Decline of protein structure rigidity with interatomic distance

Background: Protein structural rigidity was analyzed in a non-redundant ensemble of high-resolution protein crystal structures by means of the Hirshfeld test, according to which the components (uX and uY) of the B-factors of two atoms (X and Y) along the interatomic direction is related to their degree of rigidity: the atoms may move as a rigid body if uX = uY and they cannot if uX ≠ uY. Results: It was observed that the rigidity degree diminishes if the number of covalent bonds intercalated between the two atoms (d_seq) increases, while it is rather independent on the Euclidean distance between the two atoms (d): for a given value of d_seq, the difference between uX and uY does not depend on d. No additional rigidity decline is observed when d_seq ≥ ~ 30 and this upper limit is very modest, close to 0.015 Å. Conclusions: This suggests that protein flexibility is not fully described by B-factors that capture only partially the wide range of distortions that proteins can afford.