MITOCHONDRIAL GROWTH AND DIVISION DURING THE

CYCLE IN HeLa CELLS

JAMES W. POSAKONY, JAMES M. ENGLAND, and GIUSEPPE ATTARDI From the Division of Biology, California Institute of Technology, Pasadena, California 91125. Dr. England's present address is The Wistar Institute, Philadelphia, Pennsylvania 19104.

ABSTRACT The growth and division of mitochondria during the cell cycle was investigated by a morphometric analysis of electron micrographs of synchronized HeLa cells. Downloaded from The ratio of total outer membrane contour length to cytoplasmic area did not vary significantly during the cell cycle, implying a continuous growth of the mito- chondrial outer membrane. The mean fraction of cytoplasmic area occupied by mitochondrial profiles was likewise found to remain constant, indicating that the increase in total mitochondrial volume per ceil occurs continuously during inter-

phase, in such a way that the mitochondrial complement occupies a constant www.jcb.org fraction (-10-11%) of the volume of the cytoplasm. The mean area, outer mem- brane contour length, and axis ratio of the mitochondrial profiles also did not vary appreciably during the cell cycle; furthermore, the close similarity of the frequency distributions of these parameters for the six experimental time-points on August 22, 2006 suggested a stable mitochondrial shape distribution. The constancy of both the mean mitochondrial profile area and the number of mitochondrial profiles per unit of cytoplasmic area was interpreted to indicate the continuous division of mitochondria at the level of the cell population. Furthermore, no evidence was found for the occurrence of synchronous mitochondrial growth and division within individual cells. Thus, it appears that, in HeLa cells, there is no fixed temporal relationship between the growth and division of mitochondria and the events of the cell cycle. A number of statistical methods were developed for the purpose of making numerical estimates of certain three-dimensional cellular and mitochondrial pa- rameters. Mean cellular and cytoplasmic volumes were calculated for the six time- points; both exhibited a nonlinear, approx, twofold increase. A comparison of the axis ratio distributions of the mitochondrial profiles with theoretical distributions expected from random sectioning of bodies of various three-dimensional shapes allowed the derivation of an "average" mitochondrial shape. This, in turn, permitted calculations to be made which expressed the two-dimensional results in three-dimensional terms. Thus, the estimated values for the number of mitochon- dria per unit of cytoplasmic volume and for the mean mitochondrial volume were found to remain constant during the cell cycle, while the estimated number of mitochondria per cell increased approx, twofold in an essentially continuous manner.

468 THE JOURNAL OF "VOLUME 74, 1977 ' pages 468-491 Investigations of the biogenesis of mitochondria in . In the case of Saccharomyces cerevisiae several eukaryotic systems have focused consider- (15), on the other hand, it was concluded that able attention on the relative involvement of the mitochondrial division is not tightly coupled to the nuclear and mitochondrial in this process cell cycle. The results of a study on Chinese ham- (3, 6, 7, 20, 21). The evidence which has been ster ceils (37) did not point conclusively to either a obtained suggests that two phases can be distin- synchronous or a continuous mode of mitochon- guished in mitochondriogenesis (4): (a) the gross drial growth and division. formation of the organelles and (b) their differen- In the present study, it has been found that the tiation into functional sites of oxidative phospho- growth of the mitochondrial outer membrane oc- rylation. The differentiative phase is clearly de- curs continuously during interphase in HeLa cells, pendent upon products coded for by both nuclear such that the mitochondrial complement occupies and mitochondrial (3, 4, 21); the gross a constant fraction of the volume of the cytoplasm. formation of mitochondrial membranes, on the The results in addition point to a stable mitochon- other hand, seems to be entirely under nuclear drial shape distribution and to a continuous divi- control (3, 4, 21). sion of mitochondria during the cell cycle. (A In previous studies from this laboratory the cell previous histochemical study [35] has indicated cycle dependence of the expression of the mito- that the mitochondrial complement in the great chondrial in HeLa cells has been investi- majority of HeLa cells exists in the form of dis- gated. It appears that both mitochondrial DNA crete organeiles.) The conclusions from the two- replication and occur predominantly, dimensional analysis have been expressed in Downloaded from if not exclusively, in a restricted portion of the cell three-dimensional terms by making numerical cycle, covering the late S and G2 phases (33, 34). estimates of various cellular and mitochondrial The rate of mitochondrial protein synthesis, on parameters: thus, the estimated values for the the other hand, increases about twofold during number of mitochondria per unit of cytoplasmic interphase on a per cell basis, remaining fairly volume and for the mean mitochondrial volume constant throughout the cell cycle on a per unit were both found to remain constant during inter- www.jcb.org mass basis (12). In the present work, a morpho- phase, while the estimated number of mitochon- metric analysis at the ultrastructural level has been dria per cell increased approx, twofold in an applied to synchronized HeLa cells, to examine essentially continuous manner. the relationship between the growth and division on August 22, 2006 of mitochondriaI and the events of the cell cycle. MATERIALS AND METHODS Previous investigations of the temporal behav- ior of mitochondrial growth and division have cen- Conditions of and tered largely on lower . In Neurospora Synchronization (17), Saccharomyces cerevisiae (15), and Tetrahy- HeLa cells ( F-315) were grown at 37~ (-22 h mena (19), the mitochondrial to cytoplasmic vol- generation time) in roller bottles containing modified ume ratio was found to be essentially constant in Eagle's medium (11)plus 10% calf serum. Physiologi- different stages of the growth cycle, implying a cally synchronized cell populations were produced by the continuous increase in the volume of the mito- selective detachment of mitotic cells from their substrate chondrial complement. The evidence obtained in (36, 44). Three separate detachments were performed, 1 the Neurospora (17) and Tetrahymena (19) stud- h apart, the populations thus obtained containing ies, as well as in a study of Schizosaccharomyces 86.5%, 87.2%, and 87.0% mitotic cells. To avoid possi- pombe (31), was interpreted to indicate that the ble damage resulting from cold storage, the cells col- lected from each detachment were pelleted at room division of mitochondria, by contrast, occurs temperature (325 gay, 5 min) and plated immediately on within a restricted portion of the cell cycle in these 100-ram diameter plastic dishes for electron microscopy (106 cells/plate), or in the center of glass cover slips 1 The physical mechanism of mitochondrial replication is placed in 35-mm diameter plastic dishes for nuclear not understood in detail, but the evidence from studies DNA labeling and mitotic index determinations (2 • 104 on Neurospora (23, 24, 25), Tetrahymena (32), and cells/cover slip). HeLa cells (41) indicates that the number of mitochon- The timing of nuclear events in the cell cycle was dria in growing cells increases by division of preexisting determined separately for each detached population. organeUes, and that mitochondrial membranes grow by The pattern of nuclear DNA synthesis was followed by the addition of new components to old structures. counting the percentage of labeled nuclei in light micro-

POSAKONY El" AL. Mitochondrial Growth and Division in HeLa Cells 469 scope autoradiographs of cover slip cultures which were sis. In the 7-, 11-, 16-, and 19-h samples, cell profiles in a exposed at various times in the cell cycle to [methyl- given thin section were correlated with the profiles seen aH]thymidine (49.2 mCi/p,mol, 10 p.Ci/ml) for 30 min at in the corresponding thick-section autoradiograph, and 37~ rinsed with NaCl/K/Mg buffer (0.13 M NaC1, could thus be classified as having either labeled or unla- 0.005 M KC1, 0.001 M MgCl~), and fixed in acetic acid/ beled nuclei. Fig. I a shows a grid square containing a ethanol (1:3). The autoradiographs were prepared with region of a thin section from the 16-h point; the corre- Kodak AR-10 stripping film with an exposure time of sponding region of the thick-section autoradiograph is ~1 wk at 4~ The time of the peak of mitotic activity at shown in Fig. 1 b. Cells with labeled and unlabeled the end of the first cell cycle was determined by counting nuclei are clearly distinguishable. This technique pro- the percent of mitotic cells in cover slip cultures, stained vided a means of selecting a cell population for each with 1% acetic orcein, from various points in the cycle. time-point which excluded cells which were out of phase. At least 800 cells were examined in both the autoradi- Thus, only cells with unlabeled nuclei were photo- ographs and the mitotic index preparations. graphed for the 7- and 11-h points (G1 phase), and for the 19-h point (G2 phase); similarly, only cells with Electron Microscopy and Light labeled nuclei were photographed for the 16-h point (S Microscope Autoradiography phase). In a given grid square, all cell profiles containing a nuclear profile and meeting the appropriate labeling The cells to be used for electron microscopy and criterion were photographed. The micrographs were morphometric analysis were selected from the synchro- taken at original magnifications of • 975 and • 2240, nous populations at 0, 3, 7, 11, 16, and 19 h after for cell profile and mitochondrial profile measurements, plating. The 0- and 3-h samples were derived from the respectively.

third detachment, the 7-, 11-, and 16-h samples from the Downloaded from second, and the 19-h sample from the first. The cells Analysis of Electron Micrographs were labeled with [methyl-aH]thymidine (49.2 mCi/ ~mol, 10 p.Ci/ml) at 37~ for 30 min immediately pre- The electron micrographs (in the form of 35-mm ceding each nominal time-point, rinsed once with trypsin negatives) were projected onto the measuring board of a solution (0.05%), incubated in the same solution for 6- Hewlett-Packard 9864A Digitizer, connected to a Hew- lett-Packard 9820 Calculator (Hewlett-Packard Corp., 10 min at 37~ and pelleted in 4 vol of modified Eagle's medium plus 10% calf serum for 10 rain at 210ga,. The McMinnville, Oreg.). The calculator was programmed to www.jcb.org compute and record the following measurements: the pellet was fixed with 3% glutaraldehyde in 0.1 M Soren- sen's phosphate buffer (pH 7.4) for 10 min, broken into cell and nuclear profile areas, and the areas, outer mem- brane contour lengths, and axis ratios 2 of the mitochon- small pieces, and fixed for an additional 1.5 h at room drial profiles. The final magnifications, which varied temperature with frequent swirling. The tissue chunks on August 22, 2006 were then washed 3 times, 30 min each, with 0.1 M <2% from • 17,200 for measurement of cell profile parameters, and • 39,500 for measurement of mito- Sorensen's phosphate buffer, postfixed for 1 h with 1% osmium tetroxide in 0.1 M collidine buffer (pH 7.4) at chondrial profile parameters, were determined from a crossed-line diffraction grating replica photographed room temperature, and stained en bloc with uranyl ace- with each set of negatives. tate in maleate buffer (pH 5.15) (13). Finally, the mate- With this apparatus, contour lengths were measured rial was dehydrated in a graded series of ethanols fol- as the sum of many short (-2 mm) line segments, while lowed by propylene oxide, and embedded in Epon resin. In a given portion of a tissue block, about five gold-to- areas were calculated using the trapezoid rule (45). Fig. silver thin sections were cut, followed immediately by 2 b shows a computer-drawn plot of an actual tracing of the mitochondrial profiles in a typical micrograph field three to four thick (1/., /xm) sections (only thin sections were cut from the 0- and 3-h blocks). The sections (Fig. 2a). The axes used in the determinations of the profile axis ratios are also shown. prepared for a single time-point were taken from at least Due to the fact that the cells with unlabeled nuclei in two different tissue blocks. If sections were cut from the 19-h population consist of both G1 and G2 cells, a more than one portion of an individual block, the regions minimum size criterion was established for this time- cut were spaced far enough apart to avoid cutting point in order to exclude contaminating G1 cells from through a given cell more than once. the G2 cells to be analyzed. Preliminary measurements The thin sections were stained with lead citrate (46). of the cytoplasmic areas of cell profiles in the 3-h (early The thick sections were placed on glass microscope G1) point showed no values >100/xm 2. Accordingly, of slides, covered with Kodak AR-10 stripping film, ex- the cell profiles appearing in micrographs from the 19-h posed for -2 wk at 4~ developed, and stained with point, only those with cytoplasmic areas exceeding 100 0.05% toluidine blue for 10-12 min. ~.m 2 were analyzed. Thin sections were examined in a Philips EM201 electron microscope, operating at 60 kV. Only cell pro- files containing a nuclear profile (or profile Defined as the ratio of the longest axis of the profile to in the case of the 0-h sample) were considered for analy- the axis which perpendicularly bisects it.

470 THE JOURNAL OF CELL BIOLOGY" VOLUME 74, 1977 Downloaded from www.jcb.org on August 22, 2006

FIGURE 1 Recognition of nuclear DNA-synthesizing cells in thin sections. (a) Electron micrograph of a grid square containing thin-section cell profiles to be used for morphometric analysis. Cells shown are from the 16-h sample. Lead citrate stained; • 1,600. (b) Light micrograph of the same region (outlined in black) in an adjacent thick section subjected to autoradiography and stained with toluidine blue. Phase contrast; x 1,600. Nuclear DNA-synthesizing cells in (a) are readily identified by the presence of exposed silver grains over the corresponding profiles in the autoradiograph (b). (See Materials and Methods for details). Downloaded from www.jcb.org on August 22, 2006

FIGURE 2 Measurement of mitochondrial profile parameters. (a) Electron micrograph of a typical cytoplasmic field seen in a thin section. Several mitochondrial profiles are present; portions of the nuclear profile appear in the upper comers. The magnification shown is the same as that used for tracing. Lead citrate stained; x 39,000. (b) Drawing of the mitochondrial profiles in (a) as traced using the Hewlett- Packard digitizer-calculator apparatus. Axes used in the determination of the axis ratios are also shown. (See text for details.) The drawing was prepared by a Hewlett-Packard 9862A Plotter connected to the measuring apparatus. In the 0-h (mitotic) sample, cells which were not in a volumes were determined (see Appendix) for each recognizable stage of mitosis (e.g., those possessing a time-point. The values obtained, expressed rela- nuclear membrane) were excluded from analysis. tive to those of the 3-h sample, are shown in Fig. For each time-point, 28-31 cell profiles were mea- 3b. For both of these parameters, a continuous sured as described. The outer membrane contour length but nonlinear pattern of increase is observed, re- and area of all mitochondrial profiles in these cells sulting in a slightly greater than twofold change (1,100-2,200 total per time-point) were measured; for -600 mitoehondrial profiles, the axis ratio was also over the whole cell cycle. These data and their recorded. The data were punched onto computer cards derivation are discussed in more detail below. and analyzed as described below, using a Digital PDP-10 time-sharing system (Digital Equipment Corp., May- Two-Dimensional Measurements nard, Mass.) and an IBM 370/158 batch processor (IBM Table I and Figs. 4, 5, and 6a-f summarize the Corp., White Plains, N. Y.). results obtained from electron micrographic meas- urements of the cellular and mitochondrial profile Derivation of Three-Dimensional parameters selected for this study. The statistical Information treatments which were applied to these data are A number of statistical methods were developed for described in the footnotes to Table I. the purpose of estimating certain three-dimensional cel- lular and mitochondrial parameters from the two-dimen- CELL PROFILE MEASUREMENTS sional data gathered in this study. These methods and Columns 2, 3, and 4 of Table I show for each their underlying assumptions are described fully in the time-point the mean values of the cell profile Downloaded from Appendix. areas, of the cytoplasmic profile areas 3, and of the Analysis of Variance ratios of these two parameters. A general pattern of increase in the cellular and cytoplasmic profile For those parameters which appeared to remain con- areas is observed from early G1 to mitosis, no stant during the cell cycle, an F-test (8) was carried out to doubt reflecting the growth in the average cellular

test the hypothesis that the mean values for the six time- www.jcb.org points could have been obtained by random sampling and cytoplasmic volumes during the cell cycle. from the same population. The Q values (1) reported in The mean ratio of the profile areas remains rela- the Results section thus indicate the probability that the tively constant (Q > 0.1). variation observed in the six mean values is due to

MITOCHONDRIAL PROFILE on August 22, 2006 chance and not to a real variability during the cell cycle. MEASUREMENTS

RESULTS Histograms of the data on the mitochondrial profiles (area, outer membrane contour length, Cell Cycle Parameters and axis ratio) are shown in Figs. 4, 5, and 6a-f. Nuclear DNA synthesis and mitotic activity fol- There is a striking similarity from one time-point lowed similar time-courses in the synchronous cul- to another both in the shapes of these distribu- tures derived from the three separate cell detach- tions 4 and in their modal values. The only con- ments (Fig. 3a). At 37~ the cell cycle was found sistent difference appears to be a somewhat nar- to be -22 h in duration (as estimated from the rower distribution of the data for the 0-h point second peak of mitotic activity), with the maxi- than for the other time-points. The mean values of mum of nuclear DNA synthesis at 15-16 h. The the area, the outer membrane contour length, and culture samples taken for morphometric analysis the axis ratio of the mitochondrial profiles all (at 0, 3, 7, 11, 16, and 19 h after the plating of remain essentially constant (Q > 0.5) during the mitotic cells; see Fig. 3a), after cell selection by cell cycle (Table I, columns 8, 9, and 10). the criteria described above (Materials and Meth- ods), provided early, middle, and late "GI" cell 3 (Cytoplasmic profile area) = (Cell profile area) - populations (3, 7, and 11 h), an "S" population (Nuclear profile area). (16 h), a "G2" population (19 h), and an "M" 4 The histograms are well fitted by a log-normal distribu- population (0 h). tion (39, 50) with positive skewness (J. Posakony, un- published observations). A wide variety of physical and Synchronized Cell Growth biological quantities exhibit frequency distributions which are more accurately described by a skewed log- As a measure of the growth of the synchronized normal curve than by a Gaussian (normal) curve (see cell populations, mean cellular and cytoplasmic reference 39).

POSAKONY ET AL Mitochondrial Growth and Division in HeLa Cells 473 ~, (a) W, IO0 tt~ r ":)~ 8 r ~o tb c

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I I (b) 2.0 / Downloaded from O ~- 1.0 I I 1 t 8 16 24 32 Time (h) FmURE 3 (a) Temporal patterns of nuclear DNA synthesis and mitosis in the synchronous cultures. The

abscissa represents the time in hours after the plating of the harvested mitotic population. Solid symbols www.jcb.org (A, O, I) show the percentage of labeled nuclei observed in light microscope autoradiographs of cover slip cultures pulse-labeled with [3H]thymidine at various times in the cell cycle. Open symbols (&, 9 El) represent the percent of mitotic cells observed in cover slip cultures stained with acetic orcein at various

points in the cycle. Individual DNA labeling and mitotic index curves were determined for the synchronous on August 22, 2006 cultures resulting from the three separate cell detachments (see Materials and Methods). The times at which culture samples were taken for morphometric analysis are indicated by the arrows at the top of the figure. (b) Growth of the synchronized cell population. Using the method described in the Appendix, mean cellular and cytoplasmic volumes were determined from the cell profile measurements for each time- point, and are expressed here relative to the values for the 3-h point. Closed circles represent total cell volume, while open circles represent cytoplasmic volume. The values for the 0-h point are shown twice: once at the left of the panel (0 h), and again at 22 h, where dashed lines connect them with the 19-h values. (22 h is the time of the peak of mitotic activity at the end of the cell cycle; see Fig. 3 a.) The absolute values of these volumes (with standard errors) are given in Table II.

DATA DERIVED FROM CELL AND ume (47), independent of the shape of the mito- MITOCHONDRIAL PROFILE chondria. Furthermore, the ratio of total mito- MEASUREMENTS chondrial area to cytoplasmic area provides a reli- For each cell analyzed, the total area, total able estimate of the corresponding volume ratio outer membrane contour length, and number of (10, 47, 49), which is also shape-independent. the mitochondrial profiles were normalized for the cytoplasmic profile area of the cell. The mean Three-Dimensional Information values of these ratios for each time-point are pre- sented in Table I (columns 5, 6, and 7). Little An attempt was made to derive from the mor- variation (Q > 0.5; Q > 0.25; Q > 0.25) in these phometric data described above estimates of cer- ratios is observed during the cell cycle. The ratio tain parameters concerning the cells and mito- of total outer membrane contour length to cyto- chondria as three-dimensional bodies (Table II). plasmic area is directly proportional to the ratio of The statistical methods developed for this purpose outer membrane surface area to cytoplasmic voi- are discussed in the Appendix.

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TABLE 1

Two-Dimensional Measurements of Cellular and Mitochondrial Profiles* on August 22, 2006

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Tot. oute#ll Tot. mit.$ membr, cont. no. Mit.$ Time after Cyt. area profile area length profiles Outer membrane mitosis Cell area C~t. area~ Cell area Cyt. area Cyt. area v.m t Cyt. area Mit. profile area** contour length** Axis ratio** h vm t /an t % % v,'n -' mn t tan 0 197,5 -+ 9.7 133.0 • 6.8w 67.3 -+ 0.91l 11.0 • 0.4w 0.93 - 0.04w 0.57 -+ 0.03w 0.193 -+ 0.006 1.64 .+ 0.06 1.77 -+ 0.11 3 117.6 _+ 5.0 79.1 .+ 3.3 68.4 _+ 2.1 10.6 -+ 0.6 0.89 -+ 0.05 0.52 -+ 0.03 0.202 -+ 0.011 1.70 • 0.09 1.90 .+ 0,14 7 124.4 _+ 7.6 78.4 -+ 4.9 64.2 _+ 2.0 9.4 -+ 0.7 0.78 _+ 0.06 0.46 • 0.04 0.205 -+ 0.016 1.69 • 0.13 1.86 -+ 0.16 11 128.9 -+ 5.7 86.2 -+ 4.9 66.8 -+ 2.4 10.8 -+ 0.7 0.90 _+ 0.06 0.53 -+ 0.04 0.204 -+ 0.013 1.69 • 0.11 1.84 - 0.20 16 144.4 -+ 6.1 95.1 • 4.6 65.9 • 1.7 10.9 .+ 1.0 0.91 -+ 0.08 0.54 .+ 0.04 0.203 • 0.019 1.69 • 0.14 1.94 .+ 0.20 19 170.6 .+ 7.1 118.4 .+ 3.5 71.3 .+ 2.2 10.6 -+ 0.5 0.88 -+ 0.04 0.51 -+ 0.03 0.210 • 0.010 1.73 • 0.09 1.91 .+ 0.18

* Values shown are means -+ standard errors, computed from the values for individual cell profiles. At each tlme-point, measurements were made on 28-31 cell profiles and 1,100-2,200 mitochondrial profiles. :[: (Cytoplasmic profile area) = (Cell profile area) - (Nuclear profile area). e, w Computed from the corresponding value for the total cell area by using the estimated value of eyl. area/cell area (see following footnote). ta. II Since mitotic cell profiles lack defined nuclear and cytoplasmic regions, this value was estimated as the average of the values for the other time-points. 82The means and standard errors of this parameter were both computed by weighting the value of the parameter for a given cell profile according to its cytoplasmic area. ** The means and standard errors of this parameter were both computed by weighting the value of the parameter for a given cell profile according to its number of mitochondrial profiles.

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0.0 02 OA 0:6 0.8 0~0 02 014 0.6 0;8 Profile area f/j.m2} FIGURES 4-6(a-f) Frequency distributions of the mitochondrial profile data. In a given figure, each panel presents the distribution of the data from all cells for one time-point. These histograms were generated by a Digital PDP-10 computer, into which the raw measurements of the individual mitochon- drial profiles had been read. In all three sets of histograms, the final class interval shows the total number of profiles with parameter values greater than the largest value indicated on the abscissa, i.e., profile area >0.8 t~m2, outer membrane contour length >5 ,u.m, and axis ratio >5.

FIGURE 4 Frequency distributions, for each time-point, of the mitochondrial profiles with respect to profile area.

CELLULAR AND CYTOPLASMIC VOLUMES suming the cells and nuclei to be concentric spheres), and then converting the cytoplasmic The measurements of the cellular and nuclear area values into approximate cellular volumes (see profile areas at each time-point were used to ob- Appendix). The equations used (reference 26 and tain mean values of the cellular and cytoplasmic Appendix) embody statistical corrections for: (a) volumes. The procedure involves estimating the the fact that the cell profiles analyzed were derived average ratio of the cellular and nuclear radii (as- from cells sectioned only through their nuclei, and

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I 2 3 4 5 I 2 3 4 Outer membrane contour length (p.m) FIGURE 5 Frequency distributions, for each time-point, of the mitochondrial profiles with respect to outer membrane contour length.

(b) "sectioning bias", i.e., the phenomenon that a cycle, indicating a proportionate increase in nu- spherical nucleus is not in general sectioned clear and cellular volumes. through its center. Table II (columns 2, 3, and 4) presents for each THREE-DIMENSIONAL MITOCHONDRIAL time-point the estimates of the mean cellular and PARAMETERS cytoplasmic volumes, and their ratio, obtained by OUTER MEMBRANE SURFACE AREA PER this procedure. Both volume estimates appear to UNIT OF CYTOPLASMIC VOLUME: The increase continuously from early G1 to mitosis, mean ratios of total outer membrane contour exhibiting a slightly greater than twofold change length to cytoplasmic area (Table I) were con- over the entire cell cycle (Fig. 3b). The pattern of verted to mean values of outer membrane surface increase is not linear, but suggests an accelerating area per unit of cytoplasmic volume (Table II, rate of growth. The ratio of the cytoplasmic and column 5) by multiplying by the factor 4/~- (47). cellular volumes remains fairly constant over the This conversion is independent of the shape of the

POSAKONY ET AL. Mitochondrial Growth and Division in HeLa Cells 477 200 - (o) 3h - (b) 7 h - (c) II h

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I 2 ~ I 2 3 4 ,5 I 2 3 4 ,5 6 7 Axis ratio FIGURE 6(a-f) Frequency distributions, for each time-point, of the mitochondrial profiles with respect to axis ratio. FIGURE 6(g-l) Theoretical frequency distributions of profile axis ratios, expected from random section- ing of bodies of known shapes. The upper three panels (g-i) show the expected profile axis ratio distributions for three prolate spheroids with different major to minor axis ratios. Class frequencies were obtained according to Equations 18 and 21 of the Appendix. The lower panels (j-l) show the expected profile axis ratio distributions for three general ellipsoids, all with minor axis ratio 1.38:1, and with different relative major axis length. Class frequencies were determined using the Monte Carlo method described in the Appendix, with M = 10,000. The scale of both the ordinate and the abscissa of these histograms (g-l) has been adjusted to make them directly comparable to the experimental frequency distributions (a-f). mitochondria. That these values do not vary sig- employed Formulation 2 of Mayhew and Cruz nificantly (Q > 0.25) from one time-point to an- Orive [27] in applying this principle, as is appro- other implies a continuous growth of the outer pilate fbr our data.) Thus, the mean values of the membrane in terms of surface area. mitochondrial to cytoplasmic area ratios (Table I) MITOCHONDRIAL VOLUME FRACTION: are reproduced in column 6 of Table II. Their The fraction of the cytoplasmic or cellular volume relative constancy (Q > 0.5) indicates a continu- occupied by mitochondria is equivalent to the cor- ous proportionate growth of the mitochondrial responding area fraction (10, 47, 49). (We have and cytoplasmic volumes during the cell cycle, at

478 THE JOURNAL OF CELL BIOLOGY" VOLUME 74, 1977 Downloaded from www.jcb.org TAnLE II Estimates of Three-Dimensional Cellular and Mitochondrial Parameters*

(1) (2) (3) (4) (5) (6) (7) (8) (9)

~.mz Outer membr.** on August 22, 2006 Time after Cyt. vol~ surface area Tot. mit. vol:~ no. Mit.w167 no. Mit.*** O mitosis Ceil vol~ Cell vol Cyt. vol 82 /tin a Cyt. vol Cyt. vol ~,ma Cyt. vol Mit. vol82182 - " Cell ,

* Values shown are means • standard errors, computed from the values for individual cell profiles. :~ Computed according to the method described in the Appendix. w Computed using the mean value [rom the other time-points of the eyt. area/cell area ratio (see Table I) and of the parameter q (Eq. 4, Appendix). C) II Computed u~n8 the mean value from the other time-points of the parameter q (Eq. 4, Appendix). $ Computed for each time-point by multiplying the cell vol (column 2) by the cyt. vol/cell vol ratio (column 3). ** Computed by multiplying the va!ues in column 6 of Table I by 4#r (47). ~:~ Taken from column 5 of Table 1. g~ w167Computed using the estimator described in the Appendix (Eq. 1 I). A value of 3.349 was used for the shape coustant (J). This is the average of the upper and lower limits for the constant, given respectively by Equations 15 and 16 (Appendix), assuming as a model shape a general ellipsoid of axis ratios 5:1.38:1 (see Results). The means and standard errors were both computed by weighting the value of the parameter for a given cell profile according to its cytoplasmic area. I] II Computed using the mean value from the other time-points of the cyt. area/cell area ratio (see Table l). 11 Computed for each cell by dividing the mit. vol/cyt, vol ratio by the no. mit./cyt, vol. *** Computed for each cell by multiplying the no. mit./cyt, vol by the cyt. vol.

g=r-, the level of the whole cell population. As men- to the minor axes of the parent body, i.e., these tioned above, this estimate of the volume fraction distributions are bimodal (Fig. 6g-l). is independent of assumptions made about, or A comparison of Fig. 6a-f and Fig. 6g-l reveals actual changes in, mitochondrial shape. a close similarity in modal values, lack of bimodal- DERIVATION OF INFORMATION ON MI- ity (see figure legend titled "Figures 4-6[a to f]'+), TOCHONDRIAL SHAPE" m previous histo- and overall shape between the experimental axis chemical study in this laboratory (35) indicated ratio distributions and those expected from sec- that, in the great majority of HeLa F-315 cells, the tioning general ellipsoids of the chosen minor axis mitochondrial complement is divided into distinct ratio (1.38), particularly 5:1.38:1 bodies (Fig. organeiles. The work described in this and the 6k). Moreover, the mean value of the distribution following section was based on this finding. shown in Fig. 6k (5:1.38:1) is 1.97, a fairly close Unlike the case of the volume fraction, the approximation of the observed mean axis ratios estimation from two-dimensional data of the num- (1.77-1.94 [Table I]). These similarities suggest ber of mitochondria per unit of cytoplasmic vol- that the observed profile axis ratio distributions, ume requires information concerning the shape of although they clearly reflect in part a variability in the organelles. Accordingly, an attempt was made shape of individual mitochondria in the cell, can to obtain information about the class of organelle be closely approximated, at all stages of the cell shapes most consistent with the mitochondrial cycle, by the axis ratio distribution expected from profile measurements, and to derive a hypotheti- randomly sectioning a population of general ellip- Downloaded from cal "average mitochondrial shape" for purposes of soids with constant axis ratios of 5:1.38:1. calculation. The problem was approached by com- ESTIMATION OF MITOCHONDRIAL NUM- paring the experimental frequency distributions of BER DENSITY: It was shown in the previous profile axis ratios with theoretical distributions ex- section that the profile axis ratio distribution ex- pected from random sectioning of bodies of ideal- pected from sectioning a suitably chosen general ized shapes. By the use of the formulas and meth- ellipsoid, i.e., one with axis ratios of 5:1.38:1, is www.jcb.org ods described in the Appendix, distributions were very similar to the experimental distributions for generated for three representative bodies of each the mitochondrial profiles. In view of this similar- of two shape classes: prolate spheroids (one long ity, a calculation of mitochondrial number density axis and two equal, shorter axes) and general was made, based on an average shape for the on August 22, 2006 ellipsoids (three axes of any length), the former organelles of a general ellipsoid with the above- being a special case of the latter. A ratio of 1.38 to given axis ratios. In particular, the shape constant 1 was chosen for the minor axis ratio of the gen- J (see Appendix) of this ideal body was taken as eral ellipsoids. This choice was guided initially by an estimate of the mean shape constant used in the the modal values of the experimental axis ratio estimator for the number of mitochondria per unit distributions (Fig. 6a-f), and subsequently refined of cytoplasmic volume (Eq. 11, Appendix).5 by trial and error comparison of theoretical and Table II shows, for each of the six time-points, experimental distributions. The three representa- the estimated number of mitochondria per unit of tive bodies of each of the two classes of shapes cytoplasmic volume, as well as values for two were given values for the longest to shortest axis related parameters: the mean mitochondrial vol- ratio of 3:1, 5:1, and 7:1, the last corresponding ume and the number of mitochondria per cell approximately to the upper limit of the observed (columns 7, 8, and 9). While the number of mito- profile axis ratios. The theoretical distributions are presented in Fig. 6g-l as histograms, which are As discussed below (see Discussion), it has not been directly comparable to the histograms of the ex- possible to determine precisely in our system the validity perimental axis ratios (Fig. 6a-~. A common of the topological assumptions underlying this estimator; feature of all the theoretical distributions is a peak consequently, its use is to be considered tentative. More- over, the reader is to be cautioned that there is no occurring at the class interval which contains the evidence that the 5:1.38:1 ellipsoid is the only one whose ratio of the minor axes of the parent body, i.e., random sectioning would yield a profile axis ratio distri- 1.00 for the prolate spheroids and 1.38 for the bution fitting the experimental histogram, and further- general ellipsoids. The distributions for the prolate more, that there is no independent evidence that the spheroids also exhibit a significant smaller peak at shape constant of this ellipsoid is, in fact, the average the class interval containing the ratio of the major shape constant for the mitochondrial population.

480 THE JOURNAL OF CELL BIOLOGY' VOLUME 74, 1977 chondria per unit volume and the mean mitochon- gous to the bimodality in nuclear DNA content drial volume (as analyzed in the whole cell popula- observed in unsynchronized cell populations. tion) appear to be quite uniform (Q > 0.25; Q > Now assume that mitochondrial division occurs 0.5) at the various points in the cell cycle, the in a short time-interval, but that mitochondrial number of mitochondria per cell exhibits a fairly growth occurs continuously between (mitochon- regular increase from early G1 to mitosis, un- drial) divisions. Assume also that the relation dergoing a slightly greater than twofold change between mitochondrial division and mitosis is throughout the cycle.~ fixed in any individual cell, but is variable among different cells. Then, at a given point in the cell Synchrony within the Individual Cell cycle, the frequency distribution described above The results presented thus far have apparently should be essentially fiat within the range of indicated that, in a population of synchronized observed mean mitochondrial profile areas. cells, mitochondrial growth and division do not Fig. 7 shows, for each of the six time-points in occur in a restricted portion of the cell cycle. the present study, the frequency distribution of However, the type of analysis described above the cell profiles with respect to mean mitochon- would not be expected to reveal the existence of drial profile area. These distributions exhibit a synchrony of mitochondrial growth and division clear unimodal tendency. Thus, the present ob- within an individual cell, if the temporal relation servations do not appear to contain evidence for of the mitochondrial events to the cell cycle were intracellular synchrony in mitochondrial growth fixed in an individual cell, but could occur at any and division.7 Downloaded from time in the cell cycle in different cells, with equal probability. Nonetheless, most cases of strictly DISCUSSION intracellular synchrony should be detectable by an appropriate analysis of the data presented Procedures and Sources of Error above. In both of the following situations, all the The synchronized cell populations used for the cells are considered to have the same mitochon- studies reported here were prepared by the www.jcb.org drial shape distribution, independent of their method of selective detachment of mitotic cells stage in the cell cycle. from their substrate (36, 44). This procedure was Assume that in each cell, both mitochondrial chosen in order to avoid any artifactual perturba- growth and mitochondrial division occur within tion of the growth and division of mitochondria on August 22, 2006 intervals of time which are short compared to the which might have resulted from the use of tech- total length of the cell cycle. Assume further that niques for the induction of cell synchrony involv- the relation of these events to the cell cycle, i.e., ing drug treatment or nutrient deprivation. mitosis, is fixed in any individual cell, but varia- Due to variations in the cell cycle times of ble among different cells. Then, at a given point individual cells, a progressive loss of synchrony is in the cell cycle, a frequency distribution of cell found to occur in mitotically synchronized cell profiles with respect to mean mitochondrial pro- populations (28), particularly in the second part file area should be bimodal: the relative height of of the cycle. An autoradiographic correlation the peaks corresponding to the larger and smaller technique was employed in this study in order to mean profile area would be proportional to the recognize and exclude, from each culture sample, average fraction of the cell cycle spent by the cells which were out of phase as judged by the cells while containing "large" mitochondria rela- criterion of nuclear DNA synthesis (see Materi- tive to that spent while containing "small" mito- als and Methods and Fig. 1). After such selec- chondria, i.e., the average fraction of the cell tion, the cells actually analyzed at a given time- cycle intervening between mitochondrial growth point were considerably more representative of a and mitochondrial division. This would be analo- specific stage of the cell cycle than the original

6 The greater-than-twofold change in this parameter is 7 However, the sensitivity of this analysis is such that if, due to the greater-than-twofold increase observed in the in the first situation described above, the average interval mean cytoplasmic volume (Fig. 3 b), since the estimated between mitochondrial growth and mitochondrial divi- number of mitochondria per unit of cytoplasmic volume sion were very short, e.g., <5% of the cell cycle, intra- remains relatively constant. cellular synchrony would not be detectable.

POSAKONY ET AL. Mitochondrial Growth and Division in HeLa Cells 481 IO

5

E

0]2 0.18 0.24 0.50 0.18 0.24 0.30 0.18 0.24 0.30 Mean mitochondriol profile area (~zm2) FIGURE 7 Frequency distributions, for each time-point, of cell profiles with respect to mean mitochon- Downloaded from drial profile area. culture sample. The use of a size criterion for cell profiles, this corresponds to only 0.0064 ~tm. profiles in the 19-h sample (see Materials and Moreover, repeated tracings of the same plane

Methods) was similarly motivated. figure have yielded very similar values for both www.jcb.org The experimental approach employed here, contour length and area (data not shown). The i.e., morphometric analysis of electron micro- finite thickness of the tissue sections may cause graphs of thin sections, contains a number of an overestimation of the size of electron-opaque

potential sources of error. structures (e.g., mitochondria, outlined by mem- on August 22, 2006 The effect of artifacts introduced systemati- branes stained with heavy metals); this is the so- cally during the preparation of the cells for analy- called Holmes effect (47). No correction for this sis are minimized in a study such as the present effect was made in the present work, since the one, in which the conclusions rest largely upon average "diameter" of the organelles is believed relative values of experimental parameters, or to be large compared to the thickness of the upon comparison of absolute values. (The abso- sections (see reference 47). lute numbers themselves are, of course, sensitive The numerical estimates of three-dimensional to such artifacts, and must be considered less cellular and mitochondrial parameters may be in reliable). Sampling errors arising from the sec- error if the assumptions underlying these esti- tioning process and from the selection of cell mates are not valid. The formula used here to profiles containing nuclear profiles are also un- estimate mean cellular and cytoplasmic volumes likely to have had a significant effect on the (Eq. 9, Appendix) is likely to be quite sensitive conclusions of this study. Accordingly, these to deviations of the cells and nuclei from an types of errors will not be treated in detail. A assumed spherical shape or from assumed con- possible sampling error due to the other cell centricity; the net result of such deviations would selection criteria employed here is discussed in probably be an underestimate of the above pa- the next section. rameters. In the evaluation of certain mitochon- Errors in the two-dimensional measurements drial parameters (e.g., number density), the im- may come from at least two sources. The contri- portant assumption is that of convexity of the bution from the measuring apparatus used in this organelles (see Appendix). A light microscope study (see Materials and Methods) is believed to examination of HeLa F-315 cells stained with be minimal. The digitizer board of this apparatus diaminobenzidine (35) has indicated that the ma- distinguishes points 0.01 inches apart; at the jority of mitochondria in these cells are discrete magnification used for measuring mitochondrial and unbranched, but true convexity in the topo-

482 THE JOURNAL OF CELL BIOLOGY 9VOLUME 74, 1977 logical sense is difficult to establish. The effect of sured by Terasima and Tolmach (44) for the $3 a given deviation from organelle convexity on the strain of HeLa cells. It is not known to what values calculated for the mitochondrial parame- degree this is due to: (a) differences in the meth- ters is not known. ods used for estimating cellular volumes in the two studies; (b) an underestimation of F-315 cell Volume Growth of Synchronized volumes arising from the nonspherical shape of HeLa Cells cells and nuclei; or (c) a real difference between The estimates made in the present study of the $3 and F-315 cells. mean cellular and cytoplasmic volumes at various Continuous Growth and Division of times during the cell cycle of HeLa F-315 cells indicate a continuously increasing rate of volume Mitochondria during the Cell Cycle growth during interphase. This finding agrees The two-dimensional data obtained here by well with results obtained by others in the case of electron micrographic measurements of various HeLa $3 cells (44), Chinese hamster lung cells cellular and mitochondrial profile parameters (40), and CHO cells (2). In the study on CHO provide evidence in support of a number of con- cells (2), an exponential pattern of volume in- clusions. crease was observed in many of the cultures; in The observed constancy of the ratio of total the others, however, the pattern was more outer membrane contour length to cytoplasmic curved, i.e., concave, than an exponential. The area during the cell cycle indicates a continuous curve obtained in the present study (Fig. 3 b) is of growth of the mitochondrial outer membrane in Downloaded from this latter type (analysis not shown). terms of surface area. This growth results in a Due to the cell profile selection criteria applied continuous increase in total mitochondrial vol- to the 19-h (G2) and 0-h (M) samples, there is ume, as shown by the constancy of the mitochon- reason to believe that the mean cellular and cyto- drial to cytoplasmic area ratio. The volume plasmic volumes determined for these time- growth occurs in such a way that the mitochon- points may be artificially high relative to those drial complement occupies a constant fraction www.jcb.org for the other time-points. In the former case, the (-10-11%) of the volume of the cytoplasm. A size criterion employed in an attempt to exclude corollary is that the rate of mitochondrial volume contaminating G1 cells from the sample may growth increases continuously during interphase, have also excluded profiles of small G2 cells. in parallel with the rate of growth of cellular and on August 22, 2006 Similarly, in the 0-h sample, cell profile selection cytoplasmic volumes. based on the presence of a chromosome profile The present data do not provide any informa- may have increased the proportion of nearly tion as to the behavior of the inner mitochondrial equatorial sections among the cells analyzed; this membrane during the cell cycle. However, it phenomenon would apply especially to meta- should be mentioned that a limited series of phase cells, which constituted the majority of the measurements carried out on cells from the 3-h mitotic sample. The tendency of these effects and 19-h time-points showed no significant dif- would be to increase artificially the apparent ference in the contour length of the inner mem- mean cellular volumes for the respective time- brane, expressed either per mitochondrial profile points, relative to those found for the rest of or per unit of cytoplasmic area. interphase. Our observation of a greater than As noted above, a previous histochemical twofold increase in the mean cellular and cyto- study (35) has indicated that, in a large propor- plasmic volumes during the cell cycle is quite tion of HeLa F-315 cells, the mitochondrial com- possibly attributable to the effect just discussed plement is divided into distinct organelles. It is regarding the 0-h sample. Nonetheless, it should therefore appropriate in this system to discuss the be mentioned that in studies on HeLa $3 cells, cell-cycle-related division of the mitochondrial under conditions where (due to loss of syn- mass in terms of the division of discrete bodies chrony) one would expect to measure less than a and to attempt to interpret the profile measure- twofold increase in mean cellular volume, Tera- ments accordingly. A prerequisite for any such sima and Tolmach (44) observed more than a interpretation of two-dimensional data is that the doubling in this parameter. distribution of mitochondrial shapes at various The absolute values of the estimated cellular stages of the cell cycle be known or at least be and cytoplasmic volumes (Table II, columns 2 unchanged. and 4) are substantially smaller than those mea- Though the mitochondrial population in HeLa

POSAKONY ET AL. Mitochondrial Growth and Division in HeLa Cells 483 cells includes bodies of quite different shapes for the functionality of the inner membrane in (35), the present measurements have indeed sug- oxidative phosphorylation (5, 38). The finding gested that there is no appreciable change in the that their synthesis occurs continuously during distribution of shapes during the cell cycle. The the cell cycle, together with the present observa- mean area, mean outer membrane contour tion of the continuous growth of mitochondria, length, and mean axis ratio of the mitochondrial suggests that the differentiation of these organ- profiles are all observed to remain essentially elles into active sites of oxidative phosphoryla- constant during the cycle. More importantly, the tion proceeds throughout interphase. frequency distributions of these parameters are In contrast, both replication and transcription very similar from one time-point to another (with of mitochondrial DNA in HeLa cells occur the possible exception of the 0-h point). It ap- mainly in a restricted portion of the cell cycle, pears very unlikely that significant cell-cycle-de- covering the late S and the G2 phases (33, 34). pendent changes in the mitochondrial shape dis- The dissimilar temporal behavior of mitochon- tribution would not be reflected in one or more drial growth and mitochondrial DNA replication of these measurements.8 in HeLa cells represents a situation quite unlike On the basis of the foregoing considerations, that observed in , in which these two the constancy of the mean mitochondrial profile processes are closely coordinated and interde- area and of the number of profiles per unit of pendent (18). cytoplasmic area, taken together, can be inter- A constant mitochondrial to cytoplasmic vol- Downloaded from preted to indicate a continuous division of mito- ume ratio during the growth cycle has been a chondria during the cell cycle, at the level of the consistent finding in studies on mitochondrial whole cell population. Furthermore, no evidence growth and division in lower eukaryotic orga- has been found in this study for the occurrence of nisms. Mean values of -12, 14, and 11% have mitochondrial synchrony strictly within individ- been reported in Neurospora (17), Saccharomy- ual cells, though the methods of analysis availa- ces cerevisiae (15), and Tetrahyrnena (19), re- www.jcb.org ble cannot rule out all models of such synchrony. spectively, compared with -11% in the present Thus, it appears that in HeLa cells there is no study. Thus, the proportionate growth of mito- fixed temporal relationship between mitochon- chondrial and cytoplasmic volumes during the drial growth and division, on the one hand, and cell cycle appears to be a widespread phenome- on August 22, 2006 the events of the cell cycle, on the other. Rather, non. In the Neurospora (17) and Tetrahymena the picture that emerges is one of a steady-state (19) studies, as well as in a study on Schizosac- process in which the mitochondrial composition charomyces pornbe (31), morphometric data of of the cytoplasm remains relatively uniform dur- other kinds (e.g., variations in mean mitochon- ing interphase. drial profile area or in the number of profiles per With regard to the physical mode of mitochon- cell section) were interpreted to indicate that the drial division, no information was obtained in division of mitochondria occurs in a restricted this study which would support unambiguously part of the cell cycle in these organisms. In addi- either the elongation-constriction or the trans- tion, cytological evidence of a change in mito- verse partition mechanism (42, 43). chondrial shape during the cell cycle was ob- As shown in a previous study from this labora- tained in these studies (17, 19, 31). On the other tory (12), the rate of mitochondrial protein syn- hand, an examination of mitochondria in Saccha- thesis in HeLa cells increases about twofold on a romyces cerevisiae by serial thin sectioning led per cell basis during interphase, remaining fairly Grimes et al. (15) to conclude that mitochondrial constant on a per unit mass basis. The products division is not tightly coupled to the cell cycle. A of mitochondrial protein synthesis are essential study (37) in one mammalian system, Chinese hamster cells, showed parallel variations during s However, it should be mentioned that changes in the the cell cycle in the mitochondrial to cellular mean relative length of the mitochondrial bodies could profile area ratio and in the number of mitochon- conceivably escape detection by the methods used here, due to the apparent insensitivityof the profile axis ratio driat profiles per unit of cellular area. The pat- distribution to variations in the relative length of the tern observed did not point conclusively to either sectioned body above a certain value (compare, for ex- a continuous or a synchronous mode of mito- ample, Fig. 6k and l). chondrial growth and division.

484 THE JOURNAL OF CELL BIOLOGY - VOLUME 74, 1977 Numerical Evaluation of found for a significant cell-cycle-dependent Three-Dimensional Mitochondrial change in the mitochondrial shape distribution. The values calculated in this study for certain Parameters mitochondrial parameters in HeLa cells can be Our numerical estimates of various mitochon- compared with those reported by Grimes et al. drial parameters (see Table II) serve to express in (15) in their serial thin-section study on haploid three-dimensional terms the interpretation given and diploid strains of Saccharomyces cerevisiae. above to the two-dimensional results; namely, For the number of mitochondria per unit of cyto- that the constancy of the mitochondrial to cyto- plasmic volume, these authors observed values of plasmic volume ratio during the cell cycle repre- 0.85/p.m 3 and 1.02//xm 3 (haploid and diploid sents a constancy of both of its constituent fac- strains, respectively)9; for the mean mitochon- tors, the number of mitochondria per unit vol- drial volume, the values were 0.16 jzm 3 and 0.14 ume and the mean mitochondrial volume. The tzm 3, respectively. The corresponding averages apparently continuous division of mitochondria is from the present study are 0.39/~tm 3 and 0.29 reflected in the increase in the estimated number p.m z. of mitochondria per cell. Finally, it should be mentioned that the com- Although a specific estimate of the shape con- puted values for the number of mitochondria per stant J was employed here for the purpose of cell (see Table II) are quite similar to those evaluating various three-dimensional mitochon- obtained by direct count of organelles in HeLa F- drial parameters, the relative values of these pa- 315 cells stained with diaminobenzidine (J. Posa- Downloaded from rameters at the various time-points do not de- kony, unpublished observations; see reference pend upon the choice of a particular value of J. It 35). suffices that the mean shape constant for the mitochondrial population (Eq. 12, Appendix) 9 The values of 850 and 1,020 per 100 p.m3 reported in does not vary appreciably during the cell cycle. the text (15) appear to be erroneous (see Table II of

As discussed previously, no evidence has been reference 15). www.jcb.org

APPENDIX on August 22, 2006

ESTIMATION OF THREE-DIMENSIONAL CYTOLOGICAL PARAMETERS

FROM TWO-DIMENSIONAL MEASUREMENTS MADE ON CELL SECTIONS

JOEL N. FRANKLIN. From the Department of Applied Mathematics. California Institute of Technology, Pasadena, California 91125

This appendix describes a number of statistical methods which are of use in the estimation of three-dimensional cytological parameters from two-dimensional measurements made on cell sections. The full derivations of these formulas will be published elsewhere (14). For an excellent review of stereological principles and methods, see reference 47; see also references 9, 22.29, and 30.

Estimation of Cellular and last group can be estimated from measurements of Cytoplasmic Volumes the cellular and nuclear profile areas, under cer- If one makes a random section through a sam- tain assumptions concerning the shape of the cells ple of cells, a subpopulation of the cells is success- and nuclei, their size ratio, and their relative posi- fully cut by the sectioning plane; some of the cells tion (see below). However, it is necessary to apply in this subpopulation are cut through their nuclei. to the measurements statistical corrections which Mean cellular and cytoplasmic volumes I for this take into account (a) the fact that the cells under consideration were sectioned only through their The cytoplasmic volume is defined as the cellular vol- nuclei and (b) "sectioning bias", i.e., the phenom- ume minus the nuclear volume. enon that a spherical body (in this case, the nu-

POSAKONY ET AL. Mitochondrial Growth and Division in HeLa Cells 485 cleus) is not in general sectioned through its cen- p = ([(2/q) + 1]/3)89 (5) ter. The method which follows 2 is based on two Given this estimate of p, the average cytoplas- idealizing assumptions: (a) that the cells and nu- mic to cellular volume ratio for the cells analyzed clei are concentric spheres, and (b) that in the can be calculated; it is [1 - (1/p3)]. Finally, the population under study, e.g., at a given point in mean cell volume of the sectioned population can the cell cycle, all cells have the same value of the be computed as follows: In each cell, the section- parameterp, where p = R/r -- radius of cell/radius ing plane passes at a distance x from the common of nucleus; 0 -< r -< R. (Note that the cells need center of the nucleus and the cell. (Note that 0 -< x not be of the same size.) -< i.) The resulting nuclear and cellular profile The first portion of the following development, areas are, respectively, leading to Equations 1-5, is a simplified version of A. = 7r(r'-' - x 2) (6) the treatment by Mayhew and Cruz (26); Eq. 5 is the same as Correction 12 of reference 26. The and development is furthermore based on the appro- priateness of Formulation 2 of Mayhew and Cruz A c = rr(R 2 - x2). (7) Orive (27) in the analysis of our data. Suppose that a spherical cell (radius R) is ran- The cytoplasmic profile area is: domly sectioned through its nucleus (radius r). 1, The area of the resulting nuclear profile has the Ac - A, = ~(R'-' - r 2) = ~'R 2 (1 - ~) (8) Downloaded from expected value: i.e., it is independent of x. Using Eq. 8, the 2 E(A,,) = ~Trr2; (1) volume of an individual cell can be found: 4 3 the area of the resulting cell profile has the ex- ( Vc)i = ~ ~-R, = (4/3 X/N)

(9) www.jcb.org pected value: 9[p2/(p2 _ 1)F2. [(Ac)~ - (A,,),]3r-'. 1 E(Ac) = rrR 2 (1 - (2) The mean cell volume of the sectioned population 3pZ" is then: on August 22, 2006

Now consider the sectioning (through the nuclei) X of a sample of cells with R = R1, R2,... , and r = 17"~ = f-'. ~ (V~)~, (10) i=1 rl, r2 ..... in which R~ = P. The ratio of the sum r~ where N is the number of cell profiles analyzed 9 of the nuclear profile areas to the sum of the cell profile areas can be expressed using equations 1 Estimation of Number Density and 2. It is: of Organelles E(XA.) _ 2 (3) This section is concerned with counting the E(XA,) 3p 2 - 1' number Nv of organelles in a unit cellular or cytoplasmic volume. We wish to estimate this pa- An experimental value, q, of this ratio can be rameter from two-dimensional measurements computed from measurements of nuclear and cel- made on cell and organelle profiles in random lular profile areas in cell sections: sections. Currently used estimators for Nv depend .V N on a prior knowledge of the sizes of the organdies q = ~, (An)i/~ (At). (4) as three-dimensional bodies, e.g., the estimator of i=1 /=1 Weibel (47) and Weibel and Gomez (48) contains where N is the number of cell (or nuclear) profiles. a coefficient K relating to the organelle size distri- Substituting q for the left-hand side of Eq. 3 and bution. An estimator for Nv will be described (Eq. solving for p, we find 11) which does not require information concern- ing the size distribution of the intact organdies. * Another, possibly more reliable, method for estimating The estimation of Nv from cross-sectional meas- cellular and cytoplasmic volumes is given in reference urements rests on the assumption that the organ- 47. elles under study are convex bodies. The most

486 THE JOURNAL OF CELL BIOLOGY 9VOLUME 74, 1977 important implication of this term is that an indi- the unit sphere, defined by all possible u with a vidual organelle yields a single profile when sec- given origin. Taking ul = cosO sin0, u2 = sino sin0, tioned by a given plane; conversely, a given profile and Ua = cos0, derives exclusively from the sectioning of one or- ganelle. (Thus, it is apparent that branched or J = (2/r) I(u) sin0 d0dt. (14) curved organelles are nonconvex bodies 9Clearly, ff,2ff,~ it is important to determine the validity of the assumption of convexity in the system being inves- The definitions of I(u) and of J apply specifically tigated (see Discussion for evidence on this point to the estimator given in Eq. 11. Though the in the case of HeLa cells). integral in Eq. 14 cannot be evaluated analyti- If at represents the area of an individual organ- cally, it can be evaluated numerically by the use of elle profile, the number of organelles per unit the trapezoid rule (45), Simpson's rule (45), or a volume is estimated as follows: Monte Carlo method (16). When only an approxi- mate value for J is sought, the following inequali- M ties can be employed: Nv = j-l. ~ a7~ .A-1. (11) i=1 J <- rr89(ala2a3)-~ (15) In this equation, M is the number of organelle 9[(1/3) (a, z + a2 z + a32)]~4 profiles and A is the total cellular or cytoplasmic J >- rr89(a~a2az)-~ profile area in which the M organelle profiles were (16) Downloaded from observed (depending on whether Nv is being esti- 9(1/3)(a~/2 + a~12 + a]/2). mated with respect to cellular or cytoplasmic vol- ume). The number J can be called the shape Profile Axis Ratio Distributions constant. Each (convex) body of a given shape has One approach toward deriving information a corresponding value of J, which is independent about the three-dimensional shape of organelles is of the size of the body, e.g., J = ~-t for spheres of to compare experimental measurements of a given www.jcb.org all sizes. organelle profile parameter in random sections of The estimator given in Eq. 11 remains valid the organelles with theoretical predictions con- even if the organelles do not all have the same cerning the behavior of the parameter when a shape (or shape constant) 9In this case, a mean body of known shape is randomly sectioned. In on August 22, 2006 shape constant is used 9Suppose the organelle pop- this section, methods are discussed for obtaining ulation contains bodies of N different shapes, with the frequency distribution of profile axis ratios corresponding values of J: J~,J2, 9 9 9 , JN" Let f~ be expected from the sectioning of ellipsoids. the fraction of the population with shape i and The calculus of probabilities employs two types shape constant Ji (a graph of J vs. fwould describe of functions in the treatment of a random variable a shape distribution for the population) 9The mean t -> 1. One is the probability density function, shape constant is then: denoted p(t), and the other is the distribution N function, P(z). The relationship between the two J = ~ (y,.J,). (12) is as follows: i=l An equation will now be given (Eq. 14) defin- P(z) = Probability (t -< z) = p(t) dt. (17) ing J explicitly for the general ellipsoid 9Let the ellipsoid have the semi-axes al <-- a2 -< as. (Since J is independent of size, it depends on only two of In the present case, P(z) is the probability that, when an ellipsoid is sectioned by a random plane, the axis ratios of the body, say az/al and a2/a~.) Suppose that the ellipsoid is centered at the origin the profile axis ratio t is less than or equal to some and that its axes are oriented along the principal chosen z. A prolate spheroid is an ellipsoid of revolution coordinate axes 9Let u = (u~, Us, Ua) be any unit vector 9Define the dimensionless quantity with one long axis and two equal, shorter axes. Suppose a prolate spheroid is cut by a random I(u) = rrt(ala~a3)-89 (13) plane. The resulting profile is an ellipse; it has 9(a12ul 2 + az2u22 + az2u32)~4. some axis ratio t -> 1. Let the spheroid have semi- axes al = a2 < a3 and let T = aa/al, i.e., T is the The shape constant J is the average of l(u) over ratio of the long to the short axes of the spheroid 9

POSAKONV ET AL. Mitochondrial Growth and Division in HeLa Cells 48"/ If a section is cut normal to the long axis, the k = [(T 2 -1)89(2T2) -' + (2T3) -' resulting profile is a circle and t = 1; if it is cut -In [Z + (Z'-' - 1)89-I. (20) parallel to the long axis, t = T. It can be demon- strated that The construction of a frequency distribution (his- togram) of t from Eq. 18 is discussed below (Eqs. p(t) =k.t -a(T z-tz)-i (1-

Input: al, a~:, a3 M 1 = Zo ..... z. = T P(1) = 0 Downloaded from

Sets~ = 0 (i= 1 ..... n)

Define functions H(u), K(u) www.jcb.org

Compute L1, L~, La

Computey~ (i = 1 ..... n) on August 22, 2006 -+ )-Generate three random numbers xt, x2, xa (0 < x <-1)

-- Is R 2 _< 1 ? no ~ yes

Computeu/ (i = 1,2,3) M times Compute H(u), K(u)

Find m = minimum i such that H(u) >- y~.K(u) ~ (l~i~n) Add H(u) to s~ for/ = m .... ,n

Compute and print P(z~) (i = 0 ..... n) FIGURE A 1 Flow-chart diagram of the Monte Carlo method for computing profile axis ratio distribu- tions for general ellipsoids. The input information is explained in the text; equations for the other functions are given in Table A-I.

488 THE JOURNAL OF CELL BIOLOGY' VOLUME 74, 1977 Let the ellipsoid have semi-axes al -< a2 -< a3 obtained from Eq. 18 and those given by the and T = aa/az, i.e., T is the ratio of the longest to Monte Carlo method (M = 10,000), when both the shortest axis. The Monte Carlo approach to are applied to a prolate spheroid of axis ratios the calculation of P(z) is summarized in Table A-I 3:1:1 (J. Posakony, unpublished observations). and Fig. A 1. The parameter M is the desired The values are in complete agreement within the number of iterations in the calculation; the error in error of the Monte Carlo calculation. the method is of the order 1/x/~. The computing machine is given the values of at, a2, a3; M; and The generosity of Dr. Jerome Vinograd and Dr. Norman P(1) = 0. For the purposes of constructing a Davidson in making available their digitizer-calculator frequency distribution of t (see below), P(z) is apparatus is gratefully acknowledged. calculated at an appropriate number n of values of This work was supported by National Institutes of z (generally equally spaced) in the interval 1 <- z -< Health grant GM-11726, a Demham Fellowship (D 196) of the American Cancer Society to J. England, and a T. Thus, the computer is also given the desired z's, National Science Foundation Summer Research with z0 = 1 and z, = T. It is then programmed to Traineeship to J. Posakony. carry out the Monte Carlo calculation, as shown in Fig. A 1, using the equations in Table A-I, Received for publication 29 April 1976, and in revised The frequency distribution of the parameter t form 12 April 1977. may be represented by a histogram (see Fig. 6g- l). The frequency f corresponding to a given inter- REFERENCES val I~ is computed from p(t) or P(z) as follows: Downloaded from 1. ARRAMowrrz, M., and I. A. STEGUN, editors. 1964. Handbook of Mathematical Functions. Dover f~ = p(t) dt, (21) Publications, Inc., New York. zi- 1 2. ANDERSON, E. C., G. I. BELL, D. F. PETERSEN, and R. A. TOaEY. 1969. Cell growth and division. or IV. Determination of volume growth rate and divi- www.jcb.org f = P(z,) - P(z,-1), (22) sion probability. Biophys. J. 9:246-263. 3. ASHWELL,M., and T. S. WORK. 1970. The biogen- so that L is the interval from z~-i to z~. The two esis of mitochondria. Annu. Rev. Biochem. 39:251- equations are equivalent, since, 290.

4. ATrARDI, G., P. COSTANTINO, J. ENGLAND, D. on August 22, 2006 LYNCH, W. MURPHY, D. OJALA, J. POSAKONY,and f, = fzil p(t) dt = ff'p(t) dt - fl .... p(t) dt B. SromiiE. 1975. The biogenesis of mitochondria (23) in HeLa cells: a molecular and cellular study. In = P(z3 - P(zj_,). Genetics and Biogenesis of Mitochondria and Chlo- roplasts, C. W. Birky, Jr., P. S. Pedman, and T, J. A comparison has been made between the f's Byers, editors. The Ohio State University Press, Columbus, Ohio. 3-65. 5. BEArrlE, D. S. 1971. The synthesis of mitochon- TABLE A-I drial proteins. Sub-cell. Biochem. 1:1-23. 6. BmKY, C. W., JR., P. S. PERLMAN, and T. J. Equations Used in the Monte Carlo Method* for Computing Profile Axis Ratio Distribution for BYERS, editors. 1975. Genetics and Biogenesis of Mitochondria and . The General Ellipsoids Ohio State University Press, Columbus, Ohio. Lt = 1/a~ 2 7. BOARDMAN, N. K., A. W. LINNANE, and R. M. L 2 = 1/az 2 SMILLIE, editors. 1971. Autonomy and Biogenesis L3 = 1/a3 z of Mitochondria and Chloroplasts. North-Holland Publishing Co., Amsterdam. y~ = [(Z, + z71) ~]-' (i= 1,... ,n) R 2 = xl ~- + x2 2 + xa ~ 8. CRoxror~, F. E. 1953. Elementary Statistics with ut ~ = x d / li~2 Applications in Medicine and the Biological Sci- uz 2 = x2Z/R 2 ences. Dover Publications, Inc., New York. u32 = xz2/l~ 9. CRUZ OmVE, L.-M. 1976. Particle size-shape distri- H(u) = (u~2/Ll + u2~/L2 + u32/L3) 112 butions: the general spheroid problem. I. Mathe- K(u) = ul"Z(L2 + La) + u22(L1 + La) + ua2(L1 + L2) matical model. J. Microsc. ( Oxf. ). 107:235-253. P(z3 = &/s,, (i = 1,...,n);P(zo) = P(1) = 0 10. DELESSE, M. A. 1847. Proc~de m6canique pour d6terminer la composition des roches. C. R. Acad. * See Fig. A 1. Sci. 25:544.

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490 THE JOURNAL OF CELL BIOLOGY 9VOLUME 74, 1977 simplified lead citrate stain for use in electron Appl. Physiol. 17:343-348. microscopy. J. Cell Biol. 25:407-408. 49. WEIBEL, E., G. S. KISTLER, and W. F. SCHERLE. 47. WEmEL, E. 1969. Stereological principles for mor- 1966. Practical stereological methods for morpho- phometry in electron microscopic cytology. Int. metric cytology. J. Cell Biol. 30:23-38. Rev. Cytol. 26:235-302. 50. YUAN, P.-T. 1933. On the logarithmic frequency 48. WEIBEL, E., and D. M. GOMEZ. 1962. A principle distribution and the semi-logarithmic correlation for counting tissue structures on random sections. J. surface. Ann. Math. Stat. 4:30-74. Downloaded from www.jcb.org on August 22, 2006

POSAKONY ~T AL. Mitochondrial Growth and Division in HeLa Cells 491