Effects of extracellular calcium on electrical bursting and intracellular and luminal calcium oscillations in insulin secreting pancreatic beta-cells.

The extracellular calcium concentration has interesting effects on bursting of pancreatic beta-cells. The mechanism underlying the extracellular Ca2+ effect is not well understood. By incorporating a low-threshold transient inward current to the store-operated bursting model of Chay, this paper elucidates the role of the extracellular Ca2+ concentration in influencing electrical activity, intracellular Ca2+ concentration, and the luminal Ca2+ concentration in the intracellular Ca2+ store. The possibility that this inward current is a carbachol-sensitive and TTX-insensitive Na+ current discovered by others is discussed. In addition, this paper explains how these three variables respond when various pharmacological agents are applied to the store-operated model.

Electrical bursting and luminal calcium oscillation in excitable cell models.

It is shown in this paper that electrical bursting and the oscillations in the intracellular calcium concentration, [Ca2+]i, observed in excitable cells such as pancreatic beta-cells and R-15 cells of the mollusk Aplysia may be driven by a slow oscillation of the calcium concentration in the lumen of the endoplasmic reticulum, [Ca2+]lum. This hypothesis follows from the inclusion of the dynamic changes of [Ca2+]lum in the Chay bursting model. This extended model provides answers to some puzzling phenomena, such as why isolated single pancreatic beta-cells burst with a low frequency while intact beta-cells in an islet burst with a much higher frequency. Verification of the model prediction that [Ca2+]lum is a primary oscillator which drives electrical bursting and [Ca2+]i oscillations in these cells awaits experimental testing. Experiments using fluorescent dyes such as mag-fura-2-AM or aequorin could provide relevant information.

Proarrhythmic and antiarrhythmic actions of ion channel blockers on arrhythmias in the heart: model study.

We explain why 1) some class I and IV antiarrhythmia drugs could exert proarrhythmic action, 2) some class III drugs are effective in controlling reentrant arrhythmias, and 3) cycle length (CL) oscillation is involved in the termination or initiation of reentry. To explain these phenomena, we employ the following three means: bifurcation analysis, simulation, and model construction. Antiarrhythmia drugs are modeled by varying maximal conductances of Na+, Ca2+, and time-dependent delayed rectifying and time-independent inward rectifying K+ channels in the Beeler-Reuter model, where the model cells are arranged in a ring. Bifurcation analysis predicts that there is a critical ring size (CRS) at which infinite ring behavior suddenly breaks down. Channel blockers can affect CRS in different manners: Na+ and Ca2+ blockers shorten CRS, whereas delayed rectifying K+ channel blockers and the inward K+ channel blockers lengthen CRS. This differential explains why some antiarrhythmia drugs are proarrhythmic (i.e., shorten CRS) whereas others are antiarrhythmic (i.e., lengthen CRS). Simulation is then used to investigate how the drugs affect reentrant rhythms in the neighborhood of the CRS. We find that, in this region, CL, conduction velocity, and action potential duration become oscillatory. As ring size shrinks, the pattern of the oscillation becomes more complex. When the ring shrinks to a certain size, reentry can no longer be sustained, and it terminates after a few oscillatory cycles. To explain the basic mechanism involved in CL oscillation, we then construct a minimal model that contains a low-threshold fast inward current and a high-threshold slow inward current. With this model, we show that the two inward currents, with vastly different activation and inactivation kinetics, cause CL oscillations. Our results thus give theoretical explanations for the experimental finding of Frame's group in canine atrial tricuspid ring in vitro that class IC drugs can bring about stable reentry from nonsustained transient reentry, whereas class III drugs transform stable reentry to complex oscillations in CL. Our results also support the result of Frame's group, in that, in "adjustable" tricuspid rings, CL oscillation becomes more complex and its period becomes shorter as an excitable gap is shortened.

Modeling slowly bursting neurons via calcium store and voltage-independent calcium current.

Recent experiments indicate that the calcium store (e.g., endoplasmic reticulum) is involved in electrical bursting and [Ca2+]i oscillation in bursting neuronal cells. In this paper, we formulate a mathematical model for bursting neurons, which includes Ca2+ in the intracellular Ca2+ stores and a voltage-independent calcium channel (VICC). This VICC is activated by a depletion of Ca2+ concentration in the store, [Ca2+]cs. In this model, [Ca2+]cs oscillates slowly, and this slow dynamic in turn gives rise to electrical bursting. The newly formulated model thus is radically different from existing models of bursting excitable cells, whose mechanism owes its origin to the ion channels in the plasma membrane and the [Ca2+]i dynamics. In addition, this model is capable of providing answers to some puzzling phenomena, which the previous models could not (e.g., why cAMP, glucagon, and caffeine have ability to change the burst periodicity). Using mag-fura-2 fluorescent dyes, it would be interesting to verify the prediction of the model that (1) [Ca2+]cs oscillates in bursting neurons such as Aplysia neuron and (2) the neurotransmitters and hormones that affect the adenylate cyclase pathway can influence this oscillation.

Appearance of phase-locked Wenckebach-like rhythms, devil's staircase and universality in intracellular calcium spikes in non-excitable cell models.

In this paper we show that Wenckebach-like patterns of intracellular calcium concentration, [Ca2+]i, arise in non-excitable cell models when driven repetitively by the application of agonists that activate the phospholinositide-signalling pathway. These patterns are similar to action potential responses observed in excitable cells when driven periodically by external current stimuli. A model exclusively studied in this paper is based on the receptor-operated model of Cuthbertson & Chay (1991, Cell Calcium 12, 97-108), which is formulated under the assumptions that phospholipase C is a GTPase activating protein and a build-up of the GTP-bound alpha-subunit is a slow dynamic variable responsible for the refractory period. Similarities between [Ca2+]i response and action potential response make it possible to reduce the full dynamic system to a one-dimensional discrete equation designed for cardiac rhythms. The Devil's staircase constructed from both the dynamic traces and one-dimensional maps shows that the rules governing this staircase are indeed universal even in the agonist phase-locking system. This work thus provides a theoretical explanation for the appearance of blocked and delayed responses of [Ca2+]i spikes observed in the hepatocytes in response to pulsed phenylephrine agonist and, moreover, demonstrates the existence of universality in the agonist pulsed phase-locking system.

Why are some antiarrhythmic drugs proarrhythmic? Cardiac arrhythmia study by bifurcation analysis.

This study employs a bifurcation analysis approach to elucidate the effect of the key ion channels on cardiac arrhythmias and thereby explain the efficacy of antiarrhythmic drugs in controlling arrhythmias. The model used for the analysis contains the key ion channels involved in the ventricular action potential--fast sodium, slow calcium, and background potassium channels. The cardiac tissue is modeled by a ring structure. The bifurcation diagram reveals that at a certain ring size, the amplitude of the action potential suddenly shrinks and the conduction velocity (CV) becomes unstable. Instability in CV leads to termination of reentrant arrhythmias. This ring size (ie, the critical ring size [CRS]) depends of the type of channel blocker. Blocking of the sodium channel leads to a decrease in the CRS, which in turn enhances stable reentry (proarrhythmia). Although calcium channel blockers do not alter the CV, they can exert the proarrhythmic effect by drastically shortening the CRS. The potassium channel blockers, on the other hand, are effective in controlling reentry in ventricular tissues by lengthening the CRS. Near blocking of the potassium channel, however, brings about another type of arrhythmia--the formation of ectopic foci. In the neighborhood of the CRS, the cycle length oscillates with an interesting pattern that depends on ring size and drug type. Although a critical reentrant loop length for stable reentrant excitation has been investigated for a long time, this study is the first demonstration of how the key ion channels in the plasma membrane affect the loop length. Furthermore, the analysis approach provides a theoretical basis for the increased mortality associated with class I drug use in the Cardiac Arrhythmia Suppression Trial Team.

Thermal activation of a group II intron ribozyme reveals multiple conformational states.

Conformational changes often accompany biological catalysis. Group II introns promote a variety of reactions in vitro that show an unusually sharp temperature dependence. This suggests that the chemical steps are accompanied by the conversion of a folded-but-inactive form to a differently folded active state. We report here the kinetic analysis of 5'-splice-junction hydrolysis (SJH) by E1:12345, a transcript containing the 5'-exon plus the first five of six intron secondary structure domains. The pseudo-first-order SJH reaction shows (1) activation by added KCl to 1.5 M; (2) cooperative activation by added MgCl2, nHill = 4.1-4.3, and [MgCl2]vmax/2 approximately 0.040 M; and (3) a rather high apparent activation energy, Ea approximately 50 kcal mol-l. In contrast, the 5'-terminal phosphodiester bond of a domain 5 transcript (GGD5) was hydrolyzed with Ea approximately 30 kcal mol-1 under SJH conditions; the 5'-GG leader dinucleotide presumably lacks secondary structure constraints. The effect of adding the chaotropic salt tetraethylammonium chloride (TEA) was also investigated. TEA reduced the melting temperatures of GGD5 and E1:12345. TEA also shifted the profile of rate versus temperature for SJH by E1:12345 toward lower temperatures without affecting the maximum rate. TEA had little effect on the rate of hydrolysis of the 5'-phosphodiester bond of GGD5. The high apparent activation enthalpy and entropy for SJH along with the effect of TEA on these parameters imply that conversion of an inactive form of E1:12345 to an active conformation accompanies enhanced occupation of the transition state as the temperature is raised to that for maximum SJH. Analytical modeling indicates that either a two-state model (open and disordered, with open being active) or a three-state model (compact, open, and disordered) could account for the temperature dependence of kSJH. However, the three-state model is clearly preferable, since it does not require that the activation parameters for phosphodiester bond hydrolysis exhibit exceptional values or that the rates for the chemical steps of SJH respond directly to TEA addition.

Generation of periodic and chaotic bursting in an excitable cell model.

There are interesting oscillatory phenomena associated with excitable cells that require theoretical insight. Some of these phenomena are: the threshold low amplitude oscillations before bursting in neuronal cells, the damped burst observed in muscle cells, the period-adding bifurcations without chaos in pancreatic beta-cells, chaotic bursting and beating in neurons, and inverse period-doubling bifurcation in heart cells. The three variable model formulated by Chay provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable. In this paper, we use the Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the 'slow' conductance (i.e., gK, C) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as parameter gK, C is varied. Depending on the time kinetic constant of the fast variable (lambda n), however, the transition between burstings via period-adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of gK, C and lambda n. In addition, a threshold sinusoidal oscillation was observed at certain values of gK, C before triggering action potentials. Probably this explains why the neuronal cells exhibit low-amplitude oscillations before bursting.

Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses.

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)

The Hodgkin-Huxley Na+ channel model versus the five-state Markovian model.

In describing the Na+ channel-gating kinetics, it is generally believed the Hodgkin-Huxley model is inadequate and other types of Markovian models are more appropriate. In this paper, we perform detailed kinetic analyses to find out whether the Hodgkin-Huxley model is really unacceptable. Specifically, we consider two models for the analyses: A five-state Markovian model that allows inactivation to take place before opening and a Hodgkin-Huxley eight-state model. The criteria used to check the goodness of the two models are (a) Akaike's information criterion; (b) chi 2 tests on the waiting-time, open-time, and closed-time distributions, and the number of openings per record; and (c) comparison between all latency distributions and the probability of the open state predicted from the two models. In order to do this, we first develop a method of constructing probability density histograms of a specified event (e.g., waiting time, closed time, open time, number of openings per patch) from the multichannel patch-clamp recordings. The goodness of our method is checked by simulating multichannel patch recordings using a multinomial random number generator. Our kinetic analysis on the single Na+ channel recordings from the cardiac cells revealed that (a) on the basis of Akaike's information criterion, the Hodgkin-Huxley model is definitely a better model than the five-state model, but (b) on the basis of chi 2 tests on the probability density functions, the latter model is slightly better than the former. We find no evidence that the Hodgkin-Huxley model is inferior to the five-state model for this cell type.

Modelling receptor-controlled intracellular calcium oscillators.

This paper presents mathematical models for the hepatocyte calcium oscillator which follow the concepts in a class of informal models developed to account for the striking dependence on the receptor type of several features of the calcium oscillations, in particular the shape and duration of the free calcium transients. The essence of these models is that the transients should be timed by a build-up of activated GTP-binding proteins, which, combined with positive feedback processes and perhaps with cooperative effects, leads to a sudden activation of phospholipase C (PLC), followed by negative feedback processes which switch off the calcium rise and lead to a fall in free calcium back to resting levels. These models predict pulsatile oscillations in inositol (1,4,5)P3 as well as in free calcium. We show that receptor-controlled intracellular calcium oscillators involving an unknown positive feedback pathway onto PLC and negative feedback from protein kinase C (PKC) onto G-proteins and receptors, or negative feedback by stimulation of GTPase activity can simulate many of the features of observed intracellular calcium oscillations. These oscillators exhibit a dependence of frequency on agonist concentration and a dependence of transient duration on receptor and G-protein type. We also show that a PLC-dependent GTPase activating factor (GAF) could provide explanations for some otherwise puzzling features of intracellular calcium oscillations.

The effect of ATP-sensitive K+ channels on the electrical burst activity and insulin secretion in pancreatic beta-cells.

In recent years, the electrical burst activity of the insulin releasing pancreatic beta-cells has attracted many experimentalists and theoreticians, largely because of its functional importance, but also because of the nonlinear nature of the burst activity. The ATP-sensitive K+ channels are believed to play an important role in electrical activity and insulin release. In this paper, we show by computer simulation how ATP and antidiabetic drugs can lengthen the plateau fraction of bursting and how these chemicals can increase the intracellular Ca2+ level in the pancreatic beta-cell.

Effect of compartmentalized Ca2+ ions on electrical bursting activity of pancreatic beta-cells.

Patch-clamp single-channel and whole cell recordings have revealed new insights into the ionic channel properties in the pancreatic beta-cells. I have modeled the electrical events during the burst activity based on the observations that 1) the whole cell Ca2+ current has two functionally distinct components (fast and slow), 2) a fast component is inhibited by intracellular Ca2+, 3) a slow component is inactivated by depolarization, and 4) a significant fraction of the outward current is carried by the Ca2(+)-sensitive, voltage-gated K+ channels [K(Ca, V) channels]. The model contains a feature that the Ca2+ concentration in the submembrane compartment ([Ca2+]s) is higher than that in the cellular phase. At the plateau phase, [Ca2+]s is high enough to activate the K(Ca, V) channels. In addition to the K(Ca, V) channels, the model contains a voltage-activated Ca2+ channel that is quickly blocked by Ca2+ and slowly inhibited by voltage. Because the Ca2+ channel has an intracellular Ca2(+)-dependent inactivation gate, the increase in [Ca2+]s can inactivate the Ca2+ channels. According to this model, the spikes during the plateau phase are caused by a rapid movement of Ca2+ into and out of the compartment. Because of a rapid change in [Ca2+]s, the two competing currents, ICa and IK(Ca, V), fluctuate rapidly; the fluctuation leads to an emergence of spikes. The slow underlying wave is due to a voltage-dependent inactivation gate of the Ca2+ channels, which slowly closes as a result of depolarization. This model differs radically from my previous models, which featured a slowly varying intracellular Ca2+ concentration that was responsible for the underlying slow wave. Although the previous models give plateau fractions (the ratio between the plateau duration and cyclic time) to be far less than unity, the present model is the first of its kind that allows plateau fractions to be in the near-unity range.

Bursting excitable cell models by a slow Ca2+ current.

Bursting in excitable cells is a phenomenon that has attracted the interest of many electrophysiologists and non-linear dynamicists. In this paper, we present two models that give rise to bursting in action potentials. The membrane of the first model contains a voltage-activated Ca2+ channel that inactivates very slowly upon depolarization and a delayed K+ channel that is activated by voltage. This model consists of three dynamic variables--the gating variable of K+ channel (n), inactivation gating variable of the Ca2+ channel (f), and membrane potential (V). The membrane of the second model contains a voltage-activated Na+ channel that inactivates rather fast upon depolarization. This model contains altogether five dynamic variables--the Na+ inactivation gating variable (h) and Ca2+ activation variable (d), in addition to the three dynamic variables in the first model. With the first model, we show how various interesting bursting patterns may arise from such a simple three dynamic variable model. We also demonstrate that a slowly inactivating voltage-dependent Ca2+ channel may play the key role in the genesis of bursting. With the second model, we show how the participation of a quickly inactivating fast inward current may lead to a neuronal type of bursting, multi-peaked oscillations, and chaos, as the rates of the gating variables change.

Electrical bursting and intracellular Ca2+ oscillations in excitable cell models.

In bursting excitable cells such as pancreatic beta-cells and molluscan Aplysia neuron cells, intracellular Ca2+ ion plays a central role in various cellular functions. To understand the role of [Ca2+]i (the intracellular Ca2+ concentration) in electrical bursting, we formulate a mathematical model which contains a few functionally important ionic currents in the excitable cells. In this model, inactivation of Ca2+ current takes place by a mixture of voltage and intracellular Ca2+ ions. The model predicts that, although the electrical bursting patterns look the same, the shapes of [Ca2+]i oscillations could be very different depending on how fast [Ca2+]i changes in the cytosolic free space (i.e., how strong the cellular Ca2+ buffering capacity is). If [Ca2+]i changes fast, [Ca2+]i oscillates in bursts in parallel to electrical bursting such that it reaches a maximum at the onset of bursting and a minimum just after the termination of the plateau phase. If the change is slow, then [Ca2+]i oscillates out-of-phase with electrical bursting such that it peaks at a maximum near the termination of the plateau and a minimum just before the onset of the active phase. During the active phase [Ca2+]i gradually increases without spikes. In the intermediate ranges, [Ca2+]i oscillates in such a manner that the peak of [Ca2+]i oscillation lags behind the electrical activity. The model also predicts the existence of multipeaked oscillations and chaos in certain ranges of the gating variables and the intracellular Ca2+ buffer concentration.

Cardiac arrhythmias modelled by Cai-inactivated Ca2+ channels.

A model of cardiac cells incorporating the membrane potential and the intracellular calcium concentration as the two dynamical variables is developed. This model is applied to simple systems of cells to investigate its behavior as a function of the model parameters. Rational entrainment is observed in systems of two coupled pacemaker cells. The propagation of the membrane potential and intracellular calcium concentration through a sheet is simulated. Behavior suggestive of circus movement tachycardias is observed.

Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells.

To study why pancreatic beta-cells prefer to burst as a multi-cellular complex, we have formulated a stochastic model for bursting clusters of excitable cells. Our model incorporated a delayed rectifier K+ channel, a fast voltage-gated Ca2+ channel, and a slow Cai-blockable Ca2+ channel. The fraction of ATP-sensitive K+ channels that may still be active in the bursting regime was included in the model as a leak current. We then developed an efficient method for simulating an ionic current component of an excitable cell that contains several thousands of channels opening simultaneously under unclamped voltage. Single channel open-close stochastic events were incorporated into the model by use of binomially distributed random numbers. Our simulations revealed that in an isolated beta-cell [Ca2+]i oscillates with a small amplitude about a low [Ca2+]i. However, in a large cluster of tightly coupled cells, stable bursts develop, and [Ca2+]i oscillates with a larger amplitude about a higher [Ca2+]i. This may explain why single beta-cells do not burst and also do not release insulin.

Kinetic modeling for the channel gating process from single channel patch clamp data.

Allosteric interactions play a crucial role in the regulation of protein-ligand binding. The single-channel recordings of Na+ channels revealed deviations from the Hodgkin-Huxley mechanism, that may be explained in the context of protein co-operativity. In this paper, we present a Na+ channel gating model based on the allosteric nature of the channel proteins. The model contains a co-operative parameter c and a subunit interaction parameter s, in addition to four Hodgkin-Huxley kinetic parameters. Thus, in our model a deviation from the Hodgkin-Huxley mechanism is measured by these two parameters (i.e., whether they are different from unity). We then discuss how to estimate the kinetic parameters in the model from the single channel recordings. Our method of estimating the parameters is tested with Monte Carlo runs with very good results.