Analysis of cell differentiation by division tracking cytometry.

We propose a quantitative method to characterize growth and differentiation dynamics of multipotent cells from time series carboxyfluorescein diacetate, succinimidyl ester (CFDA-SE) division tracking data. The dynamics of cell proliferation and differentiation was measured by combining (CFDA-SE) division tracking with phenotypic analysis. We define division tracking population statistics such as precursor cell frequency, generation time and renewal rate that characterize growth of various phenotypes in a heterogeneous culture system. This method is illustrated by study of the divisional recruitment of cord blood CD34(+) cells by hematopoietic growth factors. The technical issue of assigning the correct generation number to cells was addressed by employing high-resolution division tracking methodology and daily histogram analysis. We also quantified division-tracking artifacts such as CFDA-SE degeneration and cellular auto-fluorescence. Mitotic activation of cord blood CD34(+) cells by cytokines commenced after 2 days of cytokine stimulation. Mean generation number increased linearly thereafter, and it was conclusively shown that CD34(+) cells cycle slower than CD34(-) cells. Generation times for CD34(+) and CD34(-) cells were 24.7 +/- 0.8 h and 15.1 +/- 0.9 h (+/-SD, n = 5), respectively. The 20-fold increase in CD34(+) cell numbers at Day 6 could be attributed to a high CD34(+) cell renewal rate (91% +/- 2% per division). Although cultures were initiated with highly purified CD34(+) cells (approximately 96%), CD34(-) numbers had expanded rapidly by Day 6. This rapid expansion could be explained by their short generation time as well as a small fraction of CD34(+) cells (approximately 5%) that differentiated into CD34(-) cells. Multitype division tracking provides a detailed analysis of multipotent cell differentiation dynamics.     

Multi-type branching models to describe cell differentiation programs

Cell proliferation and differentiation is described by a multi-type branching process, a probability model that defines the inheritance of cell type. Cell type is defined by (i) a repression index related to the time required for S-phase entry and (ii) phenotype as determined by cell markers and division history. The inheritance of cell type is expressed as the expected number and type of progeny cells produced by a mother cell given her type. Expressions for the expected number and type of cells produced by a multi-cellular (bulk culture) system are derived from the general model by making the simplifying assumption that cell generation times are independent. The multi-type Smith-Martin model (MSM) makes the further assumption that cell generation times are lag-exponentially distributed with phenotype transitions occurring just before entry into S-phase. The inheritance-modified MSM (IMSM) model includes the influence of generation time memory so that mother and daughter generation times are correlated. The expansion of human cord blood CD34⁺ cells by haematopoietic growth factors was division tracked in bulk culture using carboxyfluorescein diacetate, succinimidyl ester (CFDA-SE). The MSM model was fitted to division tracking data to indentify cell cycle length, and the rates of CD34 antigen down-regulation and apoptosis. The IMSM model was estimated for mouse granulocyte-macrophage progenitors using live cell imaging data. Multi-type branching models describe cell differentiation dynamics at both single- and multi-cell scales, providing a new paradigm for systematic analysis of stem and progenitor cell development.     

Age-dependent cell cycle models.

This paper presents age-dependent cell cycle models i.e., models where cell generation time is a random variable given by some distribution function, and the probability of cell division per unit time is a function only of cell age (not, for example, of cell mass). It is shown that there does not exist a stable mass distribution if the cells grow exponentially. In the case of linear growth, conditions for stability of the mass distribution are derived. To show these, the methods different from those considered up till now in the literature, are used. It is also shown that one can consider the cell mass growth as a linear dynamical system with a stochastic perturbation. The sister cell model as an improvement of the Transition Probability Model is derived. Statistical data are obtained for that model, and comparisons are made with some experimental data. As a verification tool, alpha and beta curves, are used.     

A computational model for telomere-dependent cell-replicative aging

Telomere shortening provides a molecular basis for the Hayflick limit. Recent data suggest that telomere shortening also influence mitotic rate. We propose a stochastic growth model of this phenomena, assuming that cell division in each time interval is a random process which probability decreases linearly with telomere shortening. Computer simulations of the proposed stochastic telomere-regulated model provides good approximation of the qualitative growth of cultured human mesenchymal stem cells.     

The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions

A deterministic/probabilistic model of the cell division cycle is analysed mathematically and compared to experimental data and to other models of the cell cycle. The model posits a random-exiting phase of the cell cycle and a minimum-size requirement for entry into the random-exiting phase. By design, the model predicts exponential "beta-curves", which are characteristic of sister cell generation times. We show that the model predicts "alpha-curves" with exponential tails and hyperbolic-sine-like shoulders, and that these curves fit observed generation-time data excellently. We also calculate correlation coefficients for sister cells and for mother-daughter pairs. These correlation coefficients are more negative than is generally observed, which is characteristic of all size-control models and is generally attributed to some unknown positive correlation in growth rates of related cells. Next we compare theoretical size distributions with observed distributions, and we calculate the dependence of average cell mass on specific growth rate and show that this dependence agrees with a well-known relation in bacteria. In the discussion we argue that unequal division is probably not the source of stochastic fluctuations in deterministic size-control models, transition-probability models with no feedback from cell size cannot account for the rapidity with which the new, stable size distribution is established after perturbation, and Kubitschek's rate-normal model is not consistent with exponential beta-curves.     

Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data

We consider an age-maturity structured model arising from a blood cell proliferation problem. This model is "hybrid", i.e., continuous in time and age but the maturity variable is discrete. This is due to the fact that we include the cell division marker carboxyfluorescein diacetate succinimidyl ester. We use our mathematical analysis in conjunction with experimental data taken from the division analysis of primitive murine bone marrow cells to characterize the maturation/proliferation process. Cell cycle parameters such as proliferative rate beta, cell cycle duration tau, apoptosis rate gamma, and loss rate micro can be evaluated from CarboxyFluorescein diacetate Succinimidyl Ester + cell tracking experiments. Our results indicate that after three days in vitro, primitive murine bone marrow cells have parameters beta = 2.2 day(-1), tau = 0.3 day, gamma = 0.3 day(-1), and micro = 0.05 day(-1).     

Elongation of rod-shaped bacteria.

Three models relating cell length to generation time are considered for rod-shaped bacteria growing under steady-state conditions; all three presuppose linear elongation. The first model assumes that the rate of elongation is proportional to the instantaneous number of chromosome replication forks per cell; the others, that it is inversely related to the generation time and doubles a fixed time prior to cell division. One of these (model 2) treats this relationship as continuous, with the doubling occurring during the last division cycle (at chromosome termination), while the other is a discrete model in which the doubling in rate takes place at chromosome initiation. Expressions are derived for mean cell length and length at birth in each case.Comparison with experimental data on E. coli B/r using non-linear least-squares techniques results in an excellent fit for model 2 and unsatisfactory ones for the others, the best estimate for the time at which the rate doubles being 15·3 min prior to cell division and for the minimum length at birth (i.e., as the growth rate of the culture tends to zero), 1·47 μm.The functional relationship between cell radius and generation time implied by model 2 is also presented. This model again produces a good fit to the experimental data and provides, for the first time, a direct estimate of the volume/origin ratio at initiation of chromosome replication 0·35 ± 0·05 μm³ (s.e.).The results obtained here are compared with various qualitative observations reported in the literature and with such numerical data as are available.     

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).     

Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model.     

A comprehensive model of the crypts of the small intestine of the mouse provides insight into the mechanisms of cell migration and the proliferation hierarchy

A comprehensive model has been formulated for the proliferative behaviour of the crypts of the small intestine based on individual cell to cell relationships rather than on the average effects of all cells. The model accommodates a wide range of cell kinetic data and provides an insight into the mechanisms involved in cell movement within the columnar sheet of cells and into the relationship between the stem cells and their progeny. The model permits the number of stem cells and transit generations to be estimated. The number of stem cells is predicted to be not less than 4 and not more than 16 per crypt with cell cycle times of between 12 and 32 h respectively. Certain conclusions can be drawn concerning the mechanisms involved in the initial cell displacements after cell division. The model also allows an estimation of parameters which cannot be measured directly such as the degree of cell generation disorder and the amount of dispersion of cells within a cell lineage.     

A growth and division model for retinoblastoma

A stochastic growth and division model for studying a two hit cancer is developed and applied to retinoblastoma. Retinoblastoma occurs if both genes coding for a tumor suppressor protein on homologous chromosomes become defective. Germinal cases occur when a patient or carrier, born with one defective gene, suffers a second insult to any progeny retinal cell. Somatic cases are far less likely as two hits to the same cell during development are required. Details of the disease, germinal or somatic, unilateral or bilateral, in combination with case data allow for the estimation of the two parameters of the model: mutation rate, estimated at p=7x10(-7) per chromosome per cell division, and carrier frequency, estimated at f=40 per million. The model indicates that carriers of the disease arise from similar mutations to germ cells; in particular, heridary transmission can occur for only a generation or two before dying out. The results show that a stochastic simulation of a multi-hit cancer is feasible and may predict tumor growth dynamics. A simulation run will have to consist of a few million cells in order to observe even a small number of mutations. And several dozens such runs will have to be simulated.     

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).     

Replication of prophage P1 is cell-cycle specific.

P1 prophage replication during the Escherichia coli division cycle has been analyzed by using the membrane-elution technique to produce cells labelled at different times during the division cycle and scintillation counting for quantitative analysis of radioactive prophage DNA. P1 prophage replicates during a restricted portion of the bacterial division cycle, like the minichromosome, but at a time during the division cycle different than the time at which the minichromosome replicates in the same cell. A high-copy mini-R6K plasmid present in the same cell replicates throughout the division cycle. Over a wide range of growth rates, the P1 prophage replicates approximately one-half generation after the minichromosome replicates. Thus, the mechanisms underlying P1 replication are similar to those for the F plasmid and the chromosome. Replication occurs when some property related to cell size or cell mass reaches a constant value per origin.     

Proliferation controls in a linear growth system: theoretical studies of cell division in the cartilage growth plate

The columnar arrangement of dividing cells in the epiphyseal cartilage plates of growing bones provides a model of a linear proliferation system. One factor which determines the rate of cell production, and hence the rate of growth, is the size of the proliferating population. In this one dimensional system this size is equal to the length of the proliferation zone. Two possible mechanisms for a differentiation control that sets a limit to the length of this zone have been tested in computer simulations. While a diffusion gradient control is consistent with cell kinetic measurements a division limit based on an inheritable growth substance is shown to require further development before the model fits experimental data.Cell division in the columns produces linear clones of cells. If the final length of a bone is set by a limit on the number of divisions that the cartilage stem cells can make, then the number of cells per clone is crucial in determining overall bone growth. The parameters that affect linear clone size have been investigated in computer simulations. Clone size depends largely on the relative division rate of stem cells to proliferation zone cells — but the data on stem cell division rates are generally unreliable.The analysis could be applied to other linear proliferating systems.     

A biological and computational model of megakaryocyte development as a stochastic branching process

The purpose of this paper is to describe a model of megakaryocytopoiesis as a branching process with stochastic processes regulating critical control points of differentiation along the stem cell megakaryocyte platelet axis. Progress of cells through these critical control points are regulated by transitional probabilities, which in turn are regulated by influences such as growth factors. The critical control points include transition of resting megakaryocytic stem cells (CFU-meg) into proliferating stem cells, the cessation of cytokinesis, and the cessation of DNA synthesis. A computerized computational method has been developed for directly fitting the stochastic branching model to colony growth data. The computational model has allowed transitional probabilities to be derived from colony size data. The model provides a unifying explanation for much of the heterogeneity of stages of maturation within populations of megakaryocytes and is fully compatible with historical data supporting the stochastic nature of hematopoietic stem cell regulation and with modern molecular concepts about control of the cell cycle.     

Analysis of a cell cycle model based on unequal division of metabolic constituents to daughter cells during cytokinesis

We demonstrate that the unequal division of RNA during cytokinesis explains the dispersion of cell generation times in CHO cell cultures. Experimental cytometric results reported previously serve as a basis for a probabilistic model of cytokinesis. Unequal RNA division to daughter cells, together with two simple laws of RNA production, are used as a source of randomness within the cell cycle. The model reproduces the experimental growth of the CHO cell population, including the observed variability in RNA content. The model has stabilizing properties which explain why a cell population with increased RNA content characteristics, a few cell cycles, to the original pattern. Other cell cycle characteristics, like sister-to-sister and mother-to-daughter generation time correlations implied by the model, are close to their experimental analogs. The conceptual basis of the model is general enough to include unequal division of factors other than RNA (cell mass, cell proteins, etc.) as sources of generation time variability. It seems that the observed dispersion of cell generation times, explained previously in the terms of random transitions in some part of the cell cycle (the Smith & Martin A and B state hypothesis), can be reduced to the single random event of unequal division. This supplies a new convenient tool in the investigation of cell cycle kinetics.     

Synthesis of photoreactivating enzyme in synchronous and asynchronous cultures of Escherichia coli

The technique of flash photolysis was used to study cellular variations in the number of photoreactivating enzyme (PRE) molecules during the cell division cycle of the UV-sensitive E. coli strain B_S?1. No variations in the number of PRE molecules per genome were observed throughout the cell division cycle when synchronized cells cultured in either glucose-minimal or succinate-minimal medium were used. This is interpreted to mean that PRE synthesis is continuous throughout the cell cycle for glucose-grown cells, but may stop at the time chromosome replication ceases prior to division, in succinate-grown cells. The effect of growth rate and stage of growth on cellular PRE content in asynchronous cultures was also determined. Variations in the number of PRE per genome were observed for both synchronous and asynchronous cells cultured in different media and occurred in a manner that suggested a dependence on growth rate. PRE per genome increased with generation time. Stationary phase cells from each culture medium (nutrient broth, glucose-minimal, succinate-minimal) had more PRE per genome than did respective log phase cells. It is suggested that PRE synthesis may be controlled by some aspect of chromosome replication.     

Interpreting CFSE Obtained Division Histories of B Cells in Vitro with Smith–Martin and Cyton Type Models

The fluorescent dye carboxyfluorescin diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time using the cyton approach that utilizes independent stochastic distributions for the time to divide and the time to die. We confirm that our analytic form for the number of cells in each division class is consistent with the numerical solution of a set of delay differential equations representing the generalized Smith–Martin model with cell death rates depending on the division class. Choosing the distribution of time to the first division to fit thymidine labeling data for B cells stimulated in vitro with lipopolysaccharide (LPS) and either with or without interleukin-4 (IL-4), we fit CFSE data to determine the dependence of B cell kinetic parameters on the presence of IL-4. We find when IL-4 is present, a greater proportion of cells are recruited into division with a longer average time to first division. The most profound effect of the presence of IL-4 was decreased death rates for smaller division classes, which supports a role of IL-4 in the protection of B cells from apoptosis.