Kinetic constants dissociation constant Equilibrium

The very slow dissociation rates for tight binding inhibitors offer some potential clinical advantages for such compounds, as described in detail in Chapter 6. Experimental determination of the value of k, can be quite challenging for these inhibitors. We have detailed in Chapters 5 and 6 several kinetic methods for estimating the value of the dissociation rate constant. When the value of kofS is extremely low, however, alternative methods may be required to estimate this kinetic constant. For example, equilibrium dialysis over the course of hours, or even days, may be required to achieve sufficient inhibitor release from the El complex for measurement. A significant issue with approaches like this is that the enzyme may not remain stable over the extended time course of such experiments. In some cases of extremely slow inhibitor dissociation, the limits of enzyme stability will preclude accurate determination of koff the best that one can do in these cases is to provide an upper limit on the value of this rate constant. 

An inflection point in a pH-rate profile suggests a change in the nature of the reaction caused by a change in the pH of the medium. The usual reason for this behavior is an acid-base equilibrium of a reactant. Here we consider the simplest such system, in which the substrate is a monobasic acid (or monoacidic base). It is pertinent to consider the mathematical nature of the acid-base equilibrium. Let HS represent a weak acid. (The charge type is irrelevant.) The acid dissociation constant, = [H ][S ]/[HS], is taken to be appropriate to the conditions (temperature, ionic strength, solvent) of the kinetic experiments. The fractions of solute in the conjugate acid and base forms are given by... 

Uncompetitive antagonism, form of inhibition (originally defined for enzyme kinetics) in which both the maximal asymptotic value of the response and the equilibrium dissociation constant of the activator (i.e., agonist) are reduced by the antagonist. This differs from noncompetitive antagonism where the affinity of the receptor for the activating drug is not altered. Uncompetitive effects can occur due to allosteric modulation of receptor activity by an allosteric modulator (see Chapter 6.4). 

Figure 15. Data from single channel experiments, plotted to show the relationship between kinetic and equilibrium parameters for several of the saxitoxins, tetrodotoxin, and Conus geographus toxin GIIIA. Compound numbering corresponds to that in Figure 1. The vertical axis is and the horizontal axis is dwell time, the reciprocal of k j. The dissociation constant, the ratio of k jj/k, therefore corresponds to distance along the diagonal. Data primarily from Ref. 95.

Equations (2.10) and (2.12) are identical except for the substitution of the equilibrium dissociation constant Ks in Equation (2.10) by the kinetic constant Ku in Equation (2.12). This substitution is necessary because in the steady state treatment, rapid equilibrium assumptions no longer holds. A detailed description of the meaning of Ku, in terms of specific rate constants can be found in the texts by Copeland (2000) and Fersht (1999) and elsewhere. For our purposes it suffices to say that while Ku is not a true equilibrium constant, it can nevertheless be viewed as a measure of the relative affinity of the ES encounter complex under steady state conditions. Thus in all of the equations presented in this chapter we must substitute Ku for Ks when dealing with steady state measurements of enzyme reactions. 

As we described in Chapter 3, the binding of reversible inhibitors to enzymes is an equilibrium process that can be defined in terms of the common thermodynamic parameters of dissociation constant and free energy of binding. As with any binding reaction, the dissociation constant can only be measured accurately after equilibrium has been established fully measurements made prior to the full establishment of equilibrium will not reflect the true affinity of the complex. In Appendix 1 we review the basic principles and equations of biochemical kinetics. For reversible binding equilibrium the amount of complex formed over time is given by the equation... 

There are a plethora of criteria that should be applied to ensure that the experimentally determined parameters provide a true reflection of the physical interactions that they represent. However, if the data are to be credible they must demonstrate an internal consistency. The equilibrium dissociation constant should, for example, be the same if it has been determined from equilibrium saturation assays or by calculation from the appropriate kinetic constants if it is not, this implies that the physical characteristics of the interaction are outside the criteria for which the equations have been developed, i.e., those rehearsed in Section 2.7. Statistical comparison of data sets must also be carefully assessed here the availability of the powerful computation facilities available on most laboratory desks has taken much of the drudgery out of such analysis. 

These equilibrium-binding relationships give rise to four different kinetic responses competitive inhibition, uncompetitive inhibition, non-competitive inhibition, mixed inhibition. Details of the kinetics of these types of inhibition and how dissociation constants for the reactions can be measured are provided in Appendix 3.6. 

This yielded a fe+i of 0.0091 + 0.002 nM mm for NO 711 binding to mGATl. Hence, the equilibrium dissociation constant calculated from kinetic MS binding experiments resulted in = 11.7 2.5 nM. This is in good accord with Kj determined in MS saturation binding experiments and confirms the validity of the new setup. 

One frequently encounters the case where the equilibrium dissociation constant (iQ, see above) is defined by microconstants with Tast rates on and off the receptor. However, any change in potency in a chemical series (affinity) must represent an increase in the on (k+i) rate or a decrease in the off rate (fe i). Occasionally, either by accident or design, the off rate is altered dramatically enough to redefine the receptor kinetics of the compound such that the rates influence the actual pharmacodynam-... 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

Rapid Equilibrium Case. In the absence of significant amounts of product (i.e., initial rate conditions thus, [P] 0), the rate expression for the rapid equilibrium random Bi Uni mechanism is v = Uniax[A][B]/(i iai b + i b[A] + i a[B] + [A][B]) where is the dissociation constant for the EA complex, and T b are the dissociation constants for the EAB complex with regard to ligands A and B, respectively, and Umax = 9[Etotai] where kg is the forward unimolecular rate constant for the conversion of EAB to EP. Double-reciprocal plots (1/v v. 1/[A] at different constant concentrations of B and 1/v v. 1/[B] at different constant concentrations of A) will be intersecting lines. Slope and intercept replots will provide values for the kinetic parameters. 

There are at least three mechanisms for the reversal of NO inhibition of an enzyme. The most obvious is that the NO can be released as the concentration of NO in solution falls or the temperature rises. This depends on the (kinetic) off constant for NO in the system in question as well as the (equilibrium) dissociation constant for most enzymes that form nitrosyl complexes, this has not been measured. Nitric oxide bound to a metal center could also undergo oxidation or reduction followed by the release of product. Finally, NO could be photodissociated from a metal center by light of the correct wavelength. 

In connection with the subject of the relation between association state and kinetic order, it is germane to mention observations of Roovers and Bywater (45). They measured the dissociation constant for the tetramer , dimer case for polyiso-prenyllithium in benzene. The technique involved a study of the electronic spectra at 272 and 320 nm. If the process they measured can be directly related to the association-dissociation equilibrium, their results can be used to calculate the dissociation constant for the correct dimer monomer system. This value is ca. 2xl0-5 at 30.5°C. If this value is accepted, then the situation is encountered where the degree of dissociation of the polyisoprenyllithium chain ends varies from about 0.10 to... 

Conductivity measurements for solutions of living poly-pPL with DBCK+ counterion in CH2CI2/8PL mixture indicated that macroions and macroion-pairs are present in the system. In Fig.l, taken from Ref. 2 the Vant Hoff>s plots are given for dissociation constants of poly-ePL macroion-pairs with DBCK+ counterion (Kp) and similar plot for dissociation constants of Ph4B DBCK+ (Kpi). Dissociation of Ph4B"DBCK+ was investigated because this salt was further used in the kinetic measurements to shift the equilibrium between macroions and macroion-pairs towards the latter ones. 

In order to explain the field effects observed for the cationic polymerizations, we have earlier proposed a kinetic scheme based on the two-state polymerization mechanism and on the field-facilitated dissociation hypothesis (11). Though the assumptions involved in the proposed interpretation turn out to be partly invalid in the light of the experimental data accumulated most recently (15), it is still necessary to give an outline of the scheme. We assumed that, by the initiation reaction between initiator molecules (C) and monomer molecules (M), active species of an ion-pair type (My) are produced, a portion of which dissociates into active species of a free ion type (Mf) and gegenions (C ). The propagation, monomer transfer and termination can be effected by the free ions and ion pairs. A dissociation equilibrium is established between the free ions and ion pairs, which can be characterized by a dissociation constant K. Then we have ... 

The constants that must be measured in scheme 2 are KT, the dissociation of the E-Tyr complex (by equilibrium dialysis or kinetics) K A, the dissociation constant of ATP. from the ternary complex, E-Tyr-ATP (from kinetics) k3, the rate... 

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