Pseudosampling Method

Proc. Nat. Acad. Sci. USA Vol. 80, pp. 1048-1052, February 1983 Genetics

Selective constraint in protein polymorphism: Study of the effectively neutral mutation model by using an improved pseudosampling method (population genetics/neutral theory of molecular evolution/polyallelic random drift) MOTOo KIMURA AND NAOYUKI TAKAHATA National Institute of Genetics, Mishima, 411 Japan Contributed by Motoo Kimura, September 30, 1982

ABSTRACT To investigate the pattern of allelic distribution tiallelic frequencies when mutation, selection, and random drift in enzyme polymorphismwith special reference to the relation- balance each other (see ref. 14, p. 394, for a review). Later, ship between the mean (H) and the variance (VH) of heterozy- Watterson (15) and Li (16) developed a method for calculating gosity, we used the model of effectively neutral mutations involv- moments of allele frequencies from Wright's formula. Theo- ing multiple alleles in which selective disadvantage of mutant retically, the formula and the method could be applied to treat alleles follows a r distribution. A simulation method was devel- the present model. It turned out, however, that when the pos- oped that enables us to study efficiently the process of random sible number of selectively different alleles involved is large or drift in a multiallelic genetic system and that saves a great deal the product of the effective population size and selection coef- ofcomputer time. Itis an improved version ofthe pseudosampling- ficient is large, or both are large, the application ofthis method variable (PSV) method [Kimura, M. (1980) Proc. NatL Acad. Sci. is very difficult. USA 77, 522-526] previously used to simulate random drift in a Here, we intend to present a method that can simulate the diallelic system. This method will be useful for simulating many process ofpolyallelic random drift very efficiently. This method models of population genetics that involve behavior of multiple (17) and alleles in a finite population. By using this method, it was shown is an improved version ofthe pseudosampling method that, as compared with the model ofstrictly neutral mutations, the it contains a device (to be called a "telescoping" method) ofsam- present model gives the reduction ofboth H and VH and an excess pling multiple alleles. As compared with other methods (18- of rare variant alleles. The results were discussed in the light of 20), it can treat a multiallelic system much more easily. Fur- recent observations on protein polymorphism with special refer- thermore, it has the merit of enabling us to incorporate a suit- ence to the functional constraint of proteins involved. able adjustment for rare alleles easily (see also ref. 21), which makes the simulation valid over all the combination of param- From the standpoint of the neutral mutation-random drift hy- eter values. This adjustment is particularly important when we pothesis (1) (or the neutral theory, for short; see refs. 2 and 3 try to obtain a satisfactory distribution for a whole domain of for review), protein polymorphism is a transient phase of mo- allele frequencies. lecular evolution (4); therefore, it is expected to be influenced by selective constraint in a similar way as in molecular evolution Telescoping method of sampling multiple alleles (5). In other words, the stronger the functional and structural constraints to which a protein is subject, the smaller the prob- The essential idea underlying the PSV method is that we sim- ability of an amino acid change by mutation being selectively ulate the diffusion model of the process of gene frequency neutral (i.e., not harmful) and, therefore, the lower the het- change by generating a sequence ofsome simple random num- erozygosity at its locus. bers, rather than faithfully following binomial or multinomial The observation that loci for substrate-specific enzymes are process by drawing, in each generation, individual gametes on the average less heterozygous than those for substrate-non- from a finite population. For example, in the simplest case of specific enzymes (6) is consistent with this expectation. Fur- two neutral alleles in a diploid population ofsize N, we generate thermore, Yamazaki (7) and more recently Gojobori (8) showed in each generation a random number from the uniform distri- that substrate specific enzymes not only have lower mean het- bution having the mean 0 and the variance x(l - x)/(2N), where erozygosity but also have smaller variance of heterozygosity x is the frequency ofone ofthe alleles at a given moment. Then, among species than expected from the standard neutral infinite this random number is added to x to form the allelic frequency allele model (9). in the next generation. This saves computer time enormously In this paper, we intend to show that these observations can and allows us to simulate easily a process of change in a very be explained by the model ofeffectively neutral mutations (10) large population. Also, we can make many replicate trials within which selective constraint (negative selection) is incorpo- out prohibitive computing time. Originally, the PSV method rated. This model is based on the idea that selective neutrality was intended to simulate the diffusion process itselfrather than is the limit when the selective disadvantage becomes indefi- the discrete binomial sampling process ("Fisher-Wright model") nitely small (5) and is an extention ofOhta's model (11) in which for which the diffusion model is usually regarded as an approx- the selection coefficients against the mutants follow an expo- imation (17). nential distribution. These models are ultimately traced to In the present paper, we shall be concerned with a multiallelic system and assume that there are K possible allelic states Ohta's hypothesis (12, 13) that very slightly deleterious muta- of ef- tions as well as the strictly neutral mutations play an important (K > 2) at a locus in a random-mating diploid population fective size Ne. Let Ak be the kth allele (k = 1, 2, .. ., K), whose role in molecular evolution and polymorphism. We first sam- One problem in our approach is that rigorous mathematical frequency in the population before sampling is Xk. study of the model is rather difficult. Previously, Wright de- rived a general formula for the equilibrium distribution ofmul- Abbreviation: PSV, pseudosampling variable. 1048 Genetics:Genetics:KimuraKimuraandandTakahataTakahata~~~~Proc.Nati. Acad. Sci. USA 80 (1983) 1049 pie allele Al so that its number follows binomial distribution we obtain Yk' by if is small, B(illn, x1 =()x~"1 -i [1] Yk' =- 7lk/nk nkYk or [6b] where n = 2Ne is the total number of gametes that contributes Yk' = 1 - ~k/nk if ~k(I - y~) is small, *to the next generation and il is the number ofgametes that contains allele Al after sampling. Actually, there is no need for the where ilk and 4k are Poisson random numbers with the mean distribution to follow faithfully the binomial distribution, but nk Yk and lk(1 - yk), respectively (see also ref. 21). we simply generate a uniform random number, U1 with mean The approximations given either by Eqs. 6a or 6b are satis- 4)~ and variance unity, and substitute the- frequency of Al after factory when nk is reasonably large, say larger than 20. If, on sampling (corresponding to il/n) by the other hand, nk 45 20, we must resort to a more exact procedure, and in this paper, we adopt the following, method: we Xr= x1 + U1 V{[xi(l - xi)]/n}. [2] generate nk random variates that follow the uniform distribution in count number of such variates that we is is the range (0, 1), and the One caution need to take here that when Al represented is a binomial a small number of individuals so that is less than happen to fall in the range (0, yb). This number, ik, by very nx, 3, random so we let say, then a more exact sampling procedure must be used (see number, below). A similar caution is also needed when x1 is near unity, Yi' = iknk (if nk -< 20), [6c] so that n(l - x1) is small, say less than 3. Next, we sample A2 so that the number of gametes (i2) containing this allele follows from which we obtain Xk' by using Eq. 5b. the distribution B0i21n, Y2), where n2 = n - il is the remainder The advantage ofthe telescoping method in saving computer of the gametes after AI is removed, and Y2 = X2/(1 - xj). Like- time comes from the fact that we can obtain each of the Yk'S wise, A3 is sampled so that the number ofgametes (i)containing (and therefore XkI, the frequency of Ak, by using Eq. 5b) by this allele follows the distribution B(iajn3, Y3), where n3 = n generating only one random number when nk is larger than 20. - il - i2 and y3 = X3/(1 - XI- X2). A similar procedure is re- In addition, the accuracy and reliability are ensured by incor- peated until all K - 1 alleles are sampled. In general, if ik (k porating -adjustments 6b and 6c for treating rare variant alleles. - 1, 2,..,K - 1) is the number ofAk-bearing gametes in the These adjustments are particularly pertinent ifwe note that, for sample, this number follows Yk =:0 or 1, application ofEq. 6a frequently produces Yk' values either ~negative or larger than 1, thus creating a possibility of serious bias in a simulation experiment. Such adjustments were not in the treatments Kimura and Ma- where = Xl- incorporated by. (17) by nk n-il-i2...--ikl1andYk Xk/(1- X2 ruyama and Takahata (22). Later, the problem was treated in . .. = n and = for k =1). Then, it can -Xk-1). (Note: nj yi x, a heuristic manner by Maruyama and Nei (19) and Takahata (20). be shown that Their heuristic treatment is concerned with allele frequency E1ik = n~,[4] changes due to mutation; thus, in the absence ofmutation, their simulation methods ofrandom drift are not satisfactory for treat- Varlik} = flXk(l - X) ing the changes of rare alleles. and Model of effectively neutral mutations CovfikiJ = flXkXl, (k # l In the original formulation (10) of the model of effectively neu- where 1 --- k K - 1 and 1 I < K - 1. Furthermore, ifwe tral mutations, it was assumed that the selective disadvantage let of mutants among different sites within a gene (cistron) ollows aF distribution. Here, we modify the model slightly and assume Yk' =ik/nk, [5a] that there are K.possible allelic states (Al, A2 ..,AK) at a locus, then, fork= -i1, and that each allele mutates with the rate v = u/(K - 1) to any 2,..K one of the remaining K - 1 alleles (23). We consider the genic selection throughout this paper and designate the selection Xk' ik = Yk'nIk/fl = Yk' - E X [5b] coefficient against Ai by si' so that the relative fitness of Ai is 1 - si'. We assume that a set of si' values are a random sample gives the frequency ofAi after sampling of gametes. extracted from the universe of s' values that follow the F dis- Eqs. 1-5b provide the procedure (telescoping method) for tribution sampling n or 2Ne gametes from the gene pool containing multiple alleles by simply repeating the binomial sampling proce- f(S') = -Stole-as [7] dure of Eq. 3. Thus, the key point of the telescoping method PAf3 is how to simulate Eq. 3 to generate Yk'- Unless nk is small, and where a = , V' (>0) is the mean selective disadvantage, and more importantly, unless none of nk Yk and - is very f/3/ nlk(l Yk) p is a such that 0 < f3 -- 1. A set small, less than we can obtain parameter of values of si's say 3, Yk' by ,allocated to the entire allelic states are assumed not to change throughout each run of simulation experiments. Thus, we can =- + - Yk'1 Yk Uk'\VYk~l Yk)/nk, [6a] treat selection deterministically once these-values are decided. where Uks are mutually independent arbitrary random vari- Under the above assumptions -of mutation and selection, the ables, each with mean 0 and variance 1, and we usually find it mean changes of the allelic frequencies per generation are convenient to choose them from the uniform distribution. Then M(sx~) = v - (u + + - iLE)/ii3, [8a] from Eq. 5-b, we can obtain xk'. v)xi (Wi When nk is large but either one of nkYk or lk(1 - Yk) is small, 'where w , =ll- s' and Th =1- >, sj' xj and i = 1, 2,.. 1050 Genetics: Kimura and Takahata Proc. Natl. Acad. Sci. USA 80 (1983) K. In a population ofeffective size Ne, the changes ofallelic fre- tioned above, the effective population size (Ne) was kept con- quencies due to random sampling ofgametes have the variances stant, such as 104, whereas we changed the mutation rate and selection coefficients a great deal for different runs. In our ap- V(8x,) = xi(1 -xi)/(2Ne), [8b] proach using the diffusion equation method, what matters are and covariances the products Nes' and Neu rather than individual parameters Ne, s' and u. Therefore, such a change is equivalent to multiplying W(8xiA8x;) =- ixj /(2Ne), (i $j), [8c] the diffusion equation by a constant factor and altering the time scale and Ne but holding u and s.' unchanged. For example, to where i, j = 1, 2, .. ., K - 1. simulate the case u = 10-6, Ne = 106, and s' = 10-4, we may choose parameters such as u = 10-4, Ne = 104, and s' = Simulation experiments 10-2. However, such a correspondence is valid when s' is suf- ficiently small (If it is relatively large, say s' = 0.1 or more, In our simulation experiments, we tried various values of i' and sampling ofsi' values may cause significant differences). In or- u, while we mostly assumed Ne = 104, K = 20, and f3 = 0.5. der to carry out a computer simulation in a relatively short time, In each run of computer simulation, we first generated K se- such a scaling is necessary, and this corresponds to measuring lection coefficients (si's) by drawing K random variates from the time in the unit of 100 generations, (see refs. 19 and 20). (We r distribution with given parameters ( and s'. The change of also checked the accuracy ofour method by examining the dis- xi due to mutation and selection was calculated deterministical- tribution of allelic frequencies, and the results were satisfac- ly by using Eq. 8a, whereas the change due to random sampling tory.) of gametes was done by the telescoping method. Starting from an arbitrary composition ofallele frequencies, RESULTS AND DISCUSSION we discarded the first 4Ne generations to ensure that an equi- The mean heterozygosity (-H or 1 - F} obtained by the exper- librium had been reached. Thereafter we observed the allele iments is shown in Fig. 1. These results were obtained for a frequencies every Ne/100 generations. Each run was continued particular set ofsi' values sampled from the F distribution. To until the total number ofobservations reached 5,000, and var- see the effect ofa different set ofsi' values on the extent ofhet- ious quantities ofinterest were obtained by taking the average erozygosity, we repeatedly simulated the case s' = 0.1, u = over them. However, we must keep in mind that these quan- 0.025, and Ne = 104 and also the case Ne = °°(no random drift), tities are still subject to statistical fluctuations, because K se- s' = 0.1, and u = 0.0002. In both cases, the mean heterozy- lection coefficients are samples from a r distribution. This cau- gosity fluctuated to a large extent because of the difference in tion is important particularly when i' is large and K is small. the set si' drawn from a given F distribution. This was due to The unit of time was taken as one generation, and, as men- the relatively large value of s'. However, this sampling effect, comingfrom adifferent set ofsi' values, was not very large when 1.0 s' c 0.01, even if K = 20. We checked this for the case Ne = 104, U = 10-4, and ' = 0.01 by performing five different runs, each with a different set ofsi' values sampled from the same F .9 distribution. The results for the mean and the variance ofhomo- zygosity (F) together with their observed standard deviations .8 .06 I .7

in .05 0 0 0 c0 .6 (A N 0 0 CYV 4, N .04 O-55 0 4, a, 0 / 4, so 0 c .4 .03 'U 0 to .C

._ .3 '4 .02 0

.2 * 0 .01 U S U .1 U

0 0 .1 .2 .3 .1 .5 1 5 10 50 100 500 1000 Mean heterozygosity Nest FIG. 2. The relationship between the mean and variance of het- assume that K = 20 and FIG. 1. The mean heterozygosity (H7) as a function of Ne~'. The erozygosity. These simulation experiments neutral mutation model with K = 20 and = 0.5 is as- ( = 0.5 (over the range of heterozygosity less than 0.3) and assess the effectively to mutation sumed. The lines were drawn based on the results of simulation ex- intensity of mean selective disadvantage s' relative the rate u: o, s' - u; *, F' = 10 ,; *, F' = 50 or 100 u. The solid curve rep- periments. , Decrease of with increasing Nei' when Neu is kept resents the theoretical for neutral mutations. constant; --, increase of when 3'/u is kept constant. relationship strictly Genetics: Kimura and Takahata Proc. Natl. Acad. Sci. USA 80 (1983) 1051 were as follows; F = 0.711 + 0.0847and VF = 0.0357 + 0.0067. in the course of evolution and if recovery from a reduced pop- It is obvious that R under effectively neutral mutations is ulation size after each bottleneck is slow, the average hetero- always lower than R in the case ofstrictly neutral mutations for zygosity will be much reduced. In nature, even ifsome species the same parameter Neu. For a given V'/u, H increases slowly are distributed widely, covering an enormous area and com- with increasing Ne However, as compared with the case in prising an immense number ofindividuals (as in some neotrop- which all ofthe deleterious alleles have the same selection coef- ical Drosophila), it is rather unlikely that they always have been ficient, the effect of selection on reducing genetic variation is so in the last millions ofyears and will continue to be so in the relatively small for effectively neutral mutations; a large value coming millions ofyears. Sooner or later, such a state would be of V'/u was required to reduce the amount of genetic variation disrupted by the process of speciation. significantly. Such low efficiency of negative selection may be The variance ofheterozygosity, VH, is very close to that pre- understood byconsidering the situation where Ne is indefinitely dicted from strictly neutral mutations (25) when3'/u is less than large so that the genetic variation is maintained by the deter- 10. A significant reduction ofvariance as comparedwith the case ministic balance between mutation and selection. In this case, of strictly neutral mutations occurs where S'/u is much larger the mean frequency of a deleterious mutant allele, Ai, is u/si' than 10 in a large population. In Fig. 2, the relationship between so that the mean frequency per allele is u/§, where s is the har- the mean and variance ofheterozygosity is shown in cases where monic mean of si' (excluding the most fit allele). Because § is 3'/u is about 1, 10, and in the range of 50-100, respectively. influenced much more by a small si' than a large one, § is always The distribution of allele frequencies also was studied, al- smaller than the arithmetic mean s. though no illustrations are shown. When 4Neu < 1, the distri- It is known that the mean heterozygosity per individual at bution is always U-shaped, and we can hardly distinguish the allozyme loci is restricted mostly to the range 0 - 0.3 in diverse case of large s'/u from that of neutral mutations. When 4Neu species (24). If the mean heterozygosity in a large population 2 1 and s'/u is large, the distribution has a peak at a frequency having Neu = 1 or more is maintained by effectively neutral very close to 1, and also there appears an excess of rare alleles mutations, it is required that 3'/u is 100 or more (Fig. 1). Ifwe compared with neutral mutations (see also ref. 26). The peak take u = 10-6 as a representative value, we must assume that becomes still closer to 1 as s'/u further increases, and the dis- i' is around 10-' in order to explain the observed level of the tribution becomes very much like U-shaped. mean heterozygosity by negative selection alone. A more plau- We also checked if K = 20 is sufficient to approximate the sible explanation is that the relevant effective size (Ne) for dis- infinite allele model for our present purpose by calculating the cussing the average heterozygosity is much smaller than the mean number of segregating alleles after random sampling ga- apparent population size due to a number of factors. Particu- metes. When Neu = 4 and Nes' = 1,000, it was about 16, but larly, if the population goes through a sequence ofbottlenecks most alleles have their frequencies less than 0.001. Therefore, 0.07

0 Est 0.06 XDH Q

4.' 0.05 0 C O FIG. 3. Theobservedrelationship N ADH between the variance of heterozygosity andthe average heterozygosity. 0 1._. 0.04 0 m, Substrate-specific enzymes; *, sub- AKD strate-specificenzymesconcernedwith 4.'2 main pathways but not with subpath- ways; o, substrate-nonspecific enzymes. A short upright bar associated 40 0.03 with these symbols means that the en- ED zyme is concerned with only one path- U way. The observed values are based on data compiled by Gojobori (8), and the co 18enzymesinvolved (fromlefttoright) 'U 0.02 are as follows: MDH, malate dehydrogenase; LDH, lactate dehydrogenase; GOT, glutamate-oxaloacetate amino- transferase; IDH, isocitrate dehydrogenase; FUM, fumerase; G6PDH, glu- 0.01 cose-6-phosphate dehydrogenase; TPI, triosephosphate isomerase; aGPDH, a-glycerophosphate dehydrogenase; HK, hexokinase; AKD, adenylate ki- nase; Pep. Lap, peptidase; ACPH, acid ---~j phosphatase; PGI, phosphoglucoiso- < merase; PGM, phosphoglucomutase; 0.0 0.1 0.2 n30.3 Amy, amylase; XDH, xanthine dehydrogenase; ADH, alcohol dehydro- Mean heterozygosity genase; Est, esterase. 1052 Genetics: Kimura and Takahata Proc.P Natl. Acad. Sci. USA 80 (1983) it is likely that, for any practical purpose, the mean and variance fit roughly to the theoretical curve. Then, downward departure ofheterozygosity thus obtained are largely independent of the ofsolid squares from the theoretical curve would become even number ofK ifK is 20 or more, although the expected number more pronounced. Comparison of these observations with ex- ofrare alleles may be somewhat less as compared with the case perimental results in Fig. 2 supports the view that the very K = o. slightly deleterious but nearly neutral mutants are playing a Finally, let us consider the bearing ofthe present simulation significant role in the maintenance of polymorphism for sub- results, particularly those pertaining to the relationship be- strate-specific enzymes. tween the mean and variance of heterozygosity (see Fig. 2) on the observed pattern of protein polymorphism. It has been This is contribution no. 1445 from the National Institute ofGenetics, shown by Nei and his associates (24, 27) that the observed re- Mishima, Shizuoka-ken, 411 Japan. This work is supported in part by lationship between the mean and variance of heterozygosity Grant-in-Aid 57120009 from the Ministry of Education, Science and agrees fairlywellwith what is predicted from the neutral infinite Culture of Japan. allele model. Yamazaki (7) pointed out, however, using Dro- enzymes, the variance 1. Kimura, M. (1968) Nature (London) 217, 624-626. sophila data, that in substrate-specific 2. Kimura, M. (1979) Sci. Am. 241(5), 94-104. ofheterozygosity (VH) tends to be lower than expected from the 3. Kimura, M. (1982) in Molecular Evolution, Protein Polymor- model of strictly neutral mutations. More recently, Gojobori phism and the Neutral Theory, ed. Kimura, M. (Jpn. Sci. Soc. (8) presented his analysis suggesting that enzymes that are sub- Press, Tokyo), pp. 3-56. ject to strong functional constraints show a similar tendency of 4. Kimura, M. & Ohta, T. (1971) Nature (London) 229, 467-469. lower-than-expected VH. He analyzed data taken from 14 Dro- 5. Kimura, M. & Ohta, T. (1974) Proc. Natl Acad. Sci. USA 71, and 31 other species. Fig. 2848-2852. sophila species, 14 Anolis species, 6. Gillespie, J. H. & Langley, C. H. (1974) Genetics 76, 837-848. 3 was constructed based on the results of his analysis presented 7. Yamazaki, T. (1977) in Molecular Evolution and Polymorphism, in his tables I and II(8). In this figure, we did not include hemo- ed. Kimura, M. (Natl. Inst. ofGenet., Mishima, Japan), pp. 127- globin and transferrin, which are carrier proteins rather than 147. enzymes. 8. Gojobori, T. (1982) in Molecular Evolution, Protein Polymor- The curve in Fig. 3 shows the theoretical relationship be- phism and the Neutral Theory, ed. Kimura, M. (Jpn. Sci. Soc. under the Press, Tokyo), pp. 137-148. tween the mean and the variance of heterozygosity 9. Kimura, M. & Crow, J. F. (1964) Genetics 49, 725-738. infinite neutral alleles model. Enzymes represented by open 10. Kimura, M. (1979) Proc. Nati Acad. Sci. USA 76, 3440-3444. circles (nonspecific, not restricted to a single main pathway) 11. Ohta, T. (1977) in Molecular Evolution and Polymorphism, ed. tend to have high average heterozygosity. Also, they tend to Kimura, M. (Natl. Inst. of Genet., Mishima, Japan), pp. 148- have higher than expected heterozygosity. On the other hand, 167. those enzymes represented by solid squares each with an up- 12. Ohta, T. (1973) Nature (London) 246, 96-98. to one main 13. Ohta, T. (1974) Nature (London) 252, 351-354. right bar (substrate specific, restricted only path- 14. Wright, S. (1969) The Theory of Gene Frequencies. Evolution way) tend to have low heterozygosity. Furthermore, they tend and the Genetics of Populations (Univ. of Chicago Press, Chica- to have lower variance than the theoretical curve. Ofthese two go), Vol. 2. groups ofenzymes, the latter must be subject to much stronger 15. Watterson, G. A. (1977) Genetics 85, 789-814. selective constraint than the former. Therefore, if the effec- 16. Li, W.-H. (1977) Proc. Nati Acad. Sci. USA 74, 2509-2513. tively neutral mutation model is applicable, enzymes in the lat- 17. Kimura, M. (1980) Proc. Nati Acad. Sci. USA 77, 522-526. 18. Itoh, Y. (1979) Inst. Stat. Math. Res. (Jpn.) Res. Memorandum ter group must have much larger s' values than those in the for- 154, 1-20. mer group. 19. Maruyama, T. & Nei, M. (1981) Genetics 98, 441-459. In comparing these observed values with the theoretical pre- 20. Takahata, N. (1981) Genetics 98, 427-440. dictions given by the curve, we must keep in mind that this 21. Pederson, D. G. (1973) Biometrics 29, 814-821. theoretical curve is based on the assumption that 4NeU values 22. Maruyama, T. & Takahata, N. (1981) Heredity 46, 49-57. are the same among different species for each enzyme locus. 23. Kimura, M. (1968) Genet. Res. 11, 247-269. 24. Fuerst, P. A., Chakraborty, R. & Nei, M. (1977) Genetics 86, In reality, however, these values must be different from species 455-483. to species because of the difference in the effective population 25. Stewart, F. M. (1976) Theor. Popul. Biol 9, 188-201. size (Ne), even ifthe mutation rate (u) is the same among them 26. Ohta, T. (1975) Proc. Natl. Acad. Sci. USA 72, 3194-3196. for a given enzyme. Thus, the true theoretical curve should be 27. Nei, M., Fuerst, P. A. & Chakraborty, R. (1976) Nature (London) raised. It is likely that, if this were done, open circles would 262, 491-493.