Estimation of the recombination fraction by the maximum likelihood method in mapping interacting genes relative to marker loci

For mapping nonlinked interacting genes relative to marker loci, the recombination fractions can be calculated by using the log-likelihood functions were derived that permit estimation of recombinant fractions by solving the ML equations on the basis of F2 data at various types of interaction. In some cases, the recombinant fraction estimates are obtained in the analytical form while in others they are numerically calculated from concrete experimental data. With the same type of epistasis the log-functions were shown to differ depending on the functional role (suppression or epistasis) of the mapped gene. Methods for testing the correspondence of the model and the recombination fraction estimates to the experimental data are discussed. In ambiguous cases, analysis of the linked marker behavior makes it possible to differentiate gene interaction from distorted single-locus segregation, which at some forms of interaction imitate phenotypic ratios.     

Epistasis interaction of QTL effects as a genetic parameter influencing estimation of the genetic additive effect

Epistasis, an additive-by-additive interaction between quantitative trait loci, has been defined as a deviation from the sum of independent effects of individual genes. Epistasis between QTLs assayed in populations segregating for an entire genome has been found at a frequency close to that expected by chance alone. Recently, epistatic effects have been considered by many researchers as important for complex traits. In order to understand the genetic control of complex traits, it is necessary to clarify additive-by-additive interactions among genes. Herein we compare estimates of a parameter connected with the additive gene action calculated on the basis of two models: a model excluding epistasis and a model with additive-by-additive interaction effects. In this paper two data sets were analysed: 1) 150 barley doubled haploid lines derived from the Steptoe × Morex cross, and 2) 145 DH lines of barley obtained from the Harrington × TR306 cross. The results showed that in cases when the effect of epistasis was different from zero, the coefficient of determination was larger for the model with epistasis than for the one excluding epistasis. These results indicate that epistatic interaction plays an important role in controlling the expression of complex traits.     

Linkage mapping of sex-specific differences

Most current linkage analyses assume identical fractions of meiotic recombination between homologous marker loci of the two sexes. This assumption is not realistic, because considerable sex-related differences have been observed in recombination fraction. In this paper, a general EM-based algorithm is presented to estimate sex-specific recombination fractions for a mixed set of molecular markers segregating differently in a full-sib family derived from two heterozygous parents. The asymptotic variances of the estimates of linkage specifically for each of the parents are evaluated using a numerical analysis based on information functions. This approach will have important implications for precise gene mapping based on sex-specific linkage maps.     

On the bias of recombination fractions, Kosambi's and Haldane's distances based on frequencies of gametes

The estimation of recombination frequencies is a crucial step in genetic mapping. For the construction of linkage maps, nonadditive recombination fractions must be transformed into additive map distances. Two of the most commonly used transformations are Kosambi's and Haldane's mapping functions. This paper reports on the calculation of the bias associated with estimation of recombination fractions, Kosambi's distances, and Haldane's distances. I calculated absolute and relative biases numerically for a wide range of recombination fractions and sample sizes. I assumed that the ratio of recombinant gametes to the total number of gametes can be adequately represented by a binomial function. I found that the bias in recombination fraction estimates is negative, i.e., the estimator is an underestimate. However, significant values were only obtained when recombination fractions were large and sample sizes were small. The relevant estimates of recombination fractions were, therefore, nearly unbiased. Haldane's and Kosambi's distances were found to be strongly biased, with positive bias for the most interesting values of recombination fractions and sample sizes. The bias of Kosambi's distance was considerably smaller than the bias of Haldane's distance.     

Efficiency of triple test cross for detecting epistasis with marker information

The triple test cross (TTC) is an experimental design for detecting epistasis and estimating the components of genetic variance for quantitative traits. In this paper, we extend the analysis to include molecular information. The statistical power of the mating design was assessed under a model assuming that a finite number of loci affect the trait in question. Formulae are developed for the analysis with or without marker information relating to the recombination fraction between loci, the genetical properties of quantitative trait controlled by the quantitative trait loci (QTL), the linkage phases of the parents and population size. Application of these formulae showed that the recombination fraction between genes and the magnitude and the types of epistasis have important interactions in their effects on power. The results demonstrate that the TTC may have increased power to detect epistasis when marker information is present. However, the simulation experiments show that the standard deviation of the estimated expected mean square was higher with one marker than that with two, whereas the corresponding value without marker information was the lowest. In addition, we demonstrate that the relative position of QTL and markers and the number of markers can both affect the power of epistatic detection.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

互作基因定位方法的研究和计算机软件的开发

质量性状中存在6种可能的基因互作类型, 即互补、重叠、累加、显性上位、隐性上位和抑制。在遗传研究中, 有时会遇到互作基因定位的问题, 但至今未见有关互作基因定位方法和计算机软件的系统的研究报道。文章给出了基于极大似然估计的互作基因定位方法及相应的计算机软件(IGMapping 1.0)。计算机模拟表明, 此方法可以无偏地估计一个共显性标记与一个互作基因之间的重组率或连锁距离。     

Doubled Haploids for Studying the Inheritance of Quantitative Characters

By using a doubled-haploid population derived from F₂ plants, additive and additive x additive genetic variances, as well as the number of segregating genes, can be estimated. An F₂-derived doubled-haploid population may contain almost 50% more of the best recombinant than an F₁-derived population. However, the best recombinant occurs in the same frequency in the two populations when there is no linkage between genes. The difference in the frequency of the best recombinant between F₂- and F₃-derived populations is small. This implies that the doubled-haploid method using F₂ plants provides only slightly less opportunity for recombination than the conventional breeding methods of self-pollinating crops. In the absence of additive epistasis, a weighted mean of recombination values can be estimated using an F₂-derived population and its parental lines. When additive epistasis is present, it can be estimated from doubled-haploid populations derived from two backcrosses. Studies on the linkage of quantitative characters are needed for determining whether doubled haploids should be produced from F₂ or from F₁ plants in a breeding program.     

Estimation of linkage in the presence of multiplicative viability effects

Log-linear analysis of contingency tables is applied to trihybrid backcross data to estimate linkage and viability. Whereas nonadditive viability differences perturb recombination estimates in the classical analysis, this statistical procedure yields maximum likelihood crossover frequency estimates in the presence of multiplicative viability effects. Other advantages of this method include: (1) estimation of viability effects of gene substitution at each locus, (2) estimation of asymptotic confidence intervals on recombination frequencies and viabilities, and (3) it tests the null hypothesis of no interference and no viability interactions. Extensions to cover more loci and to allow certain kinds of epistasis are easily made. Relative merits of the proposed and classical methods are discussed.     

Multiple-trait quantitative trait locus mapping with incomplete phenotypic data

Background

The goal of linkage analysis is to determine the chromosomal location of the gene(s) for a trait of interest such as a common disease. Three-locus linkage analysis is an important case of multi-locus problems. Solutions can be found analytically for the case of triple backcross mating. However, in the present study of linkage analysis and gene mapping some natural inequality restrictions on parameters have not been considered sufficiently, when the maximum likelihood estimates (MLEs) of the two-locus recombination fractions are calculated.

Results

In this paper, we present a study of estimating the two-locus recombination fractions for the phase-unknown triple backcross with two offspring in each family in the framework of some natural and necessary parameter restrictions. A restricted expectation-maximization (EM) algorithm, called REM is developed. We also consider some extensions in which the proposed REM can be taken as a unified method.

Conclusion

Our simulation work suggests that the REM performs well in the estimation of recombination fractions and outperforms current method. We apply the proposed method to a published data set of mouse backcross families.     

Linkage relationships and gene order around the locus for X-linked retinoschisis.

X-linked recessive retinoschisis (RS) is a hereditary disorder with variable clinical features. The main symptoms are poor sight; radial, cystic macula degeneration; and peripheral superficial retinal detachment. The disease is quite common in Finland, where at least 300 hemizygous males have been diagnosed. We used nine polymorphic DNA markers to study the localization of RS on the short arm of the X chromosome in 31 families comprising 88 affected persons. Two-point linkage results confirmed close linkage of the RS gene to the marker loci DXS43, DXS16, DXS207, and DXS41 and also revealed close linkage to the marker loci DXS197 and DXS9. Only one recombination was observed between DXS43 and RS in 59 informative meioses, giving a maximum lod score of 13.87 at the recombination fraction .02. No recombinations were observed between the RS locus and DXS9 and DXS197 (lods between 3 and 4), but at neither locus was the number of informative meioses sufficient to provide reliable estimates of recombination fractions. The most likely gene order on the basis of multilocus analysis was Xpter-DXS85-(DXS207,DXS43)-RS-DXS41-DXS 164-Xcen. Because multilocus linkage analysis indicated that the most probable location of RS is proximal to DXS207 and DXS43 and distal to DXS41, these three flanking markers are the closest and most informative markers currently available for carrier detection.     

Detection of epistatic interactions between exotic alleles introgressed from wild barley (H. vulgare ssp. spontaneum)

The expression of a quantitative phenotype can be controlled through genotype, environment and genotype by environment interaction effects. Further, genotype effects can be attributed to major genes, quantitative trait loci (QTL) and gene by gene interactions, which are also termed epistatic interactions. The present study demonstrates that two-way epistatic interactions can play an important role for the expression of domestication-related traits like heading date, plant height and yield. In the BC₂DH population S42, carrying wild barley introgressions in the genetic background of the spring barley cultivar Scarlett, 13, 8 and 12 marker by marker interaction effects could be detected for the traits heading date, plant height and yield, respectively. Significant allelic combinations at interacting loci coincided for heading date, plant height and yield suggesting the presence of pleiotropic effects rather than several linked QTL. The mode of epistasis observed was primarily characterised by either (1) compensatory effects, where allelic combinations from the same genotype buffered the phenotype, or (2) augmented effects, where only the combination of the exotic allele at both interacting loci caused an altered phenotype. The present study shows that estimates of main effects of QTL can be confounded by interactions with background loci, suggesting that the identification of epistatic effects is important for gene cloning and marker-assisted selection. Furthermore, interaction effects between loci and putative candidate genes detected in the present study reveal potential functional relationships, which can be used to further elucidate gene networks in barley.     

Genetic models to estimate additive and non-additive effects of marker-associated QTL using multiple regression techniques

Summary The development of molecular markers has recently raised expectations for their application in selection programs. However, some questions related to quantitative trait loci (QTL) identification are still unanswered. The objectives of this paper are (1) to develop statistical genetic models for detecting and locating on the genome multi-QTL with additive, dominance and epistatic effects using multiple linear regression analysis in the backcross and F_n generations from the cross of two inbred lines; and (2) to discuss the bias caused by linked and unlinked QTL on the genetic estimates. Non-linear models were developed for different backcross and F_n generations when both epistasis and no epistasis were assumed. Generation analysis of marked progenies is suggested as a way of increasing the number of observations for the estimates without additional cost for molecular scoring. Some groups of progenies can be created in different generations from the same scored individuals. The non-linear models were transformed into approximate multivariate linear models to which combined stepwise and standard regression analysis could be applied. Expressions for the biases of the marker classes from linked QTL were obtained when no epistasis was assumed. When epistasis was assumed, these expressions increased in complexity, and the biases were caused by both linked and unlinked QTL.     

Effect of varying epistasis on the evolution of recombination

Whether recombination decelerates or accelerates a population's response to selection depends, at least in part, on how fitness-determining loci interact. Realistically, all genomes likely contain fitness interactions both with positive and with negative epistasis. Therefore, it is crucial to determine the conditions under which the potential beneficial effects of recombination with negative epistasis prevail over the detrimental effects of recombination with positive epistasis. Here, we examine the simultaneous effects of diverse epistatic interactions with different strengths and signs in a simplified model system with independent pairs of interacting loci and selection acting only on the haploid phase. We find that the average form of epistasis does not predict the average amount of linkage disequilibrium generated or the impact on a recombination modifier when compared to results using the entire distribution of epistatic effects and associated single-mutant effects. Moreover, we show that epistatic interactions of a given strength can produce very different effects, having the greatest impact when selection is weak. In summary, we observe that the evolution of recombination at mutation-selection balance might be driven by a small number of interactions with weak selection rather than by the average epistasis of all interactions. We illustrate this effect with an analysis of published data of Saccharomyces cerevisiae. Thus to draw conclusions on the evolution of recombination from experimental data, it is necessary to consider the distribution of epistatic interactions together with the associated selection coefficients.     

Using molecular markers to map multiple quantitative trait loci: models for backcross,recombinant inbred,and doubled haploid progeny

Summary To maximize parameter estimation efficiency and statistical power and to estimate epistasis, the parameters of multiple quantitative trait loci (QTLs) must be simultaneously estimated. If multiple QTL affect a trait, then estimates of means of QTL genotypes from individual locus models are statistically biased. In this paper, I describe methods for estimating means of QTL genotypes and recombination frequencies between marker and quantitative trait loci using multilocus backcross, doubled haploid, recombinant inbred, and testcross progeny models. Expected values of marker genotype means were defined using no double or multiple crossover frequencies and flanking markers for linked and unlinked quantitative trait loci. The expected values for a particular model comprise a system of nonlinear equations that can be solved using an interative algorithm, e.g., the Gauss-Newton algorithm. The solutions are maximum likelihood estimates when the errors are normally distributed. A linear model for estimating the parameters of unlinked quantitative trait loci was found by transforming the nonlinear model. Recombination frequency estimators were defined using this linear model. Certain means of linked QTLs are less efficiently estimated than means of unlinked QTLs.     

A new strategy for estimating two-locus recombination fractions under some natural inequality restrictions

Linkage analysis is now being widely used to map markers on each chromosome in the human genome, to map genetic diseases, and to identify genetic forms of common diseases. Two-locus linkage analysis and multi-locus analysis have been investigated comprehensively, and many computer programs have been developed to perform linkage analysis. Yet there exists a shortcoming in traditional methods, i.e., the parameter space of two-locus recombination fractions has not been emphasized sufficiently in the usual analyses. In this paper, we propose a new strategy for estimating the two-locus recombination fractions based on data of backcross family in the framework of some natural and necessary parameter restrictions. The new strategy is based on a restricted projection algorithm, which can provide fast reasonable estimates of recombination fraction, and can therefore serve as a superior alternative algorithm. Results obtained from both real and simulated data indicate that the new algorithm performs well in the estimation of recombination fractions and outperforms current methods.     

Likelihood analysis of disequilibrium mapping, and related problems.

In this paper a theory is developed that provides the sampling distribution of alleles at a diallelic marker locus closely linked to a low-frequency allele that arose as a single mutant. The sampling distribution provides a basis for maximum-likelihood estimation of either the recombination rate, the mutation rate, or the age of the allele, provided that the two other parameters are known. This theory is applied to (1) the data of Hästbacka et al., to estimate the recombination rate between a locus associated with diastrophic dysplasia and a linked RFLP marker; (2) the data of Risch et al., to estimate the age of a presumptive allele causing idiopathic distortion dystonia in Ashkenazi jews; and (3) the data of Tishkoff et al., to estimate the date at which, at the CD4 locus, non-African lineages diverged from African lineages. We conclude that the extent of linkage disequilibrium can lead to relatively accurate estimates of recombination and mutation rates and that those estimates are not very sensitive to parameters, such as the population age, whose values are not known with certainty. In contrast, we also conclude that, in many cases, linkage disequilibrium may not lead to useful estimates of allele age, because of the relatively large degree of uncertainly in those estimates.     

Estimation of gametic frequencies from F2 populations using the EM algorithm and its application in the analysis of crossover interference in rice

The gametes produced in meiosis provide information on the frequency of recombination and also on the interdependence of recombination events, i.e. interference. Using F₂ individuals, it is not possible in all cases to derive the gametes, which have fused, and which provide the information about interference unequivocally when three or more segregating markers are considered simultaneously. Therefore, a method was developed to estimate the gametic frequencies using a maximum likelihood approach together with the expectation maximisation algorithm. This estimation procedure was applied to F₂ mapping data from rice (Oryza sativa L.) to carry out a genome-wide analysis of crossover interference. The distribution of the coefficient of coincidence in dependence on the recombination fraction revealed for all chromosomes increasing positive interference with decreasing interval size. For some chromosomes this mutual inhibition of recombination was not so strong in small intervals. The centromere had a significant effect on interference. The positive interference found in the chromosome arms were reduced significantly when the intervals considered spanned the centromere. Two chromosomes even demonstrated independent recombination and slightly negative interference for small intervals including the centromere. Different marker densities had no effect on the results. In general, interference depended on the frequency of recombination events in relation to the physical length. The strength of the centromere effect on interference seemed to depend on the strength of recombination suppression around the centromere.Electronic Supplementary Material Supplementary material is available for this article at     

WHY EPISTASIS IS IMPORTANT FOR SELECTION AND ADAPTATION

Organisms are built from thousands of genes that interact in complex ways. Still, the mathematical theory of evolution is dominated by a gene‐by‐gene perspective in which genes are assumed to have the same effects regardless of genetic background. Gene interaction, or epistasis, plays a role in some theoretical developments such as the evolution of recombination, reproductive isolation, and canalization, but is strikingly missing from our standard accounts of phenotypic adaptation. This absence is most puzzling within the field of quantitative genetics, which, despite its polygenic perspective and elaborate statistical representation of epistasis, has not found a single important role for gene interaction in evolution. To the contrary, there is a widespread consensus that epistasis is evolutionary inert, and that all we need to know to predict evolutionary dynamics is the additive component of the genetic variance. This view may have roots in convenience, but also in theoretical results showing that the response to selection derived from epistatic variance components is not permanent and will decay when selection is relaxed. I show that these results are tied to a conceptual confusion, and are misleading as general statements about the significance of epistasis for the selection response and adaptation.     

A simple method for estimating recombination percentages and linkage intensities from F2 data: examples from Triticum monococcum and other self-fertilizing diploid plant species

Summary We offer an alternative approach to the extensively used maximum likelihood and product methods for calculating recombination values and linkage intensities from F₂ data. This new method which we designate as the square root approach is simpler than the ones in current use in that it obviates the need for formulae and tables. It can be applied to autosomal F₂ data from F₁'s heterozygous in both the coupling and repulsion phases. It has greater applicability than the product method in that it can be used in all cases involving 2-, 3-, 4-, 6- and 9-class segregations regardless whether gene interaction occurs or not, provided the double recessive and other specific phenotypes are each determined by one particular genotype. The proposed method is based on the same well established genetic facts as the other two approaches. Percent recombinant gametes and therefore percent recombination are calculated by deriving the square root of the proportion of the F₂ population that expresses the double recessive or equivalent phenotype. The recombination values obtained by our method are compared with those derived by product method for 17 crosses in 7 different species and were found to be insignificantly different from the latter. The advantages and disadvantages of the square root method compared with the two most used ones are discussed.