Dual network structure of the AV node.

Biological systems, particularly the brain, are frequently analyzed as networks, conveying mechanistic insights into their function and pathophysiology. This is the first study of a functional network of cardiac tissue. We use calcium imaging to obtain two functional networks in a subsidiary but essential pacemaker of the heart, the atrioventricular node (AVN). The AVN is a small cellular structure with dual functions: a) to delay the pacemaker signal passing from the sinoatrial node (SAN) to the ventricles, and b) to serve as a back-up pacemaker should the primary SAN pacemaker fail. Failure of the AVN can lead to syncope and death. We found that the shortest path lengths and clustering coefficients of the AVN are remarkably similar to those of the brain. The network is ``small-world," thus optimized for energy use vs transmission efficiency. We further study the network properties of AVN tissue with knock-out of the sodium-calcium exchange transporter. In this case, the average shortest path-lengths remained nearly unchanged showing network resilience, while the clustering coefficient was somewhat reduced, similar to schizophrenia in brain networks. When we removed the global action potential using principal component analysis (PCA) in wild-type model, the network lost its ``small-world" characteristics with less information-passing efficiency due to longer shortest path lengths but more robust signal propagation resulting from higher clustering. These two wild-type networks (with and without global action potential) may correspond to fast and slow conduction pathways. Laslty, a one-parameter non-linear preferential attachment model is a good fit to all three AVN networks.

Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state.

Cardiac muscle contraction is initiated by an elementary Ca signal (called Ca spark) which is achieved by collective action of Ca release channels in a cluster. The mechanism of this synchronization remains uncertain. We approached Ca spark activation as an emergent phenomenon of an interactive system of release channels. We constructed a weakly lumped Markov chain that applies an Ising model formalism to such release channel clusters and probable open channel configurations and demonstrated that spark activation is described as a system transition from a metastable to an absorbing state, analogous to the pressure required to overcome surface tension in bubble formation. This yielded quantitative estimates of the spark generation probability as a function of various system parameters. We performed numerical simulations to find spark probabilities as a function of sarcoplasmic reticulum Ca concentration, obtaining similar values for spark activation threshold as our analytic model, as well as those reported in experimental studies. Our parametric sensitivity analyses also showed that the spark activation threshold decreased as Ca sensitivity of RyR activation and RyR cluster size increased.

Disorder in Ca2+ release unit locations confers robustness but cuts flexibility of heart pacemaking.

Excitation-contraction coupling kinetics is dictated by the action potential rate of sinoatrial-nodal cells. These cells generate local Ca releases (LCRs) that activate Na/Ca exchanger current, which accelerates diastolic depolarization and determines the pace. LCRs are generated by clusters of ryanodine receptors, Ca release units (CRUs), residing in the sarcoplasmic reticulum. While CRU distribution exhibits substantial heterogeneity, its functional importance remains unknown. Using numerical modeling, here we show that with a square lattice distribution of CRUs, Ca-induced-Ca-release propagation during diastolic depolarization is insufficient for pacemaking within a broad range of realistic ICaL densities. Allowing each CRU to deviate randomly from its lattice position allows sparks to propagate, as observed experimentally. As disorder increases, the CRU distribution exhibits larger empty spaces and simultaneously CRU clusters, as in Poisson clumping. Propagating within the clusters, Ca release becomes synchronized, increasing action potential rate and reviving pacemaker function of dormant/nonfiring cells. However, cells with fully disordered CRU positions could not reach low firing rates and their ß-adrenergic-receptor stimulation effect was substantially decreased. Inclusion of Cav1.3, a low-voltage activation L-type Ca channel isoform into ICaL, strongly increases recruitment of CRUs to fire during diastolic depolarization, increasing robustness of pacemaking and complementing effects of CRU distribution. Thus, order/disorder in CRU locations along with Cav1.3 expression regulates pacemaker function via synchronization of CRU firing. Excessive CRU disorder and/or overexpression of Cav1.3 boosts pacemaker function in the basal state, but limits the rate range, which may contribute to heart rate range decline with age and disease.

Mechanisms of Calcium Leak from Cardiac Sarcoplasmic Reticulum Revealed by Statistical Mechanics.

Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR), a major intracellular Ca store. However, under abnormal conditions, Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here, we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of the Ca release unit reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogs of magnetic field (h) and the inverse temperature (ß) in the Ising model, for which we have explicit formulas in terms of SR [Ca] and release channel opening and closing rates. The classification of leak regimes takes the shape of a phase ß-h diagram, with the regime boundaries occurring at h = 0 and a critical value of ß (ß∗) that we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when h > 0 and ß >ß∗, and a disordered leak occurs when ß <ß∗ and h is not too negative. The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths, an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach, we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective ryanodine receptor parameters and SR Ca load that describe and classify phase transitions and leak emergence.

Clusters of calcium release channels harness the Ising phase transition to confine their elementary intracellular signals.

Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal-to-noise ratio is achieved by a cooperative action of Ca release channels such as IP3 receptors or ryanodine receptors arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced Ca release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. Despite the positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. Here we show that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field in an Ising ferromagnet. Our simple quantitative criterion closely predicts the Ca store depletion level required for spark termination for each cluster size. We further formulate exact requirements that a cluster of release channels should satisfy in any cell type for our mapping to the Ising model and the associated formula to remain valid. Thus, we describe deterministically the behavior of a system on a coarser scale (release unit) that is random on a finer scale (release channels), bridging the gap between scales. Our results provide exact mapping of a nanoscale biological signaling model to an interacting particle system in statistical physics, making the extensive mathematical apparatus available to quantitative biology.

Modern perspectives on numerical modeling of cardiac pacemaker cell.

Cardiac pacemaking is a complex phenomenon that is still not completely understood. Together with experimental studies, numerical modeling has been traditionally used to acquire mechanistic insights in this research area. This review summarizes the present state of numerical modeling of the cardiac pacemaker, including approaches to resolve present paradoxes and controversies. Specifically we discuss the requirement for realistic modeling to consider symmetrical importance of both intracellular and cell membrane processes (within a recent "coupled-clock" theory). Promising future developments of the complex pacemaker system models include the introduction of local calcium control, mitochondria function, and biochemical regulation of protein phosphorylation and cAMP production. Modern numerical and theoretical methods such as multi-parameter sensitivity analyses within extended populations of models and bifurcation analyses are also important for the definition of the most realistic parameters that describe a robust, yet simultaneously flexible operation of the coupled-clock pacemaker cell system. The systems approach to exploring cardiac pacemaker function will guide development of new therapies such as biological pacemakers for treating insufficient cardiac pacemaker function that becomes especially prevalent with advancing age.

RyR-NCX-SERCA local cross-talk ensures pacemaker cell function at rest and during the fight-or-flight reflex.

RATIONALE: A recent study published in Circulation Research by Gao et al used sinoatrial node (SAN)-targeted, incomplete Ncx1 knockout in mice to explore the role of the Na(+)/Ca(2+) exchanger (NCX) in cardiac pacemaker. The authors concluded that NCX is required for increasing sinus rates, but not for maintaining resting heart rate. This conclusion was based, in part, on numeric model simulations performed by Gao et al that reproduced their experimental results of unchanged action potentials in the knockout SAN cells. The authors, however, did not simulate the NCX current (INCX), that is, the subject of the study. OBJECTIVE: We extended numeric examinations to simulate INCX in their incomplete knockout SAN cells that is crucial to interpret the study results. METHODS AND RESULTS: INCX and Ca(2+) dynamics were simulated using different contemporary numeric models of SAN cells. We found that minimum diastolic Ca(2+) levels and INCX amplitudes generated by remaining NCX molecules (only 20% of control) remained almost unchanged. Simulations using a new local Ca(2+) control model indicate that these powerful compensatory mechanisms involve complex local cross-talk of Ca(2+) cycling proteins and NCX. Specifically, lower NCX expression facilitates Ca(2+)-induced Ca(2+) release and larger local Ca(2+) releases that stabilize diastolic INCX. Further reduction of NCX expression results in arrhythmia and halt of automaticity. CONCLUSIONS: Remaining NCX molecules in the incomplete knockout model likely produce almost the same diastolic INCX as in wild-type cells. INCX contribution is crucially important for both basal automaticity of SAN cells and during the fight-or-flight reflex.

Synchronization of stochastic Ca²(+) release units creates a rhythmic Ca²(+) clock in cardiac pacemaker cells.

In sinoatrial node cells of the heart, beating rate is controlled, in part, by local Ca²(+) releases (LCRs) from the sarcoplasmic reticulum, which couple to the action potential via electrogenic Na(+)/Ca²(+) exchange. We observed persisting, roughly periodic LCRs in depolarized rabbit sinoatrial node cells (SANCs). The features of these LCRs were reproduced by a numerical model consisting of a two-dimensional array of stochastic, diffusively coupled Ca²(+) release units (CRUs) with fixed refractory period. Because previous experimental studies showed that ß-adrenergic receptor stimulation increases the rate of Ca²(+) release through each CRU (dubbed I(spark)), we explored the link between LCRs and I(spark) in our model. Increasing the CRU release current I(spark) facilitated Ca²(+)-induced-Ca²(+) release and local recruitment of neighboring CRUs to fire more synchronously. This resulted in a progression in simulated LCR size (from sparks to wavelets to global waves), LCR rhythmicity, and decrease of LCR period that parallels the changes observed experimentally with ß-adrenergic receptor stimulation. The transition in LCR characteristics was steeply nonlinear over a narrow range of I(spark), resembling a phase transition. We conclude that the (partial) periodicity and rate regulation of the "Calcium clock" in SANCs are emergent properties of the diffusive coupling of an ensemble of interacting stochastic CRUs. The variation in LCR period and size with I(spark) is sufficient to account for ß-adrenergic regulation of SANC beating rate.