Open Access Article

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Matthias Ballauff

Institut für Chemie und Biochemie, Freie Universität Berlin, Takustraße 3, 14195 Berlin, Germany. E-mail: matthias.ballauff@fu-berlin.de

Received
21st February 2022
, Accepted 23rd March 2022

First published on 31st March 2022

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 ΔG_{u} and a transition temperature T_{m}. For a long time, it has been known that solutes can change T_{m} profoundly. Here we present a phenomenological model that describes the change of T_{m} with the solute concentration c_{s} in terms of two effects: (i) the change of the number of correlated counterions Δn_{ci} and (ii) the change of hydration expressed through the parameter Δw and its dependence on temperature expressed through the parameter dΔc_{p}/dc_{s}. Proteins always carry charges and Δn_{ci} 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 c_{s} that defines also the activity of water. The parameter Δn_{ci} 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 lnc_{s}. Δw describes the effect on ΔG_{u} through the loss or uptake of water molecules and contains enthalpic as well as entropic effects that scale with c_{s}. It is related to the enthalpy of transition ΔH_{u} through a Maxwell relation: the dependence of ΔH_{u} on c_{s} is proportional to the dependence of Δw on temperature. While ionic effects embodied in Δn_{ci} 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.

A fundamental problem in the field 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.^{4,6,7,9–20} This effect is of obvious biological importance and can be traced back to hydration effects embodied in the Hofmeister series.^{21–24} 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.^{23,25–30} 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.^{32,35–37}

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).^{22,31,38–40} 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 (“m-value”; 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 specific heat Δc_{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 fulfilled 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 modified 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 unified 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 identified as the decisive variables and a closed expression for the free energy ΔG_{b}(T,c_{s}) of complex formation could be derived. In this model counterion release was characterized by Δn_{ci} denoting the net number of released ions during binding whereas hydration was described in terms of the parameter Δw defined already in early expositions of the problem^{1,43,44} and used frequently to describe the effect of hydration on complex formation.^{44–49} Central for the development of this model is the fact that mixed derivatives of the binding enthalpy ΔH_{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 ΔH_{b}(T,c_{s}) on c_{s} gives directly the dependence of Δw on temperature. The model thus derived is capable of describing the weak dependence of ΔG_{b}(T,c_{s}) on temperature which in turn leads to a strong compensation of enthalpy and entropy.^{34} Moreover, the values obtained for Δn_{ci} and Δw 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 exemplified 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.

(1) |

ΔG_{u} = −RTlnK_{u}
| (2) |

The basic thermodynamic analysis ΔG_{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_{±} of an added salt is given by

(3) |

(4) |

Hence, the salt concentration c_{s} is the variable on which the subsequent thermodynamic analysis is based. With the standard thermodynamic relation (∂lnK_{b}/∂T)_{cs} = ΔH_{b}/RT^{2} we obtain the differential of lnK_{u} for monovalent ions

(5) |

There is abundant experimental evidence that the parameter Δn_{ci} is independent of temperature.^{32,34,35,37,52–54} It is therefore safe to disregard the dependence of this parameter on T_{m}. With this assumption and

(6) |

(7) |

This relation demonstrates that the salt dependence of transition enthalpy is directly related to the dependence of the parameter Δw on temperature. As already lined out previously,^{34} this relation can now be used to calculate Δw as the function of temperature. In general, the transition enthalpy ΔH_{u} as the function of the melting temperature T_{m} and c_{s} can be rendered as^{34}

(8) |

(9) |

Integration leads to^{34}

(10) |

As already discussed previously,^{34} Δw 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 Δn_{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,+} = (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,−} of the anions is defined in the same way. With these definitions, Δw can be rendered by^{34}

(11) |

Evidently, the quantity Δw measures the effect of water release on the free energy of unfolding and should not be confused with the total number Δn_{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 first consider uncharged systems, that is, Δn_{ci} = 0. Integration of eqn (6) at constant temperature then leads to

lnK_{u} = lnK^{0}_{u} + 0.036Δwc_{s}
| (12) |

ΔG_{u} = ΔG^{0}_{u} − 0.036RT_{m}Δwc_{s}
| (13) |

(14) |

In many cases the difference T_{m} − 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)

(15) |

Eqn (14) may be used to calculate the m-value defined as the derivative of the free energy with regard to solute concentration at constant temperature

(16) |

This expression shows that m is given by a constant plus a term that depends quadratically on T_{m} − 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 ΔG^{0}_{u} for salt- or solute-free solutions can be derived following the prescription of Chen and Schellman:^{6} the specific heat Δc_{p,0} measured in solute-free systems can be regarded as a constant throughout the rather small temperature range under consideration here. Thus, for solute-free systems we obtain

(17) |

ΔH^{0}_{u}(T_{m}) + Δc_{p,0}(T_{m} − T^{0}_{m})
| (18) |

(19) |

(20) |

Combination with eqn (14) then leads to

(21) |

For T_{m} − T^{0}_{m} ≤ 10 K this expression can be approximated by

(22) |

Eqn (21) and (22) are the final result for the free energy of unfolding for uncharged systems.

For partially charged proteins eqn (5) shows that a term scaling with lnc_{s} must be added to eqn (21).^{1} Here it must be kept in mind that there is always a small but finite salt concentration c_{s,0} so that the integration of eqn (5) must start at this concentration. Keeping in mind that Δn_{ci} does not depend on temperature we immediately obtain from eqn (22)

(23) |

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, ΔG_{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} − c_{s,0}. The transition entropy follows as

(24) |

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 ΔΔG_{u} which gives the change of ΔG_{u} with c_{s} calculated for the transition temperature T^{0}_{m} in solute-free solution:

(25) |

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):

ΔG_{u}(c_{s}) = ΔG^{0}_{u} − RT_{m}Δβ_{23}c_{s}
| (26) |

Thus, the coefficient Δβ_{23} is identical to 0.036Δw 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 lnK_{u} as follows in the present notation as:^{6}

(27) |

The experimental data are described in terms of 3 adjustable parameters: (i) Δw(T^{0}_{m}) which is closely related to the classical m-value through eqn (15); (ii) the specific heat Δc_{p,0} in absence of solutes; and (iii) the parameter dc_{p}/dc_{s} describing the dependence of Δc_{p} on c_{s}. This parameter has been introduced by Chen and Schellman as well (the parameter 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 first 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 ΔΓ_{23}. If this parameter does not depend on c_{s}, it follows directly that

In principle, eqn (23) and eqn (26) define stability curves as defined by Becktel and Schellman^{3} inasmuch as they describe the free energy ΔG_{u} as the function of temperature and salt concentration. If Δc_{p,0} may be regarded as constant throughout a temperature range of sufficient width, the present approach could be used to construct ΔG_{u}(T,c_{s}) for all pertinent temperatures ranging from cold to thermal denaturation. Given the fact, however, that Δc_{p,0} depends on temperature,^{7} such stability curves should be regarded with caution.

A next prerequisite is the independence of Δn_{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 Maxwell-relation eqn (7) for the next step in which the salt dependence of the unfolding enthalpy ΔH_{u} is related to the dependence of the parameterΔw on temperature given through eqn (9). Hence, if ΔH_{u} turns out to depend on the concentration c_{s} of the solute, it necessarily follows that Δw 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} − T^{0}_{m}.

The above model hence makes the following predictions that can compared directly to experiments:

(1) In a first step of the analysis of experimental data, dependence of ΔH_{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 ΔH_{u} on salt concentration c_{s} gives the dependence of the quantity Δw on temperature as shown by the Maxwell-relation in eqn (7) which in turn leads to the dependence of the m-value on temperature eqn (16). Evidently, if ΔH_{u} is found to depend on salt concentration, there must be a finite dependence of m on temperature as well (eqn (16)). If, on the other hand, the dependence of ΔH_{u} on salt concentration c_{s} is small, the parameter dc_{p}/dc_{s} ≅ 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 ΔH_{u}(T,c_{s}).

(2) The term scaling with lnc_{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 Δn_{ci} assumes a finite 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 ΔΔG_{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 Δn_{ci} and the change of T_{m} by hydration may cancel each other. Thus, if Δn_{ci} < 0 as well as Δw < 0, eqn (23) demonstrates that can lead to ΔT_{m} = 0 for a finite 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 Δn_{ci} and Δw.

(3) If the term quadratic in eqn (23) and (25) can be disregarded, that is, for small ΔT, the combination of both expressions shows that in this case

(28) |

As outlined above, the analysis may start by the check of the dependence of ΔH_{u} on c_{s} (see Table 1 of ref. 18). Fig. 1a displays ΔH_{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:

Fig. 1 Evaluation of the measured transition enthalpy ΔH_{u} by eqn (8). (a) ΔH_{u} as the function of salt concentration c_{s}. The marks show the experimental data for the unfolding of ribonuclease in presence of the salts indicated in the graph. These data have been taken from Table 1 of Francisco et al.^{18} (b) ΔH_{u} as the function of ΔT_{m} = T_{m} − T^{0}_{m}. The solid lines indicate the fit of eqn (8) whereas the green dashed line indicates the transition enthalpy calculated by eqn (8) with the average value Δc_{p,0} ≅ 4.6 kJ (K^{−1} mol^{−1}) and dΔc_{p}/dc_{s} = 0. See text for further explanation. |

(1) Small ΔT_{m}; kosmotropic ions: the small dependence of ΔH_{u} on c_{s} suggests that the coefficient dΔc_{p}/dc_{s} in eqn (15), (23) and (25) can be safely neglected and the only relevant parameters are Δn_{ci} and Δw(T^{0}_{m}). Moreover, the changes ΔT = T_{m} − T^{0}_{m} are rather small so the term quadratic in ΔT in eqn (23) can hardly be determined. However, this does not imply that this term is zero for kosmotropic salts in general.

(2) Large ΔT; chaotropic ions: for NaSCN there is a marked dependence of ΔH_{u}(c_{s}) on salt concentration which immediately demonstrates that dΔc_{p}/dc_{s} assumes a finite value and the m-value (eqn (16)) in turn depends on temperature. Moreover, the observed ΔT is much larger than in case of the kosmotropic ions. Hence, fits must take into account all terms in eqn (25).

Case (1): small ΔT_{m}; kosmotropic ions: Fig. 1b gathers all data of the enthalpy ΔH_{u} as the function of the difference T_{m} − T^{0}_{m}. The error of these numbers is of appreciable magnitude and only allows us to obtain an estimate for Δc_{p,0} for which an evaluation for the data of all kosmotropic ions (NaCl, NH_{4}Cl, LiCl) gives an estimate Δc_{p,0} ≅ 4.6 kJ (K^{−1} mol^{−1}) which compares well literature (see ref. 7 and 58). Hence, the subsequent evaluation is based on dΔc_{p}/dc_{s} = 0.

Fig. 2 displays a comparison of the experimental transition temperatures T_{m} as the function of salt concentration obtained by numerical solution of eqn (23) for ΔG_{u} = 0. Here the data T_{m}(c_{s}) obtained for a given salt are fitted to eqn (23) with neglect of the term quadratic in ΔT using the MathLab routine cftool (MATLAB (2021b). Natick, Massachusetts: The MathWorks Inc.). All calculations have been done using the value of the transition enthalpy in salt-free systems ΔH^{0}_{u} = 392 kJ mol^{−1} and the transition temperature T^{0}_{m} = 326.8 K given by Francisco et al.^{18} As mentioned above, the buffer added to all solutions leads to a c_{s,0} = 0.01 M.^{18} The solid lines in Fig. 2 display the respective fits whereas Table 1 gathers the respective fit parameters. A single value of parameter Δn_{ci} turned out to describe ΔG_{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 Δw(T^{0}_{m}) can be compared to data obtained for complex formation of polyelectrolytes with proteins.

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 of ref. 18). The solid lines mark the calculated transition temperatures T_{m} calculated from the fit parameters Δn_{ci} and Δw(T^{0}_{m}) (cf. Table 1). See text for further explanation. |

System | Δn_{ci} |
Δw(T^{0}_{m}) |
dΔc_{p}/dc_{s} |
---|---|---|---|

a Δn_{ci}: number of ions released or taken up during unfolding (eqn (3) and (4)); Δw: effect of water release or uptake (eqn (3) and (4)); dΔc_{p}/dc_{s}: parameter describing the dependence of Δw on temperature (eqn (8) and (10)). |
|||

NaCl | 0.17 | −50.4 | 0 |

LiCl | 0.17 | −26.2 | 0 |

NH_{4}Cl |
0.17 | −19.4 | 0 |

NaSCN | −0.17 | 103 | 2.5 |

The parameter Δn_{ci} is positive for all kosmotropic salt analyzed herein. This finding points to the fact that a small but finite 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 Δw(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 ΔG_{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 Δw(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 ΔG_{u} but also ΔH_{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 ΔT^{2}.

Case (2): large ΔT; 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 ΔH_{u}(c_{s}) as the function of ΔT_{m} whereas the solid lines displays the fit of these data according to eqn (8). This fit can be stabilized by using the experimental value ΔH_{u}(c_{s} = 0) = 392 kJ mol^{−1} and the specific heat Δc_{p,0} = 4.6 kJ (K^{−1} mol^{−1}) estimated from the analysis of the kosmotropic systems shown in Fig. 1a. For NaSCN we obtain for the parameter dΔc_{p}/dc_{s} a value of ca. 2.5 kJ (K^{−1} mol^{−1} M^{−1}). Evidently, the small range of data and the finite 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 Δc_{p,0} = 4.6 kJ (K^{−1} mol^{−1}) and dΔc_{p}/dc_{s} = 2.5 kJ (K^{−1} mol^{−1} M^{−1}) are introduced into eqn (23) and the values of Δn_{ci} and Δw(T^{0}_{m}) are derived from a numerical solution of this equation for ΔG_{u} = 0. Input parameters are the measured T_{m} measured for different NaSCN-concentrations marked by points in Fig. 2. Table 1 again gathers the data obtained from this fit whereas the solid lines in Fig. 2 displays T_{m} calculated with the parameters Δc_{p,0} = 4.6 kJ (K^{−1} mol^{−1}), dΔc_{p}/dc_{s} = 2.5 kJ (K^{−1} mol^{−1} M^{−1}) and the values of Δn_{ci} and Δw(T^{0}_{m}). Again, a full description of the experimental transition temperatures is achieved. For the chaotropic salt NaSCN the parameter Δw(T^{0}_{m}) assumes a positive value which is directly related to the fact that SCN^{−}-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 Δn_{ci} now has assumed a negative value. This finding points to a much stronger interaction of such chaotropic ions with the unfolded protein chains. Thus, Fang and Furo could show that chaotropic ions can associate to PNIPAM chains with a Langmuir-type association behavior while NaCl is only weakly adsorbed.^{59} This effect measured through careful measurements of the electrophoretic mobility was found strongest for SCN^{−}-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 Δn_{ci} with a negative Δw 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 = 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 ΔT in eqn (23) can be disregarded. The fit of the data is shown by the solid line in Fig. 3 and leads to Δn_{ci} = −0.60 and Δw = −31.7. 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.

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 = 5 in presence of an increasing concentration of NaCl.^{16} The solid line marks the fit by theory with the parameters Δn_{ci} = −0.60 and Δw = −31.7 (eqn (23), the term quadratic in ΔT has been neglected). The green dashed line marks the temperature T^{0}_{m}. See text for further explanation. |

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