K_d Fit Model

The K_d fit model describes a molecular interaction with a 1:1 stoichiometry according to the law of mass action. Interactions in which several molecules A bind to one molecule B, but all binding sites have the same affinity for A, can still be described by this model because the binding events are independent (for the opposite situation, see Cooperativity . DI.Screening Analysis software offers two fitting models, the K_d model and the Hill Model. The K_d model is suitable for the vast majority of investigated interactions. Use the Hill model only for data evaluation if the investigated interaction is known to be cooperative.

The initial fluorescence influences the measured F_norm data and needs to be accounted for

The F_norm is calculated by dividing fluorescence values from the TRIC trace after the laser is turned on (hot region, F₁) by values that are obtained before the laser is turned on (the Initial Fluorescence or cold region, F₀).

A Ligand-Induced Fluorescence Change can influence the measured F_norm-values of the Dose Response Curve that are used to fit the K_d. A strong change in initial fluorescence not only causes a difference in amplitude, but also a shift of the Dose-response curve along the X-axis, even though the K_d is the same. Therefore the influence of the ligand induced fluorescence change, has to be corrected 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 determined 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 and 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.

K_d – model with ligand induced initial fluorescence change

With the help of mass-action kinetics we are able to 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 and 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 lower 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 induces a change in initial fluorescence. In these cases, the F_norm used for the calculation of the K_d is influenced by the change in initial fluorescence. Hence, this influence on the F_norm needs to be accounted for. The K_d no longer aligns with the inflection point of the dose-response curve of the F_norm.

Therefore, it is necessary to 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 consideration 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 software as a column in the ligands table. There are three fit parameters that have to be calculated; unbound and bound of the F₁ region and the K_d . The target concentration is a constant in the formula.

If there is no variation in the initial fluorescence, then r = 1. Inserting this in the formula of the F_norm reduces the formula to:

Outliers from fitting

A fitting method that makes the analysis robust against Outlier data points is used. Because of this outliers of the dose response curve can be identified. To determine the outliers, the fit residuals (model – data points) are calculated and the interquartile range (IQR) of the residuals is determined. Every residual that is 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 carried out 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 have a strong influence on the fit. Because it contributes quadratically to the sum of least squares (χ^2).

In order to prevent such a strong contribution of the outliers to the function that needs to be minimized, a function that is less influenced by outliers is used.

With z = χ2