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

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Proposal: DMS - 0709232 PI: Silber, MaryInstitution: Northwestern UniversityTitle: Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation ABSTRACTThe proposed research addresses two applications of bifurcation theoryand delay differential equations: (1) autoadjusting feedback controlof oscillatory patterns, and (2) mathematical modeling of cellularprotein translation. The proposed research on controlling patternsinvestigates an autoadjusting feedback control scheme aimed atstabilizing oscillatory patterned-states. The feedback control methodexploits symmetries of the targeted pattern in such a way that itbecomes noninvasive when control is achieved. Two related case studieswill be pursued: (A) stabilization of traveling plane wave solutionsof the two-dimensional complex Ginzburg-Landau equation in theBenjamin-Feir unstable regime, and (B) control of chemical travelingwave patterns of the photo-sensitive Belousov-Zhabotinsky reaction inthe oscillatory regime. The mathematical relationship between thesetwo case studies will be elucidated by the proposed analysis. Theproposed research on mathematical modeling of protein translation isaimed at deriving, by systematic approximation, a delay equation model ofprotein translation that could then be used as a component in simplemodels of synthetic gene networks involving more than one protein. Thedelay model is obtained from a continuum description of the elongationprocess, which ultimately shows up as a delay time in the reducedmathematical model. The proposed research will extend the model toincorporate the degradation of mRNA. The fidelity of the delay modelto the mechanistic one, in the case of simple gene switches andoscillators will then be investigated using bifurcation theory, aidedby a numerical continuation package that was developed for delaydifferential equations.The proposed research will contribute to the training of graduatestudents and postdoctoral fellows in interdisciplinary, appliedmathematics research. It will aid the development of feedback controlschemes for eliminating spatio-temporal chaos in chemicalreaction-diffusion systems, as well as other pattern-forming systems.Delay differential equations frequently arise in the modeling ofbiological processes, such as cellular protein translation, theprocess whereby ribosomes assemble proteins, one amino acid at a time,using the information encoded in the messenger RNA (mRNA). Theproposed research will contribute to the development of a systematicmathematical framework for deriving reduced delay models from complex,biologically-detailed mechanistic models. The proposed projectsrepresent important applications of delay differential equations andtheir analysis using bifurcation theory. The analysis of theseproposed case studies are essential to the development of thesemathematical and computational tools.