Numerical Simulation for Biochemical Kinetics

In chemical systems formed by living cells, the small numbers of molecules of a few reactant species can result in dynamical behavior that is discrete and stochastic, rather than continuous and deterministic (McAdams & Arkin, 1999; McAdams & Arkin, 1997; Arkin et al., 1998; Elowitz et al., 2002; Fedoroff & Fontana, 2002). By “discrete”, we mean the integer-valued nature of small molecular populations, which makes their representation by real-valued (continuous) variables inappropriate. By “stochastic”, we mean the random behavior that arises from the lack of total predictability in molecular dynamics. In this chapter we introduce some concepts and techniques that have been developed for mathematically describing and numerically simulating chemical systems that take proper account of discreteness and stochasticity. Throughout, we shall make the simplifying assumption that the system is well-stirred or spatially homogeneous. In practice this assumption is often justified, and it allows us to specify the state of the system simply by giving the molecular populations of the various chemical species. But in some circumstances the well-stirred assumption will not be justified; then the locations of the molecules and the dynamics of their movement must also be considered. Some approaches to this more computationally challenging situation are described in Chapter ??.