Coarse-Grained Stochastic Particle-based Reaction-Diffusion Simulation   Algorithm
  In recent years, several particle-based stochastic simulation algorithms (PSSA) have been developed to study the spatially resolved dynamics of biochemical networks at a molecular scale. A challenge all these approaches have to address is to allow for simulations at cell-biologically relevant timescales without neither neglecting important spatial and biochemical properties of the simulated system nor introducing ad-hoc assumptions not based on physical principles. Here we describe a PSSA that permits large time steps while still retaining a high degree of accuracy. The approach addresses the typical disadvantage of Brownian dynamics, namely the need to use small time steps to resolve bimolecular encounters accurately, by estimating the number of otherwise unnoticed encounters with the help of the Green's functions of the diffusion equation incorporating molecular interactions. This method has previously been proposed for purely absorbing boundary conditions and irreversible bimolecular reactions. Building on those ideas, we developed a general-purpose PSSA that is applicable to a broad class of reaction-diffusion problems by incorporating reflective and radiation boundary conditions and reversible reactions. We furthermore discuss how reaction-diffusion systems on 2D membranes can be described and derive small time expansions of the Green's functions that substantially speed up key calculations, particularly in the problematic case of molecules in close proximity. Finally, we point out the formal relationship between our and exact algorithms. The proposed algorithm may serve as an easily implementable and flexible, computationally efficient, coarse-grained description of reaction-diffusion systems in 2D and 3D that nevertheless provides a stochastic, detailed representation at the level of individual particle trajectories in space and time. 
Thorsten Pr\"ustel and Martin Meier-Schellersheim
q-bio.QM cond-mat.stat-mech physics.chem-ph