Stationary and Time-Dependent Molecular Distributions in Slow-Fast Feedback Circuits

Abstract

Biochemical reaction systems, in particular those that govern the expression of genes in single cells, involve discrete numbers of molecules and are inherently stochastic. This study concerns reaction systems with two molecular species and an arbitrary number of reactions, the rates of which conform to a scaling of the classical type in the first species (but not the second). For limiting values of the scaling parameter, the first species is highly abundant, evolves slowly, and buffers the relatively fast fluctuation of the second, scarce, species. The scale separation facilitates the construction of asymptotic approximations to the molecular distributions: the large-time behavior is described by a WKB approximation, which is further resolved into a tractable mixture of metastable modes; the earlier-time dynamics are described by quasi-stationary and linear noise approximations. The paper presents the theory and the algebraic recipes that are required to implement it. The framework is applied on two reaction systems with positive feedback: a delayed feedback circuit and a gene autoregulation model with cooperative binding of the protein to the promoter. For the former, the approximation scheme results into a mixture of Gaussian/Poisson modes for the inactive/active protein distribution. For the latter, the analysis elucidates the effects of bursty gene expression, promoter responsivity, and protein sequestration on bimodality of protein distributions.