Non-Interacting Surface (NIS) Proportions

• Non-interacting surface (NIS) proportions are defined as the fraction of an object’s surface that remains non-interactive, characterized in both protein–peptide and geometric contexts.
• They are computed using residue solvent accessibility thresholds in biomolecular systems and analytical integrals for intersecting cylinders, yielding metrics like PSI entropy.
• Applications include discriminating cognate versus non-cognate interactions and guiding design in both peptide binding optimization and computational geometry modeling.

Non-interacting surface (NIS) proportions quantify the fraction of an object's surface not engaged in specific interactions, whether these occur at protein–peptide interfaces or between intersecting geometric bodies. In molecular biophysics, NIS proportions of apolar and charged residues play a central role in peripheral surface information (PSI) entropy, a descriptor of the statistical organization of protein surfaces away from the binding interface. In geometric modeling, NIS proportions index the impervious surface remaining between intersecting bodies, such as orthogonal cylinders, with precisely defined analytical or numerical formulae. The quantification and analysis of NIS underpin predictive modeling in both physical biochemistry and computational geometry.

1. Formal Definition and Identification of NIS

The identification of non-interacting surface is context specific:

• Protein–Peptide Systems: In protein–peptide complexes, NIS regions comprise “peripheral” residues that meet two criteria: a minimum relative solvent-accessible surface area (@@@@1@@@@ ≥ 0.05, computed from per-residue SASA via NACCESS) and exclusion from the binding interface (no heavy atom within 5 Å of the partner). These peripheral residues are assigned to apolar (A: ALA, VAL, LEU, ILE, MET, PHE, TRP, PRO), charged (C: ARG, LYS, ASP, GLU), or polar (P: SER, THR, ASN, GLN, HIS, TYR, CYS) chemical classes (Grear et al., 31 Jan 2026).
• Geometric Systems: For the intersection of two equal-radius, orthogonally arranged cylinders, NIS refers to the portion of lateral surface not included within the intersection zone. This is parameterized by the normalized penetration depth $\eta = d/R$ (with $d$ the overlap and $R$ the cylinder radius), and the explicit surface calculation excludes the interacting (mutually penetrated) areas (Aschmoneit et al., 27 Dec 2025).

2. Mathematical Quantification of NIS Proportions

NIS proportions are represented by normalized metrics:

• Biomolecular context: Given $n_A$, $n_C$, $n_P$—counts of exposed, non-interface peripheral residues in apolar, charged, and polar classes, respectively—the residue-based NIS proportions are

$$\mathcal{N}_a = \frac{n_A}{n_A + n_C + n_P}, \qquad \mathcal{N}_c = \frac{n_C}{n_A + n_C + n_P}.$$

Area-weighted versions employ summed SASA over class members, normalized by the total (Grear et al., 31 Jan 2026).

• Geometric context: The NIS proportion for two cylinders is

$$P_{\rm NIS}(\eta) = 1 - \frac{A_{\rm int}(\eta)}{4\pi R^2 \eta}$$

where $A_{\rm int}(\eta)$ is the area of intersecting cylindrical shells (Aschmoneit et al., 27 Dec 2025).

3. Computational Methods for NIS Estimation

• Protein–Surface Partitioning: Residue classification, RSA thresholding, and exclusion of interface residues proceed algorithmically from atomic-resolution structures and simulation snapshots. Each structure (microstate) gives a tuple $(n_A, n_C, n_P)$; these are mapped into $(\mathcal{N}_a, \mathcal{N}_c)$ coordinates, defining a discrete “hex–cell” occupancy system in $\mathcal{N}$-space. No further clustering is used; exact counts of identical states define multiplicities (Grear et al., 31 Jan 2026).
• Cylinder Intersection Integrals: The exact NIS proportion for cylinders is computed for $\eta \leq 1$ by the single-integral formula

$$P_{\rm NIS}(\eta) = 1 - \frac{2}{\pi \eta} \int_0^{\sqrt{\eta(2-\eta)}} \cos^{-1}(2-\eta - \sqrt{1-x^2})\, dx$$

An empirical closed-form approximation,

$$P_{\rm NIS}^{\rm approx}(\eta) = 1 - \frac{4}{\pi \eta} \sin\left(\frac{\pi \eta}{4}\right)$$

achieves a maximum relative error of approximately 32% over $[0, 1]$. In the small-overlap limit $(\eta \to 0)$, both forms approach unity; at maximal penetration $(\eta = 1)$ the exact value is $\approx 0.145$, while the approximation undershoots at $\approx 0.100$ (Aschmoneit et al., 27 Dec 2025).

4. Statistical Organization and PSI Entropy

Peripheral surface information (PSI) entropy quantifies the distributional collapse or spread of NIS macrostates across ensembles of structures:

• Definition: Let $g_i$ denote microstate occupancy of macrostate $i$ (macrostate labeled by $(n_A, n_C, n_P)$), $p_i = g_i/\Omega$ the corresponding frequency, and $N$ the number of unique macrostates. The unnormalized Shannon entropy is

$S_\Psi' = -\sum_{i=1}^N p_i \log_2 p_i$

A normalization factor $K = Q/M$ (where $Q$ is the count of distinct inter-chain contact pairs and $M$ a weighted sum over all contacts) yields the final PSI entropy:

$S_\Psi = -\frac{Q}{M} \sum_{i=1}^N p_i \log_2 p_i$

Low $S_\Psi$ signifies the emergence of dominant surface modes—ensembles occupy only a small fraction of possible NIS macrostates (Grear et al., 31 Jan 2026).

5. Empirical Findings and Evolutionary Implications

• Emergence of Dominant N-states: Across WW domains bound to proline-rich peptides (PY set), PSI entropy values for semi-flexible refinement ensembles were $S_\Psi = 2.95 \pm 1.09$ bits, with the dominant macrostate near $$(\mathcal{N}_a, \mathcal{N}_c) \approx (0.41, 0.22)$$. After explicit-solvent molecular dynamics (MD), $S_\Psi$ further decreased to $2.74 \pm 0.98$ bits. Random peptide decoys (RD set) consistently produced higher $S_\Psi$ (Step 2: $3.94 \pm 1.24$ bits; Step 3: $3.62 \pm 1.10$ bits) (Grear et al., 31 Jan 2026).
• Robustness Across Systems and Protocols: The persistence of dominant NIS macrostates spans different docking (HADDOCK3) and MD protocols, as well as receptor families (WW, PDZ, MDM2). Properly matched complexes (e.g., p53 peptide with MDM2/4HFZ) show 21.6–42.8% lower PSI entropy than improper (noncognate) pairings.
• Experimental Confirmation: Analysis of 36 WW-domain PDB entries ($\Omega=657$ microstates) yielded 234 distinct NIS macrostates, with mode occupancy $g_{\max}=20$. The effective number of occupied macrostates ($N_{\rm eff} = 2^{S'_\Psi}$) was $161.8$, constituting 24.6% of the total ensemble, and centralized near a reproducible dominant mode (Grear et al., 31 Jan 2026).
• Evolutionary Implication: The consistent restriction of peripheral surfaces to narrow regions in NIS space for favored complexes implies evolutionary selection for specific apolar/charged surface fingerprints that facilitate high-affinity binding.

6. Applications and Broader Relevance

• Prediction and Design: PSI entropy serves as a scoring filter to discriminate cognate from non-cognate complexes and as an optimization objective in peptide design, favoring mutations driving ensembles toward lower $S_\Psi$ values.
• Negative Engineering: The deliberate dispersion of peripheral surface configurations (increasing $S_\Psi$) could disrupt binding, offering a rational approach to inhibiting undesirable interactions.
• Geometry and Physical Modeling: In engineering and computational geometry, exact and approximate NIS proportions for intersecting cylinders furnish reference values for numerical validation and approximation quality assessment (Aschmoneit et al., 27 Dec 2025).

7. Summary Table: NIS Proportion Formulae

Domain	NIS Proportion (NIS)	Context
Protein–peptide residues	$\mathcal{N}_a,\,\mathcal{N}_c$	Fraction of apolar/charged, peripheral, non-interface residues (Grear et al., 31 Jan 2026)
Cylinder intersection (exact)	$$P_{\rm NIS}(\eta) = 1 - \frac{2}{\pi \eta} \int_{0}^{\sqrt{\eta(2-\eta)}} \cos^{-1}(2-\eta-\sqrt{1-x^2})\, dx$$	Surface fraction not participating in mutual intersection (Aschmoneit et al., 27 Dec 2025)
Cylinder intersection (approx)	$$P_{\rm NIS}^{\rm approx}(\eta) = 1 - \frac{4}{\pi \eta} \sin\left(\frac{\pi \eta}{4}\right)$$	Closed-form with $\leq 32\%$ error on $[0, 1]$ (Aschmoneit et al., 27 Dec 2025)

In both biomolecular and geometric contexts, NIS proportions and their statistical/analytical characterization provide a rigorous, globally meaningful summary of otherwise distributed, context-dependent interfacial phenomena.