Developmental Associations Between Traits: Covariance and Beyond

Sean H. Rice1 Department of Ecology and Evolutionary Biology, Osborn Memorial Labs, Yale University, New Haven, Connecticut 06520 Manuscript received December 17, 2002 Accepted for publication September 11, 2003

ABSTRACT Statistical associations between phenotypic traits often result from shared developmental processes and include both covariation between the trait values and more complex associations between higher moments of the joint distribution of traits. In this article, an analytical technique for calculating the covariance between traits is presented on the basis of (1) the distribution of underlying genetic and environmental variation that jointly influences the traits and (2) the mechanics of how these underlying factors influence the development of each trait. It is shown that epistasis can produce patterns of covariation between traits that are not seen in additive models. Applying this approach to a trait in parents and the same trait in their offspring allows us to study the consequences of epistasis for the evolution of additive genetic variance and heritability. This analysis is then extended to the study of more complicated associations between traits. It is shown that even traits that are not correlated may exhibit developmental associations that influence their joint evolution.

NE of the most important ways in which develop- focus on the traditional decomposition of variation into O ment influences evolution is by producing statisti- additive genetic, dominance, and environmental com- cal associations between different phenotypic traits that ponents [although such a decomposition can be recov- influence the joint evolution of those traits (Maynard ered from the phenotype landscape (Wolf et al. 2001) Smith et al. 1985). Most theoretical models of develop- given a model of transmission]. In this article I assume mental associations between traits have focused on ge- that phenotype landscapes are continuous and infinitely netic covariance, resulting either from additive pleiotro- differentiable. This assumption is for illustrative pur- pic effects (Lande 1979; Roff 1997) or from particular poses only; a slight modification of the theory presented models of development, usually involving differential here applies as well to discontinuous landscapes (Rice allocation of resources to growing structures (Rendel 2002). A note on terminology: The term “epistasis” is 1963; Riska 1986; Houle 1991). These models generally used in a number of different ways in the literature assume that any developmental association between two (Phillips 1998). Throughout this article, I use the word traits can be satisfactorily captured by the genetic covari- epistasis to refer to nonadditive interactions between ance between them. underlying factors in their contributions to phenotype. φ ϭ ϩ φ In this article, I present a way to model developmental Thus, the phenotype function au 1 bu 2, where associations between traits that is amenable to any sort of is the value of a trait, u1 and u 2 are values of genetic epistatic interactions between genetic and environmental underlying factors, and a and b are constants, exhibits factors influencing those traits. I apply this approach no epistasis since the underlying factors contribute addi- to the question of how epistatic interactions influence tively to the trait value. By contrast, phenotype functions φ ϭ φ ϭ phenotypic and genetic covariation and to the study of such as the following do exhibit epistasis: u 1u 2, φ ϭ 2 developmental associations that are more complicated u1/u 2, u 1, etc. than covariance. The mathematical tools for the representation and The analysis that follows is phrased in terms of pheno- analysis of phenotype landscapes are the same as those type landscapes (Rice 2002). A phenotype landscape is presented in Rice (2002). The local geometry of the a map of some phenotypic trait as a function of all of phenotype landscape and the distribution of underlying the underlying genetic and environmental factors that genetic or environmental variation are captured with contribute to it. In this theory, environmental factors (such tensors. For our purposes, a tensor is an array of values as temperature at a particular stage of development, salin- (such as a vector or a matrix) that has the property that ity, etc.) are treated as underlying factors that may not the relationship to other such arrays is unchanged if be heritable (although they can be). We thus do not we rotate the coordinate axes (I discuss the biological meaning of this below). The rank of a tensor is the number of subscripts necessary to identify each element. Thus, a vector is a tensor of rank one and the matrices 1Address for correspondence: Department of Ecology and Evolutionary Biology, Osborn Memorial Labs, 165 Prospect St., Yale University, used in this article are tensors of rank two (nonsquare New Haven, CT 06520. E-mail: [email protected] matrices are not tensors, but we have no use for these

Genetics 166: 513–526 ( January 2004) 514 S. H. Rice in this article). (The dimension of a tensor is just the am calling the inner product is sometimes called the number of values that each subscript can take; so a 3 ϫ “n-fold inner product,” to signify that we are summing 3 matrix has rank two, because two subscripts are re- over all n indices.) quired to identify any particular element, but dimension The outer product ( ) of two tensors of rank r1 and ϩ three, because each subscript can take any of three r2 is another tensor of rank r1 r2. The outer product values). is formed by multiplying each element of one by each The notation follows that in Rice (2002). Let Pn be an element of the other. For example, the outer product 1 1 nth rank tensor with elements being the nth moments of of D1 and D2 , both of which have rank one, is a new the distribution of underlying factors (throughout this tensor of rank two, which we can write as a matrix in article, “moments” refers to central moments). Defin- which the ijth element is the product of the ith element ϭ Ϫ 1 1 ing xi (ui ui), i.e., each value of the ith trait mea- of D1 and the jth element of D2 . If each trait is influenced sured relative to the population mean for that trait, the by the same two underlying factors, then elements of Pn are φ ץ φץ φץ φץφ  ץφ  ץ 1 2 1 2 1 2 n ϭ       u ץ uץ uץ uץu  ץu  ץP (i1, i2,...,in) E[xi1xi2 ...xin]. (1)  1 ϭ 1 1 ϭ 1 ⇒ 1 1 ϭ 1 1 1 2 .φ ץ φץ φץ φץφ  D1 D2 ץφ , D2 ץD1  Because x ϭ (u Ϫ u), the vector (or tensor of rank 1)  1  2  1 2 1 2 uץ uץ uץ uץ uץ uץ i i i P1 is zero. P 2 is the standard covariance matrix, with ele-  2  2  2 1 2 2 2 ϭ ϭ 3 (4) ments P ij E(xixj) Cov(ui , uj).P is a third-order array of third moments. Higher-ranked P tensors contain the The general rule for tensors of arbitrary rank is higher-order moments; these include univariate terms 4 ϭ 4 n m ϭ n m such as kurtosis (P 1111 E[x 1]) as well as higher-order (D1 D2 )x1,...,xn,y1,...,ym (D1)x1,...,xn (D2 )y1,...,ym mixed moments, which do not generally have names. mφץ nφץ An important property of the P tensors is that they are ϭ 1 2 . (5) uץ... uץ uץ... uץ 2 highly symmetrical. For the familiar case of P (the x1 xn y1 ym covariance matrix) P 2 ϭ E(x x ) ϭ E(x x ) ϭ P 2 , this 12 1 2 2 1 21 The outer product is good for forming all combina- represents just the fact that covariances are symmetrical. tions of elements, and the inner product is good for Similarly P 4 ϭ P 4 ϭ P 4 ϭ P 4 ϭ E(x x3). In other 1222 2122 2212 2221 1 2 summing over multiples of these. Outer multiplication words, it is the number of each kind of subscript that uses the same symbol () as the Kronecker product matters, not their order. Since the rank of a tensor is and in fact is identical to the Kronecker product when the same as the number of subscripts needed to identify applied to vectors. The two operations are different, any particular element, I sometimes drop the super- though, for higher-rank tensors. For example, the Kro- script indicating rank when this is obvious from the necker product of two matrices is another matrix, while number of subscripts; thus P ϭ P4 . ijkm ijkm the outer product of two second-rank tensors (which Let Dn be a tensor of rank n, associated with pheno- a can be written as matrices) is a fourth-rank tensor, which typic trait φ , with elements defined as a cannot be written as a matrix. ,nφ An underlying factor may be any measurable valueץ n ϭ a continuous or discrete, that influences the phenotypic (2) . ץ ץ ץ (Da(i1, i2,...,in ui ui ... ui 1 2 n traits of interest. The fact that the theory puts few restric- n In words, Da contains all of the nth derivatives of pheno- tions on how these factors are defined and measured φ 1 typic trait a with respect to the underlying factors. D (see the discussion of coordinates below and in the -is just the gradient vector, ٌφ. D2 is a matrix of second appendix) is one of the virtues of the phenotype land partial derivatives of phenotype with respect to the un- scape approach, since it allows us to simultaneously con- derlying factors [this is the same as the matrix E in Rice sider the expression of individual genes, the activities (1998, 2000)]. of complex enzymes, environmental factors, and any Given two tensors of the same rank, the inner product other quantity that is biologically relevant in the system (symbolized by ͗,͘) of them is just the sum of the products of interest. Just what the underlying factors are is thus of all the corresponding elements, which is a number (the determined by the particular system under study and inner product of two tensors of different rank is not a is not necessary for the derivation of general results. number, but this is not an issue in this article). For exam- Nonetheless, it is often helpful to visualize something ple, the inner product of P 3 and D 3 is calculated as (gene sequence, enzyme, etc.) when working through the derivations below. Some areas of research that yield ͗ 3 3͘ ϭ 3 3 P , D ͚͚͚P i,j,k Di,j,k . (3) values that could be treated as underlying factors for i j k characters of interest are models of metabolic processes The familiar example of the inner product of two vectors based on enzymatic activity, quantitative trait locus anal- (tensors of rank one) is a special case of this. (What I ysis, and studies of gene expression rates. Developmental Associations Between Traits 515

The activities of different enzymes are often the main covariance between the traits. Below, I consider the special parameters in models of metabolic pathways. Such mod- case of Equation 6 appropriate to uncurved landscapes els, in which the trait being studied is the flux through and then turn to the effects of curvature (i.e., epistasis a particular pathway, allow us to derive phenotype land- and dominance) on covariation between traits. [It is scapes directly from the models (Nijhout 2002). In interesting to note that the form of the right-hand part terms of the theory presented here, we know how to of Equation 6—the inner product of a P tensor with calculate the D tensors for models involving enzymatic the outer product of a set of D tensors—is similar to the pathways. The D tensors can also be estimated, in this form of the vectors defining the direction of evolution case from empirical data, using quantitative trait locus under selection (Rice 2002).] (QTL) analysis. While quantitative trait loci are mecha- Uncurved landscapes: If both phenotype landscapes nistically ambiguous, being regions of a chromosome are uncurved (completely additive), then Dn is zero for that somehow contribute to variation in a trait, they all n Ͼ 1, so we need consider only the first term: have the advantage that the degree to which each QTL φ φ ϭ ͗ 2 1 1͘ Cov( 1, 2) P , D 1 D 2 . (8) contributes to a trait can be estimated, along with first- and second-order epistatic interactions. This equation captures the contributions to phenotypic Although metabolic models and QTL analyses allow covariance of both covariance between the underlying us to identify the underlying factors for certain traits factors (the off-diagonal elements of P2) and pleiotropy. and to calculate the D tensors, they yield no information Figure 1 illustrates how a particular covariance between about the distribution of underlying variation in a popu- two traits may be due to either covariation of the under- φ lation, which defines the P tensors. Gene expression lying factors or pleiotropy. In the case where 1 is the φ rates, while less often incorporated into mathematical value of the trait in the parents and 2 is the expected ͗ 2 1 1͘ models of phenotype, have the advantage that we can offspring value, then P , D1 D2 is the additive ge- estimate the actual population level distribution of ex- netic variance for the trait (Kojima 1959). pression rates using DNA microchip methods. The pos- Figure 1A shows the contour lines and gradient vec- ϭٌφ 1 sibility of combining QTL analysis with DNA chip analy- tors (Da a) for two traits defined by the functions sis (Schadt et al. 2003) could potentially provide a way φ ϭ ϩ 1 to fill in all the pieces. 1 u1 u2 2

φ ϭ 1 ϩ 2 u1 u2 . (9) COVARIANCE BETWEEN TRAITS 2 φ φ Two traits, 1 and 2, correspond to two different pheno- Because the phenotype functions in Equation 9 are lin- type landscapes over the set of all underlying factors ear (second and higher derivatives are all zero) we need 2 1 1 that contribute to either trait (Rice 2000, 2002; Wolf consider only P (corresponding to D D ), which is et al. 2001). Each trait thus has its own set of D tensors the same as the covariance matrix. If the two underlying i factors are uncorrelated and have unit variances, then describing the local geometry of its landscape, with Dj φ the relevant tensors are containing all of the ith derivatives of trait j. There is only one set of P tensors, though, because the distribu-     1 1   tion of underlying factors is a property of the population     10 1 ϭ 1 1 ϭ 2 2 ϭ   as a whole. The appendix shows that we can write the D1  , D2  , P     01 phenotypic covariance between the two traits as 2 1

∞ ∞  1 1 1 1  φ φ ϭ 1 ͗ iϩj Ϫ i j i j ͘ 1     1 Cov( 1, 2) ͚ ͚ P P P , D 1 D 2 . (6) 2 4 2 2 iϭ1jϭ1i!j! D1 D1 ϭ  , D1 D1 ϭ  , D1 D1 ϭ   1 1 1 1 2 2 1  1 2 1 1 1 The variance of either trait is found by calculating its covari- 2 4 2  4 2 ance with itself; thus (10) ∞ ∞ φ ϭ 1 ͗ iϩj Ϫ i j i j ͘ and, from Equations 6 and 7, we ﬁnd Var( 1) ͚ ͚ P P P , D 1 D 1 . (7) iϭ1jϭ1i!j! φ φ ϭ ͗ 2 1 1͘ ϭ 1 ϩ 1 ϭ Cov( 1, 2) P , D1 D2 1 Equation 6 describes the phenotypic covariance be- 2 2 tween two traits as a function of the distribution of underly- φ ϭ ͗ 2 1 1͘ ϭ ϩ 1 ϭ ing factors and the manner in which those factors interact Var( 1) P , D1 D1 1 1.25 4 in development of the traits. If each of the underlying factors is a gene that is passed accurately to the next φ ϭ ͗ 2 1 1͘ ϭ 1 ϩ ϭ Var( 2) P , D2 D2 1 1.25. (11) generation, then we are also calculating the genetic 4 516 S. H. Rice

φ φ Figure 1.—Three situations that produce a correlation of 0.8 between traits 1 and 2. The axes are the values of two underlying factors that contribute to the traits, the thin lines are contours of the phenotype landscapes, the boldface arrows are the gradient vectors for the traits, and the shaded regions represent the distribution of underlying variation. (A) Landscapes corresponding to Equation 9. (B) Equation 12. (C) Equation 13.

φ φ   Ϫ    So the correlation between traits 1 and 2 is 0.8 [since 1 1 0.2 0 1 ϭ 1   1 ϭ 1   2 ϭ   φ φ ϭ φ φ φ φ Ϫ1/2 D1 , D2 , P Cor( 1, 2) Cov( 1, 2) · (Var( 1)Var( 2)) ]. The √2 1 √2  1   0 1.8 case shown in Figure 1B yields the same correlation in 11  1 Ϫ1 Ϫ11 a different way. Here, the covariance between the traits 1 1 ϭ 1 1 1 ϭ 1 1 1 ϭ 1 D1 D1  , D2 D2  , D1 D2  . results from covariance between the underlying factors; 2 11 2 Ϫ11 2 Ϫ11 the relevant tensors are (13)       1 0 1 0.8 Thus, although rotating the coordinate system causes D1 ϭ  , D1 ϭ  , P 2 ϭ   1 0 2 1 0.8 1  the elements of each tensor to change, the relations between them, including the value of ͗P, D D͘,do 10 00 01 not change. This invariance under rotation is one of 1 1 ϭ   1 1 ϭ   1 1 ϭ   D1 D1 , D2 D2 , D1 D2 . the defining characteristics of tensors. For our purposes, 00 01 00 it means that we have some leeway in how we choose (12) to define the underlying factors; so long as we choose Calculating the covariance and variances as above yields factors that span the space of underlying variation, we φ φ ϭ φ ϭ φ ϭ in this case Cov( 1, 2) 0.8 and Var( 1) Var( 2) can choose a coordinate system that captures what we 1, so the correlation is again 0.8. know about the biology involved. Alternately, it is often In Figure 1A we would say that the correlation be- useful to rotate the coordinates so as to simplify the tween the traits is due to pleiotropy, whereas in the case calculations by making as many terms equal to zero as in Figure 1B we would say that the traits covary because possible (compare the analyses for Figure 1, B and C, of nonindependence of the underlying factors. It is im- above). The fact that the P and D tensors exist indepen- portant to note, though, that terms like “pleiotropy” are dently of any particular coordinate system is also impor- not necessarily inherent properties of the system, but tant when we recognize that our identification of under- rather are functions of how we choose to draw our lying factors, both genetic and otherwise, is somewhat coordinates. Figure 1C shows a system with exactly the arbitrary. same geometry as that in Figure 1B, the only difference Changes in the coordinate system that are more com- being that everything is rotated by 45Њ. In this case, both plex than simple rotation are discussed in the appendix. underlying factors influence both traits, so we would Although in this article I discuss tensors in terms of say that there is pleiotropy, even though the geometry their elements in some given coordinate system, we can is exactly the same as in Figure 1B. think of them as being independent of the way in which The relevant tensors in the case shown in Figure 1C we choose to measure things. For example, D2 captures are the phenotypic consequences of second-order interactions Developmental Associations Between Traits 517 among the underlying factors; if we change the way in 2 ϩ 1͗ 4 Ϫ 2 2 2 2͘ which we measure those factors, the elements of D P P P , D1 D2 . (18) 4 change, but its relation to the other tensors does not. We can further simplify Equation 6 in the special case Equation 18 shows that curvature of either or both in which the underlying factors are uncorrelated with of the phenotype landscapes (captured by the D2 terms) one another and are all measured in units of standard can contribute to the phenotypic covariance. To see how deviations (i.e., scaled so that the variances are all equal this occurs, consider the case of quadratic phenotype 2 ͗ 2 1 to 1); then P is just the identity matrix and P , D1 landscapes and a multivariate normal distribution of 1͘ 1 D2 is just the sum of the diagonal elements of D1 underlying variation with no covariances between the ץ φץ ץ φץ͚ 1 D2, which is 1/ ui · 2/ ui. This is the same as the underlying factors and equal variances designated by ␴2 3 ϭ 4 ϭ 4 ϭ inner product of the two gradient vectors, so for this . Under these circumstances P 0, P1111 P2222 ␴4 4 ϭ␴4 special case (zero covariances and variances equal to 3 , Pijkm when the sequence ijkm contains exactly 4 ϭ one) we have two 1’s and two 2’s, and Pelse 0. Focusing on the fourth φ φ ϭ ٌ͗φ ٌφ ͘ term on the right-hand side of Equation 18, we find Cov( 1, 2) 1, 2 . (14) (P 4 Ϫ P2 P2) ϭ 2␴4 A standard result from the definition of the inner 1111&2222 4 Ϫ 2 2 ϭ␴4 product of two vectors is (P P P )1221&2112&1212&2121 φ ٌφ ͘ ϭ ٌφ ٌφ ␪ 4 Ϫ 2 2 ϭٌ͗ 1, 2 1 · 2 ·Cos( ), (15) (P P P )else 0. (19) φ ٌφ ␪ٌ where i is the length of the vector i and is the The condition for epistasis to contribute to phenotypic angle between the two gradient vectors. Under the as- covariance given these assumptions is thus sumption of unit variances and no covariances between 2φץ 2φץ 2φץ 2φץ 2φץ 2φץ the underlying factors, the variances of the traits are 1 2 ϩ 1 2 ϩ 2 1 2 ϶ 0. (20) ץ ץ ץ ץ 2 ץ 2 ץ 2 ץ 2 ץ given by Wolf et al. (2001): u1 u1 u2 u2 u1 u2 u1 u2

φ ϭ ٌφ 2 Var( i) i . (16) Figure 3 shows a case that meets this condition. Even when the gradient vectors are at right angles and there is no Combining Equations 14, 15, and 16, we can calculate covariance between underlying factors, epistatic effects the correlation coefficient for the two traits as produce a covariance between the phenotypic traits. φ φ ϭ ␪ Phenotypic or genetic covariance induced by epistatic Cor( 1, 2) Cos( ). (17) effects differs in two important ways from covariance Thus, in an additive system with unit variances and resulting from additive pleiotropic effects or covariance no covariances between the underlying factors, the cor- between underlying factors. First, in cases such as the relation between the two traits is exactly the cosine of one in Figure 3, the correlation between the traits is a the angle between their gradient vectors. function of the total amount of underlying variation. Uni- Interestingly, there are cases in which the covariance formly reducing the variances in both underlying factors φ between the underlying factors has no effect on the in Figure 3 would reduce the correlation between 1 φ covariance between the traits. If one of the gradient and 2, because the landscapes would look more nearly vectors is a reflection of the other in one of the axes, flat over the range of variation present in the popula- then the covariance between the two traits is determined tion. This is not the case in the examples shown in by the angle between their gradient vectors and by the Figure 1, where increasing or decreasing the amount variances (but not covariances) of the underlying fac- of underlying variation, without changing the shape of tors. This is illustrated in Figure 2. This independence the distribution, has no effect on the correlation be- of phenotypic covariance and underlying covariance re- tween the phenotypic traits. sults because P 2 is symmetrical (P2 ϭ P2) but (D1 D1) φ φ ij ji 1 2 Second, the association between 1 and 2 in Figure 1 1 ϭϪ 1 1 is not. Whenever (D1 D2)ij (D1 D2)ji, then the 3 is asymmetrical. Curvature of the landscapes makes it 2 two terms containing Pij cancel one another out, so φ φ easier to simultaneously increase both 1 and 2 than it 2 φ φ Pij contributes nothing to Cov( 1, 2). In such a case, would if there were no epistasis, but it is actually harder selection for integration of the two phenotypic traits to simultaneously decrease both traits, since the propor- will not lead to covariance of the underlying factors tion of the distribution of variation that lies in the region (such as through gametic phase disequilibrium), al- in which both traits are smaller is less than it would be though it may lead to change in their variances. with the same gradient vectors on uncurved landscapes. Curved landscapes: Consider first two quadratic land- A number of studies, including Bell and Burris (1973) 1 ϭ scapes. Using the fact that P 0 (since we are using with Tribolium, Rutledge et al. (1973) with mice, and central moments), Equation 6 yields Nordskog (1977) with chickens, have observed an asymmetry in selection experiments on correlated characters φ φ ϭ ͗ 2 1 1͘ ϩ 1͗ 3 1 2͘ ϩ 1͗ 3 2 1͘ Cov( 1, 2) P , D1 D2 P , D1 D2 P , D1 D2 2 2 [e.g., body weight and tail length in mice (Rutledge et 518 S. H. Rice

Figure 2.—A case in which covariation between the underlying factors does not influence covariation between the traits. In φ φ each case the correlation between 1 and 2 is 0.8. The shaded regions represent the distribution of underlying variation in a population. In A there is no covariance between the underlying factors. In B, the underlying factors covary. al. 1973)] in which the response to selection over many variance into additive and nonadditive components and generations is significantly greater for one kind of antag- then check the magnitude of the nonadditive compo- onistic selection (e.g., small body and long tail) than for nents (Lynch and Walsh 1998). A number of authors the other (e.g., large body and short tail). This sort of have pointed out that this approach can easily underesti- pattern is not predicted by models based on only addi- mate the significance of epistatic effects (Keightley 1989; tive gene effects, but follows from a number of different Lynch and Walsh 1998). Selection experiments in scenarios involving epistasis. which the results are compared quantitatively with the Classical quantitative genetics assumes locally un- predictions of an additive model provide a more sensi- curved landscapes since this means that the local geome- tive test of whether epistasis is strong enough to influ- try is well approximated by an additive model. A com- ence evolution, and the observation of a number of mon way to test this assumption is to partition genetic cases in which the response to antagonistic selection is asymmetrical (Nordskog 1977) suggests that many traits do not have locally uncurved landscapes. Figure 4 shows a hypothetical case in which two under-

lying factors inﬂuence two traits. Factor u2 has a linear

positive effect on both traits, while factor u1 has a positive effect on one trait and a negative effect on the other.

The key is that the contribution of u1 is highly nonlinear, so that it has little effect on either trait when it is small and then rapidly increases in importance as its value increases. This creates a situation in which a relatively small part of the distribution of underlying factors lies φ in the region that produces an increased value of 1 φ and a decreased value of 2. Furthermore, moving in φ the direction necessary to increase 1 while decreasing φ 2 moves the population into a region in which the two traits are even more highly correlated. Figure 4 shows a two-dimensional case purely for the sake of drawability; the same kind of result can occur with any number of Figure 3.—A case in which covariation between two traits underlying factors. is a result of epistasis only. The gradient vectors are at right This sort of effect is apparent in the “resource alloca- angles. The shaded circles represent the distribution of underlying variation. The hatched region is the subset of that distri- tion” models studied by Houle (1991; see also de Jong bution corresponding to both traits being smaller than the and van Noordwijk 1992). Here, the development of population mean. two phenotypic traits involves the same resources, with Developmental Associations Between Traits 519

Figure 4.—A case in which the covariation induced by epistasis is asymmetrical, such that the response to selection φ φ for increasing 1 while decreasing 2 (selecting values in the hatched region) should be greater than that to selection for φ φ decreasing 2 while increasing 1.

Figure 5.—Phenotype landscapes corresponding to Equa- the values of the traits being determined by the rate at φ tion 21. The solid lines are contours of the 1 landscape and φ which resources are acquired and how they are distrib- the dashed lines are contours of the 2 landscape. uted. Letting u1 be the rate at which the resource is acquired and u2 be the proportion of the resource allo- φ 4 ϭ␴2 ␴2 cated to trait 1, the two traits are described by P1212&1221&2112&2121 1 2 and the values of the tensors in φ ϭ Equation 22 we find 1 u1u2 φ φ ϭ␴2 Ϫ Ϫ␴2 2 Ϫ␴2␴2 φ ϭ Ϫ Cov( 1, 2) 1u2(1 u2) 2u1 1 2 . (24) 2 u1(1 u2). (21) The sum of the first two terms on the right side of Here, u and u are themselves quantitative traits that 1 2 Equation 24 [␴2u (1 Ϫ u ) Ϫ␴2u2] is the covariance be- may be influenced by many genes. For simplicity, these 1 2 2 2 1 tween the additive approximations to Equations 21 and are assumed to have heritability 1 and be uncorrelated. is thus the “additive genetic covariance” between the The two phenotype landscapes for these traits are shown traits (de Jong and van Noordwijk 1992; Roff 1997). in Figure 5. The third term on the right side of Equation 24, ␴2␴2 ,is The relevant D tensors for the traits defined in Equa- 1 2 the contribution of nonadditive interactions to covari- tions 21 are ance between the traits. In traditional nomenclature this        Ϫ  u2 1 Ϫ u 01 0 1 is the “additive-by-additive” epistatic variance (Kojima 1 ϭ   1 ϭ  2 2 ϭ   2 ϭ   D1 , D2 Ϫ , D1 , D2 Ϫ . 1959; de Jong and van Noordwijk 1992). u1  u1  10  10 (22) Equation 24 has been derived elsewhere and follows From these we find from the properties of the multivariate normal distribution, so using the method presented here may seem like 2 2 ϭϪ (D1 D 2)1212&1221&2112&2121 1 overkill. In fact, quite a bit more information can be 2 2 ϭ gleaned from the D tensors in Equations 22. For example, (D1 D 2)else 0. relaxing the assumption of multivariate normality requires that we consider the third moments (measuring Given the assumption that u1 and u2 have zero covari- 2 ϭ ϶ asymmetry) of the distribution of underlying factors. ance, we know that Pij 0 for i j, so we have Simply by examining the D tensors in Equation 22, we 2 2 ϭ (P P )1212&1221&2112&2121 0. can conclude that introducing skewness in the distribu-

tion of either u1 or u2 will have no effect on the resulting ͗ 2 2 2 2͘ ϭ φ φ Thus, P P , D1 D2 0 and, given the assump- covariance between 1 and 2, but changing the mixed 3 ϭ 2 tion that P 0 (symmetrical distribution), we can write third moments (e.g., E[x1x 2]) will change the value of φ φ Cov( 1, 2). φ φ ϭ ͗ 2 1 1͘ ϩ 1͗ 4 2 2͘ Cov( 1, 2) P , D1 D2 P , D1 D2 . (23) Recall that the third moments influence covariance 4 ͗ 3 1 2͘ between the traits through the terms P , Di Dj (see 4 1 2 ϭ 1 2 Using the fact that the only nonzero elements of P are Equation 18). Since (D1 D2)111 (D1)1 ·(D2)11,weim- 520 S. H. Rice mediately see that because the diagonal elements of both   2 2 1 2 1 2 ␣2 ␣ε 0 D1 and D2 are all zero, (D1 D2)111 and (D1 D2)222   1 1 ϭ ␣εε2 must both be zero, as must the corresponding elements Dp Dp  0, (26) 1 2 000 of D2 D1 . We thus know that whatever the values of   P and P , which measure the skewness in the u and 111 222 1 from which, using Equations 6 and 7, we find u 2 directions, respectively, they have no influence on φ φ φ φ ϭ␣2 ϩ␣ε ϩ␣ε ϩ ε2 the covariance between 1 and 2, since they are both Cov( o, p) P11 P13 P21 P23 multiplied by terms that equal zero. This is relevant to φ ϭ␣2 ϩ␣ε ϩ␣ε ϩ ε2 Var( p) P11 P12 P21 P22 . (27) interpreting the generality of the model; because u1 ␤ has a lower bound at zero and no upper bound, it is The regression of offspring on parents, o,p, is then reasonable to expect that the distribution of values of φ φ ␣2 ϩ␣ε ϩ␣ε ϩ ε2 Cov( o, p) P11 P13 P21 P23 u1 would be positively skewed. An analogous argument ␤ ϭ ϭ . o,p φ ␣2 ϩ␣ε ϩ␣ε ϩ ε2 Var( p) P11 P12 P21 P22 shows that the mixed third moments, such as P112 and (28) P122, do contribute to the covariance between the traits. These moments measure, roughly, the degree to which The terms P12, P13, and P23 are, respectively, the covari- the conditional variance of one underlying factor is a ance between the parent’s genotype and the parent’s function of the value of the other factor (there is an ϭ environment (P12 P21), the covariance between the example in the final section below). parent’s genotype and the offspring’s environment

(P13), and the covariance between the parent’s environ- PARENT-OFFSPRING COVARIANCE ment and the offspring’s environment (P23). If all of AND HERITABILITY these covariances are zero, then we have One interesting application of this theory is to the calcu- ␣2P ␤ ϭ 11 , (29) o,p ␣2 ϩ ε2 lation of parent-offspring covariance. Here, instead of con- P11 P22 sidering two different traits in the same generation, we calculate the covariance between a trait in the parents, which corresponds to the standard quantitative genetics φ φ notion of heritability, with the denominator being the p, and the same trait among their offspring, o. Dividing this covariance by the variance in parental phenotype total phenotypic variance among the parents and the ␤ numerator being the component of that variance that yields the regression of offspring on parents, op, which I treat as identical to heritability. In sexually reproducing is due to heritable factors (the “additive genetic vari- diploid organisms, we can use the midparent pheno- ance”). Note, though, that when we use the full Equa- φ tion 28 the numerator contains terms that do not appear typic value as p without changing our calculations. [A number of different definitions of heritability exist in in the denominator, so we cannot represent the regres- the literature (Jacquard 1983); I focus on the regression of offspring on parents in terms of the ratio of sion of offspring on parents and the corresponding additive genetic variance to phenotypic variance, since offspring-parent covariance, since this is most directly this would require that the additive genetic variance be related to the response to selection.] either negative or greater than the phenotypic variance. Consider first the simplest case, in which phenotype In the linear case of Equation 25, the regression of is an additive function of a heritable underlying factor, offspring phenotype on parent phenotype is not a func- u 1, and an environmental factor. We assume that the tion of the values of the underlying factors. This is not expected value of u 1 is the same in offspring as in their the case, though, when there are nonlinear effects, such parents, but that the environment experienced by the as epistasis or genotype-by-environment interactions. As offspring is (potentially) independent of that of their an example, consider a trait in the parental generation, φ parents. In such a case, we really have two different p, influenced by two genetic factors, u1 and u2, and the environmental factors: the environment experienced by environment. For simplicity, we assume that u1 and u2 the parents, u 2, and the environment experienced by are transmitted without modification to the offspring their offspring, u 3. We can now write the phenotype of (i.e., assume asexual reproduction). As before, we have φ φ the parents, p, and that of their offspring, o,as two environmental factors, the value of the environment φ ϭ␣ ϩ ε experienced by the parents, u 3, and the environmental p u1 u2 value experienced by their offspring, u 4: φ ϭ␣ ϩ ε o u1 u3 . (25) φ ϭ 2 ϩ p u1 u2u3 To find the parent-offspring covariance we note that the φ ϭ u2 ϩ u u . (30) relevant D tensors are o 1 2 4 Note that the two equations are the same except that   ␣ ␣ ␣2 0 ␣ε φ φ   p contains u3, the parent’s environment, where o con- 1 ϭ ε 1 ϭ 0 1 1 ϭ ␣ε 0 ε2 Dp , Do , Dp Do  , tains u4, the offspring’s environment. The first rank D   ε 0  000 tensors are Developmental Associations Between Traits 521

2u  2u   1  1 u u 1 ϭ  3  1 ϭ  4  Dp , Do . (31)  u2   0   0   u2  Combining these according to Equation 4 yields

4u2 2u u 02u u   1 1 4 1 2 2u u u u 0 u u 1 1 ϭ  1 3 3 4 2 3  Dp Do 2 , 2u1u2 u2u4 0 u2   0000

4u2 2u u 2u u 0  1 1 3 1 2  2u u u2 u u 0 1 1ϭ  1 3 3 2 3  Dp Dp 2 . (32) 2u1u2 u2u3 u2 0  0000 The second rank D tensors are

2000 2000     0010 0001 D2 ϭ  , D2 ϭ  . (33) p 0100 o 0000 0000 0100     Figure 6.—Phenotype landscape (dashed contour lines) for the parental trait deﬁned in Equation 30 superimposed Although the outer product of these is a four-dimen- on the offspring-parent regression (heritability) for the trait, sional array, most of its elements are zero, and we can given by Equation 35 (shading). Darker shading indicates quickly ﬁnd the nonzero elements as higher heritability for the trait.

2 2 ϭ (Dp Do)1111 4

2 2 ϭ 2 2 ϭ 2 2 ϭ 2 2 ϭ (Dp Do)1124 (Dp Do)1142 (Dp Do)2311 (Dp Do)3211 2 regression of offspring phenotype on parent phenotype) for the trait φ, indicated by shading. Moving along (D2 D2) ϭ (D2 D2) ϭ (D2 D2) ϭ (D2 D2) ϭ 1 p o 2324 p o 2342 p o 3224 p o 3242 the φ ϭ 10 curve from the circle to the triangle produces 2 2 ϭ (Dp Do)else 0. (34) no change in the mean phenotype, but causes the heritability to increase from nearly zero to nearly one. Each of these will be multiplied by the corresponding Thus, when the underlying factors, both genetic and 4 element of P . We thus know that we can ignore P1122, environmental, contribute nonadditively to a trait, then 2 2 ϭ for example, because (Dp D0)1122 0. change in the values of those underlying factors tends 2 2 Note that, except for (Dp D0)1111, all nonzero ele- to lead to change in the trait’s heritability. (If all underly- 2 2 ments of (Dp D0) have subscripts with two numbers ing factors contribute additively to the trait, then herita- the same and one each of two other numbers (e.g., 3242, bility changes only as a result of change in the elements with two 2’s and a 3 and a 4). These will be multiplied of the P tensors.) by elements of the P4 tensor that are moments of the This provides one way to visualize genetic assimila- 2 ϶ ϶ form E[xi xjxk] for i j k. Fourth-order moments of tion. Moving along the dashed contour from the circle this form are zero if there are no covariances between to the triangle in Figure 6 moves a population from a the underlying factors. Thus, for a normal distribution point where variation in the trait is largely determined with all covariances equal to zero, we need consider by environmental variation to a point where most varia- only the term that multiplies P1111, which is 4. Applying tion is heritable. This phenomenon may also play a role these results to Equations 6 and 7 yields in the evolution of novel phenotypic traits. Think of a φ φ 2 ϩ ϩ trait as a function of some set of underlying factors. ␤ ϭ Cov( o, p) ϭ 4u1P11 u3u4P22 P1111 o,p (35) Even if selection acts on the function (i.e., fitness is Var(φ ) 4u2P ϩ u2P ϩ u2P ϩ P . p 1 11 3 22 2 33 1111 determined by the value of the function), there will be Figure 6 shows a slice of the phenotype landscape for no subsequent evolution if it has very low heritability the trait defined by Equation 30; the full landscape is (e.g., the circle in Figure 6). If the distribution of under- four-dimensional; this slice is produced by fixing the lying factors changes (for whatever reason) in such a ϭ ϭ values of the environmental factors, here u 3 u 4 1. way that the population comes to inhabit a region of ϭ ϭ ϭ ϭ In Figure 6, P11 P22 P33 1 and P1111 3 (which the space of underlying factors in which the function is the fourth moment for a normal distribution with in question is heritable (e.g., the triangle in Figure 6), unit variance). Figure 6 also shows the heritability (the then selection can now act to modify the value of that 522 S. H. Rice function, which now behaves in an evolutionary sense entangled if, because of developmental associations, like a distinct trait. changing some moment of the distribution of one trait leads to a change in some moment of the distribution of the other trait. Genetic covariance resulting from SELECTION DIFFERENTIALS pleiotropy is one sort of developmental entanglement; This article is concerned primarily with the statics, below I give a couple of examples of other cases. rather than the dynamics, of phenotypes, meaning that Consider two traits with phenotype landscapes de- we are concerned with concepts such as genetic and fined by phenotypic covariance, heritability, and the like, which φ ϭ 1 2 Ϫ 2 are relevant to evolution but can be defined indepen- 1 (u1 u2) dently of any particular evolutionary process. I have thus 2 φ ϭ ϩ made no formal reference to fitness. There is a link to 2 u1 u2 . (37) selection theory, though, since the selection differential for a trait (the change due only to selection, prior to The contour maps of these are shown in Figure 7. The any changes introduced by recombination, etc.) is equal relevant D tensors are to the covariance between fitness and the trait value, ϭٌφ ϭ  u1  2 ϭ 10 1 ϭ 1 2 ϭ 1 D1 1 Ϫ , D1 Ϫ , D2 , D2 [0]. divided by mean population fitness (Robertson 1966;  u2 0 1 1 Price 1970). We can thus calculate the selection differ- (38) ential for the trait φ, denoted Sφ, by substituting fitness for one of the phenotypic traits in Equation 6: If the joint distribution of u1 and u2 is uncorrelated φ φ with equal variances, then the traits 1 and 2 are also 1 1 1 iϩj i j ϭ Sφ ϭ Cov(w, φ) ϭ ͚͚ ͗P , D D φ͘. (36) uncorrelated (from Equation 6, using the fact that u1 w w w i j i!j! u2 at this point). Thus, at least locally around this point, the two traits should evolve independently under direc- Equation 36 gives the effects of selection on a trait in tional selection. The two traits are not, however, inde- terms of a fitness landscape and a phenotype landscape. pendent under other kinds of selection. Note, though, that the fitness landscape is over the space As shown in Rice (1998), stabilizing selection on trait φ of underlying factors, rather than over the space of 1 should, all else held equal, move the population along phenotypic traits, as in most quantitative genetic models the optimal contour toward a region of lower slope (as (e.g., Lande 1979). Note that Equation 36 gives the indicated by greater spacing of the contour lines). The φ φ selection differential for the trait, . It is thus different direction of maximal decrease in slope for 1 is given Ϫ 2ٌφ ϭ ϭ from the selection differential for the set of underlying by the vector D1 1 (Rice 1998). At the point u1 u2 factors derived elsewhere (Rice 2002). 2, this vector points in the exact opposite direction of φ φٌ 2. Thus, stabilizing selection on trait 1 leads to directional change of trait φ , although the two traits are OTHER DEVELOPMENTAL ASSOCIATIONS 2 BETWEEN TRAITS uncorrelated with one another. We could not identify such a system by looking at just Covariance between traits is of interest primarily be- the covariance between the traits. This does not mean cause it influences the joint evolution of the traits, iden- that there is no telltale signature, only that other mo- tifying cases in which directional selection on one trait ments of the joint distribution of the traits need to is expected to produce a change in the mean value of be considered. In the example above, a symmetrical both traits (Lande 1979). In a completely additive distribution of underlying factors would produce an model, in which all phenotype landscapes are uncurved, asymmetrical distribution of the traits, corresponding this is the only kind of developmental association that 2 to a nonzero value of the mixed third moment E[x1x 2], could influence joint evolution of the traits. When non- as shown in Figure 8. additive interactions are allowed between underlying The resource allocation model shown in Figure 5 also factors, though, other kinds of relationships become exhibits entanglement of the traits other than mere possible, because moving in the space of underlying covariance. In Figure 5, it is not possible to change the factors can change other moments of the phenotype φ degree of canalization of 1 without also changing the distribution, not just the mean. φ mean of 2. We can think of correlated evolution as being a case Any number of traits can potentially be entangled in which changing the first moment (the mean) of one with one another. For example, consider the following trait leads to a change in the first moment of the other system: trait. With epistasis, it is possible to have a situation in φ ϭ which changing the nth moment of one trait leads to 1 u1u2 a change in the mth moment of another trait or to a φ ϭ u u change in the joint moments of a set of other traits. 2 1 3 φ ϭ I shall say that two phenotypic traits are developmentally 3 u2u3 . (39) Developmental Associations Between Traits 523

Figure 7.—Case in which stabilizing selection on one trait should lead to change in the mean of another trait. The solid contours correspond φ to the phenotype landscape of trait 1. The dashed φ contours are for trait 2. Stabilizing (canalizing) φ selection on trait 1 will move the population in Ϫ 2ٌφ the direction of D1 1 .

In this system, there are three phenotype landscapes in DISCUSSION a space of three underlying factors, which means that Developmental associations between two phenotypic we would need four dimensions to draw a picture of it. traits arise when some of the same underlying genetic While this hinders illustration, it does not get in the or environmental factors influence the development of way of analysis. both traits. In this article, I have presented a formal The gradient vectors for these three traits are method for studying developmental associations on the basis of the geometry of phenotype landscapes. This u2 u3  0  ϭٌφ ϭ 1 ϭٌφ ϭ 1 ϭٌφ ϭ 1 D1 1 u1, D2 2  0  D3 3 u3. approach uses tensors to capture the distribution of  0  u1 u2 (40) If the underlying factors are symmetrically distributed with unit variances and uncorrelated, then from Equation 14 φ φ ϭ ٌ͗φ ٌφ ͘ ϭ we know that Cov( 1, 2) 1, 2 u 2u3. The direc- φ φ tion of most rapid change in Cov( 1, 2) is then  0  φ φ ϭٌٌ͗φ ٌφ ͘ ϭ ϭٌφ ٌ Cov( 1, 2) 1, 2 u3 3 . (41) u2 φ Thus, the gradient of 3 is also the gradient of the covari- φ φ ance between 1 and 2. From the symmetry of Equation 39 we can see that the same holds for any pair of these traits; the gradient of the covariance of any two traits in Equation 39 is the same as the gradient of the remaining trait. Changing any one of these traits thus leads to a consequent change in the developmental covariance of the other two. Such an association between the covariance between two traits and the value of a third trait Figure 8.—Expected distribution of phenotypic traits for has recently been observed in natural populations (Roff the system shown in Figure 7 given a symmetrical distribution et al. 2002). of underlying factors with no covariance. 524 S. H. Rice underlying factors (both genetic and environmental) Another consequence of epistasis is the possibility of and the interactions between these factors in develop- entanglement between higher-order moments of the ment. The results are thus largely independent of how distributions of the traits concerned. For example, if at we choose to measure the underlying genetic and envi- least one trait exhibits first-order epistatic effects, then ronmental factors. it is possible to have a case in which stabilizing selection This approach allows us to study general develop- on one trait leads to directional change in the mean of mental entanglement of traits. Two traits are develop- the other trait. This may occur even when the two traits mentally entangled when a developmental change that are phenotypically and genetically uncorrelated. alters some moment of the distribution of one trait leads The idea of integration of traits resulting from shared to a change in some (potentially different) moment of developmental pathways has been one of the principal the distribution of the other trait. An important special concepts linking development and evolution. Most au- case of developmental entanglement is developmental thors who have discussed integration have defined it in covariance between two traits, in which case the mean terms of covariance between the traits involved (Olsen (first moment) of one trait is entangled with the mean and Miller 1958; Wagner 1990; Magwene 2001). Un- of the other trait. Covariance is the only sort of develop- der these definitions, integration of traits would be a mental association possible when the underlying factors subset of what I am calling entanglement. The above contribute additively to both traits. In the additive case, examples show that simply saying that two traits are covariance between traits is determined completely by not integrated is not the same as saying that they are the distribution of underlying variation and the angle evolutionarily uncoupled. Many physiological and mor- between the gradient vectors of the phenotype land- phological traits are probably under strong stabilizing scapes corresponding to each trait. When underlying selection for long periods of time. Also, for some charac- factors interact epistatically, though, then these nonad- ters such as the components of articulating skeletons, ditive interactions may contribute to phenotypic and selection probably acts more strongly on the ratio of genetic covariance. different parts than on their absolute sizes. In such cases, Covariance resulting from epistatic interactions dif- entanglement between the variance or degree of inte- fers in at least two ways from that resulting from pleiot- gration of certain traits and the mean values of others ropy in an additive system. First, phenotypic and genetic may influence the direction of evolution as strongly as correlation in nonadditive systems is a function of the do genetic covariances between traits. total amount of underlying variation, because increas- This article benefited greatly from comments by Bruce Walsh and ing underlying variation exposes more of the curvature two anonymous reviewers. of the phenotype landscapes. 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Figure A1.—Construction of a covariant basis from a contravariant basis. A shows how a point is located with respect to the contravariant basis vectors u1 and u2. B shows the corresponding covariant basis vectors, u*1 and u*2 .

For spaces with more than two dimensions, we extend variant basis vectors (whatever we choose these to be) rule 1 to require that u*i is perpendicular to uj for all and describe the distribution of underlying variation ϶ j i. The vectors u1* and u*2 are referred to as the (the P tensors) in terms of the corresponding covariant “covariant” bases of our space. (The unfortunate terms basis vectors. This requires no new information, since contravariant and covariant derive from the manner in we can always ﬁnd the covariant basis vectors from the which the elements of the vectors change when we rotate contravariant vectors using the two rules listed above. the space; they have nothing whatsoever to do with the Note that if we start out with Cartesian coordinates, in statistical concept of covariance used elsewhere in this which u1 and u2 are at right angles and each of unit article.) All of the equations in this article remain cor- length, then these coordinates are both contravariant rect so long as we represent the phenotype landscapes and covariant and we need not worry about the distinc- (and thus measure the D tensors) in terms of the contra- tion.