Markus Hegland
Centre for Mathematics and its Applications
Australian National University
Canberra, Australia
Chemical reactions are stochastic processes. While in many applications the kinetic
rate equations provide an adequate model for the dynamics of the expected concentrations
this is not the case when the numbers of molecules involved in the reactions are small.
This situation occurs in many molecular biological processes including signalling and
gene regulation. In order to understand the effect of the "reaction noise" one needs
stochastic models.
The chemical master equation is a continuous time discrete state Markov model which
describes how the probabilities of the states evolve over time due to the chemical
reactions. The states are vectors of integers. Instead of the concentrations of the
kinetic rate equations here the copy numbers of the chemical species form the
components of the states. There values are typically between zero and a few hundred.
The state space grows exponentially with the number of different chemical species and,
as the probability of each state is recorded one faces the "curse of dimensionality".
This is one reason why essentially all computational approaches in this area use
stochastic simulation methods.
In this talk I will discuss methods to determine solutions of the chemical master
equations numerically. The approach we adopt uses a variant of the sparse grid technique
based on state space aggregation, which is a finite volume type approach. The
approximation order can be controlled by a method introduced by Per Loetstedt and his
collaborators which uses a piecewise polynomial approximation combined with aggregation.
A new bound for the approximation error using this approach will be given. The sparse
grid method will be illustrated for a very simple model of a signalling cascade. I will
report on some work of an ANU student who was able to solve the master equation
for a simple signalling cascade with 100 proteins species.