Protein Folding as a Game: Nash-Boltzmann Duality and Computational Complexity

We establish a formal correspondence between protein folding and finite-strategy game theory, demonstrating that the native protein state is a Nash equilibrium in an interaction game among amino acid residues. The key insight is that the grade parameter \(\rho\) from interaction-decay game theory controls both the computational complexity of finding equilibria and the thermodynamic stability of the folded protein. When \(\rho > 1\) (grade-2 dominated games), Nash equilibria can be found in polynomial time—explaining the Levinthal paradox of fast folding. The melting temperature \(T_m\) corresponds to the bifurcation point \(\rho = 1\), where the unique pure Nash equilibrium (native state) transitions to a mixed equilibrium (unfolded ensemble). We prove that amyloidogenic proteins have \(\rho \to 1\), placing their folding games at the PPAD boundary, while molecular chaperones increase \(\rho\) by reshaping the energy landscape. These results are formalized in a Lean 4 kernel with 36 verified theorems and 0 axioms. The framework unifies three previously disconnected phenomena—Levinthal's paradox, denaturation, and chaperone function—under a single game-theoretic principle.

Keywords: protein folding, Nash equilibrium, game theory, Levinthal paradox, denaturation, chaperones, computational complexity

MSC 2020: 91A10, 92C40, 82B26, 91A80