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

Abstract

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, $D\text{,}$ that is indicative of self-similar behavior with $D=1.27\pm 0.11\text{,}$ in contrast to the remaining network mitochondria whose fractal dimension is close to that of Brownian noise, $D=1.58\pm 0.10$. 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.

Significance

Scale invariance in the cardiac mitochondrial network requires rapid and flexible responses of mitochondria to external perturbations; e.g., in the form of metabolic or oxidative stress. Such responses allow the biological system to perform intricate control processes that avoid a system mode that is locked to only a few functions. Our work demonstrates that individual mitochondrial oscillators exhibit self-similar behavior that is associated with local inter-mitochondrial coupling. The results can be used to study and identify clusters of strong local coupling between mitochondrial oscillators that drive synchronized mitochondrial network oscillations. A targeted breakdown of coupling within these clusters may prevent the cardiac myocyte’s descent toward myocyte death.

Introduction

Mitochondrial networks in cardiomyocytes consist of mitochondria that are densely arranged and whose collective behavior is associated with the tight control of diverse cellular functions, such that perturbations of its activity have significant implications in arrhythmogenesis and heart failure (1,2).

The mitochondrial inner membrane potential $\Delta {\Psi }_m$ is sensitive to metabolic or oxidative stress and may enter consecutive cycles of depolarizations/repolarizations that are coordinated by the mitochondrial network (3,4). Although these oscillations can be transient and localized for individual mitochondria (5), a cluster of several mitochondria with similar oscillation dynamics can form that spans the whole cardiac myocyte (6). Such clusters may arise from a small nucleus of synchronously oscillating mitochondria that may further exert a strong control on neighboring mitochondria and eventually synchronize their oscillatory behavior until the number of oscillating mitochondria reaches a critical size to trigger myocyte-wide synchronized mitochondrial depolarizations (6,7). Mitochondrial network synchronization is thought to be caused by activation of mitochondrial membrane channels through reactive oxygen species (ROS), which trigger further ROS release to potentiate local ROS accumulations in a process called ROS-induced ROS release (8,9). Diffusive properties of ROS molecules in the peri-mitochondrial environment, mitochondrial and other cellular redox mechanisms to balance ROS, as well as density and anatomical positioning of mitochondrial ROS release channels therefore determine inter-mitochondrial coupling and communication (10,11).

The complex processes that influence mitochondrial oscillations are characterized by dynamically changing frequencies, and have been studied using the wavelet transform (12). This analysis revealed clusters of mitochondria with similarly changing frequencies, whose average frequencies were inversely correlated with the cluster’s size, presumably due to the larger number of coupled mitochondrial oscillators that adjust to a common frequency and phase (7). In parallel, the underlying mitochondrial coupling can be studied by assigning a coupling constant to each mitochondrion in a stochastic model that considers the dynamic nature of mitochondrial frequency changing and spatial arrangement within the network (13). These coupling constants quantify local inter-mitochondrial coupling; i.e., in a nearest-neighbor environment. Large clusters of synchronized mitochondrial oscillators could be shown to possess stronger local inter-mitochondrial coupling than smaller clusters (13).

However, mitochondrial network dynamics underlie not only local coupling mechanisms but also non-local, functional relations between distant clusters or pairs of mitochondria; e.g., when mitochondria respond independently to dynamic changes within the mitochondrial network (14,15). These functional relations eventually determine the mitochondrial networks’ topology, such that the functional connectedness between network mitochondria can be measured with a functional clustering coefficient, which describes the network’s robustness to functional changes, e.g., the oscillation arrest of a single mitochondrion, and the network’s ability to communicate across large intracellular distances (16). Network topologies that are susceptible to clustering have also been observed in other networks of coupled oscillators, e.g., in neuronal networks (17), and share similar dynamics of self-organization on similar timescales of seconds to minutes (18,19). A potential structural correlate of trans-mitochondrial coordination may be inter-mitochondrial junctions (20), which are presumed to play a role in electrochemical and mechanical inter-mitochondrial coupling, and which are believed to enhance intracellular bioenergetic signal propagation in analogy to electrochemical coupling at neuronal synapses across functionally connected brain areas (16,20). Furthermore, the recently found nanotunnels may provide an anatomical link that may facilitate correlative properties of inter-mitochondrial functional connectedness of locally distanced mitochondria; (21,22); however, their role in cell-wide inter-mitochondrial coupling is still unclear.(9).

Both local inter-mitochondrial coupling and functional clustering in the mitochondrial network lead to emergent scale-free features of mitochondrial network oscillations that characterize the similarity of frequency-amplitude relations across timescales (23). Scaling of mitochondrial dynamics may be due to structural or functional hubs in the mitochondrial network that increase the probability of links to other mitochondria (24). Such self-similar or fractal dynamics have been observed in whole-cell or cell-segment signal traces of mitochondrial $\Delta {\Psi }_m$ oscillations (6,25). In that case, fractal dynamics of the mitochondrial network have been characterized by a fractal dimension that describes invariant temporal scaling within the collective $\Delta {\Psi }_m$ oscillations and have been found to be distinctly different from random behavior (25). Fractal dimensions close to 1 thereby suggest that fluctuations in $\Delta {\Psi }_m$ are subject to preceding changes of $\Delta {\Psi }_m$; i.e., they reflect a long-term memory of the mitochondrial network. However, whether this fractal behavior can be observed in local, individual mitochondrial $\Delta {\Psi }_m$ oscillations has not been investigated yet. In addition, it remains unclear whether individual mitochondrial fractal behavior can be linked to local (nearest-neighbor) inter-mitochondrial coupling, local functional clustering, or local coherence; i.e., synchronicity of mitochondrial oscillations. Understanding of the local inter-mitochondrial coupling and/or functional relations may prove beneficial for a time-efficient investigation of cardiac myocyte mitochondrial networks dynamics, since a stochastic phase model analysis requires extensive computing time, involving several regularization and optimization steps (13).

This work investigates fractal dynamics of individual mitochondrial oscillators and their distribution across cardiac myocytes. Fractal dimensions will be obtained and related to functional and structural local measures of inter-mitochondrial coupling.

Material and methods

Experimental setup

All animal experiments were approved by the Johns Hopkins University Animal Care and Use Committee and carried out according to guidelines in the Guide for the Care and Use of Laboratory Animals, published by the National Institutes of Health (NIH publication no. 85-23, revised 1996). We obtained freshly isolated adult guinea pig ventricular myocytes (N = 9) with protocols that were described previously (7)⁠. Briefly, perfusion of cardiac myocytes was achieved with a Tyrode solution (pH 7.5) that contains 1 mM Ca²⁺ and 10 mM glucose, and depolarizations of mitochondrial inner membrane potential $\Delta {\Psi }_m$ were initiated with a localized (5 × 5 μm) laser flash as reported in (7). The potential $\Delta {\Psi }_m$ was visualized via the fluorescent dye tetramethylrhodamine methyl ester (TMRE) under a two-photon laser-scanning microscope (MRC-1024MP, Bio-Rad), which served to record the images with excitation at 740 nm (Tsunami Ti:Sa laser, Spectra-Physics) and collection of TMRE red emission at 605 nm utilizing 578–630 nm as a band pass filter (12)⁠. Cardiac myocytes (N = 9 from six animals) were imaged with an objective 40×/1.0 W MP with a spatial resolution of 335.5 × 335.5 nm²; a frame rate of 285.7 mHz, corresponding to one frame every 3.5 s; with an average duration of 693.70 ± 88.08 s. A detailed step-by-step description of the experimental setup can be found in (26).

To assess whether our sample size was sufficient for the subsequent analyses, we assumed “pseudoreplication” in the results due to myocytes from the same animals, and we followed the procedure proposed in Sikkel et al. that assumes a data-inherent clustering with multiple likely similar myocytes from each animal heart (27). In isolated rat cardiomyocytes, they find intraclass correlation coefficients (ICCs), which determine the degree to which data from individual myocytes is clustered in making up the data distributions in individual hearts (with ICC = 1 corresponding to highly clustered data indicating identical parameters from cells of the same animal heart, and ICC = 0 otherwise), at ICC ∼ = 0.3 (Table 2 in (27)). To calculate an effective sample size (ESS) for n animals with m myocytes each, we find $ESS=\frac{n·m}{\left[1+\left(m-1\right)·ICC\right]}\approx 8$ with m = 1.5 and ICC = 0.3 (27).

Mitochondrial TMRE signals

TMRE fluorescence signal changes in time of single mitochondria could be obtained from planar images of isolated cardiomyocytes with methods described in (7,12). In brief, an averaged image of the first $n$ subsequent images after the onset of TMRE oscillations was used to obtain a template grid delineating the anatomic location of every single mitochondrion (26). In Fig. 1 a, the first averaged image of an isolated cardiac myocytes is shown, and the dense mitochondrial network is clearly visible. Assuming an image sampling rate of $dt$, a parameter $n$ was determined with the constraint that $ndt$ was equal or smaller than the smallest period of the observed oscillations in TMRE. Also, we excluded myocytes that were shifting in and out of focus. Myocyte drift in the imaging plane was corrected by a respective shift of the template grid. The extracted signals for individual mitochondria were then examined using signal analysis tools as detailed in (7,12). In total, we analyzed 6335 individual TMRE signal traces, or 703.9 ± 156.4 mitochondria per myocyte.

Figure 1.

Mitochondrial oscillators in cardiac myocytes, wavelet analysis, and fractal dimension. (a) Cardiac cell with a lattice-like mitochondrial network. Some mitochondria are already semi-depolarized and exhibit a lower signal intensity. (b) TMRE signal of an oscillating mitochondrion and its corresponding absolute squared wavelet transform over frequency and time; the mitochondrion is represented by the green square in (a). The major frequency component (yellow) varies between 10 and 30 mHz. The fractal dimension is $D=1.24\text{.}$ (c) Non-oscillating mitochondrion, represented by the blue square in (a). No significant frequency content can be extracted from the wavelet transform, the fractal dimension is $D=1.89$, and therefore close to white noise. (d) Random permutation of signal intensity of the mitochondrion in (b), $D=2.0\text{.}$ (e) Gaussian white noise, $D=1.98$. (f) Brown noise, $D=1.54$. For further details, please see main text. To see this figure in color, go online.

Fractal dimension

The fractal dimension of a time series quantifies its self-similarity and complexity and gives a measure of the long-term memory in the signal; i.e., the influence of past $\Delta {\Psi }_m$ changes on the current $\Delta {\Psi }_m$ fluctuations. Fractal dimensions of individual mitochondrial non-linear TMRE signals were determined with the Higuchi algorithm (28), which provides stable values of the fractal dimension, even for time series with limited data points; this algorithm is less sensitive to noise than, for example, the Petrosian algorithm (29), and more computationally efficient than the box-counting algorithm (30)⁠.

For a time series $T=t_1,t_2,\dots ,t_N$ the algorithm works as follows: new time series $T_n\left(j\right)$ are formed with $T_n\left(j\right)=t_n,t_{n+j},t_{n+2j},\dots ,t_{n+j·\text{floor}\left(\frac{N-n}{j}\right)}$ where $j=\mathrm{1,2},\dots ,j_{\max }$ and $n=\mathrm{1,2},\dots ,j$ are integers that indicate the interval time and the initial time, respectively. Second, the length $L_n\left(j\right)$ of each time series is defined as $L_n\left(j\right)=\left(\frac{1}{j}\right)\left(\sum _{i=1}^{\text{floor}\left(\frac{N-n}{j}\right)}\left|t_{n+ij}-t_{n+\left(i-1\right)j}\right|\right)\frac{N-1}{j\text{floor}\left(\frac{N-n}{j}\right)}$. Third, the length $L\left(j\right)$ of the curve for time interval j is defined as the average over j sets of $L_n\left(j\right)$. The maximal value $j_{\max }$ is determined as the last value of $j$ where the change of successive slope values $\frac{\mathit{log}\left(L\left(j+1\right)\right)-\mathit{log}\left(L\left(j\right)\right)}{\mathit{log}\left(j+1\right)-\mathit{log}\left(j\right)}$ relative to the initial slope value $\frac{\mathit{log}\left(L\left(2\right)\right)-\mathit{log}\left(L\left(1\right)\right)}{\mathit{log}\left(2\right)-\mathit{log}\left(1\right)}$ is smaller than 50%. The fractal dimension D could then be obtained for these points as the negative slope of a linear regression analysis, as $L\left(j\right)\propto j^{-D}$ (28). We provide a pseudo-code implementation of the fractal dimension for mitochondrial time series in the online supplement. A typical fractal dimension for a time series that consists of Gaussian white noise is given as $D=2.0$, whereas brown noise typically features $D=1.5$; time series with a high degree of self-similarity and long-term memory have lower values that tend to approach $D\approx 1$ (see also Fig. 1).

Wavelet-based frequency analysis

The mitochondrial frequency analysis was performed using the Morlet wavelets to determine the frequency content of each mitochondrial TMRE signal; see also (7) for a detailed account of the procedure. In brief, we set the smallest wavelet scale as $s_0=4dt$ (corresponding to the smallest period that allows detecting a single oscillation), and wavelet scales were spaced apart with $dj=0.1$. When $N$ represents the number of recorded images in a myocyte, the number of scales was obtained as $j_1={\log }_2\left(\frac{N}{s_0}\right)dj+1$. Therefore, scales ranged from $s_0$ to $s_0{2}^{\left(j_1-1\right)dj}$ with each scale having $dj$ suboctaves. Cutoff frequencies ${\nu }_{\mathit{min}}$ and ${\nu }_{\mathit{max}}$ were determined with the duration $T$ of the longest synchronized myocyte-wide oscillation and the smallest possible period $s_0$ as ${\nu }_{\mathit{min}}=1/1.1T$ and ${\nu }_{\mathit{max}}=1/s_0$, respectively. Maximal scale frequencies were obtained at each point in time and for every individual mitochondrion by using power line plots between ${\nu }_{\mathit{min}}$ and ${\nu }_{\mathit{max}}$ which we interpolated with segments of 0.1 mHz for every scale. This allows extraction of the frequency segment with maximal power.

The wavelet analysis unveils the dynamic character of individual mitochondrial frequencies: Fig. 1 b shows the time series of an individual oscillating mitochondrion, indicated in the color green in Fig. 1 a, and the corresponding absolute squared wavelet transform over time and frequency, respectively. This analysis reveals a main frequency component (bright yellow color in Fig. 1 b), which shows varying frequency content between 10 and 30 mHz. The respective period, i.e., the inverse of this main frequency, corresponds to approximately 30–100 s, in agreement with the observed intervals between troughs and peaks of the oscillations (Fig. 1 b). The same time series features a higher degree of self-similarity as reflected in the fractal dimension $D=1.24$. In Fig. 1 c we further show the time series of a non-oscillating mitochondrion, indicated as a blue square in Fig. 1 a, with a fractal dimension of $D=1.89\text{,}$ which is close to white noise, and therefore likely instrument noise. Although the non-oscillating mitochondrial TMRE series does not indicate any significant lower frequency content as its oscillating counterpart in Fig. 1 b, there are numerous weaker higher frequencies, e.g., a weak frequency band at around 40–50 mHz (in light green). This frequency band is less pronounced when we compare the wavelet transform with that of a random permutation of signal intensity of the oscillating mitochondrion in Fig. 1 d or with that of Gaussian white noise in Fig. 1 e, with fractal dimensions of $D=2.0$ and $D=1.98$, respectively. There is no relevant frequency content in brown noise with a fractal dimension of $D=1.54$, see Fig. 1 f. The comparison of the mitochondrial oscillating cluster with a random TMRE signal intensity permutation, Gaussian white noise, and Brownian noise as well as their respective fractal dimensions demonstrate that the computation of fractal dimensions according to the Higuchi algorithm yield consistent and meaningful results for individual mitochondrial oscillators, and they show that an increase in the randomness of mitochondrial inner membrane potential fluctuations toward uncorrelated white noise or toward the appearance of correlations of the random walk type (brown noise) indicates a degeneration of the characteristic long-range correlation within the mitochondrial signal traces (25). The completely random behavior of random intensity permutations, white noise, or brown noise characterizes a process without memory, in contrast to mitochondrial oscillators.

We have intentionally refrained from denoising the mitochondrial signal traces to avoid eliminating higher frequency fluctuations, e.g., up to 80 mHz, as observed in (13,16). In analogy to mass spectrometric oxygen signals of self-organized multi-oscillatory continuous cultures of Saccharomyces cerevisiae, it was demonstrated in mitochondrial inner membrane potential oscillations that large-amplitude oscillations may contain a complex dynamic structure that is associated with scaling and self-similar dynamics; e.g., in the ascending slope of the large-amplitude oscillations, see Fig. 1.4 in (11) and also (25).

Coherence analysis of mitochondrial TMRE signals

The TMRE signal coherence for each mitochondrion was determined as the average of all coherences between a mitochondrion and its nearest mitochondrial neighbors, in analogy to (7)⁠. Coherence values indicate whether two mitochondrial signals show synchronous oscillations at each frequency or not, with coherence values ranging between one and zero, respectively. The frequency range for each mitochondrion was set at 0–100 mHz and divided into $\left(2^{11}/2\right)+1$ segments where a segment corresponds to approximately 0.1 mHz.

Main clusters of oscillating mitochondria

Mitochondria oscillating with similar frequencies were grouped as clusters based on a procedure detailed in (7); briefly,⁠ we obtained frequency histograms across the imaged mitochondria at each time point $t$ and found the largest peak in each histogram. Histogram peaks with amplitudes above 10% of the maximal peak were included in the subsequent analysis. We further chose a running window of size $T_W=1.1T\text{,}$ which was centered around each time point $t$, and determined the averaged TMRE signal of those mitochondria that belonged to the largest peak in each histogram. Mitochondria with frequencies adjacent to this main histogram peak were only included in the peak when their signal correlation within the running window showed a correlation coefficient of at least 95% with the averaged signal of the mitochondria comprised by the peak. They were subsequently incorporated in the cluster of mitochondria that form the main histogram peak. We repeated this process with the respective adjacent peaks until the correlation was lower than 95%. The resulting averaged TMRE signal of all mitochondria encompassed in the resulting cluster of highly correlated mitochondria was cross-correlated with the TMRE signal of every remaining mitochondrion. In an analogous procedure, remaining mitochondria were included in the cluster if the cross-correlation of their TMRE signal with that of the main cluster mitochondria was at least 95%.

Local correlation and nearest-neighbor correlation

For each mitochondrion m, the nearest neighbors were identified as in (12). Within an imaging plane, most mitochondria possessed eight nearest neighbors as for a regular arrangement of constituents in a 2D lattice. Then, the correlation coefficients $r_{\left\{m,j_m\right\}}$ for each mitochondrion $m$ with all of its $j_m$ nearest neighbors were determined for their respective signals at time $t$ in the time window $T_w$ around $t$. Finally, the mitochondrial correlation $R_m$ for each mitochondrion m was obtained as the mean correlation with its nearest neighbors: $R_m={\sum }_{jm}{r}_{\left\{m,j_m\right\}}/\left|j_m\right|$.

Functional clustering

For each pair of mitochondria within the cardiac myocyte, the functional connectedness in the mitochondrial network was determined in a procedure that analyzes the correlation coefficients of their respective TMRE signals, as previously described in (16,26). Two mitochondria are considered as functionally connected at time $t$ when their signals show a cross-correlation in the time window $T_w$ around $t$ that is above a specific cutoff value $r_c$. For the subsequent analysis, $r_c$ was chosen as $r_c=0.9$ in accordance with previous results (see Fig. 3 a and associated text in (16)⁠).

Figure 3.

Network properties in relation to fractal dimensions (N = 9 cardiac myocytes). (a) Fractal dimension versus nearest-neighbor correlation. Increased correlation coincides with low fractal dimensions; for correlation close to 1, the average fractal dimension approaches the values of the pathophysiological regime as described in (25). (b) Fractal dimension decreases with increasing local coherence between mitochondrial TMRE signals. Large coherence values, however, show a broad range of fractal dimensions, indicating that coherence is only weakly associated with self-similar properties. (c) Fractal dimensions increase for increased coupling, indicating a correlation between local coupling mechanisms and emergent fractal behavior. (d) Fractal dimensions decrease with increasing functional clustering. Fit curves are shown in green; for details, see main text. Error bars indicate standard error. To see this figure in color, go online.

The graph-theoretical concept of clustering then describes the resemblance of a set of vertices (here, mitochondria) in the functional network to a clique (corresponding to a set of mitochondria in which every subset of two mitochondria is functionally connected). Among $m_N$ topological, i.e., functionally connected, neighbors of a single vertex $m$ of the network, there are up to $m_N\left(m_N-1\right)/2$ potential undirected links. Local clustering $C_m$ is then defined as $C_m=2L_m/\left(m_N\left(m_N-1\right)\right)$, where $L_m$ represents the actual number of (undirected) links among all mitochondria neighboring mitochondrion $m$. The arithmetic mean of clustering coefficients of all $M$ network mitochondria eventually yields the mean local clustering coefficient $C=\left(\frac{1}{M}\right)\sum _m{C}_m$; see also (31). We determined time-averaged clustering coefficients for every individual mitochondrion.

Network coupling constants

Recently, we developed a stochastic phase model as an extension of the well-known Kuramoto model for mitochondrial networks (13). The model considers intrinsic mitochondrial frequencies as Ornstein-Uhlenbeck frequencies, which correspond to dynamic frequencies that drift toward the respective imaged mitochondrial frequencies; see also (32,33). For each mitochondrion, the model allows extracting a time-varying coupling constant, which describes dynamic local mitochondrial coupling of local mean field type; a detailed description and application of the model is provided in (13)⁠. The local mean field thereby extends from the mitochondrion to all its nearest neighbors; see also Fig. 2 in (13). In Fig. S1, we provide a combined dynamic visualization of the main cluster mitochondria and the local inter-mitochondrial coupling strength in a cardiac myocyte.

Statistics

All numerical analyses and fitting routines were obtained using Matlab v9.8 (R2020a). Statistics were performed in Matlab and OriginPro 9.1.0 (B215).

Results

Distribution of individual mitochondrial fractal dimensions

For each cardiac myocyte, we created maps to visualize the distribution of fractal dimensions in each individual mitochondrial oscillator’s time series; see Fig. 2. Maps were generated by placing the fractal dimension of a mitochondrion at its respective pixel position within the cardiac myocyte. We used the Matlab function griddata to interpolate over missing pixels. For some myocytes, e.g., Fig. 2 g, only a small portion of mitochondria show a low fractal dimension, whereas most mitochondria possess values >1.5. However, there are also myocytes with large areas of mitochondria with low fractal dimensions that are close to 1.0 (Fig. 2 e, f, and i). The variation of fractal dimensions is likely due to mitochondrial heterogeneity, where myocytes with mainly larger fractal dimensions similar to brown noise are presumably at an early state of myocyte-wide collective mitochondrial behavior with only weakly coupled mitochondrial oscillators, whereas others with predominantly low fractal dimensions presumably are at a later state of myocyte-wide mitochondrial TMRE oscillations, indicative of a strongly coupled cluster of mitochondrial oscillators; see also the normalized intensity videos of the respective cardiac myocytes in Fig. 2 g and i (Videos SV1 and SV2). To evaluate whether areas of low fractal dimension corresponded to mitochondria in a highly correlated cluster with similar frequencies as defined above using the wavelet-based frequency analysis, we estimated a cluster probability for each mitochondrion as the quotient of the number of image frames with the mitochondrion being part of the main cluster versus the number of all image frames during the recording. We then selected the 10 mitochondria with the highest cluster probability and likewise the 10 mitochondria with the lowest cluster probability and found the averaged fractal dimensions for all mitochondria in such a cluster over all myocytes as D = 1.27 ± 0.11. The averaged fractal dimension for all non-cluster mitochondria was found as D = 1.58 ± 0.10; i.e., closer to brown noise. Distributions of fractal dimensions for cluster and non-cluster mitochondria were significantly different (p < 0.01).

Figure 2.

Mitochondrial fractal dimensions. (a–i) Maps of fractal dimensions for the time series of single mitochondria in different cardiac myocytes. Intracellular regions with low fractal dimensions indicate self-similar dynamics, see example, myocytes in (e) and (i). For mitochondria in a major frequency cluster, we find $D=1.27\pm 0.11\text{,}$ and, for non-cluster mitochondria, $D=1.58\pm 0.10$. To see this figure in color, go online.

Fractal dimension in relation to spatio-temporal properties of mitochondrial clustering

Spatio-temporal properties of mitochondrial network oscillators have been examined previously for spatio-temporal correlation and coherence (7,16), functional clustering (16), and local coupling based on a stochastic phase model (13). The averaged correlations of mitochondria to its nearest neighbors provide a quantification of local similarity of mitochondrial oscillators.

For each mitochondrion, we determined a time-averaged correlation coefficient, a coupling constant, a clustering coefficient, and a coherence value within the image recording time, T. Time averaging was done on 127.6 ± 23.7 values (one value for each image frame) per parameter, and mitochondrion. For all parameters, this reduced the 788,624 individual parameter values to 6335 values, one for each mitochondrion. For the mitochondrial local correlation coefficients, we determined bins of size 0.01, ranging from −1 to 1. For the mitochondrial coherence values, we chose bins of size 0.01 to range from 0 to 1, and we did the same for mitochondrial clustering coefficients. For the mitochondrial coupling constants, we chose bins of size 0.001 to range from 0 to 0.4. For each bin in each parameter (local correlation, coherence, coupling constant, clustering coefficient) and each myocyte, we determined the respective contributing mitochondria, and averaged their fractal dimension to provide one fractal dimension value per bin and myocyte; e.g., for each myocyte, we obtained an averaged fractal dimension of local correlation coefficient values between 0.02 and 0.029, 0.03 and 0.039, and so on. Then, for each bin, these averaged fractal dimensions were averaged across myocytes to avoid bias from myocytes of different size.

In Fig. 3 a, one observes that the fractal dimension of individual mitochondrial time series decreases as the local correlation increases. The decreasing behavior of fractal dimension, $D\text{,}$ as a function of local correlation, $c\text{,}$ can be described by the following dependency: $D\left(c\right)=a\exp \left(-bc\right)+dc+e\text{,}$ with $a=-8.52·{10}^{-11}\pm -3.51·{10}^{-11},b=-21.65\pm 5.23,d=-0.25\pm 0.02,e=1.73\pm 0.01$ ($R^2=0.87$). Here, we keep the exponential form with tiny pre-factor $a$, since it nicely samples the behavior of fractal dimension <1.5, which is needed to adequately describe cluster mitochondrial fractal dimensions that are typically below that threshold. Otherwise, a linear approximation of the form $D\left(c\right)=dc+e$ would be sufficient to describe fractal dimensions >1.5. The resulting relation between fractal dimension D and correlation coefficient c was anticipated since mitochondria with high correlations to their nearest neighbors typically are less prone to fast frequency changes than mitochondria whose oscillating neighbors are less correlated. One observes that, as the degree of random frequency fluctuations decreases, the fractal dimension approaches 1.

To further examine the local synchrony of mitochondrial oscillations, coherence values were attributed to each mitochondrial oscillator, with coherence values close to 1 signifying synchronous oscillations. The degree of local mitochondrial coherence, $\overline{c}\text{,}$ is also reflected in individual mitochondrial fractal dimensions; see Fig. 3 b. We found that the fractal dimension decreases with local coherence as $D\left(\overline{c}\right)=a\overline{c}+b\text{,}$ where $a=-0.55\pm 0.02\text{,}$ and $b=1.95\pm 0.01\left(R^2=0.85\right)\text{.}$

Another indicator of local coupling among mitochondrial neighbors in cardiac myocytes was recently proposed employing a stochastic model that takes into account the frequency behavior of mitochondrial oscillators; that model assigns to each mitochondrion a stochastic intrinsic frequency that drifts in time toward a measured mitochondrial frequency, with the intent to obtain a local coupling constant, $k$ (13). As demonstrated in Fig. 3 c, fractal dimensions decrease for increasing local coupling between mitochondrial oscillators. The observed decrease in fractal dimension $D$ for coupling $k$ is mainly linear and can be described as $D\left(k\right)=ak+b\text{,}$ with $a=-1.46s\pm 0.06s$, and $b=1.66\pm 0.01\left(R^2=0.70\right)\text{.}$ Coupling constants k have units 1/s (13).

The mitochondrial network’s functionality can be assessed by assigning clustering coefficients to each mitochondrial oscillator; these coefficients determine the functional connectedness between mitochondrial network nodes (11,26). These functional relations are not bound to local coupling but instead span the whole mitochondrial network. Large clustering coefficients indicate a strong functional clustering; fractal dimension decreases with clustering $\tilde{c}$ as $D\left(\tilde{c}\right)=a\tilde{c}+b\text{,}$ with $a=-0.43\pm 0.01$, and $b=1.78\pm 0.01\left(R^2=0.94\right)$; see Fig. 3 d.

Discussion

Scale invariance in the cardiac mitochondrial network requires rapid and flexible responses of mitochondria to external perturbations; e.g., in the form of metabolic or oxidative stress. Such responses allow the biological system to perform intricate control processes that avoid a system mode that is locked to only a few functions (25,34). Although self-similar dynamics were observed for the whole mitochondrial network (25), they have not yet been described for individual mitochondrial oscillators.

Several conclusions can be drawn from this study based on the fractal dimensions of individual mitochondrial oscillators: first, averaged mitochondrial fractal dimensions in clusters of mitochondria with similar oscillation frequencies are lower than in non-cluster mitochondria; second, fractal dimensions decrease exponentially as local correlation between neighboring mitochondrial oscillators increases; third, fractal dimensions decrease linearly with increasing local coherence and with increasing functional connectedness between neighboring mitochondrial oscillators; and fourth, increasing local coupling between mitochondrial oscillators based on a stochastic coupling model is associated with decreasing fractal dimensions, although a plateau is reached for coupling constants >0.2. Although previous studies found a strong linear relation between functional connectedness and local coherence within a main cluster of similarly oscillating mitochondria, the present analysis also includes non-cluster mitochondria and considers local inter-mitochondrial coupling. Our work shows that, in those mitochondria that belong to a main cluster of mitochondria with similar frequencies spanning the whole myocyte, local conditions are more relevant for their dynamic behavior. This extends the potential relevance of this analysis to cells possessing non-coalescent clusters of oscillating mitochondria into a main cluster of similarly oscillating mitochondria, which may happen even transiently before such a cluster appears in cardiac myocytes exhibiting cell-wide synchronized oscillations.

Contrary to the concept of a time-series-based fractal dimension in TMRE signal traces, mitochondrial networks in cardiac myocytes have already been analyzed using a structural image-based fractal dimension of microscopy images of mitochondrial clusters that were labeled with the ROS-sensitive fluorescent probe 5-(and -6)-chloromethyl-2′,7′-dichlorohydrofluorescein diacetate (CM-H2DCFDA) in its oxidized form (see (6,35) and references therein). It was found that these fractal dimensions were close to 1.89, which is equivalent to a percolation threshold of approximately 0.6, which is valid for square lattices (6,7). A percolation threshold of a structural entity consisting of an ensemble of sites or nodes characterizes a probability threshold of occupancy for these sites, or, in our case, occupancy refers to a depolarizing mitochondrion inner membrane potential. If the average probability of a mitochondrion within the myocyte to be depolarizing is close to the percolation threshold, the mitochondrial network is at a transition point with faster and amplified enzyme reactions that may lead to a depolarization of the whole mitochondrial network. This transition is indicative of a network that spans several spatial scales, from molecular via subcellular to cellular scales. However, in our work, we focused on individual mitochondrial oscillators and their changing characteristics. Although an analysis of the structural fractal dimension of the main frequency cluster may be of potential interest, it should be noted that such clusters are dynamically changing (see also Fig. S1), and, therefore, the gain from such an analysis is of limited value.

We have demonstrated that time series of some individual mitochondrial oscillators, those that are prone to increased local inter-mitochondrial coupling, possess fractal dimensions that are indicative of a self-similar fractal process, which is distinctly different from random behavior. It was shown in earlier studies that coupled oscillators in biological systems are linked to truncated fractals (36), i.e., as in the case of mitochondrial oscillators, and the self-similarity extends at most over a few orders of magnitude. As the authors in (36) argue, this “hidden fractal property” in the network of coupled oscillators “acts as a simple structure” that limits system predictability. Furthermore, there are analytical and numerical descriptions of the classical evolution of fractal measures on a lattice-like system of non-linearly coupled oscillators (37) and for the fractal dimension in the generalized synchronization of smooth and non-smooth identical oscillators with a fixed coupling strength (38), as well as the coupling-strength-dependent appearance of chimera states in more complex coupling schemes of coupled oscillators (39). However, considering that coupling within the mitochondrial network is non-linear as well as dynamically changing, an adequate analytical description of such a system, if feasible, is immensely complex and would go far beyond this study. The relation between individual fractal dimensions and local clustering, however, has been numerically studied in complex networks with chaotic dynamics, producing a similar relation (see, e.g., Fig. 6 c in (40)), thus suggesting that mitochondrial oscillators may behave as chaotic attractors, as hypothesized earlier (41).

Although it has been observed that a decrease in oscillation frequency is associated with an increase in main cluster size (Fig. 2 in (7) or Fig. 4 in (13)), the cluster area does not necessarily increase over time since mitochondria may only be temporarily recruited into the main cluster (Fig. S1). In fact, it was shown that coupling constants in a cluster decrease over time for glucose-perfused cardiac myocytes; see Fig. 5 in (13). Strong local coupling should therefore not be considered as a hallmark for cluster affiliation but rather as part of a recruiting process for mitochondria that are adjacent to cluster mitochondria. This coincides with the notion that main cluster mitochondria do not necessarily form a contiguous area but rather a percolating, spanning cluster across the cardiac myocyte. Although it was shown that cluster area and functional clustering within the cluster, as well as cluster coherence, are strongly linearly related (Fig. 4 in (16)), this is not the case for inter-mitochondrial coupling and local coherence, see, e.g., Fig. 6 in (13), which shows a non-linear relation between coupling constants and coherence values. Also, it is not necessarily the case for mitochondria outside the cluster area. In other words, current evidence supports the conclusion that larger main clusters are also more functionally connected and coherent, but they do not form stronger oscillations. This would be the case if their oscillations were perfectly in phase, which they are not, and if their amplitudes remained constant, which they do not. In fact, it was shown that TMRE signal amplitudes from all cluster mitochondria decrease with cluster size, potentially indicating that the myocyte slowly exhausts its oxidative defense mechanisms on its way toward death; see Fig. 4 b in (7).

One should also note that denoising, based on an empirical Bayesian method with a Cauchy prior (using Matlab’s wdenoise function), on the normalized mitochondrial signals would result in an average decrease in fractal dimensions across all mitochondria in a myocyte, of −0.18 ± 0.06. This effect is small unless there is a bias relative to coherence. However, denoising may also have eliminated relevant higher frequency fluctuations not attributed to noise, as discussed above.

A potential predictive use of the fractal dimension is illustrated in Fig. 4, where we compare computed and estimated maps of local coupling constants and local coherence for a cardiac myocyte that was not included in the previous analysis. The estimations for coupling constants $k\left(D\right)$ in dependence on fractal dimension D are thereby based on the empirical relation $k\left(D\right)=\frac{D-b}{a}\text{,}$ with $a=-1.46s$ and $b=1.66\text{.}$ Likewise, estimations for local coherence $\overline{c}\left(D\right)$ are based on $\overline{c}\left(D\right)=\frac{D-b}{a}\text{,}$ with $a=-0.55$ and $b=1.95\text{.}$ One can see that areas of weak coupling agree better with the estimations (Fig. 4 a and b), as one would have assumed from Fig. 3 c, where the linear relation between fractal dimension and local coupling is strongest for low coupling constants. For local coherence (Fig. 4 c and d), the estimations tend to overestimate larger local coherence, and underestimate lower local coherence, suggesting that the empirically determined slope in Fig. 3 b is too steep for the new myocyte. However, areas of gradients in local coherence, e.g., on the right-hand-side of the myocyte, are identifiable. The mean squared error (MSE) between estimated maps and computed maps is low with an MSE of 0.0098 between the coupling maps (Fig. 4 a and b) and an MSE of 0.0652 between the coherence maps (Fig. 4 c and d).

Figure 4.

Computed and estimated maps for local inter-mitochondrial coupling and local coherence. (a) Local inter-mitochondrial coupling based on a stochastic phase model (SPM) approach (13), and (b) based on the linear empirical estimation from Fig. 3c. (c) Local coherence based on magnitude-squared coherences (MSCs) of mitochondria with all its nearest neighbors, and (d) estimated local coherence based on the empirical relation from Fig. 3b. The utilized cardiac myocyte was not included in the statistical analysis. Both estimated parameter maps tend to overestimate larger parameter values, and rather underestimate lower parameter values, suggesting a less steep slope for the empirical relations for this myocyte. However, the MSE between computed and estimated maps is low, with 0.0098 for the coupling maps and 0.0652 for the coherence maps. To see this figure in color, go online.

The presence of self-similar dynamics in mitochondria that belong to a frequency cluster, as opposed to the remaining mitochondria that exhibit a fractal dimension that is similar to brown noise, indicates that fractal behavior in the mitochondrial network is mainly driven by mitochondrial clusters, and therefore dependent on cluster properties (7,13). Large clusters show a high degree of local, nearest-neighbor correlation and coupling, presumably to maintain chemical, diffusive, or structural coupling mechanisms over longer distances (13); thus, fractal dynamics in mitochondrial $\Delta {\Psi }_m$ oscillations is associated with larger mitochondrial clusters. This is not surprising since fractal dynamics were already observed for the collective $\Delta {\Psi }_m$ oscillations of cardiac myocytes (25). Naturally, large local coupling constants are associated with large contiguous clusters of simultaneously oscillating mitochondria that involve most or at least about 60% of mitochondrial oscillators within the cardiac myocyte (6). This may be obvious when clusters are contiguous; however, this is not necessarily the case when simultaneously oscillating mitochondria form a spatially distributed cluster that, for each cluster mitochondrion, possesses a low average number of nearest neighbors that belong to the cluster; see, e.g., (13). This is likely due to depleted antioxidant defenses in the coupled mitochondria that strengthen coupling of ROS-induced ROS release, an effect linked to H₂O₂ diffusion across the myocyte (42) and consistent with the observation that ROS-sensitive fluorescent probe 5-(-6)-chloromethyl-2′,7′-dichlorofluorescein is oxidized in the cluster of oscillating mitochondria that span the myocyte (6). Oscillating mitochondria in such large clusters therefore share the same state of oxidative stress, which, according to bifurcation analysis, corresponds to a limit cycle attractor (10).

Increasingly large clusters are also associated with increasingly coherent cluster oscillations, whereby coherence measures the similarity of mitochondrial $\Delta {\Psi }_m$ oscillations. However, mitochondria with large coherence values only showed fractal dimensions close to ∼1.4 with a substantial standard deviation (see Fig. 3 b), indicating that temporal coherence of mitochondrial oscillations is only weakly associated with emergent fractal properties, although fractal dimension decreases for increasing coherence. This could indicate that smaller clusters of mitochondrial oscillators with similar frequencies, which possess a large temporal coherence, are not necessarily as self-similar as larger clusters with strong local coupling and coherence. Small clusters hereby refer to clusters that typically involve less than 50% of the myocytes’ mitochondria. This finding would concur with the observed fractal dimensions for large clustering coefficients, which are in average close to brown noise; see Fig. 3 d. Low fractal dimensions would therefore be more indicative of local, spatial coupling mechanisms as opposed to functional coupling within the mitochondrial network; see also (11). When we consider the empirically derived relation between correlation coefficient, c, and fractal dimension, D, with $D\left(c\right)=a\exp \left(-bc\right)+dc+e\text{,}$ where $a=-8.52·{10}^{-11}\pm -3.51·{10}^{-11},b=-21.65\pm 5.23,d=-0.25\pm 0.02,e=1.73\pm 0.01\text{,}$ we can solve for the local correlation coefficient: $c=\frac{\mathrm{W}\left(-\frac{ab\exp \left(\frac{b\left[e-D\right]}{d}\right)}{d}\right)}{b}+\frac{D-e}{d}$ with $W\left(x\right)$ being the product log function. Since constant a is very small compared with the other constants and values of c and D, and $W\left(x\to 0\right)=0\text{,}$ one can linearly approximate: $c\approx \frac{D-e}{d}\text{.}$ This last approximation is valid for $D>1.5\text{.}$ Using the constants d and e, we find $c\approx -4\left[D-1.73\right]\text{.}$ We derived this relation for cardiac myocytes that were laser flashed to induce large-amplitude mitochondrial inner membrane potential oscillations. This pathological state of the cardiac myocyte initiates collective mitochondrial behavior that ultimately results in myocyte death and is therefore termed the pathological regime of mitochondrial oscillations. In contrast, the physiological regime of non-laser-flashed mitochondrial oscillators, featuring high-frequency and low-amplitude mitochondrial inner membrane potential oscillations in the absence of metabolic of oxidative stress, was described in (25). In this regime, mitochondrial oscillators typically oscillate at higher frequencies (>0.3 Hz) with very small amplitudes, whereas, in this work, the sampling rate of the images was lower so that only oscillations <0.1 Hz could be observed. The above relationship within the pathological regime of mitochondrial oscillations may therefore extend the relationship derived for the physiological regime of fast frequency oscillations, when one assumes that the local correlation coefficient corresponds to the spatial correlation coefficient used in van Beek et al., where $c=2^{3-2D}-1$ (25,43)$\text{.}$

In the pathological regime of mitochondrial oscillations with low frequencies during which the main cluster of mitochondria with similar frequencies spans the whole myocyte, the fractal dimension of the whole myocyte is mostly determined by this cluster, whereas non-oscillating or differently oscillating mitochondria possess fractal dimensions that are similar to random processes. Such fractal scaling is also observed in time series of human heartbeats, as well as for cardiac arrhythmias (44,45). Although there is a link between mitochondrial energetics and the myocyte action potential morphology (46), it remains unclear whether such fractal behavior mirrors the fractal properties within the mitochondrial network, determined by its complex intracellular control mechanisms (2,47). The pathophysiological mechanistic underpinnings of the sarcolemmal membrane potential modulation by mitochondrial oscillations (48) could explain the influence of mitochondrial depolarizations on cardiac arrhythmias (49,50). However, these studies have not explored the influence of mitochondrial oscillatory activity or chaotic dynamics (41).

In summary, the findings of this study demonstrate that, during stress-induced myocyte-wide $\Delta {\Psi }_m$ oscillations, individual mitochondrial oscillators show self-similar dynamics that are correlated with local coupling mechanisms but are only weakly linked to functional properties of the mitochondrial network. The findings suggest that fractal dimensions of individual mitochondrial oscillators may serve as a simple measure of local mitochondrial coupling.

Author contributions

F.T.K. and A.A.A. designed research. F.T.K., M.A.A., and B.O’R. performed research. F.T.K. contributed analytic tools. F.T.K. and J.M.E.J. analyzed data. F.T.K., M.A.A., H.S., J.M.E.J., B.O’R., and A.A.A. wrote the article.

Editor: Arthur Sherman.