A 3D Monte Carlo analysis of the role of dyadic space geometry in spark generation

Ca2+ spark latency and control of intrinsic Ca2+ release dyssynchrony in rat cardiac ventricular muscle cells

Cardiac excitation-contraction coupling (ECC) depends on Ca2+ release from intracellular stores via ryanodine receptors (RyRs) triggered by L-type Ca2+ channels (LCCs). Uncertain numbers of RyRs and LCCs form 'couplons' whose activation produces Ca2+ sparks, which summate to form a cell-wide Ca2+ transient that switches on contraction. Voltage (Vm) changes during the action potential (AP) and stochasticity in channel gating should create variability in Ca2+ spark timing, but Ca2+ transient wavefronts have remarkable uniformity. To examine how this is achieved, we measured the Vm-dependence of evoked Ca2+ spark probability (Pspark) and latency over a wide voltage range in rat ventricular cells. With depolarising steps, Ca2+ spark latency showed a U-shaped Vm-dependence, while repolarising steps from 50 mV produced Ca2+ spark latencies that increased monotonically with Vm. A computer model based on reported channel gating and geometry reproduced our experimental data and revealed a likely RyR:LCC stoichiometry of ∼ 5:1 for the Ca2+ spark initiating complex (IC). Using the experimental AP waveform, the model revealed a high coupling fidelity (Pcpl ∼ 0.5) between each LCC opening and IC activation. The presence of ∼ 4 ICs per couplon reduced Ca2+ spark latency and increased Pspark to match experimental data. Variability in AP release timing is less than that seen with voltage steps because the AP overshoot and later repolarization decrease Pspark due to effects on LCC flux and LCC deactivation respectively. This work provides a framework for explaining the Vm- and time-dependence of Pspark, and indicates how ion channel dispersion in disease can contribute to dyssynchrony in Ca2+ release.

Electron microscopy of cardiac 3D nanodynamics: form, function, future

The 3D nanostructure of the heart, its dynamic deformation during cycles of contraction and relaxation, and the effects of this deformation on cell function remain largely uncharted territory. Over the past decade, the first inroads have been made towards 3D reconstruction of heart cells, with a native resolution of around 1 nm³, and of individual molecules relevant to heart function at a near-atomic scale. These advances have provided access to a new generation of data and have driven the development of increasingly smart, artificial intelligence-based, deep-learning image-analysis algorithms. By high-pressure freezing of cardiomyocytes with millisecond accuracy after initiation of an action potential, pseudodynamic snapshots of contraction-induced deformation of intracellular organelles can now be captured. In combination with functional studies, such as fluorescence imaging, exciting insights into cardiac autoregulatory processes at nano-to-micro scales are starting to emerge. In this Review, we discuss the progress in this fascinating new field to highlight the fundamental scientific insight that has emerged, based on technological breakthroughs in biological sample preparation, 3D imaging and data analysis; to illustrate the potential clinical relevance of understanding 3D cardiac nanodynamics; and to predict further progress that we can reasonably expect to see over the next 10 years.

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.

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.

Nitric Oxide and Mechano-Electrical Transduction in Cardiomyocytes

The ability§ of the heart to adapt to changes in the mechanical environment is critical for normal cardiac physiology. The role of nitric oxide is increasingly recognized as a mediator of mechanical signaling. Produced in the heart by nitric oxide synthases, nitric oxide affects almost all mechano-transduction pathways within the cardiomyocyte, with roles mediating mechano-sensing, mechano-electric feedback (via modulation of ion channel activity), and calcium handling. As more precise experimental techniques for applying mechanical stresses to cells are developed, the role of these forces in cardiomyocyte function can be further understood. Furthermore, specific inhibitors of different nitric oxide synthase isoforms are now available to elucidate the role of these enzymes in mediating mechano-electrical signaling. Understanding of the links between nitric oxide production and mechano-electrical signaling is incomplete, particularly whether mechanically sensitive ion channels are regulated by nitric oxide, and how this affects the cardiac action potential. This is of particular relevance to conditions such as atrial fibrillation and heart failure, in which nitric oxide production is reduced. Dysfunction of the nitric oxide/mechano-electrical signaling pathways are likely to be a feature of cardiac pathology (e.g., atrial fibrillation, cardiomyopathy, and heart failure) and a better understanding of the importance of nitric oxide signaling and its links to mechanical regulation of heart function may advance our understanding of these conditions.

Lingguizhugan decoction attenuates doxorubicin-induced heart failure in rats by improving TT-SR microstructural remodeling

BACKGROUND

Lingguizhugan decoction (LGZG), an ancient Chinese herbal formula, has been used to treat cardiovascular diseases in eastern Asia. We investigated whether LGZG has protective activity and the mechanism underlying its effect in an animal model of heart failure (HF).

METHODS

A rat model of HF was established by administering eight intraperitoneal injections of doxorubicin (DOX) (cumulative dose of 16 mg/kg) over a 4-week period. Subsequently, LGZG at 5, 10, and 15 mL/kg/d was administered to the rats intragastrically once daily for 4 weeks. The body weight, heart weight index (HWI), heart weight/tibia length ratio (HW/TL), and serum BNP level were investigated to assess the effect of LGZG on HF. Echocardiography was performed to investigate cardiac function, and H&E staining to visualize myocardial morphology. Myocardial ultrastructure and T-tubule-sarcoplasmic reticulum (TT-SR) junctions were observed by transmission electron microscopy. The JP-2 protein level was determined by Western blotting. The mRNA level of CACNA1S and RyR2 and the microRNA-24 (miR-24) level were assayed by quantitative RT-PCR.

RESULTS

Four weeks after DOX treatment, rats developed cardiac damage and exhibited a significantly increased BNP level compared with the control rats (169.6 ± 29.6 pg/mL versus 80.1 ± 9.8 pg/mL, P < 0.001). Conversely, LGZG, especially at the highest dose, markedly reduced the BNP level (93.8 ± 17.9 pg/mL, P < 0.001). Rats treated with DOX developed cardiac dysfunction, characterized by a strong decrease in left ventricular ejection fraction compared with the control (58.5 ± 8.7% versus 88.7 ± 4.0%; P < 0.001). Digoxin and LGZG improved cardiac dysfunction (79.6 ± 6.1%, 69.2 ± 2.5%, respectively) and preserved the left ventricular ejection fraction (77.9 ± 5.1, and 80.5 ± 4.9, respectively, P < 0.01). LGZG also improved the LVEDD, LVESD, and FS and eliminated ventricular hypertrophy, as indicated by decreased HWI and HW/TL ratio. LGZG attenuated morphological abnormalities and mitochondrial damage in the myocardium. In addition, a high dose of LGZG significantly downregulated the expression of miR-24 compared with that in DOX-treated rats (fold change 1.4 versus 3.4, P < 0.001), but upregulated the expression of JP-2 and antagonized DOX-induced T-tubule TT-SR microstructural remodeling. These activities improved periodic Ca²⁺ transients and cell contraction, which may underly the beneficial effect of LGZG on HF.

CONCLUSIONS

LGZG exerted beneficial effects on DOX-induced HF in rats, which were mediated in part by improved TT-SR microstructural remodeling.

Calcium Ion Fluctuations Alter Channel Gating in a Stochastic Luminal Calcium Release Site Model

Stochasticity and small system size effects in complex biochemical reaction networks can greatly alter transient and steady-state system properties. A common approach to modeling reaction networks, which accounts for system size, is the chemical master equation that governs the dynamics of the joint probability distribution for molecular copy number. However, calculation of the stationary distribution is often prohibitive, due to the large state-space associated with most biochemical reaction networks. Here, we analyze a network representing a luminal calcium release site model and investigate to what extent small system size effects and calcium fluctuations, driven by ion channel gating, influx and diffusion, alter steady-state ion channel properties including open probability. For a physiological ion channel gating model and number of channels, the state-space may be between approximately 10⁶-10⁸ elements, and a novel modified block power method is used to solve the associated dominant eigenvector problem required to calculate the stationary distribution. We demonstrate that both small local cytosolic domain volume and a small number of ion channels drive calcium fluctuations that result in deviation from the corresponding model that neglects small system size effects.

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.

Neuronal sodium channels: emerging components of the nano-machinery of cardiac calcium cycling

Excitation-contraction coupling is the bridge between cardiac electrical activation and mechanical contraction. It is driven by the influx of Ca²⁺ across the sarcolemma triggering Ca²⁺ release from the sarcoplasmic reticulum (SR) - a process termed Ca²⁺ -induced Ca²⁺ release (CICR) - followed by re-sequestration of Ca²⁺ into the SR. The Na⁺ /Ca²⁺ exchanger inextricably couples the cycling of Ca²⁺ and Na⁺ in cardiac myocytes. Thus, influx of Na⁺ via voltage-gated Na⁺ channels (Na_V ) has emerged as an important regulator of CICR both in health and in disease. Recent insights into the subcellular distribution of cardiac and neuronal Na_V isoforms and their ultrastructural milieu have important implications for the roles of these channels in mediating Ca²⁺ -driven arrhythmias. This review will discuss functional insights into the role of neuronal Na_V isoforms vis-à-vis cardiac Na_V s in triggering such arrhythmias and their potential as therapeutic targets in the context of the aforementioned structural observations.

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.

The influence of Ca²⁺ buffers on free [Ca²⁺] fluctuations and the effective volume of Ca²⁺ microdomains

Intracellular calcium (Ca(2+)) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca(2+) binding proteins (Ca(2+) buffers) play an important role in regulating Ca(2+) concentration ([Ca(2+)]). Buffers typically slow [Ca(2+)] temporal dynamics and increase the effective volume of Ca(2+) domains. Because fluctuations in [Ca(2+)] decrease in proportion to the square-root of a domain's physical volume, one might conjecture that buffers decrease [Ca(2+)] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca(2+) signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca(2+) influx with passive exchange of both Ca(2+) and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca(2+) and buffer and use these stochastic differential equations to determine the magnitude of [Ca(2+)] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca(2+) signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca(2+)] fluctuations during periods of Ca(2+) influx, whereas stationary (immobile) Ca(2+) buffers do not. Also contrary to expectations, we find that in the absence of Ca(2+) influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca(2+)] fluctuations. We derive an analytical formula describing the influence of rapid Ca(2+) buffers on [Ca(2+)] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca(2+) buffers alter the dynamics of [Ca(2+)] fluctuations in a nonintuitive manner. The finding that Ca(2+) buffers do not suppress intrinsic domain [Ca(2+)] fluctuations raises the intriguing question of whether or not [Ca(2+)] fluctuations are a physiologically significant aspect of local Ca(2+) signaling.

Control of sarcoplasmic reticulum Ca2+ release by stochastic RyR gating within a 3D model of the cardiac dyad and importance of induction decay for CICR termination

The factors responsible for the regulation of regenerative calcium-induced calcium release (CICR) during Ca(2+) spark evolution remain unclear. Cardiac ryanodine receptor (RyR) gating in rats and sheep was recorded at physiological Ca(2+), Mg(2+), and ATP levels and incorporated into a 3D model of the cardiac dyad, which reproduced the time course of Ca(2+) sparks, Ca(2+) blinks, and Ca(2+) spark restitution. The termination of CICR by induction decay in the model principally arose from the steep Ca(2+) dependence of RyR closed time, with the measured sarcoplasmic reticulum (SR) lumen Ca(2+) dependence of RyR gating making almost no contribution. The start of CICR termination was strongly dependent on the extent of local depletion of junctional SR Ca(2+), as well as the time course of local Ca(2+) gradients within the junctional space. Reducing the dimensions of the dyad junction reduced Ca(2+) spark amplitude by reducing the strength of regenerative feedback within CICR. A refractory period for Ca(2+) spark initiation and subsequent Ca(2+) spark amplitude restitution arose from 1), the extent to which the regenerative phase of CICR can be supported by the partially depleted junctional SR, and 2), the availability of releasable Ca(2+) in the junctional SR. The physical organization of RyRs within the junctional space had minimal effects on Ca(2+) spark amplitude when more than nine RyRs were present. Spark amplitude had a nonlinear dependence on RyR single-channel Ca(2+) flux, and was approximately halved by reducing the flux from 0.6 to 0.2 pA. Although rat and sheep RyRs had quite different Ca(2+) sensitivities, Ca(2+) spark amplitude was hardly affected. This suggests that moderate changes in RyR gating by second-messenger systems will principally alter the spatiotemporal properties of SR release, with smaller effects on the amount released.

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.

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.

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.

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.

Ultrastructural remodelling of Ca(2+) signalling apparatus in failing heart cells

AIMS

The contraction of a heart cell is controlled by Ca(2+)-induced Ca(2+) release between L-type Ca(2+) channels (LCCs) in the cell membrane/T-tubules (TTs) and ryanodine receptors (RyRs) in the junctional sarcoplasmic reticulum (SR). During heart failure, LCC-RyR signalling becomes defective. The purpose of the present study was to reveal the ultrastructural mechanism underlying the defective LCC-RyR signalling and contractility.

METHODS AND RESULTS

In rat models of heart failure produced by transverse aortic constriction surgery, stereological analysis of transmission electron microscopic images showed that the volume density and the surface area of junctional SRs and those of SR-coupled TTs were both decreased in failing heart cells. The TT-SR junctions were displaced or missing from the Z-line areas. Moreover, the spatial span of individual TT-SR junctions was markedly reduced in failing heart cells. Numerical simulation and junctophilin-2 knockdown experiments demonstrated that the decrease in junction size (and thereby the constitutive LCC and RyR numbers) led to a scattered delay of Ca(2+) release activation.

CONCLUSIONS

The shrinking and eventual absence of TT-SR junctions are important mechanisms underlying the desynchronized and inhomogeneous Ca(2+) release and the decreased contractile strength in heart failure. Maintaining the nanoscopic integrity of TT-SR junctions thus represents a therapeutic strategy against heart failure and related cardiomyopathies.

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.

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.

A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

BACKGROUND

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.

RESULT

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

CONCLUSION

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Control of Ca2+ release by action potential configuration in normal and failing murine cardiomyocytes

Cardiomyocytes from failing hearts exhibit spatially nonuniform or dyssynchronous sarcoplasmic reticulum (SR) Ca(2+) release. We investigated the contribution of action potential (AP) prolongation in mice with congestive heart failure (CHF) after myocardial infarction. AP recordings from CHF and control myocytes were included in a computational model of the dyad, which predicted more dyssynchronous ryanodine receptor opening during stimulation with the CHF AP. This prediction was confirmed in cardiomyocyte experiments, when cells were alternately stimulated by control and CHF AP voltage-clamp waveforms. However, when a train of like APs was used as the voltage stimulus, the control and CHF AP produced a similar Ca(2+) release pattern. In this steady-state condition, greater integrated Ca(2+) entry during the CHF AP lead to increased SR Ca(2+) content. A resulting increase in ryanodine receptor sensitivity synchronized SR Ca(2+) release in the mathematical model, thus offsetting the desynchronizing effects of reduced driving force for Ca(2+) entry. A modest nondyssynchronous prolongation of Ca(2+) release was nevertheless observed during the steady-state CHF AP, which contributed to increased time-to-peak measurements for Ca(2+) transients in failing cells. Thus, dyssynchronous Ca(2+) release in failing mouse myocytes does not result from electrical remodeling, but rather other alterations such as T-tubule reorganization.

Numerical analysis of Ca2+ signaling in rat ventricular myocytes with realistic transverse-axial tubular geometry and inhibited sarcoplasmic reticulum

The t-tubules of mammalian ventricular myocytes are invaginations of the cell membrane that occur at each Z-line. These invaginations branch within the cell to form a complex network that allows rapid propagation of the electrical signal, and hence synchronous rise of intracellular calcium (Ca²⁺). To investigate how the t-tubule microanatomy and the distribution of membrane Ca²⁺ flux affect cardiac excitation-contraction coupling we developed a 3-D continuum model of Ca²⁺ signaling, buffering and diffusion in rat ventricular myocytes. The transverse-axial t-tubule geometry was derived from light microscopy structural data. To solve the nonlinear reaction-diffusion system we extended SMOL software tool (http://mccammon.ucsd.edu/smol/). The analysis suggests that the quantitative understanding of the Ca²⁺ signaling requires more accurate knowledge of the t-tubule ultra-structure and Ca²⁺ flux distribution along the sarcolemma. The results reveal the important role for mobile and stationary Ca²⁺ buffers, including the Ca²⁺ indicator dye. In agreement with experiment, in the presence of fluorescence dye and inhibited sarcoplasmic reticulum, the lack of detectible differences in the depolarization-evoked Ca²⁺ transients was found when the Ca²⁺ flux was heterogeneously distributed along the sarcolemma. In the absence of fluorescence dye, strongly non-uniform Ca²⁺ signals are predicted. Even at modest elevation of Ca²⁺, reached during Ca²⁺ influx, large and steep Ca²⁺ gradients are found in the narrow sub-sarcolemmal space. The model predicts that the branched t-tubule structure and changes in the normal Ca²⁺ flux density along the cell membrane support initiation and propagation of Ca²⁺ waves in rat myocytes.

In cardiac muscle cells, calcium (Ca²⁺) is best known for its role in contraction activation. A remarkable amount of quantitative data on cardiac cell structure, ion-transporting protein distributions and intracellular Ca²⁺ dynamics has been accumulated. Various alterations in the protein distributions or cell ultra-structure are now recognized to be the primary mechanisms of cardiac dysfunction in a diverse range of common pathologies including cardiac arrhythmias and hypertrophy. Using a 3-D computational model, incorporating more realistic transverse-axial t-tubule geometry and considering geometric irregularities and inhomogeneities in the distribution of ion-transporting proteins, we analyze several important spatial and temporal features of Ca²⁺ signaling in rat ventricular myocytes. This study demonstrates that the computational models could serve as powerful tools for prediction and analyses of how the Ca²⁺ dynamics and cardiac excitation-contraction coupling are regulated under normal conditions or certain pathologies. The use of computational and mathematical approaches will help also to better understand aspects of cell functions that are not currently amenable to experimental investigation.

A platform for in silico modeling of physiological systems III

Physiome and systems biology have been recognized as emerging and important research areas that can integrate quantitatively growing knowledge about biological structure and physiological functions at multiple scales of time and space. For the integration, it is important to build physiologically plausible and sharable mathematical models that can be used for dynamic simulations of functions at multi-scale and multi-level. Here we describe new features of our open platform insilicoML (ISML) and insilicoIDE (ISIDE) that have been presented previously. The platform can support reuse existing mathematical models of physiological functions in the model databases, to construct brand new models, and to simulate models. The major new features of the platform include improvement of the capabilities to incorporate experimentally obtained data such as time-series and morphological data with dynamic simulation of models that may be driven by the data, and extension of variety of model types that can be described by ISML and simulated on ISIDE, such as multi agent systems and models described by partial differential equations that are solved by the finite element method.

A univariate model of calcium release in the dyadic cleft of cardiac myocytes

Local calcium sparks in the dyadic cleft of cardiac myocytes are triggered by calcium influxes via L-type calcium channels (LCCs) located on the transverse tubule (TT) membrane, and subsequently controlled by the regeneration of ryanodine receptors (RyRs) on the sarcoplasmic reticulum (SR). Calcium released from SR channels is known to be responsible for the sparks. Therefore, the activities of RyRs provide straightforward indication to the calcium concentration alteration. A method to study calcium signaling by analyzing RyR-gating statistics is described in the present study. Here we propose a univariate model with a simplified geometry of the dyadic cleft, which specifies the spatial localization of LCCs and RyRs to monitor the activity changes of RyRs. This model is used to explore two crucial aspects of local calcium signaling: the first is to disclose the tight control of calcium influxes via LCCs, and the second is to reveal the interactional impact of the self-regenerative RyRs. Patterns of active RyRs are rendered through numerous computational simulation experiments, manipulating the state initialization and the spatial localization of LCCs and RyRs to observe gating transition of RyRs.

Accurate particle-based simulation of adsorption, desorption and partial transmission

Particle-based simulators represent molecules of interest with point-like particles that diffuse and react in continuous space. These simulators are often used to investigate spatial or stochastic aspects of biochemical systems. This paper presents new particle-based simulation algorithms for modeling interactions between molecules and surfaces; they address irreversible and reversible molecular adsorption to, desorption from and transmission through membranes. Their central elements are: (i) relationships between adsorption, desorption and transmission coefficients on the one hand, and simulator interaction probabilities on the other, and (ii) probability densities for initial placements of desorbed molecules. These algorithms, which were implemented and tested in the Smoldyn simulator, are accurate, easy to implement and computationally efficient. They allow longer time steps and better address reversible processes than an algorithm that Erban and Chapman recently presented (Physical Biology 4:16-28, 2007). This paper also presents a method for simulating unbounded diffusion in a limited spatial domain using a partially absorbing boundary, as well as new solutions to the diffusion differential equation with reversible Robin boundary conditions.

Three-dimensional electron microscopy reveals new details of membrane systems for Ca2+ signaling in the heart

In the current study, the three-dimensional (3D) topologies of dyadic clefts and associated membrane organelles were mapped in mouse ventricular myocardium using electron tomography. The morphological details and the distribution of membrane systems, including transverse tubules (T-tubules), junctional sarcoplasmic reticulum (SR) and vicinal mitochondria, were determined and presumed to be crucial for controlling cardiac Ca(2+) dynamics. The geometric complexity of T-tubules that varied in diameter with frequent branching was clarified. Dyadic clefts were intricately shaped and remarkably small (average 4.39x10(5) nm(3), median 2.81x10(5) nm(3)). Although a dyadic cleft of average size could hold maximum 43 ryanodine receptor (RyR) tetramers, more than one-third of clefts were smaller than the size that is able to package as many as 15 RyR tetramers. The dyadic clefts were also adjacent to one another (average end-to-end distance to the nearest dyadic cleft, 19.9 nm) and were distributed irregularly along T-tubule branches. Electron-dense structures that linked membrane organelles were frequently observed between mitochondrial outer membranes and SR or T-tubules. We, thus, propose that the topology of dyadic clefts and the neighboring cellular micro-architecture are the major determinants of the local control of Ca(2+) in the heart, including the establishment of the quantal nature of SR Ca(2+) releases (e.g. Ca(2+) sparks).

Ryanodine receptor-mediated arrhythmias and sudden cardiac death

The cardiac ryanodine receptor-Ca²⁺ release channel (RyR2) is an essential sarcoplasmic reticulum (SR) transmembrane protein that plays a central role in excitation–contraction coupling (ECC) in cardiomyocytes. Aberrant spontaneous, diastolic Ca²⁺ leak from the SR due to dysfunctional RyR2 contributes to the formation of delayed after-depolarisations, which are thought to underlie the fatal arrhythmia that occurs in both heart failure (HF) and in catecholaminergic polymorphic ventricular tachycardia (CPVT). CPVT is an inherited disorder associated with mutations in either the RyR2 or a SR luminal protein, calsequestrin. RyR2 shows normal function at rest in CPVT but the RyR2 dysfunction is unmasked by physical exercise or emotional stress, suggesting abnormal RyR2 activation as an underlying mechanism. Several potential mechanisms have been advanced to explain the dysfunctional RyR2 observed in HF and CPVT, including enhanced RyR2 phosphorylation status, altered RyR2 regulation at luminal/cytoplasmic sites and perturbed RyR2 intra/inter-molecular interactions. This review considers RyR2 dysfunction in the context of the structural and functional modulation of the channel, and potential therapeutic strategies to stabilise RyR2 function in cardiac pathology.

Rapid creation, Monte Carlo simulation, and visualization of realistic 3D cell models

Spatially realistic diffusion-reaction simulations supplement traditional experiments and provide testable hypotheses for complex physiological systems. To date, however, the creation of realistic 3D cell models has been difficult and time-consuming, typically involving hand reconstruction from electron microscopic images. Here, we present a complementary approach that is much simpler and faster, because the cell architecture (geometry) is created directly in silico using 3D modeling software like that used for commercial film animations. We show how a freely available open source program (Blender) can be used to create the model geometry, which then can be read by our Monte Carlo simulation and visualization softwares (MCell and DReAMM, respectively). This new workflow allows rapid prototyping and development of realistic computational models, and thus should dramatically accelerate their use by a wide variety of computational and experimental investigators. Using two self-contained examples based on synaptic transmission, we illustrate the creation of 3D cellular geometry with Blender, addition of molecules, reactions, and other run-time conditions using MCell's Model Description Language (MDL), and subsequent MCell simulations and DReAMM visualizations. In the first example, we simulate calcium influx through voltage-gated channels localized on a presynaptic bouton, with subsequent intracellular calcium diffusion and binding to sites on synaptic vesicles. In the second example, we simulate neurotransmitter release from synaptic vesicles as they fuse with the presynaptic membrane, subsequent transmitter diffusion into the synaptic cleft, and binding to postsynaptic receptors on a dendritic spine.

Three-dimensional geometric modeling of membrane-bound organelles in ventricular myocytes: bridging the gap between microscopic imaging and mathematical simulation

A general framework of image-based geometric processing is presented to bridge the gap between three-dimensional (3D) imaging that provides structural details of a biological system and mathematical simulation where high-quality surface or volumetric meshes are required. A 3D density map is processed in the order of image pre-processing (contrast enhancement and anisotropic filtering), feature extraction (boundary segmentation and skeletonization), and high-quality and realistic surface (triangular) and volumetric (tetrahedral) mesh generation. While the tool-chain described is applicable to general types of 3D imaging data, the performance is demonstrated specifically on membrane-bound organelles in ventricular myocytes that are imaged and reconstructed with electron microscopic (EM) tomography and two-photon microscopy (T-PM). Of particular interest in this study are two types of membrane-bound Ca(2+)-handling organelles, namely, transverse tubules (T-tubules) and junctional sarcoplasmic reticulum (jSR), both of which play an important role in regulating the excitation-contraction (E-C) coupling through dynamic Ca(2+) mobilization in cardiomyocytes.

Ryanodine receptor-mediated Ca2+ events in atrial myocytes of patients with atrial fibrillation

BACKGROUND

Many alterations in sarcoplasmic reticulum Ca(2+) handling proteins in atrial myocytes have been associated with atrial fibrillation (AF) in clinical patients, whereas the functional consequences of these alterations mostly remain unclear.

METHODS AND RESULTS

To know whether or not ryanodine receptor (RyR)-mediated intracellular Ca(2+) events in AF atrial myocytes are affected by protein alterations, we investigated spontaneous Ca(2+) sparks and Ca(2+) waves in intact and permeabilized atrial myocytes of AF patients (n(patients) = 21) and normal sinus rhythm (NSR) patients (n(patients) = 22) by laser scanning confocal microscopy. It was found that the frequency, amplitude and rise time of Ca(2+) sparks were comparable between AF and NSR groups, while full width and full duration at half maximum intensity significantly increased in the AF group. Along with these changes, the frequency of small and global Ca(2+) waves increased in AF atrial myocytes.

CONCLUSIONS

Our results clearly indicated that the spatiotemporal properties but not the frequency of Ca(2+) sparks were affected in AF atrial myocytes. In addition, the frequency of Ca(2+) waves increased. This profile of the alterations in RyR-mediated Ca(2+) events in AF atrial myocytes was different from previous studies. The underlying mechanisms, as well as possible reasons for this discrepancy, were discussed.

Ryanodine receptor allosteric coupling and the dynamics of calcium sparks

Puffs and sparks are localized intracellular Ca(2+) elevations that arise from the cooperative activity of Ca(2+)-regulated inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or the sarcoplasmic reticulum. While the synchronous gating of Ca(2+)-regulated Ca(2+) channels can be mediated entirely though the buffered diffusion of intracellular Ca(2+), interprotein allosteric interactions also contribute to the dynamics of ryanodine receptor (RyR) gating and Ca(2+) sparks. In this article, Markov chain models of Ca(2+) release sites are used to investigate how the statistics of Ca(2+) spark generation and termination are related to the coupling of RyRs via local [Ca(2+)] changes and allosteric interactions. Allosteric interactions are included in a manner that promotes the synchronous gating of channels by stabilizing neighboring closed-closed and/or open-open channel pairs. When the strength of Ca(2+)-mediated channel coupling is systematically varied (e.g., by changing the Ca(2+) buffer concentration), simulations that include synchronizing allosteric interactions often exhibit more robust Ca(2+) sparks; however, for some Ca(2+) coupling strengths the sparks are less robust. We find no evidence that the distribution of spark durations can be used to distinguish between allosteric interactions that stabilize closed channel pairs, open channel pairs, or both in a balanced fashion. On the other hand, the changes in spark duration, interspark interval, and frequency observed when allosteric interactions that stabilize closed channel pairs are gradually removed from simulations are qualitatively different than the changes observed when open or both closed and open channel pairs are stabilized. Thus, our simulations clarify how changes in spark statistics due to pharmacological washout of the accessory proteins mediating allosteric coupling may indicate the type of synchronizing allosteric interactions exhibited by physically coupled RyRs. We also investigate the validity of a mean-field reduction applicable to the dynamics of a ryanodine receptor cluster coupled via local [Ca(2+)] and allosteric interactions. In addition to facilitating parameter studies of the effect of allosteric coupling on spark statistics, the derivation of the mean-field model establishes the correct functional form for cooperativity factors representing the coupled gating of RyRs. This mean-field formulation is well suited for use in computationally efficient whole cell simulations of excitation-contraction coupling.

Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion

Ca(2+) signaling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca(2+) ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time-steps are integrated and smoothed at the larger timescale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca(2+) signaling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca(2+) diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bimolecular binding probabilities, which we use in the RW simulations and the statistical analysis.

Protein geometry and placement in the cardiac dyad influence macroscopic properties of calcium-induced calcium release

In cardiac ventricular myocytes, events crucial to excitation-contraction coupling take place in spatially restricted microdomains known as dyads. The movement and dynamics of calcium (Ca2+) ions in the dyad have often been described by assigning continuously valued Ca2+ concentrations to one or more dyadic compartments. However, even at its peak, the estimated number of free Ca2+ ions present in a single dyad is small (approximately 10-100 ions). This in turn suggests that modeling dyadic calcium dynamics using laws of mass action may be inappropriate. In this study, we develop a model of stochastic molecular signaling between L-type Ca2+ channels (LCCs) and ryanodine receptors (RyR2s) that describes: a), known features of dyad geometry, including the space-filling properties of key dyadic proteins; and b), movement of individual Ca2+ ions within the dyad, as driven by electrodiffusion. The model enables investigation of how local Ca2+ signaling is influenced by dyad structure, including the configuration of key proteins within the dyad, the location of Ca2+ binding sites, and membrane surface charges. Using this model, we demonstrate that LCC-RyR2 signaling is influenced by both the stochastic dynamics of Ca2+ ions in the dyad as well as the shape and relative positioning of dyad proteins. Results suggest the hypothesis that the relative placement and shape of the RyR2 proteins helps to "funnel" Ca2+ ions to RyR2 binding sites, thus increasing excitation-contraction coupling gain.