Criticality in intracellular calcium signaling in cardiac myocytes

Strength in numbers: Unleashing the potential of trans-scale scope AMATERAS for massive cell quantification

Singularity biology is a scientific field that targets drastic state changes in multicellular systems, aiming to discover the key cells that induce the state change and investigate the mechanisms behind them. To achieve this goal, we developed a trans-scale optical imaging system (trans-scale scope), that is capable of capturing both macroscale changes across the entire system and the micro-scale behavior of individual cells, surpassing the cell observation capabilities of traditional microscopes. We developed two units of the trans-scale scope, named AMATERAS-1 and -2, which demonstrated the ability to observe multicellular systems consisting of over one million cells in a single field of view with sub-cellular resolution. This flagship instrument has been used to observe the dynamics of various cell species, with the advantage of being able to observe a large number of cells, allowing the detection and analysis of rare events and cells such as leader cells in multicellular pattern formation and cells that spontaneously initiate calcium waves. In this paper, we present the design concept of AMATERAS, the optical configuration, and several examples of observations, and demonstrate how the strength-in-numbers works in life sciences.

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.

SparkMaster 2: A New Software for Automatic Analysis of Calcium Spark Data

BACKGROUND

Calcium (Ca) sparks are elementary units of subcellular Ca release in cardiomyocytes and other cells. Accordingly, Ca spark imaging is an essential tool for understanding the physiology and pathophysiology of Ca handling and is used to identify new drugs targeting Ca-related cellular dysfunction (eg, cardiac arrhythmias). The large volumes of imaging data produced during such experiments require accurate and high-throughput analysis.

METHODS

We developed a new software tool SparkMaster 2 (SM2) for the analysis of Ca sparks imaged by confocal line-scan microscopy, combining high accuracy, flexibility, and user-friendliness. SM2 is distributed as a stand-alone application requiring no installation. It can be controlled using a simple-to-use graphical user interface, or using Python scripting.

RESULTS

SM2 is shown to have the following strengths: (1) high accuracy at identifying Ca release events, clearly outperforming previous highly successful software SparkMaster; (2) multiple types of Ca release events can be identified using SM2: Ca sparks, waves, miniwaves, and long sparks; (3) SM2 can accurately split and analyze individual sparks within spark clusters, a capability not handled adequately by prior tools. We demonstrate the practical utility of SM2 in two case studies, investigating how Ca levels affect spontaneous Ca release, and how large-scale release events may promote release refractoriness. SM2 is also useful in atrial and smooth muscle myocytes, across different imaging conditions.

CONCLUSIONS

SparkMaster 2 is a new, much-improved user-friendly software for accurate high-throughput analysis of line-scan Ca spark imaging data. It is free, easy to use, and provides valuable built-in features to facilitate visualization, analysis, and interpretation of Ca spark data. It should enhance the quality and throughput of Ca spark and wave analysis across cell types, particularly in the study of arrhythmogenic Ca release events in cardiomyocytes.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

The Heart's Pacemaker Mimics Brain Cytoarchitecture and Function: Novel Interstitial Cells Expose Complexity of the SAN

BACKGROUND

The sinoatrial node (SAN) of the heart produces rhythmic action potentials, generated via calcium signaling within and among pacemaker cells. Our previous work has described the SAN as composed of a hyperpolarization-activated cyclic nucleotide-gated potassium channel 4 (HCN4)-expressing pacemaker cell meshwork, which merges with a network of connexin 43⁺/F-actin⁺ cells. It is also known that sympathetic and parasympathetic innervation create an autonomic plexus in the SAN that modulates heart rate and rhythm. However, the anatomical details of the interaction of this plexus with the pacemaker cell meshwork have yet to be described.

OBJECTIVES

This study sought to describe the 3-dimensional cytoarchitecture of the mouse SAN, including autonomic innervation, peripheral glial cells, and pacemaker cells.

METHODS

The cytoarchitecture of SAN whole-mount preparations was examined by three-dimensional confocal laser-scanning microscopy of triple immunolabeled with combinations of antibodies for HCN4, S100 calcium-binding protein B (S100B), glial fibrillary acidic protein (GFAP), choline acetyltransferase, or vesicular acetylcholine transporter, and tyrosine hydroxylase, and transmission electron microscopy.

RESULTS

The SAN exhibited heterogeneous autonomic innervation, which was accompanied by a web of peripheral glial cells and a novel S100B⁺/GFAP^- interstitial cell population, with a unique morphology and a distinct distribution pattern, creating complex interactions with other cell types in the node, particularly with HCN4-expressing cells. Transmission electron microscopy identified a similar population of interstitial cells as telocytes, which appeared to secrete vesicles toward pacemaker cells. Application of S100B to SAN preparations desynchronized Ca²⁺ signaling in HCN4-expressing cells and increased variability in SAN impulse rate and rhythm.

CONCLUSIONS

The autonomic plexus, peripheral glial cell web, and a novel S100B⁺/GFAP^- interstitial cell type embedded within the HCN4⁺ cell meshwork increase the structural and functional complexity of the SAN and provide a new regulatory pathway of rhythmogenesis.

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 β-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.

Synchronization of spatially discordant voltage and calcium alternans in cardiac tissue

The heart is an excitable medium which is excited by membrane potential depolarization and propagation. Membrane potential depolarization brings in calcium (Ca) through the Ca channels to trigger intracellular Ca release for contraction of the heart. Ca also affects voltage via Ca-dependent ionic currents, and thus, voltage and Ca are bidirectionally coupled. It has been shown that the voltage subsystem or the Ca subsystem can generate its own dynamical instabilities which are affected by their bidirectional couplings, leading to complex dynamics of action potential and Ca cycling. Moreover, the dynamics become spatiotemporal in tissue in which cells are diffusively coupled through voltage. A widely investigated spatiotemporal dynamics is spatially discordant alternans (SDA) in which action potential duration (APD) or Ca amplitude exhibits temporally period-2 and spatially out-of-phase patterns, i.e., APD-SDA and Ca-SDA patterns, respectively. However, the mechanisms of formation, stability, and synchronization of APD-SDA and Ca-SDA patterns remain incompletely understood. In this paper, we use cardiac tissue models described by an amplitude equation, coupled iterated maps, and reaction-diffusion equations with detailed physiology (the ionic model) to perform analytical and computational investigations. We show that, when the Ca subsystem is stable, the Ca-SDA pattern always follows the APD-SDA pattern, and thus, they are always synchronized. When the Ca subsystem is unstable, synchronization of APD-SDA and Ca-SDA patterns depends on the stabilities of both subsystems, their coupling strengths, and the spatial scales of the initial Ca-SDA patterns. Spontaneous (initial condition-independent) synchronization is promoted by enhancing APD instability and reducing Ca instability as well as stronger Ca-to-APD and APD-to-Ca coupling, a pattern formation caused by dynamical instabilities. When Ca is more unstable and APD is less unstable or APD-to-Ca coupling is weak, synchronization of APD-SDA and Ca-SDA patterns is promoted by larger initially synchronized Ca-SDA clusters, i.e., initial condition-dependent synchronization. The synchronized APD-SDA and Ca-SDA patterns can be locked in-phase, antiphase, or quasiperiodic depending on the coupling relationship between APD and Ca. These theoretical and simulation results provide mechanistic insights into the APD-SDA and Ca-SDA dynamics observed in experimental studies.

Molecular, Subcellular, and Arrhythmogenic Mechanisms in Genetic RyR2 Disease

The ryanodine receptor (RyR2) has a critical role in controlling Ca²⁺ release from the sarcoplasmic reticulum (SR) throughout the cardiac cycle. RyR2 protein has multiple functional domains with specific roles, and four of these RyR2 protomers are required to form the quaternary structure that comprises the functional channel. Numerous mutations in the gene encoding RyR2 protein have been identified and many are linked to a wide spectrum of arrhythmic heart disease. Gain of function mutations (GoF) result in a hyperactive channel that causes excessive spontaneous SR Ca²⁺ release. This is the predominant cause of the inherited syndrome catecholaminergic polymorphic ventricular tachycardia (CPVT). Recently, rare hypoactive loss of function (LoF) mutations have been identified that produce atypical effects on cardiac Ca²⁺ handling that has been termed calcium release deficiency syndrome (CRDS). Aberrant Ca²⁺ release resulting from both GoF and LoF mutations can result in arrhythmias through the Na⁺/Ca²⁺ exchange mechanism. This mini-review discusses recent findings regarding the role of RyR2 domains and endogenous regulators that influence RyR2 gating normally and with GoF/LoF mutations. The arrhythmogenic consequences of GoF/LoF mutations will then be discussed at the macromolecular and cellular level.

Kramers Rate Theory of Pacemaker Dynamics in Noisy Excitable Media

Rhythmic activities, which are usually driven by pacemakers, are common in biological systems. In noisy excitable media, pacemakers are self-organized firing clusters, but the underlying dynamics remains to be elucidated. Here we develop a Kramers rate theory of coupled cells to describe the firing properties of pacemakers and their dependence on coupling strength and system size and dimension. The theory captures accurately the simulation results of tissue models with stochastic Hodgkin-Huxley equations except when transitions from pacemakers to spiral waves occur under weak coupling.

Critical Spatial-Temporal Dynamics and Prominent Shape Collapse of Calcium Waves Observed in Human hNT Astrocytes in Vitro

Networks of neurons are typically studied in the field of Criticality. However, the study of astrocyte networks in the brain has been recently lauded to be of equal importance to that of the neural networks. To date criticality assessments have only been performed on networks astrocytes from healthy rats, and astrocytes from cultured dissociated resections of intractable epilepsy. This work, for the first time, presents studies of the critical dynamics and shape collapse of calcium waves observed in cultures of healthy human astrocyte networks in vitro, derived from the human hNT cell line. In this article, we demonstrate that avalanches of spontaneous calcium waves display strong critical dynamics, including power-laws in both the size and duration distributions. In addition, the temporal profiles of avalanches displayed self-similarity, leading to shape collapse of the temporal profiles. These findings are significant as they suggest that cultured networks of healthy human hNT astrocytes self-organize to a critical point, implying that healthy astrocytic networks operate at a critical point to process and transmit information. Furthermore, this work can serve as a point of reference to which other astrocyte criticality studies can be compared.

Active force generation contributes to the complexity of spontaneous activity and to the response to stretch of murine cardiomyocyte cultures

Abstract

Cardiomyocyte cultures exhibit spontaneous electrical and contractile activity, as in a natural cardiac pacemaker. In such preparations, beat rate variability exhibits features similar to those of heart rate variability in vivo. Mechanical deformations and forces feed back on the electrical properties of cardiomyocytes, but it is not fully elucidated how this mechano‐electrical interplay affects beating variability in such preparations. Using stretchable microelectrode arrays, we assessed the effects of the myosin inhibitor blebbistatin and the non‐selective stretch‐activated channel blocker streptomycin on beating variability and on the response of neonatal or fetal murine ventricular cell cultures against deformation. Spontaneous electrical activity was recorded without stretch and upon predefined deformation protocols (5% uniaxial and 2% equibiaxial strain, applied repeatedly for 1 min every 3 min). Without stretch, spontaneous activity originated from the edge of the preparations, and its site of origin switched frequently in a complex manner across the cultures. Blebbistatin did not change mean beat rate, but it decreased the spatial complexity of spontaneous activity. In contrast, streptomycin did not exert any manifest effects. During the deformation protocols, beat rate increased transiently upon stretch but, paradoxically, also upon release. Blebbistatin attenuated the response to stretch, whereas this response was not affected by streptomycin. Therefore, our data support the notion that in a spontaneously firing network of cardiomyocytes, active force generation, rather than stretch‐activated channels, is involved mechanistically in the complexity of the spatiotemporal patterns of spontaneous activity and in the stretch‐induced acceleration of beating.

Key points

Monolayer cultures of cardiac cells exhibit spontaneous electrical and contractile activity, as in a natural cardiac pacemaker. Beating variability in these preparations recapitulates the power‐law behaviour of heart rate variability in vivo. However, the effects of mechano‐electrical feedback on beating variability are not yet fully understood.

Using stretchable microelectrode arrays, we examined the effects of the contraction uncoupler blebbistatin and the non‐specific stretch‐activated channel blocker streptomycin on beating variability and on stretch‐induced changes of beat rate.

Without stretch, blebbistatin decreased the spatial complexity of beating variability, whereas streptomycin had no effects.

Both stretch and release increased beat rate transiently; blebbistatin attenuated the increase of beat rate upon stretch, whereas streptomycin had no effects.

Active force generation contributes to the complexity of spatiotemporal patterns of beating variability and to the increase of beat rate upon mechanical deformation. Our study contributes to the understanding of how mechano‐electrical feedback influences heart rate variability.

Pacemaker translocations and power laws in 2D stem cell-derived cardiomyocyte cultures

Power laws are of interest to several scientific disciplines because they can provide important information about the underlying dynamics (e.g. scale invariance and self-similarity) of a given system. Because power laws are of increasing interest to the cardiac sciences as potential indicators of cardiac dysfunction, it is essential that rigorous, standardized analytical methods are employed in the evaluation of power laws. This study compares the methods currently used in the fields of condensed matter physics, geoscience, neuroscience, and cardiology in order to provide a robust analytical framework for evaluating power laws in stem cell-derived cardiomyocyte cultures. One potential power law-obeying phenomenon observed in these cultures is pacemaker translocations, or the spatial and temporal instability of the pacemaker region, in a 2D cell culture. Power law analysis of translocation data was performed using increasingly rigorous methods in order to illustrate how differences in analytical robustness can result in misleading power law interpretations. Non-robust methods concluded that pacemaker translocations adhere to a power law while robust methods convincingly demonstrated that they obey a doubly truncated power law. The results of this study highlight the importance of employing comprehensive methods during power law analysis of cardiomyocyte cultures.

Multi-Scale Computational Modeling of Spatial Calcium Handling From Nanodomain to Whole-Heart: Overview and Perspectives

Regulation of intracellular calcium is a critical component of cardiac electrophysiology and excitation-contraction coupling. The calcium spark, the fundamental element of the intracellular calcium transient, is initiated in specialized nanodomains which co-locate the ryanodine receptors and L-type calcium channels. However, calcium homeostasis is ultimately regulated at the cellular scale, by the interaction of spatially separated but diffusively coupled nanodomains with other sub-cellular and surface-membrane calcium transport channels with strong non-linear interactions; and cardiac electrophysiology and arrhythmia mechanisms are ultimately tissue-scale phenomena, regulated by the interaction of a heterogeneous population of coupled myocytes. Recent advances in imaging modalities and image-analysis are enabling the super-resolution reconstruction of the structures responsible for regulating calcium homeostasis, including the internal structure of nanodomains themselves. Extrapolating functional and imaging data from the nanodomain to the whole-heart is non-trivial, yet essential for translational insight into disease mechanisms. Computational modeling has important roles to play in relating structural and functional data at the sub-cellular scale and translating data across the scales. This review covers recent methodological advances that enable image-based modeling of the single nanodomain and whole cardiomyocyte, as well as the development of multi-scale simulation approaches to integrate data from nanometer to whole-heart. Firstly, methods to overcome the computational challenges of simulating spatial calcium dynamics in the nanodomain are discussed, including image-based modeling at this scale. Then, recent whole-cell models, capable of capturing a range of different structures (such as the T-system and mitochondria) and cellular heterogeneity/variability are discussed at two different levels of discretization. Novel methods to integrate the models and data across the scales and simulate stochastic dynamics in tissue-scale models are then discussed, enabling elucidation of the mechanisms by which nanodomain remodeling underlies arrhythmia and contractile dysfunction. Perspectives on model differences and future directions are provided throughout.

Image-Driven Modeling of Nanoscopic Cardiac Function: Where Have We Come From, and Where Are We Going?

Complementary developments in microscopy and mathematical modeling have been critical to our understanding of cardiac excitation-contraction coupling. Historically, limitations imposed by the spatial or temporal resolution of imaging methods have been addressed through careful mathematical interrogation. Similarly, limitations imposed by computational power have been addressed by imaging macroscopic function in large subcellular domains or in whole myocytes. As both imaging resolution and computational tractability have improved, the two approaches have nearly merged in terms of the scales that they can each be used to interrogate. With this review we will provide an overview of these advances and their contribution to understanding ventricular myocyte function, including exciting developments over the last decade. We specifically focus on experimental methods that have pushed back limits of either spatial or temporal resolution of nanoscale imaging (e.g., DNA-PAINT), or have permitted high resolution imaging on large cellular volumes (e.g., serial scanning electron microscopy). We also review the progression of computational approaches used to integrate and interrogate these new experimental data sources, and comment on near-term advances that may unify understanding of the underlying biology. Finally, we comment on several outstanding questions in cardiac physiology that stand to benefit from a concerted and complementary application of these new experimental and computational methods.

Lower troponin expression in the right ventricle of rats explains interventricular differences in E-C coupling

Despite distinctive functional and anatomic differences, a precise understanding of the cardiac interventricular differences in excitation–contraction (E–C) coupling mechanisms is still lacking. Here, we directly compared rat right and left cardiomyocytes (RVCM and LVCM). Whole-cell patch clamp, the IonOptix system, and fura-2 fluorimetry were used to measure electrical properties (action potential and ionic currents), single-cell contractility, and cytosolic Ca²⁺ ([Ca²⁺]_i), respectively. Myofilament proteins were analyzed by immunoblotting. RVCM showed significantly shorter action potential duration (APD) and higher density of transient outward K⁺ current (I_to). However, the triggered [Ca²⁺]_i change (Ca²⁺ transient) was not different, while the decay rate of the Ca²⁺ transient was slower in RVCM. Although the relaxation speed was also slower, the sarcomere shortening amplitude (ΔSL) was smaller in RVCM. SERCA activity was ∼60% lower in RVCM, which is partly responsible for the slower decay of the Ca²⁺ transient. Immunoblot analysis revealed lower expression of the cardiac troponin complex (cTn) in RVCM, implying a smaller Ca²⁺ buffering capacity (κ_S), which was proved by in situ analysis. The introduction of these new levels of cTn, I_to, and SERCA into a mathematical model of rat LVCM reproduced the similar Ca²⁺ transient, slower Ca²⁺ decay, shorter APD, and smaller ΔSL of RVCM. Taken together, these data show reduced expression of cTn proteins in the RVCM, which provides an explanation for the interventricular difference in the E–C coupling kinetics.

Mechanisms of Arrhythmogenicity of Hypertrophic Cardiomyopathy-Associated Troponin T (TNNT2) Variant I79N

Hypertrophic cardiomyopathy (HCM) is the most common heritable cardiovascular disease and often results in cardiac remodeling and an increased incidence of sudden cardiac arrest (SCA) and death, especially in youth and young adults. Among thousands of different variants found in HCM patients, variants of TNNT2 (cardiac troponin T—TNNT2) are linked to increased risk of ventricular arrhythmogenesis and sudden death despite causing little to no cardiac hypertrophy. Therefore, studying the effect of TNNT2 variants on cardiac propensity for arrhythmogenesis can pave the way for characterizing HCM in susceptible patients before sudden cardiac arrest occurs. In this study, a TNNT2 variant, I79N, was generated in human cardiac recombinant/reconstituted thin filaments (hcRTF) to investigate the effect of the mutation on myofilament Ca²⁺ sensitivity and Ca²⁺ dissociation rate using steady-state and stopped-flow fluorescence techniques. The results revealed that the I79N variant significantly increases myofilament Ca²⁺ sensitivity and decreases the Ca²⁺ off-rate constant (k_off). To investigate further, a heterozygous I79N^+/−TNNT2 variant was introduced into human-induced pluripotent stem cells using CRISPR/Cas9 and subsequently differentiated into ventricular cardiomyocytes (hiPSC-CMs). To study the arrhythmogenic properties, monolayers of I79N^+/− hiPSC-CMs were studied in comparison to their isogenic controls. Arrhythmogenesis was investigated by measuring voltage (V_m) and cytosolic Ca²⁺ transients over a range of stimulation frequencies. An increasing stimulation frequency was applied to the cells, from 55 to 75 bpm. The results of this protocol showed that the TnT-I79N cells had reduced intracellular Ca²⁺ transients due to the enhanced cytosolic Ca²⁺ buffering. These changes in Ca²⁺ handling resulted in beat-to-beat instability and triangulation of the cardiac action potential, which are predictors of arrhythmia risk. While wild-type (WT) hiPSC-CMs were accurately entrained to frequencies of at least 150 bpm, the I79N hiPSC-CMs demonstrated clear patterns of alternans for both V_m and Ca²⁺ transients at frequencies >75 bpm. Lastly, a transcriptomic analysis was conducted on WT vs. I79N^+/−TNNT2 hiPSC-CMs using a custom NanoString codeset. The results showed a significant upregulation of NPPA (atrial natriuretic peptide), NPPB (brain natriuretic peptide), Notch signaling pathway components, and other extracellular matrix (ECM) remodeling components in I79N^+/− vs. the isogenic control. This significant shift demonstrates that this missense in the TNNT2 transcript likely causes a biophysical trigger, which initiates this significant alteration in the transcriptome. This TnT-I79N hiPSC-CM model not only reproduces key cellular features of HCM-linked mutations but also suggests that this variant causes uncharted pro-arrhythmic changes to the human action potential and gene expression.

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.

Ca2+ and Membrane Potential Transitions During Action Potentials Are Self-Similar to Each Other and to Variability of AP Firing Intervals Across the Broad Physiologic Range of AP Intervals During Autonomic Receptor Stimulation

Ca2+ and V m transitions occurring throughout action potential (AP) cycles in sinoatrial nodal (SAN) cells are cues that (1) not only regulate activation states of molecules operating within criticality (Ca2+ domain) and limit-cycle (V m domain) mechanisms of a coupled-clock system that underlies SAN cell automaticity, (2) but are also regulated by the activation states of the clock molecules they regulate. In other terms, these cues are both causes and effects of clock molecular activation (recursion). Recently, we demonstrated that Ca2+ and V m transitions during AP cycles in single SAN cells isolated from mice, guinea pigs, rabbits, and humans are self-similar (obey a power law) and are also self-similar to trans-species AP firing intervals (APFIs) of these cells in vitro, to heart rate in vivo, and to body mass. Neurotransmitter stimulation of β-adrenergic receptor or cholinergic receptor-initiated signaling in SAN cells modulates their AP firing rate and rhythm by impacting on the degree to which SAN clocks couple to each other, creating the broad physiologic range of SAN cell mean APFIs and firing interval variabilities. Here we show that Ca2+ and V m domain kinetic transitions (time to AP ignition in diastole and 90% AP recovery) occurring within given AP, the mean APFIs, and APFI variabilities within the time series of APs in 230 individual SAN cells are self-similar (obey power laws). In other terms, these long-range correlations inform on self-similar distributions of order among SAN cells across the entire broad physiologic range of SAN APFIs, regardless of whether autonomic receptors of these cells are stimulated or not and regardless of the type (adrenergic or cholinergic) of autonomic receptor stimulation. These long-range correlations among distributions of Ca2+ and V m kinetic functions that regulate SAN cell clock coupling during each AP cycle in different individual, isolated SAN cells not in contact with each other. Our numerical model simulations further extended our perspectives to the molecular scale and demonstrated that many ion currents also behave self-similar across autonomic states. Thus, to ensure rapid flexibility of AP firing rates in response to different types and degrees of autonomic input, nature "did not reinvent molecular wheels within the coupled-clock system of pacemaker cells," but differentially engaged or scaled the kinetics of gears that regulate the rate and rhythm at which the "wheels spin" in a given autonomic input context.

β-Adrenergic Stimulation Synchronizes a Broad Spectrum of Action Potential Firing Rates of Cardiac Pacemaker Cells toward a Higher Population Average

The heartbeat is initiated by pacemaker cells residing in the sinoatrial node (SAN). SAN cells generate spontaneous action potentials (APs), i.e., normal automaticity. The sympathetic nervous system increases the heart rate commensurate with the cardiac output demand via stimulation of SAN β-adrenergic receptors (βAR). While SAN cells reportedly represent a highly heterogeneous cell population, the current dogma is that, in response to βAR stimulation, all cells increase their spontaneous AP firing rate in a similar fashion. The aim of the present study was to investigate the cell-to-cell variability in the responses of a large population of SAN cells. We measured the βAR responses among 166 single SAN cells isolated from 33 guinea pig hearts. In contrast to the current dogma, the SAN cell responses to βAR stimulation substantially varied. In each cell, changes in the AP cycle length were highly correlated (R2 = 0.97) with the AP cycle length before βAR stimulation. While, as expected, on average, the cells increased their pacemaker rate, greater responses were observed in cells with slower basal rates, and vice versa: cells with higher basal rates showed smaller responses, no responses, or even decreased their rate. Thus, βAR stimulation synchronized the operation of the SAN cell population toward a higher average rate, rather than uniformly shifting the rate in each cell, creating a new paradigm of βAR-driven fight-or-flight responses among individual pacemaker cells.

Self-Similar Synchronization of Calcium and Membrane Potential Transitions During Action Potential Cycles Predict Heart Rate Across Species

OBJECTIVES

The purpose of this study was to discover regulatory universal mechanisms of normal automaticity in sinoatrial nodal (SAN) pacemaker cells that are self-similar across species.

BACKGROUND

Translation of knowledge of SAN automaticity gleaned from animal studies to human dysrhythmias (e.g., "sick sinus" syndrome [SSS]) requiring electronic pacemaker insertion has been suboptimal, largely because heart rate varies widely across species.

METHODS

Subcellular Ca²⁺ releases, whole cell action potential (AP)-induced Ca²⁺ transients, and APs were recorded in isolated mouse, guinea pig, rabbit, and human SAN cells. Ca²⁺-Vm kinetic parameters during phases of AP cycles from their ignition to recovery were quantified.

RESULTS

Although both action potential cycle lengths (APCLs) and Ca²⁺-Vm kinetic parameters during AP cycles differed across species by 10-fold, trans-species scaling of these during AP cycles and scaling of these to APCL in cells in vitro, electrocardiogram RR intervals in vivo, and body mass (BM) were self-similar (obeyed power laws) across species. Thus, APCL in vitro, heart rate in vivo, and BM of any species can be predicted by Ca²⁺-Vm kinetics during AP cycles in SAN cells measured in any single species in vitro.

CONCLUSIONS

In designing optimal heart rate to match widely different BM and energy requirements from mice to humans, nature did not "reinvent pacemaker cell wheels," but differentially scaled kinetics of gears that regulate the rates at which the "wheels spin." This discovery will facilitate the development of novel pharmacological and biological pacemakers featuring a normal, wide-range rate regulation in animal models and the translation of these to humans to target recalcitrant human SSS.

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca2+) concentration in a cell in the present excitation may affect the Ca2+ dynamics in the next excitation via bi-directional coupling between voltage and Ca2+, forming a delayed feedback loop. Since the Ca2+ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca2+alternans in cardiac myocytes or spatially concordant or discordant Ca2+ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Cardiomyocyte calcium handling in health and disease: Insights from in vitro and in silico studies

Calcium (Ca²⁺) plays a central role in cardiomyocyte excitation-contraction coupling. To ensure an optimal electrical impulse propagation and cardiac contraction, Ca²⁺ levels are regulated by a variety of Ca²⁺-handling proteins. In turn, Ca²⁺ modulates numerous electrophysiological processes. Accordingly, Ca²⁺-handling abnormalities can promote cardiac arrhythmias via various mechanisms, including the promotion of afterdepolarizations, ion-channel modulation and structural remodeling. In the last 30 years, significant improvements have been made in the computational modeling of cardiomyocyte Ca²⁺ handling under physiological and pathological conditions. However, numerous questions involving the Ca²⁺-dependent regulation of different macromolecular complexes, cross-talk between Ca²⁺-dependent regulatory pathways operating over a wide range of time scales, and bidirectional interactions between electrophysiology and mechanics remain to be addressed by in vitro and in silico studies. A better understanding of disease-specific Ca²⁺-dependent proarrhythmic mechanisms may facilitate the development of improved therapeutic strategies. In this review, we describe the fundamental mechanisms of cardiomyocyte Ca²⁺ handling in health and disease, and provide an overview of currently available computational models for cardiomyocyte Ca²⁺ handling. Finally, we discuss important uncertainties and open questions about cardiomyocyte Ca²⁺ handling and highlight how synergy between in vitro and in silico studies may help to answer several of these issues.

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.

Proper spatial heterogeneities expand the regime of scale-free behavior in a lattice of excitable elements

Signatures of criticality, such as power law scaling of observables, have been empirically found in a plethora of real-life settings, including biological systems. The presence of critical states is believed to have many functional advantages and is associated with optimal operational abilities. Typically, critical dynamics arises in the proximity of phase transition points between absorbing disordered states (subcriticality) and ordered active regimes (supercriticality) and requires a high degree of fine tuning to emerge, which is unlikely to occur in real biological systems. In the present study we propose a rather simple, and biologically relevant mechanism that profoundly expands the critical-like region. In particular, by means of numerical simulation we show that incorporating spatial heterogeneities into the square lattice of map-based excitable oscillators broadens the parameter space in which the distribution of excitation wave sizes follows closely a power law. Most importantly, this behavior is only observed if the spatial profile exhibits intermediate-sized patches with similar excitability levels, whereas for large and small spatial clusters only marginal widening of the critical state is detected. Furthermore, it turned out that the presence of spatial disorder in general amplifies the size of excitation waves, whereby the relatively highest contributions are observed in the proximity of the critical point. We argue that the reported mechanism is of particular importance for excitable systems with local interactions between individual elements.

Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model’s necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.

The Cognitive Lens: a primer on conceptual tools for analysing information processing in developmental and regenerative morphogenesis

Brains exhibit plasticity, multi-scale integration of information, computation and memory, having evolved by specialization of non-neural cells that already possessed many of the same molecular components and functions. The emerging field of basal cognition provides many examples of decision-making throughout a wide range of non-neural systems. How can biological information processing across scales of size and complexity be quantitatively characterized and exploited in biomedical settings? We use pattern regulation as a context in which to introduce the Cognitive Lens-a strategy using well-established concepts from cognitive and computer science to complement mechanistic investigation in biology. To facilitate the assimilation and application of these approaches across biology, we review tools from various quantitative disciplines, including dynamical systems, information theory and least-action principles. We propose that these tools can be extended beyond neural settings to predict and control systems-level outcomes, and to understand biological patterning as a form of primitive cognition. We hypothesize that a cognitive-level information-processing view of the functions of living systems can complement reductive perspectives, improving efficient top-down control of organism-level outcomes. Exploration of the deep parallels across diverse quantitative paradigms will drive integrative advances in evolutionary biology, regenerative medicine, synthetic bioengineering, cognitive neuroscience and artificial intelligence. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves

Ryanodine receptors (RyRs) mediate calcium (Ca)-induced Ca release and intracellular Ca homeostasis. In a cardiac myocyte, RyRs group into clusters of variable size from a few to several hundred RyRs, creating a spatially nonuniform intracellular distribution. It is unclear how heterogeneity of RyR cluster size alters spontaneous sarcoplasmic reticulum (SR) Ca releases (Ca sparks) and arrhythmogenic Ca waves. Here, we tested the impact of heterogeneous RyR cluster size on the initiation of Ca waves. Experimentally, we measured RyR cluster sizes at Ca spark sites in rat ventricular myocytes and further tested functional impacts using a physiologically detailed computational model with spatial and stochastic intracellular Ca dynamics. We found that the spark frequency and amplitude increase nonlinearly with the size of RyR clusters. Larger RyR clusters have lower SR Ca release threshold for local Ca spark initiation and exhibit steeper SR Ca release versus SR Ca load relationship. However, larger RyR clusters tend to lower SR Ca load because of the higher Ca leak rate. Conversely, smaller clusters have a higher threshold and a lower leak, which tends to increase SR Ca load. At the myocyte level, homogeneously large or small RyR clusters limit Ca waves (because of low load for large clusters but low excitability for small clusters). Mixtures of large and small RyR clusters potentiates Ca waves because the enhanced SR Ca load driven by smaller clusters enables Ca wave initiation and propagation from larger RyR clusters. Our study suggests that a spatially heterogeneous distribution of RyR cluster size under pathological conditions may potentiate Ca waves and thus afterdepolarizations and triggered arrhythmias.

Stochastic coupled map model of subcellular calcium cycling in cardiac cells

In this study, we analyze a nonlinear map model of intracellular calcium (Ca) and voltage in cardiac cells. In this model, Ca release from the sarcoplasmic reticulum (SR) occurs at spatially distributed dyadic junctions that are diffusively coupled. At these junctions, release occurs with a probability that depends on key variables such as the SR load and the diastolic interval. Using this model, we explore how nonlinearity and stochasticity determine the spatial distribution of Ca release events within a cardiac cell. In particular, we identify a novel synchronization transition, which occurs at rapid pacing rates, in which the global Ca transient transitions from a period 2 response to a period 1 response. In the global period 2 response dyadic junctions fire in unison, on average, on alternate beats, while in the period 1 regime, Ca release at individual dyads is highly irregular. A close examination of the spatial distribution of Ca reveals that in the period 1 regime, the system coarsens into spatially out-of-phase regions with a length scale much smaller than the system size, but larger than the spacing between dyads. We have also explored in detail the coupling to membrane voltage. We study first the case of positive coupling, where a large Ca transient promotes a long action potential duration (APD). Here, the coupling to voltage synchronizes Ca release so that the system exhibits a robust period 2 response that is independent of initial conditions. On the other hand, in the case of negative coupling, where a large Ca transient tends to shorten the APD, we find a multitude of metastable states which consist of complex spatially discordant alternans patterns. Using an analogy to equilibrium statistical mechanics, we show that the spatial patterns observed can be explained by a mapping to the Potts model, with an additional term that accounts for a global coupling of spin states. Using this analogy, we argue that Ca cycling in cardiac cells exhibits complex spatiotemporal patterns that emerge via first or second order phase transitions. These results show that voltage and Ca can interact in order to induce complex subcellular responses, which can potentially lead to heart rhythm disorders.

Transverse tubular network structures in the genesis of intracellular calcium alternans and triggered activity in cardiac cells

RATIONALE

The major role of a transverse-tubular (TT) network in a cardiac cell is to facilitate effective excitation-contraction coupling and signaling. The TT network structures are heterogeneous within a single cell, and vary between different types of cells and species. They are also remodeled in cardiac diseases. However, how different TT network structures predispose cardiac cells to arrhythmogenesis remains to be revealed.

OBJECTIVE

To systematically investigate the roles of TT network structure and the underlying mechanisms in the genesis of intracellular calcium (Ca2+) alternans and triggered activity (TA).

METHODS AND RESULTS

Based on recent experimental observations, different TT network structures, including uniformly and non-uniformly random TT distributions, were modeled in a cardiac cell model consisting of a three-dimensional network of Ca2+ release units (CRUs). Our simulations showed that both Ca2+ alternans and Ca2+ wave-mediated TA were promoted when the fraction of orphaned CRUs was in an intermediate range, but suppressed in cells exhibiting either well-organized TT networks or low TT densities. Ca2+ alternans and TA could be promoted by low TT densities when the cells were small or the CRU coupling was strong. Both alternans and TA occurred more easily in uniformly random TT networks than in non-uniformly random TT networks. Subcellular spatially discordant Ca2+ alternans was promoted by non-uniformly random TT networks but suppressed by increasing CRU coupling strength. These mechanistic insights provide a holistic understanding of the effects of TT network structure on the susceptibility to arrhythmogenesis.

CONCLUSIONS

The TT network plays important roles in promoting Ca2+ alternans and TA, and different TT network structures may predispose cardiac cells differently to arrhythmogenesis.

Critical and Supercritical Spatiotemporal Calcium Dynamics in Beta Cells

A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

A computational model of spatio-temporal cardiac intracellular calcium handling with realistic structure and spatial flux distribution from sarcoplasmic reticulum and t-tubule reconstructions

Intracellular calcium cycling is a vital component of cardiac excitation-contraction coupling. The key structures responsible for controlling calcium dynamics are the cell membrane (comprising the surface sarcolemma and transverse-tubules), the intracellular calcium store (the sarcoplasmic reticulum), and the co-localisation of these two structures to form dyads within which calcium-induced-calcium-release occurs. The organisation of these structures tightly controls intracellular calcium dynamics. In this study, we present a computational model of intracellular calcium cycling in three-dimensions (3-D), which incorporates high resolution reconstructions of these key regulatory structures, attained through imaging of tissue taken from the sheep left ventricle using serial block face scanning electron microscopy. An approach was developed to model the sarcoplasmic reticulum structure at the whole-cell scale, by reducing its full 3-D structure to a 3-D network of one-dimensional strands. The model reproduces intracellular calcium dynamics during control pacing and reveals the high-resolution 3-D spatial structure of calcium gradients and intracellular fluxes in both the cytoplasm and sarcoplasmic reticulum. We also demonstrated the capability of the model to reproduce potentially pro-arrhythmic dynamics under perturbed conditions, pertaining to calcium-transient alternans and spontaneous release events. Comparison with idealised cell models emphasised the importance of structure in determining calcium gradients and controlling the spatial dynamics associated with calcium-transient alternans, wherein the probabilistic nature of dyad activation and recruitment was constrained. The model was further used to highlight the criticality in calcium spark propagation in relation to inter-dyad distances. The model presented provides a powerful tool for future investigation of structure-function relationships underlying physiological and pathophysiological intracellular calcium handling phenomena at the whole-cell. The approach allows for the first time direct integration of high-resolution images of 3-D intracellular structures with models of calcium cycling, presenting the possibility to directly assess the functional impact of structural remodelling at the cellular scale.

The organisation of the membrane and sub-cellular structures of cells in the heart closely controls the coupling between its electrical and mechanical function. Computational models of the cellular calcium handling system, which is responsible for this electro-mechanical coupling, have been developed in recent years to study underlying structure-function relationships. Previous models have been largely idealised in structure; we present a new model which incorporates experimental data describing the high-resolution organisation of the primary structures involved in calcium dynamics. Significantly, the structure of the intracellular calcium store is modelled for the first time. The model is shown to reproduce calcium dynamics in control cells in both normal and abnormal conditions, demonstrating its suitability for future investigation of structure-function relationships. Thus, the model presented provides a powerful tool for the direct integration of experimentally acquired structural data in healthy and diseased cells and assessment of the role of structure in regulating normal and abnormal calcium dynamics.

Multiscale Determinants of Delayed Afterdepolarization Amplitude in Cardiac Tissue

Spontaneous calcium (Ca) waves in cardiac myocytes underlie delayed afterdepolarizations (DADs) that trigger cardiac arrhythmias. How these subcellular/cellular events overcome source-sink factors in cardiac tissue to generate DADs of sufficient amplitude to trigger action potentials is not fully understood. Here, we evaluate quantitatively how factors at the subcellular scale (number of Ca wave initiation sites), cellular scale (sarcoplasmic reticulum (SR) Ca load), and tissue scale (synchrony of Ca release in populations of myocytes) determine DAD features in cardiac tissue using a combined experimental and computational modeling approach. Isolated patch-clamped rabbit ventricular myocytes loaded with Fluo-4 to image intracellular Ca were rapidly paced during exposure to elevated extracellular Ca (2.7 mmol/L) and isoproterenol (0.25 μmol/L) to induce diastolic Ca waves and subthreshold DADs. As the number of paced beats increased from 1 to 5, SR Ca content (assessed with caffeine pulses) increased, the number of Ca wave initiation sites increased, integrated Ca transients and DADs became larger and shorter in duration, and the latency period to the onset of Ca waves shortened with reduced variance. In silico analysis using a computer model of ventricular tissue incorporating these experimental measurements revealed that whereas all of these factors promoted larger DADs with higher probability of generating triggered activity, the latency period variance and SR Ca load had the greatest influences. Therefore, incorporating quantitative experimental data into tissue level simulations reveals that increased intracellular Ca promotes DAD-mediated triggered activity in tissue predominantly by increasing both the synchrony (decreasing latency variance) of Ca waves in nearby myocytes and SR Ca load, whereas the number of Ca wave initiation sites per myocyte is less important.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Stochastic initiation and termination of calcium-mediated triggered activity in cardiac myocytes

Cardiac myocytes normally initiate action potentials in response to a current stimulus that depolarizes the membrane above an excitation threshold. Aberrant excitation can also occur due to spontaneous calcium (Ca²⁺) release (SCR) from intracellular stores after the end of a preceding action potential. SCR drives the Na⁺/Ca²⁺ exchange current inducing a "delayed afterdepolarization" that can in turn trigger an action potential if the excitation threshold is reached. This "triggered activity" is known to cause arrhythmias, but how it is initiated and terminated is not understood. Using computer simulations of a ventricular myocyte model, we show that initiation and termination are inherently random events. We determine the probability of those events from statistical measurements of the number of beats before initiation and before termination, respectively, which follow geometric distributions. Moreover, we elucidate the origin of randomness by a statistical analysis of SCR events, which do not follow a Poisson process observed in other eukaryotic cells. Due to synchronization of Ca²⁺ releases during the action potential upstroke, waiting times of SCR events after the upstroke are narrowly distributed, whereas SCR amplitudes follow a broad normal distribution with a width determined by fluctuations in the number of independent Ca²⁺ wave foci. This distribution enables us to compute the probabilities of initiation and termination of bursts of triggered activity that are maintained by a positive feedback between the action potential upstroke and SCR. Our results establish a theoretical framework for interpreting complex and varied manifestations of triggered activity relevant to cardiac arrhythmias.

Nonlinear onset of calcium wave propagation in cardiac cells

Spontaneous calcium (Ca) waves in cardiac myocytes are known to underlie a wide range of cardiac arrhythmias. However, it is not understood which physiological parameters determine the onset of waves. In this study, we explore the relationship between Ca signaling between ion channels and the nucleation of Ca waves. In particular, we apply a master equation approach to analyze the stochastic interaction between neighboring clusters of ryanodine receptor (RyR) channels. Using this analysis, we show that signaling between clusters can be described as a barrier hopping process with exponential sensitivity to system parameters. A consequence of this feature is that the probability that Ca release at a cluster induces release at a neighboring cluster exhibits a sigmoid dependence on the Ca content in the cell. This nonlinearity originates from the regulation of RyR opening due to more than one Ca ion binding site, in conjunction with Ca mediated cooperativity between RyR channels in clusters. We apply a spatially distributed stochastic model of Ca cycling to analyze the physiological consequences of this nonlinearity, and show that it explains the sharp onset of Ca wave nucleation in cardiac cells. Furthermore, we show that this sharp onset can serve as a mechanism for Ca alternans under physiologically relevant conditions. Thus our findings identify the nonlinear features of Ca signaling which potentially underlie the onset of Ca waves and Ca alternans in cardiac cells.

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 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 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.

Long-Lasting Sparks: Multi-Metastability and Release Competition in the Calcium Release Unit Network

Calcium (Ca) sparks are elementary events of biological Ca signaling. A normal Ca spark has a brief duration in the range of 10 to 100 ms, but long-lasting sparks with durations of several hundred milliseconds to seconds are also widely observed. Experiments have shown that the transition from normal to long-lasting sparks can occur when ryanodine receptor (RyR) open probability is either increased or decreased. Here, we demonstrate theoretically and computationally that long-lasting sparks emerge as a collective dynamical behavior of the network of diffusively coupled Ca release units (CRUs). We show that normal sparks occur when the CRU network is monostable and excitable, while long-lasting sparks occur when the network dynamics possesses multiple metastable attractors, each attractor corresponding to a different spatial firing pattern of sparks. We further highlight the mechanisms and conditions that produce long-lasting sparks, demonstrating the existence of an optimal range of RyR open probability favoring long-lasting sparks. We find that when CRU firings are sparse and sarcoplasmic reticulum (SR) Ca load is high, increasing RyR open probability promotes long-lasting sparks by potentiating Ca-induced Ca release (CICR). In contrast, when CICR is already strong enough to produce frequent firings, decreasing RyR open probability counter-intuitively promotes long-lasting sparks by decreasing spark frequency. The decrease in spark frequency promotes intra-SR Ca diffusion from neighboring non-firing CRUs to the firing CRUs, which helps to maintain the local SR Ca concentration of the firing CRUs above a critical level to sustain firing. In this setting, decreasing RyR open probability further suppresses long-lasting sparks by weakening CICR. Since a long-lasting spark terminates via the Kramers’ escape process over a potential barrier, its duration exhibits an exponential distribution determined by the barrier height and noise strength, which is modulated differently by different ways of altering the Ca release flux strength.

Calcium (Ca) sparks, resulting from Ca-induced Ca release, are elementary events of biological Ca signaling. Sparks are normally brief, but long-lasting sparks have been widely observed experimentally under various conditions. The underlying mechanisms of spark duration or termination and the corresponding determinants remain a topic of debate. In this study, we demonstrate theoretically and computationally that normal brief sparks are excitable transients, while long-lasting sparks are multiple metastable states emerging in the diffusively coupled Ca release unit network, as a result of cooperativity and release competition among the Ca release units. Termination of a long-lasting spark is a Kramers’ escape process over a potential barrier, and the spark duration is the first-passage time, exhibiting an exponential distribution.

Role of connectivity and fluctuations in the nucleation of calcium waves in cardiac cells

Spontaneous calcium release (SCR) occurs when ion channel fluctuations lead to the nucleation of calcium waves in cardiac cells. This phenomenon is important since it has been implicated as a cause of various cardiac arrhythmias. However, to date, it is not understood what determines the timing and location of spontaneous calcium waves within cells. Here, we analyze a simplified model of SCR in which calcium release is modeled as a stochastic processes on a two-dimensional network of randomly distributed sites. Using this model we identify the essential parameters describing the system and compute the phase diagram. In particular, we identify a critical line which separates pinned and propagating fronts, and show that above this line wave nucleation is governed by fluctuations and the spatial connectivity of calcium release units. Using a mean-field analysis we show that the sites of wave nucleation are predicted by localized eigenvectors of a matrix representing the network connectivity of release sites. This result provides insight on the interplay between connectivity and fluctuations in the genesis of SCR in cardiac myocytes.

Delayed afterdepolarizations generate both triggers and a vulnerable substrate promoting reentry in cardiac tissue

BACKGROUND

Delayed afterdepolarizations (DADs) have been well characterized as arrhythmia triggers, but their role in generating a tissue substrate vulnerable to reentry is not well understood.

OBJECTIVE

The purpose of this study was to test the hypothesis that random DADs can self-organize to generate both an arrhythmia trigger and a vulnerable substrate simultaneously in cardiac tissue as a result of gap junction coupling.

METHODS

Computer simulations in 1-dimensional cable and 2-dimensional tissue models were performed. The cellular DAD amplitude was varied by changing the strength of sarcoplasmic reticulum calcium release. Random DAD latency and amplitude in different cells were simulated using gaussian distributions.

RESULTS

Depending on the strength of spontaneous sarcoplasmic reticulum calcium release and other conditions, random DADs in cardiac tissue resulted in the following behaviors: (1) triggered activity (TA); (2) a vulnerable tissue substrate causing unidirectional conduction block and reentry by inactivating sodium channels; (3) both triggers and a vulnerable substrate simultaneously by generating TA in regions next to regions with subthreshold DADs susceptible to unidirectional conduction block and reentry. The probability of the latter 2 behaviors was enhanced by reduced sodium channel availability, reduced gap junction coupling, increased tissue heterogeneity, and less synchronous DAD latency.

CONCLUSION

DADs can self-organize in tissue to generate arrhythmia triggers, a vulnerable tissue substrate, and both simultaneously. Reduced sodium channel availability and gap junction coupling potentiate this mechanism of arrhythmias, which are relevant to a variety of heart disease conditions.

Calcium alternans is due to an order-disorder phase transition in cardiac cells

Electromechanical alternans is a beat-to-beat alternation in the strength of contraction of a cardiac cell, which can be caused by an instability of calcium cycling. Using a distributed model of subcellular calcium we show that alternans occurs via an order-disorder phase transition which exhibits critical slowing down and a diverging correlation length. We apply finite size scaling along with a mapping to a stochastic coupled map model, to show that this transition in two dimensions is characterized by critical exponents consistent with the Ising universality class. These findings highlight the important role of cooperativity in biological cells, and suggest novel approaches to investigate the onset of the alternans instability in the heart.

Function of metabolic and organelle networks in crowded and organized media

(Macro)molecular crowding and the ability of the ubiquitous cytoskeleton to dynamically polymerize–depolymerize are prevalent cytoplasmic conditions in prokaryotic and eukaryotic cells. Protein interactions, enzymatic or signaling reactions - single, sequential or in complexes - whole metabolic pathways and organelles can be affected by crowding, the type and polymeric status of cytoskeletal proteins (e.g., tubulin, actin), and their imparted organization. The self-organizing capability of the cytoskeleton can orchestrate metabolic fluxes through entire pathways while its fractal organization can frame the scaling of activities in several levels of organization. The intracellular environment dynamics (e.g., biochemical reactions) is dominated by the orderly cytoskeleton and the intrinsic randomness of molecular crowding. Existing evidence underscores the inherent capacity of intracellular organization to generate emergent global behavior. Yet unknown is the relative impact on cell function provided by organelle or functional compartmentation based on transient proteins association driven by weak interactions (quinary structures) under specific environmental challenges or functional conditions (e.g., hypoxia, division, differentiation). We propose a qualitative, integrated structural–functional model of cytoplasmic organization based on a modified version of the Sierspinsky–Menger–Mandelbrot sponge, a 3D representation of a percolation cluster, and examine its capacity to accommodate established experimental facts.

Acute reversal of phospholamban inhibition facilitates the rhythmic whole-cell propagating calcium waves in isolated ventricular myocytes

Phospholamban (PLB) inhibits the activity of cardiac sarcoplasmic reticulum (SR) Ca(2+)-ATPase (SERCA2a). Phosphorylation of PLB during sympathetic activation reverses SERCA2a inhibition, increasing SR Ca(2+) uptake. However, sympathetic activation also modulates multiple other intracellular targets in ventricular myocytes (VMs), making it impossible to determine the specific effects of the reversal of PLB inhibition on the spontaneous SR Ca(2+) release. Therefore, it remains unclear how PLB regulates rhythmic activity in VMs. Here, we used the Fab fragment of 2D12, a monoclonal anti-PLB antibody, to test how acute reversal of PLB inhibition affects the spontaneous SR Ca(2+) release in normal VMs. Ca(2+) sparks and spontaneous Ca(2+) waves (SCWs) were recorded in the line-scan mode of confocal microscopy using the Ca(2+) fluorescent dye Fluo-4 in isolated permeabilized mouse VMs. Fab, which reverses PLB inhibition, significantly increased the frequency, amplitude, and spatial/temporal spread of Ca(2+) sparks in VMs exposed to 50 nM free [Ca(2+)]. At physiological diastolic free [Ca(2+)] (100-200 nM), Fab facilitated the formation of whole-cell propagating SCWs. At higher free [Ca(2+)], Fab increased the frequency and velocity, but decreased the decay time of the SCWs. cAMP had little additional effect on the frequency or morphology of Ca(2+) sparks or SCWs after Fab addition. These findings were complemented by computer simulations. In conclusion, acute reversal of PLB inhibition alone significantly increased the spontaneous SR Ca(2+) release, leading to the facilitation and organization of whole-cell propagating SCWs in normal VMs. PLB thus plays a key role in subcellular Ca(2+) dynamics and rhythmic activity of VMs.

Synergy Between Intercellular Communication and Intracellular Ca(2+) Handling in Arrhythmogenesis

Calcium is the primary signalling component of excitation-contraction coupling, the process linking electrical excitability of cardiac muscle cells to coordinated contraction of the heart. Understanding [Formula: see text] handling processes at the cellular level and the role of intercellular communication in the emergence of multicellular synchronization are key aspects in the study of arrhythmias. To probe these mechanisms, we have simulated cellular interactions on large scale arrays that mimic cardiac tissue, and where individual cells are represented by a mathematical model of intracellular [Formula: see text] dynamics. Theoretical predictions successfully reproduced experimental findings and provide novel insights on the action of two pharmacological agents (ionomycin and verapamil) that modulate [Formula: see text] signalling pathways via distinct mechanisms. Computational results have demonstrated how transitions between local synchronisation events and large scale wave formation are affected by these agents. Entrainment phenomena are shown to be linked to both intracellular [Formula: see text] and coupling-specific dynamics in a synergistic manner. The intrinsic variability of the cellular matrix is also shown to affect emergent patterns of rhythmicity, providing insights into the origins of arrhythmogenic [Formula: see text] perturbations in cardiac tissue in situ.

Mechanisms of ventricular arrhythmias: from molecular fluctuations to electrical turbulence

Ventricular arrhythmias have complex causes and mechanisms. Despite extensive investigation involving many clinical, experimental, and computational studies, effective biological therapeutics are still very limited. In this article, we review our current understanding of the mechanisms of ventricular arrhythmias by summarizing the state of knowledge spanning from the molecular scale to electrical wave behavior at the tissue and organ scales and how the complex nonlinear interactions integrate into the dynamics of arrhythmias in the heart. We discuss the challenges that we face in synthesizing these dynamics to develop safe and effective novel therapeutic approaches.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Excitability in a stochastic differential equation model for calcium puffs

Calcium dynamics are essential to a multitude of cellular processes. For many cell types, localized discharges of calcium through small clusters of intracellular channels are building blocks for all spatially extended calcium signals. Because of the large noise amplitude, the validity of noise-approximating model equations for this system has been questioned. Here we revisit the master equations for local calcium release, examine the multiple scales of calcium concentrations in the cluster domain, and derive adapted stochastic differential equations. We show by comparison of discrete and continuous trajectories that the Langevin equations can be made consistent with the master equations even for very small channel numbers. In its deterministic limit, the model reveals that excitability, a dynamical phenomenon observed in many natural systems, is at the core of calcium puffs. The model also predicts a bifurcation from transient to sustained release which may link local and global calcium signals in cells.

Hierarchical clustering of ryanodine receptors enables emergence of a calcium clock in sinoatrial node cells

The sinoatrial node, whose cells (sinoatrial node cells [SANCs]) generate rhythmic action potentials, is the primary pacemaker of the heart. During diastole, calcium released from the sarcoplasmic reticulum (SR) via ryanodine receptors (RyRs) interacts with membrane currents to control the rate of the heartbeat. This "calcium clock" takes the form of stochastic, partially periodic, localized calcium release (LCR) events that propagate, wave-like, for limited distances. The detailed mechanisms controlling the calcium clock are not understood. We constructed a computational model of SANCs, including three-dimensional diffusion and buffering of calcium in the cytosol and SR; explicit, stochastic gating of individual RyRs and L-type calcium channels; and a full complement of voltage- and calcium-dependent membrane currents. We did not include an anatomical submembrane space or inactivation of RyRs, the two heuristic components that have been used in prior models but are not observed experimentally. When RyRs were distributed in discrete clusters separated by >1 µm, only isolated sparks were produced in this model and LCR events did not form. However, immunofluorescent staining of SANCs for RyR revealed the presence of bridging RyR groups between large clusters, forming an irregular network. Incorporation of this architecture into the model led to the generation of propagating LCR events. Partial periodicity emerged from the interaction of LCR events, as observed experimentally. This calcium clock becomes entrained with membrane currents to accelerate the beating rate, which therefore was controlled by the activity of the SERCA pump, RyR sensitivity, and L-type current amplitude, all of which are targets of β-adrenergic-mediated phosphorylation. Unexpectedly, simulations revealed the existence of a pathological mode at high RyR sensitivity to calcium, in which the calcium clock loses synchronization with the membrane, resulting in a paradoxical decrease in beating rate in response to β-adrenergic stimulation. The model indicates that the hierarchical clustering of surface RyRs in SANCs may be a crucial adaptive mechanism. Pathological desynchronization of the clocks may explain sinus node dysfunction in heart failure and RyR mutations.