fT being the transpose of f. In order to simplify the arguments, we frequently assume in what follows that the vectors are normalized. Consequently, the scalar product between fi and fj is (fi, fj) = fxfj = 6ij, where 6~j = 1 if i=j and 6~ = 0 if i r Given an input fk, the network produces the following output: K

Mfk = ~ g~(f~, f k ) = g k i=l

9

(4)

Bulletin of Mathematical Biology Vol. 52, No. 1/2, pp. 201 207, 1990. Printed in Great Britain.

0092 8240/9053.00+0.00 Pergamon Press plc Society for Mathematical Biology

DISCUSSION: POPULATION GENETICS

1. Introduction. The three papers in this section represent the seminal work of R. A. Fisher, Sewall Wright and J. B. S. Haldane in setting the framework for the development of population genetics in the twentieth century. Their work in the 1920s and 1930s greatly stimulated efforts to quantify factors affecting gene frquencies in theory, in nature and under domestication. Mathematical population genetics has played crucial roles in sharpening qualitative arguments about forces in evolution. Theoretical and experimental population genetics are flourishing today (see, for example, Milkman, 1983; Feldman, 1989; Lewontin, 1967; 1985). This introduction is divided into two parts. The first describes the three papers and tells why they were important at the time of publication. The second part examines the implications of these papers for the later development of population genetics and why they remain pertinent today. 2. Description and Importance. Evolutionary theories grew and flourished in the 60 years after the publication of Darwin's On the Origin of Species. At the turn of the century, many books reviewing the multitudinous evolutionary theories appeared (Conn, 1900; Kellogg, 1907; Delage and Goldsmith, 1912; and for historical perspective, Bowler, 1983). But evolutionary theory remained strongly qualitative. Verbal arguments with little quantitative evidence or reasoning almost guaranteed that disputes among evolutionists could not be settled. Some evolutionists dreamed of a quantitative evolutionary theory but could not see their way to one. A quantitative, genetic evolutionary theory became possible only after the rediscovery of Mendelian heredity. The regularities of Mendelian heredity were easily quantified. One of the first papers to quantify the consequences of Mendelian heredity was by one of its most vociferous opponents, Karl Pearson, the famous biometrician. He argued (Pearson, 1903) that Mendelian heredity was inconsistent with observed correlations between parent and offspring and therefore Mendelism must be untrue. R.A. Fisher's first publication on this topic was his classic 1918 paper: "The Correlation Between Relatives on the Supposition of Mendelian Inheritance." In this paper he showed that on more realistic assumptions about dominance and other factors, the inconsistency found by Pearson disappeared. Still, exploration of the 201

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quantitative consequences of Mendelian heredity did not by itself constitute a mathematical or quantitative evolutionary theory. Trained in mathematics, physics and astronomy, Fisher loved the simplicity of major laws of physics, such as Boyle's gas laws or the second law of thermodynamics. What attracted him (and so many physicists) to these laws was that the impossibly complex individual movements of molecules disappeared in the statistical treatment and very simple and general laws with great applicability emerged. Fisher hoped to treat evolution in populations with a similar statistical analysis. The paper included here was his first attempt to construct a quantitative evolutionary theory. In the very first paragraph (Fisher, 1922, pp. 321-322; pp. 297-298 of this issue), he stated that: " . . . the whole investigation may be compared to the analytical treatment of the Theory of Gases, in which it is possible to make the most varied assumptions as to the accidental circumstances and even the essential nature of the individual molecules, and yet to develop the general laws as to the behaviour of gases, leaving but a few fundamental constants to be determined by experiment."

Fisher's conception was that an evolving population could be characterized at any time by the statistical distribution of genes in the population. Evolutionary history of the population became simply the history of the changes in this statistical distribution of genes. Mendelian heredity made possible the construction of an equation giving the statistical distribution of genes under a host of simplifying assumptions (such as infinite population size, random mating, non-overlapping generations, no mutation or selection, etc.) and one could introduce into this relatively simple distribution a large number of factors including mutation, selection, random extinction from finite population size, assortative mating, migration and many others. Fisher's insight that evolution in a population could be analysed quantitatively by construction of a statistical distribution of genes is perhaps his most fundamental and lasting contribution to evolutionary biology. The novelty was not in the mathematics he used, but in his hope and expectation that the application of relatively simple mathematical methods could extract from the hopelessly complex web of evolutionary change general quantitative laws of evolutionary change. In this paper he first analysed the problem of equilibrium under selection. He showed clearly that for a single locus with two alleles, selection favoring one homozygote would result inevitably in the elimination of the other. On the other hand, if the heterozygote were more fit than either homozygote, a stable equilibrium would result in a balanced polymorphism. He examined the survival of new mutations. He found that the survival of a mutation depended upon chance rather than selection, and the chances were very small. Thus large populations should be reservoirs of genetic variability, since recurrent

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mutations would have a greater numerical chance of surviving there in low frequencies. He calculated the magnitude of the elimination of alleles in a finite population by chance (later termed random drift) and decided that in a population of moderate size (10 000 individuals) selection was a far greater force than random extinction. Finally, Fisher showed that selection could quickly eliminate a deleterious allele only if the heterozygote were intermediate between the homozygotes. If completely recessive, however, an allele was eliminated from the population by selection only very, very slowly. Fisher concluded that natural selection operating upon large populations was the primary mechanism of evolution in nature. The deterministic results of selection operating upon single gene effects was the key to evolution in nature and under domestication. The quantitative germs of Fisher's 1930 book, The Genetical Theory of Natural Selection, can be easily seen in this paper published 8 years earlier. Of course, there is nothing in the 1922 paper about speciation or eugenics, topics that Fisher would later address in the book, or any statement of the "Fundamental Theorem" of natural selection. But this paper demonstrated forcefully that, through mathematical population genetics, it was possible to construct a truly quantitative theory of evolution. Sewall Wright was already an accomplished mathematical population geneticist at the time Fisher's 1922 paper was published. In 1921 he had published a stunning application of his newly-developed method of path coefficients (a standardized form of multiple regression coefficients) to the problem of calculating the genetic consequences of various forms of mating, especially inbreeding (Wright, 1921). Fisher read these papers on systems of mating and wanted to meet Wright, which he did in the summer of 1924 when Wright was working at the USDA in Washington, DC. At this meeting, Fisher described his 1922 paper to Wright, who had not previously heard of it, and later sent him a reprint of it. Wright was greatly impressed by Fisher's attempt in this paper to construct a statistical distribution of genes as a way of making evolutionary theory quantitative. Wright was uncomfortable with the differential and integral calculus used by Fisher and decided to construct his own quantitative theory of evolution by using his method of path coefficients to deduce the statistical distribution of genes in a population. In other words, Wright accepted the basic thrust of Fisher's paper, and wanted to use different mathematical techniques to reach basically the same statistical distribution. The manuscript of Wright's version of mathematical evolutionary theory was not published until 1931 and is the paper included here. When Wright sent a version of the paper to Fisher in 1929, there were some discrepancies in their results. A year later, however, Wright and Fisher had ironed out the quantitative differences in their models and agreed that their statistical distribution of genes were basically identical although derived by different mathematical methods.

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The qualitative theory of evolution presented in Wright's paper differed substantially from that of Fisher, however, and led in time to bitter disagreements between them. I would emphasize that the tensions between Fisher and Wright concerned differences in their beliefs about evolution in nature and not from their mathematical population genetics, which yielded the same results. Wright's paper was immediately widely read and became very influential in the experimental study of evolution in nature and in animal breeding theory as well as influencing quantitative evolutionary theory. Completely independently of Fisher's 1922 paper, so far as I am aware, J. B. S. Haldane started in 1924 a very influential series of papers under the general title: "A Mathematical Theory of Natural and Artificial Selection." The tenth and last paper in the series appeared in 1934. The first of the series is reprinted here. As Fisher had discovered in 1922, the selection term in his statistical distribution of genes was extremely complex and intractable to quantitative analysis. Much simplification was required to include the effects of selection. But Fisher, Wright and Haldane all believed that natural selection was a very important factor in evolution in nature and that in order to have a quantitative theory of evolution, a quantitative theory of selection was necessary. Haldane (1924, p. 19; p. 209 of this issue) stated this necessity at the outset of his paper: "A satisfactory theory of natural selection must be quantitative. In order to establish the view that natural selection is capable of accounting for the known facts of evolution we must show not only that it can cause a species to change, but that it can cause it to change at a rate which will account for present and past transmutations."

The express purpose of Haldane's series of papers was to quantify the theories of natural and artificial selection. The first paper in the series had a series of simplifying assumptions--infinite population size, non-overlapping generations, random mating, complete recessivity, perfect Mendelian segregation, constant selection intensities from generation to generation. His method was to derive recurrence equations from which he could calculate the proportion of gametes in one generation from the corresponding proportion in the previous generation. Haldane applied this model to cases that varied from selection in self-fertilizing populations to selection of dominant or recessive autosomal and sex-linked characters. He concluded, as had Fisher, that selection was relatively ineffective operating against rare autosomal recessives. He also calculated that in the case of the peppered moth (Biston betularia, then called Amphidasys betularia), selection for the dominant melanic form was of the order of 50% per generation, a value he called a: " . . . not very intense degree of natural selection."

In later papers in the series Haldane modified his original assumptions and then calculated the quantitative consequences of selection.

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3. Implications and Development. The mathematical models invented by Fisher, Wright and Haldane (supplemented by those of many others of the same period including especially Chetverikov and Hogben) had a great influence upon the fields of evolutionary biology and animal and plant breeding. First and foremost, mathematical population genetics, despite its many simplifying assumptions, demonstrated that a quantitative evolutionary biology and artificial breeding theory was possible. This was an enormous step from the purely qualitative arguments that had dominated these fields since the time of Darwin. The influence of mathematical population genetics went far beyond this. Population genetics arose just in time to be part of the so-called "evolutionary synthesis". I have argued recently that the evolutionary synthesis is perhaps better seen not as a synthesis but as an "evolutionary constriction". Up to the 1930s, the number of different evolutionary theories was high, and many if not most were some kind of purposive theory or dominated by one or another kind of mutation theory. Fisher, Wright and Haldane demonstrated for everyone to see that changes of gene frequencies in populations required no variables other than selection, mutation, population structure, migration, etc., and thus all the purposive forces were superfluous. Thus one major effect of the models of theoretical population genetics was to focus the attention of evolutionary biologists and breeders upon a relatively small set of variables. Agreement among population geneticists and evolutionists that a relatively small set of variables governed the evolutionary process did not, of course, mean that they agreed upon the evolutionary effect of the variables. Fisher, Wright and Haldane had many disagreements between them about substantial issues in evolutionary biology. Indeed, their disagreements stimulated much work in both theoretical population genetics and experimental field and laboratory research. Here I emphasize again that these differences came from their different qualitative conceptions of the evolutionary process rather than quantitative differences in their models. The influence of the work of Fisher, Wright and Haldane went far beyond those who could actually read and understand the mathematics in their papers and books. Many readers skimmed the papers for qualitative arguments and learned that way, as did Dobzhansky. Others were influenced by population genetics indirectly by reading such books as Dobzhansky's Genetics and the Origin of Species (1937), Julian Huxley's Evolution: The Modern Synthesis (1942), or Simpson's Tempo and Mode in Evolution (1944). During the 1930s, quantitative animal breeding emerged as a very vigorous field of research, stimulated directly by mathematical population genetics. A continued cross fertilization between models of evolution in nature and evolution in domestic populations has occurred sporadically during the past 50 years.

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To see the continuing influence of the work of Fisher, Wright and Haldane one need only glance at the recent intense controversy over the neutralistselectionist issue. In 1968 Motoo Kimura invented the neutral theory of molecular evolution, and 20 years of debate have followed. Kimura came into population genetics in the late 1940s when Fisher and Wright were arguing intensely over the possible role of random genetic drift in evolution in nature. Fisher believed that random drift played no significant role at all, and Wright thought that random drift was important in the production of novel interaction systems upon which natural selection could then act in accordance with his shifting balance theory of evolution. Kimura became very interested in this version of the random drift-selection controversy and hoped to contribute to it. He did, by first working out more fully than Wright ever had the mathematical consequences of random drift in populations and later he also worked on selection. By 1960, Kimura was the only heir apparent for the triumvirate of Fisher, Wright and Haldane, one of the few things on which all three could agree. In 1957 Haldane published his famous paper on the cost ofsdection. He had sent a copy of the manuscript to Kimura for comment. Kimura and Crow later worked on many aspects of "genetic load" of which the cost of selection in Haldane's sense was one component. Putting together Haldane's cost of selection argument with newly obtained data on gene replacements over evolutionary time periods, Kimura discovered that replacements were occurring so rapidly that the cost of selection would be prohibitive. He deduced that the vast majority of the gene replacements must be neutral or nearly neutral. The neutralist-selectionist controversy has become far more complex in the last 20 years, but I want to point out that the most visible argument in modern evolutionary biology grew directly out of the earlier population genetics work of Fisher, Wright and Ha/dane. A further extension of their work can now be seen in recent quantitative models of cultural change (Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985). Impact of the models of population genetics can also be found in modern theoretical ecology. Readers with further interests in the history of modern population genetics might wish to consult Provine (1971; 1978; 1986). LITERATURE Bowler, P. J. 1983. The Eclipse of Darwinism: The `4nti-Darwinian Theories in the Decades Around 1900. Baltimore: The Johns Hopkins University Press. Boyd, R. and P. J. Richerson. 1985. Culture and the Evolutionary Process. University of Chicago Press. Cavalli-Sforza, L. L. and M. W. Feldman. 1981. Cultural Transmission and Evolution: .4 Quantitative Approach. Princeton University Press.

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Conn, H. W. 1900. The Method of Evolution. New York: Putnam. Delage, Y. and M. Goldsmith. 1912. The Theories of Evolution. London: Frank Palmer. Dobzhansky, T. 1937. Genetics and the Origin of Species. New York: Columbia University Press. Feldman, M. W. (Ed.). 1989. Mathematical Evolutionary Theory. Princeton University Press. Fisher, R. A. 1918. The correlation between relatives on the supposition of Mendelian inheritance. Trans. R. Soc. Edinb. 52, 399-433. Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Oxford University Press. Haldane, J. B. S. 1957. The cost of natural selection. J. Genet. 55, 511-524. Huxley, J. S. 1942. Evolution: The Modern Synthesis. London: Allen and Unwin. Kellogg, V. L. 1907. Darwinism-To-Day. New York: Holt. Kimura, M. 1968. Evolutionary rate at the molecular level. Nature 217, 624-626. Lewontin, R. C. 1967. Population genetics. A. Rev. Genet. 1, 37-70. Lewontin, R. C. 1985. Population genetics. A. Rev. Genet. 19, 81-102. Milkman, R. (Ed.). 1983. Experimental Population Genetics. Stroudsburg, PA: Hutchinson Ross. Pearson, K. 1903. On a generalised theory of alternative inheritance, with special reference to Mendel's laws. Phil. Trans. R. Soc. Lond. A203, 53-87. Provine, W. B. 1971. The Origins of Theoretical Population Genetics. University of Chicago Press. Provine, W. B. 1978. The role of mathematical population genetics in the evolutionary synthesis of the 1930s and t940s. Stud. Hist. Biol. 2, 167-192. Provine, W. B. 1986. Sewall Wright and Evolutionary Biology. University of Chicago Press. Simpson, G. G. 1944. Tempo and Mode in Evolution. New York: Columbia University Press. Wright, S. 1921. Systems of mating. Genetics 6, 111-178.

WILLIAM B. PROVINE

Cornell University Ithaca, New York, U.S.A.