Binding constant intrinsic

We define the following seven intrinsic binding constants,... 

Each of the intrinsic binding constants may be interpreted either as a probability ratio or as a free-energy change for a specific binding process. Both meanings are derived directly from definitions (2.2.6). For instance,... 

It is sometimes convenient to introduce conditional intrinsic binding constants. These are defined in the same way as the conditional probabilities, namely,... 

In summary, the intrinsic binding constant to be used throughout this book always refers to a specific set of sites. They are defined in terms of the molecular properties of the system through the corresponding canonical PFs. They are also interpreted as probability ratios or as free energies of binding processes. In subsequent chapters we shall see how to extract from these quantities various correlation functions or, equivalently, cooperativities. 

Throughout the remainder of this book we shall use only the intrinsic binding constant as defined by the corresponding canonical PFs in Section 2.2. The are the quantities that are obtained directly from experiments. However, if we wish to interpret these quantities, say in terms of cooperativity, we must convert to the... 

We now define the intrinsic binding constant k as the ratio QiVfKolQid) (Section... 

In tenns of the intrinsic binding constants, we transfonn the BI into... 

Here, k and 2 replace fcj and kyy in Section 2.2. These are the more common notations for the first and second intrinsic binding constants. The latter limiting behaviors enables one to determine the correlation function... 

We denote by the first intrinsic binding constant, and by Arj the second intrinsic binding constant. The latter is the same as 1 ,/ i.e., the conditional binding on the second site, given that the first site is occupied. Figure 4.23 shows two alternative, but equivalent, ways of describing the binding of protons to a dicarboxylic acid. 

First and Second Intrinsic Binding Constants (in iiter/moi). Pair Correlations, and Corresponding Work W(l, 1) (in kcal/mol) for o,0[)-Dicarboxylic Acid COOH(CH2) COOH... 

Note that k and are the intrinsic binding constants on a and b, respectively k and kj are the conditional intrinsic binding constants, i.e.. 

Figure 4.25. The intrinsic binding constants for an amino acid.

It has been found experimentally [see Edsall and Wyman (1958), p. 485] that the monoester of dicarboxylic acid has a value of the dissociation constant nearly half of the Ki iss value of the dibasic acid. This is equivalent to saying that the binding constant ks to the monoester is nearly the same as the second intrinsic binding constant for the dicarboxylic acid, i.e., kg = k n. 

An example that conforms to this definition would be benzene-l,3,5-tricar-boxylic acid (Section 5.9). Clearly, owing to the symmetry of the molecule there is only one intrinsic binding constant k, and only one intrinsic binding constant for pairs fcj, or, equivalently, only one pair correlation function... 

We conclude this section by noting that when the subunits become identical, and A = Aj, = A = A and = q = q, the sites are, in general, not identical in either the weak or the strict sense. This is in contrast to the model discussed at the end of Section 5.5, where we found only one intrinsic constant. Here, in general, there are two different intrinsic binding constants, denoted by and for binding on the first (or third) site and on the second site, respectively. The general expression is complicated, but for q = 1 we have the following simple expressions ... 

If the system is known to have m identical sites, either in a strict or in a weak sense, then K.J. = mk, where k is the intrinsic binding constant. On the other hand, if we know that the system has m different binding sites, each having a different intrinsic binding constant k, then we must determine each of these from the limiting slope of the corresponding individual BI, i.e.. 

In terms of the two functions 7(0 and I C) and the intrinsic binding constants we define the quantity... 

For all the above reasons we have defined g(C) without reference to any hypothetical, independent-site system. One simply extracts both 1(C) and all from the experimental data, and then constructs the quantity g(C). When the sites are identical in a weak sense, i.e., all k = k, some of the correlations for a given / might differ. For example, four identical subunits arranged in a square will have only one intrinsic binding constant k, but two different pair correlation functions. For this particular example we have four nearest-neighbor pair correlations g (2), and two second-nearest-neighbor pair correlations gJJ)- The average correlation for this case is... 

Table 5.1 presents some values for the intrinsic binding constants, and also pair and triplet correlations for benzoic acid, benzene dicarboxylic acids, and benzene tricarboxylic acids. 

Intrinsic Binding Constants, Correlation Functions, and Corresponding Free Energies (in kcal/mol) for Various Carboxylic Acids ... 

The intrinsic binding constants were calculated from the experimental thermodynamic constants (see Section 2.3) as follows ... 

The BI for a system with three different binding sites is given in Eq. (5.3.4) in terms of the intrinsic binding constants k, /t, and and the various direct correlations. The more general form for the total BI, per binding system, is... 

Clearly, with this BI one cannot resolve all the seven intrinsic binding constants kfl kj, k kgf, k, k, and k. These can be obtained, however, from the individual Bis which, for this case, are... 

Ackers et al. (1982, 1986) analyzed data on the binding of the X repressor to the left and right operators obtained by footprinting titration. In order to obtain the correlations in this system, they wrote the following relations between the (overall) macroscopic binding constants Ki (i = 1,2, 3), defined in Section 2.3, and the intrinsic binding constants kf, and k ... 

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