It is now possible to use complete genetic linkage maps to locate major quantitative trait loci (QTLs) on chromosome regions. The current methods of QTL mapping (e.g., interval mapping, which uses a pair or two pairs of flanking markers at a time for mapping) can be subject to the effects of other linked QTLs on a chromosome because the genetic background is not controlled. As a result, mapping of QTLs can be biased, and the resolution of mapping is not very high. Ideally when we test a marker interval for a QTL, we would like our test statistic to be independent of the effects of possible QTLs at other regions of the chromosome so that the effects of QTLs can be separated. This test statistic can be constructed by using a pair of markers to locate the testing position and at the same time using other markers to control the genetic background through a multiple regression analysis. Theory is developed in this paper to explore the idea of a conditional test via multiple regression analysis. Various doing, we can eliminate biases in our mapping from those QTLs and potentially increase the precision of mapping. Such a test statistic can be constructed by combining Lander-Botstein's interval mapping with multiple regression analysis. In this paper, theoretical implications of multiple regression analysis in relation to QTL mapping are explored. It is shown that the partial regression coefficient of the phenotype on a marker in multiple regression depends only on those QTLs that are located in the interval bracketed by the two neighboring markers and is independent of QTLs located in other intervals. Then, using this property, we can construct a test statistic that is independent of effects of QTLs in other regions of the chromosomes. This result provides a basis for constructing an interval test for mapping QTLs. Also, by fitting multiple markers in a regression model, much background genetic variation in a population can be controlled in analysis, and, as a result, statistical power of detecting QTLs can be improved. The advantages and disadvantages of fitting multiple markers in the model for mapping QTLs are discussed, as are procedures to construct an appropriate interval test for mapping QTLs. PROPERTIES OF MULTIPLEREGRESSION ANALYSIS The Model. Let us consider, for simplicity, a backcross population that is from two inbred parental populations, Pi and P2, fixed for different alleles at m QTLs and t markers.Let the means of the P1 and P2 populations be Al and ,u2

Senescence, the decline in survivorship and fertility with increasing age, is a near-universal property of organisms. Senescence and limited lifespan are thought to arise because weak natural selection late in life allows the accumulation of mutations with deleterious late-age effects that are either neutral (the mutation accumulation hypothesis) or beneficial (the antagonistic pleiotropy hypothesis) early in life. Analyses of Drosophila spontaneous mutations, patterns of segregating variation and covariation, and lines selected for late-age fertility have implicated both classes of mutation in the evolution of aging, but neither their relative contributions nor the properties of individual loci that cause aging in nature are known. To begin to dissect the multiple genetic causes of quantitative variation in lifespan, we have conducted a genome-wide screen for quantitative trait loci (QTLs) affecting lifespan that segregate among a panel of recombinant inbred lines using a dense molecular marker map. Five autosomal QTLs were mapped by composite interval mapping and by sequential multiple marker analysis. The QTLs had large sex-specific effects on lifespan and agespecific effects on survivorship and mortality and mapped to the same regions as candidate genes with fertility, cellular aging, stress resistance and male-specific effects. Late ageof-onset QTL effects are consistent with the mutation accumulation hypothesis for the evolution of senescence, and sex-specific QTL effects suggest a novel mechanism for maintaining genetic variation for lifespan.

Adequate separation of effects of possible multiple linked quantitative trait loci (QTLs) on mapping QTLs is the key to increasing the precision of QTL mapping. A new method of QTL mapping is proposed and analyzed in this paper by combining interval mapping with multiple regression. The basis of the proposed method is an interval test in which the test statistic on a marker interval is made to be unaffected by QTLs located outside a defined interval. This is achieved by fitting other genetic markers in the statistical model as a control when performing interval mapping. Compared with the current QTL mapping method (i.e., the interval mapping method which uses a pair or two pairs of markers for mapping QTLs), this method has several advantages. (1) By confining the test to one region at a time, it reduces a multiple dimensional search problem (for multiple QTLs) to a one dimensional search problem. (2) By conditioning linked markers in the test, the sensitivity of the test statistic to the position of individual QTLs is increased, and the precision of QTL mapping can be improved. (3) By selectively and simultaneously using other markers in the analysis, the efficiency of QTL mapping can be also improved. The behavior of the test statistic under the null hypothesis and appropriate critical value of the test statistic for an overall test in a genome are discussed and analyzed. A simulation study of QTL mapping is also presented which illustrates the utility, properties, advantages and disadvantages of the method.

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A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. are many ways to define a QTL model, thus additive, A similar argument has been made for epistasis (Chevdominance, and epistatic effects. The models compared erud and Routman 1995). On the one hand, we have by Van Der Veen (1959) are all based on genotypic the model proposed by Hayman and Mather (1955) values only, so to speak. and discussed in length in Mather and Jinks (1982),The purpose of modeling QTL, of course, is to provide a way to summarize and interpret the differences between the genotypic values and also the genetic varia-1

SummaryUnderstanding and estimating the structure and parameters associated with the genetic architecture of quantitative traits is a major research focus in quantitative genetics. With the availability of a well-saturated genetic map of molecular markers, it is possible to identify a major part of the structure of the genetic architecture of quantitative traits and to estimate the associated parameters. Multiple interval mapping, which was recently proposed for simultaneously mapping multiple quantitative trait loci (QTL), is well suited to the identification and estimation of the genetic architecture parameters, including the number, genomic positions, effects and interactions of significant QTL and their contribution to the genetic variance. With multiple traits and multiple environments involved in a QTL mapping experiment, pleiotropic effects and QTL by environment interactions can also be estimated. We review the method and discuss issues associated with multiple interval mapping, such as likelihood analysis, model selection, stopping rules and parameter estimation. The potential power and advantages of the method for mapping multiple QTL and estimating the genetic architecture are discussed. We also point out potential problems and difficulties in resolving the details of the genetic architecture as well as other areas that require further investigation. One application of the analysis is to improve genome-wide marker-assisted selection, particularly when the information about epistasis is used for selection with mating.

A model of long-term correlated evolution of multiple quantitative characters is analyzed, which partitions selection into two components: one stabilizing and the other directional. The model assumes that the stabilizing component is less variable than the directional component among populations. The major result is that, within a population, the responses of characters to selection in the short term differ qualitatively from those in the long term. In the short term, the responses depend on genetic correlations between characters, but in the long term they are only determined by the fitness functions of stabilizing and directional selection, independent of genetic and phenotypic correlations. Treating the stabilizing component as a constant and assuming the directional component to vary among populations, I present formulas for the interpopulation covariation and interspecific allometry, which are functions of the intensity matrix of stabilizing selection. Particular attention is paid to the relationship between intra- and interpopulation correlations.

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 3 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 d i * that comprises the dominance effect at each QTL minus half the sum of additive 3 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.