 So in some ways I went ahead of history here. I'm going to need to tell you how we discovered these various transportation mechanisms. And the first one was in the early 20th century discovered by Hodgkin and Huxley. They were neurophysiologists and studying the nerve system and in particular they were studying electrical signals in the nerve system using squid. Squid is special and this particular squid has a giant axel. It's one gigantic nerve cell that is like 1,000 times larger than a nerve cell than a human. That made things very easy to work with in the lab because you could literally take electrical wires, attach them to this nerve cell and see what happens if you stimulate the nerve cell. Literally with that oscilloscope you could see an electrical signal. Their results were quite criticized at the time because this whole connection that something as complicated as the human nervous system could be explained by electricity. That was heretic, right? But they persisted and came up with some very simple formulations literally using idealized electrical circuits to describe how the flux of ions in and out of a cell could be explained with different types of resistors, condensers here. They also noticed that you had different concentrations of ions. You had a balance between sodium and potassium and normally had excess potassium k on the inside of a cell while you have the significant deficit in sodium Na and this deficit of sodium was greater than the excess of potassium which created this potential on the inside. They got the Nobel Prize for the discoveries in I should remember that 1963, I think it was, in physiology or medicine. But that then leads to the obvious question, how on earth do those ions get in and out of cells? And that led to the formulation. There must be some sort of channel. There must be a mechanism by which ions can enter or exit cells. That sounds easy, right? The only problem is they can't just be a there's no way an ion can go straight through the membrane. Why? You know that. Remember when you started free energy, what is the likelihood of putting an ion here? Well, that should just be the self-charge of something and then compare the self-charge in water versus this completely hydrophobic medium. And depending a little bit what epsilon you calculate, this is going to be at least 20 calories per mole, if not more. Is that the higher or lower energy? You know your Boltzmann distribution. You should compare that to 0.6 kcal. And in fact, 20 divided by 0.6, that's roughly an exponential function raised to 30. That's going to be an exceptionally low probability. It's going to be so low that it will never happen. There is no way we will ever have a free ion inside a medium. So the ions cannot simply go through the membrane. Adrian Parzigen formulated this quite beautifully in the 1960s using pure math. So we do need some sort of channel and what he showed is that for this to happen, we're going to need some sort of pore, a hole, where here you might have epsilon equals 2 or so and here we had epsilon equals 80 in the water. And one way or another, we can never put the ion in this epsilon 2 environment. So Parzigen showed that if you have a radius of a pore there and a length l, he just used fairly simple physics to calculate that the potential is proportional to the length divided by r squared. And then some other factors, epsilon and everything. And what this means is that the thinner the membrane is and the wider this pore is, the easier it's going to be for the ion to get through. And under some circumstances, this might be as cheap as just a few kcals. That's important because without that, we would never be able to transport ions through. But of course, Adrian at the time, he had no idea what the actual environment here looked like and how some sort of protein component here would make this happen. But one way or another, the protein has to create an aqueous environment so that the ion can go straight through here.