The fundamental theorem of natural selection: the end of a story.

The direction of research in population genetics theory is currently, and correctly, retrospective, that is directed toward the past. What events in the past have led to the presently observed genetic constitution of a population? This direction is inspired, first, by the large volumes of genomic data now available and, second, by the success of the classical prospective theory in validating the Darwinian theory in terms of Mendelian genetics. However, the prospective theory should not be forgotten, and in that theory, perhaps the most interesting and certainly the most controversial, is Fisher's so-called "Fundamental Theorem of Natural Selection." This article describes the history and the current status of that theorem.

On the sign of the average effect of an allele.

The concept of the average effect of an allele pervades much of evolutionary population genetics. In this context the average effect of an allele is often considered as the main component of the "fitness" of that allele. It is widely believed that, if this fitness component for an allele is positive, then the frequency of this allele will increase, at least for one generation in discrete-time models. In this note we show that this is not necessarily the case since the average effect of an allele on fitness may be different from its marginal additive fitness even in a one-locus setting in non-random-mating populations.

Mutational Load and the Functional Fraction of the Human Genome.

The fraction of the human genome that is functional is a question of both evolutionary and practical importance. Studies of sequence divergence have suggested that the functional fraction of the human genome is likely to be no more than ∼15%. In contrast, the ENCODE project, a systematic effort to map regions of transcription, transcription factor association, chromatin structure, and histone modification, assigned function to 80% of the human genome. In this article, we examine whether and how an analysis based on mutational load might set a limit on the functional fraction. In order to do so, we characterize the distribution of fitness of a large, finite, diploid population at mutation-selection equilibrium. In particular, if mean fitness is ∼1, the fitness of the fittest individual likely to occur cannot be unreasonably high. We find that at equilibrium, the distribution of log fitness has variance nus, where u is the per-base deleterious mutation rate, n is the number of functional sites (and hence incorporates the functional fraction f), and s is the selection coefficient of deleterious mutations. In a large (N=109) reproducing population, the fitness of the fittest individual likely to exist is ∼e5nus. These results apply to both additive and recessive fitness schemes. Our approach is different from previous work that compared mean fitness at mutation-selection equilibrium with the fitness of an individual who has no deleterious mutations; we show that such an individual is exceedingly unlikely to exist. We find that the functional fraction is not very likely to be limited substantially by mutational load, and that any such limit, if it exists, depends strongly on the selection coefficients of new deleterious mutations.

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.

The left-hand side of the Fundamental Theorem of Natural Selection: A reply.

In a recent paper, Grafen (2018) discussed the left-hand side in the equation stating Fisher's (1930, 1958) "Fundamental Theorem of Natural Selection" (FTNS). Fisher's original statement of the FTNS is, in effect, "The rate of increase in fitness of any organism is equal to its genetic variance in fitness at that time" with the rate of increase in fitness understood as the one "due to all changes in gene ratios" (Fisher, 1930, p. 35). For purposes of exposition, Grafen (2018) considered what is today called the analogous discrete-time model, and restated the FTNS on p. 181 as "The increase in population [mean fitness] due to changes in gene frequencies [is equal to the] additive genetic variance in fitness [divided by the] mean fitness". Allowing for the fact that Grafen's statement of the FTNS relates to a discrete-time model, his statement is in effect a discrete-time version of Fisher's. It has however been widely accepted for many years, ever since Price's (1972) deep analysis of the FTNS, that Fisher's wording does not correctly describe the content of the FTNS. The same is therefore true of Grafen's statement. The confusion caused by these misstatements is unfortunate and adds to a continuing misunderstanding of the FTNS, whose source can also be found in Fisher's (1941) own explanation. Our purpose is to review the detailed analysis of the calculations leading to the FTNS to clarify the points at issue.

Correcting for Ascertainment.

Data used to study human genetics are often not obtained by simple random sampling, which is assumed by many statistical methods, especially those that are based on likelihood for making inferences. There is a well-developed theory to correct likelihoods based on sibship data whether or not the exact mode of ascertainment is known. In the case of larger pedigrees, however, the problem is much more difficult unless they are recruited into the sample by single ascertainment. There is no one piece of software that analyzes ascertainment in general, so most of this chapter is devoted to theory. A general method by which one general genetic analysis software package corrects pedigree data for ascertainment is briefly described.

On the interpretation and relevance of the Fundamental Theorem of Natural Selection.

The attempt to understand the statement, and then to find the interpretation, of Fisher's "Fundamental Theorem of Natural Selection" caused problems for generations of population geneticists. Price's (1972) paper was the first to lead to an understanding of the statement of the theorem. The theorem shows (in the discrete-time case) that the so-called "partial change" in mean fitness of a population between a parental generation and an offspring generation is the parental generation additive genetic variance in fitness divided by the parental generation mean fitness. In the continuous-time case the partial rate of change in mean fitness is equal to the parental generation additive genetic variance in fitness with no division by the mean fitness. This "partial change" has been interpreted by some as the change in mean fitness due to changes in gene frequency, and by others as the change in mean fitness due to natural selection. (Fisher variously used both interpretations.) In this paper we discuss these interpretations of the theorem. We indicate why we are unhappy with both. We also discuss the long-term relevance of the Fundamental Theorem of Natural Selection, again reaching a negative assessment. We introduce and discuss the concept of genic evolutionary potential. We finally review an optimizing theorem that involves changes in gene frequency, the additive genetic variance in fitness and the mean fitness itself, all of which are involved in the Fundamental Theorem of Natural Selection, and which is free of the difficulties in interpretation of the Fundamental Theorem of Natural Selection.

Correcting for ascertainment.

Data used to study human genetics are often not obtained by simple random sampling, which is assumed by many statistical methods, especially those that are based on likelihood for making inferences. There is a well-developed theory to correct likelihoods based on sibship data whether or not the exact mode of ascertainment is known. In the case of larger pedigrees, however, the problem is much more difficult unless they are recruited into the sample by single ascertainment. There is no one piece of software that analyzes ascertainment in general, so most of this chapter is devoted to theory. A general method by which one general genetic analysis software package corrects pedigree data for ascertainment is briefly described.

James F. Crow and the stochastic theory of population genetics.

Research in population genetics theory has two main strands. The first is deterministic theory, where random changes in allelic frequencies are ignored and attention focuses on the evolutionary effects of selection and mutation. The second strand, stochastic theory, takes account of these random changes and thus is more complete than deterministic theory. This essay is one in the series of Perspectives and Reviews honoring James F. Crow on the occasion of his 95th birthday. It concerns his contributions to, and involvement with, the stochastic theory of evolutionary population genetics.

Adjustment for local ancestry in genetic association analysis of admixed populations.

MOTIVATION: Admixed populations offer a unique opportunity for mapping diseases that have large disease allele frequency differences between ancestral populations. However, association analysis in such populations is challenging because population stratification may lead to association with loci unlinked to the disease locus. METHODS AND RESULTS: We show that local ancestry at a test single nucleotide polymorphism (SNP) may confound with the association signal and ignoring it can lead to spurious association. We demonstrate theoretically that adjustment for local ancestry at the test SNP is sufficient to remove the spurious association regardless of the mechanism of population stratification, whether due to local or global ancestry differences among study subjects; however, global ancestry adjustment procedures may not be effective. We further develop two novel association tests that adjust for local ancestry. Our first test is based on a conditional likelihood framework which models the distribution of the test SNP given disease status and flanking marker genotypes. A key advantage of this test lies in its ability to incorporate different directions of association in the ancestral populations. Our second test, which is computationally simpler, is based on logistic regression, with adjustment for local ancestry proportion. We conducted extensive simulations and found that the Type I error rates of our tests are under control; however, the global adjustment procedures yielded inflated Type I error rates when stratification is due to local ancestry difference.

There's plenty of time for evolution.

Objections to Darwinian evolution are often based on the time required to carry out the necessary mutations. Seemingly, exponential numbers of mutations are needed. We show that such estimates ignore the effects of natural selection, and that the numbers of necessary mutations are thereby reduced to about K log L, rather than K(L), where L is the length of the genomic "word," and K is the number of possible "letters" that can occupy any position in the word. The required theory makes contact with the theory of radix-exchange sorting in theoretical computer science, and the asymptotic analysis of certain sums that occur there.

Sam Karlin and the stochastic theory of evolutionary population genetics.

Sam Karlin's role in the development of the stochastic theory of evolutionary population genetics is outlined, together with his work in developing BLAST theory.

A flexible two-stage procedure for identifying gene sets that are differentially expressed.

MOTIVATION: Microarray data analysis has expanded from testing individual genes for differential expression to testing gene sets for differential expression. The tests at the gene set level may focus on multivariate expression changes or on the differential expression of at least one gene in the gene set. These tests may be powerful at detecting subtle changes in expression, but findings at the gene set level need to be examined further to understand whether they are informative and if so how. RESULTS: We propose to first test for differential expression at the gene set level but then proceed to test for differential expression of individual genes within discovered gene sets. We introduce the overall false discovery rate (OFDR) as an appropriate error rate to control when testing multiple gene sets and genes. We illustrate the advantage of this procedure over procedures that only test gene sets or individual genes.

A review of family-based tests for linkage disequilibrium between a quantitative trait and a genetic marker.

Quantitative trait transmission/disequilibrium tests (quantitative TDTs) are commonly used in family-based genetic association studies of quantitative traits. Despite the availability of various quantitative TDTs, some users are not aware of the properties of these tests and the relationships between them. This review aims at outlining the broad features of the various quantitative TDT procedures carried out in the frequently used QTDT and FBAT packages. Specifically, we discuss the "Rabinowitz" and the "Monks-Kaplan" procedures, as well as the various "Abecasis" and "Allison" regression-based procedures. We focus on the models assumed in these tests and the relationships between them. Moreover, we discuss what hypotheses are tested by the various quantitative TDTs, what testing procedures are best suited to various forms of data, and whether the regression-based tests overcome population stratification problems. Finally, we comment on power considerations in the choice of the test to be used. We hope this brief review will shed light on the similarities and differences of the various quantitative TDTs.

Disease associations and family-based tests.

This unit describes the statistical techniques necessary for performing family-based association studies (such as the TDT) for genetic polymorphisms. Such studies have become increasingly important in the identification of genes that confer an increased risk to disease, particularly for common diseases with a complex etiology. The family-based approach avoids some of the problems often encountered when applying the traditional case-control design to genetic studies. The unit includes the Sib TDT (S-TDT) method, which allows application of the principle of the TDT to sibships without parental data, and several related tests.

Comments on the entropy-based transmission/disequilibrium test.

It has recently been claimed in this journal (Zhao et al. in Hum Genet 121:357-367, 2007) that a so-called "entropy-based" TDT test has improved power over the standard TDT test of Spielman et al. (Am J Hum Genet 52:506-516, 1993). We show that this claim is contradicted by standard statistical theory as well as by our simulation results. We show that the incorrect claim arises because of inappropriate assumptions, and also show that the entropy-based statistic has various undesirable properties.

Computing the distribution of the maximum in balls-and-boxes problems with application to clusters of disease cases.

We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time O(mn), where m is the desired maximum and n is the number of variables. We apply the method to the analysis of two situations in which an apparent clustering of cases of a disease in some locality has raised epidemiological concerns, and these concerns have been discussed in the recent literature. We conclude that one of these clusters may be explained on purely random grounds, namely the leukemia cluster in Niles, IL, in 1956-1960; whereas the other, a leukemia cluster in Fallon, NV, in 1999-2001, may not.

Common genetic variants account for differences in gene expression among ethnic groups.

Variation in DNA sequence contributes to individual differences in quantitative traits, but in humans the specific sequence variants are known for very few traits. We characterized variation in gene expression in cells from individuals belonging to three major population groups. This quantitative phenotype differs significantly between European-derived and Asian-derived populations for 1,097 of 4,197 genes tested. For the phenotypes with the strongest evidence of cis determinants, most of the variation is due to allele frequency differences at cis-linked regulators. The results show that specific genetic variation among populations contributes appreciably to differences in gene expression phenotypes. Populations differ in prevalence of many complex genetic diseases, such as diabetes and cardiovascular disease. As some of these are probably influenced by the level of gene expression, our results suggest that allele frequency differences at regulatory polymorphisms also account for some population differences in prevalence of complex diseases.