Bridging Across Scales to Model Cone Phototransduction

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Rods and cones in the retina mediate vision. While rods detect white light, cones are designed for color vision. Their role is crucial in high acuity vision as underscored by their loss in age-related macular degeneration leading to blindness. A common feature of these cells is their intricate geometrical structure, consisting of about one thousand thin pancake-like folds (discs) through which light is transformed into electrical pulses (phototransduction) for the brain to "see." Light is captured by receptors/pigments residing on these folds, and transformed into a current between interior and exterior of these cells, by a cascade of biochemical steps. These steps involve amplifiers of the light signal (transducers and effectors) and carriers of the signal (second messengers) diffusing within these folds. A mathematical understanding of these processes is complicated by the numbers of these folds (about one thousand) and their thickness (a few nanometers). The mathematical theory of homogenization seeks to replace the complex geometry with a simpler, cylindrical structure while preserving all the biochemical and biophysical functions of the original system. The transduction cascade must be reliable/stable for consistency of visual perception, and sources of instability and stabilizing mechanisms in the cascade can be broken down and analyzed via mathematical modeling. This interdisciplinary investigation involves mathematics (homogenization), computational sciences (finite element code writing), biochemistry (activation/deactivation cascades), and physiology (diffusion of second messengers). The project involves students and postdoctoral trainees and will be disseminated through training courses and seminars. This research is funded jointly by the Division of Mathematical Sciences Mathematical Biology Program and the Division of Integrative Organismal Systems Physiological Mechanisms and Biomechanics Program. Cones capture light in the red, blue and green wavelength, and as such, besides their geometrical shape, their biochemical and biophysical functions are different than rods. In particular they never saturate, they have a faster and smaller response, they are little sensitive to dim light and have a faster deactivation. The photoresponse is generated by diffusion of the second messengers calcium and CGMP in the cytoplasm within the lipidic discs. To overcame the intricate geometry of the discs (about a thousand, each a few nanometers thick) we propose an homogenization process, by which the number of discs goes to infinity and their thickness goes to zero, while the volume available for diffusion remains unchanged, and the biochemical and biophysical functions are preserved in the limit. The limiting "homogenized" cone becomes cylinder-like, with no discs, and the diffusion process is separated into interior and boundary diffusion on the homogenized domain. The activation/deactivation process is modeled by a continuous-time Markov chain tracking the various steps of the cascade. The resulting model is hybrid as it contains a deterministic part (homogenized diffusion of the second messengers) with a random input (stochastic steps of the activation/deactivation cascade). This permits one to analyze the mechanism by which random deactivation events turn into a stable photoresponse, and identify the causes of variability and variability suppression. A database of parameters will be created to populate the model and effect numerical simulations to be compared with experimental data, including lack of saturation, low sensitivity and faster deactivation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.