# Multiscale Modelling and Simulation for Spatiotemporal Master Equations

Accurate modeling of reaction kinetics is important for understanding the functionality of biological cells and the design of chemical reactors. Depending on the particle concentrations and on the relation between particle mobility and reaction rate constants, different mathematical models are appropriate. In the limit of slow diffusion and small concentrations, both discrete particle numbers and spatial inhomogeneities must be taken into account. The most detailed root model consists of particle-based reaction-diffusion dynamics (PBRD), where all individual particles are explicitly resolved in time and space, and particle positions are propagated by some equation of motion, and reaction events may occur only when reactive species are adjacent. For rapid diffusion or large concentrations, the model may be coarse-grained in different ways. Rapid diffusion leads to mixing and implies that spatial resolution is not needed below a certain lengthscale. This permits the system to be modeled via a spatiotemporal chemical Master equation (STCME), i.e. a coupled set of chemical Master equations acting on spatial subvolumes. The STCME becomes a chemical Master equation (CME) when diffusion is so fast that the entire system is well-mixed. When particle concentrations are large, populations may be described by concentrations rather than by discrete numbers, leading to a PDE or ODE formulation. Many biological processes call for detailed models (PBRD, ST-CME or CME), but these models are extremely costly to solve. Efficient mathematical and computational methods are needed in order to approximate the solutions of these models with some guaranteed accuracy level. An approach to optimal or efficient switching between different models is, as yet, missing. In this project, we will set out to develop a multiscale theory for reaction kinetics processes, starting from a consistent and well-defined formulation of PBRD models, and including spatial scaling coupled to population scaling. In particular, we aim at providing solutions for the problematic cases of having particles at diverse copy numbers and at least some slowly diffusing particles. The cascades of scales in these scenarios and efficient approximation strategies will be explored.