Further characterization of a control model for ligand-receptor interaction: phase plane geometry, stability, and oscillation.

The local concentration of an endogenous ligand that binds to a specific cellular receptor according to the mass-action law is controlled. The ligand concentration (zeta) remains bounded in an interval that contains the basal point. The current study is a more detailed examination of a control model that we previously proposed in which the feedback is a function (phi) of the concentration (y) of the receptor-bound ligand. Previously we showed that if phi' (y) less than or equal to 0 at a critical point then that point is stable. Accordingly, at an unstable critical point phi' (y) greater than 0. We now show that if phi (y) is continuous in [y1,y2], where y1 and y2 are adjacent critical points, and if phi' (y1) less than or equal to 0 then the critical point at y2 is unstable. The distance between these adjacent critical points is crucial for maintaining control in response to a dose (or release) of ligand. It is further shown that the difference in the slopes of the binding and control curves at their intersection on the phase-plane portrait is an indicator of stability and of the characteristic frequency of oscillation in the system. Together these findings provide new insights needed in the modelling of specific ligand-receptor interactions.