Evaluation of Current Population Genetics Theory1 Population ...

AMER. ZOOL., 23:111-121 (1983)

Evaluation of Current Population Genetics Theory1

OSCAR KEMPTHORNE

Iowa State University, Ames, Iowa 50011

SYNOPSIS. My aim is to give a partial evaluation or critique of the state of populationgenetics theory. A decent theory must include the following components: the developmentof concepts of fitness that have demonstrated epistemic correlations, life tables, mating,fecundity, finite (even if large) niche size, and, of course, Mendelism and mutation. Itmust in the end also include varying environment and competition between species. Theextent to which the desiderata are met is discussed. The big lacunae in the whole theoryappear to be the inadequate treatment of fitness and the ignoring of niche capacity. Sometheorems that are given as fundamental must be questioned and even discarded. Integra-tion of ideas of simple Mendelism, quantitative genetic variation, and ecology is the bigtask ahead. It is critical that more complete theory be developed.

INTRODUCTION

Population genetics theory started some60 or so years ago with work, of R. A. Fisherand S. Wright, followed shortly with workof J. B. S. Haldane. It grew extensively overthe ensuing period, with two interrelatedbranches, mathematical genetics theoryand stochastic genetics theory, and thereare many books dealing with these branchesseparately and together. My aim in thisessay is to give some views on the presentstatus of the theory. I hold the view thatconnections between theory and observa-tion, actual or potential, is essential. Toexemplify what I have in mind, in the caseof developed theory of quantitative inher-itance, that theory attempts to obtainmodels that describe quantitative varia-tion, and then attempts to validate themodels obtained by comparing predictedgains with actual gains from applied selec-tion. A conference (Pollak et al., 1977) givesan account of some attempts that have beenmade in this mode, and indicates successand failure. The main and easily under-standable aim of theory of quantitativegenetics has been to improve economicspecies. Another aim is to obtain partiallyvalidated models of the genetic variationthat we see or that is hidden behind thephenotypes we see. It is necessary to askwhat the aims of population genetics the-

1 From the Symposium on The Inter-face of Life-His-tory Evolution, Whole-Organism Ontogeny and Quantita-tive Genetics presented at the Annual Meeting of theAmerican Society of Zoologists, 27-30 December1981, at Dallas, Texas.

ory are. By chance, I happened to readparts of Darwin's Origin of species recently,and this reading suggests that the aim ofpopulation genetics theory is to give anempirically validated theoretical basis forthe processes described in verbal terms byDarwin, and for other observed evolution-ary sequences that have been observed sincethat time. It is clear that such an aim wasparticularly important to Fisher (1930).The main aim is, then, to develop validatedmodels of the dynamics of genetic popu-lations, and of the evolution, survival, andoften the disappearance of species.

ESSENTIAL INGREDIENTS OF AFULL THEORY

The theory of evolution should start, Ithink, by attempting to model how prim-itive organic entities grew out of the pri-mordial soup. How did some forms of lifeget started on planet Earth? I find it curioushow little this problem—which I call thestart-up problem—is addressed in conven-tional presentations. I was very interestedand glad that the first paper of the presentsymposium (Michod, 1982) addressed thistopic: theory of first replicators. I find itcurious that in the current public debateon creationism versus evolution, this defectof evolutionary theory receives little men-tion. I surmise that the reason is that wehave almost no ideas. In saying this, I donot wish at all to lend support to creationistviews. No scientific theory is complete.

If we set the "start-up" problem on theside, what ingredients are necessary in apopulation genetics theory? At some stage,

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112 OSCAR KEMPTHORNE

there are a number of species existing onthis finite space-ship, Earth. We want toconstruct theory, that is validated, ofcourse, of how the whole complex systemprogresses over time. The big problem inthe whole field is given by Darwin's title,The origin of species. Can one develop a well-formulated and partially validated mathe-matical theory of this whole phenomenon?I happened to have scanned recently twoclassics, Huxley's (1942) Evolution: The mod-ern synthesis, and Mayr's (1942) Systematicsand the origin of species. Naturally, I wasimpressed with the extensive verbal theorythat is presented. I was struck by the abs-ence of mathematical theory. How are thetools of mathematics and statistics to bebrought to bear?

An obvious beginning is to attempt todevelop theory of the dynamics of a singlespecies, and this has been until very recentlythe main thrust of population genetics the-ory. One starts with a single Mendelianpopulation. This population producessperm and ova which unite to form descen-dants. The gamete production is taken tofollow the simple Mendelian laws with theirstochasticity—a model that has beenstrongly validated by the work of this cen-tury. The phenomenon of mutation isintroduced, which results in new genetictypes. Naturally, one hypothesizes thatsome of the genetic types are less fit, insome sense, than others, and the forces ofselection cause the genetic structure of thissingle genetic population to change. Thewhole problem of evolution revolvesaround the production of new genetic typesand the fitness of the genetic types in thepopulation. It is natural, then, that for-malized concepts of fitness enter stronglyinto the theory that is developed. It willturn out that the ideas I express laterrevolve almost always about fitness. Fitnessclearly involves reproduction and survival.A significant part of what I shall say isrelated to my perception that every pre-sentation of population genetics uses cer-tain formalized ideas of fitness, but theactual question "what is fitness?" is littleaddressed. In making this statement, I havein mind, as will be seen, the early writingsof, say, Fisher and Haldane, two of thefounding fathers whose work I have

attempted to study in depth. If one is deal-ing with a selfing population, one can per-haps assume that survival and reproduc-tion are properties of the individual in themilieu in which it exists. But, in general,we cannot be so simplistic. We have to rec-ognize that there is competition withinspecies ("in the land of the blind, the one-eyed man is king"). There is variation inenvironments of individuals within thespecies. For most species there are phasesof life, as zygotes, as infants, as adults, withvariability in survival probabilities andlength of adult life. There will be vari-ability in reproductive performances ofmales and of females. There will be com-petition of functioning males for femalemates. Then, of course, the total environ-ment of individuals of a species consists ofindividuals of all other species that exploitsome portions of environment. In a sensethen, any species that is undergoing changeis in an environment in which other speciesare also changing, an aspect that has beenworked on by those who have exploredhost-predator and host-pathogen systems,for instance. In this abbreviated list of nec-essary aspects, it must surely be assumedthat niche size is always finite, so that thepossibility of everlasting exponentialgrowth of a species must be discountedtotally. This point has great relevance, asI shall discusss, to some of the so-calledfundamental theorems we are presentedwith. Then, finally, a population geneticstheory should have as a possible outcomethat some species simply disappear.

The modern, or neo-Darwinian, thesis isthat species evolve by the accumulationover time of extremely small changesresulting from mutation and selection atindividual loci, largely based on single locuspopulation genetics theory. It is curiousthat this theory, for which the foundingfathers gave what appear to be stronglycompelling verbal arguments, is now beingchallenged, and it will be interesting to seeif some resolution is achieved.

WHAT IS THE ROLE OF THEORY?

This question must be addressed, eventhough the answer to the question is rathersimple. If we accept as we must, Mendel-ism, mutation and selection, can we model

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POPULATION GENETICS THEORY 113

these processes and show that certain out-comes that we observe are theoretical con-sequences of our model? It is interesting tonote the very large number of referencesto Fisher, Haldane and Wright that Huxley(1942) gives as support for the verbal the-ory that he gives. To take the view, then,that population genetics theory is uselessis absurd. A verbal theory is weak, unlessit is supported by a mathematical theorythat attempts to incorporate mathematicalideas that have epistemic correlations withobservable or potentially observable aspectsof the system under discussion. Such sup-port does not entail, of course, proof of theverbal theory. Both may be failures. Whatis absolutely critical is that theory be con-fronted with data. The major defect in thewhole of population genetics theory, I sug-gest, is the lack of such confrontation. Ifthe confrontation results in the rejectionof the theory, the next step must be thedevelopment of another, hopefully better,one, which will again be confronted withdata. In general, data without theory andtheory without data are both useless.

THE DIFFICULTY OF CREATING AUSEFUL THEORY

All the necessary ingredients I mentionabove were given by Darwin. The essentialingredients he lacked were the mechanismof Mendelian segregation with gene muta-tion and the existence of a very large num-ber of segregating loci. Undoubtedly,trained population biologists are able toextend my brief list of ingredients over sev-eral pages.

It is clear that a grand theory incorpo-rating all the obvious facets of the real-world system will be impossibly compli-cated. This has led to attempts at devel-oping theory that incorporates just a fewof the facets, assuming that the others areabsent. The question must then be raisedof how reliable such limited theory is andwhether theorems that are called "funda-mental" really are as fundamental as isclaimed. I note that almost every book onpopulation genetics theory gives Fisher'sFundamental Theorem of Natural Selec-tion, on which I shall comment later.

The facets that are clearly essential fallinto two classes, those concerned with

Mendelian genetics and those connectedwith ecological concepts. Theoretical ecol-ogy, I judge, has, by and large, ignoredgenetics, life history, sex, and has used sim-ple difference and differential equations toencompass finiteness of niche and repro-duction, treating the latter as being ofclonal type. Population genetics hasignored, by and large, competition andfiniteness of niche, with some exceptions.

The main task of theoretical populationbiology is to accomplish realistic integra-tion of what is commonly regarded as pop-ulation genetics theory and ecology. Cas-well (1980) refers to the book ofRoughgarden (1979) as "A milestone inpopulation biology," a judgment withwhich I agree, even though, as will be seen,I have considerable skepticism about someparts of the presentation, such as Fisher'sFundamental Theorem, that seems to beregarded as cornerstones of the supposedintegrated theory.

BASIC POPULATION GENETICS THEORY

In forming my picture of this I was for-tunate to be able to look at the books byRoughgarden (1979) and Hard (1980).Necessarily, one starts with the basic ideasof Mendel ism. Then, obviously, one coversthe general ideas of molecular genetics. Inote in passing that Hartl (1980) tells uswhat he calls "one of the great riddles ofmodern molecular biology: a significantfraction of DNA in higher organisms is ofunknown function." There are uniquesequences, middle repetitive sequences, andhighly repetitive sequences. This mustsurely give us pause with respect to whatfollows. It also serves to illustrate a generalobservation I wish to give: a theory getsstarted at a point in time using ideas thatare well-validated at that time; then thetheory generates mathematical problemsand grows to the point at which it has anexistence on its own not related to bestpresent-day science. This phenomenonoccurs naturally and widely throughoutscience. Eventually, some ideas of the oldtheory fall into disrepute and new theoriesare developed, perhaps from a totally dif-ferent basis.

Having paid obeisance to molecular biol-ogy, the development proceeds in a very

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plausible way to the case of a Mendelianpopulation with a single segregating locus,and then one with just two alleles at thatlocus. What then are we to do about nichesize, about life table ideas, mating patterns,fecundities? It is obvious that a startingpoint is to suppose a 2-stage life, infancyand adulthood; then to suppose that thereis no genetically related variability and thatthe population niche is indefinitely large.The population can grow indefinitely large"if it wishes." The outcome of this model-ling is the famous Hardy-Weinberg law.The idea, exposited by Roughgarden(1979) for instance, is that males pour forthsperm into a huge bath, females pour fortheggs into this same bath and then sperm-egg combinations are formed at random:this is the "random union of gametes"model. Alternatively, we may supposed thatmale-female pairings take place at randomwith multiplicative fecundity.

If we have a huge number of loci, theintrinsic finiteness of niche entails that sto-chasticity must be introduced. With 106

loci, each having 2 alleles the number ofMendelian combinations is 3 raised to thepower 106, and, obviously, almost all ofthese will not occur. The expectations fromMendelian laws simply cannot be realized,and accommodating linkage in traditionalways is impossible. It seems that a theoryfor such a situation must be of statisticalform though I have no idea what that mightbe.

We can develop theory of inbreeding,e.g., selfing or full-sibbing without selectionwith a single segregating locus. But as soonas we take up the case of many loci, themathematics again defeats us. However, weshould not take the view that theory is use-less in general, even if we have to take theview that we have been unable, so far, todevelop a theory that has useful epistemiccorrelation with the real world that weknow to exist. We need more theory, notless.

If we wish to combine selection andinbreeding, again the mathematics becomesvery complicated. Again, we need reallynew mathematical formulations.

The idea that population size is finite hasbeen addressed to a limited extent. If we

take a population with a single segregatinglocus, then some mathematical theory, for-mulated initially by the founding fathers,Fisher, Haldane, and Wright, and extendedgreatly by Kimura, is possible. This theoryis difficult, but not unreasonably so whenwe realize that a very subtle stochastic pro-cess is involved.

The natural next step is to incorporatemutation and this is easy with recurrentmutation and an indefinitely large popu-lation.

Obviously, we have to take up selection.Naturally, one "tries simple things first."So one takes up the case of a single seg-regating locus, with a 2-stage life, in aninfinite niche. Roughgarden (1979, Chap-ter 3) and Hartl (1980) exposit this verynicely. One supposes that each individualsheds its genotypically determined (and sexindependent) particular number of gametesinto what may be called a huge bath, andsperm and ova unite at random. Also, eachindividual has a genotypically determined,sex independent probability of surviving tothe adult phase. This leads to what I callW-genetics theory; with each genotype isassociated an absolute value denoted by Wor S. One sees that the proportions of genesin the subsequent Hardy-Weinberg infantpopulations are affected only by relativeselective values (w), one of which is unity.One can then convert to selective valuesby forming s = 1 — w. This leads to quitesimple mathematics and one can obtain thegenotypic trajectory over generations. Theoutcome is Wright's famous equation forchange in gene frequency:

f ) 4 - ,w \ 2 / dp

and for successive population numbers (N),Nt+1 = W,Nt.

The former equation leads to the idea thatthe population moves so as to maximize w.These formulae have considerable intu-itive appeal. If Wt is less than unity, thepopulation size decreases, as we would like.

The former formula which can also bewritten with W instead of w, and suggeststhat a population moves in a path that max-imizes W, which equals the average num-

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ber of adult offspring of adults, or thatmaximizes w. The relative selective valueof a genotype is taken to be its degree ofadaptation relative to some one fixed geno-type. Then one is led to Wright's adaptivetopography. The Wright equation is oftenregarded as fundamental in populationgenetics theory. I am somewhat skeptical.I am also very skeptical about all "maxi-mum" principles.

Fisher (1930) developed his own "Fun-damental Theorem of Natural Selection"in a way that, even after much study, I amquite hesitant to accept as a proven math-ematical theorem, to the effect that therate of increase of average fitness is equalto the additive genetic variance in fitness,which is necessarily greater than or equalto zero.

So we are led by either Wright or Fisherto the view that a genetic population movesto a peak of average fitness and then staysat that peak. This proposition is given aclear unambiguous proof under certainassumptions as in Kingman's proof thatRoughgarden (1979) presents. One mustcertainly wonder how such theorems, statedin such simple, innocuous, and highlyappealing terms, can be reconciled to thehuge amount of species disappearance.Essential elements are ignored.

What is the scope and validity of these"fundamental" theorems? We have to lookat the models on which they are based andthen ask if the models are plausible approx-imations to the selection that a genetic pop-ulation may undergo. The random unionof gametes is reasonable, perhaps, for somespecies, but certainly not for "higher" ones.Even if one can accept random union ofgametes, it is surely the case that one mustincorporate finiteness of niche, in whichthe average number of adult offspring ofadults must be near unity, and if this istaken as fitness, the average fitness mustbe unity, and will not alter. The Wrightformula tells us how gene frequencieschange. I am not clear on what the Fishertheorem tells us. It does seem clear thatwith finiteness of niche size a full modellingwill include relative selective values thatdepend on population structure. For thistheorem to have real force some model of

frequency dependent selection must beincorporated in the assumptions. The der-ivations I see are based on a mathematicalmodel that is so unrealistic as to make thetheorems less than fundamental. Perhapsthe theorems are useful in indicating theprogress of a large new population in a verylarge niche.

Finiteness of niche has been takenaccount of in stochastic single-locus theorythat assumes constant selective values. Myreaction is that selective values of geno-types should be permitted in the theory todepend on population structure, just ashappens in elementary theory of quanti-tative genetics theory. This relates to theideas on "soft selection" of Wallace (1980),which I judge to be very important, andare related to the selection that the breederuses. My impression is that these ideas havenot been given appropriate consideration.

There have been massive efforts toexplore 2-locus systems in the W-geneticsmode, especially by workers at Stanfordunder the leadership of S. Karlin. Rough-garden (1979, Chapter 8) summarizes thesituation in a large diagram (pp. 130-131).The outcome is that a Mendelian popula-tion with the hypothesized dynamic modehas a trajectory in the gamete frequency-time space that is very complex, with equi-libria that depend critically on the "fit-ness" parameters that are incorporated inthe model. The outcome of this type ofmodelling is appallingly complex, and Iwonder to what extent it has led to betterunderstanding of population processes.The theory is unaccessible, I surmise,except to those prepared to spend a longtime of study, and the "pay off" from suchis not clear.

The problem of extending single-locustheory to many-locus theory, within theW-genetics mode, is huge. A simple idea isto suppose that fitness is multiplicative overloci. Then because the mathematics is soimpossible, it is natural to take recourse tonumerical examination, e.g., Franklin andLewontin (1970). This seems, indeed, tobe the only way one can use to form ideas.A concomitant virture of computer exam-ination is that one can incorporate finitepopulation size in a plausible way.

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All this bears on the problem of explain-ing by reasonable modelling, the vastamount of polymorphism that is revealedby molecular biology techniques, and leadsto neutralist theory. There is now a hugeliterature on this, from much deeply math-ematical work by Kimura, too numerousto cite, and the book by Ewens (1979),which reviews and develops the ideas, toslightly mathematical accounts by Lewon-tin (1974) and Wallace (1980). W-geneticstheory led to the theory of genetic loads.A leading critic of this is Wallace (1980and previous writings). I suggest that hisideas and criticisms be given much moreweight than I have seen in the literature.

Roughgarden (1979, Chapter 6) gives auseful brief discussion of neutralist theory,which can be developed by a single locusmodel because of absence of selection. Theproblem is to develop a plausible selectionhypothesis for a population with a largenumber of loci, as an alternative to opposethe neutralist hypothesis. One way is toassume fitness that is multiplicative overloci, but this engenders deep problems,unless one assumes some sort of soft selec-tion and theory of multilocus soft selectionis in its infancy (cf., Milkman, 1967), becauseof mathematical difficulty. I incline to theview that the so-called neutralist theory ofevolution is more like a stochastic theoryof non-evolution, having little or no bear-ing on the origin of species. The devel-opment is clearly useful with respect tosome molecular population genetics the-ory, quantification of distances betweenspecies and construction of phylogenetictrees, as described, e.g., by Nei (1975).

AGE-STRUCTURED POPULATIONS

All of the preceding that I have discusseduses a 2-stage life, but it is obvious that agestructure of populations and overlappingof generations are present. Does a theorywith 2-stage life and non-overlapping gen-erations give a useful approximation to thedynamics of a Mendelian population withage structure and overlapping genera-tions?

A common reaction to this question,influenced strongly and unduly by early-work of Haldane, is that it does. The idea

is that with possible ages of, say 0 to 5 years,then one should be able to simulate thedetailed dynamics over successive years bya discrete dynamics over the years T, 2T,3T,. . . for some T, to be determined, withabsolute or selective values of the type onesees in 2-stage life theory. If one considersthis question one has to think harder aboutfitness.

Fortunately for my present needs,Charlesworth (1980) has made a fine effortto lay out and develop appropriate theory.I shall refer to this book only, which con-tains citations of the efforts by many. It isnatural to attempt to base such theory onLotka-type theory for a population in aninfinite niche that reproduces clonally. Tohandle sex, Charlesworth says (p. 7), "Amore useful way is to attribute all offspringto the female parent." Then one has tospecify fitness, because one cannot at first,at least, represent it as a product of a gameteproducing factor and a survival-to-adult-hood factor. One introduces such factorsas age-dependent functions. If one is laterto attempt to correlate theory and obser-vation, one has the very difficult problem(except, perhaps, for humans) of deter-mining these. We see that if the niche isinfinite and there is no genetics, the pop-ulation tends to stable age distribution,growing at a logarithmic rate, which iscalled the intrinsic rate of natural increaseor the Malthusian parameter.

Fisher (1930) introduced the concept ofreproductive value of an individual andfrom this was able to discuss the totalreproductive value, V, of a population. Heobtained a theorem that has great intuitiveappeal,

dVdt

= rV.

So the total reproductive value increasesexponentially. The theorem is surely use-ful in telling us that if fecundity and lifeschedules are constant, one will have a pop-ulation explosion or will disappear, the for-mer being possible in an infinite niche. Thistheorem tells us that the modelling isdefective with respect to biological popu-lations. It seems completely clear that thedemographic parameters, the life and

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fecundity schedules, must be density-dependent, if we are to model real biolog-ical populations. One must introduce suchdependence, e.g., by the Pearl-Verhulstequation,

I F = n(a " M-In a clonally producing population thisgives logistic population growth. The firsttask with non-genetic modelling is to modeldensity-dependence of demographic pa-rameters. Charlesworth reviews attempts.My impression is that the modelling is onlyweakly successful. However, even with thisdevelopment, we see that radically differ-ent outcomes ensue, with convergence,cycling or chaos depending, critically, onthe demographic parameters. This poses avery difficult problem for the populationbiologist. It tells us that if the theory is tobe correlated with actual observations, thenthe demographic schedules and theirdependence on niche size must be esti-mated.

A footnote to these deliberations is theconsideration of a set of asexual popula-tions in an infinite niche. One then obtainsa version of Fisher's Fundamental Theo-rem, that the rate of change of populationgrowth is equal to the variance among typesin intrinsic rate of increase. My suggestionis that this version, though cute, should begiven very little weight, because of theassumptions on which it is based.

The first genetical problem is to incor-porate Mendelism, without variationamong genotypes in demographic param-eters, i.e., genotypically independent lifetables and fecundity tables. Does one finda Hardy-Weinberg law, for the new-bornpopulation, of course? With an infiniteniche and a single locus such a law holdsasymptotically in time. To incorporatefinite population size is hard, but it is pos-sible to incorporate ideas of effective pop-ulation size, that correspond to effectivepopulation sizes with a 2-stage life and toobtain a generation interval. One finds that,asymptotically, one can mimic 2-stage lifetheory without selection.

The next problem is to introduce selec-tion. Charlesworth (1980, Chapter 3)

describes our present theory. A generalproblem is that of how to specify, in math-ematical terms, the mating pattern. Onpage 126, Charlesworth gives a set ofassumptions that have been used to developdifference equations, and the reader maywish to judge the reasonableness of these.The mathematical development is not atall easy. The difference equations can beexamined, of course, on a computer. Myview is that much more computer exami-nation of models should be done. Thealternative is to use what is essentially lin-ear mathematical approximation of uncer-tain validity. Charlesworth gives an accountof equilibrium theory and of dynamicaspects under weak selection. While "nosingle parameter can be regarded as thefitness of a genotype with arbitrary selec-'tion intensities" (Charlesworth, 1980, p.154), with weak selection some conclusionslike those of 2-stage life-theory fall out, iffitness of a genotype A;Aj with fecundityfunction of age (x) equal to m^x) and sur-vival function ly(x) is defined to be rij( wherethe sum or integral of mij(x)lij(x)e~ru* is equalto unity. Curiously enough, this suggeststhat the appropriate measure of fitness witha 2-stage life is log(mjjljj) rather than myl̂ —the W that is used in W-genetics theory.

WHAT IS FITNESS?

It it obvious that population theory mustbe based on survival ability, reproductiveability and mating pattern. We see that sim-ple modelling uses a single locus, a 2-stagelife, gamete reproductive abilities, and ran-dom union of gametes. Such modelling israther unrealistic for most biological pop-ulations. Even it leads to conceptual diffi-culties (Kempthorne and Pollak, 1970).How is this modelling to be improved? Wesee, as Charlesworth exposits, how one canincorporate, to some extent, life tables andgamete fecundity tables.

Considerable difficulty occurs if we tryto model mating. Presumably, mating inall populations, apart from human ones, isbased on phenotype, particularly so in sex-ual selection. This was the basis of Fisher's(1918) work. The outcome of Fisher'smodelling is that an equilibrium popula-tion does not have Hardy-Weinberg struc-

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ture. There have been extensive effortswith 1-locus models on assortative mating,but it seems that no general theorems areobtained. We see a morass of algebra. Fish-er's (1918) paper is still rather obscure, andthe subject of continuous investigation.Even if it proves satisfactory, it deals onlywith covariances of phenotypes of relativesin a population that has reached equilib-rium, as the title of the paper tells us. Ittells us nothing about dynamics, and isstrictly a static descriptive theory.

Another huge difficulty in obtaining arealistic theory is the clear fact that a largenumber of loci are segregating. How arewe to develop theory for this? Surely, thelife tables of individuals depend on geno-type which involves a large number of loci.The same must surely be said about fecun-dity. Ability of males to corral female matesmust be of the same nature. This tells usthat if we are to try to model populationdynamics, we have to become involved inquantitative genetic variation. Naturally,then, population genetics theory mustaddress the evolution of life history, thesubject of the present symposium.

QUANTITATIVE GENETIC THEORY

For some decades, there appeared to bea bifurcation between population geneticstheory and quantitative genetic theory. Inow make a few remarks about the latter.It has been largely aimed at the improve-ment of economic species, e.g., cattle, hogs,or corn to give 3 examples. The basic ideais that one has traits that are arithmeticand are produced by individual genotypeand environment. Then one attempts tomodel the variation of single traits and thecovariation of 2 or more traits. The sepa-ration into genotypic and environmentalvariation is based on the assumption, notnecessarily valid, that genotypes and envi-ronment are associated at random, and thatthe base population is in generalized Hardy-Weinberg equilibrium. Mendelism givesthe transmission of gametes to offspring.From this one can obtain covariances ofrelatives for the same or different traits.The number of genetic loci is arbitrary,but implicitly large. One then uses a cen-tral-limit type of argument to suggest that

the individual traits follow a univariate ormultivariate distribution. From this one candevelop an idea of the effect of selection,on the lines of Fisher (1918). This thengives a theory of weak selection, as expos-ited by Falconer (1960), for instance.

This theory has a robustness, because itcan be developed partially at least, by mereconsideration of correlation and regres-sion ideas of statistics, which are rathersimple with multivariate normal distribu-tions. That the theory has been somewhatsuccessful in making choice of phenotypicselection procedures for various economicspecies can be seen from YoWaketal. (1977).

It is critical, however, to mention verysignificant weaknesses of this theory. It isvery difficult to accommodate linkage withinformative conclusions. It is non-para-metric in that the nature of determinationof phenotype by genotype is lost in the sta-tistical formulation. The theory is usefulonly in predicting the instantaneous rateof increase under selection of phenotypicmerit. The parameters of the statisticaldescription of a population are stronglydependent on the genotypic frequenciesand hence change under selection. We havebeen quite unable, except under very lim-iting assumptions, to develop theory oflong-term changes, which is what we needif we are to apply the ideas to evolution.

MERGING OF POPULATION GENETICSTHEORY AND QUANTITATIVE

GENETIC THEORY

It seems obvious that there must be somemerging of these two branches of geneticstheory. We cannot expect to characterizefitness by a single scalar variable, whichstays constant throughout the dynamicalhistory of a population. Fitness must be anoutcome of many aspects of individual andgenotypic abilities. If we confine ourselvesonly to survivability, it is obvious evenunder the simplest formulation that sur-vivability to different ages gives a. function,say l(x), the probability of surviving to agex, which is genotypically and environmen-tally determined.

If we think about this at the level ofobservation, then we have to contemplatesuch population attributes as the regres-

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sion of offspring age of death on parentalage of death. Or, we have to consider thecovariances of survival probabilities to dif-ferent ages, and attempt to get an idea ofthe extent to which such covariances aredetermined by genotype and by environ-ment. I note what I consider to be remark-ably little observational studies on this "sortof thing." This is understandable, of course,because of the difficulties of observation.The available techniques of forming someideas of this are what may be called (andreasonably so) heritability analysis, whichhas been used so widely in economic animalspecies.

The same sort of thinking must beapplied to another component of fitness,fecundity. Here the techniques will be thosestandard in quantitative genetics, herita-bility analyses, which determine statisticalattributes of a population such as regres-sion of fecundity at one age on fecundityat another age and fecundity of offspringon fecundity of parent. One example ofsuch studies is the work of Rose andCharlesworth (1980a, b). This work showsthat fecundity is a complex age-dependentattribute. Such work is related, of course,to quantitative genetic studies on litter sizein hogs, for example, but the implicationsof genotypically related differences in via-bility and fecundity with respect to popu-lation dynamics are topics that ordinaryquantitative genetic theory does not con-sider, and will be complex.

Such considerations as above lead directlyinto the problem of life history evolution.Some theory for this is described byRoughgarden (1979) and Charlesworth(1980). I judge that this theory is almostentirely of single-locus type, but it is surelya beginning.

Of different type is evolutionary theorybased on a 2-stage life based on extensiveuse of normal and multivariate normal dis-tribution theory by Lande (1976), fromwhich I quote (p. 314), "Starting from asimple formula of quantitative genetics, themethods of population genetics are usedhere to make a theory of evolution of theaverage phenotype in a population by nat-ural selection and random genetic drift."The general idea is that the fitness of an

individual is a function of some "ordinary"phenotypic trait, and a basic formula ofquantitative genetic theory for change ofthe mean phenotype in terms of heritabil-ity and selection differential is used todetermine change of fitness. This workassumes that the variances, genotypic andtotal, will not change under selection (per-haps questionably, cf., above). Curiously,this gives a formula that is similar toWright's elementary formula for changein gene frequency. Related results on thismode of theorizing are given by Kimuraand Crow (1978). This type of work is abeginning, perhaps, of integrating a scalarfitness attribute under so-called "poly-genic" control with population dynamics.It introduces, in an interesting way, theideas of "soft selection," which is the sortof selection that occurs in, say, animalbreeding, and may shed light on the seem-ingly all-pervasive occurrence of polymor-phisms.

FUSION OF MENDELIAN ANDECOLOGICAL THEORY

It seems crystal clear that if we are todevelop an informative theory of theorganic world, we have to incorporate eco-logical modelling, finiteness of niche sizewith resultant competition within species,competition between species, life andfecundity tables, and Mendelian processes,with a large number of potentially linkedMendelian loci. This is a huge task. Begin-ning ideas on such a grand theory are givenby Roughgarden (1979). The mathemati-cal systems that appear with reasonablemodelling are very difficult with respect tomathematics they engender; often indeedthey are quite impossible, mathematically,with the mathematics we now possess. Anatural first step is to attempt to find equi-libria, and then to attempt to examine localstability of these. With respect to dynamics,the main method will be, I surmise, simu-lation on a computer, even though obtain-ing general results by this route is very dif-ficult.

Very useful general reviews of evolutionof life-history traits that incorporates eco-logical type modelling is given by Stearns(1976, 1977). It is remarkable how late in

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the history of population genetics theorythis type of modelling appeared when wesee it in early verbal theory.

Much of the theory assumes randommating. How is one to validate or vitiatethis assumption? How to introduce age intomating behavior presents great difficulties.

It is not difficult to introduce ecologicalmodels without genetics and age structureand the dynamics under such models canbe explored, if necessary on a computer.The merging of ecological modelling andgenetic modelling is just beginning.

The overall task is extremely difficult,but it has to be attacked. I said early thata science cannot progress on the sole basisof data without theory. Also, I judge thattheory must be related to data. Levin (1981)has said: "Theory must be recognized asan activity closely tied to, but separate andindependent in its objectives from fieldobservation and experiment." I do notagree with the "separate and indepen-dent" qualification. As I read the litera-ture, I think I see theory that is stronglyseparated from observation, and do not seethe interplay between those that shouldprevail. It seems all too easy to developtheories that have a life of their own anddo not incorporate correlation of theoryand real-world observations. I also have theview that there must be much strongerattempts to use controlled experiments inpopulation biology.

GENERAL REACTIONS

In spite of the patent absurdity ofattempting to give an overall evaluation ofthe field, I shall attempt to do so. It is evi-dent that theory must incorporate matingpatterns, survivability and fecundity func-tions, finiteness (though variable) of niche,and parameters to characterize competi-tion within and between species, and datamust be obtained for validation of theory.In most species, this type of informationmust be obtained from both observationaland experimental ecology, and predomi-nantly the latter. Perhaps Nature has donequasi-experiments as in the case of Anolislizards in the Antilles which Roughgarden(1979) describes, or in the case of Darwin'sfinches as Grant (1981) describes. Both of

these cases illustrate the idea, it seems, thatspeciation has come about by splitting of aspecies into parts with specialized abilitiesto utilize different parts of the overall niche.

It is clear from all that I have said thatpopulation genetics theory is in its infancy.The huge effort by Wright (1969) and thebook by Crow and Kimura (1970) weredefinitive starts, and have been followedby books that start to incorporate some ofthe obviously necessary realism. I would beremiss if I did not repeat my view that wereally have no theorems that should be pro-pounded as fundamental. I hope studentsare so informed. The theorems all seem tobe based on models that leave out ingre-dients that should obviously be included.This view should not be interpreted asderogatory. One starts, obviously, withsimple models, and then attempts to adjoinadditional necessary aspects. We have noalternative but to attempt this expansionof theory. Unfortunately for the popula-tion biologist, this has the consequence thatthe mathematics becomes very difficult.The system considered generates themathematics, simple or difficult. A realdanger is that the theory is generated pri-marily by the mathematics we can do,rather than by the biological processes thatshould be included.

A message I wish to give strongly is thatwe need more theory rather than less. Itshould not be surprising that theory thatis to shed deep light on the huge panoramaof organic life will not be a matter of simplehigh-school algebra. The danger to beavoided is that theory and observationalbiology will become non-communicatingspheres, a tendency for which we canalready see far too much evidence. Stearns(1976) said: "We need experiments that testthe assumptions and predictions reviewedhere, more comprehensive theory thatmakes more readily falsifiable predictions,and examination of different definitions offitness." This brief statement conveys themain message I wish to express.

ACKNOWLEDGMENTS

I am indebted to S. C. Stearns, R. D.Milkman and M. R. Rose for importantreferences. Journal Ppaer No. J-10517 of

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the Iowa Agriculture and Home Econom-ics Experiment Station, Ames, Iowa. Proj-ect 1669. Partial support by National Insti-tutes of Health, Grant GM 13827.

REFERENCES

Caswell, H. 1980. A milestone in population biology.Ecology 61(6):1555-1556.

Charlesworth, B. 1980. Evolution in age-structured pop-ulations. Cambridge University Press, Cam-bridge.

Crow, J. F. and M. Kimura. 1970. An introduction topopulation genetics theory. Harper and Row, NewYork.

Ewens, W. J. 1979. Mathematical population genetics.Springer-Verlag, Berlin.

Falconer, D. S. 1960. An introduction to quantitativegenetics. Oliver and Boyd, Edinburgh.

Fisher, R. A. 1918. The correlation between rela-tives under the supposition of Mendelian inher-itance. Trans. Roy. Soc. Edinb. 52:399-433.

Fisher, R. A. 1930. The genetical theory of natural selec-tion. Clarendon, Oxford.

Grant, P. R. 1981. Speciation and the adaptive radia-tion of Darwin's finches. Amer. Sci. 69:653-663.

Hartl, D. L. 1980. Principles of population genetics.Sinaure, Sunderland.

Kempthorne, O. and E. Pollak. 1970. Concepts offitness in Mendelian populations. Genetics 64:125-145.

Kimura, M. and J. F. Crow. 1978. Effect of overallphenotypic selection on genetic changes at indi-vidual loci. Proc. Natl. Acad. Sci. U.S.A. 75:6168-6171.

Lande, R. 1976. Natural selection and random geneticdrift in phenotypic evolution. Evolution 30:314-334.

Levin, S. A. 1981. Theory's role in understandingpopulations. Bioscience 31:678.

Lewontin, R. C. 1974. The genetic basis of evolutionarychange. Columbia University Press, New York.

Milkman, R. D. 1967. Heterosis as a major cause ofheterozygosity in nature. Genetics 55:493-495.

Nei, M. 1975. Molecular population genetics and evo-lution. North-Holland, Amsterdam-Oxford.

Pollak, E., O. Kempthorne, and T. E. Bailey. 1977.Proceedings of the International Conference on Quan-titative Genetics. Iowa State University Press,Ames.

Rose, M. R. and B. Charlesworth. 1980a. Geneticsof life history in Drosophila melanogaster. I. Sibanalysis of adult females. Genetics 97:173-186.

Rose, M. R. and B. Charlesworth. 19806. Geneticsof life history in Drosophila melanogaster. II.Exploratory selection experiments. Genetics 97:187-196.

Roughgarden, J. 1979. Theory of population geneticsand evolutionary ecology: An introduction. Macmil-lan, New York.

Stearns, S. C. 1976. Life-history tactics: A review ofthe ideas. Q. Rev. Biol. 51:3-47.

Stearns, S. C. 1977. The evolutions of life-historytraits: A critique of the theory and a review ofthe data. Ann. Rev. Syst. Ecol. 7:145-171.

Wallace, B. 1981. Basic population genetics. ColumbiaUniversity Press, New York.

Wright, S. 1969. Evolution and the genetics of popula-tions, Vol. 2, The theory of gene frequencies. Univer-sity of Chicago Press, Chicago.

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