Population genetics and sociobiology: conflicting views of evolution

This article explores the tension between the population genetics and sociobiological approaches to the study of evolution. Whereas population geneticists, like Stanford's Marc Feldman, insist that the genetic complexities of organisms cannot be overlooked, sociobiologists (many of whom now prefer to call themselves "behavioral ecologists") rely on optimization models that are based on the simplest possible genetics.These optimization approaches have their roots in the classical result known as the fundamental theorem of natural selection, formulated by R. A. Fisher in 1930. From the start there was great uncertainty over the proper interpretation of Fisher's theorem, which became confused with Sewall Wright's immensely influential adaptive landscape concept. In the 1960s, a new generation of mathematical biologists proved that Fisher's theorem did not hold when fitness depended on more than one locus. Similar reasoning was used to attack W. D. Hamilton's inclusive fitness theory. A new theory, known as the theory of long-term evolution, attempts to reconcile the rigorous population genetics approach with the long-standing sociobiological view that natural selection acts to increase the fitness of organisms.     

An optimizing principle of natural selection in evolutionary population genetics

This paper brings together two themes in evolutionary population genetics theory. The first concerns Fisher's Fundamental Theorem of Natural Selection: a recent interpretation of this theorem claims that it is an exact result, relating to the so-called "partial" increase in mean fitness. The second theme concerns the desire to find an optimality principle in genetic evolution. Such a principle is found here: of all gene frequency changes which lead to the same partial increase in mean fitness as the natural selection gene frequency changes, the natural selection values minimize a generalized distance measure between parent and daughter generation gene frequency values.     

Fundamental theorem of natural selection and frequency-dependent selection: analysis of the matrix game diploid model

Following Ewens' interpretation about Fisher's fundamental theorem of natural selection, the matrix game model for diploid populations undergoing non-overlapping, discrete generations is investigated. The total genetic variance is decomposed and it is shown that the partial change in the mean fitness, which is equal to the additive genetic variance over the mean fitness, can be thought of as a change due only to the partial changes in the phenotypic frequencies.     

Population genetics from an information perspective

Some basic effects of population genetics are derived governing the occurrences of alleles A(i)and genotypes A(i)A(j)among its members. A principle of extreme physical information (EPI) is used. These effects are (1) the equation of genetic change, (2) Fisher's theorem of partial change, (3) a new uncertainty principle, and (4) the monotonic decrease of Fisher information with time, indicating increased disorder for the population. General conditions of population change are allowed: fitness coefficients w(ij)generally changing with time [except in effect (2)], population randomly or non-randomly mating, and a general number of loci present within each chromosome. EPI is a practical tool for deriving probability laws. It is an outgrowth of a physical process that occurs during any act of measurement. Here the measurement is the random observation of a genotype A(i)A(j). This observation is to be used to estimate the time of the observation, called "evolutionary time". The measurement activity incurs errors in the estimated observation time and fitness value of the observed genotype. By the Cramer-Rao inequality, the product of the two uncertainties must exceed unity [effect (3)]. The Fisher information I in data space is postulated to originate in the space of the genotype where it had some generally larger value J. The EPI principle extremizes the loss of information (I--J) with I=1/2 J. The solution gives rise to effects (1) and (2). Finally, it is shown that effect (4) holds when the population approaches an equilibrium state, e.g. for time values greater than a threshold if fitness coefficients w(ij)are constant. EPI provides a common framework for deriving physical laws and laws of population genetics. The new effects (3) and (4) are confirmed through computer simulation.     

Fisher'S geometrical model of fitness landscape and variance in fitness within a changing environment

The fitness of an individual can be simply defined as the number of its offspring in the next generation. However, it is not well understood how selection on the phenotype determines fitness. In accordance with Fisher's fundamental theorem, fitness should have no or very little genetic variance, whereas empirical data suggest that is not the case. To bridge these knowledge gaps, we follow Fisher's geometrical model and assume that fitness is determined by multivariate stabilizing selection toward an optimum that may vary among generations. We assume random mating, free recombination, additive genes, and uncorrelated stabilizing selection and mutational effects on traits. In a constant environment, we find that genetic variance in fitness under mutation-selection balance is a U-shaped function of the number of traits (i.e., of the so-called "organismal complexity"). Because the variance can be high if the organism is of either low or high complexity, this suggests that complexity has little direct costs. Under a temporally varying optimum, genetic variance increases relative to a constant optimum and increasingly so when the mutation rate is small. Therefore, mutation and changing environment together can maintain high genetic variance. These results therefore lend support to Fisher's geometric model of a fitness landscape.     

What was Fisher's fundamental theorem of natural selection and what was it for?

Fisher's 'fundamental theorem of natural selection' is notoriously abstract, and, no less notoriously, many take it to be false. In this paper, I explicate the theorem, examine the role that it played in Fisher's general project for biology, and analyze why it was so very fundamental for Fisher. I defend and Lessard (1997) in the view that the theorem is in fact a true theorem if, as Fisher claimed, 'the terms employed' are 'used strictly as defined' (1930, p. 38). Finally, I explain the role that projects such as Fisher's play in the progress of scientific inquiry.     

Monotonic Change of the Mean Phenotype in Two-Locus Models

It is shown that the mean phenotype monotonically approaches the optimum in a class of symmetric, two-locus, two-allele models with stabilizing selection. In this model, mean fitness does not change monotonically. Thus, Fisher's fundamental theorem does not hold, even though another quantity of evolutionary interest, the mean phenotype, can be shown to change monotonically. Using this quantity, it is proven that global stability results for this model.     

Fisher's fundamental theorem of natural selection

Fisher's Fundamental Theorem of natural selection is one of the most widely cited theories in evolutionary biology. Yet it has been argued that the standard interpretation of the theorem is very different from what Fisher meant to say. What Fisher really meant can be illustrated by looking in a new way at a recent model for the evolution of clutch size. Why Fisher was misunderstood depends, in part, on the contrasting views of evolution promoted by Fisher and Wright.     

The fundamental theorem of natural selection

R. A. Fisher's Fundamental Theorem of Natural Selection states that the rate of increase in the mean fitness of a population ascribable to gene-frequency changes is exactly equal to the additive genetic variance in fitness. It has been widely misunderstood, though clarification has gradually come about particularly through the work of G. R. Price, W. J. Ewens, and S. Lessard. Building on their interpretations we here explain the approach adopted by Fisher (1941), devising a figure as an aid to understanding this important paper.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

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.     

Adaptation and the cost of complexity

Abstract.— Adaptation is characterized by the movement of a population toward a many-character optimum, movement that results in an increase in fitness. Here I calculate the rate at which fitness increases during adaptation and describe the curve giving fitness versus time as a population approaches an optimum in Fisher's model of adaptation. The results identify several factors affecting the speed of adaptation. One of the most important is organismal complexity—complex organisms adapt more slowly than simple ones when using mutations of the same phenotypic size. Thus, as Fisher foresaw, organisms pay a kind of cost of complexity. However, the magnitude of this cost is considerably larger than Fisher's analysis suggested. Indeed the rate of adaptation declines at least as fast as n^-1 , where n is the number of independent characters or dimensions comprising an organism. The present results also suggest that one can define an effective number of dimensions characterizing an adapting species.     

A simple completion of Fisher’s fundamental theorem of natural selection

Fisher''s fundamental theorem of natural selection shows that the part of the rate of change of mean fitness that is due to natural selection equals the additive genetic variance in fitness. Fisher embedded this result in a model of total fitness, adding terms for deterioration of the environment and density dependence. Here, a quantitative genetic version of this neglected model is derived that relaxes its assumptions that the additive genetic variance in fitness and the rate of deterioration of the environment do not change over time, allows population size to vary, and includes an input of mutational variance. The resulting formula for total rate of change in mean fitness contains two terms more than Fisher''s original, representing the effects of stabilizing selection, on the one hand, and of mutational variance, on the other, making clear for the first time that the fundamental theorem deals only with natural selection that is directional (as opposed to stabilizing) on the underlying traits. In this model, the total (rather than just the additive) genetic variance increases mean fitness. The unstructured population allows an explanation of Fisher''s concept of fitness as simply birth rate minus mortality rate, and building up to the definition in structured populations.     

The stochastic edge in adaptive evolution

In a recent article, Desai and Fisher proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time tau until a new class of mutants has been established and interpret 1/tau as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine tau: Desai and Fisher back extrapolate the value of tau from the best-fit class's exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows nonexponentially during the relevant time interval. Here, we substantially extend Desai and Fisher's analysis of the stochastic edge. We show that we can apply Desai and Fisher's method to high speeds by either exponentially back extrapolating from finite time or using a nonexponential back extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al.) and agree well with numerical simulations.     

Adaptation,fitness and the selection-optimality links

We critically examine a number of aspects of Grafen’s ‘formal Darwinism’ project. We argue that Grafen’s ‘selection-optimality’ links do not quite succeed in vindicating the working assumption made by behavioural ecologists and others—that selection will lead organisms to exhibit adaptive behaviour—since these links hold true even in the presence of strong genetic and developmental constraints. However we suggest that the selection-optimality links can profitably be viewed as constituting an axiomatic theory of fitness. Finally, we compare Grafen’s project with Fisher’s ‘fundamental theorem of natural selection’, and we speculate about whether Grafen’s results can be extended to a game-theoretic setting.     

Fisher's contributions to genetics and heredity, with special emphasis on the Gregor Mendel controversy

R. A. Fisher is widely respected for his contributions to both statistics and genetics. For instance, his 1930 text on The Genetical Theory of Natural Selection remains a watershed contribution in that area. Fisher's subsequent research led him to study the work of (Johann) Gregor Mendel, the 19th century monk who first developed the basic principles of heredity with experiments on garden peas. In examining Mendel's original 1865 article, Fisher noted that the conformity between Mendel's reported and proposed (theoretical) ratios of segregating individuals was unusually good, "too good" perhaps. The resulting controversy as to whether Mendel "cooked" his data for presentation has continued to the current day. This review highlights Fisher's most salient points as regards Mendel's "too good" fit, within the context of Fisher's extensive contributions to the development of genetical and evolutionary theory.     

Secondary theorem of natural selection in biocultural populations.

The "Secondary Theorem of Natural Selection," an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.     

The heritability of fitness: Bad news for 'good genes'?

Some models of sexual selection depend on a female preference for 'good genes': females choose conspicuous males as these are advertising their possession of genes for fitness characters which can be inherited by their offspring. In contrast, Fisher's fundamental theorem of natural selection - which underlies much of population genetics theory - predicts that in a population at equilibrium there can be no additive genetic variation in fitness. Recent work on collared flycatchers in the wild shows that characters influencing fitness do indeed have a relatively low heritability. However, other studies of the inheritance of fitness in the laboratory suggest that under some circumstances a population may retain considerable genetic diversity for fitness characters. Genetically based female choice might hence have the potential to control the evolution of male sexual ornaments. More work on natural populations is needed; and birds may be a good place to start looking.     

Statistics is not enough: revisiting Ronald A. Fisher's critique (1936) of Mendel's experimental results (1866)

This paper is concerned with the role of rational belief change theory in the philosophical understanding of experimental error. Today, philosophers seek insight about error in the investigation of specific experiments, rather than in general theories. Nevertheless, rational belief change theory adds to our understanding of just such cases: R. A. Fisher's criticism of Mendel's experiments being a case in point. After an historical introduction, the main part of this paper investigates Fisher's paper from the point of view of rational belief change theory: what changes of belief about Mendel's experiment does Fisher go through and with what justification. It leads to surprising insights about what Fisher had done right and wrong, and, more generally, about the limits of statistical methods in detecting error.