Evolution, 54(1), 2000, pp. 13–20
ADAPTATION AND THE COST OF COMPLEXITY
H. ALLEN ORR
Department of Biology, University of Rochester, Rochester, New York 14627
E-mail: aorr@uhura.cc.rochester.edu
Abstract. Adaptation is characterized by the movement of a population toward a many-character optimum, movement
that results in an increase in fitness. Here I calculate the rate at which fitness increases during adaptation and describe
the curve giving fitness versus time as a population approaches an optimum in Fisher’s model of adaptation. The
results identify several factors affecting the speed of adaptation. One of the most important is organismal complexity—
complex organisms adapt more slowly than simple ones when using mutations of the same phenotypic size. Thus, as
Fisher foresaw, organisms pay a kind of cost of complexity. However, the magnitude of this cost is considerably
larger than Fisher’s analysis suggested. Indeed the rate of adaptation declines at least as fast as n21, where n is the
number of independent characters or dimensions comprising an organism. The present results also suggest that one
can define an effective number of dimensions characterizing an adapting species.
Key words.
Adaptation, complexity, Fisher’s geometric model, fundamental theorem of natural selection, pleiotropy.
Received February 25, 1999.
Many authors have argued that evolution is (or is not)
characterized by a trend of increasing complexity. Often this
is taken to mean an increase in morphological complexity,
for example, in the number of independent parts or components comprising an organism, although the precise meaning
of ‘‘independent part’’ varies from author to author (Vermeij
1974; McShea 1993, 1996; Wagner and Altenberg 1996; for
a recent review, see McShea 1998). There are, of course,
many advantages that potentially accrue to complex organisms, including the ability to invade specialized niches. But
it is not often emphasized that certain disadvantages may
also accompany increases in complexity.
Fisher (1930) was one of the first to see this. In his classic
discussion of adaptation in The Genetical Theory of Natural
Selection, Fisher calculated the probability that random mutations of a given phenotypic size would be favorable, famously showing that this probability falls rapidly: Small mutations are reasonably likely to be favorable, but large ones
are not. Indeed very large mutations suffer vanishingly small
chances of being advantageous. But Fisher further showed
that mutations of a given size are less likely to be favorable
in complex than simple organisms. The reason is intuitively
clear: A random mutation of some size is more likely to
disrupt (not improve) a complex than simple organism for
the same reason a random change of some size is more likely
to disrupt (not improve) a complex than simple machine.
Changing the length of an arbitrary mechanical part by one
inch, for instance, is more likely to derail the function of a
microscope than a hammer. Fisher quantified this effect by
considering the evolutionary fate of a random mutation of
size r in an organism comprised of n parts or dimensions.
Considering a maladapted population that presently sits a
distance d/2 from the phenotypic optimum, he showed that
the probability a mutation will be favorable attains ‘‘exceedingly small values . . . when the number of dimensions
is large, even while r is still small compared to d.’’ Indeed
‘‘probability of improvement will be determined by the ratio
which the particular change bears to [a] standard magnitude,’’
given by d/Ïn. In words, the chance that a mutation of given
size will be favorable declines (in a specifiable way) with
the complexity, n, of an organism.
This fact suggests that organisms pay a cost of complexity:
Because mutations are less likely to be favorable in complex
than simple organisms, rates of substitution of favorable mutations might, all else being equal, be lower in complex organisms.
Unfortunately, it is now clear that Fisher’s analysis was
incomplete. Kimura (1983) pointed out that Fisher overlooked an important population genetic feature of adaptation—favorable mutations must escape stochastic loss when
rare. Although small mutations enjoy greater probabilities of
being favorable, they suffer smaller probabilities of fixation
when favorable. Taking both factors into account undermines
several of Fisher’s conclusions about the size of mutations
underlying adaptation (Kimura 1983; Orr 1998). Similarly,
Fisher’s conclusions about the relationship between organismal complexity and adaptation must remain uncertain: How
much more slowly (if at all) do complex organisms adapt
when taking into account the fact that adaptation requires
both that a mutation be favorable and that it escape accidental
loss?
Here I calculate the total cost of complexity during adaptation. In particular, I calculate the rate of increase in fitness, dw̄/dt, characterizing adaptation in complex versus simple organisms in Fisher’s geometric model of evolution when
all organisms adapt via mutations of the same size. We will
see that complex organisms do pay a cost of complexity.
Indeed this cost is considerably larger than Fisher’s analysis
suggested.
THE MODEL
AND
RESULTS
Preliminary Comments
We consider Fisher’s (1930) geometric model of adaptation in which fitness is determined by n independent (orthogonal) characters. We can thus think of adaptation as occurring in n-dimensional space: If an organism is comprised
of just two characters, evolution can be pictured in a Cartesian
coordinate system in which each axis gives a trait value. In
13
q 2000 The Society for the Study of Evolution. All rights reserved.
Accepted July 16, 1999.
14
H. ALLEN ORR
the case of three characters, evolution can be pictured in a
three-dimensional space, and so on. At each of our n characters, fitness falls off as a Gaussian function of distance, zi,
from the optimum, which resides at the origin of our coordinate system. If characters are scaled such that fitness decreases at the same rate per character, total fitness is a simple
2
function of total distance from the optimum: wtot 5 e2z tot /2,
2
2
2
where ztot 5 Ïz1 1 z2 1 . . . 1 zn. For simplicity, we will
usually drop the subscript and refer to the total distance from
the optimum as z. (This equals d/2 in Fisher’s original notation.)
Organisms adapt by producing random mutations. A mutation’s phenotypic effect is represented by a vector of some
magnitude with a random direction in phenotypic space. Fisher’s model thus allows for a kind of universal pleiotropy:
Any mutation potentially affects any character, though some
characters will be far more profoundly affected than others.
During adaptation, the distance to the optimum progressively
shrinks. Considering the simplest case in which the optimum
remains fixed during the bout of adaptation studied, we hope
to find the rate at which fitness increases through time. Our
approach will be straightforward: We first find the rate at
which the distance from the optimum shrinks through time
(dz/dt). From this, it is easy to find the rate at which fitness
increases through time (dw̄/dt).
We assume that all organisms, regardless of complexity,
begin a fixed distance z from the optimum and produce mutations of fixed phenotypic size r. The ratio r/z thus remains
constant across all levels of complexity. In the case of a
perfect mutation that points directly to the optimum, the same
distance to the optimum would be traveled in both simple
and complex organisms. Although we have no guarantee that
simple and complex organisms produce mutations of the
same size, our assumption of constant r provides a convenient
way to calibrate the effects of complexity per se on adaptation, keeping all else equal.
All of the analytic results below were checked against exact
computer simulations, which were similar to those in Orr
(1998). For the sake of brevity, I do not present these results
(with one exception); in all cases, however, simulations confirmed that the analytic results are very accurate. In the course
of this work, we will have occasion to refine several of the
(rather wild) approximations used in Orr (1998). Because the
present results are more exact than those in Orr (1998), the
mathematics are more complicated. I have thus placed a good
deal of mathematical detail in Appendices 1 and 2.
The Rate of Adaptation
We consider a sexual haploid in which evolution is due to
new unique mutations. The rate of adaptive substitution is k
5 NmPaP, where N is population size, m is the total mutation
rate, Pa is the proportion of random mutations that are favorable, and P is the probability of fixation (conditional on
mutations being favorable). When a favorable substitution
occurs, the distance to the optimum changes by a mean
amount 2D¯ zfix, where the subscript indicates that we are considering a fixed, and not merely favorable, mutation. (This
distinction will prove important.) The distance to the optimum thus decreases at a rate
dz
¯ z fix .
5 2NmPa PD
dt
(1)
The probability of fixation of a favorable mutation is P ø
2sfav, assuming fairly weak selection. This selection coefficient will obviously be a function of the distance, Dzfav, a
mutation travels toward the optimum: Mutations that reach
farther to the optimum enjoy larger advantages and thus larger probabilities of fixation. (P is, therefore, a mean, where
we average over mutations having different directions and
traveling different distances to the optimum. It can be shown
that the expectation of the product Pa P Dzfix in eq. (1) equals
the product of the individual expectations.) With Gaussian
fitness, sfav 5 exp {[z2 (z 2 Dzfav)2]/2} 2 1 ø z Dzfav with
weak selection, and thus
¯ z fav .
P ø 2zD
(2)
It is important to appreciate the difference between D̄zfav
and D̄zfix. The former is the mean distance traveled to the
optimum by mutations of size r conditional on a mutation
being favorable, whereas the latter is the distance traveled to
the optimum conditional on a mutation getting fixed. D̄zfix
will clearly be larger than D̄ zfav as mutations that travel further toward the optimum are more likely to get fixed.
Substituting for P in equation (1), the distance from the
optimum changes at a rate
dz
¯ z fav D
¯ z fix .
5 22NmPa zD
dt
(3)
The Distance Traveled to the Optimum
Fisher (1930) gives the probability, Pa, that a random mu2
`
tation of size r is favorable: Pa 5 (1/Ï2p) #rÏn/2z
e2y /2 dy.
Our immediate task, therefore, is to find D¯ zfav and D¯ zfix.
D̄zfav has been elegantly derived by Hartl and Taubes (1996).
Appendix 1, which presents a slightly simplifed version of
their derivation, shows that
D̄z fav
r
5
Ïn
E
`
x
(y 2 x)e2y
E
2 /2
dy
,
`
2y 2 /2
e
(4)
dy
x
where x is Fisher’s standardized measure of the size of a
mutation (x 5 r Ïn/2z). A series expansion shows that, for
small mutations, D¯ zfav ø (Ï2/p )(r/Ïn ). D¯ zfav is, therefore,
roughly proportional to r/Ïn, which makes good intuitive
sense: Larger mutations reach farther to the optimum than
smaller and random mutations have more ways of ‘‘going
wrong’’ in complex organisms.
The quantity D̄zfix is also derived in Appendix 1, which
shows that
E
E
`
D̄z fix
r
5
Ïn
(y 2 x) 2 e2y
2 /2
dy
x
.
`
(y 2 x)e2y
2 /2
(5)
dy
x
A series expansion reveals that, for small mutations,
15
ADAPTATION AND COMPLEXITY
FIG. 1. Rate of fitness increase as a function of mutation size, r.
In the example shown, n 5 50 and z 5 1. Note that the rate of
fitness increase peaks for intermediate-sized mutations. On Fisher’s
x scale, this maximum occurs at x ø 0.925.
D̄zfix ø (Ïp/2)(r/Ïn). In other words, the ratio of integrals
is in the neighborhood of unity for mutations of small to
modest size and the distance traveled towards the optimum
by the average fixed mutation is very roughly r/Ïn , as noted
by Orr (1998). Equation (5) is, however, a considerable improvement over Orr’s approximation. Indeed exact computer
simulations (not shown) show that (5) is extremely accurate.
We are now in a position to write the rate of approach to
the optimum in purely phenotypic terms. Substituting for Pa,
¯ zfav, and D
¯ zfix, in equation (3), we get
D
dz
2Nmr 2
52
Mz,
dt
n
(6)
where M 5 (1/Ï2p) #`x (y 2 x)2 e2y /2 dy.
2
The Rate of Change in Fitness
We can now calculate the rate of change in fitness during
an adaptive walk to the optimum. Because dw̄/dt 5
(dw̄/dz) (dz/dt) and, with Gaussian fitness, (dw̄/dz) 5 2z exp
(2z2/2), it follows that
dw̄
4Nmr 2
52
Mw̄ ln w̄,
dt
n
(7a)
where, in M, x 5 rÏn/(2Ï22 ln w̄). Because we are not particularly interested in the effects of population size or total
mutation rate, we can rewrite equation (7a) in time units of
(Nm)21 generations (i.e., we measure time in the number of
mutations produced):
dw̄
4r 2
52
Mw̄ ln w̄.
dt
n
(7b)
This is our most important result. For mutations of any
size and organisms of any complexity, we can calculate the
instantaneous rate of increase in fitness that occurs during
adaptive evolution. (Eq. 7b was confirmed by simulation.)
In the case of mutations of very small effect, we can obtain
a simplification of considerable intuitive value. As the mutation size x → 0, M → 1/2 in equation (7b), and we get
dw̄
2r 2
ø2
w̄ ln w̄.
dt
n
(8)
Equations (7–8) make three important biological points.
FIG. 2. Rate of increase in fitness as a function of the number of
characters, n (log-log plot). The straight line shows the approximate
equation (8), in which dw̄/dt declines as n21. The curved line shows
the more exact equation (7b). Note that dw̄/dt declines faster than
n21. For both curves, z 5 1 and r 5 0.10.
First, and most obviously, dw̄/dt declines as w̄ increases (note
ln w̄ → 0 as w̄ → 1). Because fitness increases during adaptation, the rate of increase in fitness must slow as a population closes on a phenotypic optimum, as expected intuitively. Below we will describe the curve relating mean fitness
to time.
Second, dw̄/dt varies with mutation size, r. More subtly,
equation (7b) shows that the optimal mutation size, that is,
the size which yields the greatest gain in fitness, corresponds
to a mutation of intermediate phenotypic effect (Fig. 1). The
reason is clear. Although smaller mutations enjoy greater
probabilities of being favorable, they result in small increases
in fitness upon substitution. Conversely, although larger mutations yield larger gains in fitness, they are less likely to be
favorable. The optimal mutation walks a thin line at which
these forces balance. Surprisingly, this optimal mutation has
a constant size on Fisher’s standardized x scale. In particular,
the optimal mutation is one whose magnitude satisfies the
2
equation #`x (x 2 y)(2x 2 y)e2y /2 dy 5 0 (which depends
solely on x). The solution is
x opt ø 0.925.
(9)
It is important to note that the size of this best possible
mutation is surprisingly large. Indeed it is large enough that
its probability of being favorable is only Pa 5 0.177, compared to Pa 5 0.5 for mutations of infinitesimally small effect.
Third, when using mutations of fixed size r, the speed of
adaptation slows with greater organismal complexity, n.
There are three reasons for this. The first is that more complex
organisms pay a cost of about n21/2 in terms of probability
of fixation (because D̄ zfav roughly scales as n21/2). The second
is that more complex organisms pay a cost of about n21/2 in
terms of the gain in fitness that results when a substitution
does occur (because D̄zfix also roughly scales as n21/2). These
two factors explain the dependence of dw̄/dt on n21 in equation (8). But more exactly, equation (7a,b) shows that dw̄/dt
declines faster than n21. The reason is that more complex
organisms also pay a third cost: The probability that a random
mutation will be favorable decreases with n, as Fisher (1930)
emphasized. Figure 2 plots the exact value of dw̄/dt as a
function of n, showing the faster than n21 decline in dw̄/dt .
Indeed once n (and thus x) become very large, dw̄/dt declines
16
H. ALLEN ORR
much faster than n21: The rate of decrease becomes nearly
exponential.
It is worth noting that equation (7a) is closely related to
the fundamental theorem of natural selection, a point that is
discussed in Appendix 2. For present purposes, it suffices to
note that dw̄/dt (i.e., the right side of eq. 7a) is always nonnegative: fitness cannot decrease.
Fitness through Time
We would like to see how fitness changes with time during
adaptation. Unfortunately, it does not appear possible to solve
equation (7a,b) except in special cases (see below). Nonetheless we can numerically iterate (7b), plotting fitness
through time. Figure 3a shows the results. In the case shown,
mutations of constant size were produced throughout the walk
to the optimum (and the mutation size was chosen to correspond to the mean of mutations fixed at the first step in
adaptation given uniform mutational effects between zero and
twice the distance to the optimum [Kimura 1983; Orr 1998];
see Fig. 3 legend). As expected intuitively, fitness rises rapidly early during adaptation, but slows as the population nears
the optimum. As Figure 3a also shows, equation (7a,b) is
very accurate, nearly perfectly predicting the simulation results. (Technically, fitness in eq. (7a,b) reaches an asymptote
at w̄ 5 exp [2r 2/8], not unity, reflecting the fact that, in the
artificial case in which all mutations have size r, favorable
mutations cannot occur once z 5 r/2.)
In the special (and perhaps unbiological) case in which
mutational effects are relative to an organism’s present phenotype (i.e., r scales with z), we can solve equation (7b)
explicitly. If mutations have a constant effective size r̃ on a
standardized scale (r̃ 5 r/z), we get
˜ ln w0 )
wt 5 w01/(12Mt
,
(10)
where M̃ 5 (1/Ï2p) #r̃Ïn/2 (y 2
dy is a constant through time. Figure 3b shows that the resulting plot
of fitness through time qualitatively resembles that seen with
absolute effects. Figure 3b also shows that equation (10) is
extremely accurate.
It should be emphasized that the smooth curves shown in
Figure 3 represent fitness when averaged over many bouts
of adaptation. In any particular walk to the optimum the
population takes discrete, and sometimes quite large, steps,
as emphasized by Orr (1998).
`
2
r̃Ïn/2)2e2y /2
Effective Number of Dimensions
We turn now to a counterintuitive result. The fact that dw̄/
dt is maximized at a constant value of x but that the magnitude
of this fitness gain depends on n suggests that one might
calculate an ‘‘effective number of dimensions’’ characterizing a species, a possibility that was raised but not pursued
by Orr (1998). This effective dimensionality may, moreover,
be calculated from quantities that are in principle measurable.
To see this, note that on Fisher’s x scale, equation (7b) can
be written as Dw̄ 5 32 Mx2w̄ (ln w̄)2/ne2, where Dw̄ refers to
an observed increase in fitness and ne is the effective number
of dimensions. But when using optimally sized mutations
(xopt 5 0.925), Dw̄ 5 Dw̄max, M 5 0.0887, and rearranging,
n e 5 1.56zln w̄z
!Dw̄
w̄
max
.
(11)
FIG. 3. The pattern of fitness through time during adaptation.
Curves show theoretical predictions, whereas points show the results of computer simulations (mean of 1000 walks to the optimum).
Time is measured in (Nm)21 generations, that is, in numbers of
random mutations produced. (a) Absolute mutational size (see text).
Mutations of constant size r 5 0.424 were produced in an organism
comprised of n 5 25 dimensions. At the first step toward the optimum, mutations thus have a size of x 5 1.06 on Fisher’s standardized scale. (This value was chosen as it corresponds to the
mean size of mutations fixed at the first step towards the optimum
given uniformly distributed mutational effects [Kimura 1983; Orr
1998].) The theory curve reflects numerical iteration of equation
(7b). (b) Relative mutational size (see text). At the first step toward
the optimum, n 5 25 and x 5 1.06. The theory curve shows equation
(10).
Thus, knowing only the present fitness (relative to the optimal w̄ 5 1) and the maximal change in fitness resulting from
mutations of various sizes, ne can be deduced. In principle,
then, it might be possible to determine an organism’s effective dimensionality via the following thought experiment.
Beginning with a microbe bearing a deleterious mutation of
known fitness effect (w̄ is known), one measures the mean
fitness effects of a large number of favorable mutations belonging to various phenotypic size classes (x1 1 dx, x2 1 dx,
etc.; we temporarily postpone the problem of how to measure
ADAPTATION AND COMPLEXITY
these total phenotypic effects). Dw̄max is simply the mean
increase in fitness seen in the ‘‘best’’ mutational size class
(Fig. 1). Knowing this quantity ne follows by equation (11).
This strategy does not require direct knowledge of the total
size of a mutation summed over all characters (Fisher’s x).
Instead various proxies for x can be used. In the simplest,
the fitness effects of mutations in wild-type individuals serves
as an indicator of mutation size. In such well-adapted individuals, essentially all random mutations at sites throughout
the genome are deleterious and the magnitudes of their effects
on fitness scale monotonically with total phenotypic size (big
mutations are worse than small). (Note that a mutation’s fitness effect scales neatly with its phenotypic effect only when
a population resides at the phenotypic optimum. We exploit
this fact here.) To find ne, one simply studies the same large
collection of single mutations in an initially maladapted line
and tests the extent to which mutations belonging to known
size classes increase fitness when favorable. The largest mean
fitness increase seen among size classes is Dw̄max.
The point is not, of course, that such an experiment is
practicable; it is not. (Indeed in the ideal version of this
thought experiment, the above protocol must be repeated with
many initially unfit lines.) The point is that such an experiment, together with equation (11), shows that the notion of
an effective dimensionality is sound. Whatever the practical
difficulties, ne can in principle be determined. One can therefore speak of adaptation in an actual species as equivalent
to that in an ideal Fisherian one comprised of ne orthogonal
characters under standard Gaussian selection and with universal pleiotropy.
DISCUSSION
We have calculated the rate of fitness increase during adaptation. Not surprisingly, this rate depends on the present
fitness (i.e., on the present distance from the optimum) as
well as on the size of the random mutations produced.
Watched through time, adaptation in Fisher’s model is characterized by an initial rapid rise in fitness, followed by a
gradual slowing as the population approaches the phenotypic
optimum (Fig. 3). Such a curve makes good intuitive sense
and enjoys good, albeit rough, support from experimental
evolution studies in microbial systems, for example, see Lenski and Travisano (1994), who tracked fitness through time
in long-term selection experiments in laboratory populations
of Escherichia coli, as well as Elena et al. (1998), who performed similar experiments using RNA viruses.
Less obviously, the present results show that organisms
pay a kind of evolutionary cost of complexity: Given the
same size mutations and the same total strength of natural
selection, complex organisms cannot adapt as quickly as simpler ones (eq. 7a,b). To see this, it is important to understand
the situation modeled. We consider organisms of different
degrees of complexity that reside a fixed distance from the
optimum. These organisms adapt by producing random mutations of fixed size. Thus, the ratio of mutation size, r, to
distance from the optimum, z, remains constant across all
levels of complexity. Calculating how rapidly these organisms progress to the optimum, we find that complex ones
move more slowly than simple (for a similar, but quantitative
17
genetic, treatment of this problem, see Wagner 1988). The
resulting cost of complexity might be viewed as a special
case of Maynard Smith’s ‘‘lag load’’—the load that results
from the fact that organisms cannot invariably keep pace with
a changing environment. All else being equal, more complex
organisms suffer a greater lag load.
This lag reflects three costs of complexity, only one of
which has been previously recognized. First, random mutations of a given size have a smaller chance of being favorable
in complex organisms, a fact that Fisher (1930) emphasized
in his discussion of adaptation in The Genetical Theory of
Natural Selection. Second, the probability of fixation for mutations of a given size declines with organismal complexity.
This reflects the fact that favorable mutations travel shorter
distances to the optimum in complex organisms and so yield
smaller favorable selection coefficients. Third, when favorable mutations do get substituted, a smaller gain in fitness
accrues to complex organisms.
Although it may be obvious that complex organisms cannot
adapt as quickly as simple ones—it is, after all, always harder
to satisfy more than fewer constraints—it is not obvious how
such a cost should scale with phenotypic complexity. The
present work shows that the rate of adaptation declines at
least as fast as the inverse of the number of independent
characters (eq. 7a,b): dw̄/dt declines roughly as n21 over small
to modest n, but much faster for very large n (see Fig. 2).
The cost of complexity is therefore considerable. All else
being equal, there would seem a considerable evolutionary
advantage to reducing the number of effectively independent
characters comprising an organism, that is, to developmentally ‘‘bundling’’ characters so as to reduce ne (see below).
Wagner and colleagues have made a strong case for such an
advantage to modularity in a series of important papers (Wagner 1988; Wagner and Altenberg 1996; Baatz and Wagner
1997; see also Charlesworth 1984).
It must be emphasized, however, that the cost identified
here need not preclude the evolution of more complex forms.
The point is merely that organismal complexity is attended
by costs of calculable magnitude. But there may well be other,
and perhaps more than compensatory, ecological advantages
to complexity (e.g., the ability to invade specialized niches)
that are not reflected in such population genetic calculations.
Strictly speaking, we cannot even conclude that complex organisms pay an inevitable cost in terms of adaptive substitution rate. We can only conclude that, when using mutations
of the same size, complex organisms adapt more slowly than
simple ones, as different organisms could obviously produce
different spectra of mutations (Appendix 2). It should also
be understood that we are not concerned with the cost of
producing a more complex organism; these calculations instead tally the evolutionary cost paid by an existing complex
organism in its attempt to adaptively track a changing environment.
It is interesting to note that similar calculations have been
made in other disciplines, for example, computer science and
engineering. Rechenberg (1984, 1994), in particular, has
studied rates of progress in complex optimization problems
in which random displacements from the current state are
made and the best of all resulting solutions chosen each ‘‘generation.’’ (Remarkably, this work, which employs an n-di-
18
H. ALLEN ORR
mensional geometric model, was apparently undertaken without knowledge of Fisher’s nearly identical scheme.) Rechenberg has used this approach to solve problems in fluid dynamics, for example, optimal airfoil shape. These results from
‘‘evolutionary computation’’ differ, however, from the present ones in an important respect: Rechenberg neglects the
probability of fixation, invariably allowing the fittest individuals to seed the next generation. This difference reflects
what is surely the most fundamental distinction between engineering (in which conscious agents are free to choose the
most desirable outcome) and biological evolution (which is
constrained by chance and the laws of Mendelism, e.g., probabilities of fixation). Evolution is saddled with an additional
source of stochasticity—the accidental loss of most favorable
mutations—a fact that can profoundly change our conclusions about adaptation, as Kimura (1983) emphasized in his
critique of Fisher’s model. (Appendix 2 considers another,
and more technical, difference between the engineering and
population genetic models.)
Despite these differences, both the evolutionary computation and population genetic work show that the rate of
adaptation (dw̄/dt) is maximized for mutations of intermediate
size. This result flatly contradicts the intuition of most evolutionists (or at least those raised on micromutational dogma):
Small random mutations do not yield the fastest adaptation.
Given its robustness, this result appears to be a general feature
of adaptation in complex systems and deserves to be more
widely known. In the population genetic case, the result’s
basis is clear: Although small mutations are more likely to
be favorable, they suffer small probabilities of fixation and,
when fixed, travel small distances to the optimum. In contrast,
large mutations enjoy larger probabilities of fixation and,
when fixed, travel greater distances toward the optimum, but
are less likely to be favorable. The maximal rate of adaptation
occurs with mutations of intermediate size that strike a balance between these tendencies. Remarkably, this optimal mutation has a constant size on Fisher’s x scale and is very large
(x ø 0.925).
Although the size of the optimal mutation is constant, the
magnitude of the fitness gain resulting from it varies with
complexity. Roughly put, adaptation is a problem of motion
toward an optimum in a complex phenotypic space and the
ease of this motion falls with the number of degrees of freedom. The best mutation thus makes less progress in complex
than simple organisms. This fact suggests that one can define
an effective number of dimensions characterizing a species.
When using optimally sized mutations, adaptation in an actual species, with its complex web of character correlations,
occurs at the same rate as adaptation in an ideal Fisherian
species comprised of ne characters. (This ideal species is
comprised of orthogonal characters under standard Gaussian
selection and shows universal pleiotropy.)
The point is not, therefore, that real organisms neatly conform to Fisher’s idealization. The point is that real organisms
adapt as though they were ideal Fisherian ones of some dimensionality. The analogy with effective population size is
obvious. To point out, then, that real organisms may not show
universal pleiotropy is similar to pointing out that real populations do not show random mating among hermaphrodites.
In each case, the salient point is that a nonideal entity behaves
equivalently to an ideal one of some given complexity or
size.
The notion of an effective number of dimensions potentially clears away a good deal of confusion about adaptation.
For instance, evolutionists tend to uphold two views about
the phenotype whose mutual coherence is less than obvious.
The first is that any species is divisible into an infinite number
of characters. The second is that some species are more complex than others. But it should be clear that the fact that we
are free to define an arbitrary number of (linearly independent) characters does not mean that natural selection ‘‘sees’’
a similarly arbitrary, and thus undefinable, number of characters. The fact, in other words, that the dimensionality of
the space in which we measure organisms is arbitrary does
not mean that the dimensionality of the space in which natural
selection moves organisms is arbitrary. Some species in some
environments are genuinely more complex than others with
respect to ability to adapt.
This vector space view of phenotypic evolution is obviously somewhat alien and at first artificial. But this view, due
ultimately to Fisher, enjoys several advantages, primary
among them that particular mutations are allowed phenotypic
effects of particular sizes. In an era in which evolutionists
must confront the results of increasingly sophisticated genetic
analyses of phenotypic evolution—analyses that routinely reveal the role of factors of considerable individual effect (e.g.,
Doebley and Wang 1997)—it is not at all obvious that
Fisher’s n-character model is more artificial than his other,
and more popular, infinitesimal one.
ACKNOWLEDGMENTS
I thank N. Barton, A. Caballero, C. D. Jones, E. Leigh, L.
H. Orr, D. C. Presgraves, S. Otto, M. Turelli, G. P. Wagner,
and two anonymous reviewers for helpful criticisms, comments, and discussions. This work was supported by National
Institutes of Health grant GM51932 and by the David and
Lucile Packard Foundation.
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Corresponding Editor: A. Caballero
APPENDIX 1
Distance Traveled to the Optimum among Favorable Mutations
We wish to find the mean distance traveled to the optimum among
favorable mutations, D̄ zfav. Although the derivation sketched here
is similar to that of Hartl and Taubes (1996), it seems worth presenting as my later derivation of the critical quantity D̄ zfix hinges
on the present calculation.
Our population currently resides a distance z from the optimum.
We consider a mutation (vector) of magnitude r and angle u, where
u 5 0 points directly at the optimum and u 5 6p/2 is perpendicular
to the direction of the optimum. In the two-dimensional case (n 5
2), geometric considerations show that the distance moved toward
the optimum, Dz, is
Dz(u) ø r cos u 2 r 2 /2z.
(A1)
Because Dz is more than 0 for favorable mutations, equation (A1)
shows that the largest angle possible among favorable mutations is
u 5 ArcCos[r/2z].
As Leigh (1987) showed in his rederivation of Fisher’s probability, the higher dimension case can be considered by defining a
new variable y 5 Ïncosu. (See also Hartl and Taubes 1996.) Changing variables, we have
Dz fav (y) ø
r
Ïn
(y 2 rÏn /2z).
(A2)
The term in parentheses shows that favorable mutations are those
with y . r Ïn/2z. For simplicity, we let x 5 rÏn/2z. (This is the
origin of Fisher’s standardized size scale.)
To find the mean distance traveled to the optimum, conditioning
upon mutations being favorable, we must find
E
x
D̄z fav 5
`
Dz fav (y) f (y) dy
E
,
`
(A3)
f (y) dy
x
where the limits of integration show that we consider only favorable
displacements, and f(y) is the probability density of random mutations in Fisher’s space. Fisher provides this density in the case
of large n: f(y) 5 (1/Ï2p)exp(2y2/2) . Substituting, we get equation
(4) in the main text.
Distance Traveled to the Optimum among Fixed Mutations
I now derive the mean distance traveled to the optimum conditional on mutations being fixed. To get this distance, we weight
Fisher’s density f(y) by the probability of fixation, P ø 2sfav. As
shown in the text, probabilities of fixation can be expressed in
phenotypic terms: A particular mutation enjoys a probability of
fixation of 2zDzfav. Thus, our new probability density is f̃(y) 5
C2zDzfav(y)exp(2y2/2), where C is a normalization constant.
Replacing f(y) in equation (A3) with f̃(y), we have
E
E
`
D̄z fix 5
2z[Dz fav (y)] 2 e2y
2 /2
dy
x
,
`
2zDz fav (y)e2y
2 /2
(A4)
dy
x
which gives equation (5) in the main text. The ratio of integrals in
equation (5) of the text is in the neighborhood of unity for mutations
of small to modest size, that is, over the range of about 0 , x ,
1. Because most substitutions involve alleles of fairly small effect
(e.g., the first and largest substitution is of size E[x1] ø 1), D̄ zfix is
typically in the neighborhood of r/Ïn. This is the heuristic approximation used in Orr (1998).
Because mutations that travel a greater distance to the optimum
enjoy greater probabilities of fixation, it is obvious that D¯ zfix . D¯ zfav.
Indeed it is easy to show that
¯ z fix 5 D
¯ z fav 1
D
var[Dz fav ]
D̄z fav
.
(A5)
APPENDIX 2
The Fundamental Theorem of Natural Selection with New
Favorable Mutations
We wish to show that equation (7a,b) of the text obeys an analog
of the fundamental theorem. Consider a traditional diploid model
in which 2Nma unique favorable alleles arise per generation. Each
on average gives rise to n̄1 heterozygotes before either being lost
or going to fixation. An ergodic argument shows that 2Nman̄1 heterozygotes exist at equilibrium, that is, H 5 2man̄1. For favorable
mutations, Kimura (1983) shows that n̄1 5 4N and thus that H 5
8Nma. Because the additive genetic variance in fitness VA 5 Hs2
(where s is the heterozygous advantage), we get VA 5 8Nmas2
at equilibrium. However, the right side equals dw̄/dt when evolution is due to substitution of new mutations: dw̄/dt 5
(dw̄/dK) (dK/dt) 5 (2s)[(2Nma)P] 5 8Nmas2. Thus dw̄/dt 5 VA. An
analogous argument holds in haploids.
Engineering versus Evolution: The Production of Mutations
The engineering (Rechenberg 1984) and population genetic
models (Kimura 1983; Orr 1998) also differ in the way in which
mutations are made. In Rechenberg (1984), mutations are built
from the ‘‘character up’’, that is, mutations change the value of
each character by a random amount (drawn from a N(0, s ) distribution) and these changes are independent and identically distributed across characters. Consequently, the square of the magnitude of mutational effects is chi-square distributed, which is
nearly normal with large n. Thus the mutational effect across all
characters, r, is distributed as
f (r) 5
2r (r 2 /s 2 ) n/221 e2r
s2
2 n/2 G(n/2)
2 /2s 2
,
(A6)
20
H. ALLEN ORR
and has a mean of sÏn. Although appropriate in an engineering
context, such a distribution appears unbiological as mutational effects are maximized away from zero: small mutations are rare or
nonexistent, intermediate-sized ones common, and large ones rare
or nonexistent. This nonmonotonic distribution qualitatively differs
from the leptokurtic ones usually assumed in evolutionary biology,
in which mutations with effects close to zero are most common
(Kimura 1983; Keightley 1998). Indeed equation (A6) seems inconsistent with the nearly neutral theory (Kimura 1983; Ohta 1992):
Because fitness effects scale with mutation size, exponential or
gamma distributions (with small values of the shape parameter) of
fitness effects cannot be naturally obtained from equation (A6).
However, in Kimura (1983), Orr and Coyne (1992), and Orr
(1998), mutations are made from the ‘‘top down’’: We begin with
some biologically plausible distribution of mutational magnitudes
(e.g., exponential), but any particular mutation is forced to have a
random direction, that is, each mutation represents a random displacement in n-dimensional space (also see Hartl and Taubes 1998).
Evolution is thus modeled using mutations that, although random
with respect to direction, have a biologically plausible distribution
of magnitudes.