Spatial gene drives and pushed genetic waves.

Gene drives have the potential to rapidly replace a harmful wild-type allele with a gene drive allele engineered to have desired functionalities. However, an accidental or premature release of a gene drive construct to the natural environment could damage an ecosystem irreversibly. Thus, it is important to understand the spatiotemporal consequences of the super-Mendelian population genetics before potential applications. Here, we use a reaction-diffusion model for sexually reproducing diploid organisms to study how a locally introduced gene drive allele spreads to replace the wild-type allele, although it possesses a selective disadvantage s > 0. Using methods developed by Barton and collaborators, we show that socially responsible gene drives require 0.5 < s < 0.697, a rather narrow range. In this "pushed wave" regime, the spatial spreading of gene drives will be initiated only when the initial frequency distribution is above a threshold profile called "critical propagule," which acts as a safeguard against accidental release. We also study how the spatial spread of the pushed wave can be stopped by making gene drives uniquely vulnerable ("sensitizing drive") in a way that is harmless for a wild-type allele. Finally, we show that appropriately sensitized drives in two dimensions can be stopped, even by imperfect barriers perforated by a series of gaps.