1
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Berx J, Mashaghi A. Aggregation and structural phase transitions of semiflexible polymer bundles: A braided circuit topology approach. iScience 2024; 27:108995. [PMID: 38361617 PMCID: PMC10867648 DOI: 10.1016/j.isci.2024.108995] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2023] [Revised: 11/23/2023] [Accepted: 01/19/2024] [Indexed: 02/17/2024] Open
Abstract
We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single isolated folded linear chains, the number and arrangement of contacts within the circuitry of a folded chain give rise to increasingly complex fold topologies. Another avenue for achieving complexity is through the interaction and entanglement of two or more folded linear chains. The braided circuit topology approach describes the topology of such multiple-chain systems and offers topological measures such as writhe, complexity, braid length, and isotopy class. This extension of circuit topology to multichains reveals the interplay between collapse, aggregation, and entanglement. In this work, we show that circuit topological motif fractions are ideally suited order parameters to characterize structural phase transitions in entangled systems that can detect structural re-ordering other measures cannot.
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Affiliation(s)
- Jonas Berx
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Leiden University, Leiden 2333CC, the Netherlands
- Laboratory for Interdisciplinary Medical Innovations, Centre for Interdisciplinary Genome Research, Leiden University, Leiden 2333CC, the Netherlands
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Leiden University, Leiden 2333CC, the Netherlands
- Laboratory for Interdisciplinary Medical Innovations, Centre for Interdisciplinary Genome Research, Leiden University, Leiden 2333CC, the Netherlands
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2
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Scalvini B, Mashaghi A. Circuit Topology Analysis of Single-Cell HiC Data. Methods Mol Biol 2024; 2819:27-38. [PMID: 39028500 DOI: 10.1007/978-1-0716-3930-6_2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 07/20/2024]
Abstract
The 3D fold structure of the genome is intricately linked to its function. As a result, descriptors of 3D genome conformation are becoming increasingly important as markers for disease and therapeutic responses. Circuit topology, a theory of folds, formalizes the arrangement of contacts in an entangled chain. It is uniquely suited for the topological description of the cellular genome and changes to genomic architecture during physiological processes like cellular differentiation or pathological and therapeutic alterations. In this discussion, we will explore circuit topology and its ability to extract topological information from single-cell HiC data.
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Affiliation(s)
- Barbara Scalvini
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Leiden, The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Leiden, The Netherlands
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Leiden, The Netherlands.
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Leiden, The Netherlands.
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3
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Scalvini B, Heling LWHJ, Sheikhhassani V, Sunderlikova V, Tans SJ, Mashaghi A. Cytosolic Interactome Protects Against Protein Unfolding in a Single Molecule Experiment. Adv Biol (Weinh) 2023; 7:e2300105. [PMID: 37409427 DOI: 10.1002/adbi.202300105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2023] [Revised: 06/13/2023] [Indexed: 07/07/2023]
Abstract
Single molecule techniques are particularly well suited for investigating the processes of protein folding and chaperone assistance. However, current assays provide only a limited perspective on the various ways in which the cellular environment can influence the folding pathway of a protein. In this study, a single molecule mechanical interrogation assay is developed and used to monitor protein unfolding and refolding within a cytosolic solution. This allows to test the cumulative topological effect of the cytoplasmic interactome on the folding process. The results reveal a stabilization against forced unfolding for partial folds, which are attributed to the protective effect of the cytoplasmic environment against unfolding and aggregation. This research opens the possibility of conducting single molecule molecular folding experiments in quasi-biological environments.
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Affiliation(s)
- Barbara Scalvini
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
| | - Laurens W H J Heling
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
| | - Vahid Sheikhhassani
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
| | | | - Sander J Tans
- AMOLF, Science Park 104, Amsterdam, 1098 XG, The Netherlands
- Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Van der Maasweg 9, Delft, 2629HZ, The Netherlands
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, Leiden, 2333CC, The Netherlands
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4
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Berx J, Mashaghi A. Decoding chirality in circuit topology of a self entangled chain through braiding. SOFT MATTER 2023; 19:5888-5895. [PMID: 37477235 DOI: 10.1039/d3sm00390f] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/22/2023]
Abstract
Circuit topology employs fundamental units of entanglement, known as soft contacts, for constructing knots from the bottom up, utilizing circuit topology relations, namely parallel, series, cross, and concerted relations. In this article, we further develop this approach to facilitate the analysis of chirality, which is a significant quantity in polymer chemistry. To achieve this, we translate the circuit topology approach to knot engineering into a braid-theoretic framework. This enables us to calculate the Jones polynomial for all possible binary combinations of contacts in cross or concerted relations and to show that, for series and parallel relations, the polynomial factorises. Our results demonstrate that the Jones polynomial provides a powerful tool for analysing the chirality of molecular knots constructed using circuit topology. The framework presented here can be used to design and engineer a wide range of entangled chain with desired chiral properties, with potential applications in fields such as materials science and nanotechnology.
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Affiliation(s)
- Jonas Berx
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, 2333CC Leiden, The Netherlands.
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, 2333CC Leiden, The Netherlands.
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5
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Flapan E, Mashaghi A, Wong H. A tile model of circuit topology for self-entangled biopolymers. Sci Rep 2023; 13:8889. [PMID: 37264056 DOI: 10.1038/s41598-023-35771-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2023] [Accepted: 05/23/2023] [Indexed: 06/03/2023] Open
Abstract
Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and connected sums of these knots. This includes all proteins knots that have thus far been identified.
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Affiliation(s)
- Erica Flapan
- Mathematics and Statistics Department, Pomona College, Claremont, CA, 91711, USA.
| | - Alireza Mashaghi
- Faculty of Science, Leiden University, 2333CC, Leiden, The Netherlands
| | - Helen Wong
- Mathematical Sciences Department, Claremont McKenna College, Claremont, CA, 91711, USA
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6
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ProteinCT: An implementation of the protein circuit topology framework. MethodsX 2022; 9:101861. [PMID: 36187158 PMCID: PMC9520010 DOI: 10.1016/j.mex.2022.101861] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2022] [Accepted: 09/11/2022] [Indexed: 11/22/2022] Open
Abstract
The ability to describe the topology of a folded protein conformation is critically important for functional analysis, protein engineering, and drug design. Circuit topology is a unique topological framework which is widely applicable to protein analysis, yet a state-of-the art implementation of this concept is lacking. Here, we present an open-source Python-implemented circuit topology tool called ProteinCT. The platform provides a method for acquiring, visualizing, analyzing, and quantifying circuit topology data from proteins of interest. We mapped the universe of human proteins to a circuit topology space using conventional hardware within a few hours, demonstrating the performance of ProteinCT. In brief,•A Python-implemented circuit topology tool is developed to extract global and local topological information from a protein structure file.•Modules are developed to combine topological information with geometric and energetic information.•It is demonstrated that the method can be efficiently applied to a large set of proteins, opening a wide range of possibilities for structural proteomics research.
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7
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Woodard J, Iqbal S, Mashaghi A. Circuit topology predicts pathogenicity of missense mutations. Proteins 2022; 90:1634-1644. [PMID: 35394672 PMCID: PMC9543832 DOI: 10.1002/prot.26342] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2021] [Revised: 03/07/2022] [Accepted: 03/30/2022] [Indexed: 12/05/2022]
Abstract
The contact topology of a protein determines important aspects of the folding process. The topological measure of contact order has been shown to be predictive of the rate of folding. Circuit topology is emerging as another fundamental descriptor of biomolecular structure, with predicted effects on the folding rate. We analyze the residue‐based circuit topological environments of 21 K mutations labeled as pathogenic or benign. Multiple statistical lines of reasoning support the conclusion that the number of contacts in two specific circuit topological arrangements, namely inverse parallel and cross relations, with contacts involving the mutated residue have discriminatory value in determining the pathogenicity of human variants. We investigate how results vary with residue type and according to whether the gene is essential. We further explore the relationship to a number of structural features and find that circuit topology provides nonredundant information on protein structures and pathogenicity of mutations. Results may have implications for the polymer physics of protein folding and suggest that “local” topological information, including residue‐based circuit topology and residue contact order, could be useful in improving state‐of‐the‐art machine learning algorithms for pathogenicity prediction.
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Affiliation(s)
- Jaie Woodard
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Leiden, The Netherlands.,Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Michigan, USA
| | - Sumaiya Iqbal
- Center for the Development of Therapeutics, Broad Institute of MIT and Harvard, Cambridge, Massachusetts, USA.,Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, Massachusetts, USA.,Program in Medical and Population Genetics, Broad Institute of MIT and Harvard, Cambridge, Massachusetts, USA.,Analytic and Translational Genetics Unit, Massachusetts General Hospital, Boston, Massachusetts, USA
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Leiden, The Netherlands.,Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Leiden, The Netherlands
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8
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Sheikhhassani V, Scalvini B, Ng J, Heling LWHJ, Ayache Y, Evers TMJ, Estébanez‐Perpiñá E, McEwan IJ, Mashaghi A. Topological dynamics of an intrinsically disordered N‐terminal domain of the human androgen receptor. Protein Sci 2022; 31:e4334. [PMID: 35634773 PMCID: PMC9134807 DOI: 10.1002/pro.4334] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2022] [Revised: 05/02/2022] [Accepted: 05/03/2022] [Indexed: 12/28/2022]
Abstract
Human androgen receptor contains a large N‐terminal domain (AR‐NTD) that is highly dynamic and this poses a major challenge for experimental and computational analysis to decipher its conformation. Misfolding of the AR‐NTD is implicated in prostate cancer and Kennedy's disease, yet our knowledge of its structure is limited to primary sequence information of the chain and a few functionally important secondary structure motifs. Here, we employed an innovative combination of molecular dynamics simulations and circuit topology (CT) analysis to identify the tertiary structure of AR‐NTD. We found that the AR‐NTD adopts highly dynamic loopy conformations with two identifiable regions with distinct topological make‐up and dynamics. This consists of a N‐terminal region (NR, residues 1–224) and a C‐terminal region (CR, residues 225–538), which carries a dense core. Topological mapping of the dynamics reveals a traceable time‐scale dependent topological evolution. NR adopts different positioning with respect to the CR and forms a cleft that can partly enclose the hormone‐bound ligand‐binding domain (LBD) of the androgen receptor. Furthermore, our data suggest a model in which dynamic NR and CR compete for binding to the DNA‐binding domain of the receptor, thereby regulating the accessibility of its DNA‐binding site. Our approach allowed for the identification of a previously unknown regulatory binding site within the CR core, revealing the structural mechanisms of action of AR inhibitor EPI‐001, and paving the way for other drug discovery applications.
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Affiliation(s)
- Vahid Sheikhhassani
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Barbara Scalvini
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Julian Ng
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Laurens W. H. J. Heling
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Yosri Ayache
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Tom M. J. Evers
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
| | - Eva Estébanez‐Perpiñá
- Department of Biochemistry and Molecular Biomedicine Institute of Biomedicine (IBUB) of the University of Barcelona (UB) Barcelona Spain
| | - Iain J. McEwan
- Institute of Medical Sciences, School of Medicine, Medical Sciences and Nutrition, University of Aberdeen Scotland UK
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science Leiden University Leiden The Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science Leiden University Leiden The Netherlands
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9
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Scalvini B, Schiessel H, Golovnev A, Mashaghi A. Circuit topology analysis of cellular genome reveals signature motifs, conformational heterogeneity, and scaling. iScience 2022; 25:103866. [PMID: 35243229 PMCID: PMC8861635 DOI: 10.1016/j.isci.2022.103866] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2021] [Revised: 12/14/2021] [Accepted: 01/31/2022] [Indexed: 11/30/2022] Open
Abstract
Reciprocal regulation of genome topology and function is a fundamental and enduring puzzle in biology. The wealth of data provided by Hi-C libraries offers the opportunity to unravel this relationship. However, there is a need for a comprehensive theoretical framework in order to extract topological information for genome characterization and comparison. Here, we develop a toolbox for topological analysis based on Circuit Topology, allowing for the quantification of inter- and intracellular genomic heterogeneity, at various levels of fold complexity: pairwise contact arrangement, higher-order contact arrangement, and topological fractal dimension. Single-cell Hi-C data were analyzed and characterized based on topological content, revealing not only a strong multiscale heterogeneity but also highly conserved features such as a characteristic topological length scale and topological signature motifs in the genome. We propose that these motifs inform on the topological state of the nucleus and indicate the presence of active loop extrusion. Circuit topology quantifies heterogeneity in genomic arrangement Scale analysis reveals a characteristic length scale of 10 Mb in genome topology We identify highly conserved topological structures related to loop extrusion We suggest a topological model of chromatin arrangement for loop extrusion, the L-loop
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Affiliation(s)
- Barbara Scalvini
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
| | - Helmut Schiessel
- Cluster of Excellence Physics of Life, Technical University of Dresden, 01062 Dresden, Germany
| | - Anatoly Golovnev
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
- Centre for Interdisciplinary Genome Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, the Netherlands
- Corresponding author
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10
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Han Z, Hilburg SL, Alexander-Katz A. Forced Unfolding of Protein-Inspired Single-Chain Random Heteropolymers. Macromolecules 2022. [DOI: 10.1021/acs.macromol.1c02411] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023]
Affiliation(s)
- Zexiang Han
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Shayna L. Hilburg
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Alfredo Alexander-Katz
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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11
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Hakobyan K, Xu J, Müllner M. The challenges of controlling polymer synthesis at the molecular and macromolecular level. Polym Chem 2022. [DOI: 10.1039/d1py01581h] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
In this Perspective, we outline advances and challenges in controlling the structure of polymers at various size regimes in the context of structural features such as molecular weight distribution, end groups, architecture, composition and sequence.
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Affiliation(s)
- Karen Hakobyan
- Key Centre for Polymers and Colloids, School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia
- The University of Sydney Nano Institute (Sydney Nano), Sydney, NSW 2006, Australia
- School of Chemical Engineering, UNSW Sydney, NSW 2052, Australia
| | - Jiangtao Xu
- School of Chemical Engineering, UNSW Sydney, NSW 2052, Australia
| | - Markus Müllner
- Key Centre for Polymers and Colloids, School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia
- The University of Sydney Nano Institute (Sydney Nano), Sydney, NSW 2006, Australia
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12
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Scalvini B, Sheikhhassani V, Mashaghi A. Topological principles of protein folding. Phys Chem Chem Phys 2021; 23:21316-21328. [PMID: 34545868 DOI: 10.1039/d1cp03390e] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
What is the topology of a protein and what governs protein folding to a specific topology? This is a fundamental question in biology. The protein folding reaction is a critically important cellular process, which is failing in many prevalent diseases. Understanding protein folding is also key to the design of new proteins for applications. However, our ability to predict the folding of a protein chain is quite limited and much is still unknown about the topological principles of folding. Current predictors of folding kinetics, including the contact order and size, present a limited predictive power, suggesting that these models are fundamentally incomplete. Here, we use a newly developed mathematical framework to define and extract the topology of a native protein conformation beyond knot theory, and investigate the relationship between native topology and folding kinetics in experimentally characterized proteins. We show that not only the folding rate, but also the mechanistic insight into folding mechanisms can be inferred from topological parameters. We identify basic topological features that speed up or slow down the folding process. The approach enabled the decomposition of protein 3D conformation into topologically independent elementary folding units, called circuits. The number of circuits correlates significantly with the folding rate, offering not only an efficient kinetic predictor, but also a tool for a deeper understanding of theoretical folding models. This study contributes to recent work that reveals the critical relevance of topology to protein folding with a new, contact-based, mathematically rigorous perspective. We show that topology can predict folding kinetics when geometry-based predictors like contact order and size fail.
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Affiliation(s)
- Barbara Scalvini
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, The Netherlands.
| | - Vahid Sheikhhassani
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, The Netherlands.
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, Einsteinweg 55, 2333CC Leiden, The Netherlands.
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Abstract
The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.
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14
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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 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.
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15
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Palma CA. Topological Dynamic Matter. J Phys Chem Lett 2021; 12:454-462. [PMID: 33369418 DOI: 10.1021/acs.jpclett.0c03114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The principles of topology in condensed matter physics have expanded to areas such as photonics, acoustics, electronics, and mechanics. Their extension to dynamic (soft) matter could enable the control and design of topological thermodynamic (micro)states and nonreciprocal dynamics, potentially leading to paradigmatic applications in molecular and thermal waveguiding, logics, and energy management. This Perspective explores distinct topological concepts for dynamic matter and prospective function. Topological tools are exemplified and discussed for the study of nonlocal order parameters or invariants in dynamic molecular matter, toward the engineering of assemblies, reactions, and system chemistry with unconventional global properties-a scope which has the potential to push the frontiers of physical chemistry and transform chemical topology from form to function.
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Affiliation(s)
- Carlos-Andres Palma
- Institute of Physics, Chinese Academy of Sciences, 100190 Beijing, P.R. China
- Department of Physics & IRIS Adlershof, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
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16
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Razbin M, Mashaghi A. Elasticity of connected semiflexible quadrilaterals. SOFT MATTER 2021; 17:102-112. [PMID: 33150925 DOI: 10.1039/d0sm01719a] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Using the positional-orientational propagator of a semiflexible filament in the weakly bending regime, we analytically calculate the probability densities associated with the fluctuating tip and the corners of a grafted system of connected quadrilaterals. We calculate closed analytic expressions for the probability densities within the framework of the worm-like chain model, which are valid in the weakly bending regime. The probability densities give the physical quantities related to the elasticity of the system such as the force-extension relation in the fixed extension ensemble, the Poisson's ratio and the average of the force exerted to a confining stiff planar wall by the fluctuating tip of the system. Our analysis reveals that the force-extension relations depend on the contour length of the system (material content), the bending stiffness (chemical nature), the geometrical angle and the number of the quadrilaterals, while the Poisson's ratio depends only on the geometrical angle and the number of the quadrilaterals, and is thus a purely geometric property of the system.
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Affiliation(s)
- Mohammadhosein Razbin
- Department of Energy Engineering and Physics, Amirkabir University of Technology, 14588 Tehran, Iran.
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17
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Golovnev A, Mashaghi A. Generalized Circuit Topology of Folded Linear Chains. iScience 2020; 23:101492. [PMID: 32896769 PMCID: PMC7481252 DOI: 10.1016/j.isci.2020.101492] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2020] [Revised: 06/20/2020] [Accepted: 08/18/2020] [Indexed: 11/14/2022] Open
Abstract
A wide range of physical systems can be formally mapped to a linear chain of sorted objects. Upon introduction of intrachain interactions, such a chain can "fold" to elaborate topological structures, analogous to folded linear polymer systems. Two distinct chain-topology theories, knot theory and circuit topology, have separately provided insight into the structure, dynamics, and evolution of folded linear polymers such as proteins and genomic DNA. Knot theory, however, ignores intrachain interactions (contacts), whereas chain crossings are ignored in circuit topology. Thus, there is a need for a universal approach that can provide topological description of any folded linear chain. Here, we generalize circuit topology in order to grasp particularities typically addressed by knot theory. We develop a generic approach that is simple, mathematically rigorous, and practically useful for structural classification, analysis of structural dynamics, and engineering applications.
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Affiliation(s)
- Anatoly Golovnev
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, 2333CC Leiden, the Netherlands
| | - Alireza Mashaghi
- Medical Systems Biophysics and Bioengineering, Leiden Academic Centre for Drug Research, Faculty of Science, Leiden University, 2333CC Leiden, the Netherlands
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