A Hybrid Deterministic−Stochastic Algorithm for Modeling Cell Signaling Dynamics in Spatially Inhomogeneous Environments and under the Influence of External Fields

Cell signaling dynamics mediate myriad processes in biology. It has become increasingly clear that inter- and intracellular signaling reactions often occur in a spatially inhomogeneous environment and that it is important to account for stochastic fluctuations of certain species involved in signaling reactions. The importance of these effects enhances the difficulty of gleaning mechanistic information from observations of a few experimental reporters and highlights the significance of synergistic experimental and computational studies. When both stochastic fluctuations and spatial inhomogeneity must be included in a model simultaneously, however, the resulting computational demands quickly become overwhelming. In many situations the failure of standard coarse-graining methods (i.e., ignoring spatial variation or stochastic fluctuations) when applied to all components of a complex system does not exclude the possibility of successfully applying such coarse-graining to some components of the system. Following this approach alleviates computational cost but requires “hybrid” algorithms where some variables are treated at a coarse-grained level while others are not. We present an efficient algorithm for simulation of stochastic, spatially inhomogeneous reaction−diffusion kinetics coupled to coarse-grained fields described by (stochastic or deterministic) partial differential equations (PDEs). The PDEs could represent mean-field descriptions of reactive species present in large copy numbers or evolution of hydrodynamic variables that influence signaling (e.g., membrane shape or cytoskeletal motion). We discuss the approximations made to derive our algorithm and test its efficacy by applying it to problems that include many features typical of realistic cell signaling processes.