System size expansion

The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system. The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system. Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly decay in a physical system, or genes that are expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study of the systems statistics (for example, the mean and variance of the number of atoms or proteins as a function of time). The system size expansion allows one to obtain an approximate statistical description that can be solved much more easily than the master equation. Systems that admit a treatment with the system size expansion may be described by a probability distribution P ( X , t ) {displaystyle P(X,t)} , giving the probability of observing the system in state X {displaystyle X} at time t {displaystyle t} . X {displaystyle X} may be, for example, a vector with elements corresponding to the number of molecules of different chemical species in a system. In a system of size Ω {displaystyle Omega } (intuitively interpreted as the volume), we will adopt the following nomenclature: X {displaystyle mathbf {X} } is a vector of macroscopic copy numbers, x = X / Ω {displaystyle mathbf {x} =mathbf {X} /Omega } is a vector of concentrations, and ϕ {displaystyle mathbf {phi } } is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system. x {displaystyle mathbf {x} } and X {displaystyle mathbf {X} } are thus quantities subject to stochastic effects. A master equation describes the time evolution of this probability. Henceforth, a system of chemical reactions will be discussed to provide a concrete example, although the nomenclature of 'species' and 'reactions' is generalisable. A system involving N {displaystyle N} species and R {displaystyle R} reactions can be described with the master equation: Here, Ω {displaystyle Omega } is the system size, E {displaystyle mathbb {E} } is an operator which will be addressed later, S i j {displaystyle S_{ij}} is the stoichiometric matrix for the system (in which element S i j {displaystyle S_{ij}} gives the stoichiometric coefficient for species i {displaystyle i} in reaction j {displaystyle j} ), and f j {displaystyle f_{j}} is the rate of reaction j {displaystyle j} given a state x {displaystyle mathbf {x} } and system size Ω {displaystyle Omega } . E − S i j {displaystyle mathbb {E} ^{-S_{ij}}} is a step operator, removing S i j {displaystyle S_{ij}} from the i {displaystyle i} th element of its argument. For example, E − S 23 f ( x 1 , x 2 , x 3 ) = f ( x 1 , x 2 − S 23 , x 3 ) {displaystyle mathbb {E} ^{-S_{23}}f(x_{1},x_{2},x_{3})=f(x_{1},x_{2}-S_{23},x_{3})} . This formalism will be useful later. The above equation can be interpreted as follows. The initial sum on the RHS is over all reactions. For each reaction j {displaystyle j} , the brackets immediately following the sum give two terms. The term with the simple coefficient −1 gives the probability flux away from a given state X {displaystyle mathbf {X} } due to reaction j {displaystyle j} changing the state. The term preceded by the product of step operators gives the probability flux due to reaction j {displaystyle j} changing a different state X ′ {displaystyle mathbf {X'} } into state X {displaystyle mathbf {X} } . The product of step operators constructs this state X ′ {displaystyle mathbf {X'} } . For example, consider the (linear) chemical system involving two chemical species X 1 {displaystyle X_{1}} and X 2 {displaystyle X_{2}} and the reaction X 1 → X 2 {displaystyle X_{1} ightarrow X_{2}} . In this system, N = 2 {displaystyle N=2} (species), R = 1 {displaystyle R=1} (reactions). A state of the system is a vector X = { n 1 , n 2 } {displaystyle mathbf {X} ={n_{1},n_{2}}} , where n 1 , n 2 {displaystyle n_{1},n_{2}} are the number of molecules of X 1 {displaystyle X_{1}} and X 2 {displaystyle X_{2}} respectively. Let f 1 ( x , Ω ) = n 1 Ω = x 1 {displaystyle f_{1}(mathbf {x} ,Omega )={frac {n_{1}}{Omega }}=x_{1}} , so that the rate of reaction 1 (the only reaction) depends on the concentration of X 1 {displaystyle X_{1}} . The stoichiometry matrix is ( − 1 , 1 ) T {displaystyle (-1,1)^{T}} .