Microtubule dynamics in the spindle. II. A thermodynamic and kinetic description

We have previously presented a model for the assembly and disassembly of mitotic spindle microtubules (MTs) (Pickett-Heaps et al., 1986). In this paper, we describe the thermodynamics of such spindle MT assembly and present equations to describe the polymerization kinetics of different classes of spindle MTs. These equations are used to predict, in terms of kinetics parameters, the magnitude of forces extant on spindle MTs and to define the critical force needed to halt MT assembly. We calculate several of these forces for a hypothetical model cell; our predicted value for the force generated along kinetochore fibers is in close agreement with measured values taken from living cells. The model and its implications are discussed with reference to other recent models of spindle and MT dynamics.     

Regulation of Cell Volume by Active Cation Transport in High and Low Potassium Sheep Red Cells

A model cell which controls its cation composition and volume by the action of a K-Na exchange pump and leaks for both ions working in parallel is presented. Equations are formulated which describe the behavior of this model in terms of three membrane parameters. From these equations and the steady state concentrations of Na, K, and Cl, values for these parameters in high potassium (HK) and low potassium (LK) sheep red cells are calculated. Kinetic experiments designed to measure the membrane parameters directly in the two types of sheep red cells are also reported. The values of the parameters obtained in these experiments agreed well with those calculated from the steady state concentrations of ions and the theoretical equations. It is concluded that both HK and LK sheep red cells control their cation composition and volume in a manner consistent with the model cell. Both have a cation pump which exchanges one sodium ion from inside the cell with one potassium ion from outside the cell but the pump is working approximately four times faster in the HK cell. The characteristics of the cation leak in the two cell types are also very different since the HK cells are relatively more leaky to sodium as compared with potassium than is the case in the LK cells. Both cell types show appreciable sodium exchange diffusion but this process is more rapid in the LK than in the HK cells.     

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.     

A First-Passage-Time Theory for Search and Capture of Chromosomes by Microtubules in Mitosis

The mitotic spindle is an important intermediate structure in eukaryotic cell division, in which each of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodies located close to the two poles of the dividing cell. Several mechanisms are at work toward the formation of the spindle, one of which is the ‘capture’ of chromosome pairs, held together by kinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to 24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keep the duration of mitosis within this limit? Previous theoretical studies have suggested that the chromosome search and capture is optimized by tuning the microtubule dynamic parameters to minimize the search time. In this paper, we examine this conjecture. We compute the mean search time for a single target by microtubules from a single nucleating site, using a systematic and rigorous theoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targets and nucleating sites by physical arguments. Estimates of mitotic time scales are then obtained for different cells using experimental data. In yeast and mammalian cells, the observed changes in microtubule kinetics between interphase and mitosis are beneficial in reducing the search time. In Xenopus extracts, by contrast, the opposite effect is observed, in agreement with the current understanding that large cells use additional mechanisms to regulate the duration of the mitotic phase.     

Three-dimensional simulation of anisotropic cell-driven collagen gel compaction

Tissue equivalents (TEs), formed by entrapping cells in a collagen gel, are an important model system for studying cell behavior. We have previously (Barocas and Tranquillo in J Biomech Eng 117:161–170, 1997a) developed an anisotropic biphasic theory of TE mechanics, which comprises five coupled partial differential equations describing interaction among cells and collagen fibers in the TE. The model equations, previously solved in one or two dimensions, were solved in three dimensions using an adaptive finite-element platform. The model was applied to three systems: a rectangular isometric cell traction assay, an otherwise- acellular gel containing two islands of cells, and an idealized tissue-engineered cardiac valve leaflet. In the first two cases, published experimental data were available for comparison, and the model results were consistent with the experimental observations. Fibers and cells aligned in the fixed direction in the isometric assay, and a region of strong fiber alignment arose between the two cell islands. For the valve problem, the alignment predicted by the model was generally similar to that observed experimentally, but an asymmetry in the experiment was not captured by the model.     

Stabilization of telomeres in nonlinear models of proliferating cell lines

We analyse an age-structured model of telomere loss in a proliferating cell population. The cell population is divided into telomere classes, which shorten each round of division. The model consists of a nonlinear system of partial differential equations for the telomere classes. We prove that if the highest telomere class is exempted from mortality, then all the classes stabilize to a nontrivial equilibrium dependent on the initial state of cells in the highest telomere class.     

Oxygen, pH value, and carbon source induced changes of the mode of oscillation in synchronous continuous culture of Saccharomyces cerevisiae

Oscillations of measured process parameters occur in continuous cultures of Saccharomyces cerevisiae owing to a partial synchronization of budding. Intentional changes of the oxygen concentration, pH value, and carbon source cause effects on the period length similar to those known from variations of the dilution rate. The generation times of parent and daughter cells frequently differ in synchronous culture. To analyze the oscillation the term mode IJ of oscillation is used, which is defined as the ratio IJ of the generation times of parent and daughter cells. When the dissolved oxygen concentration was reduced to zero, the mode of oscillation changed within two periods from mode 12 to mode 11, caused by a decrease of the generation time of daughter cells and an increase of that of the parent cells. When the pH value was slowly reduced from 5.0 to 3.9, a change from mode 112 to mode 13 was observed. Mode 13, representing one parent and three daughter cell populations (the start of budding of each of the three being delayed by one period), denotes an elongated generation time of the daughter cells compared to mode 112, marked by one parent and two different daughter cell classes. When the carbon source galactose was replaced by glucose a mode change from mode 12 to mode 11 was observed. This alteration of the mode was found to be dependent on the status of the cell cycle at the time when the carbon source is changed. The population distribution in batch cultures with glucose or galactose as a substrate was analysed by dyeing the DNA and counting the bud scars. Galactose provoked higher growth rates for the older cells. According to the model for stationary synchronous growth parameters like DO, pH value or the type of carbon source can be varied within a certain range without effecting the period length. If the variation imposes a certain stress, the culture switches to a new mode. These kinds of parameters therefore provide selective measures to influence the period lengths and the modes of oscillation.     

A method for investigating hepatocyte polyploidization kinetics during postnatal development in mammals.

A method for investigating weakly-proliferating cell populations of liver parenchyma on the basis of a quantitative analysis of hepatocyte polyploidization during postnatal development is described. The method uses a mathematical model which characterizes the hepatocyte polyploidization process, and incorporates data concerning the time course for relative frequencies of hepatocytes in different ploidy classes. As a result of these measurements and calculations for rat liver, transition rates of hepatocytes (the relative number of cells during a given time unit) from one ploidy class to another, and a coefficient for the reduction of hepatocyte mitotic activity with an increase in its ploidy class were obtained. Calculated curves show a good correspondence with the real process of hepatocyte frequency changes as they relate to changes in the age of the animals. To check this method, experiments investigating time changes of autoradiographic label content in the different ploidy classes of hepatocytes were carried out. By mathematically modeling the label diluting process resulting from cell proliferation and polyploidization, transition rates of hepatocytes were calculated, and they reflect values calculated from the model according to changes in occurrence frequencies.     

New method for the analysis of flow cytometric data

A numerical method for deriving the fractions of cells in different phases of the cell cycle from a single observed DNA histogram is presented. The observed histogram is regarded as a polluted version (containing allocation errors) of the true histogram. A mathematical model is used to describe the pollution process. A theoretical histogram, representing the true histogram, is constructed so that G1 cells are put into one channel and G2M cells into another; the distribution of S cells in between is approximated with a set of harmonic functions. This theoretical histogram is subsequently disturbed with Gaussian dispersion functions to stimulate the pollution, yielding a predicted histogram. Using a maximum likelihood estimation technique, the model parameters are adjusted iteratively, matching the predicted histogram to the actually observed one. With the final parameter values substituted, the corresponding final theoretical histogram is regarded as a reliable reconstruction of the true histogram. From the latter, the required percentages can be read directly. The advantage of this approach over other mathematical analysis methods is that it allows a wide range of different, continuous distributions for relatively few model parameters (thus featuring flexibility and realism and a diminished risk of encountering computational problems). In addition, estimation errors providing a measure of accuracy can be obtained. To test the method, it was used to analyze various observed histograms from the literature that have been obtained by either simulation or actual flow cytometric measurements. The method appeared to perform well, as compared to the reported results of several other methods of analysis applied to the same data.     

Explicit Separation of Growth and Motility in a New Tumor Cord Model

We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.     

Linearized kinetic model of<Emphasis Type="Italic"> Listeria monocytogenes</Emphasis> biofilm growth

In this study the dynamics of biofilm formation on aluminum has been investigated. The process of cell growth has been observed using fluorescence microscopy. It has been confirmed that the process of biofilm formation can be represented as a sum of two separate processes: cell adhesion and colony proliferation. The derived set of equations describes kinetics of surface population growth and characteristic times for adsorption and combined growth processes, including characteristic time for the nutrient supply depletion. All equations contain variables based on the fundamental characteristics of bacterial population and can be easily determined from the experimental data or estimated theoretically. The developed theoretical model allows obtaining realistic values for population growth time and characteristic time for nutrient limitation occurrence during the biofilm development. Resulting equations qualitatively describe the biofilm formation process, and allow predicting microbial kinetics in the batch reactor system and determining critical values of the process parameters.     

Effects of asymmetric division on a stochastic model of the cell division cycle

The stochastic model of cell division formulated by Alt and Tyson is generalized to the case of imprecise binary fission. Closed-form expressions are derived for the generation-time distribution, the birth-size and division-size distributions, the beta curve, and the correlation coefficient of generation times of sister cells. The theoretical results are compared to observations of cell division statistics in a culture of fission yeast.     

Simulation of motor-driven cochlear outer hair cell electromotility.

We propose a three-dimensional (3D) model to simulate outer hair cell electromotility. In our model, the major components of the composite cell wall are explicitly represented. We simulate the activity of the particles/motor complexes in the plasma membrane by generating active strains inside them and compute the overall response of the cell. We also consider the constrained wall and compute the generated active force. We estimate the parameters of our model by matching the predicted longitudinal and circumferential electromotile strains with those observed in the microchamber experiment. In addition, we match the earlier estimated values of the active force and cell wall stiffness. The computed electromotile strains in the plasma membrane and other components of the wall are in agreement with experimental observations in trypsinized cells and in nonmotile cells transfected with Prestin. We discover several features of the 3D mechanism of outer hair cell electromotilty. Because of the constraints under which the motors operate, the motor-related strains have to be 2-3 times larger than the observable strains. The motor density has a strong effect on the electromotile strain. Such effect on the active force is significantly lower because of the interplay between the active and passive properties of the cell wall.     

The Role of Cellular Coupling in the Spontaneous Generation of Electrical Activity in Uterine Tissue

The spontaneous emergence of contraction-inducing electrical activity in the uterus at the beginning of labor remains poorly understood, partly due to the seemingly contradictory observation that isolated uterine cells are not spontaneously active. It is known, however, that the expression of gap junctions increases dramatically in the approach to parturition, by more than one order of magnitude, which results in a significant increase in inter-cellular electrical coupling. In this paper, we build upon previous studies of the activity of electrically excitable smooth muscle cells (myocytes) and investigate the mechanism through which the coupling of these cells to electrically passive cells results in the generation of spontaneous activity in the uterus. Using a recently developed, realistic model of uterine muscle cell dynamics, we investigate a system consisting of a myocyte coupled to passive cells. We then extend our analysis to a simple two-dimensional lattice model of the tissue, with each myocyte being coupled to its neighbors, as well as to a random number of passive cells. We observe that different dynamical regimes can be observed over a range of gap junction conductances: at low coupling strength, corresponding to values measured long before delivery, the activity is confined to cell clusters, while the activity for high coupling, compatible with values measured shortly before delivery, may spread across the entire tissue. Additionally, we find that the system supports the spontaneous generation of spiral wave activity. Our results are both qualitatively and quantitatively consistent with observations from in vitro experiments. In particular, we demonstrate that the increase in inter-cellular electrical coupling observed experimentally strongly facilitates the appearance of spontaneous action potentials that may eventually lead to parturition.     

Identification of the small nuclear RNAs associated with the mitotic chromosomes of Amoeba proteus

Amebas contain 7 electrophoretically distinct species of small nuclear RNAs (snRNAs), some of which are known to associate in a striking manner with mitotic chromosomes. These RNAs can be divided into 2 classes, one consisting of 4 snRNA species that shuttle in a non-random way between nucleus and cytoplasm during interphase and one consisting of 3 snRNA species that do not leave the nucleus at all during interphase. In the work reported here we sought to determine which class is associated with mitotic chromosomes. Through a series of micromanipulative procedures we arranged for the shuttling snRNAs to be the only radioactive molecules in the cell. Such cells were allowed to enter mitosis, whereupon they were fixed and subjected to autoradiography. In those cells no radioactive snRNAs were found associated with mitotic chromosomes. It is concluded, therefore, that those snRNAs that do associate with mitotic chromosomes must be one or more of the non-shuttling species. — In the Discussion, how the non-shuttling snRNAs may function in cell activities is considered.     

Estimation with correlated censored survival data with missing covariates

Incomplete covariate data are a common occurrence in studies in which the outcome is survival time. Further, studies in the health sciences often give rise to correlated, possibly censored, survival data. With no missing covariate data, if the marginal distributions of the correlated survival times follow a given parametric model, then the estimates using the maximum likelihood estimating equations, naively treating the correlated survival times as independent, give consistent estimates of the relative risk parameters Lipsitz et al. 1994 50, 842-846. Now, suppose that some observations within a cluster have some missing covariates. We show in this paper that if one naively treats observations within a cluster as independent, that one can still use the maximum likelihood estimating equations to obtain consistent estimates of the relative risk parameters. This method requires the estimation of the parameters of the distribution of the covariates. We present results from a clinical trial Lipsitz and Ibrahim (1996b) 2, 5-14 with five covariates, four of which have some missing values. In the trial, the clusters are the hospitals in which the patients were treated.     

A theoretical study of random segregation of minicircles in trypanosomatids

The kinetoplast (k) DNA network of trypanosomatids is made up of approximately 50 maxicircles and the order of 10(4) minicircles. It has been proposed, based on various observations and experiments, that the minicircles are randomly segregated between daughter cells when the parent cell divides. In this paper, this random segregation hypothesis is theoretically tested in a population dynamics model to see if it can account for the observed phenomena. The hypothesis is shown to successfully explain, in Leishmania tarentolae, the observation that there are a few major and many minor minicircle classes, the fluctuations of minicircle class copy numbers over time, the loss of non-essential minicircle classes, the long survival times of a few of these classes and that these classes are likely to be the major classes within the population. Implications of the model are examined for trypanosomatids in general, leading to several predictions. The model predicts variation in network size within a population, variation in the average network size and large-scale changes in class copy number over long time-scales, an evolutionary pressure towards larger network sizes, the selective advantage of non-random over random segregation, very strong selection for the amplified class in Crithidia fasciculata if its minicircles undergo random segregation and that Trypanosoma brucei may use sexual reproduction to maintain its viability.     

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.