EVALUATiDN 0F SiNGLE-CRGSS
SELECTION METHODS WITH
SIMULATED POPULATIONS

Thesis for the Degree of :Ph. D.
MICHIGAN STATE UNIVERSITY
BAHMAN EHDAIE
1970

 

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thesis entitled

Evaluation of Single-Cross Selection
Methods with Simulated Populations

presented by

Bahman Ehdaie

has been accepted towards fulfillment
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Ph . D degree in Agronomy

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Charles E. Cress
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ABSTRACT

EVALUATION OF SINGLE-CROSS SELECTION METHODS
'WITH SIMULATED POPULATIONS

By

Bahman Ehdaie

Three methods of breeding, namely, top-cross, reciprocal
recurrent selection (RRS), and the Hallauer method, were eval-
uated and compared under various genetic situations. Since
genotypic-environment interaction was not in the scope of this
study, environmental variation was a random normal variable”

Genetic models used were additive (A), complete domi-
nance (CD), pure overdominance (OD), Optimum number (ON),
additive by additive initial variance at gene frequencies 0.5
(AA), and additive by dominance initial variance at gene
frequencies 0.5 (AD). A range of starting gene frequencies in
population A and B were studied for each genetic model. Mild
and strong selection intensities were practiced in the gen-
erated experiments to find the effect of selection intensity on
the rate of progress. Ten cycles of selection were performed
for the (RRS) experiments, eight for the "Hallauer" experiments,
and four for the top-cross experiments. Progress in the hybrid
population was the primary objective for study for each breeding
method. Thirty independently segregating loci, each with two

alleles, were simulated to determine a single character.

 

Bahman Ehdaie

Greater reSponses were observed in the hybrid population
when stronger selection was practiced except for the situations
where the equilibrium gene frequency was present in the parental
populations with (RRS) and the top-cross methods.

The top-cross method was competitive with the Hallauer and
(RRS) method with regard to progress made in the hybrid p0pu1ation
for a wide range of gene effects except for the case when over-
dominant gene effects or selected epistatic effects were present.
The top-cross tester was a completely recessive inbred line. When
additive genetic variance was important, improvement in the hybrid
population was similar for the Hallauer and (RRS) breeding methods.
As non-additive genetic variance became notable, the Hallauer
method was superior to the (RRS) method as well as the TOp-cross
method.

Under some genetic models, (RRS) was ineffective in advanc-
ing the mean of the hybrid populations when mild selection was
practiced. Relatively small improvement was seen when stronger
selection was practiced. In contrast, the Hallauer method im-
proved the hybrid population means significantly under these
genetic models. The Hallauer method seemed to be an effective
way of producing superior single-cross hybrids under most, if not
all, genetic situations. The recommendation was made that strong
selection intensity be practiced in the first two or three cycles
with little or no further testing until the lines are completely

inbred.

Bahman Ehdaie

No inter-locus selection effects were found for the inbred
populations in the absence of epistasis for the Hallauer method.
Intra—locus as well as inter-locus selection effects were observed
for the hybrid population with the complete dominance and over-
dominance models. With epistatic models,such as optimum number
and additive by additive,both intra- and inter-locus selection
effects were found for the inbred populations and hybrid popula-

tion.

EVALUATION OF SINGLE-CROSS SELECTION METHODS
WITH SIMULATED POPULATIONS

BY

Bahman Ehdaie

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Crop and Soil Sciences

1970

662 777
7-/'70

This thesis is dedicated to my late parents,
Mr. Hamdan Ehdaie and Mrs. Robab Ehdaie (Tabib),

for their sincere support throughout my educa-

tional career.

ii

ACKNOWLEDGEMENTS

The author expresses his sincere appreciation to Dr.
C.E. Cress for his encouragement and guidance throughout the
study.

Thanks are due to the committee members for their help-
ful suggestions and to Dr. C.M. Harrison for his critical
appraisal of manuscript.

I express my deepest gratitude to my wife, Soraya,
daughter, Suzan, and son, Sasan, for their patience and devo-
tion during my years of study.

Finally, I thank the Iranian Government and the National
Science Foundation for providing financial assistance (grant

GB 6646) which made this study possible.

iii

II.

III.

IV.

V.

TABLE OF CONTENTS

INTRODUCTION
REVIEW OF LITERATURE
A - Pure-line Method
B - Top-cross Method
Testers for GCA and SCA
C - Reciprocal Recurrent Selection (RRS)
Method
D - The Hallauer Method

Non-additive gene effects

MEAN OF A SELFED POPULATION

THE MECHANISM AND LOGIC OF POPULATION
SIMULATION

A - Subprogram l: Generating the Initial

'IIIFIUOW

0

Population

Subprogram 2: Generating Gametes
Subprogram 3: Evaluating Individuals
Distribution of Random Error
Number of Runs
Genetic Models Simulated and Initial
Gene Frequencies
The Essential Features of Simulated
Experiments
1 - Reference experiment
2 - Top-cross experiment
3 - Experiments with the Hallauer method
4 - Reciprocal Recurrent Selection (RRS)

experiments

RESULTS AND DISCUSSION

A
B
C

Reference Experiment

Top-cross Experiment

"Hallauer" Experiments

1 - Mild selection intensity

2 - Strong selection intensity

Comparisons Between the Breeding Methods

1 - Top-cross method vs the Hallauer
and (RRS) methods

2 - The Hallauer method vs (RRS) method

iv

33

33
34
35
36
37

38

38
38
39
4O

41

42

42
44
49
49
55
61

VI.

VII.

SUMMARY AND GENERAL CONCLUSIONS
LITERATURE CITED
APPENDIX A (Tables)

APPENDIX B (Figures)

70
74
81

96

I. INTRODUCTION

Shull (1909) described and outlined the pure-line breeding
method for corn Clea m§y§_L.) which introduced a new era of plant
breeding.

Jones (1918) suggested a procedure in which high-yielding
single-crosses were used as parents to produce a double-cross with
sufficient hybrid seed for commercial production.

In recent years there has been renewed interest in the
commercial production of single-cross hybrids. This is due in
part to the availability of more productive inbred lines, better
field husbandry in seed production, and a better understanding of
the gene action involved in heterosis.

As the productivity of single-cross and double-cross hybrids
reached production limits, several other selection methods, top-
cross, recurrent selection, and reciprocal recurrent selection (RRS),
were developed which replaced the pure-line method. Recently,
Hallauer (1967a) has suggested a selection scheme which produces
Single-cross rather than direct production of inbred lines or
improved source material (i.e. reciprocal recurrent selection).

The corn breeders, as well as other breeders, are interested in a
selection method which produces the most productive material P0531131e

with the least amount of time, effort, and expense. This study was

undertaken to evaluate the efficiency of the Hallauer method, COP-

cross method, and reciprocal recurrent selection (RRS) methods of

l

breeding.

In comparing the efficiency of different selection proce-
dures,three different approaches have been used.

The first was to apply the various schemes for a period
of time on the same biological materials such as plants and animals
under similar conditions. Efficiency, then, was measured by the
productivity of the end materials relative to the base material
for a selection method. This system of comparing different methods
of selection was not precise enough to make totally reliable con-
clusions since the amount of biological materials are restricted
in the first place. In the second place, both biological material
and environmental factors are subjected to much variability and
take a long period of time to make valid conclusions about the
efficiency of the various selection methods.

The second technique was concerned with the mathematical
and statistical theory of selection procedures. In this case, if
the selection methods were not mathematically complicated, the
genetic gain or advance for only one cycle of selection was pre-
dicted for each selection method. The relative size of the genetic
advances predicts which selection method is the best. In most of
the mathematical developments some simplifying assumptions were
made for sake of simplicity.

Some of these assumptions are:

(l) The populations were assumed to be infinite, so as to

have a Mendelian population with its properties

(Dobzhansky 1955).

(2) The genotypic values were assumed to follow a normal
distribution.

(3) The gene effects were assumed to be very small relative
to the genotypic variance.

(4) Selection was slow in the sense that the genotypic
variance and its components did not change under
selection.

(5) Inter-allelic interactions were assumed to be of low
order.

This way of comparison is, therefore, inadequate for pre-
dicting changes in population means under different selection
schemes after an arbitrary number of generations or cycles of
selection. '

A rigorous mathematical and stochastical treatment is
required to handle the joint effect of inbreeding depression,
Epistasis, change in gene frequencies, and linkage on the mean of
a finite size pOpulation under selection pressure. The construction
and treatment of a model which contains all of these parameters is
complicated and extremely tedious from a mathematical and stochas—
tical point of view.

The third way of comparing selection schemes which has
recently been used is the Monte Carlo technique of population
simulation on high speed computers. Monte Carlo techniques not
only handle the mathematically complicated selection models but
also eliminate some of the simplifying aSSumptions restricting the

. . . me
biometrical theories. However, Simulation techniques have so

obvious disadvantages as well as merits.

No mathematical theory has been developed for the Hallauer
procedure of selecting superior single-cross hybrids due to the
mathematically complicated nature of simultaneous selfing, in-
breeding and selection.

In the present study the Monte Carlo techniques have been
used to empirically compare the method of Hallauer with the tOp—
cross and the reciprocal recurrent selection (RRS) methods.

Since the classical works of Fisher, Wright, and Haldane, who
founded the probabilistic basis of natural and artificial selec-
tion, thousands of experimental as well as natural selection
experiments have been conducted on both plants and animals. The
subsequent documentation of artificial selection, which may re-
late to this study, is so extensive that only selected papers

were discussed. Primary emphasis was placed on corn ELSE mays L.)
literature because of the origin of the selection methods

simulated.

II. REVIEW OF LITERATURE

In the present study, only certain areas of theoretical

and experimental selections on corn (Zea mays L.) will be reviewed.

Improvement

date of its

in corn has undoubtedly taken place since the earliest

cultivation, both through natural selection and

objective selection by man.

A - Pure-line Method

The

suggestion for a hybridization method in corn breeding

was made by Morrow and Gardner (1893, 1894). They presented

sufficient results to warrant the suggestion.

Shull (1908) reached the following conclusions:

a - In an ordinary field of corn the individuals are gen-

erally very complex hybrids produced by the combina-
tion of numerous elementary species (biotypes or pure-
lines);

The deleterious effects of self-fertilization is due
to the gradual build-up of homozygosity in the genetic

make-up of the individual; and

c - The goal of the corn breeders should not be to find the

best pure-line but to find and maintain the best hybrid
combination to take advantage of the hybrid vigor

(heterosis) phenomena manifested in F progenies.

l

In the following year, 1909, three articles were published
by three different authors about corn hybridization, different from
that suggested by Morrow and Gardner. These three authors were
Shull, East, and Collins. Collins' method of corn hybridization
was in some respects similar to that prOposed by Morrow and Gardner
in which it was required to search and look for a new variety or
strain each year, instead of going back to the same relatively
inbred strains for each successive crop as suggested by Shull and
East. Shull's method was more attractive to the corn breeder than
East's. Shull (1909) in his classical article outlined the pure-
line method of corn breeding, whereas East (1909) pointed out the
possibility of a line-breeding method while trying to explain
that decrease in vigor but not degeneration of character is usually
the sole effect of inbreeding.

A new era of corn breeding began when Shull described the
process of pure-line corn breeding. Shull (1909) recognized that
inbred lines have an advantage over open-pollinated varieties in
that they were homozygous and could be counted on not only to re-
produce themselves with great precision, but also to produce hybrids
of exactly the same genotypes year after year.

Shull's proposal was to use single-crosses for the cOmmer-
cial planting. These single-crosses to be made between pairs of
inbred lines selected for their superior performance in combination
With each other.

The double-cross hybrids, prOposed by Jones (1918), made

hYbrid maize economically feasible.

However, a practical problem with the pure-line method of
corn breeding was recognized when the number of productive inbred

lines increased.

n!

Considering that (3) = 2!(n-2)!

different single—cross
hybrids can be made from n inbred lines (ignoring reciprocal
crosses), it is apparent why this system of producing and testing
inbred lines broke down when a substantial number of lines became
available for testing. For instance, with only fifty inbred lines
to be tested, this direct method of testing requires measurement

of the yielding ability of 1225 F hybrids or single-crosses, pre-

1
ferably repeated in more than one season and location. To over-

come this handicap, top-cross or inbred by variety method of test-

ing the inbred lines was adopted.
B - Top-cross Method

Davis (1927) first suggested the use of inbred x variety
top-crosses to measure the general combining ability of the inbreds
under test. General combining ability (GCA) may be defined as the
comparative ability of a group of inbreds to combine with a tester
or group of testers (Sprague and Tatum 1942). Only the inbreds
with superior top-cross progeny performance are retained for further
crossing and testing. These inbreds were expected to have high
GCA. The inbreds with high GCA are combined in all possible single-
cross combinations. The following season the hybrids are grown
to measure the specific combining ability of the inbreds. Specific
combining ability (SCA) may be defined as the deviation in perfor-
mAnce of a Specific single-cross from the performance expected on

the basis of GCA (Sprague and Tatum 1942).

The most comprehensive data relating to the value of the
top-cross method were reported by Jenkins and Brunson (1932). Their
procedure was to compare the ranking of inbreds as determined by
performance in inbred-variety crosses with average performance of
the same inbreds in a number of single-crosses. Inbred lines that
produced a low yield in top-crosses were found to produce low-
yielding single-crosses. They concluded that on the basis of in-
bred x variety top-crosses, it should be possible to discard 50%
of the inbred lines without danger of losing any really superior
material. The remaining 50% may be given a more careful test in
combination with other inbred lines for SCA.

Sprague (1939) demonstrated that if the variety is used as
the seed parent it is recommended that no less than ten plants be

used to sample the gametes of the variety.
Testers for GCA and SCA

As the top-cross parent, Jenkins and Brunson (1932) suggested
the use of either the parent variety from which the inbred lines were
derived, or if the new inbred lines are intended for crossing with
inbred lines of another variety, they could be crossed with the
variety from which those lines originated.

Hull (1947a) made a statement that theoretically the most
efficient tester would be a homozygous recessive at all loci and
that homozygosity for the dominant alleles at any locus should be
avoided.

Green (1948) tested Hull's hypothesis with respect to
lodging resistance. He used a relatively high-yielding, lodging-

resistance, double-cross hybrid, and a relatively low~yielding,

lodging susceptible, Open-pollinated variety as top-cross tester
parents in crosses with 83 plants of each of three single-cross

F progenies. The data obtained indicated conclusively that the

2
susceptible tester provides greater opportunity for selection
among the segregates with which it was crossed.

Keller (1949) carried out an experiment in which a re-
lated and an unrelated single-cross were used as the tester parents

in evaluating a group of selected F plants of maize.

2

Different estimates of variability were obtained for the
agronomic characters studied. In none of the comparisons was the
difference large. Keller rejected Hull's hypothesis on the basis
of the data obtained. He reasoned that if the hypothesis is correct,
the component of variance due to the interaction of lines with
testers would be less for high combining testers than for low com-
bining testers.

Hull's hypothesis was supported theoretically by

considering the constant parent regression method of analyzing
the single-crosses developed initially by Hull (1947b) and amplified
by Griffing (1950). The regression of performance of offspring on
the performance of the variable parent was shown to be largest when
the gene frequency of the character for the constant parent was
zero. The regression coefficient was zero when the gene frequency
was One for complete dominance or at equilibrium gene frequency
for overdominance.

Matzinger (1953) conducted a study in which 16 randomly
chosen inbred lines were involved. The variance component estimates

of the interaction of inbred testers x lines, single-cross testers

10

x lines, and double-cross testers x lines for yield in bushels per
acre were 17.22, 11.90, and 6.46, respectively. The relative
magnitude of the variance component estimates of the tester x line
interaction indicated that as the genetic variation within a tester
parent increased the tester x line interaction component was de-
creased. He emphasized that when the object of an experiment is

to determine a replacement for an existing line in a certain com-
bination, SCA is of prime importance and the most apprOpriate tester
is the Opposite single-cross parent of the double cross or its
component inbred lines. When a group of new lines were to be

tested without any predetermined plan, then the ranking of lines
with reSpect to GCA could be accomplished most economically by
employing a heterozygous and heterogeneous tester.

Sprague (1955) reached the same conclusion based on a
series of experiments conducted at Iowa.

Grogan and Zuber (1957) performed a study to compare single-
crosses with double-crosses when used as top-cross parents for
measuring new lines for GCA and SCA.

They concluded:

a - A tester closely related to the lines being tested
should not be used as a top-cross parent when desiring
information on GCA.

b - Information on GCA can be obtained more economically
and with as much validity by using double-crosses rather
than the average information from single-crosses.

c - A top-cross tester with a low value for GCA is more

suited for measuring the average performance of a group

11

of lines than one having a high value for GCA.

d - A single-or a double-cross, related or unrelated to
the lines, could be acceptable as the top-cross parent
when the agronomic characters under consideration are
controlled only by a few genes. Marked differences
may be found between the testers when the traits are
controlled by many genes.

Sprague (1959) stated that a suitable tester for GCA could
be obtained by synthesizing a number of inbreds most widely used
within a maturity zone.

Thompson and Rawlings (1960) carried out an experiment to
evaluate four single-cross testers of different ear heights when
used as top-cross parents for measuring yield and ear height of
corn. A slight advantage was indicated for the two lowest yield-
ing testers for yield evaluation which was in agreement with Hull's
hypothesis.

Rawlings and Thompson (1962) further considered the role of
average gene frequency of the tester in selection for GCA in maize.
They defined a "good" tester to be one which classified correctly
in a relative sense the entries under selection,and discriminated
efficiently among the material in the test. Considering the
theoretical aspects of the experiment, they showed the following
expression as the genetic variance among test cross progenies,

half-sibs, for the ith locus.

2 2
a 91(1-pi)(1+F)[1 + (1-2qi)ai] di [1]

where

12

p = average gene frequency at the ith locus of the material
under test (test population),
q = average gene frequency at the ith locus of the material

used as the tester (tester population),

u. = half the distance between two homozygetes at the ith
i
locus,
aiui = difference between the heterozygote and the average of

the two homozygotes when ai is a measure of the
degree of dominance at the ith locus, and

F = coefficient of inbreeding of the material under test
(test population).

The following assumptions were made for the derivation of
[1] with reSpect to the ith locus:

1 - The test population is initially at Hardy-Weinberg

equilibrium,

2 - Each individual from the test population is pollinated
by a random sample of pollen from the tester population,
and

3 - no epistasis.

The total genotypic variance was obtained by summing over
all loci. All genetic variance of half-sib progeny means were
additive. This was shown by regressing the performance of half-
sib progeny means on the number of + genes of the individuals
selected from the random mating test population (Comstock £3 a;
1949, and Cress 1965).

The effect of gene frequency of the tester population, q,

is apparent. In the absence of dominance for all loci, a = 0, the

13

total genotypic variance of half-sib test cross progeny is in-
dependent of tester gene frequency. The amount of genetic variance

is proportional to
2
[1 + (l-Zqi)ai] . [2]

The quantities in [2] are always positive and pi, ai, ui,
and F are constant for a particular set of material being tested.
The homozygous recessive tester, q1 = 0, has an increasing advantage
as ai increases.

Rawling and Thompson also discussed the situation where
overdominance was large relative to partial and complete dominance
and gene frequency in the tester population was relatively high.
With several overdominant loci, a tester with high gene frequencies
could lead to more genetic variation among test cross progenies
than a tester with somewhat lower gene frequencies. From the trends
in the data, they concluded that there were differences among the
testers studied which favor poor performing testers for GCA. The
striking difference expected from the theory developed did not
appear in the data. One possible explanation was that the assump-
tion of no epistasis was inadequate.

Hays (1963) concluded, from his own experiments and other
studies, that in tests for GCA for the first isolation of inbred
lines, the tester should be genetically diverse from the lines to
be tested, adapted to the region where the inbreds are to be used
in crosses, and should consist of material that has not previously

been selected for high combining ability.

14

Early generation testing for GCA was suggested by Sprague
(1946) in which So individuals or individuals selfed for a few
generations were crossed with the top-cross parent. On the basis
of a tOp-cross progeny test a large number of individuals or
segregates were discarded from further testing. Hays (1963)
mentioned that lines proved to be high or low combiners in a top-
cross test made in S0 or S1 material, were not necessarily homo-
zygous for this condition, but might in many cases, segregate for
combining ability during the process of self-fertilization.” There-
fore, on the average, visual selection during inbreeding might be
expected in the majority of cases to lead to an improvement in

GCA of the reSultant inbred lines.
C - Reciprocal Recurrent Selection (RRS) Method

Comstock _£__l (1949) suggested the use of foundation
material from two sources that are genetically diverse and which
combined well together to give a desirable hybrid. The sources
for ease of presentation might be referred to as A and B, and the
material from these sources should each be heterozygous. Indi-
vidual plants of source A were selfed and at the same time polli-
nated by a random sample of pollen from source B. The same pro-
cedure was repeated for individual plants of source B. Thus B
used as a tester to select plants from source A that combine more
satisfactorily with source B and vice versa. The selected S1 lines
of A were intercrossed randomly to produce a synthetic. The same
procedure produced a synthetic for B. At this stage a cycle of

reciprocal recurrent selection (RRS) was completed. After comple-

tion of as many cycle as desired, or at the end of each cycle,

15

selfing and selection could be practiced, with the intention
eventually of producing single-cross hybrids of the type (A X B),
where the inbreds A were obtained from the A source and the B
inbreds from the B source. Cyclic selection could be followed as
long as there was genetic variability.

Numerous experiments were conducted by different people
using Hull's (1945) and Comstock g; fills methods not only on maize
but on many other crOp and animal species.

The results of many of these studies were discussed by
Cress (1965). Dickerson (1952) compared the two methods of re-
current selection theoretically. Schnell (1961) discussed some
aspects of (RRS). Griffing (1962) developed, theoretically, all
possible combination of recurrent selection in which one or two
random mating populations might be employed as source materials.
Recently, Cress (1966, 1967) reviewed and discussed various aspects
of recurrent selection methods, their uses and advantages in breed-
ing programs.

The selection methods described previously were developed
many years ago and are successfully used by breeders. A method
was recently proposed by Hallauer (1967a) and is in the primary
stages of testing. This method of single-cross hybrid selection

is the primary objective of the present study.
D - Hallauer Method

Hallauer (1967a) proposed a method for corn breeding to
use all genetic variance for improving a polygenic character. He
was impressed by an increasing body of experimental evidence in-

dicating the involvement of non-additive gene effects in the

l6

expression of polygenic characters and the mathematical article
by Cockerham (1961) in which the relative genetic gains due to
selection were computed for different types of hybrids. As a
background for the Hallauer method, some aspects of Cockerham's
paper and some experimental evidence concerning non—additive gene
effects in maize will be discussed.

Cockerham (1961) considered components of genetic variance
among unrelated single-, three-way—, and double-crosses, and re-
lative gains that can be expected from selecting among these
hybrids. Using the concept of identity of genes by descent de-
veloped by Malécot (1948) and defining F, the inbreeding co-
efficient of a line, as the probability of two random alleles of
the line being identica].turdescent, Cockerham demonstrated,
mathematically, that variation among the three types of hybrids
would always be in the order of single-crosses greater than three-
way and three-way greater than double crosses. The relative
advantages would be a minimum of 1 to 3/4 to 1/2 when all of the
genetic variance is additive and when the parents of the hybrids
were completely inbred, F = 1.0. The relative advantages of
selecting among the hybrids increased in favor of the single-
crosses if dominant and epistatic gene effects (nonadditive gene
effects) were important. If only additive effects were important,
selection among single-crosses would be twice as effective as
selection among double-crosses. If much of the genetic variance
was nonadditive,selection among single-crosses would be four times

as effective as among double-crosses.

17

Cockerham (1961) assumed that the lines involved in the
production of the three types of hybrids had an equal but arbitrary
degree of inbreeding and were a random sample derived from a random
mating pOpulation. The last assumption was necessary as a base
reference for making comparisons among the three types of hybrids.
He also assumed regular diploid individuals or lines having simple

Mendilian inheritance with no linkage between the loci.

Non-additive gene effects in corn population

 

Additive genetic variance has been shown to exist, at least
in moderate amounts, in most corn populations [see Gardner (1963)
for review]. Dominance variance and degree of dominance have varied
in relative magnitude for various types of corn pOpulations, and
the relative importance of the degree of dominance in heterosis is
an unsettled issue.

Although only a limited amount of data has been obtained,
epistatic variance appears to contribute little to the total genetic
variance of corn pOpulations (Eberhart g£_§l 1966, and Stuber £3 31
1966). However, evidence is accumulating for the presence of
epistasis in Specific combinations of inbred lines of maize.

Bauman (1959) used 2 corn inbreds and the single-cross
between these 2 inbreds onto an inbred tester to detect the possibility
of epistatic gene effects in determining yield, ear height, and
kernel row number in corn. If performance of the single-cross x
tester (3-way—cross) deviated significantly from the average per-
formance of the two inbred x tester single-crosses, then epistatic

gene effects were indicated. Epistatic gene effects were found to

18

be involved in the expression of the agronomic characters considered.
However, significant epistasis x year interactions were found in

some cases. The method employed proved to be relatively ineffective
in detecting epistasis.

Gorsline (1961) extended Bauman's method of detecting
epistatic gene effects for yield, grain moisture, silking, stalk
quality, plant height, ear node height, percent ear node length,
ear length, ear diameter, and ear length/diameter ratio. Epistasis
was established for all ten characters. Yield and ear length
exhibited fewer instances of epistasis than the other eight char-
acters. Epistatic gene effects appeared to be of general importance
in maize performance for the ten characters studied, and epistasis
x environment interaction was expected to be common and important.
Sprague $3.31 (1962) also used Bauman's concept to provide estimates
of the influence of epistatic gene effects on corn yield. Estimates
were obtained from comparisons involving balanced sets of single-
and three-way-crosses and between observed and predicted three-way-
cross hybrids. Significant differences in yield were observed
indicating that epistasis might be a factor of some importance in
the populations from which the inbred lines were selected for the
study.

Gamble (1962a, 1962b) outlined a procedure to separate
six genetic parameters, namely, mean, additive, and dominance gene
effects, and three types of digenic epistatic effects (additive
x additive, additive x dominance, and dominance x dominance) which
might affect genetic variation of a quantitative trait. Estimates

of the parameters were obtained using the population means of two

19

inbred lines, their crosses, and progeny from subsequent selfing
and crossing. The estimates of gene effects indicated that
dominant gene effects were quite important in the inheritance of
yield. Estimates of additive gene effects were of low magnitude
and in many cases were non-significant. Epistatic gene effects
were considered to be more important than additive gene effects
in the inheritance of yield in the crosses studied. The additive
x additive and additive x dominance gene effects were relatively
more important than the dominance x dominance effects. Hallauer
and Russell (1962) estimated the additive, dominance, and epistatic
gene effects from six population means or six generations. They
noted the relatively greater importance of some of the espitatic
components for the agronomic characters studied.

Eberhart g; _l (1964) developed a method to predict the
performance of double-cross hybrids of maize when epistasis was
present. Epistatic gene effects were detected for some of the
double-cross hybrids derived from the six inbred lines examined.

Eberhart 35.31 (1966), from a comprehensive experiment,
obtained full-sib and half-sib covariances in two open-pollinated
varieties of maize. These full- and half-sib covariances pro-
vided estimates of additive, dominant, and certain estimable
functions of epistatic variances which were useful for investigat-
ing the relative importance of the different types of genetic
variation in the two varieties. Seven characters were studied
in both varieties. The possibility of epistatic variance was

rePorted for yield of one of the varieties.

20

Stuber fig El (1966) investigated the genetic variability
and interrelationship of six economic characters in the crosses
of two maize populations. An evaluation of the epistatic com-
ponents of variation received primary consideration. However,
significant epistatic variability was not detected.

Because of Cockerham's conclusions that theoretically
greater gains could be made by selecting among single-cross hybrids
rather than double-cross hybrids or three-way-cross hybrids, and
the evidence of non-additive gene action in maize coupled with the
renewed interest in the commercial production of single-crosses,
Hallauer (1967a) described and outlined a breeding method that
isolates and tests single-cross hybrids during the inbreeding
process. The essential features of the method follows (quoted

from Hallauer):

Phase 1. Crosses are made between individual S
plants. The plants used in the crosses are also
self-pollinated to maintain the plant's genotype.
The hybrids produced by crossing the So plants are
evaluated in yield trials, and the selfed seed (or
S ) of each S plant is stored for future use. From
t e results of the yield tests, the top 30 to 50%
high yielding crosses are selected. Since the
crosses cannot be tested extensively because of in-
sufficient seed, a mild selection intensity is
suggested.

Phase 2. The pairs of 8 lines which represent
the selected S plant crosses are planted ear-to-
row. The same procedures outlined for making the
crosses and selfs between S plants are used be-
tween plants of the pairs 0 S1 progenies. Since
segregation occurs upon selfing an S plant, it
is suggested that four to six crosses be produced
within each pair of S progenies. This affords
selection for yield With S progenies, some of
which may be due to favoraSle epistatic combina-
tions. A relatively low selection intenSIty (30
to 50%) is recommended. .

Phase 3. The S seed of the selected entries
is planted ear-to-row in pairs that correSpond

21

to the original S plant crosses. Crossing and
selfing between S0 plants is continued. Yield

evaluations of the crosses are made and selec-

tions are made for continued crossing and self-
ing between plants of the S progenies.

Phase n. The procedure is repeated until the
selfed progenies of the plants used in the selected
crosses are homozygous and homogeneous. At this
time, one will have a group of selected single
crosses that have been tested for yield in each
generation of inbreeding. If favorable epistatic
combinations for yield were present, there would
have been an optimum opportunity for their
selection.

The Hallauer method was developed primarily to isolate
effectively single-cross hybrids for their SCA. This way, one
could select for the specific combination or "nick” that has
the highest performance, regardless of the relative importance
of the kind of gene effect involved. This procedure may be
applicable to any multiflowered crop species. Its usefulness
depends on the ease in which self— and cross-pollinated seed can
be obtained from one plant and whether sufficient heterosis is
obtained to warrant a hybridization program.

The only empirical result applying this procedure on some
prolific corn populations has been obtained by Hallauer (1967a,
1967b) and the summary of the data is presented in Table 1. Note
the sharply improved yield of the crosses relative to the mean
of the checks.

During the selection process the two-eared prOperty was
maintained in order to use the method. However, one could use
one-eared unrelated pOpulations of maize and produce self- and
cross-pollinated seeds on the same ear as was done by Sprague

(1939) and Williams t al (1965).

22

Table 1. Relative comparisons for 4 generations of

selection for two prolific populations of

corn using the Hallauer method.

Crossegpabove

Generations No. of X
Crosses of checks
No. Z

ASoxBSO 144 2 1.4

ASleS1 160 37 23.1

ASZXBSZ 173 131 75.7
+

AS3XB83 77 59 76.6
+

A84x384 67 59 88.0

*

Crosses 2_s

above X
of checks
No. Z
0 0.0
7 4.4
47 27.2
20 26.0
47 70.1

Original
Parents
remaining
No. Z
144 100.0
54 37.5
31 21.5
12 8.3
7 4.9

Check hybrids included 3 single-crosses (Bl4xB45, B37x0h43,

and B45xC131A) and 3 double-crosses (A.E.S.704, Ia.515, and

Ia.5116).

Personal communication.

23

The advantage of the Hallauer procedure is contingent on
being able to select for non-additive gene effects. How much non-
additive variance was present in the two pOpulations used in the
Hallauer experiment is not known at the present time. However,

a series of experiments were planned for obtaining estimates of
the relative proportions of the total genetic variance that was
additive and non-additive.

Further evidence has been obtained which indicates the
presence of non-additive gene action in corn. Eberhart and
Hallauer (1968) detected epistatic effects of genes in the maize
populations studied. Stuber and M011 (1969) used interpopulation
single-crosses (Fl's) and the selfed progenies (81's) arising
from unselected lines of Jarvis Golden Prolific and Indian Chief
varieties in an experiment to further determine the relative
importance of epistatic gene effects. Significant epistatic
effects were detected in some Specific sets of crosses. However,
the amount of the total variability that could be attributed to
epistasis was, on the average, less than 10Z. They concluded
that epistasis might be important in unique genetic combinations
but these combinations occurred either too infrequently or with

such limited effect that they were not detectable in random mating

equilibrium populations of maize.

 

III. MEAN OF A SELFED POPULATION

The genotypic mean of a selfed population will be derived,
allowing for epistasis between pairs of loci. Then, the Specific
genotypic values associated with the different genetic models used
in this study will be fitted into this genotypic mean to obtain a
set of prediction equations for calculating the expected inbred
and hybrid means in the absence of selection. This will allow
the judgment of a change in the mean due to selection in some
cases to evaluate the reciprocal selection effects of the Hallauer
method.

Consider a population with an arbitrary number of pairs of
independent loci with two alleles per locus and having an arbitrary
degree of inbreeding, F. Selection pressure is not Operating on
this population and there is no linkage between the loci. The geno-

typic mean of such a population can be derived with respect to one

pair of segregating loci as follows;

 

 

 

 

 

Second locus ‘ BB Bb bb
. ‘ 2 ._‘m
First Locus Frequency qi+q1q2F 2q1q2(1-F) q2+q1q2F Partial
Mean
2
AA 914131sz Yzz Y21 Y20 Y2.
A“ 2p1P2 (1'1”) Y12 Y11 YlO Y1
2
8" Pz+pisz Y02 Y01 Yoo Yo.

 

 

 

 

 

 

 

By definition

24

25

p1 4‘ p2 = 1.0,
q1 + (I, = 1-0.
[P2+pPF+2pp (1-F) +p2+ppFJ=1.0,
1 1 2 1 2 2 1 2
2 A 2
[(11 + (11qu + 2q1q2(1-F) + q2 + qquF] = 1.0, and

0.0 s F s 1.0.

When F = 0, the population is in the state of random mating and
the genotypic mean of the population is denoted by OR. When
F = 1, the population is composed of many inbred lines and is
referred to as an inbred population with genotypic mean uI.
When 0 < F < l, the pOpulation is partially inbred and its geno-
typic mean is ShOWn by uF.

If there are k independent sets of segregating loci in
the population, the genotypic mean is obtained by summing over the
k sets. The frequency of + alleles at the first and second loci
are shown by p1 and ql’ respectively. The frequency of - alleles
at the first and second locus are denoted by p2 and q2, respectively.
A genotypic value is shown by Y subscripted by two numbers. The
first number shows the number of + alleles at the first locus and
the second number indicates the number of + alleles at the second
locus for that genotype.

Now, consider the partial means

._ 2 2
Y2. (ql'qquF)Y22 +.(2q1q2-2q1q2F)Y21 + (qz'qlqu)Y20’
_ 2 2
= - +
Y1. (q1+q1q2F)Y12 + (2q1q2 ququ)Y11 (“2+q1q2F)Y10’ and
_. 2 2
= + 2 -2 +
Yo. (q1+q1q2F)Y02 ( q1q2 q1q2F)Y01 (qzlqlqu)Yoo’ “here

., dot, in the subscripts of T‘s represent the summation over

26

the correSponding locus (second locus).
The genotypic mean of this selfed population is obtained by
summing the partial means each weighted with the apprOpriate co-

efficient as follows

2 - -' 2
HF = (91+91p2F)Y2. + (2p1p2-2p1p2F)Y1 + (p2+p1p2F)YO .
which in terms of genotypic values extends to

’ 2 2 2 2 2
= +
“F quIYzz + plqquFYZZ + p1p2q1FY22 + p1pzq1q2F Y22

2 2 2
- - 2 F +
2P1q1q2Y21 2P1q1q2FY21 + 2P1p2q1q2FY21 p1p2q1q2 Y21
2 2 2 2 2
+ F +
p1q1Y20 + p1q1q2FY20 + p1p2q2FY20 p192q1q2 Yzo
2 F - 2 2FY - 2 FZY -+
2p1p2q1Y12 + 2p1P2q1q2 Y12 pIpzq1 12 p1p2q1q2 12
+4 FZY +
4p1pzq1q2Y11 ' 4p1p2q1q2FY11 ' 4p1p2q1q2FY11 p1pzq1°2 11
2 2Y + 2 p q FY - 2p p qZFY - 2p p q q FZY +
p1p2q2 10 p1 2q1 2 10 1 2 2 10 1 2 1 2 10
2 2 2 2 2
+ F Y +
p2q1Y02 + p2q1q2FYo2 + p1p2q1FY02 p1P2q1q2 02
2 2 Y - 2 2 FY + 2p p q q FY - 2p p q q FZY +
p2q1q2 o1 p2q1q2 01 1.2 1 2 01 1.2 1 2 01
2 2 2 2 2
‘1' FY
p2q2Yoo + p2q1q2FYoo + p1P2q2FYoo p1p2q1q2 oo
- 1+ [ 2 (y -2Y +y )+2p p q q (Y -ZY +Y )+p2q q (Y 2-
' “R p1q1q2 22 21 20 1 2 1 2 12 11 10 2 1 2 o
2
2 -2 +Y p (Y -
2YOIJ'YOOHPIPN‘1(Y22'2Y12IY02H2P1P2‘11‘12"21 Y11 01)+P1 2q2 20
- - +4Y -2Y +Y
2Y1ol¥oonF+£p1p2q1q2<Y22 2Y21+Y20 2Y12 11 1o 02

2
- . 3
2Y01+YOOJF [ 1

Kempthorne(1957) derived the genotypic mean of an inbred

Population for arbitrary epistasis, multiple alleles and no selection.

He found

27

2
= +
“F “R F D1 +'F D2, [4]

where, for two locus epistasis and two alleles per locus

2 2 1 1
D1 = .11 )3 pm.dm.m., and
1=1 mi=1 i 1 i
2 2 2 1 1' 1 1'
D = H 2 z p p (d d ) .
2 i=1 mi=1 mi.=l m1 mi' mimimi'mi'

See Kempthorne(1957) pages 442-444 and Kempthorma(l955) for notation.

Note that the D's are defined in terms of the dominance and
dominance by dominance effects of genes in the original random
mating population. In the absence of dominance by dominance and
dominance gene effects, the genotypic mean of a population with
arbitrary epistasis does not change under inbreeding. A linear
relationship exists between ”F and F if there is no dominance
by dominance effect of genes even if other types of epistatic
gene effects exist. In order to demonstrate the equality between
13] and [4], I will Show that D1 and D2 are equal to the co-
efficients of F and F2 in [3], reSpectively. The genetic para-
meters required to Show that D1 is equal to the coefficient of
F in [3] are four dominant deviations defined in a random mating
pOpulation in terms of gene frequencies and genotypic values, Yij
(1 = j = 0,1,2). The four dominant effects can be shown as follows
dll = qu22+2q1q2Y21+q3Y20+PIPZQEY12+2P192919ZY11+p1p2q2Y10+

2

pzq1Y02+2p§qlquO1+P§quOO-Piqu22’2Piqiq2Y21’plq2Y20'
2 - 2 -2 2r -4 peq'
p1P2q1Y12'2p192q1q2Y11 plpquYlO p1p2q1 22 p1 2 1 2 21

2 2 2 -2 2 ZY
2pIpzqszo'Zqu1Y12'492q1q2Y11 pzqz 10’

28

fiZY

1 2
= +
d q Y 2q1‘1'2YoiJ'q 2 oo+p21q1Y 22

+2 2
22 1 02 P

2 2
1q1q2Y21+p1q2Y20+p1p2q1Y12+
2131“:ququ 1)qu -ZquqY 12qu 22Y -

1 2 1211 1 2 10 p1 2112 1 2 1 2 11 PI 2 2 1o p2q1Y 02

2 2 2Y
2
p2q1q2Y01 pzqu 00 2p1q2Y12 4p1q1q2Y11 2p1q2Y10 2p 1p2qfY02

2
Aplpzq1q2Y01'291pzq2Yoo’

2 2Y 2Y 2 2
d11 p1 22+2p1p2Y12+p2Y02+PZ1q2Y 20+1’1p2q 2Y10+p1p2q2Y 1o+p2q2Yoo'
2 2 2 2 2 2 2
p1‘11Y22'p1p2q1Y12’p192q1Y12’p2q1Y 02 2p1q1q2Y22 4P1p2q1q2Y12-

2 2 2 2 2
2p2q192Y02'2P1q2Y21'4p1p2q2Y11'292‘12Y01’ and

+ + -
22 p1 20+291p2Y10+p +p2Y00+p1q1Y 22 2p1p2q2Y122 p2q1Y 02
2 2Y

22Y 2 2Y 4 2Y -22 Y 22Y -YZq
p1‘11 20' p1P2q2 10' p1P2q1 11 p2q1q1 oo p2‘12Y 00 p1 1 21

2 2

2
2P2“1Y01-2p1qlqu20'4p

1P2q1q2Y1o°
The superscripts of the d'S should not be confused with

powers. These superscripts refer to the first and second loci.

By definition

D =de1 + 1 d + p2 d m m m m .
1pmidmimi mi. mi'mi' mi mi 1 i. i. 1'

Substituting p1 for p , p2 for p : ql for Pm..
i

and q for p2 and expressing the d's in terms of p's, q's

m.
l.

and Y's, the equation for D1 becomes

_ 2 H2 2 Y + 2p q +
D1 - p1q1Y22+291q1q2Y21+p1q2Y 20+ph qu 12 +2p1p2q1q2 11 p1 2 2 10
F3 2

2 2 2 _ 2Y _2 3 qY %Y -

p1pzq1Y Y02+291p2q1q2Y01+P192q§Yoo3 p1q1Y 22 p1q1 2 21 W1 20

2 2 2 _ 2 -2 2 2Y -4pzp q q Y '
p192q1Y12'29192Z1q2Y11 PIquiYio p1p2q122 1 2 1 2 21

2

ZY -2 p 2(1 2Y +p q2Y+

29

2

2
2
P2q1q2 Yo1+P2q2Yoo+p1P2q2Y22+YP1p2q1q2Y21+P1P2q2Y20+

2 2 2 2 2
P1P2q1Y12+YP1P2q1q2Y11+P1P2q2Y10'P1P2qEY12':P1P2q1q2Y11‘

P1P2q2Y103 T2q1Y 02 2P2q1q2Yo1 quzYW q1Y 12 TPYP P2q1q2Y11

ZPiqugY Yio 2P1P2q2Y02 TP1P2q1q2Y Yo1’2P1:§::Yoo*P1q1Y Y22+2P1P2q1Y12 +
P2q1Y02'Piq1Y22’P1P2qYY12'P1P2qYY12 quiY 02 YPiqiquzz 4P1P2qiq 2Y12’
YPiqiquoz YPiq1q2Y21 TP1P2q q1‘12Y11 2P2q1q2Y01+P1q2Y20+2P1P2q2Y10+

2 2 pq2 3 3Y
P2q2Yoo+P1q1q2Y22+YP1P2q1q2Y12+P2q1q2Yo2P1q2Y20 P1P2q 2Y10

3 2 2 3 2 2
P1P2q2Y10'2P2q1q§Yoo'P2q2Y00'YP1q1q2Y21'4P1P2q1q2Y11'

2 2 2 2
- -4 .
2p2q1q2Y01 2p1q1q2Y20 p1P2q1q2Y10

Collecting terms

2 (1+2.
fiqu 22+p1q1q2Y22+2p1p 2q1q2Y 21 2P1q1q2Y21+P1pq 2 2 20+
2
1=:1q1q2Y20'2P1P2qu12+2P1P2q1q2Y12’8P1P2q1q2Y11‘2P1P2q2Y10+
2p p q q Y +? p qZY +vzq q Y +2p p q q Y -2p2q q Y +
l 2 l 2 10 l 2 l 02 2 l 2 02 1 2 l 2 01 2 l 2 01
2 Y

P1P2q2 Y00+P2q1q2 00

Thus, D1 is equal to the coefficient of F in [3]. To show that
2
D2 is equal to the coefficient of F in [3] the (dd) parameters
. . 2 .

are required. The equality of D2 and the coefficient of F 1n
[3] is not shown, however, the calculations are similar to those
for‘ D . Therefore, [3] is a Special case of [4]-

1

When the inbreeding coefficient, F, is unity [3] becomes

91 = “R + plqlq2 (Y22-2Y21H20)+p2q1q2 (Yoz' 2Y01 00)

- -22 ) +
P1P2q12<Y22 2Y12+Y 02) + P1 P2q2(Y20 1o"Y oo

30

+ -
Plpzq1q2(Y22 Yzo 4Y11 + Yoz +-YOO)

= +, +
qulez p1q2Y20 p2q1Y02 + p2q2Yoo [5]

Thus, the genotypic mean of an inbred population, F = l, is equal
to the summation of genotypic values of homozygous individuals,
weighted with apprOpriate gene frequencies in that population.
When the genetic model is known, the genotypic values can
be used in equation [5], to obtain an equation for “I only in
terms of gene frequencies.
The following equations were obtained for the genetic

models used in this study:

“1A = 2(91 + q1) + 1, [6]
”ICD = 2(P1 + q1) + 1, [7]
H'101) = 1 ’ [8]
”ION = 4(P1 +'ql - 291q1) + 1, [9]
HIAA = 4(291q1 - p1 - ql) + 5. and [10]
um, = 2<p1 + q1) + 1, {11]

where the second and third subscripts of uI's stand for the

genetic model used in deriving that equation.
The initial frequencies of + alleles were the same at all

loci in the populations simulated. Self-fertilization, Without

selection, does not change the initial gene frequencies. Self-

fertilization pushes the population toward a homozygous condition

With initial gene frequencies unchanged. Having considered these

31

facts, [6] to [11] become;

”IA = 4p1 + 1’ [12]
”Ian = 4p1 + 1’ [13]
”10D = 1’ [14]
u = 4392 + 8p + 1 [15]
ION 1 1 ’
2

= - + 5 d 16
”1AA 8131 891 , an [ ]

= - 17
“IAD 41)1 + 1 1 1

It is recalled that [12] to [17] were obtained in the absence of
selection pressure. Selection pressure, if effective, changes
the genotypic mean of a population through the changes in gene

frequencies.

Under condition of self-fertilization and selection pressure,

the resulting inbred population, F = 1, can be obtained whose

observed genotypic mean, 9;: can be substituted in the appropriate
equation, [12] to [17], to predict the gene frequency, pi, actually
obtained in that inbred population. Then, with the initial gene

frequencies in the random mating population and the gene frequenCies

in the inbred population, F = 1, one would be able to calculate

the efficiency of the selection pressure in terms of changes made

U
in the gene frequencies. With the predicted gene frequency P1

. I
and with the assumption that qi is equal p1 0n the average,

' ean "
one should be able to calculate a predicted genotypic m , ”I.

n d H,
The difference between pi, the observed genotypic mean, an “I

the Predicted genotypic mean, can be associated with the reciprocal

32

effects of selection. Equations [12] to [17] will be used to
separate the effects of the average gene frequencies changes due
to selection and the reciprocal effects due to selection on the

genotypic means of the hybrid populations as well as the inbred

populations.

IV. THE MECHANISM AND IOGIC OF POPULATION SIMULATION

The feasibility of setting automatic digital computers to
simulate a reproductive population with its properties was first
realized by Fraser (1957a). He saw the similarities of the
stochastic processes, i.e. Monte Carlo techniques, that can be
generated by the high speed computers and processes such as random
or nonrandom segregation of alleles, gamete formation, random
combination of gametes and selection which prevail in Mendilian
pOpulations.

The storage module of any digital computer consists of

many "words" each of which is made up of numbers of "bits", de-
pending upon the computer system. Each word can be viSulized as
a chromosome and its bits as the sites of the genes. Therefore,
a pair of words can represent a diploid individual having a pair
of homologous chromosome. The computer system used in the present
study had 60 bits in each word labeled from one to sixty; thus
each bit can be referred to if desired.

Three basic subprograms were used in the simulation of

pOpulations.

A - Subprogram l: Generating the Initial POpulation

This Subprogram generates a word consisting of 60 bits.

Each bit on the word is given a value, either one or zero, based

33

34

on the magnitude of a random number and a preassigned probability.
A bit is assigned one if the random number is greater than the
preassigned probability, the gene frequency; otherwise the bit
receives a zero.

The plus or dominant allele was represented by a one in
a bit and the minus or recessive allele by a zero.

An even number, 2N, of words, containing either one or
zero, was generated and paired to produce a diploid pOpulation
with N individuals. The diploid individuals were labeled pre-

cisely so that each could be referred to whenever desired.

B - Subprogram 2: Generating Gametes

A pair of words generated by the first subprogram (a parent)
was used in this subprogram. The member of the word pair chosen
to contribute the first bit to a third word (a gamete) was de-
termined at random. The value (0 or 1) in the first bit of the
first word chosen was placed in the first bit of the third word.
A random number was then generated and compared to a preassigned
probability, the recombination value. If the number generated
was larger than or equal to the recombination value, the same parent
word would contribute its second bit to the second bit of the gamete.
If the number generated was smaller than the recombination value,
the alternate parent word would contribute its second bit to the
gamete. This random walk procedure was continued until all bits
of the gamete was determined. The recombination value in all the

populations simulated was 0.50, i.e. free recombination (no linkage).

35

By varying the recombination value it is possible to study

the effect of linkage on selection progress.

C - Subprogram 3: Evaluating Individuals

An individual, represented here by two words, was used in
this subprogram. The adjacent pairs of bits on both words are con-
sidered simultaneously and based on the combination of ones and
zeros, a numerical value is given to each pair of bits. There are
nine possible genotypes possible for each pair of bits, if no dis-
tinction is made between the coupling and repulsion phases. Genetic
models were determined by the numerical value assigned to these
nine genotypes.

The values given to the 30 independent pairs of bits are
summed up to represent the genotypic value for that individual.

Pair-wise consideration of bits or loci, which was required
in the epistatic models, did not affect the genetic parameters in
the non-epistatic models, which behaved as if there were 60 independent
bits or loci.

A random error with a normal distribution was added to the
genotypic value of each individual to obtain the phenotypic value.

Many different problems of population genetics have recently
been attacked by means of Monte Carlo techniques. Further work by
Fraser (1957b, 1960a, 1960b, 1962), Barker (1958a, 1958b),

Crosby (1961), Gill (1965a, 1965b, 1965c),
Young (1966, 1967), Cress (1967), Pfaffenberger and Gates (1967),
and Qureshi (1968a, 1968b, 1968c) could be cited as some examples

to demonstrate the various population genetic problems that can be

36
approached by simulation.

D - Distribution of Random Error

The random errors followed a normal distribution with zero
mean and a Specific variance.

The error variance was fixed in a manner to obtain an
estimate of heritability in the broad sense, H3, in the neighbor-
hood of 25 percent. To have such an error variance, maximum geno-

typic variances, , which are a function of gene frequencies

OG,max
and the genotypic values, were calculated. Given HB = .25, the

2 .
error variance, 0E, was calculated from the following equation

for each of the genetic models simulated by Gill (1963).

2 2 2
= ;_ +
HB OG,max ° (0G,max CE)
The magnitude of error variance for a maximum H = .25

B

for additive (A), complete dominance (CD), pure overdominance
(0D), optimum number (ON), duplicate factors (DF), complementary
factors (CF), additive by additive (AA), additive by dominance
(AD), and dominance by dominance (DD) models were 90, 180, 180,
212, 348, 120, 89, 184, and 90, respectively. The median error
variance which was 180, was selected to represent a constant error
variance for the six genetic models used in this study, (A), (CD),
(OD), (0N), (AA), and (AD).

A computer tape was available which contained 10,000
standardized normal deviates. One of these normal deviates was
picked up randomly and multiplied by the square root of error
variance, 13.416, and then added to each genotypic value to

obtain the corresponding phenotypic value.

37

E - Number of Runs

Variations between independent runs of a specific simula-
tion experiment were due to chance fixation of loci and random
error. The genotypic means of two independent runs of a Specific
experiment was expected to be different due only to chance fixation
of loci or random drift. The phenotypic means for any cycle were
also expected to be different not only due to chance fixation of
loci but also due to error variation.

Since the magnitude of the variation contributed by chance
fixation of loci to the between run differences was not known, it
was decided to examine the empirical variance of the differences
between the phenotypic means of the hybrid populations in two
independent runs resulting from Hallauer selection. All genetic
models, each with three combinations of initial gene frequencies,
were run twice with separate random starting points.

The variances of individuals within cycles within runs
were pooled over cycles and the two runs. The harmonic mean of the
number of indidivduals per cycle was used in calculating an
approximate standard error of the difference between runs for any
cycle, Table 2 (Appendix A). A few differences between runs were
declared significant (a = 0.05) for one or more cycles. In no
case was the difference between runs large in magnitude (i.e.
no larger than 1:6.8 units). Thus it was decided to broaden the

sc0pe of the study at the expense of duplicate runs.

38

F - Genetic Models Simulated and Initial Gene Frequencies

The values chosen for the genetic models were such that the
maximum genotypic value was the same for each of the six models
(Table 3, Appendix A). The 7 combinations of initial gene fre-

quencies used in the base populations A and B are as follows:

Population A

Gene Frequencies 0.1 0.3 0.5 0.7

0.1 + + +
Population B 0.3 + + +
0.5 +

The seven gene frequency combinations and six models
were simulated for
l. the reference experiment,
2. the Hallauer method,
3. tOp-cross selection, and

4. reciprocal recurrent selection.

G - The Essential Features of Simulated Experiments
1 - geference experiment

This experiment was simulated to serve as a reference
method. Selection was not practiced throughout the experiment and
the population size was reduced arbitrarily. In each generation
of selfing only one selfed progeny was produced. Random production
of single-cross hybrids between populations A and B were practiced

for each generation.

39

a - Two base populations were generated with Specific
gene frequencies, each having 960 individuals.

b - Both populations were selfed for 4 generations.

c - Selfing was continued for 3 more generations
while reducing the population size by half in

each generation.

D.
I

Three more generations of selfing were followed.
2 - Top-cross experiment

A completely recessive inbred line was generated to serve

as the top-cross tester. Only 8 individuals were crossed with

1
the tester and on the basis of top-crossed progeny performance,
half of the populations were selected for a further test. The
number of top-crossed progenies and selfed progeny was five and
one, reSpectively. In each generation of selfing a number of
single-cross hybrids were produced randomly between the two popula-
tions A and B.
a - Two base pOpulations were generated with Specific
gene frequencies, each having 960 individuals.
b - Based on the tOp-cross progeny performance fifty
percent of the base populations were discarded.
c - Selection, then, was practiced on the basis of

phenotypic expression of S S 's, and S 's

I

18’ 2 3
in each population while reducing the population
size by half in each generation of selfing.

d - Seven more generations of self~fertilization

were followed without selection to produce 60

inbred lines in each of the inbred populations A and B.

40

3 - Experiments with the Hallauer method

Two classes of experiments were simulated in which dif-
ferent fractions of pOpulations were saved, 0.50 and 0.10, to
study the effect of selection intensity on selection progress.
Enough selfed progenies were produced to keep the population size
constant when necessary.

Selection was on the basis of full-sib progeny performance.
Five full-sib progeny were generated for each parental pair.

a - Two base populations were simulated with par-
ticular initial gene frequencies, each having
960 individuals.

b - 80 individuals of both populations were paired
randomly to produce 960 pairs of individuals
which were kept separately thereafter through-
out the experiments.

c - These 960 pairs were selfed, crossed and selection
between pairs practiced for 3 generations, holding
the population size constant.

d - Four generations of selfing, crossing, and selec-
tion were followed while reducing the pOpulation
size by half in each generation.

e - Selfing and crossing continued for 3 more gen-

erations without selection to obtain 60 pairs

3' le-cross h brids.
of A310 x BS10 ing y

The experiments simulated by the Hallauer method in this

Study will hereafter be denoted by "Hallauer" experiments for sake

0f Simplicity.

41

4 - Reciprocal Recurrent Selection (RRS) experiments

Two classes of experiments were generated in which dif-
ferent fractions of populations were saved, 0.50 and 0.10, to
study the effect of selection intensity on selection progress.

The essential features of these experiments were simulated accord-

ing to those given by Cress (1965).

V. RESUETS AND DISCUSSION

A - Reference Experiment

As the name indicates, these experiments were conducted to
serve as reference points. Selection was not practiced at any point
in the selfing process. Reduction in population size was at random.
A hybrid population was made for each generation of self-fertiliza-
tion using random individuals from pOpulations A and B.

The expected and observed genotypic means of the random
mating populations, F = O, and the inbred populations, F = l,
with different genetic models and gene frequencies for 30 in-
dependent pairs of loci are presented in Table 4 (Appendix A).

There existed a very close agreement between the observed and
expected genotypic mean for different populations simulated.

The genotypic means of additive (A), and additive by
additive (AA) populations, as expected, did not change under self~
fertilization. For most of the remaining genetic models, a re-
duction in the genotypic means was expected because of self-
fertilization.

However, under the additive by dominance (AD) genetic model
the genotypic mean decreased, stayed the same, and increased as the
inbreeding coefficient, F, approached unity depending on the starting
gene frequencies. The genotypic means decreased when p1 = q1 = 0.1

and p1 = q1 = 0.3, unchanged when p1 = q1 = 0.5, and increased

42

43

when p1 = q1 = 0.7.

Considering all gene frequency combinations, maximum reduction in the
genotypic mean of the populations due to selfing occurred under
the (0D) model. The relationship between the genotypic mean of
a population, average degree of dominance (a), and the inbreeding
coefficient (F) for non-epistatic models is shown by Fig. 1 (i),
(ii), and (iii), (Appendix B), when p1 = q1 = 0.1, p1 = q1 = 0.3,
and p1 = q1 = 0.7, reSpectively. The genotypic mean has a linear
relationship with both (a) and (F) under non-epistatic models.

The concept of average degree of dominance is not defined for the
epistatic models.

The relationship between the genotypic mean of a popula-
tion, frequencies of plus alleles, and the inbreeding coefficient,
F, are illustrated in Fig. 2 and 3 (Appendix B) for (ON) and (AD)
genetic models, reSpectively. The genotypic mean has a linear
relationship with the inbreeding coefficient, F, but a quadratic
one with the gene frequencies under (ON) and (AD) models. The
observed genotypic means of the inbred populations A and B and the
corresponding genotypic mean of the hybrid populations (A x B)
are presented, following 10 generations of inbreeding, in Table 5
(Appendix A).

The inbreeding coefficient, F, is zero in the hybrid popula-
tions. Therefore, their genotypic means should be equal to the
genotypic means of populations A and B when F = 0.0. These can
be checked by comparing the genotypic means of the hybrid popula-
tions of column three, six, and nine in Table 5 (Appendix A) with

columns four, six, and 8, reSpectively, in Table 4 (Appendix A)

44

for F = 0.0. The other observed genotypic means were in agree-

ment with the expected ones. Since selection was not practiced
in the Reference experiment, gene frequencies did not change in

either population A or B.

B - TOp-cross Experiment

A completely recessive inbred line was used as the top-
cross tester in testing the S1 progeny, F = 0.5. The observed
genotypic means of the parental populations A, B, and the hybrid
after 0, 2, and 10 generations of self-fertilization accompanied
with selection are presented in Table 6 (Appendix A). Recall that
phenotypic selection was practiced after discarding 50% of the
960 individuals in the base population based on the top-cross
progeny performance. Phenotypic selection was continued until
the population size was reduced to 60, with a reduction in popula-
tion size of 50% in each generation. Self-fertilization was con-
tinued without further selection.

The genotypic means of populations A, B, and (A x B) were
increased throughout the generations due to selection for all the
initial gene frequency combinations under the (A) model, Table 6
(Appendix A). The increase in the genotypic means from the S0
generation to the 82 generation was due to tOp-cross progeny
selection and was about 4-7 units, depending upon the initial
gene frequencies. An increase in the genotypic means from S

2

to S was due to phenotypic selection. Over all cycles, the

10

genotypic means of population A, B, and (A x B) increased about

45

10-12 units under the (A) model. Similar trends were seen in the
(AA) model.

Under the (CD) model both inbreeding depression and selec-
tion pressure were operating. These two forces were in the opposite
direction with respect to increase or decrease in the genotypic
means of the A and B populations. While inbreeding depression
was decreasing the genotypic means, the selection pressure was

increasing the genotypic means. With = p1 = 0.1, the geno-

Apl B

typic means of populations A and B were expected to be 42.0 after
10 generations of selfing without selection pressure. As it can
be seen from Table 6 (Appendix A), the genotypic means of pOpula-
tions A and B for this combination of initial gene frequencies
after 10 generations of selfing are 51.8 and 51.5.

Selection force was not as strong as the inbreeding de-
pression in the early generations of self~fertilization; thus a
reduction was observed in the genotypic means of populations A
and B from the So to the 82 generations. For the other com-
binations of initial gene frequencies, selection pressure over-
came part of the effects of inbreeding as can be checked by com-
paring the genotypic means of populations A and B after 10 gen-
erations of selfing with the expected genotypic mean of these
populations when F = 1.0, Table 4 (Appendix A). In the hybrid
populations (A x B) since selection pressure was present, an in-
crease in their genotypic means was expected throughout the

experiment as it can be observed from Table 6 (Appendix A).

Similar trends were observed under (ON) genetic model.

46

With the (OD) model the average degree of dominance was the
maximum;therefore,the inbreeding depression was the most intense.
In this case, no selection pressure can be effective in overcoming
the effects of inbreeding in A or B. Therefore, the genotypic
means of populations A and B were reduced to the minimum genotypic
value possible, 30 for this model, for all gene frequency combina-
tions. With this model, the phenotypic selection, which followed
the top-cross progeny selection, seemed to be incapable of increas-
ing the hybrid pOpulation means significantly for any combination
of initial gene frequency. This is indicated by the very small
progress seen in the hybrid population genotypic means after 0
and 10 generations of self-fertilization and selection, Table 6
(Appendix A).

The trends in the (AD) model are interesting. As indicated
in the Reference experiment, under (AD) model inbreeding may de-
crease, remain unchanged, or increase the genotypic mean of a
population, depending upon the frequencies of + alleles present
in that population. Therefore, because of self-fertilization,
progress was seen in the genotypic means of population A and B
from S0 to S and from S to S10 generations whenever

2 2

or were greater than or equal to 0.5. A reduction in

Apl Bpl

the genotypic means of population A and B was observed when Ap1

or Bp1 were below 0.5. The genotypic mean of the hybrid popula-
tion (A x B) was increased significantly only when the initial
gene frequencies in both populations A and B were very low, 0.1.

When the initial gene frequencies in both pOpulations A and B were

0.3, the genotypic mean of the hybrid population increased only

47

by 4 units from So to the S 0 generations. For the rest of

1
the gene frequency combinations no substantial change occurred

in the genotypic means of the hybrid populations for the (AD) model.
For this peculiar model, (AD), the progress in the hybrid popula-
tion mean was made only when the frequencies of + genes were

very low in the parental populations.

Over all the genetic models, maximum progress in the hybrid
population mean took.p1ace when the initial frequencies of the
desirable alleles in the base pOpulations A and B were very low,
0.1. Generally Speaking, the progress in the hybrid population
mean was lowest when the initial gene frequencies of populations
A and B were at different sides of 0.5, Table 6 (Appendix A).

The maximum progress in the hybrid population mean was achieved
with the (CD) model. The hybrid population mean was also increased
significantly under the (A) model, the (ON) model, and the (AA)
model but generally not under the (OD) and (AD) models.

Some discussion about the top-cross tester is in order.

The completely recessive tester was chosen in the present
study as ideal for discrimination in non-epistatic models. Such
a tester is unlikely in a practical situation. The optimum choice
of tester for the general case of epistasis is not known and is
probably not unique. When the tester is an unrelated inbred line,
selection for the dominant favorable allele would, of course, be
effective at loci recessive in the top-cross tester. However,

such loci would not necessarily be the ones at low frequency in
pOpulation under selection. LonaniSt (1968) stated that progress

in population improvement from selection based on top-cross progeny

48

performance has not been as great as would seem possible or desir-
able from the standpoint of the effort involved.

Rawlings and Thompson (1962) showed, equation [2], that
genes exhibiting a high degree of dominance contribute much more
to top—cross variance than those with low dominance, provided that
gene frequency of a dominant allele in the tester is low (below
0.5). Horner g£_al (1969) concluded, on the basis of the Rawlings
and Thompson argument, that selection based on testcross per-
formance when the parental population itself was the tester, was
expected to be most effective at loci when the frequency of the
cominant favorable allele is appreciably below 0.5 and which
exhibits both large differences between homozygotes and a high
level of dominance. However, the Rawlings and Thompson theory
was developed for situations where epistatic gene effects were
absent. Allison and Curnow (1966) pointed out that the parental
population was an effective tester because it was more likely than
any other tester to be recessive at loci where improvement is most
needed, and, if overdominance is important, gene frequency will be
stablized at optimum frequencies.

Cockerham (1963) partitioned the total genotypic variance
among selfed progeny means for a single locus with two alleles.
Horner g£_al (1969) compared equation [1] which was develOped by
Rawlings and Thompson (1962) with the additive component of the
genotypic variance among selfed progeny means which was developed
by Cockerham (1963) and concluded that selection based on selfed
progeny performance per se was much more effective than parental

toP-cross performance as well as inbred top-cross performance for

49

population improvement. Theoretical comparisons have not been
considered when epistatic gene effects are present. This is,
probably, due to the complication involved in mathematical de-
velopment and, furthermore, the concept of the average degree of

dominance is not defined when epistasis exists.

C - "Hallauer" Experiments

1 - Mild selection intensity

 

The observed genotypic means of the inbred populations A
and B with the corresponding hybrid populations (A x B) for dif-
ferent initial gene frequencies and genetic models are presented
in Table 7 (Appendix A).

The observed genotypic means of the inbred populations,
iuij (i = A, B and j = A, CD, OD, ON, AA, AD), were used to
predict the gene frequencies, ipi, in the inbred populations after
10 generations of self-fertilization and selection. Since the

observed genotypic mean of the inbred populations, iqu’ were

obtained for 30 independent pairs of loci, the right hand side of

equations [12] to [17] must be multiplied by 30 when iuij are
used in these prediction equations.
Hence, equations [12] to [17] become:
I = v
iuIA 120 ip1 + 30, [18]
l = v
iuICD 120 ip1 + 30, [19]
' =
ill-10D 30. [20]
2
l = _ l t
iuION 240 ip1 + 240 ip1 + 30, [21]

50

2
I = I _ I
iul 240 ip1 240 ip1 + 150, and [22]

ip'IAD

I
+ .
120 ip1 30 [23]

Equation [20], for this particular overdominant, (OD),
model,is independent of ipi and therefore can not be used in
predicting the gene frequency in the inbred populations. The
average gene frequencies in the inbred populations under this
model were calculated from the proportion of the fixed + alleles
and were designated by 1p; to distinguish them from ipl’ gene
frequencies predicted from the prediction equations.

The independence of equation [20] from ipi is because
the same genotypic value was given to the four homozygotes,
AB/AB, Ab/Ab, aB/aB, and ab/ab. Some overdominant model other
than pure overdominance could have been used to overcome this
problem. For instance,we could have used the following genetic
model which is overdominant and at the same time the four homo-

zygotes do not possess the same genotypic values:

A-a locus

3 4 2
B-b locus 4 5 3
2 3 1

In this example the dominant and recessive homozygotes, AB/AB
and ab/ab, are distinguished from the two other homozygotes by
their genotypic values of 3 and 1, respectively. The prediction

equation for this example is:

51

I = O I 3
p“10D 6 ip1+ O

l
which is dependent on ipi.

The prediction equations [21] and [22] are second degree
equations with respect to ipi.

with the optimum number and additive by additive models, where

These equations are associated

selection pressure accompanied with self-fertilization could,
potentially, push the inbred populations in two directions. With
the additive by additive model, selection pressure pushes each
2-1ocus pair in the inbred pOpulations toward both dominant or
both recessive homozygotes, whereas, with the optimum number
model the 2-locus pairs are directed toward the two other homo-
zygotes. Because of the reciprocal selection effects between the
interacting pairs of loci, inbred means were obtained that would
be impossible if ip1 = iql. The consequences of this ambi-
directional selection were that equations [21] and [22] gave
impossible or no solutions for ipi. We were unable to devise a
method by which the direction of the selection pressure on the
A-a and B-b loci could be determined for each of the 30 independent
pairs directly from the inbred population means. For these reasons,
the proportion of fixed loci was used to obtain 1p; for the
(ON) and (AA) models.

The initial gene frequencies (ipl)’ predicted gene fre-
quencies (ipi), and the average gene frequencies calculated from
the prOportion of + alleles (ipl) are presented in Table 8

(Appendix A) for various genetic models. This table shows that gene

frequencies in a population can be increased, stay the same, or

52

decreased, depending upon the initial gene frequencies, ip1, and
the genetic models. The observed genotypic mean of an inbred
population is a function of the final average gene frequency and
the epistatic effects of the genes. We will show that the inbred
performance was enhanced by epistasis for certain models when the
reciprocal selection method of Hallauer was used. The selective
value of an individual in population A or B is determined entirely
by the performance of the full-sib progenies of one population
with an individual of the other\population. Each population is
considered as a tester for the other. Depending upon the initial
and subsequent gene frequencies in population A and B and the
genotypic value of the homozygous individuals, selection pressure
may decrease or increase the gametic frequencies of the pOpula-
tion and thus its structure. When heritability in the narrow
sense is relatively low for a trait, the progeny test is used to
evaluate the superiority of the parents. If only the additive
component of the genotypic variance in the progeny is used, i.e.
half-sibs, no response to selection is observed when gene fre-
quencies are at equilibrium unless they are changed by some forces.
In contrast, when both additive and non-additive components of the
genotypic variance are used, i.e. full-sibs, an immediate response
to selection is observed even though the gene frequencies are at
equilibrium. The genotypic means of the inbreds, develOped by

the Hallauer method, were the indirect result of selection among
full-sibs. For the epistatic models with ambidirectional selection
pressure, (ON) and (AA), inbred means were higher than would be

expected based on the average change in gene frequency, Table 9

53

(Appendix A). The reciprocal effects of selection was reflected
in the inbred population A and B and their corresponding hybrid
populations (A x B) and can be measured by comparing the observed
genotypic means with the predicted means calculated based on the

average gene frequency, ip'

1 or ipI, where ,p = .q

i i 1

II
1.

The predicted genotypic means of the inbred pOpulations

ipl = iq
A and B and their correSponding hybrid pOpulations (A x B) are
presented in Table 9 (Appendix A). The differences between the
observed and the predicted genotypic means of the inbred popula-
tions A and B and the hybrid populations (A x B), which con-
tributed to the reciprocal effects of genes, are presented in
Table 10 (Appendix A) for different initial gene frequencies

and genetic models. If the differences in Table 10 are interpreted
as the reciprocal effects of selection, then the inbred reciprocal
effects result entirely from interlocus effects between pairs of
interacting loci. The differences for the hybrids were composed

of both an intralocus and an interlocus compatibility between the
inbreds. For the additive genetic model all differences were
expected to be zero. The non-zero differences are due to rounding
error. The differences between the observed and the predicted
means of the inbred populations for the (CD) and (CD), which are
non-epistatic models, were expected to be all zero. The reciprocal
effects of selection were reflected in the genotypic means of both
the inbred populations and hybrid pOpulation for the epistatic

models (ON) and (AA).

54

However, for the (AD) epistatic model, zero differences
were obtained between the observed and predicted genotypic means
for the inbred pOpulations. One may ask why positive reciprocal
effects of genes have been reflected in the genotypic means of
the inbred populations for the (ON) and (AA) models but not for
(AD) model. The obvious answer is that in the (AD) model selec-
tion pressure was one-sided whereas it was two-sided for (ON)
and (AA) models. For the same reason second degree prediction
equations were obtained for (ON) and (AA) models whereas a first
degree prediction equation was obtained for (AD). One expects
to obtain zero differences between the observed and predicted
genotypic means for the inbred populations when the observed geno-

typic means are used in the prediction equation to get ,p'.

1 1

The reciprocal effects of selection were all non-zero for
the genotypic means of hybrid populations (A x B) except for the
additive model as was expected. The maximum difference occurred
under the overdominant, ODD), model when the initial gene fre-
quencies, ipl’ were 0.5 for population A and B, respectively. The
maximum reciprocal effects, averaged over all gene frequencies, the
inbred populations, and the hybrid populations were obtained under
(ON) and (AA) models, respectively. Averaging over the genetic
models, the maximum reciprocal effects were obtained for the inbred
pOpulations A, B, and the hybrid population (A x B) when the initial

gene frequencies were 0.5 and 0.5, 0.5 and 0.3, and 0.7 and 0.3,

reSpectively.

55

2 - Strong selection intensity

Using the Hallauer method, a breeder is restricted by the
number of pairs of crosses produced at the beginning of the first

cycle. As soon as the number of crosses is set, the

A50 x B80
breeder has to work within and among that number of entries. New
recombination take place within a line only in the early genera-
tions. The within line variation vanishes very quickly as self-
fertilization is continued. There is no possibility of gene
exchange within and between the lines making a pair; hence no
chance of obtaining new genotypic combinations and genotypic
variation. These, although they impose some problems, provide
grounds to match the lines pairwise and select for the Specific
combinations of lines which perform the highest from the early
cycles of inbreeding. Self-fertilization leaves little chance

for selection to change gene frequency within a pair. Thus, it
would seem desirable to start with two genetically diverse base
populations to increase the chance of obtaining desirable com-
binations when the SO plants are randomly paired and crossed.

But the selection of genetically diverse populations does not,
necessarily, bring out the favorable specific combinations that

we are looking for, particularily if a different set of alleles

are presented in the base populations (Cress 1966). To overcome
these restrictions, one should start with a large number of

ASO x BSO crosses and practice relatively high selection intensity
for the first few cycles of inbreeding. Enough selfed progeny

should be produced within a pair during the early cycles of self-

fertilization to raise the opportunity of obtaining desirable

56

specific combinations when these selfed progenies are crossed.
After a few generations of self-fertilization the within line
selection is fruitless and the selfed progeny may be bulk to
represent a line.

Thetnagnitude of the selection intensity practiced within and
between ASO x BSO, A31 x B81, A32 x BS2 A83 x BS3 crosses
by Hallauer (1967b) were 37.5%, 35.0%, 10.4%, and 8.3%, respectively.

, and

Hallauer (1967a) suggested and practiced mild selection intensities
to produce sufficient seed for evaluating the pairs of crosses

extensively.
In this study two selection intensities were practiced and

kept constant for different S

A i x BSi (i = 0,1,2,3,...) by using

the Hallauer techniques to compare the effect of selection intensity
on the progress made.

The observed genotypic means of the inbred pOpulations A
and B and their hybrid populations (A x B) after 10 cycles are
presented in Table 11 (Appendix A) when selection intensity was

strong.

The same set of prediction equations was used in predicting

ipi. The proportion of fixed + alleles was used to obtain ,pi.
i

The gene frequencies and op: are given in Table 12
i

I
ipl’ ipl’
(Appendix A) for different inbred populations and genetic models.

Having obtained these gene frequencies, ipi

ass ' ' = ' d " = "
umption that ipl iql an 1P1 iql

typic means for the inbred population A and B and their hybrid

and _p", and the
1 l

the predicted geno-

populations (A x B) were computed and presented in.Table 13

(Appendix A) for various initial gene frequencies and genetic models.

57

Table 14 (Appendix A) contains the differences between the
observed and the predicted genotypic means for different populations.
These differences reflect the reciprocal effects of selection.
Similar trends for mild and strong selection may be observed. But
the non-zero values for strong selection are much greater than
those observed for mild selection.

However, none of the observed genotypic means reached the
limit value of a 150 set for all the genetic models. To observe
the effect of selection intensity on the genetic progress several
figures out of 84 possible (6 genetic models, 7 combinations of
initial gene frequencies, and 2 selection intensities) combina-
tions are presented. These figures numbered from 4 to 11 (Appendix
B) contain the phenotypic mean of selected individuals in popula-
tion A, B, and the phenotypic mean of the hybrid populations for
10 cycles of self-fertilization. Selection ceased at seventh
generation of inbreeding but self-fertilization was continued for
three more generations.

The 3.1. in the figures stands for selection intensity
and SPSC’ the standard error of the phenotypic mean of the single
crosses, calculated as follows:

11 ll

2
1.230%- US. ’3 a“. - WNW
i-l i—l

S2

PSC PSC

where NH is the harmonic mean of the number of single-cross
hybrids made for the hybrid populations (A x B) over eleven gen-

erations, S and Ni are the phenotypic variance between the

iPSC

single crosses and the number of single-crosses for ith generation,

reSpectively.

58

In Fig. 4-11 (Appendix B) there are curves for reciprocal
recurrent selection (RRS). Discussion of these curves will be
given when comparisons between the "Hallauer" and (RRS) experi-
ments are made.

The phenotypic means of A, B, and (A x B) populations in-
creased for both selection intensities in Fig. 4 (Appendix B).
This was expected since any change in gene frequency is reflected
in the inbreds and single crosses with additive gene effects. The
phenotypic mean of the (A x B) population increased from 66 to 80
under mild selection intensity, while it increased from 66 to 83
under strong selection intensity after 10 generations of self-
fertilization and 7 generations of selection. With strong selec-
tion intensity the phenotypic mean of (A x B) reached 80 at the
fifth generation while with mild selection intensity the same value
was not obtained until the tenth generation. Selection was
practiced within lines and between pairs of crosses at the early
generations of selfing while selection was entirely between pairs
of crosses after the fifth generation. This is one reason a
steeper reSponse was obtained during the early generations.

The effect of selection intensity on the mean of (A x B)
and the inbred populations is illustrated for one case for the
(CD) model in Fig. 5 (Appendix B). The interesting phenomena in
Fig. 5 are associated with means of populations A and B. Since
the initial gene frequencies for + alleles were 0.7 and 0.1 in
pOpulations A and B, respectively, the mean of A population was
expected to be greater than the mean of population B over all cycles

of inbreeding and selection. The mean of population A decreased

59

due to inbreeding for the first few generations then increased as
selection reversed this downward trend. The mean of the population
B decreased without a substantial increase at later generations.
Selection pressure increased the frequencies of + alleles in
population A considerably from the initial gene frequencies of

0.7. Selection caused the dominant homozygotes, AB/AB, to be
accumulated in population A and little increase in the dominant
alleles in pOpulation B.

When the initial gene frequencies were the same in both
population A and B, Fig. 6 (Appendix B), the means of both popula-
tions decreased without any increase in the later generations.
Again a rapid increase was observed in the hybrid pOpulation mean
in the early generations for the stronger selection intensity.

The trends in Fig. 7 (Appendix B) are typical for the
overdominant, (OD), model no matter what combination of initial
gene frequencies were present in the base populations. In Fig. 7
the initial gene frequencies were at equilibrium in both popula-
tions A and B. Therefore, the additive component of genetic
variance was zero and no response would be expected for selection
methods using only the additive component of genetic variance such
as the half-sib progeny test (Cress 1967).

Here an immediate reSponse was seen, not only because full-
sib performance was used to select individuals contributing to the
hybrid population, but also the self-fertilization in the Hallauer
method was a factor (Li 1967). As mentioned by Cress (1967), forces
such as chance fluctuations could alter the equilibrium gene fre-

quencies in populations A and B and cause a reSponse in selection

60

even when half-sibs were used as a test criterion.

The effect of selection intensity on the mean progress for
an optimum number, (0N), model with initial gene frequencies 0.1
in both populations A and B is seen in Fig. 8 (Appendix B). Even
though the initial gene frequencies were very low in the popula-
tions, a high response to selection was observed for the hybrid
population. If selection increases the frequency of the first +
alleles and decreases the frequency of the second + allele at a
locus pair in one population and the reverse situation happens
in another pOpulation, such an increase in the hybrid mean is
expected. This was observed previously as an interlocus reciprocal
effect. The phenotypic means of the inbred populations were de-
creased for one or two generations of self-fertilization. Later
selection between pairs of lines within a population caused the
means of populations A and B to increase only Slightly. As in all
models, strong selection resulted in an increased reSponse in the
hybrid during the early generations of inbreeding.

In Fig. 9 (Appendix B), both inbred population means were
greater than the hybrid population. This is not surprising for
the additive by additive, (AA), model if one notes that the initial
frequencies of the + alleles were 0.7 and 0.3 for population A and
B, respectively. In this model, both the dominant and recessive
homozygotes had the highest genotypic value relative to the other
genotypes.

In Fig. 10 (Appendix B) the hybrid population was between
the two inbred populations. By a glance at the model, (AD), it is

seen that the dominant homozygotes have the highest genotypic

 

 

61

values, recessive homozygotes have the lowest genotypic values,
and the double heterozygotes intermediate genotypic values. It
may be noted that little average change in gene frequency occurred
(Tables 8 and 12) in either population. Thus, the increase in the
hybrid mean is the result of selection for Specific combinations.
Essentially all of the increase in population A is the result of
inbreeding and not a change in gene frequency (see Table 4, (AD)
model, p1 = 0.7).

Figure 11 (Appendix B) illustrates a case when equilibrium
gene frequencies existed in the base populations A and B. That is,
the additive component of genetic variance was zero when gene fre-
quencies were 0.5. With mild selection intensity the inbred pop-
ulation means basically did not change over ten cycles of self-
fertilization and selection. With strong selection intensity the
phenotypic mean of population B increased slightly due to a little
increase in the average gene frequency. The hybrid population
means increased under both mild and strong selection intensities.
Since no substantial changes in the average gene frequencies were
observed in either population A or B for either mild or strong
selection intensity, this progress in the phenotypic mean of the
hybrid populations was related to selection for specific com-

binations.
D - Comparison Between the Breeding Methods
1 - Top;cross method vs. the Hallauer and (RRS) methods

Absolute comparisons between the top-cross method and the

Hallauer and (RRS) methods are imprOper because the goal of breeding,

62

type of selection practiced, and the cycle time were different

for these breeding procedures. Therefore, the trends observed

in the phenotypic means of hybrid populations for various breeding
methods will be compared for different initial gene frequencies,
genetic models, and only for mild selection intensity as it was
the only selection intensity practiced in the tOp-cross experi-
ment. I

The total progress made in the phenotypic mean of the
hybrid populations after 8 cycles of mild selection using the
Hallauer, (RRS), and top-cross method are presented in Table 15
(Appendix A) for different initial gene frequencies and genetic
models. Selection was practiced for four cycles in top-cross
experiment and then self-fertilization was followed. In the
"Hallauer" and (RRS) experiments selection was practiced for 8
cycles. The data in Table 15 (Appendix A) is the differences be-
tween the phenotypic means of the hybrid populations at cycle
zero and cycle eight.

Over the initial gene frequencies, greater progress was
seen in the hybrid pOpulation mean using top-cross method than
(RRS) method under the (A) model. For the same genetic model,
similar progress was observed for the top-cross and the Hallauer
methods. Very similar trends were seen for all the breeding methods
under the (CD) model. With the (OD) model, total progress made in
the phenotypic mean of the hybrid population was much smaller for
tOp-cross method compared to the (RRS) and the Hallauer methods.
Under the (ON) and (AA) models, the total progress seen for the

top-cross method was similar to that of the (RRS) method but

63

smaller than the progress made by the Hallauer method. Both (RRS)
and the top-cross methods were ineffective for the (AD) model
except when the frequency of the dominant allele was low in both
pOpulations. Except for the (OD) model, the top-cross method
resulted in greater progress in the hybrid population mean than
(RRS) but generally smaller progress as compared with the Hallauer
method. This is rather surprising because the low heritability
and epistatic models would seem to favor recurrent progeny testing
over the top-cross and phenotypic selection for 3 cycles.

When the average degree of dominance was zero, additive
gene effects, and one, complete dominant gene effects, the t0p-
cross method seemed to be the most preferred method in improving
the hybrid population mean. Its advantage in simplicity is
obvious. With the epistatic gene effects present in a population,
the top-cross method of breeding was found to be Superior to (RRS)
method but inferior to the Hallauer method in the rate of progress
of the hybrid population mean. Only when the average degree of
dominance was the highest, overdominant, the (RRS) method as well
as the Hallauer method were superior to the top-cross method.
These are not totally satisfying reSults in light of the modest
performance reported for the top~cross in corn breeding.

It should be recalled that in this study an inbred top-
cross tester, completely recessive, was used that is very unlikely

in a practical situation.

64

2 - The Hallauer method vs. (RRS) method

 

In the comparison of (RRS) with the Hallauer selection
method we must be confined to a comparison of trends. Absolute
comparison is inadvisable for several reasons. First, the
immediate goal and the end products of the two methods are not
the same. Second, the cycle time is three generations for (RRS)
and two generations for Hallauer's method. Third, the nature of
the breeding systems dictated population sizes that were not com-
parable except in a general sense. Fourth, progress by the
Hallauer method terminates in relatively few cycles, where progress
by (RRS) should be possible over a much longer period of time.

In Spite of the lack of precise comparisons, it is felt that
qaulitative contrasts, such as presence or absence of response to
selection, are meaningful.

Under an additive model and with initial frequencies of
the desirable alleles 0.3 in both pOpulations A and B, similar
progress was observed in the phenotypic means of the hybrid popula-
tion (A x B) for both types of experiments in early cycles of
breeding, Fig. 4 (Appendix B). Similar advances were observed for
the completely dominant, (CD), model with initial frequencies of
favorable alleles 0.7 and 0.1 in populations A and B, respectively,
Fig. 5 (Appendix B). The "Hallauer" and (RRS) experiments reSponded
the same throughout the experiments under the completely dominant,
(CD), model and initial frequencies of + alleles 0.3 in both
populations A and B, Fig. 6 (Appendix B). These figures are typical
of the comparison of (RRS) and the "Hallauer" methods for the (A)

and (CD) models.

65

The initial gene frequencies in the base pOpulations pre-
sented in Fig. 7 (Appendix B) were at equilibrium, 0.5. Therefore,
in the absence of chance fluctuation, which alters the gene fre-
quencies, no early response to selection was expected for the
hybrid population in the (RRS) experiments. Genetic progress or
advance is a function of selection intensity, heritability in the
narrow sense, and the phenotypic variance of the character under
consideration. No progress is expected if any one of these com-
ponents is zero. When gene frequencies are at equilibrium, the
additive component of genetic variance is zero which makes
heritability in the narrow sense zero when half-sibs are used as
the test criterion.

However, response to selection was observed in the (RRS)
experiments after the eighth generation of breeding when selection
intensity was mild. This can be attributed to the accumulation
of the additive component of genetic variance established in the
populations through random drift of the gene frequencies from 0.5.

When selection intensity was stronger, it took fewer gen—
erations for response to be observed in the (RRS) hybrid mean.
Under these same conditions of the (OD) model, larger response to
selection was observed for the Hallauer method, Tables 15 and 16
(Appendix A).

The progress in the phenotypic means of the hybrid popula-
ations (A x B) were similar for both the "Hallauer" and (RRS)
experiments for the particular (ON) gene frequencies presented in
Fig. 8 (Appendix B). This was typical response for (RRS) with

the (ON) model only when the gene frequencies of both populations

66

were low or when both populations were high (see Tables 15 and 16,
Appendix A). Progress by (RRS) was at best very modest when both
populations had gene frequencies near 0.5 or when the population
gene frequencies were on opposite sides of 0.5. The smallest
reSponse of about 11 units for the Hallauer method was obtained

= 0.7 and

when = 0.3 with mild selection. The average

Bpl

response was much greater (see Tables 15 and 16, Appendix A).

Apl

No progress was observed in the mean of the hybrid popula-
tions for the (RRS) experiments under the additive by additive
model, Fig. 9 (Appendix B), with the initial frequencies of +
alleles 0.7 and 0.3 in populations A and B, reSpectively, and a
mild selection intensity.

Griffing (1962) showed that progress due to selection from

a single cycle of (RRS) with 2-locus epistasis is

_ . 2 2 2
pl — %(l/aboh.s.){[(abOAa) + (abOAb)] + %(1 + dy)

2 2
[(abGAaAa) + (abOAbAb)]}’

where i is selection differential meaSured in terms of general

combining ability effects, is the variance of the general

aboh.s
combining ability estimates in the hybrid populations,

02 2
ab AaAa’ abCAb’

aboAa’

and are the additive and additive by

2
aboAbAb
additive component of genetic variance in the hybrid population
contributed from population A and B, respectively. The effects of

linkage is measured by dy.

In the absence of linkage, dy = 0, the formula becomes

2

_ . 2 2 2 2
p“i ' %(1/aboh.s.){[(abOAa) +'(abc’Ab) +'%L(ab°AaAa) + (abOAbAb)]}‘

67

Of the four classes of additive by additive epistatic
effects which occur in the hybrid population, only those involving
two alleles coming from the same population contribute to genetic
advance. Some progress is expected in the hybrid population even
when the additive component of genetic variance is zero, provided
that at least one of the additive by additive components of variance
associated with the genetic advance or gain not be zero. Thus, it
was surprising to find in four of the seven gene frequency com-
binations, the total progress after 8 cycles of (RRS) with mild
selection was less than 4 units (see Table 15, Appendix A). Even
with strong selection the performance of (RRS) was less than
satisfactory (see Table 16, Appendix A).

There are two adaptive peaks associated with the (AA) model.
Population A and B could be driven by selection to one of the
adaptive peaks or to different adaptive peaks or to none, depending
upon the initial gene frequencies in the pOpulations. If popula-
tion A goes to one of the adaptive peaks and pOpulation B to
another, no progress is expected in the hybrid population (A x B).
In contrast if both populations A and B go to one of the adaptive
peaks, progress is expected in the hybrid population (A x B).

With the (AA) model, when the initial frequencies of the
desirable alleles are 0.7 and 0.3 in population A and B, reSpectively,
each population is close to one of the adaptive peaks. When pOpula-
tion B is used as the tester I suggest that upward pressure is put
on the (ab) gamete in pOpulation A. This decreases the frequency
of the (AB) gamete in population A and increases the frequency of

the (ab) gamete. After a few cycles of selection, depending upon

68

the selection intensity, the (ab) gamete is as frequent as the (AB)
gamete in population A. However, when population A is used as the
tester, the frequency of the (AB) gamete increases in population B
and as a consequence the frequency of the (AB) gamete is as fre-
quent as (ab) in population B. When selection intensity is mild
changes in the gametic frequencies take place slowly in both popula-
tions A and B. However, it takes fewer cycles of selection to make
the (ab) gametes as frequent as the (AB) gamete in pOpulation A

and the (AB) gamete as frequent as the (ab) gamete in population B
when selection intensity is strong.

At this stage average gene frequencies are close to 0.5 for
both populations. If by chance fluctuation, the gene frequencies
of some of the independent pairs of loci in population A drifts
upward or downward from 0.5 and similar procedure happen for the
corresponding pairs of loci in population B some progress is
expected in the hybrid population because both populations A and
B are moving toward the same adaptive peak with reSpect to these
pairs of loci. These loci have, relatively, a greater selection
advantage with reSpect to other loci. As the proportion of these
independent pairs of loci become larger, the reSponse to selection
become greater.

With the additive by dominance, (AD), model and frequencies
of desirable alleles in a range of 0.3-0.7 in both populations A
and B, no progress in the hybrid population was made in the (RRS)
eXperiments when selection intensity was mild (see Table 15,
Appendix A). Very little reSponse was observed when selection in-

tensity became stronger, Fig. 10 and Fig. 11 (Appendix B) and Table

69

16 (Appendix A). Under this model, (AD), the (RRS) method of breed-
ing seemed to be incapable of improving the hybrid population mean
unless the frequencies of the favorable alleles were very low or
very high in both populations A and B. In contrast, when the
Hallauer method of breeding was employed, the phenotypic means of

the hybrid populations were improved considerably.

VI. SUMMARY AND GENERAL CONCLUSIONS

A set of prediction equations was developed to separate
the reciprocal effects of selection from the effects of an average
change in gene frequency on the mean of parental populations A and
B and their hybrid populations (A x B) for the "Hallauer" experi-
ments. The direct application of these prediction equations was
not possible for some genetic models following selection. Some
extreme genetic models such as pure overdominance and other epistatic
models were included in the study which may be only rarely found
in natural populations. It was thought that these models would
reflect differences that are present but not as obvious for less
extreme models.

Reciprocal recurrent selection (RRS) was found to be as
effective as the Hallauer method in advancing the hybrid population
means when most of the genetic variance was additive. However, it
was less effective than both the Hallauer and the top-cross methods
when epistatic gene effects were present. The top-cross method
was inferior to both (RRS) and the Hallauer methods under the (OD)
model.

The genetic variance within pairs of individuals was
exhausted rapidly under the Hallauer method of breeding due to its
mating system. All the genetic variance present in the later gen-

erations of selection was due to variation between pairs of inbred

7O

71

lines in population A and B. In contrast, genetic variance was
expected in population A and B and their hybrids (A x B) when the
(RRS) method of breeding is employed. This provides a basis for
improving these populations for a long period selection. However,
the Hallauer method produced a set of superior single-cross hybrids
in a shorter period of time when compared with the (RRS) method.

In the "Hallauer" experiments, population A and B consisted
of 960 80 individuals which were then reduced by selection to
60 310 individuals, each representing an inbred line. In the
(RRS) experiments, Starting population size was 90 in both pOpula-
tions and it kept constant throughout the experiments.

The effect of two different selection intensities was
studied. In the "Hallauer" experiments a sharp increase in the
hybrid population means resulted from stronger selection intensity.
Strong selection exploited variation in the early generations of
selection, and thus, an improved rate of progress was made. For
this reason as many pairs of individuals as practical should be
used in the first cycle and the amount of selfed seed produced
should be enough to maintain this population size for three cycles.
Subsequent selection is largely between pairs. Therefore,it would
seem more efficient to defer testing until the lines are essentially
inbred. In the last two or three generations of inbreeding, a
thorough evaluation of the single crosses may be made over a range
of environments. In the Hallauer method, emphasis is on specific
combining ability and the so-called "nicking" effects both within
and between loci. Individuals from A and B which contribute to a

superior hybrid are selected immediately and maintained as separate

72

pairs. The nicking effects of the alleles contributed by these selected
individuals are not lost in the Hallauer method since selection is

not followed by random mating but by self-fertilization which pre-
serves some of the genetic constitution of selected pairs.

There is no rigorous basis to compare the Hallauer method
with the (RRS) method from the vieWpoint of time and expenses in-
volved. If the purpose of breeding is to produce and maintain a
good germplasm or to generate population hybrids, some form of
recurrent selection may be indicated. But, if the goal is prop
duction of superior Single-cross or even double-cross hybrids over
a short period of time, the Hallauer method seems to be an effective
one for a wide range of situations.

Hallauer, 1967b, has Suggested modifications for situations
where multiple matings are not possible on a single genotype. As
a general view of the three selection systems Simulated:

l. The rate of progress of the hybrid population by the
top-cross and subsequent phenotypic selection was
competitive with (RRS) and the Hallauer methods for
the (A), (CD), and certain conditions of epistasis.
Because of the simplicity of the method and the good
progress under certain conditions, one should not dis—
count some form of the top-cross as a breeding tool.

2. For several reasons, reciprocal recurrent selection is
not directly comparable to the other systems simulated.
The merits of (RRS) have been established both in theory
and in practice, but the total benefits are very long

range. The poor performance of (RRS) for some conditions

73

with epitasis places some limits on its utility.

3. The reSponse to selection by the Hallauer method was
both rapid and immediate. The increase in the single
cross performance was without regard to choice of
model. For no model or gene frequency simulated did
the Hallauer method fail to respond. The terminal
nature of the method leaves very small possibility that
the maximum genotypic value will be attained when a
large number of loci are involved in determining the
trait.

Hallauer (1967a) recommended a selection intensity of 30 to

50%. However, it would seem advisable to select much more intensely

during the early generations when segregation within lines is occurring.

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Kempthorne, 0. 1955. The correlations between relatives in random
mating populations. Cold Spring Harbor Symposia on
Quantitative Biology. Vol. XX. pp. 60-78.

Kempthorne, O. 1957. An introduction to genetic statistics. John
Wiley and Sons, Inc., New York, 545 pp.

Li, C.C. 1967. Genetic equilibrium under selection. Biometrics
23:397-484.

Lonnquist, J.H. 1968. Further evidence on testcross versus line
performance in maize. Crop Sci. 8:50-53.

Malécot, G. 1948. Les mathematiques de l'hérédité. Masson et cie.
Paris.

Matzinger, D.F. 1953. Comparison of three types of testers for
the evaluation of inbred lines. Agron. J. 45:493-495.

Morrow, C.E., and F.D. Gardner. 1893. Field experiment with corn.
111. Exp. Stat. Bull. 25.

Morrow, C.E., and F.D. Gardner. 1894. Corn and oats experiments.
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Pfaffenberger, R.C., and C.E. Gates. 1967. SIMLAG: Simulation
Language Adapted for Genetics. Joint Statistical Meetings,
Washington, D.C., December 28.

Qureshi, A.W., and O. Kempthorne. 1968a. On the fixation of genes
of large effects due to continued truncation selection in
small populations of polygenic system‘with linkage. Theor.
and Appl. Genetics 38:249-255.

Qureshi, A.W., OJ Kempthorne, and L;N. Hazel. 1968b. The role of
finite population Size and linkage in reSponse to continued
truncation selection. I. Additive gene action. Theor.
and Appl. Genetics 38:256-263.

Qureshi, AHW. 1968c. The role of finite population size and linkage
in response to continued truncation selection. II. dominance
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79

Rawlings, J.O., and D.L. Thompson. 1962. Performance level as
criterion for choice of maize testers. Crop Sci. 2:217-220.

Russell, W.A. 1961. A comparison of fine types of testers in
evaluating the relationship of stalk rot resistance in
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combinations. Crop Sci. 1:393-397.

Schnell, F.W. 1961. On some aSpectS of reciprocal recurrent
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Sprague, G.F. 1966. Quantitative genetics in plant improvement.
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Sprague, G.F., W.Aw Russell, L.H. Penny, T.W. Horner, and W.D.
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testers of different ear heights of corn. Agron. J.
52:617-620.

80

Williams, J.C., L.H. Penny, and G.F. Sprague. 1965. Full-sib and
half-sib estimates of genetic variances in an open-pollinated
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Young, S.S.Y. 1966. Computer simulation of directional selection
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Young, S.S.Y. 1967. Computer simulation of directional selection
in large populations. II. The additive by additive and
mixed models. Genetics 56:73-87.

APPENDIX A

81

Table 2. Standard Errors of the Differences Between Runs
for the Phenotypic Means of the Hybrid Populations
(A x B) in any Generation Obtained from the

Hallauer Method.

Initial Gene

Frequencies (A) (CD)
AP1 = 0.1, Bp = 0.1 0.834 0.949
Apl = 0.5, Bp = 0.5 0.916 0.960
AP1 = 0.7, Bp = 0.1 0.859 0.947

Genetic Models

(0D) (0N)

0.967 1.072

1.018 0.928

0.957 0.874

(AA)

0.907

0.912

0.875

(AD)

1.038

0.924

0.872

82

Table 3. Genetic Models Simulated and the Expected Genotypic
Means for 30 Independent Pairs of Loci for Different
Combinations of Gene Frequencies.

Gene Frequencies

Genotypic
Genetic Values p 0.1 0.3 0.5 0.7
MOdels A 1 o 1 o 3 o 5 o 7
Bp1 . . . .
AA Aa aa Exp. Gen. Means

BB 5 4 3
Additive (A) Bb 4 3 2 42.0 66.0 90.0 114.0

bb 3 2 1
Complete BB 5 5 3

Dominance (CD) Bb 5 5 3 52.8 91.2 120.0 139.2
bb 3 3 1

Over- BB 1 3 1

dominance (on) Bb 3 5 3 51.6 80.4 90.0 80.4
bb 1 3 1

Optimum BB 1 4 5

Number (ON) Bb 4 5 4 62.4 105.6 120.0 105.6
bb S 4 1

Additive by BB 5 3 1

Additive (AA) Bb 3 3 3 128.4 99.6 90.0 99.6
bb l 3 5

Additive by BB 5 2 3

Dominance (AD) Bb 2 3 4 59.3 86.2 90.0 93.8

bb 3 4 1

83

Table 4. The Expected (E) and Observed (0) Genotypic Mean of
a Random Mating Population, F = 0.0, and an Inbred
Population, F = 1.0, in the Absence of Selection
for Different Gene Frequencies and Genetic Models.

Genetic Gene Frequencies

Models F p1=q1=0.1 p1=q1=0.3 p1=q1= 0.5 p1=q1=0.7
E O E O E O E O

O 42.0 42.0 66.0 66.1 90.0 89.9 114.0 114.2

(A) 1 42.0 42.3 66.0 65.8 90.0 89.3 114.0 113.6
(CD) 0 52.8 52.9 91.2 91.0 120.0 119.8 139.2 139.1
1 42.0 41.9 66.0 65.9 90.0 89.9 114.0 113.3
(00) 0 5.16 51.8 80.4 80.5 90.0 89.9 80.4 80.1
1 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0
(ON) 0 62.4 62.2 105.6 105.7 120.0 120.2 105.6 105.6
1 51.6 50.6 80.4 81.5 90.0 90.7 80.4 80.4
(AA) 0 128.4 128.4 99.6 99.8 90.0 90.1 99.6 99.6
1 128.4 128.5 99.6 99.5 90.0 89.9 99.6 98.9
(AD) 0 59.3 59.3 86.2 86.1 90.0 90.1 93.8 93.7

1 42.0 42.8 66.0 65.3 90.0 90.4 114.0 114.8

Table 5 .

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

Populations

(AKB)

42.

42.

41.

41.

41.

51.

30.

30.

51.

50.

50.

62

128.

128

129.
42.
42.

61.

84

1 0.5
1 0.1
1 89.3
0 42.7
5 65.8
7 88.9
3 42.2
6 95.6
0 30.0
0 30.0
1 91.3
6 90.6
9 50.6
.6 109.3
6 88.4
.4 128.9
0 98.4
9 89.9
4 42.4
4 89.7

0.7
0.1
112.8
42.3
77.9
113.1
42.3
117.8
30.0
30.0
107.3
81.1
50.5
127.0
99.0
128.9
91.6
114.5
43.4

95.2

0.3
0.3
66.
66.
66.
65.
67.
93.
30.
30.
81.
81.
81.
105.
99.
101.
99.
66.
65.

84.

Initial Gene Frequencies

0.5
0.3
90.
64.
76.
90.
64
108.
30.
30.
89.
88.
82.
117.
90
97
91

90.

91.

The Observed Genotypic Means of the Inbred POpulations
A and B after 10 Generations of Selfing with the Hybrid
Populations (A x B) for the Reference Experiment.

0.7
0.3

0 114.3

7 113.5

.7 65.8

7 123.7

0 30.0
0 30.0
5 98.2
6 79.8
7 81.2
8 124.9
.4 98.7
.7 99.2
.8 89.8

3 115.1

0.5
0.5
88.4
89.6
89.2
90.7
89.2
120.5
30.0
30.0
89.1
90.7
93.0
121.1
90.9
90.5
91.1
90.9
90.5

90.0

 

85

0.00
m.00~
H.¢0~
0.00
n.00H
0.00H
0.0m~
0.00H
0.5NH
5.00
0.00
0.00
0.mm~
5.00H
0.00H
~.00~
n.00H
0.00H

0H

0.00
0.00
0.50
m.q0
N.00
0.00
0.0mH
H.~0A
¢.H0~
a.00
0.mm
0.0m
n.0NH
H.H0_
n.00H
5.00
~.00
H.00

0.0
0.0

0.00
0.00
H.00
0.00
0.00
0.00
0.0mH
0.0NH
H.0Na
0.00
«.00
5.00
0.0mH
«.0NH
H.0-
0.00
~.00
H.00

n.50
n.00
0.m-
5.00
0.0HH
n.5HH
5.0Nd
0.00
H.00
0.50H
0.0m
0.00
n.5mH
0.00
0.0NH
0.~0~
0.50
5.0NH

0H

0.00
0.05
0.0aa
m.m0
H.0HH
H.005
5.5NH
0.N0
0.00
0.00
n.mm
H.mm
0.5mm
0.05
N.0NH
0.50
0.n5
0.0ma

m.0
5.0

0.00
~.00
N.q0
0.00
n.00
5.00
m.m-
5.00H
n.00H
0.00
5.00
n.00
0.0Nfi
0.50
0.0m~
0.00
H.00
H.0HH

0.00
0.00
0.m0~
0.NOH
H.0m~
5.00H
0.0ma
~.H~H
5.00~
0.00
0.0m
0.0m
5.m-
N.00
m.qoa
0.N0
0.05
0.400

0H

0.00
0.m5
0.00
0.50
0.00H
N.00
0.NNH
q.m0
m.~0H
~.a0
q.mn
H.0m
0.005
0.05
0.00H
0.m0
q.M5
5.50

m.0
0.0

0.00
~.00
~.00
¢.~0
0.00
5.00
0.5H~
0.00H
N.0NH
0.00
0.00
0.00
0.00H
N.H0
N.0m~
0.05
0.00
m.00

0.00
5.05
n.05
H.mH5
0.0HH
0.0HH
0.HNA
0.00
m.HOH
0.H0
0.00
0.0m
0.5HH
m.H0
0.00
0.00
~.H0
0.00

0H

0.00
N.m5
0.05
n.50~
H.0~H
N.0-
0.mHH
0.N0
0.m0
0.N0
«.mm
H.mm
5.N0a
~.05
0.05
0.05
m.Q5
0.M5

N
m.0
m.0

momocmsooum

.0 comumumcom cm mcomumasooa mmunhz

0.00
0.00
H.00
0.00
0.00
0.00
0.00H
5.005
0.00~
0.00
0.00
0.00
N.H0
0.50
q.~0
0.00
0.00
0.00

6:00 ficwuwcH

0.00
H.Nm
m.mma
0.50
a.m0a
5.0HH
n.0NH
0.00
0.00
0.0HH
0.0m
0.00
«.mma
~.mm
0.050
0.00
0.Hm
0.0N5

0H

5.55
6.56
5.655
6.65
5.655
5.565
5.655
6.56
6.56
5.565
5.55
6.65
5.6N5
5.66
c.5NS
5.56
5.66
6.655

N
H.0
5.0

0.m0
5.0m
5.m0
0.50
0.0NH
0.00
0.0NH
H.N0
H.00H
~.00H
5.Hm
0.00
0.5AH
0.Nm
0.000
0.05
0.m0
H.0HH

5.N0
m.#m
0.m0a
0.00H
H.m0H
n.00H
H.05H
~.00
0.00M
n.~0
0.0m
0.0m
0.mHH
5.Hm
0.00a
0.55
0.Hm
m.m0H

0a

«.50
0.00
0.00
0.00a
0.0ma
0.00
0.0HH
H.~0
5.00M
m.~0
5.~m
0.00
0.00H
H.0e
0.HOH
0.N5
0.00
5.50

0:» new mom: amen

0.00
0.00
0.00
0.00
0.0NH
0.00
0.0H~
N.N0
0.0aa
0.00
m.~n
5.00
0.00
m.~m
0.0~a
0.00
N.Nq
0.00

.muwwcoocH cofiuomamm 0552 0653 vogue:
mmouo-606 men 0:56: cowuuoaow new co5um~uflwuummuuaom 00 mcouomumcmo 0H mam
.N .0 umuwu Am x <0 0:6 .0 .< mcouumfloeom No name: umuhuocmo mo>uomoo 005

6.6a 6.66
5.55 5.66
5.55 N.56
5.565 5.655
H.565 n.655
5.56s c.65H
6.56 5.65
6.66 5.56
5.56 6.~6
5.55 6.55
6.65 5.~5
6.65 6.55
6.56 5.56
5.55 5.66
6.a5 5.66
6.65 6.66
5.65 5.66
6.55 5.66
6H 5

5.6

5.6
.6 6566a

o>mz moaao> vmuummxm a

m.0m
0.0m
n.0m
0.0NH
0.0NH
0.0NH
0.50
0.H0
0.~0
0.Hn
m.am
m.Hm
0.Nm
5.Nm
0.~m
0.Na
0.Ne
0.Nq

0

5650
.<.
6
4
Amxav
.4.
m
a
Amxav
k.

mcouuwfisaom

meowuuuocou

use
use

5040

5440

5260

5000

5666

5<0

maonoz

owowcoo

 

Table 7.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

86

The Observed Genotypic Means of the Inbred Populations
A and B after 10 Generations of Selfing with the Hybrid
Populations (A x B) using the Hallauer Method with Mild
Selection Intensity.

Initial Gene Frequencies

Populations Ap1 0.1 0.5 0.7 0.3 0.5 0.7 0.5
Bp1 0.1 0.1 0.1 0.3 0.3 0.3 0.5

A 49.6 107.3 127.1 77.8 105.1 124. 104.
B 48.6 48.0 48.4 79.0 78.7 80. 102.
(AxB) 49.1 77.6 87.7 78.4 91.7 102. 103.
A 50.3 106.7 129.6 77.7 103.4 127. 101.
B 49.5 46.4 43.7 78.0 76.3 71. 98.
(AxB) 67.6 115.0 133.4 111.2 126.3 138. 135.
A 30.0 30.0 30.1 30.1 30.1 30. 30.
B 30.0 30.0 30.0 30.1 30.0 30. 30.
(AxB) 71.5 109.5 128.4 99.2 110.9 119. 109.
A 69.1 98.1 73.6 92.8 99.3 84. 97.
B 66.3 54.9 51.8 94.3 88.0 85. 100.
(AxB) 86.6 128.4 138.1 124.8 133.1 135. 132.
A 138.3 104.2 104.0 113.4 101.2 107. 102.
B 141.3 135.6 133.8 117.8 111.2 108. 101.
(AxB) 140.2 111.6 100.6 115.9 104.7 100. 102.
A 49.7 95.2 111.0 69.3 91.5 112. 91.
B 50.7 41.0 39.7 69.1 64.0 64. 89.
(A33) 81.7 104.6 103.6 102.6 102.2 100. 102.

 

 

Table 8.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

The Initial Gene Frequencies, ip

87

1’

the Predicted Gene

Frequencies, ipl’ and the Gene Frequencies Calculated

from the Proportion of + Alleles, ipl’ of the Inbred

Pepulations A and B after 10 Generations of Self-
fertilization using the Hallauer Method with Mild
Selection Intensity.

Gen. Fre.

in Inbred Ap1

Populations Bp1

II
p1

Initial Gene Frequencies

0.1
0.1

0.1633
0.1550
0.1690
0.1625
0.1836
0.1759
0.1883
0.1667
0.0511
0.0389
0.1642

0.1725

0.5
0.1

0.6441
0.1500
0.6390
0.1366
0.6239
0.1070
0.6183
0.1133
0.3975
0.0722
0.5433

0.0917

0.7
0.1

0.8091
0.1533
0.8300
0.1140
0.8297
0.0619
0.7906
0.1003
0.6167
0.0864
0.6750

0.0808

0.3
0.3

0.3983
0.4081
0.3970
0.4000
0.3483
0.3608
0.3661
0.3864
0.2253
0.2099
0.3275

0.3258

0.5
0.3

0.6258
0.4058
0.6116
0.3858
0.5667
0.2883
0.5550
0.3161
0.4478
0.2711
0.5125

0.2833

0.7
0.3

0.7875

0.4225

0.8080

0.3490

0.7672

0.2278

0.7239

0.2828

0.6917

0.3128

0.6875

0.2875

0.5
0.5

0.6191

0.6000

0.5940

0.5717

O

.5122

O

.4933

O

.4917

0.5067

0.5072

0.5125

0.5125

0.4983

 

Table 9.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

88

The Predicted Genotypic Means of the Inbred Populations
A and B after 10 Generations Selfing with the Hybrid

Populations (A x B) Based on ipi and 1p? using the

Hallauer Method with Mild Selection Intensity.

Initial Gene Frequencies

Populations Ap1 0.1 0.5 0.7 0.3 0.5 0.7
' Bp1 0.1 0.1 0.1 0.3 0.3 0.3
A 49.6 107.3 127.1 77.8 105.1 124.5
B 48.6 48.0 48.4 79.0 78.7 80.7
(AxB) 49.1 77.7 81.7 78.4 91.9 102.6
A 50.3 106.7 129.6 77.7 103.4 127.0
B 49.5 46.4 43.7 78.0 76.2 71.9
(AxB) 66.5 112.6 131.9 106.6 121.4 135.0
A 30.0 30.0 30.0 30.0 30.0 30.0
B 30.0 30.0 30.0 30.0 30.0 30.0
(AxB) 65.4 101.7 124.7 84.9 93.4 107.5
A 66.7 86.6 69.7 85.7 89.3 78.0
B 63.3 54.1 51.7 86.9 81.9 79.2
(AxB) 82.6 121.2 133.2 114.5 120.2 125.7
A 134.4 92.5 93.3 108.1 90.7 98.8
B 141.0 133.9 131.1 110.2 102.6 98.4
(AxB) 139.7 106.9 95.3 109.1 94.7 90.0
A 49.7 95.2 111.0 69.3 91.5 112.5
B 50.7 41.0 39.7 69.1 64.0 64.5
(AxB) 72.5 91.6 94.3 87.5 90.1 90.2

0.5
0.5
104.3
102.0
103.2
101.3
98.6
129.1
30.0
30.0
90.0
90.0
90.0
120.0
90.0
90.0
90.0
91.5
89.8

90.0

 

Table 10.

Genetic

Models

(A)

(CD)

(OD)

(0N)

(AA)

(AD)

89

Differences Between the Observed and Predicted Genotypic
Means of the Inbred Populations A and B after 10 Genera-
tions of Selfing with the Hybrid Populations (A x B) using
the Hallauer Method with Mild Selection Intensity.

Populations Ap1 0.1
Bp1 0.1

A 0.0

B 0.0
(AxB) 0.0
A 0.0
B 0.0
(AxB) 1.1
A 0.0
B 0.0
(AxB) 6.1
A 2.3

B 3.0
(AxB) 4 6
A -0.1
B 0.3
(AxB) 0.5
A 0.0
B 0.0

(AxB) 9.2

Initial Gene Frequencies

0.5
0.1
0.0
0.0
-0.1
0.0
0.0
2.4
0.0
0.0
7.8
11.5
0.8
7.2
11.7
1.7
4.7
0.0
0.0

13.0

0.7
0.1
0.0
0.0
0.0
0.0
0.0
1.5
0.1
0.0
3.7
3.9
0.1
4.9
6.7
2.7
5.3
0.0
0.0

9,3

0.3

0.3

0.0

0.0

0.0

0.0

15.1

0.5 0.7
0.3 0.3
0.0 0.0
0.0 0.0
-0.2 0.0
0.0 0.0
0.0 0.0
4.9 3.9
0.1 0.1
0.0 0.1
17.5 12.3
10.0 6.7
6.1 5.9
12.9 10.2
10.5 9.1
8.6 9.9
10.7 10.4
0.0 0.0
0.0 0.0
12.1 10.6

0.5
0.5
0.0
0.0
-0.1
0.0
0.0
5.9
0.0
0.0

19.7

10.5
12.7
12.9
11.3
12.3

0.0

0.0

12.1

Table 11.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

90

The Observed Genotypic Means of the Inbred Populations

A and B after 10 Generations of Selfing with the Hybrid
Populations (A x B) using the Hallauer Method with Strong
Selection Intensity.

Initial Gene Frequencies

Populations p 0.1 0.5 0.7 0.3 0.5 0.7 0.5

A 1
Bp1 0.1 0.1 0.1 0.3 0.3 0.3 0.5
A 49.5 114.4 128.7 83. 108.2 127 1 102.8
B 56.7 49.4 52.7 81. 85.4 81.0 111.4
(AxB) 53.1 82.1 90.7 82. 96.8 104.0 107.1
A 57.2 119.8 139.7 84. 109.6 134.1 106.4
B 52.3 51.2 43.2 84. 75.5 71.8 102.3
(AxB) 76.9 128.3 142.5 119. 134.2 143.8 141.4
A 30.1 30.1 30.0 30. 30.0 30.0 30.0
B 30.1 30.0 30.0 30. 30.0 30.0 30.0
(AxB) 80.5 120.5 133.7 108. 117.8 129.0 117.3
A 72.9 97.2 62.1 98. 96.7 82.8 97.5
B 70.7 60.6 45.5 101. 93.2 80.0 97.2
(AxB) 93.9 133.7 141.2 130. 136.6 137.9 137.4
A 148.1 113.8 110.2 128. 113.7 111.6 108.5
B 146.7 140.0 138.3 116. 121.7 110.4 112.5
(AxB) 147.7 119.8 104.4 122. 112.0 102.3 107.2
A 51.7 97.1 103.5 69. 92.4 110.6 89.1
B 54.0 39.9 39.7 66. 72.6 66.6 95.1
(AxB) 91.8 107.6 109.3 106. 108.1 105.2 105.9

91

Table 12. The Initial Gene Frequencies, ,pl, the Predicted Gene
1
Frequencies, ipi, and the Gene Frequencies Calculated
from the Proportion of + Alleles, ipl’ of the Inbred

Populations A and B after 10 Generations of Self-
fertilization using the Hallauer Method with Strong
Selection Intensity.

Gen. Fre. Initial Gene Frequencies

Genetic ,
Models In Inbred AP1 0.1 0.5 0.7 0.3- 0.5 0.7 0.5
Pepulations Bp1 0.1 0.1 0.1 0.3 0.3 0.3 0.5
(A) Api 0.1625 0.7033 0.8225 0.4450 0.6517 0.8091 0.6067
Bpi 0.225 0.1658 0.1892 0.4308 0.4617 0.4250 0.6783
(CD) Api 0.2267 0.7483 0.9142 0.4500 0.6633 0.8175 0.6367
Bpi 0.1858 0.1767 0.1100 0.4542 0.3792 0.3483 0.6025
(0D) AP; 0.1878 0.7375 0.8708 0.3831 0.6292 0.7450 0.4475
3p; 0.2650 0.0656 0.0303 0.3997 0.2744 0.2714 0.5533
(0N) Apg 0.2206 0.6406 0.8556 0.4192 0.6050 0.7347 0.5200
Bpg 0.1867 0.1378 0.0644 0.3939 0.3050 0.2369 0.4961
(AA) Apg 0.0078 0.3286 0.5994 0.1428 0.3853 0.7089 0.5578
Bpg 0.0136 0.0656 0.0778 0.2097 0.1953 0.3461 0.5003
(AD) Api 0.1808 0.5592 0.6958 0.3292 0.5200 0.6717 0.4925
' 0.2000 0.0825 0.0808 0.3058 0.3550 0.3050 0.5425

Table 13.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

Populations

(AxB)

Apl

Bpl

92

Initial

0.1
0.1
49.5
56.7
53.1
57.2
52.3
74.5
30.0
30.0
72.4
71.3
66.4
88.4
148.1
146.8
147.5
51.7
54.0

75.8

0.5
0.1
114.4
49.4
82.2
119.8
51.2
125.1
30.0
30.0
114.8
85.3
58.5
123.2
97.1
135.3
112.0
97.1
39.9

92.1

ipl

and

.P
i 1
Hallauer Method with Strong Selection Intensity.

using the

Gene Frequencies

0.7
0.1
128.7
52.7
90.7
139.7
43.2
140.8
30.0
30.0
131.8
59.7
44.5
138.2
92.4
132.8
96.3
113.5

39.7

0.3

0.3

0.

0.

5

3

83.4 108.2

81.7
82.6
84.0
84.5
114.0
30.0
30.0
87.2
88.4
87.3
116.9
120.6
110.2
115.3
69.5

66.7

85

96.

109.

75.

124.

30.

30.

97.

87.

80.

122.

93.

112.

100.

92

72.

94.4 87.1 90

.4

.4

6

.1

0.7

0.3

127.1

81.0

104.1

134.1

71.8

139.6

30.0

30.0

103.4

76.8

73.4

127.4

100.5

95.7

90.2

110.6

66.6

90.2

The Predicted Genotypic Means of the Inbred Populations
A and B after 10 Generations of Selfing with the Hybrid
Populations (A x B) Based on

0.5
0.5
102.8
111.4
107.1
106.4
102.3
132.7
30.0
30.0
90.7
89.9
90.0
120.0
90.8
90.0
90.2
89.1
95.1

90.0

Table 14.

Genetic

Models

(A)

(CD)

(CD)

(0N)

(AA)

(AD)

POpulations

(AxB)

16.

Initial Gene Frequencies

0.

0.5

0.1

0.0

0.0

-0.1

0.0

0.0

3.2

0.1

0.0

5.7

11.9

2.1

10.5

16.7

4.7

7.8

0.0

0.0

15.5 14.9 18.

0.7
0.1
0.0
0.0
0.0
0.0
0.0
1.7
0.0
0.0
1.9
2.4
1.0
3.0
17.8
5.5
8.1
0.0

0.0

0.

O.

21.

l3.

l3.

0.

0.

3

3

0.

20.

15.

12.

14.

10.

11.

18.

5

3

10.

11.

14.

12.

15.

Differences Between the Observed and Predicted Genotypic
Means of the Inbred Pepulations A and B after 10 Genera?
tions of Selfing with the Hybrid Populations (A x B) Using
the Hallauer Method with Strong Selection Intensity.

0.5
0.5
0.0
0.0
0.0
0.0
0.0
8.7
0.0
0.0

26.6

17.4
17.7
22.5
17.2
0.0
0.0

15.9

94

Table 15. Total Progress made in the Phenotypic Means of Hybrid
Populations (A x B) after 8 Cycles of Mild Selection
using the Hallauer, (RRS), and top-cross* Methods.

Initial Gene Frequencies

Apl 0.1 0.5 0.7 0.3 0.5 0.7 0.5
Bpl 0.1 0.1 0.1 0.3 0.3 0.3 0.5
Gen. Mod. Hallauer
(A) 7.2 10.1 9.8 14.3 13.9 13.7 13.4
(00) 14.6 20.0 15.5 19.3 19.3 14.5 15.6
(00) 20.2 19.0 17.8 18.9 21.7 30.5 19.4
(ON) 23.1 18.2 13.3 18.5 15.9 11.1 13.9
(AA) 12.5 11.0 9.0 16.1 11.6 10.0 12.9
(AD) 25.0 15.7 10.4 16.9 13.4 11.2 12.4
Gen. Mod. (RRS)
(A) 6.5 8.6 6.2 9.4 11.0 10.1 8.8
(CD) 19.1 13.9 12.8 17.3 10.8 8.0 5.1
(00) 12.0 14.1 14.0 8.3 6.2 6.9 0.9
(ON) 23.8 13.9 7.5 11.0 4.1 3.4 0.8
(AA) 9.0 8.6 3.7 10.6 1.8 -0.4 1.7
(AD) 11.2 3.3 -0.2 2.3 0.0 -0.3 0.1
Gen. Mod. top-cross
(A) 12.0 12.7 7.4 12.2 16.3 12.9 16.8
(CD) 18.1 23.2 14.1 17.6 15.5 14.9 13.4
(CD) 2.5 2.1 1.7 0.4 1.2 0.9 -2.2
(ON) 22.8 8.0 2.2 10.1 11.5 4.1 10.7
(AA) 10.5 7.0 4.4 14.8 10.3 6.6 7.6
(AD) 14.5 4.3 -5.5 4.9 2.0 -o.5 0.9

* Four cycles of mild selection was practiced.

95

Table 16. Total Progress made in the Phenotypic Means of Hybrid
Populations (A x B) after 8 Cycles of Strong Selection
using the Hallauer and (RRS) Methods.

Initial Gene Frequencies

AP, 0.1 0.5 0.7 0.3 0.5 0.7 0.5
Bp1 0.1 0.1 0.1 0.3 0.3 0.3 0.5
Gen. Mod. Hallauer
(A) 10.1 16.8 13.0 16.3 19.0 13.4 15.5
(00) 24.1 33.4 25.0 28.8 25.6 17.7 20.3
(00) 28.9 29.8 25.2 26.1 26.6 29.1 29.2
(ON) 31.2 23.6 14.5 23.6 19.7 13.2 16.3
(AA) 18.0 21.6 11.8 24.0 19.2 11.9 18.8
(AD) 33.4 18.1 14.9 22.3 17.9 15.4 16.9
Gen. Mod. (RRS)
(A) 7.1 14.9 10.4 15.2 18.1 14.5 22.1
(00) 26.9 24.1 17.6 26.6 16.7 12.5 9.2
(00) 22.8 28.7 18.9 11.2 15.5 17.6 11.1
(ON) 34.9 18.1 7.7 22.8 7.3 5.9 3.8
(AA) 12.0 14.0 6.3 17.8 5.1 6.6 6.2

(AD) 23.8 8.5 3.0 4.7 1.5 3.1 1.8

APPENDIX B

96

Fig. 1 - Relationship among the genotypic mean of a non-epistatic

Population, the average degree of dominance (a) and the

inbreeding coefficient (F) for dominant allele frequency
equal to (1)0.1, (ii) 0.5, and (iii) 0.7.

 

6cm. Ivar»

48H“ ”an.

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r
l

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['1 1‘
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.1
A
u

 

 

     

11‘“

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Fig. 2 - Relationship

and inbreedin
model.

. 3 - Relationship amon
and inbreeding co

97

among the genotypic mean, gene frequency,
g coefficient (F) with the optimum number

8 the genotypic mean, gene frequency,

efficient (F) with the additive by
dominance model.

 

 

 

 

/ /
7 //
% / ,

w

     

./
/

 

 

égn. ”‘1‘”
I

 

 

 

 

‘4. .w—- .

Fig. 4 - Mean progress f
Hallauer method
method with ini

Bp1 = 0.3 for

intensities.

 

98

or population A, B, and (A x B) for the

, and the hybrid population for the (RRS)
tial gene frequencies Ap1 = 0.3 and

the additive model and two selection

phen- Hear!

Mean

Phen.

 

 

 

 

 

A
90‘) I

S.l.'m”d S.l,-strong

SPsc'l.308 5855“”9
5- ‘

80~ '
/
/
(3)9 /'G—-(A)
’1
S // /
/
___/
/
/
<—.J—-(AXB) 7’
) 7’ /

7o ,' ’

I, /

- \l I
6
IL 1 —r 9
v j— ' ' a —r —'
I 4 7 '0 ' h 7 '0

99

Fig. 5 - Mean progress for
Hallauer method
method with ini

= 0.1

population A, B, and (A x B) for the

, and the hybrid population for the (RRS)
tial gene frequencies Ap = 0.7 and
BP1 for the completely dominant model and two
selection intensity.

sPSc-l.352

I40:

4—-(A)

130‘

”5‘1

50 4

4—(8)

 

cycles

 

10

[ s.|,-strong
Spsc-|.323

 

 

e—-(B)

 

04

1E _. _~ M7,...__——-—'-1 “'""' '
7 1

cycles

100

Fig. 6 - Mean progress for
Hallauer method
method with ini

= 0.3

population A, B, and (A x B) for the

, and the hybrid population for the (RRS)

tial gene frequencies Ap1 = 0.3 and
Bpl for the completely dominant model and two

selection intensities.

Mean

Phen.

\
51 s.I.-mi|d
C" .1056

SP5

120‘

54

1104

100‘

 

90

 

\

S,|,-strong
Spsc"°389 I

 

 

 

101

Fig. 7 - Mean progress for population A, B, and (A x B) for the
Hallauer method, and the hybrid population for the (RRS)
method with initial gene frequencies Ap1 = 0.5 and

Bpl = 0'5 for the overdominant model and two selection
intensities.

 
      

Mean

Phen.

  

 

 

 

 

 

A '1“
mm 5";de 5°"""°"9
Spsc-l .446 5m" '“°°
5.
1104
5.
1004
S ..
’/
1,1100%)
90
(—(A)
4—(8)
5
80 r
01 - . . , 4» 1r 4 7
I 4 7 '0
Cycles

102

Fig. 8 - Mean progress for population A, B, and (A x B) for the
Hallauer method, and the hybrid population for the (RRS)
method with initial gene frequencies AP1 = 0.1 and

Bpl = 0'1 for the Optimum number model and two selection
intensities.

100‘

90‘

Mean

Phen.

 

S,l,=mild
SPSC=I.526

/
/

/

I
I
(RRS)-—-9I

 

 

S.l.-strong /
5050‘“32 /

 

 

I 4 7
Cycles

.4710
Cycles

103

Fig. 9 - Mean progress for population A, B, and (A x B) for the
Hallauer method, and the hybrid population for the (RRS)
method with initial gene frequencies Ap1 = 0.7 and
BP1 = 0.3 for the additive by additive model and two

selection intensities.

Mean

Phen.

IZO-
s,|,-mild S.|.-strong

SP (3":237 SI’SCM'ZG"5

S

1101

100

 

90*

4.1-.-

 

 

 

on‘

 

104

Fig. 10 - Mean progress for population A, B, and (A x B) for the
Hallauer method, and the hybrid pOpulation for the (RRS)
method with initial gene frequencies Ap1 = 0.7 and
Bpl = 0.3 for the additive by dominant model and two
selection intensities.

Mean

Phen.

 

At {A
5‘ S.l.-mild S.I.=strong

SPSCI-l .273

llO< +——4A)

 

 

 

 

1004
5 4
\
/
I
Iq— (RRS)
90‘ /

 

 

 

 

5
80 5.
__s_ Lf . ‘ r
0 {T1 4 '7 1'0 r ‘ " 7 '°

 

105

Fig. 11 - Mean progress for population A, B, and (A x B) for the
Hallauer method, and the hybrid population for the (RRS)
method with initial gene frequencies Ap1 = 0.5 and
Bp1 = 0.5 for the additive by dominant model and two

selection intensities.

IIOJ

 

S.l.-mild
81.269

5950

S.l.-strong

 

 

 

 

SPsc-l.26l
+—(AXB)
<—-(B)
’1'- ‘-’(RRS)
/
‘,-_/
L // 9'—"(A)
\I/
L
Li 4 . +7
I 4 7 l0

 

   

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071 2040