Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T17:29:39.337Z Has data issue: false hasContentIssue false
  • English
  • Français

Simulating Stochasticities in Chemical Reactions withDeterministic Delay Differential Equations

Published online by Cambridge University Press:  07 February 2014

H. Jin  and
J. Lei
Get access

Abstract

The stochastic dynamics of chemical reactions can be accurately described by chemicalmaster equations. An approximated time-evolution equation of the Langevin type has beenproposed by Gillespie based on two explicit dynamical conditions. However, whennumerically solve these chemical Langevin equations, we often have a small stopping time–atime point of having an unphysical solution–in the case of low molecular numbers. Thispaper proposes an approach to simulate stochasticities in chemical reactions withdeterministic delay differential equations. We introduce a deterministic Brownian motiondescribed by delay differential equations, and replace the Gaussian noise in the chemicalLangevin equations by the solutions of these deterministic equations. This modificationcan largely increase the stopping time in simulations and regain the accuracy as in thechemical Langevin equations. The novel aspect of the present study is to apply thedeterministic Brownian motion to chemical reactions. It suggests a possible direction ofdeveloping a hybrid method of simulating dynamic behaviours of complex gene regulationnetworks.

Keywords

Master equationchemical Langevin equationdeterministic Brownian motiondelay differential equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 1: Issue dedicated to Michael Mackey , 2014 , pp. 79 - 91
Copyright
© EDP Sciences, 2014

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.)

References

Anderson, D.F.. Incorporating postleap checks in tau-leaping. J. Chem. Phys., 128 (2008), 054103. CrossRefGoogle ScholarPubMed
Cao, Y., Gillespie, D.T., Petzold, L.R.. Avoiding negative populations in explicit poisson tau-leaping. J. Chem. Phys., 123 (2005), 054104. CrossRefGoogle ScholarPubMed
Cao, Y., Gillespie, D.T., Petzold, L.R.. Efficient stepsize selection for the tau-leaping simulation method. J. Chem. Phys., 124 (2006), 044109. CrossRefGoogle Scholar
Chatterjee, A., Vlachos, D., Katsoulakis, M.. Binomial distribution based τ-leap accelerated stochastic simulation. J. Chem. Phys., 122 (2005), 024112. CrossRefGoogle ScholarPubMed
Chew, L.Y., Ting, C.. Microscopic chaos and gaussian diffusion processes. Physica A., 307 (2002), 275–296. CrossRefGoogle Scholar
Cotter, S.L., Zygalakis, K.C., Kevrekidis, I.G., Erban, R.. A constrained approach to multi scale stochastic simulation of chemically reacting systems. J. Chem. Phys., 135 (2011), 094102. CrossRefGoogle Scholar
Elowitz, M.B., Levine, A.J., Siggia, E.D., and Swain, P.S., Stochastic gene expression in a single cell. Science, 297 (2002), 1183–1186. CrossRefGoogle Scholar
Gaspard, P., Briggs, M.E., Francis, M.K., Sengers, J.V., Gammon, R.W., Dorfman, J.R., Calabrese, R.V.. Experimental evidence for microscopic chaos. Nature, 394 (1998), 865–868. CrossRefGoogle Scholar
Gillespie, D.T.. The chemical langevin equation. J. Chem. Phys., 113 (2000), 297–306. CrossRefGoogle Scholar
Gillespie, D.T.. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys., 115 (2001), 1716–1733. CrossRefGoogle Scholar
Gillespie, D.T.. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58 (2007), 35–55. CrossRefGoogle Scholar
Golding, I., Paulsson, J., Zawilski, S.M., Cox, E.C., Real-time kinetics of gene activity in individual bacteria. Cell, 123 (2005), 1025–1036. CrossRefGoogle ScholarPubMed
Kærn, M., Elston, T.C., Blake, W.J., Collins, J.J., Stochasticity in gene expression: from theories to phenotypes. Nat. Rev. Genet., 6 (2005), 451–464. CrossRefGoogle ScholarPubMed
N.G. Van Kampen. Stochastic processes in physics and chemistry. 3rd ed., Elsevier, 2007.
P.E. Kloeden, E. Platen. Numerical solution of stochastic differential equations. Springer, New York, 1992.
A. Lasota, M.C. Mackey. Probabilistic properties of deterministic systems. Cambridge University Press, Cambridge, 2008.
Lei, J.. Stochasticity in gene expression with both intrinsic noise and fluctuation in kinetic parameters. J. Theor. Biol., 256 (2009), 485–492. CrossRefGoogle ScholarPubMed
Lei, J., Mackey, M.C.. Deterministic brownian motion generated from differential delay equation. Phy. Rev. E. 84 (2011), 041105. CrossRefGoogle Scholar
Mackey, M.C., Tyran-Kamińska, M.. Deterministic brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system. Physics Reports, 422 (2006), 167–222. CrossRefGoogle Scholar
G.N. Milstein. Numerical integration of stochastic differential equations. Kluwer, Dordrecht, 1995.
Munsky, B., Neuert, G., van Oudenaarden, A.. Using gene expression noise to undertand gene regulation. Science, 336 (2012), 183–187. CrossRefGoogle Scholar
Paulsson, J.. Summing up the noise in gene networks. Nature, 427 (2004), 415–418. CrossRefGoogle ScholarPubMed
Tian, T., Burrage, K.. Binomial leap methods for simulating stochastic chemical kinetics. J. Chem. Phys., 121 (2004), 10356–10364. CrossRefGoogle ScholarPubMed
T-L.To, Maheshri, N.. Noise can induce bimodality in positive transcriptional feedback loops without bistability. Science, 327 (2010), 1142–1145. Google Scholar
Trefán, G., Grigolini, P., West, B.J.. Deterministic brownian motion. Phys. Rev. A., 45 (1992), no. 2, 1249–1252.CrossRefGoogle ScholarPubMed