? 2005 The Society for the Study of Evolution. All rights reserved.
INTERNATIONAL JOURNAL OF ORGANIC EVOLUTION
THE SOCIETY FOR THE STUDY OF EVOLUTION
Vol. 59June 2005 No. 6
Evolution, 59(6), 2005, pp. 1165–1174
SIGN EPISTASIS AND GENETIC CONSTRAINT ON EVOLUTIONARY TRAJECTORIES
DANIEL M. WEINREICH,1,2RICHARD A. WATSON,1,3AND LIN CHAO4
1Department of Organismic and Evolutionary Biology, Harvard University, 16 Divinity Avenue, Cambridge, Massachusetts 02138
4Division of Biology, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093
in which it appears. Although epistasis is widely observed in nature, our understanding of its consequences for evolution
by natural selection remains incomplete. In particular, much attention focuses only on its influence on the instantaneous
rate of changes in frequency of selected alleles via epistatic contribution to the additive genetic variance for fitness.
Thus, in this framework epistasis only has evolutionary importance if the interacting loci are simultaneously segregating
in the population. However, the selective accessibility of mutational trajectories to high fitness genotypes may depend
on the genetic background in which novel mutations appear, and this effect is independent of population polymorphism
at other loci. Here we explore this second influence of epistasis on evolution by natural selection. We show that it is
the consequence of a particular form of epistasis, which we designate sign epistasis. Sign epistasis means that the
sign of the fitness effect of a mutation is under epistatic control; thus, such a mutation is beneficial on some genetic
backgrounds and deleterious on others. Recent experimental innovations in microbial systems now permit assessment
of the fitness effects of individual mutations on multiple genetic backgrounds. We review this literature and identify
many examples of sign epistasis, and we suggest that the implications of these results may generalize to other organisms.
These theoretical and empirical considerations imply that strong genetic constraint on the selective accessibility of
trajectories to high fitness genotypes may exist and suggest specific areas of investigation for future research.
Epistasis for fitness means that the selective effect of a mutation is conditional on the genetic background
Compensatory mutations, Fisher, functional epistasis, genetic recombination, statistical epistasis,Wright’s
Received April 28, 2004. Accepted March 12, 2005.
Evolutionary genetics seeks to understand how mutations
affect organismal phenotype and ultimately their influence
on fitness, or lifetime reproductive success. Genetic loci (i.e.,
units of Mendelian inheritance) commonly exhibit functional
interactions, implying that the effect of a mutation on fitness
(or any other phenotype) may be dependent on the alleles
present at other loci in the genome. Such interactions are
called epistasis and are an intrinsic property of the organism’s
mapping from genotype to phenotype. Epistasis is widely
observed in nature, but to date the evolutionary implications
of epistasis for fitness have received an incompletetheoretical
analysis owing in part to the way the problem was first for-
mulated in the modern synthesis between Darwinism and
R. A. Fisher (1930) felt that because organisms are gen-
erally ‘‘marvelously and intricately adapted’’ to their sur-
roundings, natural selection’s primary effect was the short-
3Present address: Electronics and Computer Science, Southamp-
ton University, Highfield, Southampton SO17 1BJ, United King-
term response to continual biotic and abiotic deterioration in
the environment. He asked how natural selection would cause
allele frequencies at polymorphic loci to change to offset such
environmental decline, and his fundamental theorem of nat-
ural selection gives the answer: each generation, population
mean fitness increases by an amount exactly equal to what
he called the additive component of standing genetic variance
for fitness. Because Fisher’s fundamental theorem only ad-
dresses the instantaneous selective response driven by genetic
variation now segregating in an evolving population, in this
framework epistasis only has evolutionary relevance if the
functionally interacting loci are simultaneously polymorphic.
This evolutionary influence of epistasis, dependent on the
population’s current allelic composition, is often called ‘‘sta-
tistical epistasis’’ (Fenster et al. 1997). Moreover, under the
common assumption that alleles are in linkage equilibrium,
even statistical epistasis among segregating alleles does not
contribute to additive genetic variance and so only constitutes
a form of statistical noise in the system.
Sewall Wright corroborated the mathematics underlying
Fisher’s fundamental theorem (Wright 1930; Provine 1986),
DANIEL M. WEINREICH ET AL.
and both workers recognized the evolutionary importance of
occasional novel beneficial mutations in addition to the sig-
nificance of changes in allele frequency at segregating loci.
Unlike Fisher, however, Wright was also interested in the
evolutionary consequences if the fitness effect of a novel
mutation varied as a function of the genetic background on
which it appeared, whether that background was polymorphic
or not. In the evolutionary genetics literature, this more in-
clusive concept of epistasis (i.e., independent of allele fre-
quency at the interacting loci) is sometimes referred to as
‘‘physiological’’ (Cheverud and Routman 1995) or ‘‘func-
tional’’ (Hansen and Wagner 2001) epistasis to distinguish
it from statistical epistasis (with its dependence on simul-
taneous segregation of interacting loci), and this contrast has
been highlighted by many authors (e.g., Wade 1992; Whit-
lock et al. 1995; Fenster et al. 1997; Phillips 1998; Brodie
2000). In particular, unlike statistical epistasis, which itself
changes with changing allele frequencies and thus is predic-
tive only of short-term evolutionary dynamics, functional
epistasis can also be important on the much longer time-scale
of the mutational process, over which a succession of novel
beneficial mutations may appear and reach fixation.
Wright (1932) drew particular attention to the possibility
that functional epistasis might give rise to genotypes in which
individual mutations at all loci are deleterious in spite of the
existence of higher fitness genotypes differing by mutations
at multiple loci. A hypothetical population fixed for such a
genotype would then be stuck on a local peak on the fitness
landscape because no mutational trajectory (i.e., succession
of individual mutations) would be selectively favored, and
Wright felt that under these circumstances natural selection
alone could not increase fitness (but see Carter and Wagner
2002; Iwasa et al. 2004; Weinreich and Chao 2005). Note
that this observation is outside the scope of statistical epis-
tasis because it does not depend on the interacting loci being
simultaneously polymorphic. Yet, it is functional epistasis
itself that is directly responsible for the lack of selectively
accessible mutational trajectories to higher fitness genotypes.
Thus, statistical epistasis only captures some of the evo-
lutionary importance of functional interactions between loci.
However, statistical epistasis can readily be measured in nat-
ural populations (e.g., Falconer 1994). In contrast, the ex-
istence in nature of functional epistasis sufficient to give rise
to local peaks on the fitness landscape is technically nearly
impossible to show (Whitlock et al. 1995) because it requires
examination of all genotypes in the mutational neighborhood
of a putative peak, many of which are unlikely to be present
in any population. Additionally, the existence of such peaks
is theoretically controversial (Gavrilets 2003, 2004), and the
problem of evolutionary escape by populations stuck at such
a peak have not been satisfactorily resolved (e.g., Coyne et
al. 1997, 2000; Wade and Goodnight 1998; Goodnight and
Wade 2000). These empirical and theoretical considerations
at least in part explain why many workers have followed
Fisher and focused on the evolutionary consequences of sta-
tistical epistasis among segregating alleles.
To summarize, within the framework of the modern syn-
thesis our understanding of the evolutionary implications of
epistasis for fitness is incomplete. One school of thought
focuses on statistical epistasis, asserting that only fitness in-
teractions among segregating alleles matter, but this view
excludes the important possibility of epistatic effectsbetween
new mutations and a monomorphic genetic background. The
alternative approach addresses these cases but historically has
focused on the existence of multiple peaks on the fitness
landscape, a hypothesis not easily tested empirically.
Here we explore the broader theoretical question of wheth-
er and how functional epistasis for fitness can constrain the
selective accessibility of mutational trajectories to higher fit-
ness genotypes. After carefully defining Wright’s fitness
landscape and the model of evolution by natural selection
best suited for this problem, we discuss the circumstances
under which one or more mutational trajectories leading to
the genotype of highest fitness might be selectively inacces-
sible (of which Wright’s notion of local fitness peaks is a
special case in which none are accessible). We show that
such limitations on selectively accessible trajectories arise
only as a consequence of a particular form of functional
epistasis perhaps not widely appreciated, which we denote
sign epistasis. Under sign epistasis, mutations are beneficial
on some genetic backgrounds and deleterious on others; in
other words, the sign of the fitness effect of such a mutation
is conditional on genetic background.
Additionally we show that sign epistasis exists in nature.
Circumstantial evidence exists for sign epistasis in a wide
variety of organisms and much explicit data now exist as a
consequence of recent technical innovations in microbial ex-
perimental systems that permit more exhaustive empirical
exploration of fitness values in modest mutational neighbor-
hoods. We briefly review this literature and use it to dem-
onstrate that sign epistasis is common in these organisms.
We suggest further that the implications of empirical results
derived largely from microbes may nevertheless be quite gen-
eral. Finally, we touch on the theoretical implications of ge-
netic recombination for evolutionary trajectories. We con-
clude with a summary of open questions and prospects sug-
gested by these facts.
EVOLUTION BY NATURAL SELECTION ON WRIGHT’S
Wright (1932) introduced the fitness landscape to represent
evolution by natural selection in the presence of arbitrary
epistatic interactions between loci. In fact, Wright (e.g.,
Wright 1982) advocated two related conceptions of the fitness
landscape (Provine 1986), which we denote the genotypic
fitness and population mean fitness landscapes. We briefly
describe both to motivate our focus on the former, to illustrate
that modeling evolution by natural selection on these two
landscapes recapitulates the contrast between Fisher’s and
Wright’s views of evolution outlined above, and to clarify
the very close connection between the two.
Every possible haploid genotype containing L biallelic loci
can be uniquely represented by a point in a discrete space of
dimensionality L in which each dimension represents a dif-
ferent locus and may assume two values corresponding to
the two alleles defined for that locus. We denote this the
genotype sequence space and for each point, adjacent points
in space represent genotypes that differ by a single point
mutation (Maynard Smith 1970). We shall disregard gross
SIGN EPISTASIS AND EVOLUTIONARY TRAJECTORIES
lelic loci are (L ? 1)-dimensional projections of fitness values. (A)
Genotypic fitness landscape, created by projecting the fitness value
of each genotype arranged in L-dimensional genotype sequence
space into the (L ? 1)th dimension. Left: L ? 1; right: L ? 2. In
each case, the genotype sequence space is represented below and
the genotypic fitness values are projected upward. Because genotype
sequence space is discrete, the genotypic fitness landscape is dis-
crete. (B) Population mean fitness landscape defined for genotypic
fitness values in panel A, created by plotting population mean fitness
values as a function of allele frequencies. Left: L ? 1; right: L ?
2. In each case, the allele frequency space is represented below and
the population mean fitness values are projected upward. (C) Pop-
ulation mean fitness in the presence of epistasis. Two values of D?,
a measure of linkage disequilibrium, yield two population mean
fitness landscapes for the same genotypic fitness landscape.
Fitness landscapes for haploid genomes comprising L bial-
mutational processes such as insertion/deletions and inver-
sions, whose adjacencies are not easily represented in this
space. (Formally genotype sequence space is an L-dimen-
sional hypercube whose vertices represent genotypes and
whose edges represent point mutations.) The genotypic fit-
ness landscape is created by projecting the fitness value of
each possible genotype represented in genotype sequence
space into the (L ? 1)th dimension (Fig. 1A). We assume
that genotypic fitness values are constant, and therefore dis-
regard frequency dependent selection, dominance, and tem-
porally changing environment throughout. Because genotype
sequence space is discrete, the genotypic fitness landscape is
To explore the selective accessibility of mutational trajec-
tories through genotype sequence space, we follow Gillespie
(1984) and employ the strong selection/weak mutation
(SSWM) model. Here, populations are regarded as geneti-
cally monomorphic and so can be represented by a single
point in genotype sequence space. Occasional pointmutations
generate novel progeny corresponding to adjacent points in
sequence space, and natural selection and genetic drift are
assumed to instantly fix or eliminate the mutant genotype.
Because natural selection favors higher fitness genotypes, an
evolving population will tend to follow the local gradient on
the genotypic fitness landscape through sequence space to
mutationally nearby regions of higher fitness. Thus, the ge-
notypic fitness landscape is well suited to our (and Wright’s)
interest in exploring constraints on the selective accessibility
of evolutionary trajectories.
The SSWM model represents populations as single points
in genotype sequence space and thus is unable to model either
genetic recombination (which requires polymorphism to gen-
erate novelty) or changing allele frequencies. A polymorphic
population occupies a distribution of points in this space, and
in a later section we briefly review a treatment that explores
the consequences of recombination in this framework.
Wright’s own approach to modeling changes in allele fre-
quency was to represent a population by the point corre-
sponding to its center of mass in genotype sequence space.
Note that this point lies in another L-dimensional space,
where each dimension now represents population allele fre-
quency at a different biallelic locus. We denote this the allele
frequency space and projection of the mean fitness over this
space into the (L ? 1)th dimension yields the population
mean fitness landscape (Fig. 1B). Here, beginning from some
particular population composition, natural selection will ad-
just allele frequencies in such a manner as to increase pop-
ulation mean fitness, and so an evolving population will fol-
low the local gradient on the population mean fitness land-
scape to regions of higher fitness. Thus, the population mean
fitness landscape readily addresses the questions framed by
Fisher’s fundamental theorem about selective changes to al-
lele frequencies and resultant increases in population mean
The close connection between Wright’s two fitness land-
scapes is apparent when one notes that, although the popu-
lation mean fitness landscape is continuous when population
size is infinite, it is otherwise discrete and indeed converges
to the genotypic fitness landscape as population size goes to
one. Thus, the population mean fitness landscape can be seen
simply to fill the interstices of the genotypic fitness landscape
(cf. Figs. 1A and 1B) at a granularity equal to the reciprocal
of population size. However, for fixed genotypic fitness val-
ues the mapping from allele frequency space to mean fitness
landscape is not unique in the presence of epistasis (Fig. 1C).
This one-to-many mapping from allele frequency space to
population mean fitness landscape reflects the fact that, while
all information about linkage disequilibrium between alleles
across loci is present in the representation of a population as
a distribution of points in genotype sequence space, it is lost
in the transformation to a single point in allele frequency
space. Therefore, long-term evolutionary prediction based on
the population mean fitness landscape is difficult in the pres-
ence of epistasis (Weinreich and Chao 2005). Recall, how-
DANIEL M. WEINREICH ET AL.
ever, that this is precisely the possibility that motivates our
interest in fitness landscapes.
GENETIC CONSTRAINTS ON EVOLUTIONARY TRAJECTORIES
Natural selection causes evolving populations to follow
mutational trajectories that move them upward on the ge-
notypic fitness landscape. However, as recognized early by
Wright (Provine 1986), epistasis can cause the fitness land-
scape to possess ridges and valleys that constrain the ability
of evolving populations to reach the genotype of highest
fitness. We now rigorously develop this connection.
Consider arbitrary genotypes x and y in genotype sequence
space. Without regard to fitness values, we can define a mu-
tational trajectory through sequence space from x to y as
mutations at a succession of loci that carry a population at
x first to some mutationally adjacent genotype x?, thence to
x?, and so on, eventually reaching genotype y. Writing D(x,
y) to represent the set of loci at which genotypes x and y
differ in allelic state, the shortest trajectories between x to y
are those in which mutations occur exactly once at each locus
in D(x, y) and at no others. (Allowing mutational reversions
gives rise to longer trajectories and may include mutations
at loci not in D(x, y)). If ?D(x, y)? represents the number of
loci in D(x, y), these shortest trajectories are thus ?D(x, y)?
mutations long; moreover, because mutations at the members
of D(x, y) may occur in any order, there are ?D(x, y)?! shortest
trajectories between x and y.
A fitness landscape specifies the fitness value for each pos-
sible genotype. Given an arbitrary fitness landscape f, let P(f)
represent the peak or highest fitness genotype on f. What is
the capacity of natural selection to move a population from
arbitrary genotype x along some mutational trajectory to P(f)?
We designate a trajectory as selectively accessible if and only
if each successive mutation along the trajectory is beneficial,
that is, genotypes on this trajectory are monotonically in-
creasing in fitness (but see Carter and Wagner 2002; Iwasa
et al. 2004; Weinreich and Chao 2005). If B(x, f) represents
the set of loci at which beneficial mutations are possible in
genotype x on fitness landscape f, then a selectively accessible
trajectory between x and P(f) consists first of a mutation at
some locus in B(x, f) whose fixation moves the population
to genotype x?, then by a mutation at a locus in B(x?, f) and
so on, and our interest is in the abundance and length of
selectively accessible mutational trajectories between arbi-
trary x and P(f).
All shortest mutational trajectories between arbitrary ge-
notype x and P(f) are selectively accessible if and only if all
members of D(y, P(f)) are also members of B(y, f) for all y
written D(y, P(f)) ? B(y, f). To see this, consider arbitrary
genotype x; the condition D(y, P(f)) ? B(y, f) for all genotypes
y implies that all first mutational steps along all shortest
trajectories between x and P(f) are beneficial. Moreover, be-
cause the condition applies equally to each genotype visited
along each shortest mutational trajectory, all such trajectories
between x and P(f) are selectively accessible. Thus, if D(y,
P(f)) ? B(y, f) for all genotypes y, then all shortest mutational
trajectories from all genotypes to P(f) are selectively acces-
sible, and the logical converse follows trivially from the def-
initions of D(x, y) and B(x, f).
Assuming only that D(y, P(f)) ? B(y, f) for all genotypes
y, one might suppose that for some genotype x there could
also exist loci in B(x, f) which are not in D(x, P(f)), and thus
that additionally, some longer mutational trajectories from x
to P(f) may also be selectively accessible. We now show that
this is not possible by demonstrating that if such a locus
exists for one genotype, it necessarily comes at the expense
of the selective accessibility of some shortest mutational tra-
jectories from another genotype. Put another way, if on land-
scape f all shortest mutational trajectories from all genotypes
to P(f) are selectively accessible, then no additional, longer
mutational trajectories to P(f) can be selectively accessible.
To see this, assume that D(y, P(f)) ? B(y, f) for all y, and
suppose further that for genotype x there additionally exists
a locus l that is a not member of D(x, P(f)) but is a member
of B(x, f). Let x? be the genotype produced by mutation at
locus l in genotype x. Because l is not in D(x, P(f)), it must
be in D(x?, P(f)), since the allele present at l in P(f) cannot
be present in both x and x?. Furthermore because l is in B(x,
f), it cannot be in B(x?, f), since x and x? cannot both be fitter
than the other. Therefore, because l is a member of B(x, f)
but not of D(x, P(f)), it must be a member of D(x?, P(f)) but
not of B(x?, f), contradicting the assumption that D(y, P(f))
? B(y, f) for all y. This demonstrates that under the as-
sumption that D(y, P(f)) ? B(y, f) for all y, no locus l can
exist which is a member of B(x, f) but not of D(x, P(f)) for
any genotype x, or algebraically, that if D(y, P(f)) ? B(y, f)
for all y then D(y, P(f)) ? B(y, f) for all y. (The logical
converse of this statement flows immediately from set theory:
if two sets are equal then each is also a subset of the other.)
To summarize, all shortest mutational trajectories from all
genotypes to P(f) will be selectively accessible if and only
if D(y, P(f)) ? B(y, f) for all y, and this condition has the
corollary implication that from all genotypes only the shortest
mutational trajectories to the peak can be selectively acces-
sible. On such a landscape, all mutations that take any ge-
notype closer in sequence space to the highest fitness ge-
notype are beneficial, and all beneficial mutations take a ge-
notype closer to the peak. We adopt this as a natural and
intuitive definition for a fitness landscape lacking genetic
constraint on selectively accessible mutational trajectories.
Given these facts, at the scale of individual mutational
effects there are logically only two ways that the selective
accessibility of mutational trajectories to the peak may be
constrained. First, for genotype x there may exist a locus l
which is a member of D(x, P(f)) but not of B(x, f), reducing
the number of selectively accessible trajectories to P(f). Al-
ternatively, for genotype x there may exist a locus l which
is a member of B(x, f) but not of D(x, P(f)). In this case,
natural selection will favor a mutation that nevertheless
moves the population farther from P(f). Although this second
possibility may increase the number of selectively accessible
mutational trajectories from x to P(f), such additional trajec-
tories are necessarily two mutations longer than ?D(x, P(f)?.
This follows because the mutation at l will require eventual
reversion before the trajectory reaches P(f), since l is not in
D(x, P(f)). Moreover, as we have just shown, such additional
selectively accessible trajectories from x come only at the
SIGN EPISTASIS AND EVOLUTIONARY TRAJECTORIES
genotype sequence space is represented on the x-y plane and the
log of each genotype’s fitness is represented on the z-axis. To the
right of each genotypic fitness landscape, two cross-sections
through the landscape are shown to illustrate the fitness effect of
mutation at each locus on each background: mutations at the A/a
Sign and magnitude epistasis. To the left in each panel,
locus on each background appear above; mutations at the B/b locus
on each background, below. Because log fitness values are used,
the slope of the fitness landscape cross-sections represents the se-
lection coefficient acting on the underlying mutation. (A) Sign epi-
stasis: the a → A mutation is beneficial on the -B background but
deleterious on the -b background. The b → B mutation is beneficial
on both backgrounds (i.e., on a- and A-) but necessarily exhibits
magnitude epistasis. (B) Multiple peaks: the a → A mutation is
beneficial on the -B background and deleterious on the -b back-
ground; the b → B mutation is beneficial on the A- background and
deleterious on the a- background. (C) Magnitude epistasis: the se-
lection coefficient acting on the a → A mutation is larger on the
-B background than on the -b background, and the selection co-
efficient acting on the b → B mutation is larger on the A- background
than on the a- background. However, both mutations are beneficial
on each background. (D) No epistasis: the selection coefficient act-
ing on the a → A mutation is independent of the allele present at
the B/b locus, and the selection coefficient acting on the b → B
mutation is independent of the allele present at the A/a locus. Note
that only a single fitness value on this landscape differs from those
in panel C, yet magnitude epistasis is now absent at both loci.
expense of selectively accessible trajectories to P(f) from
neighboring genotypes in sequence space.
The connection between this definition of genetic con-
straint and epistasis may be seen as follows. The condition
that B(y, f) ? D(y, P(f)) for all genotypes y is equivalent to
saying that on all genotypes y the identity of the fitter allele
at every locus is the allele found at that locus in P(f). Thus,
the selective accessibility of mutational trajectories on fitness
landscape f from all genotypes to P(f) is unconstrained if and
only if the identity of the fitter allele at every locus is un-
conditioned on genotype. Note that this does not require that
all epistasis be absent: if only the magnitude of the fitness
effect of a mutation varies with genetic background, mem-
bership in B(y, f) will remain unchanged for all genotypes y
and so no constraint on selectively accessible trajectories to
P(f) is introduced. But for the landscape to lack genetic con-
straint on selectively accessible trajectories, we require that
the sign of the fitness effect of all mutations be unconditioned
on genetic background.
It is convenient to formalize this central contrast between
forms of epistasis, and we designate the alternatives sign and
magnitude epistasis. Sign epistasis at a locus means that a
mutation there is beneficial on some genotypic backgrounds
and deleterious on others, or equivalently that the sign of the
fitness effect of a mutation is under epistatic control (Figs.
2A, B). We contrast sign epistasis with magnitude epistasis,
in which mutations are unconditionally beneficial or uncon-
ditionally deleterious, although the magnitude of their effect
may depend on genotypic background (cf. Figs. 2C and 2D).
We have shown that genetic constraint on selectively acces-
sible mutational trajectories to P(f) exists if and only if sign
epistasis is present on f. The dual manifestations of constraint
introduced by sign epistasis (the loss of selectively accessible
shortest trajectories to the peak and gain of selectively ac-
cessible longer trajectories) are illustrated in the contrast be-
tween Figure 2C (which lacks sign epistasis) and Figure 2A
(in which sign epistasis is present at the A/a locus). The sign
epistasis in Figure 2A eliminates one of the selectively ac-
cessible shortest trajectories from the ab genotype but si-
DANIEL M. WEINREICH ET AL.
multaneously introduces a novel, albeit longer selectively
accessible trajectory from Ab via ab.
A brief examination of previous classifications of epistasis
illustrates the novelty of the distinction between sign and
magnitude epistasis. Positive and negative epistasis, defined
in models used to explore the evolutionary consequence of
genetic recombination (reviewed in Kondrashov 1988; Peters
and Lively 2000) are forms of magnitude epistasis. Unidi-
mensional and multidimensional epistasis (Kondrashov and
Kondrashov 2001) can have magnitude epistasis alone or can
have both, although models of unidimensional epistasis com-
monly assume only magnitude epistasis (Kondrashov 1988)
and the exemplar model of multidimensional epistasis (Kon-
drashov and Kondrashov 2001) also possesses sign epistasis.
Finally, in the traditional two-locus quantitative genetic pa-
rameterization, if the additive genetic variance at each locus
is sufficiently small, then additive-by-additive variance gives
rise to sign epistasis. Additive-by-additive variance together
with relatively larger additive effects gives rise only to mag-
nitude epistasis between loci.
However, the consequences of sign epistasis on the selec-
tive accessibility of the shortest mutational trajectories to P(f)
have been implicitly exploited in the theoretical literature.
For example, the reduction in the number of selectively ac-
cessible trajectories from some genotypes as a consequence
of sign epistasis underlies the fitness landscape presented by
Kondrashov and Kondrashov (2001), and the capacity of sign
epistasis to lengthen trajectories is emphasized in the
root2path landscape of Horn et al. (1994; a rendering of this
landscape appears on the cover of this issue). Additionally,
although interest in neutral network fitness landscapes (Lip-
man and Wilbur 1991; Schuster et al. 1994; Huynen et al.
1996; van Nimwegen and Crutchfield 2000) traditionally fo-
cuses on the existence of neutral mutational pathways, the
rarity of beneficial mutations on such fitness landscapes ap-
pears to be a consequence of sign epistasis (not shown). In-
deed, on an exemplar neutral network (the ‘‘royal staircase
with ditches’’ fitness landscape of van Nimwegen andCrutch-
field 2000) beneficial mutations are beneficial only on a single
genotypic background and deleterious on all other back-
Finally, the most well-known theoretical example of ge-
netic constraint on the selective accessibility of mutational
trajectories, the case of fitness landscapes possessing mul-
tiple, mutationally isolated fitness peaks (e.g., Wright 1932;
Kauffman 1993), depends on sign epistasis. To see this, note
that in order for genotype x to lack beneficial mutations in
spite of the existence of higher fitness genotypes elsewhere
on f, it must be that ?B(x, f)? ? 0 while ?D(x, P(f))? ? 0.
Because we have seen that for all fitness landscapes on which
sign epistasis is absent B(y, f) ? D(y, P(f)) for all y, it follows
that f possesses sign epistasis. In this case again, sign epistasis
also increases the number of beneficial mutations available
to some mutational neighbors of local peak x, but because
these novel beneficial mutations lead to x, no additional se-
lectively accessible trajectories to P(f) are created. For ex-
ample, in Figure 2B, sign epistasis increases the number of
beneficial mutations available to genotypes aB and Ab when
compared to Figure 2C; however, this does not give rise to
novel selectively accessible trajectories to AB, the highest
To summarize, genetic constraint on the selective acces-
sibility of mutational trajectories to high-fitness genotypes
arises only as a consequence of sign epistasis, in which case
on some genetic background(s) mutation to an allele present
in a high-fitness genotype is deleterious. At the same time,
on other background(s) mutation at a locus already carrying
the allele present in a high-fitness genotype is beneficial. Put
another way, in the presence of sign epistasis the local fitness
gradient along some mutational trajectories is inconsistent
with the gradient at larger mutational scales; this is necessary
for the existence of ridges and valleys on the genotypic fitness
landscape and represents the root source of such genetic con-
straint on evolution by natural selection.
SIGN EPISTASIS IN NATURE
Sign epistasis constrains the ability of evolution by natural
selection to carry an evolving population to the genotype of
highest fitness and can even give rise to multiple peaks on
the fitness landscape. This motivates an examination of its
prevalence in nature. As noted above, the question of multiple
peaks on the fitness landscape of any organism has been a
difficult one to address experimentally because of the re-
quirement to exhaustively survey a local region of genotype
sequence space (to show that ?B(x, f)? ? 0 for some genotype
x). Fortunately, demonstration of sign epistasis in an organ-
ism requires only that for genotypes x and y, where the fitness
of y exceeds that of x, there exists some locus l that is a
member of B(x, f) but not of D(x, y) or visa versa.
Compensatory mutations fix in evolving populations in re-
sponse to fitness loss due to a mutation elsewhere in the
genome. Mutations are regarded as being conditionally ben-
eficial and thus compensatory if they do not appear in control
populations evolving in the absence of the changed genetic
background. This interpretation relies on the population ge-
netic hypothesis that, were the mutation unconditionally ben-
eficial, the control population would have sampled enough
mutants to have found and fixed it (Burch and Chao 1999).
Thus, unless such mutations are strictly neutral (or possibly
only very slightly beneficial) in the control population, they
are deleterious and so exhibit sign epistasis. Several examples
of such conditional fitness improvements have been reported,
including the bacteriophages ?6 (Burch and Chao 1999) and
?X174 (Poon and Chao 2005), HIV-1 (reviewed in Quin ˜ones-
Mateu and Arts 2001), humans (Kondrashov et al. 2002; Gao
and Zhang 2003; Kern and Kondrashov 2004), and Dro-
sophila (Kulathinal et al. 2004). In the metazoan cases, a
phylogenetic approach was employed, and an appreciable
fraction of deleterious point mutations in the focal species
were found to be the wild-type allele in one or more other
species. Here the inference is that in the other species second-
site mutations ameliorate the deleterious effect observed in
the focal species and make it beneficial (Kondrashov et al.
2002). Thus, they exhibit sign epistasis. Finally, in a recent
maximum-likelihood-based meta-analysis of the suppressor
mutation literature, Poon et al. (2005) estimated that there
are an average of 11.8 compensatory mutations per delete-
rious mutation examined. Importantly, this figure differs only
SIGN EPISTASIS AND EVOLUTIONARY TRAJECTORIES
genes of Vibrio proteolyticus and V. alginolyticus differ at four nu-
cleotides and are otherwise identical. The two naturally occurring
genes and all mutational intermediates may be represented by a
string of four characters chosen from p and a, corresponding to the
nucleotides observed in the V. proteolyticus and V. alginolyticus
genes, respectively. Nodes in the figure represent all 24? 16 pos-
sible 5S RNA genes, with the V. proteolyticus gene at top and the
V. alginolyticus gene at the bottom. Lines connect pairs of muta-
tionally adjacent genes differing at one nucleotide. Physiologically
active genes are shown in bold, and selectively accessible muta-
tional pathways connecting the V. proteolyticus and V. alginolyticus
genes are represented by solid lines. Many pathways are selectively
inaccessible, demonstrating the action of sign epistasis (see text).
(Reprinted with permission from Lee et al. 1997.) (B) An illustration
of a portion of the genotypic fitness landscape inferred for the
bacteriophage ?6. Axes as in Figure 1A, except that true adjacencies
underlying genotype sequence space cannot be accurately repre-
sented in two dimensions. Replicate lineages founded from clone
A repeatedly increased in fitness, while those founded from the
closely related clone B repeatedly declined in fitness. (Reprinted
with permission from Burch and Chao 2000.)
Empirical genotypic fitness landscapes. (A) The 5S RNA
modestly among data from viruses, prokaryotes, and eukary-
A more explicit approach to detecting sign epistasis is to
employ molecular manipulations such as site-directed mu-
tagenesis to construct genotypes occupying a number of ad-
jacent points in genotype sequence space. Recent technical
innovations in microbial systems have made this approach
practical and permit the assessment of fitness effects of in-
dividual mutations on two or more genetic backgrounds. Sev-
eral such studies are reviewed next. Moreover, in our view,
the fact that these data are derived from microbes does not
undermine their generality. Rather, we suggest that the struc-
ture of epistasis acting to constrain the evolutionary trajec-
tories of molecules encoded in microbial genomes may at
least provisionally be taken as representative of such effects
in any organism.
5S RNAs are structural components of ribosomes and are
found in nearly all organisms. In the bacterial species Vibrio
alginolyticus and V. proteolyticus, the 5S RNA genes differ
at four nucleotides. Lee et al. (1997) used site-directed mu-
tagenesis to construct all 24? 2 ? 14 mutational interme-
diates between the 5S RNA genes of these species (Fig. 3A)
and found that only five of 6! ? 24 possible mutational path-
ways between species did not require passage through a sad-
dle corresponding to a sequence of reduced physiological
activity (Fig. 3A). Thus, sign epistasis must exist among these
mutations, since we have shown that peaks and saddles in
the genotypic fitness landscape cannot exist in its absence.
Similar results showing sign epistasis were observed in the
analysis of the mutational pathways between the 5S RNA
genes in V. alginolyticus and another related bacterium, V.
nervis (Lee et al. 1997).
Molecular sign epistasis among mutations in the protease
gene of HIV-1 has also been detected using site-directed
mutagenesis. For example, the 10th amino acid in the HIV-
1 protease is a leucine, and replacing it with an isoleucine
(a mutation denoted Leu10Ile) reduces viral resistance to the
protease inhibitor saquinavir. However, the same mutation
increases saquinavir resistance in the presence of the
Gly48Val and Leu90Met mutations, either individually or
together (Mammano et al. 2000).
Both Escherichia coli (Schrag et al. 1997) and Salmonella
typhimurium (Maisnier-Patin et al. 2002) readily evolve re-
sistance to the antibiotic streptomycin. In both species, strep-
tomycin resistance mutations slow protein synthesis, thereby
lowering organismal fitness in the absence of the drug. How-
ever, this fitness loss can be offset by one of several com-
pensatory second-site mutations (Schrag and Perrot 1996;
Bjo ¨rkman et al. 1999). Importantly, when many of these sec-
ond-site mutations were placed on their respective wild-type
streptomycin-sensitive genetic backgrounds, they were found
to be deleterious (Schrag et al. 1997; Maisnier-Patin et al.
2002), explicitly showing that these compensatory second-
site mutations exhibit sign epistasis.
In both E. coli and S. typhimurium, the genotypic fitness
landscape in the absence of streptomycin may be approxi-
mated by Figure 2B, in which the wild-type genotype is rep-
resented by AB, the A → a mutation represents antibiotic
resistance and the B → b mutation represents the second-site
mutation. It is worth noting that, although both studies
DANIEL M. WEINREICH ET AL.
(Schrag et al. 1997; Maisnier-Patin et al. 2002) demonstrate
sign epistasis, formally they cannot demonstrate the existence
of isolated fitness peaks (Whitlock et al. 1995). This point
is easily overlooked: mutations at additional loci define ge-
notypes whose fitness values are in principal entirely uncon-
strained by the data in hand and that may thus span the
putative fitness valley.
This limitation of inference was overcome recently by
Burch and Chao (2000), who showed that genetically dif-
ferent clones of ?6 reside on different fitness peaks. Specif-
ically, they showed that the clones evolved toward different
equilibrium fitness values (Fig. 3B). Five lineages derived
from clone A all increased in fitness, demonstrating that high-
er fitness genotypes exist. Nevertheless, five lineages derived
from clone B were unable to attain this fitness (and actually
declined in fitness, presumably owing to the action of genetic
drift). This demonstrates that no beneficial mutations were
available to clone B and suggests that it resides on a muta-
tional peak, the existence of which implies the action of sign
epistasis in that genome as shown above. Importantly, the
fact that all five replicates failed to increase in fitness argues
that it is unlikely that the experiment failed to sample mu-
tations that would bridge the putative fitness saddle.
RECOMBINATION, SIGN EPISTASIS, AND
We have drawn attention to the influence of epistasis on
the selective availability of mutational trajectories through
genotype sequence space. For these purposes, we have adopt-
ed the SSWM assumptions (Gillespie 1984) and disregarded
population polymorphism. We have thereby also disregarded
the effects of recombination, which requires polymorphism
to generate novelty. One might ask how this approach differs
from modeling the entire genome as a single locus with a
very great number of alternate alleles (as in the infinite alleles
model; Kimura and Crow 1964). However, we recommend
the representation based on sequence space advanced here
because it makes explicit the mutational adjacency between
genotypes absent in the alternative, and no simple represen-
tation of evolutionary trajectories appears possible in the sin-
gle-locus model. Similarly, the single-locus model cannot
represent epistasis because the underlying combinations of
alleles have been subsumed into single alleles at the meta-
How can the accessibility of evolutionary trajectories in
the presence of genetic recombination be explored? To be of
evolutionary importance, recombination requires linkage dis-
equilibrium, since otherwise it will not change the frequen-
cies at which alleles coexist in the population. However, rep-
resentation of linkage disequilibrium is problematic in allele
frequency space as seen in Figure 1C. If instead populations
are represented as a distribution of points in genotype se-
quence space, this difficulty is avoided because the frequency
of each genotype in the population is preserved in the model.
In this manner, one can begin to address the influence of
genetic recombination on the selective accessibility of evo-
lutionary trajectories in the presence of arbitrary forms of
This is a long-standing problem in the computer science
literature (Holland 1975), and some results are known. Re-
combination samples genotype sequence space qualitatively
differently than does mutation (Gitchoff and Wagner 1996;
Wagner and Stadler 1998) and may produce progeny that are
not adjacent to either parental genotype in sequence space
(Watson 2005), permitting evolutionary jumps through se-
quence space. Consequently, the action of recombination may
profoundly affect the interplay between epistasis, fitness
landscape topography, and selectively accessible trajectories.
For example, fitness landscapes have been defined that lack
any selectively accessible mutational trajectories (as a con-
sequence of sign epistasis giving rise to multiple peaks) but
that have one or more accessible trajectories in the presence
of genetic recombination (Jansen and Wegener 2001; Watson
2001, 2004). The absence of any selectively accessible mu-
tational trajectories can mean that the expected time for nat-
ural selection to carry an asexual evolving population to the
maximum fitness genotype grows exponentially in genome
size, essentially because sign epistasis can render the local
fitness gradient along mutational trajectories entirely unin-
formative at longer scales. However, because recombination
may create one or more selectively accessible trajectories to
the peak genotype, the expected time to the fittest peak on
these landscapes is only linear in genome size for a sexual
Much work remains to better understand the effect of sign
epistasis on the evolution of recombining populations, but
what is clear is that these results are qualitatively different
than the interaction between recombination and magnitude
epistasis previously described (e.g., Peters and Lively 2000).
First, recombination can generate selectively accessible tra-
jectories on landscapes where none are available to an evolv-
ing asexual population. In contrast, we have shown that in
models assuming only magnitude epistasis mutation will nev-
er fail to yield selectively accessible trajectories. Addition-
ally, unlike mutation-mediated perturbations in sequence
space, recombinational perturbations are not necessarily ran-
dom with respect to fitness because the genetic material
brought together by sex has been subject to natural selection
in other individuals in previous generations (Watson 2005).
Finally, some results highlight the significance of loci that
are epistatically dependent also being physically linked on
the chromosome and show a benefit for recombination that
disappears when the physical position of loci on the chro-
mosome are reordered (Watson 2004, 2005).
Many questions remain. How common is sign epistasis in
nature? Compensatory mutations are recognized only after
becoming beneficial and fixing by natural selection. What is
the frequency of mutations that are only cryptically benefi-
cial? Site-directed mutagenesis in microbial genes such as
the Vibrio 5S RNA (Lee et al. 1997) and TEM-1 in E. coli
(Hall 2002) appear to offer an excellent opportunity to ad-
dress these questions. In these systems all possible combi-
nations of alleles at a small number of loci can be generated
by site-directed mutagenesis, their fitness (or other pheno-
type) measured, and thus the frequency of sign epistasis in
an unbiased sample of mutations may be determined. In this
SIGN EPISTASIS AND EVOLUTIONARY TRAJECTORIES
manner, the selective accessibility of all mutational pathways
through a portion of genotype sequence space can be char-
acterized. Comparable experimental work in multicellular or-
ganisms is more difficult, although we suggest that it is at
present unclear whether the structure of epistatic interactions
among loci in microbes should be qualitatively different than
that in multicellular organisms (Poon et al. 2005). Moreover,
indirect evidence of sign epistasis from phylogenetic com-
parisons in metazoans is compelling (Kondrashov et al. 2002;
Gao and Zhang 2003; Kulathinal et al. 2004), and this latter
work offers the opportunity to localize and identify the func-
tionally interacting loci that give rise to the phenomenon in
these organisms. There is theoretical reason to suppose that
if such interactions are tightly genetically linked they are
most likely to be selectively favored (Kimura 1985; Stephan
1996; Carter and Wagner 2002; Weinreich and Chao 2005),
and interactions within genes appear to be more likely than
between genes (Poon et al. 2005), as one might intuitively
suspect. Transgenics could thus be employed in Drosophila
(e.g., Siegal and Hartl 1996) to test the hypothesis that the
ameliorating second-site mutation is in the same gene as the
deleterious mutation, the effect of which it compensates.
Theoretical progress is also possible. Classifying genotypic
fitness landscapes according to the presence or absence of
sign epistasis may be more instructive than asking whether
they have multiple peaks. But still richer characterization of
the epistatic structure defined by a fitness landscape is pos-
sible. For example, the implications for the selective acces-
sibility of trajectories may differ in the case of a mutation
that is beneficial on all but one possible genetic background
compared to the case of a mutation that is deleterious on all
but one background. A still more sophisticated approach
might build on the fact that an evolving population is likely
to visit fitter genotypes more often than less-fit genotypes.
Thus, the evolutionary importance of sign epistasis may be
reduced if mutation to the allele present in the highest fitness
genotype is deleterious only on genotypes of very low fitness.
Moreover, although we have shown that magnitude epistasis
has no influence on the selective accessibility of alternative
mutational trajectories, if the magnitude of the selective ef-
fect of a mutation varies as a function of genetic background,
the rate at (and probability with) which alternative mutational
trajectories are likely to be followed by evolving populations
may be influenced by such epistasis. Finally, for simplicity
we have here focused on mutations that have fitness effects
in all genetic backgrounds, and have not explicitly addressed
conditionally neutral mutations. Work exploring this possi-
bility can also be illuminating (e.g. van Nimwegen and
In summary, prior work has recognized two mechanisms
by which epistasis for fitness can influence evolution by nat-
ural selection. Statistical epistasis among currently segre-
gating alleles can affect additive genetic variance in the pres-
ence of linkage disequilibrium and thereby influence the in-
stantaneous rate of allele frequency change. And regardless
of current population polymorphism, functional epistasis can
constrain the selective availability of mutational trajectories
to genotypes of high fitness. Previous attention to the latter
point has been focused on the possibility of mutationally
isolated fitness peaks: genotypes on which no mutations are
beneficial. Here we have emphasized that this is but one
example of a broader class of effects, in which sign epistasis
reduces the number of selectively accessible trajectories from
some genotypes. More specifically, we have shown that un-
less a system exhibits sign epistasis the fitness landscape will
be unconstrained with respect to the selective availability of
mutational trajectories to the peak genotype. Sign epistasis
is becoming experimentally tractable and we are optimistic
that attention to this phenomenon will lead to a clearer and
more sophisticated understanding of the forces that shape
evolution by natural selection.
We thank present and former members of the J. Wakeley,
D. Hartl, and L. Chao labs and B. Kerr, R. Harrison, and
anonymous reviewers for numerous contributions to the de-
velopment of this work. The figure on the cover of this issue
was prepared with the kind assistance of C. A. Muirhead.
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