Phi value analysis

Phi value analysis, formula_1 analysis, or formula_2-value analysis is an experimental protein engineering technique for studying the structure of the folding transition state of small protein domains that fold in a two-state manner. The structure of the folding transition state is hard to find using methods like protein NMR or X-ray crystallography because folding transitions states are mobile and partly unstructured by definition. In formula_1-value analysis, the folding kinetics and conformational folding stability of the wild-type protein are compared with those of point mutants to find "phi values". These measure the mutant residue's energetic contribution to the folding transition state, which reveals the degree of native structure around the mutated residue in the transition state, by accounting for the relative free energies of the unfolded state, the folded state, and the transition state for the wild-type and mutant proteins.
The protein's residues are mutated one by one to identify residue clusters that are well-ordered in the folded transition state. These residues' interactions can be checked by "double-mutant-cycle formula_1" "analysis", in which the single-site mutants' effects are compared to the double mutants'. Most mutations are conservative and replace the original residue with a smaller one ("cavity-creating mutations") like alanine, though tyrosine-to-phenylalanine, isoleucine-to-valine and threonine-to-serine mutants can be used too. Chymotrypsin inhibitor, SH3 domains, individual domains of proteins L and G, ubiquitin, and barnase have all been studied by formula_1 analysis.
Mathematical approach.
Phi is defined thus:
formula_6
formula_7 is the difference in energy between the wild-type protein's transition and denatured state, formula_8 is the same energy difference but for the mutant protein, and the formula_9 bits are the differences in energy between the native and denatured state. The phi value is interpreted as how much the mutation destabilizes the transition state versus the folded state.
Though formula_1 may have been meant to range from zero to one, negative values can appear. A value of zero suggests the "mutation" doesn't affect the structure of the folding pathway's rate-limiting transition state, and a value of one suggests the mutation destabilizes the transition state as much as the folded state; values near zero suggest the "area around the mutation" is relatively unfolded or unstructured in the transition state, and values near one suggest the transition state's local structure near the mutation site is similar to the native state's. Conservative substitutions on the protein's surface often give phi values near one. When formula_1 is well between zero and one, it is less informative as it doesn't tell us which is the case: 
Example: barnase.
Alan Fersht pioneered phi value analysis in his study of the small bacterial protein barnase. Using molecular dynamics simulations, he found that the transition state between folding and unfolding looks like the native state and is the same no matter the reaction direction. Phi varied with the mutation location as some regions gave values near zero and others near one. The distribution of formula_1 values throughout the protein's sequence agreed with all of the simulated transition state but one helix which folded semi-independently and made native-like contacts with the rest of the protein only once the transition state had formed fully. Such variation in the folding rate in one protein makes it hard to interpret formula_1 values as the transition state structure must otherwise be compared to folding-unfolding simulations which are computationally expensive.
Variants.
Other 'kinetic perturbation' techniques for studying the folding transition state have appeared recently. Best known is the psi (formula_15) value which is found by engineering two metal-binding amino acid residues like histidine into a protein and then recording the folding kinetics as a function of metal ion concentration, though Fersht thought this approach difficult. A 'cross-linking' variant of the formula_2-value was used to study segment association in a folding transition state as covalent crosslinks like disulfide bonds were introduced.
Limitations.
The error in equilibrium stability and aqueous (un)folding rate measurements may be large when values of formula_2 for solutions with denaturants must be extrapolated to aqueous solutions that are nearly pure or the stability difference between the native and mutant protein is 'low', or less than 7 kJ/mol. This may cause formula_2 to fall beyond the zero-one range. Calculated values formula_2 depend strongly on how many data point are available and lab conditions.