 ⟩ − ⟨ 1 | e 2 | r − R / 2 | | 2 ⟩ + e 2 R ⟨ 1 | 2 ⟩ {\displaystyle {{H}_{12}}=\left\langle 1|{\frac {{\mathbf {p} }^{2}}{2{{m}_{e}}}}-{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}|2\right\rangle -\left\langle 1|{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}|2\right\rangle +{\frac {{e}^{2}}{R}}\left\langle 1|2\right\rangle }
⇒ H 12 = ( E 1 + e 2 R ) ⟨ 1 | 2 ⟩ − ∫ d 3 r e 2 | r − R / 2 | ⟨ 1 | r ⟩ ⟨ r | 2 ⟩ {\displaystyle \Rightarrow {{H}_{12}}=({{E}_{1}}+{\frac {{e}^{2}}{R}})\left\langle 1|2\right\rangle -\int {{{d}^{3}}r}{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}\left\langle 1\left|\mathbf {r} \right\rangle \left\langle \mathbf {r} \right|2\right\rangle }
by inserting a complete set of states ∫ d 3 r | r ⟩ ⟨ r | {\displaystyle \int {{{d}^{3}}r}\left|\mathbf {r} \right\rangle \left\langle \mathbf {r} \right|} in the last term. ⟨ 1 | 2 ⟩ = ∫ d 3 r ⟨ 1 | r ⟩ ⟨ r | 2 ⟩ {\displaystyle \left\langle 1|2\right\rangle =\int {{{d}^{3}}r}\left\langle 1\left|\mathbf {r} \right\rangle \left\langle \mathbf {r} \right|2\right\rangle } is called the 'overlap integral'
And,
H 21 = ⟨ 2 | p 2 2 m e − e 2 | r − R / 2 | | 1 ⟩ − ⟨ 2 | e 2 | r + R / 2 | | 1 ⟩ + e 2 R ⟨ 2 | 1 ⟩ {\displaystyle {{H}_{21}}=\left\langle 2|{\frac {{\mathbf {p} }^{2}}{2{{m}_{e}}}}-{\frac {{e}^{2}}{\left|\mathbf {r} -\mathbf {R} /2\right|}}|1\right\rangle -\left\langle 2|{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}|1\right\rangle +{\frac {{e}^{2}}{R}}\left\langle 2|1\right\rangle }
⇒ H 21 = ( E 1 + e 2 R ) ⟨ 2 | 1 ⟩ − ∫ d 3 r e 2 | r + R / 2 | ⟨ 2 | r ⟩ ⟨ r | 1 ⟩ = H 12 {\displaystyle \Rightarrow {{H}_{21}}=({{E}_{1}}+{\frac {{e}^{2}}{R}})\left\langle 2|1\right\rangle -\int {{{d}^{3}}r}{\frac {{e}^{2}}{\left|\mathbf {r} +\mathbf {R} /2\right|}}\left\langle 2\left|\mathbf {r} \right\rangle \left\langle \mathbf {r} \right|1\right\rangle ={{H}_{12}}} (as the wave functions are real)
So, H 11 = H 22 and H 12 = H 21 {\displaystyle {{H}_{11}}={{H}_{22}}{\text{ and }}{{H}_{12}}={{H}_{21}}}
Because H 11 = H 22 {\displaystyle {{H}_{11}}={{H}_{22}}} as well as H 12 = H 21 {\displaystyle {{H}_{12}}={{H}_{21}}} , the linear combination of | 1 ⟩ {\displaystyle \left|1\right\rangle } and | 2 ⟩ {\displaystyle \left|2\right\rangle } that diagonalizes the Hamiltonian is | ± ⟩ = 1 2 ± 2 ⟨ 1 | 2 ⟩ ( | 1 ⟩ ± | 2 ⟩ ) {\displaystyle \left|\pm \right\rangle ={\frac {1}{\sqrt {2\pm 2\left\langle 1|2\right\rangle }}}(\left|1\right\rangle \pm \left|2\right\rangle )} (after normalization). Now as [ H , {\displaystyle [H,} i ] = 0 {\displaystyle ]=0} for H 2 + {\displaystyle {\text{H}}_{2}^{+}} , the states | ± ⟩ {\displaystyle \left|\pm \right\rangle } are also eigenstates of i . It turns out that | + ⟩ {\displaystyle \left|+\right\rangle } and | − ⟩ {\displaystyle \left|-\right\rangle } are the eigenstates of i with eigenvalues +1 and -1 (in other words, the wave functions ⟨ r | + ⟩ {\displaystyle \left\langle \mathbf {r} |+\right\rangle } and ⟨ r | − ⟩ {\displaystyle \left\langle \mathbf {r} |-\right\rangle } are gerade (symmetric) and ungerade (unsymmetric), respectively). The corresponding expectation value of the energies are E ± = 1 1 ± ⟨ 1 | 2 ⟩ ( H 11 ± H 12 ) {\displaystyle {{E}_{\pm }}={\frac {1}{1\pm \left\langle 1|2\right\rangle }}({{H}_{11}}\pm {{H}_{12}})} .
From the graph, we see that only E + {\displaystyle {{E}_{+}}} has a minimum corresponding to a separation of 1.3 Å and a total energy E + = − 15.4 eV {\displaystyle {{E}_{+}}=-15.4{\text{ eV}}} , which is less than the initial energy of the system, − 13.6 eV {\displaystyle -13.6{\text{ eV}}} . Thus, only the gerade state stabilizes the ion with a binding energy of 1.8 eV {\displaystyle 1.8{\text{ eV}}} . As a result, the ground state of H 2 + {\displaystyle {\text{H}}_{2}^{+}} is X 2 Σ g + {\displaystyle {{X}^{2}}\Sigma _{g}^{+}} and this state ( | + ⟩ ) {\displaystyle \left(\left|+\right\rangle \right)} is called a bonding molecular orbital. [ 7 ]
Thus, symmetry plays an explicit role in the formation of H 2 + {\displaystyle {\text{H}}_{2}^{+}} . | https://en.wikipedia.org/wiki/Symmetry_of_diatomic_molecules |
In mathematics , the symmetry of second derivatives (also called the equality of mixed partials ) is the fact that exchanging the order of partial derivatives of a multivariate function
does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix , is a symmetric matrix .
Sufficient conditions for the symmetry to hold are given by Schwarz's theorem , also called Clairaut's theorem or Young's theorem . [ 1 ] [ 2 ]
In the context of partial differential equations , it is called the Schwarz integrability condition .
In symbols, the symmetry may be expressed as:
Another notation is:
In terms of composition of the differential operator D i which takes the partial derivative with respect to x i :
From this relation it follows that the ring of differential operators with constant coefficients , generated by the D i , is commutative ; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to monomials , so that one can take polynomials in the x i as a domain. In fact smooth functions are another valid domain.
The result on the equality of mixed partial derivatives under certain conditions has a long history. The list of unsuccessful proposed proofs started with Euler 's, published in 1740, [ 3 ] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification. [ 4 ] Clairaut also published a proposed proof in 1740, with no other attempts until the end of the 18th century. Starting then, for a period of 70 years, a number of incomplete proofs were proposed. The proof of Lagrange (1797) was improved by Cauchy (1823), but assumed the existence and continuity of the partial derivatives ∂ 2 f ∂ x 2 {\displaystyle {\tfrac {\partial ^{2}f}{\partial x^{2}}}} and ∂ 2 f ∂ y 2 {\displaystyle {\tfrac {\partial ^{2}f}{\partial y^{2}}}} . [ 5 ] Other attempts were made by P. Blanchet (1841), Duhamel (1856), Sturm (1857), Schlömilch (1862), and Bertrand (1864). Finally in 1867 Lindelöf systematically analyzed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal. [ 6 ] [ 7 ]
Six years after that, Schwarz succeeded in giving the first rigorous proof. [ 8 ] Dini later contributed by finding more general conditions than those of Schwarz. Eventually a clean and more general version was found by Jordan in 1883 that is still the proof found in most textbooks. Minor variants of earlier proofs were published by Laurent (1885), Peano (1889 and 1893), J. Edwards (1892), P. Haag (1893), J. K. Whittemore (1898), Vivanti (1899) and Pierpont (1905). Further progress was made in 1907-1909 when E. W. Hobson and W. H. Young found proofs with weaker conditions than those of Schwarz and Dini. In 1918, Carathéodory gave a different proof based on the Lebesgue integral . [ 7 ]
In mathematical analysis , Schwarz's theorem (or Clairaut's theorem on equality of mixed partials ) [ 9 ] named after Alexis Clairaut and Hermann Schwarz , states that for a function f : Ω → R {\displaystyle f\colon \Omega \to \mathbb {R} } defined on a set Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} , if p ∈ R n {\displaystyle \mathbf {p} \in \mathbb {R} ^{n}} is a point such that some neighborhood of p {\displaystyle \mathbf {p} } is contained in Ω {\displaystyle \Omega } and f {\displaystyle f} has continuous second partial derivatives on that neighborhood of p {\displaystyle \mathbf {p} } , then for all i and j in { 1 , 2 … , n } , {\displaystyle \{1,2\ldots ,\,n\},}
The partial derivatives of this function commute at that point.
There exists a version of this theorem where f {\displaystyle f} is only required to be twice differentiable at the point p {\displaystyle \mathbf {p} } .
One easy way to establish this theorem (in the case where n = 2 {\displaystyle n=2} , i = 1 {\displaystyle i=1} , and j = 2 {\displaystyle j=2} , which readily entails the result in general) is by applying Green's theorem to the gradient of f . {\displaystyle f.}
An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case). [ 10 ] Let f ( x , y ) {\displaystyle f(x,y)} be a differentiable function on an open rectangle Ω {\displaystyle \Omega } containing a point ( a , b ) {\displaystyle (a,b)} and suppose that d f {\displaystyle df} is continuous with continuous ∂ x ∂ y f {\displaystyle \partial _{x}\partial _{y}f} and ∂ y ∂ x f {\displaystyle \partial _{y}\partial _{x}f} over Ω . {\displaystyle \Omega .} Define
These functions are defined for | h | , | k | < ε {\displaystyle \left|h\right|,\,\left|k\right|<\varepsilon } , where ε > 0 {\displaystyle \varepsilon >0} and [ a − ε , a + ε ] × [ b − ε , b + ε ] {\displaystyle \left[a-\varepsilon ,\,a+\varepsilon \right]\times \left[b-\varepsilon ,\,b+\varepsilon \right]} is contained in Ω . {\displaystyle \Omega .}
By the mean value theorem , for fixed h and k non-zero, θ , θ ′ , ϕ , ϕ ′ {\displaystyle \theta ,\theta ',\phi ,\phi '} can be found in the open interval ( 0 , 1 ) {\displaystyle (0,1)} with
Since h , k ≠ 0 {\displaystyle h,\,k\neq 0} , the first equality below can be divided by h k {\displaystyle hk} :
Letting h , k {\displaystyle h,\,k} tend to zero in the last equality, the continuity assumptions on ∂ y ∂ x f {\displaystyle \partial _{y}\partial _{x}f} and ∂ x ∂ y f {\displaystyle \partial _{x}\partial _{y}f} now imply that
This account is a straightforward classical method found in many text books, for example in Burkill, Apostol and Rudin. [ 10 ] [ 11 ] [ 12 ]
Although the derivation above is elementary, the approach can also be viewed from a more conceptual perspective so that the result becomes more apparent. [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] Indeed the difference operators Δ x t , Δ y t {\displaystyle \Delta _{x}^{t},\,\,\Delta _{y}^{t}} commute and Δ x t f , Δ y t f {\displaystyle \Delta _{x}^{t}f,\,\,\Delta _{y}^{t}f} tend to ∂ x f , ∂ y f {\displaystyle \partial _{x}f,\,\,\partial _{y}f} as t {\displaystyle t} tends to 0, with a similar statement for second order operators. [ a ] Here, for z {\displaystyle z} a vector in the plane and u {\displaystyle u} a directional vector ( 1 0 ) {\displaystyle {\tbinom {1}{0}}} or ( 0 1 ) {\displaystyle {\tbinom {0}{1}}} , the difference operator is defined by
By the fundamental theorem of calculus for C 1 {\displaystyle C^{1}} functions f {\displaystyle f} on an open interval I {\displaystyle I} with ( a , b ) ⊂ I {\displaystyle (a,b)\subset I}
Hence
This is a generalized version of the mean value theorem . Recall that the elementary discussion on maxima or minima for real-valued functions implies that if f {\displaystyle f} is continuous on [ a , b ] {\displaystyle [a,b]} and differentiable on ( a , b ) {\displaystyle (a,b)} , then there is a point c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that
For vector-valued functions with V {\displaystyle V} a finite-dimensional normed space, there is no analogue of the equality above, indeed it fails. But since inf f ′ ≤ f ′ ( c ) ≤ sup f ′ {\displaystyle \inf f^{\prime }\leq f^{\prime }(c)\leq \sup f^{\prime }} , the inequality above is a useful substitute. Moreover, using the pairing of the dual of V {\displaystyle V} with its dual norm, yields the following inequality:
These versions of the mean valued theorem are discussed in Rudin, Hörmander and elsewhere. [ 19 ] [ 20 ]
For f {\displaystyle f} a C 2 {\displaystyle C^{2}} function on an open set in the plane, define D 1 = ∂ x {\displaystyle D_{1}=\partial _{x}} and D 2 = ∂ y {\displaystyle D_{2}=\partial _{y}} . Furthermore for t ≠ 0 {\displaystyle t\neq 0} set
Then for ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} in the open set, the generalized mean value theorem can be applied twice:
Thus Δ 1 t Δ 2 t f ( x 0 , y 0 ) {\displaystyle \Delta _{1}^{t}\Delta _{2}^{t}f(x_{0},y_{0})} tends to D 1 D 2 f ( x 0 , y 0 ) {\displaystyle D_{1}D_{2}f(x_{0},y_{0})} as t {\displaystyle t} tends to 0. The same argument shows that Δ 2 t Δ 1 t f ( x 0 , y 0 ) {\displaystyle \Delta _{2}^{t}\Delta _{1}^{t}f(x_{0},y_{0})} tends to D 2 D 1 f ( x 0 , y 0 ) {\displaystyle D_{2}D_{1}f(x_{0},y_{0})} . Hence, since the difference operators commute, so do the partial differential operators D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} , as claimed. [ 21 ] [ 22 ] [ 23 ] [ 24 ] [ 25 ]
Remark. By two applications of the classical mean value theorem,
for some θ {\displaystyle \theta } and θ ′ {\displaystyle \theta ^{\prime }} in ( 0 , 1 ) {\displaystyle (0,1)} . Thus the first elementary proof can be reinterpreted using difference operators. Conversely, instead of using the generalized mean value theorem in the second proof, the classical mean valued theorem could be used.
The properties of repeated Riemann integrals of a continuous function F on a compact rectangle [ a , b ] × [ c , d ] are easily established. [ 26 ] The uniform continuity of F implies immediately that the functions g ( x ) = ∫ c d F ( x , y ) d y {\displaystyle g(x)=\int _{c}^{d}F(x,y)\,dy} and h ( y ) = ∫ a b F ( x , y ) d x {\displaystyle h(y)=\int _{a}^{b}F(x,y)\,dx} are continuous. [ 27 ] It follows that
moreover it is immediate that the iterated integral is positive if F is positive. [ 28 ] The equality above is a simple case of Fubini's theorem , involving no measure theory . Titchmarsh (1939) proves it in a straightforward way using Riemann approximating sums corresponding to subdivisions of a rectangle into smaller rectangles.
To prove Clairaut's theorem, assume f is a differentiable function on an open set U , for which the mixed second partial derivatives f yx and f xy exist and are continuous. Using the fundamental theorem of calculus twice,
Similarly
The two iterated integrals are therefore equal. On the other hand, since f xy ( x , y ) is continuous, the second iterated integral can be performed by first integrating over x and then afterwards over y . But then the iterated integral of f yx − f xy on [ a , b ] × [ c , d ] must vanish. However, if the iterated integral of a continuous function function F vanishes for all rectangles, then F must be identically zero; for otherwise F or − F would be strictly positive at some point and therefore by continuity on a rectangle, which is not possible. Hence f yx − f xy must vanish identically, so that f yx = f xy everywhere. [ 29 ] [ 30 ] [ 31 ] [ 32 ] [ 33 ]
A weaker condition than the continuity of second partial derivatives (which is implied by the latter) which suffices to ensure symmetry is that all partial derivatives are themselves differentiable . [ 34 ] Another strengthening of the theorem, in which existence of the permuted mixed partial is asserted, was provided by Peano in a short 1890 note on Mathesis :
The theory of distributions (generalized functions) eliminates analytic problems with the symmetry. The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions , which are smooth and certainly satisfy this symmetry. In more detail (where f is a distribution, written as an operator on test functions, and φ is a test function),
Another approach, which defines the Fourier transform of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously. [ a ]
The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not continuous ).
An example of non-symmetry is the function (due to Peano ) [ 36 ] [ 37 ]
This can be visualized by the polar form f ( r cos ( θ ) , r sin ( θ ) ) = r 2 sin ( 4 θ ) 4 {\displaystyle f(r\cos(\theta ),r\sin(\theta ))={\frac {r^{2}\sin(4\theta )}{4}}} ; it is everywhere continuous, but its derivatives at (0, 0) cannot be computed algebraically. Rather, the limit of difference quotients shows that f x ( 0 , 0 ) = f y ( 0 , 0 ) = 0 {\displaystyle f_{x}(0,0)=f_{y}(0,0)=0} , so the graph z = f ( x , y ) {\displaystyle z=f(x,y)} has a horizontal tangent plane at (0, 0) , and the partial derivatives f x , f y {\displaystyle f_{x},f_{y}} exist and are everywhere continuous. However, the second partial derivatives are not continuous at (0, 0) , and the symmetry fails. In fact, along the x -axis the y -derivative is f y ( x , 0 ) = x {\displaystyle f_{y}(x,0)=x} , and so:
In contrast, along the y -axis the x -derivative f x ( 0 , y ) = − y {\displaystyle f_{x}(0,y)=-y} , and so f x y ( 0 , 0 ) = − 1 {\displaystyle f_{xy}(0,0)=-1} . That is, f y x ≠ f x y {\displaystyle f_{yx}\neq f_{xy}} at (0, 0) , although the mixed partial derivatives do exist, and at every other point the symmetry does hold.
The above function, written in polar coordinates, can be expressed as
showing that the function oscillates four times when traveling once around an arbitrarily small loop containing the origin. Intuitively, therefore, the local behavior of the function at (0, 0) cannot be described as a quadratic form, and the Hessian matrix thus fails to be symmetric.
In general, the interchange of limiting operations need not commute . Given two variables near (0, 0) and two limiting processes on
corresponding to making h → 0 first, and to making k → 0 first. It can matter, looking at the first-order terms, which is applied first. This leads to the construction of pathological examples in which second derivatives are non-symmetric. This kind of example belongs to the theory of real analysis where the pointwise value of functions matters. When viewed as a distribution the second partial derivative's values can be changed at an arbitrary set of points as long as this has Lebesgue measure 0. Since in the example the Hessian is symmetric everywhere except (0, 0) , there is no contradiction with the fact that the Hessian, viewed as a Schwartz distribution , is symmetric.
Consider the first-order differential operators D i to be infinitesimal operators on Euclidean space . That is, D i in a sense generates the one-parameter group of translations parallel to the x i -axis. These groups commute with each other, and therefore the infinitesimal generators do also; the Lie bracket
is this property's reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.
The Clairaut-Schwarz theorem is the key fact needed to prove that for every C ∞ {\displaystyle C^{\infty }} (or at least twice differentiable) differential form ω ∈ Ω k ( M ) {\displaystyle \omega \in \Omega ^{k}(M)} , the second exterior derivative vanishes: d 2 ω := d ( d ω ) = 0 {\displaystyle d^{2}\omega :=d(d\omega )=0} . This implies that every differentiable exact form (i.e., a form α {\displaystyle \alpha } such that α = d ω {\displaystyle \alpha =d\omega } for some form ω {\displaystyle \omega } ) is closed (i.e., d α = 0 {\displaystyle d\alpha =0} ), since d α = d ( d ω ) = 0 {\displaystyle d\alpha =d(d\omega )=0} . [ 38 ]
In the middle of the 18th century, the theory of differential forms was first studied in the simplest case of 1-forms in the plane, i.e. A d x + B d y {\displaystyle A\,dx+B\,dy} , where A {\displaystyle A} and B {\displaystyle B} are functions in the plane. The study of 1-forms and the differentials of functions began with Clairaut's papers in 1739 and 1740. At that stage his investigations were interpreted as ways of solving ordinary differential equations . Formally Clairaut showed that a 1-form ω = A d x + B d y {\displaystyle \omega =A\,dx+B\,dy} on an open rectangle is closed, i.e. d ω = 0 {\displaystyle d\omega =0} , if and only ω {\displaystyle \omega } has the form d f {\displaystyle df} for some function f {\displaystyle f} in the disk. The solution for f {\displaystyle f} can be written by Cauchy's integral formula
while if ω = d f {\displaystyle \omega =df} , the closed property d ω = 0 {\displaystyle d\omega =0} is the identity ∂ x ∂ y f = ∂ y ∂ x f {\displaystyle \partial _{x}\partial _{y}f=\partial _{y}\partial _{x}f} . (In modern language this is one version of the Poincaré lemma .) [ 39 ] | https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives |
In mathematics , a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center , a reflection of a square across its diagonal , a translation of the Euclidean plane , or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element (a point, line or plane). [ 1 ]
In the context of molecular symmetry , a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state.
Two basic facts follow from this definition, which emphasizes its usefulness.
In the context of molecular symmetry, quantum wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.
The identity operation corresponds to doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity operation. The identity operation is denoted by E or I . In the identity operation, no change can be observed for the molecule. Even the most asymmetric molecule possesses the identity operation. The need for such an identity operation arises from the mathematical requirements of group theory.
The reflection operation is carried out with respect to symmetry elements known as planes of symmetry or mirror planes. [ 2 ] Each such plane is denoted as σ (sigma). Its orientation relative to the principal axis of the molecule is indicated by a subscript. The plane must pass through the molecule and cannot be completely outside it.
Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.
In an inversion through a centre of symmetry, i (the element), we imagine taking each point in a molecule and then moving it out the same distance on the other side. In summary, the inversion operation projects each atom through the centre of inversion and out to the same distance on the opposite side. The inversion center is a point in space that lies in the geometric center of the molecule. As a result, all the cartesian coordinates of the atoms are inverted (i.e. x,y,z to –x,–y,–z ). The symbol used to represent inversion center is i . When the inversion operation is carried out n times, it is denoted by i n , where i n = E {\displaystyle i^{n}=E} when n is even and i n = − E {\displaystyle i^{n}=-E} when n is odd.
Examples of molecules that have an inversion center include certain molecules with octahedral geometry (general formula AB 6 ), square planar geometry (general formula AB 4 ), and ethylene ( H 2 C=CH 2 ). Examples of molecules without inversion centers are cyclopentadienide ( C 5 H − 5 ) and molecules with trigonal pyramidal geometry (general formula AB 3 ). [ 3 ]
A proper rotation refers to simple rotation about an axis . Such operations are denoted by C n m , {\displaystyle C_{n}^{m},} where C n is a rotation of 360 ∘ n {\displaystyle {\tfrac {360^{\circ }}{n}}} or 2 π n , {\displaystyle {\tfrac {2\pi }{n}},} performed m times. The superscript m is omitted if it is equal to one. C 1 is a rotation through 360°, where n = 1 . It is equivalent to the Identity ( E ) operation. C 2 is a rotation of 180°, as 360 ∘ 2 = 180 ∘ , {\displaystyle {\tfrac {360^{\circ }}{2}}=180^{\circ },} C 3 is a rotation of 120°, as 360 ∘ 3 = 120 ∘ , {\displaystyle {\tfrac {360^{\circ }}{3}}=120^{\circ },} and so on.
Here the molecule can be rotated into equivalent positions around an axis. An example of a molecule with C 2 symmetry is the water ( H 2 O ) molecule. If the H 2 O molecule is rotated by 180° about an axis passing through the oxygen atom, no detectable difference before and after the C 2 operation is observed.
Order n of an axis can be regarded as a number of times that, for the least rotation which gives an equivalent configuration, that rotation must be repeated to give a configuration identical to the original structure (i.e. a 360° or 2 π rotation). An example of this is C 3 proper rotation, which rotates by 2 π 3 . {\displaystyle {\tfrac {2\pi }{3}}.} C 3 represents the first rotation around the C 3 axis by 2 π 3 , {\displaystyle {\tfrac {2\pi }{3}},} C 3 2 {\displaystyle C_{3}^{2}} is the rotation by 2 × 2 π 3 , {\displaystyle 2\times {\tfrac {2\pi }{3}},} while C 3 3 {\displaystyle C_{3}^{3}} is the rotation by 3 × 2 π 3 . {\displaystyle 3\times {\tfrac {2\pi }{3}}.} C 3 3 {\displaystyle C_{3}^{3}} is the identical configuration because it gives the original structure, and it is called an identity element ( E ). Therefore, C 3 is an order of three, and is often referred to as a threefold axis. [ 3 ]
An improper rotation involves two operation steps: a proper rotation followed by reflection through a plane perpendicular to the rotation axis. The improper rotation is represented by the symbol S n where n is the order. Since the improper rotation is the combination of a proper rotation and a reflection, S n will always exist whenever C n and a perpendicular plane exist separately. [ 3 ] S 1 is usually denoted as σ , a reflection operation about a mirror plane. S 2 is usually denoted as i , an inversion operation about an inversion center. When n is an even number S n n = E , {\displaystyle S_{n}^{n}=E,} but when n is odd S n 2 n = E . {\displaystyle S_{n}^{2n}=E.}
Rotation axes, mirror planes and inversion centres are symmetry elements , not symmetry operations. The rotation axis of the highest order is known as the principal rotation axis. It is conventional to set the Cartesian z -axis of the molecule to contain the principal rotation axis.
Dichloromethane , CH 2 Cl 2 . There is a C 2 rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the z axis as co-linear with the C 2 axis, the xz plane as containing CH 2 and the yz plane as containing CCl 2 . A C 2 rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the yz plane permutes the hydrogen atoms while reflection in the xz plane permutes the chlorine atoms. The four symmetry operations E , C 2 , σ( xz ) and σ( yz ) form the point group C 2 v . Note that if any two operations are carried out in succession the result is the same as if a single operation of the group had been performed.
Methane , CH 4 . In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular S 4 axes which pass half-way between the C-H bonds and six mirror planes. Note that S 4 2 = C 2 . {\displaystyle S_{4}^{2}=C_{2}.}
In crystals, screw rotations and/or glide reflections are additionally possible. These are rotations or reflections together with partial translation. These operations may change based on the dimensions of the crystal lattice.
The Bravais lattices may be considered as representing translational symmetry operations. Combinations of operations of the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups .
Molecular symmetry
Crystal structure
Crystallographic restriction theorem
F. A. Cotton Chemical applications of group theory , Wiley, 1962, 1971 | https://en.wikipedia.org/wiki/Symmetry_operation |
Symmorphosis is the regulation of biological units to produce an optimal outcome. [ 1 ] Symmorphosis is when a quantitative match of design and function within an organism defined within a functional system. [ 2 ] Symmorphosis can be broken down into the three predictions that are required for organs to evolve within a species.
This proposes that if organs were matched structurally and functionally, and paired with the correct energy and minerals, the body would create an organ of optimal design. [ 3 ] Some examples of this in the human body could be how the respiratory system distributes oxygen, how bones are structured to withstand stress, how blood vessels are designed to distribute blood throughout the body without using a lot of energy, or even how as a person becomes more physically fit or endures more cardio after their body has adjusted to maintain higher functioning demands. [ 2 ] The use of symmorphosis can allow for fields of science to work with the field of evolutionary biology to better understand adaptation . [ 4 ]
For symmorphosis to occur, there must be three predictions or guidelines in place and functioning at the same time. These three predictions work together to let an organ function or organ system work at full potential.
When looking at the theory of symmorphosis, one must consider if the design in the organism is fully optimized. [ 3 ] The structural design in terms of symmorphosis means that the organ is designed to allow full capacity of its function and can allow for adjustments to occur when necessary. [ 5 ] This design must contain the sufficient amount of economy material for the organ needed. In this circumstance, economy material is the careful management of resources such as tissues.
The functional capacity is when all functional units work together to determine the maximal capacity. [ 5 ] Functional capacity is overall determined by the structural design. Once the design is optimized in terms of biological materials, then the structure must be taken into account. The structure of an organ determines the maximal functional capacity and the adjustments required for morphogenesis—the process that causes an organism to create its shape—to occur. [ 3 ]
The third prediction states that if prediction two works in intermediate steps to create a function of an individual organ, then each step also helps create the upper limit of the function. [ 3 ] " This means that if multiple units work together in multiple steps, they function together to create an upper limit (e.g., V o2max) in terms of function or ability.
A common form of testing symmorphosis between species of mammals is to use comparative biology. The first system to use the proposed theory for symmorphosis is the oxygen pathway for mammals. [ 3 ]
The original experimental method for symmorphosis was used to show if the design of the organs were relative to the static demands of the mammalian respiratory system. The respiratory system is a good example to study because it has one main function, the function has a measurable limit, the limit is variable, it has a sequence of structure, and each step of the sequence has functional parameters that are not fixed. [ 2 ] A common pathway within the respiratory system is the oxygen pathway. This pathway is used because it is a good representation for mammals within most species—because it involves several organs that link together, and the overall function has a measurable upper limit. [ 3 ] In particular, this testing helps identify structural elements that differ so they can carry the maximum amount of oxygen throughout the body. [ 1 ]
The upper limit for the oxygen pathway is called the V o2max . V o2max is the maximal oxygen capacity that systems can take in, transport, and use oxygen. [ 3 ] [ 6 ] V o2max can vary among individuals due to allometric variation (the differences in body mass), adaptive variation (differences in lifestyles), and the induced variation (amount of cardio exercise). Variation of any of the three types of variation should lead researchers to expect different parameters. [ 2 ]
The oxygen cascade is one system with clear limits, and can help determine the V o2max by components such as oxygen supply to the skeletal muscle mitochondria and the demand of oxygen by these skeletal muscle mitochondria. [ 7 ] If oxygen is not transferred via skeletal muscle mitochondria, it can then be transferred across muscle capillaries. [ 4 ]
Symmophsis can be use as an analytical advancement that helps other fields of science—such as biochemistry , physiology , and astronomy —work with fields such as cell , molecular , and evolutionary biology . [ 4 ] Combining these fields helps researchers better understand past biological adaptions.
In evolution, natural selection can hinder the design when looking at the guidelines for symmorphosis. Natural selection can alter the phenotype to increase fitness of a species. In doing this, natural selection can cause adaptations that can change the optimal structural. [ 1 ] When the optimal structural design changes, it changes the amount of economy material that must be used, which changes the predictions.
An issue with symmorphosis is the problem of having an optimal design for an organ if the organ contains multiple functions. [ 1 ] An organ that performs multiple functions must compromise optimal performance of one function to perform another optimally. These complex components adding together dramatically decreases the chance that everything will optimally match. [ 1 ] An example of this in mammals is the lungs. Researchers now claim that the lungs are an exception when considering the Lungs typically are only partially adjusted to maximal oxygen capacity in terms of adaptive and allometric variation and cause a fluctuation in these values. [ 3 ]
In terms of symmorphosis, the capacity of each step of the oxygen cascade should match the demand of V max. [ 7 ] In most cases this theory holds true with the exception of when an individual exceeds the V max. When V max is exceeded there then becomes developmental constraints as well as design constraints in terms of symmorphosis. [ 4 ] When this occurs there is an unmatched capacity, although they may be similar they do not align with the predictions for symmorphosis. [ 1 ] | https://en.wikipedia.org/wiki/Symmorphosis |
Sympathetic cooling is a process in which particles of one type cool particles of another type.
Typically, atomic ions that can be directly laser cooled are used to cool nearby ions or atoms, by way of their mutual Coulomb interaction . This technique is used to cool ions and atoms that cannot be cooled directly by laser cooling, which includes most molecular ion species, especially large organic molecules. [ 1 ] However, sympathetic cooling is most efficient when the mass/charge ratios of the sympathetic- and laser-cooled ions are similar. [ 2 ]
The cooling of neutral atoms in this manner was first demonstrated by Christopher Myatt et al. in 1997. [ 3 ] Here, a technique with electric and magnetic fields were used, where atoms with spin in one direction were more weakly confined than those with spin in the opposite direction. The weakly confined atoms with a high kinetic energy were allowed to more easily escape, lowering the total kinetic energy, resulting in a cooling of the strongly confined atoms.
Myatt et al. also showed the utility of their version of sympathetic cooling for the creation of Bose–Einstein condensates .
This nuclear physics or atomic physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sympathetic_cooling |
A sympathetic detonation ( SD , or SYDET ), also called flash over or secondary/secondaries (explosion), is a detonation , usually unintended, of an explosive charge by a nearby explosion .
A sympathetic detonation is caused by a shock wave , or impact of primary or secondary blast fragments .
The initiating explosive is called the donor explosive , the initiated one is known as the receptor explosive . In case of a chain detonation, a receptor explosive can become a donor one.
The shock sensitivity , also called gap sensitivity , which influences the susceptibility to sympathetic detonations, can be measured by gap tests .
If detonators with primary explosives are used, the shock wave of the initiating blast may set off the detonator and the attached charge. However even relatively insensitive explosives can be set off if their shock sensitivity is sufficient. Depending on the location, the shock wave can be transported by air, ground, or water. The process is probabilistic, a radius with 50% probability of sympathetic detonation often being used for quantifying the distances involved.
Sympathetic detonation presents problems in storage and transport of explosives and ordnance . Sufficient spacing between adjacent stacks of explosive materials has to be maintained. [ 1 ] In case of an accidental detonation of one charge, other ones in the same container or dump can be detonated as well, but the explosion should not spread to other storage units. Special containers attenuating the shock wave can be used to prevent the sympathetic detonations; epoxy -bonded pumice liners were successfully tested. [ 2 ] Blow-off panels may be used in structures, e.g. tank ammunition compartments, to channel the explosion overpressure in a desired direction to prevent a catastrophic failure .
Other factors causing unintended detonations are e.g. flame spread , heat radiation , and impact of fragmentation .
A related term is cooking off , setting off an explosive by subjecting it to sustained heat of e.g. a fire or a hot gun barrel . A cooked-off explosive may cause sympathetic detonation of adjacent explosives.
Sympathetic detonations may occur in munitions stored in e.g. vehicles, ships (called a Magazine Explosion), gun mounts, or ammunition depot , by a sufficiently close explosion of a projectile or a bomb. Such detonations after receiving a hit have caused many catastrophic losses of vehicles. [ 3 ]
To prevent sympathetic detonations, minimal distances (specific for a given type of the mine) have to be maintained between mines when laying a minefield .
Spallation of materials after an impact on the opposite side may create fragments capable of causing sympathetic detonations of stored explosives on the opposite side of an armour plate or a concrete wall. [ 4 ] Transfer of the shock wave through the wall or armour may also be possible cause of a sympathetic detonation.
Class 1.1 solid rocket fuels are susceptible to sympathetic detonation. Conversely, class 1.3 fuels can be ignited by a nearby fire or explosion, but are generally not susceptible to sympathetic detonation. Class 1.1 fuels, however, tend to have slightly higher specific impulses , and therefore are used in those military applications where weight and/or size is at a premium, e.g. on ballistic and cruise missile submarines . [ 5 ]
Sympathetic detonation can be used for the destruction of unexploded ordnance , improvised explosive devices , land mines , or naval mines by an adjacent bulk charge.
Special insensitive explosives , such as TATB , are used in certain military applications to avoid sympathetic detonations.
During the Attack of Pearl Harbor , the USS Arizona was struck with an armor-piercing bomb which penetrated the upper deck and stopped inside the forward magazine. The bomb triggered an explosion which was powerful enough to cut the Arizona in half and is considered a sympathetic detonation as there was an apparent delay between the detonation of the bomb and the contents of the forward magazine.
Sympathetic detonation killed 320 sailors and injured 390 others in the Port Chicago Disaster of July 17, 1944 at the Port Chicago Naval Magazine in Port Chicago, California . [ 6 ] [ 7 ]
During the 1967 USS Forrestal fire , eight old Composition B based iron bombs cooked off . The last one caused a sympathetic detonation of a ninth bomb, a more modern and less cookoff-susceptible Composition H6 based one.
The Russian submarine Kursk explosion was probably caused by a sympathetic explosion of several torpedo warheads. A single dummy torpedo VA-111 Shkval exploded; 135 seconds later a number of warheads simultaneously exploded and sank the submarine.
Multiple incidents have been recorded in the more recent GWoT where airstrikes have set off explosives or ammunition caches in insurgent positions. [ 8 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ]
In rock blasting , sympathetic detonations occur when the blastholes are sufficiently close to each other, usually 24in or less, and especially in rocks that poorly attenuate the shock energy. Ground water in open channels facilitates sympathetic detonation as well. Blasthole spacing of 36in or more is suggested. However, in some ditch blasting cases sympathetic detonations are exploited purposefully. [ 13 ] Nitroglycerine -based explosives are especially susceptible. Picric acid is sensitive as well. [ 14 ] Water gel explosives , slurry explosives , and emulsion explosives tend to be insensitive to sympathetic detonations. For most industrial explosives, the maximum distances for possible sympathetic detonations are between 2–8 times of the charge diameter. [ 15 ] Uncontrolled sympathetic detonations may cause excessive ground vibrations and/or flying rocks.
The spread of shock waves can be hindered by placing relief holes – drilled holes without explosive charges – between the blastholes. [ 14 ]
The opposite phenomenon is dynamic desensitization . Some explosives, e.g. ANFO , show reduced sensitivity under pressure. A transient pressure wave from a nearby detonation may compress the explosive sufficiently to make its initiation fail. This can be prevented by introducing sufficient delays into the firing sequence. [ 14 ]
A sympathetic detonation during mine blasting may influence the seismic signature of the blast, by boosting the P-wave amplitude without significantly amplifying the surface wave . [ 16 ] | https://en.wikipedia.org/wiki/Sympathetic_detonation |
Sympatric speciation is the evolution of a new species from a surviving ancestral species while both continue to inhabit the same geographic region. In evolutionary biology and biogeography , sympatric and sympatry are terms referring to organisms whose ranges overlap so that they occur together at least in some places. If these organisms are closely related (e.g. sister species ), such a distribution may be the result of sympatric speciation . Etymologically , sympatry is derived from Greek συν (sun-) ' together ' and πατρίς (patrís) ' fatherland ' . [ 1 ] The term was coined by Edward Bagnall Poulton in 1904, who explains the derivation. [ 1 ]
Sympatric speciation is one of three traditional geographic modes of speciation. [ 2 ] [ 3 ] Allopatric speciation is the evolution of species caused by the geographic isolation of two or more populations of a species. In this case, divergence is facilitated by the absence of gene flow. Parapatric speciation is the evolution of geographically adjacent populations into distinct species. In this case, divergence occurs despite limited interbreeding where the two diverging groups come into contact. In sympatric speciation, there is no geographic constraint to interbreeding. These categories are special cases of a continuum from zero (sympatric) to complete (allopatric) spatial segregation of diverging groups. [ 3 ]
In multicellular eukaryotic organisms, sympatric speciation is a plausible process that is known to occur, but the frequency with which it occurs is not known. [ 4 ] In bacteria, however, the analogous process (defined as "the origin of new bacterial species that occupy definable ecological niches ") might be more common because bacteria are less constrained by the homogenizing effects of sexual reproduction and are prone to comparatively dramatic and rapid genetic change through horizontal gene transfer . [ 5 ]
Sympatric speciation events are quite common in plants, which are prone to acquiring multiple homologous sets of chromosomes , resulting in polyploidy . The polyploid offspring occupy the same environment as the parent plants (hence sympatry), but are reproductively isolated.
A number of models have been proposed for alternative modes of sympatric speciation. The most popular, which invokes the disruptive selection model, was first put forward by John Maynard Smith in 1966. [ 6 ] Maynard Smith suggested that homozygous individuals may, under particular environmental conditions, have a greater fitness than those with alleles heterozygous for a certain trait. Under the mechanism of natural selection , therefore, homozygosity would be favoured over heterozygosity, eventually leading to speciation. Sympatric divergence could also result from the sexual conflict . [ 7 ]
Disruption may also occur in multiple-gene traits. The medium ground finch ( Geospiza fortis ) is showing gene pool divergence in a population on Santa Cruz Island . Beak morphology conforms to two different size ideals, while intermediate individuals are selected against. Some characteristics (termed magic traits ) such as beak morphology may drive speciation because they also affect mating signals. In this case, different beak phenotypes may result in different bird calls , providing a barrier to exchange between the gene pools. [ 8 ]
A somewhat analogous system has been reported in horseshoe bats, in which echolocation call frequency appears to be a magic trait. In these bats, the constant frequency component of the call not only determines prey size but may also function in aspects of social communication. Work from one species, the large-eared horseshoe bat ( Rhinolophus philippinensis ), shows that abrupt changes in call frequency among sympatric morphs is correlated with reproductive isolation. [ 9 ] A further well-studied circumstance of sympatric speciation is when insects feed on more than one species of host plant . In this case insects become specialized as they struggle to overcome the various plants' defense mechanisms . (Drès and Mallet, 2002) [ 10 ]
Rhagoletis pomonella , the apple maggot , may be currently undergoing sympatric or, more precisely, heteropatric (see heteropatry ) speciation. The apple feeding race of this species appears to have spontaneously emerged from the hawthorn feeding race in the 1800–1850 AD time frame, after apples were first introduced into North America . The apple feeding race does not now normally feed on hawthorns , and the hawthorn feeding race does not now normally feed on apples. This may be an early step towards the emergence of a new species. [ 11 ] [ 12 ] [ 13 ]
Some parasitic ants may have evolved via sympatric speciation. [ 14 ] Isolated and relatively homogeneous habitats such as crater lakes and islands are among the best geographical settings in which to demonstrate sympatric speciation. For example, Nicaragua crater lake cichlid fishes include nine described species and dozens of undescribed species that have evolved by sympatric speciation. [ 15 ] [ 16 ] Monostroma latissimum , a marine green algae, also shows sympatric speciation in southwest Japanese islands. Although panmictic , the molecular phylogenetics using nuclear introns revealed staggering diversification of population. [ 17 ]
African cichlids also offer some evidence for sympatric speciation. They show a large amount of diversity in the African Great Lakes . Many studies point to sexual selection as a way of maintaining reproductive isolation. Female choice with regards to male coloration is one of the more studied modes of sexual selection in African cichlids. Female choice is present in cichlids because the female does much of the work in raising the offspring, while the male has little energy input in the offspring. She exerts sensory bias when picking males by choosing those that have colors similar to her or those that are the most colorful. [ 18 ] [ 19 ] [ 20 ] This helps maintain sympatric speciation within the lakes. Cichlids also use acoustic reproductive communication. The male cichlid quivers as a ritualistic display for the female which produces a certain number of pulses and pulse period. Female choice for good genes and sensory bias is one of the deciding factors in this case, selecting for calls that are within her species and that give the best fitness advantage to increase the survivability of the offspring. [ 21 ] [ 22 ] Male-male competition is a form of intrasexual selection and also has an effect on speciation in African cichlids. Ritualistic fighting among males establishes which males are going to be more successful in mating. This is important in sympatric speciation because species with similar males may be competing for the same females. There may be a fitness advantage for one phenotype that could allow one species to invade another. [ 23 ] [ 24 ] Studies show this effect in species that are genetically similar, have the capability to interbreed, and show phenotypic color variation. Ecological character displacement is another means for sympatric speciation. Within each lake there are different niches that a species could occupy. For example, different diets and depth of the water could help to maintain isolation between species in the same lake.
Allochrony offers some empirical evidence that sympatric speciation has taken place, as many examples exist of recently diverged ( sister taxa ) allochronic species. A case of ongoing sympatric divergence due to allochrony might be found in the marine insect Clunio marinus . [ 25 ]
A rare example of sympatric speciation in animals is the divergence of "resident" and "transient" orca forms in the northeast Pacific. [ 26 ] Resident and transient orcas inhabit the same waters, but avoid each other and do not interbreed. The two forms hunt different prey species and have different diets, vocal behaviour, and social structures. Some divergences between species could also result from contrasts in microhabitats. A population bottleneck occurred around 200,000 years ago greatly reducing the population size at the time as well as the variance of genes which allowed several ecotypes to emerge afterwards. [ 27 ]
The European polecat ( Mustela putorius ) exhibited a rare dark phenotype similar to the European mink ( Mustela lutreola ) phenotype, which is directly influenced by peculiarities of forest brooks. [ 28 ]
For some time it was difficult to prove that sympatric speciation was possible, because it was impossible to observe it happening. [ 3 ] It was believed by many, and championed by Ernst Mayr , that the theory of evolution by natural selection could not explain how two species could emerge from one if the subspecies were able to interbreed. [ 29 ] Since Mayr's heyday in the 1940s and 50s, mechanisms have been proposed that explain how speciation might occur in the face of interbreeding, also known as gene flow . [ 30 ] And even more recently concrete examples of sympatric divergence have been empirically studied. [ 31 ] [ 32 ] The debate now turns to how often sympatric speciation may actually occur in nature and how much of life's diversity it may be responsible for.
The German evolutionary biologist Ernst Mayr argued in the 1940s that speciation cannot occur without geographic, and thus reproductive, isolation. [ 29 ] He stated that gene flow is the inevitable result of sympatry, which is known to squelch genetic differentiation between populations. Thus, a physical barrier must be present, he believed, at least temporarily, in order for a new biological species to arise. [ 33 ] This hypothesis is the source of much controversy around the possibility of sympatric speciation. Mayr's hypothesis was popular and consequently quite influential, but is now widely disputed. [ 34 ]
The first to propose what is now the most pervasive hypothesis on how sympatric speciation may occur was John Maynard Smith , in 1966. He came up with the idea of disruptive selection. He figured that if two ecological niches are occupied by a single species, diverging selection between the two niches could eventually cause reproductive isolation . [ 35 ] By adapting to have the highest possible fitness in the distinct niches, two species may emerge from one even if they remain in the same area, and even if they are mating randomly. [ 30 ]
Investigating the possibility of sympatric speciation requires a definition thereof, especially in the 21st century, when mathematical modeling is used to investigate or to predict evolutionary phenomena. [ 34 ] Much of the controversy concerning sympatric speciation may lie solely on an argument over what sympatric divergence actually is. The use of different definitions by researchers is a great impediment to empirical progress on the matter. The dichotomy between sympatric and allopatric speciation is no longer accepted by the scientific community. It is more useful to think of a continuum, on which there are limitless levels of geographic and reproductive overlap between species. On one extreme is allopatry, in which the overlap is zero (no gene flow), and on the other extreme is sympatry, in which the ranges overlap completely (maximal gene flow).
The varying definitions of sympatric speciation fall generally into two categories: definitions based on biogeography, or on population genetics. As a strictly geographical concept, sympatric speciation is defined as one species diverging into two while the ranges of both nascent species overlap entirely – this definition is not specific enough about the original population to be useful in modeling. [ 3 ]
Definitions based on population genetics are not necessarily spatial or geographical in nature, and can sometimes be more restrictive. These definitions deal with the demographics of a population, including allele frequencies, selection, population size, the probability of gene flow based on sex ratio, life cycles, etc. The main discrepancy between the two types of definitions tends to be the necessity for "panmixia". Population genetics definitions of sympatry require that mating be dispersed randomly – or that it be equally likely for an individual to mate with either subspecies, in one area as another, or on a new host as a nascent one: this is also known as panmixia. [ 3 ] Population genetics definitions, also known as non-spatial definitions, thus require the real possibility for random mating, and do not always agree with spatial definitions on what is and what is not sympatry.
For example, micro-allopatry, also known as macro-sympatry, is a condition where there are two populations whose ranges overlap completely, but contact between the species is prevented because they occupy completely different ecological niches (such as diurnal vs. nocturnal). This can often be caused by host-specific parasitism, which causes dispersal to look like a mosaic across the landscape. Micro-allopatry is included as sympatry according to spatial definitions, but, as it does not satisfy panmixia, it is not considered sympatry according to population genetics definitions. [ 3 ]
Mallet et al. (2002) claims that the new non-spatial definition is lacking in an ability to settle the debate about whether sympatric speciation regularly occurs in nature. They suggest using a spatial definition, but one that includes the role of dispersal, also known as cruising range, so as to represent more accurately the possibility for gene flow. They assert that this definition should be useful in modeling. They also state that under this definition, sympatric speciation seems plausible. [ 33 ]
Evolutionary theory as well as mathematical models have predicted some plausible mechanisms for the divergence of species without a physical barrier. [ 30 ] In addition there have now been several studies that have identified speciation that has occurred, or is occurring with gene flow (see section above: evidence). Molecular studies have been able to show that, in some cases where there is no chance for allopatry, species continue to diverge. One such example is a pair of species of isolated desert palms. Two distinct, but closely related species exist on the same island, but they occupy two distinct soil types found on the island, each with a drastically different pH balance. [ 31 ] Because they are palms they send pollen through the air they could freely interbreed, except that speciation has already occurred, so that they do not produce viable hybrids. This is hard evidence for the fact that, in at least some cases, fully sympatric species really do experience diverging selection due to competition, in this case for a spot in the soil.
This, and the other few concrete examples that have been found, are just that; they're few, so they tell us little about how often sympatry actually results in speciation in a more typical context. The burden now lies on providing evidence for sympatric divergence occurring in non-isolated habitats. It is not known how much of the earth's diversity it could be responsible for. Some still say that panmixia should slow divergence, and thus sympatric speciation should be possible but rare (1). Meanwhile, others claim that much of the earth's diversity could be due to speciation without geographic isolation. [ 36 ] The difficulty in supporting a sympatric speciation hypothesis has always been that an allopatric scenario could always be invented, and those can be hard to rule out – but with modern molecular genetic techniques can be used to support the theory. [ 36 ]
In 2015 Cichlid fish from a tiny volcanic crater lake in Africa were observed in the act of sympatric speciation using DNA sequencing methods. A study found a complex combination of ecological separation and mate choice preference had allowed two ecomorphs to genetically separate even in the presence of some genetic exchange. [ 37 ] [ 38 ]
Heteropatric speciation is a special case of sympatric speciation that occurs when different ecotypes or races of the same species geographically coexist but exploit different niches in the same patchy or heterogeneous environment. It is thus is a refinement of sympatric speciation, with a behavioral, rather than geographical barrier to the flow of genes among diverging groups within a population. Behavioral separation as a mechanism for promoting sympatric speciation in a heterogeneous (or patchwork landscape) was highlighted in John Maynard Smith 's seminal paper on sympatric speciation. [ 39 ] In recognition of the importance of this behavioral versus geographic distinction, Wayne Getz and Veijo Kaitala introduced the term heteropatry in their extension of Maynard Smiths' analysis [ 40 ] of conditions that facilitate sympatric speciation.
Although some evolutionary biologists still regard sympatric speciation as highly contentious, both theoretical [ 41 ] and empirical [ 42 ] studies support it as a likely explanation of the diversity of life in particular ecosystems. Arguments implicate competition and niche separation of sympatric ecological variants that evolve through assortative mating into separate races and then species. Assortative mating most easily occurs if mating is linked to niche preference, as occurs in the apple maggot Rhagoletis pomonella , where individual flies from different races use volatile odors to discriminate between hawthorn and apple and look for mates on natal fruit. The term heteropatry semantically resolves the issue of sympatric speciation by reducing it to a scaling issue in terms of the way the landscape is used by individuals versus populations. From a population perspective, the process looks sympatric, but from an individual's perspective, the process looks allopatric, once the time spent flying over or moving quickly through intervening non-preferred niches is taken into account. [ citation needed ] | https://en.wikipedia.org/wiki/Sympatric_speciation |
In biology, two closely related species or populations are considered sympatric when they exist in the same geographic area and thus frequently encounter each other. [ 1 ] An initially interbreeding population that splits into two or more distinct species sharing a common range exemplifies sympatric speciation . Such speciation may be a product of reproductive isolation – which prevents hybrid offspring from being viable or able to reproduce, thereby reducing gene flow – that results in genetic divergence. [ 2 ] Sympatric speciation may, but need not, arise through secondary contact, which refers to speciation or divergence in allopatry followed by range expansions leading to an area of sympatry. Sympatric species or taxa in secondary contact may or may not interbreed .
Four main types of population pairs exist in nature. Sympatric populations (or species) contrast with parapatric populations, which contact one another in adjacent but not shared ranges and do not interbreed; peripatric species, which are separated only by areas in which neither organism occurs; and allopatric species, which occur in entirely distinct ranges that are neither adjacent nor overlapping. [ 3 ] Allopatric populations isolated from one another by geographical factors (e.g., mountain ranges or bodies of water) may experience genetic—and, ultimately, phenotypic—changes in response to their varying environments. These may drive allopatric speciation , which is arguably the dominant mode of speciation. [ citation needed ]
The lack of geographic isolation as a definitive barrier between sympatric species has yielded controversy among ecologists, biologists, botanists, and zoologists regarding the validity of the term. As such, researchers have long debated the conditions under which sympatry truly applies, especially with respect to parasitism . Because parasitic organisms often inhabit multiple hosts during a life cycle, evolutionary biologist Ernst Mayr stated that internal parasites existing within different hosts demonstrate allopatry, not sympatry. Today, however, many biologists consider parasites and their hosts to be sympatric (see examples below). Conversely, zoologist Michael J. D. White considered two populations sympatric if genetic interbreeding was viable within the habitat overlap. This may be further specified as sympatry occurring within one deme ; that is, reproductive individuals must be able to locate one another in the same population in order to be sympatric.
Others question the ability of sympatry to result in complete speciation: until recently, many researchers considered it nonexistent, doubting that selection alone could create disparate, but not geographically separated, species. In 2003, biologist Karen McCoy suggested that sympatry can act as a mode of speciation only when "the probability of mating between two individuals depend[s] [solely] on their genotypes, [and the genes are] dispersed throughout the range of the population during the period of reproduction". [ 4 ] In essence, sympatric speciation does require very strong forces of natural selection to be acting on heritable traits, as there is no geographic isolation to aid in the splitting process. Yet, recent research has begun to indicate that sympatric speciation is not as uncommon as was once assumed.
Syntopy is a special case of sympatry. It means the joint occurrence of two species in the same habitat at the same time. Just as the broader term sympatry, "syntopy" is used especially for close species that might hybridise or even be sister species . Sympatric species occur together in the same region, but do not necessarily share the same localities as syntopic species do. Areas of syntopy are of interest because they allow to study how similar species may coexist without outcompeting each other.
As an example, the two bat species Myotis auriculus and M. evotis were found to be syntopic in North America. [ 5 ] In contrast, the marbled newt and the northern crested newt have a large sympatric range in western France, but differ in their habitat preferences and only rarely occur syntopically in the same breeding ponds. [ 6 ]
The lack of geographic constraint in isolating sympatric populations implies that the emerging species avoid interbreeding via other mechanisms. Before speciation is complete, two diverging populations may still produce viable offspring. As speciation progresses, isolating mechanisms – such as gametic incompatibility that renders fertilization of the egg impossible – are selected for in order to increase the reproductive divide between the two populations.
Sympatric groups frequently show a greater ability to discriminate between their own species and other closely related species than do allopatric groups. This is shown in the study of hybrid zones . It is also apparent in the differences in levels of prezygotic isolation (by factors that prevent formation of a viable zygote ) in both sympatric and allopatric populations. There are two main theories regarding this process: 1) differential fusion, which suggests that only populations with a keen ability to discriminate between species will persist in sympatry; and 2) character displacement , which implies that distinguishing characteristics will be heightened in areas where the species co-occur in order to facilitate discrimination.
Reinforcement is the process by which natural selection reinforces reproductive isolation. In sympatry, reinforcement increases species discrimination and sexual adaptation in order to avoid maladaptive hybridization and encourage speciation. If hybrid offspring are either sterile or less-fit than non-hybrid offspring, mating between members of two different species will be selected against. Natural selection decreases the probability of such hybridization by selecting for the ability to identify mates of one's own species from those of another species.
Reproductive character displacement strengthens the reproductive barriers between sympatric species by encouraging the divergence of traits that are crucial to reproduction. Divergence is frequently distinguished by assortative mating between individuals of the two species. [ 7 ] For example, divergence in the mating signals of two species will limit hybridization by reducing one's ability to identify an individual of the second species as a potential mate. Support for the reproductive character displacement hypothesis comes from observations of sympatric species in overlapping habitats in nature. Increased prezygotic isolation, which is associated with reproductive character displacement, has been observed in cicadas of genus Magicicada , stickleback fish, and the flowering plants of the genus Phlox .
An alternative explanation for species discrimination in sympatry is differential fusion. This hypothesis states that of the many species have historically come into contact with one another, the only ones that persist in sympatry (and thus are seen today) are species with strong mating discrimination. On the other hand, species lacking strong mating discrimination are assumed to have fused while in contact, forming one distinct species.
Differential fusion is less widely recognized than character displacement, and several of its implications are refuted by experimental evidence. For example, differential fusion implies greater postzygotic isolation among sympatric species, as this functions to prevent fusion between the species. However, Coyne and Orr found equal levels of postzygotic isolation among sympatric and allopatric species pairs in closely related Drosophila . [ 8 ] Nevertheless, differential fusion remains a possible, though not complete, contributor to species discrimination. [ 9 ]
Sympatry has been increasingly evidenced in current research. Because of this, sympatric speciation – which was once highly debated among researchers – is progressively gaining credibility as a viable form of speciation.
Several distinct types of killer whale ( Orcinus orca ), which are characterized by an array of morphological and behavioral differences, live in sympatry throughout the North Atlantic, North Pacific and Antarctic oceans. In the North Pacific, three whale populations – called "transient", "resident", and "offshore" – demonstrate partial sympatry, crossing paths with relative frequency. The results of recent genetic analyses using mtDNA indicate that this is due to secondary contact, in which the three types encountered one another following the bidirectional migration of "offshore" and "resident" whales between the North Atlantic and North Pacific. Partial sympatry in these whales is, therefore, not the result of speciation. Furthermore, killer whale populations that consist of all three types have been documented in the Atlantic, evidencing that interbreeding occurs among them. Thus, secondary contact does not always result in total reproductive isolation , as has often been predicted. [ 10 ]
The parasitic great spotted cuckoo ( Clamator glandarius ) and its magpie host, both native to Southern Europe, are completely sympatric species. However, the duration of their sympatry varies with location. For example, great spotted cuckoos and their magpie hosts in Hoya de Gaudix , southern Spain, have lived in sympatry since the early 1960s, while species in other locations have more recently become sympatric. Great spotted cuckoos, when in South Africa, are sympatric with at least 8 species of starling and 2 crows, pied crow and Cape crow . [ 11 ]
The great spotted cuckoo exhibits brood parasitism by laying a mimicked version of the magpie egg in the magpie's nest. Since cuckoo eggs hatch before magpie eggs, magpie hatchlings must compete with cuckoo hatchlings for resources provided by the magpie mother. This relationship between the cuckoo and the magpie in various locations can be characterized as either recently sympatric or anciently sympatric. The results of an experiment by Soler and Moller (1990) showed that in areas of ancient sympatry (species in cohabitation for many generations), magpies were more likely to reject most of the cuckoo eggs, as these magpies had developed counter-adaptations that aid in identification of egg type. In areas of recent sympatry, magpies rejected comparatively fewer cuckoo eggs. Thus, sympatry can cause coevolution , by which both species undergo genetic changes due to the selective pressures that one species exerts on the other. [ 12 ]
Leafcutter ants protect and nourish various species of fungus as a source of food in a system known as ant-fungus mutualism . Leafcutter ants belonging to the genus Acromyrmex are known for their mutualistic relationship with Basidiomycete fungi. Ant colonies are closely associated with their fungus colonies, and may have co-evolved with a consistent vertical lineage of fungi in individual colonies. Ant populations defend against the horizontal transmission of foreign fungi to their fungal colony, as this transmission may lead to competitive stress on the local fungal garden. Invaders are identified and removed by the ant colony, inhibiting competition and fungal interbreeding. This active isolation of individual populations helps maintain the genetic purity of the fungal colony, and this mechanism may lead to sympatric speciation within a shared habitat. [ 13 ] | https://en.wikipedia.org/wiki/Sympatry |
Symphiles are insects or other organisms which live as welcome guests in the nest of a social insect (such as the ant, myrmecophily , or termite, termitophily ) by which they are fed and guarded. The relationship between the symphile and host may be symbiotic, inquiline or parasitic. [ 1 ]
This is a selection of taxa exhibiting symphilia, not a complete list.
Fibularhizoctonia , sometimes referred to as cuckoo fungus due to their adaptation to mimic termite eggs, employ chemical and morphological mimicry to benefit from the defense termites provide their brood. If termite workers are present to care for a brood which contains cuckoo fungus, the sclerotia , or "termite balls", are unlikely to germinate and their presence will increase the survival rate of the termite eggs. When worker termites were experimentally removed from brood that contained slerotia, the fungus germinated by exploiting the termite eggs. This means the termitophilic relationship between termites and Fibularhizoctonia can be parasitic or mutualistic. [ 2 ]
The large blue butterfly, Phengaris arion (formerly Maculinea arion ), exhibits a unique parasitic relationship with a single species of red ant, Myrmica sabuleti . [ 3 ]
Cuckoo bumblebees , members of the subgenus Psithyrus in the genus Bombus , are obligate brood parasites; they must use colonies of true bumblebees to rear their young. A Psithyrus female will kill or subdue the host colony's queen and then use pheromones and/or physical attacks to force the host colony to feed her and raise her brood. [ 4 ]
Many species of Staphylinidae (commonly known as “Rove Beetles”) have developed complex interspecies relationships with ants. Ant associations range from near free-living species which prey only on ants, to obligate inquilines of ants, which exhibit extreme morphological and chemical adaptations to the harsh environments of ant nests. Some species are fully integrated into the host colony, and are cleaned and fed by ants. Many of these, including species in tribe Clavigerini, are myrmecophagous , placating their hosts with glandular secretions while eating the brood. [ 5 ]
Staphylinidae is currently considered to be the largest family of beetles, with over 58,000 species described. As such, many myrmecophilous species are unknown. The majority of studied myrmecophilous Rove Beetles belong to the subfamily Aleocharinae , including the commonly studied genera Pella , Dinarda , Tetradonia , Ecitomorpha , Ecitophya , Atemeles , and Limechusa , and to the subfamily Pselaphinae , which includes Claviger and Adranes . There are also representatives of Scydmaenidae , which includes 117 myrmecophilous species in 20 genera [ 6 ] The Aleocharinae possess defensive glands on their abdomens, which are used in myrmecophilous species to prevent attacks by their host ant and in more extreme cases to integrate completely into the colony. Many Pselephinae species have trichomes , tufts of hairs which hold placating pheromones. Pselephines have evolved trichomes independently at least four times, most notably in all members of Clavigerini, but also in Attapsenius and Songius genera. [ 7 ]
Due to their large number and diversity, myrmecophilous Rove Beetles occupy an array of behaviors. Myrmecophilous interactions can be generalized into categories, in three of which Staphylinids can be found. The synecthrans, or “persecuted guests,” the synoeketes, or “tolerated guests,” and the symphiles, or “true guests.” [ 8 ]
Synecthran insects live on the periphery of the host colony and are not accepted into the colony. [ 9 ]
Synoeketetic insects live in close contact with their host ants but are not integrated into the colony. These species may be further categorized as neutral, mimetic, loricate, and symphiloid synoeketes.
Symphilic insects have been fully integrated into the host’s society. Symphilic species have undergone complex morphological adaptations, many gaining the appearance of their host's species. Most have developed trichomes, which secrete appeasement pheromones. The most extreme adaptations, found in members of tribe Clavigerini, include the reduction of mouthparts for trophallaxis and the fusing of many body and antennal segments. While most symphiles use antennal contact to stimulate food giving from their host, at least one member of Clavigerini, Claviger testaceus , secretes a chemical to induce regurgitation from its host ant Lasius flavus . [ 10 ] Symphiles typically take on many roles in the colony, raising young, feeding and grooming adults, and helping transport food and larvae. Many Staphylinids are capable of following ant pheromone trails, although they are not limited to following trails laid by their host ant. This allows symphiles of army ants to migrate with the colony. [ 11 ] Most species are trophallactic, being fed by other members of the colony. Almost all species have also been observed feeding on the brood, making them obligate parasites .
Once the larvae of the large blue butterfly ( Phengaris arion ) is brought into a Myrmica sabuleti colony, it will mimic the sounds a queen Myrmica larva would make, increasing the chances that the host ant colony will prefer to care for it over their own larvae. The caterpillar feeds on the ant grubs and is a predacious symphile.
Chemical mimicry refers to the production of one species’ chemical signals by another species. Many myrmecophilous Staphylinids have evolved chemical mimicry to deter or placate ants. For Staphylinids accepted into the host colony, chemical mimicry is used for camouflage. The majority of the chemical signals used are cuticular hydrocarbons, which are produced in the cuticle of the host ant at certain concentrations and are palpated to determine the identity of an ant. Species in close contact with their host ants are able to pick up the host’s hydrocarbons and imitate the ant’s hydrocarbon pattern, thus appearing in scent at least to be the same species as the host ant. As hydrocarbon patterns are specific to an individual colony, the rove beetles are generally restricted to one nest. The production of a new hydrocarbon pattern takes time, during which the beetle is vulnerable to detection and attack. Some species, such as Zyras comes , produce volatile pheromones as well as cuticular hydrocarbons, which may provide it more protection than contact based pheromones while traveling with its host in foraging trails. [ 12 ]
The army ants that rove beetles prey on are blind, so it is important that the rove beetles feel similar to their host species. Physical adaptation to resemble ants has evolved in rove beetles on at least twelve separate occasions. [ 13 ] | https://en.wikipedia.org/wiki/Symphiles |
In mathematical field of representation theory , a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space ( V , ω ) which preserves the symplectic form ω . Here ω is a nondegenerate skew symmetric bilinear form
where F is the field of scalars. A representation of a group G preserves ω if
for all g in G and v , w in V , whereas a representation of a Lie algebra g preserves ω if
for all ξ in g and v , w in V . Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp( V , ω ) or its Lie algebra sp ( V , ω )
If G is a compact group (for example, a finite group ), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation . Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius–Schur indicator .
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Symplectic_representation |
In mathematics , a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle \mathbb {R} } ) equipped with a symplectic bilinear form .
A symplectic bilinear form is a mapping ω : V × V → F {\displaystyle \omega :V\times V\to F} that is
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry . If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form , but not vice versa.
Working in a fixed basis , ω {\displaystyle \omega } can be represented by a matrix . The conditions above are equivalent to this matrix being skew-symmetric , nonsingular , and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix , which represents a symplectic transformation of the space. If V {\displaystyle V} is finite-dimensional , then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.
The standard symplectic space is R 2 n {\displaystyle \mathbb {R} ^{2n}} with the symplectic form given by a nonsingular , skew-symmetric matrix . Typically ω {\displaystyle \omega } is chosen to be the block matrix
where I n is the n × n identity matrix . In terms of basis vectors ( x 1 , ..., x n , y 1 , ..., y n ) :
A modified version of the Gram–Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that ω {\displaystyle \omega } takes this form, often called a Darboux basis or symplectic basis .
Sketch of process:
Start with an arbitrary basis v 1 , . . . , v n {\displaystyle v_{1},...,v_{n}} , and represent the dual of each basis vector by the dual basis : ω ( v i , ⋅ ) = ∑ j ω ( v i , v j ) v j ∗ {\displaystyle \omega (v_{i},\cdot )=\sum _{j}\omega (v_{i},v_{j})v_{j}^{*}} . This gives us a n × n {\displaystyle n\times n} matrix with entries ω ( v i , v j ) {\displaystyle \omega (v_{i},v_{j})} . Solve for its null space. Now for any ( λ 1 , . . . , λ n ) {\displaystyle (\lambda _{1},...,\lambda _{n})} in the null space, we have ∑ i ω ( v i , ⋅ ) = 0 {\displaystyle \sum _{i}\omega (v_{i},\cdot )=0} , so the null space gives us the degenerate subspace V 0 {\displaystyle V_{0}} .
Now arbitrarily pick a complementary W {\displaystyle W} such that V = V 0 ⊕ W {\displaystyle V=V_{0}\oplus W} , and let w 1 , . . . , w m {\displaystyle w_{1},...,w_{m}} be a basis of W {\displaystyle W} . Since ω ( w 1 , ⋅ ) ≠ 0 {\displaystyle \omega (w_{1},\cdot )\neq 0} , and ω ( w 1 , w 1 ) = 0 {\displaystyle \omega (w_{1},w_{1})=0} , WLOG ω ( w 1 , w 2 ) ≠ 0 {\displaystyle \omega (w_{1},w_{2})\neq 0} . Now scale w 2 {\displaystyle w_{2}} so that ω ( w 1 , w 2 ) = 1 {\displaystyle \omega (w_{1},w_{2})=1} . Then define w ′ = w − ω ( w , w 2 ) w 1 + ω ( w , w 1 ) w 2 {\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}} for each of w = w 3 , w 4 , . . . , w m {\displaystyle w=w_{3},w_{4},...,w_{m}} . Iterate.
Notice that this method applies for symplectic vector space over any field, not just the field of real numbers.
Case of real or complex field:
When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let w 1 , . . . , w m {\displaystyle w_{1},...,w_{m}} be an orthonormal basis (with respect to the usual inner product on R n {\displaystyle \mathbb {R} ^{n}} ) of W {\displaystyle W} . Since ω ( w 1 , ⋅ ) ≠ 0 {\displaystyle \omega (w_{1},\cdot )\neq 0} , and ω ( w 1 , w 1 ) = 0 {\displaystyle \omega (w_{1},w_{1})=0} , WLOG ω ( w 1 , w 2 ) ≠ 0 {\displaystyle \omega (w_{1},w_{2})\neq 0} . Now multiply w 2 {\displaystyle w_{2}} by a sign, so that ω ( w 1 , w 2 ) ≥ 0 {\displaystyle \omega (w_{1},w_{2})\geq 0} . Then define w ′ = w − ω ( w , w 2 ) w 1 + ω ( w , w 1 ) w 2 {\displaystyle w'=w-\omega (w,w_{2})w_{1}+\omega (w,w_{1})w_{2}} for each of w = w 3 , w 4 , . . . , w m {\displaystyle w=w_{3},w_{4},...,w_{m}} , then scale each w ′ {\displaystyle w'} so that it has norm one. Iterate.
Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the spectral theory of antisymmetric matrices .
There is another way to interpret this standard symplectic form. Since the model space R 2 n used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let V be a real vector space of dimension n and V ∗ its dual space . Now consider the direct sum W = V ⊕ V ∗ of these spaces equipped with the following form:
Now choose any basis ( v 1 , ..., v n ) of V and consider its dual basis
We can interpret the basis vectors as lying in W if we write x i = ( v i , 0) and y i = (0, v i ∗ ) . Taken together, these form a complete basis of W ,
The form ω defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form V ⊕ V ∗ . The subspace V is not unique, and a choice of subspace V is called a polarization . The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians .
Explicitly, given a Lagrangian subspace as defined below , then a choice of basis ( x 1 , ..., x n ) defines a dual basis for a complement, by ω ( x i , y j ) = δ ij .
Just as every symplectic structure is isomorphic to one of the form V ⊕ V ∗ , every complex structure on a vector space is isomorphic to one of the form V ⊕ V . Using these structures, the tangent bundle of an n -manifold, considered as a 2 n -manifold, has an almost complex structure , and the co tangent bundle of an n -manifold, considered as a 2 n -manifold, has a symplectic structure: T ∗ ( T ∗ M ) p = T p ( M ) ⊕ ( T p ( M )) ∗ .
The complex analog to a Lagrangian subspace is a real subspace , a subspace whose complexification is the whole space: W = V ⊕ J V . As can be seen from the standard symplectic form above, every symplectic form on R 2 n is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on C n (with the convention of the first argument being anti-linear).
Let ω be an alternating bilinear form on an n -dimensional real vector space V , ω ∈ Λ 2 ( V ) . Then ω is non-degenerate if and only if n is even and ω n /2 = ω ∧ ... ∧ ω is a volume form . A volume form on a n -dimensional vector space V is a non-zero multiple of the n -form e 1 ∗ ∧ ... ∧ e n ∗ where e 1 , e 2 , ..., e n is a basis of V .
For the standard basis defined in the previous section, we have
By reordering, one can write
Authors variously define ω n or (−1) n /2 ω n as the standard volume form . An occasional factor of n ! may also appear, depending on whether the definition of the alternating product contains a factor of n ! or not. The volume form defines an orientation on the symplectic vector space ( V , ω ) .
Suppose that ( V , ω ) and ( W , ρ ) are symplectic vector spaces. Then a linear map f : V → W is called a symplectic map if the pullback preserves the symplectic form, i.e. f ∗ ρ = ω , where the pullback form is defined by ( f ∗ ρ )( u , v ) = ρ ( f ( u ), f ( v )) . Symplectic maps are volume- and orientation-preserving.
If V = W , then a symplectic map is called a linear symplectic transformation of V . In particular, in this case one has that ω ( f ( u ), f ( v )) = ω ( u , v ) , and so the linear transformation f preserves the symplectic form. The set of all symplectic transformations forms a group and in particular a Lie group , called the symplectic group and denoted by Sp( V ) or sometimes Sp( V , ω ) . In matrix form symplectic transformations are given by symplectic matrices .
Let W be a linear subspace of V . Define the symplectic complement of W to be the subspace
The symplectic complement satisfies:
However, unlike orthogonal complements , W ⊥ ∩ W need not be 0. We distinguish four cases:
Referring to the canonical vector space R 2 n above,
A Heisenberg group can be defined for any symplectic vector space, and this is the typical way that Heisenberg groups arise.
A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative Lie algebra , meaning with trivial Lie bracket. The Heisenberg group is a central extension of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the canonical commutation relations (CCR), and a Darboux basis corresponds to canonical coordinates – in physics terms, to momentum operators and position operators .
Indeed, by the Stone–von Neumann theorem , every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one.
Further, the group algebra of (the dual to) a vector space is the symmetric algebra , and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra : one can think of the central extension as corresponding to quantization or deformation .
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym( V ) := F [ V ∗ ] , and the Weyl algebra is the group algebra of the (dual) Heisenberg group W ( V ) = F [ H ( V ∗ )] . Since passing to group algebras is a contravariant functor , the central extension map H ( V ) → V becomes an inclusion Sym( V ) → W ( V ) . | https://en.wikipedia.org/wiki/Symplectic_vector_space |
A symplectite (or symplektite ) is a material texture: a micrometre-scale or submicrometre-scale intergrowth of two or more crystals. Symplectites form from the breakdown of unstable phases, and may be composed of minerals, ceramics, or metals. Fundamentally, their formation is the result of slow grain-boundary diffusion relative to interface propagation rate. [ 1 ] [ 2 ] [ 3 ]
If a material undergoes a change in temperature, pressure or other physical conditions (e.g., fluid composition or activity), one or more phases may be rendered unstable and recrystallize to more stable constituents. If the recrystallized minerals are fine grained and intergrown, this may be termed a symplectite. A cellular precipitation reaction, in which a reactant phase decomposes to a product phase with the same structure as the parent phase and a second phase with a different structure, can form a symplectite. [ 4 ] Eutectoid reactions, involving the breakdown of a single phase to two or more phases, neither of which is structurally or compositionally identical to the parent phase, can also form symplectites. [ 5 ]
Symplectites may be formed by reaction between adjacent phases or to decomposition of a single phase. The intergrown phases may be planar or rodlike, depending on the volume proportions of the phases, their interfacial free energies, the rate of reaction, the Gibbs free energy change, and the degree of recrystallization. Lamellar symplectites are common in retrogressed eclogite . Kelyphite is a symplectite formed from the decomposition of garnet . [ 6 ] Myrmekite is a globular or bulbous symplectite of quartz in plagioclase . [ 6 ]
Examples of symplectites formed in Earth materials include
dolomite + calcite, [ 7 ] aragonite + calcite, [ 8 ] and magnetite + clinopyroxene. [ 9 ] Symplectite formation is important in metallurgy: bainite or pearlite formation from the decomposition of austenite , for example. [ 3 ] | https://en.wikipedia.org/wiki/Symplectite |
SynBio is a long-term project started in 2011 with the goal of creating innovative medicines, including what are known as Biobetters. This project is a collaborative effort of several Russian and international pharmaceutical companies. The largest private participant of SynBio is the Human Stem Cells Institute (HSCI) , a leading Russian biotech company, and Rusnano is a key investor. [ 1 ] The project is a significant example of international cooperation between researchers in Russia, England, and Germany. Special project company SynBio LLC is headquartered in Moscow. [ 2 ]
Currently, SynBio LLC is developing nine drugs based on three biotechnology platforms ( Histone , PolyXen and Gemacell) for the treatment of liver disease , cardiovascular disease , acute leukemia , growth hormone deficiency and diabetes mellitus . [ 3 ] [ 4 ]
The SynBio project also entails the creation of modern production facilities. These facilities will be dedicated to the manufacturing of the company's pharmaceutical substances and market-ready medicines once they have successfully undergone clinical testing. [ 5 ] | https://en.wikipedia.org/wiki/SynBio |
In organic chemistry , syn- and anti-addition are different ways in which substituent molecules can be added to an alkene ( R 2 C=CR 2 ) or alkyne ( RC≡CR ). The concepts of syn and anti addition are used to characterize the different reactions of organic chemistry by reflecting the stereochemistry of the products in a reaction.
The type of addition that occurs depends on multiple different factors of a reaction, and is defined by the final orientation of the substituents on the parent molecule . Syn and anti addition are related to Markovnikov's rule for the orientation of a reaction, which refers to the bonding preference of different substituents for different carbons on an alkene or alkyne. [ 1 ] In order for a reaction to follow Markovnikov's rule, the intermediate carbocation of the mechanism of a reaction must be on the more-substituted carbon, allowing the substituent to bond to the more-stable carbocation and the more-substituted carbon. [ 2 ]
Syn addition is the addition of two substituents to the same side (or face ) of a double bond or triple bond , resulting in a decrease in bond order but an increase in number of substituents. [ 3 ] Generally the substrate will be an alkene or alkyne . An example of syn addition would be the oxidation of an alkene to a diol by way of a suitable oxidizing agent such as osmium tetroxide , OsO 4 , or potassium permanganate , KMnO 4 . [ 4 ]
Anti addition is in direct contrast to syn addition. In anti addition, two substituents are added to opposite sides (or faces) of a double bond or triple bond, once again resulting in a decrease in bond order and increase in number of substituents. The classical example of this is bromination (any halogenation ) of alkenes. [ 5 ] An anti addition reaction results in a trans-isomer of the products, as the substituents are on opposite faces of the bond.
Depending on the substrate double bond, addition can have different effects on the molecule. After addition to a straight-chain alkene such as ethene ( C 2 H 4 ), the resulting alkane will rapidly and freely rotate around its single sigma bond under normal conditions (i.e. room temperature ). Thus whether substituents are added to the same side (syn) or opposite sides (anti) of a double can usually be ignored due to free rotation. However, if chirality or the specific absolute orientation of the substituents needs to be taken into account, knowing the type of addition is significant. Unlike straight-chain alkenes, cycloalkene syn addition allows stable addition of substituents to the same side of the ring, where they remain together. The cyclic locked ring structure prevents free rotation.
Syn elimination and anti elimination are the reverse processes of syn and anti addition. These result in a new double bond, such as in E i elimination . [ citation needed ]
This reaction is considered Markovnikov because the halogen substituent attaches to the more substituted carbon.
This reaction is considered Markovnikov because the hydroxyl group attaches to the more substituted carbon.
The reaction is considered Markovnikov as it results in water addition with same regiospecificity as a direct hydration reaction.
The reaction is anti-Markovnikov. Hydroxyl attaches to the less substituted carbon.
Neither Markovnikov or anti-Markovnikov because the substituents are the same.
Neither Markovnikov or anti-Markovnikov because the substituents are the same.
When alkenes undergo hydrobromination, the alkyl bromides are formed Markovnikov. | https://en.wikipedia.org/wiki/Syn_and_anti_addition |
Synantherology is a branch of botany that deals with the study of the plant family Asteraceae (also called Compositae ). The name of the field refers to the fused anthers possessed by members of the family, and recalls an old French name, synantherées , for the family. [ citation needed ]
Although many of the plants of the Asteraceae were described for the European community at least as long ago as Theophrastus , [ 1 ] an organization of the family into tribes , which remained largely stable throughout the 20th century, was published in 1873 by George Bentham . [ 2 ]
In a 1970 article titled "The New Synantherology", Harold E. Robinson advocated greater attention to microstructures (studied with the compound light microscope ). He was not the first, as Alexandre de Cassini and others of the 19th century split species based on fine distinctions of microstructure, a tendency which Bentham found excessive. [ 3 ]
Noted United States synantherologists include: [ citation needed ] | https://en.wikipedia.org/wiki/Synantherology |
A synanthrope (from ancient Greek σύν sýn "together, with" and ἄνθρωπος ánthrōpos "man") is an organism that evolved to live near humans and benefit from human settlements and their environmental modifications (see also anthropophilia for animals who live close to humans as parasites ). The term includes many animals and plants regarded as pests or weeds , but does not include domesticated species. [ 1 ] Common synanthrope habitats include houses , sheds and barns , non-building structures , gardens , parks , farms , road verges and rubbish dumps .
Examples of synanthropes are various species of insects ( ants , lice , bedbugs , silverfish , cockroaches , etc.), myriapods ( millipedes and centipedes , notably the house centipede ), arachnids ( spiders , dust mite , etc.), common house gecko , birds such as house sparrows , gulls , rock doves (pigeons), crows and magpies , honeyguides , swallows and other passerines , various rodent species (especially rats and house mice ), Virginia opossums , raccoons , [ 2 ] certain monkey species, coyotes , [ 3 ] [ 4 ] deer , and other urban wildlife . [ 1 ] [ 5 ] [ 6 ]
The brown rat is counted as one of the most prominent synanthropic animals and can be found in almost every place there are people. [ 7 ] [ 8 ]
Synanthropic plants include pineapple weed , dandelion , chicory , and plantain . Plant synanthropes are classified into two main types – apophytes and anthropophytes.
Apophytes are synanthropic species that are native in origin. They can be subdivided into the following: [ 9 ]
Anthropophytes are synanthropic species of foreign origin, whether introduced voluntarily or involuntarily. They can be subdivided into the following:
This ecology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synanthrope |
In the nervous system , a synapse [ 1 ] is a structure that allows a neuron (or nerve cell) to pass an electrical or chemical signal to another neuron or a target effector cell. Synapses can be classified as either chemical or electrical, depending on the mechanism of signal transmission between neurons. In the case of electrical synapses , neurons are coupled bidirectionally with each other through gap junctions and have a connected cytoplasmic milieu. [ 2 ] [ 3 ] [ 4 ] These types of synapses are known to produce synchronous network activity in the brain, [ 5 ] but can also result in complicated, chaotic network level dynamics. [ 6 ] [ 7 ] Therefore, signal directionality cannot always be defined across electrical synapses. [ 8 ]
Chemical synapses , on the other hand, communicate through neurotransmitters released from the presynaptic neuron into the synaptic cleft. Upon release, these neurotransmitters bind to specific receptors on the postsynaptic membrane, inducing an electrical or chemical response in the target neuron. This mechanism allows for more complex modulation of neuronal activity compared to electrical synapses, contributing significantly to the plasticity and adaptable nature of neural circuits. [ 9 ]
Synapses are essential for the transmission of neuronal impulses from one neuron to the next, [ 10 ] playing a key role in enabling rapid and direct communication by creating circuits. In addition, a synapse serves as a junction where both the transmission and processing of information occur, making it a vital means of communication between neurons. [ 11 ]
At the synapse, the plasma membrane of the signal-passing neuron (the presynaptic neuron) comes into close apposition with the membrane of the target ( postsynaptic ) cell. Both the presynaptic and postsynaptic sites contain extensive arrays of molecular machinery that link the two membranes together and carry out the signaling process. In many synapses, the presynaptic part is located on the terminals of axons and the postsynaptic part is located on a dendrite or soma . Astrocytes also exchange information with the synaptic neurons, responding to synaptic activity and, in turn, regulating neurotransmission . [ 10 ] Synapses (at least chemical synapses) are stabilized in position by synaptic adhesion molecules (SAMs) [1] projecting from both the pre- and post-synaptic neuron and sticking together where they overlap; SAMs may also assist in the generation and functioning of synapses. [ 12 ] Moreover, SAMs coordinate the formation of synapses, with various types working together to achieve the remarkable specificity of synapses. [ 11 ] [ 13 ] In essence, SAMs function in both excitatory and inhibitory synapses , likely serving as the mediator for signal transmission. [ 11 ]
Many mental illnesses are thought to be caused by synaptopathy .
Santiago Ramón y Cajal proposed that neurons are not continuous throughout the body, yet still communicate with each other, an idea known as the neuron doctrine . [ 14 ] The word "synapse" was introduced in 1897 by the English neurophysiologist Charles Sherrington in Michael Foster 's Textbook of Physiology . [ 1 ] Sherrington struggled to find a good term that emphasized a union between two separate elements, and the actual term "synapse" was suggested by the English classical scholar Arthur Woollgar Verrall , a friend of Foster. [ 15 ] [ 16 ] The word was derived from the Greek synapsis ( σύναψις ), meaning "conjunction", which in turn derives from synaptein ( συνάπτειν ), from syn ( σύν ) "together" and haptein ( ἅπτειν ) "to fasten". [ 15 ] [ 17 ]
However, while the synaptic gap remained a theoretical construct, and was sometimes reported as a discontinuity between contiguous axonal terminations and dendrites or cell bodies, histological methods using the best light microscopes of the day could not visually resolve their separation which is now known to be about 20 nm. It needed the electron microscope in the 1950s to show the finer structure of the synapse with its separate, parallel pre- and postsynaptic membranes and processes, and the cleft between the two. [ 18 ] [ 19 ] [ 20 ]
Chemical and electrical synapses are two ways of synaptic transmission.
The formation of neural circuits in nervous systems appears to heavily depend on the crucial interactions between chemical and electrical synapses. Thus these interactions govern the generation of synaptic transmission. [ 22 ] Synaptic communication is distinct from an ephaptic coupling , in which communication between neurons occurs via indirect electric fields. An autapse is a chemical or electrical synapse that forms when the axon of one neuron synapses onto dendrites of the same neuron.
An influx of Na+ driven by excitatory neurotransmitters opens cation channels, depolarizing the postsynaptic membrane toward the action potential threshold. In contrast, inhibitory neurotransmitters cause the postsynaptic membrane to become less depolarized by opening either Cl- or K+ channels, reducing firing. Depending on their release location, the receptors they bind to, and the ionic circumstances they encounter, various transmitters can be either excitatory or inhibitory. For instance, acetylcholine can either excite or inhibit depending on the type of receptors it binds to. [ 26 ] For example, glutamate serves as an excitatory neurotransmitter, in contrast to GABA, which acts as an inhibitory neurotransmitter. Additionally, dopamine is a neurotransmitter that exerts dual effects, displaying both excitatory and inhibitory impacts through binding to distinct receptors. [ 27 ]
The membrane potential prevents Cl- from entering the cell, even when its concentration is much higher outside than inside. The reversal potential for Cl- in many neurons is quite negative, nearly equal to the resting potential . Opening Cl- channels tends to buffer the membrane potential, but this effect is countered when the membrane starts to depolarize, allowing more negatively charged Cl- ions to enter the cell. Consequently, it becomes more difficult to depolarize the membrane and excite the cell when Cl- channels are open. Similar effects result from the opening of K+ channels. The significance of inhibitory neurotransmitters is evident from the effects of toxins that impede their activity. For instance, strychnine binds to glycine receptors, blocking the action of glycine and leading to muscle spasms, convulsions, and death. [ 26 ]
Synapses can be classified by the type of cellular structures serving as the pre- and post-synaptic components. The vast majority of synapses in the mammalian nervous system are classical axo-dendritic synapses (axon synapsing upon a dendrite), however, a variety of other arrangements exist. These include but are not limited to [ clarification needed ] axo-axonic , dendro-dendritic , axo-secretory, axo-ciliary, [ 28 ] somato-dendritic, dendro-somatic, and somato-somatic synapses. [ citation needed ]
In fact, the axon can synapse onto a dendrite, onto a cell body, or onto another axon or axon terminal, as well as into the bloodstream or diffusely into the adjacent nervous tissue.
Neurotransmitters are tiny signal molecules stored in membrane-enclosed synaptic vesicles and released via exocytosis. Indeed, a change in electrical potential in the presynaptic cell triggers the release of these molecules. By attaching to transmitter-gated ion channels, the neurotransmitter causes an electrical alteration in the postsynaptic cell and rapidly diffuses across the synaptic cleft. Once released, the neurotransmitter is swiftly eliminated, either by being absorbed by the nerve terminal that produced it, taken up by nearby glial cells, or broken down by specific enzymes in the synaptic cleft. Numerous Na+-dependent neurotransmitter carrier proteins recycle the neurotransmitters and enable the cells to maintain rapid rates of release.
At chemical synapses, transmitter-gated ion channels play a vital role in rapidly converting extracellular chemical impulses into electrical signals. These channels are located in the postsynaptic cell's plasma membrane at the synapse region, and they temporarily open in response to neurotransmitter molecule binding, causing a momentary alteration in the membrane's permeability. Additionally, transmitter-gated channels are comparatively less sensitive to the membrane potential than voltage-gated channels, which is why they are unable to generate self-amplifying excitement on their own. However, they result in graded variations in membrane potential due to local permeability, influenced by the amount and duration of neurotransmitter released at the synapse. [ 26 ]
Recently, mechanical tension, a phenomenon never thought relevant to synapse function has been found to be required for those on hippocampal neurons to fire. [ 29 ]
Neurotransmitters bind to ionotropic receptors on postsynaptic neurons, either causing their opening or closing. [ 27 ] The variations in the quantities of neurotransmitters released from the presynaptic neuron may play a role in regulating the effectiveness of synaptic transmission. In fact, the concentration of cytoplasmic calcium is involved in regulating the release of neurotransmitters from presynaptic neurons. [ 30 ]
The chemical transmission involves several sequential processes:
The function of neurons depends upon cell polarity . The distinctive structure of nerve cells allows action potentials to travel directionally (from dendrites to cell body down the axon), and for these signals to then be received and carried on by post-synaptic neurons or received by effector cells. Nerve cells have long been used as models for cellular polarization, and of particular interest are the mechanisms underlying the polarized localization of synaptic molecules. PIP2 signaling regulated by IMPase plays an integral role in synaptic polarity.
Phosphoinositides ( PIP , PIP2, and PIP3 ) are molecules that have been shown to affect neuronal polarity. [ 32 ] A gene ( ttx-7 ) was identified in Caenorhabditis elegans that encodes myo -inositol monophosphatase (IMPase), an enzyme that produces inositol by dephosphorylating inositol phosphate . Organisms with mutant ttx-7 genes demonstrated behavioral and localization defects, which were rescued by expression of IMPase. This led to the conclusion that IMPase is required for the correct localization of synaptic protein components. [ 33 ] [ 34 ] The egl-8 gene encodes a homolog of phospholipase C β (PLCβ), an enzyme that cleaves PIP2. When ttx-7 mutants also had a mutant egl-8 gene, the defects caused by the faulty ttx-7 gene were largely reversed. These results suggest that PIP2 signaling establishes polarized localization of synaptic components in living neurons. [ 33 ]
Modulation of neurotransmitter release by G-protein-coupled receptors (GPCRs) is a prominent presynaptic mechanism for regulation of synaptic transmission . The activation of GPCRs located at the presynaptic terminal, can decrease the probability of neurotransmitter release. This presynaptic depression involves activation of Gi/o -type G-proteins that mediate different inhibitory mechanisms, including inhibition of voltage-gated calcium channels , activation of potassium channels , and direct inhibition of the vesicle fusion process.
Endocannabinoids , synthesized in and released from postsynaptic neuronal elements and their cognate receptors , including the (GPCR) CB1 receptor located at the presynaptic terminal, are involved in this modulation by a retrograde signaling process, in which these compounds are synthesized in and released from postsynaptic neuronal elements and travel back to the presynaptic terminal to act on the CB1 receptor for short-term or long-term synaptic depression, that causes a short or long lasting decrease in neurotransmitter release. [ 35 ]
Drugs have long been considered crucial targets for transmitter-gated ion channels. The majority of medications utilized to treat schizophrenia, anxiety, depression, and sleeplessness work at chemical synapses, and many of these pharmaceuticals function by binding to transmitter-gated channels. For instance, some drugs like barbiturates and tranquilizers bind to GABA receptors and enhance the inhibitory effect of GABA neurotransmitter. Thus, reduced concentration of GABA enables the opening of Cl- channels.
Furthermore, psychoactive drugs could potentially target many other synaptic signalling machinery components. Neurotransmitter release is a complex process involving various types of transporters and mechanisms for removing neurotransmitters from the synaptic cleft. While Na+-driven carriers play a role, other mechanisms are also involved, depending on the specific neurotransmitter system. [ citation needed ] For example, Prozac is an antidepressant medication that works by preventing the absorption of serotonin neurotransmitter. Also, other antidepressants operate by inhibiting the reabsorption of both serotonin and norepinephrine. [ 26 ]
In nerve terminals, synaptic vesicles are produced quickly to compensate for their rapid depletion during neurotransmitter release. Their biogenesis involves segregating synaptic vesicle membrane proteins from other cellular proteins and packaging those distinct proteins into vesicles of appropriate size. Besides, it entails the endocytosis of synaptic vesicle membrane proteins from the plasma membrane. [ 36 ]
Synaptoblastic and synaptoclastic refer to synapse-producing and synapse-removing activities within the biochemical signalling chain. This terminology is associated with the Bredesen Protocol for treating Alzheimer's disease , which conceptualizes Alzheimer's as an imbalance between these processes. As of October 2023, studies concerning this protocol remain small and few results have been obtained within a standardized control framework.
It is widely accepted that the synapse plays a key role in the formation of memory . [ 37 ] The stability of long-term memory can persist for many years; nevertheless, synapses, the neurological basis of memory, are very dynamic. [ 38 ] The formation of synaptic connections significantly depends on activity-dependent synaptic plasticity observed in various synaptic pathways. Indeed, the connection between memory formation and alterations in synaptic efficacy enables the reinforcement of neuronal interactions between neurons. As neurotransmitters activate receptors across the synaptic cleft, the connection between the two neurons is strengthened when both neurons are active at the same time, as a result of the receptor's signaling mechanisms. Memory formation involves complex interactions between neural pathways, including the strengthening and weakening of synaptic connections, which contribute to the storage of information. [ 39 ] This process of synaptic strengthening is known as long-term potentiation (LTP) . [ 37 ]
By altering the release of neurotransmitters, the plasticity of synapses can be controlled in the presynaptic cell. The postsynaptic cell can be regulated by altering the function and number of its receptors. Changes in postsynaptic signaling are most commonly associated with a N-methyl-d-aspartic acid receptor (NMDAR)-dependent LTP and long-term depression (LTD) due to the influx of calcium into the post-synaptic cell, which are the most analyzed forms of plasticity at excitatory synapses. [ 40 ]
Moreover, Ca2+/calmodulin (CaM)-dependent protein kinase II (CaMKII) is best recognized for its roles in the brain, particularly in the neocortex and hippocampal regions because it serves as a ubiquitous mediator of cellular Ca2+ signals. CaMKII is abundant in the nervous system, mainly concentrated in the synapses in the nerve cells. Indeed, CaMKII has been definitively identified as a key regulator of cognitive processes, such as learning, and neural plasticity. The first concrete experimental evidence for the long-assumed function of CaMKII in memory storage was demonstrated
While Ca2+/CaM binding stimulates CaMKII activity, Ca2+-independent autonomous CaMKII activity can also be produced by a number of other processes. CaMKII becomes active by autophosphorylating itself upon Ca2+/calmodulin binding. CaMKII is still active and phosphorylates itself even after Ca2+ is cleaved; as a result, the brain stores long-term memories using this mechanism. Nevertheless, when the CaMKII enzyme is dephosphorylated by a phosphatase enzyme, it becomes inactive, and memories are lost. Hence, CaMKII plays a vital role in both the induction and maintenance of LTP. [ 41 ]
Beyond serving as passive relays, synapses perform complex computations on incoming signals. Models of synaptic computation describe how neurotransmitter release kinetics, receptor subunit composition, and short‑term plasticity endow individual synapses with filtering, gain control, and temporal integration capabilities. Recent connectomic and functional studies—such as those reconstructing the larval zebrafish brainstem circuit—demonstrate that synaptic wiring diagrams can predict behaviorally relevant neural codes, underscoring the computational role of synaptic networks in information processing. [ 42 ]
For technical reasons, synaptic structure and function have been historically studied at unusually large model synapses, for example:
Synaptic dysfunction and loss are now recognized as central to the pathophysiology of major neurodegenerative and neurodevelopmental disorders. In Alzheimer’s disease (AD), synapse loss correlates more strongly with cognitive decline than amyloid‑β plaque burden, and emerging biomarkers—such as the YWHAG:NPTX2 ratio in cerebrospinal fluid and plasma—offer prognostic value for AD onset and progression. Synaptic pathology in AD encompasses alterations in glutamatergic transmission, dendritic spine density, and synaptic protein turnover, highlighting synapses both as early indicators of disease and as targets for therapeutic intervention. [ 44 ] [ 45 ]
Synaptic disruptions can lead to a variety of negative effects, including impaired learning, memory, and cognitive function. [ 46 ] In fact, alterations in cell-intrinsic molecular systems or modifications to environmental biochemical processes can lead to synaptic dysfunction. The synapse is the primary unit of information transfer in the nervous system, and correct synaptic contact creation during development is essential for normal brain function. Genetic mutations can disrupt synapse formation and function, contributing to the development of neurodevelopmental and neurodegenerative disorders. [ 47 ] However, the precise relationship between specific mutations and disease phenotypes is complex and requires further investigation.
Synaptic defects are causally associated with early appearing neurological diseases, including autism spectrum disorders (ASD), schizophrenia (SCZ), and bipolar disorder (BP). Synaptic dysfunction, or synaptopathy, is often implicated in late-onset neurodegenerative diseases such as Alzheimer's, Parkinson's, and Huntington's, but the exact mechanisms contributing to this phenomenon are not fully understood. [ 48 ] These diseases are identified by a gradual loss in cognitive and behavioral function and a steady loss of brain tissue. Moreover, these deteriorations have been mostly linked to the gradual build-up of protein aggregates in neurons, the composition of which may vary based on the pathology; all have the same deleterious effects on neuronal integrity. Furthermore, the high number of mutations linked to synaptic structure and function, as well as dendritic spine alterations in post-mortem tissue, has led to the association between synaptic defects and neurodevelopmental disorders, such as ASD and SCZ, characterized by abnormal behavioral or cognitive phenotypes.
Nevertheless, due to limited access to human tissue at late stages and a lack of thorough assessment of the essential components of human diseases in the available experimental animal models, it has been difficult to fully grasp the origin and role of synaptic dysfunction in neurological disorders. [ 49 ] | https://en.wikipedia.org/wiki/Synapse |
Synapse.org is an open source platform for collaborative scientific data analysis. [ 1 ] It can store data, code, results, and descriptions research work. It is operated by nonprofit organization Sage Bionetworks . [ 2 ]
The Synapse web portal is an online registry of research projects that allows data scientists to discover and share data, models, and analysis methods.
This article related to a non-profit organization is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synapse.org |
Synapsis or Syzygy is the pairing of two chromosomes that occurs during meiosis . It allows matching-up of homologous pairs prior to their segregation, and possible chromosomal crossover between them. Synapsis takes place during prophase I of meiosis . When homologous chromosomes synapse, their ends are first attached to the nuclear envelope . These end-membrane complexes then migrate, assisted by the extranuclear cytoskeleton , until matching ends have been paired. Then the intervening regions of the chromosome are brought together, and may be connected by a protein-DNA complex called the synaptonemal complex (SC). [ 1 ] The SC protein scaffold stabilizes the physical pairing of homologous chromosomes by polymerizing between them during meiotic prophase. [ 2 ] During synapsis, autosomes are held together by the synaptonemal complex along their whole length, whereas for sex chromosomes , this only takes place at one end of each chromosome. [ 3 ]
This is not to be confused with mitosis . Mitosis also has prophase, but does not ordinarily do pairing of two homologous chromosomes. [ 4 ] In contrast to the mitosis cycle, during meiosis, the number of chromosomes is reduced by half to create haploid gametes; this reduction is called Haploidization; after fertilization, diploidy is restored. Homologous chromosomes – two copies inherited from each parent – recognize one another and pair before reductional segregation, which is essential for crossover recombination and forms chiasmata, [ 5 ] a stable physical connection that hold homologous chromosomes together until metaphase. [ 2 ] In most species, every homologous chromosome experiences at least one meiotic crossover referred to as the obligate crossover. [ 5 ]
When the non-sister chromatids intertwine, segments of chromatids with similar sequence may break apart and be exchanged in a process known as genetic recombination or "crossing-over". This exchange produces a chiasma , a region that is shaped like an X, where the two chromosomes are physically joined. At least one chiasma per chromosome often appears to be necessary to stabilise bivalents along the metaphase plate during separation. The crossover of genetic material also provides a possible defences against 'chromosome killer' mechanisms, by removing the distinction between 'self' and 'non-self' through which such a mechanism could operate. A further consequence of recombinant synapsis is to increase genetic variability within the offspring. Repeated recombination also has the general effect of allowing genes to move independently of each other through the generations, allowing for the independent concentration of beneficial genes and the purging of the detrimental.
Following synapsis, a type of recombination referred to as synthesis dependent strand annealing (SDSA) occurs frequently. SDSA recombination involves information exchange between paired non-sister homologous chromatids, but not physical exchange. SDSA recombination does not cause crossing-over. Both the non-crossover and crossover types of recombination function as processes for repairing DNA damage, particularly double-strand breaks (see Genetic recombination ).
The central function of synapsis is therefore the identification of homologues by pairing, an essential step for a successful meiosis. The processes of DNA repair and chiasma formation that take place following synapsis have consequences at many levels, from cellular survival through to impacts upon evolution itself.
Homologous chromosomes are held together by several mechanisms during meiosis, ensuring their proper pairing, alignment, and recombination. These mechanisms include:
In mammals , surveillance mechanisms remove meiotic cells in which synapsis is defective. One such surveillance mechanism is meiotic silencing that involves the transcriptional silencing of genes on asynapsed chromosomes . [ 12 ] Any chromosome region, either in males or females, that is asynapsed is subject to meiotic silencing. [ 13 ] ATR , BRCA1 and gammaH2AX localize to unsynapsed chromosomes at the pachytene stage of meiosis in human oocytes and this may lead to chromosome silencing. [ 14 ] The DNA damage response protein TOPBP1 has also been identified as a crucial factor in meiotic sex chromosome silencing. [ 12 ] DNA double-strand breaks appear to be initiation sites for meiotic silencing. [ 12 ]
In female Drosophila melanogaster fruit flies, meiotic chromosome synapsis occurs in the absence of recombination . [ 15 ] Thus synapsis in Drosophila is independent of meiotic recombination, consistent with the view that synapsis is a precondition required for the initiation of meiotic recombination. Meiotic recombination is also unnecessary for homologous chromosome synapsis in the nematode Caenorhabditis elegans . [ 16 ]
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synapsis |
Synaptic stabilization is crucial in the developing and adult nervous systems and is considered a result of the late phase of long-term potentiation (LTP). The mechanism involves strengthening and maintaining active synapses through increased expression of cytoskeletal and extracellular matrix elements and postsynaptic scaffold proteins , while pruning less active ones. For example, cell adhesion molecules (CAMs) play a large role in synaptic maintenance and stabilization. Gerald Edelman discovered CAMs and studied their function during development, which showed CAMs are required for cell migration and the formation of the entire nervous system. [ 1 ] [ 2 ] In the adult nervous system, CAMs play an integral role in synaptic plasticity relating to learning and memory . [ 3 ]
Synaptic cell adhesion molecules (CAMs) play a crucial role in axon pathfinding and synaptic establishment between neurons during neurodevelopment and are integral members in many synaptic processes including the correct alignment of pre- and post-synaptic signal transduction pathways , vesicular recycling in regards to endocytosis and exocytosis , integration of postsynaptic receptors and anchoring to the cytoskeleton to ensure stability of synaptic components. [ 4 ]
SynCAM’s (also known as Cadm or nectin-like molecules) are a specific type of synaptic CAM found in vertebrates that promotes growth and stabilization of excitatory (not inhibitory) synapses. SynCAM’s are localized primarily in the brain at both pre- and postsynaptic sites and their structures consist of intracellular FERM and PDZ binding domains, a single transmembrane domain, and three extracellular Ig-domains . During neurodevelopment, SynCAMs such as SynCAM1 act as “contact sensors” of axonal growth cones accumulating rapidly when axo-dendritic connections are made and helping to form a stable adhesion complex. [ 5 ]
synCAM1 along with neuroligin are the two CAM’s known to be sufficient to initiate the formation of presynaptic terminals, as addition of synCAM1 to media of co-cultured neuronal and non-neuronal cells lead to the establishment of presynaptic terminals. Homophillic binding of two synCAM1 molecules on the filopodia of axonal growth cone and dendritic spine allow for initial contact between pre- and postsynaptic cell to be made. [ 6 ]
synCAMs belong to the Ig superfamily of proteins. The cytosolic PDZ domains of synCAMs imbedded in the post-synaptic membrane interact with post-synaptic scaffold protein PSD-95 which helps anchor the complex to the underlying cytoskeleton. [ 7 ]
Cadherins are calcium- dependent, homophilic cell adhesion molecules that form complexes with cytosolic partners known as catenins . [ 8 ] Components of this complex bind to a number of different scaffolding proteins, phosphotases, kinases, and receptors. [ 9 ] Classical cadherins have five extracellular repeating structures which bind calcium, a single transmembrane domain, and an intracellular tail with a distal cytosolic domain that binds a catenin partner. [ 9 ] [ 10 ] Recent work has implicated the cadherin-catenin complex in a number of different central nervous system processes such as synaptic stabilization and plasticity . [ 8 ] [ 9 ] [ 10 ]
Many cadherins in the central nervous system exhibit distinct spatial and temporal expression patterns. [ 9 ] For example, N-cadherin is widely expressed at the developing synapse and later remains near the mature active zone implicating that this complex may be well-suited to provide a link between structural changes and synaptic stability. [ 9 ] In fact, local synaptic activity changes impact the expression of the cadherin-catenin complexes. [ 9 ] An increase in activity at a particular spine leads to the dimerization of N-cadherin which is then cleaved leading the repression of CBP/ CREB transcription. [ 9 ] This repression has many developmental and plasticity related implications.
In the case of dendritic spine formation and pruning , a competition hypothesis has been proposed and corroborated. [ 11 ] [ 12 ] This hypothesis suggests that relative levels of cadherin-catenin complexes, which are distributed amongst spines in a local area in an activity-dependent manner, determines the fate of individual spines. That is, the inter-spine competition for β-catenin determines whether a spine will be matured (increased number of complexes) or pruned (decreased number of complexes). [ 12 ] This is a critical mechanism during the refinement of cortical circuitry that occurs throughout development. [ 11 ]
Nectins are a distinct family of cell adhesion molecules . These CAMs are involved in the initial contact of presynaptic and postsynaptic neuronal processes during synapse formation. There are only four well characterized nectins at the synapse , they are Nectin-1, 2, 3, and 4. [ 13 ] All membrane -bound nectins possess an extracellular region with three immunoglobulin-like loops. The furthest loop from the membrane is called the V-type loop and the two loops more interior are C2-type loops. Multiple nectins on one cell membrane will bind together at the V-type loop to form a cluster of nectin proteins, a process called cis-clustering . When two cells possessing individual cis-clusters come into contact they form a strong complex called a trans-interaction which provides adhesion and, in some cases, signaling between the two cells. [ 14 ]
The most robust knowledge of nectin’s role in synaptic stabilization comes from the synapses made between mossy fiber terminals and pyramidal cell dendrites in the CA3 region of the hippocampus . [ 15 ] The nectins involved in formation and stabilization of this synapse are Nectin-1 and Nectin-3 which protrude from the plasma membrane of the postsynaptic cell and presynaptic cell, respectively, forming heterophilic extracellular contacts. The intracellular domain of all nectins directly bind to a protein called L- Afadin . L-Afadin is an actin binding protein that binds to the F-actin of the actin cytoskeleton . In this way, nectins form ridged connections of the cells actin architecture allowing for the synapse to develop in a controlled and stable environment. [ 16 ]
As synapses mature in the CA3 region, nectins and cadherins, which affiliate closely with one another in synaptic stabilization, are shifted to the periphery of the active zone and form the puncta adherens junction (PAJ). The PAJ functions much like the adherens junctions in epithelial tissues . The displacement of these CAMs and the formation of this junction provides the nascent synaptic membranes room to interact and mature while partitioning off the surrounding membrane and providing cytoskeletal fixation. [ 14 ]
Neurexin - Neuroligin interactions help establish the trans-synaptic functional asymmetry essential for the stabilization and maintenance of proper synaptic transmission . [ 17 ] Presynaptic neurexin and its postsynaptic binding partner, neuroligin, complex early in neural development and are both known to be potent inducers of synaptogenesis . [ 18 ] Non-neuronal cells that artificially express neurexin are sufficient to mobilize post-synaptic specializations in co-cultured neurons; [ 19 ] neuroligin-expressing cells are likewise able to induce markers of pre-synaptic differentiation in neighboring neurons. [ 20 ] [ 21 ] However, while both play an important role in synaptogenesis, these cell adhesion molecule are not necessary for formation of neuronal connections during development. [ 22 ] A triple knockout mouse mutant of either neurexins or neuroligins exhibit a normal number of synapses but express an embryonic lethal phenotype due to impairment of normal synaptic transmission. [ 23 ] Therefore, they are not necessary for synapse formation per se but are essential for the maturation and integration of synapses into the functional circuits necessary for survival.
Beyond their extracellular contact with each other, neurexins and neuroligins also bind intracellularly to a vast network of adaptor proteins and scaffolding structures, which in concert with the actin cytoskeleton , help localize necessary components of synaptic transmission. For example, the first neuroligin ( NLGN1 ) discovered was identified by its PDZ domain which binds to PSD95 , a well-known a scaffold protein at glutamatergic synapses that functionally links NMDA receptors to the proper post-synaptic locale. [ 21 ] [ 24 ] Similarly, another isoform of neuroligin ( NLGN2 ) interacts with gephyrin , a scaffolding protein specific to GABA-ergic synapses , and is responsible for activation of the synaptic adapator protein collybistin . [ 25 ] In the case of neurexins, their intracellular binding interactions are equally as important in recruiting the essential machinery for synaptic transmission at the active zone. Like neuroligins, neurexins possess a PDZ-domain that associates with CASK ( Calcium-calmodulin-dependent protein kinase ). [ 24 ] In addition to phosphorylating itself and neurexin, CASK promotes interactions between neurexins and actin binding proteins, thus providing a direct link by which neurexin can modulate cytoskeletal dynamics that is essential for synaptic stability and plasticity. Neurexin can also bind synaptotagmin , a protein embedded in the membrane of synaptic vesicles, and can also promote associations with voltage-gated calcium channel which mediate the ion flux required for neurotransmitter exocytosis upon synaptic stimulation. [ 26 ] [ 23 ] In this way, neurexin and neuroligin coordinate the morphological and functional aspects of the synapse which in turn permits nascent, immature contacts to stabilize into full-fledged functional platforms for neurotransmission.
Non-traditional adhesion molecules, such as the ephrins , also help stabilize synaptic contacts. Eph receptors and their membrane bound ligands, the ephrins, are involved in a variety of cellular processes during development and maturation including axon guidance , neuronal migration , synaptogenesis , and axon pruning . [ 27 ] [ 28 ] In the hippocampus , dendritic spine morphology may be regulated by astrocytes via bi-directional ephrin/EphA signaling. [ 29 ] Astrocytes and their processes express ephrin A3 , whereas the EphA4 receptor is enriched in hippocampal neurons. This interaction, mediated by ephrin A3/EphA4 signaling, induces the recruitment and activation of cyclin-dependent kinase 5 (Cdk5), which then phosphorylates the guanine exchange factor (GEF), ephexin1. [ 30 ] Phosphorylated ephexin1 can then activate the small GTPase , RhoA , leading to subsequent activation of its effector, Rho-kinase (ROCK), which results in the rearrangement of actin filaments. [ 30 ] Through this mechanism, astrocytic processes are able to stabilize individual dendritic protrusions as well as their maturation into spines via ephrin/EphA signaling. Forward signaling involving the activation of EphA4 results in the stabilization of synaptic proteins at the neuromuscular junction . [ 30 ] As in the EphA4/ephrinA3-mediated neuron–glia interaction, this process regulates dynamics of the actin cytoskeleton by activating ROCK through ephexin. [ 30 ]
Ephrin B/EphB signaling is also involved in synaptic stabilization through different mechanisms. These molecules contain cytoplasmic tails which interact with scaffolding proteins via their PDZ domains to stabilize newly formed CNS synapses. [ 28 ] For example, Ephrin B3 interacts with the adaptor protein glutamate-receptor-interacting protein 1 (GRIP-1) to regulate the development of excitatory dendritic shaft synapses. [ 28 ] This process, which was identified in cultures of hippocampal neurons, revealed that Eph/ephrin B3 reverse signaling recruits GRIP1 to the membrane of the postsynaptic shaft. [ 31 ] Once at the membrane shaft, GRIP1 helps anchor glutamate receptors below the presynaptic terminal. This process also involves the phosphorylation of a serine residue near the ephrin-B carboxyl terminus (proximal to the PDZ-binding motif) that leads to the stabilization of AMPA receptors at synapses. [ 27 ]
Another mechanism, found in hippocampal neurons, revealed that EphB signaling could promote spine maturation by modulating Rho GTPase activity, as observed with EphAs. [ 32 ] Unlike EphAs, however, the EphB2 receptor has been shown to interact with the postsynaptic N-methyl-D-aspartate receptors (NMDARs) to recruit the GEF Tiam1 to the complex upon ephrinB binding. [ 32 ] [ 30 ] [ 33 ] Phosphorylation of Tiam1 occurs in response to NMDAR activity, which allows for the influx of calcium that activates Tiam1. This mechanism also results in the modulation of the actin cytoskeleton. As a result of this stabilization, both EphB2 forward signaling and ephrin-B3 reverse signaling has been found to induce LTP via NMDARs. [ 34 ] | https://en.wikipedia.org/wiki/Synaptic_stabilization |
Synaptic tagging , or the synaptic tagging hypothesis, has been proposed to explain how neural signaling at a particular synapse creates a target for subsequent plasticity-related product (PRP) trafficking essential for sustained LTP and LTD . Although the molecular identity of the tags remains unknown, it has been established that they form as a result of high or low frequency stimulation , interact with incoming PRPs, and have a limited lifespan. [ 1 ]
Further investigations have suggested that plasticity-related products include mRNA and proteins from both the soma and dendritic shaft that must be captured by molecules within the dendritic spine to achieve persistent LTP and LTD. This idea was articulated in the synaptic tag-and-capture hypothesis. Overall, synaptic tagging elaborates on the molecular underpinnings of how L-LTP is generated and leads to memory formation.
Frey, a researcher at the Leibniz Institute for Neurobiology (later at the Medical College of Georgia and the Lund University), and Morris, a researcher at the University of Edinburgh , [ 2 ] laid the groundwork for the synaptic tagging hypothesis, stating:
"We propose that LTP initiates the creation of a short-lasting protein-synthesis-independent 'synaptic tag' at the potentiated synapse which sequesters the relevant protein(s) to establish late LTP. In support of this idea, we now show that weak tetanic stimulation, which ordinarily leads only to early LTP, or repeated tetanization in the presence of protein-synthesis inhibitors, each results in protein-synthesis-dependent late LTP, provided repeated tetanization has already been applied at another input to the same population of neurons. The synaptic tag decays in less than three hours. These findings indicate that the persistence of LTP depends not only on local events during its induction, but also on the prior activity of the neuron." [ 2 ]
L-LTP inducing stimulus induces two independent processes including a dendritic biological tag that identifies the synapse as having been stimulated, and a genomic cascade that produces new mRNAs and proteins (plasticity products). [ 3 ] While weak stimulation also tags synapses, it does not produce the cascade. Proteins produced in the cascade are characteristically promiscuous, in that they will attach to any recently tagged synapse. However, as Frey and Morris discovered, the tag is temporary and will disappear if no protein presents itself for capture. Therefore, the tag and protein production must overlap if L-LTP is to be induced by the high-frequency stimulation .
The experiment performed by Frey and Morris involved the stimulation of two different sets of Schaffer collateral fibers that synapsed on same population of CA1 cells. [ 3 ] They then recorded field EPSP associated with each stimulus on either S1 or S2 pathways to produce E-LTP and L-LTP on different synapses within the same neuron , based on the intensity of the stimulus. Results showed 1) that E-LTP produced by weak stimulation could be turned into L-LTP if a strong S2 stimulus was delivered before or after and 2)that the ability to convert E-LTP to L-LTP decreased as the interval between the two stimulations increased, creating temporal dependence. When they blocked protein synthesis prior to the delivery of strong S2 stimulation, the conversion to L-LTP was prevented, showing importance of translating the mRNAs produced by the genomic cascade.
Subsequent research has identified an additional property of synaptic tagging that involves associations between late LTP and LTD. This phenomenon was first identified by Sajikumar and Frey in 2004 and is now referred to as "cross-tagging". [ 4 ] It involves late-associative interactions between LTP and LTD induced in sets of independent synaptic inputs : late-LTP induced in one set of synaptic inputs can transform early-LTD into late-LTD in another set of inputs. The opposite effect also occurs: early LTP induced in the first synapse can be transformed into late LTP if followed by a late LTD-inducing stimulus in an independent synapse. This phenomenon is seen because the synthesis of nonspecific plasticity related proteins (PRPs) by late-LTP or -LTD in the first synapse is sufficient to transform early-LTD/LTP to late-LTD/LTP in the second synapse after synaptic tags have been set.
Blitzer and his research team proposed a modification to the theory in 2005, stating that the proteins captured by the synaptic tag are actually local proteins that are translated from mRNAs located in the dendrites. [ 3 ] This means that mRNAs are not a product of genomic cascade initiated by strong stimulus, but rather, is delivered as a result of continual basal transcription . They proposed that even weakly stimulated synapses that were tagged can accept proteins produced from a strong stimulation nearby despite lacking the genomic cascade.
Synaptic tagging/ tag-and-capture theory potentially addresses the significant problem of explaining how mRNA, proteins, and other molecules may be specifically trafficked to certain dendritic spines during late phase LTP. It has long been known that the late phase of LTP depends on protein synthesis within the particular dendritic spine, as proven by injecting anisomycin into a dendritic spine and observing the resulting absence of late LTP. [ 5 ] To achieve translation within the dendritic spine, neurons must synthesize the mRNA in the nucleus, package it within a ribonucleoprotein complex, initiate transport, prevent translation during transport, and ultimately deliver the RNP complex to the appropriate dendritic spine. [ 6 ] These processes span a number of disciplines and synaptic tagging/tag-and-capture cannot explain them all; nevertheless, synaptic tagging likely plays an important role in directing mRNA trafficking to the appropriate dendritic spine and signaling the mRNA-RNP complex to dissociate and enter the dendritic spine.
A cell's identity and the identities of subcellular structures are largely determined by RNA transcripts. Considering this premise, it follows that cellular transcription, trafficking, and translation of mRNA undergo modification at a number of different junctures. [ 7 ] Beginning with transcription, mRNA molecules are potentially modified via alternate splicing of exons and introns . The alternate splicing mechanisms allow cells to produce a diverse set of proteins from a single gene within the genome. Recent developments in next-generation sequencing have allowed for greater understanding of the diversity eukaryotic cells achieve through splice variants. [ 8 ]
Transcribed mRNA must reach the intended dendritic spine for the spine to express L-LTP. Neurons may transport mRNA to specific dendritic spines in a package along with a transport ribonucleoprotein (RNP) complex; the transport RNP complex is a subtype of an RNA granule. Granules containing two proteins of known importance to synaptic plasticity, CaMKII (Calmodulin-dependent Kinase II) and the immediate early gene Arc, have been identified to associate with a type of the motor protein kinesin, KIF5. [ 9 ] Furthermore, there is evidence that polyadenylated mRNA associates with microtubules in mammalian neurons, at least in vitro. [ 10 ] Since mRNA transcripts undergo polyadenylation prior to export from the nucleus, this suggests that the mRNA essential for late-phase LTP may travel along the microtubules within the dendritic shaft prior to reaching the dendritic spine.
Once the RNA/RNP complex arrives via motor protein to an area within the vicinity of the specific dendritic spine, it must somehow get “captured” by a process within the dendritic spine. This process likely involves the synaptic tag created by synaptic stimulation of sufficient strength. Synaptic tagging may result in capture of the RNA/RNP complex via any number of possible mechanisms such as:
Since the 1980s, it has become more and more clear that the dendrites contain the ribosomes , proteins, and RNA components to achieve local and autonomous protein translation. Many mRNAs shown to be localized in the dendrites encode proteins known to be involved in LTP, including AMPA receptor and CaMKII subunits, and cytoskeleton -related proteins MAP2 and Arc . [ 12 ]
Researchers [ 13 ] provided evidence of local synthesis, by examining the distribution of Arc mRNA after selective stimulation of certain synapses of a hippocampal cell. They found that Arc mRNA was localized at the activated synapses, and Arc protein appeared there simultaneously. This suggests that the mRNA was translated locally.
These mRNA transcripts are translated in a cap-dependent manner, meaning they use a "cap" anchoring point to facilitate ribosome attachment to the 5' untranslated region. Eukaryotic initiation factor 4 group (eIF4) members recruit ribosomal subunits to the mRNA terminus, and assembly of the eIF4F initiation complex is a target of translational control: phosphorylation of eIF4F exposes the cap for rapid reloading, quickening the rate-limiting step of translation. It is suggested that eIF4F complex formation is regulated during LTP to increase local translation. [ 12 ] In addition, excessive eIF4F complex destabilizes LTP.
Researchers have identified sequences within the mRNA that determine its final destination - called localization elements (LEs), zipcodes, and targeting elements (TEs). These are recognized by RNA binding proteins, of which some potential candidates are MARTA and ZBP1. [ 14 ] [ 15 ] They recognize the TEs, and this interaction results in formation of ribonucleotide protein (RNP) complexes, which travel along cytoskeleton filaments to the spine with the help of motor proteins. Dendritic TEs have been identified in the untranslated region of several mRNAs, like MAP2 and alphaCaMKII. [ 16 ] [ 17 ]
Synaptic tagging is likely to involve the acquisition of molecular maintenance mechanisms by a synapse that would then allow for the conservation of synaptic changes. [ 18 ] There are several proposed processes through which synaptic tagging functions. [ 19 ] One model suggests that the tag allows for local protein synthesis at the specified synapse that then leads to modifications in synaptic strength. One example of this suggested mechanism involves the anchoring of PKMzeta mRNA to the tagged synapse. This anchor would then restrict the activity of translated PKMzeta, an important plasticity related protein, to this location. A different model proposes that short-term synaptic changes induced by the stimulus are themselves the tag; subsequently delivered or translated protein products act to strengthen this change. For example, the removal of AMPA receptors due to low-frequency stimulation leading to LTD is stabilized by a new protein product that would be inactive at synapses where internalization had not occurred. The tag could also be a latent memory trace , as another model suggests. The activity of proteins would then be required for the memory trace to lead to sustained changes in synaptic strength. According to this model, changes induced by the latent memory trace, such as the growth of new filipodia , are themselves the tag. These tags require protein products for stabilization, synapse formation, and synapse stabilization. Finally, another model proposes that the required molecular products get directed into the appropriate dendritic branches and then find the specific synapses under efficacy modification, by following Ca++ microconcentration gradients through voltage-gated Ca++ channels. [ 20 ]
While the concept of the synaptic tagging hypothesis mainly resulted from experiments applying stimulation to synapses, a similar model can be established considering the process of learning in a broader - behavioral - sense. [ 21 ] Fabricio Ballarini and colleagues developed this behavioral tagging model by testing spatial object recognition, contextual conditioning, and conditioned taste aversion in rats with weak training. The applied training normally only results in alterations of short-term memory. However, they paired this weak training with a separate, arbitrary behavioral event that is assumed to induce protein synthesis. When the two behavioral events were coupled within a certain time frame, the weak training was sufficient to elicit task-related changes in long-term memory. The researchers believed that the weak training lead to a "learning tag". During the subsequent task, the cleavage of proteins resulted in the formation of long-term memory for this tag. The behavioral tagging model corresponds to the synaptic tagging model. A weak stimulation establishes E-LTP that may serve as the tag used in converting the weak potentiation to the stronger, more persistent L-LTP, once the high-intensity stimulation is applied. | https://en.wikipedia.org/wiki/Synaptic_tagging |
Synapto-pHluorin is a genetically encoded optical indicator of vesicle release and recycling. It is used in neuroscience to study transmitter release. It consists of a pH-sensitive form of green fluorescent protein (GFP) fused to the luminal side of a vesicle-associated membrane protein (VAMP). At the acidic pH inside transmitter vesicles, synapto-pHluorin is non-fluorescent ( quenched ). When vesicles get released, synapto-pHluorin is exposed to the neutral extracellular space and the presynaptic terminal becomes brightly fluorescent. Following endocytosis , vesicles become re-acidified and the cycle can start again. Chemical alkalinization of all vesicles is often used for normalization of the synapto-pHluorin signals. Synapto-pHluorin sometimes consists of yellow fluorescent protein (YFP) to monitor the cytoplasm because its pK a is higher than GFP (7.1 versus 6.0). [ 1 ]
Synapto-pHluorin was invented by Gero Miesenböck in 1998. [ 2 ] In 2006, an improved version was published, using synaptophysin to target the GFP to vesicles. [ 3 ] In 2013, a two-color release sensor (ratio-sypHy) was introduced to determine the size of the recycling pool at individual synapses. [ 4 ]
Synapto-pHluorin is mainly used by neurobiologists to study transmitter release and recycling at presynaptic terminals . [ 4 ] It has also been applied to the study of insulin secretion in beta cells of the pancreas . [ 5 ] | https://en.wikipedia.org/wiki/Synapto-pHluorin |
The synaptonemal complex ( SC ) is a protein structure that forms between homologous chromosomes (two pairs of sister chromatids ) during meiosis and is thought to mediate synapsis and recombination during prophase I during meiosis in eukaryotes . It is currently thought that the SC functions primarily as a scaffold to allow interacting chromatids to complete their crossover activities. [ 1 ]
The synaptonemal complex is a tripartite structure consisting of two parallel lateral regions and a central element. This "tripartite structure" is seen during the pachytene stage of the first meiotic prophase , both in males and in females during gametogenesis . Previous to the pachytene stage, during leptonema, the lateral elements begin to form and they initiate and complete their pairing during the zygotene stage. After pachynema ends, the SC usually becomes disassembled and can no longer be identified. [ 2 ]
In humans, three specific components of the synaptonemal complex have been characterized: SC protein-1 (SYCP1), SC protein-2 (SYCP2), and SC protein-3 ( SYCP3 ). The SYCP1 gene is on chromosome 1p13; the SYCP2 gene is on chromosome 20q13.33; and the gene for SYCP3 is on chromosome 12q. [ 3 ]
The synaptonemal complex was described by Montrose J. Moses in 1956 in primary spermatocytes of crayfish and by D. Fawcett in spermatocytes of pigeon, cat and man. [ 4 ] As seen with the electron microscope, the synaptonemal complex is formed by two "lateral elements", mainly formed by SYCP3 and secondarily by SYCP2, a "central element" that contains at least two additional proteins and the amino terminal region of SYCP1, and a "central region" spanned between the two lateral elements, that contains the "transverse filaments" composed mainly by the protein SYCP1. [ 3 ]
The SCs can be seen with the light microscope using silver staining or with immunofluorescence techniques that label the proteins SYCP3 or SYCP2.
Formation of the SC usually reflects the pairing or " synapsis " of homologous chromosomes and may be used to probe the presence of pairing abnormalities in individuals carrying chromosomal abnormalities, either in number or in the chromosomal structure. [ 5 ] The sex chromosomes in male mammals show only "partial synapsis" as they usually form only a short SC in the XY pair. The SC shows very little structural variability among eukaryotic organisms despite some significant protein differences. In many organisms the SC carries one or several "recombination nodules" associated with its central space. These nodules are thought to correspond to mature genetic recombination events or "crossovers". In male mice, gamma irradiation increases meiotic crossovers in SCs. This indicates that exogenously caused DNA damages are likely repaired by crossover recombination in SCs. [ 3 ] The finding of an interaction between a SC structural component [synaptonemal central element protein 2 (SYCE2)] and recombinational repair protein RAD51 also suggests a role for the SC in DNA repair.
In cell development the synaptonemal complex disappears during the late prophase of meiosis I. It is formed during zygotene.
Although synaptonemal complex protein 2 (SYCP2) is a meiotic protein, it is aberrantly and commonly expressed in breast and ovarian cancers . SYCP2 protein expression in these cancers is associated with broad resistance to drugs that induced DNA damage , i.e. DNA damage response (DDR) drugs. [ 6 ] SYCP2 is employed in the repair of DNA double-strand breaks by transcription-coupled homologous recombination . [ 6 ] SYCP2 appears to confer cancer cell resistance to therapeutic DNA damaging agents by stimulating R-loop mediated double strand break repair. [ 6 ] Thus inhibition of SYCP2 expression is being studied in efforts to improve therapy for breast and ovarian cancers. [ 6 ]
It is now evident that the synaptonemal complex is not required for genetic recombination in some organisms. For instance, in protozoan ciliates such as Tetrahymena thermophila and Paramecium tetraurelia genetic crossover does not appear to require synaptonemal complex formation. [ 7 ] [ 8 ] Research has shown that not only does the SC form after genetic recombination but mutant yeast cells unable to assemble a synaptonemal complex can still engage in the exchange of genetic information. However, in other organisms like the C. elegans nematode, formation of chiasmata require the formation of the synaptonemal complex. | https://en.wikipedia.org/wiki/Synaptonemal_complex |
Sync.in was a web-based collaborative real-time editor , from Cynapse. [ 1 ] It allowed multiple people to edit the same text document simultaneously. Participants could see others' changes in real-time, [ 2 ] with each author's text in their own color. A chat box in the sidebar allowed participants to communicate.
Sync.in was based [ 3 ] [ 4 ] on EtherPad that was acquired by Google [ 5 ] in December 2009, and released as open-source [ 6 ] later that month.
Sync.in used the Freemium financial model. Free service required no sign-up or registration and users could create unlimited public documents. Pro version was available in a Software as a service model and offered business-centric functions. [ 7 ]
As of May 2013, creation of public notes has been disabled on the sync.in domain. [ citation needed ]
Anyone can create a new collaborative text document, known as a "note". Each note has its own URL , and anyone who knows this URL can edit the note and participate in the document collaboration. Users can invite their friends or colleagues by sharing the note URL using the "share this note" functionality provided in each note. Note, URLs can be shared by Email or other social networking sites [ 8 ] like Twitter , Facebook , Del.ici.ous , Digg , LinkedIn , Ping.fm and Cyn.in
The software auto-saves the document at each keystroke, while participants can permanently save specific versions at any time. A “time slider” feature [ 9 ] allows anyone to explore the history of the note and how it evolved to its current state. It is a screenshot video of the creation of the note, keystroke-by-keystroke. The final document can be downloaded in HTML, plain text, or a bookmark file. [ 10 ]
Sync.in Pro allows teams to create a secure site on a sync.in subdomain with users for everyone in the team. Notes are private to the team members, [ 11 ] and individual notes can be password protected. There is also an option to selectively make notes available to the world.
An Adobe Air-based desktop client [ 12 ] is available for Windows , Mac OS X and Linux and has enhanced features for Pro users. The desktop client lets users create new notes and launch notes directly from their desktop. | https://en.wikipedia.org/wiki/Sync.in |
SyncML , or Synchronization Markup Language, was originally developed as a platform-independent standard for information synchronization . Established by the SyncML Initiative , this project has evolved to become a key component in data synchronization and device management. The project is currently referred to as Open Mobile Alliance Data Synchronization and Device Management . [ 1 ] The purpose of SyncML is to offer an open standard as a replacement for existing data synchronization solutions; which have mostly been somewhat vendor , application , or operating system specific. SyncML 1.0 specification was released on December 17, 2000, [ 2 ] and 1.1 on February 26, 2002. [ 3 ]
A SyncML message is a well-formed XML document that adheres to the document type definition (DTD) , but which does not require validation.
SyncML works by exchanging commands , which can be requests and responses. As an example:
Commands ( Alert , Sync , Status , etc.) are grouped into messages. Each message and each of its commands has an identifier, so that the pair (MsgID, CmdID) uniquely determines a command. Responses like Status commands include the pair identifying the command they are responding to.
Before commands, messages contain a header specifying various data regarding the transaction . An example message containing the Alert command for begin a refresh synchronization, like in the previous example, is:
The response from the computer could be an XML document like (comments added
for the sake of explanation):
The transaction then proceeds with a message from the mobile containing the Sync command, and so on.
This example is a refresh where the mobile sends all its data to the computer and nothing in the other way around. Different codes in the initial Alert command can be used to initiate other kinds of synchronizations. For example, in a " two-way sync", only the changes from the last synchronization are sent to the computer , which does the same.
The Last and Next tags are used to keep track of a possible loss of sync. Last represents the time of the last operation of synchronization, as measured by each device. For example, a mobile may use progressive numbers ( 1 , 2 , 3 , ...) to represent time, while the computer uses strings like " 20140112T213401Z ". Next is the current time in the same representation. This latter data is stored and then compared with Last in the next synchronization. Any difference indicates a loss of sync. Appropriate actions involving sending all data can be then taken to put the devices back in sync.
Anchors are only used to detect a loss of sync; they do not indicate which data is to be sent. Apart from the loss-of-synchronization situation, in a normal (non-refresh) synchronization, each device sends a log of changes since the last synchronization.
1 SAN = Server Alert Notification. This SyncML Push technology is based on definitions by the Open Mobile Alliance and extends the existing SyncML protocol specification by offering a method of server initiated synchronization. | https://en.wikipedia.org/wiki/SyncML |
In logic and linguistics , an expression is syncategorematic if it lacks a denotation but can nonetheless affect the denotation of a larger expression which contains it. Syncategorematic expressions are contrasted with categorematic expressions, which have their own denotations.
For example, consider the following rules for interpreting the plus sign . The first rule is syncategorematic since it gives an interpretation for expressions containing the plus sign but does not give an interpretation for the plus sign itself. On the other hand, the second rule does give an interpretation for the plus sign itself, so it is categorematic.
Syncategorematicity was a topic of research in medieval philosophy since syncategorematic expressions cannot stand for any of Aristotle 's categories despite their role in forming propositions . Medieval logicians and grammarians thought that quantifiers and logical connectives were necessarily syncategorematic. Contemporary research in formal semantics has shown that categorematic definitions can be given for these expressions in which they denote generalized quantifiers , but it remains an open question whether syncategorematicity plays any role in natural language . Both categorematic and syncategorematic definitions are commonly used in contemporary logic and mathematics . [ 1 ] [ 2 ] [ 3 ] [ 4 ]
The distinction between categorematic and syncategorematic terms was established in ancient Greek grammar. Words that designate self-sufficient entities (i.e., nouns or adjectives) were called categorematic, and those that do not stand by themselves were dubbed syncategorematic, (i.e., prepositions, logical connectives, etc.). Priscian in his Institutiones grammaticae [ 5 ] translates the word as consignificantia . Scholastics retained the difference, which became a dissertable topic after the 13th century revival of logic. William of Sherwood , a representative of terminism , wrote a treatise called Syncategoremata . Later his pupil, Peter of Spain , produced a similar work entitled Syncategoreumata . [ 6 ]
In its modern conception, syncategorematicity is seen as a formal feature, determined by the way an expression is defined or introduced in the language. In the standard semantics for propositional logic , the logical connectives are treated syncategorematically. Let us take the connective ∧ {\displaystyle \land } for instance. Its semantic rule is:
Thus, its meaning is defined when it occurs in combination with two formulas ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } . It has no meaning when taken in isolation, i.e. ‖ ∧ ‖ {\displaystyle \lVert \land \rVert } is not defined.
One could however give an equivalent categorematic interpretation using λ-abstraction : ( λ b . ( λ v . b ( v ) ( b ) ) ) {\displaystyle (\lambda b.(\lambda v.b(v)(b)))} , which expects a pair of Boolean-valued arguments, i.e., arguments that are either TRUE or FALSE , defined as ( λ x . ( λ y . x ) ) {\displaystyle (\lambda x.(\lambda y.x))} and ( λ x . ( λ y . y ) ) {\displaystyle (\lambda x.(\lambda y.y))} respectively. This is an expression of type ⟨ ⟨ t , t ⟩ , t ⟩ {\displaystyle \langle \langle t,t\rangle ,t\rangle } . Its meaning is thus a binary function from pairs of entities of type truth-value to an entity of type truth-value. Under this definition it would be non-syncategorematic, or categorematic. Note that while this definition would formally define the ∧ {\displaystyle \land } function, it requires the use of λ {\displaystyle \lambda } -abstraction, in which case the λ {\displaystyle \lambda } itself is introduced syncategorematically, thus simply moving the issue up another level of abstraction. [ citation needed ] | https://en.wikipedia.org/wiki/Syncategorematic_term |
The Synchro-Ciné was a device, invented by the French inventor Charles Delacommune in 1921, pioneer in the objective of synchronizing the projection of a film with the corresponding sounds. The sounds to be synchronized could be from the words of a narrator, the reading of a score by an orchestra conductor (or by a musician soloist ), or shooting up devices with noises-effects;. [ 1 ] But always by mechanical procedures, and with live sound by interpreters in the room (it was still a decade away for sound cinema). By assimilation, it was also known as Ciné-pupitre, its most popular and well-known device in many countries.
But in reality, under the same name are called, over time, three different things:
A) Firstly, the original synchronized reading desk, from 1921, the so-called ciné-pupitre, an isolated device for reading texts or music, for a single user (although this may be a soloist or an orchestra conductor who, since reading it, controls the complete template of instrumentalists) adjusted by an engine in synchronization with the projected images of the films.
B) Soon after, in 1922-23, the synchro-ciné, expanding the connection possibilities of its central distributor, already became a complete synchronization system, in which the reading desk is only one of the many elements around the central element, the control band , circulating in the distributor gear, and which can automatically trigger various 'noise' devices (Ciné-bruiteurs) or 'noise-makers' (some designed by Delacommune himself and incorporated into the system in the patent), shoot piano players , gramophones , and various desks that are needed, up to a maximum of, it seems, 9 devices. The proof of the change of orientation in the machinery and final objectives is the fact that, although the reading desks remain a regular part of the system; however it is already perfectly possible to perform a performance without using at all (only with the capacity of the central distributor , and the connected noises devices). Also part of the system are the various procedures and methods that are registered to improve and facilitate the use of your procedure, throughout over time. It is called in the original patent the "Synchronismes Cinématographiques" system, although it was usually abbreviated as "Synchro-ciné" system, or even more usual "Delacommune system".
c) And finally, it gave name to the production and distribution company of the films realized with this device (and directed by the own Delacommune). [ 2 ] Among others, " ElDorado ", by L'Herbier; the " Danse Macabre " by Dudley Murphy or the Ballet mécanique , by Léger. [ 3 ] With a career based on documentaries, this production company had a moment of boom at the beginning of sound cinema . [ 4 ] as this procedure and device proved to be very suitable for the synchronization of voices in dubbing (the so-called ' band rhythm '), and used until very recently by the professionals of said specialty . However, Delacommune (and his company) it ended up falling into ruin and had to be subject to a special campaign of help by his fellow inventors for their mere subsistence. [ 5 ]
The original 1921 device, the ciné-pupitre, consisted of a reading desk supplied with an aperture at which a paper text "that unfolded before the reader, by small shakes, on which was inscribed the oral accompaniment of the film. Each line corresponded to the projection time of a number of determined frames and was in the center of the reference during the projection of these frames. In other words, the fixed reference allowed the reader to know in each instant the phrase of the text to say corresponding to the projection or the appropriate musical fragment for the conductor or soloist" , say Delacommune himself in his patent. [ 5 ]
Once the paper-band was prepared, the various mismatches common to the time from multiple causes (lack of electrical current, mechanical problems, etc.) could be solved, on the fly, in two ways. Or the speed of the control band was set at the desk (or the distributor ) or by a mechanism that the orchestra conductor was placed in the hand (a rheostat), slightly vary the speed of the images. [ 5 ]
But this system also had the potential to work in another way: firing directly from its paper-band orders to the various notes of a pianola , or shooting up effects and devices of noises or records previously prepared for this purpose. It is this potentiality of automatic adjustment (once the control-band is properly recorded) that makes it a mechanical precedent to the current audiovisual mixes tables, and, perhaps, of the MIDI systems (the digital control of musical instruments ). [ 3 ]
It was not the only device of the so-called synchronization desks or synchronized reading of the time, an advance that the film industry of the moment was looking forward to, as it offered movies with live music, including large symphony orchestras in the first class halls (Cinema-Palaces) or Theater-Cinemas.
Delacommune himself says in the magazines of the time: " One immediately understands the interest of such an apparatus. Of a little expensive manufacture, with a safe operation, its use is very indicated in all the places where the words must accompany a projection. One can foresee the happy consequences in teaching, in filmed lectures, in innovations (theater-filmed, novel-movies spoken, et cetera) . [ 6 ]
Shortly after, the German industry counted with the Musikchronometer , of Carl Blum for the same thing, an analog device although with different mechanism, [ 7 ] and which was used with satisfaction by the participants (among them, Hindemith ) in the Darmstad Music encounters since 1927. [ 8 ] | https://en.wikipedia.org/wiki/Synchro-Ciné |
Synchronous coefficient of drag alteration ( SCODA ) is a biotechnology method for purifying, separating and/or concentrating bio-molecules. SCODA has the ability to separate molecules whose mobility (or drag) can be altered in sync with a driving field. This technique has been primarily used for concentrating and purifying DNA , where DNA mobility changes with an applied electrophoretic field. [ 1 ] [ 2 ] [ 3 ] Electrophoretic SCODA has also been demonstrated with RNA and proteins .
As shown below, the SCODA principle applies to any particle driven by a force field in which the particle's mobility is altered in sync with the driving field.
For explanatory purposes consider an electrophoretic particle moving (driven) in an electric field. Let:
and d v x c o s 2 ω {\displaystyle dv_{x_{cos2\omega }}}
denote an electric field and the velocity of the particle in such a field. If μ {\displaystyle \mu } is constant the time average of v → ( t ) = 0 {\displaystyle {\overrightarrow {v}}(t)=0} .
If μ {\displaystyle \mu } is not constant as a function of time and if μ {\displaystyle \mu } has a frequency component proportional to c o s ( ω t ) {\displaystyle cos(\omega t)} the time average of v → ( t ) {\displaystyle {\overrightarrow {v}}(t)} need not be zero.
Consider the following example:
Substituting (3) in (2) and computing the time average, v → ¯ {\displaystyle {\bar {\overrightarrow {v}}}} , we obtain:
Thus, it is possible to have the particle experience a non-zero time average velocity, in other words, a net electrophoretic drift, even when the time average of the applied electric field is zero.
Consider a particle under a force field that has a velocity parallel to the field direction and a speed proportional to the square of the magnitude of the electric field (any other non-linearity can be employed [ 1 ] ):
The effective mobility of the particle (the relationship between small changes in drift velocity d v → {\displaystyle d{\overrightarrow {v}}} with respect to small changes in electric field d E → {\displaystyle d{\overrightarrow {E}}} ) can be expressed in Cartesian coordinates as:
Combining (5), (6) and (7) we get:
Further consider the field E is applied in a plane and it rotates counter-clockwise at angular frequency ω {\displaystyle \omega } , such that the field components are:
Substituting (10) and (11) in (8) and (9) and simplifying using trigonometric identities results in a sum of constant terms, sine and cosine, at angular frequency 2 ω {\displaystyle 2\omega } . The next calculations will be performed such that only the cosine terms at angular frequency 2 ω {\displaystyle 2\omega } will yield non-zero net drift velocity - therefore we need only evaluate these terms, which will be abbreviated d v x c o s 2 ω {\displaystyle dv_{x_{cos2\omega }}} and d v y c o s 2 ω {\displaystyle dv_{y_{cos2\omega }}} . The following is obtained:
Let d E x {\displaystyle dE_{x}} and d E y {\displaystyle dE_{y}} take the form of a small quadrupole field of intensity d E q {\displaystyle dE_{q}} that varies in a sinusoidal manner proportional to c o s ( 2 ω t ) {\displaystyle cos(2\omega t)} such that:
Substituting (14) and (15) into (12) and (13) and taking the time average we obtain:
which can be summarized in vector notation to:
Equation (18) shows that for all positions r → {\displaystyle {\overrightarrow {r}}} the time averaged velocity is in the direction toward the origin (concentrating the particles towards the origin), with speed proportional to the mobility coefficient k, the strength of the rotating field E and the strength of the perturbing quadrupole field d E q {\displaystyle dE_{q}} .
DNA molecules are unique in that they are long, charged polymers which when in a separation medium, such as agarose gel , can exhibit highly non-linear velocity profiles in response to an electric field. As such, DNA is easily separated from other molecules that are not both charged and strongly non-linear, using SCODA [ 2 ]
To perform SCODA concentration of DNA molecules, the sample must be embedded in the separation media (gel) in locations where the electrophoretic field is of optimal intensity. This initial translocation of the sample into the optimal concentration position is referred to as "injection". The optimal position is determined by the gel geometry and location of the SCODA driving electrodes. Initially the sample is located in a buffer solution in the sample chamber, adjacent to the concentration gel. Injection is achieved by the application of a controlled DC electrophoretic field across the sample chamber which results in all charged particles being transferred into the concentration gel. To obtain a good stacking of the sample (i.e. tight DNA band) multiple methods can be employed. One example is to exploit the conductivity ratio between the sample chamber buffer and the concentration gel buffer. If the sample chamber buffer has a low conductivity and the concentration gel buffer has a high conductivity this results in a sharp drop off in electric field at the gel-buffer interface which promotes stacking.
Once the DNA is positioned optimally in the concentration gel the SCODA rotating fields are applied. The frequency of the fields can be tuned such that only specific DNA lengths are concentrated. To prevent boiling during the concentration stage due to Joule heating the separation medium may be actively cooled. It is also possible to reverse the phase of SCODA fields, so that molecules are de-focused.
As only particles that exhibit non-linear velocity experience the SCODA concentrating force, small charged particles that respond linearly to electrophoretic fields are not concentrated. These particles instead of spiraling towards the center of the SCODA gel orbit at a constant radius. If a weak DC field is superimposed on the SCODA rotating fields these particles will be "washed" off from the SCODA gel resulting in highly pure DNA remaining in the gel center.
The SCODA DNA force results in the DNA sample concentrating in the center of the SCODA gel. To extract the DNA an extraction well can be pre-formed in the gel and filled with buffer. As the DNA does not experience non-linear mobility in buffer it accumulates in the extraction well. At the end of the concentration and purification stage the sample can then be pipetted out from this well.
The electrophoretic SCODA force is gentle enough to maintain the integrity of high molecular weight DNA as it is concentrated towards the center of the SCODA gel. Depending on the length of the DNA in the sample different protocols can be used to concentrate DNA over 1 Mb in length.
DNA concentration and purification has been achieved directly from tar sands samples resuspended in buffer using the SCODA technique. DNA sequencing was subsequently performed and tentatively over 200 distinct bacterial genomes have been identified. [ 2 ] [ 4 ] SCODA has also been used for purification of DNA from many other environmental sources. [ 5 ] [ 6 ]
The non-linear mobility of DNA in gel can be further controlled by embedding in the SCODA gel DNA oligonucleotides complementary to DNA fragments in the sample. [ 7 ] [ 8 ] This then results in highly specific non-linear velocities for the sample DNA that matches the gel-embedded DNA. This artificial specific non-linearity is then used to selectively concentrate only sequences of interest while rejecting all other DNA sequences in the sample. Over 1,000,000-fold enrichment of single nucleotide variants over wild-type have been demonstrated.
An application of this technique is the detection of rare tumour-derived DNA ( ctDNA ) from blood samples. [ 9 ] | https://en.wikipedia.org/wiki/Synchronous_coefficient_of_drag_alteration |
A synchronous or synchronized culture is a microbiological culture or a cell culture that contains cells that are all in the same growth stage. [ 1 ] [ 2 ]
As numerous factors influence the cell cycle (some of them stochastic ) normal cultures have cells in all stages of the cell cycle . Obtaining a culture with a unified cell-cycle stage is useful for biological research where a particular stage in the cell cycle is desired (such as the culturing of parasitized cells [ 3 ] ). Since cells are too small for certain research techniques, a synchronous culture can be treated as a single cell; the number of cells in the culture can be easily estimated, and quantitative experimental results can simply be divided in the number of cells to obtain values that apply to a single cell. Synchronous cultures have been extensively used to address questions regarding cell cycle and growth, and the effects of various factors on these. [ 4 ]
Synchronous cultures can be obtained in several ways: | https://en.wikipedia.org/wiki/Synchronous_culture |
In electronics , a synchronous detector is a device that recovers information from a modulated signal by mixing the signal with a replica of the unmodulated carrier. This can be locally generated at the receiver using a phase-locked loop or other techniques. Synchronous detection preserves any phase information originally present in the modulating signal. With the exception of SECAM receivers, synchronous detection is a necessary component of any analog color television receiver, where it allows recovery of the phase information that conveys hue. [ 1 ] Synchronous detectors are also found in some shortwave radio receivers used for audio signals, where they provide better performance on signals that may be affected by fading .
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synchronous_detector |
A synchronous frame is a reference frame in which the time coordinate defines proper time for all co-moving observers. It is built by choosing some constant time hypersurface as an origin, such that has in every point a normal along the time line and a light cone with an apex in that point can be constructed; all interval elements on this hypersurface are space-like . A family of geodesics normal to this hypersurface are drawn and defined as the time coordinates with a beginning at the hypersurface. In terms of metric-tensor components g i k {\displaystyle g_{ik}} , a synchronous frame is defined such that
where α = 1 , 2 , 3. {\displaystyle \alpha =1,2,3.} Such a construct, and hence, choice of synchronous frame, is always possible though it is not unique. It allows any transformation of space coordinates that does not depend on time and, additionally, a transformation brought about by the arbitrary choice of hypersurface used for this geometric construct.
Synchronization of clocks located at different space points means that events happening at different places can be measured as simultaneous if those clocks show the same times. In special relativity , the space distance element dl is defined as the intervals between two very close events that occur at the same moment of time. In general relativity this cannot be done, that is, one cannot define dl by just substituting dt ≡ dx 0 = 0 in the metric . The reason for this is the different dependence between proper time τ {\displaystyle \tau } and time coordinate x 0 ≡ t in different points of space., i.e., c d τ = g 00 d x 0 . {\displaystyle cd\tau ={\sqrt {g_{00}}}dx^{0}.}
To find dl in this case, time can be synchronized over two infinitesimally neighboring points in the following way (Fig. 1): Bob sends a light signal from some space point B with coordinates x α + d x α {\displaystyle x^{\alpha }+dx^{\alpha }} to Alice who is at a very close point A with coordinates x α and then Alice immediately reflects the signal back to Bob. The time necessary for this operation (measured by Bob), multiplied by c is, obviously, the doubled distance between Alice and Bob.
The line element , with separated space and time coordinates, is:
where a repeated Greek index within a term means summation by values 1, 2, 3. The interval between the events of signal arrival and its immediate reflection back at point A is zero (two events, arrival and reflection are happening at the same point in space and time). For light signals, the space-time interval is zero and thus setting d s = 0 {\displaystyle ds=0} in the above equation, we can solve for dx 0 obtaining two roots:
which correspond to the propagation of the signal in both directions between Alice and Bob. If x 0 is the moment of arrival/reflection of the signal to/from Alice in Bob's clock then, the moments of signal departure from Bob and its arrival back to Bob correspond, respectively, to x 0 + dx 0 (1) and x 0 + dx 0 (2) . The thick lines on Fig. 1 are the world lines of Alice and Bob with coordinates x α and x α + dx α , respectively, while the red lines are the world lines of the signals. Fig. 1 supposes that dx 0 (2) is positive and dx 0 (1) is negative, which, however, is not necessarily the case: dx 0 (1) and dx 0 (2) may have the same sign. The fact that in the latter case the value x 0 (Alice) in the moment of signal arrival at Alice's position may be less than the value x 0 (Bob) in the moment of signal departure from Bob does not contain a contradiction because clocks in different points of space are not supposed to be synchronized. It is clear that the full "time" interval between departure and arrival of the signal in Bob's place is
The respective proper time interval is obtained from the above relationship by multiplication by g 00 / c {\displaystyle {\sqrt {g_{00}}}/c} , and the distance dl between the two points – by additional multiplication by c /2. As a result:
This is the required relationship that defines distance through the space coordinate elements.
It is obvious that such synchronization should be done by exchange of light signals between points. Consider again propagation of signals between infinitesimally close points A and B in Fig. 1. The clock reading in B which is simultaneous with the moment of reflection in A lies in the middle between the moments of sending and receiving the signal in B ; in this moment if Alice's clock reads y 0 and Bob's clock reads x 0 then via Einstein Synchronization condition ,
Substitute here eq. 2 to find the difference in "time" x 0 between two simultaneous events occurring in infinitesimally close points as
This relationship allows clock synchronization in any infinitesimally small space volume. By continuing such synchronization further from point A , one can synchronize clocks, that is, determine simultaneity of events along any open line. The synchronization condition can be written in another form by multiplying eq. 4 by g 00 and bringing terms to the left hand side
or, the "covariant differential" dx 0 between two infinitesimally close points should be zero.
However, it is impossible, in general, to synchronize clocks along a closed contour: starting out along the contour and returning to the starting point one would obtain a Δ x 0 value different from zero. Thus, unambiguous synchronization of clocks over the whole space is impossible. An exception are reference frames in which all components g 0α are zeros.
The inability to synchronize all clocks is a property of the reference frame and not of the spacetime itself. It is always possible in infinitely many ways in any gravitational field to choose the reference frame so that the three g 0α become zeros and thus enable a complete synchronization of clocks. To this class are assigned cases where g 0α can be made zeros by a simple change in the time coordinate which does not involve a choice of a system of objects that define the space coordinates.
In the special relativity theory, too, proper time elapses differently for clocks moving relatively to each other. In general relativity, proper time is different even in the same reference frame at different points of space. This means that the interval of proper time between two events occurring at some space point and the time interval between the events simultaneous with those at another space point are, in general, different.
Consider a rest (inertial) frame expressed in cylindrical coordinates r ′ ϕ ′ , z ′ {\displaystyle r'\,\phi ',\,z'} and time t ′ {\displaystyle t'} . The interval in this frame is given by d s 2 = c 2 d t ′ 2 − d r ′ 2 − r ′ 2 d ϕ ′ 2 − d z ′ 2 . {\displaystyle ds^{2}=c^{2}dt'^{2}-dr'^{2}-r'^{2}d\phi '^{2}-dz'^{2}.} Transforming to a uniformly rotating coordinate system ( r , ϕ , z ) {\displaystyle (r,\phi ,z)} using the relation x 0 / c = t = t ′ , x 1 = r = r ′ , x 2 = ϕ = ϕ ′ − Ω t ′ , x 3 = z = z ′ {\displaystyle x^{0}/c=t=t',\,x^{1}=r=r',\,x^{2}=\phi =\phi '-\Omega t',\,x^{3}=z=z'} modifies the interval to
Of course, the rotating frame is valid only for r < c / Ω {\displaystyle r<c/\Omega } since the frame speed would exceed speed of light beyond this radial location. The non-zero components of the metric tensor are g 00 = 1 − Ω 2 r 2 / c 2 , {\displaystyle g_{00}=1-\Omega ^{2}r^{2}/c^{2},} g 02 = − 2 Ω r 2 / c , {\displaystyle g_{02}=-2\Omega r^{2}/c,} g 11 = − 1 , {\displaystyle g_{11}=-1,} g 22 = − r 2 {\displaystyle g_{22}=-r^{2}} and g 33 = − 1. {\displaystyle g_{33}=-1.} Along any open curve, the relation
can be used to synchronize clocks. However, along any closed curve, synchronization is impossible because
For instance, when Ω r / c ≪ 1 {\displaystyle \Omega r/c\ll 1} , we have
where S {\displaystyle S} is the projected area of the closed curve on a plane perpendicular to the rotation axis (plus or minus sign corresponds to contour traversing in, or opposite to the rotation direction).
The proper time element in the rotating frame is given by
indicating that time slows down as we move away from the axis. Similarly the spatial element can be calculated to find
At a fixed value of r {\displaystyle r} and z {\displaystyle z} , the spatial element is d l = ( 1 − Ω 2 r 2 / c 2 ) − 1 / 2 r d ϕ {\displaystyle dl=(1-\Omega ^{2}r^{2}/c^{2})^{-1/2}rd\phi } which upon integration over a full circle shows that the ratio of circumference of a circle to its radius is given by
which is greater than by 2 π {\displaystyle 2\pi } .
Eq. 3 can be rewritten in the form
where
is the three-dimensional metric tensor that determines the metric, that is, the geometrical properties of space. Equations eq. 7 give the relationships between the metric of the three-dimensional space γ α β {\displaystyle \gamma _{\alpha \beta }} and the metric of the four-dimensional spacetime g i k {\displaystyle g_{ik}} .
In general, however, g i k {\displaystyle g_{ik}} depends on x 0 so that γ α β {\displaystyle \gamma _{\alpha \beta }} changes with time. Therefore, it doesn't make sense to integrate dl : this integral depends on the choice of world line between the two points on which it is taken. It follows that in general relativity the distance between two bodies cannot be determined in general; this distance is determined only for infinitesimally close points. Distance can be determined for finite space regions only in such reference frames in which g ik does not depend on time and therefore the integral ∫ d l {\textstyle \int dl} along the space curve acquires some definite sense.
The tensor − γ α β {\displaystyle -\gamma _{\alpha \beta }} is inverse to the contravariant 3-dimensional tensor g α β {\displaystyle g^{\alpha \beta }} . Indeed, writing equation g i k g k l = δ l i {\displaystyle g^{ik}g_{kl}=\delta _{l}^{i}} in components, one has:
Determining g α 0 {\displaystyle g^{\alpha 0}} from the second equation and substituting it in the first proves that
This result can be presented otherwise by saying that g α β {\displaystyle g^{\alpha \beta }} are components of a contravariant 3-dimensional tensor corresponding to metric γ α β {\displaystyle \gamma ^{\alpha \beta }} :
The determinants g and γ {\displaystyle \gamma } composed of elements g i k {\displaystyle g_{ik}} and γ α β {\displaystyle \gamma _{\alpha \beta }} , respectively, are related to each other by the simple relationship:
In many applications, it is convenient to define a 3-dimensional vector g with covariant components
Considering g as a vector in space with metric γ α β {\displaystyle \gamma _{\alpha \beta }} , its contravariant components can be written as g α = γ α β g β {\displaystyle g^{\alpha }=\gamma ^{\alpha \beta }g_{\beta }} . Using eq. 11 and the second of eqs. 8 , it is easy to see that
From the third of eqs. 8 , it follows
As concluded from eq. 5 , the condition that allows clock synchronization in different space points is that metric tensor components g 0α are zeros. If, in addition, g 00 = 1, then the time coordinate x 0 = t is the proper time in each space point (with c = 1). A reference frame that satisfies the conditions
is called synchronous frame . The interval element in this system is given by the expression
with the spatial metric tensor components identical (with opposite sign) to the components g αβ :
In synchronous frame time, time lines are normal to the hypersurfaces t = const. Indeed, the unit four-vector normal to such a hypersurface n i = ∂ t /∂ x i has covariant components n α = 0, n 0 = 1. The respective contravariant components with the conditions eq. 15 are again n α = 0, n 0 = 1.
The components of the unit normal coincide with those of the four-vector u i = dx i /ds which is tangent to the world line x 1 , x 2 , x 3 = const. The u i with components u α = 0, u 0 = 1 automatically satisfies the geodesic equations :
since, from the conditions eq. 15 , the Christoffel symbols Γ 00 α {\displaystyle \Gamma _{00}^{\alpha }} and Γ 00 0 {\displaystyle \Gamma _{00}^{0}} vanish identically. Therefore, in the synchronous frame the time lines are geodesics in the spacetime.
These properties can be used to construct synchronous frame in any spacetime (Fig. 2). To this end, choose some spacelike hypersurface as an origin, such that has in every point a normal along the time line (lies inside the light cone with an apex in that point); all interval elements on this hypersurface are space-like. Then draw a family of geodesics normal to this hypersurface. Choose these lines as time coordinate lines and define the time coordinate t as the length s of the geodesic measured with a beginning at the hypersurface; the result is a synchronous frame.
An analytic transformation to synchronous frame can be done with the use of the Hamilton–Jacobi equation . The principle of this method is based on the fact that particle trajectories in gravitational fields are geodesics. The Hamilton–Jacobi equation for a particle (whose mass is set equal to unity) in a gravitational field is
where S is the action. Its complete integral has the form:
Note that the complete integral contains as many arbitrary constants as the number of independent variables which in our case is 4 {\displaystyle 4} . In the above equation, these correspond to the three parameters ξ α and the fourth constant A being treated as an arbitrary function of the three ξ α . With such a representation for S the equations for the trajectory of the particle can be obtained by equating the derivatives ∂S / ∂ξ α to zero, i.e.
For each set of assigned values of the parameters ξ α , the right sides of equations 18a-18c have definite constant values, and the world line determined by these equations is one of the possible trajectories of the particle. Choosing the quantities ξ α , which are constant along the trajectory, as new space coordinates, and the quantity S as the new time coordinate, one obtains a synchronous frame; the transformation from the old coordinates to the new ones is given by equations 18b-18c . In fact, it is guaranteed that for such a transformation the time lines will be geodesics and will be normal to the hypersurfaces S = const. The latter point is obvious from the mechanical analogy: the four-vector ∂S / ∂x i which is normal to the hypersurface coincides in mechanics with the four-momentum of the particle, and therefore coincides in direction with its four-velocity u i i.e. with the four-vector tangent to the trajectory. Finally the condition g 00 = 1 is obviously satisfied, since the derivative − dS / ds of the action along the trajectory is the mass of the particle, which was set equal to 1; therefore | dS / ds | = 1.
The gauge conditions eq. 15 do not fix the coordinate system completely and therefore are not a fixed gauge , as the spacelike hypersurface at t = 0 {\displaystyle t=0} can be chosen arbitrarily. One still have the freedom of performing some coordinate transformations containing four arbitrary functions depending on the three spatial variables x α , which are easily worked out in infinitesimal form:
Here, the collections of the four old coordinates ( t , x α ) and four new coordinates ( t ~ , x ~ α ) {\displaystyle ({\tilde {t}},{\tilde {x}}^{\alpha })} are denoted by the symbols x and x ~ {\displaystyle {\tilde {x}}} , respectively. The functions ξ i ( x ~ ) {\displaystyle \xi ^{i}({\tilde {x}})} together with their first derivatives are infinitesimally small quantities. After such a transformation, the four-dimensional interval takes the form:
where
In the last formula, the g i k ( x ~ ) {\displaystyle g_{ik}({\tilde {x}})} are the same functions g ik ( x ) in which x should simply be replaced by x ~ {\displaystyle {\tilde {x}}} . If one wishes to preserve the gauge eq. 15 also for the new metric tensor g i k (new) ( x ~ ) {\displaystyle g_{ik}^{\text{(new)}}({\tilde {x}})} in the new coordinates x ~ {\displaystyle {\tilde {x}}} , it is necessary to impose the following restrictions on the functions ξ i ( x ´ ) {\displaystyle \xi ^{i}({\acute {x}})} :
The solutions of these equations are:
where f 0 and f α are four arbitrary functions depending only on the spatial coordinates x ~ α {\displaystyle {\tilde {x}}^{\alpha }} .
For a more elementary geometrical explanation, consider Fig. 2. First, the synchronous time line ξ 0 = t can be chosen arbitrarily (Bob's, Carol's, Dana's or any of an infinitely many observers). This makes one arbitrarily chosen function: ξ 0 = f 0 ( x ~ 1 , x ~ 2 , x ~ 3 ) {\displaystyle \xi ^{0}=f^{0}\left({\tilde {x}}^{1},{\tilde {x}}^{2},{\tilde {x}}^{3}\right)} . Second, the initial hypersurface can be chosen in infinitely many ways. Each of these choices changes three functions: one function for each of the three spatial coordinates ξ α = f α ( x ~ 1 , x ~ 2 , x ~ 3 ) {\displaystyle \xi ^{\alpha }=f^{\alpha }\left({\tilde {x}}^{1},{\tilde {x}}^{2},{\tilde {x}}^{3}\right)} . Altogether, four (= 1 + 3) functions are arbitrary.
When discussing general solutions g αβ of the field equations in synchronous gauges, it is necessary to keep in mind that the gravitational potentials g αβ contain, among all possible arbitrary functional parameters present in them, four arbitrary functions of 3-space just representing the gauge freedom and therefore of no direct physical significance.
Another problem with the synchronous frame is that caustics can occur which cause the gauge choice to break down. These problems have caused some difficulties doing cosmological perturbation theory in synchronous frame, but the problems are now well understood. Synchronous coordinates are generally considered the most efficient reference system for doing calculations, and are used in many modern cosmology codes, such as CMBFAST . They are also useful for solving theoretical problems in which a spacelike hypersurface needs to be fixed, as with spacelike singularities .
Introduction of a synchronous frame allows one to separate the operations of space and time differentiation in the Einstein field equations . To make them more concise, the notation
is introduced for the time derivatives of the three-dimensional metric tensor; these quantities also form a three-dimensional tensor. In the synchronous frame ϰ α β {\displaystyle \varkappa _{\alpha \beta }} is proportional to the second fundamental form (shape tensor). All operations of shifting indices and covariant differentiation of the tensor ϰ α β {\displaystyle \varkappa _{\alpha \beta }} are done in three-dimensional space with the metric γ αβ . This does not apply to operations of shifting indices in the space components of the four-tensors R ik , T ik . Thus T α β must be understood to be g βγ T γα + g β 0 T 0 α , which reduces to g βγ T γα and differs in sign from γ βγ T γα . The sum ϰ α α {\displaystyle \varkappa _{\alpha }^{\alpha }} is the logarithmic derivative of the determinant γ ≡ | γ αβ | = − g :
Then for the complete set of Christoffel symbols Γ k l i {\displaystyle \Gamma _{kl}^{i}} one obtains:
where λ α β γ {\displaystyle \lambda _{\alpha \beta }^{\gamma }} are the three-dimensional Christoffel symbols constructed from γ αβ :
where the comma denotes partial derivative by the respective coordinate.
With the Christoffel symbols eq. 25 , the components R i k = g il R lk of the Ricci tensor can be written in the form:
Dots on top denote time differentiation, semicolons (";") denote covariant differentiation which in this case is performed with respect to the three-dimensional metric γ αβ with three-dimensional Christoffel symbols λ α β γ {\displaystyle \lambda _{\alpha \beta }^{\gamma }} , ϰ ≡ ϰ α α {\displaystyle \varkappa \equiv \varkappa _{\alpha }^{\alpha }} , and P α β is a three-dimensional Ricci tensor constructed from λ α β γ {\displaystyle \lambda _{\alpha \beta }^{\gamma }} :
It follows from eq. 27–29 that the Einstein equations R i k = 8 π k ( T i k − 1 2 δ i k T ) {\displaystyle R_{i}^{k}=8\pi k\left(T_{i}^{k}-{\frac {1}{2}}\delta _{i}^{k}T\right)} (with the components of the energy–momentum tensor T 0 0 = − T 00 , T α 0 = − T 0α , T α β = γ βγ T γα ) become in a synchronous frame:
A characteristic feature of the synchronous frame is that it is not stationary: the gravitational field cannot be constant in such frame. In a constant field ϰ α β {\displaystyle \varkappa _{\alpha \beta }} would become zero. But in the presence of matter the disappearance of all ϰ α β {\displaystyle \varkappa _{\alpha \beta }} would contradict eq. 31 (which has a right side different from zero). In empty space from eq. 33 follows that all P αβ , and with them all the components of the three-dimensional curvature tensor P αβγδ ( Riemann tensor ) vanish, i.e. the field vanishes entirely (in a synchronous frame with a Euclidean spatial metric the space-time is flat).
At the same time the matter filling the space cannot in general be at rest relative to the synchronous frame. This is obvious from the fact that particles of matter within which there are pressures generally move along lines that are not geodesics; the world line of a particle at rest is a time line, and thus is a geodesic in the synchronous frame. An exception is the case of dust ( p = 0). Here the particles interacting with one another will move along geodesic lines; consequently, in this case the condition for a synchronous frame does not contradict the condition that it be comoving with the matter. Even in this case, in order to be able to choose a synchronously comoving frame , it is still necessary that the matter move without rotation. In the comoving frame the contravariant components of the velocity are u 0 = 1, u α = 0. If the frame is also synchronous, the covariant components must satisfy u 0 = 1, u α = 0, so that its four-dimensional curl must vanish:
But this tensor equation must then also be valid in any other reference frame. Thus, in a synchronous but not comoving frame the condition curl v = 0 for the three-dimensional velocity v is additionally needed. For other equations of state a similar situation can occur only in special cases when the pressure gradient vanishes in all or in certain directions.
Use of the synchronous frame in cosmological problems requires thorough examination of its asymptotic behaviour. In particular, it must be known if the synchronous frame can be extended to infinite time and infinite space maintaining always the unambiguous labelling of every point in terms of coordinates in this frame.
It was shown that unambiguous synchronization of clocks over the whole space is impossible because of the impossibility to synchronize clocks along a closed contour. As concerns synchronization over infinite time, let's first remind that the time lines of all observers are normal to the chosen hypersurface and in this sense are "parallel". Traditionally, the concept of parallelism is defined in Euclidean geometry to mean straight lines that are everywhere equidistant from each other but in arbitrary geometries this concept can be extended to mean lines that are geodesics . It was shown that time lines are geodesics in synchronous frame. Another, more convenient for the present purpose definition of parallel lines are those that have all or none of their points in common. Excluding the case of all points in common (obviously, the same line) one arrives to the definition of parallelism where no two time lines have a common point.
Since the time lines in a synchronous frame are geodesics, these lines are straight (the path of light) for all observers in the generating hypersurface. The spatial metric is
The determinant γ {\displaystyle \gamma } of the metric tensor is the absolute value of the triple product of the row vectors in the matrix γ α β {\displaystyle \gamma _{\alpha \beta }} which is also the volume of the parallelepiped spanned by the vectors γ → 1 {\displaystyle {\vec {\gamma }}_{1}} , γ → 2 {\displaystyle {\vec {\gamma }}_{2}} , and γ → 3 {\displaystyle {\vec {\gamma }}_{3}} (i.e., the parallelepiped whose adjacent sides are the vectors γ → 1 {\displaystyle {\vec {\gamma }}_{1}} , γ → 2 {\displaystyle {\vec {\gamma }}_{2}} , and γ → 3 {\displaystyle {\vec {\gamma }}_{3}} ).
If γ {\displaystyle \gamma } turns into zero then the volume of this parallelepiped is zero. This can happen when one of the vectors lies in the plane of the other two vectors so that the parallelepiped volume transforms to the area of the base (height becomes zero), or more formally, when two of the vectors are linearly dependent. But then multiple points (the points of intersection) can be labelled in the same way, that is, the metric has a singularity.
The Landau group [ 1 ] have found that the synchronous frame necessarily forms a time singularity, that is, the time lines intersect (and, respectively, the metric tensor determinant turns to zero) in a finite time.
This is proven in the following way. The right-hand of the eq. 31 , containing the stress–energy tensors of matter and electromagnetic field,
is a positive number because of the strong energy condition . This can be easily seen when written in components.
With the above in mind, the eq. 31 is then re-written as an inequality
with the equality pertaining to empty space.
Using the algebraic inequality
eq. 34 becomes
Dividing both sides to ( ϰ α α ) 2 {\displaystyle \left(\varkappa _{\alpha }^{\alpha }\right)^{2}} and using the equality
one arrives to the inequality
Let, for example, ϰ α α > 0 {\displaystyle \varkappa _{\alpha }^{\alpha }>0} at some moment of time. Because the derivative is positive, then the ratio 1 ϰ α α {\textstyle {\frac {1}{\varkappa _{\alpha }^{\alpha }}}} decreases with decreasing time, always having a finite non-zero derivative and, therefore, it should become zero, coming from the positive side, during a finite time. In other words, ϰ α α {\displaystyle \varkappa _{\alpha }^{\alpha }} becomes + ∞ {\displaystyle +\infty } , and because ϰ α α = ∂ ln γ / ∂ t {\displaystyle \varkappa _{\alpha }^{\alpha }=\partial \ln \gamma /\partial t} , this means that the determinant γ {\displaystyle \gamma } becomes zero (according to eq. 35 not faster than t 6 {\displaystyle t^{6}} ). If, on the other hand, ϰ α α < 0 {\displaystyle \varkappa _{\alpha }^{\alpha }<0} initially, the same is true for increasing time.
An idea about the space at the singularity can be obtained by considering the diagonalized metric tensor. Diagonalization makes the elements of the γ α β {\displaystyle \gamma _{\alpha \beta }} matrix everywhere zero except the main diagonal whose elements are the three eigenvalues λ 1 , λ 2 {\displaystyle \lambda _{1},\lambda _{2}} and λ 3 {\displaystyle \lambda _{3}} ; these are three real values when the discriminant of the characteristic polynomial is greater or equal to zero or one real and two complex conjugate values when the discriminant is less than zero. Then the determinant γ {\displaystyle \gamma } is just the product of the three eigenvalues. If only one of these eigenvalues becomes zero, then the whole determinant is zero. Let, for example, the real eigenvalue becomes zero ( λ 1 = 0 {\displaystyle \lambda _{1}=0} ). Then the diagonalized matrix γ α β {\displaystyle \gamma _{\alpha \beta }} becomes a 2 × 2 matrix with the (generally complex conjugate) eigenvalues λ 2 , λ 3 {\displaystyle \lambda _{2},\lambda _{3}} on the main diagonal. But this matrix is the diagonalized metric tensor of the space where γ = 0 {\displaystyle \gamma =0} ; therefore, the above suggests that at the singularity ( γ = 0 {\displaystyle \gamma =0} ) the space is 2-dimensional when only one eigenvalue turns to zero.
Geometrically, diagonalization is a rotation of the basis for the vectors comprising the matrix in such a way that the direction of basis vectors coincide with the direction of the eigenvectors . If γ α β {\displaystyle \gamma _{\alpha \beta }} is a real symmetric matrix , the eigenvectors form an orthonormal basis defining a rectangular parallelepiped whose length, width, and height are the magnitudes of the three eigenvalues. This example is especially demonstrative in that the determinant γ {\displaystyle \gamma } which is also the volume of the parallelepiped is equal to length × width × height, i.e., the product of the eigenvalues. Making the volume of the parallelepiped equal to zero, for example by equating the height to zero, leaves only one face of the parallelepiped, a 2-dimensional space, whose area is length × width. Continuing with the obliteration and equating the width to zero, one is left with a line of size length, a 1-dimensional space. Further equating the length to zero leaves only a point, a 0-dimensional space, which marks the place where the parallelepiped has been.
An analogy from geometrical optics is comparison of the singularity with caustics, such as the bright pattern in Fig. 3, which shows caustics formed by a glass of water illuminated from the right side. The light rays are an analogue of the time lines of the free-falling observers localized on the synchronized hypersurface. Judging by the approximately parallel sides of the shadow contour cast by the glass, one can surmise that the light source is at a practically infinite distance from the glass (such as the sun) but this is not certain as the light source is not shown on the photo. So one can suppose that the light rays (time lines) are parallel without this being proven with certainty. The glass of water is an analogue of the Einstein equations or the agent(s) behind them that bend the time lines to form the caustics pattern (the singularity). The latter is not as simple as the face of a parallelepiped but is a complicated mix of various kinds of intersections. One can distinguish an overlap of two-, one-, or zero-dimensional spaces, i.e., intermingling of surfaces and lines, some converging to a point ( cusp ) such as the arrowhead formation in the centre of the caustics pattern. [ 2 ] [ 3 ]
The conclusion that timelike geodesic vector fields must inevitably reach a singularity after a finite time has been reached independently by Raychaudhuri by another method that led to the Raychaudhuri equation , which is also called Landau–Raychaudhuri equation to honour both researchers. | https://en.wikipedia.org/wiki/Synchronous_frame |
A synchrotron light source is a source of electromagnetic radiation (EM) usually produced by a storage ring , [ 1 ] for scientific and technical purposes. First observed in synchrotrons , synchrotron light is now produced by storage rings and other specialized particle accelerators , typically accelerating electrons . Once the high-energy electron beam has been generated, it is directed into auxiliary components such as bending magnets and insertion devices ( undulators or wigglers ) in storage rings and free electron lasers .
These supply the strong magnetic fields perpendicular to the beam that are needed to stimulate the high energy electrons to emit photons .
The major applications of synchrotron light are in condensed matter physics , materials science , biology and medicine . A large fraction of experiments using synchrotron light involve probing the structure of matter from the sub- nanometer level of electronic structure to the micrometer and millimeter levels important in medical imaging . An example of a practical industrial application is the manufacturing of microstructures by the LIGA process.
Synchrotron is one of the most expensive kinds of light source known, but it is practically the only viable luminous source of wide-band radiation in far infrared wavelength range for some applications, such as far-infrared absorption spectrometry.
The primary figure of merit used to compare different sources of synchrotron radiation has been referred to as the "brightness", the "brilliance", and the "spectral brightness", with the latter term being recommended as the best choice by the Working Group on Synchrotron Nomenclature. [ 2 ] Regardless of the name chosen, the term is a measure of the total flux of photons in a given six-dimensional phase space per unit bandwidth (BW). [ 3 ]
The spectral brightness is given by
where N ˙ ph {\displaystyle {\dot {N}}_{\text{ph}}} is the number of photons per second in the beam, σ x {\displaystyle \sigma _{x}} and σ y {\displaystyle \sigma _{y}} are the root mean square values for the size of the beam in the axes perpendicular to the beam direction, σ x ′ {\displaystyle \sigma _{x'}} and σ y ′ {\displaystyle \sigma _{y'}} are the RMS values for the beam solid angle in the x and y dimensions, and d ω ω {\textstyle {\frac {d\omega }{\omega }}} is the relative bandwidth, or spread in beam frequency around the central frequency. [ 4 ] The customary value for bandwidth is 0.1%. [ 2 ]
Spectral brightness has units of time −1 ⋅distance −2 ⋅angle −2 ⋅(% bandwidth) −1 .
Especially when artificially produced, synchrotron radiation is notable for its:
Synchrotron radiation may occur in accelerators either as a nuisance, causing undesired energy loss in particle physics contexts, or as a deliberately produced radiation source for numerous laboratory applications. Electrons are accelerated to high speeds in several stages to achieve a final energy that is typically in the gigaelectronvolt range. The electrons are forced to travel in a closed path by strong magnetic fields. This is similar to a radio antenna, but with the difference that the relativistic speed changes the observed frequency due to the Doppler effect by a factor γ {\displaystyle \gamma } . Relativistic Lorentz contraction bumps the frequency by another factor of γ {\displaystyle \gamma } , thus multiplying the gigahertz frequency of the resonant cavity that accelerates the electrons into the X-ray range. Another dramatic effect of relativity is that the radiation pattern is distorted from the isotropic dipole pattern expected from non-relativistic theory into an extremely forward-pointing cone of radiation. This makes synchrotron radiation sources the most brilliant known sources of X-rays. The planar acceleration geometry makes the radiation linearly polarized when observed in the orbital plane, and circularly polarized when observed at a small angle to that plane. [ citation needed ]
The advantages of using synchrotron radiation for spectroscopy and diffraction have been realized by an ever-growing scientific community, beginning in the 1960s and 1970s. In the beginning, accelerators were built for particle physics, and synchrotron radiation was used in "parasitic mode" when bending magnet radiation had to be extracted by drilling extra holes in the beam pipes. The first storage ring commissioned as a synchrotron light source was Tantalus, at the Synchrotron Radiation Center , first operational in 1968. [ 5 ]
Bending electromagnets in accelerators were first used to generate this radiation, but to generate stronger radiation, other specialized devices – insertion devices – are sometimes employed. Current (third-generation) synchrotron radiation sources are typically reliant upon these insertion devices, where straight sections of the storage ring incorporate periodic magnetic structures (comprising many magnets in a pattern of alternating N and S poles – see diagram above) which force the electrons into a sinusoidal or helical path. Thus, instead of a single bend, many tens or hundreds of "wiggles" at precisely calculated positions add up or multiply the total intensity of the beam. [ citation needed ]
These devices are called wigglers or undulators . The main difference between an undulator and a wiggler is the intensity of their magnetic field and the amplitude of the deviation from the straight line path of the electrons. [ citation needed ]
At a synchrotron facility, electrons are usually accelerated by a synchrotron , and then injected into a storage ring , in which they circulate, producing synchrotron radiation, but without gaining further energy. The radiation is projected at a tangent to the electron storage ring and captured by beamlines . These beamlines may originate at bending magnets, which mark the corners of the storage ring; or insertion devices , which are located in the straight sections of the storage ring. The spectrum and energy of X-rays differ between the two types. The beamline includes X-ray optical devices which control the bandwidth , photon flux, beam dimensions, focus, and collimation of the rays. The optical devices include slits, attenuators, crystal monochromators , and mirrors. The mirrors may be bent into curves or toroidal shapes to focus the beam. A high photon flux in a small area is the most common requirement of a beamline. The design of the beamline will vary with the application. At the end of the beamline is the experimental end station, where samples are placed in the line of the radiation, and detectors are positioned to measure the resulting diffraction , scattering or secondary radiation.
Synchrotron light is an ideal tool for many types of research in materials science , physics , and chemistry and is used by researchers from academic, industrial, and government laboratories. Several methods take advantage of the high intensity, tunable wavelength, collimation, and polarization of synchrotron radiation at beamlines which are designed for specific kinds of experiments. The high intensity and penetrating power of synchrotron X-rays enables experiments to be performed inside sample cells designed for specific environments. Samples may be heated, cooled, or exposed to gas, liquid, or high pressure environments. Experiments which use these environments are called in situ and allow the characterization of atomic- to nano-scale phenomena which are inaccessible to most other characterization tools. In operando measurements are designed to mimic the real working conditions of a material as closely as possible. [ 8 ]
X-ray diffraction (XRD) and scattering experiments are performed at synchrotrons for the structural analysis of crystalline and amorphous materials. These measurements may be performed on powders , single crystals , or thin films . The high resolution and intensity of the synchrotron beam enables the measurement of scattering from dilute phases or the analysis of residual stress . Materials can be studied at high pressure using diamond anvil cells to simulate extreme geologic environments or to create exotic forms of matter. [ citation needed ]
X-ray crystallography of proteins and other macromolecules (PX or MX) are routinely performed. Synchrotron-based crystallography experiments were integral to solving the structure of the ribosome ; [ 9 ] [ 10 ] this work earned the Nobel Prize in Chemistry in 2009 .
The size and shape of nanoparticles are characterized using small angle X-ray scattering (SAXS). Nano-sized features on surfaces are measured with a similar technique, grazing-incidence small angle X-ray scattering (GISAXS). [ 11 ] In this and other methods, surface sensitivity is achieved by placing the crystal surface at a small angle relative to the incident beam, which achieves total external reflection and minimizes the X-ray penetration into the material. [ citation needed ]
The atomic- to nano-scale details of surfaces , interfaces, and thin films can be characterized using techniques such as X-ray reflectivity (XRR) and crystal truncation rod (CTR) analysis. [ 12 ] X-ray standing wave (XSW) measurements can also be used to measure the position of atoms at or near surfaces; these measurements require high-resolution optics capable of resolving dynamical diffraction phenomena. [ 13 ]
Amorphous materials, including liquids and melts, as well as crystalline materials with local disorder, can be examined using X-ray pair distribution function analysis, which requires high energy X-ray scattering data. [ 14 ]
By tuning the beam energy through the absorption edge of a particular element of interest, the scattering from atoms of that element will be modified. These so-called resonant anomalous X-ray scattering methods can help to resolve scattering contributions from specific elements in the sample. [ citation needed ]
Other scattering techniques include energy dispersive X-ray diffraction , resonant inelastic X-ray scattering , and magnetic scattering. [ citation needed ]
X-ray absorption spectroscopy (XAS) is used to study the coordination structure of atoms in materials and molecules. The synchrotron beam energy is tuned through the absorption edge of an element of interest, and modulations in the absorption are measured. Photoelectron transitions cause modulations near the absorption edge, and analysis of these modulations (called the X-ray absorption near-edge structure (XANES) or near-edge X-ray absorption fine structure (NEXAFS)) reveals information about the chemical state and local symmetry of that element. At incident beam energies which are much higher than the absorption edge, photoelectron scattering causes "ringing" modulations called the extended X-ray absorption fine structure (EXAFS). Fourier transformation of the EXAFS regime yields the bond lengths and number of the surrounding the absorbing atom; it is therefore useful for studying liquids and amorphous materials [ 15 ] as well as sparse species such as impurities. A related technique, X-ray magnetic circular dichroism (XMCD), uses circularly polarized X-rays to measure the magnetic properties of an element. [ citation needed ]
X-ray photoelectron spectroscopy (XPS) can be performed at beamlines equipped with a photoelectron analyzer . Traditional XPS is typically limited to probing the top few nanometers of a material under vacuum. However, the high intensity of synchrotron light enables XPS measurements of surfaces at near-ambient pressures of gas. Ambient pressure XPS (AP-XPS) can be used to measure chemical phenomena under simulated catalytic or liquid conditions. [ 16 ] Using high-energy photons yields high kinetic energy photoelectrons which have a much longer inelastic mean free path than those generated on a laboratory XPS instrument. The probing depth of synchrotron XPS can therefore be lengthened to several nanometers, allowing the study of buried interfaces. This method is referred to as high-energy X-ray photoemission spectroscopy (HAXPES). [ 17 ] Furthermore, the tunable nature of the synchrotron X-ray photon energies presents a wide range of depth sensitivity in the order of 2-50 nm. [ 18 ] This allows for probing of samples at greater depths and for non destructive depth-profiling experiments.
Material composition can be quantitatively analyzed using X-ray fluorescence (XRF). XRF detection is also used in several other techniques, such as XAS and XSW, in which it is necessary to measure the change in absorption of a particular element. [ citation needed ]
Other spectroscopy techniques include angle resolved photoemission spectroscopy (ARPES), soft X-ray emission spectroscopy , and nuclear resonance vibrational spectroscopy , which is related to Mössbauer spectroscopy . [ citation needed ]
Synchrotron X-rays can be used for traditional X-ray imaging , phase-contrast X-ray imaging , and tomography . The Ångström-scale wavelength of X-rays enables imaging well below the diffraction limit of visible light, but practically the smallest resolution so far achieved is about 30 nm. [ 19 ] Such nanoprobe sources are used for scanning transmission X-ray microscopy (STXM). Imaging can be combined with spectroscopy such as X-ray fluorescence or X-ray absorption spectroscopy in order to map a sample's chemical composition or oxidation state with sub-micron resolution. [ 20 ]
Because of the usefulness of tuneable collimated coherent X-ray radiation, efforts have been made to make smaller more economical sources of the light produced by synchrotrons. The aim is to make such sources available within a research laboratory for cost and convenience reasons; at present, researchers have to travel to a facility to perform experiments. One method of making a compact light source is to use the energy shift from Compton scattering near-visible laser photons from electrons stored at relatively low energies of tens of megaelectronvolts (see for example the Compact Light Source (CLS) [ 21 ] ). | https://en.wikipedia.org/wiki/Synchrotron_light_source |
Synchrotron radiation (also known as magnetobremsstrahlung ) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity ( a ⊥ v ). It is produced artificially in some types of particle accelerators or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization , and the frequencies generated can range over a large portion of the electromagnetic spectrum . [ 1 ]
Synchrotron radiation is similar to bremsstrahlung radiation , which is emitted by a charged particle when the acceleration is parallel to the direction of motion. The general term for radiation emitted by particles in a magnetic field is gyromagnetic radiation , for which synchrotron radiation is the ultra-relativistic special case. Radiation emitted by charged particles moving non-relativistically in a magnetic field is called cyclotron emission . [ 2 ] For particles in the mildly relativistic range (≈85% of the speed of light), the emission is termed gyro-synchrotron radiation . [ 3 ]
In astrophysics , synchrotron emission occurs, for instance, due to ultra-relativistic motion of a charged particle around a black hole . [ 4 ] When the source follows a circular geodesic around the black hole, the synchrotron radiation occurs for orbits close to the photosphere where the motion is in the ultra-relativistic regime.
Synchrotron radiation was first observed by technician Floyd Haber, on April 24, 1947, at the 70 MeV electron synchrotron of the General Electric research laboratory in Schenectady, New York . [ 5 ] While this was not the first synchrotron built, it was the first with a transparent vacuum tube, allowing the radiation to be directly observed. [ 6 ]
As recounted by Herbert Pollock: [ 7 ]
On April 24, Langmuir and I were running the machine and as usual were trying to push the electron gun and its associated pulse transformer to the limit. Some intermittent sparking had occurred and we asked the technician to observe with a mirror around the protective concrete wall. He immediately signaled to turn off the synchrotron as "he saw an arc in the tube". The vacuum was still excellent, so Langmuir and I came to the end of the wall and observed. At first we thought it might be due to Cherenkov radiation , but it soon became clearer that we were seeing Ivanenko and Pomeranchuk radiation. [ 8 ]
A direct consequence of Maxwell's equations is that accelerated charged particles always emit electromagnetic radiation. Synchrotron radiation is the special case of charged particles moving at relativistic speed undergoing acceleration perpendicular to their direction of motion, typically in a magnetic field. In such a field, the force due to the field is always perpendicular to both the direction of motion and to the direction of field, as shown by the Lorentz force law .
The power carried by the radiation is found (in SI units ) by the relativistic Larmor formula : [ 9 ] [ 10 ]
P γ = q 2 6 π ε 0 c 3 a 2 γ 4 = q 2 c 6 π ε 0 β 4 γ 4 ρ 2 , {\displaystyle P_{\gamma }={\frac {q^{2}}{6\pi \varepsilon _{0}c^{3}}}a^{2}\gamma ^{4}={\frac {q^{2}c}{6\pi \varepsilon _{0}}}{\frac {\beta ^{4}\gamma ^{4}}{\rho ^{2}}},} where
The force on the emitting electron is given by the Abraham–Lorentz–Dirac force .
When the radiation is emitted by a particle moving in a plane, the radiation is linearly polarized when observed in that plane, and circularly polarized when observed at a small angle. However, in quantum mechanics, this radiation is emitted in discrete packets of photons, which introduces quantum fluctuations in the emitted radiation and the particle's trajectory. For a given acceleration, the average energy of emitted photons is proportional to γ 3 {\displaystyle \gamma ^{3}} and the emission rate to γ {\displaystyle \gamma } . [ citation needed ]
Circular accelerators will always produce gyromagnetic radiation as the particles are deflected in the magnetic field. However, the quantity and properties of the radiation are highly dependent on the nature of the acceleration taking place. For example, due to the difference in mass, the factor of γ 4 {\displaystyle \gamma ^{4}} in the formula for the emitted power means that electrons radiate energy at approximately 10 13 times the rate of protons. [ 11 ]
Energy loss from synchrotron radiation in circular accelerators was originally considered a nuisance, as additional energy must be supplied to the beam in order to offset the losses. However, beginning in the 1980s, circular electron accelerators known as light sources have been constructed to deliberately produce intense beams of synchrotron radiation for research. [ 12 ]
Synchrotron radiation is also generated by astronomical objects, typically where relativistic electrons spiral (and hence change velocity) through magnetic fields.
Two of its characteristics include power-law energy spectra and polarization. [ 13 ] It is considered to be one of the most powerful tools in the study of extra-solar magnetic fields wherever relativistic charged particles are present. Most known cosmic radio sources emit synchrotron radiation. It is often used to estimate the strength of large cosmic magnetic fields as well as analyze the contents of the interstellar and intergalactic media. [ 14 ]
This type of radiation was first detected in the Crab Nebula in 1956 by Jan Hendrik Oort and Theodore Walraven , [ 15 ] and a few months later in a jet emitted by Messier 87 by Geoffrey R. Burbidge . [ 16 ] It was confirmation of a prediction by Iosif S. Shklovsky in 1953. However, it had been predicted earlier (1950) by Hannes Alfvén and Nicolai Herlofson. [ 17 ] Solar flares accelerate particles that emit in this way, as suggested by R. Giovanelli in 1948 and described by J.H. Piddington in 1952. [ 18 ]
T. K. Breus noted that questions of priority on the history of astrophysical synchrotron radiation are complicated, writing:
In particular, the Russian physicist V.L. Ginzburg broke his relationships with I.S. Shklovsky and did not speak with him for 18 years. In the West, Thomas Gold and Sir Fred Hoyle were in dispute with H. Alfven and N. Herlofson, while K.O. Kiepenheuer and G. Hutchinson were ignored by them. [ clarification needed ] [ 19 ]
It has been suggested that supermassive black holes produce synchrotron radiation in "jets", generated by the gravitational acceleration of ions in their polar magnetic fields. The nearest such observed jet is from the core of the galaxy Messier 87 . This jet is interesting for producing the illusion of superluminal motion as observed from the frame of Earth. This phenomenon is caused because the jets are traveling very near the speed of light and at a very small angle towards the observer. Because at every point of their path the high-velocity jets are emitting light, the light they emit does not approach the observer much more quickly than the jet itself. Light emitted over hundreds of years of travel thus arrives at the observer over a much smaller time period, giving the illusion of faster than light travel, despite the fact that there is actually no violation of special relativity . [ 20 ]
A class of astronomical sources where synchrotron emission is important is pulsar wind nebulae , also known as plerions , of which the Crab nebula and its associated pulsar are archetypal.
Pulsed emission gamma-ray radiation from the Crab has recently been observed up to ≥25 GeV, [ 21 ] probably due to synchrotron emission by electrons trapped in the strong magnetic field around the pulsar.
Polarization in the Crab nebula [ 22 ] at energies from 0.1 to 1.0 MeV, illustrates this typical property of synchrotron radiation.
Much of what is known about the magnetic environment of the interstellar medium and intergalactic medium is derived from observations of synchrotron radiation. Cosmic ray electrons moving through the medium interact with relativistic plasma and emit synchrotron radiation which is detected on Earth. The properties of the radiation allow astronomers to make inferences about the magnetic field strength and orientation in these regions. However, accurate calculations of field strength cannot be made without knowing the relativistic electron density. [ 14 ]
When a star explodes in a supernova, the fastest ejecta move at semi-relativistic speeds approximately 10% the speed of light . [ 23 ] This blast wave gyrates electrons in ambient magnetic fields and generates synchrotron emission, revealing the radius of the blast wave at the location of the emission. [ 24 ] Synchrotron emission can also reveal the strength of the magnetic field at the front of the shock wave, as well as the circumstellar density it encounters, but strongly depends on the choice of energy partition between the magnetic field, proton kinetic energy, and electron kinetic energy. Radio synchrotron emission has allowed astronomers to shed light on mass loss and stellar winds that occur just prior to stellar death. [ 25 ] [ 26 ] | https://en.wikipedia.org/wiki/Synchrotron_radiation |
Synchrotron radiation circular dichroism spectroscopy , commonly referred to as SRCD and also known as VUV-circular dichroism or VUVCD spectroscopy , is a powerful extension to the technique of circular dichroism (CD) spectroscopy , often used to study structural properties of biological molecules such as proteins and nucleic acids . The physical principles of SRCD are essentially identical to those of CD, in that the technique measures the difference in absorption (ΔA) of left (A L ) and right (A R ) circularly polarized light (ΔA=A L -A R ) by a sample in solution. To obtain a CD(SRCD) spectrum the sample must be innately optically active ( chiral ), or, in some way be induced to have chiral properties, as only then will there be an observable difference in absorption of the left and right circularly polarized light. The major advantages of SRCD over CD arise from the ability to measure data over an extended wavelength range into the vacuum ultra violet (VUV) end of the spectrum. As these measurements are utilizing a light source with a higher photon flux (quantity of light stricking a given surface area) than a bench-top CD machine it means data are more accurate at these extended wavelengths because there is a larger signal over the background noise (the signal-to-noise ratio) and, generally, less sample is needed when recording the spectra and there is more information content available in the data. [ 1 ] Many beamlines now exist around the world to enable the measurement of SRCD data.
Extending the wavelength range for CD experiments had been both considered and instigated as far back as 1970. Three research groups had created their own "in-house" CD machines, with specialist lamps as their light source, to enable measurements in this range. [ 2 ] Synchrotron radiation (SR) had been proposed for use as the light source at a meeting in Brookhaven National Laboratory on Long Island in 1972, [ 2 ] [ 3 ] however, it took a few years more before this came to fruition. Two research papers in 1980 reported the collection of CD data using SR as the light source for the experiments. Specifically, spectra were obtained in wavelength regions into the VUV range, from ~100 nanometers (nm) to ~200 nm, largely unavailable to laboratory-based bench-top spectrophotometers . Sutherland [ 4 ] et. al. focussed on the development of a versatile spectrophotometer capable of measuring CD, amongst other properties, in the VUV region of the spectrum, [ 5 ] [ 6 ] while Snyder [ 7 ] and Rowe collected CD data from a small organic compound in the wavelength range 130.5 nm to 205 nm. [ 8 ]
As shown in the diagram, a number of baffles are used throughout to remove possible stray light being reflected off the sides of the beamline tube. The use of only one mirror minimizes the loss of photon flux which is most important in the VUV region where reflectivity is poor relative to the visible wavelength range. [ 9 ]
The first constructed SRCD beamlines initially tried to utilize the intrinsic properties of the SR radiation produced, whereby there exists a "central" linearly-polarized component with, above and below this, equally opposing regions of circularly-polarized components. The premise for this was that the overall signal produced from a chiral sample would be enhanced by the absorption difference (the signal) derived from these circularly polarized features of the beam. [ 10 ] In an ideal situation this approach would work; however, this setup was modified such that all beamlines now include a linear polarizer (as shown) to remove these circularly polarized components. This was because even the minutest of movements in beam position (beam drift) led to unequal matching of the contributions of the circularly polarized components striking the sample and this, in turn, meant the SRCD signal produced was inaccurate and unreliable; often being irreproducible as a result. [ 10 ] Whereas cCD machines are purged throughout with nitrogen to minimize the absorption by oxygen of the light from the source xenon arc lamp , in an SRCD arrangement the beam passes through a calcium fluoride (CaF 2 ), or similar "VUV-wavelengths transparent", window where everything before this point is in vacuum , and everything beyond is in nitrogen. The beam interacts with a photoelastic modulator (PEM) which consequently produces an alternating right- then left-circularly polarized beam and these now interact with the sample. The resultant absorption difference by the sample is measured and amplified by a photomultiplier tube (PMT) and from this the SRCD spectrum is recorded.
The wavelength range that is utilized for SRCD studies is typically in the UV to VUV region and can go to below this; potentially from ~100 nm, up to the visible region, ~400 nm. The exact range over which data can be collected relies on the beamline set up, the sample preparation and the wavelength range of the PMT detector used. One of the primary factors limiting the lower wavelength cut off is the sample usually being in solution as a large water absorption band exists centred ~167 nm. This high absorption background swamps any possibility of measuring the very small CD difference signal, although use of deuterated water (D 2 O) as the solvent reduces the solvent absorption increasing the lower wavelength data collection range by ~10 nm. [ 11 ] [ 12 ] Removing the solvating water completely, creating a film as a result, means that data can be recorded to significantly lower wavelengths, down to around ~130 nm. [ 13 ]
The main advantages for SRCD over lab-based cCD machines arise from the use of the synchrotron light emission as the source. A number of biologically interesting absorption bands are found in the region between ~170 nm and ~350 nm. For proteins these come from their secondary and tertiary structures, while structural bands for nucleic acids, ( DNA and RNA ), and saccharides are also located in this region. However, for cCD machines the photon flux from the source reduces by around two orders of magnitude in the wavelength range from 250 nm down to 180 nm, [ 14 ] exactly in the region of most significance for these biological molecules. By contrast, typically, the photon flux for an SRCD beamline in this region is at least three orders of magnitude higher than a cCD machine, retaining that level down to ~150 nm. [ 14 ] The increased flux means the measured signals from the sample will be increased relative to the background noise, so there is a significant improvement in the signal-to-noise ratio of the sample. This will improve the accuracy of the data recorded meaning interpretation can be undertaken with more confidence in the results. A further advantage of the increased flux is that the concentration of the sample can be reduced while still retaining a significant increase in signal strength, so samples that are difficult to produce in quantity have more chance of producing usable CD data from SRCD rather than a cCD machine. Increasing the lower wavelength range provides more spectral data for analysis which means there is more information content [ 15 ] available in that data, meaning that more parameters, here secondary structure features in the protein structure, can be accurately determined. [ 15 ]
While the first reports of its use dated to 1980, it was a further two decades before the technique of SRCD took off largely due to the work of Bonnie Wallace at Birkbeck College , University of London . From around 2000, her aims in the field focused on both enhancing the collection of quality data through technical improvements, and on demonstrating "proof-of-principle" application studies, illustrating the novel information that SRCD offers. The construction on the Synchrotron Radiation Source (SRS) of the CD12 beamline [ 14 ] at Daresbury Laboratory , opened in 2005 under the auspices of the Centre for Protein and Membrane Structure and Dynamics (CPMSD) [ 1 ] [ 16 ] [ 17 ] of which Wallace was the Director, represented the first of the new, dedicated, second-generation SRCD beamlines. It was quickly identified that the high photon flux from CD12 was causing denaturation of the protein sample [ 18 ] but that this was resolvable by reducing the sample area being irradiated. [ 19 ] Later studies have identified the flux threshold limits that induce SRCD protein denaturation. [ 20 ] The input from the Wallace lab to the early years of SRCD development also included the introduction of calibration and standardization of SRCD and cCD spectrophotometers , [ 21 ] [ 22 ] the creation of software to process the spectral data using CDtool, [ 23 ] and CDtoolX, [ 24 ] and to analyse the data using DichroWeb, [ 25 ] [ 26 ] [ 27 ] [ 28 ] and the generation of reference data sets of proteins to support these data analyses. [ 29 ] [ 30 ] [ 31 ] Additionally, her lab produced sample cells with reduced pathlengths, and using material, (CaF 2 ), transparent to VUV radiation which significantly enhanced the collection of data into the SRCD lower wavelength regions. [ 32 ]
New SRCD beamlines were constructed on various synchrotrons around the world. "ASTRID" . ring, in the Department of Physics and Astronomy of Aarhus University in Denmark , became a dedicated second-generation synchrotron in 2005. Ultimately this ring had two SRCD beamlines, UV1 and CD1 , which migrated to the new third-generation ring, ASTRID2 , in 2013/14, as AU-UV and AU-CD . SOLEIL synchrotron, near Paris , France , commissioned a dedicated SRCD beamline, DISCO , around 2005. [ 33 ] At Hiroshima Synchrotron Radiation Center , also known as HiSOR, a VUVCD beamline was constructed over the same period, while a little later in 2009, an SRCD beamline was commissioned in Beijing , China . This particular beamline is unique in that the synchrotron which acts as its light source is also the electron carrying ring of the Beijing Electron Positron Collider . [ 34 ] The SRS closed in 2008 [ 35 ] being superseded in the UK by the Diamond Light Source on which an SRCD beamline opened for use in 2010. [ 36 ] With the SRS closure the CD12 SRCD beamline was moved to, and installed on, the ANKA Synchrotron Radiation Facility , (now called KARA ), part of Karlsruhe Institute of Technology (KIT), in Karlsruhe , Germany . This beamline opened for users in 2011 [ 13 ] but was closed in 2021. Currently under construction (as of June 2023) on the Sirius synchrotron light source in Campinas , Brazil , is a new SRCD beamline, CEDRO. [ 37 ]
Highlighting a few of the published works that have employed SRCD in their research studies best illustrates the power of this technique.
Cataracts are the primary cause of blindness in humans and mutations in one particular protein, γD- crystallin , have been linked to a number of congenital forms of this disease. [ 38 ] An amino acid mutation, proline (P) to threonine (T) at position 23 of the polypeptide chain has been linked to at least four different forms of this ailment. SRCD investigations were conducted on the wild-type protein and two variants, the P23T mutant found in the disease, and a related modification, P23S (proline to serine , a chemically similar amino acid to threonine), to establish the nature of the cause of cataract formation. [ 39 ] Two possible reasons were suggested as the causative factor; the reduced solubility of the mutant protein, or an instability in the structure of the protein being introduced by the mutation. Significantly, because the mutant had limited solubility, lab-based CD machines were only able to provide very noisy spectra and the data were uninterpretable as a result. However, the SRCD spectra produced had very low noise associated with their data, including the mutant, and showed clearly that the structures of the wild-type, the mutant, and the related protein all had very similar conformations. These data also established that the mutant retained stability to thermal denaturation, very similar to that of the wild-type protein. The data confirmed that the causative factor for the cataracts was the reduction in solubility associated with the P23T mutation and not changes in the stability of the protein. [ 39 ]
Because of a high degree of flexibility, it had proven difficult to determine the structure of the extramembranous C-terminal domain of bacterial voltage-gated sodium channels . Using a series of synthesised channels where this C-terminal domain had been truncated, in some cases by a single amino acid difference between the constructs , the Wallace lab used SRCD to successfully identify the structure of this region. [ 40 ]
Intrinsically disordered proteins (IDPs) have very limited innate structure in solution but gain shape specifically when interacting with partner molecules such as proteins or RNA ; however, their resultant structure is often dictated by this interaction. In addition, some proteins have sections of sequence without structure, termed intrinsically disordered regions (IDRs), that also gain structure on interaction. Having different shapes with different partners means they are functionally, as well as structurally flexible, making them centrally important to signalling pathways [ 41 ] and as regulation/control factors [ 42 ] for example. IDPs (and IDRs if capable of being isolated from the rest of the protein) have a distinct SRCD spectral appearance in solution which means that changes in their spectra that arise through interactions offer an ideal opportunity to gain insight into what is happening both structurally and functionally. In addition, SRCD studies have demonstrated that when the solvating water is removed from these proteins, generating a film, there is a gain in structure and more CD transition bands can be measured into the lower VUV wavelength region because the water absorption band is not present [ 43 ]
Myelin is the insulating sheath that is formed in the central (CNS) and peripheral nervous systems (PNS) to surround nerve cell axons thereby increasing and maintaining the electrical impulse, the action potential , sent along them. Formed mostly of lipids , there are specific proteins within the myelin components whose roles are to structure the myelin into linked layers. Two of these proteins are myelin basic protein (MBP), an IDP primarily in the CNS, and myelin protein zero (P0) which contains an IDR section (P0ct) and is key within the PNS. MBP and P0ct were employed in a study [ 44 ] which used SRCD data as a key factor to establish if there was any significance to the predictions of their IDP and IDR protein structures generated by Alphafold2 , an artificial intelligence program developed by DeepMind . PDB2CD, [ 45 ] a package that generates SRCD spectra from protein atomic coordinates, was used to calculate spectra from the Alphafold2 structures, and these spectra were then compared against SRCD experimental spectra collected from the MBP and P0ct proteins in various ambient conditions; solution, detergent and lipid-bound states. The study reported that from the SRCD comparisons, the structures predicted by Alphafold2 for MBP and P0ct bore a strong resemblance to those when they were bound to the lipid membrane. [ 44 ]
One major feature found in protein structures is the addition of sugars ( glycosylation ) to specific amino acid residues by post translational modification . Complex sugar structures can be connected to these sites, and this can substantially modify the properties of these proteins, a main reason for their presence. Attached sugars can assist in folding some proteins to their correct shape; so, affecting a proteins’ structure is a possible outcome. SRCD is ideally well suited to determining any conformational differences that might arise from different ambient environments directly because of the extended wavelength range into the VUV region which provides greater information content. However, attached sugars can contribute to the SRCD signal because their transitions are located more towards the VUV end of the spectrum. This means that their presence can cause a problem in obtaining an accurate measure of the secondary structure content of the protein as a result. Matsuo. [ 46 ] and Gekko produced the landmark study of VUVCD spectra of selected saccharides, thereby demonstrating that glycoproteins would have a contribution to their spectra from their sugar content. [ 47 ] From this and further studies [ 48 ] they demonstrated that the SRCD spectral characteristics that arose from sugars could be attributed to many factors within their conformations: the configuration of the hydroxyl group about the C1 atom of the saccharide (alpha or beta conformation, or almost axial or equatorial to the plane of the sugar ring respectively), the axial or equatorial positioning of the remaining hydroxyl groups, the trans or gauche nature of the C5 hydroxymethyl group, and the glycosidic linkage (either 1-4 or 1-6) between sugar monomers. Utilising this information, the Wallace group investigated the glycosylation of the voltage-gated sodium channel in experiments that relied on the fact that a CD(SRCD) spectrum of a mixture of components is the sum of all those components present. [ 49 ] The aim was to establish if there were differences in the three-dimensional structure of the channel with and without sugars attached to the structure; did glycosylation play any significant role in the function of these channels when sugars were attached? Three experimental sets of SRCD spectra were collected; the non-glycosylated and glycosylated channel structures and a further one of the isolated sugar components that combined to form those attached to the channel. Taking away the spectrum of the non-glycosylated channel from that of the glycosylated they demonstrated that the resultant difference spectrum corresponded to that of the sugar components. This meant that there were no structural differences between the glycosylated and non-glycosylated channel structures, so sugar attachment played no key role in their function [ 49 ]
First studied in 2010 via this method, [ 50 ] a recent investigation [ 51 ] used SRCD to examine the differences in structure in solution and when at the oil-water interface, of peptides derived from seaweed, bacteria and potatoes as potential emulsifying agents. Of these studied, the peptide from bacteria proved to be the most effective at being both an emulsifying agent and stabilising antioxidant compound. [ 51 ]
A number of SRCD beamlines exist, or are being constructed (as of 2023 [update] ), around the world as listed in the table.
a As of 2022 components from former SRCD beamline CD12 [ 13 ] (on KARA ) are now installed on the DISCO beamline
b This facility also runs as part of the Beijing Electron Positron Collider (BEPC) [ 34 ]
c Two modules (A and B) exist on this beamline
d This beamline is under construction and received its "first light" as of June 2023 [ 63 ] | https://en.wikipedia.org/wiki/Synchrotron_radiation_circular_dichroism_spectroscopy |
A syncytium ( / s ɪ n ˈ s ɪ ʃ i ə m / ; pl. : syncytia ; from Greek : σύν syn "together" and κύτος kytos "box, i.e. cell") or symplasm is a multinucleate cell that can result from multiple cell fusions of uninuclear cells (i.e., cells with a single nucleus ), in contrast to a coenocyte , which can result from multiple nuclear divisions without accompanying cytokinesis . [ 1 ] The muscle cell that makes up animal skeletal muscle is a classic example of a syncytium cell. The term may also refer to cells interconnected by specialized membranes with gap junctions , as seen in the heart muscle cells and certain smooth muscle cells, which are synchronized electrically in an action potential .
The field of embryogenesis uses the word syncytium to refer to the coenocytic blastoderm embryos of invertebrates , such as Drosophila melanogaster . [ 2 ]
In protists , syncytia can be found in some rhizarians (e.g., chlorarachniophytes , plasmodiophorids , haplosporidians ) and acellular slime moulds , dictyostelids ( amoebozoans ), acrasids ( Excavata ) and Haplozoon .
Some examples of plant syncytia, which result during plant development , include:
A syncytium is the normal cell structure for many fungi . [ 6 ]
Most fungi of Basidiomycota exist as a dikaryon in which thread-like cells of the mycelium are partially partitioned into segments each containing two differing nuclei, called a heterokaryon .
The neurons which makes up the subepithelial nerve net in comb jellies ( Ctenophora ) are fused into a neural syncytium, consisting of a continuous plasma membrane instead of being connected through synapses . [ 7 ]
A classic example of a syncytium is the formation of skeletal muscle . Large skeletal muscle fibers form by the fusion of thousands of individual muscle cells. The multinucleated arrangement is important in pathologic states such as myopathy , where focal necrosis (death) of a portion of a skeletal muscle fiber does not result in necrosis of the adjacent sections of that same skeletal muscle fiber, because those adjacent sections have their own nuclear material. Thus, myopathy is usually associated with such "segmental necrosis", with some of the surviving segments being functionally cut off from their nerve supply via loss of continuity with the neuromuscular junction .
The syncytium of cardiac muscle is important because it allows rapid coordinated contraction of muscles along their entire length. Cardiac action potentials propagate along the surface of the muscle fiber from the point of synaptic contact through intercalated discs . Although a syncytium, cardiac muscle differs because the cells are not long and multinucleated. Cardiac tissue is therefore described as a functional syncytium, as opposed to the true syncytium of skeletal muscle.
Smooth muscle in the gastrointestinal tract is activated by a composite of three types of cells – smooth muscle cells (SMCs), interstitial cells of Cajal (ICCs), and platelet-derived growth factor receptor alpha (PDGFRα) that are electrically coupled and work together as an SIP functional syncytium. [ 8 ] [ 9 ]
Certain animal immune-derived cells may form aggregate cells, such as the osteoclast cells responsible for bone resorption .
Another important vertebrate syncytium is in the placenta of placental mammals. Embryo-derived cells that form the interface with the maternal blood stream fuse together to form a multinucleated barrier – the syncytiotrophoblast . This is probably important to limit the exchange of migratory cells between the developing embryo and the body of the mother, as some blood cells are specialized to be able to insert themselves between adjacent epithelial cells. The syncytial epithelium of the placenta does not provide such an access path from the maternal circulation into the embryo.
Much of the body of Hexactinellid sponges is composed of syncitial tissue. This allows them to form their large siliceous spicules exclusively inside their cells. [ 10 ]
The fine structure of the tegument in helminths is essentially the same in both the cestodes and trematodes . A typical tegument is 7–16 μm thick, with distinct layers. It is a syncytium consisting of multinucleated tissues with no distinct cell boundaries. The outer zone of the syncytium, called the "distal cytoplasm," is lined with a plasma membrane . This plasma membrane is in turn associated with a layer of carbohydrate-containing macromolecules known as the glycocalyx , that varies in thickness from one species to another. The distal cytoplasm is connected to the inner layer called the "proximal cytoplasm", which is the "cellular region or cyton or perikarya" through cytoplasmic tubes that are composed of microtubules . The proximal cytoplasm contains nuclei , endoplasmic reticulum , Golgi complex , mitochondria , ribosomes , glycogen deposits , and numerous vesicles . [ 11 ] The innermost layer is bounded by a layer of connective tissue known as the " basal lamina ". The basal lamina is followed by a thick layer of muscle . [ 12 ]
Syncytia can also form when cells are infected with certain types of viruses , notably HSV-1 , HIV , MeV , SARS-CoV-2 , and pneumoviruses , e.g. respiratory syncytial virus (RSV). These syncytial formations create distinctive cytopathic effects when seen in permissive cells . Because many cells fuse together, syncytia are also known as multinucleated cells, giant cells , or polykaryocytes. [ 13 ] During infection, viral fusion proteins used by the virus to enter the cell are transported to the cell surface, where they can cause the host cell membrane to fuse with neighboring cells.
Typically, the viral families that can cause syncytia are enveloped, because viral envelope proteins on the surface of the host cell are needed to fuse with other cells. [ 14 ] Certain members of the Reoviridae family are notable exceptions due to a unique set of proteins known as fusion-associated small transmembrane (FAST) proteins. [ 15 ] Reovirus induced syncytium formation is not found in humans, but is found in a number of other species and is caused by fusogenic orthoreoviruses . These fusogenic orthoreoviruses include reptilian orthoreovirus, avian orthoreovirus, Nelson Bay orthoreovirus, and baboon orthoreovirus. [ 16 ]
HIV infects Helper CD4 + T cells and makes them produce viral proteins, including fusion proteins. Then, the cells begin to display surface HIV glycoproteins , which are antigenic . Normally, a cytotoxic T cell will immediately come to "inject" lymphotoxins , such as perforin or granzyme , that will kill the infected T helper cell. However, if T helper cells are nearby, the gp41 HIV receptors displayed on the surface of the T helper cell will bind to other similar lymphocytes. [ 17 ] This makes dozens of T helper cells fuse cell membranes into a giant, nonfunctional syncytium, which allows the HIV virion to kill many T helper cells by infecting only one. It is associated with a faster progression of the disease [ 18 ]
The mumps virus uses HN protein to stick to a potential host cell, then, the fusion protein allows it to bind with the host cell. The HN and fusion proteins are then left on the host cell walls, causing it to bind with neighbour epithelial cells. [ 19 ]
Mutations within SARS-CoV-2 variants contain spike protein variants that can enhance syncytia formation. [ 20 ] The protease TMPRSS2 is essential for syncytia formation. [ 21 ] Syncytia can allow the virus to spread directly to other cells, shielded from neutralizing antibodies and other immune system components. [ 20 ] Syncytia formation in cells can be pathological to tissues. [ 20 ]
"Severe cases of COVID-19 are associated with extensive lung damage and the presence of infected multinucleated syncytial pneumocytes . The viral and cellular mechanisms regulating the formation of these syncytia are not well understood," [ 22 ] but membrane cholesterol seems necessary. [ 23 ] [ 24 ]
The syncytia appear to be long-lasting; the "complete regeneration" of the lungs after severe flu "does not happen" with COVID-19. [ 25 ] | https://en.wikipedia.org/wiki/Syncytium |
In scholastic moral philosophy , synderesis ( / ˌ s ɪ n d ə ˈ r iː s ɪ s / ) or synteresis is habitual knowledge of the universal practical principles of moral action. The reasoning process in the field of speculative science presupposes certain fundamental axioms on which all science rests. Such are the principle of contradiction, "a thing cannot be and not be at the same time," and self-evident truths like "the whole is greater than its part". These are the first principles of the speculative intellect. In the field of moral conduct there are similar first principles of action, such as: "evil must be avoided, good done"; "Do not do to others what you would not wish to be done to yourself"; "Parents should be honoured"; "We should live temperately and act justly". Such as these are self-evident truths in the field of moral conduct which any sane person will admit if he understands them. According to the Scholastics, the readiness with which such moral truths are apprehended by the practical intellect is due to the natural habit impressed on the cognitive faculty which they call synderesis. While conscience is a dictate of the practical reason deciding that any particular action is right or wrong, synderesis is a dictate of the same practical reason which has for its object the first general principles of moral action. [ 1 ]
The notion of synderesis has a long tradition, including the Commentary on Ezekiel by Jerome (A.D. 347–419), where syntéresin (συντήρησιν) is mentioned among the powers of the soul and is described as the spark of conscience ( scintilla conscientiae ), [ 2 ] and the interpretation of Jerome's text given, in the 13th century, by Albert the Great and Thomas Aquinas in the light of Aristotelian psychology and ethics . An alternative interpretation of synderesis was proposed by Bonaventure , who considered it as the natural inclination of the will towards moral good.
The word synderesis is by most scholars reckoned to be a corruption of the Greek word for shared knowledge or conscience, syneidêsis ( συνείδησις ), the corruption appearing in the medieval manuscripts of Jerome's Commentary. [ 3 ]
The term is also used in psychiatric studies, with particular reference to psychopathy . [ 4 ] | https://en.wikipedia.org/wiki/Synderesis |
Syneresis (also spelled 'synæresis' or 'synaeresis'), in chemistry, is the extraction or expulsion of a liquid from a gel , such as when serum drains from a contracting clot of blood . Another example of syneresis is the collection of whey on the surface of yogurt . Syneresis can also be observed when the amount of diluent in a swollen polymer exceeds the solubility limit as the temperature changes. A household example of this is the counterintuitive expulsion of water from dry gelatin when the temperature increases. Syneresis has also been proposed as the mechanism of formation for the amorphous silica composing the frustule of diatoms . [ 1 ]
In the processing of dairy milk , for example during cheese making , syneresis is the formation of the curd due to the sudden removal of the hydrophilic macropeptides , which causes an imbalance in intermolecular forces. Bonds between hydrophobic sites start to develop and are enforced by calcium bonds, which form as the water molecules in the micelles start to leave the structure. This process is usually referred to as the phase of coagulation and syneresis. The splitting of the bond between residues 105 and 106 in the κ-casein molecule is often called the primary phase of the rennet action, while the phase of coagulation and syneresis is referred to as the secondary phase.
In cooking, syneresis is the sudden release of moisture contained within protein molecules, usually caused by excessive heat, which over-hardens the protective shell. Moisture inside expands upon heating. The hard protein shell pops, expelling the moisture.
This process is responsible for transforming juicy rare steak into dry steak when cooked thoroughly. It creates weeping in scrambled eggs, with dry protein curd swimming in the released moisture. It also causes emulsified sauces, such as hollandaise, to "break" ("split"). Additionally, it creates unsightly moisture pockets within baked custard dishes, such as flan or crème brûlée .
Gels formed from agarose are prone to syneresis, and the degree of syneresis is inversely proportional to the concentration of the agarose in the gels. [ 2 ]
In dentistry, syneresis is the expulsion of water or other liquid molecules from dental impression materials (for instance, alginate ) after an impression has been taken. Due to this process, the impression shrinks a little and therefore its size is no longer accurate. For this reason, many dental impression companies strongly recommend to pour the dental cast as soon as possible to prevent distortion of the dimension of the teeth and objects in the impression.
The opposite process of syneresis is imbibition , which is the process of a material absorbing water molecules from the surroundings. Alginate also demonstrates imbibition because it will absorb water if soaked in it.
This chemical process -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Syneresis_(chemistry) |
Synergistic catalysis is a specialized approach to catalysis whereby at least two different catalysts act on two different substrates simultaneously to allow reaction between the two activated materials. While a catalyst works to lower the energy of reaction overall, a reaction using synergistic catalysts work together to increase the energy level of HOMO of one of the molecules and lower the LUMO of another. [ 1 ] While this concept has come to be important in developing synthetic pathways, this strategy is commonly found in biological systems as well.
Synergistic catalysts have been used for a variety of reactions, especially when both substrates require some kind of significant activation either with stoichiometric amounts of an activator or through a separate reaction beforehand. Synergistic catalysts differ from other multi-catalyst systems by the nature that one catalyst activates one substrate while the other activates a different substrate. There are other types of multi-catalyst systems such as double activation catalysts where two catalysts are required to activate one substrate or cascade catalysts where one catalyst first transforms a substrate which then is activated by a second catalyst to react. [ 2 ] [ 3 ] [ 4 ]
While this field does show particular promise in affording molecules that could not be synthesized under normal synthetic strategies, there are a few issues that need to be addressed. One such issue is self quenching of the catalysts with each other. An example is if one of the catalysts is a Lewis acid and the other is a Lewis base, there is the possibility for formation of a Lewis acid base complex but this can be overcome by carefully choosing the pair. [ 5 ]
Synergistic catalysts are very common in biological systems. [ 6 ] The reactions occur by a molecule binding to a protein as a substrate and becoming active and being reacted with a coenzyme such as NADPH which is essentially an activated hydride. A specific example of this is shown by the synthesis of tetrahydrofolate via the enzyme dihydrofolate reductase . Dihydrofolate reductase catalytically activates dihydrofolate by protonating the imine, while NADPH , essentially a hydride source activated by the cofactor NADP + , can then come in and add a hydride across the imine to afford the product. [ 7 ]
Through the combination of two transition metal catalysts, synergistic catalysis has been reported to accelerate many chemical transformations, and even to induce high enantioselectivity , which could not be realized by the use individual catalysts. Sawamura et al. reported an early example of enantioselective allylic alkylation of nitriles catalyzed by a mixture of rhodium and palladium complexes. [ 8 ] The palladium catalyst with chiral ligands alone gave a high yield, but no enantioselectivity was observed. The reaction did not proceed at all using the rhodium catalyst alone. Using both together, however, gave both a high yield and enantioselectivity for the transformation.
They used trans-chelating chiral phosphine ligands (AnisTRAP) to generate chiral transition metal complexes. In their proposed mechanism schemes, an enolate is formed from an α-cyano ester and coordinates to the rhodium catalyst, while decarboxylative and oxidative addition of allyl carbonate to the palladium catalyst forms the π-allylpalladium (II) complex. Subsequently, the enolate attacks the π-allylpalladium (II) complex enantioselectively to afford the optically active product.
Besides using two transition metal catalysts, synergistic catalysis can also be carried out by utilizing one transition metal catalyst in combination with an organocatalyst . Here the synergistic α-allylation of aldehydes was accomplished by utilizing a transition metal complex in combination with a chiral amine catalyst. [ 9 ] [ 10 ] In 2013, Carreira and co-workers reported a highly enantio- and diastereoselective α-allylation of branched aldehydes. [ 11 ] They used chiral primary amines and iridium catalysts complexed with chiral ligands to afford the product with two newly formed stereocenters at the α and β position.
By matching the two chiral amines and enantiomers of the chiral ligands, they were able to access all four possible stereoisomers of the product with good yields. More importantly, their catalytic system exhibits simultaneous and almost absolute control over the stereochemical configurations of both stereocenters. | https://en.wikipedia.org/wiki/Synergistic_catalysis |
Synergy is an interaction or cooperation giving rise to a whole that is greater than the simple sum of its parts (i.e., a non-linear addition of force, energy, or effect). [ 1 ] The term synergy comes from the Attic Greek word συνεργία synergia [ 2 ] from synergos , συνεργός , meaning "working together". Synergy is similar in concept to emergence .
The words synergy and synergetic have been used in the field of physiology since at least the middle of the 19th century:
SYN'ERGY, Synergi'a , Synenergi'a , (F.) Synergie ; from συν , 'with', and εργον , 'work'. A correlation or concourse of action between different organs in health; and, according to some, in disease.
In 1896, Henri Mazel applied the term "synergy" to social psychology by writing La synergie sociale , in which he argued that Darwinian theory failed to account of "social synergy" or "social love", a collective evolutionary drive. The highest civilizations were the work not only of the elite but of the masses too; those masses must be led, however, because the crowd, a feminine and unconscious force, cannot distinguish between good and evil. [ 3 ]
In 1909, Lester Frank Ward defined synergy as the universal constructive principle of nature:
I have characterized the social struggle as centrifugal and social solidarity as centripetal. Either alone is productive of evil consequences. Struggle is essentially destructive of the social order, while communism removes individual initiative. The one leads to disorder, the other to degeneracy. What is not seen—the truth that has no expounders—is that the wholesome, constructive movement consists in the properly ordered combination and interaction of both these principles. This is social synergy , which is a form of cosmic synergy, the universal constructive principle of nature.
In Christian theology , synergism is the idea that salvation involves some form of cooperation between divine grace and human freedom.
A modern view of synergy in natural sciences derives from the relationship between energy and information . Synergy is manifested when the system makes the transition between the different information (i.e. order, complexity ) embedded in both systems. [ 4 ]
Abraham Maslow and John Honigmann drew attention to an important development in the cultural anthropology field which arose in lectures by Ruth Benedict from 1941, for which the original manuscripts have been lost but the ideas preserved in "Synergy: Some Notes of Ruth Benedict" (1969).
In the natural world, synergistic phenomena are ubiquitous, ranging from physics (for example, the different combinations of quarks that produce protons and neutrons ) to chemistry (a popular example is water, a compound of hydrogen and oxygen), to the cooperative interactions among the genes in genomes , the division of labor in bacterial colonies , the synergies of scale in multicellular organisms , as well as the many different kinds of synergies produced by socially-organized groups, from honeybee colonies to wolf packs and human societies: compare stigmergy , a mechanism of indirect coordination between agents or actions that results in the self-assembly of complex systems . Even the tools and technologies that are widespread in the natural world represent important sources of synergistic effects. The tools that enabled early hominins to become systematic big-game hunters is a primordial human example. [ 5 ] [ 6 ]
In the context of organizational behavior , following the view that a cohesive group is more than the sum of its parts, synergy is the ability of a group to outperform even its best individual member. These conclusions are derived from the studies conducted by Jay Hall on a number of laboratory-based group ranking and prediction tasks. He found that effective groups actively looked for the points in which they disagreed and in consequence encouraged conflicts amongst the participants in the early stages of the discussion. In contrast, the ineffective groups felt a need to establish a common view quickly, used simple decision making methods such as averaging, and focused on completing the task rather than on finding solutions they could agree on. [ 7 ] : 276 In a technical context, its meaning is a construct or collection of different elements working together to produce results not obtainable by any of the elements alone. The elements, or parts, can include people, hardware, software, facilities, policies, documents: all things required to produce system-level results. The value added by the system as a whole, beyond that contributed independently by the parts, is created primarily by the relationship among the parts, that is, how they are interconnected. In essence, a system constitutes a set of interrelated components working together with a common objective: fulfilling some designated need. [ 8 ]
If used in a business application, synergy means that teamwork will produce an overall better result than if each person within the group were working toward the same goal individually. However, the concept of group cohesion needs to be considered. Group cohesion is that property that is inferred from the number and strength of mutual positive attitudes among members of the group. As the group becomes more cohesive, its functioning is affected in a number of ways. First, the interactions and communication between members increase. Common goals, interests and small size all contribute to this. In addition, group member satisfaction increases as the group provides friendship and support against outside threats. [ 7 ] : 275
There are negative aspects of group cohesion that have an effect on group decision-making and hence on group effectiveness. There are two issues arising. The risky shift phenomenon is the tendency of a group to make decisions that are riskier than those that the group would have recommended individually. Group Polarisation is when individuals in a group begin by taking a moderate stance on an issue regarding a common value and, after having discussed it, end up taking a more extreme stance. [ 7 ] : 280
A second, potential negative consequence of group cohesion is group think. Group think is a mode of thinking that people engage in when they are deeply involved in cohesive group, when the members' striving for unanimity overrides their motivation to appraise realistically the alternative courses of action. Studying the events of several American policy "disasters" such as the failure to anticipate the Japanese attack on Pearl Harbor (1941) and the Bay of Pigs Invasion fiasco (1961), Irving Janis argued that they were due to the cohesive nature of the committees that made the relevant decisions. [ 7 ] : 283
That decisions made by committees lead to failure in a simple system is noted by Dr. Chris Elliot. His case study looked at IEEE-488 , an international standard set by the leading US standards body; it led to a failure of small automation systems using the IEEE-488 standard (which codified a proprietary communications standard HP-IB ). But the external devices used for communication were made by two different companies, and the incompatibility between the external devices led to a financial loss for the company. He argues that systems will be safe only if they are designed, not if they emerge by chance. [ 9 ]
The idea of a systemic approach is endorsed by the United Kingdom Health and Safety Executive . The successful performance of the health and safety management depends upon the analyzing the causes of incidents and accidents and learning correct lessons from them. The idea is that all events (not just those causing injuries) represent failures in control, and present an opportunity for learning and improvement. [ 10 ] UK Health and Safety Executive, Successful health and safety management (1997): this book describes the principles and management practices, which provide the basis of effective health and safety management. It sets out the issues that need to be addressed, and can be used for developing improvement programs, self-audit, or self-assessment. Its message is that organizations must manage health and safety with the same degree of expertise and to the same standards as other core business activities, if they are to effectively control risks and prevent harm to people.
The term synergy was refined by R. Buckminster Fuller , who analyzed some of its implications more fully [ 11 ] and coined the term synergetics . [ 11 ]
Mathematical formalizations of synergy have been proposed using information theory to rigorously define the relationships between "wholes" and "parts". [ 12 ] In this context, synergy is said to occur when there is information present in the joint state of multiple variables that cannot be extracted from the individual parts considered individually. For example, consider the logical XOR gate . If Y = X O R ( X 1 , X 2 ) {\displaystyle Y=XOR(X_{1},X_{2})} for three binary variables, the mutual information between any individual source and the target is 0 bit. However, the joint mutual information I ( X 1 , X 2 ; Y ) = 1 {\displaystyle I(X_{1},X_{2};Y)=1} bit. There is information about the target that can only be extracted from the joint state of the inputs considered jointly, and not any others.
There is, thus far, no universal agreement on how synergy can best be quantified, with different approaches that decompose information into redundant, unique, and synergistic components appearing in the literature. [ 13 ] [ 14 ] [ 15 ] [ 16 ] Despite the lack of universal agreement, information-theoretic approaches to statistical synergy have been applied to diverse fields, including climatology, [ 17 ] neuroscience [ 18 ] [ 19 ] [ 20 ] sociology, [ 21 ] and machine learning [ 22 ] Synergy has also been proposed as a possible foundation on which to build a mathematically robust definition of emergence in complex systems [ 23 ] [ 24 ] and may be relevant to formal theories of consciousness. [ 25 ]
Synergy of various kinds has been advanced by Peter Corning as a causal agency that can explain the progressive evolution of complexity in living systems over the course of time. According to the Synergism Hypothesis, synergistic effects have been the drivers of cooperative relationships of all kinds and at all levels in living systems. The thesis, in a nutshell, is that synergistic effects have often provided functional advantages (economic benefits) in relation to survival and reproduction that have been favored by natural selection. The cooperating parts, elements, or individuals become, in effect, functional "units" of selection in evolutionary change. [ 26 ] [ 27 ] [ 28 ] Similarly, environmental systems may react in a non-linear way to perturbations, such as climate change, so that the outcome may be greater than the sum of the individual component alterations. Synergistic responses are a complicating factor in environmental modeling. [ 29 ]
Pest synergy would occur in a biological host organism population, where, for example, the introduction of parasite A may cause 10% fatalities, and parasite B may also cause 10% loss. When both parasites are present, the losses would normally be expected to total less than 20%, yet, in some cases, losses are significantly greater. In such cases, it is said that the parasites in combination have a synergistic effect.
Mechanisms that may be involved in the development of synergistic effects include:
More mechanisms are described in an exhaustive 2009 review. [ 32 ]
Toxicological synergy is of concern to the public and regulatory agencies because chemicals individually considered safe might pose unacceptable health or ecological risk in combination. Articles in scientific and lay journals include many definitions of chemical or toxicological synergy, often vague or in conflict with each other. Because toxic interactions are defined relative to the expectation under "no interaction", a determination of synergy (or antagonism) depends on what is meant by "no interaction". [ 34 ] The United States Environmental Protection Agency has one of the more detailed and precise definitions of toxic interaction, designed to facilitate risk assessment. [ 35 ] In their guidance documents, the no-interaction default assumption is dose addition, so synergy means a mixture response that exceeds that predicted from dose addition. The EPA emphasizes that synergy does not always make a mixture dangerous, nor does antagonism always make the mixture safe; each depends on the predicted risk under dose addition.
For example, a consequence of pesticide use is the risk of health effects. During the registration of pesticides in the United States exhaustive tests are performed to discern health effects on humans at various exposure levels. A regulatory upper limit of presence in foods is then placed on this pesticide. As long as residues in the food stay below this regulatory level, health effects are deemed highly unlikely and the food is considered safe to consume.
However, in normal agricultural practice, it is rare to use only a single pesticide. During the production of a crop, several different materials may be used. Each of them has had determined a regulatory level at which they would be considered individually safe. In many cases, a commercial pesticide is itself a combination of several chemical agents, and thus the safe levels actually represent levels of the mixture. In contrast, a combination created by the end user, such as a farmer, has rarely been tested in that combination. The potential for synergy is then unknown or estimated from data on similar combinations. This lack of information also applies to many of the chemical combinations to which humans are exposed, including residues in food, indoor air contaminants, and occupational exposures to chemicals. Some groups think that the rising rates of cancer, asthma, and other health problems may be caused by these combination exposures; others have alternative explanations. This question will likely be answered only after years of exposure by the population in general and research on chemical toxicity, usually performed on animals. Examples of pesticide synergists include Piperonyl butoxide and MGK 264 . [ 36 ]
Synergy exists in individual and social interactions among humans, with some arguing that social cooperation requires synergy to continue. [ 37 ] One way of quantifying synergy in human social groups is via energy use, where larger groups of humans (i.e., cities) use energy more efficiently that smaller groups of humans. [ 1 ]
Human synergy can also occur on a smaller scale, like when individuals huddle together for warmth or in workplaces where labor specialization increase efficiencies. [ 38 ]
When synergy occurs in the work place, the individuals involved get to work in a positive and supportive working environment. When individuals get to work in environments such as these, the company reaps the benefits. The authors of Creating the Best Workplace on Earth Rob Goffee and Gareth Jones, state that "highly engaged employees are, on average, 50% more likely to exceed expectations that the least-engaged workers. And companies with highly engaged people outperform firms with the most disengaged folks- by 54% in employee retention , by 89% in customer satisfaction , and by fourfold in revenue growth. [ 39 ] : 100 Also, those that are able to be open about their views on the company, and have confidence that they will be heard, are likely to be a more organized employee who helps his/ her fellow team members succeed. [ 39 ]
Human interaction with technology can also increase synergy. Organismic computing is an approach to improving group efficacy by increasing synergy in human groups via technological means.
In Christian theology , synergism is the belief that salvation involves a cooperation between divine grace and human freedom. [ 40 ] Eastern Orthodox theology, in particular, uses the term "synergy" to describe this relationship, drawing on biblical language: "in Paul's words, 'We are fellow-workers ( synergoi ) with God' (1 Corinthians iii, 9)". [ 41 ]
Corporate synergy occurs when corporations interact congruently. A corporate synergy refers to a financial benefit that a corporation expects to realize when it merges with or acquires another corporation. This type of synergy is a nearly ubiquitous feature of a corporate acquisition and is a negotiating point between the buyer and seller that impacts the final price both parties agree to. There are distinct types of corporate synergies, as follows.
A marketing synergy refers to the use of information campaigns , studies, and scientific discovery or experimentation for research and development . This promotes the sale of products for varied use or off-market sales as well as development of marketing tools and in several cases exaggeration of effects. It is also often a meaningless buzzword used by corporate leaders. [ 42 ] [ 43 ]
A revenue synergy refers to the opportunity of a combined corporate entity to generate more revenue than its two predecessor stand-alone companies would be able to generate. For example, if company A sells product X through its sales force, company B sells product Y, and company A decides to buy company B, then the new company could use each salesperson to sell products X and Y, thereby increasing the revenue that each salesperson generates for the company.
In media revenue , synergy is the promotion and sale of a product throughout the various subsidiaries of a media conglomerate , e.g. films, soundtracks, or video games.
Financial synergy gained by the combined firm is a result of number of benefits which flow to the entity as a consequence of acquisition and merger. These benefits may be:
This is when a firm having a number of cash extensive projects acquires a firm which is cash-rich, thus enabling the new combined firm to enjoy the profits from investing the cash of one firm in the projects of the other.
If two firms have no or little capacity to carry debt before individually, it is possible for them to join and gain the capacity to carry the debt through decreased gearing (leverage). This creates value for the firm, as debt is thought to be a cheaper source of finance.
It is possible for one firm to have unused tax benefits which might be offset against the profits of another after combination, thus resulting in less tax being paid. However this greatly depends on the tax law of the country.
Synergy in management and in relation to teamwork refers to the combined effort of individuals as participants of the team. [ 44 ] The condition that exists when the organization's parts interact to produce a joint effect that is greater than the sum of the parts acting alone. Positive or negative synergies can exist. In these cases, positive synergy has positive effects such as improved efficiency in operations, greater exploitation of opportunities, and improved utilization of resources. Negative synergy on the other hand has negative effects such as: reduced efficiency of operations, decrease in quality, underutilization of resources and disequilibrium with the external environment.
A cost synergy refers to the opportunity of a combined corporate entity to reduce or eliminate expenses associated with running a business. Cost synergies are realized by eliminating positions that are viewed as duplicate within the merged entity. [ 45 ] Examples include the headquarters office of one of the predecessor companies, certain executives, the human resources department, or other employees of the predecessor companies. This is related to the economic concept of economies of scale .
The synergistic action of the economic players lies within the economic phenomenon's profundity. The synergistic action gives different dimensions to competitiveness, strategy and network identity becoming an unconventional "weapon" which belongs to those who exploit the economic systems' potential in depth. [ 46 ] : 3–4
The synergistic gravity equation (SYNGEq), according to its complex "title", represents a synthesis of the endogenous and exogenous factors which determine the private and non-private economic decision makers to call to actions of synergistic exploitation of the economic network in which they operate. That is to say, SYNGEq constitutes a big picture of the factors/motivations which determine the entrepreneurs to contour an active synergistic network. SYNGEq includes both factors which character is changing over time (such as the competitive conditions), as well as classics factors, such as the imperative of the access to resources of the collaboration and the quick answers. The synergistic gravity equation (SINGEq) comes to be represented by the formula: [ 46 ] : 33, 37
where:
The synergistic network represents an integrated part of the economic system which, through the coordination and control functions (of the undertaken economic actions), agrees synergies. The networks which promote synergistic actions can be divided in horizontal synergistic networks and vertical synergistic networks. [ 46 ] : 6–7
The synergy effects are difficult (even impossible) to imitate by competitors and difficult to reproduce by their authors because these effects depend on the combination of factors with time-varying characteristics. The synergy effects are often called "synergistic benefits", representing the direct and implied result of the developed/adopted synergistic actions. [ 46 ] : 6
Synergy can also be defined as the combination of human strengths and computer strengths, such as advanced chess . Computers can process data much more quickly than humans, but lack the ability to respond meaningfully to arbitrary stimuli.
Etymologically, the "synergy" term was first used around 1600, deriving from the Greek word "synergos", which means "to work together" or "to cooperate". If during this period the synergy concept was mainly used in the theological field (describing "the cooperation of human effort with divine will"), in the 19th and 20th centuries, "synergy" was promoted in physics and biochemistry, being implemented in the study of the open economic systems only in the 1960 and 1970s. [ 46 ] : 5
In 1938, J. R. R. Tolkien wrote an essay titled On Fairy Stores , delivered at an Andrew Lang Lecture, and reprinted in his book, The Tolkien Reader , published in 1966. In it, he made two references to synergy, although he did not use that term. He wrote:
Faerie cannot be caught in a net of words; for it is one of its qualities to be indescribable, though not imperceptible. It has many ingredients, but analysis will not necessarily discover the secret of the whole.
And more succinctly, in a footnote, about the "part of producing the web of an intricate story", he wrote:
It is indeed easier to unravel a single thread — an incident, a name, a motive — than to trace the history of any picture defined by many threads. For with the picture in the tapestry a new element has come in: the picture is greater than, and not explained by, the sum of the component threads.
The informational synergies which can be applied also in media involve a compression of transmission, access and use of information's time, the flows, circuits and means of handling information being based on a complementary, integrated, transparent and coordinated use of knowledge. [ 46 ] : 9
In media economics, synergy is the promotion and sale of a product (and all its versions) throughout the various subsidiaries of a media conglomerate, [ 47 ] e.g. films, soundtracks or video games. Walt Disney pioneered synergistic marketing techniques in the 1930s by granting dozens of firms the right to use his Mickey Mouse character in products and ads, and continued to market Disney media through licensing arrangements. These products can help advertise the film itself and thus help to increase the film's sales. For example, the Spider-Man films had toys of webshooters and figures of the characters made, as well as posters and games. [ 48 ] The NBC sitcom 30 Rock often shows the power of synergy, while also poking fun at the use of the term in the corporate world. [ 49 ] There are also different forms of synergy in popular card games like Magic: The Gathering , Yu-Gi-Oh! , Cardfight!! Vanguard , and Future Card Buddyfight . | https://en.wikipedia.org/wiki/Synergy |
Synexpression is a type of non-random eukaryotic gene organization. Genes in a synexpression group may not be physically linked, but they are involved in the same process and they are coordinately expressed. It is expected that genes that function in the same process be regulated coordinately. Synexpression groups in particular represent genes that are simultaneously up- or down-regulated, often because their gene products are required in stoichiometric amounts or are protein -complex subunits. [ 1 ] It is likely that these gene groups share common cis - and trans -acting control elements to achieve coordinate expression.
Synexpression groups are determined mainly by analysis of expression profiles compiled by the use of DNA microarrays . [ 1 ] The use of this technology helps researchers monitor changes in expression patterns for large numbers of genes in a given experiment. Analysis of DNA microarray expression profiles has led to the discovery of a number of genes that are tightly co-regulated. [ 1 ]
The identification of synexpression groups has affected the way some scientists view evolutionary change in higher eukaryotes. [ 1 ] Since groups of genes involved in the same biological process often share one or more common control elements, it has been suggested that the differential expression of these synexpression groups in different tissues of organisms can contribute to co-evolution tissues, organs, and appendages. [ 1 ] Today it is commonly believed that it is not primarily the gene products themselves that evolve, but that it is the control networks for groups of genes that contribute most to the evolution of higher eukaryotes. [ 1 ]
Developmental processes provide an example of how changes in synexpression control networks could significantly affect an organism's capacity to evolve and adapt effectively. In animals, it is often beneficial for appendages to co-evolve, and it has been observed that fore-and hind-limbs share expression of Hox genes early in metazoan development. [ 1 ] Thus, changes in the regulatory patterns of these genes would affect the development of both the fore- and hind-limbs, facilitating co-evolution.
One simplified example of a synexpression group is the genes cdc6, cdc3, cdc46, and swi4 in yeast , which are all co-expressed early in the G-1 stage of the cell cycle., [ 1 ] [ 2 ] These genes share one common cis -regulatory element, called ECB, which serves as a binding site for the MCM1 trans -acting protein. Although these genes are not spatially clustered, co-regulation seems to be achieved via this common cis and trans control mechanism. Most synexpression groups are more complicated than the ECB group in yeast, involving myriad cis and trans control elements. [ 1 ] [ 2 ] | https://en.wikipedia.org/wiki/Synexpression |
Syngameon refers to groups of taxa that frequently engage in natural hybridization and lack strong reproductive barriers that prevent interbreeding. [ 1 ] [ 2 ] Syngameons are more common in plants than animals, with approximately 25% of plant species and 10% of animal species producing natural hybrids. [ 3 ] The most well known syngameons include irises of the California Pacific Coast and white oaks of the Eastern United States. [ 2 ] [ 4 ] Hybridization within a syngameon is typically not equally distributed among species and few species often dominate patterns of hybridization. [ 3 ]
The term syngameon comes from the root word syngamy coined by Edward Bagnall Poulton to define groups that freely interbreed . [ 5 ] He also coined the word asyngamy referring to groups that do not freely interbreed (with the substantive noun forms Syngamy and Asyngamy ). [ 5 ] The term syngameon was first used by Johannes Paulus Lotsy , who used it to describe a habitually interbreeding community that was reproductively isolated from other habitually interbreeding communities. [ 6 ] Syngameon was used interchangeably with the term species to describe groups of closely related individuals that interbreed to varying degrees. [ 5 ] A more specific definition of syngameon has been given to groups of taxa that frequently engage in natural hybridization and lack strong morphological differences that could be used to define them. [ 1 ] [ 2 ] Taxa in syngameons may have separate species names, but evolutionary biologists often suggest they should be treated as a single species. [ 1 ] Variation among species within a syngameon can be due to a number of factors related to their biogeography , ecology , phylogeny , reproductive biology , and genetics . [ 3 ]
The terms coenospecies and syngameons are both used to describe clusters of lineages that are morphologically distinct and lack strong isolation mechanisms. [ 1 ] Coenospecies, first coined by Göte Turesson in 1922, [ 7 ] refers to the total sum of possible combinations in a genotype compound, which includes hybridization that occurs both naturally and artificially. [ 7 ] Coenospecies is often used to describe lineages that can be crossed under cultivation and only a few species pairs are found to form natural hybrids, whereas syngameons refer to species where extensive evidence of natural hybridization occurs. [ 8 ] [ 2 ] In this sense, definitions of syngameon and coenospecies correspond to the two different definitions of the Biological Species Concept proposed by Ernst Mayr; syngameon is consistent with “actually” interbreeding species, while coenospecies is consistent with “actually or potentially” interbreeding species. [ 1 ] The term ecospecies is considered a subdivision of coenospecies that refers to the genotypes within a coenospecies that hybridize and produce viable, fertile offspring. [ 7 ] | https://en.wikipedia.org/wiki/Syngameon |
Syngas , or synthesis gas , is a mixture of hydrogen and carbon monoxide , [ 1 ] in various ratios. The gas often contains some carbon dioxide and methane . It is principally used for producing ammonia or methanol . Syngas is combustible and can be used as a fuel. [ 2 ] [ 3 ] [ 4 ] Historically, it has been used as a replacement for gasoline , when gasoline supply has been limited; for example, wood gas was used to power cars in Europe during WWII (in Germany alone, half a million cars were built or rebuilt to run on wood gas). [ 5 ]
Syngas is produced by steam reforming or partial oxidation of natural gas or liquid hydrocarbons, or coal gasification . [ 6 ]
Steam reforming of methane is an endothermic reaction requiring 206 kJ/mol of energy:
In principle, but rarely in practice, biomass and related hydrocarbon feedstocks could be used to generate biogas and biochar in waste-to-energy gasification facilities. [ 7 ] The gas generated (mostly methane and carbon dioxide) is sometimes described as syngas but its composition differs from syngas. Generation of conventional syngas (mostly H 2 and CO) from waste biomass has been explored. [ 8 ] [ 9 ]
The chemical composition of syngas varies based on the raw materials and the processes. Syngas produced by coal gasification generally is a mixture of 30 to 60% carbon monoxide, 25 to 30% hydrogen, 5 to 15% carbon dioxide, and 0 to 5% methane. It also contains lesser amount of other gases. [ 10 ] Syngas has less than half the energy density of natural gas . [ 11 ]
The first reaction, between incandescent coke and steam, is strongly endothermic, producing carbon monoxide (CO) and hydrogen H 2 ( water gas in older terminology). When the coke bed has cooled to a temperature at which the endothermic reaction can no longer proceed, the steam is then replaced by a blast of air.
The second and third reactions then take place, producing an exothermic reaction —forming initially carbon dioxide and raising the temperature of the coke bed—followed by the second endothermic reaction, in which the latter is converted to carbon monoxide. The overall reaction is exothermic, forming "producer gas" (older terminology). Steam can then be re-injected, then air etc., to give an endless series of cycles until the coke is finally consumed. Producer gas has a much lower energy value, relative to water gas, due primarily to dilution with atmospheric nitrogen. Pure oxygen can be substituted for air to avoid the dilution effect, producing gas of much higher calorific value .
In order to produce more hydrogen from this mixture, more steam is added and the water gas shift reaction is carried out:
The hydrogen can be separated from the CO 2 by pressure swing adsorption (PSA), amine scrubbing , and membrane reactors . A variety of alternative technologies have been investigated, but none are of commercial value. [ 12 ] Some variations focus on new stoichiometries such as carbon dioxide plus methane [ 13 ] [ 14 ] or partial hydrogenation of carbon dioxide. Other research focuses on novel energy sources to drive the processes including electrolysis, solar energy, microwaves, and electric arcs. [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ]
Electricity generated from renewable sources is also used to process carbon dioxide and water into syngas through high-temperature electrolysis . This is an attempt to maintain carbon neutrality in the generation process. Audi , in partnership with company named Sunfire, opened a pilot plant in November 2014 to generate e-diesel using this process. [ 21 ]
Syngas that is not methanized typically has a lower heating value of 120 BTU/ scf . [ 22 ] Untreated syngas can be run in hybrid turbines that allow for greater efficiency because of their lower operating temperatures, and extended part lifetime. [ 22 ]
Syngas is used as a source of hydrogen as well as a fuel. [ 12 ] It is also used to directly reduce iron ore to sponge iron . [ 23 ] Chemical uses include the production of methanol which is a precursor to acetic acid and many acetates; liquid fuels and lubricants via the Fischer–Tropsch process and previously the Mobil methanol to gasoline process; ammonia via the Haber process , which converts atmospheric nitrogen (N 2 ) into ammonia which is used as a fertilizer ; and oxo alcohols via an intermediate aldehyde. [ citation needed ] | https://en.wikipedia.org/wiki/Syngas |
Syngas fermentation , also known as synthesis gas fermentation , is a microbial process. In this process, a mixture of hydrogen , carbon monoxide , and carbon dioxide , known as syngas , is used as carbon and energy sources, and then converted into fuel and chemicals by microorganisms . [ 1 ]
The main products of syngas fermentation include ethanol , butanol , acetic acid , butyric acid , and methane . [ 2 ] Certain industrial processes, such as petroleum refining, steel milling, and methods for producing carbon black , coke , ammonia , and methanol , discharge enormous amounts of waste gases containing mainly CO and H 2 into the atmosphere either directly or through combustion. Biocatalysts can be exploited to convert these waste gases to chemicals and fuels as, for example, ethanol. [ 3 ] In addition, incorporating nanoparticles has been demonstrated to improve gas-liquid fluid transfer during syngas fermentation. [ 4 ]
There are several microorganisms which can produce fuels and chemicals by syngas utilization. These microorganisms are mostly known as acetogens including Clostridium ljungdahlii , [ 5 ] Clostridium autoethanogenum , [ 6 ] Eubacterium limosum , [ 7 ] Clostridium carboxidivorans P7, [ 8 ] Peptostreptococcus productus , [ 9 ] and Butyribacterium methylotrophicum . [ 10 ] Most use the Wood–Ljungdahl pathway .
Syngas fermentation process has advantages over a chemical process since it takes places at lower temperature and pressure , has higher reaction specificity , tolerates higher amounts of sulfur compounds, and does not require a specific ratio of CO to H 2 . [ 2 ] On the other hand, syngas fermentation has limitations such as:
The most common utilized reactor type for syngas fermentation is the stirred-tank reactor in which the mass transfer is influenced by several factors such as geometry of the reactor, impeller configuration, the agitation speed and the gas flow rate. Additionally, less investigated reactor types like Trickle-bed reactors, bubble-column reactors and gas-lift reactors have specific drawbacks and advantages regarding the abovementioned limitations. [ 11 ] | https://en.wikipedia.org/wiki/Syngas_fermentation |
Syngas to gasoline plus (STG+) is a thermochemical process to convert natural gas , other gaseous hydrocarbons or gasified biomass into drop-in fuels, such as gasoline, diesel fuel or jet fuel, and organic solvents.
This process follows four principal steps in one continuous integrated loop, comprising four fixed bed reactors in a series in which a syngas is converted to synthetic fuels. The steps for producing high-octane synthetic gasoline are as follows: [ 1 ]
The STG+ process uses standard catalysts similar to those used in other gas to liquids technologies, specifically in methanol to gasoline processes. Methanol to gasoline processes favor molecular size- and shape-selective zeolite catalysts, [ 2 ] and the STG+ process also utilizes commercially available shape-selective catalysts, such as ZSM-5 . [ 3 ]
According to Primus Green Energy, the STG+ process converts natural gas into 90+-octane gasoline at approximately 5 US gallons per million British thermal units (65 litres per megawatt-hour). [ 4 ] The energy content of gasoline is 120,000 to 125,000 British thermal units per US gallon (9.3 to 9.7 kilowatt-hours per litre), making this process about 60% efficient, with a 40% loss of energy.
As is the case with other gas to liquids processes, STG+ utilizes syngas produced via other technologies as a feedstock. This syngas can be produced through several commercially available technologies and from a wide variety of feedstocks, including natural gas, biomass and municipal solid waste .
Natural gas and other methane-rich gases, including those produced from municipal waste, are converted into syngas through methane reforming technologies such as steam methane reforming and auto-thermal reforming .
Biomass gasification technologies are less established, though several systems being developed utilize fixed bed or fluidized bed reactors. [ 5 ]
Other technologies for syngas to liquid fuels synthesis include the Fischer–Tropsch process and the methanol to gasoline processes.
Research conducted at Princeton University indicates that methanol to gasoline processes are consistently more cost-effective, both in capital cost and overall cost, than the Fischer–Tropsch process at small, medium and large scales. [ 6 ] Preliminary studies suggest that the STG+ process is more energetically efficient and the highest yielding methanol to gasoline process. [ 7 ]
The primary difference between the Fischer–Tropsch process and methanol to gasoline processes such as STG+ are the catalysts used, product types and economics.
Generally, the Fischer–Tropsch process favors unselective cobalt and iron catalysts, while methanol to gasoline technologies favor molecular size- and shape-selective zeolites. [ 8 ] In terms of product types, Fischer–Tropsch production has been limited to linear paraffins , [ 8 ] such as synthetic crude oil, whereas methanol to gasoline processes can produce aromatics, such as xylene and toluene , and naphthenes and iso-paraffins, such as drop-in gasoline and jet fuel.
The main product of the Fischer–Tropsch process, synthetic crude oil, requires additional refining to produce fuel products such as diesel fuel or gasoline. This refining typically adds additional costs, causing some industry leaders to label the economics of commercial-scale Fischer–Tropsch processes as challenging. [ 9 ]
The STG+ technology offers several differentiators that distinguish it from other methanol to gasoline processes. These differences include product flexibility, durene reduction, environmental footprint and capital cost.
Traditional methanol to gasoline technologies produce diesel, gasoline or liquefied petroleum gas . [ 10 ] STG+ produces gasoline, diesel, jet fuel and aromatics, depending on the catalysts used. The STG+ technology also incorporates durene reduction into its core process, meaning that the entire fuel production process requires only two steps: syngas production and gas to liquids synthesis. [ 1 ] Other methanol to gasoline processes do not incorporate durene reduction into the core process, and they require the implementation of an additional refining step. [ 10 ]
Due to the additional number of reactors, traditional methanol to gasoline processes include inefficiencies such as the additional cost and energy loss of condensing and evaporating the methanol prior to feeding it to the durene reduction unit. [ 11 ] These inefficiencies can lead to a greater capital cost and environmental footprint than methanol to gasoline processes that use fewer reactors, such as STG+. The STG+ process eliminates multiple condensation and evaporation, and the process converts syngas to liquid transportation fuels directly without producing intermediate liquids. [ 7 ] This eliminates the need for storage of two products, including pressure storage for liquefied petroleum gas and storage of liquid methanol.
Simplifying a gas to liquids process by combining multiple steps into fewer reactors leads to increased yield and efficiency, enabling less expensive facilities that are more easily scaled. [ 12 ]
The STG+ technology is currently operating at pre-commercial scale in Hillsborough, New Jersey at a plant owned by alternative fuels company Primus Green Energy. The plant produces approximately 100,000 gallons of high-quality, drop-in gasoline per year directly from natural gas. [ 13 ] Further, the company announced the findings of an independent engineer’s report prepared by E3 Consulting, which found that STG+ system and catalyst performance exceeded expectations during plant operation. The pre-commercial demonstration plant has also achieved 720 hours of continuous operation. [ 14 ]
Primus Green Energy has announced plans to break ground on its first commercial STG+ plant in the second half of 2014, and the company has announced that this plant is expected to produce approximately 27.8 million gallons of fuel annually. [ 15 ]
In early 2014, the U.S. Patent and Trademark Office (USPTO) allowed Primus Green Energy’s patent covering its single-loop STG+ technology. [ 15 ] | https://en.wikipedia.org/wiki/Syngas_to_gasoline_plus |
Synizesis refers to a phenomenon sometimes observed in one of the subphases of meiosis . This phenomenon, sometimes referred to as a "synizetic knot", and contrasted with the chromosome "bouquet" more typically observed, is characterized by the localization of the meiotic chromosomes in a tight clump on one side of the nucleus. The term synizesis seems to have been coined by Clarence Erwin McClung in 1905. [ 1 ]
The synizetic knot (Synizesis) was later found to be a technical artifact induced by the feature of strong acidic fixatives used during that time (e.g., Flemming 's strong fixative) to precipitate the thread-like delicate chromosomes of the Leptotene stage of first meiotic prophase into a dark staining knot. [ 2 ] [ 3 ]
This cell cycle article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synizesis_(biology) |
A synodic day (or synodic rotation period or solar day ) is the period for a celestial object to rotate once in relation to the star it is orbiting , and is the basis of solar time .
The synodic day is distinguished from the sidereal day , which is one complete rotation in relation to distant stars [ 1 ] and is the basis of sidereal time.
In the case of a tidally locked planet, the same side always faces its parent star, and its synodic day is infinite. Its sidereal day, however, is equal to its orbital period.
Earth 's synodic day is the time it takes for the Sun to pass over the same meridian (a line of longitude ) on consecutive days, whereas a sidereal day is the time it takes for a given distant star to pass over a meridian on consecutive days. [ 2 ] For example, in the Northern Hemisphere , a synodic day could be measured as the time taken for the Sun to move from exactly true south (i.e. its highest declination ) on one day to exactly south again on the next day (or exactly true north in the Southern Hemisphere ).
For Earth, the synodic day is not constant, and changes over the course of the year due to the eccentricity of Earth's orbit around the Sun and the axial tilt of the Earth. [ 3 ] The longest and shortest synodic days' durations differ by about 51 seconds. [ 4 ] The mean length, however, is 24 hours (with fluctuations on the order of milliseconds ), and is the basis of solar time . The difference between the mean and apparent solar time is the equation of time , which can also be seen in Earth's analemma . Because of the variation in the length of the synodic day, the days with the longest and shortest period of daylight do not coincide with the solstices near the equator.
As viewed from Earth during the year, the Sun appears to slowly drift along an imaginary path coplanar with Earth's orbit , known as the ecliptic , on a spherical background of seemingly fixed stars . [ 5 ] Each synodic day, this gradual motion is a little less than 1° eastward (360° per 365.25 days), in a manner known as prograde motion .
Certain spacecraft orbits, Sun-synchronous orbits , have orbital periods that are a fraction of a synodic day. Combined with a nodal precession , this allows them to always pass over a location on Earth's surface at the same mean solar time . [ 6 ]
Due to tidal locking with Earth, the Moon 's synodic day (the lunar day or synodic rotation period) is the same as its synodic period with Earth and the Sun (the period of the lunar phases , the synodic lunar month , which is the month of the lunar calendar ).
Due to the slow retrograde rotational speed of Venus , its synodic rotation period of 117 Earth days is about half the length of its sidereal rotational period (sidereal day) and even its orbital period. [ 7 ]
Due to Mercury 's slow rotational speed and fast orbit around the Sun, its synodic rotation period of 176 Earth days is three times longer than its sidereal rotational period (sidereal day) and twice as long as its orbital period. [ 8 ] | https://en.wikipedia.org/wiki/Synodic_day |
In taxonomy , the scientific classification of living organisms, a synonym is an alternative scientific name for the accepted scientific name of a taxon . The botanical and zoological codes of nomenclature treat the concept of synonymy differently.
Unlike synonyms in other contexts, in taxonomy a synonym is not interchangeable with the name of which it is a synonym. In taxonomy, synonyms are not equals, but have a different status. For any taxon with a particular circumscription , position, and rank, only one scientific name is considered to be the correct one at any given time (this correct name is to be determined by applying the relevant code of nomenclature ). A synonym cannot exist in isolation: it is always an alternative to a different scientific name. Given that the correct name of a taxon depends on the taxonomic viewpoint used (resulting in a particular circumscription, position and rank) a name that is one taxonomist's synonym may be another taxonomist's correct name (and vice versa ).
Synonyms may arise whenever the same taxon is described and named more than once independently. They may also arise when existing taxa are changed, as when two taxa are joined to become one, a species is moved to a different genus, a variety is moved to a different species, etc. Synonyms also come about when the codes of nomenclature change, so that older names are no longer acceptable; for example, Erica herbacea L. has been rejected in favour of the conserved name of Erica carnea L. and is thus its synonym. [ 3 ]
To the general user of scientific names, in fields such as agriculture, horticulture, ecology, general science, etc., a synonym is a name that was previously used as the correct scientific name (in handbooks and similar sources) but which has been displaced by another scientific name, which is now regarded as correct. Thus Oxford Dictionaries Online defines the term as "a taxonomic name which has the same application as another, especially one which has been superseded and is no longer valid". [ 4 ] In handbooks and general texts, it is useful to have synonyms mentioned as such after the current scientific name, so as to avoid confusion. For example, if the much-advertised name change should go through and the scientific name of the fruit fly were changed to Sophophora melanogaster , it would be very helpful if any mention of this name was accompanied by "(syn. Drosophila melanogaster )". Synonyms used in this way may not always meet the strict definitions of the term "synonym" in the formal rules of nomenclature which govern scientific names (see below) .
Changes of scientific name have two causes: they may be taxonomic or nomenclatural. A name change may be caused by changes in the circumscription, position or rank of a taxon, representing a change in taxonomic, scientific insight (as would be the case for the fruit fly, mentioned above). A name change may be due to purely nomenclatural reasons, that is, based on the rules of nomenclature; [ citation needed ] as for example when an older name is (re)discovered which has priority over the current name. Speaking in general, name changes for nomenclatural reasons have become less frequent over time as the rules of nomenclature allow for names to be conserved, so as to promote stability of scientific names.
In zoological nomenclature, codified in the International Code of Zoological Nomenclature , synonyms are different scientific names of the same taxonomic rank that pertain to that same taxon . For example, a particular species could, over time, have had two or more species-rank names published for it, while the same is applicable at higher ranks such as genera, families, orders, etc. In each case, the earliest published name is called the senior synonym , while the later name is the junior synonym . In the case where two names for the same taxon have been published simultaneously, the valid name is selected according to the principle of the first reviser such that, for example, of the names Strix scandiaca and Strix noctua (Aves), both published by Linnaeus in the same work at the same date for the taxon now determined to be the snowy owl , the epithet scandiaca has been selected as the valid name, with noctua becoming the junior synonym. (Incidentally, this species has since been reclassified and currently resides in the genus Bubo , as Bubo scandiacus [ 5 ] ).
One basic principle of zoological nomenclature is that the earliest correctly published (and thus available ) name, the senior synonym, by default takes precedence in naming rights and therefore, unless other restrictions interfere, must be used for the taxon. However, junior synonyms are still important to document, because if the earliest name cannot be used (for example, because the same spelling had previously been used for a name established for another taxon), then the next available junior synonym must be used for the taxon. For other purposes, if a researcher is interested in consulting or compiling all currently known information regarding a taxon, some of this (including species descriptions, distribution, ecology and more) may well have been published under names now regarded as outdated (i.e., synonyms) and so it is again useful to know a list of historic synonyms which may have been used for a given current (valid) taxon name.
Objective synonyms refer to taxa with the same type and same rank (more or less the same taxon, although circumscription may vary, even widely). This may be species-group taxa of the same rank with the same type specimen , genus-group taxa of the same rank with the same type species or if their type species are themselves objective synonyms, of family-group taxa with the same type genus, etc. [ 6 ]
In the case of subjective synonyms , there is no such shared type, so the synonymy is open to taxonomic judgement, [ 7 ] meaning that there is room for debate: one researcher might consider the two (or more) types to refer to one and the same taxon, another might consider them to belong to different taxa. For example, John Edward Gray published the name Antilocapra anteflexa in 1855 for a species of pronghorn , based on a pair of horns. However, it is now commonly accepted that his specimen was an unusual individual of the species Antilocapra americana published by George Ord in 1815. Ord's name thus takes precedence, with Antilocapra anteflexa being a junior subjective synonym.
Objective synonyms are common at the rank of genera, because for various reasons two genera may contain the same type species; these are objective synonyms. [ 8 ] In many cases researchers established new generic names because they thought this was necessary or did not know that others had previously established another genus for the same group of species. An example is the genus Pomatia Beck, 1837, [ 9 ] which was established for a group of terrestrial snails containing as its type species the Burgundy or Roman snail Helix pomatia —since Helix pomatia was already the type species for the genus Helix Linnaeus, 1758, the genus Pomatia was an objective synonym (and useless). On the same occasion, Helix is also a synonym of Pomatia , but it is older and so it has precedence.
At the species level, subjective synonyms are common because of an unexpectedly large range of variation in a species, or simple ignorance about an earlier description, may lead a biologist to describe a newly discovered specimen as a new species. A common reason for objective synonyms at this level is the creation of a replacement name.
A junior synonym can be given precedence over a senior synonym, [ 10 ] primarily when the senior name has not been used since 1899, and the junior name is in common use. The older name may be declared to be a nomen oblitum , and the junior name declared a nomen protectum . This rule exists primarily to prevent the confusion that would result if a well-known name, with a large accompanying body of literature, were to be replaced by a completely unfamiliar name. An example is the European land snail Petasina edentula ( Draparnaud , 1805). In 2002, researchers found that an older name Helix depilata Draparnaud, 1801 referred to the same species, but this name had never been used after 1899 and was fixed as a nomen oblitum under this rule by Falkner et al. 2002. [ 11 ]
Such a reversal of precedence is also possible if the senior synonym was established after 1900, but only if the International Commission on Zoological Nomenclature (ICZN) approves an application. (Here the C in ICZN stands for Commission, not Code as it does at the beginning of § Zoology . The two are related, with only one word difference between their names.) For example, the scientific name of the red imported fire ant , Solenopsis invicta was published by Buren in 1972, who did not know that this species was first named Solenopsis saevissima wagneri by Santschi in 1916; as there were thousands of publications using the name invicta before anyone discovered the synonymy, the ICZN, in 2001, ruled that invicta would be given precedence over wagneri .
To qualify as a synonym in zoology, a name must be properly published in accordance with the rules. Manuscript names and names that were mentioned without any description ( nomina nuda ) are not considered as synonyms in zoological nomenclature.
In botanical nomenclature , a synonym is a name that is not correct for the circumscription , position, and rank of the taxon as considered in the particular botanical publication. It is always "a synonym of the correct scientific name", but which name is correct depends on the taxonomic opinion of the author. In botany the various kinds of synonyms are:
In botany, although a synonym must be a formally accepted scientific name (a validly published name): a listing of "synonyms", a "synonymy", often contains designations that for some reason did not make it as a formal name, such as manuscript names, or even misidentifications (although it is now the usual practice to list misidentifications separately [ 12 ] ).
Although the basic principles are fairly similar, the treatment of synonyms in botanical nomenclature differs in detail and terminology from zoological nomenclature, where the correct name is included among synonyms, although as first among equals it is the "senior synonym":
Scientific papers may include lists of taxa, synonymizing existing taxa and (in some cases) listing references to them.
The status of a synonym may be indicated by symbols, as for instance in a system proposed for use in paleontology by Rudolf Richter. In that system a v before the year would indicate that the authors have inspected the original material; a . that they take on the responsibility for the act of synonymizing the taxa. [ 13 ]
The accurate use of scientific names, including synonyms, is crucial in biomedical and pharmacological research involving plants. Failure to use correct botanical nomenclature can lead to ambiguity, hinder reproducibility of results, and potentially cause errors in medicine. Best practices for publication suggest that researchers should provide the currently accepted binomial with author citation, relevant synonyms, and the accepted family name according to the Angiosperm Phylogeny Group III classification. This practice ensures clear communication, allows proper linking of research to existing literature, and provides insight into phylogenetic relationships that may be relevant to shared chemical constituents or physiological effects. Online databases now make it easy for researchers to access correct nomenclature and synonymy information for plant species. [ 14 ]
The traditional concept of synonymy is often expanded in taxonomic literature to include pro parte (or "for part") synonyms. These are caused by splits and circumscriptional changes. They are usually indicated by the abbreviation "p.p." [ 15 ] For example: | https://en.wikipedia.org/wiki/Synonym_(taxonomy) |
A synonymous substitution (often called a silent substitution though they are not always silent) is the evolutionary substitution of one base for another in an exon of a gene coding for a protein , such that the produced amino acid sequence is not modified. This is possible because the genetic code is " degenerate ", meaning that some amino acids are coded for by more than one three-base-pair codon ; since some of the codons for a given amino acid differ by just one base pair from others coding for the same amino acid, a mutation that replaces the "normal" base by one of the alternatives will result in incorporation of the same amino acid into the growing polypeptide chain when the gene is translated. Synonymous substitutions and mutations affecting noncoding DNA are often considered silent mutations ; however, it is not always the case that the mutation is silent. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
Since there are 22 codes for 64 codons, roughly we should expect a random substitution to be synonymous with probability about 22/64 = 34%. The actual value is around 20%. [ 6 ]
A synonymous mutation can affect transcription , splicing , mRNA transport, and translation , any of which could alter the resulting phenotype , rendering the synonymous mutation non-silent. [ 3 ] The substrate specificity of the tRNA to the rare codon can affect the timing of translation, and in turn the co-translational folding of the protein . [ 1 ] This is reflected in the codon usage bias that is observed in many species . A nonsynonymous substitution results in a change in amino acid that may be arbitrarily further classified as conservative (a change to an amino acid with similar physiochemical properties ), semi-conservative (e.g. negatively to positively charged amino acid), or radical (vastly different amino acid).
Protein translation involves a set of twenty amino acids . Each of these amino acids is coded for by a sequence of three DNA base pairs called a codon . Because there are 64 possible codons, but only 20-22 encoded amino acids (in nature) and a stop signal (i.e. up to three codons that do not code for any amino acid and are known as stop codons , indicating that translation should stop), some amino acids are coded for by 2, 3, 4, or 6 different codons. For example, the codons TTT and TTC both code for the amino acid phenylalanine . This is often referred to as redundancy of the genetic code . There are two mechanisms for redundancy: several different transfer RNAs can deliver the same amino acid, or one tRNA can have a non-standard wobble base in position three of the anti-codon, which recognises more than one base in the codon.
In the above phenylalanine example, suppose that the base in position 3 of a TTT codon got substituted to a C, leaving the codon TTC. The amino acid at that position in the protein will remain a phenylalanine. Hence, the substitution is a synonymous one.
When a synonymous or silent mutation occurs, the change is often assumed to be neutral , meaning that it does not affect the fitness of the individual carrying the new gene to survive and reproduce.
Synonymous changes may not be neutral because certain codons are translated more efficiently (faster and/or more accurately) than others. For example, when a handful of synonymous changes in the fruit fly alcohol dehydrogenase gene were introduced, changing several codons to sub-optimal synonyms, production of the encoded enzyme was reduced [ 7 ] and the adult flies showed lower ethanol tolerance. [ 8 ] Many organisms, from bacteria through animals, display biased use of certain synonymous codons. Such codon usage bias may arise for different reasons, some selective, and some neutral. In Saccharomyces cerevisiae synonymous codon usage has been shown to influence mRNA folding stability, with mRNA encoding different protein secondary structure preferring different codons. [ 9 ]
Another reason why synonymous changes are not always neutral is the fact that exon sequences close to exon-intron borders function as RNA splicing signals. When the splicing signal is destroyed by a synonymous mutation, the exon does not appear in the final protein. This results in a truncated protein. One study found that about a quarter of synonymous variations affecting exon 12 of the cystic fibrosis transmembrane conductance regulator gene result in that exon being skipped. [ 10 ]
÷⊈⊂⊃⊅ | https://en.wikipedia.org/wiki/Synonymous_substitution |
SynqNet is an industrial automation network launched in 2001 by Danaher Corporation for meeting the performance and safety requirements of machine control applications. Synqnet is built over Ethernet link and 100BASE-TX physical layer and provides a synchronous connection between various process automation devices including motion controllers, servo drives, stepper drives and I/O modules. As of September 2008, SynqNet networks are used for controlling over 400,000 axes of motion on various motion applications worldwide. [ 1 ] SynqNet user group is formed in 2004 for enhancing the developments. [ 2 ]
This computer networking article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/SynqNet |
Synroc , a portmanteau of "synthetic rock", is a means of safely storing radioactive waste . It was pioneered in 1978 by a team led by Professor Ted Ringwood at the Australian National University , with further research undertaken in collaboration with ANSTO at research laboratories in Lucas Heights .
Synroc is composed of three titanate minerals – hollandite , zirconolite and perovskite – plus rutile and a small amount of metal alloy . These are combined into a slurry to which is added a portion of high-level liquid nuclear waste . The mixture is dried and calcined at 750 °C (1,380 °F) to produce a powder.
The powder is then compressed in a process known as hot isostatic pressing (HIP), where it is compressed within a bellows-like stainless steel container at temperatures of 1,150–1,200 °C (2,100–2,190 °F).
The result is a cylinder of hard, dense, black synthetic rock.
If stored in a liquid form, nuclear waste can enter the environment and the waterways, and cause widespread damage. As a solid, these risks are greatly minimised.
Unlike borosilicate glass , which is amorphous , Synroc is a ceramic that incorporates the radioactive waste into its crystal structure . Naturally occurring rocks can store radioactive materials for long periods. The aim of Synroc is to imitate this by converting liquid into a crystalline structure and use to store radioactive waste. Synroc-based glass composite materials (GCM) combine the process and chemical flexibility of glass with the superior chemical durability of ceramics and can achieve higher waste loadings. [ 1 ] [ 2 ]
Different types of Synroc waste forms (ratios of component minerals, specific HIP pressures and temperatures etc.) can be developed for the immobilisation of different types of waste. Only zirconolite and perovskite can accommodate actinides. The exact proportions of the main phases vary depending on the HLW composition. For example, Synroc-C is designed to contain about 20% by weight of calcined HLW and it consists of approximately (% by weight): 30 – hollandite; 30 – zirconolite; 20 – perovskite and 20 – Ti-oxides and other phases. Immobilising weapons-grade plutonium or transuranium wastes instead of bulk HLW may essentially change the Synroc phase composition to primarily zirconolite-based or a pyrochlore-based ceramic. The starting precursor for Synroc-C fabrication contains ~57% by weight TiO 2 and 2% by weight metallic Ti. The metallic titanium provides reducing conditions during ceramic synthesis and helps decrease volatilisation of radioactive cesium. [ 3 ]
Synroc is not a disposal method. [ 4 ] Synroc still has to be stored. Even though the waste is held in a solid lattice and prevented from spreading, it is still radioactive and can have a negative effect on its surroundings. Synroc is a superior method of nuclear waste storage because it minimises leaching . [ 5 ]
In 1997 Synroc was tested with real HLW using technology developed jointly by ANSTO and the US DoE's Argonne National Laboratory. [ 1 ] In January 2010, the United States Department of Energy selected hot isostatic pressing (HIP) for processing waste at the Idaho National Laboratory . [ 6 ]
In April 2008, the Battelle Energy Alliance signed a contract with ANSTO to demonstrate the benefits of Synroc in processing waste managed by Batelle as part of its contract to manage the Idaho National Laboratory . [ 7 ]
Synroc was chosen in April 2005 for a multimillion-dollar "demonstration" contract to eliminate 5 t (5.5 short tons) of plutonium -contaminated waste at British Nuclear Fuel 's Sellafield plant, on the northwest coast of England . | https://en.wikipedia.org/wiki/Synroc |
Syntactic foams are composite materials synthesized by filling a metal , polymer , [ 1 ] cementitious or ceramic matrix with
spheres as aggregates . [ 2 ] The spheres may be hollow, called microballoons [ 3 ] or cenospheres , or non-hollow, for example perlite . [ 4 ] In this context, "syntactic" means "put together." [ 5 ] The presence of hollow particles results in lower density , higher specific strength (strength divided by density), lower coefficient of thermal expansion , and, in some cases, radar or sonar transparency .
The term was originally coined by the Bakelite Company , in 1955, for their lightweight composites made of hollow phenolic microspheres bonded to a matrix of phenolic, epoxy , or polyester . [ 6 ] [ 7 ]
These materials were developed in early 1960s as improved buoyancy materials for marine applications. [ 8 ] Other characteristics led these materials to aerospace and ground transportation vehicle applications. [ 9 ]
Research on syntactic foams has recently been advanced by Nikhil Gupta .
Tailorability is one of the biggest advantages of these materials. [ 10 ] The matrix material can be selected from almost any metal, polymer, or ceramic. Microballoons are available in a variety of sizes and materials, including glass microspheres , cenospheres , carbon , and polymers. The most widely used and studied foams are glass microspheres (in epoxy or polymers), and cenospheres or ceramics [ 11 ] (in aluminium). One can change the volume fraction of microballoons or use microballoons of different effective density, the latter depending on the average ratio between the inner and outer radii of the microballoons.
A manufacturing method for low density syntactic foams is based on the principle of buoyancy. [ 12 ] [ 13 ]
The compressive properties of syntactic foams, in most cases, strongly depend on the properties of the filler particle material. In general, the compressive strength of the material is proportional to its density. Cementitious syntactic foams are reported to achieve compressive strength values greater than 30 MPa (4.4 ksi) while maintaining densities lower than 1.2 g/cm 3 (0.69 oz/cu in). [ 14 ]
The matrix material has more influence on the tensile properties. Tensile strength may be highly improved by a chemical surface treatment of the particles, such as silanization , which allows the formation of strong bonds between glass particles and epoxy matrix. Addition of fibrous materials can also increase the tensile strength. [ citation needed ]
Current applications for syntactic foam include buoyancy modules for marine riser tensioners , remotely operated underwater vehicles (ROVs), autonomous underwater vehicles (AUVs), deep-sea exploration, boat hulls , and helicopter and airplane components.
Cementitious syntactic foams have also been investigated as a potential lightweight structural composite material. These materials include glass microspheres dispersed in a cement paste matrix to achieve a closed cell foam structure, instead of a metallic or a polymeric matrix. Cementitious syntactic foams have also been tested for their mechanical performance under high strain rate loading conditions to evaluate their energy dissipation capacity in crash cushions, blast walls, etc. Under these loading conditions, the glass microspheres of the cementitious syntactic foams did not show progressive crushing. Ultimately, unlike the polymeric and metallic syntactic foams, they did not emerge as suitable materials for energy dissipation applications. [ 15 ] Structural applications of syntactic foams include use as the intermediate layer (that is, the core) of sandwich panels .
Though the cementitious syntactic foams demonstrate superior specific strength values in comparison to most conventional cementitious materials, it is challenging to manufacture them. Generally, the hollow inclusions tend to buoy and segregate in the low shear strength and high-density fresh cement paste. Therefore, maintaining a uniform microstructure across the material must be achieved through a strict control of the composite rheology . [ 16 ] In addition, certain glass types of microspheres may lead to an alkali silica reaction . Therefore, the adverse effects of this reaction must be considered and addressed to ensure the long-term durability of these composites. [ 17 ]
Other applications include; | https://en.wikipedia.org/wiki/Syntactic_foam |
In logic , syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.
The symbols , formulas , systems , theorems and proofs expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.
Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system.
In computer science , the term syntax refers to the rules governing the composition of well-formed expressions in a programming language . As in mathematical logic, it is independent of semantics and interpretation.
A symbol is an idea , abstraction or concept , tokens of which may be marks or a metalanguage of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the formation rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them.
A formal language is a syntactic entity which consists of a set of finite strings of symbols which are its words (usually called its well-formed formulas ). Which strings of symbols are words is determined by the creator of the language, usually by specifying a set of formation rules . Such a language can be defined without reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it – that is, before it has any meaning.
Formation rules are a precise description of which strings of symbols are the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the formal language which constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean).
A proposition is a sentence expressing something true or false . [ 2 ] A proposition is identified ontologically as an idea , concept or abstraction whose token instances are patterns of symbols , marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers .
A formal theory is a set of sentences in a formal language .
A formal system (also called a logical calculus , or a logical system ) consists of a formal language together with a deductive apparatus (also called a deductive system ). The deductive apparatus may consist of a set of transformation rules (also called inference rules ) or a set of axioms , or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any interpretation given to it (as being, for instance, a system of arithmetic).
A formula A is a syntactic consequence [ 3 ] [ 4 ] [ 5 ] [ 6 ] within some formal system F S {\displaystyle {\mathcal {FS}}} of a set Г of formulas if there is a derivation in formal system F S {\displaystyle {\mathcal {FS}}} of A from the set Г.
Syntactic consequence does not depend on any interpretation of the formal system. [ 7 ]
A formal system S {\displaystyle {\mathcal {S}}} is syntactically complete [ 8 ] [ 9 ] [ 10 ] [ 11 ] (also deductively complete , maximally complete , negation complete or simply complete ) iff for each formula A of the language of the system either A or ¬A is a theorem of S {\displaystyle {\mathcal {S}}} . In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an inconsistency . Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not tautologies ). Gödel's incompleteness theorem shows that no recursive system that is sufficiently powerful, such as the Peano axioms , can be both consistent and complete.
An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called formal semantics . Giving an interpretation is synonymous with constructing a model . An interpretation is expressed in a metalanguage , which may itself be a formal language, and as such itself is a syntactic entity.
Media related to Syntax (logic) at Wikimedia Commons | https://en.wikipedia.org/wiki/Syntax_(logic) |
In computer science , a syntax error is an error in the syntax of a sequence of characters that is intended to be written in a particular programming language .
For compiled languages , syntax errors are detected at compile-time . A program will not compile until all syntax errors are corrected. For interpreted languages , a syntax error may be detected during program execution , and an interpreter's error messages might not differentiate syntax errors from errors of other kinds.
There is some disagreement as to just what errors are "syntax errors". For example, some would say that the use of an uninitialized variable's value in Java code is a syntax error, but many others would disagree [ 1 ] [ 2 ] and would classify this as a (static) semantic error.
In 8-bit home computers that used BASIC interpreter as their primary user interface, the SYNTAX ERROR error message became somewhat notorious, as this was the response to any command or user input the interpreter could not parse.
A syntax error can occur or take place, when an invalid equation is being typed on a calculator. This can be caused, for instance, by opening brackets without closing them, or less commonly, entering several decimal points in one number.
In Java the following is a syntactically correct statement:
while the following is not:
The second example would theoretically print the variable Hello World instead of the words "Hello World". A variable in Java cannot have a space in between, so the syntactically correct line would be System.out.println(Hello_World) .
A compiler will flag a syntax error when given source code that does not meet the requirements of the language's grammar.
Type errors (such as an attempt to apply the ++ increment operator to a Boolean variable in Java) and undeclared variable errors are sometimes considered to be syntax errors when they are detected at compile-time. It is common to classify such errors as (static) semantic errors instead. [ 2 ] [ 3 ] [ 4 ]
A syntax error is one of several types of errors on calculators (most commonly found on scientific calculators and graphing calculators ), representing that the equation that has been input has incorrect syntax of numbers, operations and so on. It can result in various ways, including but not limited to:
This computer-programming -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Syntax_error |
Syntelic attachment occurs when both sister chromosomes are attached to a single spindle pole . [ 1 ] [ 2 ]
Normal cell division distributes the genome equally between two daughter cells, with each chromosome attaching to an ovoid structure called the spindle. During the division process, errors commonly occur in attaching the chromosomes to the spindle, estimated to affect 86 to 90 percent of chromosomes. [ 3 ]
Such attachment errors are common during the early stages of spindle formation, but they are mostly corrected before the start of anaphase . [ 4 ] Successful cell division requires identification and correction of any dangerous errors before the cell splits in two. [ 3 ] If the syntelic attachment continues, it causes both sister chromatids to be segregated to a single daughter cell. [ 5 ]
Microtubules extend from the spindle poles and attach to the first kinetochore they encounter. [ 6 ] Because this process is stochastic and not facilitated or directed, the first microtubules to come into contact with a kinetochore may not have originated at the correct spindle pole. [ 7 ] Normally, the sister kinetochores are on opposing sides of the chromosomes , facing outward toward their respective spindle poles. [ 8 ] This arrangement enhances the likelihood of properly bi-oriented chromosomes and is sometimes referred to as a mechanism for 'avoidance' of syntelic attachment. [ 8 ] [ 9 ] However, sometimes the kinetochores are found on the same side of the centromere , and this error cannot be corrected stochastically. [ 8 ] Instead, the spindle must actively exert forces on one of the two kinetochores to relocate it to the proper, outer edge of the centromere. [ 8 ] If the geometry and orientation of the two kinetochores is not corrected, the cells can still effectively achieve bi-orientation through the employment of error correction mechanisms. [ 9 ]
Polyploid cells, and tetraploids in particular, experience an increased number of syntelic attachments, which contributes to their genomic instability . [ 10 ] This phenomenon of increased rates of syntelic attachment in polyploids is thought to result from an inability to scale the mitotic spindle and kinetochore architecture to accommodate the increase in cell size. [ 10 ] Therefore, scaling defects between the genome and cellular architecture, which often occur in cancer , likely result in high rates of syntelic attachment. [ 10 ]
Error correction is closely tied to the spindle assembly checkpoint (SAC), which oversees the progression through mitosis and can halt the cell in metaphase until proper bi-orientation of all chromosomes is achieved. [ 11 ] Initial attachments occur randomly, and the cell destabilizes any incorrect microtubule-kinetochore interactions. Subsequent rounds of undirected attachment and destabilization occur until each kinetochore is attached to the correct spindle pole.
Tension was quickly identified as an important component of the error-sensing mechanism and likely of the spindle assembly checkpoint. [ 6 ] [ 11 ] Ipl1 in yeast and its functional homolog , Aurora B , in metazoans aid in tension detection and destabilization of errant attachments. [ 11 ] [ 12 ] Aurora B is found at the centromere, between the two kinetochores. [ 11 ] In the absence of tension, Aurora B can phosphorylate substrates at the kinetochores, leading to destabilization of the attached microtubules. [ 11 ] Properly attached microtubules induce tension, pulling the kinetochore far enough away from Aurora B so as to prevent phosphorylation of kinetochore components. [ 11 ] Following destabilization, the kinetochore can form new spindle attachments, and if the new attachments result in chromosome bi-orientation, they will remain. [ 12 ] Correct attachments that induce tension are more likely to occur when the kinetochores are geometrically positioned on opposite sides of the centromere. [ 7 ]
Robust destabilization by Ipl1/Aurora B in the absence of tension leads to a specific challenge: the initial establishment of bi-orientation, prior to the buildup of tension, would be sensitive to Ipl1/Aurora B activity. [ 13 ] This is referred to as the initiation problem of biorientation (IPBO), and is resolved by implementing a delay between sensing the tension and destabilizing the attachment. [ 13 ] Modeling has indicated that such a delay could be introduced if the rate of Ipl1/Aurora B kinase activity is slower than that of the counteracting phosphatase activity at the kinetochore. [ 13 ] The time delay allows for tension to be established at bi-oriented chromosomes, so that only syntelic attachments are phosphorylated and destabilized. [ 13 ]
Syntelic attachment is not uncommon in early metaphase , and can often be resolved by error correction mechanisms that are well-conserved across metazoans . [ 12 ] If syntelic attachment is left uncorrected, for example if the spindle assembly checkpoint does not successfully pause cells in metaphase, the chromosomes will not segregate correctly. [ 12 ] This failure to properly segregate results in aneuploidy , which can lead to errors in development or cancer . [ 14 ] Interestingly, segregation errors that result from syntelic attachment often occur without visible lagging . [ 15 ] In contrast, merotelic attachments will cause chromosome lagging during anaphase , but will often segregate correctly and not result in aneuploidy. [ 14 ] [ 15 ] | https://en.wikipedia.org/wiki/Syntelic |
Syntelog : a special case of gene homology where sets of genes are derived from the same ancestral genomic region. This may arise from speciation events, or through whole or partial genome duplication events (e.g. polyploidy ). This term is distinct from ortholog , paralog , in-paralog, out-paralog, and xenolog because it refers only to genes' evolutionary history evidenced by sequence similarity and relative genomic position. [ 1 ]
Comparison between two genomic regions of Arabidopsis thaliana derived from its most recent genome duplication event. Syntelogs are indicated by red lines connecting regions of sequence similarly (red boxes):
Sequence analysis and visualization of syntelogs performed by GEvo. [ 2 ] Regerate this analysis in CoGe's GEvo using this link . GEvo Sequences were compared using the BlastZ algorithm. | https://en.wikipedia.org/wiki/Syntelog |
In genetics , the term synteny refers to two related concepts:
The Encyclopædia Britannica gives the following description of synteny, using the modern definition: [ 2 ]
Genomic sequencing and mapping have enabled comparison of the general structures of genomes of many different species. The general finding is that organisms of relatively recent divergence show similar blocks of genes in the same relative positions in the genome. This situation is called synteny, translated roughly as possessing common chromosome sequences. For example, many of the genes of humans are syntenic with those of other mammals—not only apes but also cows, mice, and so on. Study of synteny can show how the genome is cut and pasted in the course of evolution.
Synteny is a neologism meaning "on the same ribbon"; Greek : σύν , syn "along with" + ταινία , tainiā "band". This can be interpreted classically as "on the same chromosome", or in the modern sense of having the same order of genes on two (homologous) strings of DNA (or chromosomes).
The classical concept is related to genetic linkage : Linkage between two loci is established by the observation of lower-than-expected recombination frequencies between them. In contrast, any loci on the same chromosome are by definition syntenic, even if their recombination frequency cannot be distinguished from unlinked loci by practical experiments. Thus, in theory, all linked loci are syntenic, but not all syntenic loci are necessarily linked. Similarly, in genomics , the genetic loci on a chromosome are syntenic regardless of whether this relationship can be established by experimental methods such as DNA sequencing /assembly, genome walking , physical localization or hap-mapping .
Students of (classical) genetics employ the term synteny to describe the situation in which two genetic loci have been assigned to the same chromosome but still may be separated by a large enough distance in map units that genetic linkage has not been demonstrated.
Shared synteny (also known as conserved synteny) describes preserved co-localization of genes on chromosomes of different species. During evolution , rearrangements to the genome such as chromosome translocations may separate two loci, resulting in the loss of synteny between them. Conversely, translocations can also join two previously separate pieces of chromosomes together, resulting in a gain of synteny between loci. Stronger-than-expected shared synteny can reflect selection for functional relationships between syntenic genes, such as combinations of alleles that are advantageous when inherited together, or shared regulatory mechanisms. [ 3 ]
In light of the more recent shift in the meaning of synteny , this conservation of gene content and linkage without preservation of order has also been termed mesosynteny . [ 4 ]
The term is currently (since ~2000) more commonly used to describe preservation of the precise order of genes on a chromosome passed down from a common ancestor, [ 5 ] [ 6 ] [ 7 ] [ 8 ] despite more "old school" geneticists rejecting what they perceive as a misappopriation of the term, [ 9 ] preferring collinearity instead. [ 10 ]
The analysis of synteny in the gene order sense has several applications in genomics. Shared synteny is one of the most reliable criteria for establishing the orthology of genomic regions in different species. Additionally, exceptional conservation of synteny can reflect important functional relationships between genes. For example, the order of genes in the " Hox cluster ", which are key determinants of the animal body plan and which interact with each other in critical ways, is essentially preserved throughout the animal kingdom. [ 11 ]
Synteny is widely used in studying complex genomes, as comparative genomics allows the presence and possibly function of genes in a simpler, model organism to infer those in a more complex one. For example, wheat has a very large, complex genome which is difficult to study. In 1994 research from the John Innes Centre in England and the National Institute of Agrobiological Research in Japan demonstrated that the much smaller rice genome had a similar structure and gene order to that of wheat. [ 12 ] Further study found that many cereals are syntenic [ 13 ] and thus plants such as rice or the grass Brachypodium could be used as a model to find genes or genetic markers of interest which could be used in wheat breeding and research. In this context, synteny was also essential in identifying a highly important region in wheat, the Ph1 locus involved in genome stability and fertility, which was located using information from syntenic regions in rice and Brachypodium. [ 14 ]
Synteny is also widely used in microbial genomics. In Hyphomicrobiales and Enterobacteriales , syntenic genes encode a large number of essential cell functions and represent a high level of functional relationships. [ 15 ]
Patterns of shared synteny or synteny breaks can also be used as characters to infer the phylogenetic relationships among several species, and even to infer the genome organization of extinct ancestral species. A qualitative distinction is sometimes drawn between macrosynteny , synteny in large portions of a chromosome, and microsynteny , synteny for only a few genes at a time.
Shared synteny between different species can be inferred from their genomic sequences. This is typically done using a version of the MCScan algorithm, which finds syntenic blocks between species by comparing their homologous genes and looking for common patterns of collinearity on a chromosomal or contig scale. Homologies are usually determined on the basis of high bit score BLAST hits that occur between multiple genomes. From here, dynamic programming is used to select the best scoring path of shared homologous genes between species, taking into account potential gene loss and gain which may have occurred in the species' evolutionary histories. [ 16 ] | https://en.wikipedia.org/wiki/Synteny |
Laboratorios Syntex SA (later Syntex Laboratories, Inc. ) was a pharmaceutical company formed in Mexico City in January 1944 by Russell Marker , Emeric Somlo, and Federico Lehmann to manufacture therapeutic steroids from the Mexican yams called cabeza de negro ( Dioscorea mexicana ) and Barbasco ( Dioscorea composita ). [ 1 ] The demand for barbasco by Syntex initiated the Mexican barbasco trade . [ 2 ]
As the American Chemistry Society later explained: “In early 1944, the new Mexican company was chartered and named Syntex, S.A. (‘Synthesis and Mexico’). Russell Marker, shortly thereafter, left Syntex on account of his ruthless cofounder. [ 3 ]
Luis E. Miramontes , George Rosenkranz and Carl Djerassi 's successful synthesis of norethisterone (also known as norethindrone ) — which was later proven to be an effective pregnancy inhibitor — led to an infusion of capital into Syntex and the Mexican steroid pharmaceutical industry. [ 4 ] George Rosenkranz and Carl Djerassi went on to synthesize cortisone from diosgenin , the same phytosteroid contained in Mexican yams used to synthesize progesterone and norethindrone. The synthesis was more economical than the previous Merck & Co. synthesis, which started with bile acids .
In 1959, Syntex moved its operating headquarters to Palo Alto, California , United States, and evolved into a transnational corporation . Its foreign scientists had become frustrated with bureaucratic delays on the part of the Mexican government in granting work visas and approving necessary imports of pharmaceutical materials for their work. After 1959, Syntex was incorporated in Panama; its administration, research and marketing were conducted from Palo Alto; its manufacturing of bulk steroid intermediates remained in Mexico; and it also manufactured finished drugs at plants in Puerto Rico and the Bahamas. [ 5 ]
Syntex agreed to be acquired by the Roche group in May 1994. [ 6 ] After the acquisition closed, Roche downsized Syntex's research and development facilities in the Stanford Research Park and finally shut down what was left of Syntex in September 2008. [ 7 ] In 2011, VMware moved into the former Syntex campus in Palo Alto. [ 8 ]
Syntex submitted its compound to a laboratory in Madison, Wisconsin , for biological evaluation, and found it was the most active, orally-effective progestational hormone of its time. Syntex submitted a patent application in November 1951. In August 1953, G.D. Searle & Co. filed for a patent for the synthesis of the double-bond isomer 13 of norethindrone called noretynodrel . Noretynodrel is converted into norethisterone under acidic conditions, such as those in the human stomach , and the new patent did not infringe on the Syntex patent. Searle obtained approval to market noretynodrel before Syntex received its approval. By 1964 three companies, including Syntex, G.D. Searle , and Johnson & Johnson under the Ortho Pharmaceutical brand, were marketing 2-mg doses of the Syntex norethindrone.
Syntex's submission of a fraudulent toxicology analysis of naproxen largely led to the Food and Drug Administration 's uncovering of extensive scientific misconduct by Industrial Bio-Test Laboratories in 1976. [ 10 ] [ 11 ] [ 12 ] [ 13 ] | https://en.wikipedia.org/wiki/Syntex |
Synthesis-dependent strand annealing ( SDSA ) is a major mechanism of homology-directed repair of DNA double-strand breaks (DSBs). Although many of the features of SDSA were first suggested in 1976, [ 1 ] the double-Holliday junction model proposed in 1983 [ 2 ] was favored by many researchers. In 1994, studies of double-strand gap repair in Drosophila were found to be incompatible with the double-Holliday junction model, leading researchers to propose a model they called synthesis-dependent strand annealing. [ 3 ] Subsequent studies of meiotic recombination in S. cerevisiae found that non-crossover products appear earlier than double-Holliday junctions or crossover products, challenging the previous notion that both crossover and non-crossover products are produced by double-Holliday junctions and leading the authors to propose that non-crossover products are generated through SDSA. [ 4 ]
In the accompanying Figure, the first step labeled “5’ to 3’ resection” shows the formation of a 3’ ended single DNA strand that in the next step invades a homologous DNA duplex. RNA polymerase III is reported to catalyze formation of a transient RNA-DNA hybrid at double-strand breaks as an essential intermediate step in the repair of the breaks by homologous recombination. [ 5 ] Formation of the RNA-DNA hybrid would protect the invading single-stranded DNA from degradation. After the transient RNA-DNA hybrid intermediate is formed the RNA strand is replaced by the Rad51 protein which catalyzes the subsequent stage of strand invasion.
In the SDSA model, repair of double-stranded breaks occurs without the formation of a double Holliday junction, so that the two processes of homologous recombination are identical until just after D-loop formation. [ 6 ] In yeast, the D-loop is formed by strand invasion with the help of proteins Rad51 and Rad52 , [ 7 ] and is then acted on by DNA helicase Srs2 to prevent formation of the double Holliday junction in order for the SDSA pathway to occur. [ 8 ] The invading 3' strand is thus extended along the recipient homologous DNA duplex by DNA polymerase in the 5' to 3' direction, so that the D-loop physically translocates – a process referred to as bubble migration DNA synthesis. [ 9 ] The resulting single Holliday junction then slides down the DNA duplex in the same direction in a process called branch migration , displacing the extended strand from the template strand. This displaced strand pops up to form a 3' overhang in the original double-stranded break duplex, which can then anneal to the opposite end of the original break through complementary base pairing. Thus DNA synthesis fills in gaps left over from annealing, and extends both ends of the still present single stranded DNA break, ligating all remaining gaps to produce recombinant non-crossover DNA. [ 10 ]
SDSA is unique in that D-loop translocation results in conservative rather than semiconservative replication , as the first extended strand is displaced from its template strand , leaving the homologous duplex intact. Therefore, although SDSA produces non-crossover products because flanking markers of heteroduplex DNA are not exchanged, gene conversion may occur, wherein nonreciprocal genetic transfer takes place between two homologous sequences. [ 11 ]
Assembly of a nucleoprotein filament comprising single-stranded DNA (ssDNA) and the RecA homolog, Rad51 , is a key step necessary for homology search during recombination . In the budding yeast Saccharomyces cerevisiae , Srs2 translocase dismantles Rad51 filaments during meiosis . [ 12 ] By directly interacting with Rad51, Srs2 dislodges Rad51 from nucleoprotein filaments thereby inhibiting Rad51-dependent formation of joint molecules and D-loop structures. This dismantling activity is specific for Rad51 since Srs2 does not dismantle DMC1 (a meiosis-specific Rad51 homolog), Rad52 (a Rad 51 mediator) or replication protein A ( RPA , a single-stranded DNA binding protein). Srs2 promotes the non-crossover SDSA pathway, apparently by regulating RAD51 binding during strand exchange. [ 13 ]
Divergence between SDSA and double-Holliday junction occurs when the D-loop is disassembled allow the nascent strand to anneal to other resected end of the DSB (in the double-Holliday junction model the strand displaced by D-loop extension anneals to the other end of the DSB in "2nd end capture"). Research in Drosophila melanogaster identified the Bloom syndrome helicase (Blm) as the enzyme promoting dissassembly of the D-loop. [ 14 ] [ 15 ] [ 16 ] Similarly, S. cerevisiae Sgs1, an ortholog of BLM, appears to be a central regulator of most of the recombination events that occur during S. cerevisiae meiosis . [ 17 ] Sgs1(BLM) may disassemble D-loop structures analogous to early strand invasion intermediates and thus promote NCO formation by SDSA. [ 17 ] The Sgs1 helicase forms a conserved complex with the topoisomerase III ( Top3 )- RMI1 heterodimer (that catalyzes DNA single strand passage). This complex, called STR (for its three components), promotes early formation of NCO recombinants by SDSA during meiosis. [ 18 ]
As reviewed by Uringa et al. [ 19 ] the RTEL1 helicase is proposed to regulate recombination during meiosis in the worm Caenorhabditis elegans . RTEL1 is a key protein in repair of DSBs. It disrupts D-loops and is thought to promote NCO outcomes through SDSA.
The number of DSBs created during meiosis can substantially exceed the number of final CO events. In the plant Arabidopsis thaliana , only about 4% of DSBs are repaired by CO recombination, [ 20 ] suggesting that most DSBs are repaired by NCO recombination. Data based on tetrad analysis from several species of fungi show that only a minority (on average about 34%) of recombination events during meiosis are COs (see Whitehouse, [ 21 ] Tables 19 and 38 for summaries of data from S. cerevisiae , Podospora anserina , Sordaria fimicola and Sordaria brevicollis ). In the fruit fly D. melanogaster during meiosis in females there is at least a 3:1 ratio of NCOs to COs. [ 22 ] These observations indicate that the majority of recombination events during meiosis are NCOs, and suggest that SDSA is the principal pathway for recombination during meiosis. | https://en.wikipedia.org/wiki/Synthesis-dependent_strand_annealing |
Bioactive glasses have been synthesized through methods such as conventional melting , quenching , the sol–gel process , flame synthesis, and microwave irradiation . The synthesis of bioglass has been reviewed by various groups, with sol-gel synthesis being one of the most frequently used methods for producing bioglass composites, particularly for tissue engineering applications. Other methods of bioglass synthesis have been developed, such as flame and microwave synthesis, though they are less prevalent in research.
The first bioactive glass , developed by Larry Hench in 1969, was produced by melting a mixture of related oxide precursors at relatively high temperatures. This original bioactive glass, named Bioglass, was melt-derived with a composition of 46.1 mol% SiO 2 , 24.4 mol% Na 2 O , 26.9 mol% CaO , and 2.6 mol% P 2 O 5 . The selection of glass composition for specific applications is often based on a comprehensive understanding of how each major component influences the properties of the glass, considering both its final use and its manufacturing process. Despite extensive research over the past 40 years, only a limited number of glass compositions have been approved for clinical use. Among these, the two melt-derived compositions approved by the U.S. Food and Drug Administration (FDA)—45S5 and S53P4—consist of four oxides: SiO 2 , Na 2 O , CaO , and P 2 O 5 . [ 1 ] [ 2 ] In general, a large number of elements can be dissolved in glasses. The effect of Al 2 O 3 , B 2 O 3 , Fe 2 O 3 , MgO , [ 3 ] SrO , [ 4 ] BaO , ZnO , Li 2 O , K 2 O , CaF 2 [ 5 ] and TiO 2 on the in vitro or in vivo properties of certain compositions of bioactive glasses has been reported. [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ excessive citations ] However, the influence of composition on the properties and compatibility of bioactive and biodegradable glasses is not fully understood. [ citation needed ]
Scaffolds fabricated by the melt quench technique have much less porosity , which causes issues with healing and tissue integration during in-vivo testing. [ citation needed ]
The sol–gel process has a long history in the synthesis of silicate systems and other oxides and has become a widely researched field with significant technological relevance. This process is used for the fabrication of thin films, coatings, nanoparticles , and fibers. Sol-gel processing technology at low temperatures, an alternative to traditional melt processing of glasses, involves the synthesis of a solution (sol), typically composed of metal-organic and metal salt precursors. This is followed by the formation of a gel through chemical reaction or aggregation, and finally, thermal treatment for drying, organic removal, and sometimes crystallization and cooling treatment. The synthesis of specific silicate bioactive glasses using the sol–gel technique at low temperatures, employing metal alkoxides as precursors, was demonstrated in 1991 by Li et al. [ 11 ] Typical precursors for bioactive glass synthesis include tetraethyl orthosilicate , calcium nitrate and triethyl phosphate . Following hydrolysis and poly-condensation reactions, a gel is formed, which is then calcined at 600–700°C to form the glass. Sol–gel derived products, such as thin films or particles, are highly porous and exhibit a high specific surface area . Recent research by Hong et al. has focused on fabricating bioactive silicate glass nanoparticles through a combination of the sol–gel route and the co-precipitation method . [ 12 ] In this process, the mixture of precursors is hydrolyzed in an acidic environment, condensed in an alkaline condition, and then subjected to freeze-drying . The morphology and size of bioactive glass nanoparticles can be tailored by varying the production conditions and the feeding ratio of reagents . [ citation needed ]
Different ions can be incorporated into bioactive glasses, including zinc , magnesium , zirconium , titanium , boron and silver , to enhance functionality and bioactivity. However, synthesizing bioactive glasses at the nanoscale with these ions can be challenging. Recently, Delben et al. developed sol–gel-derived bioactive glass doped with silver, reporting that the Si–O–Si bond number increased with higher silver concentrations, resulting in structural densification. [ 13 ] It was also observed that quartz and metallic silver crystallization increased with higher silver content, while hydroxyapatite crystallization decreased.
The sol–gel technique is widely regarded for its versatility in synthesizing inorganic materials and has proven suitable for producing various bioactive glasses. However, it is limited in the range of compositions that can be produced. Residual water or solvent content may complicate its application in biomedical fields, necessitating high-temperature calcination to eliminate organic remnants. Additionally, sol–gel processing is time-consuming and, being a batch process, can result in batch-to-batch variations. [ 14 ]
Beginning in 2006, researchers have produced alternate methods of synthesizing bioglass; these methods include flame synthesis and microwave synthesis. Flame synthesis works by baking the powders directly in a flame reactor. [ 15 ] Microwave synthesis is a rapid and low-cost method where precursors are dissolved in water, transferred to an ultrasonic bath , and then irradiated. [ 16 ] | https://en.wikipedia.org/wiki/Synthesis_of_bioglass |
Techniques have been developed to produce carbon nanotubes (CNTs) in sizable quantities, including arc discharge, laser ablation , high-pressure carbon monoxide disproportionation , and chemical vapor deposition (CVD). Most of these processes take place in a vacuum or with process gases. CVD growth of CNTs can occur in a vacuum or at atmospheric pressure. Large quantities of nanotubes can be synthesized by these methods; advances in catalysis and continuous growth are making CNTs more commercially viable. [ 1 ]
Nanotubes were observed in 1991 in the carbon soot of graphite electrodes during an arc discharge, by using a current of 100 amps , that was intended to produce fullerenes . [ 2 ] However the first macroscopic production of carbon nanotubes was made in 1992 by two researchers at NEC 's Fundamental Research Laboratory. [ 3 ] The method used was the same as in 1991. During this process, the carbon contained in the negative electrode sublimates because of the high-discharge temperatures.
The yield for this method is up to 30% by weight and it produces both single- and multi-walled nanotubes with lengths of up to 50 micrometers with few structural defects. [ 4 ] Arc-discharge technique uses higher temperatures (above 1,700 °C) for CNT synthesis which typically causes the expansion of CNTs with fewer structural defects in comparison with other methods. [ 5 ]
In laser ablation, a pulsed laser vaporizes a graphite target in a high-temperature reactor while an inert gas is led into the chamber. Nanotubes develop on the cooler surfaces of the reactor as the vaporized carbon condenses. A water-cooled surface may be included in the system to collect the nanotubes.
This process was developed by Richard Smalley and co-workers at Rice University , who at the time of the discovery of carbon nanotubes, were blasting metals with a laser to produce various metal molecules. When they heard of the existence of nanotubes they replaced the metals with graphite to create multi-walled carbon nanotubes. [ 6 ] Later that year the team used a composite of graphite and metal catalyst particles (the best yield was from a cobalt and nickel mixture) to synthesize single-walled carbon nanotubes. [ 7 ]
The laser ablation method yields around 70% and produces primarily single-walled carbon nanotubes with a controllable diameter determined by the reaction temperature . However, it is more expensive than either arc discharge or chemical vapor deposition. [ 4 ]
Single-walled carbon nanotubes can also be synthesized by a thermal plasma method, first invented in 2000 at INRS ( Institut national de la recherche scientifique ) in Varennes, Canada, by Olivier Smiljanic. In this method, the aim is to reproduce the conditions prevailing in the arc discharge and laser ablation approaches, but a carbon-containing gas is used instead of graphite vapors to supply the necessary carbon. Doing so, the growth of SWNT is more efficient (decomposing the gas can be 10 times less energy-consuming than graphite vaporization). The process is also continuous and low-cost. A gaseous mixture of argon, ethylene and ferrocene is introduced into a microwave plasma torch, where it is atomized by the atmospheric pressure plasma, which has the form of an intense 'flame'. The fumes created by the flame contain SWNT, metallic and carbon nanoparticles and amorphous carbon. [ 8 ] [ 9 ]
Another way to produce single-walled carbon nanotubes with a plasma torch is to use the induction thermal plasma method, implemented in 2005 by groups from the Université de Sherbrooke and the National Research Council of Canada . [ 10 ] The method is similar to arc discharge in that both use ionized gas to reach the high temperature necessary to vaporize carbon-containing substances and the metal catalysts necessary for the ensuing nanotube growth. The thermal plasma is induced by high-frequency oscillating currents in a coil, and is maintained in flowing inert gas . Typically, a feedstock of carbon black and metal catalyst particles is fed into the plasma, and then cooled down to form single-walled carbon nanotubes. Different single-wall carbon nanotube diameter distributions can be synthesized.
The induction thermal plasma method can produce up to 2 grams of nanotube material per minute, which is higher than the arc discharge or the laser ablation methods. [ citation needed ]
The catalytic vapor phase deposition of carbon was reported in 1952 [ 11 ] and 1959, [ 12 ] but it was not until 1993 [ 13 ] that carbon nanotubes were formed by this process. In 2007, researchers at the University of Cincinnati (UC) developed a process to grow aligned carbon nanotube arrays of length 18 mm on a FirstNano ET3000 carbon nanotube growth system. [ 14 ]
During CVD, a substrate is prepared with a layer of metal catalyst particles, most commonly nickel, cobalt, [ 15 ] iron , or a combination. [ 16 ] The metal nanoparticles can also be produced by other ways, including reduction of oxides or oxides solid solutions. The diameters of the nanotubes that are to be grown are related to the size of the metal particles. This can be controlled by patterned (or masked) deposition of the metal, annealing , or by plasma etching of a metal layer. The substrate is heated to approximately 700 °C. To initiate the growth of nanotubes, two gases are bled into the reactor: a process gas (such as ammonia , nitrogen or hydrogen ) and a carbon-containing gas (such as acetylene , ethylene , ethanol or methane ). Nanotubes grow at the sites of the metal catalyst; the carbon-containing gas is broken apart at the surface of the catalyst particle, and the carbon is transported to the edges of the particle, where it forms the nanotubes. This mechanism is still being studied. [ 17 ] The catalyst particles can stay at the tips of the growing nanotube during growth, or remain at the nanotube base, depending on the adhesion between the catalyst particle and the substrate. [ 18 ] Thermal catalytic decomposition of hydrocarbon has become an active area of research and can be a promising route for the bulk production of CNTs. Fluidized bed reactor is the most widely used reactor for CNT preparation. Scale-up of the reactor is the major challenge. [ 19 ] [ 20 ]
CVD is the most widely used method for the production of carbon nanotubes. [ 21 ] For this purpose, the metal nanoparticles are mixed with a catalyst support such as MgO or Al 2 O 3 to increase the surface area for higher yield of the catalytic reaction of the carbon feedstock with the metal particles. One issue in this synthesis route is the removal of the catalyst support via an acid treatment, which sometimes could destroy the original structure of the carbon nanotubes. However, alternative catalyst supports that are soluble in water have proven effective for nanotube growth. [ 22 ]
If a plasma is generated by the application of a strong electric field during growth ( plasma-enhanced chemical vapor deposition ), then the nanotube growth will follow the direction of the electric field. [ 23 ] By adjusting the geometry of the reactor it is possible to synthesize vertically aligned carbon nanotubes [ 24 ] (i.e., perpendicular to the substrate), a morphology that has been of interest to researchers interested in electron emission from nanotubes. Without the plasma, the resulting nanotubes are often randomly oriented. Under certain reaction conditions, even in the absence of a plasma, closely spaced nanotubes will maintain a vertical growth direction resulting in a dense array of tubes resembling a carpet or forest.
Of the various means for nanotube synthesis, CVD shows the most promise for industrial-scale deposition, because of its price/unit ratio, and because CVD is capable of growing nanotubes directly on a desired substrate, whereas the nanotubes must be collected in the other growth techniques. The growth sites are controllable by careful deposition of the catalyst. [ 25 ] In 2007, a team from Meijo University demonstrated a high-efficiency CVD technique for growing carbon nanotubes from camphor . [ 26 ] Researchers at Rice University , until recently led by the late Richard Smalley , have concentrated on finding methods to produce large, pure amounts of particular types of nanotubes. Their approach grows long fibers from many small seeds cut from a single nanotube; all of the resulting fibers were found to be of the same diameter as the original nanotube and are expected to be of the same type as the original nanotube. [ 27 ]
Super-growth CVD (water-assisted chemical vapor deposition) was developed by Kenji Hata, Sumio Iijima and co-workers at AIST , Japan. [ 28 ] In this process, the activity and lifetime of the catalyst are enhanced by the addition of water into the CVD reactor. Dense millimeter-tall vertically aligned nanotube arrays (VANTAs) or "forests", aligned normal to the substrate, were produced. The forests' height could be expressed, as
where β is the initial growth rate and τ o {\displaystyle {\tau }_{o}} is the characteristic catalyst lifetime. [ 29 ]
Their specific surface exceeds 1,000 m 2 /g (capped) or 2,200 m 2 /g (uncapped), [ 30 ] surpassing the value of 400–1,000 m 2 /g for HiPco samples. The synthesis efficiency is about 100 times higher than for the laser ablation method. The time required to make SWNT forests of the height of 2.5 mm by this method was 10 minutes in 2004. Those SWNT forests can be easily separated from the catalyst, yielding clean SWNT material (purity >99.98%) without further purification. For comparison, the as-grown HiPco CNTs contain about 5–35% [ 31 ] of metal impurities; it is therefore purified through dispersion and centrifugation that damages the nanotubes. Super-growth avoids this problem. Patterned highly organized single-walled nanotube structures were successfully fabricated using the super-growth technique.
The super-growth method is essentially a variation of CVD. Therefore, it is possible to grow material containing SWNT, DWNTs and MWNTs, and to alter their ratios by tuning the growth conditions. Their ratios change by the thinness of the catalyst. Many MWNTs are included so that the diameter of the tube is wide. [ 32 ]
The vertically aligned nanotube forests originate from a "zipping effect" when they are immersed in a solvent and dried. The zipping effect is caused by the surface tension of the solvent and the van der Waals forces between the carbon nanotubes. It aligns the nanotubes into a dense material, which can be formed in various shapes, such as sheets and bars, by applying weak compression during the process. Densification increases the Vickers hardness by about 70 times and density is 0.55 g/cm 3 . The packed carbon nanotubes are more than 1 mm long and have a carbon purity of 99.9% or higher; they also retain the desirable alignment properties of the nanotubes forest. [ 33 ]
In 2015, researchers in the George Washington University discovered a new pathway to synthesize MWCNTs by electrolysis of molten carbonates. [ 34 ] The mechanism is similar to CVD. Some metal ions were reduced to a metal form and attached on the cathode as the nucleation point for the growing of CNTs. The reaction on the cathode is
The formed lithium oxide can in-situ absorb carbon dioxide (if present) and form lithium carbonate , as shown in the equation.
Thus the net reaction is
In other words, the reactant is only greenhouse gas of carbon dioxide, while the product is high valued CNTs. This discovery was highlighted as a possible technology for carbon dioxide capture and conversion. [ 35 ] [ 36 ] [ 37 ] Later on non-lithium molten carbonate electrolytes were demonstrated or electrolyte consisting of lithium carbonate plus some other carbonate and/or additive. [ 38 ] Additionally, by changing electrolysis conditions such as electrolyte, electrode, temperature, and/or current density, a wide range of carbon nanotubes can be grown through this process including: helical; thin; thick; doped with either nitrogen, boron, sulfur, or phosphorus; bulbous; and more with multiple macrostructures being produced, some quite porous with potential uses as sponge or electrodes. [ 39 ] [ 40 ] [ 41 ] [ 42 ] [ 43 ] [ 44 ] [ 45 ] [ 46 ] [ 47 ] This method can also utilize non-gas source of carbon, such as from calcium carbonate (CaCO 3 ), in which case it produces lime/cement (CaO) free of CO 2 as that CO 2 turns into CNTs and oxygen. [ 48 ]
Fullerenes and carbon nanotubes are not necessarily products of high-tech laboratories; they are commonly formed in such mundane places as ordinary flames , [ 49 ] produced by burning methane, [ 50 ] ethylene, [ 51 ] and benzene, [ 52 ] and they have been found in soot from both indoor and outdoor air. [ 53 ] However, these naturally occurring varieties can be highly irregular in size and quality because the environment in which they are produced is often highly uncontrolled. Thus, although they can be used in some applications, they can lack in the high degree of uniformity necessary to satisfy the many needs of both research and industry. Recent efforts have focused on producing more uniform carbon nanotubes in controlled flame environments. [ 54 ] [ 55 ] [ 56 ] [ 57 ] Such methods have promise for large-scale, low-cost nanotube synthesis based on theoretical models, [ 58 ] though they must compete with rapidly developing large scale CVD production.
Nanoscale metal catalysts are important ingredients for fixed- and fluidized-bed CVD synthesis of CNTs. They allow increasing the growth efficiency of CNTs and may give control over their structure and chirality . [ 60 ] During synthesis, catalysts can convert carbon precursors into tubular carbon structures but can also form encapsulating carbon overcoats. Together with metal oxide supports they may therefore attach to or become incorporated into the CNT product. [ 61 ] The presence of metal impurities can be problematic for many applications. Especially catalyst metals like nickel , cobalt or yttrium may be of toxicological concern. [ 62 ] While unencapsulated catalyst metals may be readily removable by acid washing, encapsulated ones require oxidative treatment for opening their carbon shell. [ 63 ] The effective removal of catalysts, especially of encapsulated ones, while preserving the CNT structure is a challenge and has been addressed in many studies. [ 64 ] [ 65 ] A new approach to break carbonaceous catalyst encapsulations is based on rapid thermal annealing. [ 66 ]
Many electronic applications of carbon nanotubes crucially rely on techniques of selectively producing either semiconducting or metallic CNTs, preferably of a certain chirality. [ 67 ] Several methods of separating semiconducting and metallic CNTs are known, but most of them are not yet suitable for large-scale technological processes. The most efficient method relies on density-gradient ultracentrifugation , which separates surfactant-wrapped nanotubes by the minute difference in their density. This density difference often translates into a difference in the nanotube diameter and (semi)conducting properties. [ 59 ] Another method of separation uses a sequence of freezing, thawing, and compression of SWNTs embedded in agarose gel. This process results in a solution containing 70% metallic SWNTs and leaves a gel containing 95% semiconducting SWNTs. The diluted solutions separated by this method show various colors. [ 68 ] The separated carbon nanotubes using this method have been applied to electrodes, e.g. electric double-layer capacitor. [ 69 ] Moreover, SWNTs can be separated by the column chromatography method. Yield is 95% in semiconductor type SWNT and 90% in metallic type SWNT. [ 70 ]
In addition to the separation of semiconducting and metallic SWNTs, it is possible to sort SWNTs by length, diameter, and chirality. The highest resolution length sorting, with length variation of <10%, has thus far been achieved by size-exclusion chromatography (SEC) of DNA-dispersed carbon nanotubes (DNA-SWNT). [ 71 ] SWNT diameter separation has been achieved by density-gradient ultracentrifugation (DGU) [ 72 ] using surfactant-dispersed SWNTs and by ion-exchange chromatography (IEC) for DNA-SWNT. [ 73 ] Purification of individual chiralities has also been demonstrated with IEC of DNA-SWNT: specific short DNA oligomers can be used to isolate individual SWNT chiralities. Thus far, 12 chiralities have been isolated at purities ranging from 70% for (8,3) and (9,5) SWNTs to 90% for (6,5), (7,5) and (10,5) SWNTs. [ 74 ] Alternatively, carbon nanotubes have been successfully sorted by chirality using the aqueous two-phase extraction method. [ 75 ] [ 76 ] [ 77 ] There have been successful efforts to integrate these purified nanotubes into electronic devices, such as field-effect transistors . [ 78 ]
An alternative to separation is the development of a selective growth of semiconducting or metallic CNTs. This can be achieved by CVD that involves a combination of ethanol and methanol gases on a quartz substrate, resulting in horizontally aligned arrays of 95–98% semiconducting nanotubes. [ 79 ]
Nanotubes are usually grown on nanoparticles of magnetic metal (Fe, Co), which facilitates the production of electronic ( spintronic ) devices. In particular, control of current through a field-effect transistor by magnetic field has been demonstrated in such a single-tube nanostructure . [ 80 ] | https://en.wikipedia.org/wiki/Synthesis_of_carbon_nanotubes |
Synthesis of nucleosides involves the coupling of a nucleophilic, heterocyclic base with an electrophilic sugar. The silyl-Hilbert-Johnson (or Vorbrüggen) reaction, which employs silylated heterocyclic bases and electrophilic sugar derivatives in the presence of a Lewis acid, is the most common method for forming nucleosides in this manner. [ 1 ]
Nucleosides are typically synthesized through the coupling of a nucleophilic pyrimidine , purine , or other basic heterocycle with a derivative of ribose or deoxyribose that is electrophilic at the anomeric carbon. When an acyl-protected ribose is employed, selective formation of the β-nucleoside (possessing the S configuration at the anomeric carbon) results from neighboring group participation. Stereoselective synthesis of deoxyribonucleosides directly from deoxyribose derivatives is more difficult to achieve because neighboring group participation cannot take place.
Three general methods have been used to synthesize nucleosides from nucleophilic bases and electrophilic sugars. The fusion method involves heating the base and acetyl-protected 1-acetoxyribose to 155 °C and results in the formation of the nucleoside with a maximum yield of 70%. [ 2 ]
(1)
The metal salt method involves the combination of a metal salt of the heterocycle with a protected sugar halide. Silver [ 3 ] and mercury [ 4 ] salts were originally used; however, more recently developed methods use sodium salts. [ 5 ]
(2)
The silyl-Hilbert-Johnson (SHJ) reaction (or Vorbrüggen reaction), the mildest general method for the formation of nucleosides, is the combination of a silylated heterocycle and protected sugar acetate (such as 1-O-acetyl-2,3,5-tri-O-benzoyl-beta-D-ribofuranose ) in the presence of a Lewis acid. [ 6 ] Problems associated with the insolubility of the heterocyclic bases and their metal salts are avoided; however, site selectivity is sometimes a problem when heterocycles containing multiple basic sites are used, as the reaction is often reversible.
(3)
The mechanism of the SHJ reaction begins with the formation of the key cyclic cation 1 . Nucleophilic attack at the anomeric position by the most nucleophilic nitrogen (N 1 ) then occurs, yielding the desired β-nucleoside 2 . [ 7 ] A second reaction of this nucleoside with 1 generates bis(riboside) 3 . Depending on the nature of the Lewis acid used, coordination of the nucleophile to the Lewis acid may be significant. Reaction of this "blocked" nucleophile with 1 results in undesired constitutional isomer 4 , which may undergo further reaction to 3 . [ 8 ] Generally Lewis acid coordination is not a problem when a Lewis acid such as trimethylsilyl triflate is used; it is much more important when a stronger Lewis acid like tin(IV) chloride is employed. [ 7 ]
(4)
2-Deoxysugars are unable to form the cyclic cation intermediate 1 because of their missing benzoyl group; instead, under Lewis acidic conditions they form a resonance-stabilized oxocarbenium ion. The diastereoselectivity of nucleophilic attack on this intermediate is much lower than the stereoselectivity of attack on cyclic cation 1 . Because of this low stereoselectivity, deoxyribonucleosides are usually synthesized using methods other than the SHJ reaction. [ 9 ]
The silyl-Hilbert-Johnson reaction is the most commonly used method for the synthesis of nucleosides from heterocyclic and sugar-based starting materials. However, the reaction suffers from some issues that are not associated with other methods, such as unpredictable site selectivity in some cases (see below). This section describes both derivatives of and alternatives to the SHJ reaction that are used for the synthesis of nucleosides.
Because most heterocyclic bases contain multiple nucleophilic sites, site selectivity is an important issue in nucleoside synthesis. Purine bases, for instance, react kinetically at N 3 and thermodynamically at N 1 (see Eq. (4)). [ 4 ] Glycosylation of thymine with protected 1-acetoxy ribose produced 60% of the N 1 nucleoside and 23% of the N 3 nucleoside. Closely related triazines, on the other hand, react with complete selectivity to afford the N 2 nucleoside. [ 10 ]
(5)
The most nucleophilic nitrogen can be blocked through alkylation prior to nucleoside synthesis. Heating the blocked nucleoside in Eq. (6) in the presence of a protected sugar chloride provides the nucleoside in 59% yield. Reactions of this type are hampered by alkylation of the heterocycle by incipient alkyl chloride. [ 11 ]
(6)
Silylated heterocyclic bases are susceptible to hydrolysis and somewhat difficult to handle as a result; thus, the development of a one-pot, one-step method for silylation and nucleoside synthesis represented a significant advance. [ 12 ] The combination of trifluoroacetic acid (TFA), trimethylsilyl chloride (TMSCl), and hexamethyldisilazide (HMDS) generates trimethylsilyl trifluoroacetate in situ , which accomplishes both the silylation of the heterocycle and its subsequent coupling with the sugar. [ 13 ]
(7)
Transglycosylation, which involves the reversible transfer of a sugar moiety from one heterocyclic base to another, is effective for the conversion of pyrimidine nucleosides to purine nucleosides. Most other transglycosylation reactions are low yielding due to a small thermodynamic difference between equilibrating nucleosides. [ 14 ]
(8)
Deoxyribose-derived electrophiles are unable to form the cyclic cation 1 ; as a result, the stereoselective synthesis of deoxyribonucleosides is more difficult than the synthesis of ribonucleosides. One solution to this problem involves the synthesis of a ribonucleoside, followed by protection of the 3'- and 5'-hydroxyl groups, removal of the 2'-hydroxyl group through a Barton deoxygenation, and deprotection. [ 15 ]
(9)
A useful alternative to the methods described here that avoids the site selectivity concerns of the SHJ reaction is tandem Michael reaction/cyclization to simultaneously form the heterocyclic base and establish its connection to the sugar moiety. [ 16 ]
(10)
A second alternative is enzymatic transglycosylation, which is completely kinetically controlled (avoiding issues of chemical transglycosylation associated with thermodynamic control). However, operational complications associated with the use of enzymes are a disadvantage of this method. [ 17 ]
(11)
The sugar derivatives used for SHJ reactions should be purified, dried, and powdered before use. Heterocycles must not be too basic in order to avoid excessive complexation with the Lewis acid; amino-substituted heterocycles such as cytosine, adenine, and guanine react slowly or not at all under SHJ conditions (although their N -acetylated derivatives react more rapidly).
Silylation is most commonly accomplished using HMDS, which evolves ammonia as the only byproduct of silylation. Catalytic or stoichiometric [ 18 ] amounts of acidic additives such as trimethylsilyl chloride accelerate silylation; when such an additive is used, ammonium salts will appear in the reaction as a turbid impurity.
Lewis acids should be distilled immediately before use for best results. More than about 1.2-1.4 equivalents of Lewis acid are rarely needed. Acetonitrile is the most common solvent employed for these reactions, although other polar solvents are also common. Workup of reactions employing TMSOTf involves treatment with an ice-cold solution of sodium bicarbonate and extraction of the resulting sodium salts. When tin(IV) chloride is used in 1,2-dichloroethane, workup involves the addition of pyridine and filtering of the resulting pyridine-tin complex, followed by extraction with aqueous sodium bicarbonate. [ 19 ]
(12)
To a stirred mixture of 13.5 mL (4.09 mmol) of a 0.303 N standard solution of silylated N 2 -acetylguanine in 1,2-dichloroethane and 1.86 g (3.7 mmol) of benzoate-protected 1-acetoxy ribose in 35 mL of 1,2-dichloroethane was added 6.32 mL (4.46 mmol) of a 0.705 N standard solution of TMSOTf in 1,2-dichloroethane. The reaction mixture was heated at reflux for 1.5–4 hours, and then diluted with CH 2 Cl 2 . On workup with ice-cold NaHCO 3 solution, there was obtained 2.32 g of crude product, which was kept for 42 hours in 125 mL of methanolic ammonia at 24°. After workup, recrystallization from H 2 O gave, in two crops, 0.69 g (66%) of pure guanosine, which was homogeneous (R f 0.3) in the partition system n -butanol:acetic acid:H 2 O (5:1:4) and whose 1 H NMR spectrum at 400 MHz in D 2 O showed only traces of the undesired N 7 -anomer of guanosine. 1 H NMR (CDCl 3 ): δ 3.55, 3.63, 3.90, 4.11, 4.43, 5.10, 5.20, 5.45, 5.72, 6.52, 7.97, 10.75.
In order to understand how life arose, knowledge is required of the chemical pathways that permit formation of the key building blocks of life under plausible prebiotic conditions . Nam et al. [ 21 ] demonstrated the direct condensation of nucleobases with ribose to give ribonucleosides in aqueous microdroplets, a key step leading to RNA formation. Also, a plausible prebiotic process for synthesizing pyrimidine and purine ribonucleosides and ribonucleotides using wet-dry cycles was presented by Becker et al. [ 22 ] | https://en.wikipedia.org/wiki/Synthesis_of_nucleosides |
The synthesis of precious metals involves the use of either nuclear reactors or particle accelerators to produce these elements.
Ruthenium and rhodium are precious metals produced as a small percentage of the fission products from the nuclear fission of uranium . The longest half-lives of the radioisotopes of these elements generated by nuclear fission are 373.59 days for ruthenium and 45 days for rhodium [ clarification needed ] . This makes the extraction of the non-radioactive isotope from spent nuclear fuel possible after a few years of storage, although the extract must be checked for radioactivity from trace quantities of other elements before use. [ 1 ]
Each kilogram of the fission products of 235 U will contain 63.44 grams of ruthenium isotopes with halflives longer than a day. Since a typical used nuclear fuel contains about 3% fission products, one ton of used fuel will contain about 1.9 kg of ruthenium. The 103 Ru and 106 Ru will render the fission ruthenium very radioactive. If the fission occurs in an instant then the ruthenium thus formed will have an activity due to 103 Ru of 109 TBq g −1 and 106 Ru of 1.52 TBq g −1 . 103 Ru has a half-life of about 39 days meaning that within 390 days it will have effectively decayed to the only stable isotope of rhodium, 103 Rh, well before any reprocessing is likely to occur. 106 Ru has a half-life of about 373 days, meaning that if the fuel is left to cool for 5 years before reprocessing only about 3% of the original quantity will remain; the rest will have decayed. [ 1 ] For comparison, the activity in natural potassium (due to naturally occurring 40 K ) is about 30 Bq per gram. [ 2 ]
It is possible to extract rhodium from used nuclear fuel : 1 kg of fission products of 235 U contains 13.3 grams of 103 Rh. At 3% fission products by weight, one ton of used fuel will contain about 400 grams of rhodium. The longest lived radioisotope of rhodium is 102m Rh with a half-life of 2.9 years, while the ground state ( 102 Rh) has a half-life of 207 days. [ 1 ]
Each kilogram of fission rhodium will contain 6.62 ng of 102 Rh and 3.68 ng of 102m Rh. As 102 Rh decays by beta decay to either 102 Ru (80%) (some positron emission will occur) or 102 Pd (20%) (some gamma ray photons with about 500 keV are generated) and the excited state decays by beta decay (electron capture) to 102 Ru (some gamma ray photons with about 1 MeV are generated). If the fission occurs in an instant then 13.3 grams of rhodium will contain 67.1 MBq (1.81 mCi) of 102 Rh and 10.8 MBq (291 μCi) of 102m Rh. As it is normal to allow used nuclear fuel to stand for about five years before reprocessing, much of this activity will decay away leaving 4.7 MBq of 102 Rh and 5.0 MBq of 102m Rh. If the rhodium metal was then left for 20 years after fission, the 13.3 grams of rhodium metal would contain 1.3 kBq of 102 Rh and 500 kBq of 102m Rh. Rhodium has the highest price of these precious metals ($440,000/kg in 2022 [ 3 ] ), but the cost of the separation of the rhodium from the other metals needs to be considered [ editorializing ] , although recent high prices may create opportunity for consideration. [ 1 ]
Chrysopoeia , the artificial production of gold , is the traditional goal of alchemy . Such transmutation is possible in particle accelerators or nuclear reactors, although the production cost is estimated to be a trillion times the market price of gold. Since there is only one stable gold isotope, 197 Au, nuclear reactions must create this isotope in order to produce usable gold. [ 4 ]
Gold was synthesized from mercury by neutron bombardment in 1941, but the isotopes of gold produced were all radioactive . [ 5 ] In 1924, a German scientist, Adolf Miethe , reported achieving the same feat, but after various replication attempts around the world, it was deemed an experimental error. [ 6 ] [ 7 ] [ 8 ]
In 1980, Glenn Seaborg , K. Aleklett, and the Bevatron team transmuted several thousand atoms of bismuth into gold at the Lawrence Berkeley National Laboratory . His experimental technique using carbon-12 and neon-20 nuclei was able to remove protons and neutrons from the bismuth atoms. Seaborg's technique was far too expensive to enable the routine manufacture of gold but his work was then the closest yet to emulating an aspect of the mythical Philosopher's stone . [ 9 ] [ 10 ]
In 2002 and 2004, CERN scientists at the Super Proton Synchrotron reported producing a minuscule amount of gold nuclei from induced photon emissions within deliberate near-miss collisions of lead nuclei. [ 11 ] [ 12 ] In 2022, CERN's ISOLDE team reported producing 18 gold nuclei from proton bombardment of a uranium target. [ 13 ] In 2025, CERN's ALICE experiment team announced that over the previous decade, they had used the Large Hadron Collider to replicate the 2002 SPS mechanisms at higher energies. A total of roughly 260 billion gold nuclei were created over three experiment runs, a miniscule amount massing about 90 picograms. [ 14 ] [ 15 ] | https://en.wikipedia.org/wiki/Synthesis_of_precious_metals |
The Synthetic Biology Open Language ( SBOL ) [ 1 ] is a proposed data standard for exchanging synthetic biology designs between software packages . [ 2 ] It has been under development by the SBOL Developers Group [ 3 ] since 2008. This group aims to develop the standard in a way that is open and democratic in order to include as many interests as possible and to avoid domination by a single company. The group also aims to develop and improve the design standard over time as the field of synthetic biology reflects this development.
A graphical modeling language called SBOL Visual has also been created to visualize SBOL designs. [ 4 ]
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synthetic_Biology_Open_Language |
The Synthetic Gene Database ( SGDB ) is a database of artificially engineered genes . [ 1 ]
This Biological database -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Synthetic_Gene_Database |
Synthetic Metals is a peer-reviewed scientific journal covering electronic polymers and electronic molecular materials.
Synthetic Metals is abstracted and indexed in the following services:
According to the Journal Citation Reports , the journal has a 2020 impact factor of 3.266. It has published several highly cited papers (1 with ~1000 citations; [ 1 ] 5 with >600 citations; 30 with >200 citations, according to Web of Science ); most of them are devoted to conductive polymers (especially polyaniline ) and one to optical properties of carbon nanotubes [ 2 ] (see Kataura plot ).
This article about a materials science journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Synthetic_Metals |
Synthetic Reaction Updates was a current awareness bibliographic database from the Royal Society of Chemistry that provided alerts of recently published developments in synthetic organic chemistry .
It covered primary research in general and organic chemistry published in chemistry journals . Each record contains a reaction scheme, as well as bibliographic data and a link to the original article on the publisher's website. [ 1 ] Subscribers were able to search by topic and reaction type or register for email alerts of new content based on their search preferences. [ 2 ]
The database was established in 2015 to replace the two discontinued databases Methods in Organic Synthesis ( ISSN 0265-4245 ) and Catalysts and Catalysed Reactions ( ISSN 1474-9173 . [ 2 ]
Methods in Organic Synthesis was an online database that was established in 1998 and updated weekly with the latest developments in organic synthesis . It was also available as a monthly print bulletin.
Catalysts & Catalysed Reactions was a monthly current-awareness journal that was published from 2002 to 2014. It covered the research areas of catalysed reactions and catalysts . | https://en.wikipedia.org/wiki/Synthetic_Reaction_Updates |
Many opportunities exist for the application of synthetic biodegradable polymers in the biomedical area particularly in the fields of tissue engineering and controlled drug delivery . Degradation is important in biomedicine for many reasons. Degradation of the polymeric implant means surgical intervention may not be required in order to remove the implant at the end of its functional life, eliminating the need for a second surgery. [ 1 ] In tissue engineering , biodegradable polymers can be designed such to approximate tissues, providing a polymer scaffold that can withstand mechanical stresses, provide a suitable surface for cell attachment and growth, and degrade at a rate that allows the load to be transferred to the new tissue. [ 2 ] [ 3 ] In the field of controlled drug delivery, biodegradable polymers offer tremendous potential either as a drug delivery system alone or in conjunction to functioning as a medical device . [ 4 ]
In the development of applications of biodegradable polymers, the chemistry of some polymers including synthesis and degradation is reviewed below. A description of how properties can be controlled by proper synthetic controls such as copolymer composition, special requirements for processing and handling, and some of the commercial devices based on these materials are discussed.
When investigating the selection of the polymer for biomedical applications, important criteria to consider are;
Mechanical performance of a biodegradable polymer depends on various factors which include monomer selection, initiator selection, process conditions and the presence of additives. These factors influence the polymers crystallinity , melt and glass transition temperatures and molecular weight . Each of these factors needs to be assessed on how they affect the biodegradation of the polymer. [ 5 ] Biodegradation can be accomplished by synthesizing polymers with hydrolytically unstable linkages in the backbone. This is commonly achieved by the use of chemical functional groups such as esters , anhydrides , orthoesters and amides . Most biodegradable polymers are synthesized by ring opening polymerization.
Biodegradable polymers can be melt processed by conventional means such as compression or injection molding . Special consideration must be given to the need to exclude moisture from the material. Care must be taken to dry the polymers before processing to exclude humidity. As most biodegradable polymers have been synthesized by ring opening polymerization, a thermodynamic equilibrium exists between the forward polymerization reaction and the reverse reaction that results in monomer formation. Care needs to be taken to avoid an excessively high processing temperature that may result in monomer formation during the molding and extrusion process. It must be followed carefully. Resorbable polymers can also be 3D printed . [ 6 ]
Once implanted, a biodegradable device should maintain its mechanical properties until it is no longer needed and then be absorbed by the body leaving no trace. The backbone of the polymer is hydrolytically unstable. That is, the polymer is unstable in a water based environment. This is the prevailing mechanism for the polymers degradation. This occurs in two stages.
1. Water penetrates the bulk of the device, attacking the chemical bonds in the amorphous phase and converting long polymer chains into shorter water-soluble fragments. This causes a reduction in molecular weight without the loss of physical properties as the polymer is still held together by the crystalline regions. Water penetrates the device leading to metabolization of the fragments and bulk erosion .
2. Surface erosion of the polymer occurs when the rate at which the water penetrating the device is slower than the rate of conversion of the polymer into water-soluble materials.
Biomedical engineers can tailor a polymer to slowly degrade and transfer stress at the appropriate rate to surrounding tissues as they heal by balancing the chemical stability of the polymer backbone, the geometry of the device, and the presence of catalysts, additives or plasticisers.
Biodegradable polymers are used commercially in both the tissue engineering and drug delivery field of biomedicine. Specific applications include. | https://en.wikipedia.org/wiki/Synthetic_biodegradable_polymer |
Synthetic biological circuits are an application of synthetic biology where biological parts inside a cell are designed to perform logical functions mimicking those observed in electronic circuits . Typically, these circuits are categorized as either genetic circuits , RNA circuits , or protein circuits , depending on the types of biomolecule that interact to create the circuit's behavior. The applications of all three types of circuit range from simply inducing production to adding a measurable element, like green fluorescent protein , to an existing natural biological circuit , to implementing completely new systems of many parts. [ 1 ]
The goal of synthetic biology is to generate an array of tunable and characterized parts, or modules, with which any desirable synthetic biological circuit can be easily designed and implemented. [ 2 ] These circuits can serve as a method to modify cellular functions, create cellular responses to environmental conditions, or influence cellular development. By implementing rational, controllable logic elements in cellular systems, researchers can use living systems as engineered " biological machines " to perform a vast range of useful functions. [ 1 ]
The first natural gene circuit studied in detail was the lac operon . In studies of diauxic growth of E. coli on two-sugar media, Jacques Monod and Francois Jacob discovered that E.coli preferentially consumes the more easily processed glucose before switching to lactose metabolism. They discovered that the mechanism that controlled the metabolic "switching" function was a two-part control mechanism on the lac operon. When lactose is present in the cell the enzyme β-galactosidase is produced to convert lactose into glucose or galactose . When lactose is absent in the cell the lac repressor inhibits the production of the enzyme β-galactosidase to prevent any inefficient processes within the cell.
The lac operon is used in the biotechnology industry for production of recombinant proteins for therapeutic use. The gene or genes for producing an exogenous protein are placed on a plasmid under the control of the lac promoter. Initially the cells are grown in a medium that does not contain lactose or other sugars, so the new genes are not expressed. Once the cells reach a certain point in their growth, isopropyl β-D-1-thiogalactopyranoside (IPTG) is added. IPTG, a molecule similar to lactose, but with a sulfur bond that is not hydrolyzable so that E. coli does not digest it, is used to activate or " induce " the production of the new protein. Once the cells are induced, it is difficult to remove IPTG from the cells and therefore it is difficult to stop expression.
Two early examples of synthetic biological circuits were published in Nature in 2000. One, by Tim Gardner, Charles Cantor, and Jim Collins working at Boston University , demonstrated a "bistable" switch in E. coli . The switch is turned on by heating the culture of bacteria and turned off by addition of IPTG. They used green fluorescent protein as a reporter for their system. [ 3 ] The second, by Michael Elowitz and Stanislas Leibler , showed that three repressor genes could be connected to form a negative feedback loop termed the Repressilator that produces self-sustaining oscillations of protein levels in E. coli. [ 4 ]
Currently, synthetic circuits are a burgeoning area of research in systems biology with more publications detailing synthetic biological circuits published every year. [ 5 ] There has been significant interest in encouraging education and outreach as well: the International Genetically Engineered Machines Competition [ 6 ] manages the creation and standardization of BioBrick parts as a means to allow undergraduate and high school students to design their own synthetic biological circuits.
Both immediate and long term applications exist for the use of synthetic biological circuits, including different applications for metabolic engineering , and synthetic biology . Those demonstrated successfully include pharmaceutical production, [ 7 ] and fuel production. [ 8 ] However, methods involving direct genetic introduction are not inherently effective without invoking the basic principles of synthetic cellular circuits. For example, each of these successful systems employs a method to introduce all-or-none induction or expression. This is a biological circuit where a simple repressor or promoter is introduced to facilitate creation of the product, or inhibition of a competing pathway. However, with the limited understanding of cellular networks and natural circuitry, implementation of more robust schemes with more precise control and feedback is hindered. Therein lies the immediate interest in synthetic cellular circuits.
Development in understanding cellular circuitry can lead to exciting new modifications, such as cells which can respond to environmental stimuli. For example, cells could be developed that signal toxic surroundings and react by activating pathways used to degrade the perceived toxin. [ 9 ] To develop such a cell, it is necessary to create a complex synthetic cellular circuit which can respond appropriately to a given stimulus.
Given synthetic cellular circuits represent a form of control for cellular activities, it can be reasoned that with complete understanding of cellular pathways, "plug and play" [ 1 ] cells with well defined genetic circuitry can be engineered. It is widely believed that if a proper toolbox of parts is generated, [ 10 ] synthetic cells can be developed implementing only the pathways necessary for cell survival and reproduction. From this cell, to be thought of as a minimal genome cell, one can add pieces from the toolbox to create a well defined pathway with appropriate synthetic circuitry for an effective feedback system. Because of the basic ground up construction method, and the proposed database of mapped circuitry pieces, techniques mirroring those used to model computer or electronic circuits can be used to redesign cells and model cells for easy troubleshooting and predictive behavior and yields.
Elowitz et al. and Fung et al. created oscillatory circuits that use multiple self-regulating mechanisms to create a time-dependent oscillation of gene product expression. [ 11 ] [ 12 ]
Gardner et al. used mutual repression between two control units to create an implementation of a toggle switch capable of controlling cells in a bistable manner: transient stimuli resulting in persistent responses. [ 3 ]
Gene regulation is an essential part of developmental processes. During development, genes are turned on and off in different tissues, changes in regulatory mechanisms may result in genetic switching in a bistable system, the gene switches serve as regulatory molecule binding sites. These are proteins that activate transcription when they land on a gene switch and thereby express the gene that was expected to operate as a memory device, allowing cell fate decisions to be chosen and maintained. [ 13 ]
Toggle switch which operates using two mutually inhibitory genes, each promoter is inhibited by the repressor that is transcribed by the opposing promoter. Toggle switch design: Inducer 1 inactivates repressor 1, which means repressor 2 is produced. Repressor 2, in turn, stops transcription of the repressor 1 gene and the reporter gene. [ 14 ]
Using negative feedback and identical promoters, linearizer gene circuits can impose uniform gene expression that depends linearly on extracellular chemical inducer concentration. [ 17 ]
Synthetic gene circuits can control gene expression heterogeneity can be controlled independently of the gene expression mean. [ 18 ]
Engineered systems are the result of implementation of combinations of different control mechanisms. A limited counting mechanism was implemented by a pulse-controlled gene cascade [ 19 ] and application of logic elements enables genetic "programming" of cells as in the research of Tabor et al., which synthesized a photosensitive bacterial edge detection program. [ 20 ]
Recent developments in artificial gene synthesis and the corresponding increase in competition within the industry have led to a significant drop in price and wait time of gene synthesis and helped improve methods used in circuit design. [ 21 ] At the moment, circuit design is improving at a slow pace because of insufficient organization of known multiple gene interactions and mathematical models. This issue is being addressed by applying computer-aided design (CAD) software to provide multimedia representations of circuits through images, text and programming language applied to biological circuits. [ 22 ] Some of the more well known CAD programs include GenoCAD, Clotho framework and j5. [ 23 ] [ 24 ] [ 25 ] GenoCAD uses grammars, which are either opensource or user generated "rules" which include the available genes and known gene interactions for cloning organisms. Clotho framework uses the Biobrick standard rules. [ 22 ] | https://en.wikipedia.org/wiki/Synthetic_biological_circuit |
Synthetic biopolymers : human-made copies of biopolymers obtained by abiotic chemical routes.
Artificial polymer : human-made polymer that is not a biopolymer
Synthetic biopolymers are human-made copies of biopolymers obtained by abiotic chemical routes. [ 1 ] Synthetic biopolymer of different chemical nature have been obtained, including polysaccharides , [ 2 ] glycoproteins , [ 3 ] peptides and proteins , [ 4 ] [ 5 ] polyhydroxoalkanoates , [ 6 ] polyisoprenes . [ 7 ]
The high molecular weight of biopolymers make their synthesis inherently laborious. Further challenges can arise from specific spatial arrangement adopted by the natural biopolymer, which may be vital for its properties/activity but not easily reproducible in the synthetic copy. Despite this, chemical approaches to obtain biopolymer are highly desirable to overcome issues arising from low abundance of the target biopolymer in Nature , the need for cumbersome isolation processes or high batch-to-batch variability or inhomogeneity of the naturally-sourced species. [ 8 ]
Human-made biopolymers obtained through approaches that involve genetic engineering or recombinant DNA technology are different from synthetic biopolymers and should be referred to as artificial biopolymer ( e.g. , artificial protein, artificial polynucleotide, etc.). [ 1 ]
As their natural analogues, synthetic biopolymers find applications in numerous fields, including materials for commodities, drug delivery, tissue engineering, therapeutic and diagnostic applications. [ citation needed ] | https://en.wikipedia.org/wiki/Synthetic_biopolymer |
In 1965, Chinese scientists first synthesized crystalline bovine insulin ( Chinese : 人工合成结晶牛胰岛素 ), which was the first functional crystalline protein being fully synthesized in the world. Research on synthesizing bovine insulin started on 1958. Members in the research group were from the Chemistry Department of Beijing University ( Chinese : 北京大学化学系 ), Shanghai Institute of Biochemistry, CAS ( Chinese : 中科院上海生物化学研究所 ) and Shanghai Institute of Organic Chemistry, CAS ( Chinese : 中科院上海有机化学研究所 ). [ 1 ]
Insulin is a protein ( peptide ) consisting of two chain, A and B. Chain A consists of 21 amino acid residues while chain consists of 30 amino acid residues. The main function of insulin is to regulate the concentrate of sugar in blood. Type 1 diabetes are caused by dysfunction on the synthesis or secretory of insulin while injecting insulin can treat type 1 diabetes. [ 2 ]
In 1979, Wang Yinglai , the project's lead scientist, nominated Niu Jingyi , a team member who had made significant contributions, for the Nobel Chemistry Prize, but the nomination was unsuccessful. [ 3 ] [ 4 ] | https://en.wikipedia.org/wiki/Synthetic_crystalline_bovine_insulin |
Synthetic data are artificially generated rather than produced by real-world events. Typically created using algorithms, synthetic data can be deployed to validate mathematical models and to train machine learning models. [ 1 ]
Data generated by a computer simulation can be seen as synthetic data. This encompasses most applications of physical modeling, such as music synthesizers or flight simulators. The output of such systems approximates the real thing, but is fully algorithmically generated.
Synthetic data is used in a variety of fields as a filter for information that would otherwise compromise the confidentiality of particular aspects of the data. In many sensitive applications, datasets theoretically exist but cannot be released to the general public; [ 2 ] synthetic data sidesteps the privacy issues that arise from using real consumer information without permission or compensation.
Synthetic data is generated to meet specific needs or certain conditions that may not be found in the original, real data. One of the hurdles in applying up-to-date machine learning approaches for complex scientific tasks is the scarcity of labeled data, a gap effectively bridged by the use of synthetic data, which closely replicates real experimental data. [ 3 ] This can be useful when designing many systems, from simulations based on theoretical value, to database processors, etc. This helps detect and solve unexpected issues such as information processing limitations. Synthetic data are often generated to represent the authentic data and allows a baseline to be set. [ 4 ] Another benefit of synthetic data is to protect the privacy and confidentiality of authentic data, while still allowing for use in testing systems.
A science article's abstract, quoted below, describes software that generates synthetic data for testing fraud detection systems. "This enables us to create realistic behavior profiles for users and attackers. The data is used to train the fraud detection system itself, thus creating the necessary adaptation of the system to a specific environment." [ 4 ] In defense and military contexts, synthetic data is seen as a potentially valuable tool to develop and improve complex AI systems, particularly in contexts where high-quality real-world data is scarce. [ 5 ] At the same time, synthetic data together with the testing approach can give the ability to model real-world scenarios.
Scientific modelling of physical systems, which allows to run simulations in which one can estimate/compute/generate datapoints that haven't been observed in actual reality, has a long history that runs concurrent with the history of physics itself. For example, research into synthesis of audio and voice can be traced back to the 1930s and before, driven forward by the developments of e.g. the telephone and audio recording. Digitization gave rise to software synthesizers from the 1970s onwards [ citation needed ] .
In the context of privacy-preserving statistical analysis, in 1993, the idea of original fully synthetic data was created by Rubin . [ 6 ] Rubin originally designed this to synthesize the Decennial Census long form responses for the short form households. He then released samples that did not include any actual long form records - in this he preserved anonymity of the household. [ 7 ] Later that year, the idea of original partially synthetic data was created by Little. Little used this idea to synthesize the sensitive values on the public use file. [ 8 ]
A 1993 work [ 9 ] fitted a statistical model to 60,000 MNIST digits, then it was used to generate over 1 million examples. Those were used to train a LeNet-4 to reach state of the art performance. [ 10 ] : 173
In 1994, Fienberg came up with the idea of critical refinement, in which he used a parametric posterior predictive distribution (instead of a Bayes bootstrap) to do the sampling. [ 7 ] Later, other important contributors to the development of synthetic data generation were Trivellore Raghunathan , Jerry Reiter , Donald Rubin , John M. Abowd , and Jim Woodcock . Collectively they came up with a solution for how to treat partially synthetic data with missing data. Similarly they came up with the technique of Sequential Regression Multivariate Imputation . [ 7 ]
Researchers test the framework on synthetic data, which is "the only source of ground truth on which they can objectively assess the performance of their algorithms ". [ 11 ]
Synthetic data can be generated through the use of random lines, having different orientations and starting positions. [ 12 ] Datasets can get fairly complicated. A more complicated dataset can be generated by using a synthesizer build. To create a synthesizer build, first use the original data to create a model or equation that fits the data the best. This model or equation will be called a synthesizer build. This build can be used to generate more data. [ 13 ]
Constructing a synthesizer build involves constructing a statistical model . In a linear regression line example, the original data can be plotted, and a best fit linear line can be created from the data. This line is a synthesizer created from the original data. The next step will be generating more synthetic data from the synthesizer build or from this linear line equation. In this way, the new data can be used for studies and research, and it protects the confidentiality of the original data. [ 13 ]
David Jensen from the Knowledge Discovery Laboratory explains how to generate synthetic data: "Researchers frequently need to explore the effects of certain data characteristics on their data model ." [ 13 ] To help construct datasets exhibiting specific properties, such as auto-correlation or degree disparity, proximity can generate synthetic data having one of several types of graph structure: random graphs that are generated by some random process ; lattice graphs having a ring structure; lattice graphs having a grid structure, etc. [ 13 ] In all cases, the data generation process follows the same process:
Since the attribute values of one object may depend on the attribute values of related objects, the attribute generation process assigns values collectively. [ 13 ]
Testing and training fraud detection and confidentiality systems are devised using synthetic data. Specific algorithms and generators are designed to create realistic data, [ 14 ] which then assists in teaching a system how to react to certain situations or criteria. For example, intrusion detection software is tested using synthetic data. This data is a representation of the authentic data and may include intrusion instances that are not found in the authentic data. The synthetic data allows the software to recognize these situations and react accordingly. If synthetic data was not used, the software would only be trained to react to the situations provided by the authentic data and it may not recognize another type of intrusion. [ 4 ]
Researchers doing clinical trials or any other research may generate synthetic data to aid in creating a baseline for future studies and testing.
Real data can contain information that researchers may not want released, [ 15 ] so synthetic data is sometimes used to protect the privacy and confidentiality of a dataset. Using synthetic data reduces confidentiality and privacy issues since it holds no personal information and cannot be traced back to any individual.
Beyond privacy protection, synthetic data is also being explored for methodological innovation in drug development. For instance, synthetic data may be used to construct synthetic control arms as an alternative to conventional external control arms based on real-world data (RWD) or randomized controlled trials (RCTs). Collectively, regulatory agencies such as the FDA and EMA appear to be at various stages of recognizing and integrating AI-generated synthetic data into their methodologies. While there is growing consensus on the potential of such data to support model development and the broader lifecycle of medicinal products, to date no drug or medical device has been approved using solely or predominantly synthetic data—particularly not as a comparator arm generated entirely via data-driven algorithms. The quality and statistical handling of synthetic data are expected to become more prominent in future regulatory discussions, particularly in contexts such as predictive modeling (e.g., digital twins), where innovative approaches have already been referenced. [ 16 ]
Synthetic data is increasingly being used for machine learning applications: a model is trained on a synthetically generated dataset with the intention of transfer learning to real data. Efforts have been made to enable more data science experiments via the construction of general-purpose synthetic data generators, such as the Synthetic Data Vault. [ 17 ] In general, synthetic data has several natural advantages:
This usage of synthetic data has been proposed for computer vision applications, in particular object detection , where the synthetic environment is a 3D model of the object, [ 18 ] and learning to navigate environments by visual information.
At the same time, transfer learning remains a nontrivial problem, and synthetic data has not become ubiquitous yet. Research results indicate that adding a small amount of real data significantly improves transfer learning with synthetic data. Advances in generative adversarial networks (GAN), lead to the natural idea that one can produce data and then use it for training. Since at least 2016, such adversarial training has been successfully used to produce synthetic data of sufficient quality to produce state-of-the-art results in some domains, without even needing to re-mix real data in with the generated synthetic data. [ 19 ]
In 1987, a Navlab autonomous vehicle used 1200 synthetic road images as one approach to training. [ 20 ]
In 2021, Microsoft released a database of 100,000 synthetic faces based on (500 real faces) that claims to "match real data in accuracy". [ 20 ] [ 21 ]
In 2023, Nature (journal) published a cover of their Nature's 10 series designed by Kim Albrecht of the project "Artificial Worldviews." [ 22 ] The cover features a mapping of over 18,000 synthetically generated data points prompted from ChatGPT on the categories of knowledge. | https://en.wikipedia.org/wiki/Synthetic_data |
In algebra , synthetic division is a method for manually performing Euclidean division of polynomials , with less writing and fewer calculations than long division .
It is mostly taught for division by linear monic polynomials (known as Ruffini's rule ), but the method can be generalized to division by any polynomial .
The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, helping to prevent sign errors.
The first example is synthetic division with only a monic linear denominator x − a {\displaystyle x-a} .
The numerator can be written as p ( x ) = x 3 − 12 x 2 + 0 x − 42 {\displaystyle p(x)=x^{3}-12x^{2}+0x-42} .
The zero of the denominator g ( x ) {\displaystyle g(x)} is 3 {\displaystyle 3} .
The coefficients of p ( x ) {\displaystyle p(x)} are arranged as follows, with the zero of g ( x ) {\displaystyle g(x)} on the left:
The first coefficient after the bar is "dropped" to the last row.
The dropped number is multiplied by the number before the bar and placed in the next column .
An addition is performed in the next column.
The previous two steps are repeated, and the following is obtained:
Here, the last term (-123) is the remainder while the rest correspond to the coefficients of the quotient.
The terms are written with increasing degree from right to left beginning with degree zero for the remainder and the result.
Hence the quotient and remainder are:
The above form of synthetic division is useful in the context of the polynomial remainder theorem for evaluating univariate polynomials. To summarize, the value of p ( x ) {\displaystyle p(x)} at a {\displaystyle a} is equal to the remainder of the division of p ( x ) {\displaystyle p(x)} by x − a . {\displaystyle x-a.}
The advantage of calculating the value this way is that it requires just over half as many multiplication steps as naive evaluation. An alternative evaluation strategy is Horner's method .
This method generalizes to division by any monic polynomial with only a slight modification with changes in bold . Note that while it may not be displayed in the following example, the divisor must also be written with verbose coefficients. (Such as with 2 x 3 + 0 x 2 − 4 x + 8 {\displaystyle 2x^{3}+0x^{2}-4x+8} ) Using the same steps as before, perform the following division:
We concern ourselves only with the coefficients.
Write the coefficients of the polynomial to be divided at the top.
Negate the coefficients of the divisor.
Write in every coefficient but the first one on the left in an upward right diagonal (see next diagram).
Note the change of sign from 1 to −1 and from −3 to 3. "Drop" the first coefficient after the bar to the last row.
Multiply the dropped number by the diagonal before the bar and place the resulting entries diagonally to the right from the dropped entry.
Perform an addition in the next column.
Repeat the previous two steps until you would go past the entries at the top with the next diagonal .
Then simply add up any remaining columns.
Count the terms to the left of the bar. Since there are two, the remainder has degree one and this is the two right-most terms under the bar. Mark the separation with a vertical bar.
The terms are written with increasing degree from right to left beginning with degree zero for both the remainder and the result.
The result of our division is:
With a little prodding, the expanded technique may be generalised even further to work for any polynomial, not just monics . The usual way of doing this would be to divide the divisor g ( x ) {\displaystyle g(x)} with its leading coefficient (call it a ):
then using synthetic division with h ( x ) {\displaystyle h(x)} as the divisor, and then dividing the quotient by a to get the quotient of the original division (the remainder stays the same). But this often produces unsightly fractions which get removed later and is thus more prone to error. It is possible to do it without first reducing the coefficients of g ( x ) {\displaystyle g(x)} .
As can be observed by first performing long division with such a non-monic divisor, the coefficients of f ( x ) {\displaystyle f(x)} are divided by the leading coefficient of g ( x ) {\displaystyle g(x)} after "dropping", and before multiplying.
Let's illustrate by performing the following division:
A slightly modified table is used:
Note the extra row at the bottom. This is used to write values found by dividing the "dropped" values by the leading coefficient of g ( x ) {\displaystyle g(x)} (in this case, indicated by the /3 ; note that, unlike the rest of the coefficients of g ( x ) {\displaystyle g(x)} , the sign of this number is not changed).
Next, the first coefficient of f ( x ) {\displaystyle f(x)} is dropped as usual:
and then the dropped value is divided by 3 and placed in the row below:
Next, the new (divided) value is used to fill the top rows with multiples of 2 and 1, as in the expanded technique:
The 5 is dropped next, with the obligatory adding of the 4 below it, and the answer is divided again:
Then the 3 is used to fill the top rows:
At this point, if, after getting the third sum, we were to try and use it to fill the top rows, we would "fall off" the right side, thus the third sum is the first coefficient of the remainder, as in regular synthetic division. But the values of the remainder are not divided by the leading coefficient of the divisor:
Now we can read off the coefficients of the answer. As in expanded synthetic division, the last two values (2 is the degree of the divisor) are the coefficients of the remainder, and the remaining values are the coefficients of the quotient:
and the result is
However, the diagonal format above becomes less space-efficient when the degree of the divisor exceeds half of the degree of the dividend. Consider the following division:
It is easy to see that we have complete freedom to write each product in any row as long as it is in the correct column, so the algorithm can be compactified by a greedy strategy , as illustrated in the division below:
The following describes how to perform the algorithm; this algorithm includes steps for dividing non-monic divisors:
We interpret the results to get:
The following snippet implements Expanded Synthetic Division in Python for arbitrary univariate polynomials: | https://en.wikipedia.org/wiki/Synthetic_division |
A synthetic element is a known chemical element that does not occur naturally on Earth : it has been created by human manipulation of fundamental particles in a nuclear reactor , a particle accelerator , or the explosion of an atomic bomb ; thus, it is called "synthetic", "artificial", or "man-made". The synthetic elements are those with atomic numbers 95–118, as shown in purple on the accompanying periodic table : [ 1 ] these 24 elements were first created between 1944 and 2010. The mechanism for the creation of a synthetic element is to force additional protons into the nucleus of an element with an atomic number lower than 95. All known (see: Island of stability ) synthetic elements are unstable, but they decay at widely varying rates; the half-lives of their longest-lived isotopes range from microseconds to millions of years.
Five more elements that were first created artificially are strictly speaking not synthetic because they were later found in nature in trace quantities: 43 Tc , 61 Pm , 85 At , 93 Np , and 94 Pu ; though they are sometimes classified as synthetic alongside exclusively artificial elements. [ 2 ] The first, technetium, was created in 1937. [ 3 ] Plutonium (Pu, atomic number 94), first synthesized in 1940, is another such element. It is the element with the largest number of protons (atomic number) to occur in nature, but it does so in such tiny quantities that it is far more practical to synthesize it. Plutonium is known mainly for its use in atomic bombs and nuclear reactors. [ 4 ]
No elements with atomic numbers greater than 99 have any uses outside of scientific research, since they have extremely short half-lives, and thus have never been produced in large quantities.
All elements with atomic number greater than 94 decay quickly enough into lighter elements such that any atoms of these that may have existed when the Earth formed (about 4.6 billion years ago) have long since decayed. [ 5 ] [ 6 ] Synthetic elements now present on Earth are the product of atomic bombs or experiments that involve nuclear reactors or particle accelerators , via nuclear fusion or neutron absorption . [ 7 ]
Atomic mass for natural elements is based on weighted average abundance of natural isotopes in Earth 's crust and atmosphere . For synthetic elements, there is no "natural isotope abundance". Therefore, for synthetic elements the total nucleon count ( protons plus neutrons ) of the most stable isotope , i.e., the isotope with the longest half-life —is listed in brackets as the atomic mass.
The first element to be synthesized, rather than discovered in nature, was technetium in 1937. [ 8 ] This discovery filled a gap in the periodic table , and the fact that technetium has no stable isotopes explains its natural absence on Earth (and the gap). [ 9 ] With the longest-lived isotope of technetium, 97 Tc, having a 4.21-million-year half-life, [ 10 ] no technetium remains from the formation of the Earth. [ 11 ] [ 12 ] Only minute traces of technetium occur naturally in Earth's crust—as a product of spontaneous fission of 238 U, or from neutron capture in molybdenum —but technetium is present naturally in red giant stars. [ 13 ] [ 14 ] [ 15 ] [ 16 ]
The first entirely synthetic element to be made was curium , synthesized in 1944 by Glenn T. Seaborg , Ralph A. James , and Albert Ghiorso by bombarding plutonium with alpha particles . [ 17 ] [ 18 ]
Synthesis of americium , berkelium , and californium followed soon. Einsteinium and fermium were discovered by a team of scientists led by Albert Ghiorso in 1952 while studying the composition of radioactive debris from the detonation of the first hydrogen bomb. [ 19 ] The isotopes synthesized were einsteinium-253, with a half-life of 20.5 days, and fermium-255 , with a half-life of about 20 hours. The creation of mendelevium , nobelium , and lawrencium followed.
During the height of the Cold War , teams from the Soviet Union and the United States independently created rutherfordium and dubnium . The naming and credit for synthesis of these elements remained unresolved for many years , but eventually, shared credit was recognized by IUPAC / IUPAP in 1992. In 1997, IUPAC decided to give dubnium its current name, honoring the city of Dubna where the Russian team worked since American-chosen names had already been used for many existing synthetic elements, while the name rutherfordium (chosen by the American team) was accepted for element 104.
Meanwhile, the American team had created seaborgium , and the next six elements had been created by a German team: bohrium , hassium , meitnerium , darmstadtium , roentgenium , and copernicium . Element 113, nihonium , was created by a Japanese team; the last five known elements, flerovium , moscovium , livermorium , tennessine , and oganesson , were created by Russian–American collaborations and complete the seventh row of the periodic table.
The following elements do not occur naturally on Earth. All are transuranium elements and have atomic numbers of 95 and higher.
All elements with atomic numbers 1 through 94 occur naturally at least in trace quantities, but the following elements are often produced through synthesis.
‡ These five elements were discovered through synthesis before being found in nature. | https://en.wikipedia.org/wiki/Synthetic_element |
Exosomes are small vesicles secreted by cells that play a crucial role in intercellular communication . They contain a variety of biomolecules, including proteins , nucleic acids and lipids , which can be transferred between cells to modulate cellular processes. [ 1 ] Exosomes have been increasingly acknowledged as promising therapeutic tool and delivery platforms due to unique biological properties. [ 1 ]
However, due to exosomes being small in size (30-150 nm), present in various biological fluids (such as blood, urine, saliva), sensitivity to environmental factors (such temperature, pH), complexity of drug loading efficiency, there are challenges associated with isolation, purification, delivery and drug payload. [ 2 ] [ 5 ] [ 6 ]
While application of exosomes is still in its early stages, approaches are being explored to produce exosome-like nanovesicles (ELNs or artificial exosomes) to overcome these challenges. [ 7 ] [ 8 ]
ELNs are a type of engineered exosomes designed to modify the structure and enhance the function of natural exosomes. [ 7 ] The content of ELNs can be highly-customized to match with various medical needs, allowing for more precise control over their properties compared to natural exosomes. Additionally, ELNs can be modified with selectively expressed functional groups on the surface to enhance its targeting and uptake by cells or tissues. [ 3 ] [ 9 ] For example, ELNs can be engineered to enhance their stability in fluids, to target specific cell types, such ascytosol of brain cells. [ 10 ] Further, ELNs could consistently deliver cargo mRNA with therapeutic catalase mRNA to the brain, attenuating neurotoxicity and neuroinflammation. [ 10 ]
Above all, ELNs' properties can be tailored by researchers for specific applications with precise controlling. ELNs hold great potential as a novel approach to meet medical needs, including immunologic therapy, [ 5 ] [ 11 ] anti-tumor, [ 10 ] [ 12 ] anti-aging [ 13 ] and regeneration. [ 13 ] | https://en.wikipedia.org/wiki/Synthetic_exosome |
Synthetic genetic array analysis ( SGA ) is a high-throughput technique for exploring synthetic lethal and synthetic sick genetic interactions ( SSL ). [ 1 ] SGA allows for the systematic construction of double mutants using a combination of recombinant genetic techniques , mating and selection steps. Using SGA methodology a query gene deletion mutant can be crossed to an entire genome deletion set to identify any SSL interactions, yielding functional information of the query gene and the genes it interacts with. A large-scale application of SGA in which ~130 query genes were crossed to the set of ~5000 viable deletion mutants in yeast revealed a genetic network containing ~1000 genes and ~4000 SSL interactions. [ 2 ] The results of this study showed that genes with similar function tend to interact with one another and genes with similar patterns of genetic interactions often encode products that tend to work in the same pathway or complex. Synthetic Genetic Array analysis was initially developed using the model organism S. cerevisiae . This method has since been extended to cover 30% of the S. cerevisiae genome. [ 3 ] Methodology has since been developed to allow SGA analysis in S.pombe [ 4 ] [ 5 ] and E. coli . [ 6 ] [ 7 ]
Synthetic genetic array analysis was initially developed by Tong et al. [ 1 ] in 2001 and has since been used by many groups working in a wide range of biomedical fields. SGA utilizes the entire genome yeast knock-out set created by the yeast genome deletion project. [ 8 ]
Synthetic genetic array analysis is generally conducted using colony arrays on petriplates at standard densities (96, 384, 768, 1536). To perform a SGA analysis in S.cerevisiae , the query gene deletion is crossed systematically with a deletion mutant array (DMA) containing every viable knockout ORF of the yeast genome (currently 4786 strains). [ 9 ] The resulting diploids are then sporulated by transferring to a media containing reduced nitrogen. The haploid progeny are then put through a series of selection platings and incubations to select for double mutants. The double mutants are screened for SSL interactions visually or using imaging software by assessing the size of the resulting colonies.
Due to the large number of precise replication steps in SGA analysis, robots are widely used to perform the colony manipulations. There are a few systems specifically designed for SGA analysis, which greatly decrease the time to analyse a query gene. Generally these have a series of pins which are used to transfer cells to and from plates, with one system utilizing disposable pads of pins to eliminate washing cycles. Computer programs can be used to analyze the colony sizes from images of the plates thus automating the SGA scoring and chemical-genetics profiling.
There are six major components: | https://en.wikipedia.org/wiki/Synthetic_genetic_array |
Synthetic genome is a synthetically built genome whose formation involves either genetic modification on pre-existing life forms or artificial gene synthesis to create new DNA or entire lifeforms. [ 1 ] [ 2 ] [ 3 ] The field that studies synthetic genomes is called synthetic genomics .
Soon after the discovery of restriction endonucleases and ligases , the field of genetics began using these molecular tools to assemble artificial sequences from smaller fragments of synthetic or naturally occurring DNA . The advantage in using the recombinatory approach as opposed to continual DNA synthesis stems from the inverse relationship that exists between synthetic DNA length and percent purity of that synthetic length. In other words, as you synthesize longer sequences, the number of error-containing clones increases due to the inherent error rates of current technologies. [ 4 ] Although recombinant DNA technology is more commonly used in the construction of fusion proteins and plasmids , several techniques with larger capacities have emerged, allowing for the construction of entire genomes. [ 5 ]
Polymerase cycling assembly (PCA) uses a series of oligonucleotides (or oligos), approximately 40 to 60 nucleotides long, that altogether constitute both strands of the DNA being synthesized. These oligos are designed such that a single oligo from one strand contains a length of approximately 20 nucleotides at each end that is complementary to sequences of two different oligos on the opposite strand, thereby creating regions of overlap. The entire set is processed through cycles of: (a) hybridization at 60 °C; (b) elongation via Taq polymerase and a standard ligase; and (c) denaturation at 95 °C, forming progressively longer contiguous strands and ultimately resulting in the final genome. [ 6 ] PCA was used to generate the first synthetic genome in history, that of the Phi X 174 virus . [ 7 ]
The gibson assembly method, designed by Daniel Gibson during his time at the J. Craig Venter Institute , requires a set of double-stranded DNA cassettes that constitute the entire genome being synthesized. Note that cassettes differ from contigs by definition, in that these sequences contain regions of homology to other cassettes for the purposes of recombination . In contrast to Polymerase Cycling Assembly, Gibson Assembly is a single-step, isothermal reaction with larger sequence-length capacity; ergo, it is used in place of Polymerase Cycling Assembly for genomes larger than 6 kb.
A T5 exonuclease performs a chew-back reaction at the terminal segments, working in the 5' to 3' direction, thereby producing complementary overhangs. The overhangs hybridize to each other, a Phusion DNA polymerase fills in any missing nucleotides and the nicks are sealed with a ligase. However, the genomes capable of being synthesized using this method alone is limited because as DNA cassettes increase in length, they require propagation in vitro in order to continue hybridizing; accordingly, Gibson assembly is often used in conjunction with Transformation-Associated Recombination (see below) to synthesize genomes several hundred kilobases in size. [ 8 ]
The goal of transformation-associated recombination (TAR) technology in synthetic genomics is to combine DNA contigs by means of homologous recombination performed by the Yeast Artificial Chromosome (YAC). Of importance is the CEN element within the YAC vector , which corresponds to the yeast centromere. This sequence gives the vector the ability to behave in a chromosomal manner, thereby allowing it to perform homologous recombination . [ 9 ]
First, gap repair cloning is performed to generate regions of homology flanking the DNA contigs. Gap Repair Cloning is a particular form of the Polymerase Chain Reaction in which specialized primers with extensions beyond the sequence of the DNA target are utilized. [ 10 ] Then, the DNA cassettes are exposed to the YAC vector, which drives the process of homologous recombination, thereby connecting the DNA cassettes. Polymerase Cycling Assembly and TAR technology were used together to construct the 600 kb Mycoplasma genitalium genome in 2008, the first synthetic organism ever created. [ 11 ] Similar steps were taken in synthesizing the larger Mycoplasma mycoides genome a few years later. [ 12 ]
It is difficult to directly synthesize oligonucleotides larger than ~200 base pairs and maintain high fidelity. [ 13 ] Therefore, smaller oligonucleotides (around 5-20 base pairs) are combined to create genome-size oligonucleotides. Previous methods of stitching the smaller strands involved using T4 polynucleotide ligase . Modern techniques, like PCA/ PCR based-methods have improved on this method, increasing speed and fidelity. To further increase fidelity, PCA-based methods can include an error-reversal step in which nucleases recognize and cut mismatched base pairs. [ 14 ] Recognition is possible because errors usually cause structural budges and abnormalities in the DNA. [ 15 ] Currently, a 4-Mb E. coli genome created in May 2019 holds the record for the largest synthetic genome size. [ 16 ] | https://en.wikipedia.org/wiki/Synthetic_genomes |
Synthetic genomics is a nascent field of synthetic biology that uses aspects of genetic modification on pre-existing life forms, or artificial gene synthesis to create new DNA or entire lifeforms.
Synthetic genomics is unlike genetic modification in the sense that it does not use naturally occurring genes in its life forms. It may make use of custom designed base pair series , though in a more expanded and presently unrealized sense synthetic genomics could utilize genetic codes that are not composed of the two base pairs of DNA that are currently used by life.
The development of synthetic genomics is related to certain recent technical abilities and technologies in the field of genetics. The ability to construct long base pair chains cheaply and accurately on a large scale has allowed researchers to perform experiments on genomes that do not exist in nature. Coupled with the developments in protein folding models and decreasing computational costs the field of synthetic genomics is beginning to enter a productive stage of vitality.
Researchers were able to create a synthetic organism for the first time in 2010. [ 1 ] This breakthrough was undertaken by Synthetic Genomics, Inc. , which continues to specialize in the research and commercialization of custom designed genomes. [ 2 ] It was accomplished by synthesizing a 600 kbp genome (resembling that of Mycoplasma genitalium , save the insertion of a few watermarks) via the Gibson Assembly method and Transformation Associated Recombination. [ 3 ]
Soon after the discovery of restriction endonucleases and ligases , the field of genetics began using these molecular tools to assemble artificial sequences from smaller fragments of synthetic or naturally-occurring DNA. The advantage in using the recombinatory approach as opposed to continual DNA synthesis stems from the inverse relationship that exists between synthetic DNA length and percent purity of that synthetic length. In other words, as you synthesize longer sequences, the number of error-containing clones increases due to the inherent error rates of current technologies. [ 4 ] Although recombinant DNA technology is more commonly used in the construction of fusion proteins and plasmids , several techniques with larger capacities have emerged, allowing for the construction of entire genomes. [ 5 ]
Polymerase cycling assembly (PCA) uses a series of oligonucleotides (or oligos), approximately 40 to 60 nucleotides long, that altogether constitute both strands of the DNA being synthesized. These oligos are designed such that a single oligo from one strand contains a length of approximately 20 nucleotides at each end that is complementary to sequences of two different oligos on the opposite strand, thereby creating regions of overlap. The entire set is processed through cycles of: (a) hybridization at 60 °C; (b) elongation via Taq polymerase and a standard ligase; and (c) denaturation at 95 °C, forming progressively longer contiguous strands and ultimately resulting in the final genome. [ 6 ] PCA was used to generate the first synthetic genome in history, that of the Phi X 174 virus . [ 7 ]
The Gibson assembly method , designed by Daniel Gibson during his time at the J. Craig Venter Institute , requires a set of double-stranded DNA cassettes that constitute the entire genome being synthesized. Note that cassettes differ from contigs by definition, in that these sequences contain regions of homology to other cassettes for the purposes of recombination . In contrast to Polymerase Cycling Assembly, Gibson Assembly is a single-step, isothermal reaction with larger sequence-length capacity; ergo, it is used in place of Polymerase Cycling Assembly for genomes larger than 6 kb.
A T5 exonuclease performs a chew-back reaction at the terminal segments, working in the 5' to 3' direction, thereby producing complementary overhangs. The overhangs hybridize to each other, a Phusion DNA polymerase fills in any missing nucleotides and the nicks are sealed with a ligase. However, the genomes capable of being synthesized using this method alone is limited because as DNA cassettes increase in length, they require propagation in vitro in order to continue hybridizing; accordingly, Gibson assembly is often used in conjunction with transformation-associated recombination (see below) to synthesize genomes several hundred kilobases in size. [ 8 ]
The goal of transformation-associated recombination (TAR) technology in synthetic genomics is to combine DNA contigs by means of homologous recombination performed by the yeast artificial chromosome (YAC). Of importance is the CEN element within the YAC vector, which corresponds to the yeast centromere. This sequence gives the vector the ability to behave in a chromosomal manner, thereby allowing it to perform homologous recombination. [ 9 ]
First, gap repair cloning is performed to generate regions of homology flanking the DNA contigs. Gap Repair Cloning is a particular form of the polymerase chain reaction in which specialized primers with extensions beyond the sequence of the DNA target are utilized. [ 10 ] Then, the DNA cassettes are exposed to the YAC vector, which drives the process of homologous recombination, thereby connecting the DNA cassettes. Polymerase Cycling Assembly and TAR technology were used together to construct the 600 kb Mycoplasma genitalium genome in 2008, the first synthetic organism ever created. [ 11 ] Similar steps were taken in synthesizing the larger Mycoplasma mycoides genome a few years later. [ 12 ]
An unnatural base pair (UBP) is a designed subunit (or nucleobase ) of DNA which is created in a laboratory and does not occur in nature. In 2012, a group of American scientists led by Floyd E. Romesberg , a chemical biologist at the Scripps Research Institute in San Diego, California, published that his team designed an unnatural base pair (UBP). [ 13 ] The two new artificial nucleotides or Unnatural Base Pair (UBP) were named d5SICS and dNaM . More technically, these artificial nucleotides bearing hydrophobic nucleobases , feature two fused aromatic rings that form a (d5SICS–dNaM) complex or base pair in DNA. [ 14 ] [ 15 ] In 2014 the same team from the Scripps Research Institute reported that they synthesized a stretch of circular DNA known as a plasmid containing natural T-A and C-G base pairs along with the best-performing UBP Romesberg's laboratory had designed, and inserted it into cells of the common bacterium E. coli that successfully replicated the unnatural base pairs through multiple generations. [ 16 ] This is the first known example of a living organism passing along an expanded genetic code to subsequent generations. [ 14 ] [ 17 ] This was in part achieved by the addition of a supportive algal gene that expresses a nucleotide triphosphate transporter which efficiently imports the triphosphates of both d5SICSTP and dNaMTP into E. coli bacteria. [ 14 ] Then, the natural bacterial replication pathways use them to accurately replicate the plasmid containing d5SICS–dNaM.
The successful incorporation of a third base pair is a significant breakthrough toward the goal of greatly expanding the number of amino acids which can be encoded by DNA, from the existing 20 amino acids to a theoretically possible 172, thereby expanding the potential for living organisms to produce novel proteins . [ 16 ] The artificial strings of DNA do not encode for anything yet, but scientists speculate they could be designed to manufacture new proteins which could have industrial or pharmaceutical uses. [ 18 ]
In April 2019, scientists at ETH Zurich reported the creation of the world's first bacterial genome , named Caulobacter ethensis-2.0 , made entirely by a computer, although a related viable form of C. ethensis-2.0 does not yet exist. [ 19 ] [ 20 ] In addition to the synthetic design, the team used advanced computer algorithms to map out the entire genome's structure optimising it for stability and function. While the bacterium itself has not been successfully created yet, this achievement represents a major step toward the goal of producing fully synthetic organisms. The development of C. ethensis-2.0 exemplifies the potential of synthetic genomics to revolutionise biotechnology, offering possibilities for customised microorganisms in fields such as medicine, energy and environmental science. [ 21 ] | https://en.wikipedia.org/wiki/Synthetic_genomics |
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry ) is geometry without the use of coordinates . It relies on the axiomatic method for proving all results from a few basic properties initially called postulates , and at present called axioms .
After the 17th-century introduction by René Descartes of the coordinate method, which was called analytic geometry , the term "synthetic geometry" was coined to refer to the older methods that were, before Descartes, the only known ones.
According to Felix Klein
Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates. [ 1 ]
The first systematic approach for synthetic geometry is Euclid's Elements . However, it appeared at the end of the 19th century that Euclid 's postulates were not sufficient for characterizing geometry. The first complete axiom system for geometry was given only at the end of the 19th century by David Hilbert . At the same time, it appeared that both synthetic methods and analytic methods can be used to build geometry. The fact that the two approaches are equivalent has been proved by Emil Artin in his book Geometric Algebra .
Because of this equivalence, the distinction between synthetic and analytic geometry is no more in use, except at elementary level, or for geometries that are not related to any sort of numbers, such as some finite geometries and non-Desarguesian geometry . [ citation needed ]
The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives:
From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem .
There is no fixed axiom set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular.
Historically, Euclid's parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry , while negating it yields hyperbolic geometry . Other consistent axiom sets can yield other geometries, such as projective , elliptic , spherical or affine geometry.
Axioms of continuity and "betweenness" are also optional, for example, discrete geometries may be created by discarding or modifying them.
Following the Erlangen program of Klein , the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development.
Euclid's original treatment remained unchallenged for over two thousand years, until the simultaneous discoveries of the non-Euclidean geometries by Gauss , Bolyai , Lobachevsky and Riemann in the 19th century led mathematicians to question Euclid's underlying assumptions. [ 3 ]
One of the early French analysts summarized synthetic geometry this way:
The heyday of synthetic geometry can be considered to have been the 19th century, when analytic methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner , in favor of a purely synthetic development of projective geometry . For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models ) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry. [ 5 ]
In his Erlangen program , Felix Klein played down the tension between synthetic and analytic methods:
The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral . These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied. Gauss , Bolyai and Lobachevski independently constructed hyperbolic geometry , where parallel lines have an angle of parallelism that depends on their separation. This study became widely accessible through the Poincaré disc model where motions are given by Möbius transformations . Similarly, Riemann , a student of Gauss's, constructed Riemannian geometry , of which elliptic geometry is a particular case.
Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus , which can be considered synthetic in spirit. The closely related operation of reciprocation expresses analysis of the plane.
Karl von Staudt showed that algebraic axioms, such as commutativity and associativity of addition and multiplication, were in fact consequences of incidence of lines in geometric configurations . David Hilbert showed [ 7 ] that the Desargues configuration played a special role. Further work was done by Ruth Moufang and her students. The concepts have been one of the motivators of incidence geometry .
When parallel lines are taken as primary, synthesis produces affine geometry . Though Euclidean geometry is both an affine and metric geometry , in general affine spaces may be missing a metric. The extra flexibility thus afforded makes affine geometry appropriate for the study of spacetime , as discussed in the history of affine geometry .
In 1955 Herbert Busemann and Paul J. Kelley sounded a nostalgic note for synthetic geometry:
For example, college studies now include linear algebra , topology , and graph theory where the subject is developed from first principles, and propositions are deduced by elementary proofs . Expecting to replace synthetic with analytic geometry leads to loss of geometric content. [ 8 ]
Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms .
Ernst Kötter published a (German) report in 1901 on "The development of synthetic geometry from Monge to Staudt (1847)" ; [ 9 ]
Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines ) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem , Angle bisector theorem , Apollonius' theorem , British flag theorem , Ceva's theorem , Equal incircles theorem , Geometric mean theorem , Heron's formula , Isosceles triangle theorem , Law of cosines , and others that are linked to here .
In conjunction with computational geometry , a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory. | https://en.wikipedia.org/wiki/Synthetic_geometry |
Synthetic immunology is the rational design and construction of synthetic systems that perform complex immunological functions. [ 1 ] Functions include using specific cell markers to target cells for destruction and or interfering with immune reactions. [ 2 ] US Food and Drug Administration (FDA)-approved immune system modulators include anti-inflammatory and immunosuppressive agents, vaccines , therapeutic antibodies and Toll-like receptor (TLR) agonists. [ 1 ]
The discipline emerged after 2010 following the development of genome editing technology including TALENS and CRISPR . In 2015, one project created T cells that became active only in the presence of a specific drug, allowing them to be turned on and off in situ . Another example is a T cell that targets only cells that display two separate markers. [ 3 ]
In 2016, John Lin head of Pfizer 's San Francisco biotech unit stated, “the immune system will be the most convenient vehicle for [engineered human cells], because they can move and migrate and play such important roles.” [ 3 ]
Advances in systems biology support high-dimensional quantitative analysis of immune responses. [ 4 ] Techniques include viral gene delivery, inducible gene expression, RNA-guided genome editing, and site-specific recombinases for applications related to biotechnology and cellular immunotherapy. [ 5 ]
Researchers are exploring the creation of 'smart' organisms such as bacteriophages and bacteria that can perform complex immunological tasks. Such strategies could produce organisms that perform multistep immune functions such as presenting antigen to and co-stimulating helper T cells in a specific manner, or providing integrated signals to B cells to induce affinity maturation and isotype switching during antibody production. Such engineered organisms have the potential be as safe and as inexpensive as probiotics but precise in carrying out targeted interventions. [ 1 ]
Antibody therapeutics and other 'biologics' have proven to be effective in treating a diseases from rheumatoid arthritis to cancer . However, such agents can cause unwanted anaphylactic or inflammatory reactions, are administered by injection and are expensive. Small molecules , in contrast, are generally inexpensive to produce, orally bioavailable and are rarely allergenic. Synthetic antibody-recruiting small molecules have been created that redirect natural antibodies to pathogens for destruction. [ 1 ]
Deletion of a single transcription factor enables mature B cells to transform into T cells via dedifferentiation and redifferentiation. Technologies that can control cell fate include strategies to induce pluripotent stem cell formation and using small molecules to induce stem cells to differentiate into specific cell types. Dedifferentiation could be used to turn autoimmune cells into inactive progenitors or to suppress rejection of transplanted organs. [ 1 ]
In 2016 researchers transdifferentiated fibroblasts into induced neural stem cells. The team mixed the cells into an FDA-approved surgical glue that provided a physical support matrix. They administered the result to mice. Survival times increased from 160 to 220 percent, depending on the type of tumor. [ 6 ] [ 7 ]
Therapeutic vaccines treat and immunize patients already infected with a given disease. Provenge is an adoptive cell-transfer therapy in which a patient's antigen-presenting target autologous prostate cancer tissue. Advances in chemical biology include synthetic molecules that modulate B cell activation, structurally complex carbohydrate tumor antigen and adjuvants synthesis, immunogenic chemotherapeutic agents and chemically homogeneous, synthetic vaccines. [ 1 ] | https://en.wikipedia.org/wiki/Synthetic_immunology |
Synthetic ion channels are de novo chemical compounds that insert into lipid bilayers , form pores, and allow ions to flow from one side to the other. [ 1 ] They are man-made analogues of natural ion channels , and are thus also known as artificial ion channels . Compared to biological channels, they usually allow fluxes of similar magnitude but are
Synthetic channels, like natural channels, are usually characterized by a combination of single-molecule (e.g., voltage-clamp of planar bilayers [ 1 ] ) and ensemble techniques (flux in vesicles [ 6 ] ). The study of synthetic ion channels can potentially lead to new single-molecule sensing technologies as well as new therapeutics.
While semi-synthetic ion channels, often based on modified peptidic channels like gramicidin , had been prepared since the 1970s, the first attempt to prepare a synthetic ion channel was made in 1982 using a substituted β- cyclodextrin . [ 7 ]
Inspired by gramicidin, this molecule was designed to be a barrel-shaped entity spanning a single leaflet of a bilayer membrane , becoming "active" only when two molecules in opposite leaflets come together in an end-to-end fashion. While the compound does induce ion-fluxes in vesicles, the data does not unambiguously show channel formation (as opposed to other transport mechanisms; see Mechanism ).
Na + transport by such channels was first reported by two groups of investigators in 1989–1990. [ 8 ] [ 9 ] [ 10 ]
With the adoption of voltage clamp technique to synthetic channel research in the early 1990s, researchers were able to observe quantized electrical activities from synthetic molecules, often considered the signature evidence for ion channels. [ 1 ] This led to a sustained increase in research activity over the next two decades. In 2009, over 25 peer-reviewed papers were published on the topic, [ 11 ] and a series of comprehensive reviews are available. [ 3 ] [ 4 ] [ 5 ]
Passive transport of ions across a membrane can take place by three main mechanisms: by ferrying, through defects in a disrupted membrane, or through a defined trajectory; these corresponds to ionophore , detergent , and ion channel transporters. While synthetic ion channel research attempts to prepare compounds that show conductance via a defined path, the elucidation of mechanism is difficult and seldom unambiguous. The two main methods of characterization both have their drawbacks, and as a consequence, often function is defined but mechanism presumed.
One line of evidence for ion transport comes from macroscopic examination of statistical ensembles . All these techniques use intact vesicles with an entrapped volume, with ion channel activities reported by different spectroscopic methods. [ 6 ]
In a typical case, a dye is entrapped within the population of vesicles. This dye is selected to be respond colorimetrically or fluorometrically to the presence of an ion; this ion is typically absent from the inside of the vesicle but present in the outside. Without an ion transporter, the lipid bilayer as a kinetic barrier to block ion flux, and the dye remains "dark" indefinitely.
As an ion transporter allows ions on the outside to diffuse in, its addition will affect the color/fluorescence property of the dye. By macroscopically monitoring the dye's properties over time, and controlling outside factors, the ability of a compound to act as an ion transporter can be measured.
Observing ion transport, however, does not pin down ion channel as the mechanism. Any class of transporter can lead to the same observation, and additional corroborating evidence is usually required. Sophisticated experiments intended to probe selectivity, gating, and other channel parameters have been developed over the past two decades and recently summarized. [ 6 ]
An alternative to the ensemble-based method described above is the voltage-clamp experiment. [ 16 ] In a voltage-clamp experiment, two compartments of electrolyte are divided by an aperture, usually between 5-250 micrometres in diameter. A lipid bilayer is painted across this aperture, thus electrically separating the compartments; the molecular nature can be ascertained by measuring its capacitance .
Upon the addition of an (ideal) ion channel, a defined path between the two compartments is formed. Through this pore, ions flow down the potential and electrochemical gradient rapidly (>10 6 /second), the maximum flux limited by the geometry and dimensions of the pore. At some later instant the pore may close or collapse, whereupon the current returns to zero. This open-state current, originating and amplified from a single-molecule event, is typically on the order of pA to nA , with time-resolution of approx. millisecond. Ideal or close-to-ideal events is termed " square-tops " in the literature, and have been considered as signature for a channel-based mechanism.
It is notable that the events observed at this scale are truly stochastic - that is, they are the result of random molecular collision and conformation changes. As the membrane area is much larger than that of a pore, multiple copies may open and close independently of one another, giving rise to the staircase like appearance (Panel C in figure); these ideal events are often modelled as Markov processes .
By using the activity grid notation , [ 17 ] synthetic ion channels studied with the voltage-clamp method during the period 1982-2010 have been critically reviewed. [ 18 ] While the ideal traces are most frequently analyzed and reported in the literature, many records are decidedly non-ideal, with a subset was shown to be fractal. [ 19 ] Developing methods for analyzing these non-ideal traces and clarifying their relationship to transport mechanism is an area of contemporary research.
A diverse and large pool of synthetic molecules have been reported to act as ion transporters in lipid membranes. A selection is described here to demonstrate the breadth of feasible structures and attainable functions . Comprehensive reviews for the literature up to 2010 are available in a tripartite series. [ 3 ] [ 4 ] [ 5 ]
Most (but not all; see minimalist channels ) synthetic channels have chemical structures substantially larger than typical small molecules (molecular weights ~1-5kDa). This originates from the need to be amphiphilic , that is, have both sufficient hydrophobic portions to allow partitioning into lipid bilayer, as well as polar or charged "headgroups" to assert a defined orientation and geometry with respect to the membrane.
Ion channels containing calixarenes of ring size 3 and 4 have both been reported. For calix[4]arene, two conformations are accessible, and examples of both 1,3-alt and cone conformation have been developed.
The first synthetic ion channel was constructed by partial substitution on the primary rim of β- cyclodextrin . [ 7 ] Other substituted β-cyclodextrins have since been reported, including thiol-modified cyclodextrins, [ 20 ] an anion-selective oligobutylene channel, [ 21 ] and various poly-ethyleneoxide linked starburst oligomers. [ 22 ] Structure-activity relationships for a large suite of cyclodextrin "half-channels" prepared by "click"-chemistry has been recently reported. [ 17 ]
Alternating D/L peptide macrocycles are known to self-aggregate into nanotubes, and the resulting nanotubes have been shown to act as ion channels in lipid membranes. [ 23 ]
Other architectures use peptide helices as a scaffold to attach other functionalities, such as crown ethers of different sizes. The property of these peptide-crown channels depend strongly on the identity of the capping end-groups.
Semi-synthetic bio-hybrid channels constructed by modifications of natural ion channels had been constructed. Leveraging modern synthetic organic chemistry , these allows pinpoint modifications of existing structures to either elucidate their transport mechanisms or to graft on new functionalities.
Gramicidin and alamethicin had been popular starting points for selective modifications. [ 24 ] The above diagram illustrates one example, where a crown-ether was fixed across the mouth of the ion-passing portal. [ 25 ] This channel shows discrete conductance but different ion selectivity than wild type gramicidin in voltage-clamp experiments.
While modification of large protein channels using mutagenesis are generally considered out of the scope of synthetic channels, the demarcation is not sharp, as supramolecular or covalent bonding of cyclodextrins to alpha-hemolysin demonstrates. [ 26 ]
An ion channel can be characterized by its opening characteristics , ion selectivity , and control of flux (gating). Many synthetic ion channels show unique properties in one or more of these aspects.
An "ion-channel forming" molecule can often show multiple types of conductance activities in planar bilayer membranes. Each of these modes of action can be characterized by their
These events are not necessarily uniform throughout their durations, and as a result a variety of shapes of conducting traces are possible.
The majority of synthetic ion channels follow an Eisenman I sequence (Cs + > Rb + > K + > Na + >> Li + ) [ 27 ] in their selectivity for alkali metal cations, [ 4 ] [ 18 ] suggesting that the origin of the selectivity is governed by the difference in energy required to remove water from a fully hydrated cation. A few synthetic channels show other patterns of ion selectivity, [ 25 ] and only a single instance in which a synthetic channel following the opposite selectivity sequence (Eisenman XI; Cs + < Rb + < K + < Na + << Li + ) had been reported. [ 28 ]
Most synthetic channels are Ohmic in conductance, that is, the current passed (both individually and as an ensemble) is proportional to the potential across the membrane. Some rare channels, however, show current-voltage characteristics that is non-linear. Specifically, two different types of non-Ohmic conductance are known:
The former requires asymmetry with respect to the mid-plane of the lipid bilayer, and is realized often by introducing an overall molecular dipole. [ 29 ] [ 30 ] [ 31 ] The latter, demonstrated in natural channels such as alamethicin , is rarely encountered in synthetic ion channels. They may be related to lipid ion channels, but to date their mechanism remains elusive.
Certain synthetic ion channels have conductances that can be modulated by additional of external chemicals. Both up-modulation (channels are turned on by ligand) and down-modulation (channels are turned off by ligands) are known: different mechanisms, including formation of supramolecular aggregates, [ 32 ] [ 33 ] as well as inter- and intramolecular [ 34 ] blockage.
Regulatory elements that responds to other signals are known; examples include photomodulated conductances [ 35 ] [ 36 ] [ 37 ] as well as "thermal switches" constructed by isomerization of the carbamate group. [ 38 ] To date, no mechanosensitive synthetic ion channels have been reported. | https://en.wikipedia.org/wiki/Synthetic_ion_channels |
In fluid dynamics , a synthetic jet flow—is a type of jet flow , which is made up of the surrounding fluid . [ 1 ] Synthetic jets are produced by periodic ejection and suction of fluid from an opening. This oscillatory motion may be driven by a piston or diaphragm inside a cavity among other ways. [ 2 ] [ 3 ] [ 4 ]
A synthetic jet flow was so named by Ari Glezer since the flow is "synthesized" from the surrounding or ambient fluid. Producing a convectional jet requires an external source of fluid, such as piped-in compressed air or plumbing for water.
Synthetic jet flow can be developed in a number of ways, such as with an electromagnetic driver, a piezoelectric driver, or even a mechanical driver such as a piston. Each moves a membrane or diaphragm up and down hundreds of times per second, sucking the surrounding fluid into a chamber and then expelling it. Although the mechanism is fairly simple, extremely fast cycling requires high-level engineering to produce a device that will last in industrial applications.
For hot spot thermal management, the Synjet, commercially offered by Austin, Texas–based company Nuventix, [ 5 ] was patented in 2000 by engineers at Georgia Tech. [ 6 ] The tiny synjet module creates jets that can be directed to precise locations for industrial spot cooling. Traditionally, metallic heat sinks conduct heat away from electronic components and into the air, and then a small fan blows the hot air out. Synjet modules replace or augment cooling fans for such devices as microprocessors, memory chips, graphics chips, batteries, and radio frequency components. Additionally, SynJet technology has been used for the thermal management of high power LEDs [ 5 ] [ 7 ]
Synthetic jet modules have also been widely researched for controlling airflow in aircraft to enhance lift, increase maneuverability, control stalls, and reduce noise. [ 8 ] Problems in applying the technology include weight, size, response time, force, and complexity of controlling the flows. [ 9 ] [ 10 ] [ 11 ] [ 12 ]
A Caltech researcher has even tested synthetic jet modules to provide thrust for small underwater vehicles, modeled on the natural jets that squid and jellyfish produce. [ 13 ] Recently, research team at the School of Engineering, Taylor's University (Malaysia), successfully used synthetic jets as mixing devices. [ 14 ] Synthetic jets prove to be effective mixing devices especially for shear sensitive materials. | https://en.wikipedia.org/wiki/Synthetic_jet |
Synthetic lethality is defined as a type of genetic interaction where the combination of two genetic events results in cell death or death of an organism. [ 1 ] Although the foregoing explanation is wider than this, it is common when referring to synthetic lethality to mean the situation arising by virtue of a combination of deficiencies of two or more genes leading to cell death (whether by means of apoptosis or otherwise), whereas a deficiency of only one of these genes does not. In a synthetic lethal genetic screen , it is necessary to begin with a mutation that does not result in cell death, although the effect of that mutation could result in a differing phenotype (slow growth for example), and then systematically test other mutations at additional loci to determine which, in combination with the first mutation, causes cell death arising by way of deficiency or abolition of expression.
Synthetic lethality has utility for purposes of molecular targeted cancer therapy. The first example of a molecular targeted therapeutic agent, which exploited a synthetic lethal approach, arose by means of an inactivated tumor suppressor gene ( BRCA1 and 2), a treatment which received FDA approval in 2016 ( PARP inhibitor ). [ 2 ] A sub-case of synthetic lethality, where vulnerabilities are exposed by the deletion of passenger genes rather than tumor suppressor is the so-called "collateral lethality". [ 3 ]
The phenomenon of synthetic lethality was first described by Calvin Bridges in 1922, who noticed that some combinations of mutations in the model organism Drosophila melanogaster (the common fruit fly) confer lethality. [ 1 ] Theodore Dobzhansky coined the term "synthetic lethality" in 1946 to describe the same type of genetic interaction in wildtype populations of Drosophila . [ 4 ] If the combination of genetic events results in a non-lethal reduction in fitness, the interaction is called synthetic sickness. Although in classical genetics the term synthetic lethality refers to the interaction between two genetic perturbations, synthetic lethality can also apply to cases in which the combination of a mutation and the action of a chemical compound causes lethality, whereas the mutation or compound alone are non-lethal. [ 5 ]
Synthetic lethality is a consequence of the tendency of organisms to maintain buffering schemes (i.e. backup plans) which engender phenotypic stability notwithstanding underlying genetic variations, environmental changes or other random events, such as mutations. This genetic robustness is the result of parallel redundant pathways and "capacitor" proteins that camouflage the effects of mutations so that important cellular processes do not depend on any individual component. [ 6 ] Synthetic lethality can help identify these buffering relationships, and what type of disease or malfunction that may occur when these relationships break down, through the identification of gene interactions that function in either the same biochemical process or pathways that appear to be unrelated. [ 7 ]
High-throughput synthetic lethal screens may help illuminate questions about how cellular processes work without previous knowledge of gene function or interaction. Screening strategy must take into account the organism used for screening, the mode of genetic perturbation, and whether the screen is forward or reverse . Many of the first synthetic lethal screens were performed in Saccharomyces cerevisiae . Budding yeast has many experimental advantages in screens, including a small genome, fast doubling time, both haploid and diploid states, and ease of genetic manipulation. [ 8 ] Gene ablation can be performed using a PCR -based strategy and complete libraries of knockout collections for all annotated yeast genes are publicly available. Synthetic genetic array (SGA), synthetic lethality by microarray (SLAM), and genetic interaction mapping (GIM) are three high-throughput methods for analyzing synthetic lethality in yeast. A genome scale genetic interaction map was created by SGA analysis in S. cerevisiae that comprises about 75% of all yeast genes. [ 9 ]
Collateral lethality is a sub-case of synthetic lethality in personalized cancer therapy, where vulnerabilities are exposed by the deletion of passenger genes rather than tumor suppressor genes, which are deleted by virtue of chromosomal proximity to major deleted tumor suppressor loci. [ 3 ]
Mutations in genes employed in DNA mismatch repair (MMR) cause a high mutation rate. [ 10 ] [ 11 ] In tumors, such frequent subsequent mutations often generate "non-self" immunogenic antigens. A human Phase II clinical trial, with 41 patients, evaluated one synthetic lethal approach for tumors with or without MMR defects. [ 12 ] In the case of sporadic tumors evaluated, the majority would be deficient in MMR due to epigenetic repression of an MMR gene (see DNA mismatch repair ). The product of gene PD-1 ordinarily represses cytotoxic immune responses. Inhibition of this gene allows a greater immune response. In this Phase II clinical trial with 47 patients, when cancer patients with a defect in MMR in their tumors were exposed to an inhibitor of PD-1, 67% - 78% of patients experienced immune-related progression-free survival. In contrast, for patients without defective MMR, addition of PD-1 inhibitor generated only 11% of patients with immune-related progression-free survival. Thus inhibition of PD-1 is primarily synthetically lethal with MMR defects.
The analysis of 630 human primary tumors in 11 tissues shows that WRN promoter hypermethylation (with loss of expression of WRN protein) is a common event in tumorigenesis. [ 13 ] The WRN gene promoter is hypermethylated in about 38% of colorectal cancers and non-small-cell lung carcinomas and in about 20% or so of stomach cancers , prostate cancers , breast cancers , non-Hodgkin lymphomas and chondrosarcomas , plus at significant levels in the other cancers evaluated. The WRN helicase protein is important in homologous recombinational DNA repair and also has roles in non-homologous end joining DNA repair and base excision DNA repair . [ 14 ]
Topoisomerase inhibitors are frequently used as chemotherapy for different cancers, though they cause bone marrow suppression, are cardiotoxic and have variable effectiveness. [ 15 ] A 2006 retrospective study, with long clinical follow-up, was made of colon cancer patients treated with the topoisomerase inhibitor irinotecan . In this study, 45 patients had hypermethylated WRN gene promoters and 43 patients had unmethylated WRN gene promoters. [ 13 ] Irinitecan was more strongly beneficial for patients with hypermethylated WRN promoters (39.4 months survival) than for those with unmethylated WRN promoters (20.7 months survival). Thus, a topoisomerase inhibitor appeared to be synthetically lethal with deficient expression of WRN . Further evaluations have also indicated synthetic lethality of deficient expression of WRN and topoisomerase inhibitors. [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ]
As reviewed by Murata et al., [ 21 ] five different PARP1 inhibitors are now undergoing Phase I, II and III clinical trials, to determine if particular PARP1 inhibitors are synthetically lethal in a large variety of cancers, including those in the prostate, pancreas, non-small-cell lung tumors, lymphoma, multiple myeloma, and Ewing sarcoma. In addition, in preclinical studies using cells in culture or within mice, PARP1 inhibitors are being tested for synthetic lethality against epigenetic and mutational deficiencies in about 20 DNA repair defects beyond BRCA1/2 deficiencies. These include deficiencies in PALB2 , FANCD2 , RAD51 , ATM , MRE11 , p53 , XRCC1 and LSD1 .
ARID1A , a chromatin modifier, is required for non-homologous end joining , a major pathway that repairs double-strand breaks in DNA, [ 22 ] and also has transcription regulatory roles. [ 23 ] ARID1A mutations are one of the 12 most common carcinogenic mutations. [ 24 ] Mutation or epigenetically decreased expression [ 25 ] of ARID1A has been found in 17 types of cancer. [ 26 ] Pre-clinical studies in cells and in mice show that synthetic lethality for deficient ARID1A expression occurs by either inhibition of the methyltransferase activity of EZH2, [ 27 ] [ 28 ] by inhibition of the DNA repair kinase ATR, [ 29 ] or by exposure to the kinase inhibitor dasatinib. [ 30 ]
There are two pathways for homologous recombinational repair of double-strand breaks. The major pathway depends on BRCA1 , PALB2 and BRCA2 while an alternative pathway depends on RAD52. [ 31 ] Pre-clinical studies, involving epigenetically reduced or mutated BRCA -deficient cells (in culture or injected into mice), show that inhibition of RAD52 is synthetically lethal with BRCA -deficiency. [ 32 ]
Although treatments using synthetic lethality can stop or slow progression of cancers and prolong survival, each of the synthetic lethal treatments has some adverse side effects. For example, more than 20% of patients treated with an inhibitor of PD-1 encounter fatigue, rash, pruritus , cough, diarrhea, decreased appetite, constipation or arthralgia . [ 33 ] Thus, it is important to determine which DDR deficiency is present, so that only an effective synthetic lethal treatment can be applied, and not unnecessarily subject patients to adverse side effects without a direct benefit. | https://en.wikipedia.org/wiki/Synthetic_lethality |
An artificial membrane , or synthetic membrane , is a synthetically created membrane which is usually intended for separation purposes in laboratory or in industry. Synthetic membranes have been successfully used for small and large-scale industrial processes since the middle of the twentieth century. [ 1 ] A wide variety of synthetic membranes is known. [ 2 ] They can be produced from organic materials such as polymers and liquids, as well as inorganic materials. Most commercially utilized synthetic membranes in industry are made of polymeric structures. They can be classified based on their surface chemistry , bulk structure, morphology , and production method. The chemical and physical properties of synthetic membranes and separated particles as well as separation driving force define a particular membrane separation process. The most commonly used driving forces of a membrane process in industry are pressure and concentration gradient . The respective membrane process is therefore known as filtration . Synthetic membranes utilized in a separation process can be of different geometry and flow configurations. They can also be categorized based on their application and separation regime. [ 2 ] The best known synthetic membrane separation processes include water purification , reverse osmosis , dehydrogenation of natural gas, removal of cell particles by microfiltration and ultrafiltration , removal of microorganisms from dairy products, and dialysis .
Synthetic membrane can be fabricated from a large number of different materials. It can be made from organic or inorganic materials including solids such as metals , ceramics , homogeneous films, polymers , heterogeneous solids (polymeric mixtures, mixed glasses [ clarification needed ] ), and liquids. [ 3 ] Ceramic membranes are produced from inorganic materials such as aluminium oxides, silicon carbide , and zirconium oxide. Ceramic membranes are very resistant to the action of aggressive media (acids, strong solvents). They are very stable chemically, thermally, and mechanically, and biologically inert . Even though ceramic membranes have a high weight and substantial production costs, they are ecologically friendly and have long working life. Ceramic membranes are generally made as monolithic shapes of tubular capillaries . [ 3 ]
Liquid membranes refer to synthetic membranes made of non-rigid materials. Several types of liquid membranes can be encountered in industry: emulsion liquid membranes, immobilized (supported) liquid membranes, [ 4 ] supported molten -salt membranes, [ 5 ] and hollow-fiber contained liquid membranes. [ 3 ] Liquid membranes have been extensively studied but thus far have limited commercial applications. Maintaining adequate long-term stability is a key problem, due to the tendency of membrane liquids to evaporate, dissolve in the phases in contact with them, or creep out of the membrane support.
Polymeric membranes lead the membrane separation industry market because they are very competitive in performance and economics. [ 3 ] Many polymers are available, but the choice of membrane polymer is not a trivial task. A polymer has to have appropriate characteristics for the intended application. [ 6 ] The polymer sometimes has to offer a low binding affinity for separated molecules (as in the case of biotechnology applications), and has to withstand the harsh cleaning conditions. It has to be compatible with chosen membrane fabrication technology. [ 6 ] The polymer has to be a suitable membrane former in terms of its chains rigidity, chain interactions, stereoregularity , and polarity of its functional groups. [ 6 ] The polymers can range form amorphous and semicrystalline structures (can also have different glass transition temperatures), affecting the membrane performance characteristics. The polymer has to be obtainable and reasonably priced to comply with the low cost criteria of membrane separation process. Many membrane polymers are grafted, custom-modified, or produced as copolymers to improve their properties. [ 6 ] The most common polymers in membrane synthesis are cellulose acetate , Nitrocellulose , and cellulose esters (CA, CN, and CE), polysulfone (PS), polyether sulfone (PES), polyacrilonitrile (PAN), polyamide , polyimide , polyethylene and polypropylene (PE and PP), polytetrafluoroethylene (PTFE), polyvinylidene fluoride (PVDF), polyvinylchloride (PVC).
Polymer membranes may be functionalized into ion-exchange membranes by the addition of highly acidic or basic functional groups, e.g. sulfonic acid and quaternary ammonium, enabling the membrane to form water channels and selectively transport cations or anions, respectively. The most important functional materials in this category include proton-exchange membranes and alkaline anion-exchange membranes , that are at the heart of many technologies in water treatment, energy storage, energy generation. Applications within water treatment include reverse osmosis , electrodialysis , and reversed electrodialysis . Applications within energy storage include rechargeable metal-air electrochemical cells and various types of flow battery . Applications within energy generation include proton-exchange membrane fuel cells (PEMFCs), alkaline anion-exchange membrane fuel cells (AEMFCs), and both the osmotic- and electrodialysis-based osmotic power or blue energy generation.
Ceramic membranes are made from inorganic materials (such as alumina , titania , zirconia oxides, recrystallised silicon carbide or some glassy materials).
By contrast with polymeric membranes, they can be used in separations where aggressive media (acids, strong solvents) are present. They also have excellent thermal stability which make them usable in high temperature membrane operations .
One of the critical characteristics of a synthetic membrane is its chemistry. Synthetic membrane chemistry usually refers to the chemical nature and composition of the surface in contact with a separation process stream. [ 6 ] The chemical nature of a membrane's surface can be quite different from its bulk composition. This difference can result from material partitioning at some stage of the membrane's fabrication, or from an intended surface postformation modification. Membrane surface chemistry creates very important properties such as hydrophilicity or hydrophobicity (related to surface free energy), presence of ionic charge , membrane chemical or thermal resistance, binding affinity for particles in a solution, and biocompatibility (in case of bioseparations). [ 6 ] Hydrophilicity and hydrophobicity of membrane surfaces can be expressed in terms of water (liquid) contact angle θ. Hydrophilic membrane surfaces have a contact angle in the range of 0°<θ<90° (closer to 0°), where hydrophobic materials have θ in the range of 90°<θ<180°.
The contact angle is determined by solving the Young's equation for the interfacial force balance. At equilibrium three interfacial tensions corresponding to solid/gas (γ SG ), solid/liquid (γ SL ), and liquid/gas (γ LG ) interfaces are counterbalanced. [ 6 ] The consequence of the contact angle's magnitudes is known as wetting phenomena, which is important to characterize the capillary (pore) intrusion behavior. Degree of membrane surface wetting is determined by the contact angle. The surface with smaller contact angle has better wetting properties (θ=0°-perfect wetting). In some cases low surface tension liquids such as alcohols or surfactant solutions are used to enhance wetting of non-wetting membrane surfaces. [ 6 ] The membrane surface free energy (and related hydrophilicity/hydrophobicity) influences membrane particle adsorption or fouling phenomena. In most membrane separation processes (especially bioseparations), higher surface hydrophilicity corresponds to the lower fouling. [ 6 ] Synthetic membrane fouling impairs membrane performance. As a consequence, a wide variety of membrane cleaning techniques have been developed. Sometimes fouling is irreversible , and the membrane needs to be replaced. Another feature of membrane surface chemistry is surface charge. The presence of the charge changes the properties of the membrane-liquid interface. The membrane surface may develop an electrokinetic potential and induce the formation of layers of solution particles which tend to neutralize the charge.
Synthetic membranes can be also categorized based on their structure (morphology). Three such types of synthetic membranes are commonly used in separation industry: dense membranes, porous membranes, and asymmetric membranes. Dense and porous membranes are distinct from each other based on the size of separated molecules. Dense membrane is usually a thin layer of dense material utilized in the separation processes of small molecules (usually in gas or liquid phase). Dense membranes are widely used in industry for gas separations and reverse osmosis applications.
Dense membranes can be synthesized as amorphous or heterogeneous structures. Polymeric dense membranes such as polytetrafluoroethylene and cellulose esters are usually fabricated by compression molding , solvent casting , and spraying of a polymer solution. The membrane structure of a dense membrane can be in a rubbery or a glassy state at a given temperature depending on its glass transition temperature . [ 2 ] Porous membranes are intended on separation of larger molecules such as solid colloidal particles, large biomolecules ( proteins , DNA , RNA ) and cells from the filtering media. Porous membranes find use in the microfiltration , ultrafiltration , and dialysis applications. There is some controversy in defining a "membrane pore". The most commonly used theory assumes a cylindrical pore for simplicity. This model assumes that pores have the shape of parallel, nonintersecting cylindrical capillaries. But in reality a typical pore is a random network of the unevenly shaped structures of different sizes. The formation of a pore can be induced by the dissolution of a "better" solvent into a "poorer" solvent in a polymer solution. [ 2 ] Other types of pore structure can be produced by stretching of crystalline structure polymers. The structure of porous membrane is related to the characteristics of the interacting polymer and solvent, components concentration, molecular weight , temperature, and storing time in solution. [ 2 ] The thicker porous membranes sometimes provide support for the thin dense membrane layers, forming the asymmetric membrane structures. The latter are usually produced by a lamination of dense and porous membranes. | https://en.wikipedia.org/wiki/Synthetic_membrane |
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