Modeling the interrelations between the calcium oscillations and ER membrane potential oscillations

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.     

Stabilizing role of calcium store-dependent plasma membrane calcium channels in action-potential firing and intracellular calcium oscillations

In many biological systems, cells display spontaneous calcium oscillations (CaOs) and repetitive action-potential firing. These phenomena have been described separately by models for intracellular inositol trisphosphate (IP3)-mediated CaOs and for plasma membrane excitability. In this study, we present an integrated model that combines an excitable membrane with an IP3-mediated intracellular calcium oscillator. The IP3 receptor is described as an endoplasmic reticulum (ER) calcium channel with open and close probabilities that depend on the cytoplasmic concentration of IP3 and Ca2+. We show that simply combining this ER model for intracellular CaOs with a model for membrane excitability of normal rat kidney (NRK) fibroblasts leads to instability of intracellular calcium dynamics. To ensure stable long-term periodic firing of action potentials and CaOs, it is essential to incorporate calcium transporters controlled by feedback of the ER store filling, for example, store-operated calcium channels in the plasma membrane. For low IP3 concentrations, our integrated NRK cell model is at rest at -70 mV. For higher IP3 concentrations, the CaOs become activated and trigger repetitive firing of action potentials. At high IP3 concentrations, the basal intracellular calcium concentration becomes elevated and the cell is depolarized near -20 mV. These predictions are in agreement with the different proliferative states of cultures of NRK fibroblasts. We postulate that the stabilizing role of calcium channels and/or other calcium transporters controlled by feedback from the ER store is essential for any cell in which calcium signaling by intracellular CaOs involves both ER and plasma membrane calcium fluxes.     

Calmodulin can induce and control damping oscillations in the plasma membrane Ca2+ -ATPase activity: a kinetic model

Plasma membrane Ca2+-ATPase is the calcium pump that extrudes calcium ions from cells using ATP hydrolisis for the maintenance of low Ca2+ concentrations in the cell. Calmodulin stimulates Ca2+-ATPase by binding to the autoinhibitory enzyme domain, which allows the access of cytoplasmic ATP and Ca2+ to the active and transport cites. Our kinetic model predicts damped oscillations in the enzyme activity and interprets the known nonmonotonous kinetic behavior of the enzyme in the presence of calmodulin. For the parameters close to the experimental ones, the kinetic model explains the changes in frequency and damping factor of the oscillatory enzyme activity, as dependent on calmodulin concentration. The calculated pre-steady-state curves fit well the known experimental data. The kinetic analysis allows us to assign Ca2+-ATPase to the hysteretic enzymes exhibiting activity oscillations in open systems.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

Slow oscillations of free intracellular calcium ion concentration in human fibroblasts responding to mechanical stretch

Calcium transients in single, human gingival fibroblasts were studied after mechanical stretching of flexible culture substrates. A model system was developed to reproducibly stretch and rapidly (< 1 sec) refocus cells in the same focal plane so that changes in the concentration of free intracellular calcium ions ([Ca²⁺]_i) were monitored without delay. Attached cells were grown on flexible bottom Petriperm dishes, loaded with fura-2/AM, and stretched by 1% or 2.8% of substrate area. The stretch caused no significant cell detachment or membrane lesions. A 1% stretch induced no calcium response, but a 2.8% stretch stimulated an initial calcium transient and the subsequent generation of [Ca²⁺]_i oscillations of up to 2,000 sec. At 1% stretch, there was no calcium response. Cell shape and plating time were important determinants in the calcium response to mechanical stimulation: the responder cells were small and round without long processes. Major calcium transients were inhibited completely by 5 mM EGTA or by 10 μM gadolinium ions, by 50 μM nifedipine, or 250 μM verapamil, suggesting an influx of calcium through stretch-activated (SA) channels and L-type calcium channels. Depolarization by high KCl (144 mM) in the extracellular medium enhanced the amplitude of calcium transients by 54%. Calcium oscillations were not inhibited by preincubation with thapsigargin, caffeine, cholera toxin, staurosporine or 1-(5-isoquinolinesulfonyl)-2-methylpiperazine (H-7), indicating that IP₃ sensitive pools, IP₃ insensitive pools, G₅α subunits, and protein kinase C, respectively, were not involved in the generation of calcium oscillations. Pretreatment with genistein, a specific tyrosine kinase inhibitor or cytochalasin D, an inhibitor of actin polymerization, or pertussis toxin, an inhibitor of G_iα and G_oα subunits, completely abolished calcium transients and oscillations. These results indicate that Ca²⁺ flux due to mechanical stretching is likely mediated through SA ion channe s and is dependent on tyrosine kinases, pertussis toxin-sensitive subunits of G-proteins, and actin filaments. © 1994 Wiley-Liss, Inc.     

Visualizing the mapped ion pathway through the Na,K-ATPase pump

The Na+,K+-ATPase pump achieves thermodynamically uphill exchange of cytoplasmic Na+ ions for extracellular K+ ions by using ATP-mediated phosphorylation, followed by autodephosphorylation, to power conformational changes that allow ion access to the pump's binding sites from only one side of the membrane at a time. Formally, the pump behaves like an ion channel with two tightly coupled gates that are constrained to open and close alternately. The marine agent palytoxin disrupts this coupling, allowing both gates to sometimes be open, so temporarily transforming a pump into an ion channel. We made a cysteine scan of Na+,K+-ATPase transmembrane (TM) segments TM1 to TM6, and used recordings of Na+ current flow through palytoxin-bound pump-channels to monitor accessibility of introduced cysteine residues via their reaction with hydrophilic methanethiosulfonate (MTS) reagents. To visualize the open-channel pathway, the reactive positions were mapped onto a homology model of Na+,K+-ATPase based on the structure of the related sarcoplasmic- and endoplasmic-reticulum (SERCA) Ca2+-ATPase in a BeF3--trapped state1,2, in which the extra-cytoplasmic gate is wide open (although the cytoplasmic access pathway is firmly shut). The results revealed a single unbroken chain of reactive positions that traverses the pump from the extracellular surface to the cytoplasm, comprises residues from TM1, TM2, TM4, and TM6, and passes through the equivalent of cation binding site II in SERCA, but not through site I. Cavity search analysis of the homology model validated its use for mapping the data by yielding a calculated extra-cytoplasmic pathway surrounded by MTS-reactive residues. As predicted by previous experimental results, that calculated extra-cytoplasmic pathway abruptly broadens above residue T806, at the outermost end of TM6 which forms the floor of the extracellular-facing vestibule. These findings provide a structural basis for further understanding cation translocation by the Na+,K+-ATPase and by other P-type pumps like the Ca2+- and H+,K+-ATPases.     

Activation of a cGMP-sensitive calcium-dependent chloride channel may cause transition from calcium waves to whole cell oscillations in smooth muscle cells

In vitro, alpha-adrenoreceptor stimulation of rat mesenteric small arteries often leads to a rhythmic change in wall tension, i.e., vasomotion. Within the individual smooth muscle cells of the vascular wall, vasomotion is often preceded by a period of asynchronous calcium waves. Abruptly, these low-frequency waves may transform into high-frequency whole cell calcium oscillations. Simultaneously, multiple cells synchronize, leading to rhythmic generation of tension. We present a mathematical model of vascular smooth muscle cells that aims at characterizing this sudden transition. Simulations show calcium waves sweeping through the cytoplasm when the sarcoplasmic reticulum (SR) is stimulated to release calcium. A rise in cGMP leads to the experimentally observed transition from waves to whole cell calcium oscillations. At the same time, membrane potential starts to oscillate and the frequency approximately doubles. In this transition, the simulated results point to a key role for a recently discovered cGMP-sensitive calcium-dependent chloride channel. This channel depolarizes the membrane in response to calcium released from the SR. In turn, depolarization causes a uniform opening of L-type calcium channels on the cell surface, stimulating a synchronized release of SR calcium and inducing the shift from waves to whole cell oscillations. The effect of the channel is therefore to couple the processes of the SR with those of the membrane. We hypothesize that the shift in oscillatory mode and the associated onset of oscillations in membrane potential within the individual cell may underlie sudden intercellular synchronization and the appearance of vasomotion.     

Electrocoupling of Ion Transporters in Plants: Interaction with Internal Ion Concentrations

There are five major electroenzymes in the plasmalemma of plant cells: a driving electrogenic pump, inward and outward rectifying K⁺ channels, a Cl⁻-2H⁺ symporter, and Cl⁻-channels. It has been demonstrated previously (Gradmann, Blatt & Thiel 1993, J. Membrane Biol. 136:327–332) how voltage-gating of these electroenzymes causes oscillations of the transmembrane voltage (V) at constant substrate concentrations. The purpose of this study is to examine the interaction of the same transporter ensemble with cytoplasmic concentrations of K⁺ and Cl⁻. The former model system has been extended to account for changing internal concentrations. Constant-field theory has been applied to describe the influence of ion concentrations on current-voltage relationships of the active channels. The extended model is investigated using a reference set of model parameters. In this configuration, the system converges to stable slow oscillations with intrinsic changes in cytoplasmic K⁺ and Cl⁻ concentrations. These slow oscillations reflect alternation between a state of salt uptake at steady negative values of V and a state of net salt loss at rapidly oscillating V, the latter being analogous to the previously reported oscillations. By switching off either concentration changes or gating, it is demonstrated that the fast oscillations are mostly due to the gating properties of the Cl⁻ channel, whereas the slow oscillations are controlled by the effect of the Cl⁻ concentration on the current. The sensitivity of output results y (e.g., frequency of oscillations) to changes of the model parameters x (e.g., maximum Cl⁻ conductance) has been investigated for the reference system. Further examples are presented where some larger changes of specific model parameters cause fundamentally different behavior, e.g., convergence towards a stable state of only the fast oscillations without intrinsic concentration changes, or to a steady-state without any oscillations. The main and general result of this study is that the osmotic status of a plant cell is stabilized by the ensemble of familiar electroenzymes through oscillatory interactions with the internal concentrations of the most abundant ions. This convergent behavior of the stand-alone system is an important prerequisite for osmotic regulation by means of other physiological mechanisms, like second messengers and gating modifiers. Received: 23 February/Revised: 16 July 1998     

Calcium dynamics at a plastic synapse in APLYSIA

Because certain primitive behavioral responses in the large sea snail Aplysia have recently been linked to neurophysiological events at a synaptic level, special interest attaches to the role played by calcium ions at such synapses. Using an extended version of the model applied earlier to trace the flow of energy and information through a ganglion of the medicinal leech (Triffet & Green, 1980), the authors investigate the electropotential effects of small transient localized changes in the calcium concentration near the inner membrane surface of a neuron in the resting state.When this state is well below the firing threshold, changes in Ca²⁺ concentration less than 10⁻⁸ M are shown to result only in low-level harmonic background oscillations. When the potential of the neurons is closer to threshold, however, and/or the Ca²⁺ concentration is of the order of 10⁻⁸ M, easily recognizable graded potentials appear, and these grow into firing peaks when the calcium concentration is increased still further.Though no attempt is made to deal with the amplification effects dependent on calcium-vesicle interactions and the related release of transmitter molecules, a unified mechanism for the underlying calcium ion dynamics is proposed. Graded potentials of increasing size are associated with a progressive localized thickening of the inner and outer Debye layers. Moreover, the transverse and longitudinal calcium currents set up in such regions prove adequate to account for both the depletion of Ca²⁺ ions necessary to achieve habituation, and the increase in their concentration required for sensitization.     

The evolution of plant action potentials

Because certain primitive behavioral responses in the large sea snail Aplysia have recently been linked to neurophysiological events at a synaptic level, special interest attaches to the role played by calcium ions at such synapses. Using an extended version of the model applied earlier to trace the flow of energy and information through a ganglion of the medicinal leech (Triffet &; Green, 1980), the authors investigate the electropotential effects of small transient localized changes in the calcium concentration near the inner membrane surface of a neuron in the resting state.When this state is well below the firing threshold, changes in Ca²⁺ concentration less than 10^?8 M are shown to result only in low-level harmonic background oscillations. When the potential of the neurons is closer to threshold, however, and/or the Ca²⁺ concentration is of the order of 10^?8 M, easily recognizable graded potentials appear, and these grow into firing peaks when the calcium concentration is increased still further.Though no attempt is made to deal with the amplification effects dependent on calcium-vesicle interactions and the related release of transmitter molecules, a unified mechanism for the underlying calcium ion dynamics is proposed. Graded potentials of increasing size are associated with a progressive localized thickening of the inner and outer Debye layers. Moreover, the transverse and longitudinal calcium currents set up in such regions prove adequate to account for both the depletion of Ca²⁺ ions necessary to achieve habituation, and the increase in their concentration required for sensitization.     

A model of oscillation generation by molluscan neurons

A model describing slow oscillations of membrane potential in molluscan neurons is suggested. It is based on the view that the depolarization phase is due to the slow calcium current, whereas the hyperpolarization phase is due to the potassium current activated by intracellular Ca ions. It is shown that depending on values of the parameters of the model there are three possible types of electrical activity of the neurons: stable membrane hyperpolarization up to the resting potential which is between ?49 and ?53 mV; slow oscillations of membrane potential from ?30 to ?60 mV, with a period of 12–17 sec, and stable membrane depolarization to between ?40 and ?30 mV, which may lead to the onset of stable rhythmic activity of these neurons. Dependence of the amplitude of the oscillations of potential on the extracellular concentration of Ca, K, and Na ions was calculated and agrees qualitatively with the experimental data of Barker and Gainer [4].     

A membrane model for cytosolic calcium oscillations. A study using Xenopus oocytes.

Cytosolic calcium oscillations occur in a wide variety of cells and are involved in different cellular functions. We describe these calcium oscillations by a mathematical model based on the putative electrophysiological properties of the endoplasmic reticulum (ER) membrane. The salient features of our membrane model are calcium-dependent calcium channels and calcium pumps in the ER membrane, constant entry of calcium into the cytosol, calcium dependent removal from the cytosol, and buffering by cytoplasmic calcium binding proteins. Numerical integration of the model allows us to study the fluctuations in the cytosolic calcium concentration, the ER membrane potential, and the concentration of free calcium binding sites on a calcium binding protein. The model demonstrates the physiological features necessary for calcium oscillations and suggests that the level of calcium flux into the cytosol controls the frequency and amplitude of oscillations. The model also suggests that the level of buffering affects the frequency and amplitude of the oscillations. The model is supported by experiments indirectly measuring cytosolic calcium by calcium-induced chloride currents in Xenopus oocytes as well as cytosolic calcium oscillations observed in other preparations.     

Calcium channels in smooth muscle cells

In smooth muscle cells, the electrophysiological properties of potential-dependent calcium channels are similar to those described in other excitable cells. The calcium current is dependent on the extracellular calcium concentration; it is insensitive to external sodium removal and tetrodotoxin application. Other ions (Ba2+, Sr2+, Na+) can flow through the calcium channel. This channel is blocked by Mn2+, Co2+, Cd2+ and by organic inhibitors. The inactivation mechanism is mediated by both the membrane potential and the calcium influx. Ca2+ ions can also penetrate into the cell through receptor-operated channels. These channels show a low ionic selectivity and are generally less sensitive to organic Ca-blockers than the potential-dependent calcium channels. The finding of specific channel inhibitors as well as the study of the biochemical pathways between receptor activation and channel opening are prerequisites to further characterization of receptor-operated channels.     

A model of smooth muscle cell synchronization in the arterial wall

Vasomotion is a rhythmic variation in microvascular diameter. Although known for more than 150 years, the cellular processes underlying the initiation of vasomotion are not fully understood. In the present study a model of a single cell is extended by coupling a number of cells into a tube. The simulated results point to a permissive role of cGMP in establishing intercellular synchronization. In sufficient concentration, cGMP may activate a cGMP-sensitive calcium-dependent chloride channel, causing a tight spatiotemporal coupling between release of sarcoplasmic reticulum calcium, membrane depolarization, and influx of extracellular calcium. Low [cGMP] is associated only with unsynchronized waves. At intermediate concentrations, cells display either waves or whole cell oscillations, but these remain unsynchronized between cells. Whole cell oscillations are associated with rhythmic variation in membrane potential and flow of current through gap junctions. The amplitude of these oscillations in potential grows with increasing [cGMP], and, past a certain threshold, they become strong enough to entrain all cells in the vascular wall, thereby initiating sustained vasomotion. In this state there is a rhythmic flow of calcium through voltage-sensitive calcium channels into the cytoplasm, making the frequency of established vasomotion sensitive to membrane potential. It is concluded that electrical coupling through gap junctions is likely to be responsible for the rapid synchronization across a large number of cells. Gap-junctional current between cells is due to the appearance of oscillations in the membrane potential that again depends on the entrainment of sarcoplasmic reticulum and plasma membrane within the individual cell.     

Time-Dependent Molecular Memory in Single Voltage-Gated Sodium Channel

Excitability in neurons is associated with firing of action potentials and requires the opening of voltage-gated sodium channels with membrane depolarization. Sustained membrane depolarization, as seen in pathophysiological conditions like epilepsy, can have profound implications on the biophysical properties of voltage-gated ion channels. Therefore, we sought to characterize the effect of sustained membrane depolarization on single voltage-gated Na⁺ channels. Single-channel activity was recorded in the cell-attached patch-clamp mode from the rNa_v1.2α channels expressed in CHO cells. Classical statistical analysis revealed complex nonlinear changes in channel dwell times and unitary conductance of single Na⁺ channels as a function of conditioning membrane depolarization. Signal processing tools like weighted wavelet Z (WWZ) and discrete Fourier transform analyses attributed a “pseudo-oscillatory” nature to the observed nonlinear variation in the kinetic parameters. Modeling studies using the hidden Markov model (HMM) illustrated significant changes in kinetic states and underlying state transition rate constants upon conditioning depolarization. Our results suggest that sustained membrane depolarization induces novel nonlinear properties in voltage-gated Na⁺ channels. Prolonged membrane depolarization also induced a “molecular memory” phenomenon, characterized by clusters of dwell time events and strong autocorrelation in the dwell time series similar to that reported recently for single enzyme molecules. The persistence of such molecular memory was found to be dependent on the duration of depolarization. Voltage-gated Na⁺ channel with the observed time-dependent nonlinear properties and the molecular memory phenomenon may determine the functional state of the channel and, in turn, the excitability of a neuron.