DISEQUILIBRIA AND GENOTYPIC VARIANCE
IN A RECURRENT TRUNCATION SELECTION SYSTEM
FOR AN ADDITIVE GENETIC MODEL
1
.
',;.
-;:
by
tUGLAS LEE ROY NEELEY and J. O. RAWLINQS
f
,~
Insti.~,·'t,'. e of Statistics
M1meo~aPh Series No. 729
Ra1ei~n:",~; 1971
;;', "~.
....
•
(
I
ri. .
"
iv
TABLE OF CONTENTS
Page
LIST OF TABLES .
vi
LIST OF FIGURES
ix
INTRODUCTION . .
1
MODEL DEVELOPMENT
2
The Genotypic Model
Param~terization of the Zygotic Genotypic Variance
Parameterization of the Gametic Variance and the Covariance
between Gametes and Parental Zygotes . . . . . . . . .
Higher Order Disequilibria and Genotypic Central Moments
The Selection Model
. . . . . . . . . .
The Effect of Selection on the Genotypic Variance
THE EFFECT OF SELECTION ON GENOTYPIC VARIANCE AND GENE FREQUENCY.
Empirical Methods • . . . .
Parameters and Measures Used
General Results . . . . . . .
The Effect of Various Parameters on Gene Frequency"
Disequilibria" and Genotypic Variance
The
The
The
The
The
The
Effect
Effect
Effect
Effect
Effect
Effect
of
of
of
of
of
of
.. ..
23
33
....
..
Summary and Discussion of Parametric Effects
33
59
61
72
77
87
98
102
PROJECTION OF THE GENOTYPIC VARIANCE
Reevels Projection
Neils Projection.
Other Proj ections
Evaluation of the Projection Equations
103
107
108
110
Equal Parental Intergametic- and Intragametic-Interlocus
Digenic Disequilibria within Each Pair .
The Case of Free Recombination . . . . . . . .
Systems Involving Tight Linkage . . . . . . . .
Discussion Regarding the Projection Equations
LIST OF REFERENCES . . .
6
11
15
17
23
25
29
Linkage
Intensity of Selection
an Increased Number of Loci
Initial Hert ta bi 1i ty
Initial Gene Frequencies
Varying Initial Gene Frequencies within Runs
.
2
4
.. .
111
112
119
127
131
v
TABLE OF CONTENTS (continued)
Page
APPENDICES
Appendix A.
Appendix B.
133
Intragametic Digenic Disequilibria in the
Selected Population . . . . . . . .
....
Relations between Interloci Digenic Disequilibria
in the Selected and Test Population . . . . . .
134
137
vi
LIST OF TABLES
Page
1.
2.
3.
4.
Proportional reductions) M(t)) in genotypic variance due to
disequilibria (generations given in parentheses)
The ratio) 0-11) c ) of the reduction in genotypic variance
due to disequilibria under complete linkage to that'
under free recombination (the corresponding ratios of
time) 0
) to a given life are given in parentheses)
t)c
The ratio) 0-11) n ) of the reduction in genotypic variance
due to disequilibria at half life for n = 4 to that
for n = 2
.
35
45
62
Disequilibria) D(t)) and proportional reductions) M(t)) in
the genotypic variance due to disequilibria; number of
loci and initial heritabilities are such that ~ is a
0'&
constant for given initial gene frequencies (generations
are given in parentheses) . . .
...•
5.
6.
7.
8.
69
The response of genotypic variance to selection for
initial heritabilities of .25 and 1 and for free
recombination and complete linkage) all with an
initial gene frequency of .3
73
Initial response in change in gene frequency and
disequilibria . . . . . . . . . . . . . • . . .
83
Initial response in change in gene frequency and
intra locus disequilibria for runs involving varying
initial gene frequencies at five loci . . . . . . . •
89
Matrices of off_diagonal elements representing initial
response in interlocus disequilibria) D~~)O)) to
1.J
selection; five locus runs with varying initial gene
frequencies given in the upper and left hand margins
of the matrices (upper off-diagonal) p2(O)
off-diagonal) p2(O)
9.
= .25)
= 1;
lower
. . . . .
Proportional reductions) M(t)) in genotypic variance due to
disequilibria for varying initial gene frequencies within
runs (generations given in parentheses) . . • . . . . . •
91
94
vii
LIST OF TABLES (continued)
Page
10.
6p(0) and D(s,O) for equal (E) initial gene frequencies
within a run versus
6p~0) and D~s,O) for unequal
(U)
1.
1.
initial gene frequencies within a run; contrasts at
comparable initial gene frequencies from runs involving
2(0)
n == 5 and p
== 1 • . . . . . . . . . . 0 . . 0 0 0 .
11.
Biases associated with projections of genotypic variance
under free recombination for p(O) == 03, p2(0)
1, and
n == 4 (bias nearest zero appears with asterisk) 0 . 0 .
12.
18.
1 , an d n == 5 (b'1.as nearest zero
. . . • . . . . 0 . 0 . . . .•
117
Biases associated with projections of genotypic variance
under free recombination for varying initial gene frequen.
. h'1n runs) p2 (0)
c~es
W1t
appears with asterisk)
17.
116
Biases associated with projections of genotypic variance
under free recombination for varying initial gene frequen,
. h'1.n runs, p2 (0)
C1.es
W1.t
appears with asterisk)
16.
115
Biases associated with projections of genotypic variance
under free recombination for p(O) == .7, p2(0) == 1, and
n == 4 (bias nearest zero appears with asterisk) . . . .
15.
114
Biases associated with projections of genotypic variance
under free recombination for p(O) == .5, p2(0) == 025, and
n == 4 (bias nearest zero appears with asterisk) 0 . . .
14.
113
Biases associated with projections of genotypic variance
under free recombination for p(O) == .5, p2(0) == 1, and
n == 4 (bias nearest zero appears with asterisk) . 0 . 0
13.
97
=
25 an d n = 5 (b'1as nearest zero
..... 0 0 ...
0
e
)
.
.
Biases associated with projections of genotypic variance
(0) _ 3
2(0)
under complete linkage for p
-., p
1 and n == 4
(bias nearest zero appears with asterisk) . .
118
120
Biases associated with projections of genotypic variance
under complete linkage for p(O) == 05, p2(0)
(bias nearest zero appears with asterisk) .
1 and n == 4
121
viii
LIST OF TABLES (continued)
Page
19.
Biases associated with projections of genotypic variance
under complete linkage for p(O) = .5) p2(0) = .25 and
n = 4 (bias nearest zero appears with asterisk) . . . .
20.
Biases associated with projections of genotypic variance
under complete linkage for p(O) ~ .7) p2(0) = 1) and
n = 4 (bias nearest zero appears with asterisk) . . . .
21.
=5
(bias
124
Biases associated with projections of genotypic
variance at c )i+1
n
23.
123
Biases associated with projections of genotypic variance
under complete linkage for varying initial gene
freqtiencie& within runs) p2(0) = 1) and n
nearest zero appears with asterisk)
22.
122
=4
i
= .1 for
p(O)
= .5) p2(0) = 1) and
(bias nearest zero appears with asterisk) .
125
Biases associated with projections of genotypic variance
at c. "+1 = .1 for p(O) = .5) p2(0)
~) ~
= .25~
and n
=4
(bias nearest zero appears with asterisk) . . . . • .
126
· ix
LIST OF FIGURES
Page
1.
2.
3.
Gene frequencies under interg~nic independence (1.1.) and
various d&g~ees of linkage . . . . . . . . .
49
Intragametic digenic disequilibria as a function of
frequency of recombination and locus position
56
Proporti~na1 reduction, M(t), of genotypic variance due
to disequilibria for p(O)=.3, a=.9, and c. '+1=.1 . .
~,~
4.
Above abscis~a: intragametic digenicdisequi1ibria, n(t),
as a proportion of the lower limit; below absci.ssa:
~.9, n=3)
76
Intragametic digenic disequilibria as a function of
initial gene frequency . . . . . . . . . . . .
79
lower limit of disequilibria (p(O)=.3,
5.
6.
64
Intragameti,c di.genic disequilibria between five loci each
with five different initial gene frequencies, the
initial gene frequencies given as subscripts to the
disequilibria (:fun involving p2(O)=.25, a=.5, and
c. '+1=.5). . . . • . . . . . . . . . . . • . . . ..
1,~
.
92
INTRODUCTION
In recent years much research has been devoted to evaluating the
effect of selection on inter10cus cOrrelations.
Most of the deter-
ministic developments have dealt with maintaining equilibrium states
from one generation to the next for various selection schemes imposed
on
trai~~
(~.!.,
controlled by two loci
$ee Bodmer and Fe1senstein,
1967) •. The effect of directional selection on interlocus associations
nas been less figorous1y treated from a deterministic standpoint.
Most
studies have been concerned with finite populations and have utilized
Monte Carlosimu1qtion techniques
(~.!.,
see Qureshi
.:! ~.,
1968;
Qureshi cmd Kemptj:lOrne, ·1968; Hill and Robertson, 1966; Gill, 1965).
Howev~r,
Griffing (1960) nas approximated the effect of mass selection
on traits controlled by several loci for a deterministic model in which
the paired. locus effect was assumed to be small relative to the
phenot;yp~c
qc~urate
standarq deviati,on.
Latter (1965b)
procedures allowing for
paired-lo~us
hasdeve~opedmore
contributions of greater
magnitude.
Nonetheless, up to this time no significant efforts have been made
to defermine the effect of selection induced intergenic associations on
central
mo~ents
fQr recurrent selection schemes.
Such parameters are
importapt in projecfing the response of the genotypic mean from one
generation to the next,
eva1uat~ng
This investigation is directed towards
the effect; of recurrent mass truncation selection on
interlocus correlations for a multi-locus, additive, deterministic
model.
l!inpftasis·is placed ondescrj.bing and projecting the response of
the gepPFypic "ariance to selection;
2
MODEL
DEVELOPM~NT
The Genotypic Model
i
The genotypic value is assumed to be controlled by a completely
additive diploid
is defineq
~odel
byth~
involving n loci with two alleles per locus and
function
G ::: a Xl
( 1)
in which
X= X+fX.
ma and X are l
~ n vectors with the i th element of ~ being the contribu_
tion to the genotypic value of a favorable allele at locus i (i
2, .,., n), and the i
th
element of ...,
X being a random variable associated
with the number of favorable alleles at locus i,
derived
pat~rnal1y .and
x, = 0
m~
= 1,
fXi' \llaternally,
X. of which are
m ~
For the diploid model,
Qr 1 depending on whether the allele is unfavorable
qr favorable, respectively, and
sim~larly
for fX ,
i
Let
be a 2 x n
matri~
ofran<;lom variables representing zygotic genotypic
arrays, find denote the frequency of the specific zygotic occurrence,
by f
~oci
(V,1he~ygot;ic
arrays associated with the specific set of r
Ii, j, .,', k} (1, j,
"0'
k
Sn
and mutually unequal) is
~,
3
represented by the following 2 x
X,)
mXi) m J
= [
Z..
k
-1J. , .
£X i ) fXj'
I'
matrix of random variables:
... ,
. ..
mXk
]
fX k
}
.
. The marginal probability associated with the specific zygotic
OCCl,n;:-rence
~i'.
.k
J •••
~H: ~hose
f(~) 1'j ••• k =
loci is
f(~) )
!:
~
e
S(z)..
""" .1J •••
S(z) .....k being the set of all
"'"' 1J... .
.
at loci {i} j)
I~
an
2;
k
having the particular zygotic array
#"OJ
"'J
.
add~tive
k} specified by z-1.J
.. ... k'
model) the gametic genotypic value can be defined
in a manner analogous to the zygotic value} .!:':")
G*
x*
= .....
a X*I
-.
.
(2)
is a 1 x nvector pf random variables
arrays) the ith
- dement(i
Xi*
= O} t
= l}
repres~nting
gametic genotypic
*
. }n) being Xi)
2} •••
depending on whether the allele is unfavorable
or favc;>rable) respectively.
The probability associated with the specific g.;lmetic occurrence)
*
~}
is
r(~*).Further) the gametic arrays associated with the specific set
of
I'
loci (i) j) "'} k} (i) j) .•. ) k $. n and mutually unequal) is
represented by the following vector of
*
X..
-J.) • • •
I'
random variables:
* X,)
* ... ) XkJ
* .
k. = [X~)
...
J
Themarglln~l pro~ab:Uit:y
of the specific garneticoccurrence x.* .
-1J ...
k at
4
those loci is
f( x *) ..
-
S(x*) . .
-
1.J •••
1.J...
k=
'"
*
* e s ()
x
x- 1.J
.. ••• k
£->
f(x_*) )_
k being the set of a11~* having the particular gametic
array x:C.
-1.J •••
k at loci (i) j) ... ) k}.
Parameterization of the Zygotic Genotypic Variance
i
For the additive zygotic model) equation (1)) the genotypic
variance Can be represented in the following quadratic form:
2
~G
wherein
~lX
= -a lL,
a'
{ x-
(3)
is the variance,.covariance matrix of X.
evaluate the off_diagonal elements of
L:x'x)
In order to
it is necessary to specify
the marginal zygotic frequen,cies for each pair of loci.
And) in order
to evaluate the qiagonal elements) it is necessary to specify the
marginal frequencies for each locus.
The marginal frequencies per
locus. can be obtained directly from the marginal frequencies per pair.
The marginal frequencies for pair i) j are:
11
f(ll)
f(11)
f(11)
01
f (11)
00
(¥i-)
f({g-)
f(1O)
or
f(lO)
00
f
F ..
~1.J
=
10
(4)
ft11l )
f (01)
10
f (01)
01
f(Ol)
00
00
f(ll)
f(OO)
f(OO)
f (00)
00
10
or
ij
(Subscripts and superscripts are moved outside parentheses) braces)
5
etc. whenever those subscripts and superscripts are common to all
enclosed parameters.)
With marginal frequencies evaluated for each pair) the elements of
~
-XIX
can b e eva 1uate d .
The i-.th d'~agona1 e 1ement ~s
.
= [mp (1
- mp) + fP (1 - fP) + 2HJ.~ )
(5)
and the i) jth off-diagonal element is
(6)
The parameters in equations (5) and (6) are
.
= f (1:.). =
(7)
~
the paternal gametic gene frequency at locus i)
(8)
the maternal gametic gene frequencies at locus i) and
H
i
= [f (})
f(~)
°
f(~ f(~)
1 Ji
°
=
Cov ( X.) fXi)
m ~
D.. = [f (11) f(OO)
m ~J
f (10) f (01)] ..
.) f (.:..:...)
fD ~J
.. = [fCIT
00
f (N) f (irf) Jij = Cov (fXi) fX j )
f(~)
1R~J
.. = [f (1:.:..)
.1
.0
f (1:.:..) f (Oi) ]. . = Cov ( X,) fX )
.0
j
m ~
.
~J
2R~J
.. = [fCr 1) f (.:E-)
0.
. = Cov (fx:
X.)
f(O:) f(io)].
. ~J
V mJ
..
~J
Cov ( X,) X.)
m~ m J
(9)
6
intergenic covariances defined in a manner analogous to the determinant
(~.~.,
definition of disequilibria
see Li, 1955).
For identically
distributed parental gametic sets, F .. } equation (4), is symmetric
~~J
around the main diagonal; consequently
fPi
mPi
==
Pi
(10)
}
D..
D..
m ~J == fD ij == ~J
,
(11)
and
R..
R
2 ij == ~J
lRij
(12)
The D type covariances are the usual disequilibria between loci
within parental gametes;
disequilibria.
~'~'}
they are intragametic-interlocus
The other types measure non-random associations of
gametes (see the assortative mating parameters defined by Wright, 1921)
which have expectation zero under random mating, the R types being
intergametic-interlocus disequilibria and the H type being an
intergametic-intralocus disequilibrium.
Parameterization of the Gametic Variance and the Covariance
between Gametes and Parental Zygotes
For the gametic model presented in equation
2
()G*
in which
Lx*IX*
~
Lx*IX*
a
(2)~
the variance is
l
(13)
is the variance_covariance matrix of X*.
As with
LxIX'
the marginal frequencies for each pair of loci must be specified in
order to evaluate the off-diagonal elements of
Lx*IX*'
From these
frequencies the marginal frequencies for each locus can be obtained and
7
the diagonal elements of
~*IX*
determined.
Recall that the marginal
gametic frequencies for loci i}j are
. th d'~agona 1 e 1 ement
Th e ~--
0f
y'L
.
-X*'X ~s
Var (X~)
= p~ (1 - p~)
~
~
~
. . th
an d t h e ~}J--
0
where
(14)
f(l). }
~
ff - d'~agona 1 e 1ement ~s
.
*
D.•
~J
[r(ll) f(OO) - r(10) f(Ol)] . . .
(15)
~J
For the parental zygotic array having density function
f(~)}
the
marginal gametic frequencies for locus pair i}j are given as functions
of f(z) .. and c .. in equation (16) wherein c .. is the frequency of
~
~J
~J
recombination between the two loci.
~J
e
e
e
r(11)
r(lO)
I
r:. =
-1.J
r(OI)
r(OO)
I
ij
f(11)
11
[f(11)+f(lO)J
10
11
[ f ( 11) +f (0 1) J
01
11
00
11
01
10
[ (I_c) {f (11) +f (00) +c {f (10) +f (0 1) } J I
f(lO~
11
10
[f(1O~+f(11~ J
00
10
[f (1O~ +f (OO~ J
00
11
01
10
[c {f (11~ +f (OO~ }+( I-c) {f (1O~ +f (0 1~ } J
f (0 1)
01
[f (00) +f (01) J
01
00
[ f ( 11) +f (0 1) J
01
11
00
11
01
10
[c(f(rr)+f(OO) }+(I-c) (£(10) +f(01)} ]
f(OO)
00
[f (00) +f (0 1) J
(IT
00
00
10
[f (10) +f (00) J
00
11
01
10
[ (l-c) (£ (11) +f (00) }+c{ f (10) +f (OT) } J ..
J
10
I 1
1
"2
=1
1
I
(16)
"2
1
2
(Xl
9
With the gametic and parental zygotic frequencies so related} the
gametic parameters in equations (14) and (15) can be expressed as
functions of parental zygotic parameters;
~'~'}
1
p:'"~ = -2 ( mp + fP) ~ }
(17)
.L
and
(18)
If the previous generation were formed by the pairing of identically
distributed gametic sets} then} from equations (10) through (12)}
(19)
and
D~.
~J
= [(l-c)D
(20)
+ cR]., ,
~J
which corresponds to the expression given by Crow and Kimura (1970).
Parental disequilibria that contribute to parameters in the gametic
distribution (e,g.) D.. and R.. ) are referred to here as contributory
- -
~J
~J
disequilibria; whereas those that do not contribute
(~'~')
Hi) are
referred to as non-contributory disequilibria.
Redefining the gametic distribution as a function of pt} p;} and
*
D .. }
~J
10
*
* * + D..
PiP
j
1.J
**
Pi
qj
r~ j
-1.
*
- D..
1.J
,
::
* *
qiPj
-
D~ .
1.J
for q~ =
1.
*
l~p.,
1.
qj*
l_p~
J
,
(21)
* * + D~.
qiqj
1.J
the limits on D~. can be found.
1.J
Since no element of the above vector
can be lessthan.O nor greater than 1,
. ( ,/( * q.q.
* *) < D*.. < m1.n
. (p.q.,
* * q.p.
* *)
~m1.np.p.,
1. J
1. J
-
1.J -
1. J
which are the limits given by Lewontin (1964).
(22)
1. J
(It should be pointed
out that limits for zygotic disequilibria could have been defined in an
analpgous manner.)
~*
The lower limit on the average disequilibrium, D , is determined
by one of two restrictions:
The first being that the variance of the
number of favorable alleles cannot be less than zero, ~.~.,
!.~* Ix*
!.I
2: 0 ;
and the second being the restriction imposed by equation (22) for all
pairs of loc:/..
With these restrictions, the lower limit on the
average disequilibrium is
(23)
where
~
fJ* = ifj
*
D ••
1.J
n(n-l)
11
In the additive model the covariance between gametes and their
parental zygotes (which corresponds to the parent-offspring covariance
if the parents are randomly mated and unrelated) is half the genotypic
variance even in the presence of disequilibria and linkage;
~lX*
wherein
1
'2
~.~.,
2
(24)
<J"G
.0.
is the covariance matrix between X and x'''.
The i
th
diagonal element and the i, j th off diagonal element of L: I X~~
X
(Cov (X.X~) and Cov (X.X":), respectively) are half the corresponding
~
~
elements of
~
~IX
J
(equations (5) and (6), respectively).
Higher Order Disequilibria and Genotypic Central Moments
Referring to disequilibria between two genic sets
(~.~.,
H, D, and
R type covariances) as digenic disequilibria, disequilibria involving
three or more genic sets can be defined in terms of central moments
1
(Burrows, 1970).
And, as in the case of digenic disequilibria, there
are both contributory and non-contributory types.
Non-contributory
disequilibria are, like the H type, measures of genic associations
involving both genes at a particular locus for one or more loci.
Since
such genic associations do not exist in gametes, they can only be
reinstated from one generation to the next by some system of assortative mating or selection.
All genes involved in associations specified
by contributory disequilibria are located at different loci and
contribute to disequilibria in subsequent generations.
For zygotes formed byidentica11y distributed gametic sets,
non-contributory trigenic disequilibria are of the form
1
Burrows, P. M. 1970. Complete characterization of genic disequilibrium for two loci in diploid zygotes. Unpublished paper.
Department of Statistics, NOrth Carolina State University at Raleigh.
12
T.2.
~
~
J
E([ mX.~ - E(mX.)J[fX,
- E(fX.)J[
~
~
~
mX.J - E(mX.)]}
J
which, when expanded, yields
T
.2.
~
~
J
~
f(fr)ij
p.D ..
11
f(r)'
• ~J.
PiD ij
~
Pi Rij
~J
-
PiRij
-
2
PjHi
p.p.
~ J
PjHi
PiP j
2
(A locus subscript with a superscript "2" indicates that both genes at
that locus are involved in the disequilibrium in question.)
Anal-
ogously, contributory trigenic disequilibria,
(25)
are of the form
derived intragametically, and
T.~xJ'k
Tijxk
T.
. k
~xJx
~
~
~
:-IT ijk
PiDjk
- p.R·
J k
-
PkRij
-
PiPjPk
i ). 'k
PiRjk
-
-
PkDij
-
PiP jPk
f(ll·l)··k
. . ~J
p~RJ'k
- p.D·
.. - p.P.P
,
J ~ k - PkR ~J
~ J k
f
(1. .)
f (11
..
derived intergametically.
~J
~
~
PjRik
(An "x" between subscripts of two loci
indicates a change in the parental origin of the associated genes at
those two loci.)
13
Similarly, quadragenic disequilibria can be defined in terms of
central moments; for example the non_contributory quadragenic disequilibrium involving only loci i and j is
Q 2 2 ::: E{[ X. - E( X.) ] [ fX , - E(fX , ) J[ X. - E( X,) J[ fX' - E(fX , ) ] }
..
m~
m~
~
~
mJ
mJ
J
J
~
J
2p,T 2
J i
j
- 2p~PJ' D..
...
~J
Other non-contributory quadragenic disequilibria are Q
.2 J'k
and Q 2
~
, .JX k
'
~
and contributory types are of the form
•
(26)
Still higher order disequilibria exist for n > 2 and can be defined
in
~erms
of central moments.
The functions relating higher order gametic disequilibria to
parental zygotic disequilibria are analogous to those for digenic
disequilibria, equation (20).
For example, the following relations
hold for contributory trigenic and quadragenic disequilibria:
(27)
* ::: ~ijk~ijk~
Qijk~
in which
(28)
14
C"
-~J
k
= [C(OO), C(lO), C(Ol), C(ll)]"k
~J
C(llO), C(101), C(Oll), C(l11)]ijkt
are vectors of joint frequencies of recombination between the loci
indicated by the subscripts.
A "1" and "0" indicate the respective
states of recombination and of no recombination between a given pair
of loci.
The order in which the number appears in the argument
corresponds to the order in which that given pair of loci appears in
the subscript; e.g., C(010). 'k g is the joint probability of no re- -
~J
'1.1
combination between pair i,j, recombination between j,k, and no
recombination between k,t.
With higher order disequilibria specified, correspondingly higher
order genotypic central moments can be derived; for example, the
.
genotypic third central moment for the parental zygote is
3
(p.q.+3H.)
j.L[3]G ~ 2{L a.(q.-p.)
~
~
~
~ ~
~
2
+ 3
L a.a.[(q._p.) (D .. +R .. ) + T 2 ]
ifj ~ J
~
~
~J
~J
i j
+ 4
L
a.a.ak(T·' k + T. 'k + T.. k + T. . k)}
i>j>k ~ J
~J
~XJ
~JX
~XJX
and for the gametes
* * **
= { ~ a.(q~_p.)p.q.
3
.
~
~
~
~
~
~
+ 3
+ 4
L a . a . a T..
* k} .
i>j>k ~ J k ~J
15
Higher genotypic central moments become cumbersome expressionsj
nonetheless, once all the gene frequencies and pertinent disequilibria
are defined, all the central moments can be derived.
For the purpose
of evaluating the genotypic variance, only the gene frequencies and
digenic disequilibria need be specified.
However, when dealing with
selection systems, higher order disequilibria do affect the skewness,
kurtosis, etc. of the unselected population.
Therefore, the gene
frequencies and digenic disequilibria in the selected population will
be complicated functions of the gene frequencies and all types and
orders of disequilibria in the unselected population.
The Selection Model
At generation t, mass truncation selection is to be imposed on the
phenotypic value
yet) = y + G(t) + e .
(29)
G(t), the genotypic value at generation t, is defined in equation (1)
for the generation indicatedj e is a normally distributed environmental
effect with mean zero and variance
~;, ~~ being constant for all G(t)
and tj G(t) and e are independentj and
Y
= E(y(t)
[G(t)
= 0)
is constant for all t.
The population is assumed to be large enough that a deterministic
approach can be used.
The truncation point is always determined so as
to give q constant proportion, a, saved.
Letting w(g) be the
16
probability that an individual with genotypic value g is selected, then
the following relation holds:
L: w(g) (t) P(G=g) (t) = Ot.
g
(30)
In cases involving heritabilities of less than 1,
w(g) (t)
=1
-
F(Y6 t ) Ig)
(31)
(t)
Yo
Jy=_oo
Y6 t )
being the point of phenotypic truncation at generation t.
Since w(g) (t) is determined by truncating a normal distribution, the
selection system is not additive on the fitness scale even though the
genotypic values are controlled by an additive genotypic model.
cases for which the heritability is 1 (~2
e
= 0),
In
the fitnesses
associated with the genotypic classes are
w(g) (t) =
= gl (t)
,
(32)
gl(t) being the genotypic value for the class satisfying the equation
[1jI(gl) P(G=gl) + L:
P(G=g)] (t) = Ot
g>gl
(33)
But regardless of the value of the heritability, the probability
that an individual is selected given that it has zygotic array
17
x
m-
f~
is equal to the fitness of its genotypic value;
w(~) (t)
= w(g) (t)
if
a(
Xl
- m-
+
Xl)
f-
Therefore the probability density function of
~.~.,
= g .
~
conditional on gis the
same in the selected population as it is in the unselected population.
The proportion of individuals that are selected given that they
have the particular zygotic array
z.
.
k
~1J ...
at the specific subset of r loci (i, j, ... , k} is
(t)
w(z)
..
k =
~ 1J ...
w(z) (t) f(zl~..
L;
. . ... k
,.., e S(v 1J
~
~
1J ...
k) (t)
Z
Consequently, the marginal fitnesses associated with the above loci are
functions of gene frequencies and disequilibria involving all other
loci.
The Effect of Selection on the Genotypic Variance
Denote the marginal fitnesses of the various zygotic arrays for
(t)
loci i,j at generation t by W.. ,
~1J
18
1
W~~) = =(t)
~~J
w
11
w(ll)
11
w(lO)
11
w(Ol)
11
w(OO)
10
w(ll)
10
w(lO)
10
w(or)
10
w(OO)
01
01
w(lO)
01
w(or)
01
w(OO)
00
00
w(lO)
00
w(Ol)
00
w(OO)
w(ll~
w(l1~
(t)
ij
in which ;; (t) is the fitness of the whole population at generation t}
;;(t) =L:[w(~) f(~)J(t).
z
~
Therefore} the zygotic frequencies at loci i}j in the selected population at generation t is
(s}t)
(t)
(t)
F..
=W .. • F ..
~J
~~J
~~J
}
the dot product representing the element by element multiplication of
W~:) and F~:)} and the presence of the superscript "stl indicating that
~~J
~~J
the frequencies apply to the selected population (otherwise) they apply
to the unselected population).
It
is assumed that the test (unselected) population is formed by
random pairing of identically distributed gametic sets which are
produced by the selected population in the previous generation; there-
F~~+l) = r(t+l)
~~J
-ij
t
r(t+1)
-1j
}
r(t+l) being the marginal frequencies of gametes produced by population
-ij
With symmetry of F~~}t) around the main diagonal) it can be
(8) t) .
~~J
19
shown from equations (3)) (5)) (6) and (10) through (12) that the
variance of the selected population is
(34)
and that of the progeny is
2(~
i
2
a.p.(l_p.)
+
~ ~
~
~
ifj
a.a. D • . }(t+1)
~
J
~J
(35)
in which
(36)
from equation (19) and
D (t+1)
(37)
ij
from equation (20).
With relaxation of selection after generation t
and subsequent random mating for v generations) the disequilibrium is
D(t+v)
ij
Therefore) the limit of the genotypic variance as v goes to infinity is
2 (t+v)
lim erG
v-oo
2
= lim 2(~ a.p.(l_p.)
V-oo
i ~ ~
~
= 2 ~.
~
+ 2
a.2( p.(l_p.)} (s)t)
~
~
~
~
ifj
for
a.a.D .. }
(t-tv)
~ J ~J
0 < c .. < 1
~J
(38)
which is the variance under intergenic independence.
The functions relating the contributory parameters in the selected
group to those in the test population are far from being simple;
20
= p ~ t) + lip ~ t)
~
D(s) t)
ij
(39)
~
D.•
= (_lip. lip. +.2:l. [p.P. w(OO) .. + p.q. w(Ol) ..
~
J
W ~ J
~J
~ J
~J
+ q.p. w(lO) .. + q.q. w(ll) .. ]
~
J
~J
~
J
(40)
~J
(Appendix A) )
in which
lip ~(. t) = =(t)
1 ( qi [f (11) w(ll
- ) + f (10) w(lO)
] ..
w
~J
_ p. [f(O 1) w(O 1) + f(OO) w(OO)] . .} (t)
~
(41)
~J
and w(x*)~:) is proportional to the marginal probability that a gamete
-
~J
with genotypic array x~. at loci i and j forms a zygote which is a
-~J
~ember
of the selected population at generation t.
The intergametic-
inter10cus disequilibria in the selected population is
1
11
00
10
-(t) ([w(OO) + w(rr) - w(Ol)
2w
01
10
01
(t)
- w(lO) ]T(ll)T(OO) + [w(Ol) + w(lO) ]D} ij )
(Appendix B)
In the absence of intergenic independence} the above equations
are extremely difficult to manage since w(z)~:) and w(x) ~~) are
~
~J
-
~J
(42)
21
functions of gene frequencies and all orders of disequilibria involving
all loci.
However) with complete independence in the initial test
popula tion)
11\ (0) _ (00\ (0)
w( 00:1 ij - w Tr ij
= (10\ (0) = (01\ (0)
w 01:1 ij
W
10:1 ij
and
D~O)
1j
= 0 .
Therefore) equation (42) reduces to
= D(s)O)
ij
)
and from equation (37)
D~~)
1J
=
(l-c .. ) D~~)O) + c .. R~~)O)
1J
1J
1J 1J
which is independent of the frequency of recombination.
In fact all disequilibria and gene frequencies are independent of
linkage at t = 1.
The degree of linkage comes into play only when
recombination takes place.
Therefore) since the initial population is
in equilibrium at t = 0) the disequilibria and gene frequencies in the
(s~O)
population must be independent of linkage.
And since
(s)O)
- Pi
,equation (36») the gene frequency at t = 1 is also
(1) _
Pi
independent of linkage.
Further, under intergenic independence at
t = 0) all marginal fitnesses are equal for a specific set of r loci
(i) j, ... ) k} having xi) x.) ... , x k favorable alleles) regardless of
J
22
their coupling-repulsion relations.
With all initial disequilibria
equal to zero) these equal fitnesses operate to produce equality among
all contributory r-genic disequilibria at those loci in the (s)O)
population.
Consequently) the intragametic contributory r-genic
disequilibrium at t
=1
for that set of loci is independent of linkage.
This holds for all (i) j) ... ) k} and all r.
With the genotypic distribution at t
frequency of recombination) p~2)
~
linkage.
= 1 not a function of the
= p~s)l) is also independent of
~
But since all intergametic contributory disequilibria are
zero in the test population at t = 1 (due to random mating)) and all
intragametic disequilibria are non_zero (due to selection in the
previous generation)) selection operates differently on the r_genic
contributory disequilibria at loci i) j) ... ) k depending on whether
the disequilibria are intergametic or intragametic in nature.
With
such differences in the (8)1) population) recombination affects the
disequilibria in generation 2.
From this point on gene frequencies and
disequilibria are functions of the frequency of recombination.
In summary) as long as the fitnesses are not dependent on couplingrepulsion relations) gene frequencies are independent of linkage for
t < 2) and all disequilibria are independent of linkage for t
= 1.
The remainder of this study is directed toward empirically
evaluating the response of the genotypic variance to selection and
toward the development and evaluation of projection equations for the
genotypic variance.
23
THE EFFECT OF SELECTION ON GENOTYPIC VARIANCE
AND GENE FREQUENCY
Empirical Methods
For the selection scheme presented, the genotypic fitnesses in any
generation are functions of all gene frequencies and contributory
disequilibria of all orders; consequently, the determination of the
effects of selection and recombination becomes a formidable task.
For
this reason numerical evaluations were used, in lieu of attempting
further analytical derivations, to determine the exact results for an
infinite population.
Due to computer limitations, the number of loci
considered was kept small, n < 5.
A program was written in which each zygotic array was identified
according to its allelic composition for each gene site at each locus
and was also identified as to its genotypic value.
The initial
frequency assigned to the zygotic array
was the expected frequency under intergenic independence,
f(~) (0)
n
f
n
n
(43)
i=l k=m
In each generation the frequency of each genotypic value was
determined by adding the frequencies of all zygotic arrays which shared
that genotypic value.
The selected group was then formed by retaining
a specified upper proportion, a, of the entire population.
The
fitnesses assigned to the genotypic values satisfied equation (31)
24
for
heritabi1iti~s
less than 1 and equation (32) for heritabilities
equal to 1.
Ignoring superscripts, for heritabilities less than 1 where
w(g)
= l~F(Yolg),
an iterative technique was used to approximate the
. by
point of phenotypic t;1;'uncati,on, YO)
th~
*
value YO'
Iteration was
continued until
L;
[1 - F(y~lg)]p(G=g) _
01
1<
.0001 .
g
F(y~lg) was itself approximated for each g by the function
01;' by
in which
= [Y~
t
= .(l
....
1
-:---..,..,..."..'7"7"'~
=
+ . 23164191ZQTJ
0.319381530
= -0.356563782
b .
3
=
L 781477937
25
b
4
b
=
5
-1.821255978
1.330274429
(reference, Handbook of Mathematical Functions, 1968, page 932), the
absolute value of th~ deviation of the approximated F(y~lg) from the
actual probability being less than 7.8 x 10- 8 .
In the case of heritabilities of 1, the genotypic value, g', was
found which satisfied equation (33) and which yielded the fitnesses
given in equation (32).
Once w(g) was found for all values of g, the zygotic frequencies
in the selected population were determined (recall that
~
has genotypic value g).
w(~)
=
w(g) if
The gametic frequencies for specified
linkage conditions were then computed, and the zygotic frequencies in
the new test population evaluated assuming random pairing of gametes.
The selection procedure was then repeated until 55 generations had
lapsed or until the average gene frequency (averaged over all loci)
attained 0.999.
The following additional restrictions were imposed on the program:
(a) equal additive effects for all loci;
(b) a single chromosome;
(c) equal recombination frequencies between adjacent loci; and
(d) non-interference between recombining pairs.
Parameters and Measures Used
In order to evaluate the effect of initial gene frequency, initial
heritability, frequency of recombination, intensity of selection, and
number of loci, a complete factorial combination of the following
26
levels were used for equal initial gene frequencies at all loci:
p
(0)
(initial gene frequency) = .3, .5, .7;
p2(0) (initial heritability)
=
e
25,
0
75} 1 j
c, '+1 (frequency of recombination for adjacent loci) = .5, .1,0;
1., 1.
~
(proportion retained per generation) = .1, .5, .9;
n (number of loci) = 2, 3, 4.
The locus numbering, I, 2, ... , n, represents the sequential positions
of the n loci on a chromosome.
Additional runs were made which in_
volved some of the above levels together with some of the following
levels:
.p(O) = .1, .9;
.2, .3, .4, .6, .7, .8;
n
- 5 .
Other ·special runs were made and will be discussed later.
Besides the runs already described, runs involving 3 and 5 loci
were made for unequal initial gene frequencies over loci.
The initial
gene frequencies were chosen to be symmetric around the value
.5 so as
to yield the same variance in gene frequencies over loci for both
values of n.
With symmetry around .5, the mean and third central
moments of gene frequencies over loci were also the same,
respectively.
For five loci
.5 and 0,
27
(0)
(0)
(0)
(0)
.3) P3
• 1) P2
Pl
.5) P4
(0)
• 7) P5
:::
.9)
and for three loci
(0)
Pl
for both n
:::
(0)
.5 - '{:T2 )
P2
:::
.5)
(0)
P3
.5 + \{:Tf ;
3 and 5
L: (
(J2(0) ::: _i
P
-)2(0)
PCP
_
.08 .
n
There is no attempt here to suggest that the above levels generate
representative points on the response surfaces in question.
These
levels were simply designed to indicate possible trends in the response
variables and to aid in the evaluation of the projection equations for
the genotypic variance which will be presented later.
With the initial heritability) initial gene frequencies) and
number of loci specified) the per locus contribution of a favorable
allele relative to the environmental standard deviation) ~) is
(Je
fixed.
Since
zJ(O)
2(0)
P
(Je
it can be shown that
:e : :
I
Z(O)
~ -Z-(1 -_-pnZ:"7("'O):-)...;..P-L:-P"--;~"'O') -(l-_-p"--;;>'AO'))
i
a
~
(44)
.L
In all cases investigated --was relatively large) its smallest value
(Je
28
being attained in runs involving n = 5, p(O) = .5, and p2(O) = .25 for
a _~ .365.
It should be pointed out that for a heritability of
wh ich -O"e
1, equation (44) is undefined. However, for p2(O) = 1, the effect of
retaining a constant proportion on the gene frequencies and disequilibria is independent of the value of a.
Consequently, those runs
involving p2(O) = 1 provide information as to the asymptotic effect of
~ for specific values of
O"e
For specified values
a and c .. , the effect of
~J
selection on gene frequencies and disequilibria is invariant with
a
respect to the value of O"e so long as -- corresponds to the definition
O"e
in equation (44). Therefore, the systems investigated were treated as
though the environmental variances were constant and as if the per
locus additive contribution, a, varied in response to changes in the
parameters of equation (44).
With
0";
held constant, it was of interest
in some comparisons to vary n in equation (44), thus changing the value
of a in such a way as to keep the initial genotypic variance constant.
In other comparisons two parameters were varied simultaneously in such
a
a way as to keep -- constant.
O"e
In comparing the effects of selection for the various parameters
presented earlier, several measures will be referred to.
In addition
to gene frequencies and digenic disequilibria, the following will be
used:
D (t)
ij ,
(45)
the change due to selection in the intragametic digenic disequilibrium
within generation t; and
29
n(n-1) f>
L: p.q.
.
~
~
I
~
(t)
(46)
the absolute value of the proportional reduction in the genotypic
variance attributable to disequilibria) the actual variance in terms of
M(t) being
= 2a
2
(t)
(47)
(L: p.q.(l-M)]
~
(refer to equation (35».
~
Another measure which will be referred to in
some detail is the lower limit that j)(t) can attain (see equation (23»
and the relation of the actual value of n(t) to that minimum.
Some of the results are summarized for given selection runs at the
quarter life) half life) and three-quarter life points which are the
generations at which the average gene frequencies (averaged over loci)
had progressed one_fourth) one-half) and three-fourths the distance
from the initial average gene frequency to fixation.
Since the se1ec-
tion system operates on discrete generations) the measures given for
each of these points are linearly interpolated values based on the
average gene frequencies of the two generations bracketing the" desired
point.
General Results
By retaining
th~
same proportion in each generation) the point
of truncation shifted upwards from one generation to the next.
resulted in the fitness coefficients changing over time.
This
The manner in
which they changed sometimes caused disequilibria responses to exhibit
local minima over generations.
In runs involving a heritability of 1)
30
abrupt changes in the disequilibrium response curves corresponded to
points of loss of certain gametic types; whereas, for lowheritabilities, such local minima were rare.
As is well documented for the D type disequilibria, truncation
selection induced negative values for all types of digenic disequilib_
ria in the selec;ted population. . Therefore, as can be seen from
equation (9),
<
[f(l1~ f(OO)) ~~, t)
•.
..
~J
[ f (.!.:..) f (£.:..) ]
.0
.1
~.~ ,t)
~J
<
which, in words, says that knowing the allelic type at a given gene
site in the selected population increases the likelihood that the
allele at another specific site is of the opposite type.
For the Rand
H type diseqUilibria, no accumulation over generations was possible
since these were intergametic types and as such could not persist
through the gametic stage.
Random mating ensured that the Rand H
types were not reestablished in the test population.
But for the D
type disequilibria in the test population, the general form of the
response curve waS for D to decrease from zero to some minimum and then
r~turn
asymptotically to zero as fixation was approached.
responses for D type covariances in test populations
ha~e
many simulation studies involving directional selection
Hill and
Robe~tson,
Negative
been noted in
(~.~.,
see
1966) and have been proven theoretically for the
31
two locus model ullder the selection scheme presented here (Felsenstein,
1965).
Certain relations existed amongst the digenic disequilibria in the
selected populations.
Regarding interlocus digenic disequilibria
the following relation always held:
D~~' t)
1.J
(s, t)
··
ij
$
< 0
R
equality attained only at t
=0
(48)
or whenever both disequilibria attained
. ( p . p ., q. q..
).(s,t) .
Inequality (48) was
their lower limit, -m1.n
1. J
1. J .
expected since the R type disequilibria were always zero in the unselectcrd population whereas the D type were always negative except at
t = 0 or at fixation.
Another relation between digenic disequilibria
which usually held was the approximation
\f'H:H:
R (s, t)
ij
(s, t)
(49)
1. J
which exactly reflected the relatiolls of the lower limits of R~~' t)
. ( p.p.} q.q.. ) (s,
( ~m1.n
t» .
1.J
(s t)
(s t)
.
2 2 (s t)
to those of H.'
and H.'
(_m1.n (P., q.) ,
1.
J
1.
1.
1.J
1.J
.:
(2
an d -m1.n p ., q.2)·(s, t) , respectively) for p.,
.1.
J
J
both less than
.5.
p. both greater than or
J
The following iQequality almost always held:
< R-(s,t)
< 0 .
(50)
For Hi S 0 for all i, inequality (50) can be shown to follow directly
from equation (49) i f equation (49) were to hold exactly for all i, j.
With accumulation of the D type covariance
32
5(s,t)
< ii(s,t)
< 0
(51)
except for the first generation and near fixation in some runs.
It should be pointed out that any combination of parametric levels
that produced extremely rapid progress towards fixation resulted in a
negligible buildup in disequilibria.
Under such conditions, even i f
the minimum possible disequilibria were attained, the proportional
reduction in the genotypic variance attributable to disequilibria
(equation (46»
would be limited since the maximum negative correlation
between any given pair approaches zero near fixation.
Letting Irijl
represent the absolute value of the correlation coefficient between
loci i,j, the limit on Irijl for Dij negative as fixation is approached
is as follows:
lim (maxi r ..
~J
p._l
D ..
(max I
~J
p._l
V'Pi qi P jqj
~
p.-l
I} = lim
~
p._l
J
I}
J
=
lim (
p.-l
~
min (p . p . , q. q .)
~
J
~
J
VPiqiPjqj
p.-l
J
=
lim
p.-l
~
p._l
J
o.
}
for
D..
~J
<
0
33
The Effect of Various Parameters on Gene Frequency)
Disequilibria) and Genotypic Variance
In this interacting system) it is difficult to evaluate the effect
of anyone parameter without specifying the levels of the other
parameters.
Thus) although the general effects of each of the various
parameters are presented separately) it is often necessary to make
conditional statements regarding the contribution of the parameter in
question.
The effects of linkage) intensity of selection) number of
loci) initial heritability) and initial gene frequency will be
discussed in separate sections for runs involving equal initial gene
frequencies.
This will be followed by a discussion on the effect of
selection on populations for which the initial gene frequencies varied
over loci.
The Effect of Linkage
Tighter linkage resulted in a greater buildup in disequilibria
which in turn reduced the genotypic variance and retarded progress.
Starting with equal initial gene frequencies) for given values of
2(0)
n) p ) p
(0)
an
d
(1)
(1)
D..
~J
(1)
was constant for all i)j and p.
~
(2)
and p.
~
were constant for all i; as pointed out earlier) these measures were
independent of linkage.
In population (s)l») the following relations
held:
H~s)l) = R~~) 1) = R(s) 1)
~
~J
for all i
(52)
and
D~~) 1) = D(s) 1)
~J
for all i)j; i
f
j
(53)
34
Recall from equation (48) that
(54)
except for initially or near fixation in some runs.
Therefore}
since
= (1
) ($}l)
c .. R(s}l)
- c ij D
+ 1J
}
(55)
the disequilibria were greater for runs involving tighter linkage.
The magnitude of the negative
dis~quilibria
continued to be
greater under tighter linkage for the duration of the selection
program.
This
r~sulted
in larger reductions in the genotypic variance.
Quarter} half} and three-quarter life values of the proportional
reductions} M(t)} in the genotypic variance due to disequilibria
are given in Table 1.
Comparisons made across different values of
c . . +1 for constant values of p(O)} p2(O)} n} and ~ reveal that the
1}1
reduction due to disequilibria was consistently greater for tighter
linkage except for runs in which the particular life occurred prior to
or at generation 1 (in which case O(t) and p(t) were independent of
linkage) or for runs in which fixation was rapidly approached and the
minimum value of o(t)} equation (23)} was realized immediately after
genera tion 1.
The magnitudes of increase in M(t) with increased tightness in
linkage depended upon the levels of the other parameters.
For runs
which rapidly approached fixation} the effect of a decl;ease in c . . +1
1} 1
on M(t) was negligible or absent.
But for runs that went sufficiently
beyond the first generation prior to nearing fixation} increasing
Table 1.
Proportional reductions J M(t) j
given in parentheses)
O!
0'
= .9
.039
(8.039)
= .5
= .1
..
c=o
n=3
n=4
.050
.056
(9. 714) (11.138)
.146
.111
.139
(8.196) (10.036) (11.583)
.159
.232
.278
(8.279) (10.324) (12.154)
1/2
-.65 _
.042
.052
.056
(15.041) (18.397) (21.224)
.154
.162
.160
(15.899) (19.703) (22.780)
.297'
.427
.372
(16.907) (22.081) (26.843)
3/4
-.825-
.032
.040
.044
(22.805) (28.240) (32.808)
.104
.123
.123
(24.726) (30.831) (35.743)
.194
.340
.425
(27.810) (38.034) (47.216)
1/4
-.475-
.
in genotypic variance due to disequilibria (generations
Part a. Initial heritability = .25; initial gene frequency = .3
L1te
U
p2 t U) = .25, pt ) = .3
-Average
c-. )
c-.1
Gene
n=3
n=4
Frequencyn=2
n=2
n=3
n=4
n=2
Proportion
Retained
O!
e
e
e
1/4
-.475-
.067
(1. 828)
.092
(2.263)
.104
(2.631)
.085
(1. 828)
.124
(2.269)
.147
(2.647)
.089
(1.828)
.136
(2.272)
.169
(2.653)
1/2
-.65 -
.078
(3.582)
.099
(4.474)
.108
(5.220)
.138
(3.634)
.185
(4.599)
.205
(5.418)
.159
(3.650)
.234
(4.660)
.283
(5.559)
3/4
-.825-
.054
(5.642)
.071
(7.074)
.078
(8.296)
.109
(5.823)
.153
(7.487)
.169
(8.845)
.134
(5.893)
.220
(7.757)
.279
(9.434)
1/4
-.475-
.052
(0. 783)
.083
(0.977)
.097
(1.141)
.052
(0.783)
.083
(0.977)
.102
(1.141)
.052
(0.783)
.083
(0.977)
.103
(1.141)
1/2
_.65 _
.071
(1.589)
.107
(1.966)
.119
(2.322)
(1. 589)
.136
(1.966)
.159
(2.331)
.086
(1.589)
.145
(1.966)
.178
(2 0336)
3/4
_.825_
0056
(2.574)
.080
(3.208)
.090
(3.766)
.075
(2.574)
.120
(30247)
.145
(3.831)
.080
(2.582)
.138
(3.262)
.180
(3.866)
..
,
.083
.
continued
w
VI
e
e
e
Table 1 (continued)
Proportion
Retained
O! = .9
QI
Of
= .5
= .1
Part b. Initial heritability = .25; initial gene frequency = .5
Life
pZ(U) = .25, p(U) = .5
-Average
c=.l
c=.5
Gene
Frequencyn=2
n=3
n=4
n=2
n=3
n=4
n=2
1/4
-.625-
.036
(5.231)
.046
(6.473)
.050
(7.527)
.079
(5.297)
.104
(6.617)
.114
(7. 732)
.099
(5.322)
c=O
n=3
n=4
.146
(6. 704)
.180
(7.910)
1/2
-. 75 -
.045
.033
.042
(10.760) (13.427) (15.672)
.103
.121
.124
(11.220) (14.171) (16.594)
.162
.235
.282
(11. 547) (15.038) (18.166)
3/4
-.875-
.023
.031
.029
(17.712) (22.316) (26.190)
.068
.085
.088
(18.790) (23.926) (28.076)
.117
.203
.265
(19.842) (26.789) (33.036)
1/4
-.625-
(1. 250)
.046
(1. 558)
(1. 817)
.050
(1. 250)
.077
(1. 558)
(1. 817)
(1. 250)
.081
(1. 558)
(1. 817)
1/2
-. 75 -
.054
(2.661)
.073
(3.347)
.082
(3.917)
.078
(2.671)
.115
(3.382)
.137
(3.977)
.085
(2.674)
.135
(3.396)
.171
(4.010)
3/4
-.875-
.036
(4.536)
.049
(5.727)
.055
(6. 733)
.064
(4.597)
.096
(5.875)
.111
(6.965)
.075
(4.616)
.127
(5.951)
.164
(7.163)
1/4
-.625-
.028
(0.577)
.047
(0.703)
.062
(0.810)
.028
(0.577)
.047
(0.703)
.062
(0.810)
.028
(0.577)
.047
(0.703)
.062
(0.810)
1/2
-.75 -
(1. 217)
.067
(1. 524)
.081
(1. 766)
(1. 217)
(1. 524)
.097
(1. 766)
.049
(1. 217)
(1. 524)
(1. 766)
3/4
-.875
.037
(2.064)
.051
(2.648)
.062
(3.059)
.046
(2.065)
.070
(2.655)
.090
(3.083)
.049
(2.065)
.077
(2.657)
.104
(3.094)
.046
.066
.079
.048
.075
.096
.051
.078
.103
.103
continued
UJ
(j\
e
e
e
Table 1 (continued)
Proportion
Retained
O!
= .9
Part c. Initial heritability = .25; initial gene frequency = .7
L1te
p2 (0) = .25, p (0) = . 7
-Average
c-.l
c-.5
Gene
n=4
n=2
n=2
n=3
Frequencyn=2
n=3
n=4
1/4
-. 775-
1/2
-.85 -
3/4
-.925O!
= .5
1/4
-.775-
1/2
-.85 -
3/4
-.925O!
= .1
1/4
-.775-
1/2
-.85 -
3/4
-.925-
c=O
n=3
n=4
.116
(4.885)
.034
(3.196)
.044
(4. 049)
.048
(4.776)
.054
(3.211)
.076
(4.090)
.088
(4.845)
.060
(3.216)
.093
(4.107)
.029
(6.996)
.037
.042
(8.929) (10.558)
.068
(7.174)
.091
.098
(9.263) (11.003)
.090
(7.252)
.142
.178
(9.514) (11.505)
.027
.018
.023
(12.258) (15. 758) (18. 705)
.062
.067
.045
(12. 720) (16.582) (19. 747)
.065
.116
.156
(12.988) (17.487) (21.497)
.050
(0.997)
.062
(1. 186)
.030
(0.811)
.050
(0.997)
.064
(1. 186)
(1. 805)
.072
(2.296)
.089
(2. 714)
.048
(1. 805)
.078
(2.297)
.102
(2. 718)
.042
(4.900)
.038
(3.277)
.060
(4.193)
.074
(4.971)
.042
(3.281)
.072
(4.212)
.096
(5. 015)
.025
(0.511)
.034
(0.580)
.014
(0.430)
.025
(0.511)
.034
(0.580)
.014
(0.430)
.025
(0.511)
.034
(0.580)
.027
(0.861)
.048
(1. 038)
.057
(1. 252)
.027
(0.861)
.049
(1. 038)
.059
(1.252)
.027
(0.861)
.049
(1. 038)
.061
(1. 252)
.022
(1. 612)
.036
(1. 927)
.043
(2.30 1)
.024
(1. 612)
.043
(1. 927)
.054
(2.303)
.025
(1. 612)
.045
(1. 927)
.059
(2.304)
.030
(0.811)
.050
(0.997)
.059
(1. 186)
.030
(0.811)
.038
(1. 805)
.055
(2.292)
.064
(2. 703)
.026
(3.264)
.037
(4.146)
.014
(0.430)
.046
continued
W
""-l
e
e
e
Table 1 (continued)
Proportion
Retained
O! = .9
O! = .5
O! = .1
Part d.
Initial heritability = .75; initial gene frequency = .3
Life
p2(0) = .75; p(O) = .3
-Average
c-.l
c-.5
Gene
n=3
n=2
n=4
Frequencyn=4
n=2
n=3
n=2
-
.135
(5.991)
.145
(6.897)
.238
(5.007)
.297
(6.260)
.301
(7.309)
.292
(5. 038)
c=u
n=3
n=4
.411
(6.480)
.442
(7.759)
1/4
-.475-
.113
(4.928)
1/2
-.65 -
.147
.122
.138
(9.343) (11. 505) (13.317)
.376
.335
.340
(10.244) (12.903) (14.943)
.525
.561
.603
(11. 705) (14.700) (18.835)
3/4
-.825-
.125
.099
.134
(14.015) (17.417) (20.282)
.296
.182
.292
(16.426) (20.304) (23.407)
.213
.424
(19.374) (28.019)
1/4
-.475-
.156
(1. 086)
(1. 345)
.235
(1. 577)
(1. 086)
.234
(1. 345)
(1. 577)
.164
(1.086)
(1. 345)
.297
(1. 577)
1/2
-.65 -
.206
(2.189)
.245
(2. 760)
.262
(3.257)
.281
(2.202)
.367
(2.184)
.397
(3.371)
.301
(2.205)
.420
(2.835)
.479
(3.439)
3/4
-.825-
.148
(3.412)
.202
(4.355)
.224
(5.131)
.186
(3.506)
.310
(4.642)
.363
(5.571)
.196
(3.533)
.370
(4.814)
.511
(5.976)
1/4
-.475-
.088
(0.455)
.142
(0.567)
.180
(0.660)
.088
(0.455)
.141
(0.567)
.180
(0.660)
.088
(0.455)
.142
(0.567)
.180
(0.660)
1/2
-.65 -
.175
(0.910)
.244
(1. 166)
.275
(1. 387)
.175
(0.910)
.248
(1. 166)
.299
(1. 387)
.175
(0.910)
.250
(1. 166)
.309
(1.387)
3/4
-.825-
.113
(1. 491)
.222
(1. 869)
.241
(2.256)
.113
(1.491)
(1. 869)
.293
(2.266)
.113
(1.491)
.249
(1.869)
.317
(2.271)
.208
.162
.634
(35.149)
.242
.278
.243
continued
w
00
-
e
e
Table 1 (continued)
Part e.
Initial heritability = .75; initial gene frequency = .5
Lite
p2(0) = .75, p(O) = .5
-Average
c-.l
c-.5
Gene
Frequencyn=2
n=4
n=2
n=3
n=4
n=3
n=2
,
Proportion
Retained
~
= .9
~
= .5
= .1
n=3
n=4
.313
(4.719)
1/4
-.625-
.106
(3.080)
.129
(3.841)
.139
(4.486)
.176
(3.106)
.220
(3.926)
.241
(4.627)
.199
(3.113)
.265
(3.963)
1/2
-. 75 -
.109
(6.334)
.128
(7.949)
.136
(9.314)
.239
(6.818)
.283
.285
(8.587) (10.156)
.299
(7.110)
(9~136)
3/4
-.875Q'
c=O
.117
.085
.107
(10.102) (12.712) (14.961)
.210
.127
.241
(11.031) (14.337) (16.852)
.431
.464
(11.260)
.143
.284
.419
(11.636) (16.654) (22.239)
1/4
-.625-
.101
(0. 706)
.157
(0.870)
.197
(1.011)
.101
(0. 706)
.157
(0.870)
.197
(1.011)
.101
(0.706)
.157
(0.870)
.197
(1. 011)
1/2
-. 75 -
.146
(1.496)
.203
(1. 892)
.224
(2.257)
.154
(1. 496)
.247
(1. 892)
.288
(2.270)
.156
(1. 496)
.262
(1. 892)
.318
(2.276)
3/4
-.875-
.109
(2.447)
.162
(3.156)
.184
(3. 774)
.120
(2.452)
.206
(3.218)
.260
(3.893)
.123
(2.453)
.224
(3.241)
.310
(3.962)
1/4
-.625-
.041
(0.336)
.083
(0.408)
.110
(0.470)
.041
(0.336)
.083
(0.408)
.110
(0.470)
.041
(0.336)
.083
(0.408)
.110
(0.470)
1/2
-.75 -
.082
(0.673)
.165
(0.816)
.219
(0.939)
.082
(0.673)
.165
(0.816)
.219
(0.939)
.082
(0.673)
.165
(0.816)
.219
(0.939)
3/4
-.875-
.118
(1. 028)
.139
(1. 414)
.174
(1. 659)
.118
(1. 028)
.140
(1. 414)
.183
(1. 659)
.118
(1.028)
.140
(1. 414)
.187
(1. 659)
continued
W
\0
-
e
e
Table 1 (continued)
Proportion
Retained
Ot
Ot
= .9
= .5
Ot = .1
Part f.
Initial heritability = .75; initial gene frequency = .7
Life
p2(0) = .75; p(O) = .7
-Average
c=.5
c=.l
Gene
Frequency
n=2
n=3
n=4
n=2
n=3
n=4
n=2
c=O
n=3
n=4
1/4
-. 775-
.090
(1.839)
.118
(2.335)
.131
(2. 764)
.113
(1. 839)
.156
(2.345)
.181
(2. 787)
.118
(1.839)
.170
(2.348)
.208
(2.798)
1/2
-.85 -
.094
(3.950)
.119
(5.082)
.129
(6.031)
.137
(4.032)
.206
(5.299)
.231
(6.344)
.151
(4.059)
.258
(5.421)
.322
(6.597)
3/4
-.925-
.065
(6.516)
.093
(8.449)
.104
(10.074)
.077
(6.710)
.136
.173
(9.023) (10.932)
.081
(6.777)
.159
.235
(9.429) (11.967)
1/4
-.775-
.057
(0.465)
.093
(0.571)
.121
(0.662)
.057
(0.465)
.093
(0.571)
.121
(0.662)
.057
(0.465)
.093
(0.571)
1/2
-.85 -
.114
(0.930)
.155
(1. 206)
.180
(1. 454)
.114
(0.930)
.157
(1. 206)
(1. 454)
.114
(0.930)
(1. 206)
(1. 454)
3/4
-.925-
.068
(1. 601)
.123
(2.060)
.141
(2.530)
.069
(1. 601)
.133
(2.060)
.162
(2.540)
(1. 601)
.134
(2.061)
.176
(2.545)
1/4
-. 775-
.004
(0.264)
.026
(0.301)
.053
(0.340)
.004
(0.264)
.026
(0.301)
.053
(0.340)
.004
(0.264)
.026
(0.301)
.053
(0.340)
1/2
-.85 -
.008
(0.527)
.053
(0.603)
.107
(0.681)
.008
(0.527)
.053
(0.603)
.107
(0.681)
.008
(0.527)
.053
(0.603)
.107
(0.681)
3/4
-.925-
.012
(0. 791)
.079
(0.904)
.148
(1. 064)
.012
(0.791)
.079
(0.904)
.148
(1. 064)
.012
(0. 791)
.079
(0.904)
.148
(1. 064)
.192
.069
.158
.121
(0.662)
.197
continued
~
o
e
e
e
Table 1 (continued)
Proportion
Retained
0!=.9
O! = .5
QI
= .1
Part g. Initial heritability = 1; initial gene frequency = .3
Life
p2(0) = 1) p(O) = .3
-Average
c-.5
c-.1
Gene
Frequencyn=2
n=3
n=4
n=2
n=3
n=4
n=2
.162
(5.385)
.174
(6.187)
.358
(5.587)
.347
(6.616)
n=3
n=4
.481
(5.819)
.487
(6.996)
1/4
-.475-
.147
(4.427)
1/2
-.65 -
.142
(8.614)
3/4
-.825-
0110
(13.006)
.163
(15.801)
.165
(18.340)
1/4
-.475-
.166
(1.030)
.253
(1.209)
.282
(1. 387)
.169
(1. 030)
.272
(1. 209)
.318
(1. 387)
(1. 030)
.278
(1.209)
(1. 387)
1/2
-.65 -
.233
(2.075)
.316
(2.464)
.326
(2.874)
.314
(2.078)
.449
(2.495)
.469
(2.949)
.334
(2.079)
.505
(2.506)
.547
(2.988)
3/4
-.825-
.215
(3.100)
.267
(3.849)
.279
(4.568)
.216
(3.153)
.359
(4.157)
.454
(4.992)
.216
(3.166)
.408
(4.323)
.583
(5.370)
1/4
-.475-
.095
(0.408)
.149
(0.520)
.195
(0.607)
.095
(0.408)
.149
(0.520)
.195
(0.607)
.095
(0.408)
.149
(0.520)
.195
(0.607)
1/2
-.65 -
.190
(0.815)
.284
(1. 051)
.320
(1. 264)
.190
(0.815)
.284
(1. 051)
.334
(1. 264)
.190
(0.815)
.284
(1. 051)
.341
(1. 264)
3/4
-.825-
.151
(1. 353)
.249
(1. 729)
.312
(2.016)
(1. 353)
.367
(2.016)
.151
(1. 353)
.249
(1. 729)
.391
(2.016)
.166
.180
(10.441) (12.022)
.272
(4.517)
c~O
.441
.407
.387
(9.465) (11. 795) (13.576)
.188
.342
(15.629) (18.806)
.151
.249
(1. 729)
.374
(21. 276)
.323
(4.550)
.538
.607
.620
(11.442) (13.396) (17.863)
.212
.424
.636
(18.782) (27.233) (34.292)
.170
.333
continued
.j::'I-'
-
e
e
Table 1 (continued)
Proportion
Retained
~
Ot
Ot
= .9
= .5
= .1
Part h.
Initial heritability = 1; initial gene frequency = .5
Life
p2(0) = 1, p(O) = .5
-Average
c=.5
c=.l
Gene
Frequency
n=2
n=4
n=2
n=4
n=3
n=2
n=3
c=O
n=3
n=4
1/4
-.625-
.121
(2.862)
.154
(3.444)
.170
(4. 003)
.199
(2.875)
.253
(3.505)
.286
(4. 096)
.223
(2.879)
.299
(3.529)
.361
(4.156)
1/2
-. 75 -
.136
(5. 798)
.155
(7. 123)
.176
(8.286)
.271
(6.358)
.344
(7.675)
.356
(8.953)
.322
(6.697)
.520
(8. 036)
.531
(9.839)
3/4
-.875-
.116
.174
.137
(9.092) (11.319) (13.144)
1/4
-.625-
.lO9
(0.667)
1/2
-.75 -
.320
.139
.238
(10.020) (12.968) (14.915)
.143
.286
.429
(lO.579) (15.342) (19.764)
.173
(0.800)
.222
(0.914)
.lO9
(0.667)
.173
(0.800)
.222
(0.914)
.lO9
(0.667)
.173
(0.800)
.222'"
(0.914)
(1. 395)
.238
(1. 735)
.277
(2.009)
.171
(1. 395)
.277
(1. 735)
.339
(2.009)
.171
(1. 395)
.291
(1. 735)
.366
(2. 009)
3/4
-.875-
.147
(2.204)
.237
(2.822)
.252
(3.354)
.147
(2.204)
.253
(2.851)
.312
(3.434)
.147
(2.204)
.258
(2.862)
.342
(3.474)
1/4
-.625-
.032
(0.308)
.114
(0.348)
.135
(0.418)
.032
(0.308)
.114
(0.348)
.135
(0.418)
.032
(0.308)
.114
(0.348)
.135(0.418)
1/2
-. 75 -
• 064
(0.615)
.228
(0.696)
.270
(0.837)
.064
(0.615)
.228
(0.696)
.270
(0.837)
.064
(0.615)
.228
(0.696)
.270,
(0.837)
3/4
-.875-
.096
(0.923)
.291
(1. 111)
.201
(1. 379)
.096
(0.923)
(1. 111)
.201
(1.379)
.096
(0.923)
.291
(1. 111)
(1. 379)
.171
.291
.201
continued
.pN
-
e
e
Table 1 (continued)
Proportion
Retained
0:-.9
O!
O!
:= .5
:= .1
Part i.
Initial heritability := 1; initial gene frequency = . 7
Life
p2 (0) := 1; p (0) := . 7
-Average
c-.)
c-.l
Gene
n:=2
n:=2
n:=3
Frequency
n==4
n=3
n=4
n=2
c==U
n=3
n:=4
1/4
-. 775-
.104
(1. 695)
.137
(2.136)
.162
(2.477)
(1. 695)
.176
(2.139)
.210
(2.495)
.133
(1. 695)
.189
(2.140)
.235
(2.504)
1/2
-.85 -
.111
(3.645)
.149
(4.601)
.170
(5.386)
.146
(3.699)
.244
(4.800)
.289
(5.603)
.157
(3. 715)
.297
(4.912)
(5. 785)
3/4
-.925-
.081
(5.926)
.138
(7.497)
.145
(8.867)
.081
(6.032)
.160
(7.895)
.209
(9.587)
.081
(6.065)
.162
.242.
(8.131) (10.402)
1/4
-. 775-
.066
(0.441)
.103
(0.514)
.148
(0.585)
.066
(0.441)
.103
(0.514)
.148
(0.585)
.066
(0.441)
.103
(0.514)
1/2
-.85 -
.132
(0.882)
.198
(1. 043)
(1. 241)
.132
(0.882)
.198
(1. 043)
.257
(1. 241)
.132
(0.882)
(1. 043)
3/4
-.925-
(1. 423)
.138
(1.827)
.245
(2.091)
.086
(1.423)
(1. 827)
.245
(2.091)
(1. 423)
Fixation
.051
(0.304)
Fixation
.051
(0.304)
1/4
-. 7751/2
-.85 3/4
:';925":"
.086
in
one
Generation
.257
.102
(0.607)
.153
(0 ~ 911)
.127
.138
in
one
Generation
.102
(0.607)
.153
(0.911)
.086
.198
.138
.390
.148
(0.585)
.257
(1. 241)
(1. 827)
.245
(2.091)
Fixation
.051
(0.304)
in
one
Generation
.102
(0.607)
.153
(0.911)
+:-
w
44
linkage often had a striking effect.
_
-
M(t)/c
M
(t*)
i}i+1
Ic.
=0
_
'1-. 5
~}~+
The ratio
- (t)
=
D
- (t*)
D
Ic,~}~. +1
Ic,
'+1
~} ~
= 0
.5
is used to denote the effect of complete linkage relative to free
recombination for fixed values of n} a} p2(0)} and p(O); t and t* are
the generations at which the quarter} half} or three-quarter life
values were attained for comparable runs involving c }i+1
i
The values of n
at n
-~}c
= 3 are given in Table 2.
=
0 and .5.
n
was frequently
-~}c
of the order of 2 and} at low intensities of selection} often reached
values between 3 and 7.
The effect of heritability and selection intensity on 0-~} c is
apparent from Table 2.
0_
-11.} c
decreased monotonically with respect to
an increasing selection intensity and to an increasing heritability.
In the latter case} it should be pointed out that both the numerator
and denominator of
n
. increased with respect to an increase in
-"M} c
heritability (see Table 1)} and the decrease in n
was a function of
-~} c
the magnitude of M(t) under free recombination for the different
heritabilities.
The interaction between linkage and intensity of
selection will be discussed more thoroughly in the next subsection.
Likewise} interactions involving linkage and the initial gene frequency
will be discussed in a later subsection.
One aspect of linkage on the variance should be underscored.
With the exception of t = 2} the effect of linkage is not linear.
(t)
value of M
C,
at c. '1
~}1+
.1 tended to be intermediate between those of
'+·1 =.5 and O} especially later in the selection program (see
~) ~
The
-
e
e
The ratio) n.
, of the reduction in genotypic variance due to disequilibria under
-~)c
Table 2.
complete linkage to that under free recombination (the corresponding ratios of time,
0t ) to a given life are given in parentheses)a
,c
p(0)=.3
Proportion
Retained
CJ!
Of
Of
=.9
= .5
= .1
p (0)=. 5
Life pZ(U)=.251/(U)=.75I Z(U)=1 PZ(U) =. 251 pZ(U) =.
p
751 pZ(U) =1
pZ(U)=.25I pZ(U)=.75I p 2(0)=1
4.64
(1. 06)
3.04
(1. 08)
2.97
(1. 08)
3.17
(1. 04)
(1. 03)
1. 94
(1.03)
2.11
(1. 01)
1.44
(1. 0 1)
1. 38
(1. DO)
1/2
7.15
(1. 20)
4.07
(1. 28)
3.66
(1. 28)
5.60
(1. 12)
3.37
(1. 15)
3.36
(1. 13)
3.84
(1.07)
2.17
(L07)
1. 99
(1. 07)
3/4
8.50
(1. 35)
3.39
(1. 61)
2.60
(1. 72)
7.00
(1. 20)
2.65
(1. 31)
2.09
(1. 35)
5.04
(1. 11)
1.71
(1. 12)
L 17
(1.09)
1/4
L48
(1.00)
1. 17
(1.00)
1.10
(1.00)
1. 23
(1.00)
1.00
(1.00)
LOO
(1.00)
1.00
(1.00)
1.00
(1.00)
All
1/2
2.36··
(1.04)
1.71
(1. 03)
1. 60
(1.02)
1. 85
(1.02)
1. 29
(1. 00)
1. 22
(1.00)
1.42
(LOO)
1.02
(1. 00)
3/4
3.10
(1. 10)
1.83
(1.11)
1.53
(1. 12)
2.59
(1. 04)
1. 38
(1. 03)
1.09
(1.01)
1. 95
(1.02)
1.09
(1.00)
One
1.00
1.00
(1. 00)
All
1.00
(1.00)
LOO
(1.00)
All
1.00
(1. 00)
All
All
(1. 00)
Ratios
Ratios
1/2
1. 36
(1. 00)
1.03
(1. 00)
1. 16
(1. 00)
1.00
(1. 00)
Equal
Equal
3/4
1. 73
(1. 02)
1.12
(LOO)
1. 51
(1.00)
1.01
(1. 00)
to
to
One
One
1/4
1/4
Ratios
Equal
to
One
2.05
p (0) =.7
Ratios
Equal
to
One
(L02)
(1. 00)
1.25
(1. 00)
Ratios
Equal
to
a Data used for calculating ratios were obtained from Table 1 for n = 3.
~
\JI
•
46
Table 1).
Further, as n increased from 2 to 4 the response in M(t)
= .1 became more like that for c,~,~'+1 = .5.
for c, . 1
~,~+
Even though the value of M(t) often increased dramatically with
tighter linkage, under independent assortment it was often greater than
.20 for heritabilities of 1.
p2(0)
= .25
And, although the magnitude of M(t) for
was usually near .05 for free recombination, even under
high intensities of selection (a
= .1),
values of the order of .1 were
often approached and attained for larger number of loci (n
= 4).
Due to the decreased genotypic variance under tighter linkage, the
advance per locus was more inhibited.
Inhibition of genetic progress
due to linkage had been observed in the earliest simulation studies;
~.~.,
see Fraser (1957) and Martin and Cockerham (1960).
Retarded
progress resulting from a decreased genotypic variance is expected
since the usual prediction equation for the change in gene frequency is
Ap(t)
l;l
oc
";" (t)
~
P
(t)
(t)
(56)
CJ"G
in which I(t) is the standardized selection differential and p(t) is
the heritability at time t.
In comparable runs in which I(t) was
sufficiently independent of linkage, the approximation
(t)
(t)
CJ"G
(t*)
(t·/()
CJ"G
P
p
held quitE! well.
p2(0)
(57)
Such was the case for runs involving p (t) ::. .5 and
= .25 and for higher heritabilities for p(t) ~ .5 and a = .5.
However, for runs involving generations at which p(t) was sufficiently
different than .5 so that asymmetry affected the selection differential,
•
47
the value i(t) in equation (56) became more dependent on the linkage
parameter.
This resulted in the approximation in equation (57)
becoming less accurate,
It is possible to get a picture of the inhibition of progress due
to tighter linkage by comparing over c . . 1 the values of the time
~,~+
required to attain a given life for specified levels of p(O), p2(O),
n, and a in Table 1.
The only cases in which the time, t, to a given
life did not increase with tighter linkage were for those runs in which
t
~
2 (at which generations the gene frequency was independent of
linkage) or in which fixation was essentially reached immediately after
generation 2.
It is possible to evaluate the degree to which linkage can impede
progress by examining the magnitude of
nt, c
for the various parameters
presented in Table 2 where
nt, c
=
tic .. +1
=0
t*l c . . +1
- .5
~,~
~, ~
t,t* being the generations to a given life for comparable runs.
It is
obvious from Table 2 that linkage had its greatest effect later in the
selection program.
rapidly
(~:~.,
In runs for which fixation was not approached too
runs involving p(O)
= .3
or .5 and a
= .9), nt, c
usually
increased with respect to an increase in heritability.
The apparent
negligible retardation due to tighter linkage for p2(O)
= .75 and 1 at
a = .1 and .5 was most likely attributable to the rapid approach to
fixation caused by the large per locus effect for the small numbers of
loci investigated.
•
48
Referring again to Table I} whenever there was a striking difference in the number of generations required to reach a certain life
between runs involving free recombination and complete linkage} the
response for c, , 1
~}~+
=
.1 tended to be either more like that for free
recombination or to be intermediate between the two.
became
more pronounced at later lives.
This tendency
However} for runs in which the
approach to fixation was very rapid} the differences in the responses
for free recombination and complete linkage were small} and the
response for c, '+1 = .1 was more like that for complete linkage.
~}~
This
suggests that} whereas the effect of linkage on advance might be
approximately linear in the early generations} it becomes more strongly
non-linear as time progresses.
Even under independent assortment} negative interlocus correlations existed and progress .was slower than
it would have been if
.
intergenic independence were specified in each generation.
Figure I}
parts a through c} give examples of advance in gene frequency for
c, , 1
~}~+
=
.S} .l} and 0 and also give the advance under the state of
intergenic independence in each test population.
The state of inter_
genic independence was created by specifying the zygotic frequencies
given in equation (43) for the generation in question and by using the
gene frequency attained via selection in the previous generation.
Discussion will now be focused on the effect of the position of
loci on -a chromosome in terms of advance in gene frequencies and build_
up in disequilibria.
In the case of runs involving c. '+1
~}~
all loci are equidistant from one another;
the same number of map units.
~'~'}
= .S or 0
they are separated by
But in runs for which
•
-
e
LO
.9
.8
»
o
ffi
.7
::l
0"'
(1)
1-4
fi.l
.6
(1)
!=l
(1)
o
.5
.4
5
10
Par t a.
Figure 1.
15
Generation
20
25
.
-2
p (0)_
- .3, p2(0)_'5
- . ; , CY=::.9,
n-
Gene frequencies under intergenic independence (1.1.) and various
degrees of linkage
~
1.0
•
•
e
=0
•7
>.
o
s::
~
.6·
0"
<D
$.I
~
OJ
.5'
s::
OJ
Co'
10
5
15
Generation
Part b.
p
(0)
2 (0)
=.1, P
=·25, 0=.5, n=2
Figure 1 (continued)
VI
o
•
51
c =.1
.8
.7 .
0
5
15
10
Generation
Part c.
p (0) =. 7J p2(0)=I J
~.9J
Figure 1 (continued)
••
n=5
20
•
52
o<
c,l,1.'+1 < ,5 ,
the distances are different for different pairs, and the effect of
selection on a
part~cu1ar
loci on the chromosome.
set of loci depends on the positions of those
Under a non-interference model, the following
relation is specified:
c1.'k
= c,1.J,(l_c'k)
J
i < j < k ,
+ (l-c. ,)c'k'
1.J J
Thus, for equal recombination frequencies between adjacent loci such
that 0 < c, '+1 < .5, the following relations hold:
1., 1.
= c n- 1,n <
= c 25 =
e
••
=
C
c 13
= c 24 =
= c
n-2,n
< c 14
3
< ... < c 1n <.5 ,
n- ,n
(58)
(Since the distances between loci are the same for all adjacent pairs,
there is symmetry in response to selection for loci located at an equal
distance from the center of the chromosome; consequently,
(t)
Pn-i+1
and
n(t)
ij
= n(t)
n-j+1, n-i+1
for all t ,
In order to simplify some of the following expressions, only the
members of the symmetrically related values appearing on the left side
of the above equations will be presented,)
Under the
cond~tions
diverge in generation 2.
specified in equation (58), the disequilibria
Recalling equations (52) through (54), it can
•
53
be seen that
D(2)
12
= D(2) =
23
< D(2)
14
:::; D(2)
[ n-1] [n+3]
T'T
< ... <
D
= D(2) =
= D(2)
25
n-2] [n+4]
[T
' -2(59)
1n
in which
m .f
.
[ m_]
2 :::; 2 1m 1S even
m-1
=T if m is odd ,
Under such conditions it can be shown that the average negative
disequi1ibriwnis nearer zero for loci located more distally from the
center of the chromosome; !.~., letting
D.1.
1
=~
n_.L
E
".1.'
J, Jr,1
D' j
1
represent the avet'age disequilibrium between locus i and all other
loci, i t follows from inequality (59) that;
0(2)
1.
>
0(2)
2.
>
03(2.)
> ... > jj (2)
[n;l]
,
(60)
and also that
(61)
54
Returning to empirical findings) it was found that selection in
generation 2 favored those loci which were less inhibited by negative
disequilibria.
Thus) for t
=3
(t) > p(t) >
> (t)
Pl
2
...
p
en?]
~'~')
}.
(62)
the advance was greater for loci located further from the center
of the chromosome.
The following relation also held for t
(t)
(t)
(t)
(t)
Pi
- Pi+l ~ Pi+l - Pi+2 )
1
<
-
i
= 3:
< [n-3]
-
(63)
2
which) when viewed over all n loci) specified a convex curve opening
upwards.
Therefore) the relations among gene frequencies were similar
to those for average interloci disequilibria in the previous generation) equations (60) and (61).
The relations given by inequalities
(62) and,(63) held for the remainder of the selection program.
Figure
1 (part c) gives an example of gene frequency divergence for n = 5 and
c.1.) 1.'+1 = .1.
It should be pointed out that the differences between the
gene frequencies at different loci were negligible and became nearly
imperceptible for lower heritabilities.
Starting with generation 3) even the disequilibria between pairs
of equally spaced loci began to diverge depending on their position on
the chromosome.
Let t' be the generation at which the disequilibria
began to return to zero.· Until generation t* > t') the magnitude of
disequilibria was greater between those pairs of equally spaced loci
which enclosed segments located more distally from the center of the
chromosome.
Therefore) for the five locus model .
55
n (t)
n(t) <0
12 < 23
for 2 < t <
t~(
n (t)
for 2 < t <
t~(
(64)
and
13
<
n(t) <0
24
.
At generation t* the relation was reversed;
(65)
!:...:.:)
<the disequilibria was
of greater magnitude for loci enclosing more centrally located segments.
This continued to be the case for the remainder of the program.
Thus) in the five locus runs)
n(t) < n(t) <0
12
23
for t >
t~(
(66)
n(t) < n(t) <0
24
13
for t > t*
(67)
and
Recalling that equality in disequilibria between equally spaced loci
always existed at t = 1 and t
= 2)
if the greatest negative value was
attained at one of these generations) then inequalities (66) and (67)
held for the remainder of the program.
The latter situation existed
in runs involving higher initial gene frequencies with moderate or
high selection intensities or in runs involving low initial gene
frequencies and high intensities of selection (in other words) systems
that speeded the progress towards fixation for a given heritability).
Examples of disequilibria responses are given for c, '+1
=
in Figure 2 (parts a and b) along with those for c, , 1
= .5
~)~
~)~+
.1 and n
and O.
It should be pointed out that) for a major portion of the
selection program) the differences among disequilibria between pairs
=5
56
o
5
Generation
10
lS
.0
( t)
15
( t)
14
(t)
13
(t)
24
n(t)
.... 01
12
n(t)
23
Figure 2.
Intragametic digenic disequilibria as a function of
frequency of recombination and locus position
c=.l
e
e
e
~
Generation
1
2
4
3
7
5
to
'.-1
, n(t)
l-l
.0
15
'.-1
~-.01
n(t)
=!
14
0"
OJ
D (t)
CIl
13
•.-1
C'l
c =.1
D (t)
24
D(t)
12
D (t)
23
-.02
Part b.
p
(0)
2 (0)
=.7, P
=1, a=.5, n=5
Figure 2 (continued)
VI
"-J
58
of equally spaced loci were insignificant when compared to the
differences among disequilibria between pairs of loci that were not
equally spaced.
As expected, the following inequality existed between
disequilibria involving unequally spaced loci in runs for which n = 5:
n(t) n(t) < n(t) n(t) < n(t) < n(t)
0
12) 23 - 13) 24 - 14 - 15 <
.
(68)
Exceptions to this inequality sometimes occurred as the process neared
fixation when) in some runs) disequilibria approached their lower
limits) equation (22).
It can be shown that) when the minimum possible
values were attained)
(recall Pl(t) > p(t) = P4(t) ; inequality (62».
2
to inequality (68).
This was an exception
Actually) in such runs the response in
times crossed that of
Di~) prior to attaining a lower limit.
Di~) someWhereas)
in Figure 2 (part b) the first point at which n(t) < D(t) was the
24
12
point at which the lower limits were attained) in Figure 2 (part a) the
time of crossover between Di~) and Di~) preceded the time of attaining
the lower limits by one generation.
The effect of linkage on inter locus covariances can be discussed
in a more general manner.
Change in interlocus digenic disequilibria
from one generation to the next can be separated into two parts; the
first being attributable to changes resulting from selection per
(s)t) an d
.
( ~.e.)
R..
- -
~J
uA
(t» ) an d t h e secon db'
D..
e~ng attri b uta bl e to
~J
subsequent recombination
(~'~')
~
59
For complete linkage, the negative increase in the disequilibrium
between loci i and j continues as long as
Under less inhibited recombination, the increase in magnitude may
terminate while
~ D~~) is still negative; e.g., under free recombina_
1J
--
tion the negative increase in
therefore
D~:)
1J
will terminate whenever
~ D~:) can be less than zero.
1J
Although increased linkage results in a reduced contribution of
R(s,t) to the covariance in the next generation, this is more than
ij
(t)
(t)
and D...
1J
1J
offset by the increased carryover of 6 D..
And, although
retarded progress due to increased buildup in disequilibria tends to
reduce the magnitude of
R~~,t) and 6 D~:) at each generation, the
1J
1J
number of generations over which changes can accumulate is increased.
Interactions between linkage and some of the other parameters will
be discussed in subsequent subsections.
The Effect of Intensity of Selection
Except for runs in which fixation was reached or nearly reached in
the first generations of selection, the initial buildup in disequilibria
was greatest for runs involving higher selection intensities simply due
to the magnitude of the change in gene frequency.
But under intense
selection, there was little time in which accumulation could take
60
place, and the magnitude was soon restricted by its rapidly increasing
lower limit as fixation was approached.
Therefore, when evaluated over
the total program (see Table 1), the greatest reduction in the genotypic variance was usually associated with intermediate or low se1ection intensities.
For independent assortment the tendency was towards greater
buildup under moderate selection intensities; however, for tight 1inkage, the greater buildup was associated with low intensities of
selection.
This interaction between selection intensity and linkage
can be seen by comparing the values of M(t) over a for a given life and
given values of n, p(O), and p2(0) in Table 1.
of c . . 1
~,~+
= 0,
However, in the case
the value of M(t) was greatest for a
=
.9; the only
exceptions (Table 1, parts g through i, for the three-quarter life
values involving n = 2) appeared to be due to the failure of linear
interpolations between generations with widely different gene
frequencies to give an adequate fit for a clearly non_linear response.
In other runs investigated for which a = .1, .2, .3, ... , .9 and
c.~,~'+1
0, the greatest buildup in disequilibria was observed in those
runs involving a = .9.
This suggests that, for a given set of starting parameters
(n, p(O), and p2(0)) free recombination ultimately offsets any potentia1 buildup in disequilibria that could have been made by increasing
the number of generations over which disequilibria could accumulate
(increasing t being accomplished by decreasing the intensity of
selection).
However under tighter linkage, recombination would have
less influence and any such potential buildup (derived by increasing
a and thereby increasing t) would be more likely realized.
61
The Effect of an Increased Number of Loci
In comparing runs over the number loci, two different systems were
investigated.
The first involved changing n alone which resulted in
a reduction in the additive effect per locus, equation (44); for fixed
values of p2(O)
a
ex:
, p(O) , and ~2e:'
1
VIi
In effect, this system contrasted runs which had the same initial
genotypic variance for different values of n.
The second system
investigated involved simultaneous changes in nand p2(O) which produced the same additive effect per locus.
For the first system, increasing the number of loci almost always
reduced the buildup of negative disequilibria between loci.
The only
cases for which the disequilibria was greater in magnitude for an
increased n involved runs that approached fixation so rapidly that the
gene frequencies and disequilibria bracketing a given life differed
greatly.
This raised doubts as to the goodness of the values obtained
via linear interpolation.
Of course, a decrease in the magnitude of
disequilibria with respect to n is expected based on the lower limit of
the average disequilibria, equation (23).
But, despite the usual decrease in disequilibrium for a given
pair, with an increase from n to n + K loci resulting in ~ (2n +K - 1)
additional pairs, the total contribution of all pairs almost always
increased the proportional reduction in the genotypic variance due to
disequilibria (see Table 1).
One can get a rough idea of the magnitude
of increase in the case for which the number of loci was doubled from
62
n = 2 to n = 4 by comparing the ratio of the half life values of M(t)
for the two values of n}
= M(t) In=4
M(t*) In=2 }
t}t* being the number of generations to half life for comparable runs
(see Table 3).
The value of 0-"M} n tended to be larger for smaller
( t~o.)
values of M
Table 3.
'In=2.
The ratio}
~}n}
of the reduction in genotypic variance due
a
to disequilibria at half life for n = 4 to that for n = 2
p2 (0) =. 25
c=O
c=.l
p2 (0) =1. 0
c=O
c=.l
Proportion
Retained
c=.5
Ct'=.9
Qi=.5
Qi=.l
1. 33
1. 39
1. 68
1.04
1. 49
1. 92
1.44
1. 78
2.07
1. 27
1.40
1. 68
1. 76
1. 15
1. 64
1. 80
p(0)=.5
Qi=.9
Q'=.5
Q'=.l
1. 36
1. 52
1. 76
1. 20
1. 76
2.02
1. 74
2.01
2.10
1. 29
1. 62
4.22
1. 31
1. 98
4.22
1. 65
2.14
4.22
p (0) =. 7
Q'=.9
Q'=.5
Q'=.l
1. 45
1. 68
2.11
1.44
1. 94
1. 98
2.13
2.26
1. 53
1. 98
2.48
1. 95
1. 95
1.95
Fixation in One
Generation
Heri ta bi 11 ty
p (0) =. 3
2.19
c=.5
.92
1. 49
aData used for calculating ratios were obtained from Table 1.
Two groups of exceptions to a monotonic increase in M(t) with
respect to increasing n existed but are not apparent in Table 3.
The
first group of exceptions (see Table I) part h; a = .l} n = 3 and 4)
involved runs that went to fixation in two generations.
And} as in
similar cases cited before} the response in M(t) was far from being
linear over any interval in time.
be given to such exceptions.
Consequently} little weight should
63
The second group of exceptions to the monotonic increase inM(t)
with respect to n involved runs for which p(O)
= .3} ~ = .9)
c . . 1 = .1) and p2(O) = .25) .75) and 1 (see Table 1; parts c) f) i).
~)~+
Graphing M(t) against p(t) for these runs (see Figure 3) parts a
reve~led
through c)
that an increase in n had the following effects on
M(t) :
(a) M(t) increased more rapidly in the first few generations for
larger n.
(b) M(t) was greater in magnitude near the end of the selection
program for larger n at high heritabilities; and
(c) at intermediate generations there was usually an overlap in
the responses of M(t) for the different values of n.
Therefore) the response curves appeared to flatten out with an
increased number of loci.
This non-monotonicity of M(t) with respect to n existed at
c . . +l ::: .1) but not at c . . +1 = .5 or
~)~
~)~
o.
Such overlaps at c . . +1 = .1
~)~
can partially be explained by the fact that increasing n decreased the
average linkage;
!.!.
c = .1
for n
=2
c
~
.127
for n
=3
c
~
.151
for n = 4 .
Thus with a reduced linkage for an increased value of n} the magnitude
of n(t) was reduced more than would have been expected from an increase
in n alone.
-
e
e
.15
·10
,.....
+J
'-'
::E:
.05
.0 ,
.3
•
.4
J
.5
I
.6
I
.7
'
,
.8
.9
'\
1.0
Gene Frequency
Part a.
Figure 3.
p2 (0) =.25
Proportional reduction, M(t), of genotypic variance due
to disequilibria for p(0)=.3, a=.9, and c . . 1=.1
~,~+
0\
.+:'-
e
e
e
.J
,2
,,-..
.w
'-"
~
.1
n:....:3
n=-=2
.Of
r
I
.3
.4
.5
I
.6
,
.7
,
.8
!
.9
""
1.0
Gene Frequency
Part b.
p2 (0) =. 75
Figure 3 (continued)
0\
l.J1
•
e
e
·4
·3
,,-,.
.IJ
'-"
:::E::
.2
• Lc
.5
..6
.7
.8
Gene Frequency
Part c.
p2 (0) ==1
Figure 3 (continued)
Ci'
Ci'
67
However, this doesn't explain the whole story.
Additional runs
were made using
c,~, ~. +1 = . 1
for n = 2
c,~,~'+1 = .07805
for n = 3
c,~,~'+1 = .06402
for n = 4
,
all of which yielded an average recombination frequency of .1 correct
to the fifth decimal.
For p2(0) = .25, M(t) was consistently greater
for large n throughout the selection programs.
p2(0)
However, for
= .75 and 1, there was overlap in the responses of
M(t) for n
= 2
and n = 3; however the overlap involved far fewer generations than runs
involving equal recombination frequencies between adjacent loci.
were no overlaps in the responses for n
4 with either n
There
= 2 or n = 3.
Therefore, it was not possible to get a consistent picture as to the
effect of increasing n or M(t) for runs involving
and either c, . 1
\
~,~+
O!
= .9} p(O) = .3,
= .1 or C = .1.
But for free recombination or complete linkage, increasing n for
the range of n investigated in this paper consistently resulted in an
increase in M(t).
And this was also true for all runs involving
ci,i+l = .1 except at
O!
= .9 for p
As expected in a system
, h
w~t
(0)
a oc
= .3.
'{;i1 , the advance 'in gene
frequency was reduced with increasing n.
As selection was continued,
the advance became more retarded due to the increase in M(t) resulting
from' the increased number of loci.
Comparisons of t, the time to a
given life} for different values of n (see Table 1) give an indication
as to the magnitude of retardation attributable to larger values of n.
68
It is reasonable to ask whether the decreased negative correlation
between loci and the retarded progress was actually attributable to an
increased n per
~
or to a decreased additive effect per locus.
To
answer this question, the second system of comparisons was made in
2 (0)
which the following two combinations of nand p
.
were contrasted,
each of which yielded equal additive effects per locus for a given
initial gene frequency:
Combination I,
and combination 2,
1
n = 5, p2 (0)
For both combinations
4"
.1825
a
Table 4 gives examples of
rre - Vp(O) (l_p(O))
quarter, half, and three-quarter life values of t, n(t), and M(t) .
With only one exception, n(t) was nearer zero for the larger value
of n, and that exception (Table 4; p(O) = .5, c, '+1 = .5, and
~,~
~
= .1
at three-quarter life) appeared to be associated with the linear
interpolation at that particular interval.
greater in magnitude for n
p
(0)
=.5 and c. , 1 = .5.
~,~+
= 5.
more retarded for n
Exceptions occurred in runs involving
In those runs the three-quarter life values
of M(t) were always greater for n
values were usually less.
For most runs M(t) was
= 5, but the quarter and half life
In all runs the time to a given life was
= 5 than for n = 2.
Summarizing the effect of number of loci, whether equating
initial heritabilities or effects per locus, an increase in the number
Table 4.
e
e
e
Disequilibria, D(t), and proportional reductions, M(t), in the genotypic variance due to
disequilibria; number of loci and initial heritabilities are such that ~ is a constant
~
e
for given initial gene frequencies (generations are given in parentheses)
p
(0)
=.3
Combina tion 1
froportion
Retained
cr = .9
Of
Q'
= .5
= .1
Life
-Gene
Frequency-
Combination 2
n=2, p2(0) =.118
D(t)
n=5, p2 (0) =.25
c=O
c=.5
M(t)
D(t)
c=O
c=.5
M(t)
D(t)
M(t)
D(t)
M(t)
1/4
-.475-
.0191
-.0047
(11. 196)
':'.0266
.1068
(11. 503)
,0586
-,0037
(12.398)
-.0195
.3121
(13,839)
1/2
-.65 -
-,0044
.0196
(21. 124)
,1968
-,0448
(22.987)
,0590
-.0033
(23, 724)
_,0266
.4&73
(31. 289)
3/4
-.825-
,0143
-,0021
(32.634)
,1560
-,0225
(37.493)
,0456
-.0017
(36.839)
-,0170
.4709
(55.934)
1/4
-,475-
_,0090
.0362
(2.621)
-,0146
.0588
(2.628)
,1127
-.0070
(2.959)
_.0123
.1966
(2.995)
1/2
-.65 -
,0395
-.0090
(5,092)
.1053
-.0239
(5.196)
-,0064
.1133
(5.876)
.3209
-.0182
(6.385)
3/4
-.825-
.0265
-.0038
(8.078)
.0952
-.0138
(8.436)
-.0030
.0822
(9.357)
,3237
-.0117
(10.984)
1/4
-.475-
,0318
-,0078
(1. 159)
.0342
-.0084
(1. 159)
_.0066
.1067
(1. 286)
.1208
-.0074
(1. 286)
1/2
-,65 -
.0412
-.0092
(2.293)
.0600
-.0133
(2.297)
.1283
-.0072
(2.622)
-.0116
.2076
(2.652)
3/4
-.825-
.0300
-.0044
(3. 704)
.0578
-.0084
(3. 731)
-.0035
.0968
(4.261)
-.0076
.2108
(4.456)
continued
0\
\0
e
e
e
Table 4 (continued)
p (0) =.5
Combina tion 1
Proportion
Retained
Ol
Of
Of
= .9
= .5
= .1
Life
-Gene
Frequency-
Combination 2
n=2, p2(0) =.118
M(t)
D(t)
n=5, /(0) =. 25
c=O
c=.5
D(t)
c=O
c=.5
M(t)
D(t)
M(t)
D(t)
M(t)
1/4
-.625-
.0169
-.0040
(7.569)
-.0151
.0644
(7.696)
.0158
-.0031
(8.451)
-.0121
.2057
(10.008)
1/2
-. 75 -
-.0028
.0149
(15.733)
.1080
-.0202
(16.580)
.0283
-.0022
(17.645)
-.0149
.3174
(21.049)
3/4
-.875-
-.0010
.0095
(26.531)
.0899
-.0098
(28. 797)
.0296
-.0009
(29.598)
-.0085
.3101
(38.850)
1/4
-.625-
-.0061
.0262
(1. 841)
.0341
-.0080
(1. 841)
_.0052
.0143
(2.046)
.1215
-.0071
(2.048)
1/2
-. 75 -
.0276
-.0052
(3.914)
_.0108
.0577
(3.943)
-.0041
.0250
(4.438)
-.0092
.1976
(4.598)
3/4
-.875-
-.0017
.0173
(6. 760)
.0532
-.0058
(6.887)
-.0016
.0272
(7.621)
-.0053
.1948
(8.274)
1/4
-.625-
.0198
-.0045
(0.836)
_.0045
.0198
(0.836)
.0738
-.0043
(0.905)
-.0043
.0738
(0.905)
1/2
-. 75 -
,-.0048
.0261
(1. 797)
_.0061
.0333
(1. 797)
.0247
-.0044
(1. 976)
.1250
-.0058
(1. 976)
3/4
-.875-
-.0009
.0192
(3. 123)
.0328
-.0036
(3.134)
.0258
-.0019
(3.503)
-.0034
.1224
(3.557)
continued
......
o
e
e
e
Table 4 (cbntinued)
p (0) ==. 7
Combination 2
Combination 1
Proportion
Retained
Ci==.9
O! ;;=
.5
Ci==.l
Life
-Gene
Frequency-
n==2, p2(0) ==.118
D(t)
n==5, p2 (0) ==. 25
c=U
c-.':>
M(t)
D(t)
c-.':>
M(t)
D(t)
c=U
M(t)
D(t)
M(t)
1/4
-.775_
.0161
-.0028
(4. 798)
.0397
-.0069
(4.832)
-.0022
.0507
(5.418)
-.0059
.1346
(5.589)
1/2
-.85 -
.0129
-.0017
(10.629)
.0615
-.0078
(10.942)
.0420
-.0013
(11. 986)
-.0066
.2057
(13.316)
3/4
-.925-
-.0005
.0075
(19.119)
.0505
-.0035
(20.033)
.0265
-.0005
(21. 303)
.1882
-.0033
(25. 183)
1/4
-.775-
.0187
-.0032
(1. 222)
.0203
-.0034
(1. 222)
.0660
-.0029
(1. 348)
-.0032
.0749
(1. 348)
1/2
-.85 -
.0205
-.0026
(2. 748)
-.0041
.0326
(2. 753)
.0708
-.0023
(3.052)
-.0039
.1211
(3.085)
3/4
-.925-
-.0009
.0128
(5.010)
-.0021
.0301
(5.057)'
.0451
-.0008
(5.588)
-.0020
.1149
(5. 782)
1/4
-. 775-
.0109
-.0016
(0.601)
.0109
-.0016
(0.601)
.0421
-.0016
(0. 641)
-.0016
.0421
(0.641)
1/2
-.85 -
.0173
-.0022
(1. 310)
.0188
-.0023
(1.310)
-.0020
.0634
(1. 421)
-.0022
.0717
(1.421)
3/4
-.925-
-.0010
.0130
(2.408)
,0183
-.0013
(2.410)
-.0009
.0478
(2.609)
-.0013
.0711
(2.616)
"
I -'
72
of loci led to a decrease in the negative correlation between pairs of
loci, resulted in an increase in the proportional reduction in the
genotypic variance attributable to disequilibria for at least part (and
almost always all) of each selection program, and impeded progress when
measured in terms of time required to reach a certain life.
The
Effe~tof·Initial Herit~bility
Increase in initial heritability resulted in an increased rate of
buildup in negative disequilibria, and, consequently, resulted in
greater proportional reductions in the genotypic variance
(~.~.,
Table I compare parts a, d, and g; b, e, and h; c, f, and i).
in
These
greater reductions were never large enough to offset the greater per
locus effect; therefore, as expected, progress from selection increased
monotonically with increasing heritability.
Nonetheless, keeping in
mind that the maximum genotypic variance under intergenic independence
is expected at gene frequency .5, the proportional reduction in the
genotypic variance was such that the initial variance for p(O)
often greater than in any subsequent test population.
= .3
was
Even for runs
involving low heritabilities, as long as linkage was tight, the maximum
variance was frequently attained several generations prior to the point
at which the gene frequency of .5 was reached.
of
2 (t)
O"G
a
n
= 4.
2
Table 5 gives examples
for p(O) = .3, p2(O) = .25 and I, c . . I = .5 and 0, and
~,~+
The effect of disequilibrium can also be seen by noting the
asymmetry in the values of
2 (t)
O"G
a
equidistant from .5.
2
for attained gene frequencies
73
Table
5.
The response of genotypic variance to selection for initial
heritabilities of .25 and 1 and for free recombination and
complete linkage} all with an initial gene frequency of .3
p2(0)=1
p2 (0) =.25
c=.5
Proportion
R.etained
t
p
(t)
CY
I
Ot = .9
c=.5
c=O
2 (t)
G
2
a
P
(t)
CY
2 (t)
G
2
a
0
1
2
.300
.314
.329
1. 680
1. 682
1. 700
.300
.314
.329
1.680
1. 682
1. 679
11
12
13
14
.473
.490
.507
.524
1.883
1. 887
1. 887
1. 882
.459
.473
.486
.500
1. 477
1.446
1.415
1. 384
20
.629
1. 760
.575
1.211
..
·
·
30
. 788
1. 273
.682
40
.901
.689
.771
t
0
1
2
3
P
(t)
CY
c=O
2 (t)
G
2
a
(t)
CY
P
I
2(t)
G
2
a
.300
.327
.354
.383
1. 680
1. 588
1. 579
1. 585
.300
.327
.354
.382
1. 680
1. 588
1. 483
1. 348
.470
.499
.530
.560
1. 646
1.652
1. 631
1.610
.452
.475
.498
.519
1.089
1.023
.943
.870
.649
.679
.707
1.493
1.426
1. 369
.557
.589
.599
.609
.605
.626
.791
.816
.842
1. 090
L001
.891
.637
.652
.669
.690
.692
.678
0 .300
1 .428
2 .549
3 .665
4 .772
5 .865
6 .940
7 1.000
8
1. 680
1.458
1. 342
1. 201
.978
.692
.365
0.000
.300
.428
.549
.651
. 735
.802
.864
.928
1.000
1. 680
1. 458
1. 076
.821
.622
.440
.506
.411
0.000
·
6
7
8
9
·
12
13
.975 14
·
·
QI
= .5
0
1
2
3
4
.300
.365
.432
.500
.569
. 780 17
18
19
1. 680 .300
1. 734 .365
1. 775 .432
1. 782 .498
1. 742 .. 561
1.680
1. 734
1. 706
1. 620
1.503
.845
. 767
·
10
·
.894
.715
74
(Note that in some cases involving p2(0)
= 1 and c.~) ~'+1 = 0) the
rates of change in disequilibria and gene frequencies were such that
local maxima in the genotypic variance were sometimes realized,)
In all runs) as the gene frequencies approached fixation) the
average disequilibrium was approaching its lower limit (of course the
lower limit was itself approaching zero near fixation).
Recalling that
the lower limit could only be attained for either
O"~(t)
=0
or
(t)
D.. =
~J
. (
-m~n
p. p . ) q.q.)(t)
J
~
J
~
for all i) j
Since heterozygotic classes were always being retained prior to fixation)
O"~(t) was never zero except at fixation.
However) near the end
of the selection runs) the average disequilibrium did reach its lower
limi t whenever
(t)
-(l_p.
~
(t)
)
J
)(l-p.
for all i) j
This meant that the condition
r(OO)~jt)
...
=
[(l-p.) (l_p.) + D.. ](t) = 0
was satisfied exactly.
J
~
~J
This only occurred for runs involving p2(0)
=1
at any time when the only two genotypic values retained in the selected
parental population were g
=
(2n-1)a and g
= 2na.
For runs involving
75
= .75,
p2(0)
the average disequilibrium neared its lower limit towards
the end of the selection program.
··d
- . 2·5 was
uner f·
ree recom b'1na t'10n, n-(s,t) f or p2(0) However,
still far from its lower limit when the selection program was
terminated.
But with tighter linkage increasing the buildup in dis-
equilibria, B(s,t) was nearer to its lower limit near fixation.
An
example of the average disequilibrium as a proportion of its lower
limit is given for c. '+1
1,1
Fi~ure
4.
= .5
and 0 and p
2(0) _
-
.25, .75, and 1 in
Presented with this figure are the curves
... (n:l) [p(l_p) J (t) and ...min[p2, (l-p) 2 J (t), the shaded surfaces of
which correspond to the lower limit of the average disequilibrium.
It should be pointed out that, since ... n(n:l) p(t) (l_p(t»
neve~ attained except at fixation (because ~~(t)
f
was
0), the minimum
possible value could never be attained until the gene frequency p,(t),
representing the intersection of the two minimum response curves,
was passed.
Since
p
I
(t) _ n-1
- --.----
n
,
the gene frequency at which n(t) could attain its lower limit approached
1 as n was increased.
It should also be pointed out that, for p2(0)
= 1,
once the
selected parental population had been reduced to only the two most
favorable genotypic values, all the disequilibria within the selected
population were also at their minimum possible values;
iI(S' t)
i
for all i
~.~.,
76
/
r, 2 \0)
·=:1) c=O
________
p2(O)=.75)c=O
---------.L
p2 (0) =.25) c=O
.8
.7
2 (0), _ , ~-------I-t-_-J
P
==._) c.- .5
r~ (0) =. 75 )C=~.5~--------};t----J.~
,I p2 (0)
.w
=-=. 25
)C=;.~5:-------fJL_~I----
.;)
...J
__
__1-1
•..l
S
•..l
...:l
1-1
OJ
.5
~
. ..:1
.
2
: (t)
) (l_p) )
-m~n(p
-p (t) (1 -p (t»
n-l
Figure 4.
Above abscissa:
intragametic digenic disequilibria)
D(t») as a proportion 9f the lower limit; below
abscissa:
):Y=.
9) n=3)
lower limit of disequilibria (P(O)=.3)
77
and
for all i} j
Since there were no
coupling~repulsion
phase heterozygotes} linkage
played no role from this point on.
The Effect of
~nitial
Gene Frequencies
To determine the manner in which selection affects systems in_
volving equal values of n} ~2} and ~ but which differ in their initial
e
~
e
starting points} two sets of comparable runs were generated.
Within
each set} the initial heritability was defined in such a way as to give
~ for all values of p(O) at a given value of n.
a constant
~e
were as follows:
Set I}
. I}
p
p2 (0)
o3}
p
(0)
=
and set 2}
a
.891
vn
1
=8"
1
=4
oS}
2(0)
25
P
- 88
o7}
2 (0)
p
09 }
P
2 (0)
each with
~e
2(0)
}
1
=4
1
= 8"
The sets
78
p
(0)
=
2 (0)
,1)
P
,3)
9
=
IT
P
=
4"
,5)
p2(0)
=
32
, 7)
p2 (0)
,9)
P
2 (0)
(0)
3
25
3
=4
=
9
16
each with
a
(J"
e
-
~
2,673
V;
2 (0)
= 1
A third set) set 3) was run for which p
,9,
Within each set runs were made for
and 0) and n = 2 and 5,
~
=
and p
(0)
= , 1)
,3)
,1) ,5) and ,9; c, '+1
~)~
, , ,
=
,5
Examples of the responses in n(t) with respect
to pet) are presented for sets 1 and 3 in Figure 5 (parts a through d)
along with curves associated with the lower limit for n(t),
Under free recombination for fixed values of n and
~)
the average
disequilibria for all runs within a comparable set approached a common
curve regardless of the initial gene frequency
parts a and c),
(~'~')
see Figure 5)
The common curve was usually reached after the dis-
equilibria had passed their minima; therefore the disequilibria
responses almost always reached greater negative values for those runs
involving lower initial gene frequencies,
for p2(0)
)
(Some exceptions were noted
= 1; ~'~O) compare response curves for p(O) = ,1 and
p(O) = 03 in Table 5) part c)o
But once runs shared nearly equal
values of pet) and n(t)) subsequent changes were almost identical,
.0
•
e
e
.1
.2
.3
.4
Gene Frequency
.5
.6
•7
.8
.9
1.0
ctl
.~
I-l
~-.Ol
r-I
.~
:J
0'
Q)
(/)
.~
Cl
-p
_ (l_.p (t» 2
(t) 2
Part a.
Figure 5.
Set 1 comparisons:
a
0=.5, n~2, c i i+l=.5, ~~ .630
,
e
Intragametic digenic disequilibria as a function of initial
gene frequency
-....I
\0
Gene Frequency
-.01
-.02
-.03
-.06
-.07
-.08
Part b.
Set 1 comparisons:
Q'=.5 J n=2 J c.
Figure 5 (continued)
a
. +1 =0 J 0:-:::' .630
~J ~
e:
e
e
e
Gene Frequency
.5
co
'r-!
~
..0
.r-!
r-I
....
,
:::l
0-
Q)
(/)
-r-!
~
-.01
Part c.
Set 3 comparisons:
Q'=.
9, n=5, c _ . 1=.5,
1,1+
O'
2
e
=0
Figure 5 (continued)
00
I-'
I .
I
I
I
82
-.01.
-. o:~
ell
•.-1
l-I
,.Cl
•.-1
r-l
'M
::l
cr
rn
(1)_.03
- (1 -p
•.-1
o
•
(t).2
)
-.04
_net) (J._p(t))
n_l
-.05
Part d.
Set 3 comparisons:
Figure 5 (continued)
83
The situation was different for runs involving tighter linkage.
The response of net) for a given initial gene frequency was noticeably
more negative than all responses for higher initial gene frequencies
throughout the selection program
(~.~.,
see Figure 5, parts b and d).
Exceptions were only realized near fixation when the minimum possible
value was attained
(~.£.,
see Figure 5) part d).
In the first generation of selection) there was an obvious inter_
action between intensity of selection and initial gene frequency with
respect to their effect on
~p(O).
This corresponded to the asymmetry
in response described by Latter (1965a).
Table 6 gives values of ~p(O)
at different initial gene frequencies and different selection intensi_
ties for the comparable sets defined earlier.
Table 6.
Proportion
Saved
At ~ = .5) ~p(O) was
Initial response in change in gene frequency and
disequilibria a
Set 1
Set 2
Initial
Gene
Frequency
Set 3
~p(O)
n(s"O)
~p (0)
n(s)O)
~p (0)
n(s)O)
~
=.9
.1
.3
.5
.7
.9
.0075
.0194
.0256
.0248
.0133
-.0005
-.0034
-.0054
- .0043
-.0009
.0110
.0316
.0421
.0427
.0296
-.0012
_.0100
-.0163
-.0136
.0035
.0111
.0333
.0451
.0477
.0323
-.0012
_.0111
-.0194
-.0177
-.0046
Q'
= .5
·1
.3
.5
·7
.9
.0414
.0925
.1084
.0925
".0414
-.0027
-.0110
-.0118
-.0061
-.0008
.0861
.1612
.1802
.1612
.0861
-.0146
-.0374
-.0325
-.0146
-.0002
.1000
.1701
.1875
.1701
.1
-.0200
-.0409
-.0352
-.0169
0.0
~
= .1
·1
.3
.5
.7
.9
.1198
.2234
.2306
.1742
.0679
-.0078
-.0166
-.0107
-.0035
-.0003
.2663
.3847
.3789
.2844
.0987
-.0473
-.0416
_.0130
-.0002
0.0(-)
.2903
.4295
.4063
.3
.1
-.0523
-.0460
-.0088
0.0
0.0
aVa1ues are independent of linkage.
Number of loci
2.
84
symmetric for p(O) equidistant from .5 and was maximum at p(O)
= .5.
For any other intensity of selection, there was marked asymmetry in
response as p(O) deviated from .5.
from .5, at
For gene frequencies equidistant
~ > .5 the magnitude of ~p(O) was greater for the gene
frequency nearest 1, but the reverse was true at
~
< .5.
In terms of
the usual prediction equation, equation (56), the asymmetry is
attributable to the standardized selection differential since all other
terms on the right hand side of equation (56) are the same for any two
frequencies, p
(0)
= p, l-p.
is of opposite signs for pO
Indeed, such parameters as skewness (which
= p,
l-p) do affect the selection differen_
tial and would contribute to differences in 6p except at
~
= .5.
The asymmetrical response in 6p(0) for selection intensities ~ and
l_~
can be related in the following way.
the upper, K
= U, or lower K = L, proportion,
gene frequency p.
Subtracting
Let
~G
~
K, Q*,p
be the mean of
Q*, of a population with
The mean of the test population can be expressed as
from both sides of the equation and dividing by 2na
yields
0I6p
U, ~,p
+ (1-01) 6PL 1
, -01, P
= 0 •
(69)
For initial populations under intergenic independence
6p (0)
L, 1_01, P
= _6p(0)
U, l ... ~, l_p
Therefore, from equations (69) and (70),
(70)
85
(71)
It can be seen from the last equation that symmetry in response for
gene frequencies equidistant from p
f
For ~
= .5 must occur at
~
= .5.
.5, the following measure of asymmetry will be employed:
(6
Of,
) (0) = [ 6P U Of P ., 6PU Of 1 pJ (0), for p
P
, ,
, , -
< 1_p .
From this equation and equation (71), it can be shown that
) (0) = _
(6
l-Of,
(6
Of
1-Of
p
) (0)
Of,
(72)
p
and
1
6
1-Of, P
I
(0) <
1
6
Of,
P
I
(0)
for
Of
< .5 .
(73)
Therefore, the direction of asymmetry is reversed for selection
intensitie~
equidistant from
Of =
.5, and the absolute value of the
measure of asymmetry is less for the lower selection intensity.
All
data presented in Table 6 satisfies the restrictions imposed by
equations (71) through (73).
Although symmetry in the response of 6P(0) existed at
there was a marked asymmetry in the response of D(s,O).
Of =
.5,
For gene
frequencies equidistant from .5, D(s,O) was of greater magnitude for
the gene frequency nearer zero, and the relative difference between the
magnitudes became more pronounced as the intensity was increased to
Of
= .1.
However for
Of
= .9 the asymmetry (which was not as pronounced
as that forO! =.5 and .1) was reversed.
86
Certain analytical comparisons can be made between the aSYmmetri2
cal responses at selection intensities
~
represent the variance of the upper, K
= U, or lower, K = L, proportion,
and
Ofk, of a population with gene frequency p.
l-~.
Let O"K , Ofk ,P
The variance of the test
population can be expressed as
(74)
Since all H, R, and D type disequilibria are equal in the initial
selected population, from equation (34)
for K = U,
or for K = L,
~c
= ~
O[Ic =
l-C{ .
(75)
Since negative disequilibria are produced whether truncation is on the
upper or lower tails, the following relation holds:
(76)
"
Rea 1 1z1ng
t h at, f or t
= 0 'O"G2(0) = 2 na 2 p (0)(1 -p (0» ,and ut1'I'1zing
equations (71), (75), and (76), the following can be derived from
equation (74):
[~ D
U,C{,p
+ (l-~)D
= -[~(8PU
,
)
c{, p
2
U,l-~,l-p
] (s,O)
2 (0)
+(1-~)(8PU 1
1)]
, -~, -p
(77)
87
For
~
= .5, equation (75) reduced to
[D
U,.5,p
+ D
](s,O)
U,.5,l-p
(78)
Thus, for initial gene frequencies equidistant from .5 and for
~
= .5,
symmetry of disequilibria at (s,O) is a sufficient condition to satisfy
equation (76); however, it is clearly not necessary that symmetry
exists.
It is interesting to note that for p(O) = .5, equation (76)
reduces to
D
U,.5,.5 =-(l1p U,.5,.5 )
2
(79)
Equations (77) through (79) are satisfied by the data given in Table 6.
The Effect of Varying Initial Gene Frequencies within Runs
With gene frequencies varying over loci} the effect of selection
on change
in gene frequencies and disequilibria depended on the gene
frequencies of the loci in question.
As with runs involving equal
initial gene frequencies at all loci} the responses in l1P~O), H~s}O)}
~
and
~
D~~'O) (recall that R~~}O) = D~~}O)) were asymmetrical with respect
~J
~J
~J
to the deviation of the initial gene frequency (frequencies) of the
locus (pair of loci) from .5.
And the direction and nature of the
asymmetry were similar to the asymmetrical response described before.
For
p~O) < p~~) = l_p~O)} certain relations held.
~
~
changes in gene frequency
~
For initial
88
~p~O)
< ~p ~~~)
for
O!
~p~~)
for
O! =
> ~~~)
~
for
O!
=
~
~
< .5;
(80)
.5;
> .5
.
and, as in equation (71),
~ (0)
p.~,
O!
1-0!
=
(0)
~p i' ,l-O!
(81)
a
For the H type disequilibria in the selected population
with
H~~) being the more negative of the two at
~
a = .9.
Furthermore,
analogous to equation (77)
- [ a( ~p .
)
2,
+ O!
~,Of
(~p.
I
~,
1
2 (0)
)]
= (QH.
~,O!
_Of
+ (1- a) H., 1
~,
) (s, 0) .
-of
(83)
Table 7 gives examples of ~p~O) and H~s,O).
~
~
The D type disequilibria are not so easily characterized;
nonetheless, when contrasting the disequilibrium at loci i,j with that
at loci il,j' such that
l_p~O)
~
and
89
Table 7.
Initial response in change in gene frequency and intra10cus
disequilibria for runs involving varying initial gene
frequencies at five loci
Proportion
Saved
ti
Ot
ti
= .9
= .5
= .1
Initial
Gene
Frequency
p2 (0) =. 25
(Varying
Gene
. )a
Fre uenc~es
H(s)O)
lip (0)
p2 (0) =. 75
(Varying
Gene
Frequencies) a
H(s)O)
/},p(O)
p2(0)=1
(Varying
Gene
a
Freq uenc ies)
/},p (0) H(s)O)
.1
.005
3
-.0 27
.008
_.0 362"'(
.009
_.0 3 74
.3
.014
-.0017
.022
-.0045
.024
-.0056
.5
.019
-.0027
.032
-.0086
.036
-.0114
.7
.017
.032
-.0070
.038
-.0094
.9
.008
-.0021
_.0 3 39*
.016
-.0011
.019
-.0014
.1
.027
-.0010
.046
-.0034
.050
-.0041
.3
.065
-.0050
.1l2
-.0176
.123
-.0218
.5
.077
-.0060
.137
-.0188
.151
-.0228
.7
.065
-.0033
.1l2
-.0076
.123
.046
_.03 79 ·k
.050
-.0085
-.0 382
·k
.9
.027
_.03 46
.1
.073
-.0024
.144
-.0128
.171
-.0200
.3
.155
-.0085
.288
_.0297
.340
-.0443
.5
.167
-.0065
.284
-.0127
.324
-.0136
.7
.127
.199
.049
-.0033
_.0 326*
.220
.9
-.0026
_.0 329*
-.0030
_.0 3 23'"
.072
.079
a* Numbers of the form .00099 are expressed as .0 399.
the following relations held:
(84)
with
Di~j~)
being less than
Di~)O)
for
ti
= .9.
Thus) substituting the
*
90
initial gene frequencies at loci i,j for the subscripts in D.. , the
~J
following relations held for the five locus model at
D(S"O)
.1, . 3
< D(s,O)
.7, .9
n(s,O)
.1, .5
<
D(s ,0)
.1,. 7
< D(s,O)
.3,.9
D(s,O)
.3, .5
< D(s,O)
.5,.7
~J
.5:
D(s,O)
.5,.9
These inequalities were reversed for
D~~'O)
~ ~
~
= .9;
~.~.,
compare values of
between positions symmetrical around the secondary diagonal of
the matrices given in Table 8.
Similarly, for disequilibria having
locus i in common, the following relation held:
l_p (.O) an d
J
with reversal of the inequality at a
~
.9.
~ ~.
5 ,
Examples may be seen in
Table 8 by making appropriate comparisons within the column-row
gssociatE!dwith a specific initial gene frequency.
The response from this point on (t
expected.
~
1) was as one would have
The disequilibria at generation 1 and the gene frequencies
at generations 1 and 2 were independent of linkage, but, thereafter,
disequilibria buildup was greater and subsequent progress was more
retarded for more tightly linked systems.
Comparing digenic disequi_
1ibria having one locus in common, disequilibria responses were
eventually ordered according to their initial gene frequencies (see
Figure 6).
However, the minimum values were not so ordered, because
the greatest negative values attained for any given disequilibrium was
e
e
e
Table 8.
Matrices of off-diagonal elements representing initial response in inter10cus
disequilibria, D~~'O), to selection; five locus runs with varying initial gene
1J
frequencies given in the upper and left hand margins of the matrices (upper offdiagonal, p2(0) = 1; lower off_diagonal, p2(0) =.25)
0
cr-=.9
p
(0)
Q'=.5
.9
p (0)
.1
-.0013
•1
--
_.0 3 67 *
- - -.0081 -.0079 -.0034
3
. 5 _.0 86 -.0021
- - -.0108 -.0045
3
*
.71-.0 77 -.0019 -.0024
- - -.0036
3
3
.9 _.0 34* _.0 84* -.0010 _.0 390*
--
.3
-.0023
.5
.0025
.1
.3
.1
.3
r---
I
-.0020
.5
.7
-.0029 -.0029
oJ•
n
.7
.3
.5
.7
.9
-.0095 -.0105 -.0073 -.0025
- - -.0231 -.0152 -.0046
-.0055
- - -.0141 -.0045
-.0019 -.0042 -.0045
- - -.0027
3
.0 74* -.0016 -.0017 -.0012
.9
0'=.1
p (0)
.3
.5
.7
.9
.1
--
-.0317 -.0213 -.0108
-.0030
.3
-.0045
- - - -.0248 -.0118
-.0032
.5
_.0041
.7
_.0083
-.0027 -.0047 -.0041
3
3
_.0 91* -.0016 -.0014 _.0 87*
.9
a
.1
-.0073
- - -.0064
-.0018
-3
* Numbers of the form .00099 are expressed as .0 99.
\0
I-'
e
e
e
Generation
q
5
Generation
10
15
.v"
.(,
0
5
10
15
;~o
"1'''",
.:) , , 1
5
D:1
. , ..7
T\
J.J co
T1
co
Cd
$-I
,.0
l-l
/, . 1
.,-l
_
'"'.3,.7
.,-l
.,-l
3 J 0.5
_':l
--i
Ii. 3 ).9
,....j
,....j
.,..;
:::l
~ .•• 005
I' 7,.3
.,-l
., . 5
ry
:::l
D"7
tlJ
"~
CJ
(!J
.-1
'1'
-::
n .0
"'7,.9
.0
-" .. j / ~
1
... 00::
'-'.5,.3
D
"
-
D. L.0
• ..) , • i
r-
oll-
.), .9
_
f • ., ' \
• ·.. /..1-\.1
Figure 6.
Intragametic digenic disequili!:ria betwec~ five loci each with five different initial
gene frequeaelcs, the initial gene fre,que:,;cie~~ given as subscripts to the disequilibria (run i:lVolving p2 (0) =.25, (yo.5, ;:]"d e. ," ,:;C:. 5)
1.,1+J.
\.0
N
93
limited, in part, by the gene frequencies attained at both loci.
But
for comparisons across all disequilibria, the maximum negative value
attained was always greater between those pairs for which the initial
gene frequencies were lower at both loci.
The reduction in genotypic variance due to disequilibria was
affected by changes in the ·levels of c.. I'
~,~+
Ot,
n, and p2 (0) in a
manner similar to runs involving equal initial gene frequencies.
Comparisons in M(t) for runs involving unequal initial gene frequencies·
can be made in Table 9. . Two points should be made concerning this
table.
First, for ci,i+l ::;: .1, position effects are confounded with
the effect
o~
the initial gene frequencies at a particular locus.
But
since the effect of the position of a locus on a chromosome was minor
in runs involving equal initial gene frequencies per locus, the same is
likely to be true here also.
investigated for equal
Second, as in the set of comparable runs
in~tial
gene frequencies, a change in n affects
a .~n such a way as to yield a constant genotypic variance for given
-(J
s
2 (0)
values of p
.
First generation comparisons were made between runs involving
equal and unequal initial
Ot::;:
.1, .5, and .9.
~ene frequencies at n ::;: 5,
p2(0) ::;:
I, and
The comparisons involved 6piO) and fiis,O) at piO)
for a run involving unequal (U) gene frequencies versus 6p(0) and
D(s,O) at p(O) ::;: p~O) for a corresponding run involving equal (E)
~
initial gene frequencies.
The data are presented in Table 10.
At loci with intermediate gene frequencies, 6p was greater for the
E runs} whereas 6p was greater for the U runs in cases involving extreme
gene frequencies.
Scewness in response with respect to
Ot
at p(O) ::;: .3
Table 9.
Proportional reductions, M(t), in genotypic variance due to disequilibria for varying
initial gene frequencies within runs (generations given in parentheses)
Proportion
Retained
~
= .9
Lite
-Average
Gene
Frequency-
1/4
_.625_
1/2
-.75 -
3/4
-.875Ot
e
e
e
= .5
1/4
-.625-
1/2
-.75 -
3/4
-.875Ct'= . 1
1/4
-.625-
1/2
l(O) =.25
c=.5
c=O
c=.l
n=3
n=5
n=3
n=5
.040
(8.068)
.048
(10.447)
.105
(8.295)
.125
(10.865)
.157
. (8.432)
.229
(11.340)
.032
(17.096)
.041
(22.131)
(18.108)
.126
(23.597)
.261
(19.497)
.349
(27.072)
.014
(27.919)
.025
(37.007)
.054
(29.694)
.086
(39.619)
.140
(33.138)
.336
(49.924)
.069
(1.919)
.088
(2.528)
.087
(1. 919)
.122
(2.539)
.092
(1.919)
.138
(2.544)
.066
(4.105)
.083
(5.446)
.122
(4.177)
.164
(5.620)
.148
(4.204)
.228
(5. 739)
.033
(6.710)
.052
(9.091)
.075
(6.911)
.127
(9.607)
.101
(7.011)
.216
(10.113)
.055
(0.849)
.082
(1. 110)
.055
(0.849)
.084
(1. 110)
.055
(0.849)
.086
(1. 110)
(1. 811)
.143
(2.442)
.068
(2.987)
.137
(4.194)
.117
-.75 -
(1. 811)
.094
(2.428)
(1. 811)
.127
(2.438)
3/4
-.875-
.041
(2.971)
.064
(4.077)
.060
(2.983)
(4.155)
.066
.081
.no
n=3
.086
n=5
continued
1.0
.po.
-
e
e
Table 9 (continued)
Proportion
Retained
Q'
Q'
O!
= .9
= .5
= .1
Life
-Average
Gene
Frequency-
n=3
n=5
n=3
n=5
n=3
n=5
1/4
_.625_
.109
(4.815)
.134
(6.291)
.206
(4.943)
.265
(6.619)
.354
(4.998)
.379
(6.899)
1/2
-. 75 -
.105
(10.330)
.121
(13.633)
.305
(11.390)
.283
(14.873)
.464
(13.132)
.500
(17.619)
3/4
-.875-
.024
(17.161)
.086
(22.079)
.087
(19.282)
.246
(25.079)
.152
(22.341)
.440
(36.186)
1/4
-.625-
.167
(1. 085)
.213
(1. 454)
.173
(1. 085)
.240
(1. 454)
.174
(1.085)
(1. 454)
1/2
-.75 -
.184
(2.419)
.230
(3.224)
.259
(2.448)
.343
(3.309)
.284
(2.457)
.408
(3.355)
3/4
-.875-
.095
(3.897)
.168
(5.357)
.132
(4.013)
.296
(5. 767)
.147
(4.056)
.395
(6.132)
1/4
-.625-
.092
(0.494)
.156
(0.633)
.092
(0.494)
.156
(0.633)
.092
(0.494)
.156
(0.633)
1/2
-.75 -
.184
(0.987)
.242
(1. 366)
.184
(0.987)
.264
(1. 366)
.184
(0.987)
.272
(1. 366)
3/4
-.875-
.101
(1. 639)
.191
(2.307)
.102
(1. 639)
.238
(2.318)
.102
(1. 639)
.257
(2.323)
p2(O) =.75
c-.l
c-.5
c=O
.252
continued
\0
\.1l
e
e
e
Table 9 (continued)
Proportion
Retained
ex = .9
ex
= .5
ex = .1
Life
-Average
Gene
Frequency-
n=3
n=5
n=3
n=5
n=3
n=5
1/4
-.625-
.136
(4.364)
.175
(5.578)
.232
(4.443)
.315
(5.867)
.277
(4.477)
(6.098)
1/2
-. 75 -
.146
(9.337)
.149
(11.959)
.364
(10.550)
.328
(13.573)
.473
(12.808)
.526
(15.973)
3/4
-.875-
.021
(15.967)
.132
(19.499)
.088
(18.245)
.304
(22. 713)
.153
(21. 499)
.440
(34.948)
1/4
_.625_
.186
(1.011)
.251
(1. 315)
.187
(1. 011)
.273
(1. 315)
(1. 011)
.283
(1. 315)
1/2
-.75 -
.238
(2.215)
.289
(2.909)
.315
(2.233)
.399
(2.961)
.338
(2.238)
.455
(2.986)
3/4
-.875-
.112
(3.572)
.232
(4. 719)
.144
(3.667)
.361
(5.103)
.154
(3.697)
.442
(5.352)
1/4
_.625_
.087
(0.455)
.191
(0.552)
.087
(0.455)
.191
(0.552)
.087
(0.455)
.191
(0.552)
1/2
-.75 -
.173
(0.909)
(1. 145)
.173
(0.909)
.339
(1. 145)
.173
(0.909)
(1. 145)
3/4
-.875-
.106
(1. 444)
.264
(1. 923)
(1. 444)
.106
(1.444)
.307
(1. 923)
p2(0) =1
c=.5
c=.l
.334
.106
c=O
.295
(1. 923)
.188
.4~9
.341
\0
0\
97
Table 10.
6p(0) and D(s}O) for equal (E) initial gene frequencies
within a run versus 6p~0) and D~s}O) for unequal (U)
~
~
initial gene frequencies within a run; contrasts at
comparable initial gene frequencies from runs involving
n = 5 and p2(0) = 1
6p (0)
Initial
0'=.9
Gene
Frequency
E
U
0!=.1
0'=.5
E
U
E
U
.1
.009
.011
.050
.070
.171
.185
.3
.024
.025
.123
.112
.340
.260
.5
.036
.030
.151
.123
.324
.266
.7
.038
.029
.123
.112
.220
.228
.9
.019
.021
.050
.070
.079
.100
Average
.025
.023
.099
.097
.227
.208
D(s}O)
Initial
Q'==.5
0'=.9
Gene
Frequency
U
E
U
a=.l
E
a
E
U
.1
-.0023
-.0012
-.0075
-.0088
-.0167
-.0175
.3
-.0054
-.0059
-.0131
-.0158
-.0179
-.0215
.5
-.0067
-.0071
-.0131
-.0151
-.0136
-.0146
.7
-.0063
-.0059
-.0098
-.0073
-.0056
.9
-.0032
-.0023
-.0036
-.0093
3
-.0 91*
-.0022
0.0
Average
-.0048
-.0045
-.0094
-.0100
-.0116
-.0118
a
*
3
Numbers of the form .00099 are expressed as .0 99.
98
and p
(0)
=
.7, resulted in 6p being greater for E at p(o) = • 7 for
.1 but less at p
01=
(0)
=
. 3, whereas the opposite was true at
Of =
.9 .
The average change in gene frequency was greater for the U runs at
.1,.5 and .9, but the difference was negligible.
01
The disequilibria responses are more difficult to characterize.
D(s,O) was always greater in magnitude for the E runs at p(O) = .3 and
.5 and for the U runs at p(O)
=
more negative for the E runs at
01
at
= .9.
01
=
.7 and .9.
01 =
For p(O)
=
.1, D(s,O) was
.5 and .1, but less negative at
Likewise, the average D(s,O) was more negative for the E runs
.5 and .1 but less at
Ol
=
.9.
But once again} the differences
between the U and E averages were negligible.
Beyond the first generation of selection, the two types of runs
became so different that no meaningful comparisons could be made.
Heritabilities of less than I were not considered because of problems
associated with constant heritabilities versus constant environmental
variances for runs involving different levels of equal initial gene
frequencies.
Summary and Discussion of Parametric Effects
The effects of the various parameters on the selection process are
outlined below:
1.
Linkage:
A.
no effect of linkage on disequilibria in the first
generation;
B.
no effect of linkage on the gene frequency in the first
two generations;
99
c.
as expected, increased buildup in disequilibria with
tighter linkage after the first generation resulting in
D.
a decrease in the genotypic variance leading to
E.
retardation of genotypic progress with tighter linkage
after two generations;
F.
ultimately, greater buildup in disequilibria between those
pairs of equidistant loci enclosing more centrally located
segments on the chromosome; and
G.
greater advance for loci located at a greater distance
from the center of the chromosome.
II.
Intensity of Selection, interaction with linkage:
A.
greatest reductions in genotypic variance due to disequilibria occurred at moderate intensities of selection
for runs involving free recombination.
B.
greatest reductions occurred at low intensities for runs
involving tight linkage.
III.
Number of Loci, increasing the number of loci usually resulted
in:
A.
a decrease in the negative magnitude of the pairwise
disequilibria;
B.
a decrease in the genotypic variance attributable to the
2n(n_l) negative interloci correlations; and
c.
a retardation of genetic progress as measured by time to
quarter, half, and three_quarter lives.
100
IV.
Heritability} increase in heritability resulted in:
A.
an increased rate of buildup in negative disequilibria;
B.
but an increased advance in gene frequency attributable to
the increased per locus effe·ct.
V.
G~neFrequencies:
Initial
A.
int~factio~
1.
with
~inkage
ultimately produced similar response curves in disequiHbria with respect to attained gene frequencies
for allp (0) under free recombinations}
2.
but produced greater negative buildup in disequilibria
for smaller values ofp(O) throughout the selection
program for runs involving complete linkage.
B.Interaction with intensity of selection produced asymmetry
in the initial advance in gene frequency and disequilibria
with respect to initial gene frequencies equidistant from
.5.
Within the aystems i,nvElstigated} the effect per locus was large.
tn order to attain the value
.!.. -
.1
0"£
.
2(0)
for p
.
l;:.
25· .;lnd p
to be equal to 67.
(0)
=.5} the number of loci involved would need
Although selection operating on such a system would
produce disequilibria of negligible magnitude} the effect of those
disequilil;>ria over a114}422 pairs of loci would probably be far from
in$ignifica~t; ev~n
in systems involving free recombination.
therefore, sufficient
reaso~
There is}
to be concerned about the effect that
101
selection induced disequilibria has on progress in a recurrent selection program.
102
PROJECTION OF THE GENOTYPIC VARIANCE
The effect of selection induced intergenic correlations has been
largely ignored in equations used to project the genotypic mean and
variance from one generation to the next.
The purpose of this chapter
is to evaluate projection equations for the genotypic variance and to
discuss the circumstances under which they can best be used.
Four
different projection equations are presented here and are evaluated in
investi~ations
terms of the empirical
described earlier.
Two of the projections have been presented in the literature.
The
first, proposed by Reeve (1953), is
=
1
in which 2 p
1
2 (t) K(t)).
(1 + 2 P
2 (t)
~G
(85)
2 (t)
is the square of the correlation between the gametic
value and the parental phenotypic value in the test population and
,
(86)
the change in the parental phenotypic variance relative to the variance
in the test population.
The second projection, attributable to Nei
(1963), is the approxima tion
(87)
wherein
p. (l-p.) -,:. w= (t)
I!:.p. (t)
. :: [~
~
0
]
~
2w
op.~
(Wright, 1942) .
103
The final two projections to be considered are weighted averages
of the selected parental genotypic variance, equation (34), and the
genotypic variance under intergenic independence, equation (38); they
are
(88)
which is equal to the last projection,
(89)
as n goes to infinity.
Under free recombination these last two equa-
tions have certain desirable properties which will be discussed later.
Reeve's Projection
Reeve's projection, equation (85), can be derived via regression
theory under the assumptions that
(a) the gametic response is linearly dependent on the parental
phenotypic value,
(b) the gametic variance is constant over all values of the
phenotype,
(c) the gametic variance is half the additive variance of the
parental test population,
(d) the covariance between gametic values and parental values for
the test population is half the additive variance in the
parental population.
(Assumptions (c) and (d) are necessary
if the correlation between gametic and parental phenotypic
values is to be
1
\f2
104
(e) The offspring variance is twice the gametic
variance.
Assumption (a) would be expected to hold asymptotically with
respect to increasing the number of loci.
Ignoring superscripts, the
gametic response, G* , is linearly dependent on the parental genotypic
value, G;
G* = }
G + 5 ,
in which
E( 5) = 0
and
E(5
2
IG=g)
for the phenotypic model (29), i f the number of loci is large enough to
assume that the genotypic distribution is approximately normal, then
the regression of the genotypic value on the phenotypic value is
approximately linear with conditional mean and variance
(90)
and
(91)
And, since the gametic value is linearly related to the parental
genotypic value, the gametic value is also approximately linearly
105
dependent on the phenotypic value with conditional expectations
~
1
G* ly
=-
2
(~
G
+ p
crG
cry
(Y-~y)]
(92)
and
(93)
However) assumption (b») that the gametic variance is constant for
all phenc;>t;yp:lc values) is usually invalid.
independent and
identi~a1
For example) in the case of '
allelic distributions for all loci) the
di,stribution of the number of favorable alleles per gamete conditional
on the parental genotypic value) g) is hypergeometric with population
si~e
2n) sample $ize n) and conditional gene frequency
*1
pG g
= ..L
2na
therefore
"
ag(2n - !)
a
(94)
= ~z:(~2:-n'-'-1'"');"""
/
By using equations' (91) and (94») equation (93) can be written
J
2
CTG*ly
And) s:i,.nce
~Gly
= 1:.
4
[2(n_1)
2 2
~ly(2na_~Gly)
2n-1 (l-p )crG +
2n-1
]
2
(90) is a function of y) cr
is not constant for all
G*ly
Yj
consequently) assumption (b) does not hold except asymptotically as
p goes to zero.
106
As~umption
(e) is valid under random pairing of identical but
unrelated sets of gametes, and assumption (d) is always valid for an
additive genotypic model (see equation (24».
that the gametic variance is
ha~f
However assumption (c),
the parental, is only valid in the
case of intergenic independence or in the presence of complete linkage.
Otherwise
2
0'.
G*
>
1
2
'2O"G
due to recombination (compare equations associated with equation (14)
to those associated with equation (3); recall that Rand H type disequilibria are zero in the test population).
If the parental genotypic value is linearly dependent on the
phenotype with constant variance over all phenotypic values, then the
genotypi~
variance in the selected population at generation t is
2(t)
= (1
+ p
(t)
2 (t)
K ) O"G
,
and Reeve's projection, equation (85), can be rewritten
Using equation$ (34) and (35), the difference between this projection
and the actual variance is found to be
d(t+l)
1
"2 (t+l)
= 10"G
2 (t+1)
O"G
(95)
+
r; a.a.[(R(s,t) _ AD(t) +2c(D(s,t) _ R(S,t»]i'
ifj ~ J
J
107
wherein
AD~~) = D~~} t)
1J
1J
Thus} for negative-digenic disequilibria} the sign of d
(t+l)
can be
l
either positive or negative} depending on the sign and magnitude of
I::J)
g)
and on the magnitude of the other parameters.
A brief discussion of d(t+l) is presented here for the hypotheti1
cal case in which equal gene frequencies} equal intragametic digenic
disequilibria} etc.} are specified at each generation.
For systems in
t l
which intergenic covariances are absent} di + ) will be positive if
p(t)
>} (1_6p(t))
and negative otherwise.
The presence of intergenic
correlation will tend to push di t +l ) in the negative direction since
R(s}t) is usually more negative than l::J)(t) and since n(s}t) is either
equal to or more negative than R(s}t).
The frequency of recombination
affects the bias whenever n(s}t) and R(s}t) are unequal} the contribution being least under complete linkage since the major source of
disequilibrium} n(s}t)} did not add to the bias.
The contribution of
disequilibria to the bias may be maximized under free recombination
since n(s}t) receives its greatest weighting at that frequency of recombination; however} the buildup in n(s}t) is also the smallest under
those conditions and R(s}t) doesn't contribute at all.
Neils Projection
"2 (t+l)
.
} equat10n (87)} is only applicable to
Nei points out that Zrr
G
situations in which 6Pi is small and intergenic independence is
specified.
In terms of the additive genotypic model}
108
2
2 2: a.~ [p.q.
+
~ ~
•
(q~-p~)f!.p~J
(t)
;
.L.L.L
~
and the deviation of this projection from the actual genotypic variance
is
(t+l)
d
=
2
2 (t+l)
O"G
"2 (t+1)
20"G
=2
2
{2: a. (f!.p ~ t» 2
i
~
a.a.[(l-c .. )D .. + c .. R.. J(s}t)}
2:
irj
~
.
negat~ve}
Thus} for D and R
ij
ij
nverprojection.
(96)
d
~
J
~J
~J
~J
~J
(t+l).
"2
~s'positive and 20"G strictly an
2
In equation (96)} substituting in place of Neils
projection} the variance under intergenic independence} equation (38)}
and retaining the (f!.p)2 terms eliminates that portion of the over2
projection due to (f!.p.) .
With this redefinition)
~
(s) t)
d 2(t+l) = - 2 2: a.a.[(l-c .. )D .. + c .. R.. J
irj ~ J
~J
~J
~J ~J
(97)
The greater the buildup in D~~}t) relative to R~~}t) under tighter
~J
~J
2 (t+l)
linkage} the more of an overprojection 20"G
becomes.
Other Projections
"2
The difference between 30"G and the actual variance in generation
t+l is
d (t+l)
3
2(t+l)
O"G
"2 (t+l)
= 30"G
2
- 2n-l
2:
irj
a.a.(R .. -D .. )[n-(2n-l)c .. J
~
J
~J
~J
2
+ (n-l) 2: a.H.
- 2: a.a.R .. }(s}t)
i ~ ~
irj ~ J ~J
~J
(98)
109
which reduces to
2
2a
( 2: [n-(2n-1)c .. ](R .. -D .. ) + n(n_1) (-H-R-)} (s}t)
d (t+1) z::-;3
Ln- J.
• .L'
~J
~J
~J
~iJ
(99)
for equal additive effects.
Thus for runs in which
for all i
and
D(.S.}t)
= D(s}t)
f or a 1 1
···.
L ·J }
~}J}
~ i
~J
,,2(t+1) ~s
. an
3~G
.
.
overproJect~on
. more
wh enever D(s}t) ~s
R(s}t) and is exact whenever n(s}t) equals R(s}t).
.
negat~ve
t h an
The first term in
equation (99) is positive} and since tighter linkage results in a
greater buildup in D~~}t) and almost always results in a reduction in
~J
the
.
magn~tude
(s t)
of R } }
ij
increased linkage.
"2 (t+1)
3~G
However
becomes more of an overprojection with
"2 (t+1)
3~G
can be an underprojection whenever
the second term is negative and greater in absolute value than the
first term.
Regarding 4;~(t+1)} the deviation from the actual variance is
2 (t+1)
d (t+1)
4
~G
=
(-D'
· )(s}t) + 2: a 2.H.(s}t)
a.a. ( 1
- c2
.. · )
R..
ifj ~ J
~J
~J
~j
i ~ ~
2:
which} for equal additive effects per locus} reduces to
d (t+1) = a 2 ( 2: (1_2c .. ) (R .. -D .. ) (s) t) + nil(s} t)}
4
ifj
~J
~J
~J
(100)
110
( s t)
Thus for R.. '
~J
(s t)
= D..
'
~J
or for free recombination,
strictly an underprojection,
"'Z (t+l) .
4~G
~s
However, under tighter linkage, whenever
the D type covariances are sufficiently greater than the Rand H types,
"Z(t+1)
4~G
will be an overprojection,
Evaluation of the Projection Equations
In order to evaluate the projections for the empirical investigations described earlier, the bias measures
~ (t+1)
K
d
=
K
(7
(t+1)
,
K
=
l,Z,3,4
(101)
~G
were used in which d , d ' d , and d were the deviations of the
1
Z 3
4
projected variances from the actual as defined in equations (95), (97),
(99), and (100), respectively,
The projection equations were evaluated
on the basis of which yielded the minimum bias.
Two statements can be made for all selection runs investigated
here:
first, it can be shown analytically that in the presence of
negative disequilibria
(t+1) < Q (t+1)
(t+1)
~4
~3
< ~Z
'
(lOZ)
and second, as pointed out earlier, Nei's projection was always an
overprojection; therefore
Q
(t+1)
~Z
>
0 •
(103)
The projection equations are evaluated in three different subsections.
The first deals with generations for which the selected
parental intergametic_ and intragametic-inter1oci digenic
111
disequilibria were equal within each pair of loci.
In the second and
third subsections, the projection equations are evaluated over all
generations for runs involving free recombination and tight linkage,
respectively.
Equal Parental Intergametic- and Intragametic_Inter1ocus Digenic
Disequilibria within Each Pair
Equality of
R~~,t) and D~~,t) existed in the first generation of
~J
~J
selection or whenever disequilibria reached their minimum possible
values.
Furthermore, in runs involving equal initial gene frequencies,
all digenic disequilibria (including the H type) were equal over all
loci in the (s,O) population and were equal whenever the lower limits
in disequilibria were attained under free recombination or complete
linkage.
Under such circumstances
A2(t+1) was exact (refer to
30"G
equation (99».
However, for runs in which intergenic associations differed over
loci, R(s)t) was usually nearer to zero than H(s)t).
This resulted in
the last term of equation (99) being negative; therefore) whenever
D~~)t) was equal to R(s J t) for all pairs of loci)
ij
~J
underprojection.
"2(t+1)
30"G
,
was an
Such was the case in the first generation for runs
involving unequal initial gene frequencies and was also the case whenever the minimum possible values of R~~,t) and D~~)t) were attained for
~J
~J
runs involving unequal initial gene frequencies or for runs involving
equal initial gene frequencies and c . . 1
~)~+
= .1.
In the first generation after selection in runs involving
- (0)
p
= .5)
~
= .5,
and symmetry of initial gene frequencies around .5) Reeve's
projection was exact.
For such runs d(l)
1 ) equation (95») reduced to
112
(104)
which was zero under the restrictions imposed by equation (83) at
~ = ,5 and p(O) = ,5,
at
~
=
.1 or .5 and p
However} at ~
(0)
=
"2(1)
l~G
.3}
=
,9 and p(O)
=
,3} .5} or .7 or
was an underprojection} but at
~ = .1 and p(O) = .5 or .7 or at ~ = .5 and p(O) = , 7} 1;~(1) was an
overprojection.
Thus} systems that tended to increase the rate of
progress towards fixation by increasing p(O) or decreasing ~ in
comparison to a = .5 and
p(0)
=.5 caused
"2(1)
~
1 G
to be an overprojec-
tion;whereas systems that tended to decrease the rate of progress
caused
1~~(1) to be an underprojection,
Tables 11 through 16 provide
examples of the bias for Reeve1s projection as well as those for other
projections at t
= 1,
The Case of Free Recombination
Under free recombination p~t+1) was always less than zero (see
Tables 11 through 16)} and throughout runs involving equal initial gene
frequencies, p
( t+1)
3
was equal to or greater than zero (see Tables 11
"2 (t+1)
through 14); therefore, 3GG
and
"2 (t+1)
4~G
formed respective upper
and lower bounds on the actual variance.
(t+1)
In almost every generation P3
was nearest zero, the order of
magnitude of the bias being 10-
2
or less.
Even in runs involving un-
equal initial gene frequencies (for which p~t+1) was negative in the
first generation and near fixation)}
"2 (t+1)
3~G
was usual nearest the
actual genotypic variance,
In some runs, other projection equations were best for a few
generations.
Near fixation in runs involving unequal initial gene
113
Table 11.
Biases associated with projections of genotypic variance
under free recombination for p(O) ;:: .3, p2(0) ;:: 1, and
n ;:: 4 (bias nearest zero appears with asterisk)
Proportion
Retained
Ot ;:: .9
•
Ot ;:: .5
Ot ;:: .1
t
- (t)
p
2 (t)
erG
p(t)
2
1
a
P(t)
2
P(t)
P(t)
3
4
0
1
.3
.327
1. 680
1. 588
-.044
.109
5
.
.441
1. 619
-.137
.219
.017i(
-.017
10
.589
1.595
-.120
.214
.016*
-.017
15
.735
1. 291
-.091
.206
.014*
-.018
20
.
.867
0.752
-.044
.223
.010*
_.025
25
26
27
28
.962
.975
.988
1.0
0.255
0.178
0.093
0
.054
.162
.433
.132
.082
.038
0
1
2
3
4
5
6
7
.3
.428
.549
.665
.772
.865
.940
1.0
1.680
1. 458
1. 342
1. 201
.978
.692
.365
0
_.153
-.241
-.228
-.149
-.009*
.291
.343
.476
.485
.440
.347
.236
0
1
2
3
.3
.588
.822
1.0
1.680
1. 317
.800
-.176
.034
.472
.465
O~'(
-.018
0*
-.022
-.014
_.006
Oi<
..-.057
0*
Oi<
.028*
.031i<
.026*
.014
O~'(
o~'(
Oi<
-.046
-.045
-.043
-.042
-.039
_.079
-.058
114
Table 12.
Biases associated with projections of genotypic variance
under free recombination for p(O) = .5, p2(0) = 1, and
n = 4 (bias nearest zero appears with asterisk)
2 (t)
Proportion
Retained
0'=.9
t
0' = . 1
erG
a
2
~ (t)
~ (t)
~ (t)
~ (t)
0*
-.021
1
2
3
4
0
1
.5
.533
2.0
1. 773
-.018
.123
5
.656
1. 478
-.104
.222
.014*
-.020
10
.797
1.070
-.086
.209
.014*
-.019
15
.913
.544
-.010
.168
.008*
-.019
20
21
.990
1.0
.075
0
.561
.030
0
1
2
3
4
5
.5
.637
.749
.846
.928
1.0
2.0
1.402
1. 087
.769
.412
0
0*
_.090
-.012*
.230
.320
.383
.354
.306
0
1
2
.5
.799
1.0
2.0
.870
0
.331
.477
.
0' = .5
_(t)
p
0*
0*
.018*
.015
0*
0*
-.005
-.053
-.043
-.041
-.051
-.080
•
115
Table 13.
Biases associated with projections of genotypic variance
under free recombination for p(O) = ,5, p2(0) = ,25, and
n = 4 (bias nearest zero appears with asterisk)
2 (t)
Proportion
Retained
0'=.9
t
_(t)
(JG
p
a
2
~ (t)
~~t)
~ (t)
~ (t)
1
3
4
,5
.517
2.000
1. 942
-,004
,028
o·/(
-,005
.584
1. 847
-.025
.053
.004*
-.005
10
.665
1. 695
-,019
,052
.004*
-,004
18
19
20
21
22
. 782
.795
.808
,820
.831
1. 306
1.250
1.194
1. 137
1.080
-,005
-.003*
p,OOl*
.001*
.004
,045
.043
,042
,040
.039
.003*
.003
.003
.003
,003
-.004
_,004
-,004
-,004
-.003
30
.907
.656
.021
.026
.002
-.002
40
.962
.290
.039
.013
.001*
_.001
50
,986
.111
.050
.005
0
1
2
3
4
5
.5
.571
,637
.698
.755
,804
2.000
1. 838
1. 696
1. 539
L 360
1. 165
0
-,013
-. 005~'(
,013
,038
.066
.091
.095
,089
.079
.005*
-.011
-.010
-,009
-,008
-.007
,937
,335
,163
,025
,002*
_.002
.999
.009
,230
.001
0
1
··
·
5
·
·
•
··
0'= ,5
0(+)*
0*
.005*
.006
,006~'(
0(-)
·
,
10
,
20
0(+)'>'(
0(=)
continued
•
116
Table 13 (continued)
2 (t)
Proportion
Retained
O! =
•
.1
Table 14.
t
._ (t)
p
O"'G
a
0
1
2
3
4
5
6
7
.5
.654
.779
.871
.932
.966
.984
.992
2
2.000
1.672
1. 262
.839
.490
.258
.127
.060
~ (t)
1
~ (t)
2
~ (t)
.043
.107
.212
.332
.438
.512
.555
.082
.091
.068
.041
.022
.011
.052
0*
.005*
.004*
. 00 3~(
.001*
.001*
0(+)*
3
~ (t)
4
-.014
-.010
-.007
-.004
_.002
-.001
-.001
Biases associated with projections of genotypic variance
under free recombination for p(O) = .7) p2(0) = 1) and
n = 4 (bias nearest zero appears with asterisk)
2 (t)
Proportion
Retained
O! =
.9
t
O! =
.5
.1
O"'G
a
2
~ (t)
~ (t)
~ (t)
.011*
1
2
3
0
1
5
.7
.732
. 840
1. 680
1. 392
.900
-.019
-.056
.127
.195
··
·
10
·
.948
. 332
.059
.199
0*
. 0 (+)
0'1(
14
15
O! =
_ (t)
p
.999(+
1.0
.001
71. 49
O~'"
~ (t)
4
-.021
-.020
-.033
0(-)
0
1
2
3
4
.7
.828
.919
.989
1.0
1. 680
.850
.439
.086
0
.262
.226
2.03
.340
.362
.353
0*
0*
0*
-.057
-.060
-.006
0
1
2
.7
.947
1.0
1. 680
.334
0
1. 882
.201
O~'(
-.033
117
Table 15.
Biases associated with projections of genotypic variance
under free recombination for varying initial gene frequencies within runs) p2(0)
appears with asterisk)
= 1)
and n
=5
(bias nearest zero
2 (t)
Proportion
Retained
Ot =
.9
t
- (t)
p
erG
a
2
~ (t)
~ (t)
~ (t)
1
2
3
1.700
1.475
1. 390
=.008
-.074
.129
.175
-.003*
.004*
-.019
-.017
5
.613
1. 241
-.104
.220
· 006~'(
-.021
10
·
.714
1.034
-.100
.204
.006*
-.019
···
15
.806
.793
-.090
.180
-.001*
-.023
20
.881
.573
-.080
.130
-.008*
_.025
·
·
25
··
.947
.395
-.045
.038*
-.096
-.113
.991
.084
0
.488
.006*
-.017
-.020
-.148
-.159
-.123
-.175
.079*
.312
.386
.409
.469
.246
.138
-.008
.006*
· 00 1~(
· 002~'c'
-.059*
-.085
-.048
-.042
-.050
-.056
-.097
-.113
.116
-.081*
.531
.347
-.024*
-.086
-.093
_.141
··
·
29
30
Ot =
.5
.1
4
.5
.525
.548
0
1
2
·
··
Ot =
~ (t)
0
1
2
3
4
5
6
7
0
1
2
3
LO
.5
.599
.681
.757
.832
.892
.947
LO
.5
.727
.887
LO
1. 700
1. 207
1.028
.864
.647
.594
.391
0
1. 700
.872
.595
0
O~(
118
Table 16.
Biases associated with projections of genotypic variance
under free recombination for varying initial gene frequencies within runs) p2(0) = .25) and n
appears with asterisk)
Proportion
Retained
at = .9
t
p
2 (t)
()G
2
a
0
1
2
.5
.513
.525
=5
(bias nearest zero
~ (t)
~ (t)
~ (t)
~ (t)
1. 700
1.644
1.614
_.002
_.016
.029
.042
-.001*
.001*
-.004
-.004
.620
1.481
-.023
.051
.002~'(
-.004
.729
1. 252
_.016
.045
.001*
-.004
.821
.983
-.008
.034
.895
.709
.004
.021
-.002*
_.005
.948
.422
.024
.010
-.003*
-.004
.966
.5
.552
.601
.293
1. 700
1. 564
1. 478
.034
.006
-.002*
-.003
0*
-.027
.067
.093
-.002
.002*
-.010
-.009
.732
1. 236
-.021
.094
.003*
-.009
10
.900
.709
.051
.045
-.003*
-.009
15
0
1
2
3
4
5
6
7
.980
.189
1. 700
1.481
1. 286
1.078
.843
.578
.334
.168
.190
.009
-.002*
-.004
.017
.010
.036
.092
.198
.346
.487
.087
.108
.097
.071
.043
.022
.010
-.002*
.001*
.0 (+) *
-.003*
-.005*
-. 004~'(
-.002')'<
-.013
_.012
_.012
-.012
_.010
-.007
-.004
- (t)
1
2
3
4
·
10
·
20
·
30
•
···
40
0(+) ~'(
-.004
·
50
··
at - .5
55
0
1
2
··
5
··
0(=
.1
.S
.614
.713
.799
.871
.926
.962
.982
119
frequencies and high heritabilities) Neils projection)
"2(t+l)
sometimes slightly better than
3~G
(~'!')
fixation for the run involving
~ =
.9 in Table 15).
"2 (t+l)
in the selection program
) was
see the generations near
And at two points
) Reeve's equation) was sometimes
As pointed out earlier) ~ (1) was zero in runs for which p(O)
1
best.
and
l~G
A2(t+l)
2~G
~
= .5; consequently)
frequencies (for which
"2(1)
3~G
was an underprojection»)
(~'[')
see
~ =
However) in subsequent generations
better than
"2 (t+l)
l~G
.
.5
in runs involving unequal initial gene
best of the projection equations
and 16).
=
"2(1)
l~G
was the
.5) t = 1 in Tables 15
"2(t+l)
3~G
was almost always
An exception sometimes occurred in runs for
which ~it+l) started out negative.
In such runs) ~it+l) ultimately
passed zero and increased positively for the remainder of the program.
For generations in the neighborhood of the point at which
passed zero)
"2 (t+l)
l~G
(t+l)
~l
was sometimes the best projection equation
(examples can be seen in Tables 11) 12) 13) and 15).
But) with the exceptions mentioned above) 3;~(t+l) was the best
projection equation for traits controlled by freely recombining loci)
and the biases associated with this projection were almost always
small.
An d ) usua 11y)
"2 (t+l) was t h e next b est
4~G
.,
proJect~on)
bot h
0
f
which always strattled the actual variance in runs involving equal
initial gene frequencies.
Systems Involving Tight Linkage
Tables 17 through 23 give examples of the biases associated with
the projections for runs under tight linkage.
t he
..
proJect~ons)
"2 (t+J,)
2~G
"2 (t+l)
)3~G
) an
d
In most generations)
"2 (t+l) b
4~G
. t'
ecame overproJec ~ons
in runs involving low frequencies of recombination.
Under tight linkage
120
Table 17.
Biases associated with projections of genotypic variance
under complete linkage for p (0) = .3} p2 (0) = 1 and n = 4
(bias nearest zero appears with asterisk)
2(t)
Proportion
Retained
Ot=
.9
t
- (t)
p
erG
a
2
~ (t)
~ (t)
f3 (t)
~ (t)
1
2
3
4
-.018
.300
.327
1.680
1.588
-.044
.109
0*
5
.430
1. 146
-.026*
.711
.355
.296
··
·
10
.538
.791
-.009*
1.513
.815
.698
15
.611
.653
.051*
1.914
1.067
.926
.688
.640
-.059~""
1. 682
.886
. 753
.772
. 260
. 127"/(
4.423
2.484
2.161
30
·
·
.798
.309
-.051*
3.184
1. 809
1. 580
35
.831
.437
-.052~""
1.573
.878
.763
40
.887
.496
-.056*
. 622
. 304
. 251
45
0
1
2
3
4
5
6
7
8
0
1
2
3
.967
.300
.428
.549
.651
.735
.802
.864
.928
1.0
.300
.588
.822
1.0
.231
1.680
1.458
1.076
.821
.622
.440
.506
.411
0
1. 680
1. 317
.705
0
.083
.115
0*
-.153
-.054*
-.066*
-.011*
.057"/(
-.269
-.087
.343
.840
1. 213
1.502
1.888
.855
.304
0*
.282
.503
.712
.950
.313
0*
-.057
.189
.385
.580
.794
.223*
-.051
-.176
.172
.472
.660
0*
.152*
-.079
.068
0
1
··
··
20
··
25
··
·
Ot -
Ot -
.5
.1
-.019
121
Table 18.
Biases associated with projections of genotypic variance
.
(0)
2(0)
under complete 11nkage for p
= .5} P
= 1 and n = 4
(bias nearest zero appears with asterisk)
2 (t)
Proportion
Retained
01 =
.9
t
p- (t)
CT
G
a
2
~ (t)
( t)
~1
13 (t)
2
~ (t)
0*
.080
-.021
3
4
0
1
2
.5
.533
.565
2.000
1. 773
1. 546
-.018
_.015*
.123
.272
·
·
·
lD
.753
.692
-.021*
1. 150
.592
.499
15
.825
.425
-.044*
1. 713
.958
.832
20
.878
.500
-.055*
.714
.363
.305
23
24
25
26
27
28
29
.926
.944
.958
.971
.984
.997
1.0
.418
.350
.278
.203
.121
.027
0
-.062
-.048
.028
.121
.307
1. 720
.318
.219
.150
.097
.052
.0lD
.080
0*
0*
0*
0*
0*
0
1
2
3
4
5
6
.5
.637
. 749
.839
.915
.984
1.0
2.000
1.402
.954
.649
.448
.117
0
.320
.577
.669
.383
.050
.160
.227
0*
0*
.048
·
01 =
.5
O~(
.037*
.053*
-.031
1. 386
o~(
.041*
-.036
-.025
-.016
-.009
-.002
-.053
.090
.153
-.064
-.008
122
Tabl~
19.
Biases associated with projections of genotypic variance
under complete linkage for p(O) = .5} p2(0)
(bias nearest z~ro appears with asterisk)
= .25
and n
=4
2 (t)
J;>roportion
Retained
a
~
.9
t
p- (t)
O"G
0
1
2
"L..UUU
~
.5
.517
.534
.550
10
~(t)
~ (t)
~ (t)
3
~~t)
1.942
1.883
1.823
-.004
-.003*
-.002*
_.028
.057
.086
0*
.017
.034
-.005
.010
.025
.654
1. 429
.003*
.278
.140
.119
.769
1.013
.006*
.405
.222
.191
.854
.718
.009*
.391
.216
.187
.916
.486
.016*
.260
.143
.124
.959
.278
.030*
.126
.069
.059
55
56
0
1
2
.973
.976
.5
.571
.637
.192
.177
2.000
1.838
1. 641
.038*
.039
.082
.074
.044
.040
.038
.035*
0*
.020*
.066
.127
0*
.038
-.011
.023
··
5
··
• 796
1.066
.057*
· 220
.102
.082
. 870
.899
.923
. 752
. 617
.495
.079*
.0.93
.109
· 201
· 176
.147
.098
.086
.072
.080
.071*
.060*
.083
. 202
.026
.013
.011*
.043
.143
.227
.314
.404
.481
.534
.082
.126
.120
.087
.054
.030
.015
0'1(
.037
.045
.036
.023
.013
.007
-.014
.022*
.033*
.027'1(
.018*
.010*
.006*
2
a
1
2
,
··
20
·
·
·
·
40
30
50
·
a:".5
·
7
8
9
··
15
a -: . I
.989
0
.)
"L..UUU
1
2
3
4
5
6
7
.654
.779
.869
.929
.963
.982
.992
1.672
1. 223
.811
.488
.267
.136
.065
123
Table 20.
Biases associated with projections of genotypic variance
u~der complete linkage for p(O)
= .7) p2(0) = 1) and n = 4
(bias nearest zero appears with asterisk)
2 (t)
Proportion
Retained
a=. 9
t
a
~it)
~it)
~~t)
~~t)
.700
.732
1. 680
1. 392
.019
.127
0*
-.021
5
...
.834
.716
.008*
.548
.241
.190
10
.918
.936
.952
.966
.978
.991
1.0
.443
.380
.310
.237
.158
.070
0
-.055*
-.062
-.006
.075
.202
.608
.366
.257
.178
.119
.071
.028
.123
.023
0*
0*
0*
0*
.082
-.017*
-.030
-.020
-.012
-.005
1
2
3
4
.700
.828
.919
.989
1.0
1.680
.850
.439
.086
0
.262
.226
.203
.340
.362
.035
0*
0*
0*
-.057
-.060
-.006
0
1
2
.7
.947
1.0
1.680
.334
0
.188
.201
0*
-.034
11
12
13
14
15
16
a = •1
2
0
1
.
a= .5
p- (t)
O"G
a
124
Table 21.
Biases associated with projections of genotypic variance
under complete linkage for varying initial gene frequencies
within runs, p2(0)
with asterisk)
= 1,
and n
=5
(bias nearest zero appears
2 (t)
Proportion
Retained
ex
= .9
t
- (t)
p
O"G
a
2
13 (t)
13 (t)
f3( t)
1
2
3
13 4
.5
.533
.565
2.000
1. 773
1. 578
-.018
-.035
.123
.246
0*
.058
-.021
.026*
5
·
·
.651
1. 233
-.067*
.475
.174
.123
10
.774
.906
-.078*
.546
.218
.164
.876
.591
-.062*
.465
.186
.140
.911
.930
.948
.962
.975
.987
.494
.416
.331
.259
.182
.098
0(-)
.311
.253
.198
.135
.084
.041
0(+)
.095
.042
0(-)*
0(_)*
0(_)*
0(_)*
0(-)*
.059
.007*
-.033
-.023
-.014
-.007
0(-)
.320
.513
.539
.356
.316
0*
.113
.138
0(-)*
0(_)*
-.053
.046
.071
-.059
-.053
0
1
2
·
··
·
15
·
17·
18
19
20
21
22
23
CX ==
.5
0
1
2
3
4
5
6
1. (-)
.5
.637
.749
.841
.919
.990
1.0
-.039*
-.026
-.005
.049
.154
.407
155.0
2.000
1. 402
.994
.695
.437
.078
0
0*
-.005*
.016*
.059
2.289
125
Table 22.
Biases associated with projections of genotypic
variance at c. '+1
~)~
= .1
for p(O)
= .5)
p2(0)
= 1)
and n
=4
(bias nearest zero appears with asterisk)
2(t)
Proportion
Retained
Q'
O"G
2
~it)
~?)
~?)
~~t)
.5
.525
.548
1. 700
1.475
1.300
-.008
-.010*
.129
.256
-.003*
.074
_.019
.051
.679
.604
-.033*
1. 291
.686
.610
.802
.286
.097*
2.554
1. 379
1. 233
.845
.346
-.051*
1.645
.901
.808
.915
.928
.489
.462
-.057*
_.059
.349
.267
.134
.070
.107
.045*
.988
.998
1.0
.113
.018
0
.335
2.56
.030
.004
-.012*
-.002*
-.017
_.002
.312
.612
.838
1. 774
1.148
.372
.109
-.008
.170
.274
.767
.545
-.071*
-.042*
-.048
.114
.203
.642
.470
-.127
-.061
- (t)
t
p
a
= .9
0
1
2
·
··
10
···
20
···
30
··
·
39
40
··
45
46
:
Q'
= .5
47.·.··
0
1
2
3
4
5
6
7
8
....
.5
.599
.681
.751
.820
.855
.912
.962
1.0
...
1. 700
1.207
.884
.675
,356
.418
.494
.306
0
0*
-.009*
_.061*
.203*
-.178*
-.390
.190
126
Table 23.
Proportion
Retained
0(
= .9
Biases associated with projections of genotypic variance at
_
(0) _
2(0) _
- .5, P
- .25, and n = 4 (bias
c i ,i+1 - .1 for p
nearest zero appears with asterisk)
t
0
1
2
3
··
10
··
·
20
··
·
30
···
33
34
35
·
·
40
Q' ....
.,5
50
0
1
2
3
4
5
·
10
·
·
15
Q' -
.1
0
1
2
:3
4
5
6
7
CT
2 (t)
~it)
~ (t)
~ (t)
~~t)
,517
,534
.550
a
2.000
1,942
1.891
1.845
3
_.004
-.007
-.010*
.028
.053
.073
0*
.012
.023
_.005
.006*
.014
.659
1.577
- .014*
.140
.058
.044
.792
1.165
-.003*
.131
.056
.043
.891
.715
.015*
.087
.037
.028
.913
.920
.926
. 592
.553
.516
.021*
.023
.025
.073 .
.068
.064
.031
.029
.027
.024
.022*
.021*
,951
.354
.033
.044
.018
.014*
.981
.046
.018
.008
.006*
.571
.637
.697
.751
.799
.146
2.0UU
1.838
1. 658
1. 474
1. 291
1.111
0*
.010.*
.019*
.030
.044
.066
.116
.146
.158
.156
0*
.028
.046
.055
.057
-.011
.013
.029
.038*
.041*
.948
.368
.143
.063
.024
.017*
.991
.5
.654
.779
.870
.930
.964
.983
.992
.070
2.000
1.672
1.235
.821
.490
.265
.132
.063
.212
.012
.005
.003*
.043
.132
.219
.318
.415
.493
.543
.082
.115
.101
.069
.040
.021
.011
0*
.027
.030
.022
.013
.007
.004
-.014
.012*
.018*
.014*
.009*
.005*
.003*
p- (t)
.5
.5
G
2
2
127
there was greater buildup of
D~~,t) relative to R~~,t), and this along
1J
1J
. h foh
.
w~t
~ e sma 11 er f requenc1es
0 f recom b"1nat10n, resu 1te d'1n d4(t+1) '
equation (100), attaining positive values of great magnitude throughout
most of the selection program.
so did
"2(t+1) and
2~G
"2 (t+1)
3~G
As 4;~(t+1) became an overprojection,
(recall inequality (102».
However,
"2(t+1) was an
undeJ'pJ;'oJection near the eqd of the selection program
4 ~'G
for high heritabilities and also in the initial generations of all
runs.
Reeve's projection,
at
~
=
.9
an~
"2 (t+1)
l~G
' was usually better than the others
.5 as long as fixation was not approached too rapidly.
Under complete linkage, assumption (c) in the discussion concerning
. Reeve1s
proje~tion
was satisfied.
And Reeve1s projection was
generally better under tight linkage than under free recombination,
but, for the most part, the order of magnitude was about the same.
The fact that Reeve'spJ;'ojection was the best under tight linkage was
primarily due to tqe fact that the other projections were extremely
poor throughout most of the selection program.
Discussion Regarding the Projection Equations
Generally, projection equations are used to predict the mean
response and to determine whether the magnitude of that response would
justify the
co~t
of the selection program.
Therefore, in recurrent
selection programs, rather than using projection equations which are
corrected at each generation based on the realized parameters, a
breedel;' would usually wish to base his projections on the initial
parameters,
128
If the
n~ber
of loci controlling the trait in question is large
enough to justify the assumption that the genotypic and phenotypic
traits are bivariate normally distributed, then the prediction equation
(105)
would be appropriate in which r(t) is the ordinate of the normal
density function evaluated at the standardized point of phenotypic
truncation and divided by the proportion retained.
Recurrent relations
for equation (105) depend on developing appropriate recurrent relations
for
2(t)
CTG
•
Under one set of conditions a recurrent relation can be justified
for the additive model.
Recall that, if the trait in question is
contrQ1led by freely recombining loci with equal additive effects at
each locus, and if, in the initial generation, equal gene frequencies
"2 (t)
and intergenic independence are specified, then 3CTG
"2 (1;:)
and 4CTG . , equation
(~9),
.
,equatlon (88),
are, respectively, over-projections and
But, as n becomes large, 3~~(t) rapidly converges
under-projectipns.
to 4;~(t) anp, therefore, approaches the actual variance.
Consequent-
ly,
~
21
asymptotically with
[ CT2 + 2na 2 p (1 -p ) ] (s, t-1)
G
resp~ct
(106)
to an increasing n, where
(107)
And under the
no~a1
assumptions mentioned above,
129
2(s,t-l)
~G
- 2()J
= ([1 -ip
i-z O
2}(t_l)
~G
(108)
Equations (105) through (108) provide the recurrent relations needed
to relate the mean and variance in generation 0 to those in any
subsequent generation.
And these relations are exact asymptotically
with respect to increasing n for the conditions specified.
For n small, one might choose to use two different recurrent
BQthutilizing equation (105), but one in which
relations:
2 (t) is
~G
~2 (t)
~2(t) and the other in which ~~(t) is replaced by 4~G
t l
This would tend to yield upper and lower bounds on IJ.ci + ). However,
replaced by
3~G
•
with n small, higher central moments will affect the standardized
selection differential and the normal approximation will become less
accurate.
In any case, the projections are severely limited in that
they require the breeder to know the value of nand a.
This same problem exists when using 2a-~(t) (Neils projection)
for
~~(t)
irtequati.on (105).
Since Nei's projection is more of an
information is required for
and
A2(t)
4 crG
'
the latter two
The breeder is not faced with the problem of knowing the value of
a and n if, in equation (100») he uses
2(t)
~G
:c:
([1 _ 1
'2
7
~p
2(.
~ -
zo )J
2}(t-1)
crG
which is Reeve's projection under the assumption of bivariate normality.
The problems associated with Reeve's projection have already been
discussed in some detail.
The conditions under which Reeve's projec-
tion would best be used over time are:
130
(a) p(t) sufficiently close to .5 for the projection period
or 6p sufficiently small;
(b) negligible disequilibria, or
(c) the advance sufficiently small such that 62P(t),
·(s t)
H '
J
and Z (l-c) (6D-R)
(s t)
,
..
may be
~gnored,
or the
advance being such that H(s,t) ~ _62p(t) and
m(s,t) ~ R(s,t)
-
,
and
(d) tight linkage.
Any biases introduced will likely be magnified from one step to the
next in the recurrent process; consequently, any major deviations from
the above conditions could lead to poor estimates after a few generations.
Obviously all the projections presented have serious shortcomings;
and the breeder would need to be very familiar with the genetic system
on which he was selecting in order to justify his using anyone (if
any) of these projection equations.
131
LIST OF REFERENCES
Abramowitz} M.} and I. A. Segun (eds.). 1968, Handbook of Mathematical
Functions. Nat1. Bur. of Standards - App1. Math Series 55. U. S.
Government Printing Office} Washington} D. C.
Bodmer} W. F.} and J. Fe1senstein. 1967. Linkage and selection:
theoretical analysis of the deterministic two locus random mating
model. Genetics 57:237-265.
Crow} J. F.} and M. Kimura. 1970. An Introduction to Population
Genetics Theory. Harper and Row} New York} New York.
Fe1senstein} J. 1965. The effect of linkage on directional selection.
Genetics 62:349-363.
Fraser} A. S. 1957. Simulation of genetic systems by automatic digi_
tal computers. II. Effects of linkage on rates of advance under
se1ect~on.
Australian J. Bio1. Sci. 10:492-499.
Gill} J. L.
tions.
1965. Selection and linkage in simulated genetic popu1aAustralian J. Bio1. Sci. 18:1171-1187.
Griffing} B. 1960. Accommodation of linkage in mass selection theory.
Australian J. Bio1. Sci. 13:501-526.
Hill} W. G.} and A. Robertson. 1966. The effect of linkage on limits
to artificial selection. Genetical Res. 8:269-294.
Latter} B. D. H. 1965a. The response to artificial selection due to
autosomal genes of large effect. I. Changes in gene frequency
at an additive locus. Australian J. Bio1. Sci. 18:585-598.
Latter} B. D. H. 1965b. The response to artificial selection due to
autosomal genes of large effect. II. The effect of linkage on
limits to selection in finite populations. Australian J. Bio1.
Sci. 18:1009_1023.
Lewontin} R. c. 1964. The interaction of selection and linkage.
I. General considerations: heterotic models. Genetics 49:49-67.
Li} C. C. 1955. Population Genetics.
Press} Chicago} Illinois.
The University of Chicago
Martin} F. G.} and C. C. Cockerham. 1960. High speed selection
studies} pp. 35-45. In O. Kempthorne (ed.)} Biometrica1 Genetics.
Pergamon Press} London; England.
Nei} M, 1963. Effects of selection on the components of genetic
variance} pp. 501-515. In W. D. Hanson and H. F. Robinson (eds.)}
Statistical Genetics and-:P1ant Breeding. Nat1. Acad. Sc. - Nat1.
Res. Council Publ. 982} Washington} D. C.
132
Qureshi) A. W.) and O. Kempthorne. 1968. On the fixation of genes of
large effects due to continued truncation selection in small
populations of polygenic systems with linkage. Theor. and App1.
Genetics 38:249-255.
Qureshi) A. W.) O. Kemp thorne) and L. N. Hazel. 1968. The role of
finite size and linkage in response to continued truncation
selection. 1. Additive gene action. 2. Dominance and overdominance. Theor. and App1. Genetics 38:256-276.
Reeve) E. C. R. 1953. Studies in quantitative inheritance.
III. Heritability and genetic correlation in progeny tests using
different systems. J. of Genetics 51:520-542.
Wright) S. 1921. Systems of mating. III. Assortative mating based
on somatic resemblance. Genetics 6:144-161.
Wright) S. 1942. Statistical genetics and evolution.
Math. Soc. 48:223-246.
Bull. Am.
133
APPENDICES
134
Appendix A.
Intragametic Digenic Disequilibria in the
Selected Population
In the cases examined, the test populations were formed by random
pairing of identically distributed gametic sets.
Looking at the two
locus model, the distribution of selected zygotic arrays is
x
(s, t)
f(m~
f~
(A. 1)
in which
if
x
m-
= f-x = -x
.
And the parental gametic distribution in the selected population is
•
x (s, t)
f(x) (s, t) = f (m::)
fo~
m~ = f~ = ~ •
Therefore,
(t)
r(~ s,
w(~
= [ -;-
f(~]
(t)
w
in which
w(~ (t) =
x (t)
1:
x
f-
w(m~
f~
f(f~
( )
t
if
x
m-
=x
x (t)
=
1:
x
m-
w(m-~
f~
f( x) (t)
m-
if
f
x =x .
-
Denote the intragametic digenic disequilibria in the selected popu1ation by
•
135
=
f(ll) + £1(11)
f(1O) + £1(10)
f(0 1) + £1(01)
f(00) + £1(00)
(t)
([f(ll)f(OO) - f(1O)r(Ol)]
+ [£1(11)£1(00) -
£I(10)£I(0~]
+ [f(ll) £1(00) + f(OO) £1(11) '" f(1O) £1(01) _ f(01) £1(10)] }(t)
in which
£l(x) (t) = f(~ (s) t) _ f(~ (t) •
•
Noting that
(a)
[f(ll) f(OO) _ f(10) f(Ol)] (t)
(b) since
£1(11)
(t)
= D(t)
(t)
+ D)
£I(P1 P Z
£1(10)
)
£I(P1 (l-P Z) - D)
=
)
£1(01)
£I(1-P )P
1 Z
£1(00)
£I( (1-P ) (l-P ) + D)
1
Z
then [£1(11) £1(00) - M1O) £1(01)]
and
(t)
- D)
= -£lP £1P
1
Z
)
•
136
(c)
[r(ll)
~(OO) + f(OO) ~(11) - r(10) 8(01) _ r(01) ~(1O)] (t)
Thus equation (A.1) is
D(s, t)
= [_
8P
1
~Pz
+
~
w
_ w(OI)]}](t)
•
(D[P PZw(OO) + q1 qzw(ll) + PIqzw(Ol)
I
137
Appendix B.
Relations between Inter10ci Digenic Disequilibria
in the Selected and Test Population
In the case in which zygotes are formed by random pairing of
identically distributed gametes} the following parametric
descriptions apply to the double heterozygous frequencies in the two
locus model:
and
2
(see Burrows) 1970 ).
} [f (~) + f
(~)
Consequently}
- f
(~)
- f (U:-)] = D - R .
Recognizing that f(l1) (s) t)=~ w(l1) (t) +(11) (t) etc.} it can be seen
00
=
00
OO}
W
that
10\ + w(0 1\ ] D} (t)
+ [ w (01;1
10 ;I