Interplay of ryanodine receptor distribution and calcium dynamics

Role of ryanodine receptor cooperativity in Ca2+-wave-mediated triggered activity in cardiomyocytes

Ca2+ waves are known to trigger delayed after-depolarizations that can cause malignant cardiac arrhythmias. However, modelling Ca2+ waves using physiologically realistic models has remained a major challenge. Existing models with low Ca2+ sensitivity of ryanodine receptors (RyRs) necessitate large release currents, leading to an unrealistically large Ca2+ transient amplitude incompatible with the experimental observations. Consequently, current physiologically detailed models of delayed after-depolarizations resort to unrealistic cell architectures to produce Ca2+ waves with a normal Ca2+ transient amplitude. Here, we address these challenges by incorporating RyR cooperativity into a physiologically detailed model with a realistic cell architecture. We represent RyR cooperativity phenomenologically through a Hill coefficient within the sigmoid function of RyR open probability. Simulations in permeabilized myocytes with high Ca2+ sensitivity reveal that a sufficiently large Hill coefficient is required for Ca2+ wave propagation via the fire-diffuse-fire mechanism. In intact myocytes, propagating Ca2+ waves can occur only within an intermediate Hill coefficient range. Within this range, the spark rate is neither too low, enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during diastole of the action potential. Moreover, this model successfully replicates other experimentally observed manifestations of Ca2+-wave-mediated triggered activity, including phase 2 and phase 3 early after-depolarizations and high-frequency voltage-Ca2+ oscillations. These oscillations feature an elevated take-off potential with depolarization mediated by the L-type Ca2+ current. The model also sheds light on the roles of luminal gating of RyRs and the mobile buffer ATP in the genesis of these arrhythmogenic phenomena. KEY POINTS: Existing mathematical models of Ca2+ waves use an excessively large Ca2+-release current or unrealistic diffusive coupling between release units. Our physiologically realistic model, using a Hill coefficient in the ryanodine receptor (RyR) gating function to represent RyR cooperativity, addresses these limitations and generates organized Ca2+ waves at Hill coefficients ranging from ∼5 to 10, as opposed to the traditional value of 2. This range of Hill coefficients gives a spark rate neither too low, thereby enabling Ca2+ wave propagation, nor too high, allowing for the maintenance of a high sarcoplasmic reticulum load during the plateau phase of the action potential. Additionally, the model generates Ca2+-wave-mediated phase 2 and phase 3 early after-depolarizations, and coupled membrane voltage with Ca2+ oscillations mediated by the L-type Ca2+ current. This study suggests that pharmacologically targeting RyR cooperativity could be a promising strategy for treating cardiac arrhythmias linked to Ca2+-wave-mediated triggered activity.

Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation

The complexity of cardiac electrophysiology, involving dynamic changes in numerous components across multiple spatial (from ion channel to organ) and temporal (from milliseconds to days) scales, makes an intuitive or empirical analysis of cardiac arrhythmogenesis challenging. Multiscale mechanistic computational models of cardiac electrophysiology provide precise control over individual parameters, and their reproducibility enables a thorough assessment of arrhythmia mechanisms. This review provides a comprehensive analysis of models of cardiac electrophysiology and arrhythmias, from the single cell to the organ level, and how they can be leveraged to better understand rhythm disorders in cardiac disease and to improve heart patient care. Key issues related to model development based on experimental data are discussed, and major families of human cardiomyocyte models and their applications are highlighted. An overview of organ-level computational modeling of cardiac electrophysiology and its clinical applications in personalized arrhythmia risk assessment and patient-specific therapy of atrial and ventricular arrhythmias is provided. The advancements presented here highlight how patient-specific computational models of the heart reconstructed from patient data have achieved success in predicting risk of sudden cardiac death and guiding optimal treatments of heart rhythm disorders. Finally, an outlook toward potential future advances, including the combination of mechanistic modeling and machine learning/artificial intelligence, is provided. As the field of cardiology is embarking on a journey toward precision medicine, personalized modeling of the heart is expected to become a key technology to guide pharmaceutical therapy, deployment of devices, and surgical interventions.

T cell Ca2+ microdomains through the lens of computational modeling

Cellular Ca2+ signaling is highly organized in time and space. Locally restricted and short-lived regions of Ca2+ increase, called Ca2+ microdomains, constitute building blocks that are differentially arranged to create cellular Ca2+ signatures controlling physiological responses. Here, we focus on Ca2+ microdomains occurring in restricted cytosolic spaces between the plasma membrane and the endoplasmic reticulum, called endoplasmic reticulum-plasma membrane junctions. In T cells, these microdomains have been finely characterized. Enough quantitative data are thus available to develop detailed computational models of junctional Ca2+ dynamics. Simulations are able to predict the characteristics of Ca2+ increases at the level of single channels and in junctions of different spatial configurations, in response to various signaling molecules. Thanks to the synergy between experimental observations and computational modeling, a unified description of the molecular mechanisms that create Ca2+ microdomains in the first seconds of T cell stimulation is emerging.

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Increased SERCA2a sub-cellular heterogeneity in right-ventricular heart failure inhibits excitation-contraction coupling and modulates arrhythmogenic dynamics

The intracellular calcium handling system of cardiomyocytes is responsible for controlling excitation-contraction coupling (ECC) and has been linked to pro-arrhythmogenic cellular phenomena in conditions such as heart failure (HF). SERCA2a, responsible for intracellular uptake, is a primary regulator of calcium homeostasis, and remodelling of its function has been proposed as a causal factor underlying cellular and tissue dysfunction in disease. Whereas adaptations to the global (i.e. whole-cell) expression of SERCA2a have been previously investigated in the context of multiple diseases, the role of its spatial profile in the sub-cellular volume has yet to be elucidated. We present an approach to characterize the sub-cellular heterogeneity of SERCA2a and apply this approach to quantify adaptations to the length-scale of heterogeneity (the distance over which expression is correlated) associated with right-ventricular (RV)-HF. These characterizations informed simulations to predict the functional implications of this heterogeneity, and its remodelling in disease, on ECC, the dynamics of calcium-transient alternans and the emergence of spontaneous triggered activity. Image analysis reveals that RV-HF is associated with an increase in length-scale and its inter-cellular variability; simulations predict that this increase in length-scale can reduce ECC and critically modulate the vulnerability to both alternans and triggered activity. This article is part of the theme issue 'The cardiomyocyte: new revelations on the interplay between architecture and function in growth, health, and disease'.

Molecular, Subcellular, and Arrhythmogenic Mechanisms in Genetic RyR2 Disease

The ryanodine receptor (RyR2) has a critical role in controlling Ca²⁺ release from the sarcoplasmic reticulum (SR) throughout the cardiac cycle. RyR2 protein has multiple functional domains with specific roles, and four of these RyR2 protomers are required to form the quaternary structure that comprises the functional channel. Numerous mutations in the gene encoding RyR2 protein have been identified and many are linked to a wide spectrum of arrhythmic heart disease. Gain of function mutations (GoF) result in a hyperactive channel that causes excessive spontaneous SR Ca²⁺ release. This is the predominant cause of the inherited syndrome catecholaminergic polymorphic ventricular tachycardia (CPVT). Recently, rare hypoactive loss of function (LoF) mutations have been identified that produce atypical effects on cardiac Ca²⁺ handling that has been termed calcium release deficiency syndrome (CRDS). Aberrant Ca²⁺ release resulting from both GoF and LoF mutations can result in arrhythmias through the Na⁺/Ca²⁺ exchange mechanism. This mini-review discusses recent findings regarding the role of RyR2 domains and endogenous regulators that influence RyR2 gating normally and with GoF/LoF mutations. The arrhythmogenic consequences of GoF/LoF mutations will then be discussed at the macromolecular and cellular level.

Challenges and opportunities for the simulation of calcium waves on modern multi-core and many-core parallel computing platforms

State-of-the-art distributed-memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many-core processor has more than 60 cores with 16 GB of high-performance on-chip memory. We contrast the performance of the second-generation Intel Xeon Phi, code-named Knights Landing (KNL), with 68 computational cores to the latest multicore CPU Intel Skylake with 18 cores. A special-purpose code solving a system of nonlinear reaction-diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time-stepping, with a matrix-free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium-induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many-core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.

Cardiomyocyte calcium handling in health and disease: Insights from in vitro and in silico studies

Calcium (Ca²⁺) plays a central role in cardiomyocyte excitation-contraction coupling. To ensure an optimal electrical impulse propagation and cardiac contraction, Ca²⁺ levels are regulated by a variety of Ca²⁺-handling proteins. In turn, Ca²⁺ modulates numerous electrophysiological processes. Accordingly, Ca²⁺-handling abnormalities can promote cardiac arrhythmias via various mechanisms, including the promotion of afterdepolarizations, ion-channel modulation and structural remodeling. In the last 30 years, significant improvements have been made in the computational modeling of cardiomyocyte Ca²⁺ handling under physiological and pathological conditions. However, numerous questions involving the Ca²⁺-dependent regulation of different macromolecular complexes, cross-talk between Ca²⁺-dependent regulatory pathways operating over a wide range of time scales, and bidirectional interactions between electrophysiology and mechanics remain to be addressed by in vitro and in silico studies. A better understanding of disease-specific Ca²⁺-dependent proarrhythmic mechanisms may facilitate the development of improved therapeutic strategies. In this review, we describe the fundamental mechanisms of cardiomyocyte Ca²⁺ handling in health and disease, and provide an overview of currently available computational models for cardiomyocyte Ca²⁺ handling. Finally, we discuss important uncertainties and open questions about cardiomyocyte Ca²⁺ handling and highlight how synergy between in vitro and in silico studies may help to answer several of these issues.

Assessing Cardiomyocyte Excitation-Contraction Coupling Site Detection From Live Cell Imaging Using a Structurally-Realistic Computational Model of Calcium Release

Calcium signaling plays a pivotal role in cardiomyocytes, coupling electrical excitation to mechanical contraction of the heart. Determining locations of active calcium release sites, and how their recruitment changes in response to stimuli and in disease states is therefore of central interest in cardiac physiology. Current algorithms for detecting release sites from live cell imaging data are however not easily validated against a known “ground truth,” which makes interpretation of the output of such algorithms, in particular the degree of confidence in site detection, a challenging task. Computational models are capable of integrating findings from multiple sources into a consistent, predictive framework. In cellular physiology, such models have the potential to reveal structure and function beyond the temporal and spatial resolution limitations of individual experimental measurements. Here, we create a spatially detailed computational model of calcium release in an eight sarcomere section of a ventricular cardiomyocyte, using electron tomography reconstruction of cardiac ultrastructure and confocal imaging of protein localization. This provides a high-resolution model of calcium diffusion from intracellular stores, which can be used as a platform to simulate confocal fluorescence imaging in the context of known ground truth structures from the higher resolution model. We use this capability to evaluate the performance of a recently proposed method for detecting the functional response of calcium release sites in live cells. Model permutations reveal how calcium release site density and mitochondria acting as diffusion barriers impact the detection performance of the algorithm. We demonstrate that site density has the greatest impact on detection precision and recall, in particular affecting the effective detectable depth of sites in confocal data. Our findings provide guidance on how such detection algorithms may best be applied to experimental data and give insights into limitations when using two-dimensional microscopy images to analyse three-dimensional cellular structures.

Identification of RyR2-PBmice and the effects of transposon insertional mutagenesis of the RyR2 gene on cardiac function in mice

Ryanodine receptor 2 (RyR2) plays an important role in maintaining the normal heart function, and mutantions can lead to arrhythmia, heart failure and other heart diseases. In this study, we successfully identified a piggyBac translocated RyR2 gene heterozygous mouse model (RyR2-PBmice) by tracking red fluorescent protein (RFP) and genotyping PCR. Cardiac function tests showed that there was no significant difference between the RyR2-PBmice and corresponding wild-type mice (WTmice), regardless of whether they were in the basal state or injected with epinephrine and caffeine. However, the sarcoplasmic reticulum Ca²⁺ content was significantly reduced in the cardiomyocytes of RyR2-PBmice as assessed by measuring caffeine-induced [Ca²⁺]_i transients; the cardiac muscle tissue of RyR2-PBmice displayed significant mitochondrial swelling and focal dissolution of mitochondrial cristae, and the tissue ATP content in the RyR2-PBmice heart was significantly reduced. To further analyze the molecular mechanism behind these changes, we tested the expression levels of related proteins using RT-PCR and Western blot analyses. The mRNA level of RyR2 in RyR2-PBmice cardiac tissue decreased significantly compared with the WTmice, and the protein expression associated with the respiratory chain was also downregulated. These results suggested that the piggyBac transposon inserted into the RyR2 gene substantively affected the structure and function of mitochondria in the mouse cardiomyocytes, leading to disorders of energy metabolism.

Dynamic and Irregular Distribution of RyR2 Clusters in the Periphery of Live Ventricular Myocytes

Cardiac ryanodine receptors (RyR2s) are Ca²⁺ release channels clustering in the sarcoplasmic reticulum membrane. These clusters are believed to be the elementary units of Ca²⁺ release. The distribution of these Ca²⁺ release units plays a critical role in determining the spatio-temporal profile and stability of sarcoplasmic reticulum Ca²⁺ release. RyR2 clusters located in the interior of cardiomyocytes are arranged in highly ordered arrays. However, little is known about the distribution and function of RyR2 clusters in the periphery of cardiomyocytes. Here, we used a knock-in mouse model expressing a green fluorescence protein (GFP)-tagged RyR2 to localize RyR2 clusters in live ventricular myocytes by virtue of their GFP fluorescence. Confocal imaging and total internal reflection fluorescence microscopy was employed to determine and compare the distribution of GFP-RyR2 in the interior and periphery of isolated live ventricular myocytes and in intact hearts. We found tightly ordered arrays of GFP-RyR2 clusters in the interior, as previously described. In contrast, irregular distribution of GFP-RyR2 clusters was observed in the periphery. Time-lapse total internal reflection fluorescence imaging revealed dynamic movements of GFP-RyR2 clusters in the periphery, which were affected by external Ca²⁺ and RyR2 activator (caffeine) and inhibitor (tetracaine), but little detectable movement of GFP-RyR2 clusters in the interior. Furthermore, simultaneous Ca²⁺- and GFP-imaging demonstrated that peripheral RyR2 clusters with an irregular distribution pattern are functional with a Ca²⁺ release profile similar to that in the interior. These results indicate that the distribution of RyR2 clusters in the periphery of live ventricular myocytes is irregular and dynamic, which is different from that of RyR2 clusters in the interior.

Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves

Ryanodine receptors (RyRs) mediate calcium (Ca)-induced Ca release and intracellular Ca homeostasis. In a cardiac myocyte, RyRs group into clusters of variable size from a few to several hundred RyRs, creating a spatially nonuniform intracellular distribution. It is unclear how heterogeneity of RyR cluster size alters spontaneous sarcoplasmic reticulum (SR) Ca releases (Ca sparks) and arrhythmogenic Ca waves. Here, we tested the impact of heterogeneous RyR cluster size on the initiation of Ca waves. Experimentally, we measured RyR cluster sizes at Ca spark sites in rat ventricular myocytes and further tested functional impacts using a physiologically detailed computational model with spatial and stochastic intracellular Ca dynamics. We found that the spark frequency and amplitude increase nonlinearly with the size of RyR clusters. Larger RyR clusters have lower SR Ca release threshold for local Ca spark initiation and exhibit steeper SR Ca release versus SR Ca load relationship. However, larger RyR clusters tend to lower SR Ca load because of the higher Ca leak rate. Conversely, smaller clusters have a higher threshold and a lower leak, which tends to increase SR Ca load. At the myocyte level, homogeneously large or small RyR clusters limit Ca waves (because of low load for large clusters but low excitability for small clusters). Mixtures of large and small RyR clusters potentiates Ca waves because the enhanced SR Ca load driven by smaller clusters enables Ca wave initiation and propagation from larger RyR clusters. Our study suggests that a spatially heterogeneous distribution of RyR cluster size under pathological conditions may potentiate Ca waves and thus afterdepolarizations and triggered arrhythmias.

Computational Model of Calcium Signaling in Cardiac Atrial Cells at the Submicron Scale

In cardiac cells, calcium is the mediator of excitation-contraction coupling. Dysfunctions in calcium handling have been identified as the origin of some cardiac arrhythmias. In the particular case of atrial myocytes, recent available experimental data has found links between these dysfunctions and structural changes in the calcium handling machinery (ryanodine cluster size and distribution, t-tubular network, etc). To address this issue, we have developed a computational model of an atrial myocyte that takes into account the detailed intracellular structure. The homogenized macroscopic behavior is described with a two-concentration field model, using effective diffusion coefficients of calcium in the sarcoplasmic reticulum (SR) and in the cytoplasm. The model reproduces the right calcium transients and dependence with pacing frequency. Under basal conditions, the calcium rise is mostly restricted to the periphery of the cell, with a large concentration ratio between the periphery and the interior. We have then studied the dependence of the speed of the calcium wave on cytosolic and SR diffusion coefficients, finding an almost linear relation with the former, in agreement with a diffusive and fire mechanism of propagation, and little dependence on the latter. Finally, we have studied the effect of a change in RyR cluster microstructure. We find that, under resting conditions, the spark frequency decreases slightly with RyR cluster spatial dispersion, but markedly increases when the RyRs are distributed in clusters of larger size, stressing the importance of RyR cluster organization to understand atrial arrhythmias, as recent experimental results suggest (Macquaide et al., 2015).

Shining New Light on the Structural Determinants of Cardiac Couplon Function: Insights From Ten Years of Nanoscale Microscopy

Remodelling of the membranes and protein clustering patterns during the pathogenesis of cardiomyopathies has renewed the interest in spatial visualisation of these structures in cardiomyocytes. Coincidental emergence of single molecule (super-resolution) imaging and tomographic electron microscopy tools in the last decade have led to a number of new observations on the structural features of the couplons, the primary sites of excitation-contraction coupling in the heart. In particular, super-resolution and tomographic electron micrographs have revised and refined the classical views of the nanoscale geometries of couplons, t-tubules and the organisation of the principal calcium handling proteins in both healthy and failing hearts. These methods have also allowed the visualisation of some features which were too small to be detected with conventional microscopy tools. With new analytical capabilities such as single-protein mapping, in situ protein quantification, correlative and live cell imaging we are now observing an unprecedented interest in adapting these research tools across the cardiac biophysical research discipline. In this article, we review the depth of the new insights that have been enabled by these tools toward understanding the structure and function of the cardiac couplon. We outline the major challenges that remain in these experiments and emerging avenues of research which will be enabled by these technologies.

Beta1-adrenoceptor antagonist, metoprolol attenuates cardiac myocyte Ca2+ handling dysfunction in rats with pulmonary artery hypertension

Right heart failure is the major cause of death in Pulmonary Artery Hypertension (PAH) patients but is not a current, specific therapeutic target. Pre-clinical studies have shown that adrenoceptor blockade can improve cardiac function but the mechanisms of action within right ventricular (RV) myocytes are unknown. We tested whether the β1-adrenoceptor blocker metoprolol could improve RV myocyte function in an animal model of PAH, by attenuating adverse excitation-contraction coupling remodeling. PAH with RV failure was induced in rats by monocrotaline injection. When PAH was established, animals were given 10 mg/kg/day metoprolol (MCT + BB) or vehicle (MCT). The median time to the onset of heart failure signs was delayed from 23 days (MCT), to 31 days (MCT + BB). At 23 ± 1 days post-injection, MCT + BB showed improved in vivo cardiac function, measured by echocardiography. RV hypertrophy was reduced despite persistent elevated afterload. RV myocyte contractility during field stimulation was improved at higher pacing frequencies in MCT + BB. Preserved t-tubule structure, more uniform evoked Ca2+ release, increased SERCA2a expression and faster ventricular repolarization (measured in vivo by telemetry) may account for the improved contractile function. Sarcoplasmic reticulum Ca2+ overload was prevented in MCT + BB myocytes resulting in fewer spontaneous Ca2+ waves, with a lower pro-arrhythmic potential. Our novel finding of attenuation of defects in excitation contraction coupling by β1-adrenoceptor blockade with delays in the onset of HF, identifies the RV as a promising therapeutic target in PAH. Moreover, our data suggest existing therapies for left ventricular failure may also be beneficial in PAH induced RV failure.

Stabilization of Ca2+ signaling in cardiac muscle by stimulation of SERCA

AIMS

In cardiac muscle, phosphorylation of the RyRs is proposed to increase their Ca²⁺ sensitivity. This mechanism could be arrhythmogenic via facilitation of spontaneous Ca²⁺ waves. Surprisingly, the level of Ca²⁺ inside the SR needed to initiate such waves has been reported to increase upon β-adrenergic stimulation, an observation which cannot be easily reconciled with elevated Ca²⁺ sensitivity of the RyRs. We tested the hypothesis that this change of Ca²⁺ wave threshold could occur indirectly, subsequent to SERCA stimulation.

METHODS AND RESULTS

Cytosolic and intra-SR Ca²⁺ waves were simultaneously recorded with confocal line-scan imaging in intact and permeabilized mouse cardiomyocytes using Rhod-2 and Fluo-5-N, respectively. We analyzed changes of several Ca²⁺ signaling parameters during specific SERCA stimulation by ochratoxin A (OTA), jasmonate or the Fab fragment of a phospholamban antibody. SERCA stimulation resulted in a substantial increase of the threshold for Ca²⁺ wave initiation. Faster Ca²⁺ transient decay and SR refilling confirmed SERCA acceleration.

CONCLUSIONS

These results suggest that isolated SERCA stimulation can elevate the intra-SR threshold for the generation of Ca²⁺ waves, independently of RyR phosphorylation. Simultaneously, fractional Ca²⁺ release and wave amplitudes are reduced. Thus, SERCA stimulation appears to exert a negative feed-back on the Ca²⁺-induced Ca²⁺ release mechanisms sustaining the waves. Thereby, it may be profoundly antiarrhythmic. This may be clinically relevant when therapies are applied to stimulate the SERCA activity (e.g. SERCA overexpression with gene therapy, future small molecule SERCA stimulators).

Effects of rogue ryanodine receptors on Ca2+ sparks in cardiac myocytes

Ca2+ sparks and Ca2+ quarks, arising from clustered and rogue ryanodine receptors (RyRs), are significant Ca2+ release events from the junctional sarcoplasmic reticulum (JSR). Based on the anomalous subdiffusion of Ca2+ in the cytoplasm, a mathematical model was developed to investigate the effects of rogue RyRs on Ca2+ sparks in cardiac myocytes. Ca2+ quarks and sparks from the stochastic opening of rogue and clustered RyRs are numerically reproduced and agree with experimental measurements. It is found that the stochastic opening Ca2+ release units (CRUs) of clustered RyRs are regulated by free Ca2+ concentration in the JSR lumen (i.e. [Ca2+]lumen). The frequency of spontaneous Ca2+ sparks is remarkably increased by the rogue RyRs opening at high [Ca2+]lumen, but not at low [Ca2+]lumen. Hence, the opening of rogue RyRs contributes to the formation of Ca2+ sparks at high [Ca2+]lumen. The interplay of Ca2+ sparks and Ca2+ quarks has been discussed in detail. This work is of significance to provide insight into understanding Ca2+ release mechanisms in cardiac myocytes.

Transverse tubular network structures in the genesis of intracellular calcium alternans and triggered activity in cardiac cells

RATIONALE

The major role of a transverse-tubular (TT) network in a cardiac cell is to facilitate effective excitation-contraction coupling and signaling. The TT network structures are heterogeneous within a single cell, and vary between different types of cells and species. They are also remodeled in cardiac diseases. However, how different TT network structures predispose cardiac cells to arrhythmogenesis remains to be revealed.

OBJECTIVE

To systematically investigate the roles of TT network structure and the underlying mechanisms in the genesis of intracellular calcium (Ca2+) alternans and triggered activity (TA).

METHODS AND RESULTS

Based on recent experimental observations, different TT network structures, including uniformly and non-uniformly random TT distributions, were modeled in a cardiac cell model consisting of a three-dimensional network of Ca2+ release units (CRUs). Our simulations showed that both Ca2+ alternans and Ca2+ wave-mediated TA were promoted when the fraction of orphaned CRUs was in an intermediate range, but suppressed in cells exhibiting either well-organized TT networks or low TT densities. Ca2+ alternans and TA could be promoted by low TT densities when the cells were small or the CRU coupling was strong. Both alternans and TA occurred more easily in uniformly random TT networks than in non-uniformly random TT networks. Subcellular spatially discordant Ca2+ alternans was promoted by non-uniformly random TT networks but suppressed by increasing CRU coupling strength. These mechanistic insights provide a holistic understanding of the effects of TT network structure on the susceptibility to arrhythmogenesis.

CONCLUSIONS

The TT network plays important roles in promoting Ca2+ alternans and TA, and different TT network structures may predispose cardiac cells differently to arrhythmogenesis.

A computational model of spatio-temporal cardiac intracellular calcium handling with realistic structure and spatial flux distribution from sarcoplasmic reticulum and t-tubule reconstructions

Intracellular calcium cycling is a vital component of cardiac excitation-contraction coupling. The key structures responsible for controlling calcium dynamics are the cell membrane (comprising the surface sarcolemma and transverse-tubules), the intracellular calcium store (the sarcoplasmic reticulum), and the co-localisation of these two structures to form dyads within which calcium-induced-calcium-release occurs. The organisation of these structures tightly controls intracellular calcium dynamics. In this study, we present a computational model of intracellular calcium cycling in three-dimensions (3-D), which incorporates high resolution reconstructions of these key regulatory structures, attained through imaging of tissue taken from the sheep left ventricle using serial block face scanning electron microscopy. An approach was developed to model the sarcoplasmic reticulum structure at the whole-cell scale, by reducing its full 3-D structure to a 3-D network of one-dimensional strands. The model reproduces intracellular calcium dynamics during control pacing and reveals the high-resolution 3-D spatial structure of calcium gradients and intracellular fluxes in both the cytoplasm and sarcoplasmic reticulum. We also demonstrated the capability of the model to reproduce potentially pro-arrhythmic dynamics under perturbed conditions, pertaining to calcium-transient alternans and spontaneous release events. Comparison with idealised cell models emphasised the importance of structure in determining calcium gradients and controlling the spatial dynamics associated with calcium-transient alternans, wherein the probabilistic nature of dyad activation and recruitment was constrained. The model was further used to highlight the criticality in calcium spark propagation in relation to inter-dyad distances. The model presented provides a powerful tool for future investigation of structure-function relationships underlying physiological and pathophysiological intracellular calcium handling phenomena at the whole-cell. The approach allows for the first time direct integration of high-resolution images of 3-D intracellular structures with models of calcium cycling, presenting the possibility to directly assess the functional impact of structural remodelling at the cellular scale.

The organisation of the membrane and sub-cellular structures of cells in the heart closely controls the coupling between its electrical and mechanical function. Computational models of the cellular calcium handling system, which is responsible for this electro-mechanical coupling, have been developed in recent years to study underlying structure-function relationships. Previous models have been largely idealised in structure; we present a new model which incorporates experimental data describing the high-resolution organisation of the primary structures involved in calcium dynamics. Significantly, the structure of the intracellular calcium store is modelled for the first time. The model is shown to reproduce calcium dynamics in control cells in both normal and abnormal conditions, demonstrating its suitability for future investigation of structure-function relationships. Thus, the model presented provides a powerful tool for the direct integration of experimentally acquired structural data in healthy and diseased cells and assessment of the role of structure in regulating normal and abnormal calcium dynamics.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Mechano-chemo-transduction in cardiac myocytes

The heart has the ability to adjust to changing mechanical loads. The Frank-Starling law and the Anrep effect describe exquisite intrinsic mechanisms the heart has for autoregulating the force of contraction to maintain cardiac output under changes of preload and afterload. Although these mechanisms have been known for more than a century, their cellular and molecular underpinnings are still debated. How does the cardiac myocyte sense changes in preload or afterload? How does the myocyte adjust its response to compensate for such changes? In cardiac myocytes Ca²⁺ is a crucial regulator of contractile force and in this review we compare and contrast recent studies from different labs that address these two important questions. The 'dimensionality' of the mechanical milieu under which experiments are carried out provide important clues to the location of the mechanosensors and the kinds of mechanical forces they can sense and respond to. As a first approximation, sensors inside the myocyte appear to modulate reactive oxygen species while sensors on the cell surface appear to also modulate nitric oxide signalling; both signalling pathways affect Ca²⁺ handling. Undoubtedly, further studies will add layers to this simplified picture. Clarifying the intimate links from cellular mechanics to reactive oxygen species and nitric oxide signalling and to Ca²⁺ handling will deepen our understanding of the Frank-Starling law and the Anrep effect, and also provide a unified view on how arrhythmias may arise in seemingly disparate diseases that have in common altered myocyte mechanics.

Ryanodine receptor gating controls generation of diastolic calcium waves in cardiac myocytes

Calcium waves can form and propagate at low frequencies of spontaneous calcium sparks if the calcium dependence of spark frequency is sufficiently steep, or the number of open RyRs is sufficiently large.

The role of cardiac ryanodine receptor (RyR) gating in the initiation and propagation of calcium waves was investigated using a mathematical model comprising a stochastic description of RyR gating and a deterministic description of calcium diffusion and sequestration. We used a one-dimensional array of equidistantly spaced RyR clusters, representing the confocal scanning line, to simulate the formation of calcium sparks. Our model provided an excellent description of the calcium dependence of the frequency of diastolic calcium sparks and of the increased tendency for the production of calcium waves after a decrease in cytosolic calcium buffering. We developed a hypothesis relating changes in the propensity to form calcium waves to changes of RyR gating and tested it by simulation. With a realistic RyR gating model, increased ability of RyR to be activated by Ca²⁺ strongly increased the propensity for generation of calcium waves at low (0.05–0.1-µM) calcium concentrations but only slightly at high (0.2–0.4-µM) calcium concentrations. Changes in RyR gating altered calcium wave formation by changing the calcium sensitivity of spontaneous calcium spark activation and/or the average number of open RyRs in spontaneous calcium sparks. Gating changes that did not affect RyR activation by Ca²⁺ had only a weak effect on the propensity to form calcium waves, even if they strongly increased calcium spark frequency. Calcium waves induced by modulating the properties of the RyR activation site could be suppressed by inhibiting the spontaneous opening of the RyR. These data can explain the increased tendency for production of calcium waves under conditions when RyR gating is altered in cardiac diseases.

Ryanodine receptor cluster fragmentation and redistribution in persistent atrial fibrillation enhance calcium release

AIMS

In atrial fibrillation (AF), abnormalities in Ca(2+) release contribute to arrhythmia generation and contractile dysfunction. We explore whether ryanodine receptor (RyR) cluster ultrastructure is altered and is associated with functional abnormalities in AF.

METHODS AND RESULTS

Using high-resolution confocal microscopy (STED), we examined RyR cluster morphology in fixed atrial myocytes from sheep with persistent AF (N = 6) and control (Ctrl; N = 6) animals. RyR clusters on average contained 15 contiguous RyRs; this did not differ between AF and Ctrl. However, the distance between clusters was significantly reduced in AF (288 ± 12 vs. 376 ± 17 nm). When RyR clusters were grouped into Ca(2+) release units (CRUs), i.e. clusters separated by <150 nm, CRUs in AF had more clusters (3.43 ± 0.10 vs. 2.95 ± 0.02 in Ctrl), which were more dispersed. Furthermore, in AF cells, more RyR clusters were found between Z lines. In parallel experiments, Ca(2+) sparks were monitored in live permeabilized myocytes. In AF, myocytes had >50% higher spark frequency with increased spark time to peak (TTP) and duration, and a higher incidence of macrosparks. A computational model of the CRU was used to simulate the morphological alterations observed in AF cells. Increasing cluster fragmentation to the level observed in AF cells caused the observed changes, i.e. higher spark frequency, increased TTP and duration; RyR clusters dispersed between Z-lines increased the occurrence of macrosparks.

CONCLUSION

In persistent AF, ultrastructural reorganization of RyR clusters within CRUs is associated with overactive Ca(2+) release, increasing the likelihood of propagating Ca(2+) release.

Examination of the Effects of Heterogeneous Organization of RyR Clusters, Myofibrils and Mitochondria on Ca2+ Release Patterns in Cardiomyocytes

Spatio-temporal dynamics of intracellular calcium, [Ca²⁺]_i, regulate the contractile function of cardiac muscle cells. Measuring [Ca²⁺]_i flux is central to the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease. However, current imaging techniques are limited in the spatial resolution to which changes in [Ca²⁺]_i can be detected. Using spatial point process statistics techniques we developed a novel method to simulate the spatial distribution of RyR clusters, which act as the major mediators of contractile Ca²⁺ release, upon a physiologically-realistic cellular landscape composed of tightly-packed mitochondria and myofibrils. We applied this method to computationally combine confocal-scale (~ 200 nm) data of RyR clusters with 3D electron microscopy data (~ 30 nm) of myofibrils and mitochondria, both collected from adult rat left ventricular myocytes. Using this hybrid-scale spatial model, we simulated reaction-diffusion of [Ca²⁺]_i during the rising phase of the transient (first 30 ms after initiation). At 30 ms, the average peak of the simulated [Ca²⁺]_i transient and of the simulated fluorescence intensity signal, F/F₀, reached values similar to that found in the literature ([Ca²⁺]_i ≈1 μM; F/F₀≈5.5). However, our model predicted the variation in [Ca²⁺]_i to be between 0.3 and 12.7 μM (~3 to 100 fold from resting value of 0.1 μM) and the corresponding F/F₀ signal ranging from 3 to 9.5. We demonstrate in this study that: (i) heterogeneities in the [Ca²⁺]_i transient are due not only to heterogeneous distribution and clustering of mitochondria; (ii) but also to heterogeneous local densities of RyR clusters. Further, we show that: (iii) these structure-induced heterogeneities in [Ca²⁺]_i can appear in line scan data. Finally, using our unique method for generating RyR cluster distributions, we demonstrate the robustness in the [Ca²⁺]_i transient to differences in RyR cluster distributions measured between rat and human cardiomyocytes.

Calcium (Ca²⁺) acts as a signal for many functions in the heart cell, from its primary role in triggering contractions during the heartbeat to acting as a signal for cell growth. Cellular function is tightly coupled to its ultra-structural organization. Spatially-realistic and biophysics-based computational models can provide quantitative insights into structure-function relationships in Ca²⁺ signaling. We developed a novel computational model of a rat ventricular myocyte that integrates structural information from confocal and electron microscopy datasets that were independently acquired and includes: myofibrils (protein complexes that contract during the heartbeat), mitochondria (organelles that provide energy for contraction), and ryanodine receptors (RyR, ion channels that release the Ca²⁺ required to trigger myofibril contraction from intracellular stores). Using this model, we examined [Ca²⁺]_i dynamics throughout the cell cross-section at a much higher resolution than previously possible. We estimated the size of structural maladaptation that would cause disease-related alterations in [Ca²⁺]_i dynamics. Using our methods for data integration, we also tested whether reducing the density of RyRs in human cardiomyocytes (~40% relative to rat) would have a significant effect on [Ca²⁺]_i. We found that Ca²⁺ release patterns between the two species are similar, suggesting Ca²⁺ dynamics are robust to variations in cell ultrastructure.

Variations in local calcium signaling in adjacent cardiac myocytes of the intact mouse heart detected with two-dimensional confocal microscopy

Dyssynchronous local Ca release within individual cardiac myocytes has been linked to cellular contractile dysfunction. Differences in Ca kinetics in adjacent cells may also provide a substrate for inefficient contraction and arrhythmias. In a new approach we quantify variation in local Ca transients between adjacent myocytes in the whole heart. Langendorff-perfused mouse hearts were loaded with Fluo-8 AM to detect Ca and Di-4-ANEPPS to visualize cell membranes. A spinning disc confocal microscope with a fast camera allowed us to record Ca signals within an area of 465 μm by 315 μm with an acquisition speed of 55 fps. Images from multiple transients recorded at steady state were registered to their time point in the cardiac cycle to restore averaged local Ca transients with a higher temporal resolution. Local Ca transients within and between adjacent myocytes were compared with regard to amplitude, time to peak and decay at steady state stimulation (250 ms cycle length). Image registration from multiple sequential Ca transients allowed reconstruction of high temporal resolution (2.4 ± 1.3 ms) local CaT in 2D image sets (N = 4 hearts, n = 8 regions). During steady state stimulation, spatial Ca gradients were homogeneous within cells in both directions and independent of distance between measured points. Variation in CaT amplitudes was similar across the short and the long side of neighboring cells. Variations in TAU and TTP were similar in both directions. Isoproterenol enhanced the CaT but not the overall pattern of spatial heterogeneities. Here we detected and analyzed local Ca signals in intact mouse hearts with high temporal and spatial resolution, taking into account 2D arrangement of the cells. We observed significant differences in the variation of CaT amplitude along the long and short axis of cardiac myocytes. Variations of Ca signals between neighboring cells may contribute to the substrate of cardiac remodeling.

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.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Dantrolene suppresses spontaneous Ca2+ release without altering excitation-contraction coupling in cardiomyocytes of aged mice

Cardiac dysfunction in the aged heart reflects abnormalities in cardiomyocyte Ca(2+) homeostasis including altered Ca(2+) cycling through the sarcoplasmic reticulum (SR). The ryanodine receptor antagonist dantrolene exerts antiarrhythmic effects by preventing spontaneous diastolic Ca(2+) release from the SR. We tested the hypothesis that dantrolene prevents spontaneous Ca(2+) release without altering excitation-contraction coupling in aged myocardium. Left ventricular cardiomyocytes isolated from young (3 to 4 mo) and aged (24-26 mo) C57BL/6 mice were loaded with the Ca(2+) indicator fluo-4. Amplitudes of action potential-induced Ca(2+) transients at 1-Hz pacing were similar between young and aged mice, yet cell shortening was impaired in aged mice. Isoproterenol (1 μM) increased Ca(2+) transient amplitude and cell shortening to identical levels in young and aged; dantrolene (1 μM) had no effect on Ca(2+) transients or cell shortening during pacing. Under Ca(2+) overload conditions induced with 10 mM extracellular Ca(2+) concentration, spontaneous Ca(2+) waves were of diminished amplitude and associated with lower SR Ca(2+) content in aged versus young mice. Despite no effect in young mice, dantrolene increased SR Ca(2+) content and Ca(2+) wave amplitude in aged mice. In the presence of isoproterenol following rest from 1-Hz pacing, Ca(2+) spark frequency was elevated in aged mice, yet the time to spontaneous Ca(2+) wave was similar between young and aged mice; dantrolene decreased Ca(2+) spark frequency and prolonged the time to Ca(2+) wave onset in aged mice with no effect in young mice. Thus dantrolene attenuates diastolic Ca(2+) release in the aged murine heart that may prove useful in preventing cardiac dysfunction.

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Hierarchical clustering of ryanodine receptors enables emergence of a calcium clock in sinoatrial node cells

The sinoatrial node, whose cells (sinoatrial node cells [SANCs]) generate rhythmic action potentials, is the primary pacemaker of the heart. During diastole, calcium released from the sarcoplasmic reticulum (SR) via ryanodine receptors (RyRs) interacts with membrane currents to control the rate of the heartbeat. This "calcium clock" takes the form of stochastic, partially periodic, localized calcium release (LCR) events that propagate, wave-like, for limited distances. The detailed mechanisms controlling the calcium clock are not understood. We constructed a computational model of SANCs, including three-dimensional diffusion and buffering of calcium in the cytosol and SR; explicit, stochastic gating of individual RyRs and L-type calcium channels; and a full complement of voltage- and calcium-dependent membrane currents. We did not include an anatomical submembrane space or inactivation of RyRs, the two heuristic components that have been used in prior models but are not observed experimentally. When RyRs were distributed in discrete clusters separated by >1 µm, only isolated sparks were produced in this model and LCR events did not form. However, immunofluorescent staining of SANCs for RyR revealed the presence of bridging RyR groups between large clusters, forming an irregular network. Incorporation of this architecture into the model led to the generation of propagating LCR events. Partial periodicity emerged from the interaction of LCR events, as observed experimentally. This calcium clock becomes entrained with membrane currents to accelerate the beating rate, which therefore was controlled by the activity of the SERCA pump, RyR sensitivity, and L-type current amplitude, all of which are targets of β-adrenergic-mediated phosphorylation. Unexpectedly, simulations revealed the existence of a pathological mode at high RyR sensitivity to calcium, in which the calcium clock loses synchronization with the membrane, resulting in a paradoxical decrease in beating rate in response to β-adrenergic stimulation. The model indicates that the hierarchical clustering of surface RyRs in SANCs may be a crucial adaptive mechanism. Pathological desynchronization of the clocks may explain sinus node dysfunction in heart failure and RyR mutations.

Parallel acceleration for modeling of calcium dynamics in cardiac myocytes

Spatial-temporal calcium dynamics due to calcium release, buffering, and re-uptaking plays a central role in studying excitation-contraction (E-C) coupling in both healthy and defected cardiac myocytes. In our previous work, partial differential equations (PDEs) had been used to simulate calcium dynamics with realistic geometries extracted from electron microscopic imaging data. However, the computational costs of such simulations are very high on a single processor. To alleviate this problem, we have accelerated the numerical simulations of calcium dynamics by using graphics processing units (GPUs). Computational performance and simulation accuracy are compared with those based on a single CPU and another popular parallel computing technique, OpenMP.

Synchrony of cardiomyocyte Ca(2+) release is controlled by T-tubule organization, SR Ca(2+) content, and ryanodine receptor Ca(2+) sensitivity

Recent work has demonstrated that cardiomyocyte Ca(2+)release is desynchronized in several pathological conditions. Loss of Ca(2+) release synchrony has been attributed to t-tubule disruption, but it is unknown if other factors also contribute. We investigated this issue in normal and failing myocytes by integrating experimental data with a mathematical model describing spatiotemporal dynamics of Ca(2+) in the cytosol and sarcoplasmic reticulum (SR). Heart failure development in postinfarction mice was associated with progressive t-tubule disorganization, as quantified by fast-Fourier transforms. Data from fast-Fourier transforms were then incorporated in the model as a dyadic organization index, reflecting the proportion of ryanodine receptors located in dyads. With decreasing dyadic-organization index, the model predicted greater dyssynchrony of Ca(2+) release, which exceeded that observed in experimental line-scan images. Model and experiment were reconciled by reducing the threshold for Ca(2+) release in the model, suggesting that increased RyR sensitivity partially offsets the desynchronizing effects of t-tubule disruption in heart failure. Reducing the magnitude of SR Ca(2+) content and release, whether experimentally by thapsigargin treatment, or in the model, desynchronized the Ca(2+) transient. However, in cardiomyocytes isolated from SERCA2 knockout mice, RyR sensitization offset such effects. A similar interplay between RyR sensitivity and SR content was observed during treatment of myocytes with low-dose caffeine. Initial synchronization of Ca(2+) release during caffeine was reversed as SR content declined due to enhanced RyR leak. Thus, synchrony of cardiomyocyte Ca(2+) release is not only determined by t-tubule organization but also by the interplay between RyR sensitivity and SR Ca(2+) content.

Sensitivity of rabbit ventricular action potential and Ca²⁺ dynamics to small variations in membrane currents and ion diffusion coefficients

Little is known about how small variations in ionic currents and Ca²⁺ and Na⁺ diffusion coefficients impact action potential and Ca²⁺ dynamics in rabbit ventricular myocytes. We applied sensitivity analysis to quantify the sensitivity of Shannon et al. model (Biophys. J., 2004) to 5%-10% changes in currents conductance, channels distribution, and ion diffusion in rabbit ventricular cells. We found that action potential duration and Ca²⁺ peaks are highly sensitive to 10% increase in L-type Ca²⁺ current; moderately influenced by 10% increase in Na⁺-Ca²⁺ exchanger, Na⁺-K⁺ pump, rapid delayed and slow transient outward K⁺ currents, and Cl⁻ background current; insensitive to 10% increases in all other ionic currents and sarcoplasmic reticulum Ca²⁺ fluxes. Cell electrical activity is strongly affected by 5% shift of L-type Ca²⁺ channels and Na⁺-Ca²⁺ exchanger in between junctional and submembrane spaces while Ca²⁺-activated Cl⁻-channel redistribution has the modest effect. Small changes in submembrane and cytosolic diffusion coefficients for Ca²⁺, but not in Na⁺ transfer, may alter notably myocyte contraction. Our studies highlight the need for more precise measurements and further extending and testing of the Shannon et al. model. Our results demonstrate usefulness of sensitivity analysis to identify specific knowledge gaps and controversies related to ventricular cell electrophysiology and Ca²⁺ signaling.

The emergence of subcellular pacemaker sites for calcium waves and oscillations

Calcium (Ca(2+)) waves generating oscillatory Ca(2+) signals are widely observed in biological cells. Experimental studies have shown that under certain conditions, initiation of Ca(2+) waves is random in space and time, while under other conditions, waves occur repetitively from preferred locations (pacemaker sites) from which they entrain the whole cell. In this study, we use computer simulations to investigate the self-organization of Ca(2+) sparks into pacemaker sites generating Ca(2+) oscillations. In both ventricular myocyte experiments and computer simulations of a heterogeneous Ca(2+) release unit (CRU) network model, we show that Ca(2+) waves occur randomly in space and time when the Ca(2+) level is low, but as the Ca(2+) level increases, waves occur repetitively from the same sites. Our analysis indicates that this transition to entrainment can be attributed to the fact that random Ca(2+) sparks self-organize into Ca(2+) oscillations differently at low and high Ca(2+) levels. At low Ca(2+), the whole cell Ca(2+) oscillation frequency of the coupled CRU system is much slower than that of an isolated single CRU. Compared to a single CRU, the distribution of interspike intervals (ISIs) of the coupled CRU network exhibits a greater variation, and its ISI distribution is asymmetric with respect to the peak, exhibiting a fat tail. At high Ca(2+), however, the coupled CRU network has a faster frequency and lesser ISI variation compared to an individual CRU. The ISI distribution of the coupled network no longer exhibits a fat tail and is well-approximated by a Gaussian distribution. This same Ca(2+) oscillation behaviour can also be achieved by varying the number of ryanodine receptors per CRU or the distance between CRUs. Using these results, we develop a theory for the entrainment of random oscillators which provides a unified explanation for the experimental observations underlying the emergence of pacemaker sites and Ca(2+) oscillations.

New software developments for quality mesh generation and optimization from biomedical imaging data

In this paper we present a new software toolkit for generating and optimizing surface and volumetric meshes from three-dimensional (3D) biomedical imaging data, targeted at image-based finite element analysis of some biomedical activities in a single material domain. Our toolkit includes a series of geometric processing algorithms including surface re-meshing and quality-guaranteed tetrahedral mesh generation and optimization. All methods described have been encapsulated into a user-friendly graphical interface for easy manipulation and informative visualization of biomedical images and mesh models. Numerous examples are presented to demonstrate the effectiveness and efficiency of the described methods and toolkit.

Ca²⁺ waves in the heart

Ca(2+) waves were probably first observed in the early 1940s. Since then Ca(2+) waves have captured the attention of an eclectic mixture of mathematicians, neuroscientists, muscle physiologists, developmental biologists, and clinical cardiologists. This review discusses the current state of mathematical models of Ca(2+) waves, the normal physiological functions Ca(2+) waves might serve in cardiac cells, as well as how the spatial arrangement of Ca(2+) release channels shape Ca(2+) waves, and we introduce the idea of Ca(2+) phase waves that might provide a useful framework for understanding triggered arrhythmias.

Calcium alternans in cardiac myocytes: order from disorder

Calcium alternans is associated with T-wave alternans and pulsus alternans, harbingers of increased mortality in the setting of heart disease. Recent experimental, computational, and theoretical studies have led to new insights into the mechanisms of Ca alternans, specifically how disordered behaviors dominated by stochastic processes at the subcellular level become organized into ordered periodic behaviors. In this article, we summarize the recent progress in this area, outlining a holistic theoretical framework in which the complex effects of Ca cycling proteins on Ca alternans are linked to three key properties of the cardiac Ca cycling network: randomness, refractoriness, and recruitment. We also illustrate how this '3R theory' can reconcile many seemingly contradictory experimental observations.

Identification and characterization of calcium sparks in cardiomyocytes derived from human induced pluripotent stem cells

INTRODUCTION

Ca2+ spark constitutes the elementary units of cardiac excitation-contraction (E-C) coupling in mature cardiomyocytes. Human induced pluripotent stem cell (hiPSC)-derived cardiomyocytes are known to have electrophysiological properties similar to mature adult cardiomyocytes. However, it is unclear if they share similar calcium handling property. We hypothesized that Ca2+ sparks in human induced pluripotent stem cell (hiPSCs)-derived cardiomyocytes (hiPSC-CMs) may display unique structural and functional properties than mature adult cardiomyocytes.

METHODS AND RESULTS

Ca2+ sparks in hiPSC-CMs were recorded with Ca2+ imaging assay with confocal laser scanning microscopy. Those sparks were stochastic with a tendency of repetitive occurrence at the same site. Nevertheless, the spatial-temporal properties of Ca2+ spark were analogous to that of adult CMs. Inhibition of L-type Ca2+ channels by nifedipine caused a 61% reduction in calcium spark frequency without affecting amplitude of those sparks and magnitude of caffeine releasable sarcoplasmic reticulum (SR) Ca2+ content. In contrast, high extracellular Ca2+ and ryanodine increased the frequency, full width at half maximum (FWHM) and full duration at half maximum (FDHM) of spontaneous Ca2+ sparks.

CONCLUSIONS

For the first time, spontaneous Ca2+ sparks were detected in hiPSC-CMs. The Ca2+ sparks are predominately triggered by L-type Ca2+ channels mediated Ca2+ influx, which is comparable to sparks detected in adult ventricular myocytes in which cardiac E-C coupling was governed by a Ca2+-induced Ca2+ release (CICR) mechanism. However, focal repetitive sparks originated from the same intracellular organelle could reflect an immature status of the hiPSC-CMs.

Adding a new dimension to cardiac nano-architecture using electron microscopy: coupling membrane excitation to calcium signaling

Advances in microscopic imaging technologies and associated computational methods now allow descriptions of cellular anatomy to go beyond 2-dimensions, revealing new micro-domain dynamics at unprecedented resolutions. In cardiomyocytes, electron microscopy (EM) first described junctional membrane complexes between the sarcolemma and sarcoplasmic reticulum over a half-century ago. Since then, 3-dimensional EM technologies such as electron tomography have become successful in determining the realistic nano-geometry of membrane junctions (dyads and peripheral junctions) and associated structures such as transverse tubules (T-tubules, aka. T-system). Concomitantly, super-resolution light microscopy has gone beyond the diffraction-limit to determine the distribution of molecules, such as ryanodine receptors, with 10(-8) meter (10nm) order accuracy. This review provides the current structural perspective and functional interpretation of membrane junction complexes, which are the central machinery controlling cardiac excitation-contraction coupling via calcium signaling.

Do calcium waves propagate between cells and synchronize alternating calcium release in rat ventricular myocytes?

The aim was to investigate the propagation of Ca(2+) waves between cells and determine whether this synchronizes alternating Ca(2+) release between cells. Experiments were carried out on electrically coupled cell pairs; spontaneous Ca(2+) waves were produced by elevating external Ca(2+). There was a significant difference in the ability of these waves to propagate between cells depending on the orientation of the pairs. Although almost all pairs connected by side-to-side contacts showed propagating Ca(2+) release, this was very uncommon in end-to-end cell pairs. Confocal studies showed that there was a gap at the intercalated disc consisting of cell membranes and a region of cytoplasm devoid of sarcoplasmic reticulum. This gap was 2.3 μm in length and is suggested to interfere with Ca(2+) wave propagation. The gap measured was much smaller between side-to-side contacts: 1.5 μm and so much less likely to interfere with propagation. Subsequent experiments investigated the synchronization between cells of Ca(2+) alternans produced by small depolarizing pulses. Although this alternation results from beat-to-beat alternation of intracellular Ca(2+) wave propagation, there was no evidence that propagation of Ca(2+) waves between cells contributed to synchronization of this alternans.

Decoding myocardial Ca²⁺ signals across multiple spatial scales: a role for sensitivity analysis

Numerous studies have employed mathematical modeling to quantitatively understand release of Ca(2+) from the sarcoplasmic reticulum (SR) in the heart. Models have been used to investigate physiologically important phenomena such as triggering of SR Ca(2+) release by Ca(2+) entry across the cell membrane and spontaneous leak of Ca(2+) from the SR in quiescent heart cells. In this review we summarize studies that have modeled myocardial Ca(2+) at different spatial scales: the sub-cellular level, the cellular level, and the multicellular level. We discuss each category of models from the standpoint of parameter sensitivity analysis, a common simulation procedure that can generate quantitative, comprehensive predictions about how changes in conditions influence model output. We propose that this is a useful perspective for conceptualizing models, in part because a sensitivity analysis requires the investigator to define the relevant parameters and model outputs. This procedure therefore helps to illustrate the capabilities and limitations of each model. We further suggest that in future studies, sensitivity analyses will aid in simplifying complex models and in suggesting experiments to differentiate between competing models built with different assumptions. We conclude with a discussion of unresolved questions that are likely to be addressed over the next several years.

β-Adrenergic stimulation increases the intra-sarcoplasmic reticulum Ca2+ threshold for Ca2+ wave generation

β-Adrenergic signalling induces positive inotropic effects on the heart that associate with pro-arrhythmic spontaneous Ca(2+) waves. A threshold level of sarcoplasmic reticulum (SR) Ca(2+) ([Ca(2+)](SR)) is necessary to trigger Ca(2+) waves, and whether the increased incidence of Ca(2+) waves during β-adrenergic stimulation is due to an alteration in this threshold remains controversial. Using the low-affinity Ca(2+) indicator fluo-5N entrapped within the SR of rabbit ventricular myocytes, we addressed this controversy by directly monitoring [Ca(2+)](SR) and Ca(2+) waves during β-adrenergic stimulation. Electrical pacing in elevated extracellular Ca(2+) ([Ca(2+)](o) = 7 mM) was used to increase [Ca(2+)](SR) to the threshold where Ca(2+) waves were consistently observed. The β-adrenergic agonist isoproterenol (ISO; 1 μM) increased [Ca(2+)](SR) well above the control threshold and consistently triggered Ca(2+) waves. However, when [Ca(2+)](SR) was subsequently lowered in the presence of ISO (by lowering [Ca(2+)](o) to 1 mM and partially inhibiting sarcoplasmic/endoplasmic reticulum calcium ATPase with cyclopiazonic acid or thapsigargin), Ca(2+) waves ceased to occur at a [Ca(2+)](SR) that was higher than the control threshold. Furthermore, for a set [Ca(2+)](SR) level the refractoriness of wave occurrence (Ca(2+) wave latency) was prolonged during β-adrenergic stimulation, and was highly dependent on the extent that [Ca](SR) exceeded the wave threshold. These data show that acute β-adrenergic stimulation increases the [Ca(2+)](SR) threshold for Ca(2+) waves, and therefore the primary cause of Ca(2+) waves is the robust increase in [Ca(2+)](SR) above this higher threshold level. Elevation of the [Ca(2+)](SR) wave threshold and prolongation of wave latency represent potentially protective mechanisms against pro-arrhythmogenic Ca(2+) release during β-adrenergic stimulation.

Modeling effects of L-type ca(2+) current and na(+)-ca(2+) exchanger on ca(2+) trigger flux in rabbit myocytes with realistic T-tubule geometries

The transverse tubular system of rabbit ventricular myocytes consists of cell membrane invaginations (t-tubules) that are essential for efficient cardiac excitation-contraction coupling. In this study, we investigate how t-tubule micro-anatomy, L-type Ca²⁺ channel (LCC) clustering, and allosteric activation of Na⁺/Ca²⁺ exchanger by L-type Ca²⁺ current affects intracellular Ca²⁺ dynamics. Our model includes a realistic 3D geometry of a single t-tubule and its surrounding half-sarcomeres for rabbit ventricular myocytes. The effects of spatially distributed membrane ion-transporters (LCC, Na⁺/Ca²⁺ exchanger, sarcolemmal Ca²⁺ pump, and sarcolemmal Ca²⁺ leak), and stationary and mobile Ca²⁺ buffers (troponin C, ATP, calmodulin, and Fluo-3) are also considered. We used a coupled reaction-diffusion system to describe the spatio-temporal concentration profiles of free and buffered intracellular Ca²⁺. We obtained parameters from voltage-clamp protocols of L-type Ca²⁺ current and line-scan recordings of Ca²⁺ concentration profiles in rabbit cells, in which the sarcoplasmic reticulum is disabled. Our model results agree with experimental measurements of global Ca²⁺ transient in myocytes loaded with 50 μM Fluo-3. We found that local Ca²⁺ concentrations within the cytosol and sub-sarcolemma, as well as the local trigger fluxes of Ca²⁺ crossing the cell membrane, are sensitive to details of t-tubule micro-structure and membrane Ca²⁺ flux distribution. The model additionally predicts that local Ca²⁺ trigger fluxes are at least threefold to eightfold higher than the whole-cell Ca²⁺ trigger flux. We found also that the activation of allosteric Ca²⁺-binding sites on the Na⁺/Ca²⁺ exchanger could provide a mechanism for regulating global and local Ca²⁺ trigger fluxes in vivo. Our studies indicate that improved structural and functional models could improve our understanding of the contributions of L-type and Na⁺/Ca²⁺ exchanger fluxes to intracellular Ca²⁺ dynamics.

Criticality in intracellular calcium signaling in cardiac myocytes

Calcium (Ca) is a ubiquitous second messenger that regulates many biological functions. The elementary events of local Ca signaling are Ca sparks, which occur randomly in time and space, and integrate to produce global signaling events such as intra- and intercellular Ca waves and whole-cell Ca oscillations. Despite extensive experimental characterization in many systems, the transition from local random to global synchronous events is still poorly understood. Here we show that criticality, a ubiquitous dynamical phenomenon in nature, is responsible for the transition from local to global Ca signaling. We demonstrate this first in a computational model of Ca signaling in a cardiac myocyte and then experimentally in mouse ventricular myocytes, complemented by a theoretical agent-based model to delineate the underlying dynamics. We show that the interaction between the Ca release units via Ca-induced Ca release causes self-organization of Ca spark clusters. When the coupling between Ca release units is weak, the cluster-size distribution is exponential. As the interactions become strong, the cluster-size distribution changes to a power-law distribution, which is characteristic of criticality in thermodynamic and complex nonlinear systems, and facilitates the formation and propagation of Ca waves and whole-cell Ca oscillations. Our findings illustrate how criticality is harnessed by a biological cell to regulate Ca signaling via self-organization of random subcellular events into cellular-scale oscillations, and provide a general theoretical framework for the transition from local Ca signaling to global Ca signaling in biological cells.

Calcium alternans in a couplon network model of ventricular myocytes: role of sarcoplasmic reticulum load

Intracellular calcium (Ca) alternans in cardiac myocytes have been shown in many experimental studies, and the mechanisms remain incompletely understood. We recently developed a "3R theory" that links Ca sparks to whole cell Ca alternans through three critical properties: randomness of Ca sparks; recruitment of a Ca spark by neighboring Ca sparks; and refractoriness of Ca release units. In this study, we used computer simulation of a physiologically detailed mathematical model of a ventricular myocyte couplon network to study how sarcoplasmic reticulum (SR) Ca load and other physiological parameters, such as ryanodine receptor sensitivity, SR uptake rate, Na-Ca exchange strength, and Ca buffer levels affect Ca alternans in the context of 3R theory. We developed a method to calculate the parameters used in the 3R theory (i.e., the primary spark rate and the recruitment rate) from the physiologically detailed Ca cycling model and paced the model periodically to elicit Ca alternans. We show that alternans only occurs for an intermediate range of the SR Ca load, and the underlying mechanism can be explained via its effects on the 3Rs. Furthermore, we show that altering the physiological parameters not only directly changes the 3Rs but also alters the SR Ca load, having an indirect effect on the 3Rs as well. Therefore, our present study links the SR Ca load and other physiological parameters to whole cell Ca alternans through the framework of the 3R theory, providing a general mechanistic understanding of Ca alternans in ventricular myocytes.

Computational modeling and numerical methods for spatiotemporal calcium cycling in ventricular myocytes

Intracellular calcium (Ca) cycling dynamics in cardiac myocytes is regulated by a complex network of spatially distributed organelles, such as sarcoplasmic reticulum (SR), mitochondria, and myofibrils. In this study, we present a mathematical model of intracellular Ca cycling and numerical and computational methods for computer simulations. The model consists of a coupled Ca release unit (CRU) network, which includes a SR domain and a myoplasm domain. Each CRU contains 10 L-type Ca channels and 100 ryanodine receptor channels, with individual channels simulated stochastically using a variant of Gillespie’s method, modified here to handle time-dependent transition rates. Both the SR domain and the myoplasm domain in each CRU are modeled by 5 × 5 × 5 voxels to maintain proper Ca diffusion. Advanced numerical algorithms implemented on graphical processing units were used for fast computational simulations. For a myocyte containing 100 × 20 × 10 CRUs, a 1-s heart time simulation takes about 10 min of machine time on a single NVIDIA Tesla C2050. Examples of simulated Ca cycling dynamics, such as Ca sparks, Ca waves, and Ca alternans, are shown.

A localized meshless approach for modeling spatial-temporal calcium dynamics in ventricular myocytes

Spatial–temporal calcium dynamics due to calcium release, buffering and re-uptaking plays a central role in studying excitation–contraction (E–C) coupling in both normal and diseased cardiac myocytes. In this paper, we employ a meshless method, namely, the local radial basis function collocation method (LRBFCM), to model such calcium behaviors by solving a nonlinear system of reaction–diffusion partial differential equations. In particular, a simplified structural unit containing a single transverse tubule (T-tubule) and its surrounding half sarcomeres is investigated using the meshless method. Numerical results are compared with those generated by finite element methods, showing the capability and efficiency of the LRBFCM in modeling calcium dynamics in ventricular myocytes. The single T-tubule model is also extended to the whole-cell scale with T-tubules excluded to demonstrate the scalability of the proposed meshless method in handling very large domains. The experiments have shown that the LRBFCM is suitable to multiscale modeling of calcium dynamics in ventricular myocytes with high accuracy and efficiency.

Dynamic local changes in sarcoplasmic reticulum calcium: physiological and pathophysiological roles

Evidence obtained in recent years indicates that, in cardiac myocytes, release of Ca(2+) from the sarcoplasmic reticulum (SR) is regulated by changes in the concentration of Ca(2+) within the SR. In this review, we summarize recent advances in our understanding of this regulatory role, with a particular emphasis on dynamic and local changes in SR [Ca(2+)]. We focus on five important questions that are to some extent unresolved and controversial. These questions concern: (1) the importance of SR [Ca(2+)] depletion in the termination of Ca(2+) release; (2) the quantitative extent of depletion during local release events such as Ca(2+) sparks; (3) the influence of SR [Ca(2+)] refilling on release refractoriness and the propensity for pathological Ca(2+) release; (4) dynamic changes in SR [Ca(2+)] during propagating Ca(2+) waves; and (5) the speed of Ca(2+) diffusion within the SR. With each issue, we discuss data supporting alternative viewpoints, and we identify fundamental questions that are being actively investigated. We conclude with a discussion of experimental and computational advances that will help to resolve controversies. This article is part of a special issue entitled "Local Signaling in Myocytes."

Subcellular calcium dynamics in a whole-cell model of an atrial myocyte

In this study, we present an innovative mathematical modeling approach that allows detailed characterization of Ca(2+) movement within the three-dimensional volume of an atrial myocyte. Essential aspects of the model are the geometrically realistic representation of Ca(2+) release sites and physiological Ca(2+) flux parameters, coupled with a computationally inexpensive framework. By translating nonlinear Ca(2+) excitability into threshold dynamics, we avoid the computationally demanding time stepping of the partial differential equations that are often used to model Ca(2+) transport. Our approach successfully reproduces key features of atrial myocyte Ca(2+) signaling observed using confocal imaging. In particular, the model displays the centripetal Ca(2+) waves that occur within atrial myocytes during excitation-contraction coupling, and the effect of positive inotropic stimulation on the spatial profile of the Ca(2+) signals. Beyond this validation of the model, our simulation reveals unexpected observations about the spread of Ca(2+) within an atrial myocyte. In particular, the model describes the movement of Ca(2+) between ryanodine receptor clusters within a specific z disk of an atrial myocyte. Furthermore, we demonstrate that altering the strength of Ca(2+) release, ryanodine receptor refractoriness, the magnitude of initiating stimulus, or the introduction of stochastic Ca(2+) channel activity can cause the nucleation of proarrhythmic traveling Ca(2+) waves. The model provides clinically relevant insights into the initiation and propagation of subcellular Ca(2+) signals that are currently beyond the scope of imaging technology.

Multiscale modeling of calcium dynamics in ventricular myocytes with realistic transverse tubules

Spatial-temporal Ca(2+) dynamics due to Ca(2+) release, buffering, and reuptaking plays a central role in studying excitation-contraction (E-C) coupling in both normal and diseased cardiac myocytes. In this paper, we employ two numerical methods, namely, the meshless method and the finite element method, to model such Ca(2+) behaviors by solving a nonlinear system of reaction-diffusion partial differential equations at two scales. In particular, a subcellular model containing several realistic transverse tubules (or t-tubules) is investigated and assumed to reside at different locations relative to the cell membrane. To this end, the Ca(2+) concentration calculated from the whole-cell modeling is adopted as part of the boundary constraint in the subcellular model. The preliminary simulations show that Ca(2+) concentration changes in ventricular myocytes are mainly influenced by calcium release from t-tubules.