Approximate Values for Force Constant and Wave Number Associated with a Low-Frequency Concerted Motion in Proteins Can Be Evaluated by a Comparison of X-ray Structures

Low-frequency internal motions in protein molecules play a key role in biological functions. A direct relationship between low-frequency motions and enzymatic activity has been suggested for bovine pancreatic ribonuclease (RNase A). The flexibility−function relationship in this enzyme has been attributed to a subtle and concerted breathing motion of the β-sheet regions occurring upon substrate binding and release. Here, we calculate an approximate value for the force constant and the wave number of the low-frequency β-sheet breathing motion of RNase A, by using the Boltzmann hypothesis on a set of data derived from a simple conventional structural superimposition of an unusual large number of X-ray structures available for the protein. The results agree with previous observations and with theoretical predictions on the basis of normal-mode analysis. To the best of our knowledge, this is the first example in which the wave number and the force constant of a low-frequency concerted motion in a protein are directly derived from X-ray structures.