Deterministic and stochastic modelling of biochemical processes - an
introduction and open problems

Tomas Vejchodsky

Abstract:

   This talk introduces basic ideas of mathematical models of dynamical
chemical systems. We will compare the deterministic and stochastic
models and show that they can provide even qualitatively different
predictions. This research is motivated by biochemical processes in
living cells, where concentrations of certain chemical species are
extremely low (units of molecules). This causes strong manifestation of
stochastic effects and invalidity of deterministic models.
   Subsequent parts of the talk will discuss more advanced mathematical
tools for stochastic modelling of biochemical systems. We will discuss
the central difficulty, which is the size and complexity of biochemical
networks of intra-cellular mechanisms. We will mention reduction of
these large models. We show that solution of the stationary chemical
master equation is equivalent to finding the null-space of a large
sparse matrix. We will comment on the accuracy and boundary conditions
for chemical Fokker-Planck equation, and we will explain the need for
solving high-dimensional partial differential equations and an approach
how to achieve it.