Fisher's Theorem Research Articles

Back to the fundamentals: a reply to Basener and Sanford 2018.

Fisher's fundamental theorem of natural selection has haunted theoretical population genetic literature since it was proposed in 1930, leading to numerous interpretations. Most of the confusion stemmed from Fisher's own obscure presentation. By the 1970s, a clearer view of Fisher's theorem had been achieved and it was found that, regardless of its utility or significance, it represents a general theorem of evolutionary biology. Basener and Sanford (J Math Biol 76:1589-1622, 2018) writing in JOMB, however, paint a different picture of the fundamental theorem as one hindered by its assumptions and incomplete due to its failure to explicitly incorporate mutational effects. They argue that Fisher saw his theorem as a "mathematical proof of Darwinian evolution". In this reply, we show that, contrary to Basener and Sanford, Fisher's theorem is a general theorem that applies to any evolving population, and that, far from their assertion that it needed to be expanded, the theorem already implicitly incorporates ancestor-descendant variation. We also show that their numerical simulations produce unrealistic results. Lastly, we argue that Basener and Sanford's motivations were in undermining not merely Fisher's theorem, but the concept of universal common descent itself.

Fisher's Separation Theorem and its Limitations

When businesses make investment decisions, they often evaluate the Net Present Value (NPV) of investment projects. For investors, a project is deemed worthwhile if its NPV is greater than zero. This phenomenon can be attributed to the Fisher Separation Theorem, which asserts that in perfectly efficient capital markets without transaction costs, a single interest rate prevails. In this context, investors' production choices adhere strictly to objective market regulations aimed at maximizing attainable wealth, entirely detached from individual subjective preferences. This, in turn, enables both borrowers and lenders to base their consumption and investment decisions on the prevailing market interest rate, thereby encouraging a clear distinction between investment and financing choices. Therefore, a company's investment decisions can be discussed separately from its financing decisions. So, the enterprise's investment decision and financing decision can be discussed separately. When enterprises invest, enterprises do not have to consider the subjective consumption preferences of different investors, because investors' consumption preferences can be satisfied accordingly through perfect capital market operation. Because of Fisher's Separation Theorem, which requires the market to be transaction cost-free, it also has some limitations.

Quantifying evolution by natural selection

The theme of this paper is that Fisher's “Fundamental Theorem of Natural Selection” has various deficiencies as a quantification of the effect of natural selection in a Mendelian population which are not shared by a new different theorem described in this paper. The deficiencies in Fisher's theorem are listed in this paper. The new theorem focuses on the implications of changes in gene frequencies under natural selection and not, as does the Fundamental Theorem of Natural Selection, on changes in mean population fitness. Whereas the algebra in the new theorem corresponds in places to that in the Fundamental Theorem of Natural Selection, the approach, perspective and conclusion of the new theorem are different from those of the Fundamental Theorem of Natural Selection.

Fundamental Theorems of Evolution.

Evolutionary biology is undergirded by an extensive and impressive set of mathematical models. Yet only one result, Fisher's theorem about selection and fitness, is generally accorded the status of a fundamental theorem. I argue that although its fundamental status is justified by its simplicity and scope, there are additional results that seem similarly fundamental. I suggest that the most fundamental theorem of evolution is the Price equation, both because of its simplicity and broad scope and because it can be used to derive four other familiar results that are similarly fundamental: Fisher's average-excess equation, Robertson's secondary theorem of natural selection, the breeder's equation, and Fisher's fundamental theorem. These derivations clarify both the relationships behind these results and their assumptions. Slightly less fundamental results include those for multivariate evolution and social selection. A key feature of fundamental theorems is that they have great simplicity and scope, which are often achieved by sacrificing perfect accuracy. Quantitative genetics has been more productive of fundamental theorems than population genetics, probably because its empirical focus on unknown genotypes freed it from the tyranny of detail and allowed it to focus on general issues.

Penalized maximum likelihood estimation and effective dimension

Cet article generalise certains resultats statistiques importants dont le Theoreme de Fisher et le phenomene de Wilks a l’estimation du maximum de vraisemblance penalisee de facon quadratique. Il apparait que des developpements precis pour l’EMV penalisee $\widetilde{\boldsymbol{\theta}}_{G}$ et le maximum de vraisemblance penalise peuvent etre obtenus sans arguments asymptotiques, les resultats reposent alors seulement sur la regularite et les proprietes de la fonction de log-vraisemblance. L’erreur d’estimation est specifiee en fonction de la dimension effective $\mathtt{p}_{G}$ de l’ensemble des parametres qui peut etre beaucoup plus petite que la veritable dimension et permet ainsi de considerer un cas infini dimensionnel. Dans le cas i.i.d., le developpement de Fisher pour l’EMV penalisee peut etre etabli sous la contrainte « $\mathtt{p}_{G}^{2}/n$ est petit » tandis que le reste dans le resultat de Wilks est d’ordre $\sqrt{\mathtt{p}_{G}^{3}/n}$.

Biodiversity and models of evolution

Summary. The paper discusses the evolutionary impact of biodiversity, the backbone of noosphere, which status has been fixed by a UN convention. The examples and role of such diversity are considered the various levels of life arrangement. On the level of standalone organisms, the diversity in question manifests itself in the differentiation and separation of the key physiologic functions which significantly broaden the eco-niche for the species with the consummate type of such separation. However, the organismic level of biodiversity does not work for building any developmental models since the starting point of genetic inheritance and variability processes emerges on the minimum structural unit of the living world only, i.e. the population. It is noted that the sufficient gene pool for species development may accumulate in fairly large populations only, where the general rate of mutation does not yield to the rate of ambient variations. The paper shows that the known formal models of species development based on the Fisher theorem about the impact of genodispersion on species adjustment are not in keeping with the actual existence of the species due to the conventionally finite and steady number of genotypes within a population. On the ecosystem level of life arrangement, the key role pertains to the taxonomic diversity supporting the continuous food chain in the system against any adverse developmental conditions of certain taxons. Also, the progressive evolution of an ecosystem is largely stabilized by its multilayer hierarchic structure and the closed circle of matter and energy. The developmental system models based on the Lotka-Volterra equations describing the interaction of the open-loop ecosystem elements only insufficiently represent the position of biodiversity in the evolutionary processes. The paper lays down the requirements to such models which take into account the mass balance within a system; its trophic structure; the accumulative (buffering) properties of each ecosystem layer; the multiple interconnections among the ecosystem species; and the restricting impact of the intraspecific and interspecific competition. From this perspective, the use of bio-cybernetic concepts and mathematic tools looks quite promising.

Extension of the Fisher theorem

Extension of the Fisher theorem

Extension of the Fisher theorem

Extension of the Fisher theorem

The Prediction of the Expected Current Selection Coefficient of Single Nucleotide Polymorphism Associated with Holstein Milk Yield, Fat and Protein Contents.

Milk-related traits (milk yield, fat and protein) have been crucial to selection of Holstein. It is essential to find the current selection trends of Holstein. Despite this, uncovering the current trends of selection have been ignored in previous studies. We suggest a new formula to detect the current selection trends based on single nucleotide polymorphisms (SNP). This suggestion is based on the best linear unbiased prediction (BLUP) and the Fisher’s fundamental theorem of natural selection both of which are trait-dependent. Fisher’s theorem links the additive genetic variance to the selection coefficient. For Holstein milk production traits, we estimated the additive genetic variance using SNP effect from BLUP and selection coefficients based on genetic variance to search highly selective SNPs. Through these processes, we identified significantly selective SNPs. The number of genes containing highly selective SNPs with p-value <0.01 (nearly top 1% SNPs) in all traits and p-value <0.001 (nearly top 0.1%) in any traits was 14. They are phosphodiesterase 4B (PDE4B), serine/threonine kinase 40 (STK40), collagen, type XI, alpha 1 (COL11A1), ephrin-A1 (EFNA1), netrin 4 (NTN4), neuron specific gene family member 1 (NSG1), estrogen receptor 1 (ESR1), neurexin 3 (NRXN3), spectrin, beta, non-erythrocytic 1 (SPTBN1), ADP-ribosylation factor interacting protein 1 (ARFIP1), mutL homolog 1 (MLH1), transmembrane channel-like 7 (TMC7), carboxypeptidase X, member 2 (CPXM2) and ADAM metallopeptidase domain 12 (ADAM12). These genes may be important for future artificial selection trends. Also, we found that the SNP effect predicted from BLUP was the key factor to determine the expected current selection coefficient of SNP. Under Hardy-Weinberg equilibrium of SNP markers in current generation, the selection coefficient is equivalent to 2*SNP effect.

Finite sample behavior of a sieve profile estimator in the single index model

We apply the results of Andresen et. al. (2014) on finite sample properties of sieve M-estimators and Andresen et. al. (2015) on the convergence of an alternating maximization procedure to analyse a sieve profile maximization estimator in the single index model with linear index function. The link function is approximated with $C^{3}$-Daubechies-wavelets with compact support. We derive results like Wilks phenomenon and Fisher Theorem in a finite sample setup even when the model is miss-specified. Furthermore we show that an alternating maximization procedure converges to the global maximizer and we assess the performance of Friedman’s projection pursuit procedure. The approach is based on showing that the conditions of Andresen et. al. (2014) and (2015) can be satisfied under a set of mild regularity and moment conditions on the link function, the regressors and the additive noise. The results allow to construct non-asymptotic confidence sets and to derive asymptotic bounds for the estimator as corollaries.

The Arc‐Weighted Version of the Second Neighborhood Conjecture

AbstractSeymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher . One extension of the conjecture that has been used by several researchers is to consider vertex‐weighted digraphs. In this article we introduce a version of the conjecture for arc‐weighted digraphs. We prove the conjecture in the special case of arc‐weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.

Unified framework for quasispecies evolution and stochastic quantization

In this paper we provide a unified framework for quasispecies evolution and stochastic quantization. We map the biological evolution described by the quasispecies equation to the stochastic dynamics of an ensemble of particles undergoing a creation-annihilation process. We show that this mapping identifies a natural decomposition of the probability that an individual has a certain genotype into eigenfunctions of the evolutionary operator. This alternative approach to study the quasispecies equation allows for a generalization of the Fisher theorem equivalent to the Price equation. According to this relation the average fitness of an asexual population increases with time proportional to the variance of the eigenvalues of the evolutionary operator. Moreover, from the present alternative formulation of stochastic quantization a novel scenario emerges to be compared with existing approaches. The evolution of an ensemble of particles undergoing diffusion and a creation-annihilation process is parametrized by a variable β that we call the inverse temperature of the stochastic dynamics. We find that the evolution equation at high temperatures is simply related to the Schrödinger equation, but at low temperature it strongly deviates from it. In the presence of additional noise in scattering processes between the particles, the evolution reaches a steady state described by the Bose-Einstein statistics.

The action of evolutionary forces on metric traits

Fisher's theorem of natural selection implies that the population genetic variance of quasi-neutral traits should be mostly additive. In the case of fitness component traits, however, that variance would be characterised by a substantial contribution from non-additive loci. In parallel, Robertson's theorem states that selection will change the population mean of a trait proportionally to the magnitude of the genetic correlation between that trait and fitness, which should be weak for quasi-neutral traits or strong for the mean fitness components. Drosophila data from inbreeding and artificial selection experiments are discussed within that theoretical framework. In addition, the process of regeneration by mutation of the genetic variance of a quasi-neutral trait (abdominal bristle number) in a Drosophila population initially homozygous at all loci has been analysed. After 485 generations of mutation accumulation, the levels of additive variance found in this population closely approached those commonly observed in laboratory populations. Furthermore, these values, together with previously reported estimates for natural populations, could be jointly explained by a model assuming weak causal stabilising selection.

Theoretical Considerations of Fisher’s Theorem in Small At-Risk Populations

Small populations at risk of extinction due to threshold or competitive exclusion have a vested interest in rapid growth. Taking chance of survival into account for diploid populations, Fisher's theorem predicts a sex ratio favoring females under certain circumstances. Theoretical consideration of competitive exclusion shows such circumstances are reasonable under accepted modeling assumptions.

A Case Study of the Dynamics of In Vitro DNA Evolution Under Constant Selection Pressure

Although thousands of in vitro selection and evolution experiments have been performed to seek different types of targets, most of them have only inspected the terminal evolutionary pool for patterns. In addition, to rapidly obtain the most favorable target, many experiments have been carried out under increasing selection pressure. However, increasing selection pressure seldom occurs in natural evolution. We studied the dynamic features of DNA in vitro evolution in the presence of the Mnt repressor under sequential constant selection pressure. When evolving under a constant pressure from an initial random pool of DNA, our system showed a clear, sharp, and reproducible crossover from a random population to an advantageous population (higher binding affinities of DNA sequences to the Mnt repressor). This crossover occurs after a long latent period during which there are no obvious changes in the population phenotype. We demonstrated that the existence of the crossover is caused by a significant sequence-nonspecific binding in the repressor-DNA system. After the crossover, the population settled in a stationary distribution of genotypes, which responded immediately to a subsequent sudden increase in selection pressure. We also experimentally tested the linear correlation between the evolution speed and sequence diversity (Fisher's theorem) in our system.

Incremental parameter evaluation from incomplete data with application to the population pharmacology of anticoagulants

We develop a method for parameter evaluation from incomplete data. Improved estimates of the desired parameters are evaluated step by step, from experiment to experiment by using both Bayesian and informational methods. We make dynamical, improved predictions while the experiments are still going on and keep and interpret information about local fluctuations, which is lost on applying global techniques. The input of information in small packets leads to semi-analytic methods for data processing. An evolution criterion for parameter evaluation, similar to Fisher's theorem of population selection, is derived. We develop direct processing methods, which can be applied to low dimensional systems, semi-analytic methods based on direct or double logarithmic phase expansions, steepest descent approaches, variation and perturbation methods. The techniques are illustrated by developing a method of long-term planning of treatments with oral anticoagulants based on limited clinical data. The efficiency of treatment by oral anticoagulants depends strongly on various anthropometric and genotypic factors, which lead to large variations of the clinical response. We use the clinical data, which accumulates from medical consultations, for extracting improved, incremental information about the statistical properties of the kinetic and anthropometric parameters for a given patient, which in turn is used for making repeated, improved clinical predictions as the treatment proceeds.

The Price equation

Following the development of the Price equation, Price went on to make two other major contributions to evolutionary theory. The first of these was to formally prove and provide an interpretation for R.A. Fisher's fundamental theorem of natural selection, a mathematical result that had perplexed population geneticists for decades. The theorem states that the change in the mean fitness of the population under the action of natural selection is proportional to the variance in fitness, and Fisher claimed that it captured the directional, improving action of natural selection as the builder of organismal adaptations. Price explained that Fisher's theorem was a partial result, a description of the action of the natural selection effect with all other evolutionary effects stripped away, and the theorem is easily proven using the Price equation:(3)Price's final contribution was the first explicit application of game theory to evolutionary biology, in a seminal article co-authored with John Maynard Smith entitled The Logic of Animal Conflict. This is widely regarded as one of the most important developments of evolutionary theory, and has launched a highly successful programme of research. It is Maynard Smith who is usually attributed with this breakthrough, and indeed he played the major role in its development. But the idea originated with Price, in an unpublished manuscript that Maynard Smith had reviewed for Nature. Maynard Smith later explained that “Dr Price is better at having ideas than at publishing them. The best I can do therefore is to acknowledge that if there is anything in the idea the credit should go to Dr Price and not to me”.Price's inability to focus on publishing his theoretical insights was due to a sudden religious experience in the summer of 1970 and a shift of priorities in his life. It is not known what in particular led Price, formerly a hardline atheist, down this avenue, although he did mention to Hamilton that a series of coincidences had forced him to conclude that God existed. He came to regard his equation as a gift from God and, taking a very literal interpretation of the New Testament, gave up science in order to dedicate his life to altruism. He sheltered the homeless in his flat, and gave away all his money and possessions to the poor and needy, and his life spiralled out of control. He became deeply depressed shortly after Christmas of 1974, and was found dead in his squatter's tenement on the 6th of January 1975. He had cut his throat with nail scissors.

NATURAL SELECTION IN POPULATIONS SUBJECT TO A MIGRATION LOAD

Migration tends to oppose the effects of divergent natural selection among populations. Numerous theoretical and empirical studies have demonstrated that this migration-selection balance constrains genetic divergence among populations. In contrast, relatively few studies have examined immigration's effects on fitness and natural selection within recipient populations. By constraining local adaptation, migration can lead to reduced fitness, known as a "migration load," which in turn causes persistent natural selection. We develop a simple two-island model of migration-selection balance that, although very general, also reflects the natural history of Timema cristinae walking-stick insects that inhabit two host plant species that favor different cryptic color patterns. We derive theoretical predictions about how migration rates affect the level of maladaptation within populations (measured as the frequency of less-cryptic color-pattern morphs), which in turn determines the selection differential (the within-generation morph frequency change). Using data on color morph frequencies from 25 natural populations, we confirm previous results showing that maladaptation is higher in populations receiving more immigrants. We then present novel evidence that this increased maladaptation leads to larger selection differentials, consistent with our model. Our results provide comparative evidence that immigration elevates the variance in fitness, which in turn leads to larger selection differentials, consistent with Fisher's Theorem of Natural Selection. However, we also find evidence that recurrent adult migration between parapatric populations may tend to obscure the effects of selection.

Dynamics of DNA in vitro evolution with Mnt-repressor: simulations and analysis.

The dynamics of DNA in vitro evolution with Mnt-repressor has been studied numerically and analytically. Based on experimental data and realistic energy landscape for DNA-Mnt-repressor interaction, we investigated the dynamics of DNA in vitro evolution using stochastic simulations. The binding energy of DNA to Mnt-repressor was considered to consist of two parts: the DNA sequence specific and nonspecific. The crossover observed in real experiments is numerically recovered. We demonstrate that the evolution trajectories are drastically dispersed and no typical evolution passage exists during the evolution. Particularly, Fisher's theorem of natural selection is verified. A theoretical analysis for the evolution is also included.

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.