Synchronous cultures from the baby machine: anatomy of a model

The baby-machine system, which produces newborn Escherichia coli cells from cultures immobilized on a membrane, was developed many years ago in an attempt to attain optimal synchrony with minimal disturbance of steady-state growth. In the present article, we describe in some detail a model designed to analyse such cells with a view to characterizing the nature and quality of the synchrony in a quantitative manner; it can also serve to evaluate the methodology itself, its potential and its limitations. The model consists of five elements, giving rise to five adjustable parameters (and a proportionality constant): a major, essentially synchronous group of cells with ages distributed normally about zero; a minor, random component from a steady-state population on the membrane that had undergone only very little age selection during the elution process; a fixed background count, to account for the signals recorded by the electronic particle counter produced by debris and electronic noise; a time-shift, to allow for differences between collection time and sampling time; and the coefficient of variation of the interdivision-time distribution, taken to be a Pearson type III. The model is fitted by nonlinear least-squares to data from cells grown in glucose minimal medium. The standard errors of the parameters are quite small, making their estimates all highly significant; the quality of the fit is striking. We also provide a simple yet rigorous procedure for correcting cell counts obtained in an electronic particle counter for the effect of coincidence. An example using real data produces an excellent fit.     

Cell-cycle research with synchronous cultures: an evaluation

The baby-machine system, which produces new-born Escherichia coli cells from cultures immobilized on a membrane, was developed many years ago in an attempt to attain optimal synchrony with minimal disturbance of steady-state growth. In the present article, we put forward a model to describe the behaviour of cells produced by this method, and provide quantitative evaluation of the parameters involved, at each of four different growth rates. Considering the high level of selection achievable with this technique and the natural dispersion in interdivision times, we believe that the output of the baby machine is probably close to optimal in terms of both quality and persistence of synchrony. We show that considerable information on events in the cell cycle can be obtained from populations with age distributions very much broader than those achieved with the baby machine and differing only modestly from steady state. The data presented here, together with the long and fruitful history of findings employing the baby-machine technique, suggest that minimisation of stress on cells is the single most important factor for successful cell-cycle analysis.     

Periodic forcing of microbial cultures: a model for induction synchrony

Periodic environmental shifts have been used to induce synchrony in many different microbial populations. In this article, the induction synchrony phenomenon is analyzed using an age distribution model in which the age at which the cells divide is subjected to periodic forcing. It is found that synchrony will occur whenever the period of the forcing lies in the interval between the youngest and the oldest division age that occur in the population during the forcing. The analysis also predicts that under certain conditions it should be possible to obtain a multimodal synchrony in which cells in the population are distributed among a set of discrete, synchronized cell lines. The behavior of the age distribution when the conditions for synchrony are not satisfied is briefly explored. It is found that the age distribution model is able to exhibit a very rich spectrum of possible dynamic behavior. Many of the phenomena observed can be thought of in terms that are familiar from nonlinear analysis, such as stable and unstable limit cycles, period doubling, halving, and chaos. The richness of dynamic behavior opens the possibility that environmental shifts or periodic forcing could be used as a powerful tool in discriminating models of microbial kinetics and cell cycle control.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Investigation into the experimental kinetic support of the two-state model of the cell cycle

Five previously published cell generation-time distribution functions have been examined in an effort to elucidate the parameters of the two-state model of the cell cycle. These parameters are the fractional number of cells that bypass the G₀ state, the probability of exit from G₀ and the distribution of traversal times through the active state. To explain observed β-curve behavior of cell populations, it is necessary to define the parameters in terms of pairwise behavior of newborn sister cells. From the β-curve, we demonstrate that at least 50% of the cells must pass through the G₀ state. The α-curve is consistent with any positive fraction of newborn cells passing through the G₀ state, and provides no further information. We explore a possible method for resolving the remaining indeterminacy regarding the number of cells bypassing the G₀ state, namely, examination of the generation-time distribution functions of fast sister cells only. Such an approach, although theoretically attractive, presents formidable experimental difficulties, however. If it should turn out that indeed only 50% of the cells are apparently passing through a randomexiting phase of the cell cycle, then an alternative plausible biological mechanism for the observed variability in generation times is supplied by Prescott's hypothesis: variability is a consequence of the inequality in the metabolic content of sister cells at birth.     

A new mathematical approach predicts individual cell growth behavior using bacterial population information

A theoretical methodology has been developed for studying the growth kinetics of bacterial cells. It utilizes the steady-state cell length distribution in a bacterial population to predict the dependency of growth and division rates on cell length and age. The mathematical model has been applied to the analysis of two bacterial populations, a wild-type strain of Bacillus subtilis, and a minicell-producing strain that carries the divIVB1 mutation. The results show that our model describes the wild-type population very well and that the assumptions typically used in traditional methods are unrealistic. In the case of the minicell-producing mutant we find evidence that the rate of cell division must be a function not only of cell size but also of cell age.     

SYNCHRONOUS GROWTH OF CHLAMYDOMONAS REINHARDTII (CHLOROPHYCEAE): A REVIEW OF OPTIMAL CONDITIONS1

Chlamydomonas reinhardtii Dangeard was synchronized at optimal growth conditions under a 12:4 LD regime at 35 C and 20,000 lx with serial dilution to a standard starting cell density of (1.4 ± 0.2) × 10⁶ cells/ml. Synchronous growth and division were characterized by measuring cell number, cell volume and size distribution, dry weight, protein, carbon, nitrogen, chlorophyll, carotenoids, nucleic acids, nuclear and cytoplasmic division during the vegetative life cycle. The main properties of the present system are: Exponential growth with high productivity, high degrees of synchrony and reproducibility during repeated life cycles. The degree of synchrony of this light-dark synchronization system was evaluated and compared with those described in the literature using probit analysis of the time course of DNA synthesis, nuclear and cytoplasmic division and sporulation (increase in cell number). The results showed that the degree of synchrony is highest for cells grown under optimal conditions.     

Cell cycle model for growth rate and death rate in continuous suspension hybridoma cultures

As a result of recent advances in flow cytometry, renewed interest is shown in modeling the kinetic behavior of cells in culture on the basis of cell cycle parameters. An important but often overlooked kinetic variable in hybridoma cultures is the cell death rate. Not only the overall cell growth but also the kinetics of nutrient metabolism and monoclonal antibody production have been shown to depend on the cell death rate in continuous suspension hybridoma cultures. The present study shows that the death rate in hybridoma cultures is proportional to the fraction of cells arrested in the G(1) phase of the cell cycle. The steady-state cell age distributions in the various phases of the division cycle have been calculated analytically. A simple mathematical model has been used to produce the profiles of the cycling and arrested cell fractions with respect to the dilution rate. The calculated steady-state growth rate, death rate, and viability profiles are shown to be in agreement with recently published experimental data from continuous suspension hybridoma cultures. (c) 1992 John Wiley & Sons, Inc.     

Investigation into the experimental kinetic support of the two-state model of the cell cycle

Five previously published cell generation-time distribution functions have been examined in an effort to elucidate the parameters of the two-state model of the cell cycle. These parameters are the fractional number of cells that bypass the G0 state, the probability of exit from G0, and the distribution of traversal times through the active state. To explain observed beta-curve behavior of cell populations, it is necessary to define the parameters in terms of pairwise behavior of newborn sister cells. From the beta-curve, we demonstrate that at least 50% of the cells must pass through the G0 state. The alpha-curve is consistent with any positive fraction of newborn cells passing through the G0 state, and provides no further information. We explore a possible method for resolving the remaining indeterminacy regarding the number of cells bypassing the G0 state, namely, examination of the generation-time distribution functions of fast sister cells only. Such an approach, although theoretically attractive, presents formidable experimental difficulties, however. If it should turn out that indeed only 50% of the cells are apparently passing through a random-exiting phase of the cell cycle, then an alterative plausible biological mechanism for the observed variability in generation times is supplied by Prescott's hypothesis: variability is a consequence of the inequality in the metabolic content of sister cells at birth.     

Surface-limited growth: a model for the synchronization of a growing bacterial culture through periodic starvation

This article analyses the Surface-Limited Growth Model put forward to explain the very tight synchrony, over more than ten division cycles, obtained experimentally by subjecting a growing bacterial culture to alternating periods of starvation and dilution, using inorganic phosphate as the limiting substrate. The Model states that when an essential nutrient is in limited supply, the rate of growth of an individual cell will be proportional to its surface area (and the current concentration of the limiting substance) rather than to its volume. This decrease in dimensionality from volume to surface is expected to favor the smaller cells and so result ultimately in a narrower size distribution. The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions, and its predictions depend on how the cell replication cycle is assumed to behave under these same circumstances. Two alternatives are considered: the volume at which cells divide is the same during the starvation phase as during steady-state exponential growth, and the cells adjust immediately to the changing growth rate. In the latter case, we have tested both C + D constant with time and C + D variable (where C + D is the time between initiation of chromosome replication and the corresponding cell division), the incremental value at any instant being computed separately for each individual cell from its current effective growth rate. The simulation results are of two sorts depending on the auxiliary assumptions used. Either the dilution-starvation cycles have no effect whatsoever on the cell volume distribution, or the width of the distribution decreases gradually with time, approaching zero slowly and asymptotically, but the mean cell volume decreases as well--directly contradicting experimental observations. We conclude that the Surface-Limited Growth Model is incapable of explaining the synchronization of cells by periodic starvation of a growing bacterial culture.     

Desynchronization rate in cell populations: mathematical modeling and experimental data

We characterize the kinetics of two cancer cell lines: IGROV1 (ovarian carcinoma) and MOLT4 (leukemia). By means of flow cytometry, we selected two populations from exponentially growing in vitro cell lines, depending on the cells' DNA synthesis activity during a preceding labeling period. For these populations we determined the time course of the percentages of cells in different phases of the cycles, sampling every 3 hr for 60 hr. Initially, semi-synchronous populations quickly converged to a stable age distribution, which is typical of the cell line (at equilibrium); this desynchronization reflects the intercell variability in cell cycle duration. By matching these experimental observations to mathematical modelling, we related the convergence rate toward the asymptotic distribution (R) and the period of the phase-percentage oscillations (T), to the mean cell cycle duration and its coefficient of variation. We give two formulas involving the above-mentioned parameters. Since T and R can be drawn by fitting our data to an asymptotic formula obtained from the model, we can estimate the other two kinetic parameters. IGROV1 cells have a shorter mean cell cycle time, but higher intercell variability than the leukemia line, which takes longer to lose synchrony.     

Red cells of newborn rats have low bisphosphoglyceromutase and high pyruvate kinase activities in association with low 2,3-bisphosphoglycerate

2,3-Bisphosphoglycerate is a physiologically important regulator of red cell oxygen affinity during mammalian development. The rat has no fetal hemoglobin, but the newborn red cell has low 2,3-bisphosphoglycerate and high ATP concentrations, and high oxygen affinity. This report shows that red cell bisphosphoglyceromutase activity increases from near zero in the newborn rat to very high levels by four weeks of age. This increase roughly parallels the increase in red cell 2,3-bisphosphoglycerate concentration. Red cell pyruvate kinase activity declines ten-fold from birth to four weeks of age. This decrease is associated with a changeover in red cell populations from larger to smaller cells. The glycolytic rate is at least 50% higher in newborn than adult rat red cells. The data suggest that high pyruvate kinase activity and glycolytic rate contribute to the high ATP concentration in newborn rat red cells, but that their low 2,3-bisphosphoglycerate concentration is due primarily to low bisphosphoglyceromutase activity.     

Effect of temperature on the size of Escherichia coli cells.

The distributions of cell volumes of steady-state Escherichia coli ML30G cultures at various temperatures were measured. For cultures in a minimal medium, the distributions were indistinguishable at several temperatures between 15 and 30 C; at higher temperatures the cells were slightly smaller, and at lower temperatures they were slightly larger. For cultures in a complex medium, the cells were slightly larger at both high and low temperatures of growth. An abrupt change of temperature within the middle range led to a transient change in the distribution of cell volume, suggesting that the size of dividing cells is well regulated. No synchrony of division was induced by a change in temperature.     

Cell Synchrony Techniques. I. A Comparison of Methods

Abstract Selected cell synchrony techniques, as applied to asynchronous populations of Chinese hamster ovary (CHO) cells, have been compared. Aliquots from the same culture of exponentially growing cells were synchronized using mitotic selection, mitotic selection and hydroxyurea block, centrifugal elutriation, or an EPICS V cell sorter. Sorting of cells was achieved after staining cells with Hoechst 33258. After synchronization by the various methods the relative distribution of cells in G₁ S, or G₂+ M phases of the cell cycle was determined by flow cytometry. Fractions of synchronized cells obtained from each method were replated and allowed to progress through a second cell cycle. Mitotic selection gave rise to relatively pure and unperturbed early G₁ phase cells. While cell synchrony rapidly dispersed with time, cells progressed through the cell cycle in 12 hr. Sorting with the EPICS V on the modal G₁ peak yielded a relatively pure but heterogeneous G₁ population (i.e. early to late G₁). Again, synchrony dispersed with time, but cell-cycle progression required 14 hr. With centrifugal elutriation, several different cell populations synchronized throughout the cell cycle could be rapidly obtained with a purity comparable to mitotic selection and cell sorting. It was concluded that, either alone or in combination with blocking agents such as hydroxyurea, elutriation and mitotic selection were both excellent methods for synchronizing CHO cells. Cell sorting exhibited limitations in sample size and time required for synchronizing CHO cells. Its major advantage would be its ability to isolate cell populations unique with respect to selected cellular parameters.     

Synchronization in chains of pacemaker cells by phase resetting action potential effects

Interactions between pacemaker cells in a chain were calculated according to a "phase-reset" model. It is based on effects of action potentials in the cells on the cycle lengths of neighbouring cells. These effects were defined for each cell by a latency-phase curve (LPC), giving the latency time (L) until the onset of the next action potential in that cell, as a function of the phase (phi) at which a neighbour cell fired an action potential. Neighbour cells with simultaneous action potentials did not influence each others cycle length. We investigated how stable synchronization depends on the shape of the LPC's of the pacemaker cells and on chain length. Three types of interactive behaviour were distinguished. First, anti-phase synchrony, in which neighbouring cells fired with large phase differences with respect to the synchronized period Ps. Second, asynchrony, in which the periods of the cells did not become equal and constant. Third, in-phase synchrony, in which the phase differences between the neighbouring cells were zero or much smaller than the synchronized period Ps, depending on the differences between the intrinsic periods. Asynchrony and anti-phase synchrony may be seen as cardiophysiological arrhythmias, while in-phase synchrony represents the physiological type of synchrony in the heart. In-phase synchrony appeared to be strongly favoured by LPC's, which have a no-effect (refractory) part at early phases, a lengthened latency (or phase delay) part at intermediate phases and a shortened latency (or phase advance) part at late phases in the cycle. Such LPC-shapes are commonly found in preparations of cardiac pacemaker cells. When the pacemaker cells were identical, the synchronized period Ps during in-phase synchrony was equal to their intrinsic period P*i. For different intrinsic periods, Ps was equal to the intrinsic period of the fastest cell if the LPC's contained a sufficiently long initial no-effect period at early phases and a shortened latency part at late phases. When, on the other hand, such cell chains had a linear gradient in their intrinsic periods, "action potentials" started from the fast end and traveled along the chain. The propagation of an action potential wave slowed down as it reached the slower cells. When the gradient in the intrinsic periods was too steep, only the intrinsically fast end of the chain developed synchrony.(ABSTRACT TRUNCATED AT 400 WORDS)     

THE DISCRETE–TIME KINETIC MODEL ANALYSIS OF DNA CONTENT DISTRIBUTIONS IN EXPERIMENTAL TUMOUR CELLS

A method was developed to analyse and characterize FMF measurements of DNA content distribution, utilizing the discrete time kinetic (DTK) model for cell kinetics analysis. The DTK model determines the time sequence of the cell age distribution during the proliferation of a tumor cell population and simulates the distribution pattern of the DNA content of cells in each age compartment of the cell cycle. The cells in one age compartment are distributed and spread into several compartments of the DNA content distribution to allow for different rates of DNA synthesis and instrument dispersion effects. It is assumed that the DNA content of cells in each age compartment has a Gaussian distribution. Thus, for a given cell age distribution the DNA content distribution depends on two parameters of the cells in each age compartment: the average DNA content and its coefficient of variation. As the DTK model generates the best fit DNA content distribution to the FMF measurement data, it enables one to estimate specific values of these two parameters in each stage of the cell cycle and to determine the fraction of cells in each cycle phase. The method was utilized to fit FMF measurements of DNA content distributions and to analyse their relationship to the cell kinetic parameters, namely cell loss rate, cell cycle times and growth fraction of exponentially growing Chinese hamster ovary cells in vitro and, also, with a wide range of coefficients of variation, of the L1210 ascites tumour during the growth period.     

Technology for cell cycle research with unstressed steady-state cultures

A culture system for performing cell cycle analyses on cells in undisturbed steady-state populations was designed and tested. In this system, newborn cells are shed continuously from an immobilized, perfused culture rotating about the horizontal axis. As a result of this arrangement, the number of newborn cells released into the effluent medium each generation is identical to the number of cells residing in the immobilized population, indicating that one of the two new daughter cells is shed at each cell division. Thus, the immobilized cells constitute a continuous, steady-state culture because the concentrations, locations and microenvironments of the cells in the culture vessel do not vary with time. In tests with mouse L1210 lymphocytic leukemia cells, about 10⁸ newborn cells were produced per day. This new culture system enables a multiplicity of cell cycle analyses on large numbers of cells assured to be from populations in steady-state growth.