Claus Ernst

Analysis of the mechanism of DNA recombination using tangles

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).

The complexity of lattice knots

Abstract A family of polygonal knots Kn on the cubical lattice is constructed with the property that the quotient of length L(Kn) over the crossing number Cr(Kn) approaches zero as L approaches infinity. More precisely Cr(K n ) = O(L(K n ) 4 3 ) . It is shown that this construction is optimal in the sense that for any knot K on the cubical lattice with length L and Cr crossings Cr ⩽ 3.2L 4 3 .

Thicknesses of knots

In this paper we define a set of radii called thickness for simple closed curves denoted by K , which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

Generating equilateral random polygons in confinement III

In this paper we continue an earlier study (Diao et al 2011 J. Phys. A: Math. Theor. 44 405202) on the generation algorithms of random equilateral polygons confined in a sphere. Here, the equilateral random polygons are rooted at the center of the confining sphere and the confining sphere behaves like an absorbing boundary. One way to generate such a random polygon is the accept/reject method in which an unconditioned equilateral random polygon rooted at origin is generated. The polygon is accepted if it is within the confining sphere, otherwise it is rejected and the process is repeated. The algorithm proposed in this paper offers an alternative to the accept/reject method, yielding a faster generation process when the confining sphere is small. In order to use this algorithm effectively, a large, reusable data set needs to be pre-computed only once. We derive the theoretical distribution of the given random polygon model and demonstrate, with strong numerical evidence, that our implementation of the algorithm follows this distribution. A run time analysis and a numerical error estimate are given at the end of the paper.

Protein structure determination via an efficient geometric build-up algorithm

BackgroundA protein structure can be determined by solving a so-called distance geometry problem whenever a set of inter-atomic distances is available and sufficient. However, the problem is intractable in general and has proved to be a NP hard problem. An updated geometric build-up algorithm (UGB) has been developed recently that controls numerical errors and is efficient in protein structure determination for cases where only sparse exact distance data is available. In this paper, the UGB method has been improved and revised with aims at solving distance geometry problems more efficiently and effectively.MethodsAn efficient algorithm (called the revised updated geometric build-up algorithm (RUGB)) to build up a protein structure from atomic distance data is presented and provides an effective way of determining a protein structure with sparse exact distance data. In the algorithm, the condition to determine an unpositioned atom iteratively is relaxed (when compared with the UGB algorithm) and data structure techniques are used to make the algorithm more efficient and effective. The algorithm is tested on a set of proteins selected randomly from the Protein Structure Database-PDB.ResultsWe test a set of proteins selected randomly from the Protein Structure Database-PDB. We show that the numerical errors produced by the new RUGB algorithm are smaller when compared with the errors of the UGB algorithm and that the novel RUGB algorithm has a significantly smaller runtime than the UGB algorithm.ConclusionsThe RUGB algorithm relaxes the condition for updating and incorporates the data structure for accessing neighbours of an atom. The revisions result in an improvement over the UGB algorithm in two important areas: a reduction on the overall runtime and decrease of the numeric error.

Knot Energies by Ropes

An energy function for smooth knots is defined using the concept of thickness and proved to be a good indicator of the complexity of knots. This energy function has the advantage that it can be measured experimentally. We carry out an experiment to measure the energies for knots up to 8 crossings and compare our data with energies obtained elsewhere through numerical simulations.

Total curvature, ropelength and crossing number of thick knots

We first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let be a knot or link with a lattice embedding of minimum total curvature among all possible lattice embeddings of . We show that there exist positive constants c1 and c2 such that for any knot type . Furthermore we show that the powers of in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

THE LINEAR GROWTH IN THE LENGTHS OF A FAMILY OF THICK KNOTS

For any given knot K, a thick realization K0 of K is a knot of unit thickness which is of the same knot type as K. In this paper, we show that there exists a family of prime knots {Kn} with the property that Cr(Kn)→∞(as n→∞) such that the arc-length of any thick realization of Kn will grow at least linearly with respect to Cr(Kn).

NULLIFICATION OF KNOTS AND LINKS

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, the nullification process may be the pathway for a knotted circular DNA to unknot itself (through recombination of its DNA strands). The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. It turns out that there are several different ways to define such a number, since different conditions may be applied in the nullification process. We show that these definitions are not equivalent, thus they lead to different nullification numbers for a knot/link in general, not just one single nullification number. Our aim is to explore some mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different....

UPPER BOUNDS ON LINKING NUMBERS OF THICK LINKS

The maximum of the linking number between two lattice polygons of lengths n1, n2 (with n1 ≤ n2) is proven to be the order of n1 (n2)⅓. This result is generalized to smooth links of unit thickness. The result also implies that the writhe of a lattice knot K of length n is at most 26 n4/3/π. In the second half of the paper examples are given to show that linking numbers of order n1 (n2)⅓ can be obtained when