Maximum Entropy in the Gaussian Network Model: A Thermodynamic Reference State for Protein Dynamics

This paper presents a rigorous proof that the Gaussian Network Model (GNM) with uniform spring constants maximizes entropy subject only to constraints imposed by the native contact topology. We demonstrate that this entropic optimality establishes the GNM as a fundamental thermodynamic reference state for protein dynamics, rather than merely a computational convenience. Our analysis shows that proteins, as soft matter systems, are predominantly governed by entropic considerations, with thermal fluctuations distributed among their degrees of freedom in a manner that maximizes configurational entropy. The uniform-spring constant GNM provides the least biased representation of protein dynamics, explaining its remarkable success in predicting experimental observables including NMR-measured correlations. We discuss how applying additional constraints to this maximum-entropy reference state, such as non-uniform spring constants or specific covariance requirements, inevitably reduces entropy while increasing Helmholtz free energy. This reference state framework offers new insights into allosteric mechanisms, drug binding effects, and the evolutionary balance between thermodynamic optimality and functional specialization in proteins. Previous approaches that constrained GNM fluctuations while maximizing entropy are contextualized within this theoretical foundation, highlighting pathways for model refinement while maintaining entropic principles.