The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions

A deterministic/probabilistic model of the cell division cycle is analysed mathematically and compared to experimental data and to other models of the cell cycle. The model posits a random-exiting phase of the cell cycle and a minimum-size requirement for entry into the random-exiting phase. By design, the model predicts exponential "beta-curves", which are characteristic of sister cell generation times. We show that the model predicts "alpha-curves" with exponential tails and hyperbolic-sine-like shoulders, and that these curves fit observed generation-time data excellently. We also calculate correlation coefficients for sister cells and for mother-daughter pairs. These correlation coefficients are more negative than is generally observed, which is characteristic of all size-control models and is generally attributed to some unknown positive correlation in growth rates of related cells. Next we compare theoretical size distributions with observed distributions, and we calculate the dependence of average cell mass on specific growth rate and show that this dependence agrees with a well-known relation in bacteria. In the discussion we argue that unequal division is probably not the source of stochastic fluctuations in deterministic size-control models, transition-probability models with no feedback from cell size cannot account for the rapidity with which the new, stable size distribution is established after perturbation, and Kubitschek's rate-normal model is not consistent with exponential beta-curves.     

Modeling age-specific incidence of colon cancer via niche competition

Cancer development is a multistep process often starting with a single cell in which a number of epigenetic and genetic alterations have accumulated thus transforming it into a tumor cell. The progeny of such a single benign tumor cell expands in the tissue and can at some point progress to malignant tumor cells until a detectable tumor is formed. The dynamics from the early phase of a single cell to a detectable tumor with billions of tumor cells are complex and still not fully resolved, not even for the well-known prototype of multistage carcinogenesis, the adenoma-adenocarcinoma sequence of colorectal cancer. Mathematical models of such carcinogenesis are frequently tested and calibrated based on reported age-specific incidence rates of cancer, but they usually require calibration of four or more parameters due to the wide range of processes these models aim to reflect. We present a cell-based model, which focuses on the competition between wild-type and tumor cells in colonic crypts, with which we are able reproduce epidemiological incidence rates of colon cancer. Additionally, the fraction of cancerous tumors with precancerous lesions predicted by the model agree with clinical estimates. The correspondence between model and reported data suggests that the fate of tumor development is majorly determined by the early phase of tumor growth and progression long before a tumor becomes detectable. Due to the focus on the early phase of tumor development, the model has only a single fit parameter, the time scale set by an effective replacement rate of stem cells in the crypt. We find this effective rate to be considerable smaller than the actual replacement rate, which implies that the time scale is limited by the processes succeeding clonal conversion of crypts.     

Kinetic modelling of continuous submerged fermentation of cheese whey for single cell protein production

A mathematical model describing the kinetics of continuous production of single cell protein from cheese whey using Kluyveromyces fragilis was developed from the basic principles of mass balance. The model takes into account the substrate utilization for growth and maintenance and the effect of substrate concentration and cell death rate on the net cell growth and substrate utilization during the fermentation process. A lactose concentration below 1.91 g/L limited growth of yeast cells whereas a lactose concentration above 75 g/L inhibited the growth of the yeast. The model was tested using experimental data obtained from a continuous system operated at various retention times (12, 18 and 24 h), mixing speeds (200, 400 and 600 rpm) and air flow rates (1 and 3 vvm). The model was capable of predicting the effluent cell and substrate concentrations with R2 ranging from 0.95 to 0.99. The viable cell mass and lactose consumption ranged from 1.3 to 34.3 g/L and from 74.31% to 99.02%, respectively. A cell yield of 0.74 g cell/g lactose (close to the stoichiometric value of 0.79 g cell/g lactose) was achieved at the 12 h retention time-3 vvm air flow rate-600 rpm mixing speed combination. The total biomass output (viable and dead cells) at this combination was 37 g/L.     

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.     

Modes of Growth in Mammalian Cells

The increase of cell volume as a function of time was studied throughout the generation cycle in synchronous cultures of Chinese hamster cells using a Coulter aperture and a multichannel analyzer calibrated against known cell volumes. The experimental results were compared to a mathematical model of cell volume increase which considered the effect of the distribution of individual cell generation times on the progress of the population. Several modes of volume increase, including linear and exponential, were considered. The mean volume vs. time curve was rounded at the ends of the cycle even when linear growth was assumed. The experimental results show that cell volume increased in a smooth fashion as a function of time, with no discontinuities in rate detectable at periods when cells may have been undergoing metabolic shifts as, for example, through the phases associated with DNA synthesis, G₁, S, G₂. A statistical test on the comparison of the modal cell volume vs. time data to the predictions of linear and exponential growth models accepted both hypotheses within the resolution of these experiments. However, exponential growth was favored over linear growth in one cell line. Volume dispersion was almost constant with time in both sublines which is also consistent with exponential growth. Limitations of the electronic technique of volume measurement and indications for future experiments are discussed.     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.     

A mathematical model for analysis of the cell cycle in cell lines derived from human tumors

The growth of human cancers is characterised by long and variable cell cycle times that are controlled by stochastic events prior to DNA replication and cell division. Treatment with radiotherapy or chemotherapy induces a complex chain of events involving reversible cell cycle arrest and cell death. In this paper we have developed a mathematical model that has the potential to describe the growth of human tumour cells and their responses to therapy. We have used the model to predict the response of cells to mitotic arrest, and have compared the results to experimental data using a human melanoma cell line exposed to the anticancer drug paclitaxel. Cells were analysed for DNA content at multiple time points by flow cytometry. An excellent correspondence was obtained between predicted and experimental data. We discuss possible extensions to the model to describe the behaviour of cell populations in vivo.     

Cellular and rheological factors contributing to sickle cell microvascular occlusion

Sickle (HbSS) erythrocytes contain subpopulations that are heterogeneous in shape, size, and density and exhibit abnormal microcirculatory behavior. Their phthalate esters density distributions quantitatively distinguish subpopulations of HbSS cells from density profiles of normal (HbAA) erythrocytes. Filtration of HbSS cell suspensions, devoid of leukocytes, through 5-microns Nucleopore filters at constant flow rate (29.5 microliters/s) yields pressure-time curves that demonstrate deformability of the sickle cells to be several-fold less than equivalent suspensions of normal (HbAA) cells. For a cell flux of 6.43 X 10(5) cells/s, the rate of the rise of the pressure (Pi/t) following 1-2 s of the initial pressure reading indicates occlusion of the filter pores by the dense cell fraction. Rats exchange-transfused with human sickle (HbSS), normal (HbAA), or autologous rat erythrocytes were used to investigate the flow dynamics of these cells in the mesenteric microcirculation by intravital videomicroscopy. Time-averaged velocities of the autologous rat red cells in 16-30 microns (i.d.) arterioles ranged from 1.10 to 1.25 mm/s with varying flux and wall shear rates. Time-averaged velocities of the HbAA cells in single 15-35-microns arterioles ranged from 1.16 to 1.24 mm/s with wall shear rates similar to the estimates for the autologous cells. In contrast, sickle cells exhibited time-averaged velocities of 0.38-0.45 mm/s with lower wall shear rates in 10-35 microns single unbranched arterioles with three times less volumetric flux. In some arterioles, sickle RBCs with a high axial ratio of 3-4 and low deformability showed apparent adhesion to endothelial surfaces and occluded precapillary junctions or entry points for several seconds until dislodged by the higher flow velocity. Within single unbranched vessels or at microvascular bifurcations, sickle elliptocytes and sickle echinocytes with low deformability and axial ratios of 3-4 obstructed flow and exhibited residence times of 6-75 s at the sites of occlusion, thereby causing stasis and increasing the local apparent viscosity. Thus, both the in vitro and in vivo data demonstrate the rheological disequilibrium state induced by HbSS cells as they traverse artificial micropores or course through successive segments of the microcirculation. The specific tendency of dense cells with high axial ratio (ISCs) to manifest precapillary junctional blockade and prolonged residence times implicates this cell fraction in the initiation of microvascular occlusion.     

Cyclin and DNA Distributed Cell Cycle Model for GS-NS0 Cells

Mammalian cell cultures are intrinsically heterogeneous at different scales (molecular to bioreactor). The cell cycle is at the centre of capturing heterogeneity since it plays a critical role in the growth, death, and productivity of mammalian cell cultures. Current cell cycle models use biological variables (mass/volume/age) that are non-mechanistic, and difficult to experimentally determine, to describe cell cycle transition and capture culture heterogeneity. To address this problem, cyclins—key molecules that regulate cell cycle transition—have been utilized. Herein, a novel integrated experimental-modelling platform is presented whereby experimental quantification of key cell cycle metrics (cell cycle timings, cell cycle fractions, and cyclin expression determined by flow cytometry) is used to develop a cyclin and DNA distributed model for the industrially relevant cell line, GS-NS0. Cyclins/DNA synthesis rates were linked to stimulatory/inhibitory factors in the culture medium, which ultimately affect cell growth. Cell antibody productivity was characterized using cell cycle-specific production rates. The solution method delivered fast computational time that renders the model’s use suitable for model-based applications. Model structure was studied by global sensitivity analysis (GSA), which identified parameters with a significant effect on the model output, followed by re-estimation of its significant parameters from a control set of batch experiments. A good model fit to the experimental data, both at the cell cycle and viable cell density levels, was observed. The cell population heterogeneity of disturbed (after cell arrest) and undisturbed cell growth was captured proving the versatility of the modelling approach. Cell cycle models able to capture population heterogeneity facilitate in depth understanding of these complex systems and enable systematic formulation of culture strategies to improve growth and productivity. It is envisaged that this modelling approach will pave the model-based development of industrial cell lines and clinical studies.     

Design of a side-view particle imaging velocimetry flow system for cell-substrate adhesion studies

Experimental models that mimic the flow conditions in microcapillaries have suggested that the local shear stresses and shear rates can mediate tumor cell and leukocyte arrest on the endothelium and subsequent sustained adhesion. However, further investigation has been limited by the lack of experimental models that allow quantitative measurement of the hydrodynamic environment over adherent cells. The purpose of this study was to develop a system capable of acquiring quantitative flow profiles over adherent cells. By combining the techniques of side-view imaging and particle image velocimetry (PIV), an in vitro model was constructed that is capable of obtaining quantitative flow data over cells adhering to the endothelium. The velocity over an adherent leukocyte was measured and the shear rate was calculated under low and high upstream wall shear. The microcapillary channel was modeled using computational fluid dynamics (CFD) and the calculated velocity profiles over cells under the low and high shear rates were compared to experimental results. The drag force applied to each cell by the fluid was then computed. This system provides a means for future study of the forces underlying adhesion by permitting characterization of the local hydrodynamic conditions over adherent cells.     

Modelling cellular senescence as a result of telomere state

Telomeres in mammalian cells end in large duplex T loops. These loops protect the single-strand overhangs from degradation and/or interactions with signalling proteins. This protection is sometimes referred to as capping. At each cell division, telomeres shorten and there is a general consensus that telomere shortening triggers cell cycle exit. However, the exact mechanism by which telomere shortening causes cell cycle arrest is not known. Mathematical models of telomere shortening have been developed to help us understand the processes involved. Until now most models have assumed that the trigger for cell cycle arrest is the first telomere or a group of telomeres reaching a critically short length. However, there is evidence that cells stop cycling over a wide range of telomere lengths. This suggests that telomere length per se may not in fact be the trigger for cellular senescence. In this paper we develop a model which examines the hypothesis that uncapping of a telomere is the main trigger. By letting the probability of uncapping depend upon telomere length, we show that the hypothesized model provides a good fit to experimental data.     

Cells regulate their proliferation through alterations in transition probability

The proliferation of 3T3, 3T6 and SV3T3 cells was examined by time lapse cinephotography under a number of different growth conditions. It was found that the frequency distributions of intermitotic times of cells with widely different proliferation rates are qualitatively and quantitatively explained by the transition probability model of the cell cycle (Smith and Martin, '73). The behaviour of quiescent cells was characterized by very low values of the transition probability. No "out of cycle" or GO compartment of cells was detectable. From a consideration of these results and those in the literature it appears that the rate of cell proliferation is determined by the value of the "transition probability" (P), and that it is the biochemical manifestation of this parameter that regulates cell growth in vitro and in vivo.     

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.     

Bromodeoxyuridine labeling and flow cytometric identification of replicating Saccharomyces cerevisiae cells: lengths of cell cycle phases and population variability at specific cell cycle positions.

An immunofluorescent staining procedure has been developed to identify, with flow cytometry, replicating cells of Saccharomyces cerevisiae after incorporation of bromodeoxyuridine (BrdUrd) into the DNA. Incorporation of BrdUrd is made possible by using yeast strains with a cloned thymidine kinase gene from the herpes simplex virus. An exposure time of 4 min to BrdUrd results in detectable labeling of the DNA. The BrdUrd/DNA double staining procedure has been optimized and the flow cytometry measurements yield histograms comparable to data typically obtained for mammalian cells. On the basis of the accurate assessment of cell fractions in individual cell cycle phases of the asynchronously growing cell population, the average duration of the cell cycle phases has been evaluated. For a population doubling time of 100 min it was found that cells spend in average 41 min in the replicating phase and 24 min in the G2+M cell cycle period. Assuming that mother cells immediately reenter the S phase after cell division, daughter cells spend 65 min in the G1 cell cycle phase. Together with the single cell fluorescence parameters, the forward-angle light scattering intensity (FALS) has been determined as an indicator of cell size. Comparing different temporal positions within the cell cycle, the determined FALS distributions show the lowest variability at the beginning of the S phase. The developed procedure in combination with multiparameter flow cytometry should be useful for studying the kinetics and regulation of the budding yeast cell cycle.     

Retention of the mitochondrial probe rhodamine 123 in normal lymphocytes and leukemic cells in relation to the cell cycle

The cationic fluorochrome rhodamine 123 (R123) is specifically taken up by mitochondria of live cells where it is retained due to the mitochondrial transmembrane potential. After pulse exposure of human normal quiescent or proliferating lymphocytes, human lymphocytic leukemic MOLT cells, and mice leukemic L1210 cells to 10 micrograms/ml of R123, the dye release was studied using flow cytometry. Two distinct phases of R123 release, each following first-order kinetics, were apparent; the half-time of retention for the rapidly and slowly released fractions of R123 was 0.8-1.1 and 2.8-4.2 h, respectively. Simultaneous supravital cell staining with R123 and Hoechst 33342 made it possible to correlate retention of R123 with cell position in the cell cycle. No significant differences were observed in the rate of R123 release from cells in G1 vs S or vs G2 + M phases of the cycle. The data rule out a possibility that the release of R123 is due to periodic depolarization of the mitochondria in the cell as may be postulated by cell cycle models that assume a transient passage of cells through resting phase following division. The observed similar rates of R123 release regardless of cell type or cell cycle phase suggest that the factors affecting the exchange are similar in normal lymphocytes vs leukemic cells and unrelated to cell proliferation rate or phase of the cell cycle. Two distinct rates of R123 release indicate the presence of two kinds of binding sites differing in affinity to the dye.     

Cell-cycle-dependent protein accumulation by producer and nonproducer murine hybridoma cell lines: a population analysis

Single-cell rates of accumulation of cellular protein have been determined as a function of total protein content using flow cytometry and population balance equations for exponentially growing murine hybridoma cells in the individual G(1), S(1) and G(2) + M cell cycle phases. A novel flow cytometric technique for the identification of hybridoma cells in mitosis was developed and implemented. The data were obtained from a producer cell line which synthesizes and secretes high levels of monoclonal antibodies, and from a nonproducer clone which does not synthesize and secrete substantial amounts of antibody. The results indicate that the kinetics of single-cell protein accumulation in these two cell lines are considerably different. In particular, low protein content G(1) phase producer cells were characterized by a rate of protein accumulation which was approximately five times higher than the mean rate observed for higher protein content producer cells cycle phase. In contrast, the rate of accumulation of protein increased continuously with totalprotein content for the G(1) phase nonproducer cells. S phase hybridoma cells were characterized by a considerably lower rate of protein accumulation which did not vary much with protein content for either cell line. Finally, G(2) + M phase producer cells demonstrated a negative rate of protein accumulation which indicates that the rates of protein synthesis. It was hypothesized that these differences in total protein accumulation are caused by differences in monoclonal antibody accumulation. The distribution of rates suggests the need for a segregated approach to the modeling of the kinetics of antibody production in hybridomas.