Dynamics of Mitochondrial Redox and Energy Networks: Insights from an Experimental–Computational Synergy - Systems Biology of Metabolic and Signaling Networks - page 133

Box 5.3 (continued)
statistically, the mitochondrial network rather than exhibiting a single “char-
acteristic” frequency shows multiple frequencies, typical of dynamic fractals .
Fractals possess a two-sided nature both as geometric (spatial) and as
dynamic objects. This trait enables techniques commonly applied for character-
ization and quantification of fractals in the spatial domain to be applied to
describe dynamic behavior. The different spatial and temporal scales displayed
by an object (be it a shape or a time series) can be quantified by lacunarity, a
mass-related statistical parameter quantified by the coefficient of variation or
relative dispersion, RD ( standard deviation (SD)/mean). RD is a strong

function of scale (Aon and Cortassa 2009 ) that in the case of self-similar time
series or “dynamic fractals” remains constant (i.e., the object looks the same at
all scales) (West 1999 ). The determination of RD at successively larger intervals
from a time series constitutes the basic mathematical procedure for applying
relative dispersional analysis (RDA). RDA of

¼

ΔΨ m time series from the mito-
chondrial network revealed long-term temporal correlations (“memory”).

Box 5.4: Simulation of the Inverse Power Law Behavior Exhibited by
the Mitochondrial Network of Cardiomyocytes
From the simulations, we selected five oscillatory periods in the high-
frequency domain (between 70 and 300 ms) and one from the
low-frequency (1 min period) domain and attributed each one of them
proportionally to a network composed of 500 mitochondria (i.e., every
100 mitochondria will oscillate with the same period). This number of
mitochondria is similar to that present in a single optical slice of a

cardiomyocyte (~1 m focal depth) that we analyze by two-photon laser

scanning microscopy with 110 ms/frame time resolution. Our experimental
results could be precisely simulated with a mixture of 80 % short-period and
20 % long-period oscillations.
According to this protocol, we then constructed a matrix: with
mitochondria in columns, and time, Ti, on rows. The final matrix contained
a total of 500 columns and 6,000 rows. The time steps represented by the rows
correspond to a fixed integration step of 20 ms for the numerical integration
of the system of ODEs. The fixed integration step of 20 ms was chosen for the
simulation of all periods within the range of 70-300 ms and 1 min in order to
avoid aliasing effects.

μ

Mito1 Mito2 Mito3 Mito n

...

T 0
T 1
T 2
...

T n