Collective Fluctuations in models of adaptation

The dynamics of adaptation is difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from number fluctuations, called genetic drift, arising in the small number of particularly fit individuals of the population. Random genetic drift in this evolutionary vanguard also limits the speed of adaptation, which diverges in deterministic models that ignore these chance effects. Several approaches have been developed to analyze the crucial role of noise on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, very little is known about how genetic drift causes fluctuations to emerge on the population level, including fitness distribution variations and speed variations. Yet, these phenomena control the replicability of experimental evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we develop an exact approach to these emergent fluctuations by a combination of computational and analytical methods. We show, analytically, that the infinite hierarchy of moment equations can be closed at any arbitrary order by a suitable choice of a dynamical constraint. This constraint regulates (rather than fixes) the population size, accounting for resource limitations. The resulting linear equations, which can be accurately solved numerically, exhibit fluctuation-induced terms that amplify short-distance correlations and suppress long-distance ones. Importantly, by accounting for the dynamics of sub-populations, we provide a systematic route to key population genetic quantities, such as fixation probabilities and decay rates of the genetic diversity. We demonstrate that, for some key quantities, asymptotic formulae can be derived. While it is natural to consider the process of adaptation as a branching random walk (in fitness space) subject to a constraint (due to finite resources), we show that other noisy traveling waves likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach towards analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions or epidemics.