Models for Multi-Specie Chemical Reactions Using Polynomial Stochastic Hybrid Systems

• Published in: Proceedings of the 44th IEEE Conference on Decision and Control pp. 2969 - 2974
• Main Authors: Singh, A., Hespanha, J.P.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 2005
• Subjects: Biological materials; Chemicals; Differential equations; Kinetic theory; Nonlinear dynamical systems; Polynomials; Probability density function; Stochastic processes; Stochastic systems; Vectors
• ISBN: 9780780395671; 0780395670
• ISSN: 0191-2216
• DOI: 10.1109/CDC.2005.1582616

A procedure for constructing approximate stochastic models for chemical reactions is presented. This is done by representing the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). An important property of pSHSs is that the dynamics of all the statistical moments of its continuous states, evolves according to a infinite-dimensional linear ordinary differential equation (ODE). Under appropriate conditions, this infinite-dimensional ODE can be accurately approximated by a finite-dimensional nonlinear ODE, the state of which typically contains the moments of interest. In this paper, for a very general class of chemical reactions, we provide existence and uniqueness conditions for these finite-dimensional nonlinear ODEs. Furthermore, explicit formulas to construct them are also provided. To illustrate the applicability of our results, we construct an approximate stochastic model for a decaying and dimerizing chemical reaction set. Moment estimates obtained from the finite-dimensional nonlinear ODE are compared with estimates obtained from a large number of Monte Carlo simulations.