CAREER: Identifiability and Inference for Phylogenetic Networks using Applied Algebraic Geometry

系统发育树和网络重建不仅可以更好地理解我们的自然世界，而且在其他领域也有应用，如保护和流行病学。例如，在恢复自然植被时考虑到系统发育多样性，可以使恢复更快、更有力，而了解病原体的系统发育可以帮助追溯疾病的传播并指导流行病学干预。虽然树是代表进化组合的自然选择，但通过限制树的类别，可能会错过更复杂的事件，如杂交和水平基因转移。为了更完整的描述，系统发育网络，有向无环图，在进化生物学中越来越普遍。该项目侧重于系统发育网络及其母模型，系统发育混合模型，并将使用应用代数几何，特别是代数统计，为其推理开发新技术。该项目还对研究生进行研究培训，并向本科生和K-12学生介绍代数统计和代数生物学的教育活动。除了主持和运行一系列为期一周的代数生物学研究生研究研讨会以及后续的小组合作之外，PI还将设计一个以数据为中心的本科数学生物学课程，并为六年级学生组织年度合作和基于问题的数学活动。由于基于网络的马尔可夫模型及其混合物是用多项式参数化指定的，因此它们适用于使用代数和几何方法进行分析。本项目将利用它们的代数性质来建立可识别性，这是有意义的统计推断所必需的性质，然后利用它们的几何性质来开发网络推理程序。由于相应的模型品种是环面品种、割线品种和行动型品种，本项目不仅为生物学家开发了新的推理方法，也为代数几何和非线性代数研究的一类新问题奠定了基础和动力。该奖项由DMS数学生物学项目和环境生物学部系统与生物多样性科学集群共同资助。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。

Phylogenetic tree and network reconstructions lead not only to a better understanding of our natural world, but also have applications in other fields, such as conservation and epidemiology. For example, taking into account phylogenetic diversity in the restoration of natural vegetation can lead to restorations that establish quicker and are heartier, while understanding the phylogenetics of pathogens can aid in back-tracing the spread of a disease and guiding epidemiological interventions. While trees are a natural choice for representing evolution combinatorially, by restricting to the class of trees, it is possible to miss more complicated events such as hybridization and horizontal gene transfer. For more complete descriptions, phylogenetic networks, directed acyclic graphs, are increasingly becoming more common in evolutionary biology. This project focuses on phylogenetic networks and their parent models, phylogenetic mixture models, and will use applied algebraic geometry, in particular, algebraic statistics, to develop novel techniques for their inference. The project also has research training of graduate students and educational activities that introduce algebraic statistics and algebraic biology to undergraduate and K-12 students. In addition to hosting and running a series of one-week long graduate student research workshops in algebraic biology with follow-up small group collaborations, the PI will design a data-centered undergraduate mathematical biology course and organize annual collaborative and problem-based mathematical events for 6th graders with island-focused challenges.Since network-based Markov models and their mixtures are specified with a polynomial parameterization, they are amenable to analysis using algebraic and geometric methods. This project will utilize their algebraic properties to establish identifiability, a property necessary for meaningful statistical inference, and then utilize their geometric properties to develop a procedure for network inference. Since the corresponding model varieties are toric varieties, secant varieties, and determinental varieties, this project will not only develop new inference methods for biologists, but will also lay the foundation and motivation for a new class of problems for those working algebraic geometry and non-linear algebra. This award is jointly funded by the Mathematical Biology Program of DMS and the Cluster of Systematics and Biodiversity Sciences at the Division of Environmental Biology.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.