Dynamics of Mitochondrial Redox and Energy Networks: Insights from an Experimental–Computational Synergy - Systems Biology of Metabolic and Signaling Networks - page 122

Box 5.1 (continued)
Another signature feature of percolation processes at pc is that they are
organized as fractals. This property implies that local processes can scale to
produce macroscopic behavior. At pc , the mass of the spanning cluster
increases with the size of the lattice, L , as a power law, L Df , with D f as the
fractal dimension (Feder 1988 ; Mandelbrot 1977 ; Stauffer and Aharony
1994 ). Fractal box counting analysis of our data yielded a fractal dimension
of D f ~ 1.82, close to that exhibited by percolation clusters and cytoskeletal
lattices at p c ( D f ~ 1.90) (Aon and Cortassa 1994 , 1997 ; Aon et al. 2003 ;
Feder 1988 ; Stauffer and Aharony 1994 ).
Several interesting properties of the mitochondrial response can be
explained when the network is considered as a percolation cluster. First, the
question of the limited diffusivity and lifetime of O 2 . as the triggering
molecule is answered, since the only relevant diffusion distance is the inter-
mitochondrial spacing (~1 m). As long as there are enough neighboring

mitochondria belonging to the spanning cluster (i.e., they have accumulated
enough O 2 . to approach the percolation threshold) an universal phase tran-
sition will occur (Feder 1988 ; Schroeder 1991 ; Stauffer and Aharony 1994 )
and mitochondria will depolarize for the first time throughout the cell (Aon
et al. 2003 , 2004a ).

μ

5.4.2 Modeling Mitochondrial Network Redox Energetics

The mitochondrial oscillator model utilized in the first iterative loop corresponds to
an isolated mitochondrion representing the average behavior of the mitochondrial
population, actually (in the case of the cardiomyocyte or cardiac cell) organized as a
network. Therefore, accounting for the spatial relationships between individual
mitochondria within the network became crucial to simulating the initial depolari-
zation wave that signals the energetic collapse of the mitochondrial network. The
importance of this seminal event in the escalation of failures, from mitochondria
propagating to cells and groups of them, and finally attaining the whole heart, made
it worthwhile (and in fact crucial) to unravel the fundamental mechanisms
involved. More specifically, we wanted to explore whether a reaction-diffusion
mechanism could be responsible of the spreading of failure of individual organelles
to the whole cell. According to our model, wave propagation comprises the
nonlinear dependence of an IMAC opening on O 2 . accumulation in the matrix,
and the free radical autocatalytic release and spreading in the network; we
hypothesized that mechanistically this could be enough to reproduce the wave
phenomenon and the underlying ROS-induced ROS release (RIRR).
In order to achieve this goal we developed a mathematical model of RIRR based
on reaction-diffusion (RD-RIRR) in one- and two-dimensional mitochondrial