Phi value analysis

Energy diagrams for ${\displaystyle \phi }$ = 0 (left) or 1 (right). D is the energy of the denatured state, dagger, that of the transition state, and N, that of the native state.

Phi is defined thus:[1]

$${\displaystyle \phi ={\frac {(\Delta G_{W}^{TS-D}-\Delta G_{M}^{TS-D})}{(\Delta G_{W}^{N-D}-\Delta G_{M}^{N-D})}}={\frac {\Delta \Delta G^{TS-D}}{\Delta \Delta G^{N-D}}}}$$

${\displaystyle \Delta G_{W}^{TS-D}}$ is the difference in energy between the wild-type protein's transition and denatured state, ${\displaystyle \Delta G_{M}^{TS-D}}$ is the same energy difference but for the mutant protein, and the ${\displaystyle \Delta G^{N-D}}$ 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 ${\displaystyle \phi }$ may have been meant to range from zero to one, negative values can appear.[2] 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 ${\displaystyle \phi }$ is well between zero and one, it is less informative as it doesn't tell us which is the case:

1. The transition state itself is partly structured; or
2. There are two protein populations of near-equal numbers, one kind which is mostly-unfolded and the other which is mostly-folded.

1. Phi value analysis assumes Hammond's postulate, which states that energy and chemical structure are correlated. Though the relationship between the folding intermediate and native state's structures may correlate that between their energies when the energy landscape has a well-defined, deep global minimum, free energy destabilizations may not give useful structural information when the energy landscape is flatter or has many local minima.
2. Phi value analysis assumes the folding pathway isn't significantly altered, though the folding energies may be. As nonconservative mutations may not bear this out, conservative substitutions, though they may give smaller energetic destabilizations which are harder to detect, are preferred.
3. Restricting ${\displaystyle \phi }$ to numbers greater than zero is the same as assuming the mutation increases the stability and lowers the energy of neither the native nor the transition state. It is in the same line assumed that interactions that stabilize a folding transition state are like those of the native structure, though some protein folding studies found that stabilizing non-native interactions in a transition state facilitates folding.[3]

Alan Fersht pioneered phi value analysis in his study of the small bacterial protein barnase.[4][5] 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 ${\displaystyle \phi }$ 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 ${\displaystyle \phi }$ values as the transition state structure must otherwise be compared to folding-unfolding simulations which are computationally expensive.

Other 'kinetic perturbation' techniques for studying the folding transition state have appeared recently. Best known is the psi (${\displaystyle \psi }$) value[6][7] 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,[8] though Fersht thought this approach difficult.[9] A 'cross-linking' variant of the ${\displaystyle \phi }$-value was used to study segment association in a folding transition state as covalent crosslinks like disulfide bonds were introduced.[10]

The error in equilibrium stability and aqueous (un)folding rate measurements may be large when values of ${\displaystyle \phi }$ 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 ${\displaystyle \phi }$ to fall beyond the zero-one range.[11] Calculated values ${\displaystyle \phi }$ depend strongly on how many data point are available and lab conditions.[12]

• Chevron plot
• Denaturation midpoint
• Equilibrium unfolding