Basic Thermodynamic Relationships

Table I gives some widely accepted relationships for describing the variation of ΔG_un for a two-state 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.

1. Thermal unfolding: ΔH_un° un and ΔS_un un are the enthalpy and entropy changes for a two-state 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 Gibbs-Helmholtz 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 high-temperature and low-temperature induced unfolding.⁸
   
   
2. Denaturant-induced unfolding: The empirical relationship in Table I for chemical denaturation includes
   
   

   Temperature
   	⇒ Equation [7a] ΔGun(T )=ΔHun° − TΔSun°
   	⇒ Equation [7b] ΔGun(T )=ΔHo,un° +ΔCp(T −To) −T [ΔSo,un° +ΔCp ln(T/To)]
   where
   	ΔHo,un° is the enthalpy change at T =To.
   	ΔSun° is the entropy change at T =To.
   	ΔCp is the change in heat capacity upon unfolding.
   Chemical Denaturants
   	⇒ Equation [8] ΔGun([d])= ΔG°o ,un −m[d]h
   where
   	ΔG°o,un is the free energy change in the absence of d.
   	m =δΔGun/δ[d].
   pH
   	⇒ Equation [9] ΔGun(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.
   	Ka,U is the acid dissociation constant of a residue in the unfolded state.
   	Ka,N is the acid dissociation constant of a residue in the native state.
   Pressure
   	⇒ Equation [10] ΔGun(P)=ΔG°o ,un–ΔVun(Po–P)
   where
   	ΔVun = volume change for N ↔ U transition.
   	Po = 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 XN =1/Q, XU = exp(−ΔGun/RT)/Q, where Q =1+ exp(−ΔGun/RT) and the function for ΔGun is taken from above the average fluorescence signal, Fcalc = ΣXi (Fi + xδFi /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_unδ[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 denaturant-induced 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.
   
   
3. Acid-induced 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 acid-induced 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).
   
   
4. Pressure-induced 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. Pressure-inducedunfolding studies require a specialized high pressure cell.^12,13
   
   
5. Dissociation/unfolding of oligomeric proteins: Oligomeric proteins are interesting as models for understanding intermolecular protein-protein interactions. A general question for oligomeric proteins, including the simplest dimeric (D) proteins, is whether the protein unfolds in a two-state 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]	XU =	Kun2 + 8Kun[P]0½ − Kun	; XD = 1 − XU
   		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.