A graphical approach to multi-locus match probability computation: revisiting the product rule

The genealogical relationships of individuals in a finite population can create statistical non-independence of alleles at unlinked loci. In this paper, we introduce a flexible graphical method for computing the probabilities that two individuals in a finite, randomly mating population have the same haplotype or genotype at several loci. This method allows us to generalize the analysis of Laurie and Weir [2003. Dependency effects in multi-locus match probabilities. Theor. Popul. Biol. 63, 207-219] to cases with more loci and other models of mating. We show that monogamy increases the probabilities of genotypic matches at unlinked loci and that the effect of monogamy increases with the number L of loci. We conjecture a sharp upper bound on the effect of monogamy for a given L.     

Dependency effects in multi-locus match probabilities

For finite populations, differences in individual histories can cause between-locus allelic dependencies even for unlinked loci. The main motivation for this study is to quantify the effect of such dependencies on genotypic match probabilities. We compare the two-locus match probability, the probability that two individuals (four gametes) chosen at random will have the same genotype at both loci, with the probability computed as the product of the one-locus match probabilities. It is demonstrated that the product rule probability always underestimates the two-locus match probability. For highly mutable minisatellite loci, these probabilities can differ by an order of magnitude or more. A simplified three-locus problem is explored, providing evidence that the degree of under-estimation worsens for more loci.     

A Two-Locus Forensic Match Probability for Subdivided Populations

A two-locus match probability is presented that incorporates the effects of within-subpopulation inbreeding (consanguinity) in addition to population subdivision. The usual practice of calculating multi-locus match probabilities as the product of single-locus probabilities assumes independence between loci. There are a number of population genetics phenomena that can violate this assumption: in addition to consanguinity, which increases homozygosity at all loci simultaneously, gametic disequilibrium will introduce dependence into DNA profiles. However, in forensics the latter problem is usually addressed in part by the careful choice of unlinked loci. Hence, as is conventional, we assume gametic equilibrium here, and focus instead on between-locus dependence due to consanguinity. The resulting match probability formulae are an extension of existing methods in the literature, and are shown to be more conservative than these methods in the case of double homozygote matches. For two-locus profiles involving one or more heterozygous genotypes, results are similar to, or smaller than, the existing approaches. This revised version was published online in August 2006 with corrections to the Cover Date.     

Properties of Equilibria in Multi-Locus Genetic Systems

The classical mathematical theory of population genetics considered, for simplicity, almost exclusively one-locus systems. In the last two decades much work has been done on two-locus and, less frequently, multi-locus systems. This research has usually involved investigating properties of systems with given, and usually rather special, fitness parameters. Real genetic fitness systems are undoubtedly multi-locus and seldom will possess simplifying characteristics. One aim of this paper is to study generalized systems where no special assumptions are made about fitness structure, the number of alleles at each locus, the number of loci involved or the recombination structure between loci. A second aim is to consider marginal properties (often one-locus properties) of complex systems: the fact that many observations involve data from only on locus makes this second aim relevant.     

Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

Conditional genotypic probabilities for microsatellite loci

Modern forensic DNA profiles are constructed using microsatellites, short tandem repeats of 2-5 bases. In the absence of genetic data on a crime-specific subpopulation, one tool for evaluating profile evidence is the match probability. The match probability is the conditional probability that a random person would have the profile of interest given that the suspect has it and that these people are different members of the same subpopulation. One issue in evaluating the match probability is population differentiation, which can induce coancestry among subpopulation members. Forensic assessments that ignore coancestry typically overstate the strength of evidence against the suspect. Theory has been developed to account for coancestry; assumptions include a steady-state population and a mutation model in which the allelic state after a mutation event is independent of the prior state. Under these assumptions, the joint allelic probabilities within a subpopulation may be approximated by the moments of a Dirichlet distribution. We investigate the adequacy of this approximation for profiled loci that mutate according to a generalized stepwise model. Simulations suggest that the Dirichlet theory can still overstate the evidence against a suspect with a common microsatellite genotype. However, Dirichlet-based estimators were less biased than the product-rule estimator, which ignores coancestry.     

Inference of population subdivision from the VNTR distributions of New Zealanders

A population sample from people of diverse ethnic origins living in New Zealand serves as a database to test methods for inference of population subdivision. The initial null hypothesis, that the population sample is homogeneous across ethnic groups, is easily rejected by likelihood ratio tests. Beyond this, methods for quantifying subdivision can be based on the probability of drawing alleles identical by descent (F _ST ), probabilities of matching multiple locus genotypes, and occurrence of unique alleles. Population genetic theory makes quantitative predictions about the relation betweenF _ST , population sizes, and rates of migration and mutation. Some VNTR loci have mutation rates of 10^–2 per generation, but, contrary to theory, we find no consistent association between the degree of population subdivision and mutation rate. Quantification of population substructure also allows us to relate the magnitudes of genetic distances between ethnic groups in New Zealand to the colonization history of the country. The data suggests that the closest relatives to the Maori are Polynesians, and that no severe genetic bottleneck occurred when the Maori colonized New Zealand. One of the central points of contention regarding the application of VNTR loci in forensics is the appropriate means for estimating match probabilities. Simulations were performed to test the merits of the product rule in the face of subpopulation heterogeneity. Population heterogeneity results in large differences in estimates of multilocus genotype frequencies depending on which subpopulation is used for reference allele frequencies, but, of greater importance for forensic purposes, no five locus genotype had an expected frequency greater than 10^–6. Although this implies that a match with an innocent individual is unlikely, in a large urban area such chance matches are going to occur.Editor's commentsA side-benefit of the collection of DNA data from human populations is the light it may shed on human evolution. The authors discuss the colonization history of New Zealand in the light of such data. From a forensic viewpoint, too much should not be made of the differences between the major ethnic groups within New Zealand, as the forensic community in that country maintains separate databases for Caucasian, Maori and Pacific Islander (Buckletonet al., 1987). It will be of interest in the future to examine subdivision within these groups, as opposed to within the country as a whole. The authors' comments on testing for independence will need to read along with the findings of Zaykinet al. and Maiste and Weir in this volume. The authors had not seen the Budowleet al. (1994) rebuttal to the paper of Kraneet al. (1992).     

Ages,extinction times,and first passage probabilities for a multiallele diffusion model with irreversible mutation

We derive some new results for diffusion models in population genetics of the “infinite-alleles” type. Assuming present allelic frequencies known, we find expected values for times which may be interpreted either as the age of the kth oldest allele or the time to extinction of the “kth-to-last” allele to be lost. We also find some conditional expectations and probabilities related to the order of extinction or creation of the alleles. Computations illustrate the dependence of the expected times on the mutation rate.     

A method for distinguishing consanguinity and population substructure using multilocus genotype data

We use the patterns of homozygosity at multiple loci to distinguish between excess homozygosity caused by consanguineous mating and that due to undetected population subdivision (the Wahlund effect). Clarification of the underlying causes of excess homozygosity is of practical importance in explaining the occurrence of recessive genetic disorders and in forensic match probability calculations. We calculated a likelihood surface for two parameters: C, the proportion of the population practicing consanguinity, and theta, the genetic correlation due population subdivision. To illustrate the method, we applied it to multilocus genotypic data of two U.K. Asian populations, one practicing a high frequency of cousin marriage, and another in which caste endogamy was suspected. The method was able to successfully distinguish the different patterns of relatedness. The method also returned accurate estimates of C and theta using simulated data sets. We show how our method can be extended to allow for degrees of inbreeding closer than cousin unions, including selfing. With closer inbreeding, the relatedness of recent ancestors beyond the parents becomes an issue.     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.     

The ancestral graph and gene genealogy under frequency-dependent selection.

Minority-advantage frequency-dependent selection has been proposed as the cause for the high level of observed polymorphism in some self/nonself-recognition systems. We present a mathematically rigorous derivation of the ancestral graph for a sample of genes that evolved according to a haploid infinite-alleles model of minority-advantage frequency-dependent selection. In the case of sufficiently weak selection, the gene genealogy can be extracted from the ancestral graph. We demonstrate that the gene genealogy under this model is identical to that obtained for a diploid model with heterozygote advantage. The case of strong selection is exemplified by a one-locus haploid self-incompatibility system; in this context, we investigate the number of alleles that can be maintained in a spatial versus a non-spatial habitat. Finally, we compare gametophytic self-incompatibility to the haploid self-incompatibility model.     

Haldane,Waddington and recombinant inbred lines: extension of their work to any number of genes

In the early 1930s, J. B. S. Haldane and C. H. Waddington collaborated on the consequences of genetic linkage and inbreeding. One elegant mathematical genetics problem solved by them concerns recombinant inbred lines (RILs) produced via repeated self or brother–sister mating. In this classic contribution, Haldane and Waddington derived an analytical formula for the probabilities of 2-locus and 3-locus RIL genotypes. Specifically, the Haldane–Waddington formula gives the recombination rate R in such lines as a simple function of the per generation recombination rate r. Interestingly, for more than 80 years, an extension of this result to four or more loci remained elusive. In 2015, we generalized the Haldane–Waddington self-mating result to any number of loci. Our solution used self-consistent equations of the multi-locus probabilities ‘for an infinite number of generations’ and solved these by simple algebraic operations. In practice, our approach provides a quantum leap in the systems that can be handled: the cases of up to six loci can be solved by hand while a computer program implementing our mathematical formalism tackles up to 20 loci on standard desktop computers.     

Evolving populations with overlapping generations

In this paper, we extend a previously published model of an evolving finite population of multi-locus organisms, where the dynamics of the evolving system are described by the cumulants of the population efficacy distribution. We consider the case of overlapping generations and compare it to the previously studied case where generations are discrete. In the weak selection limit, we can solve the dynamics analytically and show that the changes in population genetic variance due to stochastic effects-genetic drift-is twice as great when generations overlap. The comparison of the dynamics of the two models shows many of the features seen in the comparison, performed by Moran, of simple one-locus genetic models with overlapping and non-overlapping generations. Studying the dynamics of the two models gives some insights into these comparisons.     

Estimation of effective population size and migration rate from one- and two-locus identity measures

Standard methods for inferring demographic parameters from genetic data are based mainly on one-locus theory. However, the association of genes at different loci (e.g., two-locus identity disequilibrium) may also contain some information about demographic parameters of populations. In this article, we define one- and two-locus parameters of population structure as functions of one- and two-locus probabilities for the identity in state of genes. Since these parameters are known functions of demographic parameters in an infinite island model, we develop moment-based estimators of effective population size and immigration rate from one- and two-locus parameters. We evaluate this method through simulation. Although variance and bias may be quite large, increasing the number of loci on which the estimates are derived improves the method. We simulate an infinite allele model and a K allele model of mutation. Bias and variance are smaller with increasing numbers of alleles per locus. This is, to our knowledge, the first attempt of a joint estimation of local effective population size and immigration rate.     

Match probabilities in racially admixed populations.

The calculation of match probabilities is the most contentious issue dividing prosecution and defense experts in the forensic applications of DNA fingerprinting. In particular, defense experts question the applicability of the population genetic laws of Hardy-Weinberg and linkage equilibrium to racially admixed American populations. Linkage equilibrium justifies the product rule for computing match probabilities across loci. The present paper suggests a method of bounding match probabilities that depends on modeling gene descent from ancestral populations to contemporary populations under the assumptions of Hardy-Weinberg and linkage equilibrium only in the ancestral populations. Although these bounds are conservative from the defendant's perspective, they should be small enough in practice to satisfy prosecutors.     

Use of unlinked genetic markers to detect population stratification in association studies.

We examine the issue of population stratification in association-mapping studies. In case-control studies of association, population subdivision or recent admixture of populations can lead to spurious associations between a phenotype and unlinked candidate loci. Using a model of sampling from a structured population, we show that if population stratification exists, it can be detected by use of unlinked marker loci. We show that the case-control-study design, using unrelated control individuals, is a valid approach for association mapping, provided that marker loci unlinked to the candidate locus are included in the study, to test for stratification. We suggest guidelines as to the number of unlinked marker loci to use.     

Prediction of identity by descent probabilities from marker-haplotypes

The prediction of identity by descent (IBD) probabilities is essential for all methods that map quantitative trait loci (QTL). The IBD probabilities may be predicted from marker genotypes and/or pedigree information. Here, a method is presented that predicts IBD probabilities at a given chromosomal location given data on a haplotype of markers spanning that position. The method is based on a simplification of the coalescence process, and assumes that the number of generations since the base population and effective population size is known, although effective size may be estimated from the data. The probability that two gametes are IBD at a particular locus increases as the number of markers surrounding the locus with identical alleles increases. This effect is more pronounced when effective population size is high. Hence as effective population size increases, the IBD probabilities become more sensitive to the marker data which should favour finer scale mapping of the QTL. The IBD probability prediction method was developed for the situation where the pedigree of the animals was unknown (i.e. all information came from the marker genotypes), and the situation where, say T, generations of unknown pedigree are followed by some generations where pedigree and marker genotypes are known.     

genecap: a program for analysis of multilocus genotype data for non‐invasive sampling and capture‐recapture population estimation

We created genecap to facilitate analysis of multilocus genotype data for use in non‐invasive DNA sampling and genetic capture‐recapture studies. genecap is a Microsoft excel macro that uses multilocus genetic data to match samples with identical genotypes, calculate frequency of alleles, identify sample genotypes that differ by one and two alleles, calculate probabilities of identity, and match probabilities for matching samples. genecap allows the user to include background data and samples with missing genotypes for multiple loci. Capture histories for each user‐defined sampling period are output in formats consistent with commonly employed population estimation programs.     

Hitchhiking of Deleterious Alleles and the Cost of Adaptation in Partially Selfing Species

Self-fertilization is generally seen to be disadvantageous in the long term. It increases genetic drift, which subsequently reduces polymorphism and the efficiency of selection, which also challenges adaptation. However, high selfing rates can increase the fixation probability of recessive beneficial mutations, but existing theory has generally not accounted for the effect of linked sites. Here, we analyze a model for the fixation probability of deleterious mutants that hitchhike with selective sweeps in diploid, partially selfing populations. Approximate analytical solutions show that, conditional on the sweep not being lost by drift, higher inbreeding rates increase the fixation probability of the deleterious allele, due to the resulting reduction in polymorphism and effective recombination. When extending the analysis to consider a distribution of deleterious alleles, as well as the average fitness increase after a sweep, we find that beneficial alleles generally need to be more recessive than the previously assumed dominance threshold (h < 1/2) for selfing to be beneficial from one-locus theory. Our results highlight that recombination aiding the efficiency of selection on multiple loci amplifies the fitness benefits of outcrossing over selfing, compared to results obtained from one-locus theory. This effect additionally increases the parameter range under which obligate outcrossing is beneficial over partial selfing.     

Multi-locus inference of population structure: a comparison between single nucleotide polymorphisms and microsatellites

Although growing numbers of single nucleotide polymorphisms (SNPs) and microsatellites (short tandem repeat polymorphisms or STRPs) are used to infer population structure, their relative properties in this context remain poorly understood. SNPs and STRPs mutate differently, suggesting multi-locus genotypes at these loci might differ in ability to detect population structure. Here, we use coalescent simulations to measure the power of sets of SNPs and STRPs to identify population structure. To maximize the applicability of our results to empirical studies, we focus on the popular STRUCTURE analysis and evaluate the role of several biological and practical factors in the detection of population structure. We find that: (1) fewer unlinked STRPs than SNPs are needed to detect structure at recent divergence times <0.3 N_e generations; (2) accurate estimation of the number of populations requires many fewer STRPs than SNPs; (3) for both marker types, declines in power due to modest gene flow (N_em=1.0) are largely negated by increasing marker number; (4) variation in the STRP mutational model affects power modestly; (5) SNP haplotypes (θ=1, no recombination) provide power comparable with STRP loci (θ=10); (6) ascertainment schemes that select highly variable STRP or SNP loci increase power to detect structure, though ascertained data may not be suitable to other inference; and (7) when samples are drawn from an admixed population and one of its parent populations, the reduction in power to detect two populations is greater for STRPs than SNPs. These results should assist the design of multi-locus studies to detect population structure in nature.