Resolving critical degrees of entanglement in Olympic ring systems

Abstract

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 threedimensional 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.

Geometric Predictors of Knotted and Linked Arcs

“…We speculate that (inter-chain) local density, 1D and 3D writhe as defined in this work may yield interesting results not only in melts of ring polymers but also in molecular (and periodic) weavings. 2,[40][41][42] We expect that different entanglement motifs are associated with distinct patterns of our geometric observables. In turn, they may be used to predict the global elastic response of the entangled network to certain perturbations.…”

“…Entanglements are often poorly defined and their unambiguous identification and quantification remains elusive. 1,2 For example, a knot is a well defined mathematical entity when tied on a closed curve, but there are many examples in physics and biology, e.g. proteins and chromatin, where knots are tied on open curves, rendering such "physical" knots much more difficult to define rigorously and unambiguously.…”

“…38,39 In parallel to these open questions, it is clear that the geometric design of systems with specific entanglements in their microstructure could in principle allow for the control of mesoscopic material properties. 2,40,41 The realisation of woven structures can now be achieved at both micro-and meso-scales using synthetic chemistry 42 or 3D printing. 41 To bypass a virtually endless trial-and-error approach, it is therefore important to be able to select the entanglement motifs to embroider in the structures in such a way that they display the desired mechanical properties.…”

“…Topological entanglements are ubiquitous and an essential feature of everyday materials and complex fluids, endowing them with viscous and elastic properties. Entanglements are often poorly defined, and their unambiguous identification and quantification remain elusive. , For example, a knot is a well-defined mathematical entity when tied on a closed curve, but there are many examples in physics and biology, e.g., proteins and chromatin, where knots are tied on open curves, rendering such “physical” knots much more difficult to define rigorously and unambiguously. − More broadly, a long-standing goal in polymer physics and the broader soft matter communities is to understand and control the topology of certain systems from the geometry of (often entangled) 1D curves. This goal encompasses many fields, such as liquid crystals, optics, , fluids, DNA, ,− proteins, , polymers, ,− soap films, , and soft matter in general. , At the same time, the unambiguous characterization of entanglements in these systems is often elusive, in turn begging for better strategies to quantify entanglements in generic soft matter systems.…”

Swelling of Tendomer Gels

“…Polyrotaxanes are just one prominent example of interlocked precursor molecules among others like polycatenanes or daisy chains − that could be used to construct topological networks by the addition of covalent bonds. An alternative strategy to prepare topological networks is the formation of Olympic gels through catenation, which was studied mainly on a theoretical basis in past decades − with a first experimental realization reported recently …”