Transition states in protein folding kinetics: Modeling Phi-values of small beta-sheet proteins

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.

💡 Research Summary

The paper addresses how to extract structural information about the transition state (TS) of small single‑domain proteins from mutational Φ‑value analysis, focusing on two β‑sheet WW domains (the PIN and the FBP WW domain). The authors propose a minimalist kinetic model in which the TS is not a single conformation but a statistical mixture of two microstates: one in which β‑hairpin 1 is formed while hairpin 2 is absent, and another in which hairpin 2 is formed while hairpin 1 is absent. This hypothesis is motivated by recent molecular‑dynamics simulations of a designed three‑strand β‑sheet protein that revealed a similar dual‑pathway TS.

The native fold of WW domains consists of two antiparallel β‑hairpins that together create a three‑strand β‑sheet, plus a small hydrophobic core that stabilizes the sheet. In the model, the free‑energy change caused by a mutation (ΔΔG) is decomposed into three additive contributions: ΔΔG₁ from hairpin 1, ΔΔG₂ from hairpin 2, and ΔΔG_core from the core. The overall activation free‑energy of folding is expressed as a weighted average of the two microstates, with a single fitting parameter p that represents the probability that hairpin 1 is formed in the TS (thus 1‑p is the probability for hairpin 2). The Φ‑value for a given mutation is then derived analytically as

Φ = p·(ΔΔG₁/ΔΔG) + (1‑p)·(ΔΔG₂/ΔΔG) + γ·(ΔΔG_core/ΔΔG),

where γ quantifies the degree to which the core is formed in the TS. In the present work γ is set to zero, reflecting the assumption that the core contributes negligibly to the TS structure.

Using published Φ‑values for a set of point mutations in each WW domain, the authors fit p by minimizing the squared deviation between experimental and calculated Φ‑values. The optimal p values are ≈0.62 for the PIN domain and ≈0.38 for the FBP domain, indicating that the PIN TS is biased toward hairpin 1 formation, whereas the FBP TS is biased toward hairpin 2. With this single parameter the model reproduces the entire experimental Φ‑value dataset, including several negative Φ‑values that have been problematic for earlier single‑structure interpretations. Negative Φ‑values arise naturally in the model when a mutation destabilizes the core (or a hairpin) that is not significantly formed in the TS, leading to a reduction in the overall activation barrier despite a destabilizing effect on the native state.

The analysis also provides a quantitative mapping of each mutation onto the structural element it most strongly perturbs. For example, a mutation that increases the polarity of a residue in hairpin 1 reduces Φ in the PIN domain because it weakens the partially formed hairpin in the TS, whereas a hydrophobic substitution at the same position raises Φ. This level of detail offers a mechanistic explanation of how specific side‑chain changes influence folding kinetics.

The strengths of the approach are its parsimony (only one adjustable parameter) and its ability to capture both positive and negative Φ‑values within a unified framework. By treating the TS as a mixture of two partially formed structures, the model accommodates the inherent heterogeneity of the folding barrier that single‑structure models overlook. However, the assumption that the hydrophobic core contributes zero to the TS (γ = 0) may not hold for all β‑sheet proteins, and extending the model to proteins with more than two hairpins or to larger β‑sandwiches will require additional parameters or a more complex state network.

In conclusion, the study demonstrates that a simple two‑state mixture model can quantitatively explain Φ‑value data for small β‑sheet WW domains, offering new insight into the nature of their transition states and providing a practical tool for interpreting mutational effects on protein folding kinetics.