BioSystems 69 (2003) 83–94
Is modularity necessary for evolvability?
Remarks on the relationship between
pleiotropy and evolvability
Thomas F. Hansen∗
Department of Biological Science, Florida State University, Tallahassee, FL 32306, USA
Evolvability is the ability to respond to a selective challenge. This requires the capacity to produce the right kind of variation
for selection to act upon. To understand evolvability we therefore need to understand the variational properties of biological
organisms. Modularity is a variational property, which has been linked to evolvability. If different characters are able to vary
independently, selection will be able to optimize each character separately without interference. But although modularity seems
like a good design principle for an evolvable organism, it does not therefore follow that it is the only design that can achieve
causes interference between the adaptation of different characters, it also increases the variational potential of those characters.
The most evolvable genetic architectures may often be those with an intermediate level of integration among characters, and in
particular those where pleiotropic effects are variable and able to compensate for each other’s constraints.
© 2002 Elsevier Science Ireland Ltd. All rights reserved.
Keywords: Evolvability; Pleiotropy; Modularity; Genetic constraint; Eye evolution; Adaptation; Adaptability; Variability; Genetic variance
One of the most fundamental, yet puzzling, prop-
erties of living organisms is their amazing ability
to evolve. In mainstream evolutionary theory this
evolvability has been taken almost for granted. Most
studies of adaptation simply assume that the organism
is variationally capable of producing the character
states that lead to the optimal phenotype. Within
certain selective constraints, such as those stemming
from limited time and energy budgets, all possible
variants of defined characters are usually assumed to
be available for selection.
E-mail address: email@example.com (T.F. Hansen).
Nowhere is this standard set of variational assump-
tions better illustrated than in Nilsson and Pelger’s
(1994) “A pessimistic estimate of the time required
for an eye to evolve”. With this elegant simulation
model, Nilsson and Pelger show how a complex eye
can evolve in a surprisingly short amount of time un-
der selection for improved visual acuity. Starting with
a sheet of photoreceptors sandwiched between a trans-
parent and a pigment layer of tissue, the eye evolves
first by changes in the size and shape of the tissue
layers to form a pinhole eye. The refractive index of
the vitreous body then change at the aperture to form
a graded-index lens. This model assumes that contin-
uous genetic variation arises in a number of defined
traits describing the size, shape and optical properties
of the involved tissues. It assumes that the variation in
each trait is independent of the variation in the other
0303-2647/02/$ – see front matter © 2002 Elsevier Science Ireland Ltd. All rights reserved.
T.F. Hansen/BioSystems 69 (2003) 83–94
traits, and of the rest of the organism. In essence, it as-
sumes that variation along the entire continuum from
the initial sheet of photoreceptors to the final com-
plex eye is made readily available for selection on a
step-by-step basis. In fact, as Nilsson and Pelger point
out, this makes the model comparable to the evolution
of a one-dimensional quantitative character with un-
limited variability (i.e. unlimited capability to produce
more extreme variants). We may ask how pessimistic
it really is to assume that an organ as complex as an
eye should have a variational basis that is no more
complex than that of simple univariate size trait?
Still, such assumptions may not be unwarranted.
Complex eyes have indeed evolved, rates of microevo-
lution can be very high (Hendry and Kinnison, 1999),
and the success of optimality analyses of adaptation
speaks to astonishing variational potential in many
characters. An adaptationist may simply take the vari-
ability as an empirical fact that justifies the focus on
selection per se. A skeptic may argue that the optimal-
ity is usually more of an assumption than a hypothesis
(Mitchell and Valone, 1990), that stasis is more com-
mon than rapid change (Williams, 1992), and that
the more advanced types of eyes have evolved only a
few times and are far from perfect (Dawkins, 1996).
Both the skeptic and the adaptationist should agree,
however, that a better understanding of the variational
capabilities of biological systems is needed. To have a
complete theory of evolution, the adaptationist needs
to explain why organisms are variationally able to re-
spond so easily to selective pressures, and the skeptic
needs a variational theory to understand and predict
constraints on evolutionary change.
The recent surge of interest in the problem of
variation and its implications for evolvability is there-
fore an important development in evolutionary the-
ory (Wagner and Altenberg, 1996; Von Dassow and
Munro, 1999; Rutherford, 2000; Stern, 2000). Evolv-
ability has been a central concern in several recent
books by authors with rather different backgrounds
and perspectives (Conrad, 1983; Kauffman, 1993;
Dennett, 1995; Maynard Smith and Szathmáry, 1995;
Dawkins, 1996; Raff, 1996; Gerhart and Kirschner,
1997), and many principles of evolvability have been
suggested. These principles include duplication and
divergence, robustness, dissociability, modularity,
symmetry, redundancy, co-option, recombination,
cryptic variation, “evolutionary cranes”, “extradi-
mensional bypass”, “the edge of chaos”, and the
emergence of new hierarchical levels of organization.
In this essay I will focus on the concept of
modularity, which is widely seen as one of the central
principles of evolvability. The idea is that modu-
lar organization favors evolvability by allowing one
module to change without interfering with the rest of
the organism (e.g. Riedl, 1977; Wagner, 1996). Fisher
(1958) demonstrated that the probability of a random
mutation being favorable was a steeply decreasing
function of the number of traits it affected. Simulta-
neous random changes in many parts of a highly inte-
grated structure are not likely to improve its function,
as the chance improvement of one part will almost
always be swamped by deleterious effects in many
other parts. But if the parts are variationally indepen-
dent, selection gets the chance to tune them one at
a time, thereby improving the probability of finding
The importance of modularity, or some similar
concept, for evolvability has been recognized for a
long time (e.g. Needham, 1933; Olson and Miller,
1958; Berg, 1960; Riedl, 1977, 1978; Bonner, 1988),
and modular thinking is deeply embedded in bi-
ology. Lewontin (1978) pointed out that the entire
adaptationistic program assumes that characters have
what he termed quasi-independent evolutionary po-
tential. In fact, as reviewed in Wagner (2001), the
concept of a biological character presupposes a de-
gree of modularity and usually carries the implicit
assumption that characters can evolve independently
of each other. Returning to Nilsson and Pelger’s anal-
ysis of the evolution of the eye, we can identify one
source of the immense evolvability of their model
as stemming from the division of the eye into a set
of characters that are variationally independent both
of each other and of the rest of the organism. This
is a seemingly innocent, but perhaps very optimistic
Indeed, the eye characters in the model are unlikely
to be modular in a strict sense. To illustrate this point,
consider a particular character, the lens. Although the
lens can be recognized as a distinct developmental
module, it has a number of interrelationships with
other characters. In amphibians, the lens is formed
through a series of inductive interactions with other
tissues such as the retina and even heart mesoderm
(Raff, 1996). The defining feature of the lens, its
T.F. Hansen/BioSystems 69 (2003) 83–94
elevated refractive index, is controlled by the secretion
of large amounts of soluble crystallines into the cell
cytoplasm. The crystallines are not strongly special-
ized for this role. Rather, they are functional proteins
of varied origin that often have a dual enzymatic func-
tion in other cells (Wistow, 1993; Raff, 1996). Thus,
the refraction of the lens is not automatically varia-
tionally independent of other organismal functions.
The recruitment or co-option of lens crystallines may
have been constrained by the need to preserve other
functions of the recruited proteins. Although, Nilsson
and Pelger may have made conservative assumptions
about the amount of genetic variation in this trait,
they were not conservative in assuming that the vari-
ation was unconstrained by pleiotropic links to other
aspects of the organism. Whether such pleiotropy
may have constrained the evolution of the lens is un-
known, and this is of central importance to assessing
A link between modularity and evolvability has also
emerged in evolutionary computer science (Wagner
and Altenberg, 1996). Indeed, modularity is a ba-
sic principle of good programming. Engineers have
learned to design robust and flexible programs by di-
viding the task up into a set of modules, subroutines
or objects, with simple well-defined interfaces. Then
appeal of modularity may stem from this engineering
analogy. It is a powerful and easily understandable
principle, which appeals to our engineering minds and
to our instinct for organizing the world into separate
But remember Jacob’s (1982) metaphor: natural se-
lection is not an engineer. Evolution does not operate
from simple and elegant blueprints of the most effi-
cient solution to a given problem. Rather it operates
like a “tinkerer”, building shortsighted ad hoc solu-
tions with whatever materials happen to be available.
Modularity may indeed be a simple, logical and ef-
ficient way of achieving evolvability, but it does not
therefore follow that it is the biological basis of evolv-
ability. Nor does the logic of modularity exclude the
possibility that evolvability is achieved in ways that
appear complex and illogical to our minds. In this es-
say I present some formal analyses of how modularity
and pleiotropy relates to evolvability, and ask whether
modularity is the only, or even the most efficient, way
of structuring an evolvable genetic architecture.
2. Evolution of modularity: integration and
Modularity enhances evolvability by allowing char-
acters to evolve without interference, but modularity
may also hamper evolvability by reducing the number
of genes that can affect the character; thereby reducing
modularity may evolve is by removing pleiotropic ef-
fects among characters (Wagner, 1996; Wagner and
Altenberg, 1996). It is clear that this will reduce the
variability of the character from which the effect is re-
moved and, as I will show below, a pleiotropic effect
is only an absolute constraint if the characters are per-
fectly correlated. Thus, removing a pleiotropic effect
may increase the evolvability of one character and re-
duce the evolvability of another. This means that it is
not clear whether the process of dissociating two char-
acters by removing pleiotropic effects will enhance or
diminish evolvability of the system as a whole.
From a different perspective, it is clear that recruit-
ing more genes to affect a character can increase its
evolvability. Genes available for recruitment will typ-
ically already have effects on other characters, and the
evolution of evolvability by recruitment of genes will
therefore, at least initially, be associated with the evo-
lution of increased integration among characters. Con-
sider a character under directional selection. An allele
that introduces a novel effect on this character may be
picked up by selection and increase in frequency. This
will lead to compensatory changes in the other char-
acters affected by this gene, and eventually the new
allele may go to fixation. If the new effect was ac-
quired through the appearance of a new enhancer that
expresses the gene on the character under directional
selection, then almost all subsequent mutations of this
gene will inherit this pleiotropic effect. Thus, through
integration, the character has acquired a new source of
mutational variability, which makes it more evolvable.
The recruitment of crystallines to the lens is a po-
tential example of such a process. Eventually, gene du-
plication may allow specialization and the pleiotropic
constraints can be removed, but it does not seem par-
ticularly likely for a duplication to be the first step in
the process. In the case of crystallines, some appear
to have duplicated and some do not (Wistow, 1993;
Raff, 1996). Nevertheless these examples demonstrate
that the evolution of modularity does not imply the
T.F. Hansen/BioSystems 69 (2003) 83–94
evolution of evolvability, and that the evolution
of evolvability does not imply the evolution of
3. Quantifying character evolvability as
independent evolutionary potential
3.1. Conditional evolvability
Consider two characters that spend most of their
time under stabilizing selection, but occasionally ex-
perience episodes of directional selection when their
selective regimes are changing. We assume that the se-
lective regimes of the two characters are independent
in the sense that the episodes of directional selection
they experience are uncorrelated. The identification of
characters may be seen as a choice of coordinate sys-
tem for the morphology. We are thus choosing a co-
ordinate system where the bases are independent with
respect to changes in selective regime. Pleiotropy is
thus defined relative to this coordinate system.
In this context, the relevant evolvability of a char-
acter is its ability to respond to directional selection
when the other character is under stabilizing selec-
tion, as in the corridor models of Wagner (1984, 1988)
and Bürger (1986). We refer to this as the conditional
lowing model, first presented in Hansen et al. (2003a),
is to derive an operational measure of the conditional
Let y designate the (vector) character under di-
rectional selection with selection gradient β, and let
x designate the (vector) character under stabilizing
selection with quadratic fitness function w0− x?Sx.
Here, w0 is a constant describing maximum fitness
when x is at its optimum (at x = 0), S is a matrix of
selection parameters that describe the decay in fitness
as x is displaced from the optimum, and?denotes
transpose. Let capital letters, Y and X, denote the
population averages of the characters. Unless the char-
acters are uncorrelated, directional selection on y will
lead to a correlated response in X. This means that
X will become displaced from its optimum and there
will appear a selection pressure to bring X back to-
wards the optimum. This selection pressure will then
constrain the further response in Y. In this situation
we describe selection on x with a local linear approx-
imation of the fitness function around the mean vector
X. This selection gradient is βx= −(S +S?)X. With
this in hand, we can use Lande’s (1979) equations for
the response to directional selection
?Y = Gyβy− Gyx(S + S?)X
?X = G?
where Gy and Gx are the additive genetic variance
matrices of y and x and Gyxis their additive genetic
covariance matrix. Now, if the genetic variances and
covariances remain constant, X will reach an equilib-
rium displacement from its optimum at the value
X∗= (S + S?)−1G−1
If this equilibrium displacement value is fed back
into the expression for the response in Y, we get
?Y = (Gy− GyxG−1
Thus, when the selective forces that act on x have
come to an equilibrium, the continued response in
Y is determined by the conditional genetic variance
matrix, Gy|x. This entity is simply a measure of the
genetic variation in the residuals from a regression of
the genetic value of y on the genetic value of x. Thus,
the asymptotic evolvability of y is determined by the
genetic variation that remains when x is held genet-
ically fixed. The conditional genetic variance is thus
a theoretical measure of the short-term conditional
evolvability of a character as long as the pattern of
genetic variances and covariances remain constant.
This is a heuristic result. The G-matrix may change
and coupling disequilibrium, but also due to epistatic
interactions among genes (Hansen and Wagner, 2001).
Furthermore, the strength of selection is unlikely to
remain constant over sufficiently long periods to reach
the asymptotic response. The conditional variance ma-
trix may still capture the qualitative dynamics of mo-
saic evolution and serve as a meaningful theoretical
parameterization of evolvability.
The concept of conditional evolvability is thus a
tool for quantifying the independent evolutionary po-
tential of a character, and may thus help operationalize
Lewontin’s (1978) notion of quasi-independence as a
prerequisite for adaptation.
In the following we will study the effects of charac-
ter integration by using the conditional G-matrix as a
yxβy− Gx(S + S?)X
T.F. Hansen/BioSystems 69 (2003) 83–94
proxy for evolvability. It is then useful to factorize the
conditional G-matrix into the genetic variance and a
coefficient of determination
Gy|x= Gy(I − ρ2
cient. In the next section we will, however, focus on
univariate characters where ρyxis simply the genetic
correlation between y and x.
yxis the squared multivariate correlation coeffi-
3.2. Optimizing evolvability
Many changes in genetic architecture will have al-
ternate effects on the evolvability of different char-
acters. If a pleiotropic link is added this will reduce
the evolvability of those characters that were previ-
of those characters that were not previously affected.
Thus, a general measure of the evolvability of a set of
characters as a whole is the sum of their conditional
evolvabilities, where each character is conditioned on
all the others. As a simplification we consider a phe-
notype consisting of only two univariate characters x
and y, so that the evolvability can be measured as
E = Gy|x+ Gx|y= (Gy+ Gx)(1 − ρ2
Consider first a simple model where we assume that
there are three underlying sources of variation: one σ2
that affects only y, one σ2
σyxthat affects them both. We assume that the sum
of these three variance components is a constant, σ2
Thus, genes can have effects either on y alone, on x
alone or on both equally. The question now is what
arrangement of genes among these possibilities will
optimize the evolvability as given by E. To answer
this, we compute the following relations:
xthat affects only x, and one
From this we find that the evolvability is given as
x+ σyx,Gyx= σyx (6)
E = σ2
It is straightforward to show that this function is
(√3 − 1)
Thus, the evolvability is maximized when approxi-
is confined to character y and 42% to character x.
Why is evolvability maximized at an intermediate
level of genetic integration? Consider first the situa-
tion where the two characters are uncorrelated. Then
the introduction of pleiotropic effects will increase
the genetic variation of the characters and this has
little cost, as there is ample genetic variation in both
characters that can compensate for the correlated
changes. However, as the correlation increases, the
genetic architecture becomes less and less able to
compensate for the correlated changes. Eventually,
the addition of further pleiotropic effects will decrease
the evolvability. In the limit as the characters become
completely correlated the evolvability drops to zero.
There is no variation that can be used for compen-
satory changes, and the characters are unable to evolve
3.3. Hypothesis: evolvability is maximized by
variable pleiotropic effects
The above model included only a single type of
positive pleiotropy where each allele had the same
effect on the two characters. However, antagonistic
pleiotropic effects, where an allele increases the effect
of one character and decrease the effect of another,
are also possible. Antagonistic pleiotropy may for ex-
ample be implemented through genes that control the
allocation of resources to the characters. Genes with
different types of pleiotropic effects may be able to
compensate for each other’s constraint, and we ex-
pect a genetic architecture with a range of different
pleiotropic interactions to be more evolvable than one
where all genes have similar effects.
In fact, the genetic architectures that maximize E, as
effects, but these are what Baatz and Wagner (1997)
called “hidden pleiotropic effects” where positive and
antagonistic pleiotropy cancel. In this situation, all
T.F. Hansen/BioSystems 69 (2003) 83–94
genes affect both characters, thus maximizing Gyand
Gx, while the positive and negative genetic covari-
ances created by alternate pleiotropic effects cancel,
making ρyx equal to zero. Thus, in this model, the
most evolvable genetic architectures are those where
all genes affect both characters, but they do so in max-
imally different ways.
Variation in pleiotropic effects may be even more
important in facilitating the evolvability of suites of
characters. The above model focused on the indepen-
dent evolvability of single traits, i.e. along two partic-
ular directions in morphospace, but adaptation to en-
vironmental change will typically require the adjust-
ment of several traits in specific combinations that will
differ from time to time. The most versatile genetic
architecture may very well be the one with maximal
variation in pleiotropic effects. This will allow selec-
tion of alleles with just the right suite of effects, and
also make a full range of compensatory changes avail-
4. Pleiotropy and variation
4.1. Pleiotropy and the maintenance of variation
under stabilizing selection
A significant negative effect of pleiotropy on
short-term evolvability is that it increases the strength
of stabilizing selection acting on individual loci,
thereby reducing the amount of variation maintained
at a locus under a balance between stabilizing se-
lection and mutation (Lande, 1980; Turelli, 1985;
Wagner, 1989a). This reduction in standing genetic
variation reduces the amount of variation available for
response to directional selection when the environ-
ment changes. The increase in the strength of selection
on individual loci must, however, be weighted against
the increased number of loci that affects the individual
characters. The analyses of Turelli (1985) and Wagner
(1989a) show that the effects of pleiotropy on variance
maintained are complicated and model dependent.
For the house-of-cards approximation, which is
valid when mutations are rare and selection not too
weak on individual loci, there seem to be no general
tendency for pleiotropy to reduce the amount of vari-
ation maintained in individual traits. In the special
case where all mutation rates, pleiotropic effects and
selection strengths are the same, the variance main-
tained in a character affected by n loci that each have
pleiotropic effects on m traits is given by
where s is strength of stabilizing selection on each
character (= eigenvalue of S), and u is per-locus mu-
tation rate (Wagner, 1989a). The same scaling with
respect to m also holds if pleiotropic effects vary, but
cancel on average, i.e. they are hidden (Turelli, 1985;
Bürger, 2000). This shows that the variation main-
tained at a locus is 1/m times the amount maintained
if there are no pleiotropic effects, but since the aver-
age number of loci affecting a trait, n, must be pro-
portional to the average number of traits affected by
each locus, m, there are no obvious scaling of genetic
variation with pleiotropy in the house-of-cards model.
With the Gaussian approximation, which is valid
when mutation rates are high and selection weak on
individual loci, there may actually be a tendency for
standing genetic variation in individual characters to
increase with integration. The reason is that in the
Gaussian equilibrium the variance scales with the
square root of selection strength, making us expect
the variance to scale with 1/√m if the pleiotropic
effects are uniform (as in Wagner, 1988), and since
the variance is still proportional to n, the average trait
variance may increase. Also, when the pleiotropic
effects are random and hidden, the amount of varia-
tion maintained in a single character is unaffected by
pleiotropy in the Gaussian model (Bürger, 2000), and
will thus increase with the number of loci affecting
Thus, although the effects of pleiotropy on variation
maintained under mutation-selection balance depend
that make integration reduce the amount of variation
maintained in individual traits; when the Gaussian ap-
proximation is valid there may even be a tendency for
individual traits to exhibit more variation when they
are integrated with other traits. The effects on evolv-
ability, however, also depend on how much genetic
correlation is created. If pleiotropic effects are vari-
able and largely hidden, the evolvability may increase,
if pleiotropic effects are largely uniform, evolvability
T.F. Hansen/BioSystems 69 (2003) 83–94
4.2. Costs to pleiotropy
Baatz and Wagner (1997) found that the response
to directional selection could be slowed down by hid-
den pleiotropic effects with a character under stabi-
lizing selection. This contrasts with the model above
and may be due to several factors. One of these is
the effect of stabilizing selection on variance. Clearly,
stabilizing selection on pleiotropic effects reduces the
evolvability of a character if gene number is kept con-
stant, but as argued above this does not mean that
a pleiotropic genetic architecture necessarily exhibit
less genetic variation in mutation-selection equilib-
rium and therefore is less evolvable. Another effect
is induced apparent stabilizing selection on the direc-
tionally selected character. Concavity of the fitness
function induce a term, −sCov[y, x2], to the selection
response in Eq. (3) (Turelli, 1988). With strong stabi-
lizing selection, this may constitute a substantial cost
to pleiotropy, and with non-Gaussian allelic distribu-
tions, it may also create a cost of hidden pleiotropic
Another important short-term cost that applies to
genetic architectures with nonzero genetic correlation
is the load that is created when correlated selection
displaces characters from their optima (see Eq. (2)).
This selection load is independent of the strength of
selection on the characters, and temporary in the sense
that it will disappear when directional selection is re-
laxed. Thus, the enhanced evolvability resulting from
more pleiotropy will usually pay the price of an in-
creased selection load. Notice, however, that this is a
cost of genetic correlation and not of pleiotropy per se.
5. Evolvability on longer time scales
5.1. Pleiotropy and the rate of advantageous
Fisher’s (1958) geometric model of adaptation
shows that the probability of a random mutation to
be advantageous is a decreasing function of the num-
ber of traits it affects (see also Kimura, 1983; Hartl
and Taubes, 1998; Orr, 1998). Fisher’s model was
a geometric attempt at capturing the notion that if
many simultaneous random changes were made to
a complex apparatus, like a microscope, this would
be extremely unlikely to improve the apparatus. This
is a devastating argument against the evolvability of
complex integrated genetic architectures, and a strong
argument in favor of modularity, as random changes
to isolated parts have a much higher probability of im-
proving function. But again, we need to weigh these
considerations against the fact that integrated genetic
architectures provide for more variability in any one
part. To do this we analyze a model similar to the
one presented by Fisher, but from the perspective of
the evolvability of a single character that is displaced
from its optimum as above.
Consider a univariate character under directional se-
lection with selection gradient β. Mutations affecting
this character are constrained by pleiotropic effects on
m other univariate characters under independent sta-
bilizing selection. The effect on fitness of a mutation
with effect y > 0 on the first character and xion the
ith constraining character is then
where siis the strength of stabilizing selection on the
ith character and the sum is over the m constraining
characters. For this mutation to be advantageous (10)
has to be positive. Now assume that the pleiotropic
effects, xi, of new mutations are normally distributed
with variance σ2, and that si= s for all i. Then we
can write (10) as
βy − sσ2χ2(m)
where ?2(m) is a chi-square random variable with m
degrees of freedom. Thus, the mutation is advanta-
geous whenever χ2(m) < r ≡ βy/sσ2; the parameter
r is a measure of the relative strength of directional
versus stabilizing selection. Clearly, the probability of
being advantageous is a decreasing function of m, but
if we assume that the number of potentially advan-
tageous mutations affecting the directionally selected
character is proportional to the level of pleiotropy, m,
then we find that the rate of appearance of advanta-
(1 + m)Prob[χ2(m) < r] > 1
This shows that pleiotropy will increase the advan-
tageous mutation rate provided directional selection is
sufficiently strong relative to stabilizing selection on
T.F. Hansen/BioSystems 69 (2003) 83–94
the constraining characters. To get at the rate of evo-
lution in this model, however, we also need to con-
sider the effects of pleiotropy on the expected selective
advantage of the mutations (Kimura, 1983). The rate
of evolution is the advantageous mutation rate multi-
plied with the fixation probability of those mutations.
The fixation probability is approximately twice the ex-
pected selective advantage of those mutations. Then
the rate of evolution becomes
2βyu(1 + m)Prob[χ2(m) < r]
1 −E[χ2(m)|χ2(m) < r]
where u is the total mutation rate. It can be shown by
Prob[χ2(m) < r] = 1 −Γ[m/2,r/2]
E[χ2(m)|χ2(m) < r]
=mΓ[m/2] − 2Γ[1 + m/2,r/2]
Γ[m/2] − Γ[m/2,r/2]
where Γ[a] is the gamma function and Γ[a, b] the
incomplete gamma function. Using the case without
pleiotropy as a benchmark, we can now compute some
examples. If there are pleiotropic effects on one other
character, m = 1, we find that that the rate of ap-
pearance of advantageous mutations is elevated pro-
vided r > 0.46, and the rate of evolution is elevated
when r > 1.08. These criteria are more likely to be
fulfilled for mutations with small effects, since it is
reasonable to assume that σ2scales with y2. Thus, if
there is strong directional selection, the addition of
a pleiotropic effect will enhance evolvability, as long
as the mutational effects are not too large or stabiliz-
ing selection too strong. As more pleiotropic effects
are added, the criteria become more difficult to ful-
fill. With m = 3, the criteria are r > 1.21 and r >
2.44, and with m = 100, they become r > 70.0 and
r > 84.7, respectively. Thus, for a given strength of
directional versus stabilizing selection (i.e. for a given
r), evolvability may be maximized at an intermediate
level of pleiotropy.
In conclusion, a certain amount of integration is
maladapted, but as the character is becoming better
adapted, the pleiotropy may become a constraint.
5.2. Short-term pessimism or long-term optimism?
The pessimism in Nilsson and Pelger’s (1994) ar-
gument refers to the conservative assumptions made
about the per generation evolvability of the eye. If
we think of the eye as a single univariate character
evolving towards the final stage, we may characterize
the evolvability of this character by observing that the
assumptions imply that the additive genetic variation
scaled by the squared trait mean was 0.01%. As de-
tailed in Hansen et al. (2003b), this is a measure of
evolvability that can be interpreted as the % response
per generation to directional selection that is as strong
as selection on fitness itself. Thus, even with this
strong selection, the response in the character would
only be a hundred of a percent per generation. Nilsson
and Pelger in fact assumed a moderately strong causal
link to fitness (their assumptions imply that the elas-
ticity of fitness with respect to trait was 0.5) to obtain
an expected response per generation of 0.005%. Still,
these are conservative assumptions from the quanti-
tative genetics perspective, and particularly so with
respect to variational properties.
Conservative, however, only from the perspective
of the short-term evolvability of an isolated character.
An evolvability of 0.01% is less conservative when it
is realized that the measure should only include the
additive variation that remains after conditioning on
other characters. How much the evolvability of differ-
ent characters is reduced by pleiotropic constraints is
an important empirical question in need of research.
Another crucial assumption is that the genetic variance
remains constant as the character is changing. If evolv-
ability over more than a few dozen generations or so
is at issue, the relevant variational property is not the
standing genetic variation, which will get exhausted,
but rather the ability to produce new genetic variation
through mutation. The amount of variation introduced
each generation by mutation is very variable across
traits, but may often be on the order of 100 to 1000
part of standing genetic variation (Lynch, 1988; Houle
et al., 1996; Houle, 1998; Lynch et al., 1999). If the
mutational variation is seen as the rate limiting pro-
cess, a per generation evolvability of 0.01% appears
This is particularly so as the estimates of novel
mutational variation still assumes that the character
is unconstrained from other characters. The relevant
T.F. Hansen/BioSystems 69 (2003) 83–94
mutational evolvability is given by the conditional mu-
tational variance matrix analogous to the conditional
G-matrix in (3). The properties of mutational variance
matrices are very poorly known, and it is hard to as-
sess the level of constraint on the mutational variance.
One would, however, expect mutational variation to
be more seriously affected by deleterious pleiotropic
side effects than what is the case for segregating varia-
tion, as the latter is already filtered by selection. There
is a need for estimates of “non-deleterious” mutation
Another optimistic aspect of the eye model is the
assumption of a continuous path of improvement that
allows the eye to evolve as if it was a single character.
This is equivalent to the assumption of a smooth adap-
tive landscape. One could argue that what the model
in fact shows is that a smooth path in the fitness land-
scape can exist (Dawkins, 1996). But note that this is
then conditional on the particular variational assump-
tions that were made. The landscape may be smooth
relative to certain quantitative eye variables, but if the
underlying genetic architecture does not support inde-
pendent quantitative variation in these variables, the
landscape is not smooth with respect to the actual ge-
netic variation. This then leads directly to one of the
core problems of evolvability theory, which is why
the landscape is smooth in the first place (Kauffman,
1993). There are in fact two functions that need to be
adaptive landscape in the Simpsonian sense), and the
mapping from genotype to phenotype. A core problem
for a variational theory is thus to explain or predict
the smoothness properties of the genotype–phenotype
Modularity is a simple way of reducing the rugged-
ness of fitness landscapes. Despite this, there are the-
oretical reasons to suspect that the simplest genetic
architectures are not necessarily those that are most
germane to evolvability. Conrad (1990) and Kauffman
(1993) have provided several reasons to expect gene
nets with intermediate connectivity to be the most fa-
vorable for evolvability. Starting with Fisher (1958),
many authors have argued that an increase in phe-
notypic dimensionality favors evolvability by reduc-
ing the likelihood of being trapped in local optima
(Conrad, 1990; Gordon, 1994; Gavrilets, 1997). From
the perspective of a single character, integration with
other aspects of the organism may provide for more
opportunities to be perturbed from an inferior local
optimum (Price et al., 1993).
To predict long-term evolvability we also need to
understand the evolution of the variational properties
themselves. This includes not only the direct changes
in genetic variances and covariances, but more fun-
damentally, changes in mutational properties. With
epistasis, both the genetic and the mutational vari-
ance matrices will be malleable, as the effects of both
new and old alleles depend on the changing genetic
background (Hansen and Wagner, 2001). If genes
reinforce each other’s effect in the direction of evo-
lutionary change, variability will increase; if genes
reduce or compensate each other, variability will de-
crease. Thus, sustained evolvability requires specific
patterns of epistasis, and an understanding of gene
interaction is therefore essential for understanding the
evolution of variability and evolvability.
6. Discussion and conclusions
For Darwin the evolution of a complex eye was the
ultimate challenge for the theory of natural selection.
The tremendous developments and successes of selec-
tion theory in this century have largely answered this
challenge. The amazing power and speed with which
variation should no longer surprise us. The remaining
puzzle is where the variations are coming from. Evolv-
ability should not be taken for granted, it is something
that needs to be explained, and this explanation must
come from a theory of organismal variation.
A core assumption that facilitated long-term evolv-
ability in the eye model was the implicit modular
structure of the genotype–phenotype map. This is
not an unusual assumption peculiar to this model,
but reflects the conceptual structure of evolutionary
biology as a whole, as the concept of a character
refers to an individualized entity with independent
evolutionary capabilities (Wagner, 1989b; Wagner
and Laubichler, 2000). Thus, the very notion of a
character presupposes a degree of modularity in the
Modularity may have come into focus in the study
of evolvability because it is an easily operationalized
property of the genotype–phenotype map. It has been
almost universally accepted that biological organisms
T.F. Hansen/BioSystems 69 (2003) 83–94
are “modular”, but the fact remains that pleiotropy
across characters is a ubiquitous property of biologi-
cal variation (see also Nagy and Williams (2001) for
a recent challenge to the modularity paradigm). There
is no agreement on how isolated a character must be
to be counted as a module (Raff and Raff, 2000).
It is therefore encouraging that explicit empirical as-
sessments of modularity are becoming more common
(e.g. Cheverud, 1996, 2001; Cheverud et al., 1997;
Armbruster et al., 1999; Mezey et al., 2000; Raff and
Sly, 2000; Klingenberg et al., 2001; Magwene, 2001;
Hansen et al., 2003a). Most of these studies are based
on comparing amount of pleiotropy or genetic correla-
tion within characters with degree among characters.
Modularity is a question of degree and can for exam-
ple be quantified by counting the number of genes that
have effects within a character versus the number that
have pleiotropic effects across characters (Cheverud,
1996; Mezey et al., 2000).
The concept of conditional evolvability may be
helpful in interpreting such studies, as it relates de-
gree of modularity directly to evolvability. The field
needs to reach some conclusions as to how much
of the variability in a character really is useful for
adaptation. This is particularly true for mutational
variation. How much of the new mutational variation
is available for adaptation in the sense that it is not
compromised by deleterious side effects? I suggest
this may be investigated by estimating mutational
variances conditional on a measure of overall fitness.
A focus on the evolution of modularity may be
too restrictive to produce a general understanding of
evolvability. A possible implication of the results of
this paper is that evolvability may be achieved in many
subsets of genetic architectures. This possibility needs
further investigation, but if true it implies that the evo-
lution of evolvability may not be as difficult as the
evolution of modularity. An important aim of the field
should be to identify and quantify genetic and devel-
opmental structures that facilitate or constrain evolv-
ability. Variation in pleiotropic effects among genes
is one structural feature that I suggest may facilitate
evolvability. The pattern of epistasis and its relation to
evolvability is another important area of research.
Modularity may refer to many different aspects of
biological organization. In this essay I have focused
on models that may most properly apply to morpho-
logical or life-history characters, but the dissociabil-
ity of developmental processes is also important, as
illustrated for example in the large literature on hete-
rochrony (see, e.g. Gould (1977), and Raff (1996) for
review). Several authors have also suggested that gene
regulation is inherently modular in the sense that reg-
ulatory elements that control, say, the expression of
the gene in a particular tissue may be added or deleted
without introducing pleiotropic effects on the gene’s
expression in other tissues or circumstances (Dynan,
1989; Dickinson, 1990; Stern, 2000).
In the developmental genetics literature, the mod-
ularity concept is often applied to gene regulatory
networks (Raff, 1996; Kirschner and Gerhart, 1998;
Von Dassow and Munro, 1999). The idea is that sets
of coregulated genes with specific functions may be
co-opted and used as modules in a number of differ-
ent circumstances. The segment–polarity system an-
alyzed by Von Dassow et al. (2000) may serve as a
paradigmatic example. This tightly regulated system
is able to set up a stable spatial pattern of expres-
sion that is surprisingly robust to initial conditions and
perturbations. In addition to its role in setting up the
parasegments in the insect embryo, the system is also
involved in a number of other circumstances including
the patterning of appendages, and even setting up wing
spots in butterflies. Homologues of the genes are also
expressed in a number of tissues in vertebrates. It is
easy to imagine that this robust system may have been
co-opted many times as a ready-made pattern-forming
Although, it makes perfect sense to think of this as
modularity from the developmental genetics point of
view, it is not clear that it implies variational modu-
larity on the morphological level. As many different
genes in different systems are presumably regulated
by members of the segment–polarity system, it may be
ting without causing some deleterious mis-expression,
and the recruitment may imply that many potential
mutations in the regulation of the system will acquire
new pleiotropic effects.
It is useful to contrast two extreme views on the
meaning of co-option. The engineering view would
be that the genetic system at large is so well regu-
lated, hierarchical, and combinatorical that genes can
easily be turned on or off in one set of circumstances
without affecting the expression of the same or other
T.F. Hansen/BioSystems 69 (2003) 83–94
genes in different circumstances. Evolvability would
then consist in the ability to recruit ready-made mod-
ules on all levels of organization into new combina-
tions. In contrast, the tinkering view would be that
co-option is associated with pleiotropic side effects
and can only happen when the selective benefits are
large. It may often be associated with the introduc-
tion of new pleiotropic effects and serve to make the
genetic architecture more complex.
On the engineering view, understanding the evo-
lution of evolvability seems like a hard problem, as
it requires explaining the evolution of the engineer-
ing properties. The arguments in this essay suggest
that an engineering perspective may not be necessary
to assure evolvability. Complex genetic architectures
may also be evolvable, even favor evolvability by in-
troducing more variation in pleiotropic interactions,
and adaptation may often have to pay the cost of a
This essay was motivated by discussions with Gün-
ter Wagner. I thank David Houle, Jason Mezey, Gün-
ter Wagner, and an anonymous reviewer for helpful
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