Dimensional Reduction of the Fokker–Planck Equation for Stochastic Chemical Reactions

Abstract

The Fokker-Planck equation models chemical reactions on a mesoscale. The solution is a probability density function for the copy number of the different molecules. The number of dimensions of the problem can be large making numerical simulation of the reactions computationally intractable. The number of dimensions is reduced here by deriving partial differential equations for the first moments of some of the species and coupling them to a Fokker-Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integro-differential equations and a Fokker-Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computatational work and memory of the original equation and with better modelling accuracy than the macroscopic reaction rate equations. Some terms in the equations are small and are ignored. Conditions are given for the influence of these terms to be small on the equations and the solutions. The difference between different models is illustrated in a numerical example.

Multiscale Modeling of Chemical Kinetics via the Master Equation

“…When the number of samples is large, as in the examples in this paper, this interval can be computed easily with the well-known formula μ ± 1.96 σ √ n by using the sample mean and sample variance. We note that other numerical methods for computing moments of the solution to the CME have also been suggested, such as in [16,22,37,12], although these are not based on the Krylov methods used here.…”

“…However, the SSA can become too slow in the presence of large molecular populations or rate constants, motivating Gillespie's τ -Leap [17] approximation and much interest in accelerated leap methods [28,34,7] and, more generally, in multiscale methods for simulating biochemical kinetics [22,8].…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php out of reach. Furthermore, there have recently been many advances in methods for approximating the CME, such as the QSSA [20,19] and dimensionality reduction [22,27], and these methods could be incorporated into the techniques described here via the embedded CME-solver. This paper has also demonstrated that extra information from the CME can be incorporated into simulation algorithms, and that this can always be done in a way that does not significantly impair their efficiency, via the FSP-Leap and hybrid methods.…”

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

“…The computational complexity is mitigated somewhat but the problem with exponential growth in work and memory remains. In [18] the dimension of the FPE is reduced by assuming that the major part of the molecular species are normally distributed with a small variance and that only a small set of species needs a full stochastic treatment with an FPE. The dimension reduction in [18] is applicable also to the master equation.…”

“…In [18] the dimension of the FPE is reduced by assuming that the major part of the molecular species are normally distributed with a small variance and that only a small set of species needs a full stochastic treatment with an FPE. The dimension reduction in [18] is applicable also to the master equation. Equations for the expected values of the majority of the species are derived and they are coupled to one FPE for the probability density function (PDF) in a low dimension.…”