2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
Electron. J. Diff. Eqns., Conference 12, 2005, pp. 117-141.

Gene regulatory networks and delay differential equations

Arcady Ponosov

This paper suggests a mathematical framework to study gene regulatory networks with time-delay effects, which is based on delay differential equations. An essential feature of the gene regulatory networks is their ``almost Boolean'' structure, where the dynamics is governed by sigmoid-type nonlinearities which are close to the step functions. This is due to the fact that genes are only activated if certain concentrations are close to the respective threshold values. Thus, any mathematical model describing such networks faces a problem of how to study the dynamics in the vicinity of the thresholds. The paper presents some properties of gene regulatory networks with delay in comparison with the non-delay model. A method of localizing stationary points near the thresholds in the presence of delays is offered. The basic technical tool, which is systematically applied in the paper, is a special modification of the well-known ``linear chain trick". The results are illustrated by a number of examples.

Published April 20, 2005.
Math Subject Classifications: 34K60, 92D10.
Key Words: Gene regulation; delay equations; stability.

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Arcady Ponosov
CIGENE - Centre for Integrative Genetics, and
Department of Mathematical Sciences and Technology
Norwegian University of Life Sciences, N-1430 As, Norway

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