[BLANK_AUDIO] Hello. Welcome back. We're now moving onto another topic in this class, which is modeling with partial differential equations. We introduced this a little bit in the last lecture the mathematical models of the action potential. And now we're going to discuss more some of the details of modeling with partial differential equations. So in this first part we're going to discuss something that's called the reaction-diffusion equation. In the last lecture on action potentials we derived this so called cable equation for voltage in excitable cells such as neurons. And we said this is an example of a reaction-diffusion equation. Now we're going to give a second example, which is how you describe ionic concentrations within a cell. How does ionic concentrations vary from one region of the cell to another region in the cell? We're going to derive this and then we're going to illustrate how this corresponds very very nicely with the cable equation that we encountered previously. This slide illustrates where we left off last time, in the last lecture on the Hodgkin-Huxon model of the action potential. We derive this equation here, which we called the cable equation, and we discussed how this describes the movement in, with, it's a function of time and space of voltage in an excitable cell such as a, such as a neuron. A partial derivative of voltage with respect to time is equal to some constants here, second derivative of voltage with respect to location minus the the ionic current. And we said that this is a reaction-diffusion equation and what I promised you is that these appear in, in other contexts. We also just briefly noted how partial differential equations could be solved numerically by converting into discrete form in both space and time. That's point number two down here. We're going to get to that in the third lecture on PDEs. What we're going to focus on today, on the, in the, in this first lecture on partial differential equations, are other examples of reaction-diffusion that are encountered in biological systems. The specific example that we're going to consider in this lecture is diffusion across an epithelial cell. We're not going to get in depth of the biology here, but the, the basics of the biological background here are that your kidneys regulate your pH and your body, in part by transporting bicarbonate across the epithelium in your kidney. If you were to lose all your bicarbonate in the urine, then your body would be less able to regulate pH. It's for acid-base balance and so you're kidney needs to make sure that that it doesn't lose, doesn't lose all your bicarbonate. It needs to take it back up, and it takes it back up by transporting the bicarbonate across an epithelial cells. And so the example we're going to discuss is bicarbonate ion concentration, that's HCO3 minus in the renal proximal tubule. This is a particular region of the kidneys where most of this bicarbonate transport takes place. And so the sort of simplest way to illustrate this is that your epithelial cell has an apex at the top of the cell and it has, the bottom has a base of the cell. And the bicarbonate is going to move from this region over here, which is the limit of your, of your nephron, to this region over here, which is your, the interstitial space of your body. And from here it goes into the blood. And so there's going to be transport of bicarbonate across the apical membrane. Then there's going to be diffusion of bicarbonate from left to right here, from the apex of epithelial cell to the base of the epithelial cells. Then there's going to be transport across the basal membrane. And we've used a capital J here for transport across the apical membrane, diffusion within the cell, and transport across the basal membrane. This is common terminology where capital J will refer to the flux of ions. So how many ions are moving? And we're going to get into the units of fluxes as they go through this derivation. And the question becomes, in this case, how do we describe the movement of bicarbonate from apex to base? When we're both in the cell here, how do we discuss, how do we describe mathematically the way that bicarbonate is moving form the left part of your diagram here, to the right part of the diagram? We're going to go through that derivation and we're going to see how we end up with a reaction-diffusion equation. It's very similar to what we saw in the last slideshow on action potentials. It's very similar to the cable equation for, for memory potential, for memory voltage. The way we're going to go through this derivation of, of diffusion of bicarbonate across an epithelial cell, is similar to the way that we went through the derivation of how voltage will vary as a function of time and location in our neuron. remember, with the neuron derivation we took a long axon and we chopped it up into discrete parts. We're going to do the same thing here. We're going to represent the cell, you epithelial cell, as a series of discrete segments. So we can think of the epithelial cell like this. We've got a cube here on the apical side. And then you've got another cube next to it and another cube next to it, et cetera, until you get to your nth cube, which is the one that's on the basal end of the cell. So you're taking your epithelium, which goes from the, from the apex down to the base and you're chopping it up into end pieces and cubes. And each cube will have a length of delta x in this case. And the terminology we're going to use the the way we're going to define our variables is we're going to say HCO3 sub i, is a concentration of bicarbonate in sub-cube i. So i in this case refers to the to the number that we're in, that we're at. And we're going to have it introduce a, a number here, a constant called capital D, sub HCO3. This is the intracellular diffusion constant for bicarbonate. And delta x, we just defined, this is the distance between adjacent sub-cubes. And the, and again the question that we're trying to get at here and what we're trying to derive is this, what are the equations that describe movement of bicarbonate from apex to base. As we derive the equations to describe movement of bicarbonate from the apex to the base, we're going to do that same thing that we did when we were deriving the cable equation for, for memory voltage. We're going to first consider what happens within three representative sub-cubes that are somewhere in the middle of the cell. So we're going to consider the ith cube here HCO3th of i. The, i minus 1 cube over here, and i plus 1 cube over here. And we're going to define fluxes that describe the transport of bicarbonate from one cube to the next. So the flux that goes from i minus 1 to i is going to be J, from i minus 1 to i. And this flux is going to be J from i to i, i plus 1. And I just, I separated the cubes here just for, for clarity just so I could draw the arrows here. But, of course, in a real cell they're going to to be right next to one another like we saw on the, on the last slide on the last diagram. We're going through just one more variable here. Capital a refers to the cross-sectional area along, along this dimension. So delta x is a distance from the center of this cube here to the center of this cube here, and capital A is the cross-sectional area along this along this other direction. In the first two equations that we can write down are J from i minus 1 to i is the the fusion constant here D HCO3 times the concentration difference, bicarbonate, i minus 1, minus bicarbonate at i, divided by the distance delta x. And then we can write a completely analogous equation for J from i to i plus 1 and these these equations describes Fick's first law of diffusion. But tou probably haven't heard of Fick's first law of diffusion before, but these equations make intuitive sense, if we think about things a little bit. First of all, what happens if the concentration in i minus 1 and the concentration in i are equal? Well, you know intuitively that if two concentrations are the same, there's not going to be any diffusion from one to the other and that they're the same and this difference will be zero and therefore the flux will be zero. So it makes sense in that regard. And intuitively what would happen if you had to travel a great distance from i minus one to i? Well, if it had to travel a great distance, then you could probably deduce that there wouldn't be that much flux and sure enough, that's why delta x is in the denominator here. So as the distance becomes greater as the ions have to travel a greater distance, the, the flux gets smaller. And then the diffusion constant here, you probably you know, haven't seen before. But this just sort of refers to how easy it is for, for the ions to move. If it's very, very difficult for them to move from, from one cube to another cube, then the diffusion constant will be very small. And if it's easy for the ions to move, then the diffusion constant will be big. And therefore, it makes sense that the diffusion constant comes in in the numerator of this equation. As the diffusion constant gets bigger the flux gets bigger. And this is the equation for flux from i minus 1 to i and then this is the equation here from flux from i to i plus 1. And what we want to do now is we want to relate these fluxes to changes in bicarbonate concentration. This is what we care about, after all, right? How much does bicarbonate concentration change as a, as a function of time? What is the derivative, the time derivative of bicarbonate concentration? And how does that relate to the, to the transport of bicarbonate from one cube to the other cube? Intuitively, we can say that this derivative should depend on the inflow versus the outflow. If we're looking at bicarbonate concentration in this cube here, in cube i, then if you have a lot coming in and you have none or just a tiny bit coming out, then its concentration is going to go up. On the other hand, if you don't have much coming in and you have a lot going out, then concentration is going to go down. So somehow or another, you're going to have to subtract these two. But in order to really look at this in quantitative terms, we're going to need to consider units in order to be able to express this precisely. So now I'll introduce some of the units of these terms, as they, as they come into this equation. Delta x is a distance, so that's in, in some sort of distance units. In this case, we'll just use centimeters for the sake of argument. But it could easily be, you could use meters. You could use micrometers, whatever. Bicarbonate concentration, because we're talking about biological ion concentrations, we usually use millimolar for this one. And millimolar means you know, 10 to the minus 3 moles per liter. And that's equivalent to a micromole per centimeter cubed. Centimeter cubed, a lot of you probably know, is equal to a, a milliliter. So, you're just taking the units here, milli- in front of the molar becomes a micro in front of the molar. And then, when you go from liters to milliliters, you're just taking the numerator, numerator and the denominator and dividing each one by a thousand. So our bicarbonate concentration, our micromoles per centimeter cubed. In the diffusion constant, diffusion constants in general are in units of length squared per unit time and the units we need here to get everything to work out are centimeters squared per second. So if we plug the units into this equation here, centimeters squared per second micromole per centimeter cubed, and then centimeter in the denominator here. Then we end up with units of micromole per centimeter squared per second. So what the flux is, is it's saying, how much substance, how many moles of something do you have, per unit time, and then also, per unit area. So if you have the same amount of stuff going across twice the area, then your flux will decrease then you, you know, your flux will be decreased be a factor of two. So that's another normalization term, is, is a cross-sectional area that your flux is travelling across. So we have our fluxes in units of micromole per centimeter squared per second, and we want to relate that to the derivative here. And the derivative that we need to get in units of micromoles per centimeters cubed per second, right. Because this needs to be in units of concentration per times. So this is the unit of concentration for time, right? Micromole per centimeter cubed is a concentration and then we have to have time in the denominator here. So somehow or another we need to get another centimeter in the denominator here. So what we're going to show next is how to relate that to the, to the geometry of our cubes that define our epithelial cell. As we just discussed, what we need to do is, we need to convert from micromole per centimeter squared per second, to micromole per centimeter cubed per second. So there's one more thing about how diffusion is treated mathematically that we need to introduce now, which is that if you want, we, we said that a flux is normalized per centimeter square per per area. So we need to multiply by the inter-cube cross-sectional area A. Then that just tells us how much total stuff, micromoles per second, is traveling into that, into that cube. Then if we want to convert that into a concentration change, what we do is we divide by the volumes. And we need to divide by the volume of the ith cube Vi. So the way that we relate our two fluxes to the concentration change, the derivative of bicarbonate concentrations, we take the difference between these two, we multiply them by the cross sectional area A, and we divide by the volume b. So remember that we need to get another centimeter in the denominator in order to get our units to work out. So we're taking something in the numerator that's going to be the nearest centimeter squared and then in, in the denominator we're going to have centimeter cubed. And therefore we're going to have another centimeter in the denominator. And our units are going to work out correctly. But, remember, we're talking about a cube here. Or not a cube, but, in general a, a, a parallel pipe guide. It's you know, it's some sort of square, squar-ish or rectangl-ish shape. So our volume is equal to our cross-sectional area times our distance. That's would be true of any such that's how you would calculate the volume of any such shape. So because our volume is equal to A times delta x, we're going to have A in the numerator here, and then we're going to have A times delta x in the denominator, so our cross-sectional area A is going to, is going to cancel out. So then we can conclude that the derivative of bicarbonate, the time derivative of bicarbonate, is the differences in the two fluxes, i minus 1 to i, minus i to i plus 1, divided by the distance, delta x. Now if we plug in our equations for for the two fluxes. Remember this is the concentration difference divided by delta x, concentration difference divided by delta x. Each one of these has diffusion constant in front of it. We took the two diffusion, diffusion constants. We factored that out, put that front here. This difference, minus this difference each of those divided by delta x, and then divided by delta x here. In the next slide we're going to, we're going to simplify this, and we're going to see how we can generalize it. What we want to do now with this equation is, we want to be able to generalize it. What's the limit as delta x goes to 0? We divided this into into discrete cubes, in order to make it conceptually, you know, something that we could wrap our heads around and in order to be able to write down some straightforward equations. But what we care about overall is what is true for all cubes on any arbitrary side. What's the limit is delta X goes to 0? Well, this term here. One concentration minus another concentration divided by the spatial difference. The limit of delta x goes to 0 here is our spatial derivative, the derivative of bicarbonate with respect to x. What do we have in our, our equation that we were just looking at here? We have a derivative here and a derivative here divided again by the spatial distance, by delta x. So we have a derivative minus a derivative, divided by distance. We've got our second derivative here. So this is, this equation here is is delta x goes to 0. This term the limit there becomes the second derivative of bicarbonate with respect to x, and we can understand that when we consider just what's the first derivative. Well, the first derivative is the concentration difference divided by delta x. So you've got one concentration difference, a second concentration difference, again divided by delta x. The limit as delta x goes to 0 of this whole thing here, is going to be the second derivative of bicarbonate with respect to location. Therefore, in the limit of small delta x, our overall equation becomes the partial derivative of the bicarbonate with respect to time, that's what we've previously had on the left hand side of the equation, is equal to the diffusion constant, times the second derivative of bicarbonate with respect to x, with respect to location. What we've just done here is we've derived a one-dimensional diffusion equation. What we said at the beginning was we were going to talk about reaction-diffusion equations and there's no reaction in here yet. And that's what we're going to, that's the final thing that we're going to introduce in order to, to fully understand in mathematical terms how ions move across cells. Now let's consider what else could be happening within the cell, rather than just ions moving from one cube to another cube. What if you also had some first order intracellular process that was consuming bicarbonate? What if we made this slightly more complicated where you could have flux by carbonate into the cube, flux of bicarbonate out of the cube, but then once the bicarbonate was within the cube, you could have some rate constant k that was consuming bicarbonate or degrading somehow. This could be binding to a buffer. In the case of bicarbonate, it's actually convergent it's combining with hydrogen ions and then being converted into water plus carbon dioxide through an enzyme called carbonic anhydrase. But we're not going to get into that level of biological detail. Really what we care about here is what if you can have flux of this species into the cube, flux out of the cube, and some first order process that's consuming bicarbonate. So, what we would have in that, in that case is, are the derivative of bicarbonate with respect to time, is this flux divided by delta x minus this flux divided delta x, minus some term, rate constant k times the bicarbonate concentration. And again we want to look at this in a continuum limit. Remember that as delta x goes to 0, this equation here simplifies to the second partial derivative of bicarbonate with respect to location. And then, again, minus k times HCO3. And now we have a reaction-diffusion equation rather than just the diffusion equation like we had on a previous slide. And this should, this should look somewhat familiar to you. So what we're going to conclude this first lecture by comparing what we just derived here for bicarbonate with respect to location, with respect to time, with what we derived in the previous lecture. Let's look at our one dimensional cable equation for voltage versus our epithelial reaction-diffusion equation for the derivative, partial derivatives of bicarbonate with respect to x and with respect to time. Our cable equation looks like this, capacitance times partial derivative of voltage with respect to time is equal to these terms that are related to geometry. Here are these constants times second derivative of voltage with respect to x, minus i ion, minus the ionic current. Why do we call this the reaction-diffusion equation? Well, this term here describes diffusion of voltage or diffusion of charge in the neuron. And this term here is a, is a reaction that either increases or decreases voltage. If you remember back to the lectures on the Hodgkin Huxley model, when you have a negative ionic current going into the cell you're going to increase voltage, when you have positive ionic current, which is cation moving out of the cell, you're going to decrease the voltage. So the ionic current can be thought of as the reaction that will either increase voltage or decrease voltage. And by analogy, what we have when we have diffusion of bicarbonate across an epithelium, this is the equation that we just derived. This term here describes diffusion and this term here describes a reaction that consumes bicarbonate. So in either case, you have the same sort of general structure. You have a term describing diffusion and you have a term that described the reaction. And in either case your, your diffusion term is the second derivative of your species with respect to location. Here we have d squared voltage over dx squared, sorry, I left off the squared here, that's a mistake, but we have d squared bicarbonate over dx squared, again this is the second derivative of bicarbonate with respect to space. So to summarize this lecture, what we've seen is that the partial differential equation, the PDE that describes transport of an ion across a cell, is analogous to the cable equation PDE that we encountered in neurons and both of these PDEs are examples of reaction-diffusion equations. What we're going to see next are some more specific examples and we're going to talk about some studies that have used these PDEs, use these reaction-diffusion equations to gain new insight into biology.
