Thermodynamic properties of ionization by common aqueous buffers
1. Introduction
Solution microcalorimetry has entrenched itself as a major technique in laboratories concerned with studying the thermodynamics of chemical systems. Recent developments in the calorimeter marketplace will undoubtedly continue to popularize microcalorimeters as mainstream instruments. The technology of microcalorimetry has in turn benefited from this trend in terms of enhanced sensitivity, signal stability, physical footprint and userfriendliness. As the popularity of solution microcalorimeters has grown, so has an impressive body of literature on various aspects of microcalorimetry, particularly with respect to biophysical characterizations. The focus of this chapter is on experimental and analytical aspects of solution microcalorimetry that are novel or represent potential pitfalls. It is hoped that this information will aid bench scientists in the formulation and numerical analysis of models that describe their particular experimental systems. This is a valuable skill, since frustrations often arise from uninformed reliance on turnkey software that accompany contemporary instruments. This chapter will cover both differential scanning calorimetry (DSC) and isothermal titration calorimetry (ITC). It targets physical chemists, biochemists, and chemical engineers who have some experience in calorimetric techniques as well as nonlinear regression (leastsquare analysis), and are interested in quantitative thermodynamic characterizations of noncovalent interactions in solution.
2. Differential scanning calorimetry
DSC measures the heat capacity (C_{p}) of a sample as the instrument “scans” up or down in temperature. For reversible systems, direct interpretation of the data in terms of thermodynamic parameters requires that chemical equilibrium be reestablished much more rapidly than the scan rate. This can be verified by comparing data obtained at different scan rates. For transitions involving a change in molecularity (
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, 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
The direction and extent of the temperature of pH for a buffered solution depends on the sign and magnitude of Δ
According to the van’t Hoff equation,
where
Thus, for a buffer with a positive (endothermic) Δ
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 Δ
Another important note relates to polyprotic species such as phosphates, citrates, and borates, whose p
Buffer  pK_{a}  ΔH°, kJ mol^{1}  ΔC_{p}°, J K^{1} mol^{1}  ΔV°, mL mol^{1} ^{a} 
Acetate  4.756  0.41  142  10.6 
Bicine  8.334  26.34  0  2.0 
Bistris  6.484  28.4  27  3.1 
Cacodylate  6.28  3  86  13.3 
Citrate  3.128 4.761 6.396 
4.07 2.23 3.38 
131 178 254 
10.7 12.3 22.3 
Glycine  2.351 9.780 
4 44.2 
139 57 
6.8 
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 
Phosphate  2.148 7.198 12.35 
8 3.6 16 
141 230 242 
16.3 25.9 36.0 
Succinate  4.207 5.636 
3.0 0.5 
121 217 

Tris  8.072  47.45  142  4.3 
^{a} Ionization volume at atmospheric pressure at infinite dilution. 
2.2. Analysis of DSC data
A complete DSC experiment consists of matched scans of a sample and a samplefree reference solution. Blanksubtracted data can be empirically analyzed to obtain modelindependent thermodynamic parameters. The difference between pre and posttransition baselines gives the change in heat capacity, Δ
Δ
Thus, a single DSC experiment yields the complete thermodynamics of a transition. Modelfree determination of thermodynamics, including the direct measurement of Δ
In general, the referencesubtracted 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,...,
where Δ
perform manual baseline subtraction before fitting a excess heat capacity function. This is intended to eliminate
2.2.1. Formulation of DSC models
The principal task in formulating DSC models is deriving expressions for α_{
i
}(
Again Δ
The two terms on the right side represent
For transitions involving changes in molecularity, α_{
i
}(
where
Variable/parameter  Per unit monomer  Per unit oligomer 
ct  [X] + n[Xn]  [X]/n + [Xn] 
α 





K 


ΔG 


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
An additional consideration for transitions involving changes in molecularity concerns the choice and interpretation of the reference temperature. In contrast with isomeric transitions,
Extension of the foregoing discussion applies readily to multistate 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:
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
3. Isothermal titration calorimetry
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
Typically, ITC is operated in incremental mode in which the titrant is injected in preset aliquots after successive reequilibration 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 with respect to the concentration of the titrant (X)
where [X_{
i
}]_{b} is the concentration of X in the
3.1. Experimental conditions for ITC
As a thermodynamic tool for studying molecular interactions, the singular strength of ITC is the direct measurement of binding enthalpies. Modelbased analysis of ITC data, the subject of Section 3.2, allows the extraction of binding affinity and additional parameters in complex systems. As a labelfree technique, ITC compares favorably with other titration techniques such as fluorescence and radioactivity. However, despite muchimproved sensitivity (minimum detectable thermal energy <0.1 µJ), baseline stability, and titrant control found in contemporary instruments, sensitivity of ITC is relatively limited. The actual limit of detection depends primarily on the intrinsic enthalpy of the binding system at the temperature of interest, and to a lesser extent, the physical configuration of the instrument. Roughly speaking, a typical ITC experiment requires at least 10^{6} M of titrate in a 1mL volume and 10^{4} M of titrant in a 100µL syringe. (Recently, socalled "low volume" instruments equipped with 200µL cells and 50µL syringes have become available.)
The sensitivity of ITC is helped considerably by the differential nature of its signal (which is proportional to
where
If the interaction under investigation occurs in a buffered solution, the earlier discussion on the temperature dependence of p
Finally, ITC has been used as an “analog” of DSC for studying the stability of complexes. A concentrated solution of complex in the syringe is titrated into pure water or buffer in the cell. The resulting dilution drives complex dissociation and the attendant enthalpy is measured. This method has been used to characterize complexes through the spectrum of stoichiometries, from dimers (Burrows et al. 1994; Lovatt et al. 1996) to higher oligomers (Lassalle et al. 1998; Luke et al. 2005) to polymeric species (Stoesser and Gill 1967; Arnaud and Bouteiller 2004). Again, given the sensitivity of ITC, relatively high concentrations are required, so this technique is limited to the measurement of relatively weak complexes. It has been shown (Poon 2010) that the ITC data can be used to diagnose a dimeric or higherorder complex based on the presence of an inflection point in the latter.
3.1.1. Baseline signals
Two types of baselines are operative in an ITC experiment. One is instrument noise. Drifts on the order of 0.02 µW/h are routinely achievable in contemporary instruments. Another source of baseline arises from the injection of titrant. At the very least, viscous mixing makes a measurable if small exothermic contribution to the observed heat. This effect can be observed in a blanktoblank injection (Figure 4), and serves as a casual useful indicator of the cleanliness of the cell and syringe between sample runs. Moreover, any mismatch in the matrices of the titrant and titrate will be manifest as a dilution enthalpy with each injection. For small molecules, solids or lyophilized samples are usually dissolved in water or buffer. To complicate matters, hydrophobic solutes often require a cosolvent such as DMSO or DMF to achieve initial solubility before addition of the aqueous solvent; dilution of the cosolvent will therefore contribute to the observed heat at each injection. In other cases, ionizable solutes can perturb solution pH due to their substantial (>10^{3} M) concentration in the syringe. Unless the solvent is adequately buffered, the pH in the cell and syringe will differ significantly and neutralization heats will contribute to the observed signal.
In the case of macromolecular titrant and titrates, the solution matrices can be closely matched by extensive codialysis in the same solution. Care must be taken, however, with ionic polymers such as nucleic acids. These solutes can alter the distribution of ions in their compartment during dialysis due to the Donnan effect. Specifically, a nondiffusible polyionic solute excludes diffusible ions of the same charge from their compartment and therefore induces an asymmetric distribution of diffusible ions across the semipermeable membrane at equilibrium. To illustrate, for a simple system consisting only of the nondiffusible polyion M and a monovalent salt AB, the ratio concentrations of A^{+} or B^{} across the membrane is (Cantor and Schimmel 1980):
where
In practical terms, none of these baselines effects are significant if the heat generated by the interaction of interest is sufficiently strong. This is not always achievable, however, for several reasons. Availability or solubility of the sample, particularly biological samples, may be limiting. It may also be desirable, for example, to perform titrations at a range of cell concentrations for binding to oligomerizing systems. Characterization of binding to a polyion at low salt concentration may require a reduced concentration. Thus, strategies for handling relatively low signaltonoise situations are helpful in many situations. The most basic of these involve inspection and, where necessary, manual editing of the power baseline to mitigate the occasion excursion due to instrumental noise. To this end, an increase in the time between injection may be indicated to unambiguously identify the restoration of baseline. In addition, the residual heats (which may be up to 10 µJ) after the equivalence point are unlikely to be negligible. Subtraction with data from a titranttoblank run would likely introduce more noise into the data and be no more helpful than taking a simple average of the final postequivalence heats. If the data is to be fitted to models, a more appropriate solution is to add a constant parameter
3.1.2. Continuous ITC titrations
While incremental titrations most commonly performed in ITC, an alternative mode of operation is a continuous titration (cITC) (Markova and Hallén 2004). In cITC, the titrant is continuously into injected the cell at a low rate (~0.1 µL/s). The primary advantage of cITC is throughput. An incremental ITC experiment typically requiring 20 injections of 5 µL at intervals of 300 s takes (neglecting time for baseline stabilization) 100 min; at 0.1 µL/s, cITC would require approximately 17 min. Another potential motivation for cITC is increased resolution in terms of modeldependent analysis. In incremental ITC, peakbypeak integration of thermal power is performed to obtain
Maintenance of quasiequilibrium conditions throughout the titration is essential to correct interpretation of derived thermodynamic parameters and is a major concern for cITC. To this end, the stirring rate in cITC must be considerably increased (up to 700 rpm) relative to incremental ITC to facilitate mixing of titrant into the titrate solution. Additionally, the instrument must be able to provide sufficient thermal compensation during the titration to maintain isothermal conditions. Finally, the kinetics of the interaction of interest must be fast relative to the injection rate. Generally speaking, these criteria are most likely met by relatively highaffinity interactions with moderate binding enthalpies.
3.2. Analysis of ITC data
The direct measurement of Δ
Estimation of Δ
The most fundamental concept in the analysis of ITC data is the differential nature of the heat signal with respect to titrant concentration (Figure 3). Recalling Eq (13),
Integration of this fundamental equation gives the is the functional form of conventional binding models:
The ideal approach to fitting ITC data is to directly fit Eq (13) (Poon 2010). For simple models, it is possible to write analytical expressions for
where
where
3.2.1. Numerical aspects of ITC data analysis
Given the usual practice of formulating models in terms of total or unbound concentrations, rather than their derivatives, conventional ITC data analysis has handled the differential nature of calorimetric data by fitting to a finitedifference version of Eq (13). Thus, for the
where δ
From the perspective of numerical analysis, Eq (22) is also entirely unnecessary. As stated previously, the most appropriate approach to handling ITC data is to fit Eq (13) directly. Posed in the form of Eq (13), ITC models (more specifically, the solution of
Recognizing that [X]_{t}, [Y]_{t} are constants (and
This approach of formulating explicit titration models for ITC has been demonstrated for many empirical models in common use, including the multisite model, homotropic cooperativity, and twostate selfassociation (Poon 2010). To further illustrate the this approach, a competitive binding model will be examined. This is a method of measuring tight binding by ITC by adding a competitive species in the cell to reduce the apparent affinity (Sigurskjold 2000). The mechanism is an example of the general multisite model involving two ligands competing for a single site (Wells 1992). The equilibrium distribution of bound titrate is a function of the affinities of the titrant X_{1} and competitor X_{2} for the titrate Y as well as the total concentrations of all three species. For the titranttitrate complex, X_{1}Y:
where
and
where
It is possible to solve the cubic equation in Eqs (24) and (25) explicitly, followed by differentiation of the solutions to obtain
The observed heat is now the sum of the enthalpy of unbinding of X_{2} from Y and the binding of X_{1} to Y:
As shown in Figure 5, a judicious choice of competitor reduces the apparent affinity of the titrant to a range more amenable for regression. In this case, the initial condition for [X_{2}Y] is
Thus, the competitive model requires prior knowledge of both the concentration and affinity of the competitor. Since the competitor would therefore require its own characterization, a weak competitor is preferred, which means it would need to be present at significant concentrations (i.e., [Y]_{t} ≥
Explicit titration models are also amenable to formulating models for continuous ITC. In the case of cITC, thermal power
Applying the chain rule and the relation
(The subscript "tot" has been used to denote total concentration to avoid ambiguity with the variable
Thus explicit expressions of
In the foregoing discussion, the need to correct for various displacement and dilution effects due to the injection process has not been considered. In the author's experience, this is best handled during preliminary data reduction, before nonlinear regression. This aspect will be discussed in Section 3.2.1.3.
3.2.1.1. Choice of dependent variable
Another benefit of formulating ITC models as explicit ODEs is the flexibility in the choice of dependent variable for implicit differentiation, as long as it is a function in [X]_{t}. For the 1:1 binding model, formulation in terms of the unbound titrant X gives the (and simple) functional form of the familiar Langmuir isotherm:
Applying the chain rule of calculus,
At the same time, using the equation of state [X] = [X]_{t}  [X],
Substituting the results from Eq (23) into Eq (33),
The simultaneous equations (32) and (34) represent another formulation of the same model, except now [X] is the explicit dependent variable instead of [X]_{b}. (The initial condition is [X] = 0 at [X]_{t} = 0.) Of course, we have previously derived Eq (23) directly, so this approach is regressive for this simple model. However, The flexibility to use any dependent variable of [X]_{t} is useful, for example, for models that are formulated in terms of the binding polynomial which is based on [L] (Schellman 1975; Freire et al. 2009).
3.2.1.2. Practical considerations in implementing explicit ITC models
Successful implementation of Eq (13) requires numerical procedures for solving IVPs. The explicit, closedform ODEs encountered in most models are typically ratios of polynomials. These functions are generally amenable any of the standard RungeKutta methods, which are widely available. A fast CPU is helpful, but not required. To this end, the tolerance for iteration should not be unnecessarily stringent in relation to the concentrations used and the value of
Several technical software suites, such as Mathematica (Wolfram Research, Champaign, IL, USA), MATLAB (the MathWorks, Natick, MA USA), Maple (Maplesoft, Waterloo, Ontario, Canada), and IgorPro (WaveMetrics, OR, USA) which have builtin numerical ODE and leastsquare minimization capabilities, represent fullfeatured, integrated solutions. Alternatively, precompiled libraries containing optimized algorithms for numerical ODEs and leastsquare minimization are available commercially (the Numerical Algorithm Group Library; Numerical Algorithms Group [NAG], Oxford, UK) or free (GNU Scientific Library [GSL]) for most computing platforms. Functions from these libraries can be called under standard programming environments (
3.2.1.3. Volume correction
The sample and reference cells are typically overfilled for both DSC and ITC. Overfilling maximizes heat transfer between the solution and the wall of the cell as air is a poor thermal conductor. In the case of ITC, overfilling also minimizes stray signal arising from mechanical agitation of the solutionaircell interface caused by the stirring paddle. However, the introduction of titrant into an overfilled ITC sample cell leads to displacement effects that need to be taken into account. Specifically, each injected volume of titrant simultaneously displaces an equal volume of titrate and any previously injected titrant out of the sample cell (into the access tube). The accounting for these displaced volumes and their effect on titrant and titrate concentration is made on the assumption that the displaced material is immediately and completely excluded from the titration. This implies that no mixing occurs between the injected and displaced materials at the time injection. The concentrations of the titrant X after the
where
There are two ways to handle volume corrections. One is to incorporate Eqs (35) and (36) as additional terms in the fitting equation. In the author’s experience (Poon 2010), it is more efficient instead to perform the volume corrections on the dataset at the outset, and simply treat [Y]_{t,i } as an additional dependent variable in the leastsquare procedure.
3.2.1.4. Error analysis in ITC
Compared to other titrations, particularly in the biochemical laboratory, that requires extensive manual manipulation
4. Conclusion
Commercial development of microcalorimetry has greatly increased the accessibility of this technique for the thermodynamic characterization of chemical systems in solution. Unfortunately, the "blackbox" nature of commercial software has engendered unwarranted reliance by many users on the turnkey software accompanying their instruments, and an attendant tendency to fit data to models of questionable relevance to the actual chemistry. This chapter discusses several novel aspects and potential pitfalls in the experimental practice and analysis of both DSC and ITC. This information should enable users to tailor their experiments and modeldependent analysis to the particular requirements.
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