## ABSTRACT

Chapter 23 starts with a probabilistic explanation and a thermodynamic derivation of the law of mass action in terms of the change in the standard free energy of reaction, ΔA
^{0}. Then, the statistical mechanics equation for ΔA
^{0 }of Boresch et al. is derived. A standard way to calculate ΔA
^{0 }is by TI, where the ligand-environment interactions are gradually eliminated in both the solvent and the protein. However, this (rigorous) approach (called DAM), might encounter convergence problems in the final TI stages, where due to weak ligand-protein interactions, the ligand might leave the active site (“the end-point problem”). This problem has been rigorously solved by an appropriate addition of harmonic restraints – a procedure known as DDM. The application of HSMD-TI consists of four steps: (1) the complex and the ligand in water are equilibrated by MD and production runs are carried out, (2) small samples of n ~ 10 are selected from each trajectory and each selected conformation, i undergoes; (3) a gradual elimination of the ligand-environment interactions in solvent and the protein, where (unlike with DAM) structure i remains fixed during TI; and (4) each structure i is reconstructed by HSMD (see Chapter 19) leading to its entropy. These terms and others averaged over the n-size samples lead to ΔA
^{0}. Since structures i are kept fixed during TI, the “end-point problem” is eliminated and the need to apply restraints is avoided! Furthermore, HSMD-TI provides (uniquely) the decrease in the ligand’s entropy in going from solvent to the protein.

HSMD-TI was applied to the ligand, FK506 (126 atoms) complexed with the protein FKBP12 – where ΔA
^{0} = −12.8 kcal/mol is known experimentally. The HSMD-TI result, −13.6 ± 1.1 kcal/mol is one of the best in the literature. The ligand’s entropy is reduced by 7.1 ± 1.2 kcal/mol. Finally, the pitfalls encountered in estimation of ΔA
^{0 }with respect to the force field and the trajectories’ length are discussed.