Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization

Long-range linkage effects in adapting sexual populations

In sexual populations, closely-situated genes have linked evolutionary fates, while genes spaced far in genome are commonly thought to evolve independently due to recombination. In the case where evolution depends essentially on supply of new mutations, this assumption has been confirmed by mathematical modeling. Here I examine it in the case of pre-existing genetic variation, where mutation is not important. A haploid population with [Formula: see text] genomes, [Formula: see text] loci, a fixed selection coefficient, and a small initial frequency of beneficial alleles [Formula: see text] is simulated by a Monte-Carlo algorithm. When the number of loci, L, is larger than a critical value of [Formula: see text] simulation demonstrates a host of linkage effects that decrease neither with the distance between loci nor the number of recombination crossovers. Due to clonal interference, the beneficial alleles become extinct at a fraction of loci [Formula: see text]. Due to a genetic background effect, the substitution rate varies broadly between loci, with the fastest value exceeding the one-locus limit by the factor of [Formula: see text] Thus, the far-situated parts of a long genome in a sexual population do not evolve as independent blocks. A potential link between these findings and the emergence of new Variants of Concern of SARS-CoV-2 is discussed.

Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian bees

The "Brownian bees" model describes an ensemble of N= const independent branching Brownian particles. The conservation of N is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of N≫1. At long times, the particle density approaches a spherically symmetric steady-state solution with a compact support of radius ℓ[over ¯]_{0}. However, at finite N, the radius of this support, L, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni et al., Phys. Rev. E 104, 054131 (2021)2470-004510.1103/PhysRevE.104.054131]. It is proportional to N^{-1}lnN at N→∞. We investigate here the tails of the probability density function (PDF), P(L), of the swarm radius, when the absolute value of the radius fluctuation ΔL=L-ℓ[over ¯]_{0} is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method. This part of the PDF displays the scaling behavior lnP∝-NΔL^{2}ln^{-1}(ΔL^{-2}), demonstrating a logarithmic anomaly at small negative ΔL. For the opposite sign of the fluctuation, ΔL>0, the PDF can be obtained with an approximation of a single particle, running away. We find that lnP∝-N^{1/2}ΔL. We consider in this paper only the case when |ΔL| is much less than the typical radius of the swarm at N≫1.

Competition in a system of Brownian particles: Encouraging achievers

We introduce and analytically and numerically study a simple model of interagent competition, where underachievement is strongly discouraged. We consider N≫1 particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of N→∞, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a nonstationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-N effects at a distance of order sqrt[N] from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin-the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as sqrt[t]. Finally, we extend the inter-agent competition from n=2 to an arbitrary number n of competing Brownian particles (n≪N). Our analytical predictions are supported by Monte Carlo simulations.

Fluctuations and first-passage properties of systems of Brownian particles with reset

We study, analytically and numerically, stationary fluctuations in two models involving N Brownian particles undergoing stochastic resetting in one dimension. We start with the well-known reset model where the particles reset to the origin independently (model A). Then we introduce nonlocal interparticle correlations by postulating that only the particle farthest from the origin can be reset to the origin (model B). At long times, models A and B approach nonequilibrium steady states. In the limit of N→∞, the steady-state particle density in model A has an infinite support, whereas in model B, it has a compact support, like the recently studied Brownian bees model. A finite system radius, which scales at large N as lnN, appears in model A when N is finite. In both models, we study stationary fluctuations of the center of mass of the system and of the radius of the system due to the random character of the Brownian motion and of the resetting events. In model A, we determine exact distributions of these two quantities. The variance of the center of mass for both models scales as 1/N. The variance of the radius is independent of N in model A and exhibits an unusual scaling (lnN)/N in model B. The latter scaling is intimately related to the 1/f noise in the radius autocorrelation. Finally, we evaluate the mean first-passage time (MFPT) to a distant target in model A, model B, and the Brownian bees model. For model A, we obtain an exact asymptotic expression for the MFPT which scales as 1/N. For model B and the Brownian bees model, we propose a sharp upper bound for the MFPT. The bound assumes an evaporation scenario, where the first passage requires multiple attempts of a single particle, which breaks away from the rest of the particles, to reach the target. The resulting MFPT for model B and the Brownian bees model scales exponentially with sqrt[N]. We verify this bound by performing highly efficient weighted-ensemble simulations of the first passage in model B.

Ancestral Lineages in Mutation Selection Equilibria with Moving Optimum

Many populations can somehow adapt to rapid environmental changes. To understand this fast evolution, we investigate the genealogy of individuals inside those populations. More precisely, we use a deterministic model to describe the phenotypic density of a population under selection when the fitness optimum moves at constant speed. We study the inside dynamics of this population using the neutral fractions approach. We then define a Markov process characterizing the distribution of ancestral phenotypic lineages inside the equilibrium. This construction yields qualitative as well as quantitative properties on the phenotype of typical ancestors. In particular, we show that in asexual populations typical ancestors of present individuals carried traits much closer to the fitness optimum than most individuals alive at the same time. We also investigate more deeply the asymptotic regime of small mutation effects. In this regime, we obtain an explicit formula for the typical ancestral lineage using the description of the solutions of the Hamilton Jacobi equation as a minimizer of an optimization problem. In addition, we compare our deterministic results on lineages with the lineages of stochastic models.

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.

Spatial structure governs the mode of tumour evolution

Characterizing the mode-the way, manner or pattern-of evolution in tumours is important for clinical forecasting and optimizing cancer treatment. Sequencing studies have inferred various modes, including branching, punctuated and neutral evolution, but it is unclear why a particular pattern predominates in any given tumour. Here we propose that tumour architecture is key to explaining the variety of observed genetic patterns. We examine this hypothesis using spatially explicit population genetics models and demonstrate that, within biologically relevant parameter ranges, different spatial structures can generate four tumour evolutionary modes: rapid clonal expansion, progressive diversification, branching evolution and effectively almost neutral evolution. Quantitative indices for describing and classifying these evolutionary modes are presented. Using these indices, we show that our model predictions are consistent with empirical observations for cancer types with corresponding spatial structures. The manner of cell dispersal and the range of cell-cell interactions are found to be essential factors in accurately characterizing, forecasting and controlling tumour evolution.

Fluctuations of a swarm of Brownian bees

The "Brownian bees" model describes an ensemble of N independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep the number of particles constant. In the limit of N→∞, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady-state solution with a compact support. Here, we study fluctuations of the "swarm of bees" due to the random character of the branching Brownian motion in the limit of large but finite N. We consider a one-dimensional setting and focus on two fluctuating quantities: the swarm center of mass X(t) and the swarm radius ℓ(t). Linearizing a pertinent Langevin equation around the deterministic steady-state solution, we calculate the two-time covariances of X(t) and ℓ(t). The variance of X(t) directly follows from the covariance of X(t), and it scales as 1/N as to be expected from the law of large numbers. The variance of ℓ(t) behaves differently: It exhibits an anomalous scaling (1/N)lnN. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of ℓ(t) can be obtained from the covariance of ℓ(t) by introducing a cutoff at the microscopic time 1/N where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.

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.

Genealogical structure changes as range expansions transition from pushed to pulled

Range expansions accelerate evolution through multiple mechanisms, including gene surfing and genetic drift. The inference and control of these evolutionary processes ultimately rely on the information contained in genealogical trees. Currently, there are two opposing views on how range expansions shape genealogies. In invasion biology, expansions are typically approximated by a series of population bottlenecks producing genealogies with only pairwise mergers between lineages-a process known as the Kingman coalescent. Conversely, traveling wave models predict a coalescent with multiple mergers, known as the Bolthausen-Sznitman coalescent. Here, we unify these two approaches and show that expansions can generate an entire spectrum of coalescent topologies. Specifically, we show that tree topology is controlled by growth dynamics at the front and exhibits large differences between pulled and pushed expansions. These differences are explained by the fluctuations in the total number of descendants left by the early founders. High growth cooperativity leads to a narrow distribution of reproductive values and the Kingman coalescent. Conversely, low growth cooperativity results in a broad distribution, whose exponent controls the merger sizes in the genealogies. These broad distribution and non-Kingman tree topologies emerge due to the fluctuations in the front shape and position and do not occur in quasi-deterministic simulations. Overall, our results show that range expansions provide a robust mechanism for generating different types of multiple mergers, which could be similar to those observed in populations with strong selection or high fecundity. Thus, caution should be exercised in making inferences about the origin of non-Kingman genealogies.

An evolution-based high-fidelity method of epistasis measurement: Theory and application to influenza

Linkage effects in a multi-locus population strongly influence its evolution. The models based on the traveling wave approach enable us to predict the average speed of evolution and the statistics of phylogeny. However, predicting statistically the evolution of specific sites and pairs of sites in the multi-locus context remains a mathematical challenge. In particular, the effects of epistasis, the interaction of gene regions contributing to phenotype, is difficult to predict theoretically and detect experimentally in sequence data. A large number of false-positive interactions arises from stochastic linkage effects and indirect interactions, which mask true epistatic interactions. Here we develop a proof-of-principle method to filter out false-positive interactions. We start by demonstrating that the averaging of haplotype frequencies over multiple independent populations is necessary but not sufficient for epistatic detection, because it still leaves high numbers of false-positive interactions. To compensate for the residual stochastic noise, we develop a three-way haplotype method isolating true interactions. The fidelity of the method is confirmed analytically and on simulated genetic sequences evolved with a known epistatic network. The method is then applied to a large sequence database of neurominidase protein of influenza A H1N1 obtained from various geographic locations to infer the epistatic network responsible for the difference between the pre-pandemic virus and the pandemic strain of 2009. These results present a simple and reliable technique to measure epistatic interactions of any sign from sequence data.

Interactions between genomic sites create a fitness landscape. The knowledge of topology and strength of interactions is vital for predicting the escape of viruses from drugs and immune response and their passing through fitness valleys. Many efforts have been invested into measuring these interactions from DNA sequence sets. Unfortunately, reproducibility of the results remains low due partly to a very small fraction of interaction pairs and partly to stochastic linkage noise masking true interactions. Here we propose a method to separate stochastic linkage and indirect interactions from epistatic interactions and apply it to influenza virus sequence data.

An Evolutionary Model of Progression to AIDS

The time to the onset of AIDS symptoms in an HIV infected individual is known to correlate inversely with viremia and the level of immune activation. The correlation exists against the background of strong individual fluctuations demonstrating the existence of hidden variables depending on patient and virus parameters. At the moment, prognosis of the time to AIDS based on patient parameters is not possible. In addition, it is of paramount importance to understand the reason of progression to AIDS in untreated patients to be able to learn to control it by means other than anti-retroviral therapy. Here we develop a mechanistic mathematical model to predict the speed of progression to AIDS in individual untreated patients and patients treated with suboptimal therapy, based on a single-time measurement of several virological and immunological parameters. We show that the gradual increase in virus fitness during a chronic infection causes slow gradual depletion of CD4 T cells. Using the existing evolution models of HIV, we obtain general expressions predicting the time to the onset of AIDS symptoms in terms of the patient parameters, for low-viremia and high-viremia patients separately. We show that the evolution model of AIDS fits the existing data on virus-time correlations better than the alternative model of the deregulation of homeostatic response.

Particle-number distribution in large fluctuations at the tip of branching random walks

We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times t, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position, but still within a distance smaller than the diffusion radius ∼sqrt[t]. Our approach consists in a study of the generating function G_{Δx}(λ)=∑_{n}λ^{n}p_{n}(Δx) for the probabilities p_{n}(Δx) of observing n particles in an interval of given size Δx from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-Δx limits, we find that the mean number of particles in the interval grows exponentially with Δx, and that the generating function obeys a nontrivial scaling law, depending on Δx and λ through the combined variable [Δx-f(λ)]^{3}/Δx^{2}, where f(λ)≡-ln(1-λ)-ln[-ln(1-λ)]. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large Δx. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.

Emergence of evolutionary driving forces in pattern-forming microbial populations

Evolutionary dynamics are controlled by a number of driving forces, such as natural selection, random genetic drift and dispersal. In this perspective article, we aim to emphasize that these forces act at the population level, and that it is a challenge to understand how they emerge from the stochastic and deterministic behaviour of individual cells. Even the most basic steric interactions between neighbouring cells can couple evolutionary outcomes of otherwise unrelated individuals, thereby weakening natural selection and enhancing random genetic drift. Using microbial examples of varying degrees of complexity, we demonstrate how strongly cell-cell interactions influence evolutionary dynamics, especially in pattern-forming systems. As pattern formation itself is subject to evolution, we propose to study the feedback between pattern formation and evolutionary dynamics, which could be key to predicting and potentially steering evolutionary processes. Such an effort requires extending the systems biology approach from the cellular to the population scale.This article is part of the theme issue 'Self-organization in cell biology'.

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.

Effective models and the search for quantitative principles in microbial evolution

Microbes evolve rapidly. Yet they do so in idiosyncratic ways, which depend on the specific mutations that are beneficial or deleterious in a given situation. At the same time, some population-level patterns of adaptation are strikingly similar across different microbial systems, suggesting that there may also be simple, quantitative principles that unite these diverse scenarios. We review the search for simple principles in microbial evolution, ranging from the biophysical level to emergent evolutionary dynamics. A key theme has been the use of effective models, which coarse-grain over molecular and cellular details to obtain a simpler description in terms of a few effective parameters. Collectively, these theoretical approaches provide a set of quantitative principles that facilitate understanding, prediction, and potentially control of evolutionary phenomena, though formidable challenges remain due to the ecological complexity of natural populations.

Antigenic evolution of viruses in host populations

To escape immune recognition in previously infected hosts, viruses evolve genetically in immunologically important regions. The host’s immune system responds by generating new memory cells recognizing the mutated viral strains. Despite recent advances in data collection and analysis, it remains conceptually unclear how epidemiology, immune response, and evolutionary factors interact to produce the observed speed of evolution and the incidence of infection. Here we establish a general and simple relationship between long-term cross-immunity, genetic diversity, speed of evolution, and incidence. We develop an analytic method fusing the standard epidemiological susceptible-infected-recovered approach and the modern virus evolution theory. The model includes the factors of strain selection due to immune memory cells, random genetic drift, and clonal interference effects. We predict that the distribution of recovered individuals in memory serotypes creates a moving fitness landscape for the circulating strains which drives antigenic escape. The fitness slope (effective selection coefficient) is proportional to the reproductive number in the absence of immunity R₀ and inversely proportional to the cross-immunity distance a, defined as the genetic distance of a virus strain from a previously infecting strain conferring 50% decrease in infection probability. Analysis predicts that the evolution rate increases linearly with the fitness slope and logarithmically with the genomic mutation rate and the host population size. Fitting our analytic model to data obtained for influenza A H3N2 and H1N1, we predict the annual infection incidence within a previously estimated range, (4-7)%, and the antigenic mutation rate of U_b = (5 − 8) ⋅ 10⁻⁴ per transmission event per genome. Our prediction of the cross-immunity distance of a = (14 − 15) aminoacid substitutions agrees with independent data for equine influenza.

Spread of many RNA viruses in a population represents a competition between host immune responses and viral evolution. RNA viruses accumulate mutations in immunologically important regions to escape immune recognition in hosts previously exposed to infection, while the immune system responds by producing new memory cells. Despite recent advances in data collection and their analysis, it remains conceptually unclear how epidemiology, immune response, and evolutionary factors interact to produce the observed speed of evolution and its incidence. By combining the standard epidemiological approach with the modern theory of viral evolution, we predict a general relationship between long-term cross-immunity, antigenic diversity of virus, its evolution speed, infection incidence, and the time to the most recent common ancestor. We apply these theoretical findings to available data on influenza virus to determine two important parameters of its evolution and confirm the model. Current strategies of vaccination against influenza should take into account stochastic fluctuations in fitness effect of mutations predicted by the theory.

Fluctuations uncover a distinct class of traveling waves

Epidemics, flame propagation, and cardiac rhythms are classic examples of reaction-diffusion waves that describe a switch from one alternative state to another. Only two types of waves are known: pulled, driven by the leading edge, and pushed, driven by the bulk of the wave. Here, we report a distinct class of semipushed waves for which both the bulk and the leading edge contribute to the dynamics. These hybrid waves have the kinetics of pushed waves, but exhibit giant fluctuations similar to pulled waves. The transitions between pulled, semipushed, and fully pushed waves occur at universal ratios of the wave velocity to the Fisher velocity. We derive these results in the context of a species invading a new habitat by examining front diffusion, rate of diversity loss, and fluctuation-induced corrections to the expansion velocity. All three quantities decrease as a power law of the population density with the same exponent. We analytically calculate this exponent, taking into account the fluctuations in the shape of the wave front. For fully pushed waves, the exponent is -1, consistent with the central limit theorem. In semipushed waves, however, the fluctuations average out much more slowly, and the exponent approaches 0 toward the transition to pulled waves. As a result, a rapid loss of genetic diversity and large fluctuations in the position of the front occur, even for populations with cooperative growth and other forms of an Allee effect. The evolutionary outcome of spatial spreading in such populations could therefore be less predictable than previously thought.

Minimal Clade Size in the Bolthausen-Sznitman Coalescent

In this article we show the asymptotics of distribution and moments of the size Xn of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman n-coalescent for n → ∞. The Bolthausen-Sznitman n-coalescent is a Markov process taking states in the set of partitions of {1, …, n}, where 1, …, n are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution of Xn. The main tool used is the connection of the Bolthausen-Sznitman n-coalescent with random recursive trees introduced by Goldschmidt and Martin (2005). With it, we show that Xn - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers.

Correlated Mutations and Homologous Recombination Within Bacterial Populations

Inferring the rate of homologous recombination within a bacterial population remains a key challenge in quantifying the basic parameters of bacterial evolution. Due to the high sequence similarity within a clonal population, and unique aspects of bacterial DNA transfer processes, detecting recombination events based on phylogenetic reconstruction is often difficult, and estimating recombination rates using coalescent model-based methods is computationally expensive, and often infeasible for large sequencing data sets. Here, we present an efficient solution by introducing a set of mutational correlation functions computed using pairwise sequence comparison, which characterize various facets of bacterial recombination. We provide analytical expressions for these functions, which precisely recapitulate simulation results of neutral and adapting populations under different coalescent models. We used these to fit correlation functions measured at synonymous substitutions using whole-genome data on Escherichia coli and Streptococcus pneumoniae populations. We calculated and corrected for the effect of sample selection bias, i.e., the uneven sampling of individuals from natural microbial populations that exists in most datasets. Our method is fast and efficient, and does not employ phylogenetic inference or other computationally intensive numerics. By simply fitting analytical forms to measurements from sequence data, we show that recombination rates can be inferred, and the relative ages of different samples can be estimated. Our approach, which is based on population genetic modeling, is broadly applicable to a wide variety of data, and its computational efficiency makes it particularly attractive for use in the analysis of large sequencing datasets.

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuation-induced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated ("pulled") waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.

Total internal and external lengths of the Bolthausen-Sznitman coalescent

In this paper we study a weak law of large numbers for the total internal length of the Bolthausen-Sznitman coalescent, thereby obtaining the weak limit law of the centered and rescaled total external length; this extends results obtained in Dhersin and Möhle (2013). An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.

Predicting evolution from the shape of genealogical trees

Given a sample of genome sequences from an asexual population, can one predict its evolutionary future? Here we demonstrate that the branching patterns of reconstructed genealogical trees contains information about the relative fitness of the sampled sequences and that this information can be used to predict successful strains. Our approach is based on the assumption that evolution proceeds by accumulation of small effect mutations, does not require species specific input and can be applied to any asexual population under persistent selection pressure. We demonstrate its performance using historical data on seasonal influenza A/H3N2 virus. We predict the progenitor lineage of the upcoming influenza season with near optimal performance in 30% of cases and make informative predictions in 16 out of 19 years. Beyond providing a tool for prediction, our ability to make informative predictions implies persistent fitness variation among circulating influenza A/H3N2 viruses.

DOI:http://dx.doi.org/10.7554/eLife.03568.001

When viruses multiply, they copy their genetic material to make clones of themselves. However, the genetic material in the clone is often slightly different from the genetic material in the original virus. These mutations can be caused by mistakes made during copying or by radiation or chemicals. Further mutations arise when the clones multiply, which means that, after many generations, there will be quite large differences in the genetic material carried by many members of the population. Most mutations have little or no effect on the ‘fitness’ of an individual - that is, on its ability to survive and multiply - but some mutations do have an influence.

Some viruses, like seasonal influenza (flu) viruses, can mutate so rapidly that the most common strains change from year to year. This is why new flu vaccines are needed every year. To date most attempts to predict the evolution of seasonal flu viruses have focused on identifying specific features within the genetic sequences that might indicate fitness. However, such approaches require lots of information about the viruses, and this information is often not available.

To address this problem, Neher, Russell and Shraiman have developed a more general method to predict fitness from virus genetic sequences. First, a ‘family tree’ for a virus population - which shows how each strain of the virus is related to other strains - was constructed by comparing the genetic sequences.

The next step was based on the observation that as long as differences in fitness arise from the accumulation of multiple mutations, the branching structure of this family tree will bear a visible imprint of the natural selection process as it unfolds. Using this insight and methods borrowed from statistical physics, Neher et al. then analyzed the shape and branching pattern of the tree to work out the fitness of the different strains relative to each other.

Neher et al. tested the method using historical influenza A virus data. In 16 of the 19 years studied, the family tree approach made meaningful predictions about which viruses were most likely to give rise to future epidemics. The ability to predict influenza virus evolution from tree shape alone suggests that influenza virus evolution may be more predictable than previously expected.

DOI:http://dx.doi.org/10.7554/eLife.03568.002

Minimal Clade Size in the Bolthausen-Sznitman Coalescent

In this article we show the asymptotics of distribution and moments of the sizeXnof the minimal clade of a randomly chosen individual in a Bolthausen-Sznitmann-coalescent forn→ ∞. The Bolthausen-Sznitmann-coalescent is a Markov process taking states in the set of partitions of {1, …,n}, where 1, …,nare referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution ofXn. The main tool used is the connection of the Bolthausen-Sznitmann-coalescent with random recursive trees introduced by Goldschmidt and Martin (2005). With it, we show thatXn- 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process withn- 1 customers.

How to infer relative fitness from a sample of genomic sequences

Mounting evidence suggests that natural populations can harbor extensive fitness diversity with numerous genomic loci under selection. It is also known that genealogical trees for populations under selection are quantifiably different from those expected under neutral evolution and described statistically by Kingman's coalescent. While differences in the statistical structure of genealogies have long been used as a test for the presence of selection, the full extent of the information that they contain has not been exploited. Here we demonstrate that the shape of the reconstructed genealogical tree for a moderately large number of random genomic samples taken from a fitness diverse, but otherwise unstructured, asexual population can be used to predict the relative fitness of individuals within the sample. To achieve this we define a heuristic algorithm, which we test in silico, using simulations of a Wright-Fisher model for a realistic range of mutation rates and selection strength. Our inferred fitness ranking is based on a linear discriminator that identifies rapidly coalescing lineages in the reconstructed tree. Inferred fitness ranking correlates strongly with actual fitness, with a genome in the top 10% ranked being in the top 20% fittest with false discovery rate of 0.1-0.3, depending on the mutation/selection parameters. The ranking also enables us to predict the genotypes that future populations inherit from the present one. While the inference accuracy increases monotonically with sample size, samples of 200 nearly saturate the performance. We propose that our approach can be used for inferring relative fitness of genomes obtained in single-cell sequencing of tumors and in monitoring viral outbreaks.

On the accumulation of deleterious mutations during range expansions

We investigate the effect of spatial range expansions on the evolution of fitness when beneficial and deleterious mutations cosegregate. We perform individual-based simulations of 1D and 2D range expansions and complement them with analytical approximations for the evolution of mean fitness at the edge of the expansion. We find that deleterious mutations accumulate steadily on the wave front during range expansions, thus creating an expansion load. Reduced fitness due to the expansion load is not restricted to the wave front, but occurs over a large proportion of newly colonized habitats. The expansion load can persist and represent a major fraction of the total mutation load for thousands of generations after the expansion. The phenomenon of expansion load may explain growing evidence that populations that have recently expanded, including humans, show an excess of deleterious mutations. To test the predictions of our model, we analyse functional genetic diversity in humans and find patterns that are consistent with our model.

Coalescence and genetic diversity in sexual populations under selection

In sexual populations, selection operates neither on the whole genome, which is repeatedly taken apart and reassembled by recombination, nor on individual alleles that are tightly linked to the chromosomal neighborhood. The resulting interference between linked alleles reduces the efficiency of selection and distorts patterns of genetic diversity. Inference of evolutionary history from diversity shaped by linked selection requires an understanding of these patterns. Here, we present a simple but powerful scaling analysis identifying the unit of selection as the genomic "linkage block" with a characteristic length, , determined in a self-consistent manner by the condition that the rate of recombination within the block is comparable to the fitness differences between different alleles of the block. We find that an asexual model with the strength of selection tuned to that of the linkage block provides an excellent description of genetic diversity and the site frequency spectra compared with computer simulations. This linkage block approximation is accurate for the entire spectrum of strength of selection and is particularly powerful in scenarios with many weakly selected loci. The latter limit allows us to characterize coalescence, genetic diversity, and the speed of adaptation in the infinitesimal model of quantitative genetics.

Genealogies of rapidly adapting populations

The genetic diversity of a species is shaped by its recent evolutionary history and can be used to infer demographic events or selective sweeps. Most inference methods are based on the null hypothesis that natural selection is a weak or infrequent evolutionary force. However, many species, particularly pathogens, are under continuous pressure to adapt in response to changing environments. A statistical framework for inference from diversity data of such populations is currently lacking. Towards this goal, we explore the properties of genealogies in a model of continual adaptation in asexual populations. We show that lineages trace back to a small pool of highly fit ancestors, in which almost simultaneous coalescence of more than two lineages frequently occurs. Whereas such multiple mergers are unlikely under the neutral coalescent, they create a unique genetic footprint in adapting populations. The site frequency spectrum of derived neutral alleles, for example, is nonmonotonic and has a peak at high frequencies, whereas Tajima's D becomes more and more negative with increasing sample size. Because multiple merger coalescents emerge in many models of rapid adaptation, we argue that they should be considered as a null model for adapting populations.