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Abstract Neurodegenerative diseases, like Alzheimer’s, are associated with the presence of neurofibrillary lesions formed by tau protein filaments in the cerebral cortex. While it is known that different morphologies of tau filaments characterize different neurodegenerative diseases, there are few metrics of global and local structure complexity that enable to quantify their structural diversity rigorously. In this manuscript, we employ for the first time mathematical topology and geometry to classify neurodegenerative diseases by using cryo-electron microscopy structures of tau filaments that are available in the Protein Data Bank. By employing mathematical topology metrics (Gauss linking integral, writhe and second Vassiliev measure) we achieve a consistent, but more refined classification of tauopathies, than what was previously observed through visual inspection. Our results reveal a hierarchy of classification from global to local topology and geometry characteristics. In particular, we find that tauopathies can be classified with respect to the handedness of their global conformations and the handedness of the relative orientations of their repeats. Progressive supranuclear palsy is identified as an outlier, with a more complex structure than the rest, reflected by a small, but observable knotoid structure (a diagrammatic structure representing non-trivial topology). This topological characteristic can be attributed to a pattern in the beginning of the R3 repeat that is present in all tauopathies but at different extent. Moreover, by comparing single filament to paired filament structures within tauopathies we find a consistent change in the side-chain orientations with respect to the alpha carbon atoms at the area of interaction. 

Abstract Entanglement of collections of filaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBCs), which create an infinite periodic system whose global entanglement may be impossible to capture and is repetitive. We introduce two new methods to assess topological entanglement in PBC: the Periodic Jones polynomial and the Cell Jones polynomial. These tools capture the grain of geometric/topological entanglement in a periodic system of open or closed chains, by using a finite link as a representative of the global system. These polynomials are topological invariants in some cases, but in general are sensitive to both the topology and the geometry of physical systems. For a general system of 1 closed chain in 1 PBC, we prove that the Periodic Jones polynomial is a recurring factor, up to a remainder, of the Jones polynomial of a conveniently chosen finite cutoff of arbitrary size of the infinite periodic system. We apply the Cell Jones polynomial and the Periodic Jones polynomial to physical PBC systems such as 3D realizations of textile motifs and polymer melts of linear chains obtained from molecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective geometric/topological entanglement complexityin such systems of physical relevance. 

We numerically estimate the superposition of the HOMFLY-PT polynomial of an open two-component link, define its spread, and describe how this quantity may be employed to quantify the degree of entanglement of confined two component open links. 

California blackworms serve as a template for the topological design of active matter