Functional Implications of Cardiac Mitochondria Clustering

Mitochondria in cardiac myocytes are densely packed to form a cell-wide network of communicating organelles that accounts for about 35% of the myocyte’s volume [1]. The lattice-like arrangement of mitochondria, mostly in long and dense rows parallel to the cardiac myocyte myofilaments, has the structural features of a highly ordered network [2]. This specific mitochondrial network architecture ensures that the large demand of adenosine triphosphate (ATP) of cardiac myocytes is met and appropriately distributed. Mitochondria synthesize approximately 30 kg of ATP each day to both provide energy for the basic cellular metabolism and to secure basic physiological functions of the cardiovascular system such as the maintenance of pulmonary and systemic blood pressure during heart contractions [3]. Their intracellular position is closely associated with the sarcoplasmic reticulum and the myofilaments to facilitate cellular distribution of ATP [4]. However, the role of mitochondrial structure and function is not only limited to ATP generation; mitochondria participate in and control numerous metabolic pathways and signaling cascades such as the calcium signaling, redox oxidation, β-oxidation of fatty acids, oxidative phosphorylation, the synthesis of aminoacids, heme and steroids, and cellular apoptosis [5]. Their structural and functional diversity thus surpasses any other cellular organelle.

Specifically, the control of mitochondrial oxidative phosphorylation is dependent on the arrangement, interplay and interdependence of mitochondrial ion channels, transporters, and pumps that are located in an inner membrane that is selectively permeable to enable the adjustment of mitochondrial volume, inner membrane potential ΔΨ_m and mitochondrial redox potential [6]. Thus, the mitochondrial membranes facilitate and balance cellular energy supply and demand.

The Mitochondrial Network in Cardiac Myocytes

Mitochondrial Inner Membrane Oscillations

When subjected to critical amounts of oxidative stress or substrate deprivation, the mitochondrial network in cardiac myocytes may pathophysiologically transition into a state where the inner membrane potential ΔΨ_m of a large amount of mitochondria depolarizes and oscillates [6–9]. Oscillations of individual mitochondria in intact myocytes were first documented in the early 1980s when Berns et al. excited quiescent cardiac myocytes with a focal laser beam to induce transient cycles of alternating ΔΨ_m depolarization and repolarization [10]. Mitochondrial inner membrane depolarizations were also detected in smooth muscle cells [11], cultured neurons [12], and in isolated mitochondria [13]. A functional link between metabolic oscillations and cardiac function was demonstrated when O’Rourke et al. [14] showed that cyclical activation of ATP-sensitive potassium currents in guinea pig cardiac myocytes under substrate deprivation was associated with low frequency oscillations in the myocyte’s action potential duration and excitation-contraction coupling. These oscillations were accompanied by a synchronous oxidation and reduction of the intra-cellular nicotinamide adenine dinucleotide (NADH) concentration.

While it was initially believed that this mechanism was due to changes in glucose metabolism, subsequent studies detected an association of the oscillations with redox transients of mitochondrial flavoprotein and waves of mitochondrial ΔΨ_m depolarization [15], therefore implying a mitochondrial origin of the phenomenon. Network-wide mitochondrial membrane potential oscillations could be initiated reliably by the application of a localized laser-flash that induced a depolarization of ΔΨ_m in just a few mitochondria [16]. The association of ΔΨ_m oscillations with reactive oxygen species (ROS) was discovered by Zorov and Sollot who showed that a localized laser-flash produced a high number of free radicals in cardiac mitochondria in an autocatalytic manner, concomitant with depolarization, referred to as ROS-induced ROS release [17, 18]. Experimentally, synchronous ΔΨ_m oscillations were observed in small contiguous mitochondrial clusters as well as in large clusters that span the whole myocyte [16], or as mitochondrial ΔΨ_m depolarization waves that could pass between neighboring cardiac myocytes through intercalated discs [15]. Mitochondrial communication may also be facilitated through the formation of intermitochondrial junctions [19], which could facilitate wave propagation.

Generally, mitochondrial network coordination is heavily influenced by the intricate balance of ROS generation and ROS scavenging within the myocyte: when a critical amount of network mitochondria exhibits a destabilized inner membrane potential, the increased presence of ROS can lead the network into a state of ROS-induced ROS release [16, 20]. It is likely that the antioxidant flux is lowered to a point at which ROS production cannot be scavenged fast enough – this causes a flux mismatch that rapidly goes out of control such that multiple mitochondria form a myocyte-wide cluster of synchronously oscillating mitochondria, also called spanning cluster; these mitochondria lock into an oscillatory mode with low frequencies and high amplitudes [6, 8, 20].

The cardiac mitochondrial network can also behave in an oscillatory mode in the absence of metabolic or oxidative stress. A computational model of the mitochondrial oscillator has predicted mitochondrial behavior that was characterized by low-amplitude, high-frequency oscillations [21], which was later demonstrated experimentally [8]. Since, the collective mitochondrial behavior happened (i) in the absence of metabolic/oxidative stress, (ii) did not involve collapse or large excursions of the membrane potential (i.e., microvolts to a few millivolts), and (iii) could be clearly distinguished from the high-amplitude, low-frequency oscillations involving strongly coupled mitochondria through elevated ROS levels, this regimen was dubbed “physiological” [6, 8]. The transition between normal and pathological behavior in mitochondrial function can be determined by the abundance and interplay between cytoplasmic (Cu, Zn SOD) and matrix (Mn SOD) superoxide dismutases with ROS generation in the respiratory chain, as recently demonstrated [22].

In intact cardiac myocytes, ROS generation from mitochondrial respiration mainly originates in complex III and complex I, and is favored by highly reduced redox potentials and high ΔΨ_m, typically corresponding to either inhibition or low respiratory flux [6, 16]. The occurrence of ΔΨ_m limit-cycle oscillations could be suppressed by adding inhibitors of mitochondrial respiration or ligands of the mitochondrial benzodiazepine receptor, but not adding inhibitors of the mitochondrial permeability transition pore (mPTP) [23]. This class of ΔΨ_m oscillations was consistent with a ROS-induced ROS release mechanism involving the activation of mitochondrial inner membrane anion channels (IMAC), see also below and [6, 16, 24].

Mitochondrial Morphology, Dynamics and Morphodynamics

Changes in the morphology of mitochondria or their dynamic characteristics can influence their collective behavior. In cardiac myocytes, mitochondria can be peri-nuclear, intermyofibrilar and subsarcolemmal, and possess different morphologies as well as proteomic and physiological status [25–29]. This (static) morphological heterogeneity is associated with different mitochondrial functions [30, 31]; for instance, mitochondria in intermyofibrillar locations show a reduced content of flavoproteins compared to those beneath the sarcolemma, thus resulting in different oxidative capacities for NADH- and FADH-generating substrates [30, 32].

The role of dynamic heterogeneity becomes important for pertubations of the mitochondrial network during oxidative or metabolic stress: the mitochondrial network’s response to stress can affect not only intracellularly clustered mitochondria, but scale to the majority of cellular mitochondria and even to the whole-organ level, thus leading to emergent phenomena, such as ventricular arrhythmias [20, 33–35]. In many non-cardiac myocytes, mitochondrial mobility and morphological plasticity or morphodynamics is affected by proteins that facilitate fusion and fission processes, by cytoskeletal proteins (through disruptive structural changes of the cytoskeleton) [36], and changes in myocyte energy metabolism or calcium homeostasis [37, 38]. Recently, it was shown that redox signaling also occurs in clusters of morphologically changing axonal mitochondria, suggesting a morphological and functional inter-mitochondrial coupling in neurons [39, 40]. However, apart from “pre-fusion” events at inter-mitochondrial junctions [19], fusion and fission events as well as mitochondrial mobility have not been observed in adult cardiac myocytes so far, although mitochondrial fusion has been observed in skeletal muscle [41]. However, there is growing evidence from mitochondrial protein gene studies that mitochondrial morphodynamics may play a role in the maintenance of mitochondrial fitness [42].

Spatio-Temporal Behavior of Individual Mitochondrial Oscillators

While many studies have investigated the response of the whole mitochondrial network in external stress stimuli (i.e. ischemia-reperfusion injury or oxidative stress) at the inner mitochondrial membrane [8, 16], more recent ones have focused on the role of individual mitochondria and their spatio-temporal behavior [9, 35, 40, 43, 44].

Alterations of the mitochondrial membrane potential ΔΨ_m can be monitored with the positively charged lipophilic fluorescent dyes tetramethylrhodamine methyl or ethyl ester (TMRM or TMRE), that redistribute from the matrix to the cytoplasm with a depolarization of ΔΨ_m [45] (see Fig. 1.1). Myocyte-wide ΔΨ_m oscillations can then be triggered by a perturbation of the mitochondrial network through thiol-oxidizing substances such as diamide, or by a highly localized laser flash [16, 46]: the ensuing sequence of locally confined ROS generation and induced autocatalytic ROS release [16, 17, 47] can lead to a critical accumulation of ROS that initiates a propagating wave of ΔΨ_m depolarization [7]. However, the reaction of each individual mitochondrion within the myocyte is highly variable: while a majority of mitochondria synchronize to cell-wide ΔΨ_m oscillations, some mitochondria do not participate whereas others participate only temporarily (see Fig. 1.1b–d) [9, 43]. This diversified mitochondrial behavior (in frequency and amplitude) is directly associated with the scavenging capacity of intracellular ROS on a local level within the myocyte, as well as the inner membrane efflux of superoxide ions.

Fig. 1.1.

Dynamic mitochondrial oscillations and wavelet transform analysis. (a) Densely packed mitochondria in an adult cardiac myocyte, monitored with the fluorescence dye TMRE. (b–d) Perturbations of the mitochondrial network can result in oscillatory patterns of individual mitochondria that differ in occurrence and frequency content. (e) Dynamic frequency content of oscillating mitochondria can be analyzed with the absolute wavelet transform: the major frequency component of mitochondrion 3 (d) varies between 15 and 20 mHz (Reprinted with permission from Ref. [81]. Copyright 2014)

A large cluster of mitochondria with synchronized ΔΨ_m oscillations can recruit additional mitochondria in analogy to a network of weakly coupled oscillators (see below), whereas some mitochondria may exit the cluster, i.e. show ΔΨ_m oscillations that are not synchronized with the cluster, because their local ROS defense mechanisms are exhausted [9, 43]. Thus, this complex interplay of ROS generation and scavenging determines the individual mitochondrial frequency and amplitude.

Since the frequencies of these oscillations change in time, they are best described by methodologies that do not assume an a priori stationarity of ΔΨ_m signals, such as wavelet transform analysis. Unlike standard methods of statistical analysis, that can only be applied if the nonstationarity of the signals is associated with the low-frequency part of the observed power spectrum, wavelet analysis allows analyzing the full range of frequencies of the nonstationary mitochondrial network. The wavelet analysis provides a time-resolved picture of the frequency content of each mitochondrial signal, see Fig. 1.1e. Earlier methodologies, e.g. relative dispersion analysis or power spectral analysis, although they provided a glimpse of the functional mitochondrial network properties such as the long-term memory of mitochondrial oscillatory dynamics [7], probed the average behavior of the whole myocyte (single signal trace) [8], without discriminating individual mitochondria. Therefore, application of wavelet analysis on individual mitochondria to characterize the dynamic behavior of the mitochondrial network across the myocyte, was a significant step forward in the systematic analysis of the whole population of mitochondrial oscillators, accounting for both their frequency dynamics and spatio-temporal correlations [9, 43].

Mitochondrial Clusters with Similar Oscillatory Dynamics

It has been shown that mitochondria whose signals are highly correlated during a specific time range, i.e. mitochondria with a similar frequency content, could be grouped into different clusters [9]. Then, the largest cluster of similarly oscillating mitochondria forms a highly correlated and coherent ΔΨ_m signal that drives the cell-wide oscillations which are observed during the recording. These major clusters do not necessarily contain mitochondria that demonstrate strong spatial contiguity (albeit the data is showing only a cross section of the network, therefore synchronized clusters that are “non-contiguous” could be connected as a 3D structure through other planes), nor do they have to coincide with local ROS levels as in the case of a “spanning cluster” as proposed in [8]. In fact, the spatial contiguity of major clusters is substrate dependent (such as glucose, pyruvate, lactate and β-hydroxybutyrate) through their influence on the cellular redox potential [35].

The size of mitochondrial clusters is related to the cluster frequency and phase: mitochondria within the cluster adjust their frequencies and phases to a common one, while large clusters comprising many (coupled) mitochondria need a longer time to synchronize [48, 49]. Similarly, clusters with large separation distances among their components or a low contiguity, take longer to synchronize compared to clusters with smaller separation distances among their components. In both cases, the common cluster frequency remains low and is characterized by an inverse relation between common cluster frequency and cluster size [9, 35].

For cardiac myocytes superfused with glucose, pyruvate, lactate or β-hydroxybutyrate, local laser-flashing still leads to synchronized cell-wide oscillations of mitochondrial clusters. However, the network’s adaptation to the specific metabolic conditions leads to differences in mitochondrial cluster synchronization and dynamics. These differences can be attributed to the altered cardiac myocyte catabolism and ROS/redox balance [35]. For substrates such as pyruvate and β-hydroxybutyrate that generally lead to strongly reduced redox potential (in comparison with glucose or lactate), the rate of change of cluster size versus averaged cluster frequency is higher than for perfusion with glucose or lactate, see Fig. 4 in [35]; yet, typical cluster frequencies for β-hydroxybutyrate were found in a much smaller range (5–20 mHz) than those for pyruvate (10–70 mHz) [35]. A large range of mitochondrial frequencies is a strong indicator of desynchronization processes that are based on dynamic heterogeneity. Likewise, an increased dispersion in cluster size, as observed for perfusion with pyruvate and β-hydroxybutyrate, is a signature of an increasingly fragmented mitochondrial network with several smaller clusters of similar size that have different frequencies. In contrast, a low mitochondrial frequency dispersion is an indicator of a rapid increase in the formation of mitochondrial network clusters during ROS-induced ROS release, such that an increasingly rapid recruitment of mitochondrial oscillators would result in a decrease of clusters with different frequencies. Since in a pathophysiological state, a slow growth in cluster size after the rapid onset of a synchronized cell-wide cluster oscillation is a sign of exhausted ROS scavenging capacity (reflected by a decrease of ΔΨ_m amplitude), quantification of cluster synchronization and contiguity properties provide important information about the myocyte’s propensity to death [9]. These findings help provide a better understanding of the altered cellular metabolism observed in diabetic cardiomyopathy or heart failure, where the myocyte’s response to the switch in substrate utilization, e.g. through an increased utilization of glucose, can be protective or maladaptive [49–51].

Apart from isolated cardiac myocytes, such a relation could also be observed in mitochondria in intact hearts preparations, where neighboring cardiac myocytes are linked electrically and chemically via gap junctions [35, 52]. In those experiments, mitochondrial populations of several myocytes show cell-wide ΔΨ_m signal oscillations following a laser flash, similar to those observed in isolated or pairs of cardiac myocytes [15, 52]. Such behavior could also be observed in rat salivary glands in vivo [53]. Generally, mitochondrial oscillations in intact tissue are faster than in isolated myocytes with an approximate frequency range of 10–80 mHz in cardiac tissue and 50–200 mHz in salivary glands [35, 53], and their frequency is not as strongly influenced by the size of the cluster as in isolated myocytes [35].

The Mitochondrial Network as a Network of Coupled Oscillators

The intricate spatio-temporal dynamics of mitochondrial networks results from many forms of inter-mitochondrial interactions that involve several metabolic pathways. Such pathways include the modification of gene expression, electrochemical interactions that are mediated by ROS through ROS-induced ROS release, other (e.g. exogenous) forms of control exerted on the local redox environment [54], and morphological changes in neighboring mitochondria [19]. In general terms, signaling or communication between (intra-cellular) organelles modulates the relation between cellular networks that involve the transformation (generation) of mass (energy), as represented by the metabolome, and those networks that carry information as represented by the proteome, genome, and transcriptome. A unique pheno-typical signature of a myocyte is then given by the fluxome which represents the network of all cellular reaction-fluxes: it interacts with all other networks and provides a description of the dynamic content of all cellular processes [55]. Emergent properties of the cardiac mitochondrial network, i.e. properties that cannot be derived from the simple sum of behavior of the network’s individual components represent a distinguishing trait of biological complexity and provide the opportunity to study the dynamical coordination of functional intra-cellular relations.

Emergent Properties of the Mitochondrial Network: Mitochondrial Criticality

Biological networks exhibit mostly scale-free dynamics and topologies: the structure of the network is represented by several network hubs, i.e. nodes with a large number of links to other nodes, and a large number of nodes with low connectivity [56]. Analysis of emergent properties of these networks, e.g. large-scale responses to external or internal stimuli (perturbations), synchronization, general collective behavior, vulnerability to damage and ability to self-repair, has been the subject of intensive research endeavors [57–59]. In those studies, the scale-free character of the networks is revealed by the presence of network constituents with different frequencies on highly dependent temporal scales [60]. However, as pointed out by Rosenfeld et al. [61], the network’s robustness to perturbations and modularity cannot be determined using a reductionist approach from knowledge of all elementary biochemical reactions at play, but instead requires network conceptualizations known from systems biology.

In the cardiac network of mitochondrial oscillators, the synchronized, myocyte-wide, low-frequency, high-amplitude depolarizations of ΔΨ_m represent an emerging self-organized spatio-temporal process in response to a perturbation of individual mitochondrial oscillators that are weakly or strongly coupled in the physiological or pathophysiological state, respectively [8, 62–64]. The oscillation period of an individual mitochondrial oscillator is therefore not static but can be modulated by long-term temporal correlations within the mitochondrial network [8, 21]. Such synchronization processes in the form of biochemical rhythms, as observed in many systems of chemically, physically or chemico-physically coupled oscillators, serve the purpose of generating and sustaining spatio-temporal organization through numerous regulatory mechanisms: the resulting dynamically changing oscillations are used by cardiac myocytes to achieve and control internal organization [62–65]. During oxidative or metabolic stress, the mitochondrial network’s response can affect not only intracellularly clustered mitochondria, but scale-up to affect the majority of cellular mitochondria and even the whole-organ, thus leading to myocardial malfunction, and ultimately to ventricular arrhythmias [24, 33, 34]. For example, during reperfusion following ischemic injury, cardiac myocytes become oxidatively stressed due to substrate deprivation (e.g., oxygen, glucose), leading to mitochondrial membrane potential collapse at the onset of oscillations [16, 46], transient inexcitability [16] and, eventually, ventricular fibrillation [24, 34, 66]. Experimental measures to prevent these effects include the application of the mBzR antagonist 4′-chlorodiazepam that modulates IMAC activity [34]. The prominent role of IMAC opening in ROS-induced ROS release and ΔΨ_m depolarization was further supported by blocking mPTP via cyclosporin A, which had no effect on ensuing post-ischemic arrhythmias [24, 34] (see also below and Fig. 1.2).

Fig. 1.2.

Schematic description of mitochondrial ROS-induced ROS release. The mitochondrial respiration complexes I and III produce superoxide anions (O₂⁻) and release them in the mitochondrial matrix. The anions can then reach the intermembrane space (IS) through IMAC, although small quantities directly exit complex III (dashed arrow that originates from complex III). Superoxide dismutases (SOD) in matrix, IS and cytoplasm convert superoxide anions into hydrogen peroxide that can diffuse freely between all compartments. Its degradation is effected by peroxidases (DEG). One positive feedback loop (PF1) is initiated by the activation of IMAC through superoxide anions in the IS to eventually depolarize the mitochondrial inner membrane potential [16, 21]. The resulting stimulation of SODs can, if passing a critical threshold, deactivate IMAC to allow electron transport and repolarization of the membrane potential. Byproducts of hydrogen peroxide-initiated redox processes can trigger mPTP opening (e.g. oxidized lipids or hydroxyl radicals), thus leading to (temporarily) increased production of superoxide anions in the respiratory chain to further open mPTPs in a positive feedback loop (PF2). (This diagram has been adapted from its original version in Yang et al. [71]).

As was previously demonstrated [9, 20], synchronization of mitochondrial oscillations can be achieved by a critical amount of spontaneously synchronized oscillators, a concept termed ‘mitochondrial criticality’ [7, 20]: a number of synchronously oscillating mitochondria recruits more mitochondria to eventually lead the majority of network mitochondria into myocyte-wide synchronized ΔΨ_m oscillations. A transition of this critical threshold into a state of collective, synchronized mitochondrial behavior is equivalent to that of a phase transition between states of matter in thermodynamics. However, in the mitochondrial network it describes a transition from a physiological state into a pathophysiological state. Such transitions among large populations of locally connected mitochondria can be described with the percolation theory which takes into consideration the long-range connectivity of (random) systems and employs a probabilistic model to assess the collective dynamics of the inter-connected network of mitochondria [67]. For ideal lattice-like networks, which is a convenient description of the dense, quasi-uniform and spatially arranged cardiac mitochondria, the theoretical percolation threshold can be identified as pc = 0.59; this threshold is close to experimentally obtained values of mitochondrial criticality where a transition into myocyte-wide ΔΨ_m oscillations was observed in about 60% of all mitochondria within the mitochondrial network with significantly increased ROS levels [20]. For mitochondria in close vicinity to the percolation threshold, small perturbations can thus provoke a transition into global limit cycle oscillations.

Stochasticity and Mitochondrial Network Modeling

Computational models may help obtain a deeper understanding of the intricate interdependencies of mitochondrial networks that lead to emergent behavior. Today, there are several computational models that describe mitochondrial network dynamics, see e.g. [68] for a detailed review.

A mathematical model that has employed ordinary differential equations to represent matrix concentration time rates of ADP, NADH, calcium, Krebs cycle intermediates and ΔΨ_m has been recently reported by Cortassa et al. [69]. In this model, the role of IMAC opening in ROS-induced ROS release and ROS diffusion dynamics were incorporated to correctly reproduce mitochondrial respiration, calcium dynamics, bioenergetics, and the generation of ΔΨ_m limit cycle oscillations [21]. The model provided evidence for the existence of synchronized clusters of mitochondria that entrain more mitochondria to participate in synchronous ΔΨ_m oscillations [47]. Other models have been developed based on the hypothesis of mPTP-mediated ROS-induced ROS release [18, 70] or a combination of mPTP- and IMAC-mediated ROS-induced ROS release [71] (see also Fig. 1.2 for physiological details). Another probabilistic, agent-based model has considered the switch of superoxide anions to hydrogen peroxide, to model ROS signaling for increasingly distant neighboring mitochondria, thus facilitating the inhibition of ROS-induced ROS release through gluthathione peroxidase 1 in the cytosol [72].

Dynamic tubular reticulum models of mitochondrial networks that involve the modeling of molecular fission and fusion events have been recently proposed [73, 74]. Some use graph-based models of the mitochondrial network with a set of simple rules of network connectivity, see e.g. [75], which predict topological properties of the mitochondrial network. However, this approach does not take into account spatial information about the proximity and positioning of neighboring mitochondria.

Overall, the dynamic response of mitochondrial networks to stimuli is due to the stochastic nature of the underlying processes. These processes can be modeled by the addition of intrinsic noise (that emerges from the stochastic processes at small timescales), into system-defining differential equations [61, 76, 77]. However, although in low-dimensional non-linear systems the introduction of random noise may lead to deterministic chaos [78], in high-dimensional non-linear systems with strong links among their components stochasticity may exhibit synchronized behavior [79], as is the case for cardiac mitochondria.

In this case, criteria for stochasticity of high-dimensional non-linear systems are easily fulfilled for strong links between components [79]. This is especially the case for mitochondria in cardiac myocytes that are morphologically and functionally coupled to reveal complex spatio-temporal dynamics [19, 80, 81], see also below.

Mitochondrial Coupling in a Stochastic Phase Model

Synchronization phenomena in networks of coupled nonlinear oscillators were first studied mathematically a few decades ago by Winfree and Kuramoto [82, 83]. The Kuramoto model assumes a (large) group of identical or nearly identical limit-cycle oscillators that are weakly coupled and whose oscillations are driven by intrinsic natural frequencies which are normally distributed around a mean frequency [84]. It provides an exact solution for the critical strength of coupling that allows a transition into a fully synchronized state [85]. The simplicity of the model and its analytical power have made it a frequently applied model in the context of circadian biology, but also in many other systems that contain biologically, physically or chemically coupled oscillators (reviewed in detail in [84, 85]).

The coupling constant in Kuramoto models is time-independent and identical for every pair of coupled oscillators [84]. In cardiac myocytes, however, the dynamic behavior of mitochondrial oscillating frequencies, as well as the presence of structurally varying areas of mitochondrial density or arrangement within the myocyte suggest a position-dependent and time-varying mitochondrial coupling.

We have recently studied the cardiac inter-mitochondrial coupling by adopting an extended Kuramoto-type model, to determine the time-dependent coupling constants of individual mitochondrial oscillators [35]. In this model, we combined wavelet analysis results with a set of stochastic differential equations that describe drifting frequencies with the intention to develop a minimum-order model that uses only two parameters to characterize inter-mitochondrial coupling: the intrinsic oscillatory frequency and the mitochondrial coupling constant [35]. By mapping and identifying each mitochondrion within the mitochondrial network, the exact nearest neighbors of each mitochondrion were used to describe the spatial variance of mitochondrial coupling, i.e. inter-mitochondrial coupling constants were considered for the local nearest-neighbor environment. The concept of local coupling acknowledges inter-mitochondrial coupling through physical connections, i.e. coupling through diffusion of chemical coupling agents such as ROS and/or forms of inter-mitochondrial communication at inter-mitochondrial junctions [19]. It also emphasizes the role of local ROS scavenging mechanisms that prevent long-range diffusion. Using this approach, we were able to determine the averaged dynamic coupling constant (of local mean field type) of each mitochondrion to all of its nearest neighbors, see Fig. 1.3. Quantitative understanding of local myocardial coupling has important implications in the study of the myocyte’s response to metabolic and oxidative challenges during its transition to a pathophysiological state.

Fig. 1.3.

Dynamic inter-mitochondrial coupling and relation to frequency cluster coherence. (a) Dynamically changing inter-mitochondrial coupling can be described with a stochastic phase model that provides local coupling constants K of mean-field form (green cloud) for each mitochondrion m to its nearest neighbors n. Inter-mitochondrial coupling is mediated by diffusive coupling agents that interact with the inner mitochondrial membrane, see also Fig. 1.2. (b) Maps of coupling constants at different time-points (from top to bottom with time intervals of 17.5 s): strong coupling is indicated in yellow and red colors, weak coupling in green and blue colors. Clustered nests of highly coupled mitochondria indicate synchronously oscillating mitochondria, whereas regions with low coupling may be associated with a local exhaustion of antioxidant defense systems. (c) Coupling constant K versus coherence for mitochondria in a large frequency cluster (details see main text) for cell-superperfusion with different substrates. Generally, cluster coupling increases with coherence, whereby the rate of increase is associated with the size of the cluster (see also Fig. 4 in [35]). Lactate-perfusion is associated with small cluster areas such that cluster coupling is not increasing with cluster coherence. Linear fit curves are shown in their respective substrate color (Reprinted with permission from Ref. [35]. Copyright 2015)

Coupling Dynamics in Mitochondrial Frequency Clusters

Inter-mitochondrial coupling within the cardiac myocyte can be studied by comparing the spatio-temporal distribution of inter-mitochondrial coupling constants of mitochondria that belong to the (largest) mitochondrial cluster with similar frequencies, against those of the remaining (“non-cluster”) mitochondria. Cellular conditions that promote low coupling constant values for the cluster mitochondria versus the remaining oscillating mitochondria (that may also be organized in clusters), indicate similar structural organization principles [35].

By relating the inter-mitochondrial coupling within a cluster with the coherence (a measure of the similarity of the mitochondrial ΔΨm oscillations) of the cluster (Fig. 1.3d), we have shown that larger clusters show stronger cluster coupling constants rendering more coherent cluster oscillations. However, for small cluster areas that occur when a myocyte is superfused with lactate (see also Fig. 4 in [35]), an increase in cluster coherence is not accompanied by an increase in cluster coupling. Such behavior is indicative of a greater influence of spatial rather than temporal coupling on the network of mitochondrial oscillators. During superfusion with pyruvate an early decrease of coupling strength (Fig. 1.3d) can be attributed to a strong drift towards the common cluster frequency of those mitochondria with the initially largest difference from the common cluster frequency. When cluster coherence levels at this point are low, which is likely is for small clusters (as experimentally verified for pyruvate perfusion (Fig. S5 in [35]), mitochondria might rapidly drop in and out of the cluster. In addition, the distribution of mitochondrial oscillator frequencies is narrowed during their rapid recruitment into the major frequency cluster, leaving fewer clusters with different frequencies. In contrast, high frequency distributions and high rates of change of cluster size compared to the common cluster frequency are a sign of desynchronization through dynamic heterogeneity.

In this system, the observed strong drift of oscillatory frequencies relates to the fundamental concept of synchronization within the mitochondrial oscillator network: its mechanism can be explained by the dynamics of local ROS generation and scavenging mechanisms that involve the mitochondrial matrix as well as extramitochondrial compartments [86]. Then, the slow drift of each mitochondrial oscillator towards the actual mitochondrial frequency is in agreement with the notion of diffusion-mediated signaling, with ROS as the primary coupling agent that is locally restricted and mainly dependent on local ROS density changes.

For small areas of structurally dense, phase-coupled mitochondrial oscillators, local ROS concentration is strongly increased through ROS-induced ROS release, thus leading to the recruitment of neighboring mitochondria. Consequently, these growing clusters lose spatial contiguity and, therefore, show a decrease in their common frequency which, in turn, leads to a decrease of the frequency of local ROS release and to declining local ROS levels, especially along the cluster perimeter. This effect naturally decreases inter-mitochondrial coupling. The formation of spanning clusters, however, provides for a slight increase in the overall basal ROS level to coincidentally increase local inter-mitochondrial coupling. Generally, these observations indicate a preference of spatial coupling over temporal coupling that affect synchronized mitochondrial oscillator networks: early cluster formation mainly drives the correlation between common cluster frequency and local mitochondrial coupling. During this time, the mean cluster coupling constant is large (see Fig. 5 in [15]). Cluster size and local coupling, however, mainly show correlative effects when the cluster of synchronized mitochondrial oscillators involves a majority of the mitochondrial network. At this point, a spanning cluster has already been formed and most mitochondria participate in cell-wide ΔΨ_m oscillations.

ROS as primary inter-mitochondrial coupling agents and products of mitochondrial respiration take a central role in cell homeostasis and its response to oxidative or metabolic stress. Under physiological conditions, mitochondria minimize their ROS emission as they maximize their respiratory rate and ATP synthesis according to the redox-optimized ROS balance hypothesis [87]. Stress alters this relationship through increased ROS release and a diminished energetic performance [88]. Thereby, the compartmentation of the redox environment within the cellular compartments such as the cytoplasm and mitochondria, is important in the control of ROS dynamics and their target processes [86]. Within physiological limits, increased mitochondrial metabolism resulting in enhanced ROS generation can function as an adaptive mechanism leading to increased antioxidant capacity and stress resistance, according to the concept of (mito)hormesis [89, 90]. Generally speaking, controlled ROS emission may serve as a robust feedback regulatory mechanism of both ROS generation and scavenging [88]. Dysfunctional mitochondria alter their redox state to elicit pathological oxidative stress, which has been implicated in ageing [91, 92], metabolic disorders like diabetes [93], as well as in cardiovascular [34, 94], neoplastic [95], and neurological diseases [40, 96]. Chronic exposure to increased ROS levels can affect excitation-contraction coupling and maladaptive cardiac remodeling that may lead or increase the propensity to cardiac arrhythmias [34, 97, 98].

Mitochondrial Scale-Free Network Topology and Functional Clustering

Self-Similar Dynamics and Fractal Behavior in Mitochondrial Networks

The defining feature of time series signals with scale-free dynamics is the similarity of frequency-amplitude relations on small, intermediate, and large time scales. In a network of coupled oscillators this translates into new links between oscillators that form in higher numbers depending upon the number of already existing links. In scale-free networks, the occurrence of hubs (i.e., nodes with multiple input and output links) increases the probability of new links being formed [99], an effect known as preferential attachment [58] or the “Matthew effect” in the sociological domain [100].

Such fractal or self-similar dynamics develop because of a multiplicative interdependence among these processes and has been observed in cardiac myocytes subjected to oxidative stress [20, 34, 101], in self-organized chemostat yeast culture [62, 102, 103] and in continuous single-layered films of yeast cells exhibiting large-scale synchronized oscillations of NAD(P)H and ΔΨ_m [62, 104]. Using wavelet analysis, and as predicted by fractal dynamics, one observes an intricate scale-free frequency content with similar features on different time scales (Fig. 1.4) [60].

Fig. 1.4.

Wavelet analysis of the O₂ signal time series obtained from the self-organized multioscillatory continuous culture of S. cerevisiae. (a) Relative membrane inlet mass spectrometric signal of O₂ versus time for S. cerevisiae. The time scale is not shown, but corresponds to hours after the start of continuous fermentor operation. Large-amplitude oscillations show substantial cycle-to-cycle variability with 13.6 ± 1.3 h (N = 8). Other evident oscillation periods are ~40 min and ~4 min. This figure was provided courtesy of Prof. David Lloyd, affiliated with Cardiff University, United Kingdom. (b) Logarithmic absolute squared wavelet transform over logarithmic frequency and time. At any time, the wavelet transform uncovers the predominant frequencies and reveals a complex and fine dynamic structure that is associated with mitochondrial fractal dynamics. Boxes and arrows show the correspondence between the time series and the wavelet plot

For cardiac mitochondrial networks in the physiological regime, it could be demonstrated that the failure or disruption of a single element beyond a critical threshold can result in a failure of the entire network; such scaling towards higher levels of organization in the mitochondrial network obeys an inverse power law behavior that implies an inherent correlation of cellular processes operating at different timescales [8, 58]. The observation of scale-free dynamics in mitochondrial oscillator networks is in agreement with their representation as weakly coupled oscillators in the physiological domain (in the presence of low ROS levels), and their role as energetic, redox and signaling hubs [105].

Functional Connectivity and Mitochondrial Topological Clustering

Dynamic changes in the network’s topology during synchronized myocyte-wide oscillations in the pathophysiological state reveal important information about functional relations between distant mitochondria displaying similar signal dynamics. These relations cannot be explained alone by local coupling through diffusion or structural connections. As in networks of communicating neurons, the spontaneous occurrence of changes in the clusters’ oscillatory patterns observed at markedly different locations, indicates strong and robust functional correlations; interestingly, the time-scales of these changes are similar in mitochondrial and neuronal networks [106, 107].

Mitochondrial network functionality can be assessed by measuring the functional connectedness between mitochondrial network nodes, i.e. by obtaining a quantitative measure of the functional relations that span the dynamically changing mitochondrial network topology [108].

Such a measure of mitochondrial functional connectedness can be the clustering coefficient, similar to that employed by Eguiluz et al. for scale-free functional brain networks [109]. In that case, for a specific network constituent, the clustering coefficient provides a measure of connectedness between neighbors [110]. For instance, clustering plays an important role in neuronal phase synchronization [111]. In the case of mitochondrial networks, dynamic functional clustering significantly differs from random Erdös-Rényi networks [112], even in the presence of similar number of links [81] (Fig. 1.5). This result supports the concept that mitochondrial networks exhibit structural-functional unity [113]. Therefore, dynamic mitochondrial clustering is consistent with its functional properties that are associated with mitochondrial positioning within the network, as previously described [30, 31]. In Fig. 1.5c, the dynamic nature of topological clustering is shown for a glucose-perfused cardiac myocyte during synchronized membrane potential oscillations. While some regions on the left part of the myocyte maintain high clustering, mitochondria on the right side show higher clustering only during the middle portion of the recording, as the spatio-temporal topology evolves.

Fig. 1.5.

Dynamic functional clustering in mitochondrial networks. (a) Comparison of dynamic functional clustering for mitochondrial networks (red line) and random networks (blue line) with the same number of links on the same set of vertices. Mitochondrial networks are scale-free networks whose functional organization is significantly different from random networks. (b) Functional clustering for a large cluster of similarly oscillating mitochondria versus the cluster’s area. Functional clustering is increasing with the number of synchronously oscillating mitochondria. (c) Consecutive maps of distributions of functional clustering coefficients in a cardiac myocyte. Some regions of the cell maintain a high functional clustering during the recording (middle left area of the cell) whereas other regions only temporarily display functional clustering (far right area of the cell) (a and b reprinted with permission from Ref. [81]. Copyright 2014)

To further investigate the mitochondrial network dynamics, future studies will need to identify and quantify functional ensembles of strongly interconnected mitochondria [80, 114]. These studies may shed new light into populations of coexistent subsarcolemmal and intermyofibrillar mitochondria that exhibit different roles in the myocyte (patho)physiology [28, 29, 115].

Concluding Remarks

Biological networks within one cell, organ or organism are not isolated entities but exhibit common nodes. Introducing higher-dimensional node degrees generates multi-layer, interdependent, networks. So far, our understanding of such networks is still in its infancy (see [116, 117] for detailed reviews). For example, within the cardiac mitochondrial network, at least two layers can be distinguished, corresponding to structural connections via inter-mitochondrial junctions, and communication through mechanisms that control local ROS balance [19, 35, 80, 118].

Stressful events can trigger synchronized mitochondrial oscillations in which mitochondria functionally self-recruit in frequency clusters [9, 16, 35, 81]. In functional terms, mitochondria can be seen as alternating between states of excitability and quiescence modulated by ROS signaling, thus modulating myocyte’s function and energy metabolism [119]. In cardiac myocytes under critical situations, e.g. at myocyte-wide ROS accumulation close to the percolation threshold, inter-mitochondrial functional, structural and dynamic coupling can lead to electrical and contractile dysfunction, and ultimately to myocyte death and whole heart dysfunction. Eventually, under criticality conditions, perturbation of a single mitochondrion could propagate to the remaining mitochondria, leading to myocyte death.

A mitochondrion’s gatekeeper role, at the brink of cardiac myocyte death, can only be understood when its function in the mitochondrial network can be predicted either experimentally or through quantitative modeling in which case the mitochondrial parameters are inferred from the available experimental data. Although an increase in computational efficiency has helped modeling make impressive progress in recent years [68], a complete description of the mitochondrial network is still evolving. Stochastic models have been well suited to provide system-wide mitochondrial parameters, such as dynamic coupling constants and functional clustering coefficients in large networks of coupled mitochondrial oscillators.

Synergistic, experimental and computational investigations have unveiled emergent properties of the complex mitochondrial oscillatory networks that describe cluster partitioning of synchronized oscillators with different oscillatory frequencies [120, 121], as well as the chimeric co-existence of synchronized and non-synchronized oscillators [122, 123]. These results coincide with our observations of mitochondrial oscillations in cardiac myocytes, thus supporting the view of network-coupled mitochondrial oscillators. The computational approach promises further discernment of the dynamic spatio-temporal relations within biological networks, while aiding our understanding of its functional implications for cell survival. Therefore, the quantitative description of inter-mitochondrial coupling and functional connectedness provides parameters whose dynamic behavior under pathological conditions may help evaluate the impact of, e.g., metabolic disorders, ischemia-reperfusion injury, heart failure, on mitochondrial networks in cardiac, cancer and neurodegenerative diseases.

To further investigate the mitochondrial network dynamics, future studies will need to identify and quantitate functional ensembles of strongly interconnected mitochondria [80, 114]. These studies may shed new light into populations of coexistent subsarcolemmal and intermyofibrillar mitochondria that exhibit different functional outputs [28, 29, 115].