Folding energetics of ligand binding proteins. I. Theoretical model1

doi:10.1006/jmbi.2000.4359 available online at http://www.idealibrary.com on

J. Mol. Biol. (2001) 306, 809±824

Folding Energetics of Ligand Binding Proteins. I. Theoretical Model JoÈrg RoÈsgen* and Hans-JuÈrgen Hinz* Institut fuÈr Physikalische Chemie, WestfaÈlische WilhelmsUniversitaÈt, Schloûplatz 4/7 48149 MuÈnster, Germany

Heat capacity curves as obtained from differential scanning calorimetry are an outstanding source for molecular information on protein folding and ligand-binding energetics. However, deconvolution of Cp data of proteins in the presence of ligands can be compromised by indeterminacies concerning the correct choice of the statistical thermodynamic ensemble. By convent, the assumption of constant free ligand concentration has been used to derive formulae for the enthalpy. Unless the ligand occurs at large excess, this assumption is incorrect. Still the relevant ensemble is the grand canonical ensemble. We derive formulae for both constraints, constancy of total or free ligand concentration and illustrate the equations by application to the typical equilibrium Nx „ N ‡ x „ D ‡ x. It is demonstrated that as long as the thermodynamic properties of the ligand can be completely corrected for by performing a reference measurement, the grand canonical approach provides the proper and mathematically signi®cantly simpler choice. We demonstrate on the two cases of sequential or independent ligand-binding the fact, that similar binding mechanisms result in different and distinguishable heat capacity equations. Finally, we propose adequate strategies for DSC experiments as well as for obtaining ®rst estimates of the characteristic thermodynamic parameters, which can be used as starting values in a global ®t of DSC data. # 2001 Academic Press

*Corresponding authors

Keywords: heat capacity; protein; deconvolution; statistical thermodynamics; ligand-binding

Introduction As a consequence of the development of highly sensitive commercially available microcalorimeters,1 differential scanning calorimetry (DSC) has been used progressively with great success for the elucidation of the energetics of protein folding. The knowledge of the parameters determining the stability of a protein is of utmost importance for the prediction of protein behaviour under a variety of different environmental conditions. First attention had been focused on the quantitative thermodynamic description of the ``classical'' transition of small proteins between a single native and a denatured state.1-3 With the establishment of more sophisticated folding models

alternative native conformations became of interest as well as folding intermediates,4-6 oligomerization7-12 and ligand-binding reactions and their in¯uence on both unfolding and refolding reactions.13-19 Until recently all theoretical solutions of these problems were based on the assumption that during the binding or conformational transition the shift of the enthalpy relative to a reference state, H(T) ÿ HN(T), is directly proportional to the shift in the population of protein species. This assumption resulted in the following equation which was the basis of all deconvolution procedures:

H…T† ÿ HN …T† ˆ

k X iˆ1

Abbreviations used: D, denatured state; DSC, differential scanning calorimetry; ITC, isothermal titration calorimetry; N, native state; Nx, liganded native state. E-mail addresses of the corresponding authors: [email protected]; [email protected] 0022-2836/01/040809±16 $35.00/0


Hi0 …T†
fi

…1†

k refers to the number of transitions, fi is the fraction of the i-th protein species in equilibrium and will also be referred to as ``population size'', and
H0i (T) is the standard enthalpy difference between the i-th state and the native state N, the # 2001 Academic Press

810

Folding Energetics

latter being the reference state. Usually the reference state is the unliganded, native (mono- or oligomeric) state of the protein. Recently we have shown that relation (1) is only true for an N to D conversion but is incorrect, if reaction stoichiometries other than 1:1 occur (e.g. Nn„ nD). This is in general the case when non-covalently linked homo-oligomeric proteins are studied.11,12 It has also been demonstrated that the correct analytical solution for the enthalpy and heat capacity changes is always obtained by taking the proper derivatives of the relative partition function Q. The equations are: @ ln Q ; H…T† ÿ HN …T† ˆ RT 2 @T   @ RT 2 @ ln Q Cp …T† ÿ Cp;N …T† ˆ @T @T

…2†

In ligand-binding experiments there arises a problem if the ligand concentration is of the same order of magnitude as the concentration of binding sites. In such a case, the concentration of free ligand changes signi®cantly during the experiment. This is especially problematic for ligands with high af®nity,17 since at low total ligand concentrations there are signi®cant differences between the concentrations of free and total ligand. This discrepancy is less dramatic for ligands having weak af®nity, since signi®cant binding occurs only at relatively high total ligand concentrations. For the case of protein-sized ligands, i.e. proteinoligomerization reactions, it has been shown, that the proportionality between enthalpy and population size is lost in the unfolding reaction.11,12,20 The question arises therefore, whether this effect plays also a role in the case of proteins complexed with small ligands. From a statistical mechanic point of view two classical cases could describe the situation correctly. On the one hand, a system exchanging particles with the surroundings, such as a ligandbinding protein, could be considered a grand canonical ensemble. On the other hand, in the calorimetric cell the number of proteins and ligands remains constant. This corresponds to the situation in a canonical ensemble. At ®rst sight both models appear to be equivalent. However, it will turn out that they yield signi®cantly different results. To address this question we shall calculate the temperature dependence of the heat capacity according to these two models. In both models the concentration of free ligand is calculated from the known total concentration of ligand and the amount of bound ligand. However, the important difference between the models is that temperature derivatives are taken either at constant total ligand or free ligand concentration. Constant total ligand concentration corresponds to the case of a canonical ensemble, constant free ligand concentration corresponds to the situation of a grand canonical ensemble. The canonical heat capacity is obtained

by a twofold temperature derivative of the partition function (equation (2)) at constant total amount of ligand. This model will be referred to as ``stoichiometry model''. To obtain the grand canonical heat capacity two temperature derivatives at constant free ligand activity are calculated. This results in a proportionality between the enthalpy and the population sizes. Therefore the model will be referred to as ``proportionality model'' (equation (1)). So far a mixture of both models was applied in the analysis of Cp curves by combining both types of derivatives. For the calculation of the enthalpy the proportionality model was used, i.e. derivatives at constant free ligand concentration. For the modelling of the heat capacity curves from the enthalpy curve numerical derivatives at constant total ligand concentration were done.13-18,21,22 In the cited studies an analytical solution for the heat capacity was never published, therefore the present study is particularly innovative with regard to the analytical solution for the canonical case (the stoichiometry model). First, we shall calculate the heat capacity according to the two different models for a protein with a single ligand-binding site. The comparison of the enthalpy equations will permit a prediction for combinations of protein concentration, ligand concentration and ligand af®nity at which the models are equivalent. At these conditions the easier to use proportionality model can be best employed for data ®tting. Next, we illustrate the general heat capacity equation for the proportionality model by providing the formula for two common examples. Following, we provide a summary of useful thermodynamic relations and discuss the optimisation of data collection of transition curves. Particular emphasis is laid upon the estimate of the binding parameters that provide good starting values for the ®nal global ®t of DSC curves.

Enthalpy Curves of Proteins Having a Single Binding Site The simplest case, involving the unfolding of a native protein N in the presence of a single ligand x, can be represented by the equation: Nx „ N ‡ x „ D ‡ x

…3†

The reaction scheme involves the ligand-protein complex Nx, the dissociation of the ligand from the native protein to yield N ‡ x, and the transformation of the native protein from N to D. The equilibrium constants Kx for ligand-binding to the native state and K0 for the denaturation reaction are de®ned in the following manner: Kx ˆ

‰NxŠ ‰NŠ‰xŠ

K0 ˆ

‰DŠ ‰NŠ

…4†

811

Folding Energetics

Their temperature dependences are given by the relations: 0 @Kx Kx
Nx N Hx ˆ 2 @T RT

and

0 @K0 K0
D NH ˆ 2 @T RT

…5†

0
Nx N Hx is the binding enthalpy of the ligand x, and D 0
NH is the unfolding enthalpy of the unliganded protein. The upper index at the
-sign refers to the ®nal, the lower index to the initial state. If we choose the native state N as reference state, the heat capacity and enthalpy functions relative to N can be calculated by differentiating the relative partition function, Q, with regard to the temperature. The expression for Q is obtained by dividing the concentrations of all protein species occurring in equilibrium by the reference state concentration [N]:

‰NŠ ‡ ‰NxŠ ‡ ‰DŠ Qˆ ‰NŠ ˆ 1 ‡ Kx
‰xŠ ‡ K0  QN ‡ Qx ‡ QD

…6†

In principle Q contains all thermodynamic information relevant to equation (3). The statistical physical foundation for this analysis was given by RoÈsgen & Hinz.12,20 The temperature derivative of ln Q at constant free ligand concentration [x] leads to the proportionality model given in equation (1). This model will be treated later. Here we ®rst derive the thermodynamic parameters for the stoichiometry model, where temperature derivatives of Q are calculated at constant total ligand concentration [x]t. This is a natural constraint for DSC studies. Analysis of the stoichiometry model The common experimental conditions of a DSC experiment on a protein in the presence of a ligand are the following. A de®ned total ligand concentration is established in the protein solution and the apparent heat capacity of the protein-ligand solution is monitored as a function of temperature. The total concentration [x]t of the ligand is therefore constant throughout the experiment: ‰xŠt ˆ ‰xŠ ‡ ‰NxŠ

…7†

where [x] and [Nx] refer to the temperature dependent concentrations of free and bound ligand, respectively. By the use of proper transformations the temperature derivative at constant total ligand concentration, (@/@T)[x]t, of the partition function Q can be expressed in terms of the much more simple derivatives, the partial derivative with regard to temperature at constant free ligand concentration, (@/@T)[x], and the partial derivative with regard to the free ligand concentration at constant temperature, (@/@[x])T. The transformations re¯ect the fact that the pro-

tein is in¯uenced by the chemical potential of the ligand x, mx, i.e. by the free ligand activity [x]. Thus, Q depends explicitly on [x], but experimentally only the total concentration of ligand, [x]t, is controlled and known. The corresponding transformation is given by the following equation: 

@ @T

 ‰xŠt



@ ˆ @T

 ‰xŠ

    c @X @ ÿ   @T ‰xŠ @‰xŠ T @X 1‡c @‰xŠ T

…8†

Its derivation is given in the appendix. c is the total protein concentration in terms of monomers  is the average number of moles of ligand and X bound per mole of protein monomer, which is equivalent to the degree of binding. For the reaction shown in equation (3) the degree of binding equals: 1
Kx ‰xŠ Kx ‰xŠ Qx  ˆ ‰NxŠ ˆ ˆ ˆ X c 1 ‡ K0 ‡ Kx ‰xŠ Q Q

…9†

Therefore we can calculate the four partial derivatives occurring in equation (8):   @X QKx ÿ Kx Qx ˆ Q2 @‰xŠ T   0 D 0 Qx
Nx @Q N H ‡ K
N H ˆ 2 @T ‰xŠ RT

…10†

  0 @X
Nx H 0 Qx …1 ‡ K† ÿ Qx K
D NH ˆ N RT 2 Q2 @T ‰xŠ   @Q ˆ Kx …11† @‰xŠ T where Qx ˆ Kx[x]. Using equations (7) and (9) the binding constant Kx can be written as:

Kx ˆ

Qx Qx QQx ˆ ˆ ‰xŠ ‰xŠt ÿ ‰NxŠ Q‰xŠt ÿ Qx c

…12†

Using relations 7 to 12 the derivative of the partition function Q with regard to temperature at constant total ligand can now be evaluated:

812 

Folding Energetics

 0 Nx 0 @Q K0
D N H ‡ Qx
N H ˆ RT 2 @T ‰xŠt 0 D 0 c
Kx ‰…1 ‡ K0 †Qx
Nx N H ÿ Q x K 0
N H Š 2 2 ‰1 ‡ …Kx Q ÿ Qx Kx †
c=Q ŠRT Q2   0
D cKx Qx NH ˆ K0 1 ‡ 2 RT 2 Q ‡ cKx …1 ‡ K†  
Nx H 0 cKx QND ‡ N 2 Qx 1 ÿ 2 RT Q ‡ cKx QND   0
D cQ2x NH K0 1 ‡ 2 ˆ RT 2 Q ‰xŠt ÿ cQ2x

ÿ

‡

0
Nx Q2 ‰xŠt ÿ QQx c N H Q x RT 2 Q2 ‰xŠt ÿ cQ2x

…13†

With these relations the enthalpy change relative to the enthalpy of the unliganded native protein N can be calculated using the equation:   2 @ ln Q
H…T† ˆ H…T† ÿ HN …T† ˆ RT @T ‰xŠt 0 ˆ
D N H …T†fD

1 1 ÿ fx2 c=‰xŠt

0 ‡
Nx N H …T†fx

1 ÿ fx c=‰xŠt 1 ÿ fx2 c=‰xŠt

…14†

In equation (14) the fractional populations of the protein in the states D and Nx, fD and fx, are de®ned by the equations: fD ˆ

K0 Q

and

fx ˆ

Qx Q

ences between the stoichiometry and proportionality model. To illustrate the effect of these factors on the transition enthalpy at temperature T,
H(T) ˆ H ÿ HN, the expressions for a and b have been plotted in Figure 1 as a function of the fraction fx of ligated protein for different ratios of the total ligand concentration per mole of protein monomers, [x]t/c. It is quite evident from Figure 1(a) and (b) that on increasing the ratio [x]t/ c both factors a and b converge very fast to unity. If the total ligand concentration [x]t is only by one order of magnitude larger than the total protein concentration c, the additional factors a and b are nearly equal to 1. This has the effect that practically no difference between the proportionality model and the stoichiometry model will be detectable once the ligand is present at tenfold or higher excess. Another useful parameter for the characterisation of the binding equilibrium is the product Kx[x]t. To obtain a rough estimate of the in¯uence of varying total ligand concentration relative to the dissociation constant 1/Kx the terms a and b are plotted in Figure 2 versus log([x]t/c) for different values of Kx[x]t. The plots were calculated for the case of a single binding site. Several important conclusions can be drawn from an inspection of the plot. First, it is quite evident that for an excess of ligand, i.e. for high ratios of [x]t/c, both terms a and b converge rapidly to unity. Second, it turns out that the term a ˆ 1/(1 ÿ f 2xc/[x]t) is of lesser

…15†

Equation (14) provides the basis for calculations of the temperature dependence of the enthalpy change
H(T) at constant total ligand concentration (stoichiometry model). Comparison of the stoichiometry and proportionality models Equation (14) must be compared with equation (16) to see the differences between the results of the stoichiometry model and the proportionality model:
H…T† ˆ H…T† ÿ H0 …T† ˆ
H10 …T†f1 ‡
H20 …T†f2 0 D 0 ˆ
Nx N H fx ‡
N H fD

…16†

The latter equation results from equation (1), when the unfolding and dissociation reactions given in equation (3) are taken into account. The comparison shows that the analytical solution for the enthalpy developed here contains the additional factors a ˆ 1/(1 ÿ fx2c/[x]t) and b ˆ (1 ÿ fxc/[x]t)/ (1 ÿ fx2c/[x]t). If these factors differ signi®cantly from unity, we have to expect signi®cant differ-

Figure 1. Magnitude of the two factors a and b that distinguish between the proportionality model and the stoichiometry model. Both factors rapidly converge to unity if an excess of ligand is used ([x]t/c > 10).

Folding Energetics

Figure 2. Dependence of the factors a and b on the af®nity of the protein to the ligand at different ratios of total ligand concentration to protein. The values are calculated for a protein with a single ligand-binding site.

importance to deviations from unity than the factor b ˆ (1 ÿ fxc/[x]t)/(1 ÿ f 2xc/[x]t). Third, the graphs demonstrate that b converges to zero, if the total concentration of ligand is much smaller than the protein concentration. This result simply re¯ects the fact that at very low molar ratios of ligand to protein concentration the in¯uence of ligand-binding is negligible. Even if ligands are bound this will not signi®cantly affect the protein population, as long as their concentration is by 2 or 3 orders of magnitude lower than the protein concentration. Thus, the region which will be critical for ®nding differences between the two models is the region around [x]t/c  1 where ligand and protein concentrations are similar. This is also the region where the ®rst, less important term a assumes values different from one. The fourth and ®nal conclusion is that differences between the models will be observable only if Kx[x]t 5 0.1, i.e. if the dissociation constant 1/Kx and the total ligand concentration [x]t are in the same order of magnitude or if the binding constant is very high. Therefore an experimentally detectable difference between the proportionality model and the stoichiometry model can only be expected at intermediate ligand concentrations that are in the same order of magnitude as the protein concentration and the dissociation constant. Since a typical protein concentration for DSC-scans is 10 mM, the stoichiometry model will apply for total ligand concentrations between [x]t  1 mM and 100 mM and dissociation constants 1/Kx 4 1 mM to 100 mM. If one or both of these conditions are not

813 met, the proportionality and stoichiometry model will yield indistinguishable results. These conclusions become immediately evident, if the changes in chemical potential resulting from the changes in ligand concentration are estimated. The additional factors in equation (14) originate from the experimental restriction of constancy of the total ligand concentration, which results in a variation of the free ligand concentration due to changing degrees of binding during the experiment. As a result of the logarithmic dependence of the chemical potential on ligand concentration a change of 10 % in concentration at millimolar ligand concentration leads to a change in the chemical potential of not more than 1 to 2 %. Thus, to a ®rst approximation such changes can be neglected. Figure 3 illustrates the differences between the two models in a three-dimensional diagram. In the H ÿ T plane enthalpy transition curves at different total ligand concentrations are shown, while in the H ÿ ln[x]t plane isothermal ligand titration curves are plotted. The 3D diagram has been calculated on the basis of equations (14) and (16). At the scale given in the diagram the signi®cant differences between the two models would be dif®cult to distinguish. However, the differences of the enthalpy values can be made visible by colour codes even at this scale and that is illustrated in Figure 3. The light yellow area in which the constraints are Kx[x]t  1 and Kx > 105 Mÿ1 indicates the experimental conditions where maximal differences between the models can be expected. This situation occurs when similar amounts of all three species, unfolded protein, folded native protein, and folded liganded protein are present in solution near the midpoint of the transition curve. In view of these constraints an unambiguous discrimination between the two models will be dif®cult in standard DSC-measurements on protein ligand complexes, where usually an excess of ligand is used. However, in cases where both the protein and ligand concentration and also the dissociation constant of the complex are in the same order of magnitude, signi®cant differences between the two models should become visible in the heat capacity curves. Furthermore in isothermal titration calorimetry (ITC) experiments, where generally a broad range of ligand concentration is covered within a single measurement, the effect should be seen more frequently than in DSC studies. On the basis of equations (8)-(13) the variation with temperature of the enthalpy difference, H ÿ HN, can be evaluated according to equation (14). Straightforward differentiation of equation (14) with respect to temperature results in equation (17) which describes the heat capacity as a function of temperature and the degree of ligandation (see Appendix I and II):

814

Folding Energetics

Figure 3. Enthalpy of a ligandbinding protein as a function of temperature and the logarithm of ligand concentration. The colour indicates the difference between the proportionality model and the stoichiometry model as given by the legend. Only in a small range of ligand concentrations and temperatures there is a signi®cant difference (red to yellow in colour codes).

Cp ˆ Cp;N ‡
D N Cp ‡
Nx N Cp fx ‡

obtained using the proportionality model. The second temperature derivative of the partition function Q (equation (6)) at constant free ligand concentration [x] results in the heat capacity equation (18) for the proportionality model, which is evidently much simpler than the equation (17) for the stoichiometry model:

1 ÿ fx c=‰xŠt 1 ÿ fx2 c=‰xŠt

0 2 …
D fD NH † RT 2 …1 ÿ fx2 c=‰xŠt †3

 ‰…1 ÿ ‡

fD 1 ÿ fx2 c=‰xŠt

fx2 c=‰xŠt †

0 2 …
Nx N H † 2 RT

ÿ fD ‰1 ‡

fx2 c=‰xŠt …1

Nx Cp ˆ Cp;N ‡
D N Cp fD ‡
N Cp fx

ÿ 2fx †ŠŠ

‡

fx …1 ÿ fx † …1 ÿ fx2 c=‰xŠt †3

ÿ2

 …1 ÿ fx c=‰xŠt †‰1 ÿ fx c=‰xŠt …2 ÿ fx †Š ÿ2

0 D 0
Nx fx fD N H
N H RT 2 …1 ÿ fx2 c=‰xŠt †3

 …1 ÿ fx c=‰xŠt †‰1 ÿ fx c=‰xŠt …2 ÿ fx †Š

2 2 …
D …
Nx N H† N H† f …1 ÿ f † ‡ fx …1 ÿ fx † D D RT 2 RT 2

…17†

As one can see, the condition of constant total ligand yields a contribution of non-proportionality both in the ®rst derivative of the relative partition function Q, i.e. the enthalpy, and the second derivative, i.e. the heat capacity. Therefore using the proportionality model for the calculation of the enthalpy curve and the constraint of constant total ligand concentration for the calculation of the heat capacity curve from the enthalpy function, mixes up both models. This was the approach that has been used so far.13-18 Although this procedure is not straightforward, it will lead only to deviations in the numerical values of the heat capacity, if signi®cant changes in the free ligand concentration occur during the course of a DSC experiment. Let us proceed now to the equation for the temperature course of the heat capacity that is

D
Nx N H
N H fx fD 2 RT

…18†

The advantage of this simpler formulation will become obvious below, where a more general formalism will be developed that allows the analysis of multi-site binding phenomena.

The molecular consequences of reference measurements in the presence of ligands Binding of small ligands In a typical DSC or ITC-measurement the signal obtained with the sample solution is corrected for by the signal of an identical solution without the protein. Since both the enthalpy and heat capacity of the protein are partial molar quantities12,23 the only relevant material that constitutes the thermodynamic system ``protein'' is the dry polypeptide chain. Coupling to the energetics of the aqueous surroundings (the heat bath) is re¯ected in the Hamiltonian of the protein. It is not manifested in any increase of the molar mass such as would be

815

Folding Energetics

obtained by a ®xed number of hydration waters.12 Therefore, subtracting the reference in DSC studies means subtracting everything that is not protein. As a consequence the thermodynamic properties derived from the DSC experiment refer to the mass of the dry protein chain. No mass-related properties of the ligand enter the observed value of the protein heat capacity but only the energetic coupling with the protein. This is the physical basis of the proportionality model. In this context the binding of a small ligand can be viewed as a simple energetic perturbation exerted on the protein by the surrounding heat bath which is qualitatively not different from other effects on the protein, such as hydration or preferential binding of a solute.24 ± 26 These latter ligands are usually present at much higher concentrations and are characterised by much lower binding constants, so that no longer the ligand-binding constant but the exchange constant with water is signi®cant. However, the same principles are operative: The mean properties of the protein are controlled by the intensive variables temperature and chemical potential of the ligand (i.e. the activity of the free ligand). Therefore when the in¯uence of the ligand on temperature stability is to be quanti®ed properly, derivatives of ln Q with respect to temperature must be done properly at constant free ligand activity. Binding of proteins and unfolding of non-covalently linked oligomeric proteins In contrast to binding equilibria involving small ligands the situation with (homo-)oligomeric proteins is completely different. In the framework of our model the non-covalently linked subunits constituting an oligomeric protein can be viewed as ligands. Then, we have to realize that in dealing with oligomeric proteins neither the enthalpy nor the heat capacity contributions of the free ligands can be corrected for, since the ligand contributions are obviously not subtracted with the reference. As a further complication changes in the particle number of the protein system occur when the oligomeric protein unfolds to monomers. Since, in this case, the ligand (subunit) belongs directly to the protein system, possible entropy contributions resulting from changes in the particle number are not subtracted with the reference. In view of these considerations Cp curves should be interpreted in the following manner from a theoretical point of view: heat capacity and enthalpy values should be calculated according to the proportionality model in all those cases where the thermodynamic state of the ligand is unaffected by the temperature rise experienced in the DSC run. This condition is obviously not met for the intrinsic interaction of homooligomeric proteins, since all subunits unfold at the same temperature However, in the majority of DSC studies involving small ligands the proportionality model is the appropriate

model for the analysis of binding equilibria. In these cases it is always clear that, as a result of the reference correction, physically the ligand does not belong to the protein system. What is observed experimentally is only the energetic effect of the interaction that results in an increase or a reduction in enthalpy of the macromolecular system. As mentioned above, in ITC-measurements one can in principle meet conditions that permit a distinction between the proportionality model and the stoichiometry model. However, we are not going to pursue this idea in the present study any further. In conclusion, we can make the following statement: it appears that theoretically as well as experimentally the proportionality model is the appropriate model for the analysis of DSC curves of protein unfolding coupled to ligand-binding equilibria, as long as the thermodynamic properties of the ligand can be corrected for by using a reference solution of the ligand. Since moreover the calculations involved in the application of the proportionality model are signi®cantly simpler than those for the stoichiometry model, the former is the method of choice. However, in cases where for example two proteins form a complex and both proteins unfold in the relevant temperature range, there is no possibility to distinguish the ``ligand'' from the binding protein. Therefore an analysis of the binding equilibrium will require different approaches.11,12,20

Multi-site Binding: A General Recipe for the Analysis of Cp-curves on the Basis of the Proportionality Model Calculation of the temperature dependence of folding energetics of proteins in terms of the proportionality model is based on the temperature derivative at constant free ligand concentration of the proper relative partition function Q ˆ iQi. Otherwise stoichiometry effects will enter the enthalpy and the heat capacity and render the formulae too complex to use for deconvolution. Similar to equations presented before4,27 the relations for enthalpy and heat capacity relevant to the proportionality model are readily calculated from equation (2). However, we shall present the heat capacity equations in a more practicable form that provides a general recipe for obtaining heat capacity formulae directly from the assumed reaction scheme: @ ln H ÿ HN ˆ RT

ˆ

k X

Qi

iˆ0

2

ˆ

@T

k X RT 2
H 0 Qi iˆ0

Q

i

RT 2

k RT 2 X @Qi Q iˆ0 @T

ˆ

k X iˆ0


Hi0 fi

…19†

816

Folding Energetics

Cp ÿ Cp;N ˆ

X


Cp;i fi ‡

X

i

ˆ

X i


Cp;i fi ‡

0 @
H 0 Qi Q i



ˆ

i

X


Cp;i fi ‡

ÿ Qi

ˆ

X

X

RT 2


Cp;i fi ‡

Qˆ

With equation (20) we get the heat capacity:

fi …1 ÿ fi †

fi fj

…20†

The fi refer to the population sizes of the i-th protein species and are de®ned by the relation fi ˆ Qi/Q. The number of protein states is k ‡ 1. Since the Qi are of the form Ki[x]n where the [x]n term stems from the overall binding of n ligands to the unliganded native protein and Ki is the respective equilibrium constant, the temperature derivatives of Qi are
HiKi[x]n/ RT2 ˆ
HiQi/RT2, as used above. In the case of a conformational change of the protein without a change in the number of ligands n equals 0. Due to the fact that the free ligand concentration [x] is assumed to be constant, the temperature derivatives of ln Q are mathematically indistinguishable for both cases n ˆ 0 and n 6ˆ 0 in the calculation of H(T) and Cp(T) from the relative partition function Q. Binding of two ligands As an example we will consider in the following two mechanisms of ligand-binding coupled to the unfolding of a protein. First, the sequential binding of two identical ligands will be treated. Then the enthalpy and heat capacity equations relevant to independent stoichiometric binding of two identical ligands will be delineated.28,29 Sequential dissociation of two identical ligands The sequential reaction scheme is: Nx2 „ Nx ‡ x „ N ‡ 2x „ D ‡ 2x The corresponding de®ned as follows: ‰DŠ ; K0 ˆ ‰NŠ

equilibrium

‰NxŠ K1 ˆ ‰NŠ‰xŠ

and

…23†

Nx2 Nx Cp ÿ Cp;N ˆ
D N Cp fD ‡
N Cp fNx ‡
N Cp fNx2

i RT 2

RT 2

‰NŠ ‡ ‰DŠ ‡ ‰NxŠ ‡ ‰Nx2 Š ‰NŠ

ˆ 1 ‡ K0 ‡ K1 ‰xŠ ‡ K2 ‰xŠ2

!

X …
H 0 †2

X X
Hi0
Hj0 j
1


Hj0 Qj A

Q2

i

i

Note that the reaction steps and, correspondingly, all thermodynamic reaction parameters have to be formulated as overall quantities relative to the reference state, which is the native, unliganded protein. It is only under these conditions that equation (20) applies. The relative partition function that is appropriate for these reactions is de®ned by equation (23):

…
Hi0 †2 Qi Q ÿ Q2i RT 2 Q2

X
Hi0
Hj0 Qi Qj

i

ÿ2

X j

i

j
@fi @T

X
H 0 i 2 RT 2 Q i

i

ÿ2


Hi0

constants

…21† are

‰Nx2 Š K2 ˆ …22† ‰NŠ‰xŠ2

‡

0 2 …
D NH † fD …1 ÿ fD † 2 RT

‡

0 2 …
Nx N H † fNx …1 ÿ fNx † 2 RT

‡

0 2 2 …
Nx N H † fNx2 …1 ÿ fNx2 † 2 RT

ÿ2

0 Nx 0
D N H
N H fD fNx RT 2

ÿ2

0 Nx2 0
D N H
N H fD fNx2 RT 2

ÿ2

0 Nx2 0
Nx N H
N H fNx fNx2 RT 2

…24†

fD, fNx and fNx2 are the respective fractions of unfolded unliganded, singly liganded folded and doubly liganded folded protein. The various
H0 0 values refer to the following transitions.
D NH : transition enthalpy of the unliganded protein from 0 2 the native to the unfolded state;
Nx N H : enthalpy of binding of one ligand to the native protein; 0 2
Nx N H : enthalpy of binding of two ligands to the native protein. Independent dissociation of two identical ligands Next, we consider the in¯uence of the independent site binding of two identical ligands on the unfolding equilibrium of a protein. The corresponding reaction scheme is shown in equation (25). In comparison to the reaction scheme of sequential binding presented in equation (21) there is an additional species xN in a parallel reaction which is a consequence of the site speci®c binding to a macromolecule containing two sites: xNx #" Nx ‡ x

„ xN ‡ x #" „ N ‡ 2x

„ D ‡ 2x

…25†

The two singly liganded native species are dis-

817

Folding Energetics

tinguished by writing the ligand x at the right or left hand side of ``N'', respectively. When de®ning the equilibrium constants as in equation (26): K0 ˆ

‰DŠ ‰NŠ

KNx ˆ

‰NxŠ ‰NŠ‰xŠ

KxN ˆ

‰xNŠ ‰NŠ‰xŠ

K2 ˆ

‰xNxŠ ‰NŠ‰xŠ2

…26†

one obtains the following expression for the relative partition function Q: Qˆ

‰NŠ ‡ ‰DŠ ‡ ‰NxŠ ‡ ‰xNŠ ‡ ‰xNxŠ ‰NŠ ˆ 1 ‡ K0 ‡ KNx ‰xŠ ‡ KxN ‰xŠ ‡ K2 ‰xŠ2

…27†

Application of equation (20) results in the following expression for the the heat capacity: Cp ÿ Cp;N ˆ
Cp;0 fD ‡
Nx N Cp fNx Nx2 ‡
xN N Cp fxN ‡
N Cp fNx2

‡

0 2 …
D NH † fD …1 ÿ fD † RT 2

‡

0 2 …
Nx N H † fNx …1 ÿ fNx † RT 2

‡

0 2 …
xN N H † fxN …1 ÿ fxN † RT 2

‡

0 2 2 …
Nx N H † fNx2 …1 ÿ fNx2 † RT 2

ÿ2

0 Nx 0
D N H
N H fD fNx RT 2

ÿ2

0 Nx2 0
D N H
N H fD fNx2 RT 2

ÿ2

0 xN 0
D N H
N H fD fxN RT 2

ÿ2

0 xN 0
Nx N H
N H fNx fxN RT 2

ÿ2

0 Nx2 0
Nx N H
N H fNx fNx2 RT 2

ÿ2

0 Nx2 0
xN N H
N H fxN fNx2 RT 2

…28†

The enthalpies are again identi®ed by the indices at the
. The upper index refers to the ®nal, the lower index to the initial state. A comparison of the two partition functions given in equations (23) and (27) and the corresponding heat capacity equations (24) and (28) shows that after two derivatives with regard to temperature the two binding mechanisms lead to distinguishable heat capacity curves. This makes DSC an outstandingly powerful tool for the discrimination of binding sites, since each chemical reaction, i.e. each ligand-binding step and each conformational change, yields in principle a heat capacity peak.

Analysis of Reaction Schemes by DSC Optimisation of data collection As the absolute value of binding enthalpies is often smaller than 100 kJ/mol, the van't Hoff equation (29) predicts only a small temperature dependence of the equilibrium constant K: @ ln K
HvH ˆÿ R @1=T

…29†

Therefore the degree of binding and, consequently, the free ligand concentration do not change signi®cantly before the protein denatures. Then, depending on ligand concentration, the heat capacity effects that appear in the DSC-curve will largely stem from the denaturation of liganded and unliganded proteins. When ligand-binding is associated with a small enthalpy change, the Nx to D heat capacity peak could be indistinguishable from a simple N to D denaturation peak. In other words, when performing a DSC-measurement at a single total ligand concentration only limited information can be gained on the reaction scheme of the protein. Thus, in general it is imperative to perform a suf®ciently large number of measurements at various ligand concentrations in order to unravel the interaction and unfolding scheme of protein-ligand reactions. Preferably, even measurements using different techniques such as CD, ¯uorescence or ITC should be combined with the DSC data and the complete data set should be subjected to a global ®tting procedure. In summary, the following experimental strategy is advisable for obtaining a maximum of information. The ®rst step will be the determination of the thermodynamic parameters of the unliganded protein. Then a variety of scans in the presence of logarithmically increasing ligand concentrations should be performed. They will show that above ligand concentrations [x]t 4 Kd (dissociation constant) the DSC curve becomes progressively in¯uenced by the presence of the ligand. This is illustrated in Figure 4. It is seen that below this ligand concentration the chemical potential of the ligand is too low for inducing a signi®cant degree

818

Folding Energetics

Figure 4. Example for the dependence of the heat capacity on the temperature and ligand concentration for (a) low and (b) high ligand af®nity. (b) At intermediate ligand concentrations two peaks may occur (continuous line). (a) At lower ligand af®nity this two-peak region cancels out and the heat capacity peak will simply start drifting to higher temperatures close to ligand concentrations that equal the pK value of dissociation. The parameters used for the calNx 0 0 culation of equation (17) are:
D NH ˆ 500 kJ/mol,
N H ˆ ÿ 50 kJ/mol, T1/2 ˆ 340 K, pK(T1/2) ˆ 3.5 (a) or 6 (b), c ˆ 10 mM.

of binding to the protein under investigation (Figure 4(a)). Employing increasingly higher ligand concentrations will result ®rst in the occupation of high af®nity binding sites and, at further increase of the chemical potential of the ligand, also in the occupancy of low af®nity sites. This results in a shift of the Cp curves towards higher transition temperatures. In this way, the reaction scheme of

the protein can be explored with respect to ligand concentration, and in principle all thermodynamic parameters of the conformational changes of the protein and the ligand-binding equilibria can be obtained. It may also happen in some cases that two heat capacity peaks occur at intermediate ligand concentrations as indicated by the red bold double-peaked

819

Folding Energetics

transition curve in Figure 4(b). Similar transition behaviour has been discussed previously by Shrake & Ross,17 however, the mathematical analysis presented there did not distinguish between the stoichiometry and the proportionality model. Estimation of the number of binding sites Overall Gibbs energy of denaturation It is of advantage to have an estimate of the number of binding sites prior to the global deconvolution of the DSC-curves. Such a knowledge reduces the computational efforts signi®cantly. A good estimate can be accomplished in the following manner. As the total Gibbs energy of denaturation is de®ned by the difference between the Gibbs energies of the denatured and native state, respectively, it can be expressed via the partition function of the denatured state, QD, and the partition function Q ÿ QD of all residual native states which evidently include native unliganded and native liganded protein molecules:
G0tot ˆ Gdenatured ÿ Gnative ˆ ÿRT ln…QD † ‡ RT ln…Q ÿ QD †   K0 ˆ ÿRT ln Q ÿ K0

…30†

Since we assume that no ligand-binding occurs in the unfolded state, QD can be equated to the equilibrium constant K0 of denaturation. The derivative of equation (30) at constant temperature with respect to ligand concentration is then:     @
G0tot @ ln…Q ÿ K0 † N ˆ RT ˆ RT X …31† @ ln‰xŠ T @ ln‰xŠ T Therefore we can in principle get an estimate of the average number of ligands bound to the native  N by plotting
G0tot versus ln[x] and calcuprotein X lating the slope. Dependence of the transition temperature on ligand concentration However, usually the value of
G0tot is not directly given by the experiment. Therefore we suggest to use another expression before continuing with the explicit calculation of
G0tot. The following transformation is required for that purpose:       @
G0tot @
G0tot @ ln x ˆ
@ ln‰xŠ
G0 ˆ0 @ ln ‰xŠ T @ ln‰xŠ
G0 ˆ0 tot tot |‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚} |‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚ ‚} ˆ0

ˆ1

    @
G0tot @T ‡
@T ln‰xŠ @ ln ‰xŠ
G0

tot ˆ0

…32†

which yields for T ˆ T1/2, the transition tempera-

ture of the protein where fD ˆ 1 ÿ fD ˆ 0.5 and
G0tot ˆ 0, the following relation:       @T1=2 @
G0tot @
G0tot = …33† ˆÿ @ ln‰xŠ @ ln ‰xŠ T @T ln‰xŠ with …@T=@ ln‰xŠ†
G0tot ˆ0 ˆ …@T1=2 =@ ln‰xŠ†. Inserting the results from equation (31) and remembering that …@
G0tot =@T†TˆT1=2 ˆ ÿ
S0tot;T1=2 one obtains for the dependence of the transition temperature on ligand concentration the expression: 

 2  N RT1=2 XN @T1=2 RT1=2 X ˆ ˆÿ 0 0 @ ln‰xŠ ÿ
Stot;…T1=2 †
Htot;…T1=2 †

…34†

In the last step the relation T1=2
S0tot;…T1=2 † = 0 has been used. Inspection of equation
Htot;…T 1=2 † (34) shows that a ®rst estimate of the average num N , can be obtained from the ber of bound ligands, X slope of a plot of 1/RT1/2 versus ln [x]:   N X @…1=RT1=2 † ˆÿ 0 @ ln‰xŠ
Htot;…T 1=2 †

…35†

provided the total transition enthalpy at T1/2 is known. This is a convenient procedure, since both 0 can be determined parameters, T1/2 and
Htot;…T 1=2 † easily in DSC experiments as a function of the ligand concentration. This method is fast and completely model-independent. Its application has been recommended previously, but usually with restrictive assumptions for the ®tting procedure.18,30,31 For many applications it is, however, useful to have a general analytical function which can be employed in a computer-based ®tting routine. Such a function will be developed in the following. At the transition temperature of the protein, T1/2, the quantity
G0tot(T1/2) de®ned by equation (30) equals zero. Then the relation: X Ki ‰xŠi …36† K0 ˆ Q ÿ K0 ˆ 1 ‡ i

holds. The equilibrium constant for the denaturation of the unliganded native state K0 refers to T1/2,[x] at the respective ligand concentration. It can be calculated from the standard Gibbs energy 0 change
D NG , which in turn is readily determined from the classical stability equation32 as explained below for
G0tot (equation (40)). Solving equation (36) for [x] is relatively simple for up to three ligands (i ˆ 1 ÿ 3). For larger values of i it is in general more cumbersome, but in principle the equation could be solved e.g. by Newton-iteration. Knowledge of the free ligand concentration [x] at T1/2 enables one to calculate the average degree of  with equation (37): ligation, X, X iKi ‰xŠi  ˆ i …37† X Q

820

Folding Energetics

and also the total ligand concentration [x]t using equation (38):  ‰xŠt ˆ ‰xŠ ‡ Xc

…38†

By solving numerically equation (36) for [x] and inserting the result into equations (37) and (38) we have established [x]t as a function of the transition temperature T1/2. Such a relationship could not be derived from equation (35). Equation (38) can be employed to ®t the experimentally determined T1/2 and [x]t pairs to model parameters of
H0i , Ki and
Cp,i. Figure 5 illustrates the dependence on ligand concentration of the transition temperature of a protein of 10ÿ4 M concentration for various theoretical binding constants. The binding enthal0 was assumed to be ÿ20 kJ/mol and py
Nx N H 0 the enthalpy of denaturation
D NH was taken as 500 kJ/mol and both values were kept constant. Only the pK values of the binding reaction were varied. Dotted curves refer to free ligand concentration [x], continuous lines to total ligand concentration [x]t. If the free ligand concentration [x] is used, the plot (broken lines) is simply shifted with the pK value of the binding equilibrium, which is located at the log[x] value, where the plot changes its slope. If, however, the total ligand concentration is known the buffer capacity of the protein with respect to the ligand alters the plot (continuous lines). Therefore up to a total ligand concentration that equals the protein concentration hardly any

additional information can be obtained from such measurements compared to experiments without ligand. One has to employ ligand concentrations that are some orders of magnitude larger than the protein concentration. This shows that a useful thermal denaturation experiment should be done under conditions that exclude any possible unexpected statistical thermodynamic effects that were discussed above. These considerations support the usefulness of the proportionality model in the context of ligand-binding particularly with regard to statistical thermodynamic implications. Only when high af®nity ligands are present and the ligand concentration is in the same order of magnitude as the protein concentration the differences between the two models will become manifest as noted above. Under such conditions more than one heat capacity peak could be observed even though there is only one unliganded and one liganded native state.17 This renders it evidently dif®cult to get an estimate of T1/2 from the experimental curves at low ligand concentrations. However an increase of the ligand concentration by one or more orders of magnitude over the protein concentration will again provide useful results. The plot of 1/RT1/2 versus ln[x] or ln[x]t employs the respective transition temperature in the presence of the ligand which is usually much higher than the physiological temperature of interest. What is, however, frequently of interest is the binding behaviour under physiological conditions.

Figure 5. Dependence of the transition temperature on the free (broken line) or total (continuous line) ligand concentration for different binding af®nities (pK ˆ 1 to 8 from the left to the right). If the free concentration is known the point where the slope changes corresponds to the pK value. If, as usual, only the total ligand concentration is known at high af®nities this point is hidden due to the buffer capacity of the protein with respect to the ligand. This is valid up to ligand concentrations around the protein concentration of c ˆ 0.1 mM. The curves were calculated using Nx 0 0 0 0 0 equations (36) and (38). The parameters
D NH (T1/2) ˆ 500 kJ/mol,
N H (Tj2) ˆ ÿ 20 kJ/mol and T1/2 ˆ 330 K were used.

821

Folding Energetics

This requires an extrapolation procedure for T1/2 which will be delineated in the following. Calculation of the total Gibbs energy of denaturation at T01/2 Integration of the DSC heat capacity peak obtained at different ligand concentrations yields the total enthalpy of denaturation,
H0tot, which can be used for a ®rst order extrapolation to the midpoint temperature of the transition, T01/2, of the protein without ligand using the following equation: 0 0 ; ‰xŠ†  ÿ
S0tot …T1=2 ÿ T1=2;‰xŠ †
G0tot …T1=2 0 …T1=2;‰xŠ † ˆ ÿ
Htot

free ligand concentration. According to equation (31) one obtains the relation:      ! @
G0tot =RT @XN @ ˆ @ ln‰xŠ @T ln‰xŠ @T ln‰xŠ   0 @
Htot =RT 2 ˆÿ @ ln‰xŠ T

T1=2;‰xŠ

tot

‡

…39†

T1/2,[x] refers to the transition temperatures in the presence of the various ligand concentrations. Such an extrapolation to T01/2 can be performed using the
G0tot(T1/2,[x]) values that are obtained at the different transition temperatures T1/2,[x] resulting from the variation in ligand concentration. From the slope of a plot of
G0tot(T01/2, ln [x]) versus ln[x] the number of binding sites at T01/2 can be estimated according to equation (31). The function Q ÿ K0 ˆ 1 ‡ iKi[x]i has the form of a polynomial in [x] with apparent overall binding constants Ki at the temperature T01/2. Therefore it is also possible to ®t the data in the
G0tot(T1/2) versus ln[x] plot to the polynomial Q ÿ K0. The ®t parameters are then the binding constants Ki at the transition temperature 0 of the ligand-free protein. T1=2

…41†

The formula is used in the transformation:       0 0 @
Htot @T @
Htot ˆ @T @T @T
G0 ˆ0
G0 ˆ0 ln‰xŠ tot

0 T1=2 ÿ T1=2;‰xŠ

T

1 

@T @ ln‰xŠ

  0 @
Htot …42† @ ln‰xŠ T


G0tot ˆ0

which yields for shifts in the transition temperature:      0 0 @
Htot
Htot …T1=2 † @X N 0 …43† ˆ
Cp;tot …T1=2 † ÿ  @T1=2 @T XN ln‰xŠ This equation shows that a
Hcal versus T1/2 plot re¯ects generally not only the heat capacity difference
C0p, resulting from unfolding of the liganded and ligand free proteins, but also the contribution due to any change in the degree of ligation with temperature. This pertains to changes in the degree of protonation as much as to changes in the degree of binding of other ligands. The latter contribution is represented by the second term on the right hand side of equation (43).

Usage of
Cp for the extrapolation of the Gibbs free energy An improvement in the accuracy of the extrapolation of
G0tot to T01/2 can be obtained, if equation (39) is modi®ed to take the heat capacity change,
C0p, into account. The value for
C0p, can be estimated from a plot of
Htot(T1/2([x])) versus T1/2,[x]. If
C0p is known, equation (40) is identical with the classical stability equation:32 0
G0tot …T1=2 ; ‰xŠ† 0 …T1=2;‰xŠ † ˆ ÿ
Htot

ÿ


C0p

0 T1=2 ÿ T1=2;‰xŠ

T1=2;‰xŠ ÿ

T1=2;…‰xŠ† 0 T1=2

‡

0 T1=2

ln

0 T1=2

T1=2;‰xŠ

!! …40†

Although the estimate of
C0p from transition curves at different ligand concentrations is not without problems, since it may be associated with errors up to 50 %33 it is still better practice to use
C0p for the extrapolation than not to use it. Finally, we calculate the contributions to
C0p from ligand-binding. We start with the temperature dependence of the average ligation at constant

Summary Two different methods for the calculation of heat capacity curves as a function of temperature and ligand concentration can be employed, depending on whether the temperature derivatives of the partition function are calculated under the constraint of: (i) constant free ligand; or (ii) constant total ligand concentration. We have shown that the application of these constraints can lead to different results. However, under excess of ligand, a condition that is commonly used in DSC studies, the numerical values will be indistinguishable. Therefore, the application of the simpler proportionality model is useful, since it renders an analytical solution possible even for very complex reaction schemes. Moreover, we showed that most likely the constraint of keeping the free ligand concentration constant in the temperature derivatives is the physically correct one. A general procedure was delineated for obtaining from DSC studies ligand-binding parameters. This included the extraction of basic thermodynamic parameters such as
H0,
Cp,

822
G0 and T1/2 and also the recipe for generating the theoretical heat capacity curves in the presence of ligands.

Acknowledgements Financial support to HJH by the DFG grants Hi 204/24-1 and GRK 234/1-96 and by the Fonds der Chemischen Industrie is gratefully acknowledged.

References 1. Privalov, P. L. & Khechinashvili, N. N. (1974). A thermodynamic approach to the problem of stabilisation of globular protein structure: a calorimetric study. J. Mol. Biol. 83, 665-684. 2. Lumry, R. & Biltonen, R. (1966). Validity of the ``two-state'' hypothesis for conformational transitions of proteins. Biopolymers, 4, 917-944. 3. Cooper, A. (1984). Protein ¯uctuations and the thermodynamic uncertainty principle. Prog. Biophys. Mol. Biol. 44, 181-214. 4. Privalov, P. L. & Potekhin, S. A. (1986). Scanning microcalorimetry in studying temperatureinduced changes in proteins. Methods Enzymol. 131, 4-51. 5. Freire, E. & Biltonen, R. L. (1978). Statistical mechanical deconvolution of thermal transitions in macromolecules. Biopolymers, 17, 463-479. 6. Griko, Y. V., Freire, E., Privalov, G., van Dael, H. & Privalov, P. L. (1995). The unfolding thermodynamics of c-type lysozymes: a calorimetric study of the heat denaturation of equine lysozyme. J. Mol. Biol. 252, 447-459. 7. Marky, L. A. & Breslauer, K. J. (1987). Calculating thermodynamic data for transitions of any molecularity from equilibrium melting curves. Biopolymers, 26, 1601-1620. 8. Kidokoro, S.-I. & Wada, A. (1987). Determination of thermodynamic functions from scanning calorimetry data. Biopolymers, 26, 213-229. 9. Kidokoro, S.-I., Uedaira, H. & Wada, A. (1988). Determination of thermodynamic functions from scanning calorimetry data. II. For the system that includes self-dissociation/association processes. Biopolymers, 27, 271-297. 10. Freire, E. (1989). Statistical thermodynamic analysis of the heat capacity function associated with protein folding-unfolding transitions. Comments Mol. Cell. Biophys. 6, 123-140. 11. RoÈsgen, J., Hallerbach, B. & Hinz, H.-J. (1998). The ``Janus'' nature of proteins. Biophys. Chem. 74, 153161. 12. RoÈsgen, J. & Hinz, H.-J. (1999). Statistical thermodynamic treatment of conformational transitions of monomeric and oligomeric proteins. Phys. Chem. Chem. Phys. 1, 2327-2333. 13. Robert, C. H., Gill, S. J. & Wyman, J. (1988). Quantitative analysis of linkage in macromolecules when one ligand is present in limited total quantity. Biochemistry, 27, 6829-6835. 14. Robert, C. H., Colosimo, A. & Gill, S. J. (1989). Allosteric formulation of thermal transitions in macromolecules, including effects of ligand binding and oligomerization. Biopolymers, 28, 1705-1729.

Folding Energetics 15. Brandts, J. F. & Lin, L.-N. (1990). Study of strong to ultratight protein interactions using differential scanning calorimetry. Biochemistry, 29, 6927-6940. 16. Straume, M. & Freire, E. (1992). Two-dimensional differential scanning calorimetry: simultaneous resolution of intrinsic protein structural energetics and ligand binding interactions by global linkage analysis. Anal. Biochem. 203, 259-268. 17. Shrake, A. & Ross, P. D. (1990). Ligand-induced biphasic protein denaturation. J. Biol. Chem. 265, 5055-5059. 18. Shrake, A. & Ross, P. D. (1992). Origins and consequences of ligand-induced multiphasic thermal protein denaturation. Biopolymers, 32, 925-940. 19. Martinez, J. C., Harrous, M. E., Filimonov, V. V., Mateo, P. L. & Fersht, A. R. (1994). A calorimetric study of the thermal stability of barnase and its interaction with 30 GMP. Biochemistry, 33, 3919-3926. 20. RoÈsgen, J. & Hinz, H.-J. (1999). Theory and practice of DSC measurements on proteins. In Handbook of Thermal Analysis and Calorimetry (Kemp, R. B., ed.), pp. 63-109, Elsevier, Amsterdam. 21. Gopal, S. & Ahluwalia, J. C. (1995). Differential scanning calorimetric studies on binding of N-acetylD-glucosamine to lysozyme. Biophys. Chem. 54, 119-125. 22. Creagh, A. L., Koska, J., Johnson, P. E., Tomme, Pl, Joshi, M. D. & McIntosh, L. P., et al. (1998). Stability and oligosaccharide binding of the N1 cellulosebinding domain of Cellulomonas ®mi Endoglucansase CenC. Biochemistry, 37, 3529-3537. 23. Privalov, P. L. (1979). Stability of proteins. Advan. Protein Chem. 33, 167-241. 24. Timasheff, S. N. (1993). The control of protein stability and association by weak interactions with water: how do solvents affect these processes? Annu. Rev. Biophys. Biomol. Struct. 22, 67-97. 25. Schellman, J. A. (1978). Solvent denaturation. Biopolymers, 17, 1305-1322. 26. Schellman, J. A. (1987). Selective binding and solvent denaturation. Biopolymers, 26, 549-559. 27. Jelesarov, I. & Bosshard, H. R. (1999). Isothermal titration calorimetry and differential scanning calorimetry as complementary tools to investigate the energetics of biomolecular recognition. J. Mol. Recog. 12, 3-18. 28. Wyman, J. & Gill, S. J. (1990). Binding and Linkage, University Science Books, Mill Vale, CA. 29. di Cera, E. (1995). Thermodynamic Theory of Sitespeci®c Binding Processes in Biological Macromolecules, Cambridge University Press, New York. 30. Schellman, J. A. (1975). Macromolecular binding. Biopolymers, 14, 999-1018. 31. Muccio, D. D., Waterhous, D. V., Fish, F. & Brouillette, C. G. (1992). Analysis of the two-state behaviour of the thermal unfolding of serum retinol binding protein containing a single retinol ligand. Biochemistry, 31, 5560-5567. 32. Becktel, W. J. & Schellman, J. A. (1987). Protein stability curves. Biopolymers, 26, 1859-1877. 33. Kidokoro, S., Miki, Y. & Wada, W. (1990). Physical and biological stability of globular proteins. In Protein Structural Analysis, Folding and Design (Hatano, M., ed.), pp. 78-85, Japan Scienti®c Societies Press and Elsevier, Amsterdam. 34. Warner, F. W. (1983). Foundations of Differentiable Manifolds and Lie Groups, Springer Verlag, Berlin.

823

Folding Energetics

Appendix I Derivative With Respect to Temperature at Constant Total Ligand Concentration In analogy to the treatment of oligomerization effects described previously by RoÈsgen & Hinz,.A1 the following relationship holds for the various partial derivatives:     @ 1 @  ˆ …AI1† @‰xŠt @‰xŠt T @‰xŠ T @‰xŠ T [x]t is the sum of free and bound ligand concentration. Since the concentration of bound ligand  where c is the protein can be expressed by c
X,  the average monomer concentration and X amount of ligand bound per mole of monomer, equation (AI1) becomes:     @ 1 @ ˆ …AI2†   @‰xŠt T @‰xŠ T @X 1‡c @‰xŠ T The temperature derivative at constant free ligand concentration can be expressed in the following manner:           @ @T @ @‰xŠt @ ˆ ‡ …AI3† @T ‰xŠ @‰xŠt T @T ‰xŠ @T @T ‰xŠt |‚‚‚‚{z‚‚‚‰xŠ ‚} ˆ1

With:        @‰xŠt @…‰xŠ ‡ cX† @X ˆ ˆc @T ‰xŠ @T @T ‰xŠ ‰xŠ

fD 0 1 ÿ fx c=‰xŠt ÿ
Nx N Cp fx 1 ÿ fx2 c=‰xŠt 1 ÿ fx2 c=‰xŠt     @K @Q Q ÿK @T ‰xg Š @T ‰xg Š

0
Cp ÿ
D N Cp

…AI4†

this results in the following equation:         @ @ c @X @ ˆ ÿ   @T ‰xŠt @T ‰xŠ @T ‰xŠ @‰xŠ T @X 1‡c @‰xŠ T …AI5† For the mathematical background of the calculations see e.g. Warner.A2

0 ˆ
D NH

Q2 …1 ÿ fx2 c=‰xŠt †

    @Qx @Q Q ÿQx @T ‰xg Š @T ‰xg Š

2fx c=‰xŠt Q2 …1 ÿ fx2 c=‰xŠt †2    2     @Qx @Q Nx 0 ‡
N H Q ÿQx Q2 @T ‰xg Š @T ‰xg Š " 1 ÿ fx c=‰xŠt fx ‡  1 ÿ fx2 c=‰xŠt …1 ÿ fx2 c=‰xŠt †2 0 ‡
D N H fD


‰ÿc=‰xg Š…1 ÿ fx2 c=‰xŠt † ‡ 2fx c=‰xŠt …1 ÿ fx c=‰xŠt †Š " # 0 2 …
D 1 ÿ fD 2fD fx2 c=‰xŠt NH † fD ÿ ˆ RT 2 1 ÿ fx2 c=‰xŠt …1 ÿ fx2 c=‰xŠt †2   @Qx @T ‰xg Š 1 ÿ fx c=‰xŠt …2 ÿ fx † 0 ÿ
D N H fD Q …1 ÿ fx2 c=xg Š†2   0 @Qx K
D NH …1 ‡ K† ÿQx 2 @T ‰xg Š RT 0 ‡
Nx N H 2 Q2 …1 ÿ fx2 c=‰xg Š†  ‰1 ÿ 2fx c=‰xg Š ‡ fx2 c=‰xg ŠŠ

…AII2†

Because of Qx ˆ Q ÿ 1 ÿ K the derivative (@Qx/ @T)[xg] is directly calculated from equation (13) by 0 2 subtraction of K
D NH /RT . Therefore, the heat capacity can be written as:

Cp ˆ Cp;N ‡
D N Cp ‡
Nx N Cp fx ‡

fD 1 ÿ fx2 c=‰xŠt

1 ÿ fx c=‰xŠt 1 ÿ fx2 c=‰xŠt

0 2 …
D fD NH † RT 2 …1 ÿ fx2 c=‰xŠt †3

Appendix II

 ‰…1 ÿ fx2 c=‰xŠt † ÿ fD ‰1 ‡ fx2 c=‰xŠt …1 ÿ 2fx †ŠŠ

Calculation of the Canonical Heat Capacity from the Canonical Enthalpy

‡

From the canonic enthalpy: fD 0 0 1 ÿ fx c=‰xŠt ‡
Nx
H ˆ
D NH N H fx 1 ÿ fx2 c=‰xŠt 1 ÿ fx2 c=‰xŠt …AII1† the heat capacity is readily calculated as:



0 2 …
Nx fx …1 ÿ fx † N H † 2 RT …1 ÿ fx2 c=‰xŠt †3

 …1 ÿ fx c=‰xŠt †‰1 ÿ fx c=‰xŠt …2 ÿ fx †Š ÿ2

0 D 0
Nx fx fD N H
N H 2 RT …1 ÿ fx2 c=‰xŠt †3

 …1 ÿ fx c=‰xŠt †‰1 ÿ fx c=‰xŠt …2 ÿ fx †Š

…AII3†

824

Folding Energetics

References A1. RoÈsgen, & Hinz, H.-J. (1999). Statistical thermodynamic treatment of conformational transitions of monomeric and oligomeric proteins. Phys. Chem. Chem. Phys. 1, 2327-2333.

A2. Warner, F. W. (1983). Foundations of Differentiable Manifolds and Lie Groups, Springer Verlag, Berlin.

Edited by R. Huber (Received 21 August 2000; received in revised form 21 November 2000; accepted 21 November 2000)