During the association phase, the free ligand (L) is binding to the immobilized receptor (R) to form the ligand-receptor complex (RL), and the binding is governed by the constant k_on. At the same time, ligand-receptor complexes are dissociating to free ligand and free receptor, governed by the constant k_off. Binding continues until the rate of free ligand binding to receptors reaches an equilibrium with the rate of RL complex dissociation. In the spr experiment, we measure time required for binding and dissociation to come to equilibrium, given by the rate constant k_obs.
Based on this info, we can write the following differential equation to represent the change in [RL] over time:
[tex]\frac{d[RL]}{dt} = k_{on}[R][L] - k_{off}[RL][/tex]
Since the total concentration of receptors is given by [R]_T = [R] + [RL], we can write:
[tex]\frac{d[RL]}{dt} = k_{on}[L]([R]_T-[RL]) - k_{off}[RL] = k_{on}[L][R]_T - (k_{on}[L]+k_{off})[RL][/tex]
where k_on, k_off, [R]_T and [L] are constants ([L] is constant because the ligand is continuously being replenished by the flow in the spr experiment).
Solving the differential equation yields [RL] as a fuction of time:
[tex][RL] = A(1-e^{-k_{obs}t})[/tex]
where A represents the fraction of receptors bound at equilibrium and k_obs = k_on [L] + k_off
Thus the observed rate constant during the association phase depends on both k_on and k_off. Determining k_on requires additional experiments to determine k_off.