An exact mathematical expression for describing competitive binding of two different ligands to a protein molecule

The dissociation constant for the binding of a spectroscopically invisible or non-radioactive ligand to its protein receptor can be determined in a competition experiment by using a structural analog that contains a reporter group. Many plotting and numerical analysis methods have been developed to calculate the binding constant of unlabeled ligand from the displacement experiments. However, a common problem with these plotting methods is that the equation transformations inevitably result in non-standard error distribution, and thus simple linear regression can not be used to extract correct values for the parameters. In the case of the numerical analysis methods, one would be faced with the possible existence of multiple solutions. In this paper, the exact mathematical expression for describing competitive binding of two different ligands to a protein molecule is presented in terms of the total concentrations of species in the system. Thus, using a commercially available non-linear regression program, all unknown parameters for describing this system can be determined by fitting the experimental data to the algebraically explicit equation without any data transformations. The distribution curves of all the species in the system can also be constructed with this equation. It is particularly useful for the cases in which the concentrations of all the species in the system are comparable to each other.