Large Deviations Analysis for Stochastic Models of Bacterial Evolution

Abstract

Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities that direct numerical simulations generally fail to evaluate accurately. In this paper, we develop a theoretical large-deviation framework for a class of Markov chains modeling the genetic evolution of bacteria such as E. coli in ``locked-box'' laboratory experiments. In particular, we develop the cost function for discrete-time Markov chains that describe the daily evolution of histograms of bacterial populations. We obtain explicit formulas for the cost function for interior histograms. We also develop explicit formulas that can be used to numerically quantify the most likely evolutionary trajectories connecting an initial histogram and the target histogram.