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Nonparametric analysis of stathmokinesis
Nonparametric MAREK
Analysis
of Stathmokinesis
KIMMEL*
Investigative Qtology Labomtoty, New York, New York Received 76 September
Memorittl Sloan - Kettering Cuncer Center,
1984; revised Z Januuy
I985
ABSTRACT The nonparametric analysis of the stathmokinetic experiment presented in this paper is an extension of procedures by Jagers and Staudte. The method allows one to estimate, under very general assumptions, the first two moments of the residence time in successive cell cycle phases. Approximate formulae for the mean square errors of the estimates are derived. Applications include experimental stathmokinetic data for various cell lines, both analyzed and not analyzed previously. Comparison proves that the nonparametric method is very accurate whenever it can be applied. Results of analysis of the stathmokinetic data are also discussed from the viewpoint of the variability of the cell cycle generation time.
1.
INTRODUCTION
The aim of this paper is to present a nonparametric method of analyzing the stathmokinetic experiment. Only very mild u priori assumptions concerning generation time distribution are necessary. The method, essentially an extension of procedures by Jagers [8,9] and Staudte [19], has been tested on a range of experimental data and proved very useful. The stathmokinetic experiment, introduced by Puck and Steffen [17], has been employed by numerous authors (e.g. Dosik et al. [7], Macdonald [14], Kimmel, Traganos, and Darzynkiewicz [13], Darzynkiewicz et al. [5], and references in Aherne, Camplejohn, and Wright [l], in Wright and Appleton [20], and in Darzynkiewicz, Traganos, and Kimmel [4]), to estimate the cell
*Author is a Visiting Investigator from the Institute of Automation, University. Gliwice, Poland. This work was supported by NC1 Grants 23296. MATHEMATICAL
BIOSCIENCES
74:111-123
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
Silesian Technical CA 28704 and CA
111
(1985)
0025-5564/85/$03.30
112
MAREK
-
G1
S
G2
KIMMEL
M \ Blocked division
FIG. 1. Generally accepted subdivision of the cell cycle. After division, the daughter cells enter phase G,, then traverse the phases S, G,, an d M, and then divide. The residence times in all the phases are treated as random. In the stathmokinetic experiment, the divisions are blocked, so that all the cells finally accumulate in M.
cycle kinetic parameters. When cell division is blocked by an external agent, the cells gradually accumulate in mitosis, emptying the postmitotic phase G, and with time also the S phase (see Figure 1). The pattern of cell accumulation in mitosis (M) depends on the kinetic parameters of the cell cycle and is used for assessment of those parameters. In the deterministic model of Puck and Steffen [17], it can be easily demonstrated that the “collection function” ln[l + fM( r)] [where &,(t) is the fraction of cells in M at time t] is, for t small enough, a straight line with slope X (see Figure 2). This slope is equal to the Malthusian parameter of the exponential steady state (ESS, proportional exponential growth in all the subcompartments of the cell cycle), assumed to exist before stathmokinesis. Stathmokinesis has also been used to determine cell cycle parameters other than A, usually the residence times in various phases of the cell cycle. Some of the corresponding methods are intuitive (Datzynkiewicz et al. [5]); others require a sophisticated mathematical approach. In the latter case, the usual procedure is to assume a parametric model of the cell cycle, to derive the resulting formulae for the stathmokinetic curves and to fit them to the empirical data, for example, with the aid of the numerical least square methods (Dosik et al [7]; Kimmel et al. [13]; Macdonald [14]; Sharpless and Schlesinger [IS]). The common disadvantage of these procedures is that the choice of the cell cycle model may influence the parameter estimates. It was shown, however, by Jagers [S, 91 that the stathmokinetic experiment can be analyzed with only a few a priori assumptions, in an astonishingly simple manner. Also, Staudte [19] proposed an error analysis for the Jagers estimators. The purpose of this paper is to apply their ideas, in a modified
ANALYSIS
g I
113
OF STATHMOKINESIS
(t)
0 ,
l
S
t
FIG. 2. Typical collection function g(r). S is the minimum residence time in phase 1. For times from the interval [0, S], the collection function is linear with slope X.
form, to a broader spectrum of biological data. A part of the original simplicity of the stathmokinetic analysis is preserved in the method presented here. An abstract model of the cell cycle (Figure 3) consisting of two successive phases, 1 and 2, will be used. How this model is related to the true cell cycle phases will become clear in the sequel. It is assumed that after mitosis, each of the daughter cells enters phase 1, staying there for a random time r, with probability density pi(t). After leaving phase 1 the cell enters phase 2 characterized by T2 and p2(f). Then the cell divides and both daughters enter phase 1 again. It is assumed that for t -c 0 (i.e. before the stathmokinesis) the cell population is in ESS, i.e., in all the subcompartments of the cycle, the expected cell number is growing proportionally to e ” . At t = 0 cell division is blocked by the stathmokinetic agent and cells begin to leave phase 1 and accumulate in phase 2, while their total number remains constant under assumption that no cell loss occurs. Let f,(t), t > 0, denote the fraction of cells present in phase i at time 1. A typical graph of fi(t) is depicted in Figure 4. Considerable information about distribution p1 (t) can be extracted from fi (1) without assuming any parametric model. Let us denote by Fe = f, (0) the ESS fraction of cells in phase 1, by Fi the area under the fi (t) graph, and by F2 the mass-center coordinate
114
MAREK
KIMMEL
I
-
f,(t)
f*(t)
-\
Blocked division FIG. 3.
Cell cycle subdivision
into two “phases.”
7; is the (random)
residence
time in
; (i = 1.2): p,(r) is its distribution density: f, (1) is the fraction of the initially cycling cells that are present in phase i at time t. phase
of the fr ( t) graph (Figure 4). Then the following holds:
(1.1)
(1.2) where E(T,), E(T:) denote the expected values of the residence time in phase 1 and of its square, respectively. Knowledge of these two quantities enables finding the standard deviation a,, of Tr. These surprisingly simple formulae are discussed in the next section. The details of the error analysis (which involve more complicated calculations), as well as an argument demonstrating that, in the present model, X is the slope of ln[l+ fi( t)] if only some minimum assumptions are satisfied, are deferred to the Appendix. The cell cycle can be subdivided into phases 1 and 2 in various ways, providing estimates of the average and variance of the sojourn time in various cell cycle phases. The estimation procedures have been applied (Section 3) to a variety of experimental stathmokinetic data. Resulting numerical estimates are compared to those obtained with other methods. Also, based on the estimated cell cycle parameters, we discuss questions related to the variability of the cell cycle. Along with other methods (see e.g. Jagers and Norrby [lo]; Cowan, Culpin, and Morris [2]; and references therein), the nonparametric analysis of the stathmokinetic experiment provides significant information about this variability.
ANALYSIS
115
OF STATHMOKINESIS
FIG. 4. Typical f,(t) exit curve. F;, is the ESS cell fraction in phase 1; 4 under f, (1): Fz is the coordinate of the mass center of the graph.
2.
is the area
ESTIMATORS
The estimation technique of this paper is based, in principle, upon a single formula for the exit curve from phase 1. Denote by P,(t) the cumulative distribution of the cell residence time in phase i:
and denote by a, ( t) the function (Y,( t) = e”‘/mp, I
( U) e-hu du
We then have the following result. PROPOSITION
1
The exit curve fi (t) from phase 1 is equal to
f,(t)=2[1-P~(t)-LYl(t)l.
(2.3)
This formula can be extracted from the formulae (5)-(7) of Macdonald [14] or derived from Kimmel [11,12]. However, the primary source seems to be Jagers ([S]; see also [9]). The reader may wish to compare these approaches.
116
MAREK
The information of a proposition. PROPOSITION
Suppose
KIMMEL
necessary to derive (1.1) and (1.2) is provided in the form
2
that the first two moments
of P,(t)
exist.
Then
F,=f,(O)=2(1-q,),
(2.4)
F;=jmjl(u)du=2[F(Tl)-y),
(2.5)
0
F,
Fz
=
jomuf,(u)du=E(Ti?)-~~(T,)+$(l-ql),
(2.6)
where q1 = a1 (0).
Solving this system for the unknowns E( T,) and E( Tf) provides the estimators (1.1) and (1.2). The proof of this proposition is reduced to formal manipulations in (2.3). Similar derivations may be found in Jagers [8,9]. Now, let h, denote the sample value of fi(t,), 0 = t,, < . . < t,,. To obtain computable expressions, function fi( .) under the integrals in (2.5) and (2.6) is replaced by its piecewise linear approximation: fl (t) G h, ~, + (t - t, 1)( h, -h,_,)/(t,-t,_,), tE[t,_,,t,][and f,(t)=O,t>t,,].Thentheestimateof the expected sojourn time E(T,) is
b(T,)=k=+
and that of the standard
The constants
a0 =
(2.7)
deviation
of T, is
a, and b, are defined as
to)?
,h(“, to) + (fl -
’
6
a,= t,~,(f,--t,~*)+t,(t,+l--t,)
+ (L-d
2
+ (t,-t,A2 6
3
i=l,...,n-1, LI(f,, a,,
=
2
Ll>
+
(t,,
-
LA2
3
’
’
ANALYSIS
117
OF STATHMOKINESIS 11 -
to
h)=2,
b,
=
b,,=
tr+1; t,,
-
It-1 ) t,,
2
-
i=l
,...,n
-1,
1 7
while 1 is an estimate of the Malthusian parameter, obtained from the collection function observations. Formally, the formula (2.7) is almost identical with that of Staudte [19] (who did not consider estimates of the standard deviation). However, the errors of both ^x and h, are calculated, in this paper, in a different way (see Appendix). Also, Staudte [19] seems to ignore the bias of fi(T,). This point is discussed in the Appendix, as well as the expressions for the mean squared errors (MSE) of estimates. Remark. An alternative way of computing the estimators & and 3 (as noted by one of the referees) is to apply trapezoidal integration directly to the whole integrated functions [e.g. to ufi (u) in Equation (2.6)]. Then, the coefficients a, are replaced, in (2.8), by t,b,, i = 0,. ., n. The method proposed in this paper seems to be more accurate, since the direct trapezoidal integration of ufi (u) does not provide the accurate result even if fi (u) is piecewise linear. However, the trapezoidal integration method seems to be worth discussing in more detail. It is possible to compare the two methods in the special case when t, = to + ih. We have then a0 - tobo = A*/6, a,, - t,,b, = - A’/6, a, = t,b, 7 i = 1,. . , n - 1. Since (h, ) is a decreasing sequence, the trapezoidal integration will yield systematically lower values of s. For the numerical examples collected in this paper, the differences in $ values for the data of Tables 2 and 4 are on the level of 1%. (Therefore, the remarks on cell cycle variability presented in the Discussion do not depend on the method of integration.) For Macdonald’s [14] data (Table 1) the difference is about 108, while for Puck and Steffen’s [17] data (Table 3) it is much higher (about 20%). 3.
RESULTS
The commonly accepted model of cell cycle includes four phases: G,, S, G,, and M (see Figure 1; also cf. Prescott [16] for the definition of the cell cycle phases). In what follows the previously defined phase 1 will be identified either with G, or with the union of G, and S. Similarly, phase 2 will be either M or G, + M. The stathmokinetic data to be considered first are those for the Chinese hamster ovary (CHO) cells analyzed by Macdonald [14]. His original method was based on the assumption that all the residence time distributions are of the gamma type.
118
MAREK TABLE Results
of Analyses
KIMMEL
1
of Stathmokinetic
Data for CHO Cells”
Method Parametric
0.0385
4.62
Nonparametric
0.0381 (0.004)
4.68 (0.65)
“Data from Macdonald Table 1, do not include errors.
2.24 (1.88)
13.5
2.19
13.36 (1.17)
2.54 (4.64)
[14]. Results of Macdonald’s procedure, published in his Br,,. The numbers in parentheses are the mean squared
Using MacdonaId’s data [14,Figure 11, the experimental values of collection function were computed and the value of x was found. In this case, G, + M was treated as our phase 2. Then the G, (phase 1) exit data were analyzed. A similar analysis was performed for the G, + S exit data (here, G, + S was phase 1). The results are presented in Table 1, compared with those of Macdonald [14]. The agreement is very good. The second, larger data set represents the stathmokinetic data for the L1210 cells published by Kimmel et al. [13]. In this reference it was assumed that the G, phase (identified here with phase 1) consisted of two disjoint subphases. The first of them, AC,, was assumed to have exponential resi-
TABLE 2 Results of Analyses
^x Experiment A B C D F G
I
(h-l) 0.0616 0.0660 0.0680 0.0653 0.0680 0.0655 0.0609
(0.004) (0.002) (0.002) (0.001) (0.003) (0.002) (0.003)
of Stathmokinetic
Data for Ll210
-m,,)
h<,,
(h)
(h)
4.27 4.03 3.95 4.10 3.X9 4.00 4.74
(0.25) (0.11) (0.11) (0.05) (0.19) (0.13) (0.26)
2.12 1.79 2.00 2.00 2.13 1.95 3.06
(0.30) (0.15) (0.13) (0.07) (0.21) (0.16) (0.24)
T+l/a
Cells” l/o
(h)
(h)
Phase 2
4.59 4.21 4.08 4.34 4.17 4.19 5.00
1.67 1.35 1.55 1.69 1.77 1.77 1.95
G>+M Gz + M G, + M GZ + M M
M M
“Data from Kimmel et al. [13]. Expectations &( q;, ) of the residence times in G,. estimated with the aid of the nonparametric method, are listed together with the corresponds values of T + l/a obtained by the least squares fit in [13]. Then 6, to l/a (see text for details). All relevant experimental data are collecied in Table 1 of [13]. The physical units as in Table 1. The numbers in parentheses are the mean squared errors.
119
ANALYSIS OF STATHMOKINESIS TABLE 3 Results of Analyses of Puck and Steffen’s Data [17]” x Method Staudte’s [19] This paper ‘Numbers
&,+s+G,)
%G,+.s+(,l
(h-‘)
(h)
0.0296 (0.0012) 0.0297 (0.0015)
24.30 (0.68)
in parentheses
(h)
22.99 (0.83)
are the mean squared
6.51 (1.34)
errors.
dence time distribution (with parameter LU),while the second, BG,, was of deterministic duration T. Under these assumptions, consistent with a modified version of the Smith-Martin model (cf. Nedelman and Rubinow [15]), the expectation of the residence time in G, is equal to T + l/a, while the standard deviation is l/a. The results obtained by the present method are compared with those obtained by the parametric method of Kimmel et al. [13] (Table 2; the last column indicates which cell cycle phase served as “phase 2” to compute the collection function). These results also show good agreement, although the nonparametric method seems to systematically provide lower estimates of the expectation E( TGl) and higher estimates of the standard deviation ar,,. Also, the estimates for the classical data of Puck and Steffen [17], obtained with the use of the method of Staudte [19], were compared with those of this paper (Table 3). There are no substantial differences. Finally, the nonparametric method was applied to the L-cell data published by Darzynkiewicz et al. [6], which had not been previously analyzed. These results are collected in Table 4. 4.
DISCUSSION
Stathmokinetic data for cell populations with substantial cell loss, as well as for the non-ESS populations, cannot be analyzed with the present version of the nonparametric method. Also, the exit function fi (t) must be observed
TABLE 4” Estimates
Obtained
for the L-cell Data of Darzynkiewicz
m, )
x 0-l)
00
0.032 (0.0008)
7.97 (0.20)
‘Numbers
in parentheses
hr,lT,s)
%<,, 00 4.78 (0.23)
are mean squared
errors.
et al. [6]
67.. +\ 1.1
00
(h)
12.34 (0.26)
4.94 (0.38)
120
MAREK
KIMMEL
until it is very close to zero (this last condition eliminated from consideration, among others, the G, + S exit data of Kimmel, Traganos, and Datzynkiewicz [13]). Within these limitations, however, the approach seems to be fruitful, providing effective (low MSE) estimates of the cell cycle parameters. Moreover, from the methodological viewpoint, it is advantageous to consider a nonparametric model, since only infrequently is any a priori information concerning the residence time distributions available. It is perhaps interesting to add that in numerical examples, the nonparametric estimators have very low bias; the bias to standard deviation ratio never exceeds 0.1. Estimates of the cell cycle phase durations are always of interest, as they provide arguments for or against various theoretical concepts of cell kinetics. Assessments of the standard deviations of the sojourn times may help in understanding the interdependences of the cell cycle phases. The case of the L cells is a good example, since the first two moments of the durations of G, and G, + S can be estimated with a satisfactory accuracy: For the L cells (Table 4) the estimated standard deviation of the sojourn time is G, is approximately equal to the standard deviation of the joint sojourn time in G, and S: 8,, = 6rc, +i. The same could have been inferred from Macdonald’s [14] CHO cell data, if the MSE errors had not been so high (see Table 1). This relation means that Var(7”)+2 Cov( TY, TG,) = 0, and since Var(T,) 2 0, this implies that either Var(T,) = 0 or the sojourn times in G, and S are negatively correlated. The first possibility is consistent with the generally accepted view concerning the relatively constant duration of S. The other would imply that cells which spend more time in G, can synthesize DNA faster. This idea is to some extent supported by the findings of Darzynkiewicz et al [3]. The L1210 cell data (Table 2) provide another interesting finding. Estimates obtained by a parametric method by Kimmel et al. [13] (assuming the Smith-Martin model; see Nedelman and Rubinow [15]) yield a coefficient of variation for the G, sojourn time of about 0.38 (average value for the seven cases of Table 2). The nonparametric method gives a markedly higher value, about 0.52. This is yet another argument that the displaced exponential distribution (implied by the Smith-Martin theory) is not an ideal model for the cell generation time randomness. To conclude, the simple nonparametric approach to the stathmokinetic analysis introduced by Jagers [8], discussed by Staudte [19], and modified in this paper proves useful for a wide range of data. The estimators introduced in this work, still fairly simple, can be of interest for biologists employing the stathmokinetic experiment. APPENDIX Three points are considered here: (a) the form of the collection function, (b) errors of ^x and h, , (c) variance and bias of the estimators of k( Tl)and 6.r-1.
ANALYSIS A.
THE
121
OF STATHMOKINESIS COLLECTION
FUNCTION
We have the following result: PROPOSITION
J
t 2 S, if and o&y ifp,(t)
g(t) = Xt +ln(2q,), Proof.
Obvious,
= 0, t s S.
if we note that fr( t)+f2( t) = 1 and that a,(t) = qleh’,
t 5 S, if and only if pr (t) = 0, t 5 S.
Thus, the collection function is a straight line for a period equal to the minimum residence time in phase 1. B.
VARIANCE
OF ^x AND
h,
Denote by g, the sample values of g(s,), 0 < sr < . . . < s,,. Here, contrary to the case of the fr (t,) samples, the first data point is not at t = 0. This is because the empirical collection function usually exhibits an initial curvature (over a period of l-2 hours) believed to be caused by a delay or imperfection of the action of the stathmokinetic agent. The error analysis in Staudte [19] is based on a binomial model which was appropriate for the counting techniques employed by Puck and Steffen [17]. Since we are dealing mainly with data obtained by flow cytometry, the error resulting from a small number of counted cells is negligible. There is, however, a new source of errors: the uncertainty in determining the threshold separating distinct cell cycle phases (see e.g. Figure 1 of Kimmel, Traganos, and Darzynkiewicz [13] for the threshold geometry of a RNA-DNA plot). We consider here this error (E, ) to be independent of the number of cells measured (which is high: - 104) and the same for all the thresholds. Obviously, the range of the error is bounded, so that its distribution has all the moments. So the statistical model of the observation is i=l,...,m,
g,=g(s,)+E,=q+Xs,+E,, h, =fi(t,)+
E,+n,+,,
i=O ,...,n,
(Al) (A2)
where 4 = ln(2qr) and (E,) is an Cd. family of random variables with E( E,) = 0. To estimate the variance (a’) of E,, we can only use (Al), since no parametric form of fi (t) is assumed. The following formulae are well known, so we will omit the derivations. We will use the generic notation x=x,+ ‘.. tx n,, XY= mxy- XJ. An unbiased estimator of h is A= GS/S=. An unbiased
estimator
(A3)
of w2 = Var(^X) is:
(A4)
122
MAREK
while an unbiased
estimator
KIMMEL
of v2 is
We may note that even when the cell counting techniques are employed, our estimates should not differ a lot from those of Staudte [19]. This is confirmed by data in Table 3. C.
VARIANCE
AND
BIAS
OF b(T,,
AND
6,,
The estimators in (2.7) and (2.8) are biased, so the appropriate their quality is the mean squared error (MSE):
measure of
MSE( ri) = [ Var( 2) + B( ~~1 1’2, where i.is equal to i = &( T,) or to s = 8rI, and B(i) is the bias of 2. To find approximate values of the variances, we use the known linearization formula, obtaining
(A6) The bias can be estimated from a less known order Taylor series expansion:
if the y, are independent.
formula
based on a second
Thus,
(A8) Equations (A6) and (A8) provide reservations. Using them, we obtain finally
and an analogous
approximations
to be used with obvious
formula for MSE(.?).
The author thanks Zhigniew Melamed for useful comments, tance.
Darzynkiewicz, Frank Traganos, and Myron R. and Robin Nager for excellent technical assis
ANALYSIS
123
OF STATHMOKINESIS
REFERENCES 1
3
W. A. Aheme, R. S. Camplejohn. and N. A. Wright, An Introduction to Cell Population Kinetm, Edward Arnold, London, 1977. R. Cowan, D. Culpin, and V. B. Morris, A method for the measurement of variability in cell lifetimes, Math. Biosci. 54: 249-263 (1981). Z. Darzynkiewicz, H. Crissman, F. Traganos. and J. Steinkamp, Cell heterogeneity
4
during the cell cycle, J. Cell Ph.vsio/. 113: 465-474 (1982). Z. Dar-zynkiewicz, F. Traganos, and M. Kimmel, Assay
2
multiparameter 5
7
X 9 10
11
cytometry,
in Cvtokinetics
of cell cycle
Techniques
(J.
kinetics
E. Gray
and
by Z.
Darzynkiewicz, Eds.), Humana, Clifton, NJ., to appear. Z. Darzynkiewicz, F. Traganos, S.-B. Xue, and M. R. Melamed, Effect of n-butyrate on cell cycle progression and in situ chromatin structure of L1210 cells. Exp. Cell Res. 136: 279-293
6
flow
(1981).
Z. Darzynkiewicz, B. Williamson, E. A. Carswell. and L. J. Old. The cell cycle effects of tumor necrosis factor, Cancer Res. 44: 83-90 (1984). G. M. Dosik, B. Barlogie, R. A. White, W. Gohde, and B. Drewinko. A rapid automated stathmokinetic method for determination of in ultra cell cycle transit times, Cell Tissue Kinet. 14: 121-134 (1981). P. Jagers, Use explicit models in the inference on population growth, Bull. Intern&. Stutrst. Inst. 47: 418-422 (1977). P. Jagers. Stochastic models for cell kinetics, Bull. Muth. Bio(. 45: 5077519 (1983). P. Jagers and K. Norrby, Estimates of the mean and variance of cycle times in cinemicrographically recorded cell populations during balanced Cell Tissue Kinet. 7:201-211 (1974). M. Kimmel, Cellular population dynamics. I: Model construction Muth. Biosci. 48:211-224 (1980).
13
M. Kimmel, F. Traganos, and Z. Danynkiewicz, Do all daughter cells enter the “indeterminate” (“A “) state of the cell cycle? Analysis of stathmokinetic experiments on L1210 cells, Cvrometr?, 4:191-201 (1983). P. D. M. Macdonald, Towards an exact analysis of stathmokinetic and continuous-
16 17 1X
19
20
dynamics,
II: Investigation
and reformulation,
M. Kimmel, 4X:225-239
15
population
growth,
12
14
Cellular (1980).
exponential
of solutions.
M&h.
Biosci.
labelling experiments, in Blomathemutm and Cell Kinetics (M. Rotenberg. Ed.), Elsevier/North Holland Biomedical Press, Amsterdam, 1981, pp. 125-142. J. Nedelman and S. I. Rubinow, Investigation into the experimental kinetic support of the two-state model of the cell cycle, Cell Bi0ph.w. 2:207-231 (19X0). D. M. Prescott, Reproduction of Eukutyotic Cells, Academic, New York. 1976. T. T. Puck and J. Steffen, Life cycle analysis of mammalian cells. Part I, Blophw. J. 3:379-397 (1963). T. K. Sharpless and F. H. Schlesinger, Flow cytometric analysis of G, exit kinetics in asynchronous L1210 cell cultures with the constant transition probability model. Cytometry 3:196-200 (1982). R. G. Staudte, On the accuracy of some estimates of cell cycle time, Bionuthemutrcs md Cell Kinetics (M. Rotenberg. Ed.), Elsevier/North Holland Biomedical Press, Amsterdam, 1981, pp. 233-239. N. A. Wright and D. R. Appleton, The metaphase arrest technique. A critical review, Cell Tissue Kinet. 13~643-663 (1980).
Analysis
of Stathmokinesis
KIMMEL*
Investigative Qtology Labomtoty, New York, New York Received 76 September
Memorittl Sloan - Kettering Cuncer Center,
1984; revised Z Januuy
I985
ABSTRACT The nonparametric analysis of the stathmokinetic experiment presented in this paper is an extension of procedures by Jagers and Staudte. The method allows one to estimate, under very general assumptions, the first two moments of the residence time in successive cell cycle phases. Approximate formulae for the mean square errors of the estimates are derived. Applications include experimental stathmokinetic data for various cell lines, both analyzed and not analyzed previously. Comparison proves that the nonparametric method is very accurate whenever it can be applied. Results of analysis of the stathmokinetic data are also discussed from the viewpoint of the variability of the cell cycle generation time.
1.
INTRODUCTION
The aim of this paper is to present a nonparametric method of analyzing the stathmokinetic experiment. Only very mild u priori assumptions concerning generation time distribution are necessary. The method, essentially an extension of procedures by Jagers [8,9] and Staudte [19], has been tested on a range of experimental data and proved very useful. The stathmokinetic experiment, introduced by Puck and Steffen [17], has been employed by numerous authors (e.g. Dosik et al. [7], Macdonald [14], Kimmel, Traganos, and Darzynkiewicz [13], Darzynkiewicz et al. [5], and references in Aherne, Camplejohn, and Wright [l], in Wright and Appleton [20], and in Darzynkiewicz, Traganos, and Kimmel [4]), to estimate the cell
*Author is a Visiting Investigator from the Institute of Automation, University. Gliwice, Poland. This work was supported by NC1 Grants 23296. MATHEMATICAL
BIOSCIENCES
74:111-123
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
Silesian Technical CA 28704 and CA
111
(1985)
0025-5564/85/$03.30
112
MAREK
-
G1
S
G2
KIMMEL
M \ Blocked division
FIG. 1. Generally accepted subdivision of the cell cycle. After division, the daughter cells enter phase G,, then traverse the phases S, G,, an d M, and then divide. The residence times in all the phases are treated as random. In the stathmokinetic experiment, the divisions are blocked, so that all the cells finally accumulate in M.
cycle kinetic parameters. When cell division is blocked by an external agent, the cells gradually accumulate in mitosis, emptying the postmitotic phase G, and with time also the S phase (see Figure 1). The pattern of cell accumulation in mitosis (M) depends on the kinetic parameters of the cell cycle and is used for assessment of those parameters. In the deterministic model of Puck and Steffen [17], it can be easily demonstrated that the “collection function” ln[l + fM( r)] [where &,(t) is the fraction of cells in M at time t] is, for t small enough, a straight line with slope X (see Figure 2). This slope is equal to the Malthusian parameter of the exponential steady state (ESS, proportional exponential growth in all the subcompartments of the cell cycle), assumed to exist before stathmokinesis. Stathmokinesis has also been used to determine cell cycle parameters other than A, usually the residence times in various phases of the cell cycle. Some of the corresponding methods are intuitive (Datzynkiewicz et al. [5]); others require a sophisticated mathematical approach. In the latter case, the usual procedure is to assume a parametric model of the cell cycle, to derive the resulting formulae for the stathmokinetic curves and to fit them to the empirical data, for example, with the aid of the numerical least square methods (Dosik et al [7]; Kimmel et al. [13]; Macdonald [14]; Sharpless and Schlesinger [IS]). The common disadvantage of these procedures is that the choice of the cell cycle model may influence the parameter estimates. It was shown, however, by Jagers [S, 91 that the stathmokinetic experiment can be analyzed with only a few a priori assumptions, in an astonishingly simple manner. Also, Staudte [19] proposed an error analysis for the Jagers estimators. The purpose of this paper is to apply their ideas, in a modified
ANALYSIS
g I
113
OF STATHMOKINESIS
(t)
0 ,
l
S
t
FIG. 2. Typical collection function g(r). S is the minimum residence time in phase 1. For times from the interval [0, S], the collection function is linear with slope X.
form, to a broader spectrum of biological data. A part of the original simplicity of the stathmokinetic analysis is preserved in the method presented here. An abstract model of the cell cycle (Figure 3) consisting of two successive phases, 1 and 2, will be used. How this model is related to the true cell cycle phases will become clear in the sequel. It is assumed that after mitosis, each of the daughter cells enters phase 1, staying there for a random time r, with probability density pi(t). After leaving phase 1 the cell enters phase 2 characterized by T2 and p2(f). Then the cell divides and both daughters enter phase 1 again. It is assumed that for t -c 0 (i.e. before the stathmokinesis) the cell population is in ESS, i.e., in all the subcompartments of the cycle, the expected cell number is growing proportionally to e ” . At t = 0 cell division is blocked by the stathmokinetic agent and cells begin to leave phase 1 and accumulate in phase 2, while their total number remains constant under assumption that no cell loss occurs. Let f,(t), t > 0, denote the fraction of cells present in phase i at time 1. A typical graph of fi(t) is depicted in Figure 4. Considerable information about distribution p1 (t) can be extracted from fi (1) without assuming any parametric model. Let us denote by Fe = f, (0) the ESS fraction of cells in phase 1, by Fi the area under the fi (t) graph, and by F2 the mass-center coordinate
114
MAREK
KIMMEL
I
-
f,(t)
f*(t)
-\
Blocked division FIG. 3.
Cell cycle subdivision
into two “phases.”
7; is the (random)
residence
time in
; (i = 1.2): p,(r) is its distribution density: f, (1) is the fraction of the initially cycling cells that are present in phase i at time t. phase
of the fr ( t) graph (Figure 4). Then the following holds:
(1.1)
(1.2) where E(T,), E(T:) denote the expected values of the residence time in phase 1 and of its square, respectively. Knowledge of these two quantities enables finding the standard deviation a,, of Tr. These surprisingly simple formulae are discussed in the next section. The details of the error analysis (which involve more complicated calculations), as well as an argument demonstrating that, in the present model, X is the slope of ln[l+ fi( t)] if only some minimum assumptions are satisfied, are deferred to the Appendix. The cell cycle can be subdivided into phases 1 and 2 in various ways, providing estimates of the average and variance of the sojourn time in various cell cycle phases. The estimation procedures have been applied (Section 3) to a variety of experimental stathmokinetic data. Resulting numerical estimates are compared to those obtained with other methods. Also, based on the estimated cell cycle parameters, we discuss questions related to the variability of the cell cycle. Along with other methods (see e.g. Jagers and Norrby [lo]; Cowan, Culpin, and Morris [2]; and references therein), the nonparametric analysis of the stathmokinetic experiment provides significant information about this variability.
ANALYSIS
115
OF STATHMOKINESIS
FIG. 4. Typical f,(t) exit curve. F;, is the ESS cell fraction in phase 1; 4 under f, (1): Fz is the coordinate of the mass center of the graph.
2.
is the area
ESTIMATORS
The estimation technique of this paper is based, in principle, upon a single formula for the exit curve from phase 1. Denote by P,(t) the cumulative distribution of the cell residence time in phase i:
and denote by a, ( t) the function (Y,( t) = e”‘/mp, I
( U) e-hu du
We then have the following result. PROPOSITION
1
The exit curve fi (t) from phase 1 is equal to
f,(t)=2[1-P~(t)-LYl(t)l.
(2.3)
This formula can be extracted from the formulae (5)-(7) of Macdonald [14] or derived from Kimmel [11,12]. However, the primary source seems to be Jagers ([S]; see also [9]). The reader may wish to compare these approaches.
116
MAREK
The information of a proposition. PROPOSITION
Suppose
KIMMEL
necessary to derive (1.1) and (1.2) is provided in the form
2
that the first two moments
of P,(t)
exist.
Then
F,=f,(O)=2(1-q,),
(2.4)
F;=jmjl(u)du=2[F(Tl)-y),
(2.5)
0
F,
Fz
=
jomuf,(u)du=E(Ti?)-~~(T,)+$(l-ql),
(2.6)
where q1 = a1 (0).
Solving this system for the unknowns E( T,) and E( Tf) provides the estimators (1.1) and (1.2). The proof of this proposition is reduced to formal manipulations in (2.3). Similar derivations may be found in Jagers [8,9]. Now, let h, denote the sample value of fi(t,), 0 = t,, < . . < t,,. To obtain computable expressions, function fi( .) under the integrals in (2.5) and (2.6) is replaced by its piecewise linear approximation: fl (t) G h, ~, + (t - t, 1)( h, -h,_,)/(t,-t,_,), tE[t,_,,t,][and f,(t)=O,t>t,,].Thentheestimateof the expected sojourn time E(T,) is
b(T,)=k=+
and that of the standard
The constants
a0 =
(2.7)
deviation
of T, is
a, and b, are defined as
to)?
,h(“, to) + (fl -
’
6
a,= t,~,(f,--t,~*)+t,(t,+l--t,)
+ (L-d
2
+ (t,-t,A2 6
3
i=l,...,n-1, LI(f,, a,,
=
2
Ll>
+
(t,,
-
LA2
3
’
’
ANALYSIS
117
OF STATHMOKINESIS 11 -
to
h)=2,
b,
=
b,,=
tr+1; t,,
-
It-1 ) t,,
2
-
i=l
,...,n
-1,
1 7
while 1 is an estimate of the Malthusian parameter, obtained from the collection function observations. Formally, the formula (2.7) is almost identical with that of Staudte [19] (who did not consider estimates of the standard deviation). However, the errors of both ^x and h, are calculated, in this paper, in a different way (see Appendix). Also, Staudte [19] seems to ignore the bias of fi(T,). This point is discussed in the Appendix, as well as the expressions for the mean squared errors (MSE) of estimates. Remark. An alternative way of computing the estimators & and 3 (as noted by one of the referees) is to apply trapezoidal integration directly to the whole integrated functions [e.g. to ufi (u) in Equation (2.6)]. Then, the coefficients a, are replaced, in (2.8), by t,b,, i = 0,. ., n. The method proposed in this paper seems to be more accurate, since the direct trapezoidal integration of ufi (u) does not provide the accurate result even if fi (u) is piecewise linear. However, the trapezoidal integration method seems to be worth discussing in more detail. It is possible to compare the two methods in the special case when t, = to + ih. We have then a0 - tobo = A*/6, a,, - t,,b, = - A’/6, a, = t,b, 7 i = 1,. . , n - 1. Since (h, ) is a decreasing sequence, the trapezoidal integration will yield systematically lower values of s. For the numerical examples collected in this paper, the differences in $ values for the data of Tables 2 and 4 are on the level of 1%. (Therefore, the remarks on cell cycle variability presented in the Discussion do not depend on the method of integration.) For Macdonald’s [14] data (Table 1) the difference is about 108, while for Puck and Steffen’s [17] data (Table 3) it is much higher (about 20%). 3.
RESULTS
The commonly accepted model of cell cycle includes four phases: G,, S, G,, and M (see Figure 1; also cf. Prescott [16] for the definition of the cell cycle phases). In what follows the previously defined phase 1 will be identified either with G, or with the union of G, and S. Similarly, phase 2 will be either M or G, + M. The stathmokinetic data to be considered first are those for the Chinese hamster ovary (CHO) cells analyzed by Macdonald [14]. His original method was based on the assumption that all the residence time distributions are of the gamma type.
118
MAREK TABLE Results
of Analyses
KIMMEL
1
of Stathmokinetic
Data for CHO Cells”
Method Parametric
0.0385
4.62
Nonparametric
0.0381 (0.004)
4.68 (0.65)
“Data from Macdonald Table 1, do not include errors.
2.24 (1.88)
13.5
2.19
13.36 (1.17)
2.54 (4.64)
[14]. Results of Macdonald’s procedure, published in his Br,,. The numbers in parentheses are the mean squared
Using MacdonaId’s data [14,Figure 11, the experimental values of collection function were computed and the value of x was found. In this case, G, + M was treated as our phase 2. Then the G, (phase 1) exit data were analyzed. A similar analysis was performed for the G, + S exit data (here, G, + S was phase 1). The results are presented in Table 1, compared with those of Macdonald [14]. The agreement is very good. The second, larger data set represents the stathmokinetic data for the L1210 cells published by Kimmel et al. [13]. In this reference it was assumed that the G, phase (identified here with phase 1) consisted of two disjoint subphases. The first of them, AC,, was assumed to have exponential resi-
TABLE 2 Results of Analyses
^x Experiment A B C D F G
I
(h-l) 0.0616 0.0660 0.0680 0.0653 0.0680 0.0655 0.0609
(0.004) (0.002) (0.002) (0.001) (0.003) (0.002) (0.003)
of Stathmokinetic
Data for Ll210
-m,,)
h<,,
(h)
(h)
4.27 4.03 3.95 4.10 3.X9 4.00 4.74
(0.25) (0.11) (0.11) (0.05) (0.19) (0.13) (0.26)
2.12 1.79 2.00 2.00 2.13 1.95 3.06
(0.30) (0.15) (0.13) (0.07) (0.21) (0.16) (0.24)
T+l/a
Cells” l/o
(h)
(h)
Phase 2
4.59 4.21 4.08 4.34 4.17 4.19 5.00
1.67 1.35 1.55 1.69 1.77 1.77 1.95
G>+M Gz + M G, + M GZ + M M
M M
“Data from Kimmel et al. [13]. Expectations &( q;, ) of the residence times in G,. estimated with the aid of the nonparametric method, are listed together with the corresponds values of T + l/a obtained by the least squares fit in [13]. Then 6, to l/a (see text for details). All relevant experimental data are collecied in Table 1 of [13]. The physical units as in Table 1. The numbers in parentheses are the mean squared errors.
119
ANALYSIS OF STATHMOKINESIS TABLE 3 Results of Analyses of Puck and Steffen’s Data [17]” x Method Staudte’s [19] This paper ‘Numbers
&,+s+G,)
%G,+.s+(,l
(h-‘)
(h)
0.0296 (0.0012) 0.0297 (0.0015)
24.30 (0.68)
in parentheses
(h)
22.99 (0.83)
are the mean squared
6.51 (1.34)
errors.
dence time distribution (with parameter LU),while the second, BG,, was of deterministic duration T. Under these assumptions, consistent with a modified version of the Smith-Martin model (cf. Nedelman and Rubinow [15]), the expectation of the residence time in G, is equal to T + l/a, while the standard deviation is l/a. The results obtained by the present method are compared with those obtained by the parametric method of Kimmel et al. [13] (Table 2; the last column indicates which cell cycle phase served as “phase 2” to compute the collection function). These results also show good agreement, although the nonparametric method seems to systematically provide lower estimates of the expectation E( TGl) and higher estimates of the standard deviation ar,,. Also, the estimates for the classical data of Puck and Steffen [17], obtained with the use of the method of Staudte [19], were compared with those of this paper (Table 3). There are no substantial differences. Finally, the nonparametric method was applied to the L-cell data published by Darzynkiewicz et al. [6], which had not been previously analyzed. These results are collected in Table 4. 4.
DISCUSSION
Stathmokinetic data for cell populations with substantial cell loss, as well as for the non-ESS populations, cannot be analyzed with the present version of the nonparametric method. Also, the exit function fi (t) must be observed
TABLE 4” Estimates
Obtained
for the L-cell Data of Darzynkiewicz
m, )
x 0-l)
00
0.032 (0.0008)
7.97 (0.20)
‘Numbers
in parentheses
hr,lT,s)
%<,, 00 4.78 (0.23)
are mean squared
errors.
et al. [6]
67.. +\ 1.1
00
(h)
12.34 (0.26)
4.94 (0.38)
120
MAREK
KIMMEL
until it is very close to zero (this last condition eliminated from consideration, among others, the G, + S exit data of Kimmel, Traganos, and Datzynkiewicz [13]). Within these limitations, however, the approach seems to be fruitful, providing effective (low MSE) estimates of the cell cycle parameters. Moreover, from the methodological viewpoint, it is advantageous to consider a nonparametric model, since only infrequently is any a priori information concerning the residence time distributions available. It is perhaps interesting to add that in numerical examples, the nonparametric estimators have very low bias; the bias to standard deviation ratio never exceeds 0.1. Estimates of the cell cycle phase durations are always of interest, as they provide arguments for or against various theoretical concepts of cell kinetics. Assessments of the standard deviations of the sojourn times may help in understanding the interdependences of the cell cycle phases. The case of the L cells is a good example, since the first two moments of the durations of G, and G, + S can be estimated with a satisfactory accuracy: For the L cells (Table 4) the estimated standard deviation of the sojourn time is G, is approximately equal to the standard deviation of the joint sojourn time in G, and S: 8,, = 6rc, +i. The same could have been inferred from Macdonald’s [14] CHO cell data, if the MSE errors had not been so high (see Table 1). This relation means that Var(7”)+2 Cov( TY, TG,) = 0, and since Var(T,) 2 0, this implies that either Var(T,) = 0 or the sojourn times in G, and S are negatively correlated. The first possibility is consistent with the generally accepted view concerning the relatively constant duration of S. The other would imply that cells which spend more time in G, can synthesize DNA faster. This idea is to some extent supported by the findings of Darzynkiewicz et al [3]. The L1210 cell data (Table 2) provide another interesting finding. Estimates obtained by a parametric method by Kimmel et al. [13] (assuming the Smith-Martin model; see Nedelman and Rubinow [15]) yield a coefficient of variation for the G, sojourn time of about 0.38 (average value for the seven cases of Table 2). The nonparametric method gives a markedly higher value, about 0.52. This is yet another argument that the displaced exponential distribution (implied by the Smith-Martin theory) is not an ideal model for the cell generation time randomness. To conclude, the simple nonparametric approach to the stathmokinetic analysis introduced by Jagers [8], discussed by Staudte [19], and modified in this paper proves useful for a wide range of data. The estimators introduced in this work, still fairly simple, can be of interest for biologists employing the stathmokinetic experiment. APPENDIX Three points are considered here: (a) the form of the collection function, (b) errors of ^x and h, , (c) variance and bias of the estimators of k( Tl)and 6.r-1.
ANALYSIS A.
THE
121
OF STATHMOKINESIS COLLECTION
FUNCTION
We have the following result: PROPOSITION
J
t 2 S, if and o&y ifp,(t)
g(t) = Xt +ln(2q,), Proof.
Obvious,
= 0, t s S.
if we note that fr( t)+f2( t) = 1 and that a,(t) = qleh’,
t 5 S, if and only if pr (t) = 0, t 5 S.
Thus, the collection function is a straight line for a period equal to the minimum residence time in phase 1. B.
VARIANCE
OF ^x AND
h,
Denote by g, the sample values of g(s,), 0 < sr < . . . < s,,. Here, contrary to the case of the fr (t,) samples, the first data point is not at t = 0. This is because the empirical collection function usually exhibits an initial curvature (over a period of l-2 hours) believed to be caused by a delay or imperfection of the action of the stathmokinetic agent. The error analysis in Staudte [19] is based on a binomial model which was appropriate for the counting techniques employed by Puck and Steffen [17]. Since we are dealing mainly with data obtained by flow cytometry, the error resulting from a small number of counted cells is negligible. There is, however, a new source of errors: the uncertainty in determining the threshold separating distinct cell cycle phases (see e.g. Figure 1 of Kimmel, Traganos, and Darzynkiewicz [13] for the threshold geometry of a RNA-DNA plot). We consider here this error (E, ) to be independent of the number of cells measured (which is high: - 104) and the same for all the thresholds. Obviously, the range of the error is bounded, so that its distribution has all the moments. So the statistical model of the observation is i=l,...,m,
g,=g(s,)+E,=q+Xs,+E,, h, =fi(t,)+
E,+n,+,,
i=O ,...,n,
(Al) (A2)
where 4 = ln(2qr) and (E,) is an Cd. family of random variables with E( E,) = 0. To estimate the variance (a’) of E,, we can only use (Al), since no parametric form of fi (t) is assumed. The following formulae are well known, so we will omit the derivations. We will use the generic notation x=x,+ ‘.. tx n,, XY= mxy- XJ. An unbiased estimator of h is A= GS/S=. An unbiased
estimator
(A3)
of w2 = Var(^X) is:
(A4)
122
MAREK
while an unbiased
estimator
KIMMEL
of v2 is
We may note that even when the cell counting techniques are employed, our estimates should not differ a lot from those of Staudte [19]. This is confirmed by data in Table 3. C.
VARIANCE
AND
BIAS
OF b(T,,
AND
6,,
The estimators in (2.7) and (2.8) are biased, so the appropriate their quality is the mean squared error (MSE):
measure of
MSE( ri) = [ Var( 2) + B( ~~1 1’2, where i.is equal to i = &( T,) or to s = 8rI, and B(i) is the bias of 2. To find approximate values of the variances, we use the known linearization formula, obtaining
(A6) The bias can be estimated from a less known order Taylor series expansion:
if the y, are independent.
formula
based on a second
Thus,
(A8) Equations (A6) and (A8) provide reservations. Using them, we obtain finally
and an analogous
approximations
to be used with obvious
formula for MSE(.?).
The author thanks Zhigniew Melamed for useful comments, tance.
Darzynkiewicz, Frank Traganos, and Myron R. and Robin Nager for excellent technical assis
ANALYSIS
123
OF STATHMOKINESIS
REFERENCES 1
3
W. A. Aheme, R. S. Camplejohn. and N. A. Wright, An Introduction to Cell Population Kinetm, Edward Arnold, London, 1977. R. Cowan, D. Culpin, and V. B. Morris, A method for the measurement of variability in cell lifetimes, Math. Biosci. 54: 249-263 (1981). Z. Darzynkiewicz, H. Crissman, F. Traganos. and J. Steinkamp, Cell heterogeneity
4
during the cell cycle, J. Cell Ph.vsio/. 113: 465-474 (1982). Z. Dar-zynkiewicz, F. Traganos, and M. Kimmel, Assay
2
multiparameter 5
7
X 9 10
11
cytometry,
in Cvtokinetics
of cell cycle
Techniques
(J.
kinetics
E. Gray
and
by Z.
Darzynkiewicz, Eds.), Humana, Clifton, NJ., to appear. Z. Darzynkiewicz, F. Traganos, S.-B. Xue, and M. R. Melamed, Effect of n-butyrate on cell cycle progression and in situ chromatin structure of L1210 cells. Exp. Cell Res. 136: 279-293
6
flow
(1981).
Z. Darzynkiewicz, B. Williamson, E. A. Carswell. and L. J. Old. The cell cycle effects of tumor necrosis factor, Cancer Res. 44: 83-90 (1984). G. M. Dosik, B. Barlogie, R. A. White, W. Gohde, and B. Drewinko. A rapid automated stathmokinetic method for determination of in ultra cell cycle transit times, Cell Tissue Kinet. 14: 121-134 (1981). P. Jagers, Use explicit models in the inference on population growth, Bull. Intern&. Stutrst. Inst. 47: 418-422 (1977). P. Jagers. Stochastic models for cell kinetics, Bull. Muth. Bio(. 45: 5077519 (1983). P. Jagers and K. Norrby, Estimates of the mean and variance of cycle times in cinemicrographically recorded cell populations during balanced Cell Tissue Kinet. 7:201-211 (1974). M. Kimmel, Cellular population dynamics. I: Model construction Muth. Biosci. 48:211-224 (1980).
13
M. Kimmel, F. Traganos, and Z. Danynkiewicz, Do all daughter cells enter the “indeterminate” (“A “) state of the cell cycle? Analysis of stathmokinetic experiments on L1210 cells, Cvrometr?, 4:191-201 (1983). P. D. M. Macdonald, Towards an exact analysis of stathmokinetic and continuous-
16 17 1X
19
20
dynamics,
II: Investigation
and reformulation,
M. Kimmel, 4X:225-239
15
population
growth,
12
14
Cellular (1980).
exponential
of solutions.
M&h.
Biosci.
labelling experiments, in Blomathemutm and Cell Kinetics (M. Rotenberg. Ed.), Elsevier/North Holland Biomedical Press, Amsterdam, 1981, pp. 125-142. J. Nedelman and S. I. Rubinow, Investigation into the experimental kinetic support of the two-state model of the cell cycle, Cell Bi0ph.w. 2:207-231 (19X0). D. M. Prescott, Reproduction of Eukutyotic Cells, Academic, New York. 1976. T. T. Puck and J. Steffen, Life cycle analysis of mammalian cells. Part I, Blophw. J. 3:379-397 (1963). T. K. Sharpless and F. H. Schlesinger, Flow cytometric analysis of G, exit kinetics in asynchronous L1210 cell cultures with the constant transition probability model. Cytometry 3:196-200 (1982). R. G. Staudte, On the accuracy of some estimates of cell cycle time, Bionuthemutrcs md Cell Kinetics (M. Rotenberg. Ed.), Elsevier/North Holland Biomedical Press, Amsterdam, 1981, pp. 233-239. N. A. Wright and D. R. Appleton, The metaphase arrest technique. A critical review, Cell Tissue Kinet. 13~643-663 (1980).