There exist naturally occurring enzymes (topoisomerases and recombinases), which, in order to mediate the vital life processes of replication, transcription, and recombination, manipulate cellular DNA in topologically interesting and non-trivial ways [24, 30]. These enzyme actions include promoting the coiling up (supercoiling) of DNA molecules, passing one strand of DNA through another via a transient enzyme-bridged break in one of the strands (a move performed by topoisomerase), and breaking a pair of strands and recombining them to different ends (a move performed by recombinase). An interesting development for topology has been the emergence of a new experimental protocol, the topological approach to enzymology [30], which directly exploits knot theory in an effort to understand enzyme action. In this protocol, one reacts artificial circular DNA substrate with purified enzyme in vitro (in the laboratory); the enzyme acts on the circular DNA, causing changes in both the euclidean geometry (supercoiling) of the molecules and in the topology (knotting and linking) of the molecules. These enzyme-caused changes are experimental observables, using gel electrophoresis to fractionate the reaction products, and rec A enhanced electron microscopy [15] to visualize directly and to determine unambiguously the DNA knots and links which result as products of an enzyme reaction. This experimental technique calls for the building of knot-theoretic models for enzyme action, in which one wishes mathematically to extract information about enzyme mechanism from the observed changes in the DNA molecules.

The DNA of all organisms has a complex and essential topology. The three topological properties of naturally occurring DNA are supercoiling, catenation, and knotting. Although these properties are denned rigorously only for closed circular DNA, even linear DNA in vivo can have topological properties because it is divided into topologically separate subdomains (Drlica 1987; Roberge & Gasser, 1992). The essentiality of topological properties is demonstrated by the lethal consequence of interfering with topoisomerases, the enzymes that regulate the level of DNA supercoiling and that unlink DNA during its replication (reviewed in Wang, 1991; Bjornsti, 1991; Drlica, 1992; Ullsperger et al. 1995).

A fundamental and interesting question in knot theory is:Question 1. How many prime knots of n crossings are there ?Over time, knot theorists have answered this question for n ≤ 13 by the method of exhaustion: one writes down a list of all possible knots of n crossings, and then works hard to eliminate duplications from the list [12]. A perhaps easier question is the following:

One challenging problem in biology is to understand the mechanism of DNA packing in a confined volume such as a cell. It is known that confined circular DNA is often knotted and hence the topology of the extracted (and relaxed) circular DNA can be used as a probe of the DNA packing mechanism. However, in order to properly estimate the topological properties of the confined circular DNA structures using mathematical models, it is necessary to generate large ensembles of simulated closed chains (i.e., polygons) of equal edge lengths that are confined in a volume such as a sphere of certain fixed radius. Finding efficient algorithms that properly sample the space of such confined equilateral random polygons is a difficult problem. In this paper we propose a method that generates confined equilateral random polygons based on their probability distribution. This method requires the creation of a large database initially. However, once the database has been created, a confined equilateral random polygon of length n can be generated in linear time in terms of n. The errors introduced by the method can be controlled and reduced by the refinement of the database. Furthermore, our numerical simulations indicate that these errors are unbiased and tend to cancel each other in a long polygon.

For a knot or link K, L(K) denotes the rope length of K and Cr(K) denotes the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well known that there exist positive constants c 1 , c 2 such that for any knot or link K, c 1 • (Cr(K)) 3/4 ≤ L(K) ≤ c 2 • (Cr(K)) 2 . In this paper, we prove that there exists a constant c > 0 such that for any knot or link K, L(K) ≤ c • (Cr(K)) 3/2 . This is done through the study of regular projections of knots and links as 4-regular plane graphs. We show that for any knot (or link) K there exists a knot (or link) K and a regular projection G of K such that K is of the same knot type as K, G has at most 4 • Cr(K) crossings, and G is a Hamiltonian graph. We then use this result to develop an embedding algorithm. Using this algorithm, we are able to embed any knot or link K into the simple cubic lattice such that the length of the embedded knot is of order at most O((Cr(K)) 3/2 ). This result in turn establishes the above mentioned upper bound on L(K) for smooth knots and links. Regular projection graphsIn this section, we introduce some basic concepts and results in knot theory and graph theory. In addition, some new terms are defined as they will be needed in our discussions.A graph G consists of a set V (G), a set E(G), and an incidence relation which says that every element of E(G) is incident with two elements of V (G). The elements of V (G) are called the vertices of G, and V (G) is called the vertex set of G. The elements of E(G) are called the edges of G, and E(G) is called the edge set of G.Let G be a graph. The degree of a vertex v of G is the number of edges of G incident with v. The graph G is called k-regular if every vertex of G is of degree k. Let e be an edge of G incident with vertices u and v. We say that e connects u and v, u and v are the ends of e, and u and v are adjacent. We also use uv or vu to denote e when there is only one edge connecting u and v. If u = v then e is called a loop edge.A graph H is a subgraph of a graph G if V (H) ⊂ V (G) and E(H) ⊂ E(G). In the case that V (H) = V (G), H is called a spanning subgraph of G. Two graphs G and H are said to be isomorphic if there exist bijections f :A path P of length k − 1, where k ≥ 2, is a graph which is isomorphic to the graph with vertex set {v 1 , v 2 , ..., v k } and edge set {v i v i+1 : i = 1, . . . , k − 1}. We say that v 1 and v k are the ends of P , P is from v 1 to v k , and P is between v 1 and v k . A cycle C of length k, where k ≥ 3, is a graph which is isomorphic to the graph with vertex set {v 1 , v 2 , ..., v k } and edge set {v i v i+1 : i = 1, . . . , k − 1} ∪ {v k v 1 }. A Hamilton cycle in a graph G is a spanning subgraph that is a cycle. A graph with a Hamilton cycle is said to be Hamiltonian.A geometric realization of a graph G in R 2 or R 3 is such that the vertices of G are represented by distinct points in the space, every edge of G is represented by a simple curve in R 2 or R 3 connecting the two points representing...

In this paper we continue our earlier studies [5, 6] on the generation methods of random equilateral polygons confined in a sphere. The first half of the paper is concerned with the generation of confined equilateral random walks. We show that if the selection of a vertex is uniform subject to the position of its previous vertex and the confining condition, then the distributions of the vertices are not uniform, although there exists a distribution such that if the initial vertex is selected following this distribution, then all vertices of the random walk follow this same distribution. Thus in order to generate a confined equilateral random walk, the selection of a vertex cannot be uniform subject to the position of its previous vertex and the confining condition. We provide a simple algorithm capable of generating confined equilateral random walks whose vertex distribution is almost uniform in the confinement sphere. In the second half of the paper we show that any process generating confined equilateral random walks can be turned into a process generating confined equilateral random polygons with the property that the vertex distribution of the polygons approaches the vertex distribution of the walks as the polygons get longer and longer. In our earlier studies, the starting point of the confined polygon is fixed at the center of the sphere. The new approach here allows us to move the the starting point of the confined polygon off the center of the sphere.

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0[les ]i[les ]3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form.

No abstract