Not necessarily math fun but, for our origin-of-life followers, this paper suggests insight:
"Harnessing energy as ion gradients across membranes is as universal as the genetic code. We leverage new insights into anaerobe metabolism to propose geochemical origins that account for the ubiquity of chemiosmotic coupling, and Na+/H+ transporters in particular. Natural proton gradients acting across thin FeS walls within alkaline hydrothermal vents could drive carbon assimilation, leading to the emergence of protocells within vent pores. Protocell membranes that were initially leaky would eventually become less permeable, forcing cells dependent on natural H+ gradients to pump Na+ ions. Our hypothesis accounts for the Na+/H+ promiscuity of bioenergetic proteins, as well as the deep divergence between bacteria and archaea." http://download.cell.com/pdf/PIIS0092867412014389.pdf --Here is something an amateur scientist might be able to figure out (if not already known). Consider a metal. It corrodes, especially in water. It produces electric potential in nature of a battery as it does so. (Similarly, certain semiconductors like iron pyrites also can do this.) Now meanwhile, bacteria use chemcial energy (ATP) to pump protons across membrane, also creating a battery with two "terminals" -- inside the cell & outside. This battery is used to fuel various cellular processes, for example the "motor" that spins flagellum, is electrically-powered. QUESTION: can bacteria "cheat" by sticking themselves to a metal surface and pirating potential difference? I have noticed bacteria forming films on metal surfaces which make me suspect this might happen.