Collaborative Research: New Methods in Phyllotaxis

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AbstractPatterns of repeated parts are frequent in Nature from the arrangement of protein hexamers in viruses to the position of leaves and flowers on stems. The so-called phyllotactic patterns of plants are particularly striking because the patterning process reaches beyond the simple steric interactions used to explain crystal symmetries. The goal of this project is to characterize mathematically the universe of all possible phyllotactic configurations and to determine formally and empirically why some configurations are more common than others in plants. To investigate the universe of all phyllotactic patterns it has been necessary to develop the concept of multilattices, a geometric framework that offers a much greater flexibility than the traditional concept of lattices. Multilattices encompass all the regular phyllotactic configurations found in Nature (including whorls and lattices). Another aspect of this project is to generate a comprehensive data-set of time-resolved phyllotactic configurations both during normal development and following perturbation so as to explore the set of stable configurations that are accessible to plants. The extensive data set will serve to calibrate the dynamical systems. The concept of multilattices is the first formalism that does not constrain observed patterns into rigid classes but in fact fully accounts for the variation found in plants and beyond. One broad impact of this project will be to provide the scientific community with a large data-set of time-resolved phyllotactic configurations as well as tools to explore these data. The project will also offer multidisciplinary training for female undergraduates from Smith College both in basic experimental techniques and mathematics. Many structures found in the living world show patterns of great regularity. It may not come as a great surprise that proteins can assemble to create patterns that emulate the beautiful symmetries of crystals. However, when the same regularity is found at the level of an entire organism, one may justly be astonished. Yet, this is a common occurrence in plants where the placement of leaves and flowers around the stem can give rise to exquisite patterns. The common geometrical features found in plants and crystals are unlikely to be explained by their molecular constituents since these diverge widely. Plants and crystals must share some general developmental rules leading to the similarities in pattern. The goal of this project is to study these rules mathematically and empirically. The types of patterns found in plants and other living structures are much richer than those of 2-D crystals, therefore their description has required a new mathematical framework called multilattices. Another aspect of this project is to generate a comprehensive data set of the types of patterns observed in plants. A formal understanding of multilattices may have many important applications. For example, the packing of flowers and seeds often determines yield in plants while the positioning of leaves around the stem determines the efficiency of a plant at capturing light which, ultimately, influences overall plant growth. Moreover, many human diseases such as Alzheimer involve the formation of protein crystals known as amyloids within cells. Understanding the rules under which proteins crystallize may help pinpoint the origin of these diseases.