RUI: Collaborative Research: DMS/NIGMS 1: The mathematical laws of morphology and biomechanics through ontogeny

尽管在20世纪，数学理论和科学实验之间有很多卓有成效的互动，但即使是现在，数学与生物学的结合仍处于起步阶段，而且大多局限于精细尺度。早在100年前，达西·汤普森（D’arcy Thompson）就提出了分析有机体发育过程中形态和功能的关键概念和工具，然而，我们仍然缺乏一个可靠的数学理论，来解释动物生长过程中宏观的、完全3D的形态和功能是如何进化的。为了彻底理解生长过程中形态和功能变化的机制，有必要推导出严格的数学规律，这些规律在发育过程中支配形态和运动。该项目将集中研究树栖蜥蜴，这是生物学许多领域的模型系统，并将结合生物力学实验和形态分析，以及数学建模和机器学习。通过利用树栖蜥蜴的快速生长和身体形态和运动行为的非凡多样性，本研究将以数学严谨的方式建立它们的形态和功能如何在发展过程中转变的综合图景。这将为我们理解面对不断变化的环境的物种的灵活性和恢复力的机制提供切实的贡献，同时刺激应用数学的新研究，如日益复杂，多尺度和非线性系统的建模。该项目将通过教育和培训高素质人才（本科生、研究生和博士后学者），特别是来自STEM领域代表性不足群体的人才，产生广泛的社区和社会影响。通过外展活动，该项目将教育学校的孩子，特别是那些来自弱势群体的孩子，以及公众，让他们知道数学和生物学可以一起工作，帮助我们理解动物的形态和运动、气候变化、人口老龄化和保护。该项目将通过数据驱动的数学规律的发现，为理解树栖蜥蜴的生物学带来新的理论和方法基础：1)它们通过个体发生的生物力学转化，以描述关节/四肢耦合运动的微分方程的形式；2)它们在形态上的变换通过个体发生，通过光滑群变换以形状之间的非线性映射的形式；3)它们的形状和力学是如何相互关联的，通过可解释的统计和机器学习模型以及跨年龄组的低维转换。将进行实验以收集全面的生物力学和形态学数据，这些数据将支持数学和机器学习模型，并将分两个阶段进行。第一阶段将利用一种孤雌树栖蜥蜴（哀悼壁虎）和一个普通的花园设计来构建模型，使我们能够通过生长来量化形式和功能的协调变化；第二阶段将在另外三种树栖蜥蜴身上测试我们模型的普遍性。所产生的纵向数据在功能形态学和比较生物力学方面的规模和范围将是无与伦比的，使其成为基准数据集，并使和刺激与发育生物学，动物行为和生态形态学相关的进一步问题的调查。这项工作将为个体发生、形态学、行为、运动、群落生态学和进化如何相互作用和融合提供一个统一的视角，通过产生一个共同的数学和计算框架，该框架将在这些学科的交叉领域广泛使用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Although fruitful interactions between mathematical theory and scientific experimentation were plentiful in the 20th century, even now the integration of math into biology is still in its infancy and mostly limited to fine scales. Already 100 years ago, D’Arcy Thompson developed critical concepts and tools for the analysis of form and function as organisms develop, yet nevertheless, we continue to lack a solid, mathematical theory of how macroscopic, fully 3D shapes and functions evolve through growth in animals. To thoroughly understand the mechanisms of changes in form and function through growth, it is necessary to derive rigorous mathematical laws that govern both morphology and locomotion through development. This project will concentrate on arboreal lizards, a model system in many fields of biology, and will combine biomechanical experiments and morphological analysis, together with mathematical modeling and machine learning. By capitalizing on the rapid growth and extraordinary diversity of body forms and locomotor behaviors of arboreal lizards, this research will build an integrative picture of how their form and function transform through development with mathematical rigor. This will provide a tangible contribution to our understanding of the mechanisms underlying the flexibility and resilience of species that face an ever-changing environment, while at the same time stimulating new research in applied mathematics such as the modeling of increasingly complex, multi-scale, and nonlinear systems. This project will have broad community and societal impacts through the education and training of highly qualified personnel (undergraduate, graduate, and postdoctoral scholars), particularly those from underrepresented groups in STEM. Through outreach initiatives, this project will educate school children, especially those from disadvantaged populations, and the public on the ways that math and biology can work together to contribute to our understanding of animal form and movement, climate change, population aging, and conservation. This project will bring new theoretical and methodological foundations to the understanding of the biology of arboreal lizards via the data-driven discovery of the mathematical laws that govern 1) their biomechanical transformation through ontogeny, in the form of differential equations describing the coupled motion of joints/limbs; 2) their transformations in morphology through ontogeny, via smooth group transformations in the form of nonlinear maps between shapes; and 3) how their shape and mechanics are related to each other, via explainable statistical and machine learning models and low-dimensional transformations across age-groups. Experiments will be conducted to collect comprehensive biomechanical and morphological data that will empower mathematical and machine learning models and will occur in two phases. Phase 1 will capitalize on a parthenogenic arboreal lizard (mourning gecko) and a common garden design to construct models that will allow us to quantify the coordinated changes of form and function through growth; Phase 2 will test the generality of our models on three additional arboreal lizard species. The longitudinal data produced will be unparalleled in its scale and scope in functional morphology and comparative biomechanics, making it a benchmark dataset and enabling and stimulating the investigation of further questions related to developmental biology, animal behavior, and ecomorphology. This work will provide a unifying perspective to how ontogeny, morphology, behavior, locomotion, community ecology, and evolution interact and merge by producing a common mathematical and computational framework that will become of broad use at the intersection of these disciplines.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.