Determination of proliferative parameters from growth curves

Abstract. Data on total cell numbers and proportions of non-proliferative cells during the development of a tissue can be obtained by experiment. This paper shows, via a simple mathematical model, that data collected over a period of time determine the time profile of two proliferative parameters. These are the rate per proliferative cell per day at which cells divide and the average number of proliferative daughters which result from the division of a proliferative cell. One dataset alone does not provide sufficient information to determine either time profile. The findings are applicable where there is no significant cell death during the growth period. The mathematical model is fitted to data from chick neural retina.     

Proliferative cell types in embryonic lineages of the central complex of the grasshopper Schistocerca gregaria

The central complex of the grasshopper Schistocerca gregaria develops to completion during embryogenesis. A major cellular contribution to the central complex is from the w, x, y, z lineages of the pars intercerebralis, each of which comprises over 100 cells, making them by far the largest in the embryonic protocerebrum. Our focus has been to find a cellular mechanism that allows such a large number of cell progeny to be generated within a restricted period of time. Immunohistochemical visualization of the chromosomes of mitotically active cells has revealed an almost identical linear array of proliferative cells present simultaneously in each w, x, y, z lineage at 50% of embryogenesis. This array is maintained relatively unchanged until almost 70% of embryogenesis, after which mitotic activity declines and then ceases. The array is absent from smaller lineages of the protocerebrum not associated with the central complex. The proliferative cells are located apically to the zone of ganglion mother cells and amongst the progeny of the neuroblast. Comparisons of cell morphology, immunoreactivity (horseradish peroxidase, repo, Prospero), location in lineages and spindle orientation have allowed us to distinguish the proliferative cells in an array from neuroblasts, ganglion mother cells, neuronal progeny and glia. Our data are consistent with the proliferative cells being secondary (amplifying) progenitors and originating from a specific subtype of ganglion mother cell. We propose a model of the way that neuroblasts, ganglion mother cells and secondary progenitors together produce the large cell numbers found in central complex lineages.     

Correlation between cell size and position within the division cycle in suspension cultures of Chang liver cells

Chang liver cells from exponentially growing suspension cultures have been separated by sedimentation at unit gravity. Determinations of the protein content per cell showed that the fractionation procedure resulted in good separation of cells of different size. On the other hand, the DNA content of individual cells from the fractions, as determined cytofluorimetrically, indicated considerable heterogeneity in the size of cells from the same stage of the division cycle. On the basis of earlier results on intermitotic growth and the variation in the length of the cell cycle in homogeneous cell populations, a mathematical model has been constructed and tested using a computer program. The present results on the size distribution of cells from the different stages of the mitotic cycle are consistent with a regeneration of size heterogeneity in each cell generation, as a result of the dispersion of intermitotic times. The variation in cell cycle times may be related to a probabilistic event in the G1 period. In the mathematical model it was necessary to include a mechanism by which the regeneration of abnormally large cells is prevented. The experimental data are compatible with a gradually increasing inhibition of growth in cells larger than a certain size (circa 400 pg protein per cell).     

CORRELATION BETWEEN CELL SIZE AND POSITION WITHIN THE DIVISION CYCLE IN SUSPENSION CULTURES OF CHANG LIVER CELLS

Chang liver cells from exponentially growing suspension cultures have been separated by sedimentation at unit gravity. Determinations of the protein content per cell showed that the fractionation procedure resulted in good separation of cells of different size. On the other hand, the DNA content of individual cells from the fractions, as determined cytofluorimetrically, indicated considerable heterogeneity in the size of cells from the same stage of the division cycle. On the basis of earlier results on intermitotic growth and the variation in the length of the cell cycle in homogeneous cell populations, a mathematical model has been constructed and tested using a computer program. The present results on the size distribution of cells from the different stages of the mitotic cycle are consistent with a regeneration of size heterogeneity in each cell generation, as a result of the dispersion of intermitotic times. The variation in cell cycle times may be related to a probabilistic event in the G₁ period. In the mathematical model it was necessary to include a mechanism by which the regeneration of abnormally large cells is prevented. The experimental data are compatible with a gradually increasing inhibition of growth in cells larger than a certain size (circa 400 pg protein per cell).     

Changes in dihydrofolate reductase (DHFR) mRNA levels can account fully for changes in DHFR synthesis rates during terminal differentiation in a highly amplified myogenic cell line.

Dihydrofolate reductase (DHFR) enzyme is preferentially synthesized in proliferative cells. A mouse muscle cell line resistant to 300 microM methotrexate was developed to investigate the molecular levels at which DHFR is down-regulated during myogenic withdrawal from the cell cycle. H- alpha R300T cells contained 540 copies of the endogenous DHFR gene and overexpressed DHFR mRNA and DHFR protein. Despite DHFR gene amplification, the cells remained diploid. As H- alpha R300T myoblasts withdrew from the cell cycle and committed to terminal differentiation, DHFR mRNA levels and DHFR synthesis rates decreased with closely matched kinetics. After 15 to 24 h, committed cells contained 5% the proliferative level of DHFR mRNA (80 molecules per committed cell) and synthesized DHFR protein at 6% the proliferative rate. At no point during the commitment process did the decrease in DHFR synthesis rate exceed the decrease in DHFR message. The decrease in DHFR mRNA levels during commitment was sufficient to account fully for the decrease in rates of DHFR synthesis. Furthermore, DHFR mRNA remained polysomal, and the average number of ribosomes per message remained constant (five to six ribosomes per DHFR mRNA). The constancy of polysome size, along with the uniform rate of DHFR synthesis per message, indicated that DHFR mRNA was efficiently translated in postreplicative cells. The results support a model wherein replication-dependent changes in DHFR synthesis rates are determined exclusively by changes in DHFR mRNA levels.     

Kinetic dimensions of cells and crypts in irradiated intestinal epithelium of mice]

The average value of axial cryptal section area and cell section area on it were studied during 8 days after total X-ray irradiation of male mice (400 rad). A small reducing of cryptal area (20%) during destructive period (1-2 days) is followed by a big overshoot (60%) during regenerative time (3-7 days). The cryptal sizes in regenerative period deviate from a steady state more than during destructive time. There are two high waves of abnormal growth of cell sizes above the steady level: the first one during destructive time and the second one during regeneration. This level seems to be near to minimal sizes of cryptal proliferative cells which are necessary for proliferation. It means that normal intestinal epithelium is a very economical and stabilized system. It is possible to evaluate quantitatively the associated with proliferation flow of substance per crypt cell for normal and irradiated intestine by means of index Iv where I is mitotical index and v - the cell volume. Cell hypertrophy at the time of regeneration on the 4th-7th days and later after irradiation (130-160%), was revealed. The crypt cell hypertrophy is the factor of destabilization of irradiated intestinal epithelium.     

Increased urinary polyamine excretion during liver regeneration

Tumor growth is a process associated with both cell proliferation and cell death. The increase in polyamine excretion observed in cancer patients may be partly due to leakage of polyamines from proliferating cells, which all contain an elevated polyamine level. However, the increased polyamine excretion may also be due to a release of polyamines from dead or damaged cells. To determine if actively proliferating cells release polyamines, the urinary polyamine excretion was measured during a proliferative event associated with minimal cell necrosis. Rats subjected to partial hepatectomy were used as an experimental model. Their 24-hr urines were collected during 6 consecutive days following the operation. Rat liver regeneration is characterized by a proliferation wave with a maximum 24 hr after the operation. The 24-hr urinary putrescine excretion reached a maximum 2 days after the operation and then decreased. The 24-hr urinary spermidine excretion increased during the second day following operation and remained essentially unchanged during the rest of the experimental period. Although there is an apparent correlation between elevated urinary polyamine excretion and the proliferative activity, concurrent permeability changes and necrotic events may contribute to the increase in polyamine excretion.     

An analysis of the growth of the retinal cell population in embryonic chicks yielding proliferative ratios, numbers of proliferative and non-proliferative cells and cell-cycle times for successive generations of cell cycles

Growth curves of the retinal cell population of embryonic chicks were fitted by a branching-process model of cell population growth, thereby estimating the proliferative ratios and mean cell-cycle times of the generations of cell cycles that underlie retinal growth. The proliferative ratio determines the proportion of cells that divides in the next generation, so the numbers of proliferative and non-proliferative cells in each generation of cell cycles were obtained. The mean cell-cycle times determine the times over which the generations are extant. Assuming growth starts from one cell in generation 0, the proliferative cells reach 3.6 × 10⁶ and the non-proliferative cells reach 1.1 × 10⁶ by generation 23. The next four generations increase the proliferative cell numbers to 13.9 × 10⁶ and produce 20.1 × 10⁶ non-proliferative cells. In the next five generations in the end phase of growth, non-proliferative cells are produced in large numbers at an average of 13.9 × 10⁶ cells per generation as the retinal lineages are completed. The retinal cell population reaches a maximum estimated here at 98.2 × 10⁶ cells. The mean cell-cycle time estimates range between 6.8 and 10.1 h in generations before the end phase of growth and between 10.6 and 17.2 h in generations in the end phase. The retinal cell population growth is limited by the depletion of the proliferative cell population that the production of non-proliferative cells entails. The proliferative ratios and the cell-cycle-time distribution parameters are the likely determinants of retinal growth rates. The results are discussed in relation to other results of spatial and temporal patterns of the cessation of cell cycling in the embryonic chick retina.     

Mathematical analysis of cancer chemotherapy

This paper presents a mathematical analysis of a tumor model first proposed by Skipper and Zubrod. The tumor model is comprised of three compartments, a proliferative compartment, a nonproliferative but viable compartment, and a dead compartment. By the suitable selection of functions describing loss of cells from the proliferative and nonproliferative compartments, the model is capable of describing tumor behavior during periods of growth and drug treatment. The loss functions during treatment are related to pharmacokinetic functions and may be chosen according to known drug properties. Tumor properties may be simulated by the appropriate choice of cell cycle parameters. It therefore seems feasible to simulate tumor behavior for scheduled treatment with chemotherapeutic agents. Another important result of this analysis is the derivation of a fraction labelled mitosis function which incorporates the nonproliferative compartments.     

Damaging agitation intensities increase DNA synthesis rate and alter cell-cycle phase distributions of CHO cells

The effects of fluid-mechanical force (agitation) on the cell cycle kinetics of Chinese hamster ovary (CHO) cells cultured in suspension in 2-L bioreactors has been examined. A two-color flow cytometry method was used to determine the fraction rate of DNA synthesis. With increased agitation intensity, cell viability decreased as a result of increased cell death. However, increased agitation induced the viable cells of the culture to a higher proliferative state relative to a control culture. The fraction of viable cells of the high-agitation culture (250 rpm) in S phase was higher (up to 45%) and in G1 phase was lower (up to 50%) compared with the viable cells of the control culture (80 rpm). The DNA synthesis rate per viable S-phase cell of the high-agitation culture was confirmed by recovery experiments, which were conducted to measure the apparent specific growth rate and the cell cycle kinetics of the high-agitation culture upon reduction in the agitation rate from 250 rpm back to 80 rpm. The apparent specific growth rate of the test culture, calculated for the first 12 h of the recovery period, was greater than the apparent specific growth rate of the control culture. Furthermore, the proliferative state of the viable cells of the test culture, which had become higher relative to the control culture during the high agitation period, gradually approached the level of the control culture during recovery. Results also show that the magnitude of the agitation intensity; the culture agitated at 250 rpm attained a greater proliferative state than a parallel culture agitated at 235 rpm. The 250-rpm culture had a higher fraction of S-phase and a lower fraction of G1-phase cells than the 235-rpm culture. The DNA sunthesis rate per viable S-phase cell of the 250-rpm culture was greater than of the 235-rpm culture. (c) 1992 John Wiley & Sons, Inc.     

Ca-alginate hydrogel mechanical transformations--the influence on yeast cell growth dynamics

A mathematical model was formulated to describe yeast cell growth within the Ca-alginate microbead during air-lift bioreactor cultivation. Model development was based on experimentally obtained data for the intra-bead cell concentration profile, after reached the equilibrium state, as well as, total yeast cell concentration per microbed and microbead volume as function of time. Relatively uniform cell concentration in the carrier matrix indicated that no internal nutrient diffusion limitations, but microenvironmental restriction, affected dominantly the dynamics of cell growth. Also interesting phenomenon of very different rates of cell number growth during cultivation is observed. After some critical time, the growth rate of cell colonies decreased drastically, but than suddenly increased again under all other experimental condition been the same. It is interpreted as disintegration of gel network and opening new free space for growth of cell clusters. These complex phenomena are modeled using the thermodynamical, free energy formalism. The particular form of free energy functional is proposed to describe various kinds of interactions, which affected the dynamics of cell growth and cause pseudo-phase transition of hydrogel. The good agreement of experimentally obtained data and model predictions are obtained. In that way the model provides both, the quantitative tools for further technological optimization of the process and deeper insight into dynamics of cell growth mechanism.     

A model of the control of cellular regeneration in the intestinal crypt after perturbation based solely on local stem cell regulation

Abstract. The control mechanisms involved in regeneration of murine intestinal crypts after perturbations are presently not well understood. The existence of some feedback signals from the cells on the villus to the cells in the crypt has been suggested. However, some recent experimental data point to the fact that regeneration in the crypt starts very early after perturbation, at a time when the villus cell population has hardly changed. In particular, this early cell proliferative activity is seen specifically at the bottom of the crypt, i.e. in the presumed stem cell zone and furthest from the villus.
The objective of this study was to investigate whether a new concept of regulation operating solely at the stem cell level could explain the present mass of accumulated data on the post-irradiation recovery, which is an extensively studied perturbation from the experimental point of view. In order to check its validity, the new concept was formalized as a mathematical simulation model thus enabling comparison with experimental data. The model describes the cellular development from stem cells to the mature villus cells. As a basic feature it is assumed that the self-maintenance and the cell cycle activity of the stem cells are controlled by the number of these cells in an autoregulatory fashion. The essential features of the experimental data (i.e. the recovery with time and the consistency between different types of measurements) can be very well reproduced by simulations using a range of model parameters. Thus, we conclude that stem cell autoregulation is a valid concept which could replace the villus crypt feedback concept in explaining the early changes after irradiation when the damage primarily affects the crypt. The question of the detailed nature of the control process requires further investigation.     

Growth and division kinetics of asymmetrically dividing Tetrahymena thermophila

The growth and division kinetics of the asymmetrically dividing mutant strain conical of Tetrahymena thermophila are discussed in terms of a simple mathematical model which predicts the relationship between the division times of the asymmetric daughter cells and the doubling time of the cell population as a whole. The average protein and RNA content per cell can be obtained from a knowledge of the time-dependence of polymer biosynthesis during the cell cycle.     

The impact of phenotypic switching on glioblastoma growth and invasion

The brain tumour glioblastoma is characterised by diffuse and infiltrative growth into surrounding brain tissue. At the macroscopic level, the progression speed of a glioblastoma tumour is determined by two key factors: the cell proliferation rate and the cell migration speed. At the microscopic level, however, proliferation and migration appear to be mutually exclusive phenotypes, as indicated by recent in vivo imaging data. Here, we develop a mathematical model to analyse how the phenotypic switching between proliferative and migratory states of individual cells affects the macroscopic growth of the tumour. For this, we propose an individual-based stochastic model in which glioblastoma cells are either in a proliferative state, where they are stationary and divide, or in motile state in which they are subject to random motion. From the model we derive a continuum approximation in the form of two coupled reaction-diffusion equations, which exhibit travelling wave solutions whose speed of invasion depends on the model parameters. We propose a simple analytical method to predict progression rate from the cell-specific parameters and demonstrate that optimal glioblastoma growth depends on a non-trivial trade-off between the phenotypic switching rates. By linking cellular properties to an in vivo outcome, the model should be applicable to designing relevant cell screens for glioblastoma and cytometry-based patient prognostics.     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.     

Cell cycle of normal bladder urothelium in developing and adult mice

The present research has employed a novel, nonradioactive technique to quantitatively study normal urothelial proliferation in foetal, neonatal, juvenile and adult mouse bladder. Using whole mount histological preparations, the total number of urothelial nuclei per mouse bladder, and per given urothelial cell layer, have been assessed to provide data of the (unstimulated) kinetic behaviour of basal urothelial cells (the proliferative population), to analyse characteristics of the normal urothelial cell cycle. The urothelial cell cycle time increases from 30.6 h (foetal) to 40 weeks (adult), the duration of mitosis from 0.23 h (foetal) to 2.71 h (adult) and the duration of DNA synthesis from 2.52 h (neonatal) to 10.83 h (adult). These are average values for the urothelial cell cycle, which do not preclude the possible existence of proliferative units. The ratio of superficial nuclei to basal and intermediate nuclei, possibly indicative of a urothelial proliferative unit, declines to reach a plateau (1:40) in adult mice. These findings indicate that rapid urothelial proliferation during early murine development was likely to be a) biologically useful, since intrauterine foetal metabolic activity may require a functional bladder urothelium at an early stage, b) kinetically similar to acutely regenerating adult urothelial cells after cytotoxic insult. During murine life, the range of durations of mitosis and DNA synthesis is much less than the range of cell cycle times. Normal unstimulated urothelium of adult mice was confirmed to proliferate slowly.     

The Spacelab 3 simulation: basis for a model of growth plate response in microgravity in the rat

Data from Spacelab 3 (SL3) suggested that spaceflight significantly reduces the activity of the rat tibial growth plate. Animal processing after SL3 began twelve hours post-landing, so data reflect post-flight re-adaptation in addition to spaceflight effects. To determine if a twelve-hour period of weight bearing after seven days of unloading could affect the physes of spaceflown rats, the present study assessed the growth plate response to unloading with or without a reloading period. Rats were subjected to hind-limb suspension for seven days and then euthanized, with or without twelve hours of reloading. Activity of the growth plate was assessed by morphometric analysis. Rats suspended without reloading had reserve zone (RZ) height greater than controls, and shorter hypertrophy/calcification zone (HCZ) with fewer cells. The greater RZ was associated with a larger cell area, indicating a possible mitotic delay or secretion defect. Twelve hours of reloading decreased RZ height and cell number, and restored the number of cells in HCZ to control values, but the number of cells in the proliferative zone and height in HCZ were reduced. These results suggest the rebound response to preserve/restore skeletal function after a period of unloading involves an acceleration of growth associated with a decreased cell cycle time in PZ. Changes during the reloading period in this simulation support our hypothesis that the effects of spaceflight on SL3 growth plates were altered by changes that occurred post-landing. The similarities in response to unloading by suspension or during spaceflight are used to propose a model of growth plate response during spaceflight.     

A neighboring-inhibition model for stomate patterning

The patternization of stomate distribution was investigated in the first leaf of the sporophyte of the fern Dryopteris thelypteris and in the leaves of the jade plant Crassula argentae. In the fern leaf, stomates arise over a period of 1.1 days (26.4 hr) and attain a frequency of 0.186 of the epidermal cells while in the jade plant the formative period is over only 0.8 days (19.2 hr) and stomates reach a frequency of 0.090 of the epidermal cells. A computer model was devised to simulate the appearance of stomates by an induction process for a new stomate which then inhibits contiguous cells from becoming stomates. The validity of the model was demonstrated in that it gave values for stomate and stomate cluster frequencies upon calculating the frequency of free cells. For the fern leaf, the model required 57 iteration intervals, six iterations for the time from stomate induction until the adjacent cells were inhibited (a period of induction plus inhibitions), and an induction rate of 0.02 cells per time interval. From these theoretical values and the measured period of 26.4 hr when stomates arise during leaf development, a period of induction/inhibitions is calculated as 2.7 hr. In the model for the jade plant, 81 iteration intervals are required with six iterations per period of induction/inhibitions along with an induction rate of 0.002 cells per interation interval. These values give the duration for a period of induction/inhibitions in jade of 1.4 hr. This study describes the patterning process of stomate formation by an explicit, mathematical algorithm, and, from measurements of actual leaves, the various periods in the model can be assigned real time values.     

Bimodal effect of phorbol ester on B cell activation. Implication for the role of protein kinase C.

The role of protein kinase C PKC in B cell activation is controversial. These studies were undertaken to determine whether protein kinase C has a stimulatory or inhibitory role in B cell activation. We found that treatment of B cells for a short period of time (30 min) with the PKC activator phorbol 12,13-dibutyrate (PDBU) primed the cells for enhanced proliferative responses to anti-immunoglobulin (anti-Ig) antibody whereas treatment for a longer period of time (3 h or more) resulted in suppression of proliferation. The enhanced proliferative response to treatment of B cells with PDBU for short periods of time was associated with inhibition of anti-Ig-stimulated increases in phosphatidyl 4,5-bisphosphate (PIP2) hydrolysis and inhibition of increases in [Ca2+]i, indicating that activation of PKC per se might be sufficient for enhancing B cell activation. The time-dependent effect of phorbol esters on the inhibition of B cell proliferation was found to be closely correlated with the kinetics of disappearance of PKC as measured by Western blot and by enzymatic activity but not with inhibition of [Ca2+]i and PIP2. These data demonstrate a bimodal time-dependent effect of PDBU on B cell activation and suggest that (a) the inhibitory effect of phorbol ester on anti-Ig-induced proliferation may be due to the disappearance of PKC rather than to the inhibition of PIP2 and Ca2+; and (b) the early activation of PKC is a stimulatory rather than an inhibitory signal in the induction of B lymphocyte proliferation by anti-Ig.