Collaborative Research: New Methods in Phyllotaxis

自然界中，从病毒中蛋白质六聚体的排列到茎上叶和花的位置，重复部分的模式是常见的。 所谓的植物叶序模式特别引人注目，因为模式化过程超越了用来解释晶体对称性的简单空间相互作用。 这个项目的目标是数学表征所有可能的叶序配置的宇宙，并正式和经验性地确定为什么有些配置比其他更常见的植物。 为了研究所有叶序图案的宇宙，有必要发展多格的概念，这是一种几何框架，比传统的格概念提供了更大的灵活性。 多重晶格包含了自然界中所有规则的叶序结构（包括螺旋和晶格）。 该项目的另一个方面是生成一个全面的数据集的时间分辨叶序配置在正常的发展和以下扰动，以探索一套稳定的配置，可访问的植物。 广泛的数据集将用于校准动力系统。 多重格的概念是第一个不将观察到的模式限制在严格的类别中的形式主义，但实际上完全解释了在植物和其他植物中发现的变化。 该项目的一个广泛影响将是为科学界提供大量的时间分辨叶序配置数据集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。在生命世界中发现的许多结构都显示出高度的规律性。 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，人们可能会感到惊讶。 然而，这在植物中很常见，在植物中，叶子和花围绕茎的位置可以产生精美的图案。 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，因为它们的分子组成差异很大。 植物和水晶必须共享一些一般的发展规则，导致模式的相似性。 这个项目的目标是从数学和经验上研究这些规则。 在植物和其他生命结构中发现的模式类型比二维晶体丰富得多，因此它们的描述需要一个新的数学框架，称为多重晶格。 该项目的另一个方面是生成在植物中观察到的模式类型的综合数据集。 对多重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，而茎周围的叶子的位置决定植物捕获光的效率，最终影响整体植物生长。 此外，许多人类疾病，如阿尔茨海默病，涉及细胞内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则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.