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Noise-induced mixing and multimodality in reaction networks

Part of: Functional-differential and differential-difference equations Time-dependent statistical mechanics (dynamic and nonequilibrium) Physiological, cellular and medical topics Applications - Dynamical systems and ergodic theory Thermodynamics and heat transfer

Published online by Cambridge University Press:  18 September 2018

TOMISLAV PLESA ,
RADEK ERBAN  and
HANS G. OTHMER
TOMISLAV PLESA
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK emails: plesa@maths.ox.ac.uk; erban@maths.ox.ac.uk
RADEK ERBAN
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK emails: plesa@maths.ox.ac.uk; erban@maths.ox.ac.uk
HANS G. OTHMER
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: othmer@umn.edu
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Abstract

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

Keywords

Chemical reaction networksstochastic dynamicsgene-regulatory networksperturbation theory

MSC classification

Primary: 92C42: Systems biology, networks
Secondary: 80A30: Chemical kinetics 37N25: Dynamical systems in biology 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) 34K26: Singular perturbations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 887 - 911
Copyright
© Cambridge University Press 2018 

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Footnotes

†

This work was supported by NIH Grant number 29123 and a Visiting Research Fellowship from Merton College, Oxford, awarded to Hans Othmer. Radek Erban would also like to thank the Royal Society for a University Research Fellowship.

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