Modelling and simulating Lenski's long-term evolution experiment

We revisit the model by Wiser, Ribeck, and Lenski (Science 342 (2013), 1364–1367), which describes how the mean fitness increases over time due to beneficial mutations in Lenski’s long-term evolution experiment. We develop the model further both conceptually and mathematically. Conceptually, we describe the experiment with the help of a Cannings model with mutation and selection, where the latter includes diminishing returns epistasis. The analysis sheds light on the growth dynamics within every single day and reveals a runtime effect, that is, the shortening of the daily growth period with increasing fitness; and it allows to clarify the contribution of epistasis to the mean fitness curve. Mathematically, we present rigorous results in terms of a law of large numbers (in the limit of infinite population size and for a certain asymptotic parameter regime), as well as approximations based on heuristics and supported by simulations for finite populations.