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The Effects of a Bottleneck on Inbreeding Depression and the Genetic Load
Author(s): Mark Kirkpatrick and Philippe Jarne
Source: The American Naturalist, Vol. 155, No. 2 (Feb., 2000), pp. 154-167
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VOL. 155, NO. 2
THE AMERICAN
NATURALIST
FEBRUARY 2000
The Effects of a Bottleneck on Inbreeding Depression
and the Genetic Load
Mark Kirkpatrick* and Philippe Jarnet
Genetique and Environnement-CC065, Institut des Sciences de
l'Evolution, Universite Montpellier II, Place Eugene Bataillon,
34095 Montpellier, CEDEX 5, France
Submitted October23, 1998; Accepted September20, 1999
ABSTRACT: We study the effects of a population bottleneck on the
inbreeding depression and genetic load caused by deleterious mutations in an outcrossing population. The calculations assume that
loci have multiplicative fitness effects and that linkage disequilibrium
is negligible. Inbreeding depression decreases immediately after a
sudden reduction of population size, but the drop is at most only
several percentage points, even for severe bottlenecks. Highly recessive mutations experience a purging process that causes inbreeding
depression to decline for a number of additional generations. On
the basis of available parameter estimates, the absolute fall in inbreeding depression may often be only a few percentage points for
bottlenecks of 10 or more individuals. With a very high lethal mutation rate and a very slow population growth, however, the decline
may be on the order of 25%. We examine when purging might favor
a switch from outbreeding to selfing and find it occurs only under
very limited conditions unless population growth is very slow. In
contrast to inbreeding depression, a bottleneck causes an immediate
increase in the genetic load. Purging causes the load to decline and
then overshoot its equilibrium value. The changes are typically modest: the absolute increase in the total genetic load will be at most a
few percentage points for bottlenecks of size 10 or more unless the
lethal mutation rate is very high and the population growth rate very
slow.
Keywords: bottlenecks, genetic load, inbreeding depression, selffertilization, deleterious mutations.
Many natural populations have unstable demographics or
* To whom
correspondence should be addressed. Present address: Section of
Integrative Biology C0930, University of Texas, Austin, Texas 78712; e-mail:
[email protected].
t Present address: Centre d'Ecologie Functionnelle et Evolutive-Centre National de la Recherche Scientifique, 1919 route de Monde, 34293 Montpellier,
CEDEX 5, France.
Am. Nat. 2000. Vol. 155, pp. 154-167. ? 2000 by The University of Chicago.
0003-0147/2000/15502-0002$03.00. All rights reserved.
experience severe reductions in size during colonization
events. Domesticated species and populations managed for
conservation also go through periods of small numbers.
Changes in population size affect patterns of genetic variation (reviewed in Jarne 1995; Lynch et al. 1995; Barrett
1998). This article considers how a brief reduction in population size affects two important genetic properties of a
population: inbreeding depression and the genetic load.
Inbreeding depression is the loss in fitness in offspring
from matings between relatives compared with offspring
from random matings within the same population. We
would like to understand how it is affected by variation
in population size for several reasons. First, inbreeding
depression plays a key role in the evolution of breeding
systems, especially of self-fertilization (Jarne and Charlesworth 1993; Uyenoyama et al. 1993; Charlesworth and
Charlesworth 1998). An allele that causes selfing automatically enjoys a large transmission advantage (Fisher
1941) that can be offset by inbreeding depression (Lloyd
1979; Charlesworth and Charlesworth 1987). Bottlenecks
cause a loss of genetic variation and therefore reduce inbreeding depression. Lande and Schemske (1985) made
the intriguing suggestion that the reduction caused by a
bottleneck might be enough to shift the inbreeding depression from a level that favors outcrossing to one that
favors selfing. Second, comparison of the inbreeding depression before and after a bottleneck can allow one to
estimate genetic parameters such as dominance coefficients and mutation rates (see Wang et al. 1998). Third,
conservation of domesticated and endangered natural
populations must take account of the genetic effects caused
by bottlenecks (Ralls and Ballou 1986; Lynch et al. 1995).
Breeding programs for captive populations that will be
reintroduced into nature, for example, need to consider
the consequences of matings between related individuals.
Another factor affected by changes in population size
is the genetic load. It measures the difference between the
average population fitness and the fitness of a mutationfree genotype (Crow 1993). The genetic load is widely
recognized as a factor that can contribute to the extinction
of small populations (Falk and Holsinger 1991; Fiedler
Bottlenecks,Depression,and Load 155
and Jain 1992; Keller et al. 1994; Lande 1994; Frankham
1995; Lynch et al. 1995; Saccheriet al. 1998).
This article examines theoreticallyhow inbreeding depression and the genetic load are affected when a large
population experiences a single population bottleneck.
Deleterious alleles maintainedby mutation are thought to
be the major source of inbreedingdepression(Crow 1993;
Charlesworthand Charlesworth1998; Dudash and Carr
1998), and so our models focus on that class of genes. We
show that neither the total inbreedingdepressionnor the
total genetic load are typically much affected by bottlenecks of 10 or more individuals. We also find that conditions where a bottleneck could favor a shift from outcrossing to selfing may be quite restricted.
Inbreeding Depression
Inbreedingdepressionis convenientlymeasuredas the average loss of relative fitness caused by self-fertilization
(Charlesworthand Charlesworth1987). We will assume
that it is caused by deleterious mutations, that loci have
multiplicativefitness effects, and that the base population
is random mating and large.
By multiplying the contributions from individual loci,
we find that the total inbreedingdepressionfor a panmictic
diploid population is
D = 1-I
i
(1 -dd),
where
wiself
di = 1
1
I(1
2
=
-ut
-
To determine the effects of a bottleneck, we first need
results for a population at equilibrium.Experimentalevidence shows that the bulk of deleteriousmutations affect
the fitness of heterozygotesand that per locus mutation
rates are typically much smaller than the strength of selection actingagainstthem (i.e., t << hs;Crow 1993;Charlesworth and Charlesworth 1998). In this situation, the
equilibrium frequency of a deleterious allele in a large
u/l(hs),where ,u is the alpopulation is approximatelyq p
lelic mutation rate (Haldane 1927). (We use hats throughout to denote quantities at equilibrium.) One can show
that the effects of fluctuations around this value caused
by drift are small in populations that are much largerthan
1/(2hs). For example, given a plausible value of hs =
0.01 (Crow 1993), this condition is satisfied if the base
population has an effectivesize of 500 or more individuals.
The inbreedingdepressioncaused by locus i at equilibrium is, therefore,
d
(1
-
2hi)li
(3)
2hi
This quantityis independentof the selection coefficientsi.
The reason is that selection holds alleles that are more
deleteriousat lower frequencies.Inbreedingdepressionis,
however, affected by dominance: deleterious mutations
that are more recessive(have smallervalues of h) generate
more inbreeding depression (Crow 1970).
(1)
An expression for the total inbreeding depression in a
population at equilibrium is found using equations (1)
and (3):
(1 - 2hi)s,iq(l - q,)
2 - 4hisiqi(1 - qi) - 2s,q2
2hi)siqi
The Base Population
-Edip 1- exp -nLE
D 1- exp
2h),
(4a)
(4a)
(2)
is the inbreeding depression from locus i, qi is the frequency of a deleterious allele at that locus, and wiel and
wi?utare, respectively,the relative fitnesses at locus i of
selfed and outcrossedoffspring (Crow 1970;Charlesworth
and Charlesworth1987). The selection coefficient si and
the dominance coefficient hi are defined such that the
relative fitnesses of the mutant heterozygote and homozygote are 1 - hisiand 1 s, respectively.A mutation is
(at least partially) recessive whenever hi< 1/2. The approximationin the second line of equation (2) is obtained
by neglecting terms involving squaresand higher powers
of qi.
D
1 - exp(-UH).
(4b)
In these expressions, U is the total rate of deleteriousmutations acrossthe entire diploid genome, and nLis the total
number of loci mutating to deleterious alleles. The function H depends on the dominance coefficients:
H=E
1 - 2h
4hi
(5)
where E[.] standsfor the expectation;note that Hincreases
as the mutants become more recessive (h becomes
156 The AmericanNaturalist
smaller). Equation (4b) assumes that mutation rates and
dominance coefficientsare statisticallyindependentacross
loci, while equation (4a) holds even if they are not. Since
we have no reason a priori to think there is a relation
between /Aand h, we will assume independencebelow for
the sake of simplicity. Both expressions assume that the
contributions of individual loci to inbreeding depression
are small (di << 1). They show that, as one expects intuitively, total inbreeding depression increases with larger
genomic mutation rates and smaller dominance
coefficients.
The expectation H could be calculated exactly if the
distributionof dominance coefficientswere known. In the
absence of that information, equation (4b) can be simplified further if variation in dominance across loci is
small:
D
1- exp -U(
+
?4f 1 +4h'].'
(6)
where h is the arithmetic average,and Uo is the variance
of the dominance coefficients. Equation (6) shows that
variation in dominance increases inbreeding depression:
loci that are more recessive than average contribute
disproportionately.
ImmediateEffectsof a Bottleneckon
InbreedingDepression
From a gene's viewpoint, a bottleneck representsa sampling event. Immediately after a bottleneck, the number
of surviving copies of a deleterious allele is a random
variablethat is multinomiallydistributed.The two parameters of this distributionare qi, the frequencyof the allele
before the bottleneck,and 2N, representingthe total number of copies of the gene to be sampled, where N is the
number of surviving individuals. Drift alone does not
change allele frequencieson average,and so the expected
frequencyof an allele immediatelyafter the bottleneck is
equal to that in the base population. The effect of the
bottleneck is to introduce variation around this expectation: the deleteriousalleleswill be lost from some loci and
increasedin frequencyat others.
What is the inbreeding depression right after a bottleneck?The situation is simple when the bottleneckis small
and deleteriousalleles are rare in the base population. In
that case, either zero or one copy of each deleteriousallele
will pass through the bottleneck; the chance that two or
more copies survive is negligible. The probabilityPi that
a deleterious allele at locus i survives the bottleneck is
simply 2N4i, and its frequencywill be qi(O)= 1/(2N) if it
does so (Lynchet al. 1995). (This approximationholds so
long as 2Nqi<< 1, which implies that N<< hisil[2,Ji] in the
base population. Taking,e.g., the plausibleparametervalues of hisi= 0.01 and ,;i= 2 x 10-6, the approximation
should be good for bottlenecks of size 250 or smaller.)
However, if a deleteriousallele does not pass through the
bottleneck, then that locus becomes fixed and makes no
contribution to the inbreeding depression.
The expected inbreeding depression contributed by a
locus just after a bottleneck is therefore
di(O)/
N(2N - 1)(1- 2hi) ,
{4N2 - [2(2N-
1-
1)h, + l]s,}h,
I
I. d.
2N
(7)
The last step in equation (7) is an approximation that
makes use of our previous assumption that h,s,< 1 and
the fact that si<< 4N2 for bottlenecks of two or more
individuals.
This result shows that a bottleneckwill alwaysdecrease
inbreeding depression and that smaller bottleneck causes
a greaterreduction. The proportional decreasein the inbreedingdepressioncontributedby a single locus is simply
1/(2N). This change is equal to the loss of heterozygosity
caused by the bottleneck.That conclusion holds (approximately) regardlessof all genetic parameters(the selection
coefficient, the mutation rate, and the dominance coefficient). There is a simple intuitive explanationfor the fact
that the drop in depressionis independentof the selection
coefficient: mutants that contribute more to inbreeding
depression in the base population are less frequent and,
therefore,less likely to pass through a bottleneck.
An expression for the total inbreeding depressionjust
afterthe bottleneckcan now be found using equations (1)
and (4a):
D(0)
1 - exp-1
I 2(O
)
- -UH,
(8a)
(8b)
Equation (8b) is a linear approximation of (8a) that is
most accurate when D is much less than one. One can
also show that (8b) is the upper limit for (8a), and so it
gives us the maximum effect that a bottleneck can have.
Together,these results show that the total inbreedingdepression immediately after a bottleneck declines with
smallervalues of N, as expected from the single-locus result. But even a small bottleneckdoes not change the total
inbreeding depression very much. With a bottleneck of
size N = 5, for example, the depression is reduced by at
Bottlenecks,Depression,and Load 157
its contributionto inbreedingdepressiondeclines.One can
show that the numbers of generationsneeded to return a
given fraction of the way to the equilibrium are approximately equal for these two loci when the mutant is not
very recessive(h x 1/2). Consequently,the inbreedingdepression contributed by the pair will return smoothly to
its original equilibrium. An example of this process is
shown in the upper panel of figure 2.
The outcome is quite different for a pair of identical
loci segregatingfor highly recessive mutations. Here, selection acting on mutant homozygotes drives down the
frequencyof the deleteriousalleleat the second locus more
quickly than mutation builds up its frequencyat the first.
As a consequence,inbreedingdepressioncontinues to decline for a number of generationsafter the bottleneckbefore the accumulationof new mutations restoresthe orig-
10%
0
O3
c)
5%
a,
a,
:0
cn
c
0
a,
0
C,
C(.)
O
10%
5%
o
a,
(D
0
0.1
1.0
10.
1.0
- \
/
\
Genomicmutationrate,U
cn
Figure 1: Proportional decrease in inbreeding depression immediately
following bottlenecks of size N = 5 and 10 as a function of the total
genomic mutation rate U. The four curves in each panel represent,from
left to right, resultsfor the dominance coefficienth = 0.05, 0.1, 0.25, and
0.4. Calculatedfrom equation (8a).
most 10%. Other numerical examples are shown in figure 1.
0
0.5
/
C,)
(a)
/
0
Mildlyrecessive
mutations
\
N.
/
La
(0
Q>
/
-
/
-
I
/
0
50
100
-,c
Recoveryof InbreedingDepressionafter a BriefBottleneck
What happens in later generations?The answer depends
on the demographyof the population. The first possibility
we consideris that the populationgrowsvery rapidlyabove
the size where drift is an important factor for deleterious
alleles (i.e., to a population size substantiallylarger than
1/[2hs]). Then drift is only important during the generation in which the bottleneck occurs, and inbreedingdepression returns to its original equilibrium under the
forces of mutation and selection alone.
What are the dynamics of the recovery?Considera pair
of identicalloci segregatingfor mildly recessivemutations.
At one of the pair, the deleterious mutation is lost in the
bottleneck, while at the other its frequency jumps to
1/(2N). In the following generations,the allele frequency
at the first locus increases as the result of mutation, and
its contribution to inbreeding depression grows. Meanwhile, the mutant frequencyat the second locus declines
towardits equilibriumunder the pressureof selection, and
0
200
400
Generationsafterbottleneck
Figure 2: Recoveryof inbreedingdepressioncontributedby a singlelocus
following a bottleneck of size 5 as a function of time (in generations).
Valuesof the depressionhave been multipliedby 106.Upperpanel,mildly
recessivemutation with h = 0.3 and s = 0.1. Lowerpanel, highly recessive
lethal with h = 0.01 and s = 1.0. For both panels, t = 10-6. Arrows on
the left of each panel indicate the inbreedingdepression at equilibrium
(upper arrow) and immediately following the bottleneck (lower arrow).
The ascending dashed curves show the contributionsfrom loci that lose
the deleteriousallele during the bottleneck, and the declining curvesthe
contributionsfrom loci wherethe bottleneckcausesthe mutantfrequency
to increase.
158
The American Naturalist
inal equilibrium. This effect is sometimes referred to as
"purging" (Lande and Schemske 1985). An example is
shown in the lower panel of figure 2.
When does this additional decline in inbreeding depression occur? This question is answered by finding when
inbreeding depression is less immediately after the bottleneck than it is one generation later. Again using the approximation that at most one copy of a mutation passes
through the bottleneck, the depression immediately after
the bottleneck is given by equation (7). The depression
one generation later is found by calculating the allele frequencies in the two classes of loci after one generation of
selection and mutation, substituting those into equation
(2), then averaging these two results by their respective
probabilities. The answer is complicated but simplifies
greatly under the reasonable assumptions that mutation
is rare and that selection against mutant heterozygotes is
weak. Some algebra then produces the rule that inbreeding
depression will continue to decline in later generations
when
h<
5N- s
(9)
(This approximation makes the plausible assumptions that
< 1, hs < N, It < h, and h2s2 < 1.) Result (9)
IONAX
shows that at least some purging occurs for all mutants
that are more recessive than approximately h = 0.2.
How long does it take for inbreeding depression to recover its equilibrium after a bottleneck? Depression caused
by mildly recessive mutations returns smoothly to its initial
equilibrium. The dynamics throughout the recovery are
inversely related to the strength of selection against heterozygotes, hs, and are largely independent of the mutation
rate ,u (Lande and Schemske 1985, p. 35). The recovery
rate is also insensitive to the size of the bottleneck N. For
highly recessive alleles, the purging process delays the return to equilibrium. When these loci finally return to near
equilibrium, however, the final rate of approach is again
inversely related to hs and roughly independent of N and
/I.
We checked these conclusions with numerical studies.
Rather than assuming that no more than a single copy of
a deleterious mutation survives the bottleneck, we did an
exact calculation. First we determined the probability fJ
that i copies (= 0, 1, ..., 2N) pass through the bottleneck.
For each value of i, we computed the allele frequencies
and inbreeding depression in later generations, assuming
that the population size grows sufficiently fast that the
bottleneck is the only point at which genetic drift acts.
Finally, for each generation we averaged the values for the
depression over all values of i, weighted by the f's.
The results are shown in table 1. The proportional drop
Table 1: Effectsof a bottleneck of size (N) on inbreeding
depression from a mutation with selection coefficient (s)
and dominance coefficient (h), assumingrapid (effectively
infinite) population growth after the bottleneck
N, s, h
dx
106
d(0)/d
dJn/d
t(dmin)
t7,5
10:
.001:
.1
.3
.01:
.1
.3
.1:
.1
.3
1.0:
.01
.03
15
3.2
.95
.95
.94
.95
2,096
0
15,124
2,611
19
3.3
.95
.95
.92
.95
422
0
2,259
281
20
3.2
.95
.95
.92
.95
46
0
239
29
.95
.96
.59
.76
32
15
189
74
15
3.2
.98
.98
.98
.98
2,303
0
15,295
2,679
19
3.3
.98
.98
.97
.98
464
0
2,324
290
20
3.2
.98
.98
.97
.98
51
0
247
30
.98
.98
.77
.89
44
18
212
80
230
76
30:
.00 1:
.1
.3
.01
.1
.3
.1:
.1
.3
1.0:
.01
.03
100:
.001:
.1
.3
.01:
.1
.3
.1:
.1
.3
1.0:
.01
.03
230
76
15
3.2
1.0
1.0
.99
1.0
2,363
0
15,335
2,704
19
3.3
1.0
1.0
.99
1.0
480
0
2,350
293
20
3.2
1.0
1.0
.99
1.0
53
0
250
31
1.0
1.0
.91
.96
55
20
231
84
231
76
Note:Theinbreedingdepressionat equilibriumis d, d(O)is its value
is the minimumdepression
immediatelyfollowingthe bottleneck,dmn,
reachedduringrecovery,t(dm,,)is the generationat whichthe minimum occurs, and t75%is the number of generations after the bottleneck
until depressionreturns75%of the way to the equilibriumfrom its
minimumvalue.The mutationrateis,u = 5 x 106. Resultsarebased
on deterministicsimulationsdescribedin the text.
in inbreeding depression immediately following the bottleneck is about 1/(2N), in agreement with equation (7).
Further purging does not occur for mutations that are
mildly recessive: when h = 0.3, the minimum depression
Bottlenecks,Depression,and Load 159
(dmin)
occurs immediately after the bottleneck (t = 0). The
recovery time is greatest for mutations that have weak
effects as heterozygotes(small values of hs). For a typical
mutation of weak effect, with s = 0.1 and h = 0.3, threequartersof the initial loss of inbreedingdepressionreturns
in about 30 generations.Purgingoccurs for mutationsthat
are highly recessive.The additional decline of inbreeding
depressionis greatestfor severebottlenecks (small N) and
small dominance coefficients(h). The times until the minimum inbreedingdepressionis reachedand until the equilibrium is reestablishedare again greatest when the heterozygouseffect is weak (hs is small). These times are quite
insensitive to the size of the bottleneck (N). For a typical
lethal that is highly recessive(h = 0.03), a severebottleneck
of 10 individualswill cause inbreedingdepressionto drop
4.5% immediately.The depression continues to fall until
15 generationslater,when its value is 76% of the equilibrium. Three-quartersof the inbreeding depression is recovered 74 generations after the bottleneck.
quite differently to a bottleneck, we will consider them
separately.The stochastic simulations focused on mildly
deleteriousgenes with s = 0.05 and h = 0.3 and on highly
recessivelethals with s = 1 and h = 0.03.
The results are shown in table 2. As anticipated, the
minimum inbreeding depression is lower when the population size recovers slowly because drift has additional
opportunityto act. The most dramaticeffectof population
growth rate seen in table 2 is for a highly recessivelethal
(h = 0.03, s = 1) experiencing a bottleneck of size N=
10. Inbreeding depression loses 24% of its equilibrium
Table2: Effectsof populationgrowthrate on
changesin the inbreedingdepressionandthe geneticload aftera bottleneck
N,s, h,andX
dmin/d
t(dmin) /max/i
t(lmax)
10:
.05:
.3:
InbreedingDepressionWhenPopulation
Size RecoversSlowly
We have been assuming that the population size rebounds
very rapidlyafterthe bottleneck,as it might when a weedy
species invadesa new habitat.In other situations,however,
population growth is much slower, which allows drift to
act for many generationsafter the bottleneck.
We investigated how population growth impacts inbreeding depressionusing stochastic simulations. Following the bottleneck,we assumed that the population grows
at a constant rate such that its size in each generation is
a factor X larger than it was in the previous one. (Eventually density dependence will cause the growth rate to
slow, of course. By assuming a constant growth rate, we
assume in effect that the slowing occurs after the effects
of drift become negligible, i.e., when the population size
becomes much larger than 1/[2hs].) We examined three
population growth rates. The first is very slow growth,
with the population increasing only 20% in each generation after the bottleneck (X = 1.2). The second is moderate growth, with the population doubling (X = 2). The
third case correspondsto the very rapidgrowthconsidered
in the previous section (X = oo), where the population
grows to a large size immediatelyafter the bottleneck.For
each generation of the simulation, inbreeding depression
was calculated. Selection was then imposed, followed by
mutation and, finally, the random sampling of alleles to
form the next generation.
Deleterious mutations tend to fall into two major categories: highly recessive lethal mutations and mutations
that are only mildly deleteriousand mildly recessive(Crow
1993; Wang et al. 1998). Since these two classes respond
1.2
2.0
.81
.93
5
0
1.16
1.15
20
7
oo
.95
0
1.03
0
.40
.64
.76
8
9
15
2.79
2.06
1.78
1
1
0
1.2
2.0
.90
.93
19
3
1.18
1.00
2
1
oo
.98
0
1.01
0
.68
.83
.89
21
18
18
1.56
1.40
1.26
1
1
0
1.2
2.0
.91
1.00
10
0
1.00
1.05
0
3
oo
1.00
0
1.00
0
.90
.91
.96
27
16
20
1.33
1.11
1.08
5
0
0
1.0:
.03:
1.2
2.0
oo
30:
.05:
.3:
1.0:
.03:
1.2
2.0
oo
100:
.05:
.3:
1.0:
.03:
1.2
2.0
oo
Note: The genetic load at equilibrium is 1, I,, is the
maximumload reachedafterthe bottleneck,and t(l,,) is
the generationat which the maximumis reached.Other
parametervaluesand symbolsaredefinedin table1. Results
arebasedon the stochasticsimulationsdescribedin the text,
with 2 x 104 replicates per parameter set.
160 The AmericanNaturalist
value when population growth is rapid, 37% when it is
moderate, and 60% when it is slow. The effect of population growth rate on inbreeding depression is less for
mildly recessivemutations. For example, the same bottleneck causes a 5% loss with rapid population growthwhen
s = 0.05 and h = 0.3, and it causesa 19%loss when growth
is slow.
These results highlight a pattern:the effects of drift on
inbreedingdepressionin later generationsare much more
pronounced for highly recessivelethalsthan for mutations
of weak effect. This may be unexpected since the frequencies of strongly selected genes are less influenced by
drift. Because they are highly recessive, however, small
changes in the frequencies of lethals are amplified into
large changes in the amount of inbreedingdepressionthat
they contribute.
Dynamicsof the TotalInbreedingDepression
So far we have focused on the response of an individual
locus to a bottleneck. The results can be extrapolatedto
find the total inbreeding depression if loci evolve independently. That will be true when there is no epistasis,
which we have already assumed. A second requirement,
however,is that there be no linkagedisequilibriumcaused
by drift. The bottleneck itself generatesrandom disequilibria,but these will decay rapidlyif the loci are not tightly
linked. Additional disequilibriawill be generatedin later
generations, however, if population growth is not rapid.
Understandingtheir impact on inbreedingdepressionis a
difficult problem, and so we will make the simplifying
assumption here that the effects of linkage disequilibria
can be neglected. We expect the conclusions that follow
to be most accuratewhen the strengthof drift,as measured
by the reciprocalof the effectivepopulation size, is much
smallerthan the recombinationratesbetween the loci carrying deleteriousmutations.
The total inbreedingdepressioncausedby a set of mildly
deleteriousalleles and a set of lethals is found by modifying equation (4b). The total depression t generations
after a bottleneck is D(t) = 1 - exp {-[ULHLdL(t)/dL][UMHMdM(t)/dM]}, where the subscript L denotes values for
lethals and M denotes values for mildly deleterious mutations. The variables d(t) are the inbreeding depression
contributedby single loci t generationsafter a bottleneck,
which we found from simulations like those reported in
tables 1 and 2. We will focus on a severe bottleneck of
N= 10 individuals.
Begin by consideringlethal mutations. Data from Drosophilaand annual plants suggest they occur at a rate of
about 0.03 per diploid genome per generation and have
an averagedominance coefficient around h = 0.03 (Crow
1993). At equilibrium,these mutationsby themselvespro-
duce an inbreeding depression of 21%. The minimum
depression reached in the purge after the bottleneck depends on population growth:it is 16%with rapid growth,
14% with moderate growth, and 9% with very slow
growth. The consequencesare more pronounced if lethals
are much more common. Lande et al. (1994) review data
suggestingthat the genomic mutation rate for lethals may
be as high as 0.2 per generationfor long-livedplants,which
lack a segregatedgermplasm.With that mutation rate,the
equilibriumvalue of inbreedingdepressionis 79%.During
the purge that follows a bottleneck, depression drops to
70% with rapid population growth, 63% with moderate
growth, and 47% with slow growth. In all cases, threefourths of the equilibrium is regained <100 generations
later. In sum, inbreeding depression from lethals drops
only several percentagepoints if population size recovers
quicklyafter a bottleneckbut can drop considerablymore
if the lethal mutation rate is very high and population
growth is very slow.
Now consider weakly recessive,mildly deleteriousmutations. Data from Drosophilaand plants suggestthat mutations with mild homozygous effects may typicallyhave
selection coefficientsof s = 0.1 and dominance coefficients
around h = 0.3 and may occur at a rate of about one per
diploid genome per generation (Crow 1993; Wang et al.
1998). (Wang et al. [1998] review some experimentsthat
suggest a much lower mutation rate. We use U = 1 here
because that value gives a greaterscope for the effects of
a bottleneck and because it may be more widely accepted
[Charlesworthand Charlesworth 1998]). These loci by
themselves produce an inbreeding depression of 28% at
equilibrium. With rapid population growth, depression
dips to 27% just after a bottleneck of size 10 and then
smoothly returns to its original equilibrium. With slow
growth, a minimum value of 22% depression is reached
a few generationsafterthe bottleneck.In brief,a bottleneck
is expected to cause the inbreedingdepressionfrom these
loci to drop by at most a few percentagepoints.
These results can now be combined to estimate the
greatest absolute decline in total inbreeding depression
after a bottleneck of 10 individuals. With the low lethal
mutation rate typical for animals and short-lived plants,
the total depressionat equilibriumwill be 43%. During a
very rapid recovery after the bottleneck and a low lethal
mutation rate, depression only drops to 39%, while with
very slow recovery it drops to 29%. With a high lethal
mutation rate, total inbreedingdepressionis 85% at equilibrium. This declines to a minimum of 78% when population growth is very rapid but falls to 59% if growth is
very slow.
In summary,even a severe bottleneck will not cause a
substantial change in inbreeding depression unless population growth afterwardis very slow. In that case, a tem-
Bottlenecks,Depression,and Load 161
porary decline on the order of 25% can occur in the absolute value of inbreeding depression.
Consequences
for the Evolutionof Selfing
One of our motivations for studying the effects of bottlenecks is to learn about their possible effects on breeding
systems. When might a bottleneck purge enough of the
inbreedingload to favor selfing in an outcrossingspecies?
A rule that seems to hold under many conditions is that
genes that increase the selfing rate are favored when the
inbreeding depression is less than one-half (reviewed in
Jarne and Charlesworth 1993; Uyenoyama et al. 1993;
Charlesworthand Charlesworth1998). By this criterion,
a bottleneck might trigger an evolutionary shift in the
breedingsystemif the inbreedingdepressionis greaterthan
one-half before the bottleneckand less than one-half after.
This is not the complete list of conditions since the inbreeding depression would have to remain low for long
enough afterthe bottleneckto allow modifiersthat increase
selfing rates to spread. It does, however, give a necessary
condition under the one-half rule. A second caveatis that
this argument ignores the effects of statisticalassociations
(disequilibria)between genes that alterthe selfing rate and
those causing inbreeding depression (Uyenoyama and
Waller 1991). Those effects may often be minor, however,
at least in large populations (Charlesworthet al. 1992).
We will return to this point in the "Discussion."
Assume for simplicity that variation between loci in
dominance coefficients and mutation rates is negligible.
While this is certainly not biologically realistic, we can
bracketthe possibilitiesby examiningextremecases.Using
equations (4a), (4b), and (5), one can find the critical
value of the genomic mutation rate that makes the total
inbreeding depression D equal to one-half in generation
t after the bottleneck:
U*(t)
4hln2
2h t)
(1 2h)d(t)ld
fall for a number of generationsafter the bottleneck.The
minimum value for d(t) after a bottleneck can be found
from simulation results like those in tables 1 and 2.
We can now find situations that might cause a bottleneck to trigger a shift in the breeding system. For the
following example, we will focus on a bottleneck of size
N = 5, even more extreme than those we discussed previously. Further,we will assume all mutations are lethals
since the earlier results show that highly recessivelethals
are most susceptibleto purging.In sum, we areconsidering
a situation that should be highly favorableto a shift from
outcrossing to selfing.
Figure 3 presents the results. The solid curve shows
parameter combinations that cause the total inbreeding
depressionin the base population to equal one-half. Combinations below it favor selfing, while those above it favor
outcrossing. The two dashed curves give parametercombinations that cause inbreeding depression to equal onehalf at the point aftera bottleneckwhen depressionreaches
its smallest value. Results for fast population growth
(X = oo) are shown by the dot-dashed curve; these were
calculatedfrom deterministicsimulationslike those of table 1. Results for very slow population growth (X = 1.2)
are shown by the dashed curve; these are from stochastic
simulations like those of table 2.
O
11
3.21-
*_ _
E-
0.320.1
recovery\,
1
Favorable to selfing
Base population
/et
/
Duringfast recovery
I
(10)
Setting d(t) = d in equation (10) gives the minimum
value of Uin the base populationthat will causeinbreeding
depression to be greter than one-half. We can also use
equation (10) to find the maximum value of U that will
cause depression to be less than one-half at some point
after a bottleneck. Just after the bottleneck, equation (7)
shows that d(O)/d= 1 - 1/(2N). If mutations that cause
inbreeding depression are all weakly recessive, that will
also be the minimum that d(t) achieves during the return
to equilibrium. But if the mutations are highly recessive,
purging will cause inbreeding depression to continue to
.>
Duringslow
0.32'//
o
(D
Favorable to outcrossing
0
0.1
I
0.2
I
0.3
Dominancecoefficient, h
Figure 3: Combinations of the genomic mutation rate and dominance
coefficient favorableto a switch from outcrossing to selfing after a bottleneck of size N = 5. Regions between the solid curve (corresponding
to the base population) and the dashed curves (correspondingto slow
and fast recovery from the bottleneck) are parametercombinations in
which the inbreeding depression is initially greaterthan one-half, then
dips below that value for some generations following a bottleneck. The
dot-dashed curve is for rapid recovery (X = co); it was calculatedfrom
equation (10), using the minimum value of d(t) after a bottleneck seen
in deterministic simulations. Open circles show stochastic simulation
results for very slow recovery (X = 1.2); the correspondingdashedcurve
was fit by eye. Other parametervalues are s = 1 and It = 5 x 10-6.
162 The AmericanNaturalist
The regions between the solid curve and the dashed
curves are conditions where a shift to selfing may be temporarilyfavoredduring recoveryfrom a bottleneck.When
population size recovers quickly, only a small range of
parameters causes inbreeding depression to shift from
greaterthan one-half before a bottleneckto less than onehalf afterward.The most likely case is when mutations are
highly recessive and only a very few individuals pass
through the bottleneck. But even there the conditions are
quite delicate.With h = 0.01, for example, a bottleneckof
five individualsrequiresthe genomic mutation rate to fall
between U = 0.028 and 0.060. When population growth
is very slow, the range of conditions is broader: U needs
to lie roughly between 0.028 and 0.2.
Even if inbreeding depression temporarilydips below
one-half after a bottleneck, its recoverywould have to be
sufficientlyslow to allow modifiersthat increaseselfing to
spread. That event may be unlikely. For example, with
rapid population growth most of the inbreeding depression from highly recessive mutations is recoveredwithin
200 generations(table 1). Apparentlythere is only a small
window of time for selfing to evolve.
li(O) (1 + 4hN
i
(13)
The total load caused by deleterious mutations across
the genome is
L(O)
,1
- exp-1
+
U,
(14)
where H is again the function of the dominance coefficients given by equation (5).
These results show that the load caused by recessive
mutations is alwaysincreasedby a bottleneck.The reason
is that the loci at which deleteriousalleles increasein frequency cause those mutations to be exposed as homozygotes. The fitness loss from these homozygotes more
than outweighs the fitness gain at those loci where the
mutant allele is lost. The increasecan be substantialwhen
the bottleneck is extremeand mutants are highly recessive
(h is small, and so His large). For example, equation (13)
shows that a bottleneck of size N = 5 will cause the load
contributedby a mutant with h = 0.01 to increaseby about
The Genetic Load
a factor of six. This behavior contrasts sharplywith that
The second impact of a bottleneckthat we considerin this of inbreeding depression, whose relative change immearticle is how it affects the genetic load. We again begin diately after a bottleneck is small and independent of genetic parameters.An algebraicexplanationfor this differby assuming the loci have independent effects on fitness.
In that case, the genetic load contributed by deleterious ence is that inbreeding depression is a relatively linear
function of allele frequencynear the equilibrium,but the
mutations across the entire genome is
load is not.
L =1- -i
(1 - I)
1- exp(-
I),
(11)
Recoveryof the Load after a Bottleneck
In the generationsfollowing a bottleneck,the genetic load
decreasestoward its equilibrium. Because of the purging
however, the return to the equilibrium is not
=
+
process,
(12)
li 2hisiq(1
qi) siq2
smooth. The load overshoots the equilibrium, and the
is the load contributed by a single locus. When the del- population enters a period where the load is actually
eteriousallelesreacha selection-mutationbalance,the load smaller than the equilibrium value. One can show anafrom a single locus is simply li 2/ui,and the genome wide lyticallythat an overshoot generallyoccurs, even for mutations that are not very recessive.Simulationsshow, howload is L 1 - exp (-U) (Haldane 1937; Crow 1993).
ever, that this overshoot is small unless the mutation is
highly recessive.Two examples are shown in figure 4.
Dynamics of the genetic load following a bottleneckare
The ImmediateEffectsof a Bottleneckon the Load
complicated, and it is difficult to extract analytic results.
We can determine the immediate effects of a bottleneck The process can be studied numerically,however, using
on the genetic load with the same approachwe used for the same exact calculationsfor allele frequencydynamics
inbreeding depression. Assume that the bottleneck is se- developed above for inbreedingdepression.Again we bevere enough that at most a single copy of a deleterious gin with the simplest case, where the population size remutation survives. Then immediately following the bot- bounds very quicklyfollowing the bottleneck.The dynamics of the load can be summarizedby four numbers. The
tleneck, the load from a single locus is
where
Bottlenecks, Depression, and Load
\
of size N = 10 will increase 78% just after the bottleneck
and decline 16% below its equilibrium value 27 generations later. At the other extreme, a bottleneck perturbs the
load contributed from slightly recessive mutations (h =
/-
Mildlyrecessive
mutations
\ /
-
2
/
/A
\
N
I
/
0O
0
0
P0
a)
50
100
Table 3: Changesin the genetic load following a bottleneck,assuming rapid (effectively
infinite) population growth after the
bottleneck
N, s, h
.001:
.1
.3
.01:
.1
.3
.1:
.1
.3
1.0:
.01
.03
20 -
10
0
100
l()/
t,
t(min)
min/i
1.17
1.03
3,703
4,426
8,099
6,680
.97
1.00
1.20
1.03
418
393
1,020
624
.96
1.00
1.20
1.03
41
34
105
57
.95
1.00
3.40
1.78
21
9
60
27
.70
.84
1.06
1.01
3,716
4,174
8,168
6,427
.99
1.00
1.06
1.01
430
373
1,060
604
.98
1.00
1.07
1.01
43
32
110
55
.98
1.00
1.80
1.26
29
11
77
31
.84
.93
1.02
1.00
3,719
4,093
8,185
6,344
1.00
1.00
1.02
1.00
435
367
1,077
597
.99
1.00
1.02
1.00
44
32
112
55
.99
1.00
1.24
1.08
34
12
91
34
.94
.98
10:
c
(D
163
200
30:
Generationsafterbottleneck
Figure 4: Recovery of the genetic load contributed by a single locus
following a bottleneck of size 5 as a function of time (in generations).
Values of the load have been multiplied by 106. Upperpanel, mildly
recessivemutation with h = 0.3 and s = 0.1. Lowerpanel, highly recessive
lethal with h = 0.01 and s = 1.0. For both panels, uc= 106. Arrows on
the left of each panel indicate the load at equilibrium(lowerarrow)and
immediately following the bottleneck (upper arrow). The ascending
dashed curves show the contributions from loci that lose the deleterious
allele during the bottleneck, and the declining curves the contributions
from loci where the bottleneck causes the mutant frequencyto increase.
first is 1'(0)/l, the proportional increase in load immediately after the bottleneck, which can be found from equation (13). The other three are determined by simulation:
ti, the number of generations after the bottleneck at which
the load returns to its equilibrium value for the first time;
t(lmin), the number of generations until the load reaches
its minimum; and Imin/l, the minimum load relative to its
equilibrium value.
Results for rapid population growth are shown in table
3. The relative increase in the load immediately after the
bottleneck and the size of the later overshoot of the equilibrium are affected by the parameters in the same way.
Both are largest when the bottleneck size is small and
mutations are highly recessive, and both are insensitive to
the mutation rate and selection coefficient. The load contributed by a lethal with h = 0.03 experiencing a bottleneck
.001:
.1
.3
.01:
.1
.3
.1:
.1
.3
1.0:
.01
.03
100:
.001:
.1
.3
.01:
.1
.3
.1:
.1
.3
1.0:
.01
.03
Note: The load at equilibriumis l, 1(0)is the load
immediatelyafterthe bottleneck,t, is the generation
at whichthe loadfirstreturnsto its equilibriumvalue,
and t(l,,m)is the generationat which it reachesits
minimum,lm,. Otherparametervalues are defined
in table 1. Resultsare basedon deterministicsimulationsdescribedin the text.
164 The AmericanNaturalist
0.3) by only a few percentagepoints. The speed with which
the population recoversits initial equilibriumincreasesas
the dominance and selection coefficients increase but is
relativelyinsensitiveto the bottlenecksize. Highlyrecessive
lethals (h = 0.01, s = 1) and mildly recessivemutations of
relativelyweak homozygous effect (h = 0.3, s = 0.1) both
produce their minimum loads about 60 generationsafter
a bottleneck.The overshootis negligiblefor mutationsthat
are mildly recessive, but the load from highly recessive
mutations can drop 30% below its equilibrium.
We investigatedhow the load is affected by the population's growth rate following the bottleneck. Again we
used stochastic simulations and compared very slow
(X = 1.2), moderate (X = 2.0), and very rapid (X = oo)
growth rates. The results are shown in table 2. As with
inbreedingdepression,highly recessivemutationsaremost
strongly affected by a bottleneck. Their contribution to
the load is greatest when the population growth rate is
slow. For a lethal with h = 0.03, after a bottleneck of size
10 the load rises 78% above its equilibriumif the population grows rapidly but 179% if the population grows
slowly.
Dynamicsof the TotalGeneticLoad
How might the total genetic load in a naturalpopulation
respond to a bottleneck?Here we will consider the combined effects of a set of lethals and a set of mildly detrimental mutations. Denoting the per locus load t generations after the bottleneck by l(t), the total load is
L(t)
1 - exp {-[ULI(t)/lL] - [UMIM(t)/lM]},where again
the subscriptsL and M denote values for lethalsand mildly
deleterious mutations. We will use simulation results for
l(t) to look at the consequences of a bottleneck of 10
individuals. Again we make the simplifying assumption
that linkage disequilibrium,which causes loci to evolve
nonindependently,is small enough that it can be ignored.
We begin with the case of rapid population growth.For
lethals (s = ), consider a genomic mutation rate of U =
0.03 and dominance coefficient of h = 0.03, as suggested
by the data on flies and annual plants. At equilibrium,
these mutations by themselves produce a load of 3.0%.
Just after the bottleneck,the load jumps to 5.2%.Purging
causesthe total load to drop to 2.5%in the twenty-seventh
generationafterward.If the genomic mutation rateto lethals is instead U = 0.2, as may be the case in long-lived
plants,the load jumps from an equilibriumof 18%to 30%
just after the bottleneck, then declines to a low of 15%
during the purge.
Now consider mildly deleteriousallelesand, again, take
their genomic mutation rate to be U = 1, their homozygous effect to be s = 0.1, and their dominance coefficient
to be h = 0.3. By themselves,they generatea total genetic
load of 63%. The bottleneck causes that load to rise to
64%. Some 57 generations afterwardthe total load will
reach a minimum that is only 0.009% below the equilibrium value.
Combining the effects of lethals and mildly deleterious
mutations, it seems that even a severe bottleneck of 10
individualswill not change the total genetic load by more
than a few percentagepoints when populationsize recovers
very quickly after the bottleneck. With a low lethal mutation rate (U = 0.03), the load increases from 64% to
66%. With a high lethal mutation rate (U = 0.2), the load
increases from 70% to 75%. The reason these increases
are so modest is that the relativecontributionfrom highly
recessive mutations is greatly affected by the bottleneck,
but the absolutecontributionis still small. Mildlyrecessive
mutations contribute substantial genetic load, but that
contribution is not much affected. The decrease in the
load below its equilibrium that occurs during the overshoot is so small as to be negligible.
A bottleneck causes larger increases in the total load
when the population size recoversslowly. We will use results from stochasticsimulations for very slow population
growth (X = 1.2) and again combine the effects of lethal
and mildly deleterious mutations. With a low lethal mutation rate (U = 0.03), the load after the bottleneck rises
from 64% to 71%, still a small change. With a high lethal
mutation rate (U = 0.2), however, it rises from 70% to
81%. Here substantialdemographicconsequencesmay be
possible, as the number of viable zygotes is reduced by
about one-third (i.e., from 30% to 19%).
We conclude that for most organismsa singlebottleneck
will typicallyproduce only small changes in the total genetic load. Species with very slow per generation population growthand high lethal mutation ratesare a possible
exception.
Discussion
A bottleneck produces random fluctuations in allele frequencies. The effects on selectivelyneutral genetic variation have been well studied (e.g., Nei et al. 1975; Maruyama and Fuerst 1985; Cornuet and Luikart 1996), but
impacts on genes under selection are less well understood.
The simple models developed here show that a bottleneck
causes the inbreeding depression from deleterious mutations to decreaseand then returnto its initial equilibrium,
in some cases after a period of further "purging."The
genetic load shows quite a different pattern after a bottleneck. It increases,then falls below its equilibrium,then
finally returns to its equilibrium.
Quantitatively,the decline of inbreedingdepressionimmediately after a bottleneck is small. The proportional
decreaseis approximately1/(2N) regardlessof genetic pa-
Bottlenecks,Depression,and Load 165
rametersunless the bottleneck is extreme and the equilibrium depression very large. The inbreeding depression
from individual loci in later generationsdepends strongly
on their dominance. The contribution from loci with
weakly recessive mutations returns smoothly to equilibrium, but a purging process causes inbreedingdepression
from highly recessive mutations to decline far below the
equilibrium. Because highly recessivemutations are quite
rare, at least in Drosophila(Crow 1993), the total change
in inbreedingdepressionis expected to be only a few percentage points for bottlenecks of 10 individuals. Much
largerchanges in inbreedingdepressionare expected only
if bottlenecks are much smaller and highly recessivemutations more abundant than in organismslike Drosophila.
These conclusions assume that loci have multiplicativefitness effects and that the effects of linkagedisequilibriacan
be ignored.
These results call into question the suggestion made by
Lande and Schemske (1985) that a very brief bottleneck
might purge enough of inbreedingdepressionto cause an
outcrossing population to evolve selfing. A surprisingly
robust rule of thumb is that genes that increasethe selfing
rate slightly are favored if inbreeding depression is less
than one-half (Jarneand Charlesworth1993; Uyenoyama
et al. 1993;Charlesworthand Charlesworth1998). By this
criterion, a bottleneck might initiate a switch from outcrossing to selfing if it causes inbreeding depression to
decreasefrom greaterthan one-half to less than one-half.
Even for bottlenecks of only five individuals, the combinations of the genomic mutation rate and dominance coefficient that allow that to happen are quite restrictive
unless population growth is very slow. The potential for
a bottleneck of a large population to trigger a transition
to selfing therefore may be quite limited.
There are severalpossibilitiesstill to be exploredfor the
role that a temporary bottleneck might play in breeding
system evolution. One involves the associations between
genotypes at differentloci. Disequilibriabetweenthe genes
that alter the selfing rate and those that cause inbreeding
depression can cause departures from the one-half rule
(Uyenoyamaand Waller1991). Charlesworthet al. (1992)
studied the impact of these disequilibriain large populations and found that departuresfrom the one-half rule
are small unless the selfing modifiers have large effects.
The picture might be differentfollowing a bottleneck,particularly when population growth is slow, because drift
also generatesdisequilibria.We simplified our analysesby
neglecting the disequilibriaamong the selected loci and
those between the selected loci and the selfing modifiers.
Accountingfor these disequilibriamight broadenthe range
of conditions under which a bottleneckcould cause selfing
to evolve.
A second situation still to be explored is where the ef-
fective population size before the bottleneckis sufficiently
small that drift impactsthe deleteriousalleles.Ronfortand
Couvet (1995) found that spatial structurecan favor intermediate selfing rates under conditions that otherwise
lead to either pure selfing or pure outcrossing. Perhapsa
bottleneck acting on a population that is alreadyexperiencing drift could cause a transition to selfing. Third, a
bottleneck can reduce population size for many generations before the recovery begins, allowing inbreeding to
purge the depression.
Genetic factors are widely recognizedas a threat to the
persistenceof small populations (Frankham1995). A potential source of confusion on this topic is that conservation biologists sometimes use "inbreedingdepression"
to mean the decline in averagefitness caused by the expression of deleterious gene in a small, randomly mating
population. We refer to that quantity as the genetic load.
The confusion is natural since the increasedgenetic load
following a bottleneck is caused by matings between related individuals,even when mating is random.Inbreeding
depression in our sense (the relative fitness loss of selfed
vs. outcrossed progeny) nevertheless does have applications in conservationbiology. An example is in the design
of controlledbreedingprograms,where one has the choice
between breeding relatedand unrelatedindividuals(Ralls
and Ballou 1986).
This study has three results that may be useful for predicting the genetichealthof a populationthat goes through
a brief phase of small numbers.The first involves the time
course of the genetic load. It increases immediatelyafter
a bottleneck but can drop later below its equilibrium as
highly deleterious mutations are purged. A second conclusion regardsthe magnitude of these changes. Even for
bottlenecksas small as 10 individuals,changes in the load
may often be negligible. Largechanges will be seen only
if highly recessive deleterious mutations are substantially
more common than values that have been estimated in
Drosophilaand some annual plant species. It is possible
that some organisms have a very high rate of lethal mutations (which tend to be highly recessive). Very small
bottlenecks in these species could cause a substantialincrease in the total genetic load, particularlyif recovery
from the bottleneck is very slow. A final point here is that
the dynamics of the genetic load after a bottleneckwill be
very differentthan those of selectivelyneutralgenetic variation. Any rational design for population management
requires an explicit model that accounts for selection as
well as random genetic drift.
Our results show that inbreeding depression and the
genetic load behave very differentlyin response to a bottleneck. Perhaps the easiest way to understand this difference is with an extreme example. Consider a recessive
deleterious allele that is at equilibriumbetween selection
166 The AmericanNaturalist
and mutation. This allele will contribute to inbreeding
depression, because selfed offspring have a higher probabilityof being homozygous for it, and to the geneticload,
because its presence reduces the population's averagefitness. Imagine now that the population size is suddenly
reduced for a number of generationsand that in the following generationsthe deleterious mutation drifts to fixation. The inbreedingdepressionmust then drop to zero:
without genetic variation, there can be no difference in
the fitnesses of selfed and outcrossed offspring. But the
same is not true of the load. The population consists only
of mutant homozygotes, and so its load is greater than
that of the base population in which homozygotes were
rare.While this example considers only one possible outcome, our resultsshow that on averagea single generation
of drift does indeed have opposing effects on inbreeding
depression and the genetic load.
We have focused on a single bottleneckacting on a large
population in order to understandits effects in isolation
from other factors. Of course, this is not the only demographic possibility.Some populations experiencebottlenecks periodically,while others are permanentlyreduced
to a small size. In those situations, deleterious alleles experience random genetic drift chronically.Qualitatively,
one anticipatesthat inbreedingdepressionwill decline and
that the load will increase.Numerical studies have shown
that drift in a small population of constant size increases
the genetic load (Kimuraet al. 1963;Hedrick 1994;Lynch
et al. 1995) and reduces inbreeding depression (Charlesworth et al. 1992;Ronfort and Couvet 1995). These trends
are also seen in analyticresultsfor a single locus (Bataillon
and Kirkpatrick,in press).An analytictheory for the effects
of drift on the total genetic load and inbreedingdepression
remains to be developed.
Acknowledgments
We are extremely grateful for support from the Centre
National de Recherche Scientifique (for a Poste Rouge
fellowship to M.K. and funds from Institut des Sciences
de l'Evolution to P.J.),the Guggenheim Foundation, and
the National Science Foundation (grant DEB-9407969to
M.K.). We are most grateful for comments from T. Bataillon, B. Charlesworth,D. Charlesworth,D. Hall, M.
Lynch,S. Otto, D. Waller,J. Wang,M. Whitlock,and three
anonymous reviewers.This is contribution 2000-28 from
Institut des Sciences de l'Evolution.
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Associate Editor:SarahP. Otto