Morphometric hydration contrast: predict shell density from contact area + curvature (universal, IDP-capable)

[mlund]

Summary

Replace the fixed per-residue hydration table (#9) with a morphometric,
geometry-derived hydration contrast: predict each surface water's excess
density δ_i from local geometric descriptors — chemical-type-resolved
water contact area plus local curvature — computed by Voronoi tessellation.
Train the coefficients once against explicit-solvent (WAXSiS) shell
decompositions, then apply universally to folded and intrinsically
disordered proteins (IDPs).

This generalises #9. #9 (fixed Linse per-amino-acid table) becomes the
simpler fallback; this issue is the universal model.

Motivation: why a fixed per-residue table fails for IDPs

A per-residue table a(Asp), a(Ala), … is a fixed average indexed by
identity. For IDPs the same residue has very different exposure/shell
across conformations, so the table mispredicts every individual frame
(cf. Linse, Fischbach & Hub, bioRxiv 2025.06.10.658889: the XAO peptide
shell is weak and strongly conformation-dependent). A geometry law
evaluated per-frame adapts automatically — and pripps already averages
per-frame over trajectories, so an IDP ensemble is the natural input.

It is also consistent with pripps' existing philosophy: the Voronoi
excluded-volume model already derives displaced volume from geometry
instead of a table. This does the same for the shell.

Proposed model

Per surface-water dummy i with parent atom of chemical class t(i):

δ_i ≈ c_area(t) · area_i  +  c_M · M_i  [ + c_χ · κ_i ]

• area_i — the parent atom's water-contact area, resolved by chemical
  class (≈4–5 classes: acidic / basic / polar / apolar / backbone).
  voronota-LT gives per-atom-pair contact areas (it is solvent-aware).
• M_i — local mean curvature of the surface at the atom.
• κ_i — optional local Gaussian/Euler term.

These are exactly the Minkowski functionals (area, integrated mean
curvature, Euler characteristic). The morphometric approach to solvation
(König, Roth & Mecke, Phys. Rev. Lett. 93, 160601, 2004) shows
solvation thermodynamics is, to good approximation, a linear combination
of these functionals with geometry-independent coefficients — i.e. the
ansatz has a theoretical backbone and a built-in transferability argument
across shapes. That is the basis for the "universal, covers IDPs" claim.

Why area alone is not enough (the apolar sign problem)

Raw contact area predicts the number of shell waters (≈ SASA), which
the Voronoi dummy placement already encodes — not the sign of the
excess density. Apolar patches have large area but negative δ
(hydrophobic dewetting), strongest on convex surfaces. The curvature
term carries exactly this: large apolar area + high mean curvature → strong
depletion. Resolving area by chemical class supplies enhancement
(acidic/polar) vs depletion (apolar); curvature supplies the geometry that
flips the sign.

Avoid double-counting area

Dummy count already scales with surface area via placement. The per-dummy
weight must therefore be the intensive δ (per water / per unit area),
not extensive shell scattering — otherwise the area dependence is squared.
Regression target = intensive shell density.

Getting curvature cheaply: two-probe finite difference

Run the tessellation at two (or three) slightly different probe radii and
finite-difference the per-atom accessible area. By the Steiner formula
V(r_p) = V₀ + A·r_p + M·r_p² + (4π/3)·χ·r_p³, so dA/dr_p ∝ M (mean
curvature) and a second difference gives the Euler/Gaussian term. Because
voronota-LT exposes per-cell sas_area, this yields a per-atom local
curvature, sign-meaningful:

• convex protrusion → area grows with probe → M > 0
• flat → ~constant → M ≈ 0
• concave pocket → probe excluded → area shrinks → M < 0

Cost ≈ 2–3× a linear-time tessellation pripps already runs (per frame for
trajectories).

Train on wet frames, predict on dry structure (the key closure)

This is what makes it deployable rather than circular:

• Training: tessellate WAXSiS MD frames with explicit waters →
  real protein-atom↔water contact areas (by class) + curvature; regress
  against the local shell density WAXSiS reports → learns c_area(t),
  c_M, c_χ.
• Prediction in pripps: the deployable predictors — per-atom solvent
  contact area (cell.sas_area, already computed in the Voronoi hydration
  stage-1 tessellation) and the two-probe curvature — are available with
  no explicit water, from the dry structure. The law transfers from wet
  training to dry prediction.

Validation

1. Regress coefficients across a diverse WAXSiS training set (folded
   proteins, varied composition/charge).
2. Test out-of-sample on a WAXSiS IDP ensemble (e.g. Hub's XAO): show
   the morphometric law tracks the conformation-dependent shell that a
   fixed per-residue table (Residue-resolved hydration shell: weight Voronoi/SASA water dummies by MD-derived per-amino-acid affinity #9) cannot, using pripps' per-frame averaging.
3. Compare the shell-only I(q) (pripps .hydration channel) directly
   against the WAXSiS shell decomposition — per-component, not a
   total-intensity fit (see Residue-resolved hydration shell: weight Voronoi/SASA water dummies by MD-derived per-amino-acid affinity #9 for the WAXSiS-decomposition protocol).
4. Report held-out R² on the δ regression as the go/no-go.

Risks

• The area+curvature↔δ correlation may still be weak; may need a local
  H-bond-capacity descriptor. Held-out R² decides viability.
• Coefficients inherit the water-model / force-field dependence of the
  WAXSiS training set.
• Keep the descriptor set minimal (≈4–5 classes × {area, M}) to stay
  morphometric, not overfit.
• Two-probe curvature needs δr tuned to balance locality vs
  finite-difference noise from the discrete icosahedral sampling — validate
  per-atom M stability first.

References

• König, P.-M., Roth, R. & Mecke, K. R. (2004). Morphological
  thermodynamics of fluids. Phys. Rev. Lett. 93, 160601. (verify DOI)
• Linse, J.-B., Fischbach, T. M. & Hub, J. S. (2025). bioRxiv
  2025.06.10.658889.
• Knight, C. J. & Hub, J. S. (2015). WAXSiS. Nucleic Acids Res. 43, W225.
• Olechnovič, K., Dapkūnas, J. & Venclovas, Č. (2025). Voronota-LT.
  J. Comput. Chem. 46, e70178.

Generalises #9.