A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

Combining Gompertzian growth and cell population dynamics

A two-compartment model of cancer cells population dynamics proposed by Gyllenberg and Webb includes transition rates between proliferating and quiescent cells as non-specified functions of the total population, N. We define the net inter-compartmental transition rate function: Phi(N). We assume that the total cell population follows the Gompertz growth model, as it is most often empirically found and derive Phi(N). The Gyllenberg-Webb transition functions are shown to be characteristically related through Phi(N). Effectively, this leads to a hybrid model for which we find the explicit analytical solutions for proliferating and quiescent cell populations, and the relations among model parameters. Several classes of solutions are examined. Our model predicts that the number of proliferating cells may increase along with the total number of cells, but the proliferating fraction appears to be a continuously decreasing function. The net transition rate of cells is shown to retain direction from the proliferating into the quiescent compartment. The death rate parameter for quiescent cell population is shown to be a factor in determining the proliferation level for a particular Gompertz growth curve.     

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

Dissociation between early loss of actin fibres and subsequent cell death in serum-deprived quiescent Balb/c-3T3 cells

Serum withdrawal from either growing or quiescent Balb/c-3T3 murine fibroblasts causes a loss of F-actin fibres and focal adhesions within 30 min. Cells that are growing survive serum deprivation, whereas the great majority of density-arrested quiescent cells die during a period of up to 5 h from serum withdrawal. During this time an approximately constant fraction of the quiescent cell population dies per unit time. The population half-life is 60–70 min during this time. Addition of an appropriate cell growth factor or second messenger agonist at the time of serum withdrawal or within 2 h after serum withdawal protects a similar fraction of viable cells. These findings suggest a model according to which withdrawal of serum (i.e. growth factors) initiates the death process in cells of the population with kinetics that approximate first-order kinetics. We postulate that appropriate growth factors or second messenger agonists block the initiating event that starts the cell death process.     

The analysis of cell kinetics using cohort fraction labelled mitosis (COFLM) curves

A method has been developed to determine the cell cycle kinetics for a quiescent population of cells which are stimulated to undergo a single transit of the division cycle. The method, known as the cohort of fraction labelled mitoses (COFLM), requires no knowledge of the proliferative fraction. The probability statements of the model were formulated and then compared by an iterative fitting procedure to experimental data to obtain estimates of the model parameters. Best fit model responses show good agreement with a set of experimental data.     

Cell kinetics in a tumour cord

In some tumours, the viable cells grow around blood vessels forming cylindrical structures called tumour cords, which are surrounded by regions of necrosis. In the present paper, we propose a mathematical model for the cell kinetics in a tumour cord at the stationary state. Both proliferating cells and quiescent cells are considered, and the proliferating cell population is structured by age. Cell migration towards cord periphery is accounted for from a continuum viewpoint. The age distribution of proliferating cells, the fraction of cells in S phase, the growth fraction and the velocity along the cord radius are computed. The predictions of the model are compared with literature data obtained from two experimental rat hepatomas. The model was used to compute the profile of the oxygen tension within the cord. Possible modifications and extensions are also presented.     

Antigen-driven T-cell turnover

A mathematical model is developed to characterize the distribution of cell turnover rates within a population of T lymphocytes. Previous models of T-cell dynamics have assumed a constant uniform turnover rate; here we consider turnover in a cell pool subject to clonal proliferation in response to diverse and repeated antigenic stimulation. A basic framework is defined for T-cell proliferation in response to antigen, which explicitly describes the cell cycle during antigenic stimulation and subsequent cell division. The distribution of T-cell turnover rates is then calculated based on the history of random exposures to antigens. This distribution is found to be bimodal, with peaks in cell frequencies in the slow turnover (quiescent) and rapid turnover (activated) states. This distribution can be used to calculate the overall turnover for the cell pool, as well as individual contributions to turnover from quiescent and activated cells. The impact of heterogeneous turnover on the dynamics of CD4(+) T-cell infection by HIV is explored. We show that our model can resolve the paradox of high levels of viral replication occurring while only a small fraction of cells are infected.     

A theory for the estimation of DNA synthesis rates by flow cytometry

A method is presented for estimating the rate of DNA synthesis of a cell population by examining the DNA histogram generated by flow cytometry (FCM). The model is based on the use renewal equations to estimate the steady-state fraction of cells in each DNA compartment. The fraction of cells in each compartment is shown to be related to the Laplace transform of the transit time through that compartment. Two methods are introduced for estimating the rate of DNA synthesis utilizing different transit time distributions. One method is shown to be a simplification of the method of Dean and Anderson. The other method allows for variability in the DNA synthesis rate. The effects of quiescent cells are considered and attention is paid to the various assumptions underlying the estimation.     

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).     

A method for the selective measurement of the radiosensitivity of quiescent cells in solid tumors--combination of immunofluorescence staining to BrdU and micronucleus assay.

C3H/He mice bearing the SCC VII tumor were irradiated after being given 10 injections of 5-bromo-2'-deoxyuridine (BrdU) to label all proliferating cells in the tumors, and the tumors were then excised and trypsinized. The tumor cell suspensions were incubated with cytochalasin-B (which blocks cytokinesis), and the micronucleus frequency in unlabeled cells was determined using immunofluorescence staining to BrdU. The micronucleus frequency was then used to calculate the surviving fraction of the unlabeled cells, using the regression line relating the micronucleus frequency to the surviving fraction determined separately for the total tumor cell population. Using this technique, a cell survival curve could be determined for the unlabeled cells, which were regarded as the quiescent cells. Assays performed both immediately after and 24 h after irradiation of normally-aerated tumors showed that unlabeled cells were more radioresistant and had a greater capacity for repair of potentially lethal damage than the tumor cell population as a whole. Moreover, when the assay was performed immediately after the irradiation of both normally-aerated and hypoxic tumors, it was found that unlabeled cells had a much higher hypoxic fraction than the tumor cell population as a whole. This appears to be a useful method for determining the responses of quiescent cells in solid tumors to various treatments.     

Asymptotic behaviour of solutions to abstract logistic equations

We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.     

A test of the punctuated-cycling hypothesis in Ambystoma forelimb regenerates: the roles of animal size, limb innervation, and the aneurogenic condition

The punctuated-cycling (PC) hypothesis [39] predicts that the proportion of actively cycling (AC) cells within the blastema influences the rate of limb regeneration in urodele amphibians. To test this, we compared the rate of regeneration and the parameters of the PC hypothesis in small and large Ambystoma mexicanum larvae and in aneurogenic limbs of Ambystoma maculatum. Aneurogenic limbs regenerated more slowly than limbs of small axolotls, but considerably faster than limbs of large axolotls. Regardless of regeneration rates, virtually all blastema cells were in the proliferative fraction (Pf) (ranging from 92.3% +/- 4.2% to 96.2% +/- 3.4%). As predicted, in the blastemata of more rapidly regenerating small axolotls, 86% of the proliferative fraction was actively cycling, but as regeneration slowed, the proportion of the proliferative fraction that was actively cycling decreased (the AC of aneurogenic limbs being 69.5%, and that of large axolotl limbs being 57.3%) and the proportion of transiently quiescent cells increased. The parameters of the PC hypothesis were also examined in small axolotls at two different times during regeneration. During dedifferentiation and initial blastema formation, 61% of the cells in the proliferative fraction were actively cycling and 34% were transiently quiescent. During the rapid-growth phase of the blastema, 88% of the cells in the proliferative fraction were actively cycling and only 7% of the cells were transiently quiescent. It therefore appears that dedifferentiated cells do not immediately begin active cycling and that the transiently quiescent population is relatively large; however, during the period of rapid growth the proportion of transiently quiescent cells is small. In amputated/denervated limbs of small axolotls, the size of the proliferative fraction decreased as the length of the denervation interval increased. Furthermore, with prolonged denervation the total proportion of actively cycling blastema cells also declined (to about 15%). The failure of denervated limbs to regenerate was correlated with an increased nonproliferative fraction and a reduced proportion of actively cycling cells.     

"Sleeping beauty": quiescence in Saccharomyces cerevisiae.

The cells of organisms as diverse as bacteria and humans can enter stable, nonproliferating quiescent states. Quiescent cells of eukaryotic and prokaryotic microorganisms can survive for long periods without nutrients. This alternative state of cells is still poorly understood, yet much benefit is to be gained by understanding it both scientifically and with reference to human health. Here, we review our knowledge of one "model" quiescent cell population, in cultures of yeast grown to stationary phase in rich media. We outline the importance of understanding quiescence, summarize the properties of quiescent yeast cells, and clarify some definitions of the state. We propose that the processes by which a cell enters into, maintains viability in, and exits from quiescence are best viewed as an environmentally triggered cycle: the cell quiescence cycle. We synthesize what is known about the mechanisms by which yeast cells enter into quiescence, including the possible roles of the protein kinase A, TOR, protein kinase C, and Snf1p pathways. We also discuss selected mechanisms by which quiescent cells maintain viability, including metabolism, protein modification, and redox homeostasis. Finally, we outline what is known about the process by which cells exit from quiescence when nutrients again become available.     

Alterations of growth fraction and DNA content in K562 cells by differentiating agents

The growth fraction, estimated by the monoclonal antibody Ki-67 labeling, and DNA content, assessed by ethidium bromide staining, were determined simultaneously in K562 leukemic cells by flow cytometry. A multiparametric analysis enabled the fraction of the cell population with G1, S, and G2 + M contents in Ki-67-positive and Ki-67-negative cells to be evaluated. Butyric acid (BUT) was used as positive control. The fraction of Ki-positive cells decreased with the BUT concentration, while the proportion of cells with G1 DNA content increased only in the Ki-negative cells. Adriamycin, aclacinomycin A, and fagaronine induced differentiation, as assessed by benzidine staining and glycophorin A expression. These drugs decreased the fraction of Ki-positive cells by more than 50% for both anthracyclines and by 25% for fagaronine. Following treatment, Ki-negative cells displayed a G1, but also a G2 and a S DNA content in different proportions, indicating that induction of quiescent cells by differentiating agents is not a uniform process and is worthy of interest.     

Quiescence in 9L cells and correlation with radiosensitivity and PLD repair

The onset of quiescence, changes in X-ray sensitivity, and changes in capacity for potentially lethal damage (PLD) repair of unfed plateau-phase 9L44 cell cultures have been systematically investigated. The quiescent plateau phase in 9L cells was the result of nutrient deprivation and was not a cell contact effect. Eighty-five to 90% of the plateau-phase cells had a G1 DNA content and a growth fraction less than or equal to 0.15. The cell kinetic shifts in the population were temporally correlated with a developing radioresistance, which was characterized by a larger shoulder in the survival curve of the quiescent cells (Dq = 5.71 Gy) versus exponentially growing cells (Dq = 4.48 Gy). When the quiescent plateau-phase cells were refed, an increase in radiosensitivity resulted which approached that of exponentially growing 9L cells. Delayed plating experiments after irradiation of exponentially growing cells, quiescent plateau-phase cells, and synchronized early to mid-G1-phase cells indicated that while significant PLD repair was evident in all three populations, the quiescent 9L cells had a higher PLD repair capacity. Although data for immediate plating indicated that 9L cells may enter quiescence in the relatively radioresistant mid-G1 phase, the enhanced PLD repair capacity of quiescent cells cannot be explained by redistribution into G1 phase. When the unfed quiescent plateau-phase 9L cells were stimulated to reenter the cell cycle by replating into fresh medium, the first G1 was extended by 6 h compared with the G1 of exponentially growing or refed plateau-phase 9L cells.(ABSTRACT TRUNCATED AT 250 WORDS)     

Isolation of a population of transiently transfected quiescent and senescent cells by magnetic affinity cell sorting.

Rous sarcoma virus (RSV) and cytomegalovirus (CMV) promoters were tested for activity in proliferating and nonproliferating (quiescent or senescent) human embryo fibroblasts. These promoters were cloned upstream of the coding sequence for the Tac subunit of the interleukin 2 receptor, and activity was calculated from the fraction of Tac antigen positive cells detected in a coupled transient transfection/magnetic affinity cell sorting assay. Differences in promoter activities are substantial in quiescent cells: the efficiency of the RSV promoter is no greater than background whereas the CMV promoter is equally active in serum concentrations ranging from 0.5 to 20%. While both promoters are functional in growing cells (WI-38 and HeLa), the CMV promoter exhibits twofold greater activity. Surprisingly, in senescent cells both promoters exhibit the same degree of activity.     

Stem Cell Proliferation and Quiescence—Two Sides of the Same Coin

The kinetics of label uptake and dilution in dividing stem cells, e.g., using Bromodeoxyuridine (BrdU) as a labeling substance, are a common way to assess the cellular turnover of all hematopoietic stem cells (HSCs) in vivo. The assumption that HSCs form a homogeneous population of cells which regularly undergo cell division has recently been challenged by new experimental results. For a consistent functional explanation of heterogeneity among HSCs, we propose a concept in which stem cells flexibly and reversibly adapt their cycling state according to systemic needs. Applying a mathematical model analysis, we demonstrate that different experimentally observed label dilution kinetics are consistently explained by the proposed model. The dynamically stabilized equilibrium between quiescent and activated cells leads to a biphasic label dilution kinetic in which an initial and pronounced decline of label retaining cells is attributed to faster turnover of activated cells, whereas a secondary, decelerated decline results from the slow turnover of quiescent cells. These results, which support our previous model prediction of a reversible activation/deactivation of HSCs, are also consistent with recent findings that use GFP-conjugated histones as a label instead of BrdU. Based on our findings we interpret HSC organization as an adaptive and regulated process in which the slow activation of quiescent cells and their possible return into quiescence after division are sufficient to explain the simultaneous occurrence of self-renewal and differentiation. Furthermore, we suggest an experimental strategy which is suited to demonstrate that the repopulation ability among the population of label retaining cells changes during the course of dilution.     

Stem Cell Proliferation and Quiescence—Two Sides of the Same Coin

The kinetics of label uptake and dilution in dividing stem cells, e.g., using Bromodeoxyuridine (BrdU) as a labeling substance, are a common way to assess the cellular turnover of all hematopoietic stem cells (HSCs) in vivo. The assumption that HSCs form a homogeneous population of cells which regularly undergo cell division has recently been challenged by new experimental results. For a consistent functional explanation of heterogeneity among HSCs, we propose a concept in which stem cells flexibly and reversibly adapt their cycling state according to systemic needs. Applying a mathematical model analysis, we demonstrate that different experimentally observed label dilution kinetics are consistently explained by the proposed model. The dynamically stabilized equilibrium between quiescent and activated cells leads to a biphasic label dilution kinetic in which an initial and pronounced decline of label retaining cells is attributed to faster turnover of activated cells, whereas a secondary, decelerated decline results from the slow turnover of quiescent cells. These results, which support our previous model prediction of a reversible activation/deactivation of HSCs, are also consistent with recent findings that use GFP-conjugated histones as a label instead of BrdU. Based on our findings we interpret HSC organization as an adaptive and regulated process in which the slow activation of quiescent cells and their possible return into quiescence after division are sufficient to explain the simultaneous occurrence of self-renewal and differentiation. Furthermore, we suggest an experimental strategy which is suited to demonstrate that the repopulation ability among the population of label retaining cells changes during the course of dilution.