Darwinian fitness and the intensity of natural selection: studies in sensitivity analysis

Darwinian fitness, the capacity of a variant type to establish itself in competition with the resident population, is determined by evolutionary entropy, a measure of the uncertainty in age of the mother of a randomly chosen newborn. This article shows that the intensity of natural selection, as measured by the sensitivity of entropy with respect to changes in the age-specific fecundity and mortality variables, is a convex function of age, decreasing at early and increasing at later ages. We exploit this result to provide quantitative evolutionary explanations of the large variation in survivorship curves observed in natural populations. Previous studies to explain variation in survivorship curves have been based on the proposition that Darwinian fitness is determined by the Malthusian parameter. Hence the intensity of natural selection will be determined by the sensitivity of the Malthusian parameter with respect to changes in the age-specific fecundity and mortality variables. This measure of the selection gradient is known to be a decreasing function of age, with implications which are inconsistent with empirical observations of survivorship curves in human and animal populations. The analysis described in this paper point to the mitigated import of sensitivity studies based on the Malthusian parameter. Our analysis provides theoretical and empirical support for the ecological and evolutionary significance of sensitivity analysis based on entropy, which is the appropriate measure of Darwinian fitness.     

The units of selection and measures of fitness

This paper presents a unified account of the properties of the measures, Malthusian parameter and entropy in predicting evolutionary change in populations of macromolecules, cells and individuals. The Malthusian parameter describes the intrinsic rate of increase of the population. The entropy describes the intrinsic variability in populations: it characterizes the variability in mutation and replication rates in populations of macromolecules; the rate of decay of synchrony in populations of cells; and the degree of iteroparity in populations of individuals. The Malthusian parameter determines ultimate population numbers: under constant environmental conditions, it is the rate of increase when equilibrium conditions are attained. Entropy determines population stability: the gain in the Malthusian parameter due to small fluctuations in the life-cycle variables is determined by entropy. These properties, which are valid for populations of macromolecules, cells and individuals, show that the Malthusian parameter and entropy act as complimentary fitness indices in understanding evolutionary change in populations.     

Mortality plateaus and directionality theory

Recent large scale studies of senescence in animals and humans have revealed mortality rates that levelled off at advanced ages. These empirical findings are now known to be inconsistent with evolutionary theories of senescence based on the Malthusian parameter as a measure of fitness. This article analyses the incidence of mortality plateaus in terms of directionality theory, a new class of models based on evolutionary entropy as a measure of fitness. We show that the intensity of selection, in the context of directionality theory, is a convex function of age, and we invoke this property to predict that in populations evolving under bounded growth constraints, evolutionarily stable mortality patterns will be described by rates which abate with age at extreme ages. The explanatory power of directionality theory, in contrast with the limitations of the Malthusian model, accords with the claim that evolutionary entropy, rather than the Malthusian parameter, constitutes the operationally valid measure of Darwinian fitness.     

Evolutionary entropy predicts the outcome of selection: Competition for resources that vary in abundance and diversity

Competition between individuals for resources which are limited and diverse in composition is the ultimate driving force of evolution. Classical studies of this event contend that the outcome is a deterministic process predicted by the growth rate of the competing types—a tenet called the Malthusian selection principle. Recent studies of competition indicate that the dynamics of selection is a stochastic process, regulated by the population size, the abundance and diversity of the resource, and predicted by evolutionary entropy—a statistical parameter which characterizes the rate at which the population returns to the steady state condition after a random endogenous or exogenous perturbation. This tenet, which we will call the entropic selection principle entails the following relations:

• (a)When resources are constant, limited and diverse, variants with higher entropy will have a selective advantage and increase in frequency.
• (b)When resources undergo large variations in abundance and are singular, variants with lower entropy will have a selective advantage and increase in frequency.

This article delineates the analytic, computational and empirical support for this tenet. We show moreover that the Malthusian selection principle, a cornerstone of classical evolutionary genetics, is the limit, as population size and resource abundance tends to infinity of the entropic selection principle. The Malthusian tenet is an approximation to the entropic selection principle—an approximation whose validity increases with increasing population size and increasing resource abundance. Evolutionary entropy is a generic concept that characterizes the interaction dynamics of metabolic entities at several levels of biological organization: cellular, organismic and ecological. Accordingly, the entropic selection principle represents a general rule for explaining the processes of adaptation and evolution at each of these levels.     

Darwinian fitness, evolutionary entropy and directionality theory

Two recent articles provide computational and empirical validation of the following analytical fact: the outcome of competition between an invading genotype and that of a resident population is determined by the rate at which the population returns to its original size after a random perturbation. This phenomenon can be quantitatively described in terms of the demographic parameter termed "evolutionary entropy", a measure of the variability in the age at which individuals produce offspring and die. The two articles also validate certain predictions of directionality theory, an evolutionary model that integrates demography and ecology with population genetics. In particular, directionality theory predicts that in populations that spend the greater part of their life cycle in the stationary growth phase, evolution will result in an increase in entropy. These species will be described by a late age of sexual maturity, small progeny sets and a broad reproductive time-span. In populations that undergo large fluctuations in size, however, the evolutionary outcome will be different. When the average size is large, evolution will result in a decrease in entropy-these populations will be described by early age of sexual maturity, large numbers of offspring and narrow reproductive span but when the average size is small, the evolutionary outcome will be random and non-directional.     

Directionality theory: a computational study of an entropic principle in evolutionary processes

Analytical studies of evolutionary processes based on the demographic parameter entropy-a measure of the uncertainty in the age of the mother of a randomly chosen newborn-show that evolutionary changes in entropy are contingent on environmental constraints and can be characterized in terms of three tenets: (i) a unidirectional increase in entropy for populations subject to bounded growth constraints; (ii) a unidirectional decrease in entropy for large populations subject to unbounded growth constraints; (iii) random, non-directional change in entropy for small populations subject to unbounded growth constraints. This article aims to assess the robustness of these analytical tenets by computer simulation. The results of the computational study are shown to be consistent with the analytical predictions. Computational analysis, together with complementary empirical studies of evolutionary changes in entropy underscore the universality of the entropic principle as a model of the evolutionary process.     

Evolutionary entropy determines invasion success in emergent epidemics

Background

Standard epidemiological theory claims that in structured populations competition between multiple pathogen strains is a deterministic process which is mediated by the basic reproduction number () of the individual strains. A new theory based on analysis, simulation and empirical study challenges this predictor of success.

Principal Findings

We show that the quantity is a valid predictor in structured populations only when size is infinite. In this article we show that when population size is finite the dynamics of infection by multi-strain pathogens is a stochastic process whose outcome can be predicted by evolutionary entropy, S, an information theoretic measure which describes the uncertainty in the infectious age of an infected parent of a randomly chosen new infective. Evolutionary entropy characterises the demographic stability or robustness of the population of infectives. This statistical parameter determines the duration of infection and thus provides a quantitative index of the pathogenicity of a strain. Standard epidemiological theory based on as a measure of selective advantage is the limit as the population size tends to infinity of the entropic selection theory. The standard model is an approximation to the entropic selection theory whose validity increases with population size.

Conclusion

An epidemiological analysis based on entropy is shown to explain empirical observations regarding the emergence of less pathogenic strains of human influenza during the antigenic drift phase. Furthermore, we exploit the entropy perspective to discuss certain epidemiological patterns of the current H1N1 swine ''flu outbreak.     

A macro-equation of natural selection

The deterministic dynamical theory of biological populations, developed widely on the basis of the classical work of Fisher and Volterra, in most cases deals with characteristics which cannot be measured directly, e.g. frequencies of various genotypes within a population, their fitness values, competition coefficients, etc. Thus, a theory dealing with a small number of simple averaged macro-characteristics, easily accessible to a direct measurement, would be of great importance. The present paper contains an attempt to establish an equation contributing to such a would-be macrotheory. It is a relationship begween the average fitness of a population (the Malthusian growth parameter), the selective delay (a new concept, introduction in section 3) and the entropy of the equilibrium structure which the population tends to under the natural selection process. A possible method of checking the proposed relationship experimentally is indicated.     

Growth rate, population entropy, and perturbation theory

This paper is concerned with the connection between two classes of population variables: measures of population growth rate--the Malthusian parameter, the net reproduction rate, the gross reproduction rate, and the mean life expectancy; and measures of demographic heterogeneity--population entropy. It is shown that the entropy functions predict the response of the growth rate parameters to perturbations in the age-specific fecundity and mortality schedule. These results are invoked to introduce the notion of environmental intensity. The intensity function, expressed in terms of the entropy parameters, is applied to give a comparative study of the effect of environmental factors on the dynamics of Swedish and French populations.     

Competitive fitness in experimental populations of Drosophila willistoni

Experiments are reported in which genetically different strains of Drosophila willistoni compete with D. pseudoobscura. The competition was studied at three temperatures, 20°, 22°, and 25°C. The outcome of the competition depends on the genetic constitution of the competing species, but at 25° and 22°C D. willistoni flies are generally stronger competitors than D. pseudoobscura, while at 20°C D. pseudoobscura generally has a competitive advantage. There is a significant interaction between genotype and temperature; the strain RP3 is the weakest competitor of all D. willistoni strains at 22° and 25°C, but not at 20°C; the strain M18 is the best competitor at 20° and 22°C but not at 25°C.The performance of the four strains of D. willistoni was measured in two more ways. First we estimated their Darwinian fitness relative to other genotypes of the same species. Second, we measured the average population size of each strain in pure culture. There is no significant correlation between population size in pure culture and either competitive fitness or Darwinian fitness. There is, however, a strong positive correlation between Darwinian fitness and interspecific competitive fitness.It is pointed out that natural selection leads to an increase in the average Darwinian fitness of a population but not necessarily to an increase in its adaptedness to the environment. Yet the synthetic theory of evolution assumes that the genes and genotypes favored by natural selection are usually those which increase the adaptedness of their carriers to the environments where they live. The correlation between Darwinian fitness and adaptedness needs to be studied experimentally.This work was supported by NSF grant GB-12562 (International Biological Program), AEC contract AT-(30-1)-3096, and PHS Career Development Award K3GM 37265 to F. J. Ayala. The senior author's stay in New York was financed in part by Research Fellowship 2-12861 from the Panamerican Union.     

Natural Selection and Age-Structured Populations

This paper studies the properties of a new class of demographic parameters for age-structured populations and analyzes the effect of natural selection on these parameters. Two new demographic variables are introduced: the entropy of a population and the reproductive potential. The entropy of a population measures the variability of the contribution of the different age classes to the stationary population. The reproductive potential measures the mean of the contribution of the different age classes to the Malthusian parameter. The Malthusian parameter is precisely the difference between the entropy and the reproductive potential. The effect of these demographic variables on changes in gene frequency is discussed. The concept of entropy of a genotype is introduced and it is shown that in a random mating population in Hardy-Weinberg equilibrium and under slow selection, the rate of change of entropy is equal to the genetic variance in entropy minus the covariance in entropy and reproductive potential. This result is an information theoretic analog of Fisher''s fundamental theorem of natural selection.     

Estimation of genetic purging under competitive conditions

Inbreeding depression for fitness traits is a key issue in evolutionary biology and conservation genetics. The magnitude of inbreeding depression, though, may critically depend on the efficiency of genetic purging, the elimination or recessive deleterious mutations by natural selection after they are exposed by inbreeding. However, the detection and quantification of genetic purging for nonlethal mutations is a rather difficult task. Here, we present two comprehensive sets of experiments with Drosophila aimed at detecting genetic purging in competitive conditions and quantifying its magnitude. We obtain, for the first time in competitive conditions, an estimate for the predictive parameter, the purging coefficient (d), that quantifies the magnitude of genetic purging, either against overall inbreeding depression (d ≈ 0.3), or against the component ascribed to nonlethal alleles (d_NL ≈ 0.2). We find that competitive fitness declines at a high rate when inbreeding increases in the absence of purging. However, in moderate size populations under competitive conditions, inbreeding depression need not be too dramatic in the medium to short term, as the efficiency of purging is also very high. Furthermore, we find that purging occurred under competitive conditions also reduced the inbreeding depression that is expressed in the absence of competition.     

Apparent competition and recovery from infection

We use mathematical models to analyse how the recovery rate from infection influences the fitness of a host in a setting of interspecific competition. We show that sub-optimal immunity against pathogens can be advantageous for the host in the presence of cross-species infection. Weaker immunity allows the parasite to be used as a biological weapon, and this increases the fitness of the host relative to a competitor. A parameter region is observed in which the outcome of competition depends on the initial conditions. We extend this model and consider the dynamics in a spatial setting and find that the outcome depends on the migration rate of the host species. At low migration rates, coexistence of the host species is possible across space. For higher migration rates, the host species characterized by a lower recovery rate can invade the territory of its competitor. Finally, we study these dynamics in an evolutionary setting. Although a lower recovery rate from infection can increase the competitive ability of a species, we find that evolution maximizes the recovery rate and minimizes parasite burden. The models presented are related to the concept of apparent competition, and our results are discussed in relation to both theoretical and empirical studies.     

Effects of competition on the fitness of wild and crop-wild hybrid sunflower from a diversity of wild populations and crop lines

Gene flow between crop fields and wild populations often results in hybrids with reduced fitness compared to their wild counterparts due to characteristics imparted by the crop genome. But the specifics of the evolutionary outcome of crop-wild gene flow may depend on context, varying due to local environmental conditions and genetic variation within and among wild populations and among crop lines. To evaluate context-dependence of fitness of F1 hybrids, sunflower crop lines were crossed with nine wild populations from across the northern United States. These crop-wild hybrids and their wild counterparts were grown under agricultural conditions in the field with and without wheat competition. Hybrids were far less fecund than wild plants, yet more likely to survive to reproduce. There was considerable variability among wild populations for fecundity and the specific crop line used to generate the crop-wild hybrid significantly affected fecundity. The fitness deficit suffered by crop-wild hybrids varied by population, as did the rankings of the crop-wild hybrids from three different crop lines. Wheat competition decreased fecundity and survival considerably and hampered seed production of wild plants more than that of hybrids. Genotype x environment interactions indicated that the response of fitness to competition differed by population. Consequently, the fitness of hybrids relative to wild plants varied considerably among wild populations and was not consistent across environments. Notably, relative fitness of hybrids was greater under competitive conditions. This research is the first study of its kind to demonstrate that the consequences of crop-wild gene flow are context dependent and contingent on the genetics of the specific wild populations and the local biotic and abiotic conditions.     

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.     

A critique of directionality theory

Directionality theory suggests that demographic entropy, defined in a way analogous to thermodynamic entropy, is as important as the Malthusian parameter in determining life history evolution in an age-structured population. In particular, it suggests that entropy should increase in equilibrium species and decrease in opportunistic species. This theory has been applied to explain the evolution of body size and of senescence. It has been claimed recently that this theory has been validated by a simulation study, but it is argued here that this study reveals substantial flaws in directionality theory and that the Malthusian parameter rather than entropy is the appropriate tool in the study of life history evolution.     

Directionality theory: an empirical study of an entropic principle in life-history evolution

Understanding the relationship between ecological constraints and life-history properties constitutes a central problem in evolutionary ecology. Directionality theory, a model of the evolutionary process based on demographic entropy, a measure of the uncertainty in the age of the mother of a randomly chosen newborn, provides an analytical framework for addressing this problem. The theory predicts that in populations that spend the greater part of their evolutionary history in the stationary growth phase (equilibrium species), entropy will increase. Equilibrium species will be characterized by high iteroparity and strong demographic stability. In populations that spend the greater part of their evolutionary history in the exponential growth phase (opportunistic species), entropy will decrease when population size is large, and will undergo random variation when population size is small. Opportunistic species will be characterized by weak iteroparity and weak demographic stability when population size is large, and random variations in these attributes when population size is small. This paper assesses the validity of these predictions by employing a demographic dataset of 66 species of perennial plants. This empirical analysis is consistent with directionality theory and provides support for its significance as an explanatory and predictive model of life-history evolution.     

Malthusian Parameters as Estimators of the Fitness of Microbes: A Cautionary Tale about the Low Side of High Throughput

The maximum exponential growth rate, the Malthusian parameter (MP), is commonly used as a measure of fitness in experimental studies of adaptive evolution and of the effects of antibiotic resistance and other genes on the fitness of planktonic microbes. Thanks to automated, multi-well optical density plate readers and computers, with little hands-on effort investigators can readily obtain hundreds of estimates of MPs in less than a day. Here we compare estimates of the relative fitness of antibiotic susceptible and resistant strains of E. coli, Pseudomonas aeruginosa and Staphylococcus aureus based on MP data obtained with automated multi-well plate readers with the results from pairwise competition experiments. This leads us to question the reliability of estimates of MP obtained with these high throughput devices and the utility of these estimates of the maximum growth rates to detect fitness differences.     

A COEVOLUTIONARY ISOMORPHISM APPLIED TO LABORATORY STUDIES OF COMPETITION

Some empirical consequences of an isomorphism between the Lotka-Volterra competitive model and a coevolutionary competitive model are developed. In both the Lotka-Volterra and coevolutionary models, four competitive outcomes are possible: 1) species one wins, 2) species two wins, 3) indeterminate outcome, and 4) stable coexistence. These two models are isomorphic in the sense that the inequalities associated with a particular competitive outcome of the Lotka-Volterra model correspond in a one-to-one manner with similar inequalities associated with the same competitive outcome of the coevolutionary model. The inequalities of the Lotka-Volterra model involve the competition coefficients themselves, while the inequalities of the coevolutionary model involve the genetic variances and covariances of the competition coefficients. The isomorphism suggests some alternative interpretations of the results of classical laboratory studies of competition. The Lotka-Volterra (or ecological) hypotheses postulate that the competition coefficients are constant and that genetic considerations play no role in determining the competitive outcome. By contrast, the evolutionary hypotheses derived from the coevolutionary model postulate that the competition coefficients are variables and that the genetic variances and covariances of the competition coefficients determine the competitive outcome. The isomorphism is applied to competitive exclusion and coexistence, and to competitive indeterminacy in Tribolium. In particular, the evolutionary hypotheses isomorphic to the two classical explanations of competitive indeterminacy, the demographic stochasticity and genetic founder effect hypotheses, are constructed. The theory developed here and in a previous paper (Pease, 1984) provides one perspective on the relation among the Lotka-Volterra competition theory, quantitative genetics, competitive exclusion, the reversal of competitive dominance, coexistence, competitive indeterminacy in Tribolium, and experiments investigating the relation between genetic variability and the rate of evolution of fitness.