Many biochemical networks involve molecular species of small copy
numbers. Of fundamental importance in systems biology is to understand
the nature of stochasticity intrinsic in these networks. The chemical
master equation (CME) provides a general framework for studying such
networks. Although Fokker-Planck/Langevin equations give useful
approximations, Gillespie Monte Carlo algorithm is often used to
simulate stochasticity. Here we describe a new method to directly
solve the CME to account for full stochasticity without either
Fokker-Planck or Gillespie for nontrivial systems. We characterize
the exact state space of a molecular network of small copy numbers
with arbitrary stochiometry at a given initial concentration
condition. We show how our method works for toggle-switch, MAPK, and
lambda-phage networks. We then show how to compute the steady state
probablistic landscape of these networks, and infer biological conditions
for bistability and the mechanism of lysis-lysogeny switch. We also
demonstrate how time evolution of concentration dynamics of molecular
species in a nework at large time span across many orders of scale can be
computed. Finally, we demonstrate a novel method for simulating
stochastic behavior of dynamic pattern formation of cell populations.
Unlike
cellular automata which provides only caricatures of cell pattern
formation, our geometric model captures the shape and size of cells
more realistically, accounts for topological events in the dynamic
re-arrangement of spatial cells, and incorporates many biochemical
forces. Example in cell differentiation will be given
(Joint work with Youfang Cao, Hsiao-Mei Lu, and Sema Kachalo. Relevant
publications can found following URL: www.uic.edu/~jliang)

A symmetric Banach space (SBS) is one whose dual ball is a bounded symmetric domain. It is a deep fundamental result that a contractively complemented subspace of an SBS is linearly isometric to another SBS. We prove the converse: if a subspace of a SBS is linearly isometric to an SBS, then that subspace is contractively complemented. Examples of SBSs are L^1-spaces and preduals of von Neumann algebras. (Joint work with B. Russo)

Since its early days in the early 1970, the field of computational rheology (the flow of complex fluids) has experienced a number of insurmountable stumbling blocks, notably, the high Weissenberg number problem (HWNP); it is a term coined to describe the breakdown of computations at a moderately high values of the elasticity parameter. In this lecture I will report a number of recent advances in an effort to elucidate the HWNP, and describe some of the remaining problems.