Phi value analysis, φ {\displaystyle \varphi } analysis, or φ {\displaystyle \varphi } -value analysis is an experimental protein engineering technique for studying structures of the transition state and intermediates in protein folding and conformational changes. The structure of the folding transition state has to be found from kinetic measurements and is not accessible by equilibrium methods such as protein NMR or X-ray crystallography and intermediates are often mobile and partly unstructured by definition. In φ {\displaystyle \varphi } -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 φ {\displaystyle \varphi } 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, WW domain, individual domains of proteins L and G, ubiquitin, and barnase have all been studied by φ {\displaystyle \varphi } analysis.
Mathematical approach
Phi is defined thus:
φ = ( Δ G W T S → D − Δ G M T S → D ) ( Δ G W N → D − Δ G M N → D ) = Δ Δ G T S → D Δ Δ G N → D {\displaystyle \varphi ={\frac {(\Delta G_{W}^{TS\rightarrow D}-\Delta G_{M}^{TS\rightarrow D})}{(\Delta G_{W}^{N\rightarrow D}-\Delta G_{M}^{N\rightarrow D})}}={\frac {\Delta \Delta G^{TS\rightarrow D}}{\Delta \Delta G^{N\rightarrow D}}}}
Δ G W T S → D {\displaystyle \Delta G_{W}^{TS\rightarrow D}} is the difference in energy between the wild-type protein's transition and denatured state, Δ G M T S → D {\displaystyle \Delta G_{M}^{TS\rightarrow D}} is the same energy difference but for the mutant protein, and the Δ G N → D {\displaystyle \Delta G^{N\rightarrow D}} 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 φ {\displaystyle \varphi } 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 φ {\displaystyle \varphi } is well between zero and one, it is less informative as it doesn't tell us which is the case:
The transition state itself is partly structured; or There are two protein populations of near-equal numbers, one kind which is mostly-unfolded and the other which is mostly-folded.
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