GENETICS | INVESTIGATION

Uneven distribution of mutational variance across thetranscriptome of Drosophila serrata revealed by

high-dimensional analysis of gene expressionEmma Hine⇤,1, Daniel E. Runcie†, Katrina McGuigan⇤ and Mark W. Blows⇤

⇤School of Biological Sciences, The University of Queensland, Australia, †Dept. of Plant Sciences, University of California Davis, USA

ABSTRACTThere are essentially an infinite number of traits that could be measured on any organism, and almost all individual traits displaygenetic variation, yet substantial genetic variance in a large number of independent traits is not plausible under basic models ofselection and mutation. One mechanism that may be invoked to explain the observed levels of genetic variance in individualtraits is that pleiotropy results in fewer dimensions of phenotypic space with substantial genetic variance. Multivariate geneticanalyses of small sets of functionally-related traits have shown that standing genetic variance is often concentrated in relativelyfew dimensions. It is unknown if a similar concentration of genetic variance occurs at a phenome-wide scale when many traitsof disparate function are considered, or if the genetic variance generated by new mutations is also unevenly distributed acrossphenotypic space. Here, we used a Bayesian sparse factor model to characterize the distribution of mutational variance of3385 gene expression traits of Drosophila serrata after 27 generations of mutation accumulation, and found that 46% of theestimated mutational variance was concentrated in just 21 dimensions with significant mutational heritability. We show that theextent of concentration of mutational variance into such a small subspace has the potential to substantially bias the response toselection of these traits.

KEYWORDS Pleiotropy; Gene expression; Mutation; Genetic constraints

The phenotype of an organism can essentially be measured inan infinite number of ways (Houle 2010), and the number

of nucleotide changes that could occur to alter the phenotypeexceeds 106 for organisms as complex as Drosophila (Johnsonand Barton 2005). There are two reasons to suspect that for anylarge number of measured traits, p, only n << p independent(orthogonal) dimensions will exhibit levels of genetic variancecomparable to the substantial levels commonly measured in indi-vidual traits. First, costs associated with organismal complexitysuggest an evolutionary advantage to limiting n (Orr 2000); theprobability of new mutations being favourable, the magnitudeof the favourable effects and the chances of fixation all decreasewith increasing n. The disadvantages of high n are predictedto increase almost exponentially as n exceeds more than a fewhundred, leading to a rapidly declining rate of adaptation in

Copyright © 2018 by the Genetics Society of Americadoi: 10.1534/genetics.XXX.XXXXXXManuscript compiled: Sunday 13th May, 2018%1e.hine@uq.edu.au

more complex organisms (Orr 2000). Second, for a populationclose to its optimum, and evolving under Gaussian stabilisingselection acting independently on each trait, genetic load willreduce mean fitness as the number of independently selectedtraits increases (Barton 1990; Johnson and Barton 2005). Basedon published estimates of heritability and stabilising selection,this basic model of mutation and selection predicts populationfitness will decrease to unsustainable levels when n exceeds afew hundred (Barton 1990; Johnson and Barton 2005).

If n << p as implied by Fisher’s geometric model (Fisher1930) and theoretical models of selection and quantitative varia-tion, much of the genetic variance will be confined to the smallern-dimensional subspace. The distribution of genetic variance ischaracterized by the empirical spectral distribution of the p ⇥ p

genetic covariance matrix (G). When n is much less than p, thefirst n eigenvalues of G will account for a greater proportion ofthe genetic variance than expected when all dimensions of phe-notypic space contain substantial genetic variance. For a set of p

traits, strengthening the covariance relationships among them

Genetics, Vol. XXX, XXXX–XXXX May 2018 1

Genetics: Early Online, published on June 8, 2018 as 10.1534/genetics.118.300757

Copyright 2018.

increases the genetic variance contained in the n-dimensionalsubspace, in turn decreasing the genetic variance contained inthe complementary (p-n)-dimensional subspace. This unevendistribution of genetic variance will affect the predicted responseto selection; as n decreases relative to p, the evolutionary re-sponse is increasingly biased towards those n directions withgreater genetic variance (Walsh and Blows 2009).

There is now considerable evidence from many empiricalstudies of small sets of functionally-related traits that a sub-stantial proportion of genetic variance is restricted to a smallersubspace of the measured phenotype (Blows and Mcguigan2015). Although this observation is generally consistent acrossdifferent trait types and organisms, there are some importantlimitations to our current understanding of the distribution ofgenetic variance. Sets of functionally related traits, which aremost often analysed in multivariate quantitative genetic stud-ies, might be expected to exhibit greater pleiotropic covariancethan typical among the disparate function traits that comprisean organism. These studies may therefore give a misleadingimpression of the extent of the concentration of genetic variancein a smaller subspace. It is also unclear how the concentrationof genetic variance may scale to a phenome-wide level, as high-dimensional genetic analyses are few (Runcie and Mukherjee2013; Blows et al. 2015).

We know much less about the extent of pleiotropy generatedby new mutations, and specifically whether mutational variance,the ultimate source of genetic variance, is concentrated in a smallsubspace or more evenly distributed than standing genetic vari-ance. Under Fisher’s model, a mutation possibly affects all traitto some extent (universal pleiotropy), but some traits are affectedmuch more than others (Orr 2000; Martin and Lenormand 2006),leading to pleiotropic covariance among traits, and consequentlya concentration of mutational covariance in a smaller subspace.While only a few studies have directly characterized the distri-bution of eigenvalues of the mutational covariance matrix (M),each has reported an uneven distribution of mutational variance,with a small number of eigenvalues accounting for the majorityof the mutational variance (Camara and Pigliucci 1999; Latimeret al. 2014; McGuigan et al. 2014). In the only empirical study weare aware of to determine the relative size of the n-dimensionalsubspace of M, (Houle and Fierst 2013) showed that for p = 20wing shape traits of Drosophila melanogaster, n ranged between 7and 19, suggesting n < p as a consequence of pleiotropy. Howthese results relate to a wider set of traits, and if the distribu-tion of mutational variance at a phenome-wide level is similarlyuneven, remains to be determined.

Here, we utilize a recently developed analytical frameworkto investigate the distribution of mutational variance across avery large number of gene expression traits with a wide rangeof putative functions. As the number of parameters requiredto estimate covariance matrices scales with p by p ⇥ (p + 1)/2,estimation of such matrices within the context of quantitativegenetic experimental designs is particularly challenging (Blowset al. 2015). The Bayesian sparse factor (BSF) model of Runcie andMukherjee (2013) is an extension of factor analytic modellingthat overcomes the estimation challenge via two key sparsityassumptions; first, few latent factors are assumed to underliethe phenotypic variance, and second, most latent factors areassumed to affect relatively few individual traits (Runcie andMukherjee 2013).

We apply the BSF model to investigate mutational covari-ance among 3385 gene expression traits after 27 generations of

mutation accumulation (MA) in 41 lines of Drosophila serrata

(McGuigan et al. 2014). Although some of the expressed genesare involved in the same biological processes (Blows et al. 2015;Collet et al. in press), there is no a priori reason to expect strongcovariance among this functionally diverse set of traits. In aprevious analysis of these data, frequent mutational covariancewas established among small p = 5 random sets of these 3385expression phenotypes, indicating that n << p to some un-known extent (McGuigan et al. 2014). The BSF model allowed usto directly investigate how many expression traits are affectedby a common latent factor (the transcriptome-wide extent ofmutational covariance), and whether the total observed muta-tional variance in this experiment is consistent with many latentfactors (mutations) affecting each trait independently, or withfew common factors affecting many traits simultaneously. Wethen investigated the potential evolutionary consequences of theextent of mutational covariance by determining the predictedbias in the response to selection that would be generated by theobserved distribution of mutational variance.

Materials and Methods

The data and the experimental design from which it was ob-tained have been described elsewhere (McGuigan et al. 2014).Briefly, 100 lines were set up from a single inbred line and main-tained for 27 generations of a mutation accumulation (MA) ex-periment. In each line one son and one daughter of a singlebrother-sister pair became the parents of the next generation. Inthe 28th generation, 11604 gene expression traits were measured(using five different probes per trait, with each probe duplicated)on two biological replicates for each of the 41 surviving MA lines.McGuigan et al. (2014) conducted univariate mixed effects mod-els on the 11604 gene expression traits and reported that 3385had non-zero estimates of among-line (mutational) variances,with 1035 significant at a = 0.05, and 533 remaining significantafter FDR correction. Of 677 randomly allocated 5-trait sets, acommon component of among-line variance was significant for245 (145 at FDR).

Statistical Analysis

A recent advance in the analysis of high dimensional geneticdata has made it possible to estimate the mutational variancesof, and covariance among, all 3385 traits in a single analysis. TheBayesian sparse factor (BSF) genetic model (Runcie and Mukher-jee 2013) uses factor analysis to partition the total phenotypicvariance into common phenotypic variance arising from latentfactors and trait-specific phenotypic variance. At the same time,the BSF model utilizes information from the experimental de-sign to further partition these variances into their genetic andresidual (environmental, technical etc) components. Two keyassumptions distinguish the BSF from traditional factor analysisand make it possible to include so many traits in the one analysiswith relatively small sample sizes. First, few latent factors areassumed to underlie the phenotypic variance. Second, most la-tent factors are assumed to affect relatively few individual traits(Runcie and Mukherjee 2013).

We refer the reader to Runcie and Mukherjee (2013) for a de-tailed description of the Bayesian sparse factor genetic model fora one-way experimental design utilizing pedigree informationin an animal model. Here, we modify this model to accommo-date the experimental design outlined above. Note that Runcieand Mukherjee (2013) use the BSF model to infer the geneticvariance-covariance matrix (G), whereas here we are estimating

2 Emma Hine et al.

mutational (co)variances, which can be summarized with thecovariance matrix M. Further, Runcie and Mukherjee’s modelassumed one probe per gene, and so we extended the modelhere to allow for multiple uncorrelated probes per gene.

Prior to analysis, we took the log10 of the mean of two mea-surements per probe on an array. A genetic variant was knownto be segregating in this set of MA lines (McGuigan et al. 2014),so we extracted the residuals from simple linear models foreach set of log 10 mean expression measures, and standardisedthem to unit variance. Let y

de f g

represent these standardisedresiduals for line d 2 1, 2, . . . , D = 41, replicate e 2 1, 2, genef 2 1, 2, . . . , F = 3385 and probe g 2 1, 2, . . . , G = 5. y

de f g

ismodeled as

y

de f g

= µf

+ Probef g

+ Lined f

+ Repde f

+ ede f g

(1)

where µf

is the global mean for gene f , Probe

f g

is the meaneffect of the two replicates per array of probe g within gene f ,Line

d f

is the mean effect of line d on gene f , Rep

de f

is the meaneffect for replicate e on gene f , and e

de f g

is the residual error.The line, replicate and residual effects are considered random(with modelled variance), and the mean and probe effects areconsidered fixed.

To model the mutational covariance among multiple gene ex-pression traits, we assume the random effects have the followingdistributions:

Lined. ⇠ MVN

F

(0, M)

Repde. ⇠ MVN

F

(0, R)

ede f g

⇠ N(0, s2f g

)

where the period in the index represents a vector formed fromall F = 3385 genes, MVN

F

is the F-dimensional multivariatenormal density, and M and R are covariance matrices of thegene expression traits caused by mutation and by microarraytechnical errors plus biological noise, respectively. Mutation andresiduals are assumed uncorrelated across lines and replicate,respectively.

Equation 1 can be re-parameterized using hierarchical cen-tering. This technique reduces correlations among parametersmaking MCMC more efficient. In particular, we introduce thelatent variable w

de f

to model the gene-level effects in each rep:

y

de f g

= w

de f

+ Probef g

+ ede f g

w

de f

= µf

+ Lined f

+ Repde f

(2)

In equation 2, ede f g

are residuals of the probe measurementsaround a “true” transcript expression, and Rep

de f

are residualsof the transcript expression around the line means. Parameter-ized as such, the model for w

de f

has an identical form to themodel presented in Runcie and Mukherjee (2013). We thereforeembedded the BSFG GIBBS sampler from Runcie and Mukherjee(2013) in a GIBBS sampler for the Probe

f g

and s2f g

parameters,sampling each parameter individually conditionally on the cur-rent sample of w

de f

.As in a traditional factor analysis, we assume any phenotypic

covariance among the observed traits is caused by k latent fac-tors, which we refer to as phenotypic common factors (PCFs),and allow specific mutational and residual variances for indi-vidual traits such that the phenotypic covariance matrix can beexpressed as:

P = LLT + ym

+ yr

where L is the F ⇥ k matrix of PCF loadings, and ym

and yr

are the diagonal matrices of specific mutational and residualvariances, respectively. Under the BSF model, the mutationalcovariance is modelled by allowing each latent PCF to have itsown heritability. This heritability is the cumulative effect of newmutations, and it should not be confused with the per generationmutational heritability often reported in mutation accumulationstudies, V

M

V

E

, where V

M

is the among-line variance divided bytwice the number of generations of mutation accumulation.

The mutational (M) covariance matrix can then be recoveredthrough the equation:

M = LSh

2 LT + ym

(3)

where Sh

2 is the k ⇥ k diagonal matrix of PCF heritabilities. Inour results we refer to the common, specific, and total muta-tional variances of the individual traits, which correspond tothe diagonals of LS

h

2 LT , ym

and M, respectively. Further, wepresent mutational variances of the estimated PCFs, rather thanattempting to interpret a spectral decomposition of M.

Parameter Expansion When assessing model convergence (seebelow), we noticed poor mixing of L = [l

ij

], and strongly nega-tive posterior correlations between individual l

ij

and the stan-dard deviation of the individual PCF scores (fj). To combat this,we implemented the parameter expansion for factor modelsproposed by Ghosh and Dunson (2009). We introduced a newparameter YF, a diagonal matrix, and replaced the parametersL, F and Fa in the original BSFG model (Runcie and Mukherjee2013) with working parameters L⇤, F⇤ and F⇤

a with the followingdistributions:

L⇤ = [l⇤ij

], lij

⇠ N(0, f�1ij

t�1j

)

F⇤ ⇠ MN(ZF⇤a , I, Y1/2

F (I � Sh2 )Y1/2F )

F⇤a ⇠ MN(0, A, Y1/2

F Sh2 Y1/2F )

YF = Diag(yF

j

), yF

j

⇠ Ga(a

F

, b

F

)

where fij

, tj

, Z, A, and Sh2 are defined as in Runcie and Mukher-jee (2013).

For each posterior sample, we rescale F⇤, F⇤a and L⇤ to recover

the original parameters and to ensure that the variance of each( f

j

) is equal to one:

L(i) = L⇤(i)Y(i)1/2F S

(i)1/2F

F(i) = F⇤(i)Y(i)�1/2F S

(i)�1/2F

F(i)a = F⇤(i)

a Y(i)�1/2F S

(i)�1/2F

where SF is a diagonal matrix of the empirical variance of F fromeach posterior sample.

Further investigations of Gibbs sampler performance iden-tified the d

j

, j 2 1 . . . k parameters as exhibiting especially poormixing and strong correlations among parameters. Since up-dates of individual d

j

parameters are relatively fast, we iteratedthrough the sampling of these parameters 100x per iteration ofthe remaining parameters in the Gibbs sampler.

Mutational variance of the transcriptome 3

Prior distributions and hyperparameters We used the sameprior distributions as in Runcie and Mukherjee (2013) (withsome changes to the hyperparameters, discussed below), andrefer the reader to their paper for a detailed description. A sum-mary of the prior distributions outlined below, and the valueswe ultimately used for their hyperparameters, are listed in Ta-ble S1. Briefly, the PCFs are modelled through a hierarchicaldistribution on the PCF loadings. Each l

ij

is assigned a normaldistribution as a prior with a unique variance drawn from aninverse-gamma distribution. Modelling the PCF loadings in thisway imposes sparsity on the PCFs and ensures that the vari-ance explained by successive PCFs decreases to zero at somesufficiently large k

⇤.The remaining parameters are estimated with more straight-

forward prior distributions. The heritability of the PCFs aremodelled with a 50% probability of h

2j

= 0 and the remaining50% distributed equally across a discrete set of evenly spacedvalues in the interval 0 < x 1. The mutational and residualspecific variances, forming the diagonal elements of y

m

and yr

are modelled with inverse gamma prior distributions. We alsoused inverse gamma prior distributions to model the residualprobe variances that form the diagonal of S

y., which were notpresent in the original BSF genetic model presented in Runcieand Mukherjee (2013).

We explored the effect of changing the hyperparameter val-ues used in Runcie and Mukherjee (2013) for three aspects ofthe prior distribution relating to the PCF trait loadings (TableS2). We ran the BSF model with five different random seedsfor each of nine sets of values for the three hyperparameters.For each analysis, we retained 1000 samples with a thinningrate of 100 after a burn-in period of 300,000 samples. The to-tal mutational variance estimated for the 3385 gene expressiontraits was consistent among the 45 analyses, with correlationsbetween pairs of analyses ranging from 0.95 to 1.00. In contrast,the number of heritable PCFs and the amount of mutationalvariance accounted for by PCFs (common mutational variance)was sensitive to the hyperparameter values (Table S2).

Model convergence From the 45 prior analyses, we identifiedthe one that detected the greatest number of heritable PCFs (Ta-ble S2) and continued the chain. After discarding an additional100,000 samples (taking the burn-in period to 400,000) then re-taining 1000 samples at a thinning rate of 100, we were satisfiedwith the convergence diagnostics of the model. Autocorrela-tion statistics for the PCF heritabilities and specific variances aresummarised in Fig. S1. A single specific mutational varianceexceeded our nominal autocorrelation threshold of 0.2. Auto-correlations for 630 of 152325 PCF trait loadings exceeded 0.2(Fig. S2). We concluded from trace plots of these parameters(specific mutational variance not shown; samples of PCF traitloadings presented in Fig. S3) that the high autocorrelationswere unlikely to be cause for concern. We note that the PCFsmost represented among the highly autocorrelated trait loadings(PCFs 1, 2, 4, 5, 7, 8, 17 and 18; Fig. S2) are among those withvery low (median posterior sample = 0), nonsignificant estimatesof PCF heritability, which we do not interpret.

Significance testing We use a local false sign rate (LFSR) ap-proach to declare significance of PCF heritabilities and trait load-ings. LFSR is the probability of assigning the incorrect sign to anestimate. For PCF heritabilities, we assign LFSR as the propor-tion of posterior samples equal to zero. For PCF trait loadings,we assign LFSR values as the proportion of posterior samples

equal to zero, or on the other side of zero from the median pos-terior sample. For example, if we declared all trait loadings with75% of the posterior samples > 0 positive, we expect to be wrong25% of the time, so LFSR = 0.25. We report the number of traitloadings per significantly heritable PCF with LFSR < 0.025 (be-ing two-tailed tests) in Table S3. We also compute an averageerror rate for each set of heritable PCF trait loadings (Stephens2017), and report the number of trait loadings for which theresulting s-value remains below 0.005 (Table S3). The s-value isanalogous to Storey’s q-value (Stephens 2017; Storey 2003), andimplies that we can expect 99% of the trait loadings declaredsignificant to actually be different from zero.

Randomisations We conducted two types of randomisations,referred to as type 1 and type 2. For type 1 randomisations,we tested whether the BSF model would allocate mutationalvariance to PCFs in the absence of a true covariance signal. Todo this, we preserved the mutational variance of individualtraits but disrupted trait covariances by shuffling line replicatepairs within traits. That is, we reassigned probe measurementy

de f g

to y

d

⇤e f g

, where all five probe measurements of the f

th

gene for both replicates of the d

th line (d = 1, 2...41) take on thecorresponding measurements for the f

th gene of the d

⇤th line,where d

⇤ is the d

th element of the shuffled vector of integers 1 to41, shuffled independently for each gene expression trait. Fortype 2 randomisations, we tested whether the BSF model wouldspuriously allocate heritability to PCFs given the observed traitcovariances but in the absence of mutational variance. We pre-served the covariance structure among traits but disrupted themutational variance of individual traits by shuffling the rows(representing individual replicates) of the 82 ⇥ 16925 data ma-trix. For each type of randomisation, we generated 100 data sets.We ran the BSF model for each data set, collecting 1000 samplesat a thinning rate of 100 after 300,000 burn-in samples.

Predicted response to selection To explore how the distribu-tion of the total mutational variance (sum of both common andspecific mutational variances as described in equation 3) mightaffect the adaptive potential of this population, we used M inplace of G in the multivariate breeders’ equation, Dz = Gb. Wecalculated the predicted response to selection, Dz, for 1,000,000random selection gradients, b, and subsequently determined theangle between b and Dz as a measure of evolutionary bias, rang-ing from zero (no bias) to 90 degrees (strong bias). Each of the3385 elements of the million random bs was drawn from a uni-form distribution between -1 and 1. For each of 1000 posteriorsamples of M, we calculated Dz for a subset of 1000 random bs,generating the total distribution of 1,000,000 predicted evolution-ary responses to random directions of selection. We comparedthe distribution of angles between b and Dz to the distributiongenerated when b was applied to M with a null covariance struc-ture (i.e., our first randomization as outlined above). For therandomized data, we applied the same set of 1,000,000 bs, allo-cating 100 bs to each of 100 posterior samples of M from each ofthe 100 randomised datasets.

Data Availability

The original dataset is available at the Gene Expression Om-nibus under accession number GSE49815. The subset ofgene expression traits used in this analysis is available at es-pace.library.uq.edu.au under accession number uql.2017.783.MATLAB code to implement the BSF analysis, MATLAB out-put, R code used to process the output and univariate REML

4 Emma Hine et al.

0 10 20 30 40

Phenotypic Common Factor

0

5

10

15

20

25 ●

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●●●●●●●● ●● ● ●● ●●● ● ● ●● ● ●● ●

Perc

enta

ge o

f Phe

noty

pic

Varia

nce

Figure 1 Phenotypic and mutational variances of the PCFs,as a proportion of the total phenotypic variance across the3385 gene expression traits. Before dividing by the total pheno-typic variance, the phenotypic variance (grey, filled circles) ofa given PCF is defined as the sum of its squared trait loadings.The phenotypic variance is multiplied by the correspondingPCF heritability to give the mutational variance attributable tothe PCF. Black filled and unfilled circles depict mutational vari-ances for PCFs with significant (s < .01) and non-significantPCF heritabilities, respectively.

estimates are also available at espace.library.uq.edu.au underaccession numbers uql.2018.121, uql.2018.111, uql.2018.101 anduql.2018.113.

Results and Discussion

After detailed investigation of the effect of several prior distribu-tion hyperparameter values, and confirmation of convergence ofthe selected final BSF model (Materials and Methods), we con-ducted two sanity checks to ensure the final BSF model provideda good representation of the known characteristics of the experi-mental design and data. First, we used the Marcenko-Pastur lawfrom random matrix theory for large sample covariance matricesto form an expectation of how many PCFs we could reasonablyexpect the BSF model to detect (Nadakuditi and Edelman 2008).Our experimental design of 82 independent experimental unitsand 3385 traits should have sufficient power to detect, withhigh probability, a minimum of 37 dimensions of phenotypicvariance (Fig. S4). The BSF model identified 45 PCFs (Fig. 1),indicating that the model had converged on a realistic numberof PCFs given our experiment. Second, the mutational variancesfor the 3385 individual expression traits previously estimatedfrom standard univariate REML analyses (McGuigan et al. 2014)were highly correlated (r=0.84, df=3383, p<0.001) with the BSFestimates (Fig. S5), indicating that the mutational variance inindividual expression traits was recovered well by the morecomplex BSF model.

Mutational heritability was significant at s < 0.01 for 21 of

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PCF 10h2 = 0.74

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PCF 19h2 = 0.69

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PCF 31h2 = 0.99

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y

PCF 32h2 = 0.93

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PCF 33h2 = 0.43

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PCF 36h2 = 0.96

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PCF 37h2 = 0.99

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PCF 38h2 = 0.93

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PCF 40h2 = 0.97

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y

PCF 41h2 = 0.99

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y

PCF 44h2 = 0.74

Replicate 1

Rep

licat

e 2

Figure 2 PCF scores of mutation accumulation line replicatepairs for the 21 heritable PCFs. Scores were estimated withinthe BSF model for the two replicate assays from each of the 41MA lines. The solid line shows the 1:1 value. Any deviationof the points from this line reflects the presence of non-genetic(other biological or technical) variation. PCFs are numberedaccording to their descending phenotypic variance (as per Fig.1).

the 45 PCFs (Fig. 1, Fig. S6), with 0.30 h

2 0.99 (Fig. 2). Wecompared the observed PCF heritabilities and their LFSR valuesto those obtained for the randomized datasets (Fig. 3). ). For thetype 1 randomisations, which simulate traits with uncorrelated

Mutational variance of the transcriptome 5

0.0

0.2

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● B

Posterior mean PCF heritability

Loca

l fal

se s

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rate

Figure 3 Comparison of significant and non-significant esti-mated PCF heritabilities from observed data (black filled andopen circles, respectively) with PCF heritabilities from ran-domised data (grey circles). Panel A: Type 1 randomisations(same mutational variance as observed data, but null phe-notypic and therefore mutational covariance). A single PCFheritability from three of these 100 randomised datasets wasin the range of the observed PCF heritabilities. Two of theseestimates sit under the single grey point in the bottom right ofPanel A. The remaining 3899 PCF heritabilites from across the100 datasets were all less than 0.1. Panel B: Type 2 randomisa-tions (phenotypic covariances as per observed data, but nullmutational (co)variance). Of the 5082 estimated PCF heritabil-ities across the 100 datasets, only 73 (1.4%) were greater thanthe smallest significant observed PCF heritability.

mutational effects, a single PCF from three of the 100 datasetswas, by chance, highly heritable (>0.8, Fig. 3A). The remaining

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x

Lam

bda_

mea

n[x,

i]

● ● ●

0 1000 2000 3000Gene expression trait

PCF

trait

load

ing

Figure 4 PCF trait loadings for the 21 heritable PCFs. PCF traitloadings with s < 0.005 are plotted with black circles. Non-significant trait loadings are plotted with grey circles. PCFs arepresented in the same order as in Fig. 2. Note that PCF traitloadings are limited to an approximate range of -1 to 1, due tothe scaling of the data to unit variance.

PCF heritabilities estimated from the type 1 randomised datasets were all less than .1, well below the smallest observed PCFheritability. For the type 2 randomisations, which simulate traitswith phenotypic covariance but no mutational variance, most ofthe PCF heritabilities were concentrated near zero. Only 1.4%of the type 2 randomized data PCF heritabilities exceeded the

6 Emma Hine et al.

smallest observed PCF heritability. These results demonstratethat, despite the limited genetic degrees of freedom, there is onlya small probability that any of the observed heritable PCFs werespuriously generated by the BSF model.

Angle in degrees

Freq

uenc

y (x

1000

)

010

020

030

040

050

0

0 10 20 30 40 50 60 70 80 90

Figure 5 Angles between random selection gradients and thecorresponding evolutionary response predicted by M. Theevolutionary responses predicted by the posterior samples ofthe observed data (open) and the randomized data sets (grey)are each plotted.

There was considerable variation in the way individual traitscontributed to the heritable PCFs. Overall, 1193 of the 3385expression traits contributed significantly at s < 0.005 to atleast one of the 21 heritable PCFs (Fig. 4). The number of traitloadings on a given PCF ranged from 3 to 266 (s < .005; Fig. 4,Table S3). Most expression traits contributed significantly to onlya single PCF, but some contributed to up to four (Table S3). Ingeneral, the extent of shared expression traits across PCFs scaledwith the size of the PCF; the three largest PCFs had the mostindividual expression traits that contributed to multiple PCFs(Table S3). Furthermore, the two heritable PCFs associated withthe greatest phenotypic and mutational variance (PCFs 9 and10) displayed individual trait loadings that were biased to oneside of zero. This is consistent with the major axis of standinggenetic variance in gene expression in this species (Blows et al.

2015), which is also characterized by a majority of trait loadingsbeing biased in one direction.

Overall, the 21 significantly heritable PCFs accounted for 46%of the total mutational variance estimated across the 3385 indi-vidual expression traits. This number dropped to 45% whenthe two least heritable PCFs (for which LFSR > 0.05) were ex-cluded from the calculation. The remaining mutational variancewas accounted for by the 24 non-heritable PCFs (12%), and the3385 trait-specific mutational variances (42%). Non-heritablePCFs might truly capture mutational covariance, but below thelevel statistically detectable with our design. Statistically, wecannot interpret the specific mutational variances, as there are3385 of these parameters and only 40 degrees of freedom at

the among-line level. Biologically, the among-line variance par-titioned into the specific mutational variance may arise fromtruly non-pleiotropic mutations, and/or the mixture of manypleiotropic mutations of relatively small effect.

To calibrate the biological consequence of the uneven distribu-tion of mutational variance generated by the PCFs, we adopteda random skewers approach using the Breeders Equation (BE).The BE is effectively a rotation of the vector of selection throughG, resulting in the potential for the response to selection to bebiased away from the direction of selection (Walsh and Blows2009). As the distribution of variance becomes more unevenin G, this bias will increase, on average, for random selectiongradients. The distribution of angles between a large number ofrandom selection gradients and their corresponding responsestherefore allows a quantification of the potential biological im-pact of the observed distribution of variance. Here, we replacedG with M, so that we could use the rotation in the BE to de-termine how the observed distribution of mutational variancemight influence evolutionary trajectories of this population.

Figure 5 compares the distribution of angles between the se-lection gradients and their resulting predicted change in traitmeans for observed and type 1 randomized data. For both theobserved and randomized data, rotating any given random se-lection gradient through M resulted in at least some bias of theresponse away from the direction of selection. This happens inthe randomized data, even though most mutational covariancesare small, because the mutational variances are not equal acrosstraits. That is, we expect some non-zero angle between the se-lection gradient and its response for any M that is not exactlyproportional to the identity matrix. The majority of angles forthe randomised data fell between 30 and 35 degrees, indicatinga small to moderate bias on the response to selection. In contrast,the bias in the predicted response to selection away from thedirection of selection was consistently greater than 76 degreesfor the observed M. Overall, these results suggest that if evolu-tion depended solely on new mutations, uneven magnitudes ofmutational variance among individual traits may have modesteffects on the direction of phenotypic evolution. In contrast, theobserved uneven distribution of mutational variance generatedby mutational covariance could strongly bias evolutionary re-sponses if it is in turn reflected in the distribution of standinggenetic variance.

Several previous experiments have all reported a similar pat-tern to that observed here, where mutation does not equallygenerate variance in all directions of phenotypic space (Camaraand Pigliucci 1999; Houle and Fierst 2013; Latimer et al. 2014). Acaveat of both the current study and those previous investiga-tions is that the distribution of mutational variances uncoveredrepresents only the effect of those mutations that occurred dur-ing the generations of experimental mutation accumulation. Itremains to be determined whether multivariate phenotypes withlow mutational variance detected in these experiments representinherent bias in the production of variation or merely stochas-tic sampling of mutations. Houle and Fierst (2013) comparedmutations affecting 20 morphological traits in two independentmutation accumulation populations. Although trait covariancesdiffered between the two populations, the general pattern thatvariance was predominantly restricted to relatively few dimen-sions was common to both, and interestingly, there was evidencethat the region of phenotypic space with low mutational vari-ance was shared between the populations (Houle and Fierst2013), suggesting a repeatable difference in the magnitude of

Mutational variance of the transcriptome 7

mutational variance across the phenotypic space.It will be important to determine whether subspaces of low

standing genetic variance can be predicted by subspaces withlow mutational variance. The G-matrices of natural popula-tions represent mutation-selection-drift balance over longer time-scales than MA experiments, where regions of phenotypic spacewith low genetic variance must represent multivariate pheno-types with either low fitness or little mutational input (Blowsand Mcguigan 2015). Recently, Houle et al. (2017) analysed sev-eral Drosophila datasets to compare patterns of among-speciesdivergence in wing shape with within species patterns of stand-ing genetic and mutational variance. Both the directions ofdivergence among species and the distribution of standing ge-netic variance within a species were strongly explained by thepatterns of mutational variance, with directions of maximumdivergence and of maximum standing genetic variance corre-sponding to maximum mutational variance. These results areconsistent with there being long-term consequences of the con-centration of mutational variance, and might also be interpretedas revealing evolution of M to match the fitness surface (Draghiand Wagner 2008; Pavlicev et al. 2011; Jones et al. 2014).

A second caveat of the current and previous experiments isthat among-line covariances in MA experiments can be gener-ated via both linkage and pleiotropy. Non-pleiotropic covariancecan arise when some lines harbour more mutations than others(Keightley et al. 2000), and through sampling when the direc-tion of mutation is biased (for example, for life history traits)(Keightley et al. 2000; Estes et al. 2005). Notably, investigationsof nucleotide variation at specific loci indicate the general po-tential for pleiotropy to generate covariance among hundreds ofexpression traits (e.g. Gerstung et al. 2015; Utrilla et al. 2016), asobserved here. We further note that the number of traits inferredto be mutationally correlated by our approach is similar to thenumber of traits inferred by other approaches to be pleiotropi-cally affected by mutation. Recent estimates from several QTLand gene knock-out/knockdown studies place the median de-gree of pleiotropy in the range 1-9% for morphological andphysiological traits, and ⇡ 0.2% for gene expression (Wagnerand Zhang 2011). Here, the proportion of traits loading signif-icantly (s<.005) onto one of the 21 heritable PCFs ranged from<0.1% to 7.9% with median 1.1% (Fig. 4, Table S3). Nonetheless,further investigation would be required to determine whetherthe observed concentration of mutational variance was gener-ated through pleiotropic mutation or linkage among individualmutations, each with independent effects on different traits.

Although our results indicate a substantial concentration ofthe mutational variance in relatively few phenotypic dimensions,the combined effects of the uneven distribution of the muta-tional variance and their selective effects may result in an evenmore limited effective dimensionality. Under the framework ofFisher’s geometric model, and assuming universal pleiotropy,the empirical distribution of fitness effects of new mutationscan be used to estimate n

e

, the effective phenotypic complexity(Martin and Lenormand 2006; Le Nagard and Tenaillon 2013) as:

n

e

=p

1 + CV(l)2 = 2E(s)2

V(s)(4)

where CV(l) is the coefficient of variation of the eigenvalues ofSM, where S is is the matrix of selective effects of the mutationson each trait, and E(s) and V(s) are the mean and variance ofthe distribution of the effects of mutations on fitness. Whenthe eigenvalues of SM are all equal, then CV(l) = 0 and all

traits are mutationally and selectively independent (ne

= p). AsCV(l) increases, the number of independent traits decreases. Inthis approach, the eigenvalues of S and M remain unobserved,and n

e

is estimated from empirical estimates of E(s) and V(s).Estimates of n

e

from the distribution of fitness effects are remark-ably small, in the range 0.2 < n

e

< 3.0 for organisms as diverseas viruses and Drosophila (Lourenco et al. 2011; Tenaillon 2014).However, these are likely to be underestimates if contrary to abasic assumption of the method, each mutation does not affectall traits (i.e., if pleiotropy is not strictly universal) (Lourencoet al. 2011). Direct estimation of M and S is likely to be requiredto determine the relationship between n

e

and n.The concentration of mutational variance in a small sub-

space has implications beyond our understanding of evolutionin natural populations. Pleiotropic effects of mutations havebeen demonstrated across different human diseases (Sivaku-maran et al. 2011; Lee et al. 2013; Chesmore et al. 2017), andthere is a growing recognition that taking pleiotropy into ac-count can greatly aid the detection of causal variants and theunderlying mechanisms of human diseases (Denny et al. 2010;Andreassen et al. 2013; Pendergrass et al. 2013; Liley and Wallace2015; Mitteroecker et al. 2016). Since we can expect n ⌧ p in hu-man populations, genetic analyses that simultaneously considermany widely disparate disease phenotypes have the potentialto greatly enhance our understanding of the impact of mutationon human health.

Acknowledgements

This work was supported by the Australian Research Council.We thank J. Thompson, C. Oesch-Lawson, D. Petfield, O. Sissa-Zubiate, and Y.H. Ye for help with the MA experiment and datacollection, A. Lund for help with Latex and navigating the Linuxenvironment, and S. Crawley for assistance using the Qriscloudfacility. This research was undertaken with the assistance ofresources from QCIF (http://www.qcif.edu.au).

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10 Emma Hine et al.

TableS1.Priordistributionsandtheirhyperparameters.Modelparameter

Description Priordistribution Hyperparametervalues

F Numberofgeneexpressiontraits

- -

G Numberofprobespergeneexpressiontrait

- -

k NumberofPCFs - -Λ FxkmatrixofPCF

loadingsljf~N(0, $%&'()%'()$+,~Ga(- 2 ,

-2)

(f=1…F,j=1…k))+ = 01

+123

03~Ga(73, 83)01~Ga(79, 89),(l=2...k).

- = 2a1=2.1b1=1/20a2=3b2=2

:;< Thejthdiagonalelementofthekxkdiagonalmatrix,:;<,containsℎ%>,theheritabilityofthejthlatentPCF.

prob(ℎ%>=0)=0.5prob(ℎ%>= ,?@)=

39(?@'3)

,(j=1…k,f=1..(nh-1)).

nh=100

Ψm AnFxFdiagonalmatrixcontainingthespecificmutationalvariances.

ψBC'3~Γ 7B, 8B ,

(f=1…F).

am=2bm=1/2

Ψr AnFxFdiagonalmatrixcontainingthespecificresidualvariances.

ψEC'3~Γ(7E, 8E),

(f=1…F).

ar=2br=1/2

Σy An(Fxk)x(Fxk)diagonalmatrixcontainingtheresidualprobevariances.

σGCH'3~Γ 7G, 8G ,

(f=1…F,g=1…G).

ay=2by=1/2

B The2xFmatrixoffixedeffects:overallmeanforeachtraitandtheeffectofthesegregatingmajorgeneoneachtrait.

I.,~K9 0,∞ ∗ O> (f=1…F).

Note:improperprior,butposteriorisproper.

TableS2.EffectofpriordistributionhyperparametersonthenumberofPCFsidentifiedandtheestimatedmutationalvariancebytheBayesianSparseFactormodel.Weranthemodelontheobserveddatawith5differentrandomseedsforeachof9differentsetsofpriordistributionhyperparameters.Fromeachofthe45models,weretained1000samplesfrom100,000(i.e.thinnedatarateof100)afteraburn-inperiodof300,000samples,andcalculatedmutationalvariancesandsignificanceofPCFsandtheirheritabilitiesusingtheaverageerrorrateofthelocalfalsesignratemethod,asdescribedinthemaintext.Wereporttherangeofestimatedparametersacrossthe5runsofeachsetofhyperparametervalues.Wefoundthetotalmutationalvarianceestimatedonatraitbytraitbasiswashighlyconsistentamongthe45analyses,withpair-wisecorrelationsbetweensetsofestimatedtraitmutationalvariancesrangingfromr=0.95-0.99.Wecontinuedthechainfromset7withthehighestnumberofheritablePCFsforafurther200,000samples,andretained1000ofthelast100,000samplesthinnedatarateof100fortheanalysispresentedinthemaintext.AdditionalPCFsweredetectedinthecontinuedrun,takingtheobservednumberofheritablePCFsto21.

Set a2 b2 - Numberof

significantPCFs

Numberofheritable

PCFs

Totalmutational

variance

Commonmutational

variance Min Max Min Max Min Max Min Max

1 2 1 2 37 45 14 16 777.0 799.7 426.7 464.72 2 1 3 35 41 12 16 775.3 785.5 405.2 419.43 2 1 9 23 27 6 8 761.9 774.3 360.6 371.94 3 1 2 31 37 11 13 789.1 793.8 407.5 437.65 3 1 3 26 35 7 12 779.0 789.4 384.0 421.86 3 1 9 22 24 5 7 765.4 782.1 353.6 370.07 3 2 2 37 41 12 18 791.4 799.1 420.1 448.98 3 2 3 34 37 10 14 771.2 784.1 393.2 424.79 3 2 9 24 27 6 9 763.4 771.9 354.9 365.4

TableS3.ThenumberofindividualgeneexpressiontraitscontributingtotheheritablePCFs.ForeachheritablePCF,wereportthenumberofsignificanttraitloadingsbasedonthelocalfalsesignrate(LFSR)andtheaverageerrorrates-value,asdescribedintheMethods.Weshowthetotalnumberandpercentageoftraitloadingsthatmeetthesignificancethresholdsforthetworelatedapproaches,andbreakthetotalnumberdownintothenumberoftraitsthatloadsignificantlyontoone,two,threeorfourPCFs.

PCF LFSR<.025 s-value<.005 1 2 3 4 Total % 1 2 3 4 Total %

9 179 63 4 1 247 7.3% 183 63 6 1 253 7.5%10 179 70 6 1 256 7.6% 183 74 8 1 266 7.9%13 94 61 3 0 158 4.7% 85 59 6 0 150 4.4%15 69 10 0 0 79 2.3% 73 13 0 0 86 2.5%16 49 11 0 0 60 1.8% 54 12 0 0 66 1.9%19 49 27 4 1 81 2.4% 46 26 4 1 77 2.3%23 61 10 0 0 71 2.1% 63 10 1 0 74 2.2%24 31 26 2 0 59 1.7% 31 27 3 0 61 1.8%25 40 2 0 0 42 1.2% 43 3 0 0 46 1.4%27 27 3 1 0 31 0.9% 29 3 1 0 33 1.0%28 24 6 1 0 31 0.9% 24 7 1 0 32 0.9%30 32 4 1 0 37 1.1% 33 4 1 0 38 1.1%31 43 4 0 0 47 1.4% 46 4 0 0 50 1.5%32 20 10 1 0 31 0.9% 20 11 1 0 32 0.9%33 20 9 0 0 29 0.9% 22 9 0 0 31 0.9%36 17 3 0 0 20 0.6% 18 3 0 0 21 0.6%37 14 0 0 0 14 0.4% 15 0 0 0 15 0.4%38 15 1 0 0 16 0.5% 15 1 0 0 16 0.5%40 16 0 0 1 17 0.5% 16 1 0 1 18 0.5%41 11 4 1 0 16 0.5% 12 4 1 0 17 0.5%44 3 0 0 0 3 0.1% 3 0 0 0 3 0.1%Bytrait 993 162 8 1 1164 34.4% 1014 167 11 1 1193 35.2%

FigureS1.Autocorrelationstatistics.Distributionsofautocorrelationvaluesareshownfor(A)mutationalspecificvariances,(B)residualspecificvariancesand(C)PCFheritabilities.Asinglespecificmutationalvarianceexceededournominalthresholdof0.2.

vm_ac

Freq

uenc

y

−0.1 0.0 0.1 0.2 0.3 0.4

050

010

0015

00A

vr_ac

Freq

uenc

y

−0.10 0.00 0.100

200

400

600

800 B

Autocorrelation value

Freq

uenc

y

−0.10 0.00 0.10 0.20

05

1015

20

CFr

eque

ncy

Autocorrelation value

FigureS2.BoxplotofPCFtraitloadingautocorrelations.Grey-filled(unfilled)boxesindicateautocorrelationsforloadingsonheritable(non-heritable)PCFs.TraceplotsoftraitloadingswiththehighestautocorrelationswithinPCFsarepresentedinFig.S4.

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1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45

0.0

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PCF sorted by phenotypic variance

Auto

corre

latio

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FigureS3.TraceplotsofhighlyautocorrelatedPCFtraitloadings.Presentedareupto10traitloadingsperPCF,forthesubsetofPCFswithatleastonetraitloadingautocorrelationexceedingournominalthresholdof0.2.NumbersincornersofpanelsindicatethePCFnumber(inorderofphenotypicvariance).

range(1:nBayes)

ylim

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Posterior sample

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FigureS4.Theoreticalphenotypicdimensionalitydetectionlimit.Thecolumnsofthe82x16925datamatrixwerestandardised,thenthedatamatrixwasreducedtoan82x3385matrixofaveragegeneexpressionmeasurementsacrossthefiveprobespergeneexpressiontrait.Thedatawasthencorrectedforthesegregatinggeneticvariantandthecorrelationmatrixofthedatathustransformedwascalculated.TheMarcenko-Pasturlawdescribesalimitforthenumberofdimensionsofasamplecorrelationmatrixthathaveahighprobabilityofdetection.Giventhenumberoftraits,p,andthesamplesize,m,thenumberofidentifiabledimensionsisnotlikelytobelessthank,wherePQisthesmallest

eigenvalueofthesamplecorrelationmatrixgreaterthanP1RS = 1 + V W(NadakuditiandEdelman2008).Inourcase37ofthe81non-zeroeigenvalues(filledcircles)ofthesamplecorrelationmatrixwereabovetheP1RSvalueof7.4(dashedline)form=82andp=3385.Consistentwiththisminimum,theBSFmodelidentified45PCFs.

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0 20 40 60 80

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FigureS5.Comparisonofthemutationalvarianceestimatedunderrestrictedmaximumlikelihood(REML)andBayesianSparseFactor(BSF)frameworks.TheBSFestimatesoftotal(common+specific)mutationalvarianceofthegeneexpressiontraits(y-axis)werehighlycorrelated(r=0.84,df=3383,p<<0.001)withtheamong-linevariancecomponentsestimatedinstandardunivariateREMLanalyses(x-axis).TheunivariateanalyseswerefirstreportedinMcGuiganetal.(2014)andhavebeenrecalculatedheretobeonthescaleenforcedbytheBSF,wherebyeachofthe16925(3385genesx5probes)setsofprobemeasurementsarestandardized.

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0.0 0.2 0.4 0.6 0.8 1.0

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REML estimate of mutational variance

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FigureS6.PosteriordistributionofPCFheritability.ForeachPCFheritability,weshowallindividualposteriorsampleswithgreycirclesandthemedianposteriorsamplewithblackcircles(closedandopencirclesforsignificantandnonsignificantPCFheritabilities,respectively).SignificantlyheritablePCFsarealsolabelledonthetopaxisforeaseofidentification.WeassignsignificancetoPCFheritabilitiesusingthelocalfalsesignrateapproachdescribedintheMethods,controllingtheaverageerrorratetoremainbelow0.01.

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PCF in order of phenotypic variance

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