关键词：
Applied Mathematics
Cellular biology
Biomedical engineering
摘要：
From foams to fruit flies, tessellations to tissues, two-dimensional (2D) and three-dimensional (3D) cellular structures have long been a subject of interest to scientists and mathematicians. Cellular networks are present in an extensive—if not innumerable—list of natural and engineered materials, and a growing number of scientific, industrial, and commercial engineering applications hinge on the development of cellular materials. Problems of accurately quantifying and robustly explaining the geometric order found in such cellular networks are, therefore, critically important to a wide variety of research problems. We address key questions regarding: (i) the interplay between randomness and determinism; (ii) correlations between continuous and discrete geometric measures, and (iii) an understanding of the dynamic and static equilibria (statistical and mechanical) that define the order within these networks in physical systems. We use a granocentric model to develop analytical approximations for the well-known Lewis' Law that correlates the first moments of the size and topology (neighbors) distributions, and we explore higher moment correlations as well. With a robust understanding these correlations from the standpoint of \pure geometry" on a firmer footing, we unveil new information about tissue morphology in the developing Drosophila wing, exploring the competing roles of disordering processes like proliferation and ordering processes like remodeling. Ultimately, we show that even for quasi-2D tissues, nontrivial processes in the third dimension may play an important role in determining the "in-plane" tissue order. Using minimal models and novel 3D imaging techniques, we highlight some of the important considerations for understanding the origins of geometric order in Drosophila epithelia.
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