POLYALLELIC MUTATIONAL EQUILIBRIA HIS Paper Is

POLYALLELIC MUTATIONAL EQUILIBRIA

JACK LESTER KING Aquatic and Population Biology Section, Department of Biological Sciences, University of California at Santa Barbara, Santa Barbara, California 93106

AND TOMOKO OHTA National Institute of Genetics, Misltima, Japan Manuscript received June 25,1974 Revised copy received November 4, 1974

ABSTRACT

A new deterministic formulation is derived of the equilibrium between mutation and natural selection, which takes into account (a) the possibility of many allelic mutation states, (b) selection coefficients of the order of magnitude of the mutation rate and (c) the possibility of further mutation of already mutant alleles. The frequencies of classes of alleles 0, 1, 2, n mutant steps removed from the type allele are shown to form a Poisson distribution, with a mean and variance of the mutation rate divided by the coefficient of selection against each incremental mutational step. -- This formulation is interpreted in terms of the expected frequencies of electromorphs, defined as classes of alleles characterized by common electrophoretic mobilities of their protein products. Electromorph frequencies are predicted to form stable unimodal dis- tributions of relatively few phenotypic classes. Common electromorph frequencies found throughout the ranges of species with large population sizes are interpreted as being a uniquely electrophoretic phenomenon; band patterns on starch and acrylamide gels are phenotypes, not genotypes. It is predicted that individual electromorphs are highly heterogeneous with regard to amino acid sequence.

HIS paper is presented in four sections: in I, a new formulation of the equi- Tl ibrium between mutation and selection is presented; in 11, this formulation is extended to predict equilibrium electrophoretic mobility class (electromorph) frequencies. In 111, the intellectual history and observational basis of the model is recounted. In IV, some implications of this model for population dynamics and for future empirical research are discussed, and some related observations are analyzed. I. Equilibrium between mutation and natural selection The classical calculation for the equilibrium frequency 4 of a semidominant mutant with an allelic selective disadvantage s relative to the alternate allele and a recurrent mutation rate U is 4 = u/s. This result has been regularly quoted and never questioned since its formulation by HALDANEin 1937. However, there are clearly instances in which the formulation is inadequate. Essential to the calcula-

Genetics 79: 681-091 April, 1975. 682 J. L. KING AXD T. OHTA tion is the assumption that the selection coefficient is much larger than the mutation rate; yet alleles with very small selection coefficients are apparently very important in evolution and, by implication, in population dynamics. The classical formulation also, and more seriously, assumes only two allelic classes, mutant and normal, which is at variance with our present knowledge of allelic variation at the molecular level. When the selection coefficient against a mutant allele greatly exceeds the mutation rate, mutants will remain rare, and any further mutational alteration of the mutant allele can safely be ignored. However, if the equilibrium is such that mutant alleles can be expected to become relatively common, one must account for further mutational alteration of both mutant and normal alleles. The following model takes account of these considerations. Although mutations with relatively large deleterious effects undoubtedly exist and are important, the present model is concerned only with nearly neutral deleterious mutations-those whose selective disadvantage is of the order of the mutation rate. These mutations are envisioned as single amino acid substitutions. For purposes of initial simplification, it will be assumed that every amino acid substitution has an identical effect in lowering the reproductive fitness of the organism, indicated by the selection coefficient F. Let the rate of amino acid substitution mutations be U per generation for a structural gene (specific back mutations and other sequential mutations within a single codon will be ignored). Since slightly deleterious alleles may persist and are subject to further mutation at other codons within the gene, there will be. in addition to the unchanged type allele, a heterogeneous class of alleles one mutant substitution removed from type. another class of alleles differing from type by two substitutions, a class three substitutions removed and so on. The model is presepted as allelic selection in a haploid species, but is readily extended to selection without dominance in a diploid species. Let it be postulated that the selection against each allele is proportional to the number of mutant substitutions from type. Thus the relative fitness of the type allele is taken as 1, while that of the first mutant class is 1-s, that of the second mutant class is 1-2s, and that of the nthmutant class is 1-m. The mutation rate for an allele of any class to the next higher class is U. This can be diagramed as follows:

U U U uuu difference from type 0 --f 1 -+ 2 -+ 3++n relative fitness 1 1-s 1-2s 1-3s 1-ns allelic class frequency The mean fitness of the population, relative to the mean fitness of the homozygote for the type allele, is m

At mutational equilibrium, so l-SZnXfn, = 1-U; (3) the equilibrium mean fitness is 1-v, as in the classical two-allele model. POLYALLELIC MUTATIONAL EQUILIBRIA 683 The equilibrium frequency of the first mutant class (ignoring terms in sv) is given by -X(1) - X(1F - X(1P + X(0P . X(1)- l-v 7 X(1j = X(0)(v/s). (4) Similarly, at equilibrium, - X(2)- X(2)U- 2SX(Z)+ X(1P X(Zj - ___ l-v , Xrz) = X(1)(v/s)/2 = X(0)(U/S)2/2. (5) In general, - X(n)- X(n)U- nsxw + X(n-1)U X(n)- l-v 7 xn = X(n-1)(v/s)/n = X(n-2) (v/s)2/(n(n--1)); (6) X(n)= X(0)(v/s)“/(n!> . (7) The sum of frequencies equals unity, m x(o)(u/s>n/(n!)= 1 (8) n=O , Hence, X, = e-(v/s). (9) and the expected frequency distribution of the mutant classes is Poisson with parameter u/s, so that X(n)= e-(v/s)( v/~)~/n! . (10) It appears probable that 99% of eukaryote genetic material does not code for polypeptides (KING and JUKES1969) ; protein-forming genes appear to be much largw than the coding portion of the messenger RNA they produce. It seems in- escapable to us that base changes in non-coding DNA sequences are often subject to selection coefficients appreciably smaller than the corresponding mutation rates-insofar as they are subject to selection at all-and that genetic variability will tend to build up until perhaps whole chromosome segments, together with their accumulated mutations, will be subject to differential selection. But it does not seem impossible that some amino-acid-specificity changes will also be SO inoffensive that their selection coefficients will be smaller than their mutation rates (OHTA 1973). The remainder of this paper explores some of the ramifica- tions of this possibility. 11. Electromorph frequencies An electromorph is a class of alleles characterized solely by a common electrophoretic mobility phenotype. An electromorph may be composed of any number of discrete alleles differing in sequence, relative fitness. and function. The “allele frequencies” so commonly reported in the literature these last nine years, when derived from gel electrophoresis data, are actually electromorph frequencies and may have only very indirect relationships to functional allele frequencies. It is beginning to appear that electromorphs in most species may frequently, and perhaps usually, be highly sequentially heterogeneous. 684 J. L. KING AND T. OEITA With respect to the model described in section I above, it can be estimated that of the class of mutant alleles that are one mutant substitution removed from the type sequence, about 70% will belong to the same electromorph as the type allele. This estimate is based on DAYHOFF'Scompilation of amino acid substitutions in molecular evolution (DAYXIOFFet al. 1972), which indicates that 70% of such substitutions involve pairs of amino acids of the same charge class. 15 % of amino acid substitutions result in an increase of one unit in the net charge of the protein involved, and 15% result in a one unit decrease in the net charge (a negligible proportion involve two-unit charge changes; see KING 1973). The distribution of electromorph frequencies within a class of mutants n amino acid substitutions removed from type is given by the summed trinomial expansion

~=n-iu

where w is the difference from type in the number of unit charges of a given electromorph, i is the number of sites at which a mutant allele is negative relative to type, (w+ i) is the number of sites at which the mutant is positive relative to type, (n- w - 2i) is the number of sites with different amino acids of the same charge class, and P(w,n) is the relative frequency of the electromorph within the class of mutants (see KING 1973, 1974). The distribution of frequencies is symmetrical about the electromorph containing the type allele, so that P(l,,,lL)= P(-ru,n).The net frequency at equilibrium of alleles n amino acids and w charge units different from type is given by the product of (10) and (11) : 2: 2: -n-tu 2 (.is)(w+zi)(.7) (n-w-zi) P(w,n,v/s)= dvls) (/InU s 1= =0 [ ] . (12) i!(w+i) ! (n-w-2z) ! The total frequency of all alleles w charge units different from type is given by (12)summed over all possible values of n:

i= -n-tu r) z (.15) (w+zi) (-7)(n-w-zi) = e-(v/s) (13) P(w,v/s) n=w (U/S)~ E ] . i =0 [ i!(w+i) ! (n-w-ei) ! In our computations the value of P(tc,a,s)was summed over increasing values of n until the increment of the function is less than some specified small amount ( A diagram of the complete model is given in Figure 1. The equilibrium electromorph frequencies are symmetrically distributed around the electromorph that includes the type allele. Calculations of equilibrium electromorph frequencies of various values of u/s are given in Table 1.

111. A history of the preserit model The equilibrium electromorph frequency model presented here is an attempt to explain some general empirical observations about real electromorph distri- butions, particularly as they have beer? reported from various Drosophila studies: 1) Electromorph frequencies at a given locus tend to be approximately con- POLYALLELIC MUTATIONAL EQUILIBRIA 685

A h 3" 3 3 v vr' a 8W%

/* hm h h n m vC Y v m a 8 w I1

' h

N N. v Y N v a 8 w I1

h -i 1 v Y rl v w I1 D. 8

n h 9 9 v Y v 0 a 8 w I1

h h 1 1 v Y rl v a a 8-H

h N hN N vC Y N v 8 w I1 PI

h n h m E v Y m v 8 w 11 PI

N n h .Lo v m v> v 0 rl h h h :la rl .VI >Lo .Lo 8 2 . > v v v I I I

(D Lo fn Lo Lo N 0 Y aa>Lo d dI rlI rlI . .d .r( Lo uaJme .-U U+.aJd

rl N C b 686 J. L. KING AND T. OHTA

TABLE 1

Expected electromorph frequencies for various ualues of v/s: P (,,,- y,s,

Charge unit difference from standard mobility

.01 ,997 ,001 - 1.01 1.28 ,787 .I ,971 ,015 - 1.06 2.56 ,414 .5 ,866 ,065 - 1.32 3.M ,384 1 ,758 .I12 .008 1.67 4.22 ,395 2 599 ,172 ,025 ,003 2.38 5.33 ,447 3 ,493 ,202 .w .m 3.04 6.09 .499 5 ,367 .219 ,0175 ,018 ,003 - 4.12 7.34 ,560 7 ,300 ,214 .C96 ,0331 ,008 .om 4.96 8.34 ,595 10 .243 ,197 ,112 ,048 .016 .05 6.0 9.59 ,626 15 .I94 ,171 .I18 ,066 ,030 ,012 7.41 11.27 .658 20 .I67 .I52 ,116 ,075 .040 ,020 8.58 12.62 ,680

U = mutation rate; s = selection coefficient per mutant substitution; ne = “effective electromorph number”, the reciprocal of the summed squares of electromorph frequencies; nlnn=the expected number of electromorphs in a random sample of 1010 genomes: nlOO= (1- (1 - P2,,)100).Note that the ratio ne/nlOOserves as an index of the evenness of the allele distribution. OHTA and KIMURA(1974) have found that, under the strict neutrality assumption with stepwise mutation scheme, this ratio becomes 0.5 - 0.6 whereas the average observed values stay around 0.4- 0.5 when more than two alleles are found in the sample. stant throughout the range of a species, showing neither random local differentiation, nor any evidence of differential adaptation to habitat, nor any effect of local population size (PRAKASH,LEWONTIN and HUBBY1969; AYALA,POWELL and DOBZHANSKY1971 ; AYALAet al. 1972). 2) Different species of Drosophila have similar levels of electromorph heterozygosity despite undoubtedly great differences in species numbers; indeed, most species of all kinds appear to have levels of electromorph heterozygosity within a surprisingly narrow range. This observation even extends to the haploid E. coli, which, while lacking the capacity for heterozygosity, closely resembles Drosophila and other species in other measures of polymorphism, such as average effective number of alleles (electromorphs) per locus (MILKMAN1973). 3) As noted by BULMER(1971), electromorph frequencies at polymorphic loci tend to contain more or less symmetrical patterns, in which the most common electromorph is flanked by progressively less common electromorphs. Rare electromorphs are seldom found with mobilities intermediate to two common electromorphs. KIMURAand CROW(1964) calculated the expected effective number of unique, selectively neutral alleles (ne)per locus in a finite population: ne = 1 3- 4Neu (15) where U is the mutation rate to unique alleles and Ne is the effective population size. The model is not appropriate io electromorph frequencies, however; KIMURA and CROWspecified that each mutation must be unique, while nearly all single POLYALLELIC MUTATIONAL EQUILIBRIA 687 mutations causing electromorph changes are of two types only: positive or negative net shift of one unit. OHTAand KIMURA(1973) presented a model appropriate to selectively neutral electromorph mutations, where

ne = 1+8Neu

(the mutation rate U is here restricted to those neutral mutations involving net charge change). While the relationship between effective number of alleles and effective population size is less linear in (16) than in (15), the formulation nevertheless pre- dicts a continuing increase in heterozygosity and effective number of alleles with increasing population size. In response to OHTA and KIMURA(1973), KING (1974) proposed a model in which the number and frequencies of selectively neutral electromorphs are limited by the number of freely variable amino acid sites, the electromorph frequencies becoming independent of population size in very large populations. In KING’S(1974) model, amino acid sites withiE proteins are conceived as being either fully variable or totally invariable, and mutations are considered to be either totally neutral or definitely deleterious. We actually have favored the views that many sites in most proteins are at least potentially subject to variability, and that there is a continuum of selective effect in the neighborhood of neutrality (cf. KIMURA1968; KING 1972; OHTAand KIMURA 1971). In fact, very slightly deleterious mutations are likely to be prevalent at the molecular level (OHTA1973). OHTAand KIMURA(1975) have recently presented several related models in which deterministic equilibria are attaincd in large populations independently of population size, through progressive very slightly deleterious mutations. In OHTA and KIMURA’S(1975) model, each electromorph is considered to be a class of alleles to which one can assign a fitness value, corresponding to the average fitness of all structural alleles of a given electrophoretic mobility, relative to the average fitness of the alleles comprising the electrotmorph to which the type allele belongs. The most common class (band) is thus assigned a mean relative fitness of one, although alleles comprising it are heterogeneous with regard to both sequence and fitness. Electromorphs removed by one, two or more steps from the mobility of the type electromorph are considered to have progressively reduced mean fitness values (average over functionally heterogeneous collections of sequence alleles). A variety of modifying assumptions in these models results in the same general equilibrium pattern: a common electromorph containing the type sequence allele, flanked by progressively less common electromorphs. This pattern is also seen in the present model and in KING’S(1974) model, and is usually approximated in nature. OHTAand KIMURA’S assignment of fitness values to specific electromorphs can be justified. It is known that some proteins (notably- cytochrome c) have strin- gent constraints on allowable net charge, such that a mutation in the direction of optimal net charge would have a high probability of increasing the fitness of the allele. while a mutation away from optimal net charge would almost cer- tainly be deleterious. However. it seems possible that, in most cases, the net 688 J. Id. KING AND T. OHTA charge of a protein is incidental to sequence changes that have occurred, and that charge changes are only indirectly related to changes in fitness. The present model does not take into account any selective force directly associated with charge as such, but nevertheless it can be shown that the net fitness of each electromorph will decrease as a function of its distance from the most common electromorph. In this the present model is essentially congruent with the previous models of OHTAand KIMURA(1975).

111. Some assumptions, cot necessarily realistic, of the present model 1) The present model presents equilibrium expectations; electromorph frequencies in real populations will be subject to effects of finite population size and finite time spans. 2) It is assumed that there is no direct effect of electrophoretic charge on the selection coefficient. 3) Only one type of sequence exists; or, if there is more than one type sequence, all such sequences have the same electrophoretic mobility. This is essentially a restatement of the “classical” view, in which a single optimal allele exists at a given locus, rather than an array of adapted types held by balancing selection (LEWONTIN1974). It is nevertheless a model in which very extensive genic heterogeneity with regard to sequence, function and fitness is the normal condition of most loci in large populations. OHTAand KIMURA(1975) investi- gated related models in which there are two “type” electromorphs; the general conclusions are unchanged. 4) There are no effects of dominance, recessivity, overdominance, frequency dependence, or of variation in environmental parameters. Such effects do. of course. exist in nature. We do not feel their existence invalidates the principal conc1us;ons of the presevt model. 5) Increasing numbers of mutational differences from type continue to decrease relative fitness; i.e., there are no nearby local adaptive peaks. Violation of assumption 5 would have interesting consequences. If a reasonably close adaptive peak did exist, mutation pressure could cause the accumulation of very slightly deleterious changes until one or more alleles would be driven into the area of the second adaptive peak; further mutational changes, combined with positive natural selection, could result in the emergence of a new “type” allele of a different electrophoretic mobility. A similar array of electromorph frequev- cies could eventually develop around the new type electromorph. Such shifts may account for some of the observed interspecific differences in electromorphs. IV. Discussion Recently BERNSTEIN,THROCKMORTON and HUBBY ( 1973) have demonstrated heretofore undetected amounts of genic heterogeneity in Drosophila enzyme polymorphisms. Using heat sensitivity as a secoEd criterion (in addition to electrophoretic mobility), they were able to demonstrate more than twice as many classes of alleles as could be demonstrated with electrophoresis alone. While this result demonstrates that electromorphs are sequentially heterogeneous, and is POLYALLELIC MUTATIONAL EQUILIBRIA 689 consistent with hypotheses of very extensive sequence heterogeneity within electromorphs, we would guess that only a second tip of the iceberg. and not the iceberg itself, has been revealed. Probably most allelic variation results neither in unique electrophoretic change nor in detectable change in thermal liability. BERNSTEIN,THROCKMORTON and HUBBY (1973) arbitrarily chose four categories of thermal liability 04 the enzyme xanthine dehydrogenase, and readily found allelic variation in all four categories. Their preliminary data indicated that allelic variation in thermal liability may exhibit marked geographic variation, in contrast to the stable equilibria seen at the level of electrophoretic mobility in the same species. Perhaps the reason for this apparent difference is that stepwise transitions of electrophoretic charge give rise to quasi-equilibria, while there is no corresponding expectation for equilibria of categories of thermal liability. We have described a model with many simplifying assumptions, and all of these assumptions are subject to strong exceptions and objections. Still we feel that this model, together with its several predecessors, gives the most adequate account of the data presently at hand, at least for very large populations such as those found in Drosophila species. Species that characteristically divide into small isolated populations, such as frogs. mice and aboriginal Homo sapiens, show a rather different pattern of geographic differentiation: typically the isolated populations fluctuate considerably with regard to electromorph frequencies (Lewontin 1974). This is consistent with the present model, in that alleles with selection coefficients in the neighborhood of the mutation rate behave as selectively neutral alleles when the effective population size is relatively small (OHTA and KIMURA1975). The differentiation of isolated frog, mouse and hu- man (small) subpopulations is the result of random drift; the lack of differentiation of isolated (large) Drosophila populations, if such populations are indeed genetically isolated, could be attributed to independent approximations of the same equilibria. An alternative explanation sometimes proposed is that the observed electromorph frequencies in all groups are the result of adaptive natural selection. But we doubt that the gelletic structure of man and Drosophila are so different that man, a behaviorally complex homeotherm, must adapt his electrophoretic allele frequencies to his immediate environment, while the fruit fly, a behaviorally rather simple poikilotherm, lacks such pervasive local adaptation. We do not feel that the present model invalidates our recent previous models of equilibrium electromorph frequencies (KING 1974; OHTAand KIMURA1975). With some variation in assumptions, all of these models predict equilibrium patterns of electromorph frequencies in large populatiorrs that are roughly consistent with published observation. All are consistent with essentially random or “neutral” fluctuations when the effective population size is smaller than the reciprocal of the mutation rate. It is likely that restrictions on the numbers of sites free to vary are important (KING1974) and that electrophoretic charge itself is a factor in weak selection (OTHAand KIMURA1975). Many other factors not considered in the present model are undoubtedly also operating, probably 690 J. L. KING AND ‘I.OHTA including some net heterozygous advantage, but they do not change the general results. For instance, it is not necessary for the present model that the sequence of selection coefficients associated with mutant classes should be exactly s, 2s, 3s, and so on, but only that they should continue to increase with increasing departure from the type allele. It is not necessary that all mutants should be semidominant with regard to fitness; in fact, it appears quite likely that a significant proportion of the electromorph variation observed in nature is due to deleterious recessive alleles. If there is any degree of dominance in any substan- tial proportion of the variant alleles in the direction of increased fitness, with any significant complementation between alleles, there would cease to be a clear operational distinction between the present model and one of generalized overdominance-except that selection coefficients would be still conceived as being infinitesimal, and effectively inoperable in closed populations with effective sizes less than the reciprocal of the mutation rate. Direct evidence for the action of natural selection in the maintenance of polymorphism, though frequently claimed, has rarely been convincing (see, for example, LEWONTIN1974). Some of the reasons for this may be better understood in the light of Ihe present model. The magnitude of selection we have been con- sidering is far too small to be readily demonstrated directly; selectionists may simply have been looking for larger effects than commonly exist. A second difficulty with demonstrating selection in studies involving electrophoresis is implicit in the hypothesis that most electromorphs are sequentially and functionally highly heterogeneous. The aspects of the ger?e most sensitive to selection may have no relationship to the electrophoretic mobility of the gene product. A typical experiment is to compare, in some fashion, the fitness of an (electromorph) heterozygote with that of (electromorph) homozygotes. While electromorph heterozygotes are unquestionably sequentially heterozygous, equally so, we believe. are most electromorph homozygotes. An example of this difficulty is found in the experiments of WILLSand NICHOLS(1972) on selection at the Drosophila pseudoobscura octanol dehydrogenase locus under severe environmental stress (in the form of excess substrate concentration). Outbred flies failed to show any significant fitness differential between electromorph heterozygotes and homozygotes, while after a period of inbreeding an apparent heterozygous advantage was sometimes found. Possibly the reason for having initially failed to find an effect was that (prior to inbreeding) the “homozygotes” were generally not homozygous at all, and that WILLS and NICHOLSwere in effect comparing one class of heterozygotes with another class of heterozygotes. Apparently inbreeding then achieved homozygosity within each electrophoretic class (the investigators’ own interpretation is different than that given here). Hopefully geneticists will cease thinking of and referrhg to electrophoretic mobility classes (electromorphs) as “alleles,” or to their frequencies as “allele frequencies.” Electrophoretic gel patterns are phenotypes.

This research was supported in part by U.S. Public Health Service Grant ROI GM21283. POLYALLELIC MUTATIONAJ, EQUILIBRIA 691

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