1022-7954/05/4106- © 2005 Pleiades Publishing, Inc.
Russian Journal of Genetics, Vol. 41, No. 6, 2005, pp. 698–704. Translated from Genetika, Vol. 41, No. 6, 2005, pp. 855–862.
Original Russian Text Copyright © 2005 by Mikhailov.
In our studies of quantitative traits in maize, we have
recorded that the frequency of manifestation of the so-
called high-parent heterosis (the advantage of offspring
over the both parents ) is quite different for different
hybrids surpass the mean parental
value in nearly all traits (for phenophases, faster devel-
opment is regarded as superiority). Along with traits
expressing high-parent heterosis nearly in all hybrid
combinations examined (productivity, ear length, ker-
nel number, plant height), there are traits that some-
times (ear diameter, stem diameter, the number of
nodes) or never (the number of kernel rows, phe-
nophases) manifest high-parent heterosis.
The same results were obtained by Chalyk  on
extensive material, including 140 maize hybrids. The
traits, in which all or nearly all hybrids were superior
to the parental plants (productivity and ear length, all
140 hybrids; ear diameter, 139 hybrids; weight of
1000 kernels, 137 hybrids), were signiﬁcantly differ-
ent in the frequency of high-parent heterosis (produc-
tivity, 140 hybrids; ear length, 128 hybrids; ear diam-
eter, 94 hybrids; weight of 1000 kernels, 76 hybrids).
This difference in trait inheritance seems to be
related to their different genetic architecture. However,
as far as we know, information on the frequency of
high-parent heterosis is not taken into account in math-
ematical models. For instance, in the model by Mather
and Jinks , the data only on an individual combina-
tion in its different variants (
etc.) are used to estimate genetic parameters.
The aim of the present work was ﬁnding a parameter
related to the frequency of high-parent heterosis which
would be suitable for transformations and rearrange-
ment and showing, at least on a simple model, how this
parameter could be used for genetic analysis.
THE CONCEPT OF ESSENTIALITY
High-parent heterosis requires condition
is the excess of the trait value in the hybrid
over the mean parental value and
is half difference
between the parental values of the trait. The higher is
the difference between mean
, the more
often is the inequality fulﬁlled.
A practical example is given in Figs. 1–4, where the
manifestation of four traits is shown in a group of ten
hybrids. The bottom part of each ﬁgure graphically pre-
values for the same hybrids. The traits
are ranged according to diminishing heterosis. For the
most heterotic trait, productivity (Fig. 1), the
varies from 4 to 57; for ear length (Fig. 2), from 1.3 to
18; for the number of kernel rows (Fig. 3), from 0.6 to
6; and, ﬁnally, for the last trait (the ear lenth/ear diam-
eter ratio, Fig. 4), from –1 to 5. The ratios of the mean
to the mean squared
for these traits are respectively
8.11, 3.84, 1.32, and 0.73, and the frequencies of high-
parent heterosis are 1, 1, 0.8, and 0.5.
ratio for an individual hybrid is referred to
as hybrid vigor  (not to mix up with level of domi-
nance) [3, 4]. However, for a group of hybrids, hybrid
vigor is an instable characteristic. When the difference
between the parents is comparable to the error of the
ratio becomes abnormally high and not
reliable. Occasionally, it even involves division by zero.
Even when the mean parental values are precisely
determined to avoid division by zero, this does not
guarantee against an abnormally high ratio that would
drastically increase the mean
for the whole group
of hybrids. This results in an unstable and meaningless
Therefore, the ratio of mean
are more suit-
able to use than the mean
ratio. The mean
convenient to calculate as a mean square, because
Statistical Parameter “Essentiality of Heterosis”
and Its Genetic Meaning
M. E. Mikhailov
Institute of Genetics, Academy of Sciences of Moldova, Chisinau, 2002 Moldova;
fax: (3732) 55-61-80; e-mail: firstname.lastname@example.org
Received June 8, 2004; in ﬁnal form January 20, 2005
—An additive dominant model of inheritance of a quantitative trait in a group of hybrids is considered.
The parameter “essentiality of heterosis,” associated with the frequency of high-parent heterosis, was proposed
to characterize the expression of a trait in a group. An expression was obtained to relate essentiality of heterosis,
mean dominance, and trait polygenity, which is fulﬁlled upon a random and independent distribution of advan-
tageous alleles among the parents. A measure of polygenity of a trait taking into account inequality of quanti-
tative effects of different loci was proposed.
MODELS AND METHODS