Spatiotemporal features of Ca2+ buffering and diffusion in atrial cardiac myocytes with inhibited sarcoplasmic reticulum

Physiology of intracellular calcium buffering

Calcium signaling underlies much of physiology. Almost all the Ca2+ in the cytoplasm is bound to buffers, with typically only ∼1% being freely ionized at resting levels in most cells. Physiological Ca2+ buffers include small molecules and proteins, and experimentally Ca2+ indicators will also buffer calcium. The chemistry of interactions between Ca2+ and buffers determines the extent and speed of Ca2+ binding. The physiological effects of Ca2+ buffers are determined by the kinetics with which they bind Ca2+ and their mobility within the cell. The degree of buffering depends on factors such as the affinity for Ca2+, the Ca2+ concentration, and whether Ca2+ ions bind cooperatively. Buffering affects both the amplitude and time course of cytoplasmic Ca2+ signals as well as changes of Ca2+ concentration in organelles. It can also facilitate Ca2+ diffusion inside the cell. Ca2+ buffering affects synaptic transmission, muscle contraction, Ca2+ transport across epithelia, and the killing of bacteria. Saturation of buffers leads to synaptic facilitation and tetanic contraction in skeletal muscle and may play a role in inotropy in the heart. This review focuses on the link between buffer chemistry and function and how Ca2+ buffering affects normal physiology and the consequences of changes in disease. As well as summarizing what is known, we point out the many areas where further work is required.

A detailed mathematical model of the human atrial cardiomyocyte: integration of electrophysiology and cardiomechanics

Mechano-electric regulations (MER) play an important role in the maintenance of cardiac performance. Mechano-calcium and mechano-electric feedback (MCF and MEF) pathways adjust the cardiomyocyte contractile force according to mechanical perturbations and affects electro-mechanical coupling. MER integrates all these regulations in one unit resulting in a complex phenomenon. Computational modelling is a useful tool to accelerate the mechanistic understanding of complex experimental phenomena. We have developed a novel model that integrates the MER loop for human atrial cardiomyocytes with proper consideration of feedforward and feedback pathways. The model couples a modified version of the action potential (AP) Koivumäki model with the contraction model by Quarteroni group. The model simulates iso-sarcometric and isometric twitches and the feedback effects on AP and Ca2+ -handling. The model showed a biphasic response of Ca2+ transient (CaT) peak to increasing pacing rates and highlights the possible mechanisms involved. The model has shown a shift of the threshold for AP and CaT alternans from 4.6 to 4 Hz under post-operative atrial fibrillation, induced by depressed SERCA activity. The alternans incidence was dependent on a chain of mechanisms including RyRs availability time, MCF coupling, CaMKII phosphorylation, and the stretch levels. As a result, the model predicted a 10% slowdown of conduction velocity for a 20% stretch, suggesting a role of stretch in creation of substrate formation for atrial fibrillation. Overall, we conclude that the developed model provides a physiological CaT followed by a physiological twitch. This model can open pathways for the future studies of human atrial electromechanics. KEY POINTS: With the availability of human atrial cellular data, interest in atrial-specific model integration has been enhanced. We have developed a detailed mathematical model of human atrial cardiomyocytes including the mechano-electric regulatory loop. The model has gone through calibration and evaluation phases against a wide collection of available human in-vitro data. The usefulness of the model for analysing clinical problems has been preliminaryly tested by simulating the increased incidence of Ca2+ transient and action potential alternans at high rates in post-operative atrial fibrillation condition. The model determines the possible role of mechano-electric feedback in alternans incidence, which can increase vulnerability to atrial arrhythmias by varying stretch levels. We found that our physiologically accurate description of Ca2+ handling can reproduce many experimental phenomena and can help to gain insights into the underlying pathophysiological mechanisms.

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.

Excitation-Contraction Coupling in the Goldfish (Carassius auratus) Intact Heart

Cardiac physiology of fish models is an emerging field given the ease of genome editing and the development of transgenic models. Several studies have described the cardiac properties of zebrafish (Denio rerio). The goldfish (Carassius auratus) belongs to the same family as the zebrafish and has emerged as an alternative model with which to study cardiac function. Here, we propose to acutely study electrophysiological and systolic Ca²⁺ signaling in intact goldfish hearts. We assessed the Ca²⁺ dynamics and the electrophysiological cardiac function of goldfish, zebrafish, and mice models, using pulsed local field fluorescence microscopy, intracellular microelectrodes, and flash photolysis in perfused hearts. We observed goldfish ventricular action potentials (APs) and Ca²⁺ transients to be significantly longer when compared to the zebrafish. The action potential half duration at 50% (APD₅₀) of goldfish was 370.38 ± 8.8 ms long, and in the zebrafish they were observed to be only 83.9 ± 9.4 ms. Additionally, the half duration of the Ca²⁺ transients was also longer for goldfish (402.1 ± 4.4 ms) compared to the zebrafish (99.1 ± 2.7 ms). Also, blocking of the L-type Ca²⁺ channels with nifedipine revealed this current has a major role in defining the amplitude and the duration of goldfish Ca²⁺ transients. Interestingly, nifedipine flash photolysis experiments in the intact heart identified whether or not the decrease in the amplitude of Ca²⁺ transients was due to shorter APs. Moreover, an increase in temperature and heart rate had a strong shortening effect on the AP and Ca²⁺ transients of goldfish hearts. Furthermore, ryanodine (Ry) and thapsigargin (Tg) significantly reduced the amplitude of the Ca²⁺ transients, induced a prolongation in the APs, and altogether exhibited the degree to which the Ca²⁺ release from the sarcoplasmic reticulum contributed to the Ca²⁺ transients. We conclude that the electrophysiological properties and Ca²⁺ signaling in intact goldfish hearts strongly resembles the endocardial layer of larger mammals.

Calcium Buffering in the Heart in Health and Disease

Changes of intracellular Ca²⁺ concentration regulate many aspects of cardiac myocyte function. About 99% of the cytoplasmic calcium in cardiac myocytes is bound to buffers, and their properties will therefore have a major influence on Ca²⁺ signaling. This article considers the fundamental properties and identities of the buffers and how to measure them. It reviews the effects of buffering on the systolic Ca²⁺ transient and how this may change physiologically, and in heart failure and both atrial and ventricular arrhythmias, as well. It is concluded that the consequences of this strong buffering may be more significant than currently appreciated, and a fuller understanding is needed for proper understanding of cardiac calcium cycling and contractility.

Simulation of calcium signaling in fine astrocytic processes: Effect of spatial properties on spontaneous activity

Astrocytes, a glial cell type of the central nervous system, have emerged as detectors and regulators of neuronal information processing. Astrocyte excitability resides in transient variations of free cytosolic calcium concentration over a range of temporal and spatial scales, from sub-microdomains to waves propagating throughout the cell. Despite extensive experimental approaches, it is not clear how these signals are transmitted to and integrated within an astrocyte. The localization of the main molecular actors and the geometry of the system, including the spatial organization of calcium channels IP3R, are deemed essential. However, as most calcium signals occur in astrocytic ramifications that are too fine to be resolved by conventional light microscopy, most of those spatial data are unknown and computational modeling remains the only methodology to study this issue. Here, we propose an IP3R-mediated calcium signaling model for dynamics in such small sub-cellular volumes. To account for the expected stochasticity and low copy numbers, our model is both spatially explicit and particle-based. Extensive simulations show that spontaneous calcium signals arise in the model via the interplay between excitability and stochasticity. The model reproduces the main forms of calcium signals and indicates that their frequency crucially depends on the spatial organization of the IP3R channels. Importantly, we show that two processes expressing exactly the same calcium channels can display different types of calcium signals depending on the spatial organization of the channels. Our model with realistic process volume and calcium concentrations successfully reproduces spontaneous calcium signals that we measured in calcium micro-domains with confocal microscopy and predicts that local variations of calcium indicators might contribute to the diversity of calcium signals observed in astrocytes. To our knowledge, this model is the first model suited to investigate calcium dynamics in fine astrocytic processes and to propose plausible mechanisms responsible for their variability.

Three dimensional coupled reaction–diffusion modeling of calcium and inositol 1,4,5-trisphosphate dynamics in cardiomyocytes

Nanoparticles have shown great promise in improving cancer treatment efficacy by changing the intracellular calcium level through activation of intracellular mechanisms. One of the mechanisms of the killing of the cancerous cell by a nanoparticle is through elevation of the intracellular calcium level. Evidence accumulated over the past decade indicates a pivotal role for the IP₃ receptor mediated Ca²⁺ release in the regulation of the cytosolic and the nuclear Ca²⁺ signals. There have been various studies done suggesting the role of IP₃ receptors (IP₃R) and IP₃ production and degradation in cardiomyocytes. In the present work, we have proposed a three-dimensional unsteady-state mathematical model to describe the mechanism of cardiomyocytes which focuses on evaluation of various parameters that affect these coupled dynamics and elevate the cytosolic calcium concentration which can be helpful to search for novel therapies to cure these malignancies by targeting the complex calcium signaling process in cardiomyocytes. Our study suggests that there are other factors involved in this signaling which can increase the calcium level, which can help in finding treatment for cancer. The cytosolic calcium level may be controlled by IP₃ signaling, leak, source influx of calcium (σ) and maximum production of IP₃ (V_P). We believe that the proposed model suggests new insight into finding treatment for cancer in cardiomyocytes through elevation of the cytosolic Ca²⁺ concentration by various parameters like leak, σ, V_P and especially by other complex cell signaling dynamics, namely IP₃ dynamics.

We propose a three-dimensional unsteady-state mathematical model to describe the mechanism of cardiomyocytes.

A composite visualization method for electrophysiology-morphous merging of human heart

BACKGROUND

Electrophysiological behavior is of great importance for analyzing the cardiac functional mechanism under cardiac physiological and pathological condition. Due to the complexity of cardiac structure and biophysiological function, visualization of a cardiac electrophysiological model compositively is still a challenge. The lack of either modality of the whole organ structure or cardiac electrophysiological behaviors makes analysis of the intricate mechanisms of cardiac dynamic function a difficult task. This study aims at exploring 3D conduction of stimulus and electrical excitation reactivity on the level of organ with the authentic fine cardiac anatomy structure.

METHODS

In this paper, a cardiac electrical excitation propagation model is established based on the human cardiac cross-sectional data to explore detailed cardiac electrical activities. A novel biophysical merging visualization method is then presented for biophysical integration of cardiac anatomy and electrophysiological properties in the form of the merging optical model, which provides the corresponding position, spatial relationship and the whole process in 3D space with the context of anatomical structure for representing the biophysical detailed electrophysiological activity.

RESULTS

The visualization result present the action potential propagation of the left ventricle within the excitation cycle with the authentic fine cardiac organ anatomy. In the visualized images, all vital organs are identified and distinguished without ambiguity. The three dimensional spatial position, relation and the process of cardiac excitation conduction and re-entry propagation in the anatomical structure during the phase of depolarization and repolarization is also shown in the result images, which exhibits the performance of a more detailed biophysical understanding of the electrophysiological kinetics of human heart in vivo.

CONCLUSIONS

Results suggest that the proposed merging optical model can merge cardiac electrophysiological activity with the anatomy structure. By specifying the respective opacity for the cardiac anatomy structure and the electrophysiological model in the merging attenuation function, the visualized images can provide an in-depth insight into the biophysical detailed cardiac functioning phenomena and the corresponding electrophysiological behavior mechanism, which is helpful for further speculating cardiac physiological and pathological responses and is fundamental to the cardiac research and clinical diagnoses.

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

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

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.

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.

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.

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.

Recruitment of the intracellular Ca2+ by ultrashort electric stimuli: the impact of pulse duration

Nanosecond-duration electric stimuli are distinguished by the ability to permeabilize intracellular membranes and recruit Ca2+ from intracellular stores. We quantified this effect in non-excitable cells (CHO) using ratiometric Ca2+ imaging with Fura-2. In a Ca(2+)-free medium, 10-, 60-, and 300-ns stimuli evoked Ca2+ transients by mobilization of Ca2+ from the endoplasmic reticulum. With 2 mM external Ca2+, the transients included both extra- and intracellular components. The recruitment of intracellular Ca2+ increased as the stimulus duration decreased. At the threshold of 200-300 nM, the transients were amplified by calcium-induced calcium release. We conclude that nanosecond stimuli mimic Ca2+ signaling while bypassing the usual receptor- and channels-mediated cascades. The recruitment of the intracellular Ca2+ can be controlled by the duration of the stimulus.

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh-Nagumo model, to get the buffered FitzHugh-Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant (K = K(a)) characterized by system excitability parameter a such that calcium buffers can be classified into two types: weak buffers (K ∈ (K(a), ∞)) and strong buffers (K ∈ (0, K(a))). We analytically show that the addition of weak buffers or strong buffers but with its total concentration b(0)(1) below some critical total concentration b(0,c)(1) into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity D1 of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to √D1 as D1 --> ∞. In contrast, the addition of strong buffers with the total concentration b(0)(1) > b(0,c)(1) into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity D1 of the added buffers is sufficiently large.

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.

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

Triggered release of Ca2+ from an individual sarcoplasmic reticulum (SR) Ca(2+) release unit (CRU) is the fundamental event of cardiac excitation–contraction coupling, and spontaneous release events (sparks) are the major contributor to diastolic Ca(2+) leak in cardiomyocytes. Previous model studies have predicted that the duration and magnitude of the spark is determined by the local CRU geometry, as well as the localization and density of Ca(2+) handling proteins. We have created a detailed computational model of a CRU, and developed novel tools to generate the computational geometry from electron tomographic images. Ca(2+) diffusion was modelled within the SR and the cytosol to examine the effects of localization and density of the Na(+)/Ca(2+) exchanger, sarco/endoplasmic reticulum Ca(2+)-ATPase 2 (SERCA), and calsequestrin on spark dynamics. We reconcile previous model predictions of approximately 90% local Ca(2+) depletion in junctional SR, with experimental reports of about 40%. This analysis supports the hypothesis that dye kinetics and optical averaging effects can have a significant impact on measures of spark dynamics. Our model also predicts that distributing calsequestrin within non-junctional Z-disc SR compartments, in addition to the junctional compartment, prolongs spark release time as reported by Fluo5. By pumping Ca(2+) back into the SR during a release, SERCA is able to prolong a Ca(2+) spark, and this may contribute to SERCA-dependent changes in Ca(2+) wave speed. Finally, we show that including the Na(+)/Ca(2+) exchanger inside the dyadic cleft does not alter local [Ca(2+)] during a spark.

Multi-Scale Continuum Modeling of Biological Processes: From Molecular Electro-Diffusion to Sub-Cellular Signaling Transduction

This article provides a brief review of multi-scale modeling at the molecular to cellular scale, with new results for heart muscle cells. A finite element-based simulation package (SMOL) was used to investigate the signaling transduction at molecular and sub-cellular scales (http://mccammon.ucsd.edu/smol/, http://FETK.org) by numerical solution of time-dependent Smoluchowski equations and a reaction-diffusion system. At the molecular scale, SMOL has yielded experimentally-validated estimates of the diffusion-limited association rates for the binding of acetylcholine to mouse acetylcholinesterase using crystallographic structural data. The predicted rate constants exhibit increasingly delayed steady-state times with increasing ionic strength and demonstrate the role of an enzyme's electrostatic potential in influencing ligand binding. At the sub-cellular scale, an extension of SMOL solves a non-linear, reaction-diffusion system describing Ca²⁺ ligand buffering and diffusion in experimentally-derived rodent ventricular myocyte geometries. Results reveal the important role for mobile and stationary Ca²⁺ buffers, including Ca²⁺ indicator dye. We found that the alterations in Ca²⁺-binding and dissociation rates of troponin C (TnC) and total TnC concentration modulate subcellular Ca²⁺ signals. Model predicts that reduced off-rate in whole troponin complex (TnC, TnI, TnT) versus reconstructed thin filaments (Tn, Tm, actin) alters cytosolic Ca²⁺ dynamics under control conditions or in disease-linked TnC mutations. The ultimate goal of these studies is to develop scalable methods and theories for integration of molecular-scale information into simulations of cellular-scale systems.

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.

Impact of sarcoplasmic reticulum calcium release on calcium dynamics and action potential morphology in human atrial myocytes: a computational study

Electrophysiological studies of the human heart face the fundamental challenge that experimental data can be acquired only from patients with underlying heart disease. Regarding human atria, there exist sizable gaps in the understanding of the functional role of cellular Ca²+ dynamics, which differ crucially from that of ventricular cells, in the modulation of excitation-contraction coupling. Accordingly, the objective of this study was to develop a mathematical model of the human atrial myocyte that, in addition to the sarcolemmal (SL) ion currents, accounts for the heterogeneity of intracellular Ca²+ dynamics emerging from a structurally detailed sarcoplasmic reticulum (SR). Based on the simulation results, our model convincingly reproduces the principal characteristics of Ca²+ dynamics: 1) the biphasic increment during the upstroke of the Ca²+ transient resulting from the delay between the peripheral and central SR Ca²+ release, and 2) the relative contribution of SL Ca²+ current and SR Ca²+ release to the Ca²+ transient. In line with experimental findings, the model also replicates the strong impact of intracellular Ca²+ dynamics on the shape of the action potential. The simulation results suggest that the peripheral SR Ca²+ release sites define the interface between Ca²+ and AP, whereas the central release sites are important for the fire-diffuse-fire propagation of Ca²+ diffusion. Furthermore, our analysis predicts that the modulation of the action potential duration due to increasing heart rate is largely mediated by changes in the intracellular Na+ concentration. Finally, the results indicate that the SR Ca²+ release is a strong modulator of AP duration and, consequently, myocyte refractoriness/excitability. We conclude that the developed model is robust and reproduces many fundamental aspects of the tight coupling between SL ion currents and intracellular Ca²+ signaling. Thus, the model provides a useful framework for future studies of excitation-contraction coupling in human atrial myocytes.

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.

Cardiac cell modelling: observations from the heart of the cardiac physiome project

In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.

Multiscale modeling in rodent ventricular myocytes

There is a growing body of experimental evidence suggesting that the Ca(2+) signaling in ventricular myocytes is characterized by a high gradient near the cell membrane and a more uniform Ca(2+) distribution in the cell interior [1]--[7]. An important reason for this phenomenon might be that in these cells the t-tubular system forms a network of extracellular space, extending deep into the cell interior. This allows the electrical signal, that propagates rapidly along the cell membrane, to reach the vicinity of the sarcoplasmic reticulum (SR), where intracellular Ca(2+) required for myofilament activation is stored [1], [8]--[11]. Early studies of cardiac muscle showed that the t-tubules are found at intervals of about 2 lm along the longitudinal cell axis in close proximity to the Z-disks of the sarcomeres [12]. Subsequent studies have demonstrated that the t-tubular system has also longitudinal extensions [9]--[11], [13].

Model of excitation-contraction coupling of rat neonatal ventricular myocytes

The neonatal rat ventricular myocyte culture is one of the most popular experimental cardiac cell models. To our knowledge, the excitation-contraction coupling (ECC) of these cells, i.e., the process linking the electrical activity to the cytosolic Ca2+ transient and contraction, has not been previously analyzed, nor has it been presented as a complete system in detail. Neonatal cardiomyocytes are in the postnatal developmental stage, and therefore, the features of their ECC differ vastly from those of adult ventricular myocytes. We present the first complete analysis of ECC in these cells by characterizing experimentally the action potential and calcium signaling and developing the first mathematical model of ECC in neonatal cardiomyocytes that we know of. We show that in comparison to adult cardiomyocytes, neonatal cardiomyocytes have long action potentials, heterogeneous cytosolic Ca2+ signals, weaker sarcoplasmic reticulum Ca2+ handling, and stronger sarcolemmal Ca2+ handling, with a significant contribution by the Na+/Ca2+ exchanger. The developed model reproduces faithfully the ECC of rat neonatal cardiomyocytes with a novel description of spatial cytosolic [Ca2+] signals. Simulations also demonstrate how an increase in the cell size (hypertrophy) affects the ECC in neonatal cardiomyocytes. This model of ECC in developing cardiomyocytes provides a platform for developing future models of cardiomyocytes at different developmental stages.

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.

Termination of cardiac Ca2+ sparks: role of intra-SR [Ca2+], release flux, and intra-SR Ca2+ diffusion

Ca(2+) release from cardiac sarcoplasmic reticulum (SR) via ryanodine receptors (RyRs) is regulated by dyadic cleft [Ca(2+)] and intra-SR free [Ca(2+)] ([Ca(2+)](SR)). Robust SR Ca(2+) release termination is important for stable excitation-contraction coupling, and partial [Ca(2+)](SR) depletion may contribute to release termination. Here, we investigated the regulation of SR Ca(2+) release termination of spontaneous local SR Ca(2+) release events (Ca(2+) sparks) by [Ca(2+)](SR), release flux, and intra-SR Ca(2+) diffusion. We simultaneously measured Ca(2+) sparks and Ca(2+) blinks (localized elementary [Ca(2+)](SR) depletions) in permeabilized ventricular cardiomyocytes over a wide range of SR Ca(2+) loads and release fluxes. Sparks terminated via a [Ca(2+)](SR)-dependent mechanism at a fixed [Ca(2+)](SR) depletion threshold independent of the initial [Ca(2+)](SR) and release flux. Ca(2+) blink recovery depended mainly on intra-SR Ca(2+) diffusion rather than SR Ca(2+) uptake. Therefore, the large variation in Ca(2+) blink recovery rates at different release sites occurred because of differences in the degree of release site interconnection within the SR network. When SR release flux was greatly reduced, long-lasting release events occurred from well-connected junctions. These junctions could sustain release because local SR Ca(2+) release and [Ca(2+)](SR) refilling reached a balance, preventing [Ca(2+)](SR) from depleting to the termination threshold. Prolonged release events eventually terminated at a steady [Ca(2+)](SR), indicative of a slower, [Ca(2+)](SR)-independent termination mechanism. These results demonstrate that there is high variability in local SR connectivity but that SR Ca(2+) release terminates at a fixed [Ca(2+)](SR) termination threshold. Thus, reliable SR Ca(2+) release termination depends on tight RyR regulation by [Ca(2+)](SR).

Sarcomere mechanics in uniform and nonuniform cardiac muscle: a link between pump function and arrhythmias

Starling's law and the end-systolic pressure-volume relationship (ESPVR) reflect the effect of sarcomere length (SL) on the development of stress (sigma) and shortening by myocytes in the uniform ventricle. We show here that tetanic contractions of rat cardiac trabeculae exhibit a sigma-SL relationship at saturating [Ca2+] that depends on sarcomere geometry in a manner similar to that of skeletal sarcomeres and the existence of opposing forces in cardiac muscle shortened below slack length. The sigma-SL -[Ca2+](free) relationships (sigma-SL-Ca relationships) at submaximal [Ca2+] in intact and skinned trabeculae were similar, although the sensitivity for Ca2+ of intact muscle was higher. We analyzed the mechanisms underlying the sigma-SL-Ca relationship by using a kinetic model assuming that the rates of Tn-C Ca2+ binding and/or cross-bridge (XB) cycling are determined by either the SL, [Ca2+], or sigma. We analyzed the correlation between the model results and steady-state sigma measurements at varied SL at [Ca2+] from skinned rat cardiac trabeculae to test the hypotheses that the dominant feedback mechanism is SL-, sigma-, or [Ca2+]-dependent, and that the feedback mechanism regulates Tn-C Ca2+ affinity, XB kinetics, or the unitary XB force. The analysis strongly suggests that the feedback of the number of strong XBs to cardiac Tn-C Ca2+ affinity is the dominant mechanism regulating XB recruitment. Using this concept in a model of twitch-sigma accurately reproduced the sigma-SL-Ca relationship and the time courses of twitch sigma and the intracellular [Ca2+]i. The foregoing concept has equally important repercussions for the nonuniformly contracting heart, in which arrhythmogenic Ca2+ waves arise from weakened areas in the cardiac muscle. These Ca2+ waves can reversibly be induced with nonuniform excitation-contraction coupling (ECC) by the cycle of stretch and release in the border zone between the damaged and intact regions. Stimulus trains induced propagating Ca2+ waves and reversibly induced arrhythmias. We hypothesize that rapid force loss by the sarcomeres in the border zone during relaxation causes Ca2+ release from Tn-C and initiates Ca2+ waves propagated by the sarcoplasmic reticulum (SR). Modeling of the response of the cardiac twitch to rapid force changes using the feedback concept uniquely predicts the occurrence of [Ca2+]i transients as a result of accelerated Ca2+ dissociation from Tn-C. These results are consistent with the hypothesis that a force feedback to Ca2+ binding by Tn-C is responsible for Starling's law and the ESPVR in the uniform myocardium and leads to a surge of Ca2+ released by the myofilaments during relaxation in the nonuniform myocardium, which initiates arrhythmogenic propagating Ca2+ release by the SR.

Three-dimensional distribution of cardiac Na+-Ca2+ exchanger and ryanodine receptor during development

Mechanisms of cardiac excitation-contraction coupling in neonates are still not clearly defined. Previous work in neonates shows reverse-mode Na(+)-Ca(2+) exchange to be the primary route of Ca(2+) entry during systole and the neonatal sarcoplasmic reticulum to have similar capability as that of adult in storing and releasing Ca(2+). We investigated Na(+)-Ca(2+) exchanger (NCX) and ryanodine receptor (RyR) distribution in developing ventricular myocytes using immunofluorescence, confocal microscopy, and digital image analysis. In neonates, both NCX and RyR clusters on the surface of the cell displayed a short longitudinal periodicity of approximately 0.7 microm. However, by adulthood, both proteins were also found in the interior. In the adult, clusters of NCX on the surface of the cell retained the approximately 0.7-microm periodicity whereas clusters of RyR adopted a longer longitudinal periodicity of approximately 2.0 microm. This suggests that neonatal myocytes also have a peri-M-line RyR distribution that is absent in adult myocytes. NCX and RyR colocalized voxel density was maximal in neonates and declined significantly with ontogeny. We conclude in newborns, Ca(2+) influx via NCX could potentially activate the dense network of peripheral Ca(2+) stores via peripheral couplings, evoking Ca(2+)-induced Ca(2+) release.

Approximate analytical time-dependent solutions to describe large-amplitude local calcium transients in the presence of buffers

Local Ca(2+) signaling controls many neuronal functions, which is often achieved through spatial localization of Ca(2+) signals. These nanodomains are formed due to combined effects of Ca(2+) diffusion and binding to the cytoplasmic buffers. In this article we derived simple analytical expressions to describe Ca(2+) diffusion in the presence of mobile and immobile buffers. A nonlinear character of the reaction-diffusion problem was circumvented by introducing a logarithmic approximation of the concentration term. The obtained formulas reproduce free Ca(2+) levels up to 50 microM and their changes in the millisecond range. Derived equations can be useful to predict spatiotemporal profiles of large-amplitude [Ca(2+)] transients, which participate in various physiological processes.

Modeling transmural heterogeneity of K(ATP) current in rabbit ventricular myocytes

To investigate the mechanisms regulating excitation-metabolic coupling in rabbit epicardial, midmyocardial, and endocardial ventricular myocytes we extended the LabHEART model (Puglisi JL and Bers DM. Am J Physiol Cell Physiol 281: C2049-C2060, 2001). We incorporated equations for Ca(2+) and Mg(2+) buffering by ATP and ADP, equations for nucleotide regulation of ATP-sensitive K(+) channel and L-type Ca(2+) channel, Na(+)-K(+)-ATPase, and sarcolemmal and sarcoplasmic Ca(2+)-ATPases, and equations describing the basic pathways (creatine and adenylate kinase reactions) known to communicate the flux changes generated by intracellular ATPases. Under normal conditions and during 20 min of ischemia, the three regions were characterized by different I(Na), I(to), I(Kr), I(Ks), and I(Kp) channel properties. The results indicate that the ATP-sensitive K(+) channel is activated by the smallest reduction in ATP in epicardial cells and largest in endocardial cells when cytosolic ADP, AMP, PCr, Cr, P(i), total Mg(2+), Na(+), K(+), Ca(2+), and pH diastolic levels are normal. The model predicts that only K(ATP) ionophore (Kir6.2 subunit) and not the regulatory subunit (SUR2A) might differ from endocardium to epicardium. The analysis suggests that during ischemia, the inhomogeneous accumulation of the metabolites in the tissue sublayers may alter in a very irregular manner the K(ATP) channel opening through metabolic interactions with the endogenous PI cascade (PIP(2), PIP) that in turn may cause differential action potential shortening among the ventricular myocyte subtypes. The model predictions are in qualitative agreement with experimental data measured under normal and ischemic conditions in rabbit ventricular myocytes.

Anomalous diffusion in Purkinje cell dendrites caused by spines

We combined local photolysis of caged compounds with fluorescence imaging to visualize molecular diffusion within dendrites of cerebellar Purkinje cells. Diffusion of a volume marker, fluorescein dextran, within spiny dendrites was remarkably slow in comparison to its diffusion in smooth dendrites. Computer simulations indicate that this retardation is due to a transient trapping of molecules within dendritic spines, yielding anomalous diffusion. We considered the influence of spine trapping on the diffusion of calcium ions (Ca(2+)) and inositol-1,4,5-triphospate (IP(3)), two synaptic second messengers. Diffusion of IP(3) was strongly influenced by the presence of dendritic spines, while Ca(2+) was removed so rapidly that it could not diffuse far enough to be trapped. We conclude that an important function of dendritic spines may be to trap chemical signals and thereby create slowed anomalous diffusion within dendrites.

Dynamic and static calcium gradients inside large snail (Helix aspersa) neurones detected with calcium-sensitive microelectrodes

We have used quartz Ca²⁺-sensitive microelectrodes (CASMs) in large voltage-clamped snail neurones to investigate the inward spread of Ca²⁺ after a brief depolarisation. Both steady state and [Ca²⁺]_i transients changed with depth of penetration. When the CASM tip was within 20 μm of the far side of the cell the [Ca²⁺]_i transient time to peak was 4.4 ± 0.5 s, rising to 14.7 ± 0.7 s at a distance of 80 μm. We estimate that the Ca²⁺ transients travelled centripetally at an average speed of 6 μm² s⁻¹ and decreased in size by half over a distance of about 45 μm. Cyclopiazonic acid had little effect on the size and time to peak of Ca²⁺ transients but slowed their recovery significantly. This suggests that the endoplasmic reticulum curtails rather than reinforces the transients. Injecting the calcium buffer BAPTA made the Ca²⁺ transients more uniform in size and increased their times to peak and rates of recovery near the membrane. We have developed a computational model for the transients, which includes diffusion, uptake and Ca²⁺ extrusion. Good fits were obtained with a rather large apparent diffusion coefficient of about 90 ± 20 μm² s⁻¹.This may assist fast recovery by extrusion.

Synapse-specific plasticity and compartmentalized signaling in cerebellar stellate cells

Here we demonstrate that cerebellar stellate cells diffusionally isolate synaptically evoked signals in dendrites and are capable of input-specific synaptic plasticity. Sustained activity of parallel fibers induces a form of long-term depression that requires opening of calcium (Ca(2+))-permeable AMPA-type glutamate receptors (CP-AMPARs) and signaling through class 1 metabotropic glutamate receptors (mGluR1) and CB1 receptors. This depression is induced by postsynaptic increases in Ca(2+) concentration ([Ca(2+)]) and is limited to activated synapses. To understand how synapse-specific plasticity is induced by diffusible second messengers in aspiny dendrites, we examined diffusion of Ca(2+) and small molecules within stellate cell dendrites. Activation of a single parallel fiber opened CP-AMPARs, generating long-lived Ca(2+) transients that were confined to submicron dendritic stretches. The diffusion of Ca(2+) was severely retarded due to interactions with parvalbumin and a general restriction of small molecule mobility. Thus stellate cell dendrites spatially restrict signaling cascades that lead from CP-AMPAR activation to endocannabinoid production and trigger the selective regulation of active synapses.

Spatial Nonuniformity of Contraction Causes Arrhythmogenic Ca2+Waves in Rat Cardiac Muscle

Landesberg and Sideman's four state model of the cardiac cross-bridge (XB) hypothesizes a feedback of force development to Ca(2+) binding by troponin C (TnC). We have further modeled this behavior and observed that the force (F)-Ca(2+) relationship as well as the F-sarcomere length (SL) relationship and the time course of F and Ca(2+) transients in cardiac muscle can be reproduced faithfully by a single effect of F on deformation of the TnC-Ca complex and, thereby, on the dissociation rate of Ca(2+). Furthermore, this feedback predicts that rapid decline of F in the activated sarcomere causes release of Ca(2+) from TnC-Ca(2+), which is sufficient to initiate arrhythmogenic Ca(2+) release from the sarcoplasmic reticulum (SR). This work investigated the initiation of Ca(2+) waves underlying triggered propagated contractions (TPCs) in rat cardiac trabeculae under conditions that simulate functional nonuniformity caused by mechanical or ischemic local damage of the myocardium. A mechanical discontinuity along the trabeculae was created by exposing the preparation to a small constant flow jet of solution that reduces excitation-contraction coupling in myocytes within that segment. Force was measured, and SL as well as [Ca(2+)](i) were measured regionally. When the jet contained caffeine, 2,3-butanedione monoxime or low-[Ca(2+)], muscle-twitch F decreased and the sarcomeres in the exposed segment were stretched by shortening the normal regions outside the jet. During relaxation, the sarcomeres in the exposed segment shortened rapidly. Short trains of stimulation at 2.5 Hz reproducibly caused Ca(2+) waves to rise from the borders exposed to the jet. Ca(2+) waves started during F relaxation of the last stimulated twitch and propagated into segments both inside and outside of the jet. Arrhythmias, in the form of nondriven rhythmic activity, were triggered when the amplitude of the Ca(2+) wave increased by raising [Ca(2+)](o). The arrhythmias disappeared when the muscle uniformity was restored by turning the jet off. These results show that nonuniform contraction can cause Ca(2+) waves underlying TPCs, and suggest that Ca(2+) dissociated from myofilaments plays an important role in the initiation of arrhythmogenic Ca(2+) waves.

Confocal imaging of CICR events from isolated and immobilized SR vesicles

The Ca2+ concentration inside the sarcoplasmic reticulum ([Ca2+]SR) is a difficult parameter to measure in ventricular cardiac myocytes. Interference from Ca2+-sensitive dye loading into cellular compartments other than the SR interferes with free Ca2+ measurement. In addition, the composition of the cytosol surrounding the SR in intact cells cannot be easily controlled. We have developed a method to measure localized [Ca2+]SR in immobilized membrane vesicles during rapid solution switches. Ca2+ uptake and release in rat SR membrane vesicles was monitored using confocal microscopy. Vesicles were immobilized on a coverslip using an agarose matrix. Perfusion with a Ca2+-containing solution supplemented with ATP initiated SR Ca2+ uptake, causing a rise in intravesicular fluorescence in vesicles containing the low-affinity Ca2+ indicator fluo-5N. Perfusion with caffeine caused SR Ca2+ release and a decrease in intravesicular flourescence. Although caffeine-dependent release was readily visible with extravesicular Ca2+-green, Ca2+ which leaked from the SR was detected only indirectly as eventless release. We conclude that SR Ca2+ uptake and release can be selectively measured in functional SR vesicles using a confocal microscope. Caffeine-dependent release is directly measurable though SR Ca2+ leak can only be inferred as subresolution events, presumably because channels in separate vesicles were not close enough to result in concerted Ca2+-induced Ca2+ release.

Functionally and structurally integrated computational modeling of ventricular physiology

Computational biology is integrative in several ways. Functionally, computational models are valuable for integrating the many interacting processes within biochemical networks and the many interacting physiological subsystems within the cell. Structurally detailed models provide a way of integrating across scales of biological organization from molecule to organism. Data integration across diverse laboratory and clinical measurements is another unique strength of computational biology. We describe examples of all three categories of integration by using recent advances in modeling cardiac excitation-contraction coupling and whole-heart electromechanics in health and disease.

The spatial pattern of atrial cardiomyocyte calcium signalling modulates contraction

We examined the regulation of calcium signalling in atrial cardiomyocytes during excitation-contraction coupling, and how changes in the distribution of calcium impacts on contractility. Under control conditions, calcium transients originated in subsarcolemmal locations and showed local regeneration through activation of calcium-induced calcium release from ryanodine receptors. Despite functional ryanodine receptors being expressed at regular (∼2 μm) intervals throughout atrial myocytes, the subsarcolemmal calcium signal did not spread in a fully regenerative manner through the interior of a cell. Rather, there was a diminishing centripetal propagation of calcium. The lack of regeneration was due to mitochondria and SERCA pumps preventing the inward movement of calcium. Inhibiting these calcium buffering mechanisms allowed the globalisation of action potential-evoked responses. In addition, physiological positive inotropic agents, such as endothelin-1 and β-adrenergic agonists, as well as enhanced calcium current, calcium store loading and inositol 1,4,5-trisphosphate infusion also led to regenerative global responses. The consequence of globalising calcium signals was a significant increase in cellular contraction. These data indicate how calcium signals and their consequences are determined by the interplay of multiple subcellular calcium management systems.

Effects of Magnesium on Cardiac Excitation-Contraction Coupling

OBJECTIVE

Magnesium regulates a large number of cellular processes. Small changes in intracellular free Mg(2+) ([Mg(2+)](i)) may have important effects on cardiac excitability and contractility. We investigated the effects of [Mg(2+)](i) on cardiac excitation-contraction coupling.

METHODS

We used our ionic-metabolic model that incorporates equations for Ca(2+) and Mg(2+) buffering and transport by ATP and ADP and equations for MgATP regulation of ion transporters (Na(+)-K(+) pump, sarcolemmal and sarcoplasmic Ca(2+) pumps).

RESULTS

Model results indicate that variations in cytosolic Mg(2+) level might sensitively affect diastolic and systolic Ca(2+), sarcoplasmic Ca(2+) content, Ca(2+) influx through L-type channels, efficiency of the Na(+)/Ca(2+) exchanger and action potential shape. The analysis suggests that the most important reason for the observed effects is a modified normal function of sarcoplasmic Ca(2+)-ATPase pump by altered diastolic MgATP levels.

CONCLUSION

The model is able to reproduce qualitatively a sequence of events that correspond well with experimental observations during cardiac excitation-contraction coupling in mammalian ventricular myocytes.

Modeling the isolated cardiac myocyte

Computer modeling of cardiac myocytes has flourished in recent years. Models have evolved from mathematical descriptions of ionic channels alone to more sophisticated formulations that include calcium transport mechanisms, ATP production and metabolic pathways. The increased complexity is fueled by the new data available in the field. The continuous production of experimental data has led to the evolution of increasingly refined descriptions of the phenomena by modelers. Integrating the numerous systems involved in cardiac myocyte homeostasis makes the use of computer models necessary due to the unreliability of intuitive approaches. However the complexity of the model should not imply a cumbersome operation of the program. As with any tool, computer models have to be easy to operate or their strength will be diminished and potential users will not benefit fully from them. The contribution of the computer modeler to their respective biological fields will be more successful and enduring if modelers devote sufficient time to implement their equations into a model with user-friendly characteristics.

Role of complexing agents in the appearance of Turing patterns

In this paper we study a four-species reaction-diffusion system where Turing patterns are stabilized by the presence of fast reversible reactions between the morphogens and two different mobile complexing agents (CAs) that are not necessarily in excess. We provide a quantitative explanation of how the interaction with the CA changes the size of the Turing space making it possible to observe patterns even in a region where the free diffusion coefficients of the relevant species are equal, as is usually the case in real systems. Our analytical treatment gives a series of mathematical relations that can be helpful for those designing experiments where Turing patterns are expected to appear. We also show how the mobility of CAs affect the characteristic size of the pattern. Finally, we provide an example of biological interest in order to illustrate the main procedures and results.

An Excel-based model of Ca2+diffusion and fura 2 measurements in a spherical cell

We wrote a program that runs as a Microsoft Excel spreadsheet to calculate the diffusion of Ca2+in a spherical cell in the presence of a fixed Ca2+buffer and two diffusible Ca2+buffers, one of which is considered to be a fluorescent Ca2+indicator. We modeled Ca2+diffusion during and after Ca2+influx across the plasma membrane with parameters chosen to approximate amphibian sympathetic neurons, mammalian adrenal chromaffin cells, and rat dorsal root ganglion neurons. In each of these cell types, the model predicts that spatially averaged intracellular Ca2+activity ([Ca2+]avg) rises to a high peak and starts to decline promptly on the termination of Ca2+influx. We compared [Ca2+]avgwith predictions of ratiometric Ca2+measurements analyzed in two ways. Method 1 sums the fluorescence at each of the two excitation or emission wavelengths over the N compartments of the model, calculates the ratio of the summed signals, and converts this ratio to Ca2+([Ca2+]avg,M1). Method 2 sums the measured number of moles of Ca2+in each of the N compartments and divides by the volume of the cell ([Ca2+]avg,M2). [Ca2+]avg,M1peaks well after the termination of Ca2+influx at a value substantially less than [Ca2+]avgbecause the summed signals do not reflect the averaged free Ca2+if the signals come from compartments containing gradients in free Ca2+spanning nonlinear regions of the relationship between free Ca2+and the fluorescence signals. In contrast, [Ca2+]avg,M2follows [Ca2+]avgclosely.

Calcium microdomains in aspiny dendrites

Dendritic spines receive excitatory synapses and serve as calcium compartments, which appear to be necessary for input-specific synaptic plasticity. Dendrites of GABAergic interneurons have few or no spines and thus do not possess a clear morphological basis for synapse-specific compartmentalization. We demonstrate using two-photon calcium imaging that activation of single synapses on aspiny dendrites of neocortical fast spiking (FS) interneurons creates highly localized calcium microdomains, often restricted to less than 1 microm of dendritic space. We confirm using ultrastructural reconstruction of imaged dendrites the absence of any morphological basis for this compartmentalization and show that it is dependent on the fast kinetics of calcium-permeable (CP) AMPA receptors and fast local extrusion via the Na+/Ca2+ exchanger. Because aspiny dendrites throughout the CNS express CP-AMPA receptors, we propose that CP-AMPA receptors mediate a spine-free mechanism of input-specific calcium compartmentalization.