Application of Large Deviations to Genetic Evolution of Bacterial Populations

University Of Houston, Houston TX

Investigators

Abstract

Populations of bacteria or viruses exhibit strong adaptivity through emergence and fixation of beneficial mutations. For instance, such mutations can be related to bacterial resistance to antibiotics or emergence of viral strains transferable from animal to humans. Currently, there exist various mathematical models which describe the evolution of bacterial or viral populations, but mathematical tools which can be used to study evolutionary trajectories of large bacterial or viral populations are still lacking. Therefore, the main goal of this work is to develop proper mathematical tools for studying rare mutational events in bacteria or viruses. In particular, the investigators will use the large deviation approach for population trajectories to develop theoretical foundation and practical computational tools to study the evolutionary trajectories of bacterial populations with concrete applications to the analysis of long term laboratory experiments on Escherichia Coli. The large deviations technique will be developed in a novel context of discrete models for population histograms. In contrast with the classical Wentzell-Freidlin approach for "small" diffusions, there is no general derivation of cost functions for discrete Markov chains with continuous state space, so that concrete analytical and numerical results for discrete models are rather scarce. However, large deviations offer significant computational advantages in the context of evolutionary dynamics described here due to the special structure of this problem. In particular, the cost function for evolutionary trajectories will be derived in an explicit form. This will lead to the development of a novel discretized difference equation for cost minimizing trajectories, thus enabling an efficient numerical "geodesic shooting" in reverse time to compute the most likely population evolution between given initial and terminal states. This novel algorithm will be used to estimate (very small) probabilities of rare genetic events such as the transitions between two successive genotype fixations and to compute the most likely sequence of genotype fixations leading from an ancestor genotype to a currently dominant genotype. Moreover, the numerical codes developed in the course of the proposed research will be used to study various epistasis scenarios. Thus, this work will generate innovative computational and conceptual tools leading to a better understanding of genetic evolution of bacterial or viral populations.