How does one derive a KD from an equilibrium titration experiment? I am definitely making a mistake somewhere

Any help would be **greatly** appreciated. Thank you in advance.


If I have an antibody A and a target B, and experimentally titrate the antibody against a single concentration of B, and then measure the % of B that is bound after the solutions reach equilibrium, I should be able to determine the KD of the interaction.


Source below (and others) shows that determining KD is as simple as determining the EC50 of such a curve. from:<https://www.sciencedirect.com/topics/nursing-and-health-professions/maximum-binding-capacity> 50% occupancy
[![enter image description here](https://i.stack.imgur.com/OBvmg.png)](https://i.stack.imgur.com/OBvmg.png)


How does that square with the following equation for KD?


KD = ([A]\*[B]) / [AB] where A is unbound antibody, B is unbound antigen, and AB is the bound complex


Let's say I have 10 pM of target antigen, 50% of which is bound when incubated to equilibrium with 10 pM of antibody. That would mean the antibody and antigen are each 50% bound and 50% unbound.


KD = 5pM \* 5pM / 5pM KD = 5 pM


So I have calculated a 5 pM KD, not the 10 pM that I would think would be similar to the graph below (assuming they had pM units on the X axis) I'm assuming these types of example graphs refer to the input concentration, right?


Even more confusing, what if the antibody is a really tight binder, such that 10pM of antibody is able to essentially bind 100%, such that 10 pM antibody is able to bind 50% of 20pM of antigen. (1:1 binding assumed)


KD = ~0.001pM \* ~10pM / ~10pM


Now you have a KD of 0.001pM, but the graph would suggest that 10 pM of antibody is necessary to bind to 50% of the target antigen.


I'm sure I'm missing something. If anyone could help it would be greatly appreciated.





> 
> So I have calculated a 5 pM KD, not the 10 pM that I would think would be similar to the graph below (assuming they had pM units on the X axis) I'm assuming these types of example graphs refer to the input concentration, right?
> 
> 
> 


Your calculation is correct. The concentration in the graph refers to the free concentration of ligand of the receptor. The receptor will be present at very low concentration, and the ligand is often cheap (if it is an approved drug), so it is fine to use it in large excess. In those cases, free ligand concentration is similar to total ligand concentration. However, the graph could be labelled more accurately.



> 
> Let's say I have 10 pM of target antigen, 50% of which is bound when incubated to equilibrium with 10 pM of antibody. That would mean the antibody and antigen are each 50% bound and 50% unbound.
> 
> 
> 


In the case of antibody and antigen, both binding partners might be expensive, so using them at a 1:1 concentration ratio might make sense. In this case, you would calculate the free concentration like you did. So when you have equal concentrations of complex and free antibody, the concentration of free antigen is equal to the dissociation constant. Or when you have equal concentrations of complex and free antigen, the concentration of free antibody is equal to the dissociation constant. In your example, that would be 5 pM.


### The graphs


The graph has a couple of issues. There should be a unit on the concentration axis (the y-axis is a percentage of full binding, so that is correct without units, but should have a percentage sign). The concentration axis should be labeled as "free ligand concentration". Often, as in the graph below, the free ligand concentration is simply written as $[\mathrm{L}]$. If you want to specify the initial (or total) concentration, the symbol you would use is $[\mathrm{L}]\_\mathrm{total}$.



> 
> [![enter image description here](https://i.stack.imgur.com/Vd64a.png)](https://i.stack.imgur.com/Vd64a.png)
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The shape of the graph looks different because the concentration axis is linear, not logarithmic as in the graph posted in the question. The source for the graph and explanation is [http://people.reed.edu/~glasfeld/Chem391/notes/ChLigand\_2015.pdf](http://people.reed.edu/%7Eglasfeld/Chem391/notes/ChLigand_2015.pdf).


If you google "graph occupancy ligand concentration", you will find that many graphs do not carefully distinguish between free ligand concentration and total ligand concentration. This is probably because these two concentration are quite similar when the ligand is in large excess over the binding protein. There is also the occasional confusion between dissociation and association constant, so you have to use your critical thinking skills when coming across figures in the wild (and elsewhere).


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