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WIREs Syst Biol Med
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Network dynamics: quantitative analysis of complex behavior in metabolism, organelles, and cells, from experiments to models and back

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Advancing from two core traits of biological systems: multilevel network organization and nonlinearity, we review a host of novel and readily available techniques to explore and analyze their complex dynamic behavior within the framework of experimental–computational synergy. In the context of concrete biological examples, analytical methods such as wavelet, power spectra, and metabolomics–fluxomics analyses, are presented, discussed, and their strengths and limitations highlighted. Further shown is how time series from stationary and nonstationary biological variables and signals, such as membrane potential, high‐throughput metabolomics, O2 and CO2 levels, bird locomotion, at the molecular, (sub)cellular, tissue, and whole organ and animal levels, can reveal important information on the properties of the underlying biological networks. Systems biology‐inspired computational methods start to pave the way for addressing the integrated functional dynamics of metabolic, organelle and organ networks. As our capacity to unravel the control and regulatory properties of these networks and their dynamics under normal or pathological conditions broadens, so is our ability to address endogenous rhythms and clocks to improve health‐span in human aging, and to manage complex metabolic disorders, neurodegeneration, and cancer. WIREs Syst Biol Med 2017, 9:e1352. doi: 10.1002/wsbm.1352 This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Metabolomics Physiology > Physiology of Model Organisms
Isolated adult ventricular cardiac myocyte from guinea pig with densely packed mitochondrial network lattice. (a) The inner mitochondrial membrane potential (ΔΨm) was monitored with the TMRE fluorescent probe. Single mitochondrial signals can be extracted as detailed in Refs ; (b–d) TMRE intensity signal (in arbitrary units) from three distinct mitochondria at different locations in the mitochondrial lattice. Mitochondria 1 and 3 show ΔΨm oscillations with different frequencies while mitochondrion 2 is not oscillating. (e) Absolute squared wavelet transform over frequency and time of mitochondrion 3. The major frequency component (dark red) varies between 15 and 20 mHz corresponding to the wavelet frequency with maximum power of the absolute squared wavelet transform at each time‐point. (Reprinted with permission from Ref . Copyright 2014 Frontiers Open Access Academic Publisher)
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From metabolomics to fluxomics. (a) Work flow diagrams leading from metabolite profile to the fluxome and the analysis of its control and regulation. (b) The table shows the metabolites concentrations determined experimentally compared with model simulations. (Reprinted with permission from Ref . Copyright 2015)
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Integrated quantitative analytical approach of a computational model for translating metabolite profiles into metabolic fluxes. Depicted are the quantitative methodologies that integrate the platform utilized in the interactive and iterative analytical work with the computational model. The method described in Ref utilizes the experimentally obtained metabolome as initial input, in terms of actual metabolite concentrations, to calculate the fluxome or set of metabolic fluxes from which the initial metabolite profile emerged (Reprinted with permission from Ref 13. Copyright 2015).
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Chaotic behavior in a self‐organized continuous culture of S. cerevisiae. (a) Relative membrane‐inlet mass spectrometry signals of the m/z = 32 and 44 corresponding to O2, and (b) to CO2 components versus time in hours after the start of fermentor continuous operation, elsewhere described. The large‐amplitude oscillation showed substantial cycle‐to‐cycle variability ranging from 11.7 to 15.5 h, giving a mean of 13.6 ± 1.3 h (SD, n = 8). The biological bases for all three oscillatory outputs of the yeast culture has been confirmed by exclusion of the possible influences of variations of aeration or stirring, pulsed medium addition, cycles of NaOH addition and pH variation, or cycles of temperature control. (c) One period of the oscillation is shown in panels (a) and (b); the inset shows the individual data points for m/z 32 (O2) for a 30‐min span starting at 730 h. (d) Shown are the values of dissolved oxygen, carbon dioxide, and hydrogen sulfide, through which this experimental system passes during the complex oscillations, depicting the strange attractor signature of chaotic behavior. The relative O2 and CO2 signals (m/z 32 and 44) are plotted on the axes, whereas the relative H2S (m/z 34) signal is mapped onto the color scale. Circulation around the attractor is in the clockwise direction.
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Chaotic dynamics of succinate in a two‐compartment mitochondrial model. (a) Chaotic behavior of succinate: model description and parameterization are described in Ref . This dynamic behavior was obtained with 21.6726976 μM MnSOD, 0.085 μM CuZnSOD, 4% Shunt, and oscillating extracellular H2O2 at amplitudes and periods range of ~1 × 10−4 μM and ~30 seconds, respectively. Number of data points 90,000 with a sampling rate of 0.1 seconds. (b) The average mutual information for the time series of succinate in panel (a); the first minimum of this function is at t = 35 seconds (black arrow), thus selected as the time lag used for phase space reconstruction. (c) The percentage of false nearest neighbors (FNN) for the same time series using t = 35 seconds. The percentage of global false nearest neighbors drops to zero at an embedding dimension of 5, indicating that phase space reconstruction can be accomplished [x(t), x(t + T), x(t +2T), x(t + 3T), x(t + 4T)]. Since visual representation only permits 3D plots, the succinate phase portrait is represented in panel (d) using only the first three dimensions and thus is not completely unfolded. (e) The correlation dimension (d = 2.2) was estimated as the slope of the double logarithmic plot of the correlation integral, C versus the radius, ε. (f) 3D phase space portrait as a function of the state variables membrane potential, extra‐mitochondrial superoxide and intramitochondrial hydrogen peroxide. Notice the complexity of the attractor.
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Mitochondrial oscillations in cardiomyocytes. (a) Sequence of changes in ΔΨm, inner membrane anion channel (IMAC) flux, Cu,ZnSOD activity, and cytosolic O2∙− during an oscillatory cycle observed in a computational model of mitochondrial energetics. At a critical level of ROS (O2∙−) accumulation in the mitochondrial matrix, the IMAC channel rapidly opens, denoted by an spike in outward current, provoking the sudden release of O2∙− from the mitochondria into the intermembrane space. The current through IMAC quickly declines due to ΔΨm loss. The rate of extra‐mitochondrial SOD increases in parallel with the burst of available O2∙− and stays high until O2∙− is consumed, at which point IMAC closes, allowing ΔΨm to repolarize and initiate the next cycle. (b) Ordinary differential equations of the mitochondrial oscillator as described in Ref . This model has been expanded to include mitochondrial and extramitochondrial compartments, and all main redox couples and antioxidant systems. ADP, adenosine diphosphate; ΔΨm, inner mitochondrial membrane potential; NADH, reduced nicotinamide adenine dinucleotide; aKG, alpha ketoglutarate; SCoA, succinyl CoA; Suc, succinate; FUM, fumarate; MAL, malate; OAA, oxaloacetate; ASP, aspartate; Ca2+, calcium; O2.‐, superoxide anion (m, mitochondrial; i, cytoplasmic); H2O2, hydrogen peroxide; CAT, catalase; SOD, superoxide dismutase. (Reprinted with permission from Ref . Copyright 2003; Ref . Copyright 2004; and Ref . Copyright 2014)
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Time series, phase space, phase portrait, and bifurcation diagram. In order to illustrate some concepts and analytic tools introduced in Emergent Self‐Organization: A Hallmark of the Dynamic Behavior of Complex Networks section, we present the simulation results obtained with an integrated model of mitochondrial energetics, calcium dynamics, ROS production, and ROS scavenging. This model (either isolated or integrated to other cellular processes) has been extensively validated against reported experimental evidence and was able to make predictions that were later confirmed under different experimental conditions. (a) Shown are the evolution of NADH and mitochondrial superoxide, O2∙−m, toward steady (i.e., a fixed‐point attractor) or (b, c) oscillatory (i.e., limit‐cycle) states, respectively. (d) Depicts the phase space plot of NADH and O2∙−m for the steady (dashed) and oscillatory (continuous) solutions. The change from a fixed‐point attractor to limit‐cycle behavior was achieved simply by increasing the concentration of respiratory chain carriers, while keeping constant all other parameters. (e) Phase portrait plot showing the relationship between mitochondrial membrane potential, ΔΨm, and the superoxide anion released to the periplasmic mitochondrial space, O2∙−c, during the depolarization phase of the oscillation. Plotted are the trajectories followed by several limit cycles corresponding to oscillatory periods ranging from 70 to 200 ms after sequentially changing only one parameter, i.e., the rate of ROS scavenging (see also panel f). (f) Bifurcation diagram of the state variable NADH as a function of ROS generation (fractional ROS production) and scavenging (superoxide dismutase, SOD, concentration). The stability and type of steady states exhibited by the model were computed by continuously varying the parameters indicated. The results obtained were represented in a bifurcation diagram, which in this case consisted of an upper branch, in which NADH was predominantly reduced, and a lower branch, in which NADH was mainly oxidized. Thick lines indicate domains of stable steady‐state behavior whereas thin lines denote either unstable or oscillatory states. A stable oscillatory domain, embedded within the upper branch, emerged as SOD concentration increased. The eigenvalues obtained from the stability analysis can be further analyzed to obtain a detailed description of the transitions at the borders of the steady states (marked by arrowheads and numbers). 1 and 2 in the upper branch indicate Hopf bifurcations delimiting the oscillatory region (thin line), and were characterized by eigenvalues with two pairs of complex conjugates, one pair showing positive real part. In the stable regions of the diagram (thick lines), all real negative eigenvalues were found, and one or two pairs of complex conjugates, but with negative real parts; 3 and 4 denote limit points. (Reprinted with permission from Ref . Copyright 2011 Wiley‐VCH Verlag GmbH & Co)
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Dynamic coupling behavior in mitochondrial networks. (a) The organization and synchronization of the mitochondrial network can be described with the help of a stochastic phase model that attributes local time‐varying coupling constants K to each mitochondrial oscillator. Local coupling of mitochondrion m to its nearest neighbors ni is schematically depicted in a mean‐field form (violet cloud). Local coupling between mitochondria is attributed to diffusive coupling agents that interact with the inner mitochondrial membrane. (b) Distribution of local coupling constants in a cardiac cell (Color bar: yellow/red: strong coupling, green/blue: weak coupling) at a specific time point, and (c) for a sequence of time‐points (from left to right with time intervals of 17.5 seconds). Loss of coupling in clustered mitochondria is associated with overwhelmed and exhausted antioxidant systems at the onset of cell death.
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Scale‐invariance and long‐range correlations in locomotor activity. (a) Cumulative locomotion activity of a quail in a home‐cage environment. Locomotion was monitored at 0.5‐second interval (xi), if the bird was ambulating, xi = 1, and if immobile xi = 0. (b) DFA of order 3 (DFA3) of the time series shown in panel (a). The double log plot of fluctuation versus window size is linear (blue line) in the region shown with blue closed circles (●) with slope ‘α’, a metric of self‐similarity. Notice that for larger scales there is no evidence of loss of fractal behavior but the α estimation in that region is unreliable. (Reprinted with permission from Ref . Copyright 2016)
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Wavelet correlation analysis between mitochondrial membrane potential and succinate concentration. Analyzed was the oscillatory dynamics obtained with the mitochondrial computational model described in Ref under the same parametric conditions described in the legend of Figure . (a) Comparison between the time series of mitochondrial membrane potential and succinate concentration. (b) Spearman correlation plot between the wavelet coefficients estimated for each time series, blue indicating the time scales where anticorrelation is observed between both time series while the red values denote a high level of correlation between them. (c) Best fit curve of the diagonal of Spearman correlation matrix represented in panel (b). Note that for small time scales significant positive correlations (>0) between the wavelet coefficients exist (orange arrow) while for larger time scales (green arrow) coefficients are anti‐correlated (<0). (d) Wavelet coefficients at the maximum (orange diamond in (c)), and (e) minimum (green diamond in panel (c)) in the scale correlation matrix were plotted as a function of time as a result of the analysis of both time series. Notice in panel (d) the high level of correlation between the wavelet coefficients time series at the small scales whereas in panel (e) a complete anticorrelation is observed. The real Morlet wavelet was used for computing the continuous wavelet transform used for correlation analysis.
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Wavelet analyses of oscillatory time series. (a, b) The same time series represented in panels (a) and (b) from Figure were subjected to wavelet analysis. Notice the qualitative similarity of the results obtained with respect to phase angle (c, d) or modulus (e, f) in the analysis of the original time series (c, e) with respect to the series with linear trend (d, f). Wavelet analysis was performed using the complex Morlet transform in MATLAB (cmor1‐1.5) on the time series binned at 1 s intervals.
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Oscillatory dynamics in a two‐compartment mitochondrial energetic‐redox computational model. (a) Oscillatory behavior of extra‐mitochondrial hydrogen peroxide (H2O2)i. The computational model description and parameterization are described in Ref (see also Figure ). The oscillations were obtained with 0.02167268011 μM MnSOD, 9.7 μM CuZnSOD, and 4% Shunt. (b) A linear trend was added to the H2O2 signal represented in panel (a). (c, d) Depicted is the power spectral analysis of the time series from panels (a) and (b), respectively. The italic blue numbers represent the oscillatory period. Notice that in (d), a strong deviation from the expected values is observed at lower frequencies likely caused by the nonstationarity in the time series. Moreover, the peak corresponding to the main oscillator at 455 seconds is not apparent. (e, f) Displayed is the autocorrelation analysis of the time series represented in panels (a) and (b), respectively. Notice that this method is practically unable to detect the oscillations in the time series. All analyses were performed with time series of 1 × 104 seconds at a constant sampling interval of 1 ms.
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