Stochastic cooperativity in non-linear dynamics of genetic regulatory networks

Abstract

Two major approaches are known in the field of stochastic dynamics of genetic regulatory networks (GRN). The first one, referred here to as the Markov Process Paradigm (MPP), places the focus of attention on the fact that many biochemical constituents vitally important for the network functionality are present only in small quantities within the cell, and therefore the regulatory process is essentially discrete and prone to relatively big fluctuations. The Master Equation of Markov Processes is an appropriate tool for the description of this kind of stochasticity. The second approach, the Non-linear Dynamics Paradigm (NDP), treats the regulatory process as essentially continuous. A natural tool for the description of such processes are deterministic differential equations. According to NDP, stochasticity in such systems occurs due to possible bistability and oscillatory motion within the limit cycles. The goal of this paper is to outline a third scenario of stochasticity in the regulatory process. This scenario is only conceivable in high-dimensional, highly non-linear systems, and thus represents an adequate framework for conceptually modeling the GRN. We refer to this framework as the Stochastic Cooperativity Paradigm (SCP). In this approach, the focus of attention is placed on the fact that in systems with the size and link density of GRN (∼25 000 and ∼100, respectively), the confluence of all the factors which are 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 concentrations, a well known effect in intracellular dynamics. We demonstrate that a high-dimensional non-linear system, despite 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 a stochastic process. This type of stochastic behavior is an inherent property of such systems and requires neither an external random force as in the Langevin approach, nor the discreteness of the process as in MPP, nor highly specialized conditions of bistability as in NDP, nor bifurcations with transition to chaos as in low-dimensional chaotic maps.