2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo,
BC, Canada.
Electron. J. Diff. Eqns., Conference 12, 2005, pp. 117141.
Gene regulatory networks and delay differential equations
Arcady Ponosov
Abstract:
This paper suggests a mathematical framework to study gene regulatory
networks with timedelay 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 sigmoidtype 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 nondelay 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 wellknown ``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, N1430 As, Norway

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