### Target Concentration

For affinity measurements we recommend that the target concentration is lower than the K_{d}. Here is a more detailed explanation why this recommendation is important and not exclusive to Spectral shift/MST experiments but other methods of equilibrium K_{d} determination as well.

**First, a practical example**

Imagine an interaction between target A and ligand/titrant B with a K_{d} of 100 nM. You choose a target concentration of 10 nM (left part of the above figure). Now consider the fraction of molecules of target A that are in a complex with ligand/titrant B. As long as the concentration of B is much lower than 100 nM (the K_{d}), the fraction of A in a complex is close to zero. As soon as we get closer to the K_{d}, while increasing the concentration of B, the fraction of molecules of A in a complex starts to increase and reaches 50% at a concentration of 100nM (the K_{d}). In other words, under this experimental condition the fraction of molecules in a complex is determined by the law of mass action. As soon as the K_{d} is reached, 50% of A-molecules (5nM in this particular case) are in a complex. The observed saturation of all molecules of A in a complex with B is also determined by the law of mass action and usually observed at a concentration ~20-fold the K_{d}, in this case 2 µM.

Now let us imagine for the same interaction with a K_{d} of 100nM you instead choose a target concentration of 10 µM (right part of the above figure). As long as the concentration of B is much lower than 100 nM, again the fraction of A in a complex is close to zero. If we now increase the concentration of B to 100 nM (the K_{d} of this interaction) one might expect 50% of all molecules of A to be in a complex.

However, there are 10µM of A-molecules, 50% of that is 5 µM but only 100 nM of B-molecules are available for binding. Therefore, the measured fraction of A-molecules in a complex is a maximum 1% (100 nM of B as a fraction of 10 µM of A). Saturation of the dose response curve will now be observed at 10 µM, rather than 2 µM because only 10 µM of B can saturate the 10 µM of A present in solution). The dose response curve that is observed under such experimental conditions is **not** determined by the law of mass action. Instead it is determined by the stoichiometry of the interaction. To interpret such curves with a K_{d} fit (which is based on the law of mass action) is only possible if the concentration of A is known with absolute accuracy (which is usually not possible, especially considering potential pipetting errors). It is therefore not recommended to determine K_{d} values under such conditions.

Of course, the experimental conditions described above are extremely different and generally one would not consider measuring in presence of 10 µM target. However, the example nicely illustrates the issues faced with target concentrations that are too high. The same issues are also faced at 10 nM K_{d} and 100 nM target concentrations, although in a more subtle way.

**Theoretical considerations**

In less practical and rather more biochemical terms, the above stated experimental considerations can also be expressed in the following way:

The goal of an MST experiment is in general to determine the dissociation constant of an interaction. In the simplest interaction experiment, one would investigate the interaction of two molecules A and B that form a complex AB. The equilibrium reaction would be described by:

Such a reaction can be well described by different parameters, for example the equilibrium constant that is described by the law of mass action:

However, it has become convention in biology and biochemistry to rather describe the stability of a complex not by the association constant (K_{a}) but rather by the dissociation constant (K_{d}).

The K_{d} has the unit molar (M) and is defined as the concentration [B] at which half of all molecules of A in the reaction are bound in the complex AB. In other words, if [B] reaches the K_{d}, then [A] = [AB].

It is, however, quite challenging to measure [A], [B], and [AB] all at the same time while maintaining an equilibrium. One can circumvent this issue by assuming that the total amount of molecules A and B in the reaction is the sum of the concentration of the complex AB and the respective free molecules, i.e. [A]_{Total} = [A] + [AB] and [B]_{Total} = [B] + [AB].

Using these assumptions, it is possible to remove the term for [A] from the above definition of the K_{d} and to rearrange the equation as follows:

Determination of the K_{d} is then achieved by keeping [A]_{Total} constant, whereas [B]_{Total} is varied from well above to well below the expected K_{d}. For each [B]_{Total}, [AB] and [B] must be determined. The results can then be visualized by plotting against [B]_{Total}, yielding a sigmoidal curve (in a semi-logarithmic plot). The concentration of B at which half of all molecules A are bound in the complex AB (i.e. where [AB]/[A]_{Total }= 0.5) is the K_{d}.

To be absolutely correct, it is necessary to measure [AB]/[A]_{Total }as a function of the fraction of molecules B that is not bound to the complex AB, i.e. [B]_{Free}. It is however, technically challenging to measure [AB] and [B]_{Free} at the same time. This problem can be overcome by keeping [A]_{Total} far below the K_{d}, because then only a negligible amount of [B]_{Total }is required for formation of the complex AB and [B]_{Total} ≈ [B]_{Free}. In this case, [AB]/[A]_{Total }can simply be plotted against the respective [B]_{Total} with negligible effect on the outcome of the experiment.

However, if it is not possible to keep [A]_{Total} far below the K_{d} (e.g. due to very high affinities in the pM range or problems with detection of the AB complex), [A]_{Total} should be kept as low as possible. In this case, an upper limit for the K_{d} can be given. Alternatively, a competition assay could be used.