Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics

In many applications, for example when computing statistics of fast subsystems in a multiscale setting, we wish to find the stationary distributions of systems of continuous-time Markov chains. Here we present a class of models that appears naturally in certain averaging approaches whose stationary distributions can be computed explicitly. In particular, we study continuous-time Markov chain models for biochemical interaction systems with non-mass action kinetics whose network satisfies a certain constraint. Analogous with previous related results, the distributions can be written in product form.

[1]  Michael A. Gibson,et al.  Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels , 2000 .

[2]  David F. Anderson,et al.  A Proof of the Global Attractor Conjecture in the Single Linkage Class Case , 2011, SIAM J. Appl. Math..

[3]  Erratum to: Models of biochemical reaction systems , 2015 .

[4]  Carsten Wiuf,et al.  Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks , 2015, Bulletin of mathematical biology.

[5]  Simon L. Cotter,et al.  Constrained approximation of effective generators for multiscale stochastic reaction networks and application to conditioned path sampling , 2015, J. Comput. Phys..

[6]  David F. Anderson,et al.  Continuous Time Markov Chain Models for Chemical Reaction Networks , 2011 .

[7]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[8]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[9]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[10]  Philipp Thomas,et al.  The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions , 2012, BMC Systems Biology.

[11]  Eric Vanden-Eijnden,et al.  Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. , 2005, The Journal of chemical physics.

[12]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[13]  Linda R Petzold,et al.  The slow-scale stochastic simulation algorithm. , 2005, The Journal of chemical physics.

[14]  N. Kampen.  The equilibrium distribution of a chemical mixture , 1976 .

[15]  E Weinan,et al.  Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales , 2007, J. Comput. Phys..

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

[17]  Ioannis G Kevrekidis,et al.  A constrained approach to multiscale stochastic simulation of chemically reacting systems. , 2011, The Journal of chemical physics.

[18]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

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

[20]  F. Horn.  Necessary and sufficient conditions for complex balancing in chemical kinetics , 1972 .

[21]  Xuan Li,et al.  Optimizing hybrid assembly of next-generation sequence data from Enterococcus faecium: a microbe with highly divergent genome , 2012, BMC Systems Biology.

[22]  T. Kurtz.  Representations of Markov Processes as Multiparameter Time Changes , 1980 .

[23]  Radek Erban,et al.  Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics , 2014, SIAM J. Sci. Comput..

[24]  T. Brill,et al.  Spectroscopy of Hydrothermal Reactions 15. The pH and Counterion Effects on the Decarboxylation Kinetics of the Malonate System , 2001 .

[25]  David F Anderson,et al.  A modified next reaction method for simulating chemical systems with time dependent propensities and delays. , 2007, The Journal of chemical physics.

[26]  David F. Anderson,et al.  Global Asymptotic Stability for a Class of Nonlinear Chemical Equations , 2007, SIAM J. Appl. Math..

[27]  Gheorghe Craciun,et al.  Toric Differential Inclusions and a Proof of the Global Attractor Conjecture , 2015, 1501.02860.

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

[29]  J. Bauer,et al.  Chemical reaction network theory for in-silico biologists , 2003 .

[30]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[31]  David F. Anderson,et al.  Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks , 2008, Bulletin of mathematical biology.

[32]  Mario di Bernardo,et al.  Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology , 2011 .

[33]  M. Feinberg.  Complex balancing in general kinetic systems , 1972 .

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

[35]  Carsten Wiuf,et al.  Product-Form Poisson-Like Distributions and Complex Balanced Reaction Systems , 2015, SIAM J. Appl. Math..