Statistical mechanics has been applied to a wide range of systems in physics, biology,
medicine and even anthropology. This theory has been recently used to model the
complex biochemical processes of gene expression and regulation. In particular, genetic
networks offer a large number of interesting phenomena, such as multistability and
oscillatory behaviour, that can be modelled with statistical mechanics tools.
In the first part of this thesis we introduce gene regulation, genetic switches, and the
colonization of a spatially structured media. We also introduce statistical mechanics
and some of its useful tools, such as the master equation and mean- field theories. We
present simple examples that are both pedagogical and also set the basis for the study
of more complicated scenarios.
In the second part we consider the exclusive genetic switch, a fundamental example
of genetic networks. In this system, two proteins compete to regulate each other's
dynamics. We characterize the switch by solving the stationary state in different limits
of the protein binding and unbinding rates. We perform a study of the bistability
of the system by examining its probability distribution, and by applying information
theory techniques. We then present several versions of a mean field theory that offers
further information about the switch. Finally, we compute the stationary probability
distribution with an exact perturbative approach in the unbinding parameter, obtaining
a valid result for a wide range of parameters values. The techniques used for this
calculation are successfully applied to other switches.
The topic studied in the third part of the thesis is the propagation of a trait inside
an expanding population. This trait may represent resistance to an antibiotic or being
infected with a certain virus. Although our model accounts for different examples in
the genetic context, it is also very useful for the general study of a trait propagating in a
population. We compute the speed of expansion and the stationary population densities
for the invasion of an established and an expanding population, finding non-trivial
criteria for speed selection and interesting speed transitions. The obtained formulae
for the different wave speeds show excellent agreement with the results provided by
simulations. Moreover, we are able to obtain the value of the speeds through a
detailed analysis of the populations, and establish the requirements for our equations
to present speed transitions. We finally apply our model to the propagation in a
position-dependent fitness landscape. In this situation, the growth rate or the maximum
concentration depends on the position. The amplitudes and speeds of the waves are
again successfully predicted in every case