In 2007, Inaba, Tsunekawa, Hirano, Yoshihiro, Kashiwagi and Sato demonstrated the first full-scale canonical, all-electron Kohn-Sham density-functional calculation on an entire protein assembly: the 4,728-atom insulin hexamer, solved with the parallel ProteinDF program and a quasi-canonical localized-orbital initial guess at the local-density-approximation level with a small split-valence Gaussian basis. The Letter reported the orbital-energy spectrum — near-gap orbitals localized on disulfide-forming cysteines and tyrosines, and a band-like region of orbitals extensively mixed across the six monomers — but deferred the detailed analytical interpretation to a companion paper (the Letter's reference list records Ref. [24] as 'in preparation' by the same six authors) and left the transfer to larger proteins untested and the choice of basis sets and exchange-correlation functionals unexplored. The scalability question has since been resolved affirmatively by the same group's third-generation density-functional method and its applications to further proteins, and by whole-protein ab initio work elsewhere; the deferred companion never appeared in the international journal literature. What remains open is: a quantitative, experimentally validated interpretation of the insulin hexamer's all-electron electronic structure — orbital assignment near the gap, statistical characterization of the band-region monomer mixing, and quantitative comparison with experimental spectroscopic data — together with a systematic basis-set and exchange-correlation-functional selection for whole-protein canonical density-functional calculations.

Background

Insulin is a 51-residue peptide hormone stored in the pancreas as a zinc-stabilized hexamer (six monomers, 306 residues, about 4,700 atoms), while the monomer is the form that binds the insulin receptor. Describing such a system from first principles means solving the Kohn-Sham equations of density functional theory (DFT) — the standard formulation in which interacting electrons are mapped onto non-interacting 'Kohn-Sham' orbitals whose self-consistent electron density minimizes an energy functional — for all electrons of the whole molecule, without fragmenting it.

Before 2007, two families of approaches existed. Fragment methods, exemplified by the fragment molecular orbital (FMO) method, divide a macromolecule into pieces, compute each piece and its pairs quantum mechanically, and recover total and inter-fragment interaction energies — but they do not yield canonical molecular orbitals (eigenvectors of the whole system's Kohn-Sham matrix) or a global orbital-energy spectrum. Full canonical calculations — solving the self-consistent-field (SCF) problem for the entire molecule and diagonalizing the whole orbital set — had been demonstrated in all-electron form (treating core electrons explicitly rather than through pseudopotentials) up to cytochrome c, about 9,600 orbitals, by the group of Fumitoshi Sato with their program ProteinDF (Sato et al., 2001).

Inaba et al. (2007) pushed this to the insulin hexamer: 4,728 atoms, 18,552 electrons, 26,790 basis functions, with Kohn-Sham and overlap matrices exceeding 5 GB that required distributed-matrix parallelism, converging in 65 hours on 64 Itanium 2 processors at 82% measured parallel efficiency (the parallel program itself is described in Inaba and Sato, 2007). Two ingredients made this feasible: the parallel ProteinDF implementation, and a quasi-canonical localized orbital (QCLO) initial guess — localized orbitals of molecular fragments that are canonical within a fragment region — constructed in five steps from single residues through tri-peptides, larger peptides and monomers up to the hexamer, so that the SCF iteration starts near convergence; the QCLO guess had been validated on the insulin monomer by the same group (Inaba et al., 2005). The electronic-structure level was deliberately modest: the local-density approximation (LDA; the Slater-VWN exchange-correlation functional, the simplest DFT level, known to misdescribe energy gaps and spectroscopic quantities) with a small split-valence Gaussian basis (a basis in which each valence atomic orbital is represented by several Gaussian functions of different widths, giving radial flexibility; S:(6321/521/1*), C,N,O:(621/41), H:(41)); 2,137 basis functions were discarded because of near-linear dependence (overlap-matrix eigenvalues below 0.007), and the crystal environment was represented by 3,832 point charges for waters and counterions.

The Letter's findings were necessarily preliminary. Near the HOMO-LUMO gap (highest occupied and lowest unoccupied molecular orbitals; computed orbital energies between −5.8-5.8 and −3.5-3.5 eV) the orbitals are sparsely distributed and localized on the cysteine residues forming insulin's disulfide bridges and on tyrosines TyrA14 and TyrA19. Deeper in the valence region the orbital energies form a quasi-continuous, band-like region in which the monomer weights wi,Mw_{i,M} — the fraction of an orbital's squared amplitude that projects onto monomer M's orthonormalized basis functions — appear randomly distributed between 0 and 1, indicating extensive mixing of orbitals across the six monomers. This reading rested on qualitative visual inspection of an energy-versus-orbital-index plot, at LDA level, with no comparison to any experiment. The authors deferred the detailed analytical interpretation (including comparison with experimental spectroscopic data and characterization of the band-region mixing) to a companion publication — the Letter's Ref. [24], listed as 'in preparation' by the same six authors in its reference list — and stated that applicability to proteins of even greater size remained to be tested and noted that the choice of basis sets and exchange-correlation functionals for such calculations was not explored.

The subsequent history determines what is now open. Canonical all-electron DFT on whole proteins became scalable through the same group's third-generation method (Hirano and Sato, 2014) and its high-performance implementations (Hirano and Sato, 2017, 2019), and whole-protein quantum chemistry matured independently elsewhere (Kulik et al., 2012; Delle Piane et al., 2016) — so feasibility at ever-larger sizes is no longer the question. What was never completed, in the international literature, is the second half of the 2007 program: turning the demonstrated computation into a validated, interpreted, and methodologically grounded electronic-structure result for the insulin hexamer, and establishing a systematic basis for the method choices that the original Letter made by default.

Problem Statement

The 2007 Letter established that a full canonical, all-electron Kohn-Sham density-functional calculation on a whole protein assembly — the 4,728-atom insulin hexamer — is feasible, but left that computation uninterpreted, unvalidated, and methodologically unjustified. The open problem is to complete this program:

For the insulin hexamer as the reference whole-protein system, produce a quantitative, experimentally validated interpretation of its all-electron canonical density-functional electronic structure, and establish the method choices such calculations require on a systematic footing. Concretely:

(i) assign the individual canonical molecular orbitals near the HOMO-LUMO gap (the energy difference between the highest occupied and lowest unoccupied molecular orbitals), determining their ordering, identities, and residue-wise localization weights;

(ii) give a quantitative characterization of the orbital mixing in the band-like region of the orbital-energy spectrum, delimiting that region explicitly and replacing the original qualitative visual inspection of the monomer weights wi,Mw_{i,M} with a statistical description carrying stated criteria able to quantify departure from the uniform-random behaviour that motivated that reading;

(iii) identify the experimental observable and the spectroscopic data against which the computed electronic structure can be tested quantitatively for the insulin hexamer, and carry out that comparison, making explicit how the ground-state orbital structure maps onto the chosen observable;

(iv) establish, by systematic study rather than by default, the choice of Gaussian basis set (including the treatment of near-linear dependence) and of exchange-correlation functional appropriate for whole-protein canonical all-electron density-functional calculations aimed at these observables — the original calculation having used the local-density approximation with a small split-valence basis.

These four results are jointly required: none of them alone resolves the open problem, and together they serve one objective — turning a demonstrated one-off computation into a validated and transferable electronic-structure account of a whole protein. Sub-result (iv) must recommend a method choice for whole-protein canonical all-electron density-functional calculations generally, grounded in the insulin hexamer as the reference study system, and must state the accuracy achieved and the basis on which the recommendation is expected to transfer; validation on further systems is not required. The further question attributed to the source — whether such calculations can be carried out on proteins even larger than the insulin hexamer — has since been answered affirmatively by subsequent work and is not part of the open core. The functional-role question attributed to the source — the roles of specific orbitals in insulin receptor binding — is likewise not part of the open core, which is scoped to the electronic structure of the insulin hexamer and its method selection rather than to the receptor-binding function of the monomer.

References

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