Estimating protein binding and disassociation

I don't have a background in the area of drugs or pharmacokinetics/Pharmacodynamics but I am trying to understand about protein binding. I was going through this paper.

If $$C_b$$ is the concentration of the bound drug

$$C_f$$ is the concentration of the free drug

$$C$$ is the total drug concentration

$$P$$ is the concentration of protein binding sites (bound and free)

$$K_d$$ is the disassociation constant of the drug protein complex

For a one compartmental model the free drug can be found through:

$$C_f=\dfrac{-(P+K_d-C)+\sqrt{(P+K_d-C)^2+4 K_d C}}{2}$$

The assumptions used in this derivation are:

1. Binding occurs only to plasma proteins and follows simple saturation kinetics. The binding process can be described by a single macroscopic dissociation constant.

2. Binding equilibrium is achieved virtually instantaneously with respect to distribution and elimination.

3. Aside from binding, all other processes (ie., distribution and elimination) are linear.

4. Distribution and elimination processes operate only on free drug.

My question is in how to determine the parameter values of $$P$$ and $$K_d$$.
The article has given certain properties of the drug but not a way in how these parameters are found.

The given details on a drug are:

1.The drug has a molecular weight of 150.

1. Drug binds only to serum albumin, whose concentration is 4.4% and whose molecular weight is 67,000.

2. At sufficiently low drug concentrations, the drug is 89% bound.

3. The distribution volume is 50 ml./kg.

4. Free drug has a half-life of 30 min.

Using these details is there a way to estimate the parameters of $$P$$ and $$K_d$$ and if so please explain how they can be derived.

1 Answer

If we call the drug L, and the binding sites S, the equilibrium is the following:

$$\ce{LS <=> L + S}$$

The equilibrium constant expression is:

$$K_d = \frac{[\ce{L}] [\ce{S}]}{[\ce{LS}]}$$

With $$[\ce{L}]_\mathrm{tot}$$ as the total concentration of ligand and $$[\ce{S}]_\mathrm{tot}$$ the total concentration of binding sites, we can solve of $$[\ce{L}]$$, the free ligand concentration to arrive at a quadratic expression like the one in the question.

If the ligand concentration is small compared to the concentration of available binding sites, you can estimate the ratio of free to bound ligand:

$$\frac{[\ce{L}]}{[\ce{LS}]} \approx \frac{K_d}{[\ce{S}]_\mathrm{tot}}\tag{1}$$

Now, we can look at the statements 1 through 5 and see what information we can extract to determine $$K_d$$ and $$[\ce{S}]_\mathrm{tot}$$:

1.The drug has a molecular weight of 150

This information is not directly relevant. I'll get back to it later.

2. Drug binds only to serum albumin, whose concentration is 4.4% and whose molecular weight is 67,000

This allows us to calculate the concentration of protein capable of binding to the drug. 4.4% refers to 0.044 kg of albumin in a solution with a volume of one liter. With a molar mass of M = 67000 g/mol, we get the following concentation:

$$c(\mathrm{albumin}) = \frac{\pu{44 g / L}}{\pu{67000 g/mol}} = \pu{0.66 mM}$$

If we knew the number of binding sites for each molecule of albumin, this would also give us the concentration of sites.

3. At sufficiently low drug concentrations, the drug is 89% bound

If 89% are bound, 11% are free, and the ratio of free to bound is 0.12. "At sufficiently low drug concentration" means the drug concentration is low with respect to the concentration of binding sites, so we can use equation (1):

$$\frac{K_d}{[\ce{S}]_\mathrm{tot}} \approx \frac{[\ce{L}]}{[\ce{LS}]} = 0.12$$

So we can estimate the ratio of dissociation constant and total concentration of sites.

4. The distribution volume is 50 ml/kg

This means for every kg of body weight, the apparent volume in which the drug dissolves is 50 mL. Picture a solvent of unknown volume. You dissolve a solute and then remove some solution to test the solute solution. You should be able to calculate the unknown volume from the amount of solute added and the concentration obtained. If, however, the solute absorbs on the container wall or diffuses through it, your answer is inaccurate.

The human body has an average of 70 mL of blood for each kilogram of body mass. A substance that remains in the blood stream should have a distribution volume of 70 ml/kg if it does not bind to anything.

I'm not sure what to do with this piece of information.

5. Free drug has a half-life of 30 min.

This ensures that the binding reaction has a chance to almost attain equilibrium. If this were not the case, we could not use equilibrium calculations, and would have to resort to non-equilibrium methods such as the ones described in the paper cited by the OP (which is behind a paywall, so I just read the abstract).

Using these details is there a way to estimate the parameters of P and Kd and if so please explain how they can be derived.

The missing piece of information is the concentration of sites. If we knew the number of sites on each albumin molecule, we could calculate this concentration because we know the concentration of albumin.

Experimentally, what we are missing is data for saturating conditions. As it is, we have one equation for two unknowns, so we only know their ratio. If you assume one binding site per albumin molecule, you would be done. However, looking at the molar masses, it is entirely possible for more than one drug molecule to bind to a single albumin molecule.

• Thanks a lot for the explanation. So, if there is only one binding site per albumin, is it correct to say, $[S]_{tot}=c(albumin)$ – sam_rox Jan 6 '20 at 4:37
• Also, is the molar mass of the drug gets used when converting mM to say mg/L? Is it the drugs molar mass that needs to be used when converting the units of K_d\$ as well? – sam_rox Jan 6 '20 at 4:46
• @sam_rox yes and yes – Karsten Theis Jan 6 '20 at 22:44