Estimating the rate constant of cyclic GMP hydrolysis by activated phosphodiesterase in photoreceptors

The early steps of light response occur in the outer segment of rod and cone photoreceptor. They involve the hydrolysis of cGMP, a soluble cyclic nucleotide, that gates ionic channels located in the outer segment membrane. We shall study here the rate by which cGMP is hydrolyzed by activated phosphodiesterase (PDE). This process has been characterized experimentally by two different rate constants $\beta_d$ and $\beta_{sub}$: $\beta_d$ accounts for the effect of all spontaneously active PDE in the outer segment, and $\beta_{sub}$ characterizes cGMP hydrolysis induced by a single light-activated PDE. So far, no attempt has been made to derive the experimental values of $\beta_d$ and $\beta_{sub}$ from a theoretical model, which is the goal of this work. Using a model of diffusion in the confined rod geometry, we derive analytical expressions for $\beta_d$ and $\beta_{sub}$ by calculating the flux of cGMP molecules to an activated PDE site. We obtain the dependency of these rate constants as a function of the outer segment geometry, the PDE activation and deactivation rates and the aqueous cGMP diffusion constant. Our formulas show good agreement with experimental measurements. Finally, we use our derivation to model the time course of the cGMP concentration in a transversally well stirred outer segment.