Hybrid modelling of stochastic chemical kinetics

It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modelled by Markov processes and, for such systems, methods such as Gillespie's algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse grained schemes, where the ``fast'' reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to significant errors in the simulation. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as computing observables of cell cycle models. In this talk, we present a hybrid scheme for simulating well-mixed stochastic kinetics, using Gillepsie--type dynamics to simulate the network in regions of low reactant concentration, and chemical langevin dynamics when the concentrations of all species is large. These two regimes are coupled via an intermediate region in which a ``blended'' jump-diffusion model is introduced. Examples of gene regulatory networks involving reactions occurring at multiple scales, as well as a cell-cycle model are simulated, using the exact and hybrid scheme, and compared, both in terms weak error, as well as computational cost.