A biological application for the oriented skein relation

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 2012 A biological application for the oriented skein relation Candice Renee Price University of Iowa Copyright 2012 Candice R. Price This dissertation is available at Iowa Research Online: Recommended Citation Price, Candice Renee. "A biological application for the oriented skein relation." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Mathematics Commons

2 A BIOLOGICAL APPLICATION FOR THE ORIENTED SKEIN RELATION by Candice Reneé Price An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa July 2012 Thesis Supervisor: Associate Professor Isabel Darcy

3 1 ABSTRACT The traditional skein relation for the Alexander polynomial involves an oriented knot, K +, with a distinguished positive crossing; a knot K, obtained by changing the distinguished positive crossing of K + to a negative crossing; and a link K 0, the orientation preserving resolution of the distinguished crossing. We refer to (K +, K, K 0 ) as the oriented skein triple. A tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K K +. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K ± K 0. The oriented triple is now viewed as K = circular DNA substrate, K + = product of topoisomerase action, K 0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K + and K, that differ by a crossing change and a link, K 0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins.

4 2 In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology, was constructed by Peter Ozsváth, Zoltán Szabó, and independently Jacob Rasmussen, denoted by HF K. It is also a refinement of a classical invariant, the Alexander polynomial. The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel links. Abstract Approved: Thesis Supervisor Title and Department Date

5 A BIOLOGICAL APPLICATION FOR THE ORIENTED SKEIN RELATION by Candice Reneé Price A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa July 2012 Thesis Supervisor: Associate Professor Isabel Darcy

6 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Candice Reneé Price has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Mathematics at the July 2012 graduation. Thesis Committee: Isabel Darcy, Thesis Supervisor J. Elisenda Grigsby Colleen Mitchell Maggy Tomova Bruce Ayati

7 To Lauren Price You got your Dr. Price. ii

8 ACKNOWLEDGEMENTS To my wonderful family who supported me throughout these long years in snowy Iowa. For my mentor Lori Holcombe who always supports me and was the first person to see my mathematical potential. For my Iowa Family for creating a place where being away from California never seemed too bad. To my EDGE family for making the last month of my graduate years so wonderful. A huge thank you to Eli Grigsby for her endless patience and always fruitful discussions. And a very special thank you to my advisor, Isabel Darcy, for her insight, candid conversations and thoughtfulness to let me be me. iii

9 ABSTRACT The traditional skein relation for the Alexander polynomial involves an oriented knot, K +, with a distinguished positive crossing; a knot K, obtained by changing the distinguished positive crossing of K + to a negative crossing; and a link K 0, the orientation preserving resolution of the distinguished crossing. We refer to (K +, K, K 0 ) as the oriented skein triple. A tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K K +. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K ± K 0. The oriented triple is now viewed as K = circular DNA substrate, K + = product of topoisomerase action, K 0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K + and K, that differ by a crossing change and a link, K 0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins. iv

10 In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology, was constructed by Peter Ozsváth, Zoltán Szabó, and independently Jacob Rasmussen, denoted by HF K. It is also a refinement of a classical invariant, the Alexander polynomial. The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel links. v

11 TABLE OF CONTENTS LIST OF TABLES viii CHAPTER LIST OF FIGURES ix CHAPTER 1 INTRODUCTION AND BACKGROUND MATERIAL Knots and Links Tangles Biology Background Transcription and Replication Topoisomerase Recombinase Tangle Model BIOLOGICAL APPLICATION Development of Our Tangle Model Application of the Skein Triple Difference Topology SMC Proteins S Condensin Experiments MukB Experiments Application of Our Model Discussion KNOT FLOER HOMOLOGY Combinatorial Description Example: Calculation of HF K m(0 1, s) BIOLOGICALLY RELEVANT KNOT FAMILIES (2, p) Torus Links Knot Floer Homology of T (2, p) Knot Floer Homology of T (2, p) Long Exact Sequence Clasp Knots, C(2, v) vi

12 4.2.1 Knot Floer Homology of T (2, p) Future Direction: 3-Strand Pretzel Links HF K of Non-Alternating 3-Strand Pretzel Knots HF K of Non-Alternating Pretzel Links REFERENCES vii

13 LIST OF TABLES Table 2.1 Exp. 1 and Exp. 2 data Gradings for Fig Differential for each generator Non-alternating 3-strand pretzel knots Non-alternating 3-strand pretzel links viii

14 LIST OF FIGURES Figure 1.1 Oriented knot triple. Triple of knots that differ only at one crossing Examples of simple knots. Courtesy of [50] Ambiguous and problematic intersections nota allowed in knot diagrams The polygonal projection and the smooth projection have eight possible knot diagrams, two are shown here Diagrams of wild knot. Courtesy of [42] Reidemeister moves: (I) twist, (II) poke, (III) slide The connected sum of knots K 1 and K 2. Courtesy of [49] knot and its mirror image Given an orientation, we can assign negative and positive crossings Simplest Tangles Equivalent tangles string tangle: There exists a homeomorphism of pairs, φ : (B, t) (B 0, t 0 ) where B 0 is the unit ball. P yz is a projection of our tangle onto the yz-plane Sum of two tangles Numerator closure of a tangle T and the numerator closure of the sum of two tangles T 1 and T 2 giving links N(T ) and N(T 1 + T 2 ) respectively Construction of a rational tangle. Used with permission from [14] Circle product of two tangles: A and (c 1,..., c n ). Used with permission from [14] Generalized Montesinos tangle ( ) (0, 1). Courtesy of [42] Sugar ring made of 5 carbon atoms. Courtesy of [51] ix

15 1.19 Deoxyribonucleic acid. Using the direction convention given to DNA strands, we read this sequence as ACT G, or equivalently CAGT. Courtesy of [52] Cartoon of negative, relaxed and positive supercoiled DNA. Reproduced with permission from [23] Two examples of supercoiled DNA seen through an electron microscope. Reproduced with permission from [23] Like DNA, ribonucleic acid (RNA) is a nucleic acid made up of a long chain of nucleotides. Courtesy of [53] Transcription-driven supercoils Topological changes to DNA during replication of circular DNA. Used with permission from [54] Schematic of topoisomerase I action. Used with permission from [9] Schematic of topoisomerase II action. Used with permission from [4] An example of a site specific recombinase mechanism where the protein makes breaks one strand of the double helix, recombines it and then does the same with the other strand Directed Repeats Recombinase action on direct repeats Inverted Repeats Recombinase action on inverted repeats Schematic of tryosine recombinase action: single stranded breaks. We model the tyrosine protein as a black ball while the double stranded DNA is modeled by red and blue rectangles Schematic of serine recombinase action: double stranded breaks. We model the serine protein as a black ball while the double stranded DNA is modeled by red and blue rectangles Tangle Model x

16 1.35 A DNA molecule before protein action is called the DNA substrate; and after, the DNA product Mathematically equivalent moves: ( 1, ) ( 1 1 1, 0, ) Oriented knot triple: Triple of knots that differ only at one crossing Model of topoisomerase action Model of recombinase action Diagram associated with equations (2.1),(2.2),(2.3) respectively An example of a topologically equivalent tangle model to Fig The top two rows show a schematic of one direction for the equivalence between the (1, 1) move and the (0, 2) move on oriented links in figure 2.4 such that U = U (1). Using U, we can find the conversion for equation 2.8, shown in the bottom row The top two rows show a schematic of one direction for the equivalence between the (1, 1) move and the (0, 2) move on oriented links in figure 2.5 such that U = U (1). Using U, we can find the conversion for equation K 0, shown in the bottom row Difference topology experiment. Used with permission from [13] General structure of an SMC protein, showing the five distinct domains along with alternative models of the structure involving two SMC proteins. Used with permission from [10] Three models for the possible conformation for 13S condesin bound DNA. DNA is modeled as a double line and 13S condensin is schematized as a V shaped object. From left to right: Model (I) shows a (+) twist, Model (II) shows a (+) wrap, adding positive writhe locally, and Model (III) shows the (+) global writhe. Figure reproduced with permission from [27] The three models of each conformation from figure 2.10 for 13S condensin on nicked DNA. Figure reproduced with permission from [27] Projection of the 4 1 knot drawn on a 6 6 grid diagram. Reproduced and modified with permission from [28] xi

17 3.2 An example of the generator (302514) for a 6 6 toroidal grid diagram. Reproduced with permission from [28] Generator p on the torus with the bottom and left edges included Generator p drawn on the torus with the top and left edges included Generator p after a complete rotation of the torus with left and bottom edges included Toroidal grid diagram with two shaded rectangles connecting p to q and two unshaded rectangles connecting q to p There are four ways to draw the two rectangles in Rect(p, j) Rect(p, k): disjoint, share an interior and share a corner, share only a corner, share only an interior Pathway between two disjoint rectangles Pathway between two rectangles that share an interior grid example Knot drawn on the grid The generators for the grid diagram in Fig labeled from left to right and top to bottom: (012), (021), (102), (120), (201) and (210) The rectangles used to calculate the differential for the generator (012) The rectangles used to calculate the differential for the generator (021) The rectangles used to calculate the differential for the generator (102) The rectangles used to calculate the differential for the generator (120) The rectangles used to calculate the differential for the generator (201) The rectangles used to calculate the differential for the generator (210) Graph, with x-axis Alexander grading and y-axis Maslov grading, that shows the generators as red circles and the generators times U 1, U 2, U 3 as blue triangles xii

18 3.20 Graph of the homology groups of the unknot Construction of torus links Parallel orientation, T (2, 4) Anti-parallel orientation, T (2, 4) Recall the oriented knot triple: (K +, K, K 0 ) Grid diagrams of T (2, 4) and T (2, 4) Clasp Knots: From left to right: v is odd and v is even K = C(2, 4), K + = C(2, 2), K 0 = K 0 when v is even K 0 when v is odd stranded Pretzel knot P (3, 4, 3) stranded Pretzel knot P (5, 3, 7). This knot has trivial Alexander polynomial xiii

19 1 CHAPTER 1 INTRODUCTION AND BACKGROUND MATERIAL There are certain proteins, topoisomerase and recombinase, that can change the topology of DNA. These changes can inhibit or aid in biological processes involving the structure of DNA including replication and transcription. The local actions of these proteins can be modeled using knot theory [20,44]. Applications of knot theory to problems involving these protein actions have been extensively studied [8, 14, 20, 44, 45, 47]. The model of the local action of these protein developed in this thesis utilizes the skein relation, which relates two knots, K +, K and one link, K 0. We define K + as an oriented knot with a distinguished positive crossing; K, a knot obtained by changing the distinguished positive crossing of K + to a negative crossing; and K 0 is a link obtained by the orientation preserving resolution of the distinguished crossing (Fig. 1.1). The triple (K +, K, K 0 ) is called an oriented knot triple. This relation is most commonly used in calculating the Alexander polynomial, an invariant for knots and links. The traditional skein relation for the Alexander polynomial also corresponds to a long exact sequence which relates the knot Floer homologies of the oriented knot triple. Knot Floer homology, denoted HF K, was constructed by Peter Ozsváth, Zoltán Szabó [32] and independently by Jacob Rasmussen [38]. It is a refinement of the Alexander polynomial and is the homology of a bigraded complex

20 2 (C, ) with Maslov grading m and Alexander grading s: HF K = m,s Z HF K m (K, s). Figure 1.1: Oriented knot triple. Triple of knots that differ only at one crossing. This dissertation is organized as follows: Chapter 1 provides background on knot theory and pertinent terminology from biology. Chapter 2 is the main focus of this thesis. Section 2.1 will develop and state our model for the local action of proteins using the oriented skein relation. This section extends the work in [15 17,46]. Section 2.2 discusses a biological application known as difference topology. In particular, we use our tangle model from section 2.1 to analyze data from difference topology experiments used to study SMC proteins. Chapter 3 will provide background on the combinatorial version of knot Floer homology as well as a detailed example calculating the knot Floer homology of the trivial knot. This chapter is a summary of known work from [28] and [29]. In chapter 4, we investigate the known knot Floer homology formulas of biologically relevant knot families (2, p)-torus knots, (2, r) clasp knots and 3-stranded pretzel knots [18,31,33]. The thesis will conclude with a section exploring

21 3 an expansion of our model to these families of knots and a discussion on finding unknown knot Floer homologies for 3-stranded pretzel links. 1.1 Knots and Links Although knots have been used since the dawn of humanity, the mathematical study of knots is only a little over 100 years old. Not only has knot theory grown theoretically, the fields of physics, chemistry and molecular biology have provided many applications of mathematical knots. A knot is defined as a closed, non-intersecting curve in R 3. Formally it is the embedding of a circle in three dimensions (we call a mapping f : X Y an embedding if the restriction mapping f : X f(x) is a homeomorphism) (Fig. 1.2). Intuitively, a knot can be thought of simply as a loop of rope with no end and no beginning, like tying a shoestring and then gluing the ends to one another. A knot that can be laid flat with no self-intersections is called a trivial knot, also known as an unknot (Fig. 1.2). A non-trivial knot can only be untangled to produce a trivial knot by breaking the curve. A link is a finite union of knots properly embedded in three-dimensional space. Each of these knots, which may be trivial, is known as a component of the link. We can view a knot as a 1-component link. A link projection is the two-dimensional image of the three-dimensional link projected onto a plane. At each double point in the projection (a crossing involving only two line segments), it is not clear which portion of the link crosses over and which

22 4 Figure 1.2: Examples of simple knots. Courtesy of [50] crosses under. To show this, gaps are left in the projection to indicate overcrossings and undercrossings. At a crossing, the strand of the knot at the at the top of the crossing, represented by a solid line segment, is called the overcrossing. The strand that is at the bottom of the crossing is called the undercrossing, represented by a broken line segment. It is known that problematic intersections (see Fig. 1.3) can be avoided so that all intersections correspond to double points. A link projection drawn with these criteria is called a link diagram. Knots and links are studied through their diagrams. Links that have diagrams that can be drawn using a finite number of polygonal circuits (i.e. closed paths) in three-dimensional space are called tame (Fig. 1.4). All other links are known as wild (Fig. 1.5). Most applications of knot theory concern only tame links, so I will only focus on this class of links. We say two links, K 1 and K 2, are equivalent if there is an ambient isotopy

23 5 Figure 1.3: Ambiguous and problematic intersections nota allowed in knot diagrams. Figure 1.4: The polygonal projection and the smooth projection have eight possible knot diagrams, two are shown here. between them. An ambient isotopy can be described as a continuous deformation from one link diagram (K 1 ) to the other (K 2 ). It allows us to stretch, bend and twist the link however we would like; we just cannot cut it. Mathematically, two links, K 1 and K 2, are ambient isotopic if there is an isotopy h : R 3 [0, 1] R 3 such that h i is a homeomorphism such that h(s, i) = h i (s) where h(k 1, 0) = h 0 (K 1 ) = K 1 and h(k 1, 1) = h 1 (K 1 ) = K 2 [12]. If two knots are equivalent, we refer to these knots as knots of the same knot type, K, where K is the equivalence class under this equivalence relation. In 1926, Kurt Reidemeister proved that if we have two distinct diagrams of the same knot, we can go from one diagram to the other using Reidemeister moves, as described in Theorem Theorem (Reidemeister [39]). Two link diagrams L 1 and L 2 are equivalent

24 6 Figure 1.5: Diagrams of wild knot. Courtesy of [42]. if and only if they can be obtained from one another by a finite sequence of planar isotopies and the three moves: twist, poke, and slide (Fig. 1.6). Given two knots, K 1 and K 2, a knot K = K 1 #K 2 can be constructed as seen in Fig This knot is known as the connected sum of K 1 and K 2. A knot that cannot be constructed in this manner using non-trivial knots is called prime. All prime knots will be referred to as they are given in Rolfsen s table of prime knots [41]. This table groups each knot type according to the number of crossings in their minimal

25 7 I II III Figure 1.6: Reidemeister moves: (I) twist, (II) poke, (III) slide.

26 8 diagram, known as the crossing index. Figure 1.7: The connected sum of knots K 1 and K 2. Courtesy of [49]. Links can be split into two groups: alternating and nonalternating. A link is called alternating if it has a diagram in which, when traveling around each component of the link, one alternates between overcrossing and undercrossings. A nonalternating link is one that is not alternating (i.e. every diagram has at least two overcrossings or two undercrossing in a row when traveling around the link). The first nonalternating knot in Rolfen s table is An oriented link is a link for which each component has been given an orientation. An oriented link is invertible if it can be deformed to be the same link diagram with the opposite orientation [1]. The first knot that is not invertible is The mirror image of a link, L, is obtained by changing every overcrossing in the link to an undercrossing and vice versa. If L is equivalent to its mirror image, then we call L amphicheiral (or achiral). If L is not equivalent to L, then it is chiral. Although not all links are achiral, most tables do not distinguish between a link and

27 9 its mirror image. One example of a knot that is amphicheiral is the 4 1 knot (Fig. 1.8). Figure 1.8: 4 1 knot and its mirror image While Reidemeister moves are helpful to see if two links are equivalent, they are not useful when showing that two links are not equivalent. Link invariants are utilized to show inequivalence between two link diagrams. A link invariant is a specific quality of a knot or link type that does not change its value under ambient isotopy. Thus, if two links are equivalent then their invariants are equal. Unfortunately, for the majority of invariants, the other direction is not usually true: equal invariant values for two link diagrams does not imply equivalent links. One basic example of a link invariant is the minimum crossing number. The minimum crossing number is the number of crossings in the minimal diagram of the knot. That is, we minimize the number of crossings over all knot diagrams in that equivalence class or knot type. A knot diagram with the minimal number of crossings is a minimum regular diagram of the knot.

28 10 Some invariants keep count of the number of topological changes made to a knot diagram. Looking at a knot diagram, exchange locally overcrossings and undercrossings. This type of alteration may change the knot type. The unknotting number is the least number of crossing changes in a diagram of a knot to get to the trivial knot, minimized over all diagrams. The linking number is a link invariant for links of two or more components. It is calculated using the crossing sign convention (Fig. 1.9). The linking number is calculating by taking the sum of the crossing signs of each crossing between the different components of the link and dividing by two. Figure 1.9: Given an orientation, we can assign negative and positive crossings. While the previous invariants give numerical quantities, the Alexander polynomial is an invariant that associates a polynomial to a knot type. While there are several ways to calculate this invariant, we will use the method involving skein relations. A skein relation requires three link diagrams be identical except at one cross-

29 11 ing (Fig. 1.1). The Alexander polynomial of a link K, denoted K (t) can be calculated using the following skein relationship: (t) = 1 2. K+ (t) K (t) = (t 1/2 t 1/2 ) K0 (t). 1.2 Tangles An n-string tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. An n-string tangle is formed by placing 2n points on the boundary of B and attaching n nonintersecting curves inside B such that B t = t. The simplest tangles are the (0)-tangle, ( )-tangle, (+1)-tangle and the ( 1)-tangle (Fig. 1.10). We consider tangles T 1 = (B, t 1 ) and T 2 = (B, t 2 ) to be equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the ball fixed (Fig. 1.11). Figure 1.10: Simplest Tangles

30 12 Figure 1.11: Equivalent tangles This work will focus on 2-string tangles. As part of the definition we consider a 2-string tangle to be a pair (B, t) and a homeomorphism sending (B, t) to the unit ball in R 3 (Fig. 1.12). We send the four endpoints of the arcs to the four equatorial points NW, NE, SE, and SW in the yz-plane described in R 3 as the points: NE : SE : ( 0, 1 2, ) 1 2 ( 1 0,, 1 ) 2 2 NW : SW : ( 0, 1 ) 1, 2 2 ( 0, 1, 1 ). 2 2 Figure 1.12: 2-string tangle: There exists a homeomorphism of pairs, φ : (B, t) (B 0, t 0 ) where B 0 is the unit ball. P yz is a projection of our tangle onto the yz-plane. We can take the sum of two tangles, T 1 and T 2, creating a new tangle, T 1 + T 2 (Fig. 1.13). Another tangle operation is the numerator closure, which connects

31 13 the northern endpoints with the shortest arc on the exterior of B and similarly the southern endpoints, resulting in a knot or link denoted N(T ). We can also perform this operation on a sum of tangles (Fig. 1.14). Figure 1.13: Sum of two tangles Figure 1.14: Numerator closure of a tangle T and the numerator closure of the sum of two tangles T 1 and T 2 giving links N(T ) and N(T 1 + T 2 ) respectively. A 2-string tangle T = (B, t) is rational if it is ambient isotopic to the (0)-tangle, allowing the boundary of the 3-ball to move. A rational tangle diagram is created by starting with the (0) tangle and interchanging the NE and SE boundary points a finite number of times creating horizontal twists. Then, continue construction

32 14 by interchanging the SW and SE boundary points a finite number of times creating vertical twists. Continue in this manner, alternating between adding vertical and horizontal twists (Fig. 1.15). John Conway associated to each rational 2-string tangle an extended rational number, m n Q { }, stating that there exists a 1-1 correspondence [11]. This number can be calculated using the Conway vector, denoted (a 1, a 2,..., a i ) where we choose i to be odd. This finite sequence of integers represents the sequence of moves performed on the (0)-tangle to produce a rational tangle. Each integer represents the number of half-twists given to the tangle, alternating between horizontal and vertical, ending with horizontal twists. We let a positive integer denote half twists whose overcrossing has a negative slope and a negative integer denote half twists whose overcrossing has a positive slope when drawn in the manner shown in Fig If a tangle T is denoted (a 1, a 2,..., a i ), then its extended rational number, is calculated as: m n = a i + a i 1 + a i a i a m The numerator closure of a rational tangle,, is referred to as a 2-bridge n knot/link denoted N ( m n ). These links are also referred to as 4-plats and rational knots/links. We will need the following definitions of circle product and Montesinos tangle. Definition [16] The circle product, A (c 1,, c n ) of two tangles A and

33 15 Figure 1.15: Construction of a rational tangle. Used with permission from [14]. (c 1,, c n ) is obtained by starting with c 1 vertical (horizontal) half twists of the SW and SE (NE and SE) endpoints of A, and alternating between c i horizontal (vertical) and vertical (horizontal) half twists when n is even (odd) (Fig.1.16). Figure 1.16: Circle product of two tangles: A and (c 1,..., c n ). Used with permission from [14]. Definition [16] A generalized Montesinos tangle is a tangle of the form ( ) a 1 b an b n (c 1,... c m ) where a i b i 1 is a rational tangle for all 1 i n and 0 c j is an integer for all 1 j m (Fig. 1.17). This definition includes all rational tangles. A Montesinos link is the numerator closure of a generalized Montesinos

34 16 Figure 1.17: Generalized Montesinos tangle ( ) (0, 1). Courtesy of [42]. 2 7 tangle. 1.3 Biology Background A crucial advancement in molecular biology was made when the structure of DNA was determined by James Watson and Francis Crick in Its structure revealed how DNA can be replicated and provided clues about how a molecule of DNA might encode directions for producing proteins [2]. Nucleic acids consist of a chain of linked units called nucleotides. Each nucleotide contains a deoxyribose, a sugar ring made of five carbon atoms which are numbered as seen in Fig This sugar ring then forms bonds to a single phosphate group between the third and fifth carbon atoms of adjacent sugar rings (Fig. 1.19). The backbone of a DNA strand is made from alternating phosphate groups and sugar rings. The four bases found in DNA are Adenine (A), Thymine (T), Cytosine (C) and Guanine (G). The shapes and chemical structure of these bases allow hydrogen bonds to form efficiently between A and T and between G and C. These bonds, along with base stacking interactions, hold the DNA strands together [2]. Each base is attached to the first carbon atom in the sugar ring to complete the nucleotide (Fig. 1.19).

35 17 Figure 1.18: Sugar ring made of 5 carbon atoms. Courtesy of [51]. The bonds between the sugars and the phosphate group give a direction to DNA strands. The asymmetric ends of the strands are called the 5 (five prime) and 3 (three prime) ends, with the 5 end having a phosphate group attached to the fifth carbon atom of the sugar ring and the 3 end with a terminal hydroxyl group attached to the third carbon atom of the sugar ring (Fig. 1.19). The direction of the DNA strands are read from 5 to 3. In a double helix the direction of one strand is opposite to the direction of the other strand: the strands are antiparallel [2]. Besides the standard linear form, a molecule of DNA can take the form of a ring known as circular DNA. One way to model circular DNA mathematically is as an annulus, R, an object that is topologically equivalent to S 1 [ 1, 1]. The axis of R is S 1 {0}. With this model we can choose an orientation for the axis of R and use the same orientation on R; thus, the axis and boundary curves of R have a parallel orientation. Note that this is a different convention than the biology/chemistry orientation. We use geometric invariants twist and writhe, denoted Tw and Wr, to describe the structure of the circular DNA molecule. Writhe can be determined by viewing the axis of R as a spatial curve and is measured as the average value of the

36 18 Figure 1.19: Deoxyribonucleic acid. Using the direction convention given to DNA strands, we read this sequence as ACT G, or equivalently CAGT. Courtesy of [52]. sum of the positive and negative crossings of the axis of R with itself, averaged over all projections [30]. The sign convention for a crossing is given in (Fig. 1.9). Twist is defined as the amount that one of the boundary curves of R twists around the axis of R [3]. One relationship between Tw and Wr is expressed in the following law:

37 19 Law (Conservation Law [21]). Lk(R) = Tw(R) + Wr(R) where Lk(R) is the linking number of the oriented link formed by the two boundary curves of R with a parallel orientation. A nick is a discontinuity in one strand of a double stranded DNA molecule. This discontinuity is a missing bond between adjacent nucleotides of the same strand. A nick can be the result of damage or enzyme action and is used experimentally to release torsion in a DNA strand, i.e. relaxation of the DNA molecule. Nicked circular DNA no longer has a linking number since at least one of the backbones is no longer a closed curve. This allows the DNA to relax to its preferred twist and writhe. We say that a DNA molecule is supercoiled when Wr 0 (Fig. 1.20). Native circular DNA appears negatively supercoiled under an electron microscope, i.e. Wr < 0 (Fig. 1.21) [3]. Recall that the structure of DNA is a double-stranded helix, where the four bases are paired and stored in the center of this helix. While this structure provides stability for storing the genetic code, Watson and Crick noted that the two strands of DNA would need to be untwisted in order to access the information stored for replication and transcription [2]. They also foresaw that there should be some mechanism to overcome this problem.

38 20 Figure 1.20: Cartoon of negative, relaxed and positive supercoiled DNA. Reproduced with permission from [23] Transcription and Replication There are three main topological forms that circular DNA can take: supercoiled, knotted, catenated (linked DNA molecules) or a combination of these. DNA is kept as compact as possible when in the nucleus, and these three states help or hinder this cause. However, when transcription or replication occur, DNA must be accessible [48]. Ribonucleic acid (RNA) is a nucleic acid made up of a chain of nucleotides (Fig. 1.22). There are three main differences between RNA and DNA: a) RNA contains the sugar ribose, while DNA contains a different sugar, deoxyribose; b) RNA contains the base uracil (U) in place of the base thymine (T), which is present in DNA; and, c) RNA molecules are single stranded, but have interesting tertiary structure. Transcription is the process of creating a complementary RNA copy of a sequence of DNA. Transcription begins with the unwinding of a small portion of the DNA double helix to expose the bases of each DNA strand. The two strands are then pulled apart

39 21 Figure 1.21: Two examples of supercoiled DNA seen through an electron microscope. Reproduced with permission from [23]. creating an opening known as the transcription bubble. During this process, DNA ahead of the transcription bubble becomes positively supercoiled, while DNA behind the transcription bubble becomes negatively supercoiled (Fig. 1.23). DNA replication is the process that starts with one DNA molecule and produces two identical copies of that molecule. During replication, the DNA molecule begins to unwind at a specific location and starts the synthesis of the new strands at this location, forming replication forks (Fig. 1.24, left). The DNA ahead of the replication fork becomes positively supercoiled, while DNA behind the replication fork

40 22 Figure 1.22: Like DNA, ribonucleic acid (RNA) is a nucleic acid made up of a long chain of nucleotides. Courtesy of [53]. becomes entangled, creating pre-catenanes, a state where the DNA molecules are beginning to form linked DNA molecules (Fig. 1.24, center). A topological problem occurs at the end of replication, when daughter chromosomes must be fully disentangled before mitosis occurs (Fig. 1.24, right) [48]. Topoisomerases play an essential role in resolving this problem Topoisomerase Topoisomerases are proteins that are involved in the packing of DNA in the nucleus and in the unknotting and unlinking of DNA links that can result from repli-

41 23 Figure 1.23: Transcription-driven supercoils. cation and other biological processes. These proteins bind to either single stranded or double stranded DNA and cut the phosphate backbone of the DNA. A type I topoisomerase cuts one strand of a DNA double helix allowing for the reduction or the introduction of stress (Fig. 1.25). Such stress is introduced or needed when the DNA strand is supercoiled or uncoiled during replication or transcription. Type II topoisomerase cuts both phosphate backbones of one DNA double helix, passes another DNA double helix through it, and then reseals the cut strands (Fig. 1.26). This action do not change the chemical composition and connectivity of DNA, but potentially changes its topology Recombinase Recombination is a process involving the genetic exchange of DNA where DNA sequences are rearranged by proteins known as recombinases [2]. Site specific re-

42 24 Figure 1.24: Topological changes to DNA during replication of circular DNA. Used with permission from [54]. Figure 1.25: Schematic of topoisomerase I action. Used with permission from [9]. combination is an operation on DNA molecules where recombination proteins, site specific recombinases, recognize short specific DNA sequences on the recombining DNA molecules. First, two sequences from the same or different DNA molecule are

43 25 Figure 1.26: Schematic of topoisomerase II action. Used with permission from [4]. drawn together. The recombinase then introduces a break near a specific site, known as a recombination site, on the double stranded DNA molecule. The protein then recombines the ends in some manner and seals the break (Fig. 1.27). The DNA sequence of a recombination site can be used to give an orientation to this site. When two sites are oriented in the same direction, the sites are called direct repeats (Fig. 1.28). Recombinase action on direct repeats normally results in a change in the number of components, taking knots to links and links to knots or a link with a higher number of components (Fig. 1.29). If the two sites are oriented in opposite directions, the sites are called inverted repeats (Fig. 1.30). The action of a recombinase on inverted repeats normally results in no change in the number of

44 26 Figure 1.27: An example of a site specific recombinase mechanism where the protein makes breaks one strand of the double helix, recombines it and then does the same with the other strand. components (Fig. 1.31). There are two families of site-specific recombinases: tyrosine recombinases and serine recombinases. Tyrosine recombinases break and rejoin one pair of DNA strands at a time (Figs. 1.27, 1.32). Serine recombinases introduce double-stranded breaks in DNA and then recombines them in some manner (Fig. 1.33) [43]. 1.4 Tangle Model In the 1980 s, Claus Ernst and DeWitt Sumners introduced a mathematical tangle model for protein bound DNA complexes [20]. One fundamental observation behind the tangle model is that pairs of DNA strands are often viewed as supercoiled through an electron microscope, and thus the DNA bound by a protein can be modeled with a rational tangle [20, 44,45]. There are two ways to use tangles to model protein action

45 27 Figure 1.28: Directed Repeats Figure 1.29: Recombinase action on direct repeats. Figure 1.30: Inverted Repeats Figure 1.31: Recombinase action on inverted repeats. Figure 1.32: Schematic of tryosine recombinase action: single stranded breaks. We model the tyrosine protein as a black ball while the double stranded DNA is modeled by red and blue rectangles. on DNA. One way is to view the protein as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. This divides the DNA into two parts: the portion of the DNA unbound by the protein, denoted U f, and the portion

46 28 Figure 1.33: Schematic of serine recombinase action: double stranded breaks. We model the serine protein as a black ball while the double stranded DNA is modeled by red and blue rectangles. of the DNA bound by the protein, denoted B (Fig. 1.34). The action of the protein is modeled by replacing the B tangle with the tangle that represents this part of the DNA after protein action, denoted E (Fig. 1.34). A second way to use tangles to model protein action on bound DNA is to observe that the action of the protein is on short sequences of DNA [45]. We then divide the DNA bound by the protein into two parts: the DNA at and very near the site where protein action occurs, denoted by P and the remaining DNA, bound by the protein but unchanged during protein action, denoted by U b (Fig. 1.34). We make the assumption that the action of a protein can be modeled by replacing the P tangle with the tangle that represents this part of the DNA after protein action, denoted R (Fig. 1.34). In both models we call the DNA before protein action DNA substrate and the DNA after protein action DNA product (Fig. 1.35). We have chosen to use the tangle model where the DNA substrate is N(U + B) and DNA product is N(U + E). We denote the action of replacing tangle B with tangle E as a (B, E) move (Eqn. (1.1) and Eqn. (1.2)).

47 29 Figure 1.34: Tangle Model N(U + B) = substrate equation (1.1) N(U + E) = product equation (1.2) Figure 1.35: A DNA molecule before protein action is called the DNA substrate; and after, the DNA product. A (B, E) move is mathematically equivalent to a (B, E ) move if and only if there exists a solution for U such that N(U + B) = K 1 and N(U + E) = K 2 if and only

48 30 if there exists a solution for U such that N(U + B ) = K 1 and N(U + E ) = K 2. The following definition [40] and theorem [16] shows how to calculate when moves are equivalent. Definition [40] Let E[x 1,..., x n ] be the Euler bracket function which equals the sum of all products obtained from the product 1 x 0 x 1 x n by omitting zero or more disjoint pairs of consecutive factors x i x j. For example E[x 0, x 1, x 2 ] = x 0 x 1 x 2 +x 0 +x 2. If n = 0 then E[x 1,..., x n ] = E[] = 1. If n < 0, define E[x 1,..., x n ] = 0. Theorem [16] Suppose f 1 g 1 = (c 1,..., c n ), n odd, where f 1 = E[c 1,..., c n ] and g 1 = E[c 1,..., c n 1 ]. Let e 1 = E[c 2,..., c n ], i 1 = E[c 2,..., c n 1 ]. If f 2 g 2 = (d 1,..., d m ), let t = (d w 1,..., d m c n, c n 1,..., c 1 ) and U = U (c n,..., c 1 ) (or equivalently, if t = w (b 1,..., b k ), let f 2 g 2 = te 1+wf 1 ti 1 +wg 1 = (b 1,..., b k + c 1,..., c n ), and U = U ( c 1,..., c n )), then for any pair of knots K 1, K 2, N ( U + f ) ( 1 = K 1 N U + f ) 2 = K 2 g 1 g 2 if and only if ( N U + 0 ) = K 1 N 1 ( U + t ) = K 2. w Hence an ( ) f 1 g 1, f 2 g 2 move is equivalent to a (0, g 1f 2 g 2 f 1 e 1 g 2 i 1 f 2 ) move, and similarly a (0, t ) move is equivalent to an ( f 1 w g 1, te 1+wf 1 ti 1 +wg 1 ) move. ( ) f Moreover if 1 g 1, f 2 g 2 is equivalent to (0, t ) then there exists e w 1 and i 1 such that g 1 e 1 f 1 i 1 = 1 and t w = g 1f 2 g 2 f 1 e 1 g 2 i 1 f 2 (or equivalently f 2 g 2 = te 1+wf 1 ti 1 +wg 1 ).

49 31 Example (Fig. 1.36) Let K 1 = N(U ) and K 2 = N(U 1 1 ). Let f 1 g 1 = 1 1 = (1) and f 2 g 2 = 1 1 = ( 1). Then, e 1 = E[] = 1, i 1 = 0 and t w = g 1f 2 g 2 f 1 e 1 g 2 i 1 f 2 = 2 1. To solve for U notice that c 1 = 1. Thus U = U (1). Figure 1.36: Mathematically equivalent moves: ( 1, ) ( 1 1 1, 0, ) Note that in the above example, we chose f 1 g 1 = (1) and f 2 g 2 = ( 1) and these particular rational expansions gave us a particular e 1 and i 1. We made this choice in order to use this example in latter work. We make note that many expansions are possible since 1 = g 1 e 1 f 1 i 1 = g 1 (e 1 + df 1 ) f 1 (i 1 + dg 1 ) for d Z, per the requirement of theorem For example, for the case where f 1 g 1 = 1 1, t w = g 1f 2 g 2 f 1 e 1 g 2 i 1 f 2 = g 1 f 2 g 2 f 1 (e 1 + df 1 )g 2 (i 1 + dg 1 )f 2. Hence ( 1, ) ( ) moves are mathematically equivalent to 0, g 1 f 2 g 2 f 1 1 (e 1 +df 1 )g 2 (i 1 +dg 1 )f 2 moves.

50 32 CHAPTER 2 BIOLOGICAL APPLICATION There are certain proteins, topoisomerase and recombinase, that can change the topology of DNA. Topoisomerases are proteins that can change the topological shape of DNA while keeping the genetic code unchanged. This work focuses on type II topoisomerase, a protein that breaks one segment of double stranded DNA and allows a second segment to pass through this break before resealing. This action is essential to regulating supercoiling in DNA, unknotting and unlinking DNA and preventing cell death [2, 3]. Recombinases are proteins that cut two segments of DNA and recombine them in some manner allowing for genetic diversity. These changes can inhibit or aid in biological processes involving the structure of DNA including replication and transcription. The local actions of both proteins can be modeled using knot theory; therefore, applications of knot theory to problems involving these proteins have been extensively studied [8, 14, 20, 44, 45, 47]. The chapter is split into two parts: the first half includes section 2.1 which develops a tangle model using the oriented skein triple for the local action of topoisomerase and site-specific recombinase on DNA. Using this model we state and prove Theorem that gives a correspondence between two 2-bridge knots that differ by a crossing change (topoisomerase action) and a link created from the oriented resolution of that crossing (recombinase action). This theorem is an extension of the results in [15 17, 46].

51 33 Theorem Let (K +, K, K 0 ) be an oriented skein triple. Assume that K +, K are 2-bridge knots and K 0 is a link. Then: K = N ( ) a b ( K + = N ( K 0 = N a 2p 2 b+2pqa b+2q 2 a 2pqb pb qa da jb ) ) #N ( ) p j where p is even or pqb is odd or (b and q are even). K = N ( ) a b ( K + = N ( K 0 = N a+2p 2 b 2pqa b 2q 2 a+2pqb a+p 2 b pqa b q 2 a+pqb ) where (pq is odd and b is even) or (q is even and b is odd). ) K + = N ( ) a b ( K = N ( K 0 = N a 2p 2 b+2pqa b+2q 2 a 2pqb a p 2 b+pqa b+q 2 a pqb ) where (pq is odd and b is even) or (q is even and b is odd). ) K + = N ( ) a b ( K = N ( K 0 = N a+2p 2 b 2pqa b 2q 2 a+2pqb pb qa da jb ) ) #N ( ) p j where p is even or pqb is odd or (b and q are even). where p and q are relatively prime integers, d and j are integers such that pd qj = 1 and p may be chosen to be positive. The last half of the chapter discusses a biological application of theorem In section 2.2 we define an experimental method known as difference topology [25, 36]. In section 2.3 we investigate difference topology experiments using a type II topoismerase to study two SMC proteins, 13S condensin and MukB [27, 37]. We conclude with section 2.4 which uses theorem to predict results from difference topology experiments using a site-specific recombinase to study 13S condensin and

52 34 MukB based on an analysis of the results from the experiments discussed in section Development of Our Tangle Model The tangle model developed in this section employs the skein relation, where K +, K, K 0 denote three links which differ only at a single crossing as indicated by the notation and seen in (Fig.2.1) Figure 2.1: Oriented knot triple: Triple of knots that differ only at one crossing. Recall from section 1.3 that DNA can take the form of a ring known as circular DNA. The trivial knot is often used to model circular unknotted DNA. Due to this perspective, a question arose: What triples can be found where one of the links in the triple is the trivial knot? Recall we denote the DNA before protein action as a DNA substrate and the DNA after protein action as a DNA product. By viewing one of the links in the triple as circular unknotted DNA substrate, the other links in the triple can then be viewed as products of topoisomerase and site specific recombinase action on unknotted DNA. This provides information on the types of knotted products each action can produce.

53 35 We first look at the skein relation and the tangle model associated to it. Recall that topoisomerases are proteins that cut one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be viewed as K K +. Recall that recombinases are proteins that cut two segments of DNA and recombine them in some manner. The actions of these proteins depends on the orientation of the protein binding sites. When these sites are oriented in the same direction, the sites are referred to as direct repeats, (Fig. 1.28). If the two sites are oriented in opposite directions, then the sites are referred to as inverted repeats (Fig. 1.30). The action of recombinase on direct repeats results in a change in the number of components. The action of recombinase on inverted repeats results in no change in the number of components. While the local action of recombinase proteins can differ, most are mathematically equivalent to resolving a crossing (Fig. 2.3). Notice that recombinase action on direct repeats is an orientation preserving action, while recombinase action on inverted repeats ignores orientation. We model the action of recombinase on direct repeats as K ± K 0 and model the action of recombinase on inverted repeats as K ± K 0i, where K 0i denotes recombinase action product on inverted repeats. We then view the oriented skein triple as K ± = circular DNA substrate, K = product of topoisomerase action, and K 0 = product of recombinase action on direct repeats. We also consider K 0i, the product of recombinase action on inverted repeats.

54 36 Figure 2.2: action. Model of topoisomerase Figure 2.3: Model of recombinase action Application of the Skein Triple Skein relations are often used to give a simple definition of knot polynomials. The traditional oriented skein relation relates a knot, K +, with a distinguished positive crossing; a knot K, obtained by changing the distinguished positive crossing of K + to a negative crossing; and a link K 0, the oriented resolution of the distinguished crossing (Fig. 2.1). Recall from section 1.4 that we can use tangles to model DNAprotein interactions. We choose a projection of our links such that the orientation of our triple matches the orientation in Fig The following system of tangle equations with orientation shown in Fig. 2.4 represents K +, K, and K 0 : K + = N K = N K 0 = N ( U 1 ) 1 ( U + 1 ) 1 ( U + 1 ) 0 (2.1) (2.2) (2.3) Since the links in our triple differ at only one crossing, U denotes the portion of the links that are equivalent while the remaining portion represents the difference in the crossing (Fig 2.4).

55 37 Since we are projecting three dimensional objects into two dimensional space, our choice of projection is arbitrary. We have chosen a projection of the knot K + such that the distinguished crossing matches the diagram for K + in Fig. 2.1 (Fig. 2.4). All possible projections are topologically equivalent to each other and will provide the same model results. For example in Fig. 2.5 we see a projection that is a 90 o rotation of the tangles in Fig Figure 2.4: Diagram associated with equations (2.1),(2.2),(2.3) respectively. Figure 2.5: An example of a topologically equivalent tangle model to Fig We solve this system of equations for the case when K + and K are 2-bridge knots, as these are the most common knot products from protein action on circular

56 38 DNA substrates. Recall that 2-bridge links, N ( m n ), are the numerator closure of rational 2-tangles. We will discuss an example of site specific recombinase action resulting in non 2-bridge links in chapter 4. For the 2-bridge case, we use results from [15, 16]. In [15 17,46] 2-bridge knots that are related by a crossing change, i.e. K + and K, were investigated. The work in this section extends the investigation by also considering K 0, the oriented resolution of the crossing relating K + to K, using the following results in [16, 19]: Theorem (Theorem 1 [16]). If N ( U + 0 1) = N ( a b ) and N ( U + t w) = N ( z v and if t > 1, then U is a generalized Montesinos tangle or equivalently, U = A C where A is a finite sum of rational tangles and C is a rational tangle. Theorem (Theorem 3 [16]). N ( U + 0 1) = N ( a b ) and N ( U + t w) = N ( z v where N ( a b ) and N ( z v) are unoriented 2-bridge links and U is a generalized Montesinos tangle if and only if the following hold: ) ) (a) If w ±1 mod t, then there exists an integer b such that b b ±1 = 1 mod a, and for any integers x and y such that b x ay = 1, ( z ) N = N v ( ) tb + wa ty + wx (2.4) In this case, U = a b for all b satisfying the above. (b) If w = ±1 mod t, then there exists relatively prime integers p and q, where p

57 39 may be chosen to be positive, such that ( z ) ( ) tp(pb qa) ± a N = N v tq(pb qa) ± b ( ) In this case, the solutions for U da jb are + j (h, 0) and pb qa p (2.5) ( ) j + da jb (h, 0) p pb qa for all p, q satisfying the above, d and j are any integers such that pd qj = 1 and h = w±1 t where the ± sign agrees with that in (2.5) (note, the choice of j and d such that pd qj = 1 has no effect on U ). To extend theorem to triples, we adjust our model slightly. As we are looking at a system of three equations, notice that we can use a ( 1, ) move to model the action of the crossing change, i.e. going from K ± to K. For theorem 2.1.2, we need a ( 0, t 1 w) move that is mathematically equivalent to a ( 1, ) ( move. From example we have that a 1, ) move is equivalent to a ( 0, ) move. Thus we have that: ( K ± = N U + 0 ) 1 ( K = N U 2 ) 1 Recall from example that U = U 1. Using this information, we adjust the equation for K 0 in our triple, giving us equation 2.8, see the bottom row of Fig Now, assuming we have oriented 2-bridge links, equations 2.1, 2.2, 2.3 are now viewed as the mathematically equivalent equations 2.6, 2.7, 2.8, shown in Fig Since Fig. 2.5 represents a topologically equivalent model to Fig 2.5, we will also find mathematically equivalent equations to 2.9, 2.10, 2.11 using U = U 1, shown in Fig. 2.7.

58 40 K + = N K = N K 0 = N ( U 2 ) 1 ( U + 0 ) 1 ( U + 1 ) 0 (2.6) (2.7) (2.8) K + = N K = N K 0 = N ( U + 0 ) 1 ( U 2 ) 1 ( U 1 ) 1 (2.9) (2.10) (2.11) Notice if we assume the substrate equation is K ± = N ( U + 1) 0, equations 2.9 and 2.10 model topoisomerase action going from K + K and equations 2.7 and 2.6 model topoisomerase action going from K K +. We will assume this substrate equation for the duration of the thesis. Figure 2.6: The top two rows show a schematic of one direction for the equivalence between the (1, 1) move and the (0, 2) move on oriented links in figure 2.4 such that U = U (1). Using U, we can find the conversion for equation 2.8, shown in the bottom row. For the ( 0, ) move, we have t = 2 > 1. Therefore, from theorem 2.1.1, U

59 41 Figure 2.7: The top two rows show a schematic of one direction for the equivalence between the (1, 1) move and the (0, 2) move on oriented links in figure 2.5 such that U = U (1). Using U, we can find the conversion for equation K 0, shown in the bottom row. is a generalized Montesinos tangle. With our new modified model, assuming that K ± = N ( ) a b and K = N ( z v) are 2-bridge knots, we solve for z, v and K 0 in terms of a and b using results from [16]. This is biologically equivalent to knowing the DNA substrate and solving for possible products of topoisomerase and site-specific recombinase action on the known DNA substrate. Theorem provides a formula for a 2-bridge knot, N( z ), in terms of a 2-bridge v knot, N( a), that is related to b N(z ) via a crossing change. It also provides a formula v for calculating U. We are assuming that K + and K are 2-bridge knots and we have that U is a Montesinos tangle. For the general case where U is any generalized Montesinos tangle (which includes

60 42 the rational case), we reference theorem [16]. Recall that for our modified model t w = 2 1, thus w = 1 mod t. From theorem 2.1.2b we obtain the following result: U = ( da jb pb qa + j ) p (h, 0) or ( ) j da jb + (h, 0) (2.12) p pb qa where d and j are integers such that pd qj = 1 and h = w±1 t = 1±1 2 where the ± sign matches equation (2.5). Theorem gives us the equation for K. We use U, in equation 2.12, and substitute into equations 2.8 and 2.11 to solve for K Let h = = 0. Then U = ( ) da jb + j (0, 0) = pb qa p ( ) da jb + j. Substituting pb qa p U into equation 2.8, we get K 0 = N ( U + 1 ) 0 = N ( da jb pb qa + j p + 1 ) 0 Note that a 90 o rotation of tangle n m is equal to tangle m n. Thus, ( ) ( pb qa K 0 = N #N p ). da jb j Substituting U into equation 2.11, we get K 0 = N ( U 1 ) 1 ( da jb = N pb qa + j p 1 ) 1 ( da jb = N pb qa + j p ) p ( ) pda pjb + pjb jqa p 2 b + pqa = N qda qjb + dpb dqa qpb + q 2 a ( ) a p 2 b + pqa = N b + q 2 a pqb Thus, K 0 = N ( ) a p 2 b + pqa. b + q 2 a pqb

61 2. Let h = = 1. Then U = 2.8, we get 43 ( ) da jb + j (1, 0). Substituting U into equation pb qa p K 0 = N ( U + 1 ) 0 (( da jb = N pb qa + j ) (1, 0) + 1 ) p 0 ( da jb = N pb qa + j p + 1 ) 1 ( ) pda pjb + pjb jqa + p 2 b pqa = N qda qjb + dpb dqa + qpb q 2 a ( ) a + p 2 b pqa = N b q 2 a + pqb Thus, K 0 = N ( ) a + p 2 b pqa. b q 2 a + pqb Substituting U into equation 2.11, we get K 0 = N ( U 1 ) 1 (( da jb = N pb qa + j ) (1, 0) 1 ) p 1 ( da jb = N pb qa + j p + 1 ) 0 This is the same as the first case, thus ( ) ( pb qa K 0 = N #N p ). da jb j Alternatively, we can use U to solve for U. Then, we can substitute U into equation 2.3 to solve for K 0 in that equation. The two formulas for K 0 will match. Since we are looking at the action of recombinase on direct repeats, we want K 0 to be a link. Note that N ( m n ) is a link if and only if m is even [11]. We now have the following theorem:

62 44 Theorem Let (K +, K, K 0 ) be an oriented skein triple. Assume that K +, K are 2-bridge knots and K 0 is a link. Then: K = N ( ) a b ( K + = N ( K 0 = N a 2p 2 b+2pqa b+2q 2 a 2pqb pb qa da jb ) ) #N ( ) p j where p is even or pqb is odd or (b and q are even). K + = N ( ) a b ( K = N ( K 0 = N a 2p 2 b+2pqa b+2q 2 a 2pqb a p 2 b+pqa b+q 2 a pqb ) where (pq is odd and b is even) or (q is even and b is odd). ) K = N ( ) a b ( K + = N ( K 0 = N a+2p 2 b 2pqa b 2q 2 a+2pqb a+p 2 b pqa b q 2 a+pqb ) where (pq is odd and b is even) or (q is even and b is odd). K + = N ( ) a b ( K = N ( K 0 = N a+2p 2 b 2pqa b 2q 2 a+2pqb pb qa da jb ) ) ) #N ( ) p j where p is even or pqb is odd or (b and q are even). where p and q are relatively prime integers, d and j are integers such that pd qj = 1 and p may be chosen to be positive. This theorem gives a relationship between two 2-bridge knots that differ by a crossing change (topoisomerase action) and a link created from the oriented resolution (recombinase action on directed repeats) of that crossing in terms of one of the knots in the triple. Note that if j or da jb = 1 then, K 0 is a 2-bridge link. The formula in theorem is consistent with a more complicated formula in [15]. Note: The formula in theorem still works for recombinase action on inverted repeats. Recall that action on inverted repeats is equivalent to a one component knot

63 45 product. If the requirements for b, p, and q in theorem do not hold, K 0 is a knot and thus corresponds to K 0i. We are currently writing a program in C that tabulates these oriented knot triples using theorem We use theorem to study an experimental method known as difference topology. 2.2 Difference Topology We use the model of skein triples for protein action to study the experimental method called difference topology. Difference topology is a technique used to find the conformation of the DNA bound by a protein. The protein under study binds to circular DNA in an interesting conformation but does not change the topology of the DNA [25, 36]. While this first protein is bound to DNA, a second protein, usually a site-specific recombinase, is added to the reaction (Fig. 2.8). After the site-specific recombinase action has taken place, the resulting product is knotted or linked where some of the DNA crossings bound by the first protein may be trapped. The first protein only binds to DNA and makes no change to the topology of the DNA substrate. Thus the conformation of the bound DNA will be lost when the protein releases the DNA (left side of Fig. 2.8). The second protein is used to trap some of the crossings bound by the first protein (right side of Fig. 2.8). In Fig. 2.8 the blue ball represents the protein that binds to the circular DNA substrate in an interesting conformation. The pink ball models the second protein, a site-specific recombinase, that makes topological changes to the DNA substrate, trapping some

64 46 of the crossings resulting in a linked product. Figure 2.8: Difference topology experiment. Used with permission from [13] The products that result from difference topology may differ from the products of the second protein acting on unbound DNA [25,36]. The types of knotted products depends on how the first protein binds to DNA and where the second protein acts on the bound substrate. Difference topology experiments using a site-specific recombinase may need many time consuming experiments since the DNA substrates need to contain specific recombination sites at various locations. Because topoisomerase is not site specific, we propose similar difference topology experiments using a type II topoisomerase as the secondary protein, in tandem with mathematical analysis via theorem The information gained from this analysis will aid in determining if difference topology experiments using a type II topoisomerase are sufficient for determining the conformation of the bound DNA substrate and/or if combining these topoisomerase results with fewer recombinase experiments will suffice. In [27, 37] difference topology experiments using type II topoisomerase were conducted to study two types of SMC proteins, 13S condensin and MukB. In the next section we will discuss the results of these experiments.

65 SMC Proteins Structural maintenance of chromosomes (SMC) proteins are a large family of proteins that play a role in the structure and function of chromosomes. Since their discovery, certain features of SMC proteins have attracted researchers. The first is the various chromosome functions in which they participate. These proteins participate in different aspects of higher-order chromosome organization and dynamics. The second feature is their protein structure. In solution, SMC proteins form a V shape composed of two molecules joined via their hinge domains (Fig. 2.9) [37]. The configuration of these proteins provides an opportunity for binding to both DNA and ATP. To carry out their function, they use energy gained by catalyzing the decomposition of ATP into ADP. Recent focus has been on the mechanisms of this particular family. Type I topoisomerase introduce supercoils in SMC protein-bound DNA in the presence of ATP [27,37]. Type II topoisomerase produce positive three nodded knots acting on SMC protein bound DNA [27,37]. We will examine experimental data from two members of this family: 13S condensin and MukB S Condensin Experiments Kimura et al. studied how the addition of an SMC protein known as 13S condensin can change the conformation of DNA [27]. Type I topoisomerase introduces positive supercoils to 13S condensin bound circular DNA in the presence of ATP. They hypothesized three possible mechanisms for the addition of these supercoils (Fig.

66 48 Figure 2.9: General structure of an SMC protein, showing the five distinct domains along with alternative models of the structure involving two SMC proteins. Used with permission from [10]. 2.10). Recall from chapter 1, the conservation law stating that there is a fixed relationship between linking number, twist and writhe: Lk(R) = Tw(R) + Wr(R). Since the linking number is fixed for a closed (non-nicked) circular DNA substrate, any change in twist will result in compensating change in writhe, and vice versa. In model I, it is hypothesized that the 13S condensin overwinds the DNA at the binding sites, increasing the helical twist, resulting in the addition of negative writhe since linking number is preserved (Fig. 2.10, left) [27]. Type I topoisomerase is then added to the reaction. Recall type I topoisomerase changes the twist of circular DNA by cutting a the backbone of the DNA allowing the other backbone to pass through before resealing (Fig. 1.25). In model 1, type I topoisomerase changes the linking number by adding negative twists. This change in the linking number is then transferred to a positive change in writhe, canceling out the negative writhe. Once all the proteins are removed, the positive twists that was induced by by 13S condensin are converted to positive supercoils.

67 49 In model II, it is hypothesized that the positive supercoils in the 13S condensin bound DNA are a result of the right-handed wrapping of DNA around the 13S condensin thus increasing the writhe (Fig. 2.10, center). Conserving the linking number, negative writhe is created to compensate. As before, relaxation of the negative writhe occurs by the addition of a type I topoisomerase, resulting in positive supercoils after the protein is removed. While the previous two models act locally on the DNA, changing the DNA only near the binding sites, model III is viewed as introducing global positive writhe producing a positive supercoiled loop in the DNA (Fig. 2.10, right) [27]. As in the first two models, the compensating negative writhe is relaxed via the action of a type I topoisomerase. Notice that in each model, the 13S condensin bound DNA results in positively supercoiled DNA after a type I topoisomerase protein relaxes compensating negative supercoils. Figure 2.10: Three models for the possible conformation for 13S condesin bound DNA. DNA is modeled as a double line and 13S condensin is schematized as a V shaped object. From left to right: Model (I) shows a (+) twist, Model (II) shows a (+) wrap, adding positive writhe locally, and Model (III) shows the (+) global writhe. Figure reproduced with permission from [27].

68 50 Figure 2.11: The three models of each conformation from figure 2.10 for 13S condensin on nicked DNA. Figure reproduced with permission from [27]. Each model is biologically possible, as discussed in [27]. Because type II topoisomerase produces knotted products in the presence of 13S condensin bound to nicked DNA, it was concluded that conformation III must be the correct hypothesized conformation as explained below [27]. Recall from our discussion of difference topology that the products resulting from topoisomerase II action on 13S condensin bound DNA may be different than products that result from topoisomerase II action on unbound DNA substrates. Type II topoisomerase is not site specific and its function is specifically to unknot DNA. An experiment was performed with circular DNA substrates in the presence of type II topoisomerase and no 13S condensin. In this experiment, only unknots were detected [27]. If the 13S condensin action is only local, then the products of type II topoisomerase action on those protein bound substrates would be similar. For model III, however, the probability of knotting by a type II topoisomerase on this conformation increases due to global writhe [27]. The action of topoisomerase on DNA can trap the crossing in this conformation, seen in Fig right, and possibly other random crossings, increasing the probability of knotted products. The

69 51 knotted products of type II topoisomerase action on nicked circular DNA bound by 13S condensin were experimentally discovered to be overwhelmingly positive trefoils (Table 2.1). Number Found Knot Type Exp. 1 Exp. 2 Total (+)3-crossing ( )3-crossing crossing (+)5-crossing torus (+)5-crossing twist ( )5-crossing twist (+)6-crossing granny Table 2.1: Exp. 1 and Exp. 2 data. Table 2.1 shows the number of knotted products identified via electron microscopy from difference topology experiments using type II topoisomerase to study 13S condensin bound to nicked DNA, used with permission from [27] In experiment 1 (Exp.1), the substrate was nicked 3.0 kb circular DNA. The DNA substrate was incubated for 60 minutes at 22 o C with purified 13S condensin, phage T2 type II topoisomerase, and ATP. The DNA was then deproteinized and analyzed

70 52 for knots using gel electrophoresis and a reference knot ladder. Gel electrophoresis is a laboratory technique which sorts DNA molecules based on their size. These molecules are separated by applying an electric field to move the molecules, which are negatively charged, through a porous gel. Smaller and/or more compact DNA molecules move faster and migrate farther than longer and/or less compact ones because smaller molecules migrate more easily through the pores of the gel. These differences in migration speed create bands in the gel, as molecules of similar size travel together. DNA knots are separated by crossing number, where knots with higher crossing number (being more compact) move faster than those of lower crossing number. A reference knot ladder is used to determine which bands are associated to which knots, grouped by crossing number. As measured by gel electrophoresis, 12% of the DNA products in Exp. 1 were knotted, with a large majority being three-crossing knots. To better determine knot type, the experiment was repeated using nicked 7.0 kb circular DNA. The DNA, extracted from the bands resulting from gel electrophoresis, was treated with Rec A protein. Rec A protein is a protein that binds to DNA, thickening the DNA strands and aiding in distinguishing overcrossings and undercrossings using electron microscopy. The DNA is then viewed under an electron microscope and the knot types of many DNA knots were determined (Table 2.1, Exp. 1). The experimental results showed that the majority of the knots were positive trefoils (Table 2.1, Exp. 1). A very small number of higher crossing number knots were found, among those were 4-crossing knots, 5-crossing torus and twist knots and

71 53 6-crossing granny knots, which are the connected sum of two 3 1 knots. 5-crossing torus and 6-crossing granny knot products require that type II topoisomerase act at least twice on the 13S bound unknotted DNA substrate [26, 27]. Experiment 2 (Exp. 2) was performed to reduce the probability of multiple actions of type II topoisomerase, i.e more than one crossing change, on the 13S condensin bound DNA. The substrate was once again nicked 7.0 kb circular DNA. In Exp. 2 the incubation time for the 13S condensin with the DNA substrate was 1 hour. The type II topoisomerase was added for an additional 5 minutes. The majority of the products in Exp. 2 were once again positive trefoils, yet still a small number of 5-crossing torus knots and 6-crossing granny knots were present (Table 2.1, Exp. 2). This gave rise to a discussion in [27] about the possibility of producing knots of this type with more that one 13S condensin bound to the DNA substrate, trapping more supercoils. Yet, even in the case of multiple 13S condensins bound to the DNA substrate, type II topoisomerase would still have to act twice to trap these supercoils. Relative to the percentage of 3-crossing knots, these higher crossing knots are a minor product for both Exp. 1 and Exp. 2. Thus, we will not include these knotted products in our analysis of difference topology experiments using type II topoisomerase to study 13S condensin. 5-crossing twist knots and 4-crossing knots are possible products of extra crossings being trapped. Recall type II topoisomerase is not a site-specific protein. Thus there may be extra crossings trapped by the action of these proteins since it cannot be controlled experimentally where type II topoisomerase action will occur. 5-crossing

72 54 twist knots can also be the result of a type II topoisomerase acting twice on the unknotted DNA substrate. Since these knots are minor products in this experiment, we do not include 5-crossing twist knots and 4-crossing knots in our analysis of 13S condensin difference topology experiments. Thus, the products from both Exp. 1 and Exp. 2 are overwhelmingly positive trefoils. We analyze this experimental data mathematically in subsection 2.4 using theorem to determine if difference topology experiments using a site-specific recombinase are needed to study 13S condensin further MukB Experiments Similar experiments were performed with the bacterial SMC protein MukB [37]. MukB was the first SMC protein discovered. It forms a complex with two other non-smc proteins, MukE and MukF, that is essential to the process of chromosome partitioning in E.coli [55]. Experiments performed using MukB in the presence of type II topoisomerase and nicked DNA also produced a majority of right handed trefoils. Differences between experiments on 13S condensin bound DNA and experiments involving MukB bound DNA include that MukB does not require ATP to change the conformation of DNA and the net supercoiling generated by MukB bound DNA in the presence of a type I topoisomerase is negative [37]. The is also some evidence that type II topoisomerase action on MukB bound nicked DNA substrates can yield 3 and 4-crossing knots [37]. Recall that type I topoisomerase introduces positive supercoils to 13S condensin

73 55 bound DNA in the presence of ATP. While the majority of the knotted products of MukB bound DNA in the presence of type II topoisomerase were positive crossing knots, the following experiment was conducted on MukB bound DNA with two types of type I topoisomerase, type IA and type IB to show that type I topoisomerase introduces negative supercoils to MukB bound DNA [37]. Type I topoisomerases cut one strand of a DNA double helix, allowing for relaxation to occur. Type IA topoisomerase binds and breaks a 5 phosphate on the DNA backbone and can relax negatively supercoiled DNA [9]. Type IB topoisomerase binds and breaks a 3 phosphate on the DNA backbone and can relax both negatively and positively supercoiled DNA [9]. If MukB traps a positive supercoil, the generated compensatory negative supercoil can be removed by either type IA or type IB topoisomerase [37]. If MukB traps a negative supercoil in DNA, only type IB topoisomerase would be able to remove the compensatory positive supercoil. Experiments were executed with MukB bound DNA substrates in the presence of each of these type I topoisomerase independently. The products of these experiments were analyzed using gel electrophoresis. It was observed that in type IA topoisomerase experiments, there was no change in supercoiling. It was noted that the type IB topoisomerase was able to relax the supercoils of the MukB bound DNA substrate [37]. Thus, the generated net supercoiling was positive. This result was also confirmed by 2D gel electrophoresis [37]. As before with 13S condensin, experiments to test the knotted products of type II topoisomerase action on MukB bound DNA were performed. In one experiment

74 56 the substrate, nicked 4.4kb circular DNA, was incubated for 30 minutes at 37 o C with purified MukB and type II topoisomerase. The DNA products were then analyzed using gel electrophoresis. A second experiment was performed with the same conditions and the same substrate, nicked 4.4kb DNA as in the first experiment. After terminating the reaction, the DNA products were then deproteinized, coated with RecA protein and viewed under an electron microscope. As with the 13S condensin experiments, the majority of the knotted products in both experiments were positive 3-crossing knots [37]. They also found a fair amount of four and five crossing knots in both experiments [37]. It has been previously mentioned that one type of 5-crossing knot may result from type II topoisomerase acting twice on the protein bound substrate. This hypothesis is strengthened by a MukB difference topology experiment where there was an increase in five crossing knots when the amount of type II topoisomerase was doubled in the reaction [37]. The other 5-crossing knot product may also be the result of the action of type II topoisomerase trapping extra crossings. It should be noted that fewer knots are produced with higher concentrations of MukB. This is most likely caused by the extensive binding of MukB to the DNA substrate, thus restricting access of the topoisomerase to the DNA [37]. A significant amount of four crossing products gives that geometry plays a role in the knotting of the SMC bound DNA substrate [37]. In [27] the authors mention that the 13S condensin bound substrates yield few 4-crossing knots in the presence of a type II topoisomerase while MukB bound substrates yield a significant amount of

75 57 4-crossing knots [37]. We can use results from the KnotPlot plug-in TopoICE to give information on the possible geometry of the SMC bound substrate [42]. We analyze the 3 and 4-crossing products in the next section. We now use theorem to mathematically analyze the experimental results discussed in subsections and 2.3.2, type II topoisomerase difference topology experiments studying SMC proteins. 2.4 Application of Our Model Using data from the experiments discussed in subsection and subsection 2.3.2, including chirality information, we use theorem to predict potential products that could result from difference topology experiments using a site specific recombinase in place of the type II topoisomerase. We have seen from the experiments in subsection2.3.1 and subsection that the primary knotted products of difference topology experiments using a type II topoisomerase are the 3-crossing knot. For MukB, our experimental data also provides a third product: 4-crossing knots. Analyzing the experimental data in [37] and [27], there was a significant number of circular unknotted DNA still present. We will included this in our analysis as a possible products: SMC protein bound nicked circular DNA substrate: K ± = 0 1 Difference topology experiment product using a type II topoisomerase as the secondary protein: K = 0 1, 3 1 or 4 1 Difference topology experiment product using a site specific recombinase as the

76 58 secondary protein: K 0 =? We will split our analysis into two cases: Case 1: K ± = 0 1 and K = 0 1. Using our tangle model from theorem 2.1.3, the previous information translates into the following equations, dependent on the orientation and the type II topoisomerase product: ( a ) K ± = N = 0 1 (2.13) b If ( z ) K ± = N = N v ( ) a + 2p 2 b 2pqa = 0 b 2q 2 1 (2.14) a + 2pqb then, ( ) ( ) e a + p 2 b pqa K 0 = N = N =? f b q 2 a + pqb ( ) ( pb qa K 0 = N #N p ) =? da jb j (2.15) If ( z ) K + = N = N v ( ) a 2p 2 b + 2pqa = 0 b + 2q 2 1 (2.16) a 2pqb then, ( ) ( ) e a p 2 b + pqa K 0 = N = N =? f b + q 2 a pqb ( ) ( pb qa K 0 = N #N p ) =? da jb j (2.17) We begin by solving for p and q and use those values to solve for K 0. We choose, without loss of generality, that a = 1 and b = 0. This is a valid choice because N ( a b ) = 01.

77 59 Using theorem we are solving the following equation for p and q: ( z ) K = N = N v ( ) a + 2p 2 b 2pqa = N b 2q 2 a + 2pqb ( ) pq = ±1 2q 2 = ±0 mod 1 From theorem 2.1.2, we assume p > 0. Thus p = 1 and q = 1 or p = 1 and q = 0. We solve the equations in 2.15 with the values we found for p and q to find possible link solutions for K 0.. Equation 1: K 0 = N ( ) e = N f ( ) a + p 2 b pqa b q 2 a + pqb Analyzing the solutions for p and q using theorem 2.1.3, we see that p = 1 and q = 0 is not valid solution in this case. Thus we will only use p = 1 and q = 1 as values for p and q. K 0 = N ( ) e f ( ) a + p 2 b pqa = N b q 2 a + pqb ( ) 0 = N = Equation 2: ( ) ( pb qa K 0 = N #N p ) da jb j Analyzing the solutions for p and q using theorem 2.1.3, we see that p = 1 and q = 1 is not valid solution in this case. Thus we will only use p = 1 and q = 0 as values for p and q. ( ) pb qa K 0 = N #N da jb ( ) ( 0 = N #N 1 ) d j ( p ) j

78 60 Note this connected sum is an unlink. We have that the possible rational link solutions for this equation are N ( 0 1). Therefore, K0 = Using theorem we are solving the following equation for p and q: ( z ) K + = N = N v ( ) a 2p 2 b + 2pqa = N b + 2q 2 a 2pqb ( ) pq = ±1 2q 2 = ±0 mod 1 From theorem 2.1.2, we assume p > 0. Thus p = 1 and q = 1 or p = 1 and q = 0. We solve for e and f to find possible link solutions for K 0. We solve the equations in 2.17 with the values we found for p and q. Equation 3: K 0 = N ( ) e = N f ( ) a p 2 b + pqa b + q 2 a pqb Analyzing the solutions for p and q using theorem 2.1.3, we see that p = 1 and q = 0 corresponds to a knot, not a link. Thus we will only use p = 1 and q = 1 as values for p and q. K 0 = N ( ) e f ( ) a p 2 b + pqa = N b + q 2 a pqb ( ) 0 = N = Equation 4: ( ) ( pb qa K 0 = N #N p ) da jb j Analyzing the solutions for p and q using theorem 2.1.3, we see that p = 1 and q = 1 corresponds to a knot, not a link. Thus we will only use

79 61 p = 1 and q = 0 as values for p and q. ( ) pb qa K 0 = N #N da jb ( ) ( 0 = N #N 1 ) d j ( p ) j Note this connected sum is an unlink. We have that the possible rational link solutions for this equation are N ( 0 1). Therefore, K0 = Then, the possible products of site specific recombinase on an SMC protein bound DNA substrate are the trivial link when the product of topoisomerase II action on the SMC protein bound DNA is the unknot. Using the possible values for p and q in each case, we can solve for the Montesinos tangle U. Notice that for the model, equations 2.6, 2.7, 2.8, U = ( da jb pb qa + j ) (h, 0) or p ( ) j da jb + (h, 0) p pb qa where pd qj = 1. Recall that a = 1 and b = 0. Thus, using the p s and q s calculated using theorem 2.1.3, for h = 0 and h = 1 the possible conformations for U for the equations associated to the topoisomerase action product of the unknot are the tangles U = (1), ( 1 0). Case 2 : K ± = 0 1 and K = 3 1. Using our tangle model from theorem 2.1.3, the previous information translates into the following equations, dependent on the orientation and the type II topoisomerase product: ( a ) K ± = N = 0 1 (2.18) b

80 62 If ( z ) K ± = N = N v ( ) a + 2p 2 b 2pqa = 3 b 2q 2 1 (2.19) a + 2pqb then, ( ) ( ) e a + p 2 b pqa K 0 = N = N =? f b q 2 a + pqb ( ) ( pb qa K 0 = N #N p ) =? da jb j (2.20) If ( z ) K + = N = N v ( ) a 2p 2 b + 2pqa = 3 b + 2q 2 1 (2.21) a 2pqb then, ( ) ( ) e a p 2 b + pqa K 0 = N = N =? f b + q 2 a pqb ( ) ( pb qa K 0 = N #N p ) =? da jb j (2.22) We begin by solving for p and q and then using those values to solve for K 0. We make the choice, without loss of generality, that a = 1 and b = 0. This is a valid choice because N ( a b ) = 01. Using theorem and equation 2.19, we are solving the following equation: ( z ) K = N = N v ( ) a + 2p 2 b 2pqa = N b 2q 2 a + 2pqb ( ) pq = ±3 2q 2 = ±1 mod 3 From theorem 2.1.2, we assume p > 0. Thus p = 1 and q = 1. Again using these values for p and q, we solve for e and f to find possible rational link solutions for K 0. We use the first equation in the set of equations in 2.20, as it is paired in theorem with the equation we used to solve for p and q.

81 63 K 0 = N ( ) e f ( ) a + p 2 b pqa = N b q 2 a + pqb ( ) 2 = N = Thus our possible link solution is 2 2 1, the Hopf link with linking number 1. For other possible rational link solution for this case, notice that since p and q are both odd, they corresponds to a knot, not a link, for our other equation for K 0 Using theorem and the equation in the set of equations in 2.21, we are solving the following equation: ( z ) K + = N = N v ( ) a 2p 2 b + 2pqa = N b + 2q 2 a 2pqb ( ) pq = ±3 2q 2 = ±1 mod 3 From theorem 2.1.2, we assume p > 0. Thus p = 1 and q = 2 or p = 2 and q = 1. Again using these values for p and q, we solve to find possible rational link solutions for K 0. We use the third equation in the set of equations in 2.22, as it is paired in theorem with the equation we used to solve for p and q. ( ) ( pb qa K 0 = N #N p ) da jb j ( ) ( 2 = N #N 1 ) d j ( ) 2 = N d

82 64 Thus our possible link solution is 2 2 1, the Hopf link with linking number 1. Therefore the possible product of site specific recombinase action on an SMC protein bound DNA substrate is the Hopf link with linking number 1 when the product of topoisomerase II action on the SMC protein bound DNA is the positive three crossing knot, 3 1. Using the possible values for p and q in each case, we can solve for the Montesinos tangle U. Notice that for the model, equations 2.6, 2.7, 2.8, U = ( da jb pb qa + j ) (h, 0) or p ( ) j da jb + (h, 0) p pb qa where pd qj = 1. Recall that a = 1 and b = 0. Thus, using the p s and q s calculated using theorem 2.1.3, for h = 0 and h = 1 the possible conformations for U associated to the topoisomerase action product being 3 1, we have that U = ( 1 2). To solve for U from the original model, equations 2.1, 2.2, 2.3, recall U = U 1. Therefore U = (0), ( 1 0) for the unknot product and U = ( 1 2) for the 31 product. From the MukB difference topology experiments, we get a possible third product, 4-crossing knots. Analyzing the 4-crossing type II topoisomerase products of difference topology using TopoICE provides two possible conformations for the unbound DNA loops for SMC bound DNA substrates: U = ( ( 2) 1 and 1 3). Comparing these results with the analysis of the 3 1 products, recall U = 1 for the knotted product. Thus we have a difference in the sign of the crossings, providing a possible difference in the binding conformation. Using this information, the possible 2-bridge link so-

83 65 lution when the topoisomerase products for the difference topology experiments are 4-crossing knots is K 0 = 2 2 1, the Hopf link with linking number +1. Thus since type II topoisomerase action on MukB bound DNA substrates yields a significant amount of 4-crossing knots, this gives insight on the possible conformations of the MukB bound substrates, but more analysis using difference topology experiments with site-specific recombinases should be done Discussion Recall that we now view the oriented skein triple as K ± = circular DNA substrate, K = product of topoisomerase action and, K 0 = product of recombinase action on direct repeats. We have chosen a projection of our DNA substrate such that we can model our oriented skein triple with the following systems of equations that depend on the orientation of our recombination sites and model a (0, 2) move: ( K + = N U 2 ) ( K + = N U + 0 ) 1 1 ( K = N U + 0 ) ( K = N U 2 ) 1 1 K 0 = N ( U + 1 ) 0 K 0 = N ( U 1 ) 1 Using this model and theorem 2.1.3, we have shown that the only possible 2- bridge products of difference topology experiments using a site specific recombinase to study a SMC protein are the 0 2 1, the unlink when the products of difference topology experiments using a type II topoisomerase to study a SMC protein are the unknot. We have also seen that if the products of difference topology experiments using a type II topoisomerase are the 3 1 knot, then the possible product of difference topology ex-

84 66 periments using a site specific recombinase to study a SMC protein are 2 2 1, the mirror image of the Hopf link. Recall that the link products are from a site specific reaction that preserves the orientation given to the recombination sites, i.e. recombinase action on direct repeats. Also recall that we proved that a (0, 2) move is mathematically equivalent to a (1, 1) move. Since these are mathematically equivalent moves, equations 2.1, 2.2, 2.3 could have been utilized to find these possible site specific recombinase products, using that U = U 1. We analyzed 4-crossing type II topoisomerase products of difference topology using TopoICE. While [37] states that there is an insignificant number of these products for 13S condensin bound DNA substrates, type II topoisomerase action on MukB bound substrates yields a significant amount of 4-crossing knots. This information gives us insight on differences in the geometry of the unbound DNA loops for the SMC bound DNA substrates. Since the percentage of 4-crossing type II topoisomerase products of difference topology with 13S condensin is negligible when compared to the percentage of 3- crossing knot products (see Table 2.1). We do not included these products in our analysis of difference topology experiments on 13S condensin. Thus, the possible geometry of the unbound DNA loops for the 13S condensin bound DNA substrates: U = (1), ( ) 1 0, from the 01 topoisomerase products, U = ( 1 2) from the 31 topoisomerase products. The possible site specific recombinase link products are the unlink, and the Hopf link, with linking number 1. Because there is no real why to tell

85 67 unlinks apart from unknots, this information provides little insight. Yet, since there are limited possible non-trivial site specific recombinase difference topology products, topoisomerase experiments give enough information to hypothesize the results of difference topology experiments using site specific recombinase to study 13S condensin bound DNA. The significant amount of 4-crossing knot type II topoisomerase products, along with the analysis of unknot and 3-crossing type II topoisomerase products of difference topology, provides information on the conformation of MukB bound DNA substrates: U = (1), ( ) 1 0 from the 01 topoisomerase products, U = ( 1 2) from the 31 topoisomerase products and U = ( 1 2), ( 1 3) from the 41 topoisomerase products. The possible site specific recombinase link products are the unlink, 0 2 1, Hopf link, with 1 linking number and the Hopf link, with +1 linking number. Therefore, while these topoisomerase experiments give some information, difference topology experiments using site specific recombinase to study MukB bound DNA should be run.

86 68 CHAPTER 3 KNOT FLOER HOMOLOGY One of the main aspects of knot theory is to provide techniques to determine whether two knots are equivalent. Thus, modern knot theory is concerned with the development of knot invariants, a property of a knot that does not change under ambient isotopy. These invariants are usually functions from the set of all knots into some object, such as polynomials or groups. In recent years, knot homology theories have become popular invariants to develop and study. These invariants associate a chain complex of abelian groups to a knot diagram and then produce a sequence of knot invariants by considering the homology groups of the associated complexes. One such invariant is knot Floer homology, denoted HF K. It is the homology of a bigraded complex (C, ) with Maslov grading m and Alexander grading s: HF K = m,s Z HF K m (K, s), and is a refinement of a pre-existing invariant, the Alexander Polynomial. Knot Floer homology was constructed by Peter Ozsváth, Zoltán Szabó [34], and independently Jacob Rasmussen [38]. It satisfies several interesting properties including that the Euler characteristic ( 1) m rankĥf K m (K, s) t s m,s is equal to the symmetrized Alexander polynomial.

87 69 When originally defined, the generators for this homology were computed combinatorially, while the differential was defined analytically. In [28], Manolescu, Ozsváth and Sarkar, give a purely combinatorial description of knot Floer homology. Note: All theorems, lemmas, proofs and definitions are from [28, 29] unless explicitly stated. This chapter will conclude with a worked example. 3.1 Combinatorial Description Combinatorial knot Floer homology is described starting from a grid presentation. Definition A planar grid diagram G lies in a square grid in R 2 with n n squares of area one, where n is known as the grid number. Each square may be left blank or decorated with an O or an X, such that every row contains exactly one X and one O. every column contains exactly one X and one O. We can label the O s and X s from 1 to n and denote the collections O = {O i } n i=1 and X = {X i } n i=1. The grid is placed such that the bottom left corner is at the origin and each cell is a square of area one. We can draw a planar projection of an oriented link in a grid diagram by drawing horizontal segments from an element in O to an element in X in each row and vertical segments from an element in X to an element in O in each column. At intersection points we force the vertical segment to be the overcrossing (Fig. 3.1). We then take this grid diagram and turn it into a torus in the usual way by associating the top and bottom lines of the grid diagram and left and right lines of the

88 70 Figure 3.1: Projection of the 4 1 knot drawn on a 6 6 grid diagram. Reproduced and modified with permission from [28] grid diagram with one another creating a new diagram known as the toroidal grid diagram. To the toroidal grid diagram, we associate the chain complex C(G). The set of generators for C(G), denoted S, are intersection points between the horizontal and vertical lines of our grid diagram, which are now horizontal and vertical circles on the toroidal diagram. Let p = {(0, c 0 ), (1, c 1, )...(n 1, c n 1 )} S. Each ordered pair, (i, c i ) in p is called a component of p. Note that each horizontal and vertical circle contain only one component of p. Each generator can also be represented as an n tuple, (c 0 c 1...c n 1 ) (Fig 3.2). The simplest version of the chain complex associated to this diagram is a chain complex over the field Z/2Z with generators in S: ĈF K(K) = p S Z/2Z p. A more enhanced version of this chain complex with generators also in S over a polynomial ring Z/2Z[U 1,..., U n ] is defined as: CF K (K) = p S Z/2Z[U 1,..., U n ] p. The summands for this chain complex are generated by expressions U d 1 1 U dn n p,

89 71 Figure 3.2: An example of the generator (302514) for a 6 6 toroidal grid diagram. Reproduced with permission from [28] where n is the grid number. These complexes are bigraded, with the Maslov grading and Alexander grading defined as follows. Given two collections A and B of finitely many points in the plane, let I(A, B) be the number of pairs (a 1, a 1 ) A and (b 1, b 2 ) B with a 1 < b 1 and a 2 < b 2. Viewing the torus once again as a square in R 2 by cutting along a horizontal and vertical circle, we view a generator p as a collection of points with integer coordinates and O = {O i } n i=1 and X = {X i } n i=1 as collections of points in the plane with half-integer coordinates. Then, the function M : S Z gives the Maslov grading of a generator: M(p) = M O (p) = I(p, p) I(p, O) I(O, p) + I(O, O) + 1. (3.1) We say that generator p has maslov grading m = M(p). Lemma The function M is well defined. Proof. We need to show that M is independent of which horizontal and vertical circles

90 72 of the torus we cut along in order to view the torus as a square in R 2 with the bottom left corner at (0, 0). Fix p = {(0, c 0 ), (1, c 1 ),..., (n 1, c n 1 )} S. Suppose that c k = 0 so that (k, 0) p (Fig. 3.3). Let p denote the same generator with the top and left edges included in the grid diagram, giving c k = n, i.e (k, n) p (Fig. 3.4). Figure 3.3: Generator p on the torus with the bottom and left edges included. Figure 3.4: Generator p drawn on the torus with the top and left edges included. Figure 3.5: Generator p after a complete rotation of the torus with left and bottom edges included.

91 73 For each i, with k < i < n, the pair (k, 0), (i, c i ) adds 1 to the count of I(p, p) while pair (k, n), (i, c i ) contributes 0 to the count of I(p, p ). For each i with 0 i < k, the pair (k, 0), (i, c i ) does not contribute to I(p, p) but (k, n), (i, c i ) contributes to I(p, p ). Thus, I(p, p) + k }{{} # of (i,c i ) that contribute to I(p,p ) but not to I(p,p) = I(p, p ) + n } {{ k 1 } # of (i,c i ) that contribute to I(p,p) but not to I(p, p ). Now, for 1 2 j n 1 2, there is an O j O with first coordinate j. For k j < n the pair (k, 0) and O j contributes 1 to I(p, O) whereas the corresponding pair (k, n) and O j does not contribute to I(p, O). For 1 j m 1, the pair 2 2 (k, 0) and O j does not contribute to I(O, p) and each pair (k, n) and O j } contributes 1 to I(O, p ). Therefore I(p, O) + I(O, p ) + n k = I(p, O) + I(O, p) + k. To complete the rotation, we change O to O by moving the bottom row of the grid diagram which contains an O with coordinates (l + 1, 1 ) to the top row, creating O 2 2 with coordinates (l + 1 2, n 1 2 ), giving generator p (Fig. 3.5). A similar argument as those used above involving the generators shows both I(p, O )+I(O, p ) = I(p, O)+I(O, p )+2l n+2 = I(p, O)+I(O, p)+2l 2n+2k+2 and I(O, O ) = I(O, O) + 2l n + 1.

92 74 Thus, M(p) = I(p, p) I(p, O) I(O, p) + I(O, O) + 1 = I(p, p ) + n 2k 1 [I(p, O ) + I(O, p ) 2l + 2n 2k 2] + I(O, O ) 2l + n = I(p, p ) I(p, O ) I(O, p ) + I(O, O ) + 1 = M(p ). Using the assumptions above for a knot, the Alexander grading of a generator, described by function A : S Z, is defined as A(p) = M O(p) M X (p) 2 n 1 2 (3.2) where M X (p) = I(p, p) I(p, X) I(X, p) + I(X, X) + 1. We say the Alexander grading of p is s = A(p). Lemma The function A is well defined. Proof. By the definition of A, this argument is analogous to the proof of lemma The gradings associated to the chain complex CF K are defined such that multiplication by the variables U i drops the Maslov grading by 2 and the Alexander grading by 1, i.e: M(U d 1 1 U dn n p) = M(p) 2 n d i (3.3) i=1

93 75 and A(U d 1 1 Un dn p) = A(p) n d i. (3.4) To define the differential associated to the chain complexes ĈF K and CF K, we first define rectangles that connect generators (Fig. 3.6). Let p and q S. Then p and q can be connected by a rectangle if all but two components of p are equal to two components of q. The rectangle r connects p to q if: i=1 1. the four corners of r are components in p or q, and 2. the bottom left corner of r is a component of p. Denote this set of rectangles connecting p to q as Rect(p, q) (Fig.3.6). Notice that if q and p can be connected by one rectangle then Rect(p, q) = Rect(q, p) = 2. Figure 3.6: Toroidal grid diagram with two shaded rectangles connecting p to q and two unshaded rectangles connecting q to p. The differential of a generator for ĈF K is: (p) = { } # r Rect(p, q) X r =,O r =,p int r = q (3.5) q S

94 76 This differential,, keeps a count of the empty rectangles connecting generators, i.e containing no elements of p or q and no elements of X or O. The differential of a generator for CF K is: (p) = q S {r Rect(p,q) p/ int(r),x r= } U #O i r 1... U #On r n q, (3.6) where #O i r denotes the number of times O i is in r. This number is either 1 or 0. Thus, keeps a count of the elements of O that are in otherwise empty rectangles connecting generators using variables U i. The differentials extend linearly over the whole complex and have the following properties: Theorem Let and be defined as above. Then, a. = 0, = 0 b. Both differentials drop Maslov grading by 1. c. Both differentials preserve the Alexander grading. Proof. a. Recall that and both count only rectangles that connect two generators. Let G be an n n toroidal grid diagram. We will calculate (p) and (p). Let q, j, k S such that Rect(p, j), Rect(p, k), Rect(j, q) and Rect(k, q) are non zero. There are four ways to draw two rectangles in Rect(p, j) Rect(p, k) and two rectangles in Rect(j, q) Rect(k, q) on G (Fig. 3.7), 1. Disjoint rectangles, 2. Two rectangles share an interior and a corner,

95 77 3. Two rectangles share only one corner, and 4. Two rectangles share only an interior. Figure 3.7: There are four ways to draw the two rectangles in Rect(p, j) Rect(p, k): disjoint, share an interior and share a corner, share only a corner, share only an interior. Recall that we are working over the field Z/2Z. We assume that the rectangles connecting our generators fit the criteria to be counted in each differential. For two disjoint rectangles, we look at possible paths from p to q (Fig. 3.8). Notice that this implies that two generators j, k, shown as open circles and filled squares, respectively, share rectangles with both the generator p shown as filled circles and the generator, q shown as open squares. Thus, when calculating (p), there are two ways to add generator q to the summand, one with the summand containing j in p and the other with the summand containing k in p. Because the differentials extend linearly over the entire complex, the summand for (p) containing q will have a coefficient of 2. This argument will also suffice two rectangles that share only an interior. Assume there are two rectangles that share an interior and a corner (Fig. 3.9).

96 78 A similar pathway works for two rectangles that share only a corner. Therefore with these two pathways, using a similar argument as above, we have that two rectangles that share an interior and a corner and two rectangles that share only a corner will give a count of zero to the differential. Figure 3.8: Pathway between two disjoint rectangles.

97 Figure 3.9: Pathway between two rectangles that share an interior. 79

98 80 b. Let r Rect(p, q) such that r is counted in the differential of p. This means Intr (p q) =. Choose the torodial grid diagram of the torus T such that the lower left corner of r coincides with the lower left corner of T. Since r Rect(p, q) we know that the lower left and upper right corners of r contain components (0, 0) and (m, l) p. Thus, the other corners contain components (0, l) and (m, 0) q. By definition p and q only differ at these components, thus we are only concerned with their effect on the calculations. Recall the formula in equation 3.1. The pair, (0, 0) and (m, l) contribute 1 to the count of I(p, p) and the pair, (0, l) and (m, 0) contribute 0 to the count of I(q, q). Thus we have, I(p, p) = I(q, q) + 1. Next, the pair (0, 0) and O contribute 1 to I(p, O) for every O i O r. While the pair (m, l) and O contributes 0 to I(p, O) and the pairs {(0, l), O}, {(m, 0), O} contribute 0 to the count of I(q, O). Therefore, I(p, O) = I(q, O) + #{O i O r}. We have that {O, (m, l)} contributes 1 to I(O, p) for every O i O r. While the pair {O, (0, 0)} contribute 0 to the count of I(O, p) and {O, (0, l)}, {O, (m, 0)} contribute 0 to the count of I(O, q). Then, I(O, p) = I(O, q) + #{O i O r}.

99 81 Thus, we have the following, M(p) = I(p, p) I(p, O) I(O, p) + I(O, O) + 1 = I(q, q) + 1 I(q, O) + #{O i O r} I(O, q) + #{O i O r} + I(O, O) + 1 = I(q, q) I(q, O) I(O, q) + I(O, O) #{O i O r} = M(q) + 1 2#{O i O r} This breaks down to two cases: Case 1 O r =. This gives M(p) = M(q) + 1 as needed. Case 2 O r. Let #{O i O r} = s. This implies that q contributes U 1,... U s q to the summand for the differential of p. From equation 3.3 we have that M(U 1,... U s q) = M(q) 2s. Now we have, M(p) = M(q) + 1 2#{O i O r} = M(q) + 1 2s = M(q) 2s + 1 = M(U 1,... U s q) + 1 Thus, since U 1,... U s q is in the summand for the differential,, we have what we need. c. Let r be as described above. Since r Rect(p, q) such that r is counted in the differential, X r =. Therefore I(p, X) = I(q, X) and I(X, p) = I(X, q). So,

100 82 M X (p) = I(p, p) I(p, X) I(X, p) + I(X, X) + 1 = I(q, q) + 1 I(q, X) I(X, q) + I(X, X) + 1 = I(q, q) I(q, X) I(X, q) + I(X, X) = M X (q) + 1 Thus we have, A(p) = M O(p) M X (p) 2 n 1 2 = (M O(q) + 1 2#{O i O r}) (M X (q) + 1) 2 ( MO (q) + 1 (M X (q) + 1) = 2 = A(q) #{O i O r} n 1 2 n 1 ) 2#{O i O r} 2 2 This is analogous to the previous argument. 3.2 Example: Calculation of HF K m(0 1, s) Figure 3.10: 3 3 grid example. Figure 3.11: Knot drawn on the grid.

101 83 In this example we calculate HF Km(0 1, s). Assume we are working over the field Z/2Z. The 3 3 grid (Fig. 3.10), is the grid diagram of 0 1 (Fig. 3.11). The chain complex associated to this grid diagram has six generators: (012), (021), (102), (120), (201) and (210) (Fig. 3.12). We calculate the two gradings (A( ), M( )) for each of the these generators using equations 3.2 and 3.1, respectively (Table 3.1). Figure 3.12: The generators for the grid diagram in Fig labeled from left to right and top to bottom: (012), (021), (102), (120), (201) and (210). To calculate the (p) for all p S, by definition we count all rectangles that connect two generators and do not contain any components of the generators or any element in X. Recall that we are looking at rectangles on a torus and that we will have two rectangles connecting one generator to another when they differ by only two components. The differential for generator (012), represented by filled circles, is calculated using the rectangles created from generators (102), (210) and

102 84 p (s = A(p), m = M(p)) (012) ( 1, 0) (021) ( 1, 1) (102) ( 1, 1) (120) (0, 0) (201) ( 2, 2) (210) ( 1, 1) Table 3.1: Gradings for Fig (021), represented by hollow squares (Fig. 3.13). Thus, (012) = (021) + (102) + (210) (Table 3.2). The differential for generator (021), represented by filled circles, is calculated using the rectangles created from generators (201), (120) and (012), represented by hollow squares (Fig. 3.14). Thus, (021) = (U 2 + U 3 )(120) (Table 3.2). The differential for generator (102), represented by filled circles, is calculated using the rectangles created from generators (012), (201) and (120), represented by hollow squares (Fig. 3.15). Thus, (102) = (U 1 + U 2 )(120) (Table 3.2). The differential for generator (120), represented by filled circles, is calculated using the rectangles created from generators (210), (021) and (102), represented by hollow squares. Thus, (120) = 0 since each rectangle contains an element of X (Table 3.2). The differential for generator (201), represented by filled circles, is calculated using the rectangles created from generators (021), (102) and (210), represented by hollow squares (Fig. 3.17). Thus, (201) = U 1 (021)+U 2 (102)+U 3 (210) (Table 3.2).

103 85 The differential for generator (210), represented by filled circles, is calculated using the rectangles created from generators (120), (012) and (201), represented by hollow squares (Fig. 3.18). Thus, (210) = (U 1 + U 3 )(120) (Table 3.2). Figure 3.13: The rectangles used to calculate the differential for the generator (012). Figure 3.14: The rectangles used to calculate the differential for the generator (021). Figure 3.15: The rectangles used to calculate the differential for the generator (102).

104 86 Figure 3.16: The rectangles used to calculate the differential for the generator (120). Figure 3.17: The rectangles used to calculate the differential for the generator (201). Figure 3.18: The rectangles used to calculate the differential for the generator (210). At the (0, 0) homology level, we see that (120) = 0. Thus the ker( (0,0) ) =< (120) >. Since there are no generators with Alexander and Maslov grading (0, i) where i 0, there are no chain complexes at levels (0, i) where i 0. Thus, we have the

105 87 (012) = (021) + (102) + (210) (021) = (U 3 + U 2 )(120) (102) = (U 1 + U 2 )(120) (120) = 0 (201) = U 1 (021) + U 2 (102) + U 3 (210) (210) = (U 1 + U 3 )(120) Table 3.2: Differential for each generator. sequence 0 Z/2Z 0, giving HF K 0 (0 1, 0) =< (120) >= Z/2Z (Fig. 3.19). Next, we look at the Alexander grading level 1. At the ( 1, 0) level there is one generator: (012). At the ( 1, 1) level there are three generators: (021), (102), (210). At the ( 1, 2) level, there are also three elements, the generator (120) times U 1, U 2, and U 3 respectively (Fig. 3.19). This gives us the following sequence: 0 3 < (012) > 2 < (021), (102), (210) > 1 < (U 1 (120), U 2 (120), U 3 (120) > Z/2Z 2 Z/2Z Z/2Z Z/2Z 1 Z/2Z Z/2Z Z/2Z 0 0 where the mapping notation is simplified. We know that this sequence begins and ends with 0 because we have no other generators with Alexander grading = 1. Thus looking at 3, we see that the img( 3 ) = 0. Notice, ker( 2 ) = 0. Thus, HF K 0 (0 1, 1) = ker( 2 )/img( 3 ) = 0.

106 88 Figure 3.19: Graph, with x-axis Alexander grading and y-axis Maslov grading, that shows the generators as red circles and the generators times U 1, U 2, U 3 as blue triangles. Calculating the image of 2 we have img( 2 ) = Z/2Z since ker( 2 ) = 0. For ker( 1 ) note 1 ((021) + (102) + (210)) = 0. Thus, ker( 1 ) contains the field generated by ((021) + (102) + (210)). Since these are the only elements that 1 sends to 0, we have ker( 1 ) = Z/2Z. Therefore, HF K 1(0 1, 1) = Z/2Z Z/2Z = 0. Next, since ker( 1 ) = Z/2Z, we have that img( 1 ) = Z/2Z Z/2Z. We also have that 0 (U i (120)) = 0 for i = 1, 2, 3. Thus we have that ker( 0 ) is equal to the

107 89 module generated by U i (120) for i = 1, 2, 3. Therefore, Z/2Z Z/2Z Z/2Z Z/2Z Z/2Z = Z/2Z HF K 2(0 1, 1) = Z/2Z[U] Continuing the calculations for the Alexander grading 2 level HF K 2(0 1, 2) = 0, HF K 3(0 1, 2) = 0, HF K 4(0 1, 2) = Z/2Z[U] (Fig. 3.19). At ( j, 2j), we have U j times the generator (120) giving a non-trivial Figure 3.20: Graph of the homology groups of the unknot. homology for j Z + 0 (Fig. 3.20). Z/2Z[U] if (s, m) = ( j, 2j) for j Z + 0 Thus: HF Ks (0 1, m) = 0 otherwise.

108 90 CHAPTER 4 BIOLOGICALLY RELEVANT KNOT FAMILIES The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular processes. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue [5]. We have seen in chapter 2 two experimental methods that aid in separating and characterizing these knotted molecules: gel electrophoresis and electron microscopy. But these methods can be time consuming, difficult and not always accurate. Gel electrophoresis separates knots based on their crossing number, but normally gives no chirality information. With electron microscopy, sometimes the determination of overcrossing versus undercrossing is difficult, if not impossible. Thus topological techniques are employed to aid in restricting the types of DNA knot and link products that arise from experiments [5]. The relationship between the knot Floer homology of links in the oriented skein triple give information of which links cannot be in the same triple. Thus, the knot Floer homology will provide an obstruction to links forming an oriented skein triple, thus providing restrictions on knotted and linked products of protein action. This chapter focuses on reviewing methods for finding the knot Floer homology for three biologically relevant subfamilies. In section 4.1, we define (2, p)-torus links, a common type of DNA knot/link product. By construction, these knots are alternating and thus their knot Floer homologies are calculated via a theorem from [31].

109 91 To illustrate, we use this theorem to create a formula for the knot Floer homology of (2, p)-torus knots and links with parallel orientation. The section concludes with an investigation of a long exact sequence between the knot Floer homologies of alternating links in an oriented skein triple. Section 4.2 discusses another common DNA knot product, (2, v) clasp knots. These knots always have an alternating diagram and thus their knot Floer homology is also known [31]. We construct a formula for the knot Floer homology of these knots using known results. The section concludes by utilizing the long exact sequence between the knot Floer homologies of alternating links in an oriented skein triple to construct a formula for (2, p)-torus links with anti-parallel orientation. The chapter concludes with a future directions, section 4.3. This section discusses 3-stranded pretzels. 3-stranded pretzels are a proposed product of site-specific recombinase action on (2, p)-torus links and (2, v)-clasp knots substrates [5,7]. While the knot Floer homology for many types of 3-stranded pretzel is known, we will discuss the limitations of the long exact sequence for finding formulas for the knot Floer homology of 3-stranded pretzel links that are non-alternating. 4.1 (2, p) Torus Links A link in S 3 is a torus link if it admits a crossing-less diagram on the trivial torus. We use the construction and definition for a torus link found in [30]. Take a cylinder in R 3 that has unit circles C 1 and C 2 at its base and top, respectively. Then, in R 3 identify C 1 and C 2 so that the central axis of the cylinder

110 92 becomes the trivial knot [30]. This is known as the trivial torus. To draw a torus link, assume that the height of the cylinder is one unit and the base, C 1, is a unit circle in the xy-plane. Assign p points, A 0, A 1,... A p 1, to C 1 and p points B 0, B 1,..., B p 1 to C 2 such that A 0 = (1, 0, 0), A 1 = A p 1 = ( cos ( cos ( 2(p 1)π p ( 2π p ), sin ) ( ) ) 2π, sin, 0,..., p ( 2(p 1)π p ) ), 0 B 0 = (1, 0, 1), B 1 = B p 1 = ( cos ( cos ( 2(p 1)π p ( 2π p ), sin ), sin ( 2π p ( 2(p 1)π p ) ), 1,..., ) ), 1 Figure 4.1: Construction of torus links. Connect the points A k and B k (k = 0, 1, 2,..., p 1) on the cylinder with segment α k as shown in Fig Now, keeping the base (C 1 ) fixed, rotate the top of the cylinder (C 2 ) about the z-axis by an angle of 2πq p where q Z, giving the cylinder

111 93 a twist. Finally, we identify the point (x, y, 0) of C 1, with the point (x, y, 1) of C 2. This creates a single trivial torus T, where the segments α 0, α 1,... α p 1 create a knot or link on the surface of T. This knot or link is a (q, p)-torus link, denoted T (q, p). Note T (q, p) is a knot if and only if gcd(p, q) = 1. For simplicity, we fix q > 0 and determine the handedness of the crossings via p. When p > 0, the torus links has left-handed crossings. When p < 0, the crossings in the torus link are right-handed crossings. For the remainder of this chapter, we assume that q = 2. Torus links are invertible. We choose an orientation on T (2, p), denoted as T (2, p), such that the segments α k are oriented from A k to B k. We refer to this orientation as the parallel orientation (Fig.4.2). Since both ways to orient a torus knot are equivalent, we will only use the notation T (2, p) when referring to torus links. Note that the previous assignment only considers one possible way to orient T (2, p) where p is even. We denoted the anti-parallel orientation as T (2, p), where the link components are not oriented in the same direction (Fig. 4.3). Results in section 4.1 are for T (2, p) torus knots and T (2, p), torus links with parallel orientation. We address T (2, p), torus links of two components with anti-parallel orientation, in section Knot Floer Homology of T (2, p). Let q = 2. Assume that p is odd. Thus T (2, p) is a knot. Notice that by our construction, T (2, p) is alternating. Since these knots are alternating, their knot Floer

112 94 Figure 4.2: Parallel orientation, T (2, 4). Figure 4.3: Anti-parallel orientation, T (2, 4). homology has been completely determined via the following theorem: Theorem (Theorem 1.3 [31]). Let K S 3 be an alternating knot in the three sphere, and write its symmetrized Alexander polynomial K (t) = n a s (t s + t s ). s=0 Let a s = a s if s < 0. Then, ĤF K m (K, s) = (Z/2Z) as is supported entirely in dimension m = s + σ 2 where σ(k) is the knot signature. Theorem states that the knot Floer homology of all alternating knots is completely supported on a diagonal (s, s + σ ). We can use this theorem to produce 2 a formula for the knot Floer homology of T (2, p) [31]. We first must calculate the symmetrized Alexander polynomial for this family of links with parallel orientation. We use the following calculation for the Alexander polynomial of T (q, p): Theorem (Theorem in [30]). Let T (q,p)(t) be the symmetrized Alexan-

113 95 der polynomial of T (q, p), p, q 0. Let gcd(q, p) = d, pq t (q 1)(p 1) d 1 (t d 1) d (t 1) 2 T (q,p) = ( 1) (t p 1)(t q 1). Let q = 2 and p > 0 be odd. Then, t p 1 2 T (2,p) (t) = (t2p 1)(t 1) (t p 1)(t 2 1) = (tp 1)(t p + 1)(t 1) (t p 1)(t 1)(t + 1) = (tp + 1) (t + 1) = t p 1 t p 2 + t p t 2 t + 1 = t p 1 2 ) (t p 1 2 t p t p t p 5 2 t p t p 1 2 Thus the symmetrized Alexander polynomial for T (2, p), is T (2,p) (t) = t p 1 2 t p t p t p 5 2 t p t p 1 2. For σ(t (2, p)), we know from [30] that σ(t (2, p)) = p 1 for p > 0. Using this signature information and the symmetrized Alexander polynomial along with theorem 4.1.1, we have the following formula for the knot Floer homology of T (2, p) when p > 0, p odd: Z/2Z ĤF K m (T (2, p), s) = if (s, m) = ( p i, ) 2p i for all odd integers 0 < i 2p 1 0 otherwise The next step is to find a formula for the knot Floer homology for T (2, p) with p < 0. Recall that p < 0 corresponds to right-handed crossings. For the formula

114 96 for the knot Floer homology of T (2, p) with p < 0, notice this is the mirror image of T (2, p). We know from [32] the following relationship between the knot Floer homology of a knot and its mirror image: ĤF K m (K, s) = ĤF K m(k, s). (4.1) Thus, assuming that p < 0, we have the following formula Z/2Z if (s, m) = ( i+p, ) i+2p ĤF K m (T (2, p), s) = for all odd integers 0 < i 2 p 1 0 otherwise Using the fact that T (2, p) is alternating and theorem 4.1.1, we have found formulas for the knot Floer homology of T (2, p), p odd. We now focus on the knot Floer homology of T (2, p), torus links of two components (i.e. p is even) with parallel orientation Knot Floer Homology of T (2, p). Assume that p > 0 is even. Note that by construction, these are alternating two component links. Thus the knot Floer homology of T (2, p) can be calculated via the following theorem [31]: Theorem Let L be an alternating l-component link. Letting L be the (onevariable) Alexander polynomial, and writing (t 1 2 t 1 2 ) l 1 L (t) = n a s (t s + t s ). s=0

115 97 Let a s = a s if s < 0. We have that ĤF K(S 3, L, s) = (Z/Z 2 ) as is supported entirely in dimension m = s + σ 2. Notice that as with alternating knots, the knot Floer homology of all alternating links is completely supported on a diagonal (s, s + σ ). We use theorem to find a 2 formula for the knot Floer homology of T (2, p) where p > 0 is even, i.e. torus links of two components with parallel orientation. First, we calculate the symmetrized one variable Alexander polynomial for T (2, p) as in theorem Letting q = 2 and p > 0 be even, d = gcd(q, p) = 2. Thus, we have the following calculation for the symmetrized Alexander polynomial: 2p t p 1 (t 2 1) 2 (t 1) 2 T (2,p) (t) = ( 1) (t p 1)(t 2 1) = (tp 1) (t + 1) = t p 1 + t p 2 t p t 2 t + 1 = t p 1 2 ) ( t p t p t p t p 1 2 Thus the symmetrized Alexander polynomial for T (2, p) for p > 0 even is T (2,p)(t) = t p t p t p t p 1 2. Recall that σ(t (2, p)) = p 1 for all p > 0. We now determine the left-hand side of the equation in theorem 4.1.3: (t 1 2 t 1 2 ) T (2,p)(t) = (t 1 2 t 1 2 )( t p t p t p t p 1 2 ) = t p 2 2t p t p 2 2 t p 2

116 98 Using theorem 4.1.3, we have the following formula for the knot Floer homology of T (2, p) where p > 0 is even: Z/2Z Z/2Z ĤF K m (T (2, p), s) = Z/2Z Z/2Z if (s, m) = ( ) p i+1, 2p i 2 2 for all odd integers 3 i 2p 3 when (s, m) = ( p, ) 2p when (s, m) = ( p, ) otherwise Using the result from [32] that ĤF K m (K, s) = ĤF K m(k, s), we now assume that p < 0 is even. Then the knot Floer homology of T (2, p) where p < 0 is even is: Z/2Z Z/2Z if (s, m) = ( ) p+i 1, 2p+i 2 2 ĤF K m (T (2, p), s) = Z/2Z Z/2Z for all odd integers 3 i 2 p 3 when (s, m) = ( p, ) 2p when (s, m) = ( p, ) otherwise Long Exact Sequence By construction, T (2, p) is alternating and thus its knot Floer homologies can be calculated via theorems and Yet, we want to explore another way of calculating the knot Floer homologies of these links. In fact, we look at an alternate way to calculate the knot Floer homology of alternating knots and links that are in an oriented triple of alternating knots and links. In [32, 35], Ozváth and Szabó prove

117 99 that there exists a long exact sequence between the knot Floer homologies of the links in the oriented skein triple (Fig. 4.4). There are two grading conventions that can be considered when studying the knot Floer homology of links. In chapter 3, we provided a grading convention such that the Malsov grading, m, for links is an integer value and the Alexander grading, s, for links is a half-integer value, i.e m Z and s 1 Z. Note that the knot Floer homology of 2 links can be normalized in such a way that alternatively m 1 Z and s Z. Ozsváth 2 and Szabó proved the existence of the following long exact sequence between the knot Floer homologies of links in the oriented skein triple (Fig. 4.4), using the above grading conventions [32]:... ĤF K(K ) ĤF K(K 0 ) ĤF K(K + )... If we include grading information, we obtain... ĤF K m (K, s) ĤF K m- 1 (K 0, s) ĤF K m 1 (K +, s) 2 ĤF K mi (K, s)... m i m 1 This sequence preserves the Alexander grading and drops the Maslov grading between the homology for K and K 0 by a half and K 0 and K + by a half. All the maps respect the splitting of ĤF K(K) into summands. However, the map from ĤF K(K + ) to ĤF K(K ) is known to only be the sum of homogeneous maps that are non-increasing for Maslov grading. Another grading convention for the knot Floer homology of links requires both gradings to assume integer values, i.e. m and s Z. This is a shift of the Alexander

118 100 grading that is described in chapter 3. The following long exact sequence between the knot Floer homologies of links in the oriented skein triple (Fig. 4.4) uses this alternate grading conventions [35]:... ĤF K m (K +, s) ĤF K m (K, s) ĤF K m-1 (K 0, s) ĤF K m-1 (K +, s)... This sequence also preserve the Alexander grading for all homologies, the Maslov grading between K + and K and the Maslov grading between K 0 and K +, but drops the Maslov grading between the homology for K and K 0 by one. Figure 4.4: Recall the oriented knot triple: (K +, K, K 0 ). Thus, to stay consistent with later work in this chapter, we use the first long exact sequence. We will now apply this long exact sequence to triples of alternating knots, using Ozváth and Szabó s characterization of the knot Floer homology of alternating knots and links as supported on a diagonal, known as thin. This thinness implies that for a given Alexander grading s, the knot Floer homology is nontrivial for at most one value of m. This gives us the following well-known result: Lemma For each s, the long exact sequence between the knot Floer homologies

119 101 of alternating knots and links can be reduced to a short exact sequence. This short exact sequence splits. Proof. Assume that K, K +, and K 0 are alternating links that differ at a single crossing as indicated by the notation and as seen in figure 4.4. Then we have the following:... ĤF K m (K, s) ĤF K m- 1 (K 0, s) ĤF K m-1 (K +, s) 2 ĤF K mi (K )... m i m 1 We know from theorems and that the knot Floer homology of an alternating link is completely supported in a diagonal. Thus for all s, the knot Floer homology for an alternating link will be non trivial for at most one m. Therefore, if we assume, without loss of generality, that ĤF K(K ) is non trivial at (s, m) then we have the following exact sequence: 0 ĤF K m+ 1 (K 0, s) ĤF K m (K +, s) ĤF K mi (K, s) 2 m i m 1 ĤF K mi 1 (K 0, s) ĤF K mi 1(K +, s) 0 m i m 1 2 m i m 1 We have the following cases: 1) ĤF K m (K +, s) 0. This implies that ĤF K mi 1(K +, s) = 0 because K + is alternating. We have the following exact sequence: 0 ĤF K m+ 1 (K 0, s) ĤF K m (K +, s) ĤF K mi (K, s) 2 m i m 1 ĤF K mi - 1 (K 0, s) 0 m i m 1 2

120 102 Notice by assumption that K 0 is alternating. Thus, ĤF K(K 0 ) cannot be non-trivial at both (s, m i 1/2) for some i and (s, m + 1/2). This gives the following cases: ĤF K m+ 1 (K 0, s) = 0 and ĤF K mi 1 (K 0, s) = 0 for all i. 2 2 This implies that 0 f 1 ĤF Km (K +, s) f 2 ĤF K mi (K, s) m i m 1 f 3 0 Notice that since our sequence is exact, img(f 1 ) = ker(f 2 ). Hence f 2 is injective. Also, img(f 2 ) = ker(f 3 ), thus f 2 is surjective. Thus, f 2 is a bijection and ĤF K m (K +, s) = ĤF K mi (K, s). m i m 1 Since K is alternating, ĤF K mi is non-zero for at most one j m. Hence ĤF K m (K +, s) = ĤF K j(k, s), for this j m. Assume that ĤF K m+ 1 (K 0, s) 0. 2 This implies that ĤF K mi 1 2 (K 0, s) = 0 for all m i m, since K 0 is alternating. Thus we have the following exact sequence: 0 ĤF K m+ 1 (K 0, s) ĤF K m (K +, s) ĤF K m (K, s) 0 2 We know from chapter 3 that the knot Floer homology of knots and links is the direct sum of copies of Z/2Z. Thus, our sequence reduces to the following 0 (Z/2Z) j f (Z/2Z) k g (Z/2Z) n 0

121 103 where f is one to one, g is onto, j, k, n Z + and (Z/2Z) i denotes Z/2Z Z/2Z... Z/2Z. By convention, we assume that }{{} i times (Z/2Z) 0 = {0}. Because the sequence is exact, k j and k n. Notice that Z/2Z is a module over itself. Also (Z/2Z) n is a free module. Since any short exact sequence 0 A B C where C is free splits by the splitting lemma [24], we have that the sequence is split exact and k = j + n. Assume that ĤF K mi 1 2 (K 0, s) 0 for some m i m. This implies that ĤF K m+ 1 (K 0, s) = 0 since K 0 is alternating. Thus we have the following exact 2 sequence: 0 ĤF K m (K +, s) ĤF K mi (K, s) m i m ĤF K mi - 1 (K 0, s) 0 m i m 2 We have shown in the previous case that a sequence similar to this sequence splits. Thus, we have a split short exact sequence in this case. 2) ĤF K m (K +, s) = 0 0 ĤF K m+ 1 (K 0, s) 0 ĤF K mi (K, s) 2 m i m ĤF K mi - 1 (K 0, s) ĤF K mi -1(K +, s) 0 m i m 2 Notice 0 f 1 ĤF Km+ 1 2 (K 0, s) f 2 0. Due to this sequence being exact, we have that img(f 1 ) = ker(f 2 ). Hence, This gives the following short split exact sequence ĤF K mi (K 0, s) = ker(f 2 ) = img(f 1 ) = 0. m i m 0 ĤF K mi (K, s) ĤF K mi - 1 (K 0, s) ĤF K mi -1(K +, s) 0 m i m m i m 2 m i m

122 104 Due to the results of theorems and 4.1.3, all cases reduce to a short exact sequence similar to one of the cases above. Thus we have that the long exact sequence between the knot Floer homologies of alternating knots and links can be reduced to a split short exact sequence. Using this lemma, if we have an oriented triple of alternating links and know the knot Floer homology of two links in the triple, we can calculate the knot Floer homology of the third link in the triple. Later in the chapter we will explore what happens when the alternating assumption is relaxed. Recall that our torus link formulas are valid for links T (2, p). The Alexander polynomial formula in theorem is not consistent for T (2, p), torus links with anti-parallel orientation. Using the information from chapter 3, we see that different orientations yield different knot Floer homologies for oriented torus links of two components (Fig. 4.5). Thus instead of using theorem to calculate the knot Floer homology of T (2, p), we employ lemma to find a formula for the knot Floer homology for T (2, p). We use the oriented triple where K + and K are clasp knots and K 0 is T (2, p). 4.2 Clasp Knots, C(2, v) Clasp knots (also known as twist knots), denoted C(2, v), are knots obtained by repeatedly twisting a closed loop and then linking the ends together into a clasp, where the integer two represents 2 crossings in the clasp and v is the number of half twists introduced before clasping (Fig. 4.6). Right-handed half twists will be denoted

123 105 Figure 4.5: Grid diagrams of T (2, 4) and T (2, 4). by positive integer v, while left-handed half twist will be denoted by negative integer, v, v > 0. Any number of half-twists may be introduced into the loop before clasping, resulting in an infinite family of knots. Figure 4.6: Clasp Knots: From left to right: v is odd and v is even. Every clasp knot has an alternating diagram since C( 2, v) = C(2, v + 1). The mirror image is created by changing all signs, i.e C(2, v) = C( 2, v). Recall the result in equation (4.1): ĤF K m (K, s) = ĤF K m(k, s). Thus, we will assume for