Large Deviations Approach for Stochastic Genetic Evolution
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Theoretical ecologists have long strived to explain how the persistence of populations depends on biotic and abiotic factors and have proposed various models to predict the long time behavior of biological populations. We are interested in modeling the effects of natural selection and adaptation in a bacterial population of $Escherichia\; coli$, one of the most intensively studied organisms on Earth. A distinctive signature of living systems is Darwinian evolution, that is, a tendency to generate as well as self-select individual diversity. Mathematical models built to describe this natural dynamics of populations must be rooted in the microscopic, stochastic description of discrete individuals characterized by one or several adaptive traits and interacting with each other. The simplest models assume asexual reproduction and haploid genetics, where an offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take a mutation step to new and different trait values and selection follows from ecological interactions among individuals. In this dissertation we borrow results from large deviation theory to predict the most likely evolutionary trajectories for genetic traits in a given bacterial population leading from known initial multi-species frequencies to terminal domination by mutants with highest fitness. To compute the most likely evolution path, we seek the trajectory with minimal large deviations cost among all genetic evolution trajectories. The goal thus reached is to compute the most likely evolutionary steps which brought an actually observed terminal overwhelming dominance by a new mutant.