Evolutionary Branching Points Research Articles

The dynamics of evolutionary branching in an ecological model

AbstractEco-evolutionary modelling involves the coupling of ecological equations to evolutionary ones. The interaction between ecological dynamics and evolutionary processes is essential to simulating evolutionary branching, a precursor to speciation. The creation and maintenance of biodiversity in models depends upon their ability to capture the dynamics of evolutionary branching. Understanding these systems requires low-dimension models that are amenable to analysis. The rapid reproduction rates of marine plankton ecosystems and their importance in determining the fluxes of climatically important gases between the ocean and atmosphere suggest that the next generation of global climate models needs to incorporate eco-evolutionary models in the ocean. This requires simple population-level models, that can represent such eco-evolutionary processes with orders of magnitude fewer equations than models that follow the dynamics of individual phenotypes. We present a general framework for developing eco-evolutionary models and consider its general properties. This framework defines a fitness function and assumes a beta distribution of phenotype abundances within each population. It simulates the change in total population size, the mean trait value, and the trait differentiation, from which the variance of trait values in the population may be calculated. We test the efficacy of the eco-evolutionary modelling framework by comparing the dynamics of evolutionary branching in a six-equation eco-evolutionary model that has evolutionary branching, with that of an equivalent one-hundred equation model that simulates the dynamics of every phenotype in the population. The latter model does not involve a population fitness function, nor does it assume a distribution of phenotype abundance across trait values. The eco-evolutionary population model and the phenotype model produce similar evolutionary branching, both qualitatively and quantitatively, in both symmetric and asymmetric fitness landscapes. In order to better understand the six-equation model, we develop a heuristic three-equation eco-evolutionary model. We use the density-independent mortality parameter as a convenient bifurcation parameter, so that differences in evolutionary branching dynamics in symmetric and asymmetric fitness landscapes may be investigated. This model shows that evolutionary branching of a stable population is flagged by a zero in the local trait curvature; the trait curvature then changes sign from negative to positive and back to negative, along the solution. It suggests that evolutionary branching points may be generated differently, with different dynamical properties, depending upon, in this case, the symmetry of the system. It also suggests that a changing environment, that may change attributes such as mortality, could have profound effects on an ecosystem’s ability to adapt. Our results suggest that the properties of the three-dimensional model can provide useful insights into the properties of the higher-dimension models. In particular, the bifurcation properties of the simple model predict the processes by which the more complicated models produce evolutionary branching points. The corresponding bifurcation properties of the phenotype and population models, evident in the dynamics of the phenotype distributions they predict, suggest that our eco-evolutionary modelling framework captures the essential properties that underlie the evolution of phenotypes in populations.

On the importance of evolving phenotype distributions on evolutionary diversification.

Evolutionary branching occurs when a population with a unimodal phenotype distribution diversifies into a multimodally distributed population consisting of two or more strains. Branching results from frequency-dependent selection, which is caused by interactions between individuals. For example, a population performing a social task may diversify into a cooperator strain and a defector strain. Branching can also occur in multi-dimensional phenotype spaces, such as when two tasks are performed simultaneously. In such cases, the strains may diverge in different directions: possible outcomes include division of labor (with each population performing one of the tasks) or the diversification into a strain that performs both tasks and another that performs neither. Here we show that the shape of the population’s phenotypic distribution plays a role in determining the direction of branching. Furthermore, we show that the shape of the distribution is, in turn, contingent on the direction of approach to the evolutionary branching point. This results in a distribution–selection feedback that is not captured in analytical models of evolutionary branching, which assume monomorphic populations. Finally, we show that this feedback can influence long-term evolutionary dynamics and promote the evolution of division of labor.

Emergence of evolutionarily stable communities through eco-evolutionary tunnelling.

Ecological and evolutionary dynamics of communities are inexorably intertwined. The ecological state determines the fate of newly arising mutants, and mutations that increase in frequency can reshape the ecological dynamics. Evolutionary game theory and its extensions within adaptive dynamics have been the mathematical frameworks for understanding this interplay, leading to notions such as evolutionarily stable states (ESS) in which no mutations are favoured, and evolutionary branching points near which the population diversifies. A central assumption behind these theoretical treatments has been that mutations are rare so that the ecological dynamics has time to equilibrate after every mutation. A fundamental question is whether qualitatively new phenomena can arise when mutations are frequent. Here, we describe an adaptive diversification process that robustly leads to complex ESS, despite the fact that such communities are unreachable through a step-by-step evolutionary process. Rather, the system as a whole tunnels between collective states over a short timescale. The tunnelling rate is a sharply increasing function of the rate at which mutations arise in the population. This makes the emergence of ESS communities virtually impossible in small populations, but generic in large ones. Moreover, communities emerging through this process can spatially spread as single replication units that outcompete other communities. Overall, this work provides a qualitatively new mechanism for adaptive diversification and shows that complex structures can generically evolve even when no step-by-step evolutionary path exists.

Effects of superinfection and cost of immunity on host-parasite co-evolution

In this paper, we investigate the cost of immunological up- regulation caused by infection in a between-host transmission dynamical model with superinfection. After introducing a mutant host to an existing model, we explore this problem in (A) monomorphic case and (B) dimorphic case. For (A), we assume that only strain 1 parasite can infect the mutant host. We identify an appropriate fitness for the invasion of the mutant host by analyzing the local stability of the mutant free equilibrium. After specifying a trade-off between the production and recovery rates of infected hosts, we employ the adaptive dynamical approach to analyze the evolutionary and convergence stabilities of the corresponding singular strategy, leading to some conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), a new fitness is introduced to measure the invasion of mutant host under the assumption that both parasite strains can infect the mutant host. By considering two trade-off functions, we can study the conditions for evolutionary stability, isoclinic stability and absolute convergence stability of the singular strategy. Our results show that the host evolution would not favour high degree of immunological up-regulation; moreover, superinfection would help the parasite with weaker virulence persist in hosts.

Evolution of dispersal under variable connectivity

The pattern of connectivity between local populations or between microsites supporting individuals within a population is a poorly understood factor affecting the evolution of dispersal. We modify the well-known Hamilton–May model of dispersal evolution to allow for variable connectivity between microsites. For simplicity, we assume that the microsites are either solitary, i.e., weakly connected through costly dispersal, or part of a well-connected cluster of sites with low-cost dispersal within the cluster. We use adaptive dynamics to investigate the evolution of dispersal, obtaining analytic results for monomorphic evolution and numerical results for the co-evolution of two dispersal strategies. A monomorphic population always evolves to a unique singular dispersal strategy, which may be an evolutionarily stable strategy or an evolutionary branching point. Evolutionary branching happens if the contrast between connectivities is sufficiently high and the solitary microsites are common. The dimorphic evolutionary singularity, when it exists, is always evolutionarily and convergence stable. The model exhibits both protected and unprotected dimorphisms of dispersal strategies, but the dimorphic singularity is always protected. Contrasting connectivities can thus maintain dispersal polymorphisms in temporally stable environments.

Evolutionary branching under multi-dimensional evolutionary constraints

The fitness of an existing phenotype and of a potential mutant should generally depend on the frequencies of other existing phenotypes. Adaptive evolution driven by such frequency-dependent fitness functions can be analyzed effectively using adaptive dynamics theory, assuming rare mutation and asexual reproduction. When possible mutations are restricted to certain directions due to developmental, physiological, or physical constraints, the resulting adaptive evolution may be restricted to subspaces (constraint surfaces) with fewer dimensionalities than the original trait spaces. To analyze such dynamics along constraint surfaces efficiently, we develop a Lagrange multiplier method in the framework of adaptive dynamics theory. On constraint surfaces of arbitrary dimensionalities described with equality constraints, our method efficiently finds local evolutionarily stable strategies, convergence stable points, and evolutionary branching points. We also derive the conditions for the existence of evolutionary branching points on constraint surfaces when the shapes of the surfaces can be chosen freely.

Adaptive evolution of foraging-related trait in intraguild predation system

This paper considers a tri-trophic food chain in which the top predator (intraguild predator) also feeds on the basal resource. We refer to the model as intraguild predation. We analyze its dynamics from an evolutionary perspective. The attack rate or foraging effort of the middle species (intraguild prey) for the basal resources is assumed to be evolvable and is also assumed to be traded off with the vulnerability to being attacked by the top predator. We focus on the analysis of the evolutionary dynamics of the attack rate using the adaptive dynamics approximation of mutation limited evolution. In particular, the critical function analysis is applied. This study reveals that the evolutionary dynamics of the intraguild predation system is completely characterized by the concavity of the trade-off function and admits trichotomous dynamic scenarios: (1) when the trade-off function is more concave than the critical function, an evolutionary singular strategy exists and is a repeller; (2) when the trade-off function is less concave than the critical function, the evolutionary singular strategy is convergence stable and turns into an evolutionary branching point, in which case the monomorphic intraguild prey will split into two different types; (3) when the trade-off function is convex, the evolutionary singular strategy turns into a continuous stable strategy and is uninvadable. Our theoretical analysis suggests that the adaptive foraging behavior may strongly influence the community stability. Consequently, it may promote the diversity of intraguild prey and the persistence of the system on the evolutionary timescale, which highlight a more comprehensive mechanistic understanding of the intricate interplay between ecological and evolutionary force. This modeling approach provides a venue for research on indirect effects from an evolutionary perspective.

Individual-based models for adaptive diversification in high-dimensional phenotype spaces

Most theories of evolutionary diversification are based on equilibrium assumptions: they are either based on optimality arguments involving static fitness landscapes, or they assume that populations first evolve to an equilibrium state before diversification occurs, as exemplified by the concept of evolutionary branching points in adaptive dynamics theory. Recent results indicate that adaptive dynamics may often not converge to equilibrium points and instead generate complicated trajectories if evolution takes place in high-dimensional phenotype spaces. Even though some analytical results on diversification in complex phenotype spaces are available, to study this problem in general we need to reconstruct individual-based models from the adaptive dynamics generating the non-equilibrium dynamics. Here we first provide a method to construct individual-based models such that they faithfully reproduce the given adaptive dynamics attractor without diversification. We then show that a propensity to diversify can be introduced by adding Gaussian competition terms that generate frequency dependence while still preserving the same adaptive dynamics. For sufficiently strong competition, the disruptive selection generated by frequency-dependence overcomes the directional evolution along the selection gradient and leads to diversification in phenotypic directions that are orthogonal to the selection gradient.

Top predators induce the evolutionary diversification of intermediate predator species

We analyze the evolutionary branching phenomenon of intermediate predator species in a tritrophic food chain model by using adaptive dynamics theory. Specifically, we consider the adaptive diversification of an intermediate predator species that feeds on a prey species and is fed upon by a top predator species. We assume that the intermediate predator׳s ability to forage on the prey can adaptively improve, but this comes at the cost of decreased defense ability against the top predator. First, we identify the general properties of trade-off relationships that lead to a continuously stable strategy or to evolutionary branching in the intermediate predator species. We find that if there is an accelerating cost near the singular strategy, then that strategy is continuously stable. In contrast, if there is a mildly decelerating cost near the singular strategy, then that strategy may be an evolutionary branching point. Second, we find that after branching has occurred, depending on the specific shape and strength of the trade-off relationship, the intermediate predator species may reach an evolutionarily stable dimorphism or one of the two resultant predator lineages goes extinct.

Structural Perspectives on the Evolutionary Expansion of Unique Protein-Protein Binding Sites

Structures of protein complexes provide atomistic insights into protein interactions. Human proteins represent a quarter of all structures in the Protein Data Bank; however, available protein complexes cover less than 10% of the human proteome. Although it is theoretically possible to infer interactions in human proteins based on structures of homologous protein complexes, it is still unclear to what extent protein interactions and binding sites are conserved, and whether protein complexes from remotely related species can be used to infer interactions and binding sites. We considered biological units of protein complexes and clustered protein-protein binding sites into similarity groups based on their structure and sequence, which allowed us to identify unique binding sites. We showed that the growth rate of the number of unique binding sites in the Protein Data Bank was much slower than the growth rate of the number of structural complexes. Next, we investigated the evolutionary roots of unique binding sites and identified the major phyletic branches with the largest expansion in the number of novel binding sites. We found that many binding sites could be traced to the universal common ancestor of all cellular organisms, whereas relatively few binding sites emerged at the major evolutionary branching points. We analyzed the physicochemical properties of unique binding sites and found that the most ancient sites were the largest in size, involved many salt bridges, and were the most compact and least planar. In contrast, binding sites that appeared more recently in the evolution of eukaryotes were characterized by a larger fraction of polar and aromatic residues, and were less compact and more planar, possibly due to their more transient nature and roles in signaling processes.

Evolution of dispersal in spatially and temporally variable environments: The importance of life cycles.

Spatiotemporal variability of the environment is bound to affect the evolution of dispersal, and yet model predictions strongly differ on this particular effect. Recent studies on the evolution of local adaptation have shown that the life cycle chosen to model the selective effects of spatiotemporal variability of the environment is a critical factor determining evolutionary outcomes. Here, we investigate the effect of the order of events in the life cycle on the evolution of unconditional dispersal in a spatially heterogeneous, temporally varying landscape. Our results show that the occurrence of intermediate singular strategies and disruptive selection are conditioned by the temporal autocorrelation of the environment and by the life cycle. Life cycles with dispersal of adults versus dispersal of juveniles, local versus global density regulation, give radically different evolutionary outcomes that include selection for total philopatry, evolutionary bistability, selection for intermediate stable states, and evolutionary branching points. Our results highlight the importance of accounting for life-cycle specifics when predicting the effects of the environment on evolutionarily selected trait values, such as dispersal, as well as the need to check the robustness of model conclusions against modifications of the life cycle.

Evolution, the loss of diversity and the role of trade-offs

We investigate how the loss of previously evolved diversity in host resistance to disease is dependent on the complexity of the underlying evolutionary trade-off. Working within the adaptive dynamics framework, using graphical tools (pairwise invasion plots, PIPs; trait evolution plots, TEPs) and algebraic analysis we consider polynomial trade-offs of increasing degree. Our focus is on the evolutionary trajectory of the dimorphic population after it has been attracted to an evolutionary branching point. We show that for sufficiently complex trade-offs (here, polynomials of degree three or higher) the resulting invasion boundaries can form closed ‘oval’ areas of invadability and strategy coexistence. If no attracting singular strategies exist within this region, then the population is destined to evolve outside of the region of coexistence, resulting in the loss of one strain. In particular, the loss of diversity in this model always occurs in such a way that the remaining strain is not attracted back to the branching point but to an extreme of the trade-off, meaning the diversity is lost forever. We also show similar results for a non-polynomial but complex trade-off, and for a different eco-evolutionary model. Our work further highlights the importance of trade-offs to evolutionary behaviour.

African great apes are naturally infected with roseoloviruses closely related to human herpesvirus 7.

Primates are naturally infected with herpesviruses. During the last 15 years, the search for homologues of human herpesviruses in nonhuman primates allowed the identification of numerous viruses belonging to the different herpesvirus subfamilies and genera. No simian homologue of human herpesvirus 7 (HHV7) has been reported to date. To investigate the putative existence of HHV7-like viruses in African great apes, we applied the consensus-degenerate hybrid oligonucleotide primers (CODEHOP) program-mediated PCR strategy to blood DNA samples from the four common chimpanzee subspecies (Pan troglodytes verus, P. t. ellioti, P. t. troglodytes, and P. t. schweinfurthii), pygmy chimpanzees (Pan paniscus), as well as lowland gorillas (Gorilla gorilla gorilla). This study led to the discovery of a novel roseolovirus close to HHV7 in each of these nonhuman primate species and subspecies. Generation of the partial glycoprotein B (1,111-bp) and full-length DNA polymerase (3,036/3,042-bp) gene sequences allowed the deciphering of their evolutionary relationships. Phylogenetic analyses revealed that HHV7 and its African great ape homologues formed well-supported monophyletic lineages whose topological resemblance to the host phylogeny is suggestive of virus-host codivergence. Notably, the evolutionary branching points that separate HHV7 from African great ape herpesvirus 7 are remarkably congruent with the dates of divergence of their hosts. Our study shows that African great apes are hosts of human herpesvirus homologues, including HHV7 homologues, and that the latter, like other DNA viruses that establish persistent infections, have cospeciated with their hosts. Human herpesviruses are known to possess simian homologues. However, surprisingly, none has been identified to date for human herpesvirus 7 (HHV7). This study is the first to describe simian homologues of HHV7. The extensive search performed on almost all African great ape species and subspecies, i.e., common chimpanzees of the four subspecies, bonobos, and lowland gorillas, has allowed characterization of a specific virus in each. Genetic characterization of the partial glycoprotein B and full-length DNA polymerase gene sequences, followed by their phylogenetic analysis and estimation of divergence times, has shed light on the evolutionary relationships of these viruses. In this respect, we conclusively demonstrate the cospeciation between these new viruses and their hosts and report cases of cross-species transmission between two common chimpanzee subspecies in both directions.

Construction of multiple trade-offs to obtain arbitrary singularities of adaptive dynamics.

Evolutionary singularities are central to the adaptive dynamics of evolving traits. The evolutionary singularities are strongly affected by the shape of any trade-off functions a model assumes, yet the trade-off functions are often chosen in an ad hoc manner, which may unjustifiably constrain the evolutionary dynamics exhibited by the model. To avoid this problem, critical function analysis has been used to find a trade-off function that yields a certain evolutionary singularity such as an evolutionary branching point. Here I extend this method to multiple trade-offs parameterized with a scalar strategy. I show that the trade-off functions can be chosen such that an arbitrary point in the viability domain of the trait space is a singularity of an arbitrary type, provided (next to certain non-degeneracy conditions) that the model has at least two environmental feedback variables and at least as many trade-offs as feedback variables. The proof is constructive, i.e., it provides an algorithm to find trade-off functions that yield the desired singularity. I illustrate the construction of trade-offs with an example where the virulence of a pathogen evolves in a small ecosystem of a host, its pathogen, a predator that attacks the host and an alternative prey of the predator.

Adaptive evolution of defense ability leads to diversification of prey species.

In this paper, by using the adaptive dynamics approach, we investigate how the adaptive evolution of defense ability promotes the diversity of prey species in an initial one-prey-two-predator community. We assume that the prey species can evolve to a safer strategy such that it can reduce the predation risk, but a prey with a high defense ability for one predator may have a low defense ability for the other and vice versa. First, by using the method of critical function analysis, we find that if the trade-off is convex in the vicinity of the evolutionarily singular strategy, then this singular strategy is a continuously stable strategy. However, if the trade-off is weakly concave near the singular strategy and the competition between the two predators is relatively weak, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the prey strategy, if the trade-off curve is globally concave, then the prey species might eventually evolve into two specialists, each caught by only one predator species. However, if the trade-off curve is convex-concave-convex, the prey species might eventually branch into two partial specialists, each being caught by both of the two predators and they can stably coexist on the much longer evolutionary timescale.

Adaptive Evolution of Virulence-Related Traits in a Susceptible-Infected Model with Treatment

Evolution problem is now a hot topic in the mathematical biology field. This paper investigates the adaptive evolution of pathogen virulence in a susceptible-infected (SI) model under drug treatment. We explore the evolution of a continuous trait, virulence of a pathogen, and consider virulence-dependent cure rate (recovery rate) that dramatically affects the outcome of evolution. With the methods of critical function analysis and adaptive dynamics, we identify the evolutionary conditions for continuously stable strategies, evolutionary repellers, and evolutionary branching points. First, the results show that a high-intensity strength drug treatment can result in evolutionary branching and the evolution of pathogen strains will tend towards a higher virulence with the increase of the strength of the treatment. Second, we use the critical function analysis to investigate the evolution of virulence-related traits and show that evolutionary outcomes strongly depend on the shape of the trade-off between virulence and transmission. Third, after evolutionary branching, the two infective species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist, and no further branching is possible, which is independent of the cure rate function.

Adaptation in a stochastic multi-resources chemostat model

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka–Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we prove a rigorous characterization of the evolutionary branching points.

Adaptive evolution of attack ability promotes the evolutionary branching of predator species

In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate the evolutionary branching phenomenon of predator species. We assume that both the prey and predators are density-dependent and the predator’s attack ability can adaptively evolve, but this has a cost in terms of its death rate. First, we identify the general properties of trade-off relationships that allow for a continuously stable strategy and evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly concave near the singular strategy, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the predator strategy, if the trade-off curve is convex–concave–convex, the predator species will eventually evolve into two different types, which can stably coexist on the much longer evolutionary timescale and no further branching is possible.

Adaptive evolution of anti-predator ability promotes the diversity of prey species: Critical function analysis

In this paper, with the method of adaptive dynamics and critical function analysis, we investigate the evolutionary diversification of prey species. We assume that prey species can evolve safer strategies such that it can reduce the predation risk, but this has a cost in terms of its reproduction. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for continuously stable strategy and evolutionary branching in the prey strategy. It is found that if the trade-off curve is globally concave, then the evolutionarily singular strategy is continuously stable. However, if the trade-off curve is concave–convex–concave and the prey's sensitivity to crowding is not strong, then the evolutionarily singular strategy may be an evolutionary branching point, near which the resident and mutant prey can coexist and diverge in their strategies. Second, we find that after branching has occurred in the prey strategy, if the trade-off curve is concave–convex–concave, the prey population will eventually evolve into two different types, which can coexist on the long-term evolutionary timescale. The algebraical analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible for the concave–convex–concave trade-off relationship.

The Ras protein superfamily: Evolutionary tree and role of conserved amino acids

The Ras superfamily is a fascinating example of functional diversification in the context of a preserved structural framework and a prototypic GTP binding site. Thanks to the availability of complete genome sequences of species representing important evolutionary branch points, we have analyzed the composition and organization of this superfamily at a greater level than was previously possible. Phylogenetic analysis of gene families at the organism and sequence level revealed complex relationships between the evolution of this protein superfamily sequence and the acquisition of distinct cellular functions. Together with advances in computational methods and structural studies, the sequence information has helped to identify features important for the recognition of molecular partners and the functional specialization of different members of the Ras superfamily.