# 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

## 1 Answer

When a reaction is at equilibrium, there will be stochastic fluctuations. These are usually not measurable (for example, if we have more than a mole of products and reactants in our reaction mixtures).

In this specific case, the fluctuations are easy to measure because the total number of antigen molecules per cell is countable, and you are separating cells (I presume) before analyzing the number of antibodies bound.

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 $$N$$.

• Is it true, that the antigens produced by one cell of the same type can be used in a neighbor cell, ie can it pass into solution? – ThermoRestart Apr 30 at 16:05
• The antigen is covalently linked to the cell surface. The antibodies are shared in this experiment. – Karsten Theis Apr 30 at 16:07
• Thanks @KarstenTheis – ThermoRestart Apr 30 at 16:07