Protein Folding as a Spectral First-Passage Problem
Abstract
We develop a spectral theory of protein folding dynamics by establishing a precise correspondence between the conformational Fokker–Planck equation and the Latent framework previously applied to financial risk, fluid regularity, and fusion plasma confinement.
The overdamped Langevin dynamics on the free energy landscape \(F(\mathbf{x})\) generates a Fokker–Planck operator \(\mathcal{L}_{\text{FP}}\) whose eigenvalue spectrum \(\{-\lambda_k\}\) determines the complete kinetic hierarchy: \(\lambda_1^{-1}\) is the slowest conformational relaxation time, \(\lambda_2^{-1}\) is the next-slowest, and so on. We prove six results:
1. Spectral Folding Theorem. The expected folding time from an unfolded initial distribution \(p_0\) to the native basin \(\Omega_N\) is \(\mathbb{E}[\tau_{\text{fold}}] = -\mathbf{1}^\top M_{\text{killed}}^{-1} A(0)\), where \(M_{\text{killed}}\) is the spectral generator with absorbing boundary on \(\Omega_N\). This is one matrix inverse, identical in form to the disruption time formula for tokamak plasmas and the default time formula for counterparty credit risk.
2. Funnel Spectral Gap Theorem. A protein has a folding funnel if and only if the spectral gap \(\Delta = \lambda_1\) of the conformational generator satisfies \(\Delta \gg k_BT / \tau_{\text{obs}}\), where \(\tau_{\text{obs}}\) is the experimental observation timescale. The funnel depth is quantified by \(\rho = \lambda_2 / \lambda_1 > 1\): the larger \(\rho\), the more separated the slowest mode is from the rest, and the more "two-state" the folding appears.
3. Grade Decomposition of the Energy Landscape. The drift field \(F(\mathbf{x}) = -\nabla U(\mathbf{x})\) decomposes by interaction order (grade): grade-1 captures the mean force toward the native state (linear restoring), grade-2 captures pairwise interactions (hydrogen bonds, van der Waals — always stabilizing as they funnel toward equilibrium), and grade-3 captures cooperative many-body effects (hydrophobic collapse, allosteric transitions — potentially misfolding-inducing). The folding funnel hypothesis becomes: evolved proteins have \(\|A^{(3)}\| / \|A^{(2)}\| \ll 1\), meaning cooperative nonlinearities are small relative to pairwise funneling.
4. Misfolding as Grade-3 Dominance. Misfolding occurs when the grade-3 component overwhelms grade-2 funneling. This is structurally identical to: (a) turbulence onset in Navier–Stokes (grade-3 advection overcomes grade-2 dissipation), (b) plasma disruption in tokamaks (3D instabilities break axisymmetric confinement), and (c) financial crises (nonlinear correlation activation produces systemic risk). The spectral signature of misfolding is the collapse of the spectral gap \(\Delta \to 0\) as grade-3 grows.
5. USRT for Conformational Dynamics. If the free energy landscape \(U(\mathbf{x})\) is analytic in a strip of width \(\delta &g
Novelty
Reframing protein folding dynamics as a spectral first-passage problem unified with the Latent framework, yielding a computable spectral ratio ρ = λ₂/λ₁ that quantifies funnel depth and a grade decomposition that gives a mechanistic misfolding criterion — the intellectual delta is the cross-domain unification and the specific ρ-based folding certificate, not the individual spectral or Fokker-Planck ingredients which are known.