Content of Protein Folding 

Table I gives some widely accepted relationships for
describing the variation of Δ G_{un} for a twostate N ↔ U transition with temperature, chemical denaturant, pH, or
pressure as the perturbations. One of the equations in
Table I, when combined with those above and Eqs. (1–3), can be used to describe data as a function of the denaturing
condition. The thermodynamic parameters related
to the relationships in Table I are briefly described
below.
 Thermal unfolding: ΔH_{un}°
un and ΔS_{un} un are the enthalpy
and entropy changes for a twostate unfolding reaction.
Both ΔH_{un}° and ΔS_{un} may be temperature dependent,
when the heat capacity change, ΔC_{p}, has a nonzero value.
In this case, Eq. (7b) in Table I (the GibbsHelmholtz
equation) should be used, where the ΔH_{o}° ,un and ΔS°_{o,un} are values at some defined reference temperature, T_{o} (e.g.,
0° or 20°C).^{6,7} The heat capacity change for unfolding of
proteins is typically found to be positive and to be related
to the increase in solvent exposure of apolar side chains
upon unfolding. That is, a positive ΔC_{p} is a result of the
hydrophobic effect. A consequence is that the ΔG_{un}°(T)
for unfoldingof a proteinwill have a parabolic dependence
on temperature and will show both hightemperature and
lowtemperature induced unfolding.^{8}
 Denaturantinduced unfolding: The empirical relationship
in Table I for chemical denaturation includes
Temperature 

⇒ Equation [7a] 
ΔG_{un}(T )=ΔH_{un}° − TΔS_{un}° 


⇒ Equation [7b] 
ΔG_{un}(T )=ΔH_{o,un}° +ΔC_{p}(T −T_{o}) −T [ΔS_{o,un}° +ΔC_{p} ln(T/T_{o})] 

where 


ΔH_{o,un}° is the enthalpy change at T =T_{o}. 



ΔS_{un}° is the entropy change at T =T_{o}. 



ΔC_{p} is the change in heat capacity upon unfolding. 

Chemical Denaturants 

⇒ Equation [8] 
ΔG_{un}([d])= ΔG°_{o
,un} −m[d]h 

where 


ΔG°_{o,un} is the free energy change in the absence of d. 



pH 

⇒ Equation [9] 
ΔG_{un}(pH)= ΔG°_{o
,un} − RT ln 
{

( 
1+ 
[H^{+}] 
^{n} 
) 
} 

Ka,U 


( 
1+ 
[H+] 
^{n} 
) 

Ka,N 




where 


ΔG°o
,un is the free energy change at neutral pH. 



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 

⇒ Equation [10] 
ΔG_{un}(P)=ΔG°_{o
,un}–ΔV_{un}(P_{o}–P) 

where 


ΔV_{un} = volume change for N ↔ U transition. 



P_{o} = reference pressure. 

For a twostate 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(−ΔG_{un}/RT)/Q,
where Q =1+ exp(−ΔG_{un}/RT) and the function for ΔG_{un} is taken from above the average fluorescence signal, F_{calc} = ΣX_{i} (F_{i} + xδF_{i} /δx ), where x is a generalized perturbant. 
ΔG°_{o
,un}, the free energy change for unfolding in the absence
of denaturant, and m, the denaturant susceptibility
parameter (= −δΔG_{un}δ[d]), where [d] is the molar
concentration of added chemical denaturant.9,10 Through
an empirical relationship, the given equation appears to adequately
describe the pattern for denaturantinduced unfolding
of a number of proteins. The ΔG°_{o
,un} value is a
direct measure of the stability of a protein at the ambient
solvent conditions, which can be moderate temperature
and pH (e.g., 20°C and pH 7). The m value also provides
structural insights, as m values have been suggested to
correlate with the change in solvent accessible apolar surface
area upon unfolding of a protein.^{11} For example, a
relatively large m value (i.e., a high susceptibility of the
unfolding reaction to denaturant concentration) indicates
that there is a large change in the exposure of apolar side
chains on unfolding, which might be the case for a protein
that has an extensive core of apolar side chains that are
exposed upon denaturation.
 Acidinduced unfolding: The relationship for acidinduced
unfolding assumes that there are n equivalent acid
dissociating groups on a protein that all have the same pK_{a,U} in the unfolded state and that they are all perturbed
to have a pK_{a,N} in the N state. If the pK_{a,N} is more than
2 pH units lower than pK_{a,U} , then the equation simplifies
with the denominator of the right term going to unity. The
simplest relationship for acidinduced unfolding includes ΔG°_{o
,un}, the free energy of unfolding at neutral pH; n, the
number of perturbed acid dissociating residues; and their pK_{a,U} in the unfolded state. Presumably, n should be an
integer and pK_{a,U} should be approximately equal to the
values for such amino acids as glutamate, aspartate (e.g., pK_{a,U} should be about 4 to 4.3) or histidine (e.g., pK_{a,U} should be around 6.5).

Pressureinduced unfolding: In the relationship for
pressure, P, induced unfolding of proteins, ΔG°_{o
,un} is
again the value of the free energy change at 1 atmosphere
pressure and ΔV_{un} = V_{U} − V_{N} is the difference in volume of the unfoldedandnative states. Pressureinducedunfolding
studies require a specialized high pressure cell.^{12,13}
 Dissociation/unfolding of oligomeric proteins: Oligomeric
proteins are interesting as models for understanding
intermolecular proteinprotein interactions. A general
question for oligomeric proteins, including the simplest
dimeric (D) proteins, is whether the protein unfolds in a
twostate manner, D ↔ 2U, or whether there is an intermediate
state, which might be either an altered dimeric
state, D´, or a folded (or partially folded) monomer
species, M. Models for these two situations are as
follows:
⇒ Equation [11a] 
D ↔ D´ ↔ 2U 
⇒ Equation [11b] 
D ↔ 2M ↔ 2U 
For a D ↔ 2M ↔ 2U model, the relationships between the
observed spectroscopic signal, S_{exp}; the mole fraction of
dimer, X_{D} , and unfolded monomer, X_{U} ; and the unfolding
equilibrium constant (K_{un} = [U]^{2} /[D]) will be given by
Eq. (5) and
⇒ Equation [12] 
X_{U} = 
K_{un}^{2} + 8K_{un}[P]_{0}^{½} − K_{un} 
; X_{D} = 1 − X_{U} 
4[P]_{0} 
where [P]_{0} is the total protein concentration (expressed as
monomeric form), where S_{i} is the relative signal of species i and where K_{un} will depend on the perturbant as given by
one of the above equations. That is, the transition should
depend on the total subunit concentration, [P]_{0}, and on
any other perturbation axis.
