Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation.

José-Luis Muñoz-Cobo,Cesar Berna

doi:10.3390/e21020181

Abstract

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

Highlights

The early studies on stochastic kinetics and the chemical master equation (CME) started in the early sixties with the works of McQuarrie and Montroll [1,2], followed two year later by the paper of Nicolis and Babloyantz [3]
The chemical reaction networks can be considered as Birth-and-Death processes without memory, which are Markov processes in a multidimensional space, this point of view was studied by Entropy 2019, 21, 181; doi:10.3390/e21020181
An alternative method, which does not require generating a large number of Monte-Carlo simulations, is the Finite State Projection method (FSP) developed originally by Munsky and Khammash; this algorithm provides a direct solution to the CME, obtaining the probability density vector at a given time directly at a desired order of accuracy [12]

Summary

Introduction

The early studies on stochastic kinetics and the chemical master equation (CME) started in the early sixties with the works of McQuarrie and Montroll [1,2], followed two year later by the paper of Nicolis and Babloyantz [3]. An alternative method, which does not require generating a large number of Monte-Carlo simulations, is the Finite State Projection method (FSP) developed originally by Munsky and Khammash; this algorithm provides a direct solution to the CME, obtaining the probability density vector at a given time directly at a desired order of accuracy [12]. Other authors such as Smadbeck, Sotiropuos and Kaznessis, [13,14] prefer to obtain from the CME, the equations for the moment evolution and solve the moment equations, reconstructing at posteriori the probability density from the moments using the maximum entropy principle (MEP). It is convenient to remind that the LNA originates from the system-size expansion first introduced by

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