Thermal motion in proteins: Large effects on the time-averaged interaction energies

As a consequence of thermal motion, inter-atomic distances in proteins fluctuate strongly around their average values, and hence, also interaction energies (i.e. the pair-potentials evaluated at the fluctuating distances) are not constant in time but exhibit pronounced fluctuations. These fluctuations cause that time-averaged interaction energies do generally not coincide with the energy values obtained by evaluating the pair-potentials at the average distances. More precisely, time-averaged interaction energies behave typically smoother in terms of the average distance than the corresponding pair-potentials. This averaging effect is referred to as the thermal smoothing effect. Here, we estimate the strength of the thermal smoothing effect on the Lennard-Jones pair-potential for globular proteins at ambient conditions using x-ray diffraction and simulation data of a representative set of proteins. For specific atom species, we find a significant smoothing effect where the time-averaged interaction energy of a single atom pair can differ by various tens of cal/mol from the Lennard-Jones potential at the average distance. Importantly, we observe a dependency of the effect on the local environment of the involved atoms. The effect is typically weaker for bulky backbone atoms in beta sheets than for side-chain atoms belonging to other secondary structure on the surface of the protein. The results of this work have important practical implications for protein software relying on free energy expressions. We show that the accuracy of free energy expressions can largely be increased by introducing environment specific Lennard-Jones parameters accounting for the fact that the typical thermal motion of protein atoms depends strongly on their local environment.