Copyright 0 1986 by the Genetics Society of America
THE SELECTION LIMIT DUE TO THE CONFLICT
BETWEEN TRUNCATION AND STABILIZING SELECTION
WITH MUTATION
ZHAO-BANG ZENG
AND
WILLIAM G. HILL
Institute of Animal Genetics, University of Edinburgh, Edinburgh EH9 3JN, Scotland
Manuscript received February 1 1, 1986
Revised copy accepted August 2 5 , 1986
ABSTRACT
Long-term selection response could slow down from a decline in genetic variance or in selection differential or both. A model of conflict between truncation
and stabilizing selection in infinite population size is analysed in terms of the
reduction in selection differential. Under the assumption of a normal phenotypic
distribution, the limit to selection is found to be a function of K , the intensity
of truncation selection, w2, a measure of the intensity of stabilizing selection, and
U*, the phenotypic variance of the character. The maintenance of genetic variation at this limit is also analyzed in terms of mutation-selection balance by the
use of the "House-of-cards" approximation. It is found that truncation selection
can substantially reduce the equilibrium genetic variance below that when only
stabilizing selection is acting, and the proportional reduction in variance is greatest when the selection is very weak. When truncation selection is strong, any
further increase in the strength of selection has little further influence on the
variance. It appears that this mutation-selection balance is insufficient to account
for the high levels of genetic variation observed in many long-term selection
experiments.
I
N artificial selection experiments and breeding programs, there may be a
conflict between artificial and natural selection. Usually, the aim of artificial
selection is to improve some particular traits; thus, it is generally directed
toward extreme phenotypic values. It is observed that natural selection often
favors intermediate expression of metric traits unless these traits are very
closely associated with fitness (e.g., LINNEY,BARNESand KEARSEY 1971). So
plateaux obtained in artificial selection experiments could result from the opposing forces of directional and natural selection, rather than from a loss of
additive genetic variance, as indicated by some long-term selection experiments
(e.g., LERNERand DEMPSTER1951; CLAYTONand ROBERTSON1957; LATTER
1966; ROBERTS1966; WILSONet al. 1971; Yoo, NICHOLAS
and RATHIE1980).
Two extreme models have been proposed for the action of natural selection
on quantitative characters. One is stabilizing (or optimum) selection, another
is overdominant selection. Stabilizing selection has received considerable attention over the last several decades (e.g., FISHER 1930; WRIGHT1935; HALDANE
1954; ROBERTSON
1956; KOJIMA 1959; LATTER1960, 1970; LEWONTIN1964;
LANDE 1975). In this selection model it is assumed that the fitness of an
Genetics 114: 1313-1328 December. 1986.
1314
Z.-B.
ZENG AND W. G. HILL
individual is solely a function of its phenotypic deviation from an intermediate
optimum. In overdominant selection, on the other hand, it is assumed that
heterozygotes at the loci affecting the quantitative character have higher fitness, and this model has been thought by many people to be important as a
possible explanation for the maintenance of genetic variability in both artificial
and natural populations (e.g., LERNER1950, 1954; ROBERTSON
1956; BULMER
1973; GILL~ESPIE
1984).
JAMES (1962) considered the case where the conflict is between truncation
and stabilizing selection. With the assumption that heritability is not greatly
altered during the course of selection, he was able to develop an approximate
expression for the selection limit that is expressed in terms of K , the intensity
of truncation selection, w2, a measure of the intensity of stabilizing selection,
and a
'
, the phenotypic variance of the character. It is not clear, however,
whether heritability is zero or not at this selection limit. In contrast, ROBERTSON ( 1 956), VERGHESE
(1974) and NICHOLAS
and ROBERTSON
(1980) investigated another model, called the homeostatic model, for which the conflict is
between directional and overdominant selection. The impact of such a model
on selection limits has been explored in depth. However, in all these studies,
mutation as a source of introducing fresh variability has been ignored.
The dynamics and maintenance of genetic variability of quantitative characters under stabilizing selection and mutation have been studied intensively
by many authors (LATTER 1960; KIMURA 1965; BULMER1972, 1980; LANDE
1975; FLEMING1979; TURELLI
1984). They examined the balance between
mutation and stabilizing selection to see whether this balance could account
for the high levels of heritable variation usually observed for many quantitative
characters in natural populations. LANDE(1 975), in particular, has forcefully
argued that high heritabilities could be maintained by this mutation-selection
balance even with strong stabilizing selection. Recently, TURELLI
(1984) has
critically reviewed this argument and numerically given the domains of applicability of the various approximations produced by him and other authors.
Based on a Lerch's zeta function analysis and numerical work, TURELLI
argued
that the approximation he used would give a better estimate of the equilibrium
genetic variance when the mutation rate per locus is of the order of
or
less. Elsewhere we have investigated the consequences of recurrent mutation
on selection response in finite populations, but in the absence of any opposing
effects of natural selection (HILL 1982).
In this paper, we examine the state of the population at the selection limit
due to the conflict between truncation and stabilizing selection, i.e., we investigate the selection differential, heritability and the population mean at this
limit. We assume that selection is on a single trait, gene effects are additive
and the population is infinitely large. First, in the section below, we present
an analysis on the phenotypic change due to selection; and then, in the following section, we investigate the genetic structure of the population at the limit.
ANALYSIS OF THE PHENOTYPIC CHANGE
Consider a metric character having phenotypic value x with the following
probability density function among juveniles (before the operation of selection)
in generation t
f ( x ) = TU^)-"^ expi-(x - u,)'/(2a2j),
(1)
1315
LIMIT T O SELECTION
where U' is the phenotypic variance which is assumed to be independent of
the mean ut. The fitness of individuals under stabilizing selection with phenotypic value x is assumed to decrease with deviation from the optimum value
according to the relation
wl(x) = exp(-x'/(2~')),
(2)
where the optimum value of x is taken to be zero, and U' is a measure of
intensity of stabilizing selection, being less intense the larger U' is in relation
to u2. Stabilizing selection may act on individuals through the whole life cycle,
by differential viability or reproductivity, or both. But here, for convenience,
it will be assumed that the stabilizing selection occurs before truncation selection. Then it is readily shown that, among the survivors after stabilizing selection, the phenotypic distribution becomes
f '(4= f(x)wl(x)/Sf(.)~l(x)dx
= ( 2 ~ c a ~ ) -exp(-(x
'/~
(3)
- cu,)'/(2cu2)],
+
where c = w'/(u2
U * ) is called the coefficient of centripetal selection by
LATTER(1970). T h e distribution (3) is still normal with mean U ; = cut and
variance U'' = cu2.
Truncation selection induces the fitness function
(4)
where T is the truncation point in absolute value. Using (3) and (4), we can
then obtain the proportion of individuals surviving the two kinds of selection
(the mean fitness of the population):
= P[u'/(u'
+ u ' ) ] ~ /exp('
u:/[2(u2
+
U')]).
where P = J: f ' ( x ) d x is the proportion of individuals surviving stabilizing
selection that are then selected by truncation. T h e change in the population
mean by truncation after stabilizing selection is
=
KWU/(U'
+
W2)1/2,
where K is the standardized selection differential of truncation selection, corresponding to P. Then the total selection differential is given by
st
= up -
U*
= ( u p - U:)
=
KU(T/(U2
+
+ (U:
- ut)
- UtU2/(U2
(7)
+
U').
1316
Z.-B. ZENG AND W. G. HILL
This relation would hold for every generation if the phenotypic distribution
remained normal before selection and U' were constant. When the second
assumption is violated, U' has to be replaced by U? in (7).
The change in the average phenotypic value in response to selection must
equal the product of heritability and selection differential, i.e.,
where hf is the heritability of the character at generation t. Thus, at a selection
limit, where Au = 0, either s or h' should be zero. If a limit is attained due
to the attenuation of selection differential, the total predicted selection advance
can be obtained by setting s = 0 in (7) and is
=
Um
+
KW(U'
(94
W2)1'2/U
or
Um
=
KW(U'
+
W2)lD/U
- Uo
if uo # 0. This relation was first obtained by
derivation.
Furthermore, from (7) and (8), we can get
JAMES
h'u'
-
= (1
%+I
=)
(9b)
(1962) by a different
h'xwu
ut
+
+ w')"'
(U'
T h e first term on the right side of (10) shows the effect of stabilizing selection
on the phenotypic change which is the same as shown in Equation 14 of LANDE
(1975). The second is the effect of truncation selection that could be constant
in standard units, if the indicated assumptions hold; that is, the phenotypic
distribution is always normal before selection, h' and U' do not change very
much during the course of experiment, and the same proportion of individuals
is selected by truncation in every generation. If uo = 0, the selection response
in the first generation of selection may be expressed as
u1 = h2Kuw/(u2
+ w')"',
(1 1)
and the ratio of the total response over the response in the first generation
becomes
um/ul =
Finally, solving (10) gives
ut = uo exp
{-
h'u't
___
U'
(U'
]+ [
+ w2
+ w')/(u'h').
1 -exp
(12)
{-~ )]
h'u't
U'
+
KU(U'
U'
+ w2)"*
U
.
(13)
Then we can get the "half-life" of the selection process, the number of generations taken to go halfway to the limit (ROBERTSON 1960). This is
to.5 = ln2(u2
+ w2)/(u'h2) generations.
(14)
There are several notable features to this analysis. First, as indicated in (13),
LIMIT TO SELECTION
1317
the response curve is exponential, which is similar to that predicted by ROBERTSON'S (1960) theory for finite populations with directional selection alone.
In this analysis the response rate is h2a2/(a2 U'), a function of h2 and 02/a2.
Therefore, the half-life is expected to be longer if selection is on a character
with a lower heritability and/or less intense stabilizing selection (i.e., higher
value of w2/a2) (14). Second, although the response rate is a function of heritability, the selection limit predicted is independent of heritability. Third, in
a large population, the total response is maximized by having the smallest
possible P (9), but in a small population, P should be 0.5 to obtain the largest
total response (ROBERTSON
1960).
+
ANALYSIS OF THE EQUILIBRIUM: DISTRIBUTION OF
ALLELIC EFFECTS
In this section, we shift our analysis from the phenotypic level to the genotypic level, and we focus attention on the distribution of allelic effects at the
selection limit. The method of analysis is similar to that of TURELLI
(1984),
but with an extension to include truncation selection.
The model: Consider a randomly mating diploid population of infinite size
and a quantitative character that is affected by n additive loci and an independent environmental effect. At each locus it is assumed that there is potentially an infinite number of allelic states, and the phenotypic effects of these
alleles are continuously distributed.
Let x denote the phenotype of this quantitative character with
x=y+e
(15)
and
where z: (z!) is the allelic effect of the maternally (paternally) inherited gene
at the ith locus in an individual, and e is the environmental effect, assumed to
be normally distributed with mean zero and variance U:.
The object is to find an approximation for the equilibrium distribution of zi
at the limit to selection. First some other assumptions need to be made. Usually, it is assumed that the phenotypic effects of mutant alleles from a given
allele state are normally distributed around the phenotypic effect of the original allele (e.g., KIMURA1965). Then, ifft(zi) denotes the distribution of allelic
effects among gametes in generation t and if mutation is assumed to occur
will be
during gametogenesis which follows selection,
1318
Z.-B. ZENG AND W. G . HILL
is the density function of allelic effects after selection, where w(zJ is the selection function on zi, and
g(zJ = (ZT")-"~
exp(-z?/(Zm?)]
(19)
is the density function of mutant effects, where m? is the variance of mutant
effects for the ith haploid locus. Interpreting (17) in words: at the beginning
of generation t
1, an allele zi has probability 1 - pi of coming from allele zi
in the previous generation, having survived selection and without mutation,
and probability pi of coming from another allele in the previous generation,
having survived selection and mutated to zi.
Generally speaking, it is difficult to analysis (17) directly without some simplification. Here we simplify (17) in the following way. By expanding g(zi - vi)
in Taylor's series about (zi - U : ) , where U{ = J vJ:(vi)dv;, we have
+
+
( Y 2 ) (U,
- U:)2g'2'(z1
-
U:)
+ . . .Id
Ut
- Pz[g(Z, - U : )
+ (%) v:g'2'
where V: = J
(U,
(z,- U : )
+ . .I,
*
- u tr )2f t (r v Z ) d v , If
. g(zJ is expressed by (19),
g(*)(zz -
U:)
- u:)'/m;
= (l/m,2)g(z, - u:)[(z,
- 11.
Then
,-
An important observation of TURELLI
(1984) is that, under the reasonable
assumption that pz 5
m,2 >> v:,
(21)
in which m,' denotes the variance of effects associated with mutation and V:
denotes the equilibrium allelic variance after selection. Then the second term
and the terms in higher order in the bracket in (20) can be ignored without
serious error, and (17) can be approximated by
fi+l(ZZ)
= (1
- Pz)fi(z*) + P*g(z*- U:)
(22)
under the condition of (21). This is the "House-of-cards" approximation used
by TURELLI
(1984) and originally introduced by KINGMAN (1978). In this approximation the effects of new mutants are assumed to be distributed around
the population mean, and to be essentially independent of their premutation
state. The density function of the equilibrium distribution of z,in this approximation is given by
fm(zz)
= Pzg(zz - ~ t > N 1 4w(~,)],
where 4 is a constant such that
(23)
1319
LIMIT TO SELECTION
S
fm(%i)dZi =
1,
(24)
and
Thus, if w(zJ and g(zi - ui)are known, f&) can be approximated by (23)
providing pi C:
Allelic effect distribution at the limit to selection: In this section, (23) is
used to find the equilibrium distribution of allelic effects zi under stabilizing
and truncation selection. First, consider w(zi), the selection function on zi. Let
ai = zi - ui be the excess of the allele zi over the mean of alleles at the ith
locus. The fitness of this allele, relative to the mean ui,is usually given by
wheref(x) is the density function of phenotype and w(x) is the selection function on phenotype. By expandingf(x - a,) in a Taylor series about x, we then
have that, to order a',
W ( Z ~ )=
1
+ Cui + (?'z)Du?,
(27)
where
D = sf(2)(x)w(x)dJsf(x)w(~)dx.
Iff(x) is the normal density function with mean
U
and variance u2, then
C = Au/u2
D = (Aa2
+ Au2)/u4,
where Au and Aa2 are the changes in the mean and variance as a result of
selection (BULMER1980).
Since there are two kinds of selection, Au and Ao2 are determined by two
components. We have already found that in (7)
Au = K
+ w ' ) ~ ' ~- uu2/(u2+
W U / ( ~ ~
(28)
U').
From (3), we have that
Aa2 (due to stabilizing) = a2(1 - w2/(u2
+ w2)) =
-U'/(.'
and it is well known that
Au2 (due to truncation) =
-K(K
- Z)U~W~+
/(U
U'),
~
+ w2),
1320
Z.-B. ZENG AND W. G . HILL
where K is the intensity of truncation selection and 2 is the standard deviate
of truncation point 7. So the total change in the variance due to selection is
+
Aa' = -(a4
+
- Z)a2w2)/(o'
K(K
(29)
U').
With ( 2 8 ) and (29), ( 2 7 ) becomes
w(z,) = 1
+
+ u ~ )-~uu2/ ~
+ - Z)w2
ai a,.
22(2+
+ U')
KUU(U'
U'
K(K
02)
62(U'
(30)
In (30) Au2 is not included in the term of a:. As shown later, this does not
influence the results. When a, = z,- U , is small in magnitude, w(zJ can also be
approxi mated by
w(z,) = exp
KUU(C2
E
U2)1'2
a'(u'
= exp(- (z,
where
+
+
- UO'
U2)
a,
-
+
a
'
K(K
2a2(a2
- B)*/(2A)Je,
- Z)W'
+
02)
(31)
is a constant,
-t
(r'(a2
A =
+
U'
K(K
U')
- 2)U'
and
KUCJ(U'
B=Ui+
'
l
c
+
+
U2)1'2
K(K
- U6'
- z)U2
'
Now inserting (31) into (23) and letting g(zi - U,) be defined by (19), we
then have the following approximation for the density function of the distribution of allelic effects at the limit to selection
It can be shown that in (32) U , = B, i.e.,
U
=K
U ( 2
+
U2)1'2/U
(33)
(see APPENDIX for proof).
The result (33) has two implications: First, since uiin (32) could, in theory,
take any value, but U = KW(C'
u ' ) ' / ~ / u ,this shows that the mean genotype,
as well as the mean phenotype (since U , = ur = U by assumption), is a fixed
value at the limit to selection, but the mean effect of the alleles at a particular
locus is not fixed. Their values at the selection limit would then largely depend
on the initial conditions, historical influences and chance events at this particular locus. As a consequence, different lines or replicates in an experiment
could be quite different in genetic constitution, even though they might show
similar phenotypic expressions (see also LANDE 1975). Second, in contrast to
the traditional argument that the maximum 'response to artificial selection is a
function of the number of loci, i.e., as the number of loci increases, the maximum response increases (e.g., ROBERTSON1960), this model predicts that the
maximum response on the phenotype is independent of the number of genes
+
1321
LIMIT TO SELECTION
responsible for the character. T h e increase in the number of loci is accompanied by a decrease in the effects of individual genes. Equation (33) for the
maximum response at the limit to selection is identical to (9).
Now let ai = zi - ui.With U = KO(CT‘
w2)1’2/u, (32) reduces to
+
pi
fm(ai) = (27rmf)’”
exp(-a?/(Zm?)}
[ l - [ exp{-a?/(2A)}] ’
(34)
equilibrium distribution of allelic effects under
which is the same as TURELLI’S
K(K - Z)w‘]. By using
stabilizing selection alone. Here A = u2(u2 w’)/[u‘
Lerch’s zeta function, TURELLI
(1984) has been able to show that
+
+
5 = exp{-p?7rA/m?]
Vi = E(a?) = 2pZA
r2
(35)
= E(a4)/[3{E(~?)}’] m?/(6pJ)
as pi + 0 for (34), where Vi is the equilibrium genetic variance due to locus i
(haploid) and r2 is the coefficient of kurtosis for this distribution. As he pointed
out, the approximations rest on the condition that
pi << m?/A
<< 1,
(36)
which will be justified numerically in the next section. In addition, it can easily
be proved that this distribution is symmetric.
Considering all relevant loci, the total genetic variance can be approximated
by
n
0,’
4piA,
=
(374
i= I
if a global linkage equilibrium is assumed. In particular, if the mutation rate
is equal for all loci,
U,’
= 4npA =
4npu2(2
U*
+
K(K
+
w2)
- z)w2.
A check on the approximations: The results of (35) were obtained by TUR(1984) from (34) as approximations, as pi + 0, i.e., p j << m?/A << 1. This
condition is internally consistent with m? >> Vl (21) which leads (17) to (34)
(TURELLI
1984). By a simulation of (17), TURELLI
provided a numerical test
of the results of (35), which clarified the conditions for the house-of-cards
approximation. In this section, we provide another numerical test of (35) directly from the moment calculation of (32) with truncation selection, which
relies on TURELLI’S
numerical calculation to support (32).
The numerical analysis was carried out to minimize five functions, each with
five real variables ([, U , V, rl, 7-2) by the Newton-Raphson method (see GILL,
MURRAYand WRIGHT1981), where rl is the coefficient of skewness. These
functions were the cumulative distribution function and the first four moment
functions of (32). T h e process of minimization was as follows: First, an initial
guess for the real variables [, U , V , rl and r2 was made, and then the increments
ELLI
1322
Z.-B. ZENG AND W. G. HILL
TABLE 1
Effects of varying the mutation rate ( p ) on the analytically predicted (Pre.)
and numerically determined (Obs.) equilibrium genetic distribution for P = 0.5,
V, = 10 and tn2 = 0.05, given pi = 0.1
€
P
IO-’
U
V
rvi
n
5.61
6.70
Pre.
Obs.
0.9999067
0.9999164
7.57
7.57
2.97 x 10-3
2.75 X lo-’
2.00 x 10-2
1.28 X lo-*
Pre.
Obs.
0.999999067
0.999999077
7.57
7.57
2.97
2.97
2.00 X lo-’
1.89 X lo-’
5.61
5.76
he.
Obs.
0.99999999067
0.99999999068
7.57
7.57
2.97 x 10-5
2.99 x 10-5
2.00 x 1 0 - ~
1.99 x
5.61 x 1 0 2
5.67 x 102
Pre.
Obs.
0.99999999991
0.9999999999 1
7.57
7.57
2.97 x 10-6
2.99 x 10-6
2.00 x
1.99 x 10-5
5.61 x 103
5.67 x 109
X
X
X
X
10
10
+
[ is a parameter of (32); U = K U ( ~ U‘)”/. is the mean genotype; V is the variance of the
distribution; [VI is the variance without truncation selection, i.e., when P = 1; and r2 = E(r, - u,)~/
{ 3 [ E ( z ,- u , ) ~ is
] ~the
) coefficient of kurtosis of the distribution.
of these variables toward the solutions were calculated. This calculation was
iterated until all increments were under the tolerance error, and the final
values of the variables were regarded as the solutions. In this study, the initial
guess was supplied by (33), (35) and r l = 0 with U * = 0.1, and the tolerance
error was IF4.
In the computations, for convenience, all measurements except p were scaled
so that U = 1. T h e results of the computations are shown in Tables 1-4, which
illustrate the effects on the analytically predicted and numerically observed
equilibrium genetic distribution of varying p, m2,V, and P separately around
p =
m2 = 0.05, V, = 10 and P = 0.5, with ui = 0.1 where V, = u2 U‘.
These values of parameters are chosen to be consistent with TURELLI’S
analysis,
so VJu2 = 10 is equivalent to V5/u: = 20 in TURELLI
(1984). The equilibrium
genetic variance [VI without truncation selection (P = 1) is also presented in
the tables for comparison. Since the distribution is symmetric, rl was found
always to be zero and thus was excluded from the tables.
T h e results of the variance [Ufor P = 1 are very consistent with those of
TURELLI.As p and V, (V, = A in this case) decrease and m 2 increases, the
approximate values of the variance became close to the observed values, and
reasonable agreement between predicted and observed variances is achieved
whenever 50p 5 m2/VSapproximately (Tables 1-3). When P = 0.5, the value
of A (A = u2V5/[u2 K ( K - Z)w2]) is severely reduced. Although V, ranges from
2 to 50 in Table 3, the value of A ranges only from 1.22 to 1.55. So the
predicted variances displayed in Tables 1-3 are a little closer to the observed
variances when P = 0.5 than when P = 1. Table 4 shows the effect of truncation selection on reducing the equilibrium genetic variance (see also Figure
1 below). It is notable how severely the genetic variance is reduced by truncation selection. In all four tables, the predicted means (KW(U’
~ ~ ) ‘ / ~ are
/a)
matched by the observed means.
+
+
+
1323
LIMIT TO SELECTION
TABLE 2
Effects of varying the variance of the effect for the mutants (m’)for 1.1
V, = 10, given 1.1~ = 0.1
= lo4, P = 0.5
and
~
E
m2
V
U
[VI
r2
0.001
Pre.
Obs.
0.99995333
0.99997010
7.57
7.57
2.97 X
2.08 x
2.00 X IO-’
5.83 x
1.12
2.14
0.005
Pre.
Obs.
0.99999067
0.99999164
7.57
7.57
2.97 x 10-4
2.73 X
2.00 x 10-3
1.28 X lo-’
5.61
6.65
0.01
Pre.
Obs.
0.99999533
0.99999559
7.57
7.57
2.97 x 1 0 - ~
2.85 X
2.00 x 10-3
1.55 X lo-’
1.12 x 10
1.23 X 10
0.05
Pre.
Obs.
0.99999907
0.99999908
7.57
7.57
2.97 x
2.97 X
2.00 x 10-3
1.89 X IO-’
5.61 x I O
5.76 X 10
0.1
Pre.
Obs.
0.999999533
0.999999536
7.57
7.57
2.97 x 10-4
3.01 x 1 0 - ~
2.00 x 1 0 - ~
1.94 x io4
1.12 x 1 0 2
1.15 x 102
0.5
Pre.
Obs.
0.9999999067
0.9999999068
7.57
7.57
2.97
3.24
2.00 x IO-’
2.01 X
5.61 x IO2
6.07 X lo2
X
X
IO-*
TABLE 3
Effects of varying the intensity of stabilizing selection (V,) for 1.1
and m 2 = 0.05, given 1.18 = 0.1
2
=
P
= 0.5
Pre.
Obs.
0.999999232
0.999999240
1.13
1.13
2.44
2.45
5
Pre.
Obs.
0.9999991 14
0,999999 124
3.57
3.57
2.82 x 10-4
2.82 X
1.00 x 10-3
9.72 X
5.91 x I O
6.07 x 10
10
Pre.
Obs.
0.99999907
0.99999908
7.57
7.57
2.97 x 10-4
2.97 X
2.00 x I O +
1.89 X IO-’
5.61 x I O
5.76 X 10
15
Pre.
Obs.
0.99999905
0.99999906
11.56
11.56
3.03 x
3.02 X
3.00 x 1 0 - ~
2.75 X lo-’
5.51 x I O
5.66 X 10
20
Pre.
Obs.
0.99999904
0.99999905
15.56
15.56
3.05
3.04
4.00
3.57
5.46
5.61
Pre.
Obs.
0.99999903
0.99999904
39.50
39.50
3.11 x 1 0 - 4
3.10 x IO-‘
50
4.00
3.98
X
X
X
X
loT4
6.82
6.99
X
X
X
IO-’
X
LOO x 10-2
7.73 x IO-’
X
X
10
10
X 10
X 10
5.37 x 10
5.52 x I O
DISCUSSION
Apart from the decline in heritability that causes plateaux in some experiments (e.g., BROWNand BELL1961; ROBERTS 1966), a reduction in selection
differential seems to be responsible for some other selection limits, particularly
in long-term selection experiments (see FALCONER 1981, chapter 12, for review). It has been observed that a decline in mean was rapid after relaxation
of selection, and that fertility and reproductivity were much poorer in many
lines-even
to the extent of causing extinction of selection lines-in
many
long-term selection experiments (e.g., LERNERand DEMPSTER
1951; CLAYTON
and ROBERTSON1957; LATTER1966; ROBERTS1966; WILSONet al. 1971; YOO,
NICHOLAS
and RATHIE 1980). This clearly indicated that additive genetic var-
1324
Z.-B. ZENG AND W. C . HILL
TABLE 4
Effects of varying the proportion of truncation selection (P) for
and m 2 = 0.05, given pi = 0.1
t
P
U
p
=
V, = 10
V
72
8.33
9.39
1 .o
Pre.
Obs.
0.9999937
0.9999942
0.00
0.00
2.00 x
1.89 x l o +
0.9
Pre.
Obs.
0.99999825
0.99999829
1.85
1.85
5.57 x 10-4
5.50 x 10-4
2.99
3.11
0.7
Pre.
Obs.
0.99999887
0,99999889
4.71
4.71
3.59 x 1 0 - ~
3.58 x 10-4
4.64 X 10
4.78 X 10
0.5
Pre.
Obs.
0.99999907
0,99999908
7.57
7.57
2.97 x 1 0 - ~
2.97 x 10-4
5.61
5.76
X
0.3
Pre.
Obs.
0.999999 176
0.999999184
1 1 .oo
11.00
2.62 x 1 0 - ~
2.63 x 10-4
6.35
6.52
X 10
X 10
0.1
Pre.
Obs.
0.999999258
0.999999265
16.65
16.65
2.36 x 1 0 - 4
2.37 x 1 0 - 4
7.06
7.24
X
X
X
X
X
10
10
10
10
10
10
iance was not exhausted in these populations at apparent limits and also suggested that natural selection opposing artificial selection might cause the reduction in the effective selection differential.
In this paper, a conflict between natural and artificial selection is examined.
It is shown that stabilizing and truncation selection can take a large population
to a selection limit where the population mean is determined by the intensities
of truncation and stabilizing selection and the phenotypic variance of the character (9). There is still genetic variance at this limit, and its amount can be
approximated by (37) in terms of the mutation-selection balance. However, it
is necessary to examine whether this balance can account for high levels of
genetic variation observed in some long-term selection experiments.
T h e balance between mutation and stabilizing selection has been examined
in detail by many authors. Both LANDE'S(1975) and TURELLI'S
(1984) mathematical calculations and parameter estimations from relevant data suggested
that high levels of variation could be maintained by mutation in the face of
stabilizing selection. However, when additional truncation selection is taken
into account in the model, this balance may or may not be sufficient to explain
the maintenance of genetic variability. Figure 1 shows the ratio of genetic
variances maintained by the balance with and without truncation selection for
different values of u2/a2 and P. It can be seen that truncation selection is a
crucial factor in quantifying the equilibrium genetic variance. Even a small
amount of selection can severely reduce the variance. But when selection is
strong, any further increase in the strength of selection has little further influence on the variance. Relevant data for estimating n p and w'/u' have been
reviewed by TURELLI
(1984), who considered that n p =: 0.01 and w2/o' =: 510 might be typical estimates for many quantitative characters. With those
values, the house-of-cards approximation predicts heritabilities (h2 = 4np(a2 +
w')/[u'
+ K ( K - Z)w']) ranging from 0.240-0.440 without truncation selection,
but 0.098-0.1 13 for P = 0.9 and 0.057-0.060 for P = 0.5. This seems to
LIMIT TO SELECTION
1325
1.
v)
al
;0.
ld
.r(
c
g
0.
CI
0
0
+-'
.r(
0.
(d
p:
FIGURE1.-The ratio of genetic variances maintained by the balance with and without truncation selection is plotted against the proportion of truncation selection (P%)for different values
of w'/u'. The ratio is equal to [ l K ( K - Z)w2/aZ]-'.
+
suggest that, in the presence of opposing truncation selection, the heritability
can only be maintained at a low level with np = 0.01. In this case the change
in the intensity of stabilizing selection does not make a significant difference
to the variance. However, if n p is 0.02, rather than 0.01, the heritability
at the selection limit would be about 0.303-0.088 for P = 0.95-0.01 and
u2/a2 = 10. Clearly, the argument relies on the estimation of the relevant
parameters, particularly the total mutation rate of the loci controlling the
character concerned.
In view of some experimental evidence that indicates that mutations in the
broad sense occurring during the experiments from whatever sources may have
contributed to the long-term responses (FRANKHAM1980) and that plateaux
have not been achieved in some long term selection experiments, for example
the Illinois corn experiments (DUDLEY1977), the present analysis is not a
complete description of the process. In due course it will be necessary to
synthesize theories in predicting responses from mutations in the absence of
natural selection (HILL 1982), the present theory incorporating stabilizing selection at the phenotypic level, and theories of natural selection acting at the
genetic level through heterozygote superiority. Nevertheless, it is hoped that
the calculation presented here will be of value in interpreting some long-term
selection experiments which reached a selection limit.
We thank M. TURELLI,R. LANDEand M. LYNCHfor their helpful comments on an earlier
for his reading of the APPENDIX.
draft, and R. THOMPSON
LITERATURE CITED
BROWN,
W. P. and A. E. BELL,1961 Genetic analysis of a "plateaued" population of Drosophila
melanogaster. Genetics 46: 407-425.
BULMER,M. G., 1972 The genetic variability of polygenic characters under optimizing selection,
mutation and drift. Genet. Res. 19: 17-25.
1326
Z.-B. ZENC AND W. G. HILL
BULMER,M. G., 1973 T h e maintenance of the genetic variability of polygenic characters by
heterozygous advantage. Genet. Res. 22: 9-1 2.
BULMER,
M. G., 1980 The Mathematical Theory of Quantitative Genetics. Clarendon Press, Oxford.
CLAYTON,G. A. and A. ROBERTSON,1957 An experimental check on quantitative genetical
theory. 11. T h e long-term effects of selection. J. Genet. 55: 152-170.
J. W., 1977 76 generations of selection for oil and protein percentage in maize. pp.
DUDLEY,
459-473. In: Proceedings of the International Conference on Quantitative Genetics, Edited by E.
POLLAK,
0. KEMPTHORNE
and T. B. BAILEY,JR. Iowa State University Press, Ames.
FALCONER,
D. S., 198 1
Introduction to Quantitative Genetics, Ed. 2. Longman, London.
FISHER,R. A., 1930 The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
FLEMING,W. H., 1979 Equilibrium distributions of continuous polygenic traits. SIAM J. Appl.
Math. 36: 148-168.
R., 1980 Origin of genetic variation in selection lines. pp. 56-68. In: Selection ExFRANKHAM,
periments in Laboratory and Domestic Animals, Edited by A. ROBERTSON.
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GILL, P. E., W. MURRAY
and M. H. WRIGHT,1981 Practical Optimization. Academic Press, London.
GILLESPIE,J. H., 1984 Pleiotropic overdominance and the maintenance of genetic variation in
polygenic characters. Genetics 107: 321-330.
J. B. S., 1954 T h e measurement of natural selection. Proc. 9th Int. Congr. Genet. (Part
HAIDANE,
1, Cargologia) 6 (Suppl.): 480-487.
HILL,W. G., 1982 Predictions of response to artificial selection from new mutations. Genet. Res.
40: 255-278.
JAMES,J. W., 1962 Conflict between directional and centripetal selection. Heredity 17: 487-499.
KIMURA,M., 1965 A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. USA 54: 731-736.
J. F. C., 1978 A simple model for the balance between selection and mutation. J.
KINGMAN,
Appl. Probab. 15: 1-12.
KOJIMA,K., 1959 Stable equilibria for the optimum model. Proc. Natl. Acad. Sci. USA 45: 989993.
LANDE,R., 1975 T h e maintenance of genetic variability by mutation in a polygenic character
with linked loci. Genet. Res. 2 6 221-235.
LATTER, B. D. H., 1960 Natural selection for an intermediate optimum. Aust. J. Biol. Sci. 13:
30-35.
LATTER,B. D. H., 1966 Selection for a threshold character in Drosophila. 11. Homeostatic behavior on relaxation of selection. Genet. Res. 8: 205-218.
LATTER, B. D. H., 1970 Selection in finite populations with multiple alleles. 11. Centripetal
selection, mutation, and isoallelic variation. Genetics 66: 165-1 86.
LERNER,I. M., 1950 Population Genetics and Animal Improvement. University Press, Cambridge.
LERNER,1. M., 1954 Genetic Homeostasis. Oliver & Boyd, Edinburgh.
LERNER,I. M. and E. R. DEMPSTER,1951 Attenuation of genetic progress under continued
selection in poultry. Heredity 5: 75-94.
R. C., 1964 T h e interaction of selection and linkage. 11. Optimum models. Genetics
LEWONTIN,
50: 757-782.
LINNEY,R., B. W. BARNESand M. J. KEARSEY,1971 Variation for metrical characters in Drosophila populations. Heredity 27: 163-1 74.
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LIMIT TO SELECTION
NICHOLAS,
F. W. and A. ROBERTSON,
1980 The conflict between natural and artificial selection
in finite population. Theor. Appl. Genet. 5 6 57-64.
ROBERTS,R. C., 1966 The limits to artificial selection for body weight in the mouse. 2. The
genetic nature of the limits. Genet. Res. 8: 361-375.
ROBERTSON,
A., 1956 The effect of selection against extreme deviants based on deviation or on
homozygosis. J. Genet. 5 4 236-248.
ROBERTSON,
A., 1960 A theory of limits in artificial selection. Proc. Roy. Soc. Lond. (Biol.) 153:
234-249.
TURELLI,
M., 1984 Heritable genetic variation via mutation-selection balance: Lerch's zeta meets
the abdominal bristles. Theor. Pop. Biol. 25: 138-193.
VERCHESE,
M. W., 1974 Interaction between natural selection for heterozygotes and directional
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1971
body weight in mice. J. Hered. 62: 228-234.
Long-term selection for
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Communicating editor: W. J. EWENS
APPENDIX
From (32), (24) and ( 2 5 ) , we have
where [ is a constant, such that
and
In this APPENDIX, we prove U ; = B in (Al).
First, we need to clarify the range of the constant
p; is a small value, this implies that
0<1
always, which suggests that 0
[
1
-t
- C; exp(-
C;.
Since ( A l ) is a density function and
(z; - B)'/(2A)}
< [ < 1. T h e n it follows that
exp(-
VI]'
exp(-
=
1.
n(zi2A
- B)*
?l=O
Therefore, (AI) can be written as
A
+ nm2
A
+ nm?
1328
Z.-B. ZENG AND W . G . HI1.L
after some calculation. Note that the last two terms in the right-hand side of (A4) define a
normal density function. By using this property and putting (A4) into (A2) and (A3), we
then obtain
and
and
Let
and
b, = a,
A
+ nm?B/u,
A
+ nmg
'
(A5) and (A6) are then equivalent to
m
m
a, =
1
b, = -.
A
n=o
"=O
cr=o
Sufficiency: It is easy to show that
a , is a monotonically decreasing series with positive
terms. Then, if U , > R , it follows that a, > b, for n = 1 , 2, . . . , except n = 0, where a,, =
bo. Consequently, c:=,) a, > cr=p=o
b, which contradicts the condition of (A7). Similarly, if
U , < B , c,"=~
a, < c,"=p=o
b,, which also contradicts the condition of (A7). While, when U , = B,
a , = b, holds for every n = 0, 1 , 2, . . .; therefore, (A7) is satisfied.
Necessity: (A5) and (AS) can be rewritten as
and
A
n ( u , - B)'
A + nm:B/u,
=
rd,,
e x p \ - 2(A 4- n m f ) j A + nm:
"=n
J
io'[
X
I
I/'
which are two power series. According to the identity theorem for power series, if the two
power series
m
2
n=o
m
yc. and
rd,
n=o
have the sanie sun1 in an interval 0 < [ < 1 (in this case) in which both of them converge,
then the t w o series are entirely identical. That is to say, for every n = 0, I , 2, . . ., c, = d,,
since (AS) and (A9) do converge with 0 < f < 1 . I t then gives U , = B always. Thus, the
proof is completed.