Stochastic Models for Genetic Evolution

Date

2016-08

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Abstract

We develop a stochastic differential equation for the purpose of studying experiments in long-term evolution of bacterial populations subject to the protocol initiated by R. E. Lenski in 1988. This sort of protocol typically consists of daily periods of free population growth between severe abrupt reductions in population size. In such experiments, thousands, or even tens of thousands, of generations may be observed allowing for the possibility of observation of the effects of evolution by natural selection.

We first take an in depth look at the dynamics of a single period of free growth. Our model of free growth builds on the classical exponential growth model by incorporating Poisson mutation events among an arbitrary number of distinct genotypes. Analysis of the model yields explicit formulae for the mean and covariance matrix of the state of the population, expressed as a vector of genotype frequencies, at the end of growth. With these quantitative tools in hand we proceed to incorporate the periodic population reduction as a multinomial random sample. This leads to the derivation of a stochastic differential equation with Gaussian noise. A key finding justifying the derivation is that the order of random variability inherent to growth is small compared to that due to the periodic random sample, as this allows for a deterministic approximation of growth.

Exhaustive simulations verify the validity of the analysis of the growth model as well as demonstrate strong empirical agreement between the stochastic differential equation model and another model taken as a "ground truth" model.

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Keywords

Stochastic differential equations (SDE), Diffusion, Long-term evolution experiment

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