Exact stochastic behavior of molecular networks and realistic simulation of cellular pattern formation

Jie Liang
Dept of Bioengineering
University of Illinois at Chicago

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 achalo)