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.     

A statistical model for the identification of genes governing the incidence of cancer with age

The cancer incidence increases with age. This epidemiological pattern of cancer incidence can be attributed to molecular and cellular processes of individual subjects. Also, the incidence of cancer with ages can be controlled by genes. Here we present a dynamic statistical model for explaining the epidemiological pattern of cancer incidence based on individual genes that regulate cancer formation and progression. We incorporate the mathematical equations of age-specific cancer incidence into a framework for functional mapping aimed at identifying quantitative trait loci (QTLs) for dynamic changes of a complex trait. The mathematical parameters that specify differences in the curve of cancer incidence among QTL genotypes are estimated within the context of maximum likelihood. The model provides testable quantitative hypotheses about the initiation and duration of genetic expression for QTLs involved in cancer progression. Computer simulation was used to examine the statistical behavior of the model. The model can be used as a tool for explaining the epidemiological pattern of cancer incidence.     

Semiparametric functional mapping of quantitative trait loci governing long-term HIV dynamics

MOTIVATION: Functional mapping has proven to be powerful for characterizing quantitative trait loci (QTL) that control complex dynamic traits. More recently, functional mapping has been extended to identify the host QTL responsible for HIV dynamics by incorporating a parametric bi-exponential function for earlier stages of viral load trajectories. However, existing functional mapping cannot be used to map long-term HIV dynamics because no mathematical functions are available for later stages of HIV dynamic changes. RESULTS: We derived a statistical model for functional mapping of dynamic QTL through characterizing HIV load trajectories during a long-term period semiparametrically. The new model was constructed within the maximum likelihood framework and implemented with the EM-simplex algorithm. It allows for the test of differences in the genetic control of short- and long-term HIV dynamics and the characterization of the effects of viral-host genome interaction. Extensive simulation studies have been performed to test the statistical behavior of this model. The new model will provide an important tool for genetic and genomic studies of human complex diseases like HIV/AIDS and their pathological progression. AVAILABILITY: Available on request from the corresponding author.     

Computational identification of genes modulating stem height–diameter allometry

The developmental variation in stem height with respect to stem diameter is related to a broad range of ecological and evolutionary phenomena in trees, but the underlying genetic basis of this variation remains elusive. We implement a dynamic statistical model, functional mapping, to formulate a general procedure for the computational identification of quantitative trait loci (QTLs) that control stem height–diameter allometry during development. Functional mapping integrates the biological principles underlying trait formation and development into the association analysis of DNA genotype and endpoint phenotype, thus providing an incentive for understanding the mechanistic interplay between genes and development. Built on the basic tenet of functional mapping, we explore two core ecological scenarios of how stem height and stem diameter covary in response to environmental stimuli: (i) trees pioneer sunlit space by allocating more growth to stem height than diameter and (ii) trees maintain their competitive advantage through an inverse pattern. The model is equipped to characterize ‘pioneering’ QTLs (piQTLs) and ‘maintaining’ QTLs (miQTLs) which modulate these two ecological scenarios, respectively. In a practical application to a mapping population of full‐sib hybrids derived from two Populus species, the model has well proven its versatility by identifying several piQTLs that promote height growth at a cost of diameter growth and several miQTLs that benefit radial growth at a cost of height growth. Judicious application of functional mapping may lead to improved strategies for studying the genetic control of the formation mechanisms underlying trade‐offs among quantities of assimilates allocated to different growth parts.     

Functional mapping of human growth trajectories

Human height is an important trait from biological and social perspectives. Genes have been widely recognized to be involved in human body growth, but their detailed controlling mechanisms are poorly understood. Here, we present a computational model for functional mapping of quantitative trait loci (QTLs) that control trajectories of human height growth through an interactive network. The model integrates mathematical equations of human growth curves into the mixture model-based functional mapping framework, allowing the identification and mapping of individual QTLs responsible for the developmental pattern of human growth. The model was derived on a random sample of subjects from a natural population, for each of which molecular markers within candidate genes or throughout the entire genome are typed and height data from childhood to adulthood are collected. A series of testable hypotheses are formulated about the genetic control of developmental timing and duration at different stages. The model was used to characterize epistatic QTLs for height growth hidden in 548 Japanese girls which is a semi-real data set with simulated the marker genotypes. With an increasing availability of genetic polymorphic data, the model will have great implications for probing the genetic and developmental mechanisms of human body growth and its associated diseases.     

Nonparametric functional mapping of quantitative trait loci underlying programmed cell death

The development of an organism represents a complex dynamic process, which is controlled by a network of genes and multiple environmental factors. Programmed cell death (PCD), a physiological cell suicide process, occurs during the development of most organisms and is, typically, a complex dynamic trait. Understanding how genes control this complex developmental process has been a long-standing topic in PCD studies. In this article, we propose a nonparametric model, based on orthogonal Legendre polynomials, to map genes or quantitative trait loci (QTLs) that govern the dynamic features of the PCD process. The model is built under the maximum likelihood-based functional mapping framework and is implemented with the EM algorithm. A general information criterion is proposed for selecting the optimal Legendre order that best fits the dynamic pattern of the PCD process. The consistency of the order selection criterion is established. A nonstationary structured antedependence model (SAD) is applied to model the covariance structure among the phenotypes measured at different time points. The developed model generates a number of hypothesis tests regarding the genetic control mechanism of the PCD process. Extensive simulation studies are conducted to investigate the statistical behavior of the model. Finally, we apply the model to a rice tiller number data set in which several QTLs are identified. The developed model provides a quantitative and testable framework for assessing the interplay between genes and the developmental PCD process, and will have great implications for elucidating the genetic architecture of the PCD process.     

Unconditional and conditional QTL mapping for the developmental behavior of tiller number in rice (<Emphasis Type="Italic">Oryza sativa</Emphasis> L.)

A single segment substitution population of 26 lines and their recipient parent Hua-jing-xian 74 (HJX74) were selected as experimental materials for analyzing the developmental behavior of tiller number in rice. By the unconditional QTL (quantitative trait locus) mapping method, a total number of 14 SSSLs were detected with QTLs controlling rice tiller number. The number of QTLs significantly affecting tiller number and their effect values estimated differed across measuring stages. More QTLs could be detected based on time-dependent measures of different stages. By the conditional QTL mapping method, it is possible to reveal net expression of gene in a time interval. 14 QTLs on tiller number expressed their effects in dynamic patterns of themselves during whole ontogeny. They exhibited mainly negative effects within 7 days after transplanting. During 7–21 days, QTLs were in active status and expressed larger positive effects. In the mid-period of 21–35 days, they had opposite genetic effects to wither tillers. Since then these QTLs expressed positive effects again to cause the appearance of noneffective tillers. The dynamics of QTL effects was in agreement with the actual change of tillers. Mapping QTL combining unconditional with conditional analysis for time-dependent measures is helpful to understand roundly the genetic bases for the development of quantitative traits.     

Genetic basis and mapping of the resistance to rice yellow mottle virus. II. Evidence of a complementary epistasis between two QTLs

 The genetic basis of resistance to rice yellow mottle virus (RYMV) was studied in a doubled-haploid (DH) population derived from a cross between the very susceptible indica variety ‘IR64’ and the resistant upland japonica variety Azucena. As a quantitative trait locus (QTL) involved in virus content estimated with an ELISA test has been previously identified on chromosome 12, we performed a wide search for interactions between this QTL and the rest of the genome, and between this QTL and morphological traits segregating in the population. Multiple regression with all identified genetic factors was used to validate the interactions. Significant epistasis accounting for a major part of the total genetic variation was observed. A complementary epistasis between the QTL located on chromosome 12 and a QTL located on chromosome 7 could be the major genetic factor controlling the virus content. Resistance was also affected by a morphology-dependent mechanism since tillering was interfering with the resistance mechanism conditioned by the epistasis between the two QTLs. Marker-assisted backcross breeding was developed to introgress the QTLs of chromosome 7 and chromosome 12 in the susceptible ‘IR64’ genetic background. First results confirmed that if both QTLs do not segregate in a backcross-derived F₂ population, then the QTL of chromosome 12 cannot explain differences in virus content. A near-isogenic line (NIL) approach is currently being developed to confirm the proposed genetic model of resistance to RYMV. Received: 20 April 1990 / Accepted: 30 April 1998     

3FunMap: full-sib family functional mapping of dynamic traits

MOTIVATION: Functional mapping that embeds the developmental mechanisms of complex traits shows great power to study the dynamic pattern of genetic effects triggered by individual quantitative trait loci (QTLs). A full-sib family, produced by crossing two heterozygous parents, is characteristic of uncertainties about cross-type at a locus and linkage phase between different loci. Integrating functional mapping into a full-sib family requires a model selection procedure capable of addressing these uncertainties. 3FunMap, written in VC++ 6.0, provides a flexible and extensible platform to perform full-sib functional mapping of dynamic traits. Functions in the package encompass linkage phase determination, marker map construction and the pattern identification of QTL segregation, dynamic tests of QTL effects, permutation tests and numerical simulation. We demonstrate the features of 3FunMap through real data analysis and computer simulation. AVAILABILITY: http://statgen.psu.edu/software.     

QTL mapping of aroma compounds analysed by headspace solid-phase microextraction gas chromatography in the apple progeny ‘Discovery’ × ‘Prima’

Improving fruit quality of apple varieties is an important but complex breeding goal. Flavour is among the key factors of apple fruit quality but in spite of the analytical and biochemical knowledge about volatiles little is known about the genetic and molecular bases of apple aroma. The aim of this study was to use a saturated molecular linkage map of apple to identify QTLs for aroma compounds such as alcohols, esters and terpenes, but also for a number of unidentified volatile compounds (non-targeted analysis approach). Two parental genetic maps were constructed for the apple cultivars ‘Discovery’ and ‘Prima’ by using mainly AFLP and SSR markers. ‘Discovery’ and ‘Prima’ showed very different volatile patterns, and ‘Discovery’ mostly had the higher volatile concentrations in comparison with the Vf-scab resistant ‘Prima’ which has its origin in the small-fruited apple species Malus floribunda. About 50 putative QTLs for a total of 27 different apple fruit volatiles were detected through interval mapping by using genotypic data of 150 F₁ individuals of the mapping population ‘C3’ together with phenotypic data obtained by head-space solid phase microextraction gas chromatography. QTLs for volatile compounds putatively involved in apple aroma were found on 12 out of the 17 apple chromosomes, but they were not evenly dispersed. QTLs were mainly clustered on linkage groups LG 2, 3 and 9. In a first attempt, a LOX (lipoxygenase) candidate gene, putatively involved in volatile metabolism, was mapped on LG 9, genetically associated with a cluster of QTLs for ester-type volatiles. Implications for aroma breeding in apple are discussed.     

Influence of dent corn genetic backgrounds on QTL detection for plant-height traits and their relationships in high-oil maize

QTL mapping for plant-height traits has not been hitherto reported in high-oil maize. A high-oil maize inbred ‘GY220’ was crossed with two dent maize inbreds (‘8984’ and ‘8622’) to generate two connected F_2:3 populations. Four plant-height traits were evaluated in 284 and 265 F_2:3 families. Single-trait QTL mapping and multiple-trait joint QTL mapping was used to detect QTLs for the traits and the genetic relationship between plant height (PH) and two other plant-height traits. A total of 28 QTLs and 12 pairs of digenic interactions among detected QTLs for four traits were detected in the two F_2:3 families. Only one marker was shared between the two populations. Joint analysis of PH with ear height (EH) and PH with top height (TH) detected 32 additional QTLs. Our results showed that QTL detection for PH was dependent on the genetic background of dent corn inbreds. Multiple-trait joint QTL analysis could increase the number of detected QTLs.     

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.     

Functional mapping of quantitative trait loci associated with rice tillering

Several biologically significant parameters that are related to rice tillering are closely associated with rice grain yield. Although identification of the genes that control rice tillering and therefore influence crop yield would be valuable for rice production management and genetic improvement, these genes remain largely unidentified. In this study, we carried out functional mapping of quantitative trait loci (QTLs) for rice tillering in 129 doubled haploid lines, which were derived from a cross between IR64 and Azucena. We measured the average number of tillers in each plot at seven developmental stages and fit the growth trajectory of rice tillering with the Wang–Lan–Ding mathematical model. Four biologically meaningful parameters in this model––the potential maximum for tiller number (K), the optimum tiller time (t ₀), and the increased rate (r), or the reduced rate (c) at the time of deviation from t ₀––were our defined variables for multi-marker joint analysis under the framework of penalized maximum likelihood, as well as composite interval mapping. We detected a total of 27 QTLs that accounted for 2.49–8.54% of the total phenotypic variance. Nine common QTLs across multi-marker joint analysis and composite interval mapping showed high stability, while one QTL was environment-specific and three were epistatic. We also identified several genomic segments that are associated with multiple traits. Our results describe the genetic basis of rice tiller development, enable further marker-assisted selection in rice cultivar development, and provide useful information for rice production management.     

Mining of Candidate Maize Genes for Nitrogen Use Efficiency by Integrating Gene Expression and QTL Data

Breeding maize varieties for high nitrogen (N) use efficiency (NUE) by marker-assisted selection using NUE quantitative trait locus (QTL) or by genetic transfer of NUE-associated genes is a viable approach for increasing grain yield in N-limited production areas. In this investigation, we evaluated a set of introgression line populations under N supply and N deficiency conditions. From 42 QTLs for grain yield and yield components, 23 were identified under N supply conditions and 33 from N limited conditions. Meta-analysis of published maize NUE QTLs revealed 37 “consensus” QTLs, of which, 18 was detected under low N conditions. In addition, 258 unique ESTs associated with low N stress response, N uptake, transport, and assimilation were aligned on the maize genome by in silico mapping. Integrating the EST map with the QTL map has resulted in the identification of candidate NUE-associated genes of the following functional categories: N uptake, transport, and assimilation; carbon (C) metabolism and assimilation; and cascades of stress response and signal transduction genes. Nine candidates that have been introgressed into Ye478 significantly altered grain yield/yield components. It is suggested that the dynamics of interactions between C and N metabolism are important for maize yield. A high NUE variety should have a highly efficient C assimilation per unit N and actively express CO₂ assimilation-related genes under N-limited conditions.     

Functional mapping imprinted quantitative trait loci underlying developmental characteristics

Background

Genomic imprinting, a phenomenon referring to nonequivalent expression of alleles depending on their parental origins, has been widely observed in nature. It has been shown recently that the epigenetic modification of an imprinted gene can be detected through a genetic mapping approach. Such an approach is developed based on traditional quantitative trait loci (QTL) mapping focusing on single trait analysis. Recent studies have shown that most imprinted genes in mammals play an important role in controlling embryonic growth and post-natal development. For a developmental character such as growth, current approach is less efficient in dissecting the dynamic genetic effect of imprinted genes during individual ontology.

Results

Functional mapping has been emerging as a powerful framework for mapping quantitative trait loci underlying complex traits showing developmental characteristics. To understand the genetic architecture of dynamic imprinted traits, we propose a mapping strategy by integrating the functional mapping approach with genomic imprinting. We demonstrate the approach through mapping imprinted QTL controlling growth trajectories in an inbred F₂ population. The statistical behavior of the approach is shown through simulation studies, in which the parameters can be estimated with reasonable precision under different simulation scenarios. The utility of the approach is illustrated through real data analysis in an F₂ family derived from LG/J and SM/J mouse stains. Three maternally imprinted QTLs are identified as regulating the growth trajectory of mouse body weight.

Conclusion

The functional iQTL mapping approach developed here provides a quantitative and testable framework for assessing the interplay between imprinted genes and a developmental process, and will have important implications for elucidating the genetic architecture of imprinted traits.     

Modeling the Genetic Control of HIV-1 Dynamics After Highly Active Antiretroviral Therapy

The progression of HIV disease has been markedly slowed by the use of highly active antiretroviral therapy (HAART). However, substantial genetic variation was observed to occur among different people in the decay rate of viral loads caused by HAART. The characterization of specific genes involved in HIV dynamics is central to design personalized drugs for the prevention of this disease, but usually cannot be addressed by experimental methods alone rather than require the help of mathematical and statistical methods. A novel statistical model has been recently developed to detect genetic variants that are responsible for the shape of HAART-induced viral decay curves. This model was employed to an HIV/AIDS trial, which led to the identification of a major genetic determinant that triggers an effect on HIV dynamics. This detected major genetic determinant also affects several clinically important parameters, such as half-lives of infected cells and HIV eradication times.Key Words: Hardy-weinberg equilibrium, bi-exponential function, quantitative trait loci, HIV dynamics, functional mapping.     

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.     

QTL analysis of resistance to sharka disease in the apricot (Prunus armeniaca L.) ‘Polonais’ × ‘Stark Early Orange’ F1 progeny

Different hypotheses on the genetic control of the resistance to the plum pox virus (PPV) have been reported in apricot, but there was a lack of agreement about the number of loci involved. In recent years, apricot genetic maps have been constructed from progenies derived from ‘Stark Early Orange’ or ‘Goldrich’, two main sources of resistance, three of these including the mapping of the PPV resistance loci. As the location of the locus was not precisely established, we mapped the PPV resistance loci using interval mapping (IM), composite interval mapping (CIM), and the Kruskal–Wallis non-parametric test in the F₁ progeny derived from a cross between the susceptible cv. ‘Polonais’ and ’Stark Early Orange’. Four genomic regions were identified as being involved in PPV resistance. One of these mapped to the upper region of linkage group 1 of ‘Stark Early Orange’, and accounted for 56% of the phenotypic variation. Its location was similar to the one previously identified in ‘Goldrich’ and Prunus davidiana. In addition, a gene strongly associated to these major quantitative trait loci (QTL) was found to be related to PPV infection. Two putative QTLs were detected on linkage groups 3 of ‘Polonais’ and 5 of both ‘Polonais’ and ‘Stark Early Orange’ with both parametric and non-parametric methods at logarithm of odds (LOD) scores slightly above the detection threshold. The last QTL was only detected in the early stage of the infection. PPV resistance is, thus, controlled by a major dominant factor located on linkage group 1. The hypothesis of recessive factors with lower effect is discussed.     

Genomic regions affecting seed shattering and seed dormancy in rice

Non-shattering of the seeds and reduced seed dormancy were selected consciously and unconsciously during the domestication of rice, as in other cereals. Both traits are quantitative and their genetic bases are not fully elucidated, though several genes with relatively large effects have been identified. In the present study, we attempted to detect genomic regions associated with shattering and dormancy using 125 recombinant inbred lines obtained from a cross between cultivated and wild rice strains. A total of 147 markers were mapped on 12 rice chromosomes, and QTL analysis was performed by simple interval mapping and composite interval mapping. For seed shattering, two methods revealed the same four QTLs. On the other hand, for seed dormancy a number of QTLs were estimated by the two methods. Based on the results obtained with the intact and de-hulled seeds, QTLs affecting hull-imposed dormancy and kernel dormancy, respectively, were estimated. Some QTLs detected by simple interval mapping were not significant by composite interval mapping, which reduces the effects of residual variation due to the genetic background. Several chromosomal regions where shattering QTLs and dormancy QTLs are linked with each other were found. This redundancy of QTL associations was explained by ”multifactorial linkages” followed by natural selection favoring these two co-adapted traits. Received: 23 November 1998 / Accepted: 27 August 1999