8. Protein Function
Hill Equation
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In this video, we're going to continue to talk about the Hill constant n h. And so we already know from our previous lesson video that when the Hill constant n h is equal to n, there's no cooperativity that takes place in the protein. And so it turns out that the hill constant n h can only equal the number of ligand binding sites n under 2 circumstances. The first is, of course, if the protein displays no cooperativity, such as a protein like myoglobin. And the second circumstance where the hill constant NH will equal n is when the protein only follows the concerted model of cooperativity. However, recall from our previous lesson videos that hemoglobin's oxygen binding behavior is explained via a combination of both the concerted and sequential models. So hemoglobin does not only follow the concerted model of cooperativity. NH is not going to equal its, n. And so we can say hemoglobin's NH is not going to equal hemoglobin's n. And so in hemoglobin's cooperative state, hemoglobin's hill constant is going to range from 2.8 to about 3. So we're we're gonna go ahead and say that hemoglobin's hill constant n h is gonna be about 3 as we move forward in our course. And this is true even though hemoglobin has 4 ligand binding sites. So hemoglobin's n h equals 3 even though it's n is equal to 4. And so if we take a look down below at our oxygen binding curve, notice on the y axis we have the fractional saturation theta or y and on the x axis, we have the partial pressure of oxygen in units of torus. And notice that we have these three curves. We have this black curve here that resembles a rectangular hyperbola. And notice that the number 1 here is indicating that it has no cooperativity. Just like up above, we could say that the protein has no cooperativity, cooperativity, such as myoglobin. And notice that this blue curve right here represents, the curve if hemoglobin, displayed the concerted model only. And so if hemoglobin displayed only the concerted model, then its NH is going to equal its n. And, of course, we know that hemoglobin's n is equal to 4. But, again, this is if hemoglobin only followed the concerted model, it would have this blue curve. But we know again that hemoglobin does not only follow the concerted model, it follows a combination of the concerted and sequential models. And so when we actually plot hemoglobins data onto this plot, what we'll get is this red curve that we see here. And notice that hemoglobin is going to have a hill constant, of 3 and NH equal to 3. And so notice that this example problem here is saying to assume hemoglobin's KD is equal to 26 tors and calculate fractional saturation the fractional saturation, we need to recall from our previous lesson video, the fractional saturation of hemoglobin. And so, what we need to recall is this equation right here from our previous lesson video where we took the fractional saturation theta equation and raise the ligand concentration to NH and the kd, we raise to the NH as well. And so notice that we're given the partial pressure of oxygen as 100 tors and that is gonna represent the concentration of ligand. So we can go ahead and start to substitute this in. So what we have is the concentration of ligand is gonna be 100, and we can raise this to the NH which again is given to us as 3. So, 100 over 3 and this is gonna be over. Again, 100 over 3. I'm sorry, 100 race to the 3rd power. Plus, the k d, which is also given to us as 26 tours. So we can put that in here, 26 6 tors, and this is also going to be raised to, the third power. And so all we need to do now is just type all of this into our calculator. So if we take 100 cubed and divide it by the sum of 100 cube plus 26 cubed, what we'll get is our answer, which is theta is equal to 0.9 83, which is, of course, equal to about 0.98. And so we can say that theta is gonna be equal to 0.98, which matches with answer option c here. And so we can indicate that c is the correct answer for this example problem. And so in our next couple of videos, we'll be able to get some practice utilizing, these equations. And, we'll also be able to talk more about Hemoglobin's Hill Constant as, we talk about the Hill plot. So I'll see you guys in our next video.
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