Evolutionary dynamics of mutants that modify population structure

Natural selection is usually studied between mutants that differ in reproductive rate, but are subject to the same population structure. Here we explore how natural selection acts on mutants that have the same reproductive rate, but different population structures. In our framework, population structure is given by a graph that specifies where offspring can disperse. The invading mutant disperses offspring on a different graph than the resident wild-type. We find that more densely connected dispersal graphs tend to increase the invader's fixation probability, but the exact relationship between structure and fixation probability is subtle. We present three main results. First, we prove that if both invader and resident are on complete dispersal graphs, then removing a single edge in the invader's dispersal graph reduces its fixation probability. Second, we show that for certain island models higher invader's connectivity increases its fixation probability, but the magnitude of the effect depends on the exact layout of the connections. Third, we show that for lattices the effect of different connectivity is comparable to that of different fitness: for large population size, the invader's fixation probability is either constant or exponentially small, depending on whether it is more or less connected than the resident.

Allelic Entropy and Times until Loss or Fixation of Neutral Polymorphisms

In the diffusion approximation of the neutral Wright-Fisher model, the expected time until fixation or loss of a neutral allele is proportional to the initial entropy of the distribution of the allele in the population. No explanation is known for this coincidence. In this paper, we show that the rate of entropy dissipation is proportional to the number of segregating alleles. Since the final fixed state has zero entropy, the expected lifetime of segregating alleles is proportional to the initial entropy in the system. We show that classical formulae on the average time to loss of segregating alleles and the expected time to fixation of the last segregating allele stem from these properties of the diffusion process. We also extend our results to the case of population size changing in time. The dissipation of heterozygosity and entropy shows that superlinear population growth leads to infinite expected fixation times, i.e., neutral alleles in fast-growing populations could segregate forever without ever becoming fixed or disappearing by genetic drift.

Weak seed banks influence the signature and detectability of selective sweeps

Seed banking (or dormancy) is a widespread bet-hedging strategy, generating a form of population overlap, which decreases the magnitude of genetic drift. The methodological complexity of integrating this trait implies it is ignored when developing tools to detect selective sweeps. But, as dormancy lengthens the ancestral recombination graph (ARG), increasing times to fixation, it can change the genomic signatures of selection. To detect genes under positive selection in seed banking species it is important to (1) determine whether the efficacy of selection is affected, and (2) predict the patterns of nucleotide diversity at and around positively selected alleles. We present the first tree sequence-based simulation program integrating a weak seed bank to examine the dynamics and genomic footprints of beneficial alleles in a finite population. We find that seed banking does not affect the probability of fixation and confirm expectations of increased times to fixation. We also confirm earlier findings that, for strong selection, the times to fixation are not scaled by the inbreeding effective population size in the presence of seed banks, but are shorter than would be expected. As seed banking increases the effective recombination rate, footprints of sweeps appear narrower around the selected sites and due to the scaling of the ARG are detectable for longer periods of time. The developed simulation tool can be used to predict the footprints of selection and draw statistical inference of past evolutionary events in plants, invertebrates, or fungi with seed banks.

Solving the stochastic dynamics of population growth

Population growth is a fundamental process in ecology and evolution. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not consider the variability of birth times, which is particularly important with small population sizes. We show that some moment-closure approximations describe the growth dynamics better than the deterministic prediction. However, they do not reduce the error satisfactorily and only apply to some population growth models. We explicitly solve the stochastic growth dynamics, and our solution applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community and population dynamics. It will allow the development of new tools for a more accurate analysis of experimental and empirical results, including the inference of important growth parameters.

Quantifying stochastic establishment of mutants in microbial adaptation

Studies of microbial evolution, especially in applied contexts, have focused on the role of selection in shaping predictable, adaptive responses to the environment. However, chance events - the appearance of novel genetic variants and their establishment, i.e. outgrowth from a single cell to a sizeable population - also play critical initiating roles in adaptation. Stochasticity in establishment has received little attention in microbiology, potentially due to lack of awareness as well as practical challenges in quantification. However, methods for high-replicate culturing, mutant labelling and detection, and statistical inference now make it feasible to experimentally quantify the establishment probability of specific adaptive genotypes. I review methods that have emerged over the past decade, including experimental design and mathematical formulas to estimate establishment probability from data. Quantifying establishment in further biological settings and comparing empirical estimates to theoretical predictions represent exciting future directions. More broadly, recognition that adaptive genotypes may be stochastically lost while rare is significant both for interpreting common lab assays and for designing interventions to promote or inhibit microbial evolution.

Population genetics of polymorphism and divergence in rapidly evolving populations

In rapidly evolving populations, numerous beneficial and deleterious mutations can arise and segregate within a population at the same time. In this regime, evolutionary dynamics cannot be analyzed using traditional population genetic approaches that assume that sites evolve independently. Instead, the dynamics of many loci must be analyzed simultaneously. Recent work has made progress by first analyzing the fitness variation within a population, and then studying how individual lineages interact with this traveling fitness wave. However, these "traveling wave" models have previously been restricted to extreme cases where selection on individual mutations is either much faster or much slower than the typical coalescent timescale Tc. In this work, we show how the traveling wave framework can be extended to intermediate regimes in which the scaled fitness effects of mutations (Tcs) are neither large nor small compared to one. This enables us to describe the dynamics of populations subject to a wide range of fitness effects, and in particular, in cases where it is not immediately clear which mutations are most important in shaping the dynamics and statistics of genetic diversity. We use this approach to derive new expressions for the fixation probabilities and site frequency spectra of mutations as a function of their scaled fitness effects, along with related results for the coalescent timescale Tc and the rate of adaptation or Muller's ratchet. We find that competition between linked mutations can have a dramatic impact on the proportions of neutral and selected polymorphisms, which is not simply summarized by the scaled selection coefficient Tcs. We conclude by discussing the implications of these results for population genetic inferences.

Dynamic sampling bias and overdispersion induced by skewed offspring distributions

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions. The resulting allele-frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the forward-in-time dynamics, especially with selection. Here, we employ an asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele-frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law.

Understanding evolutionary and ecological dynamics using a continuum limit

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual-based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright-Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi-stationary distribution in a finite population.

Modeling the selective advantage of new amino acids on the hemagglutinin of H1N1 influenza viruses using their patient age distributions

In 2009, a new strain of H1N1 influenza A virus caused a pandemic, and its descendant strains are causing seasonal epidemics worldwide. Given the high mutation rate of influenza viruses, variant strains having different amino acids on hemagglutinin (HA) continuously emerge. To prepare vaccine strains for the next influenza seasons, it is an urgent task to predict which variants will be selected in the viral population. An analysis of 24,681 pairs of an amino acid sequence of HA of H1N1pdm2009 viruses and its patient age showed that the empirical fixation probability of new amino acids on HA significantly differed depending on their frequencies in the population, patient age distributions, and epitope flags. The selective advantage of a variant strain having a new amino acid was modeled by linear combinations of patients age distributions and epitope flags, and then the fixation probability of the new amino acid was modeled using Kimura’s formula for advantageous selection. The parameters of models were estimated from the sequence data and models were tested with four-fold cross validations. The frequency of new amino acids alone can achieve high sensitivity, specificity, and precision in predicting the fixation of a new amino acid of which frequency is more than 0.11. The estimated parameter suggested that viruses with a new amino acid having a frequency in the population higher than 0.11 have a significantly higher selective advantage compared to viruses with the old amino acid at the same position. The model considering the Z-value of patient age rank-sums of new amino acids predicted amino acid substitutions on HA with a sensitivity of 0.78, specificity of 0.86, and precision of 0.83, showing significant improvement compared to the constant selective advantage model, which used only the frequency of the amino acid. These results suggested that H1N1 viruses tend to be selected in the adult population, and frequency of viruses having new amino acids and their patient ages are useful to predict amino acid substitutions on HA.

Evolution of Microbial Growth Traits Under Serial Dilution

Selection of mutants in a microbial population depends on multiple cellular traits. In serial-dilution evolution experiments, three key traits are the lag time when transitioning from starvation to growth, the exponential growth rate, and the yield (number of cells per unit resource). Here, we investigate how these traits evolve in laboratory evolution experiments using a minimal model of population dynamics, where the only interaction between cells is competition for a single limiting resource. We find that the fixation probability of a beneficial mutation depends on a linear combination of its growth rate and lag time relative to its immediate ancestor, even under clonal interference. The relative selective pressure on growth rate and lag time is set by the dilution factor; a larger dilution factor favors the adaptation of growth rate over the adaptation of lag time. The model shows that yield, however, is under no direct selection. We also show how the adaptation speeds of growth and lag depend on experimental parameters and the underlying supply of mutations. Finally, we investigate the evolution of covariation between these traits across populations, which reveals that the population growth rate and lag time can evolve a nonzero correlation even if mutations have uncorrelated effects on the two traits. Altogether these results provide useful guidance to future experiments on microbial evolution.

Multiple infection of cells changes the dynamics of basic viral evolutionary processes

The infection of cells by multiple copies of a given virus can impact viral evolution in a variety of ways, yet some of the most basic evolutionary dynamics remain underexplored. Using computational models, we investigate how infection multiplicity affects the fixation probability of mutants, the rate of mutant generation, and the timing of mutant invasion. An important insight from these models is that for neutral and disadvantageous phenotypes, rare mutants initially enjoy a fitness advantage in the presence of multiple infection of cells. This arises because multiple infection allows the rare mutant to enter more target cells and to spread faster, while it does not accelerate the spread of the resident wild-type virus. The rare mutant population can increase by entry into both uninfected and wild-type-infected cells, while the established wild-type population can initially only grow through entry into uninfected cells. Following this initial advantageous phase, the dynamics are governed by drift or negative selection, respectively, and a higher multiplicity reduces the chances that mutants fix in the population. Hence, while increased infection multiplicity promotes the presence of neutral and disadvantageous mutants in the short-term, it makes it less likely in the longer term. We show how these theoretical insights can be useful for the interpretation of experimental data on virus evolution at low and high multiplicities. The dynamics explored here provide a basis for the investigation of more complex viral evolutionary processes, including recombination, reassortment, as well as complementary/inhibitory interactions.

Selection-Like Biases Emerge in Population Models with Recurrent Jackpot Events

Evolutionary dynamics driven out of equilibrium by growth, expansion, or adaptation often generate a characteristically skewed distribution of descendant numbers: the earliest, the most advanced, or the fittest ancestors have exceptionally large number of descendants, which Luria and Delbrück called "jackpot" events. Here, I show that recurrent jackpot events generate a deterministic median bias favoring majority alleles, which is akin to positive frequency-dependent selection (proportional to the log ratio of the frequencies of mutant and wild-type alleles). This fictitious selection force results from the fact that majority alleles tend to sample deeper into the tail of the descendant distribution. The flip side of this sampling effect is the rare occurrence of large frequency hikes in favor of minority alleles, which ensures that the allele frequency dynamics remains neutral in expectation, unless genuine selection is present. The resulting picture of a selection-like bias compensated by rare big jumps allows for an intuitive understanding of allele frequency trajectories and enables the exact calculation of transition densities for a range of important scenarios, including population-size variations and different forms of natural selection. As a general signature of evolution by rare events, fictitious selection hampers the establishment of new beneficial mutations, counteracts balancing selection, and confounds methods to infer selection from data over limited timescales.

How Well Can Saliency Models Predict Fixation Selection in Scenes Beyond Central Bias? A New Approach to Model Evaluation Using Generalized Linear Mixed Models

Since the turn of the millennium, a large number of computational models of visual salience have been put forward. How best to evaluate a given model's ability to predict where human observers fixate in images of real-world scenes remains an open research question. Assessing the role of spatial biases is a challenging issue; this is particularly true when we consider the tendency for high-salience items to appear in the image center, combined with a tendency to look straight ahead (“central bias”). This problem is further exacerbated in the context of model comparisons, because some—but not all—models implicitly or explicitly incorporate a center preference to improve performance. To address this and other issues, we propose to combine a-priori parcellation of scenes with generalized linear mixed models (GLMM), building upon previous work. With this method, we can explicitly model the central bias of fixation by including a central-bias predictor in the GLMM. A second predictor captures how well the saliency model predicts human fixations, above and beyond the central bias. By-subject and by-item random effects account for individual differences and differences across scene items, respectively. Moreover, we can directly assess whether a given saliency model performs significantly better than others. In this article, we describe the data processing steps required by our analysis approach. In addition, we demonstrate the GLMM analyses by evaluating the performance of different saliency models on a new eye-tracking corpus. To facilitate the application of our method, we make the open-source Python toolbox “GridFix” available.

Prophage Provide a Safe Haven for Adaptive Exploration in Temperate Viruses

Prophage sequences constitute a substantial fraction of the temperate virus gene pool. Although subject to mutational decay, prophage sequences can also be an important source of adaptive mutations for these viral populations. Here we develop a life-history model for temperate viruses, including both the virulent (lytic) and the temperate phases of the life cycle. We then examine the survival of mutations that increase fitness during the lytic phase (attachment rate, burst size), increase fitness in the temperate phase (increasing host survival), or affect transitions between the two phases (integration or induction probability). We find that beneficial mutations are much more likely to survive, ultimately, if they first occur in the prophage state. This conclusion applies even to traits that are only expressed during the lytic phase, and arises due to the substantially lower variance in the offspring distribution during the temperate cycle. This observation, however, is balanced by the fact that many more mutations can be generated during lytic replication. Overall we predict that the prophage state provides a refuge, relatively shielded from genetic drift, in which temperate viruses can explore possible adaptive steps.

Fixation Probability in a Haploid-Diploid Population

Classical population genetic theory generally assumes either a fully haploid or fully diploid life cycle. However, many organisms exhibit more complex life cycles, with both free-living haploid and diploid stages. Here we ask what the probability of fixation is for selected alleles in organisms with haploid-diploid life cycles. We develop a genetic model that considers the population dynamics using both the Moran model and Wright-Fisher model. Applying a branching process approximation, we obtain an accurate fixation probability assuming that the population is large and the net effect of the mutation is beneficial. We also find the diffusion approximation for the fixation probability, which is accurate even in small populations and for deleterious alleles, as long as selection is weak. These fixation probabilities from branching process and diffusion approximations are similar when selection is weak for beneficial mutations that are not fully recessive. In many cases, particularly when one phase predominates, the fixation probability differs substantially for haploid-diploid organisms compared to either fully haploid or diploid species.

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.

Computational complexity of ecological and evolutionary spatial dynamics

There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).

Fate of a mutation in a fluctuating environment

Natural environments are never truly constant, but the evolutionary implications of temporally varying selection pressures remain poorly understood. Here we investigate how the fate of a new mutation in a fluctuating environment depends on the dynamics of environmental variation and on the selective pressures in each condition. We find that even when a mutation experiences many environmental epochs before fixing or going extinct, its fate is not necessarily determined by its time-averaged selective effect. Instead, environmental variability reduces the efficiency of selection across a broad parameter regime, rendering selection unable to distinguish between mutations that are substantially beneficial and substantially deleterious on average. Temporal fluctuations can also dramatically increase fixation probabilities, often making the details of these fluctuations more important than the average selection pressures acting on each new mutation. For example, mutations that result in a trade-off between conditions but are strongly deleterious on average can nevertheless be more likely to fix than mutations that are always neutral or beneficial. These effects can have important implications for patterns of molecular evolution in variable environments, and they suggest that it may often be difficult for populations to maintain specialist traits, even when their loss leads to a decline in time-averaged fitness.

Survival probability of beneficial mutations in bacterial batch culture

The survival of rare beneficial mutations can be extremely sensitive to the organism's life history and the trait affected by the mutation. Given the tremendous impact of bacteria in batch culture as a model system for the study of adaptation, it is important to understand the survival probability of beneficial mutations in these populations. Here we develop a life-history model for bacterial populations in batch culture and predict the survival of mutations that increase fitness through their effects on specific traits: lag time, fission time, viability, and the timing of stationary phase. We find that if beneficial mutations are present in the founding population at the beginning of culture growth, mutations that reduce the mortality of daughter cells are the most likely to survive drift. In contrast, of mutations that occur de novo during growth, those that delay the onset of stationary phase are the most likely to survive. Our model predicts that approximately fivefold population growth between bottlenecks will optimize the occurrence and survival of beneficial mutations of all four types. This prediction is relatively insensitive to other model parameters, such as the lag time, fission time, or mortality rate of the population. We further estimate that bottlenecks that are more severe than this optimal prediction substantially reduce the occurrence and survival of adaptive mutations.

A new approach to modeling the influence of image features on fixation selection in scenes

Which image characteristics predict where people fixate when memorizing natural images? To answer this question, we introduce a new analysis approach that combines a novel scene-patch analysis with generalized linear mixed models (GLMMs). Our method allows for (1) directly describing the relationship between continuous feature value and fixation probability, and (2) assessing each feature's unique contribution to fixation selection. To demonstrate this method, we estimated the relative contribution of various image features to fixation selection: luminance and luminance contrast (low-level features); edge density (a mid-level feature); visual clutter and image segmentation to approximate local object density in the scene (higher-level features). An additional predictor captured the central bias of fixation. The GLMM results revealed that edge density, clutter, and the number of homogenous segments in a patch can independently predict whether image patches are fixated or not. Importantly, neither luminance nor contrast had an independent effect above and beyond what could be accounted for by the other predictors. Since the parcellation of the scene and the selection of features can be tailored to the specific research question, our approach allows for assessing the interplay of various factors relevant for fixation selection in scenes in a powerful and flexible manner.

The effect of population structure on the rate of evolution

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by 'star graphs', that amplify the effects of selection by further increasing the fixation probability of advantageous mutants and decreasing the fixation probability of disadvantageous mutants. The fact that star graphs increase the fixation probability of beneficial mutations has lead to the conclusion that evolution proceeds more rapidly in star-structured populations, compared with mixed (unstructured) populations. Here, we show that the effects of population structure on the rate of evolution are more complex and subtle than previously recognized and draw attention to the importance of fixation time. By comparing population structures that amplify selection with other population structures, both analytically and numerically, we show that evolution can slow down substantially even in populations where selection is amplified.

Establishment of new mutations in changing environments

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.

Genetic linkage and natural selection

The prevalence of recombination in eukaryotes poses one of the most puzzling questions in biology. The most compelling general explanation is that recombination facilitates selection by breaking down the negative associations generated by random drift (i.e. Hill-Robertson interference, HRI). I classify the effects of HRI owing to: deleterious mutation, balancing selection and selective sweeps on: neutral diversity, rates of adaptation and the mutation load. These effects are mediated primarily by the density of deleterious mutations and of selective sweeps. Sequence polymorphism and divergence suggest that these rates may be high enough to cause significant interference even in genomic regions of high recombination. However, neither seems able to generate enough variance in fitness to select strongly for high rates of recombination. It is plausible that spatial and temporal fluctuations in selection generate much more fitness variance, and hence selection for recombination, than can be explained by uniformly deleterious mutations or species-wide selective sweeps.

Evolution of cooperation by multilevel selection

We propose a minimalist stochastic model of multilevel (or group) selection. A population is subdivided into groups. Individuals interact with other members of the group in an evolutionary game that determines their fitness. Individuals reproduce, and offspring are added to the same group. If a group reaches a certain size, it can split into two. Faster reproducing individuals lead to larger groups that split more often. In our model, higher-level selection emerges as a byproduct of individual reproduction and population structure. We derive a fundamental condition for the evolution of cooperation by group selection: if b/c > 1 + n/m, then group selection favors cooperation. The parameters b and c denote the benefit and cost of the altruistic act, whereas n and m denote the maximum group size and the number of groups. The model can be extended to more than two levels of selection and to include migration.