**2.1 Experimental conditions for DSC**

The observed or apparent thermodynamics of solution systems generally include linked contributions from other solutes in addition to the species of interest. They include buffers,

Probing Solution Thermodynamics by Microcalorimetry 873

**Buffer pKa** *H***°, kJ mol-1** *C***p°, J K-1 mol-1** *V***°, mL mol-1 <sup>a</sup>**





*H C dT* . (4)

. (5)




Acetate 4.756 -0.41 -142 -10.6 Bicine 8.334 26.34 0 -2.0 Bis-tris 6.484 28.4 27 3.1 Cacodylate 6.28 -3 -86 -13.3

HEPES 7.564 20.4 47 4.8 Imidazole 6.993 36.64 -9 1.8 MES 6.27 14.8 5 3.9 MOPS 7.184 21.1 25 4.7

Tris 8.072 47.45 -142 4.3

A complete DSC experiment consists of matched scans of a sample and a sample-free reference solution. Blank-subtracted data can be empirically analyzed to obtain modelindependent thermodynamic parameters. The difference between pre- and post-transition baselines gives the change in heat capacity, *C*p. After subtracting a suitable baseline across the transition range, the arithmetic integration of the *C*p vs. *T* trace yields the so-called

> cal p *T T f i*

*H*cal is the value of the transition enthalpy at the transition temperature, *T*m. The entropy at

cal ( ) *<sup>m</sup> m*

*T*

*<sup>H</sup> S T*

Thus, a single DSC experiment yields the complete thermodynamics of a transition. Modelfree determination of thermodynamics, including the direct measurement of *C*p, is a unique feature of DSC not possible with optical techniques (such as absorption and fluorescence spectroscopy) which access *H* via the van't Hoff equation. However, model

4.07 2.23 -3.38

4 44.2

> -8 3.6 16

3.0 -0.5

Table 1. Thermodynamic properties of ionization by common aqueous buffers

Citrate

Phosphate

Glycine 2.351

Succinate 4.207

**2.2 Analysis of DSC data** 

calorimetric enthalpy:

*T*m is

3.128 4.761 6.396

9.780

2.148 7.198 12.35

5.636

a Ionization volume at atmospheric pressure at infinite dilution.

salts, neutral cosolutes, and cosolvents. Of these, the choice of buffer, or any ionizable species in general, must take into account the change in p*K*a with respect to temperature *i.e.*, the enthalpy of ionization (*H*ion). Unless a buffer's *H*ion is negligibly small, its p*K*a will exhibit a temperature dependence, leading to a change in pH of the solution upon heating or cooling. Failure to take this fact into account may introduce significant artifacts into the observed melting behavior. Such changes in pH represent a different issue from any coupled ionization enthalpy arising from the release or uptake of protons associated with the transition of interest.

The direction and extent of the temperature of pH for a buffered solution depends on the sign and magnitude of *H*ion as well as the concentration of the buffering species. Consider the ionization of a buffer A*Z* in the direction of deprotonation to produce its conjugate base B*Z-1*:

$$\mathbf{A}^{\circ} \rightleftharpoons \mathbf{B}^{\circ - \mathbf{t}} + \mathbf{H}^{\circ} \tag{1}$$

According to the van't Hoff equation,

$$\frac{dK\_{\rm ion}}{d(1/T)} = \frac{d10^{-\rm p\%}}{d(1/T)} = -\frac{dH\_{\rm ion}}{R} \,, \tag{2}$$

where *R* is the gas constant, *T* is absolute temperature, and

$$K\_{\rm ion} = \frac{[\rm H'][\rm B^{z+}]}{[\rm A']}.\tag{3}$$

Thus, for a buffer with a positive (endothermic) *H*ion, its ionization equilibrium shifts towards deprotonation as temperature increases, leading to a drop in pH. Conversely, the pH of a solution buffered by an exothermic buffer rises with increasing pH.

Table 1 lists several common buffers for aqueous solutions (King 1969; Disteche 1972; Lo Surdo et al. 1979; Kitamura and Itoh 1987; Goldberg et al. 2002). As a group, substituted ammonium compounds exhibit substantial positive values of *H*ion, making them poor choices for DSC experiments. These compounds include the so-called "Good buffers" (Good et al. 1966) that are prevalent in biochemistry. Among these, Tris, is a particular offender: the pH of a 25 mM solution initially buffered at pH 9.0 drops by more than one pH unit from 0 to 37°C (Poon et al. 2002). In contrast, the ionization of carboxylic acids and their analogues is far less sensitive to temperature. Generally, buffers based on acetate, cacodylate, and phosphate, for example, are preferred choices for DSC experiments.

Another important note relates to polyprotic species such as phosphates, citrates, and borates, whose p*K*a also depends markedly on ionic strength. This relationship is quantitatively given by the volume changes of ionization (*V*°) and interpreted in terms of electrostriction of solvent water molecules. Thus, the addition of salts such as NaCl or guanidinium salts (the latter commonly used to denature proteins) will systematically reduce the pH of a solution buffered by polyprotic acids. The pH of a 0.1 M phosphate buffer at pH 7.2, for example, can fall by 0.5 pH unit upon addition of 0.5 M of NaCl. Molar concentrations of guanidinium hydrochloride will produce an even greater drop. On the other hand, inorganic cosolvents have the opposite effect by affecting the solution dielectric. Of course, once the pH of these buffers is adjusted to a value that is compatible with the apparent p*K*a, it will be stable with respect to temperature. As seen in Table 1, it is generally the case that a buffer is either sensitive to temperature or ionic strength in aqueous solution.

salts, neutral cosolutes, and cosolvents. Of these, the choice of buffer, or any ionizable species in general, must take into account the change in p*K*a with respect to temperature *i.e.*, the enthalpy of ionization (*H*ion). Unless a buffer's *H*ion is negligibly small, its p*K*a will exhibit a temperature dependence, leading to a change in pH of the solution upon heating or cooling. Failure to take this fact into account may introduce significant artifacts into the observed melting behavior. Such changes in pH represent a different issue from any coupled ionization enthalpy arising from the release or uptake of protons associated with

The direction and extent of the temperature of pH for a buffered solution depends on the sign and magnitude of *H*ion as well as the concentration of the buffering species. Consider the ionization of a buffer A*Z* in the direction of deprotonation to produce its conjugate base

ion ion 10

+ -1

[H ][B ] [A ] *z*

Thus, for a buffer with a positive (endothermic) *H*ion, its ionization equilibrium shifts towards deprotonation as temperature increases, leading to a drop in pH. Conversely, the

Table 1 lists several common buffers for aqueous solutions (King 1969; Disteche 1972; Lo Surdo et al. 1979; Kitamura and Itoh 1987; Goldberg et al. 2002). As a group, substituted ammonium compounds exhibit substantial positive values of *H*ion, making them poor choices for DSC experiments. These compounds include the so-called "Good buffers" (Good et al. 1966) that are prevalent in biochemistry. Among these, Tris, is a particular offender: the pH of a 25 mM solution initially buffered at pH 9.0 drops by more than one pH unit from 0 to 37°C (Poon et al. 2002). In contrast, the ionization of carboxylic acids and their analogues is far less sensitive to temperature. Generally, buffers based on acetate,

Another important note relates to polyprotic species such as phosphates, citrates, and borates, whose p*K*a also depends markedly on ionic strength. This relationship is quantitatively given by the volume changes of ionization (*V*°) and interpreted in terms of electrostriction of solvent water molecules. Thus, the addition of salts such as NaCl or guanidinium salts (the latter commonly used to denature proteins) will systematically reduce the pH of a solution buffered by polyprotic acids. The pH of a 0.1 M phosphate buffer at pH 7.2, for example, can fall by 0.5 pH unit upon addition of 0.5 M of NaCl. Molar concentrations of guanidinium hydrochloride will produce an even greater drop. On the other hand, inorganic cosolvents have the opposite effect by affecting the solution dielectric. Of course, once the pH of these buffers is adjusted to a value that is compatible with the apparent p*K*a, it will be stable with respect to temperature. As seen in Table 1, it is generally the case that a buffer is either sensitive to temperature or ionic strength in aqueous solution.

*pKa dK d dH dTdT R* 

(1 / ) (1 / )

ion

cacodylate, and phosphate, for example, are preferred choices for DSC experiments.

pH of a solution buffered by an exothermic buffer rises with increasing pH.

<sup>1</sup> A BH *z z* (1)

, (2)

*<sup>z</sup> K* . (3)

the transition of interest.

According to the van't Hoff equation,

where *R* is the gas constant, *T* is absolute temperature, and

B*Z-1*:


a Ionization volume at atmospheric pressure at infinite dilution.

Table 1. Thermodynamic properties of ionization by common aqueous buffers

#### **2.2 Analysis of DSC data**

A complete DSC experiment consists of matched scans of a sample and a sample-free reference solution. Blank-subtracted data can be empirically analyzed to obtain modelindependent thermodynamic parameters. The difference between pre- and post-transition baselines gives the change in heat capacity, *C*p. After subtracting a suitable baseline across the transition range, the arithmetic integration of the *C*p vs. *T* trace yields the so-called calorimetric enthalpy:

$$
\Delta H\_{\text{cal}} = \int\_{T\_l}^{T\_f} \mathbb{C}\_p dT \,\,. \tag{4}
$$

*H*cal is the value of the transition enthalpy at the transition temperature, *T*m. The entropy at *T*m is

$$
\Delta S(T\_{\text{w}}) = \frac{\Delta H\_{\text{cal}}}{T\_{\text{w}}} \,. \tag{5}
$$

Thus, a single DSC experiment yields the complete thermodynamics of a transition. Modelfree determination of thermodynamics, including the direct measurement of *C*p, is a unique feature of DSC not possible with optical techniques (such as absorption and fluorescence spectroscopy) which access *H* via the van't Hoff equation. However, model

Probing Solution Thermodynamics by Microcalorimetry 875

perform manual baseline subtraction before fitting a excess heat capacity function. This is

baseline subtraction requires either heuristic or semi-empirical criteria to connect the preand post-transitional states. This is both unnecessary and questionable practice, since manual editing may (and probably do) bias the data. The most appropriate approach is to fit both the excess and intrinsic heat capacities directly according to Eq (7). Since both terms are functions of *i*(*T*), the fitted baseline will objectively track the progress of each transition. Note that *C*p,0 and *C*p,*<sup>i</sup>* are taken to be constants in Eqs (6) and (7) since their temperature dependence is generally weak over the experimental range. They can be formulated, if desired, as polynomials to define nonlinear baselines. Care must be taken, however, to ensure that such curvature is not masking some low-enthalpy transition such as

The principal task in formulating DSC models is deriving expressions for *i*(*T*) from the relevant equilibrium expressions and equations of state. Implicit in this task is the computation of *K*i from its corresponding thermodynamic parameters. This in turn requires the choice of a reference temperature, the most convenient of which is the characteristic

<sup>p</sup> ( ) ( )1 ln *T T GT HT CTT T*

Again *C*p is taken to be independent of temperature in the experimental range. The simplest DSC model involves the isomeric conversion of a species in a strictly two-state manner (*i.e.*, no intermediate state is populated at equilibrium). The denaturation of many

> 2 p p 2 2 ( ) ( 1) <sup>1</sup> *KHK C T <sup>C</sup> K RT K*

> > and int

p, ( ) *C Ti* 

p, ( ) *C Ti* 

function.

*<sup>K</sup> X nX* , (10)

p, ( ) *C Ti* 

traces in Figure 1 are simulated using Eq (9) with *H* = 50 kcal/mol, *C*p = 500 cal mol-1 K-1, and *T*° = 50°C (1 cal 4.184 J). In this model, *T*° is the midpoint of the transition (*i.e.*, *K* = 1

For transitions involving changes in molecularity, *i*(*T*) includes total sample concentration, *c*t in addition to equilibrium constants. While the mechanics of formulating such models is not different, a potential source of inconsistency arises from the choice of unit in thermodynamic parameters. Specifically, every intensive thermodynamic parameter can be defined either per unit of monomer or oligomer. Either choice is correct, of course, but the resultant differences may lead to some confusion. Take for example a two-state homo-

single-domain proteins exemplifies this model. This excess heat capacity function is

) and marks the maximum of the int

*n*

oligomeric transition (Privalov and Potekhin 1986; Freire 1989):

*T T* . (8)

. (9)

, respectively. The DSC

conformation changes of proteins in the native state (Privalov and Dragan 2007).

from the fitting function. In the transition region, manual

intended to eliminate int

**2.2.1 Formulation of DSC models** 

temperature *T*° at which *G*(*T*°) = 0:

The two terms on the right side represent tr

and 0.5 1 *K K* 

p, ( ) *C Ti* 

fitting by least-square analysis can extract considerably more useful information and facilitate quantitative hypothesis testing.

In general, the reference-subtracted DSC data represent the heat capacity of the initial state 0 (*C*p,0) as well as the excess heat capacity function, <*C*p(*T*)>:

$$\mathcal{C}\_{\rm p}(T) = \mathcal{C}\_{\rm p,0} + \left\{ \Delta \mathcal{C}\_{\rm p}(T) \right\}. \tag{6}$$

Consider a general model in which the sample undergoes a transition from initial state 0 through intermediates 1, 2, ... , *i* to the final state *n*. (One can readily envisage extensions of this model in which a heterotypic complex dissociates into subunits which then go on to further, independent transitions.) The excess heat capacity function is (Privalov and Potekhin 1986)

$$\begin{aligned} \left\{\Delta \mathbf{C}\_{\mathbf{p}}(T)\right\} &= \frac{d\left\{\Delta H(T)\right\}}{dT} = \frac{d}{dT} \left[\sum\_{i=1}^{n} \Delta H\_{i}(T) \alpha\_{i}(T)\right] \\ &= \sum\_{i=1}^{n} \Delta H\_{i} \frac{d\alpha\_{i}(T)}{dT} + \sum\_{i=1}^{n} \alpha\_{i}(T) \Delta C\_{p,i} \\ &= \sum\_{i=1}^{n} \left\{\partial \mathbf{C}\_{p,i}^{\rm tr}(T)\right\} + \sum\_{i=1}^{n} \left\{\partial \mathbf{C}\_{p,i}^{\rm int}(T)\right\} \end{aligned} \tag{7}$$

where *Hi* and Cp,*i* are the enthalpy and heat capacity change, respectively, from state 0 to state *i*, and *i*(*T*) is the fractional conversion at state *i*. tr p, ( ) *C Ti* and int p, ( ) *C Ti* are called "transition" and "intrinsic" heat capacities, respectively. The intrinsic heat capacity, which represents the summed heat capacities of the various species present at *T*, is the baseline function of the observed DSC trace (Figure 1). Some analytical protocols invite the user to

Fig. 1. An excess heat capacity function and its constituent transition and intrinsic heat capacities. Integration of the transition heat capacity, tr p, ( ) *C Ti* , gives *H*cal at the transition temperature (50°C here).

fitting by least-square analysis can extract considerably more useful information and

In general, the reference-subtracted DSC data represent the heat capacity of the initial state 0

Consider a general model in which the sample undergoes a transition from initial state 0 through intermediates 1, 2, ... , *i* to the final state *n*. (One can readily envisage extensions of this model in which a heterotypic complex dissociates into subunits which then go on to further, independent transitions.) The excess heat capacity function is (Privalov and

1 1

*dT*

*i i n n*

*n n i*

1 1

*i i*

tr int p, p,

*i i*

*CT CT*

( ) ( ) () ()

*d HT <sup>d</sup> C T HT T dT dT*

where *Hi* and Cp,*i* are the enthalpy and heat capacity change, respectively, from state 0 to

"transition" and "intrinsic" heat capacities, respectively. The intrinsic heat capacity, which represents the summed heat capacities of the various species present at *T*, is the baseline function of the observed DSC trace (Figure 1). Some analytical protocols invite the user to

270 280 290 300 310 320 330 340 350 360 370 380

T, K

Fig. 1. An excess heat capacity function and its constituent transition and intrinsic heat

1

*n*

( ) ( )

*d T H T <sup>C</sup>*

*i i i*

*i*

() ()

 

p p,0 p *CT C CT* () () . (6)

p,

p, ( ) *C Ti* 

Cp

, gives *H*cal at the transition

p, ( ) *C Ti* 

and int

, (7)

p, ( ) *C Ti* 

are called

*i i*

facilitate quantitative hypothesis testing.

Potekhin 1986)

(*C*p,0) as well as the excess heat capacity function, <*C*p(*T*)>:

p

state *i*, and *i*(*T*) is the fractional conversion at state *i*. tr

3500 <Cp

> <Ctr p > <Cint p >

0

capacities. Integration of the transition heat capacity, tr

500

1000

Cp, cal mol-1 K-1

temperature (50°C here).

perform manual baseline subtraction before fitting a excess heat capacity function. This is intended to eliminate int p, ( ) *C Ti* from the fitting function. In the transition region, manual baseline subtraction requires either heuristic or semi-empirical criteria to connect the preand post-transitional states. This is both unnecessary and questionable practice, since manual editing may (and probably do) bias the data. The most appropriate approach is to fit both the excess and intrinsic heat capacities directly according to Eq (7). Since both terms are functions of *i*(*T*), the fitted baseline will objectively track the progress of each transition. Note that *C*p,0 and *C*p,*<sup>i</sup>* are taken to be constants in Eqs (6) and (7) since their temperature dependence is generally weak over the experimental range. They can be formulated, if desired, as polynomials to define nonlinear baselines. Care must be taken, however, to ensure that such curvature is not masking some low-enthalpy transition such as conformation changes of proteins in the native state (Privalov and Dragan 2007).
