Collaborative Research: New Methods in Phyllotaxis

【摘要】从病毒中蛋白质六聚体的排列到茎上叶和花的位置，自然界中重复部分的模式是常见的。植物的层状结构特别引人注目，因为这种结构过程超出了用来解释晶体对称性的简单空间相互作用。该项目的目标是用数学方法描述所有可能的层状结构，并从形式上和经验上确定为什么某些结构在植物中比其他结构更常见。为了研究所有层序模式的宇宙，有必要发展多极格的概念，这是一种比传统的晶格概念提供更大灵活性的几何框架。多晶格包含自然界中发现的所有规则的层序结构（包括螺旋和晶格）。该项目的另一个方面是生成正常发育期间和扰动后的时间分辨层状结构的综合数据集，以探索植物可获得的稳定结构集。广泛的数据集将用于校准动力系统。多极体的概念是第一个没有将观察到的模式限制为严格类别的形式主义，它实际上完全解释了在植物和其他物种中发现的变化。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层状结构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础实验技术和数学方面的多学科培训。在生物世界中发现的许多结构都显示出高度规律性的模式。蛋白质可以聚集在一起，创造出模仿晶体美丽对称的图案，这可能并不令人惊讶。然而，当在整个生物体的水平上发现同样的规律时，人们可能有理由感到惊讶。然而，这在植物中是一种常见的现象，叶子和花朵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成分来解释，因为它们差别很大。植物和晶体一定有一些共同的发展规律，从而导致图案的相似性。这个项目的目标是在数学和经验上研究这些规则。在植物和其他生物结构中发现的模式类型比二维晶体要丰富得多，因此它们的描述需要一种称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模式的综合数据集。对多极格的正式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了植物的产量，而叶子在茎周围的位置决定了植物捕捉光线的效率，而这最终会影响植物的整体生长。此外，许多人类疾病，如阿尔茨海默病，都与细胞内淀粉样蛋白晶体的形成有关。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.