Event Detail

Event Type: 
Department Colloquium
Monday, February 13, 2006 - 08:00
Stag 107

Speaker Info

University of California at Irvine

**Dr. Zhang is a candidate for an open position in the Department of Mathematics **Tea to precede talk at 330 p.m. in Kidder 302 (Faculty Room) Abstract: A morphogen is a substance whose nonuniform distribution in a field of cells differentially determines the fate and phenotype of those cells. During the embryo development of both vertebrates and invertebrates, the bone morphogenetic protein (BMP) binding with cell receptors acts as a morphogen to induce the dorsal-ventral patterning. Using experimental and computational analysis, we investigate how morphogens and other ligands cooperate to produce the desired pattern and dynamics in the Drosophila embryo. In particular, we find that the morphogen activity is much less robust than previously claimed. Then we consider the extension of the one-dimensional model to a more realistic three-dimensional reaction-diffusion system for the Zebrafish embryo development. The complex geometrical shape of the Zebrafish embryo during 30%-epiboly ~ shield stage is approximated by an open spherical ring. Computational analysis on the model reveals that two synergistic feedback loops in the zygotic control cooperate with the maternal control to regulate the complex gene-network and drive a stable BMP morphogen gradient pattern in the Zebrafish embryo. One of the major computational challenges in this study is the severe stability constraint on the time step due to the stiffness of reactions and diffusions. To overcome this difficulty, we have designed a new class of efficient semi-implicit numerical schemes which treat the linear diffusions exactly and explicitly, and the nonlinear reactions implicitly. A novel decoupling technique results in that the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points; it only depends on the number of original equations. The stability region for this new class of schemes is much larger than existing methods, and its second order version is unconditionally stable with respect to both diffusion and reaction. At last, I will talk a little bit about our new work on an efficient iterative numerical method (called fast sweeping method) for static Hamilton-Jacobi equations, which have potential applications on tissue growth. We constructed high order fast sweeping methods on rectangular meshes and extended original fast sweeping methods to unstructured meshes (triangular meshes).