First-principles electronic-structure calculations of whole proteins in aqueous solution require a cheap, faithful representation of the electrostatic effect of the nearest water molecules. Within the self-consistent cluster-embedding (SCCE) method, the standard device is an 'equivalent potential of water': each nearby water molecule is replaced by a small set of point charges tuned so that the solute's SCCE electronic structure matches the explicit-water benchmark. Zheng, Wang, Yan and Wu (2013) showed, for alanine, histidine and serine in single optimized cluster geometries, that the standard three-point-charge water models SPC and TIP4P-FQ — used with their original parameters or with per-site adjusted charges — do not outperform a tuned two-point-charge dipole; the equivalent-potential series states that such dipole-based potentials can be applied directly to SCCE electronic-structure calculations of complete proteins in solution. The open problem is to establish, with the generality the intended application requires, which simple solvent representation should serve as the equivalent potential of water: extend the comparison beyond the three tested amino acids to the remaining natural amino acids and to residues in the local environments of a protein in solution; and carry out the never-performed SCCE electronic-structure calculation of a complete solvated protein with dipole-based equivalent potentials representing the water molecules near its solvated surface regions. None of these questions has yet been answered.

Background

Computing the electronic structure of a whole protein from first principles (ab initio, i.e. without empirical parameters fitted to the molecule itself) means determining the one-electron energy eigenvalues and molecular orbitals of thousands of atoms. The self-consistent cluster-embedding (SCCE) method is a divide-and-conquer scheme of this kind introduced by Haoping Zheng: the large system is partitioned into embedded clusters, each containing a localized group of atoms, and each cluster is solved self-consistently for all electrons in the electrostatic potential (with an approximate treatment of exchange and correlation from the surroundings) produced by the rest of the system. Because each cluster is much smaller than the whole system, the cost grows roughly linearly with system size, and the method has been applied at the all-electron level to entire proteins, for example the squash-seed trypsin inhibitor (2000) and the Ascaris trypsin inhibitor (2003) in aqueous solution.

For a protein in water, a difficulty specific to first-principles calculations is the solvent itself. Macroscopic continuous-medium (dielectric) models of water average over the microscopic, position-dependent electrostatic potential of the nearest water molecules, and that local potential is precisely what shifts the localized electronic states of surface residues; on the other hand, including the very many surrounding water molecules explicitly is computationally prohibitive. The compromise adopted in this line of work is the 'equivalent potential of water': each water molecule near a residue is replaced by a small set of point charges whose electrostatic potential reproduces the effect of the real molecule on the residue's SCCE electronic structure. Between 2006 and 2012, Zheng and coworkers constructed such equivalent potentials residue by residue — the 2013 paper's abstract states that fifteen amino acids had been treated, including cysteine, histidine, alanine, serine and arginine — by replacing each nearby water molecule with a two-point-charge dipole and tuning the dipole until the SCCE eigenvalues and orbital compositions of the isolated amino acid matched those obtained with the explicit water molecules present. The series states that the resulting dipole equivalent potentials can be applied directly to SCCE electronic-structure calculations of proteins in solution.

The 2013 paper by Zheng, Wang, Yan and Wu then asked whether an off-the-shelf discrete water model could be used directly instead. Two standard models were tested: SPC ('simple point charge'), a rigid three-site model with fixed point charges on the oxygen and the two hydrogens (Berendsen et al., 1981), and TIP4P-FQ, a four-site fluctuating-charge (polarizable) model — charges on the two hydrogens and on a massless point along the HOH angle bisector, with values that in classical simulations fluctuate to represent polarization (Rick, 2001) — used here with fixed average charges. Both families are parametrized to reproduce bulk liquid-water properties, not the local electronic structure of a solute. For alanine (+7 waters), histidine (+7) and serine (+6), each in a single lowest-energy cluster geometry, the paper compared four embeddings — explicit water (the benchmark), tuned dipoles, and the original SPC and TIP4P-FQ models — using two criteria: (a) the root-mean-square deviation of the one-electron eigenvalues from the explicit-water benchmark, and (b) a mean-square charge-density deviation, whose reliability the authors themselves qualify because of the charge-density fitting procedure used. The original three-point models produced deviations substantially larger than the tuned dipoles; adjusting the point charges of each water molecule one by one to minimize criterion (a) reduced the deviations by roughly an order of magnitude but did not make the models better than the dipole representation. The authors concluded that, at least for these three amino acids, neither the original nor the adjusted water models improve on the tuned dipole, leaving the tuned dipole as the equivalent potential of choice for the series' intended SCCE calculations of complete proteins in solution.

For context outside this specific framework: the broader field of whole-protein quantum chemistry pursued solvent and environment representation along other routes after 2013, e.g. fully quantum-mechanical embedding with transferred extremely localized molecular orbitals (Macetti and Genoni, 2019) and frozen-density embedding (Wesołowski et al., 2015), none of which uses the SCCE platform or its dipole-tuning construction.

Problem Statement

The problem left open by the 2013 study is to establish, with the generality its intended application requires, which simple representation of the nearest water molecules should serve as the equivalent potential of water in SCCE electronic-structure calculations of proteins in solution. The source paper states that an equivalent potential of water is needed to calculate the electronic structure of a protein in solution, and it explicitly limits its dipole-versus-water-model conclusion: "At least for these three amino acids, neither the original water molecule model nor the adjusted one is better than the dipole in constructing the equivalent potential of water for the electronic structure of an individual amino acid." The open problem accordingly comprises two source-supported core questions serving one objective — a validated, transferable and computationally cheap solvent representation that lets SCCE calculations reproduce, for the surface regions of a whole solvated protein, the electronic structure that explicit water would produce:

(1) Core question — Generalization. The dipole-versus-water-model comparison was carried out only for alanine, histidine and serine, each in a single optimized cluster geometry. It is unresolved whether its conclusions — that tuned dipoles are at least as good as the original and the per-site adjusted SPC and TIP4P-FQ models — hold for the remaining natural amino acids (the intended whole-protein application ultimately requires coverage of all twenty), and for residues in the local environments of a protein in solution rather than in isolated optimized clusters.

(2) Core question — Demonstration on complete proteins. The intended application — performing the SCCE electronic-structure calculation of a complete protein in solution using dipole-based equivalent potentials to represent the water molecules near its solvated surface regions, such as surface side chains and the chain termini — had not been carried out, so the transferability and practical performance of the dipole equivalent potentials in real protein calculations remain undemonstrated.

A complete answer must address both source-supported core questions: the dipole-versus-water-model comparison extended beyond the three tested amino acids to cover all twenty natural amino acids, and repeated in full — tuned dipoles, original SPC, original TIP4P-FQ and per-site adjusted charges — for residues in the local environments of a protein in solution, an element that belongs to question (1) and is not by itself satisfied by the dipole-based whole-protein calculation of question (2); and the whole-protein SCCE demonstration with dipole-based equivalent potentials representing the water molecules near the protein's solvated surface regions.

References

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