Linking flickering to waves and whole-cell oscillations in a mitochondrial network model

Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise

Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with 'twist' (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators.

Fractal dynamics of individual mitochondrial oscillators measure local inter-mitochondrial coupling

Mitochondrial inner membrane potentials in cardiomyocytes may oscillate in cycles of depolarization/repolarization when the mitochondrial network is exposed to metabolic or oxidative stress. The frequencies of such oscillations are dynamically changing while clusters of weakly coupled mitochondrial oscillators adjust to a common phase and frequency. Across the cardiac myocyte, the averaged signal of the mitochondrial population follows self-similar or fractal dynamics; however, fractal properties of individual mitochondrial oscillators have not yet been examined. We show that the largest synchronously oscillating cluster exhibits a fractal dimension, D, that is indicative of self-similar behavior with D=1.27±0.11, in contrast to the remaining network mitochondria whose fractal dimension is close to that of Brownian noise, D=1.58±0.10. We further demonstrate that fractal behavior is correlated with local coupling mechanisms, whereas it is only weakly linked to measures of functional connections between mitochondria. Our findings suggest that individual mitochondrial fractal dimensions may serve as a simple measure of local mitochondrial coupling.

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.

Mechanisms and mathematical modelling of ROS production by the mitochondrial electron transport chain

Reactive oxygen species (ROS) are recognised both as damaging molecules and intracellular signalling entities. In addition to its role in ATP generation, the mitochondrial electron transport chain (ETC) constitutes a relevant source of mitochondrial ROS, in particular during pathological conditions. Mitochondrial ROS homeostasis depends on species- and site-dependent ROS production, their bioreactivity, diffusion, and scavenging. However, our quantitative understanding of mitochondrial ROS homeostasis has thus far been hampered by technical limitations, including lack of truly site- and/or ROS-specific reporter molecules. In this context, the use of computational models is of great value to complement and interpret empirical data, as well as to predict variables that are difficult to assess experimentally. During the last decades, various mechanistic models of ETC-mediated ROS production have been developed. Although these often-complex models have generated novel insights, their parameterisation, analysis, and integration with other computational models is not straightforward. In contrast, phenomenological (sometimes termed "minimal") models use a relatively small set of equations to describe empirical relationship(s) between ROS-related and other parameters, and generally aim to explore system behaviour and generate hypotheses for experimental validation. In this review, we first discuss ETC-linked ROS homeostasis and introduce various detailed mechanistic models. Next, we present how bioenergetic parameters (e.g. NADH/NAD⁺ ratio, mitochondrial membrane potential) relate to site-specific ROS production within the ETC and how these relationships can be used to design minimal models of ROS homeostasis. Finally, we illustrate how minimal models have been applied to explore pathophysiological aspects of ROS.

Mitochondrial Contributions in the Genesis of Delayed Afterdepolarizations in Ventricular Myocytes

Mitochondria fulfill the cell's energy demand and affect the intracellular calcium (Ca2+) dynamics via direct Ca2+ exchange, the redox effect of reactive oxygen species (ROS) on Ca2+ handling proteins, and other signaling pathways. Recent experimental evidence indicates that mitochondrial depolarization promotes arrhythmogenic delayed afterdepolarizations (DADs) in cardiac myocytes. However, the nonlinear interactions among the Ca2+ signaling pathways, ROS, and oxidized Ca2+/calmodulin-dependent protein kinase II (CaMKII) pathways make it difficult to reveal the mechanisms. Here, we use a recently developed spatiotemporal ventricular myocyte computer model, which consists of a 3-dimensional network of Ca2+ release units (CRUs) intertwined with mitochondria and integrates mitochondrial Ca2+ signaling and other complex signaling pathways, to study the mitochondrial regulation of DADs. With a systematic investigation of the synergistic or competing factors that affect the occurrence of Ca2+ waves and DADs during mitochondrial depolarization, we find that the direct redox effect of ROS on ryanodine receptors (RyRs) plays a critical role in promoting Ca2+ waves and DADs under the acute effect of mitochondrial depolarization. Furthermore, the upregulation of mitochondrial Ca2+ uniporter can promote DADs through Ca2+-dependent opening of mitochondrial permeability transition pores (mPTPs). Also, due to much slower dynamics than Ca2+ cycling and ROS, oxidized CaMKII activation and the cytosolic ATP do not appear to significantly impact the genesis of DADs during the acute phase of mitochondrial depolarization. However, under chronic conditions, ATP depletion suppresses and enhanced CaMKII activation promotes Ca2+ waves and DADs.

Multistability maintains redox homeostasis in human cells

Cells metabolize nutrients through a complex metabolic and signaling network that governs redox homeostasis. At the core of this, redox regulatory network is a mutually inhibitory relationship between reduced glutathione and reactive oxygen species (ROS)-two opposing metabolites that are linked to upstream nutrient metabolic pathways (glucose, cysteine, and glutamine) and downstream feedback loops of signaling pathways (calcium and NADPH oxidase). We developed a nutrient-redox model of human cells to understand system-level properties of this network. Combining in silico modeling and ROS measurements in individual cells, we show that ROS dynamics follow a switch-like, all-or-none response upon glucose deprivation at a threshold that is approximately two orders of magnitude lower than its physiological concentration. We also confirm that this ROS switch can be irreversible and exhibits hysteresis, a hallmark of bistability. Our findings evidence that bistability modulates redox homeostasis in human cells and provide a general framework for quantitative investigations of redox regulation in humans.

MitoWave: Spatiotemporal analysis of mitochondrial membrane potential fluctuations during I/R

Mitochondria exhibit unstable inner membrane potentials (ΔΨ_m) when subjected to stress, such as during ischemia/reperfusion (I/R). Understanding the mechanism of ΔΨ_m instability involves characterizing and quantifying this phenomenon in an unbiased and reproducible manner. Here, we describe a simple analytical workflow called "MitoWave" that combines wavelet transform methods and image segmentation to unravel dynamic ΔΨ_m changes in the cardiac mitochondrial network during I/R. In vitro ischemia was affected by placing a glass coverslip on a monolayer of neonatal mouse ventricular myocytes for 1 h and removing the coverslip to allow for reperfusion, revealing complex oscillatory ΔΨ_m. MitoWave analysis was then used to identify individual mitochondrial clusters within the cells and track their intrinsic oscillation frequencies over the course of reperfusion. Responses segregated into five typical behaviors were quantified by MitoWave that were corroborated by visual inspection of the time series. Statistical analysis of the distribution of oscillating mitochondrial clusters during reperfusion showed significant differences between the five different outcomes. Features such as the time point of ΔΨ_m depolarization during I/R, area of mitochondrial clusters, and time-resolved frequency components during reperfusion were determined per cell and per mitochondrial cluster. Mitochondria from neonatal mouse ventricular myocytes subjected to I/R oscillate in the frequency range of 8.6-45 mHz, with a mean of 8.73 ± 4.35 mHz. Oscillating clusters had smaller areas ranging from 49.8 ± 1.2 μm², whereas nonoscillating clusters had larger areas 66 ± 1.5 μm². A negative correlation between frequency and mitochondrial cluster area was observed. We also observed that late ΔΨ_m loss during ischemia correlated with early ΔΨ_m stabilization after oscillation on reperfusion. Thus, MitoWave analysis provides a semiautomated method to quantify complex time-resolved mitochondrial behavior in an easy-to-follow workflow, enabling unbiased, reproducible quantitation of complex nonstationary cellular phenomena.

Mitochondrial depolarization promotes calcium alternans: Mechanistic insights from a ventricular myocyte model

Mitochondria are vital organelles inside the cell and contribute to intracellular calcium (Ca2+) dynamics directly and indirectly via calcium exchange, ATP generation, and production of reactive oxygen species (ROS). Arrhythmogenic Ca2+ alternans in cardiac myocytes has been observed in experiments under abnormal mitochondrial depolarization. However, complex signaling pathways and Ca2+ cycling between mitochondria and cytosol make it difficult in experiments to reveal the underlying mechanisms of Ca2+ alternans under abnormal mitochondrial depolarization. In this study, we use a newly developed spatiotemporal ventricular myocyte computer model that integrates mitochondrial Ca2+ cycling and complex signaling pathways to investigate the mechanisms of Ca2+ alternans during mitochondrial depolarization. We find that elevation of ROS in response to mitochondrial depolarization plays a critical role in promoting Ca2+ alternans. Further examination reveals that the redox effect of ROS on ryanodine receptors and sarco/endoplasmic reticulum Ca2+-ATPase synergistically promote alternans. Upregulation of mitochondrial Ca2+ uniporter promotes Ca2+ alternans via Ca2+-dependent mitochondrial permeability transition pore opening. Due to their relatively slow kinetics, oxidized Ca2+/calmodulin-dependent protein kinase II activation and ATP do not play significant roles acutely in the genesis of Ca2+ alternans after mitochondrial depolarization, but their roles can be significant in the long term, mainly through their effects on sarco/endoplasmic reticulum Ca2+-ATPase activity. In conclusion, mitochondrial depolarization promotes Ca2+ alternans acutely via the redox effect of ROS and chronically by ATP reduction. It suppresses Ca2+ alternans chronically through oxidized Ca2+/calmodulin-dependent protein kinase II activation.

Hydrogen peroxide diffusion and scavenging shapes mitochondrial network instability and failure by sensitizing ROS-induced ROS release

The mitochondrial network of cardiac cells is finely tuned for ATP delivery to sites of energy demand; however, emergent phenomena, such as mitochondrial transmembrane potential oscillations or propagating waves of depolarization have been observed under metabolic stress. While regenerative signaling by reactive oxygen species (ROS)-induced ROS release (RIRR) has been suggested as a potential trigger, it is unknown how it could lead to widespread responses. Here, we present a novel computational model of RIRR transmission that explains the mechanisms of this phenomenon. The results reveal that superoxide mediates neighbor-neighbor activation of energy-dissipating ion channels, while hydrogen peroxide distributes oxidative stress to sensitize the network to mitochondrial criticality. The findings demonstrate the feasibility of RIRR as a synchronizing factor across the dimensions of the adult heart cell and illustrate how a cascade of failures at the organellar level can scale to impact cell and organ level functions of the heart.

A Spatiotemporal Ventricular Myocyte Model Incorporating Mitochondrial Calcium Cycling

Intracellular calcium (Ca2+) cycling dynamics in cardiac myocytes are spatiotemporally generated by stochastic events arising from a spatially distributed network of coupled Ca2+ release units that interact with an intertwined mitochondrial network. In this study, we developed a spatiotemporal ventricular myocyte model that integrates mitochondria-related Ca2+ cycling components into our previously developed ventricular myocyte model consisting of a three-dimensional Ca2+ release unit network. Mathematical formulations of mitochondrial membrane potential, mitochondrial Ca2+ cycling, mitochondrial permeability transition pore stochastic opening and closing, intracellular reactive oxygen species signaling, and oxidized Ca2+/calmodulin-dependent protein kinase II signaling were incorporated into the model. We then used the model to simulate the effects of mitochondrial depolarization on mitochondrial Ca2+ cycling, Ca2+ spark frequency, and Ca2+ amplitude, which agree well with experimental data. We also simulated the effects of the strength of mitochondrial Ca2+ uniporters and their spatial localization on intracellular Ca2+ cycling properties, which substantially affected diastolic and systolic Ca2+ levels in the mitochondria but exhibited only a small effect on sarcoplasmic reticulum and cytosolic Ca2+ levels under normal conditions. We show that mitochondrial depolarization can cause Ca2+ waves and Ca2+ alternans, which agrees with previous experimental observations. We propose that this new, to our knowledge, spatiotemporal ventricular myocyte model, incorporating properties of mitochondrial Ca2+ cycling and reactive-oxygen-species-dependent signaling, will be useful for investigating the effects of mitochondria on intracellular Ca2+ cycling and action potential dynamics in ventricular myocytes.

Temperature dependence of mitoflash biogenesis in cardiac mitochondria

Mitochondrial flashes (mitoflashes) represent fundamental biochemical and biophysical dynamics of the organelle, involving sudden depolarization of mitochondrial membrane potential (ΔΨ_m), bursting production of reactive oxygen species (ROS), and accelerated extrusion of matrix protons. Here we investigated temperature dependence of mitoflash biogenesis as well as ΔΨ_m oscillations, a subset of which overlapping with mitoflashes, in both cardiac myocytes and isolated respiring cardiac mitochondria. Unexpectedly, we found that mitoflash biogenesis was essentially temperature-independent in intact cardiac myocytes, evidenced by the constancy of frequency as well as amplitude and rise speed over 5 °C-40 °C. Moderate temperature dependence was found in single mitochondria charged by respiratory substrates, where mitoflash frequency was decreased over 5 °C-20 °C with Q₁₀ of 0.74 for Complex I substrates and 0.83 for Complex II substrate. In contrast, ΔΨ_m oscillation frequency displayed a negative temperature dependence at 5 °C-20 °C with Q₁₀ of 0.82 in intact cells, but a positive temperature dependence at 25 °C - 40 °C with Q₁₀ of 1.62 in isolated mitochondria charged with either Complex I or Complex II substrates. Moreover, the recovery speed of individual mitoflashes exhibited mild temperature dependence (Q₁₀ = 1.14-1.22). These results suggest a temperature compensation of mitoflash frequency at both the mitochondrial and extra-organelle levels, and underscore that mitoflashes and ΔΨ_m oscillations are related but distinctly different mitochondrial functional dynamics.

A Robust Model for Circadian Redox Oscillations

The circadian clock is an endogenous oscillator that controls daily rhythms in metabolism, physiology, and behavior. Although the timekeeping components differ among species, a common design principle is a transcription-translation negative feedback loop. However, it is becoming clear that other mechanisms can contribute to the generation of 24 h rhythms. Peroxiredoxins (Prxs) exhibit 24 h rhythms in their redox state in all kingdoms of life. In mammalian adrenal gland, heart and brown adipose tissue, such rhythms are generated as a result of an inactivating hyperoxidation reaction that is reduced by coordinated import of sulfiredoxin (Srx) into the mitochondria. However, a quantitative description of the Prx/Srx oscillating system is still missing. We investigate the basic principles that generate mitochondrial Prx/Srx rhythms using computational modeling. We observe that the previously described delay in mitochondrial Srx import, in combination with an appropriate separation of fast and slow reactions, is sufficient to generate robust self-sustained relaxation-like oscillations. We find that our conceptual model can be regarded as a series of three consecutive phases and two temporal switches, highlighting the importance of delayed negative feedback and switches in the generation of oscillations.

Quantitative systems models illuminate arrhythmia mechanisms in heart failure: Role of the Na+ -Ca2+ -Ca2+ /calmodulin-dependent protein kinase II-reactive oxygen species feedback

Quantitative systems modeling aims to integrate knowledge in different research areas with models describing biological mechanisms and dynamics to gain a better understanding of complex clinical syndromes. Heart failure (HF) is a chronic complex cardiac disease that results from structural or functional disorders impairing the ability of the ventricle to fill with or eject blood. Highly interactive and dynamic changes in mechanical, structural, neurohumoral, metabolic, and electrophysiological properties collectively predispose the failing heart to cardiac arrhythmias, which are responsible for about a half of HF deaths. Multiscale cardiac modeling and simulation integrate structural and functional data from HF experimental models and patients to improve our mechanistic understanding of this complex arrhythmia syndrome. In particular, they allow investigating how disease-induced remodeling alters the coupling of electrophysiology, Ca²⁺ and Na⁺ handling, contraction, and energetics that lead to rhythm derangements. The Ca²⁺ /calmodulin-dependent protein kinase II, which expression and activity are enhanced in HF, emerges as a critical hub that modulates the feedbacks between these various subsystems and promotes arrhythmogenesis. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Mechanistic Models Models of Systems Properties and Processes > Cellular Models Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

ROS-induced ROS release in plant and animal cells

Reactive oxygen species (ROS) play a key signaling role in plant and animal cells. Among the many cellular mechanisms used to generate and transduce ROS signals, ROS-induced ROS release (RIRR) is emerging as an important pathway involved in different human pathologies and plant responses to environmental stress. RIRR is a process in which one cellular compartment or organelle generates or releases ROS, triggering the enhanced production or release of ROS by another compartment or organelle. It was initially described in animal cells and proposed to mediate mitochondria-to-mitochondria communication, but later expanded to include communication between mitochondria and plasma membrane-localized NADPH oxidases. In plants a process of RIRR was demonstrated to mediate long distance rapid systemic signaling in response to biotic and abiotic stress. This process is thought to involve the enhanced production of ROS by one cell that triggers the enhanced production of ROS by a neighboring cell in a process that propagates the enhanced "ROS production state" all the way from one part of the plant to another. In contrast to the intracellular nature of the RIRR process of animal cells, the plant RIRR process is therefore primarily studied at the cell-to-cell communication level. Studies on intracellular (organelle-to-organelle, or organelle-to-NADPH oxidase) RIRR pathways are very scarce in plants, whereas studies on cell-to-cell RIRR are very scarce in animals. Here we will attempt to highlight what is known in both systems and what each system can learn from the other.

Targeting Endoplasmic Reticulum and/or Mitochondrial Ca2+ Fluxes as Therapeutic Strategy for HCV Infection

Chronic hepatitis C is characterized by metabolic disorders and by a microenvironment in the liver dominated by oxidative stress, inflammation and regeneration processes that can in the long term lead to liver cirrhosis and hepatocellular carcinoma. Several lines of evidence suggest that mitochondrial dysfunctions play a central role in these processes. However, how these dysfunctions are induced by the virus and whether they play a role in disease progression and neoplastic transformation remains to be determined. Most in vitro studies performed so far have shown that several of the hepatitis C virus (HCV) proteins also localize to mitochondria, but the consequences of these interactions on mitochondrial functions remain contradictory and need to be confirmed in the context of productively replicating virus and physiologically relevant in vitro and in vivo model systems. In the past decade we have been proposing a temporal sequence of events in the HCV-infected cell whereby the primary alteration is localized at the mitochondria-associated ER membranes and causes release of Ca²⁺ from the ER, followed by uptake into mitochondria. This ensues successive mitochondrial dysfunction leading to the generation of reactive oxygen and nitrogen species and a progressive metabolic adaptive response consisting in decreased oxidative phosphorylation and enhanced aerobic glycolysis and lipogenesis. Here we resume the major results provided by our group in the context of HCV-mediated alterations of the cellular inter-compartmental calcium flux homeostasis and present new evidence suggesting targeting of ER and/or mitochondrial calcium transporters as a novel therapeutic strategy.

Synchronism in mitochondrial ROS flashes, membrane depolarization and calcium sparks in human carcinoma cells

Mitochondria are major producers of reactive oxygen species (ROS) in many cells including cancer cells. However, complex interrelationships between mitochondrial ROS (mitoROS), mitochondrial membrane potential (ΔΨm) and Ca²⁺ are not completely understood. Using human carcinoma cells, we further highlight biphasic ROS dynamics: - gradual mitoROS increase followed by mitoROS flash. Also, we demonstrate heterogeneity in rates of mitoROS generation and flash initiation time. Comparing mitochondrial and near-extra-mitochondrial signals, we show that mechanisms of mitoROS flashes in single mitochondria, linked to mitochondrial permeability transition pore opening (ΔΨm collapse) and calcium sparks, may involve flash triggering by certain levels of external ROS released from the same mitochondria. In addition, mitochondria-mitochondria interactions can produce wave propagations of mitoROS flashes and ΔΨm collapses in cancer cells similar to phenomena of ROS-induced ROS release (RIRR). Our data suggest that in cancer cells RIRR, activation of mitoROS flashes and mitochondrial depolarization may involve participation of extramitochondrial-ROS produced either by individual mitochondria and/or by neighboring mitochondria. This could represent general mechanisms in ROS-ROS signaling with suggested role in both mitochondrial and cellular physiology and signaling.

Functional Implications of Cardiac Mitochondria Clustering

The spatio-temporal organization of mitochondria in cardiac myocytes facilitates myocyte-wide, cluster-bound, mitochondrial inner membrane potential oscillatory depolarizations, commonly triggered by metabolic or oxidative stressors. Local intermitochondrial coupling can be mediated by reactive oxygen species (ROS) that activate inner membrane pores to initiate a ROS-induced-ROS-release process that produces synchronized limit cycle oscillations of mitochondrial clusters within the whole mitochondrial network. The network's dynamic organization, structure and function can be assessed by quantifying dynamic local coupling constants and dynamic functional clustering coefficients, both providing information about the network's response to external stimuli. In addition to its special organization, the mitochondrial network of cardiac myocytes exhibits substrate-sensitive coupling constants and clustering coefficients. The myocyte's ability to form functional clusters of synchronously oscillating mitochondria is sensitive to conditions such as substrate availability (e.g., glucose, pyruvate, β-hydroxybutyrate), antioxidant status, respiratory chain activity, or history of oxidative challenge (e.g., ischemia-reperfusion). This underscores the relevance of quantitative methods to characterize the network's functional status as a way to assess the myocyte's resilience to pathological stressors.

Mitochondrial networks in cardiac myocytes reveal dynamic coupling behavior

Oscillatory behavior of mitochondrial inner membrane potential (ΔΨm) is commonly observed in cells subjected to oxidative or metabolic stress. In cardiac myocytes, the activation of inner membrane pores by reactive oxygen species (ROS) is a major factor mediating intermitochondrial coupling, and ROS-induced ROS release has been shown to underlie propagated waves of ΔΨm depolarization as well as synchronized limit cycle oscillations of ΔΨm in the network. The functional impact of ΔΨm instability on cardiac electrophysiology, Ca(2+) handling, and even cell survival, is strongly affected by the extent of such intermitochondrial coupling. Here, we employ a recently developed wavelet-based analytical approach to examine how different substrates affect mitochondrial coupling in cardiac cells, and we also determine the oscillatory coupling properties of mitochondria in ventricular cells in intact perfused hearts. The results show that the frequency of ΔΨm oscillations varies inversely with the size of the oscillating mitochondrial cluster, and depends on the strength of local intermitochondrial coupling. Time-varying coupling constants could be quantitatively determined by applying a stochastic phase model based on extension of the well-known Kuramoto model for networks of coupled oscillators. Cluster size-frequency relationships varied with different substrates, as did mitochondrial coupling constants, which were significantly larger for glucose (7.78 × 10(-2) ± 0.98 × 10(-2) s(-1)) and pyruvate (7.49 × 10(-2) ± 1.65 × 10(-2) s(-1)) than lactate (4.83 × 10(-2) ± 1.25 × 10(-2) s(-1)) or β-hydroxybutyrate (4.11 × 10(-2) ± 0.62 × 10(-2) s(-1)). The findings indicate that mitochondrial spatiotemporal coupling and oscillatory behavior is influenced by substrate selection, perhaps through differing effects on ROS/redox balance. In particular, glucose-perfusion generates strong intermitochondrial coupling and temporal oscillatory stability. Pathological changes in specific catabolic pathways, which are known to occur during the progression of cardiovascular disease, could therefore contribute to altered sensitivity of the mitochondrial network to oxidative stress and emergent ΔΨm instability, ultimately scaling to produce organ level dysfunction.

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.

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.

Cardiac mitochondria exhibit dynamic functional clustering

Multi-oscillatory behavior of mitochondrial inner membrane potential ΔΨ_m in self-organized cardiac mitochondrial networks can be triggered by metabolic or oxidative stress. Spatio-temporal analyses of cardiac mitochondrial networks have shown that mitochondria are heterogeneously organized in synchronously oscillating clusters in which the mean cluster frequency and size are inversely correlated, thus suggesting a modulation of cluster frequency through local inter-mitochondrial coupling. In this study, we propose a method to examine the mitochondrial network's topology through quantification of its dynamic local clustering coefficients. Individual mitochondrial ΔΨ_m oscillation signals were identified for each cardiac myocyte and cross-correlated with all network mitochondria using previously described methods (Kurz et al., 2010a). Time-varying inter-mitochondrial connectivity, defined for mitochondria in the whole network whose signals are at least 90% correlated at any given time point, allowed considering functional local clustering coefficients. It is shown that mitochondrial clustering in isolated cardiac myocytes changes dynamically and is significantly higher than for random mitochondrial networks that are constructed using the Erdös–Rényi model based on the same sets of vertices. The network's time-averaged clustering coefficient for cardiac myocytes was found to be 0.500 ± 0.051 (N = 9) vs. 0.061 ± 0.020 for random networks, respectively. Our results demonstrate that cardiac mitochondria constitute a network with dynamically connected constituents whose topological organization is prone to clustering. Cluster partitioning in networks of coupled oscillators has been observed in scale-free and chaotic systems and is therefore in good agreement with previous models of cardiac mitochondrial networks.

Mitochondrial network energetics in the heart

At the core of eukaryotic aerobic life, mitochondrial function like 'hubs' in the web of energetic and redox processes in cells. In the heart, these networks-extending beyond the complex connectivity of biochemical circuit diagrams and apparent morphology-exhibit collective dynamics spanning several spatiotemporal levels of organization, from the cell, to the tissue, and the organ. The network function of mitochondria, i.e., mitochondrial network energetics, represents an advantageous behavior. Its coordinated action, under normal physiology, provides robustness despite failure in a few nodes, and improves energy supply toward a swiftly changing demand. Extensive diffuse loops, encompassing mitochondrial-cytoplasmic reaction/transport networks, control and regulate energy supply and demand in the heart. Under severe energy crises, the network behavior of mitochondria and associated glycolytic and other metabolic networks collapse, thereby triggering fatal arrhythmias.

Criticality in intracellular calcium signaling in cardiac myocytes

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

Cardiac mitochondrial network excitability: insights from computational analysis

In the heart, mitochondria form a regular lattice and function as a coordinated, nonlinear network to continuously produce ATP to meet the high-energy demand of the cardiomyocytes. Cardiac mitochondria also exhibit properties of an excitable system: electrical or chemical signals can spread within or among cells in the syncytium. The detailed mechanisms by which signals pass among individual elements (mitochondria) across the network are still not completely understood, although emerging studies suggest that network excitability might be mediated by the local diffusion and autocatalytic release of messenger molecules such as reactive oxygen species and/or Ca(2+). In this short review, we have attempted to described recent advances in the field of cardiac mitochondrial network excitability. Specifically, we have focused on how mitochondria communicate with each other through the diffusion and regeneration of messenger molecules to initiate and propagate waves or oscillations, as revealed by computational models of mitochondrial network.