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1 Sulemental Information Anthony J. Greenberg, Sean R. Hacett, Lawrence G. Harshman and Andrew G. Clar Table of Contents Table S1 2 Table S2 3 Table S3 4 Figure S1 5 Figure S2 6 Figure S3 7 Figure S4 8 Text 1 (Model Descrition) 9 Text 2 (Simulation Results) 15 Data set 1 Searate file 1

2 Sulemental Information Greenberg et al. Page S2 Sulemental Table S1 Variances for each level of the data hierarchy. Numbers in arentheses reresent 95% credible intervals. relicate bloc sca gca wts 9.8 (7.9, 12.1) (14, 26) 12 (07, 17) 76 (53, 0.11) ~ 47 (39, 59) 1 (0.34, 0.51) 1.9 (0.57, 6.9) (0.19, 2) TRI 52 (44, 62) 1 (04, 19) 2.4 (8, 6.2) (29, 65) ~ 51 (43, 61) 12 (04, 26) 8.6 (3.5, 22.0) (28, 64) GLG (6, 1.3) (, 1.8) (6.3, 1) (3.2, 6.8) 10 4 ~ 8.8 (7.2, 1) (0.9, 1.6) (5.5, 9.7) (3.6, 7.4) 10 4 tprt 1.9 (1.6, 2.3) (2.9, 7.2) (1.1, 2.4) (5.4, 12.0) 10 4 ~ 1.6 (1.4, 2.0) (3.0, 8.0) (1.1, 2.4) (4.7, 1) 10 4 mprt 8.4 (7.0, 1) (0.77, 2.5) (2.5, 6.7) (1.5, 3.6) 10 3 ~ 3.9 (3.3, 4.6) (3.8, 13.0) (1.8, 6.0) (1.6, 4.0) 10 3 ADH 43 (34, 53) 13 (05, 23) 3.6 (0.96, 8.6) (81, 0.17) ~ 0.11 (88, 0.13) 8.6 (2.9, 22.0) (7, 5.5) (31, 74) ME 0.12 (98, 0.15) 21 (07, 43) 4.9 (1.5, 1) (38, 87) ~ (39, 56) 6.8 (2.7, 15.0) (3.6, 25.0) (33, 73) cmdh 2.3 (1.8, 2.9) (1.6, 3.0) (0.77, 1.5) (4.5, 9.8) 10 4 ~ 6.1 (4.9, 7.7) (3.8, 1) (1.1, 2.6) (, 2.2) 10 3 FAS 0.32 (7, 0.39) 0.11 (5, 0.19) 16 (01, 58) 6 (0.17, 0.39) ~ 0.34 (8, 1) 48 (12, 0.120) 2.6 (0.56, 15.0) (0.15, 0.36) PGI (97, 0.140) 15 (06, 28) 3.9 (0.97, 9.9) (87, ) ~ 68 (55, 84) 13 (05, 27) 1.6 (5, 5.8) (59, 0.14) PFK 85 (71, 0.100) 11 (04, 22) 1.4 (0.50, 3.7) (27, 66) ~ 61 (50, 75) 28 (15, 51) 1.2 (0, 3.4) (29, 70) G6PD 7 (0.74, 0) 37 (97, 0.100) 7.2 (1.3, 25.0) (0, 8) ~ 2 (0.35, 0.50) 33 (08, 91) 2.9 (2, 12.0) (0.16, 0.35) SPGD 82 (60, 0.130) 30 (15, 59) 2.5 (7, 5.8) (29, 70) ~ 39 (31, 49) 11 (05, 21) 1.1 (1, 2.9) (23, 54) PGM 81 (66, 98) 33 (19, 53) 2.8 (9, 6.7) (0.13, 7) ~ 41 (34, 49) 13 (06, 22) 1.4 (3, 4.5) (88, 0.18) GP 0 (6, 1.20) 39 (09, 0.130) 5.5 (0.98, 23.0) (0.35, 3) ~ 1.50 (1.30, 1.70) 71 (10, 70) 1.7 (4, 1) (0.38, 0.90) GS 3.8 (3.0, 4.9) (4.3, 6.9) (3.4, 5.5) (1.8, 3.6) 10 4 ~ 4.3 (3.5, 5.4) (4.6, 8.9) (3.1, 5.3) (1.7, 3.4) 10 4 TRE 3.7 (3.0, 4.6) (0.71, 2.1) (2.3, 6.3) (1.9, 4.4) 10 3 ~ 2.1 (1.7, 2.6) (4.6, 14.0) 2.4 (1.5, 3.9) (1.2, 2.8) 10 3 HEX 96 (79, 0.12) 24 (11, 42) 2.0 (3, 5.2) (38, 90) ~ 49 (40, 60) 15 (06, 28) 1.6 (7, 4.5) (30, 69) GPDH 17 (13, 22) 8.5 (4.4, 14.0) (2.1, 7.1) (18, 42) ~ 13 (09, 17) 3.9 (1.7, 7.8) (8, 4.3) (30, 64) GPO 7 (0.57, 0.79) 20 (05, 57) 4.8 (0.99, 16.0) (48, 0.130) ~ 0.12 (99, 0.14) 18 (06, 45) 1.3 (3, 4.2) (30, 76) PDH 24 (21, 29) 2.7 (1.2, 5.0) (4.6, 13.0) (4.5, 1) 10 3 ~ 10 (08, 12) 3.8 (1.8, 7.5) (2.3, 8.3) (6.1, 13.0) 10 3 FUM 70 (60, 83) 5.0 (2.0, 1) (3.2, 12.0) (3.3, 8.9) 10 3 ~ 8.3 (7.0, 1) (5, 3.6) (2.0, 8.2) (3.8, 9.5) 10 3 mmdh 63 (54, 74) 5.6 (2.5, 1) (2, 3.0) (09, 22) ~ 18 (15, 22) 3.8 (1.5, 8.6) (3.2, 15.0) (5.6, 14.0) 10 3 SDH 1.5 (1.3, 1.8) (1.5, 3.5) (6.6, 12.0) (2.6, 5.7) 10 4 ~ 1.8 (1.4, 2.2) (0.99, 2.0) (4.4, 7.9) (1.9, 3.9) 10 4

3 Sulemental Information Greenberg et al. Page S3 Sulemental Table S2 Genetic correlations. Values for males are above the diagonal, for females below. The middle number (in bold) is the median of the osterior distribution. The numbers above and below it delineate the 95% credible interval. weight TRI GLG tprt mprt ADH ME cmdh FAS PGI PFK G6PD 6PGD PGM GP GS TRE HEX GPDH GPO PDH FUM mmdh SDH weight TRI GLG tprt mprt ADH ME cmdh FAS PGI PFK G6PD PGD PGM GP GS TRE HEX GPDH GPO PDH FUM mmdh SDH

4 Sulemental Information Greenberg et al. Page S4 Sulemental Table S3 Environmental correlations. The table is arranged as Table S2. weight TRI GLG tprt mprt ADH ME cmdh FAS PGI PFK G6PD 6PGD PGM GP GS TRE HEX GPDH GPO PDH FUM mmdh SDH weight TRI GLG tprt mprt ADH ME cmdh FAS PGI PFK G6PD PGD PGM GP GS TRE HEX GPDH GPO PDH FUM mmdh SDH

5 Sulemental Information Greenberg et al. Page S5 matrix 1 matrix 2 (1) (2) matrix 2 matrix null distribution data values (3) (4) correlation values correlation values Sulemental Figure S1 Schematic reresentation of our matrix comarison methods. A detailed descrition is in the Methods section of the main text. In ste (1), we samle two matrices from their resective distributions. We use these samles to erform our randomsewers tests. (2) We line u the vectorized uer triangles of the matrices. At this oint, we estimate element-wise matrix correlations, comaring them to random ermutations of each vector. (3) We construct null distributions for the matrices by ermuting underlying data to destroy data-driven correlations. Any correlations that reflect artifacts of exerimental design should remain. We then assign correlation resence or absence robabilistically with one minus the robability that a given correlation comes from the null distribution. We note each correlation s sign. (4) We align the resence-absence calls as in Ste 2, and note correlations that are resent in both matrices and those that switch signs. The random exectation is calculated after randomly ermuting each vector. We reeat the whole rocess for each samle from correlation matrix distributions.

6 Sulemental Information Greenberg et al. Page S all genetic all environmental Δ degree Δ degree negative genetic tprt mprt TRI GLG PGI GPDH cmdh G6PD GPO FUM mmdh ADH ME FAS SPGD PGM PDH PFK GP TRE GS HEX SDH GLG TRI tprt mprt cmdh PGM PDH mmdh TRE HEX FUM SDH ADH ME PFK G6PD GS GPO FAS PGI GPDH SPGD GP negative environmental ositive genetic ositive environmental Δ degree Sulemental Figure S2 Degree distributions of the genetic and environmental correlation networs, corrected for relationshis with weight. The figure is laid out the same as Figure 4.

7 Sulemental Information Greenberg et al. Page S7 genetic r r 0 0 environmental N I T B Sulemental Figure S3 Posterior robabilities of the retention of correlation resence and sign between oulation-secific and overall correlation matrices. We estimated genetic (left column) and environmental (right column) correlation matrices within each oulation and robabilistically identified resence and sign of each correlation (see Methods). We then comared these with similarly identified correlations for all lines. Bar heights reresent osterior robabilities that a given correlation is resent and of the same sign in both a given oulation (rows) and in the overall correlation matrix. Color of the bars is roortional to the absolute values of osterior estimates of the correlations. The gray bars across grahs indicate the range of osterior robabilities exected for unrelated matrices. Z

8 Sulemental Information Greenberg et al. Page S8 genetic environmental r r 0 0 N I T B Sulemental Figure S4 Posterior robabilities of correlation sign switch between oulation-secific and overall correlation matrices. The figure is organized the same as Figure S3 Z

9 Sulemental Text 1 Model Descrition In this section we list the distributions underlying our model. We also detail the MCMC samling scheme we used to aroximate these distributions. Univariate versions of this model are detailed in a revious ublication (Greenberg et al., 2010), where we rovide examles of R scrits to imlement the algorithms. The model is based on ideas discussed by Gelman et al. (2004). Our samling scheme from multivariate t distributions is a trivial extension of the data augmentation method described by van Dy and Meng (2001). Indexes: j[i] index j for a higher-level variable that corresonds to the value i of a lowerlevel variable. For examle, µ cross [j] is the the value µ cross that corresonds to the value µ bl j. l all values for the lower-level variable that belong to a grou defined by the corresonding higher-level variable. For examle, j µbl j refers to the sum of all bloc relicates for cross. l[] and l[] line indexes for the female and male arents of cross. c cross tye indicator. d dimension index; equal to the number of enzyme and hysiological variables. Symbols: V max,i : Sloe (enzyme maximal rate) of the inetic curve for relicate i. σ 2 V m,i : Variance of V max for relicate i. 9

10 Sulemental Text 1 Greenberg et al. Page S10 N ( ) : Total number of relicates, blocs, etc. I : Unit diagonal matrix. ν ( ) : Degrees of freedom for the Student-t distributions. df ( ) : Degrees of freedom for Wishart distributions. β ( ) : Matrices of late and cross effect coefficients. X ( ) : Matrices of cross and late effect redictors. The redictors do not have an intercet, to imlement variable-intercet regressions (Gelman and Hill, 2007). We estimate late effects searately within each cross tye. For late effects, we use as many redictors as there are lates. That results in an over-determined model. We monitor only the re-normalized late regression coefficients (see the Samling section). This method seeds conversion and imroves mixing (chater 19 of Gelman and Hill, 2007). x ( ) i : ith row vectors of X. µ ( ) : Column vectors of bloc, cross, line and oulation means Σ ( ) : Relicate, bloc, sca and gca covariance matrices Posterior distributions: The distributions are listed as var rior lielihood. To streamline notation, neither deendencies nor flat imroer riors are listed. V max,i MVt ν re,d(µ bl j[i] + xl i βl ; Σ re c[i] ) N( V max,i ; σ 2 V m,i), Σ re c [ Inv-W ν0 (Λ 0 ) Inv-W df re [c],d q i Inv-χ 2 ν re +d ( [V max,i µ bl i c j[i] ]T [Σ re c i = 1,..., N; c[i] cross tye indicator ] 1 q i (V max,i µ bl j[i] )(V max,i µ bl j[i] )T ) ] 1 [V max,i µ bl j[i] ] [ β l N (Xl T X l) 1 Xl T (V max,i µ bl j[i] ); diag ( [Xl T X l] 1) ] σ 2 l ( [ ( ( ]) σ 2 l, c Inv-χ2 (N n l ) diag V max,i µ bl j[i] xl i βl) V max,i µ bl j[i] xl i βl) T i c µ bl j MVN d (µ cross [j] ; Σ bl c[j] ) MVN d(v max,j ; Σ re c[j] ), j = 1,..., N bl

11 Sulemental Text 1 Greenberg et al. Page S11 µ cross Σ bl c = µ sca + x β, β 1 inbreeding; β 2 outcrossing Inv-W ν0 (Λ 0 ) Inv-W df bl [c],d j c(µ bl j µ cross [j] )(µ bl j µ cross [j] ) T 1 β MVN d [ (X T X) 1 X T (µ cross µ ln ); diag ( [X T X] 1) Σ β] [ Σ β Inv-W ν0 (Λ 0 ) Inv-W df cross 1,d (µ cross µ ln x β)(µ cross µ ln ] 1 x β) T µ sca MVN d (µ ln ; Σsca ) MVN d (µ bl ; Σbl c[] ), = 1,..., N cross µ ln µ ln = l[] + µ ln l[], l = 1,..., N lines 2 [ Σ sca Inv-W ν0 (Λ 0 ) Inv-W df cross,d (µ sca ] 1 µ ln )(µsca µ ln )T µ ln l MVN d (µ o [l] ; Σgca ) MVN d (µ sca [l] ; Σsca ), = 1,..., N o [ ] 1 Σ gca Inv-W ν0 (Λ 0 ) Inv-W df ln,d (µ ln l µ o [l] )(µln l µ o [l] )T l µ o t 3 (µ; Iσ) σ Inv-χ 2 (N o 1) ( [µ o µ] 2 ) Samling: Our Gibbs samler goes through the stes described below, udating each arameter given all the others. Samling from the normal and χ 2 distributions is standard. R acages are available to samle Wishart and multivariate normal random variables. However, we imlemented our own functions based on well-established algorithms (Odell and Feiveson, 1966; Gelman et al., 2004).

12 Sulemental Text 1 Greenberg et al. Page S12 Note: denotes samle from. V max,i N 1 Iσ 2 Vm,i σ 2 V m i σ 2 V m i/χ 2 ν sloe i V max,i + 1 diag(σ re 1 Iσ 2 Vm,i c[i] )(µbl j[i] + xl i βl ) + 1 diag(σ re c[i] ) ; Iσ 2 Vm,i diag(σ re c[i] ) [ β l raw N (Xl T X l) 1 Xl T (V max,i µ bl j[i] ); diag ( [Xl T X l] 1) ] σ 2 l β l = β l raw (β l raw) c (re-normalization of an over-determined model to seed convergence) [ ( σ 2 l, c (diag ( ]) V max,i µ bl j[i] xl i βl) V max,i µ bl j[i] xl i βl) T /χ 2 (N n l ) µ bl j i c [ MVN d i j q i αc[j] 2 ] +(Σ bl c[j] ) 1 µ cross [j] ; (Σ re c[j] ) 1 + (Σ bl c[j] ) 1 [ i j q i αc[j] 2 ] 1 (α c[j] 2 Σre c[j] ) 1 q i (V max,i x l i βl )+ i j ] 1 (Σ re c[j] ) 1 + (Σ bl c[j] ) 1 [ ] q i αc[i] 2 χ2 (ν / re+d) (V max,i µ bl j[i] xl i βl ) T (Σ re c[i] ) 1 (V max,i µ bl j[i] xl i βl ) + ν re ( ) αc 2 ν re q i /χ 2 (ν redf re c ) Redundant arameter for multivariate Student-t samling i c (van Dy and Meng, 2001). ([ (Σ re c ) 1 αc 2 ν 0 Λ 0 W (ν0 +df re c ) ν 0 + df re + c + dfre c i c q i(v max,i µ bl j[i] xl i βl )(V max,i µ bl j[i] xl i βl ) T ] 1 ν 0 + df re c µ cross ( [ MVN d n (Σ bl c[] ) 1 + (Σ sca ) 1] 1 [ ] n (Σ bl c[] ) 1 µ bl + (Σsca ) 1 (µ ln + x β) ; [ n (Σ bl c[] ) 1 + (Σ sca ) 1] ) 1 µ ln µ ln = l[] + µ ln l[] 2 [ ν0 Λ 0 + df bl c ( Σ bl c ) 1 Wν0 +df bl c j c (µbl j µcross [j] ν 0 + df bl c )(µ bl j µcross [j] ) T ] 1

13 Sulemental Text 1 Greenberg et al. Page S13 [ (X β MVN T d X ) [ 1 (X X T (µ cross µ ln ); diag T X ) ] 1 Σ β] [ ( Σ β) 1 W(ν0 ν 0 Λ 0 + df β ] (µcross x β µ ln )T (µ cross x β µ ln ) 1 +df β ) ν 0 + df β µ ln l µ sca l ( [nl MVN d (Σ sca ) 1 + (Σ gca ) 1] [ 1 n l (Σ sca ) 1 µ sca l [ nl (Σ sca ) 1 + (Σ gca ) 1] ) 1 = [ ] (µ cross x β)(1 + δ h ) δ h (δµ l[] + (1 δ)µ l[]) /n l l δ = 0 if line l is the male line in the cross and 1 otherwise δ h = 0 if the cross is homozygous and 1 otherwise ( [ν0 Λ 0 + df cross (µcross (Σ sca ) 1 W (ν0 +df cross ) µ o N ( [ n diag[(σ gca ) 1 ] + 1 V [ n diag[(σ gca ) 1 ] + 1 V ] + (Σ gca ) 1 µ o [l] ; x β µ ln )(µcross ν 0 + df cross ] 1 [ n diag[(σ gca ) 1 ]µ ln + 1 ] µ ; V ] ) 1 [ ν0 (Σ gca ) 1 Λ 0 + df ln l W (µln l µ o [l] )(µln l µ o ] 1 [l] )T (ν0 +df ln ) ν 0 + df ln µ N µ o V 1 V ; 1 1 V V ( νσ 2 + (µ o µ) 2) /χ 2 ν+1 ( ) σ 2 N o ν ν 1 G + 1; 2 2 V x β µ ln )T ] 1 ) References van Dy DA, Meng XL (2001) The art of data augmentation. J Comut Grah Stat 10: 1 50

14 Sulemental Text 1 Greenberg et al. Page S14 Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian Data Analysis. London: CRC Press, 2nd edition Gelman A, Hill J (2007) Data analysis using regression and multilevel/hierarchical models. Cambridge, UK: Cambridge University Press Greenberg AJ, Hacett SR, Harshman LG, Clar AG (2010) A hierarchical Bayesian model for a novel sarse artial diallel crossing design. Genetics 185: Odell PL, Feiveson AH (1966) A numerical rocedure to generate a samle covariance matrix. J Am Satatist Ass 61:

15 Sulemental Text 2 Simulation Results The scheme we used to generate simulated data is a multivariate version of the one we ublished reviously (Greenberg et al., 2010). We started with oulation means equal to osterior most liely values estimated from the male ortion of our real data set. We then generated line means by samling from multivariate normal distributions with resective oulation means. As a covariance matrix we used the osterior most liely estimate of the GCA covariance matrix. We generated the same number of lines er oulation as we have in our real data set. Using these line values, we calculated mid-arent means for the crosses matching the scheme we followed for our real data collection. We used these values to samle cross means from multivariate normals with the SCA covariance matrix estimated from the data. Bloc and relicate means were generated in this hierarchical fashion, matching the samle size we obtained in our data, but with no missing data. Covariance matrices were generated by combining the GCA correlation matrix and variances estimated from the data for each level in the hierarchy. Underlying true environmental and genetic correlations were thus the same in our simulated data sets. We multilied the enzyme relicate means we generated by the simulated relicate values for weight and aired the resulting enzyme rates with standard deviations and degrees of freedom from real data. We then changed 48 of the oints to outliers (a number aroximately 2-fold higher than what we see in our data) by multilying their values by values samled from a uniform distribution between 5 and 10. This is the same rocedure we used reviously (Greenberg et al., 2010). We generated 100 simulated sets, each matching the real data set in structure and in the number of data oints. We analyzed each simulated data set using the same model we used for real data. We focused on the two environmental (relicate and bloc) covariance matrices and the matrix of genetic effects. Given that the correlation comonents of these three matrices were simulated to be the same, we wanted to now if the statistics we used to test their similarity would cature this feature. First, we looed at the element-wise correlations between airs of these matrices. For each simulated data set, we calculated distributions of these statistics. Grey boxes in Figure S2-1 show the range of medians of these distributions among simulated data sets. We 15

16 Sulemental Text 2 Greenberg et al. Page S16 also saved lower bounds of the element-wise correlation distributions for each simulation. White boxes in Figure S2-1 show the range of these values. Our results indicate that the estimated relicate correlation matrix is close to the genetic one, while the bloc matrix is not. This is liely because the bloc correlation matrix is not well estimated. Indeed, if we comare each matrix to the true value used in our simulations, we see that the relicate and genetic correlation matrices strongly resemble the true values, while the bloc matrix does not (Figure S2-2). Using the random sewer method of matrix comarison (see main text), we arrive at the same conclusion (Figure S2-3; here the smaller value of the angle indicates greater similarity) lower bound median correlation GCA:BL GCA:REP REP:BL Figure S2-1: Element-wise correlations between matrices estimated from simulated data. The boxes reresent ranges of values from the 100 simulations.

17 Sulemental Text 2 Greenberg et al. Page S17 correlation GCA BL REP Figure S2-2: Element-wise correlations between matrices estimated from simulated data and the true correlation matrix. The boxes reflect the range of medians of osterior distribution of the correlation statistic across simulations.

18 Sulemental Text 2 Greenberg et al. Page S angle REP:BL GCA:BL REP:GCA Figure S2-3: Random-sewer angles (in degrees) between matrices estimated from simulated data.

19 Sulemental Text 2 Greenberg et al. Page S19 References Greenberg AJ, Hacett SR, Harshman LG, Clar AG (2010) A hierarchical Bayesian model for a novel sarse artial diallel crossing design. Genetics 185: