How does the strength of selection influence genetic correlations?

Abstract Genetic correlations between traits can strongly impact evolutionary responses to selection, and may thus impose constraints on adaptation. Theoretical and empirical work has made it clear that without strong linkage and with random mating, genetic correlations at evolutionary equilibrium result from an interplay of correlated pleiotropic effects of mutations, and correlational selection favoring combinations of trait values. However, it is not entirely clear how change in the overall strength of stabilizing selection across traits (breadth of the fitness peak, given its shape) influences this compromise between mutation and selection effects on genetic correlation. Here, we show that the answer to this question crucially depends on the intensity of genetic drift. In large, effectively infinite populations, genetic correlations are unaffected by the strength of selection, regardless of whether the genetic architecture involves common small‐effect mutations (Gaussian regime), or rare large‐effect mutations (House‐of‐Cards regime). In contrast in finite populations, the strength of selection does affect genetic correlations, by shifting the balance from drift‐dominated to selection‐dominated evolutionary dynamics. The transition between these domains depends on mutation parameters to some extent, but with a similar dependence of genetic correlation on the strength of selection. Our results are particularly relevant for understanding how senescence shapes patterns of genetic correlations across ages, and genetic constraints on adaptation during colonization of novel habitats.

relates to the reduction of genetic variance of a quantitative trait caused by drift, we need to nd the probability P that two randomly chosen gametes in the ospring come from the same parent (following e.g. Crow and Kimura 1970, p345). For any given parent i leaving k i ospring, this equals k i (k i −1) 2N (2N −1) , which is then summed over parents to yield 1 The expected probability averaged over the sampling process is then where tildes again denote averages over the distribution of environmental values in the population. Then the inbreeding eective population size is simply where CV denotes a coecient of variation over the distribution of environmental values, and w = W/ W is relative tness conditional on breeding value. Note that the formula in equation (S3) (which appeared previously in e.g. Santiago and Caballero 1995) diers from more common ones based on the variance in reproductive success V ar(k) (e.g. Crow and Kimura 1970, p345-352), because we here focus on the variance in relative tness w, which determines the expected (rather than realized) number of ospring with a given phenotype (as claried by Robertson 1961). Note also that we do not allow self-fertilization in our simulations, which should increment N e by approximately half an individual (Wright 1969, p.195), but we neglect this for simplicity. With the tness function in equation (2) the mean tness of individuals with breeding value x is:W and the mean squared relative tness (conditional on breeding value) is which can be rearranged as In the simplest case where the mean breeding value is at the optimum and genetic variation is small (conditions provided below), then the exponential in equation (S6) equals 1, and we have The univariate version for a single trait is where V e is the environmental variance of the trait and ω the width of the tness peak.
However, when the distribution of breeding values is suciently broad (large genetic variance for at least some of the traits), then the eect of environmental variance on genetic drift should be averaged over the distribution of breeding values. Integrating equation (S6) over the distribution of breeding values x with mean at the optimum, and taking the reciprocal, we get where Φ is the matrix in the exponential of equation (S6) and G is the genetic covariance matrix. This shows that the inuence of genetic variation on how environmental variation aects the eective population size can be neglected as long as 2ΦG is small relative to I, in which cases equation (S6) collapses to equation (S7). For a single trait, this condition can be shown to be equivalent to 2V g 3ω 2 (with V g the additive genetic variance), that is, weak stabilizing selection on genetic variance.
To check the validity of equation (S7), we ran simple simulations with selection on nonheritable phenotypic variation, and drift at a neutral bi-allelic locus. We initiated the population by randomly drawing each gene copy of N diploid individuals with equal probability 1/2 between two alleles A and B. We then recorded the initial frequency p 0 of allele A (which may slightly dier from 1/2 because of random sampling), and the initial heterozygosity . We then iterated 10000 times the following process: 1. Randomly draw a multivariate residual component of variation e for each individual, with mean 0 and covariance matrix E.
2. Compute the tness of each individual using equation (2) with the optimum set at 0.
3. Sample N pairs of mating parents with repeat (allowing for selng), with weights given by their tnesses.
4. Randomly draw one allele from each parent (segregation), to create the diploid genotype of each individual in the ospring generation.
From this we could compute both the variance eective population size and the inbreeding eective population size where the expectations and variances are taken over the 10000 simulations (Crow and Kimura 1970). The comparison between these simulated results and the predictions from equation (S7) is shown in Figure S1.
In addition, we also ran multi-locus individual-based simulations as in the main text, except that we did not correct the population size N for the inuence of selection on environmental variation, causing N e to change with the strength of selection. Results are shown in Figures S2 and S3.  Figure S1: Inuence of selection on environmental variation of phenotypes on the effective population size. Open circles show variance eective population size (eq. (S10)) and crosses show inbreeding eective population sizes (eq. (S11)), both computed from 10000 simulations of drift at a neutral bi-allelic locus, in a population of size N = 300. The lines show predictions from equation (S7). The selection matrix was Ω = ω 2 Ω ρ , with Ω ρ = 1 0.8 and ω = 2, 3, 5, 7, 10 from black to light gray. The environmental covariance matrix was E = V e I, with V e = 0.001, 1, 2, 5, 10, 15, 20.  B Steps for solving the mutation-selection-drift equilibrium By building on the Gaussian approximation of the continuum-of-allele model (Kimura 1965) and assuming innite-sized population with non-overlapping generations, Lande (1980) derived a simple expression for genetic covariance matrix equilibrium (G) selection-mutation balance. We here extend this result to allow for random genetic drift. At equilibrium, the production of genetic variance due to new polygenic mutations is balanced by the loss of genetic variance to due both stabilizing selection and random drift (Lande 1979(Lande , 1980. For a single haploid locus we obtain : where G denotes an expectation over the stochastic evolutionary process (because of random genetic drift).
By dening V − 1 2 GV − 1 2 = X and V − 1 2 UV − 1 2 = −C, we nally have to solve the quadratic matrix equation : 0 = X 2 + BX + C A quadratic matrix equation can be solved explicitly if the following requirements are met: (i) X 2 is preceded by an identity matrix, (ii) B commutes with C, and (iii) B 2 − 4C has a square root. The solution of the quadratic matrix equation is then : In our case, an explicit solution for X exists. Indeed, X 2 is preceded by an identity matrix, B is a diagonal matrix and then always commute with C. Finally, as B 2 and C are both positive semi-denite then (B 2 − 4C) 1/2 will always have a solution. Recall that Finally by using notations (1) and (4), and summing over diploid loci (neglecting linkage disequilibrium), we obtain equation (9) C Constraints on the orientation and shape of the G matrix in the special cases where V and M matrices have the same eigenvectors C.1 Case without drift Starting from equation (8) and assuming that both matrices V ρ and M ρ have the same eigenvectors Q but dierent eigenvalues, their spectral decompositions give V ρ = QΛ s Q −1 and M ρ = QΛ m Q −1 respectively, where Λ w and Λ m are diagonal matrices of eigenvalues. Then, equation (8) can be simplied to 2 Q −1 This shows that G matrix has the same eigenvectors Q as the V and M matrices, and that its eigenvalues are the geometric means of eigenvalues of V and M.

C.2 Case with drift
Starting from equation (9) and performing a spectral decomposition of V ρ and M ρ as in the case without drift (see above), after simplication we obtain : While the eigenvalues of G are given by the equation located between the highest level of brackets, the eigenvectors of G still equal Q the same as V and M matrices.

E Inuence of the number of loci on genetic correlation
We have assumed in our analysis that linkage disequilibrium between loci does not contribute to genetic correlations between traits. As shown by Lande (1980), this assumption is generally reasonable for weak selection and unlinked loci as modeled here, and in previous theory on the topic (e.g. Jones et al. 2003). However under strong stabilizing selection, and as the number of loci becomes large, genetic covariation among loci can bias genetic correlations away from our analytical predictions above, even without linkage.
The true additive genetic covariance matrix is where G ii is the additive genetic covariance matrix at locus i, and G ij is the additive genetic cross-covariance matrix between loci i and j. The rst sum is generally described as the genic covariance matrix G g (Walsh and Lynch 2018, p.550), while the second sum G LD reects the inuence of linkage disequilibrium between loci. For equivalent loci, we further have G g = nG w and G LD = n(n − 1)G b , where G w and G b are respectively the within-locus and between-locus additive genetic covariance matrices. Our analytical results in the main text have only considered the genic covariance matrix G g , in eect neglecting linkage disequilibrium, in line with most previous theory (Bulmer 1989;Burger et al. 1989;Jones et al. 2003). But in reality, stabilizing selection produces negative linkage disequilibrium between loci, causing G LD to be negative denite. This results in hidden genetic variance, reducing the additive genetic variance of all traits below the genic variance in G g (Bulmer 1974;Lande 1980). When multiple traits are under selection, this so-called "Bulmer eect" also biases genetic correlations between traits. While each individual between-locus covariance G b is very small when loci are unlinked, these covariances are n − 1 times more numerous than within-locus covariance matrices G w , so the contribution of G LD to G can become non-negligible when n is large (Walsh and Lynch 2018). The magnitude of this eect can be quantied by which represents a multivariate equivalent to the proportional reduction of genetic variance caused by hiding in linkage disequilibrium. From (Lande 1980, eq. 17), the equilibrium between-locus additive genetic covariance matrix for equivalent and unlinked loci is The minus sign indicates that this eect overall reduces genetic variance, as expected. For a given G w (approximately determined by the mutation-selection-drift balance derived above), the magnitude of this reduction increases with the number of loci n, and with strength of selection V −1 s . We thus expect our analytical predictions for G and ρ G to become less accurate under strong selection and with many loci, because of the increasing inuence of stabilizing selection on linkage disequilibrium. Our simulations show that this is indeed the case: as the number of loci increases, the proportion of hidden genetic variance increases as predicted by equation (S14) (Fig. S6), and the inuence of the strength of selection V −1 s on genetic correlations is reduced (Fig. S7, Fig. S8). Figure S6: Inuence of the number of loci on hidden genetic variance. (a-d) The relative decrease in genetic variance caused by linkage disequilibrium between loci, measured as the mean of the diagonal elements of matrix R, is plotted against the width of the tness peak V s , for dierent eective sizes N e . Points correspond to simulation results of individual-based model (IBM), where R was estimated as in equation S13 (rightmost member), with G g computed as the sum of locus-specic covariance matrices G ii . Lines represents the analytical expectation obtained by replacing G in equation (S14) by its expectation at mutation-selection-drift equilibrium without linkage disequilibrium (eq. (9)). The parameter values for selection and mutation, corresponding to the Gaussian mutation regime, are the same as in Figure 1. Figure S7: Inuence of the number of loci on genetic correlations at mutation-selectiondrift equilibrium. (a-d) The genetic correlation is plotted against the width of the tness peak V s , for dierent eective sizes N e . The parameters values for selection and mutation, corresponding to the Gaussian mutation regime, are the same as in Figure 1. The number of loci ranges from 5 to 100. Note that as the number of loci becomes larger, genetic correlations change less with V s . Figure S8: Inuence of the number of loci on genetic correlations at mutation-selectiondrift equilibrium (Gaussian regimes, permuted correlations). All parameters are the same as in Figure S7, but with permuted correlations, ρ m = 0.8 and ρ s = −0.7. This illustrates that a larger number of loci causes ρ g to change towards ρ m , rather than become smaller.