On Borromean links and related structures

The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang & Mislow (1994). J. Math. Chem. 16, 27–35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20–25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.

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Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

10.26434/chemrxiv.12616268.v1 ◽

2020 ◽

Author(s):

Michael O'Keeffe ◽

Michael Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

<p>We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed <i>regular</i>. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.</p>

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Isogonal weavings on the sphere: knots, links, polycatenanes

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273320010669 ◽

2020 ◽

Vol 76 (5) ◽

pp. 611-621

Author(s):

Michael O'Keeffe ◽

Michael M. J. Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Materials Chemistry ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

Mathematical knots and links are described as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

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Network Routing on Regular Directed Graphs from Spanning Factorizations

Bulletin of Mathematical Sciences and Applications ◽

10.18052/www.scipress.com/bmsa.20.9 ◽

2018 ◽

Vol 20 ◽

pp. 9-24

Author(s):

Randall Dougherty ◽

Vance Faber

Keyword(s):

Parallel Machines ◽

Cayley Graphs ◽

Directed Graphs ◽

Network Routing ◽

Parallel Processor ◽

Single Node ◽

Routing Schemes ◽

Vertex Transitive ◽

High Degree ◽

Transitive Graphs

Networks with a high degree of symmetry are useful models for parallel processor networks. In earlier papers, we defined several global communication tasks (universal exchange, universal broadcast, universal summation) that can be critical tasks when complex algorithms are mapped to parallel machines. We showed that utilizing the symmetry can make network optimization a tractable problem. In particular, we showed that Cayley graphs have the desirable property that certain routing schemes starting from a single node can be transferred to all nodes in a way that does not introduce conflicts. In this paper, we define the concept of spanning factorizations and show that this property can also be used to transfer routing schemes from a single node to all other nodes. We show that all Cayley graphs and many (perhaps all) vertex transitive graphs have spanning factorizations.

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Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

10.26434/chemrxiv.12616268 ◽

2020 ◽

Author(s):

Michael O'Keeffe ◽

Michael Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

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Electron Microscopy in Molecular Biology

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100060027 ◽

1967 ◽

Vol 25 ◽

pp. 2-3

Author(s):

Cecil E. Hall

Keyword(s):

Electron Microscopy ◽

Molecular Biology ◽

Noise Level ◽

Molecular Structures ◽

Material Structure ◽

The Arts ◽

Shadow Casting ◽

Electron Image ◽

High Degree ◽

Very High

The visualization of organic macromolecules such as proteins, nucleic acids, viruses and virus components has reached its high degree of effectiveness owing to refinements and reliability of instruments and to the invention of methods for enhancing the structure of these materials within the electron image. The latter techniques have been most important because what can be seen depends upon the molecular and atomic character of the object as modified which is rarely evident in the pristine material. Structure may thus be displayed by the arts of positive and negative staining, shadow casting, replication and other techniques. Enhancement of contrast, which delineates bounds of isolated macromolecules has been effected progressively over the years as illustrated in Figs. 1, 2, 3 and 4 by these methods. We now look to the future wondering what other visions are waiting to be seen. The instrument designers will need to exact from the arts of fabrication the performance that theory has prescribed as well as methods for phase and interference contrast with explorations of the potentialities of very high and very low voltages. Chemistry must play an increasingly important part in future progress by providing specific stain molecules of high visibility, substrates of vanishing “noise” level and means for preservation of molecular structures that usually exist in a solvated condition.

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Scanning tunneling microscopy (STM) of DNA and RNA

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100152148 ◽

1989 ◽

Vol 47 ◽

pp. 34-35

Author(s):

Patricia G. Arscott ◽

Gil Lee ◽

Victor A. Bloomfield ◽

D. Fennell Evans

Keyword(s):

Molecular Structures ◽

Inorganic Materials ◽

Resolving Power ◽

Scanning Tunneling ◽

Tunneling Microscopy ◽

Form And Function ◽

Dna And Rna ◽

Calf Thymus ◽

And Function ◽

The Relationship

STM is one of the most promising techniques available for visualizing the fine details of biomolecular structure. It has been used to map the surface topography of inorganic materials in atomic dimensions, and thus has the resolving power not only to determine the conformation of small molecules but to distinguish site-specific features within a molecule. That level of detail is of critical importance in understanding the relationship between form and function in biological systems. The size, shape, and accessibility of molecular structures can be determined much more accurately by STM than by electron microscopy since no staining, shadowing or labeling with heavy metals is required, and there is no exposure to damaging radiation by electrons. Crystallography and most other physical techniques do not give information about individual molecules.We have obtained striking images of DNA and RNA, using calf thymus DNA and two synthetic polynucleotides, poly(dG-me5dC)·poly(dG-me5dC) and poly(rA)·poly(rU).

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Molecular architecture and functions of the neuronal cytoskeleton

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100157541 ◽

1990 ◽

Vol 48 (3) ◽

pp. 2-3

Author(s):

Nobutaka Hirokawa

Keyword(s):

Major Elements ◽

Molecular Structures ◽

Organelle Transport ◽

Molecular Architecture ◽

Neuronal Morphogenesis ◽

Microtubule Associated Proteins ◽

Associated Proteins ◽

And Function ◽

Small Clusters

In this symposium I will present our studies about the molecular architecture and function of the cytomatrix of the nerve cells. The nerve cell is a highly polarized cell composed of highly branched dendrites, cell body, and a single long axon along the direction of the impulse propagation. Each part of the neuron takes characteristic shapes for which the cytoskeleton provides the framework. The neuronal cytoskeletons play important roles on neuronal morphogenesis, organelle transport and the synaptic transmission. In the axon neurofilaments (NF) form dense arrays, while microtubules (MT) are arranged as small clusters among the NFs. On the other hand, MTs are distributed uniformly, whereas NFs tend to run solitarily or form small fascicles in the dendrites Quick freeze deep etch electron microscopy revealed various kinds of strands among MTs, NFs and membranous organelles (MO). These structures form major elements of the cytomatrix in the neuron. To investigate molecular nature and function of these filaments first we studied molecular structures of microtubule associated proteins (MAP1A, MAP1B, MAP2, MAP2C and tau), and microtubules reconstituted from MAPs and tubulin in vitro. These MAPs were all fibrous molecules with different length and formed arm like projections from the microtubule surface.

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