Proliferation controls in a linear growth system: theoretical studies of cell division in the cartilage growth plate

The columnar arrangement of dividing cells in the epiphyseal cartilage plates of growing bones provides a model of a linear proliferation system. One factor which determines the rate of cell production, and hence the rate of growth, is the size of the proliferating population. In this one dimensional system this size is equal to the length of the proliferation zone. Two possible mechanisms for a differentiation control that sets a limit to the length of this zone have been tested in computer simulations. While a diffusion gradient control is consistent with cell kinetic measurements a division limit based on an inheritable growth substance is shown to require further development before the model fits experimental data.Cell division in the columns produces linear clones of cells. If the final length of a bone is set by a limit on the number of divisions that the cartilage stem cells can make, then the number of cells per clone is crucial in determining overall bone growth. The parameters that affect linear clone size have been investigated in computer simulations. Clone size depends largely on the relative division rate of stem cells to proliferation zone cells — but the data on stem cell division rates are generally unreliable.The analysis could be applied to other linear proliferating systems.     

Exponential pattern of cell age distribution in dividing cells of plant meristems

In order to determine the pattern of cell age distribution in proliferating cells of Allium cepa roots we have measured by cytophotometry two cell size parameters, protein content and surface area projection, in cells that correspond to the entire proliferating population or only to the ana-telophase subpopulation. The size values of ana-telophase cells have been employed to construct theoretical size distributions for the entire proliferating cell population of the root meristem by assuming either a uniform or an exponential cell age distribution. Statistical comparison of theoretical distributions with the experimental one rules out a uniform cell age distribution and strongly favours an exponential age distribution similar to that found in bacteria.     

Dependency of size of Saccharomyces cerevisiae cells on growth rate.

The mean size and percentage of budded cells of a wild-type haploid strain of Saccharomyces cerevisiae grown in batch culture over a wide range of doubling times (tau) have been measured using microscopic measurements and a particle size analyzer. Mean size increased over a 2.5-fold range with increasing growth rate (from tau = 450 min to tau = 75 min). Mean size is principally a function of growth rate and not of a particular carbon source. The duration of the budded phase increased at slow growth rates according to the empirical equation, budded phase = 0.5 tau + 27 (all in minutes). Using a recent model of the cell cycle in which division is thought to be asymmetric, equations have been derived for mean cell age and mean cell volume. The data are consistent with the notion that initiation of the cell cycle occurs at "start" after attainment of a critical cell size, and this size is dependent on growth rate, being, at slow growth rates, 63% of the volume of fast growth rates. Previous reports are reanalyzed in the light of the unequal division model and associated population equations.     

Different growth patterns of a cancer cell population as a function of its starting growth characteristics: analysis by mathematical modelling

The growth kinetics of a cancer cell population as a function of the total number of cells and the proportion of proliferating and resting cells at the beginning of the growth has been analysed by a mathematical model. The model takes into account the processes of cell division, death and transition from proliferation to rest and backwards. It is shown that a single cell population growing under the same environmental conditions has an extremely broad spectrum of growth patterns. The whole multiplicity of possible growth patterns has been determined by the inherent cellular growth characteristics of the population, while the growth pattern actually realized of the variety of growth curves depends on the total number of cells and the proportion of proliferating and resting cells at the initial moment of growth. The model is shown to provide a good prediction of experimentally measured kinetics of regrowth of tumour cells subcultured after various times of the growth in unfed cultures, and the kinetics of tumour cell growth after severe hypoxia. The role of cell transitions between proliferating and resting stages in the problem of growth control is discussed.     

Growth Pattern of Single Fission Yeast Cells Is Bilinear and Depends on Temperature and DNA Synthesis

Cell growth and division have to be tightly coordinated to keep the cell size constant over generations. Changes in cell size can be easily studied in the fission yeast Schizosaccharomyces pombe because these cells have a cylindrical shape and grow only at the cell ends. However, the growth pattern of single cells is currently unclear. Linear, exponential, and bilinear growth models have been proposed. Here we measured the length of single fission yeast cells with high spatial precision and temporal resolution over the whole cell cycle by using time-lapse confocal microscopy of cells with green fluorescent protein-labeled plasma membrane. We show that the growth profile between cell separation and the subsequent mitosis is bilinear, consisting of two linear segments separated by a rate-change point (RCP). The change in growth rate occurred at the same relative time during the cell cycle and at the same relative extension for different temperatures. The growth rate before the RCP was independent of temperature, whereas the growth rate after the RCP increased with an increase in temperature, leading to clear bilinear growth profiles at higher temperatures. The RCP was not directly related to the initiation of growth at the new end (new end take-off). When DNA synthesis was inhibited by hydroxyurea, the RCP was not detected. This result suggests that completion of DNA synthesis is required for the increase in growth rate. We conclude that the growth of fission yeast cells is not a simple exponential growth, but a complex process with precise rates regulated by the events during the cell cycle.     

RADIATION EFFECTS ON THE GROWTH RATE AND CELL POPULATION KINETICS OF ACTIVELY GROWING AND DORMANT ROOTS OF TRADESCANTIA PALUDOSA

Actively growing and dormant roots of Tradescantia paludosa were exposed to x-rays to compare the radiosensitivity of an actively proliferating tissue with that of one which is not active but is potentially proliferative. The level of effect was ascertained by the degree of change in the rate of root growth 4 days after exposure. Cell population kinetics were measured in control and in irradiated roots to determine whether or not a change was produced either in the number of proliferating cells or in the mitotic cycle duration which was sufficient to explain the altered rate of root growth. Nuclear volumes were also measured to provide an estimate of the relative total target size in actively growing vs. dormant roots. Tritiated thymidine was used to measure the cycle duration and the proportion of cells synthesizing DNA. The results showed that 184 and 305 r respectively were required to reduce the linear root growth rate to 37 per cent of that of the control for actively growing and dormant roots. Mitotic cycle duration, measured 4 days after x-ray exposure, was the same as in the control. The number of proliferating cells, however, was reduced. The rate of cell production in the irradiated roots was reduced to approximately one-half that of the controls. The average nuclear volumes of active and dormant roots were 733 and 491 µ³ respectively; thus the difference in the number of roentgens required to reduce growth to 37 per cent of that of the control can be attributed to the different average nuclear volumes. Therefore, the experiments suggest that part if not most of the differences in sensitivity between an actively dividing and an essentially non-dividing meristematic cell population resides in their different average nuclear volumes. Thus the law of Bergonie and Tribondeau needs to be reinterpreted, since the basic reason for the differences is secondary to whether or not the meristematic cells are proliferating.     

Effects of zinc on cell proliferation and proteoglycan characteristics of epiphyseal chondrocytes

Zinc has been postulated as an important nutritional factor involved in growth promotion; however, the cellular mechanisms involved in the effects of zinc on linear growth remain to be elucidated. This study was conducted to evaluate the effects of zinc on the proliferation rate of epiphyseal growth plate chondrocytes and on the structural characteristics of the proteoglycans synthesized by these cells. For these purposes, hypertrophic and proliferating chondrocytes were isolated from the tibiae of 1- and 5-week-old chickens, respectively. Chondrocytes were cultured under serum-free conditions and primary cultures were used. The results showed that zinc stimulated proliferation by 40-50% above the baseline in the case of proliferating chondrocytes, but it had no effect on hypertrophic chondrocytes. Zinc had neither any effects on mean charge density of proteoglycans synthesized by hypertrophic chondrocytes nor in their hydrodynamic size. In contrast, zinc induced an increase in mean charge density and a decrease of hydrodynamic size of proteoglycans synthesized by proliferating chondrocytes. In both cell types zinc had no effect on the composition and hydrodynamic size of the glycosaminoglycan chains. The increased ability of proliferating chondrocytes cultured in the presence of zinc to synthesize 3'-phosphoadenosine 5'-phosphosulfate (PAPS) could be explained by the induction of enzymes participating in the sulfation pathway of proteoglycans. Therefore, the increase in mean charge density of proteoglycans observed in this study may be explained by an increase of the degree of sulfation of proteoglycan molecules. We speculate that the effect of zinc on linear growth may be explained at a cellular level by: a) an increase in proliferation rates of proliferating chondrocytes, and b) increased synthesis of highly charged proteoglycan molecules which decreases mineralization.     

Using a stem cell and progeny model to illustrate the relationship between cell cycle times of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them

There is increasing evidence that the growth of human tumours is driven by a small proportion of tumour stem cells with self-renewal properties. Multiplication of these cells leads to loss of self-renewal and after division for a finite number of times the cells undergo programmed cell death. Cell cycle times of human cancers have been measured in vivo and shown to vary in the range from two days to several weeks, depending on the individual. Cells cultured directly from tumours removed at surgery initially grow at a rate comparable to the in vivo rate but continued culture leads to the generation of cell lines that have shorter cycle times (1–3 days). It has been postulated that the more rapidly growing sub-population exhibits some of the properties of tumour stem cells and are the precursors of a slower growing sub-population that comprise the bulk of the tumour. We have previously developed a mathematical model to describe the behaviour of cell lines and we extend this model here to describe the behaviour of a system with two cell populations with different kinetic characteristics and a precursor–product relationship. The aim is to provide a framework for understanding the behaviour of cancer tissue that is sustained by a minor population of proliferating stem cells.     

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.     

Cell size homeostasis: Metabolic control of growth and cell division

Joint regulation of growth rate and cell division rate determines cell size. Here we discuss how animal cells achieve cell size homeostasis potentially involving multiple signaling pathways converging at metabolic regulation of growth rate and cell cycle progression. While several models have been developed to explain cell size control, comparison of the two predominant models shows that size homeostasis is dependent on the ability to adjust cellular growth rate based on cell size. Consequently, maintenance of size homeostasis requires that larger cells can grow slower than small cells in relative terms. We review recent experimental evidence showing that such size adjustment occurs primarily at or immediately before the G1/S transition of the cell cycle. We further propose that bidirectional feedback between growth rate and size results in cell size sensing and discuss potential mechanisms how this may be accomplished.     

Developmental Regulation of Nucleolus Size during Drosophila Eye Differentiation

When cell cycle withdrawal accompanies terminal differentiation, biosynthesis and cellular growth are likely to change also. In this study, nucleolus size was monitored during cell fate specification in the Drosophila eye imaginal disc using fibrillarin antibody labeling. Nucleolus size is an indicator of ribosome biogenesis and can correlate with cellular growth rate. Nucleolar size was reduced significantly during cell fate specification and differentiation, predominantly as eye disc cells entered a cell cycle arrest that preceded cell fate specification. This reduction in nucleolus size required Dpp and Hh signaling. A transient enlargement of the nucleolus accompanied cell division in the Second Mitotic Wave. Nucleoli continued to diminish in postmitotic cells following fate specification. These results suggest that cellular growth is regulated early in the transition from proliferating progenitor cells to terminal cell fate specification, contemporary with regulation of the cell cycle, and requiring the same extracellular signals.     

Continuous and long-term volume measurements with a commercial Coulter counter

We demonstrate a method to enhance the time resolution of a commercial Coulter counter and enable continuous and long-term cell size measurements for growth rate analyses essential to understanding basic cellular processes, such as cell size regulation and cell cycle progression. Our simple modifications to a commercial Coulter counter create controllable cell culture conditions within the sample compartment and combine temperature control with necessary adaptations to achieve measurement stability over several hours. We also wrote custom software, detailed here, to analyze instrument data files collected by either this continuous method or standard, periodic sampling. We use the continuous method to measure the growth rate of yeast in G1 during a prolonged arrest and, in different samples, the dependency of growth rate on cell size and cell cycle position in arrested and proliferating cells. We also quantify with high time resolution the response of mouse lymphoblast cell culture to drug treatment. This method provides a technique for continuous measurement of cell size that is applicable to a large variety of cell types and greatly expands the set of analysis tools available for the Coulter counter.     

Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Experimental rejection of a nonadaptive explanation for increased cell size in Escherichia coli.

Populations of Escherichia coli that have been serially propagated for thousands of generations in glucose minimal medium show heritable increases in both cell size and growth rate. We sought to test the hypothesis that the increased cell size of the derived genotypes could be explained solely by their faster growth. The regression of cell size on growth rate differed significantly between populations having ancestral and derived genotypes, with the latter producing larger cells over almost the entire range of growth rates. Thus, the physiological coupling between cell size and growth rate has been evolutionarily altered.     

Ovariectomy causes cell proliferation and matrix synthesis in the growth plate cartilage of the adult rat

The in vivo effects of ovariectomy in rats have been studied on cell proliferation and matrix synthesis in the growth plate cartilage by assessing immunohistochemically the levels of proliferating cell nuclear antigen and chondroitin sulphate proteoglycan(s). The serum levels of insulin-like growth factor-I and growth hormone were also measured by radioimmunoassay procedures. At 5 weeks after ovariectomy, the serum levels of the growth factor were significantly higher than those in sham-operated rats. In contrast, the level of growth hormone was lower. The nuclear staining of proliferating cell nuclear antigen was generally seen in the zone of proliferative chondrocytes from both groups of rats. Whereas almost all chondrocytes in the proliferative zone of ovariectomized rats expressed proliferating cell nuclear antigen immunoreactivity, fewer did so in that of the sham rats. Quantitative image analysis by ACAS 570 laser cytometry demonstrated that the n uclear antigen-positive sites in ovariectomized rats had significantly higher integrated values (staining intensity), areas and perimeters than those in sham rats. In addition, the number of chondroitin sulphate proteoglycan-immunoreactive cells in the proliferative chondrocytes was also higher in ovariectomized rats than in sham ones. These results suggest that ovariectomy significantly stimulates the cell proliferation and matrix synthesis in the growth plate cartilage, probably through the higher serum level of insulin-like growth factor-I.     

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 weight kinetics simulation in chemostat and batch culture of the rhodophyte Porphyridium cruentum

Experimental observations of cell size variations in the proliferating rhodophyte Porphyridium cruentum cultured under fully controlled conditions showed significant decreases from inoculation to a steady state in the chemostat with 0.23 d(-1) dilution rate and to a minimum in batch, dropping in size by ratios of over 10. To numerically simulate these variations, we assumed that the cell is made up of two categories of components that behave differently during the interphase and mitosis. These have been called essential (EC) and accessory (AC) components. It is assumed that the cell divides once the EC have doubled in size, regardless of the AC's state. The experimental cell weight time courses were correctly simulated by a model of synchronous cell kinetics based on these assumptions. The EC's specific growth rate was 1.5 times that of the whole cell, when no limitation occurred. The increase in cell weight observed during batch cultures after nutrient exhaustion was suitably simulated by assuming that EC growth stops when a limiting nutrient is exhausted. Several parameters characterizing the cell kinetics were defined, particularly the minimum minimorum EC or cell weight (26 and 15 pg for chemostat and batch, respectively), which was influenced by the cultivation method, and the maximum whole cell weight (224 to 244 pg), which depended on the inoculum's age. The influence of culture conditions on the amount of essential and accessory components contained in a cell was examined. A new approach was developed with respect to these compartments to determine the most suitable strategy and conduct a predictive approach for valuable molecule production.     

Evidence for a size-sensing mechanism in animal cells

Continuously proliferating cells exactly double their mass during each cell cycle. Here we have addressed the controversial question of if and how cell size is sensed and regulated. We used erythroblasts that proliferate under the control of a constitutively active oncogene (v-ErbB) or under the control of physiological cytokines (stem cell factor, erythropoietin and v-ErbB inhibitor). The oncogene-driven cells proliferated 1.7 times faster and showed a 1.5-fold increase in cell volume. The two phenotypes could be converted into each other 24 h after altering growth factor signalling. The large cells had a higher rate of protein synthesis, together with a shortened G1 phase. Additional experiments with chicken erythroblasts and mouse fibroblasts, synchronized by centrifugal elutriation, provided further evidence that vertebrate cells can respond to cell size alterations (induced either through different growth factor signalling or DNA synthesis inhibitors) by compensatory shortening of the subsequent G1 phase. Taken together, these data suggest that an active size threshold mechanism exists in G1, which induces adjustment of cell-cycle length in the next cycle, thus ensuring maintenance of a proper balance between growth and proliferation rates in vertebrates.