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.@en

Kinetoplast DNA (kDNA) is a two-dimensional Olympic-ring-like network of mutually linked DNA minicircles found in certain parasites called trypanosomes. Understanding the self-assembly and replication of this structure are not only major open questions in biology but can also inform the design of synthetic topological materials. Here, we report the first high-resolution, single-molecule study of kDNA network topology using AFM and steered molecular dynamics simulations. We map out the DNA density within the network and the distribution of linking number and valence of the minicircles. We also characterize the DNA hubs that surround the network and show that they cause a buckling transition akin to that of a 2D elastic thermal sheet in the bulk. Intriguingly, we observe a broad distribution of density and valence of the minicircles, indicating heterogeneous network structure and individualism of different kDNA structures. Finally, we estimate the 2D Young modulus of the network to be orders of magnitude smaller than that of other 2D materials. Our findings explain outstanding questions in the field and offer single-molecule insights into the properties of a unique topological material. @en