Quantitative analysis of biochemical networks often requires consideration of both spatial and stochastic aspects of chemical processes. Despite significant progress in the field, it is still computationally prohibitive to simulate systems involving many reactants or complex geometries using a microscopic framework that includes the finest length and time scales of diffusion-limited molecular interactions. For this reason, spatially or temporally discretized simulations schemes are commonly used when modeling intracellular reaction networks. The challenge in defining such coarse-grained models is to calculate the correct probabilities of reaction given the microscopic parameters and the uncertainty in the molecular positions introduced by the spatial or temporal discretization. In this paper we have solved this problem for the spatially discretized Reaction-Diffusion Master Equation; this enables a seamless and physically consistent transition from the microscopic to the macroscopic frameworks of reaction-diffusion kinetics. We exemplify the use of the methods by showing that a phosphorylation-dephosphorylation motif, commonly observed in eukaryotic signaling pathways, is predicted to display fluctuations that depend on the geometry of the system.oth spatial and stochastic aspects of chemical reactions are important for the quantitative modeling of intracellular processes (1). Spatial, because diffusion is not sufficiently fast to make the system well-stirred between individual reaction events. Stochastic, because the number of reactants within diffusion range often is small, in which case the probabilistic and nonlinear nature of chemistry invalidates mean-field descriptions. There are two different basic theoretical frameworks describing spatially heterogeneous stochastic kinetics: the continuous microscopic framework (2-5) and the spatially discretized framework, here exemplified by the canonical reaction-diffusion (or multivariate) master equation (6, 7). The spatially and temporally continuous microscopic framework resolves the exact positions of molecules as well as the finest time scales of rapid reassociation after dissociation (5). The reaction-diffusion master equation (RDME), on the other hand, is coarse-grained and will be referred to as mesoscopic, as it conventionally averages out the kinetics at microscopic length and time scales. The RDME was developed to describe and analyze the influence of chemical noise in systems with many molecules and can be considered phenomenological in the sense that it is has not been derived from a microscopic model. In fact, the RDME has been shown to diverge and give unphysical results as the discretization approaches microscopic length scales (8,9). This divergence is crucial because one way to ascertain that the chosen discretization is appropriate is to test that the results are invariant for a finer discretization. Due to the constraints on the spatial discretization, some combinations of reactions have not been possible to model using the RDME (10). In this pap...

The master equation of chemical reactions is solved by first approximating it by the Fokker-Planck equation. Then this equation is discretized in the state space and time by a finite volume method. The difference between the solution of the master equation and the discretized FokkerPlanck equation is analyzed. The solution of the Fokker-Planck equation is compared to the solution of the master equation obtained with Gillespie's Stochastic Simulation Algorithm (SSA) for problems of interest in the regulation of cell processes. The time dependent and steady state solutions are computed and for equal accuracy in the solutions, the Fokker-Planck approach is more efficient than SSA for low dimensional problems and high accuracy.

Dedicated to Björn Engquist on the occasion of his 60th birthday. Abstract.The Fokker-Planck equation on conservation form modeling stochastic chemical reactions is discretized by a finite volume method for low dimensional problems and advanced in time by a linear multistep method. The grid cells are refined and coarsened in blocks of the grid depending on an estimate of the spatial discretization error and the time step is chosen to satisfy a tolerance on the temporal discretization error. The solution is conserved across the block boundaries so that the total probability is constant. A similar effect is achieved by rescaling the solution. The steady state solution is determined as the eigenvector corresponding to the zero eigenvalue. The method is applied to the solution of a problem with two molecular species and the simulation of a circadian clock in a biological cell. Comparison is made with a Monte Carlo method. (2000): 65M20, 65M50. AMS subject classification