# Modelling and simulating Lenski's long-term evolution experiment

## TL;DR

This paper refines a model of Lenski's long-term evolution experiment by integrating a Cannings model with diminishing returns epistasis, providing new insights into fitness growth dynamics and the effects of epistasis.

## Contribution

It advances the existing model both conceptually and mathematically, clarifying the role of epistasis and the daily growth dynamics in the experiment.

## Key findings

- Identification of a runtime effect with shortening daily growth periods
- Clarification of epistasis contribution to fitness increase
- Development of approximations supported by simulations

## Abstract

We revisit the model by Wiser, Ribeck, and Lenski (Science \textbf{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 explain rigorous results in terms of a law of large numbers (in the limit of infinite population size and for a certain asymptotic parameter regime), and present approximations based on heuristics and supported by simulations for finite populations.