Matrix simulation of duodenal crypt cell kinetics. II. Cell kinetics following hydroxyurea.

The perturbed cellular kinetics of the duodenal crypt following a single injection of hydroxyurea (HU) have been simulated using matrix algebra. Following the direct effects of HU (S-phase cytotoxicity and a G1/S block) the crypt cell kinetics undergo several alterations. Previously documented alterations include: (1) a temporary partial synchronization of the surviving cells, (2) a shortening of the cell-cycle transit time, and (3) recruitment of normally non-proliferating cells into active proliferation. These conclusions have been extended by constructing several different complex but theoretically possible recovery models and the validity of each of these models has been evaluated by simulating the following biological data: the number of cells in the S and M-phase of the cell cycle, total viable cells per crypt, and the per cent labeled mitosis and the number of labeled cells following 3H-TdR injections at 9 and 21 hr after HU treatment. The model which showed visually the best overall agreement with all sets of the data was chosen as "most probable' and leads to the following interpretations. Immediately after the end of the HU block (i.e. 5 hr after HU injection) the modal cell-cycle transit time is reduced to 8 hr. By 17 hr after HU, the modal transit time is increased to 10 hr. Repopulation of the proliferating compartment, i.e. restoration of the proliferating compartment back to the control value, occurs between 12 and 17 hr after HU injection and probably consists of both recycling of the proliferating cells (i.e. they do not progress up into the non-proliferating compartment) and recruitment of the non-proliferating cells into active proliferation. Also, the rate at which the non-proliferating cells move onto the villi is reduced temporarily. The overall recovery process results in a crypt which temporarily is larger than control and produces villi cells at a rate which is faster than the control. The time when the crypt size and villus cell production rate return to normal cannot be established using the available data.     

A semi-empirical model for cell kill kinetics during hyperthermia

A three-parameter mathematical model is developed which describes the available data on cell-kill kinetics during hyperthermia. The sub-exponential behaviour of the kinetics suggests that the cell-kill is not a one-step process.     

Mathematical modeling the kinetics of cell distribution in the process of ligand-receptor binding

A statistical approach is presented to model the kinetics of cell distribution in the process of ligand-receptor binding on cell surfaces. The approach takes into account the variation of the amount of receptors on cells assuming the homogeneity of monovalent binding sites and ligand molecules. The analytical expressions for the kinetics of cell distribution have been derived in the reaction-limited approximation. In order to demonstrate the applicability of the mathematical model, the kinetics of binding the rabbit, anti-mouse IgG with Ig-receptors of the murine hybridoma cells has been measured. Anti-mouse IgG was labeled with fluorescein isothiocyanate (FITC). The kinetics of cell distribution on ligand-receptor complexes was observed during the reaction process by real-time measuring of the fluorescence and light-scattering traces of individual cells with the scanning flow cytometer. The experimental data were fitted by the mathematical model in order to obtain the binding rate constant and the initial cell distribution on the amount of receptors.     

Matrix simulation of duodenal crypt cell kinetics. I. The steady state.

Steady state crypt cell kinetics have been simulated using matrix algebra. The model crypt cell population is distributed through two proliferation compartments (P1 and P2) and a quiescent state (Q). Under steady state conditions half the daughter cells produced on completion of P1 enter G1 of P2 and half enter G1 of P1. Both P2 daughter cells enter Q. Cells in Q are non-dividing but retain the potential to divide. On completion of Q, cells lose the potential to divide and move up onto the villi. The model has been developed by simultaneously simulating the following biological data: (1) the per cent labeled mitosis (PML) curve, (2) the number of labeled cells per crypt as a function of time following an injection of 3H-thymidine, and (3) the total number of cells per crypt.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Hydrocarbon fermentations using Candida lipolytica. II. A model for cell growth kinetics

A mechanistic model is presented for the growth kinetics of a yeast grown by submerged aerobic fermentation using a liquid hydrocarbon as sole carbon source. The model is based on the assumption that cell growth is governed by the extent of probable cell attachment at the hydrocarbon oil-droplet surfaces in a four-phase dispersion. An analytical expression has been developed for the model. It is shown that for the case of relatively small oil droplets, the model predicts the present and previous experimental data for growth of yeasts (Candida species) in n-alkane systems. The model is further examined for maximal growth in terms of substrate dilution rate and agitation power consumption for a continuous fermentation process.     

Cellular kinetics,cell cycles and cell killing

Summary Techniques available for the calculation of the time variation of the number of viable mammaliar cells in a cell population are reviewed. Events in the course of the cell's growth may include one or more exposures to ionizing radiations or other cytotoxic agents. The dependence of cell killing upon the cell's position in the cell cycle is emphasized, and a unified model for calculation of cell kinetics and cell survival is discussed. For a cell population not limited in growth by contact inhibition or by nutritional factors, experimental data agree with predictions of the model.The possibility of utilizing the model to arrive at optimum treatment schedules for the management of some malignant diseases is discussed. The conclusion drawn is that the state of knowledge with respect to cellular events in solid tumors is such as to leave most such applications in the realm of speculation.This work was supported by the National Institutes of Health, United States Public Health Service, under Grants CA 5008 and CA 4542.     

Predicting the kinetics of cell spreading

We apply the wetting theory to predict the kinetics of fibroblast spreading onto an adhesive substrate, under simplifying assumptions on the cell structure and geometry. Three main parameters are used: cytoplasmic viscosity, cortical tension, and cell-to-substrate adhesion energy. The viscosity and tension values are taken from previous micromechanical studies. The adhesion energy, ill known, is adjusted by fitting the model predictions to available experimental data of contact radius versus time. The agreement is quite good, justifying such a "macroscopic" view of cell morphology.     

Stochastic kinetics of reproduction of virions inside a cell

We analyze intracellular viral kinetics in the framework of the model incorporating viral genome replication, mRNA synthesis and degradation, protein synthesis and degradation, capsid assembly, and virion release from a cell. Due to the existence of the critical concentration of viral capsid proteins and other features of reproduction of virions inside a cell, the kinetics is demonstrated to exhibit three distinct initial stages. Specifically, (i) the exponential growth of the viral genome, mRNA and protein concentrations is followed by (ii) the transient stage to (iii) the steady-state regime. The formation of mature virions starts during the transient stage. Comparison of the kinetics, obtained by using the mass-action law and Monte Carlo (MC) technique, indicates that they are nearly identical during the initial exponential growth of the viral intermediates and also during the steady-state stage. The transition from the initial stage to the steady-state regime occurs however somewhat faster in the determenistic case even if the steady-state populations of virions and genomes are appreciable (e.g., about 250 and 500, respectively).     

Quantitative description of cell cycle kinetics under chemotherapy utilizing flow cytometry.

A discrete time cell cycle kinetics model is developed to account for the effects of cytotoxic chemotherapy, particularly including the existence of cells destined to die. A model structure is determined from related experiments, leaving key parameter values undetermined. These values are found by determining the best least squares fit of the predicted to the observed DNA distribution data at a series of time intervals. The numerical methods include separable least squares, linear inequality constrained least squares and the Gauss--Newton method. This approach is applied to an experiment in which the Ehrlich ascites tumour was given a single dose of bleomycin. The results include several different parameters, including the age response function and a time series of cell age and DNA distributions, which can be used as a basis for further treatment.     

The mathematical modelling of cell kinetics in corneal epithelial wound healing

This paper considers the comparison of experimental spatial and temporal data of mitotic rates measured during corneal epithelial wound healing (CEWH) of a rat model with the predictions of a computer modelling framework. We begin by briefly showing that previous models, used in the study of corneal epithelial wound healing speeds, are inadequate for the study of cell kinetics. We proceed to formulate a new modelling framework more suited to such a study. This framework is simulated in its simplest form, and the results from this motivate a new realisation of the modelling framework, including a caricature of age structuring. Finally, a model with a simple representation of juxtacrine signalling is considered. The final model captures many, though not all, of the trends of the experimental data. This paper thus lays a foundation for the modelling of the cell kinetics of corneal epithelial wound healing, and yields valuable insight regarding the important mechanisms a model should consider in order to reproduce the observed experimental trends.     

The kinetics of E. coli RNA polymerase.

Using an assay specific for chain elongation of E. coli RNA polymerase the kinetics of this propagation reaction have been studied. The kinetic behaviour is consistent woth the mathematical model formulated for this multisubstrate enzyme. The effect of increasing salt concentration on the kinetics of the reaction indicated that DNA unwinding is probably a necessary step in the propagation step, although this may not be the rate limiting step under all conditions.     

Receptor-mediated cell attachment and detachment kinetics. I. Probabilistic model and analysis.

The kinetics of receptor-mediated cell adhesion to a ligand-coated surface play a key role in many physiological and biotechnology-related processes. We present a probabilistic model of receptor-ligand bond formation between a cell and surface to describe the probability of adhesion in a fluid shear field. Our model extends the deterministic model of Hammer and Lauffenburger (Hammer, D.A., and D.A. Lauffenburger. 1987. Biophys. J. 52:475-487) to a probabilistic framework, in which we calculate the probability that a certain number of bonds between a cell and surface exists at any given time. The probabilistic framework is used to account for deviations from ideal, deterministic behavior, inherent in chemical reactions involving relatively small numbers of reacting molecules. Two situations are investigated: first, cell attachment in the absence of fluid stress; and, second, cell detachment in the presence of fluid stress. In the attachment case, we examine the expected variance in bond formation as a function of attachment time; this also provides an initial condition for the detachment case. Focusing then on detachment, we predict transient behavior as a function of key system parameters, such as the distractive fluid force, the receptor-ligand bond affinity and rate constants, and the receptor and ligand densities.(ABSTRACT TRUNCATED AT 250 WORDS)     

Staining kinetics in single cells. Part II. Diffusion processes inside the cell

Applying hydrodynamic conditions, which certify a negligible influence of convective diffusion, the time-dependent uptake of thionin in lymphocytes, monkey kidney cells, and their separated nuclei was measured spectroscopically. Using fixed cell material the dye transport inside the cell is not hindered due to plasma membrane and cytoplasm. The staining rate depends on the dye concentration, the pretreatment of the cell, and on the electrolyte concentration of the dye solution. The mechanism of dye migration inside the cell is in accordance with a porous matrix model. The diffusion process takes place inside the pores and channels filled with liquid and is modified by adsorption of dye molecules on the walls of the pores. A dynamic reversible equilibrium exists between migrating dye molecules and the binding sites on the pore walls described by the Freundlich adsorption isotherm. The proposed model explains the observed order of reaction of the staining kinetics.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Exploring the complex folding kinetics of RNA hairpins: I. General folding kinetics analysis

Depending on the nucleotide sequence, the temperature, and other conditions, RNA hairpin-folding kinetics can be very complex. The complexity with a wide range of cooperative and noncooperative kinetic behaviors arises from the interplay between the formation of the loops, the disruption of the misfolded states, and the formation of the rate-limiting base stacks. With a rate constant model and a kinetic-cluster theory, we explore the broad landscape for RNA hairpin-folding kinetics. The model is validated through direct tests against several experimental measurements. The general kinetic folding mechanisms and the predicted great variety of folding kinetics are directly applicable and quantitatively testable in experiments. The results from this study suggest that 1), previous experimental findings based on the individual hairpins revealed only a small fraction of much broader and more complex RNA hairpin-folding landscapes; 2), even for structures as simple as hairpins, universal folding timescales and pathways do not exist; and 3), to treat the loop size as the sole factor to determine the hairpin-folding rate is an oversimplification.     

The early folding kinetics of apomyoglobin.

The folding pathway of apomyoglobin has been experimentally shown to have early kinetic intermediates involving the A, B, G, and H helices. The earliest detected kinetic events occur on a ns to micros time scale. We show that the early folding kinetics of apomyoglobin may be understood as the association of nascent helices through a network of diffusion-collision-coalescence steps G + H <--> GH + A <--> AGH + B <--> ABGH obtained by solving the diffusion-collision model in a chemical kinetics approximation. Our reproduction of the experimental results indicates that the model is a useful way to analyze folding data. One prediction from our fit is that the nascent A and H helices should be relatively more helix-like before coalescence than the other apomyoglobin helices.     

Zero order kinetics of cell wall turnover inStaphylococcus aureus

Cells exponentially grown from four strains ofS. aureus (SG 511, H, 52A5G, and248 PN-1) and uniformly labeled in their walls with³H-N-acetylglucosamine, were found to turn over their old walls at constant rates of up to 25% per generation. Wall turnover was not observed to follow first order kinetics, thus ruling out the implication that maintenance of normal wall thickness was achieved by a random distribution of new wall components in the old wall. Instead, wall turnover in all cases strictly followed zero order kinetics, indicating that newly synthesized wall material was placed layer by layer beneath the inner surface of the old cell wall. This finding correlates with evidence obtained from earlier electron microscopic investigations into the regeneration of the staphylococcal cell wall after chloramphenicol treatment. Based on the experimental data presented, a simplified model for wall turnover of the growing staphylococcal cell was proposed. The model also takes into account the finding, derived from additional experiments with strainSG 511, that the total cell wall turned over at a somewhat higher rate than the old portions of the wall. The rates of cell wall turnover found inS. aureus SG 511 are the highest reported to date for pathogenic bacteria. The medical implications of this finding were discussed.     

Growth factors and cell kinetics: A mathematical model applied to Il-3 deprivation on leukemic cell lines

We assume the existence of a specific G1 protein which is an initiator of DNA replication. This initiator is supposed to be synthesized according to Michaelis-Menten kinetics. In order to start DNA replication, it is assumed that this G1 specific protein must be produced in a required amount. Intra-cellular growth inhibitors and extra-cellular growth factors control the production of the initiator. This model allows to calculate the average G1 phase time as a function of the various chemical concentrations of nutrients, enzymes, growth inhibitors and growth factors. This model is compared to cell kinetics experiments on a leukemic cell line responding to Interleukin 3 deprivation. The curves giving the experimental average G1 phase times with respect to Interleukin-3 concentrations are fitted by the mathematical model with a quite good agreement.     

The effects of chemical kinetics on oxygen delivery to tissue.

A mathematical model has been formulated to analyze the effect of nonequilibrium kinetics on oxygen delivery to tissue. The model takes into account molecular diffusion, facilitated diffusion in the capillary blood, convection, chemical kinetics of O2 with hemoglobin, and the rate of metabolic consumption. A line iterative technique is described to solve numerically the resulting coupled system of nonlinear partial differential equations with physiologically relevant boundary and entrance conditions. With nonequilibrium kinetics the end-capillary PO2 is found to be lower than that in the venous blood. The effect is more pronounced during hypoxia and anemia. It is found that the tissue PO2 at the lethal corner decreases with the decrease in blood velocity, arterial PO2, hemoglobin concentration, P50, and increase in COHb concentration or metabolic rate, while the difference between end-capillary PO2 and venous PO2 increases, which reflects the effect of nonequilibrium kinetics on the delivery of O2 to tissue. Thus, the consideration of venous PO2 as an indicator of tissue PO2 in clinical and experimental studies may be questionable.