  Elastic network models (ENM) and constraint-based, topological rigidity
analysis are two distinct, coarse-grained approaches to study conformational
flexibility of macromolecules. In the two decades since their introduction,
both have contributed significantly to insights into protein molecular
mechanisms and function. However, despite a shared purpose of these approaches,
the topological nature of rigidity analysis, and thereby the absence of motion
modes, has impeded a direct comparison. Here, we present an alternative,
kinematic approach to rigidity analysis, which circumvents these drawbacks. We
introduce a novel protein hydrogen bond network spectral decomposition, which
provides an orthonormal basis for collective motions modulated by non-covalent
interactions, analogous to the eigenspectrum of normal modes, and decomposes
proteins into rigid clusters identical to those from topological rigidity. Our
kinematic flexibility analysis bridges topological rigidity theory and ENM, and
enables a detailed analysis of motion modes obtained from both approaches. Our
analysis reveals that collectivity of protein motions, reported by the Shannon
entropy, is significantly lower for rigidity theory versus normal mode
approaches. Strikingly, kinematic flexibility analysis suggests that the
hydrogen bonding network encodes a protein-fold specific, spatial hierarchy of
motions, which goes nearly undetected in ENM. This hierarchy reveals distinct
motion regimes that rationalize protein stiffness changes observed from
experiment and molecular dynamics simulations. A formal expression for changes
in free energy derived from the spectral decomposition indicates that motions
across nearly 40% of modes obey enthalpy-entropy compensation. Taken together,
our analysis suggests that hydrogen bond networks have evolved to modulate
protein structure and dynamics.
