On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure.

Michail Vlysidis,Yiannis Kaznessis

doi:10.3390/e20090678

Abstract

Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.

Highlights

Chemical reaction kinetics have been canonically modeled with ordinary differential equations since the pronouncement of the law of mass action kinetics, 150 years ago [1]
We present results obtained with ZI-closure scheme for the stochastic Schlögl model
We present solutions of the chemical master equation (CME) for Schlögl model systems

Summary

Introduction

Chemical reaction kinetics have been canonically modeled with ordinary differential equations since the pronouncement of the law of mass action kinetics, 150 years ago [1]. This macroscopic, continuous-deterministic modeling formalism is appropriate at the thermodynamic limit, when the volume of the system and the numbers of molecules of reactants all tend to very large values. Models are formulated in terms of discrete numbers of molecules for each of the chemical species present at time t. The system evolves stochastically, and the all-encompassing chemical master equation (CME) can model the probability distribution of the system being at a particular state at time t [3]. Kurtz [4,5] explored the relationship between stochastic and deterministic models when the macroscopic equations have a unique, asymptotically stable solution, and demonstrated that the deterministic model is the thermodynamic limit of the stochastic one

Results

Discussion

Conclusion