# Find the concentration of free receptors in a solution of 90% free ligand and 10% receptor-ligand complex, knowing only the value of kD

I could use some guidance on how to utilise the equation for the disassociation constant kD to find the concentration of free receptors [R] in a solution containing 90% free ligand [L], and 10% bound [RL] complex.

The Kd of the receptor-ligand interaction = 10^-8 mol l-1

Where Kd = ([R][L])/([RL])

I assume as a percentage I could state, [L] = 0.9 and [RL] = 0.1, but am not sure if I'm taking the right approach.

Any clarification on this would be appreciated.

One big thing you're missing is units: Concentrations here must have concentration units - they can't be dimensionless numbers or percentages.

So one of the things you need to figure out is how to convert the percentages into concentration units. In this case, it's difficult to do directly, but we can apply a bit of algebra to figure it out. We're talking about 90% and 10% of the ligand in different forms, but we don't know anything about the total amount of ligand - that's alright, we'll just represent the total concentration of the ligand as $\ce{x mol \cdot l^{-1}}$. Therefore, we can now state that:

$$\ce{ [L] = (0.9x) mol \cdot l^{-1}}$$ $$\ce{ [RL] = (0.1x) mol \cdot l^{-1}}$$

Now the other bit of information we have is the equation for Kd:

$$\ce{ K_d = \frac{[R][L]}{[RL]} }$$

So we're looking for $\ce{[R]}$ (with associated concentration units!), and we know $\ce{ K_d }$, $\ce{[L]}$ and $\ce{[RL]}$ - or at least up to the unknown of $\ce{x}$. But, if you substitute the values you do know (including the factor of $\ce{x}$) into the equation and do some algebraic manipulation, I believe you'll find it doesn't actually matter what the value of $\ce{x}$ is - it should cancel out.

Note: If the value of $\ce{x}$ doesn't matter, why did I bother with the whole rigmarole of including it? Why didn't I just substitute 0.9 and 0.1 directly?

Well, the answer to that is my lead sentence - the concentrations in the equation should have (compatible) concentration units. If you don't have concentrations, you're using the equations incorrectly. (Look into the concept of dimensional analysis for a valuable way to help make sure you're doing many (but not all) calculations like this correctly.)

While for this specific problem things worked out nicely if you substitute the fractions/percentages directly, that's not a general solution. Depending on the exact setup of the problem, the $\ce{x}$s may or may not cancel cleanly. You may have to use some other equation/knowledge to put additional constraints on the value of $\ce{x}$. The only way to know that is to properly account for them from the start.

According to the information you have provided isn't the starting equation:

1 x 10EE-8 = (0.9 x [R]) / 0.1

So you can solve for [R]?

[R] = (1 x 10EE-8 x 0.1) / 0.9

I am not saying this is correct, but by substituting the values you provided isn't this how one would algebraically solve for an unknown? Maybe I am missing something(?)

You state you wish to know the concentration of free receptor. Wouldn't you need to know the total number of moles of receptor present in the equilibrium binding reaction, and wouldn't you need to know the final volume of the reaction?

• Thanks for the input, I must admit I'm total perplexed by the question, and I agree there seems to be some data missing to solve but thats the info that's been provided. I hoped someone might have come across something similar.
– Kim
Apr 20 '16 at 3:19