Effects of Small Random Perturbations in the Extended Glass–Kauffman Model of Gene Regulatory Networks

A mathematical justification of some basic structural properties of stochastically perturbed gene regulatory networks, including those with autoregulation and delay, is offered in this paper. By using the theory of stochastic differential equations, it is, in particular, shown how to control the asy...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-04, Vol.12 (8), p.1223
Main Authors: Ponosov, Arcady, Shlykova, Irina, Kadiev, Ramazan I.
Format: Article
Language:English
Subjects:
Algorithms
Asymptotic properties
Boolean
delay effects
Differential equations
gene regulation
Genes
Networks
Numerical analysis
Ordinary differential equations
Partial differential equations
Perturbation methods
Randomness
Response functions
Robustness (mathematics)
Singular perturbation
singular perturbation analysis
stochastic differential equations
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Summary:A mathematical justification of some basic structural properties of stochastically perturbed gene regulatory networks, including those with autoregulation and delay, is offered in this paper. By using the theory of stochastic differential equations, it is, in particular, shown how to control the asymptotic behavior of the diffusion terms in order to not destroy certain qualitative features of the networks, for instance, their sliding modes. The results also confirm that the level of randomness is gradually reduced if the gene activation times become much smaller than the time of interaction of genes. Finally, the suggested analysis explains why the deterministic numerical schemes based on replacing smooth, steep response functions by the simpler yet discontinuous Heaviside function, the well-known simplification algorithm, are robust with respect to uncertainties in data. The main technical difficulties of the analysis are handled by applying the uniform version of the stochastic Tikhonov theorem in singular perturbation analysis suggested by Yu. Kabanov and S. Pergamentshchikov.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12081223