Okay. Now, I think we're ready to consider in more formal, even mathematical terms, how can we predict the equilibrium potential if we know a permeate ion and its concentration gradient? Well, the good news is that this has been worked out. And there's a formulation that explains the relationship called the Nernst Equation. Now, I don't want you to have to be, be burdened with the physical chemistry of all this. I would rather have you get an intuitive feel for what this equation could do because I think it's a powerful equation in Neurophysiology. And we will refer to it from time to time as medical neuroscience progresses, and even when we discuss clinical implications of this phenomenon. So, the Nernst Equation allows us to predict an equilibrium potential for any permeate ion. And, we will indicate that potential with this shorthand called E sub x. x being any permeate ion. Now, in the more formal version of this Nernst Equation includes a number of factors. It includes the gas constant. a measurement of absolute temperature in degrees Kelvin. knowledge of the valence of the permeate ion, that is the electrical charge on that ion. a factor reflecting Faraday's constant, which is the amount of electrical charge in a mole of univalent ion that might permeate a membrane. And, of course, we need to know the concentration gradient of the permeate ion which is a function of the concentration of ion x outside of the cell, and a concentration of ion x inside of the cell. Now, as I said, I don't intend to burden you with all this formality. Rather, what I want to be able to do for you is to give you an intuitive feel for how this Nernst Equation can be a powerful tool in understanding electrical signaling in neurons. Now, thankfully, our Nernst Equation can reduce down in fairly simple terms if we make a few assumptions. One assumption that we like to make has to do with the temperature in which we're recording. And we can either change body temperature or room temperature, depending if we're thinking about neurons in our own head. Or perhaps, in an experiment preparation, we might have on a laboratory counter top. we can also make some assumptions that reflect the impact of the fairly constant, for example the gas constant. Basically, parameters to define the world in which we operate. Well, if we were to do all that, we can express the Nernst Equation in fairly simple terms. We can say that the equilibrium potential for any given ion is equal to the number 58 divided by the balance term of that ion times the log, base 10 of the concentration gradient. That is the concentration of the ion outside of the cell divided by the concentraion of the ion inside of the cell. So, this is the form of the Nernst Equation that I do want you to know, and it's provided for you in your tutorial notes, your handout for this session. And as I've been suggesting, what I really think is important is that you gain an intuitive feel for what this means. So, for example, just looking at this formulation. This means that the equilibrium potential for an ion is going to be proportional to the concentration gradient, and that proportionality is going to be influenced by the valence of the permeon ion. Now, let's start in a very simple situation. Imagine, we're dealing with an ion-like potassium. So the valence term is actually 1. So now we can just imagine that the equilibrium potential for Potassium is going to be 58 times the log base 10, the concentration gradient. And to keep the math simple, for every 10 fold change in concentration gradient, we can expect a equilibrium potential of 58 millivolts. With the sign of that potential, having to do with whether the concentration gradient favors the inside or the outside of the cell. Now, I'd like to pause for just a moment and give you some study questions to consider. These study questions are printed for you at the bottom of your tutorial, so you can look there. And I want to encourage you to respond to the questions online now, as you work through this tutorial. Okay, while working through those problems hopefully gives you some feel for what would be the impact of a change in the concentration gradient of a permeate ion on the membrane potential of a neuron. Now, as you have surmised I'm sure we've been dealing with a fairly simple situation where we've had a membrane that allows for the passage of just one ion. And, we've only really considered one kind of permeate ion. And, of course, the solutions that are found around neurons and within the cytoplasm of neurons has multiple ions and membranes on real neurons. potentially can allow fr the permeation of a large number of different ions. So, it's useful to have some formulation that allows us to understand the equilibrium potential at any given moment in time. If we know something about what ions can permeate the neuronal membrane, and then something about their concentration gradients. Well, there's one more equation to introduce to you that allows us to do just that. This is to predict the equilibrium potential when multiple ions may permeate the neuronal plasma membrane. This equation is called the Goldman Equation. And the Goldman Equation really is an expansion of the Nernst Equation that considers the situation when there are multiple permeate ions. So, what we have in the Goldman Equations are terms that allow us to count for the permeability of ions, that's what these p terms are, and we have concentration gradients. With outside over in for Potassium, for Sodium, and then inside over out for Chloride. Now, you may be wondering why is the ratio of chloride inside to out reversed, compared to what we see here for Potassium and Sodium. So, in order to keep the math as simple as possible, we simply flip the sine of the ratio for Chloride, knowing that Chloride is an ion with a valence of minus 1. Whereas Potassium and Sodium are monovalent cations with a valence of positive 1. So, to keep the math simple, to keep the logarithmic functions aligned, we simply flip the ratio for Chloride ions with inside over outside. Okay. Well, with that said, we can use the Goldman Equation to predict at any moment in time what is the membrane potential of a neuron that might vary in it's permeability for any one of these three ions. And, of course, this equation can be expanded to account for as many permeate ions as we like. Now, let's consider the same kind of model system. But now, let's make it just a little bit more realistic by adding to the solutions on either side of the permeate membrane Sodium Chloride. So, what we have now in our model system is, on the inside, or the left-hand side, we have hop Potassium Chloride, low Sodium Chloride. And then on the outside of this membrane, we have a reverse. We have low Potassium Chloride, and high Sodium Chloride. And again, to simplify the mathematics, if we want to go there, we're going to stipulate that the concentration gradients are tenfold from one side to the other for each ion. Now, let's consider what we have in the resting state of a neuron. In the resting state, the permeability of the membrane for Potassium is very slight. There's a little bit of movement of Potassium from one side of the cell to the other, specifically from the inside to the outside. but not much just enough to allow us to establish a resting membrane potential. So essentially, the permeability of the membrane at rest for Potassium is so much larger than the permeability for Sodium that we can completely ignore that contribution. And what we find is that the resting membrane potential is very close to the nearest equilibrium potential for Potassium. Not quite, because real membranes are a bit more complicated. But in principle, the resting membrane is explained by first approximation to the Nernst prediction for a membrane that's permeable for Potassium. Now, imagine there's some mechanisms in the neuronal membrane that can flip these permeabilities. Such that the permeability to Sodium is so much greater than the permeability to Potassium, that essentially this situation dominates. Where now Sodium is so much greater than Potassium, Sodium's the only permeate ion we need to be concerned with. Well, when we do that, we would find that the membrane potential approaches the most equilibrium potential for Sodium. And likewise, if we were to flip once again those permeabilities, what we would find is that the membrane potential would fall back down towards its starting point, its resting membrane potential. Well, these considerations built upon your study with the Nernst Equation gives us an intuitive feel for what's going on with an action potential. The action potential is this explosive electrical event that happens within a neuron, where the membrane potential depolarizes. In fact, it even overshoots zero such that the inside of the cell becomes very transiently even more positive than the outside. And just as quickly as that depolarization occurs, the action potential seems to fall back down towards rest. Although, it actually undershoots the resting membrane potential. That is, the inside of the cell becomes even more negative than it typically is at rest, but just by a little bit. And then, it decays back to where we started. These considerations of permeability changes for permeate ions. Sodium and Potassium provide us with a framework for understanding the ionic basis of the action potential, and that's where we're going to go in our next tutorial. But before we get there, let's just return to the Goldman Equation and emphasize some of these considerations. Now, at the resting state, the Goldman Requate, Equation essentially reduces to the Nernst Equation for potassium. That is, the permeability for Potassium is so much greater than the permeability for these other factors, that they basically disappear. And what we have then resembles the Nernst equilibrium potential for Potassium. And predictably, the membrane potential then approaches that theoretical limit predicted by the Nernst Equation for Potassium. At the peak of the action potential, the Goldman Equation essentially reduces to the Nernst Equation for Sodium. That is to say, now, the permeability for Sodium is so much greater than the permeability for Potassium that the Potassium term essentially disappears. Now, we've not talked much about Chloride. And that's because for all practical purposes in this consideration, the permeability for Chloride is very close to zero. And what permeability there is Chloride will simply follow the movement to these positively charged ions. So we can, essentially, take Chloride out of the consideration. So again, at the peak of the action potential, the Goldman Equation reduces to the Nernst equilibrium potential for Sodium, such that the membrane potential approaches this theoretical limit. So, as we've worked with the Goldman Equation just a bit I hope that you've been able to develop a bit of an intuitive feel for how changes in membrane permeability can change membrane potential. Now, what we haven't talked about at all yet at is, how do those changes in membrane permeability come about? Well, perhaps, the surprising answer is that changes in membrane permeability are themselves influenced by the change in membrane potential. This is the ionic basis for the action potential. Which we'll have more to speak about in the next couple of tutorials. For now, I think I can summarize the main message that I've tried to convey to you in the following statement. That voltage-sensitive changes in Sodium and Potassium permeability are necessary and sufficient for the production of action potentials. Now, before we talk in more detail in forthcoming tutorials about the ionic bases of the action potential, I think it will be well worth your time to consider more deeply the nature of Electrochemical Equilibrium. This concept will come up in several more sessions as we continue to talk about the basis of neural signalling and the properties of ion channels. So, I would encourage you to view one last animation, also available at the website that supports our textbook. It is Animation 2.2, which is all about understanding Electrochemical Equilibrium. You can find a link for that at the end of our handout for this tutorial. So next time, we'll think more together about the ionic basis of the action potential.
