Adaptation and irreversibility in microevolution

Within the framework of population genetics we consider the evolution of an asexual haploid population under the effect of a rapidly varying natural selection (microevolution). We focus on the case in which the environment exerting selection changes stochastically. We derive the effective genotype and fitness dynamics on the slower time-scales at which the relevant genetic modifications take place. We find that, despite the fast environmental switches, the population manages to adapt on the fast time-scales yielding a finite positive contribution to the fitness. However, such contribution is balanced by the continuous loss in fitness due to the varying selection so that the statistics of the global fitness can be described neglecting the details of the fast environmental process. The occurrence of adaptation on fast time-scales would be undetectable if one were to consider only the effective genotype and fitness dynamics on the slow time-scales. We therefore propose an experimental observable to detect it. Introduction. – Population genetics studies the evolutionary process of a population under forces such as mutations, Darwinian selection and the random genetic drift (see [1–3] for a general exposition). A key intuition for modeling the effects of natural selection is the concept of fitness of a population in a certain environment which describes the global ability of the population to reproduce and survive [4]. The different fitness of various genotypes can be effectively visualized in terms of a fitness landscape [5] which is usually ”climbed” during the course of evolution. Such climbing is referred to as adaptation. Fisher’s fundamental theorem of natural selection [6] states that when evolution is subject only to natural selection in a constant environment, the fitness of a population increases at a positive rate equal to the variance of the population. Indeed, natural selection, by its very definition, is the force that favors fitter individuals, pushing towards the genotype configuration which maximizes the global fitness. When mutations and random drift are relevant, the stochastic nature of evolution emerges and adaptation becomes a more complex phenomenon. One of the most widely accepted model for the stochastic description of the evolutionary process is the Kimura-Ohta equation [7] that we will describe in detail in Eq. (2) below. In most natural cases the environment in which the population evolves changes in time (see for example Refs. [12,13,17–22]) and genotypes that were fit under the initial condition may successively be unfavored by selection. Consider for instance the case of a population of bacteria shaped by natural selection to metabolize a certain nutrient. If such nutrient is gradually replaced