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1 Pleiotropic Scaling of Gene Effects and the Cost of Complexity by Günter P. Wagner et al. Figure S1: Figure S1: schematic summary of findings. (a) Most QTL affect a relatively small number of traits ( 6) out of the 70 phenotypic traits measured suggesting that pleiotropic effects of gene substitutions are limited to parts of the phenotype rather than having universal pleiotropic effects as assumed in Fisher s geometric model 1 of pleiotropy. Universal pleiotropy would be incompatible with evolvability since the chance of improvement would approach zero if the number of traits affected by a gene substitution is large. In contrast our data suggests that each gene substitution only affects a limited section of the phenotype and may thus have a higher chance of being adaptive.(b) We found that the total effect of a QTL increases approximately linearly with the number of pleiotropic effects contradicting both the Euclidian Superposition Model as well as the Invariant Total Effects Model. The latter model implies a Cost of Complexity 2 because the effect of a gene substitution per character decreases as the number of characters increases. Furthermore it seems that per character effects increase with the number of characters affected. 1

2 Figure S2: Figure S2: frequency distribution of total, uncorrected QTL effects. 2

3 Figure S3: Figure S3: frequency distribution of weight-corrected QTL effects. We corrected the measured QTL effects for body weight to correct for the consequences of directional selection on body weight that was applied in generating the inbred mouse strains used in this study. The weight-corrected effects plausibly are a better representation of the mutational effect distribution than the raw QTL effects. This distribution is approximately log-normal with mean and standard deviation (Shapiro-Wilk s test on log-transformed data: W=0.9915; p=0.773). 3

4 Figure S4: Figure S4: regression of weight-corrected total QTL effects on the effective number of traits affected by the QTL. As in the other regressions (see main text) there is a positive relationship between total QTL effect and number of characters affected. 4

5 Supplementary Table T1: list of traits measured for the QTL mapping study. R 2 is repeatability of measurements. TRAIT DESCRIPTION R 2 Skull Traits Zygomatic length Max. length from malar to squamosal Zygomatic width Max. width across zygomatic arches Basicranial length Foramen magnum to posterior palate Cranial vault height Top of parietal to bottom of tympanic bulla Cranial vault length Dorsal length from bregma to inion Cranial vault width Right to left external auditory meatus Face height From diastema anterior to zygomatic root to height of nasale orthogonal to alveolus Face length Nasion to nasale Face width Anterior interorbital width Frontal bone length Bregma to nasion Interorbital width Minimum frontal width Palate length Anterior of incisor to posterior of palate Sacral traits Sacrum length Maximum dorsal along centrum Sacrum width Maximum dorsal Sacro-illium joint length Perpendicular to sacrum width Forelimb traits Superior spinous length 1 Coracoid to max. width point Superor spinous length 2 Max. width point to border-spine point Infraspinous length 1 Coracoid to max. width point Infraspinous length 2 Max. width point to border-spine point Scapula spine length Humerus length Maximum length Ulna length Maximum length Scapular width Maximum width Hindlimb traits Femur length Maximum length Obturator foramen Maximum length Iliac crest length Illium length Iliac crest to acetabulum Innominate length Iliac crest to most posterior ischium point Tibia length Maximum length Third metatarsal length Maximum length Pubis length Acetabulum to edge of symphosis Pubic symphosis length Ischiac-pubic ramus length

6 Ischium length Acetabulum to most posterior ischium point Calcaneous length Mandible Traits Sqrt of angular process area Sqrt of condyloid process area Sqrt of corpus area Sqrt of coronoid process area Sqrt of incisor alveolus area Sqrt of molar alveolus area Sqrt of masseteric area Vertebral traits Tail length st lumbar length 1 Centrum to neural spine st lumbar length 2 Neural spine to tranverse st lumbar length 3 Centrum to transverse st lumbar centrum height Height of centrum st lumbar length 4 Length along centrum rd caudal length 1 Centrum to neural spine rd caudal length 2 Neural spine to tranverse rd caudal length 3 Centrum to transverse rd caudal centrum height Height of centrum rd caudal length 4 Length along centrum rd thoracic length 1 Centrum to neural spine rd thoracic length 2 Neural spine to tranverse rd thoracic length 3 Centrum to transverse rd thoracic centrum height Height of centrum rd thoracic length Length along centrum th lumbar length 1 Centrum to neural spine th lumbar length 2 Neural spine to tranverse th lumbar length 3 Centrum to transverse th lumbar centrum height Height of centrum th lumbar length 4 Length along centrum th thoracic length 1 Centrum to neural spine th thoracic length 2 Neural spine to tranverse th thoracic length 3 Centrum to transverse th thoracic centrum length Height of centrum th thoracic length 4 Length along centrum Weights Weight at 10 weeks Weight at necropsy

7 Supplementary Notes: N1: scaling of QTL effects In order to calculate the total effect of a gene substitution we need to transform the measured QTL effects into dimensionless quantities. We considered two ways to do that. In quantitative genetics the most commonly used method is to divide the effects by the phenotypic standard deviation of the trait. This is considered appropriate since both, the standard deviations as well as the QTL effects are measures of variability. An alternative is to divide the QTL effect by the mean of the trait, leading to a measure of relative change. In order to decide which method to use we tested whether the scaling factor, either σ 1 or X 1, is correlated with the predictor variable N and thus could cause a regression artifact. To investigate this question we plotted the average standard deviation σ and the average mean X of the traits affected by a QTL over its pleiotropy N, i.e. the number of traits affected by the QTL: The x-axis in these diagrams are the number of characters affected N, the y-axis is σ on the left, and X on the right. The regression slope of σ over N is and that of X is We concluded that scaling with the mean value would introduce a stronger bias in the relationship between the total effect T and N and decided to only use scaling with the standard deviation. 7

8 N2: regression function for total effect over pleiotropy Since the relationship we found between total QTL effect and number of traits affected does not support either of the two models used in the theoretical literature we explored the shape of the relationship with a regression approach where the exponent of the predictor variable is a regression variable: T = kn α + c. Table: exponent of the regression equation T = kn α + c estimated from different data sets (+/- standard error). T is the uncorrected total QTL effect, N the number of traits affected by the QTL, Corrected T is the selection corrected total effect of a QTL and Neff is the effective number of traits affected by a QTL (see Note N4). All QTL N 25 T vs. N / / T vs. N eff / /-0.11 Corrected T vs. N / / Corrected T vs. N eff 1.83+/ / With the full data set the exponent varies between and 1.83 with an average of In order to assess whether the regression was unduly influenced by the small number of values with very high N, N>25, we recalculated the regression and obtained a range of exponents between and 1.48 (average = 0.913). Given the variability of these estimates we consider a linear regression model as parsimonious. 8

9 N3: assessing the influence of detection limits of QTL effects on the relationship between total effect T and pleiotropy N To assess whether our result in Fig. 1a can be explained as an artifact of the limitation to detect small effects, we calculated the predicted regression slope given the detection limit for QTL effects. In our data the smallest QTL effect that can be detected is 1/10th of the phenotypic standard deviation. Let us assume that all QTL have the same pleiotropy, say 30, the maximum we detected, and all of them have the same total effect T. Now let us consider a QTL where the effect on one character is less than the detection limit, say ε<0.1. Then we will detect 29 effects, i.e. pleiotropy is estimated to be 29 instead of 30, and the estimated total effect will be T ˆ = T 2 ε 2, which is less than the true total effect T. The slope of this regression will be T T ˆ. Since ε 2 is small we can write T ˆ = T 2 ε 2 T ε 2 2 ε and the predicted slope is. Note that the actual effect 2T 2T missed by the QTL scan is 0.1 rather than =0.1. For our data this slope is predicted to be if we take T for 30 characters to be about 1.5 (Fig. 2a) and ε =0.1. For N=15 the observed T is about 1.0 and the predicted slope in this area is The observed slope is about ten times larger than predicted by this model: 0.047± We thus conclude that the positive relationship between T and N cannot be explained by an artifact due to missing small effects. 9

10 N4: calculating the effective number of characters affected by a QTL If a QTL affects N traits, these traits tend to be correlated and thus represent effectively a smaller number of dimensions on average. In order to obtain an estimate of this effective number of traits, N eff, we propose the following measure. For each set of traits affected by a QTL we consider the phenotypic correlation matrix of these traits and calculate the eigenvalue variance Var(λ). The eigenvalue variance is a measure of the average correlation among variables. Var(λ) = 0 if all the correlations are zero, and Var(λ) = N 1 if all correlations are equal to one, and thus variation exists in only one dimension. We calculate the effective number of traits as N eff = N Var(λ) which becomes N eff =N if there are no correlations and N eff =1 if all the correlations are equal to 1, and linearly interpolates between these two extremes. 10

11 N5: the effect of error variance on the rank of phenotypic covariance matrix The number of dimensions in which a set of quantitative traits can vary is reflected in the rank of the co-variance matrix, where the rank is the number of linearily independent rows and columns of the matrix. Hence, in principle, the rank of a phenotypic co-variance matrix should reflect the dimensionality of the phenotype, i.e. the number of independent directions the phenotype can change. In practice, however, this estimation is difficult, because each trait measured will also contribute an independent component of error variance and thus lead to a covariance matrix of the measured variables that has full rank, i.e. the number of dimensions is equal to the number of traits measured because of error variance. For this reason implemented an adjusted bootstrap procedure which takes into account the measured error variance to separate error variance and variance of the underlying biological variable. This method is described in the Supplemental Methods section. 11

12 N6 Choice of distance measure to estimate the total effect of a QTL Measuring distances in a topological space can be done with a variety of distance measures. For vectors in Euclidian space two are used frequently: the Euclidian distance between two points x and y in n-dimensional space: d E = Manhattan distance d M = n i=1 x i y i n ( x i y i ) 2 i=1, or the so-called. Another popular distance measure, the Hamming distance, is designed for symbolic strings and not applicable to continuous variables. A priori there is no reason to prefer one distance measure over another, but here we chose Euclidian measures for two reasons. First, Euclidian distances have been used in the models predicting the effect of complexity on evolvability beginning with Fisher s geometric model, and thus the Euclidian measure is the most appropriate for testing these models. Second, among the two distance measures for continuous variables only the Euclidian distance is rotationally invariant, meaning that its value does not depend on the arbitrary choice of base vectors. In contrast, the Manhattan distance is not rotationally invariant. Never the less we performed the regressions with the Manhattan distance and also found strongly positive relationship between total effect and pleiotropy: The left figure gives the raw effects over the number of traits affected and the right figure gives the selection corrected trait values over the number of traits affected. In either case we still find a strongly positive relationship between total effect and pleiotropy. 12

13 References 1 R. A. Fisher ed., The Genetical Theory of Natural Selection. (Clarendon Press, Oxford, 1930). 2 H. A. Orr, Evolution; international journal of organic evolution 54 (1), (2000). 13