scholarly journals Persistent homology of unweighted complex networks via discrete Morse theory

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Harish Kannan ◽  
Emil Saucan ◽  
Indrava Roy ◽  
Areejit Samal
Keyword(s):  
Complex Networks ◽  
Morse Theory ◽  
Persistent Homology ◽  
Higher Order ◽  
Topological Data Analysis ◽  
Discrete Morse Theory ◽  
Order Structure ◽  
Global Topology ◽  
Minimum Number ◽  
Theoretical Minimum

Abstract Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.

scholarly journals Higher-order structure of polymer melt described by persistent homology

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yohei Shimizu ◽  
Takanori Kurokawa ◽  
Hirokazu Arai ◽  
Hitoshi Washizu
Keyword(s):  
Cell Size ◽  
Md Simulation ◽  
Persistent Homology ◽  
Polymer Melt ◽  
Higher Order ◽  
Polymer Materials ◽  
Order Structure ◽  
Materials Informatics ◽  
Optimal Method ◽  
Amorphous Nature

AbstractThe optimal method of the polymer Materials Informatics (MI) has not been developed because the amorphous nature of the higher-order structure affects these properties. We have now tried to develop the polymer MI’s descriptor of the higher-order structure using persistent homology as the topological method. We have experimentally studied the influence of the MD simulation cell size as the higher-order structure of the polymer on its electrical properties important for a soft material sensor or actuator device. The all-atom MD simulation of the polymer has been calculated and the obtained atomic coordinate has been analyzed by the persistent homology. The change in the higher-order structure by different cell size simulations affects the dielectric constant, although these changes are not described by a radial distribution function (RDF). On the other hand, using the 2nd order persistent diagram (PD), it was found that when the cell size is small, the island-shaped distribution become smoother as the cell size increased. There is the same tendency for the condition of change in the monomer ratio, the polymer chain length or temperature. As a result, the persistent homology may express the higher-order structure generated by the MD simulation as a descriptor of the polymer MI.

Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theory

2012 ◽  
Vol 28 (10) ◽  
pp. 959-969 ◽  
Author(s):  
David Günther ◽  
Jan Reininghaus ◽  
Hubert Wagner ◽  
Ingrid Hotz
Keyword(s):  
Morse Theory ◽  
Persistent Homology ◽  
Discrete Morse Theory ◽  
Efficient Computation

Macrocycles of Higher Ortho-Phenylenes: Assembly and Folding

2019 ◽  
Author(s):  
Zacharias Kinney ◽  
Viraj Kirinda ◽  
Scott Hartley
Keyword(s):  
Chemical Shift ◽  
Higher Order ◽  
Order Structure ◽  
Complex Geometries ◽  
Spatial Relationships ◽  
Predominant Species ◽  
Bite Angle ◽  
Direct Assembly

<p>Higher-order structure in abiotic foldamer systems represents an important but largely unrealized goal. As one approach to this challenge, covalent assembly can be used to assemble macrocycles with foldamer subunits in well-defined spatial relationships. Such systems have previously been shown to exhibit self-sorting, new folding motifs, and dynamic stereoisomerism, yet there remain important questions about the interplay between folding and macrocyclization and the effect of structural confinement on folding behavior. Here, we explore the dynamic covalent assembly of extended <i>ortho</i>-phenylenes (hexamer and decamer) with rod-shaped linkers. Characteristic <sup>1</sup>H chemical shift differences between cyclic and acyclic systems can be compared with computational conformer libraries to determine the folding states of the macrocycles. We show that the bite angle provides a measure of the fit of an <i>o</i>-phenylene conformer within a shape-persistent macrocycle, affecting both assembly and ultimate folding behavior. For the <i>o</i>-phenylene hexamer, the bite angle and conformer stability work synergistically to direct assembly toward triangular [3+3] macrocycles of well-folded oligomers. For the decamer, the energetic accessibility of conformers with small bite angles allows [2+2] macrocycles to be formed as the predominant species. In these systems, the <i>o</i>-phenylenes are forced into unusual folding states, preferentially adopting a backbone geometry with distinct helical blocks of opposite handedness. The results show that simple geometric restrictions can be used to direct foldamers toward increasingly complex geometries.</p>

Macrocycles of Higher ortho-Phenylenes: Assembly and Folding

2019 ◽  
Author(s):  
Zacharias Kinney ◽  
Viraj Kirinda ◽  
Scott Hartley
Keyword(s):  
Chemical Shift ◽  
Higher Order ◽  
Order Structure ◽  
Complex Geometries ◽  
Spatial Relationships ◽  
Predominant Species ◽  
Bite Angle ◽  
Direct Assembly

<p>Higher-order structure in abiotic foldamer systems represents an important but largely unrealized goal. As one approach to this challenge, covalent assembly can be used to assemble macrocycles with foldamer subunits in well-defined spatial relationships. Such systems have previously been shown to exhibit self-sorting, new folding motifs, and dynamic stereoisomerism, yet there remain important questions about the interplay between folding and macrocyclization and the effect of structural confinement on folding behavior. Here, we explore the dynamic covalent assembly of extended <i>ortho</i>-phenylenes (hexamer and decamer) with rod-shaped linkers. Characteristic <sup>1</sup>H chemical shift differences between cyclic and acyclic systems can be compared with computational conformer libraries to determine the folding states of the macrocycles. We show that the bite angle provides a measure of the fit of an <i>o</i>-phenylene conformer within a shape-persistent macrocycle, affecting both assembly and ultimate folding behavior. For the <i>o</i>-phenylene hexamer, the bite angle and conformer stability work synergistically to direct assembly toward triangular [3+3] macrocycles of well-folded oligomers. For the decamer, the energetic accessibility of conformers with small bite angles allows [2+2] macrocycles to be formed as the predominant species. In these systems, the <i>o</i>-phenylenes are forced into unusual folding states, preferentially adopting a backbone geometry with distinct helical blocks of opposite handedness. The results show that simple geometric restrictions can be used to direct foldamers toward increasingly complex geometries.</p>

Current Trends in Biotherapeutic Higher Order Structure Characterization by Irreversible Covalent Footprinting Mass Spectrometry

2019 ◽  
Vol 26 (1) ◽  
pp. 35-43 ◽  
Author(s):  
Natalie K. Garcia ◽  
Galahad Deperalta ◽  
Aaron T. Wecksler
Keyword(s):  
Mass Spectrometry ◽  
Protein Structure ◽  
Protein Interactions ◽  
Quaternary Structure ◽  
Solvent Accessibility ◽  
Higher Order ◽  
Structure Characterization ◽  
Order Structure ◽  
Protein Footprinting ◽  
Wide Range

Background: Biotherapeutics, particularly monoclonal antibodies (mAbs), are a maturing class of drugs capable of treating a wide range of diseases. Therapeutic function and solutionstability are linked to the proper three-dimensional organization of the primary sequence into Higher Order Structure (HOS) as well as the timescales of protein motions (dynamics). Methods that directly monitor protein HOS and dynamics are important for mapping therapeutically relevant protein-protein interactions and assessing properly folded structures. Irreversible covalent protein footprinting Mass Spectrometry (MS) tools, such as site-specific amino acid labeling and hydroxyl radical footprinting are analytical techniques capable of monitoring the side chain solvent accessibility influenced by tertiary and quaternary structure. Here we discuss the methodology, examples of biotherapeutic applications, and the future directions of irreversible covalent protein footprinting MS in biotherapeutic research and development. Conclusion: Bottom-up mass spectrometry using irreversible labeling techniques provide valuable information for characterizing solution-phase protein structure. Examples range from epitope mapping and protein-ligand interactions, to probing challenging structures of membrane proteins. By paring these techniques with hydrogen-deuterium exchange, spectroscopic analysis, or static-phase structural data such as crystallography or electron microscopy, a comprehensive understanding of protein structure can be obtained.

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