I am interested in modeling problems in structural and systems biology using differential equations and integral equations, and solving these problems numerically with fast algorithms and high performance computing. My main research interest is in numerically solving interface problems described by PDEs and integral equations. These problems appear when adjacent layers with different physical, chemical and biological properties are present. For example, solvated proteins have different permittivities on solutes and solvents, and metal oxide semiconductor field effect transistors (MOSFETs) have different conductivities in metal gates, semiconductors and insulators. Interface problems have discontinuities in equation coefficients and solution which require special numerical treatment. I am interested in: 1) matched interface and boundary (MIB) methods, and 2) boundary integral methods. MIB methods are high order, finite difference mesh-based algorithms used in solving elliptic PDEs, which have discontinuities in solutions and coefficients, and singularities on interfaces and sources. Boundary integral methods are more general methods to solve integral formulations of PDEs. I am interested in combining boundary integral methods with treecode, a fast algorithm for evaluating interactions of N-Body problems, to achieve high accuracy and efficiency.
I am also interested in problems from systems biology. In systems biology, a major difficulty in simulating genetic networks is combinatorial complexity. Although a network may involve a relatively low number of proteins (e.g.~10), complexes can form leading to a very large 2^10 number of species which must be simulated. Even writing down equations of each of these species is very difficult. To simulate these networks, I have been working on algorithms and codes to automatically generate and simulate genetic networks with combinatorial complexity. Our simulations demonstrate that the specific mechanisms of complex formation can drastically change the time scales and steady state behavior of these networks. Using these methods we have begun to model one of the most well studied genetic networks with combinatorial complexity: the mammalian circadian (24-hour) clock which times biological events in our body.
