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8. Protein Function
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concept

## Hill Equation 2m
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in this video, we're going to begin our lesson on the hill equation. So before we talk directly about the Hill equation, there's first some background information and a little bit of history that we need to tell you guys about cooperative log and binding in Alice Derek proteins. And so way back in 1913 before any knowledge of hemoglobin structure even existed, this scientist name Archi Bald Hill, tried to study. Hemoglobin is cooperative oxygen binding and so aren't you. Bald Hill knew what we already know from our previous lesson videos. So recall that coefficient in a reaction which are the numbers in front of a molecule, are included into the dissociation equilibrium, constant K. D as exponents. And so from this along with not knowing hemoglobin structure. At the time, it seemed Thio Archibald Hill, that for proteins with an unknown number of ligand binding sites and end number of ligand binding sites, the protein Ligon reaction and equations for Katie and Theta would be as follows down below. And so if we say that N is equal to the number of ligand binding sites on a protein, then of course the end would be included as a coefficient in front of the likened. And, of course, the end and the protein legging complex wouldn't be included as a subscript. And then so for the K D and the fractional saturation, all we would need to do is make sure to include the coefficients as exponents. And so, of course, that means that whenever we see the free lie again, which has a coefficient, we would need to include the coefficient here as an exponents. So we would need to include And here, here and here. And then, of course, for whenever we see the protein leg in complex, we need to include n as a sub script. So we would include. And here and so these equations here, it seemed to Archibald Hill would be appropriate for a protein with an end number of ligand binding sites. And so, really, this is the background information that we need, Um, as we move forward and talk about the hill equation. So I'll see you guys in our next video
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## Hill Equation 3m
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in this video, we're going to introduce the Hill equation, which is named after the scientist Archibald Hill. Now, we're not gonna talk all the details about how to derive the hill equation in this video. However, I can tell you that the equation up above from our last lesson video for the fractional saturation data. Essentially, this equation right here is the equation that can actually be algebraic lee rearranged and reformatted to get the hill equation. And so the benefit of rearranging this equation right here to get the Hill equation is that this algebraic Lee Rearranged Hill equation actually resembles the equation of a line which we all know is why equals M X plus B and so lines air just very straightforward. And any data that we can get to form a line is just gonna be overall easier toe, understand? So that is definitely the benefit of the Hill equation. Now again, because the Hill equation resembles the equation of a line, we could say that the Hill equation allows us to graph protein, leg and binding data on a linear plot called the Hill Plot, also named after Archi Bald hill. And we'll talk more details about the Hill plot later in our course. But for now, let's take a look down below at the Hill equation. And so notice over here on the left. What we have is the equation of a line. Why equals M X plus B where recall that the M here is just the slope of the line and the be here is just going to be the y intercept. And so what's important to know here is that all we need to do to get the Hill equation is literally just two substitute and variables, um, substitute in for these variables. And so for the Y value, all we need to substitute in is the log of this ratio of theta over one minus data for the slope em. All we need to substitute in is the variable n which recall is just the number of ligand binding sites on a protein for the variable X. All we need to substitute is the log of the concentration of ligand and recall for myoglobin and hemoglobin. The lie gand is actually oxygen, gas, and so the like and concentration can be replaced with the partial pressure of oxygen. When it comes to myoglobin and hemoglobin. And then, of course, be the Y intercept is just going to be replaced with the value of n this same value of n times, the log of the dissociation, equilibrium, constant k d. And so it's this equation right here that is the hill equation that again resembles the equation of a line and allows us to plot protein like in binding data on a linear plot called the Hill Plot. So now that we've introduced the Hill equation will be able to apply it mawr later in our course when we talk about the hill plot. But in our next lesson video, we're going toe introduced the Hill constant and cooperative ity, So I'll see you guys in that video.
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## Hill Equation 4m
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in this video, we're going to introduce the Hill Constant and H and how it relates to cooperative ity. And so, contrary to what it may have seemed toe Archibald Hill when he actually plotted hemoglobin experimental data onto a hill plot, he quickly realized that all of the equations above that we mentioned in our previous lesson videos must actually replace the variable end with the hill Constant and H. And so, as we'll see very shortly, this hill, Constant and H is different than the variable end. And so the hill Constant is sometimes referred to as the hill coefficient. And again it's abbreviated with an H. And so this hill constant or hill coefficient and H is really just the degree of cooperative ity of a ligand binding reaction. And so again, we have to replace the variable end and all of the equations up above with the hill constant and H, and so we could go ahead and do that now and so here with the hill equation, notice that we have the variable and showing up twice here and again, we have to replace the variable end with the Hill constant N. H. So we could go ahead and do that now so we can put an N H right here. And we can also put an age right here. And we also have to do that with the equations up above as well, again replacing all of the ends that we see with an H is. And so we could do that over here as well. And we can also do that over here. And so what you might notice is that now we have these blanks here for, uh, the K D. And the reason we have that is because it turns out that a proteins affinity for its like in represented by the K D, is actually affected by cooperative ity. And so K d is affected by cooperative ity and because the hill constant or hill coefficient and H is the degree of cooperative ity. Therefore, the K D is also affected by the hill constant and H in the fashion that we have up above. And so we can say that the K D eyes going to be raised to the power of the N. H in both of these equations here as well as here. And so this will be important later on when we're trying to determine hemoglobin is fractional saturation now down below. What I want you guys to notice is that the Hill Constant and H is always going to have a value between zero at its minimum and the maximum number of ligand binding sites on the protein. And and so essentially, what we're trying to say right here is that the Hill constant and H is always going to be greater than or equal to a value of zero and less than or equal to the value of n, which recall from our previous lesson videos and is just the number of ligand binding sites on a protein which again we can see, uh and is defined here. Number of ligand binding sites on a protein and so down below. We have this table where on the left hand side we have the hill constant and H, which we know again is going to be greater than or equal to zero and less than or equal to end. And on the right hand side. What we have is the degree of cooperative ity, and so it's important to know that when the hill constant NH is exactly equal toe one. There's absolutely no cooperative ity that takes place when the Hill constant NH is greater than one, then that means that positive cooperative it is going to take place. And then, of course, if the Hill Constant and H is less than a value of one, that means that negative cooperative ITI is taking place. And so, really, just by determining the Hill constant, NH biochemists are able to quickly determine the degree of cooperative ity off a protein. Whether there is no cooperative ity, positive cooperative ity or negative cooperative, ITI is going to depend on the value of the Hill constant. And so we'll be able to learn Maura about the Hill constant as we move forward in our course and talk about the Hill plot. And in our next lesson video, we're going to break down the hill Constant just a bit further. So I'll see you guys in that video
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## Hill Equation 5m
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