Algebraic Invariants for Phylogenetic Network Inference

生物遗传学的关键目标是能够从其活体亲属的DNA序列数据推断物种的进化历史。这在许多领域都有应用，例如追踪病毒爆发的突变，了解物种形成事件以帮助保护，甚至追踪几代人手工复制的古代手稿的历史。大多数进化历史可以用系统发育树来描述，树的“叶子”代表今天还活着的物种，树的顶点代表共同的祖先物种。然而，对于许多生物学问题，一棵树不能正确地代表所涉及的物种的进化历史。据说，这些问题已经出现了“横向演变”。一个例子发生在微生物组中，不同的微生物物种能够在一个称为水平基因转移的过程中共享其DNA的一部分。这是抗生素耐药性可以在细菌之间传播的一种机制，因此能够描述何时发生此类事件对人类健康具有重要意义。为了描述水平进化，生物学家使用所谓的系统发育网络。在这里，我们可以使用树结构作为主干，在其上绘制进一步的边缘来表示水平进化事件。推断发生水平进化的物种的进化历史的问题特别具有挑战性，并且是当今遗传学研究的焦点。系统发育推断的一种方法是使用代数不变量。这些方法在沿着一棵树推断进化方面已经有了显著的发展，在某些情况下已经显示出优于其他方法。然而，对于系统发育网络，很少有关于代数不变量的研究。本研究将开发并测试使用代数不变量进行系统发育网络推理的方法。对于特定的系统发育网络，可以使用称为马尔可夫模型的概率模型来模拟其进化过程。在这个模型下，人们可以计算在网络的叶子上观察到特定DNA模式的概率，这些概率可以用模型的数值参数表示为多项式。通过允许数值参数自由变化（即将它们视为变量），我们可以将网络表示为描述概率的方程的解的集合。这样的一组解形成了一个对象，代数学家称之为代数簇。使用这个模型的优势在于，我们能够使用代数几何的强大机制来确定观察到的DNA序列数据是否适合网络。特别地，我们可以通过使用称为代数不变量的表达式来描述对应于网络的多样性。为了确定一个特定的网络是否适合观察到的DNA序列数据，我们的想法是计算数据中模式的频率，然后将网络的代数不变量应用于这些频率。结果的数量将决定数据与网络的匹配程度。本项目将研究这种方法从DNA序列数据推断系统发育网络的有效性。为此，我们将利用该领域的最新发展来计算一类系统发育网络的不变量。接下来，我们将开发一个计算工具，通过使用我们计算出的不变量，从一组DNA序列数据中推断出最能描述进化历史的网络。然后，我们将在模拟的DNA序列数据和真实的DNA序列数据上测试我们的工具，并将结果与最先进的方法进行比较。

The key goal in phylogenetics is to be able to infer the evolutionary histories of species from DNA sequence data of their living relatives. This has applications in many fields, such as tracing the mutations of viral outbreaks, understanding speciation events to aid conservation, and even tracing the histories of ancient manuscripts that were copied by hand through generations.Most evolutionary histories can be described with a phylogenetic tree, where the "leaves" of the tree represent species that are alive today, and the vertices higher up the tree represent common ancestor species. However, for many biological problems, a tree cannot properly represent the evolutionary history of the species involved. Such problems are said to have seen "horizontal evolution". One example occurs in microbiomes, where different microbial species are able to share portions of their DNA in a process called horizontal gene transfer. This is one mechanism by which antibiotic resistance can spread between bacteria, and so being able to describe when such events have occurred has important implications for human health. To describe horizontal evolution, biologists use what's called a phylogenetic network. Here, one can use a tree structure as a backbone, onto which further edges are drawn to represent horizontal evolution events.The problem of inferring the evolutionary histories of species where horizontal evolution has occurred is particularly challenging, and is the focus of much of the research in phylogenetics today. One method of phylogenetic inference is to use algebraic invariants. These have seen significant development for inferring evolution along a tree, and in some cases have been shown to outperform other methods. For phylogenetic networks however, very little research on algebraic invariants has been done. This project will develop and test the method of using algebraic invariants for phylogenetic network inference.For a particular phylogenetic network, the process of evolution along it can be modelled using a type of probabilistic model called a Markov model. Under this model, one can calculate the probability of observing particular patterns of DNA at the leaves of the network, and these probabilities can be expressed as polynomials in the numerical parameters of the model. By allowing the numerical parameters to vary freely (i.e. treating them as variables) we can represent the network as the set of solutions to the equations describing the probabilities. Such a set of solutions forms an object that algebraists call an algebraic variety. Using this model gives us the advantage of being able to use the powerful machinery of algebraic geometry in determining whether observed DNA sequence data is a good fit for the network. In particular, we can describe the variety corresponding to a network by using expressions called algebraic invariants. To determine whether a particular network is a good fit for observed DNA sequence data, the idea is to calculate the frequencies of patterns in the data, and then apply the network's algebraic invariants to these frequencies. The resulting quantities will determine how closely the data matches the network.This project will examine how effective this method is to infer phylogenetic networks from DNA sequence data. To do this, we will utilize the most recent developments in the field to calculate the invariants for a small class of phylogenetic networks. Next, we will develop a computational tool that will infer the network that best describes the evolutionary history coming from a set of DNA sequence data, by using the invariants we have calculated. We will then test our tool on both simulated DNA sequence data and real DNA sequence data, and compare the results to state of the art methods.