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Symmetry
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Topological Methods in Chemistry and Molecular Biology
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Topological Methods in Chemistry and Molecular Biology

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 16956

Special Issue Editors

Prof. Dr. Erica Flapan

E-Mail Website
Guest Editor
Department of Mathematics, Pomona College, Claremont, CA 91711, USA
Interests: spatial graphs; knot theory; topological symmetry groups; chirality; topological approaches to DNA and proteins
Prof. Dr. Helen Wong

E-Mail Website
Guest Editor
Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA 91711, USA
Interests: quantum topology; quantum invariants; Kauffman bracket skein algebras; topological quantum computation; topological approaches to DNA and proteins

Special Issue Information

This Special Issue is devoted to topological approaches to molecular structures and their symmetries, including biopolymers, synthetic compounds, and crystallographic structures. We describe some possible topics for articles below.

DNA, RNA, proteins, and other biopolymers are flexible enough to exhibit complex topological features in their native state. Biologists believe that the topology of such biopolymers may affect their biological function, though the formation of such features and their exact role is not completely understood. Topology, geometry, dynamics, probability, statistics, and simulations have all been the basis for models of how knots, links, and entanglements may occur in biopolymers, as well as for predictions of topological features that might be discovered in biopolymers in the future.

Symmetry plays an important role in synthesizing compounds and predicting their properties. However, large molecules may not be completely rigid. Thus, their point group may not describe all of their symmetries. Rather, a topological approach which considers their deformations in addition to their rotations and reflections may be useful. For knotted and linked molecules obtained by self-assembly, comparing their topological symmetries to NMR data can help to identify their structure. Furthermore, topology can be used in describing and analyzing crystallographic and other symmetric structures.

Prof. Dr. Erica Flapan
Prof. Dr. Helen Wong
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • biopolymers
  • DNA
  • RNA
  • proteins
  • crystal structures
  • chirality
  • point group
  • topology
  • geometry
  • spatial graphs
  • knots
  • links
  • entanglements
  • topological symmetry groups
  • topological software
  • self-assembly

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (7 papers)

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Research

13 pages, 9765 KiB  
Article
Symmetric Tangling of Honeycomb Networks
by Myfanwy E. Evans and Stephen T. Hyde
Symmetry 2022, 14(9), 1805; https://doi.org/10.3390/sym14091805 - 31 Aug 2022
Cited by 2 | Viewed by 2262
Abstract
Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands [...] Read more.
Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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39 pages, 1576 KiB  
Article
A Catalog of Enumeration Formulas for Bouquet and Dipole Embeddings under Symmetries
by Mark N. Ellingham and Joanna A. Ellis-Monaghan
Symmetry 2022, 14(9), 1793; https://doi.org/10.3390/sym14091793 - 29 Aug 2022
Cited by 1 | Viewed by 1357
Abstract
Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, [...] Read more.
Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, including RNA secondary structures, Feynman diagrams, and topological graph theory. For uncolored objects, we catalog 58 distinct sequences, of which 43 have not, as far as we know, been described previously. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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13 pages, 918 KiB  
Article
Invariants of Bonded Knotoids and Applications to Protein Folding
by Neslihan Gügümcü, Bostjan Gabrovsek and Louis H. Kauffman
Symmetry 2022, 14(8), 1724; https://doi.org/10.3390/sym14081724 - 18 Aug 2022
Cited by 7 | Viewed by 2047
Abstract
In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that [...] Read more.
In this paper, we study knotoids with extra graphical structure (bonded knotoids) in the settings of rigid vertex and topological vertex graphs. We construct bonded knotoid invariants by applying tangle insertion and unplugging at bonding sites of a bonded knotoid. We demonstrate that our invariants can be used for the analysis of the topological structure of proteins. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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11 pages, 3283 KiB  
Article
On the Classification of Polyhedral Links
by Naohiro Wakayama and Koya Shimokawa
Symmetry 2022, 14(8), 1712; https://doi.org/10.3390/sym14081712 - 17 Aug 2022
Cited by 1 | Viewed by 1772
Abstract
Knots and links are ubiquitous in chemical systems. Their structure can be responsible for a variety of physical and chemical properties, making them very important in materials development. In this article, we analyze the topological structures of interlocking molecules composed of metal-peptide rings [...] Read more.
Knots and links are ubiquitous in chemical systems. Their structure can be responsible for a variety of physical and chemical properties, making them very important in materials development. In this article, we analyze the topological structures of interlocking molecules composed of metal-peptide rings using the concept of polyhedral links. To that end, we discuss the topological classification of alternating polyhedral links. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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19 pages, 493 KiB  
Article
Circuit Topology for Bottom-Up Engineering of Molecular Knots
by Anatoly Golovnev and Alireza Mashaghi
Symmetry 2021, 13(12), 2353; https://doi.org/10.3390/sym13122353 - 7 Dec 2021
Cited by 5 | Viewed by 3765
Abstract
The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions, and their entanglement is believed to [...] Read more.
The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions, and their entanglement is believed to provide them with thermal and kinetic stability. Yet, little is known about the design principles of naturally evolved molecular knots. Intra-chain contacts and chain entanglement contribute to the folding of knotted proteins. Circuit topology, a theory that describes intra-chain contacts, was recently generalized to account for chain entanglement. This generalization is unique to circuit topology and not motivated by other theories. In this conceptual paper, we systematically analyze the circuit topology approach to a description of linear chain entanglement. We utilize a bottom-up approach, i.e., we express entanglement by a set of four fundamental structural units subjected to three (or five) binary topological operations. All knots found in proteins form a well-defined, distinct group which naturally appears if expressed in terms of these basic structural units. We believe that such a detailed, bottom-up understanding of the structure of molecular knots should be beneficial for molecular engineering. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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18 pages, 7855 KiB  
Article
A Topological Selection of Folding Pathways from Native States of Knotted Proteins
by Agnese Barbensi, Naya Yerolemou, Oliver Vipond, Barbara I. Mahler, Pawel Dabrowski-Tumanski and Dimos Goundaroulis
Symmetry 2021, 13(9), 1670; https://doi.org/10.3390/sym13091670 - 10 Sep 2021
Cited by 3 | Viewed by 2700
Abstract
Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different [...] Read more.
Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins’ native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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12 pages, 249 KiB  
Article
Coloring Invariant for Topological Circuits in Folded Linear Chains
by Jose Ceniceros, Mohamed Elhamdadi and Alireza Mashaghi
Symmetry 2021, 13(6), 919; https://doi.org/10.3390/sym13060919 - 21 May 2021
Cited by 4 | Viewed by 1905
Abstract
Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural [...] Read more.
Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural symmetries stabilised by formation of intrachain contacts. Here, we adapt and apply the algebraic structure of quandles to classify and distinguish chain topologies within the framework of circuit topology. We systematically study the basic circuit topology motifs and define quandle/bondle coloring for them. Next, we explore the implications of circuit topology operations that enable building complex topologies from basic motifs for the quandle coloring approach. Full article
(This article belongs to the Special Issue Topological Methods in Chemistry and Molecular Biology)
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