Protein-Ligand Equilibrium Constants - Video Tutorials & Practice Problems
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Protein-Ligand Equilibrium Constants
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So now that we've introduced protein ligand interactions as well as protein ligand rate constants, in this video we're going to talk about the first of 2 different protein ligand equilibrium constants. So again, it turns out that there are actually 2 protein ligand equilibrium constants. And so in this video, we're really only going to talk about the very first protein ligand equilibrium constant. And then later in our next video, we'll talk about the second protein ligand equilibrium constant. Now it's important to note now that equilibrium constants are different than rate constants. And so in our last lesson video, we talked about rate constants. And in this video, we're focusing on equilibrium constants. And so the first protein ligand equilibrium constant is the protein ligand association equilibrium constant, which is capital k a. Not to be confused with lowercaseka, which is, again, the rate constant that we talk about in our last couple of lesson videos. And so recall that the equilibrium constant is abbreviated as just KEQ and really all it is is just the ratio of the concentration of over the concentration of reactants. And so this association equilibrium constant, capital KA, is really just an equilibrium constant itself. It's the exact equilibrium constant for the association of the free protein and free ligand into the protein ligand complex. Now, you might recognize this capital k a variable from way back in our previous lesson videos. And that's because we use this same exact capital k a variable to represent the acid dissociation constant, which again also uses the variable capital k a. And so it's very, very important not to confuse. So do not confuse this protein ligand association constant, k a, that we just introduced now with the acid dissociation constant that we talked about way back in our previous lesson videos. And we kinda already knew to do that anyways because we know that not all proteins are going to be acids anyways. And so this association equilibrium constant, capital K a, is actually a measure of protein affinity for a ligand. And so, the Ka and protein affinity for ligand are actually directly proportional to each other, which therefore means that the greater the value of this k a, the stronger the affinity a protein has for that particular ligand. Now this Ka also has units of inverse molarity and it turns out that it's going to be the reciprocal of the second protein ligand equilibrium constant that we'll talk more about in our next lesson kd, which k d, which again we'll talk more about in our next video. And so here what you can see is that the k a can be defined as the reciprocal of the KD. So essentially, one over KD. And again, we'll revisit this idea more when we talk about the dissociation constant, KD in our next lesson video. Now, notice down below in our image over here on the far left, what we have is a reminder of how protein ligand interactions work. And so notice that this lowercase k a and this lowercase kd represent rate constants, which we again talked about in our last few lesson videos. And so, these particular rate constants are different than the equilibrium constants that we're learning about in this video. And so over here what we have is the equilibrium constant, the association equilibrium constant, capital k A, which is again different than this rate constant over here with lowercase k a. And so again, we already know from our previous lesson videos that equilibrium constants are just the ratio of the product over the reactant. And so for the association here of the protein ligand complex, the product is just going to be PL and, the reactants are gonna be PNL. And so it's going to be, the 2 of these concentrations multiplied by each other. And so this is one way to express this association equilibrium constant KA, capital KA. But it can also be defined by the ratio of the rate constant lowercase k a over the rate constant lowercase k d. And so here what we can say is that lowercase k a, as well as lowercase kd, as a ratio just like this, k a over k d, also rep is a representation of the association equilibrium constant capital KA. And then, of course, as we said up above in this line, the KA is the reciprocal of the dissociation constant and so, we can also define the KA as, the reciprocal of the K d, the capital k d, which again, we're going to talk more about in our next lesson video. And And so over here on the far right, what we have is a reminder that the capital k a is going to have units of inverse molarity. So over here we can put in inverse molarity. And so, this here is a lot of information and as we move forward in our course and continue to talk about these protein Ligand Equilibrium Constants, we'll be able to, piece a lot of this information together and make it make even more sense. And so this here is the conclusion to our introduction of the protein ligand association equilibrium constant, capital k a, and I'll see you guys in our next lesson video where we'll talk about the dissociation Equilibrium Constant, Capital Kd.
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Protein-Ligand Equilibrium Constants
Video duration:
7m
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Video transcript
So now that we've introduced the first protein ligand association equilibrium constant, capital k a, in our last lesson video. In this video we're going to move on to our second protein ligand equilibrium constant, which is the protein ligand dissociation equilibrium constant, capital Kd, which again is different than lower case kd, which is the dissociation rate constant, not equilibrium constant. And so this dissociation equilibrium constant, capital k d, is literally an equilibrium constant itself. Except this time, instead of being the equilibrium constant for the association, it's going to be the equilibrium constant for the dissociation of the protein ligand complex backwards to form the free protein and the free ligand. Now, at this point we already know from our last lesson video that the kd and the ka are just reciprocals of each other. And so because we already know that k a has units of inverse molarity, this means that the kd is just going to have units of molarity. And so, again, we already know that both the k d as well as the k a are used to express the protein's affinity for the ligand. But it turns out that the k d is actually used much much more often than the k a to express protein affinity for ligand. But why is it that the kd would be used more often than the ka? Well, one of the reasons has to do with the fact that the ka has units of inverse molarity. And inverse molarity is a little bit tough for our brains to process, whereas the k d has units of just molarity. And we know that molarity is just a unit of concentration and it's much easier for our brains to process units of molarity than units of inverse molarity. And so, the second reason why the k d is used more often than the k a is because the k d actually ends up resembling one of the variables that we already covered in our previous lesson videos, which is the Michaelis constant, k m. And we'll talk about this a little bit more after we digest this image down below. And so notice over here on the left hand side of this image, what we have is protein ligand interaction that we had in our previous lesson videos. And notice again that lowercase k a and lowercase k d are the rate constants. Whereas capital k a and capital k d are going to be the equilibrium constants. So these are different than each other. And so over here on the right, what we have is the dissociation equilibrium constant, capital k d, and so it's going to be an equilibrium constant. So we know it's gonna be the ratio of the concentration of products over the concentration of reactants. And so for this backwards dissociation the products are going to be the free protein and the free ligand. So we can add those in here, free protein and free ligand. And then of course the reactant is going to be the protein ligand complex. And of course because the kd, we know, is really just the reciprocal of the Ka, then it's also going to be expressed as the dissociation rate constant over the association rate constant. So it's gonna be k lowercase kd over lowercase k a. And, of course, because again, the KD and the KA are reciprocals of each other, we can say that the KD is just going to be the reciprocal of the KA. So the reciprocal of the k a is just one over k a, and this is the capital k a. And, of course, as we already indicated up above, the KD is going to have units of just molarity which is again, much easier for us to process than units of inverse molarity. So moving forward in our course, we're mainly going to be talking about the k d and not so much the k a. And so as we briefly mentioned earlier in our video, it turns out that this kd is very very similar to the Michaelis constant, from our previous lesson videos which is the k m. And so the k d and the k m are going to be very very similar to each other as we'll see, here in a moment. And so, both the KD and the Kilometers are inversely proportional to the affinities that they represent and so the KD and protein affinity for the ligand are going to be inversely proportional instead of being directly proportional like the k a was. And so, therefore, what this means is that the smaller the value of the k d, the stronger the affinity the protein is gonna have for that ligand, which is a very similar relationship that the k m has to an enzyme's affinity for substrate. Now, also, similar to the way that, the Michaelis constant Kilometers is equal to an exact substrate concentration that allows the initial reaction velocity to equal half of the Vmax. The kd, because it's also in units of molarity, it also represents a specific concentration. It represents the exact concentration of ligand that allows for half of the ligand binding sites to be occupied. And so, we can actually see this down below in our image over here on the left. So, notice that we have a bunch of these proteins here and, notice that we also have some ligand that's present. And so when the ligand concentration is exactly equal to the kd, which is, units in units of molarity, that means that exactly 50% of all of the protein binding sites are going to be full. And we can see here that this is exactly true, that we have 3 of the 6 protein binding sites occupied or full of ligand when the concentration of ligand equals the k d. And so, you can see graphically here if we graph the binding percentage of the protein over here on the y axis and the ligand concentration on the x axis that the KD has a very similar, relationship to the Kilometers. So, it represents the exact ligand concentration that allows for the binding to be at 50%. And so again, this is very very similar to the Michaelis Constant Kilometers from our previous lesson videos. And so, in our next video, we'll be able to talk more about this relationship of k d with protein binding affinity. But for now, this concludes our introduction to protein ligand dissociation equilibrium constant and we'll be able to get some practice later in our course. So, I'll see you guys in our next video.
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Problem
Problem
Protein A has a binding site for ligand X with a K d of 54 mM. Protein B has a binding site for ligand X with a Kd of 58 mM. Answer the following questions based on this information:
A) Which protein has a stronger affinity for ligand X?
B) Convert the Kd to Ka for both proteins.
Ka for Protein A: __________
Ka for Protein B: __________
A
Protein A.
B
Protein B.
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Problem
Problem
You prepare a solution of protein and its ligand where the initial concentrations are [P] = 10 mM and [L] = 10 mM. At equilibrium you measure the concentration of the complex [PL] = 5 mM. If the protein-ligand reaction can be represented by P + L ⇌ PL, what is the Kd of the reaction under these conditions?