Theory

Single-site binding of a ligand to a receptor is described by the following equilibrium expression, here using nomenclature familiar from Michaelis-Menten kinetic analyses, with E representing the receptor (for example, an enzyme) and S representing the ligand (for example, an enzyme's substrate):

The protein-ligand binding affinity is usually expressed as the equilibrium dissociation constant, Kd, which is described by the following relationship between the concentrations of free receptor [E], free ligand [S], and the receptor-ligand complex [ES]:

Some titration methods utilize high ligand-to-receptor ratios to simplify data analysis, and in these cases depletion of the ligand concentration by binding to the target can be ignored, which simplifies the analysis. In the ALIS method, the receptor and ligand concentrations are comparable in magnitude and ligand depletion must be explicitly considered as the titration results are analyzed. Expressions for free receptor and ligand may be written in terms of total receptor and total ligand concentrations [E]0 and [S]0, respectively:

These values can be substituted into the original expression defining Kd. Here, no assumptions or simplifications regarding ligand depletion are made:

Solving this quadratic equation for protein-ligand receptor concentration [ES] yields the following expression. Here, the protein-ligand complex concentration [ES] is defined in terms of the Kd and the total receptor and ligand concentrations [E]o and [S]o:

[ES] (Kd + [S]o + [E]o -\J(Kd + [S]o + [E]o)2 - 4[E]o[S]o) (5)

ALIS measures the MS response of the ligand following its dissociation from the protein-ligand complex. Therefore, the magnitude of the MS response corresponds to the equilibrium concentration of the receptor-ligand complex concentration [ES] times the compound's MS calibration factor CMS, which depends on the ionization efficiency and other molecular properties of the ligand:

Substituting this expression into the equation above yields a new expression relating the MS response to four variables: the total ligand concentration [S]0, which is the known, independent variable in a titration experiment; the Kd, which is the dependent variable of interest; the total receptor concentration [E]0; and the MS response calibration factor CMS:

MS Response

= CMS (Kd + [S]o + [E]o - V(Ká + [S]o + [E]o)2 - 4[E]o[S]o) (7)

Therefore, plotting the ALIS MS response from a titration series versus the total ligand concentration yields a saturation binding curve that can be fit to this equation by nonlinear regression analysis to yield the Kd of the ligand of interest.

The MS response calibration factor CMS can be determined independently by injecting samples of known ligand concentration into the MS and correlating the response with the amount injected. This allows quantitative determination of the receptor-ligand complex concentration at each data point of the titration, and enables accurate measurement of the total receptor concentration [E]0 as the asymptote of the saturation binding curve. However, in practice it is simpler to fit the titration curve data to yield the MS calibration factor by non-linear regression, since this obviates the need to create calibration curves for each ligand under study. Another advantage of fitting the MS response factor is that any minor losses of ligand in the ALIS system (for example, due to protein-ligand complex dissociation during the SEC stage) are corrected for and do not influence the Kd estimate. In the absence of a calibration curve, solving for the MS response factor

Fig. 3.6 Simulated ALIS saturation binding experiments for ligands of varying affinity to a single-site receptor present at 5.0 mM concentration.

CMS by regression analysis may not yield an accurate value for either this variable or for [E]0, as the two variables are highly coupled in the regression results. Fortunately, the fit value of the Kd variable is not so highly coupled, and can be determined with good confidence [46].

Fig. 3.7 ALIS titration experiment for warfarin vs 5.0 mM HSA. Duplicate injections shown. (A) Fitting the data by nonlinear regression analysis yields a Kd of 5.6 G 1.0 mM. (B) Data from A, plotted as a sigmoidal curve to better show the fit at low titrant concentrations. (C) Residuals plotted as absolute and (D) as percent of signal.

Fig. 3.7 ALIS titration experiment for warfarin vs 5.0 mM HSA. Duplicate injections shown. (A) Fitting the data by nonlinear regression analysis yields a Kd of 5.6 G 1.0 mM. (B) Data from A, plotted as a sigmoidal curve to better show the fit at low titrant concentrations. (C) Residuals plotted as absolute and (D) as percent of signal.

Fig. 3.8 Examples of the ALIS-based Kd titration experiment for a variety of compounds and protein targets. (A) Compound "Merck-1" vs 5.0 |mM Akt-1, Kd = 0.3 + 0.1 |mM. (B) Staurosporine vs 4.5 |mM JNK1, Kd = 1.0 + 0.4 |M. (C) NGD-3350 vs 2.5 |M M2 receptor, Kd = 0.7 + 0.1 |M. (d) NGD-157 vs 5.0 |M DHFR, Kd = 3.5 + 1.7 |M.

Fig. 3.8 Examples of the ALIS-based Kd titration experiment for a variety of compounds and protein targets. (A) Compound "Merck-1" vs 5.0 |mM Akt-1, Kd = 0.3 + 0.1 |mM. (B) Staurosporine vs 4.5 |mM JNK1, Kd = 1.0 + 0.4 |M. (C) NGD-3350 vs 2.5 |M M2 receptor, Kd = 0.7 + 0.1 |M. (d) NGD-157 vs 5.0 |M DHFR, Kd = 3.5 + 1.7 |M.

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