Control and oscillation in ligand receptor interactions according to the law of mass action

The law of mass action is almost universally applied to interactions of both endogenous ligands and drugs with their specific receptors and results in the familiar hyperbolic equilibrium binding curve of bound (y) vs free (z) concentrations. Whereas the concentration of a drug molecule is governed by its pharmacokinetic properties and, possibly, by intrinsic control mechanisms, natural ligands are certainly controlled since their concentrations normally remain within specific limits. This paper represents a further study of control of this kinetic process in a model based on ligand production (rate F), first-order elimination (rate constant E) and a feedback function of occupancy, phi(y), that modulates these. In the controlled situation the system equilibrium occurs at states called critical points (yc,zc) at which both dy/dt and dz/dt are simultaneously zero. There are only a finite number of such points along the hyperbolic binding curve and these may be either stable or unstable. The basal state is the normal operating point of the system and is necessarily stable; that is, perturbations producing states away from it will return in time to this point. We have previously shown that phi'(y) less than or equal to 0 is a sufficient condition for stability. Accordingly, for a continuous control curve, an adjacent critical point will be unstable, and have phi'(y) greater than 0. If the system coordinates get sufficiently close to such an unstable point there is propulsion to extreme states and loss of control. The distance between the stable and unstable points determines whether a dose (or release) of the ligand will be controlled or not. The current paper focuses on the geometrical properties of the binding and control curves and how these relate to the stability of critical points and the overall control of ligand doses. In particular we show how the magnitude of the (negative) slope of the control curve at the basal point affects the frequency of oscillation about the basal state. It is further shown that high frequency control results in lower receptor occupancy, a result that may explain desensitization.