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4 Molecular Dynamics and Free Energy Calculations 914.3.3 Computed Changes in Protein Stability for theIle 96-+Ala MutationTable 4-4 lists the computed free energy values obtained by the exponential formula(EF) and thermodynamic integration (TI) procedures for the Ile 96 into Alaalchemical transformation. These are, ~lGf”~ for the solvated native protein,LIGL’~ for the solvated unfolded state, and d GI’A for the gas phase reference state.Table 4-4. Computed free energy changes (in kcal/aol) the Ile --* Ala mutation in barnase.Contribution AGf-A AGf’A AAGf,,, AAG,,, AAG,,fprotein water referenceTI -3.09 -8.3~~-7.15 - 5.21 -1.15 4.06EF -3.39 -6.81 -6.56 - 3.42 -0.25 3.17EXP -3.3a; -4.0b -0.21‘Negative values in AAG correspond to contributions in which the wild type (Ile) is stabilizedrelative to the mutant (Ala). AGf, AG, and AG, are the free energies for the alchemicaltransformation Ile + Ala, in the folded state (protein), the unfolded state (water), and the unfoldedstate in the gas phase (reference) respectively. AAGf-u is the unfolding free energy differencefor the alchemical transformation. AAG,,, and AA G,,f are respectively, the solvationfree energy differences for Ile versus Ala in water and in the protein interior. The correspondingAAG values are derived from the corresponding AG values as explained in the text.EF, TI and EXP stand for, the exponential formula Eq. (4-13), the thermodynamic integrationprocedure Eq. (4-15), and experimental results respectively. The superscripts a, b and c indicatethat values are taken from Kellis et al. [89, 901 and Wolfenden et 01. [lo61 respectively.By use of the thermodynamic cycle (Figure 4-17 a), the corresponding difference inunfolding free energy, d ~lG~,~ is obtained from the following expression [lo] :AAG~,, = AG,”~ - LIG~”~= AG&, - AG;,,, (4-16)The calculated LIL~G~+~ values are -3.41 and -5.21 kcal/mol obtained using theEF and TI procedures respectively. These values agree in sign and magnitude withthe values of - 3.3 and - 4.0 kcal/mol, measured in the experiment unfolding studiesof Kellis et al. [89, 901. The negative values indicate that unfolding the mutant isenergetically more favorable than unfolding the wild type, and hence that the wildtype folded state is more stable than the mutant folded state.Using another thermodynamic cycle the solvation free energy differencedAG,,,, is given by (Figure 4-17b):AAG,,, = AG;’* - LIG~+ = AG$, - AG:~,, (4-17)