Introduction
It is commonly thought that ITC is a typical thermodynamic technique that is not really suited to extract kinetic information. It is thus commonplace to oppose ITC to SPR since the latter is a kinetic technique par excellence. However, this is at best an oversimplification since ITC is based upon kinetic measurements. Indeed, the raw signal measured in any ITC experiment is a heat power (in µJ s-1 or µcal s-1), that is essentially the rate of heat production, and not the heat itself evolved in a reaction. Obviously, this rate of heat production is directly related to the kinetics of the reaction taking place in the measurement cell, which is the reason why a microcalorimeter is potentially much more than merely a ‘heat-meter’. This is in line with the common observation that there are systems showing after each injection a quick return to baseline, that is a quick equilibration time, and others, on the contrary, showing slow, and even very slow, return to baseline. It should not come as a surprise, therefore, that ITC has already been used to derive kinetic information. Many readers, however, will probably be surprised to learn that the first “compensation-mode” calorimeter (the ancestor of the MicroCal VP-ITC and MicroCal ITC200 instruments) was devised in 1924 and used first to measure the heat power produced by flies [1, 2]. Therefore, the first “compensation-mode” calorimeter was used for what we now call a kinITC experiment.In chemistry and physical chemistry, the link between the measured heat-power and the kinetics of the reaction has long been the subject of theoretical investigations [3-5]. However, the possibilities of the instruments of the time, particularly due to their large sample volumes of several ml and their long response times, were quite limited and the measurements were exclusively limited to slow, and sometimes very slow, reactions [6].
In the biological field, therefore, kinetic measurements have mostly been performed in enzymology after the pioneering work by Sturtevant initiated in 1937 (reviewed in [7]). Modern instruments like the MicroCal ITC200 with a 200-µl measurement cell and response times better than 10 s (3.5 s for the MicroCal ITC200 instrument used for this study) provide us with the possibility of addressing more easily biological problems not limited to enzymatic studies. Such response times can be derived from methanol dilution experiments described below. Interestingly, a recent study on slow RNA folding showed that the VP-ITC may also be valuable [8]. It is the goal of this Application note to show how and when can kinetic information be retrieved from ITC power curves. The results shown here were obtained with programs developed with Mathematica from Wolfram Research, which was employed to implement and validate the method. The underlying algorithms were all introduced into AFFINImeter and they are expected to be publicly available by April 2015.
A full account of the complete kinITC method has been given in [9]. Here, we will make a brief summary of it and emphasize on a simplified version that can yield remarkably good results as soon as a classical ITC data analysis has been performed. It has to be stressed that, in [9], we considered applications of kinITC in two different situations, first for a simple one-step kinetic scheme represented by:
A + B ↔ C with parameters kon, koff (1)
and, second, for a two-step kinetic scheme with a binding event followed by a conformational change (‘induced fit’). In the present Application Note, we focus only on one-step kinetic schemes represented by equation (1). Essentially, kinITC is based upon linking the kinetics of the reaction to heat power production in the measurement cell. Elementary kinetic considerations yield:
dC/dt=kon[A]0AB-koffC (2)
where A, B and C are simplified notations for the reduced concentrations [A]/ [A]0, [B]/ [A]0 and [C]/ [A]0, [A]0 being the concentration of the titrand in the measurement cell before any injection of the titrant B. This differential equation is valid at any step of the titration (i.e. for any ‘injection’), but only after compound B has been injected, which ensures that the system ‘is closed’ and that the following conservation equations apply: A + C = constant and B + C = constant. During the injection of a small volumeδV of compound B, on the contrary, the variations of A, B and C are also affected by the addition of B and by the resulting dilution (each added volume δV has to displace the same volume δV from the cell), which makes equation (2) insufficient to describe the system. These technical problems are fully addressed in AFFINImeter but they are not essential for the understanding of kinITC and we will thus focus only on the evolution of the system immediately after compound B has been injected.
The link between dC/dt and the heat power signal PS at any time t after injection of compound B is readily obtained as:
PS(t)=VcellΔH[A]0dC/dt (3)
where Vcell is the cell volume (200 µL for the MicroCal ITC200 and 1.4 ml for the VP-ITC). Therefore, when dC/dt is known from integration of equation (2) (see [9]), Ps(t) can be evaluated. Note that Ps(t) is not the measured heat power Pm(t) due to the finite response time (or relaxation time) τITC of the instrument. The link between and is expressed by a classical convolution equation:
Ps(t)=Pm(t)+τITCdPm/dt (4)
This equation, which has been known since 1933 as the Tian equation in the frame of calorimetry (for accessible references, see [3, 10]), is in fact a universal equation describing the influence of a finite response time on any instrument having a linear response. Obviously, the shorter τITC, the less distorted the measured signal. We will consider in the following the limits of kinITC arising from a non-null τITC value. Importantly, whatever the value of τITC, equation (4) shows that the integration of Pm(t) and Ps(t) leads to the same total heat, but only if one has left enough time between injections for the signal to return to equilibrium (see below).
Experimentally, τITC can be obtained by fitting with Pm(t)=Pmaxexp(-t/τITC) the decaying response of the instrument to a very fast thermal excitation like that following the quick injection of diluted methanol (1µl injections of 1-3% v/v MeOH). By ‘decaying response’ we mean the response after the end of the short transient signal due to the excitation itself. For the MicroCal ITC200, this allows to obtain τITC≈3.5s. Note that some variability may exist from instrument to instrument. Finally, it should be mentioned that, rigorously, a single relaxation time may not be sufficient to describe fully the response of an instrument [4, 5, 10]. We do not have to consider these refinements for the present purpose.
How can one judge the existence of a kinetic signal in heat power curves?
Quite often, when examining the successive injections, one easily discerns a significant variation of the time θ needed to return to baseline or, in other words, of the equilibration time. In particular, the injection corresponding to, or close to, mid-titration (i.e. [A] = [B] when there is one single binding site) shows the slowest equilibration time. This is illustrated with calculated data in Fig. 1. Such a feature is a clear mark of the existence of a kinetic signal. Intuitively, the explanation is that, close to mid-titration, both the concentrations of A and B are low (and even very low if the affinity is high), which slows down the kinetics of a 2nd order reaction according to equation (2). As a consequence, it is critical to leave enough time for recording in full these ‘mid-titration injections’ with longer equilibration times.
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