Recursively constructing analytic expressions for equilibrium distributions of stochastic biochemical reaction networks

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 of Kolmogorov forward equations that describe how the probability distribution of a chemically reacting system varies with time. Finding 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 characterize 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 the stationary behaviour of stochastic biochemical reaction networks. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and biochemical reactions with two-dimensional state spaces.

Identifiability of linear noise approximation models of chemical reaction networks from stationary distributions

“…In certain special cases, such as detailed or complex balance, the stationary distribution of X Ω (t) can be computed in closed form, and thus in principle one could directly analyze the approximation error when using the LNA [64,4,46]. However, for most SCRNs of interest in biology, such results are not applicable.…”

“…However, for most SCRNs of interest in biology, such results are not applicable. For example, a simple two step transcription-translation model for protein expression is not detailed or complex balanced [14], and thus the results in [64,4] cannot be used to obtain a closed form expression for the stationary distribution, whereas the results of [46] are only exact for systems where X Ω (t) has finitely many states.…”

Graph-facilitated resonant mode counting in stochastic interaction networks

“…Graph theoretical methods provide the additional advantage over the approach in [9] that they allow one to explore the bifurcation structure of the network. Despite the apparent advantage of using graph theoretical methods for the investigation of dynamical capabilities of interaction networks the graph based investigation of stochastic models is still in its infancy [16].…”