# Chemical Equilibria - Fraction Bound equation based on Kd, [R], [L], [RL], is actually an average?

Let's say 150 stem cells express exactly 150 copies of the same mebrane antigen X. While all differentiated cells express exactly 15 copies of X. To the cell mixture you add a large amount of 0.2 μM solution of antibody to X. The $$K_D=$$ 4 μM.

If you experimentally analyze the number of antibodies bound per stem cell, you will find that it is not constant. Explain why, even for cases where the number of antigens expressed is the same, the binding follows a binomial distribution?

## I found the fraction bound to be $$\frac{1}{\frac{K_D}{\left[L\right]}+1\:}=f_{bound}$$

• I scaled the [L] contribution by the number of antigens expressed per cell, so for the differentiated case, I scaled 0.2 μM by 15, and for the normal case, I scaled 0.2 μM by 150.
• the differentiated case has a lower $$f_{bound}$$ because it has less ligands.

To describe the binomial distribution for binding part

• the antigens exist in the environment of the beaker (or wherever the reaction is taking place) so they can exchange antigens? resulting in some cells $$f_{bound}$$ being lower. How exactly the interaction takes place occurs stochastically. <--- want help clarifying this part

If you expect a fluctuation of $$\sqrt{N}$$ for N objects, that is significant for $$N= 150$$ but not for $$N = \pu{6.022e23}$$ when expressed relative to $$N$$.