Ann. Hum. Genet., Lo&. (1973), 37, 21

Printed in Great Britain 21

On the origin of 4 6 , X X and 4 7 , X X Y males, 4 6 , X Y females and the position of the Xg locus

BY GILBERT B. COTB Department of Human Genetics, Birmingham Maternity Hospital,

Birmingham B15 2TG, England

Three exhaustive and exclusive combinations of parental loci can be found in X X Y Kline- felter males. If we define the genotypes at some locus on the sex chromosomes of the father as X P Y and of the mother as XMXN - where N means that the second locus is not a duplication of the first - then the three possible combinations of alleles carried are of the types X M X p Y , XMXN Y and X M X M Y . (We use the symbol X M X M Y to include also X N X N Y , which is effectively the same.) The probability of occurrence of these three types will be called a, b and c respectively, so that a + b + c = 1. The values of a , b, c depend on the locus studied.

We consider here a recessive-dominant pair of alleles, with respective frequencies q and p . We can then find maximum-likelihood estimates of a, b, c (evaluates of a , b, c in A. W. F. Edwards’ (1972) terminology) by using J. H. Edwards’ modification (1971) of Fraser’s method of maximum likelihood (1963, 1966) . If Ri and Di are the observed numbers of recessive and dominant propositi in the parental mating type i, and is the expected proportion of recessive phenotypes, as defined in Table 1, then the likelihood function is

9

i= l L = const x n (ri)Ri( l - r i )Di . ( 1 )

Evaluates are easily calculated by iteration on a computer. In large samples, their variances can also be obtained by inverting the observed information matrix. When the observed frequencies are defined as in Table 1, the four elements of this matrix are as follows

I art = - a2 In L/aa2 = {C/( 1 - a)2} + { (D + L)/a2} + {E/(a + c ) ~ } + {Fq2/( 1 + q - qa - qc)2} + {Iq2/(a?I + cl2I + {JLr4/(1 + q - aq - Pc)2} + {KP2/(1 - aPI2} + {Je12/(CP + q - aq)2} + {Na4/(1 - CPq - q2 + aq2I2} + {PP2/(a + aP + CPI2} + {&/(2 - a - C l 2 } ,

+ { J q W + q - aq2 - !w2} - {J!fPP./(CP + q - a d 2 } - {Npq3/(1 - CPq - q2 + aq2I2} + {Pp2/(q + aP + CPI2} + {&/P - a - c)2},

+ {I / (aq + cI2} + {Jq2/ (1 + p - aq2 - cd2} + {Mpz/(cp + q - aq)2} + {Np2q2/( 1 - cpq - q2 + aq2)2} + {Pp2/l(a + aP + cP)2} + (&/P - a - c)2} + ( W / l ( a + cP)2} + (fJPza”(1 - a2 - CPd2}.

I,, = Ica = - a2 In L/aaac = {E/(a + c)2} + {Fa2/( 1 + q - aq - cq)2} + {Iq/(aq + c)2}

I CC = - a2 In L/ac2 = {A/c2} + {Bq2/( 1 + q - ~ q ) ~ } + {E/(a + c)2} + {Fq2/( 1 + q - aq - ~ q ) ~ }

Following Edwards (1971)) let US define the proportions of meiotic non-disjunctions arising in the testis as t , of those taking place in the ovary at division I and I1 as u and v respectively, and of mitotic or somatic non-disjunctions as s. The proportion of X chromatids showing recombination between the centromere and the locus studied is defined as 8, so that the proportion of an odd number of crossing-overs involving the same two chromatids is equal

It is clear that all paternal non-disjunctions must have happened at division I, and that to 28.

22 GILBERT B. C ~ T E

Table 1. Expected proportions of children recessive for an X-linked character, given for all mating types in terms of the parameters a and c deJining the parental origin of the children's sex chromosomes

r r = expected proportion r Mating type Observed propositi

Father Mother of recessive individuals Recessive Dominant

+ + + + ? + ? + ?

? ? ?

- - - -

-

-

A C E G I K M P R

B D F H J L N Q S

Table 2. Proportions of the two types of ova produced by the two types of non-disjunctions, with and without crossing-over between the locus studied and the centromere of the X chromosome

Non-disjunction Non-disjunction

(u.) (v)

x~xx 2ev X M X N

a t meiosis I a t meiosis I1

Crossing-over (20) u6 XXX"

No crossing-over (1-28) u(I-2e) X X X X v(I-2e) XXX'

their proportion t must equal a. However, inferences about the type of maternal non-disjunction are not so simple because of the possibility of crossing-over between the two maternal X chromosomes. Fig. 1 shows how the X'XM and X M X N gametes are produced by the four combinations of the two types of maternal non-disjunction with or without crossing-over. By adding the gametic proportions given in Table 2 one can see that

b = - e) + 2ve, (2)

= ue+v( i -2e)+s . ( 3) The factor of 2 was missing from these equations in Edwards' paper (1971) due to overlooking the two undisturbed chromatids consequent upon a crossing-over, and the fact that non- disjunction at meiosis I accompanied by crossing-over can produce both X M X M and X M X N

gametes. Fortunately, 8 was estimated in that paper for the special case when v = 0, so that the error in formulation vanished and did not affect the result.

If known mosaics are excluded from the data, one might feel safe in assuming s to be negligible, and then solving equations (2) and (3 ) for v and 8 :

= {e(b+c)-C)/(3e-i), (4)

8 = (v-c) / (3v-b-c) . ( 5 ) Unless one knows v or is ready to make certain assumptions about it, 8 cannot be determined.

However, one can plot the likelihood of the data against 8, for each possible value of v. The peak of each curve will be defined by the solution of equation (5) when b and c are taken a t the evaluates, and all curves will intersect at 8 = 4 since this value of 6 implies that c = +(b + c), whatever v is. That is to say that, given no information on the type of non-disjunction involved, the simplest null hypothesis is that 8 between the locus and the centromere is $, which is similar

Origin of 46, XX and 47, XXY males 23

WITH Crossi ng-over

disjunction

WITHOUT C rossi ng-over

WITH Crossi ng-over

Non- < disjunction

< Normal

WITHOUT Crossi ng-over

< Normal

= 0 disjunction

= 0 disjunction

= 0 d is j u nct ion

X N

@ M X N

@ M X N

@ M X N

@ M X M

= 0 @ M X M disiunction

Fig. 1 . The genotype of an X X Y individual carrying two maternal X chromosomes can be the result of ovarian meiotic non-disjunction at either division I or 11, with or without crossing-over between the centromere and the locus under investigation. For the sake of clarity, polar bodies, fertilization and other cross-overs have been ignored, and the gene studied has been put a t the tip of the short arm of the X chromosome.

24 GILBERT B. C ~ T E

Table 3. Xg information about 506 XXY Klinefelter males, 39 XX men, and $ m e of their parents

4 7 , X X Y r A > 4 6 J X

- + Sanger --- + Father Mother Chown Sanger Chown Sanger - + + 2 5 I 2 80 I I 4 + - I 7 0 6 3* I + 2 5 6 t 36 0 4

0 6 0 0 0 0

? + I 2 0 44 I 6 ? - 0 4 0 3 I 0 + ? 0 3 0 3 0 0

? 0 2 0 2 0 0

? ? 0 39 I 234 I 7

Totals 79 427 7 32

- - -

- -- * One father was not tested but the propositus has Xg(a + ) sisters. t One mother was not tested but the propositus has an Xg(a+) sister.

to the a priori situation where 8 is most likely to equal almost 0.5 between two loci picked up at random among the entire genotype. (These arguments could be transposed if one were trying to estimate u and v when 8 were unknown: the simplest null hypothesis would be that v/(u + v) is 9.)

This method will now be used on data on the Xg phenotype of 506 X X Y Klinefelter males (table 3), taken from table I1 of Sanger, Tippet &, Gavin (1971) together with European propositi and some of their parents subsequently tested with the Xga antigen by the Medical Research Council Blood Group Unit in London (Sanger, personal communication, 1972) and 25 more families tested at the Rh Laboratory in Winnipeg, Canada (Chown, personal com- munication, 1972). Parental phenotypes and mating proportions do not significantly differ from expectation and can be regarded as samples from the normal population. The evaluates are

u = 0.41 f 0.10, (6 )

b = 0.38 0.10, (7) c = 0.21 f 0.06. (8)

Fig. 2 shows the position of this value on a Streng diagram with contours of equal relative likelihood. In Fig. 3 the likelihood ratio is plotted against 8, for various values of v. All most likely pairs of v and 8 are shown in Fig. 4. Estimation of either v or 8 from other sources would immediately give good information on the other parameter.

46,XX men constitute a group in which a similar analysis of Xga phenotypes seems worth doing. Many hypotheses have been proposed to explain the origin of these men, but no definite conclusion has been reached so far (de la Chapelle, 1972). Although the various workers who reported the known cases have different standards of diagnosis, it is hoped that most of the accumulated data describe a unique syndrome. This, of course, does not exclude the possibility that more than one mechanism produces the same condition.

Thirty-nine European propositi and some of their parents had their Xg phenotype tested at the Blood Group Research Unit (table IIIb of Sanger et al. 1971 ; and Sanger, personal com-

Origin of 46,XX and 47,XXY males

(1,0, 0)

25

Fig. 2. Likelihood distribution of the three meiotic parameters for the X X Y Klinefelter syndrome. The maximum is standardized a t 100, and the contours show values of equal relative likelihood.

e Fig. 3. Likelihood distribution of the recombination fraction 8 between the Xg locus and the centromere of the X chromosome for various values of the proportion of non-disjunction occurring a t division I1 of meiosis in mothers of 506 47,XXY males.

26 GILBERT B. C 6 ~ k

Fig. 4. Percentage of ovarian meiotic non-disjunctions occurring a t division 11, plotted against the recombination fraction 0 between the Xg locus and the centromere of the X chromosome. The curve shows all the coordinates that are most likely to happen given the data on the 506 XXP Klinefelter males.

munication, 1972) and are presented in Table 3. Parental phenotypes and mating proportions $do not significantly differ from those in the normal population. A closer look a t the data reveals that the Xga distribution simply cannot follow either of the X Y male and X X female dis- tributions since impossible observations are recorded in both cases : if we denote the phenotypes in parentheses as (father, mother, propositus), we see that one (+ + - ) and two (+ - - ) did not inherit their fathers' genes as X X females do, while one (+ - + ) did, contrary to the expectation in X Y males. Another ( ? - -) has Xg(a + ) sisters, and is presumably ( + - -). In addition, the total frequencies are significantly different from the expectations in X Y males, even when the ( + - + ) is arbitrarily removed, and an X X female distribution - expected if a gene similar to the Sxr gene in mice were causing the syndrome - is almost ruled out by the virtual absence of recurrence in siblings and of consanguinity among the parents (de la Chapelle, 1972).

This leaves a t least two mechanisms remaining and they will be studied more closely. 'The first is an X X Y origin, in which case X X men would arise from X X Y zygotes that

subsequently lose their Y , at least in those tissues where chromosomal analysis is done. This is plausible since several X X / X X Y mosaics are known to have a very low proportion of X X Y cells. These cells could go unnoticed or even completely disappear in some individuals whose Xga distribution would evidently be the same as that in X X Y men. Evaluates of a, b and c found by using equation (1) for the 39 X X men are

u = 0.20 k 0.18,

b = 0.56 & 0.24,

c = 0.25 k 0.18.

By dekition, the likelihood of these evaluates is expected to be equal to or bigger than the likelihood of the X X Y origin hypothesis defined by equations 6, 7 and 8. By standardizing the maximum likelihood at 100, the relative likelihood of the X X Y origin is 59.03 in this case,

(9)

(10)

(11)

Origin of 46,XX and 47,XXY males 27

0.0 1 .o a

Fig. 5. Likelihood ratio distribution of the data on 39 X X men plotted against the probability a of their having inherited one Xg locus from each parent. The evaluate defined by the peak is a = 0.43 0.21.

so that the odds (= 100/59.03) which are necessarily against the hypothesis are only 1.7:f. It would seem that no hypothesis should be rejected on the basis of such odds.

It must be noted, however, that the lack of maternal age effect (de la Chapelle, 1972) in XX men argues against a proportion as high as 0.81 carrying two maternal X chromosomes.

The second mechanism is a chromosomal interchange between parts of the paternal X and Y chromosomes (Ferguson-Smith, 1966). The fact that the fluorescent part of the long arm of the Y has not been seen in X X men (Philip et al. 1971 ; Fraccaro et al. 1971 ; Caspersson et al. 1971) goes well with the finding that the male determining factor that would have to take part in the interchange is located on the centromeric portion of the short arm (Krmpotic et al. 1972). This mechanism would give the observed Xg& distribution if, and only if, the Xg locus were sometimes included in the exchange.

Conveniently, one can use the same likelihood function (equation 1) to investigate this mechanism under which a is still the proportion of X X men having inherited one Xg locus from each parent, b is necessarily equal to zero, and c becomes the proportion carrying a single locus of maternal origin. This elimination of b from the function makes the total information on a or c simply

1 = 821n L/8c2 = { (A+C+M)/c2}+{Bq2/ (1 +q-cq) ,"}+((D+L)/( i -c)2} + {(I + K + WP2/(CP + d2} + {JP29,"/(1 - CPP + P d 2 } + {Nq2 / (1 - c d 2 } + {~13,"q2/( 1 - cpq -a","]-

The evaluate found by iteration is a = 0-43 & 0.21 (Fig. 5 ) and means that under this theory, the Xg locus would be included in the interchange 43 yo of the time. The exchanged portions could be so small that they would not be cytologically detectable.

The chromosomal interchange could be obtained by translocation or crossing-over . The latter is plausible since there is cytological evidence for the association of the short arms of the X and Y chromosomes during the first meiotic division (Chen & Falek, 1971) with the formation of what appears to be a single, terminal chiasma (McDermott, 1971). There is still nothing definite to say about the location of the Xg locus (Sanger et al. 1971) although it could

28 GILBERT B. C 6 ~ 6 xx P

0.9 A0.,

(X-Y interchange)

.. XYP 0.5 0.0

C a

Fig. 6 Fig. 7

Fig. 6. Evaluates a, b and c found for 47,XXY males, 46,XX males and 46,XY females; and those expected for normal 46,XY males and 46,XX females. a is the probability of having inherited one Xg locus from each parent, b is the probability of having inherited both maternal loci, c is the probability of having inherited only a single maternal locus and carrying either a single copy or two duplicates of it. The contours show values of equal relative likelihood for the XX men data. Fig. 7. Likelihood ratio distribution of the data on 14 X Y females plotted against the probability a of their having received a supplementary Xg locus from their fathers. The null hypothesis of the X-Y interchange is taken as c = 0.43. The evaluate defined by the curve is a = 0.00 + 0.48.

possibly be on the short arm of the X (de la Chapelle, Schroder & Pernu, 1972). A faulty crossing-over could thus imply the male determining factor and sometimes include the Xg locus. On that model, a similar accident involving Xg and not the male factor could explain the rare exceptions to the rules of X-linked inheritance of Xg (Race I% Sanger, 1968).

Odds can be estimated to compare the X-Y interchange and the X X Y origin hypotheses since the same likelihood function (equation 1) was used in both cases. Fig. 6 shows the different sets of parameters obtained for the X X Y and X X men. When the X X men data are used and the maximum likelihood standardized at 100, the X X Y parameters (equations 6-8) give a relative likelihood of 59-03 and the interchange theory (a = 0.43) one of 17.29. Comparison of these two values is biased in favour of the X-Y interchange because parameters for this theory were estimated by using the X X data while those for the X X Y origin were obtained from the data on X X Y men. It can thus be concluded that the odds are at least 3.4: 1 in favour of the X X Y origin.

These figures do not exclude the possibility of an interchange and it might still be worth

Origin of 46, XX and 47, XXY males 29

Table 4. Xg information about 14 pure gonadal dysgenesis females and .some of their parents

+ Father Mother - + + 2 ? + 2 ? t I

I I

7

Observed total = 14 5 9 Expectation* under 2.5 I 1 1 *49 the interchange theory ( C = 0.43)

X Y males (c = 1.00) Expectation* in 4.25 9.75

* Expected Xg(a- ) individuals in this sample are 3r1 + 3r6 + 8r, (See Table I).

looking at other consequences of this mechanism. One is that if all abnormal Y chromosomes resulting from it give viable gametes, they should produce women with pure gonadal dysgenesis with a frequency equal to that of XX men. Their Xg" distribution could be predicted from the evaluates a and c for XX men since a in X X men would correspond to c in X Y females. Table 4 gives the results for fourteen X Y females who were tested by the Blood Group Research Unit (Sanger, personal communication, 1972) and who presumably do not belong to a group whose condition is due to a recessive gene. The sample size is too small to reveal anything about the parental mating proportions or to exclude the interchange theory (see Table 4), but the male distribution receives best support (Figs. 6, 7 ) with the odds set a t 4.8: 1 against their being the counterpart of X X men on the interchange theory.

At present, neither the X X Y origin of X X men nor the X - Y interchange mechanism can be disproved although the odds are slightly against the interchange, and the apparent absence of maternal age effect argues against the X X Y origin. A mixture of both mechanisms might be the'solution. The proposed method of analysis could be used again when further data become available.

SUMMARY

A maximum-likelihood method is used to calculate odds of only 3.4: 1 in favour of the X X Y origin of 39 46 ,XX males and against the theory of an X - Y interchange in their fathers. The position of the Xg locus on the X chromosome is also considered.

The author wishes t o thank Dr Bruce Chown and Dr Ruth Sanger for their data and Professor J. H.

A post -graduate scholarship from the National Research Council of Canada is gratefully acknowledged. Edwards for suggesting this investigation.

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