Residue-based quadratic free energy relationship is a mathematical formulation of the consistency principle of protein folding

Abstract

ABSTRACTThe consistency principle represents a physicochemical condition requisite for ideal protein folding. The consistency principle assumes that any pair of amino acid residues in partially folded structures has an attractive short-range interactiononly ifthe two residues are in contact in the final native structure. This native-centric model is supported by theoretical and simulation studies, but no experimental evidence has been presented. NMR spectroscopy can determine the residue-specific equilibrium constant, K, and the residue-specific rate constant, k (forward and backward). Linear free energy relationships (LFER) in the rate equilibrium free energy relationship (REFER, log k vs. log K) plots are widely seen in protein-related phenomena. Here, we examined the theoretical basis of the residue-based LFER. First, we derived a basic expression, ρij= ½(ϕi+ ϕj), from the consistency principle, where ρijis the slope of the line segment that connects residuesiandjin the REFER plot, and ϕiand ϕjare the local fractions of the native structure in the transient state ensemble (TSE). Next, we showed that the general solution is the alignment of the (log K, log k) data points on a parabolic curve in the REFER plot. Importantly, unlike LFER, the quadratic free energy relationship (QFER) is compatible with the precedent formation of local structures in TSE. QFER provides a new insight: A foldable polypeptide chain consists of several folding units, which are loosely butconsistentlycoupled to undergo smooth structural changes. QFER also enables a new ϕ-value analysis without the need for mutations.SignificanceThe physicochemical basis of smooth protein folding has been theoretically explained by the consistency principle of protein folding. Here we propose that the consistency principle is experimentally formulated by the quadratic relationship in the double logarithm plot of residue-specific equilibrium and rate constants of a polypeptide chain. One application is a new-type ϕ-value analysis, free from the adverse effects of mutations. This study will trigger the development of innovative experimental techniques that enable the measurement of accurate residue-specific thermodynamic and kinetic parameters of foldable polypeptides.