Embedding of Markov matrices for d ⩽ 4.

The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions d ⩽ 4 , including a complete and simplified treatment of the case d = 3 , and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any d ⩾ 3 . We review the known case of d = 3 and describe the setting for future work on d = 4 .

Estimating FST and kinship for arbitrary population structures.

FST and kinship are key parameters often estimated in modern population genetics studies in order to quantitatively characterize structure and relatedness. Kinship matrices have also become a fundamental quantity used in genome-wide association studies and heritability estimation. The most frequently-used estimators of FST and kinship are method-of-moments estimators whose accuracies depend strongly on the existence of simple underlying forms of structure, such as the independent subpopulations model of non-overlapping, independently evolving subpopulations. However, modern data sets have revealed that these simple models of structure likely do not hold in many populations, including humans. In this work, we analyze the behavior of these estimators in the presence of arbitrarily-complex population structures, which results in an improved estimation framework specifically designed for arbitrary population structures. After generalizing the definition of FST to arbitrary population structures and establishing a framework for assessing bias and consistency of genome-wide estimators, we calculate the accuracy of existing FST and kinship estimators under arbitrary population structures, characterizing biases and estimation challenges unobserved under their originally-assumed models of structure. We then present our new approach, which consistently estimates kinship and FST when the minimum kinship value in the dataset is estimated consistently. We illustrate our results using simulated genotypes from an admixture model, constructing a one-dimensional geographic scenario that departs nontrivially from the independent subpopulations model. Our simulations reveal the potential for severe biases in estimates of existing approaches that are overcome by our new framework. This work may significantly improve future analyses that rely on accurate kinship and FST estimates.

Nonparametric coalescent inference of mutation spectrum history and demography.

As populations boom and bust, the accumulation of genetic diversity is modulated, encoding histories of living populations in present-day variation. Many methods exist to decode these histories, and all must make strong model assumptions. It is typical to assume that mutations accumulate uniformly across the genome at a constant rate that does not vary between closely related populations. However, recent work shows that mutational processes in human and great ape populations vary across genomic regions and evolve over time. This perturbs the mutation spectrum (relative mutation rates in different local nucleotide contexts). Here, we develop theoretical tools in the framework of Kingman's coalescent to accommodate mutation spectrum dynamics. We present mutation spectrum history inference (mushi), a method to perform nonparametric inference of demographic and mutation spectrum histories from allele frequency data. We use mushi to reconstruct trajectories of effective population size and mutation spectrum divergence between human populations, identify mutation signatures and their dynamics in different human populations, and calibrate the timing of a previously reported mutational pulse in the ancestors of Europeans. We show that mutation spectrum histories can be placed in a well-studied theoretical setting and rigorously inferred from genomic variation data, like other features of evolutionary history.

From Summary Statistics to Gene Trees: Methods for Inferring Positive Selection.

Methods to detect signals of natural selection from genomic data have traditionally emphasized the use of simple summary statistics. Here, we review a new generation of methods that consider combinations of conventional summary statistics and/or richer features derived from inferred gene trees and ancestral recombination graphs (ARGs). We also review recent advances in methods for population genetic simulation and ARG reconstruction. Finally, we describe opportunities for future work on a variety of related topics, including the genetics of speciation, estimation of selection coefficients, and inference of selection on polygenic traits. Together, these emerging methods offer promising new directions in the study of natural selection.

IndiGenomes: a comprehensive resource of genetic variants from over 1000 Indian genomes.

With the advent of next-generation sequencing, large-scale initiatives for mining whole genomes and exomes have been employed to better understand global or population-level genetic architecture. India encompasses more than 17% of the world population with extensive genetic diversity, but is under-represented in the global sequencing datasets. This gave us the impetus to perform and analyze the whole genome sequencing of 1029 healthy Indian individuals under the pilot phase of the 'IndiGen' program. We generated a compendium of 55,898,122 single allelic genetic variants from geographically distinct Indian genomes and calculated the allele frequency, allele count, allele number, along with the number of heterozygous or homozygous individuals. In the present study, these variants were systematically annotated using publicly available population databases and can be accessed through a browsable online database named as 'IndiGenomes' http://clingen.igib.res.in/indigen/. The IndiGenomes database will help clinicians and researchers in exploring the genetic component underlying medical conditions. Till date, this is the most comprehensive genetic variant resource for the Indian population and is made freely available for academic utility. The resource has also been accessed extensively by the worldwide community since it's launch.

A cautionary note on the use of unsupervised machine learning algorithms to characterise malaria parasite population structure from genetic distance matrices.

Genetic surveillance of malaria parasites supports malaria control programmes, treatment guidelines and elimination strategies. Surveillance studies often pose questions about malaria parasite ancestry (e.g. how antimalarial resistance has spread) and employ statistical methods that characterise parasite population structure. Many of the methods used to characterise structure are unsupervised machine learning algorithms which depend on a genetic distance matrix, notably principal coordinates analysis (PCoA) and hierarchical agglomerative clustering (HAC). PCoA and HAC are sensitive to both the definition of genetic distance and algorithmic specification. Importantly, neither algorithm infers malaria parasite ancestry. As such, PCoA and HAC can inform (e.g. via exploratory data visualisation and hypothesis generation), but not answer comprehensively, key questions about malaria parasite ancestry. We illustrate the sensitivity of PCoA and HAC using 393 Plasmodium falciparum whole genome sequences collected from Cambodia and neighbouring regions (where antimalarial resistance has emerged and spread recently) and we provide tentative guidance for the use and interpretation of PCoA and HAC in malaria parasite genetic epidemiology. This guidance includes a call for fully transparent and reproducible analysis pipelines that feature (i) a clearly outlined scientific question; (ii) a clear justification of analytical methods used to answer the scientific question along with discussion of any inferential limitations; (iii) publicly available genetic distance matrices when downstream analyses depend on them; and (iv) sensitivity analyses. To bridge the inferential disconnect between the output of non-inferential unsupervised learning algorithms and the scientific questions of interest, tailor-made statistical models are needed to infer malaria parasite ancestry. In the absence of such models speculative reasoning should feature only as discussion but not as results.

High-throughput Interpretation of Killer-cell Immunoglobulin-like Receptor Short-read Sequencing Data with PING.

The killer-cell immunoglobulin-like receptor (KIR) complex on chromosome 19 encodes receptors that modulate the activity of natural killer cells, and variation in these genes has been linked to infectious and autoimmune disease, as well as having bearing on pregnancy and transplant outcomes. The medical relevance and high variability of KIR genes makes short-read sequencing an attractive technology for interrogating the region, providing a high-throughput, high-fidelity sequencing method that is cost-effective. However, because this gene complex is characterized by extensive nucleotide polymorphism, structural variation including gene fusions and deletions, and a high level of homology between genes, its interrogation at high resolution has been thwarted by bioinformatic challenges, with most studies limited to examining presence or absence of specific genes. Here, we present the PING (Pushing Immunogenetics to the Next Generation) pipeline, which incorporates empirical data, novel alignment strategies and a custom alignment processing workflow to enable high-throughput KIR sequence analysis from short-read data. PING provides KIR gene copy number classification functionality for all KIR genes through use of a comprehensive alignment reference. The gene copy number determined per individual enables an innovative genotype determination workflow using genotype-matched references. Together, these methods address the challenges imposed by the structural complexity and overall homology of the KIR complex. To determine copy number and genotype determination accuracy, we applied PING to European and African validation cohorts and a synthetic dataset. PING demonstrated exceptional copy number determination performance across all datasets and robust genotype determination performance. Finally, an investigation into discordant genotypes for the synthetic dataset provides insight into misaligned reads, advancing our understanding in interpretation of short-read sequencing data in complex genomic regions. PING promises to support a new era of studies of KIR polymorphism, delivering high-resolution KIR genotypes that are highly accurate, enabling high-quality, high-throughput KIR genotyping for disease and population studies.

Dissecting the genetic history of the Roman Catholic populations of West Coast India.

Migration and admixture history of populations have always been curious and an interesting theme. The West Coast of India harbours a rich diversity, bestowing various ethno-linguistic groups, with many of them having well-documented history of migrations. The Roman Catholic is one such distinct group, whose origin was much debated. While some historians and anthropologists relating them to ancient group of Gaud Saraswat Brahmins, others relating them for being members of the Jews Lost Tribes in the first Century migration to India. Historical records suggests that this community was later forcibly converted to Christianity by the Portuguese in Goa during the Sixteenth Century. Till date, no genetic study was done on this group to infer their origin and genetic affinity. Hence, we analysed 110 Roman Catholics from three different locations of West Coast of India including Goa, Kumta and Mangalore using both uniparental and autosomal markers to understand their genetic history. We found that the Roman Catholics have close affinity with the Indo-European linguistic groups, particularly Brahmins. Additionally, we detected genetic signal of Jews in the linkage disequilibrium-based admixture analysis, which was absent in other Indo-European populations, who are inhabited in the same geographical regions. Haplotype-based analysis suggests that the Roman Catholics consist of South Asian-specific ancestry and showed high drift. Ancestry-specific historical population size estimation points to a possible bottleneck around the time of Goan inquisition (fifteenth century). Analysis of the Roman Catholics data along with ancient DNA data of Neolithic and bronze age revealed that the Roman Catholics fits well in a basic model of ancient ancestral composition, typical of most of the Indo-European caste groups of India. Mitochondrial DNA (mtDNA) analysis suggests that most of the Roman Catholics have aboriginal Indian maternal genetic ancestry; while the Y chromosomal DNA analysis indicates high frequency of R1a lineage, which is predominant in groups with higher ancestral North Indian (ANI) component. Therefore, we conclude that the Roman Catholics of Goa, Kumta and Mangalore regions are the remnants of very early lineages of Brahmin community of India, having Indo-Europeans genetic affinity along with cryptic Jewish admixture, which needs to be explored further.

The Moran process on 2-chromatic graphs.

Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where "properly colored" means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.

Human local adaptation of the TRPM8 cold receptor along a latitudinal cline.

Ambient temperature is a critical environmental factor for all living organisms. It was likely an important selective force as modern humans recently colonized temperate and cold Eurasian environments. Nevertheless, as of yet we have limited evidence of local adaptation to ambient temperature in populations from those environments. To shed light on this question, we exploit the fact that humans are a cosmopolitan species that inhabit territories under a wide range of temperatures. Focusing on cold perception-which is central to thermoregulation and survival in cold environments-we show evidence of recent local adaptation on TRPM8. This gene encodes for a cation channel that is, to date, the only temperature receptor known to mediate an endogenous response to moderate cold. The upstream variant rs10166942 shows extreme population differentiation, with frequencies that range from 5% in Nigeria to 88% in Finland (placing this SNP in the 0.02% tail of the FST empirical distribution). When all populations are jointly analyzed, allele frequencies correlate with latitude and temperature beyond what can be explained by shared ancestry and population substructure. Using a Bayesian approach, we infer that the allele originated and evolved neutrally in Africa, while positive selection raised its frequency to different degrees in Eurasian populations, resulting in allele frequencies that follow a latitudinal cline. We infer strong positive selection, in agreement with ancient DNA showing high frequency of the allele in Europe 3,000 to 8,000 years ago. rs10166942 is important phenotypically because its ancestral allele is protective of migraine. This debilitating disorder varies in prevalence across human populations, with highest prevalence in individuals of European descent-precisely the population with the highest frequency of rs10166942 derived allele. We thus hypothesize that local adaptation on previously neutral standing variation may have contributed to the genetic differences that exist in the prevalence of migraine among human populations today.

An invincible memory: what surname analysis tells us about history, health and population medical genetics in the Brazilian Northeast.

Several studies have shown that the Brazilian Northeast is a region with high rates of inbreeding as well as a high incidence of autosomal recessive diseases. The elaboration of public health policies focused on the epidemiological surveillance of congenital anomalies and rare genetic diseases in this region is urgently needed. However, the vast territory, socio-demographic heterogeneity, economic difficulties and low number of professionals with expertise in medical genetics make strategic planning a challenging task. Surnames can be compared to a genetic system with multiple neutral alleles and allow some approximation of population structure. Here, surname analysis of more than 37 million people was combined with health and socio-demographic indicators covering all 1794 municipalities of the nine states of the region. The data distribution showed a heterogeneous spatial pattern (Global Moran Index, GMI = 0.58; p < 0.001), with higher isonymy rates in the east of the region and the highest rates in the Quilombo dos Palmares region - the largest conglomerate of escaped slaves in Latin America. A positive correlation was found between the isonymy index and the frequency of live births with congenital anomalies (r = 0.268; p < 0.001), and the two indicators were spatially correlated (GMI = 0.50; p < 0.001). With this approach, quantitative information on the genetic structure of the Brazilian Northeast population was obtained, which may represent an economical and useful tool for decision-making in the medical field.

Genetic diversity and population structure of the endangered species Paeonia decomposita endemic to China and implications for its conservation.

BACKGROUND: Paeonia decomposita, endemic to China, has important ornamental, medicinal, and economic value and is regarded as an endangered plant. The genetic diversity and population structure have seldom been described. A conservation management plan is not currently available. RESULTS: In the present study, 16 pairs of simple sequence repeat (SSR) primers were used to evaluate the genetic diversity and population structure. A total of 122 alleles were obtained with a mean of 7.625 alleles per locus. The expected heterozygosity (He) varied from 0.043 to 0.901 (mean 0.492) in 16 primers. Moderate genetic diversity (He = 0.405) among populations was revealed, with Danba identified as the center of genetic diversity. Mantel tests revealed a positive correlation between geographic and genetic distance among populations (r = 0.592, P = 0.0001), demonstrating consistency with the isolation by distance model. Analysis of molecular variance (AMOVA) indicated that the principal molecular variance existed within populations (73.48%) rather than among populations (26.52%). Bayesian structure analysis and principal coordinate analysis (PCoA) supported the classification of the populations into three clusters. CONCLUSIONS: This is the first study of the genetic diversity and population structure of P. decomposita using SSR. Three management units were proposed as conservation measures. The results will be beneficial for the conservation and exploitation of the species, providing a theoretical basis for further research of its evolution and phylogeography.

Putting RFMix and ADMIXTURE to the test in a complex admixed population.

BACKGROUND: Global and local ancestry inference in admixed human populations can be performed using computational tools implementing distinct algorithms. The development and resulting accuracy of these tools has been tested largely on populations with relatively straightforward admixture histories but little is known about how well they perform in more complex admixture scenarios. RESULTS: Using simulations, we show that RFMix outperforms ADMIXTURE in determining global ancestry proportions even in a complex 5-way admixed population, in addition to assigning local ancestry with an accuracy of 89%. The ability of RFMix to determine global and local ancestry to a high degree of accuracy, particularly in admixed populations provides the opportunity for more accurate association analyses. CONCLUSION: This study highlights the utility of the extension of computational tools to become more compatible to genetically structured populations, as well as the need to expand the sampling of diverse world-wide populations. This is particularly noteworthy as modern-day societies are becoming increasingly genetically complex and some genetic tools and commonly used ancestral populations are less appropriate. Based on these caveats and the results presented here, we suggest that RFMix be used for both global and local ancestry estimation in world-wide complex admixture scenarios particularly when including these estimates in association studies.

On rare variants in principal component analysis of population stratification.

BACKGROUND: Population stratification is a known confounder of genome-wide association studies, as it can lead to false positive results. Principal component analysis (PCA) method is widely applied in the analysis of population structure with common variants. However, it is still unclear about the analysis performance when rare variants are used. RESULTS: We derive a mathematical expectation of the genetic relationship matrix. Variance and covariance elements of the expected matrix depend explicitly on allele frequencies of the genetic markers used in the PCA analysis. We show that inter-population variance is solely contained in K principal components (PCs) and mostly in the largest K-1 PCs, where K is the number of populations in the samples. We propose FPC, ratio of the inter-population variance to the intra-population variance in the K population informative PCs, and d2, sum of squared distances among populations, as measures of population divergence. We show analytically that when allele frequencies become small, the ratio FPC abates, the population distance d2 decreases, and portion of variance explained by the K PCs diminishes. The results are validated in the analysis of the 1000 Genomes Project data. The ratio FPC is 93.85, population distance d2 is 444.38, and variance explained by the largest five PCs is 17.09% when using with common variants with allele frequencies between 0.4 and 0.5. However, the ratio, distance and percentage decrease to 1.83, 17.83 and 0.74%, respectively, with rare variants of frequencies between 0.0001 and 0.01. CONCLUSIONS: The PCA of population stratification performs worse with rare variants than with common ones. It is necessary to restrict the selection to only the common variants when analyzing population stratification with sequencing data.

Motion, fixation probability and the choice of an evolutionary process.

Seemingly minor details of mathematical and computational models of evolution are known to change the effect of population structure on the outcome of evolutionary processes. For example, birth-death dynamics often result in amplification of selection, while death-birth processes have been associated with suppression. In many biological populations the interaction structure is not static. Instead, members of the population are in motion and can interact with different individuals at different times. In this work we study populations embedded in a flowing medium; the interaction network is then time dependent. We use computer simulations to investigate how this dynamic structure affects the success of invading mutants, and compare these effects for different coupled birth and death processes. Specifically, we show how the speed of the motion impacts the fixation probability of an invading mutant. Flows of different speeds interpolate between evolutionary dynamics on fixed heterogeneous graphs and well-stirred populations; this allows us to systematically compare against known results for static structured populations. We find that motion has an active role in amplifying or suppressing selection by fragmenting and reconnecting the interaction graph. While increasing flow speeds suppress selection for most evolutionary models, we identify characteristic responses to flow for the different update rules we test. In particular we find that selection can be maximally enhanced or suppressed at intermediate flow speeds.

Inside Dynamics of Integrodifference Equations with Mutations.

The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.

Exact limits of inference in coalescent models.

Recovery of population size history from molecular sequence data is an important problem in population genetics. Inference commonly relies on a coalescent model linking the population size history to genealogies. The high computational cost of estimating parameters from these models usually compels researchers to select a subset of the available data or to rely on insufficient summary statistics for statistical inference. We consider the problem of recovering the true population size history from two possible alternatives on the basis of coalescent time data previously considered by Kim et al. (2015). We improve upon previous results by giving exact expressions for the probability of correctly distinguishing between the two hypotheses as a function of the separation between the alternative size histories, the number of individuals, loci, and the sampling times. In more complicated settings we estimate the exact probability of correct recovery by Monte Carlo simulation. Our results give considerably more pessimistic inferential limits than those previously reported. We also extended our analyses to pairwise SMC and SMC' models of recombination. This work is relevant for optimal design when the inference goal is to test scientific hypotheses about population size trajectories in coalescent models with and without recombination.

General theory for stochastic admixture graphs and F-statistics.

We provide a general mathematical framework based on the theory of graphical models to study admixture graphs. Admixture graphs are used to describe the ancestral relationships between past and present populations, allowing for population merges and migration events, by means of gene flow. We give various mathematical properties of admixture graphs with particular focus on properties of the so-called F-statistics. Also the Wright-Fisher model is studied and a general expression for the loss of heterozygosity is derived.

Spatial evolutionary dynamics produce a negative cooperation-population size relationship.

Natural selection can favour cooperation, but it is unclear when cooperative populations should be larger than less cooperative ones. While experiments have shown that cooperation can increase population size, cooperation and population size can become negatively correlated if spatial processes affect both variables in opposite directions. We use a simple mathematical model of spatial common-pool resource production to investigate how space affects the cooperation-population size relationship. We find that only cooperation that is sufficiently beneficial to neighbours increases population size. However, spatial clustering variations can create a negative cooperation-population relationship between populations even when cooperation is highly beneficial, because clustering selects for cooperation but decreases population size. Individual-based simulations with variable individual movement rates produced variation in spatial clustering and the hypothesized negative cooperation-population relationships. These results suggest that variation in spatial clustering can limit the size of evolutionarily stable cooperating populations - an ecological dilemma of cooperation.

In defence of doing sums in genetics.

There has been a long history of the use of mathematics in genetics, ranging from the use of statistics to analyse genetic data to genetic models of evolutionary processes. Contemporary research into the genomic basis of disease and complex traits exemplifies the importance of statistical methods in genetics. Some examples of the development and application of population genetic models are described, which are intended to highlight the utility of such models for understanding variation and evolution in natural populations. The effects of selection on variability at sites linked to the targets of selection illustrate how fruitful interactions between theory and data can be.