Accounting for the initial fluorescence measured in F_norm:

In order to account for the influence of the initial fluorescence, the F_norm is calculated by dividing fluorescence values from the TRIC trace after the laser is turned on (hot region, F₁) using values obtained before the laser is turned on (the initial fluorescence or cold region, F₀).

A Ligand-Induced Fluorescence Change (LIFC) can influence the measured F_norm-values of the Dose-Response Curve that are used to fit the Kd. A strong change in initial fluorescence causes a difference in amplitude and a shift of the dose-response curve along the X-axis, even though the K_d is the same. This means one must correct for the influence of the ligand-induced fluorescence change when fitting the dose-response curve of the F_norm.

Example:

This example demonstrates the shift of the F_norm curve along the X-axis. We use an assumed K_d of 30 nM and a target concentration of 5 nM to simulate the F_norm values. The fluorescence of the F₁ region is furthermore described by the unbound state = 4250 and the bound state = 4500, both values kept constant.

Additionally, the initial fluorescence (F₀ region), is also described by the same K_d and target concentration, a fixed bound state of 5000 fluorescence counts. To simulate the influence of a changing initial fluorescence, the unbound state is varied between 5000 and 10000. Calculating the F_norm by using the above formula results in a varying F_norm with not only different amplitudes, but also a shift along the X-axis. This shift is best observed in the normalized F_norm values. This clearly shows that the initial fluorescence change, if present, must be accounted for when fitting F_norm data to obtain K_d values.

Kd model with ligand-induced initial fluorescence changes:

With the help of mass-action kinetics, we can derive a formula for the fraction bound in case of a binding event. The fraction bound is defined by the binding affinity K_d and the concentration of the target molecule and depends on the ligand concentration.

where 

• f(c) is the fraction bound at a given ligand concentration c

• K_d is the dissociation constant or binding affinity

• c_T is the final concentration of target in the assay

The measured fluorescence F of the F₁ and the F₀ region is defined by the fraction bound and the fluorescence of the unbound and bound state of the binding event.

Assuming that the target concentration is much smaller than the K_d, the K_d is typically identical to the inflection point (half maximal effective concentration) of the F_norm. This changes when the ligand also induces a change in initial fluorescence. In these cases, the F_norm we use to calculate the K_d is influenced by the change in initial fluorescence. Hence, we must account for this influence on the F_norm. The K_d does not align with the inflection point of the dose-response curve of the F_norm anymore.

This means we must account for this change of initial fluorescence while calculating the K_d. Using the above formulas, we derive a model that takes all of this into account and still can be easily fitted to the measured F_norm.

with

For the K_d fit, we calculate r from the measured initial fluorescence and use it as a constant in the formula fitted to the F_norm data. The factor r is also called initial fluorescence ratio and can be displayed optionally in DI.Screening Analysis as a column in the ligands table. This means we have three fit parameters that we must calculate, unbound and bound of the F₁ region and the K_d. The target concentration is typically also a constant in the formula.

In the case where there is no variation in the initial fluorescence, r = 1. Inserting this in the formula of the F_norm reduces the equation to the form below.

Outliers from fitting:

For fitting, we use a method that is robust against outliers. Because of this, we can identify any outliers in the dose-response curve. To determine the outliers, the fit residuals (model–data points) are calculated, and the interquartile range (IQR) of the residuals is determined. Every residual smaller than 2*IQR of the lower quantile or bigger than 2*IQR of the upper quantile is considered an outlier.

Robust fitting:

Fitting is done by minimizing the distances between the model and the data points. Traditional least square fitting assumes that the errors are normally distributed. A biological outlier can break this assumption and, therefore, strongly influence the fit. Because it contributes quadratically to the sum of least squares (χ^2).

To prevent such a strong contribution of the outliers to the function that needs to be minimized, we use a function that is less influenced by outliers.

where

• z = χ^2