Folding and Function as Two Sides of One Coin: Statistical Mechanics, Information Theory, and Deep Learning of Binding, Catalysis, and Allostery in an Exactly Enumerable Lattice Protein

Folding and function depend on the same conformational ensemble, but they need not favor the same population distribution. We examine this relationship in a 27-residue lattice protein with 103,346 exactly enumerated compact conformations, combining statistical mechanics, information-theoretic analysis, and variational autoencoders. Exact identities separate binding into native-site recognition and population terms, distinguish transition-state complementarity from reorganization in catalysis, and express allosteric coupling as a conformational covariance. Under composition-conserving selection, binding and turnover optimization favor a dominant compact fold, whereas strong allostery can coexist with a distributed ensemble: pure coupling optimization gives a median native population of 0.47, compared with 0.85 when designs with additional stability rewards are pooled. Composition-free binding and catalytic selection provide further counterexamples to an unconditional function-to-folding implication. Rényi divergences quantify the shortfall from the best available binding or transition-state-recognition conformation; allosteric coupling instead depends on the magnitude and alignment of two ligand-induced responses. A sequence VAE generates target-fold-compatible sequences more frequently than an independent-site baseline and partially retains binding and catalytic properties, but not allosteric coupling. Time-lagged VAEs resolve additional basin information in selected allosteric trajectories, although incomplete interconversion prevents inference of equilibrium switching kinetics. Together, these results identify a conditional relationship between functional selection and fold specificity, and distinguish population concentration for recognition from population redistribution for allosteric communication.