The use of a genetic algorithm for simultaneous mapping of multiple interacting quantitative trait loci

Here we describe a general method for improving computational efficiency in simultaneous mapping of multiple interacting quantitative trait loci (QTL). The method uses a genetic algorithm to search for QTL in the genome instead of an exhaustive enumerative ("step-by-step") search. It can be used together with any method of QTL mapping based on a genomic search, since it only provides a more efficient way to search the genome for QTL. The computational demand decreases by a factor of approximately 130 when using genetic algorithm-based mapping instead of an exhaustive enumerative search for two QTL in a genome size of 2000 cM using a resolution of 1 cM. The advantage of using a genetic algorithm increases further for larger genomes, higher resolutions, and searches for more QTL. We show that a genetic algorithm-based search has efficiency higher than or equal to a search method conditioned on previously identified QTL for all epistatic models tested and that this efficiency is comparable to that of an exhaustive search for multiple QTL. The genetic algorithm is thus a powerful and computationally tractable alternative to the exhaustive enumerative search for simultaneous mapping of multiple interacting QTL. The use of genetic algorithms for simultaneous mapping of more than two QTL and for determining empirical significance thresholds using permutation tests is also discussed.     

Efficient algorithms for quantitative trait loci mapping problems.

Rapid advances in molecular genetics push the need for efficient data analysis. Advanced algorithms are necessary for extracting all possible information from large experimental data sets. We present a general linear algebra framework for quantitative trait loci (QTL) mapping, using both linear regression and maximum likelihood estimation. The formulation simplifies future comparisons between and theoretical analyses of the methods. We show how the common structure of QTL analysis models can be used to improve the kernel algorithms, drastically reducing the computational effort while retaining the original analysis results. We have evaluated our new algorithms on data sets originating from two large F(2) populations of domestic animals. Using an updating approach, we show that 1-3 orders of magnitude reduction in computational demand can be achieved for matrix factorizations. For interval-mapping/composite-interval-mapping settings using a maximum likelihood model, we also show how to use the original EM algorithm instead of the ECM approximation, significantly improving the convergence and further reducing the computational time. The algorithmic improvements makes it feasible to perform analyses which have previously been deemed impractical or even impossible. For example, using the new algorithms, it is reasonable to perform permutation testing using exhaustive search on populations of 200 individuals using an epistatic two-QTL model.     

Use of randomization testing to detect multiple epistatic QTLs

Here, we describe a randomization testing strategy for mapping interacting quantitative trait loci (QTLs). In a forward selection strategy, non-interacting QTLs and simultaneously mapped interacting QTL pairs are added to a total genetic model. Simultaneous mapping of epistatic QTLs increases the power of the mapping strategy by allowing detection of interacting QTL pairs where none of the QTL can be detected by their marginal additive and dominance effects. Randomization testing is used to derive empirical significance thresholds for every model selection step in the procedure. A simulation study was used to evaluate the statistical properties of the proposed randomization tests and for which types of epistasis simultaneous mapping of epistatic QTLs adds power. Least squares regression was used for QTL parameter estimation but any other QTL mapping method can be used. A genetic algorithm was used to search for interacting QTL pairs, which makes the proposed strategy feasible for single processor computers. We believe that this method will facilitate the evaluation of the importance at epistatic interaction among QTLs controlling multifactorial traits and disorders.     

Empirical Bayesian elastic net for multiple quantitative trait locus mapping

In multiple quantitative trait locus (QTL) mapping, a high-dimensional sparse regression model is usually employed to account for possible multiple linked QTLs. The QTL model may include closely linked and thus highly correlated genetic markers, especially when high-density marker maps are used in QTL mapping because of the advancement in sequencing technology. Although existing algorithms, such as Lasso, empirical Bayesian Lasso (EBlasso) and elastic net (EN) are available to infer such QTL models, more powerful methods are highly desirable to detect more QTLs in the presence of correlated QTLs. We developed a novel empirical Bayesian EN (EBEN) algorithm for multiple QTL mapping that inherits the efficiency of our previously developed EBlasso algorithm. Simulation results demonstrated that EBEN provided higher power of detection and almost the same false discovery rate compared with EN and EBlasso. Particularly, EBEN can identify correlated QTLs that the other two algorithms may fail to identify. When analyzing a real dataset, EBEN detected more effects than EN and EBlasso. EBEN provides a useful tool for inferring high-dimensional sparse model in multiple QTL mapping and other applications. An R software package ‘EBEN'' implementing the EBEN algorithm is available on the Comprehensive R Archive Network (CRAN).     

The full EM algorithm for the MLEs of QTL effects and positions and their estimated variances in multiple-interval mapping

The advent of complete genetic linkage maps of DNA markers has made systematic studies of mapping quantitative trait loci (QTL) in experimental organisms feasible. The method of multiple-interval mapping provides an appropriate way for mapping QTL using genetic markers. However, efficient algorithms for the computation involved remain to be developed. In this article, a full EM algorithm for the simultaneous computation of the MLEs of QTL effects and positions is developed. EM-based formulas are derived for computing the observed Fisher information matrix. The full EM algorithm is compared with an ECM algorithm developed by Kao and Zeng (1997, Biometrics 53, 653-665). The validity of the inverted observed Fisher information matrix as an estimate of the variance matrix of the MLEs is demonstrated by a simulation study.     

Efficiency of selective genotyping for genetic analysis of complex traits and potential applications in crop improvement

Selective genotyping of individuals from the two tails of the phenotypic distribution of a population provides a cost efficient alternative to analysis of the entire population for genetic mapping. Past applications of this approach have been confounded by the small size of entire and tail populations, and insufficient marker density, which result in a high probability of false positives in the detection of quantitative trait loci (QTL). We studied the effect of these factors on the power of QTL detection by simulation of mapping experiments using population sizes of up to 3,000 individuals and tail population sizes of various proportions, and marker densities up to one marker per centiMorgan using complex genetic models including QTL linkage and epistasis. The results indicate that QTL mapping based on selective genotyping is more powerful than simple interval mapping but less powerful than inclusive composite interval mapping. Selective genotyping can be used, along with pooled DNA analysis, to replace genotyping the entire population, for mapping QTL with relatively small effects, as well as linked and interacting QTL. Using diverse germplasm including all available genetics and breeding materials, it is theoretically possible to develop an “all-in-one plate” approach where one 384-well plate could be designed to map almost all agronomic traits of importance in a crop species. Selective genotyping can also be used for genomewide association mapping where it can be integrated with selective phenotyping approaches. We also propose a breeding-to-genetics approach, which starts with identification of extreme phenotypes from segregating populations generated from multiple parental lines and is followed by rapid discovery of individual genes and combinations of gene effects together with simultaneous manipulation in breeding programs.     

A modified algorithm for the improvement of composite interval mapping

Composite interval mapping (CIM) is the most commonly used method for mapping quantitative trait loci (QTL) with populations derived from biparental crosses. However, the algorithm implemented in the popular QTL Cartographer software may not completely ensure all its advantageous properties. In addition, different background marker selection methods may give very different mapping results, and the nature of the preferred method is not clear. A modified algorithm called inclusive composite interval mapping (ICIM) is proposed in this article. In ICIM, marker selection is conducted only once through stepwise regression by considering all marker information simultaneously, and the phenotypic values are then adjusted by all markers retained in the regression equation except the two markers flanking the current mapping interval. The adjusted phenotypic values are finally used in interval mapping (IM). The modified algorithm has a simpler form than that used in CIM, but a faster convergence speed. ICIM retains all advantages of CIM over IM and avoids the possible increase of sampling variance and the complicated background marker selection process in CIM. Extensive simulations using two genomes and various genetic models indicated that ICIM has increased detection power, a reduced false detection rate, and less biased estimates of QTL effects.     

Parallel computing in interval mapping of quantitative trait loci.

Linear regression analysis is considered the least computationally demanding method for mapping quantitative trait loci (QTL). However, simultaneous search for multiple QTL, the use of permutations to obtain empirical significance thresholds, and larger experimental studies significantly increase the computational demand. This report describes an easily implemented parallel algorithm, which significantly reduces the computing time in both QTL mapping and permutation testing. In the example provided, the analysis time was decreased to less than 15% of a single processor system by the use of 18 processors. We indicate how the efficiency of the analysis could be improved by distributing the computations more evenly to the processors and how other ways of distributing the data facilitate the use of more processors. The use of parallel computing in QTL mapping makes it possible to routinely use permutations to obtain empirical significance thresholds for multiple traits and multiple QTL models. It could also be of use to improve the computational efficiency of the more computationally demanding QTL analysis methods.     

Simultaneous mapping of epistatic QTL in DU6i × DBA/2 mice

We have mapped epistatic quantitative trait loci (QTL) in an F₂ cross between DU6i × DBA/2 mice. By including these epistatic QTL and their interaction parameters in the genetic model, we were able to increase the genetic variance explained substantially (8.8%–128.3%) for several growth and body composition traits. We used an analysis method based on a simultaneous search for epistatic QTL pairs without assuming that the QTL had any effect individually. We were able to detect several QTL that could not be detected in a search for marginal QTL effects because the epistasis cancelled out the individual effects of the QTL. In total, 23 genomic regions were found to contain QTL affecting one or several of the traits and eight of these QTL did not have significant individual effects. We identified 44 QTL pairs with significant effects on the traits, and, for 28 of the pairs, an epistatic QTL model fit the data significantly better than a model without interactions. The epistatic pairs were classified by the significance of the epistatic parameters in the genetic model, and visual inspection of the two-locus genotype means identified six types of related genotype–phenotype patterns among the pairs. Five of these patterns resembled previously published patterns of QTL interactions.     

Approaches to interval mapping of QTL in a multigeneration pedigree: the example of porcine chromosome 4

Quantitative trait loci (QTLs) have been mapped in many studies of F2 populations derived from crosses between diverse lines. One approach to confirming these effects and improving the mapping resolution is genetic chromosome dissection through a backcrossing programme. Analysis by interval mapping of the data generated is likely to provide additional power and resolution compared with treating data marker by marker. However, interval mapping approaches for such a programme are not well developed, especially where the founder lines were outbred. We explore alternative approaches to analysis using, as an example, data from chromosome 4 in an intercross between wild boar and Large White pigs where QTLs have been previously identified. A least squares interval mapping procedure was used to study growth rate and carcass traits in a subsequent second backcross generation (BC2). This procedure requires the probability of inheriting a wild boar allele for each BC2 animal for locations throughout the chromosome. Two methods for obtaining these probabilities were compared: stochastic or deterministic. The two methods gave similar probabilities for inheriting wild boar alleles and, hence, gave very similar results from the QTL analysis. The deterministic approach has the advantage of being much faster to run but requires specialized software. A QTL for fatness and for growth were confirmed and, in addition, a QTL for piglet growth from weaning at 5 weeks up to 7 weeks of age and another for carcass length were detected.     

Strategies for genetic mapping of categorical traits

The search for efficient and powerful statistical methods and optimal mapping strategies for categorical traits under various experimental designs continues to be one of the main tasks in genetic mapping studies. Methodologies for genetic mapping of categorical traits can generally be classified into two groups, linear and non-linear models. We develop a method based on a threshold model, termed mixture threshold model to handle ordinal (or binary) data from multiple families. Monte Carlo simulations are done to compare its statistical efficiencies and properties of the proposed non-linear model with a linear model for genetic mapping of categorical traits using multiple families. The mixture threshold model has notably higher statistical power than linear models. There may be an optimal sampling strategy (family size vs number of families) in which genetic mapping reaches its maximal power and minimal estimation errors. A single large-sibship family does not necessarily produce the maximal power for detection of quantitative trait loci (QTL) due to genetic sampling of QTL alleles. The QTL allelic model has a marked impact on efficiency of genetic mapping of categorical traits in terms of statistical power and QTL parameter estimation. Compared with a fixed number of QTL alleles (two or four), the model with an infinite number of QTL alleles and normally distributed allelic effects results in loss of statistical power. The results imply that inbred designs (e.g. F₂ or four-way crosses) with a few QTL alleles segregating or reducing number of QTL alleles (e.g. by selection) in outbred populations are desirable in genetic mapping of categorical traits using data from multiple families. This revised version was published online in July 2006 with corrections to the Cover Date.     

A model for quantitative trait loci mapping,linkage phase,and segregation pattern estimation for a full-sib progeny

Quantitative trait loci (QTL) mapping is an important approach for the study of the genetic architecture of quantitative traits. For perennial species, inbred lines cannot be obtained due to inbreed depression and a long juvenile period. Instead, linkage mapping can be performed by using a full-sib progeny. This creates a complex scenario because both markers and QTL alleles can have different segregation patterns as well as different linkage phases between them. We present a two-step method for QTL mapping using full-sib progeny based on composite interval mapping (i.e., interval mapping with cofactors), considering an integrated genetic map with markers with different segregation patterns and conditional probabilities obtained by a multipoint approach. The model is based on three orthogonal contrasts to estimate the additive effect (one in each parent) and dominance effect. These estimatives are obtained using the EM algorithm. In the first step, the genome is scanned to detect QTL. After, segregation pattern and linkage phases between QTL and markers are estimated. A simulated example is presented to validate the methodology. In general, the new model is more effective than existing approaches, because it can reveal QTL present in a full-sib progeny that segregates in any pattern present and can also identify dominance effects. Also, the inclusion of cofactors provided more statistical power for QTL mapping.     

Improving genetic algorithms for protein folding simulations by systematic crossover

To improve protein folding simulations, we investigated a new search strategy in combination with the simple genetic algorithm on a two-dimensional lattice model. This search strategy, we called systematic crossover, couples the best individuals, tests every possible crossover point, and takes the two best individuals for the next generation. We compared the standard genetic algorithm with and without this new implementation for various chain lengths and showed that this strategy finds local minima with better energy values and is significantly faster in identifying the global minimum than the standard genetic algorithm.     

Linkage analysis of molecular markers and quantitative trait loci in populations of inbred backcross lines of Brassica napus L.

Backcross populations are often used to study quantitative trait loci (QTL) after they are initially discovered in balanced populations, such as F(2), BC(1), or recombinant inbreds. While the latter are more powerful for mapping marker loci, the former have the reduced background genetic variation necessary for more precise estimation of QTL effects. Many populations of inbred backcross lines (IBLs) have been developed in plant and animal systems to permit simultaneous study and dissection of quantitative genetic variation introgressed from one source to another. Such populations have a genetic structure that can be used for linkage estimation and discovery of QTL. In this study, four populations of IBLs of oilseed Brassica napus were developed and analyzed to map genomic regions from the donor parent (a winter-type cultivar) that affect agronomic traits in spring-type inbreds and hybrids. Restriction fragment length polymorphisms (RFLPs) identified among the IBLs were used to calculate two-point recombination fractions and LOD scores through grid searches. This information allowed the enrichment of a composite genetic map of B. napus with 72 new RFLP loci. The selfed and hybrid progenies of the IBLs were evaluated during two growing seasons for several agronomic traits. Both pedigree structure and map information were incorporated into the QTL analysis by using a regression approach. The number of QTL detected for each trait and the number of effective factors calculated by using biometrical methods were of similar magnitude. Populations of IBLs were shown to be valuable for both marker mapping and QTL analysis.     

The effects of selective genotyping on estimates of proportion of recombination between linked quantitative trait loci

Selective genotyping is the marker assay of only the more extreme phenotypes for a quantitative trait and is intended to increase the efficiency of quantitative trait loci (QTL) mapping. We show that selective genotyping can bias estimates of the recombination frequency between linked QTLs — upwardly when QTLs are in repulsion phase, and downwardly when QTLs are in coupling phase. We examined these biases under simple models involving two QTLs segregating in a backcross or F₂ population, using both analytical models and computer simulations. We found that bias is a function of the proportion selected, the magnitude of QTL effects, distance between QTLs and the dominance of QTLs. Selective genotyping thus may decrease the power of mapping multiple linked QTLs and bias the construction of a marker map. We suggest a large proportion than previously suggested (50%) or the entire population be genotyped if linked QTLs of large effects (explain > 10% phenotypic variance) are evident. New models need to be developed to explicitly incorporate selection into QTL map construction.     

Mapping quantitative trait loci controlling silking date in a diallel cross among four lines of maize

 We describe and apply an interval mapping method for quantitative trait locus (QTL) detection using F₃ and testcross progenies derived from F₂ populations obtained from a diallel cross among four elite lines of maize. Linear model-based procedures were used for the test and estimation of putative QTL effects together with genetic interactions including epistasis. We mapped QTL associated with silking date and explored their genetic effects. Ten QTL were detected, and these explained more than 40% of the phenotypic variance. Most of these QTL had consistent and stable effects among genetic backgrounds and did not show significant epistasis. QTL-by-environment interaction was important for four QTL and was essentially due to changes in magnitude of allelic effects. These results show the efficiency of our method in several genetic situations as well as the power of the diallel design in detecting QTL simultaneously over several populations. Received: 2 September 1996 / Accepted: 20 December 1996     

The role of epistasis in the manifestation of heterosis: a systems-oriented approach

Heterosis is widely used in breeding, but the genetic basis of this biological phenomenon has not been elucidated. We postulate that additive and dominance genetic effects as well as two-locus interactions estimated in classical QTL analyses are not sufficient for quantifying the contributions of QTL to heterosis. A general theoretical framework for determining the contributions of different types of genetic effects to heterosis was developed. Additive x additive epistatic interactions of individual loci with the entire genetic background were identified as a major component of midparent heterosis. On the basis of these findings we defined a new type of heterotic effect denoted as augmented dominance effect di* that comprises the dominance effect at each QTL minus half the sum of additive x additive interactions with all other QTL. We demonstrate that genotypic expectations of QTL effects obtained from analyses with the design III using testcrosses of recombinant inbred lines and composite-interval mapping precisely equal genotypic expectations of midparent heterosis, thus identifying genomic regions relevant for expression of heterosis. The theory for QTL mapping of multiple traits is extended to the simultaneous mapping of newly defined genetic effects to improve the power of QTL detection and distinguish between dominance and overdominance.     

On the use of double haploids for detecting QTL in outbred populations

The power to detect quantitative trait loci (QTL) using the double haploid (DH), full-sib (FS) and hierarchical (HI) designs implemented in outbred fish populations was assessed for interval mapping using deterministic methods. The predictions were tested using simulation. The DH design was most efficient for the range of designs and parameters considered and was most beneficial when the FS design was not very powerful. The difference between the design was largest for a low amount of residual genetic variation. Accounting for an increase of the environmental variance due to the genetic constitution of the double haploid progeny changed the magnitude of the power, but the ranking of the designs remained the same. As large full sib family sizes can be obtained in fish, the practical value of HI designs as a strategy for increasing the power of QTL mapping experiments is limited when compared with the FS design. Overall, the results suggested that the DH design could be a very useful tool for QTL mapping in fish, and of particular importance when the effect of the QTL is low and the residual genetic variation from other chromosomes can be controlled by using multiple markers.     

Bay-0 x Shahdara recombinant inbred line population: a powerful tool for the genetic dissection of complex traits in Arabidopsis

Natural genetic variation in Arabidopsis is considerable, but has not yet been used extensively as a source of variants to identify new genes of interest. From the cross between two genetically distant ecotypes, Bay-0 and Shahdara, we generated a Recombinant Inbred Line (RIL) population dedicated to Quantitative Trait Locus (QTL) mapping. A set of 38 physically anchored microsatellite markers was created to construct a robust genetic map from the 420 F6 lines. These markers, evenly distributed throughout the five chromosomes, revealed a remarkable equilibrium in the segregation of parental alleles in the genome. As a model character, we have analysed the genetic basis of variation in flowering time in two different environments. The simultaneous mapping of both large- and small-effect QTLs responsible for this variation explained 90% of the total genotypic variance. Two of the detected QTLs colocalize very precisely with FRIGIDA and FLOWERING LOCUS C genes; we provide information on the polymorphism of genes confirming this hypothesis. Another QTL maps in a region where no QTL had been found previously for this trait. This confirms the accuracy of QTL detection using the Bay-0 x Shahdara RIL population, which constitutes the largest in size available so far in Arabidopsis. As an alternative to mutant analysis, this population represents a powerful tool which is currently being used to undertake the genetic dissection of complex metabolic pathways.     

On the use of mathematically-derived traits in QTL mapping

Mathematically-derived traits from two or more component traits, either by addition, subtraction, multiplication, or division, have been frequently used in genetics and breeding. When used in quantitative trait locus (QTL) mapping, derived traits sometimes show discrepancy with QTL identified for the component traits. We used three QTL distributions and three genetic effects models, and an actual maize mapping population, to investigate the efficiency of using derived traits in QTL mapping, and to understand the genetic and biological basis of derived-only QTL, i.e., QTL identified for a derived trait but not for any component trait. Results indicated that the detection power of the four putative QTL was consistently greater than 90% for component traits in simulated populations, each consisting of 200 recombinant inbred lines. Lower detection power and higher false discovery rate (FDR) were observed when derived traits were used. In an actual maize population, simulations were designed based on the observed QTL distributions and effects. When derived traits were used, QTL detected for both component and derived traits had comparable power, but those detected for component traits but not for derived traits had low detection power. The FDR from subtraction and division in the maize population were higher than the FDR from addition and multiplication. The use of derived traits increased the gene number, caused higher-order gene interactions than observed in component traits, and possibly complicated the linkage relationship between QTL as well. The increased complexity of the genetic architecture with derived traits may be responsible for the reduced detection power and the increased FDR. Derived-only QTL identified in practical genetic populations can be explained either as minor QTL that are not significant in QTL mapping of component traits, or as false positives.