Changing preferences: deformation of single position amino acid fitness landscapes and evolution of proteins

Dynamic expedition of leading mutations in SARS-CoV-2 spike glycoproteins

The continuous evolution of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which caused the recent pandemic, has generated countless new variants with varying fitness. Mutations of the spike glycoprotein play a particularly vital role in shaping its evolutionary trajectory, as they have the capability to alter its infectivity and antigenicity. We present a time-resolved statistical method, Dynamic Expedition of Leading Mutations (deLemus), to analyze the evolutionary dynamics of the SARS-CoV-2 spike glycoprotein. The proposed L -index of the deLemus method is effective in quantifying the mutation strength of each amino acid site and outlining evolutionarily significant sites, allowing the comprehensive characterization of the evolutionary mutation pattern of the spike glycoprotein.

Phylogenetic inference of changes in amino acid propensities with single-position resolution

Fitness conferred by the same allele may differ between genotypes and environments, and these differences shape variation and evolution. Changes in amino acid propensities at protein sites over the course of evolution have been inferred from sequence alignments statistically, but the existing methods are data-intensive and aggregate multiple sites. Here, we develop an approach to detect individual amino acids that confer different fitness in different groups of species from combined sequence and phylogenetic data. Using the fact that the probability of a substitution to an amino acid depends on its fitness, our method looks for amino acids such that substitutions to them occur more frequently in one group of lineages than in another. We validate our method using simulated evolution of a protein site under different scenarios and show that it has high specificity for a wide range of assumptions regarding the underlying changes in selection, while its sensitivity differs between scenarios. We apply our method to the env gene of two HIV-1 subtypes, A and B, and to the HA gene of two influenza A subtypes, H1 and H3, and show that the inferred fitness changes are consistent with the fitness differences observed in deep mutational scanning experiments. We find that changes in relative fitness of different amino acid variants within a site do not always trigger episodes of positive selection and therefore may not result in an overall increase in the frequency of substitutions, but can still be detected from changes in relative frequencies of different substitutions.

Shifts in amino acid preferences as proteins evolve: A synthesis of experimental and theoretical work

Amino acid preferences vary across sites and time. While variation across sites is widely accepted, the extent and frequency of temporal shifts are contentious. Our understanding of the drivers of amino acid preference change is incomplete: To what extent are temporal shifts driven by adaptive versus nonadaptive evolutionary processes? We review phenomena that cause preferences to vary (e.g., evolutionary Stokes shift, contingency, and entrenchment) and clarify how they differ. To determine the extent and prevalence of shifted preferences, we review experimental and theoretical studies. Analyses of natural sequence alignments often detect decreases in homoplasy (convergence and reversions) rates, and variation in replacement rates with time-signals that are consistent with temporally changing preferences. While approaches inferring shifts in preferences from patterns in natural alignments are valuable, they are indirect since multiple mechanisms (both adaptive and nonadaptive) could lead to the observed signal. Alternatively, site-directed mutagenesis experiments allow for a more direct assessment of shifted preferences. They corroborate evidence from multiple sequence alignments, revealing that the preference for an amino acid at a site varies depending on the background sequence. However, shifts in preferences are usually minor in magnitude and sites with significantly shifted preferences are low in frequency. The small yet consistent perturbations in preferences could, nevertheless, jeopardize the accuracy of inference procedures, which assume constant preferences. We conclude by discussing if and how such shifts in preferences might influence widely used time-homogenous inference procedures and potential ways to mitigate such effects.

Episodic evolution of coadapted sets of amino acid sites in mitochondrial proteins

The rate of evolution differs between protein sites and changes with time. However, the link between these two phenomena remains poorly understood. Here, we design a phylogenetic approach for distinguishing pairs of amino acid sites that evolve concordantly, i.e., such that substitutions at one site trigger subsequent substitutions at the other; and also pairs of sites that evolve discordantly, so that substitutions at one site impede subsequent substitutions at the other. We distinguish groups of amino acid sites that undergo coordinated evolution and evolve discordantly from other such groups. In mitochondrion-encoded proteins of metazoans and fungi, we show that concordantly evolving sites are clustered in protein structures. By analysing the phylogenetic patterns of substitutions at concordantly and discordantly evolving site pairs, we find that concordant evolution has two distinct causes: epistatic interactions between amino acid substitutions and episodes of selection independently affecting substitutions at different sites. The rate of substitutions at concordantly evolving groups of protein sites changes in the course of evolution, indicating episodes of selection limited to some of the lineages. The phylogenetic positions of these changes are consistent between proteins, suggesting common selective forces underlying them.

The mode and rate of evolution of a protein site depends on the effect of its mutations on protein fitness. The fitness effect of a mutation itself can change in the course of evolution for at least two reasons. First, it can be modulated by substitutions occurring at other sites, a phenomenon called epistasis. Second, changes in selection can be non-epistatic, affecting sites independently of one another. Here, we analyse substitutions accumulated by the evolving lineages of the five proteins encoded by the mitochondrial genomes of thousands of species of metazoans and fungi. We show that substitutions at different amino acid sites occur in a coordinated fashion, and this coordination is caused both by epistasis and by episodes of selection affecting groups of sites. We partition each protein into several groups of concordantly evolving sites such that evolution of sites from different groups is discordant, and show that the proteins encoded by the mitochondrial genome consist of coevolving structural blocks. Some of these blocks have a clear functional specialization, e.g. are associated with interfaces between proteins composing respiratory complexes. Together, our results reveal a previously unrecognized complexity in the causes of variation in evolutionary rates between protein sites.

SELVa: Simulator of evolution with landscape variation

Organisms evolve to increase their fitness, a process that may be described as climbing the fitness landscape. However, the fitness landscape of an individual site, i.e., the vector of fitness values corresponding to different variants at this site, can itself change with time due to changes in the environment or substitutions at other epistatically interacting sites. While there exist a number of simulators for modeling different aspects of molecular evolution, very few can accommodate changing landscapes. We present SELVa, the Simulator of Evolution with Landscape Variation, aimed at modeling the substitution process under a changing single-position fitness landscape in a set of evolving lineages that form a phylogeny of arbitrary shape. Written in Java and distributed as an executable jar file, SELVa provides a flexible framework that allows the user to choose from a number of implemented rules governing landscape change.

Senescence and entrenchment in evolution of amino acid sites

Amino acid propensities at a site change in the course of protein evolution. This may happen for two reasons. Changes may be triggered by substitutions at epistatically interacting sites elsewhere in the genome. Alternatively, they may arise due to environmental changes that are external to the genome. Here, we design a framework for distinguishing between these alternatives. Using analytical modelling and simulations, we show that they cause opposite dynamics of the fitness of the allele currently occupying the site: it tends to increase with the time since its origin due to epistasis ("entrenchment"), but to decrease due to random environmental fluctuations ("senescence"). By analysing the genomes of vertebrates and insects, we show that the amino acids originating at negatively selected sites experience strong entrenchment. By contrast, the amino acids originating at positively selected sites experience senescence. We propose that senescence of the current allele is a cause of adaptive evolution.

Excessive Parallelism in Protein Evolution of Lake Baikal Amphipod Species Flock

Repeated emergence of similar adaptations is often explained by parallel evolution of underlying genes. However, evidence of parallel evolution at amino acid level is limited. When the analyzed species are highly divergent, this can be due to epistatic interactions underlying the dynamic nature of the amino acid preferences: The same amino acid substitution may have different phenotypic effects on different genetic backgrounds. Distantly related species also often inhabit radically different environments, which makes the emergence of parallel adaptations less likely. Here, we hypothesize that parallel molecular adaptations are more prevalent between closely related species. We analyze the rate of parallel evolution in genome-size sets of orthologous genes in three groups of species with widely ranging levels of divergence: 46 species of the relatively recent lake Baikal amphipod radiation, a species flock of very closely related cichlids, and a set of significantly more divergent vertebrates. Strikingly, in genes of amphipods, the rate of parallel substitutions at nonsynonymous sites exceeded that at synonymous sites, suggesting rampant selection driving parallel adaptation. At sites of parallel substitutions, the intraspecies polymorphism is low, suggesting that parallelism has been driven by positive selection and is therefore adaptive. By contrast, in cichlids, the rate of nonsynonymous parallel evolution was similar to that at synonymous sites, whereas in vertebrates, this rate was lower than that at synonymous sites, indicating that in these groups of species, parallel substitutions are mainly fixed by drift.

Allele-specific nonstationarity in evolution of influenza A virus surface proteins

Influenza A virus (IAV) is a major public health problem and a pandemic threat. Its evolution is largely driven by diversifying positive selection so that relative fitness of different amino acid variants changes with time due to changes in herd immunity or genomic context, and novel amino acid variants attain fitness advantage. Here, we hypothesize that diversifying selection also has another manifestation: the fitness associated with a particular amino acid variant should decline with time since its origin, as the herd immunity adapts to it. By tracing the evolution of antigenic sites at IAV surface proteins, we show that an amino acid variant becomes progressively more likely to become replaced by another variant with time since its origin-a phenomenon we call "senescence." Senescence is particularly pronounced at experimentally validated antigenic sites, implying that it is largely driven by host immunity. By contrast, at internal sites, existing variants become more favorable with time, probably due to arising contingent mutations at other epistatically interacting sites. Our findings reveal a previously undescribed facet of adaptive evolution and suggest approaches for prediction of evolutionary dynamics of pathogens.

Detecting amino acid preference shifts with codon-level mutation-selection mixture models

BACKGROUND

In recent years, increasing attention has been placed on the development of phylogeny-based statistical methodologies for uncovering site-specific changes in amino acid fitness profiles over time. The few available random-effects approaches, modelling across-site variation in amino acid profiles as random variables drawn from a statistical law, either lack a mechanistic codon-level formulation, or pose significant computational challenges.

RESULTS

Here, we bring together a few existing ideas to explore a simple and fast method based on a predefined finite mixture of amino acid profiles within a codon-level substitution model following the mutation-selection formulation. Our study is focused on the detection of site-specific shifts in amino acid profiles over a known sub-clade of a tree, using simulations with and without shifts over the sub-clade to study the properties of the method. Through modifications of the values of the amino acid profiles, our simulations show different levels of reliability under different forms of finite mixture models. Sites identified by our method in a real data set show obvious overlap with those identified using previous methods, with some notable differences.

CONCLUSION

Overall, our results show that when a site-specific shift in amino acid profile is strongly pronounced, involving two clearly different sets of profiles, the method performs very well; but shifts between profiles that share many features are difficult to correctly identify, highlighting the challenging nature of the problem.

mtProtEvol: the resource presenting molecular evolution analysis of proteins involved in the function of Vertebrate mitochondria

BACKGROUND

Heterotachy is the variation in the evolutionary rate of aligned sites in different parts of the phylogenetic tree. It occurs mainly due to epistatic interactions among the substitutions, which are highly complex and make it difficult to study protein evolution. The vast majority of computational evolutionary approaches for studying these epistatic interactions or their evolutionary consequences in proteins require high computational time. However, recently, it has been shown that the evolution of residue solvent accessibility (RSA) is tightly linked with changes in protein fitness and intra-protein epistatic interactions. This provides a computationally fast alternative, based on comparison of evolutionary rates of amino acid replacements with the rates of RSA evolutionary changes in order to recognize any shifts in epistatic interaction.

RESULTS

Based on RSA information, data randomization and phylogenetic approaches, we constructed a software pipeline, which can be used to analyze the evolutionary consequences of intra-protein epistatic interactions with relatively low computational time. We analyzed the evolution of 512 protein families tightly linked to mitochondrial function in Vertebrates and created "mtProtEvol", the web resource with data on protein evolution. In strict agreement with lifespan and metabolic rate data, we demonstrated that different functional categories of mitochondria-related proteins subjected to selection on accelerated and decelerated RSA rates in rodents and primates. For example, accelerated RSA evolution in rodents has been shown for Krebs cycle enzymes, respiratory chain and reactive oxygen species metabolism, while in primates these functions are stress-response, translation and mtDNA integrity. Decelerated RSA evolution in rodents has been demonstrated for translational machinery and oxidative stress response components.

CONCLUSIONS

mtProtEvol is an interactive resource focused on evolutionary analysis of epistatic interactions in protein families involved in Vertebrata mitochondria function and available at http://bioinfodbs.kantiana.ru/mtProtEvol /. This resource and the devised software pipeline may be useful tool for researchers in area of protein evolution.

Modeling site-specific amino-acid preferences deepens phylogenetic estimates of viral sequence divergence

Molecular phylogenetics is often used to estimate the time since the divergence of modern gene sequences. For highly diverged sequences, such phylogenetic techniques sometimes estimate surprisingly recent divergence times. In the case of viruses, independent evidence indicates that the estimates of deep divergence times from molecular phylogenetics are sometimes too recent. This discrepancy is caused in part by inadequate models of purifying selection leading to branch-length underestimation. Here we examine the effect on branch-length estimation of using models that incorporate experimental measurements of purifying selection. We find that models informed by experimentally measured site-specific amino-acid preferences estimate longer deep branches on phylogenies of influenza virus hemagglutinin. This lengthening of branches is due to more realistic stationary states of the models, and is mostly independent of the branch-length extension from modeling site-to-site variation in amino-acid substitution rate. The branch-length extension from experimentally informed site-specific models is similar to that achieved by other approaches that allow the stationary state to vary across sites. However, the improvements from all of these site-specific but time homogeneous and site independent models are limited by the fact that a protein’s amino-acid preferences gradually shift as it evolves. Overall, our work underscores the importance of modeling site-specific amino-acid preferences when estimating deep divergence times—but also shows the inherent limitations of approaches that fail to account for how these preferences shift over time.

Long-term evolution on complex fitness landscapes when mutation is weak

Understanding evolution on complex fitness landscapes is difficult both because of the large dimensionality of sequence space and the stochasticity inherent to population-genetic processes. Here, I present an integrated suite of mathematical tools for understanding evolution on time-invariant fitness landscapes when mutations occur sufficiently rarely that the population is typically monomorphic and evolution can be modeled as a sequence of well-separated fixation events. The basic intuition behind this suite of tools is that surrounding any particular genotype lies a region of the fitness landscape that is easy to evolve to, while other pieces of the fitness landscape are difficult to evolve to (due to distance, being across a fitness valley, etc.). I propose a rigorous definition for this "dynamical neighborhood" of a genotype which captures several aspects of the distribution of waiting times to evolve from one genotype to another. The neighborhood structure of the landscape as a whole can be summarized as a matrix, and I show how this matrix can be used to approximate the expected waiting time for certain evolutionary events to occur and to provide an intuitive interpretation to existing formal results on the index of dispersion of the molecular clock.

Compensatory mutations and epistasis for protein function

Adaptive protein evolution may be facilitated by neutral amino acid mutations that confer no benefit when they first arise but which potentiate subsequent function-altering mutations via direct or indirect structural mechanisms. Theoretical and empirical results indicate that such compensatory interactions (intramolecular epistasis) can exert a strong influence on trajectories of protein evolution. For this reason, assessing the form and prevalence of intramolecular epistasis and characterizing biophysical mechanisms of compensatory interaction are important research goals at the nexus of structural biology and molecular evolution. Here I review recent insights derived from protein-engineering studies, and I describe an approach for identifying and characterizing mechanisms of epistasis that integrates experimental data on structure-function relationships with analyses of comparative sequence data.

Compensatory Evolution of Intrinsic Transcription Terminators in Bacillus Cereus

Many RNA molecules possess complicated secondary structure critical to their function. Mutations in double-helical regions of RNA may disrupt Watson–Crick (WC) interactions causing structure destabilization or even complete loss of function. Such disruption can be compensated by another mutation restoring base pairing, as has been shown for mRNA, rRNA and tRNA. Here, we investigate the evolution of intrinsic transcription terminators between closely related strains of Bacillus cereus. While the terminator structure is maintained by strong natural selection, as evidenced by the low frequency of disrupting mutations, we observe multiple instances of pairs of disrupting-compensating mutations in RNA structure stems. Such two-step switches between different WC pairs occur very fast, consistent with the low fitness conferred by the intermediate non-WC variant. Still, they are not instantaneous, and probably involve transient fixation of the intermediate variant. The GU wobble pair is the most frequent intermediate, and remains fixed longer than other intermediates, consistent with its less disruptive effect on the RNA structure. Double switches involving non-GU intermediates are more frequent at the ends of RNA stems, probably because they are associated with smaller fitness loss. Together, these results show that the fitness landscape of bacterial transcription terminators is rather rugged, but that the fitness valleys associated with unpaired stem nucleotides are rather shallow, facilitating evolution.

Mapping mutational effects along the evolutionary landscape of HIV envelope

The immediate evolutionary space accessible to HIV is largely determined by how single amino acid mutations affect fitness. These mutational effects can shift as the virus evolves. However, the prevalence of such shifts in mutational effects remains unclear. Here, we quantify the effects on viral growth of all amino acid mutations to two HIV envelope (Env) proteins that differ at >100 residues. Most mutations similarly affect both Envs, but the amino acid preferences of a minority of sites have clearly shifted. These shifted sites usually prefer a specific amino acid in one Env, but tolerate many amino acids in the other. Surprisingly, shifts are only slightly enriched at sites that have substituted between the Envs—and many occur at residues that do not even contact substitutions. Therefore, long-range epistasis can unpredictably shift Env’s mutational tolerance during HIV evolution, although the amino acid preferences of most sites are conserved between moderately diverged viral strains.

The virus that causes AIDS, or HIV, has a protein called Env on its surface, which is essential for the virus to infect cells. Env can also be recognized by the immune system, which then targets the virus for destruction or blocks it from infecting cells. Unfortunately, Env evolves very quickly, which means that HIV can evade our defenses. However, there are limits to how much this protein can change, since it still needs to perform its essential role in helping viruses enter cells.

In the century since HIV first appeared in human populations, the virus has evolved considerably. There are now many HIV strains that infect people, and they bear Env proteins with substantially different sequences. However, it is not clear if these changes in sequence have resulted in Envs from distinct strains being able to tolerate different mutations.

To examine this question, Haddox et al. compared how the Envs from two strains of HIV react to modifications in their sequences. They created all possible individual mutations in the proteins, and the resulting collections of mutated viruses were then tested for their ability to infect cells in the laboratory.

Most mutations had similar effects in both Env proteins. This allowed Haddox et al. to identify portions of the protein that easily accommodate changes, and portions that must remain unchanged for viruses to remain infectious—at least in the laboratory. Some of these mutations are under different types of pressures when the virus faces the immune system, and those were identified using computational approaches.

However, some mutations were tolerated differently by the two Env proteins. Therefore, viral strains differ in how their Env proteins can evolve. The parts of Env that showed differences in mutational tolerance between the strains were not necessarily the parts that differ in sequence. This shows that changes in sequence in one part of the protein can modify how other portions evolve.

It remains to be determined whether changes in tolerance to mutations translate into differences in how the virus can escape immunity. This is an important question given that the rapid evolution of Env is a major obstacle to creating a vaccine for HIV.

Parallel Evolution of Metazoan Mitochondrial Proteins

Amino acid propensities at amino acid sites change with time due to epistatic interactions or changing environment, affecting the probabilities of fixation of different amino acids. Such changes should lead to an increased rate of homoplasies (reversals, parallelisms, and convergences) at closely related species. Here, we reconstruct the phylogeny of twelve mitochondrial proteins from several thousand metazoan species, and measure the phylogenetic distances between branches at which either the same allele originated repeatedly due to homoplasies, or different alleles originated due to divergent substitutions. The mean phylogenetic distance between parallel substitutions is ∼20% lower than the mean phylogenetic distance between divergent substitutions, indicating that a variant fixed in a species is more likely to be deleterious in a more phylogenetically remote species, compared with a more closely related species. These findings are robust to artefacts of phylogenetic reconstruction or of pooling of sites from different conservation classes or functional groups, and imply that single-position fitness landscapes change at rates similar to rates of amino acid changes.

Substitutions into amino acids that are pathogenic in human mitochondrial proteins are more frequent in lineages closely related to human than in distant lineages

Propensities for different amino acids within a protein site change in the course of evolution, so that an amino acid deleterious in a particular species may be acceptable at the same site in a different species. Here, we study the amino acid-changing variants in human mitochondrial genes, and analyze their occurrence in non-human species. We show that substitutions giving rise to such variants tend to occur in lineages closely related to human more frequently than in more distantly related lineages, indicating that a human variant is more likely to be deleterious in more distant species. Unexpectedly, substitutions giving rise to amino acids that correspond to alleles pathogenic in humans also more frequently occur in more closely related lineages. Therefore, a pathogenic variant still tends to be more acceptable in human mitochondria than a variant that may only be fit after a substantial perturbation of the protein structure.

Shifting Balance on a Static Mutation-Selection Landscape: A Novel Scenario of Positive Selection

A version of the mechanistic mutation-selection (MutSel) model that accounts for temporal dynamics at a site is presented. This is used to show that the rate ratio dN/dS at a site can be transiently >1 even when fitness coefficients are fixed or the fitness landscape is static. This occurs whenever a site drifts away from its fitness peak and is then forced back by selection, a process reminiscent of shifting balance. Shifting balance is strongest when the substitution process is not dominated by selection or drift, but admits interplay between the two. Under this condition, site-specific changes in dN/dS were inferred in 78-100% of trials, and positive selection (i.e., dN/dS>1) in 10-40% of trials, when sequence alignments generated under MutSel were fitted to two popular phenomenological branch-site models. These results demonstrate that positive selection can occur without a change in fitness regime, and that this is detectable by branch-site models. In addition, MutSel is used to show that a site can be occupied by a sub-optimal amino acid for long periods on a fixed landscape when selection is stringent. This has implications for the interpretation of constant-but-different site patterns typically attributed to changes in fitness. Furthermore, a version of MutSel with episodic changes in fitness coefficients is used to illustrate systematic differences between parameters used to generate data under MutSel and their counterparts estimated by a simple codon model. Motivated by a discrepancy in the literature, interpretation of dN/dS in the context of MutSel is also discussed.

The Impact of Selection at the Amino Acid Level on the Usage of Synonymous Codons

There are two main forces that affect usage of synonymous codons: directional mutational pressure and selection. The effectiveness of protein translation is usually considered as the main selectional factor. However, biased codon usage can also be a byproduct of a general selection at the amino acid level interacting with nucleotide replacements. To evaluate the validity and strength of such an effect, we superimposed >3.5 billion unrestricted mutational processes on the selection of nonsynonymous substitutions based on the differences in physicochemical properties of the coded amino acids. Using a modified evolutionary optimization algorithm, we determined the conditions in which the effect on the relative codon usage is maximized. We found that the effect is enhanced by mutational processes generating more adenine and thymine than guanine and cytosine, as well as more purines than pyrimidines. Interestingly, this effect is observed only under an unrestricted model of nucleotide substitution, and disappears when the mutational process is time-reversible. Comparison of the simulation results with data for real protein coding sequences indicates that the impact of selection at the amino acid level on synonymous codon usage cannot be neglected. Furthermore, it can considerably interfere, especially in AT-rich genomes, with other selections on codon usage, e.g., translational efficiency. It may also lead to difficulties in the recognition of other effects influencing codon bias, and an overestimation of protein coding sequences whose codon usage is subjected to adaptational selection.

Detecting Adaptation in Protein-Coding Genes Using a Bayesian Site-Heterogeneous Mutation-Selection Codon Substitution Model

Codon substitution models have traditionally attempted to uncover signatures of adaptation within protein-coding genes by contrasting the rates of synonymous and non-synonymous substitutions. Another modeling approach, known as the mutation–selection framework, attempts to explicitly account for selective patterns at the amino acid level, with some approaches allowing for heterogeneity in these patterns across codon sites. Under such a model, substitutions at a given position occur at the neutral or nearly neutral rate when they are synonymous, or when they correspond to replacements between amino acids of similar fitness; substitutions from high to low (low to high) fitness amino acids have comparatively low (high) rates. Here, we study the use of such a mutation–selection framework as a null model for the detection of adaptation. Following previous works in this direction, we include a deviation parameter that has the effect of capturing the surplus, or deficit, in non-synonymous rates, relative to what would be expected under a mutation–selection modeling framework that includes a Dirichlet process approach to account for across-codon-site variation in amino acid fitness profiles. We use simulations, along with a few real data sets, to study the behavior of the approach, and find it to have good power with a low false-positive rate. Altogether, we emphasize the potential of recent mutation–selection models in the detection of adaptation, calling for further model refinements as well as large-scale applications.

Epistasis and the Dynamics of Reversion in Molecular Evolution

Recent studies of protein evolution contend that the longer an amino acid substitution is present at a site, the less likely it is to revert to the amino acid previously occupying that site. Here we study this phenomenon of decreasing reversion rates rigorously and in a much more general context. We show that, under weak mutation and for arbitrary fitness landscapes, reversion rates decrease with time for any site that is involved in at least one epistatic interaction. Specifically, we prove that, at stationarity, the hazard function of the distribution of waiting times until reversion is strictly decreasing for any such site. Thus, in the presence of epistasis, the longer a particular character has been absent from a site, the less likely the site will revert to its prior state. We also explore several examples of this general result, which share a common pattern whereby the probability of having reverted increases rapidly at short times to some substantial value before becoming almost flat after a few substitutions at other sites. This pattern indicates a characteristic tendency for reversion to occur either almost immediately after the initial substitution or only after a very long time.

The tangled bank of amino acids

The use of amino acid substitution matrices to model protein evolution has yielded important insights into both the evolutionary process and the properties of specific protein families. In order to make these models tractable, standard substitution matrices represent the average results of the evolutionary process rather than the underlying molecular biophysics and population genetics, treating proteins as a set of independently evolving sites rather than as an integrated biomolecular entity. With advances in computing and the increasing availability of sequence data, we now have an opportunity to move beyond current substitution matrices to more interpretable mechanistic models with greater fidelity to the evolutionary process of mutation and selection and the holistic nature of the selective constraints. As part of this endeavour, we consider how epistatic interactions induce spatial and temporal rate heterogeneity, and demonstrate how these generally ignored factors can reconcile standard substitution rate matrices and the underlying biology, allowing us to better understand the meaning of these substitution rates. Using computational simulations of protein evolution, we can demonstrate the importance of both spatial and temporal heterogeneity in modelling protein evolution.

Causes of molecular convergence and parallelism in protein evolution

To what extent is the convergent evolution of protein function attributable to convergent or parallel changes at the amino acid level? The mutations that contribute to adaptive protein evolution may represent a biased subset of all possible beneficial mutations owing to mutation bias and/or variation in the magnitude of deleterious pleiotropy. A key finding is that the fitness effects of amino acid mutations are often conditional on genetic background. This context dependence (epistasis) can reduce the probability of convergence and parallelism because it reduces the number of possible mutations that are unconditionally acceptable in divergent genetic backgrounds. Here, I review factors that influence the probability of replicated evolution at the molecular level.