ENVIRONMENTAL ROBUSTNESS AND THE ADAPTABILITY OF …

ORIGINAL ARTICLE

doi:10.1111/j.1558-5646.2011.01526.x

ENVIRONMENTAL ROBUSTNESS ANDTHE ADAPTABILITY OF POPULATIONSAlexander J. Stewart,1,2, Todd L. Parsons,1 and Joshua B. Plotkin1,3

1Department of Biology, University of Pennsylvania, Philadelphia, 19104, Pennsylvania2CoMPLEX, University College London, Physics Building, Gower Street, London WC1E 6BT, United Kingdom

3E-mail: [email protected]

Received March 30, 2011

Accepted November 15, 2011

Recent work has shown that genetic robustness can either facilitate or impede adaptation. But the impact of environmental

robustness on adaptation remains unclear. Environmental robustness helps ensure that organisms consistently develop the same

phenotype in the face of “environmental noise” during development. Under purifying selection, those genotypes that express the

optimal phenotype most reliably will be selectively favored. The resulting reduction in genetic variation tends to slow adaptation

when the population is faced with a novel target phenotype. However, environmental noise sometimes induces the expression

of an alternative advantageous phenotype, which may speed adaptation by genetic assimilation. Here, we use a population-

genetic model to explore how these two opposing effects of environmental noise influence the capacity of a population to

adapt. We analyze how the rate of adaptation depends on the frequency of environmental noise, the degree of environmental

robustness in the population, the distribution of environmental robustness across genotypes, the population size, and the strength

of selection for a newly adaptive phenotype. Over a broad regime, we find that environmental noise can either facilitate or impede

adaptation. Our analysis uncovers several surprising insights about the relationship between environmental noise and adaptation,

and it provides a general framework for interpreting empirical studies of both genetic and environmental robustness.

KEY WORDS: Adaptation, mutations, population genetics.

Robustness and adaptability are fundamental and seemingly con-

flicting properties of biological systems (de Visser et al. 2003;

Lenski et al. 2006; Wagner 2008; Draghi et al. 2010). An organ-

ism is robust if it can reliably produce a given phenotype across

a range of environmental and genetic perturbations. But adapt-

ability requires that alternative phenotypes be expressed so that

a population can adapt to new conditions. Naively, it seems that

the more robust an organism is, the less frequently it expresses

alternative phenotypes, and so the less adaptable (or evolvable)

the population will be. However, recent work demonstrates that

genetic robustness can either facilitate or impede a population’s

capacity to adapt to a novel environment (Wagner 2008; Draghi

et al. 2010). The relationship between adaptation and environmen-

tal robustness—that is, the robustness of an individual’s expressed

phenotype against nonheritable perturbations—remains unclear.

There are reasons to believe that environmental noise may fa-

cilitate adaptation. It is well known that genetic and environmental

perturbations often have similar phenotypic effects on an organ-

ism. As a result, occasionally expressing alternative phenotypes

by environmental noise might help speed adaptation via “genetic

assimilation”—that is, by promoting those individuals who will

subsequently evolve the adaptive phenotype by mutation (Ancel

and Fontana 2000; Rutherford 2003; Earl and Deem 2004; Gibson

and Dworkin 2004; Masel 2005; Meyers et al. 2005). But there

are also reasons to believe that environmental noise may impede

adaptation: noise will select for the most environmentally robust

genotypes, thus reducing neutral standing genetic variation in a

given environment and thereby slowing adaptation to a new tar-

get phenotype (Hermisson and Wagner 2004; Wagner et al. 1997;

van Nimwegen et al. 1999; Masel and Trotter 2010). Therefore,

1 5 9 8C© 2012 The Author(s). Evolution C© 2012 The Society for the Study of Evolution.Evolution 66-5: 1598–1612

ENVIRONMENTAL ROBUSTNESS AND THE ADAPTABILITY OF POPULATIONS

we might expect a complex relationship between environmental

noise, environmental robustness, and the rate at which a popula-

tion adapts to a novel environment. It is this relationship that is

the focus of this article.

Phenocopy and Genetic Assimilation

Genetic and environmental perturbations often have similar phe-

notypic effects—a phenomenon known as “phenocopy.” Pheno-

copy was first discovered in the classic studies of Goldschmidt

(1935) and Waddington (1953a), and it has since been observed

at many different levels of biological organization, including

protein-coding sequence, mRNA and protein structures, individ-

ual gene expression patterns, regulatory or metabolic network

dynamics, and entire developmental pathways (Table 1).

The classic example of the phenocopy concept comes from

studies of heat shock in Drosophila (Waddington 1942, 1953a,b,

1959). Waddington found that the type of phenotypic variation in-

duced by heat shock could subsequently be produced by genetic

changes accrued over several generations of artificial selection

(Waddington 1953a). In other words, a phenotype that was ini-

tially produced as the result of an environmental stimulus was

subsequently produced in the absence of that stimulus as the re-

sult of genetic changes—a process called “genetic assimilation.”

The mechanistic basis for genetic assimilation in these heat shock

experiments has been traced, in part, to the behavior of the molec-

ular chaperone Hsp90 (Rutherford and Lindquist 1998; Chow and

Chan 1999; Rutherford 2003; Sangster et al. 2004), which buffers

the effects of standing genetic variation by aiding protein fold-

ing. When the function of Hsp90 is impaired, either through heat

shock or through mutation, previously cryptic genetic variation is

revealed as phenotypic variation. Any such phenotypic variation

that is adaptive can quickly become fixed through genetic as-

similation (Ancel and Fontana 2000; Gibson and Dworkin 2004;

Jarosz and Lindquist 2010; Masel and Trotter 2010).

Environmental perturbations can also lead to phenocopy in a

constant genetic background (i.e., even without standing cryptic

genetic variation). One of the most important examples of this

occurs in the secondary structures of RNA and protein molecules

(Ancel and Fontana 2000; Bloom et al. 2006). Although the major-

ity of genetically identical sequences fold to the same, minimum

energy structure, some adopt higher energy structures. The alter-

native structures that a molecule can assume generally correspond

to the minimum energy structures associated with the genotypes

available by mutation to that molecule (Ancel and Fontana 2000;

Bloom et al. 2006). The degree to which these alternative phe-

notypes are expressed depends on environmental factors, such as

temperature. In this case, phenocopy occurs on a constant genetic

background, without the need for cryptic genetic variation.

In this study, we will not specify a specific phenotype of

interest (e.g., RNA secondary structure) or a specific mechanism

of phenocopy (e.g., a chaperone-mediated capacitor), but rather

we will explore in greater generality how environmental robust-

ness influences a population’s capacity to adapt, assuming that

genetic and environmental perturbations tend to produce similar

phenotypes.

Variation in the Degree of Environmental

Robustness among Genotypes

The degree to which an individual is robust to environmental noise

may depend upon the individual’s genotype. When a population

is under stabilizing selection for a given phenotype, and when

genotypes vary in their environmental robustness, selection will

tend to concentrate the population at the highly robust genotypes.

The strength of this effect will increase as the frequency of envi-

ronmental perturbations increases. As a result of this effect, the

population will harbor less genetic diversity when environmental

perturbations occur more frequently, which in turn will reduce

the speed with which the population adapts to a new target phe-

notype. Thus, genotypic variation in environmental robustness

can influence the adaptability of a population (Masel and Trotter

2010). (We use the term adaptability, which is more accurate in

this context than the term evolvability). Standing genetic diversity

increases the speed of adaptation both by increasing the probabil-

ity that the population harbors adaptable individuals at the time

of the environmental shift and by increasing the rate at which

such individuals are produced following the environmental shift.

Therefore, standing genetic diversity is important for adaptation

when adaptable genotypes are rare.

Presentation Outline

We will construct a general, population-genetic model to explore

the effects of environmental robustness on adaptability. Our model

assumes the concept of phenocopy and it also allows for variation

in environmental robustness across genotypes. On the one hand,

the phenocopy phenomenon has been shown to aid the adaptation

of a population following an environmental shift in the presence of

environmental noise (Ancel and Fontana 2000; Rutherford 2003;

Earl and Deem 2004; Gibson and Dworkin 2004; Masel 2005;

Meyers et al. 2005). But on the other hand, as discussed above,

variation in environmental robustness across a neutral network

tends to reduce genetic variation prior to an environmental shift

(i.e., under stabilizing selection) when noise is present, and thus

slows adaptation (Wagner et al. 1997; Hermisson and Wagner

2004; Masel and Trotter 2010). These two forces therefore have

opposite influences on the capacity of a population to adapt to

a new environment. Here, we resolve how these opposing forces

interact to determine the adaptability of populations subject to

environmental noise.

The article is structured as follows. We begin by introduc-

ing a general model for the evolution of a population on an

EVOLUTION MAY 2012 1 5 9 9

ALEXANDER J. STEWART ET AL.

Table 1. Examples of phenocopy observed in different biological systems.

Phenotype Genetic perturbation Environmental perturbation References

Protein misfolding Mutations at protein-codingregions

Errors in transcription ortranslation

Goldsmith and Tawkik (2009),Meyerovich et al. (2010)

Incorrectly transcribedmRNA

Mutations at stop codons Accidental read through ofstop codons

Masel and Bergman (2003)

Protein/mRNA misfolding Mutations to coding regions Temperature induced noise Ancel and Fontana (2000),Bloom et al. (2006)

Protein/mRNA misfolding Knockout of molecularchaperone Hsp90

Heat shock Rutherford and Lindquist(1998)

Incorrect protein expressionlevel

Mutations to regulatoryregions

Intrinsic/extrinsic noise Raj and van Oudenaarden(2008)

Incorrect gene expressionpattern

Gene knockout resulting inloss of buffering in genenetwork

Large environmentalperturbation

Levy and Siegal (2008), Siegaland Bergman (2002),Lopez-Maury et al. (2008),Wagner and Wright (2007)

Incorrectly executeddevelopmental/ metabolicpathway

Mutation resulting in loss ofredundancy in pathways

Large environmentalperturbation

McLoughlin and Copley(2008), Tawkik (2010)

arbitrary neutral network, with an arbitrary distribution of en-

vironmental robustness across the network. We consider a popu-

lation initially adapted to one environment (one target phenotype)

and then subsequently exposed to a new environment (a different

target phenotype). We treat our model analytically to recover the

full probability distribution of the time required to adapt to the

new target phenotype. We next discuss a simplified neutral net-

work with constant mutational robustness in which we separate

genotypes into classes of high and low environmental robustness.

This simple model is used to develop intuition for how environ-

mental robustness affects adaptability. Finally, we summarize our

results and discuss future directions, including how our frame-

work might be used to interpret systematic studies in quantitative

genetics.

A Population-Genetic ModelOur general model is based on the Moran process from popula-

tion genetics, in which each haploid individual in a population

of constant size reproduces at a rate determined by its relative

fitness, with its offspring replacing a randomly chosen individual.

During reproduction, the offspring may acquire a mutation, which

changes its genotype from that of its parent. Different genotypes

may encode for different phenotypes; fitness is determined by

comparing an individual’s phenotype to the phenotype selected

by the current environment, as described below.

In a given environment that selects for a specific phenotype,

we will focus our analysis on the set of all genotypes that encode

that phenotype. The set of such genotypes is called a “neutral

network” because mutations among them do not change an indi-

vidual’s encoded phenotype. The structure of mutations among

the genotypes on the neutral network is defined by the adjacency

matrix M, which has entries mi j = 1, if genotype i can mutate to

genotype j without changing the phenotype, and mi j = 0 other-

wise. We define the mutational robustness, qi , of genotype i as

the proportion of all mutations that do not change its phenotype.

More specifically, we define

qi = 1

L

∑j

mij, (1)

where L denotes the total number of different mutations that could

arise in an individual (e.g., the length of its binary genome). The

mutational robustness, qi , therefore quantifies the chance that a

mutant offspring of an individual with genotype i will remain

on the neutral network (i.e., will still express the same pheno-

type), or not. Each genotype also has an associated environmental

robustness, φi , discussed in more detail below.

When an individual of genotype i is chosen to reproduce,

a mutation occurs with probability μ. The mutation rate μ is

therefore the genome wide mutation. We have typically chosen a

mutation rate of μ = 10−3 in our simulations, corresponding to a

genome of about 105 nucleotides (see Supporting Information). If

a mutation occurs, the resulting offspring remains on the neutral

network (i.e., encodes the same phenotype) with probability qi .

In this case, the offspring nonetheless has a new genotype, which

is drawn uniformly among i’s neutral neighbors (i.e., from the

set of genotypes j with mi j = 1). With probability 1 − qi , on the

other hand, the resulting offspring encodes a different phenotype

and thus lies outside of the neutral network.

Following the work of Draghi et al. (2010), we assume

that each genotype has a set of K alternative phenotypes that

1 6 0 0 EVOLUTION MAY 2012


constitute its phenotypic neighborhood—that is, the set of possi-

ble phenotypes that can be produced by a nonneutral mutation.

These K phenotypes are drawn uniformly from a total of P possi-

ble alternative phenotypes. We make the simplifying assumption

that phenotypic neighborhoods are independent, such that the

K phenotypes accessible to a genotype are redrawn whenever a

mutation occurs. When a mutation produces an individual lying

outside the neutral network, its resulting phenotype is drawn from

the K alternatives that belong to its phenotypic neighborhood. The

form of genotype–phenotype map described above corresponds to

evolution in a space of infinite genotypes, such that each mutation

results in an entirely new genotype entering the population. Our

model specifies the effects of such mutations on the phenotype,

using summary statistics such as K , q , and P . Such a statisti-

cal description of the genotype–phenotype map provides a very

good approximation to evolution on an explicit, finite genotype–

phenotype map, such as the RNA folding landscape (Draghi

et al. 2010). Furthermore, we demonstrate in the Supporting In-

formation that our results hold even when we relax the statistical

assumptions described here and allow for correlations between

the phenotypic neighborhoods of neighboring genotypes.

As described so far, our model is identical to the one explored

by Draghi et al. (2010), and it can be used to study how muta-

tional robustness influences the process of adaptation. To study

environmental robustness, we will extend the model by assuming

that when an individual produces an offspring, the offspring expe-

riences an environmental perturbation with probability ε. Such an

environmental perturbation may result in the offspring develop-

ing a different phenotype from the “intended” phenotype. In other

words, when a perturbation occurs, an offspring may express a

different phenotype than the one encoded by its genotype. The per-

turbation is specific to the development of that offspring alone,

and it does not affect other individuals in the population. The

parameter ε therefore measures the amount of noise that occurs

during the development of an individual, which we call environ-

mental noise. If such an environmental perturbation occurs during

a reproduction event, the resulting individual adopts its intended

phenotype with probability φi . With probability 1 − φi , however,

the offspring is not robust to the environmental perturbation and

the offspring adopts an alternative phenotype. The parameter φi

therefore describes the robustness of genotype i against environ-

mental noise.

We assume that the alternative phenotypes that result from

mutation in an individual and those that result from environmen-

tal perturbation are the same—so that when an environmental

perturbation produces an individual with an alternative pheno-

type, that alternative phenotype is drawn from the K phenotypes

that constitute its phenotypic neighborhood (i.e., the same set

of phenotypes the individual can produce by mutation). This as-

sumption captures the concept of phenocopy—namely that ge-

netic and environmental perturbations tend to produce a simi-

lar set of phenotypes—whose longstanding empirical support we

have described above.

We consider a haploid population of N asexuals that re-

produce according to the Moran process and occupy a neutral

network of the type described above. We assume that initially the

phenotype associated with the focal neutral network is optimal

for the environment, and we assign it fitness one without loss

of generality. All alternative phenotypes are strongly deleterious,

and assigned fitness zero (or, equivalently, any selective deficit

� 1/N [van Nimwegen et al. 1999; Draghi et al. 2010]). After a

long period of stabilizing selection, we assume that an environ-

mental shift occurs, such that one of the P alternative phenotypes

now has a selective advantage, s, over the initial phenotype. The

newly adaptive phenotype is chosen uniformly from among the P

alternative phenotypes. Once the environmental shift occurs, we

study the adaptation time—defined as the time required for the

population to produce an individual whose genotype encodes the

newly adaptive phenotype. (Once one such adaptive genotype is

produced, the subsequent probability of and time to its fixation

are well understood for asexuals, according to classical popula-

tion genetics [Ewens 2004]; however, explicit calculation of this

quantity may be difficult or impossible in the presence of clonal

interference or in sexual populations). The adaption time is a ran-

dom variable whose properties depend on how the population is

distributed across the neutral network, and on the extent to which

phenocopy speeds adaptation by occasionally producing individ-

uals with the adaptive phenotype. As the rate of environmental

perturbation, ε, increases, the first of these two effects, increased

selection for genotypes with high environmental robustness, will

reduce genetic diversity and make it more and more difficult for

the population to explore the neutral network, thus slowing adap-

tation. However, the second of these two effects, phenocopy, will

result in more frequent expression of the adaptive phenotype,

which will tend to speed adaptation by promoting those individu-

als who are a single mutation away from the adaptive phenotype.

How strongly these two opposing effects influence adaptation

time, and under what conditions, is the central problem we seek

to resolve.

Following Draghi et al. (2010), we separate the neutral

network into adaptable genotypes and nonadaptable genotypes.

Adaptable genotypes are one mutation away from the newly adap-

tive phenotype, whereas nonadaptable genotypes do not contain

the newly adaptive phenotype in their phenotypic neighborhoods

(see Fig. 1 ). In the presence of environmental perturbations ε > 0

(and provided environmental robustness is not complete, i.e., pro-

vided φ �= 1), adaptable genotypes gain a direct selective advan-

tage in our model, compared to nonadaptable individuals with

the same environmental robustness, because they occasionally

express the newly adaptive phenotype.



Figure 1. Depiction of genotypes (dots) and phenotypes (colors).

In this example, the initially adaptive phenotype is indicated by

red, and the newly adaptive phenotype following the environ-

mental shift is indicated by blue. Dark red indicates genotypes

that are adaptable (i.e., those that can express the adaptive phe-

notype either by environmental noise or genetic mutation); light

red indicates genotypes that are nonadaptable. All other pheno-

types, shown here as black or green, are deleterious. Although

some genotypes are shown as only neighboring other genotypes

of the same color, in reality, all genotypes will always have neigh-

bors that code for a different phenotype.

AnalysisWe focus on the situation in which adaptable individuals are rare

at the time of the environmental shift, meaning that the number

of adaptable individuals is O(√

N ) or smaller. We further assume

that the probability of environmental perturbation in an individual,

ε, is sufficiently small that the selective advantage of adaptable

genotypes after the environmental shift is O( 1√N

) or smaller. The

less realistic case in which adaptable individuals are common is

analyzed in Supporting Information.

In our regime of interest, following the environmental shift,

the frequencies of nonadaptable genotypes behave nearly deter-

ministically, and so their dynamics can be accurately approxi-

mated by ordinary differential equations (ODEs). In particular,

if we let Xi denote the frequency of nonadaptable genotype i

in the population, then we can approximate the dynamics by the

following system of ODEs:

d Xi

dt=

⎛⎝σi −

∑j /∈A

σ j X j

⎞⎠ Xi . (2)

Here, A denotes the set of adaptable genotypes, and σi denotes

the selective coefficient of genotype i , which we define below.

The sum∑

j /∈A σ j X j gives the mean fitness of genotypes that

are not adaptable. Loosely speaking, the nonadaptable types be-

have in a deterministic manner because we have assumed they are

common at the time of the environmental shift. We formally jus-

tify this deterministic approximation for nonadaptable genotypes

in Supporting Information, and we demonstrate below that this

approach provides a highly accurate approximation of our model

by comparing it to Monte Carlo simulations of the exact Moran

process.

Unlike the nonadaptable types, the adaptable types are as-

sumed rare at the time of the environmental shift, and so we can-

not use a deterministic approximation. Instead, we approximate

the behavior of the adaptable genotypes by deriving a diffusion

approximation of the Moran process (see Supporting Informa-

tion). The resulting diffusion approximation can be described by

a stochastic differential equation (SDE), which is similar to an

ODE except that the instantaneous dynamics involve Brownian

motion. In particular, we have derived the following SDE to ap-

proximate the (stochastic) dynamics of the frequencies of each

adaptable genotype, j :

dY j =[(

σ j −∑i /∈A

σi Xi

)Y j + θ

∑k /∈A

mk j Xk

]dt + √

2Y j dB j (t).(3)

The notation used to express equation (3) is the standard notation

for SDEs (Gardiner 2004), in which B j (t) denotes independent

Brownian motions. Intuitively, the SDE above indicates that the

change in frequency of genotype j , �Y j , in a small interval of

time, �t , is a Gaussian random variable with mean(σ j −

∑i /∈A

σi Xi

)Y j + θ

∑k /∈A

mkj Xk

and standard deviation

√2Y j .

Here, Y j denotes the number of individuals, rescaled by 1√N

, at

adaptable genotype j (see Supporting Information). A denotes

the set of adaptable genotypes, θ = Nμ denotes the population-

scaled mutation rate, and σ j denotes selection coefficients de-

fined below. The various terms in this SDE each have intuitive

interpretations. As is typical in population-genetic models, the

instantaneous mean change in the frequency of genotype j is

determined by two terms, the first due to selection and the sec-

ond due to mutation; whereas the instantaneous variance in the

change of genotype j’s frequency is determined by its current

frequency, which encodes the effect of genetic drift. The vari-

ance term arising from genetic drift has a somewhat unusual form

and it involves a√

2Y j because we have scaled the population of



adaptable genotypes by 1√N

, instead of the more typical 1N (see

Supporting Information). We formally justify this diffusion ap-

proximation for adaptable genotypes in Supporting Information,

and we demonstrate below that this approach provides a highly

accurate approximation of our model by comparing it to Monte

Carlo simulations of the exact Moran process.

The selection terms in equations (2) and (3) arise from envi-

ronmental perturbations, which cause selective differences among

genotypes on the neutral network. For a nonadaptable genotype

i , such a perturbation will either have no effect (with probabil-

ity φi ), or it will produce a strongly deleterious phenotype with

probability (1 − φi ). Thus, the expected fitness of a nonadaptable

genotype i is 1 − (1 − φi )ε, and the selection coefficient on such

individuals is:

σi√N

= −(1 − φi )ε, i /∈ A. (4)

Likewise, adaptable genotypes express a deleterious phenotype

with probability K−1K (1 − φi )ε, and they express the newly advan-

tageous target phenotype, which has fitness 1 + s, with probability1K (1 − φi )ε. Therefore, the selection coefficient for an adaptable

genotype is

σi√N

= 1 + s

K(1 − φi )ε − (1 − φi )ε, i ∈ A. (5)

The factor 1/√

N appears the two equations above because we

have assumed the selection coefficients are O(

1√N

)(see Sup-

porting Information).

Finally, we note that, following the environmental shift,

adaptable genotypes mutate to the newly adaptive phenotype at

rate μ(

1−qi

K

). This flux into the newly adaptive phenotype, along

with equations (2) and (3), can be used to solve analytically for

the distribution of waiting times for a newly adaptive genotype to

arise in the population. In particular, we can compute the mean

time it takes for a population to adapt to a novel environment, for

an arbitrary neutral network with arbitrary variation in environ-

mental and genetic robustness across the neutral genotypes. The

full analytical solution to this problem is given in the Supporting

Information.

A Simplified ModelWe now consider a simplified version of the model above, which

will allow us to develop intuition about the effects of environmen-

tal robustness on adaptability. We assume that all genotypes have

the same mutational robustness, q . Each genotype on the neutral

network is assigned either high or low environmental robustness,

denoted by φH and φL . Following a neutral mutation, genotypes

with high environmental robustness produce other genotypes with

high environmental robustness with probability πH (and there-

fore produce genotypes with low environmental robustness with

probability 1 − πH ), whereas genotypes with low environmental

robustness produce other genotypes with low environmental ro-

bustness with probability πL (and therefore produce genotypes

with low environmental robustness with probability 1 − πL ). The

neutral network now consists of four distinct classes of genotypes:

high environmental robustness and adaptable, high environmental

robustness and nonadaptable, low environmental robustness and

adaptable, and low environmental robustness and nonadaptable.

The mutation rates among these four genotypes, and between

the adaptable and the newly adaptive genotypes, are summarized

in Figure 2. Henceforth, we analyze our model in terms of the

population density at these four classes of genotypes, rather than

in terms of the frequencies of each individual genotype. Each

class represents a large number of distinct genotypes and there-

fore harbors much genetic diversity. As the rates of mutations

into a particular class increase, this corresponds to increasing the

number of distinct genotypes within that class, and thus increases

the genetic diversity within the class. Figure 2 shows the rates of

mutations between and within classes.

In our simplified model, the mutational robustness, q , is

the same for all genotypes. Therefore, we henceforth refer to

genotypes of high and low robustness, with the understand-

ing that this means genotypes of high and low environmental

robustness.

The parameters πH and πL provide a measure of how clus-

tered the genotypes of high and low robustness are on the neu-

tral network. In particular, an individual with a high-robustness

genotype will undergo 11−πH

neutral mutations, on average, be-

fore it encounters a genotype of low robustness. Similarly, an

individual with a low-robustness genotype will undergo 11−πL

mutations, on average, before it encounters a genotype of high

robustness. Moreover, the overall fraction of genotypes in the

neutral network that have high robustness is given by 1−πL2−(πL +πH )

(Fig. 3).

In the case of this simplified model, we can use equations

(3)–(5) to write down the SDEs for the diffusion process as-

sociated with the two adaptable types (those of high and low

environmental robustness):

dYH =[σH YH + qθ

K

P(πH X H + (1 − πL )X L )

]dt

+√2YH d BH (t)

(6)

dYL =[σL YL + qθ

K

P((1 − πH )X H + πL X L )

]dt

+√2YLd BL (t),

(7)

where X H and X L denote the fraction of the population at high

and low robustness nonadaptable genotypes, and YH and YL



Non-adaptable Adaptable

High ro-bustness

Low ro-bustness

High ro-bustness

Low ro-bustness

Newlyadaptive

μqπH 1 − KP

μqπHKP

μqπL 1 − KP

μqπLKP

μq(1 − πH) μq(1 − πL)

μq 1 − KP

μqKP

μ1−qK

μ1− qK

Figure 2. Rates of mutation among the four classes of genotypes, in our simplified model. Genotypes are either adaptable with

high environmental robustness (top right), adaptable with low environmental robustness (bottom right), nonadaptable with high

environmental robustness (top left), or nonadaptable with low environmental robustness (bottom left). Mutations change genotypes of

high robustness (filled circles) to genotypes of low robustness (open circles) with probability q(1 − πH ), and from low (pink) to high (red)

robustness with probability q(1 − πL ). Mutations change nonadaptable genotypes (left) to adaptable genotypes (right) with probability

q(1 − K

P

)and adaptable to nonadaptable with probability q K

P . Adaptable genotypes mutate to a genotype encoding the newly adaptive

phenotype with probability 1−qK . Mutations within classes are also indicated.

denote the number of individuals, rescaled by 1√N

(see Supporting

Information), at high- and low-robustness adaptable genotypes.

As before, the deterministic part of the system consists of a selec-

tion term, and an influx term due to mutation from nonadaptable

to adaptable types. We note that when the adaptable types are

rare, the probability of back mutation is rare, and thus it does not

appear in our limiting diffusions, as justified in the Supporting In-

formation. Following the environmental shift, the selection terms

σH and σL , for the high- and low-robustness adaptable genotypes,

respectively, are given by

σH√N

= ε1 + s

K(1 − φH ) + ε (φH − φL ) X L (8)

σL√N

= ε1 + s

K(1 − φL ) − ε (φH − φL ) X H . (9)

These selection coefficients have an intuitive interpretation. The

first term in equation (8) and equation (9) represents the selective

advantage of adaptable genotypes over the rest of the popula-

tion that results from their occasionally expressing the newly

adaptive phenotype. In particular, such individuals express an

alternative phenotype with probability ε (1 − φ), and that alter-

native phenotype will be adaptive, with selective benefit s, with

probability 1K . The second terms in equations (8) and (9) rep-

resent the strength of selection on genotypes of high and low

robustness in the population. These can be understood as follows:

high-robustness genotypes gain a selective advantage ε (φH − φL )

over low-robustness genotypes (because they express deleteri-

ous alternative phenotypes less frequently). This results in a se-

lective advantage for high-robustness genotypes in equation (8)

and a selective disadvantage for low-robustness genotypes inm

equation (9).

Since both terms in equation (8) favor high-robustness

adaptable genotypes, σH is always positive. However, for low-

robustness genotypes, one term gives a selective advantage and

the other term a selective disadvantage; therefore, σL can be either

positive or negative.

Finally, the influx terms in equation (6) and equation (7)

depend on πH and πL —that is, on how genotypes of high and

low robustness are clustered across the network.

In the Results section, we will use the analytical equations

above to study how the various factors in our model influence the



Figure 3. Illustration of mutations within and between classes

for different clusterings of individual genotypes. Left—genotypes

with high environmental robustness (filled circles) are tightly clus-

tered, and tend to neighbor one another rather than genotypes

with low environmental robustness (empty circles). This corre-

sponds to πH ∼ 1. Right—genotypes with high environmental

robustness tend not to neighbor other genotypes of the same

type. This corresponds to πH ∼ 0. When genotypes of the high-

robustness type tend to neighbor each other, the population can

spread to many different genotypes and the degree of genetic

diversity in the population increases.

capacity of a population to adapt. We will also compare our ana-

lytic approximation for the mean adaptation time to exact Monte

Carlo simulations of the Moran process. Simulations are carried

out with populations of N = 10,000 individuals. An individual is

chosen to reproduce with a probability determined by its fitness

relative the population as a whole. The chosen individual produces

a viable offspring provided the offspring does not suffer a delete-

rious mutation or a deleterious environmental perturbation. Muta-

tions occur at rate μ = 10−3, across the entire genome. A mutation

will change the offspring’s genotype, phenotypic neighborhood,

and environmental robustness as described above (see Fig. 2). The

offspring produced in such a reproduction event replaces an in-

dividual selected randomly from the population. Simulations are

stopped whenever an individual arises whose genotype encodes

the newly adaptive phenotype. The waiting time before such an

adaptation arises is averaged over 10,000 replicate simulations, to

produce the mean waiting time plotted in the figures below.

ResultsWe focus on the case in which the population has evolved for a

long time in response to selection for a given phenotype, and has

reached its equilibrium distribution on the neutral network prior

to the environmental shift. Since we are interested in the influ-

ence of environmental robustness on adaptation, we keep fixed

the population-scaled mutation rate, θ = Nμ, mutational robust-

ness, q , number of phenotypic neighbors, K , and total number

of alternative phenotypes, P; and we vary the other parameters:

the degree of environmental robustness, φH , φL , the clustering of

high- and low-robustness genotypes, πH , πL , the selective advan-

tage of the newly adaptive phenotype, s, and the probability that

an individual experiences environmental noise ε.

Prior to the environmental shift, approximately all nonadapt-

able individuals will have high robustness (i.e., X H = 1 and

X L = 0), since we have assumed high-robustness genotypes have

a selective advantage over low-robustness genotypes O(√

N ) (see

Supporting Information). Following the environmental shift, the

dynamics of the nonadaptable individuals are approximated by

the ODE in equation (2). Substituting these values in equa-

tions (6)–(9), we see that the selection terms vary with the en-

vironmental robustness of the different genotypes, φH and φL ,

with the rate of environmental perturbations, ε, and with the

strength of selection for the newly adaptive type, s. However,

the mutational influx of adaptable genotypes depends only on

how genotypes of high robustness are distributed in the neutral

network, πH .

Our central question—how do opposing influences of envi-

ronmental robustness resolve to affect adaptation time?—is most

naturally addressed by considering how adaptation time varies

with the probability that an individual experiences an environ-

mental perturbation during development, ε. However, this rela-

tionship is complex, and its form depends on the distribution of

high-robustness genotypes, πH , the degree of environmental ro-

bustness of different genotypes, φH and φL , and the strength of

selection for the newly adaptive phenotype, s. Therefore, we be-

gin by considering the relationship between these parameters and

adaptation time, before considering the frequency of environmen-

tal perturbations.

DEPENDENCE OF ADAPTATION TIME ON THE

CLUSTERING OF ROBUST GENOTYPES

We begin by considering how the clustering of high-robustness

genotypes influences adaptation time. Varying πH from 0 to 1

will increase the clustering of high-robustness genotypes. When

πH ∼ 1, the neutral neighbors of high-robustness genotypes are

almost always other high-robustness genotypes; whereas when

πH ∼ 0, the neutral neighbors of high-robustness genotypes are

almost always low-robustness genotypes (Fig. 3).

There are two distinct selective regimes that correspond to

qualitatively different ways in which adaptation time varies with

πH . When 1+sK < 1, adaptable genotypes with high robustness

are selected for more strongly than adaptable genotypes with low

robustness (i.e., σH > σL ), and adaptation time decreases as πH

increases (Fig. 4 ). In contrast, when 1+sK > 1, adaptable genotypes

with low robustness are selected for more strongly than adaptable

genotypes with high robustness (σH < σL ), and adaptation time

increases as πH increases (Fig. 4).

The first of these two regimes, 1+sK < 1, is more realistic be-

cause we expect the size of the phenotypic neighborhood, K , to

be much greater than 1 and the selective advantage, s, of the

newly adaptive phenotype to be less than 1, typically. In the

most realistic scenario, therefore, adaptation time will decrease



Figure 4. Mean adaptation time as a function of the clustering

of high-robustness genotypes, πH . As πH increases, the proba-

bility that a high-robustness genotype neighbors another high-

robustness genotype increases (i.e., more clustering). When 1+sK <

1 (red line), this causes a decrease in the mean adaptation time of

the population. When 1+sK > 1 (blue line), this causes a increase in

mean adaptation time. Plots show populations of N = 10,000 indi-

viduals, with μ = 0.001, ε = 0.1, P = 100, K = 5, q = 0.5, φH = 0.9,

and φL = 0.1 with s = 1 (red line) and s = 5 (blue line). Lines indi-

cate the analytic solution to our model, whereas dots indicate the

means of 10,000 replicate Monte Carlo simulations.

with increasing πH —that is, it takes longer for the population

to adapt when high-robustness genotypes are not clustered. The

intuition underlying this result is simple. Prior to the environ-

mental shift, high-robustness genotypes are always selectively

favored (because environmental perturbations are always dele-

terious). At the same time, after the environmental shift, high-

robustness adaptable genotypes are more selectively favored than

low-robustness adaptable genotypes, that is, σH > σL in equa-

tions 8–9. The more tightly clustered high-robustness genotypes

are (i.e., the larger πH ), the more often neutral mutations to geno-

types of high robustness will produce other genotypes of high

robustness. As a result, there is more standing neutral genetic

variation in the population prior to the environmental shift, be-

cause there are a greater number of distinct genotypes within

the class of high environmental robustness. This in turn causes

adaptable genotypes to be produced at a higher rate after the envi-

ronmental shift, speeding adaptation. In contrast, if 1+sK > 1, low-

robustness adaptable genotypes are selectively favored over high-

robustness adaptable genotypes, that is, σH < σL in equations

(8)–(9). Therefore, increasing the clustering of high-robustness

genotypes increases the neutral genetic variation in the popula-

tion prior to the environmental shift, but decreases the rate at

which advantageous adaptable genotypes are produced after the

environmental shift. These two effects oppose each other, and we

find that the overall effect is to slow adaptation. The adaptive dy-

namics in these differing regimes can be illustrated through time

series plots of the frequency of high- and low-robustness adapt-

able genotypes in the different selective regimes (see Supporting

Information).

Here, we have focused on the distribution of high-robustness

genotypes, πH , assuming that the clustering of low-robustness

genotypes, πL , remains fixed. Varying the clustering of low-

robustness genotypes does not affect adaptation time if, as we

have assumed, the population is at equilibrium prior to the envi-

ronmental shift (eqs. 6 and 7).


STRENGTH OF ENVIRONMENTAL ROBUSTNESS

We now consider how variation in the strength of environmen-

tal robustness (parameters φH and φL ) affects adaptation time.

As Figure 5 shows adaptation time always increases with in-

creasing φH (i.e., when the robustness of the high-robust type is

increased, adaptation is slower). There are two distinct reasons for

this, associated with the selection terms in equation (8) and equa-

tion (9). First, as φH increases, the (postshift) selection term for

high-robustness adaptable genotypes decreases (this can be seen

directly from eq. 8, setting X H = 1 and X L = 0, but it is also

intuitive: as φH increases, the chance that a high-robust genotype

expresses the adaptive phenotype decreases), so that the popu-

lation is less likely to flow toward these adaptable genotypes,

which slows adaptation. Second, as φH increases, the selective

term for low-robustness adaptable genotypes also decreases (this

can be seen directly from eq. 9, with X H = 1 and X L = 0 but

it is also intuitive: as φH increases, the selective disadvantage of

a low-robustness genotype compared to the rest of the popula-

tion becomes greater), which again retards flow toward adaptable

genotypes and slows adaptation.

Varying the environmental robustness of low-robustness

genotypes, φL , has a more complex effect on adaptation time.

Increasing φL has no effect on the selective advantage of high-

robustness adaptable genotypes. However, it has conflicting

effects on the selective advantage of low-robustness adaptable

genotypes (as seen in eq. 9). On the one hand, increasing φL de-

creases the probability that an environmental perturbation on a

low-robustness adaptable genotype will produce the newly adap-

tive phenotype. However, increasing φL also reduces the probabil-

ity that an environmental perturbation on a low-robustness adapt-

able genotype will produce a deleterious phenotype. Whether

the overall selection term σL is increasing or decreasing with

φL therefore depends on how advantageous it is to produce the

newly adaptive phenotype. This depends on the ratio 1+sK . When

1+sK < 1, the selection term σL for the low-robustness adaptable

genotypes increases as φL increases (eq. 9); as a result, adapta-

tion time decreases as φL increases. When 1+sK > 1, the selection

term for the low-robustness adaptable genotypes decreases as φL



Figure 5. Top—mean adaptation time as a function of the amount

of environmental robustness, φL , among low-robust genotypes.

When selection for the newly adaptive phenotype is relatively

weak ( 1+sK < 1, red line), increasing φL reduce the frequency

of adaptable individuals, which retards adaptation. When se-

lection is stronger ( 1+sK > 1, blue line), increasing φL promotes

adaptable individuals and thus facilitates adaptation. Plots show

populations of N = 10, 000 individuals, with μ = 0.001, ε = 0.1,

P = 100, K = 5, q = 0.5, φH = 1, and πH = 0.5 with s = 2 (red

line) and s = 6 (blue line). Bottom—mean adaptation time with

φH . As φH increases, the selection term for both high- and low-

robustness adaptable genotypes decreases. Plots show popula-

tions of N = 10,000 individuals, with μ = 0.001, ε = 0.1, P = 100,

K = 5, q = 0.5, φL = 0, and πH = 0.5 with s = 2 (red line) and

s = 5 (blue line). Lines indicate the analytic solution to our model,

whereas dots indicate the means of 10,000 replicate Monte Carlo

simulations.

increases (eq. 9); as a result, adaptation time increases as φL in-

creases (Fig. 5). As in the previous section, the more realistic case

occurs when 1+sK < 1. Therefore, we expect that increasing φL

will generally decrease adaptation time.

In summary, from the perspective of adaptation, the least en-

vironmentally robust genotypes in a population would generally

prefer to increase their robustness; whereas the most environ-

mentally robust genotypes would always prefer to reduce their

robustness, as we saw above. Thus, populations with less varia-

tion in environmental robustness will generally adapt faster than

those with greater variation in environmental robustness.


FREQUENCY OF ENVIRONMENTAL PERTURBATION, ε

We now turn to the central question of this study—how does

adaptation time vary with the frequency of environmental per-

turbations during development? Increasing the frequency of such

perturbations, ε, will increase the magnitude of the selection terms

associated with both high- and low-robustness adaptable geno-

types (eqs. 8 and 9). However, increasing ε does not change the

relative strength of these selection terms, the direction of selec-

tion, or the total rate of influx into the adaptable genotypes. These

depend on the factors discussed in the previous two sections—

the distribution of high-robustness genotypes, and the strength

of environmental robustness associated with both high- and low-

robustness genotypes.

There are two selective regimes of interest when considering

how adaptation time depends on the frequency of environmen-

tal perturbations. The first regime occurs when low-robustness

adaptable genotypes have lower fitness than the population as a

whole, that is, when σL is negative. According to equation (9),

this occurs when 1+sK <

φH −φL

1−φL. The second regime occurs when

low-robustness adaptable genotypes are selectively favored com-

pared to the population as a whole, that is, when σL is positive:1+sK >

φH −φL

1−φL. High-robustness adaptable genotypes are always

selectively favored compared to the population as a whole, that

is, σH > 0 (eq. 8).

In the first selective regime, increasing ε increases the

strength of selection for high-robustness adaptable genotypes,

while increasing the strength of selection against low-robustness

adaptable genotypes. As a result increasing ε has conflicting ef-

fects on the adaptation time—it increases the adaptation time from

low-robustness adaptable genotypes but decreases the adaptation

time from high-robustness adaptable genotypes. Initially, as ε in-

creases from 0, adaptation time increases, as selection against

low-robustness adaptable genotypes reduces the density of indi-

viduals at all adaptable genotypes. However, as ε continues to

increase, selection for high-robustness adaptable genotypes con-

centrates the population at these genotypes further, and causes the

overall adaptation time to decrease. This results in a nonmono-

tonic relationship between adaptation time and ε: increasing the

frequency of environmental perturbation can either facilitate or

impede adaptability (Fig. 6 ). In this regime, the population adapts

most quickly when environmental noise is either very frequent or

very infrequent.

In the second selective regime, increasing ε increases the

strength of selection for both high- and low-robustness adaptable



Figure 6. Mean adaptation time as a function of the probability

of environmental perturbation, ε. When selection for the newly

adaptive phenotype is relatively strong ( 1+sK >

φH −φL1−φL

, blue line),

increasing amounts of environmental perturbation promotes all

types of adaptable genotypes, facilitating adaptation. When se-

lection is weaker ( 1+sK <

φH −φL1−φL

, red line) increasing ε has con-

trary effects on the low-robust and high-robust adaptable indi-

viduals; in this regime, the relationship between mean adapta-

tion time and probability of environmental perturbation is non-

monotonic (increasing and then decreasing). Plots show popula-

tions of N = 10,000 individuals, with μ = 0.001, P = 100, K = 5,

q = 0.5, φH = 0.9, φL = 0.1, and πH = 0.1 with s = 2 (red line) and

s = 5 (blue line). Lines indicate the analytic solution to our model,

whereas dots indicate the means of 10,000 replicate Monte Carlo

simulations.

genotypes, and it therefore always decreases adaptation time. In

this regime, more environmental noise always speeds adaptation.

In summary, when environmental fluctuations are rare

(ε ∼ 0), increasing their frequency can either impede or facilitate

adaptation depending on the strength of selection for the newly

adaptive phenotype. However, when environmental fluctuations

are common (ε ∼ 1), increasing their frequency will always speed

adaptation.

We have also studied how the clustering of high-robustness

genotypes influences the relationship between environmental per-

turbations and mean adaptation time (see Supporting Informa-

tion).

DEPENDENCE OF ADAPTATION TIME ON

MUTATIONAL ROBUSTNESS

We have also studied how environmental noise can influence the

relationship between mutational robustness and adaptability. Pre-

vious work has shown that, in the absence of environmental noise,

intermediate levels of mutational robustness produce the fastest

rates of adaptation (Wagner 2008; Draghi et al. 2010). We find that

the addition of environmental noise does not qualitatively change

the relationship between mutational robustness and adaptation

(see Supporting Information).

DiscussionThe expression of a phenotype is inherently noisy. As a result,

when a population experiences purifying selection, those geno-

types that express the optimal phenotype more reliably will be

selectively favored. This is the key insight behind Waddington’s

concept of environmental canalization, developed over half a cen-

tury ago (Waddington 1942, 1953a,b, 1959). Nevertheless, the

extent to which environmental robustness evolves in response to

noise, and how this phenomenon impacts evolution over longer

time scales, as the optimal phenotype itself changes, is not fully

understood.

When canalization fails, environmental noise elicits the ex-

pression of an alternative phenotype in an individual. The result-

ing phenotype is often then a phenocopy—that is, a phenotype

that mimics the effect of a mutation, but is not heritable. When a

population is faced with a new environment, the phenocopy phe-

nomenon may speed adaptation, because it allows some individu-

als to express adaptive phenotypes without waiting for mutations.

However, when a population is adapted to the current environ-

ment, environmental noise tends to concentrate the population on

the most robust genotypes, which will reduce genetic diversity

and the capacity for adaptation. We have sought to elucidate how

these two conflicting effects of environmental robustness resolve

in determining the adaptability of populations.

We have employed a general population-genetic model that

was previously used to study the interplay between mutational

robustness and adaptability (Draghi et al. 2010). Our model spec-

ifies a fitness landscape in terms of the properties of a neutral

network of genotypes. Whenever an individual develops an al-

ternative phenotype as a result of environmental perturbation, it

develops into one of the K phenotypes accessible by mutation;

this assumption encodes the concept of phenocopy in our model-

ing framework.

We have studied the relationship between environmental

noise and adaptation time—that is, the time required for a pop-

ulation, initially in equilibrium, to acquire a new target pheno-

type following an environmental shift. In our simplest model, the

adaptation time depends on several parameters: the frequency of

environmental perturbations (ε), the selective advantage of the

novel target phenotype (s), the levels of environmental robustness

among genotypes on the neutral network (φH and φL ), and the

clustering of genotypes with high environmental robustness (πH ).

The interplay of these parameters gives rise to a surprisingly rich

set of behaviors, which are summarized in Table 2.

The relationship between environmental and mutational ro-

bustness may be further complicated if both are allowed to vary



Table 2. Summary of results.

Parameter varied Selective regime Mean adaptation time

High-robustness genotype clustering, πH1+sK < 1 Decreasing

High-robustness genotype clustering, πH1+sK ≥ 1 Increasing

Low-robustness genotype clustering, πL All ConstantHigh environmental robustness level, φH All IncreasingLow environmental robustness level, φL

1+sK < 1 Decreasing

Low environmental robustness level, φL1+sK ≥ 1 Increasing

Rate of environmental perturbation, ε 1+sK <

φH −φL

1−φLNonmonotonic (increasing then decreasing)

Rate of environmental perturbation, ε 1+sK ≥ φH −φL

1−φLDecreasing

Mutational robustness, q All Nonmonotonic (decreasing then increasing)

across the neutral network simultaneously. Our analysis has as-

sumed the simplest case—when environmental and mutational

robustness vary independently. Even in this simple case, we have

found a rich diversity of possible behaviors (Table 2). The as-

sumption of independence is partly supported by the few sys-

tematic empirical studies of mutational and environmental ro-

bustness. Such studies have found that polymorphisms affecting

mutational robustness map to different loci than those influencing

environmental robustness (Proulx et al. 2007; Fraser and Schadt

2010; Lehner 2010), suggesting that these traits may be largely

independent. Nonetheless, the impact of covariation in these two

types of robustness (Hermisson and Wagner 2004; Lehner 2010;

Meiklejohn and Hartl 2002; Proulx et al. 2007; Wagner et al.

1997) remains an important topic for future research.

It is interesting to contrast our results on environmental noise

with quantitative studies of the Baldwin effect, which also de-

scribe the genetic reinforcement of initially nonhereditary traits

(Ancel 1999, 2000). Whereas we have focused here on the ten-

sion between costs and benefits of randomly expressing an al-

ternative phenotype during development, studies of the Baldwin

effect have been concerned with the tension between the adaptive

advantage and inherent cost of phenotypic plasticity—that is, the

ability of an individual to express a range of phenotypes and to

deterministically choose among these the most fit phenotype in

the current environment. In other words, in our study, the pheno-

type expressed by an individual is subject to random perturbation,

whereas in studies of the Baldwin effect, the expressed pheno-

type is chosen deterministically: if an individual has the ability

to express the optimal phenotype with its range of alternatives,

it will. Models of the Baldwin effect have uncovered a complex

interplay between the rate at which the environment changes, the

degree of phenotypic plasticity in a population, and the rate of

adaptation (Ancel 1999, 2000). We have studied a different phe-

nomenon, but likewise found a complex relationship between the

amount of development noise and the rate of adaptation. Both

phenomena (phenotypic plasticity and environmental robustness)

highlight the complex relationship between environmentally in-

duced phenotypic variation and the capacity for a population to

adapt.

A phenomenon analogous to the Baldwin effect may occur

in our system if the newly adapted individual that first fixes in

the population has low robustness (analogous to high plasticity

in the models of [Ancel 1999, 2000]); whereas subsequent sta-

bilizing selection will then select for a high-robustness (adapted)

individual. This situation is more likely to occur in the regime

σL > σH .

In deriving our analytical results, we have relied on three

strong assumptions: a neutral mutation completely redraws an in-

dividual’s phenotypic neighborhood; the number of phenotypes,

K , in a genotype’s neighborhood does not vary across the neutral

network; and alternative phenotypes are generally lethal. Relaxing

each of these assumptions does not change our qualitative results

(see Supporting Information). In particular, we relax the first as-

sumption by introducing a parameter, f , which is the fraction of K

neighbors that are redrawn following a neutral mutation. Allowing

such correlations between the phenotypic neighborhoods of neu-

tral neighbors affects the rate at which individuals arrive at adapt-

able genotypes through mutation, but it preserves the qualitative

relationships between adaptability and environmental robustness

reported above. Our qualitative results are also unchanged if the

size of the phenotypic neighborhood, K , varies with genotype.

In this case, however, the selective regimes described in Table 2

now depend on the average value of 1+sK . Finally, our qualitative

results are unaltered if alternative phenotypes are moderately dele-

terious rather than lethal; however, the different selective regimes

described in Table 2 would then depend on s, K , and the size

of the fitness penalty suffered by nonadaptive alternative pheno-

types (see Supporting Information). We have limited our analysis

to the regime in which the newly adaptive type is always favored

over the initially fit phenotype, regardless of its environmental

robustness. However, it is possible to imagine scenarios in which

s is sufficiently small that newly adaptive genotypes with low

robustness are deleterious compared to high-robustness individu-

als of the initially fit phenotype. This situation will occur when s



is O(ε) or smaller (see Supporting Information). To study adapt-

ability in this case, we must determine the waiting time until

a newly adaptive individual with high robustness is produced.

However, the qualitative relationships we observe in this case

are the same as those described in the main text (see Supporting

Information).

In this article, we have considered adaptation to a single en-

vironmental shift. It would also be interesting to study repeated

environmental shifts. To do so will require analyzing the adapta-

tion time of a population which is not at equilibrium on its neutral

network when the next environmental shift occurs. Accounting

for such repeated transient dynamics should be possible in our

modeling framework, but it will certainly be nontrivial. The rela-

tionship between the rate of environmental shifts and the evolution

of the environmental robustness thus remains an interesting area

for future research.

Our modeling framework has allowed for substantial varia-

tion in robustness among a large set of genotypes. This framework

is appropriate if many sites across an organism’s genome can in-

dependently influence its robustness, such as might be expected

if networks of interacting genes jointly determine an organism’s

phenotype. In fact, the widespread contribution of many genomic

sites to robustness has been directly observed in organisms rang-

ing from yeast to plants and vertebrates (Fraser and Schadt 2010).

The framework for our study is therefore rather different from

models restricted to a specific locus (such as the HSP90 locus or a

DNA repair locus) that alone determines variation in robustness.

Despite longstanding interest in phenotypic robustness, the

role of robustness in adaptation remains a topic of active investi-

gation. In contrast to most theoretical work, which has focused on

how mutational robustness influences adaptation (de Visser et al.

2003; Lenski et al. 2006; Wagner 2008; Draghi et al. 2010), this

study provides a framework for analyzing the effects of environ-

mental robustness. Although we have uncovered a complex set of

possible behaviors (summarized in Table 2), one important gen-

eral observation emerges: environmental perturbations can either

facilitate or impede adaptation. These results have been derived

in a general theoretical framework, and so they should apply to

abroad range of biological situations and systems.

Theoretical studies of environmental robustness must even-

tually be reconciled with empirical data, either from surveys of

standing variation or, ideally, from evolution experiments. Fortu-

nately, there is an extensive literature based on quantitative ge-

netics that describes empirical patterns of environmental robust-

ness in organisms, including yeast, plants, and mammals (Mackay

and Lyman 2005; Hall et al. 2007; Nogami et al. 2007; Sangster

et al. 2008; Fraser and Schadt 2010). Such studies typically find

a broad array of traits (morphological, life-history, and gene ex-

pression levels) that are buffered against environmental noise and,

furthermore, extensive standing genetic variation in the degree of

such buffering among individuals. Thus, environmental robust-

ness is apparently selected for in many natural populations, in

accordance with theoretical expectations for a population under

stabilizing selection for a target phenotype (Wagner et al. 1997;

Hermisson and Wagner 2004); but a natural population typically

also contains variation in the degree of environmental robustness

among genotypes. Our analysis suggests that the existence of ro-

bust variants in a population, and also the variation in robustness

among individuals, will have a significant impact on the popula-

tion’s capacity to adapt to a novel environment. Thus, the concept

of canalization, first considered by Waddington for populations

under stabilizing selection, plays an important role in shaping the

evolution of populations as they adapt to new environments over

longer timescales.

ACKNOWLEDGMENTSWe thank members of the Plotkin lab for constructive comments and input.JBP acknowledges support from the James S. McDonnell Foundation, theAlfred P. Sloan Foundation, the David and Lucille Packard Foundation,the Burroughs Wellcome Fund, Defense Advanced Research ProjectsAgency (HR0011-05-1-0057), and the US National Institute of Allergyand Infectious Diseases (2U54AI057168).

LITERATURE CITEDAncel, L. W. 1999. A quantitative model of the Simpson-Baldwin effect. J.

Theor. Biol. 196:197–209.———. 2000. Undermining the Baldwin expediting effect: does phenotypic

plasticity accelerate evolution? Theor. Popul. Biol. 58: 307–319.Ancel, L. W., and W. Fontana. 2000. Plasticity, evolvability, and modularity

in rna. J. Exp. Zool. 288: 242–283.Bloom, J., S. Labthavikul, C. Otey, and F. Arnold. 2006. Protein stability

promotes evolvability. PNAS 103: 5869–5874.Chow, K., and K. Chan. 1999. Stress-induced phenocopy of C. elegans de-

fines functional steps of sensory organ differentiation. Develop. Growth.Differ.41: 629–637.

Draghi, J., T. Parsons, G. Wagner, and J. Plotkin. 2010. Mutational robustnesscan facilitate adaptation. Nature 436: 353–355.

Earl, D., and M. Deem. 2004. Evolvability is a selectable trait. PNAS 101:11531–11536.

Ewens, W. J. 2004. Mathematical population genetics, vol. v. 27.2nd ed. Springer, New York. Available at http://www.loc.gov/catdir/enhancements/fy0818/2003065728-d.html.

Fraser, H., and E. Schadt. 2010. The quantitative genetics of phenotypicrobustness. PLoS ONE 5: e8635.

Gardiner, C. W. 2004. Handbook of stochastic methods for physics, chemistry,and the natural sciences. 3rd ed. Springer, Berlin. Available at http://www.loc.gov/catdir/enhancements/fy0818/2004043676-d.html.

Gibson, G., and I. Dworkin. 2004. Uncovering cryptic genetic variation. Nat.Rev. Genet. 5: 681–690.

Goldschmidt, R. 1935. Gen und aussencharakter. Biologisches Zentralblatt55: 535–554.

Goldsmith, M., and D. Tawkik. 2009. Potential role of phenotypic mutationsin the evolution of protein expression and stability. PNAS 106: 6197–6202.

Hall, M. C., I. Dworkin, M. C. Ungerer, and M. Purugganan. 2007. Geneticsof microenvironmental canalization in Arabidopsis thaliana. Proc. Natl.Acad. Sci. USA 104: 13717–13722.



Hermisson, J., and G. P. Wagner. 2004. The population genetic theory ofhidden variation and genetic robustness. Genetics 168: 2271–2284.

Jarosz, D. F., and S. Lindquist. 2010. Hsp90 and environmental stress trans-form the adaptive value of natural genetic variation. Science 330: 1820–1824.

Lehner, B. 2010. Genes confer similar robustness to environmental, stochastic,and genetic perturbations in yeast. PLoS ONE 5: e9035.

Lenski, R., J. Barrick, and C. Ofria. 2006. Balancing robustness and evolv-ability. PLoS Biol. 4: e428.

Levy, S. F., and M. L. Siegal. 2008. Network hubs buffer environmentalvariation in Saccharomyces cerevisiae. PLoS Biol. 6: e264.

Lopez-Maury, L., S. Marguerat, and J. Bahler. 2008. Tuning gene expression tochanging environments: from rapid responses to evolutionary adaptation.Nat. Rev. Genet. 9: 583–593.

Mackay, T. F., and R. F. Lyman. 2005. Drosophila bristles and the natureof quantitative genetic variation. Philos. Trans. R. Soc. Lond. B 360:1513–1527.

Masel, J. 2005. Evolutionary capacitance may be favored by natural selection.Genetics 170: 1359–1371.

Masel, J., and A. Bergman. 2003. The evolution of the evolvability propertiesof the yeast prion [psi+]. Evolution 57: 1498–1512.

Masel, J., and M. Trotter. 2010. Robustness and evolvability. Trends Genet.26: 406–414.

McLoughlin, S., and S. Copley. 2008. A compromise required by gene sharingenables survival: implications for evolution of new enzyme activities.PNAS 105: 13497–13502.

Meiklejohn, C. D., and D. L. Hartl. 2002. A single mode of canalization.Trends Ecol. Evol. 17: 468–473.

Meyerovich, M., G. Mamou, and S. Ben-Yehuda. 2010. Visualizing higherror levels during gene expression in living bacterial cells. PNAS 107:11543–11548.

Meyers, L., F. Ance, and M. Lachmann. 2005. Evolution of genetic potential.PLoS Comput. Biol. 1: e32.

van Nimwegen, E., J. P. Crutchfield, and M. Huynen. 1999. Neutral evolutionof mutational robustness. PNAS 96: 9716–9720.

Nogami, S., Y. Ohya, and G. Yvert. 2007. Genetic complexity and quantitativetrait loci mapping of yeast morphological traits. PLoS Genet. 3: e31.

Proulx, S., S. Nuzhdin, and D. Promislow. 2007. Direct selection on geneticrobustness revealed in the yeast transcriptome. PLoS ONE 2: e911.

Raj, A., and A. van Oudenaarden. 2008. Nature, nurture, or chance: stochasticgene expression and its consequences. Cell 135: 216–226.

Rutherford, S. 2003. Between genotype and phenotype: protein chaperonesand evolvability. Nat. Rev. Genet. 4: 263–274.

Rutherford, S., and S. Lindquist. 1998. Hsp90 as a capacitor for morphologicalevolution. Nature 396: 336–342.

Sangster, T. A., S. Lindquist, and C. Queitsch. 2004. Under cover: causes,effects and implications of hsp90-mediated genetic capacitance. BioEs-says 26: 348–362.

Sangster, T. A., N. Salathia, S. Undurraga, R. Milo, K. Schellenberg,S. Lindquist, and C. Queitsch. 2008. Hsp90 affects the expression ofgenetic variation and developmental stability in quantitative traits. Proc.Natl. Acad. Sci. USA 105: 2963–2968.

Siegal, M. L., and A. Bergman. 2002. Waddington’s canalization re-visited: developmental stability and evolution. PNAS 99: 10528–10532.

Tawkik, D. 2010. Messy biology and the origins of evolutionary innovations.Nat. Chem. Biol. 6: 692–696.

de Visser, J. A., J. Hermisson, G. P. Wagner, L. Ancel Meyers, H. Bagheri-Chaichian, J. L. Blanchard, L. Chao, J. M. Cheverud, S. F. Elena, W.Fontana, et al. 2003. Perspective: Evolution and detection of geneticrobustness. Evolution 57: 1959–1972.

Waddington, C. 1942. Canalization of development and the inheritance ofacquired characteristics. Nature 150: 563–565.

Waddington, C. 1953a. The Baldwin effect, genetic assimilation and home-ostasis. Evolution 7: 386–387.

Waddington, C. 1953b. Genetic assimilation of an acquired character. Evolu-tion 7: 118–126.

———. 1959 Evolutionary systems—animal and human. Nature 183:1634–1638.

Wagner, A. 2008. Robustness and evolvability: a paradox resolved. Proc. R.Soc. Lond. B 275: 91–100.

Wagner, A., and J. Wright. 2007. Alternative routes and mutationalrobustness in complex regulatory networks. Biosystems 88: 163–172.

Wagner, G. P., G. Booth, and H. Bagheri-Chaichian. 1997. A populationgenetic theory of canalization. Evolution 51:329–347.

Associate Editor: Max Reuter



Supporting InformationThe following supporting information is available for this article:

Figure S1. Correlated phenotypic neighbourhood—Mean adaptation time as a function of the probability of environmental

perturbation, ε.

Figure S2. Variation in size of phenotypic neighbourhood—plots show populations of N = 10,000 individuals, with μ = 0.001,

P = 100, K̄ = 5, q = 0.5, φH = 0.9, φL = 0.1 and πH = 0.5 with s = 2 (red line) and s = 8 (blue line).

Figure S3. Nonlethal deleterious mutations—plots show populations of N = 10,000 individuals, with μ = 0.001, P = 100, K =10, q = 0.2, φH = 0.9, φL = 0.1 and πH = 0.5, d = 0.2 with s = 0.5 (red line) and s = 3 (blue line).

Figure S4. Deleterious low robustness adaptive genotype—plots show populations of N = 10,000 individuals, with μ = 0.001,

P = 100, K = 5, q = 0.5 and s = 0.05.

Figure S5. Mean adaptation time as a function of the probability of environmental perturbation, ε, in the regime 1+sK <

φH −φL

1−φL.

Figure S6.The mean adaptation time as a function of mutational robustness, q.

Figure S7. Most realistic selective regime with s/K = 0.2, πH = 0.5 and πL = 0.5.

Figure S8. Less realistic selective regime with s/K = 1.2, πH = 0.5 and πL = 0.5.

Figure S9. Most realistic selective regime with s/K = 0.2, πH = 0.5 and πL = 0.1.

Table S1. Model parameters.

Supporting Information may be found in the online version of this article.

Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the

authors. Any queries (other than missing material) should be directed to the corresponding author for the article.


Environmental robustness and the adaptability of

populations: Supporting Information

Alexander J. Stewart12, Todd L. Parsons1, Joshua B. Plotkin1

November 9, 2011

1Department of Biology, University of Pennsylvania, Philadelphia, PA, USA2CoMPLEX, University College London, Physics Building, Gower Street, London WC1E 6BT,

UK

1 Analytical results

In this section we discuss in detail how adaptation time is related to environmental robustness

by considering the full population dynamics and distribution of adaptation times for the

general model presented in the main text.

1.1 Model Description

We focus on the general model, described in the main text, consisting of a population of N

individuals evolving under the infinite alleles Moran model. Genotypes which belong to the

neutral network, are indexed by a set I of which a subset A ⊂ I are adaptable, meaning that

they are a single mutation away from a genotype which has a selective advantage once an

environmental shift occurs. We assume that at the time of the environmental shift individuals

with an adaptable genotype are rare, meaning that the number of adaptable individuals is

O

�√N

�or smaller. The less realistic case in which adaptable individuals are common is

analyzed in Supplementary Information.

Following Draghi et al. (2010) we adopt the concept of phenotypic neighbourhoods. A

mutation at genotype i is neutral with probability qi and results in a genotype which codes for

a new phenotype with probability 1−qi. When a mutation is non-neutral, the new phenotype

which results is drawn from K alternative phenotypes, which constitute the phenotypic

neighbourhood of the original genotype. TheK phenotypes of the phenotypic neighbourhood

are drawn uniformly form a total P possible alternatives. All alternative phenotypes are

assumed to be lethal prior to the environmental shift. Following the environmental shift

we assume that an alternative phenotype gains a selective advantage, and we calculate the

expected time for the population to produce an individual with such a genotype.

1

We also extend the concept of phenotypic neighbourhoods to describe the effects of

environmental perturbations on phenotypes. Environmental perturbations occur with rate �.

With probability φi an organism which experiences an environmental perturbation develops

the correct (target) phenotype, i. With probability 1− φi it develops the phenotype of one

of it’s K phenotypic neighbours. The parameter φi therefore measures the environmental

robustness of genotype i.

The per-genome mutation rate is µ and the genome length is L. We assume that all sites

mutate at the same rate so that mutations within the neutral network are described by the

adjacency matrix M which has entries mij = 1 if genotype i can mutate to genotype j, and

mij = 0 otherwise, with the mutational robustness qi of a genotype given by Eq. 1 in the

main text. In this section for notational convenience we define m̃ij =1Lmij.

In order to determine the adaptability of a population of the type described above we

study the continuous time Moran model in which all genotypes have the same death rate

and birth rates vary according to the fitness of the genotype. Therefore the birth rate of

an individual with genotype i is 1 + Si. If the number of individuals with genotype i is Zi,

the population dynamics of the Moran model are described by a continuous-time Markov

process with transitions

Zi(t)rij(Z(t))−−−−−−→ Zi(t+∆t) = Zi(t) + 1, Zj(t)

rij(Z(t))−−−−−−→ Zj(t+∆t) = Zj(t)− 1.

that occur at rate rij(Z(t)), for

rij(z)def=

�(1− µ)zi(1 + Si) + µ

�

k∈I

m̃kizk(1 + Sk)

��zj

N

�.

The first term describes the rate at which individuals with genotype i are born. The second

2

term describes the rate at which individuals of type j die. Individuals with genotype i are

born at a rate that depends on zi as well as on the strength of selection for that genotype, Si,

and the rate of mutation into and out of that genotype. Once an individual is born it replaces

an existing member of the population. The individual that is replaced is selected uniformly

at random from the existing individuals, and as a result the death rate of individuals with

genotype j is simply zj

N.

We note that alternative formulations of the Moran model, for example in which death

rates vary, or in the discrete time version in which the birth rates are normalised, will result

in the same limiting diffusions as those described below up to a rescaling of time (see e.g.

(Etheridge, 2011)). For the sake of improved clarity in what follows we will work with

transition rates which are described in terms of θ = Nµ and a rescaled selective coefficient

σi =√NSi, such that we have

rij(z) =

��1− θ

N

�zi

�1 +

σi√N

�+

θ

N

�

k∈I

m̃kizk

�1 +

σk√N

��zj

N

�

The notations adopted in what follows are summarised in Table 3 below:

[Table 1 about here.]

In order to recover the full population dynamics of the model we consider the infinitesimal

generator A of the continuous time stochastic process Z(t), which is given by (Ethier and

Kurtz, 2005)

(Af)(z) =�

i,j

rij(z) (f(z+ ei − ej)− f(z)) .

where ei is the unit vector in the direction of i.

3

1.2 Limit When Adaptable Types are Rare

Previous work (Draghi et al., 2010) showed that a closely related model to the one discussed

here has a diffusive limit when the number of adaptable individuals is O�√

N

�, while the

number of non-adaptable individuals is O (N), and time is rescaled by√N . To that end, we

consider the rescaled population Z̃i at each genotype i, where adaptable types are rescaled by

a factor√N and non-adaptable types are rescaled by a factor N . Since the rescaling differs

between adaptable and non-adaptable genotypes it is convenient to introduce the parameter

hi which tells us how a given genotype i is rescaled, i.e

Z̃i(t) = hiZi(t),

where

hi =

1√N

if i ∈ A

1N

otherwise.

We can now find the generator for the limiting process. Following (Ethier and Kurtz, 2005),

this is given by

limN→∞

AN

for

(ANf)(z̃) =�

i,j

√Nr̃ij(z̃) (f(z̃+ hiei − hjej)− f(z̃)) ,

where r̃ij(z̃) = rij

�(h−1

iz̃i)i∈I

�. Intuitively, rescaling time by

√N increases the rate of events

by the same factor, while increasing or decreasing Zi corresponds to changing Z̃i by ±hi.

Therefore, since hi is small in the large N limit, to find the limiting generator we Taylor

4

expand f(z̃+ hiei − hjej) about z̃ to obtain

(ANf)(z̃) =√N

��

i∈I

hi

�

j∈I

(r̃ij(z̃)− r̃ji(z̃))∂f

∂z̃i+

1

2

�

i∈I

h2i

�

j∈I

(r̃ij(z̃) + r̃ji(z̃))∂2f

∂z̃2i

−�

i �=j

hihj r̃ij(z̃)∂2f

∂z̃i∂z̃j+O

�N

− 32

��.

In what follows, we derive the advective terms; the other terms are derived in a similar

fashion.

To begin, we have

rij(z)− rji(z) =1

N32

�1− θ

N

�(σi − σj)zizj +

θ

N2

�

k∈I

(m̃kizj − m̃kjzi)zk

�1 +

σk√N

�

and, recalling�

j∈I zj = N ,

�

j∈I

(rij(z)−rji(z)) =1

N32

�

j∈I

�1− θ

N

�(σi−σj)zizj+

θ

N2

�

k∈I

�Nm̃ki − zi

�

j∈I

m̃kj

�zk

�1 +

σk√N

�

Thus,

√Nhi

�

j∈I

(r̃ij(z̃)− r̃ji(z̃)) =1

N

�

j∈I

h−1j

�1− θ

N

�(σi − σj)z̃iz̃j

+ hi

θ

N32

�

k∈I

h−1k

�Nm̃ki − h

−1iz̃i

�

j∈I

m̃kj

�z̃k

�1 +

σk√N

�

=1√N

�

j∈A

�1− θ

N

�(σi − σj)z̃iz̃j +

�

j �∈A

�1− θ

N

�(σi − σj)z̃iz̃j

+ θ

�

k∈A

�him̃ki −

1

Nz̃i

�

j∈I

m̃kj

�z̃k

�1 +

σk√N

�

+ θ

�

k �∈A

�√Nhim̃ki −

1√Nz̃i

�

j∈I

m̃kj

�z̃k

�1 +

σk√N

�.

5

If we now take the large population limit we find,

limN→∞

√Nhi

�

j∈I

(r̃ij(z̃)− r̃ji(z̃)) =

��j �∈A(σi − σj)z̃j

�z̃i + θ

�k �∈A m̃kiz̃k if i ∈ A

��j �∈A(σi − σj)z̃j

�z̃i otherwise.

Proceeding similarly,

limN→∞

√N

�

i∈I

h2i

�

j∈I

(r̃ij(z̃) + r̃ji(z̃)) =

2z̃i if i ∈ A

0 otherwise,

while

limN→∞

√N

�

i �=j

hihj r̃ij(z̃)∂2f

∂z̃i∂z̃j= 0.

Thus, the limiting process has generator

(Af)(z̃) =�

i �∈A

��

j �∈A

(σi − σj)z̃j

�z̃i

�∂f

∂z̃i

+�

i∈A

��

j �∈A

(σi − σj)z̃j

�z̃i + θ

�

k �∈A

m̃kiz̃k

�∂f

∂z̃i+ z̃i

∂2f

∂z̃2i

,

i.e. the limiting system may be decomposed into two components since the first sum describes

the dynamics of the non-adaptable genotypes (i /∈ A) and the second sum describes the

dynamics of the adaptable genotypes (i ∈ A). For readability, in what follows we distinguish

between the adaptable and non-adaptable types by defining

Z̃i =

Xi if i /∈ A

Yi if i ∈ A

Decomposing the limiting system into equations for the non-adaptable and adaptable types,

6

we have that the non-adaptable types correspond to the terms

�

i �∈A

��

j �∈A

(σi − σj)xj

�xi

�∂f

∂xi

This describes a system which evolves deterministically (i.e which has no stochastic compo-

nent), according to the dynamical system (see e.g. Gardiner (2004))

dXi

dt=

��

j �∈A

(σi − σj)Xj

�Xi,

which is Eq. 2 of the main text. However, the adaptable types correspond to

�

i∈A

��

j �∈A

(σi − σj)xj

�yi + θ

�

k �∈A

m̃kixk

�∂f

∂yi+ yi

∂2f

∂y2i

.

Therefore, given X(t), each adaptable type evolves according to an independent diffusion

processes, with non-autonomous rates dependent upon the non-adaptable types:

dYi =

��

j �∈A

(σi − σj)Xj

�Yi + θ

�

k �∈A

m̃kiXk

�dt+

�2YidBi(t),

for independent Brownian motions Bi(t), i ∈ A (Gardiner, 2004), which is Eq. ?? of the

main text.

1.3 The Waiting Time Distribution When Adaptable Types are

Rare

In the previous section we showed that the population at each adaptable genotype is de-

scribed by an independent diffusion process. We are interested in the time it takes for the

7

population to produce an individual with the newly adaptive genotype following an environ-

mental shift. Because adaptable genotypes evolve independently it suffices to consider the

waiting time problem for a single diffusion,

dY = (σY +M) dt+√2Y dB(t),

where for readability we omit the subscript i, and write σ =�

j �∈A(σi−σj)Xj for the selection

term and M = θ�

k �∈A m̃kiXk for the influx to each adaptable type. The first newly adaptive

genotype occurs when an individual at an adaptable genotype mutates to the newly adaptive

genotype. For a given adaptable genotype the waiting time T for the arrival of the first newly

adaptive mutant to arise is then given by the distribution

P {T > t} = E�e−ζV (t)

�,

where ζ is the rate at which the newly adaptive type is produced from an adaptable type

through mutation and V (t) is a stochastically varying rate function which depends on the

population at the adaptable genotype

V (t) =

�t

0

Y (s) ds (1)

We now make two observations:

1. While the process V (t) is not by itself Markovian, the pair (Y (t), V (t)) does satisfy the

Markov property. Now (1), is equivalent to the SDE dV (t) = Y (t) dt, so (Y (t), V (t))

8

satisfies the system of SDEs

dY = (σY +M) dt+√2Y dB(t),

dV = Y dt.

Thus, their joint density satisfies the Forward Kolmogorov Equation (Gardiner, 2004)

∂u

∂t= − ∂

∂y[(σy +M)u]− ∂

∂v[yu] +

∂2

∂y2[yu] (2)

with initial condition u(y, v, t) = δ(y0,0), the Dirac point mass at the single point (y0, 0).

2. If we set

φ(p, q, t)def= E

�e−pY (t)−qV (t)

�=

� ∞

0

� ∞

0

e−py−qv

u(y, v, t) dy dv,

which is the joint Laplace transform of Y (t) and V (t), we have

P {T > t} = φ(0, ζ, t).

Taking the Laplace transform of Eq. 10 yields

∂φ

∂t= (−p

2 + σp+ q)∂φ

∂p−Mpφ+ lim

y→0

�(σy +M)u− ∂

∂y[yu]− yu

�+ lim

v→0vu.

Now,

limy→0

(σy +M)u− ∂

∂y[yu]

9

is the flux, J(v, t) across the y = 0 boundary, which we assume to be 0 on biological grounds

(see Draghi et al. (2010) §1.2.2 for a more detailed discussion on boundary conditions), while

limy→0

yu = 0.

and

limv→0

vu = 0.

We are thus left with solving

∂φ

∂t+ (p2 − σp− q)

∂φ

∂p= −Mpφ,

which may be done via the method of characteristics. To that end, we solve

dp

p2 − σp− q= dt,

which, upon completing the square, p2 − σp− q =�p− σ

2

�2 −�q + σ

2

4

�, has solution

p(t) =2α(q)

1− Ce2α(q)t− α(q) +

σ

2(3)

where C is a constant of integration that identifies the characteristic curve and we have set

α(q) =

��q +σ2

4

��.

Choosing the form for p(t) given in Eq. 11, we now set φ̃(t)def= φ(p(t), q, t), so that

dφ̃

dt= −Mp(t)φ̃.

10

This linear equation has solution

φ̃(t) = e−M

�t

0 p(τ) dτF (C),

for an arbitrary function F . Integrating Eq.11 yields

�t

0

p(τ) dτ = ln

��(1− C)e2α(q)t

1− Ce2α(q)t

��+�σ

2− α(q)

�t

We can now recover φ(p, q, t) by noting from Eq. 11 that

C =p− α(q)− σ

2

p+ α(q)− σ

2

e−2α(q)t

so that, after some simplification

φ(p, q, t) =

��

e2α(q)t − p−α(q)−σ

2p+α(q)−σ

2

2α(q)p+α(q)−σ

2

��

−M

e−M(σ

2−α(q))tF

�p− α(q)− σ

2

p+ α(q)− σ

2

e−2α(q)t

�

Lastly, to determine F , we observe that

φ(p, q, 0) =

� ∞

0

� ∞

0

e−py−qv

u(y, z, 0) dy dv

=

� ∞

0

� ∞

0

e−py−qv

δ(y0,0)(y, v) dy dv = e−py0 ,

so that

e−py0 = F

�p− α(q)− σ

2

p+ α(q)− σ

2

�

i.e.

F (x) = e−( 2α(q)

1−x+σ

2−α(q))y0

11

Thus,

φ(p, q, t) =

��

e2α(q)t − p−α(q)−σ

2p+α(q)−σ

2

2α(q)p+α(q)−σ

2

��

−M

e−M(σ

2−α(q))te

−

2α(q)

1−p−α(q)−σ

2p+α(q)−σ

2e−2α(q)t

+σ

2−α(q)

y0

In order to recover the desired waiting time distribution we must set p = 0 and q = ζ, which

yields

P {T > t} =

��

�α(ζ)− σ

2

�eα(ζ)t +

�α(ζ) + σ

2

�e−α(ζ)t

2α(ζ)

��

−M

e−M

σ

2 te

−

2α(ζ)

1+α(ζ)+σ

2α(ζ)−σ

2e−2α(ζ)t

+σ

2−α(ζ)

y0

(4)

Notice that taking σ = 0 and y0 = 0, we get

P {T > t} = sechM

��ζt

�,

which is, up to factors of two owing to the difference between Wright-Fisher and Moran

models, the result given in (Draghi et al., 2010).

Eq. 12 gives the waiting time distribution for a given adaptable genotype i to give

rise to an individual with the newly adaptive genotype. Since the population at each of

the different adaptable genotypes evolves according to an independent diffusion process, we

can calculate the waiting time distribution for each adaptable genotype independently, and

recover the expected waiting time for a full network of genotypes to give rise to the newly

adaptive genotype. This is calculated as follows: take P {Ti > t} to be the probability that

we wait for longer than t for a newly adaptive genotype to arise from adaptable genotype i.

Because these distributions are independent, the probability that we wait for longer than t

for a newly adaptive genotype to arise from any adaptable genotype is just the product over

the distributions, i.e.�

iP {Ti > t}. The mean waiting time, E [T ], for a newly adaptive

12

genotype to arise anywhere in the network is then given by

E [T ] =

� ∞

0

�

i

P {Ti > t} dt (5)

In the simplified model with genotypes of only high and low robustness, this can be reduced

to the product over two distributions – the waiting time for a newly adaptive type to arise

from a high robustness adaptable genotype, P {TH > t}, and the waiting time for a newly

adaptive type to arise from a low robustness adaptable genotype, P {TL > t}, where we

must take the influx term M in each distribution to be the total influx to all high and low

robustness adaptable types respectively. These are given in Eqs. 6-7 as

MH = qθK

P(πHXH + (1− πL)XL)

and

ML = qθK

P((1− πH)XH + πLXL)

The selection terms, σH and σL used for each distribution are given in Eqs. ??–??. The

expected waiting time, E [T ], for a newly adaptive genotype to arise in the simplified model

is then given by

E [T ] =

� ∞

0

P {TH > t}P {TL > t} dt

It is this quantity, calculated for the simplified model assuming that adaptable types are suf-

ficiently rare that prior to the environmental shift no adaptable types exist in the population,

which is used in our results (Figures ??–6).

13

1.4 Limit when adaptable individuals are common

In this section we determine the adaptation time when the numer of adaptable individuals

at the time of the environmental shift is O(N). In this case we say that the adaptable

individuals are “common”, compared to the case in which they are rare (i.e. O(√N)) as

described in the main text. We assume that a subset A� ⊆ A of the adaptable genotypes

are represned by O(N) individuals, so that Zi(t) = NZ̃i(t) for i ∈ A�, Zj(t) =√NZ̃j(t) for

j ∈ A−A�.

To investigate the first arrival of the advantageous type, we will consider the process stopped

when the first advantageous type arrives: we introduce a “graveyard” state ∆, and assume

that

ri,∆(z) = µmi,∆zi for i ∈ A,

r∆,i = 0 for all i ∈ I

i.e. we assume the process jumps to the graveyard state whenever some individual produces

an offspring of the selected type, whilst the process, once in the graveyard state, remains

there. This extended process has generator

�

i,j

rij(z) (f(z+ ei − ej)− f(z))−�

i∈A

ri,∆(z) (f(∆)− f(z)) .

Proceeding as before, we take

Z̃i(t) = hiZi(t),

14

where now

hi =

1√N

if i ∈ A−A�

1N

otherwise,

and time is left unscaled. Taking the limit as N → ∞, we find that the limiting generator is

�

i∈A�

θmi,∆z̃i (f(∆)− f(z̃)) .

Thus, the process reduces to a simple Markov jump process, in which after an exponentially

distributed time with rate�

i∈A�

θmi,∆z̃i,

the process jumps to the graveyard state, i.e. an individual of the advantageous type is

produced, while the frequencies of each type remain fixed at their initial values over this

timescale. Further, the rate at which advantageous offspring are produced from an individual

of type i ∈ A� is

θmi,∆z̃i,

from which we conclude that the probability that the first advantageous individual had a

parent of type i ∈ A is

P {type i} =

θmi,∆z̃i�j∈A� θmj,∆z̃j

if i ∈ A�

0 otherwise,

while

P {T ≥ t} = e−(

�i∈A� θmi,∆z̃i)t.

This expression gives the waiting time before arrival of an individual with the newly adaptive

genotype, in the case that (some) adaptable types are common at the time of the environ-

15

mental shift. In contrast to the case in which all adaptable types are rare, we see that the

waiting time in this case is exponentially distributed, and that the individual who gives rise

to the newly adaptive genotype aways comes from an adaptable genotype i ∈ A�, i.e. the

common subset of adaptable types at the time of the environmental shift. As a result, there

is a different relationship between between environmental robustness, rate of environmental

perturbations, etc. compared to the case case in which adaptable types are rare.

2 Relaxation of model assumptions

In this section we relax each of the main model assumptions used in the main text, and we

revise our analysis accordingly. We demonstrate that our qualitative results are unchanged

by relaxing these assumptions.

2.1 Mutation rates:

In our simulations we have chose a per-genome mutation rate of µ = 10−3. The reason for

this is as follows: for a non-synonymous substitution rate of 10−8, µ = 10−3 is equivalent to

a genome of 105 nucleotides. The rate of mutations that result in a newly adaptive genotype

is not µ = 10−3 but is µ(1− q)/K (see Fig 2 of the main text). In our simulations we have

typically picked K = 5 and q = 0.5 so that µ(1 − q)/K = 10−4. It corresponds to around

104 possible adaptive mutations (given our assumed per-nucleotide substitution rate). It

therefore corresponds to the case in which many genes are initially involved in adaptation.

Mutation rates between other genotype classes can be calculated in a similar way. Smaller

mutation rates between genotype classes can be achieved by choosing different values for

other parameters,. For example, taking K = 100 results in a rate of adaptive mutations

16

of ∼ 10−6. However doing this adds no new insight since we have catalogued the way the

adaptation time varies with all of our model parameters.

2.2 Correlated phenotypic neighbourhoods:

In the main text we assume that, following a mutation, the phenotypic neighbourhood of

the resulting individual is completely redrawn – that, the set of its K phenotypic neighbours

is drawn uniformally from the set of P alternatives. This results uncorrelated phenotypic

neighbourhoods amoung different genotypes. We relax this assumption by introducing a pa-

rameter f , which is the fraction of the K phenotypes in a individual’s phenotypic neighbour-

hood that are redrawn following mutation. When f = 1 we recover the case of completely

uncorrelated phenotypic neighbourhoods. When f = 0, all genotypes have the same pheno-

typic neighbourhood. Introducing f results in a modified mutation rate between adaptable

and non-adaptable types. Specifically, the rate of mutation from non-adaptable to adapt-

able types is µqfK

Pand from adaptable to non-adaptable types is µq

�1− f

K

P

�. All other

mutation rates and selective coefficients remain unchanged from those presented in the main

text. The effect of decreasing f is therefore identical to increasing P . Decreasing the rate

of influx to adaptable types tends to slow overall adaptation time, but it does not alter the

qualitative behaviour of the model described in the main text, which depends on the selective

coefficients. This is illustrated in Fig. S1 for the variation in adaptation time with rate of

environmental perturbations, �.

[Figure 1 about here.]

17

2.3 Variation in the size of phenotypic neighbourhoods with geno-

type:

Next we relax the assumption that all genotypes have exactly K phenotypic neighbours.

To do this we instead assign each of the P alternative phenotypes to the phenotypic neigh-

bourhood of a genotype with probability k, following mutation. The size of the phenotypic

neighbourhood, K, for a given genotype is therefore drawn from a binomial distribution

with mean K̄ = kP . Because phenotypic neighbourhood size varies with genotype, differ-

ent adaptable types have different selection coefficients. Adaptable types can therefore no

longer be simply separated into categories of high and low robustness. Instead they must be

separated into categories according to their environmental robustness φ and the size of their

phenotypic neighbourhood K. The selective coefficient for a high robustness genotype with

K phenotypic neighbours is given by

σH(K)√N

= �1 + s

K(1− φH) + � (φH − φL)XL (6)

and for low robustness genotypes with K phenotypic neighbours by

σL(K)√N

= �1 + s

K(1− φL)− � (φH − φL)XH . (7)

Let Y K

His the number of individuals, rescaled by

√N , at high robustness genotypes with K

phenotypic neighbours. When adaptable types are rare, the population at such genotypes

evolves according to

18

dYK

H=

�σH(K)Y K

H+G(K)qθ

K

P((1− πH)XH + πLXL)

�dt+

�2Y K

HdBH(t) (8)

Similarly, let Y K

Lis the number of individuals, rescaled by

√N , at low robustness genotypes

with K phenotypic neighbours. The population at these genotypes evolves according to

dYK

L=

�σL(K)Y K

L+G(K)qθ

K

P(πHXH + (1− πL)XL)

�dt+

�2Y K

LdBL(t) (9)

where G(K) is the probability that the genotype has K neighbours,

G(K) =

�P

K

�kK(1− k)P−K

�P

K=1

�P

K

�kK(1− k)P−K

Where we assume that all genotypes must have at least one phenotypic neighbour (i.eK ≥ 1).

Finally, the rate at which individuals with the newly adaptive genotype are produced thor-

ough mutation at adaptable genotypes with K phenotypic neighbours is µ1−q

K.

The full distribution of adaptation times from each adaptable type can be found (see Sup-

porting Information). Let P�TK

H> t

�be the waiting time distribution from high robustness

genotypes with K phenotypic neighbours, and P�TK

L> t

�be the waiting time distribution

from low robustness genotypes with K phenotypic neighbours. The expected waiting time

for an individual with the newly adaptive phenotype to arise is then

19

E [T ] =

� ∞

0

P�

K=1

P�T

K

H> t

�P�T

K

L> t

�dt.

Allowing K to vary does not change the set of qualitative behaviours described in the main

text. However it does change the parameter values for which those behaviours occur. This

is because the observed behaviours depend on whether low robustness adaptable types are

selectively favoured compared to the population prior to the environmental shift. This in

turn depends on the ratio s

K(see Table 2, main text). Because K is distributed across

a range of values, this tends to result in some low robustness genotypes being selectively

favoured and some disfavoured. Despite this, all the qualitative behaviours described in the

main text are still observed, as shown in Figure S2.


2.4 Inclusion of non-lethal deleterious mutations:

We now relax the assumption that all deleterious mutations are lethal, i.e that alternative

phenotypes have fitness 0. To do this we introduce four new classes of genotypes. Previously

we had classes

• A – High robustness and adaptable

• B – Low robustness and adaptable

• C – High robustness and non-adaptable

• D – Low robustness and non-adaptable

20

Each of these four classes are now split into two – fit and deleterious. The phenotype coded

for by the “fit” genotypes has fitness 1 and the phenotype coded for by the “deleterious” geno-

types has fitness 1−d, where d is O�

1√N

�or larger. As before, environmental perturbations

may result in phenocopy, such that individuals develop the phenotype of their mutational

neighbours. As a result genotypes Cfit and Dfit have fitness 1−d(1−φH)� and 1−d(1−φL)�

respectively. Genotypes Cdel and Ddel are assumed to both have fitness 1−d – i.e we neglect

the influence of phenocopy on the fitness of these genotypes. Genotypes Afit and Bfit have

fitness 1− d(1−φH)�+s+d

K(1−φH)� and 1− d(1−φL)�+

s+d

K(1−φL)� respectively. Finally,

genotypes Adel and Bdel have fitness 1− d+ s+d

K(1− φH)� and 1− d+ s+d

K(1− φL)� respec-

tively. Mutations between the different classes occur as described for the previous model,

with the modification that mutations at a “fit” genotype which are non-neutral result in a

“deleterious” genotype. Mutations at a “deleterious” genotype which are non-neutral result

in a “fit” genotype with probability 1K. The resulting model has four classes of adaptable

genotype, with selective coefficients

σfit

H√N

= �d+ s

K(1− φH) + �d (φH − φL)XL (10)

for genotypes Afit,

σdel

H√N

= �d+ s

K(1− φH) + �d (φH − φL)XL − d (1− �(1− φH)) (11)

for genotypes Adel,

21

σfit

L√N

= �d+ s

K(1− φL)− �d (φH − φL)XH (12)

for genotypes Bfit, and

σdel

L√N

= �d+ s

K(1− φL)− �d (φH − φL)XH − d (1− �(1− φL)) (13)

for genotypes Bdel. We have assumed that, prior to the environmental shift, the whole pop-

ulation can be treated as being only at “fit” genotypes.

The full distribution of adaptation times from each adaptable type can be found (see Sup-

porting Information) for this model. Let P�T

fit

H> t

�and P

�T del

H> t

�be the waiting time

distribution for high robustness genotypes, and P�T

fit

L> t

�and P

�T del

L> t

�be the waiting

time distribution for low robustness genotypes. The expected waiting time for an individual

with the newly adaptive phenotype to arise is then

E [T ] =

� ∞

0

P�T

fit

H> t

�P�T

fit

L> t

�P�T

del

H> t

�P�T

del

L> t

�dt.

Once again, this model produces the same set of qualitative behaviours as described in the

main text. However it changes the parameter values for which those behaviours occur. In

particular, the different selective regimes corresponding to different behaviours, described in

Table 2 of the main text are 1+s

K< 1 and 1+s

K<

φH−φL

1−φL

. In this model the corresponding

selective regimes are d+s

K< d and d+s

K< d

φH−φL

1−φL

. This has the effect that, if d < 1, the

22

selective regime described as “unrealistic” in the main text, requiring large values of s, can

now occur for smaller values of s, making the full range of behaviours described in the main

text attainable under more realistic sets of parameters. The full range of behaviours of this

model are shown in Figure S3.


2.5 Adaptability when low robustness adaptive genotypes are se-

lected against:

Wemay also consider the case when newly adaptive genotypes with low robustness have lower

fitness than the mean population fitness. This occurs in the following scenario: Assume that

newly adaptive genotypes inherit the environmental robustness of the individual that gave

rise to them. Therefore high robustness adaptive genotypes have fitness (1 + s)(1 − (1 −

φH)�) +1K(1 − φH)� and low robustness adaptive genotypes have fitness (1 + s)(1 − (1 −

φL)�) +1K(1− φL)�. The mean fitness of the population at the environmental shift is simply

1− (1− φH)�, therefore low robustness adaptive genotypes are only selectively favoured if

(1 + s)(1− (1− φL)�) +1

K(1− φL)� > 1− (1− φH)�

i.e if

s >� ((φH − φL)K − (1− φL))

K(1− (1− φL)�)

otherwise they are selected against. When this is the case we consider that only high

23

robustness adaptive genotypes are truly adaptive, and calculate the waiting time until an

individual with this genotype arises. This is easy to do since low robustness adaptable geno-

types produce high robustness adaptive genotypes at rate (1−q)(1−πL)K

and high robustness

adaptable genotypes produce high robustness adaptive genotypes at rate (1−q)πH

K. We can

neglect the effect of low robustness adaptive genotypes mutating to produce high robustness

adaptive genotypes, since we assume that the population is initially at the non-adaptable

genotypes which cannot mutate directly to the newly adaptive genotypes. Using these modi-

fied mutation rates we determine the waiting time until a high robustness adaptive genotype

arises (assuming that s is sufficiently small that low robustness adaptive genotypes are se-

lected against). These waiting times are shown varying with the model parameters in Figure

S4 below. As can be seen the same qualitative behaviours as described for our model in the

regime 1+s

K< 1 are recovered in this case, with the addition that increasing low robustness

genotype clustering increases adaptation time (since it slows the rate of mutations from low

robustness adaptable genotypes to high robustness adaptive genotypes).


2.6 Other influences of clustering of high-robustness genotypes on

adaptation time:

We also examined how the clustering of high-robust genotypes influences the relationship

between environmental perturbations and mean adaptation time, in the regime 1+s

K<

φH−φL

1−φL

.

As described in the main text, in this regime increased clustering of high-robust genotypes

facilitates adaptation, regardless of the amount of environmental noise. In addition, Figure

5 shows that the position of the maximum adaptation time occurs at a lower value of �,

when genotypes are more clustered. In other words, clustering of robust genotypes broadens

24

the range over which increasing the frequency of environmental perturbation will facilitate

adaptation.


Dependence of adaptation time on mutational robustness:

We have also studied how environmental noise can influence the relationship between mu-

tational robustness and adaptability. Previous work has shown that, in the absence of

environmental noise, intermediate levels of mutational robustness produce the fastest rates

of adaptation (Draghi et al., 2010; Wagner, 2008). But how does adaptation time depend

on the amount of mutational robustness, q, when environmental noise is present?

We find that introducing environmental noise does not qualitatively change the way in

which adaptation time varies with mutational robustness: intermediate levels of mutational

robustness still produce the faster rates of adaptation. Nonetheless, environmental noise

does alter the quantitative aspects of this relationship (Figure 6), in two ways.

Firstly, noise can either increase or decrease overall adaptation time compared to the

case when environmental noise is absent. The direction of this effect depends on whether

adaptable individuals with low environmental robustness are selectively favored or disfavored,

that is whether 1+s

K>

φH−φL

1−φL

or 1+s

K<

φH−φL

1−φL

. As above, there is a simple intuition for

this behavior. In the first of these regimes, both types of adaptable individuals gain a

direct selective benefit in the presence of noise, speeding adaptation. In the second regime,

however, the population concentrates at genotypes with high environmental robustness, and

adaptation is generally slower (especially when noise is frequent).

25

Aside from changing the overall rate of adaptation, environmental noise also changes the

value of q that maximizes the rate of adaptation (Figure 6). That is, environmental noise

alters the level of genetic robustness that is optimal for rapid adaption.


Time series:

We plot time series data for the 4 different classes of genotype. Prior to the environmental

shift the population is in equilibrium (Fig. 7). After the environmental shift individuals

start to accumulate at adaptable genotypes due to phenocopy. Fig. 9 shows time series for

parameters identical to those shown in Fig.8, with the exception that πL has been varied

(πL = 0.5 in Fig. 7 and πL = 0.1 in Fig. 9). The fact that these two plots are the same

illustrates that piL has virtually no effect either on the equilibrium prior to the environmental

shift or on the adaptive dynamics following the environmental shift. Fig. 8 shows time series

plots for the same parameters as Fig. 7, with the exception that s has been increased (s = 1

in Fig. 8 and s = 6 in Fig. 8), such that the two cases lie in different selective regimes. We

see that in Fig. 8, in contrast to Fig. 7, individuals accumulate rapidly at low robustness

adaptable genotypes.




26

References

Draghi, J., T. Parsons, G. Wagner, and J. Plotkin, 2010. Mutational robustness can facilitate

adaptation. Nature 436:353–355.

Etheridge, A. M., 2011. Some Mathematical Models from Population Genetics, Lecture Notes

in Math, vol. 2012. Springer, Berlin Heidelberg.

Ethier, S. and T. Kurtz, 2005. Markov processes : characterization and convergence. Wiley

Interscience, Hoboken.

Gardiner, C. W., 2004. Handbook of stochastic methods for physics, chem-

istry, and the natural sciences. 3rd ed ed. Springer-Verlag, Berlin. URL

http://www.loc.gov/catdir/enhancements/fy0818/2004043676-d.html.

Wagner, A., 2008. Robustness and evolvability: a paradox resolved. Proceedings of the

Royal Society B: Biological Sciences 275:91–100.

27

List of Figures

1 Correlated phenotypic neighbourhood . . . . . . . . . . . . . . . . . . . . . . 29

2 Variation in size of phenotypic neighbourhood . . . . . . . . . . . . . . . . . 30

3 Non-lethal deleterious mutations . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Deleterious low robustness adaptive genotype . . . . . . . . . . . . . . . . . 32

5 Adaptation, environmental noise, and clustering . . . . . . . . . . . . . . . . 33

6 Adaptation and mutational robustness . . . . . . . . . . . . . . . . . . . . . 34

7 Time series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Time series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9 Time series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

28

0.0 0.2 0.4 0.6 0.8 1.00

1000

2000

3000

4000

5000

Frequency of environmental perturbations, Ε

Adaptationtime�gene

rations�

Figure 1: Correlated phenotypic neighbourhood – Mean adaptation time as a function ofthe probability of environmental perturbation, �. Plots show different levels of correlationbetween phenotypic neighbourhoods, with f = 1 (red, uncorrelated), f = 0.5 (blue) and f =0.3 (black). As phenotypic neighbourhoods become more correlated, mean adaptation timeincreases. Plots show means of 10,000 replicate Monte-Carlo simulations with populationsof N = 10, 000 individuals, with µ = 0.001, P = 100, K = 5, q = 0.5, φH = 0.8, φL = 0.2and πH = 0.9 with s = 2.

29

0.0 0.2 0.4 0.6 0.8 1.00

200400600800100012001400

High robustness genotype clustering, ΠHL


rations�

(a) Mean adaptation time as a function of high ro-bustness genotype clustering, πH

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000

Strength of low robustness genotypes, ΦL


rations�

(b) Mean adaptation time as a function of robust-ness of low robust genotypes, φL

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000

Strength of low robustness genotypes, ΦH


rations�

(c) Mean adaptation time as a function of robust-ness of high robust genotypes, φH

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

500

600

700



rations�

(d) Mean adaptation time as a function of the prob-ability of environmental perturbation, �

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000

Mutational Robustness, q


rations�

(e) Mean adaptation time as a function of muta-tional robustness, q

Figure 2: Variation in size of phenotypic neighbourhood – Plots show populations of N =10, 000 individuals, with µ = 0.001, P = 100, K̄ = 5, q = 0.5, φH = 0.9, φL = 0.1 andπH = 0.5 with s = 2 (red line) and s = 8 (blue line). Lines indicate the analytic solution toour model, whereas dots indicate the means of 10,000 replicate Monte-Carlo simulations.

30

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

High robustness genotype clustering, ΠHL


rations�


0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000



rations�

(b) Mean adaptation time as a function of robust-ness of low robust genotypes, φL

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000



rations�


0.0 0.2 0.4 0.6 0.8 1.00

200

400

600

800

1000



rations�

(d) Mean adaptation time as a function of the prob-ability of environmental perturbation, �

0.0 0.2 0.4 0.6 0.8 1.00

200400600800100012001400



rations�

(e) Mean adaptation time as a function of muta-tional robustness, q

Figure 3: Non-lethal deleterious mutations – Plots show populations of N = 10, 000 individ-uals, with µ = 0.001, P = 100, K = 10, q = 0.2, φH = 0.9, φL = 0.1 and πH = 0.5, d = 0.2with s = 0.5 (red line) and s = 3 (blue line). Lines indicate the analytic solution to ourmodel, whereas dots indicate the means of 10,000 replicate Monte-Carlo simulations.

31

0.0 0.2 0.4 0.6 0.8 1.00

1000

2000

3000

4000

5000

6000

High robustness genotype clustering, ΠH

Meanadaptationtime�gene

rations�


0.0 0.2 0.4 0.6 0.8 1.01000

1100

1200

1300

1400

1500

Low robustness genotype clustering, ΠL


rations�

(b) Mean adaptation time as a function of robust-ness of low robust genotypes, πL

0.0 0.2 0.4 0.6 0.8 1.00

200400600800100012001400



rations�


0.0 0.2 0.4 0.6 0.8 1.00

200400600800100012001400



rations�

(d) Mean adaptation time as a function of robust-ness of high robust genotypes, φL

0.0 0.2 0.4 0.6 0.8 1.09000

9500

10 000

10 500

11 000

11 500

12 000



rations�

(e) Mean adaptation time as a function of the prob-ability of environmental perturbation, �

Figure 4: Deleterious low robustness adaptive genotype – Plots show populations of N =10, 000 individuals, with µ = 0.001, P = 100, K = 5, q = 0.5 and s = 0.05. In (a), (b),(d) and (e) φH = 0.9, (a), (b), (c) and (e) in φL = 0.1, in (c) and (d) πH = 0.5 and in (e)πH = 0.9 , in (c) and (d) πL = 0.5 and in (e) πL = 0.1, in (a), (b), (c) and (d) � = 0.1. Linesindicate the analytic solution to our model.

32

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000



rations�

Figure 5: Mean adaptation time as a function of the probability of environmental pertur-bation, �, in the regime 1+s

K<

φH−φL

1−φL

. In general, adaptation is faster when high-robustgenotypes are more clustered (blue line) as opposed to less clustered (red line). Moreover,the clustering of robust genotypes broadens the range over the rate of adaptation is pos-itively correlated with the rate of environmental perturbation. Plots show populations ofN = 10, 000 individuals, with µ = 0.001, P = 100, K = 5, q = 0.5, φH = 0.9, φL = 0.1 ands = 2 with πH = 0.1 (red line) and πH = 0.3 (blue line). Lines indicate the analytic solutionto our model, whereas dots indicate the means of 10,000 replicate Monte-Carlo simulations.

33

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000

2500



rations�

Figure 6: The mean adaptation time as a function of mutational robustness, q. There isnon-monotonic relationship between mutational robustness and mean adaptation time, andthe shape of this trend is preserved even in the presence of environmental noise. However,the addition of environmental noise tends to facilitate adaptation in the regime 1+s

K>

φH−φL

1−φL

,and it retard adaptation otherwise. Moreover, the presence of environmental noise changesthe value of q that maximizes the rate of adaptation. Plots show populations of N = 10, 000individuals, with µ = 0.001, � = 0.1 P = 100, K = 5, φH = 0.9, φL = 0.1 and πH = 0.5 withs = 2 (red line) and s = 5 (blue line). The case with no environmental noise (� = 0, blackline) is also shown. Lines indicate the analytic solution to our model, whereas dots indicatethe means of 10,000 replicate Monte-Carlo simulations.

34

0 200 400 600 800 10000.925

0.93

0.935

0.94

0.945

0.95

0.955

Time (generations)

Freq

uenc

y

(a) Non-adaptable high robustness

0 200 400 600 800 10000.045

0.05

0.055

0.06

0.065

0.07

0.075

Time (generations)

Freq

uenc

y

(b) Adaptable high robustness

! "!! #!! $!! %!! &!!!"'(

"'((

"'$

"'$(

"')

"')(

"'%

"'%(

"'*

"'*( +,&!-

./01,23141567/849:

;51<=14>?

(c) Non-adaptable low robustness

! "!! #!! $!! %!! &!!!!'(

&

&'(

"

"'(

)

)'( *+&!#

,-./+01/2/345-6278

93/:;/2<=

(d) Adaptable low robustness

Figure 7: Most realistic selective regime with s/K = 0.2, πH = 0.5 and πL = 0.5 a) Non-adaptable high robustness genotypes. This shows that the prior to the environmental shiftmost of the population is at this genotype class and that following the environmental shiftthere is little change in the population at this genotype class before adaptation occurs (thereis a drop of ∼ 2%). b) Adaptable high robustness genotypes. This shows that prior tothe environmental shift the population at this genotypes class is small (due to their rarity)and that following the environmental shift and the population increases by ∼ 0.02. c) Non-adaptable low robustness genotypes. This shows that the prior to the environmental shiftthe population at this genotype class is negligible and that following the environmental shiftthere is no obvious change in the population at this genotype class before adaptation occurs.d) Adaptable high robustness genotypes. This shows that prior to the environmental shiftthe population at this genotypes class is small (due to their rarity) and that following theenvironmental shift and the population increases by ∼ 1.2 × 10−4, much less than the highrobustness adaptable genotypes (b). Plots are for N = 10, 000 individuals, with µ = 0.001,P = 100, K = 5, q = 0.5 and s = 1, φH = 0.9, φL = 0.1, πH = 0.5, πL = 0.1 and � = 0.1.Plots show averages over 1000 sample paths.

35

0 200 400 600 800 10000.84

0.86

0.88

0.9

0.92

0.94

0.96

Time (generations)

Freq

uenc

y


0 200 400 600 800 10000.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Time (generations)

Freq

uenc

y

(b) Adaptable high robustness

! "!! #!! $!! %!! &!!!"'((

"'$

"'$(

"')

"')(

"'%

"'%(

"'*

"'*( +,&!-

./01,23141567/849:

;51<=14>?


0 200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

0.03

Time (generations)

Freq

uenc

y


Figure 8: Less realistic selective regime with s/K = 1.2, πH = 0.5 and πL = 0.5 a) Non-adaptable high robustness genotypes. This shows that the prior to the environmental shiftmost of the population is at this genotype class and that following the environmental shiftthere is a larger but still small change in the population at this genotype class before adap-tation occurs (there is a drop of ∼ 10%). b) Adaptable high robustness genotypes. Thisshows that prior to the environmental shift the population at this genotypes class is small(due to their rarity) and that following the environmental shift and the population increasesby ∼ 0.07. c) Non-adaptable low robustness genotypes. This shows that the prior to theenvironmental shift the population at this genotype class is negligible and that following theenvironmental shift there is no obvious change in the population at this genotype class beforeadaptation occurs. d) Adaptable high robustness genotypes. This shows that prior to theenvironmental shift the population at this genotypes class is small (due to their rarity) andthat following the environmental shift and the population increases by ∼ 0.03, this is com-parable to the increase at the high robustness adaptable genotypes (b). The non-negligiblepopulation at low-robustness adaptable genotypes is the effect discussed in the main tex.Plots are for N = 10, 000 individuals, with µ = 0.001, P = 100, K = 5, q = 0.5 and s = 6,φH = 0.9, φL = 0.1, πH = 0.5, πL = 0.5 and � = 0.1. Plots show averages over 1000 samplepaths. 36

0 200 400 600 800 10000.925

0.93

0.935

0.94

0.945

0.95

Time (generations)

Freq

uenc

y


0 200 400 600 800 10000.045

0.05

0.055

0.06

0.065

0.07

0.075

Time (generations)

Freq

uenc

y(b) Adaptable high robustness

! "!! #!! $!! %!! &!!!"'(

"'((

"'$

"'$(

"')

"')(

"'%

"'%(

"'*

"'*( +,&!-

./01,23141567/849:

;51<=14>?


! "!! #!! $!! %!! &!!!&

&'"

&'#

&'$

&'%

"

"'"

"'#

"'$

"'% ()&!#

*+,-)./-0-123+4056

71-89-0:;


Figure 9: Most realistic selective regime with s/K = 0.2, πH = 0.5 and πL = 0.1 Theinformation here is the same as in the previous graph, but it illustrates that changing lowrobustness genotype clustering has no impact on the dynamics of adaptation. Plots are forN = 10, 000 individuals, with µ = 0.001, P = 100, K = 5, q = 0.5 and s = 1, φH = 0.9,φL = 0.1, πH = 0.5, πL = 0.1 and � = 0.1. Plots show averages over 1000 sample paths.

37

List of Tables

1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

38

Parameter Interpretation

A ⊂ I The set of adaptable types

ζi The rate mutation of the ith adaptable type to the newly adaptive type

µ = θ

NThe rate of mutation

m̃ij The probability that the mutant offspring of an individual of type i has type j

Si =σi√N

The selective advantage of the ith type

Zi(t) The number of individuals of type i at time t

Table 1: Model Parameters

39

Documents

ENVIRONMENTAL ROBUSTNESS AND THE ADAPTABILITY OF …