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Fig. 2.3 Plots of the concentration of the protein-ligand complex present at equilibrium [C]eq (mM, shown as mM) as a function of the binding constant Kd (mM), with various initial concentrations of protein [P]o and ligand [L]o. Note that the [C]eq values are the concentrations of the protein-ligand complex just prior to the GPC spin

Fig. 2.3 Plots of the concentration of the protein-ligand complex present at equilibrium [C]eq (mM, shown as mM) as a function of the binding constant Kd (mM), with various initial concentrations of protein [P]o and ligand [L]o. Note that the [C]eq values are the concentrations of the protein-ligand complex just prior to the GPC spin column experiment. When initial concentrations of ligand and protein are >5 mM, the concentration of complex produced for Kd values <20 mM is >1 mM of complex, a concentration considerably greater than the detection limit of modern ESI-Tof mass spectrometers.

tion are rapidly separated from the protein using centrifugation. At the start of GPC spin column centrifugation step (time = 0), the concentration of the protein-ligand complex is equal to the equilibrium concentration, [C]. As the complex migrates through the column during centrifugation, it dissociates. If we assume that the protein-ligand complex dissociates in the column at a rate much faster than the association rate, then the concentration of the protein-ligand complex can be expressed by the first-order rate equation:

where koff is the kinetic off-rate constant and t is the elution time. Note the use of underscores to designate concentrations not based on the law of mass action but rather based on non-equilibrium phenomena. Solving this equation, we obtain:

Ultimately in the GPC spin column screening experiment, the complex present in the eluate is dissociated and the ligand molecules liberated from the protein are detected by mass spectrometry. The amount of ligand detected is essentially equivalent to the concentration of the protein-ligand complex that eluted from the GPC spin column. Equation (7) indicates that the amount of protein-ligand

Fig. 2.4 Plots of the fraction of complex ([C]/[C]eq) eluted from the spin column from the initial equilibrium state as a function of time for a variety of off-rate constants. Assuming a GPC spin column elution time of 15 s, greater than 20% of the initial equilibrium complex concentration is recovered in the GPC spin column eluate for off-rate constants less than 0.1 s"1.

Fig. 2.4 Plots of the fraction of complex ([C]/[C]eq) eluted from the spin column from the initial equilibrium state as a function of time for a variety of off-rate constants. Assuming a GPC spin column elution time of 15 s, greater than 20% of the initial equilibrium complex concentration is recovered in the GPC spin column eluate for off-rate constants less than 0.1 s"1.

complex that survives the GPC spin column decreases exponentially as a function of the product of the off-rate constant and the elution time. Since the off-rate is controlled by the nature of the complex, the only GPC spin column parameter experimentally controllable is the spin time (t). The shorter the spin time the greater the concentration of complex eluted from the spin column. Figure 2.4 plots the fraction of complex ([C]/[C]eq ) eluted from the spin column from the initial equilibrium state as a function of time for a variety of off-rate constants. In most experiments, the spin column eluate is collected within about 15 s. Under these conditions, greater than 20% of the initial equilibrium complex concentration is recovered in the GPC spin column eluate for off rate constants less than 0.1 s"1.

Since the limit of detection for small molecule ligands, with modern ESI-Tof mass spectrometers, is approximately @0.05 mM, the concentration of the protein-ligand complex prior to the GPC spin column treatment must be about 0.25 mM. For initial protein and ligand concentrations >5 mM, this corresponds to Kd values <20 mM, as indicated in Fig. 2.3. This is a desirable region for the GPC spin column studies, since one wants to be certain to detect ligands from the stronger as well as the weakest ligand binders.

Using sub-ambient temperatures for preparing the protein-ligand equilibrium mixtures and for centrifugation of the GPC spin column, the dissociation rate constant decreases and the off-rate diminishes, thereby expanding the kinetic window observable with GPC spin column screening to even weaker binders with Kd values >20 mM.

2.1.4.3 Estimation of Relative Binding Affinities from GPC Spin-Column/ESI-MS Data

For a variety of ligands in a mixture with the same initial concentration [L]o, such that [L]o > [C] and where the equilibrium concentration of the remaining protein is [P], we can relate back to equilibrium conditions, and using Eq. 7 for computing the ratio of two components subscripted 1 and 2, we obtain:

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, koff1 = koff2, then:

i.e., the dissociation constants are inversely related to the ligand concentrations measured by mass spectrometry after elution from the GPC spin column. Equation (9) can be used to reliably estimate the unknown dissociation constants for related ligands in a mixture with a protein if the concentrations of the ligands in the GPC spin column eluate are quantitated and the dissociation constant for one of the ligands is known. Likewise for ligands, either in a mixture or as singletons, of initial equal concentrations when incubated with a protein, the relative binding affinities and relative dissociation constants for the ligands can be ranked based upon the ligand concentrations in the GPC spin column eluate as quantitated by mass spectrometry.

2.1.4.4 Experimental Determination of the Kd Value from GPC Spin-Column/ ESI-MS Data

The expression for the equilibrium concentration of the protein-ligand complex [C], described above using Eq. (5), can also be re-written in terms of the total initial protein concentration [P]o such that:

and predicts a hyperbolic, saturable dependence of the concentration of the pro-tein-ligand complex on the free ligand concentration. Equation (10) is a form of the simple Langmuir isotherm.

An experimentally most useful relationship occurs using Eq. (10), when the free ligand concentration [L] is equal to the dissociation constant Kd, namely,

i.e., the protein binding sites are half-saturated with ligand. Conversely, the free ligand concentration at 50% protein saturation is a measure of the Kd. The effective ligand concentration at 50% protein saturation is referred to as the EC50 and is equivalent to the Kd. Typically, the EC50 value is experimentally obtained by titrating various concentrations of ligand with a fixed initial protein concentration and measuring the concentration of complex formed, obtained in the GPC spin column/ESI-MS measurement. A plot of [C]/[P]o vs log10[L] produces a sigmoidal shaped curve symmetrical about log10 Kd. The Kd value can be read directly from the plot as the corresponding value of [L] where [C]/[P]o is equal to 50%, the EC50 value. See the discussion in section 2.3.3.3 for an experimental application of this methodology.

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