Topological Sparsification of Protein Contact Networks via the Inverse Spielman-Srivastava Sieve

Evaluating Cycle-Space Leverage Scores against Local Graph Baselines and Structural Ground Truth Protein domain organization, dynamic hinge localization, and allosteric communication pathways are central challenges in structural biology and computational biophysics. Coarse-grained protein contact networks provide an efficient topological representation, yet conventional cut-centric sparsification methods systematically discard redundant contacts within the hydrophobic folding core while over-preserving flexible peripheral loops. Furthermore, local triangle-counting heuristics fail catastrophically in multi-stranded beta-sheets, where paired strands form ladders of four-residue cycles with zero local triangles (T = 0). We present the Inverse Spielman–Srivastava Sieve (IS³), a closed-form algebraic framework for topological graph sparsification operating on the fundamental cycle space of protein contact networks. By projecting pairwise residue interactions onto the kernel of the simplicial boundary operator, IS³ evaluates the cycle-space leverage score (P_ee = 1 - R_eff), intrinsically preserving the dense, multi-scale cyclic packing of secondary structures while surgically isolating low-cycle mechanical bottlenecks. We prove that cycle leverage scores satisfy Foster's First Theorem to machine precision (sum P_ee = beta_1), providing a universal, parameter-free topological normalization equal to the first Betti number across folds of arbitrary size. Key findings and empirical benchmarks in Version 11 include: Ground-Truth Domain Decomposition: Evaluated across 26 crystallographically verified PDB structures, IS³ cleanly partitions modular architectures without predefined cluster counts. In Adenylate Kinase (1AKE), IS³ achieves a normalized mutual information (NMI) of 0.6456 (and 0.8276 in open 4AKE), outperforming local triangle-counting (NMI = 0.0092 to 0.2039) by preserving the central 5-stranded beta-sheet. In Calmodulin (1A29/1CFD), it achieves perfect biological lobe bipartition (NMI = 1.0000, 100% core domain purity) across open and closed physiological states. Multi-Protein Hinge Localization: Projecting effective resistance onto non-pendant residues (degree >= 3) consistently ranks biologically verified functional hinges as #1 or #2 across diverse functional families: Asp-47 (Rank 1) and Gly-77 (Rank 2) in T4 lysozyme, Phe-151 (Rank 1, omega-loop gateway) in TEM-1 beta-lactamase, Pro-92 (Rank 1, 80s flap) in FKBP12, His-268 (Rank 1, alpha-I helix hinge) in CDK2 kinase, Tyr-232 (Rank 2, P-loop gatekeeper) in c-Abl kinase, and Asp-147 (Rank 1) in Adenylate Kinase. Robustness in Complex Multi-Chain Crystals: In the highly disordered, post-cleavage Zika virus NS2B-NS3 protease (PDB ID: 5GJ4; 782 residues across 8 chains, with 29 isolated residues and 54 pendant leaves), IS³ robustly filters crystallographic noise to isolate the four functional heterodimers into identical 98-residue cores at k = 0.50, coalesces them into the two crystallographic dimers-of-heterodimers at k = 0.70, and accurately pinpoints the universal flaviviral active-site flap hinge at Gly-159 (eta = 0.87). Dynamic 3D Trajectory Sifting: Evaluated across a 21-frame morphing trajectory capturing the 23.7 Å swing of the LID domain in Adenylate Kinase, IS³ demonstrates 100% topological domain core persistence across the transition, while Foster's conservation holds to machine precision (error < 1.71 x 10^-13) and the hinge index eta(t) dynamically tracks cleft closure in real time. Linear Scaling of Molecular Disorder vs. Cycle Saturation: We demonstrate that in intrinsically disordered proteins (IDPs) and open polymer coils (beta_1 = 0), effective resistance scales strictly linearly with sequence separation (R_eff proportional to s^1, slope 1.0000, R^2 = 1.000000). In contrast, tertiary cycle shortcuts in folded native proteins collapse resistance to a bounded plateau (R_eff proportional to s^0), enabling the parameter-free Global Disorder Index Omega = (N - c) / M to distinguish disordered chains (Omega = 1.0000) from folded cores (Omega = 0.35 to 0.55). Microsecond Execution in Native C++: Implemented in hardware-agnostic C++ with OpenMP parallelization, the core cycle sifting kernel executes in 78.5 microseconds for monomeric enzymes (N = 214) and 148.5 microseconds for massive 8-chain complexes (N = 782) on commodity dual-core CPUs, delivering an ultra-high sustained throughput of 6,700 to 12,700 structures per second. This enables frame-by-frame sifting of 100,000-frame molecular dynamics trajectories in under 10–15 seconds and proteome-scale screening of the entire AlphaFold database in days on a single desktop without GPU accelerators. Package contents include the complete 24-page publication manuscript (PDF and LaTeX source), 300 DPI high-resolution figures (including 3D molecular trajectory projections, pipeline flowchart, and confusion matrix heatmaps), machine-readable benchmark datasets (CSV and JSON), technical documentation on Foster's invariance, and open-source C++ OpenMP kernels licensed under the PolyForm Noncommercial License 1.0.0.