RS Fellow - EPSRC grant (2016): Algebraic and topological approaches for genomic data in molecular biology

现代科学以前所未有的速度产生数据，通常包括及时测量基因序列信息。分子生物学的一个目标是理解产生这些数据的过程；这可以通过探索不同的假设来实现，这些假设被转化为称为模型的数学方程。我的研究的主要成果将是一系列新的方法来理解不同场景下的模型，这些场景具有不同的数据量。这一建议的重点是遗传数据。遗传水平上的分子相互作用通常涉及酶，因此可以被描述为生化反应(已知的和假设的)。在DNA中，一种名为重组酶的蛋白质家族重排DNA序列。这里的重点将放在一类特定部位的重组酶上，它们只与某些部位的DNA结合。在生物化学方面，DNA是底物，重组酶是催化这一变化的酶。研究DNA的数学模型要么关注DNA在核苷酸水平的变化，要么关注全局结构的变化。由于DNA可以被认为是一根线，当重组酶作用于DNA时，它也可以改变DNA的打结。局部层次的分析使用数学上的代数，而全局层次的分析使用拓扑学，拓扑学是一个研究形状的数学领域。通过一位目前在读的博士生最近的工作，我们有了初步的结果，即需要带状类别和新的理论来融合DNA的局部和全球观点。这个项目的目的是进一步发展数学理论和方法，开发已知位点特定重组酶和产生的DNA结(存在于不同类别的酶，称为拓扑异构酶)的数据库，然后创建预测软件。最后的扩展是如何考虑序列级别数据或全局结构实验图像数据中的不确定性/噪声。这个项目的第二部分是考虑一个结的结构如何与它的能量相关。理解纽结能量与解结有关，这与纽结理论中一个尚未解决的大问题有关：是否有一种多项式时间算法来检测解结。我将开发的方法需要将纯数学(特别是来自代数和拓扑学)的想法与计算、统计学和应用数学的技术结合起来。将传统上不相交的不同领域的思想和技术结合在一起，对于跨学科研究和新数学思想的发展来说是一个令人兴奋的机会。我有在这个十字路口进行研究项目的经验，并使用新的方法来获得对生物系统的新理解。该项目在数学方法和算法方面的进步，与数据生成技术相结合，将使我们能够以新的方式接近和理解真实世界的生物系统。

Modern science generates data at an unprecedented rate, often including the measurement of genetic sequence information in time. One aim in molecular biology is to understand the processes that generate these data; this can be achieved by exploring different hypotheses that are translated into mathematical equations called models. The main outcome of my research will be a range of new methods to understand models in different scenarios with varying amounts of data. The focus of this proposal is genetic data.The molecular interactions at the genetic level often involve enzymes and therefore can be described as biochemical reactions (known and hypothesised). In DNA, a family of proteins called recombinases rearrange DNA sequences. The focus here will be on the class of site-specific recombinases, which only bind to the DNA at certain sites. Biochemically, the DNA is the substrate and the recombinase is the enzyme that catalyses the change. The mathematical models that study DNA either focus on the changes of the DNA at the nucleotide level or the global structure. Since DNA can be thought of as a string, when a recombinase acts on the DNA, it can also change the knotting of the DNA. The local level analysis mathematically employs algebra, while the global level analysis using topology, a field of mathematics that studies shapes. With recent work by a current PhD student, we have preliminary results that ribbon categories and new theory is required to merge between the local and global view of DNA. The aim of this project is to develop the mathematical theory and methods further, develop a database of known site-specific recombinases and resulting DNA knots (which exists for a different class of enzymes called topoisomerases) and then create prediction software. Final extensions are how to take into account uncertainty/noise in either the sequence level data or the global structure experimental image data. The second part of this project is to consider how a knot's configuration relates to its energy. Understanding the knot energies relates to unknots, which relates to a large unsolved problem in knot theory: Is there a polynomial-time algorithm to detect the unknot.The methods that I will develop require marrying ideas from pure mathematics (in particular from algebra and topology) with computing, statistics, and techniques from applied mathematics. To combine ideas and techniques from different fields that traditionally do not intersect is an exciting opportunity for interdisciplinary research, and the development of new mathematical ideas. I have experience conducting research projects at this intersection, and employing new methods to gain a new understanding of biological systems. The advances in mathematical methods and algorithms that result from this project, in combination with data-generating technologies, will enable to approach and understand real-world biological systems in new ways.