Permeability and membrane potentials

So I've got four empty cells here. Let's imagine these are four cells in someone's body. And the inside of the cells are the same, and the outside of the cells are the same. So they're really completely the same except for one thing. And that one thing is that, let's say that these cells, each of them, is only permeable to one ion. And I'm going to write the ion that they're permeable to underneath them. So each of them can only let in or let out one ion. So in the first cell, you know that there's a lot of potassium that wants to get out, so it's going to have potassium leaving. In the second cell, you know that we generally have more sodium on the outside that wants to get in. And the same is true for chloride that wants to get in. And the same is true for calcium that wants to get in. So this is how the four cells are going to have a movement of ions, and these are the concentration gradients. And so then, of course, if you want to figure out what the membrane potentials are, you have to think, OK, well, if it's a positive ion leaving, then it's going to make the membrane potential negative. In fact, we even calculated it to be negative 92 millivolts. And for sodium, it turns out to be positive 67 millivolts. And these are estimates based on rough concentrations. Of course, concentrations aren't exact everywhere. Different cell types have different concentrations. But these are kind of rough approximations. And chloride would be somewhere around, let's say, negative 86 millivolts. And you know it's negative because a negative ion is entering. And finally, calcium is going to be positive 123 millivolts. And just to remind us, the reason that calcium isn't simply sodium times two-- because you might think that because calcium has two positive charges. The reason it's not just sodium's number times two is that these numbers are actually based on concentration gradients, and the concentration gradient for calcium might be very different. In fact, it is very different than it is for sodium. So that's how these numbers are made, using that Nernst equation that we went over previously. So now we know that these are the resting potentials for each individual ion. But what is the potential for a cell, a real cell? And you know we're not actually using a real cell as an example, because real cells are permeable to multiple ions. And so let me actually give you an example of what a real cell might look like. Of course, it looks slightly different than that because you actually have potassium leaving. And at the same time, you might have sodium entering, you might have chloride entering, and you might have calcium entering. So this is what a real cell would look like. And let's figure out, maybe using an example, how to actually work through calculating the membrane potential for a real cell like that. So I'm going to write out the four ions and make it really clear so that we don't get confused about the four. We're talking about, again, potassium, sodium, chloride, and calcium. And the reason I chose these four-- I could have chosen others as well-- but that these four probably contribute the majority to the resting potential. In particular, I would say potassium, but you'll see that all of them have a tiny little role in contributing to it. So let's first get to the idea of permeability. So until now, we've been assuming that each cell is, in the first example, only permeable to one ion. And now the whole difference is that now we have cells that have permeability to multiple ions. So here's how you think about it. Think about the fact that all ions crossing back and forth along the membrane make up permeability. So permeability is all ions crossing back and forth. Permeability is all ions crossing-- and I'll just write crossing, and you get the idea-- crossing back and forth along the border. So what percentage is going to be from potassium? What percentage from sodium, chloride, and calcium? And of course, the total permeability has to be 100% right? We have to add up to 100%. And let's assume right now that we only have four ions going back and forth. So I'm just going to make up some quick numbers. So let's say that of the 100%, potassium is 95%, meaning that 95% of all border crossings-- if we think about our cell border or cell membrane, 95% of the crossings are with the ion potassium. And that means that only 5% of the crossings are with the other three ions. So let's say that it's 1% from sodium, and 2% from chloride, and 2% from calcium. So really, in terms of dominating the permeability, in this case, I've set it up so that potassium is dominating the permeability, right? And actually, in most cells that's about right. Potassium is the dominant ion in most cells. In fact, sometimes even more than 95%. So how do you actually calculate the membrane potential based on this? So we've started with good information. We've got the permeability. And now we need to multiply it by potassium's ideal membrane potential. What would potassium like it to be? It would like it to be negative 92 millivolts, right? And sodium would like it to be positive 67 millivolts. And chloride would like the membrane potential to be negative 86 millivolts. And calcium would like it to be positive 123. I mean, that's ideally where those ions would like to be. But again, 95% of the voting, in a sense, for what the cell is going to agree upon comes from one ion. It comes from potassium. And so we just have to add all of this up and get a total. So this part right here, 95% of the ions being permeable, multiplied by the membrane potential for potassium. 95% times negative 92-- I'm just going to quickly do the math on a calculator-- works out to negative 87.4 millivolts. And this bit right here, 1% of the 67, well, that's easy. That's 0.7 millivolts. That's just 1%. And then this bit, 2% times negative 86 millivolts, that works out to about negative 1.7 millivolts. And finally, this part right here for the calcium ends up being positive 2.5 millivolts. So if you add up all the stuff, what do you get? You get a total of negative 85.9 millivolts. So this would be the membrane potential for a cell that ended up having 95% permeability to potassium, and only 1% and 2% to the other three ions. So if it's going to be dominated by potassium, you can see that this final number is going to be really close to what potassium would like it to be, that negative 92, because 95% of it came from there. Now that would be one way of doing it. Let me do it one more time, and you'll see how it can actually change. So just as before, let's say that in the second case-- so again, this is case one. And let's say case two, now I make potassium not very permeable at all. Let's say I drop it all the way down to 16%. And I raise sodium all the way up to 80%. So now, all of a sudden, our same cell as before is very permeable to sodium. And you might think, well, how would that be the case? Let's imagine that sodium channels get put into the cell membrane so that sodium can just go right through those channels, so something like that. But let's say that the other two ions stay about the same. 2% and 2%. So you've got a similar setup as before, and this time you've got-- let me do the math. So we've got negative 92 millivolts. And again, the permeability-- this is actually an important point-- adds up to 100% again, right? Because you've got 16 plus 80 plus 2 plus 2. So overall, we're still talking about 100% permeability, but in this case, most of that permeability is going to sodium. So negative 92 millivolts for the potassium, you've got positive 67 for the sodium, negative 86 over here, and you've got positive 123 over here. So I'm going to do the last two first, because those are going to be the same as before, right? And then we add them all together. So here, of course, as before, I have negative 1.7. That doesn't change. And as before, I have positive 2.5. That doesn't change. So some of it's going to be the same. But some of it's going to be different. So these two numbers-- let me just check out what my math works out to be-- this is negative 14.7 millivolts, so a lot less than the negative 87. And now this is going to be a huge number compared to that measly 0.7 that we had before. Now we have 53.6 coming from sodium. So now sodium is playing a much bigger role than it was last time. And if I was to add these four numbers, my overall permeability was 100%, and my overall membrane potential as a result of that is going to be 39.7. So let me just-- and that's positive 39.7. So we went from negative 85.9 to positive 39.7. And here, the dominant thing was this. So you can see how it's starting to approach positive 67, simply because we just had so much of the permeability coming from sodium. So in a way, the permeability becomes almost like a vote. Like the more permeable one of the ions is relative to the other ones, the more votes it gets in terms of what the final membrane potential is going to be. And in this case, the more an ion votes, the more it's going to be close-- the final membrane potential will be close-- to what it wants, which is its resting potential. So we have these two membrane potentials. And finally, I'm just going to show you on a little graph what this might look like. So let's say you've got a little graph here. And I'm going to draw a positive and negative. So this is positive, this is negative, and this is millivolts. And what I'm drawing for you is the cell's membrane potential in millivolts. So at time point 1-- let's say time point one was right here, and time point 2 is right here. At time point 1, we had a very negative-- I forget the number. I think it was like negative 80 something. I'll just say it was negative 86 or something down here. And then at time point 2, we had a number up here. This is our 39.7. And actually let me just double check the number, I don't want to get it wrong here. Yup, negative 86. This is negative 86, or 85.9. So really, what you had is, in just a matter of switching the permeabilities, you can actually change the membrane potential from something very low to something very high. So let's stop there and we'll pick up.