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where [MATH] and [MATH] are functions of [MATH] but independent of [MATH] [EQUATION] Performing the interior integration in Eq. A-9 yields |
[EQUATION] Expanding out the various [MATH] s as well as [MATH] , we now have for the first term of [MATH] [EQUATION] which has an indirect dependence on [MATH] . Performing the same integration for the second term in Eq. A-2 yields analogous indirect [MATH] dependence. As a result, when [MATH] , the first term is grea... |
The dependence is illustrated in Fig. A. Here, we assume a uniform distribution of [MATH] on [MATH] and construct [MATH] [MATH] ’s for the underlying population [MATH] from which we draw, independently, a sample [MATH] of size [MATH] and 200 samples [MATH] |
from which we estimate [MATH] under the null hypothesis. Sample [MATH] is drawn i.i.d. from [MATH] with sample sizes ranging from [MATH] |
to [MATH] , permitting us to plot [MATH] as [MATH] is varied. The simulation results are shown as circles, overlayed with a plot of Eq. A-2 using the result in Eq. A-15 and assuming the uniform distribution of [MATH] . The values for [MATH] obtained from the simulation closely matches those derived from Eq. A-15 In Fig... |
Acknowledgments This research was supported by the Intramural Research Program of the National Cancer Institute, National Institutes of Health, Bethesda, MD. RB was supported by the Cancer Prevention Fellowship Program, National Cancer Institute, National Institutes of Health, Bethesda, MD. |
# Source: arxiv 0903.0137 # Title: p-Adic numbers in bioinformatics: from genetic code to PAM-matrix # Sections: all # Downloaded: 2026-03-02T08:42:26.914064+00:00 |
[MATH] -Adic numbers in bioinformatics: from genetic code to PAM-matrix Abstract In this paper we denonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, generacy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic di... |
We obtain the following results. We show that the PAM matrix [MATH] allows the expansion into the sum of the two matrices [MATH] , where the matrix [MATH] is 2–adically regular (i.e. matrix elements of this matrix are close to locally constant with respect to the discussed earlier by the authors 2–adic parametrization ... |
[MATH] is sparse. We discuss the structure of the matrix [MATH] in relation to the side chain properties of the corresponding amino acids. |
Introduction Various clustering procedures play a crucial role in bioinformatics, in particular, genetics, see, e.g., or An important class of such procedures is based on introduction of various metrics on the space information strings, see e.g. |
. A metric with new interesting features was recently used in theoretical physics (from string theory to theory of disordered systems, spin glasses), see e.g. |
in cognitive science, psychology and image analysis . This is so called [MATH] –adic metric (in fact, a class of metrics depending on the parameter [MATH] |
— a prime number). The main distinguishing feature of this metric is its sensitivity to hierarchic patterns in information having a special structure matching with [MATH] –adic encoding of information. |
A few years ago 2–adic metric was applied to study the problem of degeneration of the genetic code, see These [MATH] –adic models can be considered as new development in the approach to investigation of the structure of the genetic code from the point of view of coding theory, see |
In the present paper we discuss the structure of the PAM matrix used in bioinformatics (see for example ) from the point of view of [MATH] –adic analysis. We use the 2–adic parametrization of the genetic (amino acid) code obtained in |
(see also for the different [MATH] –adic parametrization). In it was shown that, after some special parametrization of the space of codons (triples of nucleotides) the genetic code becomes a locally constant map of [MATH] –adic argument. Moreover, the degeneracy of the genetic code in this language takes the form of lo... |
Let us also mention the application of the [MATH] –adic parametrization to the description of the Parisi matrix from the replica symmetry breaking approach to spin glasses |
. After the [MATH] –adic parametrization of the numbers of the lines and the columns the Parisi matrix becomes a locally constant block matrix. |
It is natural to check, using the [MATH] –adic parametrization approach, the structure of the PAM matrix. The PAM matrix is used in bioinformatics for sequence alignment and is constructed using a Markov chain model of point mutations for a protein chain. |
We assume that the structure of the PAM matrix has some relation to the structure of the genetic code. Using this idea we enumerate the lines and the columns of the PAM matrix using the 2–adic parametrization of the genetic code. After this parametrization the PAM matrix becomes more regular, namely, the dependence of ... |
[MATH] is close to locally constant with respect to the 2–adic norm for the majority of matrix elements. We have some exceptions from this rule. It is easy to see that these exceptions are related to several amino acids, namely to Y, W, C, F, L. In order to describe this deviations from 2–adicity we introduce the follo... |
[EQUATION] The matrix in this expansion [MATH] is 2–adically regular (close to locally constant). The matrix [MATH] is sparse (the majority of matrix elements are zero, non zero matrix elements are mainly concentrated of the lines and columns related to the amino acids Y, W, C, F, L). |
One can see that the deviations from 2–adicity (i.e. non–zero matrix elements of [MATH] ) are related to amino acids which are in some sense special — to the aromatic amino acids Y, W, F, and to Cysteine C which contains the SH group. |
We also mention that the 2–adic structure of the genetic code is related to some chemical properties of the amino acids. In particular, hydrophobic amino acids are clustered in two ball with respect to the 2–adic norm. Therefore the 2–adic parametrization allows to separate the impact of the chemical and geometrical pr... |
The structure of the present paper is the following. In Section 2 we discuss some family of ultrametric spaces. In Section 3 we describe the 2–adic 2–dimensional parametrization of the genetic code of |
In Section 4 we put the PAM250 matrix. In Section 5 we describe the reshuffling of the lines and the columns of the PAM matrix, corresponding to the 2–adic parametrization of the genetic code of Section 2. |
In Section 6 we introduce the expansion of the PAM matrix into the sum of the two matrices, one of which is 2–adically regular (close to locally constant) and the other is sparse (majority of matrix elements are equal to zero). |
Sections 7 and 8 are appendices where the definitions of PAM matrices and the eucaryotic genetic code are exposed. Ultrametric spaces |
An ultrametric space is a metric space where the metric [MATH] satisfies the strong triangle inequality: [EQUATION] The strong triangle inequality can be stated geometrically: each side of a triangle is at most as long as the longest one of the two other sides. Such a triangle is quite restricted when considered in the... |
An ultrametric space is a natural mathematical object for description of a hierarchical system. On ultrametric spaces there exist many locally constant functions, i.e., functions which are constant on some vicinity of any point, but not necessarily constant on the whole space. In particular, we show that the genetic co... |
Let [MATH] be an ultrametric space. We consider balls [MATH] So, [MATH] is the center of the ball [MATH] having radius [MATH] We mention a few unusual (from the viewpoint of usual Euclidean geometry) properties of ultrametric balls: |
a). Each point of [MATH] can be chosen as its center. So, inside a ball all points have ”equal rights”. b). Any two balls either do not intersect or one of the balls contains the other ball. In this framework, clustering into disjoint balls is a very natural operation. |
We remark that ultrametric spaces were widely used in bioinformatics, see . However, in this paper we plan to elaborate new applications of ultrametric spaces to biology (genetics) which are different from mentioned ones. |
We are interested in the following special class of ultrametric spaces [MATH] Every point [MATH] is the infinite sequence of digits |
[EQUATION] Each digit yields the finite number of values, [EQUATION] where [MATH] is a natural number, the base of the alphabet [MATH] |
If the sequence [MATH] contains only finite number of non-zero terms [MATH] , then we can consider [MATH] as the natural number [EQUATION] |
Moreover, this formula defines the one to one correspondence between natural numbers (with zero) and the space of final sequences |
[MATH] We denote the space of sequences ( ), ( ) by the symbol [MATH] . Ultrametric is introduced on this set in the following way. For two points |
[EQUATION] we set [EQUATION] The ultrametric space [MATH] is called the space of [MATH] -adic integers. The ultrametric [MATH] describes the following hierarchical structure. If [MATH] |
[MATH] , is a vector encoding information on some object, then digits [MATH] have different weights. The digit [MATH] is the most important, [MATH] dominates over |
[MATH] and so on. Such hierarchic information vectors can be created by living systems, e.g., by the brain to process information. Applications of [MATH] -adic numbers to information theory and, in particular, to description of cognitive processes and complex social systems were developed in |
In applications we will use not only ”one dimensional” [MATH] -adic spaces, but also cartesian products of a few spaces, e.g., [MATH] -adic plane [MATH] and so on. Our aim is to show that 2-adic plane structure was embedded in the genetic code, see |
We remark that [MATH] -adic numbers for [MATH] , where [MATH] is a prime number were intensively used (during last 20 years) in mathematical physics |
. The number theoretic definition is as follows. Let us fix a prime number [MATH] . For example, fix [MATH] or fix [MATH] . The example of ultrametric space is the field of [MATH] -adic numbers [MATH] which is the completion of the field of rational numbers with respect to [MATH] -adic norm [MATH] , defined as follows:... |
[MATH] and [MATH] [MATH] are non-zero and are not divisible by [MATH] , its [MATH] -adic norm is [EQUATION] [MATH] -Adic norm is widely used in number theory and algebraic geometry. |
If a rational number is divisible by [MATH] , where [MATH] is very large, then its [MATH] -adic norm is very small. This hierarchy of the degrees of [MATH] (i.e. of values of the [MATH] –adic norm) gives the hierarchical (ultrametric) structure of the [MATH] –adic norm. [MATH] -Adic numbers are in one to one correspond... |
[EQUATION] where [MATH] is integer and [MATH] (this expansion is the analogue of ( )). The unit ball in [MATH] with center at [MATH] zero coincides with the space of [MATH] -adic integers [MATH] |
[MATH] -Adic numbers were actively used (since pioneer papers of I.Volovich, ) in high energy physics (superstring theory, cosmology) and theory of disordered systems (spin glasses), see, e.g., review |
and the book 2-Adic parametrization of the genetic code In the 2–adic parametrization approach of we enumerate in some special way the set of 64 codons (triples of nucleotides) by pairs of digits [MATH] [MATH] . These digits are in one to one correspondence with the triples |
[MATH] and [MATH] of 0 and 1 (the expansions of [MATH] and [MATH] over degrees of 2): [EQUATION] 2–Adic norm of [MATH] is equal to [MATH] , where [MATH] is the number of the first non zero [MATH] in the above expansion (analogously for |
[MATH] ). Each pair of digits [MATH] is defined by a nucleotide using the rule [EQUATION] Since the nucleotides [MATH] and [MATH] are purines, [MATH] |
and [MATH] are pyrimidines, the different first digits in the above binary representation corresponds to the different chemical types of the nucleotides. Namely, the nucleotide [MATH] with [MATH] |
is a purine, and the nucleotide [MATH] with [MATH] is a pyrimidine. The second digit [MATH] in the considered parametrization describes the [MATH] –bonding character (weak for [MATH] and strong for [MATH] ). |
Using the above correspondence between the nucleotides and the digits 0 and 1, we introduce the correspondence between the codons (triples of nucleotides) and the triples [MATH] and |
[MATH] of 0 and 1 (equivalently, the corresponding [MATH] ) by the following prescription. The second nucleotide in the codon defines the pair [MATH] , the first nucleotide in the codon defines the pair [MATH] , and the third nucleotide in the codon defines the pair [MATH] |
This rule is related to the following hierarchy of nucleotides in the codon [EQUATION] i.e the second nucleotide in the codon is the most important (and the largest in the 2–adic norm) and the third nucleotide is the least important. |
After that we make the special reshuffling of the values of [MATH] and [MATH] [EQUATION] similar to made in Then we enumerate codons using the described above rule. Namely we put the codons in the table [MATH] with the natural 2–adic norm (here the numbers of the lines and columns are [MATH] and [MATH] |
defined above): [EQUATION] The [MATH] quadrates here are 2–dimensional 2–adic balls (or clusters) of the diameter [MATH] After application of the eucaryotic genetic (amino acid) code (described in the Appendix) to the above table we get the table of amino acids on the 2–adic plane |
[EQUATION] where Ter is the stop codon. Each square in the above table is the 2 by 2 square in the 2-adic plane of codons. In particular, the genetic code map acts as follows on the 2–adic balls |
[EQUATION] Here we use the standard notations for the nucleotides A, U, G, C and the amino acids. We see, that the degeneracy of the genetic code in the above 2-adic parametrization is described by the 2-adic proximity — the codons which encode the same amino acid are 2-adically close. Moreover, the domains with the di... |
The described here 2-adic 2-dimensional parametrization of the genetic code is related to physical–chemical properties of the amino acids. Namely, the hydrophobic amino acids are clustered in the following two 2-adic balls in the 2-adic plane: |
[EQUATION] Here the hydrophobic amino acids are listed according to the book Using the 2-adic parametrization of the genetic code we can divide all the set of amino acid in the groups [MATH] K, N, E, D, Y, Q, H [MATH] |
[MATH] R, G, W, C [MATH] [MATH] I, M, V, L, F [MATH] [MATH] T, A, S, P [MATH] . These groups are the images with respect to the map of the genetic code of the four quadrants (2–adic balls of the diameter [MATH] ) of the 2–adic plane of codons. |
The PAM matrix The following table describes the PAM250 matrix: [EQUATION] Because the matrix [MATH] is symmetrical we put only the half of matrix elements. This matrix looks irregular and has no any obvious structure. In the next Section we will show that some reshuffling of the numbers of the lines and columns will p... |
Reshuffling of matrix elements of the PAM matrix Let us enumerate the lines and the columns of the PAM matrix [MATH] of the previous Section using the 2–adic parametrization of the genetic code. We get for [MATH] the following: |
[EQUATION] Here we divided all the set of amino acids to the groups [MATH] K, N, E, D, Y, Q, H [MATH] [MATH] R, G, W, C [MATH] [MATH] I, M, V, L, F [MATH] [MATH] T, A, S, P [MATH] . These four groups correspond to the second nucleotide in the codons which encode (through the amino acid genetic code) amino acids in the ... |
Compared to the PAM matrix from the previous Section, this reshuffled matrix is more regular. It has large positive matrix elements at the main diagonal, and off diagonal terms are close to block constant at least at some of the 16 blocks, corresponding to matrix elements with the indices from one of the 4 groups of am... |
In particular, in the block corresponding to the matrix elements [MATH] [MATH] T, A, S, P [MATH] , all off diagonal matrix elements are equal to 1 (10 matrix elements) or 0 (2 matrix elements). |
In the block [MATH] [MATH] K, N, E, D, Y, Q, H [MATH] [MATH] T, A, S, P [MATH] we have matrix elements equal to 0 (13 matrix elements), |
[MATH] (10 matrix elements), 1 (1 matrix element) and anomalous matrix elements equal to [MATH] and [MATH] corresponding to the amino acid Y. The analogous situation we will have in the other blocks of the matrix [MATH] |
We arrive to the following picture: the PAM matrix [MATH] will be a block matrix with matrix elements close to locally constant (i.e. constant in the 16 blocks) if we will exclude matrix elements corresponding to some amino acids, namely, to the amino acids Y, W, C, L, F, and R. |
Expansion for the PAM matrix In the present Section we introduce the main construction of this paper: we will expand the PAM matrix [MATH] in the sum of the matrices |
[MATH] and [MATH] [EQUATION] where the matrix [MATH] will be 2–adically regular (matrix elements are close to locally constant), and the matrix |
[MATH] will be sparse (i.e. majority of matrix elements of this matrix will be equal to zero). We propose the following choice for matrices [MATH] and |
[MATH] [EQUATION] [EQUATION] Non zero matrix elements of [MATH] are mainly concentrated on the lines and columns corresponding to Y, W, C, L, F. There are also several non–zero matrix elements corresponding to R and some other amino acids. |
We see that the non zero matrix elements of [MATH] are mainly concentrated on aromatic amino acids, such as Y, F, W, and on C which contains the SH group. Therefore deviations from 2–adic regularity (i.e. the block structure of the [MATH] matrix) can be discussed as related to the geometric properties of the side chain... |
The above pictures show the amino acids W, R, C, F, Y corresponding to non-zero matrix elements of the matrix [MATH] . Let us mention that amino acids F and Y for which the corresponding matrix element [MATH] is very large (and majority of the other matrix elements of [MATH] for these amino acids are negative) are very... |
Of course, our analysis of PAM matrix based on the 2-adic plane representation of the genetic code is only the first step in using |
[MATH] -adic numbers in genetics and bioinformatics in general. We hope to proceed towards other important problems, cf., e.g., . Finally, we mention the famous ”In Silico Biology” project, see, e.g., Yamato et al. |
. We point out that, in fact, operation of any computer can be represented as 2-adic dynamical system, see . Therefore 2-adic representation of the genetic code might be useful in realization of the ”In Silico Biology” project. |
Appendix: the PAM matrix In this section we discuss the construction of the Dayhoff PAM matrix, which can be found for example in |
. We start with blocks — ungapped multiple alignments of proteins from existing databases. Any sequence in the block is no more than [MATH] |
different from any other sequence in this block. Then, the Markov model, which reproduces the mentioned blocks of proteins was constructed. This Markov model is defined by the amino acid substitutions (point mutations). We have the stationary distribution [MATH] of the probabilities of the amino acids, |
[MATH] , and the transition probability [MATH] normalized by the condition that the probability of a point mutation (substitution of the amino acid) at one step of the Markov model is equal to [MATH] |
[EQUATION] Then we take the matrix given by the [MATH] steps of the Markov model, i.e. the [MATH] -th degree [MATH] of the matrix [MATH] , and consider the matrix with the matrix elements |
[EQUATION] This matrix is known as the PAM matrix (usually [MATH] is taken to be equal to 250 and the matrix elements are approximated by integers). |
Appendix: the genetic code The following table describes the eucaryotic genetic code — the correspondence between codons (triples of nucleotides and amino acids): |
CGG R AGC S UGC C GGC G CGC R ACA T UCA S GCA A CCA P ACU T UCU S GCU A CCU P ACG T UCG S GCG A CCG P ACC T UCC S GCC A CCC P Acknowledgments The authors would like to thank B.Dragovich, I.V.Volovich for fruitful discussions and valuable comments. The authors were partially supported by the grant of International cente... |
# Source: arxiv 0903.5178 # Title: Collective Langevin Dynamics of Flexible Cytoskeletal Fibers # Sections: all # Downloaded: 2026-03-03T05:14:32.313733+00:00 |
Collective Langevin Dynamics of Flexible Cytoskeletal Fibers (May 2, 2007) Abstract We develop a numerical method to simulate mechanical objects in a viscous medium at a scale where inertia is negligible. Fibers, spheres and other voluminous objects are represented with points. Different types of connections are used t... |
Keywords : Cytoskeleton, Simulation, Flexible fibers, Langevin dynamics, Mechanics. Introduction The internal architecture of living cells relies largely on microscopic fibers, which form the cytoskeleton with their associated proteins. These fibers have remarkable mechanical properties. Microtubules and actin filament... |
. Yet these fibers can also be broken down quickly, because they are formed by the non-covalent assembly of protein monomers. Filament ends can grow or shrink, or even alternate between those two states in a remarkable process called dynamic instability |
Structurally, the monomers in microtubules and actin filaments assemble head to tail in a regular manner. On the resulting polar lattices, mechano-enzymes called molecular motors (for example kinesin on microtubules or myosin on actin-filaments) use chemical energy to move directionally |
or to organize the filaments in space . Furthermore, specific enzymes control the filaments by regulating nucleation, assembly/disassembly or even by severing the filaments. |
The cytoskeleton is involved in multiple cellular processes such as cytokinesis, motility, polarization and mitosis. These functions are accomplished by many filaments working together. In this way, a set of dynamic or short-lived filaments may form a stable larger assembly, as exemplified by the mitotic spindle |
Many of the enzymes involved in the assembly of these structures are part of multi-functional entities For example, motors form oligomers that can actively connect filaments together |
; motors may be able to disassemble filaments ; nucleation can be controlled such that it occurs on existing filaments ; crosslinkers may be polarity-specific |
and motors are sometimes linked to proteins that track the tips of growing microtubules Generally speaking, modularity allows the cytoskeleton to be reprogrammed, for example at different stages of the cell cycle. It allows cells to reuse the same functional elements to achieve different tasks and multiplies the number... |
. In any case, the cytoskeleton in addition to fibers contains a kit of activities which can be combined in many ways. Biological systems are hard to understand, and theory is necessary to approach the non-intuitive aspects |
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