# Propensity approach to nonequilibrium thermodynamics of a chemical reaction network: controlling single E-coli β-galactosidase enzyme catalysis through the elementary reaction steps.

26 Dec 2013-Journal of Chemical Physics (American Institute of Physics)-Vol. 139, Iss: 24, pp 244104-244104

TL;DR: This work develops an approach to nonequilibrium thermodynamics of an open chemical reaction network in terms of the elementary reaction propensities, and thoroughly analyzes the temporal as well as the steady state behavior of various thermodynamic quantities for each elementary reaction.

Abstract: In this work, we develop an approach to nonequilibrium thermodynamics of an open chemical reaction network in terms of the elementary reaction propensities. The method is akin to the microscopic formulation of the dissipation function in terms of the Kullback-Leibler distance of phase space trajectories in Hamiltonian system. The formalism is applied to a single oligomeric enzyme kinetics at chemiostatic condition that leads the reaction system to a nonequilibrium steady state, characterized by a positive total entropy production rate. Analytical expressions are derived, relating the individual reaction contributions towards the total entropy production rate with experimentally measurable reaction velocity. Taking a real case of Escherichia coli β-galactosidase enzyme obeying Michaelis-Menten kinetics, we thoroughly analyze the temporal as well as the steady state behavior of various thermodynamic quantities for each elementary reaction. This gives a useful insight in the relative magnitudes of various energy terms and the dissipated heat to sustain a steady state of the reaction system operating far-from-equilibrium. It is also observed that, the reaction is entropy-driven at low substrate concentration and becomes energy-driven as the substrate concentration rises.

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TL;DR: A simple theory is developed which predicts that certain classes of enzyme pathways can be distinguished by studying the turnover rate, V, as a function of substrate concentration, [S], and it is found to depend sensitively on the manner in which substrate molecules in the bath are replenished.

Abstract: The ability to dynamically probe single enzymes allows the experimental investigation of enzyme kinetics with unprecedented resolution. In this paper we develop a simple theory which predicts that certain classes of enzyme pathways can be distinguished by studying the turnover rate, V, as a function of substrate concentration, [S]. In particular, we study the steady state of a single enzyme interacting with a bath of substrate molecules, and analyse it as a stochastic process. The V([S]) relation is found to depend sensitively on the manner in which substrate molecules in the bath are replenished. We focus on a gedanken experiment in which the average substrate concentration is kept fixed by allowing molecules to enter the bath at a constant rate. We derive the exact relationship between V and [S], which has a relatively simple form, though different to that of the Michaelis–Menten (MM) equation. Interestingly, the MM equation is exactly recovered if the substrate concentration is instantaneously maintained with molecular precision. We examine the new V([S]) relation for a number of enzyme pathways and find that it differentiates between enzyme reactions involving one or many intermediate enzyme–substrate complexes. This, in principle, allows one to probe the internal conformations of enzymes by careful measurement of V([S]) curves in appropriately designed single enzyme experiments.

19 citations

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TL;DR: Dissipation plays a guiding role in the optimization of the catalytic rate in the tRNA selection network of protein synthesis, and the network tends to maximize both the EPR and catalytic rates, but not the accuracy.

Abstract: Major biological polymerization processes achieve remarkable accuracy while operating out of thermodynamic equilibrium by utilizing the mechanism known as kinetic proofreading. Here, we study the interplay of the thermodynamic and kinetic aspects of proofreading by exploring the dissipation and catalytic rate, respectively, under the realistic constraint of fixed chemical potential difference. Theoretical analyses reveal no-monotonic variations of the catalytic rate and total entropy production rate (EPR), the latter quantifying the dissipation, at steady state. Applying this finding to a tRNA selection network in protein synthesis, we observe that the network tends to maximize both the EPR and catalytic rate, but not the accuracy. Simultaneously, the system tries to minimize the ratio of the EPRs due to the proofreading steps and the catalytic steps. Therefore, dissipation plays a guiding role in the optimization of the catalytic rate in the tRNA selection network of protein synthesis.

5 citations

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TL;DR: Using some special properties of the Legendre transformation, here, a relation between the fluctuations of fluxes and dissipation rates is provided, and among them, the fluctuation of the turnover rate is routinely estimated but the fluctuations in the dissipation rate is yet to be characterized for small systems.

Abstract: In the framework of large deviation theory, we have characterized nonequilibrium turnover statistics of enzyme catalysis in a chemiostatic flow with externally controllable parameters, like substrate injection rate and mechanical force. In the kinetics of the process, we have shown the fluctuation theorems in terms of the symmetry of the scaled cumulant generating function (SCGF) in the transient and steady state regime and a similar symmetry rule is reflected in a large deviation rate function (LDRF) as a property of the dissipation rate through boundaries. Large deviation theory also gives the thermodynamic force of a nonequilibrium steady state, as is usually recorded experimentally by a single molecule technique, which plays a key role responsible for the dynamical symmetry of the SCGF and LDRF. Using some special properties of the Legendre transformation, here, we have provided a relation between the fluctuations of fluxes and dissipation rates, and among them, the fluctuation of the turnover rate is routinely estimated but the fluctuation in the dissipation rate is yet to be characterized for small systems. Such an enzymatic reaction flow system can be a very good testing ground to systematically understand the rare events from the large deviation theory which is beyond fluctuation theorem and central limit theorem.

3 citations

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03 Sep 2015

TL;DR: In this paper, an approach to nonequilibrium thermodynamics of an open chemical reaction network in terms of the propensities of individual elementary reactions and corresponding reverse reactions is introduced.

Abstract: We have introduced an approach to nonequilibrium thermodynamics of an open chemical reaction network in terms of the propensities of the individual elementary reactions and the corresponding reverse reactions. The method is a microscopic formulation of the dissipation function in terms of the relative entropy or Kullback-Leibler distance which is based on the analogy of phase space trajectory with the path of elementary reactions in a network of chemical process. We have introduced here a fluctuation theorem valid for each opposite pair of elementary reactions which is useful in determining the contribution of each sub-reaction on the nonequilibrium thermodynamics of overall reaction. The methodology is applied to an oligomeric enzyme kinetics at a chemiostatic condition that leads the reaction to a nonequilibrium steady state for which we have estimated how each step of the reaction is energy driven or entropy driven to contribute to the overall reaction.

2 citations

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TL;DR: In this article, entropy change along a single stochastic trajectory of a biomolecule is discussed for two different sources of non-equilibrium entropy, and the total entropy change obeys an integral fluctuation theorem and a class of further exact relations.

Abstract: Entropy production along a single stochastic trajectory of a biomolecule is discussed for two different sources of non-equilibrium. For a molecule manipulated mechanically by an AFM or an optical tweezer, entropy production (or annihilation) occurs in the molecular conformation proper or in the surrounding medium. Within a Langevin dynamics, a unique identification of these two contributions is possible. The total entropy change obeys an integral fluctuation theorem and a class of further exact relations, which we prove for arbitrarily coupled slow degrees of freedom including hydrodynamic interactions. These theoretical results can therefore also be applied to driven colloidal systems. For transitions between different internal conformations of a biomolecule involving unbalanced chemical reactions, we provide a thermodynamically consistent formulation and identify again the two sources of entropy production, which obey similar exact relations. We clarify the particular role degenerate states have in such a description.

1 citations

##### References

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TL;DR: In this article, a simulation algorithm for the stochastic formulation of chemical kinetics is proposed, which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of a given chemical system.

Abstract: There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly predictable process which is governed by a set of coupled, ordinary differential equations (the “reaction-rate equations”); the stochastic approach regards the time evolution as a kind of random-walk process which is governed by a single differential-difference equation (the “master equation”). Fairly simple kinetic theory arguments show that the stochastic formulation of chemical kinetics has a firmer physical basis than the deterministic formulation, but unfortunately the stochastic master equation is often mathematically intractable. There is, however, a way to make exact numerical calculations within the framework of the stochastic formulation without having to deal with the master equation directly. It is a relatively simple digital computer algorithm which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of the given chemical system. Like the master equation, this “stochastic simulation algorithm” correctly accounts for the inherent fluctuations and correlations that are necessarily ignored in the deterministic formulation. In addition, unlike most procedures for numerically solving the deterministic reaction-rate equations, this algorithm never approximates infinitesimal time increments df by finite time steps At. The feasibility and utility of the simulation algorithm are demonstrated by applying it to several well-known model chemical systems, including the Lotka model, the Brusselator, and the Oregonator.

9,512 citations

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TL;DR: In this paper, an exact method is presented for numerically calculating, within the framework of the stochastic formulation of chemical kinetics, the time evolution of any spatially homogeneous mixture of molecular species which interreact through a specified set of coupled chemical reaction channels.

Abstract: An exact method is presented for numerically calculating, within the framework of the stochastic formulation of chemical kinetics, the time evolution of any spatially homogeneous mixture of molecular species which interreact through a specified set of coupled chemical reaction channels. The method is a compact, computer-oriented, Monte Carlo simulation procedure. It should be particularly useful for modeling the transient behavior of well-mixed gas-phase systems in which many molecular species participate in many highly coupled chemical reactions. For “ordinary” chemical systems in which fluctuations and correlations play no significant role, the method stands as an alternative to the traditional procedure of numerically solving the deterministic reaction rate equations. For nonlinear systems near chemical instabilities, where fluctuations and correlations may invalidate the deterministic equations, the method constitutes an efficient way of numerically examining the predictions of the stochastic master equation. Although fully equivalent to the spatially homogeneous master equation, the numerical simulation algorithm presented here is more directly based on a newly defined entity called “the reaction probability density function.” The purpose of this article is to describe the mechanics of the simulation algorithm, and to establish in a rigorous, a priori manner its physical and mathematical validity; numerical applications to specific chemical systems will be presented in subsequent publications.

5,354 citations

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TL;DR: The results provide the first direct experimental evidence of the biochemical origin of phenotypesic noise, demonstrating that the level of phenotypic variation in an isogenic population can be regulated by genetic parameters.

Abstract: Stochastic mechanisms are ubiquitous in biological systems. Biochemical reactions that involve small numbers of molecules are intrinsically noisy, being dominated by large concentration fluctuations 1‐3 . This intrinsic noise has been implicated in the random lysis/lysogeny decision of bacteriophage-λ 4 , in the loss of synchrony of circadian clocks 5,6 and in the decrease of precision of cell signals7. We sought to quantitatively investigate the extent to which the occurrence of molecular fluctuations within single cells (biochemical noise) could explain the variation of gene expression levels between cells in a genetically identical population (phenotypic noise). We have isolated the biochemical contribution to phenotypic noise from that of other noise sources by carrying out a series of differential measurements. We varied independently the rates of transcription and translation of a single fluorescent reporter gene in the chromosome of Bacillus subtilis, and we quantitatively measured the resulting changes in the phenotypic noise characteristics. We report that of these two parameters, increased translational efficiency is the predominant source of increased phenotypic noise. This effect is consistent with a stochastic model of gene expression in which proteins are produced in random and sharp bursts. Our results thus provide the first direct experimental evidence of the biochemical origin of phenotypic noise, demonstrating that the level of phenotypic variation in an isogenic population can be regulated by genetic parameters. We selected as our reporter system a single-copy chromosomal gene with an inducible promoter. As an estimated 50‐80% of bacterial genes are transcriptionally regulated 8 , this system typifies the majority of naturally occurring genes, allowing our results to be extended to natural systems. We incorporated a single copy of our reporter, the green fluorescent protein gene (gfp), into the chromosome of B. subtilis. We chose to integrate gfpinto the chromosome itself, rather than in the form of plasmids, as variation in plasmid copy number 9,10 can act as an additional and unwanted source of noise. Transcriptional efficiency was regulated by using an isopropyl-β-D-thiogalactopyranoside (IPTG)‐inducible promoter, Pspac, upstream of gfp, and varying the concentration of IPTG in the growth medium. Translational

1,623 citations

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TL;DR: A molecular memory phenomenon, in which an enzymatic turnover was not independent of its previous turnovers because of a slow fluctuated of protein conformation, was evidenced by spontaneous spectral fluctuation of FAD.

Abstract: Enzymatic turnovers of single cholesterol oxidase molecules were observed in real time by monitoring the emission from the enzyme's fluorescent active site, flavin adenine dinucleotide (FAD). Statistical analyses of single-molecule trajectories revealed a significant and slow fluctuation in the rate of cholesterol oxidation by FAD. The static disorder and dynamic disorder of reaction rates, which are essentially indistinguishable in ensemble-averaged experiments, were determined separately by the real-time single-molecule approach. A molecular memory phenomenon, in which an enzymatic turnover was not independent of its previous turnovers because of a slow fluctuation of protein conformation, was evidenced by spontaneous spectral fluctuation of FAD.

1,296 citations