Michael Nivala

Alternans and Arrhythmias: From Cell to Heart

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.

Computational Modeling and Numerical Methods for Spatiotemporal Calcium Cycling in Ventricular Myocytes

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.

Calcium-Voltage Coupling in the Genesis of Early and Delayed Afterdepolarizations in Cardiac Myocytes

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 others 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 ICa,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.

Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws

We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws of nonlinear systems of partial differential equations in multi-dimensions.

Molecular Basis of Hypokalemia-Induced Ventricular Fibrillation

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 &mgr;mol/L). Spontaneous early afterdepolarizations (EADs) and ventricular tachycardia/fibrillation 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. Pretreatment with the Ca-calmodulin kinase II (CaMKII) inhibitor KN-93, but not its inactive analogue KN-92, abolished EADs and hypokalemia-induced ventricular tachycardia/fibrillation, 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 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.

Elliptic solutions of the defocusing NLS equation are stable

The stability of the stationary periodic solutions of the integrable (one-dimensional, cubic) defocusing nonlinear Schrodinger (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.

The emergence of subcellular pacemaker sites for calcium waves and oscillations.

•  Calcium (Ca2+) is fundamental to biological cell function, and Ca2+ waves generating oscillatory Ca2+ signals are widely observed in many cell types. •  Some experimental studies have shown that Ca2+ waves initiate from random locations within the cell, while other studies have shown that waves occur repetitively from preferred locations (pacemaker sites). •  In both ventricular myocyte experiments and computer simulations of a heterogeneous model of coupled Ca2+ release units (CRUs), we show that Ca2+ waves occur randomly in space and time when the Ca2+ level is low, but as the Ca2+ level increases, waves occur repetitively from the same sites. •  Ca2+ waves are self‐organized dynamics of the CRU network, and the wave frequency strongly depends on CRU coupling. •  Using these results, we develop a theory for the entrainment of random oscillators, which provides a unified explanation for the experimental and computational observations.

Linking Flickering to Waves and Whole-Cell Oscillations in a Mitochondrial Network Model

It has been shown that transient single mitochondrial depolarizations, known as flickers, tend to occur randomly in space and time. On the other hand, many studies have shown that mitochondrial depolarization waves and whole-cell oscillations occur under oxidative stress. How single mitochondrial flickering events and whole-cell oscillations are mechanistically linked remains unclear. In this study, we developed a Markov model of the inner membrane anion channel in which reactive-oxidative-species (ROS)-induced opening of the inner membrane anion channel causes transient mitochondrial depolarizations in a single mitochondrion that occur in a nonperiodic manner, simulating flickering. We then coupled the individual mitochondria into a network, in which flickers occur randomly and sparsely when a small number of mitochondria are in the state of high superoxide production. As the number of mitochondria in the high-superoxide-production state increases, short-lived or abortive waves due to ROS-induced ROS release coexist with flickers. When the number of mitochondria in the high-superoxide-production state reaches a critical number, recurring propagating waves are observed. The origins of the waves occur randomly in space and are self-organized as a consequence of random flickering and local synchronization. We show that at this critical state, the depolarization clusters exhibit a power-law distribution, a signature of self-organized criticality. In addition, the whole-cell mitochondrial membrane potential changes from exhibiting small random fluctuations to more periodic oscillations as the superoxide production rate increases. These simulation results may provide mechanistic insight into the transition from random mitochondrial flickering to the waves and whole-cell oscillations observed in many experimental studies.

A unified theory of calcium alternans in ventricular myocytes.

Intracellular calcium (Ca2+) alternans is a dynamical phenomenon in ventricular myocytes, which is linked to the genesis of lethal arrhythmias. Iterated map models of intracellular Ca2+ cycling dynamics in ventricular myocytes under periodic pacing have been developed to study the mechanisms of Ca2+ alternans. Two mechanisms of Ca2+ alternans have been demonstrated in these models: one relies mainly on fractional sarcoplasmic reticulum Ca2+ release and uptake, and the other on refractoriness and other properties of Ca2+ sparks. Each of the two mechanisms can partially explain the experimental observations, but both have their inconsistencies with the experimental results. Here we developed an iterated map model that is composed of two coupled iterated maps, which unifies the two mechanisms into a single cohesive mathematical framework. The unified theory can consistently explain the seemingly contradictory experimental observations and shows that the two mechanisms work synergistically to promote Ca2+ alternans. Predictions of the theory were examined in a physiologically-detailed spatial Ca2+ cycling model of ventricular myocytes.

Symbolic integration using homotopy methods

The homotopy algorithm is a powerful method for indefinite integration of total derivatives. By combining these ideas with straightforward Gaussian elimination, we construct an algorithm for the optimal symbolic integration that contain terms that are not total derivatives. The optimization consists of minimizing the number of terms that remain unintegrated. Further, the algorithm imposes an ordering of terms so that the differential order of these remaining terms is minimal. A different method for the summation of difference expressions is presented in Appendix B.