Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

Radek Erban; S. Jonathan Chapman; Ioannis G. Kevrekidis; Tomáš Vejchodský

doi:10.1137/080731360

Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

Radek Erban, S. Jonathan Chapman, Ioannis G. Kevrekidis, Tomáš Vejchodský

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Abstract

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

Original language	English (US)
Pages (from-to)	984-1016
Number of pages	33
Journal	SIAM Journal on Applied Mathematics
Volume	70
Issue number	3
DOIs	https://doi.org/10.1137/080731360
State	Published - Jan 2009
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: Received by the editors July 28, 2008; accepted for publication ( in revised form) June 9, 2009; published electronically August 21, 2009. This work is based on work supported by St. John's College, Oxford; Linacre College, Oxford; Somerville College, Oxford; and by award KUK-C1-013-04, given by King Abdullah University of Science and Technology (KAUST) (RE).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1137/080731360

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title = "Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model",

abstract = "A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. {\textcopyright} 2009 Society for Industrial and Applied Mathematics.",

author = "Radek Erban and Chapman, {S. Jonathan} and Kevrekidis, {Ioannis G.} and Tom{\'a}{\v s} Vejchodsk{\'y}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: Received by the editors July 28, 2008; accepted for publication ( in revised form) June 9, 2009; published electronically August 21, 2009. This work is based on work supported by St. John's College, Oxford; Linacre College, Oxford; Somerville College, Oxford; and by award KUK-C1-013-04, given by King Abdullah University of Science and Technology (KAUST) (RE). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

AB - A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

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Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model

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