Thermodynamics of the binding of ligands by macromolecules

The thermodynamics of the binding of ligands by proteins and other biological macromolecules has been treated by Wyman and others on the basis of the binding polynomial and the binding potential. However, the thermodynamics of the binding of ligands by small molecules and the effects of ligands on the apparent equilibrium constants of biochemical reactions has been developed on the basis of Legendre transformed Gibbs energies of formation. This article brings these seemingly disparate approaches together by considering simple systems and the binding of oxygen by hemoglobin. When the ligand is H+, examples involving small molecules show that the standard transformed Gibbs energy of formation of a reactant at a specified pH is equal to the negative of the binding potential plus a term related to the standard thermodynamic properties of the elements. The standard transformed Gibbs energies of formation of eight forms of deoxygenated and oxygenated hemoglobin are calculated here for a specific set of conditions. This is the most efficient way to store the information from the seven independent apparent equilibrium constants involved. In a second step, a Legendre transform is used to introduce the concentration of molecular oxygen as a natural variable and calculate the apparent equilibrium constant K" for 2TotD = TotT at specified values of [O2], where TotD is the sum of the concentrations of the dimer and its oxygenated forms and TotT is the sum of the concentrations of the tetramer and its oxygenated forms.