Broadly speaking, pattern formation refers to phenomena wherein a physical or biological system produces a large-scale, ordered structure from a combination of complex, small-scale interactions. Pigmentation patterns on fauna, such as the colored stripes on a zebrafish, provide a typical example. In this case the small-scale interactions refer to the dynamics of two types of cells, namely blueish cells called melanophores and yellowish cells called xanthrophores. The large-scale patterned structure, namely the stripes themselves along the skin of the fish, emerge from these small-scale interactions. A similar process also occurs in physical chemistry. The complex, small-scale forces arising between individual atoms or molecules can combine to produce a larger, ordered  "supra-molecular" structure involving hundreds or thousands of individual molecules. The term "self-assembly" refers to this process. For example, bilayers and micelles consist of a large number of surfactant molecules arranged into either flat sheets (a bilayer) or a hollow sphere (a micelle) with molecules coating the surface. The fundamental forces (electrostatic and hydrophobic/hydrophilic forces) between surfactant molecules drive them to self-assemble into bilayers or micelles. 

The projects funded under this grant focus on furthering our current understanding of both pattern forming processes and self-assembly processes. A common mathematical model for pattern formation is the reaction-diffusion system, i.e. a set of partial differential equations that describe the dynamics of a (typically) large number of different chemical compounds. Although this class of models successfully reproduce a wide variety of pattern forming behaviors observed in nature, they do have their drawbacks. Specifically, these models tend to be difficult to analyze --- one partial differential equation can be hard or easy, but dealing with a large number of them is almost always hard. Our work provides a means to address this issue. We focus on understanding the interfaces or "co-dimension one" structures that drive pattern formation. To give an oversimplified example, we consider how the stripes themselves evolve on a zebrafish rather than inferring it from the dynamics of individual cells. In other words, we directly tackle the dynamics of the large-scale structure. This approach allows us to study pattern formation from a more mathematical (rather than computer simulation) point-of-view. Our self-assembly work uses a similar set of techniques, in that we mathematically describe large-scale, co-dimension one structures, but generally aims at a different set of problems. At times the actual physical forces driving a particular self-assembly are unknown, while at other times simplifications are made in the underlying physical equations. Our work provides a mathematical means for exploring these issues. In particular, we provide results that scientists can use to decide whether certain effects, such as directional/anisotropic forces, are required when trying to model a given self-assembly process. These results also place limits on when such anisotropic effects can be safely neglected, thereby simplifying a given model. Finally, our self-assembly work opens up a new approach to modeling and understanding those collective behaviors where directional effects do indeed play a major role.

In addition to the interdisciplinary nature of its results, the projects funded under this grant impact the broader community through mentoring and the development of undergraduate research. Portions of these projects provided undergraduates with the opportunity to participate in a mentored research experience. Undergraduate research opportunities of this sort are ideal for encouraging and supporting young mathematicians in their pursuit of graduate degrees and careers in STEM disciplines.

Last Modified: 08/30/2017
Modified by: James Von Brecht