Hybrid modelling of stochastic chemical kinetics
Abstract
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
modeled 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.
This is joint work with A. Duncan (Imperial) and R. Erban (Oxford)