Beyond the chemical master equation: Stochastic chemical kinetics coupled with auxiliary processes.

Davin Lunz,Jakob Ruess,J Frédéric Bonnans,Gregory Batt,Vassily Hatzimanikatis

doi:10.1371/journal.pcbi.1009214

Abstract

The chemical master equation and its continuum approximations are indispensable tools in the modeling of chemical reaction networks. These are routinely used to capture complex nonlinear phenomena such as multimodality as well as transient events such as first-passage times, that accurately characterise a plethora of biological and chemical processes. However, some mechanisms, such as heterogeneous cellular growth or phenotypic selection at the population level, cannot be represented by the master equation and thus have been tackled separately. In this work, we propose a unifying framework that augments the chemical master equation to capture such auxiliary dynamics, and we develop and analyse a numerical solver that accurately simulates the system dynamics. We showcase these contributions by casting a diverse array of examples from the literature within this framework and applying the solver to both match and extend previous studies. Analytical calculations performed for each example validate our numerical results and benchmark the solver implementation.

Highlights

The chemical master equation (CME) [1] governs the evolution of the probability distribution of the configuration, or state, of a reaction network
Many important processes act on this cellular heterogeneity at the population level, leading to an intricate coupling between the singlecell and the population-level dynamics
We propose a unifying framework that extends the classical chemical master equation to faithfully capture the single-cell variability alongside the population-level evolution

Summary

Introduction

The chemical master equation (CME) [1] governs the evolution of the probability distribution of the configuration, or state, of a reaction network. The CME and its continuum approximations, in particular the Fokker–Planck approximation [2], are fundamental modeling tools used for describing chemical reaction networks These reaction networks find broad application across the quantitative sciences, providing accurate descriptions of a wide variety of chemical, biological and social phenomena [3]. When a state-dependent selection pressure is exerted upon the population, for example, phenotypic growth or cell-fate decisions, individuals are affected heterogeneously due to this cell-to-cell variability [6, 7]. The dynamics of such phenomena cannot be captured by a stochastic reaction network [8].

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