Interfering waves of adaptation promote spatial mixing

Graph-structured populations elucidate the role of deleterious mutations in long-term evolution

Birth-death models are used to understand the interplay of genetic drift and natural selection. While well-mixed populations remain unaffected by the order of birth and death and where selection acts, evolutionary outcomes in spatially structured populations are affected by these choices. We show that the choice of individual moving to vacant sites-parent or offspring-controls the initial mutant placement on a graph and hence alters its fixation probability. Moving parent individuals introduces, to our knowledge, previously unexplored update rules and fixation categories for heterogeneous graphs. We identify a class of graphs, amplifiers of fixation, where fixation probability is larger than in well-mixed populations, regardless of the mutant fitness. Under death-Birth parent moving, the star graph is an amplifier of fixation, with a non-zero fixation probability for deleterious mutants, in contrast to very large well-mixed populations. Most Erdős-Rényi graphs of size 8 are amplifiers of fixation under death-Birth parent moving, but suppressors of fixation under Birth-death offspring moving. Surprisingly, amplifiers of fixation attain lower fitness in long-term evolution, despite favouring beneficial mutants, while suppressors of fixation attain higher fitness. These counterintuitive findings are explained by the fate of deleterious mutations and highlight the crucial role of deleterious mutants for adaptive evolution.

Clonal interference, genetic variation and the speed of evolution in structured populations

When it comes to understanding the role that population structure plays in shaping rates of evolution, it is commonly accepted that interference between evolutionary innovations is more prevalent in structured populations compared to well-mixed, and that population structure reduces the rate of evolution, while simultaneously promoting maintenance of genetic variation. Prior models usually represent population structure using two or more connected demes or lattices with periodic boundary conditions. Fundamentally, the observed spatial evolutionary slow-down is rooted in the fact that these types of structures increase the time it takes for a selective sweep and therefore, increase the probability that multiple beneficial mutations will coexist and interfere. Here we show that systematically introducing more heterogeneity in population structure can reshape these prior conclusions and lead to a much wider range of observed evolutionary outcome, including increased rates of evolution. At a big picture level, our results showcase that the evolutionary effects of population structure crucially depend on the properties of the topology considered. Understanding these spatial properties is therefore requisite for making meaningful evolutionary comparisons across different topologies, for forming sensible null expectations about experimental or observational data or for developing an intuition for when well-mixed approximations, or approximations done using spatial models with a high degree of symmetry, might not apply.

Selective sweep probabilities in spatially expanding populations

Evolution during range expansions shapes biological systems from microbial communities and tumours up to invasive species. A fundamental question is whether, when a beneficial mutation arises during a range expansion, it will evade clonal interference and sweep through the population to fixation. However, most theoretical investigations of range expansions have been confined to regimes in which selective sweeps are effectively impossible, while studies of selective sweeps have either assumed constant population size or have ignored spatial structure. Here we use mathematical modelling and analysis to investigate selective sweep probabilities in the alternative yet biologically relevant scenario in which mutants can outcompete and displace a slowly spreading wildtype. Assuming constant radial expansion speed, we derive probability distributions for the arrival time and location of the first surviving mutant and hence find surprisingly simple approximate and exact expressions for selective sweep probabilities in one, two and three dimensions, which are independent of mutation rate. Namely, the selective sweep probability is approximately 1 - c w t / c m d , where c w t and c m are the wildtype and mutant radial expansion speeds, and d the spatial dimension. Using agent-based simulations, we show that our analytical results accurately predict selective sweep frequencies in the two-dimensional spatial Moran process. We further compare our results with those obtained for alternative growth laws. Parameterizing our model for human tumours, we find that selective sweeps are predicted to be rare except during very early solid tumour growth, thus providing a general, pan-cancer explanation for findings from recent sequencing studies.

Modeling the evolution of recombination plasticity: A prospective review

Meiotic recombination is one of the main sources of genetic variation, a fundamental factor in the evolutionary adaptation of sexual eukaryotes. Yet, the role of variation in recombination rate and other recombination features remains underexplored. In this review, we focus on the sensitivity of recombination rates to different extrinsic and intrinsic factors. We briefly present the empirical evidence for recombination plasticity in response to environmental perturbations and/or poor genetic background and discuss theoretical models developed to explain how such plasticity could have evolved and how it can affect important population characteristics. We highlight a gap between the evidence, which comes mostly from experiments with diploids, and theory, which typically assumes haploid selection. Finally, we formulate open questions whose solving would help to outline conditions favoring recombination plasticity. This will contribute to answering the long-standing question of why sexual recombination exists despite its costs, since plastic recombination may be evolutionary advantageous even in selection regimes rejecting any non-zero constant recombination.

Biofilm antimicrobial susceptibility through an experimental evolutionary lens

Experimental evolution experiments in which bacterial populations are repeatedly exposed to an antimicrobial treatment, and examination of the genotype and phenotype of the resulting evolved bacteria, can help shed light on mechanisms behind reduced susceptibility. In this review we present an overview of why it is important to include biofilms in experimental evolution, which approaches are available to study experimental evolution in biofilms and what experimental evolution has taught us about tolerance and resistance in biofilms. Finally, we present an emerging consensus view on biofilm antimicrobial susceptibility supported by data obtained during experimental evolution studies.

Population genetic models for the spatial spread of adaptive variants: A review in light of SARS-CoV-2 evolution

Theoretical population genetics has long studied the arrival and geographic spread of adaptive variants through the analysis of mathematical models of dispersal and natural selection. These models take on a renewed interest in the context of the COVID-19 pandemic, especially given the consequences that novel adaptive variants have had on the course of the pandemic as they have spread through global populations. Here, we review theoretical models for the spatial spread of adaptive variants and identify areas to be improved in future work, toward a better understanding of variants of concern in Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) evolution and other contemporary applications. As we describe, characteristics of pandemics such as COVID-19-such as the impact of long-distance travel patterns and the overdispersion of lineages due to superspreading events-suggest new directions for improving upon existing population genetic models.

Lineage dynamics in growing biofilms: Spatial patterns of standing vs. de novo diversity

Microbial biofilms show high phenotypic and genetic diversity, yet the mechanisms underlying diversity generation and maintenance remain unclear. Here, we investigate how spatial patterns of growth activity within a biofilm lead to spatial patterns of genetic diversity. Using individual-based computer simulations, we show that the active layer of growing cells at the biofilm interface controls the distribution of lineages within the biofilm, and therefore the patterns of standing and de novo diversity. Comparing biofilms of equal size, those with a thick active layer retain more standing diversity, while de novo diversity is more evenly distributed within the biofilm. In contrast, equal-sized biofilms with a thin active layer retain less standing diversity, and their de novo diversity is concentrated at the top of the biofilm, and in fewer lineages. In the context of antimicrobial resistance, biofilms with a thin active layer may be more prone to generate lineages with multiple resistance mutations, and to seed new resistant biofilms via sloughing of resistant cells from the upper layers. Our study reveals fundamental "baseline" mechanisms underlying the patterning of diversity within biofilms.

Application of simultaneous selective pressures slows adaptation

Beneficial mutations that arise in an evolving asexual population may compete or interact in ways that alter the overall rate of adaptation through mechanisms such as clonal or functional interference. The application of multiple selective pressures simultaneously may allow for a greater number of adaptive mutations, increasing the opportunities for competition between selectively advantageous alterations, and thereby reducing the rate of adaptation. We evolved a strain of Saccharomyces cerevisiae that could not produce its own histidine or uracil for ~500 generations under one or three selective pressures: limitation of the concentration of glucose, histidine, and/or uracil in the media. The rate of adaptation was obtained by measuring evolved relative fitness using competition assays. Populations evolved under a single selective pressure showed a statistically significant increase in fitness on those pressures relative to the ancestral strain, but the populations evolved on all three pressures did not show a statistically significant increase in fitness over the ancestral strain on any single pressure. Simultaneously limiting three essential nutrients for a population of S. cerevisiae effectively slows the rate of evolution on any one of the three selective pressures applied, relative to the single selective pressure cases. We identify possible mechanisms for fitness changes seen between populations evolved on one or three limiting nutrient pressures by high-throughput sequencing. Adding multiple selective pressures to evolving disease like cancer and infectious diseases could reduce the rate of adaptation and thereby may slow disease progression, prolong drug efficacy and prevent deaths.

Evolutionary Rescue from a Wave of Biological Invasion

Evolution can potentially rescue populations from being driven extinct by biological invasions, but predictions for this occurrence are generally lacking. Here I derive theoretical predictions for evolutionary rescue of a resident population experiencing invasion from an introduced competitor that spreads over its introduced range as a traveling spatial wave that displaces residents. I compare the likelihood of evolutionary rescue from invasion for two modes of competition: exploitation and interference competition. I find that, all else equal, evolutionary rescue is less effective at preventing extinction caused by interference-driven invasions than by exploitation-driven invasions. Rescue from interference-driven invasions is, surprisingly, independent of invader dispersal rate or the speed of invasion and is more weakly dependent on range size than in the exploitation-driven case. In contrast, rescue from exploitation-driven invasions strongly depends on range size and is less likely during fast invasions. The results presented here have potential applications for conserving endemic species from nonnative invaders and for ensuring extinction of pests using intentionally introduced biocontrol agents.

Evolutionary Dynamics in Structured Populations Under Strong Population Genetic Forces

In the long-term neutral equilibrium, high rates of migration between subpopulations result in little population differentiation. However, in the short-term, even very abundant migration may not be enough for subpopulations to equilibrate immediately. In this study, we investigate dynamical patterns of short-term population differentiation in adapting populations via stochastic and analytical modeling through time. We characterize a regime in which selection and migration interact to create non-monotonic patterns of population differentiation over time when migration is weaker than selection, but stronger than drift. We demonstrate how these patterns can be leveraged to estimate high migration rates using approximate Bayesian computation. We apply this approach to estimate fast migration in a rapidly adapting intra-host Simian-HIV population sampled from different anatomical locations. We find differences in estimated migration rates between different compartments, even though all are above [Formula: see text] = 1. This work demonstrates how studying demographic processes on the timescale of selective sweeps illuminates processes too fast to leave signatures on neutral timescales.

Evolution of Stress-Induced Mutagenesis in the Presence of Horizontal Gene Transfer

Stress-induced mutagenesis has been observed in multiple species of bacteria and yeast. It has been suggested that in asexual populations, a mutator allele that increases the mutation rate during stress can sweep to fixation with the beneficial mutations it generates. However, even asexual microbes can undergo horizontal gene transfer and rare recombination, which typically interfere with the spread of mutator alleles. Here we examine the effect of horizontal gene transfer on the evolutionary advantage of stress-induced mutator alleles. Our results demonstrate that stress-induced mutator alleles are favored by selection even in the presence of horizontal gene transfer and more so when the mutator alleles also increase the rate of horizontal gene transfer. We suggest that when regulated by stress, mutation and horizontal gene transfer can be complementary rather than competing adaptive strategies and that stress-induced mutagenesis has important implications for evolutionary biology, ecology, and epidemiology, even in the presence of horizontal gene transfer and rare recombination.

Hitchhiking in space: Ancestry in adapting, spatially extended populations

Selective sweeps reduce neutral genetic diversity. In sexual populations, this "hitchhiking" effect is thought to be limited to the local genomic region of the sweeping allele. While this is true in panmictic populations, we find that in spatially extended populations the combined effects of many unlinked sweeps can affect patterns of ancestry (and therefore neutral genetic diversity) across the whole genome. Even low rates of sweeps can be enough to skew the spatial locations of ancestors such that neutral mutations that occur in an individual living outside a small region in the center of the range have virtually no chance of fixing in the population. The fact that nearly all ancestry rapidly traces back to a small spatial region also means that relatedness between individuals falls off very slowly as a function of the spatial distance between them.

Persistence of antibiotic resistance plasmids in bacterial biofilms

The emergence and spread of antibiotic resistance is a crisis in health care today. Antibiotic resistance is often horizontally transferred to susceptible bacteria by means of multidrug resistance plasmids that may or may not persist in the absence of antibiotics. Because bacterial pathogens often grow as biofilms, there is a need to better understand the evolution of plasmid persistence in these environments. Here we compared the evolution of plasmid persistence in the pathogen Acinetobacter baumannii when grown under antibiotic selection in biofilms versus well-mixed liquid cultures. After 4 weeks, clones in which the plasmid was more stably maintained in the absence of antibiotic selection were present in both populations. On average plasmid persistence increased more in liquid batch cultures, but variation in the degree of persistence was greater among biofilm-derived clones. The results of this study show for the first time that the persistence of MDR plasmids improves in biofilms.

Tumor evolution: Linear, branching, neutral or punctuated?

Intratumor heterogeneity has been widely reported in human cancers, but our knowledge of how this genetic diversity emerges over time remains limited. A central challenge in studying tumor evolution is the difficulty in collecting longitudinal samples from cancer patients. Consequently, most studies have inferred tumor evolution from single time-point samples, providing very indirect information. These data have led to several competing models of tumor evolution: linear, branching, neutral and punctuated. Each model makes different assumptions regarding the timing of mutations and selection of clones, and therefore has different implications for the diagnosis and therapeutic treatment of cancer patients. Furthermore, emerging evidence suggests that models may change during tumor progression or operate concurrently for different classes of mutations. Finally, we discuss data that supports the theory that most human tumors evolve from a single cell in the normal tissue. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer?, edited by Dr. Robert A. Gatenby.

Evolution at 'Sutures' and 'Centers': Recombination Can Aid Adaptation of Spatially Structured Populations on Rugged Fitness Landscapes

Epistatic interactions among genes can give rise to rugged fitness landscapes, in which multiple “peaks” of high-fitness allele combinations are separated by “valleys” of low-fitness genotypes. How populations traverse rugged fitness landscapes is a long-standing question in evolutionary biology. Sexual reproduction may affect how a population moves within a rugged fitness landscape. Sex may generate new high-fitness genotypes by recombination, but it may also destroy high-fitness genotypes by shuffling the genes of a fit parent with a genetically distinct mate, creating low-fitness offspring. Either of these opposing aspects of sex require genotypic diversity in the population. Spatially structured populations may harbor more diversity than well-mixed populations, potentially amplifying both positive and negative effects of sex. On the other hand, spatial structure leads to clumping in which mating is more likely to occur between like types, diminishing the effects of recombination. In this study, we use computer simulations to investigate the combined effects of recombination and spatial structure on adaptation in rugged fitness landscapes. We find that spatially restricted mating and offspring dispersal may allow multiple genotypes inhabiting suboptimal peaks to coexist, and recombination at the “sutures” between the clusters of these genotypes can create genetically novel offspring. Sometimes such an offspring genotype inhabits a new peak on the fitness landscape. In such a case, spatially restricted mating allows this fledgling subpopulation to avoid recombination with distinct genotypes, as mates are more likely to be the same genotype. Such population “centers” can allow nascent peaks to establish despite recombination. Spatial structure may therefore allow an evolving population to enjoy the creative side of sexual recombination while avoiding its destructive side.

For a novel genotype to establish in a population, it must (1) be created, and (2) not be subsequently lost. Recombination is a double-edged sword in this process, potentially fostering creation, but also hastening loss as the novel genotype is being recombined with other genotypes, especially when rare. In this study, we find that spatial structure may affect both the creative and destructive aspects of recombination in rugged fitness landscapes. By slowing the spread of high-fitness genotypes, spatially restricted mating and dispersal may allow diverse subpopulations to arise. Reproduction across the borders of these subpopulations—at “sutures”—may create genetic novelty. Depending on the topography of the fitness landscape, such novelty may be in the domain of attraction of a new, higher peak; the population may “peak-jump” to an area of genotype space unlikely to be explored by mutation alone. Lineages founded by peak-jumping events are particularly prone to early extinction, as recombination with unlike genotypes may disrupt the rare allele combination and thereby produce low-fitness offspring. However, these fledgling peak lineages may be protected from early extinction by mating within small homotypic clusters—in “centers”. Thus, spatial structure may allow a population to create rare genotypes via recombination, and allow those rare genotypes to persist despite recombination.

Spatial Moran models, II: cancer initiation in spatially structured tissue

We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This investigation contributes to an understanding of how the selective advantage of cancer cells together with the rates of mutations driving cancer, impact the process and timing of carcinogenesis. These results can be used to give insights into tumor heterogeneity and the "cancer field effect," the observation that a malignancy is often surrounded by cells that have undergone premalignant transformation.

Experimental evolution in biofilm populations

Biofilms are a major form of microbial life in which cells form dense surface associated communities that can persist for many generations. The long-life of biofilm communities means that they can be strongly shaped by evolutionary processes. Here, we review the experimental study of evolution in biofilm communities. We first provide an overview of the different experimental models used to study biofilm evolution and their associated advantages and disadvantages. We then illustrate the vast amount of diversification observed during biofilm evolution, and we discuss (i) potential ecological and evolutionary processes behind the observed diversification, (ii) recent insights into the genetics of adaptive diversification, (iii) the striking degree of parallelism between evolution experiments and real-life biofilms and (iv) potential consequences of diversification. In the second part, we discuss the insights provided by evolution experiments in how biofilm growth and structure can promote cooperative phenotypes. Overall, our analysis points to an important role of biofilm diversification and cooperation in bacterial survival and productivity. Deeper understanding of both processes is of key importance to design improved antimicrobial strategies and diagnostic techniques.

This review paper provides an overview of (i) the different experimental models used to study biofilm evolution, (ii) the vast amount of diversification observed during biofilm evolution (including potential causes and consequences) and (iii) recent insights in how growth in biofilms can lead to the evolution of cooperative phenotypes.

Gene expression variability in clonal populations: Causes and consequences

During the last decade it has been shown that among cell variation in gene expression plays an important role within clonal populations. Here, we provide an overview of the different mechanisms contributing to gene expression variability in clonal populations. These are ranging from inherent variations in the biochemical process of gene expression itself, such as intrinsic noise, extrinsic noise and bistability to individual responses to variations in the local micro-environment, a phenomenon called phenotypic plasticity. Also genotypic variations caused by clonal evolution and phase variation can contribute to gene expression variability. Consequently, gene expression studies need to take these fluctuations in expression into account. However, frequently used techniques for expression quantification, such as microarrays, RNA sequencing, quantitative PCR and gene reporter fusions classically determine the population average of gene expression. Here, we discuss how these techniques can be adapted towards single cell analysis by integration with single cell isolation, RNA amplification and microscopy. Alternatively more qualitative selection-based techniques, such as mutant screenings, in vivo expression technology (IVET) and recombination-based IVET (RIVET) can be applied for detection of genes expressed only within a subpopulation. Finally, differential fluorescence induction (DFI), a protocol specially designed for single cell expression is discussed.

The population genetics of drug resistance evolution in natural populations of viral, bacterial and eukaryotic pathogens

Drug resistance is a costly consequence of pathogen evolution and a major concern in public health. In this review, we show how population genetics can be used to study the evolution of drug resistance and also how drug resistance evolution is informative as an evolutionary model system. We highlight five examples from diverse organisms with particular focus on: (i) identifying drug resistance loci in the malaria parasite Plasmodium falciparum using the genomic signatures of selective sweeps, (ii) determining the role of epistasis in drug resistance evolution in influenza, (iii) quantifying the role of standing genetic variation in the evolution of drug resistance in HIV, (iv) using drug resistance mutations to study clonal interference dynamics in tuberculosis and (v) analysing the population structure of the core and accessory genome of Staphylococcus aureus to understand the spread of methicillin resistance. Throughout this review, we discuss the uses of sequence data and population genetic theory in studying the evolution of drug resistance.

The Role of Standing Variation in Geographic Convergent Adaptation

The extent to which populations experiencing shared selective pressures adapt through a shared genetic response is relevant to many questions in evolutionary biology. In this article, we explore how standing genetic variation contributes to convergent genetic responses in a geographically spread population. Geographically limited dispersal slows the spread of each selected allele, hence allowing other alleles to spread before any one comes to dominate the population. When selectively equivalent alleles meet, their progress is substantially slowed, dividing the species range into a random tessellation, which can be well understood by analogy to a Poisson process model of crystallization. In this framework, we derive the geographic scale over which an allele dominates and the proportion of adaptive alleles that arise from standing variation. Finally, we explore how negative pleiotropic effects of alleles can bias the subset of alleles that contribute to the species' adaptive response. We apply the results to the malaria-resistance glucose-6-phosphate dehydrogenase-deficiency alleles, where the large mutational target size makes it a likely candidate for adaptation from deleterious standing variation. Our results suggest that convergent adaptation may be common. Therefore, caution must be exercised when arguing that strongly geographically restricted alleles are the outcome of local adaptation. We close by discussing the implications of these results for ideas of species coherence and the nature of divergence between species.

Acceleration of evolutionary spread by long-range dispersal

The spreading of evolutionary novelties across populations is the central element of adaptation. Unless populations are well mixed (like bacteria in a shaken test tube), the spreading dynamics depend not only on fitness differences but also on the dispersal behavior of the species. Spreading at a constant speed is generally predicted when dispersal is sufficiently short ranged, specifically when the dispersal kernel falls off exponentially or faster. However, the case of long-range dispersal is unresolved: Although it is clear that even rare long-range jumps can lead to a drastic speedup--as air-traffic-mediated epidemics show--it has been difficult to quantify the ensuing stochastic dynamical process. However, such knowledge is indispensable for a predictive understanding of many spreading processes in natural populations. We present a simple iterative scaling approximation supported by simulations and rigorous bounds that accurately predicts evolutionary spread, which is determined by a trade-off between frequency and potential effectiveness of long-distance jumps. In contrast to the exponential laws predicted by deterministic "mean-field" approximations, we show that the asymptotic spatial growth is according to either a power law or a stretched exponential, depending on the tails of the dispersal kernel. More importantly, we provide a full time-dependent description of the convergence to the asymptotic behavior, which can be anomalously slow and is relevant even for long times. Our results also apply to spreading dynamics on networks with a spectrum of long-range links under certain conditions on the probabilities of long-distance travel: These are relevant for the spread of epidemics.

Non-equilibrium physics and evolution--adaptation, extinction, and ecology: a key issues review

Evolutionary dynamics in nature constitute an immensely complex non-equilibrium process. We review the application of physical models of evolution, by focusing on adaptation, extinction, and ecology. In each case, we examine key concepts by working through examples. Adaptation is discussed in the context of bacterial evolution, with a view toward the relationship between growth rates, mutation rates, selection strength, and environmental changes. Extinction dynamics for an isolated population are reviewed, with emphasis on the relation between timescales of extinction, population size, and temporally correlated noise. Ecological models are discussed by focusing on the effect of spatial interspecies interactions on diversity. Connections between physical processes--such as diffusion, turbulence, and localization--and evolutionary phenomena are highlighted.

Clonal interference and Muller's ratchet in spatial habitats

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller's ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio [Formula: see text] between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using, again, an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.

Multifocality and recurrence risk: a quantitative model of field cancerization

Primary tumors often emerge within genetically altered fields of premalignant cells that appear histologically normal but have a high chance of progression to malignancy. Clinical observations have suggested that these premalignant fields pose high risks for emergence of recurrent tumors if left behind after surgical removal of the primary tumor. In this work, we develop a spatio-temporal stochastic model of epithelial carcinogenesis, combining cellular dynamics with a general framework for multi-stage genetic progression to cancer. Using the model, we investigate how various properties of the premalignant fields depend on microscopic cellular properties of the tissue. In particular, we provide analytic results for the size-distribution of the histologically undetectable premalignant fields at the time of diagnosis, and investigate how the extent and the geometry of these fields depend upon key groups of parameters associated with the tissue and genetic pathways. We also derive analytical results for the relative risks of local vs. distant secondary tumors for different parameter regimes, a critical aspect for the optimal choice of post-operative therapy in carcinoma patients. This study contributes to a growing literature seeking to obtain a quantitative understanding of the spatial dynamics in cancer initiation.

Quantifying the role of population subdivision in evolution on rugged fitness landscapes

Natural selection drives populations towards higher fitness, but crossing fitness valleys or plateaus may facilitate progress up a rugged fitness landscape involving epistasis. We investigate quantitatively the effect of subdividing an asexual population on the time it takes to cross a fitness valley or plateau. We focus on a generic and minimal model that includes only population subdivision into equivalent demes connected by global migration, and does not require significant size changes of the demes, environmental heterogeneity or specific geographic structure. We determine the optimal speedup of valley or plateau crossing that can be gained by subdivision, if the process is driven by the deme that crosses fastest. We show that isolated demes have to be in the sequential fixation regime for subdivision to significantly accelerate crossing. Using Markov chain theory, we obtain analytical expressions for the conditions under which optimal speedup is achieved: valley or plateau crossing by the subdivided population is then as fast as that of its fastest deme. We verify our analytical predictions through stochastic simulations. We demonstrate that subdivision can substantially accelerate the crossing of fitness valleys and plateaus in a wide range of parameters extending beyond the optimal window. We study the effect of varying the degree of subdivision of a population, and investigate the trade-off between the magnitude of the optimal speedup and the width of the parameter range over which it occurs. Our results, obtained for fitness valleys and plateaus, also hold for weakly beneficial intermediate mutations. Finally, we extend our work to the case of a population connected by migration to one or several smaller islands. Our results demonstrate that subdivision with migration alone can significantly accelerate the crossing of fitness valleys and plateaus, and shed light onto the quantitative conditions necessary for this to occur.

Experimental evidence has recently been accumulating to suggest that fitness landscape ruggedness is common in a variety of organisms. Rugged landscapes arise from interactions between genetic variants, called epistasis, which can lead to fitness valleys or plateaus. The time needed to cross such fitness valleys or plateaus exhibits a rich dependence on population size, since stochastic effects have higher importance in small populations, increasing the probability of fixation of neutral or deleterious mutants. This may lead to an advantage of population subdivision, a possibility which has been strongly debated for nearly one hundred years. In this work, we quantitatively determine when, and to what extent, population subdivision accelerates valley and plateau crossing. Using the simple model of an asexual population subdivided into identical demes connected by gobal migration, we derive the conditions under which crossing by a subdivided population is driven by its fastest deme, thus giving rise to the maximal speedup. Our analytical predictions are verified using stochastic simulations. We investigate the effect of varying the degree of subdivision of a population. We generalize our results to weakly beneficial intermediates and to different population structures. We discuss the magnitude and robustness of the effect for realistic parameter values.

Catastrophic shifts and lethal thresholds in a propagating front model of unstable tumor progression

Unstable dynamics characterizes the evolution of most solid tumors. Because of an increased failure of maintaining genome integrity, a cumulative increase in the levels of gene mutation and loss is observed. Previous work suggests that instability thresholds to cancer progression exist, defining phase transition phenomena separating tumor-winning scenarios from tumor extinction or coexistence phases. Here we present an integral equation approach to the quasispecies dynamics of unstable cancer. The model exhibits two main phases, characterized by either the success or failure of cancer tissue. Moreover, the model predicts that tumor failure can be due to either a reduced selective advantage over healthy cells or excessive instability. We also derive an approximate, analytical solution that predicts the front speed of aggressive tumor populations on the instability space.

The impact of natural transformation on adaptation in spatially structured bacterial populations

BACKGROUND

Recent studies have demonstrated that natural transformation and the formation of highly structured populations in bacteria are interconnected. In spite of growing evidence about this connection, little is known about the dynamics of natural transformation in spatially structured bacterial populations.

RESULTS

In this work, we model the interdependency between the dynamics of the bacterial gene pool and those of environmental DNA in space to dissect the effect of transformation on adaptation. Our model reveals that even with only a single locus under consideration, transformation with a free DNA fragment pool results in complex adaptation dynamics that do not emerge in previous models focusing only on the gene shuffling effect of transformation at multiple loci. We demonstrate how spatial restriction on population growth and DNA diffusion in the environment affect the impact of transformation on adaptation. We found that in structured bacterial populations intermediate DNA diffusion rates predominantly cause transformation to impede adaptation by spreading deleterious alleles in the population.

CONCLUSION

Overall, our model highlights distinctive evolutionary consequences of bacterial transformation in spatially restricted compared to planktonic bacterial populations.

The rate of adaptation in large sexual populations with linear chromosomes

In large populations, multiple beneficial mutations may be simultaneously spreading. In asexual populations, these mutations must either arise on the same background or compete against each other. In sexual populations, recombination can bring together beneficial alleles from different backgrounds, but tightly linked alleles may still greatly interfere with each other. We show for well-mixed populations that when this interference is strong, the genome can be seen as consisting of many effectively asexual stretches linked together. The rate at which beneficial alleles fix is thus roughly proportional to the rate of recombination and depends only logarithmically on the mutation supply and the strength of selection. Our scaling arguments also allow us to predict, with reasonable accuracy, the fitness distribution of fixed mutations when the mutational effect sizes are broad. We focus on the regime in which crossovers occur more frequently than beneficial mutations, as is likely to be the case for many natural populations.

The impact of spatial structure on viral genomic diversity generated during adaptation to thermal stress

BACKGROUND

Most clinical and natural microbial communities live and evolve in spatially structured environments. When changes in environmental conditions trigger evolutionary responses, spatial structure can impact the types of adaptive response and the extent to which they spread. In particular, localized competition in a spatial landscape can lead to the emergence of a larger number of different adaptive trajectories than would be found in well-mixed populations. Our goal was to determine how two levels of spatial structure affect genomic diversity in a population and how this diversity is manifested spatially.

METHODOLOGY/PRINCIPAL FINDINGS

We serially transferred bacteriophage populations growing at high temperatures (40°C) on agar plates for 550 generations at two levels of spatial structure. The level of spatial structure was determined by whether the physical locations of the phage subsamples were preserved or disrupted at each passage to fresh bacterial host populations. When spatial structure of the phage populations was preserved, there was significantly greater diversity on a global scale with restricted and patchy distribution. When spatial structure was disrupted with passaging to fresh hosts, beneficial mutants were spread across the entire plate. This resulted in reduced diversity, possibly due to clonal interference as the most fit mutants entered into competition on a global scale. Almost all substitutions present at the end of the adaptation in the populations with disrupted spatial structure were also present in the populations with structure preserved.

CONCLUSIONS/SIGNIFICANCE

Our results are consistent with the patchy nature of the spread of adaptive mutants in a spatial landscape. Spatial structure enhances diversity and slows fixation of beneficial mutants. This added diversity could be beneficial in fluctuating environments. We also connect observed substitutions and their effects on fitness to aspects of phage biology, and we provide evidence that some substitutions exclude each other.

Competing islands limit the rate of adaptation in structured populations

Beneficial mutations can co-occur when population structure slows down adaptation. Here, we consider the process of adaptation in asexual populations distributed over several locations ("islands"). New beneficial mutations arise at constant rate ub, and each mutation has the same selective advantage s>0. We assume that populations evolve within islands according to the successional mutations regime of Desai and Fisher (2007), that is, the time to local fixation of a mutation is short compared to the expected waiting time until the next mutation occurs. To study the rate of adaptation, we introduce an approximate model, the successional mutations (SM) model, which can be simulated efficiently and yields accurate results for a wide range of parameters. In the SM model, mutations fix instantly within islands, and migrants can take over the destination island if they are fitter than the residents. For the special case of a population distributed equally across two islands with population size N, we approximate the model further for small and large migration rates in comparison to the mutation rate. These approximations lead to explicit formulas for the rate of adaptation which fit the original model for a large range of parameter values. For the d island case we provide some heuristics on how to extend the explicit formulas and check these with computer simulations. We conclude that the SM model is a good approximation of the adaptation process in a structured population, at least if mutation or migration is limited.

Pre-tumour clones, periodic selection and clonal interference in the origin and progression of gastrointestinal cancer: potential for biomarker development

Classically, the risk of cancer progression in premalignant conditions of the gastrointestinal tract is assessed by examining the degree of histological dysplasia. However, there are many putative pro-cancer genetic changes that have occurred in histologically normal tissue well before the onset of dysplasia. Here we summarize the evidence for such pre-tumour clones and the existing technology that can be used to locate these clones and characterize them at the genetic level. We also discuss the mechanisms by which pre-tumour clones may spread through large areas of normal tissue, and highlight emerging theories on how multiple clones compete and interact within the gastrointestinal mucosa. It is important to gain an understanding of these processes, as it is envisaged that certain pre-tumour changes may be powerful predictive markers, with the potential to identify patients at high risk of developing cancer at a much earlier stage.

Cutting through the complexity of cell collectives

Via strength in numbers, groups of cells can influence their environments in ways that individual cells cannot. Large-scale structural patterns and collective functions underpinning virulence, tumour growth and bacterial biofilm formation are emergent properties of coupled physical and biological processes within cell groups. Owing to the abundance of factors influencing cell group behaviour, deriving general principles about them is a daunting challenge. We argue that combining mechanistic theory with theoretical ecology and evolution provides a key strategy for clarifying how cell groups form, how they change in composition over time, and how they interact with their environments. Here, we review concepts that are critical for dissecting the complexity of cell collectives, including dimensionless parameter groups, individual-based modelling and evolutionary theory. We then use this hybrid modelling approach to provide an example analysis of the evolution of cooperative enzyme secretion in bacterial biofilms.

Genetic hitchhiking in spatially extended populations

When a mutation with selective advantage s spreads through a panmictic population, it may cause two lineages at a linked locus to coalesce; the probability of coalescence is exp(-2rT), where T∼log(2Ns)/s is the time to fixation, N is the number of haploid individuals, and r is the recombination rate. Population structure delays fixation, and so weakens the effect of a selective sweep. However, favourable alleles spread through a spatially continuous population behind a narrow wavefront; ancestral lineages are confined at the tip of this front, and so coalesce rapidly. In extremely dense populations, coalescence is dominated by rare fluctuations ahead of the front. However, we show that for moderate densities, a simple quasi-deterministic approximation applies: the rate of coalescence within the front is λ∼2g(η)/(ρℓ), where ρ is the population density and ℓ=σ2/s is the characteristic scale of the wavefront; g(η) depends only on the strength of random drift, η=ρσs/2. The net effect of a sweep on coalescence also depends crucially on whether two lineages are ever both within the wavefront at the same time: even in the extreme case when coalescence within the front is instantaneous, the net rate of coalescence may be lower than in a single panmictic population. Sweeps can also have a substantial impact on the rate of gene flow. A single lineage will jump to a new location when it is hit by a sweep, with mean square displacement σeff(2)/σ(2)=(8/3)(L/ℓ)(Λ/R); this can be substantial if the species' range, L, is large, even if the species-wide rate of sweeps per map length, Λ/R, is small. This effect is half as strong in two dimensions. In contrast, the rate of coalescence between lineages, at random locations in space and on the genetic map, is proportional to (c/L)(Λ/R), where c is the wavespeed: thus, on average, one-dimensional structure is likely to reduce coalescence due to sweeps, relative to panmixis. In two dimensions, genes must move along the front before they can coalesce; this process is rapid, being dominated by rare fluctuations. This leads to a dramatically higher rate of coalescence within the wavefront than if lineages simply diffused along the front. Nevertheless, the net rate of coalescence due to a sweep through a two-dimensional population is likely to be lower than it would be with panmixis.

Cancer in light of experimental evolution

Cancer initiation, progression, and the emergence of therapeutic resistance are evolutionary phenomena of clonal somatic cell populations. Studies in microbial experimental evolution and the theoretical work inspired by such studies are yielding deep insights into the evolutionary dynamics of clonal populations, yet there has been little explicit consideration of the relevance of this rapidly growing field to cancer biology. Here, we examine how the understanding of mutation, selection, and spatial structure in clonal populations that is emerging from experimental evolution may be applicable to cancer. Along the way, we discuss some significant ways in which cancer differs from the model systems used in experimental evolution. Despite these differences, we argue that enhanced prediction and control of cancer may be possible using ideas developed in the context of experimental evolution, and we point out some prospects for future research at the interface between these traditionally separate areas.

Mutational pathway determines whether drug gradients accelerate evolution of drug-resistant cells

Drug gradients are believed to play an important role in the evolution of bacteria resistant to antibiotics and tumors resistant to anticancer drugs. We use a statistical physics model to study the evolution of a population of malignant cells exposed to drug gradients, where drug resistance emerges via a mutational pathway involving multiple mutations. We show that a nonuniform drug distribution has the potential to accelerate the emergence of resistance when the mutational pathway involves a long sequence of mutants with increasing resistance, but if the pathway is short or crosses a fitness valley, the evolution of resistance may actually be slowed down by drug gradients. These predictions can be verified experimentally, and may help to improve strategies for combating the emergence of resistance.

A framework for estimating the fixation time of an advantageous allele in stepping-stone models

Determining how population subdivision increases the fixation time of an advantageous allele is an important problem in evolutionary genetics as this influences many processes. Here, I lay out a framework for calculating the fixation time of a positively selected allele in a subdivided population, as a function of the number of demes present, the migration rate between them and the manner in which they are connected. Using this framework, it becomes clear that a beneficial allele's fixation time is significantly reduced through migration continuously introducing copies of the allele into a newly colonized subpopulation, increasing its frequency within these demes. The effect that migration has on allele frequency needs to be explicitly taken into account to produce a realistic estimate of fixation time. This behaviour is most prominent when demes are arranged on a two-dimensional torus, in comparison with populations where demes are arranged in a circle. This is because each subpopulation is connected to several neighbours over a torus, so that there are multiple paths that an allele can take in order to fix. As a consequence, some demes experience a greater influx and efflux of migrants than others. Analytical results are found to be very accurate when compared to stochastic simulations, and are generally robust if there are a large number of demes, or if the allele is weakly selected for.

Limits to the rate of adaptive substitution in sexual populations

In large populations, many beneficial mutations may be simultaneously available and may compete with one another, slowing adaptation. By finding the probability of fixation of a favorable allele in a simple model of a haploid sexual population, we find limits to the rate of adaptive substitution, Λ, that depend on simple parameter combinations. When variance in fitness is low and linkage is loose, the baseline rate of substitution is Λ₀ = 2NU , where N is the population size, U is the rate of beneficial mutations per genome, and is their mean selective advantage. Heritable variance v in log fitness due to unlinked loci reduces Λ by e⁻⁴(v) under polygamy and e⁻⁸ (v) under monogamy. With a linear genetic map of length R Morgans, interference is yet stronger. We use a scaling argument to show that the density of adaptive substitutions depends on s, N, U, and R only through the baseline density: Λ/R = F (Λ₀/R). Under the approximation that the interference due to different sweeps adds up, we show that Λ/R ~(Λ₀/R) / (1 +2Λ₉/R) , implying that interference prevents the rate of adaptive substitution from exceeding one per centimorgan per 200 generations. Simulations and numerical calculations confirm the scaling argument and confirm the additive approximation for Λ₀/R ~ 1; for higher Λ₀/R , the rate of adaptation grows above R/2, but only very slowly. We also consider the effect of sweeps on neutral diversity and show that, while even occasional sweeps can greatly reduce neutral diversity, this effect saturates as sweeps become more common-diversity can be maintained even in populations experiencing very strong interference. Our results indicate that for some organisms the rate of adaptive substitution may be primarily recombination-limited, depending only weakly on the mutation supply and the strength of selection.

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Population subdivision and adaptation in asexual populations of Saccharomyces cerevisiae

Population subdivision limits competition between individuals, which can have a profound effect on adaptation. Subdivided populations maintain more genetic diversity at any given time compared to well-mixed populations, and thus "explore" larger parts of the genotype space. At the same time, beneficial mutations take longer to spread in such populations, and thus subdivided populations do not "exploit" discovered mutations as efficiently as well-mixed populations. Whether subdivision inhibits or promotes adaptation in a given environment depends on the relative importance of exploration versus exploitation, which in turn depends on the structure of epistasis among beneficial mutations. Here we investigate the relative importance of exploration versus exploitation for adaptation by evolving 976 independent asexual populations of budding yeast with several degrees of geographic subdivision. We find that subdivision systematically inhibits adaptation: even the luckiest demes in subdivided populations on average fail to discover genotypes that are fitter than those discovered by well-mixed populations. Thus, exploitation of discovered mutations is more important for adaptation in our system than a thorough exploration of the mutational neighborhood, and increasing subdivision slows adaptation.

Spatial structure increases the waiting time for cancer

Cancer results from a sequence of genetic and epigenetic changes which lead to a variety of abnormal phenotypes including increased proliferation and survival of somatic cells, and thus, to a selective advantage of pre-cancerous cells. The notion of cancer progression as an evolutionary process has been experiencing increasing interest in recent years. Many efforts have been made to better understand and predict the progression to cancer using mathematical models; these mostly consider the evolution of a well-mixed cell population, even though pre-cancerous cells often evolve in highly structured epithelial tissues. In this study, we propose a novel model of cancer progression that considers a spatially structured cell population where clones expand via adaptive waves. This model is used to assess two different paradigms of asexual evolution that have been suggested to delineate the process of cancer progression. The standard scenario of periodic selection assumes that driver mutations are accumulated strictly sequentially over time. However, when the mutation supply is sufficiently high, clones may arise simultaneously on distinct genetic backgrounds, and clonal adaptation waves interfere with each other. We find that in the presence of clonal interference, spatial structure increases the waiting time for cancer, leads to a patchwork structure of non-uniformly sized clones, decreases the survival probability of virtually neutral (passenger) mutations, and that genetic distance begins to increase over a characteristic length scale L(c). These characteristic features of clonal interference may help to predict the onset of cancers with pronounced spatial structure and to interpret spatially-sampled genetic data obtained from biopsies. Our estimates suggest that clonal interference likely occurs in the progression of colon cancer, and possibly other cancers where spatial structure matters.