Modelling the effect of antibody depletion for dose-response behavior of common immunostaining protocols

Antibody binding properties for immunostaining applications are often characterized by dose-response curves, which describe the amount of bound antibodies as a function of the initially applied antibody concentration. A common model for the dose-response curve is the Langmuir isotherm, which assumes an equilibrium between the binding and unbinding of antibodies. However, for common immunostaining protocols, the equilibrium assumption is violated, and the dose-response behavior is governed by an accumulation of permanently bound antibodies. Assuming a constant antibody concentration, the resulting accumulation model can easily be solved analytically. However, in many experimental setups the total amount of antibodies is fixed, such that antibody binding reduces the concentration of free antibodies. Solving the corresponding depletion accumulation model is more difficult and seems to be impossible for heterogeneous epitope landscapes. In this paper, we first solve the depletion accumulation model analytically for a homogeneous epitope landscape. From the obtained solution, we derive inequalities between the depletion accumulation model, the depletion-free accumulation model, and the Langmuir isotherm. This allows us to characterize the depletion effect for homogeneous epitope landscapes. Next, we generalize the problem to heterogeneous epitope landscapes, where we prove the existence and uniqueness of a solution that behaves as expected from the experimental setting. These natural properties define bounds for the depletion accumulation model. We conclude this paper by applying the bounds to characterize the depletion effect for heterogeneous epitope landscapes.