State vector models of the cell cycle II: The first three moments of the transit time distribution.

A discrete time state vector model (the Hahn model) has been used to simulate many experiments in cell kinetics. In the first paper in this series the authors described a new method to define the parameters of the Hahn model suitable for use in automatic fitting of fraction of labelled mitoses (FLM) experiments. In this paper it is shown how to compute the first three moments of the transit time distribution which arises from a Hahn model. These moments are compared analytically and numerically to the corresponding moments of the distribution the authors used to define the Hahn model. Finally, the problems involved in estimating the moments of the transit time distribution observed in fitting FLM data using a Hahn model are discussed.     

Temperature response of the cell cycle of Haplopappus gracilis in suspension culture and its significance to the G1 transition probability model

The effects of temperature on the cell cycle of Haplopappus gracilis suspension cultures were analysed by the fraction of labelled mitoses method. Sphase in these cultures shows a different temperature optimum as compared to optima derived for G₂ and mitosis. G₁ phase has a much lower Q₁₀ than the other cell cycle phases and shows no temperature optimum between 22 and 34° C. These results are discussed in relation to a transition probability model of the cell cycle proposed by Smith and Martin (Proc. Natl. Acad. Sci. USA 70, 1263–1267, 1973), in which each cell has a time independent probability of initiating the transition into another round of DNA replication and division. The implications of such a model for cell cycle analysis are discussed and a tentative model for a probabilistic transition trigger is advanced.Abbreviations FLM Fraction of labelled mitoses - T_B Total B-phase     

TRANSIT TIMES THROUGH THE CYCLE PHASES OF JEJUNAL CRYPT CELLS OF THE MOUSE ANALYSIS IN TERMS OF THE MEAN VALUES AND THE VARIANCES

Mean transit times as well as variances of the transit times through the individual phases of the cell cycle have been determined for the crypt epithelial cells of the jejunum of the mouse. To achieve this the fraction of labelled mitoses (FLM) technique has been modified by double labelling with [³H] and [¹⁴C]thymidine. Mice were given a first injection of [³H]thymidine, and 2 hr later a second injection of [¹⁴C]thymidine. This produces a narrow subpopulation of purely ³H-labelled cells at the beginning of G₂-phase and a corresponding subpopulation of purely ¹⁴C-labelled cells at the beginning of the S-phase. When these two subpopulations progress through the cell cycle, one obtains FLM waves of purely ³H- and purely ¹⁴C-labelled mitoses. These waves have considerably better resolution than the conventional FLM-curves. From the temporal positions of the observed maxima the mean transit times of the cells through the individual phases of the cycle can be determined. Moreover one obtains from the width of the individual waves the variances of the transit times through the individual phases. It has been found, that the variances of the transit times through successive phases are additive. This indicates that the transit times of cells through successive phases are independently distributed. This statistical independence is an implicit assumption in most of the models applied to the analysis of FLM curves, however there had previously been no experimental support of this assumption. A further result is, that the variance of the transit time through any phase of the cycle is proportional to the mean transit time. This implies that the progress of the crypt epithelial cells is subject to an equal degree of randomness in the various phases of the cycle.     

State vector models of the cell cycle. III: Continuous time cell cycle models

In this paper the elements of the matrix of the Hahn cell-cycle model are identified with the infinitesimal transition probabilities of a Markov process, and as a limiting process a differential equation analogue is derived. The probability density function of the discrete time model is derived and used to obtain the density function for transit times of the continuous time model. It is shown that the mean transit time remains constant and that the variances of the discrete and continuous time models are the same to the order of the time increment. Finally, it is shown how to derive the Takahashi model from the continuous time Hahn model.     

A Rapid Automated Stathmokinetic Method For Determination of In Vitro Cell Cycle Transit Times

To provide a rapid method for examining cell cycle dynamics, we utilized continuous exposure of Chinese hamster ovary cells and human colon cancer cells to colcemid to block cycling cells in metaphase, suppressing re-entry into G₁. Changes in cell cycle compartment distribution were monitored by DNA flow cytometry. Analysis of the rate of G₂+ M compartment accumulation after addition of colcemid permitted calculation of all cycle transit parameters. These compared favorably with data in the same cell lines determined by the fraction of labeled mitoses technique. Serial assessment of DNA flow cytometry after addition of colcemid permits rapid quantitation of cycle traverse rates.     

CIRCADIAN INFLUENCE ON THE WAVE FORM OF THE FREQUENCY OF LABELED MITOSES IN MOUSE CORNEAL EPITHELIUM

The FLM was studied in mouse corneal epithelium at two different phases of the mouse circadian system. The following results were obtained: (1) a circadian rhythm was found in the corneal mitotic index of both experimental groups, with the peak near 09.00 hours, the trough near 21.00 hours and a 10-fold difference in values between the peak and the trough; (2) when ³H-TdR was injected at 09.00 hours, the first wave of labelled mitoses rose earlier, attained a broader peak and fell later when compared to the first wave of labeled mitoses obtained when ³H-TdR was injected at 21.00 hours; and (3) in both experimental groups there was a suggestion of a second wave of labeled mitoses, especially in the group injected with ³H-TdR at 21.00 hours, where a flat second wave appeared in the region of 60–90 hr. Transit times derived from FLM curves obtained from such synchronous populations of cells are of questionable validity. Circadian rhythmicity and other causes of synchronization must be ruled out before accepting a single FLM curve as a valid indication of transit times.     

Cell kinetic studies on the JB-1 ascites tumour of the mouse

Abstract. The FLM method, modified by double labelling with [³H]- and [¹⁴C]-thymidine, has been applied to the 4-day old JB-1 ascites tumour of the mouse. It results in well separated waves of purely [³H]- and purely [¹⁴C]-labelled mitoses, which show a remarkable asymmetry with long tails to the right. The following values for the mean transit times of the cells have been derived from this FLM curve, for a tumour age of 4–6 days: T_C= 32.5 hr, T_S= 16.7 hr, T_G1= 3.7 hr, T_G1= 11.0 hr and T_M= 1.1 hr. A further evaluation of the FLM curve, however, is difficult, due to the non-stationary growth of the tumour. A number of other experimental findings (growth curve, decrease of the labelling and mitotic index with increasing tumour age, two single-labelled FLM curves starting 4 and 6 days after tumour inoculation) indicate that the cell cycle time increases during the experimental period of the double-labelled FLM curve (about 2 days). A lengthening of the cycle time should result in an increasing enlargement of the areas under the waves of the modified FLM curve. However, such an increase in area has not been found; the areas are constant. All the results of the present cell kinetic studies would be consistent if it were postulated that the cell cycle time lengthens with increasing tumour age up to about 4 days after inoculation, then remains relatively constant at between 4 and 6 days and thereafter increases again. Short-term double labelling experiments suggest that this is actually the case. Under the assumption of nearly constant phase durations during the 5th and 6th day of tumour growth further conclusions can be drawn from the modified FLM curve. In particular, it follows that the transit times of the cells through successive cycle phases are uncorrelated and the variances of the transit times through a cycle phase are proportional to the duration of this phase.     

The Automatic Analysis of Flm Curves

Abstract. Seven parameters of the cell cycle are extracted from experimental FLM data by computer using a completely automated, least-squares, regression analysis. the procedure is based on a model of the cell cycle with four phases, three coefficients of variation, and a Poisson process of cell progression. T _M is estimated separately, using mitoses per labeled cells over the first cycle. Beginning with raw data, the computer calculates initial estimates of the parameters, uses these estimates to generate a synthetic FLM curve, and then measures a weighted mean square deviation of fit between the data and the curve. By a process of iteration, involving a three-dimensional Newton-Raphson method for mean transit times and an orthogonal search for coefficients of variation, the measure of fit is progressively minimized. Eighteen experimental data sets have been analysed successfully. Several procedures for the evaluation of the analysis are described.     

Stability analysis of the two compartment model of cell proliferation.

The behavior of the G₀ model upon perturbation of its equilibrium state is investigated in some detail. Algebraic expressions for stability conditions of the linearized model are derived. A distribution of mitotic times as well as population-dependent transition probabilities are incorporated into the analysis.     

Comparison of in vitro and in vivo cell cycle parameters of lymphoid cells in Pig spleens

Summary Parameters of the cell cycle of lymphoid cells were estimated by analyzing percent labeled mitoses curves after a ³H-thymidine flash. Either anaesthetized pigs were labeled and multiple biopsies taken from the spleen in vivo or isolated perfused pig spleens were labeled in vitro. The data from in vivo and in vitro experiments were very similar.The mean values for cell cycle parameters were: 20.2 to 20.5 hours for the generation time, about 0.5 to 1 hour for G₂, about 1.2 to 1.3 hours for M; about 17 to 16.5 hours for S and about 1.5 to 1.7 hours for G₁. The mean grain count halving time of labeled mitoses was in accordance with the measured generation time. The isolated perfused spleen seems to give results equal to in vivo data and could, therefore, be employed as a model for studying cell cycle parameters not only in animal but also in human lymphoid tissue.The expert technical assistance of Mrs. A. Fischer is gratefully acknowledged. This study was supported by the Deutsche Forschungsgemeinschaft, SFB 112.     

RAPID CELL CYCLE ANALYSIS BY MEASUREMENT OF THE RADIOACTIVITY PER CELL IN A NARROW WINDOW IN S PHASE (RCSi)

A new rapid method for the cell cycle analysis of asynchronously growing cells is presented. The new method is an alternative to the more time consuming and subjective fraction of labeled mitoses (FLM) method. Like the FLM method, all cells in the S phase of the cell cycle are marked by pulse labeling with a radioactive DNA precursor. The subsequent progress of the cohort of cells thus labeled is monitored through a narrow window in the cell cycle. The window is defined by a narrow range of DNA contents corresponding to cells in mid-S phase and is designated S_i. The cellular DNA content is measured by flow cytometry and the cells in the window S_i are selected by electronic cell sorting. The radioactivity per cell in S_i (RCS_i) is determined by liquid scintillation counting. The duration of S phase and of the total cycle and the dispersions therein are determined from the oscillation of the RCS_i values with time. The complete cell cycle analysis can be accomplished in as little as 1 day following the collection of samples. Exponentially growing Chinese hamster ovary (CHO) cells were analyzed according to the RCS_i method and the FLM method. It is demonstrated that the two techniques give essentially the same results.     

STUDIES ON LYMPHOCYTES

Some of the time parameters of the cell cycle in bovine thoracic duct lymphocytes have been estimated by analysing labeled mitoses curves, and by double labeling. The two methods gave similar estimates of T_s. Thus, T_s measured directly from labeled mitoses curves varied from 4 to 6 hr, while the estimates from double labeling were 4.8 and 4.5 hr. T _G measured directly from labeled mitoses curves was 5 hr, and estimates of T_G from the values of T_s ranged from 6.3 to 7.7 hr. The present data confirm the short generative cycle of large thoracic duct lymphocytes. Extracorporeal irradiation of the lymph (ECIL) had no detectable effect on the cell cycle or the fractional production rate of the lymphocytes. However, the calculated absolute production was reduced following ECIL, due to a decrease in the absolute number of cells present. The grain count over mitoses after ECIL was approximately one-half that found before ECIL.     

A COMPARISON OF COMPUTER METHODS FOR THE ANALYSIS OF FRACTION LABELLED MITOSES CURVES

Computer methods developed by the authors for analysis of fraction labelled mitoses curves (FLM curves) have been compared. Four test examples were used in the study; the first example was the synthesis of a FLM curve with fixed parameters and the others involved fitting actual data. Experimental FLM curves showing various degrees of damping were used in the curve fitting tests. In each test example the comparison was based on the assumptions of exponential growth, a growth fraction of unity and no cell loss. In three of the test examples good agreement between the methods was observed but in one example some important discrepancies arose in the analysis of a heavily damped FLM curve.     

Analytical expressions for PLM(t), CL(t) and CLM(t) without use of Laplace transforms

Based on the age density functions for each phase of the cell life cycle (G₁, S, G₂ and M) in an exponentially growing steady state population derived by Trucco &; Brockwell (1968), the expressions for the percentage labeled mitoses curve [PLM(t)], the continuous labeling curve [CL(t)] and the continuous labeled mitotic curve [CLM(t)] are obtained explicitly without use of Laplace transforms. This approach is useful in describing the cell population when the steady state is disturbed due to, for example, irradiation. The mitotic index [MI(t)] for this case is considered.     

Transit times through the cycle phases of jejunal crypt cells of the mouse. Analysis in terms of the mean values and the variances.

Mean transit times as well as variances of the transit times through the individual phases of the cell cycle have been determined for the crypt epithelial cells of the jejunum of the mouse. To achieve this the fraction of labelled mitoses (FLM) technique has been modified by double labelling with [3H] and [14C]thymidine. Mice were given a first injection of [3H]thymidine, and 2 hr later a second injection of [14C]thymidine. This produces a narrow subpopulation of purely 3H-labelled cells at the beginning of G2-phase and a corresponding subpopulation of purely 14C-labelled cells at the beginning of the S-phase. When these two subpopulations progress through the cell cycle, one obtains FLM waves of purely 3H- and purely 14C-labelled mitoses. These waves have considerably better resolution than the conventional FLM-curves. From the temporal positions of the observed maxima the mean transit times of the cells through the individual phases of the cycle can be determined. Moreover one obtains from the width of the individual waves the variances of the transit times through the individual phases. It has been found, that the variances of the transit times through successive phases are additive. This indicates that the transit times of cells through successive phases are independently distributed. This statistical independence is an implicit assumption in most of the models applied to the analysis of FLM curves, however there had previously been no experimental support of this assumption. A further result is, that the variance of the transit time through any phase of the cycle is proportional to the mean transit time. This implies that the progress of the crypt epithelial cells is subject to an equal degree of randomness in the various phases of the cycle.     

A CHARACTERIZATION OF THE BOUNDARY BETWEEN THE CYCLING AND RESTING STATES IN ASCITES TUMOR CELLS

Growth deceleration of an Ehrlich ascites tumor with increasing mass is associated with a prolongation of the cell cycle and a decline in the growth fraction. These effects are reversed upon transfer of cells from an older tumor into a new host. Studies were made to locate the stages at which a cell cycle could be suspended or resumed. Transplantation caused a prompt rise in both mitotic and flash H³TdR labeling indices. When all the cells in cycle including mitoses were prelabeled with H³TdR in older tumors, the fraction of labeled mitoses did not decline for a considerable period after transplantation into new hosts. This suggests that the early rise in mitoses is not due to a flow of resting (G_o) cells from a G² store (G²-G^o transition). It appears rather to be a reflection of a lag of the mitotic process relative to other stages during the initial readjustment of the cycle. A prompt rise in flash H₃TdR indices in the transplants suggested cell entry into S from either a suspended G_I (G₁-G_o transition) or a suspended S (S-G_o transition). These possibilities were examined by relating micro-spectrophotometric estimates of DNA to the cell cycle stage as revealed by H³TdR autoradiography. Since G_o cells had DNA values corresponding to G_I, it was concluded that decycling or recycling could occur only after mitosis and before DNA synthesis.     

KINETIC ANALYSIS OF CELL SIZE AND DNA CONTENT DISTRIBUTIONS DURING TUMOR CELL PROLIFERATION: EHRLICH ASCITES TUMOR STUDY

In order to study the growth dynamics of proliferating and non-proliferating cells utilizing discrete-time state equations, the cell cycle was divided into a finite number of age compartments. In analysing tumor growth, the kinetic parameters associated with a retardation in the growth rate of tumors were characterized by computer simulation in which the simulated results of the growth curve, the growth fraction, and the mean generation time were adjusted to fit the experimental data. The cell age distribution during the period of growth was obtained and by a linear transformation of the state transition matrices, was employed to specify the cell size and DNA content distributions. In an application of the model, the time-course behavior of cell cycle parameters of Ehrlich ascites tumor is illustrated, and the parameters important for the transition of cells in the proliferating compartment to the non-proliferating compartment are discussed, particularly in relation to the G₁-G₀ and G₂-G₀ transitions of non-cycling cells as revealed by the variation of cell size distribution.     

CELL KINETICS OF IRRADIATED EXPERIMENTAL TUMORS: CELL TRANSITION FROM THE NON-PROLIFERATING TO THE PROLIFERATING POOL

Parenchymal tumor cells of murine mammary carcinomas can be divided into two pools, using nucleoli as morphological ‘markers’. Cells with dense nucleoli traverse the cell cycle and divide, thus constituting the proliferating pool. Cells with trabeculate or ring-shaped nucleoli either proceed slowly through G₁ phase or are arrested in it. The role of these non-proliferating, G₁ phase-confined cells in tumor regeneration was studied in vivo after a subcurative dose of X-irradiation in two transplantable tumor lines. Tumor-bearing mice were continuously injected with methyl[³H]thymidine before and after irradiation. Finally, the labeling was discontinued, mice injected with vincristine sulfate and cells arrested in metaphase were accumulated over a 10-hr period. Two clearly delineated groups of vincristinearrested mitoses emerged in autoradiograms prepared from tumor tissue at the time of starting tumor regrowth: one group with the silver-grain counts corresponding to the background level, the other with heavily labeled mitoses. As the only source of unlabeled mitoses was unlabeled G₁ phase-confined cells persisting in the tumor, this observation indicated cell transition from the non-proliferating to the proliferating pool, which took place in the initial phase of the tumor regrowth. Unlabeled progenitors have apparently remained in G₁ phase for at least 5–12 days after irradiation.     

CHARACTERISTICS OF THE ISOPRENALINE STIMULATED PROLIFERATIVE RESPONSE OF RAT SUBMAXILLARY GLAND

A simple stochastic model has been developed to determine the cell cycle kinetics of the isoprenaline stimulated proliferative response in rat acinar cells. The response was measured experimentally, using ³H-TdR labelling of interphase cells and cumulative collections of mitotic cells with vincristine. The rise and fall of the fraction of labelled interphase cells and of metaphase cells is expressed by the product of the proliferative fraction and a difference of probability distributions. The probability statements of the model were formulated and then compared by an iterative fitting procedure to experimental data to obtain estimates of the model parameters. The model when fitted to the combined fraction labelled interphase (FLIW) and fraction metaphase (FMW,) waves gave a mean G_is transit time of 21-2 hr, mean G_is+ S transit time of 270 hr, and mean G_is+ S + G₂ transit time of 35-8 hr for a single injection of isoprenaline, where G_is is the initiation to S phase time. When successive injections of isoprenaline were given at intervals of 24 and 28 hr the corresponding values after the third injection were 12-4 hr, 20-8 hr and 25-7 hr respectively. The variance of the G_is phase dropped from 18-1 to 1–3 while the other variances remained unchanged. The estimated proliferative fraction was 0–24 after a single injection of isoprenaline, and 0–31 after three injections of the drug. Independently determined values of the proliferative fraction, obtained from repeated ³H-TdR injections, were 0–21 and 0–36 respectively.