Epistasis and natural selection shape the mutational...

ARTICLE

Received 19 Aug 2013 | Accepted 24 Mar 2014 | Published 14 May 2014

Epistasis and natural selection shape themutational architecture of complex traitsAdam G. Jones1, Reinhard Burger2 & Stevan J. Arnold3

The evolutionary trajectories of complex traits are constrained by levels of genetic variation

as well as genetic correlations among traits. As the ultimate source of all genetic variation is

mutation, the distribution of mutations entering populations profoundly affects standing

variation and genetic correlations. Here we use an individual-based simulation model to

investigate how natural selection and gene interactions (that is, epistasis) shape the evolution

of mutational processes affecting complex traits. We find that the presence of epistasis

allows natural selection to mould the distribution of mutations, such that mutational effects

align with the selection surface. Consequently, novel mutations tend to be more compatible

with the current forces of selection acting on the population. These results suggest that in

many cases mutational effects should be seen as an outcome of natural selection rather than

as an unbiased source of genetic variation that is independent of other evolutionary

processes.

DOI: 10.1038/ncomms4709

1 Department of Biology, Texas A&M University, 3258 TAMU, College Station, Texas 77843, USA. 2 Institut fur Mathematik, Universitat Wien, Wien 1090,Austria. 3 Department of Integrative Biology, Oregon State University, Corvallis, Oregon 97331, USA. Correspondence and requests for materials should beaddressed to A.G.J. (email: [email protected]).

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The relationship between the genotype and the phenotype,sometimes called the genotype–phenotype map, has takencentre stage in the study of complex traits for very good

reasons1,2. For instance, many important human disorders, suchas susceptibility to heart disease or Alzheimer’s disease, aredetermined by numerous genetic loci as well as environmentaleffects, thrusting these traits directly into the realm of quantitativegenetics3,4. An understanding of how genes and the environmentconspire to shape these traits might lead to better screening andtreatment options. The complexity of the problem calls for anapproach based on correlations of genetic variants with traitvalues, either in the context of genome-wide association studiesor quantitative trait locus mapping5,6, but these approachestypically identify genetic loci that explain only a small fraction ofthe genetic variance in these sorts of complex traits7. Thisinsufficiency problem has led to a widespread appreciation thatinteractions among genes, a phenomenon called epistasis inthe quantitative genetics literature, could make a substantialcontribution to the genetic variation in complex traits5–9,although the matter is still hotly debated10–12.

An added wrinkle to these considerations is that traits do notexist in isolation from other traits. Individuals who express onetrait, such as hypertension, may display a tendency to expressother traits, such as diabetes13. In other words, different traits canbe genetically correlated, and from an evolutionary standpoint wewould like to understand how such genetic correlations arise andconstrain population-level processes14. Genetic correlations canarise from a number of factors, and chief among them are naturalselection and mutation15,16. If certain trait combinations confer afitness advantage relative to others, then the variants that workwell in combination will tend to be inherited together due to theincreased fitness of their bearers17. From a mutation standpoint,if a mutation that affects one trait in a positive fashion also affectsa second trait in a similar direction due to pleiotropy, then thesenew mutations will contribute to a genetic correlation betweentraits. This source of genetic correlations can be very strongindeed18–20. Given the central role of this ‘mutationalarchitecture’ in the evolution of complex traits and theapparent importance of epistasis as revealed by studies ofquantitative trait loci9,21–25, our goal in the present study is toinvestigate how epistasis influences the spectrum of mutationsentering populations and how the evolution of mutational effectsin turn constrains the genetic architecture of complex traits atthe population level. Our results show that epistasis allowsthe mutational architecture of the multivariate phenotype to beshaped by natural selection and that the evolution of themutational architecture in turn affects standing levels of geneticvariance and the ability of a population to respond to selection.

ResultsThe epistasis model. We model epistasis using an individual-based Monte Carlo approach to simulate a population of Nindividuals, each of which has a two-trait phenotype determinedby both genetic and environmental effects. The genetic effectsarise from a suite of n loci, each of which is pleiotropicand potentially epistatic. Epistasis is included using the multi-linear approach26–28, which has been employed extensively tostudy the effects of epistasis on a single-trait phenotype29–31. Ourimplementation allows pairwise interactions among all loci. Asthe loci are pleiotropic, the epistatic effects can occur within orbetween trait effects. Our model accommodates both types ofepistasis. An individual’s phenotype is determined by the sum ofadditive effects and epistatic terms (see Methods), plus anenvironmental effect drawn from a normal distribution with amean of zero and variance of one. The life cycle consists of (1)

random mating, (2) production of offspring, including mutationand recombination, (3) natural selection specified by a bivariateGaussian individual selection surface (summarized by the x-matrix) and (4) population regulation (see Methods for moredetails). We start with a core set of parameter values (Table 1)and vary numerous combinations of parameters to investigate theevolution of the genetic variance and mutational architectureunder a wide range of biologically plausible conditions. For eachcombination of parameters, we run the simulation for 5,000initial generations to reach a balance between selection, mutationand genetic drift. These initial generations are followed by 5,000experimental generations, during which we calculate variables ofinterest (see Methods). For each parameter combination, weconduct 20 independent runs of the complete simulation,including the 5,000 initial and 5,000 experimental generations.We average values of interest across these 20 independent runs.Our main variables of interest in the present model are themutational variances (M11 and M22) and mutational correlation(rM), which together describe the distribution of the phenotypiceffects of new mutations entering the population and can bethought of as the mutational architecture of the two-traitphenotype, which we will also refer to as the M-matrix. We arealso interested in the variables describing the distribution ofgenetic variation in the population, including the total geneticvariances and covariance (11VG, 22VG and 12VG), the additivegenetic variances and covariance (11VA, 22VA and 12VA), and theepistatic genetic variances and covariance (11VAA, 22VAA and12VAA). The additive genetic variances and covariances determinethe response of the population mean to selection, and are oftenorganized into a matrix known as the G-matrix.

The evolution of the genetic variance and mutation matrix.Several key results emerge from our analysis. The first majorresult is that epistasis affects the evolution of the genetic andmutational architecture of quantitative traits under a very widerange of parameter combinations. In particular, the mutationalvariances (that is, measures of the absolute size of the phenotypiceffects of new mutations entering populations) of the quantitativetrait loci show apparently adaptive changes in response toselection when epistasis is present (Table 2), and these changes inmutational variances carry implications for the standing levels ofgenetic variance. One striking result is that the magnitude ofmutational variances is negatively related to population size(Table 3).

The evolution of the mutational variances has a profound effecton the standing levels of genetic variation in our simulatedpopulations. For instance, in a strictly additive model, mutationalvariances cannot evolve, and larger populations tend to harbourgreater amounts of genetic variance compared with smallerpopulations due to reduced losses of variation because of a less-important role of genetic drift in the large populations (Fig. 1).In the presence of epistasis, however, the situation changesdramatically. Smaller populations evolve larger mutationalvariances, and this pattern becomes more pronounced as theaverage absolute values of epistatic parameters increase (Fig. 1;Table 2). These larger mutational variances increase the amountof genetic variance introduced by mutation each generation,which in turn increases the standing level of genetic variation.For moderately strong epistasis (that is, epistatic parametervariance, s2

e � 0:5), this increase in the mutational varianceresults in a tendency for larger populations to harbour lessadditive genetic variance than their smaller counterparts (Fig. 1;Table 3). However, when populations become exceptionallysmall, the variance-reducing effects of drift become strongenough to overcome the increase in mutational variances,resulting in a non-monotonic relationship between population

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4709

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size and additive genetic variance under moderate to strongepistasis (Fig. 1). Thus, the evolution of the mutational variance,as a consequence of evolving epistatic effects, has importantimplications at the population level in terms of standing levels ofgenetic variation.

Triple alignment. Our second major result is that the mutationalcovariance evolves in a way that causes adaptive alignment withthe individual selection surface. If selection favors certain com-binations of traits, then the presence of epistasis allows themutational architecture to evolve in a way that new mutationstend to reinforce these favourable trait combinations. Thisalignment result is very general, and it occurs under almost allinvestigated parameter combinations, as evidenced by the evo-lution of a positive mutational correlation whenever we imposecorrelational selection (Tables 2 and 3). We investigate theveracity of the alignment between the individual selection surface

(the x-matrix), the additive genetic architecture (the G-matrix)and the mutational architecture (the M-matrix), by conductingsimulation runs involving selection surfaces oriented differentlyin phenotypic space, but otherwise of identical shape, andtracking the evolutionary responses of the G-matrix andM-matrix. When we perform this exercise, we find remarkablealignment between the x-matrix, the G-matrix and the M-matrix(Fig. 2). Under our parameter combinations, the elongate selec-tion surface results in a somewhat less-elongate G-matrix, and inturn an even less-eccentric M-matrix, but the leading eigenvectorsof all three matrices align almost perfectly in phenotypic space.These aligned M-matrices tend to remain stable within a run,and while different runs sometimes produce quantitatively dif-ferent M-matrices, nearly all of them evolve toward alignmentwith the selection surface (Supplementary Table 1). Figure 2shows results from a large population (N¼ 4,096), but this sort oftriple alignment also occurs in much smaller populations

Table 1 | Key parameters and core parameter values for the multilivariate epistasis model.

Parameter Symbol Core value Range investigated

Adult carrying capacity K 512 64–4,096Adult population size N 512 64–4,096Number of offspring per female 2B 4 4Number of loci n 20 4–50Mutational variances of reference effects a2

1 ; a22 0.05 0.01–0.50

Mutational correlation of reference effects rm 0 �0.9–0.9Mutation rate per locus m 0.0005 0.0001–0.001Selection surface variance o11, o22 49 4–199Selectional correlation ro 0 0–0.9Variance of epistatic coefficients s2

e 1 0.1–10

Table 2 | The effects of the epistatic parameter variances on the genetic variance and the M-matrix.

rx r2e 11VG 22VG rG 11VA 22VA 12VA (rA) 11VAA 22VAA 12VAA M11 M22 rM

0 0 0.53 0.51 0.018 0.52 0.50 0.010 (0.02) 0.007 0.004 0.000 0.05 0.05 00.50 0 0.48 0.49 0.126 0.48 0.48 0.065 (0.14) 0.004 0.003 0.001 0.05 0.05 00.75 0 0.41 0.39 0.266 0.41 0.39 0.113 (0.28) 0.002 0.002 0.000 0.05 0.05 00.90 0 0.30 0.30 0.420 0.30 0.30 0.132 (0.44) 0.002 0.002 0.000 0.05 0.05 0

0 0.1 0.59 0.59 0.014 0.54 0.54 0.009 (0.02) 0.047 0.045 0.000 0.108 0.107 0.0070.50 0.1 0.53 0.50 0.137 0.49 0.45 0.077 (0.16) 0.038 0.040 0.002 0.103 0.097 0.0380.75 0.1 0.44 0.44 0.311 0.40 0.41 0.148 (0.37) 0.029 0.029 0.000 0.098 0.097 0.0950.90 0.1 0.30 0.31 0.455 0.27 0.29 0.147 (0.53) 0.019 0.018 0.002 0.078 0.083 0.120

0 0.5 0.70 0.69 0.019 0.58 0.57 0.014 (0.02) 0.111 0.106 0.000 0.196 0.186 0.0030.50 0.5 0.62 0.59 0.198 0.53 0.50 0.121 (0.24) 0.091 0.086 0.004 0.191 0.186 0.0640.75 0.5 0.47 0.49 0.318 0.40 0.41 0.162 (0.40) 0.064 0.067 0.002 0.152 0.163 0.1120.90 0.5 0.31 0.31 0.453 0.27 0.27 0.146 (0.54) 0.041 0.041 0.004 0.126 0.131 0.151

0 1.0 0.75 0.79 0.002 0.60 0.63 0.001 (0.00) 0.144 0.147 0.002 0.267 0.276 �0.0020.50 1.0 0.67 0.64 0.185 0.54 0.51 0.124 (0.24) 0.114 0.113 0.002 0.251 0.248 0.0570.75 1.0 0.45 0.44 0.278 0.36 0.35 0.131 (0.37) 0.082 0.082 0.003 0.195 0.189 0.0660.90 1.0 0.31 0.31 0.441 0.25 0.26 0.141 (0.55) 0.054 0.048 0.004 0.168 0.167 0.141

0 10.0 0.75 0.74 0.008 0.46 0.44 0.003 (0.01) 0.277 0.280 0.002 0.565 0.553 0.0050.50 10.0 0.59 0.60 0.134 0.35 0.36 0.078 (0.22) 0.227 0.224 0.010 0.496 0.491 0.0470.75 10.0 0.42 0.40 0.191 0.25 0.25 0.078 (0.31) 0.151 0.142 0.007 0.411 0.389 0.0430.90 10.0 0.23 0.23 0.260 0.14 0.14 0.059 (0.42) 0.081 0.085 0.007 0.282 0.284 0.074

Average s.e.m. 0.016 0.017 0.019 0.015 0.016 0.011 0.004 0.004 0.001 0.009 0.009 0.017

These results are derived from a population evolving under the core parameter set, except we use selectional correlations (ro) ranging from 0 to 0.9 (first column), and we use variances in our epistaticparameters (s2

e ) ranging from 0 (no epistasis) to 10.0 (second column). The variables reported in this table include the total genetic variances for traits one and two (11VG, 22VG), the total geneticcorrelation between the traits (rG), the additive genetic variances and covariance (11VA, 22VA, and 12VA) with the additive genetic correlation shown parenthetically (rA), the additive-by-additive epistaticvariances and covariance (11VAA, 22VAA, and 12VAA), and the average mutational variances and mutational correlation across loci (M11, M22, and rM). The last row shows the average s.e.m. across all entriesin the corresponding column to provide a rough guide to the dispersion of the data. In the first four rows of the table, the values for VAA are not precisely zero, despite the absence of epistasis, due to asmall amount of statistical error that arises from our breeding-design approach to estimating genetic variance components.

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(Supplementary Table 2), even though the alignment isdisrupted somewhat in smaller populations by the operation ofgenetic drift.

The evolution of larger mutational variances in smallpopulations can be understood by considering the relationshipbetween average allelic effects at the quantitative trait loci and theaverage epistatic coefficients for each locus. As the epistaticcoefficients are parameters in the multilinear model, they do notchange during a given simulation run (see Methods). Rather,epistatic contributions, and hence genotypic values, evolve as theallelic effects of individual loci change over evolutionary time.Loci with favourable epistatic coefficients can evolve larger alleliceffects that enhance their epistatic effects. Alternatively loci canevolve allelic effects in opposition to their epistatic coefficients toreduce the phenotypic effects of new mutations. Figure 3 showsthe relationship between average allelic effects and averageepistatic coefficients in a small population (N¼ 128) in whichlarge mutational variances evolve, and in a large population(N¼ 2,048) in which small mutational variances evolve. In thesmall population, we see a very weak, non-significant negativerelationship between average allelic effects and average epistaticcoefficients. However, in the large population, we see a strong andhighly significant negative relationship. Thus, in the largepopulation, loci with negative epistatic effects on average tendto have positive allelic effects and loci with positive epistaticeffects tend to evolve negative allelic effects, with the consequencethat the reference effects of most new mutations are largelycounteracted by their opposing epistatic effects. This maskingprocess due to opposing reference and epistatic effects is moreimportant in larger populations than smaller populations,resulting in a negative relationship between mutational variancesand population size. In Fig. 3, for the sake of simplicity, weaddress only the evolution of the mutational variance at one trait,but the evolution of mutational covariances arises from similarprocesses. In short, the mutational architecture evolves as a

consequence of the quantitative trait loci evolving allelic effectsthat interact with their epistatic coefficients according to thecurrent regime of selection and drift.

One other consequence of the evolution of the mutationalarchitecture under epistasis is that the alignment of the M-matrixwith the G-matrix will tend to strengthen any additive geneticcorrelations that exist in the population (Tables 2 and 3). Exceptunder very strong epistasis or large population size and strongepistasis, the majority of genetic variance arising from epistasis isadditive (Tables 2 and 3; Supplementary Table 3), which meansthat this genetic variance can contribute to a response toselection. Indeed, our results indicate that a mutationalarchitecture evolving under epistasis can enhance a population’sability to respond to selection. For instance, in small populationswith very high mutational variances (caused by epistasis), we see astronger response to selection compared with larger populations(Supplementary Table 3). Much of the genetic variance in thesmaller population is attributable to the large mutationalvariance, which is a product of the evolution of the mutationalarchitecture made possible by epistasis. However, the majority ofthe genetic variance is nonetheless additive (compare VA with VG

in Supplementary Table 3), and therefore available for naturalselection. Similarly, the genetic covariance, which is strengthenedby the alignment of the G-matrix and M-matrix, is mainlyadditive genetic in nature and thus produces a correlatedresponse to selection (Supplementary Table 3).

Evolution of the M-matrix and triple alignment of the type wedescribe here occurs under almost all parameter combinations.Regardless of the strength of epistasis (Table 2), populationsize (Table 3), mutational variance of reference effects(Supplementary Table 4) or strength of stabilizing selection(Supplementary Table 5), we see a tendency for the M-matrix toalign with the selection surface, as evidenced by the evolution of apositive mutational correlation in the presence of positivecorrelational selection. However, we do observe that the evolution

Table 3 | The effects of population size on the evolution of the genetic variance and the M-matrix.

rx N 11VG 22VG rG 11VA 22VA 12VA (rA) 11VAA 22VAA 12VAA M11 M22 rM

0 128 0.77 0.79 0.05 0.71 0.72 0.034 (0.05) 0.040 0.049 0.004 0.710 0.818 0.0180.50 128 0.63 0.66 0.22 0.58 0.61 0.151 (0.25) 0.036 0.035 0.005 0.620 0.628 0.1090.75 128 0.52 0.51 0.30 0.48 0.47 0.174 (0.37) 0.024 0.026 0.004 0.567 0.561 0.0290.90 128 0.33 0.32 0.41 0.30 0.30 0.155 (0.52) 0.016 0.017 0.002 0.425 0.423 0.081

0 256 0.80 0.84 �0.02 0.72 0.75 �0.014 (�0.02) 0.065 0.070 0.000 0.463 0.498 �0.0110.50 256 0.70 0.65 0.19 0.62 0.58 0.136 (0.23) 0.061 0.059 0.004 0.461 0.407 0.0310.75 256 0.51 0.52 0.33 0.46 0.46 0.185 (0.40) 0.046 0.048 0.005 0.326 0.319 0.1020.90 256 0.33 0.33 0.45 0.29 0.29 0.160 (0.55) 0.035 0.032 0.004 0.236 0.259 0.135

0 512 0.74 0.74 0.01 0.60 0.59 0.002 (0.00) 0.132 0.141 0.003 0.259 0.261 0.0200.50 512 0.64 0.64 0.23 0.51 0.50 0.152 (0.30) 0.132 0.131 0.004 0.220 0.223 0.1080.75 512 0.49 0.49 0.32 0.39 0.40 0.164 (0.42) 0.084 0.082 0.002 0.205 0.221 0.1130.90 512 0.30 0.30 0.44 0.26 0.25 0.141 (0.55) 0.052 0.054 0.005 0.154 0.150 0.147

0 1,024 0.70 0.70 0.00 0.44 0.44 0.003 (0.01) 0.240 0.246 �0.002 0.168 0.164 �0.0090.50 1,024 0.60 0.60 0.17 0.39 0.40 0.100 (0.25) 0.200 0.191 0.005 0.158 0.156 0.0690.75 1,024 0.43 0.43 0.28 0.32 0.30 0.127 (0.41) 0.128 0.125 0.004 0.142 0.135 0.1050.90 1,024 0.28 0.28 0.40 0.20 0.21 0.111 (0.54) 0.066 0.069 0.005 0.112 0.117 0.163

0 2,048 0.68 0.65 �0.02 0.37 0.34 �0.013 (�0.04) 0.297 0.305 0.000 0.139 0.135 �0.0120.50 2,048 0.56 0.56 0.15 0.32 0.32 0.085 (0.27) 0.235 0.238 0.002 0.126 0.128 0.0650.75 2,048 0.40 0.39 0.24 0.25 0.24 0.095 (0.39) 0.147 0.146 0.003 0.112 0.108 0.0970.90 2,048 0.26 0.27 0.40 0.19 0.19 0.105 (0.55) 0.074 0.074 0.002 0.095 0.097 0.177

Average s.e.m. 0.019 0.018 0.020 0.018 0.016 0.013 0.004 0.004 0.002 0.015 0.016 0.020

Other than population size (N) and selectional correlation (ro), the parameter values used to generate this table are from the core set. The variables reported in this table include the total geneticvariances for traits one and two (11VG, 22VG), the total genetic correlation between the traits (rG), the additive genetic variances and covariance (11VA, 22VA and 12VA) with the additive genetic correlationshown parenthetically (rA), the additive-by-additive epistatic variances and covariance (11VAA, 22VAA and 12VAA), and the average mutational variances and mutational correlation across loci (M11, M22 andrM). The last row shows the average s.e.m. across all entries in the corresponding column to provide a rough guide to the dispersion of the data.





of the mutational correlation becomes less pronounced asmutations become sufficiently rare (Supplementary Table 6).

DiscussionOur analysis of the two-trait version of the multilinear model ofepistasis provides several important insights into the evolution ofthe genetic variance of quantitative traits. The first key insight isthat epistasis allows the evolution of mutational effects and inparticular larger mutational effects and variances in smallerpopulations. These larger mutational variances, in turn, causesmaller populations to harbour greater amounts of additivegenetic variance than larger populations, a counterintuitive resultthat is nevertheless consistent with the broader literature on theevolution of mutational effects and mutational robustness32–34,

which is defined as the ability of an organism or trait to maintainits function despite the occurrence of novel mutations33. Thesecond major insight from our model is that epistasis allowsthe mutational matrix to evolve towards alignment with theindividual selection surface. As both the mutational matrix andthe individual selection surface influence the shape andorientation of the G-matrix, we find that epistasis produces asituation of triple alignment, in which patterns of mutation,genetic variation and selection evolve towards a commonorientation in phenotypic space. These results illuminate thepotential importance of epistasis and the evolution of mutationaleffects in evolutionary processes.

The most counterintuitive result in our study is that smallerpopulations evolve larger mutational variances than largerpopulations to such a degree that smaller populations tend toharbour greater levels of additive genetic variance in quantitativetraits. This pattern likely arises from three main processes, onedriven by epistasis and the other two arising from genetic drift.

M =0.089 0.016

0.016 0.090

G =0.174 0.096

0.096 0.175

� = 49 44.1

44.1 49

M =0.097 0.008

0.008 0.075

G =0.239 0.060

0.060 0.102

� = 82 30.0

30.0 16

M =0.103 0.000

0.000 0.072

G =0.262 0.000

0.000 0.079

� = 93 0

0 5

Figure 2 | Triple alignment of natural selection, genetic variation and

mutation. Epistasis promotes alignment of the individual selection surface

(the x-matrix), the additive genetic architecture (the G-matrix) and the

mutational architecture (the M-matrix) of a two-trait phenotype. The actual

matrices are shown to the left, and graphical depictions of the overlapping

matrices are shown to the right. These results are from simulations using

our core parameter set, except that population size is 4,096 and the

individual selection surface (described by x) is held at a constant shape but

oriented with its long axis turned in a different direction in phenotypic space

for different simulation runs (but note that within a run the individual

selection surface is always constant). The ellipses are 95-percent

confidence ellipses, and the angle of the long axis of each ellipse is given by

the leading eigenvector of the corresponding matrix (green for M, blue for

G and orange for x) in a plot with trait one on the x axis and trait two on the

y axis. The x-matrix is not drawn to scale, but its orientation and

proportions are correct. As the selection surface rotates, both the G-matrix

and the M-matrix evolve to align with the selection surface in phenotypic

space. This alignment result is extremely general and it occurs under almost

all investigated parameter combinations.

0

0.2

0.4

0.6

0.8

1

0 500 1,000 1,500 2,000 2,500

Variance in epistatic coefficients=0.0Variance in epistatic coefficients=0.1Variance in epistatic coefficients=0.5Variance in epistatic coefficients=2.0

0

0.1

0.2

0.3

0.4

0.5

0 500 1,000 1,500 2,000 2,500

Population size (N)

Add

itive

gen

etic

var

ianc

e ( 1

1VA)

Mut

atio

nal v

aria

nce

(M11

)a

b

Figure 1 | The additive genetic variance and the mutational variance of a

trait evolve as a function of underlying levels of epistasis. These

simulation results were produced using our core set of parameters, except

we imposed correlational selection, ro¼0.9, and varied the population size

from 64 to 2,048 across different runs. The top panel (a) shows the

relationship between the equilibrium additive genetic variance for trait one

and the population size. In a strictly additive model, larger populations

maintain larger amounts of additive genetic variance (red diamonds), but

with moderate-to-strong epistasis (green squares, closed circles) the

pattern is reversed (with the exception of the smallest populations). The

bottom panel (b) reveals the cause of this reversal. In an additive model, the

mutational variance has no way to evolve, so small populations have the

same equilibrium mutational variance as large populations (red diamonds).

In the presence of epistasis, however, smaller populations evolve larger

mutational variances than large populations (triangles, squares, circles),

and these larger mutational variances in small populations contribute to a

greater level of standing genetic variance, except when the effects of

genetic drift are extremely strong (that is, when N¼64). In (a) and (b),

error bars show one s.e.m. across 20 independent simulation runs; if error

bars are not visible, then they are smaller than the symbol.





First, because epistasis is non-directional on average in our model(that is, positive and negative interactions potentially balance),mutational variance tends to increase in the absence of otherevolutionary forces (see Methods). Second, the efficacy ofstabilizing selection is lower in smaller populations, allowingthese populations to maintain alleles with larger ‘reference’effects, which can be thought of as the phenotypic effect the allelewould have in the absence of epistasis. These larger referenceeffects in turn increase the absolute magnitude of epistaticcontributions. In particular, large reference effects at some locican be compensated for by large reference effects of opposite signat other loci, thus leading to more compensatory evolution insmall populations. For a seemingly similar observation in adifferent model, see the recent report from Rajon and Masel35.The third cause of a higher mutational variance in smallerpopulations is that stabilizing selection favours the evolution ofsmaller additive genetic variances36, a phenomenon that has beenobserved in other studies as an increase in mutational robustnessin large populations34,37. As smaller populations tend to havetheir phenotypic means displaced away from the bivariateoptimum by the action of genetic drift, they experience largerabsolute forces of directional selection, which favors an increasein additive genetic variation38, and less stabilizing selectioncompared with larger populations. Therefore, the evolution ofmutational robustness is less effective in the smaller populations.Together, these factors produce large mutational and additivegenetic variances in small populations.

Another possibility is that multiple quasi-stable equilibria existin our simulated populations, a phenomenon that has beenobserved in other studies of the multilinear model of epistasis26,and that genetic drift allows smaller populations to shift betweenthese equilibria more often than larger populations39. Such ascenario could also produce an increase in genetic variance andmutational variances in the small populations. Regardless, despitethe fact that selection is less efficient in small populations due tothe effects of genetic drift, our results show that the combinedaction of drift and selection can allow these small populations tomaintain large mutational variances. Thus, these conflictingevolutionary pressures combine to produce a negative correlationbetween effective population size and mutational variances, andwe see a pronounced manifestation of this expectation in ourresults (Table 3).

Our analysis of the evolution of the orientation of themutational matrix extends previous work, which focusedexclusively on the evolution of the mutational correlation. Whenthe mutational correlation is treated as a quantitative trait in anadditive model, it has a tendency to evolve towards alignmentwith the selection surface20. However, a much more realistic wayto model the evolution of the M-matrix is by using an explicit,general model of epistasis, as we have done here. Our results showthat, indeed, the mutational correlation does evolve towardsalignment with the selection surface, and more importantly, themutational variances and covariances evolve in tandem toproduce a mutational matrix nearly perfectly aligned with theselection surface. Under these circumstances, the selection surfaceand mutational matrix both influence the standing geneticvariance in the population, which also results in triplealignment of the M-matrix, G-matrix and selection surface.This alignment scenario is important from an evolutionarystandpoint, because constraints imposed by the M-matrixcan be quite strong18,19, but here we see that these mutationalconstraints are shaped in part by natural selection. Triplealignment means that new mutations entering the populationwill tend to fall along the ridge of the selection surface, if there isone, thus mitigating their deleterious impacts. Furthermore, triplealignment will facilitate evolution along both genetic and selective

lines of least resistance40,41. Thus, the evolution of robustness andof evolvability occur simultaneously in our model33,42,43.

Our study has several limitations that offer fodder for futurework on how epistasis affects multivariate trait evolution.Importantly, we follow the convention of other individual-basedstudies of quantitative genetic phenomena, including epistasis,and use unrealistically high per-locus mutation rates. Althoughthis rate inflation is likely to facilitate the evolution of geneticarchitecture34, this device is necessary because realistic per-locusmutation rates in such simulations tend to produce unrealistically

–2

–1

0

1

2

–1 –0.5 0 0.5 1

–2

–1

0

1

2

–1 –0.5 0 0.5 1Per-locus mean epistatic coefficent

Per

-locu

s m

ean

alle

lic e

ffect

a

b

N=128

N=2,048

Figure 3 | Epistasis allows the mutational variance to evolve as function

of population size. The average allelic effect can evolve to be correlated

with the average epistatic coefficient, and the strength of this relationship

varies with population size. These data are from 20 independent simulation

runs using our core parameter set, except with only 10 quantitative trait

loci. In addition, we allow only within-trait epistasis affecting trait one and

no epistasis involving trait two, with population sizes of (a) N¼ 128 and

(b) N¼ 2,048. Each point represents a single quantitative trait locus. The

x axis shows the magnitude of epistasis (mean epistatic effect of a locus,

averaged across all of its epistatic coefficients), and the y axis presents the

mean allelic effect (or reference effect) of alleles at the corresponding

locus, averaged across all alleles segregating at the locus. In small

populations, large mutational variances are maintained by the evolution of a

large range in allelic effects; we see a slightly negative but non-significant

relationship between epistatic coefficients and allelic effects (linear

regression, N¼ 200, R2¼0.01, P¼0.09). In large populations (b), which

evolve lower mutational variances than small populations, we see a much

smaller range in allelic effects and these effects show a strong negative

relationship with the mean epistatic coefficients across loci (linear

regression, N¼ 200, R2¼0.22, Poo0.0001). Thus, the allelic effects of a

particular locus tend to evolve values that are largely counteracted by the

epistatic effects of the locus in question. This figure is concerned with the

evolution of the mutational variance, but a similar effect explains the

evolution of mutational covariances.





low levels of additive genetic variance. For instance, withmutation rates of the order of 10� 6 or 10� 7, our populationslose all genetic variation and all interesting evolutionaryphenomena cease to occur. This result illustrates that we stilldo not fully understand the mechanisms maintaining geneticvariation in natural populations, but it also represents a realconstraint for the type of model we employ here36. It is worthnoting, however, that our mutation rates are much more realisticthan those used in some univariate studies of the multilinearmodel. Le Rouzic et al.30, for instance, employed a per-locusmutation rate of 0.01, arguing that the mutation rate haslittle effect on the qualitative dynamics of the system beyondaffecting the timescale of evolution44,45. In addition, our modeltypically focused on quantitative traits determined by a smallnumber of loci, typically 20, due to computational constraints.Actual traits in living systems may be affected by hundreds orthousands of loci, which would give them a much largermutational footprint than the traits considered here. Moreover,if quantitative traits are sometimes affected by suites of physicallylinked genes, then mutations at these supergenes could occurmore frequently than they would occur for any single gene in thegenome. These sorts of tightly linked gene clusters areappropriately simulated by the type of model we used here,where each simulated locus could be interpreted as a group ofphysically linked genes affecting the phenotype. Regardless,progress in reconciling simulation-based models and real datawill require additional data on the genetic details of multivariate,quantitative phenotypes.

Our results also show that the type of epistasis influences theevolution of mutational architecture. Most of our simulationsallow all possible pairwise epistatic effects. However, in our modelall loci are pleiotropic, meaning each locus has an effect on traitone and an effect on trait two, so epistatic effects can potentiallyoccur within or between trait effects across loci. If only within-trait effects are allowed (for example, the trait-one effect at onelocus interacts with the trait-one effect at another locus to affectonly the trait-one phenotype), then the mutational correlationcannot evolve (Supplementary Table 7). Thus, the alignment ofthe M-matrix with the selection surface requires at least somebetween trait epistasis. Recent empirical studies indicate that thistype of epistasis, necessary for the evolution of mutationalcovariances, does exist in natural populations. This type ofepistasis has been termed ‘differential epistasis’ by Cheverudet al.46 and has been shown to occur for morphological andphysiological traits in mice24,25,47.

The results of the present model should provide a foundationfor studies involving more realistic assumptions, and severalobvious directions for future studies emerge from our results. Forinstance, our model ignores complications such as dominance,directional epistasis and higher-order epistasis, all of which caninfluence the evolution of the mutational architecture29,48. Wealso allow all pairwise epistatic interactions among loci, whereasthe genetic architectures of actual traits are probably determinedby gene networks with far fewer epistatic interactions.Furthermore, we constrain the epistatic parameters to remainconstant within a simulation run, a feature that we retain fromthe univariate version of the multilinear model. However, inactual biological systems, the strengths of epistatic interactionsamong loci may evolve. A model with evolving epistaticcoefficients would require assumptions about the genetic basisand inheritance of epistatic effects and is well beyond the scope ofthe model we present here, but such a model could be veryenlightening with respect to the evolution of mutational andgenetic architectures of complex traits.

In summary, the application of the multilinear model ofepistasis to a two-trait phenotype results in several startling

insights into the evolutionary process. The most importantinsight is that natural selection, embodied by the individualselection surface, causes mutational architectures to evolve in anadaptive way. This result contradicts the simplistic view ofmutation presented in most texts in which mutation is claimed tobe random with respect to adaptation. Our results reinforce andextend the results of other studies that have addressed variousaspects of the evolution of the mutational architecture byexploring the effects of epistasis in the univariate case28–31,39,by examining the evolution of mutational correlations20,47, andby addressing the effects of phenotypic plasticity on the evolutionof mutational processes49. Our approach is unique in thatwe allow the mutational variances and covariance to evolvesimultaneously, and our results show a striking pattern ofthree-way alignment across levels of biological organization. Inparticular, the M-matrix, which describes the distribution ofmutational effects entering the population, evolves to align withthe individual selection surface. This alignment increasesmutational robustness in the sense that it is expected to reducethe fitness impacts of novel mutations32,34. In turn, the G-matrix,which describes the standing levels of additive genetic variance inthe population, evolves to align with both the M-matrix and theselection surface. This three-way alignment of mutation, geneticvariation and selection is significant for a number of reasons.First, the mutational architecture of traits should not be seenas something that is independent of natural selection. Rather,the mutational architecture is partially a product of naturalselection50. Second, the evolution of the M-matrix will tend toreinforce any genetic correlations produced by selection, andthis reinforcement increases the efficacy of correlated responsesto selection, which determine evolutionary trajectories inphenotypic space. Finally, the extent of alignment between themutational architecture and the selection surface will influencethe fitness effects of new mutations. Stronger alignment reducesthe deleterious impacts of new mutations. In general, our resultssuggest that the already impressive forces of natural selection mayextend to the very roots of the evolutionary process by shapingthe nature of variation that enters populations as a consequenceof novel mutations.

MethodsThe multivariate multilinear model. Our Monte Carlo simulation is an extensionof the models used by Jones et al.18–20 to study the evolution of additive geneticvariances and covariances in sexually reproducing populations. These modelsexplicitly simulate all individuals in the population. Every individual has a two-traitphenotype determined by its genotype and random environmental effects. In theoriginal models, all loci are assumed to be additive, so an individual’s genetic valueis determined by simply summing across all alleles at all loci. As in the originalmodel, all loci are assumed to be pleiotropic, so each allele has an effect on bothtraits and both effects for a particular allele are inherited together.

The most important difference between the present model and previous modelsis the addition of epistasis. Our implementation of epistasis follows the multilinearmodel28, which has been used successfully to study the effects of epistasis on aunivariate phenotype29–31. The addition of epistasis to the model changes the way amultilocus genotype is converted into a phenotype. The multilinear model simplyextends the additive model by specifying additional terms, which describe theeffects of epistasis. Thus, in the univariate multilinear model, the phenotype isgiven by

X¼x0 þX

i

yðiÞ þX

i

Xj:j4i

eði;jÞyðiÞyðjÞ; ð1Þ

where X is the individual’s genotypic value for the quantitative trait, x0 is the valueof an arbitrary reference genotype, which for our model with a stationaryintermediate optimum can be assumed to be zero, y(i) is the reference effect of anindividual’s genotype at locus i (the two allelic values in the diploid organism aresummed to obtain the genotype’s reference effect) and e(i,j) is an epistaticcoefficient, which determines the nature of the interaction between locus i andlocus j. Clearly, if all epistatic coefficients are zero, then this model reduces to astrictly additive model and the reference effects correspond to additive effects. Thisdescription of the multilinear model includes only pairwise interactions.





In principle, higher-order interactions can be included in the model, but in thepresent study we allow only pairwise interactions between loci.

The multiple trait version of this multilinear model requires additional notationand additional epistatic terms. In the present paper, we restrict attention to thetwo-trait case, which is simple enough to understand yet complex enough tocapture the essence of the evolution of the multivariate phenotype. In our model,every locus is potentially pleiotropic, in the sense that it has a reference effect onboth traits. In addition, every locus is potentially epistatic, as specified by themultilinear model. In this model, then, every individual has two genotypic values,one for each trait, specified by

aX¼ax0 þX

iayðiÞ þ

Xi

Xj:j4i

Xb

Xc

abceði;jÞbyðiÞcyðjÞ; ð2Þ

where aX is an individual’s genotypic value for trait a, and ax0 is the value of thereference genotype, which will be zero for our analysis. As in the univariate case,ay(i) is the individual’s reference genotypic value on trait a at locus i, and in theabsence of epistasis, this value would be the additive effect of the locus. The finalsummation term represents the epistatic interactions among loci, where abce(i,j)

gives the epistatic effect on trait a of the interaction between the effects of locus i ontrait b and locus j on trait c. We assume that no locus interacts with itself, soabce(i,i)¼ 0 and that interactions are symmetric in the sense that abce(i,j)¼ acbe(j,i).Each epistatic term is simply the product of the relevant epistatic coefficient andthe reference effects at the two interacting loci. However, this model allows thereference effects of the two loci on one trait to affect an individual’s genotypic valueat another trait, so the model is general, and most forms of epistasis can berepresented as special cases of this multivariate multilinear model.

Equation (2) allows us to calculate an individual’s genotypic value across all lociat both traits, taking into account all possible pairwise epistatic interactions. Wesimulate environmental variance by drawing a value from a normal distributionwith a mean of zero and a variance of one independently for each trait. Theseenvironmental effects are added to the genotypic values to determine anindividual’s phenotypic value for each quantitative trait.

The life cycle: mating, recombination and mutation. Each generation of thesimulated life cycle begins with the adults of the previous generation mating andproducing zygotes. The epistasis model employs a mating system in which eachfemale mates with exactly two males and produces a total of four offspring, twofrom each father. Mates are chosen at random, and individual males can mate asmany times as they are chosen. This breeding design facilitates the estimation ofquantitative genetic values, as described below. Alleles are inherited in a Mendelianfashion, and we assume that all loci are physically unlinked.

Each gamete contributing to a zygote has a probability of nm of carrying a newmutation, where n is the number of loci affecting the quantitative traits and m is theper-locus mutation rate. Recalling that each locus affects both quantitative traits, wedraw mutational effects at random from a bivariate normal distribution withmutational variances of a2

1 (for trait one) and a22 (for trait two) and a mutational

correlation specified by rm. These mutational effects are then added to the existingreference effects of the allele undergoing the mutation. Hence, each time a mutationoccurs, it alters the pleiotropic allele’s effects on both traits. The changes are at thelevel of reference effects, which will be additive effects if all the epistatic parameters arezero. However, in the presence of epistasis, changes in reference effects do notnecessarily translate directly into changes in additive effects. Even though the epistaticcoefficients remain constant throughout a simulation run, the epistatic interactionscan evolve as the allelic effects present at various loci change over time. As theepistatic interactions also determine the mapping of reference effects to the genotypicvalue of an individual, this model allows the M-matrix, which summarizes thedistribution of new mutations entering the population, to evolve as well.

The life cycle: selection. We impose selection by assuming an individual selectionsurface with the shape of a bivariate Gaussian function. Assuming z is a vector ofan individual’s phenotypic values at the traits under consideration, the probabilityof surviving selection is

WðzÞ¼ exp � 12ðz� hÞTx� 1ðz� hÞ

� �; ð3Þ

where h is a vector of trait optima, T represents matrix transposition and x is amatrix that describes the shape of the selection surface. In our two-trait case, x is asymmetric 2� 2 matrix with diagonal elements, analogous to variances, indicatingthe strength of stabilizing selection on each trait. Smaller values result in a steepersurface with stronger stabilizing selection. The off-diagonal element, analogous tothe covariance, indicates the strength of correlational selection, and can beconveniently summarized as the selectional correlation, ro, with larger absolutevalues corresponding to stronger correlational selection (see below).

We impose viability selection by choosing a uniformly distributedpseudorandom number between 0 and 1 for each individual. If the numberis less than W(z), then the individual survives to the next phase of the life cycle,population regulation.

The life cycle: population regulation. In this evolutionary model, we assume thata population is near its carrying capacity, K. We restrict attention to cases in whichthe population invariably produces more than K offspring, and we imposepopulation regulation by choosing K individuals at random from the survivors ofselection. We also impose an equal sex ratio on these adults. These individuals arethe adults of the new generation, and they will go on to mate as described above toproduce the next generation of progeny.

Important parameter values. In this model, we restrict attention to a populationevolving in response to a stationary individual selection surface. Thus, genetic driftand stabilizing selection are the main sources of evolutionary change. Somedirectional selection occurs when drift moves the population away from thebivariate optimum, and this directional selection moves the population backtoward the optimum.

In the present analysis, we explore a large swath of parameter space, and wereport results from the most important parameters. An exhaustive exploration ofparameter space is intractable for this sort of model, so we start with a core set ofparameter values and examine how deviations from this core set affect evolutionarydynamics. The core parameter set is given in Table 1.

Several of these parameters require some explanation. As noted above, theparameters describing the selection surface, x, whose elements are o11, o22 ando12, are critically important because they determine the strength of selection andthe extent to which correlational selection acts on the population. For convenience,we use the selectional correlation (ro) rather than o12, because ro, which isconstrained to fall between � 1 and 1, is more conceptually understandable. Ofcourse, ro is simply o12=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffio11o22p

, so the conversion between the selectionalcorrelation and the selectional covariance is trivial. The mutational variances(a2

1; a22) and mutational correlation (rm) for reference effects determines the

distribution of new mutations entering the population. Epistasis can cause thereference effects to be only loosely connected to the effects of the mutation on thegenotype, so we specify the latter as the M-matrix (see below). The M-matrix isthus a variable (of considerable interest) in this model rather than a parameter,whereas the effects of mutations on the reference effects of alleles are trueparameters that can be specified and remain constant for a given simulation run.As we are interested in the tendency for epistasis to generate mutationalcorrelations, we use a value of zero for the mutational correlation of referenceeffects (rm).

Another feature of the epistasis model is that there are many epistaticcoefficients. For instance, in a single-trait multilinear model, there will be a total ofn(n� 1)/2 such coefficients, where n is the number of loci. In the two-traitmultilinear model, there are six times as many coefficients, because all interactionsbetween reference effects within and between traits must be considered. Thus, amodel with 20 loci has a total of 1140 unique epistatic coefficients. The onlyfeasible way to model epistasis, then, is to draw these coefficients from adistribution. We draw them from a normal distribution with a mean of zero and avariance of s2

e . This approach allows a mixture of positive and negative epistasis.One key aspect of this model is that the epistatic parameters are set at thebeginning of each independent simulation run, and they do not change during therun. As the epistatic coefficients remain constant, the epistatic effects evolve as aconsequence of changes in the reference effects of loci, which do evolve as aconsequence of mutation, drift and selection. The other parameters listed inTable 1 also remain constant throughout a particular simulation run.

Evolution of mutational effects. We show for the univariate case that themutational variance will increase in the absence of other evolutionary forces if thereis no directional epistasis, that is, if E[e]¼ 0, as assumed throughout this paper.Consider just two loci. Then, by equation (1), the effect of a mutation of size a atthe first locus is

Dm¼ðx0 þ yð1Þ þ aþ yð2Þ þ eðyð1Þ þ aÞyð2ÞÞ � ðx0 þ yð1Þ þ yð2Þ þ eyð1Þyð2ÞÞ¼að1þ eyð2ÞÞ;

ð4Þ

where we write e¼ e(1,2). As E[a]¼ 0, also E[Dm]¼ 0. Taking expectations withrespect to mutational effects, epistasis parameters and locus effects, the variance ofmutational effects becomes

E Dmð Þ2� �

¼E a2 1þ eyð Þ2� �

¼E a2� �

1þ 2E e½ �E y½ � þ E e2� �

E y2� ��

; ð5Þ

where we used pairwise independence of a, e and locus effects. Now, theassumption E[e]¼ 0 yields

E Dmð Þ2� �

¼E a2� �ð1þs2

eE y2� �Þ � E a2

� �: ð6Þ

However, at each particular locus mutational variances may increase ordecrease, depending on the particular choice of epistatic parameters.

Statistical and estimation issues. The addition of epistasis to our model carrieswith it a number of challenges regarding the estimation of variables of interest. Inthis study, we are interested in the distribution of total and additive genetic var-iance in the population at any given time, and we are also keenly interested in theevolution of the M-matrix, which serves as the central source of motivation of this





study. We represent the elements of the M-matrix as M11, M22 and M12, and wealso specify the mutational correlation as rM (which is equal to M12=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM11M22p

).We estimate the genetic variance components by building a half-sib breeding

design into the model. By having each simulated female mate twice, we generate anumber of half-sib families equal to the number of females in the population. Theanalysis of this breeding design can be accomplished through a standard analysis ofvariance approach51. Our population lacks dominance, so the total genetic variancecan be partitioned into parts arising from additive genetic variance and additive-by-additive epistatic variance. Even when epistatic effects are large, much of thegenetic variance arising from the epistatic terms in equation (2) is additive and thuscontributes to parent–offspring resemblance.

The M-matrix is prohibitively difficult to estimate analytically, due to themany epistatic interactions and the possible presence of linkage disequilibriumamong loci, so we use an empirical approach to determine the distribution ofmutational effects. Every 100 or 200 generations, we make a copy of all progenyproduced and induce individual mutations 50 times per locus for each individual.After each single-locus mutation, we refigure each individual’s genotypic valuefor the two traits, as described in equation (2), and compare this new genotypicvalue to the value before mutation. The individual’s genotype is then set backto its original value before the next mutation. The change in the genotypicvalue is the effect of the mutation, and we use this approach to compile adistribution for each locus separately. In most cases, we report the averageM-matrix, which we calculate as the mean mutational variances and mutationalcorrelation across loci.

For each simulation run, we start with an initial population of adults withpopulation size (N) equal to the carrying capacity (K), and indeed N¼K for theduration of each run. Each locus starts with five equally frequent alleles with alleliceffects drawn from a bivariate normal distribution with a mean of zero, s.d. of thecorresponding mutational s.d. divided by the number of loci, and covariance ofzero. This initial population is then permitted to evolve for a period of 5,000generations to reach a state of quasi-equilibrium between genetic drift, selectionand mutation. These initial generations are followed by 5,000 experimentalgenerations during which we calculate values of interest. For each combination ofparameter values, we typically conduct 20 independent simulation runs. Variablesare often averaged across generations within a run and then these means areaveraged across runs to give the values we report.

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AcknowledgementsThis work was supported by National Science Foundation grants to S.J.A., R.B. and A.G.J.(DEB-0447554 and DEB-0448268) and by funds from the Austrian Science Fund (FWF)to R.B. (projects P21305 and P25188). We thank Joachim Hermisson for valuablediscussion and suggestions for improvement of the manuscript.

Author contributionsAll authors contributed to the design and interpretation of this study. The simulation-based model was developed primarily by A.G.J. and the mathematical results werederived mainly by R.B. All three authors contributed to the writing of the manuscript.

Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Competing financial interests: The authors declare no competing financial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

How to cite this article: Jones, A. G. et al. Epistasis and natural selection shapethe mutational architecture of complex traits. Nat. Commun. 5:3709doi: 10.1038/ncomms4709 (2014).






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1

Supplementary Table 1. Variation within and between runs in the elements of the M-matrix. This table

shows estimates of the mutational variances, covariance and correlation from four randomly chosen runs

with parameter values equal to those used for the top panel of Figure 2. For each run, we show the single-

generation M-matrix every 1000 generations. Each run was entirely independent of the other runs, with

unique starting conditions and epistasis parameters. These runs used a selection surface with positive

correlational selection, and the data show that the tendency to evolve a positive mutational correlation is

quite robust. Within a run, the M-matrix tends to be stable, but different runs do show some variation in

their average M-matrices, likely due to differences in the extent to which their randomly chosen epistatic

parameters facilitate the evolution of mutational variances and covariances. Regardless, the evolution of

triple alignment, as shown in Figure 2, occurs in nearly all runs. The last two rows show the means over

all runs and the standard deviation of the within-run means across 20 independent runs.

Run-Generation M11 M22 M12 rµ

R5-G1000 0.089 0.075 0.012 0.143 R5-G2000 0.101 0.080 0.016 0.192 R5-G3000 0.099 0.081 0.015 0.170 R5-G4000 0.094 0.080 0.015 0.177 R5-G5000 0.099 0.075 0.015 0.180 Run Mean 0.095 0.079 0.014 0.164 Run St. Dev. 0.005 0.003 0.002 0.020 R6-G1000 0.084 0.080 0.004 0.066 R6-G2000 0.081 0.080 0.006 0.084 R6-G3000 0.087 0.081 0.006 0.082 R6-G4000 0.087 0.079 0.008 0.099 R6-G5000 0.096 0.090 0.012 0.120 Run Mean 0.086 0.083 0.007 0.091 Run St. Dev. 0.004 0.004 0.003 0.024 R11-G1000 0.095 0.092 0.024 0.247 R11-G2000 0.103 0.096 0.025 0.241 R11-G3000 0.103 0.093 0.032 0.287 R11-G4000 0.098 0.093 0.026 0.242 R11-G5000 0.095 0.086 0.024 0.238 Run Mean 0.099 0.093 0.026 0.254 Run St. Dev. 0.004 0.004 0.003 0.017 R16-G1000 0.089 0.098 0.028 0.276 R16-G2000 0.093 0.100 0.028 0.272 R16-G3000 0.110 0.104 0.038 0.336 R16-G4000 0.102 0.099 0.033 0.336 R16-G5000 0.098 0.106 0.037 0.350 Run Mean 0.099 0.103 0.033 0.313 Run St. Dev. 0.006 0.003 0.005 0.033 Overall Mean 0.089 0.090 0.016 0.169 St. Dev. Run Means 0.009 0.006 0.008 0.076

2

Supplementary Table 2. Evolution of the additive genetic variances, additive genetic covariances, and

M-matrix under different orientations of the selection surface and various population sizes. In these

simulations, the selection surface retains the same shape but is oriented differently in phenotypic space for

different sets of runs. Note, however, that within a run the individual selection surface does not change.

Both the M-matrix and the G-matrix (the additive genetic variance-covariance matrix) tend to evolve to

align with the individual selection surface. The parameter values used to generate this table are identical

to those used for Table 2, except as indicated. The first four columns describe the individual selection

surface, as described in Table 1, where is the angle of its leading eigenvector. The additive genetic

variances and additive genetic correlation are given by 11VA, 22VA, and rA, while M11, M22, and rM represent

the mutational variances and the mutational correlation for the two traits. The angles of the leading

eigenvectors of the G-matrix and M-matrix are given by A and M , respectively.

ω11 ω22 rω N 11VA 22VA rA

A M11 M22 rM M

93 5 0 0 256 0.451 0.144 0.005 0.2 0.271 0.216 0.008 2.0

88 10 0.69 13.8 256 0.469 0.161 0.283 13.4 0.298 0.237 0.078 17.1

82 16 0.80 20.6 256 0.407 0.187 0.380 21.8 0.284 0.259 0.078 29.4

71 27 0.87 30.0 256 0.403 0.223 0.469 28.7 0.327 0.285 0.042 15.7

60 38 0.89 37.7 256 0.333 0.259 0.500 38.0 0.254 0.234 0.112 35.0

49 49 0.90 45.0 256 0.321 0.298 0.562 43.1 0.269 0.264 0.170 43.3

93 5 0 0 1024 0.305 0.097 -0.006 -0.3 0.130 0.094 0.007 1.2

88 10 0.69 13.8 1024 0.296 0.110 0.301 15.1 0.124 0.097 0.084 17.5

82 16 0.80 20.6 1024 0.287 0.122 0.385 20.6 0.125 0.097 0.085 17.0

71 27 0.87 30.0 1024 0.254 0.155 0.472 31.1 0.117 0.108 0.135 36.8

60 38 0.89 37.7 1024 0.227 0.167 0.482 36.2 0.115 0.104 0.114 33.4

49 49 0.90 45.0 1024 0.207 0.198 0.532 43.9 0.112 0.108 0.157 41.6

93 5 0 0 4096 0.262 0.079 0.000 0.0 0.103 0.072 0.002 0.4

88 10 0.69 13.8 4096 0.258 0.090 0.264 12.9 0.100 0.074 0.062 11.0

82 16 0.80 20.6 4096 0.239 0.102 0.382 20.5 0.098 0.075 0.093 18.0

71 27 0.87 30.0 4096 0.210 0.123 0.460 29.8 0.094 0.081 0.129 29.7

60 38 0.89 37.7 4096 0.186 0.145 0.505 38.2 0.091 0.083 0.148 36.4

49 49 0.90 45.0 4096 0.174 0.175 0.551 45.1 0.089 0.090 0.183 45.9

3

Supplementary Table 3. An analysis of the response to selection in the presence of epistasis. The

values in this table were generated by allowing populations to evolve under individual selection surfaces

shaped like those in Supplemental Table S2. Every 100 generations, we made a copy of the entire

population and imposed directional selection on trait one by culling all individuals below the mean. The

value of the selection gradient on trait one, β1, is then the slope of the least-squares regression of relative

fitness on trait values. There is no selection on trait two (i.e., β2 = 0). Results for population sizes of 128,

512, and 4096 are shown here. We report the total genetic variances and covariance (11VG, 22VG, and

12VG), the additive genetic variances and covariance (11VA, 22VA, and 12VA), and several quantities related to

the response of the population mean to selection. The values Exp. 11RG and Exp. 22RG are the responses to

selection that would be expected if all genetic variance were additive, whereas Exp. 11RA and Exp. 22RA are

the expected responses to selection for trait one and trait two based on the observed additive genetic

variances and covariance and the magnitude of β1. Note that the absence of selection on trait two means

that the response to selection for this trait is due entirely to a correlated response to selection. Obs. 11R

and Obs. 22R are the observed responses to selection in the simulated population. We also relaxed

selection for two generations after the single generation of strong directional selection. The remaining

responses to selection after this two-generation relaxation of selection are shown in the last two columns.

The standard errors of the means for the values in this table are much, much smaller than the means

reported here.

N

β1

11VG

22VG

12VG

11VA

22VA

12VA

Exp.

11RG

Exp.

22RG

Exp.

11RA

Exp.

22RA

Obs.

11R

Obs.

22R 11R,

relaxed 22R,

relaxed

128 0.657 0.567 0.182 -0.001 0.531 0.155 -0.001 0.372 -0.001 0.349 0.000 0.317 -0.002 0.313 -0.003

128 0.660 0.508 0.201 0.079 0.474 0.175 0.079 0.335 0.052 0.313 0.052 0.299 0.047 0.298 0.050

128 0.672 0.471 0.225 0.105 0.441 0.199 0.103 0.317 0.070 0.296 0.069 0.276 0.066 0.274 0.068

128 0.674 0.446 0.266 0.151 0.413 0.240 0.149 0.300 0.102 0.278 0.100 0.269 0.090 0.264 0.089

128 0.693 0.373 0.311 0.154 0.344 0.281 0.151 0.259 0.106 0.239 0.105 0.228 0.096 0.225 0.096

128 0.700 0.327 0.312 0.149 0.297 0.283 0.145 0.229 0.104 0.208 0.102 0.201 0.090 0.199 0.092

512 0.669 0.431 0.168 0.005 0.373 0.118 0.005 0.289 0.004 0.250 0.003 0.257 0.004 0.245 0.003

512 0.672 0.416 0.183 0.072 0.360 0.132 0.069 0.279 0.048 0.242 0.047 0.249 0.045 0.239 0.047

512 0.677 0.396 0.202 0.090 0.339 0.150 0.088 0.268 0.061 0.230 0.060 0.239 0.061 0.227 0.060

512 0.679 0.388 0.236 0.119 0.330 0.182 0.116 0.264 0.081 0.224 0.079 0.236 0.078 0.223 0.079

512 0.694 0.323 0.275 0.125 0.265 0.216 0.123 0.224 0.087 0.184 0.085 0.196 0.084 0.186 0.085

512 0.705 0.283 0.286 0.121 0.229 0.232 0.119 0.199 0.085 0.161 0.084 0.177 0.083 0.165 0.084

4096 0.684 0.354 0.166 0.000 0.262 0.079 0.000 0.242 0.000 0.179 0.000 0.205 0.000 0.181 0.001

4096 0.688 0.343 0.174 0.041 0.258 0.090 0.040 0.236 0.028 0.177 0.028 0.203 0.029 0.185 0.025

4096 0.692 0.326 0.185 0.061 0.239 0.102 0.060 0.226 0.042 0.165 0.041 0.190 0.040 0.170 0.042

4096 0.700 0.295 0.207 0.076 0.210 0.123 0.074 0.207 0.053 0.147 0.052 0.173 0.050 0.152 0.051

4096 0.707 0.273 0.203 0.086 0.186 0.145 0.083 0.193 0.061 0.131 0.059 0.159 0.060 0.138 0.057

4096 0.710 0.260 0.263 0.100 0.174 0.175 0.096 0.185 0.071 0.124 0.068 0.152 0.068 0.132 0.068

4

Supplementary Table 4. The effects of the mutational variance for reference effects on the evolution of

the genetic variance and the M-matrix in the multivariate model of epistasis. The parameter values used

to generate this table are identical to those indicated in Table 1, except as noted. Variables and symbols

are described in Table 2.

rω 2

2

2

1

11VG 22VG rG 11VA 22VA 12VA (rA) 11VAA 22VAA 12VAA M11 M22 rM

0 0.01 0.43 0.45 -0.01 0.40 0.41 -0.004 (-0.01) 0.034 0.032 0.000 0.051 0.054 -0.002

0.50 0.01 0.37 0.36 0.11 0.34 0.32 0.043 (0.13) 0.030 0.030 0.001 0.046 0.045 0.012

0.75 0.01 0.30 0.27 0.29 0.27 0.25 0.090 (0.35) 0.025 0.024 0.001 0.041 0.037 0.095

0.90 0.01 0.20 0.21 0.37 0.18 0.19 0.080 (0.43) 0.017 0.017 0.001 0.035 0.037 0.116

0 0.02 0.54 0.56 0.02 0.47 0.49 0.008 (0.02) 0.067 0.066 0.000 0.098 0.106 0.017

0.50 0.02 0.49 0.48 0.14 0.43 0.42 0.075 (0.18) 0.059 0.058 -0.001 0.099 0.094 0.027

0.75 0.02 0.42 0.40 0.32 0.37 0.35 0.138 (0.38) 0.044 0.044 0.001 0.091 0.090 0.108

0.90 0.02 0.25 0.27 0.40 0.22 0.23 0.114 (0.51) 0.030 0.031 0.002 0.068 0.072 0.136

0 0.05 0.78 0.70 0.01 0.63 0.56 0.013 (0.02) 0.134 0.132 0.001 0.275 0.245 -0.007

0.50 0.05 0.63 0.64 0.21 0.49 0.49 0.132 (0.27) 0.133 0.132 0.004 0.227 0.222 0.077

0.75 0.05 0.49 0.48 0.32 0.40 0.39 0.164 (0.42) 0.081 0.079 0.004 0.209 0.201 0.114

0.90 0.05 0.30 0.30 0.41 0.24 0.24 0.126 (0.53) 0.049 0.048 0.002 0.159 0.156 0.131

0 0.25 0.79 0.82 -0.01 0.50 0.52 -0.013 (-0.03) 0.260 0.272 0.007 0.912 0.956 -0.007

0.50 0.25 0.67 0.66 0.16 0.44 0.43 0.103 (0.24) 0.211 0.214 0.005 0.867 0.856 0.041

0.75 0.25 0.46 0.46 0.23 0.31 0.31 0.104 (0.34) 0.134 0.135 0.006 0.695 0.740 0.054

0.90 0.25 0.28 0.28 0.33 0.20 0.20 0.092 (0.46) 0.071 0.074 0.006 0.595 0.597 0.076

0 1.00 0.87 0.86 0.02 0.47 0.47 0.011 (0.02) 0.330 0.328 0.001 3.536 3.596 0.010

0.50 1.00 0.72 0.74 0.15 0.39 0.42 0.101 (0.25) 0.273 0.258 0.007 3.241 3.341 0.039

0.75 1.00 0.55 0.55 0.25 0.33 0.32 0.128 (0.39) 0.174 0.172 0.010 3.097 3.101 0.052

0.90 1.00 0.35 0.34 0.35 0.21 0.20 0.114 (0.56) 0.094 0.098 0.008 2.581 2.471 0.067

Ave SEM 0.016 0.015 0.020 0.015 0.014 0.011 0.004 0.004 0.001 0.03 0.03 0.02

5

Supplementary Table 5. The effects of the strength of stabilizing selection on the evolution of the

genetic variance and the M-matrix in the multivariate model of epistasis. The parameter values used to

generate this table are identical to those indicated in Table 1, except as noted. Variables and symbols are

described in Table 2.

rω 2211

11VG 22VG rG 11VA 22VA 12VA (rA) 11VAA 22VAA 12VAA M11 M22 rM

0 9 0.18 0.18 0.00 0.14 0.14 -0.001 (-0.01) 0.034 0.034 0.000 0.138 0.127 0.008

0.50 9 0.15 0.15 0.18 0.12 0.12 0.027 (0.23) 0.028 0.027 0.001 0.119 0.120 0.051

0.75 9 0.12 0.12 0.29 0.09 0.10 0.035 (0.37) 0.022 0.021 0.001 0.107 0.103 0.079

0.90 9 0.09 0.09 0.39 0.08 0.08 0.036 (0.45) 0.014 0.014 0.001 0.093 0.093 0.095

0 29 0.45 0.46 0.00 0.35 0.36 0.003 (0.01) 0.095 0.096 -0.001 0.189 0.195 0.011

0.50 29 0.41 0.43 0.20 0.32 0.34 0.087 (0.26) 0.076 0.081 0.004 0.185 0.206 0.087

0.75 29 0.31 0.33 0.33 0.25 0.26 0.108 (0.42) 0.055 0.058 0.003 0.166 0.168 0.113

0.90 29 0.21 0.20 0.43 0.17 0.17 0.092 (0.54) 0.035 0.033 0.002 0.131 0.120 0.134

0 49 0.72 0.73 0.01 0.57 0.58 0.007 (0.01) 0.141 0.140 0.002 0.255 0.270 0.003

0.50 49 0.62 0.61 0.17 0.49 0.48 0.114 (0.24) 0.119 0.120 0.002 0.231 0.215 0.043

0.75 49 0.50 0.49 0.31 0.40 0.39 0.159 (0.40) 0.086 0.085 0.003 0.218 0.200 0.085

0.90 49 0.31 0.30 0.41 0.26 0.24 0.133 (0.53) 0.048 0.050 0.003 0.170 0.166 0.110

0 89 1.19 1.22 -0.01 0.95 0.98 -0.010 (-0.01) 0.219 0.214 -0.002 0.309 0.324 -0.004

0.50 89 1.03 1.02 0.18 0.83 0.83 0.185 (0.22) 0.189 0.179 0.007 0.299 0.311 0.040

0.75 89 0.79 0.76 0.29 0.63 0.61 0.235 (0.38) 0.142 0.143 0.003 0.249 0.247 0.102

0.90 89 0.46 0.46 0.39 0.38 0.38 0.194 (0.51) 0.075 0.077 0.006 0.193 0.189 0.105

0 149 1.88 1.96 -0.02 1.56 1.63 -0.044 (-0.03) 0.290 0.312 0.002 0.427 0.440 -0.023

0.50 149 1.56 1.53 0.13 1.28 1.26 0.214 (0.17) 0.256 0.256 0.006 0.392 0.384 0.026

0.75 149 1.18 1.19 0.29 0.98 0.99 0.368 (0.37) 0.191 0.192 0.005 0.313 0.324 0.088

0.90 149 0.72 0.72 0.40 0.59 0.59 0.299 (0.51) 0.118 0.120 0.006 0.238 0.230 0.113

Ave SEM 0.021 0.023 0.019 0.020 0.023 0.015 0.005 0.005 0.002 0.011 0.011 0.019

6

Supplementary Table 6. The effects of the mutation rate (µ) on the evolution of the genetic variance

and the M-matrix in the multivariate model of epistasis. The parameter values used to generate this table

are identical to those indicated in Table 1, except as noted. Variables and symbols are described in Table

2.

rω µ 11VG 22VG rG 11VA 22VA 12VA (rA) 11VAA 22VAA 12VAA M11 M22 rM

0 0.0001 0.16 0.16 0.01 0.15 0.15 0.004 (0.03) 0.014 0.014 0.001 0.171 0.175 0.000

0.50 0.0001 0.13 0.13 0.19 0.12 0.11 0.026 (0.23) 0.012 0.011 0.001 0.137 0.135 0.028

0.75 0.0001 0.11 0.10 0.24 0.10 0.09 0.034 (0.36) 0.009 0.008 0.000 0.133 0.124 0.005

0.90 0.0001 0.06 0.06 0.35 0.06 0.06 0.027 (0.45) 0.005 0.005 0.000 0.096 0.089 0.013

0 0.0002 0.30 0.29 0.00 0.26 0.25 -0.001 (0.00) 0.033 0.034 0.000 0.203 0.210 0.009

0.50 0.0002 0.24 0.25 0.11 0.21 0.21 0.033 (0.16) 0.029 0.031 0.001 0.169 0.176 0.004

0.75 0.0002 0.20 0.20 0.29 0.17 0.18 0.063 (0.36) 0.024 0.024 0.002 0.164 0.168 0.063

0.90 0.0002 0.12 0.11 0.35 0.10 0.10 0.043 (0.43) 0.013 0.013 0.001 0.119 0.112 0.042

0 0.0005 0.75 0.75 0.04 0.59 0.58 0.029 (0.05) 0.149 0.149 0.002 0.234 0.242 0.036

0.50 0.0005 0.67 0.65 0.21 0.54 0.52 0.145 (0.27) 0.120 0.116 0.002 0.255 0.243 0.074

0.75 0.0005 0.46 0.45 0.28 0.37 0.36 0.129 (0.35) 0.089 0.085 0.006 0.195 0.185 0.072

0.90 0.0005 0.29 0.30 0.40 0.23 0.24 0.122 (0.52) 0.051 0.050 0.001 0.153 0.160 0.129

0 0.001 1.34 1.35 -0.03 0.90 0.91 -0.035 (-0.04) 0.407 0.400 -0.007 0.255 0.256 -0.022

0.50 0.001 1.17 1.15 0.18 0.81 0.80 0.207 (0.26) 0.329 0.326 0.009 0.251 0.241 0.062

0.75 0.001 0.91 0.93 0.35 0.64 0.66 0.322 (0.50) 0.247 0.251 0.012 0.209 0.216 0.170

0.90 0.001 0.58 0.61 0.46 0.44 0.46 0.273 (0.61) 0.132 0.131 0.007 0.178 0.184 0.201

0 0.002 2.37 2.37 0.01 1.23 1.21 0.014 (0.01) 1.067 1.073 0.001 0.271 0.268 0.006

0.50 0.002 2.09 2.14 0.18 1.12 1.17 0.376 (0.33) 0.896 0.896 0.018 0.255 0.260 0.090

0.75 0.002 1.67 1.61 0.34 1.00 0.95 0.546 (0.56) 0.609 0.611 0.020 0.237 0.230 0.188

0.90 0.002 1.08 1.06 0.44 0.70 0.68 0.460 (0.67) 0.340 0.346 0.020 0.196 0.191 0.225

Ave SEM 0.018 0.019 0.021 0.017 0.018 0.013 0.006 0.005 0.003 0.008 0.008 0.020

7

Supplementary Table 7. Evolution of the additive genetic variances, additive genetic covariances, and

M-matrix under different types of epistasis. As in Supplementary Table 2, the selection surface retains

the same shape but rotates in the two-dimensional parameter space. The parameter values used to

generate this table are identical to those indicated in Table 1, except as noted. The first four columns

describe the individual selection surface, as described in Table 1, where is the angle of its leading

eigenvector. The next two columns indicate the variances for the distributions from which the epistatic

parameters were drawn. In the first six rows of the table, the within-trait epistasis parameters (i.e., 1ε11

and 2ε22) were drawn from a distribution with a variance of one, while all other epistatic parameters (i.e.,

1ε22, 2ε11, 1ε21, and 2ε12) were set at zero. In the last six rows of the table, the within-trait epistasis

parameters (i.e., 1ε11 and 2ε22) were set at zero and the remaining epistasis parameters were drawn from a

normal distribution with a variance of one (i.e., 1ε22, 2ε11, 1ε21, and 2ε12). The additive genetic variances

and additive genetic correlation are given by 11VA, 22VA, and rA, while M11, M22, and rM represent the

mutational variances and the mutational correlation for the two traits. The angles of the leading

eigenvectors (in degrees in two-dimensional trait space) of the G-matrix and M-matrix are given by A

and M , respectively.

ω11 ω22 rω

Var.

222111 ,

Var. other ε

parameters 11VA 22VA rA

A M11 M22 rM M

93 5 0 0 1 0 0.625 0.098 -0.004 -0.1 0.166 0.071 0.000 0.0

88 10 0.69 13.8 1 0 0.530 0.106 0.186 5.9 0.146 0.074 0.000 0.0

82 16 0.80 20.6 1 0 0.392 0.119 0.215 9.4 0.138 0.082 0.000 0.0

71 27 0.87 30.0 1 0 0.270 0.135 0.266 18.5 0.105 0.084 0.000 0.0

60 38 0.89 37.7 1 0 0.226 0.155 0.308 29.1 0.101 0.083 0.000 0.0

49 49 0.90 45.0 1 0 0.181 0.201 0.325 49.7 0.088 0.108 0.000 90.0

93 5 0 0 0 1 0.371 0.125 0.001 0.1 0.161 0.128 0.011 7.5

88 10 0.69 13.8 0 1 0.384 0.142 0.291 14.6 0.160 0.125 0.048 7.9

82 16 0.80 20.6 0 1 0.361 0.149 0.374 19.6 0.160 0.125 0.094 20.5

71 27 0.87 30.0 0 1 0.337 0.202 0.508 31.4 0.157 0.141 0.121 33.9

60 38 0.89 37.7 0 1 0.306 0.221 0.527 36.3 0.163 0.141 0.140 31.4

49 49 0.90 45.0 0 1 0.273 0.275 0.553 45.2 0.143 0.150 0.161 48.2

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