Oscillation Criteria in Large-Scale Gene Regulatory Networks with Intrinsic Fluctuations

TL;DR

This paper develops a practical method to determine when large-scale gene regulatory networks with feedback can sustain oscillations despite intrinsic fluctuations, highlighting the impact of network size on oscillatory behavior.

Contribution

It introduces a second-moment approach to analyze stochastic dynamics in large GRNs, identifying the critical size needed for oscillations to persist amid fluctuations.

Findings

Small systems with high fluctuations do not oscillate.

The method determines the minimum network size for stable oscillations.

Analysis of repressilator and five-node systems illustrates the approach.

Abstract

Gene Regulatory Networks(GRNs) with feedback are essential components of many cellular processes and may exhibit oscillatory behavior. Analyzing such systems becomes increasingly complex as the number of components increases. Since gene regulation often involves a small number of molecules, fluctuations are inevitable. Therefore, it is important to understand how fluctuations affect the oscillatory dynamics of cellular processes, as this will allow comprehension of the mechanisms that enable cellular functions to remain even in the presence of fluctuations or, failing that, to determine the limit of fluctuations that permits various cellular functions. In this study, we investigated the conditions under which GRNs with feedback and intrinsic fluctuations exhibit oscillatory behavior. Our focus was on developing a procedure that would be both manageable and practical, even for extensive regulatory networks, that is, those comprising numerous nodes. Using the second-moment approach, we described the stochastic dynamics through a set of ordinary differential equations for the mean concentration and its second central moment. The system can attain either a stable equilibrium or oscillatory behavior, depending on its scale and, consequently, the intensity of fluctuations. To illustrate the procedure, we analyzed two relevant systems: a repressilator with three nodes and a system with five nodes, both incorporating intrinsic fluctuations. In both cases, it was observed that for very small systems, which therefore exhibit significant fluctuations, oscillatory behavior is inhibited. The procedure presented here for analyzing the stability of oscillations under fluctuations enables the determination of the critical minimum size of GRNs at which intrinsic fluctuations do not eliminate their cyclical behavior.