By what mechanism does the phosphate bind to ADP in the electron transport chain?
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It doesn't happen in the electron transport system, but in the next stage, chemiosmosis.
In mitochondria, the electron transport system creates an electrochemical gradient of protons across the inner mitochondrial membrane. This is a form of potential energy, similar to having a dam with a lake of water built up behind it.
The potential energy of the proton gradient is converted into kinetic energy as the protons flow back into the mitochondrial matrix, through membrane-spanning enzymes called ATP synthase. The flow of protons through ATP synthase is used to combine ADP + Pi into ATP.
A bit more technical.
The flow of protons through ATP synthase causes a subunit composing a central shaft to spin. This central shaft is asymmetric; it has a sort of bump along one edge. The central shaft spins through 3 other regions that undergo conformational changes as the bulge of the shaft spins. Each of the 3 different conformations has a different affinity for ADP and ATP. In the first step, ADP and Pi bind; in the second step, the bound ADP and Pi are "squeezed" together to form ATP; and in the third step, the ATP is released. Now the cycle starts over, with ADP and Pi binding again.
In mitochondria, the electron transport system creates an electrochemical gradient of protons across the inner mitochondrial membrane. This is a form of potential energy, similar to having a dam with a lake of water built up behind it.
The potential energy of the proton gradient is converted into kinetic energy as the protons flow back into the mitochondrial matrix, through membrane-spanning enzymes called ATP synthase. The flow of protons through ATP synthase is used to combine ADP + Pi into ATP.
A bit more technical.
The flow of protons through ATP synthase causes a subunit composing a central shaft to spin. This central shaft is asymmetric; it has a sort of bump along one edge. The central shaft spins through 3 other regions that undergo conformational changes as the bulge of the shaft spins. Each of the 3 different conformations has a different affinity for ADP and ATP. In the first step, ADP and Pi bind; in the second step, the bound ADP and Pi are "squeezed" together to form ATP; and in the third step, the ATP is released. Now the cycle starts over, with ADP and Pi binding again.