A three-dimensional model of intercellular calcium signaling in epithelial cells

We have developed a fully three-dimensional (3D) model of calcium signaling in epithelial cells based on a set of reaction diffusion equations that are solved on a large-scale finite-element code in three dimensions. We have explicitly included the cellular compartments including the cell nucleus, cytoplasm, and gap junctions. The model allows for buffering of free Ca2+, calcium-induced calcium release, and the explicit inclusion of mobile buffers. To make quantitative comparisons to experimental results, we used fluorescence microscopy images of cells to generate an accurate mesh describing cell morphology. We found that Ca2+ wave propagation through the tissue is a function of both initial conditions used to start the wave and various geometrical parameters that affect propagation such as gap junction density and distribution, and the presence of nuclei. The exogenous dyes used in experimental imaging also affect wave propagation.     

Wave propagation in bone media

The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.     

Intercellular Ca2+ wave propagation through gap-junctional Ca2+ diffusion: a theoretical study

Intercellular regenerative calcium waves in systems such as the liver and the blowfly salivary gland have been hypothesized to spread through calcium-induced calcium release (CICR) and gap-junctional calcium diffusion. A simple mathematical model of this mechanism is developed. It includes CICR and calcium removal from the cytoplasm, cytoplasmic and gap-junctional calcium diffusion, and calcium buffering. For a piecewise linear approximation of the calcium kinetics, expressions in terms of the cellular parameters are derived for 1) the condition for the propagation of intercellular waves, and 2) the characteristic time of the delay of a wave encountered at the gap junctions. Intercellular propagation relies on the local excitation of CICR in the perijunctional space by gap-junctional calcium influx. This mechanism is compatible with low effective calcium diffusivity, and necessitates that CICR can be excited in every cell along the path of a wave. The gap-junctional calcium permeability required for intercellular waves in the model falls in the range of reported gap-junctional permeability values. The concentration of diffusive cytoplasmic calcium buffers and the maximal rate of CICR, in the case of inositol 1,4,5-trisphosphate (IP3) receptor calcium release channels set by the IP(3) concentration, are shown to be further determinants of wave behavior.     

Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.     

Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines

We present a robust and computationally inexpensive method to estimate the lengths and three-dimensional moment arms for a large number of musculotendon actuators of the human lower limb. Using a musculoskeletal model of the lower extremity, a set of values was established for the length of each musculotendon actuator for different lower limb generalized coordinates (joint angles). A multidimensional spline function was then used to fit these data. Muscle moment arms were obtained by differentiating the musculotendon length spline function with respect to the generalized coordinate of interest. This new method was then compared to a previously used polynomial regression method. Compared to the polynomial regression method, the multidimensional spline method produced lower errors for estimating musculotendon lengths and moment arms throughout the whole generalized coordinate workspace. The fitting accuracy was also less affected by the number of dependent degrees of freedom and by the amount of experimental data available. The spline method only required information on musculotendon lengths to estimate both musculotendon lengths and moment arms, thus relaxing data input requirements, whereas the polynomial regression requires different equations to be used for both musculotendon lengths and moment arms. Finally, we used the spline method in conjunction with an electromyography driven musculoskeletal model to estimate muscle forces under different contractile conditions, which showed that the method is suitable for the integration into large scale neuromusculoskeletal models.     

Quantitative Decomposition of Dynamics of Mathematical Cell Models: Method and Application to Ventricular Myocyte Models

Mathematical cell models are effective tools to understand cellular physiological functions precisely. For detailed analysis of model dynamics in order to investigate how much each component affects cellular behaviour, mathematical approaches are essential. This article presents a numerical analysis technique, which is applicable to any complicated cell model formulated as a system of ordinary differential equations, to quantitatively evaluate contributions of respective model components to the model dynamics in the intact situation. The present technique employs a novel mathematical index for decomposed dynamics with respect to each differential variable, along with a concept named instantaneous equilibrium point, which represents the trend of a model variable at some instant. This article also illustrates applications of the method to comprehensive myocardial cell models for analysing insights into the mechanisms of action potential generation and calcium transient. The analysis results exhibit quantitative contributions of individual channel gating mechanisms and ion exchanger activities to membrane repolarization and of calcium fluxes and buffers to raising and descending of the cytosolic calcium level. These analyses quantitatively explicate principle of the model, which leads to a better understanding of cellular dynamics.     

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.     

Inositol 1,4,5-trisphosphate mass changes from fertilization through first cleavage in Xenopus laevis.

After fertilization in Xenopus laevis, inositol 1,4,5-trisphosphate (IP3) mass increased from 53 to 261 fmol/cell and returned to near basal by 10 min after insemination. IP3 was also elevated over control egg levels during first mitosis and first cleavage. Because IP3 levels and the fertilization calcium wave decline at about the same time and because calcium ionophore or pricking the egg increased IP3, the fertilization calcium wave may be due to calcium-induced IP3 production. In addition, the onset of sperm motility was associated with an increase, whereas the acrosomal reaction was accompanied by a decrease in IP3 mass. Combining our published data with this report, the first chronology of the levels of IP3 from the induction of meiosis (maturation) through fertilization and cleavage in one cellular system is summarized. These data suggest an in vivo dose response for IP3 and calcium release. A small (17 fmol/cell) IP3 change during the induction of meiosis may not be associated with a calcium change. Larger IP3 changes at cleavage (40 fmol/cell) and mitosis (125 fmol/cell) are associated with localized small calcium increases, whereas the largest IP3 change (208 fmol/cell) is associated with the large calcium increase at fertilization.     

SALTANT PRODUCTION BY WAVE LENGTHS OF VISIBLE AND LONG ULTRAVIOLET MONOCHROMATIC IRRADIATION,AND A COMPARISON WITH SALTANTS PRODUCED BY SHORT WAVE LENGTHS OF MONOCHROMATIC ULTRAVIOLET IRRADIATION IN THE FUNGUS CHAETOMIUM GLOBOSUM

1. Saltants have been produced in the fungus Chaetomium globosum by longer wave lengths than previously reported—by 365 mµ and by a visible line 404 mµ. 2. Absence at these wave lengths of the K saltant, which is so abundant at short wave lengths, is marked. 3. Ratio of percentage irradiated spores germinating to control spores germinating decreases from 83 per cent at 265 mµ, a short ultraviolet wave length, to 57 per cent at 404 mµ, a visible violet wave length.     

Calcium waves

Waves through living systems are best characterized by their speeds at 20 degrees C. These speeds vary from those of calcium action potentials to those of ultraslow ones which move at 1-10 and/or 10-20 nm s(-1). All such waves are known or inferred to be calcium waves. The two classes of calcium waves which include ones with important morphogenetic effects are slow waves that move at 0.2-2 microm s(-1) and ultraslow ones. Both may be propagated by cycles in which the entry of calcium through the plasma membrane induces subsurface contraction. This contraction opens nearby stretch-sensitive calcium channels. Calcium entry through these channels propagates the calcium wave. Many slow waves are seen as waves of indentation. Some are considered to act via cellular peristalsis; for example, those which seem to drive the germ plasm to the vegetal pole of the Xenopus egg. Other good examples of morphogenetic slow waves are ones through fertilizing maize eggs, through developing barnacle eggs and through axolotl embryos during neural induction. Good examples of ultraslow morphogenetic waves are ones during inversion in developing Volvox embryos and across developing Drosophila eye discs. Morphogenetic waves may be best pursued by imaging their calcium with aequorins.     

Calcium Regulation of an Actin Spring

Calcium is essential for many biological processes involved in cellular motility. However, the pathway by which calcium influences motility, in processes such as muscle contraction and neuronal growth, is often indirect and complex. We establish a simple and direct mechanochemical link that shows how calcium quantitatively regulates the dynamics of a primitive motile system, the actin-based acrosomal bundle of horseshoe crab sperm. The extension of this bundle requires the continuous presence of external calcium. Furthermore, the extension rate increases with calcium concentration, but at a given concentration, we find that the volumetric rate of extension is constant. Our experiments and theory suggest that calcium sequentially binds to calmodulin molecules decorating the actin filaments. This binding leads to a collective wave of untwisting of the actin filaments that drives bundle extension.     

Excitation wave propagation as a possible mechanism for signal transmission in pancreatic islets of Langerhans

In response to glucose application, beta-cells forming pancreatic islets of Langerhans start bursting oscillations of the membrane potential and intracellular calcium concentration, inducing insulin secretion by the cells. Until recently, it has been assumed that the bursting activity of beta-cells in a single islet of Langerhans is synchronized across the whole islet due to coupling between the cells. However, time delays of several seconds in the activity of distant cells are usually observed in the islets of Langerhans, indicating that electrical/calcium wave propagation through the islets can occur. This work presents both experimental and theoretical evidence for wave propagation in the islets of Langerhans. Experiments with Fura-2 fluorescence monitoring of spatiotemporal calcium dynamics in the islets have clearly shown such wave propagation. Furthermore, numerical simulations of the model describing a cluster of electrically coupled beta-cells have supported our view that the experimentally observed calcium waves are due to electric pulses propagating through the cluster. This point of view is also supported by independent experimental results. Based on the model equations, an approximate analytical expression for the wave velocity is introduced, indicating which parameters can alter the velocity. We point to the possible role of the observed waves as signals controlling the insulin secretion inside the islets of Langerhans, in particular, in the regions that cannot be reached by any external stimuli such as high glucose concentration outside the islets.     

An image-based model of calcium waves in differentiated neuroblastoma cells

Calcium waves produced by bradykinin-induced inositol-1,4, 5-trisphosphate (InsP(3))-mediated release from endoplasmic reticulum (ER) have been imaged in N1E-115 neuroblastoma cells. A model of this process was built using the "virtual cell," a general computational system for integrating experimental image, biochemical, and electrophysiological data. The model geometry was based on a cell for which the calcium wave had been experimentally recorded. The distributions of the relevant cellular components [InsP(3) receptor (InsP(3)R)], sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA) pumps, bradykinin receptors, and ER] were based on 3D confocal immunofluorescence images. Wherever possible, known biochemical and electrophysiological data were used to constrain the model. The simulation closely matched the spatial and temporal characteristics of the experimental calcium wave. Predictions on different patterns of calcium signals after InsP(3) uncaging or for different cell geometries were confirmed experimentally, thus helping to validate the model. Models in which the spatial distributions of key components are altered suggest that initiation of the wave in the center of the neurite derives from an interplay of soma-biased ER distribution and InsP(3) generation biased toward the neurite. Simulations demonstrate that mobile buffers (like the indicator fura-2) significantly delay initiation and lower the amplitude of the wave. Analysis of the role played by calcium diffusion indicated that the speed of the wave is only slightly dependent on the ability of calcium to diffuse to and activate neighboring InsP(3) receptor sites.     

Does the Potential for Chaos Constrain the Embryonic Cell-Cycle Oscillator?

Although many of the core components of the embryonic cell-cycle network have been elucidated, the question of how embryos achieve robust, synchronous cellular divisions post-fertilization remains unexplored. What are the different schemes that could be implemented by the embryo to achieve synchronization? By extending a cell-cycle model previously developed for embryos of the frog Xenopus laevis to include the spatial dimensions of the embryo, we establish a novel role for the rapid, fertilization-initiated calcium wave that triggers cell-cycle oscillations. Specifically, in our simulations a fast calcium wave results in synchronized cell cycles, while a slow wave results in full-blown spatio-temporal chaos. We show that such chaos would ultimately lead to an unpredictable patchwork of cell divisions across the embryo. Given this potential for chaos, our results indicate a novel design principle whereby the fast calcium-wave trigger following embryo fertilization synchronizes cell divisions.     

A PKC wave follows the calcium wave after activation of Xenopus eggs

Calcium waves are well-known hallmarks of egg activation that trigger resumption of the cell cycle and development of the embryo. These waves rapidly and efficiently assure that activation signals are transmitted to all regions of the egg. Although the mechanism by which the calcium wave propagates across an egg as large as that of Xenopus is not known, two models prevail. One model is a wave of calcium-induced calcium release (CICR) and the other is propagation by inositol-induced calcium release (IICR). IICR requires a wave of phosphatidylinositol 4,5-bisphosphate (PIP2) hydrolysis, generating two second messengers, IP3, which then releases calcium and DAG, which activates protein kinase C (PKC). We show here that a wave of PKC-green fluorescent protein travels across the egg immediately following, and at the same velocity as, the calcium wave. This is the first example of a PKC wave in a vertebrate egg and supports the IICR model of wave propagation.     

Calcium signalling in wound-responsive normal human urothelial cell monolayers

Epithelial tissue repair requires coordination of migratory and proliferative activity both adjacent to and remote from the wound edge. Although calcium signalling is implicated, the specific mechanisms are poorly understood. This study characterises the calcium signal invoked in response to scratch wounding of normal human urothelial (NHU) cells and relates it to the localised cellular response. Immediately after wounding of confluent NHU cell monolayers, cells adjacent to the wound edge showed a sustained (>30 min) rise in [Ca(2+)](i), while there was an independent, but simultaneous calcium wave that propagated out from the wound edge. The transient signal involved release of calcium from intracellular stores and was not mediated via gap junctions, but by diffusion of extracellular agonists. We demonstrated that ATP was partially responsible for the initiation and propagation of the calcium wave and showed that the calcium release mechanism was mediated in part via activation of inositol-1,4,5-triphosphate (IP(3)) receptors. By contrast, the sustained calcium signal originated from the extracellular milieu and correlated with an increased rate of migration by these cells. The work presented here provides supportive evidence that the calcium signature, defined by its temporal and amplitude characteristics, is important in co-ordinating the response of cells within an epithelial cell monolayer after wounding.     

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.     

Photoreversal of nuclear and cytoplasmic effects of short ultraviolet radiation on Paramecium caudatum

1. Irradiation with three short ultraviolet (UV) wave lengths, 226, 233, and 239 mµ rapidly immobilizes Paramecium caudatum, the dosage required being smaller the shorter the wave length. 85 per cent of paramecia immobilized with wave length 226 mµ recover completely. Recovery from immobilizing doses is less the longer the wave length. 2. Irradiation continued after immobilization kills the paramecia in a manner which is markedly different for very short (226, 233, and 239 mµ) and longer (267 mµ) wave lengths. 3. An action spectrum for immobilization in P. caudatum was determined for the wave lengths 226, 233, 239, 248, and 267 mµ, and found to resemble the absorption of protein and lipide in the wave length region below 248 mµ. Addition of these data to those of Giese (1945 b) gives an action spectrum resembling the absorption by albumin-like protein. 4. Division of P. caudatum is delayed by doses of wave lengths 226, 233, and 239 mµ which cause immobilization, the longest wave length being most effective. 5. Immobilization at any of the wave lengths tested (226, 233, 239, 248, 267 mµ) is not photoreversible when UV-treated paramecia are concurrently illuminated. 6. Division delay resulting from immobilizing doses of 226, 233, and 239 mµ is photoreversible by exposure to visible light concurrently with the UV. 7. Division delay induced by exposure to wave length 267 mµ is reduced by exposure to visible light applied concurrently with UV or immediately afterwards. 8. The data suggest that the shortest UV wave length tested (226 mµ) affects the cytoplasm selectively, because it is absorbed superficially as indicated by unilateral fluorescence in UV. Consequently it immobilizes paramecia rapidly but has little effect on the division rate because little radiation reaches the nucleus. 9. The data support the view that nuclear effects of UV are readily photoreversed but cytoplasmic effects are not.     

The initiation and propagation of the fertilization wave in sea urchin eggs

Calcium waves sweep across most eggs of the deuterostome lineage at fertilization. The precise timing of the initiation and propagation of a fertilization calcium wave has been best studied in sea urchin embryos, since the rapid depolarization caused by sperm egg fusion can be detected as a calcium influx using confocal imaging of calcium indicator dyes. The time between sperm egg fusion and the first sign of the calcium increase that constitutes the calcium wave is comparable to the time it takes for the wave to sweep across the egg, once initiated. The latency and rise time of the calcium response is sensitive to inhibitors of the InsP3 signalling pathway, as reported previously. Using calcium green dextran and confocal microscopy, we confirm that the propagation time of the calcium wave is lengthened and that initiation of the calcium wave involves activation of calcium release at hot spots that may represent clusters of calcium release channels, as has been seen in other cell types.     

An activatable molecular spring reduces muscle tearing during extreme stretching

The purpose of this study was to determine failure stresses and failure lengths of actively and passively stretched myofibrils. As expected, myofibrils failed at average sarcomere lengths (about 6–7 μm) that vastly exceeded sarcomere lengths at which actin–myosin filament overlap ceases to exist (4 μm) and thus actin–myosin-based cross-bridge forces are zero at failure. Surprisingly, however, actively stretched myofibrils had much greater failure stresses and failure energies than passively stretched myofibrils, thereby providing compelling evidence for strong force production independent of actin–myosin-based cross-bridge forces. Follow-up experiments in which titin was deleted and cross-bridge formation was inhibited at high and low calcium concentrations point to titin as the regulator of this force, independent of calcium. The results of this study point to a mechanism of force production that reduces stretch-induced muscle damage at extreme length and limits injury and force loss within physiologically relevant ranges of sarcomere and muscle lengths.