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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Page 218
Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Page 219
Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Suggested Citation:"TOPOLOGICAL TOOLS FOR DNA ANALYSIS." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 212 new precision in the determination of the topology of the reaction product spectrum opens the door for the building of detailed topological models for enzyme action. TOPOLOGICAL TOOLS FOR DNA ANALYSIS In this section, we will describe the parts of knot theory and tangle calculus of biological relevance. We give intuitive definitions that appeal to geometric imagination. For a rigorous mathematical treatment we refer the reader to Burde and Zieschang (1985), Kauffman (1987), and Rolfsen (1990) for knot theory and Ernst and Sumners (1990) for tangle calculus. Knot theory is the study of the entanglement of flexible circles in 3-space. The equivalence relation between topological spaces is that of homeomorphism. A homeomorphism h: X → Y between topological spaces is a function that is one-to-one and onto, and both h and h−1 are continuous. An embedding of X in Y is a function f: X → Y such that f is a homeomorphism from X onto f(X) ⊂ Y. An embedding of X in Y is the placement of a copy of X into the ambient space Y. We will usually take Euclidean 3-space R3 (xyz-space) as our ambient space. A knot K is an embedding of a single circle in R3; a link L is an embedding of two or more circles in R3. For a link, each of the circles of L is called a component of L. In chemistry and biology a nontrivial link is called a catenane, from the Latin cataena for ''chain," since the components of a catenane are topologically entangled with each other like the links in a chain. In this excursion, we will restrict attention to dimers, that is, links of two components, because dimers are the only links that turn up in topological enzymology experiments. We regard two knots (links) to be equivalent if it is possible to continuously and elastically deform one embedding (without breaking strands or passing strands one through another) until it can be superimposed upon the other. More precisely, if K1 and K2 denote two knots (links) in R3, they are equivalent (written K1 = K2 ) if and only if there is a homeomorphism of pairs h: (R3, K1) → (R3,K2) that preserves orientation on the ambient space R3. We take our ambient space R3 to have a fixed (right-handed)

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 213 orientation, where the right-hand thumb corresponds to the X-axis, the right-hand index finger corresponds to the Y-axis, and the right-hand middle finger corresponds to the Z-axis. R3 comes locally equipped with this right- handed orientation at all points. A homeomorphism from R3 to R3 is orientation-preserving if the local right- handed frame at each point of the domain maps to a local right-handed frame in the range. Reflection in a hyperplane (such as reflection in the xy-plane by f: (x,y,z) → (x,y,z)) reverses the orientation of R3. We might also require that the circular subspace K come equipped with an orientation (usually indicated by an arrow). If so, we say that our knot or link K is oriented; if not, we say that it is unoriented. Unless otherwise specified, all of our knots will be unoriented. The homeomorphism of pairs h superimposes K1 on K2; in this case the knots (links) can be made congruent by a flexible motion or flow (ambient isotopy) of space. An ambient isotopy is a 1-parameter family of homeomorphisms of R3 that begins with the identity and ends with the homeomorphism under consideration: H0 = identity and H1 = h. An equivalence class of embeddings is called a knot (link) type. A knot (link) is usually represented by drawing a diagram (projection) in a plane. This diagram is a shadow of the knot (link) cast on a plane in 3-space. By a small rigid rotation of the knot (link) in 3-space, it can be arranged that no more than two strings cross at any point in the diagram. For short, crossing points in a diagram are called crossings. In the figures in this chapter, at each crossing in a diagram, the undercrossing string is depicted with a break in it, so that the three-dimensional knot (link) type can be uniquely re-created from a two- dimensional diagram. Figure 8.4a-e shows standard diagrams (Rolfsen, 1990) for the knots and links that turn up in Tn3 recombination experiments. In the definition of knot type, we insisted that the transformation that superimposes one knot on another must be orientation-preserving on the ambient space. This restriction allows us to detect a property of great biological significance: chirality. The mirror image of a knot (link) is the configuration obtained by reflecting the configuration in a plane in R3. Starting with a diagram for a knot (link), one can obtain a diagram for the mirror image by reversing each crossing; the underpass becomes the overpass and vice versa (compare

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 214 Figures 8.4d and 8.4e). If K denotes a knot (link), let K* denote the mirror image. If K ≠ K*, then we say that K is achiral; if K¹ K*, then we say that K is chiral. For example, the Hopf link (Figure 8.4a) and the figure eight knot (Figure 8.4b) are achiral, and the (+) Whitehead Figure 8.4 (a) Hopf link, (b) figure eight knot, (c) (+) Whitehead link, (d) , and (e) 62 (mirror image of ).

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 215 link (Figure 8.4c) and the knot (Figure 8.4d) and its mirror image 62 (Figure 8.4e) are chiral. Moreover, all the knots and links in Figure 8.4 are prime, that is, they cannot be formed by the process of tying first one knot in a string and then another. By moving the knot around in space and then projecting it, it is clear that every given knot (link) type admits infinitely many "different" diagrams, and so the task of recognizing that two completely different diagrams represent the same knot type can be exceedingly difficult. In order to make this job a bit easier, one usually seeks diagrams for the knot type with a minimal number of crossings. This minimal number is called the crossing number of the knot (link) type. The projections in Figures 8.4 are minimal. Crossing number is a topological invariant of knot type. A topological invariant is a number, algebraic group, polynomial, and so on that can be unambiguously attached to a knot (link) type. Most invariants can be algorithmically computed from diagrams (Burde and Zieschang, 1985; Crowell and Fox, 1977; Lickorish, 1988; Kauffman, 1987). If any invariant differs for two knots (links), then the two knots (links) are of different types. If all known invariants are identical, the only conclusion that can be reached is that all known invariants fail to distinguish the candidates. One must then either devise a new invariant that distinguishes the two or prove that they are of the same type by construction of the homeomorphism that transforms one to the other (often by direct geometric manipulation of the diagram or by manipulation of string models). Nevertheless, it is possible to devise invariants (algebraic classification schemes) that uniquely classify certain homologous subfamilies of knots and links, for example, torus knots, two-bridge knots (4-plats), and so on. The algebraic classification schemes for these homologous subfamilies can be used to describe and compute enzyme mechanisms in the topological enzymology protocol. Fortunately for biological applications, most (if not all) of the circular DNA products produced by in vitro enzymology experiments fall into the mathematically well-understood family of 4-plats. This family consists of knot and link configurations produced by patterns of plectonemic supercoiling of pairs of strands about each other. All "small" knots and links are members of this family—more precisely, all prime knots with crossing number less than 8 and all prime (twocomponent) links with crossing number less than 7 are 4-plats. A 4-plat

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 216 is a knot or two-component link that can be formed by platting (or braiding) four strings. All of the knots and links of Figure 8.4 are 4-plats; their standard 4-plat diagrams are shown in Figure 8.5. Each standard 4-plat diagram consists of four horizontal strings, numbered 1 through 4 from top to bottom. The standard pattern of plectonemic interwinding for a 4-plat is encoded by an odd-length classifying vector with positive integer entries < c1,c2,. . .,c2k+1 >, as shown in Figure 8.5. Beginning from the left, strings in positions 2 and 3 undergo c1 left- handed plectonemic interwinds (half-twists), then strings in positions 1 and 2 undergo c2 right-handed plectonemic interwinds, then strings in positions 2 and 3 undergo c3 left-handed plectonemic interwinds, and this process continues until at the right the strings in positions 2 and 3 undergo c2k+1 left-handed plectonemic interwinds. In the standard diagram for a 4-plat, the string in position 4 is not involved in any crossing. The vector representation for the standard diagram of a 4-plat is unique up to reversal of the symbol. That is, the vector < c2k+1 c2k,. . .,c1 > represents the same type as the vector < c1,c2,. . .,c2k+1 >, because turning the 4-plat 180° about the vertical axis reverses the pattern of supercoiling. The standard 4-plat diagram is alternating; that is, as one traverses any strand in the diagram, one alternately encounters over- and undercrossings. Also, standard 4-plat diagrams (with the exception of the unknot < 1 >) are minimal (Ernst and Sumners, 1987). For in vitro topological enzymology, we can regard the enzyme mechanism as a machine that transforms 4- plats into other 4-plats. We need a mathematical language for describing and computing these enzyme-mediated changes. In many enzyme-DNA reactions, a pair of sites that are distant on the substrate circle are juxtaposed in space and bound to the enzyme. The enzyme then performs its topological moves, and the DNA is then released. We need a mathematical language to describe configurations of linear strings in a spatially confined region. This is accomplished by means of the mathematical concept of tangles. Tangles were introduced into knot theory by J.H. Conway (1970) in a seminal paper involving construction of enumeration schemes for knots and links. The unit 3-ball B3 in R3 is the set of all vectors of length ≤ 1. The boundary 2-sphere S2 = ∂ B3 is the set of all vectors of length 1. The equator of this 3-ball is the intersection of the boundary S2 with

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 217 Figure 8.5 Standard 4-plats. (a) < 2 > Hopf link, (b) < 2,1,1 > figure eight knot, (c) < 1,1,1,1,1 > (+) figure eight catenane, (d) < 1,2,1,1,1 > 62*, and (e) < 3,1,2 > 62. the xy-plane; the equatorial disk is the intersection of B3 with the xy-plane. On the unit 3-ball, select four points on the equator (called NW, SW, SE, NE). A 2-string tangle in the unit 3-ball is a configuration of two disjoint strings in the unit 3-ball whose endpoints are the four special points {NW,SW,SE,NE}. Two tangles in the unit 3-ball are equivalent if it is possible to elastically transform the strings of one tangle into the strings of the other without moving the endpoints {NW,SW, SE,NE} and without breaking a string or passing one string

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 218 through another. A class of equivalent tangles is called a tangle type. Tangle theory is knot theory done inside a 3-ball with the ends of the strings firmly glued down. Tangles are usually represented by their projections, called tangle diagrams, onto the equatorial disk in the unit 3ball, as shown in Figure 8.6. In all figures containing tangles, we assume that the four boundary points {NW,SW,SE,NE} are as in Figure 8.6a, and we suppress these labels. All four of the tangles in Figure 8.6 are pairwise inequivalent. However, if we relax the restriction that the endpoints of the strings remain fixed and allow the endpoints of the strings to move about on the surface (S2) of the 3-ball, then the tangle of Figure 8.6a can be trans Figure 8.6 Tangles. (a) Rational, (b) locally knotted, (c) prime, and (d) trivial.

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 219 formed into the trivial tangle of Figure 8.6d. This can be accomplished by rotating (on S2) the {NE,SE} endpoints one left half-turn (180°) about each other, then rotating the {SW,SE} endpoints three right halfturns about each other, and finally rotating the {NE,SE} endpoints two left half-turns about each other. The tangles in Figures 8.6b and 8.6c cannot be transformed to the trivial tangle by any sequence of such turning motions of the endpoints on S2. The family of tangles that can be converted to the trivial tangle by moving the endpoints of the strings on S2 is the family of rational tangles. Equivalently, a rational tangle is one in which the strings can be continuously deformed (leaving the endpoints fixed) entirely into the boundary 2-sphere of the 3-ball, with no string passing through itself or through another string. Rational tangles form a homologous family of 2-string configurations in B3 and are formed by a pattern of plectonemic supercoiling of pairs of strings. Like 4-plats, rational tangles look like DNA configurations, being built up out of plectonemic supercoiling of pairs of strings. More specifically, enzymes are often globular in shape and are topologically equivalent to our unit defining ball B3. Thus, in an enzymatic reaction between a pair of DNA duplexes, the pair {enzyme, bound DNA} forms a 2-string tangle. Since the amount of bound DNA is small, the enzyme-DNA tangle so formed will admit projections with few nodes and therefore is very likely rational. For example, all locally unknotted 2-string tangles having less than five crossings are rational. There is a second, more natural argument for rationality of the enzymeDNA tangle. In all cases studied intensively, DNA is bound to the surface of the protein. This means that the resulting protein-DNA tangle is rational, since any tangle whose strings can be continuously deformed into the boundary of the defining ball is automatically rational. A classification scheme for rational tangles is based on a standard form that is a minimal alternating diagram. The classifying vector for a rational tangle is an integer-entry vector (a1,a2,. . .,an) of odd or even length, with all entries (except possibly the last) nonzero and having the same sign, and with |a1| > 1. The integers in the classifying vector represent the left-to-right (west-to-east) alternation of vertical and horizontal windings in the standard tangle diagram, always ending with horizontal windings on the east side of the diagram. Horizontal winding

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 220 is the winding between strings in the top and bottom (north and south) positions; vertical winding is the winding between strings in the left and right (west and east) positions. By convention, positive integers correspond to horizontal plectonemic right-handed supercoils and vertical left-handed plectonemic supercoils; negative integers correspond to horizontal left-handed plectonemic supercoils and vertical righthanded plectonemic supercoils. Figure 8.7 shows some standard tangle diagrams. Two rational tangles are of the same type if and only if they have identical classifying vectors. Due to the requirement that |al| > 1 in the classifying vector convention for rational tangles, the corresponding tangle projection must have at least two nodes. There are four rational tangles {(0), (0,0),(1),(−1)} that are exceptions to this convention (|a1| = 0 or 1) and are displayed in Figure 8.7c-f. The classifying vector (a1,a2,. . .,an) can be converted to an (extended) rational number b/a ∈ Q ∞ by means of the following continued fraction calculation: b/a = an + l / (an−1 + (1 / (an −2 +. . .))) Two rational tangles are of the same type if and only if these (extended) rational numbers are equal (Conway, 1970), which is the reason for calling them "rational" tangles. In order to use tangles as building blocks for knots and links, and mathematically to mimic enzyme action on DNA, we now introduce the geometric operations of tangle addition and tangle closure. Given tangles A and B, one can form the tangle A + B as shown in Figure 8.8a. The sum of two rational tangles need not be rational. Given any tangle C, one can form the closure N(C) as in Figure 8.8b. In the closure operation on a 2-string tangle, ends NW and NE are connected, ends SW and SE are connected, and the defining ball is deleted, leaving a knot or a link of two components. Deletion of the defining B3 is analogous to deproteinization of the DNA when the synaptosome dissociates. One can combine the operations of tangle addition and tangle closure to create a tangle equation of the form N(A + B) = knot (link). In such a tangle equation, the tangles A and B are said to be summands of the resulting knot (link). An example of this phenomenon is the tangle equation N((−3,0) + (1)) = <2 >, shown in Figure 8.8c. In general, if A and B are any two rational tangles, then

LIFTING THE CURTAIN: USING TOPOLOGY TO PROBE THE HIDDEN ACTION OF ENZYMES 221 Figure 8.7 Tangle diagrams. (a) (2,3,1), (b) (−3,0), (c) (0), (d) (0,0), (e) (1), and (f) (−1). N(A + B) is a 4-plat. Given these constructions, rational tangles are summands for 4-plats.

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