**2.2.1 Formulation of DSC models**

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

$$
\Delta \mathbf{G}(T) = \Delta H(T^{\odot}) \left( 1 - \frac{T}{T^{\odot}} \right) + \Delta C\_p \left( T - T^{\odot} + T \ln \frac{T^{\odot}}{T} \right). \tag{8}
$$

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

$$
\left\langle \Delta \mathcal{C}\_{p}(T) \right\rangle = \frac{K}{(K+1)^{2}} \frac{\Delta H^{2}}{RT^{2}} + \frac{K}{K+1} \Delta \mathcal{C}\_{p} \,. \tag{9}
$$

The two terms on the right side represent tr p, ( ) *C Ti* and int p, ( ) *C Ti* , respectively. The DSC 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 and 0.5 1 *K K* ) and marks the maximum of the int p, ( ) *C Ti* function.

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

$$X\_\* \xrightleftharpoons{K} nX\_\* \tag{10}$$

Probing Solution Thermodynamics by Microcalorimetry 877

An additional consideration for transitions involving changes in molecularity concerns the choice and interpretation of the reference temperature. In contrast with isomeric transitions, *T*° is neither the midpoint of a transition (in the context of concentrations) nor does it mark the maximum of the transition heat capacity function. Both of the latter temperatures are lower than *T*°. In addition, the midpoint of the transition, *T*50, is below the temperature of the transition heat capacity maximum, *T*m. These relationships are illustrated for the twostate dissociation model in Figure 2. The non-equivalence of *T*m and *T*50 also introduces a systematic difference between the calorimetric and van't Hoff enthalpies (*H*vH, reported at Tm) (Freire 1989; Freire 1995). Moreover, the actual values of *T*50 and *T*m are concentrationdependent, and this serves as a diagnostic for a change in molecularity in the transition. For data fitting purposes, *T*° remains the most efficient choice because it is independent of concentration. After data fitting, estimates of *T*m and *T*50 can also be easily obtained from the

Extension of the foregoing discussion applies readily to multi-state transitions. However, an explicit, statistical thermodynamic approach is generally used to derive the required equations for each state as a function of the partition function (Freire and Biltonen 1978). Details in deriving these models have been discussed extensively by Privalov's and Freire's groups (Privalov and Potekhin 1986; Freire 1994). From the standpoint of numerical analysis, it is worth noting that the excess enthalpy is the summed contributions from each

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

Depending on the number of states considered, the expansion of the derivative on the right side of Eq (11) can be formidable. Commercial programs such as Mathematica (Wolfram Research, Champaign, IL, USA) are thus recommended for symbolic manipulation for all but the most trivial derivations. Less preferably, one can numerically integrate the raw *C*<sup>p</sup> vs. *T* data and fit *H T*( ) directly. There are generally enough data points (at 0.1°C

As its name indicates, ITC measures the heat change accompanying the injection of a titrant into titrate at constant temperature. In contemporary instruments, this is accomplished by compensating for any temperature difference between the sample and reference cells (the latter lacking titrate, usually just water). The raw ITC signal is therefore power *P*, a timedependent variable. Integration with respect to time therefore yields heat *q* which is the

> 0 ( ) *<sup>t</sup> q t Pdt*

Typically, ITC is operated in incremental mode in which the titrant is injected in preset aliquots after successive re-equilibration periods. A feature of ITC that distinguishes it from most titration techniques is that the measured heat does not accumulate from one injection to the next, but dissipates as the instrument measures the heat signal by returning the sample and reference cells to isothermal conditions. ITC is therefore a differential technique

resolution) in an experiment that any loss of resolution should be negligibly small.

primary dependent variable that tracks the progress of the titration of interest:

1 ( ) () () *<sup>n</sup>*

*i*

*i i*

(11)

. (12)

p

**3. Isothermal titration calorimetry** 

fitted curve.

state:


where *K* is the equilibrium dissociation constant. Table 2 shows the subtle differences in accounting arising out of the two definitions.

Table 2. Equivalent formulations of a two-state homo-oligomeric transition

In the author's experience (Poon et al. 2007), the choice of per unit monomer is more convenient, particularly when oligomers of different molecularities are compared. In addition, it can be seen that *K* is a polynomial in of order *n*. Even in cases where can be solved explicitly in terms of *K* and *c*t (*n* ≤ 4), it is advisable to use numerical procedures such as Newton's method instead to minimize potential algebraic errors and avoid a loss of significance in the fitting procedure.

Fig. 2. Two-state dissociation of an homo-oligomer. The traces are simulated for a pentamer (*n* = 5) where *H* = 50 kcal mol-1, *C*p = 250 cal mol-1 K-1 and *T*° = 100°C. All thermodynamics parameters are per unit *monomer*. Note the asymmetry in both heat capacity functions.

where *K* is the equilibrium dissociation constant. Table 2 shows the subtle differences in

*c*t [X] + *n*[Xn] [X]/*n* + [Xn]

2

*c* <sup>t</sup>

[X] *nc*

(1 ) ( 1)

 

 <Cp > <Cint p >

*n n n ct*

*n n RT*

1 1 *n*

 

2 2

*H*

**Variable/parameter Per unit monomer Per unit oligomer**  

> t [X]

> > 1 *n*

*G nRT K* ln *RT K* ln

In the author's experience (Poon et al. 2007), the choice of per unit monomer is more convenient, particularly when oligomers of different molecularities are compared. In addition, it can be seen that *K* is a polynomial in of order *n*. Even in cases where can be solved explicitly in terms of *K* and *c*t (*n* ≤ 4), it is advisable to use numerical procedures such as Newton's method instead to minimize potential algebraic errors and avoid a loss of

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

Tmax

T, K

Fig. 2. Two-state dissociation of an homo-oligomer. The traces are simulated for a pentamer

thermodynamics parameters are per unit *monomer*. Note the asymmetry in both heat

(*n* = 5) where *H* = 50 kcal mol-1, *C*p = 250 cal mol-1 K-1 and *T*° = 100°C. All

<sup>T</sup> T° <sup>50</sup>

(1 ) ( 1) *n H n n RT*

> 

 

*<sup>n</sup> nct*

Table 2. Equivalent formulations of a two-state homo-oligomeric transition

<sup>2</sup>

*K* <sup>1</sup>

accounting arising out of the two definitions.

*d dT* 

significance in the fitting procedure.

0

500

1000

1500

Cp, cal mol-1 K-1

capacity functions.

2000

2500

3000

3500

An additional consideration for transitions involving changes in molecularity concerns the choice and interpretation of the reference temperature. In contrast with isomeric transitions, *T*° is neither the midpoint of a transition (in the context of concentrations) nor does it mark the maximum of the transition heat capacity function. Both of the latter temperatures are lower than *T*°. In addition, the midpoint of the transition, *T*50, is below the temperature of the transition heat capacity maximum, *T*m. These relationships are illustrated for the twostate dissociation model in Figure 2. The non-equivalence of *T*m and *T*50 also introduces a systematic difference between the calorimetric and van't Hoff enthalpies (*H*vH, reported at Tm) (Freire 1989; Freire 1995). Moreover, the actual values of *T*50 and *T*m are concentrationdependent, and this serves as a diagnostic for a change in molecularity in the transition. For data fitting purposes, *T*° remains the most efficient choice because it is independent of concentration. After data fitting, estimates of *T*m and *T*50 can also be easily obtained from the fitted curve.

Extension of the foregoing discussion applies readily to multi-state transitions. However, an explicit, statistical thermodynamic approach is generally used to derive the required equations for each state as a function of the partition function (Freire and Biltonen 1978). Details in deriving these models have been discussed extensively by Privalov's and Freire's groups (Privalov and Potekhin 1986; Freire 1994). From the standpoint of numerical analysis, it is worth noting that the excess enthalpy is the summed contributions from each state:

$$\left\langle \Delta \mathcal{C}\_{\text{p}} \right\rangle = \frac{d \left\langle \Delta H(T) \right\rangle}{dT} = \frac{d}{dT} \left[ \sum\_{i=1}^{n} \alpha\_{i}(T) \Delta H\_{i}(T) \right] \tag{11}$$

Depending on the number of states considered, the expansion of the derivative on the right side of Eq (11) can be formidable. Commercial programs such as Mathematica (Wolfram Research, Champaign, IL, USA) are thus recommended for symbolic manipulation for all but the most trivial derivations. Less preferably, one can numerically integrate the raw *C*<sup>p</sup> vs. *T* data and fit *H T*( ) directly. There are generally enough data points (at 0.1°C resolution) in an experiment that any loss of resolution should be negligibly small.
