An estimate on the effect of point mutation and natural selection on the rate of amino acid replacement in proteins

We outline a method for estimating quantitatively the influence of point mutations and selection on the frequencies of codons and amino acids. We show how the mutation rate, i.e., the rate of amino acid replacement due to point mutation, can be affected by the codon usage as well as by the rates of the involved base exchanges. A comparison of the mutation rates calculated from reliable values of codon usage and base exchange probabilities with those that would be expected on the basis of chance reveals a notable suppression of replacements leading to tryptophan, glutamate, lysine, and methionine, and particularly of those leading to the termination codons. If selection constraints are neglected and only mutations are taken into account, the best agreement between expected and observed frequencies of both codons and amino acids is obtained for alpha = 1.13-1.15, where (Formula: see text). The "selection values" of codons and amino acids derived by our method show a pattern that partially deviates from others in the literature. For example, the selection pressure on methionine and cysteine turns out to be much more pronounced than expected if only the discrepancies between their observed and expected occurrences in proteins are considered. To estimate to what extent randomly occurring amino acid replacements are accepted by selection, we constructed an "acceptability matrix" from the well-established matrix of accepted point mutations. On the basis of this matrix "acceptability values" of the amino acids can be defined that correlate with their selection values. We also examine the significance of mutations and selection of amino acids with respect to their physicochemical properties and functions in proteins. The conservatism of amino acid replacements with respect to certain properties such as polarity can be brought about by the mutational process alone, whereas the conservatism with respect to other relevant properties--among them all measures of bulkiness--obviously is the result of additional selectional constraints on the evolution of protein structures.     

A stochastic gene evolution model with time dependent mutations

We develop here a new class of gene evolution models in which the nucleotide mutations are time dependent. These models allow to study nonlinear gene evolution by accelerating or decelerating the mutation rates at different evolutionary times. They generalize the previous ones which are based on constant mutation rates. The stochastic model developed in this class determines at some time t the occurrence probabilities of trinucleotides mutating according to 3 time dependent substitution parameters associated with the 3 trinucleotide sites. Therefore, it allows to simulate the evolution of the circular code recently observed in genes. By varying the class of function for the substitution parameters, 1 among 12 models retrieves after mutation the statistical properties of the observed circular code in the 3 frames of actual genes. In this model, the mutation rate in the 3rd trinucleotide site increases during gene evolution while the mutation rates in the 1st and 2nd sites decrease. This property agrees with the actual degeneracy of the genetic code. This approach can easily be generalized to study evolution of motifs of various lengths, e.g., dicodons, etc., with time dependent mutations.     

A metric model of amino acid substitution

MOTIVATION: We address the question of whether there exists an effective evolutionary model of amino-acid substitution that forms a metric-distance function. There is always a trade-off between speed and sensitivity among competing computational methods of determining sequence homology. A metric model of evolution is a prerequisite for the development of an entire class of fast sequence analysis algorithms that are both scalable, O(log n) and sensitive. RESULTS: We have reworked the mathematics of the point accepted mutation model (PAM) by calculating the expected time between accepted mutations in lieu of calculating log-odds probabilities. The resulting substitution matrix (mPAM) forms a metric. We validate the application of the mPAM evolutionary model for sequence homology by executing sequence queries from a controlled yeast protein homology search benchmark. We compare the accuracy of the results of mPAM and PAM similarity matrices as well as three prior metric models. The experiment shows that mPAM significantly outperforms the other three metrics and sufficiently approaches the sensitivity of PAM250 to make it applicable to the management of protein sequence databases.     

Quantifying Clonal and Subclonal Passenger Mutations in Cancer Evolution

The vast majority of mutations in the exome of cancer cells are passengers, which do not affect the reproductive rate of the cell. Passengers can provide important information about the evolutionary history of an individual cancer, and serve as a molecular clock. Passengers can also become targets for immunotherapy or confer resistance to treatment. We study the stochastic expansion of a population of cancer cells describing the growth of primary tumors or metastatic lesions. We first analyze the process by looking forward in time and calculate the fixation probabilities and frequencies of successive passenger mutations ordered by their time of appearance. We compute the likelihood of specific evolutionary trees, thereby informing the phylogenetic reconstruction of cancer evolution in individual patients. Next, we derive results looking backward in time: for a given subclonal mutation we estimate the number of cancer cells that were present at the time when that mutation arose. We derive exact formulas for the expected numbers of subclonal mutations of any frequency. Fitting this formula to cancer sequencing data leads to an estimate for the ratio of birth and death rates of cancer cells during the early stages of clonal expansion.     

Joint evolution of dispersal and inbreeding load

Inbreeding avoidance is often invoked to explain observed patterns of dispersal, and theoretical models indeed point to a possibly important role. However, while inbreeding load is usually assumed constant in these models, it is actually bound to vary dynamically under the combined influences of mutation, drift, and selection and thus to evolve jointly with dispersal. Here we report the results of individual-based stochastic simulations allowing such a joint evolution. We show that strongly deleterious mutations should play no significant role, owing to the low genomic mutation rate for such mutations. Mildly deleterious mutations, by contrast, may create enough heterosis to affect the evolution of dispersal as an inbreeding-avoidance mechanism, but only provided that they are also strongly recessive. If slightly recessive, they will spread among demes and accumulate at the metapopulation level, thus contributing to mutational load, but not to heterosis. The resulting loss of viability may then combine with demographic stochasticity to promote population fluctuations, which foster indirect incentives for dispersal. Our simulations suggest that, under biologically realistic parameter values, deleterious mutations have a limited impact on the evolution of dispersal, which on average exceeds by only one-third the values expected from kin-competition avoidance.     

Molecular Clock of Neutral Mutations in a Fitness-Increasing Evolutionary Process

The molecular clock of neutral mutations, which represents linear mutation fixation over generations, is theoretically explained by genetic drift in fitness-steady evolution or hitchhiking in adaptive evolution. The present study is the first experimental demonstration for the molecular clock of neutral mutations in a fitness-increasing evolutionary process. The dynamics of genome mutation fixation in the thermal adaptive evolution of Escherichia coli were evaluated in a prolonged evolution experiment in duplicated lineages. The cells from the continuously fitness-increasing evolutionary process were subjected to genome sequencing and analyzed at both the population and single-colony levels. Although the dynamics of genome mutation fixation were complicated by the combination of the stochastic appearance of adaptive mutations and clonal interference, the mutation fixation in the population was simply linear over generations. Each genome in the population accumulated 1.6 synonymous and 3.1 non-synonymous neutral mutations, on average, by the spontaneous mutation accumulation rate, while only a single genome in the population occasionally acquired an adaptive mutation. The neutral mutations that preexisted on the single genome hitchhiked on the domination of the adaptive mutation. The successive fixation processes of the 128 mutations demonstrated that hitchhiking and not genetic drift were responsible for the coincidence of the spontaneous mutation accumulation rate in the genome with the fixation rate of neutral mutations in the population. The molecular clock of neutral mutations to the fitness-increasing evolution suggests that the numerous neutral mutations observed in molecular phylogenetic trees may not always have been fixed in fitness-steady evolution but in adaptive evolution.     

The evolution of stress-induced hypermutation in asexual populations

Numerous empirical studies show that stress of various kinds induces a state of hypermutation in bacteria via multiple mechanisms, but theoretical treatment of this intriguing phenomenon is lacking. We used deterministic and stochastic models to study the evolution of stress-induced hypermutation in infinite and finite-size populations of bacteria undergoing selection, mutation, and random genetic drift in constant environments and in changing ones. Our results suggest that if beneficial mutations occur, even rarely, then stress-induced hypermutation is advantageous for bacteria at both the individual and the population levels and that it is likely to evolve in populations of bacteria in a wide range of conditions because it is favored by selection. These results imply that mutations are not, as the current view holds, uniformly distributed in populations, but rather that mutations are more common in stressed individuals and populations. Because mutation is the raw material of evolution, these results have a profound impact on broad aspects of evolution and biology.     

The frequency of shifts between alternative equilibria

We derive a formula giving the frequency with which random drift shifts a population between alternative equilibria. This formula is valid when such shifts are rare (Ns much greater than 1), and applies over a wide range of mutation rates. When the number of mutations entering the population is low (4 N mu much less than 1), the rate of stochastic shifts reduces to the product of the mutation rate and the probability of fixation of a single mutation. However, when many mutations enter the population in each generation (4 N mu much greater than 1), the rate is higher than would be expected if mutations were established independently, and converges to that given by a gaussian approximation. We apply recent results on bistable systems to extend this formula to the general multidimensional case. This gives an explicit expression for the frequency of stochastic shifts, which depends only on the equilibrium probability distribution near the saddle point separating the alternative stable states. The plausibility of theories of speciation through random drift are discussed in the light of these results.     

Lévy flights in evolutionary ecology

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge's model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.     

Mutual information analysis of mutation,nonlinearity, and triple interactions in proteins

Mutations are the cause of several diseases as well as the underlying force of evolution. A thorough understanding of their biophysical consequences is essential. We present a computational framework for evaluating different levels of mutual information (MI) and its dependence on mutation. We used molecular dynamics trajectories of the third PDZ domain and its different mutations. Nonlinear MI between all residue pairs are calculated by tensor Hermite polynomials up to the fifth order and compared with results from multivariate Gaussian distribution of joint probabilities. We show that MI is written as the sum of a Gaussian and a nonlinear component. Results for the PDZ domain show that the Gaussian term gives a sufficiently accurate representation of MI when compared with nonlinear terms up to the fifth order. Changes in MI between residue pairs show the characteristic patterns resulting from specific mutations. Emergence of new peaks in the MI versus residue index plots of mutated PDZ shows how mutation may change allosteric pathways. Triple correlations are characterized by evaluating MI between triplets of residues. We observed that certain triplets are strongly affected by mutation. Susceptibility of residues to perturbation is obtained by MI and discussed in terms of linear response theory.     

Effects of pollen availability and the mutation bias on the fixation of mutations disabling the male specificity of self‐incompatibility

The evolution of self‐compatibility (SC) by the loss of self‐incompatibility (SI) is regarded as one of the most frequent transitions in flowering plants. SI systems are generally characterized by specific interactions between the male and female specificity genes encoded at the S‐locus. Recent empirical studies have revealed that the evolution of SC is often driven by male SC‐conferring mutations at the S‐locus rather than by female mutations. In this study, using a forward simulation model, we compared the fixation probabilities of male vs. female SC‐conferring mutations at the S‐locus. We explicitly considered the effects of pollen availability in the population and bias in the occurrence of SC‐conferring mutations on the male and female specificity genes. We found that male SC‐conferring mutations were indeed more likely to be fixed than were female SC‐conferring mutations in a wide range of parameters. This pattern was particularly strong when pollen availability was relatively high. Under such a condition, even if the occurrence of mutations was biased strongly towards the female specificity gene, male SC‐conferring mutations were much more often fixed. Our study demonstrates that fixation probabilities of those two types of mutation vary strongly depending on ecological and genetic conditions, although both types result in the same evolutionary consequence—the loss of SI.     

The impact of single substitutions on multiple sequence alignments

We introduce another view of sequence evolution. Contrary to other approaches, we model the substitution process in two steps. First we assume (arbitrary) scaled branch lengths on a given phylogenetic tree. Second we allocate a Poisson distributed number of substitutions on the branches. The probability to place a mutation on a branch is proportional to its relative branch length. More importantly, the action of a single mutation on an alignment column is described by a doubly stochastic matrix, the so-called one-step mutation matrix. This matrix leads to analytical formulae for the posterior probability distribution of the number of substitutions for an alignment column.     

Analysis of distributions of amino acids and amino acid pairs in human tumour necrosis factor precursor and its eight mutations according to a random mechanism

In this study, we use the random principle to analyse the distributions of amino acids and amino acid pairs in human tumour necrosis factor precursor (TNF-!) and its eight mutations, to compare the measured distribution probability with the theoretical distribution probability and to rank the measured distribution probability against the theoretical distribution probability. In this way, we can suggest that distributions with a high random rank should not be deliberately evolved and conserved and those with a low random rank should be deliberately evolved and conserved in human TNF-!. An increased distribution probability in a mutation means probabilistically that the mutation is more likely to occur spontaneously, whereas a decreased distribution probability in a mutation means probabilistically that the mutation is less likely to occur spontaneously and perhaps is more related to a certain cause. The results, for example, show that the distributions of 30% of the amino acids are identical with their probabilistic simplest distributions, and the distributions of some of the remaining amino acids are very close to their probabilistic simplest distributions. With respect to probabilities of distributions of amino acids in mutations, the results show that mutations lead to an increase in eight probabilities, which are thus more likely to occur. Eight probabilities decrease and are thus less likely to occur. With respect to the random ranks against the theoretical probabilities of distributions of amino acids, the results show that mutations lead to an increase in seven and a decrease in seven probabilities, with two probabilities unchanged.     

Mutations and the value of information.

On the basis of the literary data the relative probabilities of the point mutations are evaluated in the proteins and in RNA's. The relative probabilities of the nonsense mutations are estimated. The probability of the nonsense mutation of the codon UGG (Trp) is especially high. The notion of the value of information is introduced as the measure of the irreplaceability of an element of a message. Using the data on replaceabilities of the amino-acidic residues the tentative values of information of the codons and of the amino-acidic residues are determined. The value, i.e. the irreplaceability of the information increases in the course of biological development. The increase of the summary value of the protein chain of cytochrome c in phylogenesis is shown. The increase of the value of information correlates with the increase of the entropy of a protein chain.     

On the Origin of Plasmid-borne, Extended-spectrum, Antibiotic Resistance Mutations in Bacteria

Many antibiotic resistance mutations arise in pathogenic bacteria that harbor plasmids (R-plasmids). Resistance to third generation cephalosporins, for instance, largely occurs by one or more point mutations in plasmidblagenes that expand the resistance spectrum of β-lactamases. Here I review relevant evidence underlying the worldwide emergence of extended spectrum β-lactamases (ESBLs). The conclusion reached is that the origin of these resistance-conferring mutations cannot be explained by a series of single point mutation and selection events. Instead, highly advantageous stochastic processes might exist that generate alterations in the sequence or the conformation of particular regions in chromosomal or plasmid genomes such asbla, i.e., recombination or mutation. Several explanations for the origin of ESBLs are reviewed but direct experimental evidence to support or to invalidate them is still lacking. The cellular conditions under which ESBLs arise are unknown; however, involvement of nutritional stresses inside natural animal hosts and of plasmid conjugal functions appear likely.     

A stochastic model of gene evolution with chaotic mutations

We develop here a new class of stochastic models of gene evolution in which the mutations are chaotic, i.e. a random subset of the 64 possible trinucleotides mutates at each evolutionary time t according to some substitution probabilities. Therefore, at each time t, the numbers and the types of mutable trinucleotides are unknown. Thus, the mutation matrix changes at each time t. The chaotic model developed generalizes the standard model in which all the trinucleotides mutate at each time t. It determines the occurrence probabilities at time t of trinucleotides which chaotically mutate according to three substitution parameters associated with the three trinucleotide sites. Two theorems prove that this chaotic model has a probability vector at each time t and that it converges to a uniform probability vector identical to that of the standard model. Furthermore, four applications of this chaotic model (with a uniform random strategy for the 64 trinucleotides and with a particular strategy for the three stop codons) allow an evolutionary study of the three circular codes identified in both eukaryotic and prokaryotic genes. A circular code is a particular set of trinucleotides whose main property is the retrieval of the frames in genes locally, i.e. anywhere in genes and particularly without start codons, and automatically with a window of a few nucleotides. After a certain evolutionary time and with particular values for the three substitution parameters, the chaotic models retrieve the main statistical properties of the three circular codes observed in genes. These applications also allow an evolutionary comparison between the standard and chaotic models.     

Selection, adaptation, and bacterial operons

Bacteria are especially useful as systems to study the molecular basis of adaptive evolution. Selection for novel metabolic capabilities has allowed us to study the evolutionary potential of organisms and has shown that there are three major "strategies" for the evolution of new metabolic functions. (i) Regulatory mutations may allow a gene to be expressed under unusual conditions. If the product of that gene is already active toward a novel resource, then a regulatory mutation alone may confer a new metabolic capability. (ii) Structural gene mutations may alter the catalytic properties of enzymes so that they can act on novel substrates. These structural gene mutations may dramatically improve catalytic capabilities, and in some cases they can confer entirely new capabilities upon enzymes. In most cases both regulatory and structural gene mutations are required for the effective evolution of new metabolic functions. (iii) Operons that are normally silent, or cryptic, may be activated by either point mutations or by the action of mobile genetic elements. When activated, these operons can provide entirely new pathways for the metabolism of novel resources. Selection can also play a role in modulating the probability that a particular adaptive mutation will occur. In this paper I present evidence that a specific adaptive mutation, reversion of the metB1 mutation, occurs 60 to 80 times more frequently during prolonged selection on plates under conditions where the members of the population are not growing than it does in growing cells under nonselective conditions. This selective condition, methionine starvation, does not increase the frequency of other mutations unrelated to methionine biosynthesis.(ABSTRACT TRUNCATED AT 250 WORDS)     

Role of gene duplication in evolution

It is now known that many multigene and supergene families exist in eukaryote genomes: multigene families with uniform copy members like genes for ribosomal RNA, those with variable members like immunoglobulin genes, and supergene families such as those for various growth factor and hormone receptors. Many such examples indicate that gene duplication and subsequent differentiation are extremely important for organismal evolution. In particular, gene duplication could well have been the primary mechanism for the evolution of complexity in higher organisms. Population genetic models for the origin of gene families with diverse functions are presented, in which natural selection favors those genomes with more useful mutants in duplicated genes. Since any gene has a certain probability of degenerating by mutation, success versus failure in acquiring a new gene by duplication may be expressed as the ratio of probabilities of spreading of useful versus detrimental mutations in redundant gene copies. Also examined are the effects of gene duplication on evolution by compensatory advantageous mutations. Results of the analyses show that both natural selection and random drift are important for the origin of gene families. In addition, interaction between molecular mechanisms such as unequal crossing-over and gene conversion, and selection or drift is found to have a large effect on evolution by gene duplication.