DCA for genome-wide epistasis analysis: the statistical genetics perspective

Statistical genetics in and out of quasi-linkage equilibrium

This review is about statistical genetics, an interdisciplinary topic between statistical physics and population biology. The focus is on the phase ofquasi-linkage equilibrium(QLE). Our goals here are to clarify under which conditions the QLE phase can be expected to hold in population biology and how the stability of the QLE phase is lost. The QLE state, which has many similarities to a thermal equilibrium state in statistical mechanics, was discovered by M Kimura for a two-locus two-allele model, and was extended and generalized to the global genome scale byNeher&Shraiman (2011). What we will refer to as the Kimura-Neher-Shraiman theory describes a population evolving due to the mutations, recombination, natural selection and possibly genetic drift. A QLE phase exists at sufficiently high recombination rate (r) and/or mutation ratesµwith respect to selection strength. We show how in QLE it is possible to infer the epistatic parameters of the fitness function from the knowledge of the (dynamical) distribution of genotypes in a population. We further consider the breakdown of the QLE regime for high enough selection strength. We review recent results for the selection-mutation and selection-recombination dynamics. Finally, we identify and characterize a new phase which we call the non-random coexistence where variability persists in the population without either fixating or disappearing.

Coordinated evolution at amino acid sites of SARS-CoV-2 spike

SARS-CoV-2 has adapted in a stepwise manner, with multiple beneficial mutations accumulating in a rapid succession at origins of VOCs, and the reasons for this are unclear. Here, we searched for coordinated evolution of amino acid sites in the spike protein of SARS-CoV-2. Specifically, we searched for concordantly evolving site pairs (CSPs) for which changes at one site were rapidly followed by changes at the other site in the same lineage. We detected 46 sites which formed 45 CSP. Sites in CSP were closer to each other in the protein structure than random pairs, indicating that concordant evolution has a functional basis. Notably, site pairs carrying lineage defining mutations of the four VOCs that circulated before May 2021 are enriched in CSPs. For the Alpha VOC, the enrichment is detected even if Alpha sequences are removed from analysis, indicating that VOC origin could have been facilitated by positive epistasis. Additionally, we detected nine discordantly evolving pairs of sites where mutations at one site unexpectedly rarely occurred on the background of a specific allele at another site, for example on the background of wild-type D at site 614 (four pairs) or derived Y at site 501 (three pairs). Our findings hint that positive epistasis between accumulating mutations could have delayed the assembly of advantageous combinations of mutations comprising at least some of the VOCs.

Temporal epistasis inference from more than 3 500 000 SARS-CoV-2 genomic sequences

We use direct coupling analysis (DCA) to determine epistatic interactions between loci of variability of the SARS-CoV-2 virus, segmenting genomes by month of sampling. We use full-length, high-quality genomes from the GISAID repository up to October 2021 for a total of over 3 500 000 genomes. We find that DCA terms are more stable over time than correlations but nevertheless change over time as mutations disappear from the global population or reach fixation. Correlations are enriched for phylogenetic effects, and in particularly statistical dependencies at short genomic distances, while DCA brings out links at longer genomic distance. We discuss the validity of a DCA analysis under these conditions in terms of a transient auasilinkage equilibrium state. We identify putative epistatic interaction mutations involving loci in spike.

Epistasis Creates Invariant Sites and Modulates the Rate of Molecular Evolution

Invariant sites are a common feature of amino acid sequence evolution. The presence of invariant sites is frequently attributed to the need to preserve function through site-specific conservation of amino acid residues. Amino acid substitution models without a provision for invariant sites often fit the data significantly worse than those that allow for an excess of invariant sites beyond those predicted by models that only incorporate rate variation among sites (e.g., a Gamma distribution). An alternative is epistasis between sites to preserve residue interactions that can create invariant sites. Through computer-simulated sequence evolution, we evaluated the relative effects of site-specific preferences and site-site couplings in the generation of invariant sites and the modulation of the rate of molecular evolution. In an analysis of ten major families of protein domains with diverse sequence and functional properties, we find that the negative selection imposed by epistasis creates many more invariant sites than site-specific residue preferences alone. Further, epistasis plays an increasingly larger role in creating invariant sites over longer evolutionary periods. Epistasis also dictates rates of domain evolution over time by exerting significant additional purifying selection to preserve site couplings. These patterns illuminate the mechanistic role of epistasis in the processes underlying observed site invariance and evolutionary rates.

Inferring genetic fitness from genomic data

The genetic composition of a naturally developing population is considered as due to mutation, selection, genetic drift, and recombination. Selection is modeled as single-locus terms (additive fitness) and two-loci terms (pairwise epistatic fitness). The problem is posed to infer epistatic fitness from population-wide whole-genome data from a time series of a developing population. We generate such data in silico and show that in the quasilinkage equilibrium phase of Kimura, Neher, and Shraiman, which pertains at high enough recombination rates and low enough mutation rates, epistatic fitness can be quantitatively correctly inferred using inverse Ising-Potts methods.

A dual process for the coupled Wright-Fisher diffusion

The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

Global analysis of more than 50,000 SARS-CoV-2 genomes reveals epistasis between eight viral genes

Genome-wide epistasis analysis is a powerful tool to infer gene interactions, which can guide drug and vaccine development and lead to deeper understanding of microbial pathogenesis. We have considered all complete severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) genomes deposited in the Global Initiative on Sharing All Influenza Data (GISAID) repository until four different cutoff dates, and used direct coupling analysis together with an assumption of quasi-linkage equilibrium to infer epistatic contributions to fitness from polymorphic loci. We find eight interactions, of which three are between pairs where one locus lies in gene ORF3a, both loci holding nonsynonymous mutations. We also find interactions between two loci in gene nsp13, both holding nonsynonymous mutations, and four interactions involving one locus holding a synonymous mutation. Altogether, we infer interactions between loci in viral genes ORF3a and nsp2, nsp12, and nsp6, between ORF8 and nsp4, and between loci in genes nsp2, nsp13, and nsp14. The paper opens the prospect to use prominent epistatically linked pairs as a starting point to search for combinatorial weaknesses of recombinant viral pathogens.

Epistatic contributions promote the unification of incompatible models of neutral molecular evolution

Mathematical models of evolution help us understand mechanisms driving protein-sequence change. Previous models recapitulate a disjoint subset of statistical features of natural sequences. We present a neutral evolution model that unifies features including extreme variance of the molecular clock’s tick rate and the observation of an evolutionary Stokes shift, an irreversible effect of mutations in the fitness landscape during sequence evolution. We show that interactions between amino acid sites, which inform our fitness metric, are required to observe these features. These interactions are inferred by using direct coupling analysis, which has been successfully utilized to predict protein structures, dynamics, and complexes from coevolutionary information. We anticipate our model will have applications in phylogenetics, ancestral reconstruction of sequences, and protein design.

We introduce a model of amino acid sequence evolution that accounts for the statistical behavior of real sequences induced by epistatic interactions. We base the model dynamics on parameters derived from multiple sequence alignments analyzed by using direct coupling analysis methodology. Known statistical properties such as overdispersion, heterotachy, and gamma-distributed rate-across-sites are shown to be emergent properties of this model while being consistent with neutral evolution theory, thereby unifying observations from previously disjointed evolutionary models of sequences. The relationship between site restriction and heterotachy is characterized by tracking the effective alphabet dynamics of sites. We also observe an evolutionary Stokes shift in the fitness of sequences that have undergone evolution under our simulation. By analyzing the structural information of some proteins, we corroborate that the strongest Stokes shifts derive from sites that physically interact in networks near biochemically important regions. Perspectives on the implementation of our model in the context of the molecular clock are discussed.

Genome-wide epistasis and co-selection study using mutual information

Covariance-based discovery of polymorphisms under co-selective pressure or epistasis has received considerable recent attention in population genomics. Both statistical modeling of the population level covariation of alleles across the chromosome and model-free testing of dependencies between pairs of polymorphisms have been shown to successfully uncover patterns of selection in bacterial populations. Here we introduce a model-free method, SpydrPick, whose computational efficiency enables analysis at the scale of pan-genomes of many bacteria. SpydrPick incorporates an efficient correction for population structure, which adjusts for the phylogenetic signal in the data without requiring an explicit phylogenetic tree. We also introduce a new type of visualization of the results similar to the Manhattan plots used in genome-wide association studies, which enables rapid exploration of the identified signals of co-evolution. Simulations demonstrate the usefulness of our method and give some insight to when this type of analysis is most likely to be successful. Application of the method to large population genomic datasets of two major human pathogens, Streptococcus pneumoniae and Neisseria meningitidis, revealed both previously identified and novel putative targets of co-selection related to virulence and antibiotic resistance, highlighting the potential of this approach to drive molecular discoveries, even in the absence of phenotypic data.