Electron. J. Diff. Equ., Vol. 2013 (2013), No. 24, pp. 1-32.

Nonlinear perturbations of piecewise-linear systems with delay and applications to gene regulatory networks

Valeriya Tafintseva, Arcady Ponosov

Abstract:
We study piecewise-linear delay differential systems which describe gene regulatory networks with Boolean interactions. Under very general assumptions put on the regulatory functions it is shown how to construct the limit dynamics of the systems by applying singular perturbation analysis. The obtained results are compared with those based on the multilinear representation of the regulatory functions usually considered in the literature. It is shown that sliding modes may be quite different in the multilinear and general case. Polynomial representations of the systems are proposed to describe generic cases of the dynamics. The results presented in this paper may give the insight into the theory of gene regulatory networks.

Submitted November 14, 2012. Published January 27, 2013.
Math Subject Classifications: 34K07, 34K26, 34K60.
Key Words: Delay differential equations; singular perturbation analysis; polynomial representation; gene regulatory networks.

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Valeriya Tafintseva
CIGENE - Center for Integrative Genetics at UMB
Department of Mathematical Sciences and Technology
Norwegian University of Life Sciences (UMB)
P.O. Box 5003, Drobakveien 31, As, N-1432, Norway
email: valeta@umb.no, tel.: +4796851337; fax: +4764965401
Arcady Ponosov
CIGENE - Center for Integrative Genetics at UMB
Department of Mathematical Sciences and Technology
Norwegian University of Life Sciences (UMB)
P.O. Box 5003, Drobakveien 31, As, N-1432, Norway
email: arkadi@umb.no

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