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Do quantum mechanical energies calculated for small models of protein-active sites converge?

A common approach for the computational modeling of enzyme reactions is to study a rather small model of the active site (20-200 atoms) with quantum mechanical (QM) methods, modeling the rest of the surroundings by a featureless continuum with a dielectric constant of approximately 4. In this paper, we discuss how the residues included in the QM model should be selected and how many residues need to be included before reaction energies converge. As a test case, we use a proton-transfer reaction between a first-sphere cysteine ligand and a second-sphere histidine group in the active site of [Ni,Fe] hydrogenase. We show that it is not a good approach to add groups according to their distance to the active site. A better approach is to add groups according to their contributions to the QM/MM energy difference. However, the energies can still vary by up to 50 kJ/mol for QM systems of sizes up to 230 atoms. In fact, the QM-only approach is based on the hope that a large number of sizable contributions will cancel. Interactions with neutral groups are, in general, short-ranged, with net energy contributions of less than 4 kJ/mol at distances above 5 A from the active site. Interactions with charged groups are much more long-ranged, and interactions with buried charges 20 A from the active site can still contribute by 5 kJ/mol to the reaction energy. Thus, to accurately model the influence of the surroundings on enzyme reaction energies, a detailed and unbiased atomistic account of the surroundings needs to be included.