1
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Goychuk A, Kannan D, Kardar M. Delayed Excitations Induce Polymer Looping and Coherent Motion. PHYSICAL REVIEW LETTERS 2024; 133:078101. [PMID: 39213554 DOI: 10.1103/physrevlett.133.078101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2024] [Revised: 06/25/2024] [Accepted: 07/12/2024] [Indexed: 09/04/2024]
Abstract
We consider inhomogeneous polymers driven by energy-consuming active processes which encode temporal patterns of athermal kicks. We find that such temporal excitation programs, propagated by tension along the polymer, can effectively couple distinct polymer loci. Consequently, distant loci exhibit correlated motions that fold the polymer into specific conformations, as set by the local actions of the active processes and their distribution along the polymer. Interestingly, active kicks that are canceled out by a time-delayed echo can induce strong compaction of the active polymer.
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2
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Luo MB, Shen YF. Langevin dynamics simulations for the critical adsorption of end-grafted active polymers. SOFT MATTER 2024; 20:5113-5121. [PMID: 38894642 DOI: 10.1039/d4sm00526k] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/21/2024]
Abstract
The critical adsorption of end-grafted active polymer chains on an attractive surface is studied using Langevin dynamics simulations. The active polymers are composed of an active Langevin particle located at the head and a sequential passive chain. Results show that the active force exerted by the active head pulls the active polymer away from the surface. Consequently, the adsorption of the active polymer is hindered, and the critical surface attraction strength, , increases proportionally to the square of the active force, Fa2. The increase in depends on the rotation behavior of the active head. Specifically, for the restricted rotating active polymer (RRAP) chain with a longer rotational persistence time as the rotation of the active head is restricted, increases significantly with Fa. On the other hand, for the freely rotating active polymer (FRAP) chain with a shorter rotational persistence time as the rotation of the active head is free, shows a weak dependence on Fa. The results show that the active force has a significantly stronger pulling effect on the RRAP chain than on the FRAP chain. Furthermore, knotted conformations are observed for the adsorbed RRAP chain at large Fa. These knots reduce the adsorption of monomers near the grafted end. In contrast, no knotted conformations are observed for the FRAP chains due to the comparatively weaker pulling effect of the active force.
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Affiliation(s)
- Meng-Bo Luo
- School of Physics, Zhejiang University, Hangzhou 310027, China.
| | - Yi-Fan Shen
- School of Physics, Zhejiang University, Hangzhou 310027, China.
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3
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Li J, Zhang B, Wang ZY. Activity-induced stiffness, entanglement network and dynamic slowdown in unentangled semidilute polymer solutions. SOFT MATTER 2024; 20:5174-5182. [PMID: 38895794 DOI: 10.1039/d4sm00341a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/21/2024]
Abstract
Active polymers possess numerous unique properties that are quite different from those observed in the system of small active molecules due to the intricate interplay between their activity and topological constraints. This study focuses on the conformational changes induced by activity, impacting effective stiffness and crucially influencing entanglement and dynamics. When the two terminals of a linear chain undergo active modification through coupling to a high-temperature thermal bath, there is a substantial increase in chain size, indicating a notable enhancement in effective stiffness. Unlike in passive semiflexible chains where stiffness predominantly affects local bond angles, activity-induced stiffness manifests at the scale of tens of monomers. While activity raises the ambient temperature, it significantly decreases diffusion by over an order of magnitude. The slowdown of the dynamics observed can be attributed to increased entanglement due to chain elongation.
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Affiliation(s)
- Jing Li
- School of Physical Science and Technology, Southwest University, Chongqing 400715, China.
- Chongqing Key Laboratory of Micro-Nano Structure Optoelectronics, Chongqing 400715, China
| | - Bokai Zhang
- School of Physical Science and Technology, Southwest University, Chongqing 400715, China.
- Chongqing Key Laboratory of Micro-Nano Structure Optoelectronics, Chongqing 400715, China
| | - Zhi-Yong Wang
- School of Physical Science and Technology, Southwest University, Chongqing 400715, China.
- Chongqing Key Laboratory of Micro-Nano Structure Optoelectronics, Chongqing 400715, China
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4
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Yadav RS, Sharma S, Metzler R, Chakrabarti R. A passive star polymer in a dense active bath: insights from computer simulations. SOFT MATTER 2024; 20:3910-3922. [PMID: 38700098 DOI: 10.1039/d4sm00144c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2024]
Abstract
Using computer simulations in two dimensions (2D), we explore the structure and dynamics of a star polymer with three arms made of passive monomers immersed in a bath of active Brownian particles (ABPs). We analyze the conformational and dynamical changes of the polymer as a function of activity and packing fraction. We also study the process of motility induced phase separation (MIPS) in the presence of a star polymer, which acts as a mobile nucleation center. The presence of the polymer increases the growth rate of the clusters in comparison to a bath without the polymer. In particular, for low packing fraction, both nucleation and cluster growth are affected by the inclusion of the star polymer. Clusters grow in the vicinity of the star polymer, resulting in the star polymer experiencing a caged motion similar to a tagged ABP in the dense phase. Due to the topological constraints of the star polymers and clustering nearby, the conformational changes of the star polymer lead to interesting observations. Inter alia, we observe the shrinking of the arm with increasing activity along with a short-lived hairpin structure of one arm formed. We also see the transient pairing of two arms of the star polymer, while the third is largely separated at high activity. We hope our findings will help in understanding the behavior of active-passive mixtures, including biopolymers of complex topology in dense active suspensions.
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Affiliation(s)
- Ramanand Singh Yadav
- Department of Chemistry, Indian Institute of Technology Bombay, Mumbai 400076, India.
| | - Sanaa Sharma
- Department of Chemistry, Indian Institute of Technology Bombay, Mumbai 400076, India.
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, Germany.
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
| | - Rajarshi Chakrabarti
- Department of Chemistry, Indian Institute of Technology Bombay, Mumbai 400076, India.
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5
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Vatin M, Kundu S, Locatelli E. Conformation and dynamics of partially active linear polymers. SOFT MATTER 2024; 20:1892-1904. [PMID: 38323323 DOI: 10.1039/d3sm01162c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2024]
Abstract
We perform numerical simulations of isolated, partially active polymers, driven out-of-equilibrium by a fraction of their monomers. We show that, if the active beads are all gathered in a contiguous block, the position of the section along the chain determines the conformational and dynamical properties of the system. Notably, one can modulate the diffusion coefficient of the polymer from active-like to passive-like just by changing the position of the active block. Further, we show that a slight modification of the self-propulsion rule may give rise to an enhancement of diffusion under certain conditions, despite a decrease of the overall polymer activity. Our findings may help in the modelisation of active biophysical systems, such as filamentous bacteria or worms.
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Affiliation(s)
- Marin Vatin
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy.
- INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
| | - Sumanta Kundu
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy.
- INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
- International School for Advanced Studies (SISSA), 34136, Trieste, Italy
| | - Emanuele Locatelli
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy.
- INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
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6
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Li C, Chen Q, Ding M. Escape dynamics of active ring polymers in a cylindrical nanochannel. SOFT MATTER 2024; 20:1719-1724. [PMID: 38284326 DOI: 10.1039/d3sm01524f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/30/2024]
Abstract
We explore the escape dynamics of active ring polymers confined in a cylindrical nanochannel using Brownian dynamics. Our simulation results show that the escape time decreases with the increase of the Péclet number, which is not noticeable between the two stages of the escape process, based on whether the center of mass of the polymer is inside or outside the nanochannel. However, the monomer motion trajectory of the active polymer is very different from that of the passive polymer, similar to the snake-like motion with uniform velocity. The passive polymer, however, is in constant fugitive motion with increased velocity at the tail end of the escape. Our work is vital for understanding the escape dynamics of active ring polymers in the confined nanochannel, which provides new perspectives on their characterization and analysis.
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Affiliation(s)
- Chuqiao Li
- Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matter Physics, College of Physical Science and Technology, Yili Normal University, Yining 835000, China
| | - Qiaoyue Chen
- Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matter Physics, College of Physical Science and Technology, Yili Normal University, Yining 835000, China
| | - Mingming Ding
- Xinjiang Laboratory of Phase Transitions and Microstructures in Condensed Matter Physics, College of Physical Science and Technology, Yili Normal University, Yining 835000, China
- School of Chemical Engineering and Light Industry, Guangdong University of Technology, Guangzhou 510006, China.
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7
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Rudyak VY, Lopushenko A, Palyulin VV, Chertovich AV. Long-range ordering of velocity-aligned active polymers. J Chem Phys 2024; 160:044905. [PMID: 38275191 DOI: 10.1063/5.0181252] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 12/26/2023] [Indexed: 01/27/2024] Open
Abstract
In this work, we study the effect of covalent bonding on the behavior of non-equilibrium systems with the active force acting on particles along their velocity. Self-ordering of single particles does not occur in this model. However, starting from some critical polymerization degree, the ordered state is observed. It is homogeneous and exhibits no phase separation. In the ordered state, the chains prefer a near-two-dimensional configuration and all move in one direction. Importantly, the self-ordering is obtained only at intermediate active force magnitudes. At high magnitudes, the transition from the disordered to ordered state is suppressed by the swelling of the chains during the transition, as we show by the transition kinetics analysis. We demonstrate the bistable behavior of the system in a particular range of polymerization degrees, amplitudes of active force, densities, and thermostat temperatures. Overall, we show that covalent bonding greatly aids the self-ordering in this active particle model, in contrast to active Brownian particles.
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Affiliation(s)
- Vladimir Yu Rudyak
- Semenov Federal Research Center for Chemical Physics, Kosygina, 4, 119991 Moscow, Russia
| | - Alexander Lopushenko
- Semenov Federal Research Center for Chemical Physics, Kosygina, 4, 119991 Moscow, Russia
| | - Vladimir V Palyulin
- Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, 121205 Moscow, Russia
| | - Alexander V Chertovich
- Semenov Federal Research Center for Chemical Physics, Kosygina, 4, 119991 Moscow, Russia
- Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia
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8
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Panda A, Winkler RG, Singh SP. Characteristic features of self-avoiding active Brownian polymers under linear shear flow. SOFT MATTER 2023; 19:8577-8586. [PMID: 37905462 DOI: 10.1039/d3sm01334k] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/02/2023]
Abstract
We present Brownian dynamics simulation results of a flexible linear polymer with excluded-volume interactions under shear flow in the presence of active noise. The active noise strongly affects the polymer's conformational and dynamical properties, such as the stretching in the flow direction and compression in the gradient direction, shear-induced alignment, and shear viscosity. In the asymptotic limit of large activities and shear rates, the power-law scaling exponents of these quantities differ significantly from those of passive polymers. The chain's shear-induced stretching at a given shear rate is reduced by active noise, and it displays a non-monotonic behavior, where an initial polymer compression is followed by its stretching with increasing active force. The compression of the polymer in the gradient direction follows the relation ∼WiPe-3/4 as a function of the activity-dependent Weissenberg number WiPe, which differs from the scaling observed in passive systems ∼WiPe-1/2. The flow-induced alignment at large Péclet numbers Pe ≫ 1, where Pe is the Péclet number, and large shear rates WiPe ≫ 1 displays the scaling behavior WiPe-1/2, with an exponent differing from the passive value -1/3. Furthermore, the polymer's zero-shear viscosity displays a non-monotonic behavior, decreasing in an intermediate activity regime due to excluded-volume interactions and increasing again for large Pe. Shear thinning appears with increasing Weissenberg number with the power-laws WiPe-1/2 and WiPe-3/4 for passive and active polymers, respectively. In addition, our simulation results are compared with the results of an analytical approach, which predicts quantitatively similar behaviors for the various aforementioned physical quantities.
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Affiliation(s)
- Arindam Panda
- Department of Physics, Indian Institute of Science Education and Research, Bhopal 462 066, Madhya Pradesh, India.
| | - Roland G Winkler
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, 52428 Jülich, Germany.
| | - Sunil P Singh
- Department of Physics, Indian Institute of Science Education and Research, Bhopal 462 066, Madhya Pradesh, India.
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9
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Sahoo R, Chakrabarti R. Structure and dynamics of an active polymer chain inside a nanochannel grafted with polymers. SOFT MATTER 2023; 19:5978-5988. [PMID: 37497754 DOI: 10.1039/d3sm00618b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/28/2023]
Abstract
We use computer simulations to investigate the complex dynamics of a polymer, made of active Brownian particles, inside a channel grafted internally with passive polymer chains. Our simulations reveal that this probe-polymer, if passive, exhibits a compact structure when its interaction is repulsive with the grafted chains as it tends to stay within the hollow space created along the axis of the channel. On increasing the attractive interaction, the passive probe-polymer is pulled towards the grafted polymeric region and adopts an extended structure. By contrast, switching on the activity helps the probe-polymer to escape from the local traps caused by the sticky grafted chains. The interplay between the activity of the probe-polymer and its sticky interaction with the grafted chains results in shrinking, followed by swelling as the activity is increased. To elucidate the dynamics we compute the mean square displacement (MSD) of the center of mass of the probe-polymer, which increases monotonically with activity and displays superdiffusive behavior at an intermediate time and enhanced diffusion at a long time period. In addition, compared with the attractive interaction, the active probe-polymer shows faster dynamics when the interaction is repulsive to the grafted polymers. We believe that our current study will provide insights into the structural changes and dynamics of active polymers in heterogeneous media and will be useful in designing polymer-based drug delivery vehicles.
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Affiliation(s)
- Rajiblochan Sahoo
- Department of Chemistry, Indian Institute of Technology Bombay, Mumbai 400076, India.
| | - Rajarshi Chakrabarti
- Department of Chemistry, Indian Institute of Technology Bombay, Mumbai 400076, India.
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10
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Goychuk A, Kannan D, Chakraborty AK, Kardar M. Polymer folding through active processes recreates features of genome organization. Proc Natl Acad Sci U S A 2023; 120:e2221726120. [PMID: 37155885 PMCID: PMC10194017 DOI: 10.1073/pnas.2221726120] [Citation(s) in RCA: 17] [Impact Index Per Article: 17.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/25/2022] [Accepted: 04/02/2023] [Indexed: 05/10/2023] Open
Abstract
From proteins to chromosomes, polymers fold into specific conformations that control their biological function. Polymer folding has long been studied with equilibrium thermodynamics, yet intracellular organization and regulation involve energy-consuming, active processes. Signatures of activity have been measured in the context of chromatin motion, which shows spatial correlations and enhanced subdiffusion only in the presence of adenosine triphosphate. Moreover, chromatin motion varies with genomic coordinate, pointing toward a heterogeneous pattern of active processes along the sequence. How do such patterns of activity affect the conformation of a polymer such as chromatin? We address this question by combining analytical theory and simulations to study a polymer subjected to sequence-dependent correlated active forces. Our analysis shows that a local increase in activity (larger active forces) can cause the polymer backbone to bend and expand, while less active segments straighten out and condense. Our simulations further predict that modest activity differences can drive compartmentalization of the polymer consistent with the patterns observed in chromosome conformation capture experiments. Moreover, segments of the polymer that show correlated active (sub)diffusion attract each other through effective long-ranged harmonic interactions, whereas anticorrelations lead to effective repulsions. Thus, our theory offers nonequilibrium mechanisms for forming genomic compartments, which cannot be distinguished from affinity-based folding using structural data alone. As a first step toward exploring whether active mechanisms contribute to shaping genome conformations, we discuss a data-driven approach.
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Affiliation(s)
- Andriy Goychuk
- Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA02139
| | - Deepti Kannan
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA02139
| | - Arup K. Chakraborty
- Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA02139
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA02139
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139
- Ragon Institute of Massachusetts General Hospital, Massachusetts Institute of Technology and Harvard University, Cambridge, MA02139
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA02139
| | - Mehran Kardar
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA02139
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11
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Yan R, Tan F, Wang J, Zhao N. Conformation and dynamics of an active filament in crowded media. J Chem Phys 2023; 158:114905. [PMID: 36948796 DOI: 10.1063/5.0142559] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/04/2023] Open
Abstract
The structural and dynamical properties of active filamentous objects under macromolecular crowding have a great relevance in biology. By means of Brownian dynamics simulations, we perform a comparative study for the conformational change and diffusion dynamics of an active chain in pure solvents and in crowded media. Our result shows a robust compaction-to-swelling conformational change with the augment of the Péclet number. The presence of crowding facilitates self-trapping of monomers and, thus, reinforces the activity mediated compaction. In addition, the efficient collisions between the self-propelled monomers and crowders induce a coil-to-globulelike transition, indicated by a marked change of the Flory scaling exponent of the gyration radius. Moreover, the diffusion dynamics of the active chain in crowded solutions demonstrates activity-enhanced subdiffusion. The center of mass diffusion manifests rather new scaling relations with respect to both the chain length and Péclet number. The interplay of chain activity and medium crowding provides a new mechanism to understand the non-trivial properties of active filaments in complex environments.
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Affiliation(s)
- Ran Yan
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Fei Tan
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Jingli Wang
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu 610064, China
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12
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Tejedor AR, Carracedo R, Ramírez J. Molecular dynamics simulations of active entangled polymers reptating through a passive mesh. POLYMER 2023. [DOI: 10.1016/j.polymer.2023.125677] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
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13
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Hu HX, Shen YF, Wang C, Luo MB. Dynamics of a two-dimensional active polymer chain with a rotation-restricted active head. SOFT MATTER 2022; 18:8820-8829. [PMID: 36367147 DOI: 10.1039/d2sm01139e] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
Abstract
The dynamics of a two-dimensional active polymer composed of an active Brownian particle (ABP) at the head and a passive polymer chain is investigated using Langevin dynamics simulation. The ABP experiences a self-propulsion force fs and a resistance torque M as the passive polymer chain is bonded to the edge of the ABP. M restricts the rotation of the ABP, and thus the dynamics of the ABP and that of the whole active polymer are influenced significantly. Due to this restriction, the persistence time τr, which characterizes the random rotation of the ABP, is increased significantly and changes non-monotonically with the rotational friction coefficient ηr. Our simulation results show that the effect of M on the dynamics of the active polymer can be characterized mainly by the change of τr. Moreover, the propulsive diffusion coefficient DP of the whole polymer chain originated from the self-propulsion force can be described by a scaling relation DP ∝ fs2τr/N2ηt2 with ηt the translational friction coefficient and N the polymer length. Our results show that the diffusion is promoted by the resistance torque M and τr is a key factor for the diffusion of active polymers.
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Affiliation(s)
- Han-Xian Hu
- Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China.
| | - Yi-Fan Shen
- Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China.
| | - Chao Wang
- Department of Physics, Taizhou University, Taizhou 318000, Zhejiang, China.
| | - Meng-Bo Luo
- Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China.
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14
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Wang C, Zhou Y, Yang X, Chen Y, Shen Y, Luo M. Conformation and dynamics of a tethered active polymer chain. Phys Rev E 2022; 106:054501. [PMID: 36559343 DOI: 10.1103/physreve.106.054501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2022] [Accepted: 10/13/2022] [Indexed: 11/05/2022]
Abstract
The conformational and dynamical properties of a tethered semiflexible polymer chain under tangential active force (f_{a}) are studied by using the Langevin dynamics simulation method. The head of the polymer is fixed near an infinite flat surface at z=0. The polymer is equilibrated first at f_{a}=0 and then subjected to the active force. Under the influence of the active force, the polymer is gradually compressed. Specially, for large f_{a} and large bending rigidity (k_{b}), the polymer is buckled into a quasihelical structure rotating around the z axis at the steady state. It is found that both the radius of the quasihelical structure (R) and the angular velocity of the rotation (ω) are nearly independent of the polymer length (N), but show scaling relations with f_{a} and k_{b}, i.e., R∝f_{a}^{-1/3}k_{b}^{1/3} and ω∝f_{a}^{4/3}k_{b}^{-1/3}, which are explained by simple dynamical models. Before reaching the steady state, it is further found that the buckling velocity of the polymer is proportional to f_{a} but roughly independent of k_{b} and N, then the buckling time (t_{b}) can be described by a scaling relation t_{b}∝Nf_{a}^{-1}. The underlying mechanism of the buckling process is revealed.
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Affiliation(s)
- Chao Wang
- Department of Physics, Taizhou University, Taizhou 318000, Zhejiang, China
| | - Yanli Zhou
- Department of Physics, Taizhou University, Taizhou 318000, Zhejiang, China
| | - Xiao Yang
- Department of Physics, Taizhou University, Taizhou 318000, Zhejiang, China
| | - Yingcai Chen
- Department of Physics, Taizhou University, Taizhou 318000, Zhejiang, China
| | - Yifan Shen
- Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China
| | - Mengbo Luo
- Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China
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15
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Paul S, Majumder S, Janke W. Activity mediated globule to coil transition of a flexible polymer in a poor solvent. SOFT MATTER 2022; 18:6392-6403. [PMID: 35979819 DOI: 10.1039/d2sm00354f] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Understanding the role of self-propulsion on the conformational properties of active filamentous objects has relevance in biology. In this work, we consider a flexible bead-spring model for active polymers with both attractive and repulsive interactions among the non-bonded monomers. The activity for each monomer works along its intrinsic direction of self-propulsion which changes diffusively with time. We study its kinetics in the overdamped limit, following quenching from good to poor solvent conditions. We observe that with low activities, though the kinetic pathways remain similar, the scaling exponent for the relaxation time of globule formation becomes smaller than that for the case with no activity. Interestingly, for higher activities when self-propulsion dominates over interaction energy, the polymer conformation becomes extended coil-like. There, in the steady state, the variation of the spatial extension of the polymer, measured via its gyration radius, shows two completely different scaling regimes: the corresponding Flory exponent ν changes from 1/3 to 3/5 similar to a transition of the polymer from a globular state to a self-avoiding walk. This can be explained by an interplay among the three energy scales present in the system, viz., the "ballistic", thermal, and interaction energy.
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Affiliation(s)
- Subhajit Paul
- Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany.
- International Center for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore-560089, India
| | - Suman Majumder
- Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany.
- Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida 201313, India
| | - Wolfhard Janke
- Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany.
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16
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Kozhukhov T, Shendruk TN. Mesoscopic simulations of active nematics. SCIENCE ADVANCES 2022; 8:eabo5788. [PMID: 36001669 PMCID: PMC9401632 DOI: 10.1126/sciadv.abo5788] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/11/2022] [Accepted: 07/11/2022] [Indexed: 06/15/2023]
Abstract
Coarse-grained, mesoscale simulations are invaluable for studying soft condensed matter because of their ability to model systems in which a background solvent plays a substantial role but is not the primary interest. Such methods generally model passive solvents; however, far-from-equilibrium systems may also be composed of complex solutes suspended in an active fluid. Yet, few coarse-grained simulation methods exist to model an active medium. We introduce an algorithm to simulate active nematics, which builds on multiparticle collision dynamics (MPCD) for passive fluctuating nematohydrodynamics by introducing dipolar activity in the local collision operator. Active nematic MPCD (AN-MPCD) simulations not only exhibit the key characteristics of active nematic turbulence but, as a particle-based algorithm, also reproduce crucial attributes of active particle models. Thus, mesoscopic AN-MPCD is an approach that bridges microscopic and continuum descriptions, allowing simulations of composite active-passive systems.
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17
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Philipps CA, Gompper G, Winkler RG. Dynamics of active polar ring polymers. Phys Rev E 2022; 105:L062501. [PMID: 35854564 DOI: 10.1103/physreve.105.l062501] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 05/23/2022] [Indexed: 06/15/2023]
Abstract
The conformational and dynamical properties of isolated semiflexible active polar ring polymers are investigated analytically. A ring is modeled as a continuous Gaussian polymer exposed to tangential active forces. The analytical solution of the linear non-Hermitian equation of motion in terms of an eigenfunction expansion shows that ring conformations are independent of activity. In contrast, activity strongly affects the internal ring dynamics and yields characteristic time regimes, which are absent in passive rings. On intermediate timescales, flexible rings show an activity-enhanced diffusive regime, while semiflexible rings exhibit ballistic motion. Moreover, a second active time regime emerges on longer timescales, where rings display a snake-like motion, which is reminiscent to a tank-treading rotational dynamics in shear flow, dominated by the mode with the longest relaxation time.
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Affiliation(s)
- Christian A Philipps
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
- Department of Physics, RWTH Aachen University, 52056 Aachen, Germany
| | - Gerhard Gompper
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
| | - Roland G Winkler
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
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18
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Bera P, Wasim A, Mondal J. Hi-C embedded polymer model of Escherichia coli reveals the origin of heterogeneous subdiffusion in chromosomal loci. Phys Rev E 2022; 105:064402. [PMID: 35854496 DOI: 10.1103/physreve.105.064402] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 05/10/2022] [Indexed: 06/15/2023]
Abstract
Underneath its apparently simple architecture, the circular chromosome of Escherichia coli is known for displaying complex dynamics in its cytoplasm, with past investigations hinting at inherently diverse mobilities of chromosomal loci across the genome. To decipher its origin, we simulate the dynamics of genome-wide spectrum of E. coli chromosomal loci, via integrating its experimentally derived Hi-C interaction matrix within a polymer-based model. Our analysis demonstrates that, while the dynamics of the chromosome is subdiffusive in a viscoelastic media, the diffusion constants are strongly dependent of chromosomal loci coordinates and diffusive exponents (α) are widely heterogenous with α ≈ 0.36-0.60. The loci-dependent heterogeneous dynamics and mean first-passage times of interloci encounter were found to be modulated via genetically distant interloci communications and is robust even in the presence of active, ATP-dependent noises. Control investigations reveal that the absence of Hi-C-derived interactions in the model would have abolished the traits of heterogeneous loci diffusion, underscoring the key role of loci-specific genetically distant interaction in modulating the underlying heterogeneity of the loci diffusion.
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Affiliation(s)
- Palash Bera
- Tata Institute of Fundamental Research, Hyderabad 500046, India
| | - Abdul Wasim
- Tata Institute of Fundamental Research, Hyderabad 500046, India
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19
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Mao DJ, Qin CR, Tian WD. Kick effect of enzymes causes filament compression. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:284003. [PMID: 35477158 DOI: 10.1088/1361-648x/ac6b09] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2021] [Accepted: 04/27/2022] [Indexed: 06/14/2023]
Abstract
We investigate the influence of enzymes on the structure and dynamics of a filament by dissipative particle dynamics simulations. Enzyme exerts a kick force on the filament monomer. We pay particular attention to two factors: the magnitude of kick force and enzyme concentration. Large kick force as well as high enzyme concentration prefers a remarkable compression of the filament reminiscent of the effective depletion interaction owing to an effective increase in enzyme size and the reduction of solvent quality. Additionally, the kick effect gives rise to an increase of enzyme density from the center-of-mass of the filament to its periphery. Moreover, the increase of enzyme concentration and kick force also causes a decrease in relaxation time. Our finding is helpful to understand the role of catalytic force in chemo-mechano-biological function and the filament behavior under chemical reaction via kick-induced change of solvent quality.
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Affiliation(s)
- Dan-Jian Mao
- Center for Soft Condensed Matter Physics & Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
| | - Chao-Ran Qin
- Center for Soft Condensed Matter Physics & Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
| | - Wen-de Tian
- Center for Soft Condensed Matter Physics & Interdisciplinary Research, School of Physical Science and Technology, Soochow University, Suzhou 215006, People's Republic of China
- Department of Chemical Engineering, Macromolecular Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States of America
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20
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Affiliation(s)
- Namita Jain
- Department of Physics, Indian Institute of Science Education and Research, Bhopal 462066, India
| | - Snigdha Thakur
- Department of Physics, Indian Institute of Science Education and Research, Bhopal 462066, India
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21
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Wu JC, Lin FJ, Ai BQ. Absolute negative mobility of active polymer chains in steady laminar flows. SOFT MATTER 2022; 18:1194-1200. [PMID: 35037681 DOI: 10.1039/d1sm01664d] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We investigate the transport of active polymer chains in steady laminar flows in the presence of thermal noise and an external constant force. In the model, the polymer chain is worm-like and is propelled by active forces along its tangent vectors. Compared with inertial Brownian particles, active polymer chains in steady laminar flows exhibit richer movement patterns due to their specific spatial structures. The simulation results show that the velocity-force relation is strongly dependent on the system parameters such as the chain length, bending rigidity, active force and so on. The polymer chain may move in some preferential movement directions and exhibits absolute negative mobility within appropriate parameter regimes, i.e., the polymer chain can move in a direction opposite to the external constant force. In particular, we can observe giant negative mobility in a broad range of parameter regimes.
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Affiliation(s)
- Jian-Chun Wu
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China.
- School of Physics and Electronic Information, Shangrao Normal University, Shangrao 334001, China
| | - Fu-Jun Lin
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China.
| | - Bao-Quan Ai
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China.
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22
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Eisenstecken T, Winkler RG. Path integral description of semiflexible active Brownian polymers. J Chem Phys 2022; 156:064105. [DOI: 10.1063/5.0081020] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - Roland G. Winkler
- Institute for Advanced Simulation, Forschungszentrum Jülich, Germany
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23
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Mahajan A, Saintillan D. Self-induced hydrodynamic coil-stretch transition of active polymers. Phys Rev E 2022; 105:014608. [PMID: 35193223 DOI: 10.1103/physreve.105.014608] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2021] [Accepted: 01/10/2022] [Indexed: 06/14/2023]
Abstract
We analyze the conformational dynamics and statistical properties of an active polymer model. The polymer is described as a freely jointed bead-rod chain subject to stochastic active force dipoles that act on the suspending solvent where they drive long-ranged fluid flows. Using Langevin simulations of isolated chains in unconfined domains, we show how the coupling of active flows with polymer conformations leads to emergent dynamics. Systems with contractile dipoles behave similarly to passive Brownian chains with enhanced fluctuations due to dipolar flows. In systems with extensile dipoles, however, our simulations uncover an active coil-stretch transition whereby the polymer spontaneously unfolds and stretches out in its own self-induced hydrodynamic flow, and we characterize this transition in terms of a dimensionless activity parameter comparing active dipolar forces to thermal fluctuations. We discuss our findings in the context of the classic coil-stretch transition of passive polymers in extensional flows and complement our simulations with a simple kinetic model for an active trimer.
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Affiliation(s)
- Achal Mahajan
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, California 92093, USA
| | - David Saintillan
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, California 92093, USA
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24
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Anand SK, Singh SP. Migration of active filaments under Poiseuille flow in a microcapillary tube. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:150. [PMID: 34910263 DOI: 10.1140/epje/s10189-021-00153-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2021] [Accepted: 11/20/2021] [Indexed: 06/14/2023]
Abstract
We present a comprehensive study of active filaments confined in a cylindrical channel under Poiseuille flow. The activity drives the filament towards the channel boundary, whereas external fluid flow migrates the filament away from the boundary. This migration further shifts towards the centre for higher flow strength. The migration behaviour of the filaments is presented in terms of the alignment order parameter that shows the alignment grows with shear and activity. Further, we have also addressed the role of length of filament on the migration behaviour, which suggests higher migration for larger filaments. Moreover, we discuss the polar ordering of filaments as a function of distance from the centre of channel that displays upstream motion near the boundary and downstream motion at the centre of the tube.
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Affiliation(s)
- Shalabh K Anand
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh, 462066, India
| | - Sunil P Singh
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh, 462066, India.
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25
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Starkov D, Parfenyev V, Belan S. Conformational statistics of non-equilibrium polymer loops in Rouse model with active loop extrusion. J Chem Phys 2021; 154:164106. [PMID: 33940823 DOI: 10.1063/5.0048942] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Motivated by the recent experimental observations of the DNA loop extrusion by protein motors, in this paper, we investigate the statistical properties of the growing polymer loops within the ideal chain model. The loop conformation is characterized statistically by the mean gyration radius and the pairwise contact probabilities. It turns out that a single dimensionless parameter, which is given by the ratio of the loop relaxation time over the time elapsed since the start of extrusion, controls the crossover between near-equilibrium and highly non-equilibrium asymptotics in the statistics of the extruded loop, regardless of the specific time dependence of the extrusion velocity. In addition, we show that two-sided and one-sided loop extruding motors produce the loops with almost identical properties. Our predictions are based on two rigorous semi-analytical methods accompanied by asymptotic analysis of slow and fast extrusion limits.
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Affiliation(s)
- Dmitry Starkov
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av., 142432 Chernogolovka, Russia and National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia
| | - Vladimir Parfenyev
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av., 142432 Chernogolovka, Russia and National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia
| | - Sergey Belan
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av., 142432 Chernogolovka, Russia and National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia
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26
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Mousavi SM, Gompper G, Winkler RG. Active bath-induced localization and collapse of passive semiflexible polymers. J Chem Phys 2021; 155:044902. [PMID: 34340385 DOI: 10.1063/5.0058150] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
The conformational and dynamical properties of a passive polymer embedded in a bath of active Brownian particles (ABPs) are studied by Langevin dynamics simulations. Various activities and ABP concentrations below and above the critical values for motility-induced phase separation (MIPS) are considered. In a homogeneous ABP fluid, the embedded polymer swells with increasing bath activity, with stronger swelling for larger densities. The polymer dynamics is enhanced, with the diffusion coefficient increasing by a power-law with increasing activity, where the exponent depends on the ABP concentration. For ABP concentrations in the MIPS regime, we observe a localization of the polymer in the low-density ABP phase associated with polymer collapse for moderate activities and a reswelling for high activities accompanied by a preferred localization in the high-density ABP phase. Localization and reswelling are independent of the polymer stiffness, with stiff polymers behaving similarly to flexible polymers. The polymer collapse is associated with a slowdown of its dynamics and a significantly smaller center-of-mass diffusion coefficient. In general, the polymer dynamics can only partially be described by an effective (bath) temperature. Moreover, the properties of a polymer embedded in a homogeneous active bath deviate quantitatively from those of a polymer composed of active monomers, i.e., linear chains of ABPs; however, such a polymer exhibits qualitatively similar activity-dependent features.
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Affiliation(s)
- S Mahdiyeh Mousavi
- Theoretical Physics of Living Matter, Institute for Advanced Simulation and Institute of Biological Information Processing, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Gerhard Gompper
- Theoretical Physics of Living Matter, Institute for Advanced Simulation and Institute of Biological Information Processing, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Roland G Winkler
- Theoretical Physics of Living Matter, Institute for Advanced Simulation and Institute of Biological Information Processing, Forschungszentrum Jülich, D-52425 Jülich, Germany
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27
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Jose F, Anand SK, Singh SP. Phase separation of an active colloidal suspension via quorum-sensing. SOFT MATTER 2021; 17:3153-3161. [PMID: 33616149 DOI: 10.1039/d0sm02131h] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We present the Brownian dynamics simulation of an active colloidal suspension in two dimensions, where the self-propulsion speed of a colloid is regulated according to the local density sensed by it. The role of concentration-dependent motility in the phase-separation of colloids and their dynamics is investigated in detail. Interestingly, the system phase separates at a very low packing fraction (Φ≈ 0.125) at higher self-propulsion speeds (Pe), into a dense phase coexisting with a homogeneous phase and attains a long-range crystalline order beyond the transition point. The transition point is quantified here from the local density profiles and local and global-bond order parameters. We have shown that the characteristics of the phase diagram are qualitatively akin to the active Brownian particle (ABP) model. Moreover, our investigation reveals that the density-dependent motility amplifies the slow-down of the directed speed, which facilitates phase-separation even at low packing fractions. The effective diffusivity shows a crossover from quadratic rise to a power-law behavior of exponent 3/2 with Pe in the phase-separated regime. Furthermore, we have shown that the effective diffusion decreases exponentially with packing fraction in the phase-separated regime, while it shows a linear decrease in the single phase regime.
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Affiliation(s)
- Francis Jose
- Department of Physics, Indian Institute of Science Education and Research, Bhopal 462 066, Madhya Pradesh, India.
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28
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Shee A, Gupta N, Chaudhuri A, Chaudhuri D. A semiflexible polymer in a gliding assay: reentrant transition, role of turnover and activity. SOFT MATTER 2021; 17:2120-2131. [PMID: 33439187 DOI: 10.1039/d0sm01181a] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We consider a model of an extensible semiflexible filament moving in two dimensions on a motility assay of motor proteins represented explicitly as active harmonic linkers. Their heads bind stochastically to polymer segments within a capture radius, and extend along the filament in a directed fashion before detaching. Both the extension and detachment rates are load-dependent and generate an active drive on the filament. The filament undergoes a first order phase transition from the open chain to spiral conformation and shows a reentrant behavior in both the active extension and the turnover, defined as the ratio of attachment-detachment rates. Associated with the phase transition, the size and shape of the polymer change non-monotonically, and the relevant autocorrelation functions display a double-exponential decay. The corresponding correlation times show a maximum signifying the dominance of spirals. The orientational dynamics captures the rotation of spirals, and its correlation time decays with activity as a power law.
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Affiliation(s)
- Amir Shee
- Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India. and Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India
| | - Nisha Gupta
- Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678557, India
| | - Abhishek Chaudhuri
- Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, SAS Nagar 140306, Punjab, India
| | - Debasish Chaudhuri
- Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India. and Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India
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29
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Das S, Kennedy N, Cacciuto A. The coil-globule transition in self-avoiding active polymers. SOFT MATTER 2021; 17:160-164. [PMID: 33164018 DOI: 10.1039/d0sm01526a] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We perform numerical simulations of an active fully flexible self-avoiding polymer as a function of the quality of the embedding solvent described in terms of an effective monomer-monomer interaction. Specifically, by extracting the Flory exponent of the active polymer under different conditions, we are able to pin down the location of the coil-globule transition for different strengths of the active forces. Remarkably, we find that a simple rescaling of the temperature is capable of qualitatively capturing the dependence of the Θ-point of the polymer on the amplitude of active fluctuations. We discuss the limits of this mapping and suggest that a negative active pressure between the monomers, not unlike the one that has already been found in suspensions of active hard spheres, may also be present in active polymers.
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Affiliation(s)
- S Das
- Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027, USA.
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30
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Affiliation(s)
- Roland G. Winkler
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Gerhard Gompper
- Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
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