ABSTRACT
As described in Chapter 3, a chemical reaction model can be transformed into a set of first order ODEs. An ODE model, however, assumes that concentrations vary continuously and deterministically. Unfortunately, chemical systems satisfy neither of these assumptions. The number of molecules of a chemical species is a discrete quantity in that it is always a natural number (0, 1, 2, . . . ). Furthermore, chemical reactions typically occur after two molecules collide. Unless one tracks the exact position and velocity of every molecule (something one obviously does not wish to do), it is impossible to know when a reaction may occur. Therefore, it is preferable to consider the occurrence of chemical reactions to be a stochastic process. Despite the violation of these assumptions, in systems which involve large molecular counts, ODE models give a quite accurate picture of their behavior. If the molecular counts are small, however, the discrete and stochastic nature may have significant influence on the observable behavior. Genetic circuits typically involve small molecule counts, since there are often only a few 10s or 100s of molecules of each transcription factor and one strand of DNA. Therefore, accurate analysis of genetic circuits often requires a stochastic process description. This chapter presents one such description. First, Section 4.1 describes
the stochastic chemical kinetic (SCK) model. Next, Section 4.2 describes the chemical master equation (CME), the formal representation that describes the time evolution of the probabilities for the states of an SCK model. Sections 4.3 to 4.5 present algorithms to numerically analyze the CME. Section 4.6 describes the relationship between the CME and the reaction rate equations. Section 4.7 introduces stochastic Petri nets (SPN), an alternative modeling formalism for stochastic models. Section 4.8 presents a stochastic model of the phage λ decision circuit. Finally, Section 4.9 briefly describes a method to consider spatial issues in a stochastic model.
