Ann. Hum. Genet., Lond.(1976),39, 89
89
Printed in Qreat Britain
Probability and genetic relationship :two loci*?$ BY CARTER DENNISTON Laboratory qf Genetics, University of Wisconsin, Madison, U.S.A.
It is the purpose of this paper (1) to define a set of probabilities (k-coefficients)which specify in detail the genetic relationship between two individuals with respect to two loci, (2) to outline a procedure for calculating these probabilities for any pedigree to which they apply, and (3) to present some examples of how these probabilities may be used. A comparison to the work of Campbell & Elston (1971) is included. See also Gallais, 1974. IDENTITY BY DESCENT
Consider two diploid individuals, X and Y , and two autosomal loci, A and B, with recombination fraction, c. We label the two homologous chromosomes of X, 1 and 2, and those of Y , 3 and 4 , s o that the individual genes of X and Y are (a,b,/a,b,) and (a,b,/u,b,), respectively. To avoid circumlocution later on, we call such designations zygotypes; that is, the zygotype of X is (a,b,/ a,b,) and this is simply a labelling of the genes and chromosomesof X and has no other significance. The total number of identity (bydescent)relations (Cotterman, 1940; Malecot, 1948) on the eight genes in two individuals is 1Ei2 = 225. In this paper we will restrict our attention to the 7, = 49 relations applicable to two individuals neither of whom is inbred. (It will be convenient to introduce some terminology at this point. According to Cotterman, 1960, two non-inbred relatives are called regular relatives; if a pair of homologous genes in an individual are identical by descent, that individual is said to be autozygous at that locus; otherwise, the individual is allozygous. Regular relatives are thus allozygous.) The restriction of this paper to regular relatives, then eliminates all relationships for which a, = a, or b, = b, or a, E a, or b, = b,, where ' = ' denotes identity by descent. The 49 identity relations for regular relatives are displayed in Fig. 1. A solid line in Fig. 1 denotes identity by descent at the A locus, and a dotted line denotes identity by descent at the B-locus. Another useful way of depicting these 49 events is the phylotype notation of Cotterman (1960). Genes which are identical by descent are assigned identical subscripts. For example, the event in the fifth row, second column of Fig. 1 is written as (a,pl/a2f3,,a11,/a4,8,).Greek base letters are used to distinguish the phylotype notation from the zygotype notation, above. In array ( l ) ,the 49 identity relations are represented in a modified phylotype notation in which the expression (, ij/kl) is shorthand for (a,/3,/~,/3~, a&/a,,/3,); the arrangement of events is the same as in Fig. 1. (, 33/44) (, 13/44) (, 33/14) (, 33/24) (, 23/44) (, 13/24) (, 23/14) (, 31/44) (, 11/44) (, 31/14) (, 31/24) (, 21/44) (, 11/24) (, 21/14)
(, 33/41) (, 13/41) (, 33/11) (, 33/21) (, 23/41) (, 13/21) (, 23/11)
* t
Paper No. 1593 of the Laboratory of Genetics, University of Wisconsin, Madison, Wisconsin 53706. Taken from a thesis submitted in partial fulfillment of the requirements for the Ph.D. degree at. the University of Wisconsin, 1967. $ This work was supported in part by NIH Grant GM15422.
90
C. DENNISTON (3
33/42) (, 13/42) (, 33/12) (, 33/22) (, 23/42) (, 13/22) (, 23/12)
(3
32/44) (, 12/44) (, 32/14) (, 32/24) (, 22/44) (, 12/24) (, 22/14)
(, 31/42)
(3
(, 32/41)
(3
11/42) 12/41)
( 9
(9
31/12) ( 9 31/22) (, 21/42) (, 11/22) (, 21/22) 32/11) (, 32/21) (, 22/41) (, 12/21) (, 22/11).
Fig. 1. The 49 gene identity relations for regular relatives: two loci.
now define the 15 k-coeficients as follows :
k,,
= Pr (, 33/44),
kio = Pr (, 13/44)+Pr (, 33/14) = Pr (, 33/24) +Pr (, 23/44), k20 = Pr (, 13/24) +Pr (, 23/14), k o ~= Pr 31/44) +Pr (, 33/41) = Pr (, 33/42) +Pr (, 32/44), LO, = pr (, 31/42)+Pr (, 32/41), kCC 11 (, 11/44)+- (, 33/11) = Pr (, 33/22)+Pr (, 22/44), kyl = pr 31/14) +PI' (, 13/41) = Pr (, 23/42)+Pr (, 32/24), k?C 1 1 - Pr 21/44)+Pr (, 33/21) = Pr (, 33/12)+Pr (, 12/44), kz = Pr (, 31/24) +Pr (, 23/41) = Pr (, 13/42) +Pr (, 32/14), 4 1 = Pr 11/24) +fi (, 23/11) = Pr (, 13/22) +Pr (, 22/14), (3
(3
(5
(9
(1)
Probability and genetic relationship
91
kzl = Pr (, 21/14)+Pr (, 13/21) = Pr (, 23/12)+Pr (, 12/24), ki2 = Pr (, 11/42)+Pr (, 32/11) = P r ( , 31/22)+Pr (, 22/41), k f 2 = Pr (, 31/12)+Pr (, 12/41) = Pr (, 21/42)+Pr (, 32/21), ki2 = Pr (, 11/22)+Pr (, 22/11), kf2 = Pr (, 21/12) + P r (, 12/21). (2) The form of these 15 k-coefficients has mnemonic value. For example, kyl is the probability that X and Y (both allozygous a t both loci) share exactly one gene in common at each locus and that the shared genes are in coupling in X and in repulsion in Y .kf2 is the probability that X and Y share exactly one gene in common a t the A-locus, exactly two genes in common at the B-locus, and that the A-locus gene shared by X and Y is linked t o non-identical (different) B-locus genes in the t'wo individuals. The following equat,ions should be noted:
+ +
+
koo + 2k1o k2o 2ko1+ ko2 2kE",+ 2ky1+ 2k:
+ 2k; + 2kl1+ 2kf1+ 2kS,2+ 2kf2+ k,8, + kf2 = 1,
(3)
koo + 2klO + k20 = ooko,
(4)
koo + 2kOl+ k02 = ooko,
(5)
+
+ +
kol k s k:C, kl;, + k",, + kt1 = OOkl, kl0 + ky1+ @I+ k;", + k;72 + + kf2 = OOkl,
e2
+
+ +
(6)
(7)
koz 2kf2 + k8,, + kf2 = 00k2, (8) k,o 2k8,1+ 2k;1+ k8,2 + k f p = 00k2, (9) (k8,2+ k8,1+ kS,2) (c2+ d 2 )+ (&I) d2+ ( k c ) c2+ (2k& + 2kt1+ 2kf2 + kC:, + k);e) cd = 2F,,, (10) where ooko,oak, and ookz are the one-locus k-coefficients for regular relatives (Cotterman, 1940; Denniston, 1967, 1974; Crow & Kimura, 1970), and F y y is the coefficientof consanguinity of X and Y defined for two loci, i.e. the probability of a n (hypothetical) offspring of X and Y being autozygous a t both the A-locus and the B-locus (Haldane, 1949). It should be noted that the Haldane article just cited adumbrates both MalBcot's concept of identity by descent and the twoloci extension of the inbreeding coefficient. A method for calc,ulating the k-coeficients We shall denote the probabilities of the 49 identity relations of Fig. 1 as
p* =
A* A* A* A* B*
C* G* BT H* B*
G* C* H* BT I*
BT H* C* G*
H* Bt G* C* D* I *
D* I*
I*
D* D* I* I* D* E*
I
J*
where A*
pr (, 13/44) = Pr (, 33/14) = Pr (, 33/24) = Pr (, 23/44) = Pr (, 31/44) = Pr (, 33/41) = Pr (, 33/42) = Pr (, 32/44), =
B* = Pr (, 13/24) = Pr (, 23/14)
BT and so on.
=
Pr (, 31/42) = Pr (, 32/41),
= Pr (, 31/24) = Pr (, 23/41) = Pr (, 13/42) = Pr (, 32/14),
C. DENNISTON
92
In addition, we define a second array
1 A
A C
B I
A A A B B G B H D I
D I
D
J
E
where
A = Pr (a, = a,) = Pr (a, = a,) = Pr (aoE a,) = Pr (a, = a,) = Pr (b, = b,) = Pr (b, =- b,) = Pr (b, = b4) = Pr (b, = b3), B = Pr (al = a3,a23 a,) = Pr (a, = a,, a, 3 a,) = Pr (b, = b,, b, = b,) = Pr (b, = b,, b, = b,) = Pr (b, = b,, a, = a,) = Pr (b, = b,, a, = a,) = Pr (al = a,, b, = b,) = Pr (a, = a,, b, z b,), and so on. Note that although the arrangements of arrays (l), (1 1)and (12) all correspond to that of Fig. 1, the lack of a line in Fig. 1 is interpreted as nonidentity in arrays (1) and ( 1 1) but as noncommital in array (12). The relations connecting arrays (11) and (12) are
J* = J E* = E I* = I - J* D* = D-E* B: = B-2D*-E* H* = H-21*- J* C* = C-2D*-E* Q* = G - 2 1 * -
J*
B* = B - 2 0 * - 2 J * - E * - J* A* = A - B * - B : - C * - C * - H * - 3 0 * - 3 I * - E * -
K*
= 1 - 8A* - 4B* - 4B: -4C*
J*
- 4G* -4H* - 8D* - 81* - 2E* - 2J*,
and the k-coefficients are given by
Loo = K*, klo = 2A*, k20= 2B*, kol = 2A*, kO2= 2B*,
ky, = 2c*,
el= 2G*, ky,
= 2H*,
k'&
= 2BT,
6, = 2D*, k& = 21*, k:, = 2D*, kf2 = 21*, k;, = 2E*, ki2 = 2J*.
(13)
Probability and genetic relationship C
A
93
D
Five of the probabilities of array (12) are given by the following equations:
A = (fX,Y,+fX,P,+fX,Y, +fX2Y,)/4, B = (fX,Y,fX,Y, +fX1Y,fX,Y,)/2~ c = (Fx1,1+~xlYg+~x,Y,+Fx~Y*)/4, = (~Y,Y,fX,Y,+FX,Y,fX,
E
= (KY1YlFX&
I
(15)
Y,+~x.*PlfxlY*+~x*Y~fxlY1~/~~
+ FX1Y2flX2Y1)/2,
where X, and X, are the parents of X , Yl and Y, are the parents of Y ,and fzivjFxi pj,is the probability that the gamete received by X from X i is identical by descent at a particular locus or, respectively, a t both of two loci) to the gamete received by Y from Y;.. These five equations utilize certain independence assumptions, the correctness of which depends upon the fact that we have limited our discussion to regular relatives. The remaining four probabilities cannot be formulated quite so neatly, but the same pathcounting arguments which allow us to calculatefx, and Fxiy j in equations (15) can be applied directly to the calculation of G, H , I and J .
The path-counting method We must calculate twelve probabilities by the path-counting method; however, eight of them are straightforward coefficients of consanguinity. They are: ( 1 ) fxc (2)
Fxi
for i
=
1 , 2 ; j = 1, 2,
for i = 1 , 2 ; j = 1,2,
(3) G = Pr (al = a3,b,
b,), (4)H = Pr (a, = a3,b, = b3), (5) I = Pr (a, = a3,a2 = a4,b, ( 6 ) J = Pr (a, = a3,a, = a,, b, 3
= b4), E
b,, b,
= b3).
From the fay, y j and Fyiyj, calculated in steps 1 and 2, we obtain A , B, C , D, E , through equations (15). G, H , I and J are calculated directly in steps 3, 4, 5 and 6. The following two examples exemplify the path-counting method and the general procedure for computing the k-coefficients for regular relatives.
Example 1. Double first cousins Consider the pedigree in Fig. 2. The zygotype of X is (albl/a,b,) and the zygotype of Y is (a,b3/a4b4). Each gamete has been labelled; for example, the gamete connecting X, with X is
94
C. DENNISTON
h 0
0
*-.
\
‘\
.
*.
*.
...‘’.
O\\
.
*.A‘*. \
O 0
Fig. 3. Phylograms of the event (awbw= a&) (see text).
awb,, Y, with Y is a,b,, A with X , is a5b, (a,b,, a2b2,a3b3and a4b4are reserved for the zygotypes of X and Y , and the zygotype notation does not indicate from which parent each chromosome is derived). Clearly, fx,y, = fx, y, = P r
=
and fx,y, = fx2y, = 0. To find the Fxiy j we proceed as follows:
[a,b,
= a,b,I
= a,, b, = b,] = Pr [a,Ra,, a5 = a,, a7R’uy, b,Rb5, b, = b,, b,R’b,] +Pr [a,Ra,, a, = a,, a, R’a,, b, Rb,, b, = b,, b, R’b,] = P r [a,
+ Pr [a,Ra,, a, = a,, a, R’a,, + Pr [a, Rae, a, E a,, a, R’a,, = Pr [a,Ra,,
b, Rb,, b,
= b,,
b, Rb,, b,
= b,, b, R’b,]
b, Rb,] Pr [a, = a,, b,
b, R’b,]
= b,] P r [a,R’uy,b, R’b,]
+ Pr [u,Ra,, b,Rb,] P r [a, = a,, b, = be]P r [u,IZ’a,, b, R’b,] +Pr [a, Ra,, b,Rb,] Pr [a, = a,] Pr [b, = b,] Pr [a,R’a,, b, R’b,] +Pr [ a ,Ra,, b, Rb,] Pr [a, = a,] Pr [b5 = b,] P r [a, R’a,, b, R‘b,] = (d/2)([c2 +d21/2) + (d/2)([@+d21/2) + ( C P ) (1/2)(W(c/2)+ (c/2)(1/2)(W(c/2)
+
+
= (d2[c2 d 2 ] / 4 ) (c2/8).
where the expression ‘pRq’ means ‘ p is an immediate replicate of q’ and the expression ‘pR‘q’ means ‘ q is an immediate replicate of p ’ . The four mutually exclusive events whose union makes up the event (a, b, = a, b,) are depicted in Fig. 3 in the form of partial pedigrees or phylograms. The relevant phylograms were obtained by drawing all genetic paths connecting a, with a, (solid lines), drawing all genetic paths connecting b, with b, (dotted lines), and combining these A-locus and B-locus paths in all possible ways.
95
Probability and genetic relationship
;d
1 -
+C
1 -
f (c2+d2)
4
4
Fig. 4. The five basic meiotic events.
There are only five basic meiotic events with two loci (and regular relatives) as shown in Fig. 4, so it is a simple matter to determine the probability of any phylogram by inspection. +d2]/4) ( ~ ~ 1and 8 ) Fxl y z = Fx, y, = 0. It is clear from an inspecSimilarly, Fx, y, = (d2[c2 tion of Fig. 2 that G = H = I = J = 0. From equations (15) we get A = 118, B = 1/32, C = F / 2 , D = F / 8 and E = F2/2, where
+
F
=
+
(d2[c2 d2]/4)+ ( ~ ~ 1 8 ) .
We have then 1
118 118 0 F / 2 118 F/2 118 0 1/32 118 0 1/32 118 0 F / 8 1/32 0 1/32 F/8 0
P=
1/32 118 118 1/32 0 F/8 0 1/32 0 F/2 0 F/8 F/2 0 0 F/8 0 F2/2 0 F/8 0
1/32 0 F/8 0 F/8 0 F2/2
From equation ( 1 3 ) , UVllS
F U/4 0 vy32 8 F V/8 0
V2/32
UV/l6 UV/l6 0 vy32 FU/4 0 0 FUI4 0 vy32 0 FV/8 FV/8 0
U V / l S V2/32 0 FV/8 vy32 0 0 FV/8 FU/4 0 0 F2/2
V2/32 0 FV/8 0 FV/8 0
FV/8
F2/2
0
whereb'= 1 + 2 F a n d V = 1-4F. Finally, from equations (14)we get
k,,
k; = V2/16,
= u2/4,
k,, = k,,
=
U V/8,
k2, = V2/16,
k,,
=
&C,
= FU/2,
krC 11
-
e l
=
k;2 = kil = F V / 4 , kd - kd d 1 2 - 2 1 - 0 = k22, G2= F2.
0,
Example 2 ; Uncle-nephew Consider the pedigree in Fig. 5 . The zygotype of X is (a,bl/a2b2)and that o f Y is (a3b,/a,b,). The four equally probable sources of these four chromosomes are shown in Fig. 5. We have then fx,17, = fs, y l = 114,fx,y, = fx,y, = 0, F~~y, = G, = d(c2 + d v 4 , F~~y, = F~~y, = 0, H=c/16
and G = I = J = O .
C. DENNISTON
96
Sources of chromosomes
Proba bi I ities
Total probability
-1
f
0
f
1
\ yo%$ 0
\ \
\\
0
i d (c2+dz) None
qoqo*q \ \
\ \
0
0
0
fd(c2+dz)
0
H
None
0
hc
0
None
0
J c
G I
None
0
J
Fig. 6. Phylogrtlms for the relation uncle-nephew.
From equations (15) we have A = 118, B = 0, C = d(c2+d2)/8, D = 0 and E = 0. We have then 1/8 118 P = 1/8 118 0
where R = c2 + d2.
dR/8 0 0 c/16 0 0 dR/8 c/16 0 0 0 ~ 1 1 6 dR/8 0 0 c/16 0 0 dR/8 0 0 0 0 0 0
0 0 O), 0 0
Probability and genetic relationship
97
From equation (13), (2R+c)/4 S / l S S / l S S/l6 dR/8 0 8/16 0 dR/8 p* = 27/16 0 c/l6 S/l6 c/16 0 0 0 0 0 0 0
S/l6 S/lS 0 c/16 c/l6 0 dR/8 0 0 dR/8 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
where S = 1 - d( 1 - 4[c2 + d 2 ] ) . Finally from equation (14) we obtain the fifteen k-coefficients,
koo = (2R + c)/4, ko1 = k,o = S/S, k;: = CIS, k:; = dR/4, kzo = kO2= kyl = kTl = ki1 = kt1 = ki2 = kf, = ki2 = k,: = 0. Genotype Pair probabilities Consider a pair of regular relatives, X and Y . Consider, further, two loci, the A-locus and the B-locus, each segregating two alleles. The two alleles a t the A-locus, A and a, have frequencies p1 and q,,respectively ;the two alleles a t the B-locus, B and b, have frequenciesp, and q2, respectively. Linkage equilibrium is assumed throughout. What is the probability that X is of genotype ABIAB and that Y is of genotype ABIAB? The derivation of this probability can be made more succinct by combining genotype and phylotype notation (Cotterman, 1960). Allelic states are distinguished by capital and lower case letters ( A ,a ; B, b) as stated above. Genes which are identical by descent are assigned the same subscript. For example, ‘A,A,’ denotes an allozygote (a1a2) who is also a homozygote ( A A ) ;‘a,a,’ denotes an autohomozygote; ‘Ala2’denotes an alloheterozygote, and so on. We may then write
Pr(X = AB/AB, Y = AB/AB) = Pr(X = A B / A B ) P r ( Y = AB/AB(X = AB/AB) = pTpg. Pr
( Y = AB/AB(X = AB/AB) = p,?pi[Pr(, 33/44) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) Pr (p, = B )
+ Pr (, 13/44) Pr (a,= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 33/14) Pr (a,= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 33/24) Pr (a3= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 23/44) Pr (a,= A ) Pr (p3= B ) Pr (PI = B ) + Pr (, 13/24) Pr (p3= B ) Pr (p, = B )+ Pr (, 23/14) Pr (p3= B ) Pr (p, = B ) +Pr (, 31/44 Pr (a,= A ) Pr (a,= A ) Pr (p, = B ) + P r (, 33/41) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) + Pr (, 33/42) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) +Pr (, 32/44) Pr (a,= A ) Pr (a,= A ) Pr (p, = B ) +Pr (, 31/42) Pr (a3= A ) Pr (a,= A ) +Pr (, 32/41) Pr (a3= A ) Pr (a,= A ) + P r ( , 11/44)Pr(a,= A ) P r ( p , = B ) + P r ( ,33/11)Pr(a3=A)Pr(,8, = B ) + P r ( , 33/22)Pr(a3=A)Pr(,8,=B)+Pr(,22/44)Pr(a4=A)Pr(,8,=BB) 7
HGE
39
98
C. DENNISTON +Pr (, 31/14) Pr (a3= A ) Pr (p4= B ) +Pr (, 13/41) Pr (a4= A ) Pr (p3= B) +Pr(, 23/42) Pr(a, = A ) Pr(p3= B)+Pr(,32/24) Pr(a3= A ) Pr(p4= B) + P r ( , 2 1 / 4 4 ) P r ( a 4 = A ) P r ( p 4= B ) + P r ( , 3 3 / 2 1 ) P r ( a , = A ) P r ( p 3 = B )
+Pr (, 33/12) Pr (a3= A ) Pr (p3= B ) +Pr (, 12/44) Pr (a4= A ) Pr (pa= B) +Pr(, 31/24) Pr (a3= A ) Pr(p4= B)+Pr(,23/41) Pr(a4= A ) Pr(p3= B) +Pr (, 13/42) Pr (a4= A ) Pr (p3= B ) +Pr (, 32/14) Pr (a3= A ) Pr (p4= B) +Pr (, 11/24) Pr(p4= B)+Pr(,23/11) Pr(p3= B) + P r ( , 13/22) Pr(p3= B)+Pr(,22/14) Pr(P4= B) + P r ( , 21/14)Pr(p4= B ) + P r ( , 13/21) Pr(p3= B)
+Pr (, 23/12) Pr (/I3 = B )+ Pr (, 12/24) Pr (p4= B) +Pr(, 11/42) Pr(a4= A ) + P r ( , 32/11) Pr(a3= A )
+Pr(,31/22) Pr(a3= A ) + P r ( , 22/41) Pr(a4= A ) +Pr(, 31/12) Pr(a3= A)+Pr(, 12/41) Pr(a4= A ) +Pr (, 21/42) Pr (a4= A ) +Pr (, 32/21) Pr (a3= A )
In a similar manner we may obtain all of the 100 conditional genotype-pair probabilities. For example, the 10 genotype-pair probabilities, Pr
, are displayed in Table 1.
Genetic counselling problems The following is an example of the type of problem in genetic counselling which the techniques of this paper enable one to handle. Suppose an individual, Y, learns that a relative of his, X, is suffering from a recessive disease of the B-locus. Y is known not to be homozygous recessive, but he may be a carrier of the disease gene. I n addition, genotypic information is available on both individuals for the A-locus, which is linked to the B-locus with recombination fraction, c. For example, X is of genotype Aa and Y is of genotype A A . What is the probability that Y is a carrier of the B-locus disease gene? In symbols, the problem and its solution are as follows:
Pr(Y = BblX = Aabb, Y
= AAB-)
- Pr(X = Aabb, Y = AABb) Pr(X = Aabb, Y = A A B - ) -
Pr(X = Aabb, Y = AABb) Pr (X = Aabb, Y = AABB) +Pr (X = Aabb, Y = AABb) '
-
Pr ( Y = AABbIX = Aabb) Pr ( Y = AABBIX = Aabb) +Pr ( Y = AABblX = Aabb)
Probability and genetic relationship
99
4 Q)
2E
$
o
$
o
o
o
o
o
o
0
$
0
~
0
0
0
0
0
00
$
~
o
o
o
o
o
o
0 o
o
o
$
$
o
o
o
o
o
o
7-2
C. DENNISTON
100
Table 2. The k-coeficients as polynomials in c, the recombination fraction (E.g. for full sibs, koo = i - c + zca - zc* + c4; ki? = c* - zc8 + c4.) A. Unilineal relatives Parent- Grandoffparentspring grandchild koo k10
kE
= ko,
0
( 4 3
0
( 0 9
(49
4:
t) t)
(0,
( 0 9
-4) f) -494) 4, -4)
Half sibs
(t,- 1 9 1 ) (03
(fv
First cousins
Uncle (aunt)nephew (niece) (&,
-:,2,
-I)
($9
3, - 4) (0,i, - j) -&, &) (f, -&I - 3 9
First cousins
13
(0,+,
19
($9
once removed
( Z , -*,43,
-s,2, - f) - ; 9 3
*:+
-a,
- 4,i)
3:. -8:. t , -+)
(-+i*
0
(0,
-%I,
-+$,
(0,
-4, %, -$, f)
3
0
--I,%i,
0
B. Bilineal relatives Full sibs
(4,
koo k10
4: 4:
= ko,
-132, (0, -$,
(4,
kZ
4,
-I,
-2, I ) 2, - I )
2, - 2 , 0
Double first cousins
ciQ,-,
-8
(0, $9
-gsY ,-'eY, - I , Pa - 2L 148
21
,"iC
I)
k20 = k,, k;, = ka12 k:l = 4 2 k:,
9
64
9
8
9
-f,t)
5, - f ) - y ,y , f) -!$,
- 5 9
0
-Y,%?, -?,Y,-:,a,
2, - I )
(2-
-2,I)
16,
- 3 *ar -B 2a.a ie, z* e4
0
4 2
4
88
8
3
(0,011, ( O , O , I , - y , y " z , -a,",?, (09 ft + *-: $a, -'ez,* 0
0
(a, - 1 , 2 ,
9
84
0
(o,o, I , - 2 , I ) lo,o, I , - 2 , I )
(o,&,-:,
is,
8
0
4:
Ql - 6 , a 4 1 -ha
-3,f) -?, $-f)
-418 9
8
9
aa 8
-8 9
z,t)
0
which from equationssimilar to those in Table 1, kllPlP2 + 2klOPlP29.2 + 2kOlP2,P2+ 2kOOP?P 2 Q 2 kllPlP2 + 2klOP1 2329.2 + 2kOlP1PZ + 2kO"P1P2Q2+ kloPlP; + kooP1P2, where k,, = kl;, + k;",+ kt: + kyl. Suppose that X and Y are full sibs for which the k-coefficients are given in Table 2.
-
We have after some manipulation, where 5 = cd, P r ( Y = B b / X = Aabb, Y = A A B - )
-
-w2
(1 (1 + 2PlQ2)+45(1- 25) (24 +!la) + at2 (1 - 2EI2(1 + 22319.2+ P l P 2 ) + 25) P P l + 2q2 +P2) + 4E2'
into which we can substitute the appropriate recombination fraction and gene frequencies. In the special case of no linkage (c = 1/2)the expression above reduces to 2 + 29.2 2 + 2P2 +P2 ' which is independent of any consideration of the A-locus as expected. Comparison with Campbell and Elston Campbell & Elston (1971) treat the two-locus problem dealt with here in a different way. The correspondences between their notation and that in the present paper are: c1 = leS,,+k,d, c2 = G1 ki1 = k:, k f 2 , cg CS = k,CP kEE k;; k;, c4 = k20 = ko,,
+
+
+
+ + +
c6
= k10 = k O l ,
c7
= koo,
Probability and genetic relationship
101
although in all examples given in their paper kt2 = k& = k$ = 0 . Campbell & Elston distinguish ordered heterozygotes; I do not. I distinguish coupling and repulsion; they do not. The coefficients of their Table 3 agree with my Table 2 if one makes the appropriate transformation of notation. (There is one exception: c6, grandparent or grandchild, which reads 114 in their table, is an obvious misprint; it should be ( h / 4 )(Elston, personal communication.)
Covariance between relatives with arbitrary linkages and epistasis Cockerham (1954) and Kempthorne (1954) have shown how the total variance of a trait may be partitioned in a random mating population and how this partition relates to the correlation between relatives. Cockerham (1956) extended this work to include linkage, but his results applied only to certain relationships. It is the purpose of this section t o derive an expression for the covariance between relatives, neither of whom is inbred, for a trait involving two loci, with multiple alleles a t each, and arbitrary linkage and epistasis. It is assumed that the population is randomly mating and in linkage equilibrium. Consider two loci, A and B, each segregating multiple alleles with frequencies, pi and qkr respectively. Denote the average genotypic value of genotype AiAjBkBIby Kjkland the average value of the trait in the entire population by P. It will be convenient to work entirely in terms of deviations from the mean, so let yiikl = &jkl - y. We will also distinguish between Ai A, and A , Ai and between B, B, and BlB,; however, it is assumed throughout that yijkl= yjik1= yiilk = Y j i l k . Linkage equilibrium is assumed so that the frequency of genotype A,A,BkBlispipjqkq,. Dot notation will be used to denote means. For example, yijkl= the average deviation of genotype AiAjB,Bl, yijk. = & qIyijkl = the average deviation of the partial genotype, Ai A jB,., I
y i j , .= C 2
qkqlyijkl =
the average deviation of the partial genotype, A i A j . . ,etc.
k l
Following Kempthorne (1957, chapter 19), the model is additive effects of Ai and A j dominance deviation of A locus additive effects of B, and Bl dominance deviation of B locus additive x additive interaction dominant x additive interaction additive x dominant interaction dominant x dominant interaction where
102
Now
since
E[aiai.] = Pr (i = i f )E(a2)+Pr (i + i f )Pr (axat) = Pr (i = if)E(az),
Probability and genetic relationship
103
Similarly
E[pkpk’)= Pr (k 3 k’)E ( p 2 ) , E[d$)d&,] = Pr (i = i‘,j = j’)E(d@)’)+Pr (i ~ j ‘ , ji’)E E[dj$df!;,] = P r (k = k‘, 1
E [ ( L X (/c~@))~~ ,~~=] Pr (i = i’,k
= 1’) E(cPa)+ Pr ( k = 1‘, l = k’)E(d@)’), 3
k’)E[(a,8)2],
and so on. The most complicated set of terms involves the additive x additive interaction which will be presented in detail. Writing E[ik,i’k‘] for E[(a/3)ik(aP)i.k,], the 16 terms are as follows:
E(ik,i’k’)+ E(ik,i’l’)+ E(ik,j’k’)+ E(ik,j’l’) + E(iZ,i’k’)+ E(il,i’l’)+ E(il,j’k’)+ E(il,j’l‘)
+E ( j k ,i’k’)+E(jk,i’l’) + E ( j k , j ’ k ’ )+E(jk,j’l’) +E(jZ,i’k’)+E(jl,i’l’)+ E(jZ,j’k’)+E(jZ,j’l‘) = Pr (i = i’) k = k’)E(a/3)&+ Pr (i = i’, k = 1’) E(aP)& + Pr (i = j ’ , k = k’)E(ap)&+ P r (i = j’,k = 1’) E(aP)& +Pr(i = i‘,Z = k’)E(aP)i2,+Pr(i=i’,Z = 1’) E(ap)b
plus ten more terms.
By similar arguments the entire covariance between relatives can be written as the sum of variance components each weighted by the appropriate two-locus k-coefficients (in the case of interactions) and one locus coefficients (in the case of single locus effects). We have then that the covariance between two individuals, X and Y , is
C. DENNISTON
104 say, where
k(d0 = Theprobability of sharing at least one gene in common at both loci, kgo = The probability of sharing two genes of the A locus and at least one at the D locus, and so on. In the case of no linkage (c = 4). This reduces to the well known formula
+
+
CV,, = (k, k2) ~5 k, g& For full sibs we get CV8 =
+ (hi+ k2)2
+ (k1+k2) k2
+
V&A
+ (ki+ k2) k2 U ~ +Dk%&.
+&a&+Q[2 (1 - 2c)2] & +Q[1+ (1 - 2c)2]a& +Q[1+ (1 - 2c)2] U5D
++x[ 1 + (1 -
2c)2]2
u;g,
where c = the recombination frequency. This result agrees with Cockerham (1956). For uncle-niece we get cv,, = &a5+'iij[2d( 1 - 2cd)+ c] .&c SUMMARY
A method has been described for calculating the 15 k-coefficientsrequired to completely specify the genetic identity relations between two individuals at two linked loci. These k-coefficients are then used to derive a general expression for the covariance between relatives for a trait involving two linked loci and arbitrary epistasis. REFERENCES
CAMPBELL, M. A. & ELSTON,R. C. (1971). Relatives of Probands: models for preliminary genetic analysis. Annals of Human Genetics 35, 225-236. COCKEREAM, C. C. (1954). An extension of the concept of partitioning hereditary variances for analysis of covariance among relatives when epistasis is present. Genetics 39, 859-882. COCKERHAM,C. C. (1956). Effects of linkage on the covarirtnces between relatives. Genetics 41, 138-141. COTTERMAN,C. W. (1940). A calculus for statistical genetics. Ph.D. thesis, Ohio State University, Columbus. COTTERMAN, C. W. (1960). Relationship and probability in Mendelian Populations. Unpublished notes. CROW, J. F. & MOTOOKIMURA (1970). A n Introduction to Population Genetics Theory. New York: Harper and Row. CARTER(1967). Probability and Genetic Relationship. Unpublished thesis, University of WisconDENNISTON, sin, Madison. DENNISTON, CARTER (1974). An extension of the probability approach to genetic relationships: One locus. Theoretical Population Biology 6 , 68-75. GALLAIS, A. (1974). Covariances between arbitrary relatives with linkage and epistasis in the case of linkage disequilibrium. Biometrics 30,429-446. HALDANE, J. B. S. (1949). The association of characters as a result of inbreeding and linkage. Av~naleof Eugenia 15, 17-23. KEMPTHORNE, 0. (1954). The correlation between relatives in a random mating population. Proceedings of the Royal Society B, 143, 103-113. KEMPTEORNE, 0 . (1967). An introduction to Genetic Statietics. New York: John Wiley & Sons. MALI~COT, G. (1948). Lm Mathdrnatiques de l'hdrdditd. Paris: Masson.
89
Printed in Qreat Britain
Probability and genetic relationship :two loci*?$ BY CARTER DENNISTON Laboratory qf Genetics, University of Wisconsin, Madison, U.S.A.
It is the purpose of this paper (1) to define a set of probabilities (k-coefficients)which specify in detail the genetic relationship between two individuals with respect to two loci, (2) to outline a procedure for calculating these probabilities for any pedigree to which they apply, and (3) to present some examples of how these probabilities may be used. A comparison to the work of Campbell & Elston (1971) is included. See also Gallais, 1974. IDENTITY BY DESCENT
Consider two diploid individuals, X and Y , and two autosomal loci, A and B, with recombination fraction, c. We label the two homologous chromosomes of X, 1 and 2, and those of Y , 3 and 4 , s o that the individual genes of X and Y are (a,b,/a,b,) and (a,b,/u,b,), respectively. To avoid circumlocution later on, we call such designations zygotypes; that is, the zygotype of X is (a,b,/ a,b,) and this is simply a labelling of the genes and chromosomesof X and has no other significance. The total number of identity (bydescent)relations (Cotterman, 1940; Malecot, 1948) on the eight genes in two individuals is 1Ei2 = 225. In this paper we will restrict our attention to the 7, = 49 relations applicable to two individuals neither of whom is inbred. (It will be convenient to introduce some terminology at this point. According to Cotterman, 1960, two non-inbred relatives are called regular relatives; if a pair of homologous genes in an individual are identical by descent, that individual is said to be autozygous at that locus; otherwise, the individual is allozygous. Regular relatives are thus allozygous.) The restriction of this paper to regular relatives, then eliminates all relationships for which a, = a, or b, = b, or a, E a, or b, = b,, where ' = ' denotes identity by descent. The 49 identity relations for regular relatives are displayed in Fig. 1. A solid line in Fig. 1 denotes identity by descent at the A locus, and a dotted line denotes identity by descent at the B-locus. Another useful way of depicting these 49 events is the phylotype notation of Cotterman (1960). Genes which are identical by descent are assigned identical subscripts. For example, the event in the fifth row, second column of Fig. 1 is written as (a,pl/a2f3,,a11,/a4,8,).Greek base letters are used to distinguish the phylotype notation from the zygotype notation, above. In array ( l ) ,the 49 identity relations are represented in a modified phylotype notation in which the expression (, ij/kl) is shorthand for (a,/3,/~,/3~, a&/a,,/3,); the arrangement of events is the same as in Fig. 1. (, 33/44) (, 13/44) (, 33/14) (, 33/24) (, 23/44) (, 13/24) (, 23/14) (, 31/44) (, 11/44) (, 31/14) (, 31/24) (, 21/44) (, 11/24) (, 21/14)
(, 33/41) (, 13/41) (, 33/11) (, 33/21) (, 23/41) (, 13/21) (, 23/11)
* t
Paper No. 1593 of the Laboratory of Genetics, University of Wisconsin, Madison, Wisconsin 53706. Taken from a thesis submitted in partial fulfillment of the requirements for the Ph.D. degree at. the University of Wisconsin, 1967. $ This work was supported in part by NIH Grant GM15422.
90
C. DENNISTON (3
33/42) (, 13/42) (, 33/12) (, 33/22) (, 23/42) (, 13/22) (, 23/12)
(3
32/44) (, 12/44) (, 32/14) (, 32/24) (, 22/44) (, 12/24) (, 22/14)
(, 31/42)
(3
(, 32/41)
(3
11/42) 12/41)
( 9
(9
31/12) ( 9 31/22) (, 21/42) (, 11/22) (, 21/22) 32/11) (, 32/21) (, 22/41) (, 12/21) (, 22/11).
Fig. 1. The 49 gene identity relations for regular relatives: two loci.
now define the 15 k-coeficients as follows :
k,,
= Pr (, 33/44),
kio = Pr (, 13/44)+Pr (, 33/14) = Pr (, 33/24) +Pr (, 23/44), k20 = Pr (, 13/24) +Pr (, 23/14), k o ~= Pr 31/44) +Pr (, 33/41) = Pr (, 33/42) +Pr (, 32/44), LO, = pr (, 31/42)+Pr (, 32/41), kCC 11 (, 11/44)+- (, 33/11) = Pr (, 33/22)+Pr (, 22/44), kyl = pr 31/14) +PI' (, 13/41) = Pr (, 23/42)+Pr (, 32/24), k?C 1 1 - Pr 21/44)+Pr (, 33/21) = Pr (, 33/12)+Pr (, 12/44), kz = Pr (, 31/24) +Pr (, 23/41) = Pr (, 13/42) +Pr (, 32/14), 4 1 = Pr 11/24) +fi (, 23/11) = Pr (, 13/22) +Pr (, 22/14), (3
(3
(5
(9
(1)
Probability and genetic relationship
91
kzl = Pr (, 21/14)+Pr (, 13/21) = Pr (, 23/12)+Pr (, 12/24), ki2 = Pr (, 11/42)+Pr (, 32/11) = P r ( , 31/22)+Pr (, 22/41), k f 2 = Pr (, 31/12)+Pr (, 12/41) = Pr (, 21/42)+Pr (, 32/21), ki2 = Pr (, 11/22)+Pr (, 22/11), kf2 = Pr (, 21/12) + P r (, 12/21). (2) The form of these 15 k-coefficients has mnemonic value. For example, kyl is the probability that X and Y (both allozygous a t both loci) share exactly one gene in common at each locus and that the shared genes are in coupling in X and in repulsion in Y .kf2 is the probability that X and Y share exactly one gene in common a t the A-locus, exactly two genes in common at the B-locus, and that the A-locus gene shared by X and Y is linked t o non-identical (different) B-locus genes in the t'wo individuals. The following equat,ions should be noted:
+ +
+
koo + 2k1o k2o 2ko1+ ko2 2kE",+ 2ky1+ 2k:
+ 2k; + 2kl1+ 2kf1+ 2kS,2+ 2kf2+ k,8, + kf2 = 1,
(3)
koo + 2klO + k20 = ooko,
(4)
koo + 2kOl+ k02 = ooko,
(5)
+
+ +
kol k s k:C, kl;, + k",, + kt1 = OOkl, kl0 + ky1+ @I+ k;", + k;72 + + kf2 = OOkl,
e2
+
+ +
(6)
(7)
koz 2kf2 + k8,, + kf2 = 00k2, (8) k,o 2k8,1+ 2k;1+ k8,2 + k f p = 00k2, (9) (k8,2+ k8,1+ kS,2) (c2+ d 2 )+ (&I) d2+ ( k c ) c2+ (2k& + 2kt1+ 2kf2 + kC:, + k);e) cd = 2F,,, (10) where ooko,oak, and ookz are the one-locus k-coefficients for regular relatives (Cotterman, 1940; Denniston, 1967, 1974; Crow & Kimura, 1970), and F y y is the coefficientof consanguinity of X and Y defined for two loci, i.e. the probability of a n (hypothetical) offspring of X and Y being autozygous a t both the A-locus and the B-locus (Haldane, 1949). It should be noted that the Haldane article just cited adumbrates both MalBcot's concept of identity by descent and the twoloci extension of the inbreeding coefficient. A method for calc,ulating the k-coeficients We shall denote the probabilities of the 49 identity relations of Fig. 1 as
p* =
A* A* A* A* B*
C* G* BT H* B*
G* C* H* BT I*
BT H* C* G*
H* Bt G* C* D* I *
D* I*
I*
D* D* I* I* D* E*
I
J*
where A*
pr (, 13/44) = Pr (, 33/14) = Pr (, 33/24) = Pr (, 23/44) = Pr (, 31/44) = Pr (, 33/41) = Pr (, 33/42) = Pr (, 32/44), =
B* = Pr (, 13/24) = Pr (, 23/14)
BT and so on.
=
Pr (, 31/42) = Pr (, 32/41),
= Pr (, 31/24) = Pr (, 23/41) = Pr (, 13/42) = Pr (, 32/14),
C. DENNISTON
92
In addition, we define a second array
1 A
A C
B I
A A A B B G B H D I
D I
D
J
E
where
A = Pr (a, = a,) = Pr (a, = a,) = Pr (aoE a,) = Pr (a, = a,) = Pr (b, = b,) = Pr (b, =- b,) = Pr (b, = b4) = Pr (b, = b3), B = Pr (al = a3,a23 a,) = Pr (a, = a,, a, 3 a,) = Pr (b, = b,, b, = b,) = Pr (b, = b,, b, = b,) = Pr (b, = b,, a, = a,) = Pr (b, = b,, a, = a,) = Pr (al = a,, b, = b,) = Pr (a, = a,, b, z b,), and so on. Note that although the arrangements of arrays (l), (1 1)and (12) all correspond to that of Fig. 1, the lack of a line in Fig. 1 is interpreted as nonidentity in arrays (1) and ( 1 1) but as noncommital in array (12). The relations connecting arrays (11) and (12) are
J* = J E* = E I* = I - J* D* = D-E* B: = B-2D*-E* H* = H-21*- J* C* = C-2D*-E* Q* = G - 2 1 * -
J*
B* = B - 2 0 * - 2 J * - E * - J* A* = A - B * - B : - C * - C * - H * - 3 0 * - 3 I * - E * -
K*
= 1 - 8A* - 4B* - 4B: -4C*
J*
- 4G* -4H* - 8D* - 81* - 2E* - 2J*,
and the k-coefficients are given by
Loo = K*, klo = 2A*, k20= 2B*, kol = 2A*, kO2= 2B*,
ky, = 2c*,
el= 2G*, ky,
= 2H*,
k'&
= 2BT,
6, = 2D*, k& = 21*, k:, = 2D*, kf2 = 21*, k;, = 2E*, ki2 = 2J*.
(13)
Probability and genetic relationship C
A
93
D
Five of the probabilities of array (12) are given by the following equations:
A = (fX,Y,+fX,P,+fX,Y, +fX2Y,)/4, B = (fX,Y,fX,Y, +fX1Y,fX,Y,)/2~ c = (Fx1,1+~xlYg+~x,Y,+Fx~Y*)/4, = (~Y,Y,fX,Y,+FX,Y,fX,
E
= (KY1YlFX&
I
(15)
Y,+~x.*PlfxlY*+~x*Y~fxlY1~/~~
+ FX1Y2flX2Y1)/2,
where X, and X, are the parents of X , Yl and Y, are the parents of Y ,and fzivjFxi pj,is the probability that the gamete received by X from X i is identical by descent at a particular locus or, respectively, a t both of two loci) to the gamete received by Y from Y;.. These five equations utilize certain independence assumptions, the correctness of which depends upon the fact that we have limited our discussion to regular relatives. The remaining four probabilities cannot be formulated quite so neatly, but the same pathcounting arguments which allow us to calculatefx, and Fxiy j in equations (15) can be applied directly to the calculation of G, H , I and J .
The path-counting method We must calculate twelve probabilities by the path-counting method; however, eight of them are straightforward coefficients of consanguinity. They are: ( 1 ) fxc (2)
Fxi
for i
=
1 , 2 ; j = 1, 2,
for i = 1 , 2 ; j = 1,2,
(3) G = Pr (al = a3,b,
b,), (4)H = Pr (a, = a3,b, = b3), (5) I = Pr (a, = a3,a2 = a4,b, ( 6 ) J = Pr (a, = a3,a, = a,, b, 3
= b4), E
b,, b,
= b3).
From the fay, y j and Fyiyj, calculated in steps 1 and 2, we obtain A , B, C , D, E , through equations (15). G, H , I and J are calculated directly in steps 3, 4, 5 and 6. The following two examples exemplify the path-counting method and the general procedure for computing the k-coefficients for regular relatives.
Example 1. Double first cousins Consider the pedigree in Fig. 2. The zygotype of X is (albl/a,b,) and the zygotype of Y is (a,b3/a4b4). Each gamete has been labelled; for example, the gamete connecting X, with X is
94
C. DENNISTON
h 0
0
*-.
\
‘\
.
*.
*.
...‘’.
O\\
.
*.A‘*. \
O 0
Fig. 3. Phylograms of the event (awbw= a&) (see text).
awb,, Y, with Y is a,b,, A with X , is a5b, (a,b,, a2b2,a3b3and a4b4are reserved for the zygotypes of X and Y , and the zygotype notation does not indicate from which parent each chromosome is derived). Clearly, fx,y, = fx, y, = P r
=
and fx,y, = fx2y, = 0. To find the Fxiy j we proceed as follows:
[a,b,
= a,b,I
= a,, b, = b,] = Pr [a,Ra,, a5 = a,, a7R’uy, b,Rb5, b, = b,, b,R’b,] +Pr [a,Ra,, a, = a,, a, R’a,, b, Rb,, b, = b,, b, R’b,] = P r [a,
+ Pr [a,Ra,, a, = a,, a, R’a,, + Pr [a, Rae, a, E a,, a, R’a,, = Pr [a,Ra,,
b, Rb,, b,
= b,,
b, Rb,, b,
= b,, b, R’b,]
b, Rb,] Pr [a, = a,, b,
b, R’b,]
= b,] P r [a,R’uy,b, R’b,]
+ Pr [u,Ra,, b,Rb,] P r [a, = a,, b, = be]P r [u,IZ’a,, b, R’b,] +Pr [a, Ra,, b,Rb,] Pr [a, = a,] Pr [b, = b,] Pr [a,R’a,, b, R’b,] +Pr [ a ,Ra,, b, Rb,] Pr [a, = a,] Pr [b5 = b,] P r [a, R’a,, b, R‘b,] = (d/2)([c2 +d21/2) + (d/2)([@+d21/2) + ( C P ) (1/2)(W(c/2)+ (c/2)(1/2)(W(c/2)
+
+
= (d2[c2 d 2 ] / 4 ) (c2/8).
where the expression ‘pRq’ means ‘ p is an immediate replicate of q’ and the expression ‘pR‘q’ means ‘ q is an immediate replicate of p ’ . The four mutually exclusive events whose union makes up the event (a, b, = a, b,) are depicted in Fig. 3 in the form of partial pedigrees or phylograms. The relevant phylograms were obtained by drawing all genetic paths connecting a, with a, (solid lines), drawing all genetic paths connecting b, with b, (dotted lines), and combining these A-locus and B-locus paths in all possible ways.
95
Probability and genetic relationship
;d
1 -
+C
1 -
f (c2+d2)
4
4
Fig. 4. The five basic meiotic events.
There are only five basic meiotic events with two loci (and regular relatives) as shown in Fig. 4, so it is a simple matter to determine the probability of any phylogram by inspection. +d2]/4) ( ~ ~ 1and 8 ) Fxl y z = Fx, y, = 0. It is clear from an inspecSimilarly, Fx, y, = (d2[c2 tion of Fig. 2 that G = H = I = J = 0. From equations (15) we get A = 118, B = 1/32, C = F / 2 , D = F / 8 and E = F2/2, where
+
F
=
+
(d2[c2 d2]/4)+ ( ~ ~ 1 8 ) .
We have then 1
118 118 0 F / 2 118 F/2 118 0 1/32 118 0 1/32 118 0 F / 8 1/32 0 1/32 F/8 0
P=
1/32 118 118 1/32 0 F/8 0 1/32 0 F/2 0 F/8 F/2 0 0 F/8 0 F2/2 0 F/8 0
1/32 0 F/8 0 F/8 0 F2/2
From equation ( 1 3 ) , UVllS
F U/4 0 vy32 8 F V/8 0
V2/32
UV/l6 UV/l6 0 vy32 FU/4 0 0 FUI4 0 vy32 0 FV/8 FV/8 0
U V / l S V2/32 0 FV/8 vy32 0 0 FV/8 FU/4 0 0 F2/2
V2/32 0 FV/8 0 FV/8 0
FV/8
F2/2
0
whereb'= 1 + 2 F a n d V = 1-4F. Finally, from equations (14)we get
k,,
k; = V2/16,
= u2/4,
k,, = k,,
=
U V/8,
k2, = V2/16,
k,,
=
&C,
= FU/2,
krC 11
-
e l
=
k;2 = kil = F V / 4 , kd - kd d 1 2 - 2 1 - 0 = k22, G2= F2.
0,
Example 2 ; Uncle-nephew Consider the pedigree in Fig. 5 . The zygotype of X is (a,bl/a2b2)and that o f Y is (a3b,/a,b,). The four equally probable sources of these four chromosomes are shown in Fig. 5. We have then fx,17, = fs, y l = 114,fx,y, = fx,y, = 0, F~~y, = G, = d(c2 + d v 4 , F~~y, = F~~y, = 0, H=c/16
and G = I = J = O .
C. DENNISTON
96
Sources of chromosomes
Proba bi I ities
Total probability
-1
f
0
f
1
\ yo%$ 0
\ \
\\
0
i d (c2+dz) None
qoqo*q \ \
\ \
0
0
0
fd(c2+dz)
0
H
None
0
hc
0
None
0
J c
G I
None
0
J
Fig. 6. Phylogrtlms for the relation uncle-nephew.
From equations (15) we have A = 118, B = 0, C = d(c2+d2)/8, D = 0 and E = 0. We have then 1/8 118 P = 1/8 118 0
where R = c2 + d2.
dR/8 0 0 c/16 0 0 dR/8 c/16 0 0 0 ~ 1 1 6 dR/8 0 0 c/16 0 0 dR/8 0 0 0 0 0 0
0 0 O), 0 0
Probability and genetic relationship
97
From equation (13), (2R+c)/4 S / l S S / l S S/l6 dR/8 0 8/16 0 dR/8 p* = 27/16 0 c/l6 S/l6 c/16 0 0 0 0 0 0 0
S/l6 S/lS 0 c/16 c/l6 0 dR/8 0 0 dR/8 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
where S = 1 - d( 1 - 4[c2 + d 2 ] ) . Finally from equation (14) we obtain the fifteen k-coefficients,
koo = (2R + c)/4, ko1 = k,o = S/S, k;: = CIS, k:; = dR/4, kzo = kO2= kyl = kTl = ki1 = kt1 = ki2 = kf, = ki2 = k,: = 0. Genotype Pair probabilities Consider a pair of regular relatives, X and Y . Consider, further, two loci, the A-locus and the B-locus, each segregating two alleles. The two alleles a t the A-locus, A and a, have frequencies p1 and q,,respectively ;the two alleles a t the B-locus, B and b, have frequenciesp, and q2, respectively. Linkage equilibrium is assumed throughout. What is the probability that X is of genotype ABIAB and that Y is of genotype ABIAB? The derivation of this probability can be made more succinct by combining genotype and phylotype notation (Cotterman, 1960). Allelic states are distinguished by capital and lower case letters ( A ,a ; B, b) as stated above. Genes which are identical by descent are assigned the same subscript. For example, ‘A,A,’ denotes an allozygote (a1a2) who is also a homozygote ( A A ) ;‘a,a,’ denotes an autohomozygote; ‘Ala2’denotes an alloheterozygote, and so on. We may then write
Pr(X = AB/AB, Y = AB/AB) = Pr(X = A B / A B ) P r ( Y = AB/AB(X = AB/AB) = pTpg. Pr
( Y = AB/AB(X = AB/AB) = p,?pi[Pr(, 33/44) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) Pr (p, = B )
+ Pr (, 13/44) Pr (a,= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 33/14) Pr (a,= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 33/24) Pr (a3= A ) Pr (p3= B ) Pr (p, = B ) + Pr (, 23/44) Pr (a,= A ) Pr (p3= B ) Pr (PI = B ) + Pr (, 13/24) Pr (p3= B ) Pr (p, = B )+ Pr (, 23/14) Pr (p3= B ) Pr (p, = B ) +Pr (, 31/44 Pr (a,= A ) Pr (a,= A ) Pr (p, = B ) + P r (, 33/41) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) + Pr (, 33/42) Pr (a3= A ) Pr (a,= A ) Pr (p3= B ) +Pr (, 32/44) Pr (a,= A ) Pr (a,= A ) Pr (p, = B ) +Pr (, 31/42) Pr (a3= A ) Pr (a,= A ) +Pr (, 32/41) Pr (a3= A ) Pr (a,= A ) + P r ( , 11/44)Pr(a,= A ) P r ( p , = B ) + P r ( ,33/11)Pr(a3=A)Pr(,8, = B ) + P r ( , 33/22)Pr(a3=A)Pr(,8,=B)+Pr(,22/44)Pr(a4=A)Pr(,8,=BB) 7
HGE
39
98
C. DENNISTON +Pr (, 31/14) Pr (a3= A ) Pr (p4= B ) +Pr (, 13/41) Pr (a4= A ) Pr (p3= B) +Pr(, 23/42) Pr(a, = A ) Pr(p3= B)+Pr(,32/24) Pr(a3= A ) Pr(p4= B) + P r ( , 2 1 / 4 4 ) P r ( a 4 = A ) P r ( p 4= B ) + P r ( , 3 3 / 2 1 ) P r ( a , = A ) P r ( p 3 = B )
+Pr (, 33/12) Pr (a3= A ) Pr (p3= B ) +Pr (, 12/44) Pr (a4= A ) Pr (pa= B) +Pr(, 31/24) Pr (a3= A ) Pr(p4= B)+Pr(,23/41) Pr(a4= A ) Pr(p3= B) +Pr (, 13/42) Pr (a4= A ) Pr (p3= B ) +Pr (, 32/14) Pr (a3= A ) Pr (p4= B) +Pr (, 11/24) Pr(p4= B)+Pr(,23/11) Pr(p3= B) + P r ( , 13/22) Pr(p3= B)+Pr(,22/14) Pr(P4= B) + P r ( , 21/14)Pr(p4= B ) + P r ( , 13/21) Pr(p3= B)
+Pr (, 23/12) Pr (/I3 = B )+ Pr (, 12/24) Pr (p4= B) +Pr(, 11/42) Pr(a4= A ) + P r ( , 32/11) Pr(a3= A )
+Pr(,31/22) Pr(a3= A ) + P r ( , 22/41) Pr(a4= A ) +Pr(, 31/12) Pr(a3= A)+Pr(, 12/41) Pr(a4= A ) +Pr (, 21/42) Pr (a4= A ) +Pr (, 32/21) Pr (a3= A )
In a similar manner we may obtain all of the 100 conditional genotype-pair probabilities. For example, the 10 genotype-pair probabilities, Pr
, are displayed in Table 1.
Genetic counselling problems The following is an example of the type of problem in genetic counselling which the techniques of this paper enable one to handle. Suppose an individual, Y, learns that a relative of his, X, is suffering from a recessive disease of the B-locus. Y is known not to be homozygous recessive, but he may be a carrier of the disease gene. I n addition, genotypic information is available on both individuals for the A-locus, which is linked to the B-locus with recombination fraction, c. For example, X is of genotype Aa and Y is of genotype A A . What is the probability that Y is a carrier of the B-locus disease gene? In symbols, the problem and its solution are as follows:
Pr(Y = BblX = Aabb, Y
= AAB-)
- Pr(X = Aabb, Y = AABb) Pr(X = Aabb, Y = A A B - ) -
Pr(X = Aabb, Y = AABb) Pr (X = Aabb, Y = AABB) +Pr (X = Aabb, Y = AABb) '
-
Pr ( Y = AABbIX = Aabb) Pr ( Y = AABBIX = Aabb) +Pr ( Y = AABblX = Aabb)
Probability and genetic relationship
99
4 Q)
2E
$
o
$
o
o
o
o
o
o
0
$
0
~
0
0
0
0
0
00
$
~
o
o
o
o
o
o
0 o
o
o
$
$
o
o
o
o
o
o
7-2
C. DENNISTON
100
Table 2. The k-coeficients as polynomials in c, the recombination fraction (E.g. for full sibs, koo = i - c + zca - zc* + c4; ki? = c* - zc8 + c4.) A. Unilineal relatives Parent- Grandoffparentspring grandchild koo k10
kE
= ko,
0
( 4 3
0
( 0 9
(49
4:
t) t)
(0,
( 0 9
-4) f) -494) 4, -4)
Half sibs
(t,- 1 9 1 ) (03
(fv
First cousins
Uncle (aunt)nephew (niece) (&,
-:,2,
-I)
($9
3, - 4) (0,i, - j) -&, &) (f, -&I - 3 9
First cousins
13
(0,+,
19
($9
once removed
( Z , -*,43,
-s,2, - f) - ; 9 3
*:+
-a,
- 4,i)
3:. -8:. t , -+)
(-+i*
0
(0,
-%I,
-+$,
(0,
-4, %, -$, f)
3
0
--I,%i,
0
B. Bilineal relatives Full sibs
(4,
koo k10
4: 4:
= ko,
-132, (0, -$,
(4,
kZ
4,
-I,
-2, I ) 2, - I )
2, - 2 , 0
Double first cousins
ciQ,-,
-8
(0, $9
-gsY ,-'eY, - I , Pa - 2L 148
21
,"iC
I)
k20 = k,, k;, = ka12 k:l = 4 2 k:,
9
64
9
8
9
-f,t)
5, - f ) - y ,y , f) -!$,
- 5 9
0
-Y,%?, -?,Y,-:,a,
2, - I )
(2-
-2,I)
16,
- 3 *ar -B 2a.a ie, z* e4
0
4 2
4
88
8
3
(0,011, ( O , O , I , - y , y " z , -a,",?, (09 ft + *-: $a, -'ez,* 0
0
(a, - 1 , 2 ,
9
84
0
(o,o, I , - 2 , I ) lo,o, I , - 2 , I )
(o,&,-:,
is,
8
0
4:
Ql - 6 , a 4 1 -ha
-3,f) -?, $-f)
-418 9
8
9
aa 8
-8 9
z,t)
0
which from equationssimilar to those in Table 1, kllPlP2 + 2klOPlP29.2 + 2kOlP2,P2+ 2kOOP?P 2 Q 2 kllPlP2 + 2klOP1 2329.2 + 2kOlP1PZ + 2kO"P1P2Q2+ kloPlP; + kooP1P2, where k,, = kl;, + k;",+ kt: + kyl. Suppose that X and Y are full sibs for which the k-coefficients are given in Table 2.
-
We have after some manipulation, where 5 = cd, P r ( Y = B b / X = Aabb, Y = A A B - )
-
-w2
(1 (1 + 2PlQ2)+45(1- 25) (24 +!la) + at2 (1 - 2EI2(1 + 22319.2+ P l P 2 ) + 25) P P l + 2q2 +P2) + 4E2'
into which we can substitute the appropriate recombination fraction and gene frequencies. In the special case of no linkage (c = 1/2)the expression above reduces to 2 + 29.2 2 + 2P2 +P2 ' which is independent of any consideration of the A-locus as expected. Comparison with Campbell and Elston Campbell & Elston (1971) treat the two-locus problem dealt with here in a different way. The correspondences between their notation and that in the present paper are: c1 = leS,,+k,d, c2 = G1 ki1 = k:, k f 2 , cg CS = k,CP kEE k;; k;, c4 = k20 = ko,,
+
+
+
+ + +
c6
= k10 = k O l ,
c7
= koo,
Probability and genetic relationship
101
although in all examples given in their paper kt2 = k& = k$ = 0 . Campbell & Elston distinguish ordered heterozygotes; I do not. I distinguish coupling and repulsion; they do not. The coefficients of their Table 3 agree with my Table 2 if one makes the appropriate transformation of notation. (There is one exception: c6, grandparent or grandchild, which reads 114 in their table, is an obvious misprint; it should be ( h / 4 )(Elston, personal communication.)
Covariance between relatives with arbitrary linkages and epistasis Cockerham (1954) and Kempthorne (1954) have shown how the total variance of a trait may be partitioned in a random mating population and how this partition relates to the correlation between relatives. Cockerham (1956) extended this work to include linkage, but his results applied only to certain relationships. It is the purpose of this section t o derive an expression for the covariance between relatives, neither of whom is inbred, for a trait involving two loci, with multiple alleles a t each, and arbitrary linkage and epistasis. It is assumed that the population is randomly mating and in linkage equilibrium. Consider two loci, A and B, each segregating multiple alleles with frequencies, pi and qkr respectively. Denote the average genotypic value of genotype AiAjBkBIby Kjkland the average value of the trait in the entire population by P. It will be convenient to work entirely in terms of deviations from the mean, so let yiikl = &jkl - y. We will also distinguish between Ai A, and A , Ai and between B, B, and BlB,; however, it is assumed throughout that yijkl= yjik1= yiilk = Y j i l k . Linkage equilibrium is assumed so that the frequency of genotype A,A,BkBlispipjqkq,. Dot notation will be used to denote means. For example, yijkl= the average deviation of genotype AiAjB,Bl, yijk. = & qIyijkl = the average deviation of the partial genotype, Ai A jB,., I
y i j , .= C 2
qkqlyijkl =
the average deviation of the partial genotype, A i A j . . ,etc.
k l
Following Kempthorne (1957, chapter 19), the model is additive effects of Ai and A j dominance deviation of A locus additive effects of B, and Bl dominance deviation of B locus additive x additive interaction dominant x additive interaction additive x dominant interaction dominant x dominant interaction where
102
Now
since
E[aiai.] = Pr (i = i f )E(a2)+Pr (i + i f )Pr (axat) = Pr (i = if)E(az),
Probability and genetic relationship
103
Similarly
E[pkpk’)= Pr (k 3 k’)E ( p 2 ) , E[d$)d&,] = Pr (i = i‘,j = j’)E(d@)’)+Pr (i ~ j ‘ , ji’)E E[dj$df!;,] = P r (k = k‘, 1
E [ ( L X (/c~@))~~ ,~~=] Pr (i = i’,k
= 1’) E(cPa)+ Pr ( k = 1‘, l = k’)E(d@)’), 3
k’)E[(a,8)2],
and so on. The most complicated set of terms involves the additive x additive interaction which will be presented in detail. Writing E[ik,i’k‘] for E[(a/3)ik(aP)i.k,], the 16 terms are as follows:
E(ik,i’k’)+ E(ik,i’l’)+ E(ik,j’k’)+ E(ik,j’l’) + E(iZ,i’k’)+ E(il,i’l’)+ E(il,j’k’)+ E(il,j’l‘)
+E ( j k ,i’k’)+E(jk,i’l’) + E ( j k , j ’ k ’ )+E(jk,j’l’) +E(jZ,i’k’)+E(jl,i’l’)+ E(jZ,j’k’)+E(jZ,j’l‘) = Pr (i = i’) k = k’)E(a/3)&+ Pr (i = i’, k = 1’) E(aP)& + Pr (i = j ’ , k = k’)E(ap)&+ P r (i = j’,k = 1’) E(aP)& +Pr(i = i‘,Z = k’)E(aP)i2,+Pr(i=i’,Z = 1’) E(ap)b
plus ten more terms.
By similar arguments the entire covariance between relatives can be written as the sum of variance components each weighted by the appropriate two-locus k-coefficients (in the case of interactions) and one locus coefficients (in the case of single locus effects). We have then that the covariance between two individuals, X and Y , is
C. DENNISTON
104 say, where
k(d0 = Theprobability of sharing at least one gene in common at both loci, kgo = The probability of sharing two genes of the A locus and at least one at the D locus, and so on. In the case of no linkage (c = 4). This reduces to the well known formula
+
+
CV,, = (k, k2) ~5 k, g& For full sibs we get CV8 =
+ (hi+ k2)2
+ (k1+k2) k2
+
V&A
+ (ki+ k2) k2 U ~ +Dk%&.
+&a&+Q[2 (1 - 2c)2] & +Q[1+ (1 - 2c)2]a& +Q[1+ (1 - 2c)2] U5D
++x[ 1 + (1 -
2c)2]2
u;g,
where c = the recombination frequency. This result agrees with Cockerham (1956). For uncle-niece we get cv,, = &a5+'iij[2d( 1 - 2cd)+ c] .&c SUMMARY
A method has been described for calculating the 15 k-coefficientsrequired to completely specify the genetic identity relations between two individuals at two linked loci. These k-coefficients are then used to derive a general expression for the covariance between relatives for a trait involving two linked loci and arbitrary epistasis. REFERENCES
CAMPBELL, M. A. & ELSTON,R. C. (1971). Relatives of Probands: models for preliminary genetic analysis. Annals of Human Genetics 35, 225-236. COCKEREAM, C. C. (1954). An extension of the concept of partitioning hereditary variances for analysis of covariance among relatives when epistasis is present. Genetics 39, 859-882. COCKERHAM,C. C. (1956). Effects of linkage on the covarirtnces between relatives. Genetics 41, 138-141. COTTERMAN,C. W. (1940). A calculus for statistical genetics. Ph.D. thesis, Ohio State University, Columbus. COTTERMAN, C. W. (1960). Relationship and probability in Mendelian Populations. Unpublished notes. CROW, J. F. & MOTOOKIMURA (1970). A n Introduction to Population Genetics Theory. New York: Harper and Row. CARTER(1967). Probability and Genetic Relationship. Unpublished thesis, University of WisconDENNISTON, sin, Madison. DENNISTON, CARTER (1974). An extension of the probability approach to genetic relationships: One locus. Theoretical Population Biology 6 , 68-75. GALLAIS, A. (1974). Covariances between arbitrary relatives with linkage and epistasis in the case of linkage disequilibrium. Biometrics 30,429-446. HALDANE, J. B. S. (1949). The association of characters as a result of inbreeding and linkage. Av~naleof Eugenia 15, 17-23. KEMPTHORNE, 0. (1954). The correlation between relatives in a random mating population. Proceedings of the Royal Society B, 143, 103-113. KEMPTEORNE, 0 . (1967). An introduction to Genetic Statietics. New York: John Wiley & Sons. MALI~COT, G. (1948). Lm Mathdrnatiques de l'hdrdditd. Paris: Masson.
