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CHAPTER 6. QUANTITATIVE GENETICS–EMPIRICAL RESULTS RELEVANT TO PLANT BREEDING
R. H. MOLL AND C. W. STUBER
breeding, crossbreeding, and selection are essential features of any plant
breeding program. A primary objective of quantitative genetic research is
an understanding of the genetic consequences of such manipulations.
A basic premise of quantitative genetics is that the genes that affect quantitative traits follow the same laws of transmission as genes that affect qualitative traits. Usually many loci with small individual effects are involved;
therefore, it is necessary to study these traits through statistics appropriate
for continuous variables, such as means, variances, and covariances. Fisher
( I 918) provided the initial framework for the study of quantitative inheritance. Since that time, his developments have been clarified, elaborated,
and extended by numerous geneticists and statisticians. Unfortunately, the
experimental aspects of quantitative genetics have lagged behind theory.
Because it is difficult to design quantitative genetic experiments with definitive alternative hypotheses, many of the experimental conclusions have
been reached from the experience of numerous individual empirical investigations that have shown similar results.
Most of our emphasis will be concentrated on reviewing and interrelating
recent research results in areas of most significance to plant breeders, such
as ( 1 ) kinds of genetic variability found, (2) effects of inbreeding and
crossbreeding, ( 3 ) genotype-environmental interaction, and (4) selection
methodology and response. It is not our purpose to present a detailed description of quantitative genetic theory. Cursory summaries (using nomenclature from Falconer, 1960) are included to provide background
for readers untrained in quantitative genetics and to aid in understanding
results from experimental research.
Evaluations of inheritance mechanisms in quantitative genetics research
depend on valid assessments of genotypic values. However, the genotypic
value of an individual must be ascertained from measurements made on
its phenotype. Phenotypic value then, is defined as the performance of
a particular genotype in the environment in which it is grown. The two
components of the phenotypic value (P)-genotypic value ( G ) and environmental deviation (E)-are
usually represented in the equation for
phenotypic value as: P = G E.
A genotype is considered as the particular assemblage of genes possessed
by an individual, and genotypic value for a given genotype is defined as
the average of all possible phenotypic values, expressed as a deviation from
the population mean. In other words, it is the average phenotypic value
when genotypes are grown over all possible environments (the mean environmental deviation is zero) . Diagrammatically, the relationship between
genotypes and genotypic values for a single locus may be represented as
In this representation, the origin, or zero point, is midway between the
value of the two homozygotes, and B , represents the allele that increases
the genotypic value. The value, d, of the heterozygote depends on the
degree of dominance. In order to determine the contribution of this locus
to the population mean, values of the genotypes must be weighted by their
respective frequencies. Contributions of each individual locus must be combined in the computation of the population mean.
Estimation of genetic effects and variances requires some type of family
structure. To analyze the properties of a population composed of various
family structures, it.is necessary to deal with concepts concerning transmission of value from parent to offspring. This cannot be done with only genotypic values, because parents pass on their genes, not their genotypes, to
the next generation. Genotypes are created anew by uniting gametes in
each generation. Therefore, a measure is required that allows the assignment of values associated with the genes carried by an individual and transmitted to its offspring, i.e., a measure is needed that reflects the average
effects of genes. The average effect of a gene at a locus is defined as the
average difference resulting from the substitution of one allele for the other.
For example, if B , genes could be changed at random in a population
to B , genes (see the diagram above), the resulting change in value is the
average effect of the gene substitution. Because the average effect of a gene
substitution depends on the gene frequency, it is a property of the population as well as of the gene.
Although the average effects of genes cannot be measured directly,
breeding values, which are weighted functions of average effects of genes,
can be measured experimentally. If an individual is mated with a number
of random individuals from the population, the breeding value is estimated
as twice the mean deviation of its progeny from the population. This deviation must be doubled because only one-half the genes in its progeny are
contributed by the individual being evaluated.
In addition to the breeding value, another component of the genotypic
value that must be considered is referred to as the dominance deviation.
For a single locus, it is defined as the difference between the genotypic
value and the breeding value. Although dominance deviations are within-
R. H. MOLL AND C . W. STUBER
locus interactions, their importance depends on gene frequencies in the
population, so they are not simply measures of the degree of dominance.
Most quantitative traits are conditioned by genes at many loci. However,
the aggregate genotypic value may or may not be simply an additive combination of the genotypic values for individual loci. When this combination
is not additive, genes are said to interact or to show epistasis.
The phenotypic value has been expressed as P = G E. Therefore,
the phenotypic variance, up2,may be expressed as:
U E ~
~ U G E
where uG2= genotypic variance, uK2= environmental variance,
covariance of genotypic and environmental effects.
The genotypic partition may be further divided as follows:
where uA2= additive genetic variance, uD2= dominance variance, u12=
The additive variance, which is the variance of breeding values, is the
primary measure of the resemblance between relatives and is relevant to
the effectiveness of selection. The objective of selection is to increase the
frequency of favorable alleles in a population by substituting favorable
genes for unfavorable genes. The effectiveness of gene substitution in
changing the population mean is directly related to the average effects of
genes. Average effects, in turn, are reflected in breeding values. The average effect of a gene substitution is actually a weighted regression coefficient
(weighted by gene frequency). Therefore, additive genetic variance at a
single locus may be considered as variance caused by the weighted linear
regression of genotypic values on number of favorable alleles. Dominance
variance, then, is variance attributed to deviations from regression. If the
total variation over loci is larger than the summation of additive and dominance variances for individual loci, the differences will be variance caused
Resemblance between relatives is reflected in similarities of expression
of quantitative traits. The degree of resemblance expected provides the
basis for estimating genetic variance components. Estimation procedures
require systematic mating schemes that result in different types of relatives.
Using appropriate experimental designs and statistical analyses, design variance components can then be calculated. Genetic interpretations of these
design components are facilitated by translating them into covariances
among relatives. Theoretical considerations of the genetic causes for resemblances between relatives permit the translation of these covariances into
functions of genetic variance components. For example, with the assump-
tions of no epistasis and an inbreeding level of zero in the population,
the covariance between a parent and its offspring produced from mating
at random in the population is 1/2 uA2,the covariance among half-sibs is
% uA2,and the covariance among full-sibs is '/2 uA2 1/4 uBZ,These relationships change with different levels of inbreeding.
A comprehensive summary of methods for estimating genetic variances
was presented by Cockerham (1963). He also discussed many of the problems and limitations inherent in the definition and estimation of genetic
variances when inbreeding is present. For most cross-pollinated species,
genetic variances can be estimated with mating designs that do not depend
on inbred relatives. However, in many self-pollinators, production of sufficient seed for replicated evaluation trials is nearly impossible without the
use of inbred generations. The development by Stuber (1970) of methods
using inbred relatives has facilitated the estimation of genetic variances
in self-pollinated species.
Diallels and modified diallels are often used to estimate genetic variances. In this type of design, general and specific combining ability components of variance are routinely estimated. The general combining ability
component is primarily a function of additive genetic variance. However.
if epistasis is present, it may include functions of additive types of epistasis.
The specific combining ability component is primarily a function of dominance variance, but it may include all types of epistatic components. The
relative proportions of genetic variances in the two combining ability components depends on the inbreeding level of the parents. Although diallels
can be generated with parents chosen at random from some random mating
reference population, this type of mating design generally has been used
for specific sets of parents. With specific sets, the variance estimates must
be interpreted as characteristic of only the set of parents involved and
should not be used to characterize more broadly defined reference
Excellent reviews of genetic variance estimates available before 1962
for important crop species are given by Gardner (1963) and Matzinger
(1963). Since 1962, a large number of experiments have been reported
which cover essentially all major crop species and many different kinds
of traits. Most of the data reported points to one general conclusion: genetic variability of important agronomic traits is predominantly additive
genetic variance. Nonadditive variance also exists in nearly all species and
for many important traits, but it is generally smaller than additive genetic
R. H. MOLL AND
C . W. STUBER
Quantitative inheritance of various traits of maize has been studied most
extensively, and studies reported in the literature deal with a wide range
of maize populations. Estimates of genetic variability summarized by
Gardner (1963) and Moll and Robinson (1967), as well as evidence in
several more recent reports, indicate that additive genetic variance exceeds
dominance variance in many different kinds of populations, including
open-pollinated varieties, synthetics, variety hybrids, and variety composites. Furthermore, when sampling errors of variance estimates are considered, it appears that locally adapted open-pollinated varieties have genetic variances of essentially the same order of magnitude. Important
differences in genetic variability seem to occur only between populations
of distinctly different kinds. For example, composite populations formed
by intermating a number of varieties tend to have greater variability than
the parental varieties themselves. Composites of more genetically diverse
populations have greater variances than composites of less diverse populations. Even so, estimates reported for such composites show additive variance to be larger than nonadditive variance for most traits.
Although a number of extensive variability studies have been reported,
epistatic variability has not been shown to be an important component
of genetic variances of maize populations (Chi et al., 1969; Eberhart el
al., 1966; Stuber et al., 1966). Studies of inbred line hybrids of various
kinds, however, frequently reveal significant epistatic effects (Wright et
al., 1971; Russell and Eberhart, 1970; Stuber and Moll, 1971; and others).
Joint consideration of these two kinds of evidence leads to the conclusion
that epistatic interactions must occur in maize populations, but they contribute very little variability beyond that accounted for by additive and
The patterns seen in variance estimates in many other cross-pollinators
is similar to that reported for maize; however, the data are much less extensive. Although sugarcane presents difficulties for quantitative genetic
studies because of irregular meiosis and mating incompatibilities, estimates
reported by Hogarth (1 97 1 ) indicate that additive genetic variance exceeds
dominance variance, Variance estimates in alfalfa also provide evidence
that additive genetic variance exceeds nonadditive variance, but there is
some evidence for variance caused by trigenic, quadragenic, or epistatic
effects (Dudley et al., 1969). Hill et al. (1972) also report a preponderance of variance caused by general combining ability (which wouId be
largely additive genetic variance) and, although variance caused by specific
combining ability was detected, it was relatively small in magnitude. Significant general combining ability has also been found in several forage
grasses, including Bromus inermis Leyss. (Mishra and Drolsom, 1972;
Dunn and Wright, 1970), Dactylis glumerata L. (Kalton and Leffel,
1955), and Lolium perenne L. (Hayward and Lawrence, 1972).
A series of studies of genetic variability in flue-cured tobacco (a selfpollinator) have also shown that additive genetic variance is predominant
for a number of traits in several different populations. There appears to
be some epistatic variability in certain populations for certain traits, especially plant height and leaf measurements. Dominance variance tends to
be small and usually nonsignificant (Matzinger, 1968; Matzinger et al.,
1960, 1966, 1971). Changes in means after several generations of random
mating were interpreted as evidence for epistasis. Segregation of epistatic
gene complexes may have caused a breakdown of internal balance by
forced intercrossing in a naturally self-pollinated species (Humphrey et
Genetic variability for a number of traits of soybeans has also been
shown to be predominantly additive, but nonadditive variability is significant for many of the traits (Brim and Cockerham, 1961; Hanson et al.,
1967; Weber et al., 1970). A series of diallel studies involving a number
of pulse crops, such as mungbean, cowpeas, blackgram, and lentil, have
shown significant general combining ability for a number of traits. Specific
combining ability appears to be important for certain traits, but it is usually
less important than general combining ability (Singh and Jain, 1971, 1972;
K. B. Singh and Singh, 1971; T. P. Singh and Singh, 1972). Diallel studies
in small grains, particularly in wheat and oats, and in sorghum have also
indicated that general combining ability for yield and related traits is more
important than specific combining ability, even though specific combining
ability is statistically significant in some instances (Collins and Pickett,
1972; Lee and Kaltsikes, 1972; Ohm and Patterson, 1973a,b; Gyawali et
al., 1968; Walton, 1972; Widner and Lebsock, 1973; and others).
A genetic study of pearl millet by Bains (1971), which used the analysis
proposed by Kearsey and Jinks (1968), found that additive genetic variance was important for three agronomic traits, but epistasis also appeared
to be prevalent. In diallel studies, however, Gupta and Singh (1971) found
the additive component to be nonsignificant for grain yield and ear number
in a study of eight diverse lines of pearl millet. General combining ability
was either smaller than specific combining ability or nearly the same size
in four traits studied by Ahluwalia et al. ( 1962).
Although many diallel studies in cotton indicate that additive variance
is more important than nonadditive types, there are several examples of
deviation from this pattern. For example, Gupta and Singh (1970) reported dominance variance to be larger than additive variance in several
seed and fiber traits. Their study involved eight diverse strains of upland
cotton. Baker and Verhalen (1973) reported similar results in a study of
10 selected upland cotton lines. They also presented an extensive literature
review in which they documented several instances in which additive variances predominated and several in which dominance predominated.
R. H. M O L L AND C. W. STUBER
Inbreeding Depression and Heterosis
Inbreeding results from matings between related individuals. The degree
of inbreeding is measured by the inbreeding coefficient, which is the probability that two genes at a locus (in a diploid) are identical by descent.
The effect of inbreeding upon the population mean can be shown to be
a function of the gene frequency, dominance effects, and the coefficient
of inbreeding. If dominance is directional, i.e., the majority of loci show
dominance for the favorable allele, inbreeding will result in a decrease in
the mean proportional to the inbreeding coefficient.
Heterosis, which results from crossing unrelated strains, is the reverse
of inbreeding depression. It also depends upon directional dominance for
its expression. Therefore, inbreeding depression and heterosis both refer
to differences in mean performance directly related to differences in
heterozygosity, and in diploids the level of heterozygosity is directly related
to the coefficient of inbreeding. Inbreeding depression is the decline in trait
expression with decreased heterozygosity, and heterosis is the enhancement
of trait expression with increased heterozygosity.
The relationship between the mean expression and the coefficient of inbreeding tends to be linear for most traits of maize, and yield of grain
shows a steeper rate of depression than other agronomic traits (Sing et
al., 1967). Papers by Levings (1964) and Busbice (1969) show that in
autotetraploids the loss of heterozygosity is not directly related to the inbreeding coefficient, and is much slower than in diploids. Levings et al.
( 1967) reported that the relationship between heterozygosity and performance in autotetraploid maize was linear for three quantitative traits. The
decrease in ear weight relative to the inbreeding coefficient was slightly
less than half as rapid in the autotetraploid as it was for yield of ear corn
in diploid maize studied by Sing et al. ( 1967). Comparison of these results
agrees qualitatively, at least, with theoretical expectations.
A direct comparison of inbreeding effects in crested wheatgrass [Agropyron cristatum (L. ) Gaertn.] suggests that inbreeding depression is
greater than expected at higher levels of ploidy (Dewey, 1966). Forage
yields of self-pollinated progeny for diploids, tetraploids,, and hexaploids
were 35.9, 50.4, and 67.4%, respectively, less than yields of noninbred
progenies. Also contrary to theoretical expectations, diploid and tetraploid
sugar beets (Beta vulgaris L.) were reported to show the same rate of
inbreeding depression (Hecker, 1972). In a study involving diploid and
autotetraploid rye (Secale cereale L.), Lundqvist (1969) found less inbreeding depression in the autotetraploids than in the diploids, but not
as much less as expected from genetic theory.
Inbreeding in alfalfa is accompanied by impaired reproductive fertility,
as well as by loss of vigor and productivity. The relationship between inbreeding coefficient and inbreeding depression was linear for several traits,
and most drastic for yield and spring vigor. The inbreeding rates for yield
and spring vigor approach a theoretical curve for tetragenic inheritance,
whereas plant height approximates the curve for duplex inheritance
(Aycock and Wilsie, 1968).
1 . Genetic Diversify and Heterosis
Heterosis, in quantitative genetic terminology, is usually measured as
the superiority of a hybrid over the average of its parents, and has been
reported for a wide range of crop species, which include both self- and
cross-pollinators.The expression of heterosis is greatly influenced by the
magnitude of genetic differences for some traits, but not for others. For
example, several recent reports of diallel crosses among strains of wheat
show greater heterosis associated with crosses of more distantly related
parents (Fonesca and Patterson, 1968; Widner and Lebsock, 1973; Sun
1972). On the other hand, Gyawali et al. (1968) found no evidence
for an increase in heterosis associated with interclass differences between
soft red and hard red parents. A study of crosses of nine strains of tall
fescue (Festuca arundinaceae Schreb) suggests that heterosis increased
with genetic divergence with respect to morphological traits and flowering
time, and also with respect to geographical origin of the parents (Moutray
and Frakes, 1973).
Comparisons of inter- and intraspecific hybrids of alfalfa and cotton
show greater heterosis associated with greater diversity (Sriwatanapongse
and Wilsie, 1968; Marani, 1963, 1968). Heterosis in lint yield of cotton
tended to be associated with a greater number of bolls rather than boll size,
especially in interspecific hybrids, in which boll size was often less than
the average of the parents. A relationship of heterosis to diversity is also
reported for several traits of cotton relating to plant growth, such as plant
height, leaf area index, and dry matter accumulation (Marani and Avieli,
Studies involving interracial crosses of maize and interspecific crosses
of tobacco indicate that the relationship between diversity and heterosis
may not be linear over very wide ranges of diversity. There is considerable
evidence that increased genetic differences between inbred lines of maize