In Guillawme/rfret: Analyze FRET Binding Data

This document explains how the quadratic model equation used to fit binding data from a FRET experiment is derived from the law of mass action.

Binding equilibrium

In this document, we will call M the macromolecule, L the ligand, ML the macromolecule/ligand complex, and [M], [L] and [ML] their molar concentrations at equilibrium. Moreover, we will call $M_{tot}$ the total concentration of macromolecule, and $L_{tot}$ the total concentration of ligand. We have, by definition because of mass conservation:

1: $M_{tot} = [M] + [ML]$ 2: $L_{tot} = [L] + [ML]$

The binding equilibrium is represented by the following reaction:

$M + L \rightleftharpoons ML$

The equilibrium binding constant $K_D$ is defined by the following equation (law of mass action):

3: $K_D = \frac{[M] \times [L]}{[ML]}$

Determining $K_D$ is the goal of a binding assay, like a FRET titration experiment.

Deriving the quadratic binding equation

To determine $K_D$, we need to measure [ML] at equilibrium. [M] and [L] can be expressed as a function of [ML] from equations 1 and 2, and $M_{tot}$ and $L_{tot}$ are known parameters of the experiment.

Rewriting equation 3 gives us:

4: $K_D \times [ML] - [M] \times [L] = 0$

By substituting into equation 4 [M] and [L] using equations 1 and 2, we have:

$K_D \times [ML] - (M_{tot} - [ML]) \times (L_{tot} - [ML]) = 0$

We can now solve this equation for [ML]:

$K_D \times [ML] - (M_{tot} \times L_{tot} - M_{tot} \times [ML] - L_{tot} \times [ML] + [ML]^2) = 0$

$K_D \times [ML] - M_{tot} \times L_{tot} + M_{tot} \times [ML] + L_{tot} \times [ML] - [ML]^2 = 0$

5: $[ML]^2 - (K_D + M_{tot} + L_{tot}) \times [ML] + M_{tot} \times L_{tot} = 0$

Equation 5 is of the form $ax^2 + bx + c = 0$, of solutions $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

This gives us [ML] expressed as a function of $K_D$ and the known parameters $M_{tot}$ and $L_{tot}$:

6: $[ML] = \frac{(K_D + M_{tot} + L_{tot}) \pm \sqrt{(-(K_D + M_{tot} + L_{tot}))^2 - 4 \times M_{tot} \times L_{tot}}}{2}$

Using the quadratic binding equation to fit a binding curve

If we have a detectable signal $S$ proportional to [ML] (like FRET), we can express it as a function of its minimal and maximal values ($S_{min}$ and $S_{max}$) and the fraction of ligand bound $\frac{[ML]}{M_{tot}}$:

7: $S = S_{min} + (S_{max} - S_{min}) \times \frac{[ML]}{M_{tot}}$

With this expression, $S = S_{min}$ when $\frac{[ML]}{M_{tot}} = 0$ and $S = S_{max}$ when $\frac{[ML]}{M_{tot}} = 1$.

Substituting equation 6 into equation 7 gives:

8: $S = S_{min} + (S_{max} - S_{min}) \times \frac{(K_D + M_{tot} + L_{tot}) \pm \sqrt{(-(K_D + M_{tot} + L_{tot}))^2 - 4 \times M_{tot} \times L_{tot}}}{2 \times M_{tot}}$

In equation 8, $S_{min}$ and $S$ can be measured experimentally: they are, respectively, the signal observed without ligand, and across the titration series at given values of $L_{tot}$. The observed signal is, indeed, a function of the total ligand concentration. $M_{tot}$ and $L_{tot}$ are known experimental parameters. Therefore, $K_D$ and $S_{max}$ can be determined by fitting equation 8 to the experimental data. In practice, $S_{min}$ is also obtained from fitting the equation to the experimental data.

Guillawme/rfret documentation built on Dec. 17, 2021, 10:24 p.m.