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Polymers are a primary building block in many biomaterials, often interacting with anisotropic backgrounds. While previous studies have considered polymer dynamics within nematic solvents, rarely are the effects of anisotropic viscosity and polymer elongation differentiated. Here, we study polymers embedded in nematic liquid crystals with isotropic viscosity via numerical simulations to explicitly investigate the effect of nematicity on macromolecular conformation and how conformation alone can produce anisotropic dynamics. We employ a hybrid multi-particle collision dynamics and molecular dynamics technique that captures nematic orientation, thermal fluctuations and hydrodynamic interactions. The coupling of the polymer segments to the director field of the surrounding nematic elongates the polymer, producing anisotropic diffusion even in nematic solvents with isotropic viscosity. For intermediate coupling, the competition between background anisotropy and macromolecular entropy leads to hairpins – sudden kinks along the backbone of the polymer. Experiments of DNA embedded in a solution of rod-like fd viruses qualitatively support the role of hairpins in establishing characteristic conformational features that govern polymer dynamics. Hairpin diffusion along the backbone exponentially slows as coupling increases. Better understanding two-way coupling between polymers and their surroundings could allow the creation of more biomimetic composite materials. 

Abstract Knots in open strands such as ropes, fibers, and polymers, cannot typically be described in the language of knot theory, which characterizes only closed curves in space. Simulations of open knotted polymer chains, often parameterized to DNA, typically perform a closure operation and calculate the Alexander polynomial to assign a knot topology. This is limited in scenarios where the topology is less well-defined, for example when the chain is in the process of untying or is strongly confined. Here, we use a discretized version of the Second Vassiliev Invariant for open chains to analyze Langevin Dynamics simulations of untying and strongly confined polymer chains. We demonstrate that the Vassiliev parameter can accurately and efficiently characterize the knotted state of polymers, providing additional information not captured by a single-closure Alexander calculation. We discuss its relative strengths and weaknesses compared to standard techniques, and argue that it is a useful and powerful tool for analyzing polymer knot simulations. 

In this study, we simulate mechanically interlocked semiflexible ring polymers inspired by the minicircles of kinetoplast DNA (kDNA) networks. Using coarse-grained molecular dynamics simulations, we investigate the impact of molecular topological linkage and nanoconfinement on the conformational properties of two- and three-ring polymer systems in varying solvent qualities. Under good-quality solvents, for two-ring systems, a higher number of crossing points lead to a more internally constrained structure, reducing their mean radius of gyration. In contrast, three-ring systems, which all had the same crossing number, exhibited more similar sizes. In unfavorable solvents, structures collapse, forming compact configurations with increased contacts. The morphological diversity of structures primarily arises from topological linkage rather than the number of rings. In three-ring systems with different topological conformations, structural uniformity varies based on link types. Extreme confinement induces isotropic and extended conformations for catenated polymers, aligning with experimental results for kDNA networks and influencing the crossing number and overall shape. Finally, the flat-to-collapse transition in extreme confinement occurs earlier (at relatively better solvent conditions) compared to non-confined systems. This study offers valuable insights into the conformational behavior of mechanically interlocked ring polymers, highlighting challenges in extrapolating single-molecule analyses to larger networks such as kDNA. 

The last years have witnessed remarkable advances in our understanding of the emergence and consequences of topological constraints in biological and soft matter. Examples are abundant in relation to (bio)polymeric systems and range from the characterization of knots in single polymers and proteins to that of whole chromosomes and polymer melts. At the same time, considerable advances have been made in the description of the interplay between topological and physical properties in complex fluids, with the development of techniques that now allow researchers to control the formation of and interaction between defects in diverse classes of liquid crystals. Thanks to technological progress and the integration of experiments with increasingly sophisticated numerical simulations, topological biological and soft matter is a vibrant area of research attracting scientists from a broad range of disciplines. However, owing to the high degree of specialization of modern science, many results have remained confined to their own particular fields, with different jargon making it difficult for researchers to share ideas and work together towards a comprehensive view of the diverse phenomena at play. Compelled by these motivations, here we present a comprehensive overview of topological effects in systems ranging from DNA and genome organization to entangled proteins, polymeric materials, liquid crystals, and theoretical physics, with the intention of reducing the barriers between different fields of soft matter and biophysics. Particular care has been taken in providing a coherent formal introduction to the topological properties of polymers and of continuum materials and in highlighting the underlying common aspects concerning the emergence, characterization, and effects of topological objects in different systems. The second half of the review is dedicated to the presentation of the latest results in selected problems, specifically, the effects of topological constraints on the viscoelastic properties of polymeric materials; their relation with genome organization; a discussion on the emergence and possible effects of knots and other entanglements in proteins; the emergence and effects of topological defects and solitons in complex fluids. This review is dedicated to the memory of Marek Cieplak. 

Abstract We use graph theory simulations and single molecule experiments to investigate percolation properties of kinetoplasts, the topologically linked mitochondrial DNA from trypanosome parasites. The edges of some kinetoplast networks contain a fiber of redundantly catenated DNA loops, but previous investigations of kinetoplast topology did not take this into account. Our graph simulations track the size of connected components in lattices as nodes are removed, analogous to the removal of minicircles from kinetoplasts. We find that when the edge loop is taken into account, the largest component after the network de‐percolates is a remnant of the edge loop, before it undergoes a second percolation transition and breaks apart. This implies that stochastically removing minicircles from kinetoplast DNA would isolate large polycatenanes, which is observed in experiments that use photonicking to stochastically destroy kinetoplasts fromCrithidia fasciculata. Our results imply kinetoplasts may be used as a source of linear polycatenanes for future experiments.