Mathematical Modeling of Mechanically Modulated Rhythm Disturbances in Homogeneous and Heterogeneous Myocardium with Attenuated Activity of Na<Superscript>+</Superscript>–K<Superscript>+</Superscript> Pump

A mathematical model of the cardiomyocyte electromechanical function is used to study contribution of mechanical factors to rhythm disturbances in the case of the cardiomyocyte calcium overload. Particular attention is paid to the overload caused by diminished activity of the sodium-potassium pump. It is shown in the framework of the model, where mechano-calcium feedback is accounted for that myocardium mechanics may significantly enhance arrhythmogenicity of the calcium overload. Specifically, a role of cross-bridge attachment/detachment processes, a role of mechanical conditions of myocardium contractions (length, load), and a role of myocardium viscosity in the case of simulated calcium overload have been revealed. Underlying mechanisms are analyzed. Several approaches are designed in the model and compared to each other for recovery of the valid myocardium electrical and mechanical performance in the case of the partially suppressed sodium-potassium pump.     

Mechanistic systems models of cell signaling networks: a case study of myocyte adrenergic regulation

Signal transduction networks coordinate a wide variety of cellular functions, including gene expression, metabolism, and cell fate processes. Understanding biological networks quantitatively is a major challenge to post-genomic biology, and mechanistic systems models will be crucial for this task. Here, we review approaches towards developing mechanistic systems models of established cell signaling networks. The ability of mechanistic system models to generate testable biological hypotheses and experimental strategies is discussed. As a case study of model development and analysis, we examined the functional roles of phospholamban, the L-type calcium channel, the ryanodine receptor, and troponin I phosphorylation upon β-adrenergic stimulation in the rat ventricular myocyte. Model analysis revealed that while protein kinase A-mediated phosphorylation of the ryanodine receptor greatly increases its calcium sensitivity, calcium autoregulation may adapt quickly by negating potential increases in contractility. Systematic combinations of in silico perturbations supported the conclusion that phospholamban phosphoregulation is the primary mechanism for increased sarcoplasmic reticulum load and calcium relaxation rate during β-adrenergic stimulation, while both phospholamban and the L-type calcium channel contribute to increased systolic calcium. Combined with detailed experimental studies, mechanistic systems models will be valuable for developing a quantitative understanding of cell signaling networks.     

Cellular calcium and atherosclerosis: A brief review

Evidence for and against the theory that cell calcium is causally involved in the pathogenesis of atherosclerosis is presented and evaluated. In particular, it is argued that: (1) arterial calcium is increased in atherosclerosis; (2) this increase in tissue calcium content is largely intracellular; (3) this increased intracellular calcium content is caused by increased plasma membrane calcium permeability; (4) the increased calcium content is causally related to atherogenesis; (5) many of the cell physiological, cell biological, biochemical, and molecular biological processes, known to function abnormally in atherosclerosis, are also known to be calcium regulated; and (6) these processes are activated or inactivated in atherosclerosis in a manner consistent with increased cell calcium. It is concluded that the calcium-atherogenesis hypothesis has the potential to unify macroscopic clinical risk factors in terms of intracellular mechanisms that are controlled by cell calcium, and that this hypothesis deserves further experimental tests.     

The blocking of capacitative calcium entry by 2-aminoethyl diphenylborate (2-APB) and carboxyamidotriazole (CAI) inhibits proliferation in Hep G2 and Huh-7 human hepatoma cells

Calcium entry is a component of the processes regulating the proliferative phenotype of some types of cancer. In non-excitable cells, capacitative calcium entry (CCE) and non-capacitative calcium entry (NCCE) are thought to be the main pathways of Ca2+ influx into cells. Thus, blocking calcium entry may prevent normal and pathological cell proliferation and there is evidence to suggest that molecules blocking calcium entry also have antiproliferative properties. Carboxyamidotriazole (CAI), a novel inhibitor of the non-voltage-dependent calcium entry has been shown to have such properties in model systems in vitro and in vivo. We used Hep G2 and Huh-7 human hepatoma cells to investigate the effects of calcium entry blockers on cell proliferation. CAI (10 microM) and 2-APB (20 microM) completely blocked CCE in thapsigargin-treated Huh-7, and CAI and 2-APB inhibited cell proliferation with IC50 of 4.5 and 43 microM, respectively. The plateau phase of the [Ca2+]i increases triggered by 10% FCS were abolished in the absence of external Ca2+ and in the presence of CAI or 2-APB. We, therefore, suggest that CCE is the main pathway involved in regulation of the processes leading to cell proliferation.     

Transport-dependent calcium signaling in spatially segregated cellular caveolar domains

We developed a two-dimensional model of transport-dependent intracellular calcium signaling in endothelial cells (ECs). Our purpose was to evaluate the effects of spatial colocalization of endothelial nitric oxide synthase (eNOS) and capacitative calcium entry (CCE) channels in caveolae on eNOS activation in response to ATP. Caveolae are specialized microdomains of the plasma membrane that contain a variety of signaling molecules to optimize their interactions and regulate their activity. In ECs, these molecules include CCE channels and eNOS. To achieve a quantitative understanding of the mechanisms of microdomain calcium signaling and the preferential sensitivity of eNOS to calcium entering the cell through CCE channels, we constructed a mathematical model incorporating the cell morphology and cellular physiological processes. The model predicts that the spatial segregation of calcium channels in ECs can create transport-dependent sharp gradients in calcium concentration within the cell. The calcium concentration gradient is affected by channel density and cell geometry. This transport-dependent calcium signaling specificity effect is enhanced in ECs by increasing the spatial segregation of the caveolar signaling domains. Our simulation significantly advances the understanding of how Ca2+, despite its many potential actions, can mediate selective activation of signaling pathways. We show that diffusion-limited calcium transport allows functional compartmentalization of signaling pathways based on the spatial arrangements of Ca2+ sources and targets.     

Mathematical modeling of transmembrane calcium transport kinetics in smooth muscle cells

A mathematical model, which describes kinetics of transmembrane calcium transport in a smooth muscular cell, has been elaborated and investigated taking into account that the change of calcium cations concentration within a cell is determined by two mutually opposite processes: an increase of a carrying capacity of calcium channels of plasma membrane under signal substance action and calcium removal from the intracellular space by Mg2+, ATP-dependent calcium pump localized on the plasma membrane. The fundamental difference of the proposed model against the models analyzed in literature before is that the cellular system returns to the initial stationary state after enzyme-catalysed transformation of the signal substance. The results of calculations showed that this model really described the experimental kinetics of the transmembrane calcium transport. In this paper the influence of different parameters (Michaelis constant and ultimate rate of calcium pump, initial concentrations of signal substance and enzyme decomposing it, rate constants) on kinetics of calcium transport through the plasma membrane has been investigated in detail.     

Influence of calcium and other cations on surface adhesion of bacteria and diatoms: A review

Association with a surface is an important aspect of survival for microorganisms in natural and manmade environments/Both bacteria and diatoms are involved in such associations. In many cases, this leads to surface fouling, which often results in surface deterioration and mechanical failure in industrial systems. We now know that microorganisms exploit many strategies to establish associations with surfaces. As in the case of other cellular processes, calcium ions seem to play an important role in adhesion of cells to surfaces. Calcium is involved in non-specific interactions such as neutralization of the electrical double layer between cell and substratum surface as well as specific adhesive interactions that cannot be replaced by other cations. The unique properties of calcium ions promote both specific and non-specific interactions with protein and polysaccha-ride adhesin molecules at the cell surface. As important, but less well understood, calcium ions also influence the way microbial cells interact with different substrata.     

Calcium regulation in crustaceans during the molt cycle: a review and update

Epithelial cells of the gut, gills, antennal glands and integument regulate calcium concentrations in crustaceans during the molt cycle. A cellular calcium transport model has been proposed suggesting the presence of calcium pumps, cation antiporters and calcium channels in transporting epithelial membranes that regulate the movements of this cation across the cell layer. Basolateral calcium transport during postmolt appears mainly regulated by the low affinity NCX antiporter, while calcium regulating 'housekeeping' activities of these cells in intermolt are controlled by the high affinity calcium ATPase (PMCA). A model is proposed for the involvement of the epithelial ER in the massive transepithelial calcium fluxes that occur during premolt and postmolt. This model involves the endoplasmic reticulum SERCA and RyR proteins and proposed cytoplasmic unstirred layers adjacent to apical and basolateral plasma membranes where calcium activities may largely exceed those in the bulk cytoplasmic phase. A result of the proposed transepithelial calcium transport model is that large quantities of calcium can be moved through these cells by these processes without affecting the low, and carefully controlled, bulk cytoplasmic calcium activities.     

Rearrangement of the endoplasmic reticulum and calcium transient formation: the computational approach

Experiments affecting calcium signaling often lead to changes in the calcium transient height. The present work is designed to approach this effect theoretically. Use of computational model let us to follow results of precisely designed changes in the endoplasmic reticulum distribution as a possible cause of cytoplasmic free calcium ion level. Obtained results suggest that indeed, rearrangement of the endoplasmic reticulum elements may be responsible for modulation of calcium signal's strength. We have also noticed that even if the endoplasmic reticulum concentration levels are local, the resulting changes in free calcium concentration are global and evenly distributed throughout the cell. The used mathematical method proved to be a powerful tool which made us understand the chemical dynamics of nonequilibrium processes of calcium transient formation. Presented data show how Ca2+ signal resulting from IP3 provoked release of calcium from the endoplasmic reticulum may depend on the cytoskeleton structure.     

Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study)

Computer simulation using Luo-Rudy I1 model of ventricular myocyte showed that intracellular calcium dynamics become irregular in case of high rate stimulation. This causes the transition from stationary to nonstationary spiral wave and its breakup in 2D model of cardiac tissue. Obtained results suggest how ventricular fibrillation may occur due to the abnormalities of intracellular calcium dynamics. The short review of existing cardiac cell models with calcium dynamics is presented.     

Spatial Wavelet Analysis of Calcium Oscillations in Developing Neurons

Calcium signals play a major role in the control of all key stages of neuronal development, and in particular in the growth and orientation of neuritic processes. These signals are characterized by high spatial compartmentalization, a property which has a strong relevance in the different roles of specific neuronal regions in information coding. In this context it is therefore important to understand the structural and functional basis of this spatial compartmentalization, and in particular whether the behavior at each compartment is merely a consequence of its specific geometry or the result of the spatial segregation of specific calcium influx/efflux mechanisms. Here we have developed a novel approach to separate geometrical from functional differences, regardless on the assumptions on the actual mechanisms involved in the generation of calcium signals. First, spatial indices are derived with a wavelet-theoretic approach which define a measure of the oscillations of cytosolic calcium concentration in specific regions of interests (ROIs) along a cell, in our case developing chick ciliary ganglion neurons. The resulting spatial profile demonstrates clearly that different ROIs along the neuron are characterized by specific patterns of calcium oscillations. Next we have investigated whether this inhomogeneity is due just to geometrical factors, namely the surface to volume ratio in the different subcompartments (e.g. soma vs. growth cone) or it depends on their specific biophysical properties. To this aim correlation functions are computed between the activity indices and the surface/volume ratio along the cell: the data thus obtained are validated by a statistical analysis on a dataset of different cells. This analysis shows that whereas in the soma calcium dynamics is highly correlated to the surface/volume ratio, correlations drop in the growth cone-neurite region, suggesting that in this latter case the key factor is the expression of specific mechanisms controlling calcium influx/efflux.     

Nonuniformity of calcium efflux from pancreatic acinar cells and its analysis by mathematical model of calcium diffusion and buffering in extracellular solution

We studied spatial and temporal patterns of Ca²⁺ extrusion from pancreatic acinar cells evoked by acetylcholine(ACh)-induced activation of plasma membrane calcium pumps. Using a modification of an earlier developed model, we estimated the time course of extracellular calcium concentration changes near the basal pole of a cell in the case, when calcium ions are released from the same site on the cell surface, and in the case when they are extruded from the apical pole and diffuse to the basal one. It is concluded that at the first stage of ACh-induced Ca²⁺ extrusion the appearance of Ca²⁺ elevation near the basal pole of the cells cannot be explained as a result of diffusion, but is mainly determined by Ca²⁺ efflux from this pole. The results also show that there are plasma membrane calcium pumps in both apical and basal parts of pancreatic acinar cells, but the activity of the pumps is substantially higher in the apical region.     

A realistic model of biphasic calcium transients in electrically nonexcitable cells.

In many electrically nonexcitable cells, the release of calcium from internal stores is followed by a much slower phase in which the intracellular calcium concentration decreases gradually to a sustained value higher than the concentration before stimulation. This elevated calcium plateau has been shown to be the result of calcium influx. The model presented in this work describes a system consisting of a cytoplasmic calcium store and a plasma membrane calcium channel, both excitable by a membrane receptor; a fast cytoplasmic calcium buffer; and calcium pumps in both the calcium store and cellular membranes. Inherent difficulties in the numerical evaluation of the model, caused by very large calcium fluxes across the store membrane, were overcome by analytically separating the fast processes of calcium release from the slower processes of calcium cycling across the plasma membrane. This enabled the simulation of realistic biphasic calcium transients similar to those observed experimentally. The model predicted 1) a strong correlation between the rate of calcium cycling across the plasma membrane and the rate of calcium decay; and 2) a dependence on the level of cell excitation of the maximum rise in cytoplasmic calcium concentration, the level of the elevated calcium plateau, and the rate of calcium decay. Using the model, we simulated the washout of agonist from the bathing solution and the depletion of the calcium store by a pharmacological agent (such as thapsigargin) under several experimental conditions.     

Electroporation-induced formation of individual calcium entry sites in the cell body and processes of adherent cells.

Electroporation is a widely used method for introducing macromolecules into cells. We developed an electroporation device that requires only 1 microl of sample to load adherent cells in a 10-mm2 surface area while retaining greater than 90% cell survivability. To better understand this device, field-induced permeabilization of adherent rat basophilic leukemia and neocortical neuroblastoma cells was investigated by using fluorescent calcium and voltage indicators. Rectangular field pulses led to the formation of only a few calcium entry sites, preferentially in the hyperpolarized parts of the cell body and processes. Individual entry sites were formed at the same locations when field pulses were repeated. Before calcium entry, a partial breakdown of the membrane potential was observed in both polar regions. Based on our results, a model is proposed for the formation and closure of macromolecule entry sites in adherent cells. First, the rapid formation of a large number of small pores leads to a partial membrane potential breakdown in both polar regions of the cell. Second, over tens of milliseconds, a few entry sites for macromolecules are formed, preferentially in the hyperpolarized part of cell body and processes, at locations defined by the local membrane structure. These entry sites reseal on a time scale of 50 ms to several seconds, with residual small pores remaining open for several minutes.     

Convective-diffusive mass transfer of agonist and the intracellular calcium response of endothelial cell

Endothelial cell layer regulates several crucial physiological processes of the vascular system. The mechanism of the response of this cell layer to the flow of surrounding fluid is still largely unclear. In the present article, a comparison of the available experimental results for the intracellular calcium ion concentration and theoretical results for the extra-cellular ATP concentration obtained using a convective-diffusive mass transfer model, has been made which supports the mass transfer model for the endothelial cell response to the fluid flow. The experimental results are in excellent agreement with the calculated values assuming the effect of ATP concentration alone.     

Cyclic AMP oscillations in suspensions of Dictyostelium discoideum

A model developed previously for signal relay and adaptation in the cellular slime mould Dictyostelium discoideum is shown to account for the observed oscillations of calcium and cyclic AMP in cellular suspensions. A qualitative argument is given which explains how the oscillations arise, and numerical computations show how characteristics such as the period and amplitude of the periodic solutions depend on parameters in the model. Several extensions of the basic model are investigated, including the effect of cell aggregation and the effect of time delays in the activation and adaptation processes. The dynamics of mixed cell populations in which only a small fraction of the cells are capable of autonomous oscillation are also studied.     

Model for external influences on cellular signal transduction pathways including cytosolic calcium oscillations

Experiments on the effects of extremely-low-frequency (ELF) electric and magnetic fields on cells of the immune system, T-lymphocytes in particular, suggest that the external field interacts with the cell at the level of intracellular signal transduction pathways. These are directly connected with changes in the calcium-signaling processes of the cell. Based on these findings, a theoretical model for receptor-controlled cytosolic calcium oscillations and for external influences on the signal transduction pathway is presented. We discuss the possibility that the external field acts on the kinetics of the signal transduction between the activated receptors at the cell membrane and the G-proteins. It is shown that, depending on the specific combination of cell internal biochemical and external physical parameters, entirely different responses of the cell can occur. We compare the effects of a coherent (periodic) modulation and of incoherent perturbations (noise). The model and the calculations are based on the theory of self-sustained, nonlinear oscillators. It is argued that these systems form an ideal basis for information-encoding processes in biological systems. © 1995 Wiley-Liss, Inc.