The Binding Isotherm

The grand PF of the system (open with respect to G only) is 

The four terms on the rhs of (3.1.2) correspond to the four states of the polymer (Fig. 3.1). Here, the polymer is unaffected by the binding of ligands therefore we may factor out the quantity 2(0, 0), the PF of the empty polymer. In this model the two states (0, 1) and (1,0) are distinguishable, but the corresponding PFs are identical. Thus we write  

Clearly, when 5=1, i.e., there is no interaction between ligands, Eq. (3.1.8) reduces to the simple Langmuir isotherm. If the ligands are at equilibrium with an ideal-gas phase, then  

FIGURE 3.1. Four states of a polymer, corresponding to the four terms in Eq. (3.1.2). 

We denote by K the Langmuir binding constant, in the same way as we did in the case of the Langmuir isotherm i.e., we define 

Normally the experiment is performed holding the concentration of one species relatively constant (either strictly constant or with slight dilutions during the experiment), while varying the concentration of the other species. For example, when holding the concentration of the host constant, it is common to employ a relationship such as Eq. 4.24, where [H]o is the initial concentration of H. This equation is derived from Eq. 4.2 using the relationship that (H)o = (H GJ -I- [H]. If we can measure the H G concentration as a function of the concentration of free G in solution, we can calculate K. Most commonly, some parameter related to the concentration of H G is plotted along the y axis (see examples below), not actually IH G] itself. 

Importantly, [G] in Eq. 4.24 is not the amount of G that is added to the solution ([G]o), but instead it is free G. Since the experimentalist controls [H]o and [G]o, in a graphical analysis it is one of these values that would be used for the x axis. Routinely, the analysis is done for many concentrations of Go with Hq constant, where [G]o = [G] + [H G]. A plot of experimentally determined [H G]s as a function of many [G]o values gives the curve to be analyzed. 

What is a high enough concentration of G to give saturation High relative to what Just as in our analysis of dilution (Section 4.1.1), the concentration is relative to the Kj for the 

obtained from Eq. 4.3 by substitution of [H] = [H]o - [H G], gives quick insight into predicting saturation behavior, especially when [H]o = K. For example, when [H]o = and [G] Ka, the denominator of Eq. 4.26 can be approximated by K, and consequently we find that [H G] = [G]. Under these conditions the host is not saturated but the concentration of H G tracks with how much G has been added. Conversely, when [H]o = Ki and [G] Xd, the denominator can be approximated by [G], and [H G] = [H]o, meaning that the host is now saturated. If we now use a low [H]o relative to Xj, it will take more G to saturate H, and if we start with a high [H]o relative to Xd, it will take less G to saturate H. 

In the discussions above we focused upon measuring [H G] as a function of [GJo- In practice, it is arbitrary which compound we call H and which we call G, so we can plot [H G] as a function of either the added concentration of host or guest while keeping the other constant. Remembering that this is a completely general discussion, you can plot against either of the two components that come together to create a complex. 

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Figure 2. Influence of the ionic strength and the polymer concentration on the binding isotherms of Pb2+ by sugar-beet pectins in water (empty symbols) and in 0.1 M NaNOs (full symbols) at 25°C ( ) 2 mequiv. COO. l-, ( ) 8 mequiv. COO-.l-i (—) total binding of added Pbz+.

Figure 3. Influence the metal ion type on the binding isotherms of sugar-beet (A) and citrus (B) pectins at 2 mequiv. COO-.l- in 0.1 M NaNOs and at 25 °C. Symbols as in figure 1.

By comparing the level of the binding isotherms (figure 3) for both metals and pectins, it became possible to set up an affinity order of pectins, whatever their origin, for the five metal ions Cu2+ Pb + Zn2+ Ni2+ > Ca2+. This scale, already found by pH-measurements, confirmed that Cu2+ and Pb2+ were more strongly bound than the other thi cations with no difference between pectins. 

Note that Equations (A2.14) and (A2.18) do not take into account any influence of substrate concentration on the apparent value of Kd. As described in Chapter 5, this can be accounted for most generally by replacing the term Kd in these equations with the observed value of Kfp or IC50. Making this substitution in Equation (A2.18), we obtain the binding isotherm equation that has been used throughout this book ... 

Both Reynolds and Karim worked at neutral pH, with denatured proteins, and with reduced disulfide bonds. Under these conditions, proteins are in a random coil conformation (Mattice et al., 1976), so that their hydrodynamic radius is monotoni-cally related to their molar mass. Takagi et al. (1975) reported that the binding isotherm of SDS to proteins strongly depends upon the method of denaturing disulfide bonds. Presumably, protein-SDS complexes are not fully unfolded when disulfide bonds are left intact, which breaks the relationship between molar mass and hydrodynamic... 

When DsPheno1 Dsmlcelle, significant binding of phenol to micelles occurs, )Pheno1 is largely reduced. From the binding isotherm, an estimation of phenol molecules per surfactant molecule can be obtained. 

The experimentally observed pseudo-first order rate constant k is increased in the presence of DNA (18,19). This enhanced reactivity is a result of the formation of physical BaPDE-DNA complexes the dependence of k on DNA concentration coincides with the binding isotherm for the formation of site I physical intercalative complexes (20). Typically, over 90% of the BaPDE molecules are converted to tetraols, while only a minor fraction bind covalently to the DNA bases (18,21-23). The dependence of k on temperature (21,24), pH (21,23-25), salt concentration (16,20,21,25), and concentration of different buffers (23) has been investigated. In 5 mM sodium cacodylate buffer solutions the formation of tetraols and covalent adducts appear to be parallel pseudo-first order reactions characterized by the same rate constant k, but different ratios of products (21,24). Similar results are obtained with other buffers (23). The formation of carbonium ions by specific and general acid catalysis has been assumed to be the rate-determining step for both tetraol and covalent adduct formation (21,24). 

Figure 13.10 Calorimetric titration response showing the exothermic raw (downward-projecting peaks, upper panel) heats of the binding reaction over a series of injections titrating 0.061 mM RNase A (sample) with 2.13 mM 2CMP at 30°C. Bottom panel shows the binding isotherm obtained by plotting the areas under the peaks in the upper panel against the molar ratio of titrant added. The thermodynamic parameters were estimated (shown in the inlay of the upper panel) from a fit of the binding isotherm.

In order to compute the binding isotherm (Section 2.1) of any system, one must know all the microstates of the system. This cannot be done for even the smallest binding system. However, in order to understand the origin of cooperativity and the mechanism by which ligands cooperate, it is sufficient to consider simple models having only a few macrostates. This understanding will be helpful for the selection of methods to extract information from experimental data, and for the meaningful interpretation of this information. 

The second property is that each term of the PF is proportional to the probability of occurrence of the particular state it represents when the system is at equilibrium. We shall use mainly the second property of the PF. The next section is devoted to this aspect of the theory. Once we have the probabilities of all possible events we can compute average quantities pertaining to the system at equilibrium. Of these, the average occupation number, or the binding isotherm, will be the central quantity to be examined and analyzed in this book. 

The binding isotherm (BI) of any binding system was originally referred to as a curve of the amount of ligands adsorbed as a function of the concentration or partial pressure of the ligand at a fixed temperature. A typical curve of this kind is shown in Fig. 2.1. Numerous molecular models have been studied that simulate particular experimental Bis. 

The binding isotherm is a monotonous increasing function of X. This follows from the thermodynamic stability of the macroscopic system. Thus... 

The partition function and the binding isotherm of a general two-site model were discussed in Section 4.2. Here, we examine a special case of direct correlation only. The model is essentially the same as in the previous section, except that now... 

It is assumed that we have experimental data on the binding isotherm of a system known to consist of m binding sites. We denote by the average number... 

Figure 5.12 shows the BI and the quantities g(C) - 1 for this model. This illustration shows that although the binding isotherms seem to belong to a negative cooperative system, it is, in fact, meaningless in general to refer to the cooperativity of the system where there exists more than one type of cooperativity. In Fig. 5.12a, the curve starts with positive cooperativity, mainly due to the indirect part, i.e.,... 

We note that in spite of the large differences in the two sets of results reported in Table 5.3, the binding isotherms computed with these three sets of results were almost indistinguishable on the scale of Fig. 5.19. The most important differences between the calculated correlations and those reported by Senear et al. are, first, there is a large negative cooperativity between sites a and c, while Senear et al. assumed from the outset that no long-range cooperativity exists, and second, the triplet correlation is not additive, i.e., neither Sj nor 62 is zero, while Senear et al. assumed from the outset that 82 = 0. 

Figure 6.3. The binding isotherm and the average correlation [as f(G) -1] for the square and tetrahedral models discussed in Section 6.6. No direct ligand-ligand interactions are assumed. The parameters for the indirect correlations are A = 0.01, AT = 1, = 4Cu, and t = 0.01. The lower two...

In general, one can define other correlations, say between two unoccupied sites, or one occupied and one unoccupied site, etc. We shall not require these here. Also, as always, we shall need only the X, —> 0 limit of the correlations, since only these enter into the binding isotherm. 

THE CONNECTION BETWEEN THE KINETIC EQUATION AND THE BINDING ISOTHERM... 

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