Interbeat Interval Modulation in the Sinoatrial Node as a Result of Membrane Current Stochasticity—A Theoretical and Numerical Study

A single isolated sinoatrial pacemaker cell presents intrinsic interbeat interval (IBI) variability that is believed to result from the stochastic characteristics of the opening and closing processes of membrane ion channels. To our knowledge, a novel mathematical framework was developed in this work to address the effect of current fluctuations on the IBIs of sinoatrial pacemaker cells. Using statistical modeling and employing the Fokker-Planck formalism, our mathematical analysis suggests that increased stochastic current fluctuation variance linearly increases the slope of phase-4 depolarization, hence the rate of activations. Single-cell and two-dimensional computerized numerical modeling of the sinoatrial node was conducted to validate the theoretical predictions using established ionic kinetics of the rabbit pacemaker and atrial cells. Our models also provide, to our knowledge, a novel complementary or alternative explanation to recent experimental observations showing a strong reduction in the mean IBI of Cx30 deficient mice in comparison to wild-types, not fully explicable by the effects of intercellular decoupling.     

Threshold fluctuations in an N sodium channel model of the node of Ranvier.

Computer simulations of stochastic single-channel open-close kinetics are applied to an N sodium channel model of a node of Ranvier. Up to 32,000 voltage-gated sodium channels have been simulated with modified amphibian sodium channel kinetics. Poststimulus time histograms are obtained with 1000 monophasic pulse stimuli, and measurements are made of changes in the relative spread of threshold (RS) with changes in the model parameters. RS is found to be invariant with pulse durations from 100 microseconds to 3 ms. RS is approximately of inverse proportion to square-root of N. It decreases with increasing temperature and is dependent on passive electrical properties of the membrane as well as the single-channel conductance. The simulated RS and its independence of pulse duration is consistent with experimental results from the literature. Thus, the microscopic fluctuations of single, voltage-sensitive sodium channels in the amphibian peripheral node of Ranvier are sufficient to account for the macroscopic fluctuation if threshold to electrical stimulation.     

Dynamical description of sinoatrial node pacemaking: improved mathematical model for primary pacemaker cell

We developed an improved mathematical model for a single primary pacemaker cell of the rabbit sinoatrial node. Original features of our model include 1) incorporation of the sustained inward current (I(st)) recently identified in primary pacemaker cells, 2) reformulation of voltage- and Ca(2+)-dependent inactivation of the L-type Ca(2+) channel current (I(Ca,L)), 3) new expressions for activation kinetics of the rapidly activating delayed rectifier K(+) channel current (I(Kr)), and 4) incorporation of the subsarcolemmal space as a diffusion barrier for Ca(2+). We compared the simulated dynamics of our model with those of previous models, as well as with experimental data, and examined whether the models could accurately simulate the effects of modulating sarcolemmal ionic currents or intracellular Ca(2+) dynamics on pacemaker activity. Our model represents significant improvements over the previous models, because it can 1) simulate whole cell voltage-clamp data for I(Ca,L), I(Kr), and I(st); 2) reproduce the waveshapes of spontaneous action potentials and ionic currents during action potential clamp recordings; and 3) mimic the effects of channel blockers or Ca(2+) buffers on pacemaker activity more accurately than the previous models.     

Pacemaker activity of the rabbit sinoatrial node. A comparison of mathematical models.

In the past decade, three mathematical models describing the pacemaker activity of the rabbit sinoatrial node have been developed: the Bristow-Clark model, the Irisawa-Noma model, and the Noble-Noble model. In a comparative study it is demonstrated that these models, as well as subsequent modifications, all have several drawbacks. A more accurate model, describing the pacemaker activity of a single pacemaker cell isolated from the rabbit sinoatrial node, was constructed. Model equations, including equations for the T-type calcium current, are based on experimental data from voltage clamp experiments on single cells that were published during the last few years. In contrast to the other models, only a small amount of background current contributes to the overall electrical charge flow. The action potential parameters of the model cell, its responses to voltage clamp steps and its current-voltage relationships have been computed. The model is used to discuss the relative contribution of membrane current components to the slow diastolic depolarization phase of the action potential.     

Effects of stochastic channel gating and distribution on the cardiac action potential

Ion channels exhibit stochastic conformational changes determining their gating behavior. In addition, the process of protein turnover leads to a natural variability of the number of membrane and gap junctional channels. Nevertheless, in computational models, these two aspects are scarcely considered and their impacts are largely unknown. We investigated the effects of stochastic current fluctuations and channel distributions on action potential duration (APD), intercellular conduction delays (ICDs) and conduction blocks using a modified ventricular cell model (Rudy et al.) with Markovian formulations of the principal ion currents (to simulate their stochastic open-close gating behavior) and with channel counts drawn from Poisson distributions (to simulate their natural variability). In single cells, APD variability (coefficient of variation: 1.6% at BCL=1000 ms) was essentially caused by stochastic channel gating of I_Ks, persistent I_Na and I_Ca,L. In cell strands, ICD variability induced by stochastic channel gating and Poissonian channel distributions was low under normal conditions. Nonetheless, at low intercellular coupling levels, Poissonian gap junctional channel distribution resulted in a large ICD variability (coefficient of variation >20%), highly heterogeneous conduction patterns and conduction blocks. Therefore, the stochastic behavior of current fluctuations and channel distributions can contribute to the heterogeneity of conduction patterns and to conduction block, as observed previously in experiments in cardiac tissue with altered intercellular coupling.     

Bifurcation analysis and effects of changing ionic conductances on pacemaker rhythm in a sinoatrial node cell model

The electrical excitation (action potential generation) of sinoatrial node (cardiac pacemaker) cells is directly related to various ion channels (pore-forming proteins) in cell membranes. In order to analyze the relation between action potential generation and ion channels, we use the Yanagihara-Noma-Irisawa (YNI) model of sinoatrial node cells, which is described by the Hodgkin-Huxley-type equations with seven variables. In this paper, we analyze the global bifurcation structure of the YNI model by varying various conductances of ion channels, and examine the effects of these conductance changes on pacemaker rhythm (frequency of action potential generation). The coupling effect on pacemaker rhythm is also examined approximately by applying external current to the YNI model.     

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.     

Three types of ion channels are present on the plasma membrane of Friend erythroleukemia cells

In Friend murine erythroleukemia cells the presence of ion channels was investigated with the patch-clamp technique. During the first 48 hours after cell seeding, three types of ion channels, with the following order of membrane density, were found: i) a Ca2+-dependent K+ channel, fully activated at a cytosolic Ca2+ concentration of 10(-6) M and moderately activated at 10(-7)M; ii) a monovalent cation channel non voltage-activated, with an open-close kinetics dependent on the pressure gradient across the patch; iii) a chloride channel with a slow open-close kinetics. The latter two channels were labile and did not survive during intracellular perfusion. The membrane potential of the leukemia cells was not constant, but underwent large (tens of millivolts) fluctuations due to the opening of a few channels. The average resting membrane potential recorded in this study agrees with that measured in these cells by means of the accumulation ratio of the lipophilic cation Tetraphenylphosphonium.     

Simulation analysis of excitation conduction in the heart: Propagation of excitation in different tissues

The normal excitation and conduction in the heart are maintained by the coordination between the dynamics of ionic conductance of each cell and the electrical coupling between cells. To examine functional roles of these two factors, we proposed a spatially-discrete model of conduction of excitation in which the individual cells were assumed isopotential. This approximation was reasoned by comparing the apparent space constant with the measured junctional resistance between myocardial cells. We used the four reconstruction models previously reported for five kinds of myocardial cells. Coupling coefficients between adjacent cells were determined quantitatively from the apparent space constants. We first investigated to what extent the pacemaker activity of the sinoatrial node depends on the number and the coupling coefficient of its cells, by using a one-dimensional model system composed of the sinoatrial node cells and the atrial cells. Extensive computer simulation revealed the following two conditions for the pacemaker activity of the sinoatrial node. The number of the sinoatrial node cells and their coupling coefficients must be large enough to provide the atrium with the sufficient electric current flow. The number of the sinoatrial node cells must be large so that the period of the compound system is close to the intrinsic period of the sinoatrial node cell. In this simulation the same sinoatrial node cells produced action potentials of different shapes depending on where they were located in the sinoatrial node. Therefore it seems premature to classify the myocardial cells only from their waveforms obtained by electrical recordings in the compound tissue. Second, we investigated the very slow conduction in the atrioventricular node compared to, for example, the ventricle. This was assumed to be due to the inherent property of the membrane dynamics of the atrioventricular node cell, or to the small value of the coupling coefficient (weak intercellular coupling), or to the electrical load imposed on the atrioventricular node by the Purkinje fibers, because the relatively small atrioventricular node must provide the Purkinje fibers with sufficient electric current flow. Relative contributions of these three factors to the slow conduction were evaluated using the model system composed of only the atrioventricular cells or that composed of the atrioventricular and Purkinje cells. We found that the weak coupling has the strongest effect. In the model system composed of the atrioventricular cells, the propagation failure was not observed even for very small values of the coupling coefficient.(ABSTRACT TRUNCATED AT 400 WORDS)     

Selected contribution: axial stretch increases spontaneous pacemaker activity in rabbit isolated sinoatrial node cells.

Isolated, spontaneously beating rabbit sinoatrial node cells were subjected to longitudinal stretch, using carbon fibers attached to both ends of the cell. Their electrical behavior was studied simultaneously in current-clamp or voltage-clamp mode using the perforated patch configuration. Moderate stretch ( approximately 7%) caused an increase in spontaneous beating rate (by approximately 5%) and a reduction in maximum diastolic and systolic potentials (by approximately 2.5%), as seen in multicellular preparations. Mathematical modeling of the stretch intervention showed the experimental results to be compatible with stretch activation of cation nonselective ion channels, similar to those found in other cardiac cell populations. Voltage-clamp experiments validated the presence of a stretch-induced current component with a reversal potential near -11 mV. These data confirm, for the first time, that the positive chronotropic response of the heart to stretch is, at least in part, encoded on the level of individual sinoatrial node pacemaker cells; all reported data are in agreement with a major contribution of stretch-activated cation nonselective channels to this response.     

Computational evaluation of the roles of Na+ current, iNa, and cell death in cardiac pacemaking and driving

Voltage-dependent sodium (Na(+)) channels are heterogeneously distributed through the pacemaker of the heart, the sinoatrial node (SA node). The measured sodium channel current (i(Na)) density is higher in the periphery but low or zero in the center of the SA node. The functional roles of i(Na) in initiation and conduction of cardiac pacemaker activity remain uncertain. We evaluated the functional roles of i(Na) by computer modeling. A gradient model of the intact SA node and atrium of the rabbit heart was developed that incorporates both heterogeneities of the SA node electrophysiology and histological structure. Our computations show that a large i(Na) in the periphery helps the SA node to drive the atrial muscle. Removal i(Na) from the SA node slows down the pacemaking rate and increases the sinoatrial node-atrium conduction time. In some cases, reduction of the SA node i(Na) results in impairment of impulse initiation and conduction that leads to the SA node-atrium conduction exit block. Decrease in active SA node cell population has similar effects. Combined actions of reduced cell population and removal of i(Na) from the SA node have greater impacts on weakening the ability of the SA node to pace and drive the atrium.     

Hyperpolarization-activated cyclic nucleotide-gated channel 1 is a molecular determinant of the cardiac pacemaker current I(f)

The pacemaker current I(f) of the sinoatrial node (SAN) is a major determinant of cardiac diastolic depolarization and plays a key role in controlling heart rate and its modulation by neurotransmitters. Substantial expression of two different mRNAs (HCN4, HCN1) of the family of pacemaker channels (HCN) is found in rabbit SAN, suggesting that the native channels may be formed by different isoforms. Here we report the cloning and heterologous expression of HCN1 from rabbit SAN and its specific localization in pacemaker myocytes. rbHCN1 is an 822-amino acid protein that, in human embryonic kidney 293 cells, displayed electrophysiological properties similar to those of I(f), suggesting that HCN1 can form a pacemaker channel. The presence of HCN1 in the SAN myocytes but not in nearby heart regions, and the electrophysiological properties of the channels formed by it, suggest that HCN1 plays a central and specific role in the formation of SAN pacemaker currents.     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.     

Changes in conduction within the diffuse sinoatrial node of the chicken following premature atrial stimulation

Premature atrial stimulation was used to estimate sinoatrial conduction within the diffuse sinoatrial node of the bird (chicken), and compare its conduction with that reported for mammals. While sinoatrial conduction could not be determined in the chicken because reset did not occur, the premature wavefront did have an effect on the sinoatrial node because the recovery interval following the premature stimulus became less than compensatory with shortening of the premature stimulus interval. This less than compensatory non-reset recovery interval is interpreted as a conduction dependent response in which the intrinsic wavefront leading to the first recovery atrial activation conducts out of the node faster than normal. This conduction dependent recovery interval is seen infrequently in mammals (rabbit, dog and man). The absence of reset and the presence of a less than compensatory non-reset response in the chicken suggests that while the general organization of the sinoatrial node of the chicken is similar to that in mammals, a larger transitional cell network in the chicken prevents a premature wavefront from reaching the pacemaker cells and resetting them.     

Thermodynamic and kinetic studies of the gating behavior of a K+- selective channel from the sarcoplasmic reticulum membrane

A voltage-dependent, K+-selective ionic channel from sarcoplasmic reticulum of rabbit skeletal muscle has been studied in a planar phospholipid bilayer membrane. The purpose [corrected] of this work is to study the mechanism by which the channel undergoes transitions between its conducting and nonconducting states. Thermodynamic studies show that the "open" and "closed" states of the channel exist in a voltage-dependent equilibrium, and that the channel displays only a single open state; the channel conductance is 120 pmho in 0.1 M K+. The channel's gating process follows single exponential kinetics at all voltages tested, and the individual opening and closing rate constants are exponentially dependent on voltage. The individual rate constants may also be determined from a stochastic analysis of channel fluctuations among multiple conductance levels. Neither the thermodynamic nor the kinetic parameters of gating depend on the absolute concentration of channels in the bilayer. The results are taken as evidence that the channel gates by an unusually simple two-state conformational mechanism in which the equivalent of 1.1 net charges are moved across the membrane during the formation of the open channel.     

超极化激活的环核苷酸门控性阳离子通道（HCN）与疾病的相关性

超极化激活的环核苷酸门控的阳离子通道(hyperpolarization-activated cyclic nucleotide-gate cation channel,HCN)是一种特殊的阳离子通道,存在于神经细胞、小肠间质细胞、窦房结细胞或心脏细胞等具有自律性的细胞膜上,是产生过度激活正离子电流的结构基础,被认为是起搏细胞的重要特征。HCN离子蛋白通道不但与细胞凋亡以及电流传导有着密切关系,而且还与多种生命活动过程密切相关,近年来,已涉及到疼痛、癫痫、心律失常、消化道系统等许多疾病,特别是有关神经系统方面的疾病,下面将超极化激活的环核苷酸门控性阳离子通道(HCN)与疾病的关系综述如下。     

Regulation of Hyperpolarization-activated Cyclic Nucleotide-gated (HCN) Channel Activity by cCMP

Activation of hyperpolarization-activated cyclic nucleotide-gated (HCN) channels is facilitated in vivo by direct binding of the second messenger cAMP. This process plays a fundamental role in the fine-tuning of HCN channel activity and is critical for the modulation of cardiac and neuronal rhythmicity. Here, we identify the pyrimidine cyclic nucleotide cCMP as another regulator of HCN channels. We demonstrate that cCMP shifts the activation curves of two members of the HCN channel family, HCN2 and HCN4, to more depolarized voltages. Moreover, cCMP speeds up activation and slows down deactivation kinetics of these channels. The two other members of the HCN channel family, HCN1 and HCN3, are not sensitive to cCMP. The modulatory effect of cCMP is reversible and requires the presence of a functional cyclic nucleotide-binding domain. We determined an EC(50) value of ～30 μm for cCMP compared with 1 μm for cAMP. Notably, cCMP is a partial agonist of HCN channels, displaying an efficacy of ～0.6. cCMP increases the frequency of pacemaker potentials from isolated sinoatrial pacemaker cells in the presence of endogenous cAMP concentrations. Electrophysiological recordings indicated that this increase is caused by a depolarizing shift in the activation curve of the native HCN current, which in turn leads to an enhancement of the slope of the diastolic depolarization of sinoatrial node cells. In conclusion, our findings establish cCMP as a gating regulator of HCN channels and indicate that this cyclic nucleotide has to be considered in HCN channel-regulated processes.     

Calcium-permeable channel in sarcoplasmic reticulum of rabbit skeletal muscle

Membrane vesicles from sarcoplasmic reticulum of rabbit skeletal muscle were incorporated into a bilayer lipid membrane. With this system, single current fluctuation was observed in the presence of 50 mM Ba-gluconate. This channel activity was observed only in vesicles from terminal cisternae. The single channel conductance was 14.1 pS, and the channel state was almost wholly open. The open-close transition of the channel obeyed simple two-state kinetics and was voltage-independent. The ionic selectivity was also studied, and the channel showed no selectivity among Ba, Ca, Mn, and Mg. On the other hand, it was less permeable to Cs than to Ba. Based on these results, the relation of the Ca channel to excitation-contraction coupling is discussed.