Denaturation of proteins: electrostatic effects vs. hydration

The unfolding transition of proteins in aqueous solution containing various salts or uncharged solutes is a classical subject of biophysics. In many cases, this transition is a well-defined two-stage equilibrium process which can be described by a free energy of transition ΔGu and a transition temperature Tm. For a long time, it has been known that solutes can change Tm profoundly. Here we present a phenomenological model that describes the change of Tm with the solute concentration cs in terms of two effects: (i) the change of the number of correlated counterions Δnci and (ii) the change of hydration expressed through the parameter Δw and its dependence on temperature expressed through the parameter dΔcp/dcs. Proteins always carry charges and Δnci describes the uptake or release of counterions during the transition. Likewise, the parameter Δw measures the uptake or release of water during the transition. The transition takes place in a reservoir with a given salt concentration cs that defines also the activity of water. The parameter Δnci is a measure for the gain or loss of free energy because of the release or uptake of ions and is related to purely entropic effects that scale with ln cs. Δw describes the effect on ΔGu through the loss or uptake of water molecules and contains enthalpic as well as entropic effects that scale with cs. It is related to the enthalpy of transition ΔHu through a Maxwell relation: the dependence of ΔHu on cs is proportional to the dependence of Δw on temperature. While ionic effects embodied in Δnci are independent of the kind of salt, the hydration effects described through Δw are directly related to Hofmeister effects of the various salt ions. A comparison with literature data underscores the general validity of the model.


Introduction
The denaturation of proteins by a globule to coil transition is a classical subject of biophysics. 1 The thermal denaturation in which the protein goes from natural folded state to a random coil in aqueous solution occurs with raising temperature. Cold denaturation, 2 which has been known for a long time, is the transition to denatured state taking place with decreasing temperature. It is well-established that for many proteins chain denaturation is a two state transition [3][4][5][6] in which the globular and the denatured form of the protein are well-dened thermodynamic states in equilibrium with each other. Hence, an equilibrium constant K u can be dened between the globular and denatured state which allows us to treat the denaturation as a fully thermodynamic problem relating the melting temperature T m to the transition enthalpy DH u and the transition entropy DS u . 7 A fundamental problem in the eld is the change T m of a given protein with solutes in the aqueous phase. Up to now, there have been an enormous number of experimental studies that started out in the sixties of the last century. 8 There are many investigations that study the change of T m in the presence of various salts and non-charged solutes which can stabilize or destabilize the globular state. 4,6,7,[9][10][11][12][13][14][15][16][17][18][19][20] This effect is of obvious biological importance and can be traced back to hydration effects embodied in the Hofmeister series. [21][22][23][24] The collapse transition of poly(n-isopropylacrylamide) (PNIPAM) in aqueous solution is another well-studied and fundamental problem where a coiled polymer undergoes a transition from the coiled to the globular state with raising temperature. Here too there is a large number of fundamental and detailed studies on this transition in solutions of various ions. 23,[25][26][27][28][29][30] Taken together the folding/unfolding transition of proteins and polymers in general is problem of fundamental importance.
Early studies of protein denaturation clearly revealed the central role of charge-charge interaction. 1 The unfolding of the globular protein exposes charged groups to water and this interaction leads to an important contribution to the free energy of unfolding that scales with the logarithm of the salt concentration in solution. 1 This term is due to the release or uptake of ions during unfolding and play an important role both for unfolding of proteins as well as for denaturing of DNA in presence of various salts (see the discussion in ref. 31). A similar process takes place when polyelectrolytes form a complex with a protein (counterion release force; see the discussions in ref. 1 and 32-34 and further citations given there). Here a wealth of experimental data demonstrates that this effect is purely entropic and therefore independent of temperature. 32,[35][36][37] The unfolding of proteins also exposes hydrophobic amino acids to water. As mentioned above, hydration therefore plays an important role which has been the subject of exhaustive investigations by Record and coworkers in the frame of the solute partitioning model (SPM). 22,31,[38][39][40] This model treats the partitioning of the solute ions or solutes between the hydrate and the bulk water. Kosmotropic ions are depleted from the hydrate water whereas chaotropic ions are enriched in this phase. Moreover, these investigations have clearly revealed that effects due to the partitioning of solutes scale linearly with salt concentration which is in full agreement with the analysis by Schellman using Kirkwood-Buff integrals. 4,5 Thus, for many kosmotropic salts in the Hofmeister series, a linear relation between the free energy and salt concentration is found ("mvalue"; see the discussion in ref. 29). In many cases the m-value is found to be independent of temperature. Based on these considerations, Chen and Schellman developed a phenomenological model that is based on a m-values that do not depend on temperature 6,41 ("linear model"; cf. ref. 18). A fact overlooked in later expositions of this theory is the linear dependence of the specic heat Dc p on salt concentration. Chen and Schellman could demonstrate that this dependence is a direct consequence of the assumption of a constant m-value. 6 The notion of a m-value independent of temperature, however, is a stringent condition that may not be fullled for a given system. 42 Hence, a general model should avoid this prerequisite.
Surveying the literature on denaturation of proteins, it becomes clear that exchange of water and counterions during unfolding present two important factors that determines the stability of proteins in aqueous solution to a large extend. Both are modied by the added solute. Hence, a quantitative treatment of the effect of ions and water is a necessary prerequisite for a quantitative evaluation of data related to the unfolding of proteins in presence of various solutes. In a recent paper we have presented a unied approach for the free energy of complex formation between proteins and polyelectrolytes that comprises both effects. 34 Temperature T and salt concentration c s were identied as the decisive variables and a closed expression for the free energy DG b (T,c s ) of complex formation could be derived. In this model counterion release was characterized by Dn ci denoting the net number of released ions during binding whereas hydration was described in terms of the parameter Dw dened already in early expositions of the problem 1,43,44 and used frequently to describe the effect of hydration on complex formation. [44][45][46][47][48][49] Central for the development of this model is the fact that mixed derivatives of the binding enthalpy DH b (T,c s ) with regard to T and c s must be the same. Hence, this Maxwell-relation leads to prediction that the dependence of DH b (T,c s ) on c s gives directly the dependence of Dw on temperature. The model thus derived is capable of describing the weak dependence of DG b (T,c s ) on temperature which in turn leads to a strong compensation of enthalpy and entropy. 34 Moreover, the values obtained for Dn ci and Dw obtained by the present model for the denaturation of a given protein can directly be compared to data deriving from studies of complex formation of polyelectrolytes with proteins. 33,46,47,50,51 Based on this model we here present a phenomenological approach to unfolding transitions of proteins that are partially charged. A closed expression for the free energy of unfolding will be presented that contains both the effect of electrostatics as well as of hydration. The consequences of the model for data evaluation will be discussed and exemplied using recent experimental data. 16,18 The entire discussion presented here aims at a systematic analysis of experimental data obtained on polymeric unfolding transitions of various systems in aqueous phase.

Theory
We consider the transition of a single chain of a polypeptide from a folded to an unfolded state in sufficiently dilute solution.
In each stage of this transition the unfolded state is in equilibrium with the still folded part of the chain. This two-state mechanism is well-established for a great number of systems (see the discussion of this point in ref. 1, 16 and 18). Experimentally, the unfolding transition can be monitored e.g. by measurements of the circular dichroism leading to a fraction a of unfolded protein. The equilibrium constant K u for the process of unfolding is related to a by whereas the free energy of unfolding DG u is related to K u through The basic thermodynamic analysis DG u was already discussed a long time ago by Record, Anderson, and Lohman. 1 In general, the change of the equilibrium constant K u with the activity a AE of an added salt is given by where Dn ci denotes the total number of released or taken-up ions during the process of unfolding. The parameter Dw treats the release or uptake of water in the course of the unfolding transition while p ¼ 2 for monovalent salt with molality m. By denition, Dw is independent of salt concentration. The factor 55.6 is the molality of water and the parameters g f and g u are the activity coefficients of the chain in the folded and unfolded state, respectively. Note that this equation with necessary adaptions has been the basis of our recent discussion of complex formation of polyelectrolytes with proteins. 34 In the following, the same approximations will be made: (i) the change of the activity coefficients g f and g u with the activity a AE of the added salt give a small but non-negligible contribution of the term Dn ci (see the discussion in ref. 1), (ii) the mean activity coefficient of the salt ions will be set to unity, and (iii) the molality m of the salt will be equated to its concentration c s . With these approximations the justication of which will be discussed below eqn (3) becomes Hence, the salt concentration c s is the variable on which the subsequent thermodynamic analysis is based. With the standard thermodynamic relation (vln K b /vT) c s ¼ DH b /RT 2 we obtain the differential of ln K u for monovalent ions where DH u denotes the enthalpy change at the unfolding transition and T m the respective temperature of unfolding. Thus, in the following the unfolding transition will be treated as the function of the two decisive variables, namely temperature T m and salt concentration c s . There is abundant experimental evidence that the parameter Dn ci is independent of temperature. 32,34,35,37,[52][53][54] It is therefore safe to disregard the dependence of this parameter on T m . With this assumption and vln K u vc s Dw (6) we obtain the Maxwell-relation 34 This relation demonstrates that the salt dependence of transition enthalpy is directly related to the dependence of the parameter Dw on temperature. As already lined out previously, 34 this relation can now be used to calculate Dw as the function of temperature. In general, the transition enthalpy DH u as the function of the melting temperature T m and c s can be rendered as 34 Here, the quantity Dc p,0 denotes the change of the specic heat in absence of added salt whereas the coefficient dc p /dc s describes the change of the specic heat with salt or solute concentration. 34 T 0 m denotes the melting temperature for saltfree solutions. Together with eqn (7), this relations leads to Integration leads to 34 where the quantity Dw(T 0 m ) denotes the magnitude of Dw at T 0 m in salt-free solution.
As already discussed previously, 34 Dw can be interpreted in terms of the solute partitioning model as follows. Both the polyelectrolyte as well as the protein are hydrated in aqueous solution. During the unfolding a certain number Dn w of water molecules of both reactants is taken up or released. Furthermore, it is assumed that there is a partitioning of the ions between the bulk solution and the hydration water on the surface of the protein described by the partition coefficient K p,+ ¼ (m loc + /m bulk + ) for the cations where m loc + denotes the molality of the cations in the hydrated shell whereas m bulk + is the respective quantity in bulk. The partition coefficient K p,À of the anions is dened in the same way. With these denitions, Dw can be rendered by 34 Dw y 1 2 Evidently, the quantity Dw measures the effect of water release on the free energy of unfolding and should not be confused with the total number Dn w taken up or released during unfolding. For an equal distribution of the ions between the hydrate and the bulk phase, this contribution will vanish.
In the following, we rst consider uncharged systems, that is, Dn ci ¼ 0. Integration of eqn (6) at constant temperature then leads to where K 0 u is the equilibrium constant in salt-free solution. Therefore Here, DG 0 u denotes the free energy of unfolding at c s ¼ 0. Hence, the dependence of DG u on c s can be written down as In many cases the difference T m À T 0 m does not exceed 10 degrees so that the last term in eqn (14) can be expanded to yield (see the derivation of eqn (11) of ref. 55) Eqn (14) may be used to calculate the m-value dened as the derivative of the free energy with regard to solute concentration at constant temperature This expression shows that m is given by a constant plus a term that depends quadratically on T m À T 0 m . For small temperature differences the second term will be small and the m-value is a constant in good approximation. However, it should be noted that m is in general a quantity that depends explicitly on temperature.
Eqn (14) and (15) contain only the dependence of the free energy on c s . The quantity DG 0 u for salt-or solute-free solutions can be derived following the prescription of Chen and Schellman: 6 the specic heat Dc p,0 measured in solute-free systems can be regarded as a constant throughout the rather small temperature range under consideration here. Thus, for solutefree systems we obtain and which gives Combination with eqn (14) then leads to For T m À T 0 m # 10 K this expression can be approximated by Eqn (21) and (22) are the nal result for the free energy of unfolding for uncharged systems.
For partially charged proteins eqn (5) shows that a term scaling with ln c s must be added to eqn (21). 1 Here it must be kept in mind that there is always a small but nite salt concentration c s,0 so that the integration of eqn (5) must start at this concentration. Keeping in mind that Dn ci does not depend on temperature we immediately obtain from eqn (22) In many cases the concentration c 0 s is small and can be disregarded in eqn (23) except for the last term, of course. Eqn (23) also shows that for small concentrations c 0 s the free energy of unfolding may contain an appreciable contribution originating from the release or uptake of ions during denaturation. Hence, DG u will be dominated by the last term for small c s . The respective transition enthalpy is given by eqn (8) where c s is replaced by c s À c s,0 . The transition entropy follows as In many cases it is only possible to deduct the change of the free energy of unfolding with increasing solute concentration. Thus, we require the quantity DDG u which gives the change of DG u with c s calculated for the transition temperature T 0 m in solute-free solution: It is interesting to compare eqn (21) and (23) to phenomenological approach of Chen and Schellman 6 (cf. also ref. 41). The generalized van't Hoff equation used by these authors is based on eqn (17)- (19). Moreover, the dependence of the free energy of unfolding is assumed to be linear in c s as derived above in eqn (13): Thus, the coefficient Db 23 is identical to 0.036Dw in eqn (13). In the linear model of Chen and Schellman, 6 this linear dependence has be deduced from experiments whereas the above considerations leading to eqn (13) demonstrate that this relation is a direct consequence of eqn (1). Based on these premises Chen and Schellman formulate ln K u as follows in the present notation as: 6 where all thermodynamic quantities DH u , DS u , are explicit function of solute concentration and temperature whereas Dc p is only a function of c s . All parameters will be treated as adjustable parameters for each c s in a comparison with experimental data. The present approach, on the other hand, reveals the interrelation between the various quantities and the concentration of solute which is based on the Maxwell-relation eqn (7). The experimental data are described in terms of 3 adjustable parameters: (i) Dw(T 0 m ) which is closely related to the classical mvalue through eqn (15); (ii) the specic heat Dc p,0 in absence of solutes; and (iii) the parameter dc p /dc s describing the dependence of Dc p on c s . This parameter has been introduced by Chen and Schellman as well (the parameter Dc p23 in eqn (8) and (9) of ref. 6) but not used further. Its application to complex formation of polyelectrolytes with proteins has been discussed recently. 34 The rst two parameters are directly measurable and have an obvious physical meaning. The newly introduced parameter dc p /dc s describes the dependence of hydration effects on temperature.
A comprehensive phenomenological analysis of the denaturation temperature for uncharged polymers was presented some time ago by Heyda and Dzubiella. 29 Here, the hydration effects are described in terms of the preferential interaction parameter DG 23 . If this parameter does not depend on c s , it follows directly that where DG 0 23 is dened as the preferential interaction parameter independent of c s . The analysis of the changes effected by kosmotropic salts showed indeed that this equation provides a very good approximation of the experimental data obtained for the collapse transition of PNIPAM-chains in aqueous solution. 26,29 Thus, these data could be compared directly to the prediction of the SPM with moderate success (cf.    (16)) becomes independent of temperature and the K p,AE as dened through eqn (11) are much smaller than unity. If, on the other hand, the K p,AE are approximately unity, the m-value will be small but exhibit a considerable dependence on temperature (cf. eqn (11)). In this situation the dependence of the free energy of unfolding should depend quadratically on DT m which has been found previously for the complex formation of polyelectrolytes with proteins. 34 It should be kept in mind, that these considerations disregard the counterion release term in eqn (23). The m-value observed for charged systems where Dn ci s 0, will differ considerably and the predictions of the SPM are related only to the parameter Dw as dened through eqn (10).
In principle, eqn (23) and eqn (26) dene stability curves as dened by Becktel and Schellman 3 inasmuch as they describe the free energy DG u as the function of temperature and salt concentration. If Dc p,0 may be regarded as constant throughout a temperature range of sufficient width, the present approach could be used to construct DG u (T,c s ) for all pertinent temperatures ranging from cold to thermal denaturation. Given the fact, however, that Dc p,0 depends on temperature, 7 such stability curves should be regarded with caution.

Results and discussion
Basic predictions of the model The basis of the present model is eqn (5) which is general except for the neglect of the activity coefficients of the solute. Previous discussions, however, have shown that this approximation is inconsequential and will only change slightly the resulting parameters. 4,33,34 Eqn (5) or its integrated from has been used very oen to analyze the release of water upon complex formation of highly charged macromolecules as e.g. DNA with various proteins. 33,46,47,50,51 It is thus interesting to compare its magnitude for complex formation with values deriving from protein unfolding. Evidently, the parameter Dw introduced by this equation does not give the number of released water molecules dened as Dn w but measures the thermodynamic effect of this release (see the discussion of eqn (11) above). 34 A next prerequisite is the independence of Dn ci on temperature. As already mentioned above, this fact is well-borne out of a large bulk of experimental data and can safely be assumed here as well (see e.g. the discussion by Privalov et al. 32 and in ref. 34, 37 and 52-54). This fact allows us to use the Maxwellrelation eqn (7) for the next step in which the salt dependence of the unfolding enthalpy DH u is related to the dependence of the parameterDw on temperature given through eqn (9). Hence, if DH u turns out to depend on the concentration c s of the solute, it necessarily follows that Dw is not a constant but depends on temperature. This fact is one of the central points inasmuch it shows that in this case the m-value given here by eqn (16) contains a term depending quadratically on the difference T m À T 0 m . The above model hence makes the following predictions that can compared directly to experiments: (1) In a rst step of the analysis of experimental data, dependence of DH u on salt concentration c s can be checked. Eqn (8) demonstrates that this quantity is a function of temperature and salt concentration c s . Moreover, the dependence of DH u on salt concentration c s gives the dependence of the quantity Dw on temperature as shown by the Maxwellrelation in eqn (7) which in turn leads to the dependence of the m-value on temperature eqn (16). Evidently, if DH u is found to depend on salt concentration, there must be a nite dependence of m on temperature as well (eqn (16)). If, on the other hand, the dependence of DH u on salt concentration c s is small, the parameter dc p /dc s y 0 and the terms in eqn (23) and (25) depend only on T m , that is, the quadratic term can be dismissed. Hence, the evaluation of experimental data can begin by a critical check of DH u (T,c s ).
(2) The term scaling with ln c s will profoundly change the dependence of the free energy on salt concentration and this dependence will be most marked for small c s (cf. eqn (6)). The dependence of T m on c s will therefore be non-linear at small c s if Dn ci assumes a nite value. Since the effect embodied in this parameter is of entirely entropic origin, the non-linear dependence on c s thus effected should be independent of the nature of the added salt of same valency, that is, T m should be a universal function of c s for small c s . Hofmeister effects are expected to come into play only for higher salt concentrations where DDG u scales linearly with c s . Hence, T m is expected to be independent on the nature of the salt ions if the salt concentration is small. The observation of this effect, however, requires a small c s,0 and precise measurements at concentrations only slightly larger than c s,0 . Evidently, the ionic effect embodied in Dn ci and the change of T m by hydration may cancel each other. Thus, if Dn ci < 0 as well as Dw < 0, eqn (23) demonstrates that can lead to DT m ¼ 0 for a nite salt concentration. This problem has already been discussed by Chudoba et al. 30 and is seen directly in the study of the unfolding of RNase A. 16 Similar observations have also been made for thermophilic proteins. 56,57 The present theory allows us to model this effect in terms of the parameters Dn ci and Dw.
(3) If the term quadratic in eqn (23) and (25) can be disregarded, that is, for small DT, the combination of both expressions shows that in this case which predicts that DDG u /DH 0 u z DT/T m should be an universal function. Hence, for small DT, this equation may be used to check the internal consistency of data (cf. the discussion of this point by Senske et al. 16 ).

Evaluation of data
Thus, the evaluation of the experimental data should proceed in the following steps: The unfolding is usually determined by microcalorimetric studies in which the heat change during this process is measured precisely. These measurements yield the heat of transition DH u (c s ) at different concentrations of the solute c s and the melting temperature T m at the respective salt concentration c s (see e.g. the discussion in ref. 15 and 18). In the following, the comprehensive set of data of Francisco et al. 18 on the unfolding of ribonuclease A in presence of sodium salts will be used to exemplify the steps of evaluation. Here, the unfolding of RNase A has been observed at a pH of 4 in 10 mM acetate buffer. Therefore, the concentration c s,0 ¼ 0.01 M in the subsequent analysis.
As outlined above, the analysis may start by the check of the dependence of DH u on c s (see Table 1 of ref. 18). Fig. 1a displays DH u (c s ) for a typical kosmotropic salt as NaCl as well as for NaSCN which provides a good example for a chaotropic system. The enthalpy of denaturation in presence of NaCl hardly depends on salt concentration whereas a marked dependence is found for NaSCN. This test splits up the experimental data sets into two classes: (1) Small DT m ; kosmotropic ions: the small dependence of DH u on c s suggests that the coefficient dDc p /dc s in eqn (15), (23) and (25) can be safely neglected and the only relevant parameters are Dn ci and Dw(T 0 m ). Moreover, the changes DT ¼ T m À T 0 m are rather small so the term quadratic in DT in eqn (23) can hardly be determined. However, this does not imply that this term is zero for kosmotropic salts in general.
(2) Large DT; chaotropic ions: for NaSCN there is a marked dependence of DH u (c s ) on salt concentration which immediately demonstrates that dDc p /dc s assumes a nite value and the m-value (eqn (16)) in turn depends on temperature. Moreover, the observed DT is much larger than in case of the kosmotropic ions. Hence, ts must take into account all terms in eqn (25).
Case (1): small DT m ; kosmotropic ions: Fig. 1b gathers all data of the enthalpy DH u as the function of the difference T m À T 0 m . The error of these numbers is of appreciable magnitude and only allows us to obtain an estimate for Dc p,0 for which an evaluation for the data of all kosmotropic ions (NaCl, NH 4 Cl, LiCl) gives an estimate Dc p,0 y 4.6 kJ (K À1 mol À1 ) which compares well literature (see ref. 7 and 58). Hence, the subsequent evaluation is based on dDc p /dc s ¼ 0.  T m (c s ) obtained for a given salt are tted to eqn (23) with neglect of the term quadratic in DT using the MathLab routine cool (MATLAB (2021b). Natick, Massachusetts: The MathWorks Inc.). All calculations have been done using the value of the transition enthalpy in salt-free systems DH 0 u ¼ 392 kJ mol À1 and the transition temperature T 0 m ¼ 326.8 K given by Francisco et al. 18 As mentioned above, the buffer added to all solutions leads to a c s,0 ¼ 0.01 M. 18 The solid lines in Fig. 2 display the respective ts whereas Table 1 gathers the respective t parameters. A single value of parameter Dn ci turned out to describe DG u for all systems under consideration here in agreement with the above general considerations. This fact has already been observed by Francisco et al. 18 and the presence analysis compares well with eqn (21) of ref. 18 inasmuch T m can be described by the combination of a linear and a logarithmic term (see eqn (23)). Pegram et al. also found that a single parameter was sufficient to describe the dependence of the unfolding of DNA as well as for the DNA-binding domain of the lac repressor at small salt concentrations. 31 Hence, an important prediction of the present model is fully corroborated by the experimental data and the parameter Dw(T 0 m ) can be compared to data obtained for complex formation of polyelectrolytes with proteins.
The parameter Dn ci is positive for all kosmotropic salt analyzed herein. This nding points to the fact that a small but nite number of ions attached closely to the surface of the protein is released during the unfolding transition. With increasing c s these ions are released into a reservoir with increasing activity which requires additional free energy during the unfolding transition. Hence, this effect stabilizes the folded state and leads to a higher transition temperature.
The parameter Dw(T 0 m ) is negative which means that the water molecules needed for the hydration of the unfolded protein must have a higher activity as the bulk water since addition of these salts increases the magnitude of DG u . Hence, free energy is needed to transport water from a state of lower activity in bulk to a state of higher activity in the hydrate shell upon unfolding of the protein. This effect is due to a partial depletion of these kosmotropic ions from the hydrate shell of the protein and leads to a stabilization of the folded state. The magnitude of Dw(T 0 m ) found here is in the same range as found previously for complex formation of proteins with DNA. 47 It should be noted that the present analysis not only treats DG u but also DH u at the same time. Thus, the independence of the m-value of temperature follows here from an analysis of the latter quantity. Only this analysis allows us to disregard the term in eqn (25) that depends quadratically on DT 2 .
Case (2): large DT; chaotropic ions: in the following, the evaluation of the respective parameters will be shown using the data for NaSCN (Table 1 of ref. 18). Fig. 1b shows experimental DH u (c s ) as the function of DT m whereas the solid lines displays the t of these data according to eqn (8). This t can be stabilized by using the experimental value DH u (c s ¼ 0) ¼ 392 kJ mol À1 and the specic heat Dc p,0 ¼ 4.6 kJ (K À1 mol À1 ) estimated from the analysis of the kosmotropic systems shown in Fig. 1a. For NaSCN we obtain for the parameter dDc p /dc s a value of ca. 2.5 kJ (K À1 mol À1 M À1 ). Evidently, the small range of data and the nite accuracy of the data allows for an estimate of these quantities only. However, since these parameters present only corrections in eqn (25) and (23) and not leading terms, this error is inconsequential for the purpose at hand.
In the next step, the parameters Dc p,0 ¼ 4.6 kJ (K À1 mol À1 ) and dDc p /dc s ¼ 2.5 kJ (K À1 mol À1 M À1 ) are introduced into eqn (23) and the values of Dn ci and Dw(T 0 m ) are derived from a numerical solution of this equation for DG u ¼ 0. Input parameters are the measured T m measured for different NaSCNconcentrations marked by points in Fig. 2. Table 1 again gathers the data obtained from this t whereas the solid lines in Fig. 2 displays T m calculated with the parameters Dc p,0 ¼ 4.6 kJ (K À1 mol À1 ), dDc p /dc s ¼ 2.5 kJ (K À1 mol À1 M À1 ) and the values of Dn ci and Dw(T 0 m ). Again, a full description of the experimental transition temperatures is achieved. For the chaotropic salt NaSCN the parameter Dw(T 0 m ) assumes a positive value which is directly related to the fact that SCN À -ions are adsorbed on the unfolded protein chain thus lowering the activity of the hydrate water molecules. Hence, free energy is gained when hydrating the unfolded chain by bulk water having a higher activity. The parameter Dn ci now has assumed a negative value. This nding Fig. 2 Comparison of theory and experimental data taken from the denaturation of RNase A for the 3 kosmotropic salt NaCl, NH 4 Cl, LiCl and for the chaotropic salt NaSCN. 18 The points show the transition temperatures taken in presence of different salts as indicated in the graph (see Table 1 59 This effect measured through careful measurements of the electrophoretic mobility was found strongest for SCN À -ions. Hence, adsorption of chaotropic ions can diminish or even reverse the effective charge of unfolded proteins. However, further investigations of T m at very low ion concentrations are needed to clarify this problem. As mentioned above, the combination of a negative Dn ci with a negative Dw value should lead to a non-monotonic dependence of T m on salt concentration. This effect is seen in a careful study of the unfolding of RNase A in NaCl solutions by Senske et al. 16 These data have been taken using a 50 mM citrate buffer at pH ¼ 5 and are hence not directly comparable to the data of Francisco et al. 18 discussed above. Fig. 3 displays the data obtained for solutions with varying concentration of NaCl. Since the range of temperature is rather small, the term quadratic in DT in eqn (23) can be disregarded. The t of the data is shown by the solid line in Fig. 3 and leads to Dn ci ¼ À0.60 and Dw ¼ À31.7. At small salt concentrations, the logarithmic term in eqn (23) dominates the transition temperature. In this regime, it stabilizes the unfolded state which takes up ions from solution more easily at higher salt concentration. At higher salt concentration, the term linear in salt concentration in eqn (23) takes over and the unfolded state is now destabilized leading to a higher T m again.

Conclusions
A phenomenological model describing the unfolding transition of proteins has been presented. Within this model, the change of T m with the solute concentration c s is captured by two effects: (i) the change of the number of correlated counterions Dn ci during the unfolding transition, and (ii) the change of hydration expressed through the parameter Dw. The latter parameter is not directly the number of water molecules released or taken up during transition but described the change of the free energy by the release or uptake of water (see the discussion of eqn (11)). The model can be cast in terms of the closed expression eqn (23) giving the free energy of unfolding in terms of the salt/solute concentration c s . The enthalpy DH u can directly be related to the parameter Dw by the Maxwell-relation eqn (7) leading to eqn (8) in which a new parameter dDc p /dc s describes the direct dependence of DH u on salt concentration. The model allows us to discuss the classical m-value in terms of these parameters (eqn (16)) and predicts that m is depending on temperature if the parameter dDc p /dc s assumes a nite value. A rst comparison with experimental data taken from literature shows the general validity of the model.

Conflicts of interest
There are no conicts to declare. Fig. 3 Reversal of T m through a competition of counterion release and preferential hydration. The data marked by red points have been measured by Senske et al. for the unfolding of RNase A at pH ¼ 5 in presence of an increasing concentration of NaCl. 16 The solid line marks the fit by theory with the parameters Dn ci ¼ À0.60 and Dw ¼ À31.7 (eqn (23), the term quadratic in DT has been neglected). The green dashed line marks the temperature T 0 m . See text for further explanation.