Discussion

In this paper explicit genetic models are considered for the evolution of a quantitative polygenic character under directional selection. We assumed that the character is coupled to an uncorrelated quantitative character via `hidden' pleiotropic effects. The term `hidden pleiotropic effects' means that the pleiotropic effects do not contribute to genetic correlations between the characters (Turelli, 1985). It is shown above that the evolution of the mean value is influenced by selection on a pleiotropically coupled character even in the absence of genetic correlations. This effect is due to apparent stabilizing selection caused by deleterious pleiotropic effects. Dynamically the effect is mediated through an association between the mean value of one character and the genetic variance of the pleiotropically linked character. This association was first predicted by Turelli (1988a, his term in Eq. (32)).

The intuitive explanation of the effects described in this paper is the following: the two characters are uncorrelated, the first is under directional selection and the second under stabilizing selection. Consequently the first character responds to directional selection by a change in the mean value, if there is additive genetic variation for this character. Because of pleiotropic effects the selection response of the first character can be associated with a change in the genetic variance of the second character, which is under stabilizing selection. The sign of the change in genetic variance depends on the frequency of the genes. If the genes are rare, then directional selection on the first character increases their frequencies and the variance of the second character increases due to pleiotropic effects. However, this increase in variance has a negative effect on fitness since the second character is under stabilizing selection. Consequently, the evolution of the first character is inhibited by stabilizing selection on hidden pleiotropic effects and the selection response of the first character is smaller than predicted from models ignoring pleiotropic effects. If stabilizing selection on pleiotropic effects is strong relative to directional selection, the evolution of the first character may be completely inhibited, leading to a genetic constraint on adaptation. This effect is reminiscent of Winnie-the-Pooh getting stuck in the rabbit hole because he ate too much honey (the `Pooh-effect').

However, stabilizing selection on hidden pleiotropic effects is not universally inhibiting to the rate of adaptation. If the frequencies of the genes are high, i.e. during late phases of gene substitution, the variance of the second character decreases. This effect leads to an additional increase of mean fitness beyond the increase caused by the change of the mean value of the first character, which accelerates evolution.

The main results have been shown analytically for two as well as for more loci. In the case of more than two genes additional terms than the ones discussed above appear in the equation for the evolution of the first character which slightly mitigate the Pooh-effect. Selection creates a linkage disequilibrium in favor of gametes hiding some of the pleiotropic effects (so-called Bulmer effect, Bulmer, 1980). But this effect is small in the range investigated so far (n 8).

The results can also be interpreted in terms of apparent stabilizing selection. Apparent stabilizing selection can be caused by deleterious pleiotropic effects (Robertson, 1956; Barton, 1990; Keightley and Hill 1990; Gavrilets and deJong, 1983; Kondrashov and Turelli, 1992; Caballero and Keightley 1994). Besides the direct selection on the first character the deleterious pleiotropic effects induce an apparent stabilizing selection. This apparent stabilizing selection can be stronger than the direct selection. Hence, the Pooh-effect results if apparent stabilizing selection dominates over the directional selection on the first character. However, if directional selection is strong relative to apparent selection, there is no selection limit predicted by this model. This is in contrast to other models of the interaction between directional selection and pleiotropy (Keightley and Hill 1990; Gavrilets and deJong, 1983; Kondrashov and Turelli, 1992; Caballero and Keightley 1994). The main difference in this model is that deleterious effects are not unconditionally deleterious. Deleterious pleiotropic effects are modeled as additive effects on a character which is under stabilizing selection. Additive effects of one gene can be compensated by additive effects of opposite sign at another locus. This is to say that the pleiotropic effects are epistatic with respect to fitness. Whether a pleiotropic effect increases or decreases fitness depends on the current breeding value for the second character of the genotype. If the breeding value of the second character is at the optimum for this character, every pleiotropic effect is deleterious. However, if the breeding value of the second character does not coincide with its optimum, the fitness effect might be positive or negative. If the pleiotropic effect brings the breeding value closer to the optimum of the second character, fitness increases, if it leads away from the optimum, the fitness decreases.

Unconditionally deleterious effects are a good model for short term selection responses, if the likelihood of compensatory mutations is low (Caballero and Keightley, 1994). However, unconditionally deleterious mutations are not a good model of sustained selection responses or evolutionary adaptation. If each mutation is unconditionally deleterious, long term evolutionary adaptation and indeed the sustained existence of life would be impossible. We therefore prefer models which allow compensatory pleiotropic mutations to occur.

The effect on asymptotic rates of evolution in finite populations of hidden pleiotropic effects was investigated by stochastic simulations. It was found that the asymptotic rate of evolution is reduced by stabilizing selection on hidden pleiotropic effects. The magnitude of the reduction depends on the strength of directional selection as compared to the stabilizing selection. If this ratio is below the limit for the Pooh-effect, predicted by the deterministic theory, then the rate of evolution is more than 50% of the rate predicted by the standard one character theory. Under these conditions the influence on the rate of evolution is moderate. Considerable influence is found once the apparent stabilizing selection dominates over directional selection and can then be as low as 5% of the value predicted by one character theory.

The biological implications of these results can be divided into two main topics: the potential of the Pooh-effect to happen in real populations, and the implications for long-term selection response. The Pooh-effect is the complete inhibition of selection response on a character due to stabilizing selection on the hidden pleiotropic effects. The plausibility of this effect in nature is closely related to a frequently discussed problem in theoretical quantitative genetics, namely the dynamics of the genetic variance in response to directional selection. The Pooh-effect in the models is mediated through an inflation of the genetic variance of the second character via pleiotropic effects. Hence the Pooh-effect can only occur if the genetic variance is increasing when a population experiences a shift from stabilizing selection to directional selection. This is to be expected if genetic variance in the initial population is caused by rare alleles with large effects (Barton and Turelli, 1987). However, in selection experiments a significant increase in genetic variance has not been observed (see discussion in Turelli, 1988a; Bürger, 1993). There are several possibilities to explain this fact (see Turelli 1988a). One is that genetic variance is not maintained by mutation selection equilibrium and is thus caused by more frequent alleles than predicted by mutation-selection theory. This is possibly the case for abdominal bristle number (Lai et al., 1994; Long et al., 1995). Other possibilities are that the rare genes are not present in the comparatively small experimental populations, or that linkage disequilibrium masks a buildup of genic variance. Finite population size models predict that a significant increase in genetic variance can only be expected in populations larger than 500 (Bürger 1993). Hence it seems unlikely that the Pooh-effect is a frequent cause of a genetic constraints to the adaptation of quantitative characters in small populations. However, in large populations and with strong stabilizing selection on pleiotropically linked characters a genetic constraint on phenotypic adaptation can be expected. The data required to assess the plausibility of the Pooh-effects consist of multivariate selection surfaces for ecologically important characters and data on the average mutational effects on these characters.

Another metaphor that could be invoked is `inner friction' which resists phenotypic change. The metaphor is actually relatively close, since the effects of hidden pleiotropic effects depend on the `atomic' (Mendelian) structure of quantitative variation, much as inner friction of fluid flow is a reflection of the molecular structure of fluids. This effect is not predicted by infinite loci models.

Deviations from the standard prediction of selection response have been modeled as a conflict between artificial truncation selection and stabilizing natural selection (Zeng and Hill, 1986). These models are not directly comparable with the one analyzed here. The present model is a model of natural selection on two characters and does not represent a conflict between artificial and natural selection.

Without pleiotropic effects the asymptotic rate of evolution under sustained directional selection has been shown to be proportional to the effective population size, the mutational variance and the selection intensity: (Bürger, 1993; Hill, 1982; Keightley and Hill, 1987). The results in this paper show that this prediction is a good upper limit to the rate of evolution under most circumstances. Recently Barton has developed a method for predicting the substitution rate under the influence of deleterious mutations (Barton, 1995). He showed that the impact can be modeled as a change in effective population size. In our simulation it was found that the reduced rate of evolution was primarily determined by a smaller asymptotic genetic variance than expected without selection against pleiotropic effects. Intuitively one can explain this effect by a lower fraction of mutations that become selected due to selection against pleiotropic effects. This effect can be understood as a lower effective mutational variance , feeding the selection response.

It has been shown that the observed rate of phenotypic evolution is many orders of magnitude lower than expected on the basis of phenotypic evolution equations (Lande, 1976; Lynch, 1990). There are many explanations for this fact. One of them are scaling effects on the estimation of evolutionary rates (Gingerich, 1993). Another obvious explanation is stabilizing selection in the respective traits (Charlesworth et al., 1982; Bürger & Lynch, 1995). The present analysis suggests that apparent stabilizing selection induced by hidden pleiotropic effects is also causing adaptive inertia to weak directional selection. If the basic assumptions of our model are true (i.e. extensive pleiotropy and mosaic evolution) one can expect adaptive intertia to be a common phenomenon of phenotypic evolution.