Abstract

This presentation is joint work with Pamela Burrage.  The standard approach to simulating chemical kinetics is to assume a well-stirred homeogeneous environment and this leads to the use of the stochastic simulation algorithm (in which the waiting time to the next reaction is exponentially distributed), the chemical master equation (descrbing its pdf), the chemical langevin equation and the use of the Law of Mass Action to derive a system of ordinary differential equations.  However, if the chemical kinetics operate in a spatially crowded heterogeneous environment then the modelling and simulation becomes more complex.  If a parameter alpha between 0 and 1 captures the broad features of this heterogeneity, then we must deal with the so-called Mittag-Leffler (heavy tailed function) in all of the above settings along with the concept of time change.  We briefly discuss these issues.  We also show how to adapt the approach of Jahnke and Huisinga, who have given the exact solution of the CME for unimolecular reactions, in terms of convolutions of multinomial and Poisson distributions in the homogeneous setting, together with the use of iterated Brownian path algorithms when alpha takes the form of (1/2)^k to accuately simulate the fractional kinetics of some simple systems.