The Mathematics of Evolution: Adaptive Dynamics in Theory and Experiment

A Fields workshop at the University of Ottawa

Speakers, titles and abstracts


Invited Speakers

Martin Ackermann: Evolutionary Causes and Consequences of Phenotypic Variation

We are using experiments with bacteria to study evolutionary causes and consequences of phenotypic variation. We are interested in phenotypic variation between asexual individual that is not a consequence of environmental variation, and that arises either from new mutations or from noise during the expression of the phenotype. The level of phenotypic variation is partially under genetic control; genes control the mutation rate, and they influence the amount of phenotypic noise. We are interested in how selection acts on the mechanisms that control the level of variation produced, and in how the level of phenotypic variation influences the course of evolution.
 
I will first discuss how phenotypic variation in the distribution of cellular damage can be advantageous, and how this variation forms the basis of aging in unicellular organism. Then I will move on to more general aspects of evolutionary causes and consequences of phenotypic variation: I will present preliminary results of how organisms respond to artificial selection for increased phenotypic variation, and of how genetic manipulations that lead to an increase in the level of variation influence the course of experimental evolution. One of the main conclusions from our work is that internal properties of the evolving organism can have a substantial influence on the evolutionary dynamics, and that a better understanding of the organism can thus benefit our understanding of the evolutionary process.


Michael Doebeli: Adaptive diversification: theory and experiments

Understanding the origin of diversity is a fundamental problem in biology. Traditional evolutionary theory predicts uniformity: natural selection, acting on organisms under given environmental conditions and developmental constraints, produces a unique, optimally adapted phenotype. According to this view, different types only come about through a change in conditions over space or time. In particular, the process of diversification, that is, the split of an ancestral population into distinct descendent lineages, is a by-product of geographical separation. This traditional view misses out on the important perspective that diversification itself can be an adaptive process. In this talk I will review recent theoretical work showing that diversification as an adaptive response to biological interactions is a plausible evolutionary process. This work is based on the mathematical framework of adaptive dynamics, and in particular on the phenomenon of evolutionary branching due to frequency-dependent ecological interactions. I will describe basic models for evolutionary branching based on resource competition, as well as models of diversification in spatially structured populations. I will then describe ongoing efforts to test the theory of evolutionary branching in evolving Escherichia coli populations, which provide promising experimental model systems for studying adaptive diversification at the genetic, physiological and ecological level.

Richard Law: Adaptive Dynamics: concerns of an ecologist

The use of Adaptive Dynamics (AD) for the study of phenotypic evolution is controversial.  On the one hand, AD brings the selection generated by ecological dynamics to the centre of the evolutionary stage: you do not have a fully specified AD model until these dynamics have been specified.  On the other hand, AD as currently used, does not deal with the intricate machinery of inheritance on which the evolutionary process depends and which forms the core of contemporary evolutionary biology.  The intention of this talk is to generate discussion about the interface between AD and mainstream evolutionary biology:
•    Does AD provide understanding of ecological patterns in the natural world?  Some people have argued that AD is a theoretical framework which does not address the real properties of natural systems.
•    In what contexts should AD modellers be especially cautious in interpreting the the results of their work?  Under what circumstances are the predictions of AD likely to be inconsistent with predictions from other frameworks for analysis of evolution?
•    Does AD have to ignore the genetic machinery of inheritance?  To put it another way, can AD develop closer to mainstream evolutionary biology without losing its capacity to address the intricacies of ecological interactions?
•    What part can AD play in furthering understanding of evolutionary processes?  Are there classes of problems which can be addressed by AD which would be hard to tackle using more traditional methods?


Peter Taylor: Thoughts on allele-frequency change in finite populations

I will review the different measures of selective advantage in a finite population (average allele frequency change, fixation probability, inclusive fitness effect) and discuss the equivalences between them, illustrating my remarks with examples from structured populations.

Lindi Wahl: Adapting models to make testable predictions about fixation

Fixation, the process through which the frequency of an initially rare beneficial mutation ultimately approaches one, is fundamental to adaptation.  Although estimates of the fixation probability, $\pi$, have a rich history in theoretical population genetics, experimental estimates of $\pi$ have only recently become feasible.  Since the mathematical techniques are quite different from more traditional adaptive dynamics, I will begin with a brief historical overview of theoretical estimates of $\pi$.  I will then describe recent progress in adapting these classical models to deliver concrete, testable predictions for laboratory selection experiments.  Finally, I will describe our first set of experimental results, demonstrating that $\pi$ sensitively depends on both the life-history of the organism, and the details of the experimental protocol.


Contributed Talks

Alizon, Samuel: Virulence evolution in an embedded model allowing for multiple infections

Understanding multiple infections is essential to predict and to control virulence evolution. There is a strong debate among theoreticians and empiricists to know whether multiple infections select for increased levels of virulence. Here, we link within-host and epidemiological dynamics to study adaptive dynamics of parasite virulence. We show that the classical assumption that within-host and ecological time scales do not overlap does not apply in this case. We develop an original means to take into account this overlap and show that coinfections create a host heterogeneity (with susceptible hosts and infected hosts) which can lead to evolutionary branching in the parasite population with emergence of a supervirulent parasite strategy. These results could have applications in a virulence management perspective and call for further developments on the overlap between ecological and evolutionary dynamics.


Duffy, Meghan: Parasite-mediated disruptive selection followed by assortative mating in a natural Daphnia population

Because parasites can have large impacts on host fitness, they have long been thought to play an important role in the evolution of their host populations. While examples of parasite-mediated selection on host populations are quite limited, existing examples document parasite epidemics that caused directional selection for resistant host genotypes, leading to a winnowing of host diversity. Yet, theory suggests that parasitism can actually promote diversity, through mechanisms such as disruptive selection. Here, we document parasite-mediated disruptive selection on a natural population of Daphnia by a virulent yeast parasite, leading to increased genetic variance and a bimodal distribution of susceptibilities in the host population. This same bimodal distribution was retained after one generation of sexual reproduction, suggesting assortative mating. We are using adaptive dynamics to explore the factors promoting disruptive selection as well as to explore the effects of rapid host evolution on disease dynamics.

Forde, Samantha: Context-dependent coevolution: fitness of resistant hosts depends on resources, time, and mutation-type.

Coevolution between hosts and parasites, including bacteria and bacteriophage, has been modeled using an adaptive dynamics approach. These models can be strengthened by empirically-derived parameters of trait values for the host and parasite. Here I present data on how the fitness of bacterial mutants that are resistant to bacteriophage can vary as a function of resource concentration, time, and mutation type. In contrast to most theoretical predictions, there was no cost of resistance to bacteriophage, on average, for the bacteria. However, variation in fitness of resistance mutants relative to the ancestral type decreased over time. Relative fitness also depended on resource availability, as indicated by significant genotype by environment interactions. Finally, there were differences in the frequencies of the types of mutations dominating resistant populations over time and under different resource concentrations. These results demonstrate that the host and parasite trait values can be highly context-dependent, and can be used to inform future theoretical models of coevolutionary interactions.

Foster, Jacob: Modeling Adaptive Radiation in Pseudomonads

Adaptive radiation is a ubiquitous phenomenon in evolution. Much recent experimental attention has focussed on Pseudomonas fluorescens in a static microcosm as a microbial model system of adaptive radiation. In this talk, we will review these experiments and present a mathematical model of adaptive radiation in P. fluorescens, emphasizing the role of spatial structure in diversification.  We will also discuss preliminary experimental work in progress with Mike Surette's lab at the University of Calgary on adaptive radiation in P. aeruginosa and  P fluorescens.  The talk will elucidate the importance of microbial models in understanding the interaction of factors driving adaptive radiation.

Gauduchon, Mathias: Evolution and persistence of obligate mutualists and exploiters: The Partner Competition Hypothesis

Mutualisms are ubiquitous in nature, as is their exploitation by both conspecific and heterospecific cheaters Yet, evolutionary theory predicts that cheating should be favored by natural selection. Here we show theoretically that asymmetrical competition for partners generally determines the evolutionary fate of obligate mutualisms facing exploitation by thirdspecies invaders. When asymmetry in partner competition is relatively weak, mutualists may either exclude exploiters or coexist with them, in which case their co-evolutionary response to exploitation is usually benign. When asymmetry is strong, the mutualists evolve toward evolutionary attractors where they become extremely vulnerable to exploiter invasion. However, exploiter invasion at an early stage of the mutualism’s history can deflect mutualists’co-evolutionary trajectories towards slightly different attractors that confer long-term stability against further exploitation. Thus, coexistence of mutualists and exploiters may often involve an historical effect whereby exploiters are co-opted early in mutualism history and provide lasting ‘evolutionary immunization’ against further invasion.

Gudelij, Ivana: Constraints on microbial metabolism drive evolutionary diversification in homogeneous environments

In this talk we investigate the role of two established biochemical trade-offs in microbial diversification in the chemostat: 1) a trade-off between the rate and affinity of substrate transport and (2) a trade-off between the rate and yield of ATP production.

We take a standard chemostat competition model and introduce evolution by natural selection through a Markov mutation operator. We show that the long term diversity of a population will depend on an ecological, biochemical and an evolutionary component, namely the chemostat dilution rate, the geometry of the trade-off functions and the mutations that result from clonal reproduction. Excluding mutations for the model eliminates any chance of observing diversity.

Finally we relate the results of a number of microbial selection experiments to the predictions of our model.

Alex Hall: Decay of unused characters by selection and drift

The reduction and loss of redundant phenotypic characters is a common feature of evolution in changing environments. However, the mechanisms that drive deterioration of unused characters remain unclear. We outline a simple framework where the relative importance of selective and neutral processes varies with environmental factors, because the energetic cost associated with unused traits does not always reduce fitness. We tested our hypotheses using experimental evolution of the bacterium Pseudomonas fluorescens in spatially uniform environments. Results show that selection against an unused character, swimming motility, was most effective in environments with limited resources where we detected the greatest fitness cost of motility. This supports previous evidence that the availability of resources can critically affect the outcomes of evolution.

Jessup, Christine: The shape of an ecological trade-off in a microbial experimental system

Central to most theories that explain the diversity of life is the concept that organisms face trade-offs. Theoretical work has shown that the precise shape of a trade-off relationship can affect evolutionary predictions. One common trade-off is that between competitive ability and resistance to predators, pathogens or herbivores. The shape of the relationship between cost of resistance and the degree of resistance has not been directly determined for any predator-prey interaction. We used a microbial experimental system to elucidate the shape of the relationship between bacteriophage resistance and competitive ability. We selected 70 phage-resistant isolates of Escherichia coli B that exhibited varying levels of resistance to phage T2. For each of the isolates, we measured the degree of resistance to T2 and relative competitive ability in both the resource environment in which strains were isolated and in two alternate resource environments. The relationship between phage resistance and competitive ability was linear in the selected resource environment. Hump-shaped and U-shaped relationships were observed in the alternate environments. Further analysis of the isolates revealed that the relationships were made up of clusters of similar mutations. The results also show that different environmental conditions can impose different trade-off relationships on the same pairwise interaction. The observed trade-off shapes have important implications for ecological interactions and evolutionary dynamics which can be further tested using microbial experimental systems.

Amir Kermany: Intraspecies Competition Causes Negative Epistasis

It has proved difficult to define a set of plausible conditions that provide a selective advantage for genetic recombination, especially over a broad range of biological species and over a wide range of environmental conditions. While it is generally agreed that negative epistatic effects on fitness for gene interactions between loci would provide an advantage for recombination, it has not been easy to see why such negative epistasis should be the general rule in nature.

Here, we show that competitive interactions between genotypes within a population can result in negative epistasis at the level of realized fitness of the genotypes. We consider a population with finite resources and we assume that each advantageous mutation results in an increased ability to compete for the limited resources in the population. In particular, we consider the case in which there is no inherent epistatic effect at the phenotypic level, i.e. competitive ability. The dynamics of the population are described as solutions to a set of Lotka-Volterra equations.

Our results show that even when the competitive interaction are non-epistatic (i.e., either additive or multiplicative) there are nevertheless negative epistatic interactions at the level of realized fitness. This negative epistasis leads to negative linkage disequilibrium between the positively selected alleles, thus providing an advantage for genetic recombination. We confirm these results using computer simulations. Our conclusion is that competition between individuals within a species favors the maintenance of genetic recombination.


Lambert, Amaury: Genetic Drift versus Weak Selection in Adaptive Dynamics

The theory of adaptive dynamics proposes a description of evolution which relies on three assumptions : large populations, rare mutations and small mutation steps. Under these assumptions, the evolution of a quantitative dominant trait in an isolated population is described by a deterministic differential equation called "canonical equation of adaptive dynamics". In this work, we consider instead finite, randomly fluctuating populations to allow for the action of genetic drift and weak selection. Our model includes general ecological interactions (e.g. competition), with density-dependent reproduction depending on the individual trait (selection), featuring possible mutations. Applying a limit of rare mutations leads to a jump process on Dirac measures with oscillating mass, where evolution proceeds by successive invasions and fixations of mutants. Rescaling mutation steps then yields a diffusion on the trait space christened "canonical diffusion of adaptive dynamics", in which genetic drift (diffusive term) is combined with directional selection (deterministic term) driven by the gradient of the mutant invasion fitness. In the logistic branching case, invasion fitness factorises as a function of the initial mutant frequency times "invasibility coefficients" of the resident (fertility, defence, aggressiveness, isolation, survival).

Lion, Sebastien: Adaptive dynamics in space  

Over the last twenty years, the role of spatial self-structuring as a template for evolution has drawn much attention. Spatial structure can be an important component of the eco-evolutionary feedback loop: the evolution of a trait (e.g. migration) can shape the local structure of the population, which in turn creates new selective pressures on the evolving trait. As a consequence, the evolution of spatially structured populations often displays very different features from the evolution of well-mixed populations.

The theory of adaptive dynamics provides a general framework to investigate the interplay between ecological and evolutionary dynamics, and is would thus be natural to use this approach to make the connection between spatial ecological dynamics and evolution. Yet, with notable exceptions, the majority of models consider populations without spatial structure. While analytical tractability is often the main motivation of that assumption, the recent development of spatial moment equations in ecology paves the way for an intermediate approach between individual-based spatial simulations and non-spatial analytical models.

We review several contributions of this modelling approach to the study of the evolution of spatially-structured populations. We discuss the definition of invasion fitness in spatial models, and argue that the unit of selection in viscous populations is best viewed as a cluster of invading individuals. We then present several game-theoretical models played in space, and show that spatial structure has an important impact on the evolutionary outcomes, compared with the mean-field version. In particular, we present examples where spatial structure can either inhibit or promote evolutionary branching.

Rueffler, Claus: The evolution of phenotype determination

Many heterogeneous environments favour different phenotypes in different places or at different times. Phenotypic diversity can either result from genetic diversity of from a single genotype capable of producing different phenotypes. A single genotype might produce different phenotypes for example in response to an environmental cue (phenotypic plasticity), through a randomization mechanism (bet-hedging), or through a combination of the two. A large part of the existing theoretical literature attempts to give conditions under which one of these specific mechanisms is favoured over a phenotypically monomorphic population. However, in many circumstances different evolutionary responses are favoured simultaneously and the real question becomes which of these different responses might evolve first and possibly pre-empt any selection driving one of the alternative responses. I will address this question by analysing a lottery-model designed to study evolution in a temporarily heterogeneous environment.

Schreiber, Sebastian:  Cycling in space: Persistence of rock-scissor-paper polymorphisms in patchy landscapes.

In the past decade, several empirical studies have shown that the children's game of rock-scissor-paper occurs in natural systems. For instance, different strains of E.coli that produce and/or resist toxins can generate an intransitive dominance relationship. For this system, persistence of all three strains requires that the strains have localized dispersal. To better understand how spatial heterogeneity, dispersal rates, and strategic asymmetries influence local and global persistence, I introduce a general class of spatial evolutionary games and an approach to analyze them. The analysis applied to the rock-scissor-paper version of these games suggests how different forms of spatial and stategic heterogeneity can promote persistence at different rates of dispersal.

Gilman Tucker: Changes in resource availability along an ecological gradient can result in the collapse of sympatric speciation processes through selective introgression

Changes in resource availability along an ecological gradient can result in the collapse of sympatric speciation processes through selective introgression. A growing body of theory suggests that assortative mating coupled with evolution in response to intraspecific resource competition can lead to sympatric speciation.  We ask whether changes in resource distribution along a gradient corresponding to an ecological trait under selection can interrupt the speciation process increasing hybridization and ultimately causing the collapse of incipient species pairs. We begin with Dieckmann and Doebeli's (1999) seminal model of sympatric speciation.

Once stochastic processes have initiated the speciation process, speciation resists changes in resource distribution.  Incipient species pairs resist hybridization, but under appropriate resource distributions species pairs converge ecologically.  We introduce a novel and biologically reasonable mating system to capture the cost of selectivity in a female limited system.  Preliminary results of the enhanced model suggest that changes in resource distribution may lead to the collapse of species pairs in a process that includes both convergence and selective introgression.