Speed of evolution in large asexual populations with diminishing returns

The adaptive evolution of large asexual populations is generally characterized by competition between clones carrying different beneficial mutations. Interference slows down the adaptation speed and makes the theoretical description of the dynamics more complex with respect to the successional occurrence and fixation of beneficial mutations typical of small populations. A simplified modeling framework considering multiple beneficial mutations with equal and constant fitness advantage is known to capture some of the essential features of laboratory evolution experiments. However, in these experiments the relative advantage of a beneficial mutation is generally dependent on the genetic background. In particular, the general pattern is that, as mutations in different loci accumulate, the relative advantage of new mutations decreases, a trend often referred to as "diminishing return" epistasis. Here, we propose a phenomenological model that generalizes the fixed-advantage framework to include this negative epistasis in a simple way. We evaluate analytically as well as with direct simulations the quantitative consequences of diminishing returns on the evolutionary dynamics. The speed of adaptation decreases in time and reaches a limit value corresponding to neutral evolution in the long time limit. This corresponds to an increase of the diversity in terms of "classes of mutation" in the population. Finally, we show how the model can be compared with dynamic data on fitness and number of beneficial mutations from laboratory evolution experiments.