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.