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.     

Match probabilities in a finite, subdivided population

We generalize a recently introduced graphical framework to compute the probability that haplotypes or genotypes of two individuals drawn from a finite, subdivided population match. As in the previous work, we assume an infinite-alleles model. We focus on the case of a population divided into two subpopulations, but the underlying framework can be applied to a general model of population subdivision. We examine the effect of population subdivision on the match probabilities and the accuracy of the product rule which approximates multi-locus match probabilities as a product of one-locus match probabilities. We quantify the deviation from predictions of the product rule by R, the ratio of the multi-locus match probability to the product of the one-locus match probabilities. We carry out the computation for two loci and find that ignoring subdivision can lead to underestimation of the match probabilities if the population under consideration actually has subdivision structure and the individuals originate from the same subpopulation. On the other hand, under a given model of population subdivision, we find that the ratio R for two loci is only slightly greater than 1 for a large range of symmetric and asymmetric migration rates. Keeping in mind that the infinite-alleles model is not the appropriate mutation model for STR loci, we conclude that, for two loci and biologically reasonable parameter values, population subdivision may lead to results that disfavor innocent suspects because of an increase in identity-by-descent in finite populations. On the other hand, for the same range of parameters, population subdivision does not lead to a substantial increase in linkage disequilibrium between loci. Those results are consistent with established practice.     

A complete enumeration and classification of two-locus disease models

There are 512 two-locus, two-allele, two-phenotype, fully penetrant disease models. Using the permutation between two alleles, between two loci, and between being affected and unaffected, one model can be considered to be equivalent to another model under the corresponding permutation. These permutations greatly reduce the number of two-locus models in the analysis of complex diseases. This paper determines the number of nonredundant two-locus models (which can be 102, 100, 96, 51, 50, or 58, depending on which permutations are used, and depending on whether zero-locus and single-locus models are excluded). Whenever possible, these nonredundant two-locus models are classified by their property. Besides the familiar features of multiplicative models (logical AND), heterogeneity models (logical OR), and threshold models, new classifications are added or expanded: modifying-effect models, logical XOR models, interference and negative interference models (neither dominant nor recessive), conditionally dominant/recessive models, missing lethal genotype models, and highly symmetric models. The following aspects of two-locus models are studied: the marginal penetrance tables at both loci, the expected joint identity-by-descent (IBD) probabilities, and the correlation between marginal IBD probabilities at the two loci. These studies are useful for linkage analyses using single-locus models while the underlying disease model is two-locus, and for correlation analyses using the linkage signals at different locations obtained by a single-locus model.     

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.     

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.     

Protein local structure prediction from sequence

A basis set of protein canonical fragments, or centroids, represents the range of local structure found in globular proteins. We develop a methodology to predict centroids from the amino acid sequence. The predictor gives the probability of each centroid in the basis set, at each loci along the backbone. The predictor selects the best-fit centroid at about 40% of the loci. The predicted probabilities are accurate and can be used to judge the confidence of each centroid prediction. For example, when filtering out centroids with <0.50 probability, the predictor is 65% accurate, although such high-probability centroids occur at only 28% of the loci. Centroids with high probability can be interpreted as segments that are highly influenced by the amino acid sequence, whereas centroids with low probability can be interpreted as segments that are more likely influenced by tertiary contacts. Low-resolution, starting point structures, can be generated by fitting the predicted centroids together.     

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.     

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.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

Prediction of multilocus identity-by-descent

Previous studies have enabled exact prediction of probabilities of identity-by-descent (IBD) in random-mating populations for a few loci (up to four or so), with extension to more using approximate regression methods. Here we present a precise predictor of multiple-locus IBD using simple formulas based on exact results for two loci. In particular, the probability of non-IBD X(ABC) at each of ordered loci A, B, and C can be well approximated by X(ABC) = X(AB)X(BC)/X(B) and generalizes to X(123...k) = X(12)X(23...)X(k)(-1,k)/X(k-2), where X is the probability of non-IBD at each locus. Predictions from this chain rule are very precise with population bottlenecks and migration, but are rather poorer in the presence of mutation. From these coefficients, the probabilities of multilocus IBD and non-IBD can also be computed for genomic regions as functions of population size, time, and map distances. An approximate but simple recurrence formula is also developed, which generally is less accurate than the chain rule but is more robust with mutation. Used together with the chain rule it leads to explicit equations for non-IBD in a region. The results can be applied to detection of quantitative trait loci (QTL) by computing the probability of IBD at candidate loci in terms of identity-by-state at neighboring markers.     

How to model a complex trait. 2. Analysis with two disease loci

Complex traits are often governed by more than one trait locus. The first step towards an adequate model for such diseases is a linkage analysis with two trait loci. Such an analysis can be expected to have higher power to detect linkage than a standard single-trait-locus linkage analysis. However, it is crucial to accurately specify the parameters of the two-locus model. Here, we recapitulate the general two-locus model with and without genomic imprinting. We relate heterogeneity, multiplicative, and additive two-locus models to biological or pathophysiological mechanisms, and give the corresponding averaged ("best-fitting") single-trait-locus models for each of the two loci. Furthermore, we derive the two-locus penetrances from the averaged single-locus models, under the assumption of one of the three model classes mentioned above. Using these formulae, if the best-fitting single-locus models are available, investigators may perform a two-trait-locus linkage analysis under a realistic model. This procedure will maximize the power to detect linkage for traits which are governed by two or more loci, and lead to more accurate estimates of the disease-locus positions.     

Probability of matching RFLP patterns from unrelated individuals.

RFLP patterns from more than 3,700 individuals, collected by several southeastern laboratories, were compared using a series of mathematical match criteria to determine the probability of randomly matching two samples from different individuals. Data from five loci (D1S7, D2S44, D4S139, D10S28, and D17S79) were compared in all possible combinations. The probability of declaring a false match under any of the match criteria decreased by at least one order of magnitude for each locus added to the comparison. No false matches were declared across four or more loci when match criteria approximately equivalent to that of forensic laboratories were used.     

Stepping-stone spatial structure causes slow decay of linkage disequilibrium and shifts the site frequency spectrum

The symmetric island model with D demes and equal migration rates is often chosen for the investigation of the consequences of population subdivision. Here we show that a stepping-stone model has a more pronounced effect on the genealogy of a sample. For samples from a small geographical region commonly used in genetic studies of humans and Drosophila, there is a shift of the frequency spectrum that decreases the number of low-frequency-derived alleles and skews the distribution of statistics of Tajima, Fu and Li, and Fay and Wu. Stepping-stone spatial structure also changes the two-locus sampling distribution and increases both linkage disequilibrium and the probability that two sites are perfectly correlated. This may cause a false prediction of cold spots of recombination and may confuse haplotype tests that compute probabilities on the basis of a homogeneously mixing population.     

Accounting for non-independence of extinction probabilities in the derivation of conservation priorities based on Weitzman’s diversity concept

In the original formulation of Weitzman’s diversity concept, extinction probabilities of different operational taxonomic units (OTUs) are implicitly defined to be independent. In this study, it is shown, that dependencies like concurrence or synergistic relations between OTUs can be accounted for by assuming that joint extinction probabilities differ from the product of the extinction probabilities of the interacting OTUs. A fully analytical treatment is provided for the case of two interacting OTUs, and all cases are illustrated with an example data set. For the case of concurring OTUs, the following general results are derived: (a) concurrence between OTUs always reduces the amount of diversity expected to be conserved in the future; (b) concurrence has a more adverse effect on the more endangered OTUs involved; (c) concurrence can change conservation priorities between OTUs; and (d) with high levels of concurrence, investments in the conservation of some of the concurring OTUs can have a negative effect on the overall diversity conserved. In addition to conservation activities targeted towards some of the OTUs, reduction of the causes of concurrence may be a valid and cost-efficient alternative. For the case of synergistic dependencies between OTUs it was found, that such dependencies always increase the conservation potential of the involved OTUs but may lead to a change of priority ranking in extreme cases. Finally possible extensions and generalisations of the suggested approach are discussed and it is argued, that by adopting these extensions Weitzman’s diversity concept becomes an even more powerful and flexible tool to derive conservation priorities both in an ecological and in a farm animal context.     

DNA fingerprinting for forensic identification: potential effects on data interpretation of subpopulation heterogeneity and band number variability

Some methods of statistical analysis of data on DNA fingerprinting suffer serious weaknesses. Unlinked Mendelizing loci that are at linkage equilibrium in subpopulations may be statistically associated, not statistically independent, in the population as a whole if there is heterogeneity in gene frequencies between subpopulations. In the populations where DNA fingerprinting is used for forensic applications, the assumption that DNA fragments occur statistically independently for different probes, different loci, or different fragment size classes lacks supporting data so far; there is some contrary evidence. Statistical association of alleles may cause estimates based on the assumption of statistical independence to understate the true matching probabilities by many orders of magnitude. The assumptions that DNA fragments occur independently and with constant frequency within a size class appear to be contradicted by the available data on the mean and variance of the number of fragments per person. The mistaken use of the geometric mean instead of the arithmetic mean to compute the probability that every DNA fragment of a randomly chosen person is present among the DNA fragments of a specimen may substantially understate the probability of a match between blots, even if other assumptions involved in the calculations are taken as correct. The conclusion is that some astronomically small probabilities of matching by chance, which have been claimed in forensic applications of DNA fingerprinting, presently lack substantial empirical and theoretical support.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

Two-stage two-locus models in genome-wide association

Studies in model organisms suggest that epistasis may play an important role in the etiology of complex diseases and traits in humans. With the era of large-scale genome-wide association studies fast approaching, it is important to quantify whether it will be possible to detect interacting loci using realistic sample sizes in humans and to what extent undetected epistasis will adversely affect power to detect association when single-locus approaches are employed. We therefore investigated the power to detect association for an extensive range of two-locus quantitative trait models that incorporated varying degrees of epistasis. We compared the power to detect association using a single-locus model that ignored interaction effects, a full two-locus model that allowed for interactions, and, most important, two two-stage strategies whereby a subset of loci initially identified using single-locus tests were analyzed using the full two-locus model. Despite the penalty introduced by multiple testing, fitting the full two-locus model performed better than single-locus tests for many of the situations considered, particularly when compared with attempts to detect both individual loci. Using a two-stage strategy reduced the computational burden associated with performing an exhaustive two-locus search across the genome but was not as powerful as the exhaustive search when loci interacted. Two-stage approaches also increased the risk of missing interacting loci that contributed little effect at the margins. Based on our extensive simulations, our results suggest that an exhaustive search involving all pairwise combinations of markers across the genome might provide a useful complement to single-locus scans in identifying interacting loci that contribute to moderate proportions of the phenotypic variance.     

Linkage analysis of a large African family segregating stuttering suggests polygenic inheritance

We describe a pedigree of 71 individuals from the Republic of Cameroon in which at least 33 individuals have a clinical diagnosis of persistent stuttering. The high concentration of stuttering individuals suggests that the pedigree either contains a single highly penetrant gene variant or that assortative mating led to multiple stuttering-associated variants being transmitted in different parts of the pedigree. No single locus displayed significant linkage to stuttering in initial genome-wide scans with microsatellite and SNP markers. By dividing the pedigree into five subpedigrees, we found evidence for linkage to previously reported loci on 3q and 15q, and to novel loci on 2p, 3p, 14q, and a different region of 15q. Using the two-locus mode of Superlink, we showed that combining the recessive locus on 2p and a single-locus additive representation of the 15q loci is sufficient to achieve a two-locus score over 6 on the entire pedigree. For this 2p + 15q analysis, we show LOD scores ranging from 4.69 to 6.57, and the scores are sensitive to which marker is chosen for 15q. Our findings provide strong evidence for linkage at several loci.     

On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppe's urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.Supported in part by NSF grants DMS-8704369 and DMS-8902991