📒 Nivala 2010

Linking flickering to waves and whole-cell oscillations in a mitochondrial network model1



  • In contrast to the cell-wide waves, spontaneous transient depolarizations of individual mitochondria, called flickers, have been widely observed, and tend to occur randomly with variable depolarization durations

  • In previous studies Aon et al. coined the term mitochondrial criticality, in which the mitochondrial network becomes very sensitive to small perturbations of ROS at the critical state


Single-mitochondrion model

  • (A) The single-mitochondrion model

  • (B) Four-state Markov model of the IMAC: closed (C), open (O), inactivated (I), and refractory (R)

  • (C) The network model

  • (D) Representation of a single mitochondrion as a 3 × 3 grid

  • (E) Two-state Markov model of shunt parameter.

  • Including matrix, intermembrane space, and cytosol

  • Ordinary differential equations (ODEs) are used to describe the membrane potential (Ψ) across the inner membrane and the superoxide concentrations

  • superoxide released into the matrix and intermembrane spaces at 85% and 15%, respectively

  • Superoxide diffuses freely across the outer membrane at the rate VSO,IC. However, superoxide cannot diffuse across the inner membrane except through the opening of IMACs

Superoxide production and diffusion rate

  • the Nerst potential component has been replaced with the more general formulation of the Goldman-Hodgkin-Katz equation

$$ \begin{aligned} V_{sox} &= shunt * 0.85 * (1 - 0.4 \cdot Hill(SO_m, 0.45, 2)) * 6 * Hill(\Delta\Psi, 250, 5) \cr V_{IMAC} &= 800 \frac{N_O}{N_{IMAC}} * (0.0782 + \frac{7.82}{1 + e^{0.07 (4 + \Delta\Psi)}}) * \Delta\Psi \cr V_{ROS}^{tr} &= 0.5 * V_{IMAC} * (SO_{i} * \frac{SO_{m} - e^{F \Psi / RT}}{1 - e^{F \Psi / RT}}) \cr V_{S O, I C}&=k_{S O I, C}\left(4 S O_{I}-\sum_{j=1}^{4} S O_{C, j}\right) \end{aligned} $$

  • we modeled the rate r->c to have a second-order dependence on the concentration of superoxide in the matrix: when SOM is high, the IMAC is more likely to be in state C

Markov model of IMAC dynamics

Units in Hz. α=0.4 and β=10,000 (1/ mM^2 s) $$ \begin{array}{l}{k_{R \rightarrow C}=k_{I \rightarrow O}=\alpha^{*} S O_{M}^{2}} \cr {k_{C \rightarrow O}=k_{R \rightarrow I}=\beta * S O_{I}^{2}} \cr {k_{O \rightarrow I}=k_{C \rightarrow R}=0.1} \cr {k_{I \rightarrow R}=k_{O \rightarrow C}=100.0} \end{array} $$

$$ \begin{array}{l}{\frac{d p_{C}}{d t}=-\left(k_{C \rightarrow O}+k_{C \rightarrow R}\right) * p_{C}+k_{R \rightarrow C} * p_{R}+k_{O \rightarrow C} * p_{O}} \cr {\frac{d p_{O}}{d t}=-\left(k_{O \rightarrow I}+k_{O \rightarrow C}\right) * p_{O}+k_{C \rightarrow O} * p_{C}+k_{I \rightarrow O} * p_{I}} \cr {\frac{d p_{I}}{d t}=-\left(k_{I \rightarrow O}+k_{I \rightarrow C}\right) * p_{I}+k_{O \rightarrow I} * p_{O}+k_{R \rightarrow I} * p_{R}} \cr {\frac{d p_{R}}{d t}=-\left(k_{R \rightarrow I}+k_{R \rightarrow C}\right) * p_{R}+k_{I \rightarrow R} * p_{I}+k_{C \rightarrow R} * p_{C}} \end{array} $$

$$ p_{C}+p_{O}+p_{I}+p_{R}=1 $$

  • It should be noted that the Markov model of the IMAC and the reaction rates are phenomenological, as quantitative information is limited.


$g_{IMM}$$19.2$$V/s$Conductance of IMM
$V_{P, \Psi}$$3.5$$V/s$Charge rate of ETC
$k_{SOD,m}$$0.08$$mM/s$Rate constant of SOD, mitochondrial
$k_{SOD,i}$$0.006$$mM/s$Rate constant of SOD, intermembrane space
$k_{SOD,c}$$0.006$$mM/s$Rate constant of SOD, cytosolic
$K_{m,SOD}$$0.002$$mM$Michaelis constant for superoxide of SOD
$V_{P, \Psi}$$3.5$$V/s$Charge rate of ETC
$G_L$$0.0782$$mM/V$Leak Conductance of IMAC
$G_{max}$$7.82$$mM/V$Integral conductance of IMAC
$\alpha$$0.4$$Hz \cdot mM^{-2}$Transition rate constant of IMAC
$\beta$$10000$$Hz \cdot mM^{-2}$Transition rate constant of IMAC
$k_{SO, IC}$$5$$Hz$Diffusion rate of superoxide between intermembrane space and cytosol
$D_{SO}$$1$$\mu m^2 / s$Diffusion constant of superoxide in the cytosol

2D mitochondrial network model

  • 100 × 20 grid of mitochondria
  • Because each individual mitochondrion has the dimensions 0.9 μm × 0.9 μm, the spatial scale of the network model is 90 μm × 18 μm, which are approximately the dimensions of an average myocyte
  • mitochondrial superoxide production displays a bistable behavio: production switches between high and low states
  • matrix superoxide buildup is required for IMACs to open to result in membrane potential oscillations
  • transient depolarizations tend to occur randomly in space and time

Numerical methods

  • ODE & PDE: FE method, Δt = 0.0002s
  • Markov model: Gillespie’s algorithm
  • The IMAC channel dynamics occur on a millisecond timescale
  • Model is written in CUDA


Dynamics of a single IMAC

  • the channel exhibits a bursting behavior: remains closed for a long period and then enters into a mode of flipping quickly between closing and opening
  • The 108-pS channel in mitoplasts (a candidate for IMAC) showed similar behavior: bursting type, time scale in ms

Depolarization dynamics of a single mitochondrion

  • During low shunt conditions: both SOM and SOI very low => No RIRR
  • Under high shunt conditions, ROS accumulate in matrix => higher steady state of superoxide in the intermembrane space => positive-feedback effect of RIRR
  • the transient depolarizations of a single mitochondrion are not periodic, but more or less random

  • The onset of RIRR is extremely fast and accounts for the steepness during the initial depolarization

  • interspike intervals (ISIs)
  • When NIMAC is increased to 100 (same total conductance), the average ISI increases and the distribution is slightly broadened
  • an increase in the number of channels increases the oscillation frequency and narrows the distribution (more periodic)
  • Increasing shunt increases the oscillation frequency and narrows the distribution

Spatiotemporal dynamics of the mitochondrial network

  • p = 0.25, the distribution was close to an exponential distribution (y = 8951e^−0.6256x)
  • for p = 0.4, the distribution exhibited a stronger power-law (y = 12847e^−0.0914x x−1.5713)
  • for p = 0.6, this distribution was close to a pure power-law distribution (y = 4585x^−1.6455)
  • The power-law distribution indicates that the system can exhibit all size scales, a property of critical phenomena in statistical physics
  • the single mitochondrion flickering is more random and less frequent than the depolarization of the network
  • coupling between mitochondria through RIRR causes synchronized mitochondrial depolarizations, and thus periodicity in the average membrane potential
  • ACF = autocorrelation function, the correlation between a signal at time t and a later time (t + lag time)

The conduction velocity of mitochondrial depolarization waves

  • Aon : 22 μm/s (IMAC)

  • Brady: <0.1 μm/s (mPTP)

  • This paper: both ΨM oscillations and slow waves, and wave velocities ranged from 0.1 to 2.2 μm/s

  • enhancing superoxide dismutase activity decreases wave velocity

  • Stronger coupling causes an earlier transition from flickering to waves and oscillations.


  • a Markov model of the IMAC in a single mitochondrion and a mitochondrial network to simulate mitochondrial flickering
  • The periodicity of the flickering depends on the total number of IMACs in the mitochondrial membrane and the superoxide production rate (coupling strength)
  • The mitochondrial criticality is reached via the dynamics of self-organized criticality


  • mPTP opening triggered by sarcoplasmic reticulum (SR) calcium release or ROS release
  • transient depolarizations or flickering may be caused by different mechanisms under different conditions (single blocker ineffective)
  • Our IMAC model was partially based on the experimental observations of Borecky et al (108-pS channel)
  • This model depends strongly on the random behavior of the channels, the depolarizations occur nonperiodically, simulating the flickering behavior. Deterministic ODEs cannot attain this.
  • the spatiotemporal dynamics in our model is similar to the dynamics of calcium (Ca) signaling in experiments and computer modeling: Ca signaling hierarchy of quarks, sparks, macrosparks (or spark clusters), abortive (short lived) waves, and persistent waves are also governed by the dynamics of self-organized criticality
  • oscillations are not governed by clocks or pacemakers but are stochastic in nature

Model critique

A simple mitochondrial model only includes the RIRR of the IMAC and ROS production and dismutation

  1. Nivala M, Korge P, Nivala M, Weiss JN, Qu Z. Linking flickering to waves and whole-cell oscillations in a mitochondrial network model. Biophys J. 2011;101(9):2102-11. PMC3207179 ↩︎