Delay chemical master equation: direct and closed-form solutions

Andre Leier,Tatiana T Marquez-Lago

doi:10.1098/rspa.2015.0049

Abstract

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

Highlights

The Markov jump formalism has been widely used to describe the stochastic nature of chemical reactions [1,2], gene regulation [3] and other systems involving randomly fluctuating population sizes [4]
In the terminology of chemical reactions, the number of molecules of all present chemical species determines the state of the system, and the system dynamics are governed by a set of reactions involving these species
We will show how an exact solution can be derived in certain cases, and portray cases in which the delay CME (DCME) can be equivalently solved by a chemical master equation (CME) with time-varying parameters

Summary

Introduction

The Markov jump formalism has been widely used to describe the stochastic nature of chemical reactions [1,2], gene regulation [3] and other systems involving randomly fluctuating population sizes [4]. In the terminology of chemical reactions, the number of molecules of all present chemical species determines the state of the system, and the system dynamics are governed by a set of reactions involving these species This can be modelled as a continuoustime, discrete space Markov process and represented. We will show how an exact solution can be derived in certain cases, and portray cases in which the DCME can be equivalently solved by a CME with time-varying parameters. These two observations have never been described in the literature before, opening up both applications and alternative methodologies to solve stochastic chemical kinetics with prescribed delays

Delay stochastic simulation algorithms

The delay chemical master equation

Exact closed-form solution of a delay chemical master equation

Conclusion