Binding Assay Quadratic

For the Binding Assay, with P = Initial Protein and L = Initial Ligand, and PL = x = amount of bound protein, we have the equilibrium expression:

\(K_d = \frac{([P]-x)([L]-x)}{x}\)

if we solve for x and divide by L, we will get fraction bound.

\(K_d = \frac{[P][L] -[P]x -x[L] + x^2}{x}\)

\(K_dx = [P][L] -[P]x -x[L] + x^2\)

\([P][L] -[P]x -x[L] -K_dx + x^2 = 0\)

\([P][L] -(K_d + [P] + [L])x + x^2 = 0\)

The quadratic solution, divlded by [L],

\(\frac{x}{L} = \frac{(K_d + [P] + [L]) - \sqrt{(K_d + [P] + [L])^2-4([P]+[L])}}{2[L]}\)

source("samscript.R", local = knitr::knit_global())