9th April 2009, 1325 MC206

**Dr. Marcus Hegland**

from

Centre Mathematics and its Applications ANU

will speak on

Approximating the Solution of the Chemical Master Equation

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.