What is life if not extracting electrons from something and use them to generate an electrochemical potential that drives energy transduction ? Physical separation of protons though membranes allow this. I like to think that Rothko’s paintings illustrate the electrochemical gradient with different chromatic fields. I see the thick orange lines as the membranes, the top orange field as the cytoplasm and the white space as the lumen full of protons. Illustrated below in the inner membrane with the two separate motors is, of course, the ATP synthase (PDB: 6CP3).

ATPase