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Mathematical Model of the Spatio-Temporal Dynamics of Second Messengers in Visual Transduction

D. Andreucci ^*, P. Bisegna , G. Caruso , H.E. Hamm  and E. DiBenedetto ^¶

^* Dipartimento di Metodi e Modelli Matematici, Università di Roma La Sapienza, 00161 Rome, Italy; Dipartimento di Ingegneria Civile, Università di Roma Tor Vergata, 00133 Rome, Italy; ITC-CNR, Rome, Italy; Department of Pharmacology, Medical Center, Vanderbilt University, Nashville, Tennessee 37232; and ^¶ Biomathematics Study Group, Department of Mathematics, Stevenson Center, Vanderbilt University, Nashville, Tennessee 37240

Correspondence: Address reprint requests to E. DiBenedetto, E-mail: em.diben{at}Vanderbilt.edu.

A model describing the role of transversal and longitudinal diffusion of cGMP and Ca²⁺ in signaling in the rod outer segment of vertebrates is developed. Utilizing a novel notion of surface-volume reaction and the mathematical theories of homogenization and concentrated capacity, the diffusion of cGMP and Ca²⁺ in the interdiscal spaces is shown to be reducible to a one-parameter family of diffusion processes taking place on a single rod cross section; whereas the diffusion in the outer shell is shown to be reducible to a diffusion on a cylindrical surface. Moreover, the exterior flux of the former serves as a source term for the latter, alleviating the assumption of a well-stirred cytosol. A previous model of visual transduction that assumes a well-stirred rod outer segment cytosol (and thus contains no spatial information) can be recovered from this model by imposing a "bulk" assumption. The model shows that upon activation of a single rhodopsin, cGMP changes are local, and exhibit both a longitudinal and a transversal component. Consequently, membrane current is also highly localized. The spatial spread of the single photon response along the longitudinal axis of the outer segment is predicted to be 3–5 µm, consistent with experimental data. This approach represents a tool to analyze pointwise signaling dynamics without requiring averaging over the entire cell by global Michaelis-Menten kinetics.

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