Model reduction of multi-scale chemical Langevin equations

This paper addresses the model reduction problem for a class of stiff chemical Langevin equations that arise as models of biomolecular networks with fast and slow reactions and can be described as continuous Markov processes. Initially, a coordinate transformation is sought that allows the decoupling of fast and slow variables in the model equations. Necessary and sufficient conditions are derived for such a linear transformation to exist, along with an explicit change of variables which achieves the desired decoupling. For the systems for which this step is applicable, the method of adiabatic elimination is applied to determine a representation of the slow dynamics. Theoretical concepts and results are illustrated with simple examples.

[1]  Vassilios Sotiropoulos,et al.  Synthetic tetracycline-inducible regulatory networks: computer-aided design of dynamic phenotypes , 2007, BMC Systems Biology.

[2]  Richard B. Dickinson,et al.  Transport Equations and Indices for Random and Biased Cell Migration Based on Single Cell Properties , 1995, SIAM J. Appl. Math..

[3]  Prodromos Daoutidis,et al.  Nonlinear model reduction of chemical reaction systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[4]  J. Goutsias.  Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. , 2005, The Journal of chemical physics.

[5]  G. S. Ladde,et al.  Averaging principle and systems of singularly perturbed stochastic differential equations , 1990 .

[6]  Prodromos Daoutidis,et al.  Non-linear reduction for kinetic models of metabolic reaction networks. , 2004, Metabolic engineering.

[7]  D. Elworthy.  ASYMPTOTIC METHODS IN THE THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS , 1992 .

[8]  D. Gillespie.  The chemical Langevin equation , 2000 .

[9]  S. Nakajima.  On Quantum Theory of Transport Phenomena Steady Diffusion , 1958 .

[10]  Janusz Golec.  Stochastic averaging principle for systems with pathwise uniqueness , 1995 .

[11]  J. Rawlings,et al.  Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics , 2002 .

[12]  Tatsuo Shibata,et al.  Reducing the master equations for noisy chemical reactions , 2003 .

[13]  F. Shibata,et al.  Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion , 1979 .

[14]  Y. Kabanov,et al.  Two-scale stochastic systems , 2002 .

[15]  C. Rao,et al.  Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm , 2003 .

[16]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[17]  J. Gillis,et al.  Asymptotic Methods in the Theory of Non‐Linear Oscillations , 1963 .

[18]  J. A. M. Janssen,et al.  The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case) , 1989 .

[19]  J. A. M. Janssen,et al.  The elimination of fast variables in complex chemical reactions. II. Mesoscopic level (reducible case) , 1989 .

[20]  T. Elston,et al.  Stochasticity in gene expression: from theories to phenotypes , 2005, Nature Reviews Genetics.

[21]  Prodromos Daoutidis,et al.  Model Reduction of Multiscale Chemical Langevin Equations: A Numerical Case Study , 2009, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[22]  Rafail Z. Khasminskii,et al.  Asymptotic Series for Singularly Perturbed Kolmogorov-Fokker-Planck Equations , 1996, SIAM J. Appl. Math..

[23]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[24]  R. Z. Khasminskij.  On the principle of averaging the Itov's stochastic differential equations , 1968, Kybernetika.

[25]  R. Zwanzig.  Ensemble Method in the Theory of Irreversibility , 1960 .

[26]  U. M. Titulaer,et al.  The systematic adiabatic elimination of fast variables from a many-dimensional Fokker-Planck equation , 1985 .

[27]  Rafail Z. Khasminskii,et al.  On Averaging Principles: An Asymptotic Expansion Approach , 2004, SIAM J. Math. Anal..

[28]  Hans Christian Öttinger,et al.  Stochastic Processes in Polymeric Fluids , 1996 .

[29]  Vassilios Sotiropoulos,et al.  An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: chemical Langevin equation, a proof of concept. , 2008, The Journal of chemical physics.

[30]  Adam P Arkin,et al.  Fifteen minutes of fim: control of type 1 pili expression in E. coli. , 2002, Omics : a journal of integrative biology.

[31]  Nikolaos Kazantzis,et al.  Model reduction and coarse-graining approaches for multiscale phenomena , 2006 .

[32]  George Yin,et al.  On Transition Densities of Singularly Perturbed Diffusions with Fast and Slow Components , 1996, SIAM J. Appl. Math..