Quantitative trait loci: individual gene effects on quantitative characters

J. evol. Biol. 6: 463 480 ( 1993) 1010 061X:93,040463-185 1.50+0.2010

c‘ 1993 BirkhHuser Verlag. Base1

Mini-review

Quantitative trait loci: individual gene effects on quantitative characters

James M. Cheverud and Eric Routman

Dc~ppartment of Anutomy & Neurobiology, Wmhington Unicersity School of’ Medicine, St. Louiss, MO, 6.3110, /ISA

Key ,rords: Quantitative genetics; molecular markers; QTLs

Over the past few years great progress has been made in detecting and mapping genes of major effect, including the genes responsible for cystic fibrosis. Hunting- ton’s chorea, and Duchenne Muscular dystrophy. Now, due to innovations in quantitative and molecular genetics, it is becoming possible to study the genetic basis of polygenic traits; to identify, map, and measure the effects of quantitative trait loci (QTLs). Quantitative trait loci are those loci with relatively minor, quantitative, effects on phenotypes. In the past, these individual gene effects have not been readily accessible to evolutionary geneticists. The impact of quantitative trait locus studies on evolutionary genetics has yet to be felt. However, in the future, the effect is likely to be profound, opening new avenues for research into persistent problems in genetics and evolution.

Access to the effects of individual genes on quantitative characters will finally allow us to address the underlying genetics of quantitative traits, rather than rely- ing on untested and previously untestable assumptions concerning the nature of allelic effects on quantitative characters (Barton and Turelli, 1989). For example, Barton and Turelli (1987) have suggested that evolutionary quantitative genetic models are sometimes based on unrealistic assumptions concerning the effects of individual genes. However, we know very little about genie effects on quantitative characters, making it difficult to judge the relative dependability of various assumptions used in theoretical quantitative genetic models (Barton and Turelli, 1989). Measuring the effects of QTLs will also provide important new insights into the genetic bases of speciation and species differences, the role of epistasis in evolutionary change, the genetic response to phenotypic selection, and the effects of natural selection on the frequency of individual genetic variants. These problems have long been of interest in evolutionary genetics and the subject of much theoretical

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464 Chcverud and Routman

conjecture. Now we will be able to address them with empirical data on a broad scale.

Quantitative and evolutionary geneticists have long wanted to uncover the relatively minor effects of single genes on complex quantitative phenotypes but success has been limited. Complex experimental designs required to isolate small portions of chromosomes on a homogeneous genetic background have been suc- cessful to some extent (Thompson and Thoday, 1979; Milkman, 1979; Chai, 1961, 1968) but are quite laborious and restricted to measuring the effect of one chromosomal region at a time in a manipulable experimental population. This approach has never been widespread and extrapolation of these results to natural populations is difficult at best.

Alternatively. as soon as inherited phenotypic and biochemical markers became available, attempts were made to statistically associate them with phenotypes of interest (Sax, 1923). Again, some limited success has been obtained associating blood groups (e.g. Vogel and Motulsky, 1986) and immunological markers (e.g. Bonner and Tyan, 1982, 1983, 1984; Vogel and Motulsky, 1986) with disease phenotypes, and associating enzyme electrophoretic variants with measures of physiology and fitness (e.g. Aquadro et al., 1986: Smith and Small, 1982; Pember- ton et al., 1988). However, these studies have been limited in the numbers of markers included and their potential phenotypic effects. Enzyme electrophoretic variants, while common, tend to represent a functional subset of genes and are frequently considered neutral, having no biologically significant effects on higher level phenotypes. Correlation of a given character with a specific electrophoretic marker must often be considered fortuitous since electrophoretic markers were most often chosen for availability and variability rather than a specified u priori functional relationship to the phenotype of interest, although important counterexam- pies also exist (e.g. Watt, 1986).

Recent developments in molecular genetics have fundamentally altered this situation by uncovering ever more highly variable molecular genetic systems useful for genetic mapping, such as restriction fragment length polymorphisms (RFLPs), simple sequence repeat polymorphisms (SSRPs), and random amplified polymorphic DNAs (RAPDs). These highly variable systems make nearly any specific gene available for population study and provide enough genetic variants that variation can be considered across the entire genome, rather than being restricted to enzyme loci.

Coupled with the new availability of molecular genetic variation, quantitative genetics has responded by developing the analytical tools necessary to make use of this new data (Lander and Botstein, 1989; Knapp et al., 1990; Luo and Kearsey, 1989, 1990; Haley and Knott, 1992). With quantitative trait loci we can begin an empirical approach to long-standing questions in evolutionary genetics, such as the role of epistasis in speciation, the number and relative effects of loci involved in population differentiation and speciation, and the strength of natural selection on minor adaptive allelic variants. Answers to these questions have proved elusive in the past due to a lack of empirical methods for measuring the minor effects of single genes on phenotypic characters.

Quantitative trait loci

Single locus quantitative genetics

465

Before proceeding further it will be useful to review some of the basic quantitative genetic concepts utilized in QTL studies (see Falconer, 1989 for a more detailed description). Quantitative genetics is concerned with the inheritance of quantitative differences between individuals. This inheritance is conceived of as being due to many genes, each of small effect, which together are responsible for inherited differences between individuals. Inherited variation provides the basis for evolutionary response to selection (Falconer, 1989).

In quantitative genetics, a phenotypic value (P). or measurement made on an individual, is conceived of as being composed of two parts, a genotypic value (G) and an environmental deviation (E),

P=G+E. (1)

The total genotypic value (G) is the result of the combination of all the genes an individual carries. At a single locus, the genotypic value is the average phenotype for individuals of the specified genotype. By convention for two alleles, the genotypic values are spread along a number line (Fig. IA). An example of the genotypic values at the renin locus for blood pressure and heart weight (Rapp et al., 1989) is given in Figure 1 B and IC.

Although useful for quantifying the effect of the genotype on the phenotype (see equation I), genotypic values do not bear directly on inheritance in sexually reproducing organisms. For studying inheritance, we need to know the effects of inherited units, alleles not genotypes. The inherited effect of an allele is its average effect, defined as the mean deviation from the population mean of individuals which received that allele from one parent, the other allele (from the other parent) coming at random from the population at large (Falconer, 1989). With two alleles in a population, the average effect of allele A, is given by

“I =,&[a +4/?-P,)l (2)

where a, is the average effect, p, is the frequency of allele A,, p2 is the frequency of allele A?, ‘a’ is the genotypic value of the homozygotes, and ‘rf’ is the genotypic value of the heterozygote (see Fig. I). A similar equation holds for the average effect of A?. Examples of average effects of alleles at the renin locus for blood pressure and heart weight (Rapp et al., 1989) are given in Table I for varying allele frequencies. Note that the average effect is not a property of an allele itself but of an allele in a particular population because gene frequencies and dominance (d) contribute to the average effect (see equation 2 and Tab. I).

In the same way that we can partition the phenotypic value, the total genotypic value (G) can be subdivided into heritable and non-heritable parts. At a single locus, the genotypic value can be divided into a heritable breeding value (A) and a non-heritable dominance deviation (D). The breeding value is the sum of the average effects of the alleles carried by an individual (breeding values of A,A, = 2a,; A,A, = a, + a,; A,A, = 24) and is twice the expected value of an individual’s offspring given random mating. The genotypic value at a locus equals

466 Cheverud and Routman

Fig. 1. Genotypic values ( IA) for the general case based on Falconer (1989). The origin is taken as the

mid-point of the homozygote classes, so A,A, has a genotypic value of “ +u” and A,A, has a value of

“-u”. The genotypic value of the heteroLygote is given as “ri”. The value of “8 relative to “u” is used to classtfy the dominance of one allele relative to another for the phenotype in question [d = (I, A, is

dominant to A2; d = --(I, A, is recessive to A>; d = 0, A, and A, arc co-dominant; 0 < d < (1, A, is parttally dominant to AZ: d > a, the heterozygote is overdominant]. The genotypic values for male blood pressure ( I B) and heart weight ( IC) are from the FL cross of salt-hypertension sensitive (S) and resistant (R) strains of rat at the renin locus (Rapp et al., 1989).

the breeding value unless intralocus allelic interactions (dominance) are present. These interactions result in a dominance deviation, or the deviation of the genotypic value from the breeding value. The total genotypic value over all loci is simply the sum of the breeding values and dominance deviations over the loci unless interlocus genotypic interactions (epistasis) are present. Interlocus interactions result in an epistatic deviation (I) from interlocus additive expectations. Thus the total genotypic value is

G=A+D+I. (3)

Only the breeding value is considered heritable within a random mating population, although minor portions of epistatic effects can also be inherited (see equation 5 below).

These concepts can be used to derive the inherited variation between individuals (V, ). At a single locus this is

v,4 = 2P,Pz[U + u’(P, -P, )I’ = 2P,Pz(T - %12. (4)

Quantitative trait loci 467

Table 1. Genotypic and genie values and variances at the renin locus for male blood pressure and heart weight in the F, generation of a cross between salt-hypertension sensitive (S) and resistant (R) rats (Rapp et al., 1989).

Genotypic value Blood pressure (mmHg) Heart weight (mg)

RR RS ss

z

Genie ( ps = 0.5)

m/i “s VA VII

Genie ( P,~ = 0.9)

ER x.5 VA VII

148.7 1314 165.6 1364 171.0 1438

II.5 62 5.7 -12

-5.6 -31 5.6 31

62.7 1922 8.4 36

-5.9 0.7 7.1 1.0

-64.4 7.2

922.x 4.8

The dominance variance (V,,) is (2~,p~d)~ (see Tab. 1 for a numerical example). Assuming no epistasis, the total inherited variance is the sum of the inherited variance at each independently segregating locus. Therefore, to obtain the heritable genetic variance for a character, we need to measure the variance at each of the separate loci affecting the character and sum the variances. This is not a practical approach because to achieve it we would need to identify all the segregating loci affecting a character. Furthermore, epistasis causes the combined effects of genes to be greater or less than the sum of their individual effects.

In a major contribution to the study on inheritance, Fisher (1918) derived the relationship between the total inherited variance, V,, and the similarity among relatives

cov(relatives) = rVA + uV, + r’VAn + ruV,, + u~V,,, f. (5)

where cov(relatives) is the phenotypic covariance among relatives, r is the genetic correlation among relatives (Wright’s coefficient of relationship), u is the coefficient of double coancestry (probability that both alleles in two individuals are identical by descent), V, is the additive variance, Vn is the dominance variance, and V,,, V An, and VIjn are the additive by additive, additive by dominance, and dominance by dominance epistatic variances (Crow and Kimura, 1970). Given the muhiplica- tive nature of their coefficients, the epistatic terms are usually considered to contribute only a minor amount to the covariance among relatives. Also in a non-inbred population, the coefficient of double coancestry (u) is zero unless individuals share both parents in common, as full sibs do. Thus, in the case of non-inbred parents and offspring, where r = 0.5 and u = 0, equation 5 can be simplified to cov(relatives) = 0.5V.,, and an indirect estimate of heritable variance


obtained. Unfortunately, dominance, epistasis, and the potential for environmen- tally-based phenotypic covariance among relatives allow only rough estimates of heritable genetic variance.

The relationship between phenotypic resemblance of relatives and heritable variance derived by Fisher allowed quantitative genetics to proceed without the measurement of genotypic values at all loci affecting a character. Instead, the phenotypic resemblance of relatives has been extensively used to estimate levels of inherited variation. After all, it is the level and pattern of heritable variation (V,), not total genotypic variation, which together with the level and pattern of natural selection determines the rate and direction of evolution. The covariance among relatives provides the summary measure of heritable variance which is necessary in evolutionary investigations.

However, the use of equation 5 to measure heritable variance leaves several questions unanswered. The genetic system is treated as a black box and we don’t really know what V, is composed oT:

(1) How many loci contribute to heritable variance? (2) What are the relative contributions of dominance? (3) Is there epistasis between loci? (4) What is the distribution of phenotypic effects in terms of major versus minor

genes? (5) What is the frequency distribution of allelic effects at individual loci?

These questions can be summarized by asking, How do single loci contribute to inherited variation or what is the genetic basis of inherited differences? Fisher’s theory conveniently allows us to bypass these questions but the outcome of long-term evolutionary models depends on the answers (Wagner, 1989; Turelli, 1988; Barton and Turelli, 1987, 1989; Lande, 1976, 1979, 1980).


Quantitative trait loci (QTLs) are the individual genes with relatively small, quantitative, effects on a phenotype. These genes have been referred to as polygenes and are not likely to be a separate class of gene loci but rather the minor alleles at the same loci as alleles of major effect (Pedersen and Berg, 1988; Edwards et al., 1992). There are two major approaches to quantitative trait locus studies, the candidate gene approach and the marker locus approach. Each approach has its own advantages and disadvantages making them complementary rather than mutually exclusive. The advantages of one approach are the limitations of the other.

Cmdidutc lorus qyroach

In the candidate locus approach genotypic values are measured at loci with known physiological and biochemical relationships to the phenotype of interest.

Quantilativc trait loci 469

For example, in a study of individual genes affecting murine growth, Winkelman and Hodgetts (1992) scored molecular variants at the insulin-like growth factor 2 (IGF-2) and growth hormone (GH) loci since these growth factors are known to be important in somatic growth. There are several advantages to the candidate locus approach:

(I) It focuses on relevant genomic regions for the characters of interest; (2) The results are interpretable in relation to trait physiology; (3) It provides direct measures of genotypic values; and (4) It is relatively easily applied to variation within natural populations. To carry out a candidate locus study one first identifies a locus or loci of interest

based on prior knowledge of trait development and physiology. An excellent example of a candidate locus study is the work of Sing and colleagues (1988) and Pedersen and Berg ( 1988, 1989) on the individual genes affecting heart disease and cholesterol level in humans. They have identified a large number of candidate genes but here we focus on their studies of Apolipoprotein E (ApoE) and the low density lipoprotein receptor (LDL-R) as proteins which affect the transfer of cholesterol from the bloodstream into cells. It is thought that these loci may play a role in controlling variation in blood cholesterol levels.

After identifying loci of interest, molecular variants must be identified at or very near these loci. These variants are often RFLPs but can also be electrophoretic variants. The alleles at the ApoE locus are identified by isoelectric focusing (Sing et al., 1988). Once molecular variants are identified, the phenotype of interest is measured and the genotypes identified in a sample drawn from a defined population. No genealogical information is necessary for these individuals, just that they be representative of a population of interest.

The phenotypic mean of each genotypic class is obtained and tested for statistically significant differences using analysis of variance or regression methods. Equations 2 and 4 given above along with those in Falconer (1989) can be used to measure the average effect of a gene and the additive and dominance genetic variance at each candidate locus (see Tab. 1 for an example). In their studies of ApoE effects on serum cholesterol levels, Sing et al. (1988) found that in most human populations Apoc2 lowers cholesterol levels while Apoc4 raises cholesterol levels relative to the population mean. Since Apoc3 is the most common allele, its effects are largely responsible for the mean value rather than deviations from the mean and it has a relatively small average effect. In nearly all human populations studied to date, the genetic variance at this locus is largely additive. Approximately 7 to 10% of the variation in total cholesterol level is due to variation at the ApoE locus, after correcting for known environmental risk factors. As more and more candidate loci are studied, the sum total of the variation explained by individual loci is likely to exceed 100%. We obtain this logically impossible result by assuming that epistasis is absent.

Quantitative trait locus studies can detect epistatic interactions between loci. Pedersen and Berg (1989) found epistatic interactions between the ApoE and LDL-R (Low Density Lipoprotein Receptor) loci. The LDL-R locus has a significant effect on serum cholesterol levels (Pedersen and Berg, 1988) and a major

470 Chcvcrud and Routman

variant at this locus is responsible for severe familial hypercholesterolcmia (Goldstein and Brown, 1983). ApoB normally binds to the LDL receptor as an early step in bringing cholesterol into cells. However, ApoE also has an affinity for the LDL receptor and competes with ApoB for binding sites. The ApoE-LDL receptor complex does not lead to incorporation of cholesterol into cells. The differential effects of Apoc2 and Apoc4 on serum cholesterol levels are only apparent in individuals with the A,A, genotype at the LDL-R locus due to differential binding of the A, isoform of LDL with the isoforms of Apoc (Pedersen and Berg, 1989).

Sing et al. (1988) found that not only did means differ between genotypes at the ApoE locus but that serum cholesterol variances and intertrait correlations also varied across genotypes. In nearly all evolutionary genetic models phenotypic variances and covariances are assumed to be constant across genotypes. Sing et al.‘s (1988) results indicate that this may be a fallacious assumption in many instances. If phenotypic variances and covariances vary between genotypes, changes in genotype frequencies will result in changes in phenotypic variance, even if the genotypic means do not vary. Genotypically-based differences in phenotypic variances and covariances provide an additional factor which needs to be considered in the evolution of variance/covariance patterns.

As demonstrated by the work of Sing and his colleagues, the candidate locus approach can be very powerful in measuring the etfects of genes on characters of interest. However, it also carries several disadvantages relative to the marker locus approach. Negative results may be fairly common because functionally distinct alleles may not be segregating at the candidate locus even if the locus could produce functionally variant mutations. It may also be difficult to identify molecular variants even though functional variants are segregating in the population.

The ability to detect statistically significant results in a candidate locus study depends on sufhcient sample sizes for each genotype present in a population. In a natural population, it is possible that one allele will be in high frequency, coupled with several rare variant alleles. In this case, the sample sizes for heterozygotes and especially for homozygotes for rare alleles may be too small to obtain accurate genotypic values (e.g Pedersen and Berg, 1988). Also, if there are many alleles segregating at appreciable frequencies, the sample sizes of individual genotypic classes will be small, leading to unreliable estimates of genotypic values.

Finally, no new loci are discovered using the candidate locus approach. The physiological role of the locus of interest must already be known and the locus identified, limiting the approach to well-known physiological and biochemical systems. Only the limited portion of the genome surrounding the candidate locus is tested even though many other identified and unidentified loci may also affect the character. Finally, we must assume that the effect is due to the candidate locus itself rather than being due to linked loci. Functionally-related loci can sometimes be found in close linkage resulting in high levels of linkage disequilibrium and difficulty in separating their effects in a statistical study.


Marker locus approach

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In the marker locus approach genotypes are measured at a large number of marker loci of no particular phenotypic effect, such as RFLPs or SSRPs. Ideally, the loci included are spread across the entire genome so that variable markers are available every 5 to 10 centiMorgans. With such a high density of marker loci, it is likely that marker loci will be in linkage disequilibrium with quantitative trait loci resulting in a correlation between quantitative trait values and marker genotypes. The advantages of this approach are that it:

(1) Surveys the entire genome; (2) Is effectively used with phenotypically and genetically wide crosses such as

between subspecies or closely-related species; and (3) Allows mapping and quantitative assessment of unknown quantitative trait

loci. By using marker loci scattered throughout the genome it is very likely that multiple QTLs will be uncovered and that some of these may be in locations not previously known to affect the phenotype of interest.

As a first step in carrying out a marker locus study one needs to compare an appropriate pair of genetically and phenotypically divergent populations, such as two subspecies or species (Paterson et al., 1988) two inbred strains (Edwards et al., 1987, 1992), or the end results of high and low selection lines (Lander and Botstein, 1989). High levels of genetic divergence assure that a large number of markers will differ between populations. The populations are then crossed to produce an F, generation which is either intercrossed to produce an F, generation or backcrossed to one parent population or the other. The phenotypes of interest and marker genotypes are measured in the backcross or F2 generation and the phenotypes are then correlated with marker genotypes. The crossing design assures intermediate allele frequencies and maximal samples for each genotype. Several studies, especially with agricultural plants, have used the marker locus approach and have typically detected many likely QTLs of moderate phenotypic effect (Tanksley et al., 1982; Weller, 1987; Weller et al., 1988; Paterson et al., 1988, 1990; Martin et al., 1989; Edwards et al., 1987, 1992; Doebley et al., 1990; Doebley and Stec, 1991). One difficulty in using marker locus correlations to measure the effects of QTLs is that genotypic values measured at a marker locus are biased downwards relative to the true genotypic values at the linked quantitative trait locus due to recombination between marker and quantitative trait loci (Edwards et al., 1987).

Lander and Botstein (1989) developed a method, interval mapping, which accounts for this bias, thus providing unbiased estimates of the genotypic values at the QTL itself. Interval mapping uses maximum likelihood procedures to measure genotypic values and linkage odds (LOD scores) along the whole length of marked chromosome and identifies the location of the QTL as the position with the highest LOD score. The linkage odds for the markers and QTL must exceed 1 in 1000 for a QTL to be located. Similar flanking marker methods have been described by Knapp et al. ( 1990) and Haley and Knott ( 1992).

Interval mapping methods have been used in several intercross and backcross


experiments (Paterson et al., 1988, 1990; Doebley and Stec, 1991). The first such experiment illustrates the utility of the marker locus method. Paterson et al. (1988) analyzed tomato production characters, including fruit mass, soluble solids concen- tration, and pH, in 237 backcrosses of Lycoprrsicon esculentum by L. chmidewkii hybrids to L. esculentum. Their genetic markers included 63 RFLPs and 5 enzyme electrophoretic variants spread across I2 chromosomes. They found evidence for QTLs throughout the genome, including chromosomes 1, 3, 4, and 6 through I I. The QTLs detected account for approximately 50% of the variation in these characters in the backcross population. Thus, the study successfully detected a large number of loci affecting these production characters. Furthermore, several of these quantitative trait loci, on chromosomes 3, 4, 6, and 7 seem likely to affect more than one character although no formal test of pleiotropy was performed.

Another variant of the random marker approach compares sets of recombinant inbred lines originally formed by inbreeding separate lines from the F, generation of an inbred cross (Bailey, 1971). These strain sets have proven very useful in quantitative trait locus analyses (Gelman et al., 1988; Plomin et al., 1991; Neumann et al., 1990; Neumann and Collins, 1991; Rousset et al., 1992; Belknap et al., 1992). Recombinant inbred strains have the advantage that individuals within strains are genetically identical so that only one animal per strain needs to be scored for molecular markers. However, this approach also carries the disadvantage of limited power and small sample size due to the restricted number of strains in any given set. Recombinant inbred strains bear many similarities to the isofemale lines used so extensively in Drosophila genetics.

While the marker locus approach is very powerful in detecting quantitative trait loci, it has several disadvantages relative to the candidate locus approach. Only indirect measures of QTL genotypic values are available through the interval mapping method. The measures of genetic effect obtained may be difficult to interpret in specific physiological and biochemical terms because the loci discovered are of unknown function. While a large number of QTLs are typically found, the large number of marker loci used leads to multiple comparisons problems. The false positives among the multiple comparisons can be weeded out by replicating the experiment. Finally, since the approach relies on divergent populations, crosses, and backcrosses, it is not well suited to the study of quantitative trait loci in natural populations.

Application of QTI, studies to natural populations

Both the candidate and random marker approaches can be applied to evolutionary questions in natural populations but they are perhaps best applied in different circumstances. The candidate locus approach will be most powerful in intrapopulation studies while the random marker approach is perhaps best applied in analyzing interpopulation differences, such as subspecific and interspecific differences.

To analyze intrapopulation variation in candidate loci such loci have to be identified and the population surveyed for variations at the candidate loci and in the phenotype(s) of interest. Candidate loci can be identified on physiological grounds or on the basis of earlier random marker analyses. With an increasing number of physiological genes being cloned in a variety of species, DNA probes based on conserved genetic regions can be used to detect RFLPs in related species.

One important concern associated with measuring variations at candidate loci lies in the methods which are used to detect alleles. For example, protein electrophoresis probably detects only about 40% (Ramshaw et al., 1979) of the variation in protein sequence and genes coding for enzymes. Electrophorcsis detects none of the variation in regulatory regions up and downstream from the structural locus. Electrophoresis pools different alleles which happen to have the same electrical charge. Given that the electrical charge of an clcctromorph may have little relation to its physiological function, it is likely that functionally divergent alleles with similar electric charges will be pooled masking the effect of the locus on the phenotype of interest. Alcohol dehydrogenase in Drosophilu is an example with some inappropriate pooling because slow and fast electromorphs can cut across functional categories (Aquadro et al., 1986). However, as discussed above, some pooling of alleles is likely to be necessary in order to obtain accurate genotypic values with reasonable total sample sires but pooling by electrical charge is not necessarily a suitable criterion.

Templeton and colleagues (Templeton et al., 1987, 1988, 1992; Sing et al., 1992; Templeton and Sing, 1993) have shown that when the various alleles at a locus (or haplotypes in a restricted chromosomal region) can be cladistically related to one another, it is possible to pool historically-related alleles without masking phenotypic efTects. This method even allows one to identify when and where in the evolution of a locus a functionally important mutation has occurred and which allelic contrasts will be most informative regarding the molecular basis of the functional change. The cladistic approach requires detailed information on molecular variation, either a restriction map or nucleotide sequence variation, in order to determine the cladistic relationships among alleles. Nonetheless, it allows much more powerful genetic interpretations of gene effects than methods which do not pool alleles, pool them by functional effect, or arbitrarily, as in protein electromorphs.

Once molecular variation at the candidate locus is defined, the population can be surveyed for variations at this locus and for the phenotypes of evolutionary interest. In practice, these surveys would involve much the same field work as in current surveys of neutral markers for studies of population structure. No genealogical data is necessary. The genotypic values, average effects, and the heritable and non-heritable variances and covariances of the characters of interest due to the candidate loci can be estimated from the phenotypic and genetic data. Selection on the candidate locus genotypes can be statistically and biologically related to selection on characters of interest.


If the genealogical relationships among the sampled individuals can be ascertained, one can analyze the effect of the candidate locus in the context of the total additive genetic variation affecting a phenotype (Hopper and Mathews, 19X2; Boerwinkle and Sing, 1987; Blangero et al., 1992). With this approach it is possible to determine the proportion of both the total phenotypic variation and its heritable and non-heritable genetic components accounted for by variation at the candidate locus. For example, Blangero et al. (1992) analyzed the contribution of variation at the ApoAI locus on serum levels of apo Al using a sample of pedigreed baboons. They found that 7.5% of the total phenotypic variance and 20% of the total additive genetic variance in apo Al serum levels was due to the ApoAI locus variants they detected.

IfltclrlJolJul~Ition wriution using random markers

To analyze interpopulation variation using the random marker approach one needs to identify a large number of molecular markers which differ across the populations being crossed. These markers need to be arranged in a genetic map using an interpopulation crossing experiment and then the mapped molecular variants correlated with the phenotype(s) of interest.

/dent$~-in~ rundom markers: Random marker studies of quantitative trait loci have been performed largely in agricultural and experimental species in which a large body of molecular markers are mapped and available for scoring. Creating fine scale maps based on enzyme electrophoresis and RFLPs has been a laborious procedure. New techniques, however, allow the generation of a marker map in a relatively short period of time. Love et al. (1990) Aitman et al. (1991) Hearne et al. ( 1991) and Dietrich et al. (1992) have mapped simple sequence repeat polymorphisms (SSRPs or variation in the length of microsatellites) in mice while Serikawa et al. (1992b) has mapped SSRPs in rats. Some of these microsatellites appear to be conserved between rats and mice (Stallings et al., 1991). Random amplified polymorphic DNAs (RAPDs) have also been used in making genetic maps in mice and rats (Nadeau et al., 1992; Serikawa et al., 1992a), in conifers (Carlson et al., 1991) and in tomatoes (Klein-Lankhorst et al., 1991) although these markers are phenotypically dominant making heterozygotes difficult to detect. Mapping with RAPD markers is just beginning and also may suffer from potentially high levels of PCR artifacts (Devos and Gale, 1992; Reidy et al., 1992). Genetic maps for any particular species of interest can be generated in the same fashion as they have been for mice and rats (Dietrich et al., 1992). Currently, simple sequence repeat loci appear most promising for constructing genetic maps, although further development of RAPDs as a mapping tool is ongoing.

Mupping random markers and testing,for QTLs: Once a series of molecular markers are developed they need to be tested for polymorphism and mapped. The mapping requires individuals of known genealogical relationship, either families within a population or F?s or backcrosses between distinct parental populations. The

Quantitative trait 10~1 475

individuals are scored both for the phenotypes of interest and the random markers. The genotypes of progeny can then be analyzed by any of several genetic mapping algorithms (Lander et al., 1987) to identify linkage groups and the map distances for the markers along each chromosome.

Once a genetic map is obtained for the variable molecular markers, the markers can be used with an interpopulation cross to measure the effects of QTLs on the phenotypes which differentiate the populations. This approach allows the genetic basis for phenotypic differences between populations to be identified. If the two populations are separate species, one can dissect the genetic basis for phenotypic differences important in speciation. One example of this is Doebley’s analysis of genetic differences between a phenotypically primitive Central American maize and a teosinte thought to represent a likely ancestor for agricultural maize (Doebley et al., 1990; Doebley and Stec, 1991). He found that most of the phenotypic differences between teosinte and maize were due to multiple genetic factors. rather than the product of a small number of major genes (Beadle, 1980).

Potential contributions of quantitative trait locus studies to evolutionary biology

The recent availability of quantitative trait loci finally allows access to the underlying genetics of polygenic characters rather than having to rely soley on measurement of genetic variances through the relationship between genetic variance and the phenotypic similarity among relatives. The underlying genetics of phenotypes plays an important role in their evolutionary dynamics (Barton and Turelli, 1989) and in several evolutionary controversies, such as the basis for the maintc- nance of genetic variation (Turelli, 1988).

Quantitative genetic evolutionary models are based on the assumption that there are many loci of primarily additive effect contributing to a character so that the distribution of breeding values is Gaussian. However, models explicity based on the dynamics of gene frequencies indicate that after several generations of directional selection, the distribution of breeding values should become skewed, especially if the number of “variance effective” loci is limited (Barton and Turelli, 1987; Turelli, 1988). Additionally, the accuracy of long-term evolutionary predictions concerning population means and especially variances and covariances based on standard biometrical models (Lande, 1976, 1979) depends on a Gaussian distribution of allelic effects at single loci. The number of loci segregating for a character or potentially affecting a character (Wagner, 1989) and the distribution of allelic effects at loci are important parameters in long-term evolutionary models. The accumulation of QTL results over many comparisons will allow estimation of these parameters and begin to resolve these controversies.

Pleiotropic genie effects also play an important role in evolutionary genetic models, especially whether specific alleles at a locus show similar or very different pleiotropic effects. Wagner’s (1989) evolutionary genetic model assumes constrained pleiotropy, i.e. that allelic effects on a set of phenotypes at a single locus will have a correlation of one or minus one. The measurement of the quantitative effects of


several alternate alleles at a locus for a series of characters provides a test of this hypothesis. Antagonistic pleiotropy, when genes have opposite effects on two traits both affecting fitness (Rose, 1982) can maintain genetic variation at single loci. However, there is a paucity of studies of pleiotropic effects at single loci. The presence of antagonistic pleiotropy at a locus can be uncovered by quantitative trait locus anlayses.

Pleiotropy for loci affecting morphological traits is also important for Wright’s shifting balance theory of evolution (Wright, 1931). In Wright’s model, interaction among morphological traits creates fitness epistasis for the pleiotropic loci that affect the traits. This fitness epistasis causes the adaptive landscape to have multiple peaks. Determining the presence or absence of pleiotropy at QTLs would test one of the major assumptions of this influential theory.

The accessibility of measured genotypic effects will also allow identification of epistatic effects on fitness within populations, another important aspect of Wright’s shifting balance theory of evolution (Wright, 1931). Although whenever traditional genetic studies have looked for epistasis it has been found, it has proven very ditfcult to measure the level of epistatic variance in a population. This is not surprising since epistatic variance is of relatively minor importance for the phenotypic covariance among relatives as compared to additive genetic variance (see equation 5). With measurement of gene effects at single loci it becomes possible to measure epistasis and its effect on genotypic variance (Edwards et al., 1987; Doebley and Stec, 1991; Rousset et al., 1992).

Quantitative trait locus studies, especially those using the random marker locus approach, will be quite valuable in dissecting the genetic basis of speciation. There has been much discussion over the genetic processes involved in species formation and maintenance. For example. models which claim to support (Mayr, 1963; Templeton, 1980) or deny (Barton and Charlesworth, 1984) the importance of founder effects in speciation differ in their assumptions about the genetic architec- ture of fitness. Specifically, proponents of founder effect speciation assume that the loci affecting fitness or fitness components in the ancestral population exist in epistatically interacting adaptive gene complexes. These complexes are polymorphic and in selective equilibrium in the ancestral population but are disrupted by the changes in gene frequencies caused by the founder event. The artificial conditions of a random marker QTL study would also disturb equilibrium, and many of the loci involved in adaptive gene complexes would be detected, if they exist. These loci could be surveyed in the ancestral population to determine if they are polymorphic and act epistatically. Alternatively, crosses between species could be used to see whether species isolating mechanisms are due to numerous, additively acting loci (as predicted by the divergence models of Barton and Charlesworth, 1984) or due to fewer epistatically acting loci (as the founder effect models would predict).

When combined with studies of phenotypic selection, as suggested by Lande and Arnold ( 1983), candidate locus studies will allow population geneticists to ascertain the importance of natural selection in maintaining genetic variation and in genotypic evolution. While there have been a few studies of the response of allele frequencies to selection (Endler, 1986), the use of QTLs in such studies will make


explicit the relationship between fitness and other phenotypes on the one hand and phenotypes and genotypes on the other. Population geneticists have successfully studied population structure and random genetic drift for many decades using neutral genetic variants. With the QTL approach, the population genetics of adaptive variants will finally be accessible to researchers, so that population genetic studies of adaptation can take their place alongside studies of population structure to give a much more complete picture of genetic evolution than has been available to date.

Acknowledgements

We would especially like to thank Allen Moore, Luci Kohn, Alan Templeton. and Giinter Wagner for

their valuable insights and discussion of the issues presented here. This work was supported by NSF grant BSR 9106565.

References

Aitman, T. J.. C. M. Hearne, M. A. McAlcer and J. A. Todd. 1991. Mononucleotide repcats are an abundant source of length variants in mouse genomic DNA. Mammalian Genome I: 206 210.

Aquadro, C. F., S. F. Desse, M. Bland, C. Langley and C. C. Laurie-Ahlberg. 1986. Molecular

population genetics of the alcohol dehydrogenase gene region of Drosophilu melanogc~ster. Genetics 114: 116551190.

Bailey, D. W. 1971. Recombinant inbred strains, an aid to finding identtty, linkage, and function of

histocompatibility and other genes. Transplantation 11: 325 -327. Barton N. H. and B. Charlesworth. 1984. Genetic revolutions, founder effects, and speciation. Ann. Rev.

Ecol. Syst. 15: 133-m164.

Barton, N. and M. Turelli. 1987. Adaptive landscapes. genetic distance and the evaluation of quantitative characters. Genet. Res., Camb. 49: 157 173.

Barton, N. and M. Turelli. 1989. Evolutionary quantitative genetics: How little do we know? Ann. Rev.

Genet. 23: 337 370. Beadle, G. W. 1980. The ancestry of corn. Sci. Am. 242: 112 119. Belknap, J. K., T. J. Phillips and L. A O’Toole. 1992. Quantitative trait loci associated with brain weight

in the BXD/Ty recombinant inbred mouse strains. Brain Res. Bull. 29: 3377344. Blangero, J., S. Williams-Blangero and J. E. Hixson. 1992. Assessing the effects of candidate genes on

quantitative traits in primate populations. Am. J. Primatol. 27: 119- 132. Boerwinkle, E. and C. F. Sing. 1987. The use of measured genotype information in the analysis of

quantitative phenotypes in man. III. Simultaneous estimation of the frequencies and effects of the

apohpoprotein E polymorphism and residual polygenic effects on cholesterol, betalipoprotein, and triglyceride levels. Ann. Hum. Genet. 51: 21 1~~226.

Bonner, J. J. and M. L. Tyan. 1982. Backcross test demonstrates linkage of glucocorticoid-induced cleft

palate susceptibility to H-2. Teratology 26: 2133216. Bonner, J. J. and M. L. Tyan. 1983. Glucocorticoid-induced cleft palate in the mouse: Two major

histocompatibility complex (H-2) loci with different mechanisms. Genetics 103: 263 276. Bonnet-, J. J. and M. L. Tyan. 1984. Glucocorticoid-induced cleft palate genes in chromosome 17:

Genetic linkage and mapping analyses. Immunogenetics 20: 169-~ 183. Carlson, J. E., L. K. Tutsierdm, J. C. Glaubitr, V. W. K. Luk, C. Kauffeldt and R. Rutledge. 1991.

Segregation of random amplified DNA markers in Fl progeny of conifers. Theor. Appl. Genet. 83:

194 200.


Chai, C. 1961. Analysis of quantitative inheritance of body sire in mice. IV. An attempt to isolate polygenes. Gcnet. Res., Camb. 2: 25 32.

Chai. C. 1968. Analysis of quantitative inheritance of body size in mice. V. Erects of small numbers of polygenes on similar genetic backgrounds. Genet. Res.. Camb. I I: 239 246.

Chapman, V. M.. J. Klein, J. H. Nadeau and L. M. Silver. 1991. Encyclopedia of the Mouse Genome I. Mammalian Genome, Volume I. Special Issue, pp. Sl -S515.

Crow. J. F. and M. Kimura. 1970. An Introduction to Population Genetics Theory. Burgess Publishing Company, Minneapolis, Minnesota.

Devos. K. M. and M. D. Gale. 1992. The USC of random amplified polymorphic DNA markers m wheat.

Thcor. Appl. Genet. X4: 567-572. Dletrlch, W., H. Katz, S. E. Lincoln. H. Shin, J. Friedman, N. Dracopoli and E. S. Lander. 1992. A

genetIc map suitable for typing intraspecific crosses. Genetics 131: 423--447. Docbley. J. and A. Stcc. 1991. Genetic analysis of the morphological differences between maize and

teosintc. Genetics 129: 2X5--295. Doehlcy, J., A. Stec, J. Wendel and M. Edwards. 1990. Genetic and morphological analysis of a

maiLe-teosinte F1 population: Implications for the origin of maize. Proc. Nat. Acad. Sci. X7: 9XXX 9892.

Edwards. M.. T. Ilelentjaris. S. Wright and C. Stuber. 1992. Molecular-marker-facilitated investigations of quantitative trait IOCI in maize. 4. Analysis based on genome saturation with isozyme and restriction fragment length polymorphism markers. Theor. Appl. Genet. X3: 765 -774.

Edwjards, M., C. Stuber and J. Wendel. 1987. Molecular-marker-l’xilitated investigations of quantita-

tlvc-trait loci in maize. 1. Numbers. genomic distribution and types of gene action. Genetics 116: II3 125.

Endler. J. A. 1986. Natural Selection in the Wild. Monographs in Population Biology. No. 21. Princeton Ilnivcrsity Press, Princeton. New Jersey.

f.alconer. D. S. 1989. Introduction to Quantitative Genetics. Longman Press, New York.

Fisher. R A. 1918. The correlations between relalives on the supposition of Mendelian inheritance. Trans. Roy. Sot. Edinb. 52: 399 433.

Gclman. R.. A. Wason. R. Bronson and E. Yunis. 1988. Murine chromosomal regions correlated with longevity. Genetics I IX: 693 704.

Goldstein. J. and M. Brown. 1983. Familial hypercholestcrolcmla, pp. 672 712. /n J. Stanbury. J.

Wyngaarden, D. Fredrickson. J. Goldstein and M. Brown (eds.). The Metabolic Basis of Inherited Dlxase. McGraw-Hill Co., New York.

Haley. C. S. and S. A. Knott. 1992. A simple regression method for mapping quantitative trait loci in hnc crosses using flanking markers. Heredity 69: 315 324.

flearne. C. M.. M. A. McAleer, J. M. Love, T. J. Aitman. R. J. Cornall, S. Ghosh. A. M. Knight, J-B. Prms and J. A. Todd. 1991. Additional microsatellite markers for mouse genome mapping. Mammalian Genome I: 273-282.

Hopper. J. L. and J. D. Mathews. 1982. Extensions to multivariatc normal models for pedigree analysis. Ann. Hum. Gcnet. 46: 373 383.

Klein-Lankhorst, R. M., A. Vermunt, R. Weide, T. Liharska and P. Zdbel. 1991. Isolation of molecular markers for the tomato (L~coper.~icon r.s~&ntum) using random amplified polymorphic DNA (RAPD). Theor. Appl. Genct. X3: 108 114.

Knapp, S. J., W. C. Bridges and D. Birkes. 1990. Mapping quantitative trait loci using marker linkage maps. Theor. Appl. Genet. 79: 5X3 592.

Landc. R. 1976. Natural selection and random genetic drift in phenotypic evolution. Evolution 30: 314 334.

Lande. R. 1979. Quantitative genetic analysis of multivariate evolution applied to brain : body allometry. Evolution 33: 402-416.

Landc. R. 1980. The genetic covariance between characters maintained by pleiotropic mutation. Genetics 94: 203 215.

Lande, R. and S. J. Arnold. 1983. The measurement of selection on correlated characters. Evolution 37: 1210 1226.


Lander. E. and D. Botstein. 1989. Mapping Mend&an factors underlying quantttative traits using RFLP linkage maps. Genetics 121: 185 199.

Lander, E. S., P. Green, J. Abrahamson, A. Barlow, M. Daly, S. Lincoln and L. Newburg. 1987. MAPMAKER: An interactive computer package for constructing primary genetic linkage maps of

experimental and natural populations. Genomics I: 174- 181. Love, J. M., A. M. Knight, M. A. McAlcer and J. A. Todd. 1990. Towards construction of a high

resolution map of the mouse genome using PCR analysed microsatellites. Nucleic Acids Research 18: 4123 4130.

Luo, Z. W. and M. J. Kearsey. 1989. Maximum likelihood estimation of linkage between a marker gene

and a quantitative locus. Heredity 63: 401 408. Luo, Z. W. and M. J. Kearscy. 1990. Maximum likelihood estimation of linkage between a marker gene

and a quantitattvc locus. II. Application to backcross and doubled hapohd populations. Heredity 66: II7 ~124.

Martin, B., J. Nienhuis, C. King and A. Schaefer. 1989. Restriction fragment length polymorphisms associated with water use efficiency in tomato. Science 243: 1725-172X.

Mayr, E. 1963. Animal Species and Evolution. Belknap Press, Cambridge, Massachusetts. Milkman, R. 1979. The posterior crossvein in Drosophila as a model phenotype, pp. 157- 176. In J. N.

Thompson and J. M. Thoday (eds.), Quantitative Genetic Variation. Academic Press, New York. Nadeau, J. H., H. G. Bcdigian, G. Bouchard, T. Denial, M. Kosowsky. R. Norberg, S. Pugh, E. Sargeant,

R. Turner and B. Paigen. 1992. Multilocus markers for mouse genome analysis: PCR amplification based on single primers of arbitrary nuclcotide sequence. Mammalian Genome 3: 55-64.

Neumann, P. E. and R. L. Collins. 1991. Genetic dissection of susceptibility to audiogenic seizures m inbred mice. Proc. Nat. Acad. Sci. 88: 54085412.

Neumann, P. E., G. G. Mueller and R. L. Sidman. 1990. Identification and mapping of a mouse gene influencing cerebellar folial pattern. Brain Res. 524: 85 X9.

Paterson, A., E. Lander, J. Hewitt, S. Peterson, S. Lincoln and S. Tankslcy. 1988. Resolution of

quantitative traits into Mendelian factors by using a complete linkage map of restriction length polymorphisms. Nature 335: 721~ 726.

Paterson. A. H., J. W. DeVerna, B. Lanini and S. D. Tanksley. 1990. Fine mapping of quantitative trait loci using selected overlapping recombinant chromosomes in an interspecies cross of tomato. Genetics 124: 735 742.

Pcdersen. J. C. and K. Berg. 1988. Normal DNA polymorphism at the low density lipoprotein receptor

(LDLR) locus associated with serum cholesterol level. Clinical Genetics 34: 3066312. Pedersen, J. C. and K. Berg. 1989. Interaction between low density lipoprotein receptor (LDLR) and

Apolipoprotein-E (APOE) alleles contributes to normal variation in liptd-level. Clinical Genetics 35: 331-337.

Pemberton. J. M., S. D. Albon, F. E. Guinness, T. H. Glutton-Brock and R. J. Berry. 1988. Genetic variation and juvenile survival in red deer. Evolution 42: 921-934.

Plomin, R., G. E. McClearn. G. Gors-Maslak and J. M. Neiderhiser. 1991. Use of recombinant inbred strains to detect quantitative trait loci associated with behavior. Behav. Genet. 21: 99 116.

Ramshaw, J. A., J. A. Coyne and R. C. Lewontin. 1979. The sensitivity of gel electrophoresis as a detector of genetic variation. Genetics 93: 1019 1037.

Rapp, J., S. Wang and H. Dene. 1989. A genetic polymorphism in the renin gene of Dahl rats cosegregates

with blood pressure. Science 243: 5422544. Reidy, M. F., W. J. Hamilton and C. F. Aquadro. 1992. Excess of non-parental bands in offspring from

known primate pedigrees assayed using RAPD PCR. Nucleic Acids Research 20: 918. Rose, M. 1982. Antagonistic pleiotropy, dominance, and genetic variation. Heredity 48: 63378. Rousset, M., J. M. Carrillo, C. 0. Qualset and D. D. Kasarda. 1992. Use of recombinant inbred lines

of wheat for study of associations of high-molecular-weight glutenin subunit alleles to quantttative

traits. 2. Milling and bread-baking quality. Theor. Appl. Genet. 83: 403- 412. Sax, K. 1923. The association of size differences with seed-coat pattern and pigmentation in Phaseolus

vulgaris. Genetics 8: 5522556. Serikawa, T., X. Montagutelli, D. Simon-Chazottes and L-L. Guenet. 1992a. Polymorphisms revealed by


PCR wtth single, short-sized, arbitrary primers are reliable markers for mouse and rat gene mapping. Mammalian Genomc 3: 65 72.

Sertkavva, T., T. Kuramoto. P. Hilhert. M. Mori. J. Yamada. C. J. Duhay, K. Lindpainter, D. Gantcn. J-L. Guenet, G. M. Ldthrop and J. S. Beckmann. 1992h. Rat gene mapping using PCR-analyzed microsatellites. Genettcs I3 I : 201 721.

Sing. C. F., E. Boerwinkle, P. P. Mall and A. R. Templeton. 1988. Charactcrtzatton of genes utfecting

quantitative traits in humans, pp. 250 269. In B. S. Weir. E. J. Eisen. M. M. Goodman and G. Namkoong (eds.), Proceedings of the Second International Conference on Quantitative Genetics. Sinauer Associates Inc., Sunderland, Massachusetts.

Sing, C. F.. M. B. Haviland, K. E. Zcrva and A. R. Templeton. 1992. Applicatton of cladisttcs to the analysis of genotype-phenotype relationships. European Journal of Epidemiology 8: 3 9.

Smith, D. G. and M. F. Small. 1982. Selection and the transferrin polymorphism in rhesus monkeys

(MUUUI mulur/cr). Folia Primatol. 37: 127 136. Stallings, R. L., A. F. Ford, D. Nelson, D. C. Torney. C. E. Hildehrand and R. K. Moyzis. 1991.

Evolution and distribution of (CT),,. Genomics IO: X07 8 IS.

Tanksley, S., H. Medina-Filho and C. Rick. 19X2. LJse of naturally-occurring enzyme variatton to detect and map gcncs controlling quantitative traits in the intcrspecitic hackcross of tomato. Heredity 49:

II 25. Templeton. A. R. 1980. The theory of speciation via the founder princtplc. Genetics 94: IO1 I 103X. Templeton, A. R., E. Boerwinkle and C. F. Sing. 1987. A cladistic analysis of phenotype associations

with haplotypes inferred from restriction endonuclease mapping. I. Basic theory and an analysis 01 alchohol dehydrogenase activity in Drosophila. Genetics I 17: 343 351.

Templeton, A., C. Sing, A. Kcssling and S. Humphries. 1988. A cladistic analysis of phenotype associations with haplotypcs inferred from restriction endonuclcase mapping. II. The analysis of natural populations. Genetics 120: I 145 I 154.

Templeton. A. R., K. A. Crandall and C. F. Sing et al. 1992. A cladistic analysis of phenotype

associations wtth haplotypes inferred from restriction cndonuclease mapping. 111. Cladogram estimation. Genetics 132: 619~ 633.

Templeton, A. R. and C. F. Sing. 1993. A cladistic analysis of phenotype associations with haplotypes inferred from restriction endonuclease mapping. IV. Nested analyses with cladogrdm uncertainty and recombination. Genetics 134: 659 -669.

Thompson, J. N. and J. M. Thoday. 1979. Quantitative Genetic Variation. Academic Press. New York.

Turelli, M. 19X8. Population genetic models for polygenic variation and cvolutton, pp. 601 618. 1?1 B Weir, E. Eisen, M. Goodman and G. Namkoong (eds.), Proceedings of the Second International Conference on Quantitative Genetics. Sinauer Associates, Sundcrland, Massachusetts,

Vogel, F. and A. Cr. Motulsky. 1986. Human Genetics: Problems and Approaches. Springer-Verlag. New York.

Wagner. G. 1989. Multivariate mutation-selection balance with constrained pleiotropic effects. Genetics 122: 223 234.

Watt, W. 1986. Power and efficiency as indexes of fitness in metabolic organization. Am. Nat. 127: 629 653.

Weller, J. 1987. Mapping and analysis of quantitative trait loci in Lycoper.sicon (tomato) with the atd of genetic markers using approximate maximum likelihood methods. Heredity 59: 413 421.

Weller, J., M. Soiler and T. Brody. 1988. Linkage analysis of quantitative traits in an interspecific cross of tomato (Lyc’opmsicon esculentum x Lvcopersicon pimpinrll~folium) by means of genetic markers. Genetics 118: 3299339.

Winkelman, D. C. and R. B. Hodgetts. 1992. RFLPs for somatotropic genes identify quantitative trait

loci for growth in mice. Genetics 131: 929 937. Wright, S. 1931, Evolution in Mendelian populations. Genetics 16: 97 159.

Received 29 March 1993; accepted 6 April 1993.

Corresponding Editor: G. Wagner

Quantitative trait loci: individual gene effects on quantitative characters

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