Statistical mechanics of gene competition
Ortiz, Juan Venegas
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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.