THE GENETIC STRUCTURE
OFNATURAL POPULATIONS
OF
DROSOPHILA MELANOGASTER. XIV.
EFFECTS
OFTHE
INCOMPLETE DOMINANCE
OFTHE
ZN(2LR)SMI ( C y )CHROMOSOME
ONTHE
ESTIMATES O F VARIOUSGENETIC
PARAMETERS*TERUMI MUKAI
Department of Biology, Kyushu University 33, Fukuoka 812, Japan
Manuscript received December 12, 1978 Revised copy received May 31, 1979
ABSTRACT
Recent reports (MUKAI et al. 1974; KATZ and CARDELLINO 1978; COCKER HAM and MUEAI 1978) have indicated that the Cy chromosome is not always dominant over its homologous chromosome with respect to viability. Thus, the genetic parameters previously estimated using viabilities determined by the Cy method are biased. In the present paper, the biases of the estimates for the polygenic mutation rate, the degree of dominance and the homozygous load are examined. The results indicate that the biases for the mutation rate and the degree of dominance are small and that the estimate of the homozygous load relative to the average viability of the population is not biased.
INCE some experimental evidence suggested that the rearranged second chromcsome
,
Zn(2LR)SMI, which is marked with the Cy mutation, almost completely suppresses the effect of deleterious mutations located in homologous chromosome (MUKAI 1964;MUKAI
et al. 1972). The object was to measure the method(WALLACE
1956) or its equivalent to estimate the relative viabilities of second or third chromosomes of Drosophila melanoigaster(MUKAI
and YAMAGUCHI 1974, and earlier). However, recent careful experiments conducted by
COCKERHAM and
MUKAI
(1978) clearly showed that the C y chromosome is not always completely dominant. Thus, it is necessary to examine how the estimates of genetic parameters such as the polygenic mutation rate, the degree of domi nance and the homozygous genetic load, all of which are based on the assumed complete dominance of the Cy chromosome ( MUKAI 1964;MUKAI
and YAMAZAKI 1968;
MUKAI
andYAMAGUCHI
1974;WATANABE,
YAMAGUCHI and MUKAI 1976, etc.), are biased. Fortunately, the results of the examination show that the biases are small. This article presents the results, along with the effect of syner gistic interaction (MUKAI 1969a) on the estimate of the polygenic mutation rate.* Paper No. 4 from the Laboratory of Population Genetics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan. This work was mainly supported by Grants 144001 and 348003 from the Ministry of Education, Science and Culture of Japan.
170
ANALYSIS
Direction of the bias in the estimates of mutation rate of Viability polygenes und their average effects: Mutations were accumulated under minimum selec tion pressure on many chromosomes that originated from a single wild second chromosome (MUKAI 1964; MUKAI et al. 1972b). The object was to measure the rate at which these occurred, as well as their average effect on viability.
To
do this, we assumed that the ith wildtype (+) chromosome carried polygenic mutations whose cumulative homozygous effects reduced viability by a propor tion, S, and that the genotypic variance of the viability effects among the chromosome lines was V ( S ) . Then, the mean number of polygenic mutations affecting viability (p) and the average homozygous effect of single mutations! S ) could be expressed as follows (BATEMAN 1959;
MTJKAI
1964; MUKAI et aZ. 1972) :where
s
is the mean of S.genes estimated by the C y method (;) can be expressed as:
The relative viability of a
+/+
homozygote carrying mutant viability polyA
ls
(3)
In this formula, H is the degree of dominance of the mutant chromosome hetero zygous with Cy.
If
H S
and S are much smaller than 1, then (3')Thus, the ratio
(R,)
of the minimum mutation rate of viability polygenes, assuming complete dominance of the Cy chromosome ( p ),
to the value estimated allowing for incomplete dominance ($) is:z 1  S S H S , 1
 
so that lv
=
sHS
.
(4)
Assuming that S and H are independent,
 V ( S ) .V (1 H)
+sz
V (
l  H )+
( I d ) 2 V ( S )
V ( S ) (1H)2
2

V(1H) + S . V ( l  H )
= I f
INCOMPLETE DOMINANCE OF
In(2LR)SMI
171where V ( S  S H ) and V ( H ) are variances of ( S  S H ) and
H ,
respectively. 
I n the workOF
MUKAI andYAMAZAKI
(1968) andMUKAI
(1969b), H , S,V ( H )
andV ( S ) have been estimated to be 0.43, 0.1462,0.00504 and 0.00524 after accumu lating spontaneous mutations for 32 generations.
If
these estimates are applied to the C y chromosome, thenRp = 1.08
.
(6)Formula (6) implies that the previously published values for the mutation rates of viability polygenes (0.140.1 7 per second chromosome per generation.
MUKAI
1964;
MUKAI
et al. 1972h) are slightly underestimated.The ratio ( R A ) of the true maximum selection coefficient assuming complete dominance of the Cy chromosome to the estimated value allowing for incomplete dominance is:
V(S)
.
V(SSH)R,=;
S
(SSH)

Under the same assumption as above, i.e., S and H are independent,

V(S) (1R)
v
(S) .V(1
H)
+V(S). (1R)Z+V
(lH)3


V(H)V
(H)8’
( l   H ) +  r +
lH
V(S)(1P)
I
(7)
s’
I n the experiments cited above, was much larger than V ( H ) [1+].
(1
4 7 )
v
(S) Thus, the estimates of S reported previously r0.027 ~ M U K A I 1964); 0.023(MUKAI
et al. (1972b)l are underestimates. Using the estimates of appropriate parameters shown above. R becomes I .63. This estimate is not highly reliable. since V ( H ) andE
were estimated not in the genetic background of the C yheterozygotes, but in the wildtype genetic background.
Synergistic interaction between mutant viability polygenes was found in one experiment (MUKAI 1964). but not in another
(MUKAI
et al. 1972).72
expressed as a function of the number of mutant polygenes (z) as f o l l ~ w s (
MUKAI
1969a) :y = az+bz2 (a$
>
0 ).
(8)
=
f [ a + b ( f + l ) ],
(9)
Then j i = E+=
= E + b [ f 2 + V ( z ) ]
assuming that mutant polygenes are distributed on the chromosomes according to a Poisson distribution. The variance of y , V ( y ) , was as follows:
V ( y ) =
a2V(z)+b2V(z2)+2abCov(x,z2)
= a 2 f + b 2 [ f ( 1+62+4f2)]+2ab(2i?2+x”)=
Z[
( a + b ) 2 + b 2 ( 6 + 4 f ) f + 4 a b f ] . ( 1 0 )The mutation rate ( p ) was estimated as
If we take the difference
(D)
between the numerator and the denominator in the bracket of formula ( l l ) , then,D
= [ a + b ( f + l ) ] 2 [ ( a + b ) 2+2b2 (3 + 2 f ) f + 4 a b f ]This difference is strengthened if the variance of viabilities among individuals carrying the same numbers of mutations is considered, since the denominator in formula ( 1 1 ) becomes larger under the above condition.
From formula (12), it can be concluded that quadratic synergistic interaction causes a n underestimation of the mutation rate. Therefore, the estimate of the mutation rate of viability polygenes (MUKAI 1964) is a n underestimate because of the incomplete dominance of the C y chromosome and synergistic interaction among mutant viability polygenes. For the sake of reference, a and b were esti mated to be 0.009813 and 0.005550, respectively
(MUKAI
1969a).Direction of the bias of the estimate of auerage degree of dominance of viabil ity polygenes: It is assumed that, under ideal conditions, the numbers of Cy flies and wildtype flies in the offspring of a cross using the C y method are 2b
and a, respectively, and that the selection coefficient of a mutant viability poly gene, its degree of dominance in wildtype genetic background and the C y het erozygote genetic background are s, h and
k,
respectively. These relationships are shown in Figure 1 , where A and a stand for the wildtype and mutant alleles, respectively.For the present, it is assumed that the C y chromosome carries wildtype alleles
( A ) at all loci.
Relative viabilities of A A , Aa and aa can be expressed as v ( A A )
=  [ 3 b 2 f 2 + 2 ( 2 b 2 + a b ) f ]
<
0.
_{(12) }a
a(1s) a
b
,
and
is nearly equal to 1 . u ( A a ) =EEx,
and v ( a a ) =ks b(1ks)
b ( 1  y ) L,
INCOMPLETE DOMINANCE OF
In(2LR)SMI
173Par en t s
9 x 3
Offspring
9
a
4
a
fi
A
Wr
B
PA A A
a
a a a aRatio b b a b b(lks) 41hs) b(lks) Mlks) a(1s)
Za(1hs)
”=
Zd(1S)v = a V =
b+b(l ks) b(1 kS)+b(l kS)
Viabi Ii t y
j ndex 2b
FIGURE 1.Relative viabilities of phenotypically Cy and wildtype flies in the case of incom plete dominance of the Cy chromosome with ths Cy method.
A A : 1 7
Aa : ls(h:)
4
I
aa : Is(1k) J
Assuming that the gene frequency of A is p and that of a is q ( p f q 1 ) in a random mating population, the following relationship can be obtained on the basis of ( 1 3 ) .
Genotype A A Aa aa
Frequency PZ 2Pq q2
Reduction of relative viability ( y ) 0 s ( h   ) s ( 1  k )
Reduction of relative viability ( 5 ) 0 s(1k) 2 s ( l  k )
The means of 5 and y (2 and
y,
respectively) can be calculated as follows: k2
Corresponding homozygotes AA+AA A.4+aa aa+m
f = 2 w s ( 1
k)
+2q% ( 14 )
= 2gs( 1k)
= qs (2ph+qk)
.
y =
2 p q s ( h  3 +q2(lk)sThe variances of
x
[V(x)] and y [V(y)] are: V(x) = 2pqs2(lk)2+4q2s2( l  k ) 2  [ 2 q s ( l  k ) ] 2= 2pqs22pqsZk ( 2 4 )
V ( y ) = 2pqs2 (h;)2+q2s2(lk)zy
The covariance of
x
and y, Cov(s,y), is:(A) Equilibrium random mating population
If h and s are positive and appreciably larger than 0, then q2 is much less than one and can be considered to be 0. Thus, V(z), V ( y ) and Cov(x,y) can be approximately expressed as follows:
1
V ( s ) s 2qs2 (Ik)? (a: p
=
1)V ( y ) Y 2qs2 (h$)
Then, the regression coefficient of Y on
X
(&X)becomes as follows:
zgs2 (1
k)
(h%)
zqs2( 1k)
P Y . X
=
(on a chromosome basis)
(14)
It is assumed that the population is in genetic equilibrium as in the case of
D.
melanogaster in Raleigh, North Carolina, described byMUKAI
andYAMA
GUCHI (1974). Then, the equilibrium gene frequency of CL at locus i may be
where pi is the mutation rate from
A
to a and c i is aexpressed as Qi z

factor by which the selection coefficient for the viability of a heterozygote can be related to the selection coefficient for its fitness as a whole (cihisi is the selection coefficient of heterozygote for fitness as a whole).
P i CihiSi
Thus,
zs2
P (1k)
(h$)chs
P Y X =
P
ZSZ ( 1
k)
2 chsz x ( l  k ) 1 (hT) k


4
1
2( 1 k)2
h
w
(assuming that

is not correlated withk
and h.)C
(151)

INCOMPLETE DOMINANCE OF In(2LR)SMI 175
 
I  py.x(12k+k2+u;) (,+ kkZ,o;)
A
h =
( 153)Several typical cases for the relationship between
k
and h should be considered: (1) Case wherek
is not correlated with h and = 0 (U; # 0) .This situation may be close to that of WALLACE andDOBZHANSKY
(1962). Theh
value, which is the arithmetic mean of h values in an equilibrium population (seeMORTON,
CROW and MULLER 1956),
becomes:
1 h
2
Thus, in the case of
PYX
<
,
ppx is an overestimate of
h. For instance, if i p x 0.3 and U:= 0.044 (This is an actual estimate for vi, seeMUKAI
1969b),then
h
= 0.29. The magnitude of bias is very small.(2) Case where
k
is equal to h.In
this case, h becomes as follows:

2 P Y X

h =
1,h+2p,.x (2h)

If
~ Y= X0.3 andz=
0.43 (seeMUKAI
1969b), then,h=
0.40. The relationshipsbetween
pyx
andh
whenK =
0.43 are shown below:P Y X 0.3 0.2 0.1 0.05
h 0.40 0.33 0.23 0.14

U
Generally speaking, in this case,
by.y
underestimates h.constant).
In this case, formula (15) turns out to be:
P Y X
(3) Case where there is a relationship of k = a h between h and
k
( a is a(18)

h =
l+a
{
p y . x (2ah)7i
(I;);
}
*
Assuming = 0.3, the relationships between a and h are calculated as follows:
‘2 1 0.8 0.6 0.5 0.4 0.2 0 0.1
11 0.40 0.38 0.36 0.35 0.34 0.32 0.30 0.29


In so far as a is positive,
(B) Population with newly arisen mutations only. In this case, the gene frequency of a ( q ) can be expressed as (i. = m p , where m is the number of gen erations during which mutations were accumulated. If it is assumed that (i. is much less than one, then formula (14) can be applied to the present case. Thus,
176
formulae (15'1) and (15'2) can be obtained, which correspond to formulae ( 151 ) and ( 1 52), respectively.

I   1  hkkh+k2_{e }
2 1 2k+ki P Y X =
1
2 2
( 1 %)
(d%)
Cov (h, k ) +a;

( l  X ) 2 f o ;
From formula (15'2), the following relationship can be obtained:
(15'1)
( 1 5'2)
Ppx[
(l~)2+u;]+b(l~)+Cov(h,k)~;
1(
193)
2 2
E =
l  k
Several typical cases for the relationship between k and h should be considered:
( 1 ) Case where
k
is not correlated with h andE
= 0 (U: # 0). From (15'2),the following formula can be derived:
1

h = j3P.X+u2(
pyxT)
.
1 " 2
Thus, in the case of
PYX
< ,
Py..y
is a n overestimate of%.
This relation is the same as formula ( 1 6 ) .I n this case, the following relationship can be obtained: (2) Case where
k
is equal to h.If a i = 0.044 and
pyx
= 0.43(MUKAI
1969b), then,2
becomes 0.563. Thisestimate is not realistic at face value, since it is larger than 0.5. This may be caused by using a n inappropriate assumption ( h = k ) o r by using
ai
= 0.044. Anyway, from this result it may be seen that.z is nearly 0.5. If U: is 0, then6
lurns out to be

h = 2pp.x 2p,.,+1 a
( 1 7'a)
( 3 ) Case where there is a relationship of
k
= ah between h and k ( a is a constant).The following formula can be obtained from (15'2) :

h= 4aPpX+2ad (272) _{2a }_{(2apP.x+eU) }24u2 ( a22appx) ' U z h ( 18')
INCOMPLETE DOMINANCE OF
h ( 2 L R ) S M I
177Assuming pyx =
0.43
and U; = 0.044, the relationships between a andh
are calculated as follows:a 1 0.8 0.6 0.5 0.2 0 0.1
h 0.57 0.52 0.49 0.47 0.45 0.43 0.43
From all the above results, it may be said that the bias due to incomplete dominance of the Cy chromosome in estimating
E
is trivial, and the unbiased estimate must be very close to0.5
or additivity.(C) Case where the Cy chromosome carries deleterious genes. There still remains one unsolved problem. The Cy chromosome has deleterious genes (a) a t several loci. The effects of these on the estimate of the average degree of domi nance were examined in a way similar to the preceding analysis, but assuming that the Cy chromosome is completely dominant over its homologous chromo some. This assumption is not restrictive.
Let us assume that there are m loci at which wildtype alleles
( A )
exist andn loci where there are mutant alleles (a) in the Cy chromosome. Then, the following relationship can be obtained:

(m+n)pyXnh E
m  n

In general, m is much greater than n, so that h is nearly equal to
pyx,
In fact, according to the simulation conducted byMUKAI
andMARUYAMA
(1971 ),
only six to 13 mutant polygenes, on the average, can be expected on the second chromosome in an equilibrium population of D. metan_ogaster. Thus, the effect of mutant loci in the Cy chromosome on the estimate ofh
is trivial.The same explanation may be applied to the case where the Cy chromosome is not completely dominant.
(D)
Estimation of arithmetic mean of h for newly arisen mutations from an equilibrium population. We have also estimated the average degree of dominance of newly arisen polygenic mutations by the inverse ofpZp
(the regression co efficient of the sum of the two homozygote viabilities on the viability of the corresponding heterozygote on chromosomal basis). The amount of bias due to incomplete dominance of the Cy chromosome was examined.PEP
is as follows:N
chs c
(Using q =
!!?
and no correlation between
and h.)___
1
(&k+
k
()
k2

k2
hx+
(=)
 2h
I$
[
cov(k,;)
+&]
E++
[
COV(P,+)
+
(<+U;).I
1Kk+;
[
cov(A+)
1+
(u,z+K 2).I
1h h
.
(15”2) 
h
These formulae were derived in the same way as formulae (151) and (152). Several typical cases for the relationship between
k
and h should be considered:(1) Case where
k
is independent ofh
andck = 0 (U: # 0).Prom formula (15#’2) the following formula can be obtained (p’ = I/pxY) :
( 16”)
q+r;/X)
1 (1272)Theref ore,
pz
( 16“a)1
1 ++/h
2
Thus, when is less than 0.5, p’ overestimates
E.
Otherwise, it EnderestimatesA.
For example, if%.
= 0.43, u2, = 0.044(MUKAI
1969b), and h = 0.25, then/?
= 0.44, a very small bias.The formula corresponding to (1 7) is as follows: (2) Case where
k
is equal toh.
h p’=
2(1h)

( 17”)Thus, when
1969b), t h e n b = 0.38, and the amount of bias is only 12%. constant).
The following relationship can be obtained:
<
0.5, p’ underestimatesh.
For example, ifK
= 0.43(MUKAI
(3) Case where there is a relationship of
k
= ah between h andk
( a is a( 18“)
1
When
%
< 
and a>
0, p’ underestimatesz.
For example, the following table 2can be obtained, assuming = 0.43:
a 1 0.8 0.6 0.5 0.4 0.2 0 1
INCOMPLETE DOMIN.4NCE O F In(2LR)SMI 179
MUKAI and YAMAGUCHI (1974) proposed the following method by which the existence of overdominance in an equilibrium natural population can be tested: In addition to the above notations, let us define the following parameters: Cov(x,y)’ and V ( y ) ’ for covariance between
x
and y and variance of y at over dominant loci. Then,where V , ( y ) and CovD(x,y) are for dominant loci. For overdominant loci Cov(s,y)’ = 0 for fitness as a whole in an equilibrium population (MUKAI and YAMAGUCHI 1974). Since viability is positively correlated with fitness as a whole (MUKAI 1977), Cov(x,y)‘ for viability must be close to 0. Thus, /3’ overestimates
h if there are some overdominant loci. In fact, the existence of overdominance at a few loci was suggested in the Raleigh, North Carolina, population by the experimental results of MUKAI and YAMAGUCHI (1974) for the second chromo somes and of WATANABE, YAMAGUCHI and MUKAI (1976) f o r the third chromo somes.
If the C y chromosome does not suppress the deleterious effects of mutations in an homologous chromosome, the above method might become ineffective. Thus, the following calculation was conducted: Under the assumption of incom plete dominance of the C y chromosome, the genetic covariance between
x
and ycan be expressed in general as:
(21)
for an overdominant locus. Hence,
h
In an equilibrium population,
4
=
2h1
(21’)
This quantity is positive, if
k
is negative, namely, that overdominant genes in wildtype genetic background also manifest heterotic effects in the heterozygous condition with the C y chromosome. In such a case, w i l l be underestimated in comparison with the case ofk
= 0, and the detection of overdominance (or some form of balancing selection) becomes difficult. Thus, the detectability of over dominance on the basis of the comparison between p’ and decreases whenk
<
0. However, ifk
is positive, namely, that overdominant genes in wildtype genetic background do not manifest heterotic effects in the C y heterozygotes(cf. MUKAI, CHIGUSA and YOSHIKAWA 1965), the detectability of overdominance increases using the present method.
Effect of the incomplete dominance of the Cy chromosome on the estimated homozygous load: Let us assume that the C y chromosome carries wildtype alleles ( A ) at m loci and mutant alleles (a) at n loci in the chromosome. The former situation is called case 1, and the latter case 2.
Assuming the random combination of the chromosomes, the relationship shown in Table 1 can be obtained with respect to a single locus (a and b stand for the expected number of flies counted).

COV ( ~ , y ) = ( 1
k)
{ 2p@ [ h+q ( 1 2h) ] pqk~’}.
MUKAI
TABLE 1
The frequencies of matings in the C y method for the estimation of viabilities of wildtype chromosomes collected from a random mating population and their viability indices
[A] Heterozygote:
Offspring and, Frequency Viability
Parents segregatlon ran0 of mating index
9 6
A CY
A

A
CY C y / A X C y / A

A
Case 1 b b a
Case 2 b(1ks) b(1ks) a
Case 1 b(1ks) b a(1hs)
Case 2 b(1s) b(1ks) u(1hs)
P2
a

b
a
i/zb( lks+lks)
a(1hs)
% b ( 1
+
1 4 s ) a(1hs)CY A

CY
A U a
C Y / a x C y / A

~1/2 b ( 1 s+ 1  4 s )

Case 1 b b(1ks) a(1hs) a ( 1  h )
%b ( 1 +I 4 s )
Case 2 b(1ks) b(1s) u(1hs) a(1hs)
1/2 b ( 1 ks+ 1s)
CY a

CY
C y / a X C y / a

__a a a
Case 1 b(1ks) b(1ks) a(1s)
Case 2 b (1s) b (1s) a ( 1 s)
a (1s)
a ( 1s) i/e b( lk~+lks)
1/2 b (1 s+l s)
[B] Homozygote:
Offspring and Frequency
Parents segregation ratio of mating
A
CY
CY C Y / A x C y / A

A

A
ACase 1 b b a
Case 2 b(1ks) b(1ks) a
a
~ __
CY CY
C y / a X Cy/a

U a
Case 1 b(1ks) b(1ks) a(1s)
Case 2 b ( 1 s) b (1s) a (1s)
P Viability mdex a b

ag b ( lk~+lks)
a (Is) %b( lk~+lks)
INCOMPLETE DOMINANCE O F In(2LR)SMl 181
(1 ) Case 1 where the C y chromosome has a wildtype allele
( A )
: The average viability in the “random mating” population ( U o ) is approximately as follows:U,  a[p*xl+2pq (lhs+;) +92(1’s+ks)]
b
U
=  [ 2pqhsq2s+kqs]
.
bThe average viability of homozygote population ( 5,) is approximately as follows:
(21
1
n
=  [ 1 qs+qks]
.
bhomozygous load
(e,)
becomes as follows:Thus, from formulae ( 2 0 ) and ( 2 1 ) , the expected value of the estimated
= In ( 1 qs+qks) +In ( 1 2pqhsq2s+ksq) q ~  q k ~  2 p q h ~  q ~ ~ + k s q
= qs (2pqhs+q2s)
=LILR
.
(22)Namely, the effect of incomplete dominance disappears.
( 2 ) Case 2 where the C y chromosome has a mutant allele ( a ) : The average viability in the “random mating” population ( U o ) is approximately as follows:
(23
1
ab
=  [ 1 2pghs+pqs+pks]
.
The average viability of homozygote population ( V I ) in this case is approxi mately as follows:
(24)
 U
v r ~  ( l + p l c s )
.
bThe expected value of estimated homozygous load
(k,)
canbe
obtained from formulae ( 2 3 ) and ( 2 4 ) as follows:
VI
U 0
2,
= 1n == In ( 1 S p k s ) S l n (1 2pqhsfpqsfpks)
= pq~2pqhs
pk~2pqhsSpqs+pks
U qs2qhs
zz L,Ln
.
( 2 5 )(*: p cz1 )
182
previous publications (
MUKAI
andYAMAGUCHI
1974;WATANABE,
YAMAGUCHI
and
MUKAI
1976).The same conclusion is obtained f o r the case where mutant genes show over dominance.
DISCUSSION A N D CONCLUSION
In the present paper, the effects of incomplete dominance of the Cy chromo some on the estimates of various genetic parameters on the basis of the Cy method
for the estimation of relative viability of Drosophila melanogaster were dis cussed. In general, the effects are small and do not invalidate previous conclusions about these genetic parameters; in fact, if anything, the conclusions are strengthened. The mutation rate of viability polygenes appears to have been slightly Underestimated previously, as was the average effect of these mutant polygenes. The analysis presented here also shows that the mutation rate may have been underestimated because there is synergistic interaction among mutant polygenes with respect to viability. Thus, the rate published earlier (0.14 per second chromosome per generation,
MUKAI
1964) is likely to be a bit on the low side.The average degree of dominance of newly arisen mutant viability polygenes is most probably underestimated, but if the mean value of
k
is 0 with some variance,pyx
slightly overestimates the average degree of dominance. However, this situation is unlikely (COCKERHAM andMTJKAI
1978). The estimate of the average degree of dominance of newly arisen viability polygenes has been re ported to be 0.40.43(MUKAI
andYAMAZAKI
1964<, 1968; MUKAI 1969b). The unbiased estimate may be approximately 0.5 using formula (18’). An approxi mate additivity holds in this case.In equilibrium populations, the average degree of dominance is smaller than that for newly arisen mutations; the expected value is the harmonic mean of the h values of newly arisen mutations
(MORTON,
CROW andMULLER
1956).We have estimated this to be approximately 0.20.3
(MUKAI
et al. 1972;MUKAI
andYAMAGUCHI
19741;WATANABE,
YAMAGUCHI
andMUKAI
1976).According to the present investigation, these values are underestimates. The amount of underestimation can be calculated theoretically. If h and
k
are the same, the average degree of dominance in equilibrium population turns out to be 0.330.40.A method was devised by which the existence of some forms of balancing selection including overdominance can be tested (MUKAI and YAMAGUCHI 1974).
In this method, the regression coefficients of the sum of the homozygote viabili ties on the corresponding heterozygote viability
(pZy)
is employed.If
there are no overdominant loci, the inverse ofpXy
or p’ underestimates the average degreeDf dominance of newly arisen mutant viability polygenes
( h )
whenk
>
0. Ifthere are some overdominant loci in addition to incompletely dominant loci, and if both
k
and h are negative, then the increment of p’ becomes small in compari son with the case ofk
= 0, and the test for detecting overdominance becomes weak. However, ifk
is positive in contrast to h being negative, the increment ofINCOMPLETE DOMINANCE OF
In(2LR)SMI
183The estimate of homozygous load relative to the average viability of hetero zygotes obtained by the method of GREENBERG and CROW (1960) is not biased even if the C y chromosome is not completely dominant. This is intuitively understandable, because both the estimates of homozygote and heterozygote viabilities are biased due to the incomplete dominance of the Cy chromosome and the bias in the estimate of the homozygous load relative to the average heterozygote viability disappears by taking the difference in their logarithms.
I n conclusion, it can be said that the incomplete dominance of the Cy chromo some does not seriously change our estimates of the genetic parameters discussed above. Of course, the entire problem could be circumvented by designing the experiments differently; it would be possible to use the Cy/Pm method
(WALLACE
1956), in which a fourclass segregation occurs, i.e., Cy/Pm : Cy/+j : Pm/+i :
+i/+j, as the offspring of Cy/+i females X Pm/+j males. This method is better than the Cy method in that the numbers of
Fi/Fj
flies can be compared to the Cy/Pm flies, which are common to all cultures. However, at present, this method is not always better than the Cy method since the Pm chromosome is somewhat peculiar. It appears to show meiotic drive (COCKERHAM andMUKAI
1978), and sometimes its phenotype is close to the wild type and the chance of misclassifi cation is not small. In addition, the classification into the four classes is much more tedious than the classification into the two classes using the Cy method. Thus, before marker chromosomes (with multiple inversions) better than the Pm chromosome appear, it might be better to use the Cy method in the estima tion of relative viability.I would like to thank H. TACHIDA and S. KUSAKABE, who checked some formulae in the present studies, and one of the reviewers for his valuable suggestions.
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