Grafen,the Price equations,fitness maximization,optimisation and the fundamental theorem of natural selection

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory.     

The formal darwinism project in outline: response to commentaries

The broader context for the formal darwinism project established by two of the commentators, in terms of reconciling the Modern Synthesis with Darwinian arguments over design and in terms of links to other types of selection and design, is discussed and welcomed. Some overselling of the project is admitted, in particular of whether it claims to consider all organic design. One important fundamental question raised in two commentaries is flagged but not answered of whether design is rightly represented by an optimisation program, and another from one commentary of whether the coreplicon dissolves in the face of multi-generational imprinting. Calls for the project to be extended to design at levels above and below the individual are considered sympathetically, but judged impractical at the high level of abstraction of the project. All claims of substantive technical error are emphatically rejected. Close technical readings are welcomed that, among other things, represent the project as 'axiomatizing fitness'. The prospects for the project are set out in the light of this highly varied set of commentaries.     

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.     

“The Theory was Beautiful Indeed”: Rise,Fall and Circulation of Maximizing Methods in Population Genetics (1930–1980)

Describing the theoretical population geneticists of the 1960s, Joseph Felsenstein reminisced: “our central obsession was finding out what function evolution would try to maximize. Population geneticists used to think, following Sewall Wright, that mean relative fitness, W, would be maximized by natural selection” (Felsenstein 2000). The present paper describes the genesis, diffusion and fall of this “obsession”, by giving a biography of the mean fitness function in population genetics. This modeling method devised by Sewall Wright in the 1930s found its heyday in the late 1950s and early 1960s, in the wake of Motoo Kimura’s and Richard Lewontin’s works. It seemed a reliable guide in the mathematical study of deterministic effects (the study of natural selection in populations of infinite size, with no drift), leading to powerful generalizations presenting law-like properties. Progress in population genetics theory, it then seemed, would come from the application of this method to the study of systems with several genes. This ambition came to a halt in the context of the influential objections made by the Australian mathematician Patrick Moran in 1963. These objections triggered a controversy between mathematically- and biologically-inclined geneticists, with affected both the formal standards and the aims of population genetics as a science. Over the course of the 1960s, the mean fitness method withered with the ambition of developing the deterministic theory. The mathematical theory became increasingly complex. Kimura re-focused his modeling work on the theory of random processes; as a result of his computer simulations, Lewontin became the staunchest critic of maximizing principles in evolutionary biology. The mean fitness method then migrated to other research areas, being refashioned and used in evolutionary quantitative genetics and behavioral ecology.     

The simplest formal argument for fitness optimization

The Formal Darwinism Project aims to provide a formal argument linking population genetics to fitness optimization, which of necessity includes defining fitness. This bridges the gulf between those biologists who assume that natural selection leads to something close to fitness optimization and those biologists who believe on theoretical grounds that there is no sense of fitness that can usefully be said to be optimized. The current paper’s main objective is to provide a careful mathematical introduction to the project, and it also reflects on the project’s scope and limitations. The central argument is the proof of close ties between the mathematics of motion, as embodied in the Price equation, and the mathematics of optimization, as represented by optimization programmes. To make these links, a general and abstract model linking genotype, phenotype and number of successful gametes is assumed. The project has begun with simple dynamic models and simple linking models, and its progress will involve more realistic versions of them. The versions given here are fully mathematically rigorous, but elementary enough to serve as an introduction.     

A first formal link between the price equation and an optimization program

The Darwin unification project is pursued. A meta-model encompassing an important class of population genetic models is formed by adding an abstract model of the number of successful gametes to the Price equation under uncertainty. A class of optimization programs are defined to represent the "individual-as-maximizing-agent analogy" in a general way. It is then shown that for each population genetic model there is a corresponding optimization program with which formal links can be established. These links provide a secure logical foundation for the commonplace biological principle that natural selection leads organisms to act as if maximizing their "fitness", provides a definition of "fitness", and clarifies the limitations of that principle. The situations covered do not include frequency dependence or social behaviour, but the approach is capable of extension.     

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.     

Ecological theatre and the evolutionary game: how environmental and demographic factors determine payoffs in evolutionary games

In the standard approach to evolutionary games and replicator dynamics, differences in fitness can be interpreted as an excess from the mean Malthusian growth rate in the population. In the underlying reasoning, related to an analysis of “costs” and “benefits”, there is a silent assumption that fitness can be described in some type of units. However, in most cases these units of measure are not explicitly specified. Then the question arises: are these theories testable? How can we measure “benefit” or “cost”? A natural language, useful for describing and justifying comparisons of strategic “cost” versus “benefits”, is the terminology of demography, because the basic events that shape the outcome of natural selection are births and deaths. In this paper, we present the consequences of an explicit analysis of births and deaths in an evolutionary game theoretic framework. We will investigate different types of mortality pressures, their combinations and the possibility of trade-offs between mortality and fertility. We will show that within this new approach it is possible to model how strictly ecological factors such as density dependence and additive background fitness, which seem neutral in classical theory, can affect the outcomes of the game. We consider the example of the Hawk–Dove game, and show that when reformulated in terms of our new approach new details and new biological predictions are produced.     

Foundations of a mathematical theory of darwinism

This paper pursues the ‘formal darwinism’ project of Grafen, whose aim is to construct formal links between dynamics of gene frequencies and optimization programmes, in very abstract settings with general implications for biologically relevant situations. A major outcome is the definition, within wide assumptions, of the ubiquitous but problematic concept of ‘fitness’. This paper is the first to present the project for mathematicians. Within the framework of overlapping generations in discrete time and no social interactions, the current model shows links between fitness maximization and gene frequency change in a class-structured population, with individual-level uncertainty but no uncertainty in the class projection operator, where individuals are permitted to observe and condition their behaviour on arbitrary parts of the uncertainty. The results hold with arbitrary numbers of loci and alleles, arbitrary dominance and epistasis, and make no assumptions about linkage, linkage disequilibrium or mating system. An explicit derivation is given of Fisher’s Fundamental Theorem of Natural Selection in its full generality.     

Optimization of inclusive fitness

The first fully explicit argument is given that broadly supports a widespread belief among whole-organism biologists that natural selection tends to lead to organisms acting as if maximizing their inclusive fitness. The use of optimization programs permits a clear statement of what this belief should be understood to mean, in contradistinction to the common mathematical presumption that it should be formalized as some kind of Lyapunov or even potential function. The argument reveals new details and uncovers latent assumptions. A very general genetic architecture is allowed, and there is arbitrary uncertainty. However, frequency dependence of fitnesses is not permitted. The logic of inclusive fitness immediately draws together various kinds of intra-genomic conflict, and the concept of 'p-family' is introduced. Inclusive fitness is thus incorporated into the formal Darwinism project, which aims to link the mathematics of motion (difference and differential equations) used to describe gene frequency trajectories with the mathematics of optimization used to describe purpose and design. Important questions remain to be answered in the fundamental theory of inclusive fitness.     

Mort ou persistance du darwinisme ? Regard d’un épistémologue

The article examines why evolutionary biologists have been haunted by the question whether they are “Darwinian” or “non-Darwinian” ever since Darwin's Origin of species. Modern criticisms addressed to Darwinism are classified into two categories: those concerning Darwin's hypothesis of “descent with modification” and those addressed to the hypothesis of natural selection. In both cases, although the particular models that Darwin proposed for these two hypotheses have been significantly revised and expanded, Darwin's general framework has constrained and canalized evolutionary research, in the sense that it has settled an array of possible theoretical choices. Gould's changing attitudes regarding Darwinism is taken as a striking illustration of this interpretation.     

Does evolution lead to maximizing behavior?

A long‐standing question in biology and economics is whether individual organisms evolve to behave as if they were striving to maximize some goal function. We here formalize this “as if” question in a patch‐structured population in which individuals obtain material payoffs from (perhaps very complex multimove) social interactions. These material payoffs determine personal fitness and, ultimately, invasion fitness. We ask whether individuals in uninvadable population states will appear to be maximizing conventional goal functions (with population‐structure coefficients exogenous to the individual's behavior), when what is really being maximized is invasion fitness at the genetic level. We reach two broad conclusions. First, no simple and general individual‐centered goal function emerges from the analysis. This stems from the fact that invasion fitness is a gene‐centered multigenerational measure of evolutionary success. Second, when selection is weak, all multigenerational effects of selection can be summarized in a neutral type‐distribution quantifying identity‐by‐descent between individuals within patches. Individuals then behave as if they were striving to maximize a weighted sum of material payoffs (own and others). At an uninvadable state it is as if individuals would freely choose their actions and play a Nash equilibrium of a game with a goal function that combines self‐interest (own material payoff), group interest (group material payoff if everyone does the same), and local rivalry (material payoff differences).     

The elastic between genes and culture

A society's culture may come to include information tending to lead to fitness-reducing behavior on the part of some or all of its members. This phenomenon results from conflict among factions within each society, from transmitted misinformation (e.g., cupping restores health), from natural and human-caused environmental change so that previously adaptive information becomes maladaptive, and from the long and short-term negative “side effects” of information that may otherwise be fitness enhancing. Because some cultural information may be fitness reducing, we apparently have been selected for individual-level traits that often result in our revising socially transmitted information that might otherwise have maladaptive consequences. Two examples of such traits are adolescent “rebelliousness” and the tendency to learn most readily from those higher than ourselves in status. Such leading-to-culture-revision traits are very imperfect mechanisms, however, so that some likely-to-be maladaptive cultural information, such as medical cupping or denying infants the colostrum, remains part of the culture. It is doubtful, given the structure of modern human populations and the ubiquity of culture change, that such maladaptive socially transmitted information leads to natural selection for genetic “direct biases” against accepting the practices in question.     

What is hierarchical selection?

It has been proposed that natural selection occurs on a hierarchy of levels, of which the organismic level is neither the top nor the bottom. This hypothesis leads to the following practical problem: in general, how does one tell if a given phenomenon is a result of selection on level X or level Y. How does one tell what the units of selection actually are? It is convenient to assume that a unit of selection may be defined as a type of entity for which there exists, among all entities on the same “level” as that entity, an additive component of variance for some specific component F of fitness which does not appear as an additive component of variance in any decomposition of this F among entities at any lower level. But such a definition implicitly assumes that if f(x, y) depends nonadditively on its arguments, there must be interaction between the quantities which x and y represent. This assumption is incorrect. And one cannot avoid this error by speaking of “transformability to additivity” instead of merely “additivity”. A general mathematical formulation of the concepts of interaction and non-interaction is proposed, followed by a correspondingly modified approach to the definition of a unit of selection. The practical difficulty of verifying the presence of hierarchical selection is discussed.     

On the status of mean fitness maximization as a governing principle in evolution

Mean fitness is not always maximized under natural selection. In particular, in two locus models, recombination and epistasis may combine to prevent the operation of a maximizing priciple for mean fitness. If inversion phenomena are considered, however, there exist fully polymorphic equilibria which maximize the mean fitness and moreover the initial progress of an inversion can proceed if and only if it gives rise to an increase in mean fitness. While not applicable to all models, the principle of mean fitness maximization is still useful heuristically.     

Models of the evolution of altruism

There are two ways of calculating the spread of a gene for altruism. One, originally proposed by Hamilton, is to allow for the effects of the gene on the survival and reproduction of collateral relatives of the individual carrying it (i.e., “inclusive fitness”); this leads to the condition k > 1/r for the spread of the gene, where k is a benefit/cost ratio. The other is to count only the direct offspring of a carrier, but to allow for the altruistic acts performed toward the carrier by its relatives (“neighbour modulated fitness” or “personal fitness”). A recent personal fitness model (L. L. Cavalli Sforza and M. W. Feldman, 1978, Theor. Pop. Biol.14, 268–280) analyses parent-offspring and sib-sib altruism and concludes that k > 1/r is applicable only when fitness components are combined additively. The present paper analyses some simple models in which the phenotypic effects are carefully specified. It is concluded that it is sometimes, but not always, appropriate to combine fitness components additively. The relative roles of inclusive and personal fitness models are compared. The former have the virtue of being easier to think about in causal terms; and the latter of incorporating the evolution of altruism into the corpus of population genetics as an example of frequency-dependent selection.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

Differential migration from high fitness demes in the shining fungus beetle, Phalacrus substriatus

Abstract.— Using data from three years (1994–1996), I tested whether differential migration occurs from demes of high mean fitness in the shining fungus beetle, Phalacrus substriatus . The results show evidence for differential migration, thus providing evidence from a natural population for a critical demographic assumption of many interdemic selection models. To predict the evolutionary response to interdemic selection through differential migration, the genetic basis of the variation among demes in mean fitness must be known because the observed patterns could also be explained by some demes having an intrinsically favorable habitat. Thus, the importance of differential migration through interdemic selection in natural populations cannot be unequivocally answered without experiments specifically addressing the question of what causes differences in mean fitness among demes.     

Simplification,Innateness, and the Absorption of Meaning from Context: How Novelty Arises from Gradual Network Evolution

How does new genetic information arise? Traditional thinking holds that mutation happens by accident and then spreads in the population by either natural selection or random genetic drift. There have been at least two fundamental conceptual problems with imagining an alternative. First, it seemed that the only alternative is a mutation that responds “smartly” to the immediate environment; but in complex multicellulars, it is hard to imagine how this could be implemented. Second, if there were mechanisms of mutation that “knew” what genetic changes would be favored in a given environment, this would have only begged the question of how they acquired that particular knowledge to begin with. This paper offers an alternative that avoids these problems. It holds that mutational mechanisms act on information that is in the genome, based on considerations of simplicity, parsimony, elegance, etc. (which are different than fitness considerations). This simplification process, under the performance pressure exerted by selection, not only leads to the improvement of adaptations but also creates elements that have the capacity to serve in new contexts they were not originally selected for. Novelty, then, arises at the system level from emergent interactions between such elements. Thus, mechanistically driven mutation neither requires Lamarckian transmission nor closes the door on novelty, because the changes it implements interact with one another globally in surprising and beneficial ways. Finally, I argue, for example, that genes used together are fused together; that simplification leads to complexity; and that evolution and learning are conceptually linked.     

Plasticity of plant defense and its evolutionary implications in wild populations of Boechera stricta

Phenotypic plasticity is thought to impact evolutionary trajectories by shifting trait values in a direction that is either favored by natural selection (“adaptive” plasticity) or disfavored (“nonadaptive” plasticity). However, it is unclear how commonly each of these types of plasticity occurs in natural populations. To answer this question, we measured glucosinolate defensive chemistry and reproductive fitness in over 1500 individuals of the wild perennial mustard Boechera stricta, planted in four common gardens across central Idaho, United States. Glucosinolate profiles—including total glucosinolate concentration as well as the relative abundances and overall diversity of different compounds—were strongly plastic both among habitats and within habitats. Patterns of glucosinolate plasticity varied greatly among genotypes. Plasticity among sites was predicted to affect fitness in 27.1% of cases; more often than expected by chance, glucosinolate plasticity increased rather than decreased relative fitness. In contrast, we found no evidence for within‐habitat selection on glucosinolate reaction norm slopes (i.e., plasticity along a continuous environmental gradient). Together, our results indicate that glucosinolate plasticity may improve the ability of B. stricta populations to persist after migration to new habitats.