The Nonstationary Dynamics of Fitness Distributions: Asexual Model with Epistasis and Standing Variation

Adaptation of a quantitative trait to a changing environment: New analytical insights on the asexual and infinitesimal sexual models

Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimal phenotype, whose value is shifted continuously through time. In this context, the population fate results from the equilibrium distribution of the trait, relative to the moving optimum. Such a distribution may vary with the shape of selection, the system of reproduction, the number of loci, the mutation kernel or their interactions. Here, we develop a methodology that provides quantitative measures of population maladaptation and potential of survival directly from the entire profile of the phenotypic distribution, without any a priori on its shape. We investigate two different systems of reproduction (asexual and infinitesimal sexual models of inheritance), with various forms of selection. In particular, we recover that fitness functions such that selection weakens away from the optimum lead to evolutionary tipping points, with an abrupt collapse of the population when the speed of environmental change is too high. Our unified framework allows deciphering the mechanisms that lead to this phenomenon. More generally, it allows discussing similarities and discrepancies between the two systems of reproduction, which are ultimately explained by different constraints on the evolution of the phenotypic variance. We demonstrate that the mean fitness in the population crucially depends on the shape of the selection function in the infinitesimal sexual model, in contrast with the asexual model. In the asexual model, we also investigate the effect of the mutation kernel and we show that kernels with higher kurtosis tend to reduce maladaptation and improve fitness, especially in fast changing environments.

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.

Forecasting emergence of COVID-19 variants of concern

We consider whether one can forecast the emergence of variants of concern in the SARS-CoV-2 outbreak and similar pandemics. We explore methods of population genetics and identify key relevant principles in both deterministic and stochastic models of spread of infectious disease. Finally, we demonstrate that fitness variation, defined as a trait for which an increase in its value is associated with an increase in net Darwinian fitness if the value of other traits are held constant, is a strong indicator of imminent transition in the viral population.

Adaptation in a heterogeneous environment I: persistence versus extinction

Understanding how a diversity of plants in agroecosystems affects the adaptation of pathogens is a key issue in agroecology. We analyze PDE systems describing the dynamics of adaptation of two phenotypically structured populations, under the effects of mutation, selection and migration in a two-patch environment, each patch being associated with a different phenotypic optimum. We consider two types of growth functions that depend on the n-dimensional phenotypic trait: either local and linear or nonlocal nonlinear. In both cases, we obtain existence and uniqueness results as well as a characterization of the large-time behaviour of the solution (persistence or extinction) based on the sign of a principal eigenvalue. We show that migration between the two environments decreases the chances of persistence, with in some cases a 'lethal migration threshold' above which persistence is not possible. Comparison with stochastic individual-based simulations shows that the PDE approach accurately captures this threshold. Our results illustrate the importance of cultivar mixtures for disease prevention and control.

Genetic Paths to Evolutionary Rescue and the Distribution of Fitness Effects Along Them

The past century has seen substantial theoretical and empirical progress on the genetic basis of adaptation. Over this same period, a pressing need to prevent the evolution of drug resistance has uncovered much about the potential genetic basis of persistence in declining populations. However, we have little theory to predict and generalize how persistence-by sufficiently rapid adaptation-might be realized in this explicitly demographic scenario. Here, we use Fisher's geometric model with absolute fitness to begin a line of theoretical inquiry into the genetic basis of evolutionary rescue, focusing here on asexual populations that adapt through de novo mutations. We show how the dominant genetic path to rescue switches from a single mutation to multiple as mutation rates and the severity of the environmental change increase. In multi-step rescue, intermediate genotypes that themselves go extinct provide a "springboard" to rescue genotypes. Comparing to a scenario where persistence is assured, our approach allows us to quantify how a race between evolution and extinction leads to a genetic basis of adaptation that is composed of fewer loci of larger effect. We hope this work brings awareness to the impact of demography on the genetic basis of adaptation.

Natural selection in compartmentalized environment with reshuffling

The emerging field of high-throughput compartmentalized in vitro evolution is a promising new approach to protein engineering. In these experiments, libraries of mutant genotypes are randomly distributed and expressed in microscopic compartments-droplets of an emulsion. The selection of desirable variants is performed according to the phenotype of each compartment. The random partitioning leads to a fraction of compartments receiving more than one genotype making the whole process a lab implementation of the group selection. From a practical point of view (where efficient selection is typically sought), it is important to know the impact of the increase in the mean occupancy of compartments on the selection efficiency. We carried out a theoretical investigation of this problem in the context of selection dynamics for an infinite non-mutating subdivided population that randomly colonizes an infinite number of patches (compartments) at each reproduction cycle. We derive here an update equation for any distribution of phenotypes and any value of the mean occupancy. Using this result, we demonstrate that, for the linear additive fitness, the best genotype is still selected regardless of the mean occupancy. Furthermore, the selection process is remarkably resilient to the presence of multiple genotypes per compartments, and slows down approximately inversely proportional to the mean occupancy at high values. We extend out results to more general expressions that cover nonadditive and non-linear fitnesses, as well non-Poissonian distribution among compartments. Our conclusions may also apply to natural genetic compartmentalized replicators, such as viruses or early trans-acting RNA replicators.

Transcriptional noise and exaptation as sources for bacterial sRNAs

Understanding how new genes originate and integrate into cellular networks is key to understanding evolution. Bacteria present unique opportunities for both the natural history and experimental study of gene origins, due to their large effective population sizes, rapid generation times, and ease of genetic manipulation. Bacterial small non-coding RNAs (sRNAs), in particular, many of which operate through a simple antisense regulatory logic, may serve as tractable models for exploring processes of gene origin and adaptation. Understanding how and on what timescales these regulatory molecules arise has important implications for understanding the evolution of bacterial regulatory networks, in particular, for the design of comparative studies of sRNA function. Here, we introduce relevant concepts from evolutionary biology and review recent work that has begun to shed light on the timescales and processes through which non-functional transcriptional noise is co-opted to provide regulatory functions. We explore possible scenarios for sRNA origin, focusing on the co-option, or exaptation, of existing genomic structures which may provide protected spaces for sRNA evolution.

Evolution of bacteria specialization along an antibiotic dose gradient

Antibiotic and pesticide resistance of pathogens are major and pressing worldwide issues. Resistance evolution is often considered in simplified ecological contexts: treated versus nontreated environments. In contrast, antibiotic usually present important dose gradients: from ecosystems to hospitals to polluted soils, in treated patients across tissues. However, we do not know whether adaptation to low or high doses involves different phenotypic traits, and whether these traits trade‐off with each other. In this study, we investigated the occurrence of such fitness trade‐offs along a dose gradient by evolving experimentally resistant lines of Escherichia coli at different antibiotic concentrations for ∼400 generations. Our results reveal fast evolution toward specialization following the first mutational step toward resistance, along with pervasive trade‐offs among different evolution doses. We found clear and regular fitness patterns of specialization, which converged rapidly from different initial starting points. These findings are consistent with a simple fitness peak shift model as described by the classical evolutionary ecology theory of adaptation across environmental gradients. We also found that the fitness costs of resistance tend to be compensated through time at low doses whereas they increase through time at higher doses. This cost evolution follows a linear trend with the log‐dose of antibiotic along the gradient. These results suggest a general explanation for the variability of the fitness costs of resistance and their evolution. Overall, these findings call for more realistic models of resistance management incorporating dose‐specialization.

Evolutionary Rescue over a Fitness Landscape

Evolutionary rescue describes a situation where adaptive evolution prevents the extinction of a population facing a stressing environment. Models of evolutionary rescue could in principle be used to predict the level of stress beyond which extinction becomes likely for species of conservation concern, or, conversely, the treatment levels most likely to limit the emergence of resistant pests or pathogens. Stress levels are known to affect both the rate of population decline (demographic effect) and the speed of adaptation (evolutionary effect), but the latter aspect has received less attention. Here, we address this issue using Fisher's geometric model of adaptation. In this model, the fitness effects of mutations depend both on the genotype and the environment in which they arise. In particular, the model introduces a dependence between the level of stress, the proportion of rescue mutants, and their costs before the onset of stress. We obtain analytic results under a strong-selection-weak-mutation regime, which we compare to simulations. We show that the effect of the environment on evolutionary rescue can be summarized into a single composite parameter quantifying the effective stress level, which is amenable to empirical measurement. We describe a narrow characteristic stress window over which the rescue probability drops from very likely to very unlikely as the level of stress increases. This drop is sharper than in previous models, as a result of the decreasing proportion of stress-resistant mutations as stress increases. We discuss how to test these predictions with rescue experiments across gradients of stress.

Beneficial mutation-selection dynamics in finite asexual populations: a free boundary approach

Using a free boundary approach based on an analogy with ice melting models, we propose a deterministic PDE framework to describe the dynamics of fitness distributions in the presence of beneficial mutations with non-epistatic effects on fitness. Contrarily to most approaches based on deterministic models, our framework does not rely on an infinite population size assumption, and successfully captures the transient as well as the long time dynamics of fitness distributions. In particular, consistently with stochastic individual-based approaches or stochastic PDE approaches, it leads to a constant asymptotic rate of adaptation at large times, that most deterministic approaches failed to describe. We derive analytic formulas for the asymptotic rate of adaptation and the full asymptotic distribution of fitness. These formulas depend explicitly on the population size, and are shown to be accurate for a wide range of population sizes and mutation rates, compared to individual-based simulations. Although we were not able to derive an analytic description for the transient dynamics, numerical computations lead to accurate predictions and are computationally efficient compared to stochastic simulations. These computations show that the fitness distribution converges towards a travelling wave with constant speed, and whose profile can be computed analytically.