Collaborative Research: Nonlocal Interfaces in Biological Systems

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This award supports the development of mathematical tools for understanding phenomena caused by the interactions among biological interfaces. An interface is a transition between regions in a system, each exhibiting different biological properties. The natural world displays a vast gallery of organized systems characterized by complex interface dynamics, including the edges of growing tumors or bacterial colonies, the surface of a growing, hollow embryo, or the transition region between high and low electrical activity in a beating heart. This project will investigate biological systems that involve interfaces with interactions on long length-scales; the behaviors of these interfaces have properties which cannot be predicted by purely local dynamics, as when a bubble popping far away allows other bubbles to grow. The new tools developed will be validated through experimentation with bacterial biofilms and human organoids. Undergraduate and graduate students will be involved in the research at both California State University-Long Beach and Montana State University, and Native American students will be recruited using collaborative outreach programs in existence at Montana State University.The investigation will focus on studying complex systems characterized by a collection of mutually interacting curves or surfaces. These nonlocal mutual interactions can produce a variety of pattern forming behaviors that typically do not occur in classical interfacial models. For example, nonlocal interfacial models exhibit arrested fronts, which are stationary, dynamically stable interfaces between quasi-homogeneous regions in a complex system. The interactions between certain strains of bacteria provide a concrete instance of such fronts. The evolution of genetically identical sibling colonies lead to competition between these colonies, and this competition produces a collection of mutually interacting interfaces (i.e. the boundary of each colony) that eventually arrest in regions of sibling-sibling competition. The project will introduce a first-principle modeling framework for nonlocal interfaces that offers reduced model complexity from both an analytical and computational perspective. Despite the relative simplicity of this framework, its specific instances still capture the essential pattern forming mechanisms in a variety of biological systems. As a consequence, research under the award will focus on validating this overall approach via experimental predictions in several biological systems. This approach to understanding nonlocal pattern formation appears new at the level of generality introduced in the project, and it opens the door to numerous explorations of the mathematical behavior of the framework and of its applications for different experimental systems. The major aim is to provide researchers with a comparatively flexible, minimalistic framework to study arrested fronts and other interfacial phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.