Although a number of studies have shown that natural and laboratory populations initially well adapted to their environment can evolve rapidly when conditions suddenly change, the dynamics of rapid adaptation are not well understood. Here a population genetic model of polygenic selection is analyzed to describe the short-term response of a quantitative trait after a sudden shift of the phenotypic optimum. We provide explicit analytical expressions for the timescales over which the trait mean approaches the new optimum. We find that when the effect sizes are small relative to a scaled mutation rate, small to moderate allele frequency changes occur in the short-term phase in a synergistic fashion. In contrast, selective sweeps, i.e., dramatic changes in the allele frequency, may occur provided the size of the effect is sufficiently large. Applications of our theoretical results to the relationship between QTL and selective sweep mapping and to tests of fast polygenic adaptation are discussed. The rapid changes are responses to natural and humaninduced shifts in the environment. The genetic architecture underlying these traits ranges from few genes of major effect to highly polygenic systems (van't Hof et al. 2011;Lamichhaney et al. 2012Lamichhaney et al. , 2015Linnen et al. 2013). In this article, we study a model that encompasses a wide range of genetic architectures. Our aim is to understand the genomic signatures of positive selection in these systems that occur after environmental changes, with an emphasis on rapid adaptation.There is a growing body of literature on the detection of adaptive signatures in molecular population genetics. Following the pioneering work of Maynard Smith and Haigh (1974), the impact of positive selection on neutral DNA variability (selective sweeps) has attracted much interest. This theory has been applied to huge data sets that emerge from modern high-throughput sequencing. A large number of statistical tests have been developed to detect sweep signals and estimate the frequency and strength of selection (Kim and Stephan 2002;Nielsen et al. 2005;Pavlidis et al. 2010). To find sweep signatures in the genome, strong positive selection is required (with N e s 1; where N e is the effective population size and s is the selective advantage of a beneficial allele) (Kaplan et al. 1989;Stephan et al. 1992). Thus, sweeps are characteristic signals of fast adaptation. They are expected to be found at individual genes or at major loci if a trait is controlled by multiple genes.This raises the question whether fast evolution is also possible in highly polygenic systems and, if so, which genomic signatures arise. A number of genome-wide association studies (GWAS) have shown that quantitative traits are typically highly polygenic (e.g., Turchin et al. 2012) and there is growing evidence that the molecular scenario of sweeps only covers part of the adaptive process and needs to be revised to include polygenic selection nucleotide polymorphisms (SNPs) relevant to quantitative traits (Viss...

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We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance d eff up to which the population can spread in genotype space. For d eff ¼ L, the deterministic quasi-species theory operates while for d eff , 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T 3 while above T 3 the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime d eff , 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.