Spurious cooperativity

The term cooperativity, as defined qualitatively in Section 2.1 and in mote detail in Chapter 4, requires at least two ligands that can conununicate on the same adsorbent molecule. Clearly since this chapter is devoted to single-site molecules, the term cooperativity is not even definable in such systems. Nevertheless, we discuss in this section a phenomenon referred to as spurious cooperativity, which 

One should be careful about the order in which we let and K- 0. Here, we first let h be very 

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 this section we find it more convenient to start with an ensemble of M independent and indistinguishable systems (i.e., the systems are identical but not localized, as assumed in Section 2.4), each of which has a single binding site. We stress from the outset that the concept of cooperativity, as defined in Section 4.2, does not apply to such systems. What we shall show is that under certain conditions a single-site system can exhibit behavior that is similar to the behavior of a cooperative system. 

the GPF of the equilibrated system, defined by the variables (T, M, A), is always larger than the GPF of a system (T, M, Mu, A) with any arbitrary but fixed 

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Figure 3.5. The BI and corresponding slope for the case = 1 and = 10 but different values of xj and X. Curves 1,2, and 3 correspond to X = X = 1/2 . = 2/3, X = 1/3 and X = 3/4, X = 1/4, respectively. Note that the distance between the locations of the maximal slopes of Qf is the same for the three cases. The relative steepness of the curves is a measure of higher-order spurious cooperativities (Appendix F).

We shall discuss in Section 4.6 and Appendix F a generalization of this behavior where higher-order spurious cooperativities can be observed in noncooperative systems. 

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 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. 

This is also equivalent to a mixture of two different binding systems. However, here we stress the case of a mixture that is obtained from an equilibrated system. It is only in such a case that one might misinterpret spurious cooperativity as genuine see Section 4.8 for an experimental example. 

Spurious Cooperativity in Some Alkylated Succinic Acids... 

As we have noted in previous subsections, at present it is not possible to reproduce the experimental values of fcj, kj, and g(l, 1) for these compounds. The main difficulty is to account for the solvent effects, which cannot be ignored in these molecules. In spite of this limitation we shall see below that the phenomenon of spurious cooperativity can explain the two major observations first, the decrease in g(l, 1) by five to six orders of magnimde upon increasing the alkyl substituent in the racemic series, and second, that these changes occur only in the racemic and not in the meso form. 

Thus, the occmrence of spurious cooperativity in single-site systems is unlikely to deceive us into thinking that the system is genuinely cooperative. 

There are several directions along which one can generalize the concept of spurious cooperativity discussed in Section 3.5. (1) Instead of equal mole fractions = o = l/2, one can start with any composition of the two components. 

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