Low-frequency localized spin-dynamical coupling in proteins

We show that the magnetic field dependence of the proton spin–lattice relaxation rates 1/T₁=Aω^-b₀ in non-crystalline proteins may be quantitatively related to structural fluctuations localized along the backbone that modulate proton–proton dipolar couplings. The parameter A is related to the temperature, the dipolar coupling strength and the energy for the highest vibrational frequency in the polymer backbone. The parameter b is related to the fractal dimensionality of the spatial distribution of protons and to the spectral dimensionality that characterizes the anomalous diffusion. Extension of the theory is presented to treat the case of hydrated proteins. This theory is satisfactorily compared with field cycling experiments realized on lysozyme protein.