Proteins as Hierarchies of Modules: Exact Elastic-Network Thermodynamics of Folding and Function

Lattice models suggested information-theoretic rules for protein foldability and function but cannot represent the architecture of real proteins. Using Gaussian and anisotropic elastic network models of 183 globular proteins and 77 functional systems, an Ising–Gaussian network model of folding in which every native-contact state carries its exact chain entropy, and statistical mechanics, information theory and variational autoencoders, we show that native proteins are hierarchies of modules. At every level of a recursive bisection, native modules are more weakly coupled to one another than those of densitymatched compact globules, whereas compact lattice chains have no such structure; the hierarchy is structural rather than dynamically autonomous. The folding free energy decomposes exactly over it, from 1.84 kBT per contact in the smallest modules to 0.47 kBT for the whole protein, and contact formation is weakly biased towards it. Catalytic residues are rigid and lie on the boundaries between the largest modules, and large functional motions are rigid-body displacements of modules. Function adds a local level of its own: binding free energies reside in site coordinates that slow-mode truncation, rigid modules and a variational autoencoder fail to preserve, whereas exact Schur elimination around explicit sites preserves them. This defines an exact hierarchical model of a protein, with nested modules carrying cooperativity, slow motion and communication between sites, and functional sites retained explicitly.