Some Mathematical Problems in Visual Trans- duction

Abstract. We present a mathematical model for the phototransduction cas-cade, taking into account the spatial localization of the dierent reaction processes. The geometric complexity of the problem (set in the rod outer segment) is simplied by a process of homogenization and concentration of capacity. 1. The Phototransduction Cascade Signal transduction in living cells occurs by precise, highly regulated localization of key enzimes. The relevant diusion and reaction processes take place in sub-compartments of the cell. The standard ODE approach, based on Michaelis-Menten well-stirred kinetics averages concentrations within the whole cell volume, and therefore is unable to predict the spatial localization of the signaling processes. A PDE approach seems more suitable as takes into account the spatial depen-dence of the signal. However, the geometry of the domain open to diusion may be so complex as to prevent in practice any aective qualitative or numerical analysis of the problem. This occurs whenever the cell contains structures which impede, or however in uence, the diusion of the signaling molecules, but whose detailed geometry cannot be directly taken into account in the mathematical model. In [2, 3, 4] we have addressed these issues in the context of the phototrans-duction cascade in the rod outer segment (ROS) of vertebrates (Figure 1). The ROS is a cylinder of height H and cross section a disc of radius R + ", housing a longitudinal stack of equispaced, parallel, coaxial cylinders Cj, each of radius R and thickness ", called discs. The stack cuts between any two contiguous discs C