Effects of subpopulation structure on probability calculations of DNA profiles from forensic PCR analysis

DNA typing for forensic identification is a two-step process. The first step involves determining the profiles of samples collected at the crime scene and comparing them with the profiles obtained from suspects and the victims. In the case of a match that includes the suspect as the potential source of the material collected at the crime scene, the last step in the process is to answer the question, what is the likelihood that someone in addition to the suspect could match the profile of the sample studied? This likelihood is calculated by determining the frequency of the suspect's profile in the relevant population databases. The design of forensic databases and the criteria for comparison has been addressed by the NRC report of 1996 (National Research Council, 1996). However, the fact that geographical proximity, migrational patterns, and even cultural and social practices have effects on subpopulation structure establishes the grounds for further study into its effects on the calculation of probability of occurrence values. The issue becomes more relevant in the case of discrete polymorphic markers that show higher probability of occurrence in the reference populations, where several orders of magnitude difference between the databases may have an impact on the jury. In this study, we calculated G values for all possible pairwise comparisons of allelic frequencies in the different databases from the races or subpopulations examined. In addition, we analyzed a set of 24 unrelated Caucasian, 37 unrelated African-American, and 96 unrelated Sioux/Chippewa individuals for seven polymorphic loci (DQA1, LDLR, GYPA, HBGG, D7S8, GC, and D1S80). All three sets of individuals where sampled from Minnesota. The probability of occurrence for all seven loci were calculated with respect to nine different databases: Caucasian, Arabic, Korean, Sioux/Chippewa, Navajo, Pueblo, African American, Southeastern Hispanic, and Southwestern Hispanic. Analysis of the results demonstrated marked differences in the probabilities of occurrence when individuals were compared to the different populations and subpopulation databases. The possible genetic and forensic consequences of subpopulation structure on probability calculations are discussed.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.     

Genetic management of captive populations: the advantages of circular mating

We performed computer simulations to evaluate the effectiveness of circular mating as a genetic management option for captive populations. As a benchmark, we used the method proposed by Fernández and Caballero according to which parental contributions are set to produce minimum coancestry among the offspring and matings are performed so as to minimize mean pairwise coancestry (referred to as the Gc/mc method). In contrast to other methods, fitness does not vary with population size in the case of circular mating, and can be higher than under random mating. Whether circular mating is an effective method in conserving captive populations depends on the trade-off between different considerations. On the one hand, circular mating shows the highest allelic diversity and the lowest mean pairwise coancestry for all population sizes. It also shows a relatively higher efficiency of purging deleterious alleles. More importantly, circular mating can significantly increase the success probability of populations released to the wild relative to the Gc/mc method. On the other hand, circular mating has the drawback of showing high inbreeding rates and low fitness in early generations, which can result to an increase in the extinction probability of the captive populations. However, this increase is slight unless population size and litter size are both very low. Overall, if the slight increase in extinction probability can be tolerated then circular mating fulfils the primary goals of a captive breeding program, i.e., it maintains high levels of genetic diversity and increases the success probability of reintroduced populations.     

Fixed-bin analysis for statistical evaluation of continuous distributions of allelic data from VNTR loci, for use in forensic comparisons.

The detection of DNA polymorphisms by RFLP analysis is having a major impact on identity testing in forensic science. At present, this approach is the best effort a forensic scientist can make to exclude an individual who has been falsely associated with an evidentiary sample found at a crime scene. When an analysis fails to exclude a suspect as a potential contributor of an evidentiary sample, a means should be provided to assess suitable weight to the putative match. Most important, the statistical analysis should not place undue weight on a genetic profile derived from an unknown sample that is attributed to an accused individual. The method must allow for limitations in conventional agarose-submarine-gel electrophoresis and Southern blotting procedure, limited sample population data, possible subpopulation differences, and potential sampling error. A conservative statistical method was developed based on arbitrarily defined fixed bins. This approach permits classification of continuous allelic data, provides for a simple and portable data-base system, and is unlikely to underestimate the frequency of occurrence of a set of alleles. This will help ensure that undue weight is not placed on a sample attributed to an accused individual.     

Marker-based estimation of the coefficient of coancestry in hybrid breeding programmes

Molecular markers allow to estimate the pairwise relatedness between the members of a breeding pool when their selection history is no longer available or has become too complex for a classical pedigree analysis. The field of population genetics has several estimation procedures at its disposal, but when the genotyped individuals are highly selected inbred lines, their application is not warranted as the theoretical assumptions on which these estimators were built, usually linkage equilibrium between marker loci or even Hardy–Weinberg equilibrium, are not met. An alternative approach requires the availability of a genotyped reference set of inbred lines, which allows to correct the observed marker similarities for their inherent upward bias when used as a coancestry measure. However, this approach does not guarantee that the resulting coancestry matrix is at least positive semi-definite (psd), a necessary condition for its use as a covariance matrix. In this paper we present the weighted alikeness in state (WAIS) estimator. This marker-based coancestry estimator is compared to several other commonly applied relatedness estimators under realistic hybrid breeding conditions in a number of simulations. We also fit a linear mixed model to phenotypical data from a commercial maize breeding programme and compare the likelihood of the different variance structures. WAIS is shown to be psd which makes it suitable for modelling the covariance between genetic components in linear mixed models involved in breeding value estimation or association studies. Results indicate that it generally produces a low root mean squared error under different breeding circumstances and provides a fit to the data that is comparable to that of several other marker-based alternatives. Recommendations for each of the examined coancestry measures are provided.     

The effect of relatedness on likelihood ratios and the use of conservative estimates

DNA profiling can be used to identify criminals through their DNA matching that left at the scene of a crime. The strength of the evidence supplied by a match in DNA profiles is given by the likelihood ratio. This, in turn, depends upon the probability that a match would be produced if the suspect is innocent. This probability could be strongly affected by the possibility of relatedness between the suspect and the true source of the scene-of-crime DNA profile. Methods are shown that allow for the possibility of such relatedness, arising either through population substructure or through a family relationship. Uncertainties about the likelihood ratio have been taken as grounds for the use of very conservative estimates of this quantity. The use of such conservative estimates can be shown to be neither necessary nor harmless.Editor's commentsThe author makes a good case for the use of likelihood ratios in presenting evidence. The author cites Nichols and Balding (1991) for the match probability involvingF _ST , and readers will also need to consult Balding and Nichols (1994 and this volume) and Weir (1994). In light of current concern over laboratory errors or fraud, the author's comment that Undetected and unsuspected human error or mischief is potentially a problem in almost all criminal cases, yet no convictions could follow if this possibility was consistently deemed to lead to reasonable doubt is well taken.     

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 note on the rationale for estimating genealogical coancestry from molecular markers

Background

Genetic relatedness or similarity between individuals is a key concept in population, quantitative and conservation genetics. When the pedigree of a population is available and assuming a founder population from which the genealogical records start, genetic relatedness between individuals can be estimated by the coancestry coefficient. If pedigree data is lacking or incomplete, estimation of the genetic similarity between individuals relies on molecular markers, using either molecular coancestry or molecular covariance. Some relationships between genealogical and molecular coancestries and covariances have already been described in the literature.

Methods

We show how the expected values of the empirical measures of similarity based on molecular marker data are functions of the genealogical coancestry. From these formulas, it is easy to derive estimators of genealogical coancestry from molecular data. We include variation of allelic frequencies in the estimators.

Results

The estimators are illustrated with simulated examples and with a real dataset from dairy cattle. In general, estimators are accurate and only slightly biased. From the real data set, estimators based on covariances are more compatible with genealogical coancestries than those based on molecular coancestries. A frequently used estimator based on the average of estimated coancestries produced inflated coancestries and numerical instability. The consequences of unknown gene frequencies in the founder population are briefly discussed, along with alternatives to overcome this limitation.

Conclusions

Estimators of genealogical coancestry based on molecular data are easy to derive. Estimators based on molecular covariance are more accurate than those based on identity by state. A correction considering the random distribution of allelic frequencies improves accuracy of these estimators, especially for populations with very strong drift.     

Likelihood ratios for evaluating DNA evidence when the suspect is found through a database search

A crime has been committed, and a DNA profile of the perpetrator is obtained from the crime scene. A suspect with a matching profile is found. The problem of evaluating this DNA evidence in a forensic context, when the suspect is found through a database search, is analysed through a likelihood approach. The recommendations of the National Research Council of the U.S. are derived in this setting as the proper way of evaluating the evidence when finiteness of the population of possible perpetrators is not taken into account. When a finite population of possible perpetrators may be assumed, it is possible to take account of the sampling process that resulted in the actual database, so one can deal with the problem where a large proportion of the possible perpetrators belongs to the database in question. It is shown that the last approach does not in general result in a greater weight being assigned to the evidence, though it does when a sufficiently large amount of the possible perpetrators are in the database. The value of the likelihood ratio corresponding to the probable cause setting constitutes an upper bound for this weight, and the upper bound is only attained when all but one of the possible perpetrators are in the database.     

A vectorized method of importance sampling with applications to models of mutation and migration

An importance-sampling method is presented for computing the likelihood of the configuration of population genetic data under general assumptions about population history and transitions among states. The configuration of the data is the number of chromosomes sampled that are in each of a finite set of states. Transitions among states are governed by a Markov chain with transition probabilities dependent on one or more parameters. The method assumes that the joint distribution of coalescence times of the underlying gene genealogy is independent of the genetic state of each lineage. Given a set of coalescence times, the probability that a pair of lineages is chosen to coalesce in each replicate is proportional to the contribution that the coalescence event makes to the probability of the data. This method can be applied to gene genealogies generated by the neutral coalescent process and to genealogies generated by other processes, such as a linear birth-death process which provides a good approximation to the dynamics of low-frequency alleles. Two applications are described. In the first, the fit of allele frequencies at two microsatellite loci sampled in a Sardinian population to the one-step mutation model is tested. The one-step model is rejected for one locus but not for the other. The second application is to low-frequency alleles in a geographically subdivided population. The geographic location is the allelic state, and the alleles are assumed to be sufficiently rare that their dynamics can be approximated by a linear birth-death process in which the birth and death rates are independent of geographic location. The analysis of eight low-frequency allozyme alleles found in the glaucous-winged gull, Larus glaucescens, illustrates how geographically restricted dispersal can be detected.     

随机漂移亚群体等位基因频率分布的假定及在DNA指纹数据中的应用

研究影响计算两ＤＮＡ指纹偶然匹配率的亚群体结构,首次提出随机漂移各亚群体单个位点等位基因频率服朋参数为（ｐ１（１－θ）／θ,ｐ２（１－θ）／θ,…ｐｍ（１－θ）／θ）的Ｄｉｒｉｃｈｌｅｔ分布的假定（ｐ１,ｐ２,…,ｐｍ为随机漂移初始频率,θ为近交系数）,证明依分布产生亚群体,其结构、性质与群体遗传理论、样本理论一致;将该分布应用于ＤＮＡ指纹数据,得到了其它方法的类似结论。     

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.     

Using genomic tools to maintain diversity and fitness in conservation programmes

Conservation programmes aim at maximizing the survival probability of populations, by minimizing the loss of genetic diversity, which allows populations to adapt to changes, and controlling inbreeding increases. The best known strategy to achieve these goals is optimizing the contributions of the parents to minimize global coancestry in their offspring. Results on neutral scenarios showed that management based on molecular coancestry could maintain more diversity than management based on genealogical coancestry when a large number of markers were available. However, if the population has deleterious mutations, managing using optimal contributions can lead to a decrease in fitness, especially using molecular coancestry, because both beneficial and harmful alleles are maintained, compromising the long‐term viability of the population. We introduce here two strategies to avoid this problem: The first one uses molecular coancestry calculated removing markers with low minor allele frequencies, as they could be linked to selected loci. The second one uses a coancestry based on segments of identity by descent, which measures the proportion of genome segments shared by two individuals because of a common ancestor. We compare these strategies under two contrasting mutational models of fitness effects, one assuming many mutations of small effect and another with few mutations of large effect. Using markers at intermediate frequencies maintains a larger fitness than using all markers, but leads to maintaining less diversity. Using the segment‐based coancestry provides a compromise solution between maintaining diversity and fitness, especially when the population has some inbreeding load.     

A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity

A method is proposed for allowing for the effects of population differentiation, and other factors, in forensic inference based on DNA profiles. Much current forensic practice ignores, for example, the effects of coancestry and inappropriate databases and is consequently systematically biased against defendants. Problems with the ‘product rule’ for forensic identification have been highlighted by several authors, but important aspects of the problems are not widely appreciated. This arises in part because the match probability has often been confused with the relative frequency of the profile. Further, the analogous problems in paternity cases have received little attention. The proposed method is derived under general assumptions about the underlying population genetic processes. Probabilities relevant to forensic inference are expressed in terms of a single parameter whose values can be chosen to reflect the specific circumstances. The method is currently used in some UK courts and has important advantages over the ‘Ceiling Principle’ method, which has been criticized on a number of grounds. Editor's comments The authors' work offers a sound approach to accommodating the effects of population structure, based on use of Wright'sF _ST . Their equations 1 and 2 are very convenient, and are good approximations to the exact results given by Weir (1994). As they point out, good estimates ofF _ST are needed. The comments about the ‘generally mixed’ results of independence tests may be met, in part, by the paper of Maiste and Weir in this volume. The authors cite Kraneet al. (1992) but had not seen the subsequent rebuttal by Budowleet al. (1994). The work of Wallet al. (1993) contained errors, as noted in Greenhalghet al. (1994). An erratum to this article is available at .     

A nonparametric Bayesian basket trial design

Targeted therapies on the basis of genomic aberrations analysis of the tumor have shown promising results in cancer prognosis and treatment. Regardless of tumor type, trials that match patients to targeted therapies for their particular genomic aberrations have become a mainstream direction of therapeutic management of patients with cancer. Therefore, finding the subpopulation of patients who can most benefit from an aberration‐specific targeted therapy across multiple cancer types is important. We propose an adaptive Bayesian clinical trial design for patient allocation and subpopulation identification. We start with a decision theoretic approach, including a utility function and a probability model across all possible subpopulation models. The main features of the proposed design and population finding methods are the use of a flexible nonparametric Bayesian survival regression based on a random covariate‐dependent partition of patients, and decisions based on a flexible utility function that reflects the requirement of the clinicians appropriately and realistically, and the adaptive allocation of patients to their superior treatments. Through extensive simulation studies, the new method is demonstrated to achieve desirable operating characteristics and compares favorably against the alternatives.     

The probability of duplicate gene preservation by subfunctionalization

It has often been argued that gene-duplication events are most commonly followed by a mutational event that silences one member of the pair, while on rare occasions both members of the pair are preserved as one acquires a mutation with a beneficial function and the other retains the original function. However, empirical evidence from genome duplication events suggests that gene duplicates are preserved in genomes far more commonly and for periods far in excess of the expectations under this model, and whereas some gene duplicates clearly evolve new functions, there is little evidence that this is the most common mechanism of duplicate-gene preservation. An alternative hypothesis is that gene duplicates are frequently preserved by subfunctionalization, whereby both members of a pair experience degenerative mutations that reduce their joint levels and patterns of activity to that of the single ancestral gene. We consider the ways in which the probability of duplicate-gene preservation by such complementary mutations is modified by aspects of gene structure, degree of linkage, mutation rates and effects, and population size. Even if most mutations cause complete loss-of-subfunction, the probability of duplicate-gene preservation can be appreciable if the long-term effective population size is on the order of 10(5) or smaller, especially if there are more than two independently mutable subfunctions per locus. Even a moderate incidence of partial loss-of-function mutations greatly elevates the probability of preservation. The model proposed herein leads to quantitative predictions that are consistent with observations on the frequency of long-term duplicate gene preservation and with observations that indicate that a common fate of the members of duplicate-gene pairs is the partitioning of tissue-specific patterns of expression of the ancestral gene.     

Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

Inference from gene trees in a subdivided population

This paper studies gene trees in subdivided populations which are constructed as perfect phylogenies from the pattern of mutations in a sample of DNA sequences and presents a new recursion for the probability distribution of such gene trees. The underlying evolutionary model is the coalescent process in a subdivided population. The infinitely-many-sites model of mutation is assumed. Ancestral inference questions that are discussed are maximum likelihood estimation of migration and mutation rates; detection of population growth by likelihood techniques; determining the distribution of the time to the most recent common ancestor of a sample of sequences; determining the distribution of the age of the mutations on the gene tree; determining in which subpopulation the most recent common ancestor of all the sequences was; determining subpopulation ancestors, where they were, and times to them; and determining in which subpopulations mutations occurred. A computational technique of Griffiths and Tavaré used is a computer intensive Markov chain simulation, which simulates gene trees conditional on their topology implied by the mutation pattern in the sample of DNA sequences. The software GENETREE, which implements these ancestral inference techniques, is available.     

Effects of coancestry on accuracy of individual assignments to population of origin: examples using Great Lakes lake trout (Salvelinus namaycush)

Methods for assigning individuals to population of origin are widely used in ecological genetics, resources management, and forensics. Characteristics of genetic data obtained from putative source populations that enhance accuracy of assignment are well established. How non-independence within and among unknown individuals to be classified [i.e., gene correlations within individual (inbreeding) and gene correlations among individuals within group (coancestry)] affect assignment accuracy is poorly understood. We used empirical data for six microsatellite loci and offspring from full-sib crosses of hatchery strains of lake trout (Salvelinus namaycush; Salmonidae) representing known levels of coancestry (mean θ = 0.006 and 0.06) within families to investigate how gene correlations can affect assignment. Additional simulations were conducted to further investigating the influence of allelic diversity (2, 6 or 10 alleles per locus) and inbreeding (F = 0.00, 0.05, and 0.15) on assignment accuracy for cases of low and high inter-population variance in allele frequency (mean F _st = 0.01 and 0.1, respectively). Inbreeding had no effect on accuracy of assignments. In contrast, variance in assignment accuracy across replicated simulations, and for each empirical case study increased with increasing coancestry, reflecting non-independence of probabilities of correct assignment among members of kin groups. Empirical estimates of assignment error rates should be interpreted with caution if appreciable levels of coancestry are suspected. Additional emphasis should be placed on sampling designs (spatially and temporally) that define or minimize the potential for sampling related individuals.