Finding analytic stationary solutions to the chemical master equation by gluing state spaces at one or two states recursively

Abstract

Noise is often indispensable to key cellular activities, such as gene expression, necessitating the use of stochastic models to capture its dynamics. The chemical master equation (CME) is a commonly used stochastic model that describes how the probability distribution of a chemically reacting system varies with time. Knowing analytic solutions to the CME can have benefits, such as expediting simulations of multiscale biochemical reaction networks and aiding the design of distributional responses. However, analytic solutions are rarely known. A recent method of computing analytic stationary solutions relies on gluing simple state spaces together recursively at one or two states. We explore the capabilities of this method and introduce algorithms to derive analytic stationary solutions to the CME. We first formally characterise state spaces that can be constructed by performing single-state gluing of paths, cycles or both sequentially. We then study stochastic biochemical reaction networks that consist of reversible, elementary reactions with two-dimensional state spaces. We also discuss extending the method to infinite state spaces and designing stationary distributions that satisfy user-specified constraints. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and chemical reactions with two-dimensional state spaces.

Coupling from the Past for the Stochastic Simulation of Chemical Reaction Networks

“…A common approach to modelling chemical reaction networks involves representing the time-rates-of-change in species' concentrations with a set of coupled differential equations known as the reaction rate equations. Underpinning these models are the assumptions that the molecular concentrations vary continuously over time and evolve in a deterministic way [1]. Many homogeneous chemical systems, particularly reaction systems involving large numbers of molecules, can be well approximated by such models [2].…”

“…Many homogeneous chemical systems, particularly reaction systems involving large numbers of molecules, can be well approximated by such models [2]. However, the processes underlying chemical reactions are inherently statistical in nature and the assumption that a reaction network can be represented as a continuous process may be invalid for some reactions such as those involving low molecular counts [1,3,4,5,6]. Such networks should be modelled stochastically.…”

“…The stochastic modelling of a chemical reaction network, dating back to Delbrück [7], is based upon the idea that molecular concentrations are subject to statistical fluctuation. Rather than treat the molecular counts as continuous quantities that evolve deterministically in time, the stochastic representation follows the time-evolution of the probability distributions of discrete states of a stochastic process [1,3] where the collections of states is the set of all permissible combinations of molecular counts of the chemical species in the reaction network at a time t ≥ 0 [4,8,9].…”

“…While the the chemical master equation deals more carefully with the probabilistic nature of dilute reaction mixtures, analytic solutions to (1) are extremely difficult to compute and are known for only a handful of special cases [1,10,11]. Alternate methods for computing analytical solutions, such as the novel "gluing" methods introduced by [6] and explored in [1,12], have been proposed.…”

“…While the the chemical master equation deals more carefully with the probabilistic nature of dilute reaction mixtures, analytic solutions to (1) are extremely difficult to compute and are known for only a handful of special cases [1,10,11]. Alternate methods for computing analytical solutions, such as the novel "gluing" methods introduced by [6] and explored in [1,12], have been proposed. Such solution methods decompose the state space into subsets whose solutions are straightforward to compute and then build up the full analytical solution by recursively "gluing" solutions from each subset together at one or two states.…”