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An Exploration Into Knot Theory Application Regarding Enzymatic AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfllment Of the Requirements for the Degree Master of Science In Mathematics By Albert Joseph Jefferson IV 2019 SIGNATURE PAGE THESIS: AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES AUTHOR: Albert Joseph Jefferson IV DATE SUBMITTED: Fall 2019 Department of Mathematics and Statistics Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. John Rock Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics ii ACKNOWLEDGMENTS I would frst like to thank my mother, who tirelessly continues to support every goal of mine, regardless of the feld, and whose guidance has been invaluable in my life. I also would like to thank my 3 sisters, each of whom are excellent role models and have given me so much inspiration. A giant thank you to my advisor, Robin Wilson, who has dealt with my procrastination gracefully, as well to my thesis committee for being so fexible with the defense (each of you are forever my mentors and I owe so much to you all). Finally, I’d like to thank my boyfriend who has been my rock through this entire process, I love you so so much. iii ABSTRACT This thesis will explore the applications of knot theory in biology at the graduate level, specifcally knot theory’s application on DNA super coiling, site specifc recombination, and the overall topology of DNA. We will start with a short introduction to knot theory, introduce rational tangles (the foundation for our tangle model for enzymatic reactions on DNA), review a very specifc overview of DNA and site specifc recombination as it pertains to our motivations, and fnally introduce the tangle model for a Tyrosine Recom- binase and its infuence on the topology of DNA following its action on the molecule. iv Contents 1 Knot Theory Introduction 1 1.1 Defnition of a Knot and Link . 1 1.2 Properties of Knots and Links . 3 1.3 Linking number and Writhe . 7 1.4 Prime and Composite Knots . 9 2 Knot Classifcations 11 2.1 Bridge Knots . 11 2.2 2-Bridge Knot . 12 3 Manifolds 14 3.1 Torus Knots . 15 3.2 Dehn Surgery . 17 3.3 Covering Spaces . 18 4 Rational Tangles 20 4.1 Tangles . 20 4.2 Conway Number . 22 4.3 Tangle Operations . 24 v 4.4 Rational Tangles, 4-Plats and Lens Spaces . 24 5 DNA Overview 26 5.1 Structure of DNA . 27 5.2 DNA Supercoiling . 28 5.3 Topoisomerases and Site-Specifc Recombination . 30 6 The Tangle Model 31 7 Tangle Model for a Tyrosine Recombinases 33 7.1 Flp Protien Model . 33 7.2 Xer Protein Model . 34 8 Conclusion 37 Bibiliography 38 vi Chapter 1 Knot Theory Introduction What is a knot and why do we study knots? The motivation originated from physi- cists in the early twentieth century attempting to understand the nature of atoms. From a suggestion by Scottish scientist Lord Kelvin, for whom the SI base unit of thermo- dynamic temperature ‘Kelvin’ is named, Peter Guthrie Tait through experimentation concluded that atoms were knotted elements of aether (aether being the incorrect idea of the medium needed for the propagation of electromagnetic forces, disproved by the Michelson-Morley experiment) [6]. Although the result was ultimately incorrect, the two scientists inadvertently birthed the theory of knots which now have important appli- cations in mathematics, biology and chemistry. 1.1 Defnition of a Knot and Link Take a shoestring and tie a knot in it. Now, somehow fuse the ends of the string together to form a knotted loop. Unless we cut the string, there is no way to untangle the string to become an unknotted loop. A knot is simply a knotted loop of string, except mathematically we assume that the string has zero thickness, does not intersect, and has 1 (a) Unknot (c) Trefoil knot (b) Figure-8 knot Figure 1.1: Often Referenced Knots a cross-section of a single point. With this simple idea, a barrage of mathematics can be observed, much of which has applications in a variety of different felds. Defnition 1.1.1 If given points x and y of topological space S, if every neighborhood U of x in S meeting every neighborhood V of y in S then requires x = y, then S is a Hausdorff space. Defnition 1.1.2 Let X and Y be Hausdorff spaces. A function f : X ! Y is an embedding if f : X ! f (X) is a homeomorphism. Defnition 1.1.3 A knot K is a smooth embedding of a circle S1 into R3 , or in the 3- sphere S3 , such that f : S1 ! K ⊂ R3 or S3 where K = f (S1) [5]. The simplest example of a knot is the unknot or trivial knot: an unknotted loop in R3 with no crossings. Two often referenced knots include the trefoil knot and fgure-8 knot, as seen in fgure 1.1. 2 Figure 1.2: Hopf Link Defnition 1.1.4 A link is defned to be a fnite union of knots (Note that a knot can be considered a link with only one component.) The simplest example of a link is the Hopf link, which is two unknots linked together in R3. Most practical links are two or more knots where the components are knotted together. However, a link can consist of distinct knots lying on different sides of a plane. Such a link is called splittable. 1.2 Properties of Knots and Links We have defned knots as an embedding of S1 in R3, but for study we will fnd it much easier to use projections. The most common projection map, which we will denote by p, sends the points P(x;y;z) on a knot K in R3 to the points Pˆ(x;y;0) on the xy-plane. Note that p can be a projection onto any two dimensional plane. We will follow [1] and denote p(K) = Kˆ projected onto an arbitrary plane [1]. While easier to study, note that Kˆ is not a simple closed curve on the plane chosen for projec- tion since Kˆ will posses several intersection points following projection (i.e the crossing points in the projection loses information as two or more points may be mapped to the same point on a chosen plane). 3 Figure 1.3: A regular trefoil knot projection Defnition 1.2.1 If Kˆ satisfes the below conditions it is said to be a regular projection: 1. Kˆ has at most a fnite number of points of intersection. 2. If Q is a point of intersection of Kˆ , then the inverse image, p−1(Q) \ K; of Q in K has exactly two points. 3. A vertex of K is never mapped onto a double point of Kˆ . From now on, we will assume all further projections to be regular. Even with the above defnition, we still have ambiguity of double points in a regular projection due to lost information on whether the knot passes over or under itself during a crossing. To remove this ambiguity, we further change the projection by adding small breaks before and after double/intersecting points where a crossing occurs. We call this altered regular projection a regular diagram [1]. From now on, we will assume all further diagrams to be regular. The knots in Figure 1.1 are examples of regular diagrams. One of the diffculties in knot theory is the deceptive nature of knots. It is easy to deform a knot, without cutting it, and cause the knot to resemble a seemingly different knot. If the deformation was not seen, one might mistakenly assume the two knots are distinct. However, intrinsically, it is natural to see the original knot and the deformed 4 (a) 1st Reidemeister Move (b) 2nd Reidemeister Move (c) 3rd Reidemeister Move Figure 1.4: Reidemeister Moves knot as the same even though the two knot diagrams may be markedly unrelated, so we must develop equivalent classes of knots as a check towards confusion between the infnitely many deformations of one knot. Defnition 1.2.2 We say two knots are equivalent if one can be deformed into the other in R3 , and non-equivalent if there is no such deformation [6]. Determining whether two knots are equivalent can be extremely diffcult. For exam- ple, there are many knots which seem to be very knotted but are in fact the unknot [8]. Another famous example of this diffculty is the Perko Pair. Since 1885 until Perko’s proof, many mathematicians believed two knots (labeled 10161 and 10162) to be non- equivalent due to a misprint classifcation in the Little knot table. They were shown to be equivalent by Kenneth Perko in 1974 using his famous Perko move [15]. The Perko Pair is just one example of the diffculty in distinguishing between two equivalent knots. One important theorem in this area was proved by German mathe- matician Kurt Reidemeister in 1926. Reidemeister hoped to fnd the minimum number of deformations needed to relate all equivalent knots, and proved there were only three distinct deformations needed called Reidemeister moves:. [8] The 1st Reidemeister move entails twisting or untwisting a loop in a knot (fgure 1.4a). The 2nd Reidemeister move entails moving one strand under or over another strand (fg- ure 1.4b). 5 The 3rd Reidemeister move entails moving one strand over or under a crossing. (fgure 1.4c). Theorem 1.2.1 Two link diagrams are equivalent if and only if there exists a fnite num- ber of Reidemeister moves between them.
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