Two reaction-transport problems for Bill Deen

Bill Deen's work as an educator and scientist provides a great example for anyone interested in engineering analysis of complex systems. His remarkable ability to identify important problems, from transport in kidneys to Nitric Oxide dynamics in cells, and provide solutions with clarity fitting for teaching in graduate and undergraduate courses is a hard act to match. The elegance of his research can be approached only asymptotically. I will present two examples from our recent work in this direction. Both problems are related to mechanisms of cell communication and lead to reaction-diffusion problems that are difficult to solve at first glance. In the first of these problems, which comes from ligand binding to cell surface receptors, one must solve reaction-diffusion PDEs with fine-grained boundary conditions. In the second problem, related to tissue patterning by chemical gradients, one is interested in low-dimensional approximation of concentration dynamics. In both of these cases, we could derive simple analytical results that relate quantities of interest to parameters of the original problems, very much in the style of Chapter 3 of Bill's classical textbook.