Binding equilibrium

The equilibrium binding constant for this 1 1 association is Xu = ki/lLi. The Xu values were measured spectrophotometrically, and the rate constants were determined by the T-jump method (independently of the X,j values), except for substrate No. 6, which could be studied by a conventional mixing technique. Perhaps the most striking feature of these data is the great variability of the rate constants with structure compared with the relative insensitivity of the equilibrium constants. This can be accounted for if the substrate must undergo desolvation before it enters the ligand cavity and then is largely resolvated in the final inclusion complex. ... 

The data in the upper and lower panels were fit simultaneously with a single association rate constant (k = 3.23 x lO s ) and separate dissociation rate constants (k = 0.0108/s, upper panel 0.083/s, lower panel). The kinetic aspects of the fit were verified by the agreement with the equilibrium binding (see Figure 4 caption). 

Equilibrium binding consfanfs can be determined for ligand binding to DNA or RNA, as revealed either from absorbance or fluorescence spectroscopy by keeping a constant concentration of ligand and varying the nucleic acid concentration [125], using the Scatchard equation ... 

Figure 5 An example calibration curve. Absorbance is plotted against log (concentration of analyte). The competitive equilibrium binding process results in a sigmoidal curve that is fitted using a four-parameter fit. The IC50 is defined as the concentration of analyte that results in a 50% inhibition of the absorbance...

Thus, as described by Equation (2.1), the equilibrium dissociation constant depends on the rate of encounter between the enzyme and substrate and on the rate of dissociation of the binary ES complex. Table 2.1 illustrates how the combination of these two rate constants can influence the overall value of Kd (in general) for any equilibrium binding process. One may think that association between the enzyme and substrate (or other ligands) is exclusively rate-limited by diffusion. However, as described further in Chapter 6, this is not always the case. Sometimes conformational adjustments of the enzyme s active site must occur prior to productive ligand binding, and these conformational adjustments may occur on a time scale slower that diffusion. Likewise the rate of dissociation of the ES complex back to the free... 

In practice, measurement of the individual rate constants or equilibrium constants for these various chemical steps requires specialized methodologies, such as transient state kinetics (see Johnson, 1992, Copeland, 2000, and Fersht, 1999, for discussion of such methods) and/or a variety of biophysical methods for measuring equilibrium binding (Copeland, 2000). These specialized methods are beyond the scope of the present text. More commonly, the overall rate of reaction progress after ES complex formation is quantified experimentally in terms of a composite rate constant given the symbol km. 

Figure 3.9 Apparent value of the dissociation constant (K,) for a labeled inhibitor, I, as a function of the concentration of a second inhibitor, J when measured by equilibrium binding methods. The solid circles represent the behavior expected when compounds I and J bind in a mutually exclusive fashion with one another. The other symbols represent the behavior expected when compounds I and J bind in a nonexclusive, but antagonistic (i.e., noncompetitive, a > 1) fashion, to separate binding sites. The data for mutually exclusive binding were fit to the equation (apparent)K, = A, 1 + ([f ] A",) I and that for nonexclusive binding were fit to the equation (apparent)Kt = ( [J] + Kj / Kj + f[I]/y) ) for y values of 5 (closed triangles), 10 (open squares), 20 (closed squares), and 50 (open circles).

Note that in some cases one may follow the time course of covalent E-A formation by equilibrium binding methods (e.g., LC/MS, HPLC, NMR, radioligand incorporation, or spectroscopic methods) rather than by activity measurements. In these cases substrate should also be able to protect the enzyme from inactivation according to Equation (8.7). Likewise a reversible competitive inhibitor should protect the enzyme from covalent modification by a mechanism-based inactivator. In this case the terms. S and Ku in Equation (8.7) would be replaced by [7r] and K respectively, where these terms refer to the concentration and dissociation constant for the reversible inhibitor. 

Let us now consider the situation where [/] [E], We have here a situation that is analogous to our discussion of pseudo-first-order kinetics in Appendix 1. When [/] E in equilibrium binding studies, the diminution of [/]f due to formation of El is so insignificant that we can ignore it and therefore make the simplifying assumption that [/]f = [/]T. Combining this with the mass balance Equations (A2.1) and (A2.2), and a little algebra, we obtain... 

Bauerle, H.-D. Seelig, J., Interaction of charged and uncharged calcium channel antagonists with phospholipid membranes. Binding equilibrium, binding enthalpy, and membrane location, Biochemistry 30, 7203-7211 (1991). 

The study of receptor-ligand binding is one of the most important applications of free energy simulations [1]. To approach this problem theoretically, one must first partition the conformational space into bound and unbound states. There is no unique way to do this, but in practical situations there is often a natural choice. The equilibrium binding constant is... 

Dower, S.K., PeLisi, C., Titus, J.A., and Segal, D.M. (1981) Mechanism of binding of multivalent immune complexes to Fc receptors. 1. Equilibrium binding. Biochemistry 20, 6326-6334. 

The observed hyperbolic dependences suggest a mechanism that involves a pre-equilibrium binding of two pyridine carboxylates to the Fem of lm, followed by the intramolecular proton transfer from the coordinated acid (Scheme 3). This option has been supported by measurements of the binding constants for py and related ligands. The data in Fig. 8 agree with the mechanism in Scheme 3. The reactive intermediate for robust lm is the diaxially coordinated species, one of the two ligands being picolinic acid (L). 

The generally accepted process for metal ion-catalyzed reactions of the sort we consider here involves pre-equilibrium binding with the substrate, followed by a reaction of the complex as schematized in Equation (1). Whether the metal ion is free or complexed by ligands, or bears an associated lyate, or whether the substrate is neutral or anionic, these appear to be just the sort of processes one might expect to experience large rate accelerations in passing from water to a medium of reduced dielectric constant such as alcohols or other lower polarity solvents. 

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