THE DISCRETE–TIME KINETIC MODEL ANALYSIS OF DNA CONTENT DISTRIBUTIONS IN EXPERIMENTAL TUMOUR CELLS

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMF measurements of DNA content distributions and to analyse their relationship to the cell kinetic parameters, namely cell loss rate, cell cycle times and growth fraction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coefficients of variation, of the L1210 ascites tumour during the growth period.     

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 distibution 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.     

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.     

Kinetics of protein and DNA synthesis studied by mathematical modelling of flow cytometric protein and DNA histograms

Abstract. Mathematical models for histograms of cellular protein content as measured by flow cytometry were developed, based on theoretical protein distributions. These were derived from the age distribution of cells and the accumulation function for cellular protein content as a function of age within the cell cycle. A model assuming an exponential age distribution and an exponential protein. accumulation function was found to give the best representation of protein histograms of exponentially growing NHIK 3025 cells. This is in good agreement with the known kinetic behaviour of such cells. By the combined use of the protein histogram model and a similar model for DNA content, and assuming linear DNA accumulation during S, the fraction of cells in S, as a function of cellular protein content, was simulated. This function showed good agreement with values of the [³H]TdR labelling index scored in cells sorted by flow cytometry from 5-channel intervals of the protein histogram. The protein and DNA histogram models were combined into a two-dimensional model for correlated protein/DNA measurements. Comparison between simulated data and experimentally derived two-dimensional protein/DNA histograms gave further support to the cell kinetic assumptions underlying the models, but also identified some minor deviations which could not be recognized in the analysis of the one-dimensional histograms.     

ANALYSIS OF THE PROLIFERATION KINETICS OF BURKITT'S LYMPHOMA CELLS

The in vitro proliferation kinetics of a cell line derived from a patient with American Burkitt's lymphoma were investigated at three different growth phases: lag (day 1), exponential (day 3) and plateau (day 5). The growth curve, labeling and mitotic indices, percentage labeled mitosis (PLM) curves and DNA content distributions were determined. The data obtained have been analysed by the previously developed discrete-time kinetic (DTK) model by which a time course of DNA distributions during a 10-day growth period was characterized in terms of other cell kinetic parameters. The mean cell cycle times, initially estimated from PLM curves on days 1, 3 and 5, were further analysed by the DTK model of DNA distributions and subsequently the mean cell cycle times with respect to DNA distributions during the entire growth period were determined. The doubling times were 39·6, 31·2 and 67·2 hr, respectively, at days 1, 3 and 5. The mean cell cycle time increased from 23·0 to 37·7 hr from day 3 to day 5 mainly due to an elongation of the G₁ and G₂ phases. A slight increase in the cell loss rate from 0·0077 to 0·0081 fraction/hr was accompanied by a decrease in the cell production rate from 0·0299 to 0·0184 fraction/hr. This calculated cell loss rate correlated significantly with the number of dead cells determined by trypan blue exclusion. Analysis of the number of dead cells in relation to the cell cycle stage revealed that a majority of cell death occurred in G₁ (r= 0·908; P < 0·0001). There was a good correlation between the in vitro proliferation kinetics at plateau phase of this Burkitt's lymphoma derived cell line and the in vivo proliferation kinetics of African Burkitt's lymphoma (Iversen et al., 1974), suggesting the potential utility of information obtained by in vitro kinetic studies.     

Flow cytometric analysis of protein thiol groups in relation to the cell cycle and the intracellular content of glutathione in rat hepatocytes

The relationship between protein thiols (PSH) and cell proliferation was examined in ethanol-fixed rat hepatocytes. A new protocol was developed for simultaneous measurement of protein thiol vs. DNA content by flow cytometry. The fluorescent dye o-phthalaldehyde (OPT) was used for flow cytometric measurements of protein thiol groups. The influence of nonprotein thiols was examined by monitoring the cell cycle of cells in which the glutathione content (GSH) was modified by treatment with buthionine sulphoximine (BSO). Three rat liver cell lines (IAR 20, IAR 6.1, IAR 6.1RT7) were used: these cell lines possess different growth characteristics and degrees of tumorigenicity, which made it possible to analyse changes in PSH during normal and deranged cell proliferation. The effects on the cell cycle of the changes in PSH due to the depletion of GSH were measured by 5-bromo-2'-deoxyuridine (BrdUrd) incorporation and flow cytometry. The data obtained can be summarised as follows: a) OPT fluorescence increases with increasing DNA content in all rat liver cell lines examined; b) the greatest variation in PSH content occurs in G1. There is a smaller variation in G2 + M, and PSH levels are relatively invariant throughout S-phase; c) a higher content of PSH is found in the tumorigenic cell lines; d) the amount and distribution of PSH is not affected by BSO treatment; e) kinetic studies indicate that BSO treatment has no effect on the ability of the IAR rat liver cell lines to progress through the cycle.     

Estimating the variation in S phase duration from flow cytometric histograms

A stochastic model for interpreting BrdUrd DNA FCM-derived data is proposed. The model is based on branching processes and describes the progression of the DNA distribution of BrdUrd-labelled cells through the cell cycle. With the main focus on estimating the S phase duration and its variation, the DNA replication rate is modelled by a piecewise linear function, while assuming a gamma distribution for the S phase duration. Estimation of model parameters was carried out using maximum likelihood for data from two different cell lines. The results provided quite a good fit to the data, suggesting that stochastic models may be a valuable tool for analysing this kind of data.     

An easy-to-use simulation program demonstrates variations in bacterial cell cycle parameters depending on medium and temperature

Many studies are performed on chromosome replication and segregation in Escherichia coli and other bacteria capable of complex replication with C phases spanning several generations. For such investigations an understanding of the replication patterns, including copy numbers of origins and replication forks, is crucial for correct interpretation of the results.Flow cytometry is an important tool for generation of experimental DNA distributions of cell populations. Here, a Visual Basic based simulation program was written for the computation of theoretical DNA distributions for different choices of cell cycle parameters (C and D phase durations, doubling time etc). These cell cycle parameters can be iterated until the best fit between the experimental and theoretical DNA histograms is obtained. The Excel file containing the simulation software is attached as supporting information.Cultures of Escherichia coli were grown at twelve different media and temperature conditions, with following measurements by flow cytometry and simulation of the DNA distributions. A good fit was found for each growth condition by use of our simulation program. The resulting cell cycle parameters displayed clear inter-media differences in replication patterns, but indicated a high degree of temperature independence for each medium. The exception was the poorest medium (acetate), where the cells grew with overlapping replication cycles at 42 °C, but without at the lower temperatures.We have developed an easy-to-use tool for determination of bacteria's cell cycle parameters, and consequently the cells' chromosome configurations. The procedure only requires DNA distribution measurements by flow cytometry. Use of this simulation program for E. coli cultures shows that even cells growing quite slowly can have overlapping replication cycles. It is therefore always important not only to assume cells' replication patterns, but to actually determine the cell cycle parameters when changing growth conditions.     

Modeling T Cell Proliferation and Death in Vitro Based on Labeling Data: Generalizations of the Smith–Martin Cell Cycle Model

The fluorescent dye carboxyfluorescein diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time based on the distribution of times to the first division. Choosing the distribution of time to the first division to fit thymidine labeling data for T cells stimulated in vitro under different concentrations of IL-2, we fit CFSE data to determine the dependence of T cell kinetic parameters on the concentration of IL-2. As the concentration of IL-2 increases, the average cell cycle time is shortened, the death rate of cells is decreased, and a higher fraction of cells is recruited into division. We also find that if the average cell cycle time increases with division class then the qualify of our fit to the data improves.     

Quantitative cytology in leukemia research

A study was undertaken to determine the usefulness of flow cytometric analysis of bone marrow cells as an objective means for diagnosis, classification and prognosis in patients with leukemia. Abnormal DNA content as a marker of neoplastic disease was found in only 15% of 264 adult patients with acute leukemia (13% in AML, 26% in ALL/AUL). Alternative means of tumor cell detection in heterogeneous marrow samples include determination of nucleolar antigen density and double-stranded RNA content. Phenotypic characterization of leukemia subtypes can be afforded by RNA content analysis of acridine orange-stained cells, demonstrating significantly higher mean RNA content values in AML, compared to ALL/AUL. Cytokinetic parameters amenable to flow cytometric analysis include measurements of cell cycle compartment distribution by DNA content, of cycle traverse rate by BUdR-induced modification of fluorescence intensity of DNA specific dyes and of growth fraction employing the method of in situ DNA denaturation and subsequent acridine orange staining. Determination of cell cycle distribution and RNA content pretreatment and serially during remission induction in 82 patients demonstrated a significantly lower pretreatment biopsy S phase proportion in responding patients with AML compared to individuals failing treatment whereas an opposite trend was noted in patients with ALL/AUL. While of no prognostic impact pretreatment, serial determinations of the RNA content during the first chemotherapy induction course revealed significant differences between responding and failing patients with AML. Also, patients attaining remission demonstrated a rise in marrow biopsy S phase compartment size by day 10 to 14 of treatment, thus, predicting remission during marrow hypoplasia. We conclude that quantitative cytologic examination of marrow cells from patients with acute leukemia provides useful diagnostic and prognostic information that should aid in the stratification of patients with poor prognosis to receive new agents.     

BrdU-Hoechst flow cytometry reveals regulation of human lymphocyte growth by donor-age-related growth fraction and transition rate

An improved BrdU-Hoechst flow assay was applied to cell kinetic studies of human lymphocyte cultures during a 24-96 hr interval after PHA stimulation. The assay shows that the duration of the initial lag phase and the proportions of noncycling cells increase as a function of donor age, whereas the rates of transition from each cell cycle compartment to the next decrease. Cell cycle arrest occurs in the first S and G2 phase after stimulation of lymphocytes from a 75-year-old donor but not from younger donors. The data are consistent with several models of cell cycle kinetics, so long as these models are modified to include a fraction of noncycling cells in each cell cycle compartment.     

Analysis of a model of cell cycle in eukaryotes

A dynamic model of the cell cycle for eukaryotic cells, which takes into account the rates of ribosome and protein synthesis and the discontinuous events of DNA replication and cell division, is analyzed. It is shown that, by changing the values of the parameters, three different cell cycle regimens are possible, which are similar to cell cycle patterns experimentally observed and which show the action of different control mechanisms. The model allows the determination of the macromolecular levels as a function of the cycle time. Taking into consideration the age distribution function of the cells in an ideal exponentially growing population, mathematical relations are calculated that link the levels of macromolecular components (protein, ribosomes and DNA) to the temporal parameters of the cell cycle, such as the relative duration of the S phase. It is also shown that the relative length of all cell cycle phases may be determined if the labelling index and the relative DNA content of the cell population are known. All these relations suggest new and convenient procedures to determine cell cycle parameters.     

Flow cytometric BrdUrd-pulse-chase study of heat-induced cell-cycle progression delays

The flow cytometric, bromodeoxyuridine (BrdUrd)-pulse-chase method was extended by analysing five kinetic parameters to study perturbed cell progression through the cell cycle. The method was used to analyse the cell-cycle perturbations induced by heat shock. Exponentially growing, asynchronous Chinese hamster ovary (CHO) cells were pulse labelled with BrdUrd and simultaneously heated at 43°C for 5,10 or 15 min. The cells were then incubated in a BrdUrd-free medium and, at various times thereafter, were prepared for flow cytometry. Five compartments (BrdUrd-labelled divided and undivided, and unlabelled G₁, G₁S, and G₂) were defined in the resulting dual-parameter histograms. The fraction of cells and the mean DNA content, when appropriate, were calculated for each compartment. The rates of cell-cycle progression were assessed as time-dependent changes in the fraction of cells in a given compartment and/or the relative DNA content of cells within a given compartment. Linear regression analysis of the data revealed two distinct modes of alteration in cell progression: 1 a delay in cell transit (either out of or into a given compartment), and 2 a decrease in the rate of cell transit. Hyperthermia produced a delay in the exit of cells from the G₁ compartment of ≈ 16 min per minute of heat at 43°C with no threshold. In contrast, the delay in the exit of cells from all other compartments showed a threshold of from 3 to 5 min at 43°C. Above this threshold the delay in exit of cells from the BrdUrd-labelled, undivided compartment was 25 min per minute of heat at 43°C. The more complex dose-response function of this latter compartment may reflect the fact that it includes two cell-cycle phases, S and G₂+ M. The decrease in the rate of transit out of G₂ for cells heated in G₂ was significantly larger than that for any other compartment, consistent with previous studies, which showed a G₂ accumulation following hyperthermia. These results indicate that heat exposure induces very complex alterations in cell-cycle progression and that this flow cytometric method offers a straightforward approach for observing such alterations.     

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.     

Desynchronization rate in cell populations: mathematical modeling and experimental data

We characterize the kinetics of two cancer cell lines: IGROV1 (ovarian carcinoma) and MOLT4 (leukemia). By means of flow cytometry, we selected two populations from exponentially growing in vitro cell lines, depending on the cells' DNA synthesis activity during a preceding labeling period. For these populations we determined the time course of the percentages of cells in different phases of the cycles, sampling every 3 hr for 60 hr. Initially, semi-synchronous populations quickly converged to a stable age distribution, which is typical of the cell line (at equilibrium); this desynchronization reflects the intercell variability in cell cycle duration. By matching these experimental observations to mathematical modelling, we related the convergence rate toward the asymptotic distribution (R) and the period of the phase-percentage oscillations (T), to the mean cell cycle duration and its coefficient of variation. We give two formulas involving the above-mentioned parameters. Since T and R can be drawn by fitting our data to an asymptotic formula obtained from the model, we can estimate the other two kinetic parameters. IGROV1 cells have a shorter mean cell cycle time, but higher intercell variability than the leukemia line, which takes longer to lose synchrony.     

Molecular quantification of cell cycle-related gene expression at the protein level

BACKGROUND: Immunofluorescence cytometry of antigen and DNA content provides relative measurements of the cell cycle phase distribution of a specific epitope. Measurement of correlated expression of epitopes on signaling and regulatory proteins will be useful in the study of the complex pathways involved in cell cycle regulation and carcinogenesis. However, to formulate regulatory pathway models, measurements of molecules per cell would be more useful than relative measurements of intensity. Here, we report on a system in which the relationship between molecules and fluorescence is determined for a reference set of cell lines that are then used to directly calculate the number of molecules for unknowns. To demonstrate the process, we calculated the cell cycle phase distribution of SV40 large T antigen (Tag) in the reference cells. METHODS: A set of cell line clones expressing different levels of Tag were isolated. Quantitative Western blots of these cells and purified, recombinant Tag were performed. Cells from the same sample were stained and analyzed by flow cytometry for Tag and DNA. The relationship between molecules and fluorescence was established and calculations were performed for the phase distributions of Tag. RESULTS: The five cell lines had 0.11, 0.27, 1.06, 2.44, and 2.63 x 10(6) molecules of Tag per cell, determined by Western blot. The average coefficient of variation was 10.6%. The relationship of molecules to fluorescence fit a linear equation (r(2) = 0.96) over the range, 0.11 - 2.63 x 10(6) molecules, however, the same equation did not fit the relationship between 0 molecules, defined by isotype staining controls, and the lowest expressing cell line. To calculate the phase distributions of molecules in the lowest cell line, a second linear equation from 0 to 110,000 molecules was used. CONCLUSIONS: This work describes a system where fixed cells expressing various levels of a target antigen quantified by Western blots can be used to standardize flow cytometric measurements of gene expression in absolute terms.     

THE KINETICS OF GRANULOSA CELLS IN DEVELOPING FOLLICLES IN THE MOUSE OVARY

This investigation describes the kinetics of the granulosa cells in medium-sized follicles type 3b, 4 and 5a in ovaries of 28-day-old Bagg mice. the method of labelling with ³H-thymidine followed by high resolution autoradiography is used in the experimental work, which consist of determining percentage labelled mitosis (PLM-) and continuous labelling (CL-) curves. In order to analyse the data by computer two alternative hypotheses A and B are set up. Both include the assumptions of no cell loss, exponential growth and a resting compartment Q. In hypothesis A cells from Q re-enter the mitotic cycle via the normal DNA-synthesis compartment S_p. Hypothesis B includes beside compartment S_p a special DNA-synthesis compartment S_q where only cells from Q are synthesizing DNA, and these cells re-enter the mitotic cycle via the G₂ compartment. the mean transit time in S_q is considered to be longer than the mean transit time in S_q. On the basis of the hypothesis mathematical expressions for the PLM- and CL-curves are obtained, and by means of a computer the theoretical curves are fitted to the experimental values: thereby all relevant cell kinetical parameters are estimated. Hypothesis B seems to give the best fit between the theoretical and experimental curves. the estimated parameters are: mean cycle times, μ_c= (56.1 hr, 56.1 hr and 22.3 hr for type 3b, 4 and 5a respectively), doubling times, T _D= (96.4 hr, 118.6 hr and 59.1 hr) and the proportion of cells in Q, p _Q = (0.60, 0.71 and 0.69).     

Mathematical model analysis of mouse epidermal cell kinetics measured by bivariate DNA/anti-bromodeoxyuridine flow cytometry and continuous [3H]-thymidine labelling

In a previous study the epidermal cell kinetics of hairless mice were investigated with bivariate DNA/anti-bromodeoxyuridine (BrdU) flow cytometry of isolated basal cells after BrdU pulse labelling. The results confirmed our previous observations of two kinetically distinct sub-populations in the G2 phase. However, the results also showed that almost all BrdU-positive cells had left S phase 6-12 h after pulse labelling, contradicting our previous assumption of a distinct, slowly cycling, major sub-population in S phase. The latter study was based on an experiment combining continuous tritiated thymidine [( 3H]TdR) labelling and cell sorting. The purpose of the present study was to use a mathematical model to analyse epidermal cell kinetics by simulating bivariate DNA/BrdU data in order to get more details about the kinetic organization and cell cycle parameter values. We also wanted to re-evaluate our assumption of slowly cycling cells in S phase. The mathematical model shows a good fit to the experimental BrdU data initiated either at 08.00 hours or 20.00 hours. Simultaneously, it was also possible to obtain a good fit to our previous continuous labelling data without including a sub-population of slowly cycling cells in S phase. This was achieved by improving the way in which the continuous [3H]TdR labelling was simulated. The presence of two distinct subpopulations in G2 phase was confirmed and a similar kinetic organization with rapidly and slowly cycling cells in G1 phase is suggested. The sizes of the slowly cycling fractions in G1 and G2 showed the same distinct circadian dependency. The model analysis indicates that a small fraction of BrdU labelled cells (3-5%) was arrested in G2 phase due to BrdU toxicity. This is insignificant compared with the total number of labelled cells and has a negligible effect on the average cell cycle data. However, it comprises 1/3 to 1/2 of the BrdU positive G2 cells after the pulse labelled cells have been distributed among the cell cycle compartments.     

Multi-user system for analysis of data from flow cytometry.

A new program is described for the analysis of DNA histograms from flow cytometry. The fundamental model representing the cell population is similar to one described previously. It assumes the population is grouped into compartments, each consisting of cells having approximately the same DNA content. After staining the cells with an appropriate fluorochrome, the fluorescence distribution of cells within each compartment is assumed to be Gaussian. In the present algorithm, the parameters of the model can either be computed directly by the program from the data, or can be specified as input by the user. When synchronous cell populations lacking distinct G1 and G2/M phases are analyzed, the parameter values must first be obtained using an appropriate control. Percentages of cells in the various compartments are computed using a gradient search method described by Bevington.     

Quantitative Description of Cell Cycle Kinetics Under Chemotherapy Utilizing Flow Cytometry

A discrete time cell cycle kinetics model is developed to account for the effects of cytotoxic chemotherapy, particularly including the existence of cells destined to die. A model structure is determined from related experiments, leaving key parameter values undetermined. These values are found by determining the best least squares fit of the predicted to the observed DNA distribution data at a series of time intervals. the numerical methods include separable least squares, linear inequality constrained least squares and the Gauss-Newton method. This approach is applied to an experiment in which the Ehrlich ascites tumour was given a single dose of bleomycin. the results include several different parameters, including the age response function and a time series of cell age and DNA distributions, which can be used as a basis for further treatment.