Deterministic model of steady-state cell proliferation

A new approach to the kinetics of cell proliferation, based on the postulated restriction of the number of cell divisions in an organism gives the possibility to determine the individual lifetimes of cells. In the model, a necessary condition for a steady-state population is that two sister cells have distinct lifetimes. A steady state was obtained as a consequence of constant rate of cell production in each generation, when sister cell divisions alternated. The mean value of the generation time of cells is in the direct proportion to the number of cells in each generation and connected (with a coefficient of 2) with the generation number. In consequence, we attach great importance to the identification of cells belonging to distinct generations. Corresponding mathematical method to determine the cell population parameters is given and conclusions about stem cells' organization have been drawn.     

Temperature and guanidine hydrochloride dependence of the structural stability of ribonuclease T1.

The thermal unfolding of ribonuclease T1 has been studied by high-sensitivity differential scanning calorimetry as a function of temperature, [GuHCl], and scanning rate. The destabilizing effect of GuHCl has revealed that the kinetics of the unfolding transition become extremely slow as the transition temperature decreases. At pH 5.3 and zero GuHCl, the unfolding transition is centered at 59.1 degrees C; upon increasing the GuHCl concentration, the transition occurs at lower temperatures and exhibits progressively slower kinetics; so, for example, at 3 M GuHCl, the transition temperature is 40.6 degrees C and is characterized by a time constant close to 10 min. Under all conditions studied (pH 5.3, pH 7.0, [GuHCl] < 3 M), the transition is thermodynamically reversible. The slow kinetics of the transition induce significant distortions in the shape of the transition profiles that can be mistakenly interpreted as deviations from a two-state mechanism. Determination of the thermodynamic parameters from the calorimetric data has required the development of an analytical formalism that explicitly includes the thermodynamics as well as the kinetics of the transition. Using this formalism, it is shown that a two-state slow-kinetics model is capable of accurately describing the structural stability of ribonuclease T1 as a function of temperature, GuHCl concentration, and scanning rate. Multidimensional analysis of the calorimetric data has been used to estimate the intrinsic thermodynamic parameters for protein stability, the interaction parameters with GuHCl, and the time constant for the unfolding transition and its temperature dependence.     

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.     

DETERMINATION OF STEADY STATE GENERATION TIME DISTRIBUTIONS FROM LABELLED MITOSIS EXPERIMENTAL DATA

The proper application of detailed deterministic cell kinetic models depends on the way in which cells are assigned their generation times. A method is presented for the determination of population generation time distributions from labelled mitoses experiments. the model assumes that the generation time of each new cell is a function of both the steady-state generation time distribution function of the population, and also the generation time frequency-function of the previous generation of cells. This approach is applied to two different cell types to successfully simulate extended labelled mitoses curves using a population balance model with constant maturation rates.     

Equilibrium and kinetics of the allosteric transition of GroEL studied by solution X-ray scattering and fluorescence spectroscopy

We have studied the ATP-induced allosteric structural transition of GroEL using small angle X-ray scattering and fluorescence spectroscopy in combination with a stopped-flow technique. With X-ray scattering one can clearly distinguish the three allosteric states of GroEL, and the kinetics of the transition of GroEL induced by 85 microM ATP have been observed directly by stopped-flow X-ray scattering for the first time. The rate constant has been found to be 3-5s(-1) at 5 degrees C, indicating that this process corresponds to the second phase of the ATP-induced kinetics of tryptophan-inserted GroEL measured by stopped-flow fluorescence. Based on the ATP concentration dependence of the fluorescence kinetics, we conclude that the first phase represents bimolecular non-cooperative binding of ATP to GroEL with a bimolecular rate constant of 5.8 x 10(5)M(-1)s(-1) at 25 degrees C. Considering the electrostatic repulsion between negatively charged GroEL (-18 of the net charge per monomer at pH 7.5) and ATP, the rate constant is consistent with a diffusion-controlled bimolecular process. The ATP-induced fluorescence kinetics (the first and second phases) at various ATP concentrations (< 400 microM) occur before ATP hydrolysis by GroEL takes place and are well explained by a kinetic allosteric model, which is a combination of the conventional transition state theory and the Monod-Wyman-Changeux model, and we have successfully evaluated the equilibrium and kinetic parameters of the allosteric transition, including the binding constant of ATP in the transition state of GroEL.     

ON THE EXISTENCE OF A Go-PHASE IN THE CELL CYCLE

The model is based on the assumption that the cell cycle contains a G_o-phase which cells leave randomly with a constant probability per unit time, γ. After leaving the G_o-phase, the cells enter the C-phase which ends with cell division. The C-phase and its constituent phases, the‘true’G₁-phase, the S-phase, the G₂-phase and mitosis are assumed to have constant durations of T, T₁T_s, T₂ and T_m, respectively. For renewal tissue it is assumed that the probability per unit time of being lost from the population is a constant for all cells irrespective of their position in the cycle. The labelled mitosis curve and labelling index for continuous labelling are derived in terms of γ, T, and T_s. The model generates labelled mitosis curves which damp quickly and reach a constant value of twice the initial labelling index, if the mean duration of the G_o-phase is sufficiently long. It is shown that the predicted labelled mitosis and continuous labelling curves agree reasonably well with the experimental curves for the hamster cheek pouch if T has a value of about 60 hr. Data are presented for the rat dorsal epidermis which support the assumption that there is a constant probability per unit time of a cell being released from the Go-phase.     

Dependence of Miniature Endplate Current on Kinetic Parameters of Acetylcholine Receptors Activation: A Model Study

Mathematical modeling was applied to study the dependence of miniature endplate current (MEPC) amplitude and temporal parameters on the values of the rate constants of acetylcholine binding to receptors (k₊) when cholinesterase was either active or inactive. The simulation was performed under two different sets of parameters describing acetylcholine receptor activation–one with high and another with low probability (Po_high and Po_low) of receptor transition into the open state after double ligand binding. The dependence of model MEPC amplitudes, rise times, and decay times on k₊ differs for set Po_low and set Po_high. The main outcome is that for set Po_high, the rise time is significantly longer at low values of k₊ because of the prolongation of ACh diffusion time to the receptor. For the set Polow, the rise time is shorter at low values of k1, which can be explained by the small probability of AChR forward isomerization after ACh binding and faster MEPC's peak formation.     

Continuous models for frequency-density-dependent selection

The growth of a panmictic monoecious diploid population with two alleles at one locus is modeled by making fitnesses depend on the genotypes' abundance. This implies an implicit dependence of fitnesses on both density and gene frequency. Equations are derived for the gene frequency and for the population size in the overlapping generation case. A diffusion model for the gene frequency is finally obtained, and the gene frequency transition probability density function is determined in the case of no dominance.     

The transition from the native to the acid-state characterized by multi-spectroscopy approach: Study for the holo-form of bovine α-lactalbumin

The transition of the holo-form of bovine α-lactalbumin from the native (N) to the pH-generated acidic-state (A-state) was analyzed by probing its tertiary and secondary structure using a concerted spectroscopic approach combining near- and far-UV circular dichroism (CD), electrospray ionization ion mobility mass spectrometry (ESI-IM-MS), vibrational circular dichroism (VCD), and Fourier transform infrared spectroscopy (FTIR) in the attenuated total reflection (ATR) and transmission (TR) modes. The spectroscopic results, which relied on the interaction of an electromagnetic field with different molecular targets, confirmed the decay of extensive rigid side-chain packing interactions during the pH-induced N → A-state transition and revealed the targets' dependence on secondary structural changes. Independent analyses of the spectral changes using two methods of multivariate analysis, such as principal component analysis and two-dimensional correlation spectroscopy, revealed small but significant differences in the secondary structure as a result of the all-or-none transition. The cooperativity of the transition was quantitatively described using values corresponding to the mid-point (t_m) and width of the transition (Δt_m). The averages of the two parameters, calculated using the data collected by the different probes, were equal to 3.5 ± 0.2 and 0.6 ± 0.1(SE), respectively. The variable two-state nature of the cooperative N → A-state transition confirmed that the protonation of the side chain carboxyl groups on the Asp and Glu residues and that the release of a Ca^2 + ion induced structural changes on both the secondary and tertiary levels. The changes have been confirmed by results obtained from the concerted spectroscopic approach.     

Stathmokinetic Analysis of Human Epidermal Cells in vitro

Proliferation kinetics of cultured human epidermal cells is characterized in quantitative terms. Three distinct subpopulations of keratinocytes, two of which are cycling have been discriminated by two parameter DNA/RNA flow cytometry. Based on mathematical modelling, the cell cycle parameters of the cycling subpopulations have been assessed from stathmokinetic data collected at different time points after initiation of cultures (7–15 days). the first subpopulation is composed of low-RNA cells which resemble basal keratinocytes of epidermis and which show some characteristics of stem cells; these cells have a mean generation time of approximately 100 hr. the second subpopulation consists of high-RNA cells, resembling stratum spinosum cells of epidermis, which have an average generation time of approximately 40 hr. the third subpopulation consists of non-cycling cells with G_o/G₁ DNA content, with cytochemical features similar to those of cells in granular layer of epidermis. The results based on modelling can reproduce with acceptable accuracy the actual growth curve of the cultured cell population. Analysis of kinetics and differentiation of human keratinocytes is of interest in view of the recent application of cultured epidermal cell sheets for transplantation onto burn wounds. the results of this study also reveal the existence of regulatory mechanisms associated with proliferation and differentiation in the cultured epidermal cell population.     

Mass,length and growth rate in single cells

A heuristic model for the time course of elongation and mass increase of a population of single cells with generation times different from the mean is considered. The fundamental assumption is that the dynamics governing the fluctuations around the mean are the same as those describing the mean itself. Cell length is assumed to depend on cell mass by a $\text{1}\text{3}$ power law. The final cell length and mass are obtained by demanding that a daughter cell's initial length and mass obey the same equation as those of the mother cell when the mother cell was born. An exponential time course is assumed for simplicity, although most of the results depend only on the initial and final values, not the actual time behavior. The aspect ratio (length/radius) is found to be a constant for all cells at birth, and twice that value at maturity. Thus the individual generation time is a doubling time for the aspect ratio, even though neither mass nor length double in general. Oscillations in ancestor/progeny generation times are derived, and their stability considered. The model yields a non-hereditary determinant of individual generation time. The well-known skewness in the distribution of individual generation times and the “β-plots” of the transition probability model find a natural explanation in the inequality of cell division.     

Size control models of Saccharomyces cerevisiae cell proliferation.

By using time-lapse photomicroscopy, the individual cycle times and sizes at bud emergence were measured for a population of saccharomyces cerevisiae cells growing exponentially under balanced growth conditions in a specially constructed filming slide. There was extensive variability in both parameters for daughter and parent cells. The data on 162 pairs of siblings were analyzed for agreement with the predictions of the transition probability hypothesis and the critical-size hypothesis of yeast cell proliferation and also with a model incorporating both of these hypotheses in tandem. None of the models accounted for all of the experimental data, but two models did give good agreement to all of the data. The wobbly tandem model proposes that cells need to attain a critical size, which is very variable, enabling them to enter a start state from which they exit with first order kinetics. The sloppy size control model suggests that cells have an increasing probability per unit time of traversing start as they increase in size, reaching a high plateau value which is less than one. Both models predict that the kinetics of entry into the cell division sequence will strongly depend on variability in birth size and thus will be quite different for daughters and parents of the asymmetrically dividing yeast cells. Mechanisms underlying these models are discussed.     

Effects of subculturing and physical condition of medium on the nuclear behavior of a plant tissue culture

A callus of the common garden peony, Paeonia suffruticosa, was subcultured on solid and liquid media and analyzed intensively for a period of 153 days in order to test the effects of subculturing and the physical conditions of culture on the mitotic cycle kinetics of a population of cells, particularly in relation to the degree of heteroploidy. The parameters investigated in the kinetic studies included mitotic index values, cell generation time, and the time required for the cell population to double. The mitotic index of Paeonia cells cultured in liquid medium was found to be about two and a half times higher than for those cultured on solid; successive subculturing did not affect the mitotic index on either type of medium. The most significant results of the study came from the chromosome count data, in which diploid and tetraploid cells fluctuated in predominance in successive subcultures, and the apparent earlier manifestation of polyploidy on liquid medium. Mitotic index, cell generation, and population doubling times remained constant throughout the study.     

Effects of temperature on aggregation and the mitogen-induced exit of lymphocytes from the resting state

We have determined the temperature dependence of the kinetics of entry into the first S phase of phytohemagglutinin-stimulated lymphocytes under conditions varying the stability of substrata over which the cells have settled. An exponential model was used to characterize entry into S phase. This model yields as parameters duration of lag period, t₀, apparent first order rate constant for entry, k, and the number of cells committed to enter the first S phase, N_A(t₀). Values of t₀ and N_A(t₀) show a 1.5-fold and 2.0-fold decrease and increase, respectively, over a 4°C temperature range and are independent of variation in substrate stability. The temperature dependence of the apparent first-order rate constant, k, however, is strongly influenced by stability. The observed activation energy increases from 3.0 kcal to 37 kcal when the substratum is agitated. This correlates well with reduced adherence of multicellular aggregates in agitated samples. The temperature dependencies for these three parameters are all numerically different, indicating that these parameters are determined by different rate-limiting processes. We propose that the mechanism mirrored by k is linked to the adherence of multicellular aggregates to the substratum.     

Uncoupling of Methanogenesis from Growth of Methanosarcina barkeri by Phosphate Limitation

Production of methane by Methanosarcina barkeri from H₂-CO₂ was studied in fed-batch culture under phosphate-limiting conditions. A transition in the kinetics of methanogenesis from an exponentially increasing rate to a constant rate was due to depletion of phosphate from the medium. The period of exponentially increasing rate of methanogenesis was extended by increasing the initial concentration of phosphate in the medium. Addition of phosphate during the constant period changed the kinetics to an exponentially increasing rate of methanogenesis, indicating the reversibility of phosphate depletion. The relation between methanogenesis and growth of M. barkeri was investigated by measuring the incorporation of phosphorus, supplied as KH₂³²PO₄, in the medium. At a low (1 μM) initial concentration of phosphate in the medium and during the constant period of methanogenesis, there was no net cell growth. At a higher (10 μM) initial concentration of phosphate, cell growth proceeded linearly with time after phosphate had been removed from the medium by uptake into cells.     

The alkaline transition of turnip peroxidases.

A stopped-flow kinetic study of the spectral changes which appear during the transition between the neutral and the alkaline form of two turnip peroxidases P₁ and P₇ was conducted. With P₁, the kinetics of the spectral changes present three distinct steps: One is fast while the other two are slow. From the pH dependence of the observed rate constant for the fast step, it is proposed that this step might represent deprotonation by the hydroxide anions of a heme-linked group with a pK of 10.1. For P₇, only one step was detected which was fast. In this case, the pH dependence of the observed rate constant indicates that a heme-linked group is deprotonated either by water solvent molecules or by hydroxide anions. The simplest explanation for the observed results is that the groups titrated represent a water molecule in the sixth coordination position of the iron for both peroxidases. The small values of the rate constants found for these deprotonation reactions are explained in terms of hydroxide anion binding to the sixth coordination position of the iron or by the existence of a negatively charged electrostatic gate which prevents negatively charged ligands from entering the heme pocket. A reanalysis of the results reported for a similar study with horseradish peroxidase shows that the alkaline transition of this hemoprotein can also be explained in the same manner as for turnip peroxidases.     

Time-resolved infrared spectroscopy of pH-induced aggregation of the Alzheimer Abeta(1-28) peptide

Aggregation of the Alzheimer's disease-related Aβ_1-28 peptide was induced by a rapid, sub-millisecond pH jump and monitored by time-resolved infrared spectroscopy on the millisecond to second time-scale. The release of protons was induced by the photolysis of a caged compound, 1-(2-nitrophenyl)ethyl sulfate (NPE-sulfate). The pH jump generated in our experimental setup is used to model the Aβ peptide structural conversions that may occur in the acidic endosomal/lysosomal cell compartment system. The aggregation of the Aβ_1-28 peptide induced by the pH jump from 8.5 to < 6 yields an antiparallel β-sheet structure. The kinetics of the structural transition is biphasic, showing an initial rapid phase with a transition from random coil to an oligomeric β-sheet form with a time constant of 3.6 s. This phase is followed by a second slower transition, which yields larger aggregates during 48.0 s.     

CHARACTERISTICS OF THE ISOPRENALINE STIMULATED PROLIFERATIVE RESPONSE OF RAT SUBMAXILLARY GLAND

A simple stochastic model has been developed to determine the cell cycle kinetics of the isoprenaline stimulated proliferative response in rat acinar cells. The response was measured experimentally, using ³H-TdR labelling of interphase cells and cumulative collections of mitotic cells with vincristine. The rise and fall of the fraction of labelled interphase cells and of metaphase cells is expressed by the product of the proliferative fraction and a difference of probability distributions. 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. The model when fitted to the combined fraction labelled interphase (FLIW) and fraction metaphase (FMW,) waves gave a mean G_is transit time of 21-2 hr, mean G_is+ S transit time of 270 hr, and mean G_is+ S + G₂ transit time of 35-8 hr for a single injection of isoprenaline, where G_is is the initiation to S phase time. When successive injections of isoprenaline were given at intervals of 24 and 28 hr the corresponding values after the third injection were 12-4 hr, 20-8 hr and 25-7 hr respectively. The variance of the G_is phase dropped from 18-1 to 1–3 while the other variances remained unchanged. The estimated proliferative fraction was 0–24 after a single injection of isoprenaline, and 0–31 after three injections of the drug. Independently determined values of the proliferative fraction, obtained from repeated ³H-TdR injections, were 0–21 and 0–36 respectively.