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Characterization of centromere arrangements and test for random distribution in Go, G1, S, G2, G1, and early S’ phase in human lymphocytes

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Summary Introduction Methods Results Discussion References Appendix

Hum Genet 88:673-682 © Springer-Verlag

R. Weimer 1, T. Haaf 2, J. Krüger 3, M. Poot 1, and M. Schmid 1

Characterization of centromere arrangements and test for random distribution in Go, G1, S, G2, G1, and early S’ phase in human lymphocytes

Department of Genetics, Stanford University School of Medicine, Stanford, CA 94305, USA (2)

Institut für Humangenetik der Universität, Biozentrum, Am Hubland, 97074 Würzburg, Germany (1)

Institut für Anthropologie und Humangenetik der Universität, Im Neuenheimer Feld 328, 69120 Heidelberg, Germany (3)

Summary

The arrangement of centromeres, cluster formation and association with the nucleolus and the nuclear membrane were characterized in human lymphocytes during the course of interphase in a cell-phase-dependent manner. We evaluated 3 893 cell nuclei categorized by five parameters. The centromeres were visualized by means of indirect immunofluorescent labeling with anticentromere antibodies (ACA) contained in serum of patients with CREST syndrome. The cell nuclei were classified as Go, G1, S, G2, G1′ and early S’ phase by comparing microscopically identified groups of cell nuclei with flow cytometric determination of cell cycle stage of synchronized and unsynchronized lymphocyte cell cultures. Based on a discrimination analysis, a program was devised that calculated the probability for any cell nucleus belonging to the Go, G1, S, G2, G1′ and early S’ phase using only two microscopic parameters. Various characteristics were determined in the Go, S, and G2 stages. A transition stage to S phase within G1 was detected. This stage shows centromere arrangements not repeated in later cell cycles and which develop from the dissolution of centromere clusters in the periphery of the nucleus during Go and G1. S phase exhibits various non-random centromere arrangements and associations of centromeres with the nucleolus. G1′ and early S’ phase of the second cell cycle display no characteristic centromere arrangement. The duplication of centromeres in G2 is asynchronous in two phases. For all cell phases a test for random distribution of the centromeres in the cell nucleus was performed. There is a distinct tendency for centromeres to be in a peripheral position during Go and G1; this tendency becomes weaker in S phase. Although the visual impression is a seemingly random distribution of centromeres in G2 and G1′, statistical analysis still demonstrates a significant deviation from random distribution in favor of a peripheral location. Only the early S phase of the second cell cycle shows no significant deviation from a random distribution. zurück zum Seitenanfang >>>

Introduction

The interphase nucleus is a highly organized structure in which chromosomes not only reserve specific regions, but in which they also change their positions throughout the entire course of the interphase itself. This higher spatial order is related to gene expression, DNA replication and cell division, and it is maintained by very fundamental ordering principles involving the inner cell membrane, the nuclear skeleton and the nucleolus (Haaf et al. 1990). Additional ordering principles are described by the position of the chromosomes relative to each other (Comings 1968; Vogel and Schroeder 1974, 1980; Hubert and Bourgeois 1986). The centromere region, which contains constitutive and pericentromeric heterochromatin along with the trilaminar kinetochore plate, is invisible to microsocopic observation during interphase due to the decondensation of the chromosomes. These regions can, however, be recognized in interphase by the use of special techniques. In addition to thin-layer serial reconstruction of osmium tetroxide-stained interphase nuclei (Moens and Church 1977), 3H-thymidine labeling of late-replicating DNA of the centromere region (Fussell 1975), or in situ hybridization with centromere-specific probes, immunoperoxidase staining and indirect immunofluorescene (Coons and Kaplan 1950) can be employed to examine the spatial arrangement of centromere regions within the cell nucleus. Using anti-centromere antibodies (ACA) from the serum of patients with CREST syndrome (a special form of progressive systemic scleroderma PSS), centromeres can be labeled specifically. The characterization and localization of antigenic determinants recognized by ACA (Moroi et al. 1980; Van Venrooij et al. 1985) provides evidence for the high degree of specificity of the antibody to certain kinetochore antigens (Moroi et al. 1980; Brenner et al. 1981; Cox et al. 1983; Earnshaw and Rothfield 1985). These studies showed that the autoimmune serum binds to exactly localizable components of the kinetochore: the inner region and the outer kineto674 chore plate together with the immediately adjacent chromatin (omitting, however, the pericentromeric heterochromatin). This has formed the basis for many studies describing centromere arrangements in mitosis and in interphase nuclei. The investigations of centromere arrangements in the interphase nucleus of different cell lines, which employed ACA-labeling and indirect immunofluorescence, has provided both confirmations and novel insights (Moroi et al. 1981; Brinkley et al. 1984; Hadlaczky et al. 1986; Haaf and Schmid 1989; Brenner et aI. 1981). These studies, however, pose the question of whether specific centromere arrangements occur in a cell-phase-dependent manner in the Go, G1, S, G2, G1′, and early S’ phase. Flow cytometric determination of cell cycle stage by means of DNA-specific fluorescence measurements permits the classification of centromere arrangements observed by fluorescent microscopy at specific cell phases. This yields deeper, cell-stage-specific insights on the internal order of the interphase nucleus. zurück zum Seitenanfang >>>

Materials and methods

Cell culture and preparations

Human lymphocytes, under stimulation with phytohemagglutinin, were isolated by density gradient centrifugation with Ficoll and cultivated in RPMI 1640 medium with fetal calf serum and penicillin G additives. The cultures were harvested after Oh (directly from the blood), 33 h, 46 h, and 72 h and carefully centrifuged onto a slide with a cytocentrifuge (Cytospin 2, Shandon). A part of each of the samples was frozen in 10% dimethylsulfoxide/RPMI-solution and kept for flow cytometry (ICP 22, Ortho Diagnostic Systems, Westwood, Mass.).

Synchronization of cell culture

The synchronization and enrichment of cell nuclei in various phases was accomplished by adding thymidine (0.3 mg/ml) to the medium for 15 h at 51h of cell culture (Xeros 1962). The block was removed after washing the cells in PBS buffer and recultivation in a thymidine-free medium.

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Isolation of nuclei

Nuclei from human lymphocytes were isolated by centrifugation through a sucrose cushion basically as described by Birnie (1978). Unfixed nuclei were stained with Hoechst 33 258 and kept in suspension for microscopical determination of nuclear diameter and calculation of the deformation resulting from cytocentrifugation.

Flow cytophotometry

Using the principle of DNA-specific fluorescence measurements, the fractions of the phases Go, G1, S, and G2 in interphase were determined by flow cytometry. The cell samples were previously stained with Hoechst 33258 and analyzed by an arc-lamp flow cytometer (ICP-2; Phywe, Göttingen, FRG) interfaced to a PDP 11/23 microcomputer (Digital Equipment Corp., Maynard, Mass.). The obtained histograms of fluorescence intensity give percentages of the cell fractions (Kubbies and Rabinovich 1983).

Indirect immunofluorescence

Indirect immunofluorescence labeling of the cell nuclei adhering to the slides was performed with ACA-containing serum and FITCconjugated antibodies (goat anti-human IgG, Dianova), as described in detail by Schmid et al. (1989). DNA counterstaining was subsequently performed with Hoechst 33 258 and ethidium bromide. The preparations were mounted with Mowiol (Calbiochem) and examined under a fluorescence microscope equipped with an HBO 50 W-AC mercury lamp and epifluorescence optics. Specific FITC and ethidium bromide fluroescence were obtained by exciting with 485 nm UV light. Hoechst 33 258 fluorescence was observed under excitation with UV light in the 360-400 nm range. The negligible emission overlapping of FITC and Hoechst staining with ethidium bromide permits an unencumbered evaluation of nuclear diameter and labelings.

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Determination of cell cycle stage

Interphase nuclei were grouped according to the following microscopic parameters: nuclear diameter, number of centromeres, termination time, type of nucleus and number of duplicated centromeres. The cell phases tlhese groups correspond to (Go, G1, S, G2, G1′ and early S’ phase) are determined by establishing the time of harvest of each cell culture and by comparing the percentage fractions of these groups of cell nuclei with flow cytometric determination of cell cycle stage fractions. If the classification of interphase nuclei into these different groups is correct, then a high degree of correlation should be evident using both methods of determination of cell-phase fractions (microscopic for nuclei group classification and flow-cytometric for cell phase determination). This should be maintained also after synchronizing the cell cultures with thymidine, which shifts the cell-phase fractions. In this way, identification of cell phase can be experimentally proved. Furthermore, the existence of microscopic cell-phase-specific parameters (e.g., number of duplicated centromeres indicates G2 phase) facilitated correlation of a group of nuclei with a specific cell cycle stage.

Statistical methods

Test for random distribution of the centromeres. In order to test for random distribution of the centromeres in the interphase nucleus (i.e., all positions are equally probable for each eentromere, see Appendix for an exact definition), 2323 (randomly chosen) cell nuclei were photographed and projected orthogonally with a magnifier (Kaiser) onto circular areas containing a second concentric circle of half the radius in their interior. For each projected nucleus, both the total number of centromeres (g) as well as the number of centromeres positioned in the inner circle or centrum (c) were determined. Under the assumption of random distribution of centromeres in the nucleus, the relative frequency c/g at a constant g is a binomially distributed random variable with an expected value of P = 0.35 (see Appendix ). Further, each nucleus was classified according to the phase to which it belongs, i.e. Go, G1, S, G2, G1′ and early S’. Within each phase, the experimentally established distribution (histogram) of the c/g values of all nuclei was divided into classes with widths of 0.05 and compared with the distribution that would be expected if the centromeres were randomly distributed. Since the value g varies for individual nuclei, the exact distribution for comparison is not binomial, but is instead produced by the combination of the binomial distributions belonging to the different values of g, where each component of the distribution is weighted with the relative frequency of the corresponding g-value. This exact distribution proved to be relatively well aprroximated by the binomial distribution corresponding to the average value N of all centromere numbers in the respective phase. Its probability function Px(x = 0,…, N) was calculated by means of the wellknown recursion formula (see Geigy Tables, p. 184) where (q = l-p). For the graphic comparison with the experimental c/g histograms the separate values were connected to form a smooth curve. For testing the statistical hypothesis of a random distribution of the centromeres within the nucleus, we used the statistic (C, G = sum of the c- or g-values for all nuclei), which under this hypothesis is approximately distributed as a chi-square with one degree of freedom.

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Test for homogeneity.

Testing the hypothesis of random distribution of means of equation (2) for all phases combined only makes sense if the c/g distributions in the individual phases are not significantly different. This can be determined by means of the Kruskal- Waltis test. If there is a significant global difference between the c/g distributions in the phases (heterogeneity of the phases with regard to c/g), the hypothesis of random distribution is tested b means of equation (2) separately within the phases. Furthermore Mann and Whitney’s U-test was employed to find those pairs of phases with true differences in their c/g distribution.

Discriminant analysis.

Microscopically defined types of cell nuclei were grouped and discriminant analyses (Cooley 1971) were performed with the data obtained for the individual nuclei (nuclear diameter, number of centromeres, termination time, type of nucleus and number of duplicated centromeres). This provided a test for the hypothetical group classifications and a confirmation of the association of these groups with cell phases.

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Results

Characterization of the observed centromere arrangement groups

The data (centromere number, nuclear diameter, number of duplicated centromeres) of a total of 1629 labeled cell nuclei (0 h, 33 h, 46 h and 72 h) were obtained. Figure i shows the histograms of centromere number. After a symmetrical distribution at 0 h, three distinct positively (left-) skewed distributions develop at 33 h. At 46 h overlapping distributions of Go and G1 are clearly visible, as well as a distinct S phase distribution including G1′. At 72 h, most cell nuclei in Go were already stimulated for division, and the cells in G1′ are found in the distribution with a modal class at 41.

Fig. 1. Count histograms of the kinetochores labeled in the cell nuclei of lymphocytes after various growth times in cell culture. A total of 1629 nuclei were evaluated. The numerical value in the square boxes denotes the most frequent kinetochore number; in parentheses is indicated the cell phase to which the distribution was attributed

Two accumulations are predicted in the region of higher centromere number, indicating asynchronous duplication at the G2 phase. In addition to the above measurements, centromere arrangements were characterized for each cell nucleus. We were thus able to differentiate six groups of nuclei which could be classified as to interphase stage. Figure 2 shows the average measured values of these interphase stages after summing the cell nuclei of the same phase over all culture times (0 h, 33 h, 46 h, 72 h). The cell nuclei of Go and G1′ exhibit the same average nuclear diameter, which is increased in the Gt, G2, and S phases. G1′ and S have the same average centromere number. In Fig. 3, the nuclei at Go (Fig. 3 a-c) never exhibit duplication of the centromeres and fluorescence appears intense. There is no specific arrangement. The fluorescence of the centromeres in early Gt phase (Fig. 3a) has more intensity and covers a large area. With surprising frequency, the cell nuclei at late G1 (Fig. 3a) exhibit clusters of centromeres with intense fluorescence. There are rarely more than four centromeres in one cluster. No other arrangement was noted. G1′ (Fig. 3b, d, e) presents cell nuclei with a small diameter (< 10 gm), weak fluorescence and a seemingly random distribution of the centromeres. At S phase (Fig. 3 a-e), centromeres have duplicated an average of 3.5 times and exhibit stronger fluorescence.

Five centromere arrangements are distinguishable: SA = accumulation in various nuclear regions (Fig. 3b, d, e), SB = accumulation in the center (Fig. 3b, c), SC = chain-like arrangement (Fig. 3a), SD = not classifiable (Fig. 3b), SE (early S’) = relatively uniform distribution across the cell nucleus (nuclear diameter > 10 ~tm) (Fig. 3d, e). The cell nucleus at G2 (Fig. 3f) is distinguished by a high centromere number, a large nuclear diameter and six apparently duplicated centromeres. The centromeres appear uniformly distributed throughout the nucleus. The best differentiation between the various cell types is achieved either by direct microscopic analysis or by examination of color prints. The classification into Go, G1, G1′, S, G2, and early S’ phase was made by comparing the percentage fractions of the characterized groups with the fractions calculated by flow cytometry.

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Figure 4 a, b show the fractions of the microscopically characterized groups of nuclei in percent after 0 h, 33 h, 46 h and 72 h. These fractions were added over the Go/G1, S, and G2 phases and subtracted from the corresponding percentages determined by flow cytometry.

The following result were obtained: The relativley small differences (absolute values) in this comparison confirm the high degree of precision in the classification of the characterized types of cell nuclei to the phases of interphase. To experimentally test this comparison, we synchronized cell cultures with thymidine in order to obtain enhanced numbers of cell nuclei in specific cell phases. After the block was removed, the cell cultures were harvested immediately and after 6 h. Both microscopic and flow cytometric determinations of cell cycle stage were again performed. The results are shown below:

The synchronization experiments show that the high degree of correlation of the various methods of determination is also maintained when the phase fractions are shifted.

Program for identification of nuclear phase.

A computer program based on discriminant analysis was developed, which permits the classification of cell nuclei under the fluroescence microscope into Go, G1, S, and G2 after the centromeres are FITC-tabeled and the cell nucleus is counterstained with ethidium bromide. The centromere number and the nuclear diameter are used to calculate the percentage probability with which the cell nucleus belongs to one of these phases (the program requires the operating system MS-DOS 3.2 and can be obtained from the author). The discriminant analysis is based on the parameters centromere number, nuclear diameter, number of apparently duplicated centromeres, growth time of the culture, and type of cell nucleus. The phases Go and G2 are sharply distinct from the other cell phases. G1 overlaps slightly with Go. G1′ overlaps with early S’-phase. With the exception of G1′, S phase can be distinguished easily from all other cell phases. G2 is always separated.

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Test for random distribution of centromeres in the interphase of human lymphocytes

To answer the question of whether characteristic groups of cell nuclei have a random distribution of the centromeres in interphase, the relative frequencies c/g of 2323 labeled lymphocyte nuclei harvested at 0 h, 33 h, 46 h and 72 h were determined (see Materials and methods). The cell nuclei were further classified into cell cycle phases, according to the previously established principles. Before testing the hypothesis of random distribution of the centromeres, the Kruskal-Wallis test for homogeneity was applied. According to the test result, the c/g distributions of the six phases are heterogeneous at a high level of significance (P < 0.01). The more detailed examination with the U-test of Mann and Whitney results in four (partially overlapping) groups of phases, i.e., Go, (G1, G2), (G2, S, G1′), and (G1′, early S’), within which there is no significant heterogeneity, whereas two phases belonging to different groups (e.g., G1 and S) show significant differences with regard to the c/g values of their nuclei. Due to the heterogeneity of the c/g values among the different phases, the hypothesis for random distribution was tested by means of statistic (2) only within phases. The quotients (C/G for the six phases are 0.225 (Go), 0.244 (G1), 0.290 (G2), 0.294 (S), 0.33 (G1′), and 0.348 (early S’). All these values are smaller than the value P = 0.35 expected under the hypothesis of random distribution, and the deviation, as determined by the chisquare value (2) is, with the exception of early S’, highly significant for all phases (P < 0.01). The graphical comparison (Figs. 5 a-f) of the histograms of c/g values in the individual phases with the theoretical distribution of randomly distributed centromeres, confirms the tendency for a peripheral localization of the centromeres in the nucleus. This tendency is strongest in Go, corresponding to the lowest value of C/G, and decreases as the C/G values increase. Only early S’ phase graphically shows no deviation of the c/g distribution from the hypothetical random distribution, as expressed by the non-significance of the chi-square value for this phase.

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Deformation factor of cytocentrifugation

The deformation factor of cytocentrifugation was determined for 200 isolated nuclei by measuring the nuclear diameter before and after cytocentrifugation: the values obtained were Oh= 1.08, 33h = 1.2, 46h = 1.3, 72h = 1.16. The deformation of the cell nuclei is always considerably lower than the maximum deformation of time-dependent flattening of cell nuclei on adhesion slides. Here, the maximum deformation factor is 2.27 after 15 min adhesion time. Thus, with regard to deformation, cytocentrifugation represents a suitable method for studies describing the three-dimensional structures within lymphocyte cell nuclei by means of a flat projection of an area.

Discussion

Figure 6 shows an interpretative summary of all the results obtained.

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Go stage and G1 phase

The definitive separation of Go from all other phases of the cell cycle is confirmed by all parameters of the investigation, particularly by the significant result of Mann and Whitney’s U-test with the c/g values, and by discriminant analysis. The average value of C/G of 0.225 (0.35 for random distribution) indicates a positioning of the centromeres tending very distinctly towards the nuclear periphery. The number of centromeres labeled in Go averages 15 and is thus much lower than that expected for the human genome. This is in contrast to all other cell phases and to all previous studies, where agreement has been reported concerning the number of centromeres and the number of chromosomes in the genome of a given cell type (Brenner et al. 1981; Moroi et al. 1981; Brinkley et al. 1984; Hadlaczky 1986; Haaf and Schmid 1989). This poses the question of the presence of a cell phase-dependent antigen, i.e., does the CREST serum contain numerous antibodies which bind to antigens found only in mitotic cells as well as to antigens which in interphase represent “prekinetochores” (Rieder 1983). This question can be extended to the interphase stages and may be resolved with the synthesis of monoclonal antibodies. The method of nuclear preparation employed, modified according to Merry et al. (1985), and the lack of acetic acid fixation, rules out a destruction of the antigen in the Go stage. Thus, the very intense fluorescence in Go and G1 (Figs. 3a, 2a, b) leads to the assumption that the centromeres are grouped in clusters localized in the nuclear periphey which then separate into individual centromeres on transition to S phase. The increased nuclear diameter in G1 can be explained by the synthesis of primarily soluble proteins, and thus by a growth-related instability of the structuring of the nuclear matrix, during cytocentrifugation.

Dissolution of the peripheral clusters and transition to S phase

Late G1 phase (2 b in Fig. 3a, 3d) is distinguished by the dissolution of the clusters into separate centromers, which are still peripherally located. This is indicated by the C/ G value, which is higher for G1 than for Go. The proportion of late G1 nuclei is 33% relative to the total G1, and, on average, 22 labels are visible compared with 18 in total G1. These features characterize the transition to S phase. The localization of centromeres and clusters at the nuclear membrane and the potential association with lamina structures at the inner nuclear membrane, particularly in Go and G1, is demonstrated by the C/G value, which is much lower than the expected value and is in agreement with the studies by Haaf and Schmid (1989) and Moroi et al. (1981).

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Tendency towards the nuclear center during S phase

S phase is much more heterogeneous than other cell phases. There is a strong tendency of the centromeres to relocate from the periphery toward the center of the nucleus. Comings (1980) states: “the inactive late-replicating DNA is condensed below the surface of the inner nuclear membrane and nuclear lamina and around the nucleolus, while the genetically active DNA is found in the inner nuclear matrix.” The centromere arrangements S(A) and S(B) (Fig. 3b, d, e) are explained by the association of chromosomes 13, 14, 15, 21, and 22, which carry the NORs and are obligatorily arranged around the nucleolus. The S(C) form observed by Haaf and Schmid (1989) in GLC1 nuclei as pearl-necklace formations does not appear in such a distinct manner.

G2 phase, asynchronous duplication of the kinetochore

The complete duplication of the centromeres occurs in late G2, as seen by Haaf and Schmid (1989) and Brenner et al. (1981). The non-synchronous duplication of sister chromatids during G2 is indicated by the histogram of the centromere numbers after 72h. The number of cell nuclei examined in Gz in this study is not sufficient to show significant differences between the observed subdistributions.

G1′ and early S’ phase

The 72 h and 46 h cultures contained small and large nuclei whose centromere distribution closely approximate the hypothetical random distribution. They correspond to GI’ (Fig. 3d-5) and early S’ phase (Fig. 3d-6) respectively. Once these types of cell nuclei occur, the G1 fractions decrease considerably down to a value of 1% after 72h (see Fig. 4b).

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The non-random, peripheral localization of centromeres in interphase

With the exception of morphological descriptions (e.g., associations around the nucleolus), most studies use a comparison with model distributions in order to demonstrate non-random distribution or association of chromosomes, or chromatin regions, in the cell nucleus. The various approaches are based on three different models with different assumptions. Spaeter (1974) assumes a random distribution of the centromeres within a circular projection area of the nucleus using the distribution of distances between two random points within a circle (Kendall and Moran 1963) to compare with the experimentally obtained distributions of distances between the centromeres of homologous chromosomes. The probability that the position of a labeled point falls into a specific region of this circular area is the same for all points and is proportional to the area of the region. The model by Rappold et al. (1984), used to study the position of sex chromosomes in the interphase nucleus of human lymphocytes, starts from a random distribution of the chromosome regions in the nucleus (assumed to be spherical) itself: the probability of a point’s being in any nuclear region is the same for all points and proportional to the volume of that region. The theoretical distribution of the distances between two points (Rappold et al. 1974), which was used to compare with the distribution of measured distances, is calculated on the basis of this model. In the current study, the model chosen for the comparison with the experimental data assumes an ellipsoid nuclear shape (a sphere is included as a special case) in which the centromeres are randomly distributed as in the model of Rappold et al. (1984). Under this assumption, the expected value of relative frequency of the centromere sites falling into a central region of the projection area of the nucleus is calculated. As in the model by Rappold et al. (1984), the prerequisite is that the photographs used for the determination of positions represent orthogonal projections on the image plane. The non-random distribution of centromeres in the interphase nucleus and the association with the nucleolus and the nuclear membrane has been repeatedly confirmed (Moroi et al. 1981; Brinkley et al. 1984; Hadlaczky et al. 1986; Haaf and Schmid 1989; Brenner et al. 1981). In certain interphase stages, clusters of centromeres or centromeric associations and special arrangements have been described (Brinkley et al. 1984; Hadlaczky et al. 1986; Haaf and Schmid 1989; Kirsch-Volders et al. 1980). The present investigation proves that centromeres, and therefore also chromosomes, in the Go, G1, G2, S, GI’, and S’ phases go through specific cell-phase dependent movements, which are related to the manifold functions of the interphase nucleus. The exact microscopic determination of the interphase stages has been monitored through a computer program for identification of nuclear phase. The possibility of unequivocal microscopic determination of cell cycle phase through processing of data obtained from indirect immunofluorescence opens new applications in the field of immunocytogenetics.

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Acknowledgement.

We would like to express our special gratitude to Dr. I. Haubitz of the computing center of the University of Wgrzburg for her help in developing a program for the identification of nuclear phases.

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Recent:

Stanislav Kozubek (2005) Spatial arrangement of the human genome and its possible functional role Institute of Biophysics, Academy of Sciences, Kralovopolska 135, 612 65 Brno, Czech Republic

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Appendix

Mathematical model for test of random distribution of centromeres in interphase After determination of central point M, the projection area of the cell nucleus is divided into regions BI, B2, . . . . Bm in such a way that the m-1 lines separating these regions divide any straight line from M to a boundary point of the projection area into m parts of the same length. In the case of m = 2 and a circular projection area, the single separating line between B1 (“centrum”) and B2 (“periphery”) is the circle concentric to the boundary circle, whose radus is half that of the latter edge. Assuming (assumption of independence) that the projection of each centromere has, independently of all others, the same probability P of being in area Bi (i = 1, . . . , m), then the relative frequency of centromeres whose projections fall into Bi among all centromeres “in the cell nucleus is a random variable with the expected value P. This value will be calculated under the further assumption that for each centromere the probability of being in a specific part of the nucleus is proportional to its volume (hypothesis of random distribution of centromeres in the nucleus). In this case, Pi is equal to the volume ratio of that part of the nucleus which is projected into Bi to that of the entire nucleus. This volume ratio can only be calculated when the exact three-dimensional shape of the cell nucleus is known. However, the probabilities P1, P2 . . . . ,Pm depend only on m if the nucleus is an ellipsoid with one axis parallel to the direction of projection. In a rectangular coordinate system, with its origin at the center of the ellipsoid and the z-axis in the direction of projection, the interior of the ellipsoid is described by the inequality:

Thus, under the model assumption of random distribution and a circular projection area the projection of a centromere falls with the probability P = 0.350481 into the interior of the circle with a radius half that of the boundary of the projection area (the area of this interior is only one quarter of the entire projection area!). It is not difficult to probe that the probability P is smaller than the value 0.350481, when the centromere distribution deviates from the random distribution in such a way” that the centromere takes a specific position in the nucleus with a higher probability the more this position is removed from the nuclear centrum [i.e., the greater (x/a) 2 + (y/b) a + (z/c) 2 is, where x, y, and z are the coordinates of the position in the above coordinate system]. Conversely, the probability that a centromere projection will fall in the inner circle of the projection area is higher than the value 0.350481 if positions in the nucleus are more likely for the centromeres with increasing closeness to the centrum of the nucleus.

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