Scaling of Mutational Effects in Models for Pleiotropy

Corresponding author: Ned S. Wingreen, 4 Independence Way, Princeton, NJ 08540., wingreen{at}nec-labs.com (E-mail)

Communicating editor: M. W. FELDMAN

	ABSTRACT

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

Mutation-selection 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 mutation-selection 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 SCHLOTTERER 2001 ; WLOCH et 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 (SHAW et al. 2000 ; ZHANG et al. 2002 ) and the role of complexity in evolution (WAXMAN and PECK 1998 ; WAGNER et al. 1999 ; COPPERSMITH et al. 1999 ; HARTWELL et 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 steady-state 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 continuum-of-alleles model, such an atom of probability at the upper limit of fitness was first reported by KINGMAN 1978 and considered more generally by BURGER 1988 , BURGER 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 discrete-alleles models in which there is a minimum fitness difference between the most-fit and next-most-fit phenotypes. Indeed, in EIGEN 1971 discrete phenotype model for molecular quasi-species, one finds a finite fraction of the population with the optimal phenotype, below a certain error threshold (EIGEN et al. 1988 ). The Waxman-Peck model behaves in exactly the same way: There is a mutation-rate threshold below which one finds an "atom" of probability in the most-fit phenotype. However, the discrete phenotype model employed by Eigen is not necessarily appropriate to treat pleiotropy in proteins. In fact, discrete-alleles models, or the Turelli continuum-of-alleles model that suppresses mutations of small overall effect, appear to be inconsistent with observed distributions of mutational effects in proteins (WLOCH et al. 2001 ) and with the evidence that many amino-acid substitutions are nearly silent (LIM and SAUER 1989 ; MATTHEWS 1995 ). Moreover, the prediction of the Waxman-Peck 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 amino-acid 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 steady-state 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 single-locus mutants generated from a single ancestral line.

	THE MODEL

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

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 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 distinct characteristics (x₁, ... , x). Selection acts on the phenotype (x₁, ... , x) through the square magnitude

(1)

with the fitness (probability of survival at each generation) being given by

(2)

so that the width of the fitness function is V^1/2_s. WAXMAN and PECK 1998 explicitly included an environmental contribution to phenotype. The effect can be entirely absorbed into a redefinition of V_s, so we neglect the environmental contribution without loss of generality.

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 byEquation 1 andEquation 2.

The distribution of mutations is pleiotropic—that is, each mutation may affect all separate characteristics. The way in which mutations are modeled is extremely important in determining the steady-state 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 x₁, the second character by x₂, etc., is given by the multivariate Gaussian distribution

(3)

where m² is the variance of mutant effects for a single character. Mutations with effects of small total magnitude are suppressed in any pleiotropy model, such as (3), in which the effects on different characters are uncorrelated. Specifically, in the Turelli model (3) the probability of a change of magnitude is

(4)

Consider, for example, the case = 3. A single mutation changes the phenotype by (x₁, x₂, x₃), yielding a total magnitude of change . Given that each of x₁, x₂, x₃ is chosen independently from a Gaussian distribution, what is the probability density (r) for a particular value of r? The result is given byEquation 4. As a simple derivation, note that the integral over all r of (r) must be the same as an integral over all x₁, x₂, and x₃ of f(x₁, x₂, x₃). So,

The factor of 4r² comes from the transformation to spherical coordinates, leading to (r) r²exp[-r²/(2m²)].

When pleiotropy is present, i.e., when 2, the factor of r^-1 inEquation 4 strongly suppresses the probability that a mutation has an effect of small total magnitude. This effect is shown graphically in Fig 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.

View larger version (36K): [in this window] [in a new window] [Download PPT slide]   Figure 1. (a) Schematic diagram of our mutation model, shown for the case that three characteristics may be affected by each mutation ( = 3). First, the direction of the mutation r is chosen at random. Second, the magnitude of the effect of the mutation is chosen from the distribution p(r). The new phenotype is given by r* + r, where r* is the parent phenotype. (b) (—) Distribution of mutational effect magnitudes p(r) given byEquation 5. Within our model, p(r) is the same for all numbers of characteristics . (- - -) Distribution of mutational effect magnitudes for the Turelli-Waxman-Peck model (r) (r)-1 exp[-(r)2/(2m2)], for = 3. When pleiotropy is present, > 1, the model of TURELLI 1985 and WAXMAN and PECK 1998 results in a suppression of mutations with effects of small overall magnitude.

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 Fig 1A: First a random direction r is chosen in the space of phenotypes. Second, a magnitude for the effect of the mutation is chosen from the half-Gaussian distribution

(5)

This mutation model is identical to Turelli's in the case with no pleiotropy, i.e., with only a single characteristic ( = 1). However, since the same distribution p(r) is also used for > 1, there is no suppression of mutational effects of small magnitude when pleiotropy is present. Within this model, the probability distribution of mutations as a function of the change of phenotype is

(6)

where . Numerically, it is straightforward to generate mutations with this distribution of phenotypic effects: A random direction r can be obtained using Turelli's multivariate Gaussian distribution (3). Then a magnitude r can be generated from the half-Gaussian distribution (5). Within this model, magnitudes of mutational effects on different characters are correlated. For example, in Fig 2, we plot the expectation of the square magnitude of one character x²_i as a function of the square magnitude of another character x²₁ for the case = 3,

(7)

where . It can be seen in Fig 2 that if x²_i is small, then the other x²₁ are also likely to be small.

View larger version (11K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Correlation of mutational effects on different characters within our model for the case that three characters may be affected by each mutation ( = 3). The expectation <x2i>/m2 is plotted as a function of log(x21/m2), according toEquation 7. Mutations with small effect on x1 are likely to also have small effect on the other xi. Within the Turelli-Waxman-Peck model, mutational effects on different characters are uncorrelated, so <x2i>/m2 = 1, independent of x21.

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 steady-state distribution of phenotypes qualitatively by extinguishing the function at perfect phenotype. Our choice of a half-Gaussian 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/(m²))^1/2, will yield essentially the same steady-state 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 (WLOCH et al. 2001 ), with detailed studies of amino-acid 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

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

Using our model for mutations, we find that the steady-state distribution of phenotypes as a function of is independent of the number of characteristics . The distributions with respect to a single characteristic do depend on , but there is never an atom, i.e., never a finite fraction of the steady-state population with the perfect phenotype.

To derive these results, we use the method employed by WAXMAN and PECK 1998 . Briefly, the steady-state distribution of phenotypes (r) must satisfy the recursion relation

(8)

where is the mutation rate, f(r) is the distribution of the effects of mutations (6), w(r) is the fitness function (2), and

(9)

is the fraction of the population that survives viability selection each generation at steady state. [Dividing by normalizes the population, (r)dr = 1, following selection and mutation.] Within our model for mutations, > 1 - , which allows us to rewrite the recursion relation as

(10)

To solve for the steady-state distribution, we apply the house-of-cards approximation (KINGMAN 1978 ; TURELLI 1984 ). This approximation is valid provided = V_s/m² << 1 and consists of replacing f(r - r')w(r')(r')dr' inEquation 10 by . Essentially, the house-of-cards approximation exploits the fact that the steady-state distribution is narrow compared to the width of the distribution of the effects of mutations for << 1. Hence, the phenotype after mutation is largely independent of the parent phenotype. The house-of-cards substitution yields

(11)

where the distribution of mutational effects f(r) is given byEquation 6. To obtain , we integrate both sides ofEquation 11 over all phenotypes. Making the reasonable assumption V_s/m² >> 1 (TURELLI 1984 ), the Gaussian w(r) can be replaced by 1 - r²/(2V_s), with the result

(12)

This is the same result obtained by WAXMAN and PECK 1998 for = 1. Within our model for mutations, it now applies for all numbers of characteristics .

Within the house-of-cards approximation, the steady-state distribution as a function of phenotype magnitude is independent of and is given by

(13)

(14)

where = V_s/m². The steady-state distribution of phenotype magnitudes (r) as given byEquation 14 is plotted as insets in Fig 3. The distribution (r) is obtained by integrating (r) inEquation 11 over angles at fixed r, and (r) satisfies the normalization . The variance of (r) is given by r² 2m² = 2V_s for << 1 and is independent of the variance of mutational effects m². However, since the distribution has a Lorentzian form, (r) 1/[r² + 2²m²] for r << m, and since the variance diverges for a pure Lorentzian distribution, the variance r² depends strongly on the behavior of (r) at large r. A more informative characterization of the distribution is therefore the half-width-at-half-maximum r_HWHM (2)^1/2m = (2)^1/2V_s/m for << 1. The narrowing of r_HWHM with increasing variance of mutational effects m² reflects the decrease in the density of mutations with effects of small overall magnitude, p(r = 0) = (2/(m²))^1/2. The Lorentzian form for (r) for r << m can be traced to the quadratic maximum of the fitness function w(r) = exp(-r²/2V_s). The fall off of (r) as 1/r² therefore corresponds to an inverse dependence of the steady-state population on the death rate 1 - w(r) r²/2V_s. While we have taken the distribution of mutant effects p(r) to be half-Gaussian, any smoothly decreasing distribution with the same value near zero would yield essentially the same steady-state distribution of phenotypes (r). This is because (r) falls off rapidly, as 1/r², over a narrow region where p(r) is approximately constant at its r = 0 value.

View larger version (41K): [in this window] [in a new window] [Download PPT slide]   Figure 3. (a) Steady-state distribution of phenotypes for the nonpleiotropic case of a single characteristic x1 ( = 1). The distribution is given byEquation 15 with = Vs/m2 = 0.05. For this nonpleiotropic case, the model is identical to that of WAXMAN and PECK 1998 . (b) Solid curve, steady-state distribution of phenotypes for one characteristic x1 in a pleiotropic case with three total characteristics ( = 3) for our model. The distribution is given byEquation 16 for = 0.05 and has a weak, logarithmic singularity around the perfect phenotype. Dotted curve, steady-state distribution of phenotypes for the Waxman-Peck model using theirEquation 11 for = 3, also with = 0.05. The apparent loss of normalization occurs because 90% of the population has the "perfect" phenotype x1 = 0 and is not shown. [Insets, steady-state distributions of phenotype magnitudes (r). (—) Results of our model; note that (r) is independent of the degree of pleiotropy and so is the same in both insets. (- - -) Steady-state distribution of phenotype magnitudes (r) = [4/(2m2)1/2]· exp(-r2/2m2) for the Waxman-Peck model with = 3. Approximately 90% of the distribution has the perfect phenotype r = 0 and is not shown.]

It is nevertheless apparent from (13) that radically different choices of the distribution of the effects of mutations p(r) can yield qualitatively different steady-state distributions (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 (r) vanishes at r = 0. This leads to the essential difference between the steady-state results: For 3 and a high enough mutation rate, Turelli's model (WAXMAN and PECK 1998 ) produces a finite fraction of the steady-state population with the perfect phenotype, and our model does not. Implicit in their distribution of the effects of mutations (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 (r) r^-1exp(-r²/2m²). It is precisely the factor of r^-1 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 Waxman-Peck 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 BURGER 1988 , BURGER 2000 for a more general statement of this condition]. In our model for mutations, p(0) = (2/(m²))^1/2. Consequently, there is no preservation of the perfect phenotype by part of the population.

The steady-state distribution with respect to a single characteristic ₁(x₁) can be found from the full distribution (11). When there is only a single characteristic ( = 1), the result is the same as that obtained by Waxman and Peck:

(15)

This distribution is plotted in Fig 3A. When there are multiple characteristics, ₁(x₁) is obtained by integrating (11) over x₂, ... , x. For the case of three or more characteristics ( 3), the result is qualitatively different from that of Waxman and Peck. Instead of a finite fraction of the steady-state population obtaining the perfect phenotype x₁ = 0, our model yields only a continuous distribution of phenotypes, which narrows with increasing . Specifically, the variance with respect to a single characteristic is given by for << 1. As noted above, the variance <x²₁> is independent of the variance of mutant effects m², which reflects the Lorentzian behavior of (r) for r << m.

As an example of the distribution with respect to one characteristic in the presence of pleiotropy, we consider the case of three characteristics, = 3. The steady-state distribution for a single characteristic is

(16)

where . The distribution ₁(x₁) for = 3 is plotted in Fig 3B. There is a weak, logarithmic divergence of ₁(x₁) around the most-fit value x₁ = 0.

	DISCUSSION

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

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 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 ( = 1), a Gaussian distribution of the phenotypic effects of mutations is generally regarded as plausible. The Turelli-Waxman-Peck 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. Fig 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 distribution of mutations of selective disadvantage s. [In terms of the mutation-selection 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 distribution is that the frequency of mutations of small selective disadvantage obeys a power law

(17)

Kimura concluded that the distribution with ß = ¹/₂ [hence F(s) s^-1/2] was consistent with observed rates of heterogeneity, and large deviations from ß = ¹/₂ were unlikely; i.e., Kimura inferred a distribution strongly peaked for small selective disadvantages s. In the model of WAXMAN and PECK 1998 , in contrast, one finds

(18)

since the selective disadvantage s satisfies s r²/(2V_s) for s << 1. Hence in the presence of pleiotropy ( 2), the model of WAXMAN and PECK 1998 is incompatible with the rates of heterogeneity in populations found by Kimura. In fact, for > 2 Waxman and Peck predict a strong suppression of mutations of small selective disadvantage—opposite to the conclusions of Kimura. Observe that it is precisely the suppression of mutations of small effect in the TURELLI 1985 model, (r) r^-1, that leads to the discrepancy with Kimura's observations. In the absence of pleiotropy, = 1, the Turelli model yields F(s) s^-1/2, which is consistent with Kimura's inferred distribution of selective disadvantages s. Interestingly, Kimura's choice of ß = ¹/₂ corresponds to a "flat" distribution of mutational effects of small magnitude—p(r) constant as r 0. This suggests that models with a smooth nonvanishing distribution of mutations of small effect may give the best account of evolutionary data. Our half-Gaussian distribution of the magnitudes of mutational effects p(r) is just one possible choice that is consistent with Kimura's observations.

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^(-2)/2 predicted by WAXMAN and PECK 1998 , except in the nonpleiotropic case = 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 mutation-selection model of WAXMAN and PECK 1998 and that these particular proteins have strongly suppressed within-population 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 amino-acid 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 amino-acid sequence changes have been correlated with protein structure and function, demonstrate clearly that a wide variety of amino-acid 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 repressor mutants, finding that amino-acid 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 amino-acid substitution. For example, approximately half of the mutants displayed wild-type 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 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 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 ¹/₂ 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 out-of-focus 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 Turelli-Waxman-Peck predicts F(s) s^(-2)/2, for a gene with pleiotropic dimension . 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 = 2 and equal to zero for > 2 in the Turelli-Waxman-Peck 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 , with a smaller set of well-studied repressors, is also a good test system because the available mutants are easily engineered into standard laboratory strains. Moreover, phage can be propagated both sexually and asexually, in lytic and lysogenic modes and under a wide variety of environmental conditions (LITTLE et 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

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

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 amino-acid substitutions in proteins. Direct experimental measurement of the mutational distribution of selective disadvantages of single-locus 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 mutation-selection models that allow for a realistic probability of mutations of small effect do not produce fixation.

Manuscript received November 25, 2002; Accepted for publication March 28, 2003.

	LITERATURE CITED

TOP ABSTRACT THE MODEL CALCULATIONS AND RESULTS DISCUSSION SUMMARY AND CONCLUSIONS LITERATURE CITED

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