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.
Limit cycles and oscillations are fundamental features of numerous biological systems, including circadian rhythms 1,2 . In this study, we focused specifically on gene regulatory networks (GRNs) with negative feedback 3 . Deterministic models of these networks may have limit cycles and thus exhibit oscillatory behavior if delays or intermediate processes are present 4,5 . Within these systems, the cyclical dynamics of protein concentrations are pivotal for gene modules to perform their functions.
GRNs play crucial roles in cellular functions and development, enabling cells to respond to environmental stimuli by regulating gene expression 6 . Examples of such systems include the p53-Mdm2 feedback loop 7,8 , where p53 acts as a tumour suppressor and the mutations that are frequently found in cancer 9,10 . The synthesis of Hes-1, a transcription factor involved in stem cell maintenance, differentiation, and the inhibition of cancer progression 11,12 , is also an example of a gene module with a feedback loop.
However, these networks operate in regimes characterized by low molecule numbers and are thus significantly affected by stochastic fluctuations. In this a) manuel.hernandezgarcia@viep.com.mx b) jorge.velazquezcastro@correo.buap.mx study, we focus only on intrinsic fluctuations arising from their discrete nature and the randomness of molecular interactions [13][14][15] . Several formalisms exist to study such stochastic systems, including the chemical master equation 15 , stochastic simulation algorithms such as Gillespie’s algorithm 16 , and approximate approaches such as the Langevin equation 14,17 and Fokker-Planck equations 18 . An alternative is the moment approach [19][20][21] , particularly the second-moment approach, which describes the dynamics of the system via a system of ordinary differential equations (ODEs) for the mean concentrations and its second central moments. Higherorder moments can be included as needed to capture nonlinear effects, such as those arising from Hill-type interactions 22 . Furthermore, Hernandez et al. 19 show that the dynamics of chemical networks consisting of first-order reactions or even nonlinear reaction rates such as Hill functions are described exactly by this method. In contrast with other methods, this ODE-based formulation enables the direct application of dynamical-systems theory, such as stability and bifurcation analysis 23 , to stochastic systems, including results that classify system behavior in terms of equilibrium points or periodic orbits 24 .
The goal of this work is to investigate large-scale GRNs with negative feedback under stochastic conditions driven by intrinsic fluctuations 25,26 and to identify the conditions, including minimum protein concentrations, under which such systems can exhibit oscillatory behavior. Analyzing large chemical networks, as GRNs normally are, is particularly challenging due to the complexity of determining oscillation criteria in highdimensional spaces. Prior studies employed controltheory graphical tools, based on the transfer function, to address these challenges in deterministic contexts 27,28 . However, because the moment approach reformulates a stochastic system as a set of ODEs, in the present analysis these methods are adapted to assess oscillatory conditions in stochastic systems.
We illustrate the methodology using two representative systems: a repressilator, a synthetic oscillator originally described by Elowitz et al. 29 and previously studied primarily in deterministic frameworks 30 , and a five-node network consisting solely of repressors. These examples show the applicability of the framework to cyclic GRNs with negative feedback and an arbitrary number of nodes and highlight the impact of system size and component concentrations on the emergence and attenuation of oscillations.
The remainder of this paper is organized as follows. In Section II we present the chemical master equation and a set of ODEs for the mean and second central moments using the moment approach. In Section III, we analyze a cyclic gene regulatory network with negative feedback, and we identify the conditions under which oscillations appear. In Section IV we apply the framework to two representative systems and examine the impact of intrinsic fluctuations on the oscillations emergence. Finally, in Section V we present our results and conclusions.
In this work, we consider only intrinsic fluctuations; for this purpose, we followed the methodology in 15 to derive the Chemical Master Equation (CME). Let be N chemical species, S l (l ∈ {1, 2, ..N}) and r reactions R i (i ∈ {1, 2, …, r}) where species are transformed as follows:
k i is the kinetic parameter of the reaction rate, and coefficients α il and β il are non-negative integers. From these, we derive the stoichiometric matrix of the system Γ il = β liα li (indices are exchanged because it is the transpose). Through collisions (or interactions) between different elements, the
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