Chronic Sympathetic Hyperactivity Triggers Electrophysiological Remodeling and Disrupts Excitation-Contraction Coupling in Heart.

The sympathetic nervous system is essential for maintenance of cardiac function via activation of post-junctional adrenergic receptors. Prolonged adrenergic receptor activation, however, has deleterious long-term effects leading to hypertrophy and the development of heart failure. Here we investigate the effect of chronic adrenergic receptors activation on excitation-contraction coupling (ECC) in ventricular cardiomyocytes from a previously characterized mouse model of chronic sympathetic hyperactivity, which are genetically deficient in the adrenoceptor α2A and α2C genes (ARDKO). When compared to wild-type (WT) cardiomyocytes, ARDKO displayed reduced fractional shortening (~33%) and slower relaxation (~20%). Furthermore, ARDKO cells exhibited several electrophysiological changes such as action potential (AP) prolongation (~50%), reduced L-type calcium channel (LCC) current (~33%), reduced outward potassium (K+) currents (~30%), and increased sodium/calcium exchanger (NCX) activity (~52%). Consistent with reduced contractility and calcium (Ca2+) currents, the cytosolic Ca2+ ([Ca2+]i) transient from ARDKO animals was smaller and decayed slower. Importantly, no changes were observed in membrane resting potential, AP amplitude, or the inward K+ current. Finally, we modified our existing cardiac ECC computational model to account for changes in the ARDKO heart. Simulations suggest that cellular changes in the ARDKO heart resulted in variable and dyssynchronous Ca2+-induced Ca2+ release therefore altering [Ca2+]i transient dynamics and reducing force generation. In conclusion, chronic sympathetic hyperactivity impairs ECC by changing the density of several ionic currents (and thus AP repolarization) causing altered Ca2+ dynamics and contractile activity. This demonstrates the important role of ECC remodeling in the cardiac dysfunction secondary to chronic sympathetic activity.

Quantitative tests reveal that microtubules tune the healthy heart but underlie arrhythmias in pathology.

KEY POINTS: Our group previously discovered and characterized the microtubule mechanotransduction pathway linking diastolic stretch to NADPH oxidase 2-derived reactive oxygen species signals that regulate calcium sparks and calcium influx pathways. Here we used focused experimental tests to constrain and expand our existing computational models of calcium signalling in heart. Mechanistic and quantitative modelling revealed new insights in disease including: changes in microtubule network density and properties, elevated NOX2 expression, altered calcium release dynamics, how NADPH oxidase 2 is activated by and responds to stretch, and finally the degree to which normalizing mechano-activated reactive oxygen species signals can prevent calcium-dependent arrhythmias. ABSTRACT: Microtubule (MT) mechanotransduction links diastolic stretch to generation of NADPH oxidase 2 (NOX2)-dependent reactive oxygen species (ROS), signals we term X-ROS. While stretch-elicited X-ROS primes intracellular calcium (Ca2+ ) channels for synchronized activation in the healthy heart, the dysregulated excess in this pathway underscores asynchronous Ca2+ release and arrhythmia. Here, we expanded our existing computational models of Ca2+ signalling in heart to include MT-dependent mechanotransduction through X-ROS. Informed by new focused experimental tests to properly constrain our model, we quantify the role of X-ROS on excitation-contraction coupling in healthy and pathological conditions. This approach allowed for a mechanistic investigation that revealed new insights into X-ROS signalling in disease including changes in MT network density and post-translational modifications (PTMs), elevated NOX2 expression, altered Ca2+ release dynamics (i.e. Ca2+ sparks and Ca2+ waves), how NOX2 is activated by and responds to stretch, and finally the degree to which normalizing X-ROS can prevent Ca2+ -dependent arrhythmias.

Real-time local oxygen measurements for high resolution cellular imaging.

Single-cell metabolic investigations are hampered by the absence of flexible tools to measure local partial pressure of O2 (pO2) at high spatial-temporal resolution. To this end, we developed an optical sensor capable of measuring local pericellular pO2 for subcellular resolution measurements with confocal imaging while simultaneously carrying out electrophysiological and/or chemo-mechanical single cell experiments. Here we present the OxySplot optrode, a ratiometric fluorescent O2-micro-sensor created by adsorbing O2-sensitive and O2-insensitive fluorophores onto micro-particles of silica. To protect the OxySplot optrode from the components and reactants of liquid environment without compromising access to O2, the micro-particles are coated with an optically clear silicone polymer (PDMS, polydimethylsiloxane). The PDMS coated OxySplot micro-particles are used alone or in a thin (~50 µm) PDMS layer of arbitrary shape referred to as the OxyMat. Additional top coatings on the OxyMat (e.g., fibronectin, laminin, polylysine, special photoactivatable surfaces etc.) facilitate adherence of cells. The OxySplots report the cellular pO2 and micro-gradients of pO2 without disrupting the flow of extracellular solutions or interfering with patch-clamp pipettes, mechanical attachments, and micro-superfusion. Since OxySplots and a cell can be imaged and spatially resolved, calibrated changes of pO2 and intracellular events can be imaged simultaneously. In addition, the response-time (t0.5 = 0.7 s, 0-160 mmHg) of OxySplots is ~100 times faster than amperometric Clark-type polarization microelectrodes. Two usage example of OxySplots with cardiomyocytes show (1) OxySplots measuring pericellular pO2 while tetramethylrhodamine methyl-ester (TMRM) was used to measure mitochondrial membrane potential (ΔΨm); and (2) OxySplots measuring pO2 during ischemia and reperfusion while rhod-2 was used to measure cytosolic [Ca2+]i levels simultaneously. The OxySplot/OxyMat optrode system provides an affordable and highly adaptable optical sensor system for monitoring pO2 with a diverse array of imaging systems, including high-speed, high-resolution confocal microscopes while physiological features are measured simultaneously.

Axial tubule junctions control rapid calcium signaling in atria.

The canonical atrial myocyte (AM) is characterized by sparse transverse tubule (TT) invaginations and slow intracellular Ca2+ propagation but exhibits rapid contractile activation that is susceptible to loss of function during hypertrophic remodeling. Here, we have identified a membrane structure and Ca2+-signaling complex that may enhance the speed of atrial contraction independently of phospholamban regulation. This axial couplon was observed in human and mouse atria and is composed of voluminous axial tubules (ATs) with extensive junctions to the sarcoplasmic reticulum (SR) that include ryanodine receptor 2 (RyR2) clusters. In mouse AM, AT structures triggered Ca2+ release from the SR approximately 2 times faster at the AM center than at the surface. Rapid Ca2+ release correlated with colocalization of highly phosphorylated RyR2 clusters at AT-SR junctions and earlier, more rapid shortening of central sarcomeres. In contrast, mice expressing phosphorylation-incompetent RyR2 displayed depressed AM sarcomere shortening and reduced in vivo atrial contractile function. Moreover, left atrial hypertrophy led to AT proliferation, with a marked increase in the highly phosphorylated RyR2-pS2808 cluster fraction, thereby maintaining cytosolic Ca2+ signaling despite decreases in RyR2 cluster density and RyR2 protein expression. AT couplon "super-hubs" thus underlie faster excitation-contraction coupling in health as well as hypertrophic compensatory adaptation and represent a structural and metabolic mechanism that may contribute to contractile dysfunction and arrhythmias.

Ryanodine receptor sensitivity governs the stability and synchrony of local calcium release during cardiac excitation-contraction coupling.

Calcium-induced calcium release is the principal mechanism that triggers the cell-wide [Ca(2+)]i transient that activates muscle contraction during cardiac excitation-contraction coupling (ECC). Here, we characterize this process in mouse cardiac myocytes with a novel mathematical action potential (AP) model that incorporates realistic stochastic gating of voltage-dependent L-type calcium (Ca(2+)) channels (LCCs) and sarcoplasmic reticulum (SR) Ca(2+) release channels (the ryanodine receptors, RyR2s). Depolarization of the sarcolemma during an AP stochastically activates the LCCs elevating subspace [Ca(2+)] within each of the cell's 20,000 independent calcium release units (CRUs) to trigger local RyR2 opening and initiate Ca(2+) sparks, the fundamental unit of triggered Ca(2+) release. Synchronization of Ca(2+) sparks during systole depends on the nearly uniform cellular activation of LCCs and the likelihood of local LCC openings triggering local Ca(2+) sparks (ECC fidelity). The detailed design and true SR Ca(2+) pump/leak balance displayed by our model permits investigation of ECC fidelity and Ca(2+) spark fidelity, the balance between visible (Ca(2+) spark) and invisible (Ca(2+) quark/sub-spark) SR Ca(2+) release events. Excess SR Ca(2+) leak is examined as a disease mechanism in the context of "catecholaminergic polymorphic ventricular tachycardia (CPVT)", a Ca(2+)-dependent arrhythmia. We find that that RyR2s (and therefore Ca(2+) sparks) are relatively insensitive to LCC openings across a wide range of membrane potentials; and that key differences exist between Ca(2+) sparks evoked during quiescence, diastole, and systole. The enhanced RyR2 [Ca(2+)]i sensitivity during CPVT leads to increased Ca(2+) spark fidelity resulting in asynchronous systolic Ca(2+) spark activity. It also produces increased diastolic SR Ca(2+) leak with some prolonged Ca(2+) sparks that at times become "metastable" and fail to efficiently terminate. There is a huge margin of safety for stable Ca(2+) handling within the cell and this novel mechanistic model provides insight into the molecular signaling characteristics that help maintain overall Ca(2+) stability even under the conditions of high SR Ca(2+) leak during CPVT. Finally, this model should provide tools for investigators to examine normal and pathological Ca(2+) signaling characteristics in the heart.

Mitochondrial calcium and the regulation of metabolism in the heart.

Consumption of adenosine triphosphate (ATP) by the heart can change dramatically as the energetic demands increase from a period of rest to strenuous activity. Mitochondrial ATP production is central to this metabolic response since the heart relies largely on oxidative phosphorylation as its source of intracellular ATP. Significant evidence has been acquired indicating that Ca(2+) plays a critical role in regulating ATP production by the mitochondria. Here the evidence that the Ca(2+) concentration in the mitochondrial matrix ([Ca(2+)]m) plays a pivotal role in regulating ATP production by the mitochondria is critically reviewed and aspects of this process that are under current active investigation are highlighted. Importantly, current quantitative information on the bidirectional Ca(2+) movement across the inner mitochondrial membrane (IMM) is examined in two parts. First, we review how Ca(2+) influx into the mitochondrial matrix depends on the mitochondrial Ca(2+) channel (i.e., the mitochondrial calcium uniporter or MCU). This discussion includes how the MCU open probability (PO) depends on the cytosolic Ca(2+) concentration ([Ca(2+)]i) and on the mitochondrial membrane potential (ΔΨm). Second, we discuss how steady-state [Ca(2+)]m is determined by the dynamic balance between this MCU-based Ca(2+) influx and mitochondrial Na(+)/Ca(2+) exchanger (NCLX) based Ca(2+) efflux. These steady-state [Ca(2+)]m levels are suggested to regulate the metabolic energy supply due to Ca(2+)-dependent regulation of mitochondrial enzymes of the tricarboxylic acid cycle (TCA), the proteins of the electron transport chain (ETC), and the F1F0 ATP synthase itself. We conclude by discussing the roles played by [Ca(2+)]m in influencing mitochondrial responses under pathological conditions. This article is part of a Special Issue entitled "Mitochondria: From BasicMitochondrial Biology to Cardiovascular Disease."

Superresolution modeling of calcium release in the heart.

Stable calcium-induced calcium release (CICR) is critical for maintaining normal cellular contraction during cardiac excitation-contraction coupling. The fundamental element of CICR in the heart is the calcium (Ca(2+)) spark, which arises from a cluster of ryanodine receptors (RyR). Opening of these RyR clusters is triggered to produce a local, regenerative release of Ca(2+) from the sarcoplasmic reticulum (SR). The Ca(2+) leak out of the SR is an important process for cellular Ca(2+) management, and it is critically influenced by spark fidelity, i.e., the probability that a spontaneous RyR opening triggers a Ca(2+) spark. Here, we present a detailed, three-dimensional model of a cardiac Ca(2+) release unit that incorporates diffusion, intracellular buffering systems, and stochastically gated ion channels. The model exhibits realistic Ca(2+) sparks and robust Ca(2+) spark termination across a wide range of geometries and conditions. Furthermore, the model captures the details of Ca(2+) spark and nonspark-based SR Ca(2+) leak, and it produces normal excitation-contraction coupling gain. We show that SR luminal Ca(2+)-dependent regulation of the RyR is not critical for spark termination, but it can explain the exponential rise in the SR Ca(2+) leak-load relationship demonstrated in previous experimental work. Perturbations to subspace dimensions, which have been observed in experimental models of disease, strongly alter Ca(2+) spark dynamics. In addition, we find that the structure of RyR clusters also influences Ca(2+) release properties due to variations in inter-RyR coupling via local subspace Ca(2+) concentration ([Ca(2+)]ss). These results are illustrated for RyR clusters based on super-resolution stimulated emission depletion microscopy. Finally, we present a believed-novel approach by which the spark fidelity of a RyR cluster can be predicted from structural information of the cluster using the maximum eigenvalue of its adjacency matrix. These results provide critical insights into CICR dynamics in heart, under normal and pathological conditions.

Tachycardia-induced silencing of subcellular Ca2+ signaling in atrial myocytes.

Atrial fibrillation (AF) is characterized by sustained high atrial activation rates and arrhythmogenic cellular Ca2+ signaling instability; however, it is not clear how a high atrial rate and Ca2+ instability may be related. Here, we characterized subcellular Ca2+ signaling after 5 days of high atrial rates in a rabbit model. While some changes were similar to those in persistent AF, we identified a distinct pattern of stabilized subcellular Ca2+ signaling. Ca2+ sparks, arrhythmogenic Ca2+ waves, sarcoplasmic reticulum (SR) Ca2+ leak, and SR Ca2+ content were largely unaltered. Based on computational analysis, these findings were consistent with a higher Ca2+ leak due to PKA-dependent phosphorylation of SR Ca2+ channels (RyR2s), fewer RyR2s, and smaller RyR2 clusters in the SR. We determined that less Ca2+ release per [Ca2+]i transient, increased Ca2+ buffering strength, shortened action potentials, and reduced L-type Ca2+ current contribute to a stunning reduction of intracellular Na+ concentration following rapid atrial pacing. In both patients with AF and in our rabbit model, this silencing led to failed propagation of the [Ca2+]i signal to the myocyte center. We conclude that sustained high atrial rates alone silence Ca2+ signaling and do not produce Ca2+ signaling instability, consistent with an adaptive molecular and cellular response to atrial tachycardia.

Calcium movement in cardiac mitochondria.

Existing theory suggests that mitochondria act as significant, dynamic buffers of cytosolic calcium ([Ca(2+)]i) in heart. These buffers can remove up to one-third of the Ca(2+) that enters the cytosol during the [Ca(2+)]i transients that underlie contractions. However, few quantitative experiments have been presented to test this hypothesis. Here, we investigate the influence of Ca(2+) movement across the inner mitochondrial membrane during both subcellular and global cellular cytosolic Ca(2+) signals (i.e., Ca(2+) sparks and [Ca(2+)]i transients, respectively) in isolated rat cardiomyocytes. By rapidly turning off the mitochondria using depolarization of the inner mitochondrial membrane potential (ΔΨm), the role of the mitochondria in buffering cytosolic Ca(2+) signals was investigated. We show here that rapid loss of ΔΨm leads to no significant changes in cytosolic Ca(2+) signals. Second, we make direct measurements of mitochondrial [Ca(2+)] ([Ca(2+)]m) using a mitochondrially targeted Ca(2+) probe (MityCam) and these data suggest that [Ca(2+)]m is near the [Ca(2+)]i level (∼100 nM) under quiescent conditions. These two findings indicate that although the mitochondrial matrix is fully buffer-capable under quiescent conditions, it does not function as a significant dynamic buffer during physiological Ca(2+) signaling. Finally, quantitative analysis using a computational model of mitochondrial Ca(2+) cycling suggests that mitochondrial Ca(2+) uptake would need to be at least ∼100-fold greater than the current estimates of Ca(2+) influx for mitochondria to influence measurably cytosolic [Ca(2+)] signals under physiological conditions. Combined, these experiments and computational investigations show that mitochondrial Ca(2+) uptake does not significantly alter cytosolic Ca(2+) signals under normal conditions and indicates that mitochondria do not act as important dynamic buffers of [Ca(2+)]i under physiological conditions in heart.

Mitochondrial calcium uptake.

Calcium (Ca(2+)) uptake into the mitochondrial matrix is critically important to cellular function. As a regulator of matrix Ca(2+) levels, this flux influences energy production and can initiate cell death. If large, this flux could potentially alter intracellular Ca(2+) ([Ca(2+)]i) signals. Despite years of study, fundamental disagreements on the extent and speed of mitochondrial Ca(2+) uptake still exist. Here, we review and quantitatively analyze mitochondrial Ca(2+) uptake fluxes from different tissues and interpret the results with respect to the recently proposed mitochondrial Ca(2+) uniporter (MCU) candidate. This quantitative analysis yields four clear results: (i) under physiological conditions, Ca(2+) influx into the mitochondria via the MCU is small relative to other cytosolic Ca(2+) extrusion pathways; (ii) single MCU conductance is ∼6-7 pS (105 mM [Ca(2+)]), and MCU flux appears to be modulated by [Ca(2+)]i, suggesting Ca(2+) regulation of MCU open probability (P(O)); (iii) in the heart, two features are clear: the number of MCU channels per mitochondrion can be calculated, and MCU probability is low under normal conditions; and (iv) in skeletal muscle and liver cells, uptake per mitochondrion varies in magnitude but total uptake per cell still appears to be modest. Based on our analysis of available quantitative data, we conclude that although Ca(2+) critically regulates mitochondrial function, the mitochondria do not act as a significant dynamic buffer of cytosolic Ca(2+) under physiological conditions. Nevertheless, with prolonged (superphysiological) elevations of [Ca(2+)]i, mitochondrial Ca(2+) uptake can increase 10- to 1,000-fold and begin to shape [Ca(2+)]i dynamics.

NCLX: the mitochondrial sodium calcium exchanger.

The free Ca(2+) concentration within the mitochondrial matrix ([Ca(2+)]m) regulates the rate of ATP production and other [Ca(2+)]m sensitive processes. It is set by the balance between total Ca(2+) influx (through the mitochondrial Ca(2+) uniporter (MCU) and any other influx pathways) and the total Ca(2+) efflux (by the mitochondrial Na(+)/Ca(2+) exchanger and any other efflux pathways). Here we review and analyze the experimental evidence reported over the past 40years which suggest that in the heart and many other mammalian tissues a putative Na(+)/Ca(2+) exchanger is the major pathway for Ca(2+) efflux from the mitochondrial matrix. We discuss those reports with respect to a recent discovery that the protein product of the human FLJ22233 gene mediates such Na(+)/Ca(2+) exchange across the mitochondrial inner membrane. Among its many functional similarities to other Na(+)/Ca(2+) exchanger proteins is a unique feature: it efficiently mediates Li(+)/Ca(2+) exchange (as well as Na(+)/Ca(2+) exchange) and was therefore named NCLX. The discovery of NCLX provides both the identity of a novel protein and new molecular means of studying various unresolved quantitative aspects of mitochondrial Ca(2+) movement out of the matrix. Quantitative and qualitative features of NCLX are discussed as is the controversy regarding the stoichiometry of the NCLX Na(+)/Ca(2+) exchange, the electrogenicity of NCLX, the [Na(+)]i dependency of NCLX and the magnitude of NCLX Ca(2+) efflux. Metabolic features attributable to NCLX and the physiological implication of the Ca(2+) efflux rate via NCLX during systole and diastole are also briefly discussed.

Stimulated emission depletion live-cell super-resolution imaging shows proliferative remodeling of T-tubule membrane structures after myocardial infarction.

RATIONALE: Transverse tubules (TTs) couple electric surface signals to remote intracellular Ca(2+) release units (CRUs). Diffraction-limited imaging studies have proposed loss of TT components as disease mechanism in heart failure (HF). OBJECTIVES: Objectives were to develop quantitative super-resolution strategies for live-cell imaging of TT membranes in intact cardiomyocytes and to show that TT structures are progressively remodeled during HF development, causing early CRU dysfunction. METHODS AND RESULTS: Using stimulated emission depletion (STED) microscopy, we characterized individual TTs with nanometric resolution as direct readout of local membrane morphology 4 and 8 weeks after myocardial infarction (4pMI and 8pMI). Both individual and network TT properties were investigated by quantitative image analysis. The mean area of TT cross sections increased progressively from 4pMI to 8pMI. Unexpectedly, intact TT networks showed differential changes. Longitudinal and oblique TTs were significantly increased at 4pMI, whereas transversal components appeared decreased. Expression of TT-associated proteins junctophilin-2 and caveolin-3 was significantly changed, correlating with network component remodeling. Computational modeling of spatial changes in HF through heterogeneous TT reorganization and RyR2 orphaning (5000 of 20 000 CRUs) uncovered a local mechanism of delayed subcellular Ca(2+) release and action potential prolongation. CONCLUSIONS: This study introduces STED nanoscopy for live mapping of TT membrane structures. During early HF development, the local TT morphology and associated proteins were significantly altered, leading to differential network remodeling and Ca(2+) release dyssynchrony. Our data suggest that TT remodeling during HF development involves proliferative membrane changes, early excitation-contraction uncoupling, and network fracturing.

Dynamics of calcium sparks and calcium leak in the heart.

We present what we believe to be a new mathematical model of Ca(2+) leak from the sarcoplasmic reticulum (SR) in the heart. To our knowledge, it is the first to incorporate a realistic number of Ca(2+)-release units, each containing a cluster of stochastically gating Ca(2+) channels (RyRs), whose biophysical properties (e.g., Ca(2+) sensitivity and allosteric interactions) are informed by the latest molecular investigations. This realistic model allows for the detailed characterization of RyR Ca(2+)-release properties, and shows how this balances reuptake by the SR Ca(2+) pump. Simulations reveal that SR Ca(2+) leak consists of brief but frequent single RyR openings (~3000 cell(-1) s(-1)) that are likely to be experimentally undetectable, and are, therefore, "invisible". We also observe that these single RyR openings can recruit additional RyRs to open, due to elevated local (Ca(2+)), and occasionally lead to the generation of Ca(2+) sparks (~130 cell(-1) s(-1)). Furthermore, this physiological formulation of "invisible" leak allows for the removal of the ad hoc, non-RyR mediated Ca(2+) leak terms present in prior models. Finally, our model shows how Ca(2+) sparks can be robustly triggered and terminated under both normal and pathological conditions. Together, these discoveries profoundly influence how we interpret and understand diverse experimental and clinical results from both normal and diseased hearts.

Stochastic simulation of cardiac ventricular myocyte calcium dynamics and waves.

A three dimensional model of calcium dynamics in the rat ventricular myocyte was developed to study the mechanism of calcium homeostasis and pathological calcium dynamics during calcium overload. The model contains 20,000 calcium release units (CRUs) each containing 49 ryanodine receptors. The model simulates calcium sparks with a realistic spontaneous calcium spark rate. It suggests that in addition to the calcium spark-based leak, there is an invisible calcium leak caused by the stochastic opening of a small number of ryanodine receptors in each CRU without triggering a calcium spark. The model also explores the mechanism of calcium wave propagation between release sites under the conditions of calcium overload.

Models of cardiac excitation-contraction coupling in ventricular myocytes.

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Moment closure for local control models of calcium-induced calcium release in cardiac myocytes.

In prior work, we introduced a probability density approach to modeling local control of Ca2+-induced Ca2+ release in cardiac myocytes, where we derived coupled advection-reaction equations for the time-dependent bivariate probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum (SR) [Ca2+] conditioned on Ca2+ release unit (CaRU) state. When coupled to ordinary differential equations (ODEs) for the bulk myoplasmic and network SR [Ca2+], a realistic but minimal model of cardiac excitation-contraction coupling was produced that avoids the computationally demanding task of resolving spatial aspects of global Ca2+ signaling, while accurately representing heterogeneous local Ca2+ signals in a population of diadic subspaces and junctional SR depletion domains. Here we introduce a computationally efficient method for simulating such whole cell models when the dynamics of subspace [Ca2+] are much faster than those of junctional SR [Ca2+]. The method begins with the derivation of a system of ODEs describing the time-evolution of the moments of the univariate probability density functions for junctional SR [Ca2+] jointly distributed with CaRU state. This open system of ODEs is then closed using an algebraic relationship that expresses the third moment of junctional SR [Ca2+] in terms of the first and second moments. In simulated voltage-clamp protocols using 12-state CaRUs that respond to the dynamics of both subspace and junctional SR [Ca2+], this moment-closure approach to simulating local control of excitation-contraction coupling produces high-gain Ca2+ release that is graded with changes in membrane potential, a phenomenon not exhibited by common pool models. Benchmark simulations indicate that the moment-closure approach is nearly 10,000-times more computationally efficient than corresponding Monte Carlo simulations while leading to nearly identical results. We conclude by applying the moment-closure approach to study the restitution of Ca2+-induced Ca2+ release during simulated two-pulse voltage-clamp protocols.

Modeling local and global intracellular calcium responses mediated by diffusely distributed inositol 1,4,5-trisphosphate receptors.

Considerable insight into intracellular Ca2+ responses has been obtained through the development of whole cell models that are based on molecular mechanisms, e.g., single channel kinetics of the inositol 1,4,5-trisphosphate (IP3) receptor Ca2+ channel. However, a limitation of most whole cell models to date is the assumption that IP3 receptor Ca2+ channels (IP3Rs) are globally coupled by a "continuously stirred" bulk cytosolic [Ca2+], when in fact open IP3Rs experience elevated "domain" Ca2+ concentrations. Here we present a 2N+2-compartment whole cell model of local and global Ca2+ responses mediated by N=100,000 diffusely distributed IP3Rs, each represented by a four-state Markov chain. Two of these compartments correspond to bulk cytosolic and luminal Ca2+ concentrations, and the remaining 2N compartments represent time-dependent cytosolic and luminal Ca2+ domains associated with each IP3R. Using this Monte Carlo model as a starting point, we present an alternative formulation that solves a system of advection-reaction equations for the probability density of cytosolic and luminal domain [Ca2+] jointly distributed with IP3R state. When these equations are coupled to ordinary differential equations for the bulk cytosolic and luminal [Ca2+], a realistic but minimal model of whole cell Ca2+ dynamics is produced that accounts for the influence of local Ca2+ signaling on channel gating and global Ca2+ responses. The probability density approach is benchmarked and validated by comparison to Monte Carlo simulations, and the two methods are shown to agree when the number of Ca2+ channels is large (i.e., physiologically realistic). Using the probability density approach, we show that the time scale of Ca2+ domain formation and collapse (both cytosolic and luminal) may influence global Ca2+ oscillations, and we derive two reduced models of global Ca2+ dynamics that account for the influence of local Ca2+ signaling on global Ca2+ dynamics when there is a separation of time scales between the stochastic gating of IP3Rs and the dynamics of domain Ca2+.

A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes.

We present a probability density approach to modeling localized Ca2+ influx via L-type Ca2+ channels and Ca2+-induced Ca2+ release mediated by clusters of ryanodine receptors during excitation-contraction coupling in cardiac myocytes. Coupled advection-reaction equations are derived relating the time-dependent probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum [Ca2+] conditioned on "Ca2+ release unit" state. When these equations are solved numerically using a high-resolution finite difference scheme and the resulting probability densities are coupled to ordinary differential equations for the bulk myoplasmic and sarcoplasmic reticulum [Ca2+], a realistic but minimal model of cardiac excitation-contraction coupling is produced. Modeling Ca2+ release unit activity using this probability density approach avoids the computationally demanding task of resolving spatial aspects of global Ca2+ signaling, while accurately representing heterogeneous local Ca2+ signals in a population of diadic subspaces and junctional sarcoplasmic reticulum depletion domains. The probability density approach is validated for a physiologically realistic number of Ca2+ release units and benchmarked for computational efficiency by comparison to traditional Monte Carlo simulations. In simulated voltage-clamp protocols, both the probability density and Monte Carlo approaches to modeling local control of excitation-contraction coupling produce high-gain Ca2+ release that is graded with changes in membrane potential, a phenomenon not exhibited by so-called "common pool" models. However, a probability density calculation can be significantly faster than the corresponding Monte Carlo simulation, especially when cellular parameters are such that diadic subspace [Ca2+] is in quasistatic equilibrium with junctional sarcoplasmic reticulum [Ca2+] and, consequently, univariate rather than multivariate probability densities may be employed.