Acta Biotheoretica 40: 131-137, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

FLOW C Y T O M E T R I C ANALYSIS OF THE CELL CYCLE: M A T H E M A T I C A L M O D E L I N G A N D BIOLOGICAL INTERPRETATION

Josd Pierrez + & Xavier Ronot ++ ÷ L a b o r a t o i r e de C y t o m d t r i e Appliqude, 22, B o u l e v a r d T z a r e w i t c h , F - 0 6 0 0 0 Nice, F r a n c e + ÷Equipe de R e c o n n a i s s a n c e des F o r m e s et M i c r o s c o p i c Quantitative, L a b o r a t o i r e T I M 3 , I M A G , U S R C N R S B 6 9 0 , Universitd Joseph F o u r i e r , C E R M O , B.P. 53X, F-38041 Grenoble Cedex, France.

ABSTRACT Estimation of the repartition of asynchronous cells in the cell cycle can be explained by two hypotheses: - the cells are supposed to be distributed into three groups: cells with a 2c DNA content (G0/1 phase), cells with a 4c DNA content (G2+M phase) and cells with a DNA content ranging from 2c to 4c (S phase); - there is a linear relationship between the amount of fluorescence emitted by the fluorescent probe which reveals the DNA and the DNA content. According to these hypotheses, the cell cycle can be represented by the following equation: DNA~(y) = I D N A T ( X ) ' P ( x ' y ) ' d x All the solutions for this equation are approximations. Non parametric methods (or graphical methods: rectangle, peak reflect) only use one or two phase(s) of the cell cycle, the remaining phase(s) being estimated by exclusion. In parametric methods (Dean & Jett, Baisch lI, Fried), the DNAr(x) distribution is supposed to be known and is composed of two gaussians (representative of G0/1 and G2+M) and a P(x,y) function representative of S phase. Despite the generality, these models are not applicable to all sample types, particularly heterogeneous cell populations with various DNA content. In addition, the cell cycle is dependent on several regulation points (transition from quiescence to proliferation, DNA synthesis initiation, mitosis induction) and biological perturbations can also lead to cytokinesis perturbations. Before the emergence of flow cytometry, the current view of cell cycle resided in the assessment of cell proliferation (increase in cell number) or the kinetic of molecules incorporation (DNA precursors). The widespread development of flow cytometry has revealed the concept of cell cycle distribution and this snapshot distribution required methods to analyze this new information; this aspect led to the notion of cell cycle Mathematical Modeling.

132

1. MATHEMATICAL MODELING OF THE CELL CYCLE 1.1. The basic equation (Pierrez & Guerci, 1988) The equation which supports the cell cycle can be established from common hypotheses. These are: 1) The cells are supposed to be distributed into three groups: cells with a 2c DNA content (GO/1 phase), cells with a 4c DNA content ( G 2 + M phase) and cells with a DNA content ranging from 2c to 4c (S phase). This hypothesis leads to the theoretical distribution as shown in figure 1A. 2) The fluorescence of the cells stained for DNA content, using a fluorescent probe, is directly proportional to their DNA content. The variabilities due to the cytometer and to the staining techniques are responsible of a dispersion o f the measured fluorescence compared to the theoretical fluorescence. The variabilities include especially: 1) the cell position in the exciting beam, ii) the instability related to the fiuorochrome/staining used (fixation, temperature . . . . ) and iii) the instability of the detectors, o f electronics and of the light source. This induces a spreading of the theoretical distribution, leading to the typical and well known cell cycle curve (figure 1B).

GOII 111

"5

A

u oJ "O

o

L

)

G2+M E 0 z

rNlt||m mm||I r

Num~ro de canal

Quantit~ d'ADN

Fig. 1: Theoretical (A) and experimental (B) DNA content distribution of an asynchronous cell population.

For a cell with an x DNA content, the detected fluorescence in relation with the fluorochrome bound to the DNA will be y. Due to the previously described instabilities, the y value cannot be predicted. Only the probability to detect a y fluorescence intensity, P(y) for a given x value, can be anticipated. For a current x value, the probability for a cell with an x DNA content to be detected as any fluorescence intensity is P(x,y). If this concept is applied to all the cells whose DNA distribution is DNAT(x), the observed fluorescence will be DNAE(y) calculated according to the following equation: DNAE(Y) -- / DNAT(X)'P(x'y)'dx d 0

Remarks on P(x,y): The information concerning P(x,y) can be obtained by analyzing cells with known

133 DNAE and D N A T distributions. Fluorescent beads simulate quiescent G0/1 cells appropriately. This simulation leads to the following findings: - P(x,y) takes on a unique maximum, P(x,y) rapidly decreases around this maximum, - if the signal of the detected fluorescence is enhanced (by electronics or laser adjustments), the modal channel of the distribution ( < y > ) will be increased as well as the width of the distribution (or), proportionally to < y > ; a~ < y > remains constant and is defined as the coefficient of variation (CV). P(x,y) is assimilated by the flow cytometry users to a gaussian function (Baisch & Beck, 1978), with x - a . y and a parameters in such a way that: < x > - a . < y > = 0 and a = CV. < y > . Therefore P(x,y) = P ( x - a . y , CV. < y > ) . -

Other functions such as a Log-normal distribution (Barrett, 1964) or a gamma function (Pearson, type III (Kendall, 1948)) could also be suitable, since experimental statistics do not provide significant differences. The gaussian distribution is widely used. Basic equation: Assuming that P(x,y) is gaussian, the basic equation is expressed as: DNAE(Y) -- l DNAT(X)" P(x-c~ y, CV. < y > ) . dx It is worth noting that this equation cannot be solved at the present time and this explains the existence of numerous models. This equation is the convolution result of two functions DNAT and P(x,y). P(x,y) can be interpretable as a transfer function from the cytometer and leads to another way for the solving, in so far as P(x,y) is accurately known.

2. MATHEMATICAL MODELS Two ways are proposed to circumvent the impossibility in solving the equation: - either the equation is fully ignored, only the hypotheses are retained and theses conditions lead to graphical methods. - or the equation is fully or partially used and this condition leads to parametric methods.

2.1. Graphical methods The common feature of the graphical methods consists in making an hypothesis on one (or two) phase(s) of the cell cycle, the remaining phase(s) being estimated by exclusion. The commonly used are: - The rectangle method (model I of Baisch et al., 1975) which supposes that the S phase is a rectangle from the modal peak of G0/1 to the modal peak of G 2 + M . The G0/1 and G 2 + M phases are estimated by exclusion, respectively. - The peak reflects method (Barlogie et al., 1976) which considers that the left half

134 of the GO/1 peak and the right half of the G 2 + M peak are not contaminated with S cells. These peak halves are folded around their respective mode forming two symmetric peaks whose areas yield the fraction of cells in each of these two phases. S phase is calculated by exclusion. These two models have a tendency to underestimate S phase.

2.2. Parametric methods In all these methods, the DNAT(X) distribution is supposed to be known and is composed of two gaussians: G0/I(FG1, ,or1) and G2+M(FG2, < x2 > ,trz) which are respectively representative of GO/1 and G 2 + M and a P(x) function representative of S phase. The parameters are solved by a least square method. The differences between the various methods rely upon the hypotheses for P(x) and the parameters estimated. These are: - D E A N a n d J E T T m o d e l (Christensen et al., 1978): P(x) consists in a sum of gaussian the amplitudes of which are modulated by a branch of parabola whose standard deviations are equivalent to: P(x) = ~ (axi=+bxi+c). EXP(-(x-xi)2/2.~) i

(Krishan & Frei, 1976): P(x) is assimilated to a branch of parabola ranging from < xl > to < x~ > . - F O X m o d e l (Baisch & Beck, 1978): is an adaptation of DEAN and JETT model, which superimpose to P(x,y) a gaussian representative of partially synchronous cells. - B A 1 S C H m o d e l I1 (Baisch & Beck, 1978): P(x,y) function supposes that the fraction cells entering S phase is constant as a function time. P(x,y) is assimilated to the sum of the integral of two gaussians respectively adjusted on < xl > and < x2 > . - F R I E D m o d e l (1976): GO/I, G 2 + M and P(x,y) are assimilated to delta functions with varying amplitudes Delta(x~). P(x,y) is assimilated to a gaussian. The basic equation can be expressed as: - Simplified DEAN andJETl'model

ADNE(y) -- ~ DELTA(xj).P(x,y) J These models are widely used but do not fit with some distributions. An intricate S phase which requires a suitable P(x,y) or a population made of several proliferating clones with various DNA contents are two examples. In the last one, the superposition of one or several mathematical models is not adapted.

3. BIOLOGICAL INTERPRETATION DERIVED FROM MATHEMATICAL ANALYSIS OF DNA DISTRIBUTIONS Most of the methods for estimating cell-cycle-traverse concentrate on estimating G0/1, S and G 2 + M phase fractions from single DNA histograms. The errors in the calculation of phase fractions also depend on the magnitude of the GO/l, S and G 2 + M phase fractions. Moreover, when the position of the GO/1 peak is unknown, the calculated results deviated much more from the simulation input. This shows that extreme caution is required when analyzing DNA histograms for partially synchronous cell populations. The

135 error in the estimate of a phase fraction in a single histogram from such a histogram may be especially large. An internal standard should be used to prevent this problem. No single method can be shown clearly to be superior and/or more or less accurate for all histograms. Accuracy seems to depend much more on the details of the fitting program rather than on the mathematical model used. The development of a mathematical analysis procedure consists in a three-step process: i) a conceptual model of the biological process being studied must be established and include all that is known and observable about the process, ii) a mathematical formalism is developed to describe the process and consists of one or more equations with several parameters. The independent variable is the DNA content in cell kinetics, iii) the parameters of the model are adjusted until the data computed from the model match the experimentally obtained data. If the computed and experimental data match, then the parameters are the correct constants that describe the biological process. Simple mathematical models providing only the fraction of cells in each phase are even not suitable when phase duration and dispersions in the duration, cell loss and perturbations in cell growth are observed. For example, the fraction of a cell population in each phase of the cell cycle is often used to monitor the effects of a particular drug. This can be done using flow cytometry to measure the DNA content distribution and a relatively simple mathematical model to resolve the distribution into its compartments. Although this method of obtaining cell kinetic information is simple and rapid, it is very limited in its application, since it provides so little information, and this information can be misleading. Commonly, only total DNA associated fluorescence per cell is measured by flow cytometry. If doublets discrimination can be achieved using pulse shape analysis (peak/area), the flow cytometer recognizes binucleate G0/1 cells as G2 cells. While DNA replication and mitosis take place in the cells, cytokinesis does not occur, leading therefore to the formation of binucleate (or multinucleate) cell, but the accumulation of cells with G2 DNA content. However little attention is paid to the common finding of binucleation. The central dogma of flow cytometry is that fluorescence from stained fluorescently for DNA content is proportional to cellular dye content. Thus, G0/1 cells are expected to give equal fluorescent signals despite significant variability in cell volume and/or morphology. G2 and M cells are also expected to give fluorescence despite the large differences in the geometric distributions of chromatin between interphase and mitosis (Kerker et al., 1982). Applications to most biological and biomedical problems tacitly assume the validity of the central dogma. Because of similar fluorescence intensity, distinguishing G2 cells from cells containing two GO/1 nuclei is not easy using single parameter DNA content measurement. On the contrary, the greater the potential of a drug to induce damage or alterations of DNA, the higher the frequency of G2 arrest (Ronot et al., 1986). Drug DNA interaction could result in the inactivation or alteration of some genes, which in turn could lead to the failure of synthesis of certain proteins required for the transition of cells through G2 to mitosis, leading to an unbalanced growth. Flow cytometry gives the possibility of gaining a number of rapid and valuable insights into the intracellular events controlling cell proliferation kinetics. However, a single parameter DNA histogram is only a snapshot of the population distribution of DNA content at the time of measurement, and this approach does not solve all the kineticist's problems, since G2, M and binucleate cells are not differentiated. For example, the

136 treatment of articular chondrocytes, HeLa or L 929 fibroblasts cells by an anti-rheumatic drug (D-penicillamine) induced a cell type-dependent cycle perturbation. For chondrocytes, D-penicillamine induced a partial accumulation in G0/1 and to a lesser extent, in the G 2 + M phase. Using transformed cell types (HeLa and L 929), D-penicillamine led principally to a marked accumulation of cells in G 2 + M , with a transient effect on the S phase. Measurement o f the mitotic index and the percentage of binucleate cells allowed to determine the precise cell distribution in G2 and/or M phase, since flow cytometric analysis cannot be used for this purpose as G0/1 binucleate cells and G 2 + M cells give equal fluorescent intensities after specific DNA staining. No significant modification of the mitotic index was observed for these three cell types. In contrast, the determination of binucleate cells showed an increase for treated HeLa and L 929 ceils, but not for chondrocytes (Table 1). The accumulation of cells in G 2 + M for these two cellular types resulted in two simultaneous phenomena: i) an increase of cells in G2 and ii) a cytokinesis perturbation leading to an increase in the number of binucleate cells. This cannot be solved by any type of mathematical models (Friteau et al., 1988) Table 1: Percentages of cells in each phase of the cell cycle (estimated by the peak reflects method), following a 72 hr in vitro D-penicillamine treatment.

% Chondroeytes Control 5 x 10.3 M HeLa Control 5 × 10 .3 M L 929 Control 5 × 103 M

G0/1

S

G2+M

G2

M

BN

87.0 80.0

2.5 2.0

10.5 18.0

8.9 17.2

1.2 0.1

0.4 0.7

57.0 33.0

24.0 20.0

19.0 47.0

9.4 25.3

4.1 6.2

5.5 15.5

67.0 34.0

21.0 19.0

12.0 47.0

9.0 32.2

2.3 2.1

0.7 12.7

BN: binucleate cells.

4. INFLUENCE OF G1-ARREST AND CONTINUUM MODELS The dominant and current view of the mitotic period consists into the subdivision: G1, S, G2, and mitosis/cytokinesis. The conventional cell cycle regulation model resides in the G1 phase, characterized by specific events and controls. An other proposal suggests that there are no G1 specific events or controls and the statement of the importance of G1 phase consists in a continuous process that occurs in all phases of the division cycle, leading to a simplification o f the cell cycle analysis, namely the concept of the continuum model (Okuda & Cooper, 1989). In the conventional division cycle, the end of each particular phase initiates a new phase of the division cycle, where M phase starts a new G1 phase. In the continuum model, the end of the S and G2 phases are the end of a process but this ending does not start any new processes. Cell preparation for division is

137 c o n t i n u o u s . D e s p i t e the theoretical a n d e x p e r i m e n t a l basis for this c o n t i n u u m m o d e l , it i n t r o d u c e s c o n t r o v e r s i a l a n d c o n f u s i n g i n f o r m a t i o n r e g a r d i n g the use o f m a t h e m a t i c a l m o d e l s a n d biological i n t e r p r e t a t i o n o f cell cycle kinetics.

ACKNOWLEDGEMENTS T h e a u t h o r s are grateful to Dr. Jacques A u g e r for critical reading and V i c t o r i a v o n H a g e n for E n g l i s h r e v i s i o n o f the manuscript.

REFERENCES Baisch, H., W. Gohde & W.A. Linden (1975). Mathematical analysis of pulse-cytophotometric data to determine the fraction of cells in the various phases of cell cycle. Radiat. Environ. Biophys. 12: 31. Baisch, H. & H.P. Beck (1978). Comparison of cell kinetics parameters obtained by flow cytometry and autoradiography. In: A.J. Valleron & P.D.M. McDonald, eds. Biomathematics and Cell Kinetics. Amsterdam, Elsevier/North Holland Biomedical Press, 411-422. Barlogie, B., B. Drewinko, D.A. Johnston, T. Buchner, W.H. Hauss & E.J. Freidreich (1976). Pulse cytophotometric analysis of synchronized cells in vitro. Cancer Res. 36:1176. Barrett, J.C. (1964). A mathematical model of the mitotic cycle and its application to the interpretation of percentage labeled mitoses data. J. Natl. Cancer Inst. 37: 443. Christensen, I., N.R. Hartmann, N. Keiding, J.K. Larsen, H. Noer & L. Vinddov (1978). Statistical analysis of DNA distribution from cell populations with partial synchrony. In: D. Lutz, ed., Pulse Cytophotometry m. Gent, European Press, p. 71-78. Fox, M.H. (1980). A model for computer analysis of synchronous DNA distributions obtained by flow cytometry. Cytometry 71 : 25. Fried, J. (1976). Method for the quantitative evaluation of data from flow microfluorometry. Comp. Biomed. Res. 9: 263. Friteau, L., P. Jaffray, X. Ronot & M. Adolphe (1988). Differential effect of D-penicillamine on the cell kinetic parameters of various normal and transformed cellular types. J. Cell. Physiol. 136: 514. Kendall, D.G. (1948). On the role of variable generation time in the development of a stochastic birth process. Biometrika 35: 316. Kerker, M., M.A. van Dilla, A. Brunsting, J.P. Kratohvil, P. Hsu, D.S. Wang, J.W. Gray & Langlois, R.G. (1982). Is the central dogma of flow cytometry true: That fluorescence intensity is proportional to cellular dye content? 3: 71. Krsihan, A. & E. Frei (1976). Effect of Adriamycin on the cell cycle traverse and kinetics of cultured human lymphoblasts. Cancer Res. 36: 143. Okuda, A. & S. Cooper (1989). The continuum model: an experimental and theoretical challenge to the G 1 model of cell cycle regulation. Exp. Cell. Res. 185: 1. Pierrez, J. & A. Guerci (1988). Les mod61isations math6matiques. In: P. M6t6zeau, X. Ronot, G. Le Noan Merdrignac & M.H. Ratinaud, eds., La Cytom6trie en Flux. Paris, Medsi/McGraw-Hill, p. 31-40. Ronot, X., C. Hecquet, S. Larno, B. Hainque & M. Adoiphe (1986). G2 arrest, binucleation, and singleparameter DNA flow cytometric analysis. Cytometry 7: 286.