ISSN 10227954, Russian Journal of Genetics, 2015, Vol. 51, No. 8, pp. 818–826. © Pleiades Publishing, Inc., 2015.
Original Russian Text © M.E. Mikhailov, 2015, published in Genetika, 2015, Vol. 51, No. 8, pp. 953–962.
The influence of linkage on genetic estimations is a
longlasting and still unsolved problem of biometric
genetics. Because of the possibility of linkage, biomet
ric data often do not allow unambiguous interpreta
tion, and theoretical solutions obtained for unlinked
genes are useless or are only conditionally useful.
It primarily affects such an important value as the
degree of dominance. Adopted in the classical analy
sis, a method of estimating the average degree of dom
inance based on the ratio of the variances was devel
oped for unlinked genes. The application of this
method to the analysis of experimental data is usually
accompanied by the conditions that the obtained value
may be overestimated because of linkage (e.g., [1, 2]).
These estimations may have a comparative value for
comparisons but they provide limited information,
since they indicate only the upper limit of the degree
The knowledge of the average degree of dominance
is very important for practical breeding and for the
theory of heterosis. For example, the question of the
reality of overdominance is still open because of the
lack of unbiased estimates . It should be noted that
the problem of the distorting effect of linkage on the
estimation of the degree of dominance persists in quan
titative trait analysis with molecular markers [4, 5].
The number of loci affecting the value of a trait is
among the other variables. Wright’s estimator is also
valid for unlinked genes. In the general case, this esti
mator does not show the number of loci but indicates
the number of socalled effective factors—groups of
linked genes that recombine as a single gene .
The distortion introduced by the linkage in each
individual case depends on the location of the genes
involved on the map. If the location of the genes is
unknown, it is not possible to calculate a correction for
their linkage, but it is possible to estimate their most
probable value as expected based on a random
arrangement. On the basis of this assumption, in this
study we developed unbiased estimates for the average
degree of dominance and the number of major loci. In
some cases overestimates and underestimates are pos
sible, but it is expected that the incidence of overesti
mations and underestimations will be approximately
the same in wider application.
EXISTING ESTIMATES OF THE AVERAGE
DEGREE OF DOMINANCE
For a single pair of alleles, the degree of dominance
characterizes the manifestation of a quantitative trait
in the heterozygote. It is equal to zero when the quan
titative trait intermediately manifests in the heterozy
gote; it is equal to one for complete domination and
higher than one in the case of overdominance. It is
mathematically expressed as
is the dom
inant effect (excess of heterozygotes over the average
value of homozygous) and
is an additive effect (half
the difference between homozygotes).
nonnegative, whereas the
value can have any sign.
In the case of heterozygosity, the most correct
expression at many loci for the average degree of dom
inance would be
. However, since
estimated by biometric methods in classical analysis,
the average degree of dominance is expressed as
Accounting for Expected Linkage
in Biometric Analysis of Quantitative Traits
M. E. Mikhailov
Institute of Genetics, Physiology and Plant Protection, Academy of Sciences of Moldova, Kishinev, MD 2002 Moldova
Received July 30, 2014
—The problem of accounting for a genetic estimation of expected linkage in the disposition of ran
dom loci was solved for the additivedominant model. The Comstock–Robinson estimations for the sum of
squares of dominant effects, the sum of squares of additive effects, and the average degree of dominance were
modified. Also, the Wright’s estimation for the number of loci controlling the variation of a quantitative trait
was modified and its application sphere was extended. Formulas that should eliminate linkage, on average,
were derived for these estimations. Nonbiased estimations were applied to the analysis of maize data. Our
result showed that the most likely cause of heterosis is dominance rather than overdominance and that the
main part of the heterotic effect is provided by dozens of genes.