The anomalous effects of biased mutation

A model is presented in which alleles at a number of loci combine to influence the value of a quantitative trait that is subject to stabilizing selection. Mutations can occur to alleles at the loci under consideration. Some of these mutations will tend to increase the value of the trait, while others will tend to decrease it. In contrast to most previous models, we allow the mean effect of mutations to be nonzero. This means that, on average, mutations can have a bias, such that they tend to either increase or decrease the value of the trait. We find, unsurprisingly, that biased mutation moves the equilibrium mean value of the quantitative trait in the direction of the bias. What is more surprising is the behavior of the deviation of the equilibrium mean value of the trait from its optimal value. This has a nonmonotonic dependence on the degree of bias, so that increasing the degree of bias can actually bring the mean phenotype closer to the optimal phenotype. Furthermore, there is a definite maximum to the extent to which biased mutation can cause a difference between the mean phenotype and the optimum. For plausible parameter values, this maximum-possible difference is small. Typically, quantitative-genetics models assume an unconstrained model of mutation, where the expected difference in effect between a parental allele and a mutant allele is independent of the current state of the parental allele. Our results show that models of this sort can easily lead to biologically implausible consequences when mutations are biased. In particular, unconstrained mutation typically leads to a continual increase or decrease in the mean allelic effects at all trait-controlling loci. Thus at each of these loci, the mean allelic effect eventually becomes extreme. This suggests that some of the models of mutation most commonly used in quantitative genetics should be modified so as to introduce genetic constraints.     

Stabilizing selection,mutational bias,and the evolution of sex*

Stabilizing selection around a fixed phenotypic optimum is expected to disfavor sexual reproduction, since asexually reproducing organisms can maintain a higher fitness at equilibrium, while sex disrupts combinations of compensatory mutations. This conclusion rests on the assumption that mutational effects on phenotypic traits are unbiased, that is, mutation does not tend to push phenotypes in any particular direction. In this article, we consider a model of stabilizing selection acting on an arbitrary number of polygenic traits coded by bialellic loci, and show that mutational bias may greatly reduce the mean fitness of asexual populations compared with sexual ones in regimes where mutations have weak to moderate fitness effects. Indeed, mutation and drift tend to push the population mean phenotype away from the optimum, this effect being enhanced by the low effective population size of asexual populations. In a second part, we present results from individual‐based simulations showing that positive rates of sex are favored when mutational bias is present, while the population evolves toward complete asexuality in the absence of bias. We also present analytical (QLE) approximations for the selective forces acting on sex in terms of the effect of sex on the mean and variance in fitness among offspring.     

THE EVOLUTION OF COSTLY MATE PREFERENCES I. FISHER AND BIASED MUTATION

Fisher's runaway process is the standard explanation of the evolution of exaggerated female preferences. But mathematical formulations of Fisher's process (haploid and additive diploid) show it cannot cause stable exaggeration if female preference carries a cost. At equilibrium female fitness must be maximized. Our analysis shows that evolutionary stable exaggeration of female preference can be achieved if mutation pressure on the male character is biased, that is, mutation has a directional effect. At this equilibrium female fitness is not maximized. We discuss the reasons and evidence for believing that mutation pressure is typically biased. Our analysis highlights the previously unacknowledged importance of biased mutation for sexual selection.     

An Improved Codon Modeling Approach for Accurate Estimation of the Mutation Bias

Phylogenetic codon models are routinely used to characterize selective regimes in coding sequences. Their parametric design, however, is still a matter of debate, in particular concerning the question of how to account for differing nucleotide frequencies and substitution rates. This problem relates to the fact that nucleotide composition in protein-coding sequences is the result of the interactions between mutation and selection. In particular, because of the structure of the genetic code, the nucleotide composition differs between the three coding positions, with the third position showing a more extreme composition. Yet, phylogenetic codon models do not correctly capture this phenomenon and instead predict that the nucleotide composition should be the same for all three positions. Alternatively, some models allow for different nucleotide rates at the three positions, an approach conflating the effects of mutation and selection on nucleotide composition. In practice, it results in inaccurate estimation of the strength of selection. Conceptually, the problem comes from the fact that phylogenetic codon models do not correctly capture the fixation bias acting against the mutational pressure at the mutation–selection equilibrium. To address this problem and to more accurately identify mutation rates and selection strength, we present an improved codon modeling approach where the fixation rate is not seen as a scalar, but as a tensor. This approach gives an accurate representation of how mutation and selection oppose each other at equilibrium and yields a reliable estimate of the mutational process, while disentangling the mean fixation probabilities prevailing in different mutational directions.     

Mutation-biased adaptation in a protein NK model

Evolutionary trends responsible for systematic differences in genome and proteome composition have been attributed to GC:AT mutation bias in the context of neutral evolution or to selection acting on genome composition. A possibility that has been ignored, presumably because it is part of neither the Modern Synthesis nor the Neutral Theory, is that mutation may impose a directional bias on adaptation. This possibility is explored here with simulations of the effect of a GC:AT bias on amino acid composition during adaptive walks on an abstract protein fitness landscape called an "NK" model. The results indicate that adaptation does not preclude mutation-biased evolution. In the complete absence of neutral evolution, a modest GC:AT bias of realistic magnitude can displace the trajectory of adaptation in a mutationally favored direction, to such a degree that amino acid composition is biased substantially and persistently. Thus, mutational explanations for evolved patterns need not presuppose neutral evolution.     

Exact solution of the multi-allelic diffusion model

We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus, this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.     

Can mutation or fixation biases explain the allele frequency distribution of human single nucleotide polymorphisms (SNPs)?

One of the most abiding controversies in evolutionary biology concerns the role of neutral processes in molecular evolution. A main focus of the debate has been the evolution of isochores, the strong and systematic variation of base composition in mammalian genomes. One set of hypotheses argue that regions of similar GC are owing to localised mutational biases coupled with neutral evolution. The alternatives point to either selection or biased gene conversion as mechanisms to preferentially remove A or T bases, favouring G and C instead. Using a novel method, we compare models including such fixation biases to models based on mutation bias alone, under the assumption that non-coding, non-repetitive human DNA is at compositional equilibrium. While failing to fully explain the allele frequency distributions of recent single nucleotide polymorphism data, we show that the data are best fitted if the mutation bias is assumed to be constant across the genome, while fixation bias varies with GC content. We also attempt to estimate the strength of fixation bias, which increases linearly with increasing GC. Our approximation suggests that this force exists within the necessary parameter range: it is not so weak as to be drowned by random drift, but not so strong as to lead to exclusive use of G and C alone. Together these results demonstrate that mutation bias fails to explain the evolution of isochores, and suggest that either selection or biased gene conversion are involved.     

How does mutation affect the distribution of phenotypes?

The potential for mutational processes to influence patterns of neutral or adaptive phenotypic evolution is not well understood. If mutations are directionally biased, shifting trait means in a particular direction, or if mutation generates more variance in some directions of multivariate trait space than others, mutation itself might be a source of bias in phenotypic evolution. Here, we use mutagenesis to investigate the affect of mutation on trait mean and (co)variances in zebrafish, Danio rerio. Mutation altered the relationship between age and both prolonged swimming speed and body shape. These observations suggest that mutational effects on ontogeny or aging have the potential to generate variance across the phenome. Mutations had a far greater effect in males than females, although whether this is a reflection of sex‐specific ontogeny or aging remains to be determined. In males, mutations generated positive covariance between swimming speed, size, and body shape suggesting the potential for mutation to affect the evolutionary covariation of these traits. Overall, our observations suggest that mutation does not generate equal variance in all directions of phenotypic space or in each sex, and that pervasive variation in ontogeny or aging within a cohort could affect the variation available to evolution.     

The Effects of Mutation and Natural Selection on Codon Bias in the Genes of Drosophila

Codon bias varies widely among the loci of Drosophila melanogaster, and some of this diversity has been explained by variation in the strength of natural selection. A study of correlations between intron and coding region base composition shows that variation in mutation pattern also contributes to codon bias variation. This finding is corroborated by an analysis of variance (ANOVA), which shows a tendency for introns from the same gene to be similar in base composition. The strength of base composition correlations between introns and codon third positions is greater for genes with low codon bias than for genes with high codon bias. This pattern can be explained by an overwhelming effect of natural selection, relative to mutation, in highly biased loci. In particular, this correlation is absent when examining fourfold degenerate sites of highly biased genes. In general, it appears that selection acts more strongly in choosing among fourfold degenerate codons than among twofold degenerate codons. Although the results indicate regional variation in mutational bias, no evidence is found for large scale regions of compositional homogeneity.     

Evidence That Mutation Is Universally Biased towards AT in Bacteria

Mutation is the engine that drives evolution and adaptation forward in that it generates the variation on which natural selection acts. Mutation is a random process that nevertheless occurs according to certain biases. Elucidating mutational biases and the way they vary across species and within genomes is crucial to understanding evolution and adaptation. Here we demonstrate that clonal pathogens that evolve under severely relaxed selection are uniquely suitable for studying mutational biases in bacteria. We estimate mutational patterns using sequence datasets from five such clonal pathogens belonging to four diverse bacterial clades that span most of the range of genomic nucleotide content. We demonstrate that across different types of sites and in all four clades mutation is consistently biased towards AT. This is true even in clades that have high genomic GC content. In all studied cases the mutational bias towards AT is primarily due to the high rate of C/G to T/A transitions. These results suggest that bacterial mutational biases are far less variable than previously thought. They further demonstrate that variation in nucleotide content cannot stem entirely from variation in mutational biases and that natural selection and/or a natural selection-like process such as biased gene conversion strongly affect nucleotide content.     

Epistasis in polygenic traits and the evolution of genetic architecture under stabilizing selection

We consider the effects of epistasis in a polygenic trait in the balance of mutation and stabilizing selection. The main issues are the genetic variation maintained in equilibrium and the evolution of the mutational effect distribution. The model assumes symmetric mutation and a continuum of alleles at all loci. Epistasis is modeled proportional to pairwise products of the single-locus effects. A general analytical formalism is developed. Assuming linkage equilibrium, we derive results for the equilibrium mutation load and the genetic and mutational variance in the house of cards and the Gaussian approximation. The additive genetic variation maintained in mutation-selection balance is reduced by any pattern of the epistatic interactions. The mutational variance, in contrast, is often increased. Large differences in mutational effects among loci emerge, and a negative correlation among (standard mean) locus mutation effects and mutation rates is predicted. Contrary to the common view since Waddington, we find that stabilizing selection in general does not lead to canalization of the trait. We propose that canalization as a target of selection instead occurs at the genic level. Here, primarily genes with a high mutation rate are buffered, often at the cost of decanalization of other genes. An intuitive interpretation of this view is given in the discussion.     

Mutation Patterns at Dinucleotide Microsatellite Loci in Humans

Microsatellites are a major type of molecular markers in genetics studies. Their mutational dynamics are not clear. We investigated the patterns and characteristics of 97 mutation events unambiguously identified, from 53 multigenerational pedigrees with 630 subjects, at 362 autosomal dinucleotide microsatellite loci. A size-dependent mutation bias (in which long alleles are biased toward contraction, whereas short alleles are biased toward expansion) is observed. There is a statistically significant negative relationship between the magnitude (repeat numbers changed during mutation) and direction (contraction or expansion) of mutations and standardized allele size. Contrasting with earlier findings in humans, most mutation events (63%) in our study are multistep events that involve changes of more than one repeat unit. There was no correlation between mutation rate and recombination rate. Our data indicate that mutational dynamics at microsatellite loci are more complicated than the generalized stepwise mutation models.     

Stabilizing Selection,Purifying Selection,and Mutational Bias in Finite Populations

Genomic traits such as codon usage and the lengths of noncoding sequences may be subject to stabilizing selection rather than purifying selection. Mutations affecting these traits are often biased in one direction. To investigate the potential role of stabilizing selection on genomic traits, the effects of mutational bias on the equilibrium value of a trait under stabilizing selection in a finite population were investigated, using two different mutational models. Numerical results were generated using a matrix method for calculating the probability distribution of variant frequencies at sites affecting the trait, as well as by Monte Carlo simulations. Analytical approximations were also derived, which provided useful insights into the numerical results. A novel conclusion is that the scaled intensity of selection acting on individual variants is nearly independent of the effective population size over a wide range of parameter space and is strongly determined by the logarithm of the mutational bias parameter. This is true even when there is a very small departure of the mean from the optimum, as is usually the case. This implies that studies of the frequency spectra of DNA sequence variants may be unable to distinguish between stabilizing and purifying selection. A similar investigation of purifying selection against deleterious mutations was also carried out. Contrary to previous suggestions, the scaled intensity of purifying selection with synergistic fitness effects is sensitive to population size, which is inconsistent with the general lack of sensitivity of codon usage to effective population size.     

The distribution of allelic effects under mutation and selection

The Price (1970, 1972) equation is applied to the problem of describing the changes in the moments of allelic effects caused by selection, mutation and recombination at loci governing a quantitative genetic character. For comparable assumptions the resulting equations are the same as those obtained by different means by Barton & Turelli (1987; Turelli & Barton, 1989). The Price equation provides a natural framework within which to examine certain kinds of non-additive allelic effects, recombination and assortative mating. The use of the Price equation is illustrated by finding the equilibrium genetic variance under multiplicative dominance and epistasis and under assortative mating at an additive locus. The limitations of the use of recursion equations for the moments of allelic effects are also discussed.     

Apparent directional selection by biased pleiotropic mutation

Pleiotropic effects of deleterious mutations are considered to be among the factors responsible for genetic constraints on evolution by long-term directional selection acting on a quantitative trait. If pleiotropic phenotypic effects are biased in a particular direction, mutations generate apparent directional selection, which refers to the covariance between fitness and the trait owing to a linear association between the number of mutations possessed by individuals and the genotypic values of the trait. The present analysis has shown how the equilibrium mean value of the trait is determined by a balance between directional selection and biased pleiotropic mutations. Assuming that genes act additively both on the trait and on fitness, the total variance-standardized directional selection gradient was decomposed into apparent and true components. Experimental data on mutation bias from the bristle traits of Drosophila and life history traits of Daphnia suggest that apparent selection explains a small but significant fraction of directional selection pressure that is observed in nature; the data suggest that changes induced in a trait by biased pleiotropic mutation (i.e., by apparent directional selection) are easily compensated for by (true) directional selection.     

The effects of demography and linkage on the estimation of selection and mutation parameters

We explore the effects of demography and linkage on a maximum-likelihood (ML) method for estimating selection and mutation parameters in a reversible mutation model. This method assumes free recombination between sites and a randomly mating population of constant size and uses information from both polymorphic and monomorphic sites in the sample. Two likelihood-ratio test statistics were constructed under this ML framework: LRTγ for detecting selection and LRTκ for detecting mutational bias. By carrying out extensive simulations, we obtain the following results. When mutations are neutral and population size is constant, LRTγ and LRTκ follow a chi-square distribution with 1 d.f. regardless of the level of linkage, as long as the mutation rate is not very high. In addition, LRTγ and LRTκ are relatively insensitive to demographic effects and selection at linked sites. We find that the ML estimators of the selection and mutation parameters are usually approximately unbiased and that LRTκ usually has good power to detect mutational bias. Finally, with a recombination rate that is typical for Drosophila, LRTγ has good power to detect weak selection acting on synonymous sites. These results suggest that the method should be useful under many different circumstances.     

Unconstrained evolution in short introns? – An analysis of genome‐wide polymorphism and divergence data from Drosophila

An unconstrained reference sequence facilitates the detection of selection. In Drosophila, sequence variation in short introns seems to be least influenced by selection and dominated by mutation and drift. Here, we test this with genome‐wide sequences using an African population (Malawi) of D. melanogaster and data from the related outgroup species D. simulans, D. sechellia, D. erecta and D. yakuba. The distribution of mutations deviates from equilibrium, and the content of A and T (AT) nucleotides shows an excess of variance among introns. We explain this by a complex mutational pattern: a shift in mutational bias towards AT, leading to a slight nonequilibrium in base composition and context‐dependent mutation rates, with G or C (GC) sites mutating most frequently in AT‐rich introns. By comparing the corresponding allele frequency spectra of AT‐rich vs. GC‐rich introns, we can rule out the influence of directional selection or biased gene conversion on the mutational pattern. Compared with neutral equilibrium expectations, polymorphism spectra show an excess of low frequency and a paucity of intermediate frequency variants, irrespective of the direction of mutation. Combining the information from different outgroups with the polymorphism data and using a generalized linear model, we find evidence for shared ancestral polymorphism between D. melanogaster and D. simulans, D. sechellia, arguing against a bottleneck in D. melanogaster. Generally, we find that short introns can be used as a neutral reference on a genome‐wide level, if the spatially and temporally varying mutational pattern is accounted for.     

Mutation-selection balance: ancestry,load, and maximum principle

We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.     

Reducing mutational bias in random protein libraries

The success of protein optimization through directed molecular evolution depends to a large extent on the size and quality of the displayed library. Current low-fidelity DNA polymerases that are commonly used during random mutagenesis and recombination in vitro display strong mutational preferences, favoring the substitution of certain nucleotides over others. The result is a biased and reduced functional diversity in the library under selection. In an effort to reduce mutational bias, we combined two different low-fidelity DNA polymerases, Taq and Mutazyme, which have opposite mutational spectra. As a first step, random mutants of the Bacillus thuringiensis cry9Ca1 gene were generated by separate error-prone polymerase chain reactions (PCRs) with each of the two polymerases. Subsequent shuffling by staggered extension process (StEP) of the PCR products resulted in intermediate numbers of AT and GC substitutions, compared to the Taq or Mutazyme error-prone PCR libraries. This strategy should allow generating unbiased libraries or libraries with a specific degree of mutational bias by applying optimal mutagenesis frequencies during error-prone PCR and controlling the concentration of template in the shuffling reaction while taking into account the GC content of the target gene.     

Logarithmic transformation bias in allometry

It has been known for some time (DJ Finney, J. Roy. Stat. Soc. Suppl. 7:155–161, 1941) that transformation of an arithmetic data set to logarithms results in biased estimates when predicted values from a leastsquares regression are detransformed back to arithmetic units. Predicted values are estimates of the geometric mean of the dependent variable at that value of the independent variable, rather than the arithmetic mean. Since the geometric mean is always less than the arithmetic mean, detransformed predictions will underestimate the value in question. This bias affects the interpretations of allometric equations used for estimation, such as predicting fossil body mass from skeletal dimensions, and applications of allometry as a “criterion of subtraction,” in which residual variation is evaluated. A number of parametric and nonparametric corrections for transformation bias have been developed. Although this problem is relatively unexplored in mammalian morphometrics, it has received considerable attention in other disciplines that use power functions structurally identical to the allometric equation. Insights into transformation bias and the use of correction terms from economics, limnology, forestry, and hydrology are reviewed and interpreted for application to mammalian allometry. © 1993 Wiley-Liss, Inc.