  Small single-domain proteins often exhibit only a single free-energy barrier,
or transition state, between the denatured and the native state. The folding
kinetics of these proteins is usually explored via mutational analysis. A
central question is which structural information on the transition state can be
derived from the mutational data. In this article, we model and structurally
interpret mutational Phi-values for two small beta-sheet proteins, the PIN and
the FBP WW domain. The native structure of these WW domains comprises two
beta-hairpins that form a three-stranded beta-sheet. In our model, we assume
that the transition state consists of two conformations in which either one of
the hairpins is formed. Such a transition state has been recently observed in
Molecular Dynamics folding-unfolding simulations of a small designed
three-stranded beta-sheet protein. We obtain good agreement with the
experimental data (i) by splitting up the mutation-induced free-energy changes
into terms for the two hairpins and for the small hydrophobic core of the
proteins, and (ii) by fitting a single parameter, the relative degree to which
hairpin 1 and 2 are formed in the transition state. The model helps to
understand how mutations affect the folding kinetics of WW domains, and
captures also negative Phi-values that have been difficult to interpret.
