M11: Stochastic modelling of reaction-diffusion processes

Background

Diffusion processes are an essential mechanism for the evolution of many processes in science. Stochastic diffusion is especially important in molecular and systems biology due to the typically low copy numbers that are involved. There are two common methods to simulate stochastic diffusion: Brownian simulation is often chosen when precision in the microscopic description of the molecules is important, and compartment-based simulation is used when larger copy numbers need to be simulated and computational time will be an issue.

Techniques and Challenges

We have developed the two-regime method (TRM), an algorithm that combines the best parts of both of these simulation regimes to optimise both the precision and computational efficiency of simulations of stochastic diffusion [11/15]. We have also implemented this simulation approach in a model for calcium release from the endoplasmic reticulum, a phenomena that would be problematic to model using either of the two approaches individually.

Results

The TRM algorithm has been shown to converge the expected concentration of a reaction-diffusion simulation over an interface between compartment-based and molecular-based regimes to that of a Partial Differential Equation (PDE) mean-field description. We have used the TRM successfully to modelling IP3-gated calcium ion channels. This model uses a three-dimensional two-regime method to give accurate approximations to the frequency of calcium puffs from the channels and is controlled by calcium ion concentrations in the cytosol. Furthermore we have completed a number of tests on the TRM in cases of irregular 1D lattices and their convergence in small compartment size and time step limits.

The Future

We plan to improve the current TRM framework so that it can be used for more general problems. We will be generalising the TRM so that it can be used in higher dimensions with structured and unstructured meshes and arbitrary domains. We will also be showing how the simulations of molecules are affected if there is drift as well as diffusion in the molecular dynamics.

References

[11/15] Flegg M.B., Chapman S.J., Erban R.: The two-regime method for optimizing stochastic reaction-diffusion simulations, J R Soc Interface 9(70): 859-868, 2012

[10/14] Lipkova J., Zygalakis K.C., Chapman S.J., Erban R.: Analysis of Brownian Dynamics Simulations of Reversible Bimolecular Reactions, SIAM J. Appl. Math., 71(3), 714–730, 2011

Erban R., Chapman S.J.: Reactive boundary conditions for stochastic simulations of reaction-diffusion processes, Physical Biology, Volume 4, Number 1, pp. 16-28, 2007

Erban R., Chapman S.J.: Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions, Physical Biology, Volume 6, Number 4, 046001, 2009

Erban R., Chapman S.J.: Time scale of random sequential adsorption, Physical Review E, Volume 75, Number 4, 041116, 2007

Erban R., Chapman S.J., Maini P.K.: A practical guide to stochastic simulations of reaction-diffusion processes, Lecture Notes, 2007

This project is funded by the European Research Council Starting Independent Researcher Grant awarded to Dr Erban.