We investigate the interplay between the structure of biological networks and their spatial and temporal dynamical behavior. A deep understanding of this interplay is essential not only for the analysis of existing systems, but also for the design of novel synthetic networks. We have thus far investigated cyclical interconnection structures that are commonly found in biological oscillators, and developed control-theoretic criteria to predict oscillations or convergence to steady-states. We are currently studying other recurrent structures and, in addition, identifying novel synthetic networks for prescribed dynamical behaviors. One such research direction, pursued in collaboration with the Maharbiz and Arkin groups, is designing networks that generate spatial patterns of gene expression. Motivated by naturally occurring developmental processes where specific genes are activated in specific regions in the embryo, our goal is to design gene networks that have the ability to generate patterns with a robust set of parameters. Our investigations have led to a new a class of networks that we call “quenched oscillator” systems. These systems consist of a primary feedback loop that serves as an oscillator, and a secondary feedback loop that quenches the oscillations and incorporates a diffusible molecule. Diffusion releases the quenching effect in higher spatial frequencies, thus generating patterns.
-- Master's Thesis: Exploration of Turing Phenomena in Quenched Oscillator Systems
-- Proposed Dissertation: Design and Engineering of Pattern Formation in Gene Expression in Escherichia coli.
Research project for the class "Control and Optimization of Distributed Systems and Partial Differential Equations" in the Spring of 2007. Worked under the graduate student Edgar Lobaton to help model and simulate bacterial motion in environments with low Reynolds number. Framed as an adjoint optimization problem over the elasticities of the spring connecting the particles that made up the model of the flagellum. Continued work on this project through the Summer of 2007. Acknowledged in the resulting paper.