## Proton Motive Force

Peter Mitchell introduced an entity proton motive force as the intermediate betweeen the electron transport chain (driving force ) and ATP synthesis (driven force) so that the overaall efficiency of the prohsophorylation process is given by:

1. \begin{align} \eta&= - \frac{J_P \Delta G_P}{J_O \Delta G_o} \\ &\le 1 \\ \end{align} (Eq.1)\\

Equation (1) seems to give the impression that $$J_P$$ and $$J_o$$ are directly coupled. The thermodynamic translation of the Mitchell picture is as follows:

1. \begin{align} -\Delta G_o & \rightarrow \Delta \hat\mu_H\\ \Delta \hat\mu_H& \rightarrow -\Delta G_P\\ \end{align} (Eq.2)\\

The actual flow and flux equation that follows can be written as:

1. \begin{align} J_O& = l_{OO} (-\Delta G_o) + l_{OH}\Delta\hat\mu_{H}\\ J_P&= l_{PP} (-\Delta G_P) + l_{PH}\Delta \hat\mu_{H}\\ J_H&= l_{PH}(-\Delta G_P) +l_{OH} (-\Delta G_o) +l_{HH}\Delta \hat\mu_{H}\\ \end{align} (Eq.3)\\

Let us now assume that there is no overall acidification of the medium. In other words we can substitute the steady state condition $$J_H=0$$ in the equation (3). What follows is a linear relation between the three free energy terms.

1. \begin{align} \Delta \hat\mu_{H}&=-\frac{l_{PH}(-\Delta G_P) +l_{OH} (-\Delta G_o)}{l_{HH}} \\ \end{align} (Eq.4)\\

Substituting (4) in equation (3) we obtain,

1. \begin{align} J_O& = l_{OO} (-\Delta G_o) -l_{OH}\frac{l_{PH}(-\Delta G_P) +l_{OH} (-\Delta G_o)}{l_{HH}}\\ J_P&= l_{PP} (-\Delta G_P) - l_{PH}\frac{l_{PH}(-\Delta G_P) +l_{OH} (-\Delta G_o)}{l_{HH}}\\ \end{align} (Eq.5) Equation (5) can be re-written as,
2. \begin{align} J_O = \frac{l_{OO}l_{HH}-l_{OH}^2}{l_{HH}}\cdot (-\Delta G_{O})+ \frac{l_{OH}.l_{PH}}{l_{HH}}\cdot (-\Delta G_{P})\\ J_P= \frac{l_{OH}.l_{PH}}{l_{HH}}\cdot (-\Delta G_{O})+\frac{l_{PP}l_{HH}-l_{PH}^2}{l_{HH}}\cdot (-\Delta G_{P})\\ \end{align} (Eq.6)

The equation reduces an even simpler form:

1. \begin{align} J_O& = L_{OO}\cdot (-\Delta G_{O})+ L_{OP}\cdot (-\Delta G_{P})\\ J_P&= L_{PO}\cdot (-\Delta G_{O})+L_{PP} (-\Delta G_{P})\\ L_{PO}&=L_{OP}&\\ & = \frac{l_{OH}.l_{PH}}{l_{HH}} \end{align} (Eq.7) One may find in https://books.google.co.in/books?id=WSmcAAAAQBAJ&pg=PA545&lpg=PA545&dq=oxidative+phosphorylation+phenomenological+equation&source=bl&ots=QOMJHQXfwM&sig=cSXy4BvEGno3xGpDRhrObgA9W_U&hl=en&sa=X&ved=0ahUKEwj-sK_txJ3XAhVJKY8KHVvQB_sQ6AEILDAA#v=onepage&q=oxidative%20phosphorylation%20phenomenological%20equation&f=false a brief discussion on the above.

## Interpretation of the coupling equation

The equation (7) clearly indicates that as the proton conductance $$l_{HH}\rightarrow \infty$$ the cross diagonal-coefficient $$L_{OP}\rightarrow 0$$. The coupling coefficient is expressed by the relation:

1. \begin{align} q&= \sqrt{\frac{L_{OP}^2}{L_{PP}.L_{OO}}}\\ \end{align} (Eq.8)

The uncoupled condition corresponds to $$q\rightarrow 0$$ andthis happens at higg value of proton conductance , this being the basic thermodynamic implication of the Mitchell hyopthesis.

## Static Head and Level Flow

We can define two classes of steady states for oxidative phosphorylation.
* Static head - ATP production is 0 ,i.e. $$J_P=0$$
* Level Flow - $$\Delta G_P=0$$ and no net work is done by the mitochondria.