FunMap: functional mapping of complex traits

SUMMARY: FunMap is a Web-based user interface designed to map quantitative trait loci (QTL) affecting function-valued traits or infinite-dimensional traits in well-structured pedigrees or natural populations. User input includes three files: longitudinal trait data, marker genotypes and/or a linkage map. This software allows for a systematic genome-wide scan and significance test of QTL throughout the map. The dynamic change of QTL effects during the time course of growth is automatically drawn, from which specific biological hypotheses regarding the genetic control mechanisms of growth and development can be tested. AVAILABILITY: http://web.biostat.ufl.edu/~cma/genetics/software.html     

Bayesian functional mapping of dynamic quantitative traits

Without consideration of other linked QTLs responsible for dynamic trait, original functional mapping based on a single QTL model is not optimal for analyzing multiple dynamic trait loci. Despite that composite functional mapping incorporates the effects of genetic background outside the tested QTL in mapping model, the arbitrary choice of background markers also impact on the power of QTL detection. In this study, we proposed Bayesian functional mapping strategy that can simultaneously identify multiple QTL controlling developmental patterns of dynamic traits over the genome. Our proposed method fits the change of each QTL effect with the time by Legendre polynomial and takes the residual covariance structure into account using the first autoregressive equation. Also, Bayesian shrinkage estimation was employed to estimate the model parameters. Especially, we specify the gamma distribution as the prior for the first-order auto-regressive coefficient, which will guarantee the convergence of Bayesian sampling. Simulations showed that the proposed method could accurately estimate the QTL parameters and had a greater statistical power of QTL detection than the composite functional mapping. A real data analysis of leaf age growth in rice is used for the demonstration of our method. It shows that our Bayesian functional mapping can detect more QTLs as compared to composite functional mapping.     

A semiparametric approach for composite functional mapping of dynamic quantitative traits

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.     

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.     

Model selection for quantitative trait loci mapping in a full-sib family

Statistical methods for mapping quantitative trait loci (QTLs) in full-sib forest trees, in which the number of alleles and linkage phase can vary from locus to locus, are still not well established. Previous studies assumed that the QTL segregation pattern was fixed throughout the genome in a full-sib family, despite the fact that this pattern can vary among regions of the genome. In this paper, we propose a method for selecting the appropriate model for QTL mapping based on the segregation of different types of markers and QTLs in a full-sib family. The QTL segregation patterns were classified into three types: test cross (1:1 segregation), F₂ cross (1:2:1 segregation) and full cross (1:1:1:1 segregation). Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Laplace-empirical criterion (LEC) were used to select the most likely QTL segregation pattern. Simulations were used to evaluate the power of these criteria and the precision of parameter estimates. A Windows-based software was developed to run the selected QTL mapping method. A real example is presented to illustrate QTL mapping in forest trees based on an integrated linkage map with various segregation markers. The implications of this method for accurate QTL mapping in outbred species are discussed.     

Functional mapping of dynamic traits with robust t-distribution

Functional mapping has been a powerful tool in mapping quantitative trait loci (QTL) underlying dynamic traits of agricultural or biomedical interest. In functional mapping, multivariate normality is often assumed for the underlying data distribution, partially due to the ease of parameter estimation. The normality assumption however could be easily violated in real applications due to various reasons such as heavy tails or extreme observations. Departure from normality has negative effect on testing power and inference for QTL identification. In this work, we relax the normality assumption and propose a robust multivariate t-distribution mapping framework for QTL identification in functional mapping. Simulation studies show increased mapping power and precision with the t distribution than that of a normal distribution. The utility of the method is demonstrated through a real data analysis.     

A likelihood approach for mapping growth trajectories using dominant markers in a phase-unknown full-sib family

Dominant markers have been commonly used in mapping quantitative trait loci (QTLs) in outcrossing species, in which not much prior genome information is available. But the dominant nature of these markers may lead to reduced QTL mapping precision and power. A new statistical method is proposed to incorporate growth laws into a QTL mapping framework, under which the use of the efficiency of dominant markers can be increased. This new method can be used to identify specific QTLs affecting differentiation in growth trajectories, and further estimate the timing of a QTL to turn on, or turn off, affecting growth during the entire ontogeny of a species. Using this method based on dominant markers we have successfully mapped a QTL for stem height growth trajectories to a linkage group in a forest tree. The implications of this method for the understanding of the genetic architecture of growth using dominant markers are discussed.Communicated by F. Salamini     

A fast algorithm for functional mapping of complex traits

By integrating the underlying developmental mechanisms for the phenotypic formation of traits into a mapping framework, functional mapping has emerged as an important statistical approach for mapping complex traits. In this note, we explore the feasibility of using the simplex algorithm as an alternative to solve the mixture-based likelihood for functional mapping of complex traits. The results from the simplex algorithm are consistent with those from the traditional EM algorithm, but the simplex algorithm has considerably reduced computational times. Moreover, because of its nonderivative nature and easy implementation with current software, the simplex algorithm enjoys an advantage over the EM algorithm in the dynamic modeling and analysis of complex traits.     

A non-stationary model for functional mapping of complex traits

SUMMARY: Understanding the genetic control of growth is fundamental to agricultural, evolutionary and biomedical genetic research. In this article, we present a statistical model for mapping quantitative trait loci (QTL) that are responsible for genetic differences in growth trajectories during ontogenetic development. This model is derived within the maximum likelihood context, implemented with the expectation-maximization algorithm. We incorporate mathematical aspects of growth processes to model the mean vector and structured antedependence models to approximate time-dependent covariance matrices for longitudinal traits. Our model has been employed to map QTL that affect body mass growth trajectories in both male and female mice of an F2 population derived from the Large and Small mouse strains. The results from this model are compared with those from the autoregressive-based functional mapping approach. Based on results from computer simulation studies, we suggest that these two models are alternative to one another and should be used simultaneously for the same dataset.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behavior of a system, differential equations have proven to be powerful for modeling and unraveling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge-Kutta fourth order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

QTL mapping of potato chip color and tuber traits within an autotetraploid family

Cultivated potato (Solanum tuberosum L.) is a highly heterozygous autotetraploid crop species, and this creates challenges for traditional line development and molecular breeding. Recent availability of a single-nucleotide polymorphism (SNP) array with 8303 features and software packages for linkage and association mapping in autotetraploid species present new opportunities for the identification of genomic regions that contribute to high-value traits in cultivated potato. A biparental tetraploid potato population was evaluated across three field seasons and storage trials in order to identify quantitative trait loci (QTL) for multiple tuber traits including fried chip color after 5.5–7.2 °C storage. Tetra-allelic dosage information was used to construct a genetic linkage map that covered 1041 cM and contained 2095 SNP markers with a median marker interval of 0.4 cM. A total of 41 QTL were identified for flower color, tuber yield, tuber number per plant, tuber weight, tuber size, and chip color after various storage regimes. Moderate effect QTL for chip color at 3 months were identified that co-localized with candidate genes vacuolar invertase (VInv), invertase inhibitor (INH2), and apoplastic invertase (Inv _ap -b). A separate QTL for chip color after 6 months of storage was identified in the short arm of chromosome 2, and this locus may contribute to variation in senescent sweetening resistance. QTL for tuber weight, length, and width co-localized with a known QTL for plant maturity on chromosome 5. Genome-wide association mapping using a polyploid model detected the tuber size QTL and identified a number of candidate SNPs, but was unable to detect markers significantly associated with chip color.     

Geographic distributions shape the functional traits in a large mammalian family

Traits of organisms are shaped by their living environments and also determined in part by their phylogenetic relationships. For example, phylogenetic relationships often affect the geographic distributions of animals and cause variation in their living environments, which usually play key roles in the life history and determine the functional traits of species. As an ancient family of mammals, bears widely distribute and have evolved some specific strategies for survival and reproduction during their long‐term evolutionary histories. Many studies on the ecology of bears have been conducted in recent decades, but few have focused on the relationships between their geographic distributions and ecological adaptations. Here, using bears as a model system, we collected and reanalyzed data from the available literatures to explore how geographic distributions and phylogenetic relationships shape the functional traits of animals. We found a positive relationship between phylogenetic relatedness and geographic distributions, with bears distributed in adjacent areas applying more similar strategies to survive and reproduce: (a) Bears living at high latitudes consumed a higher proportion of vertebrates, which may provide more fat for adaptation to low temperatures, and (b) their reproduction rhythms follow fluctuations in seasonal forage availability and quality, in which bears reach mating status from March to May and give birth in approximately November or later.     

Niche overlap estimates based on quantitative functional traits: a new family of non-parametric indices

The concept of niche overlap appears in studies of the mechanisms of the maintenance of species diversity, in searches for assembly rules, and in estimation of within-community species redundancy. For plant traits measured on a continuous scale, existing indices are inadequate because they split the scale into a number of categories thus losing information. An index is easy to construct if we assume a normal distribution for each trait within a species, but this assumption is rarely true. We extend and apply an index, NO_K, which is based on kernel density functions, and can therefore work with distributions of any shape without prior assumptions. For cases where the ecologist wishes to downweight traits that are inter-correlated, we offer a variant that does this: NO_Kw. From either of these indices, an index of the mean niche overlap in a community can be calculated: NO_K,community and NO_Kw,community. For all these indices, the variance can be calculated and formulae for this are given. To give examples of the new indices in use, we apply them to a coastal fish dataset and a sand-dune plant dataset. The former exhibits considerable non-normality, emphasising the need for kernel-based indices. Accordingly, there was a considerable difference in index values, with those for an index based on a normal distribution being significantly higher than those from an index which, being based on kernel fitting, is not biased by an assumption for the distribution. The NO_K values were ecologically consistent for the fish species concerned, varying from 0.02 to 0.53. The sand-dune plant data also showed a wide range of overlap values. Interestingly, the least overlap was between two graminoids, which would have been placed in the same functional group in the coarse classification often used in functional-type/ecosystem-function work.     

Linkage information content and efficiency of full-sib and half-sib designs for gene mapping

The accuracy of a genetic map depends on the amount of linkage information contained in the data set used for construction of the map. The amount of linkage information is related to the designs employed for linkage analysis. The purpose of this study was to provide general formulations for various genotyping schemes and family structures in order to evaluate the amount of linkage information in a data set. Linkage information content (LIC) was defined as the frequency of fully informative gametes, which are gametes from doubly heterozygous parents with known linkage phases. Depending on the design, LIC is based on two generations if the parental phases are determined statistically, or three generations if the parental phases are determined genetically. Different schemes were considered in deriving LIC: (1) genotyping of one parent or two parents, and (2) genotyping of two or three generation families. The LIC for a full-sib design was found to be generally greater than for a half-sib design but requires typing a large number of individuals when at least one locus has only two alleles. The efficiency of the full-sib design is reduced significantly if a sex-specific linkage map is sought.     

A differential equation model for functional mapping of a virus-cell dynamic system

The dynamic pattern of viral load in a patient’s body critically depends on the host’s genes. For this reason, the identification of those genes responsible for virus dynamics, although difficult, is of fundamental importance to design an optimal drug therapy based on patients’ genetic makeup. Here, we present a differential equation (DE) model for characterizing specific genes or quantitative trait loci (QTLs) that affect viral load trajectories within the framework of a dynamic system. The model is formulated with the principle of functional mapping, originally derived to map dynamic QTLs, and implemented with a Markov chain process. The DE-integrated model enhances the mathematical robustness of functional mapping, its quantitative prediction about the temporal pattern of genetic expression, and therefore its practical utilization and effectiveness for gene discovery in clinical settings. The model was used to analyze simulated data for viral dynamics, aimed to investigate its statistical properties and validate its usefulness. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the molecular genetic mechanism of virus dynamics and disease progression.     

Genetic mapping of EST-derived simple sequence repeats (EST-SSRs) to identify QTL for leaf morphological characters in a Quercus robur full-sib family

The availability of genomic resources such as expressed sequence tag-derived simple sequence repeat (EST-SSR) markers in adaptive genes with high transferability across related species allows the construction of genetic maps and the comparison of genome structure and quantitative trait loci (QTL) positions. In the present study, genetic linkage maps were constructed for both parents of a Quercus robur × Q. robur ssp. slavonica full-sib pedigree. A total of 182 markers (61 AFLPs, 23 nuclear SSRs, 98 EST-SSRs) and 172 markers (49 AFLPs, 21 nSSRs, 101 EST-SSRs, 1 isozyme) were mapped on the female and male linkage maps, respectively. The total map length and average marker spacing were 1,038 and 5.7 cM for the female map and 998.5 and 5.8 cM for the male map. A total of 68 nuclear SSRs and EST-SSRs segregating in both parents allowed to define homologous linkage groups (LG) between both parental maps. QTL for leaf morphological traits were mapped on all 12 LG at a chromosome-wide level and on 6 LG at a genome-wide level. The phenotypic effects explained by each single QTL ranged from 4.0 % for leaf area to 15.8 % for the number of intercalary veins. QTL clusters for leaf characters that discriminate between Q. robur and Quercus petraea were mapped reproducibly on three LG, and some putative candidate genes among potentially many others were identified on LG3 and LG5. Genetic linkage maps based on EST-SSRs can be valuable tools for the identification of genes involved in adaptive trait variation and for comparative mapping.     

Multivariate eQTL mapping uncovers functional variation on the X-chromosome associated with complex disease traits

Very few studies have investigated the associations between genetic polymorphisms and gene expression on the X-chromosome. This is a major bottleneck when conducting functional follow-up studies of trait-associated variants, as those identified in genome-wide association studies (GWAS). We used a multivariate approach to test the association between individual single nucleotide polymorphisms (SNPs) and exon expression levels measured in 356 Epstein–Barr virus-transformed lymphoblastoid cell lines (LCLs) from the Geuvadis RNA sequencing project to identify SNPs associated with variation in gene expression on the X-chromosome, which we refer to as eSNPs. At an FDR of 5 %, we discovered 548 independent [linkage disequilibrium (LD) r ² < 0.1] eSNPs on the X-chromosome. Of these, 35 were in LD (r ² > 0.8) with previously published disease- or trait-associated variants identified through GWAS. One of the strongest eSNPs identified was rs35975601, which was associated with F8A1 expression (p value = 3 × 10^?20) and was in LD with a type 1 diabetes risk variant. Additionally, we identified a number of genes for which eSNPs were in LD with multiple diseases or traits, including DNASE1L1 which was mapped to bilirubin levels, type 1 diabetes and schizophrenia. Our results also indicate that multivariate exon-level analysis provides a more powerful approach than univariate gene-level analysis, particularly when SNPs influence the expression of different exons with different magnitude and/or direction of effect. The associations identified in our study may provide new insights into the molecular process by which gene expression may contribute to trait variation or disease risk in humans.     

Functional mapping - how to map and study the genetic architecture of dynamic complex traits

The development of any organism is a complex dynamic process that is controlled by a network of genes as well as by environmental factors. Traditional mapping approaches for analysing phenotypic data measured at a single time point are too simple to reveal the genetic control of developmental processes. A general statistical mapping framework, called functional mapping, has been proposed to characterize, in a single step, the quantitative trait loci (QTLs) or nucleotides (QTNs) that underlie a complex dynamic trait. Functional mapping estimates mathematical parameters that describe the developmental mechanisms of trait formation and expression for each QTL or QTN. The approach provides a useful quantitative and testable framework for assessing the interplay between gene actions or interactions and developmental changes.     

Combined S2 and crossbred family selection in full-sib reciprocal recurrent selection

Summary A method (CRRS) that combines S₂ and crossbred family selection in full-sib reciprocal recurrent selection (FSRRS) is proposed. The method requires four generations per cycle in single-eared maize populations. Selection is based on performance of S₂ and full-sib families by applying selection index theory. Equations to estimate the coefficients included in the index are given. These estimates are functions of the genetic and phenotypic variances and covariances among and between the two kinds of families. Comparisons of FSRRS and CRRS under equivalent amount of effort show that CRRS has some advantage over FSRRS for low heritability of the trait being selected (e.g., maize yield) and when only one or two locations with two replications are involved in the selection experiment.Joint contribution: Institute Nacional de Investigaciones Agrarias, La Coruna, Spain; and Agricultural Research, Science and Education Administration, U.S. Department of Agriculture, and Journal Paper No. J-10118 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA 50011. Project 2194