Perturbation theory of Phi-value analysis of two-state protein folding: relation between p fold and Phi values

In protein folding, the transition state ensemble is defined as the set of conformations with p(fold)=12, where the p(fold) of a conformation is the probability that starting from this conformation the protein folds before it unfolds. Experimentally, this ensemble is probed by the Phi-value analysis, where Phi is the ratio of the changes in the logarithms of the folding rate and the equilibrium constant when the system is perturbed by a mutation. We show that for a two-state protein the Phi value can be expressed in terms of the perturbation and only the first two eigenfunctions of the evolution operator (e.g., a rate matrix) of the wild-type protein. The first eigenfunction is the equilibrium probability distribution while the second is proportional to p(fold), thus establishing a formal relation between p(fold) and Phi values. In addition to providing insight into the theoretical foundation of the Phi-value analysis, our results may prove practically useful in performing such analyses within the framework of models containing a large number of states.