The Mathematics of Darwin's Legacy

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Reinhard Buerger (U. Vienna, AT)

Mathematical Models in Evolutionary Genetics. Natural selection and inheritance are the central agencies driving biological evolution. Whereas selection acts on phenotypes, inheritance passes the genetic material that determines the phenotype from one generation to the next, however, not without reshuffling and change. Many phenotypic traits can be measured, often on a continuous scale. Then they are called quantitative traits. Most quantitative traits have a complex genetic basis because they are determined by many genes. In order to understand, reconstruct, or predict evolution in a quantitative way, it is necessary to have a theory that can describe the response to selection and the evolution of such traits. For several reasons, this is a difficult enterprise, but it can be traced back to the Biometricians at the end of the 19th century. Although the modern foundations, based on Mendelian genetics, were laid by R.A. Fisher about 90 years ago, this is still a very active research area. We shall review both classical and recent approaches to study the evolution and architecture of quantitative traits, and we shall point out some of the most interesting unresolved questions.

Warren J. Ewens (U. Pennsylvania, USA)

What changes have mathematics and genetics made to the Darwinian theory? The Darwinian theory of evolution by natural selection was put forward by Darwin in complete ignorance of a key requirement for the theory, namely the nature of the hereditary mechanism. Indeed the hereditary mechanism most frequently assumed at the time that he wrote “The Origin of Species” was the blending theory, whereby any characteristic in a child is assumed to be a blending, or mixture, of the corresponding characteristics in the two parents. However, it is easily seen that this mechanism would soon destroy the very variation in a population upon which selection has to act. A mathematical analysis of the Mendelian hereditary mechanism overcomes this problem. This analysis also reveals modifications that are necessary to the Darwinian paradigm. For example, not all the variation in some character is available for evolutionary purposes: only that variation which derives from “genes within genotypes” is so available. A detailed mathematical analysis is also needed to assess the effects of the random transmission of genes from parents to offspring. This talk will survey a number of aspects where mathematics provides a more refined version of the Darwinian theory than was available in Darwin’s time.

Mats Gyllenberg (U. Helsinki, FI)

Optimisation principles in evolution.

Vincent Jansen (Royal Holloway, UK)

On group and kin selection. The evolution of helping and altruism has had a very prominent position in the study of evolution. To describe the evolution of such social characteristics a number of different theoretical approached have been used. There is an ongoing debate in how far these different modelling perspectives can be used to understand the evolution of social interactions in structured populations. Where some argue that the inclusive fitness approach, which includes kin selection, provides a generic framework to study the evolution of social interactions, others prefer description in terms of groups. We will compare and contrast the mathematics underlying some of these approaches and clarify where some of the different interpretations of these two approaches arise from by concentrating on the genetical and demographic structure in these models. We will use this analysis to identify open questions and new challenges in the study of social interactions.
This is joint work with Sebastien Lion and Troy Day.

Sylvie Méléard (E. Polytechnique, FR)

Polymorphic trait substitution sequence and evolutionary branching. We are interested in the study of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. Our modeling is based on a (stochastic) individual-based model that details the ecological dynamics of each individual and on the biologically motivated assumptions of rare mutations and large population. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems.

Hans Metz (U. Leiden, NL)

The Geometry of Macro-evolution: links between Adaptive Dynamics and Evo-Devo. Adaptive dynamics (AD) is a recently developed framework geared towards making the transition from micro-evolution to long-term evolution based on a time scale separation approximation. This assumption allows defining the fitness of a mutant as the rate constant of initial exponential growth of the mutant population in the environment created by the resident community dynamics. This definition makes that all resident types have fitness zero. If in addition it is assumed that mutational steps are small, evolution can be visualized as an uphill walk in a fitness landscape that keeps changing as a result of the evolution it engenders. AD arguments are largely local in character. As such they can only deal with what might be called meso-evolution. For longer timescales (macro-evolution) it becomes necessary to look at general morphological and developmental arguments that bear on the larger scale geometry of fitness landscapes. From this enlarged perspective the low dimensional fitness landscapes studied in AD can be seen as the surfaces at the top of ridges in a much higher dimensional landscape over potential morphologies, with the abyss around the ridges created by the lack of a proper development, or functioning. The location of the ridges and abysses is grossly the same for large sets of possible environmental conditions. Biological parlance expresses this constancy by referring to the corresponding selective processes as internal. High dimension and ridgyness turn out to conspire in a number of ways: 1. Developmental systems leading to mutation distributions that are in some way aligned with the ridges evolve much faster than systems where such is not the case. 2. The stabilizing selection in the off-ridge directions has a great robustness of the developmental system as evolutionary consequence. Yet, the high dimension of genotype space makes that this robustness can never lead to a lack of suitable mutational variation, and thereby to the conservation of features. Hence, the fact that evolution largely proceeds through the quantitative variation in the size and shape of homologous parts should be due to the stabilizing internal selection that arises as a consequence of organismal organization. 3. So-called allopatric speciation supposedly occurs by separated populations wandering around along the high fitness maze, so that after a while their mixed offspring, having intermediate properties, ends up in the abyss. As random fitness landscapes almost never engender allopatric speciation the question arises whether, and if so for what reason, evolved genotype to phenotype maps may be more speciation prone. 4. Large mutational steps far more often than not make an individual land in the fitness abyss, and only the much rarer very small steps keep it on the top. This may provide a theoretical justification for the assumptions made in AD.

Masayasu Mimura (Meiji U., JP)

Model aided understanding of secrets in biological systems. Over the past ten years, our understanding of how spatio-temporal patterns in biological systems has been deepened. Especially, collaborative research of experimental and theoretical works has gradually discovered the mechanism how self-organized complex patterns are generated in far from equilibrium systems. The term of self-organizing was originally introduced by Ross Ashby [1] in 1947, who was an English psychiatrist and also a pioneer in cybernetics, and a great contributor to this filed is ILya Prigogine who received the Nobel prize in chemistry in 1977 [2]. Nowadays, self-organizing behavior has been observed in many disciplines in not only natural sciences but also social sciences such as economics [3]. Turning to biology, especially to developmental biology, it has been reported that genetics does not always reveal the occurrence of such patterns and even simple systems may generate regular as well as chaotic irregular patterns in a self-organized way. In this talk, I would like to focus on bacterial colonies experimented by Budrene and Berg [1,2]. They observed diverse complex but regulated patterns generated by chemotactic bacteria of E.coli, depending on concentration of nutrients. The aim of this lecture is to understand whether such patterns are generated by genetic control or self-organization by using a model –aided approach.
[1] Ashby R.: Journal of General Psychology 37, pp.125~128 (1947); [2] Nicolis G. and Prigogine I.: Self-Organization in Non-Equilibrium Systems, Wiley (1977); [3] Krugman P. R.: Self-organizing Economy, Cambridge, Mass., and Oxford: Blackwell Publ. (1996); and Budrene E. O and Berg, H. C.: Nature 349, pp. 630~633, (1991); [4] Budrene E. O and Berg, H. C.: Nature 376, pp. 49~53 (1995).

Jorge Pacheco (U. Lisboa and U. Minho, PT)

The Evolution of Group Cooperation. Cooperation has played a major role in evolution. In Humans, cooperation is mostly related to problems of collective action, which often require the participation of groups of individuals who should decide whether to participate or not in a joint enterprise. Public Goods Games provide the appropriate mathematical tool to address these types of problems, which may deal with situations ranging from family issues to global warming. Evolutionary game theory predicts that the temptation to forgo the public good mostly wins over collective cooperative action, something which is often confirmed in behavioral economic experiments. Here I will address 2 important aspects of evolutionary dynamics which have been neglected so far in the context of public goods games: On one hand, the fact that often there is a threshold above which a public good is reached. On the other hand, the fact that individuals often participate in several games, related to their social context and pattern of social ties, defined by a social network. In the first case, the existence of a threshold dictates a rich evolutionary pattern: in finite populations, whenever public goods require participation of nearly the entire community, the direction of natural selection can be inverted compared to standard expectations. In networked games, cooperation blooms whenever the act of contributing is more important than the effort contributed.

Benoit Perthame (U. Paris, FR)

Adaptive evolution: a population view. The two processes of mutations and selection, proposed by C. Darwin, can be written in mathematical words. In a very simple, general and idealized description, the environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to multiply faster because they are better adapted to use the environment. This leads to select the 'fittest trait' in the population. On the other hand, the new-born individuals undergo small variations of the trait under the effect of mutations. In these circumstances, is it possible to observ 'speciation' and to describe the dynamical evolution of the 'fittest' traits?
We will give a class of self-contained mathematical models of such population dynamics and show that an asymptotic view allows us to formalize precisely the concepts of monomorphic or polymorphic populations. We can describe the evolution of the 'fittest traits' and various forms of branching points. We will also show that numerical solutions are consistant with individual based stochastic simulations.
Eventhough the model is very simple, its analysis relates to remarkable recent progresses of nonlinear analysis.

Peter Schuster (U. Vienna, AT)

The mathematics of Darwin's theory of evolution -- 1859 and 150 years later. Charles Darwin’s ‘Origin of Species’ published 150 years ago does not contain a single mathematical expression. Therefore, we can only speculate how Darwin might have cast natural selection into a mathematical language. One reason for the wide-spread applicability of Darwin’s theory, which ranges from evolution of molecules to economics and social sciences, is the fact that there are no requirements for the evolving entities except the capability to replicate and an inherent possibility of variation, for example through replication errors or recombination. Then, selection follows automatically from finite resources. Evolution can be studied not only with relatively simple organisms like bacteria or even simpler entities like viruses and viroids, it can also be investigated with replicating molecules in entirely cell-free systems. In the talk an overview of the state of the art of modeling evolution in simple systems by rigorous mathematical techniques will be presented and present day challenges coming form molecular genetics will be discussed.

Peter Taylor (Queens U., CA)

Homogeneous population structure: putting the “inclusive” into inclusive fitness and then taking it out again (and rescuing Darwin from the mud). Consider a behavioural trait that affects the fecundity of the actor and of certain “neighbours.” To calculate the selective advantage of such a trait using an inclusive fitness analysis one has to take a weighted sum of all its fitness effects, not only the direct fecundity effects, but also mortality effects due to altered competitive pressures. It has been known for close to 20 years that if the population structure has “perfect” internal symmetry, there is in this sum an almost complete cancellation of the mortality effects with the fecundity effects, leaving us with only the fecundity effect of the actor on herself. This striking result is significant in that it gives us a “base case” from which we can work to understand what types of inhomogeneity are required to give cooperative or altruistic behaviours a selective edge. Here we use mathematical group theory to provide an elegant and more general version of the basic result.

Franjo Weissing (U. Groningen, NL)

The dynamics of sexual selection. Not surprisingly, it was Charles Darwin who (in 1871) laid the foundations of sexual selection theory. After more than a century of stagnation, the theory got off the ground in the 1980s. Although sexual selection is still an immensely popular research field, theory formation has again stagnated in the last two decades. Many researchers think that the theory has now been firmly established. In my talk, I will argue that, on the contrary, many fundamental issues have not yet been resolved and that even the standard models of sexual selection may yield surprising results. I will demonstrate this with three examples from our own research. First, I will show that the dynamics of sexual selection will often not lead to equilibrium. In other words, mate preferences are much more “dynamic” than one used to think. This finding is in line with phylogenetic evidence, and it has major theoretical implications. In fact, many of the standard results of sexual selection theory do apply to populations in equilibrium. Second, I will address the interaction of sexual selection and sex allocation. It has been argued that theory predicts that attractive males should overproduce sons, but the evidence from field and lab studies is inconclusive. I will show that theoretical predictions are also much less conclusive than verbal arguments seem to suggest. Interestingly, the ability to make the sex ratio dependent on the attractiveness of one’s mate undermines sexual selection, the driving force of this very process. Third (if there is time), I will discuss the role disruptive sexual selection plays as a driving force of sympatric speciation. Sexual selection may induce diverging evolution and sympatric speciation, but only under highly restrictive conditions. Moreover, virtually all models on sympatric speciation through sexual selection rely on Fisher’s runaway process of sexual selection. I will argue that good genes sexual selection may be more important for speciation than hitherto acknowledged.