[Thesis]. Manchester, UK: The University of Manchester; 2013.
The topic of this thesis is the stochastic dynamics of biochemical reaction systems.
The importance of the intrinsic fluctuations in these systems has become more widely
appreciated in recent years, and should be accounted for when modelling such systems
mathematically. These models are described as continuous time Markov processes and
their dynamics defined by a master equation. Analytical progress is made possible
by the use of the van Kampen system-size expansion, which splits the dynamics into
a macroscopic component plus stochastic corrections, statistics for which can then
be obtained. In the first part of this thesis, the terms obtained from the expansion
are written down for an arbitrary model, enabling the expansion procedure to be automated
and implemented in the software package COPASI. This means that the fluctuation analysis
may be used in tandem with other tools in COPASI, in particular parameter scanning
and optimisation. This scheme is then extended so that models involving multiple compartments
(e.g. cells) may be studied. This increases the range of models that can be evaluated
in this fashion. The second part of this thesis also concerns these multi-compartment
models, and examines how oscillations can synchronise across a population of cells.
This has been observed in many biochemical processes, such as yeast glycolysis. However,
the vast majority of modelling of such systems has used the deterministic framework,
which ignores the effect of fluctuations. It is now widely known that the type of
models studied here can exhibit sustained temporal oscillations when formulated stochastically,
despite no such oscillations being found in the deterministic version of the model.
Using the van Kampen expansion as a starting point, multi-cell models are studied,
to see how stochastic oscillations in one cell may influence, and be influenced by,
oscillations in neighbouring cells. Analytical expressions are found, indicating whether
or not the oscillations will synchronise across multiple cells and, if synchronisation
does occur, whether the oscillations synchronise in phase, or with a phase lag.