Effect of Backbone Cyclization on Protein Folding Stability: Chain Entropies of both the Unfolded and the Folded States are Restricted

Circular versions of a large number of proteins have been designed by connecting the N and C termini via peptide linkers. A motivation for these designs is the assumed enhancement in folding stability, because backbone cyclization reduces the chain entropy of the unfolded state. Here, it is recognized that backbone cyclization also reduces the chain entropy of a flexible peptide linker in the folded state. Specifically, the end-to-end distance of the linker is restricted to fluctuations around the average displacement between the N and C termini of the folded protein. The balance of the chain-entropy reductions in the folded and unfolded states is used to predict the change in the unfolding free energy, ΔΔGcycl, by backbone cyclization. Predicted values of ΔΔGcycl are in quantitative agreement with results of a careful study on cyclizing the 34 residue PIN1 WW domain by linkers with two to seen residues. The experimental results of an optimal linker length l=4 and a maximum stabilization of 1.7 kcal/mol are reproduced. Calculations of ΔΔGcycl for a broad selection of circular proteins suggest that the stabilizing effect of backbone cyclization is modest, reflecting entropy reductions in both the unfolded and the folded states.