Contents

πŸ“’ Mathematical modeling of mitochondrial respiratory chain: a summary

Mathematical modeling of mitochondrial respiratory chain.

Electron transport chain (ETC)

  • Consumes hydrogen carriers (NADH, FADHβ‚‚) and oxygen
  • Generates NAD, FAD, water, and proton motive force (pumping) for ATP synthase the produce ATP
  • Reactive oxygen species (ROS) generation by complex I and III
  • Dependence of mitochodnrial membrane potential, pH, and substrate / product concentrations

Magnus and Keiser model

  • Islet beta cell mitochondrial model1
  • Also used in the ECME model2 and the Nguyen-2007 model3 for cardiomyocytes
  • 6-state proton pump using King-Altman method for steady-state rates (integrating complex I-III-IV)
  • ECME model have similar expression for complex II-III-IV rates

Demin model

  • Complete Q cycle of complex III with ROS side reactions45 with Semi-forward mechanism

https://i.imgur.com/xjPJgdW.png
Fig. 1: Q cycle

Gauthier model

  • Guinea pig heart mitochondria6
  • Complex II / III / IV from Demin’s model5
  • Complex I : 7-state steady-state flux similiar to the Magnus model1
  • Complex V: The same as the ECME one2, similiar to the Magnus model1
  • Problem of equilibrium constant in the original parameters

Guillaud model

  • Semi-reverse mechanism for complex III ROS generation7
  • Different set of parameters for Antimycin A absence / prescence

Basil and Beard model

  • Complex I8, complex III910, and integrated11 kinetic models for bovine heart mitochondria.
  • Heavy use of binding polynomials and Gibbs free energies (partly from midpoint potentials)
  • Redox and ROS generation fluxes in various conditions with supportive data

Berndt model

Neuron cell model12 stydying KGDH inhibition and ROS generation3.

Model description (pdf)

Hoek model

  • ROS generation model for complex I and III13
  • Detailed step by step electron transfer, law of mass expressions

Pannala CcO model

  • mitochondrial cytochrome c oxidase model14
  • nitric oxide inhibition
  • Steady-state flux expression

$$ \begin{aligned} Q &= 1.5 \cdot \Delta p \cr \phi_Q &= exp(-QF / RT) \cr \phi_H &= [H^+]_m / K_H \cr \end{aligned} $$

$$ \begin{aligned} K_{eq}^{CcO} &= exp(4(E_{\mathrm{m}, \mathrm{O_2}}^{0} - E_{\mathrm{m, cytc}}^{0})F/RT) \cdot 10^{-3} \cr K_{eq}^{12} &= \phi_Q \cdot exp((E_{\mathrm{m, CuB}}^{0} - E_{\mathrm{m, cytc}}^{0})F/RT) \cr K_{eq}^{23} &= \frac{k_2^+}{k_2^-} \cr K_{eq}^{34} &= K_3 \cdot \phi_H \cdot exp((E_{\mathrm{m, a3}}^{0} - E_{\mathrm{m, CuB}}^{0})F/RT) \cr K_{eq}^{45} &= \phi_H \phi_Q exp((2E_{\mathrm{m}, \mathrm{O}_{2}}^{0} - E_{\mathrm{m, a3}}^{0} - E_{\mathrm{m, cytc}}^{0})F/RT) \cr K_{eq}^{35} &= K_{eq}^{34}K_{eq}^{45} \cr K_{eq}^{51} &= \phi_H^2 \phi_Q^2 \left( \frac{c^{2+}}{c^{3+}} \right)^2 \frac{K_{eq}^{CcO}}{K_{eq}^{12}K_{eq}^{23}K_{eq}^{34}K_{eq}^{45}} \cr k_1^+ &= k_{10}^+ \phi_Q^\beta \cr k_1^- &= k_1^+ / K_{eq}^{12} \cr k_{3a}^+ &= k_{3a0}^+ \phi_H \phi_Q^\beta \cr k_{3a}^- &= k_{3a}^+ / K_{eq}^{35} \cr k_{3a}^+ &= k_{3b0}^+ \phi_H \phi_Q^\beta \cr k_{3b}^- &= k_{3b}^+ / K_{eq}^{45} \cr \Delta_1 &= 1 + \frac{NO}{K_{i1}} + \frac{1}{K_{eq}^{51}} \cr f_1^+ &= 1 / \Delta_1 \cr f_1^- &= f_2^+ = \frac{K_{i2}}{K_{i2} + NO} \cr f_2^- &= f_{3a}^+ = \frac{1}{1 + K_{eq}^{34}} \cr f_{3b}^+ &= 1 - f_{3a}^+ \cr f_3^- &= \frac{1}{K_{eq}^{51}} \frac{1}{\Delta_1} \cr \alpha_1 &= f_1^+ k_1^+ c^{2+} \cr \alpha_2 &= f_2^+ k_2^+ O_2 \cr \alpha_3 &= (f_{3a}^+ k_{3a}^+ + f_{3b}^+ k_{3b}^+) c^{2+} \cr \beta_1 &= f_1^- k_1^- c^{3+} \cr \beta_2 &= f_2^- k_2^- \cr \beta_3 &= f_3^- (k_{3a}^- + k_{3b}^-) c^{3+} \cr J_{cco} &= V_{O_2} = \frac{\rho_{C4}}{\Sigma}(\alpha_1 \alpha_2 \alpha_3 - \beta_1 \beta_2 \beta_3) \cr \Sigma &= \alpha_1 \alpha_2 + \alpha_2 \alpha_3 + \alpha_3\alpha_1 + \beta_1\beta_2 + \beta_2\beta_3 + \beta_3\beta_1 + \alpha_1\beta_2 + \alpha_2\beta_3 + \alpha_3\beta_1 \end{aligned} $$

Parameters

SymbolValueUnitsDescription
$E_{m,CuB}^0$$0.35$$\text{V}$Midpoint potential of copper center
$E_{m,a3}^0$$0.385$$\text{V}$Midpoint potential of cytochrome a3
$E_{m,cytc}^0$$0.25$$\text{V}$Midpoint potential of cytochrome c
$E_{m,O_2}^0$$0.85$$\text{V}$Midpoint potential of oxygen
$k_{1, 0}^+$$8 \cdot 10^{6}$$\text{Hz/mM}$Rate constant
$k_{2}^+$$6 \cdot 10^{5}$$\text{Hz/mM}$Rate constant
$k_{2}^-$$10$$\text{Hz}$Rate constant
$K_3$$17.5$Equlibrium constant for E3 = E4
$k_{3a, 0}^+$$1.3 \cdot 10^{7}$$\text{Hz/mM}$Rate constant
$k_{3b, 0}^+$$6.1 \cdot 10^{5}$$\text{Hz/mM}$Rate constant
$K_{i1}$$40$$\text{nM}$Inhibition constant for NO in E1
$K_{i2}$$7 $$\text{nM}$Inhibition constant for NO in E2
$\beta$$0.3$Voltage dependence factor

  1. Magnus, G., & Keizer, J. (1997). Minimal model of beta-cell mitochondrial Ca2+ handling. The American Journal of Physiology, 273(2 Pt 1), C717-33. ↩︎

  2. Cortassa, S., Aon, M. A., MarbΓ‘n, E., Winslow, R. L., & O’Rourke, B. (2003). An integrated model of cardiac mitochondrial energy metabolism and calcium dynamics. Biophysical journal, 84(4), 2734–2755. doi:10.1016/S0006-3495(03)75079-6 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1201507/ ↩︎

  3. Nguyen, M.-H. T., Dudycha, S. J., & Jafri, M. S. (2007). Effect of Ca2+ on cardiac mitochondrial energy production is modulated by Na+ and H+ dynamics. American Journal of Physiology. Cell Physiology, 292(6), C2004-20. https://www.physiology.org/doi/full/10.1152/ajpcell.00271.2006 ↩︎

  4. Demin, O. V., Kholodenko, B. N., & Skulachev, V. P. (1998). A model of O2.-generation in the complex III of the electron transport chain. Molecular and Cellular Biochemistry, 184(1–2), 21–33. ↩︎

  5. Demin, O. V., Gorianin, I. I., Kholodenko, B. N., & Westerhoff, H. V. (2001). [Kinetic modeling of energy metabolism and generation of active forms of oxygen in hepatocyte mitochondria]. Molekuliarnaia Biologiia, 35(6), 1095–1104. http://www.ncbi.nlm.nih.gov/pubmed/11771135 ↩︎

  6. Gauthier, L. D., Greenstein, J. L., Cortassa, S., O’Rourke, B., & Winslow, R. L. (2013). A computational model of reactive oxygen species and redox balance in cardiac mitochondria. Biophysical Journal, 105(4), 1045–1056. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3752118 ↩︎

  7. Guillaud, F., DrΓΆse, S., Kowald, A., Brandt, U., & Klipp, E. (2014). Superoxide production by cytochrome bc1 complex: a mathematical model. Biochimica et Biophysica Acta, 1837(10), 1643–1652. https://www.sciencedirect.com/science/article/pii/S0005272814005076?via%3Dihub ↩︎

  8. Bazil, J. N., Pannala, V. R., Dash, R. K., & Beard, D. A. (2014). Determining the origins of superoxide and hydrogen peroxide in the mammalian NADH:ubiquinone oxidoreductase. Free Radical Biology & Medicine, 77, 121–129. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4258523 ↩︎

  9. Bazil, J. N., Vinnakota, K. C., Wu, F., & Beard, D. A. (2013). Analysis of the kinetics and bistability of ubiquinol:cytochrome c oxidoreductase. Biophysical Journal, 105(2), 343–355. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3714890 ↩︎

  10. Bazil, J. N. (2017). Analysis of a functional dimer model of ubiquinol cytochrome c oxidoreductase. Biophysical Journal, 113(7), 1599–1612. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5627346 ↩︎

  11. Bazil, J. N., Beard, D. A., & Vinnakota, K. C. (2016). Catalytic coupling of oxidative phosphorylation, ATP demand, and reactive oxygen species generation. Biophysical Journal, 110(4), 962–971. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4776027 ↩︎

  12. Berndt, N., Bulik, S., & HolzhΓΌtter, H.-G. (2012). Kinetic Modeling of the Mitochondrial Energy Metabolism of Neuronal Cells: The Impact of Reduced Ξ±-Ketoglutarate Dehydrogenase Activities on ATP Production and Generation of Reactive Oxygen Species. International journal of cell biology, 2012, 757594. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3376505 ↩︎

  13. Markevich, N. I., & Hoek, J. B. (2015). Computational modeling analysis of mitochondrial superoxide production under varying substrate conditions and upon inhibition of different segments of the electron transport chain. Biochimica et Biophysica Acta, 1847(6–7), 656–679. ↩︎

  14. Pannala, V. R., Camara, A. K. S., & Dash, R. K. (2016). Modeling the detailed kinetics of mitochondrial cytochrome c oxidase: Catalytic mechanism and nitric oxide inhibition. Journal of Applied Physiology, 121(5), 1196–1207. ↩︎