Stochastic computer model of the cell microtubule dynamics

To study the effect of various factors on the microtubule system, one of the main cytoskeletal elements in the cell, which organizes the intracellular transport of different organelles and is necessary for mitosis and meiosis, a computer model of this system is created. Using a stochastic approach, the model describes the microtubule assembly/disassembly as a set of chemical reactions with certain rate constants. Microtubules are visualized in the computer program field, which makes the model vivid. The program imitates the dynamics and structure of the microtubule system with high reliability. The parameters calculated by the model correlate with the corresponding parameters of microtubules in living cells. This approach to modeling microtubules and similar systems continues to be developed so that the models would better describe living systems and the effect of a still broader range of factors could be studied.     

A computer model for epidermal cellular interactions

A model has been constructed of an assembly of mammalian epidermal cells. The model, simulated by computer, has served as a framework by which experiments can be performed “in machina”. Relationships between the various parameters considered have been established. Hypotheses referring to the time at which cells leave the basal layer, and to the simultaneous occurrence of folded basal layer and of a significant number of vertical mitosis in some hyperplastic conditions have been implemented and examined.     

Stochastic disease dynamics of a hospital infection model

A stochastic model for hospital infection incorporating both direct transmission and indirect transmission via free-living bacteria in the environment is investigated. We examine the long term behavior of the model by calculating a stationary distribution and normal approximation of the distribution. The quasi-stationary distribution of the model is studied to investigate the models’ behavior before extinction and the time to extinction. Numerical results show agreement between the calculated distributions and results of event-driven simulations. Hand hygiene of volunteers is more effective in terms of reducing the mean (or standard deviation) of the stationary distribution of colonized patients and the expected time to extinction compared to hand hygiene of health care workers (HCWs), on the basis of our parameter values. However, the indirect (or direct) transmission rate can lead to either increase or decrease in the standard deviation of the stationary distribution, but the impact of the indirect transmission is much greater than that of the direct transmission. The findings suggest that isolation of new admitted colonized patients is most effective in reducing both the mean and standard deviation of the stationary distribution and measures related to indirect transmission are secondary in their effects compared to other interventions.     

Cadherin-mediated regulation of microtubule dynamics

Epithelial polarization and neuronal outgrowth require the assembly of microtubule arrays that are not associated with centrosomes. As these processes generally involve contact interactions mediated by cadherins, we investigated the potential role of cadherin signalling in the stabilization of non-centrosomal microtubules. Here we show that expression of cadherins in centrosome-free cytoplasts increases levels of microtubule polymer and changes the behaviour of microtubules from treadmilling to dynamic instability. This effect is not a result of cadherin expression per se but depends on the formation of cell-cell contacts. The effect of cell-cell contacts is mimicked by application of beads coated with stimulatory anti-cadherin antibody and is suppressed by overexpression of the cytoplasmic cadherin tail. We therefore propose that cadherins initiate a signalling pathway that alters microtubule organization by stabilizing microtubule ends.     

Modulation of microtubule dynamics by drugs: a paradigm for the actions of cellular regulators

Microtubules are intrinsically dynamic polymers. Two kinds of dynamic behaviors, dynamic instability and treadmilling, are important for microtubule function in cells. Both dynamic behaviors appear to be tightly regulated, but the cellular molecules and the mechanisms responsible for the regulation remain largely unexplored. While microtubule dynamics can be modulated transiently by the interaction of regulatory molecules with soluble tubulin, the microtubule itself is likely to be the primary target of cellular molecules that regulate microtubule dynamics. The antimitotic drugs that modulate microtubule dynamics serve as excellent models for such cellular molecules. Our laboratory has been investigating the interactions of small drug molecules and stabilizing microtubule-associated proteins (MAPs) with microtubule surfaces and ends. We find that drugs such as colchicine, vinblastine, and taxol, and stabilizing MAPs such as tau, strongly modulate microtubule dynamics at extremely low concentrations under conditions in which the microtubule polymer mass is minimally affected. The powerful modulation of the dynamics is brought about by the binding of only a few drug or MAP molecules to distinct binding sites at the microtubule surface or end. Based upon our understanding of the well-studied drugs and stabilizing MAPs, it is clear that molecules that regulate dynamics such as Kin 1 and stathmin could bind to a large number of distinct tubulin sites on microtubules and employ an array of mechanisms to selectively and powerfully regulate microtubule dynamics and dynamics-dependent cellular functions.     

Stochastic model of chemoattractant receptor dynamics in leukocyte chemosensory movement

Mammalian white blood cells are known to bias the direction of their movement along concentration gradients of specific chemical stimuli, a phenomenon called chemotaxis. Chemotaxis of leukocyte cells is central to the acute inflammatory response in living organisms and other critical physiological functions. On a molecular level, these cells sense the stimuli termed chemotactic factor (CF) through specific cell surface receptors that bind CF molecules. This triggers a complex signal transduction process involving intracellular biochemical pathways and biophysical events, eventually leading to the observable chemotactic response. Several investigators have shown theoretically that statistical fluctuations in receptor binding lead to “noisy” intracellular signals, which may explain the observed imperfect chemotactic response to a CF gradient. The most recent dynamic model (Tranquillo and Lauffenburger,J. Math. Biol. 25, 229–262. 1987) couples a scheme for intracellular signal transduction and cell motility response with fluctuations in receptor binding. However, this model employs several assumptions regarding receptor dynamics that are now known to be oversimplifications. We extend the earlier model by accounting for several known and speculated chemotactic receptor dynamics, namely, transient G-protein signaling, cytoskeletal association, and receptor internalization and recycling, including statistical fluctuations in the numbers of receptors among the various states. Published studies are used to estimate associated constants and ensure the predicted receptor distribution is accurate. Model analysis indicates that directional persistence in uniform CF concentrations is enhanced by increasing rate constants for receptor cytoskeletal inactivation, ternary complex dissociation, and binary complex dissociation, and by decreasing rate constants for receptor internalization and recycling. For most rate constants, we have detected an optimal range that maximizes orientation bias in CF gradients. We have also examined different desensitization and receptor recycling mechanisms that yield experimentally documented orientation behavior. These yield novel insights into the relationship between receptor dynamics and leukocyte chemosensory movement behavior.     

AKAP350 modulates microtubule dynamics

AKAP350 is a multiply spliced type II protein kinase A-anchoring protein that localizes to the centrosomes in most cells and the Golgi apparatus in epithelial cells. Multiple studies suggest that AKAP350 is involved in microtubule nucleation at the centrosome. Our previous studies demonstrated that AKAP350 was necessary for the maintenance of Golgi apparatus integrity. These data suggested that AKAP350 might be necessary for normal cytoskeletal interactions with the Golgi. To examine the relationship of AKAP350 with the microtubule cytoskeleton, we analyzed the effect of the depletion of AKAP350 on microtubule regrowth after nocodazole treatment in HeLa cells. The decrease in AKAP350 expression with short interfering RNA induced a delay in microtubule elongation with no effect on microtubule aster formation. In contrast, overexpression of the centrosomal targeting domain of AKAP350 elicited alterations in aster formation, but did not affect microtubule elongation. RNA interference for AKAP350 also induced an increase in cdc42 activity during microtubule regrowth. Our data suggest that AKAP350 has a role in the remodeling of the microtubule cytoskeleton.     

Measuring nanometer scale gradients in spindle microtubule dynamics using model convolution microscopy

A computational model for the budding yeast mitotic spindle predicts a spatial gradient in tubulin turnover that is produced by kinetochore-attached microtubule (kMT) plus-end polymerization and depolymerization dynamics. However, kMTs in yeast are often much shorter than the resolution limit of the light microscope, making visualization of this gradient difficult. To overcome this limitation, we combined digital imaging of fluorescence redistribution after photobleaching (FRAP) with model convolution methods to compare computer simulations at nanometer scale resolution to microscopic data. We measured a gradient in microtubule dynamics in yeast spindles at approximately 65-nm spatial intervals. Tubulin turnover is greatest near kinetochores and lowest near the spindle poles. A beta-tubulin mutant with decreased plus-end dynamics preserves the spatial gradient in tubulin turnover at a slower time scale, increases average kinetochore microtubule length approximately 14%, and decreases tension at kinetochores. The beta-tubulin mutant cells have an increased frequency of chromosome loss, suggesting that the accuracy of chromosome segregation is linked to robust kMT plus-end dynamics.     

Regulation of microtubule dynamics by kinesins

The simple mechanistic and functional division of the kinesin family into either active translocators or non-motile microtubule depolymerases was initially appropriate but is now proving increasingly unhelpful, given evidence that several translocase kinesins can affect microtubule dynamics, whilst non-translocase kinesins can promote microtubule assembly and depolymerisation. Such multi-role kinesins act either directly on microtubule dynamics, by interaction with microtubules and tubulin, or indirectly, through the transport of other factors along the lattice to the microtubule tip. Here I review recent progress on the mechanisms and roles of these translocase kinesins.     

Direct observation of steady-state microtubule dynamics

Different types of unusual dynamic behavior have been reported for steady-state microtubules. While almost all earlier reports relied on kinetic measurements of bulk polymerization, we have directly visualized the steady-state addition of subunits to individual microtubules through the use of tubulin derivitized with biotin. Biotinylated tubulin was used both as an internal seed for polymerization and as a marker for assembly onto the ends of microtubules composed of purified tubulin. Biotinylated segments were distinguished from unmodified tubulin by double-label immunofluorescence. Microtubule lengths, number concentrations, and segment lengths have been monitored with time at steady state under two buffer conditions. The results indicate that the microtubule steady state under these conditions is a balance between a majority of slowly growing microtubules and a minority of rapidly depolymerizing ones as described by the dynamic instability model (Mitchison T., and M. Kirschner, 1984, Nature (Lond.)., 312:232-242). Microtubules show no evidence of treadmilling; instead most show progressive growth off both ends at steady state. Although solvent conditions markedly influence the growth rates, qualitatively the behavior is unchanged.     

How cellular membranes can regulate microtubule network

Microtubule array in eukaryotic cells supports directed transport of various cargoes driven by motor proteins. The arrangement of microtubules in cytoplasm is not stochastic; they are organized in a certain way setting a system of coordinates for intracellular transport. Most cultured fibroblast-like cells possess a radial microtubule array with the minus ends of microtubules gathered on the centrosome and plus ends directed towards the periphery of the cell. Mechanisms that regulate the formation of radial microtubule system remain unclear. Usually centrosome works as a microtubule-organizing center; however, the radial system of microtubules can be formed without centrosome participation. At least in some cases microtubule network can be organized by dynein-dynactin complexes associated with membrane vesicles. Membrane vesicles can nucleate microtubules, anchor them and move along them. However, the role of membrane organelles in microtubule organization began to attract attention of researches only recently. It this review we summarize the data indicating that membrane organelles can organize microtubules, providing “tracks” for their subsequent transport.     

The effect of podophyllotoxin on microtubule dynamics

We have investigated the effects of podophyllotoxin on the dynamic properties of microtubules assembled from pure tubulin dimer. Excess podophyllotoxin causes the complete disassembly of microtubules, through formation of a tubulin-GTP-podophyllotoxin ternary complex with a dissociation rate constant of 160 s-1 at 37 degrees C, similar to that found upon extensive isothermal dilution in this buffer system. Addition of substoichiometric concentrations of podophyllotoxin causes partial disassembly of microtubules through production of an equivalent amount of the ternary complex. Microtubule length measurements and incorporation of [3H]GTP-tubulin dimer show that podophyllotoxin can suppress the dynamic instability of tubulin dimer microtubules and that it acts substoichiometrically in so doing. We interpret the action of substoichiometric podophyllotoxin on microtubule ends in terms of effects on interconversion of growing and shrinking microtubules in a dynamic system in which tubulin-GTP-podophyllotoxin is kinetically analogous to tubulin-GTP in addition and to tubulin-GDP in dissociation. The ability to suppress dynamic instability may be one way in which drugs such as podophyllotoxin, acting at relatively low concentrations, are able to arrest cell growth and development in a selective way, without necessarily affecting the integrity of the major part of the cytoskeletal microtubule network.     

Stochastic dynamics of metastasis formation

Tumor metastasis accounts for the majority of deaths in cancer patients. The metastatic behavior of cancer cells is promoted by mutations in many genes, including activation of oncogenes such as RAS and MYC. Here, we develop a mathematical framework to analyse the dynamics of mutations enabling cells to metastasize. We consider situations in which one mutation is necessary to confer metastatic ability to the cell. We study different population sizes of the main tumor and different somatic fitness values of metastatic cells. We compare mutations that are positively selected in the main tumor with those that are neutral or negatively selected, but faster at forming metastases. We study whether metastatic potential is the property of all (or the majority of) cells in the main tumor or only the property of a small subset. Our theory shows how to calculate the expected number of metastases that are formed by a tumor.     

Stochastic models of neuronal dynamics

Cortical activity is the product of interactions among neuronal populations. Macroscopic electrophysiological phenomena are generated by these interactions. In principle, the mechanisms of these interactions afford constraints on biologically plausible models of electrophysiological responses. In other words, the macroscopic features of cortical activity can be modelled in terms of the microscopic behaviour of neurons. An evoked response potential (ERP) is the mean electrical potential measured from an electrode on the scalp, in response to some event. The purpose of this paper is to outline a population density approach to modelling ERPs.We propose a biologically plausible model of neuronal activity that enables the estimation of physiologically meaningful parameters from electrophysiological data. The model encompasses four basic characteristics of neuronal activity and organization: (i) neurons are dynamic units, (ii) driven by stochastic forces, (iii) organized into populations with similar biophysical properties and response characteristics and (iv) multiple populations interact to form functional networks. This leads to a formulation of population dynamics in terms of the Fokker-Planck equation. The solution of this equation is the temporal evolution of a probability density over state-space, representing the distribution of an ensemble of trajectories. Each trajectory corresponds to the changing state of a neuron. Measurements can be modelled by taking expectations over this density, e.g. mean membrane potential, firing rate or energy consumption per neuron. The key motivation behind our approach is that ERPs represent an average response over many neurons. This means it is sufficient to model the probability density over neurons, because this implicitly models their average state. Although the dynamics of each neuron can be highly stochastic, the dynamics of the density is not. This means we can use Bayesian inference and estimation tools that have already been established for deterministic systems. The potential importance of modelling density dynamics (as opposed to more conventional neural mass models) is that they include interactions among the moments of neuronal states (e.g. the mean depolarization may depend on the variance of synaptic currents through nonlinear mechanisms).Here, we formulate a population model, based on biologically informed model-neurons with spike-rate adaptation and synaptic dynamics. Neuronal sub-populations are coupled to form an observation model, with the aim of estimating and making inferences about coupling among sub-populations using real data. We approximate the time-dependent solution of the system using a bi-orthogonal set and first-order perturbation expansion. For didactic purposes, the model is developed first in the context of deterministic input, and then extended to include stochastic effects. The approach is demonstrated using synthetic data, where model parameters are identified using a Bayesian estimation scheme we have described previously.     

Stochastic dynamics of cancer initiation

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.     

A model of microtubule oscillations

Simulations of microtubule oscillations have been obtained by a kinetic model including nucleation of microtubules, elongation by addition of GTP-loaded tubulin dimers, disassembly into oligomers, and dissolution of oligomers followed by nucleotide exchange at the free dimers. Dynamic instability is described by the on and off rates for dimer association in the growth phase, the rate of rapid shortening, and the transition rates for catastrophe and rescue. The latter are assumed to be completely determined by the current state of the system (short cap hypothesis). Microtubule oscillations and normal polymerizations measured by time-resolved X-ray scattering were used to test the model. The model is able to produce oscillations without further assumptions. However, in order to obtain good fits to the experimental data one requires an additional mechanism which prevents rapid desynchronization of the microtubules. One of several possible mechanisms that will be discussed is the destabilization of microtubules by the products of disassembly.Abbreviations MT(s) microtubule(s) - G-MT/S-MT microtubule in the state of growth/shortening - GTP guanosine 5-triphosphate - GDP guanosine 5-diphosphate - TU · GDP/TU · GTP tubulin dimer with GDP/GTP bound to the exchangeable nucleotide binding site - MAP(s) microtubule-associated protein(s) - PC tubulin phosphocellulose-purified tubulin - PIPES piperazine-1,4-bis(2-ethane sulfonic acid) - DDT dithiothreitol - EGTA ethylene glycol-O,O-bis(2-amino ethyl ether)-N,N,N,N-tetraacetic acid     

Kinetochore microtubule dynamics and the metaphase-anaphase transition

We have quantitatively studied the dynamic behavior of kinetochore fiber microtubules (kMTs); both turnover and poleward transport (flux) in metaphase and anaphase mammalian cells by fluorescence photoactivation. Tubulin derivatized with photoactivatable fluorescein was microinjected into prometaphase LLC-PK and PtK1 cells and allowed to incorporate to steady-state. A fluorescent bar was generated across the MTs in a half-spindle of the mitotic cells using laser irradiation and the kinetics of fluorescence redistribution were determined in terms of a double exponential decay process. The movement of the activated zone was also measured along with chromosome movement and spindle elongation. To investigate the possible regulation of MT transport at the metaphase-anaphase transition, we performed double photoactivation analyses on the same spindles as the cell advanced from metaphase to anaphase. We determined values for the turnover of kMTs (t1/2 = 7.1 +/- 2.4 min at 30 degrees C) and demonstrated that the turnover of kMTs in metaphase is approximately an order of magnitude slower than that for non-kMTs. In anaphase, kMTs become dramatically more stable as evidenced by a fivefold increase in the fluorescence redistribution half-time (t1/2 = 37.5 +/- 8.5 min at 30 degrees C). Our results also indicate that MT transport slows abruptly at anaphase onset to one-half the metaphase value. In early anaphase, MT depolymerization at the kinetochore accounted, on average, for 84% of the rate of chromosome movement toward the pole whereas the relative contribution of MT transport and depolymerization at the pole contributed 16%. These properties reflect a dramatic shift in the dynamic behavior of kMTs at the metaphase-anaphase transition. A release-capture model is presented in which the stability of kMTs is increased at the onset of anaphase through a reduction in the probability of MT release from the kinetochore. The reduction in MT transport at the metaphase-anaphase transition suggests that motor activity and/or subunit dynamics at the centrosome are subject to modulation at this key cell cycle point.     

Profilin connects actin assembly with microtubule dynamics

Profilin controls actin nucleation and assembly processes in eukaryotic cells. Actin nucleation and elongation promoting factors (NEPFs) such as Ena/VASP, formins, and WASP-family proteins recruit profilin:actin for filament formation. Some of these are found to be microtubule associated, making actin polymerization from microtubule-associated platforms possible. Microtubules are implicated in focal adhesion turnover, cell polarity establishment, and migration, illustrating the coupling between actin and microtubule systems. Here we demonstrate that profilin is functionally linked to microtubules with formins and point to formins as major mediators of this association. To reach this conclusion, we combined different fluorescence microscopy techniques, including superresolution microscopy, with siRNA modulation of profilin expression and drug treatments to interfere with actin dynamics. Our studies show that profilin dynamically associates with microtubules and this fraction of profilin contributes to balance actin assembly during homeostatic cell growth and affects micro­tubule dynamics. Hence profilin functions as a regulator of microtubule (+)-end turnover in addition to being an actin control element.     

Stathmin regulates microtubule dynamics and microtubule organizing center polarization in activated T cells

Polarization of T cells involves reorientation of the microtubule organizing center (MTOC). Because activated ERK is localized at the immunological synapse, we investigated its role by showing that ERK activation is important for MTOC polarization. Suspecting that ERK phosphorylates a regulator of microtubules, we next focused on stathmin, a known ERK substrate. Our work indicates that during T cell activation, ERK is recruited to the synapse, allowing it to phosphorylate stathmin molecules near the immunological synapse. Supporting an important role of stathmin phosphorylation in T cell activation, we showed that T cell activation results in increased microtubule growth rate dependent on the presence of stathmin. The significance of this finding was demonstrated by results showing that CTLs from stathmin(-/-) mice displayed defective MTOC polarization and defective target cell cytolysis. These data implicate stathmin as a regulator of the microtubule network during T cell activation.