The evolutionary divergence of protein structure is mostly contained within a subspace spanned by the evolving protein’s lowest vibrational normal modes, a remarkable recent result, so far unexplained. To understand the mechanism underlying this behavior, here I introduce a linearly forced elastic network model (LFENM) of protein structural evolution. For a test case of globins, LFENM results are in very good agreement with observations. Moreover, in contrast with tentative biological explanations, the model predicts that protein structures will evolve along the lowest normal modes even under unselected random mutations, as a result of the chemical physics of the response of elastic networks of oscillators to perturbations.