The application of a linear algebra to the analysis of mutation rates.

Cells and bacteria growing in culture are subject to mutation, and as this mutation is the ultimate substrate for selection and evolution, the factors controlling the mutation rate are of some interest. The mutational event is not observed directly, but is inferred from the phenotype of the original mutant or of its descendants; the rate of mutation is inferred from the number of such mutant phenotypes. Such inference presumes a knowledge of the probability distribution for the size of a clone arising from a single mutation. We develop a mathematical formulation that assists in the design and analysis of experiments which investigate mutation rates and mutant clone size distribution, and we use it to analyse data for which the classical Luria-Delbrück clone-size distribution must be rejected.     

Representation of mutation pressure and selection pressure by PAM matrices

This paper analyses the relationship between the mutation data matrix 1PAM/PET91, representing the effect of both mutation and selection pressures exerted on 16130 homologous proteins of different organisms, and a mutation probability matrix (1PAM/MPM) representing the effect of pure mutation pressure on protein coding of the Borrelia burgdorferi genome. The 1PAM/PMP matrix was derived with the help of computer simulations, which used empirical nucleotide substitution rates found for the B. burgdorferi genome. Here, it is shown that the frequency of amino acid occurrence is strongly related to their effective survival time. We found that the shorter the turnover time of an amino acid under pure mutation pressure, the lower its fraction in the proteins coded by the genome and the more protected by selection pressure is its position in proteins. Results of analyses suggest that during evolution the mutational pressure has been optimised to some extent to the selection requirements.     

The Effect of Linkage on Establishment and Survival of Locally Beneficial Mutations

We study invasion and survival of weakly beneficial mutations arising in linkage to an established migration–selection polymorphism. Our focus is on a continent–island model of migration, with selection at two biallelic loci for adaptation to the island environment. Combining branching and diffusion processes, we provide the theoretical basis for understanding the evolution of islands of divergence, the genetic architecture of locally adaptive traits, and the importance of so-called “divergence hitchhiking” relative to other mechanisms, such as “genomic hitchhiking”, chromosomal inversions, or translocations. We derive approximations to the invasion probability and the extinction time of a de novo mutation. Interestingly, the invasion probability is maximized at a nonzero recombination rate if the focal mutation is sufficiently beneficial. If a proportion of migrants carries a beneficial background allele, the mutation is less likely to become established. Linked selection may increase the survival time by several orders of magnitude. By altering the timescale of stochastic loss, it can therefore affect the dynamics at the focal site to an extent that is of evolutionary importance, especially in small populations. We derive an effective migration rate experienced by the weakly beneficial mutation, which accounts for the reduction in gene flow imposed by linked selection. Using the concept of the effective migration rate, we also quantify the long-term effects on neutral variation embedded in a genome with arbitrarily many sites under selection. Patterns of neutral diversity change qualitatively and quantitatively as the position of the neutral locus is moved along the chromosome. This will be useful for population-genomic inference. Our results strengthen the emerging view that physically linked selection is biologically relevant if linkage is tight or if selection at the background locus is strong.     

Clone selection and the evolution of modifying features

A statistical mechanism of long-run selection is formulated in order to explain the evolution of modifying features governing mutation, recombination, sexual behavior, demographic mobility, and other factors that do not directly increase the individual fitness of their carrier but which are, supposedly, essential for future evolution of a population. Survival probability of a clone, rather than that of the individual, is shown to play the main role in this mechanism. Changing environment proves to be the main factor affecting it.A theoretical possibility of a long-run adaptation to the dynamics of an environmental change—rather than to a static situation resulting from it—is demonstrated.     

Ancestral Processes with Selection

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call theancestral selection graph. This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavaré (1994).     

Evolutionary Invasion and Escape in the Presence of Deleterious Mutations

Replicators such as parasites invading a new host species, species invading a new ecological niche, or cancer cells invading a new tissue often must mutate to adapt to a new environment. It is often argued that a higher mutation rate will favor evolutionary invasion and escape from extinction. However, most mutations are deleterious, and even lethal. We study the probability that the lineage will survive and invade successfully as a function of the mutation rate when both the initial strain and an adaptive mutant strain are threatened by lethal mutations. We show that mutations are beneficial, i.e. a non-zero mutation rate increases survival compared to the limit of no mutations, if in the no-mutation limit the survival probability of the initial strain is smaller than the average survival probability of the strains which are one mutation away. The mutation rate that maximizes survival depends on the characteristics of both the initial strain and the adaptive mutant, but if one strain is closer to the threshold governing survival then its properties will have greater influence. These conclusions are robust for more realistic or mechanistic depictions of the fitness landscapes such as a more detailed viral life history, or non-lethal deleterious mutations.     

Is the Initial Event in Carcinogenesis An Enhancement of the Mutation Rate?

The age-specific incidence rates of adult cancer indicate that the carcinogenic process is a power function of elapsed time. If the malignant clone arises by a series of mutations, and each mutation is regarded as an independent event with a small instantaneous probability of occurrence, then the slope of the age-specific incidence gives an indication of the number of genes involved. In most cases, the epidemiological data exhibit an age-specific incidence the slope of which is between a fourth and a seventh power of age. Assuming that the mutations involved are deletions and must occur in otherwise viable and proliferative cells, the mutation rate required to generate enough mutant cells to fit the cancer incidence data must be remarkably high in the pre-malignant cell population. The initiation process may thus be an event that results in a raised mutation rate in the affected cell and its progeny. It is proposed that the process of induction involves one or more mutations that induce a metabolic lesion that has the effect of increasing the subsequent mutation rate in the affected clone. A possible mechanism involving free radical generation is suggested and some of the biological implications of the proposal examined.     

Is the Initial Event in Carcinogenesis An Enhancement of the Mutation Rate?

The age-specific incidence rates of adult cancer indicate that the carcinogenic process is a power function of elapsed time. If the malignant clone arises by a series of mutations, and each mutation is regarded as an independent event with a small instantaneous probability of occurrence, then the slope of the age-specific incidence gives an indication of the number of genes involved. In most cases, the epidemiological data exhibit an age-specific incidence the slope of which is between a fourth and a seventh power of age. Assuming that the mutations involved are deletions and must occur in otherwise viable and proliferative cells, the mutation rate required to generate enough mutant cells to fit the cancer incidence data must be remarkably high in the pre-malignant cell population. The initiation process may thus be an event that results in a raised mutation rate in the affected cell and its progeny. It is proposed that the process of induction involves one or more mutations that induce a metabolic lesion that has the effect of increasing the subsequent mutation rate in the affected clone. A possible mechanism involving free radical generation is suggested and some of the biological implications of the proposal examined.     

Sexual selection and the evolution of evolvability

Here we show that sexual selection can have an effect on the rate of mutation. We simulated the fate of a genetic modifier of the mutation rate in a sexual population with and without sexual selection (modelled using a female choice mechanism). Female choice for 'good genes' should reduce variability among male subjects, leaving insufficient differences to maintain female preferences. However, female choice can actually increase genetic variability by supporting a higher mutation rate in sexually selected traits. Increasing the mutation rate will be selected against because of the resulting decline in mean fitness. However, it also increases the probability of rare beneficial mutations arising, and mating skew caused by female preferences for male subjects carrying those beneficials with few deleterious mutations ('good genes') can lead to a mutation rate above that expected under natural selection. A choice of two male subjects was sufficient for there to be a twofold increase in the mutation rate as opposed to a decrease found under random mating.     

Another set of responses and correlated responses to selection on age at reproduction in Drosophila melanogaster

Ageing is the decline in survival probability and fertility later in adult life. It can evolve through mutation accumulation and pleiotropy. Artificial selection by age at reproduction is a useful method for detecting the effects of pleiotropy, and for producing lines that differ in their rate of ageing for further analysis. However, the approach has encountered difficulties from gene-environment interaction and inadvertent selection. We have produced a new set of selection lines in Drosophila melanogaster, breeding from either 'young' or 'old' adults, and avoiding some of the difficulties present in previous studies. Breeding from older adults resulted in an evolutionary increase in survival but, contrary to all previous studies using this method, in no increase in late-life fertility. The increase in survival was accompanied by an evolutionary decline in fertility early in adult life, confirming the importance of pleiotropy in the evolution of ageing. Contrary to previous studies, there were no correlated responses to selection in the pre-adult period; development time, larval competitive ability and adult size achieved did not differ between the lines from the two selection regimes.     

Coexistence and error propagation in pre-biotic vesicle models: a group selection approach

Compartmentalization of unlinked, competing templates is widely accepted as a necessary step towards the evolution of complex organisms. However, preservation of information by templates confined to isolated vesicles of finite size faces much harder obstacles than by free templates: random drift allied to mutation pressure wipe out any template that does not replicate perfectly, no matter how small the error probability might be. In addition, drift alone hinders the coexistence of distinct templates in a same compartment. Here, we investigate the conditions for group selection to prevail over drift and mutation and hence to guarantee the maintenance and coexistence of distinct templates in a vesicle. Group selection is implemented through a vesicle survival probability that depends on the template composition. By considering the limit case of an infinite number of vesicles, each one carrying a finite number of templates, we were able to derive a set of recursion equations for the frequencies of vesicles with different template compositions. Numerical iteration of these recursions allows the exact characterization of the steady state of the vesicle population-a quasispecies of vesicles-thus revealing the values of the mutation and group selection intensities for which template coexistence is possible. Within the main assumption of the model-a fixed, finite or infinite, number of vesicles-we find no fundamental impediment to the coexistence of an arbitrary number of template types with the same replication rate inside a vesicle, except of course for the vesicle capacity. Group selection in the form of vesicle selection is a must for compartmentalized primordial genetic systems even in the absence of intra-genomic competition of different templates.     

The mutation patterns in B-cell immunoglobulin receptors reflect the influence of selection acting at multiple time-scales

During the several-week course of an immune response, B cells undergo a process of clonal expansion, somatic hypermutation of the immunoglobulin (Ig) genes and affinity-dependent selection. Over a lifetime, each B cell may participate in multiple rounds of affinity maturation as part of different immune responses. These two time-scales for selection are apparent in the structure of B-cell lineage trees, which often contain a ‘trunk’ consisting of mutations that are shared across all members of a clone, and several branches that form a ‘canopy’ consisting of mutations that are shared by a subset of clone members. The influence of affinity maturation on the B-cell population can be inferred by analysing the pattern of somatic mutations in the Ig. While global analysis of mutation patterns has shown evidence of strong selection pressures shaping the B-cell population, the effect of different time-scales of selection and diversification has not yet been studied. Analysis of B cells from blood samples of three healthy individuals identifies a range of clone sizes with lineage trees that can contain long trunks and canopies indicating the significant diversity introduced by the affinity maturation process. We here show that observed mutation patterns in the framework regions (FWRs) are determined by an almost purely purifying selection on both short and long time-scales. By contrast, complementarity determining regions (CDRs) are affected by a combination of purifying and antigen-driven positive selection on the short term, which leads to a net positive selection in the long term. In both the FWRs and CDRs, long-term selection is strongly dependent on the heavy chain variable gene family.     

Fixation probability of rare nonmutator and evolution of mutation rates

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation‐selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift‐barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.     

When can a clonal organism escape senescence?

Abstract Some clonal organisms may live for thousands of years and show no signs of senescence, while others consistently die after finite life spans. Using two models, we examined how stage-specific life-history rates of a clone's modules determine whether a genetic individual escapes senescence by replacing old modules with new ones. When the rates of clonal or sexual reproduction and survival of individual modules decline with age, clones are more likely to experience senescence. In addition, the models predict that there is a greater tendency to find senescence in terms of a decline in the rate of sexual reproduction with clone age than in terms of an increase in the probability of clone mortality, unless rates of sexual reproduction increase dramatically with module stage. Using a matrix model modified to represent the clonal lifestyle, we show how a trade-off between sexual and clonal reproduction could result in selection for or against clonal senescence. We also show that, in contrast to unitary organisms, the strength of selection on life-history traits can increase with the age of a clone even in a growing population, countering the evolution of senescence.     

Selection in Finite Populations with Multiple Alleles. III. Genetic Divergence with Centripetal Selection and Mutation

Natural selection for an intermediate level of gene or enzyme activity has been shown to lead to a high frequency of heterotic polymorphisms in populations subject to mutation and random genetic drift. The model assumes a symmetrical spectrum of mutational variation, with the majority of variants having only minor effects on the probability of survival. Each mutational event produces a variant which is novel to the population. Allelic effects are assumed to be additive on the scale of enzyme activity, heterosis arising whenever a heterozygote has a mean level of activity closer to optimal than that of other genotypes in the population.-A new measure of genetic divergence between populations is proposed, which is readily interpreted genetically, and increases approximately linearly with time under centripetal selection, drift and mutation. The parameter is closely related to the rate of accumulation of mutational changes in a cistron over an evolutionary time span.-A survey of published data concerning polymorphic loci in man and Drosophila suggests than an alternative model, based on the superiority of hybrid molecules, is not of general importance. Thirteen loci giving rise to hybrid zones on electrophoresis have a mean heterozygote frequency of 0.22 +/-.06, compared with a value of 0.23 +/-.04 for 16 loci classified as producing no hybrid enzyme.     

DNA fingerprinting in clonal organisms

The use of DNA fingerprinting to identify members of the same clone in completely or partially asexual organisms requires that the individuals within a clone share a recent common ancestor. By considering the expected distributions of band–sharing values in asexual and sexual organisms, it is shown that DNA fingerprinting may be effective in distinguishing members of the same clone, provided that the frequency of sexual reproduction is considerably greater than the minisatellite mutation rate.     

Hypermutable Non-Synonymous Sites Are under Stronger Negative Selection

Mutation rate varies greatly between nucleotide sites of the human genome and depends both on the global genomic location and the local sequence context of a site. In particular, CpG context elevates the mutation rate by an order of magnitude. Mutations also vary widely in their effect on the molecular function, phenotype, and fitness. Independence of the probability of occurrence of a new mutation''s effect has been a fundamental premise in genetics. However, highly mutable contexts may be preserved by negative selection at important sites but destroyed by mutation at sites under no selection. Thus, there may be a positive correlation between the rate of mutations at a nucleotide site and the magnitude of their effect on fitness. We studied the impact of CpG context on the rate of human–chimpanzee divergence and on intrahuman nucleotide diversity at non-synonymous coding sites. We compared nucleotides that occupy identical positions within codons of identical amino acids and only differ by being within versus outside CpG context. Nucleotides within CpG context are under a stronger negative selection, as revealed by their lower, proportionally to the mutation rate, rate of evolution and nucleotide diversity. In particular, the probability of fixation of a non-synonymous transition at a CpG site is two times lower than at a CpG site. Thus, sites with different mutation rates are not necessarily selectively equivalent. This suggests that the mutation rate may complement sequence conservation as a characteristic predictive of functional importance of nucleotide sites.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

Selection under random mutations in stochastic eigen model

The concept of natural selection is examined, which is one of basic principles for the Darwinian interpretation of evolution. In this model selection is defined as a solution of the deterministic Eigen equation. Next, the random effect is introduced through the mutation term. However, the probability of finding the solution expressing the selection is shown to be smallest. The validity of the model and its applicability to polynucleotide replication are discussed.     

AN ADAPTIVE WALK BY HUMAN IMMUNODEFICIENCY VIRUS TYPE 1 THROUGH A FLUCTUATING FITNESS LANDSCAPE

The mutational landscape model of adaptive sequence evolution has been used to explain an unexpected strong positive linear relationship between marginal fitness and mean site‐specific amino acid frequency in the functionally important HIV‐1 gp120 V3 protein region. The model predicts a positive linear relationship between the probability that a particular beneficial allele, among several, is the next to spread to fixation during an adaptive walk, its transition probability, and the allele's selection coefficient. Here, stochastic simulation is used to confirm the intuition that the linear relationship between transition probabilities and selection coefficients, predicted by the model, should, under fluctuating selection, produce a linear relationship between allele frequency, averaged across populations, and fitness. In addition, these relationships hold for the effective population size and mutation rate of HIV‐1 and for the moderately strong selection observed for V3. A survey of the strength of mutation for diverse organisms suggests that these relationships may be widely applicable.