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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 magni- tude 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 re- moved 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 inter- preted 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 fre- quencies found throughout the ranges of species with large population sizes are interpreted as being a uniquely electrophoretic phenomenon; band pat- terns on starch and acrylamide gels are phenotypes, not genotypes. It is pre- dicted 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 muta- tion 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 muta- tions with relatively large deleterious effects undoubtedly exist and are impor- tant, the present model is concerned only with nearly neutral deleterious muta- tions-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 electro- phoretic 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 per- haps 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 frequen- cies 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 ] .
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