Peripheral Surface Information Entropy

• Peripheral Surface Information (PSI) entropy is a thermoinformatic descriptor that measures the statistical variability of a protein’s non-interacting surface to assess binding specificity.
• It employs molecular docking and molecular dynamics to generate conformational ensembles and computes a normalized Shannon entropy reflecting peripheral residue patterns.
• The metric distinguishes specific from non-specific interactions and offers insights for enhancing machine-learning models and designing targeted mutations.

Peripheral Surface Information (PSI) entropy is a thermoinformatic descriptor designed to quantify the statistical variability of a protein’s non-interacting surface (NIS) in relation to binding specificity. PSI entropy captures how the ensemble-level diversity of apolar and charged residue exposure on the periphery of a receptor protein correlates with favorable, focused protein-peptide interactions. By leveraging conformational ensembles generated via molecular docking and molecular dynamics (MD), this framework reveals emergent, low-entropy surface patterns that distinguish cognate from non-cognate binders, and proposes a bridge between peripheral surface architecture, energetic specificity, and evolutionary selection (Grear et al., 31 Jan 2026).

1. Non-Interacting Surface (NIS): Definition and Chemical Classification

In a protein–peptide complex, the non-interacting surface (NIS) is defined as the set of receptor residues that are both solvent-exposed (relative solvent accessibility, RSA $>0.05$) and located at least 5 Å (heavy-atom distance) from any peptide atom. Residues comprising the interface (buried or within 5 Å) are excluded from the NIS analysis. Each NIS residue is further classified by side-chain type:

Class	Residue Set	Label
Apolar	A, V, I, L, M, F, W, Y	A
Charged	D, E, K, R	C
Polar	All other uncharged, polar residues	P

The NIS is described for each microstate (conformation) by the count tuple $(n_A, n_C, n_P)$, where $n_A$ is the number of apolar residues, $n_C$ charged, and $n_P$ polar. Because the total number of NIS residues per complex is fixed, specifying $(n_A, n_C)$ uniquely indexes a macrostate $\mathcal{N}_i$ of peripheral chemical composition.

2. Mathematical Formulation of PSI Entropy

Given a conformational ensemble of $\Omega$ microstates, the distribution of NIS macrostates $\{\mathcal{N}_1, ..., \mathcal{N}_N\}$ is quantified by enumerating the frequency $g(\mathcal{N}_i)$ of each tuple:

• Empirical probability: $p_i = g(\mathcal{N}_i) / \Omega$, with $\sum_{i=1}^N p_i = 1$.

A base diversity metric is the Shannon entropy over $p_i$:

$$S_{\Psi}' = -\sum_{i=1}^N p_i \log_2 p_i \quad \text{(bits)}$$

To adjust for energetic specificity, $S_{\Psi}'$ is normalized by an interface-focused factor $K$. For all inter-chain residue–residue contacts $(i,j)$:

• Assign unnormalized “mass” $m_{ij}$ proportional to contact probability.
• Apply a chemical pair weighting: $\tilde m_{ij} = \gamma(c_i, c_j) m_{ij}$, where $\gamma \in [0.61, 1.65]$ (see Supplement Table S2 in (Grear et al., 31 Jan 2026)).
• Compute total contact mass $M = \sum_{(i,j)} \tilde m_{ij}$ and total distinct pairs $Q$.

Define $K = Q / M$, and the normalized PSI entropy:

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

This normalization ensures $S_{\Psi}$ reflects the diversity of peripheral chemical surface patterns per unit of favorable interface contact mass. Low $S_{\Psi}$ values indicate strong, recurring NIS patterns accompanying energetically focused interfaces; high values indicate diffuse or polyspecific interactions.

3. Computational Workflow

The PSI entropy workflow is as follows:

1. Docking and MD Ensemble Generation:
   • Rigid-body docking with HADDOCK3 (~3,000 poses) is followed by semi-flexible refinement and explicit-solvent MD, yielding hundreds of microstates per complex.
2. NIS Macrostate Assignment:
   • For each microstate:
     1. Compute RSA per residue.
     2. Identify NIS residues (RSA $>0.05$, $>5$ Å from peptide).
     3. Classify each as A, C, or P; count $(n_A, n_C, n_P)$.
     4. Assign macrostate $\mathcal{N}_i$ using $(n_A, n_C)$.
3. Probability Estimation and Contact Statistics:
   • Tabulate $g(\mathcal{N}_i)$, compute $p_i$.
   • For each ensemble, determine $m_{ij}$, apply $\gamma(c_i,c_j)$, sum to $M$, count $Q$.
4. Entropy Calculation:
   • Insert all values into the normalized formula for $S_{\Psi}$.

4. Patterns of PSI Entropy in Cognate and Non-Cognate Complexes

Examination of numerous protein–peptide systems demonstrates the discriminative power of PSI entropy:

• WW–Smad7 (PDB 2LTW): 227 microstates collapse to $N = 67$ macrostates; a dominant mode at $$(\mathcal{N}_a, \mathcal{N}_c) \approx (0.40, 0.22)$$ indicates peripheral compositional focusing.
• Robustness to Parameters: Across alternate initial peptide conformations (NMR, AF2, AF3), sublinear scaling of macrostate numbers is accompanied by a persistent $\mathcal{N}_a$ mode, indicating insensitivity to sampling details.
• Cognate vs. Random Decoys: For the PPxY WW–domain system, cognate binders yield $S_{\Psi}=2.74\pm0.98$ versus $3.62\pm1.10$ for random decoys, even when unnormalized diversity is similar, underscoring the effect of $Q/M$ rescaling.
• Cross-System Specificity: Docking cognate peptides to MDM2 (4HFZ) and PDZ (1ZUB) gives $S_\Psi$ reductions of 21.6% and 42.8%, respectively, over non-cognate binders. Lower $S_{\Psi}$ is consistently associated with favorable binding and dense contact maps, exemplifying “Regime I”.

5. Cross-System and Experimental Meta-Ensemble Analysis

The robustness and biological relevance of PSI entropy are supported by both cross-system in silico and experimental data:

• System Generality: Across MDM2 and PDZ domains, the descriptor consistently distinguishes cognate from non-cognate complexes under uniform workflow parameters.
• Experimental Validation: An aggregate of 36 high-resolution WW-domain structures (34 NMR, 2 X-ray; $\Omega=657$ microstates) reveals a dominant NIS fingerprint, with 234 discrete macrostates (103 singletons, maximal occupancy 20). The effective macrostate fraction, $N_{\mathrm{eff}}/\Omega \approx 0.25$, indicates that about 25% of peripheral patterns dominate, qualitatively supporting an evolutionary preference for select NIS organizations.

6. Applications, Limitations, and Future Directions

PSI entropy serves as an ensemble-level readout of macromolecular recognition distinct from interface-centric metrics:

• Applications:
  • Integration of $S_{\Psi}$ into machine-learning scoring functions to enhance binder/discriminator models.
  • Time-resolved monitoring of NIS-state trajectories—for instance, to study allosteric coupling or responses to environmental perturbations (e.g., osmotic stress).
  • “Anti-directed” mutation design strategies that increase $S_{\Psi}$ to destabilize unwanted or aberrant complexes by suppressing dominant NIS modes.
• Limitations:
  • Results depend on docking/MD sampling depth, choice of RSA and distance cutoffs, and the tripartite residue coarsening (A, C, P).
• Prospects:
  • Refinement through finer chemical partitioning or use of continuous properties (e.g., electrostatic potential).
  • Extension to protein–protein and protein–nucleic acid complexes.
  • Dynamical analysis of $S_{\Psi}(t)$ to probe transition pathways in NIS space.

PSI entropy thus offers a quantitative and thermoinformatic perspective on the interplay between peripheral surface organization, interface energetics, and evolutionary selection, supplementing conventional metrics of biomolecular recognition (Grear et al., 31 Jan 2026).