ARTICLE IN PRESS
Journal of Theoretical Biology 238 (2006) 279–285
www.elsevier.com/locate/yjtbi
The distribution of fitness effects among beneficial mutations
in Fisher’s geometric model of adaptation
H. Allen Orr!
Department of Biology, University of Rochester, Rochester, NY 14627, USA
Received 4 March 2005; received in revised form 10 May 2005; accepted 19 May 2005
Available online 28 June 2005
Abstract
Recent models of adaptation at the DNA sequence level assume that the fitness effects of new mutations show certain statistical
properties. In particular, these models assume that the distribution of fitness effects among new mutations is in the domain of
attraction of the so-called Gumbel-type extreme value distribution. This assumption has not, however, been justified on any
biological or theoretical grounds. In this note, I study random mutation in one of the simplest models of mutation and adaptation—
Fisher’s geometric model. I show that random mutation in this model yields a distribution of mutational effects that belongs to the
Gumbel type. I also show that the distribution of fitness effects among rare beneficial mutations in Fisher’s model is asymptotically
exponential. I confirm these analytic findings with exact computer simulations. These results provide some support for the use of
Gumbel-type extreme value theory in studies of adaptation and point to a surprising connection between recent phenotypic- and
sequence-based models of adaptation: in both, the distribution of fitness effects among rare beneficial mutations is approximately
exponential.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Adaptation; Extreme value theory; Geometric model; Mutational landscape
1. Introduction
Recent work in the theory of adaptation has focused
on DNA sequence models. Real adaptation in real
organisms must, after all, occur in a space of alternative
DNA sequences. These recent efforts build on pioneering work by John Maynard Smith (1962, 1970) and John
Gillespie (1983, 1984, 1991), who emphasized that, with
realistically low mutation rates, natural selection can
only ‘‘see’’ mutant sequences that differ from wild-type
by a single base-pair change: double and triple, etc.
mutants are too rare to be of much significance to
molecular evolution. Gillespie further emphasized that,
because the wild-type allele typically enjoys high fitness
and adaptation involves the substitution of sequences
!Tel.: +585 275 3838; fax: +585 275 2070.
E-mail address: [email protected].
0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2005.05.001
having yet higher fitness, almost all adaptive evolution
occurs among the fittest few alleles locally available at a
locus or small genome. Put differently, almost all
adaptation occurs within the right-hand tail of the
distribution of allelic fitnesses (Gillespie, 1991; Orr,
2003, 2005). As Gillespie further argued, this means that
we can import extreme value theory—a body of
probability theory that characterizes extreme draws
from distributions (Gumbel, 1958; Leadbetter et al.,
1983; Embrechts et al., 1997)—into the study of
adaptation.
Gillespie (1983, 1984, 1991) used extreme value theory
to characterize the statistical properties of molecular
evolution in his ‘‘mutational landscape model,’’ a model
of adaptation over rugged fitness landscapes. More
recent work has used extreme value theory to study the
genetics of adaptation in this model. Orr (2002), for
instance, showed that if the wild-type allele represents
the ith fittest allele (more precisely, single base-pair
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H.A. Orr / Journal of Theoretical Biology 238 (2006) 279–285
changes to the wild-type yield i ! 1 beneficial mutations), natural selection will on average substitute a
mutant allele having fitness rank ði þ 2Þ=4 at the next
step in adaptation. It has also been shown that the mean
selection coefficient, s, fixed at subsequent steps in
adaptation falls off as an approximate geometric
sequence (Orr, 2002), and that parallel evolution should
be common at the DNA sequence level; indeed parallel
evolution should occur about twice as often under
positive selection as under neutrality (Orr, 2004a).
Finally, it has been shown that new beneficial mutations
have exponentially distributed fitness effects (Orr, 2003).
These results, like most that depend on extreme value
theory, are robust to many biological details. Most
important, these results hold for many possible distributions of allelic fitnesses—a distribution that is almost
always unknown. Studies of the mutational landscape
model do, however, depend on certain assumptions
about the tail behavior of the distribution of allelic
fitnesses.
In particular, all studies of the mutational landscape
model assume that the right tail of the distribution of
allelic fitnesses falls within the domain of attraction of
the so-called Gumbel-type extreme value distribution
(EVD), which has cumulative distribution function
LðxÞ ¼ exp½! expð!xÞ'. An EVD describes the distribution of maxima (or a linear transformation of maxima)
drawn from a distribution. In reality, there are three
different types of extreme value distribution (Gumbel,
1958; Leadbetter et al., 1983; Embrechts et al., 1997).
The Gumbel type holds for almost all ‘‘ordinary’’
distributions, including the exponential, gamma, normal, lognormal, and logistic. The Frechet type holds for
very heavy-tailed distributions, like the Cauchy, that
lack all or higher moments. The Weibull type holds for
many (though not all) distributions that are truncated
on the right.
There are good reasons why the theory of adaptation
has, so far, assumed that the distribution of allelic
fitnesses is of the Gumbel type. For one thing, the
Gumbel type was the focus of classical extreme value
distribution and arguably is better understood than the
alternatives; indeed the Gumbel type is often referred to
as the EVD. More important, the Gumbel type holds for
a wider range of distributions than the Frechet and
Weibull types (Embrechts et al., 1997). Although it is
sometimes claimed that the Gumbel EVD holds only for
exponential-like distributions, this is misleading. In
reality, distributions having infinite or finite (truncated)
right end-points can belong to the Gumbel type
(Leadbetter et al., 1983; Embrechts et al., 1997).
Moreover, distributions whose tails are lighter than
exponential (‘‘subexponential,’’ like the lognormal), or
whose tails are heavier than exponential (‘‘superexpoential,’’ like the normal) can belong to the Gumbel type
(Embrechts et al., 1997, pp. 138, 145, 277).
The other EVD types may also be inappropriate
biologically. The Weibull type, for instance, appears
inappropriate as it is hard to see why there should, in
principle, be a ceiling on the highest fitness possible at a
gene. (In any given case, i.e. given a particular wild-type
allele, there is a best possible mutant allele, but that is a
different matter; see Section 3.) The situation may be
worse for the Frechet type, which does not easily allow
weak selection (extreme draws from heavy-tailed distributions are separated by large spacings). Also,
because the Frechet type holds for distributions lacking
all or higher moments, we would have no guarantee that
mean fitness at a gene could even be defined.
These arguments are, however, obviously not decisive. In
this note, I present some support for the Gumbel
assumption. In particular, I show that random mutation
in Fisher’s (1930) geometric model of adaptation gives rise
to a distribution of mutational fitness effects of the Gumbel
type. Fisher’s geometric model represents one of the
simplest and best studied models of mutation and
adaptation. The model pictures a population as a point
in a high-dimensional phenotypic space, in which each axis
represents a trait. The population is assumed to be
presently off the (local) phenotypic optimum and moves
closer to it by producing random mutations. These
mutations are represented by vectors having some magnitude and random direction in phenotypic space. Mutations
that fall closer to the optimum are beneficial, while those
that fall farther away from the optimum are deleterious;
because fitness declines monotonically with distance from
the optimum (i.e. the landscape is locally smooth), one can
calculate the fitness effect of any mutation.
Fisher (1930) used this geometric model to calculate
the probability that a mutation of some phenotypic size
will be beneficial. He showed that this probability falls
off very rapidly with the size of a mutation; Fisher
interpreted this to mean that mutations of very small
phenotypic effect must be the stuff of adaptation.
Kimura (1983) showed, however, that, when taking into
account the stochastic loss of beneficial mutations, the
distribution of phenotypic effects among mutations
fixed at the first step in adaptation is bell-shaped, with
mutations of intermediate effect getting substituted most
often (also see Otto and Jones, 2000). Finally, Orr (1998)
showed that, when integrating over entire adaptive
walks (which may involve many substitutions), the
distribution of phenotypic effects among mutations
fixed during adaptation is nearly exponential.
Here I show that random mutation in Fisher’s model
gives rise to a distribution of mutational fitness effects of
the Gumbel type. I also show that the distribution of
fitness effects among beneficial mutations in Fisher’s
model is approximately exponential.
My approach is mostly analytic. But because this
work involves several approximations, I check the
accuracy of all results with exact computer simulations.
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H.A. Orr / Journal of Theoretical Biology 238 (2006) 279–285
2. Model and results
2.1. Preliminaries
Following Fisher (1930, pp. 39, 40), I first consider
‘‘changes of a given magnitude’’ that ‘‘occur at random
in all directions’’ in the geometric model. To see what
this means, consider for the moment a simple organism
comprised of just two characters. Evolution in such a
species can be studied in a two-dimensional space
(Fig. 1). A large population currently resides at point
A, which is distance z from the phenotypic optimum
(point O). For convenience, the optimum sits at the
origin of the x ! y coordinate system shown. Fitness
falls off from the optimum at the same rate in all
directions. The fitness function describing this decline
might be linear, quadratic, Gaussian, etc. Mutations are
vectors of fixed magnitude r having variable angle y,
where y ¼ 0 points directly to the optimum and y ¼
(p=2 points perpendicular to the optimum. Because
mutations are random in direction, the density of
mutations on the circumference of the small ‘‘mutation
circle’’ shown in Fig. 1 is uniform.
We are most interested in the higher dimensional case
since, as Fisher emphasized, the essence of adaptation is
that organisms must conform to their environment in
many ways. Fortunately evolution in many, n, dimensions can be collapsed mathematically to evolution in
two dimensions. To do so, we rotate axes such that the
organism is perfectly adapted at n ! 1 axes and
A
z
r
O
Fig. 1. Fisher’s geometric model. In the case shown, an organism is
comprised of only two characters (x and y axes). The locally optimal
combination of trait values is at point O, while the population
currently sits at point A, a distance z from the optimum. The
population produces mutations of fixed magnitude r that have random
direction in phenotypic space (represented by the smaller mutation
circle; one mutation is shown). Any mutation that falls within the large
circle is favorable, while any mutation that falls outside the large circle
is deleterious.
maladapted at one. Only displacements onto this special
axis are then relevant and we can replace the problem of
keeping track of the distribution of n ! 1 angles in n
dimensions with that of finding a single distribution of
angles in two dimensions. Fisher (1930), Leigh (1987)
and Hartl and Taubes (1996) showed how this can be
done (the first implicitly, the latter two explicitly). In
particular, these authors showed that the quantity
pffiffi
y ¼ n cos y
(1)
is a standard normal random variable for large n, where
y is the angle between the mutation vector and the axis
that leads to the optimum. This means that most
mutations point perpendicular to the optimum (since the
density peaks at y ¼ 0, which occurs when y ¼ (p=2)
and fewppoint
to the optimum (since the density is small
ffiffi
at y ¼ n, which occurs when y ¼ 0). This captures the
biological fact that a random mutation is unlikely to
point towards the optimum in a complex organism as
there are many ways of ‘‘going wrong’’ in a high
dimensional space. Appendix 1 presents a heuristic
derivation of Eq. (1) that is based on Hartl and Taubes’s
(1996) more rigorous one.
Throughout the following analysis, I assume that
adaptation occurs in a high dimensional space (n is
large) and that beneficial mutations are rare. In practice,
this means that r is fairly large relative to z.
2.2. Distribution of distances traveled to the optimum
The fitness effect of a mutation depends on how far it
travels to or away from the optimum. As Hartl and
Taubes (1996) first showed, geometric considerations in
two dimensions reveal that a mutation having angle y
travels a distance
Dz ) r cos y ! r2 =2z,
(2)
2
where the derivation ignores terms of ðDzÞ . (These
terms can be ignored only when mutations travel small
to modest distances or from the optimum. As Hartl and
Taubes (1996) show, and as noted above, this is
generally true with large n.) As the historical convention
is that Dz40 for beneficial mutations, Eq. (2) shows that
the largest angle possible among such mutations is
y0 ¼ arccos½r=2z'.
pffiffi
Changing variables (y ¼ n cos y), we have
pffiffi
r
Dz ) pffiffi ðy ! r n=2zÞ
n
r
¼ pffiffi ðy ! xÞ,
ð3Þ
n
pffiffi
where x ¼ r n=ð2zÞ. The quantity x corresponds to
Fisher’s standardized measure of mutational size.
We know the approximate distribution of distances
traveled to the optimum. Because (3) shows that Dz is
approximately linear in y—and y is normal—f ðDzÞ is
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also approximately normal. In particular,
"
#
pffiffi
n
1
n½Dz þ r2 =ð2zÞ'2
exp !
f ðDzÞ ) pffiffiffiffiffi
,
2r2
2p r
(4)
which has mean and variance
r2
,
2z
r2
Var½Dz' ¼ .
n
2.3. Distribution of fitness effects among mutations
E½Dz' ¼ !
ð5Þ
Because Eq. (4) has a negative mean, random
displacements in phenotypic space yield more deleterious than beneficial mutations, as emphasized by Fisher.
I checked the accuracy of the approximation in
Eq. (4) with exact computer simulations of Fisher’s
model. These simulations are identical to those described in Orr (1998, 1999, 2000). Fig. 2 shows the
results and confirms that Dz is approximately normally
distributed, although the linear approximation in
Eqs. (2) and (3) is clearly imperfect. Fortunately, the
results below will only depend on the right tail of Dz,
which involves very small Dz, and where our approximations in Eqs. (2) and (3) are quite accurate. The figure
shows the case in which n ¼ 100 dimensions, z ¼ 1 unit
to the optimum, and mutations have magnitude r ¼ 0:5.
With these parameter values, Fisher’s standardized
mutational size x ¼ 2:5 and beneficial mutations are rare
(Fisher’s probability of being favorable is p ¼ 0:0062).
We have thus arrived at a pleasingly simple, albeit
approximate, picture of mutation in Fisher’s model: Dz
is approximately normally distributed with a negative
mean and beneficial mutations are those rare displacements in which Dz40. Because the mean Dz is
0.06
0.05
Frequency
The fact that the tail of the distribution of Dz is given
by the tail of a normal distribution has an important
implication: because the normal distribution is in the
domain of attraction of the Gumbel-type EVD, the
distribution of mutational effects in Fisher’s model, as
measured by Dz, is of the Gumbel type.
It also follows that the right tail of the distribution of
fitness effects ðDW Þ among mutations is of the Gumbel
type. This is trivially true if fitness declines linearly with
distance from the optimum. But it remains true to a
good approximation if the fitness function is, for
example quadratric or Gaussian. The reason is that
EVD type depends only on the tail of a distribution. But
the right tail of f ðDzÞ involves very small beneficial
displacements Dz (Fig. 2), reflecting the fact that it is
very hard to travel a considerable distance to the
optimum in a high dimensional space. But with small
displacements DW is nearly linear in Dz. With a
Gaussian fitness function, for example DW ¼
exp½!ðz ! DzÞ2 =2' ! exp½!z2 =2', which, with small Dz,
is DW ) zDz expð!z2 =2Þ. Random mutation in Fisher’s
model thus yields a distribution of mutational fitness
effects that, to the order of our approximations, is of the
Gumbel type; we will see far clearer evidence of this
below. This is our first main conclusion.
2.4. Asymptotic distribution of fitness effects among
beneficial mutations
0.07
The distribution of distances traveled to the optimum
by beneficial mutations is given by the right tail of a
normal distribution. As Fig. 2 makes clear, if we focus
on the case in which beneficial mutations are rare (which
they surely are), we can think of beneficial mutations as
rare excesses above a high threshold of Dz ¼ 0.
An important result in extreme value theory, first
proved by Pickands (1975), shows that the distribution
of excesses above a high threshold, u, assumes a limiting
form given by the Generalized Pareto Distribution
(GPD). The GPD has cumulative distribution function
obs
exp
0.04
0.03
0.02
0.01
0
-0.01
-0.3
independent of n, while the variance in Dz shrinks with
increasing n, the proportion of mutations that cross the
threshold of Dz ¼ 0 and so are beneficial decreases as n
increases. In words, beneficial mutations are rarer in
more complex organisms.
-0.25
-0.2 -0.15 -0.1 -0.05
0
Distance to/from Optimum (delZ)
0.05
Fig. 2. Distance traveled to or from the optimum in Fisher’s model.
The figure compares the results of exact computer simulations
(observed) to the approximation in Eq. (4) (expected). The case of
n ¼ 100 dimensions, z ¼ 1 unit to the optimum, and r ¼ 0:5 is shown.
Plot reflects approximately 32,000 random mutations.
G e;bðuÞ ðyÞ ¼ 1 ! ½1 þ ey=bðuÞ'!1=e ,
(6)
where bðuÞ is a positive function, e is a shape parameter,
and the GPD is defined for yX0 when eX0 and
0pyp ! bðuÞ=e when eo0. The three cases e40, eo0,
and e ¼ 0 yield three different types of ‘‘excess distribution’’ that correspond to the three types of extreme value
distributions discussed earlier: Frechet, Weibull, and
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H.A. Orr / Journal of Theoretical Biology 238 (2006) 279–285
Gumbel, respectively. In the case of e ¼ 0, the GPD is
given by the limit of Eq. (6) as e ! 0, i.e. the GPD is
exponential. (There is, in practice, no sharp cutoff at
which a distribution switches from being in the domain
of attraction of the Gumbel EVD to one of the other
EVDs. For e sufficiently close to zero, a distribution
behaves as though it is of the Gumbel type; agreement
with Gumbel theory gradually worsens as e deviates
farther from zero.)
The important point is that, since f ðDzÞ is nearly
normal, e*0 and f ðDzÞ is of the Gumbel type.
Consequently, the distribution of Dz among rare
beneficial mutations is approximately exponential:
F ðDzjDz40Þ ) 1 ! exp½!Dz=bDz ðuÞ'.
(7)
Again, because DW is linear in Dz for rare beneficial
mutations, the asymptotic distribution of fitness effects
among rare beneficial mutations in Fisher’s model is
also approximately exponential:
F ðDW jDW 40Þ ) 1 ! exp½!DW =bDW ðuÞ'.
(8)
This is our second main conclusion.
Fig. 3 shows the results of exact computer simulations
that confirm that rare beneficial mutations in Fisher’s
model are approximately exponentially distributed.
While Fig. 3 shows the case in which n ¼ 100, z ¼ 1,
and r ¼ 0:5, different parameter values yielded similarly
good agreement with theory. (Varying the precise shape
of the monotonic fitness function also did not affect this
result; not shown.) So long as beneficial mutations are
rare, their fitness effects in Fisher’s model are nearly
exponential.
1
Frequency
0.1
0.01
0.001
0.0001
0
0.01
0.02
0.03 0.04 0.05
Fitness Effect
0.06
0.07
Fig. 3. Beneficial fitness effects in Fisher’s model are approximately
exponentially distributed (straight line on a semilog plot). Fitness
declined as a Gaussian function with distance from the optimum.
Parameter values are the same as in Fig. 2. Plot reflects 5000 beneficial
mutations.
283
The exponentiality shown in Fig. 3 provides good
evidence that mutational fitness effects in Fisher’s model
are of the Gumbel type—exponential excess distributions are characteristic of distributions in the domain of
attraction of the Gumbel EVD (Embrechts et al., 1997).
2.5. Distributions of mutational magnitudes
So far, we have considered the simple case in which all
mutations have fixed magnitude, r. It is worth briefly
considering the more complex case in which mutations
have a distribution of magnitudes: r might, for example
be exponentially, gamma, or uniformly distributed. In
this case, the distributions of Dz and DW become
complicated mixture distributions. While it is unclear if
we can make much analytic progress with these
distributions, it is trivial to answer our main question
by computer simulation: Does the distribution of
beneficial fitness effects remain approximately exponential even when mutations have a distribution of
magnitudes?
The answer appears to be yes. Fig. 4a shows the excess
distribution (Dz40) given gamma distributed mutational magnitudes, r; Fig. 4b shows analogous results
given uniform r. Computer simulations with many other
mutational distributions yielded similar results. So long
as beneficial mutations are rare, the distribution of their
fitness effects is approximately exponential. This is our
third main conclusion.
3. Discussion
I have studied Fisher’s geometric model in the case in
which a small proportion of random mutations is
beneficial. (More precisely, I have assumed that the
number of dimensions, n, is large and that the
magnitude of mutational effects, r, is modest to large
relative to the distance to the optimum, z. Under these
conditions, few random mutations are beneficial.) This
rare-beneficial-mutation scenario is the same as that
considered in the mutational landscape model of
adaptation at the DNA level (Gillespie, 1983, 1984,
1991; Orr, 2002, 2003). I have arrived at three
conclusions.
First, the distribution of fitness effects among random
mutations of fixed magnitude r is of the Gumbel type.
This is perhaps not surprising given that a wider variety
of distributions fall within the domain of attraction of
the Gumbel-type EVD than within the domains of
attraction of the Frechet or Weibull types (Embrechts et
al., 1997). The present result was not, however, obvious
a priori. Indeed, given that the distance to the optimum
in Fisher’s model is finite, one might have guessed that
mutational effects would be of the Weibull type, which
holds for many truncated distributions (beneficial
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1
Frequency
0.1
0.01
0.001
0.0001
0
(a)
0.01
0.02
0.03
0.04
0.05
Fitness Effect
1
Frequency
0.1
0.01
0.001
0.0001
0
(b)
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Fitness Effect
Fig. 4. Beneficial fitness effects in Fisher’s model are approximately
exponentially distributed even when random mutations have a
distribution of magnitudes, r. (a) Mutations were gamma distributed
with scale parameter a ¼ 2 and shape parameter b ¼ 2. The
mutational distribution was thus bell-shaped with E½r' ¼ 1 and
Var½r' ¼ 1=2; (b) mutations were uniformly distributed between
0prp2, i.e. mutations spanned the diameter of Fisher’s ‘‘sphere.’’ In
both a and b, beneficial mutations were rare (Fisher’s probability of
being beneficial was p ¼ 0:025 in a, and p ¼ 0:030 in b and fitness
declined as a Gaussian function with distance from the optimum. Each
plot reflects 5000 beneficial mutations.
mutations in Fisher’s model cannot be larger than twice
the distance to the optimum). It appears, however, that
because beneficial mutations travel a very short distance
to the optimum, beneficial displacements are so small
relative to the distance to the optimum that this ceiling
on favorable effects is essentially irrelevant. From the
fact that the distribution of mutational effects ðDW Þ in
Fisher’s model is of the Gumbel type, it follows that the
distribution of fitnesses among mutant alleles (W ) is
also of the Gumbel type (as these fitnesses are the sum of
a constant wild-type fitness and DW ). The present
results may thus provide some support for the assump-
tion that fitness distributions are of the Gumbel type
(Gillespie, 1983, 1984, 1991; Orr, 2002, 2003). Although
we obviously do not know the distribution of fitness
effects at any locus in nature, we at least know that
Gumbel-type distributions arise in a simple and reasonably natural model of mutation.
Second, the distribution of fitness effects among rare
beneficial mutations in Fisher’s geometric model is
approximately exponential. More formally, it can be
shown that the excess distribution above a high threshold (e.g. wild-type fitness) is given by the GPD and that
the GPD collapses to an exponential for distributions of
the Gumbel type (Pickands, 1975; Embrechts et al.,
1997). The present result obviously mirrors the earlier
finding that fitness effects among beneficial mutations in
the mutational landscape model are exponential (Orr,
2003). In both cases, exponentiality of beneficial effects
holds only when beneficial mutations are rare.
Third, computer simulations show that, even when
mutations have a distribution of mutational magnitudes,
the distribution of fitness effects among rare beneficial
mutations remains roughly exponential, at least for the
range of distributions studied.
The fact that rare beneficial mutations in Fisher’s
model have approximately exponentially distributed
effects suggests a connection between the phenotypic
and sequence-based models of adaptation. Although it is
important not to exaggerate the closeness of this
connection—the two models differ in several respects—it is clear that favorable mutations show similar
statistical properties in the geometric and mutational
landscape models of adaptation. In retrospect, this
similarity is perhaps not surprising given that the models
yield several similar results (Orr, 2004b). It seems
plausible that these similarities at least partly reflect
the fact that the raw material of adaptive evolution—
beneficial mutations—have the same nearly exponential
distribution of fitness effects in both Fisher’s phenotypic
and Gillespie’s sequence-based models. Put differently,
the present findings may suggest that many of the
admittedly artificial features of Fisher’s geometric model
(e.g. isotropic mutational effects over orthogonal
characters) may not be of much significance. The
model’s behavior may, to a considerable extent, depend
only on the key fact that random mutation in Fisher’s
phenotypic space gives rise to an approximately
exponential distribution of fitness effects among beneficial mutations.
Acknowledgments
I thank A. Betancourt, Y. Kim, and two anonymous
reviewers for helpful comments. This work was supported by NSF grant DEB-0449581.
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Appendix 1
While Hartl and Taubes (1996) show rigorously that y
in Eq. (1) is standard normal, the following heuristic
argument may be of some value.
Consider a population that sits at point A in an ndimensional space away from the optimum O. Rotate
axes so that the population is perfectly adapted at all
characters but one. The distance between A and O along
this special axis gives the distance to the optimum. The
population makes random mutations of magnitude r.
The resulting mutations can be pictured as a ‘‘sphere’’
(actually a hypersphere) of radius r originating at A.
Picture a new coordinate system centered at A.
Mutations have displacements onto each of the n axes
of this coordinate system, including the axis leading to
the optimum. The distribution of displacements onto
any axis of the mutant hypersphere is determined by a
large number of random variables, in particular by the
aggregate effects of n ! 1 angles. A central limit
argument thus suggests that the distribution of displacements, r cos yi , onto each axis should p
beffiffi roughly
normal. That the particular quantity
n cos y is
standard normal, follows from the fact that the sum of
squared
displacements on each axis equals r2 :
Pn
2
2
2
i¼1 ðr cos yi Þ ¼ r or E½ðcos yÞ ' ¼ 1=n. But this ex2
pectation is just E½cos y' þ Var½cos y' ¼ 1=n. Because
displacements are symmetric about zero, E½cos y' ¼ 0
and Var½cos y' ¼ 1=n. But if
pfficos
ffi y is normal with the
above mean and variance, n cos y must be standard
normal.
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