Abstract:The goal of systems biology is to relate events at the molecular level to more integrated scales from organelle to cell, tissue, and living organism. Here, we review how normal and abnormal excitation-contraction coupling properties emerge from the protein scale, where behaviors are dominated by randomness, to the cell and tissue scales, where heart has to beat with reliable regularity for a lifetime. Beginning with the fundamental unit of excitation-contraction coupling, the couplon where L-type Ca channels in the sarcolemmal membrane adjoin ryanodine receptors in the sarcoplasmic reticulum membrane, we show how a network of couplons with 3 basic properties (random activation, refractoriness, and recruitment) produces the classic physiological properties of excitation-contraction coupling and, under pathophysiological conditions, leads to Ca alternans and Ca waves. Moving to the tissue scale, we discuss how cellular Ca alternans and Ca waves promote both reentrant and focal arrhythmias in the heart. Throughout, we emphasize the qualitatively novel properties that emerge at each new scale of integration. (Circ Res. 2011;108:98-112.) Key Words: cardiac excitation-contraction coupling Ⅲ ventricular arrhythmias Ⅲ Ca signaling Ⅲ cardiac electrophysiology Ⅲ heart failure I t is a fundamental tenet of evolutionary biology that new properties that confer a survival advantage to a species are preserved. The properties endowing the greatest advantages are qualitatively novel ones in which "the whole becomes greater than the sum of the parts." Indeed, these "emergent" properties, which arise nonintuitively from the nonlinear interactions between the parts, are the very basis of life. Without nonlinear interactions, the most fundamental aspects of living systems, such as oscillations underlying the cell cycle and the heart beat, could not exist (because in linear systems, the whole can be no more nor less than the sum of the parts). Yet recognizing this fact as a guiding principle in biology comes at a cost: if the whole is greater than the sum of the parts, then the whole cannot be understood simply by studying the parts in isolation. This premise defines the rationale for systems biology.The most fundamental "parts" in biological systems are the genes and proteins that they encode for. Genes and proteins have fascinating properties in their own right, but they do not, by themselves, explain the qualitatively new properties that arise as these molecules self-organize themselves, stage by stage, into the components of living systems, from supramo- http://circres.ahajournals.org/ Downloaded from lecular complexes to organelles to cells to tissues to organs. For example, individual proteins are dominated by stochastic behaviors, as illustrated by the random-appearing opening and closing of ion channels ( Figure 1A). However, when integrated into a cardiac myocyte, the highly irregular stochastic behavior of many ion channels produces a very regular cardiac action potential (AP), intracellular Ca (Ca i ) transient, a...

The stability of the stationary periodic solutions of the integrable (onedimensional, cubic) defocusing nonlinear Schrödinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additional argument using an appropriate Krein signature allows us to conclude the (nonlinear) orbital stability of all stationary solutions of the defocusing NLS equation with respect to so-called subharmonic perturbations: perturbations that have period equal to an integer multiple of the period of the amplitude of the solution. All results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles.

Calcium (Ca) is a ubiquitous second messenger that regulates many biological functions. The elementary events of local Ca signaling are Ca sparks, which occur randomly in time and space, and integrate to produce global signaling events such as intra- and intercellular Ca waves and whole-cell Ca oscillations. Despite extensive experimental characterization in many systems, the transition from local random to global synchronous events is still poorly understood. Here we show that criticality, a ubiquitous dynamical phenomenon in nature, is responsible for the transition from local to global Ca signaling. We demonstrate this first in a computational model of Ca signaling in a cardiac myocyte and then experimentally in mouse ventricular myocytes, complemented by a theoretical agent-based model to delineate the underlying dynamics. We show that the interaction between the Ca release units via Ca-induced Ca release causes self-organization of Ca spark clusters. When the coupling between Ca release units is weak, the cluster-size distribution is exponential. As the interactions become strong, the cluster-size distribution changes to a power-law distribution, which is characteristic of criticality in thermodynamic and complex nonlinear systems, and facilitates the formation and propagation of Ca waves and whole-cell Ca oscillations. Our findings illustrate how criticality is harnessed by a biological cell to regulate Ca signaling via self-organization of random subcellular events into cellular-scale oscillations, and provide a general theoretical framework for the transition from local Ca signaling to global Ca signaling in biological cells.

Background Hypokalemia is known to promote ventricular arrhythmias, especially in combination with Class III antiarrhythmic drugs like dofetilide. Here we evaluated the underlying molecular mechanisms. Methods and Results Arrhythmias were recorded in isolated rabbit and rat hearts or patch-clamped ventricular myocytes exposed to hypokalemia (1.0-3.5 mmol/l) in the absence or presence of dofetilide (1 μmol/l). Spontaneous early afterdepolarizations (EADs) and ventricular tachycardia/fibrillation (VF/VF) occurred in 50% of hearts at 2.7 mmol/l [K] in the absence of dofetilide, and 3.3 mmol/l [K] in its presence. Pre-treatment with the CaMKII inhibitor KN-93, but not its inactive analogue KN-92, abolished EADs and hypokalemia-induced VT/VF, as did the selective late Na current (INa) blocker GS-967. In intact hearts, moderate hypokalemia (2.7 mmol/l) significantly increased tissue CaMKII activity. Computer modeling revealed that EAD generation by hypokalemia (with or without dofetilide) required Na-K pump inhibition to induce intracellular Na and Ca overload with consequent CaMKII activation enhancing late INa and the L-type Ca current. K current suppression by hypokalemia and/or dofetilide alone in the absence of CaMKII activation were ineffective at causing EADs. Conclusions We conclude that Na-K pump inhibition by even moderate hypokalemia plays a critical role in promoting EAD-mediated arrhythmias by inducing a positive feedback cycle activating CaMKII and enhancing late INa. Class III antiarrhythmic drugs like dofetilide sensitize the heart to this positive feedback loop.

Early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs) are voltage oscillations known to cause cardiac arrhythmias. EADs are mainly driven by voltage oscillations in the repolarizing phase of the action potential (AP), while DADs are driven by spontaneous calcium (Ca) release during diastole. Because voltage and Ca are bidirectionally coupled, they modulate each other's behaviors, and new AP and Ca cycling dynamics can emerge from this coupling. In this study, we performed computer simulations using an AP model with detailed spatiotemporal Ca cycling incorporating stochastic openings of Ca channels and ryanodine receptors to investigate the effects of Ca-voltage coupling on EAD and DAD dynamics. Simulations were complemented by experiments in mouse ventricular myocytes. We show that: 1) alteration of the Ca transient due to increased ryanodine receptor leakiness and/or sarco/endoplasmic reticulum Ca ATPase activity can either promote or suppress EADs due to the complex effects of Ca on ionic current properties; 2) spontaneous Ca waves also exhibit complex effects on EADs, but cannot induce EADs of significant amplitude without the participation of I Ca,L ; 3) lengthening AP duration and the occurrence of EADs promote DADs by increasing intracellular Ca loading, and two mechanisms of DADs are identified, i.e., Ca-wave-dependent and Ca-wave-independent; and 4) Ca-voltage coupling promotes complex EAD patterns such as EAD alternans that are not observed for solely voltage-driven EADs. In conclusion, Ca-voltage coupling combined with the nonlinear dynamical behaviors of voltage and Ca cycling play a key role in generating complex EAD and DAD dynamics observed experimentally in cardiac myocytes, whose mechanisms are complex but analyzable.

Calcium alternans is associated with T-wave alternans and pulsus alternans, harbingers of increased mortality in the setting of heart disease. Recent experimental, computational, and theoretical studies have led to new insights into the mechanisms of Ca alternans, specifically how disordered behaviors dominated by stochastic processes at the subcellular level become organized into ordered periodic behaviors. In this article, we summarize the recent progress in this area, outlining a holistic theoretical framework in which the complex effects of Ca cycling proteins on Ca alternans are linked to three key properties of the cardiac Ca cycling network: randomness, refractoriness, and recruitment. We also illustrate how this ‘3R theory’ can reconcile many seemingly contradictory experimental observations.

Intracellular calcium (Ca) cycling dynamics in cardiac myocytes is regulated by a complex network of spatially distributed organelles, such as sarcoplasmic reticulum (SR), mitochondria, and myofibrils. In this study, we present a mathematical model of intracellular Ca cycling and numerical and computational methods for computer simulations. The model consists of a coupled Ca release unit (CRU) network, which includes a SR domain and a myoplasm domain. Each CRU contains 10 L-type Ca channels and 100 ryanodine receptor channels, with individual channels simulated stochastically using a variant of Gillespie’s method, modified here to handle time-dependent transition rates. Both the SR domain and the myoplasm domain in each CRU are modeled by 5 × 5 × 5 voxels to maintain proper Ca diffusion. Advanced numerical algorithms implemented on graphical processing units were used for fast computational simulations. For a myocyte containing 100 × 20 × 10 CRUs, a 1-s heart time simulation takes about 10 min of machine time on a single NVIDIA Tesla C2050. Examples of simulated Ca cycling dynamics, such as Ca sparks, Ca waves, and Ca alternans, are shown.

Nivala M, Qu Z. Calcium alternans in a couplon network model of ventricular myocytes: role of sarcoplasmic reticulum load. Am J Physiol Heart Circ Physiol 303: H341-H352, 2012. First published June 1, 2012; doi:10.1152/ajpheart.00302.2012.-Intracellular calcium (Ca) alternans in cardiac myocytes have been shown in many experimental studies, and the mechanisms remain incompletely understood. We recently developed a "3R theory" that links Ca sparks to whole cell Ca alternans through three critical properties: randomness of Ca sparks; recruitment of a Ca spark by neighboring Ca sparks; and refractoriness of Ca release units. In this study, we used computer simulation of a physiologically detailed mathematical model of a ventricular myocyte couplon network to study how sarcoplasmic reticulum (SR) Ca load and other physiological parameters, such as ryanodine receptor sensitivity, SR uptake rate, Na-Ca exchange strength, and Ca buffer levels affect Ca alternans in the context of 3R theory. We developed a method to calculate the parameters used in the 3R theory (i.e., the primary spark rate and the recruitment rate) from the physiologically detailed Ca cycling model and paced the model periodically to elicit Ca alternans. We show that alternans only occurs for an intermediate range of the SR Ca load, and the underlying mechanism can be explained via its effects on the 3Rs. Furthermore, we show that altering the physiological parameters not only directly changes the 3Rs but also alters the SR Ca load, having an indirect effect on the 3Rs as well. Therefore, our present study links the SR Ca load and other physiological parameters to whole cell Ca alternans through the framework of the 3R theory, providing a general mechanistic understanding of Ca alternans in ventricular myocytes. calcium cycling; calcium spark; modeling T-WAVE ALTERNANS HAVE BEEN closely associated with cardiac arrhythmias and sudden cardiac death (3,50,51,66,69). Intracellular calcium (Ca) alternans, as a potential cause of T-wave alternans (15,46), have been shown in many experimental studies (2, 5, 7, 16 -19, 25, 35, 41, 68, 71). However, the underlying mechanism of Ca alternans remains incompletely understood. Adler et al. (1) proposed a theoretical mechanism linking a nonlinear sarcoplasmic reticulum (SR) Ca release function to mechanical alternans. Evidence from experimental studies (17)(18)(19)71) shows that Ca alternans could be explained by a steep nonlinear dependence of SR Ca release upon the diastolic SR Ca load (Ca DSRL ) immediately preceding the release (a steep fractional release-load relationship). This mechanism requires that Ca DSRL alternate concomitantly with SR Ca release, but the refractoriness of SR Ca release is not explicitly required. Subsequent mathematical analyses and simulation studies either potentiated this theory (30,45,57,64,69,71) (41), and the importance of refractoriness in Ca alternans has been demonstrated in later studies (32,58). Therefore, the above experimental studies raise the issue of the roles of SR Ca an...