Reducing a chemical master equation by invariant manifold methods.

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.

[1]  George H. Weiss,et al.  Stochastic Processes in Chemical Physics: The Master Equation , 1977 .

[2]  Marc R. Roussel,et al.  Invariant manifold methods for metabolic model reduction. , 2001, Chaos.

[3]  Marc R. Roussel,et al.  Geometry of the steady-state approximation: Perturbation and accelerated convergence methods , 1990 .

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

[5]  Epaminondas Mastorakos,et al.  An algorithm for the construction of global reduced mechanisms with CSP data , 1999 .

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

[7]  Adel Blouza,et al.  Reduction of linear kinetic systems with multiple scales , 2000 .

[8]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[9]  Hans G. Kaper,et al.  Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics , 2004, J. Nonlinear Sci..

[10]  An algorithm for targeted convergence of Euler or Newton iterations. , 2001, Comptes rendus de l'Academie des sciences. Serie III, Sciences de la vie.

[11]  Stefan Katzenbeisser,et al.  MATLAB 6.5 , 2002 .

[12]  D. Chapman,et al.  LV.—The interaction of chlorine and hydrogen. The influence of mass , 1913 .

[13]  K. Shuler,et al.  Stochastic processes in chemical physics , 1969 .

[14]  Marc R. Roussel,et al.  On the geometry of transient relaxation , 1991 .

[15]  D Lebiedz,et al.  Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics. , 2004, The Journal of chemical physics.

[16]  Iliya V. Karlin,et al.  Method of invariant manifold for chemical kinetics , 2003 .

[17]  János Tóth,et al.  Fast variables elimination in stochastic kinetics , 1993 .

[18]  F. Battelli,et al.  On the pseudo-steady-state approximation and Tikhonov theorem for general enzyme systems , 1985 .

[19]  D. Gillespie.  A rigorous derivation of the chemical master equation , 1992 .

[20]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[21]  A. Bartholomay,et al.  A stochastic approach to statistical kinetics with application to enzyme kinetics. , 1962, Biochemistry.

[22]  M. Roussel.  Forced‐convergence iterative schemes for the approximation of invariant manifolds , 1997 .

[23]  P J Staff.  A stochastic development of the reversible Michaelis-Menten mechanism. , 1970, Journal of theoretical biology.

[24]  M. Pilling.  Low-temperature combustion and autoignition , 1997 .

[25]  Samuel Paolucci,et al.  On slow manifolds of chemically reactive systems , 2002 .

[26]  Simon J. Fraser,et al.  Geometrical picture of reaction in enzyme kinetics , 1989 .

[27]  Linda R. Petzold,et al.  Improved leap-size selection for accelerated stochastic simulation , 2003 .

[28]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[29]  Michael J. Davis,et al.  Geometric investigation of low-dimensional manifolds in systems approaching equilibrium , 1999 .

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

[31]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics , 1993 .