Residue-based correlation between equilibrium and rate constants is an experimental formulation of the consistency principle for smooth structural changes of proteins.

The consistency principle represents a physicochemical condition requisite for ideal protein folding. It assumes that any pair of amino acid residues in partially folded structures has an attractive short-range interaction the two residues are in contact within the native structure. The residue-specific equilibrium constant, , and the residue-specific rate constant, (forward and backward), can be determined by NMR and hydrogen-deuterium exchange studies. Linear free energy relationships (LFER) in the rate-equilibrium free energy relationship (REFER) plots (i.e., log vs. log ) are widely seen in protein-related phenomena, but our REFER plot differs from them in that the data points are derived from one polypeptide chain under a single condition. Here, we examined the theoretical basis of the residue-based LFER. First, we derived a basic equation, ρ=½(φ+φ), from the consistency principle, where ρ is the slope of the line segment that connects residues and in the REFER plot, and φ and φ are the local fractions of the native state in the transient state ensemble (TSE). Next, we showed that the general solution is the alignment of the (log , log ) data points on a parabolic curve in the REFER plot. Importantly, unlike LFER, the quadratic free energy relationship (QFER) is compatible with the heterogeneous formation of local structures in the TSE. Residue-based LFER/QFER provides a unique insight into the TSE: A foldable polypeptide chain consists of several folding units, which are coupled to undergo smooth structural changes.