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Encyclopedia of Physical Science and Technology EN013H-614 July 27, 2001 10:29

184 Protein Folding

TABLE I Relationships Describing Two-State Transitions in Proteins
Temperature
o
G un (T ) = H un − T S o un (7a)
o
o
G un (T ) = H o,un + C p (T − T o ) − T [ S o,un + C p ln(T/T o )] (7b)
where
o
H o,un is the enthalpy change at T = T o .
o
S un is the entropy change at T = T o .
C p is the change in heat capacity upon unfolding.
Chemical Denaturants
G un ([d]) = G o o,un − m[d] (linear extrapolation model) (8)
where
G o o,un is the free energy change in the absence of d.
m = δ G un /δ[d].
pH
 n 
+
[H ]
 
 
 1 + 
 
K a,U
 
G un (pH) = G o o,un − RT • ln n (9)
+
 [H ] 
 
 1 + 


 
K a,N
where
G o is the free energy change at neutral pH.
o,un
K a,U is the acid dissociation constant of a residue in the unfolded state.
K a,N is the acid dissociation constant of a residue in the native state.
Pressure
G un (P) = G o o,un – V un (P o –P) (10)
where
V un = volume change for N ↔ U transition.
P o = reference pressure.
For a two-state transition, A ↔ B (or N ↔ U for the unfolding of a native, N, to an unfolded, U, state
of a protein) the mole fractions of the N and U states are given as X N = 1/Q, X U = exp(– G un /RT )/Q,
where Q = 1 + exp(– G un /RT ) and the function for G un is taken from above the average fluorescence signal,

F calc = X i (F i + xδF i /δ x ), where x is a generalized perturbant.
G o o,un , the free energy change for unfolding in the ab- 3. Acid-induced unfolding: The relationship for acid-
sence of denaturant, and m, the denaturant susceptibility induced unfolding assumes that there are n equivalent acid
parameter (= –δ G un /δ[d]), where [d] is the molar dissociating groups on a protein that all have the same
concentration of added chemical denaturant. 9,10 Through pK a,U in the unfolded state and that they are all perturbed
anempiricalrelationship,thegivenequationappearstoad- to have a pK a,N in the N state. If the pK a,N is more than
equately describe the pattern for denaturant-induced un- 2 pH units lower than pK a,U , then the equation simplifies
folding of a number of proteins. The G o o,un value is a with the denominator of the right term going to unity. The
direct measure of the stability of a protein at the ambient simplest relationship for acid-induced unfolding includes
solvent conditions, which can be moderate temperature G o , the free energy of unfolding at neutral pH; n, the
o,un
o
and pH (e.g., 20 C and pH 7). The m value also provides number of perturbed acid dissociating residues; and their
structural insights, as m values have been suggested to pK a,U in the unfolded state. Presumably, n should be an
correlate with the change in solvent accessible apolar sur- integer and pK a,U should be approximately equal to the
face area upon unfolding of a protein. 11 For example, a values for such amino acids as glutamate, aspartate (e.g.,
relatively large m value (i.e., a high susceptibility of the pK a,U should be about 4 to 4.3) or histidine (e.g., pK a,U
unfolding reaction to denaturant concentration) indicates should be around 6.5).
that there is a large change in the exposure of apolar side 4. Pressure-induced unfolding: In the relationship for
chains on unfolding, which might be the case for a protein pressure, P, induced unfolding of proteins, G o is
o,un
that has an extensive core of apolar side chains that are again the value of the free energy change at 1 atmosphere
exposed upon denaturation. pressure and V un = V U − V N is the difference in volume