The association rate (or "on-rate") can be likewise defined as the product of the association rate constant kon, in units of s_1, the concentration of free pro-
144 | 3 ALIS: An Affinity Selection-Mass Spectrometry System for the Discovery and Characterization tein [E], and the concentration of free ligand [S]:
The overall rate of change in the concentration of protein-ligand complex with time is the sum of its rate of formation and its rate of depletion:
It should also be noted that when the rate of change in the protein-ligand complex concentration is zero (by definition, when the system is at equilibrium), this equation reduces to the equilibrium expression below, with the binding affinity constant Kd defined as the ratio of the dissociation rate koff to the association rate kon:
As mentioned above in a qualitative sense, it can be seen from this equation that, for a given association rate constant kon, a lower value of dissociation rate constant koff yields a smaller value of Kd and hence a higher equilibrium concentration of the desired protein-ligand complex.
The half-life (t1=2) of binding is another convenient metric for comparing dissociation rates. For a first-order process such as protein-ligand complex dissociation, the half-life is defined from the dissociation rate constant koff as follows:
In the absence of any protein-ligand re-association (for example, under hypothetical conditions of infinite dilution of the complex), the half-life is the time required for half of the complex to decay to unbound protein and ligand.
As mentioned previously, the concentrations of free protein [E] and free ligand [S] are related to the total protein and ligand concentrations [E]0 and [S]0 by the following relationships:
Substituting these expressions into the equations above yields a new expression that enables the interaction kinetics to be readily modeled for single-site, reversible binding between a protein and a single ligand:
In analogy to the single-ligand, single-site equilibrium described above, competitive binding between a ligand S and an inhibitor ligand I is described by the following equation:
Here, free protein E can react either with ligand S to form the complex ES, or react with free inhibitor I to form complex EI. It follows that the overall rate of change in the concentrations of protein-ligand complexes [ES] and [EI] is described by the following simultaneous differential equations:
® - feS-on([E]0 - [ES] - [EI])([S]0 - [ES]) - kS-ff [ES] (17)
d[EI] dt kI.on([E]o - [ES] - [EI])([I]o - [EI]) - k,0#[EI] (18)
The kinetics of a system of competing ligands can be modeled by simultaneous numerical solution of these two equations given initial values for the system parameters, including the total protein concentration [E]0, the total ligand and total inhibitor concentrations [S]0 and 0, the rates of association and dissociation for the interacting components of the mixture ks-on, ks-off, kI-on, and kI-off, and initial values for [ES] and [EI]. Note that simultaneous solution of these equations where the initial value of [ES] is not zero allows the behavior of the system to be modeled versus time upon addition of an excess of inhibitor.
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