Methods for the accurate estimation of confidence intervals on protein folding phi-values

Phi-values provide an important benchmark for the comparison of experimental protein folding studies to computer simulations and theories of the folding process. Despite the growing importance of phi measurements, however, formulas to quantify the precision with which phi is measured have seen little significant discussion. Moreover, a commonly employed method for the determination of standard errors on phi estimates assumes that estimates of the changes in free energy of the transition and folded states are independent. Here we demonstrate that this assumption is usually incorrect and that this typically leads to the underestimation of phi precision. We derive an analytical expression for the precision of phi estimates (assuming linear chevron behavior) that explicitly takes this dependence into account. We also describe an alternative method that implicitly corrects for the effect. By simulating experimental chevron data, we show that both methods accurately estimate phi confidence intervals. We also explore the effects of the commonly employed techniques of calculating phi from kinetics estimated at non-zero denaturant concentrations and via the assumption of parallel chevron arms. We find that these approaches can produce significantly different estimates for phi (again, even for truly linear chevron behavior), indicating that they are not equivalent, interchangeable measures of transition state structure. Lastly, we describe a Web-based implementation of the above algorithms for general use by the protein folding community.