Forecast: written 2026-10-15 23:50 CEST, before any number below was computed

Question: can the orbital layer, built for molecules of up to 120 atoms, draw the electronic structure of a metal site inside a protein of any size, fast enough for a 2-core server? And can it say something true about the site: where the electron pairs that hold the zinc point, and where the zinc's empty orbitals point?

The approach: never compute the whole protein. Find the motif in the sequence, cut out the matched stretch (backbone included, capped where it is cut), add hydrogens and the metal, and run the orbital layer on that fragment of a few hundred atoms. Mol* shows the whole protein; the fragment's orbitals are drawn in the protein's own coordinates.

Test structures, both from the test data of open-source structural-biology libraries:

• 5A7U: the ADR1 zinc finger folded inside the ribosome exit tunnel, cryo-EM at 4.8 Å (Nilsson et al., Cell Reports 2015). Zinc ligands Cys5, Cys8, His21, His26 (pattern C-x(2)-C-x(12)-H-x(4)-H).
• 7DDO: human ACE2 with a coronavirus RBD, cryo-EM at 3.4 Å. Catalytic zinc held by His374, His378 (the HEXXH motif) and Glu402.

What is already known, and therefore not a forecast: the deposited distances. 5A7U has Zn–N 1.86 and 1.89 Å and Zn–S 1.88 Å, with the fourth ligand (Cys5 Sγ) at 3.2 Å. 7DDO has Zn–O 2.06, Zn–N 2.14 and 2.20 Å. Target distances are Harding's (Acta Cryst. D62, 678, 2006): Zn–S(Cys) 2.31, Zn–N(His) 2.03, Zn–O(Asp/Glu) 1.99, Zn–O(water) 2.09 Å.

#Zinc's radial functions

Zn²⁺ is [Ar]3d¹⁰. The orbitals that accept the ligands' electron pairs are its empty 4s and 4p. They are taken as the virtual orbitals of a closed-shell Hartree–Fock calculation on Zn²⁺: the orbitals one added electron would occupy in the field of the ion, which is the field the ligands' pairs see.

• F1. The solver, unchanged except for allowing zinc, reproduces the numerical Hartree–Fock limit of the neutral atom (−1777.848116 hartree) to better than 1 mhartree, as it does for Kr and Xe.
• F2. Virtual 4s of Zn²⁺: ε = −0.62 ± 0.04 hartree. The measured ionisation energy of Zn⁺ (NIST: 144 892.6 cm⁻¹) corresponds to −0.660 hartree. Hartree–Fock binds the 4s electron slightly less, because it lacks the correlation with the 3d shell.
• F3. Virtual 4p of Zn²⁺: ε = −0.41 ± 0.04 hartree. Measured: Zn⁺ 4p ²P lies at 49 064 cm⁻¹ (J-weighted), so it is bound by −0.437 hartree.
• F4. Sizes ⟨r⟩: neutral-atom 4s 1.55 ± 0.15 Å (much smaller than calcium's 2.23 Å, because the 3d shell screens the nucleus poorly); Zn²⁺ virtual 4s 1.35 ± 0.15 Å; virtual 4p 1.75 ± 0.25 Å. The 4p is larger than the 4s.
• F5. As for every other element, 2–6 Slater terms reproduce each function with an overlap of at least 0.9999 with Hartree–Fock.

#The site

• F6. Placing zinc where the model has none. AlphaFold models carry no metal. The zinc is placed by least squares to the target distances, with the donor geometry deciding between mirror images. This is tested by deleting the deposited zinc and placing it again.
  • 7DDO (three ligands): the placed zinc lands within 0.30 Å of the deposited one, with an rms distance residual under 0.12 Å. With three ligands there are two mirror solutions; the donor directions (the imidazole N's in-plane lone pair, the carboxylate O's in-plane lone pairs) pick the right one.
  • 5A7U (four ligands, distorted): no position fits all four target distances; the rms residual is at least 0.30 Å and the site is flagged. The placed zinc lies 0.3–0.8 Å from the deposited one.
• F7. Choosing the ligands. Without a metal, the builder tries every assignment of the motif's Cys/His/Asp/Glu residues: which His nitrogen (Nδ1 or Nε2), which carboxylate oxygen, three or four ligands. On 7DDO it finds His374 Nε2, His378 Nε2 and Glu402, and leaves Glu375 out (it is part of the HEXXH motif but is the catalytic base, not a ligand). On 5A7U it takes all four residues, with Nε2 on both histidines.
• F8. Do the lone pairs point at the zinc? The donor's hybrid toward the zinc is computed with Coulson's method, as for any bond. Its angle to the donor→Zn line:
  • on 7DDO, at most 12° for both histidines and at most 25° for the glutamate oxygen;
  • on 5A7U, at least one ligand above 20°.
• F9. Zinc's acceptor orbitals. With four ligands they form a nearly equivalent sp³-like set, s character 0.25 ± 0.05 each. With three (7DDO), the fourth orbital is empty and unused: it points at the open coordination site, where the catalytic water binds. Its direction lies within 25° of the direction opposite the mean of the three Zn→ligand directions.

#Scale

• F10. Size. The full zinc-finger stretch of ADR1, Cys5 to His26 (22 residues) with ACE/NME caps and hydrogens, has 360 ± 40 atoms. The sequence the user searched for (H-x(3)-H) does not match ADR1, whose histidines are H-x(4)-H; a range x(3,4) matches it.
• F11. Time. As it is today, the orbital layer finds rings by a breadth-first search from every bond, once per call, and it is called once per atom. On 360 atoms that is far slower than linear: more than 20 s. Cached and depth-limited ring search makes it linear. After that the whole layer, including Hückel π for every amide and ring, takes under 1.5 s on one core of this sandbox. Doubling the stretch multiplies the time by less than 2.3.
• F12. Payload. The JSON for the 22-residue site, including the π-field data the browser needs, is under 1.5 MB uncompressed.
• F13. Drawing. Every hybrid of the 22-residue site (about 600 orbitals) meshes in the browser in under 4 s at low quality, under 10 s at medium. The 120-orbital cap of the 3D viewer is therefore a cap on patience, not on correctness, and can become a budget on grid points.

#What would change the plan

• If F2–F4 miss by more than twice the stated range, the virtual-orbital choice is wrong for drawing acceptor orbitals, and the neutral-atom 4s/4p would be used instead.
• If F6 fails on 7DDO, placement without a deposited metal is not trustworthy, and the tool must say "no metal in this model" instead of placing one.
• If F11 cannot be met, sites stay restricted to the ligand side chains.