Fit Function for K_d from the law of mass action

The K_d fit model describes a molecular interaction with a 1:1 stoichiometry according to the law of mass action. Interactions where 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).

NanoTemper Technologies 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 in MST data evaluation when the investigated interaction is known to be cooperative.

The K_d is estimated by fitting the equation

where f(c_ligand) is the F_norm value at a given ligand concentration c_ligand;
Unbound is the F_norm signal of the target alone;
Bound is the F_norm signal of the complex;
K_d is the dissociation constant or binding affinity;
and c_target is the final concentration of target in the assay.

Fit Function for EC₅₀ from the Hill equation

The Hill Fit is used for affinity quantification of multivalent interactions and provides information about the degree of cooperativity. For monovalent interactions and interactions without cooperativity, the K_d model is used instead.

The Hill fit model uses the equation

where f(c_ligand) is the fraction bound at a given ligand concentration c_ligand;
Unbound is the F_norm signal of the target;
Bound is the F_norm signal of the complex;
EC₅₀ is the half-maximal effective concentration;
and n_Hill is the Hill coefficient.

The reason why the Hill Fit yields an EC₅₀ instead of a K_d is that the Hill Fit is used for interactions with cooperativity and non-1:1 interactions, where the individual binding sites differ in affinity. In these situations, the affinity (K_d) of the individual binding sites cannot easily be determined but instead the EC₅₀ is a useful parameter.

The Hill coefficient n_Hill describes the degree of cooperativity of an interaction: n_Hill>1 indicates positive cooperativity (e.g. binding of O₂ to hemoglobin), while n_Hill<1 indicates negative cooperativity (e.g. for some dimeric GPCRs or metabolic enzymes). The Hill coefficient is also used as an indicator for unspecific / promiscuous binders in small molecule research. Here, n_Hill>1 for protein – small molecule interactions is often used as an indicator for non-1:1 binding stoichiometry, suggesting non-specific interaction effects. Note that the Hill coefficient does not describe the stoichiometry of an interaction but rather it's cooperativity. In general, when working with Hill coefficients one should be cautious with their interpretation.

Importantly, the Hill Fit should only be used in MST data evaluation when the investigated interaction is known to be cooperative. For all other interactions, use the K_d model. Due to its additional variable (compared to the K_d Fit), binding curves calculated with the Hill Fit will often fit the measurement data very well. This does not necessarily mean that the interaction shows cooperativity, or several binding events. While a Hill coefficient of n_Hill≠1 may indicate cooperativity, this has to be confirmed with appropriate experiments. Stoichiometry, for example, can be a factor in this that can be investigated with MST using a special setup. See FAQ on Stoichiometry determination for further details.

Derivation of the Fit Function for K_d

Here we describe how the fit formula for the K_d fit can be derived. Note that in in this part of this document we derive an equation to describe the fraction of bound target as a function of ligand concentration. In the above part we describe the formula for F_norm as a function of ligand concentration. Knowing the F_norm values for bound and unbound states these two expressions of the same curve are easily convertible.

Definition Law of Mass Action:

where K_d is the dissociation constant or binding affinity;
 is the concentration of free target molecules;
 is the concentration of free ligand molecules;
c_complex is the concentration of complex formed

Important: the law of mass action uses “free concentrations” (fraction of concentration which is not bound)! These are typically unknown and a K_d can thus not be calculated based on this equation. We therefore need expressions for the concentrations of free molecules as functions of the concentration used in the experiment and the amount of complex formed:

c_complex is the provided concentration of target molecule in the experiment (fluorescently labeled)

C_ligand is the provided concentration of ligand molecule in the experiment (non-fluorescent)

To put this in words, the free molecule concentrations equal the known concentrations of the molecules used in the binding experiment minus the concentration of those molecules that form a complex.

To derive an expression for the fraction of target molecules as a function of ligand concentration the first step is to replace the free concentrations for free target and ligand in Formula 1 with the according expressions provided in formulas 2 and 3:

We then need to rearrange formula 4 and divide the result by c_target. This yields a quadratic equation with c_complex as a variable:

Solving this quadratic equation and dividing the result by c_target results in an equation that describes the fraction of bound target molecules as a function of the titrated ligand concentration c_ligand and the target concentration c_target used in the experiment.