Rare events in population genetics: stochastic tunneling in a two-locus model with recombination

Conflicting effects of recombination on the evolvability and robustness in neutrally evolving populations

Understanding the benefits and costs of recombination under different scenarios of evolutionary adaptation remains an open problem for theoretical and experimental research. In this study, we focus on finite populations evolving on neutral networks comprising viable and unfit genotypes. We provide a comprehensive overview of the effects of recombination by jointly considering different measures of evolvability and mutational robustness over a broad parameter range, such that many evolutionary regimes are covered. We find that several of these measures vary non-monotonically with the rates of mutation and recombination. Moreover, the presence of unfit genotypes that introduce inhomogeneities in the network of viable states qualitatively alters the effects of recombination. We conclude that conflicting trends induced by recombination can be explained by an emerging trade-off between evolvability on the one hand, and mutational robustness on the other. Finally, we discuss how different implementations of the recombination scheme in theoretical models can affect the observed dependence on recombination rate through a coupling between recombination and genetic drift.

Getting higher on rugged landscapes: Inversion mutations open access to fitter adaptive peaks in NK fitness landscapes

Molecular evolution is often conceptualised as adaptive walks on rugged fitness landscapes, driven by mutations and constrained by incremental fitness selection. It is well known that epistasis shapes the ruggedness of the landscape’s surface, outlining their topography (with high-fitness peaks separated by valleys of lower fitness genotypes). However, within the strong selection weak mutation (SSWM) limit, once an adaptive walk reaches a local peak, natural selection restricts passage through downstream paths and hampers any possibility of reaching higher fitness values. Here, in addition to the widely used point mutations, we introduce a minimal model of sequence inversions to simulate adaptive walks. We use the well known NK model to instantiate rugged landscapes. We show that adaptive walks can reach higher fitness values through inversion mutations, which, compared to point mutations, allows the evolutionary process to escape local fitness peaks. To elucidate the effects of this chromosomal rearrangement, we use a graph-theoretical representation of accessible mutants and show how new evolutionary paths are uncovered. The present model suggests a simple mechanistic rationale to analyse escapes from local fitness peaks in molecular evolution driven by (intragenic) structural inversions and reveals some consequences of the limits of point mutations for simulations of molecular evolution.

Ninety years ago, Wright translated Darwin’s core idea of survival of the fittest into rugged landscapes—a highly influential metaphor—with peaks representing high values of fitness separated by valleys of lower fitness. In this picture, once a population has reached a local peak, the adaptive dynamics may stall as further adaptation requires crossing a valley. At the DNA level, adaptation is often modelled as a space of genotypes that is explored through point mutations. Therefore, once a local peak is reached, any genotype fitter than that of the peak will be away from the neighbourhood of genotypes accessible through point mutations. Here we present a simple computational model for inversion mutations, one of the most frequent structural variations, and show that adaptive processes in rugged landscapes can escape from local peaks through intragenic inversion mutations. This new escape mechanism reveals the innovative role of inversions at the DNA level and provides a step towards more realistic models of adaptive dynamics, beyond the dominance of point mutations in theories of molecular evolution.

Lanza V, Rodríguez-Beltrán J, Galán JC, San Millán A, Cantón R, Coque TM. Evolutionary Pathways and Trajectories in Antibiotic Resistance

Evolution is the hallmark of life. Descriptions of the evolution of microorganisms have provided a wealth of information, but knowledge regarding "what happened" has precluded a deeper understanding of "how" evolution has proceeded, as in the case of antimicrobial resistance. The difficulty in answering the "how" question lies in the multihierarchical dimensions of evolutionary processes, nested in complex networks, encompassing all units of selection, from genes to communities and ecosystems. At the simplest ontological level (as resistance genes), evolution proceeds by random (mutation and drift) and directional (natural selection) processes; however, sequential pathways of adaptive variation can occasionally be observed, and under fixed circumstances (particular fitness landscapes), evolution is predictable. At the highest level (such as that of plasmids, clones, species, microbiotas), the systems' degrees of freedom increase dramatically, related to the variable dispersal, fragmentation, relatedness, or coalescence of bacterial populations, depending on heterogeneous and changing niches and selective gradients in complex environments. Evolutionary trajectories of antibiotic resistance find their way in these changing landscapes subjected to random variations, becoming highly entropic and therefore unpredictable. However, experimental, phylogenetic, and ecogenetic analyses reveal preferential frequented paths (highways) where antibiotic resistance flows and propagates, allowing some understanding of evolutionary dynamics, modeling and designing interventions. Studies on antibiotic resistance have an applied aspect in improving individual health, One Health, and Global Health, as well as an academic value for understanding evolution. Most importantly, they have a heuristic significance as a model to reduce the negative influence of anthropogenic effects on the environment.

Epidemics with mutating infectivity on small-world networks

Epidemics and evolution of many pathogens occur on similar timescales so that their dynamics are often entangled. Here, in a first step to study this problem theoretically, we analyze mutating pathogens spreading on simple SIR networks with grid-like connectivity. We have in mind the spatial aspect of epidemics, which often advance on transport links between hosts or groups of hosts such as cities or countries. We focus on the case of mutations that enhance an agent’s infection rate. We uncover that the small-world property, i.e., the presence of long-range connections, makes the network very vulnerable, supporting frequent supercritical mutations and bringing the network from disease extinction to full blown epidemic. For very large numbers of long-range links, however, the effect reverses and we find a reduced chance for large outbreaks. We study two cases, one with discrete number of mutational steps and one with a continuous genetic variable, and we analyze various scaling regimes. For the continuous case we derive a Fokker-Planck-like equation for the probability density and solve it for small numbers of shortcuts using the WKB approximation. Our analysis supports the claims that a potentiating mutation in the transmissibility might occur during an epidemic wave and not necessarily before its initiation.

Recombination and mutational robustness in neutral fitness landscapes

Mutational robustness quantifies the effect of random mutations on fitness. When mutational robustness is high, most mutations do not change fitness or have only a minor effect on it. From the point of view of fitness landscapes, robust genotypes form neutral networks of almost equal fitness. Using deterministic population models it has been shown that selection favors genotypes inside such networks, which results in increased mutational robustness. Here we demonstrate that this effect is massively enhanced by recombination. Our results are based on a detailed analysis of mesa-shaped fitness landscapes, where we derive precise expressions for the dependence of the robustness on the landscape parameters for recombining and non-recombining populations. In addition, we carry out numerical simulations on different types of random holey landscapes as well as on an empirical fitness landscape. We show that the mutational robustness of a genotype generally correlates with its recombination weight, a new measure that quantifies the likelihood for the genotype to arise from recombination. We argue that the favorable effect of recombination on mutational robustness is a highly universal feature that may have played an important role in the emergence and maintenance of mechanisms of genetic exchange.

Key issues review: evolution on rugged adaptive landscapes

Adaptive landscapes represent a mapping between genotype and fitness. Rugged adaptive landscapes contain two or more adaptive peaks: allele combinations with higher fitness than any of their neighbors in the genetic space. How do populations evolve on such rugged landscapes? Evolutionary biologists have struggled with this question since it was first introduced in the 1930s by Sewall Wright. Discoveries in the fields of genetics and biochemistry inspired various mathematical models of adaptive landscapes. The development of landscape models led to numerous theoretical studies analyzing evolution on rugged landscapes under different biological conditions. The large body of theoretical work suggests that adaptive landscapes are major determinants of the progress and outcome of evolutionary processes. Recent technological advances in molecular biology and microbiology allow experimenters to measure adaptive values of large sets of allele combinations and construct empirical adaptive landscapes for the first time. Such empirical landscapes have already been generated in bacteria, yeast, viruses, and fungi, and are contributing to new insights about evolution on adaptive landscapes. In this Key Issues Review we will: (i) introduce the concept of adaptive landscapes; (ii) review the major theoretical studies of evolution on rugged landscapes; (iii) review some of the recently obtained empirical adaptive landscapes; (iv) discuss recent mathematical and statistical analyses motivated by empirical adaptive landscapes, as well as provide the reader with instructions and source code to implement simulations of evolution on adaptive landscapes; and (v) discuss possible future directions for this exciting field.

Effect of Population Size and Mutation Rate on the Evolution of RNA Sequences on an Adaptive Landscape Determined by RNA Folding

The dynamics of populations evolving on an adaptive landscape depends on multiple factors, including the structure of the landscape, the rate of mutations, and effective population size. Existing theoretical work often makes ad hoc and simplifying assumptions about landscape structure, whereas experimental work can vary important parameters only to a limited extent. We here overcome some of these limitations by simulating the adaptive evolution of RNA molecules, whose fitness is determined by the thermodynamics of RNA secondary structure folding. We study the influence of mutation rates and population sizes on final mean population fitness, on the substitution rates of mutations, and on population diversity. We show that evolutionary dynamics cannot be understood as a function of mutation rate µ, population size N, or population mutation rate Nµ alone. For example, at a given mutation rate, clonal interference prevents the fixation of beneficial mutations as population size increases, but larger populations still arrive at a higher mean fitness. In addition, at the highest population mutation rates we study, mean final fitness increases with population size, because small populations are driven to low fitness by the relatively higher incidence of mutations they experience. Our observations show that mutation rate and population size can interact in complex ways to influence the adaptive dynamics of a population on a biophysically motivated fitness landscape.

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.

The Role of Recombination in Evolutionary Rescue

How likely is it that a population escapes extinction through adaptive evolution? The answer to this question is of great relevance in conservation biology, where we aim at species' rescue and the maintenance of biodiversity, and in agriculture and medicine, where we seek to hamper the emergence of pesticide or drug resistance. By reshuffling the genome, recombination has two antagonistic effects on the probability of evolutionary rescue: it generates and it breaks up favorable gene combinations. Which of the two effects prevails depends on the fitness effects of mutations and on the impact of stochasticity on the allele frequencies. In this article, we analyze a mathematical model for rescue after a sudden environmental change when adaptation is contingent on mutations at two loci. The analysis reveals a complex nonlinear dependence of population survival on recombination. We moreover find that, counterintuitively, a fast eradication of the wild type can promote rescue in the presence of recombination. The model also shows that two-step rescue is not unlikely to happen and can even be more likely than single-step rescue (where adaptation relies on a single mutation), depending on the circumstances.

Fitness-valley crossing with generalized parent-offspring transmission

Simple and ubiquitous gene interactions create rugged fitness landscapes composed of coadapted gene complexes separated by "valleys" of low fitness. Crossing such fitness valleys allows a population to escape suboptimal local fitness peaks to become better adapted. This is the premise of Sewall Wright's shifting balance process. Here we generalize the theory of fitness-valley crossing in the two-locus, bi-allelic case by allowing bias in parent-offspring transmission. This generalization extends the existing mathematical framework to genetic systems with segregation distortion and uniparental inheritance. Our results are also flexible enough to provide insight into shifts between alternate stable states in cultural systems with "transmission valleys". Using a semi-deterministic analysis and a stochastic diffusion approximation, we focus on the limiting step in valley crossing: the first appearance of the genotype on the new fitness peak whose lineage will eventually fix. We then apply our results to specific cases of segregation distortion, uniparental inheritance, and cultural transmission. Segregation distortion favouring mutant alleles facilitates crossing most when recombination and mutation are rare, i.e., scenarios where crossing is otherwise unlikely. Interactions with more mutable genes (e.g., uniparental inherited cytoplasmic elements) substantially reduce crossing times. Despite component traits being passed on poorly in the previous cultural background, small advantages in the transmission of a new combination of cultural traits can greatly facilitate a cultural transition. While peak shifts are unlikely under many of the common assumptions of population genetic theory, relaxing some of these assumptions can promote fitness-valley crossing.

The competition between simple and complex evolutionary trajectories in asexual populations

BACKGROUND

On rugged fitness landscapes where sign epistasis is common, adaptation can often involve either individually beneficial "uphill" mutations or more complex mutational trajectories involving fitness valleys or plateaus. The dynamics of the evolutionary process determine the probability that evolution will take any specific path among a variety of competing possible trajectories. Understanding this evolutionary choice is essential if we are to understand the outcomes and predictability of adaptation on rugged landscapes.

RESULTS

We present a simple model to analyze the probability that evolution will eschew immediately uphill paths in favor of crossing fitness valleys or plateaus that lead to higher fitness but less accessible genotypes. We calculate how this probability depends on the population size, mutation rates, and relevant selection pressures, and compare our analytical results to Wright-Fisher simulations.

CONCLUSION

We find that the probability of valley crossing depends nonmonotonically on population size: intermediate size populations are most likely to follow a "greedy" strategy of acquiring immediately beneficial mutations even if they lead to evolutionary dead ends, while larger and smaller populations are more likely to cross fitness valleys to reach distant advantageous genotypes. We explicitly identify the boundaries between these different regimes in terms of the relevant evolutionary parameters. Above a certain threshold population size, we show that the probability that the population finds the more distant peak depends only on a single simple combination of the relevant parameters.

Stochastic tunneling and metastable states during the somatic evolution of cancer

Tumors initiate when a population of proliferating cells accumulates a certain number and type of genetic and/or epigenetic alterations. The population dynamics of such sequential acquisition of (epi)genetic alterations has been the topic of much investigation. The phenomenon of stochastic tunneling, where an intermediate mutant in a sequence does not reach fixation in a population before generating a double mutant, has been studied using a variety of computational and mathematical methods. However, the field still lacks a comprehensive analytical description since theoretical predictions of fixation times are available only for cases in which the second mutant is advantageous. Here, we study stochastic tunneling in a Moran model. Analyzing the deterministic dynamics of large populations we systematically identify the parameter regimes captured by existing approaches. Our analysis also reveals fitness landscapes and mutation rates for which finite populations are found in long-lived metastable states. These are landscapes in which the final mutant is not the most advantageous in the sequence, and resulting metastable states are a consequence of a mutation-selection balance. The escape from these states is driven by intrinsic noise, and their location affects the probability of tunneling. Existing methods no longer apply. In these regimes it is the escape from the metastable states that is the key bottleneck; fixation is no longer limited by the emergence of a successful mutant lineage. We used the so-called Wentzel-Kramers-Brillouin method to compute fixation times in these parameter regimes, successfully validated by stochastic simulations. Our work fills a gap left by previous approaches and provides a more comprehensive description of the acquisition of multiple mutations in populations of somatic cells.

Multidimensional epistasis and the transitory advantage of sex

Identifying and quantifying the benefits of sex and recombination is a long-standing problem in evolutionary theory. In particular, contradictory claims have been made about the existence of a benefit of recombination on high dimensional fitness landscapes in the presence of sign epistasis. Here we present a comparative numerical study of sexual and asexual evolutionary dynamics of haploids on tunably rugged model landscapes under strong selection, paying special attention to the temporal development of the evolutionary advantage of recombination and the link between population diversity and the rate of adaptation. We show that the adaptive advantage of recombination on static rugged landscapes is strictly transitory. At early times, an advantage of recombination arises through the possibility to combine individually occurring beneficial mutations, but this effect is reversed at longer times by the much more efficient trapping of recombining populations at local fitness peaks. These findings are explained by means of well-established results for a setup with only two loci. In accordance with the Red Queen hypothesis the transitory advantage can be prolonged indefinitely in fluctuating environments, and it is maximal when the environment fluctuates on the same time scale on which trapping at local optima typically occurs.

Stochastic Tunneling of Two Mutations in a Population of Cancer Cells

Cancer initiation, progression, and the emergence of drug resistance are driven by specific genetic and/or epigenetic alterations such as point mutations, structural alterations, DNA methylation and histone modification changes. These alterations may confer advantageous, deleterious or neutral effects to mutated cells. Previous studies showed that cells harboring two particular alterations may arise in a fixed-size population even in the absence of an intermediate state in which cells harboring only the first alteration take over the population; this phenomenon is called stochastic tunneling. Here, we investigated a stochastic Moran model in which two alterations emerge in a cell population of fixed size. We developed a novel approach to comprehensively describe the evolutionary dynamics of stochastic tunneling of two mutations. We considered the scenarios of large mutation rates and various fitness values and validated the accuracy of the mathematical predictions with exact stochastic computer simulations. Our theory is applicable to situations in which two alterations are accumulated in a fixed-size population of binary dividing cells.

Stability of localized wave fronts in bistable systems

Localized wave fronts are a fundamental feature of biological systems from cell biology to ecology. Here, we study a broad class of bistable models subject to self-activation, degradation, and spatially inhomogeneous activating agents. We determine the conditions under which wave-front localization is possible and analyze the stability thereof with respect to extrinsic perturbations and internal noise. It is found that stability is enhanced upon regulating a positional signal and, surprisingly, also for a low degree of binding cooperativity. We further show a contrasting impact of self-activation to the stability of these two sources of destabilization.

The effect of bacterial recombination on adaptation on fitness landscapes with limited peak accessibility

There is ample empirical evidence revealing that fitness landscapes are often complex: the fitness effect of a newly arisen mutation can depend strongly on the allelic state at other loci. However, little is known about the effects of recombination on adaptation on such fitness landscapes. Here, we investigate how recombination influences the rate of adaptation on a special type of complex fitness landscapes. On these landscapes, the mutational trajectories from the least to the most fit genotype are interrupted by genotypes with low relative fitness. We study the dynamics of adapting populations on landscapes with different compositions and numbers of low fitness genotypes, with and without recombination. Our results of the deterministic model (assuming an infinite population size) show that recombination generally decelerates adaptation on these landscapes. However, in finite populations, this deceleration is outweighed by the accelerating Fisher-Muller effect under certain conditions. We conclude that recombination has complex effects on adaptation that are highly dependent on the particular fitness landscape, population size and recombination rate.

The emergence and persistence of recombination is a long-standing open question in evolutionary biology. Most previous theoretical studies assumed relatively simple fitness landscapes, i.e., simple relationships between allelic states at different loci and fitness. By contrast, empirically determined bacterial and viral fitness landscapes reveal pervasive complex interactions between alleles at different loci. In this study, we explore the effect of recombination on adaptation on fitness landscapes where some trajectories leading to a global fitness peak are interrupted by genotypes of very low fitness. We find that in infinitely large populations, recombination generally reduces the rate of adaptation. However, in finite populations and under certain conditions, recombination can substantially speed up adaptation. Our study provides insights into the effect of recombination on more realistic fitness landscapes. Moreover, it helps gain a better understanding of the dynamics of the spread of adaptive genes in recombining bacterial populations during niche expansion and colonization of new habitats.