A MODEL ALLOWING CONTINUOUS VARIATION IN ELECTROPHORETIC MOBILITY OF NEUTRAL ALLELES

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A MODEL ALLOWING CONTINUOUS VARIATION IN ELECTROPHORETIC MOBILITY OF NEUTRAL ALLELES GARY COBBS Department of Biology, University of Louisville, Louisville, Keniucky 40208 Manuscript received April

1 A MODEL ALLOWING CONTINUOUS VARIATION IN ELECTROPHORETIC MOBILITY OF NEUTRAL ALLELES GARY COBBS Department of Biology, University of Louisville, Louisville, Keniucky Manuscript received April 25, 1978 Revised copy received January 22, 1979 ABSTRACT The infinite-sites model with no recombination is extended to include mutations that affect electrophoretic mobility. The model allows the effect of a single-site mutation to have a continuous effect on mobility. Formulae are obtained for the variance of electrophoretic mobility of alleles after an arbitrary length of time. A special case of the general model is the case of stepwise production of neutral alleles with an arbitrary number of steps. MODELS accounting for the existence of genetically determined variation in electrophoretic mobility of proteins have developed extensively during the past decade. A persistent problem has been that of allowing for the limited resolving power of gel electrophoresis. This problem was dealt with, at least in part, by the model of stepwise production of neutral mobility classes introduced by OHTA and KIMURA (1973). The essential assumption of this step moldel is that only amino acid changes that involve charged residues will be detectable by gel electrophoresis. The model assumes that electrophoretic mobility is changed by discrete jumps, all of equal magnitude, in either the positive or negative direction. Some justification for this assumption is given in their original paper (OHTA and KIMURA 1973). Since this initial paper, there have been numerous extensions and refinements of the mathematical treatment of this and closely related models (KIMURA and OHTA 1975; MORAN 1975; WEHRHAHN 1975). Results for various kinds of step models have been derived from assumptions about the relationship between mutations and electrophoretic mobility and the effect of genetic drift due to finite population size (OHTA and KIMURA 1973; WEHRHAHN 1975; MORAN 1975). We will show here how some of the same results can be obtained by starting irom results for the infinite-sites model with no recombination (WATTERSON 1975; LI 1977). We will also generalize the model to include the case where the effect of a site mutation on electrophoretic mobility is a continuous variable. GENERAL THEORY We start with the results of Lr (1977) for the infinite-sites model with no recombination. Let Pk(t) = Prob(dt = k) denote the probability that at genera- Genetics 92: June. 1979

2 6 70 G. COBBS tion time t the number (d) of site differences between two randomly chosen cistrons is k. The probability-generating function for the &(t) values is then LI (1 977) has shown, for the infinite-sites model with no recombination in which it is assumed that no two mutations ever occur at the same site, that -1 G(Z,t) = 2Na(Z) + 2Na(Z) ea'z) t + ea(z)tg(z,o), where a(2) = vZ and A = 1/2N + 2v. Here, N is the effective population size and v is the mutation rate. For t infinitely large, we have 1 G(Z,m) = 1+6'-02 ' where 6' = 4Nv. In order to obtain results pertaining to protein polymorphism, we will henceforth let k be the number of site differences in cistrons that cause amino acid differences in the proteins coded for by the cistrons. We also let v be restricted to mutations causing amino acid changes. Now let the random variable x1 be the difference in electrophoretic mobility of the protein products of two randomly selected cistrons that have a single-site difference. This random variable has range (- 00, f a) and may be discrete or continuous. Denote the probability density function (or probability function for the discrete case) of x1 by f(xl) and its characteristic function by +(s). The moments of x1 can be found from +(s). We will make most use of the fact that E(x12) = -+"(O). We will assume that the effect of a single-site mutation on mobility has a symmetrical distribution about zero so that its distribution is identical to that of xl. Should this not be true in nature, then. our model may still be used and we simply take f(xl) to be the symmetricized distribution of the effect of a single random-site mutation on electrophoretic mobility. This can be done since z1 is defined using two randomly selected cistrons. Now let the random variable xh be the effect of k site differences on electrophoretic mobility. The variable xk is the sum of k independent and identically distributed values of xl. This is an important assumption underlying this model, as well as all the step models. Essentially, we are assuming that an amino acid substitution in one part of a protein does not alter the effect on mobility of another substitution in another part of the same protein. Some data on the electrophoretic and mobilities of hemoglobin alleles does support this assumption (see COBBS PRAKASH ). Now regard k as a random variable with probability-generating function given by (1) and let the random variable x be the mobility difference between the protein products of two randomly selected cistrons. Then by a lemma

3 ELECTROPHORETIC MOBILITY 671 given by FELLER (1966, p. 477) and using (1), we may write the characteristic function of the distribution of x as From the manner of definition of zk and x, we know that they must have symmetrical distributions and expectation of zero. This results from the fact that we define ztgand x using two randomly selected cistrons. The two sequences of selection of any two cistrons are equally likely. Since E(z) = 0, the central moments of x are the same as the moments about the origin. We may obtain the moments of the distribution of z from the derivatives of (3), using well known properties of derivatives of characteristic functions. Denote the nta moment of x at time t by u,(t), which may be found from (3) as where 2 = -1 and H(")(O,t) denotes the nth derivative of H(s,t) with respect to s evaluated at s = 0 (see for example EISEN 1969, p. 334). In theory, the probability-density function of z could be obtained by inversion of (3), though we shall not do so here. Substitution of (3) into (4) yields Var (qt) = 4NvVar (z,) (1 - et/sn) + Var(x,O), (5) where Var (XJ) denotes the variance of x at time t. In principle, we could obtain results analogous to (5) for higher moments of x. Now let mi and pi be the mobility and frequency in a population at time t of allelic state i. Here, we will suppress the variable t for convenience. The values u,~ may be written in terms of mi and pi as Some algebra shows that (6) is equivalent to U,, =Z Z pipj(mi - mj)". (6) a? where U, = z pjmir and is the rth moment about the origin of the electrophoretic mobilities, Equation (7) reveals that u,(t) = 0 for k = 1, 3, 5,... and also we have I Var(z,t) = 2Var(m,t), (8) where Var(m,t) denotes the variance of electrophoretic mobilities at time t. Substitution of (8) into (5) gives Var (m,t) = 2NvVar (x,) (I - &IZN) + Var (m,o) e-t/zx, (9) from which we obtain the steady state value Var(m,m) = 2NvVar(z,). (10)

4 6 72 G. COBBS Equation (9) reveals that the rate of approach to the equilibrium variance is slow. If a population is initially monomorphic for electrophoretic mobility, then Var(m,O) = 0, and it takes 4.6N generations to attain 90% of the equilibrium variance in electrophoretic mobility. SPECIAL CASES Step models: Here we assume that changes in mobility occur in discrete jumps of unit value and that a single-site difference may cause any number, up to a maximum of r, of such jumps. The probability that a single-site difference causes a mobility change of j units in the positive and negative directions will be denoted by c+? and c-j, respectively, for j = 1, 2,..,r. The probability that the site difference causes no change in mobility is co. In terms of our general theory, we write Prob(x, = i) = cj. This set of 2r + I values constitutes an empirical probability distribution of xl. Here, we shall assume that c+j = cq for all i in order to obtain symmetry of the distribution of xl. Some information on the actual values of cj for the case I = 2 may be found in genetic literature (MARSHALL and BROWN 1975; LI 1977). The variance of the distribution of x1 is easily found from the values of cj and use of (9) yields Var (m,t) = 2Nv ( ] =-I Z Pcj) ( 1 -&IzN) + Var (m,o) ctien. (11) For a population at equilibrium, the variance of mobility classes is then Var(m,w) = 2Nv I: jzcj. (12) If we allow only jumps of a single unit in either direction (r = 1). then (12) reduces to Var (m, 00) = 2Nv ( 1 -c,). This is very close to the result given by MORAN (1975) of 2(N-l)~. which he obtains from a slightly different model in which he assumes c, = 0. If r = 2 then (12) becomes Var(m, w) = ~NV[C-~+C+,+~(C-,+C+?)], which is the same as a result given by WEHR- HAHN (1975) for a model allowing one or two jumps. I we assume that the steps are of length w rather than one, we then have v J=-r Var(m,w) = 2Nv I =-r I: (jo)zcj. (13) Here, o may be regarded as the amount of change in the relative mobility of proteins. This is the most common method of expressing mobility differences of alloz ymes. Continuous models: Here we assume that changes in mobility are continuous in nature. The model follows closely that of the step model except that f(xl) is now considered to be a density function of a continuous random variable. We assume that f(xl) has the form

5 ELECTROPHORETIC MOBILITY 6 73 where fj (x,) is a density function and cj is the probability that z1 follows the density fj (xl). As in the step models, we assume that c-j = c+j for all j. The cj values here have essentially the same interpretation as they did in the step models. We further assume the density fj (zl) has mean pi and variance uj2 and that fj(xl) =f-j(-xl) for j = 1,2,...,I. (15) Thus. pj = -,p-j and ai2 = uq2. The characteristic function of fj (xl) and f(x1) are denoted by +] (s) and (~(s), respectively, and from (14) we have from which we obtain E(x,) = 0 and Var(x,) = 7. 1 =-r cj(pj2 +U,'). (17) Further simplification results from the reasonable assumption that pi = io. Here, is the expected value of the absolute value of the change in relative mobility caused by a random amino acid substitution that changes the number of ionizable groups per monomeric subunit by exactly one. This random variable will be denoted by w. The parameter pj represents the average effect on relative mobility of an amino acid substitution that changes i ionization groups per monomeric subunit of the protein. We may arbitrarily regard values of j as the number of acidic groups added minus the number of basic groups added. For example, consider the substitution lys + glu. This is a change from a monobasic amino acid to a monoacidic one; thus, j = 2. Using pli = io and (1 7) in (9) gives Var (m,t) = 2Nv, cj [ (io) + uj2] (1 + Var (m,o), ( 18) 3 =-r and at equilibrium 1- Var(m,m) =2Nv,E =-r cj [(jo)2+uj2]. (19) If = 1 and uj2 = 0 for all j, then the continuous model becomes the step model with steps of unit value, as (18) and (19) are then identical to (11) and (12). DISCUSSION Our results give a relation between mutations in the infinite-sites model with no recombination and mutations affecting electrophoretic mobility. These results may be used to gain some idea as to whether or not, for a given locus, the observed variation in mobility is in agreement with what is expected for neutral alleles. Here we will consider only the predicted variance for an equilibrium population. In order to calculate a predicted variance with (19), we need estimates of 0.0, and ci for i = 0, i.1, *2,.. fr. Values of cli for an average protein inay be estimated from knowledge of the genetic code, structure of amino acids and the amino acid composition of an average protein. MARSHALL and BROWN (1975) estimate (G. C-~, co, c+~, c+~) to be

6 74 G. COBBS (0.0059, 0.1393, 0.7365, 0.1092, 0.0091) for an average protein. They also suggest that cj = 0 for lii > 2.

6 6 74 G. COBBS (0.0059, , , , ) for an average protein. They also suggest that cj = 0 for lii > 2. Our niodel assumes that c+j = c-j, so that we average the estimates of c+j and c-j to obtain our estimates of c+j and c-j to obtain our estimates of c+j and c+ to obtain our estimates of c+, and c-j. This procedure gives the estimates (0.0075, , , , ), which we will use in our calculations. The value of 0 = 4Nu may be estimated from the number olf alleles occurring in a sample. The estimating procedure is described by EWENS (1972) for the infinite-alleles model, which. for our purposes, is equivalent to the infinite-sites model with no recombination (see WATTERSON 1975). Here, it is necessary to assume that all alleles present in a sample are detected even if they have negligible effect on electrophoretic mobility. Methods are available that detect at least some of the alleles that are not detected by routine methods of gel electro- phoresis (see MCDOWELL and PRAKASH 1976; COBBS and PRAKASH 1977a). In any case, we may obtain a minimum estimate of 6' from the number of alleles in a sample. According to the theory given by EWENS (1972), if more alleles are detected in a sample by some additional procedure, then the estimate of 6' will be increased; it cannot be decreased. We shall estimate 6' for the esterase-5 locus of Drosophila pseudoolbscura from the data o MCDOWELL and PRAKASH (1976) and for D. miranda from the data of COBBS and PRAKASH (1977a). The values of w and ai2 are unknown for nearly all loci. Preliminary estimates of these quantities for the esterase-5 locus of the Drosophila obscura species group are given by COBBS and PRAKASH (1977b). Their data suggests that the relation is approximately true, and we shall use it here. We shall also assume ao2 ui2. COBBS and PRAKASH (1977b) report data that give the estimates o = and ai2 = 0.000,841 for dimer mobilities at the esterase-5 locus. Using procedures discussed in the APPENDIX, an unbiased estimate of is found, using the data of COBBS and PRAKASH (1977b), to be With these values in (20) and using our values of cj equation (17) gives the estimate Var(xl) = 0.001,465 for the esterase-5 locus. The estimates of 0 and the predicted variance for D. pseudo- TABLE 1 Comparison of predicied and observed variance of mobility for the esterase-5 locus of Drosophila pseudoobscura and D. miranda D. pseudoobscura D. miranda ,006, , , ,998 0,002, ,

7 ELECTROPHORETIC MOBILITY 675 obscurg and D. miranda are given in Table 1. 'The observed variance of electrophorectic mobility is calculated from the data of PRAKASH (1977). It is desirable to have some idea ob the accuracy with which we can predict the mobility variance for an equilibrium population. The accuracy will depend on the error in estimation of all the underlying parameters. Here, we will give only an approximate method that may be used to analyze the agreement between the observed and predicted values for the mobility variance. The predicted variance and its sampling variance will be denoted by V, and Var (Vp), respectively. The observed mobility variance will be denoted by V,. Using (20) in (19) and substituting estimates for the population quantities yields A V, = BY (21) Var (v,) + ~zvar(6) +var(e^) Var (Y). Here 6 is the estimate of 8 obtained using methods given by EWENS (1972) and where U(Ul2) and U(d) are unbiased estimates of the population variance and the square of the population mean, respectively, of the random variable w. The quantities U(u12) and U(d) may be obtained from sample values of w. A formula for U ( ui2) is well known and a formula for U (Uz) is given in the APPENDIX. A sample of 13 observations of w for the esterase-5 locus may be obtained by multiplying each of the 13 observations in the rightmost column of Table 3 of COBBS and PRAKASH (1977b) by 2. These numbers may then be used to e-t' imate u12 and d. From (22) we obtain where Cov (.,.) denotes the covariance. Formulas for unbiased estimates of Var [U( u12) 1, Var [ U(d) 1 and Cov [U(U{), U (0')] are given in the APPENDIX. The sampling variance of the estimate 8 is found numerically using formula of EWENS (1972). The value of e is used in EWEN'S eq. (23) to generate the distribution ni, which is the probability that the sample has i alleles conditioned

8 676 G. COBBS on 0. For each value of i = 1,2,.., n - 1, we then calculate the value 0i, which is the estimate of 0 if i alleles occurred in a sample of n genes. The values ii are then considered a random variable with probabilities ~ i and, the variance is calculated and taken to be the sample variance of h. We now have the ability to calculate V, and Var (V,) using (21), (22) and (23). As a measure of disagreement between the predictions of the model and the observed mobility variation, we will use the difference V, - Vp. In order to roughly assess the significance of the difference V, - V,, we will examine the variable If the value of L falls within the interval (-2, +2), we shall conclude there is no significant difference between V, and V,. This is equivalent to the condition that V, fall withir? two standard deviation units of either side of V,. The expected value of L is probably not zero. but it should be close to zero if the model is being followed in nature. The procedure outlined here is approximate in that, among other things, it does not include the sample variance of V, nor error in estimates of the ci values. However, if a nonsignificant deviation is found with the above procedure, it would almost certainly bc nonsignificant in any procedure that did take into account the variance of Vo. The results of this analysis for the esterase-5 locus of D. pseudoobscura and D. miranda are given in Table 1. Here, we see that both species have a lower rariance than the model predicts, though in neither case is this deviation statistically significant by our test. This analysis is not intended to be a rigorous test of the model, and more precise statistical procedures are clearly needed. Further improvements of the analysis would also result from more accurate estimates of Q and ui2, The estimates used here are derived from the data of COBBS and PRAKASH (1977b). Their procedure used chemically modified proteins as an attempt to simulate the effect of amino acid substitutions in proteins. The accuracy of this procedure should be checked by estimating 0 and u12 from naturally occurring amino acid substitutions. We are currently using this procedure in our laboratory to obtain improved estimates of 0 and ut2. The models presented here should be useful in determining whether or not the processes of neutral mutation and genetic drift are important causes of genetic variation for electrophoretic mobility at protein-producing loci. LITERATURE CITED COBBS, 6. and S. PRAKASH, 1977a A comparatiie study of the esterase-5 locus in Drosophila pseudoobscura, D. persmilis and D. miranda. Genetics 85: , 1977b An experimental investigation of the unit charge model of protein polymorphism and its relation to the esterase-5 locus of Drosophila pseudoobscura, D. persimilis and D. miranda. Genetics 87: CRAMER, H., 1946 New Jersey. Mathematical Methods of Statistics. Princeton University Press, Princeton,

9 ELECTROPHORETIC MOBILITY 677 EISEN, W. J., 1972 Introduction to Mathematical Probability Theory. Prentice-Hall, Englewood Cliffs, New Jersey. EWENS, W. J., 1972 The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3: FELLER, W., 1966 Introduction to Mathemrrtical Probability Theory and its Applications, vol. 11. Wiley, New York. KIMURA, M. and T. OHTA, 1975 Distribution of allelic frequencies in a finite population under stepwise production of neutral alleles. Proc. Natl. Acad. Sci. U.S. 72: LI, W.-H., 1977 Distribution of nucleotide differences between two randomly chosen cistrons in a finite population. Genetics 85: MARSHALL, D. R. and A. H. D. BROWN, 1975 The charge-state model of protein polymorphism in natural populations. J. Mol. Evol. 6: MCDOWELL, R. and S. PKAKASH, 1976 Allelic heterogeneity within allozymes separated by electrophoresis in Drosophila pseudoobscura. Proc. Natl. Acad. Sci. US. 73: MORAN, P. A. P., 1975 Wandering distributions and the electrophoretic profile. Theoret. Pop. Biol. 8: OHTA, T. and M. KIMURA, 1973 A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Genet. Res. 22 : PRAK~SH, S., 1977 Genetic divergence in closely related sibling species Drosophila pseudoobscura, D. persimilis, and D. miranda. Evolution 31: WATTERSON, G. A., 1975 On the number of segregating sites in genetical models without recombination. Theoret. Pop. Biol. 7 : WEHRHAHN, C. F, 1975 The evolution of selectively similar electrophoretically detectable alleles in finite natural populations. Genetics 80: Corresponding editor: W. W. ANDERSON APPENDIX Consider a sample of n independent observations zl, z2, ----, I, from a population whose kth moment about zero is ak and whose kt central moment is Pk. It is possible to estimate the quantities ak and Pk, using the sample values. Unbiased estimates of a6 and PI, will be denoted by U(a,) and U(&), respectively. In most statistical literature U(al) and U(p,) are denoted by Z and s2, respectively. In general we shall denote an unbiased estimate of any population quantity, say z, by U(z). The sample moment about the origin will be denoted by 1 k a7( = - Z ~,k. It is well known that U(a,) = ak. Formulae for U(P,) far k= 1, 2, 3, 4 in n 2=1 terms of sample quantities are given by CRAMER (1946, p. 352). The purpose of this APPENDIX is to provide formulae for U(a12) and unbiased estimates of Var[U(a12)], Var[U(p,)] and Cov[U(a,Z), U(P,)]. Here Var(.) and Cov(.,.) denote the 1-ariance and covariance of the quantities within the brackets. The formula U(p,) = (L) n-1 (a2-alz) is well established and needs no further mention here. Given below are the expected values of some sample quantities that we shall find useful. These results may be obtained using procedures discussed in CRAMER (1946, p. 347). (n-i) a12 E (a12) = - + (AI 1 n (n-1) (n-2) (n-3) a14f6 (n-1 ) (n-2) a12 a,f3 (n-l)a22+4 (n-1) al a3+a4 E(a,*) =- (A21 n3

10 G 78 G. COBBS E(a,2 a,) = - (n-i) (n-2) a12 a2 + 2(n-1) a1 as + (n-i) a 2 + a4 n2 From equations (AI) through (A5), we obtain the results given below. U(a12) = a12-3- n n3 a14 - U(a4) 6U(alZa2) 3U(a22) + 4U(a, a,) U(a14) = (n-i) (n-2) (n-3) n - 3 (n-2) (n-3) n a2, - a4 U(a2*) = n-i ('46) (A71 (-48) na cz -a4 U((Y1a3) =-l n,- 1 These unbiased estimates will be useful in estimating the quantities Var ( U(al2)), Var(U (p2)) and cov (U(a,2), U(P2)). Beginning with the relation Var [U(al*)] = E[U(a,Z) - al2i2, substituting in (A6) and expanding and taking expectation, using (AI) through (A5), yields (n-i ) (n-2) ( n-3)a14+6 Var(U(a,z)) = (n-i ) (n-2) a12a2+3 (n-1 ) 01,~+4 (n-1 )a1a3+a4 ~ _. (All) n3 We may now obtain an unbiased estimate of Var(U(n,z)) by substituting unbiased estimates for the terms appearing on the right side of (All). Beginning with the relation Var(V(P,)) = E[U(P,) - P21z, substituting formulae for U@,) in terms of ak and P2 in terms of ak, expanding and then taking expectstions by using (AI) through (A5) gives n 2 ~ar[~(p,)l = (--) [~(a,2) - 2~(a,za,) + ~(a,4)1- ((Y2-a12)2. n-i In order to estimate Var[U(P2)], we use (A2), (A3) and (A4) to obtain an expression for Var[U(P2)] in terms of population quantities. We then substitute the unbiased estimates of the terms into the expression to obtain an unbiased estimate of Var[U(P,)]. Beginning with the relation covru(cu,~), U(P,)l = E[U(C12) U(P2)I P? and inserting the formulae for U(,,?) and U(P2) in terms of ak and for P2 in terms of afg and then expanding and taking expectations yields Cov(U(a12), U(P2)) = (E-)[? E(a,za,) -E(a14) --] E(a2') -in12a2+a14. (A121 n-i n We obtain an unbiased estimate of Cov[U(a12), U(P,)] from (Ale) using a procedure similar to that used for estimating Var[U(aIz)] and Var[U(p,)].

DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS 1N A F'INITE POPULATION'

DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS 1N A F'INITE POPULATION' WEN-HSIUNG LI Center for Demographic and Population Genetics, University of Texas Health Science Center,