Thermal protein unfolding by differential scanning calorimetry and circular dichroism spectroscopy Two-state model versus sequential unfolding

Abstract Thermally-induced protein unfolding is commonly described with the two-state model. This model assumes only two types of protein molecules in solution, the native (N) and the denatured, unfolded (U) protein. In reality, protein unfolding is a multistep process, even if intermediate states are only sparsely populated. As an alternative approach we explore the Zimm–Bragg theory, originally developed for the α-helix-to-random coil transition of synthetic polypeptides. The theory includes intermediate structures with concentrations determined by the cooperativity of the unfolding reaction. We illustrate the differences between the two-state model and the Zimm–Bragg theory with measurements of apolipoprotein A-1 and lysozyme by differential scanning calorimetry (DSC) and CD spectroscopy. Nine further protein examples are taken from the literature. The Zimm–Bragg theory provides a perfect fit of the calorimetric unfolding transitions for all proteins investigated. In contrast, the transition curves and enthalpies predicted by the two-state model differ considerably from the experimental results. Apolipoprotein A-1 is ~50% α-helical at ambient temperature and its unfolding follows the classical α-helix-to-random coil equilibrium. The unfolding of proteins with little α-helix content, such as lysozyme, can also be analyzed with the Zimm–Bragg theory by introducing the concept of ‘folded’ and ‘unfolded’ peptide units assuming an average unfolding enthalpy per peptide unit. DSC is the method of choice to measure the unfolding enthalpy, $\Delta H_{\rm exp} ^0 $, but CD spectroscopy in combination with the two-state model is often used to deduce the unfolding enthalpy. This can lead to erroneous result. Not only are different enthalpies required to describe the CD and DSC transition curves but these values deviate distinctly from the experimental result. In contrast, the Zimm–Bragg theory predicts the DSC and CD unfolding transitions with the same set of parameters.


Introduction
Proteins can fold spontaneously into their native conformations. The folding/unfolding transition is a highly cooperative process characterised by the presence of no or very few thermodynamically stable intermediate states. 'Small globular proteins of molecular weight less than 20 kDa usually undergo folding/unfolding transitions in which the only two conformations that become significantly populated at any point during the transition are the folded and unfolded states. The entire protein essentially behaves as a single cooperative unit' (Privalov et al. 1989). However, even when intermediate states are not significantly populated, the folding/unfolding transition is a sequential process, as, for example, the α-helix-to-coil transition. The cooperative interaction of protein domains may then constitute a next higher level of cooperativity (Freire & Murphy, 1991).
Thermal unfolding of a protein requires heat and leads to a reorganization of the protein structure. Different physicalchemical methods are employed to quantify the unfolding transition. Differential scanning calorimetry (DSC) measures the heat consumed and constitutes the reference method for the thermodynamic analysis. Spectroscopic methods, in particular circular dichroism (CD) spectroscopy, provide structural data.
As an example, Fig. 1 shows DSC and CD spectroscopy unfolding transitions of apolipoprotein A-1 (Apo A-1). Identical protein solutions were used in the two experiments. The heat uptake is reflected in an increase of the molar heat capacity, C p . The area under the C p (T) versus temperature curve yields the total unfolding enthalpy, ΔH 0 exp , which comprises the enthalpy of the conformational change and the enthalpy change brought about by the increased heat capacity of the unfolded protein.
The mean residue ellipticity at 222 nm, ε 222nm , is shown in parallel and is a quantitative measure of the α-helix content of Apo A-1. The structural change is tightly linked to the thermodynamic equilibrium, but thermodynamic information can be derived only indirectly by applying an unfolding model. In contrast, DSC is a model-free thermodynamic method.
The two-state model is commonly applied in spectroscopic studies, be it for heat denaturation or for isothermal denaturation with chemical reagents. The model recognizes just two conformations, the native N and the unfolded U conformation (for reviews see references (Bolen & Yang, 2000;Konermann, 2012;Zhou et al. 1999). The two-state model simplifies, however, the reaction pathway as the unfolding of a protein is truly a sequential process illustrated, for example, by the zipper-like opening of an α-helical domain. A sequential cooperative model such as described by the Zimm-Bragg theory can be expected to provide a more realistic physical insight (Zimm & Bragg, 1959). The present review addresses two questions. (i) We compare DSC and CD spectroscopy with respect to the thermodynamic information derived. (ii) We compare two-state unfolding with sequential unfolding. Which model provides a consistent description of both the calorimetric and spectroscopic experiments? Can the Zimm-Bragg theory, originally developed for the α-helix-to-coil transition, be extended to globular proteins with little α-helix content?
In the following, the basic features of the two-state model and the Zimm-Bragg theory are introduced and applied to the analysis of CD spectroscopy and DSC unfolding experiments. Human Apo A-1, ∼50% α-helical protein, and egg white lysozyme, a globular protein with little α-helical content, are chosen to represent two proteins of opposite character. CD-measured unfolding transitions of both proteins are quantified by spectral simulation and analysed with the two models. Identical protein solutions are also measured with DSC and evaluated accordingly. A quantitative thermodynamic comparison of (i) CD spectroscopy versus DSC and (ii) two-state model versus Zimm-Bragg theory is thus possible.
In addition, literature data are analysed with both the two-state model and the Zimm-Bragg theory. A 50-amino acid peptide and 6 proteins with rather well-documented DSC and CD transition curves were selected. In addition, several high-quality DSC scans of proteins for which no spectroscopic data are available, were also analysed with both models.

Two-state model versus sequential protein unfolding
The predominant measuring parameter in protein unfolding experiments is the fraction of unfolded protein, Θ U (T). It is commonly deduced from changes in the fluorescence intensity or ellipticity and is tightly linked to the thermodynamics of the unfolding reaction. In DSC the physical readout is the heat capacity C p,NU (T) yielding simultaneously the enthalpy of unfolding ΔH cal (T). The latter is given by the area underneath the C p,NU (T) versus T curve and can be evaluated without the need for a specific unfolding model. In contrast, spectroscopic experiments provide thermodynamic information only by applying an appropriate folding model. Thermodynamic results derived from spectroscopic data are thus indirect and model-dependent.

Temperature course of heat capacity and enthalpy
Protein unfolding is not a true physical phase transition where the total heat is absorbed at a fixed temperature. Instead, the heat of unfolding, ΔH cal (T), is consumed over an extended temperature range ΔT = T end − T ini and its temperaturedependence follows the change in heat capacity, C p,NU (T) ΔH cal (T) comprises contributions from (i) the conformational enthalpy ΔH 0 NU , associated with the conformational change proper, and (ii) the increased heat capacity of the unfolded protein given by The heat capacity of the unfolded protein, C 0 p,U , is larger than that of the native protein, C 0 p,N , because additional water molecules bind to the open structure (Myers et al. 1995;Privalov & Dragan, 2007). ΔC 0 p,NU is detected only with DSC but not with CD spectroscopy (unless the unfolding transition at low temperature ('cold denaturation') can also be measured (Nicholson & Scholtz, 1996)). The increase in heat capacity makes an important contribution to the total unfolding enthalpy (Privalov & Dragan, 2007). Consequently, the conformational enthalpy ΔH 0 NU alone is insufficient to describe the thermodynamics of the Fig. 1. Differential scanning calorimetry and CD spectroscopytwo methods to study protein unfolding. Thermal unfolding of apolipoprotein A-1 (100 µM in PBS buffer).
3 unfolding process and must be complemented by a temperature-dependent heat capacity term For reasons given below the midpoint of the conformational transition is chosen as reference temperature T 0 .
The theoretical expression for the experimentally accessible enthalpy ΔH exp (T) is obtained by multiplying ΔH NU (T) (Eq. (3)) with the fraction of unfolded protein, Θ U (T).
In a perfect simulation ΔH(T) is identical with the experimental H exp (T).
The predicted heat capacity is the derivative of H(T) with respect to temperature The first term on the right side of Eq. (5) is the specific contribution of the conformational change, the second term that of the increased heat capacity. The enthalpy change caused by ΔC 0 p,NU alone is given by Θ U (T) must be calculated with either the two-state model or the Zimm-Bragg theory (see Sections 2.2 and 2.3).
Knowledge of ΔH NU (T) leads to further thermodynamic parameters. The protein unfolding equilibrium is determined by the change in free energy, ΔG NU (T).
Taking into account the change in heat capacity, ΔC 0 p,NU , the unfolding entropy ΔS NU (T) is given by ΔS 0 NU is not known independently. However, at the midpoint T 0 of the phase transition, equal fractions of native and unfolded protein exist in the solution, the equilibrium constant is K NU (T 0 ) = Θ U (T 0 )/(1 − Θ U (T 0 )) = 1, and the free energy is ΔG NU (T 0 ) = 0, yielding The Gibbs free energy is thus simplified to with ΔH 0 NU and ΔC 0 p,NU as the only free parameters. In spectroscopic experiments ΔC 0 p,NU cannot be measured. A truncated formula is used as ΔC 0 p,NU = 0

Two-state model
The quantitative analysis of protein unfolding is almost exclusively based on the two-state model (Bolen & Yang, 2000;Konermann, 2012;Zhou et al. 1999). It assumes that a protein in solution adopts only two conformational states, the native (N) and the unfolded (U) state. The equilibrium N U is described with a temperature-dependent equilibrium constant K NU (T).
[N] and [U] are the equilibrium concentrations of the native and the unfolded protein, respectively, and Θ U (T) and Θ N (T) are 4 the fractions of unfolded and native protein, respectively.
With this definition a stable native protein is characterised by a small equilibrium constant. K NU (T) is related to the Gibbs free energy according to The temperature dependence of K NU (T) is thus where ΔG NU (T) is given by Eqs. (10) or (11). Eq. (15) allows the calculation of the fraction of unfolded protein, Θ U (T), over the whole temperature range.
An approximation of Eq. (15) is the van't Hoff equation, which is commonly used in the evaluation of spectroscopic experiments. ln It is based on Eq. (11) and assumes that the unfolding enthalpy is independent of temperature. (ΔC 0 p,NU = 0). A plot of ln K(T) versus the reciprocal absolute temperature, 1/T, yields a straight line with slope −ΔH 0 NU /R. The total free energy change associated with the unfolding reaction is a physical property of general interest. If K ini at temperature T ini and K end at temperature T end denote beginning and end of the conformational change, then the free energy change is given by 2.3 Zimm-Bragg theory. Sequential protein unfolding The advantage of the two-state model is its simplicity. On the other hand, it ignores or even contradicts the molecular mechanism of unfolding. A physically more realistic model for a sequential process is provided by the Zimm-Bragg theory (Zimm & Bragg, 1959), which successfully describes the α-helix-to-coil transition of synthetic polypeptides, but is not commonly applied to protein unfolding. A review on recent advances in helix-coil theory is available (Doig, 2002).
The two-state model requires the total unfolding enthalpy, ΔH NU (T), as the input parameter. In contrast, the sequential Zimm-Bragg theory is based on the much smaller enthalpy 'h' of the elementary step. The change of a single peptide unit from α-helix ('folded') to coil ('unfolded') conformation is the basic two-state equilibrium. It nevertheless involves several peptide units and leads to the cooperative reorganization of all torsion angles between peptide units i and i + 3.
Typically, a protein solution will contain a mixture of molecules, all with different helix contents, ranging from all helix to all coil. 'In order to interpret experiments on helical peptides and make theoretical predictions on helices, it is therefore essential to use a helix-coil theory that considers every possible location of the helix within a sequence' (Doig, 2002). In the following, we will use the Zimm-Bragg formalism in its simplest form (Davidson, 1962).
In this model the fractions of helix and random coil amino acid residues are determined by three parameters: the nucleation parameter σ, the enthalpy of helix formation h, and the chain length N.
The nucleation parameter σ defines the cooperativity of the folding/unfolding equilibrium. The smaller σ, the steeper is the cooperative conformational transition. σ is assumed to be temperature-independent and typical values are σ ∼ 10 −3 -10 −6 . The nucleation parameter σ is strictly defined for homopolymers. If the peptide chain contains different amino acids, σ is an average value over the different amino acids involved.
The enthalpy h determines the growth parameter, s(T) 5 s(T) is the equilibrium constant for the addition of an α-helical peptide unit to an existing α-helix. The equilibrium constant for the formation of a helical peptide unit within a stretch of random coil peptide units is smaller by the factor σ and is σs(T).
The characteristic temperature, T ∞ , is identical with the midpoint of the unfolding transition for a sufficiently long peptide chain with N ≫ σ −1/2 .
The mathematical formalism of the Zimm-Bragg theory can be summarised as follows. A polypeptide chain of N peptide units adopts a maximum of 2 N conformations as each segment can be either coil (c) or helix (h). If a polypeptide chain of length i ends on c or h, the addition of a further segment at position i + 1 leads to the combinations cc, hc, ch and hh. The conditional probabilities of occurrence are combined in the matrix M (Davidson, 1962) from which the helix fraction can be calculated In combination with Eqs. (4) and (5), Eq. (21) predicts the thermodynamics of sequential unfolding.
The Gibbs free energy of the unfolding transition, ΔG 0 NU , is determined by the growth parameter s(T). With T ini and T end denoting the beginning and the end of the unfolding transition, respectively, the total free energy change of N peptide units is given by The Zimm-Bragg theory allows the calculation of probabilities of specific conformations. Of particular interest is n σ , the number of nuclei within the linear sequence. It can be calculated according to

Energetics of 'folded' peptide units in globular proteins
α-Helices and β-sheets are the dominant structural elements in proteins. It is usually assumed that the formation of these structures is driven by the formation of peptide hydrogen bonds (Privalov & Makhatadze, 1993). This predicts an only marginal stability of helices in water because hydrogen bonds between peptide units and water appear to be more favorable. Unexpectedly and in contrast to this classical view, short alanine-based peptides showed stable α-helix formation in H 2 O (Marqusee et al. 1989). It was concluded that individual alanine residues had a high helix-forming potential and that hydrophobic interactions played an important role in stabilising isolated α-helices (Marqusee et al. 1989). Helix formation was induced by the gain in free energy upon burial of hydrophobic groups from water in adopting the helical conformation. It was concluded that 'hydrophobic interaction may be an important determinant of α-helix stability' (Marqusee et al. 1989). Earlier literature supporting this finding is listed in the same reference.
Free energy calculations using the CHARMM potential function and accounting for solvation effects with various continuum solvation models also argue against a dominant energetic role of hydrogen bonds for α-helix and β-sheet stability (Yang & Honig, 1995a, b). Hydrogen bond formation was found to contribute little to α-helix stability because the internal hydrogen bonding energy is largely canceled by the large free energy cost associated with removing polar groups from water. The major driving force favoring helix formation was associated with enhanced van der Waals interactions in the close-packed helix conformation and the hydrophobic effect (Yang & Honig, 1995b).
involving only backbone-backbone interactions (including β-carbons) is not sufficient to overcome the loss in configurational entropy that accompanies sheet formation. Rather we suggest that cross-strand non-polar side chainside chain interactions are essential for sheet formation, explaining why large non-polar amino acids have the greatest sheet forming propensities' (Yang & Honig, 1995a).
It follows from these experimental and theoretical studies that (i) specific hydrogen bonds are not the dominant energetic factors in secondary structures of proteins, and (ii) the energetics is determined by van-der-Waals and hydrophobic interactions in the folded protein. A decrease in non-polar accessible surface area favors helix formation.
These theoretical and experimental results suggest a conceptual extension of the Zimm-Bragg theory for globular proteins. The sequential unfolding is considered to be a change from 'folded' to 'unfolded' peptide units. We postulate an average energetic difference (enthalpy h; free energy, g) between 'folded' and 'unfolded' peptide units, identical for all residues. Moreover, the free energy of a peptide unit is not connected to the formation of specific hydrogen bond but requires simply the well-defined burial of a 'folded' residue in the native protein structure.
At the present stage it is helpful to summarise the energetic parameters obtained for α-helix and β-sheet formation as they serve as energetic markers for the generalised concept. A large number of calorimetric studies has shown that the enthalpy of α-helix formation is h helix ∼ −1·1 kcal mol −1 (Chou & Scheraga, 1971;Rialdi & Hermans, 1966;Scholtz et al. 1991a, b). The h-parameter is smaller if α-helix formation occurs in a hydrophobic environment. A value of h helix = −0·7 kcal mol −1 was reported for α-helix formation in tri-fluoroethanol-water mixtures (Luo & Baldwin, 1997).
Less information is available on the thermodynamic and kinetic parameters of the random coil ⇆ β-structure transition. The contribution of the β-sheet formation to the overall folding process was determined with analogues of the KIGAKI repeat where two adjacent amino acids were replaced by their D-enantiomers (Meier & Seelig, 2007). The thermodynamic parameters were h β = −0·23 kcal mol −1 per residue and a free energy change of g β = −0·15 kcal mol −1 residue (Meier & Seelig, 2007).

CD experiments with human Apo A-1. Unfolding of an α-helical protein
Human Apo A-1 is a 28·2 kDa protein (243 aa) involved in the reverse transport and metabolism of cholesterol. It has been widely investigated for its role in reducing cardiovascular risks. CD spectroscopy reports an α-helix content of ∼50% for Apo A-1 in solution (at room temperature) (Arnulphi et al. 2004;Gursky & Atkinson, 1996;Saito et al. 2003bSaito et al. , 2004Schulthess et al. 2015;Suurkuusk & Hallen, 1999;Zehender et al. 2012). The 2·2-Å crystal structure of a truncated Δ(185-243)Apo A-1 reveals long stretches of α-helix (∼80% helix) (Mei & Atkinson, 2011). When heated, Apo A-1 displays an unfolding transition centered at about 55°C, which is fully reversible both in CD and DSC experiments. The thermal unfolding of Apo A-1 is a true chemical equilibrium if the molecule is not heated above 90°C. Figure 2a shows CD spectra of recombinant Apo A-1 (245 aa, 2 additional N-terminal glycine residues) as a function of temperature. The two bold lines define the spectra recorded at the lowest (5°C, black line) and highest temperature (90°C, olive line), i.e. the spectra with the highest and lowest α-helix content, respectively. An apparent isodichroic point is observed at 203 nm. Increasing the resolution around 203 nm reveals however small deviations from a single cross-over point.
Different methods can be used to derive quantitative information on the progress of the protein unfolding reaction. The simplest protocol is to evaluate the mean residue ellipticity at λ = 222 nm since the unfolded protein is assumed to have no absorbance at this wavelength. The α-helix fraction, f α , is determined according to (Morriset et al. 1973) The corresponding analysis of the spectra of Fig. 2a is displayed in Fig. 2b (□). Equation (24) tends to overestimate the α-helix content, in particular if the α-helix content is low.
A more precise method is to deconvolute the CD spectra in terms of their α-helix-, β-sheet, β-turn, and random coil content. A CD fitting procedure based on 44 reference spectra (Reed & Reed, 1997) was applied to the spectra of Fig. 2a and the results are also shown in Fig. 2b. The α-helix content decreases from 50% to 4%, the random coil structure increases from 28% to 49%, and the β-structure (β-sheet + β-turn) increases from 22% to 47% (in the interval 5-90°C). The isodichroic point seen in Fig. 2 is not exactly confirmed in this structural analysis, which can be traced back to the 25% increase in β-structure with temperature.
If Apo A-1 unfolding is interpreted as a two-state process, it should be possible to quantitate the equilibrium by a linear combination of two limiting spectra. We assume that these are the 5 and 90°C spectra, representing the folded and the unfolded Apo A-1, respectively. We define the fraction of native Apo A-1 as Θ N = 1·0 at 5°C and Θ N = 0 at 90°C. Spectra at intermediate temperatures are then simulated by the weighted superposition of the two limiting spectra. An excellent fit of all spectra is indeed possible by this analysis. Figure 3a shows the variation of Θ N with temperature (red dots).
Alternatively, the α-helix fraction, f α (Fig. 2b), can also be interpreted in terms of a two-state equilibrium. We define the maximum α-helix content at 5°C as f N and the minimum α-helix content at 90°C as f U . The fraction of native protein is then and is also shown in Fig. 3a (black squares). A perfect agreement of the two methods is obtained even though f α (α-helix fraction) considers only a single structural element, whereas the linear combination of spectra includes all structural elements.
The observation of an isodichroic point is usually taken as evidence for equilibrium between just two protein confirmations, the native and the denatured, unfolded protein. However, an isodichroic point can also be generated by an intramolecular two-state equilibrium. Each peptide unit within a protein can exist in a native 'folded' and a denatured 'unfolded' conformation. The two conformations have different CD spectra. An increase in temperature shifts the intramolecular equilibrium of peptide units from folded to unfolded. This intramolecular equilibrium generates an isodichroic point. The protein solution may contain protein molecules with many different conformations, ranging from all-helical to all-unfolded but the constituting peptide units fluctuate between just two conformations.

Two-state model applied to Apo A-1 unfolding
The binding constant K NU (T) = (1 − Θ N )/Θ N (Eq. (12)) can be calculated from the data of Fig. 3a for a limited temperature range of 35-80°C. The equilibrium constant increases with temperature from K NU = 0·03 at 35°C to 16 at 80°C. Figure 3b shows the corresponding van't Hoff plot of lnK versus 1/T. The slope of the linear regression analysis yields an unfolding enthalpy of ΔH 0 NU = 30.5 kcal mol −1 , in agreement with earlier studies (Gursky & Atkinson, 1996;Saito et al. 2003a;Tanaka et al. 2008).
The foregoing analysis is limited to the central region of the transition curve. The whole temperature range can be simulated with Eqs. (11) and (15) as shown in Fig. 4. The solid black line is the two-state model calculated with ΔH 0 NU = 30 kcal mol −1 . The predicted transition curve fits the experimental data quite well in the transition region but deviates considerably in the region of high α-helix content.
As an aside it may be noted that a ΔC 0 p,NU > 0 term in Eq. (10) has no influence on the shape of the CD unfolding transition curve. This is explained by the fact that the contribution of the conformational enthalpy ΔH 0 NU is much larger than that of ΔC 0 p,NU . A ΔC 0 p,NU > 0 leads to a second unfolding transition at low temperatures ('cold denaturation') (Nicholson & Scholtz, 1996).

Zimm-Bragg theory applied to Apo A-1 unfolding
The recombinant human Apo A-1 has a chain length of 245 amino acids. Its α-helix content varies between f α ∼53 ± 5 % at 10°C and ∼5 ± 2% at 90°C. The number of peptide units involved in the α-helix-to-random coil transition is N ∼ 115 ± 10. Figure 4 shows the simulation of the CD unfolding curve with the Zimm-Bragg theory. The enthalpy for the formation of a helical peptide unit was set at h = −1·1 kcal mol −1 . The other parameters were σ = 6 × 10 −4 , T ∞ = 335 K, and N = 120. The Zimm-Bragg theory leads to a better fit to the spectroscopic data over the whole temperature range than the two-state model. The Zimm-Bragg theory provides thermodynamic insight via the h-parameter. As N ∼ 120 α-helical residues are unfolded, the expected enthalpy change is ΔH 0 NU = N × (−h) ∼120 × 1·1 kcal mol −1 = 132 kcal mole −1 . This is much larger than the 30 kcal mol −1 deduced with the two-state model but very close to the calorimetric enthalpy of ΔH 0 exp = 138 kcal mol −1 obtained with DSC (see Section 4.1).
The CD spectra of a 10 µM lysozyme solution are shown as a function of temperature in Fig. 5a. An isodichroic point is observed at 203 nm. The structural analysis is given in Fig. 5b. The α-helix content of the native protein is only 22% at 20°C and decreases to 8% at 87°C. As lysozyme is composed of 129 amino acids, at most ∼20-30 peptide units are involved in the α-helix-to-random coil transition. The extent of β-structure is about 40% and decreases only little above the unfolding temperature.
The quantitative analysis of the CD spectra is based on the α-helix content (•) and the linear combination of the 20 and 85°C spectra (■) (Fig. 6). The fraction of native protein, Θ N , is plotted as a function of temperature. The midpoint of the unfolding  transition is at T 0 = 335 K = 62°C, in agreement with earlier reports (Privalov et al. 1995). The unfolding of lysozyme occurs between 40 and 70°C and the transition is clearly sharper than that of Apo A-1.
Application of the two-state model (Fig. 6, solid blue line) yields an unfolding enthalpy of ΔH 0 NU = 90·5 kcal mol −1 . In order to apply the Zimm-Bragg theory, we consider the unfolding of lysozyme as a change from 'folded' peptide units to 'unfolded' peptide units involving all 129 peptide units. The caloric heat of unfolding as measured with DSC and discussed below is ΔH 0 exp = 138 kcal mol −1 , yielding an enthalpy change of 138/129 = 1·07 kcal mol −1 per peptide unit. This is close to the enthalpy for the opening of a hydrogen bond. However, as discussed above, energy calculations show that the major energy of α-helix and β-structure formation comes from van-der-Waals interactions and the hydrophobic effect (Yang & Honig, 1995a, b). We therefore postulate an enthalpy difference between 'unfolded' and 'folded' peptide units of h = −1·1 kcal mol −1 , independent of the specific nature of the protein. The total enthalpy of unfolding is determined by the number of involved peptide units. The quantitative analysis of lysozyme is then based on the full length protein (red line in Fig. 6) with σ = 1·1 × 10 −6 and h = −1·1 kcal mol −1 . These simulation parameters are identical to those derived from DSC measurements (discussed in detail in Fig. 8). The Zimm-Bragg theory provides an excellent fit of the CD unfolding curve of lysozyme.

Sloping baselines in CD spectroscopy
Inspection of Figs 1-5 reveals sloping CD baselines at the beginning and the end of the unfolding transition. This effect is quite pronounced for Apo A-1 with its rather broad transition but occurs also to a lesser extent for the highly cooperative lysozyme. Special algorithms have been proposed to fit 'sloping baselines' (Gursky, 2015;Santoro & Bolen, 1988). However, sloping baselines could also be the result of the cooperativity of the system. It may be noted that sloping baselines are better accounted for by the Zimm-Bragg theory than by the two-state model (see Supplementary Information S1, S5, S6, S9, S12). The definition of the beginning and end of the conformational transition has a distinct influence on the van't Hoff enthalpy determined with the two-state model. The narrower the selected temperature range, the steeper is the slope of the transition curve and the larger the van't Hoff enthalpy.

Calorimetry of Apo A-1
The thermal unfolding of lipid-free recombinant Apo A-1 was measured between 10 and 90°C. Figure 7 (black line) shows the DSC scan of a 100 µM Apo A-1 solution after subtracting the buffer baseline. The molar heat capacity reaches a maximum at the midpoint of the unfolding transition (T 0 = 52·5°C). The unfolded protein has a larger heat capacity than the native protein with ΔC 0 p,NU = 2·508 kcal molK −1 . The unfolding process is completely reversible in the temperature interval of 10-90°C. Three consecutive DSC scans gave identical results (Schulthess et al. 2015;Zehender et al. 2012).
The conformational change occurs between 30 and 70°C. The area under the C P (T) versus T curve in this interval yields an unfolding enthalpy of ΔH 0 exp = 138·4 kcal mole −1 . It includes the conformational change proper and the contribution of the increased heat capacity. As seen by CD spectroscopy ∼120 amino acids participate in the transition and the contribution of each peptide unit is 1·13 kcal mol −1 .
The change in the molar heat capacity, ΔC 0 p,NU , is usually not detected in spectroscopic measurements. It is however essential for the analysis of the thermodynamic equilibrium. The total heat of unfolding, not only the conformational enthalpy, must be considered for the molecular interpretation of the folding unfolding equilibrium. 'It is clear that in considering the energetic characteristics of protein unfolding one has to take into account all energy which is accumulated upon heating and not only the very substantial heat effect associated with gross conformational transitions, that is, all the excess heat effects must be integrated' (Privalov & Dragan, 2007).
We analyzed the DSC unfolding experiment with the two models introduced above, taking into account the contribution of ΔC 0 p,NU in both cases. The simulation with the two-state model is presented in Fig. 7a. The fit parameters are T 0 = 52·4°C, ΔH 0 NU = 66·9 kcal mol −1 and ΔC 0 p,NU = 2·508 kcal molK −1 and are in agreement with previous DSC measurements (Schulthess et al. 2015;Tall et al. 1975Tall et al. , 1976. The contribution of the ΔC 0 p,NU term to the unfolding process is given by the area under the dashed magenta line in Fig. 7a and is ΔH 0 Cp,NU = 45·2 kcal mol −1 . This yields a total unfolding enthalpy of 112·0 kcal mol −1 , which is 20% smaller than the experimental result ΔH 0 exp = 138·4 kcal mole −1 . The conformational enthalpy of ΔH 0 NU,DSC = 66·9 kcal mol −1 deduced with the two-state model is more than twice as large as that deduced with the same model from CD spectroscopic (ΔH 0 NU,CD = 30 kcal mol −1 ). A much sharper transition is recorded in DSC than in CD spectroscopy.

A 50-amino acid peptide
Remarkable differences between the two-state model and the Zimm-Bragg theory are found for a short synthetic 50-amino acid peptide with sequence Ac-Y (AEAAKA) 8 F-NH 2 . The peptide is almost completely α-helical at 0°C and shows a very broad unfolding transition centered at 42°C (Scholtz, 1991;Scholtz et al. 1991a, b). Figure 9a displays the temperaturedependence of the ellipticity at 222 nm (Scholtz et al. 1991a). The α-helix fraction, calculated according to Eq. (24), is f α = 0·98 at 0°C and 0·23 at 80°C. No plateau is reached at 80°C and the shape of the transition curve suggests f α = 0 at high temperatures. We therefore assume that all 50 amino acid residues participate in the α-helix-to-random coil transition. The ellipticity is linearly proportional to the helix fraction according to ε 222nm (degM −1 cm −1 ) = −39000 Θ helix . The helix fraction can then be calculated with either the two-state model (blue line in Fig. 9a) or the Zimm-Bragg theory (red line in Fig. 9a). An almost perfect fit of the CD transition curve is obtained with both models. The two-state model requires an unfolding enthalpy of ΔH 0 NU = 12 kcal mol −1 . The Zimm-Bragg theory uses N = 50, σ = 4 × 10 −3 , h = −0·93 kcal mol −1 , and T ∞ = 325 K (ΔC 0 p,NU = 0). Figure 9b displays the calorimetric data ( Fig. 4 in (Scholtz, 1991;Scholtz et al. 1991a)). The large width of the transition requires assumptions about the baseline and restricts the accuracy of the calorimetric analysis. The calorimetric heat deduced from the experimental data is ΔH cal = 45·8 kcal mol −1 ( Table 2 in (Scholtz et al. 1991a)). Fig. 9b is the interpretation of the experimental data with the Zimm-Bragg theory. Exactly the same parameters are used for the DSC transition as listed above for the CD transition curve. The calculated enthalpy of unfolding is ΔH 0 NU = 46·1 kcal mol −1 , in excellent agreement with the calorimetric result. The solid blue line in Fig. 9b shows the fit with the two-state model with ΔH 0 NU = 27 kcal mol −1 . The maximum C P value is reached, but the fit of the whole transition curve is poor. It is obvious that the two-state model fails to reproduce the DSC data.

The solid red line in
Scholtz et al. compared the CD spectra of the 50-amino acid peptide with those of shorter fragments (Scholtz et al. 1991b). They used σ = 3·3 × 10 −3 , h = −0·955 kcal mol −1 and T ∞ = 329 K, consistent with the present analysis.

Proteins selected from the literature
The comparison between the two-state model and the Zimm-Bragg theory was extended to protein unfolding experiments reported in the literature. Most useful were studies where both CD and DSC data were available and where the increased heat capacity of the unfolded protein was also measured. Examples are pseudo wild-type T4 lysozyme and its mutant S44[A] T4-lysozyme (Carra et al. 1996), the aspartate receptor C-fragment and its mutant S461L (Seeley et al. 1996;Wu et al. 1995) andβ-lactoglobulin (Garcia-Hernandez et al. 1998).
In all publications the authors used the two-state model for the interpretation of the experimental results. As the data were only available as printed figures, the corresponding traces were enlarged and digitized manually (see Supplementary Information Figs S1-S17) Table 1 summarises the experimental parameters deduced from these figures. Some protein    Carra et al. (1996) Sup. Info S17 (lysozyme, ubiquitin, and β-lactoglobulin) were measured by different groups. Table 1 illustrates the variability of the different measurements. Table 1 lists the width of the transition, ΔT, the calorimetric and spectroscopic midpoint temperatures, T 0,DSC and T 0,CD, and the total heat of unfolding, ΔH 0 exp . ΔH 0 exp was determined by numerical integration of the digitized DSC curve in the transition interval. Table 1 further includes the change in the molar heat capacity, ΔC 0 p,NU . The proteins are ordered according to decreasing width of the unfolding transition, indicating increasing cooperativity.
The extracted DSC and CD spectroscopy unfolding transitions were analysed with the two models. The results are summarized in Table 2 for the two-state model and in Table 3 for the Zimm-Bragg theory.
In Table 2 the enthalpy ΔH 0 NU,DSC is compared with available literature data, ΔH DSC lit , also obtained with the two-state model. Table 2 also contains ΔH 0 Cp,NU , the contribution of the ΔC 0 p,NU term to the unfolding transition. The total unfolding enthalpy is ΔH 0 NU,DSC + ΔH 0 Cp,NU and must be compared with ΔH 0 exp . The unfolding transitions measured with CD spectroscopy were also analysed and the corresponding conformational enthalpies are listed as ΔH 0 NU,CD . Table 3 contains the parameters of the Zimm-Bragg theory such as the number of peptide units N ZB , the enthalpy per peptide unit h, the nucleation parameters σ DSC and σ CD , and the characteristic temperature T ∞ . The predicted total enthalpy of unfolding is ΔH 0 calc,ZB and the corresponding free energy ΔG 0 NU,ZB .

Cooperative unfolding and two-state model applied to DSC and CD spectroscopy
DSC is the method of choice for the thermodynamic analysis of protein unfolding. The integration of the C p versus T calorimetric transition curve yields the total enthalpy of the unfolding reaction, ΔH 0 exp . It comprises the so-called conformational enthalpy, ΔH 0 NU , and the enthalpy contribution ΔH 0 Cp,NU , caused by the increase in heat capacity. The unfolding 'process results in a significant increase in heat capacity, by a value that does not depend noticeably either on temperature or on environmental conditions and is specific for the given protein' (Privalov & Makhatadze, 1993). No unfolding model is required to evaluate ΔH 0 exp .

5.1
The total heat of unfolding ΔH exp 0 A first criterion for the quality of the two-state model and the Zimm-Bragg theory is their capability to reproduce the shape and the total enthalpy of the DSC transition curve. In Fig. 10 the enthalpies calculated with the Zimm-Bragg theory and the two-state model are plotted against the experimental result, ΔH 0 exp . Linear regression analysis of the Zimm-Bragg data yields a straight line through the origin with slope m = 1 (average deviation 2·1 ± 1·7%). The two-state model shows a larger scatter and regression analysis that yields a slope of m = 0·89 only. The two-state model systematically underestimates the total heat of unfolding, ΔH 0 exp , by about 11%. The results of the two-state model are particularly poor for broad transitions. For the 50-amino acid peptide a fit with a single two-state model was not possible (see Fig. 9).

Equivalence of DSC and CD spectroscopy unfolding transitions?
A second quality criterion follows from a comparison of DSC and CD spectroscopy unfolding transitions obtained for a given protein under identical experimental conditions. If DSC and CD spectroscopy indeed report the same physical process, the simulation of the two unfolding transitions should be possible with identical thermodynamic parameters. The critical parameter for this comparison is the cooperativity, measured by the nucleation parameter σ of the Zimm-Bragg theory and the conformational enthalpy ΔH 0 NU of the two-state model. Figure 11a shows a plot of the nucleation parameter σ CD , deduced from CD measurements, versus σ DSC , obtained from DSC. The 50-amino acid peptide and the aspartate receptor fragment have broad transitions with nucleation parameters of σ = 4 × 10 −3 and σ = 1 × 10 −3 , respectively. At the other extreme, pseudo WT T4 lysozyme has the sharpest transition with σ = 5 × 10 −7 . For 6 of the 7 proteins identical nucleation parameters σ DSC and σ CD are obtained. Linear regression analysis yields a straight line through the origin with slope m = 1. It can be concluded that (i) DSC and CD spectroscopy report the same transition, and (ii) the Zimm-Bragg theory describes both transitions with the same set of thermodynamic parameters. The only exception in Fig. 11a is Apo A-1, which shows a distinctly broader transition in CD spectroscopy than in DSC.
The two-state model uses the conformational enthalpy, ΔH 0 NU (van't Hoff enthalpy) as fit parameter. The larger ΔH 0 NU , the sharper is the predicted transition curve. In Fig. 11b the CD parameter, ΔH 0 NU,CD , is plotted against the DSC parameter, ΔH 0 NU,DSC . The scatter of the data is considerable and the average deviation is 24 ± 18%. The straight line through the origin has a slope of m = 0·81, indicating that the enthalpy deduced with CD spectroscopy systematically underestimate the  (Gursky, 2015;Santoro & Bolen, 1988).
The two-state model provides the best agreement between DSC and CD spectroscopy for highly cooperative transitions. Pseudo wild-type T4 lysozyme with its very sharp transition has very similar fit parameters for DSC and CD with ΔH 0 NU,DSC = 125·4 kcal mol −1 and ΔH 0 NU,CD = 120·6 kcal mol −1 , respectively. The two-state model leads to the following conclusions. (i) The conformational enthalpy deduced from CD data, ΔH 0 NU,CD , is typically 20% smaller than that determined by DSC, ΔH 0 NU,DSC . The difference is large for a broad transition and narrows to a few percent for a very cooperative transition. (ii) CD spectroscopy is limited to the so-called conformational enthalpy and cannot detect the change in protein heat capacity, ΔC p , unless cold-denaturation is measured in the same experiment (Nicholson & Scholtz, 1996;Privalov et al. 1986). The heat capacity change upon protein unfolding is an important thermodynamic parameter of the unfolding reaction. It accounts for about 20-50% of the total enthalpy of the unfolding reaction (see Tables 2 and 3).
The unfolding transition of lysozyme was measured with CD spectroscopy (Fig. 5) and DSC (Fig. 8). The fit parameters h, σ, N and T ∞ of the Zimm-Bragg theory were determined from the DSC experiment. Using exactly the same set of parameters a perfect fit of the CD unfolding curve was obtained. A similar result is shown in Fig. 9 for the 50-residue peptide (Scholtz et al. 1991a). Again the same set of parameters explains both the DSC and the CD transition. Good agreement between DSC and  CD using the Zimm-Bragg theory was obtained for pseudo wild-type T4 lysozyme (Carra et al. 1996), S[44]A T4 lysozyme, (Carra et al. 1996) and the aspartate receptor fragment (Wu et al. 1995) (cf. Supplementary Information).

Zimm-Bragg theory applied to globular proteins
The Zimm-Bragg theory was originally developed for the reversible α-helix-coil transition of synthetic peptides (Zimm & Bragg, 1959). The formalism is however applicable to any linear sequential process with two energetically different states. This suggests that the theory can also be applied to proteins with low α-helix content provided the unfolding is a sequential transition between 'folded' and 'unfolded' peptide units with enthalpy difference h. As discussed in Section 2.4 'hydrophobic and close-packed interactions provide protein stability, while hydrogen bond formation constitutes a structural constraint imposed by the high free energy cost associated with the burying of unsatisfied hydrogen bonding groups' (Yang & Honig, 1995b). We therefore assume 'folded' and 'unfolded' peptide units with an enthalpy parameter h = −1100 cal mol −1 (except for the 50 amino acid peptide with h = −930 cal mol −1 ).
6.1 Zimm-Bragg theory. An excellent thermodynamic approach to protein unfolding  Table 3 lists the proteins according to increasing cooperativity. The width of the transition, ΔT, is reflected in the nucleation parameter σ. The smaller ΔT, the smaller is the nucleation parameter σ and the sharper and more cooperative is the unfolding process.
The steepness of the transition is determined not only by the nucleation parameter, σ, but also by the number of peptide units, N, participating in the unfolding transition. Only for long chains with N ≫ 1/ σ √ is the steepness independent of N and determined exclusively by σ. In contrast, if N is small (as is the case for the 50-amino acid peptide) even very small values of σ cannot generate a steep transition. Proper knowledge of N ZB is hence important for chains, which are shorter than the cooperative length N coop = 1/ σ √ .
The proteins in Table 3 can be divided into three categories. A first group includes proteins with a high helix content and a rather broad transition so that N ≫ 1/ σ √ (50-amino acid peptide, Apo A-1, aspartate receptor fragment). The CD spectra of the 50-amino acid peptide (Scholtz et al. 1991a) and of recombinant human ApoA-1 (245aa) report an α-helix content of ∼98% and 53 ± 5%, respectively, at 20°C. Upon heating to 90°C the α-helix content drops to nearly 0% for both molecules. The number of peptide units involved in the unfolding transition is thus N ≈ 50 and N ≈ 120 ± 10, respectively, which is clearly larger than the cooperative chain length. Each helical peptide unit contributes ∼1·1 kcal mol −1 (50-amino acid peptide: 0·95 kcal mol −1 ) to the unfolding enthalpy, and the total enthalpies are estimated to be 47·5 kcal mol −1 for the 50-residue peptide and 132 kcal mol −1 for ApoA-1. The experimental enthalpies of 46 and 138·4 kcal mol −1 , respectively, are in excellent agreement with these estimates. The aspartate receptor fragment (297aa) is the third member of this group. It is 43% α-helical (∼114 α-helix peptide units) (Wu et al. 1995) and has an enthalpy of unfolding of 107·2 kcal mol −1 . It can be estimated that N = 107·2/1·1 = 97 amino acid residues participate in the unfolding transition, which is consistent with the change in ellipticity ε 222nm (see Supplementary Information S1).
A second group comprises pseudo-wild type T4 lysozyme (164aa) and its mutant S44[A] lysozyme (165aa) (Carra et al. 1996). Both are 61% α-helical according to their ellipticity at 222 nm, corresponding to ∼100 α-helical peptide units. The total unfolding enthalpies are 158·2 and 131·8 kcal mol −1 , respectively, and are clearly larger than the expected ∼110 kcal mol −1 . Additional amino acids residues appear to be involved in the unfolding reaction. Assuming an average unfolding enthalpy of 1·1 kcal mol −1 per peptide unit, 120 peptide units of pseudo WT T4 lysozyme and 140 of S44[A] lysozyme are estimated to participate in unfolding. This allows an excellent fit of the DSC cal mol −1 unfolding curves. A similar analysis can be made for myoglobin, which is about 60% α-helical in solution at 30°C (Privalov et al. 1986) (see Supplementary  Information).
The third group comprises proteins with a low α-helix content. Lysozyme (129aa) has at most 30 α-helical peptide units. The α-helix content of ubiquitin (79aa) (Ibarra-Molero et al. 1999a) and RNAse (129aa) (Kurapkat et al. 1997) can be estimated to be less than 30%. Nevertheless the unfolding enthalpies of these three enzymes are large. We again assume a sequential unfolding with an average enthalpy of 1·1 kcal mol −1 per peptide unit. Consequently, ∼120 peptide units are estimated to 20 become unfolded in lysozyme, ∼70 in ubiquitin, and ∼90 in RNAse. With these numbers the Zimm-Bragg theory provides a perfect fit of the DSC data of the three proteins.
6.2 The free energy change upon thermal and chemical denaturation Table 1 lists the width of the unfolding transition, ΔT, for the different proteins investigated. The change of the Gibbs free energy in this temperature interval is the free energy of unfolding ΔG 0 NU . It can be calculated with either the two-state model (Eq. (17)) or the Zimm-Bragg theory (Eq. (22)). Tables 2 and 3 summarise the corresponding ΔG 0 NU -values. The two-state model predicts numerical values between 5 and 10 kcal mol −1 . The variability is larger for the Zimm-Bragg theory with ΔG 0 NU in the range of 6 kcal mol −1 ⩽ ΔG 0 NU ⩽ 22 kcal mol −1 . The Gibbs free energy per peptide unit, g NU = ΔG 0 NU /n, calculated with the Zimm-Bragg theory, is linearly correlated with the width of the transition ΔT (g NU (cal mol −1 ) = 3·17ΔT + 3·9; Supplementary Information S18). It changes from g NU = 440 cal mol −1 for the broad transition of the 50-amino acid peptide to g NU = 57 cal mol −1 for the highly cooperative transition of pseudo WT T4 lysozyme. In contrast, no systematic variation of ΔG 0 NU can be recognized for the two-state model. Thermal protein unfolding can be compared with chemical denaturation. In this protocol, the fraction of unfolded protein is measured with spectroscopic techniques at different concentrations of denaturant. The equilibrium constant K NU and the corresponding free energy ΔG NU are calculated with the two-state model. ΔG NU is then plotted as a function of denaturant concentration and extrapolated to zero denaturant concentration (linear extrapolation method, LEM) (Bolen & Yang, 2000;Konermann, 2012) The free energy at zero denaturant concentration is assumed to be equivalent to the free energy change produced by heat denaturation. Chemical denaturation is however a multi-site equilibrium and not a two-state process. The denaturants (guanidineHCl, urea) bind to a large number of units by electrostatic and/or hydrophobic forces (Makhatadze & Privalov, 1992). It has been noted: 'Given that ΔG 0 NU values are often determined from a method that is empirical in origin (the linear extrapolation method) it is doubtful that the quantity we call ΔG 0 NU deserves the complete credibility it is often given' (Bolen & Yang, 2000). The thermal denaturation of lysozyme under conditions described in Figs 6 and 8 occurs between 50 and 73°C. The free energy of unfolding is ΔG 0 NU = 6·3 kcal mol −1 , calculated with the two-state model and the CD-parameter ΔH 0 NU = 90·8 kcal mol −1 . Using the DSC parameters ΔH 0 NU = 106·9 kcal mol −1 and ΔC 0 p,NU = 2·269 kcal molK −1 results in ΔG 0 NU = 7·3 kcal mol −1 . The Zimm-Bragg theory predicts ΔG 0 NU = 9·8 kcal mol −1 based on the DSC experiment. Chemical denaturation experiments at 20°C with guanidine HCl resulted in ΔG 0 NU = 8·9 kcal mol −1 (pH 7·0) (Ahmad et al. 1982) and 8·2 kcal mol −1 (pH 6·0) (Laurents & Baldwin, 1997).
The Zimm-Bragg theory can easily be modified to describe chemical denaturation. This is explained briefly in Supplementary Information S19.

Conclusions
The two-state model treats protein unfolding as a single global event. The Zimm-Bragg theory, in contrast, sees it as a sequence of local processes. Thermally-induced protein unfolding is a sequential multi-state process. Even if intermediate states are only sparsely populated a sequential model is physically more realistic than a two-state model.

The Zimm-Bragg theory predicts a sequential unfolding and accounts for intermediate states. A 'folded'
'unfolded' equilibrium of peptide units with an average unfolding enthalpy of 1·1 kcal mol −1 is valid also for proteins with low α-helix content. The Zimm-Bragg theory provides a perfect fit to the DSC data of all proteins investigated and describes equally well the CD transition curves with the same parameters. It predicts the calorimetric heat of unfolding, ΔH 0 exp , with an error of less than 5%. In contrast, the two-state model requires different parameters for DSC-and CD-unfolding transitions. The calculated unfolding enthalpy ΔH 0 calc,2−state is typically 10-20% smaller than the experimental result ΔH 0 exp . Moreover, the enthalpies determined from CD spectroscopy data are even smaller than those accused is the two-state model from DSC experiments.
The observation of an isodichroic point is not sufficient evidence for an equilibrium between just two unique protein conformations ('native' and 'unfolded'). An isodichroic point can also be generated by an intramolecular equilibrium between 'folded' and 'unfolded' peptide units. The solution can thus contain a manifold of protein conformations, each protein containing a different fraction of "folded" and "unfolded" peptide units.

Supplementary material
The supplementary material for this article can be found at http://dx.doi.org/10.1017/S0033583516000044. 21