How does feedback from phage infections influence the evolution of phase variation in Campylobacter?

Campylobacter jejuni (C. jejuni) causes gastroenteritis following the consumption of contaminated poultry meat, resulting in a large health and economic burden worldwide. Phage therapy is a promising technique for eradicating C. jejuni from poultry flocks and chicken carcasses. However, C. jejuni can resist infections by some phages through stochastic, phase-variable ON/OFF switching of the phage receptors mediated by simple sequence repeats (SSR). While selection strength and exposure time influence the evolution of SSR-mediated phase variation (PV), phages offer a more complex evolutionary environment as phage replication depends on having a permissive host organism. Here, we build and explore several continuous culture bacteria-phage computational models, each analysing different phase-variable scenarios calibrated to the experimental SSR rates of C. jejuni loci and replication parameters for the F336 phage. We simulate the evolution of PV rates via the adaptive dynamics framework for varying levels of selective pressures that act on the phage-resistant state. Our results indicate that growth reducing counter-selection on a single PV locus results in the stable maintenance of the phage, while compensatory selection between bacterial states affects the evolutionary stable mutation rates (i.e. very high and very low mutation rates are evolutionarily disadvantageous), whereas, in the absence of either selective pressure the evolution of PV rates results in mutation rates below the basal values. Contrastingly, a biologically-relevant model with two phase-variable loci resulted in phage extinction and locking of the bacteria into a phage-resistant state suggesting that another counter-selective pressure is required, instance, the use of a distinct phage whose receptor is an F336-phage-resistant state. We conclude that a delicate balance between counter-selection and phage-attack can result in both the evolution of phase-variable phage receptors and persistence of PV-receptor-specific phage.

Searching for fat tails in CRISPR-Cas systems: Data analysis and mathematical modeling.

Understanding CRISPR-Cas systems-the adaptive defence mechanism that about half of bacterial species and most of archaea use to neutralise viral attacks-is important for explaining the biodiversity observed in the microbial world as well as for editing animal and plant genomes effectively. The CRISPR-Cas system learns from previous viral infections and integrates small pieces from phage genomes called spacers into the microbial genome. The resulting library of spacers collected in CRISPR arrays is then compared with the DNA of potential invaders. One of the most intriguing and least well understood questions about CRISPR-Cas systems is the distribution of spacers across the microbial population. Here, using empirical data, we show that the global distribution of spacer numbers in CRISPR arrays across multiple biomes worldwide typically exhibits scale-invariant power law behaviour, and the standard deviation is greater than the sample mean. We develop a mathematical model of spacer loss and acquisition dynamics which fits observed data from almost four thousand metagenomes well. In analogy to the classical 'rich-get-richer' mechanism of power law emergence, the rate of spacer acquisition is proportional to the CRISPR array size, which allows a small proportion of CRISPRs within the population to possess a significant number of spacers. Our study provides an alternative explanation for the rarity of all-resistant super microbes in nature and why proliferation of phages can be highly successful despite the effectiveness of CRISPR-Cas systems.

Revisiting the Role of Hyperparasitism in the Evolution of Virulence.

AbstractHyperparasitism denotes the natural phenomenon where a parasite infecting a host is in turn infected by its own parasite. Hyperparasites can shape the dynamics of host-parasite interactions and often have a deleterious impact on pathogens, an important class of parasites, causing a reduction in their virulence and transmission rate. Hyperparasitism thus could be an important tool of biological control. However, host-parasite-hyperparasite systems have so far been outside the mainstream of modeling studies, especially those dealing with eco-evolutionary aspects of species interactions. Here, we theoretically explore the evolution of life-history traits in a generic host-parasite-hyperparasite system, focusing on parasite virulence and the positive impact that hyperparasitism has on the host population. We also explore the coevolution of life-history traits of the parasite and hyperparasite, using adaptive dynamics and quantitative genetics frameworks to identify evolutionarily singular strategies. We find that in the presence of hyperparasites, the evolutionarily optimal pathogen virulence generally shifts toward more virulent strains. However, even in this case the use of hyperparasites in biocontrol could be justified, since overall host mortality decreases. An intriguing possible outcome of the evolution of the hyperparasite can be its evolutionary suicide.

Coevolution of female fidelity and male help in populations with alternative reproductive tactics.

In socially monogamous species, pair-bonded males often continue to provide care to all offspring in their nests despite some degree of paternity loss due to female extra-pair copulation. Previous theoretical models suggested that females can use their within-pair offspring as 'hostages' to blackmail their social mates, so that they continue to provide care to the brood at low levels of cuckoldry. These models, however, rely on the assumption of sufficiently accurate male detection of cuckoldry and the reduction of parental effort in case of suspicion. Therefore, they cannot explain the abundant cases where cuckolded males continue to provide extensive care to the brood. Here we use an analytical population genetics model and an individual-based simulation model to explore the coevolution of female fidelity and male help in populations with two genetically determined alternative reproductive tactics (ARTs): sneakers that achieve paternity solely via extra-pair copulations and bourgeois that form a mating pair and spend some efforts in brood care. We show that when the efficiency of mate guarding is intermediate, the bourgeois males can evolve to 'specialize' in providing care by spending more than 90% of time in helping their females while guarding them as much as possible, despite frequent cuckoldry by the sneakers. We also show that when sneakers have tactic-specific adaptations and thus are more competitive than the bourgeois in gaining extra-pair fertilizations, the frequency of sneakers and the degrees of female fidelity and male help can fluctuate in evolutionary cycles. Our theoretical predictions highlight the need for further empirical tests in species with ARTs.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator.

The use of predator-prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka-Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator-prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator-prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Exploring the role of spatial and stoichiometric heterogeneity in the top-down control in eutrophic planktonic ecosystems.

Understanding the impact of eutrophication on the dynamics of aquatic food webs, remains a long-term challenge in ecology. Mathematical models generally predict the destabilisation of such webs, under increasing eutrophication levels, with large oscillations of species densities that can result in their extinction. This is at odds with a number of ecological observations that show stable dynamics even for high nutrient loads. The apparent discrepancy between theory and observations is known as the Rosenzweig's 'paradox of enrichment' and various solutions to the problem have been proposed over the years. In this study, we explore the stabilisation of dynamics of a tri-trophic plankton model in a eutrophic environment which occurs as a result of interplay of space heterogeneity, ecological stoichiometry, and food taxis of predators. We build a variety of models of increasing complexity, to explore various scenarios of phytoplankton growth, zooplankton food-dependent vertical movement, and different stoichiometric limitations of zooplankton. We show that the synergy among the vertical gradient in phytoplankton growth, phytoplankton structuring in terms of their stoichiometric ratio, and food-dependent vertical movement of zooplankton, would result in a postponing of destabilisation of eutrophic systems as compared to a well-mixed system. Our approach reveals a high complexity of the bifurcation structure of the system when key model parameters, such as the degree of eutrophication and light shading, are varied. We find coexistence of limit cycles and stable equilibria and that the possibility of multiple attractors in the system can result in hysteresis phenomena when the nutrient load is manipulated. These results are relevant and should be taken into account in lake restoration programs.

Modelling evolution of virulence in populations with a distributed parasite load.

Modelling evolution of virulence in host-parasite systems is an actively developing area of research with ever-growing literature. However, most of the existing studies overlook the fact that individuals within an infected population may have a variable infection load, i.e. infected populations are naturally structured with respect to the parasite burden. Empirical data suggests that the mortality and infectiousness of individuals can strongly depend on their infection load; moreover, the shape of distribution of infection load may vary on ecological and evolutionary time scales. Here we show that distributed infection load may have important consequences for the eventual evolution of virulence as compared to a similar model without structuring. Mathematically, we consider an SI model, where the dynamics of the infected subpopulation is described by a von Förster-type equation, in which the infection load plays the role of age. We implement the adaptive dynamics framework to predict evolutionary outcomes in this model. We demonstrate that for simple trade-off functions between virulence, disease transmission and parasite growth rates, multiple evolutionary attractors are possible. Interestingly, unlike in the case of unstructured models, achieving an evolutionary stable strategy becomes possible even for a variation of a single ecological parameter (the parasite growth rate) and keeping the other parameters constant. We conclude that evolution in disease-structured populations is strongly mediated by alterations in the overall shape of the parasite load distribution.

Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods.

Here, we propose a computational approach to explore evolutionary fitness in complex biological systems based on empirical data using artificial neural networks. The essence of our approach is the following. We first introduce a ranking order of inherited elements (behavioral strategies or/and life history traits) in considered self-reproducing systems: we use available empirical information on selective advantages of such elements. Next, we introduce evolutionary fitness, which is formally described as a certain function reflecting the introduced ranking order. Then, we approximate fitness in the space of key parameters using a Taylor expansion. To estimate the coefficients in the Taylor expansion, we utilize artificial neural networks: we construct a surface to separate the domains of superior and interior ranking of pair inherited elements in the space of parameters. Finally, we use the obtained approximation of the fitness surface to find the evolutionarily stable (optimal) strategy which maximizes fitness. As an ecologically important study case, we apply our approach to explore the evolutionarily stable diel vertical migration of zooplankton in marine and freshwater ecosystems. Using machine learning we reconstruct the fitness function of herbivorous zooplankton from empirical data and predict the daily trajectory of a dominant species in the northeastern Black Sea.

Towards the Construction of a Mathematically Rigorous Framework for the Modelling of Evolutionary Fitness.

Modelling of natural selection in self-replicating systems has been heavily influenced by the concept of fitness which was inspired by Darwin's original idea of the survival of the fittest. However, so far the concept of fitness in evolutionary modelling is still somewhat vague, intuitive and often subjective. Unfortunately, as a result of this, using different definitions of fitness can lead to conflicting evolutionary outcomes. Here we formalise the definition of evolutionary fitness to describe the selection of strategies in deterministic self-replicating systems for generic modelling settings which involve an arbitrary function space of inherited strategies. Our mathematically rigorous definition of fitness is closely related to the underlying population dynamic equations which govern the selection processes. More precisely, fitness is defined based on the concept of the ranking of competing strategies which compares the long-term dynamics of measures of sets of inherited units in the space of strategies. We also formulate the variational principle of modelling selection which states that in a self-replicating system with inheritance, selection will eventually maximise evolutionary fitness. We demonstrate how expressions for evolutionary fitness can be derived for a class of models with age structuring including systems with delay, which has previously been considered as a challenge.

Revealing Evolutionarily Optimal Strategies in Self-Reproducing Systems via a New Computational Approach.

Modelling the evolution of complex life history traits and behavioural patterns observed in the natural world is a challenging task. Here, we develop a novel computational method to obtain evolutionarily optimal life history traits/behavioural patterns in population models with a strong inheritance. The new method is based on the reconstruction of evolutionary fitness using underlying equations for population dynamics and it can be applied to self-reproducing systems (including complicated age-structured models), where fitness does not depend on initial conditions, however, it can be extended to some frequency-dependent cases. The technique provides us with a tool to efficiently explore both scalar-valued and function-valued traits with any required accuracy. Moreover, the method can be implemented even in the case where we ignore the underlying model equations and only have population dynamics time series. As a meaningful ecological case study, we explore optimal strategies of diel vertical migration (DVM) of herbivorous zooplankton in the vertical water column which is a widespread phenomenon in both oceans and lakes, generally considered to be the largest synchronised movement of biomass on Earth. We reveal optimal trajectories of daily vertical motion of zooplankton grazers in the water column depending on the presence of food and predators. Unlike previous studies, we explore both scenarios of DVM with static and dynamic predators. We find that the optimal pattern of DVM drastically changes in the presence of dynamic predation. Namely, with an increase in the amount of food available for zooplankton grazers, the amplitude of DVM progressively increases, whereas for static predators DVM would abruptly cease.

Towards developing a general framework for modelling vertical migration in zooplankton.

Diel vertical migration (DVM) of zooplankton is a widespread phenomenon in both oceans and lakes, and is generally considered to be the largest synchronized movement of biomass on Earth. Most existing mathematical models of DVM are based on the assumption that animals maximize a certain criterion such as the expected reproductive value, the venturous revenue, the ratio of energy gain/mortality or some predator avoidance function when choosing their instantaneous depth. The major shortcoming of this general point of view is that the predicted DVM may be strongly affected by a subjective choice of a particular optimization criterion. Here we argue that the optimal strategy of DVM can be unambiguously obtained as an outcome of selection in the underlying equations of genotype/traits frequency dynamics. Using this general paradigm, we explore the optimal strategy for the migration across different depths by zooplankton grazers throughout the day. To illustrate our ideas we consider four generic DVM models, each making different assumptions on the population dynamics of zooplankton, and demonstrate that in each model we need to maximize a particular functional to find the optimal strategy. Surprisingly, patterns of DVM obtained for different models greatly differ in terms of their parameters dependence. We then show that the infinite dimensional trait space of different zooplankton trajectories can be projected onto a low dimensional space of generalized parameters and the genotype evolution dynamics can be easily followed using this low-dimensional space. Using this space of generalized parameters we explore the influence of mutagenesis on evolution of DVM, and we show that strong mutagenesis allows the coexistence of an infinitely large number of strategies whereas for weak mutagenesis the selection results in the extinction of most strategies, with the surviving strategies all staying close to the optimal strategy in the corresponding mutagenesis-free system.

Imperfect prey selectivity of predators promotes biodiversity and irregularity in food webs.

Ecological communities are often characterised by many species occupying the same trophic level and competing over a small number of vital resources. The mechanisms maintaining high biodiversity in such systems are still poorly understood. Here, we revisit the role of prey selectivity by generalist predators in promoting biodiversity. We consider a generic tri-trophic food web, consisting of a single limiting resource, a large number of primary producers and a generalist predator. We suggest a framework to describe the predator functional response, combining food selectivity for distinctly different functional prey groups with proportion-based consumption of similar prey species. Our simulations reveal that intermediate levels of prey selectivity can explain a high species richness, functional biodiversity, and variability among prey species. In contrast, perfect food selectivity or purely proportion-based food consumption leads to a collapse of prey functional biodiversity. Our results are in agreement with empirical phytoplankton rank-abundance curves in lakes.

Revisiting the role of behavior-mediated structuring in the survival of populations in hostile environments.

Increasing the population density of target species is a major goal of ecosystem and agricultural management. This task is especially challenging in hazardous environments with a high abundance of natural enemies such as parasites and predators. Safe locations with lower mortality have been long considered a beneficial factor in enhancing population survival, being a promising tool in commercial fish farming and restoration of threatened species. Here we challenge this opinion and revisit the role of behavior structuring in a hostile environment in shaping the population density. We build a mathematical model, where individuals are structured according to their defensive tactics against natural enemies. The model predicts that although each safe zone enhances the survival of an individual, for an insufficient number of such zones, the entire population experiences a greater overall mortality. This is a result of the interplay of emergent dynamical behavioral structuring and strong intraspecific competition for safe zones. Non-plastic structuring in individuals' boldness reduces the mentioned negative effects. We demonstrate emergence of non-plastic behavioral structuring: the evolutionary branching of a monomorphic population into a dimorphic one with bold/shy strains. We apply our modelling approach to explore fish farming of salmonids in an environment infected by trematode parasites.

Top-down control in a patchy environment: revisiting the stabilizing role of food-dependent predator dispersal.

In this paper, we revisit the stabilizing role that predator dispersal and aggregation have in the top-down regulation of predator-prey systems in a heterogeneous environment. We consider an environment consisting of sites interconnected by dispersal, and propose a novel mechanism of stabilization for the case with a non-sigmoid functional response of predators. We assume that the carrying capacity of the prey is infinitely large in each site, and show that successful top-down regulation of this otherwise globally unstable system is made possible through an interplay between the unevenness of prey fitness across the sites and the rapid food-dependent migration of predators. We argue that this mechanism of stabilization is different from those previously reported in the literature: in particular, it requires a high degree of synchronicity in local oscillations of species densities across the sites. Prey outbreaks take place synchronously, but the unevenness of prey growth rates across the sites results in a pronounced difference in the species densities, and so the predator quickly disperses to the sites with the highest prey abundances. For this reason, the consumption of prey mostly takes place in the sites with high densities of prey, which assures an efficient suppression of outbreaks. Furthermore, when the total size of prey population is low, the distribution of both species among the sites becomes more even, and this prevents overconsumption of the prey by the predator. Finally, we put forward the hypothesis that this mechanism, when considered in a tri-trophic plankton community in the water column, can explain the stability of the nutrient-rich low-chlorophyll open ocean regions.

Predation on infected host promotes evolutionary branching of virulence and pathogens' biodiversity.

Traditionally, theoretical works on the evolution of virulence of wildlife infections have focused on interactions between just the host and its parasite. In a large number of study cases, however, infected host individuals also incur severe mortality due to predation of higher trophic levels. Such mortality should be virulence-dependent since the population size of predators is determined by the available amount prey they consume, which, in turn, is a function of pathogen virulence. The potential role of trophic pressure by predators in the evolution of virulence of their prey remains largely unaddressed in the literature. Here we investigate the possible role of predators in promoting biodiversity and disruptive evolution (evolutionary branching) of pathogen strains infecting the prey that those predators consume. Our theoretical study is based on principles of adaptive dynamics and evolutionary game theory. With the help of a fairly simple model we demonstrate that predation on infected prey can result in evolutionary branching of pathogen virulence, which would be impossible in the same system without predators. We show that predator-mediated evolutionary branching can occur within a large range of species life-history traits and for various types of transmission-virulence trade-off relation. We argue that predation can play an important role in explaining the existing polymorphism and biodiversity of pathogen strains in wildlife.

Assassin snails (Anentome helena) as a biological model for exploring the effects of individual specialisation within generalist predators.

Quantifying feeding behaviour of generalist predators at the population and individual levels is crucial for understanding the structure and functioning of food webs. Individual predator/consumer feeding niches can be significantly narrower than that of the population across animal taxa. In such species, the population of a generalist predator becomes essentially an ensemble of specialist individuals and this often highly affects the dynamics of the prey-predator interactions. Currently, few experimental systems exist that are both easily technically manipulated in a lab and are reliable to accurately assess effects of individual specialisation within generalist predators. Here we argue that a freshwater predaceous snail, Anentome helena (also known as an 'assassin snail'), is a convenient and reliable experimental system to study feeding of a generalist predator on multiple food types which exhibits well-pronounced specialisation of foraging individuals. Using A. helena we experimentally test: (i) how relative prey abundances in the environment affect the feeding patterns, (ii) whether the feeding patterns are consistent over the duration of the experimental period, and (iii) compare the feeding niche breadth of individuals to that of the laboratory population. By offering four different prey snail species, at a range of relative abundances, we show that there are consistent patterns in feeding. Importantly, the consumption of each prey was independent of the relative abundance at which they were present. Individual predators showed selectivity to a particular prey, i.e. the population of assassin snails seems to be formed of individuals that specialise on different prey. Our findings would contribute to the recent revision and the ongoing debate on the classification of predator species into generalists and specialists.

Revisiting implementation of multiple natural enemies in pest management.

A major goal of biological control is the reduction and/or eradication of pests using various natural enemies, in particular, via deliberate infection of the target species by parasites. To enhance the biological control, a promising strategy seems to implement a multi-enemy assemblage rather than a single control agent. Although a large body of theoretical studies exists on co-infections in epidemiology and ecology, there is still a big gap in modelling outcomes of multi-enemy biological control. Here we theoretically investigate how the efficiency of biological control of a pest depends on the number of natural enemies used. We implement a combination of eco-epidemiological modelling and the Adaptive Dynamics game theory framework. We found that a progressive addition of parasite species increases the evolutionarily stable virulence of each parasite, and thus enhances the mortality of the target pest. However, using multiple enemies may have only a marginal effect on the success of biological control, or can even be counter-productive when the number of enemies is excessive. We found the possibility of evolutionary suicide, where one or several parasite species go extinct over the course of evolution. Finally, we demonstrate an interesting scenario of coexistence of multiple parasites at the edge of extinction.

Structural sensitivity of biological models revisited.

Enhancing the predictive power of models in biology is a challenging issue. Among the major difficulties impeding model development and implementation are the sensitivity of outcomes to variations in model parameters, the problem of choosing of particular expressions for the parametrization of functional relations, and difficulties in validating models using laboratory data and/or field observations. In this paper, we revisit the phenomenon which is referred to as structural sensitivity of a model. Structural sensitivity arises as a result of the interplay between sensitivity of model outcomes to variations in parameters and sensitivity to the choice of model functions, and this can be somewhat of a bottleneck in improving the models predictive power. We provide a rigorous definition of structural sensitivity and we show how we can quantify the degree of sensitivity of a model based on the Hausdorff distance concept. We propose a simple semi-analytical test of structural sensitivity in an ODE modeling framework. Furthermore, we emphasize the importance of directly linking the variability of field/experimental data and model predictions, and we demonstrate a way of assessing the robustness of modeling predictions with respect to data sampling variability. As an insightful illustrative example, we test our sensitivity analysis methods on a chemostat predator-prey model, where we use laboratory data on the feeding of protozoa to parameterize the predator functional response.