Signal transduction in cells occurs by precise, highly-regulated localization of key enzymes in subcompartments cells. Classical methods assume a well-stirred environment and thus do not describe the local effects emanating from these highly-organized structures sometimes called signalsomes. Such a complexity calls for mathematical models of signal transduction for critical evaluation of the data, for quantitative understanding of the processes, and for designing discriminating experiments. A team of mathematicians, biologists, and computational scientists will develop spatio-temporal mathematical models of prototype signal transduction pathways. The models will be formulated, analyzed mathematically, implemented into computer codes, and predicted results will be compared with known biological data. In this iterative process, the model will be critically tested and modified accordingly. Our long term goal is to produce validated computational models of signal transduction processes broadly useful to biologists. We initially will focus on one of the best-studied eukaryotic signal transduction pathways, photoreceptor visual transduction in retinal rod cells. This pathway embodies key processes common to a variety of signaling systems. Rod photoreceptors are composed of disk membranes containing high concentrations of the photoreceptor rhodopsin, the rod G protein (Gt), and the effector enzyme cGMP phosphodiesterase (PDE). Light activation of rhodopsin leads to activation of PDE, breakdown of cGMP, and closure of cGMP-gated channels in the plasma membrane, which hyperpolarizes the cell and leads to a drop in membrane potential. This signaling system has provided key insights into basic mechanisms of G protein-coupled signal transduction. We have begun modeling the diffusion of the second messengers, cGMP and Ca2+, within the cytosol of the rod cell. PDE*-cGMP interactions, which physically occur on the surface of the discs, are modeled as flux sources located on the discs. Evolution of Ca2+ is effected by influx through cGMP--gated channels, and as such is described by source terms supported on the lateral boundary of the rod. This results in a system of evolution partial differential equations with non linear boundary conditions on the boundary of the outer segment and on each of the internal discs. The intricate geometry of the rod cell (about 800 disc membranes, of diameter 11 mu m, 14 nm apart from each other) presents severe limitations for diffusion in the cytosol and mathematical and numerical difficulties. We propose to model the phototransduction cascade, including the recovery phase, by the mathematical techniques of homo-genization and concentrated capacity. This casts the diffusion problem into one in a much simpler homogenized geometry, for which computation becomes easier and more efficient. The diffusion problem will simulated numerically, in the full as well as the homogenized geometries in order to compare the two, and assess the effectiveness of the latter. Further developments will incorporate the effects of additional messenger proteins, aiming at a complete description of the entire cascade of biochemical events constituting a signal transduction pathway.
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