Dvoretzky’s Theorem as a Geometric Framework for Protein Frustration

Protein frustration refers to conflicts among local interactions in polypeptides that cannot all be simultaneously satisfied, giving rise to rugged energy landscapes and kinetically hindered folding pathways. Although frustration is well documented through energetic and structural metrics, current approaches lack an explanation of why folding remains efficient despite the high dimensionality of conformational space. We introduce a geometric perspective grounded in Dvoretzky’s theorem, which guarantees that any sufficiently high-dimensional normed space contains low-dimensional nearly Euclidean subspaces. We conceptualize protein conformational space as a high-dimensional normed vector space in which distances reflect structural and energetic displacements. Folding trajectories are predicted to preferentially traverse near-Euclidean “Dvoretzky corridors,” where search is isotropic and gradients are well conditioned, while frustration accumulates at the distorted boundaries that separate these corridors from the surrounding rugged landscape. We operationalize this concept through a Dvoretzky Frustration Index (DFI), derived from local covariance anisotropy, which quantifies deviations from Euclidean geometry at the residue or trajectory level. In our simulations, high DFI regions overlapped with areas corresponding to frustration hotspots, allosteric residues and sites of heightened mutational sensitivity. Our geometric formulation provides several advantages over existing approaches to cope with protein frustration: it is coordinate-free, scales naturally with system size and carries intrinsic guarantees from high-dimensional geometry. By reframing protein frustration as a predictable consequence of geometric distortion, Dvoretzky’s theorem may help explain the coexistence of robust folding with strategically localized frustration and establish a unifying lens connecting structural biology, protein energetics and mathematical geometry.