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

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

Slow Ca²âº sparks de-synchronize Ca²âº release in failing cardiomyocytes: evidence for altered configuration of Ca²âº release units?

In heart failure, cardiomyocytes exhibit slowing of the rising phase of the Ca(2+) transient which contributes to the impaired contractility observed in this condition. We investigated whether alterations in ryanodine receptor function promote slowing of Ca(2+) release in a murine model of congestive heart failure (CHF). Myocardial infarction was induced by left coronary artery ligation. When chronic CHF had developed (10 weeks post-infarction), cardiomyocytes were isolated from viable regions of the septum. Septal myocytes from SHAM-operated mice served as controls. Ca(2+) transients rose markedly slower in CHF than SHAM myocytes with longer time to peak (CHF=152 ± 12% of SHAM, P<0.05). The rise time of Ca(2+) sparks was also increased in CHF (SHAM=9.6 ± 0.6 ms, CHF=13.2 ± 0.7 ms, P<0.05), due to a sub-population of sparks (≈20%) with markedly slowed kinetics. Regions of the cell associated with these slow spontaneous sparks also exhibited slowed Ca(2+) release during the action potential. Thus, greater variability in spark kinetics in CHF promoted less uniform Ca(2+) release across the cell. Dyssynchronous Ca(2+) transients in CHF additionally resulted from T-tubule disorganization, as indicated by fast Fourier transforms, but slow sparks were not associated with orphaned ryanodine receptors. Rather, mathematical modeling suggested that slow sparks could result from an altered composition of Ca(2+) release units, including a reduction in ryanodine receptor density and/or distribution of ryanodine receptors into sub-clusters. In conclusion, our findings indicate that slowed, dyssynchronous Ca(2+) transients in CHF result from alterations in Ca(2+) sparks, consistent with rearrangement of ryanodine receptors within Ca(2+) release units.

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 excitation­contraction 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.

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.

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.

Computational modeling of subcellular transport and signaling.

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

Nonequilibrium reactivation of Na+ current drives early afterdepolarizations in mouse ventricle.

BACKGROUND: Early afterdepolarizations (EADs) are triggers of cardiac arrhythmia driven by L-type Ca(2+) current (ICaL) reactivation or sarcoplasmic reticulum Ca(2+) release and Na(+)/Ca(2+) exchange. In large mammals the positive action potential plateau promotes ICaL reactivation, and the current paradigm holds that cardiac EAD dynamics are dominated by interaction between ICaL and the repolarizing K(+) currents. However, EADs are also frequent in the rapidly repolarizing mouse action potential, which should not readily permit ICaL reactivation. This suggests that murine EADs exhibit unique dynamics, which are key for interpreting arrhythmia mechanisms in this ubiquitous model organism. We investigated these dynamics in myocytes from arrhythmia-susceptible calcium calmodulin-dependent protein kinase II delta C (CaMKIIδC)-overexpressing mice (Tg), and via computational simulations. METHODS AND RESULTS: In Tg myocytes, ß-adrenergic challenge slowed late repolarization, potentiated sarcoplasmic reticulum Ca(2+) release, and initiated EADs below the ICaL activation range (-47 ± 0.7 mV). These EADs were abolished by caffeine and tetrodotoxin (but not ranolazine), suggesting that sarcoplasmic reticulum Ca(2+) release and Na(+) current (INa), but not late INa, are required for EAD initiation. Simulations suggest that potentiated sarcoplasmic reticulum Ca(2+) release and Na(+)/Ca(2+) exchange shape late action potential repolarization to favor nonequilibrium reactivation of INa and thereby drive the EAD upstroke. Action potential clamp experiments suggest that lidocaine eliminates virtually all inward current elicited by EADs, and that this effect occurs at concentrations (40-60 µmol/L) for which lidocaine remains specific for inactivated Na(+) channels. This strongly suggests that previously inactive channels are recruited during the EAD upstroke, and that nonequilibrium INa dynamics underlie murine EADs. CONCLUSIONS: Nonequilibrium reactivation of INa drives murine EADs.

A three-dimensional finite element model of human atrial anatomy: new methods for cubic Hermite meshes with extraordinary vertices.

High-order cubic Hermite finite elements have been valuable in modeling cardiac geometry, fiber orientations, biomechanics, and electrophysiology, but their use in solving three-dimensional problems has been limited to ventricular models with simple topologies. Here, we utilized a subdivision surface scheme and derived a generalization of the "local-to-global" derivative mapping scheme of cubic Hermite finite elements to construct bicubic and tricubic Hermite models of the human atria with extraordinary vertices from computed tomography images of a patient with atrial fibrillation. To an accuracy of 0.6 mm, we were able to capture the left atrial geometry with only 142 bicubic Hermite finite elements, and the right atrial geometry with only 90. The left and right atrial bicubic Hermite meshes were G1 continuous everywhere except in the one-neighborhood of extraordinary vertices, where the mean dot products of normals at adjacent elements were 0.928 and 0.925. We also constructed two biatrial tricubic Hermite models and defined fiber orientation fields in agreement with diagrammatic data from the literature using only 42 angle parameters. The meshes all have good quality metrics, uniform element sizes, and elements with aspect ratios near unity, and are shared with the public. These new methods will allow for more compact and efficient patient-specific models of human atrial and whole heart physiology.

uPy: a ubiquitous CG Python API with biological-modeling applications.

The uPy Python extension module provides a uniform abstraction of the APIs of several 3D computer graphics programs (called hosts), including Blender, Maya, Cinema 4D, and DejaVu. A plug-in written with uPy can run in all uPy-supported hosts. Using uPy, researchers have created complex plug-ins for molecular and cellular modeling and visualization. uPy can simplify programming for many types of projects (not solely science applications) intended for multihost distribution. It's available at http://upy.scripps.edu. The first featured Web extra is a video that shows interactive analysis of a calcium dynamics simulation. YouTube URL: http://youtu.be/wvs-nWE6ypo. The second featured Web extra is a video that shows rotation of the HIV virus. YouTube URL: http://youtu.be/vEOybMaRoKc.

Instabilities of the resting state in a mathematical model of calcium handling in cardiac myocytes.

We analyze a recently published model of calcium handling in cardiac myocytes in order to find conditions for the presence of instabilities in the resting state of the model. Such instabilities can create calcium waves which in turn may be able to initiate cardiac arrhythmias. The model was developed by Swietach, Spitzer and Vaughan-Jones in order to study the effect, on calcium waves, of varying ryanodine receptor (RyR)-permeability, sarco/endoplasmic reticulum calcium ATPase (SERCA) and calcium diffusion. We study the model using the extracellular calcium concentration c(e) and the maximal velocity of the SERCA-pump v(SERCA) as control parameters. In the (c(e),v(SERCA))-domain we derive an explicit function v∗=v∗(c(e)), and we claim that any resting state based on parameters that lie above the curve (i.e. any pair (c(e),v(SERCA)) such that with v(SERCA) > v∗(c(e))) is unstable in the sense that small perturbations will grow and can eventually turn into a calcium wave. And conversely; any pair (c(e),v(SERCA)) below the curve is stable in the sense that small perturbations to the resting state will decay to rest. This claim is supported by analyzing the stability of the system in terms of computing the eigenmodes of the linearized model. Furthermore, the claim is supported by direct simulations based on the non-linear model. Since the curve separating stable from unstable states is given as an explicit function, we can show how stability depends on other parameters of the model.

Slow Calcium-Depolarization-Calcium waves may initiate fast local depolarization waves in ventricular tissue.

Intercellular calcium waves in cardiac myocytes are a well-recognized, if incompletely understood, phenomenon. In a variety of preparations, investigators have reported multi-cellular calcium waves or triggered propagated contractions, but the mechanisms of propagation and pathological importance of these events remain unclear. Here, we review existing experimental data and present a computational approach to investigate the mechanisms of multi-cellular calcium wave propagation. Over the past 50 years, the standard modeling paradigm for excitable cardiac tissue has seen increasingly detailed models of the dynamics of individual cells coupled in tissue solely by intercellular and interstitial current flow. Although very successful, this modeling regime has been unable to capture two important phenomena: 1) the slow intercellular calcium waves observed experimentally, and 2) how intercellular calcium events resulting in delayed after depolarizations at the cellular level could overcome a source-sink mismatch to initiate depolarization waves in tissue. In this paper, we introduce a mathematical model with subcellular spatial resolution, in which we allow both inter- and intracellular current flow and calcium diffusion. In simulations of coupled cells employing this model, we observe: a) slow inter-cellular calcium waves propagating at about 0.1 mm/s, b) faster Calcium-Depolarization-Calcium (CDC) waves, traveling at about 1 mm/s, and c) CDC-waves that can set off fast depolarization-waves (50 cm/s) in tissue with varying gap-junction conductivity.

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 Ca(2+) channel (LCC) clustering, and allosteric activation of Na(+)/Ca(2+) exchanger by L-type Ca(2+) current affects intracellular Ca(2+) 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(+)/Ca(2+) exchanger, sarcolemmal Ca(2+) pump, and sarcolemmal Ca(2+) leak), and stationary and mobile Ca(2+) 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 Ca(2+). We obtained parameters from voltage-clamp protocols of L-type Ca(2+) current and line-scan recordings of Ca(2+) concentration profiles in rabbit cells, in which the sarcoplasmic reticulum is disabled. Our model results agree with experimental measurements of global Ca(2+) transient in myocytes loaded with 50 µM Fluo-3. We found that local Ca(2+) concentrations within the cytosol and sub-sarcolemma, as well as the local trigger fluxes of Ca(2+) crossing the cell membrane, are sensitive to details of t-tubule micro-structure and membrane Ca(2+) flux distribution. The model additionally predicts that local Ca(2+) trigger fluxes are at least threefold to eightfold higher than the whole-cell Ca(2+) trigger flux. We found also that the activation of allosteric Ca(2+)-binding sites on the Na(+)/Ca(2+) exchanger could provide a mechanism for regulating global and local Ca(2+) 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(+)/Ca(2+) exchanger fluxes to intracellular Ca(2+) dynamics.