Geometric singular perturbation theory for a dimerisation problem in mathematical pharmacology.

We study the pharmacological model of dimerisation where a receptor binds to two ligand molecules. The dimerisation model is an adaptation of the well studied target-mediated drug disposition (TMDD) model, in which a receptor binds to one ligand molecule. In these systems the drug binds to its target with high affinity. In both models, it is assumed that the receptor binding is the fastest process. This leads to a separation of time scales, allowing us to use techniques from singular perturbation theory to analyse these models. Techniques from singular perturbation theory such as asymptotic analysis, geometric singular perturbation theory and geometric desingularisation work well for the TMDD model where the underlying critical manifold consists of two two-dimensional submanifolds. However, the known techniques can not be applied to the dimerisation problem, as the underlying critical manifold is degenerate and consists of a two-dimensional and a three-dimensional submanifold. The intersection of these manifolds is important for the Dimerisation problem as there is a type of rebound in the dimerisation problem occurring at the specified bifurcation point. Motivated by the dimerisation problem, we consider a general two parameter slow-fast system in which the critical manifold consists of a one-dimensional and a two-dimensional submanifold. These submanifolds intersect transversally at the origin. Using geometric desingularisation, we show that for a particular subset of parameters the continuation of the slow manifold connects the attracting components of the critical set. We also show that the direction of this continuation on the two-dimensional manifold can be expressed in terms of the model parameters. This method is then applied to the dimerisation problem in order to understand and approximate the rebound