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Sheung JY, Garamella J, Kahl SK, Lee BY, McGorty RJ, Robertson-Anderson RM. Motor-driven advection competes with crowding to drive spatiotemporally heterogeneous transport in cytoskeleton composites. FRONTIERS IN PHYSICS 2022; 10:1055441. [PMID: 37547053 PMCID: PMC10403238 DOI: 10.3389/fphy.2022.1055441] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 08/08/2023]
Abstract
The cytoskeleton-a composite network of biopolymers, molecular motors, and associated binding proteins-is a paradigmatic example of active matter. Particle transport through the cytoskeleton can range from anomalous and heterogeneous subdiffusion to superdiffusion and advection. Yet, recapitulating and understanding these properties-ubiquitous to the cytoskeleton and other out-of-equilibrium soft matter systems-remains challenging. Here, we combine light sheet microscopy with differential dynamic microscopy and single-particle tracking to elucidate anomalous and advective transport in actomyosin-microtubule composites. We show that particles exhibit multi-mode transport that transitions from pronounced subdiffusion to superdiffusion at tunable crossover timescales. Surprisingly, while higher actomyosin content increases the range of timescales over which transport is superdiffusive, it also markedly increases the degree of subdiffusion at short timescales and generally slows transport. Corresponding displacement distributions display unique combinations of non-Gaussianity, asymmetry, and non-zero modes, indicative of directed advection coupled with caged diffusion and hopping. At larger spatiotemporal scales, particles in active composites exhibit superdiffusive dynamics with scaling exponents that are robust to changing actomyosin fractions, in contrast to normal, yet faster, diffusion in networks without actomyosin. Our specific results shed important new light on the interplay between non-equilibrium processes, crowding and heterogeneity in active cytoskeletal systems. More generally, our approach is broadly applicable to active matter systems to elucidate transport and dynamics across scales.
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Affiliation(s)
- Janet Y. Sheung
- W. M. Keck Science Department, Scripps College, Claremont, CA, United States
- W. M. Keck Science Department, Pitzer College, Claremont, CA, United States
| | - Jonathan Garamella
- Physics and Biophysics Department, University of San Diego, San Diego, CA, United States
| | - Stella K. Kahl
- W. M. Keck Science Department, Scripps College, Claremont, CA, United States
| | - Brian Y. Lee
- W. M. Keck Science Department, Pitzer College, Claremont, CA, United States
| | - Ryan J. McGorty
- Physics and Biophysics Department, University of San Diego, San Diego, CA, United States
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2
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Peddireddy KR, Clairmont R, Neill P, McGorty R, Robertson-Anderson RM. Optical-Tweezers-integrating-Differential-Dynamic-Microscopy maps the spatiotemporal propagation of nonlinear strains in polymer blends and composites. Nat Commun 2022; 13:5180. [PMID: 36056012 PMCID: PMC9440072 DOI: 10.1038/s41467-022-32876-y] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2022] [Accepted: 08/15/2022] [Indexed: 11/08/2022] Open
Abstract
How local stresses propagate through polymeric fluids, and, more generally, how macromolecular dynamics give rise to viscoelasticity are open questions vital to wide-ranging scientific and industrial fields. Here, to unambiguously connect polymer dynamics to force response, and map the deformation fields that arise in macromolecular materials, we present Optical-Tweezers-integrating-Differential -Dynamic-Microscopy (OpTiDMM) that simultaneously imposes local strains, measures resistive forces, and analyzes the motion of the surrounding polymers. Our measurements with blends of ring and linear polymers (DNA) and their composites with stiff polymers (microtubules) uncover an unexpected resonant response, in which strain alignment, superdiffusivity, and elasticity are maximized when the strain rate is comparable to the entanglement rate. Microtubules suppress this resonance, while substantially increasing elastic storage, due to varying degrees to which the polymers buildup, stretch and flow along the strain path, and configurationally relax induced stress. More broadly, the rich multi-scale coupling of mechanics and dynamics afforded by OpTiDDM, empowers its interdisciplinary use to elucidate non-trivial phenomena that sculpt stress propagation dynamics-critical to commercial applications and cell mechanics alike.
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Affiliation(s)
- Karthik R Peddireddy
- Department of Physics and Biophysics, University of San Diego, San Diego, CA, 92110, USA
| | - Ryan Clairmont
- Department of Physics and Biophysics, University of San Diego, San Diego, CA, 92110, USA
| | - Philip Neill
- Department of Physics and Biophysics, University of San Diego, San Diego, CA, 92110, USA
| | - Ryan McGorty
- Department of Physics and Biophysics, University of San Diego, San Diego, CA, 92110, USA
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3
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Sasanpour M, Achiriloaie DH, Lee G, Leech G, Hendija M, Lindsay KA, Ross JL, McGorty RJ, Robertson-Anderson RM. Reconstituting and Characterizing Actin-Microtubule Composites with Tunable Motor-Driven Dynamics and Mechanics. J Vis Exp 2022:10.3791/64228. [PMID: 36094259 PMCID: PMC10290881 DOI: 10.3791/64228] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2023] Open
Abstract
The composite cytoskeleton, comprising interacting networks of semiflexible actin filaments and rigid microtubules, restructures and generates forces using motor proteins such as myosin II and kinesin to drive key processes such as migration, cytokinesis, adhesion, and mechanosensing. While actin-microtubule interactions are key to the cytoskeleton's versatility and adaptability, an understanding of their interplay with myosin and kinesin activity is still nascent. This work describes how to engineer tunable three-dimensional composite networks of co-entangled actin filaments and microtubules that undergo active restructuring and ballistic motion, driven by myosin II and kinesin motors, and are tuned by the relative concentrations of actin, microtubules, motor proteins, and passive crosslinkers. Protocols for fluorescence labeling of the microtubules and actin filaments to most effectively visualize composite restructuring and motion using multi-spectral confocal imaging are also detailed. Finally, the results of data analysis methods that can be used to quantitatively characterize non-equilibrium structure, dynamics, and mechanics are presented. Recreating and investigating this tunable biomimetic platform provides valuable insight into how coupled motor activity, composite mechanics, and filament dynamics can lead to myriad cellular processes from mitosis to polarization to mechano-sensation.
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Affiliation(s)
| | - Daisy H Achiriloaie
- Department of Physics and Biophysics, University of San Diego; W. M. Keck Science Department, Scripps College, Pitzer College, and Claremont McKenna College
| | - Gloria Lee
- Department of Physics and Biophysics, University of San Diego
| | - Gregor Leech
- Department of Physics and Biophysics, University of San Diego
| | - Maya Hendija
- Department of Physics and Biophysics, University of San Diego
| | | | | | - Ryan J McGorty
- Department of Physics and Biophysics, University of San Diego
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4
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Topological digestion drives time-varying rheology of entangled DNA fluids. Nat Commun 2022; 13:4389. [PMID: 35902575 PMCID: PMC9334285 DOI: 10.1038/s41467-022-31828-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2021] [Accepted: 07/04/2022] [Indexed: 11/17/2022] Open
Abstract
Understanding and controlling the rheology of polymeric complex fluids that are pushed out-of-equilibrium is a fundamental problem in both industry and biology. For example, to package, repair, and replicate DNA, cells use enzymes to constantly manipulate DNA topology, length, and structure. Inspired by this feat, here we engineer and study DNA-based complex fluids that undergo enzymatically-driven topological and architectural alterations via restriction endonuclease (RE) reactions. We show that these systems display time-dependent rheological properties that depend on the concentrations and properties of the comprising DNA and REs. Through time-resolved microrheology experiments and Brownian Dynamics simulations, we show that conversion of supercoiled to linear DNA topology leads to a monotonic increase in viscosity. On the other hand, the viscosity of entangled linear DNA undergoing fragmentation displays a universal decrease that we rationalise using living polymer theory. Finally, to showcase the tunability of these behaviours, we design a DNA fluid that exhibits a time-dependent increase, followed by a temporally-gated decrease, of its viscosity. Our results present a class of polymeric fluids that leverage naturally occurring enzymes to drive diverse time-varying rheology by performing architectural alterations to the constituents. Understanding and controlling the rheology of polymeric complex fluids is of fundamental importance in both industry and biology. Here, Michieletto et al. show how to achieve time-dependent rheology of DNA solutions via enzymatically-driven architectural alterations by restriction endonucleases.
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5
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Verwei HN, Lee G, Leech G, Petitjean II, Koenderink GH, Robertson-Anderson RM, McGorty RJ. Quantifying Cytoskeleton Dynamics Using Differential Dynamic Microscopy. J Vis Exp 2022:10.3791/63931. [PMID: 35781524 PMCID: PMC10398790 DOI: 10.3791/63931] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/05/2023] Open
Abstract
Cells can crawl, self-heal, and tune their stiffness due to their remarkably dynamic cytoskeleton. As such, reconstituting networks of cytoskeletal biopolymers may lead to a host of active and adaptable materials. However, engineering such materials with precisely tuned properties requires measuring how the dynamics depend on the network composition and synthesis methods. Quantifying such dynamics is challenged by variations across the time, space, and formulation space of composite networks. The protocol here describes how the Fourier analysis technique, differential dynamic microscopy (DDM), can quantify the dynamics of biopolymer networks and is particularly well suited for studies of cytoskeleton networks. DDM works on time sequences of images acquired using a range of microscopy modalities, including laser-scanning confocal, widefield fluorescence, and brightfield imaging. From such image sequences, one can extract characteristic decorrelation times of density fluctuations across a span of wave vectors. A user-friendly, open-source Python package to perform DDM analysis is also developed. With this package, one can measure the dynamics of labeled cytoskeleton components or of embedded tracer particles, as demonstrated here with data of intermediate filament (vimentin) networks and active actin-microtubule networks. Users with no prior programming or image processing experience will be able to perform DDM using this software package and associated documentation.
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Affiliation(s)
- Hannah N Verwei
- Cell Biology, Neurobiology and Biophysics, Department of Biology, Faculty of Science, Utrecht University
| | - Gloria Lee
- Department of Physics and Biophysics, University of San Diego
| | - Gregor Leech
- Department of Physics and Biophysics, University of San Diego
| | - Irene Istúriz Petitjean
- Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology
| | - Gijsje H Koenderink
- Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology
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Yadav I, Al Sulaiman D, Soh BW, Doyle PS. Phase Transition of Catenated DNA Networks in Poly(ethylene glycol) Solutions. ACS Macro Lett 2021; 10:1429-1435. [PMID: 35549007 DOI: 10.1021/acsmacrolett.1c00463] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
Abstract
Conformational phase transitions of macromolecules are an important class of problems in fundamental polymer physics. While the conformational phase transitions of linear DNA have been extensively studied, this feature of topologically complex DNA remains unexplored. We report herein the polymer-and-salt-induced (Ψ) phase transition of 2D catenated DNA networks, called kinetoplasts, using single-molecule fluorescence microscopy. We observe that kinetoplasts can undergo a reversible transition from the flat phase to the collapsed phase in the presence of NaCl as a function of the crowding agent poly(ethylene glycol). The nature of this phase transition is tunable through varying ionic strengths. For linear DNA, the coexistence of coil and globule phases was attributed to a first order phase transition associated with a double well potential in the transition regime. Kinetoplasts, however, navigate from the flat to the collapsed phase by passing through an intermediate regime, characterized by the coexistence of a multipopulation with varying shapes and sizes. Conformations of individual molecules in the multipopulation are long-lived, which suggests a rugged energy landscape.
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Affiliation(s)
- Indresh Yadav
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Dana Al Sulaiman
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
| | - Beatrice W. Soh
- Institute of Materials Research and Engineering, 2 Fusionopolis Way, Singapore 138634, Singapore
| | - Patrick S. Doyle
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States
- Harvard Medical School Initiative for RNA Medicine, Boston, Massachusetts 02215, United States
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7
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Cerbino R, Giavazzi F, Helgeson ME. Differential dynamic microscopy for the characterization of polymer systems. JOURNAL OF POLYMER SCIENCE 2021. [DOI: 10.1002/pol.20210217] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Roberto Cerbino
- Faculty of Physics University of Vienna Vienna Austria
- Department of Medical Biotechnology and Translational Medicine University of Milan Segrate Italy
| | - Fabio Giavazzi
- Department of Medical Biotechnology and Translational Medicine University of Milan Segrate Italy
| | - Matthew E. Helgeson
- Department of Chemical Engineering University of California Santa Barbara Santa Barbara California USA
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8
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Smrek J, Garamella J, Robertson-Anderson R, Michieletto D. Topological tuning of DNA mobility in entangled solutions of supercoiled plasmids. SCIENCE ADVANCES 2021; 7:eabf9260. [PMID: 33980492 PMCID: PMC8115916 DOI: 10.1126/sciadv.abf9260] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2020] [Accepted: 03/26/2021] [Indexed: 05/04/2023]
Abstract
Ring polymers in dense solutions are among the most intriguing problems in polymer physics. Because of its natural occurrence in circular form, DNA has been extensively used as a proxy to study the fundamental physics of ring polymers in different topological states. Yet, torsionally constrained-such as supercoiled-topologies have been largely neglected so far. The applicability of existing theoretical models to dense supercoiled DNA is thus unknown. Here, we address this gap by coupling large-scale molecular dynamics simulations with differential dynamic microscopy of entangled supercoiled DNA plasmids. We find that, unexpectedly, larger supercoiling increases the size of entangled plasmids and concomitantly induces an enhancement in DNA mobility. These findings are reconciled as due to supercoiling-driven asymmetric and double-folded plasmid conformations that reduce interplasmid entanglements and threadings. Our results suggest a way to topologically tune DNA mobility via supercoiling, thus enabling topological control over the (micro)rheology of DNA-based complex fluids.
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Affiliation(s)
- Jan Smrek
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Jonathan Garamella
- Department of Physics and Biophysics, University of San Diego, San Diego, CA 92110, USA
| | | | - Davide Michieletto
- School of Physics and Astronomy, University of Edinburgh Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK.
- MRC Human Genetics Unit, Institute of Genetics and Molecular Medicine University of Edinburgh, Edinburgh EH4 2XU, UK
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9
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Anderson SJ, Garamella J, Adalbert S, McGorty RJ, Robertson-Anderson RM. Subtle changes in crosslinking drive diverse anomalous transport characteristics in actin-microtubule networks. SOFT MATTER 2021; 17:4375-4385. [PMID: 33908593 PMCID: PMC8189643 DOI: 10.1039/d1sm00093d] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Anomalous diffusion in crowded and complex environments is widely studied due to its importance in intracellular transport, fluid rheology and materials engineering. Specifically, diffusion through the cytoskeleton, a network comprised of semiflexible actin filaments and rigid microtubules that interact both sterically and via crosslinking, plays a principal role in viral infection, vesicle transport and targeted drug delivery. Here, we elucidate the impact of crosslinking on particle diffusion in composites of actin and microtubules with actin-actin, microtubule-microtubule and actin-microtubule crosslinking. We analyze a suite of transport metrics by coupling single-particle tracking and differential dynamic microscopy. Using these complementary techniques, we find that particles display non-Gaussian and non-ergodic subdiffusion that is markedly enhanced by cytoskeletal crosslinking, which we attribute to suppressed microtubule mobility. However, the extent to which transport deviates from normal Brownian diffusion depends strongly on the crosslinking motif - with actin-microtubule crosslinking inducing the most pronounced anomalous characteristics. Our results reveal that subtle changes to actin-microtubule interactions can have complex impacts on particle diffusion in cytoskeleton composites, and suggest that a combination of reduced filament mobility and more variance in actin mobilities leads to more strongly anomalous particle transport.
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Affiliation(s)
- S J Anderson
- Department of Physics & Biophysics, University of San Diego, San Diego, CA 92110, USA.
| | - J Garamella
- Department of Physics & Biophysics, University of San Diego, San Diego, CA 92110, USA.
| | - S Adalbert
- Department of Physics & Biophysics, University of San Diego, San Diego, CA 92110, USA.
| | - R J McGorty
- Department of Physics & Biophysics, University of San Diego, San Diego, CA 92110, USA.
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10
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Xu Z, Dai X, Bu X, Yang Y, Zhang X, Man X, Zhang X, Doi M, Yan LT. Enhanced Heterogeneous Diffusion of Nanoparticles in Semiflexible Networks. ACS NANO 2021; 15:4608-4616. [PMID: 33625839 DOI: 10.1021/acsnano.0c08877] [Citation(s) in RCA: 24] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The transport of nanoparticles in semiflexible networks, which form diverse principal structural components throughout living systems, is important in biology and biomedical applications. By combining large-scale molecular simulations as well as theoretical analysis, we demonstrate here that nanoparticles in polymer networks with semiflexible strands possess enhanced heterogeneous diffusion characterized by more evident hopping dynamics. Particularly, the hopping energy barrier approximates to linear dependence on confinement parameters in the regime of moderate rigidity, in contrast to the quadratic dependence of both its soft and hard counterparts. This nonmonotonic feature can be attributed to the competition between the conformation entropy and the bending energy regulated by the chain rigidity, captured by developing an analytical model of a hopping energy barrier. Moreover, these theoretical results agree reasonably well with previous experiments. The findings bear significance in unraveling the fundamental physics of substance transport confined in network-topological environments and would provide an explanation for the dynamics diversity of nanoparticles within various networks, biological or synthetic.
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Affiliation(s)
- Ziyang Xu
- State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China
| | - Xiaobin Dai
- State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China
| | - Xiangyu Bu
- School of Science, Beijing Jiaotong University, Beijing, 100044, China
| | - Ye Yang
- State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China
| | - Xuanyu Zhang
- State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China
| | - Xingkun Man
- Center of Soft Matter Physics and its Applications, Beihang University, Beijing, 100191, China
- School of Physics, Beihang University, Beijing, 100191, China
| | - Xinghua Zhang
- School of Science, Beijing Jiaotong University, Beijing, 100044, China
| | - Masao Doi
- Center of Soft Matter Physics and its Applications, Beihang University, Beijing, 100191, China
- School of Physics, Beihang University, Beijing, 100191, China
| | - Li-Tang Yan
- State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing, 100084, China
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11
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Garamella J, Regan K, Aguirre G, McGorty RJ, Robertson-Anderson RM. Anomalous and heterogeneous DNA transport in biomimetic cytoskeleton networks. SOFT MATTER 2020; 16:6344-6353. [PMID: 32555863 PMCID: PMC7388685 DOI: 10.1039/d0sm00544d] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
The cytoskeleton, a complex network of protein filaments and crosslinking proteins, dictates diverse cellular processes ranging from division to cargo transport. Yet, the role the cytoskeleton plays in the intracellular transport of DNA and other macromolecules remains poorly understood. Here, using single-molecule conformational tracking, we measure the transport and conformational dynamics of linear and relaxed circular (ring) DNA in composite networks of actin and microtubules with variable types of crosslinking. While both linear and ring DNA undergo anomalous, non-Gaussian, and non-ergodic subdiffusion, the detailed dynamics are controlled by both DNA topology (linear vs. ring) and crosslinking motif. Ring DNA swells, exhibiting heterogeneous subdiffusion controlled via threading by cytoskeleton filaments, while linear DNA compacts, exhibiting transport via caging and hopping. Importantly, while the crosslinking motif has little effect on ring DNA, linear DNA in networks with actin-microtubule crosslinking is significantly less ergodic and shows more heterogeneous transport than with actin-actin or microtubule-microtubule crosslinking.
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Affiliation(s)
- Jonathan Garamella
- Department of Physics & Biophysics, University of San Diego, San Diego, CA 92110, USA.
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12
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Cho JH, Cerbino R, Bischofberger I. Emergence of Multiscale Dynamics in Colloidal Gels. PHYSICAL REVIEW LETTERS 2020; 124:088005. [PMID: 32167319 DOI: 10.1103/physrevlett.124.088005] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 02/05/2020] [Indexed: 06/10/2023]
Abstract
To gain insight into the kinetics of colloidal gel evolution at low particle volume fractions ϕ, we utilize differential dynamic microscopy to investigate particle aggregation, geometric percolation, and the subsequent transition to nonergodic dynamics. We report the emergence of unexpectedly rich multiscale dynamics upon the onset of nonergodicity, which separates the wave vectors q into three different regimes. In the high-q domain, the gel exhibits ϕ-independent internal vibrations of fractal clusters. The intermediate-q domain is dominated by density fluctuations at the length scale of the clusters, as evidenced by the q independence of the relaxation time τ. In the low-q domain, the scaling of τ as q^{-3} suggests that the network appears homogeneous. The transitions between these three regimes introduce two characteristic length scales, distinct from the cluster size.
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Affiliation(s)
- Jae Hyung Cho
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Roberto Cerbino
- Dipartimento di Biotecnologie Mediche e Medicina Traslazionale, Università degli Studi di Milano, Via. F.lli Cervi 93, Segrate (MI) I-20090, Italy
| | - Irmgard Bischofberger
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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