Topological Mechanics of Protein Recognition Dataset

Abstract

Molecular protein recognition is governed by discrete algebraic topology rather than continuous geometry or sequence distributions. The core hypotheses state: Topological Novelty: Simply connected complexes ($\beta_{1} = 0$) generate sequences orthogonal to known biologics ($< 20\%$ identity), escaping CDR loop entrapment ($\beta_{1} \ge 1$). Nilpotent Realizability: Boundary nilpotency ($B_{1}B_{2} = 0$) is mandatory for 3D Euclidean realization and foldability. Target Invariance: Sequence orthogonality of $\beta_{1} = 0$ designs is invariant across indications and drug target density. Critical Percolation: Environmental pH responsiveness is driven by homological phase transitions where protonating $N_{\text{His}} \ge 4$ causes boundary rank collapse and unbinding ($\Delta\Delta G \ge +14.5\text{ kcal/mol}$). Shannon Entropy: Topological novelty stems from high structural order rather than sequence randomness. Data Overview and Gathering Methodology The dataset (`topological_protein_discovery.db`) evaluates $N = 10,000$ de novo binder candidates across 24 disease targets in 6 therapeutic areas. Candidates were compiled via Directed Acyclic Cell Complexes (DACCs), checked for boundary nilpotency ($B_1 B_2 = 0$), and validated in silico (AlphaFold2) and against novelty databases. Key Findings 100% vs. 0% Foldability Separation (Theorem 1): Nilpotent candidates ($B_1 B_2 = 0$) achieved a 100.0% folding success rate (mean pLDDT 90.33), whereas non-nilpotent controls yielded 0.0% success (mean pLDDT 51.56). First Betti Number Governs Sequence Novelty (Theorem 2): Simply connected designs ($\beta_1 = 0$) averaged 15.00% sequence identity to known therapeutics, while loop-bearing designs ($\beta_1 \ge 1$) matched canonical germlines (67.9%–72.6% identity). Strict Target Invariance (Theorem 3): Sequence identity for $\beta_1 = 0$ designs remained flat (~15%) across all therapeutic areas regardless of existing target density ($\beta = 0.0008, p = 0.421$). Discontinuous Histidine Switch Threshold (Theorem 4): Protonating $N_{\text{His}} \ge 4$ triggers a homological rank collapse, switching $\Delta G_{6.0}$ positive and forcing complete dissociation ($\Delta\Delta G \ge +14.5\text{ kcal/mol}$). Lower Shannon Entropy in Novel Binders (Theorem 5): $\beta_1 = 0$ designs showed lower Shannon entropy (3.72 bits) than standard antibody frameworks (4.10 bits), confirming novelty is driven by structured helical packing. Universal Druggability Genus (Theorem 6): Druggable contact patches across all targets satisfied the 2-disk Euler characteristic $\chi = \vert{}K_0\vert{} - \vert{}K_1\vert{} + \vert{}K_2\vert{} = 1$.