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The importance of the environmental variance-covariance structure in predicting evolutionary responses.

Natural selection acts on phenotypes and does not, in itself, guarantee evolution. Lande and Arnold (1983) showed that correlations (phenotypic and/or genetic) among characters can make the extension from within-generation selective effects (i.e., observed selection differentials) to cross-generational (evolutionary) responses quite complicated. In particular, they showed that if negative correlations exist, the evolutionary response to selection can be of dramatically different magnitude, even opposite in sign, to the within-generation selection differential. In this note, we develop a scenario, inherent in the Lande-Arnold model, that serves to further point out the complications that can arise with selection on correlated characters.

We begin with a question: given positive selection differentials for two characters that share positive phenotypic and genetic correlations, what are the expected evolutionary responses (assuming that there are no other traits under selection)? Answer: in the next generation, the mean value of both characters will increase - sometimes. We show that the evolutionary response to selection can be negative even if all observed selection differentials and all variance-covariance terms (both phenotypic and genetic) are positive. (Note that throughout, in using the terms "negative" and "positive" evolution, we do not refer to progress in evolution or even to adaptation, rather these qualitative terms refer to the direction of evolutionary response relative to the within-generation effect of selection on the mean phenotype.)

The Lande-Arnold Model

Lande and Arnold (1983) produced a multivariate analogue of the breeder's equation. For a set of characters, [z.sub.1], [z.sub.2], . . . , [z.sub.n].

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the column vector of cross-generation selection responses [[Mathematical Expression Omitted]], G is the variance-co-variance matrix of breeding values, and [Beta] is the column vector of partial-regression coefficients of relative fitness on the characters (further definitions of these parameters are given in Lande and Arnold [1983]). [Beta] is known as the directional selection gradient - its elements describe the path of steepest ascent up the multivariate fitness surface. [Beta] can be further decomposed:

[Beta] = [P.sup.-1]s, (2)

where [P.sup.-1] is the inverse of the phenotypic variance-covariance matrix, and s is a column vector of observed (within-generation) selection differentials [[Mathematical Expression Omitted]]; [Mathematical Expression Omitted] denotes the vector of mean phenotypes after selection at generation t. Substituting back into (1) yields

[Mathematical Expression Omitted].

To show that a negative evolutionary response can result under purely positive phenotypic and genetic correlations and positive observed selection differentials, we analyze (3) for the case of two correlated characters.

For two characters, [z.sub.1] and [z.sub.2], let

[Mathematical Expression Omitted],

where [G.sub.11] is the additive genetic variance of character 1, [G.sub.22] is the additive genetic variance of character 2, and [G.sub.12] = [G.sub.21] is the additive genetic covariance between the two characters; the phenotypic variances and covariances of the P matrix are subscripted in the same manner; the elements of the s vector were described from equation (2). Assume all elements in these three matrices are positive. To analyze equation (3), the P matrix must be inverted yielding

[Mathematical Expression Omitted].

Note that in performing this inversion, negative terms are introduced into the matrix. Substituting (4) into equation (2),

[Mathematical Expression Omitted].

The scalar term of (5) must always be positive. To see this, recall that, by definition, the square of the (phenotypic) correlation coefficient [Mathematical Expression Omitted] and 0 [less than or equal to] [[Rho].sup.2] [less than or equal to] 1. Thus, [Mathematical Expression Omitted] can never be greater than [P.sub.22][P.sub.11] - the constant fractional portion of equation (5) must always be positive. However, one of the elements of the directional selection gradient can be negative if

[[Beta].sub.1] = ([P.sub.22][s.sub.1] - [P.sub.12][s.sub.2]) [less than] 0


[[Beta].sub.2] = (-[P.sub.12][s.sub.1] + [P.sub.11][s.sub.2]) [less than] 0.

Rearranging these equations, [[Beta].sub.1] or [[Beta].sub.2] will be negative when

[s.sub.1] [less than] [P.sub.12]/[P.sub.22] [s.sub.2] (6A)


[s.sub.2] [less than] [P.sub.12]/[P.sub.11] [s.sub.1], (6B)

respectively. These two conditions are mutually exclusive. To see this, let [s.sub.1 max] = ([P.sub.12]/[P.sub.22])[s.sub.2]. Substituting into (6B) yields [Mathematical Expression Omitted]; again [[Rho].sup.2] is constrained to be less than or equal to 1, and this inequality cannot hold; both characters cannot show reversals of sign.

Substituting (5) into (3) and performing matrix multiplication yields the vector of evolutionary responses

[Mathematical Expression Omitted].

Thus, characters 1 or 2 will evolve negatively when

k[[s.sub.1]([G.sub.11][P.sub.22] - [G.sub.12][P.sub.12]) + [s.sub.2]([G.sub.12][P.sub.11] - [G.sub.11][P.sub.12])] [less than] 0 (8A)


k[[s.sub.1]([G.sub.12][P.sub.22] - [G.sub.22][P.sub.12]) + [s.sub.2]([G.sub.22][P.sub.11] - [G.sub.12][P.sub.12])] [less than] 0, (8B)

respectively, where k is the scalar component of equation (7).

Under our starting condition that all genetic variance-covariance terms are positive, the negative terms in equation (8) can lead to a negative evolutionary response in one of the characters. A simple numerical example may add clarity. Imagine two characters in which the phenotypic variances are standardized to unity ([P.sub.11] = [P.sub.22] = 1.0), the genetic variances (equal to the heritabilities) are [G.sub.11] = [G.sub.22] = 0.5, the phenotypic covariance [P.sub.12] = 0.5, and the genetic co-variance [G.sub.12] = 0.05. What values of [s.sub.1] and [s.sub.2] can lead to a negative evolutionary response in one of the characters? Plugging these values into (8), a negative response of character 1 will occur when [s.sub.1] [less than] 0.42[s.sub.2]. Conversely, character 2 will show a negative response when [s.sub.2] [less than] 0.42[s.sub.1]. Two things can be taken from this example. First, the range over which the evolutionary response can be opposite in sign to the within-generation selection differential can be quite broad. Second, it is not possible for both characters simultaneously to have positive selection differentials and negative selection responses. The three other outcomes are all possible: character 1 shows a negative response; character 2 shows a negative response; both characters show positive responses.

These results are, at first glance, counterintuitive. Positive correlations seem to imply that selection on one character should simply induce further change in the positive direction for the other character. Why does this not always occur? A second example may make the results more intuitive. Again, assume that all phenotypic and genotypic variance-covariance terms are positive. Suppose that there is no observable selection on character 1 ([s.sub.1] = 0) and positive selection on character 2 ([s.sub.2] [greater than] 0). The [s.sub.1] comprises a direct selection component, and an indirect component that arises from selection on the phenotypically correlated character 2. Positive selection on character 2 implies that the indirect component of [s.sub.1] should be positive. Because [s.sub.1] = 0, we must conclude that direct selection on character 1 is exactly equal and opposite in sign to the indirect component. The evolutionary response of trait 1, from equation (8A) is [ks.sub.2]([G.sub.12][P.sub.11] - [G.sub.11][P.sub.12]). Setting this value less than 0 and rearranging yields the condition for a negative evolutionary response of character 1

[G.sub.12]/[P.sub.12] [less than] [G.sub.11]/[P.sub.11]. (9)

Thus, character 1 shows a negative evolutionary response when the heritable fraction of the phenotypic variation that experiences (negative) direct selection (the variance terms) outweighs the heritable proportion experiencing (positive) indirect selection (the covariance terms). Extending this to the case in which both characters show positive selection differentials (i.e., when the positive indirect component is larger than the negative direct component), a negative evolutionary response will occur when the heritable fraction of the (positive) indirect component of [s.sub.1] is less than the heritable fraction of the counterbalancing (negative) direct component.


Natural selection changes the phenotypic distribution within a generation. Reliable predictions about evolutionary responses based on these changes require detailed knowledge not only of the genetic architecture but the phenotypic as well. When the evolutionary response of a character does not correspond with the within generation-selection differentials, the intuitive reaction is to doubt the completeness of the analysis. Indeed, if there are characters, not included in the analysis, that share negative correlations with the character of interest, then the evolutionary response can be opposite to thai expected from observed, within-generation changes. We have shown that such negative genetic correlations are not necessary (nor even sufficient) to fully explain these results.

The conditions for such reversals can be further characterized. By partitioning the phenotypic variances into their genetic and environmental components, the inequalities of equation (8) become

[s.sub.1]/[s.sub.2] [less than] [G.sub.11]([G.sub.12] + [E.sub.12]) - [G.sub.12]([G.sub.11] + [E.sub.11])/[G.sub.11]([G.sub.22] + [E.sub.22]) - [G.sub.12]([G.sub.12] + [E.sub.12]) (10a)


[s.sub.2]/[s.sub.1] [less than] [G.sub.22]([G.sub.12] + [E.sub.12]) - [G.sub.12]([G.sub.22] + [E.sub.22])/[G.sub.22]([G.sub.11] + [E.sub.11]) - [G.sub.12]([G.sub.12] + [E.sub.12]). (10b)

Under our starting assumptions that the genetical terms are fixed and positive, the determinants of the direction of selection (and the resultant response to selection) are the environmental variance and covariance terms. Rausher (1992) discusses the bias that unmeasured environmental covariance between a trait and fitness (or between one trait and another, unanalyzed fitness-affecting trait) can introduce in the estimation of selection differentials and proposes an alternative method of analysis that does not suffer from these biases. We emphasize that even if assumptions of the Lande-Arnold model hold, predictions of evolutionary responses based on within-generation phenotypic changes can be difficult.

For example, if one repeats a bivariate selection experiment, using the same selection intensities (i.e., values for [s.sub.i] are identical in the two experiments) on the same initial population (i.e., the elements of the G matrix are also identical), but maintaining slightly different environmental conditions (such that the elements of the E matrix differ between the two populations), the resultant responses in the two experiments may not only differ quantitatively, but qualitatively. In other words, the two experimental populations may actually diverge, even though they are initially identical genetically and even though the selection regimes are the same. Returning to the numerical example given above, suppose these conditions are maintained in a selection experiment in which the experimentally determined selection differentials are [s.sub.1] = 1 and [s.sub.2] = 3. In this case, character 1 will show a negative response since [s.sub.1] [less than] 0.42[s.sub.2]. Now imagine a second selection experiment on this same population where [E.sub.12] = 0.35 rather than 0.45 (i.e., [P.sub.12] = 0.4 instead of 0.5). In this case, the evolutionary response of character 1 is positive [[Mathematical Expression Omitted], from equation (7)]. Thus, these two genetically replicated populations will exhibit character divergence on the basis of a difference in the environmental covariance between the two characters. The response of character 2 is positive in both of these hypothetical environments, but it is easy to show that both characters can show reversals under more exaggerated environmental differences.

Correlations among characters seriously complicate the extension from observable within-generation selection differentials to cross-generational selection responses. Some such correlations (i.e., negative genetic) are considered very important in constraining adaptive evolution. A lack of correspondence between the direction of selection differentials, the directional selection gradient, and selection responses is commonly taken as evidence for correlations between characters in the analysis and characters that are missing. We have shown, through a simple example, that the evolutionary responses of correlated characters can take on wide ranges of values, even within limited areas of parameter space. To reemphasize the conclusions of Lande and Arnold (1983), to reliably predict the direction of evolution, detailed quantitative knowledge of phenotypic and genetic variances, and strengths of correlations and selection are required. This is true even when all relevant characters are included in the analysis.


We thank M. Lynch, R. Lande, D. Houle, D. Price, J. Willis, and S. Via for helpful discussion. R. Lande and three anonymous reviewers provided comments on an earlier draft. Financial support was provided by a National Institutes of Health (NIH) genetics training fellowship (PHS GM0741317) to T.T.K. and NIH grant PHS GM36827 and National Science Foundation grant BSR8911038 to M. Lynch.


Lande, R., and S. J. Arnold. 1983. The measurement of selection on correlated characters. Evolution 37:1210-1226.

Rausher, M. D. 1992. The measurement of selection on quantitative traits: biases due to environmental covariances between the traits and fitness. Evolution 46:608-626.
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Author:Deng, Hong-Wen; Kibota, Travis T.
Date:Jun 1, 1995
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