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Hiltner L, Carme Calderer M, Arsuaga J, Vázquez M. Chromonic liquid crystals and packing configurations of bacteriophage viruses. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200111. [PMID: 34024128 DOI: 10.1098/rsta.2020.0111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 12/01/2020] [Indexed: 06/12/2023]
Abstract
We study equilibrium configurations of hexagonal columnar liquid crystals in the context of characterizing packing structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria-resistant antibiotics. The energy that we propose consists of the Oseen-Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the core and the energy of the unknown interface between the outer ordered region of the capsid and the inner disordered core. The problem becomes of free boundary type, with constraints. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer. Moreover, we present examples of toroidal structures formed by DNA in free solution and compare them with the analogous ones occurring in experiments with other types of lyotropic liquid crystals, such as food dyes and additives. This article is part of the theme issue 'Topics in mathematical design of complex materials'.
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Affiliation(s)
- Lindsey Hiltner
- School of Mathematics, University of Minnesota, Minneapolis, MN 55442, USA
| | - M Carme Calderer
- School of Mathematics, University of Minnesota, Minneapolis, MN 55442, USA
| | - Javier Arsuaga
- Department of Cellular and Molecular Biology, Briggs Hall, Davis, CA 09
- Department of Mathematics, MSB, 2115, University of California Davis, One Shields Avenue, Davis, CA 95616, USA
| | - Mariel Vázquez
- Department of Mathematics, MSB, 2150, University of California Davis, One Shields Avenue, Davis, CA 95616, USA
- Department of Microbiology and Molecular Genetics, Briggs Hall, Davis, CA 09
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Curk T, Farrell JD, Dobnikar J, Podgornik R. Spontaneous Domain Formation in Spherically Confined Elastic Filaments. PHYSICAL REVIEW LETTERS 2019; 123:047801. [PMID: 31491267 DOI: 10.1103/physrevlett.123.047801] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2019] [Indexed: 06/10/2023]
Abstract
Although the free energy of a genome packing into a virus is dominated by DNA-DNA interactions, ordering of the DNA inside the capsid is elasticity driven, suggesting general solutions with DNA organized into spool-like domains. Using analytical calculations and computer simulations of a long elastic filament confined to a spherical container, we show that the ground state is not a single spool as assumed hitherto, but an ordering mosaic of multiple homogeneously ordered domains. At low densities, we observe concentric spools, while at higher densities, other morphologies emerge, which resemble topological links. We discuss our results in the context of metallic wires, viral DNA, and flexible polymers.
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Affiliation(s)
- Tine Curk
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Faculty of Chemistry and Chemical Engineering, University of Maribor, 2000 Maribor, Slovenia
| | | | - Jure Dobnikar
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
- Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom
| | - Rudolf Podgornik
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
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Popadić A, Svenšek D, Podgornik R, Praprotnik M. Density–Nematic Coupling in Isotropic Linear Polymers: Acoustic and Osmotic Birefringence. ADVANCED THEORY AND SIMULATIONS 2019. [DOI: 10.1002/adts.201900019] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Affiliation(s)
- Aleksandar Popadić
- Laboratory for Molecular ModelingNational Institute of ChemistrySI‐1001 Ljubljana Slovenia
| | - Daniel Svenšek
- Department of Physics, Faculty of Mathematics and PhysicsUniversity of LjubljanaSI‐1000 Ljubljana Slovenia
| | - Rudolf Podgornik
- Department of Physics, Faculty of Mathematics and PhysicsUniversity of LjubljanaSI‐1000 Ljubljana Slovenia
- Department of Theoretical PhysicsJ. Stefan InstituteSI‐1000 Ljubljana Slovenia
- School of Physical Sciences and Kavli Institute for Theoretical SciencesUniversity of Chinese Academy of SciencesBeijing 100049 China
- CAS Key Laboratory of Soft Matter Physics, Institute of PhysicsChinese Academy of SciencesBeijing 100190 China
| | - Matej Praprotnik
- Laboratory for Molecular ModelingNational Institute of ChemistrySI‐1001 Ljubljana Slovenia
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Popadić A, Svenšek D, Podgornik R, Daoulas KC, Praprotnik M. Splay-density coupling in semiflexible main-chain nematic polymers with hairpins. SOFT MATTER 2018; 14:5898-5905. [PMID: 29972386 DOI: 10.1039/c8sm00835c] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
A main-chain nematic polymer melt/solution exhibits macroscopic orientational order of main polymer chains, i.e., a preferred (nematic) direction. It has long been known that in such polymeric liquid crystals spatial density/concentration variations and distortions of the nematic direction are coupled, obeying a vectorial continuity constraint whose rigidity increases with chain length. Its vectorial nature precludes the application to flexible chains, where backfolds (hairpins) are present and apolar nematic symmetry is manifest, which has been its puzzling feature from the beginning. We now establish a description of the splay-density coupling in the case of arbitrary backfolding, devising a continuity constraint for the "recovered" polar order of the chain tangents and introducing hairpins as its new type of sources. Performing detailed Monte Carlo simulations of nematic monodomain melts of "soft" worm-like chains with variable length and flexibility, we show via their structure factors that the weakening of the coupling due to the backfolding can be consistently quantified on the macroscopic level.
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Affiliation(s)
- Aleksandar Popadić
- Laboratory for Molecular Modeling, National Institute of Chemistry, SI-1001 Ljubljana, Slovenia
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Lu BS. Multiscale approach to nematic liquid crystals via statistical field theory. Phys Rev E 2017; 96:022709. [PMID: 28950485 DOI: 10.1103/physreve.96.022709] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2017] [Indexed: 06/07/2023]
Abstract
We propose an approach to a multiscale problem in the theory of thermotropic uniaxial nematics based on the method of statistical field theory. This approach enables us to relate the coefficients A, B, C, L_{1}, and L_{2} of the Landau-de Gennes free energy for the isotropic-nematic phase transition to the parameters of a molecular model of uniaxial nematics, which we take to be a lattice gas model of nematogenic molecules interacting via a short-ranged potential. We obtain general constraints on the temperature and volume fraction of nematogens for the Landau-de Gennes theory to be stable against molecular orientation fluctuations at quartic order. In particular, for the case of a fully occupied lattice, we compute the values of the isotropic-nematic transition temperature and the order parameter discontinuity predicted by (i) a continuum approximation of the nearest-neighbor Lebwohl-Lasher model and (ii) a Lebwohl-Lasher-type model with a nematogenic interaction of finite range. We find that the predictions of (i) are in reasonably good agreement with known results of Monte Carlo simulation.
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Affiliation(s)
- Bing-Sui Lu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore
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Svenšek D, Podgornik R. Generalized conservation law for main-chain polymer nematics. Phys Rev E 2016; 93:052703. [PMID: 27300956 DOI: 10.1103/physreve.93.052703] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2016] [Indexed: 11/07/2022]
Abstract
We explore the implications of the conservation law(s) and the corresponding so-called continuity equation(s), resulting from the coupling between the positional and the orientational order in main-chain polymer nematics, by showing that the vectorial and tensorial forms of these equations are in general not equivalent and cannot be reduced to one another, but neither are they disjoint alternatives. We analyze the relation between them and elucidate the fundamental role that the chain backfolding plays in the determination of their relative strength and importance. Finally, we show that the correct penalty potential in the effective free energy, implementing these conservation laws, should actually connect both the tensorial and the vectorial constraints. We show that the consequences of the polymer chains' connectivity for their consistent mesoscopic description are thus not only highly nontrivial but that its proper implementation is absolutely crucial for a consistent coarse-grained description of the main-chain polymer nematics.
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Affiliation(s)
- Daniel Svenšek
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Rudolf Podgornik
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia.,Department of Theoretical Physics, Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia.,Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
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Podgornik R, Aksoyoglu MA, Yasar S, Svenšek D, Parsegian VA. DNA Equation of State: In Vitro vs In Viro. J Phys Chem B 2016; 120:6051-60. [DOI: 10.1021/acs.jpcb.6b02017] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Rudolf Podgornik
- Department
of Physics, University of Massachusetts, Amherst, Massachusetts 01003, United States
- Department
of Theoretical Physics, Jožef Stefan Institute, SI-1000 Ljubljana, Slovenia
- Department
of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - M. Alphan Aksoyoglu
- Department
of Physics, University of Massachusetts, Amherst, Massachusetts 01003, United States
| | - Selcuk Yasar
- Department
of Physics, University of Massachusetts, Amherst, Massachusetts 01003, United States
| | - Daniel Svenšek
- Department
of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - V. Adrian Parsegian
- Department
of Physics, University of Massachusetts, Amherst, Massachusetts 01003, United States
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Svenšek D, Podgornik R. Correlation functions of main-chain polymer nematics constrained by tensorial and vectorial conservation laws. J Chem Phys 2015; 143:114902. [DOI: 10.1063/1.4930920] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Gemünden P, Daoulas KC. Fluctuation spectra in polymer nematics and Frank elastic constants: a coarse-grained modelling study. SOFT MATTER 2015; 11:532-544. [PMID: 25418080 DOI: 10.1039/c4sm02075h] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Monte Carlo simulations of uniaxial nematic polymer melts are performed, based on a discrete worm-like chain model combined with soft, anisotropic non-bonded potentials. Different chain lengths are considered, the contour length of the longest being an order of magnitude larger than the persistence length. From equilibrated melt configurations, density and director fluctuation spectra are calculated and compared with analytical predictions available in literature. The latter typically correspond to hydrodynamic treatments of correlations and assume that there is no chain backfolding along the nematic director. Nevertheless, it is demonstrated that the analytical theories capture several features of the spectra obtained in the current simulations, where moderate backfolding of polymer chains is observed. Based on the available analytical expressions for density and director fluctuation spectra, material properties, such as Frank elastic constants, are extracted. Their dependence on polymerisation degree is studied and found to reproduce theoretically expected trends. For example, evidence is provided that the splay constant increases linearly with chain length, when effects of hairpins are negligible.
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Affiliation(s)
- Patrick Gemünden
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany.
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Svenšek D, Grason GM, Podgornik R. Tensorial conservation law for nematic polymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052603. [PMID: 24329288 DOI: 10.1103/physreve.88.052603] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2013] [Indexed: 06/03/2023]
Abstract
We derive the "conservation law" for nematic polymers in tensorial form valid for quadrupolar orientational order, in contradistinction to the conservation law in the case of polar orientational order. Due to microscopic differences in the coupling between the orientational field deformations and the density variations for polar and quadrupolar order, we find that the respective order parameters satisfy fundamentally distinct constraints. Being necessarily scalar in its form, the tensorial conservation law is obtained straightforwardly from the gradients of the polymer nematic tensor field and connects the spatial variation of this tensor field with density variations. We analyze the differences between the polar and the tensorial forms of the conservation law, present some explicit orientational fields that satisfy the tensorial constraint, and discuss the role of singular "hairpins," which do not affect the local quadrupolar order of polymer nematics, but nevertheless influence its gradients.
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Affiliation(s)
- Daniel Svenšek
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Gregory M Grason
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - Rudolf Podgornik
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia and Department of Theoretical Physics, J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia and Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
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Saper G, Kler S, Asor R, Oppenheim A, Raviv U, Harries D. Effect of capsid confinement on the chromatin organization of the SV40 minichromosome. Nucleic Acids Res 2013; 41:1569-80. [PMID: 23258701 PMCID: PMC3561987 DOI: 10.1093/nar/gks1270] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2012] [Revised: 10/26/2012] [Accepted: 11/05/2012] [Indexed: 01/10/2023] Open
Abstract
Using small-angle X-ray scattering, we determined the three-dimensional packing architecture of the minichromosome confined within the SV40 virus. In solution, the minichromosome, composed of closed circular dsDNA complexed in nucleosomes, was shown to be structurally similar to cellular chromatin. In contrast, we find a unique organization of the nanometrically encapsidated chromatin, whereby minichromosomal density is somewhat higher at the center of the capsid and decreases towards the walls. This organization is in excellent agreement with a coarse-grained computer model, accounting for tethered nucleosomal interactions under viral capsid confinement. With analogy to confined liquid crystals, but contrary to the solenoid structure of cellular chromatin, our simulations indicate that the nucleosomes within the capsid lack orientational order. Nucleosomes in the layer adjacent to the capsid wall, however, align with the boundary, thereby inducing a 'molten droplet' state of the chromatin. These findings indicate that nucleosomal interactions suffice to predict the genome organization in polyomavirus capsids and underscore the adaptable nature of the eukaryotic chromatin architecture to nanoscale confinement.
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Affiliation(s)
- Gadiel Saper
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
| | - Stanislav Kler
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
| | - Roi Asor
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
| | - Ariella Oppenheim
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
| | - Uri Raviv
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
| | - Daniel Harries
- Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel, The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel and Department of Hematology, Hebrew University–Hadassa Medical School, Jerusalem 91120, Israel
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Siber A, Božič AL, Podgornik R. Energies and pressures in viruses: contribution of nonspecific electrostatic interactions. Phys Chem Chem Phys 2011; 14:3746-65. [PMID: 22143065 DOI: 10.1039/c1cp22756d] [Citation(s) in RCA: 110] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
We summarize some aspects of electrostatic interactions in the context of viruses. A simplified but, within well defined limitations, reliable approach is used to derive expressions for electrostatic energies and the corresponding osmotic pressures in single-stranded RNA viruses and double-stranded DNA bacteriophages. The two types of viruses differ crucially in the spatial distribution of their genome charge which leads to essential differences in their free energies, depending on the capsid size and total charge in a quite different fashion. Differences in the free energies are trailed by the corresponding characteristics and variations in the osmotic pressure between the inside of the virus and the external bathing solution.
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Cherstvy AG. Electrostatic interactions in biological DNA-related systems. Phys Chem Chem Phys 2011; 13:9942-68. [DOI: 10.1039/c0cp02796k] [Citation(s) in RCA: 120] [Impact Index Per Article: 9.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
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