The Spatial Mixing of Genomes in Secondary Contact Zones

Abstract

history of admixture produces many smaller parental tracts. We assume that population differentiation within the parental populations is weak relative to that between them and so consider only admixture LD (and not mixture LD that is covariance between loci induced by substructure) in the focal populations. Data from many (potentially weakly) differentiated markers allow for the identification and quantification of admixture in individuals (e.g., Pritchard et al. 2000) and the inference of the ancestral origin of a given chromosomal region (e.g., Falush et al. 2003; Price et al. 2009; Hellenthal et al. 2014). The continued mixing of differentiated genotypes, as described above, produces predictable population genomic patterns that change through time, and these signals can be used to not only detect past admixture in an extant population, but also learn about the timing and history of these admixture events (e.g., Harris and Nielsen 2013; Loh et al. 2013; Hellenthal et al. 2014). Such inferences have been used to reconstruct historical population movements, highlighting the importance of admixture in shaping patterns of diversity in human populations (Reich et al. 2009; Patterson et al. 2012; Loh et al. 2013; Moorjani et al. 2013; Hellenthal et al. 2014). These studies have utilized powerful methods that first identify stretches of chromosome inherited from a particular parental population [admixture tracts (Gravel 2012; Hellenthal et al. 2014)] or measure the covariance, over spatial scales, of variants that are diagnostic of parental populations [admixture LD (Patterson et al. 2012; Loh et al. 2013)] and then infer the genetic scale over which this measured coancestry decays. Commonly this is done by assuming a model of admixture in which one isolated population is formed by a single admixture event in time, with subsequent random mating. Under this simple model, the distribution of admixture tract lengths and the decay of admixture LD with respect to genetic distance are approximately exponential, with the rate parameter corresponding to the time in generations since admixture. However, violations of the assumptions of the single-pulse model can result in substantial departure between expected and observed rates of decay of coancestry with respect to time. Models incorporating multiple admixture times, or sustained migration (Pool and Nielsen 2009; Gravel 2012; Hellenthal et al. 2014; Liang and Nielsen 2014b), have been built to address more complex admixture scenarios in single populations. However, these do not incorporate the fact that admixture often occurs in a geographic context—beginning at a given point in time, then spreading across space. Most current models treat each admixed population as an independent event, not accounting for this spatial context, even when admixture in spatially distributed populations is potentially attributable to a single historical event. In this article we build an alternative model of diffusion of ancestry across geography in time. Specifically, we consider a scenario in which two populations spread back into contact, generating a gradient of admixture across space with the greatest variance in ancestry at the point of initial contact. We refer to this mixture across space, where migration is