Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-11T04:26:07.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Noise-induced mixing and multimodality in reaction networks

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

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Anderson, D. F. & Kurtz, T. G. (2015) Stochastic Analysis of Biochemical Systems, Springer International Publishing, Switzerland.CrossRefGoogle Scholar
Anderson, D. F., Craciun, G. & Kurtz, T. G. (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Bio. 72(8), 1947–1970.CrossRefGoogle ScholarPubMed
Cotter, S. & Erban, R. (2016) Error analysis of diffusion approximation methods for multiscale systems in reaction kinetics. SIAM J. Sci. Comput. 38(1), B144–B163.CrossRefGoogle Scholar
Cotter, S., Vejchodský, T. & Erban, R. (2013) Adaptive finite element method assisted by stochastic simulation of chemical systems. SIAM J. Sci. Comput. 35(1), B107–B131.CrossRefGoogle Scholar
Cotter, S., Zygalakis, K., Kevrekidis, I. & Erban, R. (2011) A constrained approach to multiscale stochastic simulation of chemically reacting systems. J. Chem. Phys. 135(9), 094102.CrossRefGoogle ScholarPubMed
Craciun, G. (2015) Toric Differential Inclusions and a Proof of the Global Attractor Conjecture. Available at: http://arxiv.org/abs/1501.02860.Google Scholar
Duncan, A., Liao, S., Vejchodský, T., Erban, R. & Grima, R. (2015) Noise-induced multistability in chemical systems: discrete vs continuum modelling. Phys. Rev. E 91, 042111.CrossRefGoogle Scholar
Erban, R., Chapman, J., Kevrekidis, I. & Vejchodský, T. (2009) Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model. SIAM J. Appl. Math. 70(3), 984–1016.CrossRefGoogle Scholar
Erban, R., Kevrekidis, I., Adalsteinsson, D. & Elston, T. (2006) Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation. J. Chem. Phys. 124(8), 084106.CrossRefGoogle ScholarPubMed
Érdi, P. & Tóth, J. (1989) Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models, Manchester University Press, Manchester, UK.Google Scholar
Feinberg, M. (1979) Lectures on Chemical Reaction Networks. Lecture Notes, Mathematics Research Center, University of Wisconsin.Google Scholar
Fraser, D. & Kaern, M. (2009) A chance at survival: gene expression noise and phenotypic diversification strategies. Mol. Microbiol. 71(6), 1333–1340.CrossRefGoogle ScholarPubMed
Gadgil, C., Lee, C. H. & Othmer, H. G. (2005) A stochastic analysis of first-order reaction networks. Bull. Math Biol. 67(5), 901–946.CrossRefGoogle ScholarPubMed
Gillespie, D. T. (1977) Exact stochastic simulation of coupled chemical reactions. J. Phy. Chem. 81(25), 2340–2361.CrossRefGoogle Scholar
Kan, X., Lee, C. H. & Othmer, H. G. (2016) A multi-time-scale analysis of chemical reaction networks: II Stochastic systems. J. Math. Biol. 73, 1081–1129.CrossRefGoogle ScholarPubMed
Kepler, T. B. & Elston, T. C. (2001) Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. Biophys. J. 81, 3116–3136.CrossRefGoogle ScholarPubMed
Kuwahara, H. & Gao, X. (2013) Stochastic effects as a force to increase the complexity of signaling networks. Sci. Rep. 3, 2297.CrossRefGoogle ScholarPubMed
Kuthan, H. (2001) Self-organisation and orderly processes by individual protein complexes in bacterial cell. Prog. Biophys. Mol. Biol. 75(1–2), 1–17.CrossRefGoogle ScholarPubMed
Levsky, J. M. & Singer, R. H. (2003) Gene expression and the myth of the average cell. Trends Cell Biol. 13(1), 4–6.CrossRefGoogle ScholarPubMed
Liao, S., Vejchodský, T. & Erban, R. (2015) Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks. J. Roy. Soc. Interface 12(108), 20150233.CrossRefGoogle ScholarPubMed
Othmer, H. G. (1981) A Graph-Theoretic Analysis of Chemical Reaction Networks. Lecture Notes, Rutgers University.Google Scholar
Ozbudak, E. M., Thattai, M., Kurtser, I., Grossman, A. D. & van Oudenaarden, A. (2002) Regulation of noise in the expression of a single gene. Nat. Genet. 31(1), 69–73.CrossRefGoogle ScholarPubMed
Pavliotis, G. A. & Stuart, A. M. (2008) Multiscale Methods: Averaging and Homogenization, Springer, New York, NY.Google Scholar
Prigogine, I. & Lefever, R. (1968) Symmetry breaking instabilities in dissipative systems, II. J. Chem. Phys. 48(4), 1695–1700.CrossRefGoogle Scholar
Plesa, T., Vejchodský, T. & Erban, R. (2017) Test models for statistical inference: two-dimensional reaction systems displaying limit cycle bifurcations and bistability. In: Stochastic Dynamical Systems, Multiscale Modeling, Asymptotics and Numerical Methods for Computational Cellular Biology, Springer.Google Scholar
Plesa, T., Zygalakis, K. C., Anderson, D. F. & Erban, R. (2018) Noise control for molecular computing. J. Roy. Soc. Interface 15(144), 20180199.CrossRefGoogle ScholarPubMed
Raj, A. & van Oudenaarden, A. (2008) Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135(2), 216–226.CrossRefGoogle ScholarPubMed
Samoilov, M. S. & Arkin, A. P. (2006) Deviant effects in molecular reaction pathways. Nat. Biotechnol. 24(10), 1235–1240.CrossRefGoogle ScholarPubMed
Senecal, A., Munsky, B., Proux, F., Ly, N., Braye, F. E., Zimmer, C., Mueller, F. & Darzacq, X. (2014) Transcription factors modulate c-fos transcriptional bursts. Cell Rep. 8(1): 75–83.CrossRefGoogle ScholarPubMed
Spudich, J. L. & Koshland, D. E. Jr. (1976) Non-genetic individuality: chance in the single cell. Nature 262, 467–471.CrossRefGoogle ScholarPubMed
Van Kampen, N. G. (2007) Stochastic Processes in Physics and Chemistry, 3rd ed., Elsevier, North-Holland.Google Scholar
Wickramasinghe, V. O. & Laskey, R. A. (2015) Control of mammalian gene expression by selective mRNA export. Nat. Rev. Mol. Cell Biol. 16(7), 431–442.CrossRefGoogle ScholarPubMed