Relationship of Leffler (Brønsted) α values and protein folding Φ values to position of transition-state structures on reaction coordinates

Abstract

The positions of transition states along reaction coordinates (r ‡) for simple chemical reactions are often estimated from Leffler α values, the slope of plots of ΔG‡ (activation energy) versus ΔG0 (equilibrium free energy) for a series of structural variants. Protein folding is more complex than simple chemical reactions and has a multitude of reaction coordinates. Φ-Value analysis measures degree of structure formation at individual residues in folding transition states from the ratio ΔΔG‡/ ΔΔG0 for mutations. α values are now being used to analyze protein folding by lumping series of Φ values into single plots. But, there are discrepancies in the values of α for folding with more classical measures of the extent of structure formation, which I rationalize here. I show for chemical reactions with just a single reaction coordinate that α = r‡ only for limiting cases, such as for reactants and products being in parabolic energy wells of identical curvature. Otherwise, α can differ radically from r‡, with α being determined just by the angles of intersection of reactant and product energy surfaces. Φ is an index of the progress of a local, energy-based reaction coordinate at the global transition state: Φ <0.5 corresponds to <50% progress of the local coordinate at the global transition state and Φ >0.5 means >50%. Protein Leffler plots can force different local indexes to a single fit and give skewed underestimates of the extent of global structure formation in transition states that differ from other measures of structure formation.