Natural selection maximizes Fisher information

In biology, information flows from the environment to the genome by the process of natural selection. However, it has not been clear precisely what sort of information metric properly describes natural selection. Here, I show that Fisher information arises as the intrinsic metric of natural selection and evolutionary dynamics. Maximizing the amount of Fisher information about the environment captured by the population leads to Fisher's fundamental theorem of natural selection, the most profound statement about how natural selection influences evolutionary dynamics. I also show a relation between Fisher information and Shannon information (entropy) that may help to unify the correspondence between information and dynamics. Finally, I discuss possible connections between the fundamental role of Fisher information in statistics, biology and other fields of science.     

Fluctuations in the Size of Isolated Single-Species Populations and Natural Selection

This paper discusses the basic types of dynamical behavior of populations obtained in discrete models, such as monotonous dynamics, stable limited cycles, and chaotic variations. All these modes are shown to have possibly arisen in the evolution of limited populations under the effect of density-independent selection. This effect together with that of density-dependent non-selective factors has been termed F-selection, which is characterized by independence of relative fitnesses from population density, whereas populations may be ecologically limited; in other words, absolute fitnesses prove to be a function of population size. The characteristic of F-selection is to be not sensitive to changes in population size but to lead to fluctuations, that create conditions for achieving density-dependent selection.     

The effect of trait type and strength of selection on heritability and evolvability in an island bird population

The heritability (h²) of fitness traits is often low. Although this has been attributed to directional selection having eroded genetic variation in direct proportion to the strength of selection, heritability does not necessarily reflect a trait's additive genetic variance and evolutionary potential (“evolvability”). Recent studies suggest that the low h² of fitness traits in wild populations is caused not by a paucity of additive genetic variance (V_A) but by greater environmental or nonadditive genetic variance (V_R). We examined the relationship between h² and variance‐standardized selection intensities (i or β_σ), and between evolvability (I_A:V_A divided by squared phenotypic trait mean) and mean‐standardized selection gradients (β_μ). Using 24 years of data from an island population of Savannah sparrows, we show that, across diverse traits, h² declines with the strength of selection, whereas I_A and I_R (V_R divided by squared trait mean) are independent of the strength of selection. Within trait types (morphological, reproductive, life‐history), h², I_A, and I_R are all independent of the strength of selection. This indicates that certain traits have low heritability because of increased residual variance due to the age at which they are expressed or the multiple factors influencing their expression, rather than their association with fitness.     

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.     

Extending the range of additivity in using inclusive fitness

Inclusive fitness is a concept widely utilized by social biologists as the quantity organisms appear designed to maximize. However, inclusive fitness theory has long been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we articulate a set of modeling assumptions that extend the range of scenarios in which inclusive fitness can be applied. We reanalyze recent formal analyses that searched for, but did not find, inclusive fitness maximization. We show (a) that previous models have not used Hamilton''s definition of inclusive fitness, (b) a reinterpretation of Hamilton''s definition that makes it usable in this context, and (c) that under the assumption of probabilistic mixing of phenotypes, inclusive fitness is indeed maximized in these models. We also show how to understand mathematically, and at an individual level, the definition of inclusive fitness, in an explicit population genetic model in which exact additivity is not assumed. We hope that in articulating these modeling assumptions and providing formal support for inclusive fitness maximization, we help bridge the gap between empiricists and theoreticians, which in some ways has been widening, demonstrating to mathematicians why biologists are content to use inclusive fitness, and offering one way to utilize inclusive fitness in general models of social behavior.     

Fisher’s fundamental theorem of inclusive fitness and the change in fitness due to natural selection when conspecifics interact

Competition and cooperation is fundamental to evolution by natural selection, both in animals and plants. Here, I investigate the consequences of such interactions for response in fitness due to natural selection. I provide quantitative genetic expressions for heritable variance and response in fitness due to natural selection when conspecifics interact. Results show that interactions among conspecifics generate extra heritable variance in fitness, and that interacting with kin is the key to evolutionary success because it translates the extra heritable variance into response in fitness. This work also unifies Fisher’s fundamental theorem of natural selection (FTNS) and Hamilton’s inclusive fitness (IF). The FTNS implies that natural selection maximizes fitness, whereas Hamilton proposed maximization of IF. This work shows that the FTNS describes the increase in IF, rather than direct fitness, at a rate equal to the additive genetic variance in fitness. Thus, Hamilton’s IF and Fisher’s FTNS both describe the maximization of IF.     

Natural selection. IV. The Price equation

The Price equation partitions total evolutionary change into two components. The first component provides an abstract expression of natural selection. The second component subsumes all other evolutionary processes, including changes during transmission. The natural selection component is often used in applications. Those applications attract widespread interest for their simplicity of expression and ease of interpretation. Those same applications attract widespread criticism by dropping the second component of evolutionary change and by leaving unspecified the detailed assumptions needed for a complete study of dynamics. Controversies over approximation and dynamics have nothing to do with the Price equation itself, which is simply a mathematical equivalence relation for total evolutionary change expressed in an alternative form. Disagreements about approach have to do with the tension between the relative valuation of abstract versus concrete analyses. The Price equation's greatest value has been on the abstract side, particularly the invariance relations that illuminate the understanding of natural selection. Those abstract insights lay the foundation for applications in terms of kin selection, information theory interpretations of natural selection and partitions of causes by path analysis. I discuss recent critiques of the Price equation by Nowak and van Veelen.     

Joint phenotypes,evolutionary conflict and the fundamental theorem of natural selection

Multiple organisms can sometimes affect a common phenotype. For example, the portion of a leaf eaten by an insect is a joint phenotype of the plant and insect and the amount of food obtained by an offspring can be a joint trait with its mother. Here, I describe the evolution of joint phenotypes in quantitative genetic terms. A joint phenotype for multiple species evolves as the sum of additive genetic variances in each species, weighted by the selection on each species. Selective conflict between the interactants occurs when selection takes opposite signs on the joint phenotype. The mean fitness of a population changes not just through its own genetic variance but also through the genetic variance for its fitness that resides in other species, an update of Fisher''s fundamental theorem of natural selection. Some similar results, using inclusive fitness, apply to within-species interactions. The models provide a framework for understanding evolutionary conflicts at all levels.     

The formal Darwinism project: a mid-term report

For 8 years I have been pursuing in print an ambitious and at times highly technical programme of work, the 'Formal Darwinism Project', whose essence is to underpin and formalize the fitness optimization ideas used by behavioural ecologists, using a new kind of argument linking the mathematics of motion and the mathematics of optimization. The value of the project is to give stronger support to current practices, and at the same time sharpening theoretical ideas and suggesting principled resolutions of some untidy areas, for example, how to define fitness. The aim is also to unify existing free-standing theoretical structures, such as inclusive fitness theory, Evolutionary Stable Strategy (ESS) theory and bet-hedging theory. The 40-year-old misunderstanding over the meaning of fitness optimization between mathematicians and biologists is explained. Most of the elements required for a general theory have now been implemented, but not together in the same framework, and 'general time' remains to be developed and integrated with the other elements to produce a final unified theory of neo-Darwinian natural selection.     

The genetical theory of multilevel selection

The theory of multilevel selection (MLS) is beset with conceptual difficulties. Although it is widely agreed that covariance between group trait and group fitness may arise in the natural world and drive a response to ‘group selection’, ambiguity exists over the precise meaning of group trait and group fitness and as to whether group selection should be defined according to changes in frequencies of different types of individual or different types of group. Moreover, the theory of MLS has failed to properly engage with the problem of class structure, which greatly limits its empirical application to, for example, social insects whose colonies are structured into separate age, sex, caste and ploidy classes. Here, I develop a genetical theory of MLS, to address these problems. I show that taking a genetical approach facilitates a decomposition of group‐level traits – including reproductive success – into the separate contributions made by each constituent individual, even in the context of so‐called emergence. However, I uncover a novel problem with the group‐oriented approach: in many scenarios, it may not be possible to express a meaningful covariance between trait and fitness at the level of the social group, because the group's constituents belong to separate, irreconcilable classes.     

LONG-TERM EXPERIMENTAL EVOLUTION IN ESCHERICHIA COLI. VII. MECHANISMS MAINTAINING GENETIC VARIABILITY WITHIN POPULATIONS

Six replicate populations of the bacterium Escherichia coli were propagated for more than 10,000 generations in a defined environment. We sought to quantify the variation among clones within these populations with respect to their relative fitness, and to evaluate the roles of three distinct population genetic processes in maintaining this variation. On average, a pair of clones from the same population differed from one another in their relative fitness by approximately 4%. This within-population variation was small compared with the average fitness gain relative to the common ancestor, but it was statistically significant. According to one hypothesis, the variation in fitness is transient and reflects the ongoing substitution of beneficial alleles. We used Fisher's fundamental theorem to compare the observed rate of each population's change in mean fitness with the extent of variation for fitness within that population, but we failed to discern any correspondence between these quantities. A second hypothesis supposes that the variation in fitness is maintained by recurrent deleterious mutations that give rise to a mutation-selection balance. To test this hypothesis, we made use of the fact that two of the six replicate populations had evolved mutator phenotypes, which gave them a genomic mutation rate approximately 100-fold higher than that of the other populations. There was a marginally significant correlation between a population's mutation rate and the extent of its within-population variance for fitness, but this correlation was driven by only one population (whereas two of the populations had elevated mutation rates). Under a third hypothesis, this variation is maintained by frequency-dependent selection, whereby genotypes have an advantage when they are rare relative to when they are common. In all six populations, clones were more fit, on average, when they were rare than when they were common, although the magnitude of the advantage when rare was usually small (~1% in five populations and ~5% in the other). These three hypotheses are not mutually exclusive, but frequency-dependent selection appears to be the primary force maintaining the fitness variation within these experimental populations.     

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.     

Adaptation as organism design

The problem of adaptation is to explain the apparent design of organisms. Darwin solved this problem with the theory of natural selection. However, population geneticists, whose responsibility it is to formalize evolutionary theory, have long neglected the link between natural selection and organismal design. Here, I review the major historical developments in theory of organismal adaptation, clarifying what adaptation is and what it is not, and I point out future avenues for research.     

Natural selection. VII. History and interpretation of kin selection theory

Kin selection theory is a kind of causal analysis. The initial form of kin selection ascribed cause to costs, benefits and genetic relatedness. The theory then slowly developed a deeper and more sophisticated approach to partitioning the causes of social evolution. Controversy followed because causal analysis inevitably attracts opposing views. It is always possible to separate total effects into different component causes. Alternative causal schemes emphasize different aspects of a problem, reflecting the distinct goals, interests and biases of different perspectives. For example, group selection is a particular causal scheme with certain advantages and significant limitations. Ultimately, to use kin selection theory to analyse natural patterns and to understand the history of debates over different approaches, one must follow the underlying history of causal analysis. This article describes the history of kin selection theory, with emphasis on how the causal perspective improved through the study of key patterns of natural history, such as dispersal and sex ratio, and through a unified approach to demographic and social processes. Independent historical developments in the multivariate analysis of quantitative traits merged with the causal analysis of social evolution by kin selection.     

Interpretations arising from Wrightian and Malthusian fitness under strong frequency dependent selection

Fitness is the central concept in evolutionary theory. It measures a phenotype's ability to survive and reproduce. There are different ways to represent this measure: Malthusian fitness and Wrightian fitness. One can go back and forth between the two, but when we characterize model properties or interpret data, it can be important to distinguish between them. Here, we discuss a recent experiment to show how the interpretation changes if an alternative definition is used.     

Genetic variance in fitness and its cross-sex covariance predict adaptation during experimental evolution

The additive genetic variation (V_A) of fitness in a population is of particular importance to quantify its adaptive potential and predict its response to rapid environmental change. Recent statistical advances in quantitative genetics and the use of new molecular tools have fostered great interest in estimating fitness V_A in wild populations. However, the value of V_A for fitness in predicting evolutionary changes over several generations remains mostly unknown. In our study, we addressed this question by combining classical quantitative genetics with experimental evolution in the model organism Tribolium castaneum (red flour beetle) in three new environmental conditions (Dry, Hot, Hot-Dry). We tested for potential constraints that might limit adaptation, including environmental and sex genetic antagonisms captured by negative genetic covariance between environments and female and male fitness, respectively. Observed fitness changes after 20 generations mainly matched our predictions. Given that body size is commonly used as a proxy for fitness, we also tested how this trait and its genetic variance (including nonadditive genetic variance) were impacted by environmental stress. In both traits, genetic variances were sex and condition dependent, but they differed in their variance composition, cross-sex and cross-environment genetic covariances, as well as in the environmental impact on V_A.     

Natural selection. VI. Partitioning the information in fitness and characters by path analysis

Three steps aid in the analysis of selection. First, describe phenotypes by their component causes. Components include genes, maternal effects, symbionts and any other predictors of phenotype that are of interest. Second, describe fitness by its component causes, such as an individual's phenotype, its neighbours’ phenotypes, resource availability and so on. Third, put the predictors of phenotype and fitness into an exact equation for evolutionary change, providing a complete expression of selection and other evolutionary processes. The complete expression separates the distinct causal roles of the various hypothesized components of phenotypes and fitness. Traditionally, those components are given by the covariance, variance and regression terms of evolutionary models. I show how to interpret those statistical expressions with respect to information theory. The resulting interpretation allows one to read the fundamental equations of selection and evolution as sentences that express how various causes lead to the accumulation of information by selection and the decay of information by other evolutionary processes. The interpretation in terms of information leads to a deeper understanding of selection and heritability, and a clearer sense of how to formulate causal hypotheses about evolutionary process. Kin selection appears as a particular type of causal analysis that partitions social effects into meaningful components.     

On the logic of Fisherian sexual selection*

In Fisher's model of sexual selection, a female preference for a male trait spreads together with the trait because their genetic bases become correlated. This can be interpreted as a “greenbeard” system: a preference gene, by inducing a female to mate with a trait-bearing male, favors itself because the male is disproportionately likely also to carry the preference gene. Here, we use this logic to argue that Fisherian sexual selection in diploids proceeds via two channels: (i) trait-bearing males are disproportionately the product of matings between preference-bearing mothers and trait-bearing fathers, and thus trait and preference genes are correlated “in trans”; (ii) trait and preference genes come into gametic phase disequilibrium, and thus are correlated “in cis.” Gametic phase disequilibrium is generated by three distinct mechanisms that we identify. The trans channel does not operate when sexual selection is restricted to the haploid phase, and therefore represents a fundamental difference between haploid and diploid models of sexual selection. We show that the cis and trans channels contribute equally to the spread of the preference when recombination between the preference and trait loci is free, but that the trans channel is substantially more important when linkage is tight.     

Rapid evolution of the intersexual genetic correlation for fitness in Drosophila melanogaster

Sexual antagonism (SA) arises when male and female phenotypes are under opposing selection, yet genetically correlated. Until resolved, antagonism limits evolution toward optimal sex‐specific phenotypes. Despite its importance for sex‐specific adaptation and existing theory, the dynamics of SA resolution are not well understood empirically. Here, we present data from Drosophila melanogaster, compatible with a resolution of SA. We compared two independent replicates of the “LH_M” population in which SA had previously been described. Both had been maintained under identical, controlled conditions, and separated for around 200 generations. Although heritabilities of male and female fitness were similar, the intersexual genetic correlation differed significantly, being negative in one replicate (indicating SA) but close to zero in the other. Using population sequencing, we show that phenotypic differences were associated with population divergence in allele frequencies at nonrandom loci across the genome. Large frequency changes were more prevalent in the population without SA and were enriched at loci mapping to genes previously shown to have sexually antagonistic relationships between expression and fitness. Our data suggest that rapid evolution toward SA resolution has occurred in one of the populations and open avenues toward studying the genetics of SA and its resolution.     

Natural selection and the genetics of adaptation in threespine stickleback

Growing knowledge of the molecular basis of adaptation in wild populations is expanding the study of natural selection. We summarize ongoing efforts to infer three aspects of natural selection—mechanism, form and history—from the genetics of adaptive evolution in threespine stickleback that colonized freshwater after the last ice age. We tested a mechanism of selection for reduced bony armour in freshwater by tracking genotype and allele frequency changes at an underlying major locus (Ectodysplasin) in transplanted stickleback populations. We inferred disruptive selection on genotypes at the same locus in a population polymorphic for bony armour. Finally, we compared the distribution of phenotypic effect sizes of genes underlying changes in body shape with that predicted by models of adaptive peak shifts following colonization of freshwater. Studies of the effects of selection on genes complement efforts to identify the molecular basis of adaptive differences, and improve our understanding of phenotypic evolution.