Thermodynamics of the Two-State Folding Model

Within this model, the folded state is assumed to be in equilibrium with the unfolded state at each temperature. The equilibrium constant K_u defined in this way yields the standard value of the free enthalpy  of unfolding^4, 32, 55

()

 depends on temperature by

()

where  denotes the enthalpy of unfolding whereas  denotes the change of specific heat during the transition taking place at  . The degree of unfolding α i. e. the molar fraction of unfolded protein is related to the equilibrium constant through

()

and can be expressed by

()

where  is given by eq.(2). The question arises whether the accuracy of the degree of unfolding is good enough to allows the determination not only of  but also of  . In the following we will for brevity refer to the thermodynamic properties as  and  .

Given the validity of the two-state folding model, the experimental data can be evaluated in terms of the degree of unfolding α. We shall first discuss the evaluation of the data obtained by DSF and subsequently the information obtained from DSC. Special emphasis is put on the determination of Δc_p which presents a central piece of thermodynamic information.

Evaluation of DSF-Data

The evaluation of the degree of unfolding α from fluorescence spectroscopy was recently discussed in detail by Zoldak et al.⁵⁶ In principle, α could be obtained from the ratio of the intensities measured at 350 nm and at 330 nm. The rationale behind this method is the observation that the strong dependence of the intensities on temperature (see below) can be mostly removed in this way. However, Zoldak et al. demonstrated that a determination of α solely from the ratio of the intensities measured at 350 nm and 330 nm can lead to considerably errors.^56, 57 Hence, we adopted an interactive procedure which allows for an assessment of the different steps in the analysis:

Within the two-state folding model, the measured intensity F₃₅₀ and F₃₃₀ can be split into⁵⁷

()

()

where F_f,350 and F_u,350 denote the intensities of fluorescence at a wavelength of 350 nm in the folded and the unfolded state, respectively, whereas F_f,330 and F_u,330 refer to the respective quantities measured at a wavelength of 330 nm. Hence, four different intensities F_f,350, F_u,350, F_f,330, and F_u,330 must be determined. These intensities have to be obtained from fits of the measured intensities F₃₅₀ and F₃₃₀. Within the two-state folding model the system will be in the folded and the unfolded state well below or above the transition temperature, respectively. Hence, the temperature dependence of the fluorescence in each state which oftentimes exhibits a pronounced non-linear behavior as discussed extensively by Eftink³⁰ can be determined if the curves have been determined for a sufficiently wide temperature ranges below and above the transition temperature. In case a small range of temperatures is considered, this dependence may be approximated by a linear function.³⁰ However, the intensities F_f(T) in the folded state and of F_u(T) of the unfolded state must be extrapolated over a comparably wide range of the temperature below and above the transition. Therefore, the extrapolation of the fluorescence of the two states into the region in which their concentration is minute becomes a critical factor for the precision of the subsequent analysis. Here we found that in many cases an exponential provides an accurate description of the observed intensities:

()

In some cases, however, a linear fit proves to be a better description of the data which stresses the necessity for an interactive procedure in which the quality of each step is critically assessed. In all cases an excellent fit of F_f(T) and F_u(T) in the entire one-phase regions, respectively, is key for an accurate determination of thermodynamic quantities as this is a mandatory requirement for a meaningful extrapolation.

With a good fit of F_f(T) and F_u(T), the degree of unfolding α can be determined from the data obtained at a single wavelength e. g. 350 nm by:

()

In this way the unfolding can be monitored for the two available wavelength and compared to each other. In addition, from eq.(1) and (2) it follows that the ratio R of the two measured intensity is given by

()

Hence,

()

from which the degree of unfolding can be calculated through eq.(3). Application of eq.(10) has the advantage that data taken at two different wavelengths are evaluated together which minimizes possible statistical errors of the data. On the other hand, solutes like salts or sucrose may shift the fluorescence spectra considerably so that the intensities at 330 nm and at 350 nm refer to different states. In this case, the evaluation must proceed through eq.(8).

As an alternative, a global fit can be done by fitting α to the entire intensity F₃₅₀. Here we combine

()

with eq.(4), (2) and (7) so that all parameters including a and b in eq.(7) are fitted in one global, non-linear least square fit using e. g. a Levenberg-Marquard algorithm (the analysis of the data shown here were done using its implementation in Matlab). It should be kept in mind that this fit has four additional adjustable parameters, namely the parameters a and b for the folded and the unfolded regime, respectively, which may allow for a compensation of errors. A comparison of both ways of evaluation may hence give direct information about possible errors incurred for the various thermodynamic parameters.

Evaluation of DSC-Data

Here we follow mostly the established way of evaluation specified in two important points.^{1, 47, 48, 58, 59} The normalized heat signal C_p can be approximated linearly in the folded state by

()

For the unfolded state we have similarly

()

With these definitions, the background can be obtained as follows

()

The entire C_p(T) now follows as

()

The quantity  defines the net caloric effect of protein unfolding.

From general thermodynamics we have

()

Please note that the enthalpy of the process usually termed van’t Hoff enthalpy (ΔH_vH) are identical to the enthalpy of transition ΔH_u if the two-state folding model is adopted. Hence, in a self-consistent treatment of protein denaturation, both quantities must agree within prescribed limits of error. With eq.(1) we have¹

()

In the course of an analysis of DSC-data, the necessary assumption is that

()

where ΔH_cal is the measured caloric enthalpy of transition derived from experimental data through

()

There is no reason whatsoever that ΔH_cal should agree in all cases with ΔH_u. The caloric enthalpy ΔH_cal contains all effects as e.g the heat of ionization of the buffer whereas ΔH_vH=ΔH_u is precisely defined by eq.(16). Taking together eq.(17) and (18) and expressing the temperature dependence of  by its value at the melting temperature ( ) and the change in heat capacity associated with the process ( ), we obtain for  the following expression:

()

Evidently,  is related to the degree of unfolding α by two well-defined and different enthalpies. Note that  is a constant while  depends on temperature.

Another way of calculating  follows directly from eq.(17): For T=T_m and α=0.5 and we get⁷

()

Evidently, both ways eq.(20) and eq.(21) must come to the same value of ΔH_u.

Evaluation of the Hydrodynamic Radius

Within the frame of the two-step folding model, the overall dimensions of a protein must be characterized by exactly two hydrodynamic radii, namely the hydrodynamic radius  in the folded state and  characterizing the unfolded state. Since the two states are always at equilibrium during the transition, the actually measured hydrodynamic radius R_H must be a linear superposition of both radii weighed by the degree of unfolding:

()

A comparison of the measured R_H with eq.(23) provides another way of checking the assumption of a two-state folding model.

nanoDSF

All measurements have been done using a nanoDSF device Prometheus Panta PNT-00203 (Nanotemper technologies, Germany). For these measurements capillaries were typically filled with 20 μL protein solution with the typical concentration of 10 μM. The measurement of the unfolding and refolding of the protein has been done by heating and cooling between 25 °C and 77 °C using a rate of 0.5 °C per minute. This resulted in 18.18 points/°C. The signal was obtained by setting the power level of the excitation laser (280 nm) to 100 %. The measurements were always done at least with double repetitions.

DSC

DSC experiments were performed using a Nano DSC (TA Instruments) with 0.3 mL capillary cell volume. Samples were equilibrated at 20 °C and then heated to 85 °C with a scan rate of 1 °C per minute and at 3 atm. Samples were degassed for 10 min before loading cells and pressurized at 3 atm to avoid bubble formation.^12, 23

Analysis of the nanoDSF-Data

We first discuss the unfolding of lysozyme in buffer solution at pH 2. Figure 2 displays the intensities of the intrinsic fluorescence measured at 330 nm and at 350 nm as a function of temperature. It should be kept in mind that no fluorescent dye has been added. The accuracy and reproducibility of the intensities is very good despite the fact that all data have been obtained at a protein concentration of just 10 μM. The change of fluorescence upon unfolding is more pronounced for the data obtained at 350 nm as expected for the fluorescence of tryptophan. In principle, the transition temperature T_m could be estimated directly from the point of inflection of the curve in Figure 1a. Alternatively, the ratio F₃₅₀/F₃₃₀ can be plotted against temperature and evaluated. As mentioned above, this procedure may lead to considerable errors as demonstrated by Zoldak and coworkers.⁵⁶ Therefore we fit the intensity of fluorescence as the function of temperature below and above the transition point by eq.(7) allowing to extrapolate the intensities F_f(T) and F_u(T) into the two-phase region. Evidently, the dependence of both F_f(T) and F_u(T) on T is non-linear and linear fits would lead to erroneous results. The dashed lines in Figure 1a and b show the exponential fits (eq.(7)) of the intensities below and above the transition. The fits lead to an accurate determination of F_f(T) and F_u(T) so that both functions can be securely extrapolated into the temperature region in which the transition takes place.

For the evaluation of the degree of unfolding α one may use either eq.(8) or eq.(10) in conjunction with eq.(3). As already discussed above, an evaluation of both sets of data obtained at 350 nm and 330 nm at the same time would be preferable over evaluating the fluorescence data obtained at a single wavelength only. Figure 2 displays the respective data. Both sets agree except for the region in which α>0.95. Here an analysis of K_u as determined by eq.(10) shows that this constant exhibits an appreciable error in this region and may become even negative. This observation points to problems of the evaluation using the results of both wavelength whereas no numerical problems are seen when evaluating the data through eq.(8). Moreover, these difficulties are even aggravated in presence of solutes as NaCl or sucrose which may lead to a considerable shift of the fluorescence spectra. Hence, the data obtained from a single wavelength through eq.(8) are more reliable and used in all subsequent experiments.

In the next step eq.(4) together with eq.(2) can be used for a thermodynamic evaluation of the degree of unfolding α. This fit is done using the MatLab routine cftool and leads to the unfolding temperature T_m, the enthalpy of unfolding  and the change of the specific heat  . The red solid line in Figure 2 marks the resulting theoretical α derived from the fit. Full agreement of the fit with the experimental data is seen over the entire range of temperatures. This observation indicates that the two-step folding model as expressed through eq.(2)–(4) gives a good description of the experimental data at pH 2 as expected.

It should be noted that the determination of F_f(T) and F_u(T) is done independently of this thermodynamic analysis. Hence, the accuracy of the description of both partial intensities can be assessed in each point. As an alternative, the fit of F_f(T) and F_u(T) and the analysis of α could be done in one step by use of eq.(11). Such a procedure has been applied frequently to experimental data obtained by UV/VIS or fluorescence.⁴¹ However, this evaluation may mask inconsistencies in the fits of F_f(T) and F_u(T) and the resulting parameters should be compared in all cases with the results of the step-wise fit. Figure 1a displays such a global fit to the data shown in Figure 1a. Good agreement is found and a quantitative description is possible for all sets of data under consideration here. From the fit of the degree of unfolding α (see red solid line in Figure 1a) we obtain T_m=327.6 K,  =313 kJ/mol, and Δc_p=10 kJ/(K mol). From the global fit (cf. Figure 3) we get T_m=327.5,  =307 kJ/mol, and Δc_p=13 kJ/(K mol). The differences between the thermodynamic results from both methods of evaluation can give a good assessment of the limits of error of the thermodynamic parameters: In general, the transition temperature T_m can be determined with excellent accuracy (±0.2 K). The temperature of the unfolding transition is hence the most secure parameter to be obtained from the experiment. The enthalpy of transition  is afflicted by a considerably larger error of the order of ±15 kJ/mol (5–10 %) which needs to be discussed for each set of data (see below). The change of specific heat Δc_p, however, is afflicted by an error of more than 30 %.

Reversibility

The foregoing thermodynamic analysis requires reversibility in the range of temperatures in which α is raising from 0 to 1. This important point can be checked conveniently by monitoring the intensity of fluorescence during a cooling run. Figure 3a,b display two typical examples for this analysis. Figure 3a shows the signal measured during heating in blue and the respective signal upon cooling (red) when the sample was heated to just above the transition only. Full reversibility is seen. Figure 3b, on the other hand, shows the runs if the sample was heated to a higher temperature above the transition point. Here small changes are seen but the fluorescence signal is largely recovered. If the sample is heated to 363 K, however, no reversibility is seen and the signal of the cooling curve differs strongly from the signal of the heating curve (data not shown). Heating of the protein solution to temperatures near the boiling point of water obviously destroys a part of the lysozyme structure by e. g. partial hydrolysis so that no refolding results upon cooling. A similar analysis done for pH 2.8, 4, 5, and 6 is shown in Figures S1–S4 of the supplementary information. It demonstrates that the transition is reversible except for pH 6. This finding is in full agreement with literature as shown in the extensive discussion by Eftink³⁰ and by Blumlein and McManus.⁶⁰

Dependence on Protein Concentration in DSF

The obvious advantage of nanoDSF as measured by the Panta-device is ability to investigate proteins at low concentrations as compared to the protein concentrations necessary for a DSC-experiment. While 10 μM protein concentration is sufficient to obtain high quality data, it is important to analyze to which extent the resulting thermodynamic parameters are depending on this parameter. To this end, we have performed nanoDSF measurements at different concentrations and comparative DSC experiments to elucidate the comparability of the two methods in more detail.

In principle, F₃₅₀/c_protein measured for different concentrations of protein should overlap if there is no influence of this parameter. The plot of F₃₅₀/c_protein vs. T shown in Figure 4, however, demonstrates that there is a strong influence of c_protein on the measured fluorescence below the unfolding transition. However, the data taken above T_m virtually agree. While it is possible to describe the temperature dependent decay of the fluorescence at 350 nm (F₃₅₀) by an exponential for small protein concentrations significant deviations are observed at 80 μM leading to an almost a linear dependence for c_protein=100 μM below the transition. This finding already demonstrates that raising the concentration of lysozyme leads to profound changes in the interaction of the protein.

The data was analyzed by fitting F₃₅₀ below and above the transition using exponentials as described above for all but the curve for c_protein=100 μM. Please note that the data obtained at 80 μM exhibits a clear change in curvature around 316 K which renders a reasonable fitting of this data set within the given model impossible. The data obtained for c_protein=100 μM cannot be described using an exponential decay, instead F_f,350 was fitted using a linear approximation. Table 1 gather the respective thermodynamic parameters. The strong dependence of F₃₅₀ below the transition is reflected in a marked increase of  with protein concentration.

Table 1. Thermodynamic parameters for the unfolding of lysozyme.

Method	cp /μM	pH	Tm /K	ΔHu /kJ/mol	Δcp /kJ/(K mol)	ΔHcal /kJ/mol	ΔHu/ΔHcal
DSF	10	2	327.6 0.2	313 20	5	–	–
DSF	30	2	327.00.2	339 16	11±4	–	–
DSF	50	2	327.0±0.2	339±20	(14±6)	–	–
DSF	100	2	328.4±0.2	424±15	(10.2±4)	–	–
DSF	10	2.8	339.7 0.4	393 21	14.22.2	–	–
DSF	10	4	348.3 0.5	463 57	15.83.3	–	–
DSF	10	5	348.10.1	467 18	15.91.5	–	–
DSF	10	6	345.60.3	420 27	9.8 14.7	–	–
DSC	69	2	325.7±0.2	375 ±10	2.78±0.5	344.8±5	1.09
DSC	173	2	325.7±0.2	400±10	4.46±0.5	304.3±5	1.31

Since the protein concentrations are rather high, the question arises whether the measurements of the intensity are disturbed by an inner filter effect.⁶¹ However, the extinction coefficient of lysozyme is ~36000 which at a concentration of 100 micromolar for a capillary tube having an ID of 0.5 mm translates into an optical density of less than 0.2 at the highest concentrations studied. This leads to a small contribution of the inner filter effect at the concentrations studied. Also, because the optical density of the folded and unfolded states of the protein are essentially the same, any contribution of the inner filter effect would manifest as a constant baseline effect throughout the experiment, which has been corrected for.

Analysis of the DSC-Data

For a comparison with the nanoDSF results discussed above, DSC measurements were conducted at c_protein=69 μM (0.964 mg/ml) and 173 μM (2.418 mg/ml)) as shown in Figure 5a and b, respectively. Here the heat capacity below and above the transition is approximated linearly according to eq.(12) and (13). The red and green dashed lines show the fit below (C_p,u) and above (C_p,f) the transition, respectively. Evidently, the linear approximation leads to a good excellent fit of the data. In principle, this fit could be done together with the fit of the entire C_p(T).⁵⁹ The present procedure, however, ensures that the underlying assumptions and possible errors of the fit can be assessed separately at each stage. The change of the specific heat Δc_p can directly been read off from the difference of the heat capacities at T_m:⁷

()

This determination is afflicted by an appreciable error. The values Δc_p=2.78 kJ/(K mol) derived from the data displayed in Figure 5a, and Δc_p=4.76 kJ/(K mol) from the data in Figure 5b, however, are good enough for the subsequent evaluation of the data.

For the evaluation of the DSC-data we developed an iterative procedure which will be exemplified for the data shown in Figure 5a: For the calculation of ΔH_cal according to eq.(10)  must be evaluated first through eq.(15) which requires the subtraction of the background signal C_bg. The evaluation of this quantity according to eq.(14), however, requires the degree of unfolding α which is not available at this stage. A trial value of ΔH_cal can be obtained through subtraction of C_p,f(T) from C_p for T ≤ T_m. For T > T_m, C_p,u(T) is subtracted from the measured heat capacity C_p. Hence, we obtained trial values for  that yield ΔH_cal=344.5 kJ/mol upon numerical integration. In the next step, ΔH_u is determined by use of eq.(21).  can be approximated by the mean value of the trial values of this quantity obtained at step 2. Here we obtain ΔH_u=389.6 kJ/mol and T_m=325.7 K. The latter value is taken as the improved transition temperature and used in all subsequent steps.

Given T_m, ΔH_u and Δc_p, the free energy ΔG_u can be calculated according to eq.(2). This allows us to calculate C_bg according to eq.(14). The pink line in Figures 1 shows this new background line. Subsequently, C_bg can be subtracted from C_p to yield an improved value of C_p^ex (eq.(15)). Numerical integration of the improved C_p^ex leads to ΔH_cal=344.8 kJ/mol which agrees very well with the value determined in step 2. Hence, ΔH_cal=344.8 kJ/mol is used in all subsequent calculations, no further iteration is done. C_p^ex shown in Figure 2 can now be fitted by eq.(20). Here ΔH_u is treated as the only adjustable parameter. The red line in Figure 2 shows the resulting fit which leads to ΔH_u=375.2 kJ/mol. Application of eq.(22) yields 374.9 kJ/mol so that in the following a value ΔH_u=375 kJ/mol is used. Since these data indicate self-consistency, no further iteration must be done. Table 1 summarizes the results obtained for both protein concentrations.

A comparison of the data gathered in Table 1 shows that the transition temperatures measured by DSC are slightly lower than the ones measured by nanoDSF. Moreover, the enthalpies determined by both methods demonstrate clearly that higher protein concentrations are followed by higher enthalpies of unfolding. This finding is in accord with results of Kitamura and Sturtevant⁴⁷ who studied systematically the dependence of thermodynamic parameters of the unfolding of lysozyme of the T4-bacteriophage on protein concentration (cf. Table 1 of ref.⁴⁷). With respect to Δc_p the DSC results show a significant increase with increasing concentration. This is not mirrored in the nanoDSF results. However, as already discussed above, this parameter is afflicted by a considerable error. In particular, its determination based on fits of F₃₅₀ is sensitive to the quality of the description in the one-state regions which is worse for the data set at c_protein=100 μM.

It is interesting to note that the ratio of ΔH_u/ΔH_cal=1.09 obtained from DSC-measurements at c_protein=69 μM is in a range expected for the two-state folding model of protein unfolding.^6-8 A much higher value results for the highest concentration under consideration here (see Table 1). Thus, a value of ΔH_u/ΔH_cal=1.31 is already outside the range in which the two-state folding model can be applied safely.⁸ Taking all these results together, it becomes obvious that a higher protein concentration leads to an increase of the unfolding enthalpy while the caloric enthalpy ΔH_cal is lowered significantly.

A possible reason for this result may be due to association of lysozyme below the temperature of the transition. Since a long time it is known that lysozyme forms small clusters in the unfolded state in aqueous solution due to a balance of short-range attractive and long-range repulsive forces.^62, 63 A comprehensive review of this work has been given by Stradner and Schurtenberger.⁶⁴ Hence, formation of clusters is followed by a higher ΔH_u but a lower caloric enthalpy. In addition to this, the parameter ΔH_U/ΔH_cal deviates from unity. This finding indicates that the unfolding of lysozyme may not be described safely in terms of a two-state process anymore. Figure 4, on the other hand, clearly indicates that cluster formation plays no significant role anymore above the transition temperature.

The Panta device also measures dynamic light scattering simultaneously which allows to extract the temperature dependence of the hydrodynamic radius R_H. This set of data allows to monitor the overall dimensions during the unfolding transition. Attempts to measure R_H at low concentrations as 10 μM failed. For these conditions the scatter of the data was too high which is typically found if the signal of a weakly scattering sample is obscured by the presence of small amounts of strongly scattering objects such as dust particles. A meaningful signal could only be obtained for a protein concentration of 100 μM. In this concentration range, however, the effect of association may make itself already felt and the analysis of these data must proceed with caution.

Figure S5 of the SI displays the raw data that still exhibit a strong scatter of the data. To compare these data with theory, outliers of R_H, that is, values of R_H tenfold larger than the average were removed first. Then a Savitzky-Golay filtering was applied to the data (2^nd order polynomial, ±5 points). The resulting R_H as the function of temperature is shown in Figure 6. The scatter of the data is still quite high but the transition is clearly visible. The solid line shows the fit of these data according to eq.(22) where R_H,f and R_H,u denote the hydrodynamic radius of the folded and the unfolded state of lysozyme, respectively. The degree of unfolding was calculated using eq.(4) using the parameters obtained for a protein concentration of 100 μM (see Table 1). The radii R_H,f and R_H,u have been treated as fit parameters yielding R_H,f =1.75 nm and R_H,u=1.91 nm. Full agreement of theory and experiment is seen which further underscores the validity of the two-state folding model.

The hydrodynamic radii obtained herein compare favorably with the radii of gyration R_g determined by Hirai et al. analyzing small-angle x-ray scattering data.⁶⁵ These authors studied the unfolding of lysozyme by measuring R_g as the function of temperature and pH. For a pH of 1.2 they found a radius of gyration of ca. 1.6 nm whereas a value of 1.9 nm was deduced for the unfolded state at 80 °C. Kamatari et al. found a radius of gyration in the native state of 1.57 nm in their study, also done by small-angle X-ray scattering.⁶⁶ Since the hydrodynamic radius for a compact object should be larger than its radius of gyration, full agreement with the present data can be seen for the unfolded state. Unfolding of lysozyme in presence of methanol led to a radius of 2.87 nm as found by Kamatari and coworkers.⁶⁷ However, Katamari et al. demonstrated that denaturation by alcohols lead to an extended helical structure which is larger than a globular conformation. The rather small value for the dimensions in the unfolded state of 1.9 nm found here and by Hirai et al.⁶⁵ shows that the unfolded protein assumes a rather dense structure which is in general agreement with early theoretical deductions^68, 69 and a more recent analysis.⁷⁰

Dependence on pH

In literature thermodynamic properties of the folding/unfolding transition of lysozyme were also deduced from pH-dependent measurements (see the discussion e. g. in ref.^{47, 48, 71, 72}). In this way the transition enthalpy was obtained for different temperatures which can be used as an alternative approach to determine the change of Δc_p.⁴⁴ Moreover, the dependence of T_m on pH gives information about the number of exchanged protons during the transition. In this way the dependence of T_m and  gives highly valuable information on the unfolding transition. It is thus interesting to compare these results with pH-dependent nanoDSF measurements. The latter were performed at pH 2 and 2.8 using a glycine buffer whereas the pH range 4–6 was adjusted by a citrate buffer. All buffers had a total salt concentration of 50 mM. Based on the discussion presented above (see the discussion of Figure 3 above) a reliable extraction of thermodynamic data was possible up to a pH of 5. The respective intensities F₃₅₀ measured at different pH are gathered in Figures S6–S10.

Figure 7a displays the measured transition temperatures as the function of pH whereas Figure 7b shows the respective transition enthalpies as the function of T_m. Table 1 gathers all data derived from this analysis in the present work. The literature data in Figure 7a have been obtained by DSC,^{1, 7, 43, 48} and by US/VIS-spectroscopy.^53, 54 The agreement of data in general is very good at low pH whereas differences are seen beyond a pH of 4. In this region of higher pH-values, however, the transition is only partially reversible which may impede the entire thermodynamic analysis. In general, the survey shown in Figure 7a suggests that T_m is a rather robust quantity that may be obtained by quite different methods in a secure fashion. This finding is in full agreement with the findings discussed above.

Figure 7a shows that T_m seems to go through a shallow maximum which is located around a pH of 4. Such a maximum of the transition temperature as the function of pH is a very common feature observed for many proteins.⁶⁰ Often it is related to the pH of the environment in which the protein or enzyme is working. The dependence of T_m gives direct information on the number of protons Δν released or taken up during the unfolding transition according to

()

The dependence of T_m between pH 2–4 is indicated by a dashed line in Figure 7a and leads to a value of Δν ∼ 2 for this pH-range, i. e., two protons are taken up during unfolding. This value agrees approximately with data obtained earlier on different mutants of lysozyme by the Sturtevant group.^72, 73

Figure 7b displays the corresponding transition enthalpies obtained from the analysis within the two-state folding model. Apart from the nanoDSF and DSC results presented above, results from literature are also included. The latter can be grouped in two sets which have been obtained by DSC and spectroscopic methods (UV/Vis or fluorescence). In the former case we have direct measurements of  whereas the second set contains data derived from the application of eq.(2). The enthalpies obtained here by fluorescence spectroscopy (filled blue circles) agree quite well with the ones obtained by UV/Vis-spectroscopy (triangles up and down; ref.^53, 54). The data obtained by DSC here agree well with the corresponding DSC-data from literature.^{1, 7, 43, 48} Figure 7b demonstrates that data obtained by DSC are in general systematically higher than the ones obtained by spectroscopic methods. This is in full agreement with the DSC-data obtained here (see the blue diamonds in Figure 7b). As already discussed above, this finding can be traced back to the higher protein concentrations necessary for a typical DSC-experiment which is followed by the formation of clusters of native lysozyme.

The observation of  for different pH has been the classical method to obtain the change of specific heat Δc_p.^7, 58 The enthalpy of transition raises with the pH of the solution (cf. Figure 7b) and the increase of  is traced back to Δc_p. The dashed line in Figure 7b shows the analysis of the enthalpies derived from nanoDSF. A linear fit to  as the function of T_m has a slope of 7.1 kJ/(K mol) which agrees quite well with the value for Δc_p=6.5 kJ/(K mol) found by Velicelebi and Sturtevant⁷ in the course of DSC-experiments. It shows that nanoDSF leads to accurate values for the unfolding enthalpy. The value of Δc_p thus derived is considerably smaller, however, than the value found here by analyzing α(T) (Table 1). The reason for this obvious discrepancy is not yet clear. It should be kept in mind that the values of  plotted in Figure 7b refer to widely different states because they were obtained different pH. A direct determination from the fit of the degree of unfolding α for a given pH may thus appear as the more reliable procedure. From this procedure we get a value of Δc_p ~10 kJ/(K mol) which is considerably larger than the value found from Figure 7b. Evidently, Δc_p determined from α(T) is afflicted by a large error but the discrepancy seen here seems to be outside of this error. Hence, this problem is certainly worth reconsidering.