The change of protein vibrations oil ligand binding is of functional and thermodynamic importance. Here, this process is characterized using a simple analytical "ball-and-spring" model and all-atom normal-mode analysis (NMA) of the binding of the cancer drug, methotrexate (MTX) to its target, dihydrofolate reductase (DHFR). The analytical model predicts that the Coupling between protein vibrations and ligand external motion generates entropy-rich, low-frequency vibrations in the complex. This is consistent with the atomistic NMA which reveals vibrational softening in forming the DHFR-MTX complex, a result also in qualitative agreement with neutron-scattering experiments. Energy minimization of the atomistic bound-state (13) structure while gradually decreasing the ligand interaction to zero allows the generation of a hypothetical "intermediate" (1) state, without the ligand force field but With a structure similar to that of B. In going from I to 13, it is found that the vibrational entropies of both the protein and MTX decrease while the complex structure becomes enthalpically stabilized. However, the relatively weak DHFR:MTX interaction energy results in the net entropy gain arising from Coupling between the protein and MTX external motion being larger than the loss of vibrational entropy oil complex formation. This. together with the I structure being more flexible than the unbound Structure, results in the observed vibrational softening on ligand binding.