Quantum Mechanical Free Energy Barrier for an Enzymatic Reaction

We discuss problems related to in silico studies of enzymes and show that accurate and converged free energy changes for complex chemical reactions can be computed if a method based on a thermodynamic cycle is employed. The method combines the sampling speed of molecular mechanics with the accuracy of a high-level quantum mechanics method. We use the method to compute the free energy barrier for a methyl transfer reaction catalyzed by the enzyme catechol $O$-methyltransferase at the level of density functional theory. The surrounding protein and solvent are found to have a profound effect on the reaction, and we show that energies can be extrapolated easily from one basis set and exchange-correlation functional to another. Using this procedure we calculate a barrier of $69\mathrm{kJ}/\mathrm{mol}$, in excellent agreement with the experimental value of $75\mathrm{kJ}/\mathrm{mol}$.

Figure 1

The thermodynamic cycle used to calculate the QM/MM free energy change, $\Delta F_{\mathrm{QM}}^{\mathrm{A}\to \mathrm{B}}$, between system A and system B on the basis of classical sampling of the phase space for systems A and B. The free energy change is calculated as $\Delta F_{\mathrm{QM}}^{\mathrm{A}\to \mathrm{B}}=-\Delta F_{\mathrm{MM}\to \mathrm{QM}}^{\mathrm{A}}+\Delta F_{\mathrm{MM}}^{\mathrm{A}\to \mathrm{B}}+\Delta F_{\mathrm{MM}\to \mathrm{QM}}^{\mathrm{B}}$.

Figure 2

Setup employed in the simulations. The left panel shows the unit cell employed in the calculations. Solvent molecules are depicted with dots, whereas nonsolvent atoms are illustrated by spheres with the active site in a lighter color. The right panel shows a close up of the active site. Atoms in the QM region are depicted with thick sticks and the methyl transfer reaction is indicated with arrows. The QM system consists of the catecholate molecule, the ${\mathrm{Mg}}^{2+}$ ion coordinated by catecholate, an $\mathrm{S}({\mathrm{CH}}_3{)}_3^{+}$ molecule to model $S$-adenosylmethionine, ${\mathrm{HCOO}}^{-}$, ${\mathrm{HCOO}}^{-}$, and ${\mathrm{HCONH}}_2$ to mimic the ${\mathrm{Mg}}^{2+}$ ligands: Asp-141, Asp-169, and Asn-170, respectively, and a Mg coordinating water molecule.

Figure 3

Calculated barrier for the methyl transfer reaction catalyzed by catechol $O$-methyltransferase using the B3LYP hybrid functional and a large basis set. Forward reaction is from right to left. The dashed line indicates the contribution to the free energy from the MM-QM interaction. Error bars are not shown, but the statistical standard errors are less than $1\mathrm{kJ}/\mathrm{mol}$ (for comparison, the height of a dot corresponds to $\sim 4\mathrm{kJ}/\mathrm{mol}$). Inset: Correlation between QM/MM energy fluctuations calculated with the B3LYP functional and a large basis set (vertical axis) and the PBE functional and a medium sized basis set (horizontal axis).