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II The model We develope here a theoretical model for the polymerization of a protein by ribosomes using an artificially synthesized mRNA template that consists of a homogeneous sequence (i.e., all the codons of which are identical). In the surrounding medium, |
two species of amino-acids are available only one of which is the correct one according to the genetic code. The ribosome deploys a quality control system that rejects the amino acid monomer if it is incorrect. If this quality control system never fails, perfect fidelity of translation would result in a homopolymer who... |
hetero-polymer . We’ll study the effects of the quality control system on the DTD of the ribosomes. A ribosome consists of two interconnected subunits which are designated as “large” and “small” (see fig. ). The small subunit binds with the mRNA track and decodes the genetic message of the codon whereas the polymerizat... |
The three main stages in each mechano-chemical cycle of a ribosome, shown in fig. , are as follows: (i) selection of the cognate (i.e., correct) aa-tRNA, (ii) formation of the peptide bond between the amino acid brought in by the selected aa-tRNA and the elongating protein, (iii) translocation of the ribosome by one co... |
Let us begin with the state labelled by “1”; both the E and A sites are empty while the site P is occupied as shown in fig. An incoming aa-tRNA molecule, bound to an elongation factor EF-Tu, occupies the site A; the resulting state is labelled by “2”. The transition [MATH] takes place at the rate [MATH] . However, not ... |
rodnina01 rodnina06 ogle05 zaher09 the ribosome deploys a quality control mechanism to ensure that the aa-tRNA selected is, indeed, cognate (i.e., carries the correct amino acid as dictated by the mRNA template). A non-cognate tRNA is rejected on the basis of codon-anticodon mismatch; the corresponding transition |
[MATH] takes place at the rate [MATH] . However, if the aa-tRNA is not a non-cognate one, the GTP molecule bound to it is then hydrolyzed to GDP causing the irreversible transition [MATH] at the rate |
[MATH] . At this stage, kinetic proofreading can reject the tRNA if it is a near-cognate one; the transition [MATH] captures this phenomenon and takes place at the rate [MATH] . In our model we do not explicitly treat those sub-steps in which EF-TU and the products of the hydrolysis of GTP leave the ribosome. We’ll lat... |
If the selected aa-tRNA is not rejected, it does not necessarily imply that it is cognate because, occasionally, even non-cognate or near-cognate aa-tRNA escapes the ribosomes quality control mechanism. The amino acid supplied by the selected aa-tRNA is then linked to the growing polypeptide by a peptide bond; this pep... |
[MATH] , however, ends a “futile” cycle. The only difference between the states [MATH] and [MATH] is that the last amino acid in the polypeptide is correct in [MATH] but incorrect in [MATH] . While the polypeptide gets elongated by one amino acid, a fresh molecule of GTP enters bound with an elongation factor EF-G. The... |
Next, spontaneous Brownian (relative) rotation of the two subunits of the ribosome coincides with the back-and-forth transition between the so-called “classical” and “hybrid” configurations of the two tRNA molecules. |
noller89 noller98 noller02 noller07 noller08 frank00 frank03 frank07 pan07 moran08 shoji09 In the classical configuration, both ends of the two tRNA molecules correspond to the locations of [MATH] and [MATH] sites. In contrast, in the hybrid configuration the ends of tRNA molecules interacting with the large subunit ar... |
[MATH] along the wrong branch) whereas the reverse transition [MATH] takes place at the rate [MATH] along the correct branch (and [MATH] along the wrong branch). The reversible transition [MATH] , which is caused by spontaneous Brownian fluctuations, does not need any free energy input from GTP hydrolysis. |
Finally the hydrolysis of GTP drives the irreversible transition [MATH] which involves the translocation of the ribosome on its track by one codon and, simultaneously, that of the tRNAs inside the ribosome by one binding site following which the deacetylated (i.e., denuded of amino acid) tRNA exits from the E site. The... |
The rate constant [MATH] can be made identical for both species of amino acid monomers by maintaining their concentrations in the medium appropriately. The rate constant [MATH] refers only to the wrong amino acids (i.e., non-cognate tRNA) because, we assume, cognate tRNA is not rejected at all. Since the transition [MA... |
accounts for the hydrolysis of a GTP molecule by the GTPase EF-Tu, the corresponding rate constant is [MATH] , irrespective of the identity of the attached aa-tRNA. In kinetic proofreading, the rate of rejection of the non-cognate tRNA is much higher than that of a cognate tRNA. For the sake of simplicity, we assume th... |
[MATH] referers to the rejection of only non-cognate tRNA. For the remaining steps of the mechano-chemical cycle, the rate constants are [MATH] [MATH] [MATH] and |
[MATH] provided a cognate tRNA has been selected finally. On the other hand, the corresponding rate constants are [MATH] [MATH] [MATH] and [MATH] respectively, if a non-cognate tRNA escapes rejection by the quality control system of the ribosome. Since the last step involves not only hydrolysis of GTP by the GTPase EF-... |
[MATH] need not be equal, in general. Thus our model is an extension of the generic models for molecular motors based on stochastic chemical kinetic approach which was pioneered by Fisher and Kolomeisky fishkolo kolorev Following Hill hillbook the mechano-chemical kinetics of a ribosome in our model can be regarded as ... |
All the numerical data for the graphical plots have the following set of values of the rate constants (except in the figures where [MATH] has been plotted for several different values of the parameters |
[MATH] and [MATH] ) : [MATH] -1 [MATH] -1 [MATH] -1 [MATH] -1 [MATH] -1 [MATH] -1 [MATH] -1 [MATH] -1 The value of [MATH] -1 is identical to that used in our earlier papers (see basu07 gccr and the references therein). For the purpose of plotting our results, the magnitude of the other parameters have been chosen to be... |
III Dwell time distribution For the convenience of mathematical calculations, we assume that, after reaching the state [MATH] (or [MATH] ) at location [MATH] the system makes a transition to a hypothetical state [MATH] at location [MATH] which, then, relaxes to the state [MATH] , at the same location, at the rate |
[MATH] . At the end of the calculation our model is recovered by setting [MATH] Suppose [MATH] denote the probability at time [MATH] that the ribosome is in the “chemical” state [MATH] and is decoding the |
[MATH] -th codon. Let us use the symbol [MATH] to denote the probability of finding the ribosome in the hypothetical state [MATH] at time [MATH] while decoding the [MATH] -th codon. The time taken by the ribosome to reach the state [MATH] at |
[MATH] , starting from the initial state [MATH] at [MATH] , defines its time of dwell at the [MATH] -th codon. Since, in this context, all the “chemical” states, except [MATH] refer to the [MATH] -th codon while [MATH] corresponds to the [MATH] -th codon, from now onwards we drop the site index [MATH] (and [MATH] ) to ... |
The master equations governing the time evolution of the probabilities [MATH] can be written as [EQUATION] [EQUATION] [EQUATION] |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] Because of the normalization condition [EQUATION] not all the equations ( )-( ) above are independent of each other. |
For the calculation of the dwell time, we impose the initial conditions [EQUATION] Suppose, [MATH] is the probability density of the dwell times. Then, the probability of adding one amino acid to the growing polypeptide in the time interval between [MATH] and [MATH] is [MATH] |
where [EQUATION] The calculation of the DTD chemla08 shaevitz05 liao07 linden07 fisher07 garai10 is essentially that of a distribution of first-passage times redner |
The exact probability density of the dwell times is given by [EQUATION] where [MATH] [MATH] and [MATH] are solution of the cubic equation |
[EQUATION] [MATH] and [MATH] are the solution of the quadratic equation [EQUATION] and [MATH] and [MATH] are the solution of the quadratic equation |
[EQUATION] Some of the details of this derivation are given in the appendix. Note that the problem of determining the rates [MATH] |
[MATH] ) in terms of the rate constants of the model is similar to that of expressing the normal modes of vibration of a set of coupled harmonic oscillators. It is the “backward” reactions in the mechano-chemical cycle of our model which play the role of coupling of the harmonic oscillators. For example, if [MATH] and ... |
[MATH] , the expressions for [MATH] [MATH] ) reduce to the simple form [MATH] [MATH] [MATH] [MATH] , and [MATH] The distribution ( LABEL:eq-ftfinal ) is plotted in Fig. |
for a few different values of the parameter [MATH] . Note that [EQUATION] is a measure of the fidelity of translation. Therefore, increasing |
[MATH] , keeping [MATH] fixed, enhances translational fidelity. Moreover, a higher [MATH] also corresponds to faster peptidyl transferase reaction. Therefore, the trend of variation of the most probable dwell time with [MATH] is consistent with the intuitive expectation that the slower is the peptidyl transferase react... |
The kinetic parameter [MATH] is a measure of the rate of rejection of the aa-tRNA by kinetic proofreading. Consequently, the effect of the variation of the [MATH] on the most probable dwell time is opposite to that of [MATH] (see Fig. ); the higher is the frequency of tRNA rejection by kinetic proofreading, the longer ... |
IV Mean rate of polymerization: a Michaelis-Menten-like equation The average Dwell time can be calculated by substituting ( LABEL:eq-ftfinal into the definition |
[EQUATION] Hence, the expression for [MATH] [EQUATION] written in terms of [MATH] and [MATH] has a clear physical meaning. In the special case [MATH] |
[MATH] . However, if [MATH] , then the sum [MATH] is replaced by the last two terms of ( 18 ) where [MATH] and [MATH] are the weight factors associated with the two paths emanating from the state labelled by [MATH] |
Finally, in terms of the rate constants for the mechano-chemical transitions in the model depicted in fig. [EQUATION] Next we’ll show that the equation ( LABEL:eq-avtfinal ) can be re-expressed in a form that resembles the Michaelis-Menten equation for simple enzymatic reactions dixon . Consider an enzymatic reaction o... |
[EQUATION] where [MATH] is the enzyme, [MATH] is the substrate and [MATH] is the product of the reaction catalyzed by [MATH] . Given that the total initial concentration of the enzyme is [MATH] , the rate |
[MATH] of this reaction is given by [EQUATION] where the maximum possible rate of the reaction is [EQUATION] and the Michaelis constant [MATH] is given by |
[EQUATION] In the case of our model, we assume that the “pseudo” first order rate constant [MATH] can be written as [MATH] where [MATH] is the concentration of tRNA molecules in the solution. Treating tRNA molecules as the analogues of the substrates in an enzymatic reaction, equation ( LABEL:eq-avtfinal ) can be re-ex... |
[EQUATION] where [EQUATION] and the Michaelis constant [EQUATION] with [EQUATION] Thus, the mean dwell time for the ribosomes follows a Michaelis-Mentan-like equation. This result is consistent with the experimental observations in recent years |
qian02 english06 kou05 min05 min06 basu09 that, in spite of the fluctuations of an enzymatic reaction catalyzed by a single enzyme molecule, the average rate of the reaction is, most often, given by the Michaelis-Menten equation. What makes the Michaelis-Menten-like equation for the ribosome even more interesting is th... |
Effects of crowding on the dwell time distribution It is well known that most often a large number of ribosomes simultaneously move on the same mRNA track each polymerizing one copy of the same protein. This phenomenon is usually referred to as ribosome traffic because of its superficial similarity with vehicular traff... |
macdonald68 macdonald69 lakatos03 shaw03 shaw04a shaw04b chou03 chou04 dong1 dong2 cook ciandrini basu07 mitarai08 . Suppose, [MATH] denotes the number of codons that a ribosome can cover simultaneously. Extending the prescription used in our earlier works on ribosome traffic basu07 for capturing the steric interaction... |
[EQUATION] [EQUATION] [EQUATION] where [MATH] is the conditional probability that, given a ribosome in site i, site j is empty. All the other equations for [MATH] remain unchanged. Note that these equations have been written under the mean-field approximation |
In the limit [MATH] , all the sites are treated on the same footing so that the site-dependence of [MATH] drops out. In this limit, [MATH] takes the simple form basu07 |
[EQUATION] where [MATH] is the number density of the ribosomes (i.e., number of ribosomes per unit length of the mRNA track). Therefore, in ribosome traffic the distribution of the dwell times of the ribosomes is given by the expression ( LABEL:eq-ftfinal ) where |
[MATH] and [MATH] are replaced by [MATH] and [MATH] , respectively. Using the expression ( 31 ), we get the DTD for the given number density [MATH] of the ribosomes. For the purpose of graphical demostration of the effects of ribosome crowding on the DTD, we use the coverage density |
[MATH] . In Fig. we plot the DTD for a few different values of [MATH] . The higher is the density, the stronger is the hindrance and longer is the dwell time. Moreover, higher coverage density introduces stronger correlations which our mean-field equations ignore. Consequently, our analytical predictions, based on equa... |
VI Comparison with experimental data The distribution of dwell times involves essentially five different rate-determining parameters [MATH] [MATH] ) if the translational fidelity is perfect. This is consistent with the earlier observation wen09 that the best fit to the simulation data was obtained with five rate-determ... |
wen08 . This strongly indicates the possibility that only two of the five rate-determining parameters were rate-limiting under the conditions maintained in those experiments. However, for a quantitative testing of the predictions of our theoretical model, one should use a mRNA template with a homogeneous codon sequence... |
In order to clarify the proposed experimental set up, let us consider the concrete example shown schematically in fig. This example is essentially the protocols used by Uemura et al. |
uemura10 in their single-molecule studies of translation in real time. The actual coding sequence to be translated consists of [MATH] number of identical codons; in fig. |
[MATH] and each codon is UUU which codes for the amino acid Phenylalanine (abbreviated Phe or ). The coding sequence is preceeded by a start codon AUG and is followed by a stop codon UAA. The start codon itself is preceeded by an untranslated region (UTR) at the 5’-end of the mRNA; this is required for assembling the r... |
The effects of kinetic proofreading on the rate of translational error has been investigated experimentally by several different methods in the last three decades. However, most of those methods (see, for example, ref. hopfield1 hopfield2 ) require bulk samples. But, the more recent single-molecule FRET technique used ... |
The cluster of ribosomes translating the same mRNA simultaneously is usually referred to as a polysome . For studying the effects of ribosome crowding on the DTD, one has to measure simultaneously the DTD and the polysome size arava03 |
VII Summary and conclusion In this paper we have presented a theoretical model of translation that captures all the main steps in the mechano-chemical cycle of a ribosome during the elongation stage. This model also accounts for translational fidelity, kinetic proofreading and the crowding of the ribosomes on the same ... |
In spite of the details already incorporated in this model, we have suceeded in carrying out an analytical calculation of the distribution of the dwell times of the ribosomes at the successive codons. We have compared this theoretical estimate of the DTD with the corresponding numerical data which we have obtained from... |
However, because of the mean-field approximation made in capturing the effects of crowding of the ribosomes the corresponding analytical expression is approximate. Therefore, the higher is the coverage density, the larger is the deviation of the analytical estimate from the simulation data which are averaged over many ... |
We have analyzed the dependence of the DTD on some of the crucially important kinetic parameters of the model to elucidate the physical implications of the result. Our results are in good qualitative agreement with the experimental data reported in the literature |
wen08 wen09 . However, for the reasons explained in the sections and II , it is not possible to compare these experimental data quantitatively with the analytical expressions of DTD which we have reported in this paper. We hope our theoretical predictions will stimulate further experimental studies along the lines sugg... |
VI although some technical hurdles may hinder quick progress. A combination of the single-ribosome experiments and bulk measurements may be required for comparing the theoretically predicted variation of the DTD with the concentrations of tRNA, GTP and other key molecules involved in translation as well as with the inc... |
It would also be desirable to extend our model in future to account for some important features of translation; some of these possible extentions are listed below. (i) Capturing the sequence heterogeneity of real mRNA templates will open up larger number of branched pathways in Fig. , each corresponding to a distinct s... |
Acknowledgements: One of the authors (DC) thanks Joachim Frank and Joseph D. Puglisi for useful correspondences. Appendix Solving the equations ( )-( ) under the initial condition ( 10 ) we get |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] where [MATH] [MATH] and [MATH] are the solutions of the cubic equation ( 13 ) while [MATH] [MATH] |
are the solutions of the quadratic equation ( 14 ) and [MATH] [MATH] are the solutions of the quadratic equation 15 ). The coefficients [MATH] [MATH] ) and [MATH] [MATH] which are determined by the initial condition ( 10 ), are as follows: [EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] [EQUATION] |
# Source: arxiv 1009.2044 # Title: Computational Modalities of Belousov-Zhabotinsky Encapsulated Vesicles # Sections: all # Downloaded: 2026-03-02T08:58:22.372503+00:00 |
Computational Modalities of Belousov-Zhabotinsky Encapsulated Vesicles Abstract We present both simulated and partial empirical evidence for the computational utility of many connected vesicle analogs of an encapsulated non-linear chemical processing medium. By connecting small vesicles containing a solution of sub-exc... |
Keywords: Belousov-Zhabotinsky reaction, computation, logic gates, half adder, excitable media, unconventional computing Introduction |
The last half of the twentieth century has been witness to huge leaps in technology spanning all areas of science. One of the most noticeable areas has been the dramatic success of the vonn Neumann |
architecture electronic digital computer. Although modern digital computers or the software has not advanced to a point where one computer could independently create another technologically superior computer, or where software could compose more advanced software , one could argue that from a purely technological persp... |
. This is an area of study that explores alternative computational representation, substrates and strategies, the results of which not only create novel experimental processing devices |
, but also contribute towards algorithms operating on conventional serial digital computers. One direction in this genre is the study of reaction diffusion (RD) computing |
. Where the innate behaviour of a chemical reaction and subsequent diffusion in space and time can be used to present and manipulate information. A suitable and convenient chemical reaction for such processing is the Belousov-Zhabotinsky (BZ) reaction, a type of reaction that is subject to non-equilibrium thermodynamic... |
. In certain formulations the BZ reaction can produce visible travelling waves which can be used to represent information . Wave development is effected not only by the reaction conditions, but by geometric obstacles and collisions with other waves. Computation circuits |
analogous to electronic circuits can be created with chemical pathways (conductors) routed through a passive substrate (insulator) with waves representative of signals (electron flow). |
In order to illustrate the possibility of computation in a BZ substrate some of the key components that are used to create electronic digital computers have been created, such as; diodes |
, coincidence detector and logic gates . These components have been combined to create more complex circuits such as memory counters |
and binary adders These circuits serve to demonstrate that it is possible to create computational devices by modelling existing digital components and functions within the RD frame work, this approach amounts to conventional computing on an unconventional substrate. More interesting are some systems that exemplify unco... |
shortest path calculation , image processing , information encoding and direction detection Contrary to these previous computation approaches in a BZ medium we have focused on exploring the utility of connecting small spherical processing elements containing BZ medium (vesicles) into functional networks |
. Vesicles can be created by surrounding a solution of BZ reactant with a mono-layer of lipids . This cell like structure has some interesting parallels with real neurons. When two or more vesicles are pressed together in solution the gap between the lipid layer forms a chemical junction similar to a synaptic cleft. Tr... |
In terms of connection, self adaptation and longevity the vesicle [MATH] neuron analogy does not hold. Real neurons are typified by their distributed connectedness, ability to learn and self sustain. Vesicles under consideration in this work are only connected locally, also they cannot be sustained beyond exhaustion of... |
The remainder of the paper is comprised of the following: Section 2.1 details the method of BZ numerical computer simulation and graphical presentation. Section 2.2 introduces vesicle simplification, geometry and networking. Simulation results exploring the vesicle geometry, connectivity and membrane function are prese... |
3.3 . The results are discussed, future directions considered and a summary presented in the remaining sections Methods 2.1 Computer simulations |
We have employed a two variable version of the Oregonator model as a model of the BZ reaction adapted for photo-sensitive modulation of the |
\cf Ru-catalysed reaction [EQUATION] Variables [MATH] and [MATH] are the local instantaneous dimensionless concentrations of the bromous acid autocatalyst activator \cf HBrO2 and the oxidised form of the catalyst inhibitor \cf Ru(bpy)3^3+. |
[MATH] symbolises the rate of bromide production proportional to applied light intensity. Bromide \cf Br^- is an inhibitor of the |
\cf Ru-catalysed reaction, therefore excitation can be modulated by light intensity; high intensity light inhibits the reaction. Dependant on the rate constant and reagent concentration |
[MATH] represents the ratio of the time scales of the two variables [MATH] and [MATH] [MATH] is a scaling factor dependent on the reaction rates alone. The diffusion coefficients [MATH] and [MATH] of |
[MATH] and [MATH] were set to unity and zero respectively. The coefficient [MATH] is set to zero because it is assumed that the diffusion of the catalyst is limited. |
Numerical simulations were achieved by integrating the equations using the Euler-ADI method with a time step [MATH] and a spatial step [MATH] . Experimental parameters are given in Tab. |
Networks of discs where created by mapping 2 different [MATH] values (proportional to light intensity) onto a rectangle of homogeneous simulation substrate. To improve simulation performance the rectangle size was automatically adapted depending on the size of the network, but the simulation point density remained cons... |
Discs are always separated by a single simulation point wide boundary layer. Connection apertures between discs are created by superimposing another small link disc at the point of connection (typically a 2 |
[MATH] 6 simulation point radius), simulation points have a 1:1 mapping with on screen pixels. The reagent concentrations are represented by a red and blue colour mapping; the activator, [MATH] is proportional to red level and inhibitor, [MATH] proportional to blue. The colour graduation is automatically calibrated to ... |
Wave fragment flow is represented by a series of superimposed time lapse images (unless stated otherwise), the time lapse is 50 simulation steps. To improve clarity, only the activator ( [MATH] ) wave front progression is recorded. Figure |
illustrates the same wave fragment in both colour map ( [MATH] [MATH] ) and time lapse versions ( [MATH] ). Inputs are created by perturbing a small circular area of the activator ( [MATH] ) set to a value of 1 with a radius of 2 simulation points in the center of the disc. All discs representing inputs and outputs are... |
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