Modelling cardiac calcium sparks in a three-dimensional reconstruction of a calcium release unit

Spatial modeling algorithms for reactions and transport in biological cells

Biological cells rely on precise spatiotemporal coordination of biochemical reactions to control their functions. Such cell signaling networks have been a common focus for mathematical models, but they remain challenging to simulate, particularly in realistic cell geometries. Here we present Spatial Modeling Algorithms for Reactions and Transport (SMART), a software package that takes in high-level user specifications about cell signaling networks and then assembles and solves the associated mathematical systems. SMART uses state-of-the-art finite element analysis, via the FEniCS Project software, to efficiently and accurately resolve cell signaling events over discretized cellular and subcellular geometries. We demonstrate its application to several different biological systems, including yes-associated protein (YAP)/PDZ-binding motif (TAZ) mechanotransduction, calcium signaling in neurons and cardiomyocytes, and ATP generation in mitochondria. Throughout, we utilize experimentally derived realistic cellular geometries represented by well-conditioned tetrahedral meshes. These scenarios demonstrate the applicability, flexibility, accuracy and efficiency of SMART across a range of temporal and spatial scales.

Electrophysical cardiac remodeling at the molecular level: Insights into ryanodine receptor activation and calcium-induced calcium release from a stochastic explicit-particle model

We present the first-ever, fully discrete, stochastic model of triggered cardiac Ca2+ dynamics. Using anatomically accurate subcellular cardiac myocyte geometries, we simulate the molecular players involved in Ca2+ handling using high-resolution stochastic and explicit-particle methods at the level of an individual cardiac dyadic junction. Integrating data from multiple experimental sources, the model not only replicates the findings of traditional in silico studies and complements in vitro experimental data but also reveals new insights into the molecular mechanisms driving cardiac dysfunction under stress and disease conditions. We improve upon older, nondiscrete models using the same realistic geometry by incorporating molecular mechanisms for spontaneous, as well as triggered calcium-induced calcium release (CICR). Action potentials are used to activate L-type calcium channels (LTCC), triggering CICR through ryanodine receptors (RyRs) on the surface of the sarcoplasmic reticulum. These improvements allow for the specific focus on the couplon: the structure-function relationship between LTCC and RyR. We investigate the electrophysical effects of normal and diseased action potentials on CICR and interrogate the effects of dyadic junction deformation through detubulation and orphaning of RyR. Our work demonstrates the importance of the electrophysical integrity of the calcium release unit on CICR fidelity, giving insights into the molecular basis of heart disease. Finally, we provide a unique, detailed, molecular view of the CICR process using advanced rendering techniques. This easy-to-use model comes complete with tutorials and the necessary software for use and analysis to maximize usability and reproducibility. Our work focuses on quantifying, qualifying, and visualizing the behavior of the molecular species that underlie the function and dysfunction of subcellular cardiomyocyte systems.

Purinoreceptors and ectonucleotidases control ATP-induced calcium waveforms and calcium-dependent responses in microglia: Roles of P2 receptors and CD39 in ATP-stimulated microglia

Adenosine triphosphate (ATP) and its metabolites drive microglia migration and cytokine production by activating P2X- and P2Y- class purinergic receptors. Purinergic receptor activation gives rise to diverse intracellular calcium (Ca2+ signals, or waveforms, that differ in amplitude, duration, and frequency. Whether and how these characteristics of diverse waveforms influence microglia function is not well-established. We developed a computational model trained with data from published primary murine microglia studies. We simulate how purinoreceptors influence Ca2+ signaling and migration, as well as, how purinoreceptor expression modifies these processes. Our simulation confirmed that P2 receptors encode the amplitude and duration of the ATP-induced Ca2+ waveforms. Our simulations also implicate CD39, an ectonucleotidase that rapidly degrades ATP, as a regulator of purinergic receptor-induced Ca2+ responses. Namely, it was necessary to account for CD39 metabolism of ATP to align the model's predicted purinoreceptor responses with published experimental data. In addition, our modeling results indicate that small Ca2+ transients accompany migration, while large and sustained transients are needed for cytokine responses. Lastly, as a proof-of-principal, we predict Ca2+ transients and cell membrane displacements in a BV2 microglia cell line using published P2 receptor mRNA data to illustrate how our computer model may be extrapolated to other microglia subtypes. These findings provide important insights into how differences in purinergic receptor expression influence microglial responses to ATP.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Prolonged β-adrenergic stimulation disperses ryanodine receptor clusters in cardiomyocytes

Ryanodine receptors (RyRs) exhibit dynamic arrangements in cardiomyocytes, and we previously showed that ‘dispersion’ of RyR clusters disrupts Ca²⁺ homeostasis during heart failure (HF) (Kolstad et al., eLife, 2018). Here, we investigated whether prolonged β-adrenergic stimulation, a hallmark of HF, promotes RyR cluster dispersion and examined the underlying mechanisms. We observed that treatment of healthy rat cardiomyocytes with isoproterenol for 1 hr triggered progressive fragmentation of RyR clusters. Pharmacological inhibition of Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) reversed these effects, while cluster dispersion was reproduced by specific activation of CaMKII, and in mice with constitutively active Ser2814-RyR. A similar role of protein kinase A (PKA) in promoting RyR cluster fragmentation was established by employing PKA activation or inhibition. Progressive cluster dispersion was linked to declining Ca²⁺ spark fidelity and magnitude, and slowed release kinetics from Ca²⁺ propagation between more numerous RyR clusters. In healthy cells, this served to dampen the stimulatory actions of β-adrenergic stimulation over the longer term and protect against pro-arrhythmic Ca²⁺ waves. However, during HF, RyR dispersion was linked to impaired Ca²⁺ release. Thus, RyR localization and function are intimately linked via channel phosphorylation by both CaMKII and PKA, which, while finely tuned in healthy cardiomyocytes, underlies impaired cardiac function during pathology.

Nanoscale organization of ryanodine receptor distribution and phosphorylation pattern determines the dynamics of calcium sparks

Super-resolution imaging techniques have provided a better understanding of the relationship between the nanoscale organization and function of ryanodine receptors (RyRs) in cardiomyocytes. Recent data have indicated that this relationship is disrupted in heart failure (HF), as RyRs are dispersed into smaller and more numerous clusters. However, RyRs are also hyperphosphorylated in this condition, and this is reported to occur preferentially within the cluster centre. Thus, the combined impact of RyR relocalization and sensitization on Ca2+ spark generation in failing cardiomyocytes is likely complex and these observations suggest that both the nanoscale organization of RyRs and the pattern of phosphorylated RyRs within clusters could be critical determinants of Ca2+ spark dynamics. To test this hypothesis, we used computational modeling to quantify the relationships between RyR cluster geometry, phosphorylation patterns, and sarcoplasmic reticulum (SR) Ca2+ release. We found that RyR cluster disruption results in a decrease in spark fidelity and longer sparks with a lower amplitude. Phosphorylation of some RyRs within the cluster can play a compensatory role, recovering healthy spark dynamics. Interestingly, our model predicts that such compensation is critically dependent on the phosphorylation pattern, as phosphorylation localized within the cluster center resulted in longer Ca2+ sparks and higher spark fidelity compared to a uniformly distributed phosphorylation pattern. Our results strongly suggest that both the phosphorylation pattern and nanoscale RyR reorganization are critical determinants of Ca2+ dynamics in HF.

Multi-Scale Computational Modeling of Spatial Calcium Handling From Nanodomain to Whole-Heart: Overview and Perspectives

Regulation of intracellular calcium is a critical component of cardiac electrophysiology and excitation-contraction coupling. The calcium spark, the fundamental element of the intracellular calcium transient, is initiated in specialized nanodomains which co-locate the ryanodine receptors and L-type calcium channels. However, calcium homeostasis is ultimately regulated at the cellular scale, by the interaction of spatially separated but diffusively coupled nanodomains with other sub-cellular and surface-membrane calcium transport channels with strong non-linear interactions; and cardiac electrophysiology and arrhythmia mechanisms are ultimately tissue-scale phenomena, regulated by the interaction of a heterogeneous population of coupled myocytes. Recent advances in imaging modalities and image-analysis are enabling the super-resolution reconstruction of the structures responsible for regulating calcium homeostasis, including the internal structure of nanodomains themselves. Extrapolating functional and imaging data from the nanodomain to the whole-heart is non-trivial, yet essential for translational insight into disease mechanisms. Computational modeling has important roles to play in relating structural and functional data at the sub-cellular scale and translating data across the scales. This review covers recent methodological advances that enable image-based modeling of the single nanodomain and whole cardiomyocyte, as well as the development of multi-scale simulation approaches to integrate data from nanometer to whole-heart. Firstly, methods to overcome the computational challenges of simulating spatial calcium dynamics in the nanodomain are discussed, including image-based modeling at this scale. Then, recent whole-cell models, capable of capturing a range of different structures (such as the T-system and mitochondria) and cellular heterogeneity/variability are discussed at two different levels of discretization. Novel methods to integrate the models and data across the scales and simulate stochastic dynamics in tissue-scale models are then discussed, enabling elucidation of the mechanisms by which nanodomain remodeling underlies arrhythmia and contractile dysfunction. Perspectives on model differences and future directions are provided throughout.

Image-Driven Modeling of Nanoscopic Cardiac Function: Where Have We Come From, and Where Are We Going?

Complementary developments in microscopy and mathematical modeling have been critical to our understanding of cardiac excitation-contraction coupling. Historically, limitations imposed by the spatial or temporal resolution of imaging methods have been addressed through careful mathematical interrogation. Similarly, limitations imposed by computational power have been addressed by imaging macroscopic function in large subcellular domains or in whole myocytes. As both imaging resolution and computational tractability have improved, the two approaches have nearly merged in terms of the scales that they can each be used to interrogate. With this review we will provide an overview of these advances and their contribution to understanding ventricular myocyte function, including exciting developments over the last decade. We specifically focus on experimental methods that have pushed back limits of either spatial or temporal resolution of nanoscale imaging (e.g., DNA-PAINT), or have permitted high resolution imaging on large cellular volumes (e.g., serial scanning electron microscopy). We also review the progression of computational approaches used to integrate and interrogate these new experimental data sources, and comment on near-term advances that may unify understanding of the underlying biology. Finally, we comment on several outstanding questions in cardiac physiology that stand to benefit from a concerted and complementary application of these new experimental and computational methods.

Magnesium Ions Moderate Calcium-Induced Calcium Release in Cardiac Calcium Release Sites by Binding to Ryanodine Receptor Activation and Inhibition Sites

Ryanodine receptor channels at calcium release sites of cardiac myocytes operate on the principle of calcium-induced calcium release. In vitro experiments revealed competition of Ca2+ and Mg2+ in the activation of ryanodine receptors (RyRs) as well as inhibition of RyRs by Mg2+. The impact of RyR modulation by Mg2+ on calcium release is not well understood due to the technical limitations of in situ experiments. We turned instead to an in silico model of a calcium release site (CRS), based on a homotetrameric model of RyR gating with kinetic parameters determined from in vitro measurements. We inspected changes in the activity of the CRS model in response to a random opening of one of 20 realistically distributed RyRs, arising from Ca2+/Mg2+ interactions at RyR channels. Calcium release events (CREs) were simulated at a range of Mg2+-binding parameters at near-physiological Mg2+ and ATP concentrations. Facilitation of Mg2+ binding to the RyR activation site inhibited the formation of sparks and slowed down their activation. Impeding Mg-binding to the RyR activation site enhanced spark formation and speeded up their activation. Varying Mg2+ binding to the RyR inhibition site also dramatically affected calcium release events. Facilitation of Mg2+ binding to the RyR inhibition site reduced the amplitude, relative occurrence, and the time-to-end of sparks, and vice versa. The characteristics of CREs correlated dose-dependently with the effective coupling strength between RyRs, defined as a function of RyR vicinity, single-channel calcium current, and Mg-binding parameters of the RyR channels. These findings postulate the role of Mg2+ in calcium release as a negative modulator of the coupling strength among RyRs in a CRS, translating to damping of the positive feedback of the calcium-induced calcium-release mechanism.

A Stochastic Spatiotemporal Model of Rat Ventricular Myocyte Calcium Dynamics Demonstrated Necessary Features for Calcium Wave Propagation

Calcium (Ca2+) plays a central role in the excitation and contraction of cardiac myocytes. Experiments have indicated that calcium release is stochastic and regulated locally suggesting the possibility of spatially heterogeneous calcium levels in the cells. This spatial heterogeneity might be important in mediating different signaling pathways. During more than 50 years of computational cell biology, the computational models have been advanced to incorporate more ionic currents, going from deterministic models to stochastic models. While periodic increases in cytoplasmic Ca2+ concentration drive cardiac contraction, aberrant Ca2+ release can underly cardiac arrhythmia. However, the study of the spatial role of calcium ions has been limited due to the computational expense of using a three-dimensional stochastic computational model. In this paper, we introduce a three-dimensional stochastic computational model for rat ventricular myocytes at the whole-cell level that incorporate detailed calcium dynamics, with (1) non-uniform release site placement, (2) non-uniform membrane ionic currents and membrane buffers, (3) stochastic calcium-leak dynamics and (4) non-junctional or rogue ryanodine receptors. The model simulates spark-induced spark activation and spark-induced Ca2+ wave initiation and propagation that occur under conditions of calcium overload at the closed-cell condition, but not when Ca2+ levels are normal. This is considered important since the presence of Ca2+ waves contribute to the activation of arrhythmogenic currents.

In silico simulations reveal that RYR distribution affects the dynamics of calcium release in cardiac myocytes

The dyads of cardiac myocytes contain ryanodine receptors (RYRs) that generate calcium sparks upon activation. To test how geometric factors of RYR distribution contribute to the formation of calcium sparks, which cannot be addressed experimentally, we performed in silico simulations on a large set of models of calcium release sites (CRSs). Our models covered the observed range of RYR number, density, and spatial arrangement. The calcium release function of CRSs was modeled by RYR openings, with an open probability dependent on concentrations of free Ca2+ and Mg2+ ions, in a rapidly buffered system, with a constant open RYR calcium current. We found that simulations of spontaneous sparks by repeatedly opening one of the RYRs in a CRS produced three different types of calcium release events (CREs) in any of the models. Transformation of simulated CREs into fluorescence signals yielded calcium sparks with characteristics close to the observed ones. CRE occurrence varied broadly with the spatial distribution of RYRs in the CRS but did not consistently correlate with RYR number, surface density, or calcium current. However, it correlated with RYR coupling strength, defined as the weighted product of RYR vicinity and calcium current, so that CRE characteristics of all models followed the same state-response function. This finding revealed the synergy between structure and function of CRSs in shaping dyad function. Lastly, rearrangements of RYRs simulating hypothetical experiments on splitting and compaction of a dyad revealed an increased propensity to generate spontaneous sparks and an overall increase in calcium release in smaller and more compact dyads, thus underlying the importance and physiological role of RYR arrangement in cardiac myocytes.

Computational modeling approaches to cAMP/PKA signaling in cardiomyocytes

The cAMP/PKA pathway is a fundamental regulator of excitation-contraction coupling in cardiomyocytes. Activation of cAMP has a variety of downstream effects on cardiac function including enhanced contraction, accelerated relaxation, adaptive stress response, mitochondrial regulation, and gene transcription. Experimental advances have shed light on the compartmentation of cAMP and PKA, which allow for control over the varied targets of these second messengers and is disrupted in heart failure conditions. Computational modeling is an important tool for understanding the spatial and temporal complexities of this system. In this review article, we outline the advances in computational modeling that have allowed for deeper understanding of cAMP/PKA dynamics in the cardiomyocyte in health and disease, and explore new modeling frameworks that may bring us closer to a more complete understanding of this system. We outline various compartmental and spatial signaling models that have been used to understand how β-adrenergic signaling pathways function in a variety of simulation conditions. We also discuss newer subcellular models of cardiovascular function that may be used as templates for the next phase of computational study of cAMP and PKA in the heart, and outline open challenges which are important to consider in future models.

An Open-Source Mesh Generation Platform for Biophysical Modeling Using Realistic Cellular Geometries

Advances in imaging methods such as electron microscopy, tomography, and other modalities are enabling high-resolution reconstructions of cellular and organelle geometries. Such advances pave the way for using these geometries for biophysical and mathematical modeling once these data can be represented as a geometric mesh, which, when carefully conditioned, enables the discretization and solution of partial differential equations. In this work, we outline the steps for a naïve user to approach the Geometry-preserving Adaptive MeshER software version 2, a mesh generation code written in C++ designed to convert structural data sets to realistic geometric meshes while preserving the underlying shapes. We present two example cases: 1) mesh generation at the subcellular scale as informed by electron tomography and 2) meshing a protein with a structure from x-ray crystallography. We further demonstrate that the meshes generated by the Geometry-preserving Adaptive MeshER software are suitable for use with numerical methods. Together, this collection of libraries and tools simplifies the process of constructing realistic geometric meshes from structural biology data.

Recruiting RyRs to Open in a Ca2+ Release Unit: Single-RyR Gating Properties Make RyR Group Dynamics

In cardiac myocytes, clusters of type-2 ryanodine receptors (RyR2s) release Ca²⁺ from the sarcoplasmic reticulum (SR) via a positive feedback mechanism in which fluxed Ca²⁺ activates nearby RyRs. Although the general principles of this are understood, less is known about how single-RyR gating properties define the RyR group dynamics in an array of many channels. Here, we examine this using simulations with three models of RyR gating that have identical open probabilities: the commonly used two-state Markov gating model, one that utilizes multiple exponentials to fit single-channel open time (OT) and closed time (CT) distributions, and an extension of this multiexponential model that also includes experimentally measured correlations between single-channel OTs and CTs. The simulations of RyR clusters that utilize the multiexponential gating model produce infrequent Ca²⁺ release events with relatively few open RyRs. Ca²⁺ release events become even smaller when OT/CT correlations are included. This occurs because the correlations produce a small but consistent bias against recruiting more RyRs to open during the middle of a Ca²⁺ release event, between the initiation and termination phases (which are unaltered compared to the uncorrelated simulations). In comparison, the two-state model produces frequent, large, and long Ca²⁺ release events because it had a recruitment bias in favor of opening more RyRs. This difference stems from the two-state model's single-RyR OT and CT distributions being qualitatively different from the experimental ones. Thus, the details of single-RyR gating can profoundly affect SR Ca²⁺ release even if open probability and mean OTs and CTs are identical. We also show that Ca²⁺ release events can terminate spontaneously without any reduction in SR [Ca²⁺], luminal regulation, Ca²⁺-dependent inactivation, or physical coupling between RyRs when Ca²⁺ flux is below a threshold value. This supports and extends the pernicious attrition/induction decay hypothesis that SR Ca²⁺ release events terminate below a threshold Ca²⁺ flux.

Multiscale Modeling of Dyadic Structure-Function Relation in Ventricular Cardiac Myocytes

Cardiovascular disease is often related to defects of subcellular components in cardiac myocytes, specifically in the dyadic cleft, which include changes in cleft geometry and channel placement. Modeling of these pathological changes requires both spatially resolved cleft as well as whole cell level descriptions. We use a multiscale model to create dyadic structure-function relationships to explore the impact of molecular changes on whole cell electrophysiology and calcium cycling. This multiscale model incorporates stochastic simulation of individual L-type calcium channels and ryanodine receptor channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We found action potential duration, systolic, and diastolic [Ca²⁺] to respond most sensitively to changes in L-type calcium channel current. The ryanodine receptor channel cluster structure inside dyadic clefts was found to affect all biomarkers investigated. The shape of clusters observed in experiments by Jayasinghe et al. and channel density within the cluster (characterized by mean occupancy) showed the strongest correlation to the effects on biomarkers.

Stochastic and deterministic approaches to modelling calcium release in cardiac myocytes at different spatial arrangements of ryanodine receptors

Calcium release sites (CRSs) play a key role in excitation-contraction coupling of cardiac myocytes. Recent studies based on electron tomography and super-resolution imaging revealed that CRSs are not completely filled with ryanodine receptors (RyRs) and that the spatial arrangement of RyRs is neither uniform nor static. In this work, we studied the effect of spatial arrangement of RyRs on RyR activation using simulations based on Monte Carlo (MC) and mean-field (MF) approaches. Both approaches showed that activation of RyRs is sensitive to the arrangement of RyRs in the CRS. However, the MF simulations did not reproduce results of MC simulations for non-compact CRSs, suggesting that the approximations used in the MF approach are not suitable for simulation studies of RyRs arrangements observed experimentally. MC simulations revealed the importance of realistic spatial arrangement of RyRs for adequate modelling of calcium release in cardiac myocytes.

Calcium-Mediated Oscillation in Membrane Potentials and Atrial-Triggered Activity in Atrial Cells of Casq2R33Q/R33Q Mutation Mice

Aim: We investigated the underlying mechanisms in atrial fibrillation (AF) associated with R33Q mutation and Ca²⁺-triggered activity. Methods and Results: We examined AF susceptibility with intraesophageal burst pacing in the sarcoplasmic reticulum (SR) Ca²⁺ leak model calsequestrin 2 R33Q (Casq2^R33Q/R33Q) mice. Atrial trigger appeared in R33Q mice but not WT mice (17.24%, 5/29 vs. 0.00%, 0/32, P < 0.05). AF was induced by 25 Hz pacing in R33Q mice (48.27%, 14/29 vs. 6.25%, 2/32, P < 0.01). The mice were given 1.5 mg/kg isoproterenol (Iso), and the incidences of AF increased (65.51%, 19/29 vs. 9.21%, 3/32, P < 0.01). Electrophysiology experiments and the recording of intracellular Ca²⁺ indicated significant increases in the Ca²⁺ sparks (5.24 ± 0.75 100 μM^-1.s^-1 vs. 0.29 ± 0.04 100 μM^-1.s^-1, n = 20, P < 0.05), intracellular free Ca²⁺ (0.238 ± 0.009 μM vs. 0.172 ± 0.006 μM, n = 20, P < 0.05), Ca²⁺ wave (11.74% vs. 2.24%, n = 20, P < 0.05), transient inward current (ITi) (-0.56 ± 0.02 pA/pF vs. -0.42 ± 0.01 pA/pF, n = 10, P < 0.05), and oscillation in membrane potentials (10.71%, 3/28 vs. 4.16%, 1/24, P < 0.05) in the R33Q group, but there was no significant difference in the L-type calcium current. These effects were enhanced by Iso, and the inhibition of calmodulin-dependent protein kinase II (CaMKII) by 1 μM KN93 reversed the effects of Iso on Ca²⁺ sparks (5.01 ± 0.66 100 μm^-1.s^-1 vs. 11.33 ± 1.63 100 μm^-1.s^-1, P < 0.05), intracellular Ca²⁺ (0.245 ± 0.005 μM vs. 0.324 ± 0.008 μM, P < 0.05), Ca²⁺ wave (12.35% vs. 17.83%, P < 0.05), ITi (-0.61 ± 0.02 pA/pF vs. -0.78 ± 0.03 pA/pF, n = 10, P < 0.05), and oscillation in membrane potential (17.85% 5/28 vs. 32.17% 9/28, P < 0.05). The reduction of ryanodine receptor 2 (RyR2) stable subunits (Casq2, triadin, and junctin) rather than RYR2 and the increase in CaMKII, phosphor-CaMKII, phosphor-RyR2 (Ser 2814), SERCA, and NCX1.1 was reflected in the R33Q group. Conclusion: This study demonstrates that the increase in spontaneous calcium elevations corresponding to ITi that may trigger the oscillation in membrane potentials in the R33Q group, thereby increasing the risk of AF. The occurrence of spontaneous calcium elevations in R33Q atrial myocytes is due to the dysfunction of RyR2 stable subunits, CaMKII hyperactivity, and CaMKII-mediated RyR phosphorylation. An effective therapeutic strategy to intervene in Ca²⁺-induced AF associated with the R33Q mutation may be through CaMKII inhibition.

Ryanodine receptor dispersion disrupts Ca2+ release in failing cardiac myocytes

Reduced cardiac contractility during heart failure (HF) is linked to impaired Ca²⁺ release from Ryanodine Receptors (RyRs). We investigated whether this deficit can be traced to nanoscale RyR reorganization. Using super-resolution imaging, we observed dispersion of RyR clusters in cardiomyocytes from post-infarction HF rats, resulting in more numerous, smaller clusters. Functional groupings of RyR clusters which produce Ca²⁺ sparks (Ca²⁺ release units, CRUs) also became less solid. An increased fraction of small CRUs in HF was linked to augmented ‘silent’ Ca²⁺ leak, not visible as sparks. Larger multi-cluster CRUs common in HF also exhibited low fidelity spark generation. When successfully triggered, sparks in failing cells displayed slow kinetics as Ca²⁺ spread across dispersed CRUs. During the action potential, these slow sparks protracted and desynchronized the overall Ca²⁺ transient. Thus, nanoscale RyR reorganization during HF augments Ca²⁺ leak and slows Ca²⁺ release kinetics, leading to weakened contraction in this disease.

The muscle cells of the heart coordinate how they contract and relax in order to produce the heartbeat. During heart failure, these cells become less able to contract. As a result the heart becomes inefficient, pumping less blood around the body.

For the cardiac muscle cells to contract, the levels of calcium ions in the cells needs to rapidly increase. In failing hearts, these increases in calcium ion levels are smaller, slower and less well coordinated. It was not known what causes these changes, making it difficult to treat heart failure.

Calcium ions are released in cardiac muscle cells through protein channels called ryanodine receptors. These receptors form clusters that allow them to synchronize when they open and close. Could the reorganization of ryanodine receptors account for the problems seen in failing hearts? To investigate, Kolstad et al. examined rat hearts using a technique called super-resolution microscopy. This showed that the clusters of ryanodine receptors break apart during heart failure to form smaller clusters. Further experiments showed that calcium ions ‘leak’ from these smaller clusters, reducing the amount of calcium that can be released into cardiac muscle cells during each heartbeat. Released calcium also spreads between the dispersed clusters, resulting in a slower rise of the calcium levels in the cells. Both changes contribute to weakened contractions of cells in failing hearts.

Therefore, heart failure can be traced back to very small rearrangements of the ryanodine receptors. This understanding will help researchers as they investigate new ways to treat heart failure.

The Subcellular Distribution of Ryanodine Receptors and L-Type Ca2+ Channels Modulates Ca2+-Transient Properties and Spontaneous Ca2+-Release Events in Atrial Cardiomyocytes

Spontaneous Ca²⁺-release events (SCaEs) from the sarcoplasmic reticulum play crucial roles in the initiation of cardiac arrhythmias by promoting triggered activity. However, the subcellular determinants of these SCaEs remain incompletely understood. Structural differences between atrial and ventricular cardiomyocytes, e.g., regarding the density of T-tubular membrane invaginations, may influence cardiomyocyte Ca²⁺-handling and the distribution of cardiac ryanodine receptors (RyR2) has recently been shown to undergo remodeling in atrial fibrillation. These data suggest that the subcellular distribution of Ca²⁺-handling proteins influences proarrhythmic Ca²⁺-handling abnormalities. Here, we employ computational modeling to provide an in-depth analysis of the impact of variations in subcellular RyR2 and L-type Ca²⁺-channel distributions on Ca²⁺-transient properties and SCaEs in a human atrial cardiomyocyte model. We incorporate experimentally observed RyR2 expression patterns and various configurations of axial tubules in a previously published model of the human atrial cardiomyocyte. We identify an increased SCaE incidence for larger heterogeneity in RyR2 expression, in which SCaEs preferentially arise from regions of high local RyR2 expression. Furthermore, we show that the propagation of Ca²⁺ waves is modulated by the distance between RyR2 bands, as well as the presence of experimentally observed RyR2 clusters between bands near the lateral membranes. We also show that incorporation of axial tubules in various amounts and locations reduces Ca²⁺-transient time to peak. Furthermore, selective hyperphosphorylation of RyR2 around axial tubules increases the number of spontaneous waves. Finally, we present a novel model of the human atrial cardiomyocyte with physiological RyR2 and L-type Ca²⁺-channel distributions that reproduces experimentally observed Ca²⁺-handling properties. Taken together, these results significantly enhance our understanding of the structure-function relationship in cardiomyocytes, identifying that RyR2 and L-type Ca²⁺-channel distributions have a major impact on systolic Ca²⁺ transients and SCaEs.

Creating a Structurally Realistic Finite Element Geometric Model of a Cardiomyocyte to Study the Role of Cellular Architecture in Cardiomyocyte Systems Biology

With the advent of three-dimensional (3D) imaging technologies such as electron tomography, serial-block-face scanning electron microscopy and confocal microscopy, the scientific community has unprecedented access to large datasets at sub-micrometer resolution that characterize the architectural remodeling that accompanies changes in cardiomyocyte function in health and disease. However, these datasets have been under-utilized for investigating the role of cellular architecture remodeling in cardiomyocyte function. The purpose of this protocol is to outline how to create an accurate finite element model of a cardiomyocyte using high resolution electron microscopy and confocal microscopy images. A detailed and accurate model of cellular architecture has significant potential to provide new insights into cardiomyocyte biology, more than experiments alone can garner. The power of this method lies in its ability to computationally fuse information from two disparate imaging modalities of cardiomyocyte ultrastructure to develop one unified and detailed model of the cardiomyocyte. This protocol outlines steps to integrate electron tomography and confocal microscopy images of adult male Wistar (name for a specific breed of albino rat) rat cardiomyocytes to develop a half-sarcomere finite element model of the cardiomyocyte. The procedure generates a 3D finite element model that contains an accurate, high-resolution depiction (on the order of ~35 nm) of the distribution of mitochondria, myofibrils and ryanodine receptor clusters that release the necessary calcium for cardiomyocyte contraction from the sarcoplasmic reticular network (SR) into the myofibril and cytosolic compartment. The model generated here as an illustration does not incorporate details of the transverse-tubule architecture or the sarcoplasmic reticular network and is therefore a minimal model of the cardiomyocyte. Nevertheless, the model can already be applied in simulation-based investigations into the role of cell structure in calcium signaling and mitochondrial bioenergetics, which is illustrated and discussed using two case studies that are presented following the detailed protocol.

Ambiguous interactions between diastolic and SR Ca2+ in the regulation of cardiac Ca2+ release

Sobie et al. highlight unresolved issues concerning the regulation of sarcoplasmic reticulum calcium release in cardiac myocytes.

Ultrastructural Analysis of Self-Associated RyR2s

In heart, type-2 ryanodine receptor (RyR2) forms discrete supramolecular clusters in the sarcoplasmic reticulum known as calcium release units (CRUs), which are responsible for most of the Ca(2+) released for muscle contraction. To learn about the substructure of the CRU, we sought to determine whether RyR2s have the ability to self-associate in the absence of other factors and if so, whether they do it in a specific manner. Purified RyR2 was negatively stained and imaged on the transmission electron microscope, and RyR2 particles closely associated were further analyzed using bias-free multivariate statistical analysis and classification. The resulting two-dimensional averages show that RyR2s can interact in two rigid, reproducible configurations: "adjoining", with two RyR2s alongside each other, and "oblique", with two partially overlapped RyR2s forming an angle of 12°. The two configurations are nearly identical under two extreme physiological Ca(2+) concentrations. Pseudo-atomic models for these two interactions indicate that the adjoining interaction involves contacts between the P1, SPRY1 and the helical domains. The oblique interaction is mediated by extensive contacts between the SPRY1 domains (domains 9) and P1 domains (domains 10) of both RyR2s and not through domain 6 as previously thought; in addition its asymmetric interface imposes steric constrains that inhibit the growth of RyR2 as a checkerboard, which is the configuration usually assumed, and generates new configurations, i.e., "branched" and "interlocked". This first, to our knowledge, structural detailed analysis of the inter-RyR2 interactions helps to understand important morphological and functional aspects of the CRU in the context of cardiac EC coupling.

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.

Mechanotransduction and Metabolism in Cardiomyocyte Microdomains

Efficient contractions of the left ventricle are ensured by the continuous transfer of adenosine triphosphate (ATP) from energy production sites, the mitochondria, to energy utilization sites, such as ionic pumps and the force-generating sarcomeres. To minimize the impact of intracellular ATP trafficking, sarcomeres and mitochondria are closely packed together and in proximity with other ultrastructures involved in excitation-contraction coupling, such as t-tubules and sarcoplasmic reticulum junctions. This complex microdomain has been referred to as the intracellular energetic unit. Here, we review the literature in support of the notion that cardiac homeostasis and disease are emergent properties of the hierarchical organization of these units. Specifically, we will focus on pathological alterations of this microdomain that result in cardiac diseases through energy imbalance and posttranslational modifications of the cytoskeletal proteins involved in mechanosensing and transduction.

Calcium Dynamics of Ex Vivo Long-Term Cultured CD8+ T Cells Are Regulated by Changes in Redox Metabolism

T cells reach a state of replicative senescence characterized by a decreased ability to proliferate and respond to foreign antigens. Calcium release associated with TCR engagement is widely used as a surrogate measure of T cell response. Using an ex vivo culture model that partially replicates features of organismal aging, we observe that while the amplitude of Ca2+ signaling does not change with time in culture, older T cells exhibit faster Ca2+ rise and a faster decay. Gene expression analysis of Ca2+ channels and pumps expressed in T cells by RT-qPCR identified overexpression of the plasma membrane CRAC channel subunit ORAI1 and PMCA in older T cells. To test whether overexpression of the plasma membrane Ca2+ channel is sufficient to explain the kinetic information, we adapted a previously published computational model by Maurya and Subramaniam to include additional details on the store-operated calcium entry (SOCE) process to recapitulate Ca2+ dynamics after T cell receptor stimulation. Simulations demonstrated that upregulation of ORAI1 and PMCA channels is not sufficient to explain the observed alterations in Ca2+ signaling. Instead, modeling analysis identified kinetic parameters associated with the IP3R and STIM1 channels as potential causes for alterations in Ca2+ dynamics associated with the long term ex vivo culturing protocol. Due to these proteins having known cysteine residues susceptible to oxidation, we subsequently investigated and observed transcriptional remodeling of metabolic enzymes, a shift to more oxidized redox couples, and post-translational thiol oxidation of STIM1. The model-directed findings from this study highlight changes in the cellular redox environment that may ultimately lead to altered T cell calcium dynamics during immunosenescence or organismal aging.

A Computational Modeling and Simulation Approach to Investigate Mechanisms of Subcellular cAMP Compartmentation

Subcellular compartmentation of the ubiquitous second messenger cAMP has been widely proposed as a mechanism to explain unique receptor-dependent functional responses. How exactly compartmentation is achieved, however, has remained a mystery for more than 40 years. In this study, we developed computational and mathematical models to represent a subcellular sarcomeric space in a cardiac myocyte with varying detail. We then used these models to predict the contributions of various mechanisms that establish subcellular cAMP microdomains. We used the models to test the hypothesis that phosphodiesterases act as functional barriers to diffusion, creating discrete cAMP signaling domains. We also used the models to predict the effect of a range of experimentally measured diffusion rates on cAMP compartmentation. Finally, we modeled the anatomical structures in a cardiac myocyte diad, to predict the effects of anatomical diffusion barriers on cAMP compartmentation. When we incorporated experimentally informed model parameters to reconstruct an in silico subcellular sarcomeric space with spatially distinct cAMP production sites linked to caveloar domains, the models predict that under realistic conditions phosphodiesterases alone were insufficient to generate significant cAMP gradients. This prediction persisted even when combined with slow cAMP diffusion. When we additionally considered the effects of anatomic barriers to diffusion that are expected in the cardiac myocyte dyadic space, cAMP compartmentation did occur, but only when diffusion was slow. Our model simulations suggest that additional mechanisms likely contribute to cAMP gradients occurring in submicroscopic domains. The difference between the physiological and pathological effects resulting from the production of cAMP may be a function of appropriate compartmentation of cAMP signaling. Therefore, understanding the contribution of factors that are responsible for coordinating the spatial and temporal distribution of cAMP at the subcellular level could be important for developing new strategies for the prevention or treatment of unfavorable responses associated with different disease states.

Subcellular compartmentation of the ubiquitous second messenger cAMP has been widely proposed as a mechanism to explain how this one signaling molecule produces unique receptor-dependent functional responses. But, how exactly compartmentation occurs, is unknown. This is because there has been no way to measure the regulation and movement of cAMP in cells with intact subcellular structures. In this study, we applied novel computational approaches to predict whether PDE activity alone or in conjunction with restricted diffusion is sufficient to produce cAMP gradients in submicroscopic signaling domains. We also used the models to test the effect of a range of experimentally measured diffusion rates on cAMP compartmentation. Our simulations suggest that PDE activity alone is not sufficient to explain compartmentation, but if diffusion of cAMP is limited by potential factors such as molecular crowding, PKA buffering, and anatomical barriers, then compartmentation is predicted to occur.

Modeling Na+-Ca2+ exchange in the heart: Allosteric activation, spatial localization, sparks and excitation-contraction coupling

The cardiac sodium (Na⁺)/calcium (Ca²⁺) exchanger (NCX1) is an electrogenic membrane transporter that regulates Ca²⁺ homeostasis in cardiomyocytes, serving mainly to extrude Ca²⁺ during diastole. The direction of Ca²⁺ transport reverses at membrane potentials near that of the action potential plateau, generating an influx of Ca²⁺ into the cell. Therefore, there has been great interest in the possible roles of NCX1 in cardiac Ca²⁺-induced Ca²⁺ release (CICR). Interest has been reinvigorated by a recent super-resolution optical imaging study suggesting that ~18% of NCX1 co-localize with ryanodine receptor (RyR2) clusters, and ~30% of additional NCX1 are localized to within ~120nm of the nearest RyR2. NCX1 may therefore occupy a privileged position in which to modulate CICR. To examine this question, we have developed a mechanistic biophysically-detailed model of NCX1 that describes both NCX1 transport kinetics and Ca²⁺-dependent allosteric regulation. This NCX1 model was incorporated into a previously developed super-resolution model of the Ca²⁺ spark as well as a computational model of the cardiac ventricular myocyte that includes a detailed description of CICR with stochastic gating of L-type Ca²⁺ channels and RyR2s, and that accounts for local Ca²⁺ gradients near the dyad via inclusion of a peri-dyadic (PD) compartment. Both models predict that increasing the fraction of NCX1 in the dyad and PD decreases spark frequency, fidelity, and diastolic Ca²⁺ levels. Spark amplitude and duration are less sensitive to NCX1 spatial redistribution. On the other hand, NCX1 plays an important role in promoting Ca²⁺ entry into the dyad, and hence contributing to the trigger for RyR2 release at depolarized membrane potentials and in the presence of elevated local Na⁺ concentration. Whole-cell simulation of NCX1 tail currents are consistent with the finding that a relatively high fraction of NCX1 (~45%) resides in the dyadic and PD spaces, with a dyad-to-PD ratio of roughly 1:2. Allosteric Ca²⁺ activation of NCX1 helps to "functionally localize" exchanger activity to the dyad and PD by reducing exchanger activity in the cytosol thereby protecting the cell from excessive loss of Ca²⁺ during diastole.

Enzyme localization, crowding, and buffers collectively modulate diffusion-influenced signal transduction: Insights from continuum diffusion modeling

Biochemical reaction networks consisting of coupled enzymes connect substrate signaling events with biological function. Substrates involved in these reactions can be strongly influenced by diffusion "barriers" arising from impenetrable cellular structures and macromolecules, as well as interactions with biomolecules, especially within crowded environments. For diffusion-influenced reactions, the spatial organization of diffusion barriers arising from intracellular structures, non-specific crowders, and specific-binders (buffers) strongly controls the temporal and spatial reaction kinetics. In this study, we use two prototypical biochemical reactions, a Goodwin oscillator, and a reaction with a periodic source/sink term to examine how a diffusion barrier that partitions substrates controls reaction behavior. Namely, we examine how conditions representative of a densely packed cytosol, including reduced accessible volume fraction, non-specific interactions, and buffers, impede diffusion over nanometer length-scales. We find that diffusion barriers can modulate the frequencies and amplitudes of coupled diffusion-influenced reaction networks, as well as give rise to "compartments" of decoupled reactant populations. These effects appear to be intensified in the presence of buffers localized to the diffusion barrier. These findings have strong implications for the role of the cellular environment in tuning the dynamics of signaling pathways.

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 (Ca²⁺) spark, which arises from a cluster of ryanodine receptors (RyR). Opening of these RyR clusters is triggered to produce a local, regenerative release of Ca²⁺ from the sarcoplasmic reticulum (SR). The Ca²⁺ leak out of the SR is an important process for cellular Ca²⁺ management, and it is critically influenced by spark fidelity, i.e., the probability that a spontaneous RyR opening triggers a Ca²⁺ spark. Here, we present a detailed, three-dimensional model of a cardiac Ca²⁺ release unit that incorporates diffusion, intracellular buffering systems, and stochastically gated ion channels. The model exhibits realistic Ca²⁺ sparks and robust Ca²⁺ spark termination across a wide range of geometries and conditions. Furthermore, the model captures the details of Ca²⁺ spark and nonspark-based SR Ca²⁺ leak, and it produces normal excitation-contraction coupling gain. We show that SR luminal Ca²⁺-dependent regulation of the RyR is not critical for spark termination, but it can explain the exponential rise in the SR Ca²⁺ leak-load relationship demonstrated in previous experimental work. Perturbations to subspace dimensions, which have been observed in experimental models of disease, strongly alter Ca²⁺ spark dynamics. In addition, we find that the structure of RyR clusters also influences Ca²⁺ release properties due to variations in inter-RyR coupling via local subspace Ca²⁺ concentration ([Ca²⁺]_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.

Electron tomography of rabbit cardiomyocyte three-dimensional ultrastructure

The field of cardiovascular research has benefitted from rapid developments in imaging technology over the last few decades. Accordingly, an ever growing number of large, multidimensional data sets have begun to appear, often challenging existing pre-conceptions about structure and function of biological systems. For tissue and cell structure imaging, the move from 2D section-based microscopy to true 3D data collection has been a major driver of new insight. In the sub-cellular domain, electron tomography is a powerful technique for exploration of cellular structures in 3D with unparalleled fidelity at nanometer resolution.

Electron tomography is particularly advantageous for studying highly compartmentalised cells such as cardiomyocytes, where elaborate sub-cellular structures play crucial roles in electrophysiology and mechanics. Although the anatomy of specific ultra-structures, such as dyadic couplons, has been extensively explored using 2D electron microscopy of thin sections, we still lack accurate, quantitative knowledge of true individual shape, volume and surface area of sub-cellular domains, as well as their 3D spatial interrelations; let alone of how these are reshaped during the cycle of contraction and relaxation. Here we discuss and illustrate the utility of ET for identification, visualisation, and analysis of 3D cardiomyocyte ultrastructures such as the T-tubular system, sarcoplasmic reticulum, mitochondria and microtubules.

Multi-scale Modeling of the Cardiovascular System: Disease Development, Progression, and Clinical Intervention

Cardiovascular diseases (CVDs) are the leading cause of death in the western world. With the current development of clinical diagnostics to more accurately measure the extent and specifics of CVDs, a laudable goal is a better understanding of the structure-function relation in the cardiovascular system. Much of this fundamental understanding comes from the development and study of models that integrate biology, medicine, imaging, and biomechanics. Information from these models provides guidance for developing diagnostics, and implementation of these diagnostics to the clinical setting, in turn, provides data for refining the models. In this review, we introduce multi-scale and multi-physical models for understanding disease development, progression, and designing clinical interventions. We begin with multi-scale models of cardiac electrophysiology and mechanics for diagnosis, clinical decision support, personalized and precision medicine in cardiology with examples in arrhythmia and heart failure. We then introduce computational models of vasculature mechanics and associated mechanical forces for understanding vascular disease progression, designing clinical interventions, and elucidating mechanisms that underlie diverse vascular conditions. We conclude with a discussion of barriers that must be overcome to provide enhanced insights, predictions, and decisions in pre-clinical and clinical applications.

Long-Lasting Sparks: Multi-Metastability and Release Competition in the Calcium Release Unit Network

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength.

Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Modeling calcium regulation of contraction, energetics, signaling, and transcription in the cardiac myocyte

Calcium (Ca(2+)) plays many important regulatory roles in cardiac muscle cells. In the initial phase of the action potential, influx of Ca(2+) through sarcolemmal voltage-gated L-type Ca(2+) channels (LCCs) acts as a feed-forward signal that triggers a large release of Ca(2+) from the junctional sarcoplasmic reticulum (SR). This Ca(2+) drives heart muscle contraction and pumping of blood in a process known as excitation-contraction coupling (ECC). Triggered and released Ca(2+) also feed back to inactivate LCCs, attenuating the triggered Ca(2+) signal once release has been achieved. The process of ECC consumes large amounts of ATP. It is now clear that in a process known as excitation-energetics coupling, Ca(2+) signals exert beat-to-beat regulation of mitochondrial ATP production that closely couples energy production with demand. This occurs through transport of Ca(2+) into mitochondria, where it regulates enzymes of the tricarboxylic acid cycle. In excitation-signaling coupling, Ca(2+) activates a number of signaling pathways in a feed-forward manner. Through effects on their target proteins, these interconnected pathways regulate Ca(2+) signals in complex ways to control electrical excitability and contractility of heart muscle. In a process known as excitation-transcription coupling, Ca(2+) acting primarily through signal transduction pathways also regulates the process of gene transcription. Because of these diverse and complex roles, experimentally based mechanistic computational models are proving to be very useful for understanding Ca(2+) signaling in the cardiac myocyte.

An integrated finite element simulation of cardiomyocyte function based on triphasic theory

In numerical simulations of cardiac excitation-contraction coupling, the intracellular potential distribution and mobility of cytosol and ions have been mostly ignored. Although the intracellular potential gradient is small, during depolarization it can be a significant driving force for ion movement, and is comparable to diffusion in terms of net flux. Furthermore, fluid in the t-tubules is thought to advect ions to facilitate their exchange with the extracellular space. We extend our previous finite element model that was based on triphasic theory to examine the significance of these factors in cardiac physiology. Triphasic theory allows us to study the behavior of solids (proteins), fluids (cytosol) and ions governed by mechanics and electrochemistry in detailed subcellular structures, including myofibrils, mitochondria, the sarcoplasmic reticulum, membranes, and t-tubules. Our simulation results predicted an electrical potential gradient inside the t-tubules at the onset of depolarization, which corresponded to the Na(+) channel distribution therein. Ejection and suction of fluid between the t-tubules and the extracellular compartment during isometric contraction were observed. We also examined the influence of t-tubule morphology and mitochondrial location on the electrophysiology and mechanics of the cardiomyocyte. Our results confirm that the t-tubule structure is important for synchrony of Ca(2+) release, and suggest that mitochondria in the sub-sarcolemmal region might serve to cancel Ca(2+) inflow through surface sarcolemma, thereby maintaining the intracellular Ca(2+) environment in equilibrium.

A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles

Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca²⁺ content within tens of seconds. Our concept for a multiscale mathematical model of Ca²⁺ -induced Ca²⁺ release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca²⁺-concentration profiles using three previously published RyR-channel Markov schemes.

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.

Regulation of RYR2 by sarcoplasmic reticulum Ca(2+)

Ca(2+) is arguably the most important ion involved in the contraction of the heart. The cardiac ryanodine receptor (RyR2), the major Ca(2+) release channel located in the sarcoplasmic reticulum (SR) membrane, is responsible for releasing the bulk of Ca(2+) required for contraction. Moreover, RyR2 is also crucial for maintaining SR Ca(2+) homeostasis by releasing Ca(2+) from the SR when it becomes overloaded with Ca(2+) . During normal contraction, RyR2 is activated by cytosolic Ca(2+) , whereas during store overload conditions, the opening of RyR2 is governed by SR Ca(2+) . Although the process of the cytosolic control of RyR2 is well established, the molecular mechanism by which SR luminal Ca(2+) regulates RyR2 has only recently been elucidated and remains controversial. In addition to the activation of RyR2, SR luminal Ca(2+) also determines when the RyR2 channel closes. RyR2-mediated Ca(2+) release from the SR does not continue until the SR is completely depleted. Rather, it ceases when SR luminal Ca(2+) falls below a certain level. Given the importance of SR Ca(2+) , it is not surprising that the SR luminal Ca(2+) level is tightly controlled by SR Ca(2+) -buffering proteins. Consequently, the opening and closing of RyR2 is heavily influenced by the presence of such proteins, particularly those associated with RyR2, such as calsequestrin and the histidine-rich Ca(2+) -binding protein. These proteins appear to indirectly alter RyR2 activity by modifying the microdomain SR Ca(2+) level surrounding RyR2.

A model study of sequential enzyme reactions and electrostatic channeling

We study models of two sequential enzyme-catalyzed reactions as a basic functional building block for coupled biochemical networks. We investigate the influence of enzyme distributions and long-range molecular interactions on reaction kinetics, which have been exploited in biological systems to maximize metabolic efficiency and signaling effects. Specifically, we examine how the maximal rate of product generation in a series of sequential reactions is dependent on the enzyme distribution and the electrostatic composition of its participant enzymes and substrates. We find that close proximity between enzymes does not guarantee optimal reaction rates, as the benefit of decreasing enzyme separation is countered by the volume excluded by adjacent enzymes. We further quantify the extent to which the electrostatic potential increases the efficiency of transferring substrate between enzymes, which supports the existence of electrostatic channeling in nature. Here, a major finding is that the role of attractive electrostatic interactions in confining intermediate substrates in the vicinity of the enzymes can contribute more to net reactive throughput than the directional properties of the electrostatic fields. These findings shed light on the interplay of long-range interactions and enzyme distributions in coupled enzyme-catalyzed reactions, and their influence on signaling in biological systems.

T-tubule disruption promotes calcium alternans in failing ventricular myocytes: mechanistic insights from computational modeling

In heart failure (HF), T-tubule (TT) disruption contributes to dyssynchronous calcium (Ca) release and impaired contraction, but its role in arrhythmogenesis remains unclear. In this study, we investigate the effects of TT disruption and other HF remodeling factors on Ca alternans in ventricular myocytes using computer modeling. A ventricular myocyte model with detailed spatiotemporal Ca cycling modeled by a coupled Ca release unit (CRU) network was used, in which the L-type Ca channels and the ryanodine receptor (RyR) channels were simulated by random Markov transitions. TT disruption, which removes the L-type Ca channels from the associated CRUs, results in "orphaned" RyR clusters and thus provides increased opportunity for spark-induced Ca sparks to occur. This effect combined with other HF remodeling factors promoted alternans by two distinct mechanisms: 1) for normal sarco-endoplasmic reticulum Ca ATPase (SERCA) activity, alternans was caused by both CRU refractoriness and coupling. The increased opportunity for spark-induced sparks by TT disruption combined with the enhanced CRU coupling by Ca elevation in the presence or absence of increased RyR leakiness facilitated spark synchronization on alternate beats to promote Ca alternans; 2) for down-regulated SERCA, alternans was caused by the sarcoplasmic reticulum (SR) Ca load-dependent mechanism, independent of CRU refractoriness. TT disruption and increased RyR leakiness shifted and steepened the SR Ca release-load relationship, which combines with down-regulated SERCA to promote Ca alternans. In conclusion, the mechanisms of Ca alternans for normal and down-regulated SERCA are different, and TT disruption promotes Ca alternans by both mechanisms, which may contribute to alternans at different stages of HF.

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.

Molecular and subcellular-scale modeling of nucleotide diffusion in the cardiac myofilament lattice

Contractile function of cardiac cells is driven by the sliding displacement of myofilaments powered by the cycling myosin crossbridges. Critical to this process is the availability of ATP, which myosin hydrolyzes during the cross-bridge cycle. The diffusion of adenine nucleotides through the myofilament lattice has been shown to be anisotropic, with slower radial diffusion perpendicular to the filament axis relative to parallel, and is attributed to the periodic hexagonal arrangement of the thin (actin) and thick (myosin) filaments. We investigated whether atomistic-resolution details of myofilament proteins can refine coarse-grain estimates of diffusional anisotropy for adenine nucleotides in the cardiac myofibril, using homogenization theory and atomistic thin filament models from the Protein Data Bank. Our results demonstrate considerable anisotropy in ATP and ADP diffusion constants that is consistent with experimental measurements and dependent on lattice spacing and myofilament overlap. A reaction-diffusion model of the half-sarcomere further suggests that diffusional anisotropy may lead to modest adenine nucleotide gradients in the myoplasm under physiological conditions.

Computational modeling of subcellular transport and signaling

Numerous signaling processes in the cell are controlled in microdomains that are defined by cellular structures ranging from nm to μm in size. Recent improvements in microscopy enable the resolution and reconstruction of these micro domains, while new computational methods provide the means to elucidate their functional roles. Collectively these tools allow for a biophysical understanding of the cellular environment and its pathological progression in disease. Here we review recent advancements in microscopy, and subcellular modeling on the basis of reconstructed geometries, with a special focus on signaling microdomains that are important for the excitation contraction coupling in cardiac myocytes.

Calcium and IP3 dynamics in cardiac myocytes: experimental and computational perspectives and approaches

Calcium plays a crucial role in excitation-contraction coupling (ECC), but it is also a pivotal second messenger activating Ca²⁺-dependent transcription factors in a process termed excitation-transcription coupling (ETC). Evidence accumulated over the past decade indicates a pivotal role of inositol 1,4,5-trisphosphate receptor (IP₃R)-mediated Ca²⁺ release in the regulation of cytosolic and nuclear Ca²⁺ signals. IP₃ is generated by stimulation of plasma membrane receptors that couple to phospholipase C (PLC), liberating IP₃ from phosphatidylinositol 4,5-bisphosphate (PIP₂). An intriguing aspect of IP₃ signaling is the presence of the entire PIP₂-PLC-IP₃ signaling cascade as well as the presence of IP₃Rs at the inner and outer membranes of the nuclear envelope (NE) which functions as a Ca²⁺ store. The observation that the nucleus is surrounded by its own putative Ca²⁺ store raises the possibility that nuclear IP₃-dependent Ca²⁺ release plays a critical role in ETC. This provides a potential mechanism of regulation that acts locally and autonomously from the global cytosolic Ca²⁺ signal underlying ECC. Moreover, there is evidence that: (i) the sarcoplasmic reticulum (SR) and NE are a single contiguous Ca²⁺ store; (ii) the nuclear pore complex is the major gateway for Ca²⁺ and macromolecules to pass between the cytosol and the nucleoplasm; (iii) the inner membrane of the NE hosts key Ca²⁺ handling proteins including the Na⁺/Ca²⁺ exchanger (NCX)/GM1 complex, ryanodine receptors (RyRs), nicotinic acid adenine dinucleotide phosphate receptors (NAADPRs), Na⁺/K⁺ ATPase, and Na⁺/H⁺ exchanger. Thus, it appears that the nucleus represents a Ca²⁺ signaling domain equipped with its own ion channels and transporters that allow for complex local Ca²⁺ signals. Many experimental and modeling approaches have been used for the study of intracellular Ca²⁺ signaling but the key to the understanding of the dual role of Ca²⁺ mediating ECC and ECT lays in quantitative differences of local [Ca²⁺] in the nuclear and cytosolic compartment. In this review, we discuss the state of knowledge regarding the origin and the physiological implications of nuclear Ca²⁺ transients in different cardiac cell types (adult atrial and ventricular myocytes) as well as experimental and mathematical approaches to study Ca²⁺ and IP₃ signaling in the cytosol and nucleus. In particular, we focus on the concept that highly localized Ca²⁺ signals are required to translocate and activate Ca²⁺-dependent transcription factors (e.g., nuclear factor of activated T-cells, NFAT; histone deacetylase, HDAC) through phosphorylation/dephosphorylation processes.

Influence of neighboring reactive particles on diffusion-limited reactions

Competition between reactive species is commonplace in typical chemical reactions. Specifically the primary reaction between a substrate and its target enzyme may be altered when interactions with secondary species in the system are substantial. We explore this competition phenomenon for diffusion-limited reactions in the presence of neighboring particles through numerical solution of the diffusion equation. As a general model for globular proteins and small molecules, we consider spherical representations of the reactants and neighboring particles; these neighbors vary in local density, size, distribution, and relative distance from the primary target reaction, as well as their surface reactivity. Modulations of these model variables permit inquiry into the influence of excluded volume and competition on the primary reaction due to the presence of neighboring particles. We find that the surface reactivity effect is long-ranged and a strong determinant of reaction kinetics, whereas the excluded volume effect is relatively short-ranged and less influential in comparison. As a consequence, the effect of the excluded volume is only modestly dependent on the neighbor distribution and is approximately additive; this additivity permits a linear approximation to the many-body effect on the reaction kinetics. In contrast, the surface reactivity effect is non-additive, and thus it may require higher-order approximations to describe the reaction kinetics. Our model study has broad implications in the general understanding of competition and local crowding on diffusion-limited chemical reactions.

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.

Evoked centripetal Ca(2+) mobilization in cardiac Purkinje cells: insight from a model of three Ca(2+) release regions

Despite strong suspicion that abnormal Ca(2+) handling in Purkinje cells (P-cells) is implicated in life-threatening forms of ventricular tachycardias, the mechanism underlying the Ca(2+) cycling of these cells under normal conditions is still unclear. There is mounting evidence that P-cells have a unique Ca(2+) handling system. Notably complex spontaneous Ca(2+) activity was previously recorded in canine P-cells and was explained by a mechanistic hypothesis involving a triple layered system of Ca(2+) release channels. Here we examined the validity of this hypothesis for the electrically evoked Ca(2+) transient which was shown, in the dog and rabbit, to occur progressively from the periphery to the interior of the cell. To do so, the hypothesis was incorporated in a model of intracellular Ca(2+) dynamics which was then used to reproduce numerically the Ca(2+) activity of P-cells under stimulated conditions. The modelling was thus performed through a 2D computational array that encompassed three distinct Ca(2+) release nodes arranged, respectively, into three consecutive adjacent regions. A system of partial differential equations (PDEs) expressed numerically the principal cellular functions that modulate the local cytosolic Ca(2+) concentration (Cai). The apparent node-to-node progression of elevated Cai was obtained by combining Ca(2+) diffusion and 'Ca(2+)-induced Ca(2+) release'. To provide the modelling with a reliable experimental reference, we first re-examined the Ca(2+) mobilization in swine stimulated P-cells by 2D confocal microscopy. As reported earlier for the dog and rabbit, a centripetal Ca(2+) transient was readily visible in 22 stimulated P-cells from six adult Yucatan swine hearts (pacing rate: 0.1 Hz; pulse duration: 25 ms, pulse amplitude: 10% above threshold; 1 mm Ca(2+); 35°C; pH 7.3). An accurate replication of the observed centripetal Ca(2+) propagation was generated by the model for four representative cell examples and confirmed by statistical comparisons of simulations against cell data. Selective inactivation of Ca(2+) release regions of the computational array showed that an intermediate layer of Ca(2+) release nodes with an ~30-40% lower Ca(2+) activation threshold was required to reproduce the phenomenon. Our computational analysis was therefore fully consistent with the activation of a triple layered system of Ca(2+) release channels as a mechanism of centripetal Ca(2+) signalling in P-cells. Moreover, the model clearly indicated that the intermediate Ca(2+) release layer with increased sensitivity for Ca(2+) plays an important role in the specific intracellular Ca(2+) mobilization of Purkinje fibres and could therefore be a relevant determinant of cardiac conduction.

Calcium movements inside the sarcoplasmic reticulum of cardiac myocytes

Sarcoplasmic reticulum (SR) Ca content ([Ca]SRT) is critical to both normal cardiac function and electrophysiology, and changes associated with pathology contribute to systolic and diastolic dysfunction and arrhythmias. The intra-SR free [Ca] ([Ca]SR) dictates the [Ca]SRT, the driving force for Ca release and regulates release channel gating. We discuss measurement of [Ca]SR and [Ca]SRT, how [Ca]SR regulates activation and termination of release, and how Ca diffuses within the SR and influences SR Ca release during excitation-contraction coupling, Ca sparks and Cac waves. The entire SR network is connected and its lumen is also continuous with the nuclear envelope. Rapid Ca diffusion within the SR could stabilize and balance local [Ca]SR within the myocyte, but restrictions to diffusion can create spatial inhomogeneities. Experimental measurements and mathematical models of [Ca]SR to date have greatly enriched our understanding of these [Ca]SR dynamics, but controversies exist and may stimulate new measurements and analysis.