This chapter provides an introduction to stochastic methods for modelling spatially homogeneous systems of chemical reactions. The Gillespie stochastic simulation algorithm and the chemical master equation are presented using simple examples of chemical systems. The chemical master equation is analysed for chemical systems containing zeroth-order, first-order and second-order chemical reactions. For zeroth-order and first-order chemical reactions, the average behaviour of the stochastic chemical system is described by the ordinary differential equations (ODEs) given by the standard deterministic model. However, when we consider higher-order chemical reactions, for which the deterministic description is nonlinear, the deterministic ODE model does not provide an exact description of the average behaviour of the stochastic system.

This chapter presents microscopic models of diffusion (Brownian motion). The discussed diffusion models explicitly describe the dynamics of solvent molecules. Such molecular dynamics models provide many more details than the models discussed in Chapter 4 (which simply postulate that the diffusing molecule is subject to a random force) and can be used to assess the accuracy of the stochastic diffusion models from Chapter 4. The analysis starts with theoretical solvent models, including a simple “one-particle” description of the solvent (heat bath), which is used to introduce the generalized Langevin equation and the generalized fluctuation–dissipation theorem. Analytical insights are provided by theoretical models with short- and long-range interactions. The chapter concludes with less analytically tractable, but more realistic, computational models, introducing molecular dynamics (molecular mechanics) and applying it to the Lennard-Jones fluid and to simulations of ions in aquatic solutions.

This chapter shows how active transport (for example, by an electrical field, molecular and cellular motors, running, swimming or flying, all in response to external cues) can be incorporated into the stochastic diffusion and reaction–diffusion algorithms we have introduced in Chapters 4 and 6. The resulting stochastic diffusion–advection and reaction–diffusion–advection models are analysed. Applications include systems consisting of many interacting “particles”, where individual particles can range in size from small ions and molecules to individual cells and animals. Three examples illustrate this: mathematical modelling of ions and ion channels, modelling bacterial chemotaxis, and studying collective behaviour of social insects. The chapter concludes with the discussion of the Metropolis–Hastings algorithm, which can be used to compute stationary (equilibrium) properties of complicated diffusion–advection problems.

This chapter introduces stochastic differential equations (SDEs) from the computational point of view, starting with several examples to illustrate the computational definition of the SDE that is used throughout the book. The Fokker–Planck and Kolmogorov backward equations are then derived and their consequences presented. They are used to compute the mean transition time between favourable states of SDEs. The SDE formalism is then applied to a chemical system by deriving the chemical Fokker–Planck equation and the corresponding chemical Langevin equation. They are used to further analyse the chemical systems from Chapter 2, including the system with multiple favourable states and the self-induced stochastic resonance.

This chapter presents stochastic models of molecular diffusion together with some important properties of the diffusion process. Diffusion models based on both position-jump processes and velocity-jump processes are analysed. The position-jump processes include the SDE-based approach (introduced in Chapter 3) together with the compartment-based (lattice-based) modelling, described by the diffusion master equation. Boundary conditions for the diffusion process are then discussed, including systems in which molecules are adsorbed by surfaces and models with a chemically reactive boundary. The Einstein–Smoluchowski relation between the diffusion constant and the size of the molecule is derived.

This chapter discusses multi-resolution simulation methods for modelling reaction–diffusion processes. They use a detailed modelling approach only in certain parts of the computational domain, whilst in the remainder of the domain a coarser, less detailed, method is used. Two examples of multi-resolution methods are presented. The first example couples the Brownian dynamics with the corresponding compartment-based description. The second example couples molecular dynamics together with its coarser Langevin description. The chapter concludes with an overview of related multi-resolution approaches in the literature.

This chapter presents examples of chemical systems where deterministic modelling fails and a stochastic approach is necessary. They include a chemical system with stochastic switching between favourable states of a system, and systems close to the bifurcation points of the corresponding deterministic ordinary differential equation (ODE) models. It is shown that the stochastic model might have qualitatively different properties than its deterministic counterpart for some parameter regimes. A chemical reaction system can also be redesigned, by adding extra reactions, in such a way that its stochastic behaviour qualitatively changes, while its deterministic ODEs do not change at all. In particular, there exist many stochastic reaction networks that correspond to the same deterministic ODE model.

This chapter presents approaches for efficient modelling of stochastic chemical systems. These methods are useful when a computational model based on the Gillespie stochastic simulation algorithm (SSA), introduced in Chapter 1, cannot be executed in a reasonably short time. Scenarios in which the Gillespie SSA can be computationally intensive are discussed, starting with an example involving fast and slow chemical reactions. This example is used to introduce the multiscale SSA, which speeds up simulations by assuming that the fast processes are at equilibrium, and other approximate approaches for problems with multiple time scales. Exact approaches to decrease the computational intensity of the Gillespie SSA are then discussed, including the next-reaction SSA.

This chapter discusses stochastic approaches for modelling chemical reactions (introduced in Chapter 1) and molecular diffusion at the same time. The presented stochastic reaction–diffusion processes add chemical reactions to the two position-jump models of molecular diffusion that are introduced in Chapter 4: the compartment-based approach (described by the reaction–diffusion master equation) and the SDE-based approach, which gives the Brownian dynamics. Basic principles of each approach are explained using an example that includes only zeroth- and first-order chemical reactions. This is followed by discussion of more complicated systems when some chemical species are subject to higher-order chemical reactions. The reaction radius, reaction probability and the choice of the compartment size are studied in detail. The chapter concludes with the discussion of applications to pattern formation in biology, including stochastic French flag model and stochastic Turing patterns.