DNA Knotting and Linking: Applications of 3-Manifold Topology to DNA-Protein Interactions

DNA是现代分子生物学中为数不多的几乎人人都熟悉的部分之一。我们都知道DNA负责我们的基因遗传，并且都看到DNA的模型是一个双链分子，形状像双螺旋楼梯，即所谓的双螺旋。在所有常见的DNA图片中，双螺旋的轴线看起来都是笔直的。然而，在所有细胞中，任何DNA分子的轴都远不是直的，实际上是令人难以置信的扭曲；这样它占用的空间要小得多。有时，与异性恋的偏离甚至更为明显。例如，在细菌细胞中，DNA分子的两端可以连接起来形成环状DNA。如果我们把一根绳子的两端连在一起，有时会在绳子上打结。当DNA分子的两端连接在一起时也会发生同样的事情所以DNA结就产生了。更一般地说，如果我们有两条或更多的绳子，把所有绳子的末端绑起来，我们就会得到许多结，这些结可能会连接在一起——就像奥林匹克五环一样。所以如果我们有不止一个而是很多DNA分子那么我们就可以形成DNA链和DNA结。自20世纪60年代末发现以来，人们发现DNA结和链接在细胞过程的宿主中起着关键作用。因为它们无处不在，所有生物体都进化出了特殊的蛋白质——拓扑异构酶，其功能是帮助解开DNA结和连接。还有其他重要的蛋白质——被称为重组酶和转座酶——可以改变DNA碱基对的序列顺序。虽然重组酶和转座的主要功能是重新排列碱基对的顺序，但在此过程中，它们经常导致DNA打结或连接的变化。由于所有这些原因，分子生物学家开始对学习结和链接感兴趣。自19世纪后期以来，数学家们一直在研究结，因为他们有自己的原因，与DNA无关。从数学上讲，结是三维空间中的一维物体，就像标准圆一样，但我们不能将其平滑地变形为标准圆。结和连杆的数学理论是非常丰富和令人惊讶的复杂，并且与一般的三维空间密切相关，称为3流形。（对这些空间的研究被称为3流形拓扑。）虽然这个主题非常深奥，但一些最简单的问题仍然没有答案：即使在今天，如果你交给世界上最顶尖的结理论家两个足够复杂的结，他们也没有已知的算法可以用来判断一个结是否可以变形成另一个结。利用结理论的工具，数学家已经能够帮助生物学家更好地理解一些蛋白质与DNA相互作用的方式。例如，包括申请人在内的数学家已经建立了重组酶和转座酶蛋白质如何重组DNA序列的模型。然后，这些模型可以预测这些相互作用的各种新特征——例如，当蛋白质附着时DNA的特定几何结构，或者反应进行的生化途径。DNA可以形成非常复杂的结。但在所有可能的非常复杂的结中，只有一小部分以DNA结的形式出现。最近，我描述了重组酶作用于DNA结结构的初始家族后会出现哪些结。在这个建议中，我们将探讨这个问题，更广泛的家族的初始DNA构型，以及类似的问题转座酶反应。我们还将考虑解结和解连DNA分子。我们特别希望了解两个DNA结何时通过交叉变化联系在一起。为了回答这些问题，我们将使用来自3流形拓扑的尖端技术。这些答案将帮助我们更全面地了解这些重要的蛋白质，它们是抗生素和一些抗癌药物的主要靶点。

DNA is one of the very few parts of modern molecular biology familiar to almost everyone. We all know that DNA is responsible for our genetic inheritance and have all seen models of DNA as a two-stranded molecule with a shape like a double spiral staircase, a so-called double helix. In all the usual pictures of DNA the axis of the double helix looks nice and straight. However, in all cells the axis of any DNA molecule is far from straight and is in fact incredibly twisted up; it occupies much less space this way. Sometimes, the deviation from straight is even more pronounced. For example in bacterial cells, the two ends of a DNA molecule can be joined up to form circular DNA. If we take a piece of string and join the ends we sometimes get a knot in the string. Exactly the same thing can happen when the 2 ends of a DNA molecule get joined up and so DNA knots are born. More generally if we have two or more pieces of string and tie up the ends of all the pieces of string then we get many knots that might be linked together-like the Olympic rings. So if we have not just one but many DNA molecules then we can form DNA links as well as DNA knots.Since their discovery in the late 1960s, DNA knots and links have been found to play key roles in hosts of cellular processes. Because they are so ubiquitous all organisms have developed special proteins--topoisomerases--whose function is to help untie DNA knots and links. There are also other important proteins--called recombinases and transposes--that can alter the order of the sequence of the DNA basepairs. While the main function of recombinases and transposes is to rearrange the order of basepairs, in the process of doing this they often cause changes to DNA knotting or linking. For all these reasons molecular biologists became interested in learning about knots and links.Mathematicians have studied knots since the late 19th century for their own reasons, having nothing to do with DNA. Mathematically a knot is a one-dimensional object sitting inside 3-space, just like a standard circle does, but which we cannot smoothly deform to a standard circle. The mathematical theory of knots and links turns out to be very rich and surprisingly complicated, and intimately related to general 3-dimensional spaces, called 3-manifolds. (The study of these spaces is called 3-manifold topology.) Although the subject is very deep, some of the simplest questions remain unanswered: even today if you hand the world's top knot theorists two sufficiently complicated knots there is no known algorithm they can use to always tell whether one knot can be deformed into the other. Using tools from knot theory, mathematicians have been able to help biologists better understand the ways some proteins interact with DNA. For example, mathematicians, including the applicant, have developed models of how the recombinase and transposase proteins reshuffle the DNA sequence. These models can then predict various new features of these interactions -- e.g. particular geometric configuration the DNA takes when the protein is attached or what biochemical pathway the reactions proceeds through. DNA can form very complicated knots. But only a small fraction of all possible very complicated knots appear as DNA knots. Recently I characterized which knots can show up after a recombinase acts on an initial family of DNA knot configurations. In this proposal we will explore this question for a much wider family of initial DNA configurations, and also the analogous question for transposase reactions. We will also consider unknotting and unlinking DNA molecules. In particular we hope to understand when two DNA knots are related by a crossing change. To answer these questions, we will use cutting-edge techniques from 3-manifold topology. The answers will help us understand these important proteins, the main targets of antibiotics and some anti-cancer drugs, more completely.