Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation

Northwestern University, Evanston IL

Investigators

Abstract

Proposal: DMS - 0709232 PI: Silber, Mary Institution: Northwestern University Title: Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation ABSTRACT The proposed research addresses two applications of bifurcation theory and delay differential equations: (1) autoadjusting feedback control of oscillatory patterns, and (2) mathematical modeling of cellular protein translation. The proposed research on controlling patterns investigates an autoadjusting feedback control scheme aimed at stabilizing oscillatory patterned-states. The feedback control method exploits symmetries of the targeted pattern in such a way that it becomes noninvasive when control is achieved. Two related case studies will be pursued: (A) stabilization of traveling plane wave solutions of the two-dimensional complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime, and (B) control of chemical traveling wave patterns of the photo-sensitive Belousov-Zhabotinsky reaction in the oscillatory regime. The mathematical relationship between these two case studies will be elucidated by the proposed analysis. The proposed research on mathematical modeling of protein translation is aimed at deriving, by systematic approximation, a delay equation model of protein translation that could then be used as a component in simple models of synthetic gene networks involving more than one protein. The delay model is obtained from a continuum description of the elongation process, which ultimately shows up as a delay time in the reduced mathematical model. The proposed research will extend the model to incorporate the degradation of mRNA. The fidelity of the delay model to the mechanistic one, in the case of simple gene switches and oscillators will then be investigated using bifurcation theory, aided by a numerical continuation package that was developed for delay differential equations. The proposed research will contribute to the training of graduate students and postdoctoral fellows in interdisciplinary, applied mathematics research. It will aid the development of feedback control schemes for eliminating spatio-temporal chaos in chemical reaction-diffusion systems, as well as other pattern-forming systems. Delay differential equations frequently arise in the modeling of biological processes, such as cellular protein translation, the process whereby ribosomes assemble proteins, one amino acid at a time, using the information encoded in the messenger RNA (mRNA). The proposed research will contribute to the development of a systematic mathematical framework for deriving reduced delay models from complex, biologically-detailed mechanistic models. The proposed projects represent important applications of delay differential equations and their analysis using bifurcation theory. The analysis of these proposed case studies are essential to the development of these mathematical and computational tools.