Numerical Simulation for Biochemical Kinetics
Numerical Simulation for Biochemical Kinetics
This chapter discusses concepts and techniques for mathematically describing and numerically simulating chemical systems that into account discreteness and stochasticity. The chapter is organized as follows. Section 16.2 outlines the foundations of “stochastic chemical kinetics” and derives the chemical master equation (CME)—the time-evolution equation for the probability function of the system’s state. The CME, however, cannot be solved, for any but the simplest of systems. But numerical realizations (sample trajectories in state space) of the stochastic process defined by the CME can be generated using a Monte Carlo strategy called the stochastic simulation algorithm (SSA), which is derived and discussed in Section 16.3. Section 16.4 describes an approximate accelerated algorithm known as tau-leaping. Section 16.5 shows how, under certain conditions, tau-leaping further approximates to a stochastic differential equation called the chemical Langevin equation (CLE), and then how the CLE can in turn sometimes be approximated by an ordinary differential equation called the reaction rate equation (RRE). Section 16.6 describes the problem of stiffness in a deterministic (RRE) context, along with its standard numerical resolution: implicit method. Section 16.7 presents an implicit tau-leaping algorithm for stochastically simulating stiff chemical systems. Section 16.8 concludes by describing and illustrating yet another promising algorithm for dealing with stiff stochastic chemical systems, which is called the slow-scale SSA.
Keywords: stochastic chemical kinetics, chemical master equation, stochastic simulation algorithm, tau-leaping, chemical Langevin equation, reaction rate equation
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