Na Li and Matthew Stephens on Modeling Linkage Disequilibrium

Probabilistic models have played an indispensable role in population genetics for close to a century. They provide a powerful lens through which one can investigate how various evolutionary forces interact and produce the intricate patterns of genetic variation in a population. Furthermore, through statistical inference, one can estimate evolutionary parameters from population genetic data and draw important biological conclusions. There is a formidable challenge in this effort, however. Although the models (e.g., diffusion processes and coalescent theory) are relatively straightforward to describe, computing the relevant likelihoods for statistical inference is often intractable. This is especially true when recombination is taken into account. Indeed, a far-reaching consequence of recombination is that different loci can have different evolutionary histories, and this complication leads to an overwhelming explosion in the dimensionality of the model.

In their influential work, Li and Stephens (2003) proposed a simple and elegant approach to circumvent this computational challenge, leading to a paradigm shift in modeling genetic relatedness with large-scale data. Their key methodological insight was to construct an approximate probabilistic model that captures the essential features of a genealogical process with recombination but produces a dramatic computational speed-up. Specifically, they considered the problem of approximating the conditional sampling probability (CSP) of the next haplotype given the haplotypes that have already been observed. A useful approximation had been proposed by Stephens and Donnelly (2000) for the simpler case of completely linked loci. They suggested approximating the next haplotype $h_k$ as an imperfect copy of one of the first $k-1$ haplotypes, $h_1,\dots ,h_{k-1}$, with copying errors corresponding to mutation. Fearnhead and Donnelly (2001) generalized this approach to incorporate recombination, assuming that haplotype $h_k$ is generated by copying segments from $h_1,\dots ,h_{k-1}$, where recombination can change the haplotype from which copying is performed. The associated CSP can be computed efficiently using standard methods (dynamic programming applied to a hidden Markov model). Li and Stephens (2003) proposed a modification to the Fearnhead and Donnelly approximation to obtain a simpler generative model, thus providing a computational speed-up. More important, they showed how the CSPs could be combined to approximate the likelihood of the haplotype data under a model that allowed recombination rates to vary over short distances. This permitted effective inference of the fine structure of recombination rate variation from population genomic data. This clever approach opened up new avenues of statistical inference in population genetics.

The modeling framework proposed by Li and Stephens has had a profound impact. Their “copying” model has been extended and applied to a wide range of problems, including inference of gene conversion parameters (Gay et al., 2007; Yin et al., 2009), recombination rates in admixed populations (Hinch et al., 2011; Wegmann et al., 2011), human colonization history (Hellenthal et al., 2008), fine population structure (Lawson et al., 2012), and local ancestry in admixed populations (Sundquist et al., 2008; Price et al., 2009). The model has also been used to phase genotype sequence data into haplotypes and impute missing data to improve the power of genome-wide association studies (Stephens and Scheet 2005; Marchini et al., 2007; Howie et al., 2009; Li et al., 2010).

In addition to these impressive applications, the work of Li and Stephens has also fueled theoretical research on deriving improved approximations for CSPs directly from the underlying population genetic models (Griffiths et al., 2008; Paul and Song 2010). This research has led to genealogically interpretable approximations (Paul et al., 2011) that can be applied to more complex models, allowing inference of population demographic histories (Sheehan et al., 2013; Steinrücken et al., 2013, 2015), including variable population sizes, divergence times, and gene flow between populations. A recent extension (Rasmussen et al., 2014) shows how related ideas can be used to develop a Monte Carlo algorithm to sample genealogical histories, which have applications in association mapping and detecting signatures of natural selection.

In summary, the paper by Li and Stephens contains groundbreaking work employing biologically motivated approximations to allow efficient statistical inference from genomic data using informative models. Their innovative modeling approach has facilitated the development of numerous useful analytical tools that scale up to whole genomes while capturing important features of realistic population genetic models. It is a must-read for anyone interested in developing inference methods in population genetics and computational biology.