This thesis presents the results of mathematical modeling of both individual genes and small
networks of genes. The regulation of gene activity is essential for the proper functioning of
cells, which employ a variety of molecular mechanisms to control gene expression. Despite
this, there is considerable variation in the precise number and timing of protein molecules
that are produced. This is because gene expression is fundamentally a noisy process, subject
to a number of sources of randomness, including
uctuations in metabolite levels, the
environment and ampli ed by the very low number of molecules involved.
I have developed a probabilistic model of the burst size distribution (the number of
proteins produced by the binding of one promoter) of a single gene. Recent experimental
data provides excellent agreement with the model, but also reveals limitations of currently
available data in determining the origin of variations in expression.
A second strand of my work has addressed the dynamics of networks of genes. A
network motif is a sub-graph that occurs more often in the network than would be expected
by chance. The recurrent presence of certain motifs has been linked to systematic di erences
in the functional properties of networks.
I have developed models of the possible dynamical behaviour, in particular for the bi-fan
motif, a small sub-network with four genes. This motif has been identi ed as the most prevalent
in the regulatory networks of both the bacterium Escherichia coli and Saccharaomyces
cerevisiae. The results of this work show that the microscopic details of the interactions
are of paramount importance, with few inherent constraints on the network dynamics from
consideration of network structure alone. This result is relevant to all attempts to model
gene networks without su ciently detailed knowledge of the mechanisms of interaction.