Determination and Interpretation of the Titration Curves

Quenching and desalting cause the ligand(s) to dissociate, liberating the protein for molecular mass measurements by mass spectrometry to afford the number of deuteriums taken up by solvent-accessible amides. A plot of the mass difference between the deuterated and nondeuterated protein (deuterium uptake) versus the total ligand concentration (or the ratio of total ligand concentration to the total protein concentration) gives the PLIMSTEX curve (Fig. 11.2). Usually, the deute-

Protein Ligand Binding

Fig. 11.2 Schematic illustration of a PLIMSTEX curve for 1:1 protein:ligand binding. P is protein, L is ligand. b1 is the binding constant for binding one ligand; Do is the deuterium level for apoprotein, and AD1 is the difference between the average deuterium level of one-ligand-bound protein complex and that of the apo-protein.

Fig. 11.2 Schematic illustration of a PLIMSTEX curve for 1:1 protein:ligand binding. P is protein, L is ligand. b1 is the binding constant for binding one ligand; Do is the deuterium level for apoprotein, and AD1 is the difference between the average deuterium level of one-ligand-bound protein complex and that of the apo-protein.

rium uptake values decrease with an increase in the ligand concentration, and a decrease reflects increased hydrogen bonding (overall deuterium shift AD) of some backbone amide protons upon forming the complex. Intermediate states for multiple ligand binding may also be monitored when a species deuterium shift (AD;) can be related to a specific binding species.

PLIMSTEX curves are sensitive to the total protein concentration. When the protein is titrated at high concentration (@100 times the 1/Ka or Kd), a ''sharp-break'' curve is obtained (see curve B in Fig. 11.2), and the ratio of [ligand]total to [protein]total at the break clearly indicates binding stoichiometry. To quantify affinity, PLIMSTEX requires that a change occurs in the extent of H/D exchange during a titration at a protein concentration comparable to the dissociation constant Kd (see curve A in Fig. 11.2). The change may be conformational and/or a stability difference between the apo- and ligand-bound proteins.

To extract the binding affinity, the titration data are fitted using a 1:n protein:li-gand sequential binding model, where n is the number of binding sites for the same ligand. There are two assumptions for the modeling: (1) the ligand binds the protein in a stepwise fashion, and (2) H/D exchange of each amide is independent (i.e., does not depend on exchange at any other site of the protein). A nonlinear least squares (NLLS) regression [32] is involved in a calculation of the extent of change in the H/D exchange during the titration (AD) as a function [Eq. (1)] of the total ligand concentration ([LigT]), the overall binding constants (b;, which is the product of the stepwise macroscopic binding constants K;, where ; = 1 to n), and the species deuterium shifts (D0 and AD;, ; = 1 to n). D0 is the shift in the molecular weight of the apo protein caused by H/D exchange (deuterium uptake). To minimize experimental errors, we do not accept the experiment value of D0 (the deuterium uptake of the apo-protein) but rather take it as a variable or unknown parameter. AD; is the difference between the average deuterium level of each protein-ligand complex and that of the apo-form (Fig. 11.2). It is weighted by its binding fraction [Prot-Lig;]/[ProtT], which is a function of [LigT] and b; (i = 1 ^ n), the latter of which is the product of all the stepwise equilibrium binding constants (b; = K1K2 ■ ■ ■ K;). A positive AD; indicates that binding of i ligand(s) to the protein leads to more hydrogen bonding and less D uptake as compared to the apo-form. A negative AD; points to the formation of a more open structure with less hydrogen bonding relative to its apo form. When AD; is approximately zero, little conformational change apparently occurs upon binding although changes in one part of the protein may be compensated by changes in another. If no net change occurs, PLIMSTEX may not be appropriate for determining the corresponding equilibrium constant (b;).

AD(b1,..., bn; Do, AD1,..., ADn, [LigT]) = Do - £ AD; ^Lp (1)

The best fit is obtained by a search, which iterates through a sequence of trials to minimize the error between the calculated overall deuterium shift, AD, and the experimentally measured shift by varying the unknown parameters (b;, D0, AD;). The average data (at least two runs) are used for the curve fitting to give mean values for the unknown parameters (b;, AD;, D0). The macroscopic K; values are calculated from bi values. Finally, a resampling statistical analysis is used to evaluate the precision for each parameter in the search.

We described previously the detailed modeling procedure for analyzing PLIM-STEX data [24]. For fitting the protein self-association data, we modified the modeling to acknowledge that both ligand and protein are the same, and these modifications were described at a recent conference [33]. All modeling procedures were implemented with Mathcad 2001 Professional (MathSoft, Cambridge, Mass.). This modeling process is not only a new tool for analyzing H/D exchange data acquired by electrospray ionization-mass spectrometry (ESI-MS), but also possesses some novel aspects in modeling experimental titration data to determine the affinity of ligand binding.

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