3

CALORIMETRY

3.1. ISOTHERMAL REACTION MICROCALORIMETRY

3.1.1. The Heat of Mixing Reaction

In the simplest binding reaction, macromolecule M has one binding site for the ligand L:

The association constant of this reaction is

 (3.1)

where [ML], [M], and [L] are the corresponding concentrations, which in dilute solutions are close to the activities.

In considering the dissociation process

the dissociation constant is

 (3.2)

Thus, Kd = 1/Ka. Correspondingly, the dimension of Ka is M−1, and of Kd is M.

The fraction F of macromolecules with the bound ligand is

 (3.3)

The fraction of macromolecules without ligands is

 (3.4)

From Equations (3.3) and (3.4) one gets

 (3.5)

At the concentration where F = 1/2,

 (3.6)

Thus, the dissociation constant, which is the reciprocal of the association constant, is just the concentration of free ligand at which half of the macromolecules have a bound ligand and half do not. The larger the binding constant, the lower the dissociation constant—and the lower the concentration of ligand at which the half-saturation is reached.

The heat of the association reaction depends on the molar enthalpy of association ΔHa, the association constant Ka, and the ligand concentration [L]. The heat effect of binding can be positive or negative, but usually for protein–protein or protein–DNA interactions it is not large in magnitude, less than 100 kJ/mol. If Ka is large, then all ligands would be bound even at small concentrations. Therefore, to get the complete binding curve, the isotherm of reaction, we need to use a very low concentration of macromolecules. A high association constant is specific for many reactions of biological macromolecules, which requires precise recognition of the partner; for example, the binding constant of gene regulating proteins is above 108 M−1 (i.e., the dissociation constant is below 10−8 M). Correspondingly, to get a complete picture of their binding, one needs to work with very dilute solutions of macromolecules, of the order of 10−8 M. On the other hand, there are many nonspecific binding reactions, which are characterized by low binding constants. For example, the interaction of denaturants (urea or guanidine hydrochloride) with proteins is characterized by a very low binding constant. Therefore, a significant effect of these reagents on proteins is observed only in concentrated solutions. However, the solubility of denaturants is limited—less than 8 M. Therefore, to achieve a high concentration of denaturant in a solution with protein, one has to mix a comparable amount of the protein solution with the concentrated solution of the denaturant. In contrast, when the binding constant is high, we have to add a very small volume of the reagent to a large volume of protein solution and repeat that many times to get a binding isotherm. Correspondingly, for studying biopolymers two different types of reaction calorimetric instruments are needed: one that permits mixing two solutions in comparable volumes, and another that permits mixing a small portion of one solution in a large portion of another one and repeating that many times.

3.1.2. Mixing of Reagents in Comparable Volumes

There are two methods of mixing reagents in comparable volumes: batch and flow-mix.

In batch calorimeters comparable volumes of two reagents are placed in the two equal compartments of the rotating calorimetric cell and are mixed by turning this cell upside down. The heat of mixing is measured by the electric signal that is produced by the heat flowing through the Peltier element (a battery of semiconductors) to the thermostat. The main disadvantage of batch calorimeters is that the noise resulting from the rotating cell and from mixing the reagents is too high and does not reproduce well. The main reason is the air phase in the calorimetric cell—which, upon turning the cell upside-down, produces bubbles in the mixture that is formed. These bubbles induce some heat effects, which cannot be taken into account. Therefore these calorimeters are no longer much used for studying reactions with biological macromolecules.

In the flow-mix calorimeter continuous laminar flows of two reagents are mixed in the reactor (Fig. 3.1). The mixture flows through a heat exchanger with the Peltier element, which converts the heat flowing to the thermostat into an electric current that is amplified and measured. The advantage of this instrument is that the liquids being studied do not contact the gas phase and the laminar constant flow of the reagent does not produce much noise. Efficient mixing in this case requires a certain turbulence; however, the Joule heat resulting from this turbulence is rather constant and can be taken into account. The main disadvantage of the existing commercial flow-mix calorimeters is that they require quite a considerable amount of the reagents to realize their constant flow through the instrument. Nevertheless, the most thorough calorimetric study of the thermodynamics of protein interaction with urea and guanidinium chloride was done using an LKB FLOW-MIX calorimeter (Pfeil and Privalov, 1976; Makhatadze and Privalov, 1992).

Figure 3.1. Schematic of the flow-mix calorimeter. The solution of one reactant is injected continuously into the calorimetric channel by the syringe F or a peristaltic pump; syringe G injects the other reagent into the channel at a point after the first heat-measuring part A and before the second heat-measuring part B. The difference between the heat effects recorded by B and A represents the heat effect of mixing of these two solutions, which is recorded.

3.1.3. Isothermal Titration Microcalorimeter

Isothermal titration calorimeters (ITCs), which permit calorimetric titration of one reagent by another, were widely used in physical chemistry. However, the reactors of conventional titration calorimeters were of rather large volume (of the order of deciliters) needed to place a mechanical stirrer to homogenize the mixture and also to minimize the influence of the surroundings on the measured heat effect. Therefore these instruments cannot be used for studying the heats of reactions between biological molecular objects, which are available in very limited amounts. The first micro-scale modification of the ITC that could be practically used for studying biological reactions was designed by S.J. Gill (McKinnon et al., 1984) and is shown in Figure 3.2. In this differential instrument two identical cells of 0.5 mL volume are placed into the thermal equilibrator block and a few microliters of titrant are injected periodically into the reactor cell by the rotating microsyringe, which also serves as a stirrer. The heat effect of mixing thus produced is detected by the sensor (the copper–constantan battery) placed between the reactor and the thermal equilibrator block and is compared with the heat effect between the reference cell and this block. After the appearance of this first micro-ITC, several companies (Microcal, Calorimetry Science Corporation CSC, TA Instruments) started production of various modifications differing from the original mainly in the sensors, electronics, and thermostatization system used.

Figure 3.2. The first twin isothermal titration microcalorimeter, designed by Stanley Gill: (a) submarine container; (b) thermal equilibration block; (c) thermoelectric module; (d) calorimetric cell (glass bulb embedded in copper cylinder with low-melting metal); (e) electric heaters; (f) close-fitting stainless-steel tube with inside glass tube; (g) glass capillary stirrer and injection needle; (h) stainless-steel centering sleeve; (i) syringe holder; (j) microsyringe; (k) sleeve bearing assembly for syringe centering; (l) stirrer drive gear (McKinnon et al., 1984).

The operational volume of commercial ITC instruments is usually on the order of 1 mL. A smaller operational volume is not appropriate since the most critical part for an ITC experiment is not the sample volume but the concentration of the reagents, whereas decreasing the operational volume increases the optimal concentrations for the ITC experiment.

Figure 3.3 shows the construction of some important parts of the commercial Nano-ITC of TA Instruments: (a) the calorimetric block, showing the locations of the twin reaction cells within the thermostatization system; (b) the reactor cell with the syringe/stirrer inside; and (c) the block rotating syringe/stirrer and its piston, which moves stepwise to inject precise portions of the titrant into the reactor cell. Perfect thermostatization and electronics of this instrument permitted reduction of the noise level to 2.5 nW and the baseline stability to ±20 nW per hour. Other important characteristics of the instrument are its short response time and the high accuracy of the portions of reagent injected into the reactor cell. The cells are made from pure gold, which is the best material for calorimetric cells not only because of its high chemical inertness but also because of its high thermal conductivity, which is important for the temperature uniformity in the cell. Figure 3.4 shows the Nano-ITC of TA Instruments.

Figure 3.3. Construction of the Nano-ITC of...