Mathematical Determination of Cell Population Doubling Times for Multiple Cell Lines

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.     

Computing multiple cell kinetic properties from a single time point

New developments in experimental procedures have made it necessary to extend the theory for describing the movement of a population of cells and estimating the kinetic properties of the population. The new procedures are based on the use of fluorescent monoclonal antibodies to halogenated analogues of thymidine, which are incorporated as a label into cells during DNA synthesis. These populations may be examined by dual-parameter flow cytometry to discriminate between the labelled and unlabelled populations of cells and define their position within the DNA reproductive cycle. A particular need exists for a theory that can be used for measurements of tumors in which many cells are not actively cycling and only a single time point can be obtained. In order to develop a useful theory for evaluating the kinetic properties of the cells observed by these techniques, the standard methods of theoretical cell kinetics have been recast in a form that is amenable to the type of analysis demanded by these constraints and a novel method for the rapid analysis of the kinetic properties of the cell population is presented. The method is shown to yield a direct measurement for the population doubling time from a single time point as well as estimates for the transit times through each phase of the cell cycle. The method which is approximately linear is shown to be robust to the effects of different assumptions about the distribution of transit times as well as being insensitive to the effects of variation in the transit times of the cells. The methodology developed in this paper may also be used to examine other theoretical methods of computing kinetic properties.     

Effect of serum on the growth of Balb oT3 A31 mouse fibroblasts and an SV40-transformed derivative.

The effect of serum on the growth properties of non-transformed Balb 3T3 A31 and SV40-transformed Balb 3T3 A31 was studied. The concentration of serum in the growth medium of non-transformed cells had little effect on the initial population doubling time, but did regulate the cell density at which the population became quiescent in G1. The doubling time of transformed cells, however, was increased significantly as the concentration of serum was decreased below 4%. This effect on the growth of transformed cells was seen at serum concentrations so low that non-transformed cells did not complete one population doubling. Flow microfluorometric analysis of these populations indicated that the primary effect of different serum concentrations on the non-transformed cells was to modulate the average residence time in G1, whereas, all the cell cycle phases of the transformed cells were affected by serum. At saturation densities, the non-transformed cells became quiescent in G1, but the transformed cells still traversed the cell cycle and their saturation density appeared to be a balance between cell production and cell death occurring primarily in the G1 phase of the cell cycle.     

Cell elongation and division probability during the Escherichia coli growth cycle.

A new method is presented for determining the growth rate and the probability of cell division (separation) during the cell cycle, using size distributions of cell populations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific size during its life history. This method was used to analyze cell length distributions of six cultures of Escherichia coli, for which doubling times varied from 19 to 125 min. The results for each culture are in good agreement with a single model of growth and division kinetics: exponential elongation of cells during growth phase of the cycle, and normal distributions of length at birth and at division. The average value of the coefficient of variation was 13.5% for all strains and growth rates. These results, based upon 5,955 observations, support and extend earlier proposals that growth and division patterns of E. coli are similar at all growth rates and, in addition, identify the general growth pattern of these cells to be exponential.     

A COMPARATIVE STUDY OF THE REPOPULATING POTENTIAL OF GRAFTS FROM VARIOUS HAEMOPOIETIC SOURCES: CFU REPOPULATION

The growth rate of the CFU populations in spleens and femora has been studied in irradiated mice injected with cell suspensions, containing equivalent number of CFU, from various sources. The doubling times are shown to be dependent upon the source of the cells. Grafts of bone marrow, spleen and foetal liver cells produced doubling times in the spleen of approximately 25, 19 and 16 hr respectively. Grafts of marrow-derived and spleen-derived spleen colony cells both gave rise to CFU doubling times lower than those of the corresponding primary grafts (approx. 33 and 26 hr respectively in the spleen). In the case of both bone marrow and spleen grafts the CFU population growth was shown to be independent of the size of the graft. A hypothesis is advanced which invokes at least a dual population of CFU, having different doubling times, different seeding capacities in the haemopoietic tissues following i.v. injection and present in different proportions in the various haemopoietic tissues.     

VARIATIONS IN GROWTH AND CELL CYCLE TIME OF ARTERIAL SMOOTH MUSCLE CELLS CULTURED AT LOW DENSITY

The population kinetics of cultured rat arterial smooth muscle cells replated at various low densities were studied by direct counting and observation of the cells. Population doubling time decreases with increasing initial density of the culture. These variations in population doubling times depend on both the variation in the percentage of quiescent cells and on the variation of the mean cell cycle time of non-quiescent cells.     

Estimation of cell cycle parameters from double labeling experiments

Double labeling of cell populations with radioactive thymidine yields two types of differently labeled nuclei. Their numbers and the number of unlabeled nuclei can be used to estimate doubling times, T, and S-phase lengths, S. As of yet, such estimations have been performed either for stationary populations in which proliferation and losses are in balance, or for exponentially growing populations in which all cells have the same cycle duration. We calculate S and T for the more general type of cell population with arbitrarily distributed frequencies of cycle durations. The calculations do not require more mathematical or computational effort. We obtain three main results: (i) The estimation of T and S does not require explicit knowledge of the frequency distribution of cycle durations; (ii) in particular, equivalent estimates for T and S are obtained for both types of growing cell populations without losses, one with arbitrarily distributed cycle durations and one with the same cycle duration for all cells; and (iii) for small labeling indices, the estimate for S from the general model approaches the S-phase length of a stationary population and the estimate for T from the general model approaches the generation time of a stationary population, multiplied by the constant factor 1n(2). These relationships are valuable tools for reinterpreting results derived under the assumption of stationarity, which are considerably easier to obtain.     

Self-Cycling Fermentation Applied to Acinetobacter calcoaceticus RAG-1

The self-cycling fermentation (SCF) technique was applied to a culture of Acinetobacter calcoaceticus RAG-1. This method was shown to result in synchronization of the cells, achieving a 77% improvement in cell synchrony over that of the batch case. Cellular occurrences, averaged out by asynchronous batch cultures, were magnified by the temporal alignment of metabolic events brought about by the synchronization associated with SCFs. The cell population doubled only once per cycle, thus establishing an equality between cycle time and doubling time. Parameters of interest were biomass concentration, total bioemulsifier (emulsan) production, cycle time, and residual carbon concentration. Cycle-to-cycle variation of these parameters was, in most cases, insignificant. Repeatability of doubling time estimates (based on 95% confidence intervals) was roughly 7 to 10 times better between cycles in an SCF than between batch replicates. The carbon substrate was completely utilized in all cases in which it was measured, giving this technique an advantage over chemostat-type fermentations. The dissolved-oxygen profiles monitored throughout a cycle were found to be repeatable. A characteristic shape, which can be related to the growth of the organism, was associated with each carbon source. The specific emulsan productivity of SCFs was found to be approximately 50 times greater than that of the batch process and 2 to 9 times greater than that of the chemostat, depending on the dilution rate considered. With respect to specific emulsan production, a 25-fold improvement over that in an immobilized cell system recently introduced was obtained. Thus, SCFs are a viable alternative to established fermentation techniques.     

Regrowth kinetics of cells from different regions of multicellular spheroids of four cell lines

A basic understanding of the recruitment of quiescent tumor cells into the cell cycle would be an important contribution to tumor biology and therapy. As a first step in pursuing this goal, we have investigated the regrowth kinetics of cells from different regions in multicellular spheroids of rodent and human origin. Cells were isolated from four different depths within the spheroids using a selective dissociation technique. The outer cells were proliferating and resumed growth after replating with a 0-8-hour lag period, similar to cells from exponentially growing monolayers. With increasing depth of origin, the lag periods prior to regrowth increased to 2-3 times the monolayer doubling time; cells from plateau-phase monolayers showed a lag period of 1-1.5 times the doubling period. After resuming growth, all cells of a given cell line grew with the same doubling time and achieved the same confluency level. The inner spheroid cells and cells from plateau-phase monolayers had reduced clonogenic efficiencies. The inner cells were initially 1.5-3 times smaller than the outer cells, but began to increase in volume within 4 hours of replating. The fractions of S-phase cells were greatly decreased with increasing depth of origin in the spheroids; there were long delays prior to S-phase recovery after plating, to a maximum of 1-1.5 times the normal doubling time. These results suggest that those quiescent cells from spheroids and monolayers which are able to reenter the cell cycle are predominantly in the G1-phase. However, quiescent cells from the innermost spheroid region require approximately twice as long to enter normal cell cycle traverse as cells from plateau-phase monolayers. The selective dissociation method can isolate very pure populations of proliferating and quiescent cells in a rapid and nonperturbing manner; this system will be valuable in further characterizing quiescent cells from spheroids.     

Proliferation kinetics of endothelial and tumour cells in three mouse mammary carcinomas

Abstract. An autoradiographic study of three corded mouse tumours is reported. The proliferation characteristics of both tumour cells and endothelial cells were studied. The doubling time of these three tumours differed by a factor of 2.6 but there was only a small difference in the intermitotic time. All three tumours showed a very high cell loss factor (˜0.80) and the differences in growth rate resulted mainly from differences in the growth fraction .
The endothelial cell proliferation rates differed markedly in the three tumours, with labelling indices ranging from 18% in the faster tumours to 4.5% in the slowest. The potential doubling times for endothelium, calculated from these values, were much slower than the tumour cell cycle time or the tumour potential doubling time, but were two to four times faster than the volume doubling time of the tumour.
It appears likely that the endothelial proliferation rate influences the growth fraction, but similar high cell loss factors can occur in tumours with a four-fold difference in endothelial cell production rates. Inadequate branching of blood vessels seems likely to be at least as important as inadequate production of endothelial cells. It is not possible to determine whether slow tumour cell production evokes a slower endothelial growth or vice versa.     

Neutron irradiation of human melanoma cells

The biological characteristics and in vitro radiosensitivity of melanoma cells to thermal neutrons were investigated as a guide to the effectiveness of boron neutron capture therapy. Plateau phase cultures of three human malignant melanoma-established cell lines were examined for cell density at confluence, doubling time, cell cycle parameters, chromosome constitution, and melanin content. Cell survival dose-response curves, for cells preincubated in the presence or absence of p-boronophenylalanine. HCl (10B1-BPA), were measured over the dose range 0.6-8.0 Gy (N + gamma). The neutron fluence rate was 2.6 x 10(9) n/cm2/s and the total dose rate 3.7 Gy/h (31% gamma). Considerable differences were observed in the morphology and cellular properties of the cell lines. Two cell lines (96E and 96L) were amelanotic, and one was melanotic (418). An enhanced killing for neutron irradiation was found only for the melanotic cells after 20 h preincubation with 10 micrograms/ml 10B1-BPA. In view of the doubling times of the cell lines of about 23 h (96E and 96L) or of 36 h (418), it seems likely that an increased boron uptake, and hence increased radiosensitivity, might result if the preincubation period with 10B1-BPA is extended to several hours longer than the respective cell cycle times.     

Modelling complex populations formed by proliferating,quiescent and quasi-quiescent cells: application to plant root meristems

A proliferating population of cells may be considered complex when its proliferative or growth fraction P is lower than 1 and/or when it is formed by subpopulations with different mean cycle times. The present paper shows that in such complex populations exponential growth is consistent with a steady-state distribution of cells. Obviously, when P=1 then cell distribution is only a function of cell age. An analytical model has been developed to study complex populations including both quiescent fractions formed by cells with unreplicated genome (G(0) cells) and cells with fully duplicated chromosomes (Q(2) cells). The model also considers those quasi-quiescent cells in their last transit through G(1) and S (Q(1) and Q(s) cells) before becoming quiescent. In order to solve the difficulties of a direct analysis of the whole population, its kinetic parameters have been obtained by studying the negative exponential distribution of two subpopulations: one formed by the proliferating cells and another formed by the quasi-quiescent cells. Additionally, the model could be applied when quiescence is initiated at any other cycle phase different from G(1) and G(2), for instance, cells in the process of replicating their DNA or being at any other mitotic phases. The utility of the method was illustrated in populations which constitute the root meristems of both Allium cepa L. and Pisum sativum L. Three facts should be stressed: (1) the method seems to be rather powerful because it can be carried out from different sets of experimentally measured parameters; (2) the rate of division and, therefore, the population doubling time can be easily estimated by this method; and (3) it also allows the determination of the amount of cells that had become quiescent either before they had replicated their DNA (G(0)) or after having completed their replication (Q(2)), as well as those quasi-quiescent cells which are progressing throughout their last pre-replicative and replicative periods (thus Q(1) and Q(s), respectively).     

The generation and characterization of a radiation-resistant model system to study radioresistance in human breast cancer cells.

To systematically study the selection of radioresistant cells in clinically advanced breast cancer, a model system was generated by treating MDA-MB231 breast cancer cells with fractionated gamma radiation. A clonogenic assay of the surviving cell populations showed that 2-6 Gy per fraction resulted in a rapid selection of radioresistant populations, within three to five fractions. Irradiation with additional fractions after this initial increase did not increase the radioresistance of the surviving population significantly. Doses of 0.5 and 8 Gy per fraction were not effective in selecting radioresistant cells. To further determine the cause of the changes in radiosensitivity, 15 clones were isolated from the cell populations treated with 40 or 60 Gy with 2 or 4 Gy per fraction, respectively, and were analyzed for radiosensitivity. The average D(10) for these clones was 6.75 +/- 0.36 Gy, which was higher than that for the parental cell population (D(10) = 6.0 +/- 0.2 Gy). The operation of cell cycle checkpoints and the doubling time were similar for both the nonirradiated parental population and the isolated radioresistant subclones. In contrast, a decrease in the apoptotic potential was correlated (r = 0.7, P < 0.01) with increased survival after irradiation, suggesting that apoptosis is an important factor in determining radioresistance under our experimental conditions. We also isolated several subclones from the nonirradiated parental cell population and analyzed them to determine their radiosensitivity after fractionated irradiation. Ten fractions of 4 Gy (40 Gy in total) did not result in a significant increase in the radioresistance of these subclones compared to the irradiated cell populations. The possible mechanisms of the increased radioresistance after fractionated irradiation are discussed.     

Early and late volume changes during erythroid differentiation of cultured Friend leukemic cells.

Friend erythroleukemic cells (FLC) can be induced to differentiate in vitro by addition of dimethylsulfoxide (DMSO). We have studied the kinetics of induction by measuring cell volume, volume coefficient of variation and cell doubling time. Two distinct volume changes (early and late) are observed after the addition of the inducing agent. The early change occurs after ten hours and consist of a 10-20% decrease in volume compared to an untreated control population. This shift persists for two days and its magnitude is proportional to both the concentration of DMSO and the number of differentiated cells seen on day 5. FLC lines which induce weakly or not all with DMSO exhibit a reduced or absent early volume shift. Inclusion of a local anaesthetic in the culture prevents the appearance of differentiated cells and also counteracts the early volume shift. The exact time of the early volume change is a function of cell growth rate and appears to be cell cycle related. Synchronized cell populations exposed to DMSO during G2 and S phase undergo one round of mitosis before expression of the volume change whereas cells in G2-M express the change only after a second mitosis. A later, more gradual decrease in volume is observed in those cultures which begin to produce hemoglobin. It occurs after approximately five doubling times and coincides with the first appearance of hemoglobin-containing cells. Volume distribution parameters indicate that only a proportion of the population becomes smaller in size.     

INFLUENCE OF ENVIRONMENTAL FACTORS AND POPULATION COMPOSITION ON THE TIMING OF CELL DIVISION IN THALASSIOSIRA FLUVIATILIS (BACILLARIOPHYCEAE) GROWN ON LIGHT/DARK CYCLES1

Cell division patterns in Thalassiosira fluviatilis grown in a cyclostat were analyzed as a function of temperature, photoperiod, nutrient limitation and average cell size of the population. Typical cell division patterns in populations doubling more than once per day had multiple peaks in division rate each day, with the lowest rates always being greater than zero. Division bursts occurred in both light and dark periods with relative intensities depending on growth conditions. Multiple peaks in division rate were also found, when population growth rates were reduced to less than one doubling per day by lowering temperature, nutrients, or photoperiod and the degree of division phasing was not enhanced. Temperature and nutrient limitation shifted the timing of the major division burst relative to the light/dark cycle. Average cell volume of the inoculum was found to be a significant determinant of the average population growth rate and the timing and magnitude of the peaks in division rate. The results are interpreted in the context of a cell cycle model in which generation times are “quantized” into values separated by a constant time interval.     

Effect of serum step-down on protein metabolism and proliferation kinetics of NHIK 3025 cells

Human NHIK 3025 cells growing exponentially in 30% or 3% serum had population doubling times of 19.1 and 27.6 hours, respectively. These values were equal to the calculated protein doubling times (17.6 and 26.5 hours, respectively), showing that the cells were in balanced growth at both serum concentrations. Stepdown from 30% to 3% serum reduced the rate of protein synthesis within 1–2 hours, from 5.7% hour to 4.3% hour, while the rate of protein degradation was unchanged (1.7%/hour). In cells synchronized by mitotic selection from an exponentially growing population, the median cell cycle durations in 30% and 3% serum were 17.2 and 23.6 hours, respectively, which were also in good agreement with the protein doubling times. The median G₁ durations were 7.1 and 9.6 hours, respectively. Thus the duration of G₁ relative to the total cell cycle duration was the same in the two cases. Complete removal of serum for a period of 3 hours resulted in a 3-hour prolongation of the cell cycle regardless of the time after mitotic selection at which the serum was removed. For synchronized cells, the rate of entry into both the S phase and into the subsequent cell cycle were reduced in 3% serum as compared to 30% serum, the former rate being significantly greater than the latter at both serum concentrations. Our results thus indicate that these cells are continuously dependent upon serum throughout the entire cell cycle.     

First generation synchrony of isolated Hyphomicrobium swarmer populations

A method is described for obtaining synchronously growing swarmer cell populations of Hyphomicrobium sp. strain B-522. This was accomplished by isolating young swarmers from random cultures by centrifugation and filtration. Cell multiplication occurred during 38% of the growth cycle in populations synchronized in this manner. Observations were made of the changes in cellular morphology which occurred during the growth cycle. Of the 14.25 h required for the doubling in cell numbers, an average of 5 h passed before the swarmer cells began to develop their hyphae. This time varied over a range of 10 h. The time interval between the beginning of hyphal development and the beginning of bud formation was 3.5 to 4.5 h. The maturation of the first buds and their separation from the mother cells were completed in 5.5 h. The duration of these steps is compared to those measured previously in agar slide cultures.     

Evaluation of flow cytometric methods for determining population potential doubling times using cultured cells

Various methods have been proposed for determining the potential doubling times (Tpot) of mammalian cell populations by using flow cytometric techniques after labeling the cells with bromodeoxyuridine (BrdUrd). We show here that, in a well-defined in vitro system where multiple time measurements are possible, all the methods give similar results that are close to the true population doubling time. Of ultimate interest, however, is the accuracy of determination of Tpot from a single time point. In this paper we compare the accuracy and precision of the methods in making such determinations at different times after labeling. The relative movement (RM) of BrdUrd-labeled cells that have not divided at the time of assay allows for computation of the length of S phase (Ts). The precision of estimation of Ts was enhanced when a quantity, v (a function of the fraction of BrdUrd-labeled divided and the fraction of BrdUrd-labeled undivided cells), was used to estimate the initial intercept of RM. Furthermore, calculation of Tpot from the formula, Tpot = ln(2) Ts/v, gave values closest to the observed population doubling time. It is suggested that the use of RM with v be the analytical method of choice for the calculation of Tpot from single time-point observations, preferably made at times between the length of the G2 and M phases (TG2M) and Ts.