Eleni Panagiotou

The linking number and the writhe of uniform random walks and polygons in confined spaces

Random walks and polygons are used to model polymers. In this paper we consider the extension of the writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form . Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.

A Study of the Entanglement in Systems with Periodic Boundary Conditions

We define the local periodic linking number, LK, between two oriented closed or open chains in a system with three-dimensional periodic boundary conditions. The properties of LK indicate that it is an appropriate measure of entanglement between a collection of chains in a periodic system. Using this measure of linking to assess the extent of entanglement in a polymer melt we study the effect of CReTA algorithm on the entanglement of polyethylene chains. Our numerical results show that the statistics of the local periodic linking number observed for polymer melts before and after the application of CReTA are the same.

The linking number in systems with Periodic Boundary Conditions

Periodic Boundary Conditions (PBC) are often used for the simulation of complex physical systems. Using the Gauss linking number, we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing PBC. In the case of closed chains in PBC, the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system. For open chains, the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system. In this paper we give a rigorous proof that the periodic linking number is defined for the infinite system, i.e., that it converges for one, two, and three PBC models. It gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains. Similarly, for a single oriented chain, we define the periodic self-linking number and prove that it also is defined for open chains. In addition, we define the cell periodic linking and self-linking numbers giving localizations of the periodic linking numbers. These can be used to give good estimates of the periodic linking numbers in infinite systems. We also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers. Finally, we study and compare these measures when applied to a PBC model of polyethylene melts.

Resolving critical degrees of entanglement in Olympic ring systems

Olympic systems are collections of small ring polymers whose aggregate properties are largely characterized by the extent (or absence) of topological linking in contrast with the topological entanglement arising from physical movement constraints associated with excluded volume contacts or arising from chemical bonds. First, discussed by de Gennes, they have been of interest ever since due to their particular properties and their occurrence in natural organisms, for example, as intermediates in the replication of circular DNA in the mitochondria of malignant cells or in the kinetoplast DNA networks of trypanosomes. Here, we study systems that have an intrinsic one, two, or three-dimensional character and consist of large collections of ring polymers modeled using periodic boundary conditions. We identify and discuss the evolution of the dimensional character of the large scale topological linking as a function of density. We identify the critical densities at which infinite linked subsystems, the onset of percolation, arise in the periodic boundary condition systems. These provide insight into the nature of entanglement occurring in such course grained models. This entanglement is measured using Gauss linking number, a measure well adapted to such models. We show that, with increasing density, the topological entanglement of these systems increases in complexity, dimension, and probability.

Linking in Systems with One-Dimensional Periodic Boundaries

With a focus on one-dimensional periodic boundary systems, we describe the application of extensions of the Gauss linking number of closed rings to open chains and, then, to systems of such chains via the periodic linking and periodic self-linking of chains. These lead to the periodic linking matrix and its associated eigenvalues providing measures of entanglement that can be applied to complex systems. We describe the general one-dimensional case and applications to one-dimensional Olympic gels and to tubular filamental structures.