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

TL;DR

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.

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.

Figures (6)

• Figure 1: Cyclic gene regulatory network. In this figure, there is a cyclic gene regulatory network with negative feedback, where the arrows represent positive regulation, and the bars represent negative regulation, but the complete cycle has negative feedback. At each node, there is a module of transcription-translation processes that synthesizes a protein that acts as a transcription factor for the next module.
• Figure 2: Block diagram. In the left, the overall GRN. In the right panel, the modified GRN wich is equivalent to the left panel.
• Figure 3: Repressilator. This is a cycle gene regulatory network with three modules, in which the protein represses mRNA synthesis in the next module, and the system has negative feedback.
• Figure 4: Criteria of oscillation in the repressilator. In this figure, we show that whether $\Omega$ increases, the mean concentration oscillates, which is based on graphical criteria. In panel (a), the results of the mean concentrations show that when $\Omega$ increases, the eigenvalues reach the region $\Lambda_1^{+}$, and then the concentrations oscillate. In panel (b), we analyze the second central moment, and it is observed that when $\Omega$ increases, the eigenvalues move away from the region $\Lambda_2^{+}$, and then the second central moment does not oscillate. In panel (c), we observe the behavior of the oscillation dynamics as $\Omega$ increases (we used the initial conditions of Table \ref{['tabla3']}).
• Figure 5: Penta-silador. This is a cycle gene regulatory network with five modules in which each module represses the synthesis of mRNA of the next.