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Theoretical basis for cell cycle analysis
J. Theoret. Biol. (i968) 18, 195-209
Theoretical Basis for Cell Cycle Analysis II. Further Studies on Labelled Mitosis Wave Method MANABU
Dcpartmerrt
of Pathology,
TAKAHASHI
Yamaguchi University Ube, Japan
School of Medicine,
(Received 13 June 1967) The method of interpreting experimental data on lahelled mitosis wave (Takahashi, 1966) is reconfirmed using a less restrictive model which has the following characteristics: (i) duration of each stage (Cl, S, G2 and M) is scattered according Pearson Type III distribution; (ii) their coefficient of variation can he changed independently of the mean, even if variation of a cycle time is fixed; (iii) it can he accommodated to a system which involves death process and, as special cases, to situations where labelled cells are more liable to die than unlahelled counterparts or some fraction of lahelled cells are missed undetected autoradiographically. A method for computation of labelled mitosis wave which is expected theoretically from the mode: was shown. Numerical solutions of some equations had to be obtained by digital computer. As a consequence of the present study, it was stressed that, when the first peak of the wave is helow 80% (or the second below SO%), coincidence of the experimental curve with the theoretical one obtained, by putting presumed estimates into the parameters, should he reached by trial and error before the estimates are finally trusted. The model and the computational method were so designed as to be applicable to other methods of cell cycle analysis.
1. Introduction It seems to be a rule in some reported data (e.g. McCarter 8c Quastler, 1962; Post & Hoffman, 1964; Reiskin t Mendelsohn, 1964) that a change of cell cycle occurs during the course of carcinogenesis, as manifested in the labelled mitosis wave by a decrease in the slope of its declining limb. This fact indicates, according to Wimber’s way of interpretation (Wimber, 1960), that a variation of S stage duration increases in association with carcinogenesis. It is probably due to loss of control of over DNA synthesis particularly over its termination. Since this assertion has important implications, it is desirable to preclude every conceivable error of interpretation.
196
M.
TAKAHASHI
In the preceding paper (Takahashi, 1966), quantitative relations between labelled mitosis wave and parameters of cell cycle were examined, using a stochastic model. The results closely paralleled Wimber’s notion (Wimber, 1960). However, the model had restrictions in three respects; i.e. (i) coeiticient of variation of stage duration could not be altered independently of its mean, provided that total variation of a cycle time is fixed; (ii) death process was not appropriately taken into account; and (iii) preferential death of labelled cells was not assumed. The purpose of this report is to assure the previous results by using a generalized model and further to establish a foundation for testing the validity of various methods for cell cycle analysis. As will be seen below, the first restriction was relaxed by assuming different phase transition den@es for different stages and the second by introducing a probability of death which may occur in any of the reproductive phases. The third was eliminated by assigning smaller reproductive coefficients to labelled cells. Several investigators (Barret, 1966; Trucco, 1967) treated the related problems: the latter author by making use of Van Foerster equations (Von Foerster, 1959). However, their theories were not indispensable since they ultimately resorted to Monte Carlo procedure, a simulation method which is unrelated in principle to the theories they described. The present model is a realistic, though not ideal, representation of the cellular system, and the following ilIustrations will enable the readers to prepare by themselves a computer programme for obtaining theoretical values of “percentage of mitosis labelled”. 2. Preliminary Considerations A cell cycle is an example of stochastic processes in which the durations of Cl, S, G2 and M stages are probabilistic variables. The distribution of a cycle time is known to be positively skewed (e.g. Kelly & Rahn, 1932; Powell, 1955; Burns, 1956; Prescott, 1959; I-&u, 1960; Kubitschek, 1962a; Dawson, Madoc-Jones & Field, 1965; Sisken & Morasca, 1965). Among a variety of expressions for cycle time distribution such as Yule’s distribution (Rahn, 1932), Pearson Type III distribution (Kendall, 1948), normal distribution of generation rate (Kubitschek, 1962u), and log-normal distribution (Barret, 1966), the author adopted the second simply for ease of mathematical treatment. This choice may be justified because of mutual similarity of these distributions: reciprocal or logarithm of variables in Pearson Type III distribution yields fairly linear plots on a probit paper. For the same reason, the data of Sisken & Morasca (1965) seem to support a view that the variation of component stages can be approximated by Pearson Type III distribution,
LABELLED
MITOSIS
WAVE
METHOD
197
It has been reported that in bacterial cell populations there are positive correlations of individual generation times between sisters and slightly negative correlations between mother and daughters (Powell, 1955; Schaechter, Williamson, Hood & Koch, 1962; Kubitschek, 1962~). Sisken & Morasca (1965) and Killander & Zetterberg (1965) suggested, moreover, that lengths of successive stages are negatively correlated in mammalian cells. However, what is really important in relation to labelled mitosis wave is not the correlations but the distributions of combined length of consecutive stages (i.e. Gl, Gl +S, Gl +S+G2, etc.) which are also close to Pearson Type III. Indeed, unless these are less variable than the component stages, the cell cycle can be simulated by this model reasonably well, but otherwise not. 3. Model
Mathematical structure of the generalized model is essentially the same as the previous one except that some restrictions are relaxed. However, its thorough description may be helpful to understand the papers to be published later. As illustrated in Fig. 1, the model is specified by the following postulates. (1) A cell passes through hypothetical (mathematical) phases, k in number, as it proceeds on a cell cycle. Transition of the phase constitutes a Poisson process so that phase duration is distributed exponentially. (2) Of k phases, the first gr phases represent Gl stage which is cytologically discernible. Succeeding s phases [i.e. from (gri- 1)th to (gr +s)th phases] correspond to S stage. The other stages are likewise specified so that g,+s+g2+m = k. (1) A transition to (k + l)th phase signifies a birth of daughters in their first phase. (3) Two daughters are yielded at each cell division (binary fission). However, this is not an essential postulate. (4) A cell transits from the ith phase (i = 1,2,. . . , k) to the next with probability densities (phase transition densities) Ri which may differ for Gl, S, G2 and M stages and will be equalized to &i, A,, A,, and AM, respectively. (5) A cell cycle may be interrupted by an incidence of death (reproductive or metabolic) in any of the phases. Probability densities (death densities) in Gl, S, G2 and M stages will be equalized to pcl, p,, p,, and ~1~. This is a device that is analogous to Kendall’s way of simulating variation of individual generation time of bacterial cells (Kendall, 1948). However, this differs slightly from Kendall’s in that simulation extended to the component stages within a generation and also from the previous one (Taka-
198
M.
TAKAHASHI
Fro. 1. Schematic illustration of model. A cell passes through k hypothetical phases sequentially as cell cycle evolves clockwise. At a turn of the cycle the cell is divided into two daughters.
hashi, 1966) in that a condition for uniformity of transition density was relaxed and death density introduced. The present model will be called general&d multiple-phase birth-and-death process. 4. Some Features of the Model 4.1. DURATION OF A STAGE PASSED SUCCESSFULLY From the model (section 3) can be derived’s frequency function of stage duration which for S stage, for example, is of the form
&+hYT.-l (s-l)!
expC-&+hH,
(2)
(see Appendix for proof). This is the Pearson Type III distribution which we looked for (section 2). The mean and coefficient of variation of S stage duration are
and
T-S s- a-Ps
(3)
cv, = s-f.. (4) Here, the statistics were assumed to be taken with exclusion of cells which die in the course of S stage. As evident from (7), unless a large fraction of cells die in the S stage, the mean (3) can be approximated by
LABELLED
Comparable
MITOSIS
WAVE
199
METHOD
relations hold for other stages. 4.2.
DURATION
OF A STAGE
TERMhATING
IN
DEATH
The interval from the beginning of S stage to death varies according to frequency function :
rs-l/“-l 8 -~~-~exp[-(L,+&r], 4 -c l-r, i=l (i-l)!
where r, is the probability fully, hence
(6)
with which a cell passes through S stage success-
(see Appendix for proof). Therefore, the mean and coefficient of variation of the interval are (8)
and
(9) Relations similar to (6), Q, (8) and (9) can be applied to other stages by changing the parameters and suffixes. 4.3.
DISCBETE
VARIATIONS
OF
STAGE
DURATION
The variable in (4) being an integer, the CY of stage duration can take only discrete values. However, this does not b&come a serious drawback. Kubitschek (1962b) called attention to discrete variations in the CV of generation rate. It would not be unreasonable to expect that the same holds true for the component stages also. 5. Adjustment of the Model Hypothetical parameters of the model are connected with observable quantities of cell population as follows: s = cvp, (10) 1 Ps = (11) and (12)
lkl = rc1 t-11 -rol Tm+T
1
gl = fix (W&q
rQi
Cl
I I
-.
CV;
T; and
ps---
1 __~
s = fix (CV,-‘)
rs
CV,
Ts and
S
t Reproductive coefficient is equal to rol rSrcarnd. 2 The symbol fix ( ) represents an integer closest to real number in the parenthesis.
Phase transition density
3z ‘3 $j Death density SE OCd
Number of phasest
Mean and CV of stage duration Bm I, .P E .s Mean and CV of entryto-death interval a4 0= Rate of successful passa&
Name of stage
1
Adjustment of parameters
TABLE
rG2
and
CV’,
, TcaST a
1
ga = fix (CV,-,)
T’,
Tcz and CV,,
G2
CV,
p*, _. - __-__- 1
m = fix (CVi2)
rM
TLandCV&
Tnl and
M
2 7c * X * ?
z.
LABELLED
MITOSIS
WAVE
METHOD
201
Parameters specifying the other stages can likewise be determined (Table 1). If the CVof a stage duration is given experimentally, an integer closest to (CV)-2 should be adopted as the number of phases. When labelled cells obey different kinetics, their parameters should be determined separately (see section 7). 6. Kinetics of Cell Population
Although renewal equation is frequently applied to analysis of cell population kinetics (e.g. Bellman & Harris, 1952; Harris, 1959, 1963), it does not suit the present problem in which the growth of subpopulations (cells in different phases) should be treated separately (see section 7). Therefore, Kendall’s method was modified for the purpose. According to the model described in Section 3, every cell is in either one of k phases. A subpopulation composed of ith-phased cells is called ith phase fraction (or compartment), and the mean number of ith-phased cells denoted by nJt>. Conceptionally, all the cells move sequentially through discrete compartments which as a whole form a closed circuit. As they pass through the compartments, some fraction of cells may be eliminated by death but on entering a new round of cycle they are doubled in number by ceil division. Thus, the model leads to a set of relations (13) and (19, which are similar to continuity equation of hydrodynamics (Fig. 2). The influx into the ith phase fraction (i # 1) in At is li-lni-l(t)At (13) and the eflux from it (J-i + pi> ni(t)At, (14) so that the increment Ani is obtained by subtraction: Art;(t) = Ri-lni-I(t)At-(li+~t3ni(t)At (i = 2,3,. : !, k)
(15)
and An&) = 2Q,(t)At-(11
+&n,(t)At.
(16)
Fro. 2. Growth of subpopulations. T.B.
D
202
M.
TAKAHASHI
Thus, simultaneous differential equations describing “compartment are derived : -I dn,(O ^~__ = 2l,n,(t)-(1,+~(,)n,(t) dt t dd) __ = ~i-lni-,(t)-(~i+c1j)ni(t)l dt
kinetics”
(17)
(i = 2,3,. . ,, k). ,
7. Percentage of Lab&d Mitoses Suppose that a cell population is pulse-labelled at t = 0. Only S-staged cells [i.e. cells in s compartments from (gI + 1)th to (gl +s)th phases] are labelled so that their number amounts to 8, +r (18) i=J+ $O). Any labelled mitotic cell which will appear later must, therefore, be their progeny. The descendants of those having been in other stages have no radioactivity. Hence, the number of labelled mitoses at t is equal to the sum M*(t) =
i i=k-m+
n*(t), 1
(19)
where n:(t) is giveri by solving the simultaneous differential equations (17), with the initial conditions that all the compartments other than those corresponding to S stage are nullified, i.e. n’(0) = 0 if lsisg, or y,+s
n:(t). i=k-m+l
(22)
The latter term can be calculated by solving the equations (17) using “rate of passage through stages”, rgl, I$, rz2 and r& under different initial conditions, i.e. no(o) = 0 if gl< i=
LABELLED
MITOSIS
WAVE
METHOD
203
Therefore, the formula that describes time-dependent change of “percentage of labelled mitoses” is obtained by dividing (19) with (22):
PLM(r) = M*(t) M(r) ’
(25)
This will be called “PLM function”. In a special case where labelled cells are more liable to die than unlabelled ones (or some fraction of labelled cells is missed undetected), smaller “rates of passage through stages” T*‘S (5 ,“s) must be used at the calculation of M*(r) (see Table 1). 8. Limiting Phase Structure Behaviours of cell population depend generally on both age distribution and distribution of stage duration. The labelled mitosis wave is not an exception to this. However, when the latter distribution is defined, relative cell number in each phase fraction converges, as cell cycle continues to develop indefinitely and reaches a steady state, to a limit which is determined uniquely by TG1, T,,. . ., TM, CV,,,. . ., and CV,. For simplicity, we will consider a case where there is a single cell in the first phase at t = 0: n,(O) = 1, (26) i?i(O) = 0, (i # 1).> This is not an essential condition because a population starting growth from different initial condition attains the same limiting phase structure. Laplace transform of the equations (17) with the initial condition (26) is (4 +/.h +P)N,w-2w,w = 1, -Ri-,N,-,(p)+(~i+~li+p)Ni(p),=O, (27) (i = 2,3,. . .,k) where N,(p) is Laplace transform of ni(r), i.e* N,(P) c ni(t)*
(28)
Therefore, the solutions of (17) in complex domain are Ni(p) =J$
(i = 1,2,. . ., k),
(29)
where i- 1
A(P) = nnj j=O
fi (lj+Pj+P) j=i+l
IIn0= II
(30)
*
(31)
and g(P) =,fil(ij+rj+P)m2,filAj
P
204
M.
TAKAHASHI
According to Heaviside’s expansion theorem (e.g. Jaeger, 1961), the inverse transform of (29) becomes (32) where e,, e2,. . . , e, are the roots of characteristic equation c?(P) = 0. (33) A predominant term in the series (32) at large t is that corresponding to the largest real root E of (33). The relative cell number of the ith-phase fraction converges, as t increases, to a limit (limiting phase fraction): (34) Consequently, in a steady state there exist relations (35) Results of numerical calculation disclosed a curious fact that limiting phase fractions Fl depend not only on the relative stage durations but also on their coefficients of variation. 9. Limiting Age Distribution
The age distribution of a population will also converge to a limit (limiting age distribution) as proliferation continues. This distribution can be calculated, if required, from the limiting phase fractions because the age distribution for a whole population is equal to the weighted sum of those for the subfractions. 10. Digital Computer Analysis
The foregoing considerations paved the way to analysis of labelled mitosis wave. However, in order to draw theoretical PLM curve of cell population for which a set of parameters is given, numerical calculation must be performed such that can be accomplished only with the aid of digital computer. The computation was performed by FACOM 231 of Yamaguchi University. The largest real root of characteristic equation (33) was obtained by Newton’s method, and the simultaneous differential equations (17) solved by RungeKutta-Gill method. 10.1.
ESTIMATION
OF MEAN
STAGE
DURATION
By changing the parameters TGI, TS, TG2 and TM in various combinations, it was examined what parts of the wave indicate the mean stage durations. The results obtained by using this generalized model supported the statement made previously (Takahashi, 1966): a choice of 50 % intercepts is satisfactory
LABELLED
MITOSIS
WAVE
METHOD
205
as far as the first peak of the wave is above 80% level. Labelled mitosis wave with the first peak lower than 70% level must be interpreted with caution, since in this case the mean S stage duration tends to be underestimated. To avoid this risk it is advisable to examine each time whether or not the experimental PLM curve coincides with that calculated, by putting presumed values into computational programmes. 10.2. ESTIMATION OF STANDARD DEVIATION OF STAGE DURATION The present model can be used to examine what part of the wave reflects CV of a stage duration. The relation was studied by varying one of the parameters gr, s, g2 and m. The results were in good agreement with the previous report (Takahashi, 1966) especially when the peaks are sufficiently high. 10.3. EFFECTS OF DOSES OF RADIOISOTOPE When cell population is brought into contact with [3H]thymidine of high specific activity, adverse radiobiological effects, such as increased cell death, chromosome breakage and decreased uptake of [3H]thymidine, become manifest (Drew & Painter, 1959; Johnson & Cronkite, 1959; Wimber, 1959; Post & Hoffman, 1961; Zajicek & Gross, 1964). The effect on’ the labelled mitosis wave can be examined by assigning smaller reproductive coefficients to labelled cells in the model than to unlabelled counterparts. As anticipated intuitively, height of the second wave became lower and the trough more depressed. If, on the contrary, [3H]thymidine is of low specific activity (or photographic exposure time is relatively insufficient), the label may be missed undetected particularly after the radioactivity is halved by cell division. The false-negative mitoses (labelled mitoses scored erroneously as unlabelled) add only to the denominator. The decrease of PLM value due to this failure is expected to be more pronounced than in the reverse, because in the latter condition death of labelled cells leads to the decreases of both numerator and denominator. 11. Errors due to Protracted Labelliog Before applying the results mentioned above, a question will be asked: whether the mode of labelling which, at mathematical treatment, was assumed to be virtually instantaneous is acceptable or not. Many workers allow cells to remain in contact with labelled thymidine for some 10 minutes or more to secure 100% detection of S-staged cells. Under such a circumstance, PLM curve takes such a configuration as shown by a bold line in Fig. 3.
206
M. TAKAHASHI
Time oiler
pulse-lobellinq
FIG. 3. Effect of protracted labelling. (A) PLM curve of cells labelled at the beginning of exposure; (B) PLM curve of cells labelled at the end of exposure; (C) experimentally observed PLM curve; @) duration of exposure to label.
12. Implication of Delayed Labelliog Several investigators (e.g. Diderholm, Fichtellius & Linder, 1962; Robinson & Brecher, 1963; Bryant, 1965) presented evidence that labelled thymidine is released from dying cells which have once incorporated it into DNA and then re-utilized by growing cells, the cells with rapid turnover rate becoming transient reservoir of the label. However, the re-utilization phenomenon is expected not to exert any appreciable effect on labelled mitosis wave, at least, of the first or second generation. 13. Stage of Tbymidine Uptake Important is the question whether or not the so-called “DNA synthetic period” may be accepted as such in a strict sense of the word. Recently, Moffat & Pelt (1966) demonstrated, by autoradiography, the presence of enormous amount of label in cryostat-sectioned tissue of [3H]thymidineinjected animal, most part of which was dissolved away in the course of tissue fixation and embedding. Basing on this observation they claimed that the label is pooled in the cells at the time of flashing without necessarily being incorporated into macromolecular DNA. Thus, it will serve as a source of labelled precursor for later synthesis of DNA, though not all of it is fixed and utilized. Similar observations have been made by Feinendegen & Bond (1962) and Cleaver & Holford (1965). For this reason, a new terminology a “stage of thymidine uptake” seems more appropriate than a “stage of DNA synthesis”. It is evident from cyclic change of PLM value that thymidine uptake takes place only in a limited period within a cell cycle. This fact suggests that a specific enzymatic process might be a likely mechanism for intracellular fixation of this substance.
LABELLED
MITOSIS
WAVE
METHOD
207
The enzymes necessary to accomplish the phosphorylation of thymidine, which are present in the liver of the newborn, are lost as the animal grows older (Hiatt & Bojarski, 1960) but reappear after partial hepatectomy at the time at which DNA synthesis recommences (Bollum & Potter, 1959; Canellakis, Jaffe, Montsavinos & Krakow, 1959; Gray ei al., 1960; Hiatt & Bojarski, 1960). Weissman, Smellie & Paul (1960) postulated a sequential induction of thymidine phosphorylating enzymes. It is not unlikely that the activity of these enzymes has some bearings with cellular uptake of thymidine. 14. Discussion A mathematical model is the only system that fits a purpose of examining relations between the shape of PLM curve and parameters of cell cycle. The model used here is satisfactory in that it is conformable to such wide variety of situations as enumerated below: (i) cell population may be growing exponentially, normally renewing, or in the intermediate state: a constancy of mean generation time being the only prerequisite; (ii) a stage duration may be variable, though its distribution should belong (or be similar) to Pearson Type III distribution and the length of a stage is independent of the others; (iii) mode of cellular differentiation or of metabolic death can be defined as desired ; (iv) growth kinetics of labelled cells can be made different from that of non-labelled counterparts and the possibility can be taken into account that a certain fraction of labelled cells may be missed in the autoradiogram ; and (v) arbitrary phase structure can be chosen as initial condition although limiting phase fraction can be calculated. It is hoped that widespread use of this model will lead to a srandardization of interpreting method thus facilitating comparison of data of different investigators on a common basis. Extensive computation of theoretical PLM curves corresponding to various combinations of parameters will yield sufficient information as to enable one to interpret every feature of the curve (height of the peaks, depth of the trough, gradient of the slope, etc.) without the aid of computer. The results and their application to the reported data will be published elsewhere. Of special interest is to find the characteristics of labelled mitosis waves of primary neoplasm or of cells under the influence of carcinogen which generally show much slower decline of descending limb as contrasted
208
M.
TAKAHASHI
to a rather abrupt rise of ascending portion of the first wave, though there are some exceptions to this rule (e.g. DBrmer, Tulinius & Oehlert, 1964). In the light of present investigation, the above fact is interpreted as indicating that the period of thymidine uptake is remarkably variable in the early stage of neoplastic change. I would like to thank Professor K. Inouye for his interest in this work and for much encouragement.
REFERENCES BARRET, J. C. (1966). J. natn. Cancer Inst. 37,443. BELLMAN, R. & HARRIS, T. (1952). Ann. Math. 55, 280. B~LLUM, F. J. & POTIZR, V. R. (1959). Cancer Res. 19, 561. BRYANT. B. J. (1965). ‘Exol. CeN Res. 37. 490. BURNS, i. W. (1956j. J. iel[. camp. Physiol. 47, 357. CANELLAKIS, E. S., JAFFE, J. J., MONTSAMNOS, R. & KRAKOW,
J. S. (1959).
J. biol. Chem.
234,2096. CLEAVER, J. E. & HOLFORD, R. M. (1965). Biophys. biochim. Acta, 103,657. DAW~ON, K. B., MADOC-JONES, H. & FIELD, E. 0. (1965). Expl. Cell Res. 38, 75. DIDERHOLM, H., FICHTELLIUS, K. E. & LINDER, 0. (1962). Expl. Cell Res. 27, 431. DORMER, P., TUUNIUS, H. & OEHLERT, W. (1964). 2. Krebsforsch. 66, 11. DREW, R. M. & PAINTER, R. B. (1959). Rad. Res. 11, 635. FEINENDEGEN, L. E. & Born), V. P. (1962). Expl. Cell Res. 27,474. GRAY, E. D., WEISSMAN, S. M., RICHARDS, J., BELL, D., KEIR, H. M., SHELLIE, R. M. S. & DAVIDSON, J. N. (1960). Biochim. biophys. Actu, 45, 111. HARRIS, T. E. (1959). In “The Kinetics of Cellular Proliferation” (F. Stohlman, Jr., ed.), p. 368. New York & London: Grune & Stratton. HARRIS, T. E. (1963). “The Theory of Branching Process”. Berlin; Giittingen; Heidelberg: Springer-Verlag. HIATT. H. H. & BOJARSKI, T. B. (1960). Biochem. biophFs. Res. Cornmun. 2, 35. Hsu, ?. C. (1960). Texas Rep. Biol. Med. 18,31. JAEGER, J. C. (1961). “Laplace Transformation”. London: Methuen. JOHNSON, H. A. & CRONKTTE, E. P. (1959). Rud. Res. 11, 325. KELLY, C. D. & RAHN, 0. (1932). J. Butt. 23, 147. KENDALL. D. G. (1948). Biomefrika. 35, 316. KILLAND~R, D. i ZE&ERBERG, A. (1963). Expl. Celi Res. 40, 12. KUBITXHEK, H. E. (1962u). Expl. CeN Res. 26,439. KLJBITXHEK, H. E. (19626). Nature, Land. 195, 350. MCCARTER, J. A. & QUASTLER, H. (1962). A’ature, Lond. 194, 873. MOFFAT, G. H. & PELC, S. R. (1966). Expl. Cell. Res. 42, 460. POST. J. & HOFFMAN. J. (19611. Expl. Cell Res. 14. 713. POST; J. & HOFFMAN; J. (1964). J. ??eUBiol. 22, 341. POWELL. E. 0. (1955). Biometrika, 42, 16. Pmcorr, D. M. (1959). Expf. Cell Res. 16, 279. RAHN, 0. (1932). J. gen. Physiol. 15, 257. REISKIN. A. B. & MENDELSOHN, M. L. (1964). Cancer Res. 24, 1131. ROBIN&N, S. H. & BRECXER, 6. (1963): Sci&ce, N. Y. 142, 392. SCHAECHTER, M., WILLIAMSON, J. P., Hook, J. R., JR. & KOCH, A. L. (1962). J. gen.
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L. (1965).
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LABELLED
MITOSIS
TAKAHASHI.M. (1966).J. Theoret. Biol. TRUCCO, E. (1967). VON FOERSTER, H.
WAVE
METHOD
209
13, 202.
J. Theoret. Biol. 15, 180.
(1959). In “The Kinetics of Cellular Proliferation”, (F. Stohlman, Jr., ed.), p. 382. New York 8cLondon: Grune & Stratton. WEISSMAN, S. M., SMELLIE, R. M. S. & PAUL, J. (1960). Biochim. biophys. Actu, 45, 101. WIMBER. WIMBER; ZMCEK,
D. E. (1959). D. E. (1960). G. &G~oss,
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Appendix A.1. DISTRIBUTION OF STAGE DURATION A special case where s = 1. will be considered first. A necessary and sufficient condition for the S stage *being equal to T is that neither a phase transition nor a death takes place in every infinitesimal interval until the former occurs in the last (7-Ar, 7). The probability density for realization of such an event is lim
IsA7[l-(is+ps)A7]i
=
&exp[-&+~s)~l~
Ar-0
(A4
Since S stage is composed of a series of s phases, frequency function of its duration is given by a convolution of equation (A.l) so that it becomes
4 ---t5-1exp[-(&+&t]. (s-l)! Derivation of equation (A.2) from equation (A.l) is facilitated by the use of Laplace transform. By normalizing (A.2) with co r, =/(A.2)dr
= (+-j=,
(A.31
0
one can obtain a frequency function (2) of the text. A.2. DURATION OF A STAGE TERMINATING IN DEATH In order that a cell dies in ith phase of the S stage, neither phase transition nor death should occur until the cell dies finally in the ith phase. Therefore, by making reference to equations (A.1) and (A.3), it becomes evident that the frequency function (6) holds true.
Theoretical Basis for Cell Cycle Analysis II. Further Studies on Labelled Mitosis Wave Method MANABU
Dcpartmerrt
of Pathology,
TAKAHASHI
Yamaguchi University Ube, Japan
School of Medicine,
(Received 13 June 1967) The method of interpreting experimental data on lahelled mitosis wave (Takahashi, 1966) is reconfirmed using a less restrictive model which has the following characteristics: (i) duration of each stage (Cl, S, G2 and M) is scattered according Pearson Type III distribution; (ii) their coefficient of variation can he changed independently of the mean, even if variation of a cycle time is fixed; (iii) it can he accommodated to a system which involves death process and, as special cases, to situations where labelled cells are more liable to die than unlahelled counterparts or some fraction of lahelled cells are missed undetected autoradiographically. A method for computation of labelled mitosis wave which is expected theoretically from the mode: was shown. Numerical solutions of some equations had to be obtained by digital computer. As a consequence of the present study, it was stressed that, when the first peak of the wave is helow 80% (or the second below SO%), coincidence of the experimental curve with the theoretical one obtained, by putting presumed estimates into the parameters, should he reached by trial and error before the estimates are finally trusted. The model and the computational method were so designed as to be applicable to other methods of cell cycle analysis.
1. Introduction It seems to be a rule in some reported data (e.g. McCarter 8c Quastler, 1962; Post & Hoffman, 1964; Reiskin t Mendelsohn, 1964) that a change of cell cycle occurs during the course of carcinogenesis, as manifested in the labelled mitosis wave by a decrease in the slope of its declining limb. This fact indicates, according to Wimber’s way of interpretation (Wimber, 1960), that a variation of S stage duration increases in association with carcinogenesis. It is probably due to loss of control of over DNA synthesis particularly over its termination. Since this assertion has important implications, it is desirable to preclude every conceivable error of interpretation.
196
M.
TAKAHASHI
In the preceding paper (Takahashi, 1966), quantitative relations between labelled mitosis wave and parameters of cell cycle were examined, using a stochastic model. The results closely paralleled Wimber’s notion (Wimber, 1960). However, the model had restrictions in three respects; i.e. (i) coeiticient of variation of stage duration could not be altered independently of its mean, provided that total variation of a cycle time is fixed; (ii) death process was not appropriately taken into account; and (iii) preferential death of labelled cells was not assumed. The purpose of this report is to assure the previous results by using a generalized model and further to establish a foundation for testing the validity of various methods for cell cycle analysis. As will be seen below, the first restriction was relaxed by assuming different phase transition den@es for different stages and the second by introducing a probability of death which may occur in any of the reproductive phases. The third was eliminated by assigning smaller reproductive coefficients to labelled cells. Several investigators (Barret, 1966; Trucco, 1967) treated the related problems: the latter author by making use of Van Foerster equations (Von Foerster, 1959). However, their theories were not indispensable since they ultimately resorted to Monte Carlo procedure, a simulation method which is unrelated in principle to the theories they described. The present model is a realistic, though not ideal, representation of the cellular system, and the following ilIustrations will enable the readers to prepare by themselves a computer programme for obtaining theoretical values of “percentage of mitosis labelled”. 2. Preliminary Considerations A cell cycle is an example of stochastic processes in which the durations of Cl, S, G2 and M stages are probabilistic variables. The distribution of a cycle time is known to be positively skewed (e.g. Kelly & Rahn, 1932; Powell, 1955; Burns, 1956; Prescott, 1959; I-&u, 1960; Kubitschek, 1962a; Dawson, Madoc-Jones & Field, 1965; Sisken & Morasca, 1965). Among a variety of expressions for cycle time distribution such as Yule’s distribution (Rahn, 1932), Pearson Type III distribution (Kendall, 1948), normal distribution of generation rate (Kubitschek, 1962u), and log-normal distribution (Barret, 1966), the author adopted the second simply for ease of mathematical treatment. This choice may be justified because of mutual similarity of these distributions: reciprocal or logarithm of variables in Pearson Type III distribution yields fairly linear plots on a probit paper. For the same reason, the data of Sisken & Morasca (1965) seem to support a view that the variation of component stages can be approximated by Pearson Type III distribution,
LABELLED
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197
It has been reported that in bacterial cell populations there are positive correlations of individual generation times between sisters and slightly negative correlations between mother and daughters (Powell, 1955; Schaechter, Williamson, Hood & Koch, 1962; Kubitschek, 1962~). Sisken & Morasca (1965) and Killander & Zetterberg (1965) suggested, moreover, that lengths of successive stages are negatively correlated in mammalian cells. However, what is really important in relation to labelled mitosis wave is not the correlations but the distributions of combined length of consecutive stages (i.e. Gl, Gl +S, Gl +S+G2, etc.) which are also close to Pearson Type III. Indeed, unless these are less variable than the component stages, the cell cycle can be simulated by this model reasonably well, but otherwise not. 3. Model
Mathematical structure of the generalized model is essentially the same as the previous one except that some restrictions are relaxed. However, its thorough description may be helpful to understand the papers to be published later. As illustrated in Fig. 1, the model is specified by the following postulates. (1) A cell passes through hypothetical (mathematical) phases, k in number, as it proceeds on a cell cycle. Transition of the phase constitutes a Poisson process so that phase duration is distributed exponentially. (2) Of k phases, the first gr phases represent Gl stage which is cytologically discernible. Succeeding s phases [i.e. from (gri- 1)th to (gr +s)th phases] correspond to S stage. The other stages are likewise specified so that g,+s+g2+m = k. (1) A transition to (k + l)th phase signifies a birth of daughters in their first phase. (3) Two daughters are yielded at each cell division (binary fission). However, this is not an essential postulate. (4) A cell transits from the ith phase (i = 1,2,. . . , k) to the next with probability densities (phase transition densities) Ri which may differ for Gl, S, G2 and M stages and will be equalized to &i, A,, A,, and AM, respectively. (5) A cell cycle may be interrupted by an incidence of death (reproductive or metabolic) in any of the phases. Probability densities (death densities) in Gl, S, G2 and M stages will be equalized to pcl, p,, p,, and ~1~. This is a device that is analogous to Kendall’s way of simulating variation of individual generation time of bacterial cells (Kendall, 1948). However, this differs slightly from Kendall’s in that simulation extended to the component stages within a generation and also from the previous one (Taka-
198
M.
TAKAHASHI
Fro. 1. Schematic illustration of model. A cell passes through k hypothetical phases sequentially as cell cycle evolves clockwise. At a turn of the cycle the cell is divided into two daughters.
hashi, 1966) in that a condition for uniformity of transition density was relaxed and death density introduced. The present model will be called general&d multiple-phase birth-and-death process. 4. Some Features of the Model 4.1. DURATION OF A STAGE PASSED SUCCESSFULLY From the model (section 3) can be derived’s frequency function of stage duration which for S stage, for example, is of the form
&+hYT.-l (s-l)!
expC-&+hH,
(2)
(see Appendix for proof). This is the Pearson Type III distribution which we looked for (section 2). The mean and coefficient of variation of S stage duration are
and
T-S s- a-Ps
(3)
cv, = s-f.. (4) Here, the statistics were assumed to be taken with exclusion of cells which die in the course of S stage. As evident from (7), unless a large fraction of cells die in the S stage, the mean (3) can be approximated by
LABELLED
Comparable
MITOSIS
WAVE
199
METHOD
relations hold for other stages. 4.2.
DURATION
OF A STAGE
TERMhATING
IN
DEATH
The interval from the beginning of S stage to death varies according to frequency function :
rs-l/“-l 8 -~~-~exp[-(L,+&r], 4 -c l-r, i=l (i-l)!
where r, is the probability fully, hence
(6)
with which a cell passes through S stage success-
(see Appendix for proof). Therefore, the mean and coefficient of variation of the interval are (8)
and
(9) Relations similar to (6), Q, (8) and (9) can be applied to other stages by changing the parameters and suffixes. 4.3.
DISCBETE
VARIATIONS
OF
STAGE
DURATION
The variable in (4) being an integer, the CY of stage duration can take only discrete values. However, this does not b&come a serious drawback. Kubitschek (1962b) called attention to discrete variations in the CV of generation rate. It would not be unreasonable to expect that the same holds true for the component stages also. 5. Adjustment of the Model Hypothetical parameters of the model are connected with observable quantities of cell population as follows: s = cvp, (10) 1 Ps = (11) and (12)
lkl = rc1 t-11 -rol Tm+T
1
gl = fix (W&q
rQi
Cl
I I
-.
CV;
T; and
ps---
1 __~
s = fix (CV,-‘)
rs
CV,
Ts and
S
t Reproductive coefficient is equal to rol rSrcarnd. 2 The symbol fix ( ) represents an integer closest to real number in the parenthesis.
Phase transition density
3z ‘3 $j Death density SE OCd
Number of phasest
Mean and CV of stage duration Bm I, .P E .s Mean and CV of entryto-death interval a4 0= Rate of successful passa&
Name of stage
1
Adjustment of parameters
TABLE
rG2
and
CV’,
, TcaST a
1
ga = fix (CV,-,)
T’,
Tcz and CV,,
G2
CV,
p*, _. - __-__- 1
m = fix (CVi2)
rM
TLandCV&
Tnl and
M
2 7c * X * ?
z.
LABELLED
MITOSIS
WAVE
METHOD
201
Parameters specifying the other stages can likewise be determined (Table 1). If the CVof a stage duration is given experimentally, an integer closest to (CV)-2 should be adopted as the number of phases. When labelled cells obey different kinetics, their parameters should be determined separately (see section 7). 6. Kinetics of Cell Population
Although renewal equation is frequently applied to analysis of cell population kinetics (e.g. Bellman & Harris, 1952; Harris, 1959, 1963), it does not suit the present problem in which the growth of subpopulations (cells in different phases) should be treated separately (see section 7). Therefore, Kendall’s method was modified for the purpose. According to the model described in Section 3, every cell is in either one of k phases. A subpopulation composed of ith-phased cells is called ith phase fraction (or compartment), and the mean number of ith-phased cells denoted by nJt>. Conceptionally, all the cells move sequentially through discrete compartments which as a whole form a closed circuit. As they pass through the compartments, some fraction of cells may be eliminated by death but on entering a new round of cycle they are doubled in number by ceil division. Thus, the model leads to a set of relations (13) and (19, which are similar to continuity equation of hydrodynamics (Fig. 2). The influx into the ith phase fraction (i # 1) in At is li-lni-l(t)At (13) and the eflux from it (J-i + pi> ni(t)At, (14) so that the increment Ani is obtained by subtraction: Art;(t) = Ri-lni-I(t)At-(li+~t3ni(t)At (i = 2,3,. : !, k)
(15)
and An&) = 2Q,(t)At-(11
+&n,(t)At.
(16)
Fro. 2. Growth of subpopulations. T.B.
D
202
M.
TAKAHASHI
Thus, simultaneous differential equations describing “compartment are derived : -I dn,(O ^~__ = 2l,n,(t)-(1,+~(,)n,(t) dt t dd) __ = ~i-lni-,(t)-(~i+c1j)ni(t)l dt
kinetics”
(17)
(i = 2,3,. . ,, k). ,
7. Percentage of Lab&d Mitoses Suppose that a cell population is pulse-labelled at t = 0. Only S-staged cells [i.e. cells in s compartments from (gI + 1)th to (gl +s)th phases] are labelled so that their number amounts to 8, +r (18) i=J+ $O). Any labelled mitotic cell which will appear later must, therefore, be their progeny. The descendants of those having been in other stages have no radioactivity. Hence, the number of labelled mitoses at t is equal to the sum M*(t) =
i i=k-m+
n*(t), 1
(19)
where n:(t) is giveri by solving the simultaneous differential equations (17), with the initial conditions that all the compartments other than those corresponding to S stage are nullified, i.e. n’(0) = 0 if lsisg, or y,+s
n:(t). i=k-m+l
(22)
The latter term can be calculated by solving the equations (17) using “rate of passage through stages”, rgl, I$, rz2 and r& under different initial conditions, i.e. no(o) = 0 if gl< i=
LABELLED
MITOSIS
WAVE
METHOD
203
Therefore, the formula that describes time-dependent change of “percentage of labelled mitoses” is obtained by dividing (19) with (22):
PLM(r) = M*(t) M(r) ’
(25)
This will be called “PLM function”. In a special case where labelled cells are more liable to die than unlabelled ones (or some fraction of labelled cells is missed undetected), smaller “rates of passage through stages” T*‘S (5 ,“s) must be used at the calculation of M*(r) (see Table 1). 8. Limiting Phase Structure Behaviours of cell population depend generally on both age distribution and distribution of stage duration. The labelled mitosis wave is not an exception to this. However, when the latter distribution is defined, relative cell number in each phase fraction converges, as cell cycle continues to develop indefinitely and reaches a steady state, to a limit which is determined uniquely by TG1, T,,. . ., TM, CV,,,. . ., and CV,. For simplicity, we will consider a case where there is a single cell in the first phase at t = 0: n,(O) = 1, (26) i?i(O) = 0, (i # 1).> This is not an essential condition because a population starting growth from different initial condition attains the same limiting phase structure. Laplace transform of the equations (17) with the initial condition (26) is (4 +/.h +P)N,w-2w,w = 1, -Ri-,N,-,(p)+(~i+~li+p)Ni(p),=O, (27) (i = 2,3,. . .,k) where N,(p) is Laplace transform of ni(r), i.e* N,(P) c ni(t)*
(28)
Therefore, the solutions of (17) in complex domain are Ni(p) =J$
(i = 1,2,. . ., k),
(29)
where i- 1
A(P) = nnj j=O
fi (lj+Pj+P) j=i+l
IIn0= II
(30)
*
(31)
and g(P) =,fil(ij+rj+P)m2,filAj
P
204
M.
TAKAHASHI
According to Heaviside’s expansion theorem (e.g. Jaeger, 1961), the inverse transform of (29) becomes (32) where e,, e2,. . . , e, are the roots of characteristic equation c?(P) = 0. (33) A predominant term in the series (32) at large t is that corresponding to the largest real root E of (33). The relative cell number of the ith-phase fraction converges, as t increases, to a limit (limiting phase fraction): (34) Consequently, in a steady state there exist relations (35) Results of numerical calculation disclosed a curious fact that limiting phase fractions Fl depend not only on the relative stage durations but also on their coefficients of variation. 9. Limiting Age Distribution
The age distribution of a population will also converge to a limit (limiting age distribution) as proliferation continues. This distribution can be calculated, if required, from the limiting phase fractions because the age distribution for a whole population is equal to the weighted sum of those for the subfractions. 10. Digital Computer Analysis
The foregoing considerations paved the way to analysis of labelled mitosis wave. However, in order to draw theoretical PLM curve of cell population for which a set of parameters is given, numerical calculation must be performed such that can be accomplished only with the aid of digital computer. The computation was performed by FACOM 231 of Yamaguchi University. The largest real root of characteristic equation (33) was obtained by Newton’s method, and the simultaneous differential equations (17) solved by RungeKutta-Gill method. 10.1.
ESTIMATION
OF MEAN
STAGE
DURATION
By changing the parameters TGI, TS, TG2 and TM in various combinations, it was examined what parts of the wave indicate the mean stage durations. The results obtained by using this generalized model supported the statement made previously (Takahashi, 1966): a choice of 50 % intercepts is satisfactory
LABELLED
MITOSIS
WAVE
METHOD
205
as far as the first peak of the wave is above 80% level. Labelled mitosis wave with the first peak lower than 70% level must be interpreted with caution, since in this case the mean S stage duration tends to be underestimated. To avoid this risk it is advisable to examine each time whether or not the experimental PLM curve coincides with that calculated, by putting presumed values into computational programmes. 10.2. ESTIMATION OF STANDARD DEVIATION OF STAGE DURATION The present model can be used to examine what part of the wave reflects CV of a stage duration. The relation was studied by varying one of the parameters gr, s, g2 and m. The results were in good agreement with the previous report (Takahashi, 1966) especially when the peaks are sufficiently high. 10.3. EFFECTS OF DOSES OF RADIOISOTOPE When cell population is brought into contact with [3H]thymidine of high specific activity, adverse radiobiological effects, such as increased cell death, chromosome breakage and decreased uptake of [3H]thymidine, become manifest (Drew & Painter, 1959; Johnson & Cronkite, 1959; Wimber, 1959; Post & Hoffman, 1961; Zajicek & Gross, 1964). The effect on’ the labelled mitosis wave can be examined by assigning smaller reproductive coefficients to labelled cells in the model than to unlabelled counterparts. As anticipated intuitively, height of the second wave became lower and the trough more depressed. If, on the contrary, [3H]thymidine is of low specific activity (or photographic exposure time is relatively insufficient), the label may be missed undetected particularly after the radioactivity is halved by cell division. The false-negative mitoses (labelled mitoses scored erroneously as unlabelled) add only to the denominator. The decrease of PLM value due to this failure is expected to be more pronounced than in the reverse, because in the latter condition death of labelled cells leads to the decreases of both numerator and denominator. 11. Errors due to Protracted Labelliog Before applying the results mentioned above, a question will be asked: whether the mode of labelling which, at mathematical treatment, was assumed to be virtually instantaneous is acceptable or not. Many workers allow cells to remain in contact with labelled thymidine for some 10 minutes or more to secure 100% detection of S-staged cells. Under such a circumstance, PLM curve takes such a configuration as shown by a bold line in Fig. 3.
206
M. TAKAHASHI
Time oiler
pulse-lobellinq
FIG. 3. Effect of protracted labelling. (A) PLM curve of cells labelled at the beginning of exposure; (B) PLM curve of cells labelled at the end of exposure; (C) experimentally observed PLM curve; @) duration of exposure to label.
12. Implication of Delayed Labelliog Several investigators (e.g. Diderholm, Fichtellius & Linder, 1962; Robinson & Brecher, 1963; Bryant, 1965) presented evidence that labelled thymidine is released from dying cells which have once incorporated it into DNA and then re-utilized by growing cells, the cells with rapid turnover rate becoming transient reservoir of the label. However, the re-utilization phenomenon is expected not to exert any appreciable effect on labelled mitosis wave, at least, of the first or second generation. 13. Stage of Tbymidine Uptake Important is the question whether or not the so-called “DNA synthetic period” may be accepted as such in a strict sense of the word. Recently, Moffat & Pelt (1966) demonstrated, by autoradiography, the presence of enormous amount of label in cryostat-sectioned tissue of [3H]thymidineinjected animal, most part of which was dissolved away in the course of tissue fixation and embedding. Basing on this observation they claimed that the label is pooled in the cells at the time of flashing without necessarily being incorporated into macromolecular DNA. Thus, it will serve as a source of labelled precursor for later synthesis of DNA, though not all of it is fixed and utilized. Similar observations have been made by Feinendegen & Bond (1962) and Cleaver & Holford (1965). For this reason, a new terminology a “stage of thymidine uptake” seems more appropriate than a “stage of DNA synthesis”. It is evident from cyclic change of PLM value that thymidine uptake takes place only in a limited period within a cell cycle. This fact suggests that a specific enzymatic process might be a likely mechanism for intracellular fixation of this substance.
LABELLED
MITOSIS
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207
The enzymes necessary to accomplish the phosphorylation of thymidine, which are present in the liver of the newborn, are lost as the animal grows older (Hiatt & Bojarski, 1960) but reappear after partial hepatectomy at the time at which DNA synthesis recommences (Bollum & Potter, 1959; Canellakis, Jaffe, Montsavinos & Krakow, 1959; Gray ei al., 1960; Hiatt & Bojarski, 1960). Weissman, Smellie & Paul (1960) postulated a sequential induction of thymidine phosphorylating enzymes. It is not unlikely that the activity of these enzymes has some bearings with cellular uptake of thymidine. 14. Discussion A mathematical model is the only system that fits a purpose of examining relations between the shape of PLM curve and parameters of cell cycle. The model used here is satisfactory in that it is conformable to such wide variety of situations as enumerated below: (i) cell population may be growing exponentially, normally renewing, or in the intermediate state: a constancy of mean generation time being the only prerequisite; (ii) a stage duration may be variable, though its distribution should belong (or be similar) to Pearson Type III distribution and the length of a stage is independent of the others; (iii) mode of cellular differentiation or of metabolic death can be defined as desired ; (iv) growth kinetics of labelled cells can be made different from that of non-labelled counterparts and the possibility can be taken into account that a certain fraction of labelled cells may be missed in the autoradiogram ; and (v) arbitrary phase structure can be chosen as initial condition although limiting phase fraction can be calculated. It is hoped that widespread use of this model will lead to a srandardization of interpreting method thus facilitating comparison of data of different investigators on a common basis. Extensive computation of theoretical PLM curves corresponding to various combinations of parameters will yield sufficient information as to enable one to interpret every feature of the curve (height of the peaks, depth of the trough, gradient of the slope, etc.) without the aid of computer. The results and their application to the reported data will be published elsewhere. Of special interest is to find the characteristics of labelled mitosis waves of primary neoplasm or of cells under the influence of carcinogen which generally show much slower decline of descending limb as contrasted
208
M.
TAKAHASHI
to a rather abrupt rise of ascending portion of the first wave, though there are some exceptions to this rule (e.g. DBrmer, Tulinius & Oehlert, 1964). In the light of present investigation, the above fact is interpreted as indicating that the period of thymidine uptake is remarkably variable in the early stage of neoplastic change. I would like to thank Professor K. Inouye for his interest in this work and for much encouragement.
REFERENCES BARRET, J. C. (1966). J. natn. Cancer Inst. 37,443. BELLMAN, R. & HARRIS, T. (1952). Ann. Math. 55, 280. B~LLUM, F. J. & POTIZR, V. R. (1959). Cancer Res. 19, 561. BRYANT. B. J. (1965). ‘Exol. CeN Res. 37. 490. BURNS, i. W. (1956j. J. iel[. camp. Physiol. 47, 357. CANELLAKIS, E. S., JAFFE, J. J., MONTSAMNOS, R. & KRAKOW,
J. S. (1959).
J. biol. Chem.
234,2096. CLEAVER, J. E. & HOLFORD, R. M. (1965). Biophys. biochim. Acta, 103,657. DAW~ON, K. B., MADOC-JONES, H. & FIELD, E. 0. (1965). Expl. Cell Res. 38, 75. DIDERHOLM, H., FICHTELLIUS, K. E. & LINDER, 0. (1962). Expl. Cell Res. 27, 431. DORMER, P., TUUNIUS, H. & OEHLERT, W. (1964). 2. Krebsforsch. 66, 11. DREW, R. M. & PAINTER, R. B. (1959). Rad. Res. 11, 635. FEINENDEGEN, L. E. & Born), V. P. (1962). Expl. Cell Res. 27,474. GRAY, E. D., WEISSMAN, S. M., RICHARDS, J., BELL, D., KEIR, H. M., SHELLIE, R. M. S. & DAVIDSON, J. N. (1960). Biochim. biophys. Actu, 45, 111. HARRIS, T. E. (1959). In “The Kinetics of Cellular Proliferation” (F. Stohlman, Jr., ed.), p. 368. New York & London: Grune & Stratton. HARRIS, T. E. (1963). “The Theory of Branching Process”. Berlin; Giittingen; Heidelberg: Springer-Verlag. HIATT. H. H. & BOJARSKI, T. B. (1960). Biochem. biophFs. Res. Cornmun. 2, 35. Hsu, ?. C. (1960). Texas Rep. Biol. Med. 18,31. JAEGER, J. C. (1961). “Laplace Transformation”. London: Methuen. JOHNSON, H. A. & CRONKTTE, E. P. (1959). Rud. Res. 11, 325. KELLY, C. D. & RAHN, 0. (1932). J. Butt. 23, 147. KENDALL. D. G. (1948). Biomefrika. 35, 316. KILLAND~R, D. i ZE&ERBERG, A. (1963). Expl. Celi Res. 40, 12. KUBITXHEK, H. E. (1962u). Expl. CeN Res. 26,439. KLJBITXHEK, H. E. (19626). Nature, Land. 195, 350. MCCARTER, J. A. & QUASTLER, H. (1962). A’ature, Lond. 194, 873. MOFFAT, G. H. & PELC, S. R. (1966). Expl. Cell. Res. 42, 460. POST. J. & HOFFMAN. J. (19611. Expl. Cell Res. 14. 713. POST; J. & HOFFMAN; J. (1964). J. ??eUBiol. 22, 341. POWELL. E. 0. (1955). Biometrika, 42, 16. Pmcorr, D. M. (1959). Expf. Cell Res. 16, 279. RAHN, 0. (1932). J. gen. Physiol. 15, 257. REISKIN. A. B. & MENDELSOHN, M. L. (1964). Cancer Res. 24, 1131. ROBIN&N, S. H. & BRECXER, 6. (1963): Sci&ce, N. Y. 142, 392. SCHAECHTER, M., WILLIAMSON, J. P., Hook, J. R., JR. & KOCH, A. L. (1962). J. gen.
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MITOSIS
TAKAHASHI.M. (1966).J. Theoret. Biol. TRUCCO, E. (1967). VON FOERSTER, H.
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(1959). In “The Kinetics of Cellular Proliferation”, (F. Stohlman, Jr., ed.), p. 382. New York 8cLondon: Grune & Stratton. WEISSMAN, S. M., SMELLIE, R. M. S. & PAUL, J. (1960). Biochim. biophys. Actu, 45, 101. WIMBER. WIMBER; ZMCEK,
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Appendix A.1. DISTRIBUTION OF STAGE DURATION A special case where s = 1. will be considered first. A necessary and sufficient condition for the S stage *being equal to T is that neither a phase transition nor a death takes place in every infinitesimal interval until the former occurs in the last (7-Ar, 7). The probability density for realization of such an event is lim
IsA7[l-(is+ps)A7]i
=
&exp[-&+~s)~l~
Ar-0
(A4
Since S stage is composed of a series of s phases, frequency function of its duration is given by a convolution of equation (A.l) so that it becomes
4 ---t5-1exp[-(&+&t]. (s-l)! Derivation of equation (A.2) from equation (A.l) is facilitated by the use of Laplace transform. By normalizing (A.2) with co r, =/(A.2)dr
= (+-j=,
(A.31
0
one can obtain a frequency function (2) of the text. A.2. DURATION OF A STAGE TERMINATING IN DEATH In order that a cell dies in ith phase of the S stage, neither phase transition nor death should occur until the cell dies finally in the ith phase. Therefore, by making reference to equations (A.1) and (A.3), it becomes evident that the frequency function (6) holds true.