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A1. Reversible Binding of a Ligand to a Macromolecule

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  • CASE 2: WHEN FREE \(L\) IS NOT KNOWN (i.e., \(L_o \cancel{\gg} M_o\)) OR YOU WISH TO CALCULATE \(ML\) FROM JUST \(L_o\), \(M_o\) AND \(K_D\)

    Substitute Equations \ref{2} and \ref{3} into Equation \ref{1}:

    \[\begin{align} K_d &= \dfrac{[M][L]}{[ML]} = \dfrac{[M_o-ML][L_o]-[M_L]}{[ML]} \nonumber \\[5pt] [ML]K_d &= ([M_o] - [ML])([L_o] - [ML]) \nonumber \\[5pt] [M_L]K_d &= [M_o][L_o] - [ML][L_o] - [ML][M_o] + [ML]^2 \label{8} \end{align} \]


    \[ [M_L]^2 - (L_o + M_o +K_d)[ML] + [M_o][L_o] = 0 \label{9} \]

    which is of the form \(ax^2 + bx + c = 0\), where

    • \(a = 1\)
    • \(b = - ([L_o] + [M_o] +K_d)\)
    • \(c = [M_o][L_o]\)

    which are all constants, and

    \[x = \dfrac{-b \pm (b^2 - 4ac)^{1/2}}{2a} \]


    \[M_L = \dfrac{(L_o+M_o+K_d) - ((L_o+M_o+K_d)^2 - 4M_oL_o)^{1/2}}{2} \label{10}\]

    Wolfram Mathematica CDF Player - Interactive Graph of ML at various Lo, Mo, and \(K_d\) values (free plugin required)

    A graph of ML calculated from this formula vs free L (or Lo if Lo >> Mo) give a A HYPERBOLA.

    Play around with the sliders. If you set \(K_d\) to a very low number and vary Mo, you will see a curve very much like a titration curve with a sharp rise and abrupt plateau that occurs when \(M_o\) is approximately equal to \(L_o\).


    In the derivations, we came up with two equations for ML:

    • one (Equation \ref{5}) using mass conservation on M, which gave: \[ML = \dfrac{M_oL}{[K_d +L]} \nonumber\]
    • one (Equation \ref{10}) using mass conservation on M and L, which gave ML = quadratic equation as function of Mo, Lo, and Kd: \[ML = \dfrac{-(L_o+M_o+K_d) \pm ((L_o+M_o+K_d)^2 - 4M_oL_o)^{1/2}}{2} \nonumber\]

    Both equations are valid. In the first you must known free L which is often Lo if Mo << Lo. In the second, you don't need to know free M or L at all. At a given Lo, Mo, and Kd, you can calculate ML, which should be the same ML you get from the first equation if you know free L.

    Equations \ref{5} and \ref{10} are useful in several circumstances. They can be used to

    • calculate the concentration of ML if Kd, Mo, and L (for Equation \ref{5}) or if Kd, Mo, and Lo (for Equation \ref{10}) are known. This is analogous to the use of the Henderson-Hasselbach equation to calculate the protonation state (HA) and hence charge state of an acid at various pH values. In the former case we are measuring the concentration of bound ligand (ML) and in the later case, the concentration of bound protons (HA).
    • calculate \(K_d\) if ML, Mo, and L (for Equation \ref{5}) or if ML, Mo, and Lo (for Equation \ref{10}) are known. Techniques to extract the \(K_d\) from binding data will be discussed in the next chapter section.