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The topology of biological polymers such as proteins and nucleic acids is an important aspect of their 3D structure. Recently, two applications of topology to molecular chains have emerged as important theoretical developments that are beginning to find utility in heteropolymer characterization and design: namely, circuit topology (CT) and knot theory. Here, we review the application of these two theories to protein, RNA, and DNA/genome structure, focusing on connections to conventional 3D structural information and relevance to function and highlighting recent experimental findings. We conclude with a discussion of recent applications to molecular origami and engineering.

Considerable mechanistic insight has been gained into amyloid aggregation; however, a large class of non-amyloid protein aggregates are considered 'amorphous,' and in most cases little is known about their mechanisms. Amorphous aggregation of {\gamma}-crystallins in the eye lens causes a widespread disease of aging, cataract. We combined simulations and experiments to study the mechanism of aggregation of two {\gamma}D-crystallin mutants, W42R and W42Q - the former a congenital cataract mutation, and the latter a mimic of age-related oxidative damage. We found that formation of an internal disulfide was necessary and sufficient for aggregation under physiological conditions. Two-chain all-atom simulations predicted that one non-native disulfide in particular, between Cys32 and Cys41, was likely to stabilize an unfolding intermediate prone to intermolecular interactions. Mass spectrometry and mutagenesis experiments confirmed the presence of this bond in the aggregates and its necessity for oxidative aggregation under physiological conditions in vitro. Mining the simulation data linked formation of this disulfide to extrusion of the N-terminal \b{eta}-hairpin and rearrangement of the native \b{eta}-sheet topology. Specific binding between the extruded hairpin and a distal \b{eta}-sheet, in an intermolecular chain reaction similar to domain swapping, is the most probable mechanism of aggregate propagation., Comment: *equal contribution; {\dag}corresponding authors