Frequency-dependent viability selection (a single-locus, multi-phenotype model)

Frequency-dependent natural selection models are examined where the viability of an individual in the diploid population is determined by its phenotype and the frequency of other phenotypes present. The equilibria of the multi-phenotypic system are characterized through local mean fitness functions. It is shown that stability can best be analyzed by combining the principles of maximization of population mean fitness with the evolutionary stability conditions that apply when phenotypic fitnesses relative to the genetic constraints are equal.     

Alternative formulations of multilevel selection

Hierarchical expansions of the theory of natural selection exist in two distinct bodies of thought in evolutionary biology, the group selection and the species selection traditions. Both traditions share the point of view that the principles of natural selection apply at levels of biological organization above the level of the individual organism. This leads them both to considermultilevel selection situations, where selection is occurring simultaneously at more than one level. Impeding unification of the theoretical approaches of the multilevel selection traditions are the different goals of investigators in the different subdisciplines and the different types of data potentially available for analysis. We identify two alternative approaches to multilevel situations, which we termmultilevel selection [1] andmultilevel selection [2]. Of interest in the former case are the effects of group membership onindividual fitnesses, and in the latter the tendencies for the groups themselves to go extinct or to found new groups (i.e., group fitnesses). We argue that: neither represents the entire multilevel selection process; both are aspects of any multilevel selection situation; and both are legitimate approaches, suitable for answering different questions. Using this formalism, we show that: multilevel selection [2] does not require emergent group properties in order to provide an explanatory mechanism of evolutionary change; multilevel selection [1] is usually more appropriate for neontological group selection studies; and species selection is most fruitfully considered from the point of view of multilevel selection [2]. Finally we argue that the effect hypothesis of macroevolution, requiring, in selection among species, both the absence of group effects on organismic fitness (multilevel selection [1]), and the direct determination of species fitnesses by those of organisms, is untestable with paleontological data. Furthermore, the conditions for the effect hypothesis to hold are extremely restrictive and unlikely to apply to the vast majority of situations encountered in nature.     

Genetic and strategic models for the evolution of mating systems

Male and female fitnesses in the Shaw-Mohler equation are partitioned into components which putatively determine mating systems. The resultant genetic models provide criteria for evolutionary stable population states and yield strategic models based on maximization principles and fitness sets.     

A self-organizing neural network with a function of associative memory: Feedback-type cognitron

Previously, one of the authors proposed a new hypothesis on the organization of synaptic connections, and constructed a model of self-organizing multi-layered neural network cognitron (Fukushima, 1975). the cognitron consists of a number of neural layers with similar structure connected in a cascade one after another. We have modified the structure of the cognitron, and have developed a new network having an ability of associative memory. The new network, named a feedback-type cognitron, has not only the feedforward connections as in the conventional cognitron, but also modifiable feedback connections from the last-layer cells to the front-layer ones. This network has been simulated on a digital computer. If several stimulus patterns are repeatedly presented to the network, the interconnections between the cells are gradually organized. The feedback connections, as well as the conventional feedforward ones, are self-organized depending on the characteristies of the externally presented stimulus patterns. After adequate number of stimulus presentations, each cell usually acquires the selective responsiveness to one of the stimulus patterns which have been frequently given. That is, every different stimulus pattern becomes to elicit an individual response to the network. After the completion of the self-organization, several stimulus patterns are presented to the network, and the responses are observed. Once a stimulus is given to the network, the signal keeps circulating in the network even after cutting off the stimulus, and the response gradually changes. Even though an imperfect or an ambiguous pattern is presented, the response usually converges to one of the patterns which have been frequently given during the process of self-organization. In some cases, however, a new pattern which has never been presented before, emerges. It is seen that this feedback-type cognitron has characteristics quite similar to some functions of the brain, such as the associative recall of memory, or the creation of a new idea by intuition.     

Estimating relative fitness in asexually reproducing plant pathogen populations

Summary A mathematical model is presented and analysed to find the conditions under which changes in gene frequencies can be used in asexually reproducing populations for estimating fitness of single genes, for example, for estimating the fitnesses of unnecessary virulence genes relative to their corresponding avirulence genes. It is concluded that the underlying distribution of relative fitness of clones (genotypes) has to be unimodal and that many populations consisting of a mixture of distinguishable clones then provide the best experimental data for estimating relative fitness of single genes. An improved statistical test procedure, i.e. generalized logistic regression, is suggested for analysing changes in gene frequencies in population experiments with a mixture of distinguishable clones. A population study of Erysiphe graminis f.sp. hordei (Klug-Andersen 1980) provides data to illustrate the procedure in the case where the population consists of a large number of genotypes. A bimodal distribution of genotypes possessing the virulence gene is indicated here.     

The fitness threshold model: Random environmental change alters adaptive landscapes

We used a probabilistic optimization model to explore the joint evolutionary effects of random phenotypic and environmental variation. Two forms of environmental noise were defined in which the optimal phenotype remained constant but all organisms experienced either the same proportionate or the same absolute fitness gains and losses. There was no evolutionary effect of proportionate fitness fluctuations. In contrast, the optimal genotype varied with absolute fitness fluctuations, despite the environmental effect being phenotype-independent. We refer to such phenotype-independent fluctuation in absolute fitness as the fitness threshold model, because shared fitness effects determine the zero-fitness points (i.e. the baseline) on an intrinsic fitness function. Thus, environmental effects that are unrelated to a focal trait can cause peak shifts in the genetic optimum for the trait. Changes in the fitness threshold not only changed peak locations, but also altered the slopes defining the peaks, and so should alter the rate of evolution towards optima. This model pertains to evolution in any system, unless there is no phenotypic or environmental variance, or the selection function and distribution of phenotypic error assume similar shapes. Our results have many basic and applied implications for topics such as the maintenance of genetic variation, the canalization of development and the management of natural populations.     

Cross entropy minimization in uninvadable states of complex populations

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an order stabilizing effect.     

Unifying life-history analyses for inference of fitness and population growth

The lifetime fitnesses of individuals comprising a population determine its numerical dynamics, and genetic variation in fitness results in evolutionary change. This dual importance of individual fitness is well understood, but empirical fitness records generally violate the assumptions of standard statistical approaches. This problem has undermined comprehensive study of fitness and impeded empirical synthesis of the numerical and genetic dynamics of populations. Recently developed aster models remedy this problem by explicitly modeling the dependence of later-expressed components of fitness (e.g., fecundity) on those expressed earlier (e.g., survival to reproduce). Moreover, aster models employ different sampling distributions for different components of fitness (e.g., binomial for survival over a given interval and Poisson for fecundity). Analysis is done by maximum likelihood, and the resulting distributions for lifetime fitness closely approximate observed data. We illustrate the breadth of aster models' utility with three examples demonstrating estimation of the finite rate of increase, comparison of mean fitness among genotypic groups, and analysis of phenotypic selection. Aster models offer a unified approach to addressing the breadth of questions in evolution and ecology for which life-history data are gathered.     

A self-organizing neural network sharing features of the mammalian visual system

This paper describes a neural network model whose structure is designed to closely fit neuroanatomical and-physiological data, and not to be most suitable for rigorous mathematical analysis.It is shown by computer simulation that a process of self-organization that departs from a fixed retinotopic order at peripheral layers and includes hebbian modifications of synaptic connectivity at higher processing levels leads to a system that is capable of mimicking various functions of visual systems:In the initial state the overall structure of the network is preset, individual connections at higher levels are randomly selected and their strength is initialized with random numbers.For this model the outcome of the self-organization process is determined by the stimulation during the developmental phase. Depending on the type of stimuli used the model can either develop towards a featureselective preprocessor stage in a complex vision system or towards a subsystem for associative recall of abstract patterns.This flexibility supports the hypothesis that the principles embodied are rather universal and can account for the development of various nervous system structures.Presented at teh 9th Cybernetics-Congress, Göttingen, March 1986     

Development of simple fitness landscapes for peptides by artificial neural filter systems

The applicability of artificial neural filter systems as fitness functions for sequence-oriented peptide design was evaluated. Two example applications were selected: classification of dipeptides according to their hydrophobicity and classification of proteolytic cleavage-sites of protein precursor sequences according to their mean hydrophobicities and mean side-chain volumes. The cleavage-sites covered 12 residues. In the dipeptide experiments the objective was to separate a selected set of molecules from all other possible dipeptide sequences. Perceptrons, feedforward networks with one hidden layer, and a hybrid network were applied. The filters were trained by a (1,) evolution strategy. Two types of network units employing either a sigmoidal or a unimodal transfer function were used in the feedforward filters, and their influence on classification was investigated. The two-layer hybrid network employed gaussian activation functions. To analyze classification of the different filter systems, their output was plotted in the two-dimensional sequence space. The diagrams were interpreted as fitness landscapes qualifying the markedness of a characteristic peptide feature which can be used as a guide through sequence space for rational peptide design. It is demonstrated that the applicability of neural filter systems as a heuristic method for sequence optimization depends on both the appropriate network architecture and selection of representative sequence data. The networks with unimodal activation functions and the hybrid networks both led to a number of local optima. However, the hybrid networks produced the best prediction results. In contrast, the filters with sigmoidal activation produced good reclassification results leading to fitness landscapes lacking unreasonable local optima. Similar results were obtained for classification of both dipeptides and cleavage-site sequences.     

Correlated and uncorrelated fitness landscapes and how to tell the difference

The properties of multi-peaked fitness landscapes have attracted attention in a wide variety of fields, including evolutionary biology. However, relaively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's N-k model), that is shown to be a generic AR(1) landscape. We close by discussing how these ideas might be useful in the tuning of combinatorial optimization algorithms, and in modelling in the experimental sciences.     

Chromosome Interactions in DROSOPHILA MELANOGASTER. II. Total Fitness

In a large experiment, using nearly 200 population cages, we have measured the fitness of Drosophila melanogaster homozygous (1) for the second chromosome, (2) for the third chromosome, and (3) for both chromosomes. Twentyfour second chromosomes and 24 third chromosomes sampled from a natural population were tested. The mean fitness of the homozygous flies is 0.081 ± 0.014 for the second chromosome, 0.080 ± 0.017 for the third chromosome, and 0.079 ± 0.024 for both chromosomes simultaneously. Assuming that fitnesses are multiplicative (the additive fitness model makes no sense in the present case because of the large selection coefficients involved), the expected mean fitness of the homozygotes for both chromosomes is 0.0066; their observed fitness is more than ten times greater. Thus, it appears that synergistic interactions between loci are considerable; and that, consequently, the fitness function substantially departs from linearity. Two models are tentatively suggested for the fitness function: a "threshold" model and a "synergistic" model.—The experiments reported here confirm previous results showing that the concealed genetic load present in natural populations of Drosophila is sufficient to account for the selective maintenance of numerous polymorphisms (of the order of 1000).     

Nuclear–mitochondrial epistasis: a gene's eye view of genomic conflict

We use population genetic models to investigate the cooperative and conflicting synergistic fitness effects between genes from the nucleus and the mitochondrion. By varying fitness parameters, we examine the scope for conflict relative to cooperation among genomes and the utility of the “gene's eye view” analytical approach, which is based on the marginal average fitness of specific alleles. Because sexual conflict can maintain polymorphism of mitochondrial haplotypes, we can explore two types of evolutionary conflict (genomic and sexual) with one epistatic model. We find that the nuclear genetic architecture (autosomal, X‐linked, or Z‐linked) and the mating system change the regions of parameter space corresponding to the evolution by sexual and genomic conflict. For all models, regardless of conflict or cooperation, we find that population mean fitness increases monotonically as evolution proceeds. Moreover, we find that the process of gene frequency change with positive, synergistic fitnesses is self‐accelerating, as the success of an allele in one genome or in one sex increases the frequency of the interacting allele upon which its success depends. This results in runaway evolutionary dynamics caused by the positive intergenomic associations generated by selection. An inbreeding mating system tends to further accelerate these runaway dynamics because it maintains favorable host–symbiont or male–female gene combinations. In contrast, where conflict predominates, the success of an allele in one genome or in one sex diminishes the frequency of the corresponding allele in the other, resulting in considerably slower evolutionary dynamics. The rate of change of mean fitness is also much faster with positive, synergistic fitnesses and much slower where conflict is predominant. Consequently, selection rapidly fixes cooperative gene combinations, while leaving behind a slowing evolving residue of conflicting gene combinations at mutation–selection balance. We discuss how an emphasis on marginal fitness averages may obscure the interdependence of allelic fitness across genomes, making the evolutionary trajectories appear independent of one another when they are not.     

The “averaging fallacy” and the levels of selection

This paper compares two well-known arguments in the units of selection literature, one due to , the other due to . Both arguments concern the legitimacy of averaging fitness values across contexts and making inferences about the level of selection on that basis. The first three sections of the paper shows that the two arguments are incompatible if taken at face value, their apparent similarity notwithstanding. If we accept Sober and Lewontin's criterion for when averaging genic fitnesses across diploid genotypes is illegitmate, we cannot accept Sober and Wilson's criterion for when averaging individual fitnesses across groups is illegitimate, and vice versa. The final section suggests a possible way of reconciling the two arguments, by invoking an ambiguity in the concept of genic selection.     

Genetic and phenotypic models of natural selection

The following theorem is proposed: when two phenotypes differ in attributes affecting their relative fitness, selection will cease to cause further evolutionary change when the two phenotypes have the same fitness, provided that certain modes of inheritance apply; in particular, all genotypes specifying the same phenotype must have the same average fitness. If these conditions of “uniform fitness” patterns of inheritance are not met, particular genetic models of natural selection should replace an analysis of phenotypes. If the conditions are met, an analysis of the stationary conditions when the phenotypes have equal fitnesses permits quantitative statements about the outcome of selection without recourse to genetic models. Phenotypic analyses of natural selection are illustrated by models of sex ratios in plants, sexual versus asexual reproduction in plants, and parental investment by animals.     

Maynard Smith,optimization, and evolution

Maynard Smith’s defenses of adaptationism and of the value of optimization theory in evolutionary biology are both criticized. His defense does not adequately respond to the criticism of adaptationism by Gould and Lewontin. It is also argued here that natural selection cannot be interpreted as an optimization process if the objective function to be optimized is either (i) interpretable as a fitness, or (ii) correlated with the mean population fitness. This result holds even if fitnesses are frequency-independent; the problem is further exacerbated in the frequency-dependent context modeled by evolutionary game theory. However, Eshel and Feldman’s new results on “long-term” evolution may provide some hope for the continuing relevance of the game-theoretic framework. These arguments also demonstrate the irrelevance of attempts by Intelligent Design creationists to use computational limits on optimization algorithms as evidence against evolutionary theory. It is pointed out that adaptation, natural selection, and optimization are not equivalent processes in the context of biological evolution. It is a pleasure to dedicate this paper to the memory of John Maynard Smith. Thanks are due to James Justus and Samir Okasha for comments on an earlier draft.     

The immune system as a neural network: a multi-epitope approach.

The term "neural network" has been applied to arrays of simple activation units linked by weighted connections. If the connections are modified according to a defined learning algorithm, such networks can be trained to store and retrieve patterned information. Memories are distributed throughout the network, allowing the network to recall complete patterns from incomplete input (pattern completion). The major biological application of neural network theory to date has been in the neurosciences, but the immune system may represent an alternative organ system in which to search for neural network architecture. Previous applications of parallel distributed processing to idiotype network theory have focused upon the recognition of individual epitopes. We argue here that this approach may be too restrictive, underestimating the power of neural network architecture. We propose that the network stores and retrieves large, complex patterns consisting of multiple epitopes separated in time and space. Such a network would be capable of perceiving an entire bacterium, and of storing the time course of a viral infection. While recognition of solitary epitopes occurs at the cellular level in this model, recognition of structures larger than the width of an antibody binding site takes place at the organ level, via network architecture integration of, i.e. individual epitope responses. The Oudin-Cazenave enigma, the sharing of idiotypic determinants by antibodies directed against distinct regions of the same antigen, suggests that some network level of integration of the individual clonal responses to large antigens does occur. The role of cytokines in prior neural network models of the immune system is unclear. We speculate that cytokines may influence the temperature of the network, such that changes in the cytokine milieu serve to "anneal" the network, allowing it to achieve the optimum steady-state in the shortest period of time.     

What is wrong with absolute individual fitness?

One of the most basic facts about evolution is that fitness is a relative concept. It does not matter how well an organism survives and reproduces, only that it does so better than other organisms bearing alternative traits. Nevertheless, many evolutionary arguments are framed in terms of absolute individual fitness. The absolute fitness criterion (AFC) can be justified in terms of relative fitness only given certain assumptions that are frequently violated in nature. In particular, interactions must occur in groups that are randomly formed and phenotypic variation among groups must be tightly coupled to genetic variation. Complicating the genotype-phenotype relationship can cause phenotypic variation among groups to become nonrandom, even when the groups are randomly formed, favoring traits that do not maximize absolute individual fitness. Complex genotype-phenotype relationships and complex population structures require explicit models of evolutionary change based on relative fitness differences within and among groups.     

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.