Elevated Frequency of Tay-Sachs Disease Among Ashkenazic Jews Unlikely by Genetic Drift Alone

Am J Hum Genet 30:256-261, 1978

Elevated Frequency of Tay-Sachs Disease among Ashkenazic Jews Unlikely by Genetic Drift Alone

ARAVINDA CHAKRAVARTI1 AND RANAJIT CHAKRABORTY

INTRODUCTION Tay-Sachs disease (TSD) or infantile amaurotic idiocy has a significantly higher incidence among Ashkenazic Jews and the Pennsylvania Dutch semi-isolates compared to other populations [1, 2]. The autosomal recessive lethal gene causing this disease in homozygotes has a frequency of approximately .0133 among Ashkenazic Jews and .0015 in non-Jewish populations [2]. Controversy still persists as to why the TSD gene, despite its mass elimination through homozygotes, is found in such high frequency among some restricted populations. Chase and McKusick [3] offered the argument of genetic drift and founder effect in the Ashkenazic Jews, while others [2, 4-6] favor the hypothesis of heterozygote advantage, at least in an early period of Jewish population expansions. Their arguments are based mainly on empirical grounds; analytical results are from deterministic models of mutation-selection balance. A more reasonable approach, on the other hand, is to use a stochastic theory of gene frequency change in such an event, which forms the rationale of this paper. In a general set up, Rao and Morton [7] approached this problem to give a stochastic treatment, but they used the gene frequency distribution for a selectively neutral gene to approximate the behavior of TSD genes (which are known to be recessive lethals) in a finite population and then went on to derive the large deviation probabilities. In this paper, we evaluate the chance of finding the elevated gene frequency among Ashkenazic Jews, assuming no heterozygote advantage and using the steady-state gene frequency distribution of a recessive lethal. In recent years, an increasing rate of gene exchange between different Jewish communities has been reported [8]. We, therefore, also indicate the effect such migration has on the probability we compute.

PROBABILITY OF LARGE DEVIATION IN THE TSD GENE FREQUENCY Consider a randomly mating population with variance effective size N in which a lethal gene a (e.g., TSD gene) and its allelic normal gene A are segregating. Let the frequency ofa beX and the fitness of the three possible genotypes AA, Aa, and aa be 1,

Received October 18, 1976; revised November 22, 1977. This work was supported by grants from the U.S. National Institutes of Health and the National Science Foundation. 1 Both authors: Center for Demographic and Population Genetics, P.O. Box 20334, University of Texas Health Science Center, Houston, Texas 77025. Address reprint requests to R. Chakraborty. © 1978 by the American Society of Human Genetics. All rights reserved. 256 TAY-SACHS AMONG ASHKENAZIC JEWS 257 1 + h, and 0, respectively. Then the gene frequency distribution ofa in an equilibrium population is given by Wright [9] approximately as kh(X) X exp[-2N(1 - 2h)x2 + 4Nhx/X4NUl, (1) where u represents the mutation rate per generation from A to a, and h represents the selective advantage to heterozygotes. Thus, assuming no heterozygote advantage, the distribution (1) reduces to 4() 2(2N)6101/N2 F()/2) , (2) where F(,.) is a gamma function, and 6 = 4Nu. To compute the probability that X, the gene frequency in general, is greater than XT, the frequency found in a restricted population, we have 1 Prob(X . xT) =I0 XT XT =I - J(Px& 0 = 1 - y(2NXT2, 0/2)/r(0/2) , (3) where y (., .) is an incomplete gamma function defined as y(x,p) = l e -4P-Wt. Thus, for given values of the parameters N and u, the probability of a large deviation can be computed by equation (3). In theory, the parameters are estimable from the density +(x), if we have data on a series of population surveys. It is easy to show that the rth raw moment of the density ¢(x) is

= E(Xr) - (2N()r(/2) ; r = 1,2, (4) and hence using the method of moments, we obtain estimates of u and N, from the second and fourth moments as u = A2', and N = .2'/[2(L4' - A22)ij Large sample variances of these estimates are obtained as V(u) = ul(2Nn), and V(N) - N[(4Nu - I)n2 + 6n + 1 ]1(2un 3), where n is the number of surveys used to estimate the parameters. In practice, at present we have only three or four large surveys for several non-Jewish populations (see Aronson and Myrianthopoulos [10] for review). This estimation approach, therefore, is quite inefficient if applied to the limited body of available data. On the other hand, one might choose to examine the relationship between the mean (,u l') and N or ,u 1' and u to obtain reasonable estimates of the parameters in question. From the relationship pul' = r[(6 + 1)/2]1[V2W F(0/2)], we plot pul' againstN and u in figures 1 and 2. For u = 1O-5, Nei [11 ] has demonstrated the relationship of pLl' with N in a comparative study with other models. From ,ul' = .0015 (average gene frequency in non-Jewish populations), the values of u and N as obtained from figures 1 and 2 are shown in table 1. Then using equation (3), with XT = .0133 (average gene frequency among Ashkenazic Jews), the probability of large deviation for different estimated 258 CHAKRAVARTI AND CHAKRABORTY

0041. 104

0~~~~~~~~~~~~~~~~~0 -

C C3~~~I

10020

FIG. 1. -Relationship between mean gene frequency (1Ar) and effective population size (N) for a completely recessive lethal gene in equilibrium populations. The u values for different curves represent mutation rates. The dotted lines indicate N values corresponding to ,U 1' = .0015. values of u and N are also obtained using the built-in incomplete gamma function of a WANG 62 electronic calculator (see table 1, last column). In every case the elevated frequency is found to be significantly higher than the mean (the probability of elevation by chance being less then 5%). Although at present we do not have any precise estimate of u or N, the values N = 5,000 and u = 9.17 x 10-6 may be taken as

.000

*006

.004

EZ[X.002

10 6 lo-5 10-4 FIG. 2.-Relationship between mean gene frquency (AMl') and mutation rate (u) for a completely recessive lethal gene in equilibrium populations. The N values for different curves represent effective population sizes. The dotted lines indicate u values corresponding to ,t 1' = .0015. TAY-SACHS AMONG ASHKENAZIC JEWS 259 TABLE 1 PROBABILITY OF LARGE DEVIATION FOR TAY-SACHS GENE FREQUENCY AMONG ASHKENAZI JEWS FOR SEVERAL ESTIMATED VALUES OF PARAMETERS

FROM FIGURE 1 FROM FtGuRE 2 u N Prob(X 2XT) N u Prob(X xr)

10-4 ...... 42 .0304 100 ...... 6.03 x 10-5 .0334

5 X 10-5 ...... 178 .0396 500 ...... 2.71 X 10-5 .0358 2 x 10-5 ...... 832 .0309 1,000 ...... 1.92 x 10-5 .0308 10-5 ...... 5,623 .0067 5,000 ...... 9.17 X 10-6 .0072 5 X 10-6 ...... 25,704 6.2 x 10 10,000 ...... 6.80 X 10-6 .0012

NOTE. -XT = .0133 refers to the average frequency of the TSD gene among Ashkenazi Jews (see text for details).

representative of the population concerned. These values ofN and u yield 6 .183 and Prob(X . .0133) .007, as given in table 1. The effective size N, in this context, refers to the average (harmonic mean) effective size during the entire history of the population [12].

DISCUSSION The above computations do not constitute a proof of the hypothesis of selective advantage of Tay-Sachs heterozygotes. The low probabilities only signify that the elevated TSD gene frequency in Ashkenazic Jews is an unlikely event by drift alone. Selective advantage and founder effect are the only other viable alternative hypotheses which can possibly explain this phenomenon. McKusick [13] lists 22 genetic disorders which are found in elevated frequencies among the Ashkenazim. Hence, if there had been any founder effect, the early ancestors of Ashkenazic Jews must have been carrying several detrimental genes all in heterozygous form. This is again a low probability event making founder effect an untenable explanation. Some critics may view our analysis as one that is too dependent on the assumption that distribution (2) is appropriate. This criticism is surely a valid one. In fact, particularly in a population with the size fluctuations of the Ashkenazim, it is difficult to ascertain if equilibrium has been reached. But the question that is really being asked here is whether or not the observed TSD gene frequency among the Ashkenazim can be recognized as just a high value of the gene frequency distribution given by equation (2). It is important, therefore, to establish whether or not the TSD gene frequency distribution can be assumed to be in steady state in an average population. Thus, our assumption of a steady-state distribution is not exactly invalidated by recent size fluctuations of Ashkenazic Jewish populations. Furthermore, even the deterministic treatments of the present problem assume a mutation-selection balance. We only incorporated the stochastic changes in gene frequencies. The exact treatment needs evaluation of a transient distribution which is rather involved even in the case of selectively neutral alleles [14]. Under heterozygote advantage (h > 0), although the mean gene frequency is always elevated, analytically it is not easy to establish that the large deviation probability will be more than the one shown in equation (3). Some numerical computations seem to 260 CHAKRAVARTI AND CHAKRABORTY indicate that the average selection coefficient for heterozygotes (h) needs to be larger than a critical value in order to make Prob(X 2 XT) appreciable enough when XT is several times larger than the average gene frequency (tkl') in a general population. (Note that in the present case XT is approximately 10 times as large as Al'.) This is indirect evidence that if the hypothesis of heterozygote advantage is true, the selection coefficient (h) must have been very strong, at least temporarily, before the complete redistribution of Ashkenazic Jews throughout the world. A relationship to urbanization and the possibility that heterozygotes for some sphingolipid disorders are resistant to tuberculosis [4, 6] in these environments may suggest this as well. It may be conjectured that a major part of the gene frequency elevation in this population may have occurred under those circumstances, and since then, the selective advantage has reduced to a level which is statistically undetectable from the present day reproductive performance of heterozygotes. Mayo's study [15] may be taken as indirect evidence of this; in his simulation experiment with constant selection coefficients, the 95% confidence limits of the gene frequency distribution for an overdominant (h = .0101) recessive lethal gene did not embrace the value q = .0133 even when the mean was about .0015, although the value of N and u used by Mayo may not be considered as representative of Tay-Sachs in our present context. So far in our arguments, we considered the Ashkenazic Jews to have remained isolated from other populations. Recent evidence [8], however, shows that some gene exchange does occur between the Jewish populations. If m is the rate of such gene exchange per generation between the Ashkenazic Jews and other populations in which the average frequency of the TSD gene is x0, then the probability density (2) becomes

m(X) ac e 2NX4x(u+mxo)-1X - X)4Nm(l-xo)-1 (5)

Numerical integration of equation (5) indicates that the large deviation probability in fact decreases as m increases. For example, if N = 1,000, and u = 1.92 x 10-5 (cf. table 1), there is a 4.4% reduction in the large deviation probability when m is 0. 1% per generation. Such a migration rate decreases the large deviation probability by 15.2% when N = 5,000, and u = 9.17 x 10-6. So, our computations of table 1 can be regarded as the maximal estimates of the large deviation probability. In this context, it is worthwhile to note that the elevated TSD gene frequency of at least .0051 in the Pennsylvania Dutch [1] may be explained by random genetic drift alone (Prob(X 2 .0051) = .1016) as suggested by the original investigators themselves. A philosophical issue may be raised by some critics arguing that the above statistical test is biased since the gene frequency in Ashkenazic Jewish populations represents an extreme value and is therefore not a random observation from the same distribution. In theory, this criticism may not be totally ignored. However, the usual theory of order statistics may not have sufficient statistical power here since the surveyed gene frequency estimates are not totally independent because of their historical correlations derived from the common ancestry of the different populations under consideration. In view of this, the conclusion reached in this paper should be treated as a tentative one. TAY-SACHS AMONG ASHKENAZIC JEWS 261 SUMMARY Using the steady-state distribution of a recessive lethal gene the probability of finding the elevated frequency of Tay-Sachs (TSD) gene among Ashkenazic Jews is computed. For various estimated values of mutation rate and population size, this probability is found to be statistically significant. This probability, in fact, becomes even smaller if a steady influx of foreign genes into the Ashkenazic Jewish populations is considered. It is suggested that heterozygote advantage together with random genetic drift should be considered as the most probable mechanism for the elevation of TSD gene frequency among the Ashkenazic Jews.

ACKNOWLEDGMENTS The authors are grateful to Drs. M. Nei, A. Knudson, Jr., and W.-H. Li for their comments and suggestions. The comments of Dr. W. J. Ewens were also constructive in revising the paper.

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