Protein-Ligand Fractional Saturation - Online Tutor, Practice Problems & Exam Prep

Fractional saturation (θ or y) represents the fraction of occupied ligand binding sites on a protein, calculated as the ratio of bound proteins to total protein content. It ranges from 0 (no binding) to 1 (fully saturated). The dissociation equilibrium constant (K_d) indicates the ligand concentration at which θ equals 0.5, reflecting protein affinity. The maximum θ value is always 1 for protein-ligand interactions. An alternative expression for θ relates it to K_d and resembles the Michaelis-Menten equation, emphasizing the connection between binding affinity and saturation dynamics.

Protein-Ligand Fractional Saturation

In this video, we're going to begin our discussion on protein ligand fractional saturation. So first, we need to define fractional saturation, which can actually be abbreviated with 2 different variables. The first is the Greek letter theta, and the second is the capital letter Y. And so some textbooks use theta, other textbooks use Y, but on your exam or in some practice problems, you might find that either theta or Y could be used to represent the fractional saturation. So it's important for us to be able to recognize both of these variables. However, moving forward in our course, I'm mainly going to be using theta to represent the fractional saturation only because it's a little bit more unique and visually distinguishable. But I'll also be trying to remind you guys that every time you see theta in our lesson, we could easily replace the theta with capital Y. And so what exactly is this fractional saturation anyway? Well, it's just the fraction of occupied, or in other words, the fraction of saturated ligand binding sites in a protein sample. And so when a ligand binding site on a protein is occupied or saturated, it just means that that ligand binding site is bound to a ligand. And so this fractional saturation theta or Y is really just a ratio. And really it's just the ratio of occupied proteins over the total protein content. And so notice that's exactly what we're seeing down below in our image in our first box. So, we can see that the fractional saturation, theta or Y, is really just equal to this ratio of the occupied proteins, which are just the proteins that are bound by Ligand, over the total protein content. And so, notice that that's exactly what this ratio over here is expressing as well. So, the proteins that are bound by ligand is just going to be the concentration of protein Ligand complex. And then the total protein is just going to be the concentration of protein Ligand complex, plus the concentration of free protein.

Now, it's also important to recognize that this expression right here for the fractional saturation actually resembles the rectangular hyperbola equation, which recall that we covered way back when we talked about the Michaelis-Menten equation. And so we can see that in this rectangular hyperbola equation, the x's here represent the protein ligand concentration, and, the b represents the concentration of free protein, and then, of course, this A right here must be equal to a value of 1 in our equation up above because it's not being shown. And so notice that because it resembles a rectangular hyperbola that when we actually plot this data, we get a rectangular hyperbola like we see in this plot. Now, before we actually get down to this plot, it's important to note that this fractional saturation, theta or Y, can actually help us reveal the exact percentage of occupied ligand binding sites on a protein. And the values of the fractional saturation theta or Y are going to range from 0 at its minimum when there's absolutely no ligand that's bound to the protein. All the way up to the highest value of 1 when all of the protein binding sites are bound by ligand. And so, at this point, we already know. So, recall from our previous lesson videos that the dissociation equilibrium constant, capital Kd, has units of molarity and it's equal to the exact Ligand concentration that allows for 50% of all of the available Ligand binding sites to be occupied. And so, 50% corresponds with theta or the fractional saturation equaling to a value of 0.5. And so down below, this plot that we see over here on the right is actually referred to as a saturation curve or a protein ligand binding graph. And what it does is it plots the fractional saturation, theta or Y, on the Y axis. So, we can put theta here or Y. And so that's what's plotted on the Y axis and then on the X axis, notice that it plots the ligand concentration. And so, you can see that when the value of the fractional saturation theta or Y is equal to 0.5, this is a very unique and critical point on these saturation curves or protein ligand binding graphs. And that's because when theta is equal to 0.5, the ligand concentration that's associated with that reveals the Kd, which is a measure of the affinity that the protein has for the ligand. And so in our next video, we'll be able to continue to talk more about the fractional saturation and how it applies directly to protein ligand interactions. And so I'll see you guys in that video.

Protein-Ligand Fractional Saturation

So now that we've introduced protein-ligand fractional saturation, in this video we're going to focus on the maximum theta or the maximum value of the fractional saturation. It's important to note that for all protein-ligand interactions, the equivalent of the theoretical maximal reaction velocity \( v_{max} \) is always going to be 100% ligand binding. You can't have any more than 100% ligand binding. Therefore, 100% ligand binding is always going to correlate with the maximum value of theta, which is always equal to 1.

What we need to recall from our previous lesson videos is that the \( v_{max} \) can vary between different enzymes. Some enzymes have a smaller \( v_{max} \), while others have a much larger \( v_{max} \). However, when it comes to protein-ligand interactions, all protein-ligand interactions are always going to have a maximum theta value equal to 1, since you can never have more than 100% ligand binding.

Also recall from our previous lesson videos that the Michaelis constant \( K_M \) is very similar to the dissociation equilibrium constant \( K_d \). This means that the smaller the value of \( K_d \), the stronger the protein's affinity for that ligand. Taking a look at our image below, notice that we have this saturation curve here where we have the fractional saturation on the y-axis (either theta or y) and it ranges from 0 all the way up to 1. On the x-axis, we have the ligand concentration. Notice that we have these two different curves: a black curve for protein A and a blue curve for protein B. Both protein A and protein B have the same maximum value for theta, which is 1. This shows that regardless of the protein-ligand interaction we're looking at, all protein-ligand interactions are going to have a max theta of 1.

We also need to note that the example problem here is asking which protein has a stronger affinity to the ligand: protein A or protein B? Recall that the smaller the value of \( K_d \), the stronger the protein's affinity for that ligand. Therefore, if we're looking for a stronger affinity, we want to look for the smaller \( K_d \) value. Notice that the \( K_d \) value corresponding with the black curve is smaller than the \( K_d \) value corresponding with the blue curve. Thus, we can say that the smaller \( K_d \) belongs to protein A, and protein A therefore has a stronger affinity. We can indicate that option A here is the correct answer for this example problem. In our next lesson video, we'll talk about a different way to express the fractional saturation. I'll see you guys in that video.

Protein-Ligand Fractional Saturation

So now that we know that the maximum θ value is equal to 1, in this video we're going to talk about an alternative way to express θ. Through algebraic rearrangements and substitutions of previous equations, θ can also be defined in another way. This way mathematically relates θ to the kd, and it also resembles the Michaelis-Menten equation from our previous lesson videos.

Below on the right-hand side, you can see that we have the fractional saturation which is θ or y, and we already know that it's equal to the ratio of the proteins bound by ligand over the total concentration of protein. Everything up to this point is a review from our previous lesson video.

In this video, we are saying that this equation, through algebraic rearrangements and substitutions of previous equations, can actually be rearranged to this equation right here. This equation mathematically relates θ to the kd. So, we can relate θ to the value of the kd. Notice that this equation here actually resembles the Michaelis-Menten equation from our previous lesson videos.

All we need to do is substitute the substrate concentrations with ligand concentrations, substitute the Michaelis constant with the dissociation equilibrium constant kd, and then, of course, the maximum θ, we know that the equivalent of the maximum θ is the max θ. From our last lesson video, we know max θ is equal to 1. Technically here, we're supposed to have max θ, but we know max θ is equal to 1. So we could just put a one here. And of course, one times anything is just going to be that thing. So, we can cross this off here because it's being multiplied by 1, so it doesn't really matter.

Really, this equation right here is the alternative way to express the fractional saturation. In our next few videos, we'll be able to get some practice utilizing all of these concepts. So, I'll see you guys in our next video.

Protein-Ligand Fractional Saturation Example 1

Alright, so here we have an example problem that's asking us about antibodies which we haven't yet talked about, but we will talk about them more later in our course. But antibodies are just proteins that display protein-ligand interactions. And so this example problem says that if an antibody binds to an antigen, which is its ligand, with a kd=5×10-8 molar, what concentration of antigen or what concentration of ligand, will θ equal 0.2? And we've got these 4 potential answer options down below. And so notice down below here in the box, what we've got is the equation from our last lesson video for the fractional saturation.

Notice here we are actually given the value of θ and we're also given the value of kd. And so really what we need to solve for is the concentration of antigen or the concentration of ligand which is this variable down below. And so, essentially, what we can do is just use our algebra skills to essentially isolate this concentration of ligand all by itself. And so we can do that, by first, moving this entire denominator up to the left-hand side of the equation by multiplying both sides of the equation by the entire denominator. So, on the left-hand side, what we end up getting is θ×Y and then, on the right-hand side of the equation, we're getting rid of this whole thing. So, we're just left with the concentration of ligand and so we can fill in these ligands here.

Alright, and then, next, what we can do is distribute this θ here, right here and right here. So, just multiply it through. So, that's pretty simple. So, we can go ahead and do that. So, we'll get θ×Y+θ×kd and then on the right-hand side of the equation, we have the same exact thing. And so now, what we can do is, again, we're trying to isolate for this ligand. So, we want to put it on the same side of the equation. So, we can take this and subtract it from both sides of the equation. So, when we do that, we get θ×kd all by itself is equal to the concentration of ligand minus this portion right here that we're subtracting off. θ times the concentration of and so now what we have is the ability to factor out this concentration of ligand from this term and so when we do that, what we get is the concentration of ligand times when we factor it out from this component, it's just going to be 1, and then when we factor out from this component, we're just left with θ. And so the left-hand side of the equation is going to be exactly the same. θ×kd. And so, essentially, what you'll see when we factored out this concentration of ligand, to get it out front, you can check to see if it's the same as the top here by distributing it here.

So, concentration of ligand times 1 gives you the concentration of ligand and then the concentration of ligand times θ gives you θ×Y. So, this is the same thing as this, just a different format. And so now, you can see we have one variable for the concentration of ligand and so we just want to get rid of this whole thing. So, we can do that by dividing both sides of the equation by 1-θ and so what we end up getting is, we get θ×kd/1-θ is equal to the concentration of ligand. And so now we've got our equation rearranged so that all we need to do is plug in the variables that we're given and we'll get the concentration of ligand. Let's take this and, we're going to plug in up here. So, θ we're given as 0.2. So we can go ahead and plug that in, 0.2, and then the kd=5×10-8 molar, and that is given to us as 0.2 and this is all going to be equal to the concentration of ligand that we're looking for.

And so really all we need to do is use our calculators at this point and just crunch these numbers. So if you do 0.2 times 5 times 10 to the 8th and take the answer of that and divide it by 1 minus 0.2, what you'll get is our concentration of Ligand which is going to be equal to 1.25×10-8 molar. And of course, this is equal to the concentration of ligand. So this is the answer to our example problem and that matches with answer option A. So, we can indicate that A is the correct answer for this example problem and basically, you can just see how this equation here can be used and just rearranged to solve for whatever variable that we need to solve for. In this case, the concentration of ligand. So we'll be able to get some practice utilizing these concepts that we've learned moving forward. So I'll see you guys in our next video.

Consider the following graph for parts A-C.

A) What is the protein-ligand dissociation constant (K_d) for protein X?

a)  2 µM.      b) 4 µM.      c) 6 µM.      d) 8 µM.

B) What is the protein-ligand dissociation constant (K_d) for protein Y?

a)  2 µM.      b) 4 µM.      c) 6 µM.     d) 8 µM.

C) Which protein has a greater affinity for ligand A?

a) Protein X.       b) Protein Y.

• A. 2 µM.
• B. 4 µM.
• C. 6 μM.
• D. 8 μM.

Match the dissociation constants in the table below to the appropriate curves on the graph.

Alright. So this practice problem wants us to match the dissociation constants in the table down below over here to the appropriate curves on the graph. And so notice that this table has the protein names of 5 different proteins, A, B, C, D, and E, and then we're given the KD in units of molarity for each of these 5 proteins. And then on the right over here, we have our saturation curve or our protein ligand binding plot that has the fractional saturation θ on the y-axis and the concentration of ligand on the x-axis and we've got these 5 different colored curves that we see here in our plot that correspond with each of these 5 different proteins and their KDs. And so what I want you guys to notice first is that this x-axis down below for the concentration of ligand is not actually labeled with values. And we've actually done that intentionally because we want you to solve this problem conceptually more so, than trying to solve it mathematically with some kind of equation. And so, we actually do not need values at all in order to solve this problem for this X-axis. And so what we need to recall is that the KD is going to be the exact ligand concentration that allows for fractional saturation θ to be 0.5, which is going to be a value right in between 0.6 and 0.4. And so we can go ahead and draw in our little dotted line going across for 0.5 here. And, of course, where our curves meet this dotted line, where it corresponds on the x-value is going to represent the KD. And so, essentially, what we can see is that this blue curve is going to have the smallest KD followed by, this curve here, then this curve, and then, of course, the green curve, and last but not least, this pink curve, which will have the greatest KD. And so, really, that's all we need to do is order these KDs from smallest to greatest. And so the smallest KD is going to be protein D. So it has a value of 4×10-8. So that's the smallest KD, so we can put D here by the blue curve, which is going to have the smallest KD on this axis. The next smallest is going to be, protein B, which has a K D value of 1×10-7. So we can put B as next. Then followed, is going to be protein E, which has a slightly greater, K D than protein B at 9×10-7. So we can put protein E here. And then, the next one is going to be protein C, which has a value of 1×10-6. So, we can put that here. And then last but not least, the greatest value of the KD is actually going to be protein A with 2×10-6. So, we can put A down below. And so, really this is the correct answer that we were looking for matching each of the proteins their KDs with the appropriate curves that we have over here. And so, that concludes this practice and I'll see you guys in our next video.

Use the table below to answer questions A, B & C below.

A) Which protein has a greater affinity for their ligand?

B) According to the data in the table, what is the dissociation constant (K_d) for Protein 1? 

Protein 1 K_d = ________

C. According to the data in the table, what is the association constant (K _a) for Protein 2? 

Protein 2 K_a = ________

• A. Protein 1.
• B. Protein 2.

A sample of cells has a total protein-receptor concentration of 10 mM. 25% of the protein-receptors are occupied with ligand when the concentration of free ligand is 15 mM. Calculate the K_d for the receptor-ligand interaction.

• A. 5 mM.
• B. 67 mM.
• C. 45 mM.
• D. 7.5 mM.
• E. 2.5 mM.

Here’s what students ask on this topic:

What is fractional saturation in protein-ligand interactions?

Fractional saturation (θ or y) represents the fraction of ligand binding sites on a protein that are occupied by ligands. It is calculated as the ratio of the concentration of the protein-ligand complex to the total protein concentration, which includes both bound and unbound proteins. Mathematically, it can be expressed as:

$\frac{\mathrm{[PL]}}{\mathrm{[P]}+\mathrm{[PL]}}$

where [PL] is the concentration of the protein-ligand complex and [P] is the concentration of free protein. Fractional saturation ranges from 0 (no binding) to 1 (fully saturated).

How is the dissociation equilibrium constant (Kd) related to fractional saturation?

The dissociation equilibrium constant (Kd) is a measure of the affinity between a protein and its ligand. It is defined as the ligand concentration at which the fractional saturation (θ) is 0.5, meaning 50% of the ligand binding sites are occupied. A lower Kd value indicates a higher affinity, as less ligand is needed to achieve 50% saturation. The relationship between θ and Kd can be expressed as:

$\frac{\mathrm{[L]}}{\mathrm{Kd}+\mathrm{[L]}}$

where [L] is the ligand concentration.

What is the maximum value of fractional saturation (θ) in protein-ligand interactions?

The maximum value of fractional saturation (θ) in protein-ligand interactions is always 1. This corresponds to 100% ligand binding, meaning all available ligand binding sites on the protein are occupied. Unlike the maximum reaction velocity (Vmax) in enzyme kinetics, which can vary between enzymes, the maximum θ value is a constant 1 for all protein-ligand interactions.

How does the fractional saturation curve relate to the Michaelis-Menten equation?

The fractional saturation curve resembles the Michaelis-Menten equation used in enzyme kinetics. By substituting substrate concentration with ligand concentration, the Michaelis constant (Km) with the dissociation equilibrium constant (Kd), and Vmax with the maximum θ (which is 1), the equation for fractional saturation can be expressed as:

$\frac{\mathrm{[L]}}{\mathrm{Kd}+\mathrm{[L]}}$

This equation shows how θ varies with ligand concentration, similar to how enzyme velocity varies with substrate concentration in the Michaelis-Menten model.

What does a lower Kd value indicate about a protein's affinity for its ligand?

A lower dissociation equilibrium constant (Kd) indicates a higher affinity between a protein and its ligand. This means that a smaller concentration of ligand is required to achieve 50% fractional saturation (θ = 0.5). In other words, the protein binds more tightly to the ligand, making it more likely to form a protein-ligand complex even at lower ligand concentrations.