A model for translocation inheritance in man, segregation patterns for a single centric-fusion translocation

THEORETICAL

POPULATION

BIOLOGY

9,

151-177 (1976)

A Model for Translocation Inheritance in Man, Segregation Patterns for a Single Centric-Fusion Translocation JON STENE Institute of Statistics,

University

of Copenkagen, Copenhagen, Denmark

Received March

27, 1975

A probability model is developed for transmission of a centric-fusion translocation from one generation to the next, giving probability distributions for gametes, zygotes, and liveborn children of different types. The biological assumptions of the model and different alternatives to them are thoroughly discussed. Models of this type form an important part of a theory of population genetics for inherited structural chromosome rearrangements. The treatment is devoted to human populations only. With reference to two segregation analyses carried out by means of the model, the types of insights in the biological mechanisms obtainable by using such a model are demonstrated.

1. INTRODUCTION Mendelian inheritance and the theory based on it is devoted to genes and assumes that the chromosomes are kept unchanged, except for some point mutations and crossing-overs. The existing theory of population genetics is devoted exclusively to this type of inheritance. This theory mainly refers either to organisms other than man, or to no living organisms at all. Problems important in that theory, e.g., expected number of generations that a certain character will remain in the population for certain types of mating, are often of less significance for the study of the genetics of human populations and vice versa. Important problems regarding man are, e.g., expected number of different types of abnormal individuals within a certain period. Such questions relate to a number of social problems unique for man, e.g., controversies about legal abortion, or planning of institutions and other services for handicapped individuals. Owing to social patterns and regulations in human societies, a number of special conditions are automatically imposed on any theory devoted to human populations. The aim of this paper is to consider some important aspects regarding a population genetical theory for inherited structural chromosome rearrangements in human populations. 151 Copyright 0 1976by Academic Press, Inc. All rights of reproduction in any form reserved.

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Such rearrangements are much more common in man than usually recognized. In 43,558 consecutive live births compiled by Jacobs et al. (1974), 102 individuals or 0.24q/b had such aberrations. This figure is likely to be an underestimate. These aberrations will often give rise to children with serious congenital malformations or to reduced fertility for couples carrying such rearrangements. As pointed out by Cavalli-Sforza and Bodmer (1971, p. 44) we need three basic types of information on the characteristics of a population for constructing models of the evolutionary process. These are with references to the area considered here, 1.

Patterns of inheritance.

(i) Rules phenotypes.

of correspondence

(ii)

Types of inheritance

(iii)

Mutation

2.

constitution

and

abnormalities

and structural

patterns (when more than one numerical

abnormalities). or structural

Selective forces. (i)

3.

chromosome

(segregation types and segregation probabilities).

rates (numerical

(iv) Recombination abnormality is present).

between

Gametic selection.

(ii)

Zygotic

(iii)

Viability

of individuals.

(iv)

Fertility

of different

Population

or foetal selection. phenotypes.

structure.

(i)

Probabilities

(ii)

Fertility

for different

of different

types of matings.

types of matings.

In the present paper, we will mainly consider I(ii), 2(i), 2(ii), and mention some problems connected with 2(iv) and 3(ii). Human population cytogenetics is a term coined by Court-Brown (1967) and it seems to include the relevant subject area. Court-Brown and Smith (1969) have provided this definition: “Human population cytogenetics is concerned with frequencies of chromosome aberrations, their causation, and the selective forces that may be exerted on their carriers.” By this definition, it covers also empirical and theoretical studies of aberrations not being inherited as well as aberrations other than structural rearrangements. Only a limited part of this rather wide area hitherto has been covered. Research in human population cytogenetics has mainly been concentrated on collecting data and establishing rules of correspondence between phenotypes and chromosome constitutions, on incidence and mutation rates and on variations

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in the form and size of the different chromosomes, i.e., l(i) and l(iii) above. See Jacobs et al. (1970) and Jacobs (1972). The inheritance of chromosome rearrangements has mainly been considered qualitatively. Among the few papers devoted to quantitative studies of this aspect, most of them are not based on specified probability models, and contradictory results have arisen. (See sections 5.4 and 5.5). In the present paper, a probability model will be developed for the transmittance of a certain, important type of chromosome rearrangement, namely, centric-fusion translocation, from one generation to the next. The assumptions of the model as well as reasonable alternatives to these model assumptions will be discussed. A thorough knowledge of these assumptions is necessary to formulate relevant statistical hypotheses, to develop suitable statistical test procedures, and to understand the interpretations of the test results. Similar models can be developed for other inherited structural arrangements by much the same types of arguments. The discussion will be given in such a general form that it will be able to be applied to other arrangements with slight modifications.

2. ELEMENTARY

CYTOGENETICAL

CONCEPTS

In this section, a number of cytogenetical terms and concepts are collected for easy reference. Only the most basic definitions are given here. More complete information is provided by a number of textbooks in cytogenetics and genetics, see, e.g., Hamerton (1971), Lewis and John (1963), and White (1973). To see the relation between centric-fusion translocations and other structural rearrangements, the main types of these latter ones and other types of translocations are defined. 2.1. DEFINITIONS. Structural rearrangements occur when chromosome segments are transferred from their normal positions to other positions in the same or other chromosomes. Structural heterozygotes are individuals who are heterozygous for structural rearrangements of a part of their chromosome material. Balanced structural gametes OY heterozygotes are, respectively, gametes or individuals possessing almost all their usual chromosome material, but in a rearranged way. Chromosomally unbalanced gametes or individuals are either lacking some chromosome material or have some in excess or both. The terms “balanced structural” and “chromosomally” will often be omitted. Meiosis is a type of cell division found in gamete production. It consists of two

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divisions, one of which reduces the number of chromosomes usually from 2n to n (in chromosomally normal individuals). Homologous chromosomes pair and assort at random to produce gametes with the haploid (n) number of chromosomes. Centromere is the region keeping the arms of the chromosome together. Usually a differentiated region somewhere along the length of the chromosome which seems to act as the point where force is exerted in the separation of dividing chromosomes. Acrocentric chromosome is a chromosome with the centromere near an end. Metacentric chromosome is a chromosome with the centromere not near any end. Homologous chromosomes are chromosomes that pair in meiosis and are similar in size, shape, structure and function. They have alleles of the same genes. Homologous (chromosome) segments are similarly defined. Chromosome complement is the chromosome material in the cells of an organism. Karyotype is a systematized array of the chromosomes of a single cell prepared either by drawing or by photography with the extension in meaning that the chromosomes of a single cell can typify the chromosomes of an individual or even a species (Denver Convention 1961). Monosomic organisms have a full set of chromosomes minus one, (2n - 1). The cells are said to be monosomic for the pair where one chromosome is lacking. Trisomic organisms have one extra chromosome, (2n + 1). NuZZisomic organisms lack one pair of chromosomes. Tetrasomic organisms have two extra chromosomes of a given kind, making four of the kind in question, (2n + 2). The terms monosomic etc. will also refer to chromosome segments or the chromosome material for a whole chromosome, but in a possibly rearranged manner. 2.2.

Types of Chromosome Rearrangements

Chromosomes can break and recombine in a number of ways involving only a single chromosome or two or more chromosomes. The rearrangements produced in this way will often be transmitted to the offspring. Only chromosomes with centromeres will be considered here. In this section we will denote a normal chromosome by the following sequence of letters AB * CDEF, the . indicating the centromere. Deletion and dublication. A chromosome missing a segment is said to have a deletion, e.g., AB . CD or AB . EF. A chromosome having the same segment twice is said to have a dublication, e.g., AB . CDCDEF. If a chromosome segment occurs in the opposite order than in the Inversion. standard case, the chromosome is said to have an inversion, e.g., ADC * BEF or AB 9 CEDF.

TRANSLOCATION

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Shift. Transference of a chromosome segment in one chromosome to another position in the same chromosome, e.g., AEDB . CF. A transference of a chromosome Translocation (often denoted interchange). segment from one chromosome to another nonhomologous chromosome. Human chromosomes and the structural arrangements now can be identified with relatively high precision by special techniques. The classification and nomenclature of human chromosomes and their rearrangements have been established with greater and greater accuracy successively in the Denver report (I 961), the Chicago Conference (1966), and the Paris Conference (1971). In the latter, bands within the chromosomes were classified as well. All the abovementioned types of structural rearrangements have been detected in man (see Hamerton (1971)). 2.3. Types of translocations Translocations them.

can be classified after the number of breaks that have caused

(a) One break. The only theoretical possibility, a transference of a terminal segment of one chromosome to the end of another one does not seem to occur. (b)

Two breaks.

(i) Reciprocal tramlocations are caused by exchange of terminal segments in two nonhomologous chromosomes. See Fig. 1 where the arrows indicate the break points.

:: {b ::: ::: :: BEFORE

FIG. 1. Formation of a reciprocal translocation, reunion. The arrows indicate breaking points.

AFTER

before

and after

breakage

and

(ii) Centric-fusion (or Robertsonian) translocations are formed by two acrocentric chromosomes having dropped their small arms and are forming a single metacentric chromosome together. (It can be considered as a reciprocal translocation where the break points are either within the centromere or proximate to it.) The small chromosome will disappear. See Fig. 2.

156

JON

STENE

BEFORE

FIG. 2. disappear.

Formation

AFTER

of a centric-fusion

translocation.

The

small

chromosome

will

(iii) Tandem tramlocation. A translocation formed by a break just below the centromere in an acrocentric chromosome and a break near the end of another nonhomologous chromosome and rejoining. See Fig. 3.

;::++ :.i -IL AFTER

BEFORE

FIG.

3.

Formation

of a tandem translocation.

The small chromosome;will

disappear.

(c) Thee breaks. When two chromosomes are involved we get an insertion, which is a transference of an interior segment of one chromosome to an interior position in another. See Fig. 4. Three break translocations involving three chromosomes might also occur. (d) Four OYmore breaks. possible.

A

number

of

different

:: ;: 36 ,- :. :: ;+ .

A B C $iD

AFTER

BEFORE

FIG.

4.

Formation

of an insertion.

recombinations

are

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MAN

These different types of translocations are all occurring in man. A review of reciprocal and centric-fusion translocations is given by Hamerton (1971). More complicated translocations have been reported in the last few years. 2.4. Karyotypes

Connected with Centric-Fusion

Translocations

Let the chromosome pairs in the normal karyotype be A, A, B, B, C, C, etc. with 46 chromosomes altogether in man. Let A and B be acrocentric chromosomes. A centric-fusion translocation between A and B is denoted A/B. An individual with the karyotype A, A/B, B, C, C, etc. where the other chromosomes are supposed to be normal, is a balanced heterozygote. We will assume that all genetic material in A and B is contained in A/B. (This is not always the case, and then, the heterozygote has some deficiency and may be less viable than a normal one). It has then the complete chromosome material, but one chromosome less than usual, since one of the A chromosomes and one of the B chromosomes have formed a single chromosome together, the translocation A/B. Such an individual has 45 chromosomes. An individual with the karyotype A, A, A/B, B, C, C, etc. is unbalanced since it has the material of an A chromosome in excess. It has 46 chromosomes, but is in fact trisomic for A. An individual with karyotype A/B, B, C, C, etc. is unbalanced as well. It is monosomic for A and has 44 chromosomes. An individual can possess more than one centric-fusion translocation and still be balanced. If A, B, C, D are acrocentric chromosomes, we can have, e.g., -4, A/B, B/C, C, D, D, etc. Here the two translocations A/B and B/C have one arm in common. In the case A, A/B, B, C, C/D, E, E, etc. the two translocations are independent. In both cases the individuals have 44 chromosomes, but are balanced. An example of this in man is mentioned by Hamerton (1971, p. 273). All the balanced karyotypes with translocations mentioned hitherto, are translocation heterozygotes. An individual with the karyotype A/B, A/B, C, C, D, D, etc. will have 44 chromosomes and be a translocation homozygote. It will be balanced if no chromosome material have disappeared by the formation of the translocation. No such individual has been found in man yet. Segregation problems for individuals and matings where more than one translocation is involved, will be discussed elsewhere. Centric-fusion translocations as well as inversions have a very important evolutionary significance for formation of new species. A thorough discussion of this aspect is given by White (1973). 2.5. Mechanisms Heterozygotes

Connected

with

the Gamete

Production

of

Translocation

Here we will only make some brief comments. More thorough explanations are given in textbooks in genetics and cytogenetics, e.g., Hamerton (1971).

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JON STENE

During meiosis, when the gametes are formed, homologous chromosome segments will pair and make crossing-over with possible exchange of homologous chromosome segments. When no structural rearrangements are present, all chromosomes will pair in the usual way with two chromosomes belonging to the same homologous pair, laying straight along each other except for some crossing-overs. When structural rearrangements are present, more complicated configurations will usually occur with more than two chromosomes usually taking part. The following stages in the gamete formation for translocation heterozygotes are important from our point of view, (i) The configuration connected with the pairing, whether the maximal number of chromosomes are present, i.e., three for centric-fusion and tandem translocations and four for reciprocal translocations, or less than the maximal number of chromosomes are involved. (ii) Whether crossing-over takes place between the centromere breakpoints. (This does not apply to centric-fusion). (iii) The orientation and segregation of the centromeres, combination of centromeres move into the two daughter cells.

and the

i.e., which

We will assume that the other chromosome pairs behave normally. This might not be the case always. The presence of a translocation or another chromosome abnormality may cause other disturbances in the division mechanism. (See Mikkelsen and Stene (1970) and Mikkelsen (1971).)

3. TYPES OF GAMETES, ZYGOTES, AND OFFSPRING PRODUCED WHEN A SINGLE

CENTRIC-FUSION

TRANSLOCATION

IS INHERITED

3.1. GametesProduced by a Translocation Heterozygote As mentioned above, maximum three chromosomes can be involved in the configuration connected with the pairing during meiosis. This pairing is illustrated in Fig. 5. The two acrocentric chromosomes are A and B and the translocation they form together is A/B.

In the sequel, A and B will always be two acrocentric chromosomes. The centromeres may segregate such that two of them are moving into one daughter cell and one into the other. Such segregations will be called (2 : 1) segregations. The three possible (2 : 1) segregations are given in Fig. 6. Occassionally all three centromeres are moving into the same daughter cell. This type will be called (3 : 0) segregation. Both (2 : 1) and (3 : 0) segregations can occur when, e.g., A and A/B are pairing and B is left over or none of them are pairing.

TRANSLOCATION

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4ll............ f 4.*. -. ...:.. .i ‘..‘..~~& a@ 1 ~:::::::::::: :i

FIG. 5.

Pairing of the chromosomes

A, B and the translocation

A/B.

ALTERNATE

t

ADJACENT

t *..*.-.. *. .‘;.::y& F ADJACENT

& t

1

2

b

FIG. 6. The three possible (2 : 1)-segregations of the trivalent tion. The arrows indicate to which daughter cells the centromeres

or maximal configuraare moving.

160

JON

STENE

In Fig. 6, the arrows indicate which of the two daughter cells the different centromeres move into in the first meiotic division. The end products of a division involving a maximal configuration or configurations with less than three chromosomes are indistinguishable. The gametes arising from the different types of segregations are given in Table I. TABLE

I

Gametes Segregation type

Alternate

Adjacent

Gametes

(A, Bh (A/B)

(m-6

Deviations from balanced 6% B)

to, 0) (0, 0)

(0, +I)

m

1 (4

(0, -1)

Adjacent (4 A/B), (+i,o)

2 @I

(-1,O)

(3 : 0) (4 A/B, Bh (6) (+I,

+1)(--l,

--I)

It will be seen that only in case of alternate segregation, balanced gametes will be produced. All others have a A or a B in dublicate or are deficient for one. As mentioned before, the gametes are assumed to have a single one from each of the other pairs. The amount of deviation from balanced ((A, B) or (A/B)) is indicated in the last row. It should be mentioned that crossing-overs take place between the long arms of the acrocentric chromosomes and the homologous segments of the translocation. These crossing-overs will have no effect on the chromosome constitution of the gametes, except for keeping the configuration together, but linkage between alleles in the chromosomes involved might be heavily affected. This problem will be considered elsewhere.

3.2. Mating Types and Zygotes When one centric-fusion translocation is inherited there may occur three types of chromosomally balanced individuals as far as A and B are concerned, namely (1) chromosomally normal individuals with karyotype (A, A, B, B), (2) translocation heterozygotes (A, A/B, B), and (3) translocation homozygotes (A/B, A/B). We will denote them by N, T, and H, respectively. In this paper, we will for simplicity assume that only balanced individuals reproduce. If some types of unbalanced individuals may reproduce as well, the theory can easily be extended to such cases. By this restriction we have the mating types (N x N), (N x T), (N x H), (T x T), (T x H), and (H x H). The offspring of the matings (N x N), (N x H) and (H x H) will be exclusively N, T, and H, respectively. The

Probabilities

Mother

T 4 6 8 10 14 18

(A, A/B)

(AI&

(4

(B)

(A, A/B, B)

(-)

(0, -t-l)

+l)

-1)

(0, -1)

(-I,@

(+I,

(--I,

B)

N

(A/B)

(A, B)

(090)

Pl

(4 B)

Karyotype

Deviation from balanced

Probabilities

(090) (0, 0) (+I, 0)

Deviation from balanced

Father Gamete

II

19

15

11

9

I

5

H

T

(A/B)

K40)

P2

0)

9

22

T

12

15

16

5

4

(A, A/B)

(+I,

PLS

Zygotes in a (T X T)-mating

TABLE

11

23

13

T

17

15

7

6

(A/B, B)

(0, +l)

CL4

24

4

18

20

T

12

9

8

(4 .___

(0, -1)

t%

25

6

21

18

13

T

11

10

(B)

(-1,O)

CL6

11)

(-1,

T

26

6

4

23

22

15

14

(A, A/B, B)

(+I,

P7

21

T

25

24

11

9

19

18

C-1

-1)

CL.9

162

JON STENE

possible zygotes of the (T x T)-matings are given in Table II with the karyotypes of the different zygotes listed in Table III. From Tables II and III we notice that 27 different zygotes may occur. The zygotes 4-27 have all unbalanced karyotypes. In Table III the amount of deviation in A or B from the balanced types (2A, 2B) are given. It should be noticed that the different zygotes are formed in one, two, four5 or eight different ways. The balanced T-zygotes can be produced by two unbalanced gametes in six different ways.

TABLE Karyotypes

Zygote

No.

H(3) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Karyotype

A/B, A/B A, A, A/B, B A, AI& AIB A,A/B,B,B AI& AIB, B A, A, B A, A/B A,B,B AI& B A,A,A/B AIB,B,B A,A,A]B,B,B A, A/B, A/B, B A, A, A/B, A/B

AIB, AI&&B A,B -W A, A B, B A, A, -44 AI& B A, A/B, A/& B, B A B A, A, A/B, A/B, B, B -

III

of Zygotes in Table II

Deviation from balanced

(090) WI, 0) (+I, 0) (0, +I) (0, +I) (0, --I) (0, -1) (--I, 0) C-1,0) (+I, -1) (k-1, +I) (+I, +I) (+I, + I) (+2,0) (0, -t-2) C-1, -1) c-1, -1) (0, -2) C--2,0) t+2, +I) (+I, +a (E-1, -2)

(F-2, --I) t+2, +a C-2, -9

Zygotic probabilities

Probabilities assuming random fertilization after

TRANSLOCATION INHERITANCE IN MAN

163

Zygotes having the same amount of chromosome material from A and B, but rearranged, may develop to phenotypically indistinguishable individuals, e.g., 4 and 5. The possible zygotes in the (N x T)- and the (T x H)-matings also can be found in Tables II and III. Since N-individuals produce (A, B)-gametes only, and H-individuals (A/B)-gametes only, it is seen that in the (N x T)-mating we may get the zygotes N, T, 4, 6, 8, 10, 14, and 18 and in the (T x H)-mating T, H, 5, 7, 9, 11, 15, and 19. The zygotes 12, 13, 16, 17, and 20-27 occur only in the (T x T)-mating. This large amount of types not occurring among the parents is a characteristic feature of the inheritance of structural rearrangements. A similar pattern does not occur in Mendelian (or gene) inheritance. Since unbalanced pollen will usually be aborted a number of these zygotes will not be produced in plants. In other organisms than man, zygotes with unbalanced karyotypes will usually be aborted in an early stage of development (see, e.g., Ford (1970), White (1973)). This fact is, e.g., the background for using translocations for insect pest control (see Curtis and Hill (1971) and references given there). In man, however, liveborn individuals with an unbalanced karyotype occur with a not negligible frequency. This is especially the case for trisomics of different kinds, i.e., individuals having a chromosome (or a segment of it) in excess. Liveborn individuals monosomic for a whole chromosome are very rare. The two possible (N x T)-matings may prove to be different with respect to the types of unbalanced individuals being born. For D/21- and 21/22translocations in man, most families have got no unbalanced offspring from (N x T)-matings, where the father is T, but quite frequent when the mother is T.

4. PROBABILITIES FOR DIFFERENT TYPES 4.1. Probabilities for Gametesand Zygotes In the preceding section we have discussed the possible types of gametes and zygotes. The discussion in the literature on formal aspects of segregation in connection with translocation inheritance does not seem to have moved far beyond that stage. Exceptions are papers by Rickards (1964) and Searle et aZ. (1971), which contain some brief considerations about the probability distributions for the segregation patterns. Some comments on this topic are given by Curtis and Hill (1971) as well. However, all these papers are devoted solely to reciprocal translocations. Referring to Table I, we will let the probabilities for the four segregation types be, respectively, n,, for alternate, rrr for adjacent 1, 7rzfor adjacent 2, and 7s

164

JON STENE

for (3 : O)-segregation, with n0 + rr + rs + ?r, = 1. For each of the segregation types the two possible gametes have the same conditional probability, i.e., $. The probabilities 7r0,..., ms are different for different translocations, for the two sexes and may be different for different individuals, e.g., be age dependent, may follow some type of periodicity, or may be more or less unstable. In a number of cases 71s seems to be approximately equal to unity, such that almost only balanced gametes are produced. In Table IV, the gametes are numbered in the same way as in Table II. The probabilities for a gamete of type i to be formed is qa , to be formed and fertilized is (piei . The probabilities ci refer to a possible gametic selection. If there is no such selection, all Q’S are equal. The conditional probability that a gamete that is fertilized, belongs to type i, is

&

=

piei

I

C TiCf

2

i = l,..., 8

Weseethath,+.~*+h, = 1. Analogously to what is mentioned before, the his may vary in different ways. To distinguish between the two sexes in the (T x T)-mating, we will let the hi’s defined above refer to the mother and let the corresponding probabilities for the father be pr ,..., ps . They are given in Table II. The gametic selection mentioned above depends on the chromosome constitution of the gamete itself. The probability that a given gamete will be fertilized may depend on the chromosome constitution of the gametes of the mate. It might occur that, e.g., one type of unbalanced gamete can be fertilized only by a certain other type of unbalanced gamete such that these together form a balanced T-zygote. This mechanism takes place in some plants. In man we will assume that this type of selection does not occur such that the different gametes are fertilized independently of the chromosome constitution of the gametes of the mate. When this assumption holds, the probabilities for the different zygotes are the functions of the &‘s and pj’s given in the last column of Table III. Otherwise, we may have no specification of the &‘s in Table III. We see that & + *a* +
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The functionfJt) depends heavily on the phenotype of the zygote and foetus and perhaps also on the karyotype, e.g., in mice monosomic foetuses are aborted very early in the pregnancy and trisomic ones at a later stage (C. E. Ford, personal communication). A similar pattern may be present in man as far as spontaneous abortions are concerned. The functionsfi(t) are likely to be dependent of the karyotype of the mother, her age and other properties, e.g., etnic origin. Naylor (1974) has found some increase in total abortion rate with age for different etnic groups. Many pregnancies are also terminated by induced abortions, either legal or illegal ones, sometimes caused by information about the karyotype of the foetus through prenatal diagnosis, by suspicion caused by the karyotypes of the parents, or for other reasons. The f<(t)‘s, and hence, the oh’s, may vary from familmy to family because of inherited properties. In some families, the T~‘S are likely to approach zero for unbalanced zygotes. (If no or few abortions are reported and karyotyped in these families, we cannot decide whether no _N 1 or ri F 0.) If the variations in the f,(t)‘s are likely to be small for matings of the same type, thef,(t)‘s can be estimated on data of reasonable size. Then it should be possible to estimate with some accuracy estimates of the functions for second and third trimester, given, e.g., that the foetus has passed its thirteenth week. Because of difficulties in ascertaining early abortuses, the functions are not likely to be estimated too accurately for the first part of the pregnancy. Carr (1970, 1971) and Pawlowitzki (1972) have discussed some problems connected with the estimation of these functions for normal mothers, and they have provided some results derived from long series of spontaneous abortions.

4.3. Dependency of Sex and other Karyotypical Foetus

Variations in the Zygote and

In the previous sections we have assumed that the probabilities are depending only on the chrdmosome constitution with regard to A and B. The probabilities in Table IV may depend on whether the gamete is carrying a X-chromosome or a Y-chromosome and whether other structural rearrangements are involved, as well. Similarly, the zygotic probabilities in Table II and the survival probabilities fi(t) may also depend on the same variations. The theory could be extended to cover all these cases, but in this paper, we will confine ourselves to consider dependency of the sex of the zygote or foetus, only. The notation is changed by introducing an additional index j for sex (j = I, female), (j = 2, male), such that we will write lij , f&t), and 7u , where i = I,..., 27; j = 1,2 with the obvious modifications for the matings (N x T) and (T x H).

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5. A PROBABILITY MODEL AND ITS DERIVATION 5.1. Assumptions for a Probability

Model

As mentioned before, the probabilities introduced in the preceding sections may vary in many ways, from family to family, from individual to individual, or for the same individual, by being, e.g., age dependent. No stabilities in the probability distributions have been assumed. To develop a probability model we have to make a number of assumptions. Such a set of assumptions will be given in a biological and a probabilistic version. We will first develop a model for a family consisting of a single sibship and their parents since this is a basic unit in human populations. Biological

version.

1. (a) The possible karyotypes of the zygotes depend on the karyotypes of the parents. (b) The possible zygotes formed by a (T x T)-mating are given in Table III, the possible zygotes of a (N x T)- and a (T x H)-mating are given in Section 3.2. (c) The parents are identical for each birth. (d) Each pregnancy has a single foetus. (e) The karyotypes can be distinguished at birth or at abortion. 2. (a) The probability that a conception gives rise to a zygote with a given chromosomal constitution and sex keeps constant during the fertile period of the parents. This holds for all karyotypes and both sexes separately. (b) The probability for each type of zygote to develop to a liveborn child is assumed to keep constant during the fertile period of the mother, but depends on karyotype and sex of the zygote and the karyotype of the mother. 3. (a) The probability th at a certain zygote has a given karyotype, is independent of the karyotypes of the zygotes having been formed previously. (b) The probability that a certain zygote develops to a liveborn child is independent of the fate of zygotes formed previously. 4. The number of zygotes for the couple is determined in advance and is independent of the number of abortuses and liveborn children of different types. 5. (a) The determinations of sex and chromosome constitution as far as A and B are concerned, are independent.

(b) The proportion between the probabilities for a male foetus and a female foetus to develop at least to a certain age t is independent of the karyotype. And this holds for all ages including livebirth.

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Probabilistic version. 1.

Each trial has a given set of possible outcomes. This set keeps constant.

2. (a) The probabilities cii , i = I,..., 27; j = 1, 2, where &j Jii = 1, for (T x T), (and similarly for (N x T) and (T x H)), keep constant and depend on the mating type only. (b) The functions fij(t)

remains unchanged.

3.

Independent

trials.

4.

The number

z of zygotes is fixed in advance and remains unchanged.

5. (a) I& = &ai for all i = l,..., 27; j = 1, 2. (b) fidWi&)

= 4%

0 < t < co; i = l,..., 27.

Comments ofz the assumptions. A number of the assumptions have been discussed in previous sections. From these discussions it follows that many of them hold only approximately. Some of them are difficult or impossible to check in case of man. For other animals, e.g., the mouse, such possibilities exist. Re 1. (a, b) We assume that the possible karyotypes of the zygotes are completely determined by the karyotypes of the parents as demonstrated in Sections 3.1 and 3.2. As mentioned at the end of Section 2.5, this may not be completely true. We consider only (T x T)-, (N x T)-, and (T x H)-matings in order to simplify the situation. Some cases of unbalanced individuals with offspring are reported, see Mikkelsen (1971). Re 1. (c) The assumption not essential.

is introduced

to simplify

the situation.

It is

Re 1. (d) This holds in principle. In some cases technical difficulties have precluded the determination of karyotypes of especially early abortuses, see Carr (1970) and Pawlowitzki (1972). Re 2. (a, b) No data seems to be against for chromosome abnormalities. To test the hypotheses against more or less specified alternatives, large amounts of data are necessary. Difficult ascertainment problems are involved, as mentioned before. Re 3. (a, b) These assumptions

seem to be genetically

without

objections.

Re 4. This assumption might be a critical one. Little is known about the reproductive behavior connected with miscarriages and births of children with these abnormalities. A number of problems are involved, e.g., selective limitation of family size because of births of abnormal children or because of several abortions, compensations for malformed and abortions, and different effects on time intervals between conceptions. These forces are opposing each other. Naylor (1974) has done some empirical studies indicating that there are ethnic

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differences in the reproductive behavior in connections with abortions. Theoretical studies of the effects of the probability distributions by limiting the family size have been done by Goodman (1963) and of different reproductive behavior connected with abortions by Peritz (1964) and Das Gupta (1973). If thef<(t)‘s -+ 0 already for small values of t, for i’s referring to abnormal foetuses, the effects are likely to be small also on the duration of the reproductive histories of the families. Re 5. (a, b) No data seem to be against these assumptions. They do not seem to be tested either. Among liveborn children usual sex ratios seem to be present. If there are any deviations, they are likely to be small. 5.2. Construction of the model

Let the number of zygotes of type (&j) being aborted be aij and the number of the same type developing to a liveborn child be bij . Then

From 2(a) and 2(b) it follows that the probability for zygote (i, j) to deveIop to a liveborn child is &rii and to be aborted lij(l - Tag).It follows that

1 [&jTPj+ 5ij(l - Qi)l = 1. i.j

(1)

Then we have the probability for aij abortions and bij liveborn children of type (i, j), i = l,..., 27; j = 1, 2 is according to assumptions 1, 2, 3, and 4 multinomially distributed (2) Let the number of liveborn children be M = Ci,j bi3 . The probability of n liveborn and aii , i = l,..., 27; j = 1,2 abortions of the different types is

Hence, we get from (2) and (3) the conditional probability of bij , i = I,..., 27, j = 1,2 liveborn of the different types, given 71livebom in all

(4)

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Usually only a restricted number of types of unbalanced zygotes develop to a liveborn child, e.g., in the families considered in Stene (197Oc, d) only one unbalanced type occured, none of the five other types were registrated in liveborn children. Let the set of (i, j)‘s for which there are registrated liveborn children be B. This set might be different for, e.g., (N x T)-matings where the father is T and where the mother is T. See comment at the end of Section 3.2. Then it is easily found from (4) that the probability distribution of the bij’s, (i, j) E B, is

Consider now assumption 5. From 5(b) we have 7ij = riaj, i = I,..., 27, j = 1,2. Introducing this expression and iii = &Q from 5(a) in (5) we get

(6) Assuming now that the set B is the same for .both sexes of the foetus, i.e., for j = 1, 2, we can write (6) in the form (7) where bi. = &r bij , and b+ = xieB bij . From (7) we get the marginal distribution

of the bi.‘s

where iEB. Here, pi is the conditional probability that a liveborn child is of type i, where i is one of the karyotypes being registrated among liveborn children. This probability is independent of the sex of the child. After obvious modifications the same arguments apply for (N x T)- and (T x H)-matings. For the (N x Qmating with three registrated karyotypes among liveborn, formula (8) was given in Stene (197Oc, Formula (3.2)) and was based mainly on the same arguments as here. Because that paper was devoted to problems connected with data collection and data analysis, as well, these arguments were given only briefly. Probability distributions similar to (8) can be easily put up if abortuses of at least a certain gestational age are included.

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5.3. Extensions of the Model to Kindreds and Collections of Kindreds Model (8) gives the probability distribution of the number of liveborn children of different types in a sibship of given size. By introducing some additional assumptions and statements this model can be extended to kindreds, i.e., families consisting of several sibships, and to collections of kindreds with similar translocations inherited. Assumptions Regarding TransEocation Inheritance in a Kindred Biological 1. 2. possible sex are Section 3.

version.

Only one translocation

is inherited

in each kindred.

All 9 types of matings between N-, T-, and H-individuals are a priori in a kindred (e.g., (N x T)-matings, where the T-parent is of opposite different). Possible zygotes of the different mating types are specified in 3.2. (i)

The same translocation

is inherited

in the same kindred.

(ii) For matings of the same type the probabilities with the same karyotypes. Probabilistic

are equal for zygotes

version.

I.

As above.

2.

As above.

3. The probabilities for the different the translocation and the mating type.

zygotes are uniquely

determined

by

Comments. Re (1) This assumption is only included to simplify the situation, such that we avoid recombination problems and other changes in the segregation patterns and probabilities caused by the presence of another translocation. Such problems will be discussed elsewhere. Re (2) (T x T)-matings may occur kindred and may produce H-children.

as consanguineous

matings

in a

Re (3) We will assume that each translocation is uniquely characterized by the zygotic probabilities in the different mating types. A slightly different translocation might have different zygotic probabilities. (The survival probabilities might vary between different matings of the same type in the kindred). The model for a kindred. When the number of conceptions (or alternatively, the number of livebirths) in each sibship is determined in advance, then the numbers of the individuals of the different types within each sibship are stochastically independent of the corresponding numbers in other sibships in the kindred. The joint probability distribution for the whole kindred is the

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product of the distributions for the different sibships. From the assumptions given above, it follows that the &‘s are fixed for all sibships belonging to the same mating type. Extension to a collection of kindreds. Analyses of segregation patterns of inherited translocations of a certain type will usually be based on data from several kindreds with perhaps slightly different translocations. Some statements about the relative importance of different causes of variation are, however, necessary to carry out such an analysis. These statements will form the basis for the statistical hypotheses to be tested and will determine the order in which they should be considered. Statements about Translocation Biological

Inheritance

version.

1. Within the same kindred type varies only slightly.

the abortion

pattern for matings of the same

2. If both chromosomes involved in a number of centric-fusion translocations are fixed, but the translocations theirselves might be slightly different (as translocations formed and inherited independently in different kindreds usually will be) the segregation patterns may vary slightly for matings of the same type, but in different kindreds. The size of this variation is likely to be somewhat larger than that described in statement 1. 3. Let one of the chromosomes involved in a number of centric-fusion translocations be kept fixed (as in Stene (197Oc), no. 21). Let the other one belong to the same chromosome group (the D-group in Stene (1970~)) and let the phenotypes of the liveborn offspring be the same for all matings of the same type for all translocations in the study. Then the segregation pattern of the liveborn types is assumed to vary not very much for matings of the same type. This variation is supposed to be larger than that described in statement 2. 4. Differences in segregation patterns between (N x T)-matings and between (T x H)-matings with T-parents of opposite sex are larger than the variation described in statement 3. Probabilistic

version.

1. The parameters pi , 7s ,... vary only slightly type within the same kindred. 2.

of the same

The &-i’s vary slightly

3. The probabilities of the described type. 4.

for matings

between matings of the described type. p, , p, ,... in (8) may vary somewhat between matings

For (N x T)- and (T x H)-matings be quite different.

Pl > P2 >‘.. might

with

T-parents

of opposite

sex,

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Comments. Re 1. The different mating types separately. Variations in segregation patterns between are in our model assumed to be caused by variations (Compare assumption 3 above). These variations are

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have to be considered matings of the same type in abortion patterns only. likely to be small.

Re 2. Differences between translocations of the described type might be caused by slightly different breakpoints and centromeres of different origin. (See section 5.4). offspring from Re 3. This statement presupposes that unbalanced matings of the considered type can be unbalanced only for the fixed chromosome. Individuals unbalanced for the other chromosomes must have phenotypes depending on the actual chromosome involved in the translocation, and we will assume that individuals of these types are not being born. Since survival probabilities especially depend on the phenotype of the foetus, phenotypically equal foetuses are likely to develop to liveborn children with about the same probabilities, although the karyotypes may be different. Hence, phenotypically equal foetuses carrying translocations with one chromosome different will have about the same probabilities to be liveborn children, i.e., the corresponding oh’s are approximately equal. Centric-fusion translocations of one fixed chromosome and the other one being any chromosome belonging to the same chromosome group, have all a similar form as far as the relative size of the two chromosomes involved in the translocation is concerned. For such translocations no , for T-carriers =17 n2 P and =a are likely to be of the same order of magnitude of the same sex. If we now consider matings of the same type, then if the gametic selections are of the same order, the zygotic probabilities i& , [a ,..., should be of the same relative order of magnitude for zygotes belonging to the types that will develop to liveborn children. Hence, we get the statement that Pl, P2 T... are not too different for matings of the same type. Re 4. The gametes are formed in quite different ways in the two sexes, and this can cause quite different segregation patterns on the gametic level and zygotic level. Abortion pattern for a N-female and a T-female might be different, as well. 5.4. Segregation Analyses and Their Results Before any segregation analysis can be carried out we have to take into account how the data are brought into our record. We have to consider how the kindred are detected, how informative sibships within kindreds are found and effects of incomplete diagnosis of the phenotypes and karyotypes. These problems have been discussed in Stene (197Oa, b, c; 1975, 1977) where relevant methods have been developed. The statistical procedures possess the property that each possible statistical conclusion can easily be interpreted in the biomedical frame of reference.

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In Stene (197Oc, d) segregation analyses are carried out for respectively 38 kindreds with D/21-translocations and 7 kindreds with 21/22-translocations, all detected through at least one M-individual. Only (N x T)- and (N x M)matings occurred in these data. In the liveborn offspring only phenotypically normal individuals and patients with Down’s syndrome were detected. The phenotypically normal individuals were either T or N. At the time when these analyses were carried out, the different D-chromosomes could not be identified, so they had to be considered as a single group. The segregation patterns turned out to be very similar for the two types of translocations. Because of identification problems caused by incomplete diagnosis, the segregation in M versus phenotypically normal and the segregation among the latter type in T and N had to be considered separately. In the latter segregation type, no heterogeneities were detected in any of the materials. The (N x T)-matings with T-father and with T-mother showed the same pattern. The conditional probability for a phenotypically normal individual to be T was not found to be significantly different from 0.5. As far as the former segregation type is concerned, no heterogeneities were found in the offspring of (N x T)-matings with T-mother. For these matings the probability for a liveborn child to be M was estimated to respectively 10.1 y/, for the D/21 and 8.9% for the 21/22-translocations. In the (N x T)-matings with T-father, the corresponding probability is usually much lower than these figures. For these matings the segregation M versus phenotypically normal was heterogeneous for the D/21-data. A similar analysis could not be undertaken for the 21/22-data, because the only Mindividuals with T-father where those by whom their kindreds were brought into our record. Discussion. In this connection it should be mentioned that no M-individuals were born in (N x T)-matings with T-mother in kindreds detected through exclusively M-children with T-father. Another observation might also be reported. In D/21-pedigrees collected in Edinburgh and brought into the record through T-individuals, no M-individuals have been recorded (P. A. Jacobs, personal communication). These further data are still too small to allow any significant conclusion, but at least the heterogeneity detected for matings with T-father, suggests that more than one mechanism is operating. Differences in the behavior of these translocations might perhaps be caused by different centromeres, whether they originate from chromosome 21 or not. The striking similarity between the segregation patterns in the D/21- and the 21/22-data has some interesting theoretical implications. As mentioned before, the phenotypes of the liveborn offspring of (N x T)matings in the two materials are the same. Referring to the comments on statement 3 about survival probabilities, the similar pattern suggests that r,, ,

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rrr , us and ns are of the same order of magnitude for these two types of translocations given matings of the same type. This suggestion has another interesting implication. Since D-chromosomes are large, but 21 and 22 are small acrocentric chromosomes, the maximal configuration during meiosis (see Figs. 5 and 6) will for 21/22-translocations be symmetric, but asymmetric for D/21-translocations. DID-translocations are symmetric as well. Hamerton (1971, p. 284) supposes that the reason why the proportion of unbalanced progeny is much lower in the offspring of D/Dcarriers than in that of D/21 carriers, can be sought in the different symmetry conditions for the maximal configurations of these two translocation types. The discussion above indicates that an explanation should be sought elsewhere. One reasonable explanation is, with reference to the model in this paper, that the survival probabilities for unbalanced foetuses of D/D-carriers are much lower than the survival probabilities for unbalanced foetuses of D/21- and 21/22translocations. 5.5. Concluding Remarks The discussion in the preceding section should demonstrate the importance of a suitable probability model when studying the segregation patterns of inherited translocations. The different parts of the analysis are carried out in reasonable order, and the conclusions obtained give more details than other types of segregation analysis. Analyses without access to models of the type considered in this paper have been carried out by Dutrillaux and Lejeune (1969), Hamerton (1966,1968,1970, 1971), Jacobs et al. (1970), and Jacobs (1972). These papers demonstrate clearly how difficult it is to reach logically satisfactory conclusions without such a model.

ACKNOWLEDGMENT I am indebted to C. E. Ford, J. H. Edwards, P. A. Jacobs, M. Mikkelsen, C. A. B. Smith, and E. Stene for a number of valuable comments during the preparation of this paper. A large part of the work was done during my visits at The Galton Laboratory, University College London in winter 1973-74 and in January 1975. I want to thank Professor Cedric A. B. Smith for providing me with good working facilities.

REFERENCES CARR, D. H. 1970. Chromosome abnormalities and spontaneous abortions, in “Human Population Cytogenetics,” (P. A. Jacobs, W. H. P rice, . and P. Law, Eds.), pp. 103-118. Pfizer Medical Monographs 5. Edinburgh Univ. Press, Edinburgh. CARR, D. H. 1971. Chromosomes and abortion, in “Advances in Human Genetics 2,” (H. Harris and K. Hirschhorn, Eds.), pp. 201-257. Plenum Press, New York. 6531912-4

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CAVALLI-SFORZA, L. L. AND BODMER, W. F. 1971. “The Genetics of Human Populations,” Freeman, San Francisco. CHICAGO CONFERENCE. 1966. Standardization cytogenetics. Birth Defects: Original Article Series. II, 2. Also in Hamerton (1971, 327-343). COURT-BROWN, W. M. 1967. “Human Population Cytogenetics,” North-Holland, Amsterdam. COURT-BROWN, W. M. AND SMITH, P. G. 1969. Human population cytogenetics. Brit. Med. Bull. 25.1, 74-80. CURTIS, C. F. AND HILL, W. G. 1971. Theoretical studies on the use of translocations for the control of tsetse flies and other disease vectors. Theor. Popul. Biol. 2, 71-90. DAS GUPTA, P. 1973. “A stochastic Model of Human Reproduction,” Population Monograph. Sot. Inst. Int. Stud. Univ. Calif. Berkeley. DENVER CONFERWCE. 1960. A proposed standard system of nomenclature of human mitotic chromosomes. Ann. Hum. Genet. 24, 319-325; 25, 105. Also in Hamerton (1971, 310-319). DUTRILLAUX, B. AND LEJEUNE, J. 1969. Etude de la descendance de porteurs d’une translocation t(2lqDq). Ann. GLm?t. Hum. 12, 77-82. FORD, C. E. 1970. The population cytogenetics of other mammalian species, in “Human Population Cytogenetics,” (P. A. Jacobs, W. H. Price, and P. Law, Eds.), pp. 221-239. Pfizer Medical Monographs 5. Edinburgh Univ. Press, Edinburgh. GOODMAN, L. A. 1963. Some possible effects of birth control on the incidence of disorders and on the influence of birth order. Ann. Hum. Genet. 27, 41-52. HAMERTON, J. L. 1966. Chromosome segregation in three human interchanges, in “Chromosomes Today 1,” (C. D. D ar 1‘m gt on and K. R. Lewis, Eds.), pp. 237-252. Oliver & Boyd, Edinburgh. HAMERTON, J. L. 1968. Robertsonian translocations in man: Evidence for prezygotic selection. Cytogenetics 7, 260-276. HAMERTON, J. L. 1970. Robertsonian translocations: Evidence on segregation from family studies, in “Human Population Cytogenetics,” (P. A. Jacobs, W. H. Price, and P. Law, Eds.), pp. 64-80. Edinburgh Univ. Press, Edinburgh. HAMERTON, J. L. 1971. “Human Cytogenetics. I.” Academic Press, New York. JACOBS, P. A. 1972. Human population cytogenetics, in “Human Genetics,” (J. de Grouchy, F. J. G. Ebling, and I. W. Henderson, Eds.), pp. 232-242. Exerpta Medica, Amsterdam. JACOBS, P. A., AIKEN, J., FRACKIEWICZ, A., LAW, P., NEWTON, M. S., AND SIWTH, P. G. 1970. The inheritance of translocations in man: Data from families ascertained through a balanced heterozygote. Ann. Hum. Genet. 34, 119-136. JACOBS, P. A., MELVILLE, M., RATCLIFFE, S., KEAY, A. J., AND SYME, J. 1974. A cytogenetic survey of 11680 newborn infants. Ann. Hum. Genet. 37, 359-376. LEWIS, K. R. AND JOHN, B. 1963. “Chromosome Marker,” Churchill, London. MIKKELSEN, M. 1971. Down’s syndrome. Current stage of cytogenetic research. Humangenetik. 12, l-28. MIKKELSEN, M. AND STENE, J. 1970. Genetic counselling in Down’s syndrome. Hum. Hered. 20, 457-464. NAYLOR, A. F. 1974. Sequential aspects of spontaneous abortion: Maternal age, parity and pregnancy compensation artifact. Sot. Biol. 21, 195-204. PARIS CONFERENCE. 1971. Standardization in human cytogenetics. Birth Defects: Original Article Series VIII. 7, 1972. The National Foundation, New York. PAWLOWITZKI, I. H. 1972. Frequency of chromosome abnormalities in abortions. Humangenetik. 16, 13 l-l 36.

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PERITZ, E. 1966. On some models for segregation analysis. Ann. Hum. Genet. 30, 183-192. RICKARDS, G. K. 1964. Some theoretical aspects of selective segregation in interchange complexes. Chromosoma 15, 140-155. SEARLE, A. G., FORD, C. E., AND BEECHEY, C. V. 1971. Meiotic disjunction in mouse translocations and the determination of centromere position. Genet. Res. 18, 215-235. SMITH, C. A. B. 1968. Testing segregation ratios, in “Haldane and Modern Biology,” (K. R. Dronamraju, Ed.), pp. 99-130. Johns Hopkins Press, Baltimore. STENE, J. 1970a. Analysis of segregation patterns between sibships within families ascertained in different ways. Ann. Hum. Genet. 33, 261-283. STENE, J. 1970b. Comparison of segregation ratios for families ascertained in different ways. Ann. Hum. Genet. 33, 395-412. STENE, J. 1970~. Statistical inference on segregation ratios for D/G-translocations ascertained in different ways. Ann. Hum. Genet. 34, 93-115. STENE, J. 1970d. A statistical segregation analysis of (2lq22q)-translocations. Hum. Hered. 20, 465-472. STENE, J. 1975d. Sampling of pedigrees. Adv. Appl. Probab. 7, 18-23. STENE, J. 1977. Multinomial sampling with partially categorized data. J. Amer. Stat. Ass. to appear. WHITE, M. J. D. 1973. “Animal Cytology and Evolution,” 3rd ed. Cambridge Univ. Press, London/New York.