Selective sweep probabilities in spatially expanding populations.

Evolution during range expansions is an important feature of many biological systems including tumours, microbial communities, and invasive species. A selective sweep is a fundamental process, in which an advantageous mutation evades clonal interference and spreads through the population to fixation. However, most theoretical investigations of selective sweeps have assumed constant population size or have ignored spatial structure. Here we use mathematical modelling and analysis to investigate selective sweep probabilities in populations that grow with constant radial expansion speed. We derive probability distributions for the arrival time and location of the first surviving mutant and hence find simple approximate and exact expressions for selective sweep probabilities in one, two and three dimensions, which are independent of mutation rate. Namely, the selective sweep probability is approximately (1 -c wt / c m ) d , where c wt and c m are the wildtype and mutant radial expansion speeds, and d is the spatial dimension. Using agent-based simulations, we show that our analytical results accurately predict selective sweep frequencies in the two-dimensional spatial Moran process. We further compare our results with those obtained for alternative growth laws. Parameterizing our model for human tumours, we find that selective sweeps are predicted to be rare except during very early solid tumour growth, thus providing a general, pan-cancer explanation for findings from recent sequencing studies.