Multi-constituent inhibitory systems with self-organizing properties

George Washington University, Washington DC

Investigators

Abstract

This project investigates several important binary and ternary systems with self-organizing properties, with an emphasis on ternary systems and the longer ranging confinement mechanism through nonlocal interaction or inhibitor variables. One intriguing feature that sets ternary systems apart from binary systems is the triple junction phenomenon. The three constituents of the system may meet at a point in the two dimensional case or at a curve in the three dimensional case. The PI proposes to show the existence of a double bubble assembly pattern, where the triple junction phenomenon occurs in each double bubble. Two novel techniques, restricted perturbation classes and internal variables, will be used in the proof. The second feature in ternary systems is the complexity of the long range interaction manifested in a two by two matrix of parameters. For instance it will be shown that the core-shell pattern appears only if the 2-2 entry is greater than the 1-2 entry of the matrix. When multi-constituent inhibitory systems appear in a curved space, such as a vesicle of lipid membranes, the role played by the Riemann curvature will also be investigated. Patterns and their possible defects will be shown as a balance and compromise of growth, inhibition, and curvature. Exquisitely structured patterns arise in many multi-constituent physical and biological systems as orderly outcomes of self-organization principles. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. On its own, it would lead to an unlimited increase and spreading. Pattern formation requires in addition a longer ranging confinement of the locally self-enhancing process. This project derives geometrical patterns from self-organization principles in various binary and ternary physical and biological systems. Analysis of these geometric structures is a fundamental step in understanding the mechanical, optical, electrical, ionic, barrier and other properties of these systems. This project supports a culturally diverse environment that fosters critical and independent thinking both in classroom and research settings in the ethnically diverse metropolitan area of Washington, DC. It produces mathematics of depth and beauty, and offers profound insights into the natural world.