In Guillawme/rfret: Analyze FRET Binding Data

This document explains how the hyperbolic 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 hyperbolic model 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, $M_{tot}$ is a known parameter of the experiment. [L] is impossible to measure in a FRET assay, but under conditions where $L_{tot} >>> M_{tot}$ we can safely approximate [L] by $L_{tot}$.

From equation 3, we can express [ML] as a function of all other parameters:

$[ML] = \frac{[M] \times L_{tot}}{K_D}$

And from equation 1, we can replace [M] by $M_{tot} - [ML]$, giving us:

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

Solving for [ML] gives us:

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

4: $[ML] = \frac{M_{tot} \times L_{tot}}{K_D + L_{tot}}$

Equation 4 defines a hyperbola, which is the shape of the saturation curve showing the concentration of complex [ML] as a function of the ligand concentration $L_{tot}$.

Using the hyperbolic model 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 $\frac{[ML]}{M_{tot}}$:

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

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

Substituting equation 4 into equation 5 gives us:

6: $S = S_{min} + (S_{max} - S_{min}) \times \frac{L_{tot}}{K_D + L_{tot}} $

In equation 6, $S_{min}$ 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. $L_{tot}$ is a known experimental parameter. Therefore, $K_D$ and $S_{max}$ can be determined by fitting equation 6 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.