Speciation in multidimensional evolutionary space

Nonmonotone invasion landscape by noise-aware control of metastasis activator levels

A major pharmacological assumption is that lowering disease-promoting protein levels is generally beneficial. For example, inhibiting metastasis activator BACH1 is proposed to decrease cancer metastases. Testing such assumptions requires approaches to measure disease phenotypes while precisely adjusting disease-promoting protein levels. Here we developed a two-step strategy to integrate protein-level tuning, noise-aware synthetic gene circuits into a well-defined human genomic safe harbor locus. Unexpectedly, engineered MDA-MB-231 metastatic human breast cancer cells become more, then less and then more invasive as we tune BACH1 levels up, irrespective of the native BACH1. BACH1 expression shifts in invading cells, and expression of BACH1's transcriptional targets confirm BACH1's nonmonotone phenotypic and regulatory effects. Thus, chemical inhibition of BACH1 could have unwanted effects on invasion. Additionally, BACH1's expression variability aids invasion at high BACH1 expression. Overall, precisely engineered, noise-aware protein-level control is necessary and important to unravel disease effects of genes to improve clinical drug efficacy.

Eco-Geography and Phenology Are the Major Drivers of Reproductive Isolation in the Royal Irises, a Species Complex in the Course of Speciation

The continuous nature of speciation implies that different species are found at different stages of divergence, from no- to complete reproductive isolation. This process and its underlying mechanisms are best viewed in incipient species. Moreover, the species complex can offer unique insight into how reproductive isolation (RI) has evolved. The royal irises (Iris section Oncocyclus) are a young group of species in the course of speciation, providing an ideal system for speciation study. We quantified pre- and post-zygotic reproductive barriers between the eight Israeli species of this complex and estimated the total RI among them. We tested for both pre-pollination and post-pollination reproductive barriers. Pre-pollination barriers, i.e., eco-geographic divergence and phenological differentiation were the major contributors to RI among the Iris species. On the other hand, post-pollination barriers, namely pollen-stigma interactions, fruit set, and seed viability had negligible contributions to total RI. The strength of RI was not uniform across the species complex, suggesting that species may have diverged at different rates. Overall, this study in a young, recently diverged group of species provides insight into the first steps of speciation, suggesting a crucial role of the pre-zygotic barriers.

The evolution of niche overlap and competitive differences

Competition can result in evolutionary changes to coexistence between competitors but there are no theoretical models that predict how the components of coexistence change during this eco-evolutionary process. Here we study the evolution of the coexistence components, niche overlap and competitive differences, in a two-species eco-evolutionary model based on consumer-resource interactions and quantitative genetic inheritance. Species evolve along a one-dimensional trait axis that allows for changes in both niche position and species intrinsic growth rates. There are three main results. First, the breadth of the environment has a strong effect on the dynamics, with broader environments leading to reduced niche overlap and enhanced coexistence. Second, coexistence often involves a reduction in niche overlap while competitive differences stay relatively constant or vice versa; in general changes in competitive differences maintain coexistence only when niche overlap remains constant. Large simultaneous changes in niche overlap and competitive difference often result in one of the species being excluded. Third, provided that the species evolve to a state where they coexist, the final niche overlap and competitive difference values are independent of the system's initial state, although they do depend on the model's parameters. The model suggests that evolution is often a destructive force for coexistence due to evolutionary changes in competitive differences, a finding that expands the paradox of diversity maintenance.

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.

The dynamics of trait variance in multi-species communities

In this paper, we establish the explicit connection between deterministic trait-based population-level models (in the form of partial differential equations) and species-level models (in the form of ordinary differential equations), in the context of eco-evolutionary systems. In particular, by starting from a population-level model of density distributions in trait space, we derive what amounts to an extension of the typical models at the species level known from adaptive dynamics literature, to account not only for abundance and mean trait values, but also explicitly for trait variances. Thus, we arrive at an explicitly polymorphic model at the species level. The derivations make precise the relationship between the parameters in the two classes of models and allow us to distinguish between notions of fitness on the population and species levels. Through a formal stability analysis, we see that exponential growth of an eigenvalue in the trait covariance matrix corresponds to a breakdown of the underlying assumptions of the species-level model. In biological terms, this may be interpreted as a speciation event: that is, we obtain an explicit notion of the blow-up of the variance of (possibly a linear combination of) traits as a precursor to speciation. Moreover, since evolutionary volatility of the mean trait value is proportional to trait variance, this provides a notion that species at the cusp of speciation are also the most adaptive. We illustrate these concepts and considerations using a numerical simulation.

Lotka-Volterra approximations for evolutionary trait-substitution processes

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka–Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster’s total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka–Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community’s Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the ‘invasion implies substitution’ result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

Evolutionary branching in distorted trait spaces

Biological communities are thought to have been evolving in trait spaces that are not only multi-dimensional, but also distorted in a sense that mutational covariance matrices among traits depend on the parental phenotypes of mutants. Such a distortion may affect diversifying evolution as well as directional evolution. In adaptive dynamics theory, diversifying evolution through ecological interaction is called evolutionary branching. This study analytically develops conditions for evolutionary branching in distorted trait spaces of arbitrary dimensions, by a local nonlinear coordinate transformation so that the mutational covariance matrix becomes locally constant in the neighborhood of a focal point. The developed evolutionary branching conditions can be affected by the distortion when mutational step sizes have significant magnitude difference among directions, i.e., the eigenvalues of the mutational covariance matrix have significant magnitude difference.

Clade diversification dynamics and the biotic and abiotic controls of speciation and extinction rates

How ecological interactions, genetic processes, and environmental variability jointly shape the evolution of species diversity remains a challenging problem in biology. We developed an individual-based model of clade diversification to predict macroevolutionary dynamics when resource competition, genetic differentiation, and landscape fluctuations interact. Diversification begins with a phase of geographic adaptive radiation. Extinction rates rise sharply at the onset of the next phase. In this phase of niche self-structuring, speciation and extinction processes, albeit driven by biotic mechanisms (competition and hybridization), have essentially constant rates, determined primarily by the abiotic pace of landscape dynamics. The final phase of diversification begins when intense competition prevents dispersing individuals from establishing new populations. Species’ ranges shrink, causing negative diversity-dependence of speciation rates. These results show how ecological and microevolutionary processes shape macroevolutionary dynamics and rates; they caution against the notion of ecological limits to diversity, and suggest new directions for the phylogenetic analysis of diversification.

The history and patterns of species diversity are shaped by a variety of ecological and evolutionary factors. Here, the authors develop a computational model to predict clade diversification dynamics and rates of speciation and extinction under the influences of resource competition, genetic differentiation, and random landscape fluctuation.

Adaptive diversification in heterogeneous environments

The role of environmental heterogeneity in the evolution of biological diversity has been studied only for simple types of heterogeneities and dispersals. This article broadens previous results by considering heterogeneities and dispersals that are structured by several environmental factors. It studies the evolution of a metapopulation, living in a network of patches connected by dispersal, under the effects of mutation, selection and migration. First, it is assumed that patches are equally connected and that they carry habitats characterized by several factors exerting selection pressures on several individual traits. Habitat factors may vary in the environment independently or they may be correlated. It is shown that correlations between habitat factors promote adaptive diversification and that this effect may be modified by trait interactions on survival. Then, it is assumed that patches are structured by two crossed factors, called the row and column factors, such that patches are more connected when they occur in the same row or in the same column. Environmental patterns in which each habitat appears in each row the same number of times and appears in each column the same number of times are found to hinder adaptive diversification.

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set D in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to D. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to D, just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.

Evolutionary branching under slow directional evolution

Evolutionary branching is the process by which ecological interactions induce evolutionary diversification. In asexual populations with sufficiently rare mutations, evolutionary branching occurs through trait-substitution sequences caused by the sequential invasion of successful mutants. A necessary and sufficient condition for evolutionary branching of univariate traits is the existence of a convergence stable trait value at which selection is locally disruptive. Real populations, however, undergo simultaneous evolution in multiple traits. Here we extend conditions for evolutionary branching to bivariate trait spaces in which the response to disruptive selection on one trait can be suppressed by directional selection on another trait. To obtain analytical results, we study trait-substitution sequences formed by invasions that possess maximum likelihood. By deriving a sufficient condition for evolutionary branching of bivariate traits along such maximum-likelihood-invasion paths (MLIPs), we demonstrate the existence of a threshold ratio specifying how much disruptive selection in one trait direction is needed to overcome the obstruction of evolutionary branching caused by directional selection in the other trait direction. Generalizing this finding, we show that evolutionary branching of bivariate traits can occur along evolutionary-branching lines on which residual directional selection is sufficiently weak. We then present numerical analyses showing that our generalized condition for evolutionary branching is a good indicator of branching likelihood even when trait-substitution sequences do not follow MLIPs and when mutations are not rare. Finally, we extend the derived conditions for evolutionary branching to multivariate trait spaces.

Adaptive radiation driven by the interplay of eco-evolutionary and landscape dynamics

We investigate an individual-based model of adaptive radiation based on the biogeographical changes of the Great African Lakes where cichlid fishes radiated. In our model, the landscape consists of a mosaic of three habitat types which may or may not be separated by geographic barriers. We study the effect of the alternation between allopatry and sympatry called landscape dynamics. We show that landscape dynamics can generate a significantly higher diversity than allopatric or sympatric speciation alone. Diversification is mainly due to the joint action of allopatric, ecological divergence, and of disruptive selection increasing assortative mating and allowing for the coexistence in sympatry of species following reinforcement or character displacement. Landscape dynamics possibly increase diversity at each landscape change. The characteristics of the radiation depend on the speed of landscape dynamics and of the number of geographically isolated regions at steady state. Under fast dynamics of a landscape with many fragments, the model predicts a high diversity, possibly subject to the temporary collapse of all species into a hybrid swarm. When fast landscape dynamics induce the recurrent fusion of several sites, diversity is moderate but very stable over time. Under slow landscape dynamics, diversification proceeds similarly, although at a slower pace.

How is the equilibrium of continuous strategy game different from that of discrete strategy game?

Cooperation in the prisoner's dilemma (PD) played on various networks has been explained by so-called network reciprocity. Most of the previous studies presumed that players can offer either cooperation (C) or defection (D). This discrete strategy seems unrealistic in the real world, since actual provisions might not be discrete, but rather continuous. This paper studies the differences between continuous and discrete strategies in two aspects under the condition that the payoff function of the former is a linear interpolation of the payoff matrix of the latter. The first part of this paper proves theoretically that for two-player games, continuous and discrete strategies have different equilibria and game dynamics in a well-mixed but finite population. The second part, conducting a series of numerical experiments, reveals that such differences become considerably large in the case of PD games on networks. Furthermore, it shows, using the Wilcoxon sign-rank test, that continuous and discrete strategy games are statistically significantly different in terms of equilibria. Intensive discussion by comparing these two kinds of games elucidates that describing a strategy as a real number blunts D strategy invasion to C clusters on a network in the early stage of evolution. Thus, network reciprocity is enhanced by the continuous strategy.

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

A new mechanism for recurrent adaptive radiations

Models of adaptive radiation through intraspecific competition have attracted mounting attention. Here we show how extending such models in a simple manner, by including a quantitative trait under weak directional selection, naturally leads to rich macroevolutionary patterns involving recurrent adaptive radiations and extinctions. Extensive tests demonstrate the robustness of this finding to a wide range of variations in model assumptions. In particular, recurrent adaptive radiations and extinctions readily unfold both for asexual and for sexual populations. Since the mechanisms driving the investigated processes of endogenous diversification result from generic geometric features of the underlying fitness landscapes--frequency-dependent disruptive selection in one trait and weak directional selection in another--the reported phenomena can be expected to occur in a wide variety of eco-evolutionary settings.

Competitive exclusion and limiting similarity: A unified theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its impact and sensitivity niches are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niche overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a continuum of populations is structurally unstable.

Link between population dynamics and dynamics of Darwinian evolution

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the singular points of the evolutionary state space. Here "evolutionary branching" may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.