Equilibrium ligand concentration

Note that, in Eq. (8), the concentrations for the complex [C] and related ligand [L] are equal because the ligand is liberated from the complex by denaturing the complex. These non-equilibrium ligand concentration values are obtained by mass spectrometry from the denatured GPC spin column eluate. If the off-rates for the different compounds are the same, k gi = koff2j then ... 

Once the equilibrium ligand concentration is known, then the aj values and, therefore, the for all the species can be obtained. This represents an exact analogy to the procedure in finding the concentrations of all the species in a Bronsted acid-base system when the pH is known. 

Scheme VIII has the form of Scheme II, so the relaxation time is given by Eq. (4-15)—appjirently. However, there is a difference between these two schemes in that L in Scheme VIII is also a participant in an acid-base equilibrium. The proton transfer is much more rapid than is the complex formation, so the acid-base system is considered to be at equilibrium throughout the complex formation. The experiment can be carried out by setting the total ligand concentration comparable to the total metal ion concentration, so that the solution is not buffered. As the base form L of the ligand undergoes coordination, the acid-base equilibrium shifts, thus changing the pH. This pH shift is detected by incorporating an acid-base indicator in the solution.

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

Exchange of complex cations. Complexation of transition metal cations with uncharged ligands such as with amines and with amino acids results in a selectivity enhancement compared to the selectivity of the aqueous metal cation (27, 65-72). Fig. 3 shows an example for the Cu(ethylenediamine) adsorption in montmorillonites of different charge density. Standard thermodynamic data for other cases are given in table IV. In all cases the free ligand concentration in equilibrium solution was... 

At very low ligand concentrations, Ka2bi (pco)2/Q. 1 becomes the predominant expression in the inhibition term and Eq. 6 can eventually be simplified to (kg modified constant including the equilibrium constants)... 

There are two important conclusions that can be drawn from this equation. Firstly, when the ligand concentration, [L], is very much greater that the equilibrium dissociation constant, Kj ([L] Kd), Eq. (5) simplifies to ... 

In some cases more complex reaction schemes may give rise to linear Scatchard plots (Conners, 1987), and nonlinear plots may arise from a number of experimental artefacts, e.g., failure to reach equilibrium at low ligand concentrations. The interpretation of this particular linearisation approach has been the subject of many articles to which the reader is referred for further insight (Boeynaems and Dumont, 1975 Norby et al., 1980 Klotz, 1982, 1983 Hulme, 1992). 

Clearly throughout the kinetic experiments the concentration of each of the components RL, R and L are changing but, if the initial ligand concentration [Lq] is very much greater than the total receptor concentration [Rq], the free ligand concentration will suffer little depletion as the association continues to equilibrium. This is referred to as pseudo first-order kinetics which are easier to analyse ... 

At equilibrium, the concentration of free ligand, [L], is the same on both sides of the dialysis membrane. Thus, the difference in the amount of radioactivity measured in aliquots taken from both sides of the membrane can be used to estimate [RL] (see Protocol 4.1)... 

The general method for ASMS is shown in Fig. 4.1. In ASMS, the target concentration is generally set at 5-10 xM, so that at equilibrium, ligands with affinities of no weaker than Ku 10 xM will be significantly bound and, therefore, retained in the ultrafiltration steps. The minimal concentration of each small molecule is dictated by the eventual need to detect ligands by mass spectrometry after several cycles of ultrafiltration and subsequent extraction. In order to ensure detection just above baseline for the vast majority of compounds, which vary in inherent ionization properties and efficiency of mass spectrometric visibility, the starting compound concentration is set at 1.5 pM per compound. The mixture... 

One asset of mass spectrometry in protein science is that ESI and MALDI [11, 75] can introduce noncovalent complexes to the gas phase [12, 76, 77]. If one can assume that the gas-phase ion abundances (peak intensities) for the complex, apo protein, and ligand are directly related to their equilibrium concentrations in solution, the relative and absolute binding affinities can be deduced [78-81]. Extended methods are now available that also make use of the intensity of the complex and the protein at high ligand concentration to determine binding constants [78, 82-84]. 

Kd is the equilibrium dissociation constant and corresponds to that ligand concentration at which 50 % of binding sites are occupied. The values given in (A) and used for plotting the concentration-binding graph (B) result when Kd = 10. 

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