Abstract
Mutationselection models provide a framework to relate the parameters of microevolution to properties of populations. Like all models, these must be subject to test and refinement in light of experiments. The standard mutationselection model assumes that the effects of a pleiotropic mutation on different characters are uncorrelated. As a consequence of this assumption, mutations of small overall effect are suppressed. For strong enough pleiotropy, the result is a nonvanishing fraction of a population with the “perfect” phenotype. However, experiments on microorganisms and experiments on protein structure and function contradict the assumptions of the standard model, and Kimura’s observations of heterogeneity within populations contradict its conclusions. Guided by these observations, we present an alternative model for pleiotropic mutations. The new model allows mutations of small overall effect and thus eliminates the finite fraction of the population with the perfect phenotype.
SOME of the most important problems in microevolution—overall rates and types of mutation and the distribution of mutant effects—are yielding to direct experiments on microorganisms (Imhof and Schlötterer 2001; Wlochet al. 2001). To relate the results of these experiments to properties of populations currently requires a theoretical framework. In principle, theory combined with experiments can address such large issues as the maintenance of genetic variation (Shawet al. 2000; Zhanget al. 2002) and the role of complexity in evolution (Waxman and Peck 1998; Wagneret al. 1999; Coppersmithet al. 1999; Hartwellet al. 1999; Wagner and Mezey 2000; Weiss and Fullerton 2000). Theoretical interest in these questions has been spurred by Waxman and Peck (1998), who reported an intriguing property of a standard model of mutation and selection with pleiotropy (Turelli 1985). As the pleiotropy—the number of phenotypic characters that can be affected by each mutation—is increased, the steadystate distribution of phenotypes progressively narrows. When three or more characters can be affected simultaneously, Waxman and Peck (1998) found that in steady state a finite fraction of the population acquires the “perfect” phenotype. For a continuumofalleles model, such an atom of probability at the upper limit of fitness was first reported by Kingman (1978) and considered more generally by Bürger (1988, 2000). Waxman and Peck (1998) were the first to report that a plausible fitness function when combined with pleiotropy can also lead to an atom of probability.
Here we argue that this reduction of variation (the atom) arises from an unrealistic property of the standard model for pleiotropic mutations (Turelli 1985). Namely, effects of a mutation on different characters are uncorrelated and, hence, the probability of a mutation of small overall effect is strongly suppressed. In this way, the Turelli model for pleiotropic mutations is similar to discretealleles models in which there is a minimum fitness difference between the mostfit and nextmostfit phenotypes. Indeed, in Eigen’s (1971) discrete phenotype model for molecular quasispecies, one finds a finite fraction of the population with the optimal phenotype, below a certain error threshold (Eigenet al. 1988). The WaxmanPeck model behaves in exactly the same way: There is a mutationrate threshold below which one finds an “atom” of probability in the mostfit phenotype. However, the discrete phenotype model employed by Eigen is not necessarily appropriate to treat pleiotropy in proteins. In fact, discretealleles models, or the Turelli continuumofalleles model that suppresses mutations of small overall effect, appear to be inconsistent with observed distributions of mutational effects in proteins (Wlochet al. 2001) and with the evidence that many aminoacid substitutions are nearly silent (Lim and Sauer 1989; Matthews 1995). Moreover, the prediction of the WaxmanPeck model that a finite fraction of the population attains the perfect phenotype seems to be incompatible with Kimura’s observations of allelic heterogeneity within populations (Kimura 1979).
Here, we present an alternative model for how the magnitudes of the phenotypic effects of mutations scale with the degree of pleiotropy. Data from experiments on microorganisms, from aminoacid substitutions, and from studies of populations all appear to be consistent with the alternative model. Its main new feature is that the distribution of the effects of mutations includes a finite probability for mutations of small overall effect, even as the degree of pleiotropy increases. A similar model for pleiotropy has been used recently by Orr to study rates of adaptation (Orr 2000). A result of the new distribution of mutations is that the steadystate distribution of fitnesses is universal and independent of the degree of pleiotropy. In particular, at steady state, there is no preservation of the perfect phenotype; i.e., there is no atom of probability at the upper limit of fitness.
Resolution of the issues surrounding pleiotropic genes will likely depend on more input from experiments. The existence or nonexistence of an atom of probability at the upper limit of fitness and the suppression or nonsuppression of mutational effects of small magnitude for pleiotropic genes are important tests of theory. In light of our present results, we consider how experiments might further probe the scaling of mutational effects with pleiotropy. Of particular value would be information on the distribution of fitnesses among singlelocus mutants generated from a single ancestral line.
THE MODEL
In the model, an asexual species evolves under opposing pressure of selection and mutation. At each discrete generation, the entire population is first subject to selection, and then a fraction Θ of the population is subject to mutation. Finally the entire population is renormalized to its original size.
To incorporate pleiotropy, organisms are considered to have Ω distinct characteristics (x_{1},..., x_{Ω}). Selection acts on the phenotype (x_{1},..., x_{Ω}) through the square magnitude
In the model, all characteristics x_{i} appear on the same footing. We consider each x_{i} to represent a normalized measure of some physical characteristic, with the normalization chosen to yield the simple fitness relation expressed by Equations 1 and 2.
The distribution of mutations is pleiotropic—that is, each mutation may affect all Ω separate characteristics. The way in which mutations are modeled is extremely important in determining the steadystate distribution of phenotypes. To treat pleiotropy, Turelli (1985) introduced a mutation model in which the probability of a simultaneous change of the first character by δx_{1}, the second character by δx_{2}, etc., is given by the multivariate Gaussian distribution
When pleiotropy is present, i.e., when Ω≥ 2, the factor of r^{Ω1} in Equation 4 strongly suppresses the probability that a mutation has an effect of small total magnitude. This effect is shown graphically in Figure 1b. In general, mutational effects of small total magnitude will be suppressed in any pleiotropy model in which effects on different characters are uncorrelated for a single mutation.
We introduce an alternative model for the effects of a mutation, in which there is no suppression of mutations of small total effect. The mutation model is shown schematically in Figure 1a: First a random direction
In short, we propose a mutation model that uses a Gaussian along a random direction, rather than the multivariate Gaussian used by Turelli. As we show in calculations and results, this simple alteration to the Turelli model changes the steadystate distribution of phenotypes qualitatively by extinguishing the δ function at perfect phenotype. Our choice of a halfGaussian distribution of mutational effects is mathematically convenient and reduces to the standard form in the absence of pleiotropy. However, any smoothly decreasing distribution with the same value near zero, p(r = 0) = (2/(πm^{2}))^{1/2}, will yield essentially the same steadystate distribution of phenotypes.
Which is more realistic, Turelli’s original model or our modified version? It is shown in the discussion that the alternative model is consistent with quantification of mutational effects in microorganisms (Wlochet al. 2001), with detailed studies of aminoacid mutations (Lim and Sauer 1989; Matthews 1995) and with Kimura’s observations on allele variation in populations (Kimura 1979), whereas the Turelli model is in conflict with these same data. This observation is, in our view, a strong argument that our alternative mutation model is more realistic.
CALCULATIONS AND RESULTS
Using our model for mutations, we find that the steadystate distribution of phenotypes as a function of
To derive these results, we use the method employed by Waxman and Peck (1998). Briefly, the steadystate distribution of phenotypes Φ(r) must satisfy the recursion relation
Within the houseofcards approximation, the steadystate distribution as a function of phenotype magnitude
It is nevertheless apparent from (13) that radically different choices of the distribution of the effects of mutations p(r) can yield qualitatively different steadystate distributions ϕ(r). The essential difference between our model and that of Turelli (1985) is that in cases with pleiotropy his distribution of magnitudes for the effects of each mutation p˜(r) vanishes at r = 0. This leads to the essential difference between the steadystate results: For Ω≥ 3 and a high enough mutation rate, Turelli’s model (Waxman and Peck 1998) produces a finite fraction of the steadystate population with the perfect phenotype, and our model does not. Implicit in their distribution of the effects of mutations p˜(_{r}_{) is} the assumption that the probability for a mutation in one of the perfect organisms to leave it almost perfect is strongly reduced by pleiotropy. Specifically, for an organism with the perfect phenotype (0,..., 0), the probability of a mutation yielding a phenotype of total magnitude r is p˜(_{r}_{)} ∝ r^{Ω1}exp(r^{2}/2m^{2}). It is precisely the factor of r^{Ω1} that reduces the probability of almost perfect mutations [cf. the discussion by Waxman and Peck (1998) under the heading “[O]rigin and explanation of the results”]. Put intuitively, the WaxmanPeck model produces almost perfect mutants at such a slow rate that the entire continuum of mutants cannot outcompete the single perfect phenotype. In contrast, when there is a nonvanishing probability density for perfect organisms to mutate into nearly perfect ones—that is, if p(0) ≠ 0—the result of Waxman and Peck (1998) does not hold: No fraction of the population has the perfect phenotype [see Bürger (1988, 2000) for a more general statement of this condition]. In our model for mutations, p(0) = (2/(πm^{2}))^{1/2}. Consequently, there is no preservation of the perfect phenotype by part of the population.
The steadystate distribution with respect to a single characteristic Φ_{1}(x_{1}) can be found from the full distribution (11). When there is only a single characteristic (Ω= 1), the result is the same as that obtained by Waxman and Peck:
As an example of the distribution with respect to one characteristic in the presence of pleiotropy, we consider the case of three characteristics, Ω= 3. The steadystate distribution for a single characteristic is
DISCUSSION
Here, we have proposed an alternative to the standard model for pleiotropic mutations (Turelli 1985). At issue is the scaling of mutational effects with pleiotropy. That is, as the number of phenotypic characters Ω that can be affected by a single mutation increases, how does the distribution of fitness effects change? The importance of this question lies in the role of complexity in evolution. Increased pleiotropy represents an increase in complexity, and the scaling of mutational effects determines the balance between mutation and selection, which eventually determines the genetic variation within a population. The Turelli model (3) suggests that increased complexity (pleiotropy) leads to a collapse of genetic variation, with the perfect phenotype dominating for large enough pleiotropy (Waxman and Peck 1998). In contrast, our model for mutations (5) produces, at steady state, a smooth, universal distribution of phenotypes, independent of pleiotropy.
Fundamentally, the two models differ regarding the scaling of mutational effects with pleiotropy. For a single character without pleiotropy (Ω= 1), a Gaussian distribution of the phenotypic effects of mutations is generally regarded as plausible. The TurelliWaxmanPeck approach to pleiotropy simply takes a superposition of these Gaussians, one for each character. Underlying this approach is the assumption that a single mutation has completely uncorrelated effects on the various characters. The consequence of this assumption is a suppression of mutations of small overall effect, leading to a finite fraction of the population with the perfect phenotype (an atom). Indeed, for strong enough pleiotropy, this atom dominates the population. In contrast, our assumption is that mutational effects on different characters are correlated—a mutation with a small effect on one character is likely to have a small effect on the other characters as well (cf. Figure 2). There is thus always a finite probability of mutations of small overall effect, and, as a result, there is no preservation of the perfect phenotype by part of the population. To assess which model for mutations is preferable to describe natural populations, we consider evidence at several levels: populations, organisms, and individual proteins.
Populations: At the level of populations, the scaling of mutational effects with pleiotropy can be addressed via the neutral theory of molecular evolution (Ohta and Gillespie 1996). The neutral theory accounts consistently for the statistics of variations of alleles. To explain observed allelic heterogeneity in populations, Kimura (1979) was led to hypothesize a Γ distribution of mutations of selective disadvantage s. [In terms of the mutationselection model of Waxman and Peck (1998), selective disadvantage is given by the complement of the fitness s = 1  w.] In Kimura’s analysis, the essential property of the Γ distribution is that the frequency of mutations of small selective disadvantage obeys a power law
Organisms: Additional evidence for the scaling of mutational effects with pleiotropy comes from direct studies of mutations in microorganisms. For the yeast Saccharomyces cerevisiae, Wloch et al. (2001) determined the overall rate and distribution of selective disadvantages by measuring growth rates of tetrads of spores from single homozygous diploid cells. For spontaneous mutations, they observed a distribution of selective disadvantages strongly peaked at the smallest observable magnitudes. The observed distribution is consistent with the form F(s) ∝ s^{1/2} inferred by Kimura and predicted by our model. The observed distribution is inconsistent with the distribution F(s) ∝ s^{(Ω2)/2} predicted by Waxman and Peck (1998), except in the nonpleiotropic case Ω= 1.
In the experiments there is no characterization of the genes responsible for the deleterious mutations. In principle, therefore, the observed distribution of selective disadvantages could be dominated by nonpleiotropic genes. If the number of pleoitropic genes is very small, then a suppression of mutations of small effect for these genes could be hidden in the data. Therefore, until the genes responsible for mutations can be determined, experiments of this nature cannot absolutely rule out a suppression of mutations of small effect for pleiotropic genes.
Single proteins: One may argue, as above, that only a small set of pleiotropic proteins is addressed by the mutationselection model of Waxman and Peck (1998) and that these particular proteins have strongly suppressed withinpopulation variation. Nevertheless, one can reject this argument on the basis of protein chemistry. Even for proteins with pleiotropic effects, some amino acids are more important for function than others. In particular, similar aminoacid substitutions within the core, or far from the active site, are generally silent or nearly silent with respect to function. Examples abound. Two extensive studies (Lim and Sauer 1989; Matthews 1995), in which aminoacid sequence changes have been correlated with protein structure and function, demonstrate clearly that a wide variety of aminoacid substitutions alter activity in a continuous fashion: from no detectable change, through modest reductions in function, to destruction of function.
In one study, Lim and Sauer (1989) carefully documented the relationship between the structure and function of λ repressor mutants, finding that aminoacid substitutions perturb both in a continuous fashion. In another, summarized by Matthews (1995), the crystal structure and activity of ∼300 lysozyme mutants were examined. This study shows unambiguously that protein function is a continuous function of aminoacid substitution. For example, approximately half of the mutants displayed wildtype activity, and the others spanned the range from no detectable activity to three times that of wild type.
Both studies dealt with proteins for which mutants have profound pleiotropic effects on viral fitness. Bacteriophage λ repressor directly or indirectly controls the entire viral life cycle, including such important contributors to organismal fitness as host range, DNA packaging efficiency, DNA replication rates, and burst size. T4 lysozyme is directly involved in burst size and latency. The studies suggest that for these pleiotropic proteins many substitutions of amino acids exist, which will have small effect on all the characters influenced by the protein. Of course, these studies were carried out entirely in vitro, with purified proteins. However, in a recent article by Ghaemmaghami and Oas (2001) it was shown that the stability of λ repressor is the same in vivo and in vitro. Moreover, this was true as well for a mutant form, Q33Y, which stabilizes the repressor in both environments. Thus there is good reason to believe that in vitro measures of function reflect in vivo activity.
By contrast, in the standard model for pleiotropic mutations (Turelli 1985), the magnitudes of the effects of a single mutation on distinct characters are uncorrelated. The absence of correlation leads directly to (a) the suppression of mutations of small overall effect and (b) the preservation of the perfect phenotype (Waxman and Peck 1998). As we have shown here, a is in conflict with protein chemistry and data from studies of mutations in microorganisms and b is in conflict with Kimura’s analysis of heterogeneity in populations.
We can highlight the problems inherent in the mutation model of Turelli (1985) by appealing to the metaphor used by Fisher (1958) in his discussion of pleiotropy. He used a geometric argument in three dimensions, to show that the probability of very small, i.e., nearly neutral, pleiotropic mutations leading to increased fitness was ½ and that as the effect of the mutations became stronger, the probability of increased fitness rapidly fell to zero. He likened this to the chances that an outoffocus microscope could be focused by small vs. large random adjustments. A series of small adjustments has some chance of correctly focusing the instrument, whereas very large excursions have none, with the probability of correct focus falling off rapidly with the magnitude of the excursions. Viewed in terms of proteins, the Turelli model disallows small adjustments, and this is contrary to the available evidence on protein structure and function.
Perspective: Advances in molecular biology may soon make it possible to directly test the distribution of mutant effects for pleiotropic genes. One approach to probe the distribution of pleiotropic mutations would be to measure the fitness w of various mutants at a single locus in an otherwise fixed genetic background. By focusing directly on fitness, this approach avoids the difficulty of quantifying phenotypes. Different models for pleiotropy can be contrasted by their different predictions for the distribution of selective disadvantages F(s). The model of TurelliWaxmanPeck predicts F(s) ∝ s^{(Ω2)/2}, for a gene with pleiotropic dimension Ω. In contrast, the model we have presented predicts F(s) ∝ s^{1/2} independent of pleiotropy. For genes with any degree of pleiotropy, these predictions are strikingly different. Specifically, the density of small disadvantages F(0) is finite for Ω= 2 and equal to zero for Ω> 2 in the TurelliWaxmanPeck model, while the same quantity F(0) always diverges in our alternative model.
Bacteriophage T4 would be an ideal candidate for experiments to probe F(s), given the wealth of information on structure and function mentioned above. Unfortunately, T4 is difficult to engineer, so that the many hundred mutants studied by the Matthews group cannot now be studied in vivo. Bacteriophage λ, with a smaller set of wellstudied repressors, is also a good test system because the available mutants are easily engineered into standard laboratory strains. Moreover, phage λ can be propagated both sexually and asexually, in lytic and lysogenic modes and under a wide variety of environmental conditions (Littleet al. 1999).
The general advantage of direct experimental determination of mutant effects, compared to population studies, is that many confounding effects can be avoided. For example, population measurements that attempt to infer the distribution of mutant effects from observed heterogeneity may be confounded by uncertainties in effective population sizes, mutation rates, and steepness of local fitness functions. Even attempts to contrast evolution rates for pleiotropic vs. nonpleiotropic genes may be impeded by intrinsic differences in mutation rates between genes. The interplay of species history and fitness landscape can lead to many subtle complications in population analysis—for example, a recent theoretical study of RNA evolution indicates that environmental history can “canalize” genes into regions of little genetic variability (Ancel and Fontana 2000).
SUMMARY AND CONCLUSIONS
In summary, we have presented a new model for the effects of pleiotropic mutations on an evolving asexual species. In contrast to previous models (Turelli 1985; Waxman and Peck 1998), we do not assume that the effects of a mutation on different characters are uncorrelated. Consequently, and in contrast to the results of Waxman and Peck (1998), there is no suppression of mutations of small overall effect and no preservation of the perfect phenotype by a finite fraction of the population. Our model appears to represent an improvement over previous treatments insofar as it is consistent both with Kimura’s observations of heterogeneity in populations and with experimental results both on mutations in microorganisms and on aminoacid substitutions in proteins. Direct experimental measurement of the mutational distribution of selective disadvantages of singlelocus mutations would clearly differentiate the two models.
The role of complexity in fixing characteristics of a species or group of species remains an open question. Our work suggests that simple mutationselection models that allow for a realistic probability of mutations of small effect do not produce fixation.
Footnotes

Communicating editor: M. W. Feldman
 Received November 25, 2002.
 Accepted March 28, 2003.
 Copyright © 2003 by the Genetics Society of America