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

Arcady Ponosov (), Irina Shlykova and Ramazan I. Kadiev
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Arcady Ponosov: Department of Mathematics, Norwegian University of Life Sciences, 1432 Aas, Norway
Irina Shlykova: Department of Mathematics, Norwegian University of Life Sciences, 1432 Aas, Norway
Ramazan I. Kadiev: Dagestan Research Center of the Russian Academy of Sciences, Department of Mathematics, Dagestan State University, 367005 Makhachkala, Russia

Mathematics, 2024, vol. 12, issue 8, 1-17

Abstract: 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.

Keywords: gene regulation; stochastic differential equations; singular perturbation analysis; delay effects (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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