Protein-Ligand Equilibrium Constants: Study with Video Lessons, Practice Problems & Examples

The protein-ligand interaction involves two key equilibrium constants: the association constant (K_A) and the dissociation constant (K_D). K_A measures the affinity of a protein for a ligand, with higher values indicating stronger affinity, while K_D is its reciprocal, representing the concentration of ligand at which half of the binding sites are occupied. K_D is often preferred due to its simpler molarity units and similarity to the Michaelis constant (K_M), which also indicates substrate affinity in enzymatic reactions.

Protein-Ligand Equilibrium Constants

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 lowercase k_a, 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 just the ratio of the concentration of over the concentration of reactants. And this association equilibrium constant, capital K_a, 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 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 anyway because we know that not all proteins are going to be acids anyway. And so this association equilibrium constant, capital K_a, is actually a measure of protein affinity for a ligand. And so, the K_a 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 K_a 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 k_d, 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 K_d. So essentially, 1/K_d. And again, we'll revisit this idea more when we talk about the dissociation constant, K_d 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 k_d 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 going to be P and L. 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 K_a, capital K_a. 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 k_d, as a ratio just like this, k_a over k_d, also represents the association equilibrium constant capital K_a. And then, of course, as we said up above in this line, the K_a is the reciprocal of the dissociation constant and so, we

Protein-Ligand Equilibrium Constants

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 \( K_d \), which again is different than lowercase \( k_d \), 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 \( K_d \) and the \( K_a \) are just reciprocals of each other. And so, because we already know that \( K_a \) has units of inverse molarity, this means that the \( K_d \) 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 more often than the \( K_a \) to express the protein affinity for the ligand. But why is it that the \( K_d \) would be used more often than the \( K_a \)? Well, one of the reasons has to do with the fact that \( K_a \) 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 from 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 going to 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 \( K_d \), we know, is really just the reciprocal of the \( K_a \), then it's also going to be expressed as the dissociation rate constant over the association rate constant. So it's going to be \( k_{kd} \) over \( k_{ka} \). And, of course, because again, the \( K_d \) and the \( K_a \) are reciprocals of each other, we can say that the \( K_d \) is just going to be the reciprocal of the \( K_a \). So the reciprocal of the \( K_a \) is just \( \frac{1}{K_a} \), and this is the capital \( K_a \). And, of course, as we already indicated up above, the \( K_d \) 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 \( K_d \) is 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 similar to each other as we'll see, here in a moment. And so, both the \( K_d \) and the \( K_m \) are inversely proportional to the affinities that they represent and so the \( K_d \) 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 going to 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 \( K_m \) is equal to an exact substrate concentration that allows the initial reaction velocity to equal half of the \( V_{max} \). The \( K_d \), 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 \( K_d \), which is 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 \( K_d \) has a very similar relationship to the \( K_m \). So, it represents the exact ligand concentration that allows for the binding to be at 50%. And so again, this is very similar to the Michaelis Constant \( K_m \) from our previous lesson videos.

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 the 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.

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 K_d of 58 mM. Answer the following questions based on this information:

A) Which protein has a stronger affinity for ligand X?

B) Convert the K_d to K_a for both proteins.

K_a for Protein A: __________ 

K_a for Protein B: __________

• A. Protein A.
• B. Protein B.

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 K_d of the reaction under these conditions?

• A. 0.05 mM.
• B. 20 nM.
• C. 5 mM.
• D. 25 uM.
• E. 20 mM.
• F. 25 mM.

Here’s what students ask on this topic:

What is the difference between protein-ligand association constant (KA) and dissociation constant (KD)?

The protein-ligand association constant (KA) and dissociation constant (KD) are both equilibrium constants but represent opposite processes. KA measures the affinity of a protein for a ligand, with higher values indicating stronger affinity. It is defined as the ratio of the concentration of the protein-ligand complex to the product of the concentrations of the free protein and free ligand. In contrast, KD is the reciprocal of KA and represents the concentration of ligand at which half of the binding sites are occupied. KD is often preferred because it has units of molarity, making it easier to interpret. Mathematically, KA = 1/KD.

Why is the dissociation constant (KD) used more often than the association constant (KA)?

The dissociation constant (KD) is used more often than the association constant (KA) for two main reasons. First, KD has units of molarity, which are easier for our brains to process compared to the inverse molarity units of KA. Second, KD is similar to the Michaelis constant (KM) used in enzymology, which also has units of molarity and represents substrate affinity. This similarity makes KD more intuitive and easier to understand in the context of biochemical reactions.

How are the equilibrium constants KA and KD related to protein affinity for a ligand?

The equilibrium constants KA and KD are directly related to protein affinity for a ligand but in opposite ways. KA is directly proportional to the protein's affinity for the ligand; a higher KA indicates a stronger affinity. Conversely, KD is inversely proportional to the protein's affinity for the ligand; a lower KD indicates a stronger affinity. Mathematically, KA = 1/KD, meaning that as one increases, the other decreases.

What units are used for the association constant (KA) and dissociation constant (KD)?

The association constant (KA) has units of inverse molarity (M^-1), while the dissociation constant (KD) has units of molarity (M). These units reflect the nature of the constants: KA measures how readily a protein and ligand form a complex, whereas KD measures the concentration of ligand needed to occupy half of the protein's binding sites.

How does the dissociation constant (KD) relate to the Michaelis constant (KM)?

The dissociation constant (KD) is similar to the Michaelis constant (KM) in that both are used to describe affinities in biochemical reactions and have units of molarity. KD represents the concentration of ligand at which half of the protein's binding sites are occupied, while KM represents the substrate concentration at which the reaction velocity is half of its maximum (Vmax). Both constants provide a measure of how tightly a protein or enzyme binds to its ligand or substrate.