Relationship of Leffler (Bronsted) alpha values and protein folding Phi values to position of transition-state structures on reaction coordinates

The positions of transition states along reaction coordinates (r++) for simple chemical reactions are often estimated from Leffler alpha values, the slope of plots of DeltaG++ (activation energy) versus DeltaG(0) (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. Phi-Value analysis measures degree of structure formation at individual residues in folding transition states from the ratio DeltaDeltaG++/DeltaDeltaG(0) for mutations. alpha values are now being used to analyze protein folding by lumping series of Phi values into single plots. But, there are discrepancies in the values of alpha 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 alpha = r++ only for limiting cases, such as for reactants and products being in parabolic energy wells of identical curvature. Otherwise, alpha can differ radically from r++, with alpha being determined just by the angles of intersection of reactant and product energy surfaces. Phi is an index of the progress of a local, energy-based reaction coordinate at the global transition state: Phi <0.5 corresponds to <50% progress of the local coordinate at the global transition state and Phi >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.