Eleni Panagiotou’s research while affiliated with Arizona State University and other places

What is this page?


This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.

Publications (12)


Figure 1: (Left) Examples of (polygonal) knotoids (open simple arc diagrams). Notice that knotoids refer to projections of open chains, while knots refer to closed chains in 3-space. (Right) Forbidden moves on knotoids. Knotoids are classified via Reidemeister moves and the forbidden moves.
Figure 2: Generalised Reidemeister moves on virtual knots/links : the classical Reidemeister moves, R 1 , R 2 , R 3 ; the virtual moves, V 1 , V 2 , V 3 ; and the semi-virtual move SV . (Figure from [7].)
Figure 5: Parts of projections of two mutant open curves in the neighborhood of two mutant knot embeddings. Outside the circle, both projections are identical. Inside the circle, the two diagrams may not be related by rotation.
On the neighborhood of knots
  • Preprint
  • File available

October 2024

·

23 Reads

Eleni Panagiotou

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the knot type via its knot-type knotoids. We prove that the pure knotoids in the knotoid spectra of a knot, which are individually agnostic of the knot type, can distinguish knots of Gordian distance greater than one. We also prove that the neighborhood of some embeddings of the unknot can be distinguished from any embedding of any non-trivial knot that satisfies the cosmetic crossing conjecture. Topological invariants of knots can be extended to their open curve neighborhood to define continuous functions in the neighborhood of knots. We discuss their properties and prove that invariants in the neighborhood of knots may be able to distinguish more knots than their application to the knots themselves. For example, we prove that an invariant of knots that fails to distinguish mutant knots (and mutant knotoids), can distinguish them by their neighborhoods, unless it also fails to distinguish non-mutant pure knotoids in their spectra. Studying the neighborhood of knots opens the possibility of answering questions, such as if an invariant can detect the unknot, via examining possibly easier questions, such as whether it can distinguish height one knotoids from the trivial knotoid.

Download

The Virtual Spectrum of Linkoids and Open Curves in 3-space

April 2024

·

25 Reads

·

2 Citations

Journal of Knot Theory and Its Ramifications

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.


The Jones polynomial in systems with periodic boundary conditions

April 2024

·

35 Reads

·

3 Citations

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 (PBC), 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 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 frommolecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective entanglement complexity in such systems of physical relevance.


Mathematical topology and geometry-based classification of tauopathies

March 2024

·

57 Reads

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.


The Virtual Spectrum of Linkoids and Open Curves in 3-space

October 2023

·

120 Reads

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.


The Jones polynomial in systems with periodic boundary conditions

September 2023

·

76 Reads

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 (PBC), 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 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 entanglement complexity in such systems of physical relevance.


HOMFLY-PT polynomials of open links

May 2023

·

2 Reads

Journal of Knot Theory and Its Ramifications

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.




Figure 2. (a) The distribution of the local writhe values in the PDB sample. (b) The local topological free energy in writhe, Π Wr , in the PDB sample. Figure from [44]. The distribution of local torsion and Π T can be found in [44].
Figure 3. (a) From left to right, snapshots of SARS-CoV-2 pre-fusion Spike protein at four stages: uncleaved closed (6ZGE, all 3 RBD down), cleaved closed (6ZGI, all 3 RBD down, proteolytically cleaved), cleaved open (6ZGG, one RBD up), and intermediate (6ZGH, RBD has been removed). (b) The normalized total local Π Wr for SARS-CoV-2 protein at the four pre-fusion stages. (c) The distribution of the normalized total local Π Wr -values for SARS-CoV-2 protein domains at the four pre-fusion stages. Crystal structure images were pulled from the Protein Data Bank [9].
High local topological free energy conformations of SARS-CoV-2 (WT).
The Local Topological Free Energy of the SARS-CoV-2 Spike Protein

July 2022

·

28 Reads

·

6 Citations

Polymers

The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its Spike protein. Understanding protein rearrangement and identifying specific amino acids where mutations affect protein rearrangement has attracted much attention for drug development. In this manuscript, we use a mathematical method to characterize the local topology/geometry of the SARS-CoV-2 Spike protein backbone. Our results show that local conformational changes in the FP, HR1, and CH domains are associated with global conformational changes in the RBD domain. The SARS-CoV-2 variants analyzed in this manuscript (alpha, beta, gamma, delta Mink, G614, N501) show differences in the local conformations of the FP, HR1, and CH domains as well. Finally, most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function. Namely, the residues 484, 570, 614, 796, and 969, which are present in variants of concern and are targeted as important in protein function, are predicted as such from our model.


Citations (4)


... Then ϕ is not surjective nor injective [16,37]. Similar results exist for linkoids [1,7]. Definition 2.4 (height, virtual crossing number). ...

Reference:

On the neighborhood of knots
The Virtual Spectrum of Linkoids and Open Curves in 3-space

Journal of Knot Theory and Its Ramifications

... In this direction, several numerical, polynomial, and finite-type invariants have been constructed for doubly periodic textile structures, such as woven and knitted fabrics (e.g., works initiated by Grishanov et al., cf. [11][12][13][14][15][16][17][18]), which form a particular subclass of DP tangles. The equivalence relation that these invariants respect is based on assumptions of minimal motifs, that is motifs that are minimal for reproducing the DP tangles under two-periodic boundary conditions. ...

The Jones polynomial in systems with periodic boundary conditions

... in. 55,56 The conformational properties of the polymer chain, among others its mean squared end-to-end distance ⟨R 2 ⟩, and mean squared radius of gyration ⟨R 2 g ⟩, describe the coil transformation into a stretched string of segments subject to applied shear, as well as the globule transformation into pearl-necklace conformations. ...

A computational package for measuring Topological Entanglement in Polymers, Proteins and Periodic systems (TEPPP)
  • Citing Article
  • May 2023

Computer Physics Communications

... Building on this, Baldwin and Panagiotou [3] introduced a new measure of local topological and geometrical free energy based on the writhe and torsion of protein chains, highlighting its critical role in the rate-limiting steps of protein folding. Furthermore, Baldwin et al. [4] extended these topological concepts to the study of the SARS-CoV-2 Spike protein, showing how local geometric features such as writhe and torsion influence its stability and behavior. ...

The Local Topological Free Energy of the SARS-CoV-2 Spike Protein

Polymers