Abstract
Evolution during range expansions shapes biological systems from microbial communities and tumours to invasive species. A fundamental question is whether, when a beneficial mutation arises during a range expansion, it will evade clonal interference and sweep to fixation. However, most theoretical investigations of range expansions have considered regimes in which selective sweeps are effectively impossible, while studies of selective sweeps have assumed constant population size or ignored spatial structure. Here we use mathematical modelling and analysis to investigate selective sweep probabilities and timings in biologically relevant scenarios, including the case in which mutants can displace a slowly spreading wildtype. Assuming constant expansion speed, we find surprisingly simple approximate and exact expressions for sweep probabilities in one, two and three dimensions, which are independent of mutation rate. Agent-based simulations confirm that our predictions are accurate for the spatial Moran process and remain informative when mutation effects on fitness are random and multiplicative. We further compare and synthesise our results with those obtained for alternative growth laws. Parameterised for human tumours, our model predicts that selective sweeps are rare except during early solid tumour growth, thus providing a general, pan-cancer explanation for findings from recent sequencing studies.