Pseudo-Random Fluctuations, Stochastic Cooperativity and Burstiness in Dynamically Unstable High-Dimensional Biochemical Networks

Abstract

The goal of this paper is to outline a scenario of emerging stochasticity in high-dimensional highly nonlinear systems, such as genetic regulatory networks (GRN). We focus attention on the fact that in such systems confluence of all the factors necessary for gene expression is a comparatively rare event, and only massive redundancy makes such events sufficiently frequent. An immediate consequence of this rareness is burstiness in mRNA and protein copy numbers, a well known experimentally observed effect. We introduce the concept of stochastic cooperativity and show that this phenomenon is a natural consequence of high dimensionality coupled with highly nonlinearity of a dynamical system. In mathematical terms, burstiness is associated with heavy-tailed probability distributions of stochastic processes describing the dynamics of the system. The sequence of stochastic cooperativity events allows for transition from continuous deterministic dynamics expressed in terms of ordinary differential equations (ODE) to discrete stochastic dynamics expressed in terms of Langevin and Fokker-Plank equations. We demonstrate also that high-dimensional nonlinear systems, even in the absence of explicit mechanisms for suppressing inherent instability, may nevertheless reside in a state of stationary pseudo-random fluctuations which for all practical purposes may be regarded as stochastic process. This type of stochastic behavior is an inherent property of such systems and requires neither an external random force, nor highly specialized conditions of bistability.