State vector models of the cell cycle. III: Continuous time cell cycle models

In this paper the elements of the matrix of the Hahn cell-cycle model are identified with the infinitesimal transition probabilities of a Markov process, and as a limiting process a differential equation analogue is derived. The probability density function of the discrete time model is derived and used to obtain the density function for transit times of the continuous time model. It is shown that the mean transit time remains constant and that the variances of the discrete and continuous time models are the same to the order of the time increment. Finally, it is shown how to derive the Takahashi model from the continuous time Hahn model.     

Control models of cell cycle transit, exit, and arrest

Several distinct cycles mediate the events which occur between one cell division and the next. In micro-organisms there are generally two cycles. One governs biomass growth, the other DNA synthesis and cell division. In higher eukaryotes there can be as many as four distinct cycles, with growth, DNA synthesis, cell division, and nuclear division each possessing its own functional sequence of events. These cycles are controlled and coordinated by several different regulatory mechanisms. Restriction points are specific steps in the cycle whose completion is governed by external regulatory agents. One set of restriction points requires nutrients and growth hormones for step completion. Another set serves as receptors for differentiating factors which cause cycle arrest and initiate cellular differentiation. There is currently a debate as to whether restriction point inhibition involves permanent arrest or temporary arrest with a stochastic arrested-state residence time controlled by a transition probability mechanism. Tissue sizing is a process of negative feedback inhibition mediated by intercellular communication via cell surface contact and the extracellular matrix. Sizers commonly operate throughout broad portions of the cycle and appear to cause a slowing of cycle transit velocity rather than arrest. Sizers are probably the major regulatory mechanism for cell growth under conditions of nutrient and growth factor excess. They also generate compensatory proliferation following wounding or cell death. A growing body of evidence suggests that both the transit velocity, with which cells move through their several cycles, and the coordination of the cycles are controlled by intracellular regulatory mechanisms which behave as biological oscillators. These oscillators trigger complex sequences of events such as DNA synthesis and cell division.     

The theory of sliding filament models for muscle contraction. II. Biochemically-based models of the contraction cycle

A seven-state sliding filament model is proposed which differs from the model of Eisenberg & Greene. It is based on a simplified version of the in-vitro contraction cycle of Stein et al., and also has some desirable dynamical features of the empirical three-state model of Nishiyama & Murase. Appropriate x-dependences for all reaction rates are derived from the transition-state theory. The seventh-state is assumed to be a high-tension intermediate of A.M.ATP, from which direct but x-dependent dissociation can occur. If the final A.M.ATP state has a sufficiently lower tension than that of A.M.ADP.Pi, then the dominant escape path from the intermediate state is shown to be direct dissociation of the actin-myosin bond. This leads to an approximate five-state model for active and relaxed muscle in which A.M and the final A.M.ATP state are omitted.     

Molecular aspects of cadherin-mediated cell adhesion: the first moments of the interaction

Cadherins play a major role in the development and maintenance of all solid tissues. These transmembrane glycoproteins are responsible for calcium-dependent homophilic cell interactions. Recently, many different experimental approaches have been used to untangle the molecular basis of cadherin-mediated adherence. Various models have been suggested, particularly from high-resolution structures. Whilst the adherence mechanism is still under controversy, it is widely accepted that the specificity of the adherent interaction is localized to the N-terminal domain. New biophysical techniques together with biological approaches will allow a better understanding of how cadherins regulate cell-cell adherence. Integrating kinetics properties of cadherin interaction at the single molecule level has led to a greater understanding of cadherin molecular regulations.     

Analysis of distribution of S. cerevisiae cells for protein content using cell cycle models

Histograms of cell distributions according to protein content obtained by means of flow cytofluorometry characterize the physiological state of the population as a whole and permit to calculate the velocity of protein accumulation in the cell in the course of the cell cycle. Dependence of population heterogeneity on culturing conditions is considered. Mathematical analysis of histograms of continuous cultures of S. cerevisiae is carried out at dilution rates 0.4 hours-1 and 0.05 hours-1. Calculations are carried out on condition that the protein content in the cell rises a) exponentially and b) linearly in the course of the cell cycle. At low growth rate (0.05 hours-1) the distribution is bimodal and therefore it is highly informative. The assumption concerning linear accumulation of the protein allows good approximation of the experimental distributions by the theoretical ones.     

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle

Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for _n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, _*(x). For the special case of the transition probability model, we display _*(x) explicitly.This work was supported by the National Science Foundation under grants MCS8301104 (to J.J.T.) and MCS8300559 (to K.B.H.), and by the National Institutes of Health under grant GM27629 (to J.J.T.).     

Stimulation of WI-38 cell cycle transit: Effect of serum concentration and cell density

Flow microfluorometry has been used to characterize the effects of serum concentration and cell density on the initiation of cell cycle transit of stationary phase (G₀) human diploid fibroblasts (strain WI-38). The concentration of serum used to stimulate these cultures had no effect on the time cells began appearing in S (the DNA synthetic period), nor on the synchrony with which they moved around the cell cycle. However, as the serum concentration increased, the fraction of the stationary phase population released from G₀ increased. Cell density modulated the ability of serum to stimulate cell cycle traverse. For example, at a cell density of 1.81 × 10⁴ cells/cm², 78% of the population was sensitive to serum stimulation; whereas, when the density was increased to 7.25 × 10⁴ cells/cm², only 27% of the population could be stimulated. This effect of cell density on the serum response is not simply the result of changing the ratio of serum concentration to cell density, but appears to reflect a true modulation of the population's sensitivity to serum stimulation. These results are consistent with the interpretation that the primary action of serum is to determine the transition of cells from a non-cycling G₀ state to a cycling state and that cell density determines the proportion of the population capable of undergoing this transition.     

The ORC1 cycle in human cells: II. Dynamic changes in the human ORC complex during the cell cycle

The origin recognition complex (ORC) plays a central role in regulating the initiation of DNA replication in eukaryotes. The level of the ORC1 subunit oscillates throughout the cell cycle, defining an ORC1 cycle. ORC1 accumulates in G1 and is degraded in S phase, although other ORC subunits (ORCs 2-5) remain at almost constant levels. The behavior of ORC components in human cell nuclei with respect to the ORC1 cycle demonstrates that ORCs 2-5 form a complex that is present throughout the cell cycle and that associates with ORC1 when it accumulates in G1 nuclei. ORCs 2-5 are found in both nuclease-insoluble and -soluble fractions. The appearance of nuclease-insoluble ORCs 2-5 parallels the increase in the level of ORC1 associating with nuclease-insoluble, non-chromatin nuclear structures. Thus, ORCs 2-5 are temporally recruited to nuclease-insoluble structures by formation of the ORC1-5 complex. An artificial reduction in the level of ORC1 in human cells by RNA interference results in a shift of ORC2 to the nuclease-soluble fraction, and the association of MCM proteins with chromatin fractions is also blocked by this treatment. These results indicate that ORC1 regulates the status of the ORC complex in human nuclei by tethering ORCs 2-5 to nuclear structures. This dynamic shift is further required for the loading of MCM proteins onto chromatin. Thus, the pre-replication complex in human cells may be regulated by the temporal accumulation of ORC1 in G1 nuclei.     

Prolongation of cell cycle transit time and the presence of non-cycling cells in human lymphoblastoid cells cultured under adverse conditions

The growth characteristics of B lymphocytes infected with Epstein-Barr virus (lymphoblastoid cells) have been investigated by flow cytometric analysis of DNA content and by estimation of cell culture doubling times. It was found that the manipulative procedures involved in the cell cycle analysis resulted in a slowing of the growth rate. This slowing of growth was brought about by the prolongation of cell cycle transit times and by the entry of cells into a short-lived non-cycling pool. The entry of a proportion of the cells into the non-cycling pool may be the normal response of lymphoblastoid cells to non-optimal conditions. The non-cycling cells survived in culture with a T 1/2 of approximately 30-60 hr and continued to secrete immunoglobulin. Their surface transferrin receptors were considerably reduced, which suggests that the failure to divide may have resulted from a failure of growth factor receptors to reach a threshold value following mitosis.     

Prolongation of cell cycle transit time and the presence of non-cycling cells in human lymphoblastoid cells cultured under adverse conditions

Abstract. The growth characteristics of B lymphocytes infected with Epstein-Barr virus (lymphoblastoid cells) have been investigated by flow cytometric analysis of DNA content and by estimation of cell culture doubling times. It was found that the manipulative procedures involved in the cell cycle analysis resulted in a slowing of the growth rate. This slowing of growth was brought about by the prolongation of cell cycle transit times and by the entry of cells into a short-lived non-cycling pool. The entry of a proportion of the cells into the non-cycling pool may be the normal response of lymphoblastoid cells to non-optimal conditions. The non-cycling cells survived in culture with a T _1/2 of approximately 30–60 hr and continued to secrete immunoglobulin. Their surface transferrin receptors were considerably reduced, which suggests that the failure to divide may have resulted from a failure of growth factor receptors to reach a threshold value following mitosis.     

Single cell studies of the cell cycle and some models

Analysis of growth and division often involves measurements made on cell populations, which tend to average data. The value of single cell analysis needs to be appreciated, and models based on findings from single cells should be taken into greater consideration in our understanding of the way in which cell size and division are co-ordinated. Examples are given of some single cell analyses in mammalian cells, yeast and other microorganisms. There is also a short discussion on how far the results are in accord with simple models.     

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.     

On the distribution of cell cycle generation times

The problem of whether the cell cycle is a deterministic or probabilistic process is widely discussed in the current literature (P. Nurse, Nature, 286, pp. 9–10, 1980). In this report the question of fluctuations of cell cycle period is treated in the limits of the membrane model of cell division regulation. The parametric analysis of the equations set both for normal and tumour cells is carried out. We describe the bifurcation parameters in the neighbourhood of which the system can amplify the small fluctuations. The presence of white noise in parameters describing the lipids and antioxidants influxes into membrane is examined by methods of Marcovian processes and also by direct stochastic computer simulation. The equation for the distribution function of generation times is obtained and the increase of dispersion and mean cycle time during the changes of those parameters which would be connected with cell culture density is calculated.The influence of parameter fluctuations upon the cycle period for both normal and tumour cells is compared in the framework of model assumptions. The ratio of dispersion of generation time distribution to mean period value for an ensemble of tumour cells is shown to be several times greater than that for normal ones.In the discussion the problem of the presence of a premitotical (G₀₂) resting state and of the possibility of its experimental detection is considered.     

The role of lymphokine-activated cell-associated antigen. II. Distribution and correlation with cell cycle

It has previously been shown that killer-blocking monoclonal antibody (KBA MAb) recognizes lymphokine-activated cell-associated antigen (LAA) involved in broad-reactive killer. (BRK) cell-mediated cytotoxicity. We now report that LAA is expressed on all lymphoid cells, though the amount of LAA on unstimulated lymphocytes is low. In contrast, lymphocytes activated in vitro with either concanavalin A, alloantigens, lipopolysaccharide, or recombinant interleukin 2 express high levels of LAA. In addition, in vivo activated lymphocytes, such as OK-432-activated lymphocytes and tumor-infiltrating lymphocytes express higher levels of LAA than unstimulated lymphocytes. We also demonstrate that the expression of LAA is restricted in T-cell lymphomas and a M phi cell line, while myelomas, fibrosarcomas, and carcinomas do not express LAA. Cell cycle analysis using propidium iodide and KBA MAb showed that LAA expression was closely correlated with the transition of cells from G1a to G1b phase.