Quantum mechanics as a theory was invented in order to explain why Newtonian mechanics and classical electrodynamics did not work in certain situations: small scales, spectra of atoms , black body radiation. Newtonian mechanics and classical electrodynamics cannot explain or predict the behavior of atoms (their interactions) , neither that of molecules.
Thus the answer is proteins interact with quantum mechanics. It can be mathematically demonstrated that classical mechanics and electrodynamics emerges from the underlying quantum nature when dimensions are large.