.
THE TWO-~CUS INBREEDING FUNCTION
by
Bruce S. Weir and C. C. Cockerham
Public Health Service Grant GM-J.J.,ttt6
Quantitative Genetics Research ProgrJ'ProJect
Institute of Statistics
Mimeograph Series No. 584
June 1968
•
.
'
ERRATA
THE TWO LOCUS INBREEDING FUNCTION
Weir and Cockerham
~
Page iv:
omit 4.3.10
Page vi:
Omit 4.6
Page 7, line 11:
Narain (1965)
Page 10, line 6:
X
10
Page 18, line 1:
1_;\.2
+~ (:l.~
= Prob (a
Page 24:
Omit section 4.3.10
Page 25:
Omi t Table 4.6
;; a', b ~ b' )
Page 25, equation (4.3.13):
21 ~ 2HG
Page 25, equation (4.3.14):
~ ~ ~HG
Page 28, line 17: replace "function" by "coefficient"
Page 28, line 24:
replace t by t-l
Page 28, line 25:
replace t-l by t
Page 41, line 9:
Page 42, line 14:
add "The offspring of U and V is generation 0."
Cl
t+2
Page 42, line 18: (generation 0)
,
Page 43, line 2:
Omit F00
Page 43, line 9:
0Ini t F
Page 43, line 10:
O
FO
Page 43, line 12: F'
.;Iii-
ERRATA (continued)
-2-
Page 43, line 16: F2
Page 44, line 14:
(6.2.1)
Page 44, line 15:
l:f- F~~l
Page 45, line 5:
Page 45:
8
F
Generation 2, A = 1.0: 0.2500
Page 49, Generation 3, A = 0.8: 8.2800
Page 54, line 4:
2QAA', p..A"
Page 54, line 5: QAA', A"A" I
Page
56,
line 19: vB
- X'
Page 63, line 4:
DyEz
PB = 6PB
7
1
Page 66, last equation is
Page 67, line 2:
-
12PB + 6PB - 2PB
246
(7.4.1~)
(7.4.19)
Page 83, line 9: I B, B'B"
Page 84, element (1, 4): (l-A 2)(N_l)/2N
Page 84:
element (4, 9): (1-A)(N-l)/8N, element (8, 1): 1/8N(N-l), element (8, 2):
1/8N(N-l)
before If mf
1
'
Page 93, line 3:
insert P
f
Page 94, line 4:
first divisor is 2.
Page 94, line 7:
f, ,
Omit Pf (6 I rom" + -m
m, mllm)
7 '"""1DlIl ,
,.
.
'
.
-
.
ERRATA (continued)
-3-
(1-).) (1+P2-2Pl )/8
Page SS, element (6, 9): Q4/8, element (8, 4): (2Pl -P2 )/4, element (8, 8):
(2+:J?2-3Pl )/4
Page 99, element (8, 9): (1-3Pl +2P )/8, element (9, 1): P /l2 + P4/4
2
6
Page 99, element (9, 6): (1+3P4-4P2 )/4 - (P6+P )/12
7
.
Page 100, l~ne
5:
Tlt+l
Page 127, generation 7,
Page 132: Wright, S.
= Tl / 2
~Nll:
1931.
tj 2, Tt+l
+ T~
2
= Tlt/ 2
t j2
+ (l-Pl)T~
0.0192
Evolution in Mendelian Populations
iv
TABLE OF CONTENTS
Page
•
LIST OF TABLES • •
vi
LIST OF FIGURES
• viii
1.
INTRODUCTION .
1
2.
REVIEW OF LITERATURE •
4
3.
ONE LOCUS MEASURES •
8
4.
TWO LOCUS MEASURES .
•
•
•
411
•
•
10
4.2
14
14
•.•.
General Expansions • • • • • •
~BC
4.2.2 IB, GH •
.........
...........
o
4.2.3 -DE,
GH
4.3 Special
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6
4.3.7
4.3.8
4.3.9
IB, cc . . . . .
.£BB, cc
20
Expansions
~BB
.••••••
IB, BB •
.£BB, BB
.....
•••••••••••••• . . . . .
• •••••••
.•..
IB, ~ •.
21
mr
22
.£BB, Gii
23
23
.£BB,
.£BC, BC
4.3.11 .£BG, BH
• •
•• , ••
...
27
Specific Pedigrees • • • • •
Pedigree Systems of Mating
Rate of Inbreeding • • • • •
EXAMPLES OF INDIVIDUAL MATING SYSTEMS
6.1 Specific Pedigree • • • • . • •
6.2 Repeated Parent-Offspring Mating • • • •
6.3 Successive Parent-Offspring Mating
24
25
• • • ••
GENERAL ANALYSIS OF INDIVIDUAL MATING SYSTEMS
6.
14
15
16
16
18
19
19
4.3.10 .£BC, CB
•
10
4.1 Definitions
4,2.1
.e
•
~
...
.
. . ..
27
27
30
36
36
41
45
v
TABLE OF CONTENTS (continued)
Page
7.
GROUP MEASURES • •
· 51
7.1 Definitions.
7.2 General Expansions
7.3 Special Expansions
7.3.1 IB, CC'
7.3.2 ~AA.' • •
7.3.3 -BBB , CC
I
· . . . 51
.54
· . . . . . . 55
• • • • • 58
• • • •
I
· . . . . . . • . . . 58
· . . . . . . . 59
.59
Gametic Sampling Probabilities
8.
8.1
8.2
8.3
8.4
8.5
8.6
.e
9.
-.
.
EXAMPLES OF GROUP MATING SYSTEMS . • ••
10.
11-
· 69
· · · · 69
74
· ·· · 80
·
· · 89
97
· · ·· ·.106
Monoecious Popula tions: General Sampling Scheme
Monoecious Populations: Combined Sampling Scheme •
Monoecious Populations: Independent Sampling Scheme
Dioecious Populations: General Sampling Scheme •
Dioecious Populations: Combined Sampling Scheme
Dioecious Populations: Independent Sampling Scheme •
....
.····.• ·
.119
SUMMARY . . . .
. . . . . . . · · · · . · · . · · · · .130
LIST OF REFERENCES .
....
. · · · · .131
····
DISCUSSION
.
vi
LIST OF TABLES
..
Page
4.1 Marginal totals of measures • •
.e
4.2
Expansion of 8
4.3
Expansion of
4.4
Expansion of 111B, BC •
4.5
Expansion of 0llBC, BC
4.6
Expansion of ~BC, CB
4.7
Expansion of 0llBG, BH
6.1
Values of F
6.2
Dominant root of transition matrix for successive parentoffspring mating
• • • • • • • • • • . . • • 50
7.1
Expansion of
8.1
Expansion of individual measures in terms of gametic set
measures for a monoecious population • • • • • . . • • . • . 73
8.2
Dominant root of transition matrix for monoecious populations with equal chance gamete formation
. • • • • • • 78
8.3
Dominant root of transition matrix for monoecious populations with combined sampling and two gametes per parent . • • 81
8.4
Dominant root of transition matrix for monoecious populations with independent sampling and two gametes per
paren t
8.5
•
. 12
•
IB,
11
• • • • 20
CC • • • • •
• 21
• . • 23
. • 25
• • • •
.............
· . . 26
for successive parent-offspring mating
~AA I
.
• • • 17
11BB
.
.
•
.
•
.
.
.
•
.
•
•
.
.
.
.
•
•
. • 49
•
•
•
.
Expansion of individual measures in terms of gametic set
measures for a dioecious population . • • . • • • • . •
lI'
.
•
.
.58
•
86
• 95
8.6
Dominant root of transition matrix for dioecious populations
with equal chance gamete formation • • • • • . • • • • • • • 102
8.7
Dominant root of transition matrix for dioecious populations
with combined sampling and two gametes per parent • • • • • • 107
8.8
Dominant root of transition matrix for dioecious populations
with independent sampling and two gametes per parent • • • • 115
9.1
Maximum number of measures needed for regular pedigrees
and monoecious populations ••
•• • • • • . • • • • •• 123
vii
LIST OF TABLES (continued)
Page
•
Maximum number of measures needed for dioecious
populations
••••••••
•••••
• . • • 123
9.3 Maximum value of identity disequilibrium and generation
of attainment for successive parent-offspring mating
• • • • 1~5
9.4 Identity disequilibrium functions for a monoecious
population of size eight with equal chance gamete
formation . . . . . . . . . . . . . . . . . . . . . . . . . .127
.e
•
viii
LIST OF FIGURES
Page
•
1.1
The general pedigree
6.1
Specific pedigree • • .
6.2
Repeated parent-offspring mating
3
·.
36
· 41
6.3 Successive parent-offspring mating
· . . • 46
69
8.1 Monoecious mating • • • • • • • • •
8.2
F
Eor monoecious populations with equal chance gamete
11
formation . . . . . . . . . . . .
· . .. . .
• • • • 79
8.3 F11 for monoecious populations with combined sampling and
two gametes per parent
8.4 F
.e
•
• • • • • • • . • • •
• • • 82
for monoecious populations with independent sampling
11
and two gametes per parent
····
....····
· · 89
8.5
Dioecious mating
8.6
F
for dioecious populations with equal chance gamete
11
formation
.........·······
· 87
· · · · · · · .103
8.7
F
for dioecious populations with combined sampling and
11
two gametes per parent
8.8
F
for dioecious populations with independent sampling
11
and two gametes per parent
· · · · · · · . . · · · · · · · ,108
· · · · · · · . . · · · · · · · .116
1.
•
INTRODUCTION
A consequence of all forms of inbreeding is that all homologous
loci of the inbred individual tend to have identical genes.
Such genes
will not only be identical in nature ("identical in state"), but also
will be "identical by descent" through having originated from a single
ancestral gene.
This work is concerned with determining the prob-
ability that two linked autosomal loci of a disomic individual carry
genes identical by descent.
Such probability statements will be con-
sidered relative to some base population where there was no identity by
descent.
For an individual a two locus inbreeding function will be defined.
.e
This will be a vector valued function with four components which measure the probability that both, the first, the second, or neither of the
two loci respectively carry genes identical by descent.
The first
component, corresponding to an individual with double identity, will be
termed the two locus inbreeding coefficient and the fourth component,
corresponding to an individual with double non-identity, will be termed
the two locus panmictic function.
This terminology is a natural exten-
sion of that for the one locus case.
While similar two locus inbreeding functions have been discussed
before (Haldane,
1947; Schnell, 1961; Shikata, 1962), no correct method
has been given for their determination in all generations of systems
other than self mating.
•
The difficulty arises because linkage between
loci prevents genes from each locus being transmitted independently .
2
This work shows that recurrence formulae can be found when the
•
inbreeding function is regarded as a special case of a more general
measure.
Instead of insisting that the two genes at each locus are in
one individual, the general measure is defined for any two genes at
each of two loci.
As with the inbreeding function it is a vector func-
tion with four components giving the probabilities that both, the
first, the second or neither of the two loci carry genes identical by
descent.
Three broad classes of this measure arise according to wheth-
er the four genes are carried on two, three or four gametes.
three will be called digamet ic, trigamet ic
respectively.
These
or quadrigametic measures
For two uniting gametes then, the digametic measure is
just the inbreeding function.
For two gametes from two parents, the
digametic measure is called the coancestry function.
For the general pedigree (Figure 1.1) general transition equations
for the three types of measure will be derived.
•
When any specific
inbreeding system is to be analyzed each of the three types must be
further subdivided into special cases according to whether the corresponding gametes came from separate or common individuals.
The method
is also applied to finite populations when there is random pairing of
gametes.
For all systems studied, transition equations linking values between the measures in successive generations are given.
The equations
for the fourth ("panmictic") component of the measures are usually
written in matrix form, and the characteristic equation of this transi-
•
tion matrix employed to give a recurrence formula for the two locus
panmictic coefficient.
Tables of values of the dominant root of the
3
transition matrix and graphs showing values of the two locus inbreed-
,
ing coefficient for various linkage
~ialues
will be given.
Effects such as selection, mutat JO;; or interference ('
-e
assumed
to be absent, but no restriction will be made on the number of alleles
for a locus •
•
.
,':"igur~:
I
-.1
The ""enf:ral pedigree
4
2.
REVIEW OF LITERATURE
The effects of inbreeding on one locus have been studied extensively by many authors.
Calculation of the
coefficient has been by three methods.
C'rie
Wright
locus inbreeding
(1921)
used the concept
of correlation between uniting gametes and gave a general formula)
Wright
(1922))
which gives the value of the inbreeding coefficient for
any individual in terms of those of ancestors
parents.
Malecot
(1948)
C(l·.ITl'wn
to both of its
used probability arguments to derive the prob-
ability that two genes at a locus were identical by descent.
(1949)
unified the method of using a generation matrix of mating types.
Kempthorne
.e
Fisher
(1957)
has evaluated these approaches.
Malecot were extended by Cockerham
(1967)
The methods of
to include pedigrees of
groups.
The study of two loci) when the loci may be linked) has not
•
progressed as far.
Early approaches almost always specified an initial
array of zygotes) calculated the gametic array formed by these zygotes
and then) according to the mating scheme being studied) calculated the
zygotic array in the next generation.
(1917)
The first paper was by Jennings
and showed such calculations for random mating) selfing) sib
mating and some forms of assortative mating.
The unwieldy nature of
the method is indicated by the fact that sib mating required
types".
55
"family
Only for selfing could Jennings display the zygotic array
after n generations.
Robbins
(1918)
Haldane and Waddington
extended this work for random and
(1931)
.,
self mating.
attempted to complete the
e
.
work of Jennings by setting up transition equations for mating types.
They were able to give the equilibrium frequency of crossover zygotes
5
for sib mating (100 mating types) and successive parent~ffspring
mating (20 mating types).
They estimated it would take 10,000 mating
types to treat double first cousing mating, and the method could not
be applied to dioecious systems with unequal numbers of males and
females.
Bennett (1954a) just considered those mating types that gave
offspring heterozygous at both loci.
He listed the dominant roots of
the appropriate generation matrix for sib mating (9 mating types) and
successive parent-offspring mating (8 mating types).
His object was to
determine the fraction of map length expected to be heterogeneous after
a given amount of inbreeding.
While all the papers listed thus far
have considered only two alleles per locus, Bennett (1954b) used the
zygote-gamete-zygote approach to study random mating for an arbitrary
number of alleles at an arbitrary number of loci.
Quite different methods have also been used.
•
Wright (1933) by
using path coefficients obtained the results of Haldane and Waddington
with much less effort, and could also analyze the dioecious systems.
For each system his result was the equilibrium frequency of crossover
zygotes.
Geiringer (1944) proved that, for random mating and an arbi-
trary number of alleles at any number of loci, the limiting gametic
distribution was the product of the appropriate marginal gene distributions, unless linkage was complete.
three probabilities.
Kimura (1963) worked with just
For any two nonallelic genes he considered the
probabilities that they were on one chromosome, or two chromosomes in
the same or different individuals.
rium crossover frequencies.
He derived expressions for equilib-
6
Mixtures of self and random mating have received a lot of atten-
...
tion.
Bennett and Binet (1956) and Ghai (1964a) used a zygote-gamete-
zygote approach.
Kimura (1958) gave equilibrium frequencies of double
homozygotes and single and double heterozygotes.
Other papers by Ghai
and Narain are mentioned below.
The techniques discussed in all of the above papers were only able
to describe some aspect of the equilibrium population, or else specify
one generation in terms of the immediately preceding (non-inbred) one
for any system other than self mating.
The first paper to extend the
concept of inbreeding function to two loci, and give properties of
generations between initial and final was by Haldane (1947).
He de-
fined a function giving the probability that two loci carried genes
identical by descent.
Restricting himself to two alleles per locus he
gave this probability for the first generation of inbreeding for half
first cousins, full sibs and double half first cousins.
•
He also gave
a formula, analogous to that of Wright for one locus, for his function
for an offspring of two individuals related by a single chain of m
links through a single common outbred ancestor.
The requirement of an
outbred ancestor precluded its general use.
A general inbreeding function for any number of alleles at any
number of loci was defined by Schnell (1961)--it gave the probability
that a certain set of loci carried genes identical by descent.
As with
Haldane though, he could give its value only for the first generation
of inbreeding.
•
Both Haldane and Schnell said that, in the absence of
linkage, the n-Iocus inbreeding coefficient must be the nth power of
the one locus coefficient.
This is only true for regular inbreeding
"e
7
pedigrees of individuals where all matings are specified, and does not
hold for pedigrees of finite groups where matings are at random.
Shikata (1962) defined a function very similar to that of Schnell
and gave recurrence formulae for it for several systems.
These formu-
lae were correct for self mating, but not for other systems such as
sib mating and parent-offspring mating (Shikata, 1965a), which he
acknowledged (Shikata, 1965b).
Rajagopalan (1958) used what was essentially one component of the
measure of Schnell and Shikata and gave its value for self mating.
Ghai (1964b) extended this work for a mixture of self and random mating
for the case of zero linkage.
Narain (1963) used the function of
Schnell to give a completely general analysis of selfing (arbitrary
.e
numbers of alleles and loci), and a mixture of self and random mating
(Narain, 1966).
A new class of measures was defined by Cockerham and Weir (1968),1
•
which included the measure used by Schnell and Shikata.
For an arhi-
trary number of alleles at two loci they found the probability that
two loci carried genes identical by descent for any generation of a
sib mating scheme, and any degree of linkage.
•
1Cockerham, C. Clark, and B. S. Weir. 1968. Sib mating with two
linked loci. Department of Experimental Statistics, North Carolina
State University at Raleigh, North Carolina. (Submitted to Genetics
for publication.)
--
8
3.
ONE LOCUS MEASURES
The one locus case will be reviewed briefly to introduce the
concept of general measure which forms the basis of the two locus work.
For any two distinct genes a, a' at a locus a general measure !(a, a')
is defined as:
Prob(a _ a')
x1(a, a' )
!(a, a' )
=
(3.1)
=
Prob(a ~ a')
xO(a, a' )
where the identity sign _ means identity by descent.
Thus !(a, a') is
a vector with two components, the sum of which is unity.
Note that
there has been no restriction on the number of possible alleles for
either locus.
Now a, a' will necessarily be carried on distinct gametes between
•
generations, but two cases of the digametic measure will be distinguished according to whether or not the two gametes unite.
These two
cases are given special symbols:
E:.
A
~BC
=
!(a, a'
a 3A, a ' 3A)
(3.2)
!(a, a'
a
B, a'
(3.3)
E
E
C) •
Equation (3.2) expresses the fact that a, a' are on gametes that
unite to form individual A, while
(3.3)
formed by individuals B, C respectively •
•
states that a, a' are on gametes
9
The usual terminology for these measures is:
F
LA
the inbreeding coefficient of A,
F
OA
the panmictic coefficient of A,
8 1BC :
the coefficient of parentage (coancestry) of Band C.
In the general pedigree (Figure 1.1), the inbreeding coefficient
of A is just the coancestry of its parents:
so that
.e
There is no need in the one locus case to consider both components
of the measures in analyzing any system.
The recurrence equations for
systems of mating are most simply written in terms of the panmictic
•
coefficient, while the inbreeding coefficient is the simplest component
to calculate from specific pedigrees .
•
10
4.
TWO LOCUS MEASURES
4.1
Definitions
The general measure must now take account of genes at two loci.
For any two distinct genes a, a' at one locus (the a-locus), and any
two distinct genes b, b' at another locus (the b-locus), a general
measure ~(ab, a'b') is defined as:
XII (ab, a'b')
Prob(a - a ' , b - b' )
X (ab, a 'b ')
10
Prob(a
-
b', bib')
a 'b')
Prob (a
i
a', b - b' )
XOO(ab, a 'b')
Prob(a
i
a' , b 1= b')
~(ab, a 'b')
=
x01 (ab,
.e
(4.1.1)
Once again there has been no restriction on the number of possible
.
alleles for each locus.
There is no ordering intended between the two
pairs of genes, so that:
~(ab, a'b') = ~(a'b', ab) .
Other relations will be indicated below for the special cases.
Now the four genes in the argument of
three, or four gametes.
~
may be carried on two,
In the digametic case, once again, a distinc-
tion will be made for the case when the two gametes unite.
There are
thus four symbols, two digametic, one trigametic and one quadrigametic,
to be defined:
!:.A
=
~(ab, a'b': ab 3A, a'b' 3A) ,
(4.1.2)
11
,
~BC
;:::
!(ab, a 'b': abEB, a'b'EC)
IB, DE
;:::
!(ab, a'b': abEB, a 'ED, b' EE),
(4.1.4)
~BC,DE
;:::
!(ab, a'b': aEB, bEC, a' ED, b'EE).
(4.1. 5)
~A
is the inbreeding function of A and
tion of Band C.
(4.1.3)
~BC
the coancestry func-
The symbols used have the same meaning as for the one
For example, equation (4.1.4) states that a, b are on one
locus case.
gamete from B, a' is on a gamete from individual D and b' is on a
gamete from individual E.
Just as in the one locus case, from Figure 1.1,
~A
.e
;:::
(4.1.6)
~BC
The four components of each measure may be arranged in two by two
tables as in Table 4.1 and marginal totals taken.
totals will always be one locus measures.
These marginal
This relation between one
and two locus measures may be expressed as:
Xl. (ab, a'b')
;:::
X . (ab, a'b')
o
(4.1. 7)
and
!(b, b')
(4.1.8)
;:::
X.o(ab, a'b')
..
12
Table 4.1
Marginal totals of measures
XII (ab, a'b' )
X (ab, a 'b I)
10
Xl (a, a ')
X (ab, a 'b ')
Ol
XOO(ab, a 'b ')
X (a, a' )
Xl (b, b I)
F
IOA
Fl. A
°llBC
°10 Be
° 1 . Be
F,_,j A.
~.,'"
F
OOA
F O. A
°OlBC
°OOBC
°0. BC
F. lA
F. OA
1
O.lBC
O.OBC
1
IllB , DE
°llBC, DE
100B
°.1BE
...
1
FIlA
,,j
-e
XO(b, b ' )
o
,
° .OBE
DE
~\OBC, DE
°OlBC, DE
1
°.1CE
O.OCE
1
13
The sum of the four components of ~(ab, alb'), which may be written as
X
(ab, alb'), is equal to unity.
One further convention is necessary
to distinguish between the two loci.
For individual A, for example,
F 1.A and F .
O A are the inbreeding and panmictic coefficients for the a-
locus, while F.
1A
and F.
OA
are those for the b-locus.
As mentioned above, certain relations hold among the measures
because of the lack of order between the two pairs of genes.
These
relations are:
= ~B
~BC
IDE
,
IB
B
,
DE
~DE, BC = ~BE , DC = -°DC , BE
b
..- gC, DE
.
There will also be need for the following averages:
~BC , 1m'
=
The linkage parameter A is defined so that the gametic array
produced by an individual with genotype a b / alb' is:
ab ,
Hence A
(1+A)
4
alb',
(I-A)
4
ab' ,
...;..(1....,.-_21..:.-) a I b)
4
•
= 1 corresponds to the one locus case, for the two loci are
completely linked and a and b are transmitted as one gene.
transmission of a and b occurs when A
=
O.
Independent
Often a measure X will be
"-
14
written as !(A) to emphasize that it is indeed a function of the
linkage parameter.
4.2
General Expansions
The general expansions of the measures will be derived for the
general pedigree in Figure 1.1.
As an abbreviation, !(abEB, a'b'EC)
will replace !(ab, a'b': abEB, a'b'EC) for example.
4.2.1
~BC
From Figure 1.1 it can be seen that the gamete alb' from indi-
vidual C is a parental type from either G or H with probability (I+A)/4
in each case, or a recombinant type with a' from G and b' from H or
vice versa, each with probability (1- A)/4.
The following expansion is
thus possible:
!(abEB), a'b'EC)
~BC
4.2.2
1 + A
4
~(abEB, a'b'EG) +
1 + A
4
= (1 4+ A) (~BG + ~BH ) + (1 2- A) IB,
~(abEB, a'b'EH)
em .
(4.2.1)
IB, GH
Here the gamete ab is replaced by its values in the previous generation:
--
15
IB , GH =
(1 + l)
A similar equation for IB
IB , GH =
(1 - l)
4
GH) +
,
2
~,
GH·
HG leads to;
(1 + l)
(4.2.2)
4
Combining equations
(4.2.1) and (4.2.2) leads to the complete ex-
~BC;
pansion of
2
~BC
= (1 4+ l
+
-e
4.2.3 ~DE,
-"IE,
)
~
lJn
+ "I
+ "I
-G, DE
-H,
~)
uc
+ (1
ii
(4.2.3)
2
l)
~, GH •
GH
Each of the four genes are now on separate gametes, so each has
probability of one half of coming from one of two individuals •
•
~DE,
GH
=
~ (~KE "
GH +
~LE
(4.2.4)
GH) •
Proceeding in this fashion leads to:
~DE,
GH =
~ (~KM,
GH +
~KN,
GH +
~LM,
GH +
The parents of G and H may also be brought in.
~LN,
GH) •
This method of treating
one gene at a time may also be used to complete the expansion of
IB,
GH:
16
IB, GH
=
1
+ A
8
(In , PH
+ I'
-n, QH
+ I'
+ -E,
I'
-E, PH
~)
(4.2.6)
+
1 - A
4
(~, l'lI + ~, QH)
.
Similarly the parents of H may be brought in.
Note that digametic functions expand back to digametic, trigametic,
and quadrigametic functions, that trigametic functions expand back to
trigametic and quadrigametic functions, and that quadrigametic functions expand only to quadrigametic functions.
4.3
Special Expansions
The original definition of
b, b' should be distinct.
.e
~
specified that the genes a, a' and
No restrictions were placed on the indi-
viduals used as subscripts for
~,
I,
and~.
Whenever such a subscript
does appear more than once (indicating that more than one gamete per
parent is being considered) special expansions are required since the
a and b genes may no longer be distinct.
Such non-distinct genes (or
copies of the same gene) are automatically identical by descent.
4.3.1
~BB
The two gametes being considered, ab and alb', each from individual B, are replaced by their respective gametic arrays.
These arrays
are identical here and equal to
where subscripts indicate which parent the corresponding gene comes
from.
..
These arrays are arranged as headings in a two-way table
17
(Table 4.2), and the probabilities that the two gametes carry genes
identical by descent are put into the body of the table.
It should be noted that this approach could have been used to
derive the previous expansions.
While it would have provided the same
results, it has the disadvantage of being cumbersome, for it gives
only one component of the measure at one time.
Table 4.2
Expansion of e
llBB
ab
e
llBB
.e
1 + A
1 + A
4
4
aEb E
aDb D
1
1 + A
4
1 - A
1 - A
aEb D
aDb E
e
. lED
e1. ED
4
4
aDb D
e
llED
1
e1..ED
e. lED
aEb D
e. lED
e
1.ED
1
e
e . lED
e
arb'
1 - A
4
llED
1
llED
18
8
11BB
=
=
1 +
2
1..
4
1 +
4
+ 1 +
2
1..
8
4
1..
llED
+
1 -
2
1 -
1..
4
81. ED
llED
+ 8
l0ED
+ 8
01ED
+ 8
2
4
2
(8
1..
)
00ED +
8. lED
1 +
4
2
1..
Applying the same procedure for the other three components of
8
l1ED
~BB
leads
to the following equation.
1
1
~e
2
1
0
~BB
2
1 +
1..
0
1 -
1..
1
1 -
1..
1 +
1..
1
2
4
2
2
4
=
=
0
0
"2
o
o
o
e
2
~ED
4
2
4
~B •
BB is the same as that for ~BB except
The two-way table for IB
)
1
that the frequencies for the gametic array for alb' are all equal to 4
Since a l } b l are on separate gametes} linkage cannot affect these
.
19
frequencies.
The expansion, then, is obtained directly from
(4.3.1)
by removing A.
=
where
IT
IT ~B
,
1
1
1
1
"2
"2
1+
0
"2
0
"4
0
0
1
1
"2
"4
o
o
o
1+
1
1
=
1
4.3.3 ~BB, BB
Linkage does not affect the gametic arrays of either ab or alb'
now, otherwise the argument is just the same as for
The expansion will be the same as for
~BB, BB
IB,
~BB
or
IB,
BB.
BB.
(4.3.3)
=
The gametic arrays for ab, alb' are now different, and are arrayed
along the sides of a two-way table (Table
4.3).
Linkage appears in the
frequencies for that of ab, but not alb' since a' and b' are on separate gametes.
The difference to
(4.2.6) arises because in the previ-
ous generation a' and b' may have been on the same gamete, which was
not the case for
IB ,
CD in the general pedigree.
20
Table
4.3 Expansion of IB, CC
alb'
1 + A
4
1
1
1
1
"4
"4
"4
~DG
~DH
ID,
GH
ID,
HG
aEb E
~EG
~EH
IE ,
GH
IE ,
HG
aEb D
IG ,
ED
IG,
DE
"4
ab
1 - A
~e
4
..
The expansion is given in equation
IB,
CC
1 + A
= 16
+
,
IH
ED
8
I
G,
~ED, HG
~DE , GH
-DE, HG
5
(4.3.4).
(~DG + ~DH + ~EG + ~EH) +
l-A(
~ED, GH
ED + I H, ED
)
+
1 + A
8
I-A
4
(ID,
GH +
Q......"...
IE, Gil)
(4.3.4)
-ED, GH
4.3. 5 ~BB, CC
The argument is the same as for IB
,
CC except that linkage does
not now affect the frequencies for ab or alb'.
..
The appropriate
21
expansion, equation
(4.3.5)
is obtained by removing A from
(4.3.4) .
.
For
l-A
4
IB,
Be' the gametic array for ab is
l-A
4
aDb E,
)
aEb D ·
(~
aDb D, 1
4A
aEb E,
The gene a l may be aD or a E with equal prob-
ability, and there is nothing to be gained by specifying b l , which is
just written bC'
4.4).
(Table
Table
The two-way table for the first component is given
4.4
Expansion of r llB , BC
ab
1 + A
1 + A
4
4
8. 1DC
r llE , DE
1 -
A
4
1 - A
--r-
8. lED
alb'
8. 1EC
8. 1CD
·-
22
Similar tables for the other three components lead to the expansion:
t
IB , BC =
where
l'1
l'1
(~CD
+
~E)
+
t
1
0
1
0
0
1
0
1
0
0
0
0
0
0
0
0
+ I'
(I'
-E, CD)
-D, EC
(4.3.6)
=
By symmetry the equation for IB, CB is:
~ f(~D
IB, CB =
-e
where
+
~E)
+
~(ID ,
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
I'
CE)
CE + -E,
(4.3.7)
!l
•
Combining
(4.3.6) and (4.3.7) leads to the required expansion:
(4.3.8)
4.3.7 ~BB, OC
In this case linkage does not affect the frequencies of ab, and
, OC.
the argument is as for IB
that expansion, equation
As linkage did not play any part in
(4.3.9) follows directly.
23
4.3.8
~BB, GH
The argument is just as for
part.
IB,
GH except that linkage plays no
Hence (4.3.10) follows from (4.2.6) by removing A.
(4.3.10)
As stated in section 4.2, the parents of H may also be brought in, but
the expansion will still be in terms of trigametic and quadrigametic
measures.
·e
24
Similar tables for the other three components lead to equation
(4.3.11) .
•
1
~ 4> ~GH
1
1
~BC , BC -2 E~DE +
+ 1 D
'4 -DG, EH
(4.3.11)
=2
2
+ 24>
!B
!c
+ 1 D
'4 -DG, EH '
where
.-.
........
1
1
1
1
2
2
1
2
0
0
0
0
0
1
1
=
.e
4>
1
1
1
2
2
0
2
1
1
=
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
4. 3.10 ~BC, CB
Table
4.6 shows calculations for
~BC,
CB' and the expansion is
given in equation (4.3.12).
(4.3.12)
4.3.11 ~BG, BH
Calculations for this final special case are indicated in Table
4.7.
There is no point in expanding the two b genes there.
pansion is given in equation
~BG, BH
1
(4.3.13).
1
=
"2 A ~DG +"2 ~DH, EG .
=
"2
By symmetry:
~GB, HB
l,rr
I
~DG
+1.,.
"2 ~HD, GE .
The ex-
26
Table
4.7 Expansion of
~llBGJ BH
•
ab
1
'2
1
'2
8. 1GH
alb'
8. 1GH
.
27
5.
GENERAL ANALYSIS OF INDIVIDUAL MATING SYSTEMS
5.1
Specific Pedigrees
With the measures defined generally) and all possibilities
accounted for in the expansion equations) it is possible to determine
the inbreeding function for any mating system.
The first type of pedigree to be considered is a specific one.
Here an initial array of individuals) of known relationship and degree
of inbreeding is specified.
inbred.
Usually they will be unrelated and non-
It is required to determine the inbreeding function of some
specified descendent of this initial array.
There is no requirement
of any pattern to the matings between the initial population and the
indiv~dual
of interest.
The method is straightforward.
The inbreeding function of the
individual is expanded back to digametic) trigametic) and quadrigametic
measures of its parents.
These measures) in turn) are further expanded
back to measures involving their parents.
This process continues until
the inbreeding function under study has been expressed entirely in
terms of measures involving the initial population) whereupon the
numerical values for these measures are substituted.
Section 6.1 contains an example of the procedure.
5.2
Pedigree Systems of Mating
When there is a constant mating pattern between every generation)
the above approach is neither practicable for long pedigrees) nor
necessary.
established.
Instead) recurrence relations for the various measures are
--
28
Such relations are well established for the (one locus) inbreeding
and panmictic coefficients.
Moreover these coefficients form the
marginal totals to the two by two tables of the components of the inbreeding function (Table 4.1).
It is thus evident that, to determine
the four components of the inbreeding function, the only information
needed is one of the components in addition to the one-locus coefficients.
It will be convenient to work with the two locus panmictic
function, and corresponding (fourth) components of other measures.
Because the pedigree now has a recurring pattern, a new notational
device is introduced.
Instead of being subscripted according to indi-
vidual, the measures will be superscripted according to generation.
The subscripts refer to the individuals providing the gametes in the
.e
.
argument of the measures; the superscripts refer to the generation
which is formed by the gametes.
The object is to find a minimal set of simultaneous transition
equations for a set of (fourth components of) measureS.
The two locus
panmictic function is necessarily a set of this smallest, or complete,
set of measures.
Any member of a complete set of measures in genera-
tion (t + 1) can be expressed in terms of the members of the set in
generation t.
·
Th e f ~rst
step is to expand
vious generation.
Faa t+1
b ac k '~nto measures
0
f t h e pre-
The types of additional measures necessary on the
right hand side of this equation are noted, and transition equations
established relating their values in generation t to values of measures
in generation t-l.
(s) equations.
The process continues until there are just enough
The complete set of measures, then, of order s, is such
29
that if any measure is removed from the set, the transition equations
of the remaining s - 1 cannot be expressed in terms of only themselves,
in the previous generation.
t 1
+ is of principal interest, the object is to use the s
As F
oo
t+1
simultaneous transition equations to find F
. If s is sufficiently
OO
small this can be done directly.
For selfing, s
= 1 and for parent-
offspring mating with one parent used repeatedly, s = 2 (see section
6.2).
For larger s, matrix techniques must be used.
The s transition equations are written in matrix form:
u
-t+1
=
nu ,
-t
] t of complete meas-
where ~~ is the s vector [F OO ' XOO ' • • ., XOO
.e
ures) and n is the s x s transition matrix.
s-l
Cockerham and Weir 2 showed
that the required recurrence formula for Foot is given directly by the
minimal equation f(x) of n.
Generally the minimal equation and char-
acteristic equations of n coincide.
equation generally factors.
currence formula for F
OO
t
For A
=0
or A
= 1,
this minimal
The factor g(x) to be used for the re-
is the factor of smallest order for which
the vector g(n)~t has zero first component.
The transition matrix thus provides a recurrence formula for F t
OO
which goes back s generations.
s
= 6;
For sib mating (Cockerham and Weir)
and for parent-offspring mating (section
used once, s
components of
= 4.
!)
6.3) with each parent
Algebraic expressions for F
(and hence the other
OO
in any generation will thus be extremely complicated.
Numerical methods will generally be used to calculate values of F
OO
for various linkage values in successive generations .
•
In programming a computer to evaluate F ' it is easiest to
OO
compute each of s measures for each generation--a procedure equivalent
to taking successive powers of the transition matrix,
5.3 Rate of Inbreeding
All the inbreeding systems studied in this work have known initial
and final inbreeding functions--generally !O
F
oo
= [1,0,0,
OJ'.
= [0, 0, 0, 1J' and
The systems differ however according to the rate
at which double non-identity is replaced by double identity, and this
rate provides a convenient characterization of a system.
For any quantity yt associated with generations, the usual rate
of increase, pt, of yt at time t is given by:
.e
t
=
P
(lim
yt) _ yt-1
t....;> 00
In the one locus case, the same rate is found whether the inbreeding or panmictic coefficients are used, since:
P
t
F t _ F t-1
F t _ F t-1
1
1
1 _ F t-1
° F °t-l
-°
1
This "rate of inbreeding" will be constant for all generations
t 1
t
o = kFo -
only if F
(k is a constant), which is the case for finite
monoecious populations with random pairing of gametes.
such a case.
Self mating is
For other systems however, an approximation can be made
which does provide a constant rate of inbreeding.
•
formula for F
t
o
is written as:
If the recurrence
31
s-l
F t+s
a
a
L:
=
r=O
Ft+r
a
r
then the solution of this difference equation is:
F t
=
a
where x
s
L:
b (x ) t ,
r=O
r
r
are the roots of the polynomial
r
x
8-1
L: a
t+s
r=O
t+r
x
o .
r
Subs ti tu ting Fat from (5.3.4) into (5.3.2) and dividing throughout by
the (t-1)
.e
..
th
p
power of the largest root, x ' of (5.3.5) leads to:
R
x
8
t
(2:)
r X
R
r=O
t-1
b
L:
=
x
8
(l-x r )/
(2:)
r x
R
r=O
L:
t-1
b
x t-1
r)
Hence, wh en t is I arge enough for (_
to be negligible for r
x
R
P
t
= 1 - x
R
f
R:
•
The same approximate (or limiting) rate of inbreeding would have resulted by working with F
t
instead of Fat.
1
In the one locus case
therefore, systems of mating may be compared by considering the quantity x
R
for that system.
Using the constant inbreeding rate also
provides:
F
t
a
.
=
X
R
Fa
t-1
or
F
t
a
=
if
a
a
F
1 .
32
A similar theory is desired for the two locus case.
can be defined for the inbreeding function, the
4x
reduced to a scalar by means of some suitable norm.
Before a rate
1 vector F must be
The norm will
also allow distances to be defined between vectors--so that, using the
difference between FO and !t, the degree of inbreeding can be specified.
The norm to be used is the two locus panmictic coefficient (the
fourth component of !).
II!t
II
=
Hence II!O II = 1 and II!oo"
function of t.
.e
° and
II!t II is a mono tone dec reas ing
The "degree of inbreeding", or distance from FO to F
t
t
is just 1 - F
' and the rate of inbreeding is:
OO
P
t
The limiting constant value of this rate is found just as in the
one locus case.
For pedigree systems of mating, when matrix techniques
are used, the polynomial corresponding to
(5.3.5) is the minimal poly-
nomial of the transition matrix for the fourth components of the
complete set of measures.
~(A)
R
of the transition matrix gives the rate of inbreeding:
p(t)
..
Corresponding to x ' the maximum eigenvalue
=
1 - ~(A)
33
For complete linkage, Foot
= FO. t = F .0 t = FOt ,
so that equation
(5.3.7) gives:
=
~ ( 1 ) ( F00
=
~
t-l
+ F10
t-l)
'
( 1 ) ( F t-l + F t-l) .
OO
01
These results, together with the requirement that F
t
1, lead to,
for sufficient large t:
F t +1
1 - ~(1)
0
~(1)
1- ~(1)
1 + ~(A) - 2~(1)
~(1) - ~(A)
0
=
F
.e
O
If F
1
= [0,
0
0
~(1)
0
0
0
t
~(1) - ~(A)
~(A)
0, 0, 1J', the inbreeding function can thus be approximated
by:
(~(l))t
F
t
_
(~(A))t
(5.3.8)
=
(~(l))t
_
(~(A))t
When two linked loci are considered, then,the inbreeding of any
system of mating is characterized by the two quantities
•
~(l)
and
~(A)
.
The maximum eigenvalue ~(l) is a monotone increasing function of l}
so that ~(l) ~ ~(1).
While this monotonicity will be demonstrated numerically} it will
not be proven analytically.
A verbal proof can be given though.
In any inbreeding system} when there is independent assortment of
genes at the two loci (l
= a)} the fact that uniting gametes carry
copies of a single ancestral gene at one locus does not imply that they
do for the other locus also.
(This is quite apart from the fact that
the probability that each locus) considered separately} carried identical genes is the same for both loci).
On the other hand} as linkage
increases} there is greater tendency for genes originally on one
chromosome to stay together.
.e
Hence identity at one locus is more
likely to imply simultaneous identity at the other locus.
tion of double identity (and hence F
11
The propor-
) is thus a monotone increasing
function of the linkage parameter l.
Equation
(5.3.8) thus implies that
~(l)(and
Faa) is a monotone
increasing function of l.
By its very nature} inbreeding implies a decrease in double
non-identity regardless of the degree of linkage} so that ~(l) < 1.
On the other hand} the fastest form of inbreeding is selfing} for
which
~(l)
=
(1 +l2)/4.
Finally) for pedigree systems of mating} the
gametes ab and alb' uniting to form an individual are specified and Faa
may be written:
Faa
=
Pr (a ~ a') b ~ b')
=
Pr (a ~ a'
I
b ~ b') P (b ~ b') •
For complete linkage:
pr (a
~
aI
Ib
~
b I)
=
1,
which implies
and for zero linkage:
pr(a ~ a l
I
b ~ bl)
= P(a
~
at) ,
so that:
and
The previous paragraph
leads to two inequalities, the first for
all inbreeding systems and the second for pedigree systems.
•
They are:
36
6.
EXAMPLES OF INDIVIDUAL MATING SYSTEMS
6.1
•
Specific Pedigree
An example of a specific pedigree will be given to illustrate the
procedures outlined in section
5.1.
The pedigree is shown in Figure
6.1.
For the one locus inbreeding coefficient, direct application of
Wright's general formula gives:
=F
A. 1
= 5/32,
if G, K, E are each non-inbred and unrelated.
.e
A
Figure 6.1
.
Specific pedigree
37
For the two locus case, !A will be expanded back until all
measures involve gametes from G, K, E only •
•
__ 1 + A
-r:'+
(1
==
(e
-DC
+ ~EC
4A)2 (~DG
2
1- A (
+
8
+
)
1 - A
+ -2- ..!..C,
l
'I'n'!I'
ur:.
~DH + ~EG + ~EH
ID,
HG +
IE,
)
IG,
HG +
(6.1.1)
DE +
This first stage follows directly from equation
.e
IH,
)
DE
(4.2.3).
Each of the
measures on the right hand side still needs expanding though •
~DG
l+A(
==
+
~KG
)
1-A
+ -2-
I G,
GK '
(*'+) 2 (~GK + ~GE + ~KE + ~KK )
~DH
~EH
~ ~GG
l+A(
==
~ ~EK + ~EE
1- A (
)
1_A
+ -2-
IE,
KE '
)
+~~GK+~GE'
,
,
•
+
~,
DE
=~
1- A (
--r:;:- ~,
GE
(~,
~,
GE
+
+ ~,KE
KE
)
'
~,
+
GE
+
0eE,
KE) •
Substitution of the expansions of these last eight equations into
equation
(6.1.1) leads to:
.e
2
+ (1 + A) 2 (17 + 2 A + A )
•
--r:;:-
16
2
+ (1 - A) 2 (5 + 2A + A
16
4
.
•
1
~GE
n....'::GK,
+ "4~, GE +
1 +A (
)
KE' + --g- ~, GK + ~, EK
3 +A
(
--g- ~,
EG
+ ~, TIE)) •
39
With !A now expanded back as far as possible, it remains to replace each of the initial measures in the expansion by its numerical
value.
Since E, K, G are each non-inbred:
~EE
==
~GG == ~KK == [
from equation
(4.3.1).
2
1 +:A.
(6.1.3)
4
If the three initial individuals are also
assumed to be unrelated:
~GK
==
~EK
==
~GE
==
(6.1.4)
[0, 0, 0, I]'
since there is no chance of gametes from different individuals being
identical by descent.
.e
Any other measure with distinct subscripts has
the same value:
IG,
KE
= IE, KG = IK,
GE
= [0,0,0, I]'
The remaining measures have repeated subscripts, and their expansions
follow from section
4.3.
From equation
(4.3.8):
(6.1.6)
These values could have been written down immediately.
In
IG,
GE for
example there is a probability of one half that the two a genes were
copies of one gene in G and hence identical.
the two b genes being identical.
Hence
..
Ie ,
1
GE
1
[0, '2 ' 0, '2 ]
There is no chance of
By synunetry:
IG, EG
1
=
1
[0, 0, 2 ' 2]
r
,
and hence the equation (6.1.6).
For the quadrigametic measures, equations (4.3.10), (4.3.13) and
(4.3.14) give:
(6.1. 7)
and equations (4.3.11) and (4.3.12) lead to:
_
~ , "GR - ~ ,
_
ER -
_ 1
1
1
5
rrE - [g , g , rr
, rr]
0
0
DGE
-,
r
•
(6.1.8)
Combining equations (6.1.2), • • • , (6.1.8) then gives the value
of the inbreeding function of individual A to be:
.e
When linkage is complete
(l
1), it can be seen from this result, and
=
that obtained above for the one locus coefficient that:
F
-F
llA -
-F
1.A -
-
5
.lA - 32 '
and that for independently segregating loci
•
•
(l
=
0),
41
The effect of linkage is thus seen to be to increase the relative
frequency of double identity and double non-identity (F
11
and Faa) at
the expense of the frequencies of single identities.
6.2
Repeated Parent-Offspring Mating
Consider the parent-offspring pedigree of Figure 6.2 where
parent (U) is used repeatedly.
t + 2 J t + lJ t respectively.
Individuals A, B, C are in generations
The initial pair U and V are not re-
lated.
The case where they are also non-inbred will be considered
first.
The method of section 5.2 will be used.
V
?
I
I
I
.e
I
I
I
,
I
~
I
I
I
I
I
Figure 6.2
•
on~
Repeated parent-offspring mating
42
From equation (4.2.1):
1 + A
4
(800UU + 800eu ) +
1 - A
2
,
I'OOU
eu'
(6.2.1)
and from equation (4.3.1):
so that (6.2.1) may be written:
F00 t+2
==
2
(1 + A)(l + A ) + 1 + A F t+1
1 - A t+1
16
4
00
+ 2
a
.
(6.2.2)
From equation (4.3.8):
1
)'oou, BU ==
'2
I'OOU,
uu
+ 1
'2
I'OOU, eu
(6.2.3)
'
and from equation (4.3.2):
,
I'OOU
1
UU ==
'4 '
so that (6.2.3) may be written:
(6.2.4)
The use of at for I'OOU
,
BU is an abbreviation consistent with remarks
on notation in section 5.2.
Eliminating a from (6.2.3), (6.2.4) pro-
vides the recurrence relation for the two locus panmictic coefficient:
.
==
3 + A
4
FOO
t+1
1 + AFt
---'8~
00
+
2
3 - A + A + A3
32
Provided the first individual (generation 1) to mate with U
•
43
(generation 0) was not inbred:
From previous authors
tution of A
(~.~.,
Kempthorne, 1957), or from direct substi-
1 into (6.2.2), the recurrence formula for the (one
=
locu~
coefficient of parentage is known to be:
F t+l
°
:=
1 + ~ F t
4
2
0'
F
°
°
(6.2.6)
= 1 .
Using (6.2.5), (6.2.6) and the fact that F
O
= FO• = F. O'
inbreeding functions for this system are found to be:
O
F
.e
=
[0, 0, 0, I]'
F- - [0, 0, 0, I]'
2
(1 + A + A + A3)/16
F
2
C3
_ A - A2
C3
_ A - A2 _ A3 ) /16
A3 )/16
=
2
(9 + A + A + )..3) /16
•
•
the following
44
(3 - }... + }...2 + }...3) /4 (3-}...)
-
}...2 _ }...3) /4 (3-}...)
(3 - }... -
}...2 _ }...3) /4 (3-}...)
(3
F
oo
-
}...
=
(3 - }... + }...2 + }...3) /4 (3-}...)
In each generation therefore, when}...
= I,
F
OO
= FO. = F. O which
is to be expected as completely linked genes are transmitted as one
unit.
When}...
= 0, FOO = FO.F. O' which shows that in the
~bsence
of
linkage the two genes are transmitted independently.
A more practical case may now be considered.
is ofcen used when the initial pair U and
.e
double identity.
V
Each is completely inbred.
This mating scheme
are unrelated but each has
Equation (6.2.1) still
holds, but because any two genes from U at one locus must now be
•
identical, 8
00UU
= YOOU, eu = O.
The desired recurrence formula is
then, from (6.2.2):
F OO
t+2 _ 1+}... F t
- T
00
(6.2.7)
The corresponding one locus equation is thus:
F t +2
o
=
..!
F t +1
2
0
(6.2.8)
These last two equations may now be combined to provide the recurrence relation for the vector F.
..
.
t, t+1 are used .
For convenience, the superscripts
45
~e'
F t +1
F
1
1
1
l+A
"2
"2
4""
0
"2
0
T
1
l-:A
F
=
1
=
0
0
'2
I-A
4""
o
o
o
4""
1
t
(6.2.9)
1+:A
[OJ OJ OJ 1]'
This system is very similar to self mating.
The appropriate
transition equation for selfing follows immediately from (4.3.1):
.e
-F
t +1 _
Ft
- ®-
•
The matrix ® differs from that in
pace
1
d by
,2
1\.
(6.2.9) only by having each
:A re-
This form of repeated parent-offspring mating differs
from selfing then only when linkage is neither absent nor complete
(0<:A<1).
6.3 Successive Parent-Offspring Mating
The pedigree is shown in Figure 6.3 and the procedure of section
5.2 is used.
Individual A belongs to generation (t+l).
=
•
•
i.e. FOO t+l
=~
4
(~t + F t) + 1 - :A at
~
00
---2---
(6.3.1)
46
From equation (4.3.1):
"
i.e.
<j>t+1
=
1 + )..2 F t
4
00
(6.3.2)
From equation (4.3.8):
YOOB , "Be = ~ (YOOC , C15 + Yoon , CC)
i. e. ex
t+1
1
t
='4 (cl + I3 )
.
(6.3.3)
c
,B
I
A
Figure 6.3
*'
,
I
Successive parent-offspring mating
-e
(4.3.3):
From equation
1
BB
= '4
(eooee + e OoeD ) +1:
2
=1
(<I> t
+ F t) + 1: at
2
00
looe ,
i. e. 13t+1
'4
The complete set, (F
OO
'
<1>,
0,
looe ,
eD
(3) is thus of order
4,
and the
transition matrix equation is:
F
t+1
1 +
)..
1 + )..
4
OO
1 + )..2
4
4
1 - )..
2
o
o
o
o
o
1
1
'4
'4
o
=
~e
0
0
t
1
1
1
'4
'4
'2
t
For non-inbred unrelated initial individuals, the initial vector is
~
,
[1, 1, 1, 1].
=
The characteristic function for the transition matrix is:
4
x
2
+)..
4
:)
x
This leads directly to the following recurrence relation for the two
locus panmictic coefficient.
F
OO
t+4
2 + )..
4
F
OO
t+:)
+ )..2 + )..3 F . t+2
+ 2
16
00
(6.3.6)
•
1 + 5).. + )..2 + )..3 F t+1
64
00
)..(1 + )..2)
t
F
128
OO
48
-e
When A
=1
this reduces to:
t +2
= 1:.
F
F00
2
t +1
00
1
+ '4 F00
t
'
which is the usual one locus equatior: (Jennings., 1916) so that
F .... (1)
O....1
F o.
When A = 0, it reduc':>;,; ':0:
1
t)1~
which is satisfied by FO.F.
.
t
too -'
F 2 and is the same as the corresponding
o
o
one for rd.b mating as given by Cockerhan and Weir (1968)3.
The dif-
ferenc'? hetween full sib mati.ng and this scheme of parent-offspring
mating
ftHir
.e
i
~ith
each parent used once, is only reflected in the inbreeding
i.on when linkage is nei.ther complete nor zero •
~ble 6.1 gives the value of F
11
, for various generations and
linkage values.
Table 6.2 lists ~(A), the dominant root of the transition matrix.
Values there marked with
All agree,
exef~pt
all
asterisk
that. for t,
0.77L")11- instead of the
c'o.n'f'i'i
o. ~)O
•
5 t' I
'1'1c.
I
'lpcn~a8"~ !Ii
n
given by Bennett (195l.ja).
for which Bennett gave the value
lJ.'T(I}iE.\.
ferent, the dominant r"at (If ILLs
the limiting rate of
Wel"e
i{
While his approach was dif-
£3 matrix also gave the value of
doublt' non-identity •
•
e
•
e
•
•
e
Table 6.1
Values of F
11
fot' successive parent-offspring mating
Linkage Parameter, A
Generation 0.0
0.1
0.2
0.,
0.4
0.5
0.6
0.7
0.8
0·9
1.0
.l..
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
2
.0625
.0694
.0780
.0886
.1015
.1172
.1360
.1583
.1845
.2149
.1~0
3
.1406
.1448
.1514
.1611
.1745
.1924
.2154
.2443
.1800
.3233
.37~
4
.25)0
.2536
.2595
.268~
.2807
.2978
.3207
.3506
.3892
.4383
.~oo
co,
'"
.3525
.3553
.3599
~3670
.3774
.3922
.4130
.440;
.4781
.5280
·5938
10
.7385
.7391
.7401
.7419
.7446
. 7489
.7556
.7662
.7835
.8120
.8594
15
.9049
.90~
.9052
.9055
.9061
.9010
.9085
.9114
.9169
.9280
.9513
20
.9665
.9665
.9666
,9666
.9667
.9669
.9672
.9679
.9694
.9732
.9831
30
.9960
.9960
-'9960
:-9960.
.9960
.9960
~9g60
.9960
.9961
.9965
.9980
~
.9999
.9999
'~9999
.9999"
.9999
'.9999
.9999
.9999
.9999
.9999
.9999
"
.
..
-=---~...,...-
.... -
--
_. - _.
~"~"----_,-",,,,-
,
..
~-----------------"'.
$
Table 6.2
Dominant root of transition matrix for successive parentoffspring mating
A
fl
0.00
.65451*
.6561:;
0.10
0.20
0.30
0.60
0.70
0.80
.67131
.68161
.69587*
.71508*
.74013*
0.90
1.00
.77148*
.80902*
0.40
0.50
.e
•
•
.65919*
.66407
*Values also found by Bennett (1954a) .
51
·e
7.
GROUP MEASURES
7.1
•
Definitions
The previous work on inbreeding functions will now be extended to
include groups of individuals using the concepts and notations of
Cockerham (1967) who extended one locus inbreeding measures for
groups.
The general pedigree of Figure 1.1 is still appropriate
here, except that the letters now refer to groups of individuals
rather than single individuals.
Groups A, B, . . . have N , N ,
A
B
individuals in them.
Corresponding to the definitions of section 4.1, group measures
are defined as:
.e
E-LA
= !(ab:3 A, a'b'3A),
(7.1.1)
~.fBC = !(ab € B, a'b' €C),
lLB , DE
= !(ab € B, a' € D, b' € E) ,
~LBC , DE = !(a € B, b € C, a' € D, b' € E)
Equation
(7.1.2)
(7.1.3),
(7.1.3)
.
(7.1.4)
for example, states that ab is a random gamete
from a random individual of group B and a', h' are on random gametes
from random individuals in groups D and E respectively.
the group inbreeding function of group A, and
function of groups Band C.
~LBC
~LA
is called
the group coancestry
The relationship between the two is no
longer as simple as that for individual measures, and is considered
•
in the next section•
52
Each of the group measures is related directly to the individual
measures.
The group inbreeding function F
--fA
•
demonstrates this.
It is
the average of 2N (2N -1) probabilities, one for each pairing of the
A A
2N
A
gametes that united to form group A; of these pairs, N are within
A
individuals and the remainder are among individuals.
Hence equation
The quantity !A is the average of the N inbreeding functions of
A
individuals in group A.
As it is usual to assign the expected value
of the inbreeding function to all members of the group, the bar will
The quantity ~AA is the average of the N (N -1)/2 coA A
be dropped.
.e
ancestry functions of distinct individuals in group A.
Once again,
the expected value of the coancestry for each pair is the same so
that the bar may be dropped.
written
~A~'
In this case though, the measure will be
to emphasize that distinct individuals are involved in
the measure.
This notation is extended to the other group measures in the
equations below.
to groups.
The subscripts on each of the group measures refer
The subscripts on each individual measure refer to indi-
viduals within groups, and primes denote distinct individuals.
Each
individual measure is an average value--the averages being for all
individuals within groups.
•
e
•
~fBC
= ~BC
~fAA
= -N1 -AA
e + N- ~AA'
(7.1.6)
NA-l
A
A
(7.1.7)
•
liB, GH
= -B,
)'
liB, CC
= -N1 -B,
)'
N -1
C
+ --)'
CC
N
-B, CC '
C
liB J Be
= -N1 -B,
)'
N -1
B
-+--)'
BC
N
-B J BiC
B
GH
C
B
liA J AA
(N -1)(N -2)
A
A
+
lA, A'A"
N2
A
~lDE,
GH
=
~LDE,
lm
=.l..
N
~iBBJ
~DE,
GH
N -1
D
CC
0
-DB,
TIC +
NBN
C
-BB J
(N
_
BC
C
NBNC
1
(N
•
D
0
-DE,
~
-1) (N -1)
B
--- 0
NBN -Be,
C
+
-E..N
•
N -1
N -1
+ _C_ 0
+ B
0
CC
NBN -BB, CC '
NBN -BB', CC
C
C
= -1 - 0
+
~iBe,
(7.1.12)
•
BC
~BB' J
N -1
+ _C_ 0
NBN -BC,
C
CC'
BC'
.
(7.1.14)
N -1
+_B_ 0
NBN -BC, B'C
C
-1) (N -1)
B
C
NBNC
~BC ,
B'C'
(7.1.15)
~LBB,
BC
•
(N -1) (N -2)
B
B
+
N2
~BB', ~
(7.1.16)
B
7.2
General Expansions
Two rules given by Cockerham (1967) will be restated here.
They
are:
1.
The numbers of gametes in uniting input sets are equal,
2.
Gametes from one input set unite at random with gametes from
the other input set.
With these rules it is clear that the probability of a random
~amete
in one uniting set being identical by descent with a random
gam~te
in the other uniting set is the same as the inbreeding function
of an individual in the offspring group, and as the coancestry function of the parental groups.
.
•
!:.A =
~LBC •
For figure 1.1 for example:
-e
-
55
For the same pedigree, the expansion of the
measures proceeds just as in section 4.2.
GH + lIE , GH )
'Y I J;3, GH =
re~aining
three group
For example:
+l-A(
4
)
~IDE, GH + ~IED, GH '
These general expansions only apply for group measures for different groups.
This is evident from equations (7.1.6), (7.1.8) and
(7.1.12), where it is seen that these group measures are the averages
of individual measures that could be expanded by the general methods
.e
of section 4.2 •
When the group measures involve gametes from the same group however, the remaining equations at the end of the last section show that
they are functions of the special individual measures which were expanded in section
4.3.
Whenever one group provides more than one
gamete for a measure, there is the possibility that these gametes come
from the same individual and thus may be automatically identical.
The
next section will consider these cases.
7.3 Special Expansions
Unlike the individual measures, group measures with repeated
subscripts depend upon the gametic sampling plan appropriate to the
system.
Two basic sampling plans will be considered:
combined sampling of gametic sets from the same group •
•
independent and
-e
For the independent sampling plan, the probability that any
individual of a parental group contributes a random gamete in an out-
•
put set of gametes is the same for all members and not dependent on
gametes in the other output sets from the same group.
In the combined
sampling plan, all output gametes from one group are considered together, rather than in separate output sets.
In independent sampling situations it may happen that every
parent contributes two gametes to the next generation.
For a dioecious
system with equal numbers of males and females this would be the case
if every parent gave exactly one gamete to one offspring of each sex.
Thus, if A is a group of males, and B is its group of fathers, the
two gametes for
~AA'
cannot both have originated from one member of B
--each father has only one son.
include the group measure
~LBB'
The expansion of
~AA'
then cannot
Instead a new class of measures--
gametic set measures--which take account of the gametic sampling plan,
must be defined.
he
x y
=
These are:
!(ab, a'b' are two gametes received by groups x, y
from parental groups B,
!B e E
=
e
respectively),
!(ab, a', b ' are three gametes received by groups
x, y z
x, y, z from parental groups B, 0, E respectively),
~Bx ey Dz Ew
=
!(a, b, a', b' are four gametes received by groups
x, y, z, w from parental groups B,
e,
D, E
respectively).
These gametic set measures can differ from the three group measures only when at least one parental group gives more than one gamete
•
57
to an offspring group.
While the expansion of
~BC
may be made in
terms of group measures, such as ~lDH (Figure 1.1), the expansion of
~BB'
must involve gametic set measures, such as !p D' For combined
BB
sampling of gametes however, all output gametes are considered to~
gether, so that, if the subdivision of the combined set into input
sets for various offspring groups is random with respect to parents,
~he
gametic set measures must be equal to the corresponding group
measures.
For combined sampling schemes in this work then, for example,
= ~lBC
!B C
x y
for all x, y.
Notwithstanding this sometime equivalence of gametic set
measur~s
and group measures, expansion of the individual measures appearing in
equations (7.1.6), . . . , (7.1.17) will be given in terms of gametic
set measures.
Once again tables showing identity relations between
gametic arrays may be used to obtain the expansions.
The subscripts
on gametic arrays for ab, alb' now refer to groups instead of individuals though.
The pedigree of Figure 1.1 is still appropriate.
From Table 7.1, the expansion of
~AA'
=
for a group pedigree is:
+ A)2(,tr
)
(1 4
'I' B B
+ 2YB C + \jr ro C
- AA
A A -"-1\ A
1 _ A2 (
+
•
~AA'
8
) + (1 _ A)2 I'
~ B Be + ~"if:'C':"
4
~B:"C':" TC:'A
A•
A' A A
A' A A
A A'
--
7.3.1 IB, ee'
The expansion of the trigametic measure
equation
(7.3.2).
IB, ee' is shown in
In that equation the dot subscript for indiv~dual
B means that the particular offspring group that the gamete fJ;"om
group B goes to can have no influence on the measure concerned.
IB, ee'
=
i
(v
-B ,
The measure ~BB,
similar one.
7·3.2
~AA'
+v
GeGe -B, HeHe
ee' has the same
+2v
-B., ~
)
.
expa~sion, and ~BB',
(7.3.2)
ee has a
-e
59
~BB'
7.3.3
, eel
The expansion of a quadrigametic measure
shown in equation
fo~
a group pedigree is
(7.3.3).
-e
There is little point in listing all possible transition equations.
In the next chapter, where examples are considered, those
expansions that are needed will be shown.
The next step is to derive
expressions for the various probabilities of gametes originating from
the same or different parents.
7.4
Gametic Sampling Probabilities
For a group B with N members, there are seven probabilities to
B
be defined.
P
These are for the possible origins of the gametes:
B
= Prob(two
B
= Prob(three
gametes from one individual in group B),
B
= Prob(three
gametes from two individuals in group B),
gametes from one individual in group B),
1
P
2
P
3
•
60
Prob(four gametes from one individual in group B),
P
B
= Prob(three
5
gametes from one individual and one ga~ete from
another individual in group B),
PB
6
= Prob(two
gametes from each of two individuals in gro~p B),
P
= Prob(two
gametes from one individual, and one gamete from
B
7
each of two other individuals in group B).
The j:emaining three possible origins for the
gamete~
are; two
gametes from two individuals in group B (with probability l-P
),
B
1
three gametes from three individuals in group B (with probability
.;J..-P
-e
- P ), and four gametes from four individuals in group B (with
B
B
2
3
probability l-P B - PB - P B - P ).
B
4
5
6
7
For independent sampling of gametic sets, in one set let k be
1
th
the number of gametes contributed by the i
parent. If this set is
an input set for group A, the total number of gametes 1s the same as
the number of offspring:
with a mean of
For combined sampling, k. will refer to the total number of
~
gametes contributed by the i
th
individual, and A will refer to the
total collection of offspring individuals.
Each summation in the expressions for the seven probabilities
listed below is over the integers 1 to N .
B
61
,
,
PB
3
=
3 I:. 1
k. (k.-l)
(N B KAB-k.)
1
1
1
,
NB KAB(N B KAB-l)(N B KAB -2)
I: k.(k.-l)(k.-2)(k.-3)
PB
4
.1111
1
NB KAB(N B KAB-l)(N B KAB-2)(N B KAB-3)
(k.-2)(N B kAB
-k.)
4 I:.k.(k.-l)
1 1 1
1
-e
PB
5
1
,
,
NB KAB(N B KAB-l)(N B KAB-2)(N B KAB -3)
,
If each of the k. are identically distributed, each of the
1
probabilities can be replaced by its expected value, making use of
these central moments:
m
22 AB
= E(I: ~
i j
(k.-K AB )
1
2
2
(k.-K AB) /N B) ,
J
62
-e
The probabilities now take the form:
,
,
•
Note that the reciprocal of P
B
is usually called the (two gamete)
1
effective population number (Wright, 1931).
Similarly, the reciprocals
of P
and P
could be called three and four gamete effective
B
B
2
4
tion numbers respectively for group B.
PQPula~
When two groups are considered, additional probabilities are
needed.
P
BC
For groups Band C with N and N members respectively:
C
B
=
Prob(two gametes from one individual in group B, and two
1
gametes from one individual in group C),
P
BC 2
=
Prob(two gametes from one individual in group B, and two
gametes from two individuals in group C),
P
BC
Prob(two gametes from two individuals in group B, and two
3
gametes from one individual in group
C).
Hence, for two gametes from two individuals in group B, and two
g~etes
from two individuals in group C, the probability is
1-P BC -P BC -P BC '
123
Sampling within one group is supposed to be independent of that
in another group, so:
~e
64
P
P
P
BC
BC
BC
1
= PB
1
= P
2
Pc
B (I-PC )
1
1
= (I-P
3
B
1
(7.4.1)
'
1
}P
,
(7.4.2)
(7.4.3)
C
1
Two gametic distributions commonly considered are "equal chance"
and a distribution as close as possible to a constant number of
gametes per parent.
For equal chance, each member of a group has equal probability of
providing any of the gametes in an output set of that group.
For the gametic set transmitted from B to A then, the k
distributed binomially Bi(N
B
KAB ,
i
are
NIB) and the central moments are:
Substitution of these values into the above definitions leads to:
P
P
P
e
B
1
= I/N
B
,
2
B
2
= l/N '
B
2
B
3
= 3(N -l) /N '
B
B
The other special case considered may be where every member of B
contributes exactly the same number of gametes.
All moments about
the mean of the k. distribution are then zero, and the probabilities
1.
become:
KAB-I
P
BI
=
P
=
B2
,
NB KAB-I
(KAB-I) (K AB-2)
,
(N B kAB-I)(N B kAB-2)
,
..
.
--
66
,
PB
7
6K~B(KAB-1)(NB-1)(NB-2)
=------------
(N B kAB-1) (N B KAB-2) (N B k AB -3)
In combined sampling plans, and equal sized groups, K
AB
= 2,
so
that the probabilities are:
P
B
=
P
= 0 ,
1
--
B
2
1/ (2N B-l) ,
(7.4.11)
(7.4.12)
(7. 4.13)
PB = 3/(2N B-1) ,
3
,
(7. 4.14)
P
B =0 ,
5
(7. 4.15)
PB
6
(7. 4.16)
P
B4
=0
=
3/(2N B-1) (2N B-3) ,
P B = 12(N-2)/(2N B-l) (2N B-3)
7
.
(7.4.17)
For independent sampling plans, each parent generally gives just
one gamete to an output set, so
i
= 1,
KAB = 1, and:
•.• ,
7 .
A result that follows almost trivially from definitions can now
be shown for the three measures
•
I,
~,1.
Consider
IB B
for example.
AA
From its definition, and that of the above probabilities:
(7.4.l8)
•
-e
For equal chance gamete distribution, P
B
1
that
(7.4.18)
just reduces to
(7.3.1).
= l/N B
(7.4.4)
from
so
In other words:
The same procedure for the other two measures leads to:
=
LiB, BB'
_e
+ 2~BB', ~)/3
+ (l-P B -P B -P B
DD
4
=
5
6
~iBB , BB·
Similar equations lead to the result that, for equal chance gamete
formation, there is no difference between group measures and gametic
set measures, and hence no difference between various sampling plans.
For one gamete per parent to a particular offspring group,
-e
68
There is now sufficient theory developed to consider some examples,
as in the next chapter.
_e
•
-e
69
8.
8.1
EXAMPLES OF GROUP MATING SYSTEMS
Monoecious Populations: General Sampling Scheme
The group measures are now applied to finite monoecious populations.
(group).
Inbreeding arises because of the finite size of each generation
The mating scheme is shown in Figure 8.1, where B, C are the
groups of individuals comprising generations t, t+l respectively.
With sexuality of gametes, gametic sets are involved, and the numbers
of each) male and female, must be the same.
There is only one group in each generation, and hence only three
types of group measures:
~£BB' Z£B,BB
gametic set measures however is nine.
and ~£BB, BB.
These, together with an abbre-
viated notation, are now listed.
8
Figure 8.1
The number of
t
Monoecious mating
70
r t +1
-1
v
= -B
v
B B = -B
m, mm
f , BfB f
)
t+1
= -B
v
= -B
v
, 'if"13 )
,
m BmB f
f
f m
t+1
= -B
v
=v
B
,
-Be Bmm
m BfB f
£2
£)
~+1 - S
- -B B ) B B
mm
mm
",t+1
~
= -BfB
S
f , BfB f
=S
~+1 =
-)
,
-B B
,
,
m m'
S
-B B
m m'
In these equations, an m subscript indicates a male gamete and
an f denotes a female gamete.
Because there are the same number of
male and female gametes formed each generation, there is nothing to
be gained by distinguishing between, say, IB Band !B B • Instead
mm
f f
.
t h ey are g1ven
t h e common va 1ue!2t+1 . As any male and any female
gamete from group B may unite to form an individual in generation
(t+1):
71
·e
Hence the measure !1 is just the inbreeding function> and this is the
function the value of which is to be determined.
Each of the nine gametic set measures will now be expressed in
terms of individual measures, making use of the probabilities defined
in section
7.4.
The cwo digametic measures each have the form of equation (8.1.1),
where Band B' are any distinct individuals in group B.
gametic sampling plan is specified, the values of P
!1 and !2 cannot be given.
P B ~BB
t+l
appropriate for
They may of course be different.
+ (l-P B ) ~BB'
1
B
Until the
t+l
(8.1.1)
1, 2 .
i
1
If B, B', B" are any three distinct individuals of group B, then
equation (8.1.2) holds for each of the trigametic measures.
r.t+l
P B YB BB
2
'
-~
+ (l-P
t+l
(
+ P B IB
1
B2
_p
)"1
B'B'
t+l
'
+ 2 IB, ~
t+l
i
BIB
3 - , B' B"
t+l /
) 3
3.
::::; 1, 2,
(8.1.2)
The four quadrigametic measures have the form of equation (8.1.3),
where B, B'
•
,
B"
,
B' " are any four distinct individuals of group B.
+ ~BB' , BB'
t+l
)/3 + P B (2~BB B'B"
7'
t+l
t+l
B"B'"
+
~BB'
t+l
BB"
)/3
i ::::; 1, 2,
3, 4.
,
(8.1.3)
72
There are thus thirteen individual measures to expand back in
terms of gametic set measures of the previous generation.
The methods
for obtaining such expansions were outlined in the previous chapter.
They are all listed in Table 8.1 where, for convenience, just the
fourth component of individual and gametic set measures are shown.
The double zero subscript has been dropped.
When the sampling plan is specified, values of the probabilities
P . can be given, and equations (8.1.1), (8.1.2) and (8.1.3) together
B
~
with Table 8.1 lead to simultaneous transition equations for the
gametic set measures.
At this stage the methods of Chapter 4 are
employed to give values of the inbreeding function for the system.
The next two sections consider combined and independent sampling of
gametic sets.
Regardless of the sampling plan however, when the
groups are
single individuals, the system reduces to a selfing pedigree.
For any
value of t then:
F t +1
-
=
e t+l
-BB
=(8)
from equation (4.3.1).
_F t
Hence
°
and if the initial individual was not inbred, so that! =[0,0,0, I]':
-e
73
Table 8.1
Expansion of individual measures in terms of gametic set
measures for a monoecious population
t
T
1
2
(lt )
t+1
eBB
t+l
e
(1+:>-.)2
----g--
BB'
'Y B, BB
t+1
1
t+1
'Y B, Blf'"
t+1
'Y B, BIB"
~e
,
t+1
BBB, B'B'
BBB'
,
.
t+1
t+1
BB'
t+1
,
BBB
rt
~
~
~
~
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-----s-
(1-:>-')
(It)
0
0
0
0
(\;:>-.)
cr
(lH)2
----g--
B'B"
t+1
BBB'
,
BB"
BBB'
,
BUB'"
t+1
0
(1_:>-.2)
-2-
3
0
0
-----s-
(It)
0
'2
0
0
0
1
1
'4
'4
0
0
(1+:>-.)
1
BB
BBB mP"
r 2t
t+l
t+1
,
BBB
r 1t
0
'4
'Y B, B'B'
t
T
2
(lH)
cr
1
(1 :>-')
4
'4
0
0
0
0
0
0
'4
1
(1+:>-')
cr
0
0
(1-:>-')
~
(\3:>-.)2
(1_:>-.)2
----g--
(1-:>-')
0
0
0
0
'4
0
0
0
0
1
1
1
'2
0
0
0
1
'8
0
1
'8
'8
'8
0
0
0
0
0
0
0
'4
0
0
1
1
'8
'4
g
0
'4
0
0
0
0
0
0
1
1
'4
'4
0
0
0
0
0
g
1
1
1
1
1
1
g
1
1
'2
'4
0
1
g
0
1
g
1 + )..2 t
- ( 4
)
=
_
(
(
1 +
4
,2 t
11.
)
1 + )..2 t
4
)
8.2 Monoecious Populations: Combined Sampling Scheme
Great simplification arises when the combined sampling of gametes
is considered--the sex of each gamete being random with respect to its
parents.
From section
7.3
it is known that the gametic set measures
are not different from the group measures in this case.
The following
relations thus hold:
.
T t +1
= T t +1
_f t+1
t+1
t+1
= -2
f
= -3
f
--
-1
1
-2
e
(8.2.1)
= -PBB '
(8.2.2)
~
..LPB, BB '
t 1
t 1
tH
6 + = 6
= ~+1 = 6 + = _0 nBB
=1
-2
-)
-4
~, BB •
(8.2.3)
(8.2.1), (8.2.2) and (8.2.3) will be
1.
and ~t +
respect1vely.
Table 8 .1 may be modified
The common values of equations
labelled Tt+1 ,
I t+1
simply by adding the first two columns and labelling the resulting
co1umn T t , and labelling the sum of the third, fourth and fifth
columns by
f
t
t
and that of the last four columns by 6.
The table now,
-e
75
together with equations (8.1.1), (8.1.2) and (8.1.3) leads to the
transition equations:
(8.2.4)
~t+1 = ~t '
where u'
-t
= [T,
f, 6]t and the transition matrix
n
has the form below
(equation 8.2.5).
4
2
(1 +A) (l-P
B
1
)
12
2
2
Now if each generation is constrained to be of the same size, so
that the mean number of gametes per parent is K
P
B
'
., P
= 2, the probabilities
are constant for all generations, so that
B
n is
the
7
1
same for all generations and (8.2.4) holds for all consecutive generations t, t+l.
P
B
1
,
The initial vector is
~
= [1,
1, 1].
For an equal chance gametic distribution, (K
= 2), the values of
were given in equations (7. 4• 4),
., (7.4.10) and
. , P
B
7
substitution of these into equation (8.2.5) gives the following transition matrix (equation 8.2.6).
-e
76
n =
(1+1)2
4
1
2N
(1+1)
4N
1
4N2
(2N-1)
4N 3
(N-1) (1_12)
2N
(N-1) (1_1)2
4N
(N-1) (N+1+1(N-2))
2N 2
(N-1) (1-1) (2N-3)
4N2
(N-1) (2N-1)
N3
(N-1) (2N-1) (2N-3)
4N 3
The common size of each generation has been written as
(8.2.6)
N.
The
recurrence formula for the two locus panmictic coefficient goes back
three generations.
For complete linkage however it reduces to:
or:
t+1 = (2N-1)
t
2N Faa '
FOO
•
which is the usual (one locus) panmictic coefficient equation (Wright,
1931).
Setting 1 = a however does not lead to an equation satisfied
by Fa. F.
O
as was the case for pedigree systems of mating (section 5.3~
For any group mating scheme, for an offspring A of parental groups
Band C:
where the sum is over a number of individual measures X. which may be
~
di-, tri-, or quadrigametic.
The coefficients Pi are just the prob-
abilities introduced in section
7.4.
Since
77
•
then
This matter will be discussed later.
The equal chance system was studied numerically, and values of
the dominant root of the transition matrix are listed in Table 8.2.
There the root is clearly a monotone function of A as discussed
viously.
pre~
It is also a monotone increasing function of N, agreeing
with the general result that larger populations are less inbred.
Values of the two locus inbreeding function F
ll are displayed in
Figure 8.2.
If the gametic distribution is such that every parent gives
exactly two gametes, the probabilities PB "
equations
1
7
(7.4.11), . . . , (7.4.17), and the transition matrix takes
the following form (equation
(1+1..)2
4
n
=
• ., PB are given by
A
2(2N-l)
1+1..
4(2N-l)
1
2(2N~I)(2N~3)
8.2.7).
(I~A2) (N-l)
2N-l
(I_A)2(N_l)
2(2N-l)
1 + A(N-2)
2N-l
(I-A) (4N-7)
4(2N-l)
"2
4N-7
(2N~I)(2N~3r
•
(8.2,7)
16N2 - 56N + 51
4(2N-l) (2N-3)
For complete linkage, the recurrence formula for the two locus
panmictic coefficient reduces to:
..
-e
•
78
Table 8.2
Dominant root of transition matrix for monoecious populations with equal chance gamete formation
Population Size, N
2
4
8
16
32
0.0
.57908
.76956
.87959
.93858
.96901
0.1
.58104
·77021
.87972
.93860
.96901
0.2
.5842 5
.77119
.87992
.93863
.96902
0.3
.58912
.77265
.88023
.93868
.96902
0.4
.59626
.77485
.88070
.93877
.96904
0.5
.6065)
.77824
.88147
.93890
.96906
0.6
.62100
.78361
.88277
.93915
.96910
0·7
.64120
.79246
.88521
.93965
.96918
0.8
.66865
.80759
.89037
.94084
.96940
0.9
.70468
.83353
.90329
.94411
.97031
1.0
.75)00
.875)0
.9375)
.96875
.98438
Linkage Parameter, A
.e
..
,
..
•
e
e
e
I
1.0
08
~1
0.6
0.4
0.2
0.00
20
10
30
40
50
Generation
Figure 8.2
F
11
for monoecious populations with equal chance gamete formation
-..J
\0
·-e
but for the other values of A, it goes back three generations.
Numerical values of the dominant root of
8.3 where, once again,
F
11
t
are shown in
~(A)
Fi~re
n
are listed in Table
increases with both A and N.
Values of
8.3.
8.3 Monoecious Populations: Independent Sampling Scheme
From section
7.4
it is known that there can be no difference
between gametic set measures and group measures for an equal chance
gametic distribution.
However, the expansion of group measures is
independent of the gametic sampling plan, so that independent sampling
of gametic sets is the same as combined sampling for an equal chance
.e
distribution.
The results of the previous section for equal chance thus also
hold for this section.
In particular, equation (8.2.6) gives the
appropriate transition matrix, and Table 8.2 and Figure 8.2 give
numerical characteristics of the system.
The two gametic sampling plans do differ however when the gametic
distribution is such that every parent gives exactly two gametes.
In
this independent sampling plan, every parent is supposed to contribute
one male and one female gamete to the next generation.
It is this
system that shows up the differences in the general equations, (8.1.1),
(8.1.2) and (8.1.3) according to which particular measure is being
discussed.
The differences arise between measures when the gametes
fo~
which the measure is defined are located in the same or different output sets.
81
Table
8.3
root of transition matrix for monoecious populations with combined sampling and two gametes per parent
Do~inant
Population Size, N
2
4
8
16
32
0.0
.69769
.97670
.93445
.96800
.984;1.9
0.1
.69800
.86236
.93445
.96800
.98419
0.2
.69909
.86249
.93447
.96800
.98419
0.3
.70126
.86276
.93450
.96801
.98419
0.4
·705)1
.86326
.93456
.96802
.98419
0·5
.71106
.86418
.93468
.96803
.98419
0.6
.72057
.86588
.93493
.96807
.98420
0.7
.73523
.86916
,93547
.96814
.98421
0.8
.75734
.87607
.93683
.96837
.98424
0.9
.78944
.89213
.94155
.96938
.98444
1.0
.83333
.92857
.96667
.98387
.99206
Linkage Parameter, A
.e
•
.
-
e
•
e
lO,
08
I
N=2
----------::::::=
/~O
N=4
~~
~
0.6
N=16
0.4
0.2
0.00
10
20
30
40
50
Generation
Figure
8.3 F11 for monoecious populations with combined sampling and two gametes per parent
OJ
I\)
83
For gametes going to the same set (the set of male or female
gametes), the probabilities for one gamete per parent (equation
(7.4.18»
are appropriate.
probabilities
(7.4.17»
For gametes going to different sets, the
for two gametes per parent (equation
are appropriate.
. .,
(7.4.11), .
Combinations of the two sets of prob-
abilities must be used when the gametes
~or
the measure are such
some go to one output set, and some to the other.
tha~
As an example,
consider the three cases of equation (8.1.2):
1 y
.!:2 = -N -B,
.e
.!:3
mrr
2
y
W
N -B,
= -
1
N-2
-y
+ NIB, B'B' +N
-B, BIB"
N..2
,
+NIB
B'B"
,
.
With the proper probabilities in equations
(8.1.1), (8.1.2),
(8.1.3), using the expansions of Table 8.1 leads to the 9 x 9 transition matrix in equation
(8.3.1).
With complete linkage, there are only two equations to conetder
since the complete set of measures is then just (T , T ).
1
seen from the matrix by substituting A = 1.
~ions
aJ;'e:
t
T1t+~ = T1t j2 + (N)
-1 T2/2N '
so that
2
This can be
The two transition
eq~a_
tl
e
e
e
1+A2
4ir +
2
(l A)
g
0
n =1
,
J
(1+A) 2 (N-l)
BN
2
(1+A) (N-l)
BN
2
(l A)
g
0
0
(I-A2fr~N-l)
0
0
0
0
l_A2
-2-
0
0
0
0
""11
I-A
I-A
T
T
T
4N
1 + (1+A§ (N-2)
N
0
(I-A) (N-2)
4N
(I-A) (N-2)
BN
(I-A§~N-2)
1 + (1+A) (N-2)
BN
0
(I-A) (N-2)
4N
(I-A) (N-2)
BN
(I-A§~N-2)
I+A
T
(I+A§~N-2)
I+A
I+A
""11
t
I-A
I+A
IH
EN
EN
0
0
0
0
0
0
0
4N
2N
4N
EN
(:;)
(N-~)
N N-l
~
1
2N
N-2
4N(N-l)
(N-2 (N-3)
BN N-l)
t
2N
1
4N(N-l)
1
4N(N-l)
N-2
2N(N-l)
N
1
N-2
2N(N-l)
(N-2~(N-3)
BN( -1)
2N
8N
(I+A§~N-2)
1
3
+ (IH (N-2)
N
4N
!... + (1+Afr (N-2)
2N
N
2N
0
0
1
1
1
1
g
'2
2N-3
N-3
(I-A) 2(N-l)
BN
(I A)2
S
1
Ii
N-l
(I_ A)2(N_l)
BN
(I_A)2
er-
1
N-l
liN
EN
N-2
2~-4N+3
2
N _3N+3
BN(N-l)
N-2
~-3N+3
4N(N-l)
(8.3.1)
g
~
BN(N-l)
I
1
g
CP
+=-
85
1. e.
F
OO
t+l
= FOO t
- F
OO
t-l
/4N,
which is the result given by Cockerham
populations, F
OO
for the
O
=1
and FOOl
= 1/2N.
(1967).
For non inbred initial
Values of the dominant ropt
9 x 9 transition matrix are displayed in Table 8.4,
values of F
ll
Tables
t
are given in Figure
whi~e
8.4.
8.2, 8.3 and 8.4, together with Figures 8.2, 8.3 and 8.4
allow conclusions to be drawn for monoecious mating systems and comparisons to be made between the three systems studied here.
In each
system inbreeding starts in the first generation.
Firstly, the graphs show immediately that an equal chance gamete
formation scheme leads to a more inbred population than do either of
the two gametes per parent (controlled mating) schemes.
For any
generation, in populations of the same size and between loci for which
there is the same amount of linkage, the value of F
ll
for equal
chanc~
gamete formation is greater than that for either scheme of two gametes
per parent.
For these two controlled systems, F
ll
for the combined
sampling plan is initially less but later greater than F
for the
11
independent sampling plan.
The generation in which inbreeding
increases with population size.
i~
equal
Since these comments apply for all
linkage values, including complete linkage, they are also true for the
one locus coefficient F , and as such may be termed "first order"
l
effects.
The "second order", or linkage, effects are indicated by the
three lines for each population size in the graphs.
•
The greatest
86
-e
Table 8.4
Dominant root of transition matrix for monoecious populations with independent sampling and two gametes per parent
Population Size, N
2
4
8
0.0
.73689
.87067
0.1
·73717
0.2
16
32
.93646
.968~
.98431
.87072
.93647
.968~
.98431
.73817
.87085
.93648
.968~
.984;1
0.3
.74015
.87112
.93651
.968~
.984;1
0.4
.74354
.87161
.93658
.96851
.984;1
0·5
.74897
.872~
.93670
.96853
.98432
0.6
·75739
.87413
.93694
.96856
.98432
0·7
.77014
.87727
.93746
.96864
.98433
0.8
.78908
.88385
.93879
.96886
.98437
0.9
.81628
.89897
.94338
.96984
.98457
1.0
.85355
.93301
.96771
.98412
.9921;
Linkage
Parameter, A
,
;
,e
•
•
•
e
e
•
•
e
1.0
N=4
0.8
~1
0.6
N=16
0.4
0.2
0.0
0
10
20
30
L()
rsJ
Generation
Figure
8.4 F 11 for monoecious populations with independent sampling and two gametes per parent
co
--J
effect due to linkage, Fll(l) - Fll(O), is seen to have a maximum in
an early generation and decrease to zero at complete double identity.
For any population size, this maximum (effect of linkage) is least for
equal chance gamete formation and greatest for two gametes per parent
with independent sampling.
The curve for 1
midway between the two extreme lines (1
= 0.9
is seen to lie about
= 0.0, 1 = 1.0) in the first
few generations and then tend towards the 1
= 0.0 line.
This demon-
strates the non-linear way in which linkage effects inbreeding:
90
per cent linkage gives only about half the effect of 100 per cent
linkage.
Whereas the rate of increase of Fll(l) is monotone, the
curve for Fll(O) first has a decreasing rate and then an increasing
(or at least non-decreasing) rate.
.e
linkage values.
The same is true for other small
This results in populations of different sizes being
able to be equally inbred at the points where, say the Fll(O) line of
one intersects the other lines of another.
The generation at which combined and independent sampling plans
with two gametes per parent have equal values of F
linkage.
is later.
ll
is altered by
As linkage decreases from complete to zero, this generation
In other words, as linkage decreases the combined sampling
controlled plan avoids inbreeding for a longer time than does the
independent sampling controlled plan.
The final behavior is unchangAd.
Limiting behavior for the three systems is summarized by the
values of the dominant roots of the transition matrices.
In keeping
with the higher rate of inbreeding for equal chance gamete formation
shown in the graphs, the dominant roots are less than the corresponding ones for either system with two gametes per parent.
Of these two
89
systems, combined sampling gives smaller roots than does independent
sampling.
•
The difference between the latter two sets of values how-
ever is less than that between either one and that of equal chance
gamete formation--which is reflected in the greater similarity of the
graphs for the two controlled systems.
8.4 Dioecious Populations: General Sampling Scheme
The next group pedigree involves two groups (males and females)
per generation.
Population size is assumed constant at N males and
m
N females per generation.
f
The mating scheme is shown in Figure
where groups are labelled with m or f for males or females.
8.5,
Measures
will be superscripted according to generation.
.e
When N
m
= Nf = 1, the system is a full sib pedigree, which has
been studied by Cockerham and Weir
(1968)4.
m
m
Figure
4
•
Ibid •
f
f
t
t+1
8.5 Dioecious mating
90
With two groups per generation, there are nine group measures
(assuming equal inbreeding of males and females), and twenty-three
•
gametic set measures.
These latter, together with abbreviated nota-
tion, are now listed.
=
(!m m
mm
= (v-m ,
.e
f
+ v
f
-m , f f f f )/2 ,
mm
=v
-mm' mf f • '
•
v
+ -m
~2
•
=
(v
=
(~m
-m f' mm
mm
m m'
f f
f'
m m )/2 ,
f f
+ v
)/2
-mm' mm
'
f f
mm
f f )/2 ,
+~ m
m m' f f
91
~l = 1n f , mfff '
mm
•
S , m f )/2 ,
~2= (1n f
m f + ..:mff
m m' mm
f f
f
S , m f m)/2 ,
~3 = (1nmf m' mmff + ..:mff
f
f
= (~ m
S , m::r)/2
iilT + ..:mfm
m m' m.
f mf •
,
m)
f /2
~2 = (~mmm' mff. + ~fmf' m •
,
S
(1n m m f + ..:mfmm'
m f.)/2
m f' m.
f
,
+ S
~41 = (~mmm' mm
..:mfm f , mf mf )/2
mm
,
~l
~3
.e
+ 1n? mm) /2 ,
~2 = (~mmm' iiiIil
mf
f' f m
~43 =1nm
m m' mfmf
•
,
Every quantity in these equations is understood to have the same
(t+l) superscript indicating generation.
Because there may be unequal numbers of males and females in each
and
mm
However, each has a very similar expansion, so it is convenient
generation, it is not possible to equate such measures as '"
-m m
'"
.
-'-mfm f
to work with their average.
There is another average implicit,
not expressed, in the above equations.
bu~
Each gametic set measure listed
there represents the average of itself and that measure obtained by
placing m by f and f by m in all positions of the subscripts.
•
re~
92
A dot subscript indicated that the gamete from that group goes to
either male or female offspring groups.
As any male gamete and any
female gamete--!.=. a gamete from the male group, and a gamete from
the female group--from a generation may unite to form an individual
in the next generation.
1jrt+1
~f
= T t +1
-11
Hence the measure T 1 is just the inbreeding function, and this is the
-1
quantity to be determined for all generations.
Each of the twenty-three gametic set measures will now be expressed in terms of individual measures, making use of the probabilities defined in section
.e
7.4.
As for the monoecious case, some of the
digametic, trigametic or quadrigametic set measures have the same
general expansions.
Differences do not appear until the gametic
sampling plan is specified •
•
For the digametic measures,
!21 and !22 have that of (8.4.2).
!11 has the form of (8.4.1) and both
In these equations, m, m' are any
distinct males and f, f' are any distinct females in generation (t).
(8.4.1)
-T2 ·
~
i
= 1, 2 .
For the trigametic measures, where m, m', mil are any three
.
distinct males and f, f', f" are any three distinct females in generation( t):
93
f
-
,
L. ff + P
I'f
)/2 + «l-P f h
ff' + (l-P )I'f nun,)/2,
1 1 = (p f 1 -"',
m1 - , nun
1 ~,
m1 - ,
f
-
1m
,
2 1 = (Pml '
f
,
31
1m
= (Pm
ilif + (l-P f ) I'f' =)/2 ,
m
1 ,mI
I' ,
~,
(8.4.4)
nun + Pf
2'
i
.e
iii! + 1f ilif)/2 + «l-P )
, m
m1
= 1, 2 .
i
-
(8.4.3)
1, 2 .
i
1f
2'
= 1, 2,
ff
)/2 + (p
m
I'
3
~,
m'm'
3.
For the quadrigametic measures, where primes indicate distinct individuals in generation
(t):
6-11, = Pmf 1 -mIll,
0
ff + Pmf 2 -mIll
0 , , ff + Pmf
"
+ (l-Pmf -Pmf -Pmf ) -tDID.,
0 , ff"
2
1
6- 2 1,
= Pmf
1
0
3
-tDID.,
ff'
1, 2,
i
3.
3
(0
0
fm) /2 + P
o,
0 f , mf'
-mf, mf + -fm,
mf -mf,
m f + Pmf -m
2
3
+ (I-P f -P f -P f )(0 f
If' +of
f'm,)/2,
m 1 m 2 m 3 -m, m
- m,
i
•
= 1, 2, 3•
(8.4.6)
94
+ «I-P
m2
-P
m
3
) 0
I mIff + (I-P f -Pf ) ~ff', f"m)/2 ,
2
3
-mm,
i
:;: 1, 2,
(8.4.8)
3 .
0
+ Pf off ff)/4 + (p
5
mm' + Pf ~ff ~)/2
~4i = (pm4 -mm,
mm
4 - ,
m -mn1,
5 ' 1:L·
5
mImI
+ (p
~
(0 I ,,+ 0 I
" ) + Pf (0 I mm" + 0 I
m"m)
-mm.mm
-m m, m m
7 -mm,
-m m,
+ Pf (~ffl, ff" + Of If f"f))/12 + «1-P -P -P -P )0 I ""I
7
'
m4 m5 m6 m7 -rom ,m m
i
= 1,
2, 3, 4.
(8.4.9)
In these nine equations, every group and individual measure 1s
understood to have the superscript (t+l).
!here are thus 57 individual measures to expand back in terms of
gametic set measures of the previous generation.
However, there are
only 28 distinct expansions--a result of the averaging that took place
in defining the gametic set measures.
These 28 expansions are listed
in Table 8.5 for the fourth component of each measure.
•
Of the 28
individual measures listed therej 3 have subscripts symmetric in m and
f, 23 represent the average of that measure and that obtained by
•
e
•
Table 8.5
T
e t +1
mf
e t +1
mm
e t +1
mm'
t+1
1 m, ff
t+1
1 m, ff'
t+1
1 m, iii!
t+1
1 m, iii"Tf
t+1
1 m iiiffiT
t+1
1 m m'mI!
t+1
1 m mim'
t+1
1 m mm
Bt +1
mm ff
t+1
Bmm " ff'
Bt +1
mm', ff
t
11
e
T
(1+A)
~
~
(1+A)
0
2
~
0
1+A
1+A
0-
0-
0
0
0
0
0
1+A
0
0
0
0
0
t
22
f
t
n
t
12
f
2
(1+A)2
1+1,.2
e
Expansion of individual measures in terms of gametic set measures for a dioecious
population
t
T
21
~
•
•
\9
t
21
1_1,.2
-2f
0
0
0
0
0
0
0
0
0
0
(1+A)2
~
0
0
0
0
0
0
HI,.
0
0
1
'4
1+1,.
00
0
1
0
"2
1+A
1+A
0
II
0
0
1
II
l+A
0-
1
f
t
22
f
t
32
f
t
33
~1
(1_1,.)2
0
0
0
0
0
0
0
0
0
0
0
0
1_1,.2
-2-
0
0
0
0
0
'4
0
1+A
0-
0-
0
t
31
0
1+A
0
f
1
1-1,.
0
0
0
0
0
0
0-
0
0-
0
Il
0
(1_1,.)2
~
0-
0
0
1+A
0
0
0
II
0
1-1,.
0
0
0
0
(1_1,.)2
0
0-
0-
~1
0
1+A
1+1,.
t
""13
0
0
0
0
l+A
1
~
t
""12
0
0
~
1-1,.
01-1,.
0
0
0
0
0
0
0
0
0
0
(1_1,.)2
~
0
0
0
0
0
1-1,.
0
0
0
0
1-1,.
0
0-
0-
0-
t
""23
0
0
1-A
~2
0
0
0
0
0
0
1-1,.
0
0-
0
0
1-A
01-1,.
t
""41
t
D42
t
""43
t\4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1-1,.
1-1,.
0
0
0
0
~1 ~2 ~3
0
0
1-A
1-1,.
Il
0- 00
0
0
0
0
0
0
Il
0
0
0
0
0
0
0
0-
0
0
0
"2
0
0
0
0
0-
0
0
0-
0
0
0
0
0
0
0
0
"4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
01
g
0
0
g
0
0
0
0
0
0
0
0
0
1
"2
0
0
1
1
g
"4
0
0
0
0
0
0
0
0
0
0
1
g
0
1
1
0
g
1
1
g
g
g
0
0
0
0
1
g
1
g
1
g
0
0
0
0
0
0
1
2"
1
"4
1
0
0
0
g
0
0
0
0
0
0
\0
\J1
•
•
e
Table
e t +1
mf mf
e t +1
mf, m'f'
tH
e
mf, m'f
et+l
ann iii£
e t +1
mml, m"f
e t +1
iii'1'
ann,
et+l
ann '
e t +1
ann
e t +1
ann
e t +1
ann
e t +1
ann '
et+l
mf
ann
mimI!
mimI
ann"
am', nan'
e t +1t
mm
e t +1
ann
t
mltm ll1
iiiiii""
•
•
e
•
e'
e
8.5 (continued)
t
Tn
t
T
21
t
T
22
0
0
0
f
t
ll
0
f
t
12
0
f
t
21
0
f
t
22
0
f
t
31
0
f
t
32
0
f
t
33
0
6
t
n
1
'4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
1
0
1
0
0
1
'4
1
0
0
0
g
0
g
g
0
g
0
0
0
E
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
1
'4
0
1
g
0
0
0
0
0
0
0
0
0
0
0
1
g
0
0
0
0
0
0
0
0
0
0
1
E
0
0
0
0
1
'4
0
0
0
0
'4
1
2
0
0
0
0
0
1
'4
g
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
g
0
0
1
0
'4
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
~3 ~4
0
0
'4
0
0
0
0
0
0
'4
0
0
0
2
0
'4
1
0
t
42
0
0
0
0
6
0
0
'4
0
~3 ~1 ~2 ~3 ~1
0
0
0
~2
g
0
0
0
~1
0
0
0
t
13
0
0
1
0
6
0
0
1
L\2
0
0
0
0
1
E
1
g
1
'4
1
'4
1
0
0
g
1
g
1
1
0
g
0
0
E
0
0
0
0
'4
g
1
1
0
0
E
0
E
'4
0
0
0
0
'4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
E
1
E
1
1
0
'4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
'4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
'4
0
0
g
0
2
0
0
g
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
\D
0\
·e
97
replacing m with f and f with m in the subscript, and the remaining
are the averages of four measures.
~
These last two, and the averages
they represent are:
~I, mf: (~" mf +~ff', fm +~tm, fm +~f'f, mf)/4,
~I,
nun":
(~I,
nun
+~ff', ff" +0-m1m, m"m +~f'f, f"f)/4.
As in the monoecious case, values for the Pm' P , P
probabilimf
f
ties can be given when the gametic sampling plan is specified.
Special
cases are considered in the next two sections.
8.5 Dioecious Populations: Combined Sampling Scheme
As in the monoecious case, great simplification arises when
.e
combined sampling of gametes from each group is considered.
Since the
gametic set measures are equivalent to the group measures in this
case, the following relations hold.
..
The first term in each of these
nine equations will be the notation used for the remainder of this
section.
t+1
Tt +1
-1
= .'!.11 = ~£mf '
Tt +1
-2
= Tt +1 = (e t +1 + et +1 )/2
-Lmm
-iff
-2i
t+1
)
i
= 1,
2.
2.
t+1
( t+1
= .£11 = Lim,
ff + Llf, nun) 2,
i
= 1,
t+1
t+1
/
t+1
= -r 2t+1
, = (Lim iiiI + Llf
1
,
, Iiii) 2,
i
= 1, 2.
t+1
t+1
= r t +1 = (Lim
,
-3i
i
= 1,
1::1
1::2
t+1
/
..
e
•
1::3
t+l
nun +Llf
,
ff) /2 ,
2, 3.
·e
98
•
~+1
= .6t ;-1 = 5 t +1
i = 1, 2,
3.
.6t +1
= .6t +1 = 5 t +1
-2i
-imf, mf '
i
= 1, 2,
3.
-2
-inun, ff'
-11
+ 5 t +1
~+1 = ~+1 = (5 t +1
- i
-inun, iii! -iff, mr) /2 ,
i =
.6t +1 _ .6t +1
- =4i
-4
i
=
(5 t+l
+ 5 t +1 ff)/2 •
iff,
inun, nun
1, 2, 3.
:;: 1, 2,
3, 4.
By adding together appropriate columns as for the monoecious
case, Table 8.5 will now have only nine columns.
The expansions
represented by that table, together with equations (8.4.1),
. . .,
(8.4.9) lead to nine simultaneous transition equations:
.e
~t+l
where
=
n ~t
~~ = [T 1 , T2 ,
1, 1, 1, 1, 1].
•
'
f
1,
f
2,
f
y .61 ' ~, ~, L\]t and ~ = [1, 1, 1, 1,
The two locus panmictic function is T •
1
this transition matrix is displayed.
First however, some new notatipn
must be introduced to reduce the size of each element.
•
Let:
PI
= (p +P f ) /2,
P2
= (p +P f )/2
,
P
4
= (pm4 +Pm ) /2,
P
6
= (p +P f ) /2
,
P
7
= (p +P f )/2,
Q1 = Pmf
m1
1
4
m7
7
m2
,P
m3
ures, as given in section 7.4.
2
m6
The matrix follows in equation (8.5.1).
been made of the relations between P
•
The form of
f3
, P
6
,
1
In the matrix,
f5
u~e
has
and the other meas.
•
e
.
•
(1+1.)2
(1+A)2
----0-
----0-
2
1+1. P
2
+ (1+1.) (l-P )
~ 1 ----01
~(1-P1)
0
1+1. P
T
1
1+1.
T
P1
~
0
0
1+1.
P1
"4 + T
(1-P )
1
1+A
P2
T
(P -P 2 ) +"4
1
1+A
T
(P -P 2 )
1
~
0
2
(1-P )
1
P -P
1 2
+
Q
1
Q
1
Q -+<1
e
•
e
2
1_1.
-2-
(1_1.)2
(1_1.)2
0
----0-
----0-
2
1 1.
y(1-P )
1
0
2
¥-(1-P )
1
2
¥-(1-P )
1
1+1.
T
(1-P )
1
1-1.
T
--0
.!:? (l-P
¥
P1
1+1.
2 +T
(1-P 1 )
P
~
+.!:? (l-P1)
1+1.
T(1+2P
1)
1+1.
T(1+2P 2 -3P )
1
(P -P ) +
1 2
1+1.
--o(1+2P 2 -3P )
1
n ; I
•
2
1-1.
(1-P )
1
T1-A( 1+2P2 -3 P )
1
¥
(l-P 1 )
1-A( 1+2P -3 P )
T
2
1
0
0
0
0
¥
(1-P )
1
0
¥
(1-P )
1
0
¥(1+2P 2 -3P )
1
0
I
+3P )
1
1-+<1 1 -Q4
Q -+<1
1-+<14
1
0-
ff
1-Q1-+<14
1-Q1-+<1 4
~
~
0
0
2 3
.,-
~
2 3
.,-
Q4
1-Q1 -+<14
Q4
0
0
0
0
0
"4
0
~
0
0
0
P
1
"4
P -P
1 2
~
1-P
1
I;-
0-
P
P
P
7
12 + P2 -P 4
1 +3 P4 - 4P 2
1-4P +3P
2
4
1-4p2 +3P
b7 +"3 + P 2 -P 4
6
8
2
6
12
6
12
P -2P
2
l
P
P
~
6
P
7
12
1
2P +P
6
1-P
7
-~
2+2P -3P
2
1
1
P +P
6
4
7
-~
2P +P
6
7
4
-~
(8.5.1)
1-3P +P
1 2
------g
1-4P +3P
2
4
------g
P +P
6 7
- .,-
~
100
Two checks on this matrix are possible.
When A
=
1, the system
should reduce to a one locus dioecious scheme for the combined sampling
•
plan.
This value substituted into the matrix shows that (T , T ) form
2
1
a complete set, and that the two transition equations are:
so that:
or, using the symbol F for (one locus) panmictic function:
O
(8.5.2)
This is the form of the result given by Wright (1931), where 2/P1 is
•
the "overall (digametic) effective number."
The other check comes from setting N =N =l.
m f
P1=Q1~1
and that the other Pi and Q are zero.
i
This implies that
These values in the
matrix show that the complete set is now (T , T2 , f , f , 6 , ~),
2
1
1
1
and the 6 x 6 matrix for these measures is just that given by Cockerham
and Weir
(1968)
5
for a sib mating pedigree--as it should be.
For an equal chance gametic distribution (~=2), the values of
P,
m1
., P
~
., (7.4.10).
,
•
5
Ibid.
and P
f1
' ••. , P
f7
are given by equations
(7. 4. 4),
From equations (7.4.1), (7.4.2) and (7.4.3) then the
101
values of Pmf ' Pmf ' Pmf can be found. The following quantities,
2
1
3
needed for the matrix, are given values then:
P - 3(P -P )
2 4
6
P
7
= 6p1-18p2+12P
4
Ql
,
l/NmN f '
Q2 = (NCl)/NmN f
,
Q = (Nm-l)/NmNf '
3
t
Q4 = (Nm-l)(Nf-l)/NmN f •
From the value of PI' i t can be seen that the "overall (digametic)
effective number" is 4N N/(N +N ), as given by Wright (1931).
m
m f
Now
equation (8.5.2) shows that the limiting rate of inbreeding for
complete linkage (l-~(l)) is a monotone increasing function of Pl.
Hence
~(l)
will be an increasing function of the overall effective
number N =4N Nf/(N +N ).
e
m
m f
The system was studied numerically and
dominant roots displayed in Table 8.6, where the relation between the
.
root and effective population number is seen to be the same for all
linkage values.
The values of F
t
ll
for four populations with equal
numbers, N/2, of males and females are displayed in Figure 8.6 •
•
.
'
e
.
•
•
•
e
e
1.0
0.8
~1
06
0.4
02
00'
o
-
nI
I
~
I
~
I
~
I
~
Generation
Figure
8.6 F11 for dioecious populations with equal chance gamete formation
b
\>l
·e
.
104
Table
8.6 shows two cases where different numbers of males and
females give the same effective population number.
= 30
10.909.
both giving N
e
These combinations
For complete linkage the
dominant roots must be equal for each pair because of this equality of
effective number.
The roots also stay close for other linkage values
but the relation is not a simple one, and must depen.d on the three and
four gamete effective numbers also.
These latter numbers are dif-
ferent for the different pairs of N , N •
m f
With unequal numbers in each sex, N < N say, it is not possible
m
f
for every parent to contribute the same number of gametes--ma1es contribute an average of (Nm+Nf)/N and females an average of (Nm+Nf)/N f •
m
.e
For integral values of (N +Nf)/N however, the moments of the gametic
m
m
distribution for males may be made zero.
...,
N
f
= ':":N-r::(N:--+~Nf--~l~)
,
m m
2
N (N +N -l)(N +N -2)
m m f
m f
,
N3 (N +N -l) (N +N -2) (N +N - 3)
m m f
m f
m f
•
From equations (7.4.11),
105
,
•
P
~
One distribution of female gametes in this case is for N of the
m
females to contribute two gametes each, and the remaining (Nf-N ) to
m
contribute just one each.
The probabilities P
f
' P
f
' P
f
and P
234
f
are
5
all zero, and the remaining probabilities found using the moments:
.e
..
It is then possible to calculate the Pi and Q needed for the transii
tion matrix.
Matters are simplified for an equal number, N, of males and
females in each generation.
Each parent may be supposed to contribute
exactly two gametes, so that:
P. =
1.
•
and the common value comes from setting N
•
= NB
in equations (7.4.11),
w6
, (7.4.17).
From equations (7.4.1), (7.4.2), (7.4.3) then:
•
Q2
=
Q
3
=
(2N-2)/(2N-1)
2
,
so that
2
Q4 = (2N-2) /(2N-1)
2
•
This case of equal numbers of males and females was studied
numerically, and the dominant root of the transition matrix shown in
Table 8.7.
In that table, the population size is the
11
of males
Figure 8.7 shows values of the two locus
plus the number of females.
inbreeding coefficient F
n~ber
for N/2 males and females.
8.6 Dioecious Populations: Independent Sampling Scheme
For equal chance gamete
formatio~,
both the combined and the in-
dependent sampling schemes lead to the same equations.
This case was
discussed in the last section and the results displayed in equation
(8.5.1), Table 8.6 and Figure 8.6.
For a system with as little variation as possible in the number
of gametes per parent the algebra for independent sampling is very
complex.
The complexity was suggested for combined sampling, even
when it was possible to work with 9 group measures instead of 23
gametic set measures.
To reduce the complexity the simplifying assumption of equal
numbers, N/2, of each sex will be made.
Each male and each female is
supposed to contribute one male and one female gamete.
•
The
107
•
Table 8.7
Dominant root of transition matrix for dioecious populations
with combined sampling and two gametes per parent
Population Size, N
2
4
8
16
32
0.0
.65451
.85299
.93231
.96749
.98407
0.1
.65484
.85302
.93231
.96749
.98407
0.2
.65600
.85315
.93232
.96749
.98407
0.3
.65832
.85343
.93236
. 967'JJ
.98407
0.4
.66231
.85395
.93242
.96751
.98407
0.5
.66879
.85491
.93255
.96752
.98407
0.6
.67906
.85668
.93280
.96756
.98407
0.7
.69524
.86011
.93335
.96764
.98408
0.8
.72031
.86734
.93475
.96786
.984;1.2
0.9
.75758
.88445
.93961
.96889
.98432
1.0
.80902
.92356
.96556
.98361
.99200
Linkage Parameter, A
~e
..
•
>
,
,.
e
.
e
1.0
•
•
e
N=2
0.8
~1
0.6
N=16
0.4
0.2
0.0
0
10
20
30
40
50
Generation
Figure
8.7
F
11
for dioecious populations with combined sampling and two gametes per parent
b
(X)
-e
prQbabilities needed in equations (8.4.1), • • . , (8.4.9) may be
determined from equation (7.4.11), . • • , (7.4.18).
•
As in the
monoecious case, distinction must be made, on the basis of whether
the gametes in question are in the same or different output sets,
between using a mean number of one ((7.4.18)) or two ((7.4.11) . . •
(7.4.17)) gametes per parent.
By way of illustration consider the two cases of equation
(8.4.3):
2
N-2
ff + ~
ill
= N 1m,
r
-12
= ~,ff"
y
1m,
ff' ,
Substitution of the values of these probabilities, and the
combining of the equations with the expansions of Table 8.5 leads to
23 simultaneous transition equations.
The corresponding transition
matrix, 0, will be displayed, but because of its size will be subdivided:
o
23 x 23
=
-----------~----------
Each of the four sub-matrices is now
disp~ayed
measures listed along the edges of each •
•
with the appropriate
.
e
•
t
t
Tn
T t +1
n
Tt +1
21
Tt +1
22
f
f
"1
f
f
f
f
f
T
21
(H),l
(1;1.)2
---cr2
1+A
(1+1.)2 N-2
2N +-O--N
o
(It)2
o
t
e
)
t
•
t
12
t
T22
f
o
o
o
o
o
o
----c:r-
o
o
o
(1+A)2 N_2
-O--N
(1+A)
2
f
n
f
f
2l
1_1.2
t
f
22
o
~
t
o
o
o
o
o
o
o
o
o
o
o
o
1-~
N-2
-2-N
1_1.2
~2~
1+1.
1+1.
4N
o
o
o
o
o
o
t+1
21
o
o
o
t+1
o
o
o
o
o
o
o
o
o
4
t+1
1+A
4N
o
1+1.
32
4N
o
1
1+A N-4
2N +0N'"
o
3
1+1. N-4
~+4N'"
oN
o
o
o
o
N+oN
1
o
1 + 1+1. N-4
1+1. N-4
N
oN
t+1
12
22
t+1
31
t+1
33
1
1+1.
1+1. N-2
4
1 +1+A N-2
o
1+1.
NoN
1+1.
o
o
4N
1+1.
o
N+ON
1 + 1+1. N-2
o
1+A
1+A N-4
t
33
o
4N
N
f
o
t+1
1+A N-2
t
32
o
n
oN
f
31
e
•
1+A N-2
oN
1+1.
4N
oN
o
o
1+A
o
o
1+1.
o
o
1+1.
o
o
1+A N-4
o
o
o
o
o
1+A
1+A N-2
o
1+A N_2
oN
I-'
b
•
•
e
L\1
Sl
(l A
r t +1
11
r t +1
0
21
t 1
r 22
+
r t +1
11
I
I
t+l
"2
6
0
0
(1_}..)2 N-2
"--N
t
13
~1
(l- Al
.,--
1-)..
0
0
(1-}..)
.,--
e
~3
~1
~2
~3
0
0
0
0
0
0
0
0
0
0
0
TN
0
0
0
T
0
1-}.. N-2
1-}.. N-2
oN
oN
0
1-}..
1-}..
0
0
(1_}..)2 N-2
0
"--N
0
0
---g--
1-}..
0
0
0
0
0
0
1-}..
e
~2
0
2
0
1-}..
r 12
L\2
•
)
(1-}..) 2
6
t
44
t\1
t\2
t\3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1-}.. N-2
1-}..
=
t+1
r 21
r t +1
22
I
0
oN
0
0
0
r t +1
0
r t +1
0
oN
r t +1
0
oN
31
32
33
1-}.. N-2
0
1-}..
0
1-}.. N-2
1-}.. N-4
1-}..
1-}.. N-2
0
0
oN
0
0
0
1-}..
0
0
0
0
oN
0
0
oN
0
1-}.. N-2
1-}.. N-4
1-}..
1-}..
0
T
0
TN
0
TN
1-}.. N-4
1-}.. N-4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
~
I-'
.
e
t
T
ll
°3
=
.
,.
•
t
T
21
b,t+1
1
1
11
2;
2;
'
t
T
22
f
0
0
ft
l2
N-2
2
2N
0
0
0
0
b,t+1
13
0
0
0
0
e
e
t
U
t\;1
.
0
t
f 21
J.
t
f 22
f
t
31
f
0
0
t
32
f
-
N-2
N
0
0
0
0
0
0
0
0
0
4N
1
1
4N
2N
2;
1
~~1
I
0
0
0
0
0
0
0
0
0
0
~;1
I
0
0
0
0
0
0
0
0
0
0
~?
I
0
0
0
0
0
0
0
0
0
0
~~1
0
0
0
0
0
0
0
0
0
0
~;1
0
0
0
4N
0
4N
4N
0
4N
~?
0
0
0
0
1
2N
2N
1
2N
0
2N
0
~~1
0
0
0
0
0
0
0
0
0
0
~;1
0
0
0
0
N
0
if
0
0
0
N-2
2N(N-2)
0
1
1
N
N(N-2)
0
0
0
N-4
N(N-2)
0
N
2
1
N(N:2T
0
0
1
2N(N-2)
b,t+1
43
b,t+l
44
I
1
NtN=2Y
0
0
1
1
1
2N(N-2)
1
N{N:2)
1
t
33
1
1
1
1
~
1
1
0
I
I-'
~
.,
e
•
4
t\~l
I E1
e
~l
~2
~3
~l
~2
~3
~l
'\2
0
E
1
0
0
0
1
N
0
0
0
87"
L\?
E
L\?
E
1
0
E
0
0
0
0
0
0
E
1
~
N-2
0
E
0
0
0
N-l
2N
0
0
0
EN
0
0
0
(N_2)2
2
8N
0
0
0
N-2
2N
0
0
0
--mr-
0
0
0
E
0
0
0
2
0
0
0
E
0
~
0
0
0
N-l
2N
0
0
0
~
0
0
E
0
E
0
0
0
0
EN
N-2
0
EN
N-4
0
0
0
E
1
0
N-4
~
~
0
0
1
0
2
0
0
0
N-2
0
0
0
0
(N_2)2
87"
1
E
1
1
1
~~l
1+
~;l
1+
~?
1+
0
0
0
0
1+
~;l
0
0
TiN'"""
~;l
0
0
TiN'"""
0
i\~l
0
1+
0
0
E
i\;l
0
W
0
0
W
~;l
0
if-4N-tO
4N(N-2)
0
0
0
2
N -6N+12
4N(N-2)
0
0
= ~~l
~:l
1
1
•
I
1
N-2
1
N-l
N-2
1
N-2
N2 _6N+12
8N(N-2)
1
E
1
N-2
1
.
•
~3
~2
~l
.11
•
~
1
E
N-4
EN
N-4
1
2
1
1+
N-l
TiN'"""
N-2
1
1
E
EN
1
'\3
(N_2)2
1
N-2
ti4
0
0
0
(N_2)2
1
N-2
W
E
0
0
E
N-3
2N
0
0
EN
0
N-4
2N
0
0
(N-4) (N-6)
8N(N-2)
0
0
0
0
N-4
2N
0
0
(N-4) (N-6)
8N(N-2)
0
0
0
1
1
N-6
e
I
f-'
I-'
VI
114
Here again, two checks are possible for these equations.
complete linkage (A~l), the complete set is (T
~orresponding
1
1
2
1
o
1
o
2
, T~l' T ) and the
22
transition matrix is:
2
2
11
For
o
(N-2)
2N
1
2
This leads to the following recurrence formula for the (one locus)
panmictic coefficient:
.e
F t+l ~ F t _ F t-2/ 4N
000
'
which is just that given by Cockerham
By
•
aqd the
and Weir
(1967).
setting N=2, the complete set is (T
11, T21 ,
f
11,
f
21, 6 11,
6 x 6 transition matrix is the same as that given by
~1)
Co~kerham
(1968)6 for a sib mating pedigree.
Table
8.8 shows the dominant root of the 23 x 23 transition matrix
while values of F
are graphed in Figure 8.8.
11
The three tables of dominant r90ts (Tables
8.6, 8.7 and 8.8) and
11 (Figures 8.6, 8.7 and 8.8) may now be employed
the three graphs of F
to evaluate the three dioecious systems here.
ponclusions apply as were made in section
8.3
Substantially the same
for monoecious mating
systems, although inbreeding now starts in the second generation, and
•
115
•
Table
8.8 Dominant root of transition
matri~ for dioecious populat~ons
with independent sampltng and two gametes per parent
Population
~inkage
.e
•
•
•
Si~e,
N
2
4
8
16
32
0.0
.65451
.85617
.93324
.96773
.98413
0.1
.65484
.85641
.93327
.96774
.98413
0.2
.65600
.85676
.93332
.96774
.98413
0.3
.65832
.85729
.93338
.96775
.98413
0.4
.66231
.85810
.93349
.96776
.98413
0.5
.66879
.85938
.93367
.96779
.98413
0.6
.67906
.86152
.93398
.96783
.98414
0·7
.69524
.86537
.93462
.96793
.98415
0.8
.72031
.87304
.93612
.96818
.98419
0.9
.75758
.89005
.94111
.96925
.98441
1.0
.80902
·92732
.96655
.98386
.99206
Parameter, A
•
e
•
•
•
e
1.0
•
e
"
N=2
08
~1
0.6
N=16
04
02
0.00
10
2J
3)
40
50
Generation
Figure
8.8 F11 for dioecious populations with independent sampling and two gametes per parent
I-'
I-'
0"\
In
all systems are identical for N
=2
(just one male and one female ea~h
generation).
As far as first order effects are concerned, once again the
equal chance gamete formation scheme has smaller dominant roots and
greater F
Il
values than do the controlled mating schemes.
Althougq the
combined sampling plan with two gametes per parent has greater late
v~lues
(and hence smaller dominant roots) than does the corre~
l1
sponding independent sampling plan, initially it does not. To a
of F
greater extent than even in the monoecious systems though these two
plans are very similar.
Linkage effects also present a similar story.
The logistic.
type curve for FII(O) provides an early maximum for Fl1(1) ~ FIl(O)
.e
and enables populations of unequal sizes and unequal linkage values to
be equally inbred in early generations.
Finally, comparisons can be made between monoecious and
•
dioecious populations of the same total size.
For equal chance gamete formation schemes, the smaller dominant
roots for the monoecious populations indicate larger values of F
for dioecious populations.
The graphs of F
II
ll
than
values confirm this, and
show that the relation holds for all generations.
This relation is ip
no way affected by linkage, and holds equally well for F ,
l
When each parent is constrained to produce exactly two gametes
(with either sampling plan) the dioecious populations have smaller
dominant roots and greater final rates of inbreeding.
•
not sufficiently large to illustrate this relation, for in the
populations the roots are very close in value.
•
The graphs are
1ar~er
However the graphs do
~.
lill
illustrate the higher initial rate of inbreeding for monoecious popula.
tions ••which is largely a result of inbreeding starting a generation
•
earlier.
These observations are also independent of linkage.
The
generation in which a dioecious population becomes as inbred as a
corresponding monoecious one is a function of the linkage parameter
though.
.e
•
As linkage decreases, this generation becomes later •
9.
DISCUSSION
The machinery to allow the analysis of any inbreeding system
wh~n
two linked autosomal loci are considered has now been developed and
demonstrated.
In contrast to the one locus case where only two gametes
need be taken account of, genes for two loci are necessarily carried
on two, three, or four gametes between generations.
The analyses thus
make use of digametic, trigametic or quadrigametic measures--each being
a probability statement concerning the identity of descent of genes at
the two loci on the appropriate gametes.
For pedigrees of individuals, just these three types of individual
measures are needed.
When pedigrees of groups are involved though,
additional analogous quantities--group measures, and gametic set
measures--must be used and related to the individual measures.
The methods of determining one of the digametic measures, the iq•
breeding function, in given situations were specified in Chapter
5.
At
any stage, it is the inbreeding function for one individual that is to
be determined, and the analysis of all systems depends upon the expansion of this measure into a combination of measures involving individuals of the previous generations.
In specific pedigrees the
expansions are taken back to the initial
populati~n
while for pedigree
systems of mating a set of simultaneous transition equations connecting
values in successive generations of a minimal, complete, set of measures is derived.
Detailed instructions for expanding the measures were given in
•
Chapter
•
4.
Each of the pairs of genes ab, alb' in the argument of
120
!(ab, alb') may be replaced by the gametic array which identifies the
in4ividuals from which the pairs came.
These arrays are used as
~~ad-
ings in a two-way table, the cells of which are filled with the measures appropriate to the marginal genes for that cell.
This method is
formally correct, and is the basis for all of the work of Chapter
4.
In particular cases, the results given there may be consulted and
applied.
Measures which do not involve repeated subscripts (or one
individ.ual supplying more than one gamete) may be expanded directly
however.
Such direct expansions are performed for one gamete at a time.
The individual, B say, from which it came is replaced by its two
parents, (DE) in parentheses in the subscript.
becomes ~(DE)C and
IG,
HB becomes
IG,
H(DE)·
For example, ~BC
If both genes on the
gamete are being considered, as in ~BC' the bracketed pair (DE) is replaced by D, E, DE or ED with probabilities (1+A)/4, (1+A)/4, (1-A)/4,
•
or (1-A)/4,
.!.. ~. ,
~(DE)C
=
1+A(
)
I-A (
4"" ~DC + ~EC + 4"" Ie, DE + Ie, ED
)
.
This is merely a statement that a gene from B has two ways of being
parental and two of being recombinant.
If only one gene on the gamete
is being considered, the pair (DE) is replaced by D or E with probability 1/2 each,
IG,
•
1
H(DE) ="2
(IG,
HD +
IG,
HE) •
Each of the subscripts is treated in the same way.
cannot be given though when a subscript is repeated.
•
Such general rules
121
For the specific pedigrees, expansion through all generations back
to the original population completes the problem--the required inbreed-
•
ing function has been found.
For the pedigree systems of mating, the
transition matrix for the fourth components of the complete set of
measures also essentially completes the problem.
a vector whose first component is
are assumed known.
Faa'
The matrix acts upon
and the one locus coefficients
Hence the matrix leads to the whole function
!,
but the particular way in which it is used can vary.
If only the first few generations are of interest, the first few
powers of the matrix can be calculated to provide algebraic expressions
oa '
for F
and hence F.
This is only practicable for small matrices.
When the whole course of the inbreeding process is to be studied the
characteristic equation of the matrix provides a recurrence formula for
Faa
and thus all values of
Faa
can be found.
The number of generations
this formula goes back is given immediately by the number of measures
in the complete set, or the size of the transition matrix.
There is
some question as to the purpose to be served by displaying the recurrence formula, especially when the order of the equation is large.
For example, calculating all the coefficients in the equation of degree
23 for the dioecious system of section
8.6, would be of little use.
On
the other hand an adequate picture of the system can be drawn by calculating
Faa
numerically for specific linkage values by taking succes_
sive powers of the matrix with a computer.
A final use for the matrix
is in using it to characterize the limiting behavior of the system.
•
Any inbreeding system has its limiting behavior completely specified
by ~(A) and ~(1) when A f 1j no two systems will have the same values
•
122
for both of these quantities.
The stage at which equation
becomes appropriate varies with the linkage parameter.
•
(5.,.8)
In the
monoecious population with equal chance gamete formation for example,
the ratio between successive values of F
is constant to 5 decimal
OO
places (and hence equal to ~(l)) after no more than 10 generations for
l
= O.
For 0 < l < 0.5, the number of generations is not greater than
20, but for higher l the number rapidly increases.
generations to reach the steady decrease when l
It can take 60
= 0.9.
It has been shown that the complexity of the analysis, as indicated by the size of the transition matrix, depends on the number of
measures needed to form a complete set.
There seems to be no way in
which this total number of measures can be written down for any mating
system, although the maximum number needed depends only on the size of
each generation.
These maxima are shown in Table 9.1.
The table shows, for example, that if each generation has only two
•
members, no mating system can lead to a matrix greater than
9 x 9.
The
number of measures for a dioecious system is shown in Table 9.2.
The results of applying these techniques and determining the
effects of linkage on inbreeding can now be discussed, although some
comments have already been made earlier.
The obvious quantity to characterize the effects of linkage on the
identity of two pairs of linked genes in an individual of genotype
ab/a'b' is:
T"J ij (l)
F .. (l) - F.F. ,
1J
1 J
•
i.e. T"J11(l)
•
= Prob(a=a',b=b')
- Prob(a=a')Prob(b=b') •
123
Table 9.1
Maximum number of measures needed for regular pedigrees and
monoecious populations
•
Population Size, N
Table
1
2
3
4+
Number of Digametic Measures
1
2
2
2
Number of Trigametic Measures
0
3
4
4
Number of Quadrigametic Measures
0
4
6
7
Total Number of Measures
1
9
12
13
9.2 Maximum number of measures needed for dioecious populations
Number in Each Sex, NL2
1
2
3
4+
Number of Digametic Measures
2
3
3
3
Number of Trigametic Measures
3
7
8
8
Number of Quadrigametic Measures
4
13
16
17
Total Number of Measures
9
23
27
28
•
•
•
124
This quantity, the "identity disequilibrium function" was used in
discussing sib mating by Cockerham and Weir (1968)7.
The four COmpO-
nents of the function have the same numerical value:
For pedigree systems of mating, as discussed in section 5.3,
Fij(O) has the same value as FiF , so that the identity disequilibrium
j
function may also be written as F .. (A)-F .. (0).
~J
finally this quantity is zero.
~J
Both initially and
It reaches a maximum quickly and then
decreases, but not at a constant rate.
The maximum value and genera-
tion of its attainment for successive parent-offspring mating is shown
in Table 9.3.
For complete linkage, ~11(1)
= F1 (1-F 1 )
which has a
maximum of 0.25.
For group mating systems it was shown in section 8.2 that Fij(O)
is not equal to FiF •
j
..
It becomes convenient then to divide the iden-
tity disequilibrium function into two components:
The first component gives the disequilibrium in the absence of linkage
and will be designated
N.
~N'"
~J
It is a function of the population size
The second term, written as ~Lij(A), shows the effects of linkage
on the identity disequilibrium.
Hence:
or:
•
F
11
2
(A) = F + TI"NIl + TI'ILll (A) .
1
7 Ibid.
125
•
Table 9.3
Maximum value of identity disequilibrium and generation of
attainment for successive parent-offspring mating
Maximum Value
Generation
Linkage Parameter, A
.e
0.0
0
0
0.1
.0069
2
0.2
.0155
2
0.3
.0261
2
0.4
.0390
2
0·5
.0550
2
0.6
.0748
3
0.7
.1037
3
0.8
.1394
3
0.9
.1883
4
1.0
.2500
4
126
For any group mating system the family of F
11
(A) curves for a
particular population size has the same characteristics.
is bounded above by the curve for FII(I)
= FI ,
The family
and bounded below by
FII(O).
Now FII(O) is at a distance DNII above Fi, and this difference
2
has an early maximum after which FII(O) converges to Fl' The maximum
value of DLII(I) (the difference between F
but at this stage D
is small.
NII
I
and Fi) occurs when F
I
=O.5,
This accounts for the divergence of
F and FII(O) in the graphs of Chapter 8 until about the time when
1
F
I
= 0.5, whereupon they converge.
As linkage increases, DLII(l)
becomes increasingly greater than D
and the generation of its maxiNII
mum value moves later in time towards that of DLII(I).
Finally it
should be noted that the overall identity disequilibrium DII(l) has a
.e
maximum value greater than either of those of its two components and
at a time between those of the components.
trated in Table
•
These concepts are illus-
9.4 where DII(l), DNII and DLII(l) are shown for a
monoecious population of size eight with equal chance gamete formation.
In that table asterisks denote maximum values.
Table
9.4, the graphs, and tables of dominant roots in Chapter 8
all show that the effects of linkage do not increase linearly with the
linkage parameter l--which was not to be expected since both !(l) and
~(l)
involve polynomials in l of degree greater than one.
It is only
for high values of l (say 0.9) that linkage produces significant
effects, and even then the effects are not as great as those of population size.
•
At best then, linkage plays a transient role, and is most
important at high values, small populations, and early generations •
-_
127
Table 9.4
Identity disequilibrium functions for a monoecious population of size eight with equal chance gamete formation
A = 0.5
.e
•
Generation
Tl N11
1
.0274
2
,.
e
a
a
TIll
Tl L11
.0078
.0352
.0308*
.0187
3
.0292
4
= 0.9
a
A
a
= 1.0
a
a
TIll
Tl L11
.0253
.052 7
.0312
.0586
.0495
.0600
.0908
.0756
.1064
.0248
.0540*
.0886
.1178
.1158
.1450
.0266
.0 2 71
.0537
.1094
.1360
.1491
•L757
5
.0240
.0272*
.0512
.1237
.1477
.1758
.1998
6
.0215
.0261
.0476
.1335
.1550
.1965
.2180
7
.1092
.1247
.0439
.1373
.1565*
.2122
.2314
8
.0171
.0229
.0400
.1388*
.1559
.2235
.2406
9
.0154
.0210
.0364
.1379
.1533
.2311
.2465
10
.0138
.0292
.0330
.1352
.1490
.2356
.2494
11
.0122
.0176
.0298
.1313
.1435
.2377*
.2499*
12
.0110
.0160
.0 2 70
.1263
.1373
.2375
.2485
13
.0097
.0146
.0243
.1207
.1304
.2357
.2454
14
.0087
.0132
.0219
.1147
.1234
.2323
.2410
15
.0078
.0120
.1098
.1086
.1164
.2278
.2356
20
.0044
.0073
.0117
.0783
.0827
.1950
.1994
30
.0015
.0025
.0040
.0354
.0369
.1220
.1235
40
.0005
.0009
.0014
.0148
.0153
.0695
.0700
50
.0001
.0003
.0004
.0059
.0060
.0380
.0381
a
III
a
A
Tl L11
* - Maximum values.
TIll
~8
The discussion of genotype frequencies (or at least the frequencies of double identity and double non-identity) in the presense of
linkage is one use for the inbreeding function found in this work,
Another was discussed by Schnell
(1961),
For a random member of a group of individuals with the same history
of inbreeding, each with n loci under consideration, the number, m, of
loci carrying genes identical by descent has expected value Em
= nFl'
If each pair of loci, p and q, has linkage parameter A , the covaripq
ance between the numbers of the two loci identical by descent among
individuals is:
~
pq
=
so that, as given by Schnell, the variance of m among individuals is:
2
m
~
=
~
pq
The summation is over all distinct pairs p, q and thus has n(n-l)
terms.
As Schnell says, this variance is a desirable supplement to F
l
in describing the effect of a mating scheme on the homozygosity by
descent in resulting individuals.
The variance can be seen to increase
with the amount of linkage.
Complete descriptions though can only be given by an n-locus inbreeding function.
Certainly a vector valued measure
nents could be defined.
~
with 2n compo-
As discussed in Cockerham and Weir
(1968)8,
such measures must now be x-gametic, where x may range from 2 to n
subject to x < 2N for a population of size N.
The preliminary algebra
would follow in the same manner as in this work but would be very extensive, and a computer could be employed to solve the resulting
transition equations numerically.
Haldane and Waddington (1931) do
show how some three loci results can follow from these for two loci •
•
130
10.
..
SUMMARY
Various systems of inbreeding, including specific pedigrees,
pedigree mating systems and group mating systems are studied when two
linked autosomal loci of a diploid organism are considered.
A general function with four components, each of which is a probability statement concerning the identity by descent of any two pairs
of genes is defined.
Three broad classes -' digametic., trigametic and
quadrigametic, of the functioI!.
;;'C'.'
(::Lscu.:"sed.
W"ht.:-n the two pairs of
genes are on uniting gametes, the digametic function is called the two
locus inbreeding function, and this is the quantity to be determined
for the inbreeding systems.
For pedigree and group mating systems transition equations connecting values in successive generations for one component of each member
of a complete set of functions are derived.
the inbreeding function.
The complete set
The equations are displayed in
for each system of inbreeding.
include~
matri~
form
The characteristic equation of the
transition matrix furnishes a recurrence formula for the two locus
panmictic function, and the largest eigenvalue, which is tabulated,
affords a characterization of the limiting values of the inbreeding
function.
Values of the two locus inbreeding coefficient for the
systems studied are presented in either tables or graphs.
Some general conclusions regarding the effects of linkage on inbreeding are drawn and some applications of the inbreeding function
are discussed •
•
131
11.
I'
Bennett, J. H.
breeding.
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Group inbreeding and coancestry.
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.e
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,.
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