In two-site systems

In multiple-site systems, spurious cooperativity can occur along with genuine cooperativity (as defined in subsequent chapters). It is only in the single-site system that any apparent cooperativity is necessarily spurious, and therefore we place the discussion of this phenomena in this section. We shall return to spurious cooperativity in two-site systems in Section 4.6. The reader should keep in mind the possibility of spurious cooperativity whenever processing and interpreting experimental data, especially when one has reason to suspect that the two or more conformations might not be in equilibrium. 

In Section 3.5 we discussed the phenomenon of spurious cooperativity in single-site systems. Since cooperativity, as defined in this book, is undefinable for single-site systems, any apparent cooperative behavior must be due to the presence of different and independent sites. In Section 4.4 we encounter the same phenomenon in two-site systems with different sites. This was shown to be equivalent to the system in Section 3.5. 

In two-site systems, there is only one correlation function which characterizes the cooperativity of the system. In systems with more than two identical sites, for which additivity of the higher-order correlations is valid, it is also true that the pair correlation does characterize the cooperativity of the system. This is no longer valid when we have different sites or nonadditivity effects. In these cases there exists no single correlation that can be used to characterize the system, hence the need for a quantity that measures the average correlation between ligands in a general binding system. There have been several attempts to define such a quantity in the past. Unfortunately, these are valid only for additive systems, as will be shown below. 

The kinetic information for NMR experiments is contained in the line broadening observed for a nucleus that resides in two different magnetic environments, and values for rate constants can be obtained using line-shape analysis.28,68,69 Line broadening experiments obtained using ID NMR is the method of choice when analyzing the kinetics of a molecule in two sites. 2D NMR techniques, such as 2D EXYS are employed when the kinetics are sequential, i.e. more than one step, or multiple sites are analyzed.69 For example in the case of supramolecular systems this technique was employed to measure the exchange kinetics in capsules.70... 

Langmuir isotherm. We shall see in Chapter 4 that this is a typical BI for a two-site system with negative cooperativity. We shall also discuss in Sections 3.5 and 4.6 how experimental data could be misinterpreted in such cases. At this stage we stress again that our system consists of M independent adsorbent molecules, and cooperativity in the sense defined in this book (see Chapter 4) is not definable in this system. The generalization to any number of types of sites is quite straightforward. The BI per site of the mixture is simply... 

THE GENERAL DEFINITION OF CORRELATION AND COOPERATIVITY IN A TWO-SITE SYSTEM... 

In this section we start with two-site systems, where genuine (positive or negative) cooperativity exists in each molecule. We explore the emergence of additional spurious cooperativity due to freezing-in of an equilibrium between two forms L H. As we shall see below, in this case it is not always possible to distinguish spurious from genuine cooperativity. 

Since each site has a diiferent value of (a = a, b, c), the sign of the different correlations g(a, p) depends on the product (/ - 1) (Ji - 1). This is the same as in the two-site system, discussed at the end of Section 4.5. Perhaps the most important aspect of these correlations is their independence of the ligand-ligand distance. This is true for yia, b) and y(b, c), as well as for y(a, c). In fact, when the sites are identical in the weak sense, then h = hi, = h = h.ln this case all the indirect... 

Figure 11 shows the contour plots of the 2D 51V EXSY spectrum of a two-site system (40 mM total vanadate at pH 10.9) in which V exchanges with V2. Quantification of the EXSY spectrum and calculation of the error propagated to the rate constants from the integration precision gives a 25% error on the rate constant. The results (both the rate constants and the errors) correspond nicely to the results obtained from a ID magnetization transfer experiment on the same sample (27). The EXSY spectrum of a sample containing 12.5 mM total vanadate at 1.0... 

Figure 11. 2D 51V EXSY NMR spectra recorded at 23° C of three different vanadate solutions. A two-site system was observed in a solution containing 40 mM total vanadate at ionic strength of 0.4 M and pH 10.9. Two four-site systems were recorded at 10 and 12.5 mM total vanadate. The calculated rate constants are shown next to the off-diagonal resonances in all three maps. The errors were propagated to the rate constants from the integration precision and are also shown next to the off-diagonal resonance.

The total polaron bandwidth can be estimated by calculating in finite-site systems where some discrete values of k s, are available. In the two-site problem, if we write the ground-state wavefunction for a polaron localized at site j as I j, the... 

Fig. 2 Inverse of the polaron mass enhancement factor, m/m, as a function of for the T (8) a (HP Holstein polaron) and the (8) e JT polaron. In the latter, the result in the infinite chain d = 1) is compared with that in the two-site system as weU as the analytic result in (26). The anti-adiabatic condition of (Uo// = 5 is assumed...

In this illustration (Fig. 2), the imaginary part, I, is plotted for the same catalyst systems illustrated in Fig. 1. In the two-site system the I... 

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