Collaborative Research: New Methods in Phyllotaxis

摘要从病毒中蛋白质六聚体的排列到茎上的叶和花的位置，自然界中经常出现重复部分的模式。植物的所谓叶系模式特别引人注目，因为图案化过程超越了用来解释晶体对称性的简单空间相互作用。这个项目的目标是从数学上描述所有可能的叶系构型，并正式和经验地确定为什么某些构型在植物中比其他构型更常见。为了研究所有种系模式的宇宙，有必要发展多晶格的概念，这是一个提供了比传统晶格概念更大灵活性的几何框架。多重晶格包括自然界中发现的所有规则的叶系构型(包括螺纹和晶格)。该项目的另一个方面是生成在正常发育期间和在扰动之后的时间分辨叶系构型的综合数据集，以探索植物可获得的稳定构型集。大量的数据集将用于校准动力系统。多晶格的概念是第一个形式主义，它没有将观察到的模式限制在严格的类别中，但实际上完全解释了在植物和其他地方发现的变化。该项目的一个广泛影响将是向科学界提供一个大型数据集，其中包括时间分辨的类群结构以及探索这些数据的工具。该项目还将为史密斯学院的女本科生提供基本实验技术和数学方面的多学科培训。在生活世界中发现的许多结构都显示出极具规律性的图案。蛋白质可以组装成模仿晶体美丽对称性的图案，这可能并不令人惊讶。然而，当在整个有机体的水平上发现同样的规律性时，人们可能会理所当然地感到惊讶。然而，这在植物中很常见，在茎周围放置叶子和花可以产生精致的图案。在植物和晶体中发现的共同几何特征不太可能用它们的分子组成来解释，因为它们有很大的分歧。植物和晶体必须有共同的一般发育规律，才能产生相似的图案。这个项目的目标是从数学和经验上研究这些规律。在植物和其他生物结构中发现的图案类型比二维晶体中的图案要丰富得多，因此对它们的描述需要一种名为多晶格的新数学框架。该项目的另一个方面是生成在植物中观察到的各种类型的模式的综合数据集。对多晶格的形式化理解可能有许多重要的应用。例如，花和种子的包装通常决定植物的产量，而叶片在茎周围的位置决定植物捕捉光线的效率，最终影响植物的整体生长。此外，许多人类疾病，如阿尔茨海默病，都涉及细胞内被称为淀粉样蛋白的蛋白质晶体的形成。了解蛋白质结晶的规律可能有助于查明这些疾病的起源。

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.