Formalism for coupled equilibria: statistical mechanics and applications*

The thermodynamics of a system of coupled equilibria is determined by the free
energy of the system's states. While experimental data and computer simulation
can tell us the free energy difference between a system's states, physical
constraints, such as energy conservation, may not be accounted for when
building a thermodynamic cycle from these data. We develop a method for the
aggregation of these data in order to determine a system's state free energies
up to an arbitrary constant. Given a thermodynamic cycle defined by
experimental or computationally determined equilibrium constants, maximum
likelihood estimation yields the free energy difference between the system's
states without having to sidestep data points with large variances. The use of
this method enforces that any closed path within the cycle sums to zero free
energy difference. The method was applied considering the problem of
competitive binding of sodium and protons in the binding site of a
sodium/proton antiporter membrane protein. The method was also applied to a
set of molecules to determine macroscopic pKa values where the deprotonation
energies were determined quantum mechanically.