The formal darwinism project in outline

The formal darwinism project aims to provide a mathematical framework within which important fundamental ideas in large parts of biology can be articulated, including Darwin's central argument in The Origin (that mechanical processes of inheritance and reproduction can give rise to the appearance of design), modern extensions of evolutionary theory including ESS theory and inclusive fitness, and Dawkins' synthesis of them into a single structure. A new kind of argument is required to link equations of motion on the one hand to optimisation programs on the other, and a major point is that the biologist's concept of fitness maximisation is not represented by concepts from dynamical systems such as Lyapunov functions and gradient functions. The progress of the project so far is reviewed, though with only a brief glance at the rather complicated mathematics itself, and the centrality of fitness maximisation ideas to many areas of biology is emphasised.     

Formalizing Darwinism and inclusive fitness theory

Inclusive fitness maximization is a basic building block for biological contributions to any theory of the evolution of society. There is a view in mathematical population genetics that nothing is caused to be maximized in the process of natural selection, but this is explained as arising from a misunderstanding about the meaning of fitness maximization. Current theoretical work on inclusive fitness is discussed, with emphasis on the author''s ‘formal Darwinism project’. Generally, favourable conclusions are drawn about the validity of assuming fitness maximization, but the need for continuing work is emphasized, along with the possibility that substantive exceptions may be uncovered. The formal Darwinism project aims more ambitiously to represent in a formal mathematical framework the central point of Darwin''s Origin of Species, that the mechanical processes of inheritance and reproduction can give rise to the appearance of design, and it is a fitting ambition in Darwin''s bicentenary year to capture his most profound discovery in the lingua franca of science.     

Group adaptation, formal darwinism and contextual analysis

We consider the question: under what circumstances can the concept of adaptation be applied to groups, rather than individuals? Gardner and Grafen (2009, J. Evol. Biol. 22 : 659–671) develop a novel approach to this question, building on Grafen's ‘formal Darwinism’ project, which defines adaptation in terms of links between evolutionary dynamics and optimization. They conclude that only clonal groups, and to a lesser extent groups in which reproductive competition is repressed, can be considered as adaptive units. We re‐examine the conditions under which the selection–optimization links hold at the group level. We focus on an important distinction between two ways of understanding the links, which have different implications regarding group adaptationism. We show how the formal Darwinism approach can be reconciled with G.C. Williams’ famous analysis of group adaptation, and we consider the relationships between group adaptation, the Price equation approach to multi‐level selection, and the alternative approach based on contextual analysis.     

Capturing the superorganism: a formal theory of group adaptation

Adaptation is conventionally regarded as occurring at the level of the individual organism. However, in recent years there has been a revival of interest in the possibility for group adaptations and superorganisms. Here, we provide the first formal theory of group adaptation. In particular: (1) we clarify the distinction between group selection and group adaptation, framing the former in terms of gene frequency change and the latter in terms of optimization; (2) we capture the superorganism in the form of a ‘group as maximizing agent’ analogy that links an optimization program to a model of a group‐structured population; (3) we demonstrate that between‐group selection can lead to group adaptation, but only in rather special circumstances; (4) we provide formal support for the view that between‐group selection is the best definition for ‘group selection’; and (5) we reveal that mechanisms of conflict resolution such as policing cannot be regarded as group adaptations.     

Evolutionarily stable communication between kin: a general model

At present, the most general evolutionary theory of honest communication is Grafen''s model of Zahavi''s ''handicap'' signalling system, in which honesty of signals about the signaller''s quality (e.g. mate suitability or fighting ability) is maintained by the differentially high cost of signals to signallers having lower quality. The latter model is here further generalized to include any communication between signallers and receivers that are genetically related (e.g. parents and begging offspring, cooperative or competing siblings). Signalling systems involving relatives are shown to be evolutionarily stable, despite a potential pay-off for false signalling, if the Zahavian assumption of differential signal costs holds and there are diminishing reproductive returns to the signaller as the receiver''s assessed value of its attribute increases, or if, regardless of whether the Zahavian assumption holds, signallers with high values of the attribute benefit more from a given receiver assessment than signallers with low values (e.g. begging chicks that are hungrier benefit more from being fed). In stable systems of signalling among kin, it is also shown to be generally true that (i) levels of signalling and thus observed signal costs will decline as relatedness increases or as the receiver''s reproductive penalty for erroneous assessment increases, and (ii) receivers will consistently, altruistically overestimate the true value of the signalled attribute.     

Hamilton's inclusive fitness in finite-structured populations

Hamilton''s formulation of inclusive fitness has been with us for 50 years. During the first 20 of those years attention was largely focused on the evolutionary trajectories of different behaviours, but over the past 20 years interest has been growing in the effect of population structure on the evolution of behaviour and that is our focus here. We discuss the evolutionary journey of the inclusive-fitness effect over this epoch, nurtured as it was in an essentially homogeneous environment (that of ‘transitive’ structures) having to adapt in different ways to meet the expectations of heterogeneous structures. We pay particular attention to the way in which the theory has managed to adapt the original constructs of relatedness and reproductive value to provide a formulation of inclusive fitness that captures a precise measure of allele-frequency change in finite-structured populations.     

Are reproductive skew models evolutionarily stable?

Reproductive skew theory has become a popular way to phrase problems and test hypotheses of social evolution. The diversity of reproductive skew models probably stems from the ease of generating new variations. However, I show that the logical basis of skew models, that is, the way in which group formation is modelled, makes use of hidden assumptions that may be problematical as they are unlikely to be fulfilled in all social systems. I illustrate these problems by re-analysing the basic concessive skew model with staying incentives. First, the model assumes that dispersal is an all-or-nothing response: all subordinates disperse as soon as concessions drop below a certain value. This leads to a discontinuous 'cliff-edge' shape of dominant fitness, and it is not clear that selection will balance a population at such an edge. Second, it is assumed that subordinates have perfect knowledge of their benefits if they stay in the group. I examine the effects of relaxing these two assumptions. Relaxing the first one strengthens reproductive skew theory, but relaxing the latter makes evolutionary stability disappear. In cases where subordinates cannot accurately measure benefits provided by the individual dominant with which they live, so that their behaviour instead evolves as a response to population-wide average benefits, the logic of reproductive skew models does not apply. This warns against too indiscriminate an application of reproductive skew theory to problems in social evolution: for example, transactional models of extra-pair paternity assume perfect knowledge of paternity, which is unlikely to hold true in nature. It is recommended that models specify the mechanisms by which individuals can adjust their behaviour to that of others, and pay attention to changes that occur in evolutionary versus behavioural time.     

Evolving cell models for systems and synthetic biology

This paper proposes a new methodology for the automated design of cell models for systems and synthetic biology. Our modelling framework is based on P systems, a discrete, stochastic and modular formal modelling language. The automated design of biological models comprising the optimization of the model structure and its stochastic kinetic constants is performed using an evolutionary algorithm. The evolutionary algorithm evolves model structures by combining different modules taken from a predefined module library and then it fine-tunes the associated stochastic kinetic constants. We investigate four alternative objective functions for the fitness calculation within the evolutionary algorithm: (1) equally weighted sum method, (2) normalization method, (3) randomly weighted sum method, and (4) equally weighted product method. The effectiveness of the methodology is tested on four case studies of increasing complexity including negative and positive autoregulation as well as two gene networks implementing a pulse generator and a bandwidth detector. We provide a systematic analysis of the evolutionary algorithm’s results as well as of the resulting evolved cell models.     

A formal theory of the selfish gene

Adaptation is conventionally regarded as occurring at the level of the individual organism. In contrast, the theory of the selfish gene proposes that it is more correct to view adaptation as occurring at the level of the gene. This view has received much popular attention, yet has enjoyed only limited uptake in the primary research literature. Indeed, the idea of ascribing goals and strategies to genes has been highly controversial. Here, we develop a formal theory of the selfish gene, using optimization theory to capture the analogy of 'gene as fitness-maximizing agent' in mathematical terms. We provide formal justification for this view of adaptation by deriving mathematical correspondences that translate the optimization formalism into dynamical population genetics. We show that in the context of social interactions between genes, it is the gene's inclusive fitness that provides the appropriate maximand. Hence, genic selection can drive the evolution of altruistic genes. Finally, we use the formalism to assess the various criticisms that have been levelled at the theory of the selfish gene, dispelling some and strengthening others.     

A demographic model for sex ratio evolution and the effects of sex‐biased offspring costs

The evolution of the primary sex ratio, the proportion of male births in an individual's offspring production strategy, is a frequency‐dependent process that selects against the more common sex. Because reproduction is shaped by the entire life cycle, sex ratio theory would benefit from explicitly two‐sex models that include some form of life cycle structure. We present a demographic approach to sex ratio evolution that combines adaptive dynamics with nonlinear matrix population models. We also determine the evolutionary and convergence stability of singular strategies using matrix calculus. These methods allow the incorporation of any population structure, including multiple sexes and stages, into evolutionary projections. Using this framework, we compare how four different interpretations of sex‐biased offspring costs affect sex ratio evolution. We find that demographic differences affect evolutionary outcomes and that, contrary to prior belief, sex‐biased mortality after parental investment can bias the primary sex ratio (but not the corresponding reproductive value ratio). These results differ qualitatively from the widely held conclusions of previous models that neglect demographic structure.     

Phylodynamic Inference and Model Assessment with Approximate Bayesian Computation: Influenza as a Case Study

A key priority in infectious disease research is to understand the ecological and evolutionary drivers of viral diseases from data on disease incidence as well as viral genetic and antigenic variation. We propose using a simulation-based, Bayesian method known as Approximate Bayesian Computation (ABC) to fit and assess phylodynamic models that simulate pathogen evolution and ecology against summaries of these data. We illustrate the versatility of the method by analyzing two spatial models describing the phylodynamics of interpandemic human influenza virus subtype A(H3N2). The first model captures antigenic drift phenomenologically with continuously waning immunity, and the second epochal evolution model describes the replacement of major, relatively long-lived antigenic clusters. Combining features of long-term surveillance data from the Netherlands with features of influenza A (H3N2) hemagglutinin gene sequences sampled in northern Europe, key phylodynamic parameters can be estimated with ABC. Goodness-of-fit analyses reveal that the irregularity in interannual incidence and H3N2''s ladder-like hemagglutinin phylogeny are quantitatively only reproduced under the epochal evolution model within a spatial context. However, the concomitant incidence dynamics result in a very large reproductive number and are not consistent with empirical estimates of H3N2''s population level attack rate. These results demonstrate that the interactions between the evolutionary and ecological processes impose multiple quantitative constraints on the phylodynamic trajectories of influenza A(H3N2), so that sequence and surveillance data can be used synergistically. ABC, one of several data synthesis approaches, can easily interface a broad class of phylodynamic models with various types of data but requires careful calibration of the summaries and tolerance parameters.     

A theory of Fisher's reproductive value

The formal Darwinism project aims to provide a mathematically rigorous basis for optimisation thinking in relation to natural selection. This paper deals with the situation in which individuals in a population belong to classes, such as sexes, or size and/or age classes. Fisher introduced the concept of reproductive value into biology to help analyse evolutionary processes of populations divided into classes. Here a rigorously defined and very general structure justifies, and shows the unity of concept behind, Fisher's uses of reproductive value as measuring the significance for evolutionary processes of (i) an individual and (ii) a class; (iii) recursively, as calculable for a parent as a sum of its shares in the reproductive values of its offspring; and (iv) as an evolutionary maximand under natural selection. The maximand is the same for all parental classes, and is a weighted sum of offspring numbers, which implies that a tradeoff in one aspect of the phenotype can legitimately be studied separately from other aspects. The Price equation, measure theory, Markov theory and positive operators contribute to the framework, which is then applied to a number of examples, including a new and fully rigorous version of Fisher's sex ratio argument. Classes may be discrete (e.g. sex), continuous (e.g. weight at fledging) or multidimensional with discrete and continuous components (e.g. sex and weight at fledging and adult tarsus length).     

Hamilton's rule meets the Hamiltonian: kin selection on dynamic characters

Many biological characters of interest are temporal sequences of decisions. The evolution of such characters is often modelled using dynamic optimization methods such as the maximum principle. A quantity central to these analyses is the ''Hamiltonian'' function, named after the mathematician William R. Hamilton. On the other hand, evolutionary models in which individuals interact with relatives are usually based on Hamilton''s rule, named after the evolutionary biologist William D. Hamilton. In this article we present a generalized maximum principle that includes the effects of interactions among relatives and we show that a time-dependent (dynamic) version of Hamilton''s rule holds involving the Hamiltonian. This result brings together the power and generality of both the maximum principle and Hamilton''s rule thereby providing a natural framework for understanding the evolution of ''dynamic'' characters under kin selection.     

Inclusive fitness and the sociobiology of the genome

Inclusive fitness theory provides conditions for the evolutionary success of a gene. These conditions ensure that the gene is selfish in the sense of Dawkins (The selfish gene, Oxford University Press, Oxford, 1976): genes do not and cannot sacrifice their own fitness on behalf of the reproductive population. Therefore, while natural selection explains the appearance of design in the living world (Dawkins in The blind watchmaker: why the evidence of evolution reveals a universe without design, W. W. Norton, New York, 1996), inclusive fitness theory does not explain how. Indeed, Hamilton’s rule is equally compatible with the evolutionary success of prosocial altruistic genes and antisocial predatory genes, whereas only the former, which account for the appearance of design, predominate in successful organisms. Inclusive fitness theory, however, permits a formulation of the central problem of sociobiology in a particularly poignant form: how do interactions among loci induce utterly selfish genes to collaborate, or to predispose their carriers to collaborate, in promoting the fitness of their carriers? Inclusive fitness theory, because it abstracts from synergistic interactions among loci, does not answer this question. Fitness-enhancing collaboration among loci in the genome of a reproductive population requires suppressing alleles that decrease, and promoting alleles that increase the fitness of its carriers. Suppression and promotion are effected by regulatory networks of genes, each of which is itself utterly selfish. This implies that genes, and a fortiori individuals in a social species, do not maximize inclusive fitness but rather interact strategically in complex ways. It is the task of sociobiology to model these complex interactions.     

Intergenerational and Sibling Conflict Under Patrilocality

Here we argue that models developed to examine cooperation and conflict in communal breeders, using a “tug-of-war” model of reproductive skew generated by incomplete control, are an appropriate way to model human kinship systems. We apply such models to understand the patterns of effort put into competition between father and son and between brothers in conflict over family resources in a patrilineal kinship system. Co-resident kin do not necessarily emerge with equal shares of the cake in terms of reproductive output. The models show that, depending on the efficiency with which they can gain more control of the resource, on the marriage system, and on the relatedness of the partners in conflict, individuals can do better to help their relatives breed rather than fight each other for the resources needed to reproduce. The models show that when a son’s father is still breeding with his mother, sons should not compete for any share of reproduction. However, under polygyny, increased effort is spent on father/son and brother/brother conflict. Fathers will win the majority of reproduction if dominant to sons (in contrast to the finding that daughters-in-law win in conflict over mothers-in-law in patrilocal kinship systems, which has been suggested as explaining the evolution of menopause). Hence who wins in the sharing of reproduction depends not just on which sex disperses but also on the relative competitive ability of all individuals to exploit family resources. Anthropologists have long argued that cultural norms can reduce conflict. These formal evolutionary models help us to quantify the effects of reproductive conflict in families, throwing light on the evolutionary basis not just of patterns of reproductive scheduling, but also human kinship and marriage systems.     

Human behavioral ecology and niche construction

We examine the relationship between niche construction theory (NCT) and human behavioral ecology (HBE), two branches of evolutionary science that are important sources of theory in archeology. We distinguish between formal models of niche construction as an evolutionary process, and uses of niche construction to refer to a kind of human behavior. Formal models from NCT examine how environmental modification can change the selection pressures that organisms face. In contrast, formal models from HBE predict behavior assuming people behave adaptively in their local setting, and can be used to predict when and why people engage in niche construction. We emphasize that HBE as a field is much broader than foraging theory and can incorporate social and cultural influences on decision‐making. We demonstrate how these approaches can be formally incorporated in a multi‐inheritance framework for evolutionary research, and argue that archeologists can best contribute to evolutionary theory by building and testing models that flexibly incorporate HBE and NCT elements.     

Effects of allometry,productivity and lifestyle on rates and limits of body size evolution

Body size affects nearly all aspects of organismal biology, so it is important to understand the constraints and dynamics of body size evolution. Despite empirical work on the macroevolution and macroecology of minimum and maximum size, there is little general quantitative theory on rates and limits of body size evolution. We present a general theory that integrates individual productivity, the lifestyle component of the slow–fast life-history continuum, and the allometric scaling of generation time to predict a clade''s evolutionary rate and asymptotic maximum body size, and the shape of macroevolutionary trajectories during diversifying phases of size evolution. We evaluate this theory using data on the evolution of clade maximum body sizes in mammals during the Cenozoic. As predicted, clade evolutionary rates and asymptotic maximum sizes are larger in more productive clades (e.g. baleen whales), which represent the fast end of the slow–fast lifestyle continuum, and smaller in less productive clades (e.g. primates). The allometric scaling exponent for generation time fundamentally alters the shape of evolutionary trajectories, so allometric effects should be accounted for in models of phenotypic evolution and interpretations of macroevolutionary body size patterns. This work highlights the intimate interplay between the macroecological and macroevolutionary dynamics underlying the generation and maintenance of morphological diversity.     

A quantitative test of population genetics using spatiogenetic patterns in bacterial colonies

It is widely accepted that population-genetics theory is the cornerstone of evolutionary analyses. Empirical tests of the theory, however, are challenging because of the complex relationships between space, dispersal, and evolution. Critically, we lack quantitative validation of the spatial models of population genetics. Here we combine analytics, on- and off-lattice simulations, and experiments with bacteria to perform quantitative tests of the theory. We study two bacterial species, the gut microbe Escherichia coli and the opportunistic pathogen Pseudomonas aeruginosa, and show that spatiogenetic patterns in colony biofilms of both species are accurately described by an extension of the one-dimensional stepping-stone model. We use one empirical measure, genetic diversity at the colony periphery, to parameterize our models and show that we can then accurately predict another key variable: the degree of short-range cell migration along an edge. Moreover, the model allows us to estimate other key parameters, including effective population size (density) at the expansion frontier. While our experimental system is a simplification of natural microbial community, we argue that it constitutes proof of principle that the spatial models of population genetics can quantitatively capture organismal evolution.     

Physical work causes suppression of ovarian function in women.

The suppression of reproductive function is known to occur in women engaging in activities that require high energetic expenses, such as sport participation and subsistence work. It is still unclear, however, if reproductive suppression is a response to high levels of energy expenditure, or only to the resulting state of negative energy balance. To our knowledge, this study provides the first evidence that work-related energy expenditure alone, without associated negative energy balance, can lead to the suppression of reproductive function in women. We document suppression of ovarian function expressed as lowered salivary progesterone levels in women from an agricultural community who work hard, but remain in neutral energy balance. We propose two alternative evolutionary explanations (the ''pre-emptive ovarian suppression'' hypothesis and the ''constrained down-regulation'' hypothesis) for the observed results.     

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms'' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin''s arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.