1
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Caraglio M, Micheletti C, Orlandini E. Unraveling the Influence of Topology and Spatial Confinement on Equilibrium and Relaxation Properties of Interlocked Ring Polymers. Macromolecules 2024; 57:3223-3233. [PMID: 38616813 PMCID: PMC11008367 DOI: 10.1021/acs.macromol.3c02203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2023] [Revised: 02/27/2024] [Accepted: 03/01/2024] [Indexed: 04/16/2024]
Abstract
We use Langevin dynamics simulations to study linked ring polymers in channel confinement. We address the in- and out-of-equilibrium behavior of the systems for varying degrees of confinement and increasing topological and geometrical complexity of the interlocking. The main findings are three. First, metric observables of different link topologies collapse onto the same master curve when plotted against the crossing number, revealing a universal response to confinement. Second, the relaxation process from initially stretched states is faster for more complex links. We ascribe these properties to the interplay of several effects, including the dependence of topological friction on the link complexity. Finally, we show that transient forms of geometrical entanglement purposely added to the initial stressed state can leave distinctive signatures in force-spectroscopy curves. The insight provided by the findings could be leveraged in single-molecule nanochannel experiments to identify geometric entanglement within topologically linked rings.
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Affiliation(s)
- Michele Caraglio
- Institut
für Theoretische Physik, Universität
Innsbruck, Technikerstraße 21A, Innsbruck A-6020, Austria
| | - Cristian Micheletti
- Scuola
Internazionale Superiore di Studi Avanzati—SISSA, Via Bonomea 265, Trieste 34136, Italy
| | - Enzo Orlandini
- Department
of Physics and Astronomy, University of
Padova, Via Marzolo 8, Padova I-35100, Italy
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2
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Chen X, Soh BW, Ooi ZE, Vissol-Gaudin E, Yu H, Novoselov KS, Hippalgaonkar K, Li Q. Constructing custom thermodynamics using deep learning. NATURE COMPUTATIONAL SCIENCE 2024; 4:66-85. [PMID: 38200379 PMCID: PMC10808076 DOI: 10.1038/s43588-023-00581-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2023] [Accepted: 11/30/2023] [Indexed: 01/12/2024]
Abstract
One of the most exciting applications of artificial intelligence is automated scientific discovery based on previously amassed data, coupled with restrictions provided by known physical principles, including symmetries and conservation laws. Such automated hypothesis creation and verification can assist scientists in studying complex phenomena, where traditional physical intuition may fail. Here we develop a platform based on a generalized Onsager principle to learn macroscopic dynamical descriptions of arbitrary stochastic dissipative systems directly from observations of their microscopic trajectories. Our method simultaneously constructs reduced thermodynamic coordinates and interprets the dynamics on these coordinates. We demonstrate its effectiveness by studying theoretically and validating experimentally the stretching of long polymer chains in an externally applied field. Specifically, we learn three interpretable thermodynamic coordinates and build a dynamical landscape of polymer stretching, including the identification of stable and transition states and the control of the stretching rate. Our general methodology can be used to address a wide range of scientific and technological applications.
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Affiliation(s)
- Xiaoli Chen
- Department of Mathematics, National University of Singapore, Singapore, Singapore
- Institute for Functional Intelligent Materials, National University of Singapore, Singapore, Singapore
| | - Beatrice W Soh
- Institute of Materials Research and Engineering, A*STAR (Agency for Science), Singapore, Singapore
| | - Zi-En Ooi
- Institute of Materials Research and Engineering, A*STAR (Agency for Science), Singapore, Singapore
| | - Eleonore Vissol-Gaudin
- School of Materials Science and Engineering, Nanyang Technological University, Singapore, Singapore
| | - Haijun Yu
- LSEC and ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
| | - Kostya S Novoselov
- Institute for Functional Intelligent Materials, National University of Singapore, Singapore, Singapore.
| | - Kedar Hippalgaonkar
- Institute for Functional Intelligent Materials, National University of Singapore, Singapore, Singapore.
- Institute of Materials Research and Engineering, A*STAR (Agency for Science), Singapore, Singapore.
- School of Materials Science and Engineering, Nanyang Technological University, Singapore, Singapore.
| | - Qianxiao Li
- Department of Mathematics, National University of Singapore, Singapore, Singapore.
- Institute for Functional Intelligent Materials, National University of Singapore, Singapore, Singapore.
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3
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Mao R, Dorfman KD. Diffusion of knots in nanochannel-confined DNA molecules. J Chem Phys 2023; 158:2890486. [PMID: 37184024 DOI: 10.1063/5.0151025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2023] [Accepted: 05/03/2023] [Indexed: 05/16/2023] Open
Abstract
We used Langevin dynamics simulations without hydrodynamic interactions to probe knot diffusion mechanisms and the time scales governing the evolution and the spontaneous untying of trefoil knots in nanochannel-confined DNA molecules in the extended de Gennes regime. The knot untying follows an "opening up process," wherein the initially tight knot continues growing and fluctuating in size as it moves toward the end of the DNA molecule before its annihilation at the chain end. The mean knot size increases significantly and sub-linearly with increasing chain contour length. The knot diffusion in nanochannel-confined DNA molecules is subdiffusive, with the unknotting time scaling with chain contour length with an exponent of 2.64 ± 0.23 to within a 95% confidence interval. The scaling exponent for the mean unknotting time vs chain contour length, along with visual inspection of the knot conformations, suggests that the knot diffusion mechanism is a combination of self-reptation and knot region breathing for the simulated parameters.
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Affiliation(s)
- Runfang Mao
- Department of Chemical Engineering and Materials Science, University of Minnesota-Twin Cities, 421 Washington Ave. SE, Minneapolis, Minnesota 55455, USA
| | - Kevin D Dorfman
- Department of Chemical Engineering and Materials Science, University of Minnesota-Twin Cities, 421 Washington Ave. SE, Minneapolis, Minnesota 55455, USA
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4
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Li F, Luo Y, Xi G, Fu J, Tu J. Single-Molecule Analysis of DNA structures using nanopore sensors. CHINESE JOURNAL OF ANALYTICAL CHEMISTRY 2022. [DOI: 10.1016/j.cjac.2022.100089] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/07/2023]
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5
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Soh BW, Doyle PS. Equilibrium Conformation of Catenated DNA Networks in Slitlike Confinement. ACS Macro Lett 2021; 10:880-885. [PMID: 35549205 DOI: 10.1021/acsmacrolett.1c00299] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
A kinetoplast is a planar network of catenated DNA rings with topology that resembles that of chain mail armor. In this work, we use single-molecule experiments to probe the conformation of kinetoplasts confined to slits. We find that the in-plane size of kinetoplasts increases with degree of confinement, akin to the slitlike confinement of linear DNA. The change in kinetoplast size with channel height is consistent with the scaling prediction from a Flory-type approach for a 2D polymer. With an increase in extent of confinement, the kinetoplasts appear to unfold and take on more uniform circular shapes, in contrast to the broad range of conformations observed for kinetoplasts in bulk.
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Affiliation(s)
- Beatrice W. Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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6
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Polson JM, Hastie CG. Free energy of a knotted polymer confined to narrow cylindrical and conical channels. Phys Rev E 2020; 102:052502. [PMID: 33327190 DOI: 10.1103/physreve.102.052502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2020] [Accepted: 10/16/2020] [Indexed: 06/12/2023]
Abstract
Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology 3_{1}, 4_{1}, or 5_{1}, as well as unknotted polymers that are capable of forming S loops. We measure the variation of the free energy F with the end-to-end polymer extension length X and examine the effect of varying the polymer topology, persistence length P, and cylinder diameter D on the free-energy functions. Similarly, we characterize the behavior of the knot span along the channel. We find that increasing the knot complexity increases the typical size of the knot. In the regime of low X, where the knot/S-loop size is large, the conformational behavior is independent of polymer topology. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of F with the position of a knot in conical channels for various values of the cone angle α. The free energy decreases as the knot moves in a direction where the cone widens, and it also decreases with increasing α and with increasing knot complexity. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in F is the change in the size of the hairpins as the knot moves to the wider region of the channel.
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Affiliation(s)
- James M Polson
- Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island, C1A 4P3, Canada
| | - Cameron G Hastie
- Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island, C1A 4P3, Canada
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7
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Soh BW, Klotz AR, Doyle PS. Topological Simplification of Complex Knots Untied in Elongational Flows. Macromolecules 2020. [DOI: 10.1021/acs.macromol.0c01322] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Beatrice W. Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Alexander R. Klotz
- Department of Physics and Astronomy, California State University, Long Beach, California 90840, United States
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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8
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Affiliation(s)
- Zixue Ma
- Department of Chemical Engineering and Materials Science, University of Minnesota, Twin Cities, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, United States
| | - Kevin D. Dorfman
- Department of Chemical Engineering and Materials Science, University of Minnesota, Twin Cities, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, United States
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9
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Soh BW, Doyle PS. Deformation Response of Catenated DNA Networks in a Planar Elongational Field. ACS Macro Lett 2020; 9:944-949. [PMID: 35648605 DOI: 10.1021/acsmacrolett.0c00360] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
Abstract
A kinetoplast is a complex catenated DNA network that bears resemblance to a two-dimensional polymeric system. In this work, we use single-molecule experiments to study the transient and steady-state deformation of kinetoplasts in a planar elongational field. We demonstrate that kinetoplasts deform in a stagewise manner and undergo transient deformation at large strains, due to conformational rearrangements from an intermediate metastable state. Kinetoplasts in an elongational field achieve a steady-state deformation that depends on strain rate, akin to the deformation of linear polymers. We do not observe an abrupt transition between the nondeformed and deformed states of a kinetoplast, in contrast to the coil-stretch transition for a linear polymer.
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Affiliation(s)
- Beatrice W. Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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10
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Charles N, Gazzola M, Mahadevan L. Topology, Geometry, and Mechanics of Strongly Stretched and Twisted Filaments: Solenoids, Plectonemes, and Artificial Muscle Fibers. PHYSICAL REVIEW LETTERS 2019; 123:208003. [PMID: 31809094 DOI: 10.1103/physrevlett.123.208003] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2018] [Revised: 03/11/2019] [Indexed: 06/10/2023]
Abstract
Soft elastic filaments that can be stretched, bent, and twisted exhibit a range of topologically and geometrically complex morphologies. Recently, a number of experiments have shown how to use these building blocks to create filament-based artificial muscles that use the conversion of writhe to extension or contraction, exposing the connection between topology, geometry, and mechanics. Here, we combine numerical simulations of soft elastic filaments that account for geometric nonlinearities and self-contact to map out the basic structures underlying artificial muscle fibers in a phase diagram that is a function of the extension and twist density. We then use ideas from computational topology to track the interconversion of link, twist, and writhe in these geometrically complex physical structures to explain the physical principles underlying artificial muscle fibers and provide guidelines for their design.
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Affiliation(s)
- Nicholas Charles
- Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Mattia Gazzola
- Department of Mechanical Science and Engineering and National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - L Mahadevan
- Paulson School of Engineering and Applied Sciences, Department of Physics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
- Kavli Institute for Nanobio Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA
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11
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Affiliation(s)
- Liang Dai
- Department of Physics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
| | - Beatrice W. Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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12
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Soh BW, Klotz AR, Dai L, Doyle PS. Conformational State Hopping of Knots in Tensioned Polymer Chains. ACS Macro Lett 2019; 8:905-911. [PMID: 35619478 DOI: 10.1021/acsmacrolett.9b00462] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We use Brownian dynamics simulations to study the conformational states of knots on tensioned chains. Focusing specifically on the 81 knot, we observe knot conformational state hopping and show that the process can be described by a two-state kinetic model in the presence of an external force. The distribution of knot conformational states depends on the applied chain tension, which leads to a force-dependent distribution of knot untying pathways. We generalize our findings by considering the untying pathways of other knots and find that the way knots untie is generally governed by the force applied to the chain. From a broader perspective, being able to influence how a knot unties via external force can potentially be useful for applications of single-molecule techniques in which knots are unwanted.
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Affiliation(s)
- Beatrice W Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Alexander R Klotz
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Liang Dai
- Department of Physics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
| | - Patrick S Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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13
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Soh BW, Klotz AR, Robertson-Anderson RM, Doyle PS. Long-Lived Self-Entanglements in Ring Polymers. PHYSICAL REVIEW LETTERS 2019; 123:048002. [PMID: 31491263 DOI: 10.1103/physrevlett.123.048002] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2019] [Revised: 04/03/2019] [Indexed: 06/10/2023]
Abstract
The entanglement of ring polymers remains mysterious in many aspects. In this Letter, we use electric fields to induce self-entanglements in circular DNA molecules, which serve as a minimal system for studying chain entanglements. We show that self-threadings give rise to entanglements in ring polymers and can slow down polymer dynamics significantly. We find that strongly entangled circular molecules remain kinetically arrested in a compact state for very long times, thereby providing experimental evidence for the severe topological constraints imposed by threadings.
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Affiliation(s)
- Beatrice W Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Alexander R Klotz
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | | | - Patrick S Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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14
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Bleha T, Cifra P. Force-displacement relations at compression of dsDNA macromolecules. J Chem Phys 2019; 151:014901. [PMID: 31272182 DOI: 10.1063/1.5099522] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
The elasticity of dsDNA molecules is investigated by Monte Carlo simulations based on a coarse-grained model of DNA. The force-displacement (f-r) curves are computed under the constraints of the constant force (Gibbs) or the constant length (Helmholtz) ensemble. Particular attention was paid to the compressional (negative) and weak tensile forces. It was confirmed that simulations using the vector Gibbs ensemble fail to represent the compression behavior of polymers. Simulations using the scalar Gibbs protocol resulted in a qualitatively correct compressional response of DNA provided that the quadratic averages of displacements were employed. Furthermore, a well-known shortcoming of the popular Marko-Siggia relation for DNA elasticity at weak tensile forces is elucidated. Conversely, the function f-r from the simulation at the constant length constraint, as well as the new closed-form expressions, provides a realistic depiction of the DNA elasticity over the wide range of negative and positive forces. Merely a qualitative resemblance of the compression functions f-r predicted by the employed approaches supports the notion that the elastic response of DNA molecules may be greatly affected by the specifics of the experimental setups and the kind of averaging of the measured variable.
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Affiliation(s)
- Tomáš Bleha
- Polymer Institute, Slovak Academy of Sciences, 84541 Bratislava, Slovakia
| | - Peter Cifra
- Polymer Institute, Slovak Academy of Sciences, 84541 Bratislava, Slovakia
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15
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Zhou Y, Hsiao KW, Regan KE, Kong D, McKenna GB, Robertson-Anderson RM, Schroeder CM. Effect of molecular architecture on ring polymer dynamics in semidilute linear polymer solutions. Nat Commun 2019; 10:1753. [PMID: 30988290 PMCID: PMC6465312 DOI: 10.1038/s41467-019-09627-7] [Citation(s) in RCA: 41] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2018] [Accepted: 03/11/2019] [Indexed: 11/09/2022] Open
Abstract
Understanding the dynamics of ring polymers is a particularly challenging yet interesting problem in soft materials. Despite recent progress, a complete understanding of the nonequilibrium behavior of ring polymers has not yet been achieved. In this work, we directly observe the flow dynamics of DNA-based rings in semidilute linear polymer solutions using single molecule techniques. Our results reveal strikingly large conformational fluctuations of rings in extensional flow long after the initial transient stretching process has terminated, which is observed even at extremely low concentrations (0.025 c*) of linear polymers in the background solution. The magnitudes and characteristic timescales of ring conformational fluctuations are determined as functions of flow strength and polymer concentration. Our results suggest that ring conformational fluctuations arise due to transient threading of linear polymers through open ring chains stretching in flow.
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Affiliation(s)
- Yuecheng Zhou
- Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
- Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
| | - Kai-Wen Hsiao
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
| | - Kathryn E Regan
- Department of Physics, University of San Diego, San Diego, CA, 92110, USA
| | - Dejie Kong
- Department of Chemical Engineering, Texas Tech University, Lubbock, TX, 79409, USA
| | - Gregory B McKenna
- Department of Chemical Engineering, Texas Tech University, Lubbock, TX, 79409, USA
| | | | - Charles M Schroeder
- Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA.
- Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA.
- Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA.
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16
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Soh BW, Klotz AR, Doyle PS. Untying of Complex Knots on Stretched Polymers in Elongational Fields. Macromolecules 2018. [DOI: 10.1021/acs.macromol.8b01879] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Affiliation(s)
- Beatrice W. Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Alexander R. Klotz
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
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17
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Klotz AR, Soh BW, Doyle PS. Motion of Knots in DNA Stretched by Elongational Fields. PHYSICAL REVIEW LETTERS 2018; 120:188003. [PMID: 29775326 DOI: 10.1103/physrevlett.120.188003] [Citation(s) in RCA: 42] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2017] [Revised: 02/07/2018] [Indexed: 06/08/2023]
Abstract
Knots in DNA occur in biological systems, serve as a model system for polymer entanglement, and affect the efficacy of modern genomics technologies. We study the motion of complex knots in DNA by stretching molecules with a divergent electric field that provides an elongational force. We demonstrate that the motion of knots is nonisotropic and driven towards the closest end of the molecule. We show for the first time experimentally that knots can go from a mobile to a jammed state by varying an applied strain rate, and that this jamming is reversible. We measure the mobility of knots as a function of strain rate, demonstrating the conditions under which knots can be driven towards the ends of the molecule and untied.
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Affiliation(s)
- Alexander R Klotz
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02142, USA
| | - Beatrice W Soh
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02142, USA
| | - Patrick S Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02142, USA
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