# ðŸ“’ Krab 2011
> Explaining the enigmatic KMfor oxygen in cytochrome c oxidase: A kinetic model[^Krab2011]
[Sciwheel](https://sciwheel.com/work/#/items/6109541)
## Introduction
* Complex IV (cytochrome c oxidase) reaction:
4cytc2+ + 8H_M+ + O2 â†’ 4cytc3+ + 4H_C+ + 2H2O
* Apparent high affinity (low Km,app) for oxygen (<1Î¼M). But the dissociation constant is 0.28mM?!
=> fast trapping of the already bound O2
* High pmf or low c2+/c3+: higher Km,app = 1~10 Î¼M
* Kinetic models for this enzyme that incorporate all features of *Gibbs energy transduction* are scarce
## The model
### The reaction cycle
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr1_lrg.jpg)
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-fx1.jpg)
![eqn2](https://user-images.githubusercontent.com/40054455/86699812-2091f680-c043-11ea-9495-daeb03faa30d.png)
![eqn3](https://user-images.githubusercontent.com/40054455/86699816-212a8d00-c043-11ea-98a4-2c45c20509dc.png)
* The number of charges transported across the inner mitochondrial membrane is given in parentheses.
* Division of reaction 5, the O to E transition, into 6 steps to describe the proton pumping mechanism
![eqn4](https://user-images.githubusercontent.com/40054455/86699819-21c32380-c043-11ea-95df-7d524eb59ab1.png)
![eqn5](https://user-images.githubusercontent.com/40054455/86699820-225bba00-c043-11ea-8323-b73e0388c648.png)
![eqn6](https://user-images.githubusercontent.com/40054455/86699823-225bba00-c043-11ea-8602-1a3842dc463b.png)
![eqn7](https://user-images.githubusercontent.com/40054455/86699825-22f45080-c043-11ea-9a9c-29c44be77c6e.png)
![eqn8](https://user-images.githubusercontent.com/40054455/86699827-238ce700-c043-11ea-9387-0d5f0355f0ba.png)
![eqn9](https://user-images.githubusercontent.com/40054455/86699829-238ce700-c043-11ea-97dc-2c49a4507b27.png)
* 6-fold divisions of reactions 3, 4 and 6 are also modelled on this
### Values of model parameters
![tbla1](https://user-images.githubusercontent.com/40054455/86699844-2687d780-c043-11ea-854e-72351e43e2c9.png)
### Variables in the model
* fractions for cytc2+ vs cytc3+
* free proton concentrations at the C-side and M-side
* transmembrane electric potential difference (C-side minus M-side): membrane potential dependence of both kinetic constants is equal but opposite
$$
k_{\text { forward }}=k_{\text { forward }}^{\circ} \cdot e^{\frac{-q \cdot F \cdot \Delta \psi}{2 R \cdot T}}
$$
$$
k_{\mathrm{backward}}=k_{\mathrm{backward}}^{\circ} \cdot e^{\frac{q F \Delta \psi}{2 R T}}
$$
* q2 = 0.33, q3 = 0.67
### Calculation of steady state
![tbl1](https://user-images.githubusercontent.com/40054455/86699830-24257d80-c043-11ea-8bc6-b88262bb5f62.png)
* King and Altman method
* 26 * 26 matrix for transitions rates
### Calculation of the KM for O2
* R00 -> A00 is proportional to [O2]
* cycle flux to [O2]
$$
v=\frac{\alpha \cdot\left[O_{2}\right]-\beta}{\gamma \cdot\left[O_{2}\right]+\delta}
$$
* Î² << Î± â‹… [O2]
$$
v \approx \frac{V_{\max } \cdot\left[O_{2}\right]}{\left[O_{2}\right]+K_{\mathrm{M}}}
$$
$$
V_{\max }=\frac{\alpha}{\gamma} \quad K_{\mathrm{M}}=\frac{\delta}{\gamma}
$$
### Control coef fi cients
$$
C_{i}^{Z}=\frac{\partial \ln Z}{\partial \ln \lambda_{i}}
$$
* Î»_i is the factor by which both forward and backward rate constant of step i are changed simultaneously
* this modulation does not alter the Î” G0
## Results and discussion
### Rate and enzyme states as a function of the energy state of the membrane
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr2_lrg.jpg)
* state E04 accumulates (especially under low energy conditions)
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr3_lrg.jpg)
* depends on pmf, absolute H+, Î”pH, cytochrome c redox state
### Energy state and KM for O2
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr4_lrg.jpg)
* KM values are less than 1 Î¼M at lower values of Î”Î¼H+, increase into the micromolar range when Î”Î¼H+ is increased
* a lower reduction level of cytochrome c brings about an increase in KM
* as in experiments
### Control of cycle flux and kinetic properties by individual steps in the mechanism
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr5_lrg.jpg)
* there is no individual step limiting turnover
* the same steps (13, 14, 17, 19, 25 and 26) contribute signi fi cantly to control of all three of these entities,
![](https://ars.els-cdn.com/content/image/1-s2.0-S0005272810008029-gr6_lrg.jpg)
* 8 main contributions (control values > 0.05) to control of KMas a function of energy state ( Î”Î¼H+)
![tbl6](https://user-images.githubusercontent.com/40054455/86699842-25ef4100-c043-11ea-9819-354bb9e48e2f.png)
* Apart from the R00â†’ A00and A00â†’ P00transitions, all these steps are electrogenic proton movements
### Sensitivity of KMincrease to arbitrarily chosen parameters
* The main prediction of our model is the increase of KMfor oxygen upon energisation
* A number of parameters in the model (midpoint potentials of heme a3 in the P, F, E states, pKaof protonation of the Fe-ligand in the F, E states and 6 bias factors) have arbitrarily assigned values.
* explored how dependent the KMchange predicted by the model is on the values of these parameters, and on the charge separation parameters q2and q3
* these small variations of the parameter values did not abolish the KMdifference
### Utility of the model
* dependence of the electron transport rate through cytochrome oxidase on **membrane energisation** is modelled realistically and explicitly.
* analysis and prediction of the separate effects of membrane potential, matrix pH and intermembrane space pH on the rate
* Not included: branches of catalytic function (slips), dimerization, regulations
* It is possible to use this model to calculate transient kinetics of the enzyme
## Reference
[^Krab2011]: Krab K, Kempe H, WikstrÃ¶m M. Explaining the enigmatic K(M) for oxygen in cytochrome c oxidase: a kinetic model. Biochim. Biophys. Acta 2011;1807(3):348-358. doi:10.1016/j.bbabio.2010.12.015.