Molecular understanding of calorimetric protein unfolding experiments

Abstract

Testing and predicting protein stability gained importance because proteins, including antibodies, became pharmacologically relevant in viral and cancer therapies. Isothermal scanning calorimetry is the principle method to study protein stability. Here, we use the excellent experimental heat capacity C_p(T) data from the literature for a critical inspection of protein unfolding as well as for the test of a new cooperative model. In the relevant literature, experimental temperature profiles of enthalpy, H_cal(T), entropy, S_cal(T), and free energy, G_cal(T) are missing. First, we therefore calculate the experimental H_cal(T), S_cal(T), and G_cal(T) from published C_p(T) thermograms. Considering only the unfolding transition proper, the heat capacity and all thermodynamic functions are zero in the region of the native protein. In particular, the free energy of the folded proteins is also zero and G_cal(T) displays a trapezoidal temperature profile when cold denaturation is included. Second, we simulate the DSC-measured thermodynamic properties with a new molecular model based on statistical-mechanical thermodynamics. The model quantifies the protein cooperativity and predicts the aggregate thermodynamic variables of the system with molecular parameters only. The new model provides a perfect simulation of all thermodynamic properties, including the observed trapezoidal G_cal(T) temperature profile. Importantly, the new cooperative model can be applied to a broad range of protein sizes, including antibodies. It predicts not only heat and cold denaturation but also provides estimates of the unfolding kinetics and allows a comparison with molecular dynamics calculations and quasielastic neutron scattering experiments.

Why it matters

Protein stability is an important issue in the development of pharmaceutical biologics. Because biologics are often large molecules (e.g., antibodies), and as folding is a highly cooperative process, a new model is proposed that takes into account cooperativity and protein size. The model is based on a consistent set of thermodynamic parameters, derived from differential scanning calorimetry (DSC) experiments. Here, we apply the model to published DSC data of four very different proteins to demonstrate its predictive power. It not only yields heat denaturation but also cold denaturation, as well as the kinetics of unfolding. Moreover, the results of the cooperative model can be compared with molecular dynamics calculations.

Introduction

Protein stability is a most relevant issue in the development of pharmaceutical biologics. Thermodynamic aspects of protein unfolding have acquired significant practical importance because they provide the theoretical framework for rational protein design and protein modifications (1). Differential scanning calorimetry (DSC) is the method of choice for thermodynamic studies of protein folding/unfolding equilibria (2,3). Analysis of DSC experiments with simple thermodynamic models has been key for developing our understanding of protein stability (4). So far, the reversible denaturation reaction has been analyzed with a two-state model (5).

However, the protein folding $\rightleftarrows$ unfolding equilibrium is a dynamic reaction with many short-lived intermediates (6). A multistate cooperative algorithm is therefore a physically more realistic alternative. We have shown for several proteins that the cooperative multistate Zimm-Bragg theory is such an alternative (7, 8, 9, 10, 11, 12, 13, 14). The theory yields a quantitated measure of cooperativity, is not limited in protein size, and provides excellent simulations of protein unfolding thermodynamics. Here, we present a significant improvement of the model by combining the Zimm-Bragg theory with statistical mechanics. As a result, the macroscopic thermodynamic properties of heat unfolding and cold denaturation can be explained with molecular parameters of well-defined physical meaning. The new model is validated by analyzing published DSC measurements of lysozyme (10), the classic example of protein unfolding, gpW62 (15), an ultrafast folding protein, mAb, a large monoclonal antibody (13), metmyoglobin (16), a protein exhibiting cold denaturation, and ubiquitin, an α-helical protein (17). In the long-standing history of DSC of protein unfolding, the heat capacity function C_p(T) was mainly used to evaluate the enthalpy change ΔH_cal of the unfolding transition. Entropy and free energy were ignored, although both properties are easily derived from C_p(T). Here, we systematically evaluate the complete temperature profiles H_cal(T), S_cal(T), and G_cal(T) from experimental C_p(T) data and analyze them with the cooperative multistate Zimm-Bragg theory. In particular, the free energy is found to display a trapezoidal shape with a zero free energy in the region around the native protein. This unique thermodynamic signature is precisely predicted by the Zimm-Bragg theory, but disagrees with the parabolic shape predicted by the two-state model. The Zimm-Bragg analysis covers DSC experiments in the temperature range of 20–90°C. The same fit parameters can be used to calculate the entropies of completely denatured proteins at a denaturation temperature of 225°C. Excellent agreement with the predictions of the Dynameomics Entropy Dictionary is obtained (18).

Materials and methods

This study describes a new model for protein unfolding, which is based on the multistate, cooperative Zimm-Bragg theory. The most significant accomplishment of this model is that it enables us to calculate the aggregate thermodynamic variables from molecular parameters. The total energy, entropy, and free energy are derived in terms of a continuous canonical partition function or its derivatives. It is an analytical model expressing the macroscopic thermodynamic properties of the system with molecular parameters only.

The potential of the model is illustrated with published DSC experiments of protein unfolding. In DSC the heat capacity ${\text{C}}_{\text{p}}\left(\text{T}\right)=(\partial \mathrm{H}/\partial \mathrm{T})\mathrm{p}$ is recorded as a function of temperature. In protein unfolding, C_p(T) is composed essentially of two parts, the basic heat capacity of the native protein and that of the phase transition proper. Native lysozyme, for example, has a basic heat capacity of ∼5 kcal/molK, which increases to ∼25 kcal/molK at the midpoint of unfolding, and decreases again to ∼7.2 kcal/molK for the unfolded protein (19,20). The basic heat capacity makes a large contribution to the thermodynamic properties and dominates enthalpy, entropy, and free energy (cf. Fig. 2). Therefore, it is common practice to apply a baseline correction such that the heat capacity of the native protein is zero (4,21). As a consequence of this subtraction it follows that all thermodynamic properties are zero in the region of the native protein, including the free energy. The model was tested with published DSC data of proteins ranging in size between 62 and 1200 amino acid (aa) residues. In three examples the published data were already baseline corrected. In one example we present both raw and baseline-corrected data.

Figure 2.

gpW62 DSC unfolding (7). (▲) Original C_p(T) data (taken from Fig. 4 A in reference (15)). (▪) Baseline-corrected data. Data points in (B–D) were calculated from data points in (A) with Eqs. 10, 11, and 12. Red lines: simulations of unfolding transition proper with the Zimm-Bragg theory (h₀ = 1.26 kcal/mol, c_v = 5 cal/molK, σ = 5.0 × 10⁻⁴, ν = 62 residues). Blue lines: two-state model (ΔH₀ = 49 kcal/mol, $\Delta {\text{C}}_{\text{p}}^0$= 0.9 kcal/molK). The red dotted lines in (A) are the differences between the experimental DSC data and the simulations. For better visibility, the blue dotted line is shifted by −1 kcal/molK. Green lines: sum of Zimm-Bragg theory and the contribution of the basic heat capacity of the native protein C_p = 2.7 kcal/molK (cf. Eqs. 7, 8, and 9).

The C_p(T) thermograms were integrated (Eqs. 10, 11, and 12), yielding the temperature profiles H_cal(T), S_cal(T), and G_cal(T) of the unfolding transition. Such experimental profiles are missing in the relevant literature. Here, they are evaluated from experimental data for all four proteins investigated.

The theoretical analysis proceeds as follows. The heat capacity C_p(T) is simulated with the cooperative model presented below. The fit parameters are rather independent of each other and are determined sequentially. The unfolding enthalpy h₀ per amino acid residue is ∼1.1 ± 0.2 kcal/mol (10). The number of amino acid residues in the unfolding reaction is $\nu \sim \Delta {\text{H}}_{\text{cal}}/{\text{h}}_0$. Here, ΔH_cal is the total heat of unfolding. The cooperativity parameter σ is typically $\sigma \sim {10}^{-3}-{10}^{-7}$. σ is easily adjusted by fitting the sharp transition peak. Finally, the heat capacity c_v is determined by fitting the observed increase in heat capacity $\Delta {\text{C}}_{\text{p}}^0$. No global fit procedure is needed as all four parameters can be determined precisely by a judicial inspection of the experimental data. The same parameters are then used to calculate enthalpy H_cal(T), entropy S_cal(T), and free energy G_cal(T).

The popular two-state model is briefly discussed as far as the free energy is concerned. The model is restricted to the unfolding transition proper and is applied to baseline-corrected thermograms. It assumes a zero heat capacity for the native protein (3,22,23). However, in disagreement with DSC, the model predicts a positive free energy for the region of the native protein. An alternative two-state model with correct thermodynamic predictions has been published (24).

All calculations presented in this study were performed with MathCad 15. In this program, all equations can be written as shown in the text.

Theory

Zimm-Bragg theory extended to cold denaturation

Protein unfolding is a dynamic process in which individual amino acid residues flip from their native (n) to their unfolded (u) state. Rapid equilibria between many short-lived intermediates can be expected. An early example of cooperative unfolding is the α-helix to random coil transition of synthetic peptides described with the Zimm-Bragg theory (25, 26, 27). The cooperative folding theory distinguishes between a growth process with an equilibrium constant q(T) and a nucleation step with an equilibrium constant σq(T), where σ is the cooperativity or nucleation parameter. Growth is defined as the addition of a new helical segment to an already existing α-helix. Nucleation is the formation of a new helical segment within an unstructured region. The steepness of the transition is determined by the cooperativity parameter σ.

The central element of the original Zimm-Bragg theory is a discrete canonical partition function of the form $\text{Z}\left(\text{T}\right)=\sum {\text{e}}^{\frac{-{\text{E}}_{\text{i}}}{\text{RT}}}$, which collects all the energetic states of the folding process. The system is allowed to exchange heat with its environment, but its volume and the number of particles are fixed. The volume changes in protein unfolding are indeed very small (28,29) and can be neglected in the present analysis. The rather lengthy expressions of the partition function Z(T) in the initial Zimm-Bragg theory (25) can be written in compact form (27).

$$\text{Z(T)}=\left(\begin{array}{ll}1& 0\end{array}\right){\left(\begin{array}{ll}1& \mathrm{\sigma }\text{q}\left(\text{T}\right)\\ 1& \text{q}\left(\text{T}\right)\end{array}\right)}^{\text{ν}}\left(\begin{array}{l}1\\ 1\end{array}\right).$$ (1)

ν denotes the number of amino acid residues involved in unfolding. The cooperativity parameter is small with σ ∼10⁻³–10⁻⁷. The larger σ, the less cooperative is the system and the broader is the transition peak. The limit of no cooperativity is reached with $\mathrm{\sigma }=1$. At this condition the Zimm-Bragg theory reduces to a two-state model, similar to the chemical two-state model, but providing a more stringent thermodynamic interpretation of DSC thermograms. The chain length ν also has some influence on the width of the transition. The smaller ν, the broader is the unfolding transition. Short chains lead a broad transition (26).

The equilibrium constant q(T) is given by

$$\text{q}\left(\text{T}\right)={\text{e}}^{\frac{-\text{h}\left(\text{T}\right)}{\text{R}}\left(\frac{1}{\text{T}}-\frac{1}{{\text{ T}}_{\infty }}\right)}.$$ (2)

h(T) is the enthalpy needed to induce the n $\to$ u conformational change. Up until now, this parameter was assumed to be temperature-independent with h₀ ∼ 0.9–1.3 kcal/mol (26,30, 31, 32, 33). Here, we introduce a temperature-dependent unfolding enthalpy, which changes Eq. 1 to a continuous canonical partition function

$$\text{h}\left(\text{T}\right)={\text{h}}_0+{\text{c}}_{\text{v}}(\text{T}-{\text{T}}_0).$$ (3)

T₀ is the midpoint temperature of heat-induced unfolding. The heat capacity c_v refers to the unfolding of a single amino acid residue. The reference temperature ${\text{T}}_{\infty }$ in Eq. 2 determines the position of the heat capacity maximum on the temperature axis. It is identical with T₀ if the number of amino acid residues ν is much larger than σ^−1/2.

The fraction of unfolded protein Θ_U(T) is

$${\Theta }_{\text{U}}\left(\mathrm{T}\right)=\frac{\text{q}\left(\text{T}\right)}{\text{ν}}\frac{\text{d}\left(\ln \text{Z}\left(\text{T}\right)\right)}{\text{dT}}{\left(\frac{\text{dq}\left(\text{T}\right)}{\text{dT}}\right)}^{-1}.$$ (4)

For a noncooperative system with σ = 1 the multistate Zimm-Bragg theory reduces to a two-state model as mentioned above (14).

Cold denaturation

The heat capacity term c_v has two consequences. First, c_v produces the heat capacity increase $\Delta {\text{C}}_{\text{p}}^0$ of the unfolded protein. Second, c_v leads to an additional transition at low temperature (cold denaturation). The exponent in the equilibrium constant q(T) (Eq. 2) has zeros at T₀ and at h(T) = 0. The latter relation leads to

$${\text{T}}_{\text{c}}={\text{T}}_0-\frac{{\text{h}}_0}{{\text{c}}_{\text{v}}}.$$ (5)

T₀ is the midpoint of heat denaturation, T_c that of cold denaturation. The temperature difference between heat and cold denaturation is ΔT = h₀/c_v.

Partition function and statistical-mechanical thermodynamics

We show that the thermodynamics of protein unfolding and, consequently, the DSC experiments can be simulated without macroscopic fit parameters. According to statistical-mechanical thermodynamics, the partition function Z(T) is the sum of all conformational energies and is sufficient to determine the thermodynamic system parameters, that is, the inner energy E(T), the entropy S(T), and the Helmholtz free energy F(T) (34,35). The relevant equations are as follows

$$\text{F}\left(\text{T}\right)=-\text{RT}\ln \text{Z}\left(\text{T}\right),$$ (6)

$$\text{E}\left(\text{T}\right)={\text{RT}}^2\frac{\text{d}\ln \text{Z}\left(\text{T}\right)}{\text{dT}}+\underset{{\text{T}}_{\text{ini}}}{\overset{\text{T}}{\int }}{\text{C}}_{\text{p}}\left(\text{T}\right)\text{dT},$$ (7)

$$\text{S}\left(\text{T}\right)=\frac{\text{E}\left(\text{T}\right)-\text{F}\left(\text{T}\right)}{\text{T}}+{\int }_{{\text{T}}_{\text{ini}}}^{\text{T}}\frac{{\text{C}}_{\text{p}}\left(\text{T}\right)}{\text{T}}\text{dT},$$ (8)

$${\text{C}}_{\text{V}}\left(\text{T}\right)=\frac{\text{dE}\left(\text{T}\right)}{\text{dT}}+{\text{C}}_{\text{p}}\left(\text{T}\right).$$ (9)

The first term on the right side determines the phase transition proper. The second terms is the contribution of the basic protein heat capacity C_p(T), which is independent of the unfolding transition. The experimental data can thus be analyzed without any arbitrary baseline correction. In baseline-corrected thermograms the heat capacity of the native protein is C_p(T) = 0 kcal/molK and only the first terms need to be considered. For ${\text{C}}_{\text{p}}\left(\text{T}\right)\ne 0\text{ kcal}/\text{molK }$Eq. 6 must be replaced by $\text{F}\left(\text{T}\right)=\text{E}\left(\text{T}\right)-\text{TS}\left(\text{T}\right)$, where E(T) and S(T) are given by Eqs. 7 and 8, respectively. Integration starts at T_ini, the temperature at the beginning of the DSC measurement. In almost all published examples, a baseline correction is performed and a zero heat capacity is assigned to the native protein. This is necessary as neither the chemical two-state model nor the Zimm-Bragg theory account for the basic protein heat capacity. The inclusion of a C_p(T) term obscures the phase transition proper. This is illustrated in Fig. 2, where the original DSC data are compared with the baseline-corrected results.

DSC

DSC is the method of choice to determine the thermodynamic properties of protein unfolding. DSC measures the temperature course of the heat capacity C_p(T), which includes not only the conformational enthalpy proper but also the increase in heat capacity $\Delta {\text{C}}_{\text{p}}^0$ between native and denatured protein (36,37). Stepwise integration of C_p(T) provides unfolding enthalpy, entropy, and Gibbs free energy:

$$\text{H}\left({\text{T}}_{\text{i}}\right)=\sum _1^{\text{i}}\left[\frac{{\text{C}}_{\text{p}}\left({\text{T}}_{\text{i}}\right)+{\text{C}}_{\text{p}}\left({\text{T}}_{\text{i}+1}\right)}{2}\right]\left[{\text{T}}_{\text{i}+1}-{\text{T}}_{\text{i}}\right],$$ (10)

$$\text{S}\left({\text{T}}_{\text{i}}\right)=\sum _1^{\text{i}}\frac{{\text{C}}_{\text{p}}\left({\text{T}}_{\text{i}+1}\right)+{\text{C}}_{\text{p}}\left({\text{T}}_{\text{i}}\right)}{2{\text{T}}_{\text{i}}}\left[{\text{T}}_{\text{i}+1}-{\text{T}}_{\text{i}}\right],$$ (11)

$$\text{G}\left({\text{T}}_{\text{i}}\right)=\text{H}\left({\text{T}}_{\text{i}}\right)-{\text{T}}_{\text{i}}\text{S}\left({\text{T}}_{\text{i}}\right).$$ (12)

"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" (23). The DSC-measured thermodynamic parameters characterizing the total unfolding transition are denoted ΔH_cal, ΔS_cal, and ΔG_cal.

DSC measurements are made at constant pressure. As mentioned above, the volume changes in protein unfolding are very small (<5%) and the following relations are valid without loss of accuracy. Heat capacity C_p(T) $\cong$ C_v(T), enthalpy H(T) $\cong$ inner energy E(T), entropy S_p(T) $\cong$ S_v(T), Gibbs free energy G(T) $\cong$ Helmholtz free energy F(T).

Results

Our earlier simulations of DSC experiments with the Zimm-Bragg theory required molecular as well as macroscopic fit parameters. The extent of unfolding Θ_U(T) was calculated with molecular parameters (h₀, σ) and was then multiplied with macroscopic parameters, that is, the unfolding enthalpy ΔH₀ and the heat capacity increase $\Delta {\text{C}}_{\text{p}}^0$. Here, we eliminate ΔH₀ and $\Delta {\text{C}}_{\text{p}}^0$ by combining the partition function Z(T) with statistical-mechanical thermodynamics (Eqs. 6, 7, 8, and 9). All macroscopic thermodynamic properties are now predicted exclusively with molecular parameters (h₀, c_v, σ, ν). Of particular interest is the free energy of unfolding G(T) $\cong$ F(T). Proposed as a parabolic function in the theory of the two-state model, the DSC-accessible free energy G_cal(T) (Eq. 12) appears to be completely ignored in the relevant DSC literature. In the following we compare proteins of different size and structure, which were carefully studied with DSC. The analysis of the C_p(T) thermograms provides the caloric, model-independent results for enthalpy H_cal(T), entropy S_cal(T), and free energy G_cal(T). The measured thermodynamic properties are then compared with the predictions by the cooperative Zimm-Bragg theory and the two-state model.

Lysozyme unfolding: DSC and molecular multistate partition function

Lysozyme is a 129-residue protein composed of ∼25% α-helix, ∼40% β-structure, and ∼35% random coil in solution at room temperature (10). Upon unfolding, the α-helix is almost completely lost and the random coil content increases to ∼60%. Thermal unfolding is completely reversible and lysozyme is the classical example to demonstrate two-state unfolding (2,20,22). Fig. 1 A shows the C_p(T) thermograms of lysozyme unfolding with a resolution of 0.17°C (10) Unfolding takes place in the temperature range of 45°C $\le$ T $\le$ 73°C (midpoint temperature T₀ = 61.7°C). The unfolding enthalpy is ΔH_cal = 147 kcal/mol (Eq. 10) and the entropy ΔS_cal = 0.437 kcal/molK (Eq. 11). The enthalpy/entropy ratio ΔH_cal/ΔS_cal = 335.5 K = 62.5°C agrees with the midpoint temperature T₀. The molar heat capacity of unfolded lysozyme increases by $\Delta {\text{C}}_{\text{p}}^0$ = 2.269 kcal/molK, in agreement with literature data (22,36,37).

Figure 1.

Differential scanning calorimetry of 50 μM lysozyme in 20% glycine buffer, pH 2.5. Black data points: DSC thermograms obtained with a heating rate of 1°C min⁻¹ and a step size of 0.17°C. Red lines: simulations with the Zimm-Bragg theory with h₀ = 1.0 kcal/mol, c_v = 6 cal/molK, σ = 1.0 × 10⁻⁶, ν = 129. (A) Heat capacity C_p(T). (B) Unfolding enthalpy H(T). (C) Unfolding entropy S(T). (D) Free energy of unfolding G(T). Blue line: ΔG(T) (Eq. 13) predicted by the two-state model calculated with a conformational enthalpy ΔH₀ = 107 kcal/mol and a heat capacity increase $\Delta {\text{C}}_{\text{p}}^0$ = 2.269 kcal/mol. Data taken from reference (10).

The high cooperativity of lysozyme unfolding (cooperativity parameter σ = 1.0 × 10⁻⁶) leads to a sharp unfolding transition, which is well approximated by a two-state equilibrium. Two-state model and Zimm-Bragg theory both provide excellent simulations of the heat capacity C_p(T) (Fig. 1 A).

Differences between the two models exist, however, in predicting the temperature dependence of the free energy. The two-state model defines the free energy as

$$\Delta \text{G}\left(\text{T}\right)=\Delta {\text{H}}_0\left(1-\frac{\text{T}}{{\text{T}}_0}\right)+\Delta {\text{C}}_{\text{p}}^0(\text{T}-{\text{T}}_0)-\text{T}\Delta {\text{C}}_{\text{p}}^0\ln \left(\frac{\text{T}}{{\text{T}}_0}\right),$$ (13)

yielding the parabolic shape in Fig. 1 D (blue line, same parameters as in Fig. 1 A). The free energy has a positive maximum of 7.18 kcal/mol for the native protein at 20°C and becomes zero at T₀ = 62°C, where lysozyme is 50% unfolded. However, this is not what is seen in the DSC experiment (Fig. 1 D, black data points). The free energy of the native protein is zero and becomes immediately negative upon unfolding. Between 27°C and T₀ the DSC experiment reports a negative free energy change ΔG_cal = −0.7 kcal/mol, which increases rapidly beyond T₀ to ΔG_cal = −6.06 kcal/mol at 73°C (90% unfolding). The Zimm-Bragg theory (Eq. 7) reproduces precisely the DSC result (Fig. 1 D, red line). Extending this simulation to low temperatures yields cold denaturation at T_cold = −103°C. The Zimm-Bragg theory thus predicts a trapezoidal shape of the free energy and the flat, near zero region extends over almost 140°C. An experimental proof for the free energy trapezoidal shape is given in Fig. 4.

Figure 4.

Unfolding of metmyoglobin at acidic pH. (▪) Experimental data taken from reference (16). Red lines: simulations with the multistate cooperative Zimm-Bragg theory (fit parameters listed in Table 1). (red ▪) Differences between DSC data and the Zimm-Bragg simulation (shifted in B by −1 kcal/molK for better visibility). Blue lines: two-state model. (A) C_p(T) at pH 4.1 (Fig. 13 in reference (16)). (B) C_p(T) at pH 3.83 (Fig. 12 in reference (16)). (C and D) Free energies calculated using the heat capacity data shown in (A) and (B).

gpW62, an ultrafast folding protein

gpW62 is a 62-residue α+β protein that belongs to a group of ultrafast folding proteins (4,15). gpW62 folding was investigated with a variety of techniques, including DSC. Fig. 2 A shows the original heat capacity C_p(T) (7) from Fig. 4 A of reference (15)) and the baseline-corrected C_p(T) (▪). Baseline correction was achieved by simply subtracting the heat capacity of the native protein of 2.7 kcal/molK. Both data sets were simulated with Eq. 9, either with C_p (green line) or without C_p (red line). The blue line is the best fit with the chemical two-state model (calculated with Eq. 14 of reference (3)). The dotted lines in Fig. 2 A are the differences between experimental results and simulations. Equations 10, 11, and 12 were applied to both the original and the baseline-corrected C_p(T) data yielding experimental results for enthalpy (Fig. 2 B), entropy (Fig. 2 C), and Gibbs free energy (Fig. 2 D). In the uncorrected data (7) the unfolding transition is masked by the very substantial contribution of the basic heat capacity C_p = 2.7 kcal/molK. Still, it should be noted that the Gibbs free energy of the native protein is zero, even for the noncorrected heat capacity data. In the following we only discuss the baseline-corrected data.

The midpoint temperature is T₀ = 67°C. The unfolding enthalpy is ΔH_cal = 91.7 kcal/mol, measured between 37 and 102°C. This is a large enthalpy for a 62-residue protein. The free energy change between native and unfolded gpW62 is ΔG_cal = −8.11 kcal/mol. The free energy per amino acid residue g_cal = −131 cal/mol is almost three times larger than that of lysozyme with g_cal = −47 cal/mol.

Several aspects of the gpW62 folding equilibrium are unusual. The cooperativity is low with σ = 5 × 10⁻⁴ and unfolding extends over a broad temperature range of ΔT ∼ 65°C. This may be compared with lysozyme with σ = 1 × 10⁻⁶ and ΔT ∼ 30°C. As the transition is broad, the Zimm-Bragg theory provides a better fit than the two-state model (Fig. 2 A). The low cooperativity facilitates fast folding by a low nucleation free energy (see Discussion). Fast folding of gpW62 could also be promoted by the large free energy change of the unfolding reaction. According to the thermodynamics of irreversible processes, the chemical reaction rate is proportional to the affinity, i.e., the free energy, of the reaction (38, 39, 40). The ultrafast folding of gpW62 could thus arise from the combination of a low nucleation energy and a large unfolding affinity.

Monoclonal antibody mAb

The two-state model works best for enthalpies of 50–200 kcal/mol. No such limitation exists for the Zimm-Bragg theory. This is demonstrated for the recombinant monoclonal IgG1 antibody mAb (143 kDa, ∼1280 aa), the main transition of which has an enthalpy of ∼1000 kcal/mol (13). The antibody is formed of two identical heavy chains of ∼450 residues each and two identical light chains of ∼200 residues. The heavy and light chains fold into domains of ∼110 aa residues. The secondary structure of mAb is composed of 7–11% α-helix and 40–45% β-sheet (41).

The DSC experiment (Fig. 3 A) reveals a pretransition at 73°C and a main transition at 85°C. The unfolding enthalpy of the pretransition is ΔH_cal = 290 kcal/mol involving ν_pre = ΔH_cal/h₀ = 263 aa residues. The main transition has an enthalpy of ΔH_cal $\simeq$ 1000 kcal/mol with ν_main $\simeq$ 880 aa residues. Taken together, pre- and main transition account for ∼90% of all amino acid residues. A molecular interpretation of pre- and main transition based on the mAb structure has been given (13). The pretransition results from the unfolding of two C_H2 domains, whereas the main transition represents the unfolding of eight domains of the Fab fragment and two domains of the Fc fragment. In Fig. 3 the pretransition (green) and main transition (violet) are superimposed (red). The pretransition is slightly less cooperative with σ = 5 × 10⁻⁵ than the main transition with σ = 2 × 10⁻⁵. The same molecular parameters h₀ = 1.1 kcal/mol and c_v = 7.0 cal/molK were used in all simulations. The theory predicts the heat capacity increase upon unfolding as $\Delta {\text{C}}_{\text{p}}^0$ = 6.34 kcal/molK for the pretransition and $\Delta {\text{C}}_{\text{p}}^0$ = 17.19 kcal/molK for the main transition, consistent with the number of amino acids involved. The DSC-measured temperature profile of the free energy follows the same pattern as observed for lysozyme and gpW62. The free energy of the native mAb is zero up to about 65°C followed by a biphasic decrease. As shown in Fig. 3 D, the contributions of the pretransition (green) and the main transition (violet) are well separated. The multistate partition function Z(T) precisely predicts this biphasic behavior. The free energy per residue is g_cal = −30 ± 1 cal/mol for both pre- and main transition and thus clearly smaller than those of lysozyme (−47 cal/mol) and gpW62 (−136 cal/mol). Neither the pretransition nor the main transition can be simulated with the two-state model.

Figure 3.

Thermal unfolding of monoclonal antibody mAb at pH 6.2. (▪) DSC experiment. Solid lines are simulations with the Zimm-Bragg theory: green, pretransition; violet, main transition; red, sum of pre- and main transition. (A) Molar heat capacity. (B) Unfolding enthalpy. (C) Unfolding entropy. (D) Free energy. Simulation parameters: h₀ = 1.1 kcal/mol, c_v = 7.0 cal/molK. Pretransition: T₀ = 73°C, ν_pre = 263, σ = 5 × 10⁻⁵. Main transition: T₀ = 85.4°C, ν_main = 880, σ = 2 × 10⁻⁵. Data taken from reference (13).

Heat and cold denaturation of metmyoglobin

A protein that is stable at ambient temperature can be unfolded by heating or, less commonly, by cooling. Cold denaturation of most proteins occurs at subzero temperatures. Rather drastic conditions are needed to shift cold denaturation above 0°C (e.g., addition of denaturants, low or high pH). DSC experiments reporting cold denaturation or at least partial cold denaturation are available for metmyoglobin (16), staphylococcus nuclease (42), β-lactoglobulin (43), streptomyces subtilisin inhibitor (44), or thioredoxin (45).

Metmyoglobin (153 residues) consists of 8 α-helical regions connected by loops (46). DSC at pH 4.1 (Fig. 4 A) displays heat denaturation at T₀ = 69°C and cold denaturation starting at 3°C (data from Fig. 13 of reference (16)). Heat denaturation of the already partially destabilized protein is characterized by ΔH_cal = 146 kcal/mol, ΔS_cal = 0.431 kcal/mol, and ΔG_cal = −6.4 kcal/mol. The ratio ΔH_cal/ΔS_cal = 339 K = 66°C is consistent with the C_p(T) maximum. The unfolding entropy per residue is s_cal = 2.81cal/molK. Quasielastic neutron scattering (QENS) yields s_QENS = 10 J/molK = 2.39 cal/molK for the unfolding of myoglobin with 66% α-helix content (Fig. 10 in reference (47), Fig. 9 in reference (48)).Extrapolation to 100% α-helix results in s_QENS = 3.63 cal/molK. Inspection of Fig. 4 A shows that the Zimm-Bragg theory (fit parameters in Table 1) provides a clearly better fit of C_p(T) than the two-state model (fit parameters ΔH₀ = 112 kcal/mol, $\Delta {\text{C}}_{\text{p}}^0$ = 2.9 kcal/molK).

Table 1.

Thermal unfolding. Differential scanning calorimetry (DSC) and parameters of the Zimm-Bragg theory.

Protein	Ν	DSC	DSC	DSC	DSC	DSC	ZB	Zimm-Bragg theory	ZB	ZB
		ΔHcal	hcal = ΔHcal/ν	scal = ΔScal/ν	gcal = ΔGcal/ν	$\Delta {\text{C}}_{\text{p}}^0$i	Hconk	h0	cv	σ
		kcal/mol	kcal/mol	cal/molK	kcal/mol	kcal/molK	kcal/mol	kcal/mol	cal/molK
Native protein$\to$unfolded protein
gpW62a	62	91.7	1.48	4.34	−0.131	0.9	66.2	1.26	5	5 × 10−4
Ubiquitinb	74	89.7	1.21	3.41	−0.067	1.04	68.1	1.06	4.3	1 × 10−6
Lysozymec	129	147.2	1.14	3.39	−0.047	2.28	113.3	0.99	6	1 × 10−6
Metmyoglobind	153	178.1	1.16	3.32	−0.039	2.89	142	1.08	7	7 × 10−6
mAb pree	263	290.6	1.10	3.11	−0.031	4.7	244	1.1	7	5 × 10−5
mAb maine	880	1020	1.16	3.23	−0.033	17	900	1.1	7	2 × 10−5
Destabilized protein$\to$unfolded protein
Metmyoglobin pH 12f	145	144.7	1.00	3.04	−0.048	2.54	93.7	0.75	7	1 × 10−5
Metmyoglobin pH 4.1g	145	141.0	0.97	2.90	−0.044	3.58	70.5	0.6	10	2 × 10−6
Metmyoglobin pH 3.84h	100	96.0	0.96	2.91	−0.031	3.74	43.8	0.6	15	7 × 10−5
			average = 1.13					average = 0.95
			SD = 0.16					SD = 0.3

^aReference (15), Fig. 4 A (this work: Figs. 2 and 5A).

^bReference (17), Fig. 1 (this work: Fig. 5B).

^cReference (10), Fig. 8 (this work: Figs. 1 and 5C).

^dReference (16), Fig. 3 B, pH 10.7 (this work: Fig. 5D).

^eReference (13) (this work: Fig. 3).

^fReference (16), Fig. 3 B (no simulation shown).

^gReference (16) Fig. 13 (this work: Fig. 4A and C).

^hReference (16) Fig. 12 (this work: Fig. 4B and D).

ⁱDSC-measured increase in molar heat capacity upon protein unfolding.

^kConformational unfolding enthalpy proper, calculated with the Zimm-Bragg theory by setting c_v = 0.

At pH 3.83, metmyoglobin is even more destabilized. DSC reports two unfolding transitions with C_p(T) maxima at T_cold = 8°C and T₀ = 56.5°C (Fig. 4 B). The thermodynamic properties of heat unfolding are ΔH_cal = 96 kcal/mol, ΔS_cal = 0.291, and ΔH_cal/ΔS_cal = 329.9 K = 56.9°C. Cold denaturation is not the mirror image of heat denaturation as unfolding enthalpy and entropy are much smaller with ΔH_cold ∼ 53.8 kcal/mol and ΔS_cold ∼ 0.193 kcal/molK, respectively. ΔH_cold/ΔS_cold = 274 K = 1°C is lower than the experimental T_cold = 8°C. The free energy profile is displayed in Fig. 4 D. DSC yields a zero free energy for the native protein and negative free energies for heat and cold denaturation. This trapezoidal temperature profile is correctly predicted by the Zimm-Bragg theory. The specifics in the simulation of cold denaturation are a large heat capacity c_v and a small unfolding enthalpy h₀ (cf. Table 1). The two-state model predicts a parabolic profile and assigns a large positive free energy to the native protein.

Metmyoglobin at pH 3.83 was analyzed previously by a hierarchical algorithm defining a partition function in terms of multiple levels of interacting folding units (49). The model reproduces an idealized, symmetrical shape of the heat capacity C_p(T) and a parabolic free energy function.

Table 1 summarizes the DSC results and the Zimm-Bragg fit parameters of all proteins discussed. The table includes additional measurements of metmyoglobin at pH 10 and 12 and of ubiquitin (17), for which no simulations are shown.

Discussion

Proteins in solution do not show a simple, two-state equilibrium between a fully folded and a fully unfolded conformation. Depending on temperature, they form a complex mixture of many short-lived intermediates. Here, we present a new model, which predicts the important thermodynamic functions, enthalpy, entropy, and free energy, on the basis of molecular parameters only. The performance of the model is demonstrated by comparison with DSC experiments. The Zimm-Bragg theory provides excellent simulations of the temperature course of all thermodynamic functions reported by DSC. With this model we obtain insights into the cooperativity and dynamics of protein folding.

Protein stability

The most relevant parameter of protein stability is the midpoint temperature of unfolding T₀. Protein unfolding can be approximated by a first-order phase transition, and T₀ is determined by T₀ = ΔH_cal/ΔS_cal. ΔH_cal and ΔS_cal are the DSC-measured unfolding enthalpy and entropy, respectively. Minor changes in ΔH_cal or ΔS_cal produce distinctive shifts in T₀. The ultrafast folder gpW62 (62 aa) and ubiquitin (74 aa) are short proteins with almost identical unfolding enthalpies of 91.7 and 89.7 kcal/mol, respectively. Nevertheless, their transition temperatures are ∼20°C apart with gpW62 at 67°C and ubiquitin at 89.5°C. The difference is caused by the larger gpW62 entropy ΔS_cal = 0.269 kcal/molK compared with ΔS_cal = 0.25 kcal/molK of ubiquitin. The difference becomes even more obvious on a per residue basis with s_cal = ΔS_cal/ν = 4.34 kcal/mol for gpW62 and s_cal = 3.41 kcal/mol for ubiquitin. Likewise, very small differences in enthalpy and entropy cause the 12°C difference in the midpoint temperatures of the two mAb domains. A priori predictions of T₀ therefore require highly precise molecular dynamics (MD) calculations of unfolding enthalpy and entropy.

In a true first-order phase transition (e.g., melting of ice) the heat ΔH_cal is absorbed at the constant temperature T₀ and the heat capacity is a sharp peak (singularity). In contrast, ΔH_cal in protein unfolding is absorbed over 20–60°C and the heat capacity C_p(T) is a broad peak. In particular, and not generally recognized, the relation ΔH_cal = T₀ΔS_cal is limited to the overall reaction, but not applicable to the measured heat H(T₀) and entropy S(T₀). Considering lysozyme as an example, the DSC-measured heat absorbed up to T₀ is H(T₀) = 63.4 kcal/mol (Eq. 10) and is less than half of the total heat of 147.2 kcal/mol). The corresponding entropy is S(T₀) = 0.191 kcal/molK (Eq. 11). As H(T₀) < T₀S(T₀) this results in a negative free energy G(T₀) = −0.677 kcal/mol (Eq. 12), more realistic for a ∼50% unfolded protein than the zero free energy predicted by the two-state model. Analogous results are obtained for all proteins discussed here.

The two hallmarks of the two-state model, that is, the positive free energy of the native protein and the zero free energy of the 1:1 folded/unfolded mixture, are thus not confirmed by DSC. Instead, the free energy shows a trapezoidal shape that is precisely predicted by the new Zimm-Bragg folding model.

Molecular unfolding enthalpy h(T)

The Zimm-Bragg parameter h₀ is an average value of all types of interactions, independent of specific conformations (α-helix, β-sheet, ionic interactions, etc.). h₀ is close to the calorimetric average h_cal = ΔH_cal/ν. Metmyoglobin is an α-helical protein and the average enthalpy of the native protein (pH 10) is h_cal = 178/153 = 1.16 kcal/mol (Fig. 3 in reference (16)) while h₀ = 1.08 kcal/mol in the Zimm-Bragg simulation. For the ∼50% α-helical apolipoprotein A1 the DSC result is h_cal = 1.08 ± 0.07 kcal/mol and the Zimm-Bragg parameter h₀ = 1.1 kcal/mol (9). Lysozyme, a globular protein with mainly β-sheet structure, yields h_cal = 1.14 kcal/mol and h₀ = 0.99 kcal/mol (pH 2.5) (10). A comparison of a larger set of proteins has led to the conclusion that the Zimm-Bragg parameter is h₀ = 1.1 ± 0.2 kcal/mol (10).

The enthalpy h₀ is usually associated with the opening of an α-helix hydrogen bond (26,30, 31, 32). However, MD calculations have led to the conclusion that “hydrogen bond formation contributes little to helix stability […] The major driving force for helix is associated with interactions, including van der Waals interactions, in the close packed helix conformation and the hydrophobic effect” (50). This is supported by experimental results obtained with short alanine-based peptides, where hydrophobic interactions play the dominant role in stabilizing isolated α-helices (33).

Cold denaturation

Cold denaturation has been proposed as a tool to measure protein stability (51). The heat capacity c_v is a new parameter in the Zimm-Bragg theory leading to a second unfolding transition at low temperature. The temperature difference between heat and cold denaturation is given by ΔT = h₀/c_v (Eq. 5). Cold denaturation near ambient temperature thus requires a small h₀ and a large c_v. This is confirmed by metmyoglobin at pH 3.83, where h₀ is distinctly reduced to h₀ = ∼0.6 kcal/mol and c_v is increased to c_v = 10–15 cal/molK. In general, proteins with h₀/c_v $\ge$ 100 K display cold denaturation at very low subzero temperatures.

Molecular unfolding entropy

MD calculations consider all possible degrees of dihedral freedom of each amino acid residue in sampling the conformational space. In contrast, the Zimm-Bragg theory is an algorithm that differentiates only between folded and nonfolded amino acid residues, independent of the specific conformation. However, the theory makes predictions, solidly based on experimental data, that can be compared with MD calculations. The temperature course of the entropy is a good example. The entropy S(T) can be calculated with the Zimm-Bragg theory according to Eq. 8. The excellent agreement with the experimental data is displayed in Figs. 1, 2, and 3. In Fig. 5 we have repeated these calculations, including also ubiquitin, for the much larger temperature range of 298–498 K as this is the temperature interval of the “Dynameomics Entropy Dictionary” (18).

Figure 5.

Unfolding entropy. (▪) Experimental data. (magenta ▪) Dynameonics Entropy Dictionary (18). (A) gpW62 (15). (B) Ubiquitin (17) (Fig. 1 in reference (17)). (C) Lysozyme (10) (D) Metmyoglobin (16) (Fig. 3 in reference (16), pH 10). Red lines: Zimm-Bragg total entropy. Brown lines: conformational entropy proper. Green lines: contribution of the heat capacity c_v.

By averaging some 800 MD calculations the Dynameomics Entropy Dictionary provides the unfolding entropies of all amino acids when heated from the native state (298 K) to the fully denatured state (498 K). Using Table 2 in reference (18), we calculated the MD unfolding entropies at 498 K for the specific amino acid sequences of, gpW62 (62 aa), ubiquitin (74 aa), lysozyme (129 aa), and metmyoglobin (153 aa). The results are shown in Fig. 5 by the magenta data points at 498 K. The error bars were also calculated for each amino acid sequence using reference (18). In parallel, the Zimm-Bragg simulations were extended to 498 K, with the same parameters as deduced from C_p(T) at low temperature. This extrapolation is in excellent agreement with the Dynameomics Entropy Dictionary and supports the relevance of the molecular parameters of the Zimm-Bragg theory in protein unfolding.

The entropy S(T) (red) can be divided into the conformational entropy proper (brown line, h₀ = const., c_v = 0) and the contribution of the heat capacity term c_v (green line, h₀ = 0, c_v = const.). The h₀ term determines S(T) up to the end of the conformational transition, where Θ_U ∼ 0.85–0.9. However, it takes another temperature increase of more than 100°C to reach complete denaturation.

The average entropy is s_cal = ΔS_cal/ν = 3.25 ± 0.25 cal/molK per residue (Table 1) and involves at least three single bonds. The entropy per single bond is ∼1.1 cal/molK and can be compared with other phase transitions. The solid-fluid transition of long-chain paraffins yields a much larger entropy of 1.8–2.7 cal/molK per C-C bond. The gel-to-liquid crystal transition of phospholipid bilayers leads 1.25–1.6 cal/molK per C-C bond (Table 2.7, p. 47 in reference (52)). As judged by the small entropy of 1.1 cal/molK, the unfolded proteins are still characterized by a restricted motion of their molecular constituents. For metmyoglobin it was noted that “the unfolded state retains some residual ellipticity, which may be caused by the fluctuating α-helical conformation of the unfolded polypeptide chain” (16). This conclusion is supported by neutron spin-echo spectroscopy studies on apo-myoglobin where secondary structural elements and α-helices are found to be transiently formed at denaturation or molten globule conditions (53). Likewise, the combination of FRET, NMR, and SAX techniques has revealed a residual structure in denatured ubiquitin (54).

Protein cooperativity

The folding/unfolding transition of proteins is a cooperative process. The Zimm-Bragg theory provides a quantitated measure of cooperativity. In fact, the cooperativity parameter σ plays an important role in the energy and kinetics of the folding process as it determines the free energy to start a new fold within an unfolded region (nucleation) (55). The corresponding free energy is

$$\Delta {\text{G}}_{\mathrm{\sigma }}=-{\text{RT}}_0\ln \left(\mathrm{\sigma }\right).$$ (14)

A large σ corresponds to low cooperativity and to a small nucleation energy ΔG_σ. gpW62 has a cooperativity parameter σ = 5 × 10⁻⁴ and hence a nucleation energy of ΔG_σ = 5.13 kcal/mol. In contrast, lysozyme unfolding is highly cooperative with σ = 1 × 10⁻⁶ and ΔG_σ = 9.21 kcal/mol. The low gpW62 nucleation energy makes gpW62 folding easier than that of lysozyme. If ΔG_σ is assumed to be correlated with the kinetic activation energy, then gpW62 folding should be ∼500 times faster than that of lysozyme. In infrared T-jump experiments of gpW62, the return to equilibrium followed a single exponential with a relaxation time of τ = 15.7 μs at 57°C (15). In contrast, lysozyme was found to fold in a two-step mechanism with a slow nucleation (τ = 14 ms) followed by a fast growth step (τ = 300 μs) at room temperature (56).

Free energy of unfolding

The free energy of unfolding ΔG_cal scales with the size of the protein, and a per residue free energy g_cal = ΔG_cal/ν is better suited for comparative purposes. According to the Zimm-Bragg theory g_cal can be approximated by

$${\text{g}}_{\text{cal}}\simeq \text{h}\left({{\text{T}}_{\text{en}}}_{\text{d}}\right)(1-{\text{T}}_{\text{end}}/{\text{T}}_0)=-({\text{h}}_0+{\text{c}}_{\text{v}}\Delta \text{T})(\Delta \text{T}/{\text{T}}_0).$$ (15)

T_end denotes the temperature at the end of the conformational transition and ΔT = T_end − T₀ is the halfwidth of the transition. The approximation Eq. 15 agrees within 2% with the DSC measurement for the proteins listed in Table 1. The free energy per amino acid residue varies between g_cal = −131 cal/mol for gpW62 to g_cal = −33 cal/mol for mAb. In parallel, the width of the transition decreases from 65°C for gpW62 to 20°C for mAb (large domain). An approximately linear relationship between g_cal and ΔT is predicted by Eq. 15 and is confirmed by DSC, that is, a large g_cal correlates with a broad transition. Considering the three parameters T₀, g_cal, and ΔT, gpW62 is the least stable protein discussed here.

Conclusions

The protein folding/unfolding transition is a highly cooperative process which cannot be adequately simulated by the noncooperative two-state model. A multistate cooperative model is provided by the Zimm-Bragg theory. Here, we combined the partition function of the Zimm-Bragg theory with statistical-mechanical thermodynamics. The model predicts the DSC-measured enthalpy, entropy, and free energy of protein unfolding with molecular parameters, which have well-defined physical meanings. We analyzed the DSC thermograms of proteins of different chain length and structure. We show that the temperature profile of the free energy is characterized by a trapezoidal shape. The new model is in excellent agreement with this experimental finding. The present model reveals whether a protein is a fast or a slow folder and predicts heat as well as cold denaturation. The results of the new model can be compared with MD calculations. Using the molecular parameters derived from DSC at relatively low temperature the entropy at complete unfolding at 498 K was calculated for four different proteins. The results are in excellent agreement with the predictions of the Dynameomics Entropy Dictionary and quasi elastic neutron scattering.

Author contributions

J.S. developed the theory. J.S. and A.S. wrote the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Jörg Enderlein.