On the genetics of prelingual deafness.

In view of the many discordant findings in previous studies regarding the genetics of prelingual deafness, family data (133 nuclear families and 25 pedigrees) were gathered from India. Analysis of these data has revealed that the defect is primarily genetic, which is in agreement with earlier findings. Segregation analysis was performed to compare various autosomal diallelic one-locus and multilocus models. Our analysis revealed that the most parsimonious model for prelingual deafness is that it is controlled by recessive genes at a pair of unlinked diallelic autosomal loci. Individuals are affected if and only if they are recessive homozygous at both loci. The likelihood of the present data under this two-locus multiple recessive homozygosis model is at least 10(8) times higher than that of the one-locus models that were examined in previous studies. This model is also the best-fitting model among other plausible two-locus models.     

Schizophrenia: the testing of genetic models by pedigree analysis.

Simulated pedigrees of schizophrenia generally show a clear peak in their likelihood surface corresponding to analysis by the genetic models, which served as the basis for the simulation. The likelihood surface obtained with real data permits determination of the allelic frequency and the selection of an optimal one-locus, two-locus, and four-locus model. These three models have certain features in common, notably, a relatively high frequency of the allele predisposing to schizophrenia (about 20%) and a relatively low index of genetic determination (23%--34%). However, direct likelihood comparisons do not permit distinctions between the one-locus, two-locus, and four-locus models. The most likely interpretation of this finding is that the etiology of schizophrenia is heterogeneous or even nongenetic. However, a simple model with a single completely recessive locus and incomplete penetrance in the homozygote also produces a flat likelihood surface closely resembling that obtained with the real data. With reservation, this single-locus model may be put forward as a potentially useful working hypothesis.     

Evidence for two unlinked loci regulating total serum IgE levels.

Studies investigating the genetic control of total serum IgE levels are of major importance in understanding basic pathophysiologic mechanisms in atopy and asthma, since IgE levels predict onset and correlate with the clinical expression of these disorders. Previous analysis of data from 92 families, ascertained through a parent with asthma, showed evidence for recessive inheritance of high IgE levels with linkage to chromosome 5q. Since there was significant residual familial correlation in the one-locus segregation analysis, two-locus segregation and linkage analyses were performed. Segregation analyses provided evidence for a second major locus unlinked to the locus on 5q. Utilization of this two-locus model corroborates the previous evidence for linkage between this trait and markers on 5q31-q33. The LODs for the most informative marker D5S436 increased from 3.00 at 10% recombination to 4.67 at 9% recombination, when the two-locus model was used. Additional linkage studies are needed to map this second locus. These results demonstrate the importance of performing multilocus segregation and linkage analyses for quantitative traits that are related to the phenotype of a complex disorder. This approach has given further insight into the genetics of allergy and asthma by providing evidence for a two-locus model.     

Further analysis of complex allozyme polymorphisms in a barley population

New theory has recently been developed for two-locus models. In the light of this theory, an earlier analysis of esterase allozyme data from an experimental barley population has been modified to take proper account of initial gametic phase (linkage) disequilibria. The results show that the directions in which two-locus genotypic frequencies deviated from products of one-locus frequencies in this population followed those predicted by neutral descent theory. The observed departures were, however, much larger in size than predicted by the new descent measure theory, indicating that selection is operating in the population.     

Variational principles in evolution

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of then-locus genetic system moves on the surface of the product space ofn higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.     

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.     

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.     

On coding genotypes for genetic markers with multiple alleles in genetic association study of quantitative traits

Background

In genetic association study of quantitative traits using F_∞ models, how to code the marker genotypes and interpret the model parameters appropriately is important for constructing hypothesis tests and making statistical inferences. Currently, the coding of marker genotypes in building F_∞ models has mainly focused on the biallelic case. A thorough work on the coding of marker genotypes and interpretation of model parameters for F_∞ models is needed especially for genetic markers with multiple alleles.

Results

In this study, we will formulate F_∞ genetic models under various regression model frameworks and introduce three genotype coding schemes for genetic markers with multiple alleles. Starting from an allele-based modeling strategy, we first describe a regression framework to model the expected genotypic values at given markers. Then, as extension from the biallelic case, we introduce three coding schemes for constructing fully parameterized one-locus F_∞ models and discuss the relationships between the model parameters and the expected genotypic values. Next, under a simplified modeling framework for the expected genotypic values, we consider several reduced one-locus F_∞ models from the three coding schemes on the estimability and interpretation of their model parameters. Finally, we explore some extensions of the one-locus F_∞ models to two loci. Several fully parameterized as well as reduced two-locus F_∞ models are addressed.

Conclusions

The genotype coding schemes provide different ways to construct F_∞ models for association testing of multi-allele genetic markers with quantitative traits. Which coding scheme should be applied depends on how convenient it can provide the statistical inferences on the parameters of our research interests. Based on these F_∞ models, the standard regression model fitting tools can be used to estimate and test for various genetic effects through statistical contrasts with the adjustment for environmental factors.     

Constraints on the origin and maintenance of genetic kin recognition

Kin-recognition mechanisms allow helping behaviors to be directed preferentially toward related individuals, and could be expected to evolve in many cases. However, genetic kin recognition requires a genetic polymorphism on which recognition is based, and kin discriminating behaviors will affect the evolution of such polymorphism. It is unclear whether genetic polymorphisms used in kin recognition should be maintained by extrinsic selection pressures or not, as opposite conclusions have been reached by analytical one-locus models and simulations exploring different population structures. We analyze a two-locus model in a spatially subdivided population following the island model of dispersal between demes of finite size. We find that in the absence of mutation, selection eliminates polymorphism in most cases, except with extreme spatial structure and low recombination. With mutation, the population may reach a stable limit cycle over which both loci are polymorphic; however, the average frequency of conditional helping can be high only under strong structure and low recombination. Finally, we review evidence for extrinsic selection maintaining polymorphism on which kin recognition is based.     

GLIDE: GPU-Based Linear Regression for Detection of Epistasis

Due to recent advances in genotyping technologies, mapping phenotypes to single loci in the genome has become a standard technique in statistical genetics. However, one-locus mapping fails to explain much of the phenotypic variance in complex traits. Here, we present GLIDE, which maps phenotypes to pairs of genetic loci and systematically searches for the epistatic interactions expected to reveal part of this missing heritability. GLIDE makes use of the computational power of consumer-grade graphics cards to detect such interactions via linear regression. This enabled us to conduct a systematic two-locus mapping study on seven disease data sets from the Wellcome Trust Case Control Consortium and on in-house hippocampal volume data in 6 h per data set, while current single CPU-based approaches require more than a year's time to complete the same task.     

Secondary theorem of natural selection in biocultural populations.

The "Secondary Theorem of Natural Selection," an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.     

A single-locus minisatellite discriminates chinook salmon (Oncorhynchus tshawytscha) populations

A knowledge of genetic structure in natural populations is often necessary for conservation and management purposes, especially in declining Pacific salmon populations. To test for genetic differentiation between nine populations of chinook salmon, Oncorhynchus tshawytscha, from south-western British Columbia, Canada, DNA was extracted from 603 fish and hybridized with a single-locus minisatellite probe. Multivariate statistical analyses of the resulting allele size data permitted successful overall population identification of 52% (individual population range: 24–78%; P < 0.005), indicating a high level of genetic differentiation among the nine populations. Two of the nine populations were further analysed using data from a second minisatellite locus. The discrimination success rate improved from 81.1% (one-locus analyses) to 90.0% (two-locus analyses), indicating the potential for greatly increased resolution gained by the addition of more loci. These results indicate that variation at minisatellite loci can be used for assessing population-level genetic structure, even with artificial gene flow.     

The effect on neutral gene flow of selection at a linked locus

The effect on gene flow at a neutral locus of a selective cline at a linked locus is investigated. A diffusion approximation for a two-locus island model is derived in which only one locus is subject to selection. The moments of the stationary distribution are obtained and compared to the corresponding moments from a one-locus, neutral island model. This comparison yields an effective migration rate. The effective migration rate is always less than the actual migration rate, but this effect is seen to be small for weak selection and loose linkage in the case of adult migration. The importance of selection at linked loci to the question of genetic differentiation in a subdivided population is discussed.     

Genetic epidemiology of vitiligo: multilocus recessivity cross-validated.

Vitiligo is a dermatological disorder characterized by hypopigmentary patches that tend to become progressive over time. There are reports of extensive familial aggregation. A genetic model for this disorder was earlier proposed by us. This model postulates that recessive alleles at multiple unlinked autosomal loci interact epistatically in the pathogenesis of vitiligo. The present family study was primarily undertaken to cross-validate the proposed genetic model. Data on 194 families from the United States were collected. Each family was ascertained through an affected proband. Analyses of these data reveal that approximately 20% of probands have at least one first-degree relative afflicted with vitiligo. All types of first-degree relatives of probands show a significant risk of developing vitiligo. Results of segregation and robustness analyses reveal that the genetic model postulated by us previously is the most parsimonious model for the present family data set.     

The genetics of generalized vitiligo and associated autoimmune diseases

Vitiligo is an acquired disorder in which patches of depigmented skin and often overlying hair, and mucous membranes, are the result of progressive autoimmune loss of melanocytes from the involved areas. Considered the most common pigmentary disorder, vitiligo involves complex interaction of environmental and genetic factors that ultimately contribute to melanocyte destruction, resulting in the characteristic depigmented lesions. In the past few years, studies of the genetic epidemiology of vitiligo have led to the recognition that generalized vitiligo is part of a broader autoimmune disease diathesis. Attempts to identify genes involved in susceptibility to generalized vitiligo have involved gene expression studies, genetic association studies of candidate genes, and genome-wide linkage analyses to discover new genes. These studies have begun to yield results that shed light on the mechanisms of vitiligo pathogenesis. It is anticipated that the discovery of biological pathways of vitiligo pathogenesis will provide novel targets for future approaches to the treatment and prevention of vitiligo and its associated autoimmune diseases.     

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.     

Partial selfing and linkage: the effect of a heterotic locus on a neutral locus

Equilibria are determined for the two-locus model in a partially selfing population when one locus is neutral and the other locus is heterotic. At an equilibrium point, the frequency of heterozygotes at the neutral locus is greater than that expected from one-locus theory, even if the heterotic locus is on a different chromosome. Thus, the neutral locus also appears to be heterotic. The magnitude of this effect is determined for several different proportions of selfing and amounts of recombination.     

Inclusive fitness arguments in genetic models of behaviour

My purpose here is to provide a coherent account of inclusive fitness techniques, accessible to a mathematically literate graduate student in evolutionary biology, and to relate these to standard one-locus genetic models. I begin in Sect. 2 with a general formulation of evolutionary stability; in Sect. 3 and Sect. 4 I interpret the basic stability conditions within genetic and inclusive fitness models. In Sect. 5 I extend these concepts to the case of a class-structured population, and in Sect. 6 I illustrate these notions with a sex ratio example. In Sect. 7 I give a proof of the result that under additive gene action and weak selection, an inclusive fitness argument is able to verify an important stability condition (2.5) for one-locus genetic models. Most of these results have been published.     

A sib-pair strategy for the use of restriction fragment length polymorphisms to study the mode of transmission of type II diabetes.

Three models for explaining the joint distribution of DNA insertional elements and type II diabetes in sibships are given as potential candidates for resolving the mode of transmission for this common non-Mendelian disorder. While the distributions predicted from all models are subject to the same rigid marginal constraints, only two models can be considered genetic (a combined major locus multifactorial background model and a two-locus model). A sequential probability ratio test is proposed to distinguish between these two models. Once the probands have been ascertained and using realistic parameter estimates, it is shown that, on the average, fewer than 100 affected sib-pairs are required to reach a decision at the 1% significance level with 99% power.     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.