Graph-Spectral Protein Folding: Simulating Peptide Self-Assembly via Laplacian Fiedler Vector Optimization

Abstract

Protein folding is traditionally described as a trajectory over a complex energy landscape governed by empirical physical forcefields (electrostatics, van der Waals, torsional potentials, and implicit/explicit solvent interactions). However, evaluating all atomic force interactions demands massive computational power, and long-standing questions remain regarding how peptides efficiently navigate the vast conformational space (Levinthal’s paradox) to reach their native structures. Here, we present a forcefield-free, graph-spectral paradigm for peptide folding. By mapping a peptide and its hydration shell onto a weighted graph, we demonstrate that protein self-assembly can be driven solely by maximizing the Fiedler value (λ2)—the second smallest eigenvalue of the graph Laplacian matrix, which measures algebraic connectivity. Incorporating a Contact-Locking mechanism to model topological cooperativity, our algorithm successfully folds the benchmark 10-residue peptide Chignolin (PDB ID: 1UAO) into its native β-hairpin conformation with a radius of gyration (Rg ≈ 5.12–5.29 Å) closely matching experimental values (5.17 Å) without calculating any potential energy. Furthermore, by introducing a Polar-Priority Phase Model, we quantitatively reproduce the classical "Framework Model" of biophysics, wherein backbone hydrogen-bond scaffolds form prior to hydrophobic packing. We extend this model to the stabilized variant CLN025 and the 20-residue Trp-cage (PDB ID: 1L2Y), achieving less than 1.9% error in compaction. Finally, we establish a theoretical duality between solvent Fiedler maximization and Proton-Coupled Electron Transfer (PCET) pathways (Proton Wires). This graph-spectral framework provides a novel alternative to molecular dynamics (MD) simulations and establishes λ2 as a fundamental topological reaction coordinate in structural biology. Keywords: Protein Folding, Spectral Graph Theory, Fiedler Value, Algebraic Connectivity, Contact-Locking, Chignolin, Framework Model, Solvent Network.