Two-locus inbreeding measures for recurrent selection

Summary For a population undergoing recurrent selection, a method is presented for determining the average inbreeding coefficients at the end of each breeding cycle. The coefficients are derived in terms of probability measures that genes are identical by descent. For the one-locus case, two digametic measures are defined and employed in the derivation of a recurrence formula for the inbreeding coefficient. Two further classes of measures, trigametic and quadrigametic, are required for transition from one cycle to the previous one to allow the calculation of the inbreeding function for the two-locus case. Numerical values of the average probability of double identity by descent for populations with various imposed assumptions are listed to illustrate the effects of linkage and population size on the accrual of inbreeding and hence of homozygosity.Paper number 5018 of the Journal Series of the North Carolina Agricultural Experiment Station, Raleigh, North Carolina. This investigation was supported in part by NIH research grant number GM 11546 from the National Institute of General Medical Sciences.     

Cellular genomics for complex traits

Recent developments in the collection and analysis of cellular multilayered data in large cohorts with extensive organismal phenotyping promise to reveal links between genetic variation and biological processes. The use of these cellular resources as models for human biology - known as 'cellular phenotyping' - is likely to transform our understanding of the genetic and long-term environmental influences on complex traits. I discuss the advantages and caveats of a deeper analysis of cellular phenotypes in large cohorts and assess the methodological advances, resource needs and prospects of this new approach.     

Linkage strategies for genetically complex traits. I. Multilocus models

In order to investigate linkage detection strategies for genetically complex traits, multilocus models of inheritance need to be specified. Here, two types of multilocus model are described: (1) a multiplicative model, representing epistasis (interaction) among loci, and (2) an additive model, which is shown to closely approximate genetic heterogeneity, which is characterized by no interlocus interaction. A ratio lambda R of risk for type R relatives that is compared with population prevalence is defined. For a single-locus model, lambda R - 1 decreases by a factor of two with each degree of relationship. The same holds true for an additive multilocus model. For a multiplicative (epistasis) model, lambda R - 1 decreases more rapidly than by a factor of two with degree of relationship. Examination of lambda R values for various classes of relatives can potentially suggest the presence of multiple loci and epistasis. For example, data for schizophrenia suggest multiple loci in interaction. It is shown in the second paper of this series that lambda R is the critical parameter in determining power to detect linkage by using affected relative pairs.     

High-resolution genetic mapping of complex traits.

Positional cloning requires high-resolution genetic mapping. To plan a positional cloning project, one needs to know how many informative meioses will be required to narrow the search for a disease gene to an acceptably small region. For a simple Mendelian trait studied with linkage analysis, the answer is straightforward. In this paper, we address the situation of a complex trait studied with affected-relative-pair methods. We derive mathematical formulas for the size of an appropriate confidence region, as a function of the relative risk attributable to the gene. Using these results, we provide graphs showing the number of relative pairs required to narrow the gene hunt to an interval of a given size. For example, we show that localizing a gene to 1 cM requires a median of 200 sib pairs for a locus causing a fivefold increased risk to an offspring and 700 sib pairs for a locus causing a twofold increased risk. We discuss the implications of these results for the positional cloning of genes underlying complex traits.     

Pedigree models for complex human traits involving the mitochondrial genome.

Recent biochemical and molecular-genetic discoveries concerning variations in human mtDNA have suggested a role for mtDNA mutations in a number of human traits and disorders. Although the importance of these discoveries cannot be emphasized enough, the complex natures of mitochondrial biogenesis, mutant mtDNA phenotype expression, and the maternal inheritance pattern exhibited by mtDNA transmission make it difficult to develop models that can be used routinely in pedigree analyses to quantify and test hypotheses about the role of mtDNA in the expression of a trait. In the present paper, we describe complexities inherent in mitochondrial biogenesis and genetic transmission and show how these complexities can be incorporated into appropriate mathematical models. We offer a variety of likelihood-based models which account for the complexities discussed. The derivation of our models is meant to stimulate the construction of statistical tests for putative mtDNA contribution to a trait. Results of simulation studies which make use of the proposed models are described. The results of the simulation studies suggest that, although pedigree models of mtDNA effects can be reliable, success in mapping chromosomal determinants of a trait does not preclude the possibility that mtDNA determinants exists for the trait as well. Shortcomings inherent in the proposed models are described in an effort to expose areas in need of additional research.     

Combining information within and between pedigrees for mapping complex traits.

This paper is concerned with efficient strategies for gene mapping using pedigrees containing small numbers of affecteds and identity-by-descent data from closely spaced markers throughout the genome. Particular attention is paid to additive traits involving phenocopies and/or locus heterogeneity. For a sample of pedigrees containing a particular configuration of affecteds, e.g., pairs of siblings together with a first cousin, we use a likelihood analysis to find 1-df statistics that are very efficient over a broad range of penetrances and allele frequencies. We identify configurations of affecteds that are particularly powerful for detecting linkage, and we show how pedigrees containing different numbers and configurations of affecteds can be efficiently combined in an overall test statistic.     

From complex traits to complex alleles

Several recent studies using natural populations of Drosophila show that one must be very careful when sorting among the large number of molecular polymorphisms found at most loci to identify the nucleotide changes responsible for phenotypic variation in complex traits. Indeed, several mutations within a single allele can interact to generate the overall observed effect. The results are instructive both for those interested in the genetics of evolutionary change and for those attempting to ferret out the genetic basis of complex human diseases.     

A dissection model for mapping complex traits

Many quantitative traits are composites of other traits that contribute differentially to genetic variation. Quantitative trait locus (QTL) mapping of these composite traits can benefit by incorporating the mechanistic process of how their formation is mediated by the underlying components. We propose a dissection model by which to map these interconnected components traits under a joint likelihood setting. The model can test how a composite trait is determined by pleiotropic QTLs for its component traits or jointly by different sets of QTLs each responsible for a different component. The model can visualize the pattern of time‐varying genetic effects for individual components and their impacts on composite traits. The dissection model was used to map two composite traits, stemwood volume growth decomposed into its stem height, stem diameter and stem form components for Euramerican poplar adult trees, and total lateral root length constituted by its average lateral root length and lateral root number components for Euphrates poplar seedlings. We found the pattern of how QTLs for different components contribute to phenotypic variation in composite traits. The detailed understanding of the genetic machineries of composite traits will not only help in the design of molecular breeding in plants and animals, but also shed light on the evolutionary processes of quantitative traits under natural selection.     

Mapping of complex traits by single-nucleotide polymorphisms.

Molecular geneticists are developing the third-generation human genome map with single-nucleotide polymorphisms (SNPs), which can be assayed via chip-based microarrays. One use of these SNP markers is the ability to locate loci that may be responsible for complex traits, via linkage/linkage-disequilibrium analysis. In this communication, we describe a semiparametric method for combined linkage/linkage-disequilibrium analysis using SNP markers. Asymptotic results are obtained for the estimated parameters, and the finite-sample properties are evaluated via a simulation study. We also applied this technique to a simulated genome-scan experiment for mapping a complex trait with two major genes. This experiment shows that separate linkage and linkage-disequilibrium analyses correctly detected the signals of both major genes; but the rates of false-positive signals seem high. When linkage and linkage-disequilibrium signals were combined, the analysis yielded much stronger and clearer signals for the presence of two major genes than did two separate analyses.     

Prospects for admixture mapping of complex traits

Admixture mapping extends to human populations the principles that underlie linkage analysis of an experimental cross. For detecting genes that contribute to ethnic variation in disease risk, admixture mapping has greater statistical power than family-linkage studies. In comparison with association studies, admixture mapping requires far fewer markers to search the genome and is less affected by allelic heterogeneity. Statistical-analysis programs for admixture mapping are now available, and a genomewide panel of markers for admixture mapping in populations formed by West African-European admixture has been assembled. Some of the remaining technical challenges include the ability to ensure that the statistical methods are robust and to develop marker panels for other admixed populations. Where admixed populations and panels of markers informative for ancestry are available, admixture mapping can be applied to localize genes that contribute to ethnic variation in any measurable trait.     

Maximization principles for frequency-dependent selection I: the one-locus two-allele case

In this article we study the one-locus two-allele version of the pairwise-interaction model of frequency-dependent selection in discrete and continuous time. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles. We provide a systematic approach that covers all possible facets of the dynamical behavior of the model, and we illustrate our results by concrete examples. We show that the mean fitness of the population is nondecreasing if the interaction coefficients are symmetric and positive. Moreover, monotonic convergence to the set of equilibria always occurs, which is not true if we also consider negative interaction coefficients. For asymmetric interaction, we provide necessary conditions when the mean fitness is nondecreasing and sufficient conditions when it is not. Furthermore, in discrete time, we show that limit cycles cannot occur, unless some interaction coefficients are negative.     

Use of population isolates for mapping complex traits

Geneticists have repeatedly turned to population isolates for mapping and cloning Mendelian disease genes. Population isolates possess many advantages in this regard. Foremost among these is the tendency for affected individuals to share ancestral haplotypes derived from a handful of founders. These haplotype signatures have guided scientists in the fine mapping of scores of rare disease genes. The past successes with Mendelian disorders using population isolates have prompted unprecedented interest among medical researchers in both the public and private sectors. Despite the obvious genetic and environmental complications, geneticists have targeted several population isolates for mapping genes for complex diseases.     

Linkage strategies for genetically complex traits. II. The power of affected relative pairs.

The power to detect disease-susceptibility loci through linkage analysis using pairs of affected relatives and affected-unaffected pairs is examined. Allelic identity by descent (ibd) for a completely polymorphic marker for sibling, uncle-nephew, grandparent-grandchild, half-sib, and first-cousin pairs is considered. Affected-unaffected pairs generally represent a poor strategy. For single-locus models, ibd depends on lambda R, the risk ratio for type R relatives compared with population prevalence, and the recombination fraction theta. The ibd for grandparent-grandchild pairs is least affected by recombination, followed by sibs, half-sib, uncle-nephew, and first-cousin pairs. For diseases with large lambda values and for small theta values, distant relatives offer greater power. For larger theta values, grandparent-grandchild pairs are best; for small lambda values, sibs are best. Additive and multiplicative multilocus models are considered. For the multiplicative model, the same formulas as in the single-locus model apply, except that lambda iR (for the ith contributing locus) is substituted for lambda R. For the additive model, the deviation from null expectation for ibd is divided among all contributing loci. Compared with the multiplicative model, for an additive model there is usually greater advantage in distant relationships. Multipoint analysis using linked marker loci for affected relative pairs is described. Simultaneous use of multiple markers diminishes the effect of recombination and allows for localization of the disease-susceptibility locus.     

Two-trait-locus linkage analysis: a powerful strategy for mapping complex genetic traits.

Recent advances in molecular biology have provided geneticists with ever-increasing numbers of highly polymorphic genetic markers that have made possible linkage mapping of loci responsible for many human diseases. However, nearly all diseases mapped to date follow clear Mendelian, single-locus segregation patterns. In contrast, many common familial diseases such as diabetes, psoriasis, several forms of cancer, and schizophrenia are familial and appear to have a genetic component but do not exhibit simple Mendelian transmission. More complex models are required to explain the genetics of these important diseases. In this paper, we explore two-trait-locus, two-marker-locus linkage analysis in which two trait loci are mapped simultaneously to separate genetic markers. We compare the utility of this approach to standard one-trait-locus, one-marker-locus linkage analysis with and without allowance for heterogeneity. We also compare the utility of the two-trait-locus, two-marker-locus analysis to two-trait-locus, one-marker-locus linkage analysis. For common diseases, pedigrees are often bilineal, with disease genes entering via two or more unrelated pedigree members. Since such pedigrees often are avoided in linkage studies, we also investigate the relative information content of unilineal and bilineal pedigrees. For the dominant-or-recessive and threshold models that we consider, we find that two-trait-locus, two-marker-locus linkage analysis can provide substantially more linkage information, as measured by expected maximum lod score, than standard one-trait-locus, one-marker-locus methods, even allowing for heterogeneity, while, for a dominant-or-dominant generating model, one-locus models that allow for heterogeneity extract essentially as much information as the two-trait-locus methods. For these three models, we also find that bilineal pedigrees provide sufficient linkage information to warrant their inclusion in such studies. We also discuss strategies for assessing the significance of the two linkages assumed in two-trait-locus, two-marker-locus models.     

Two-locus linkage analysis of cutaneous malignant melanoma/dysplastic nevi.

Previous linkage analyses of 19 cutaneous malignant melanoma/dysplastic nevi (CMM/DN) kindreds showed significant evidence of linkage and heterogeneity to both chromosomes 1p and 9p. Five kindreds also showed evidence of linkage (Z>0.7) to both regions. To further examine these findings, we conducted two-trait-locus, two-marker-locus linkage analysis. We examined one homogeneity and one heterogeneity single-locus model (SL-Hom and SL-Het), and two-locus (2L) models: an epistatic model (Ep), in which CMM was treated as a genuine 2L disease, and a heterogeneity model (Het), in which CMM could result from disease alleles at either locus. Both loci were modeled as autosomal dominant. The LOD scores for CMM alone were highest using the SL-Het model (Z = 8.48, theta = .0). There was much stronger evidence of linkage to chromosome 9p than to 1p for CMM alone; the LOD scores were approximately two times greater on 9p than on 1p. The change in LOD scores from an evaluation of CMM alone to CMM/DN suggested that a chromosome 1p locus (or loci) contributed to both CMM and CMM/DN, whereas a 9p locus contributed more to CMM alone. For both 2L models, the LOD scores from 1p were greater for CMM/DN than for CMM alone (Ep: Z=4.63 vs. 3.83; Het: 4.94 vs. 3.80, respectively). In contrast, for 9p, the LOD scores were substantially lower with CMM/DN than with CMM alone (Ep: 4.64 vs. 7.06; Het: 5.38 vs. 7.99, respectively). After conditioning on linkage to the other locus, only the 9p locus consistently showed significant evidence for linkage to CMM alone. Thus, the application of 2L models may be useful to help unravel the complexities of familial melanoma.     

Two-locus theory in recurrent selection for general combining ability in maize

Summary Two-locus theory for recurrent selection for general combining ability in maize was developed. The theory featured: (a) recombination of the selfed progeny of selected parents; and (b) linkage disequilibrium in the initial gametic array. The theory indicated: (a) that initial linkage disequilibrium exerts a permanent influence upon selection progress; (b) that interposition of one or more generations of random mating before each cycle reduces the permanent effect in ensuing cycles; and (c) that random mating done before initiation of selection is more efficient in removing the influence of linkage disequilibrium on selection progress than random mating done between subsequent cycles.     

Polygenic disease: methods for mapping complex disease traits

Improved genotyping technology has made it feasible to use a genetic approach to map genes involved in the etiology of common human diseases. We discuss here recent developments in several different statistical approaches to linkage analysis of these traits, including affected-sib-pair methods, the affected-pedigree-member method, regressive models and linkage-disequilibrium-based approaches. We discuss advantages and disadvantages of the various approaches, as well as factors influencing study design and the ability to detect loci. Statistical methodology in this area is advancing rapidly and will help enable the mapping and cloning of loci involved in susceptibility to common multifactorial diseases.     

Linkage disequilibrium and the mapping of complex human traits.

The potential value of haplotypes defined by several single nucleotide polymorphisms has attracted recent interest. With sufficient linkage disequilibrium (LD), haplotypes could be used in association studies to map common alleles that might influence the susceptibility to common diseases, as well as for reconstructing the evolution of the genome. It has been proposed that a globally useful resource need only be based on high frequency variants, identified from a few modest samples. Rapid progress has been made in quantifying the pattern of human LD and haplotypes defined by such common variants within and among populations. However, the quality and utility of the proposed LD-based resource could be seriously compromised if important sampling and analytical factors are overlooked in its design. The LD map should be based on adequately justified criteria defined by sound population genetic principles.