Homogenization and concentration of capacity in the rod outer segment with incisures

Abstract

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, Séries 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

Acknowledgement

This research was supported by NIH grant RO1 GM68953-01.

Notes

¹ For the Salamander H≈22 μm, R≈5.5 μm, ϵ _o ≈14 nm, νϵ _o ≈14 nm, σϵ _o ≈15 nm, n_o ≈1,000; see 5. The discs in the ROS of the Salamander exhibit up to 18 incisures, each of largest width of the order of 10 nm. We refer to the review article 3 for a detailed description of the rod anatomy.