Stochastic switching in biology: from genotype to phenotype

Abstract

There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1-1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/ transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a systemsize expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker-Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel-Kramers-Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a selfcontained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels.

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

“…The reactivity is thus related to the probability of reaction event at the encounter [12][13][14]. Robin boundary condition with homogeneous (constant) reactivity κ was often employed to describe many chemical and biochemical reactions and permeation processes [15][16][17][18][19][20][21][22][23][24][25][26], to model stochastic gating [27][28][29][30], or to approximate the effect of microscopic heterogeneities in a random distribution of reactive sites [31][32][33] (see a recent overview in [34]). In spite of its practical importance, diffusion-controlled reactions with heterogeneous surface reactivity κ(s) remain much less studied.…”

Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback

“…Friedman et al [12] proposed the continuous master equation model and it was pointed out later that the stochastic process underlying this model is a stochastic differential equation (SDE) driven by a compound Poisson process [35]. In addition, many authors [32,33,37,42,43] modeled stochastic gene expression kinetics as ordinary differential equations (ODEs) or SDEs with hybrid Markov switching in the regime of relatively slow promoter switching. In our recent work [42], we have unified most discrete and continuous models by regarding the latter as various macroscopic limits of the former.…”

Accelerated Bayesian inference of gene expression models from snapshots of single-cell transcripts

“…Piecewise-deterministic Markov processes (PDMP) have become a useful, coarse-grained description of stochastic gene dynamics, where the underlying discrete variable s(t) captures the stochastic dynamics of gene states and the continuous variable λ(t) captures the first moment of downstream gene products [36][37][38][39][40][41][42]. The key FIG.…”