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We investigated theoretically and experimentally the Ca2+-contraction coupling in rat tracheal smooth muscle. [Ca2+]i, isometric contraction and myosin light chain (MLC) phosphorylation were measured in response to 1 mM carbachol. Theoretical modeling consisted in coupling a model of Ca2+-dependent MLC kinase (MLCK) activation with a four-state model of smooth muscle contractile apparatus. Stimulation resulted in a short-time contraction obtained within 1 min, followed by a long-time contraction up to the maximal force obtained in 30 min. ML-7 and Wortmannin (MLCK inhibitors) abolished the contraction. Chelerythrine (PKC inhibitor) did not change the short-time, but reduced the long-time contraction. [Ca2+]i responses of isolated myocytes recorded during the first 90 s consisted in a fast peak, followed by a plateau phase and, in 28% of the cells, superimposed Ca2+ oscillations. MLC phosphorylation was maximal at 5 s and then decreased, whereas isometric contraction followed a Hill-shaped curve. The model properly predicts the time course of MLC phosphorylation and force of the short-time response. With oscillating Ca2+ signal, the predicted force does not oscillate. According to the model, the amplitude of the plateau and the frequency of oscillations encode for the amplitude of force, whereas the peak encodes for force velocity. The long-time phase of the contraction, associated with a second increase in MLC phosphorylation, may be explained, at least partially, by MLC phosphatase (MLCP) inhibition, possibly via PKC inhibition.

The mathematical model presented here describes the interactions among Ca2+, calmodulin (CaM), and myosin light chain kinase (MLCK) and consists of a kinetic scheme taking into account 7 reactions instead of 12 as proposed previously. We derive a system of 5 nonlinear ordinary differential equations. Solving it yields the prediction of active MLCK as a function of [Ca2+] whereby the active MLCK is defined to be proportional to the Ca4CaM.MLCK complex concentration. The model predictions are compared with other theoretical and experimental predictions of active MLCK as well as with the results of our previously proposed complex model.

Active Ca2+/calmodulin (CaM)-dependent myosin light chain kinase (MLCK) plays an important role in the process of MLC phosphorylation and consecutive smooth muscle contraction. Here, we propose a mathematical model of a detailed kinetic scheme describing interactions among Ca2+, CaM and MLCK and taking into account eight different aggregates. The main model result is the prediction of the Ca2+ dependent active form of MLCK, which is in the model taken as proportional to the concentration of Ca4CaM · MLCK complex. Wegscheider’s condition is additionally applied as a constraint enabling the prediction of some parameter values that have not yet been obtained by experiments.

Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority ( approximately 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.

Intracellular calcium oscillations provide a natural clock that may be of crucial importance for the timing of many cellular processes. Elucidating of the mechanisms underlying these oscillations is of particular interest. The theoretical description presented here extends existing models of calcium oscillations by allowing for two types of proteins differing in their calcium-binding properties. This model reflects experimental findings by considering both a fast calcium-binding process to low-affinity protein binding sites such as found in the N-domains of calmodulin or troponin C and a class of high-affinity calcium binding proteins with slow binding kinetics (e.g., parvalbumin or the C-domains of calmodulin and troponin C). Furthermore, recalling that calcium is mainly stored in small subcompartments of the ER, it is argued that only a small fraction of its overall volume participates in the rapid release and uptake of calcium. The effect of the size of this fraction is studied. The hypothesis saying that any electric potential difference across the ER membrane would be dissipated by the highly permeant ions is critically examined by an analytical estimation based on the electroneutrality condition and by numerical integration of the complete model equations. It is predicted theoretically that the transmembrane potential of the ER calcium stores, which is up to now virtually impossible to determine in experiment, builds up in the millivolt range at physiological concentrations of monovalent ions. The phenomenology of oscillations is studied by numerical integration. The model reproduces experimentally observed values of frequency and amplitude as well as the typical spike-like shape of oscillations. The model reveals also the time course of a shift of the bound Ca2+ population from the low-affinity binding sites to the high-affinity binding sites.

We investigated theoretically and experimentally the role of Rho kinase (RhoK) in Ca(2+)-contraction coupling in rat airways. Isometric contraction was measured on tracheal, extrapulmonary and intrapulmonary bronchial rings. Intracellular [Ca(2+)] was recorded in freshly isolated tracheal myocytes. Stimulation by carbachol (0.3 and 10 μm) and 50 mm external KCl induced a short-time, Hill-shaped contraction obtained within 90 s, followed by a sustained or an additional delayed contraction. Responses of [Ca(2+)](i) to acetylcholine consisted in a fast peak followed by a plateau and, in 42% of the cells, superimposed Ca(2+) oscillations. The RhoK inhibitor Y27632 (10 μm) did not alter the [Ca(2+)](i) response. Whatever the agonist, Y27632 did not modify the basal tension but decreased the amplitude of the short-duration response, without altering the additional delayed contraction. The Myosin Light Chain Phosphatase (MLCP) inhibitor calyculin A increased the basal tension and abolished the effect of RhoK. KN93 (Ca(2+)-calmodulin-dependent protein kinase II inhibitor) and DIDS (inhibitor of Ca(2+)-activated Cl(-) channels) had no influence on the RhoK effect. We built a theoretical model of Ca(2+)-dependent active/inactive RhoK ratio and subsequent RhoK-dependent MLCP inactivation, which was further coupled with a four-state model of the contractile apparatus and Ca(2+)-dependent MLCK activation. The model explains the time course of the short-duration contraction and the role of RhoK by Ca(2+)-dependent activation of MLCK and RhoK, which inactivates MLCP. Oscillatory and non-oscillatory [Ca(2+)](i) responses result in a non-oscillatory contraction, the amplitude of which is encoded by the plateau value and oscillation frequency. In conclusion, Ca(2+)-dependent but CaMK II-independent RhoK activation contributes to the early phase of the contractile response via MLCP inhibition.

In airway smooth muscles, kinase/phosphatase-dependent phosphorylation and dephosphorylation of the myosin light chain (MLC) have been revealed by many authors as important steps in calcium (Ca(2+)) signalling pathway from the variation of Ca(2+) concentration in cytosol to the force development. Here, a theoretical analysis of the control action of MLC-kinase (MLCK) and MLC-phosphatase (MLCP) in Ca(2+) signalling is presented and related to the general control principles of these enzymes, which were previously studied by Reinhart Heinrich and his co-workers. The kinetic scheme of the mathematical model considers interactions among Ca(2+), calmodulin (CaM) and MLCK and the well-known 4-state actomyosin latch bridge model, whereby a link between them is accomplished by the conservation relation of all species of MLCK. The mathematical model predicts the magnitude and velocity of isometric force in smooth muscles upon transient biphasic Ca(2+) signal. The properties of signal transduction in the system such as the signalling time, signal duration and signal amplitude, which are reflected in the properties of force developed, are studied by the principles of the metabolic control theory. The analysis of our model predictions confirms as shown by Reinhart Heinrich and his co-workers that MLCK controls the amplitude of signal more than its duration, whereas MLCP controls both. Finally, the simulations of elevated total content of MLCK, a typical feature of bronchial muscles of asthmatic subjects and spontaneously hypertensive rats as well as potentiation of MLCP catalytic activity, are carried out and are discussed in view of an increase in the force magnitude.

Theoretical models of intracellular calcium oscillations have hitherto focused on the endoplasmic reticulum (ER) as an internal calcium store. These models reproduced the large variability in oscillation frequency observed experimentally. In the present contribution, we extend our earlier model [Marhl et al., Biophys. Chem., 63 (1997) 221] by including, in addition to the ER, mitochondria as calcium stores. Simple plausible rate laws are used for the calcium uptake into, and release from, the mitochondria. It is demonstrated with the help of this extended model that mitochondria are likely to act in favour of frequency encoding by enabling the maintenance of fairly constant amplitudes over wide ranges of frequency.

Oscillations of the cytosolic calcium concentration have turned out to be an important phenomenon in a variety of living cells and have recently been intensely investigated (Woods et al., 1986; De Young and Keizer, 1992; Dupont and Goldbeter, 1993; Li and Rinzel, 1994; Jafri and Gillo, 1994; Jouaville et al., 1995; Goldbeter, 1996). They play an important role in intracellular information processing (Cuthbertson, 1989; Goldbeter et al., 1990). The risk of overloading the cell with calcium in the process of signalling is believed to be avoided by oscillatory mechanisms instead of adjustable stationary messenger concentrations (Berridge, 1989). Another advantage is that a pulsatile signal can carry two types of information: a digital signal (oscillation or stationarity) acting as a switch, for example, between two modes of metabolism, and an analogue signal which may determine the flux of some metabolic pathway (Cuthbertson, 1989).

A refined electrochemical model accounting for intracellular calcium oscillations and their interrelations with oscillations of the potential difference across the membrane of the endoplasmic reticulum (ER) or other intracellular calcium stores is established. The ATP dependent uptake of Ca2+ from the cytosol into the ER, the Ca2+ release from the ER through channels following a calcium-induced calcium release mechanism, and a potential-dependent Ca2+ leak flux out of the ER are included in the model and described by plausible rate laws. The binding of calcium to specific proteins such as calmodulin is taken into account. The quasi-electroneutrality condition allows us to express the transmembrane potential in terms of the concentrations of cytosolic calcium and free binding sites on proteins, which are the two independent variables of the model. We include monovalent ions in the model, because they make up a considerable portion in the balance of electroneutrality. As the permeability of the endoplasmic membrane for these ions is much higher than that for calcium ions, we assume the former to be in Nernst equilibrium. A stability analysis of the steady-state solutions (which are unique or multiple depending on parameter values) is carried out and the Hopf bifurcation leading from stable steady states to self-sustained oscillations is analysed with the help of appropriate mathematical techniques. The oscillations obtained by numerical integration exhibit the typical spike-like shape found in experiments and reasonable values of frequency and amplitude. The model describes the process of switching between stationary and pulsatile regimes as well as changes in oscillation frequency upon parameter changes. It turns out that calcium oscillations can arise without a permanent influx of calcium into the cell, when a calcium-buffering system such as calmodulin is included.

Ionic concentrations in the close proximity of a carrier may be different from those in the bulk solution. An immediate layer in the solution in which this situation occurs is known as a diffusion layer. Such diffusion layers were calculated using general diffusion equations and postulating a membrane to be homogeneous in the plane with respect to its permeability. In contrast, the present mathematical model considers single-carrier mediated transport of ions across the membrane and their diffusion away from the carrier site into the electrolyte solution. In particular, the transport of Ca2+ ions is considered. The diffusion of electrolyte ions (Na+ and Cl-) and of Ca2+ ions is described by the Nernst-Planck electrodiffusion equation. The relation between the local electric potential and the ion concentrations is taken into account by the Poisson equation. The equations are solved numerically for radial symmetry by the relaxation method. The model predicts concentration and potential profiles in dependence of the flux rate of Ca2+ ions. It is shown that for fluxes mediated by a single carrier, a diffusion layer becomes significant if the flux is larger than 10(5) Ca2+ ions per second.