Modeling Chemical Reactions with Single Reactant Specie Abhyudai Singh and Jo˜ ao Pedro Hespanha Abstract—A procedure for constructing approximate stochastic models for chemical reactions involving a single reactant specie 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 some moments of interest. In this paper we provide existence and uniqueness conditions for these finite-dimensional nonlinear ODEs. Furthermore, explicit formulas to construct them are also provided. I. INTRODUCTION The time evolution of a spatially homogeneous mixture of chemically reacting molecules is often modeled using a stochastic formulation, which takes into account the inherent randomness of thermal molecular motion. This formulation is superior to the traditional deterministic formulation of chemical kinetics and is motivated by complex reactions inside living cells, where small populations of key reactants can set the stage for significant stochastic effects [1], [2], [3]. Although most of the mathematical modeling of genetic networks represents gene expression and regulation as deterministic processes, recent observations of gene expression in individual cells illustrate the stochastic nature of transcription [4], [5]. Furthermore, studies of engineered genetic circuits designed to act as toggle switches or oscillators have revealed large stochastic effects [2], [6]. Stochasticity is therefore an inherent feature of biological dynamics and developing models which capture this have become increasingly important. In the stochastic formulation of chemical reactions, the time evolution of the system is described by a single equation for a grand probability function, where time and species populations appear as independent variables, called the Master equation [7]. However, this equation can only be solved for relatively few, highly idealized cases and generally Monte Carlo simulation techniques [8]-[9], are used. Since one is often interested in only the first and This material is based upon work supported by the Institute for Collab- orative Biotechnologies through grant DAAD19-03-D-0004 from the U.S. Army Research Office and by the National Science Foundation under Grant No. CCR-0311084. A.Singh and J.P.Hespanha are with the Center for Control Engineering and Computation University of California, Santa Barbara, CA 93101. abhi@engineering.ucsb.edu, hespanha@ece.ucsb.edu second order moments for the number of molecules of the different species involved, much effort can be saved by applying approximate methods to estimate the evolution of these low-order moments, without actually having to solve for the probability density function. Various such approximate methods have been developed, for example, using the Fokker-Plank approximation or expanding the Master equation [7]. In [10], an alternative approximate method for estimating lower-order moments was introduced. This was done by representing the dynamics of a chemical reaction as a Stochastic Hybrid System (SHS) whose state x, is the population of all the species involved in the reaction. In order to fit the framework of our problem, this class of SHS had trivial continuous dynamics ˙ x = 0, with reset maps and transitional intensities defined by the stoichiometry and the reaction rates of the reactions, respectively. In essence, if no reaction takes place, the population of species remain constant and whenever the reaction takes place, the reset map is “activated” and the population is reset according to the stoichiometry of the reaction. Details for the stochastic modeling of chemical reactions are presented in Sec. II. It was also shown in [10] that these SHSs used to model chemical reactions are actually polynomial Stochastic Hybrid Systems (pSHS). An important property of pSHSs is that, if one creates an infinite vector containing all the statistical moments of x, the dynamics of this vector is governed by an infinite-dimensional linear ordinary differential equation (ODE) which we call the infinite-dimensional moment dynamics. In general, the infinite-dimensional moment dynamics cannot be solved analytically, however, as shown in Sec. III, under appropriate conditions, they can be approximated by a finite-dimensional nonlinear ODE, which we call the truncated moment dynamics. The state of the truncated moment dynamics μ , typically contains some lower-order moments of interest. A procedure to construct these truncated moment dynamics was outlined in [10]. The procedure proposed in [10] is general but not systematic. Moreover, [10] provided no conditions under which the truncated moment dynamics could be found. In this paper we resolve these issues for a general class of K chemical reactions involving one specie. More specifically, we show in Sec. IV, that given a vector μ , containing the first n moments of x, one can find truncated moment dynamics, if one drops some lower order moments from the time derivatives of μ . These dropped moments are at least of