Applicable Analysis Vol. 85, No. 1–3, January–March 2006, 303–331 Homogenization and concentration of capacity in the rod outer segment with incisures DANIELE ANDREUCCIy, PAOLO BISEGNAz and EMMANUELE DIBENEDETTO*x yDepartment of Mathematical Methods and Models in Applied Sciences, University of Rome, La Sapienza, Via A. Scarpa 16, 00161 Rome, Italy zDepartment of Civil Engineering, University of Rome, Tor Vergata, Via del Politecnico 1, 00133 Rome, Italy xDepartment of Mathematics, Vanderbilt University, Nashville, TN 37240, USA Communicated by R. Magnanini (Received 12 October 2004; in final form 26 June 2005) Dedicated to the memory of Carlo Pucci We present a quantitative model of the spatio-temporal dynamics of second messengers mediating phototransduction in retinal rods. The spatial domain (the rod outer segment) has a quite complex geometry, involving different ‘‘thin’’ domains, whose thickness is three orders of magnitude smaller than the other dimensions. The model relies on a ‘‘pointwise’’ application of first principles leading to a system of evolution equations set in such a structured geometry. Then, exploiting an idea first presented in [Andreucci, D., Bisegna, P. and DiBenedetto, E., 2002, Homogenization and concentrated capacity in reticular almost disconnected structures. Comptes Rendus Mathematique. Academie des Sciences. Paris, Se ´ries I, 335, 329–332], the diffusion problem is reduced to one with a simpler geometry, still preserving the essential features of the original one. This is achieved by an homogenization and concentration limit. However, here we take into account for the first time the presence of ‘‘incisures’’, which are important for phototransduction, and introduce new mathematical features mainly in the concentration limit. Keywords: Homogenization; Signal transduction; Concentration of capacity; Disconnected structure; Reticular structure 2000 Mathematics Subject Classifications: 35B27; 35K50; 92C37 1. Introduction Figure 1 is a schematic rendering of the anatomy of a rod outer segment (ROS) in the retina of vertebrates. It consists of a right circular cylinder, of height and diameter *Corresponding author. Email: em.diben@vanderbilt.edu Applicable Analysis ISSN 0003-6811 print: ISSN 1563-504X online ß 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00036810500276381