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Fiasconaro A, Díez-Señorans G, Falo F. End-pulled polymer translocation through a many-body flexible pore. POLYMER 2022. [DOI: 10.1016/j.polymer.2022.125305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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Strand NE, Vroylandt H, Gingrich T. Using tensor network states for multi-particle Brownian ratchets. J Chem Phys 2022; 156:221103. [DOI: 10.1063/5.0097332] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet's current. To surmount the many-body problem, we employ the time-dependent variational principle (TDVP) applied to binary tree tensor networks (BTTN). Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases.
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Affiliation(s)
- Nils E Strand
- Chemistry, Northwestern University, United States of America
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Abstract
In this work we study the assisted translocation of a polymer across a membrane nanopore, inside which a molecular motor exerts a force fuelled by the hydrolysis of ATP molecules. In our model the motor switches to its active state for a fixed amount of time, while it waits for an ATP molecule which triggers the motor, during an exponentially distributed time lapse. The polymer is modelled as a beads-springs chain with both excluded volume and bending contributions, and moves in a stochastic three dimensional environment modelled with a Langevin dynamics at a fixed temperature. The resulting dynamics shows a Michaelis-Menten translocation velocity that depends on the chain flexibility. The scaling behavior of the mean translocation time with the polymer length for different bending values is also investigated.
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Levien E, Bressloff PC. Quasi-steady-state analysis of coupled flashing ratchets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042129. [PMID: 26565190 DOI: 10.1103/physreve.92.042129] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2015] [Indexed: 06/05/2023]
Abstract
We perform a quasi-steady-state (QSS) reduction of a flashing ratchet to obtain a Brownian particle in an effective potential. The resulting system is analytically tractable and yet preserves essential dynamical features of the full model. We first use the QSS reduction to derive an explicit expression for the velocity of a simple two-state flashing ratchet. In particular, we determine the relationship between perturbations from detailed balance, which are encoded in the transitions rates of the flashing ratchet, and a tilted-periodic potential. We then perform a QSS analysis of a pair of elastically coupled flashing ratchets, which reduces to a Brownian particle moving in a two-dimensional vector field. We suggest that the fixed points of this vector field accurately approximate the metastable spatial locations of the coupled ratchets, which are, in general, impossible to identify from the full system.
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Affiliation(s)
- Ethan Levien
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
| | - Paul C Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
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Fiasconaro A, Mazo JJ, Falo F. Active polymer translocation in the three-dimensional domain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022113. [PMID: 25768464 DOI: 10.1103/physreve.91.022113] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2014] [Indexed: 06/04/2023]
Abstract
In this work we study the translocation process of a polymer through a nanochannel where a time dependent force is acting. Two conceptually different types of driving are used: a deterministic sinusoidal one and a random telegraph noise force. The mean translocation time presents interesting resonant minima as a function of the frequency of the external driving. For the computed sizes, the translocation time scales with the polymer length according to a power law with the same exponent for almost all the frequencies of the two driving forces. The dependence of the translocation time with the polymer rigidity, which accounts for the persistence length of the molecule, shows a different low frequency dependence for the two drivings.
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Affiliation(s)
- A Fiasconaro
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain
- Instituto de Ciencia de Materiales de Aragón, C.S.I.C.-Universidad de Zaragoza, 50009 Zaragoza, Spain
- School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
| | - J J Mazo
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain
- Instituto de Ciencia de Materiales de Aragón, C.S.I.C.-Universidad de Zaragoza, 50009 Zaragoza, Spain
| | - F Falo
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, 50018 Zaragoza, Spain
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Tutu H, Nagata S. Robust unidirectional rotation in three-tooth Brownian rotary ratchet systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022144. [PMID: 23496496 DOI: 10.1103/physreve.87.022144] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2012] [Indexed: 06/01/2023]
Abstract
We apply a simple Brownian ratchet model to an artificial molecular rotary system mounted in a biological membrane, in which the rotor always maintains unidirectional rotation in response to a linearly polarized weak ac field. Because the rotor and stator compose a ratchet system, we describe the motion of the rotor tip with the Langevin equation for a particle in a two-dimensional three-tooth ratchet potential of threefold symmetry. Unidirectional rotation can be induced under the field and optimized by stochastic resonance, wherein the mean angular momentum (MAM) of the rotor exhibits a bell-shaped curve for the noise strength. We obtain analytical expressions for the MAM and power loss from the corresponding Fokker-Planck equation, via a Markov transition model for coarse-grained states (six-state model). The MAM expression reveals a significant effect depending on the chirality of the ratchet potential: in achiral cases, the MAM approximately vanishes with respect to the polarization angle φ of the field; in chiral cases, the MAM does not crucially depend on φ, but depends on the direction of the ratchet; i.e., the parity of the unidirectional rotation is inherent in the ratchet structure. This feature is useful for artificial rotary systems to maintain robust unidirectional rotation independent of the mounting condition.
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Affiliation(s)
- Hiroki Tutu
- Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
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Speer D, Eichhorn R, Evstigneev M, Reimann P. Dimer motion on a periodic substrate: spontaneous symmetry breaking and absolute negative mobility. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:061132. [PMID: 23005076 DOI: 10.1103/physreve.85.061132] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2012] [Indexed: 06/01/2023]
Abstract
We consider two coupled particles moving along a periodic substrate potential with negligible inertia effects (overdamped limit). Even when the particles are identical and the substrate spatially symmetric, a sinusoidal external driving of appropriate amplitude and frequency may lead to spontaneous symmetry breaking in the form of a permanent directed motion of the dimer. Thermal noise restores ergodicity and thus zero net velocity, but entails arbitrarily fast diffusion of the dimer for sufficiently weak noise. Moreover, upon application of a static bias force, the dimer exhibits a motion opposite to that force (absolute negative mobility). The key requirement for all these effects is a nonconvex interaction potential of the two particles.
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Affiliation(s)
- David Speer
- Universität Bielefeld, Fakultät für Physik, 33615 Bielefeld, Germany
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Polson JM, Bylhouwer B, Zuckermann MJ, Horton AJ, Scott WM. Dynamics of a polymer in a Brownian ratchet. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051931. [PMID: 21230524 DOI: 10.1103/physreve.82.051931] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2010] [Revised: 09/29/2010] [Indexed: 05/30/2023]
Abstract
We have used Brownian dynamics simulations to study the dynamics of a bead-and-spring polymer subject to a flashing ratchet potential. To elucidate the role of hydrodynamic (HD) interactions, simulations were carried out for the cases where HD interactions are present and when they are absent. The average speed of the polymer and its conformational properties were examined upon variation in the polymer length, N, and the ratchet spatial period, L. Two distinct dynamical regimes were evident. In the low-N/high-L regime, the velocity decreases with increasing N, and center-of-mass diffusion is a key part of the motional mechanism. By contrast, in the high-N /low-L regime, the velocity is insensitive to variation in N, and motion is achieved via the coupling of internal modes to the cycling of the ratchet potential. The location of the regimes is correlated with the average conformational state of the polymer. Incorporating HD interactions increases the average polymer velocity for all polymer lengths and ratchet spatial periods considered. The dynamical behavior of polymers in the low-N/high-L regime can be understood using simple a theoretical model that yields quantitatively reasonable predictions of the polymer velocity.
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Affiliation(s)
- James M Polson
- Department of Physics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3
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Fiasconaro A, Mazo JJ, Falo F. Translocation time of periodically forced polymer chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:031803. [PMID: 21230097 DOI: 10.1103/physreve.82.031803] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2010] [Revised: 07/05/2010] [Indexed: 05/30/2023]
Abstract
In this work we study the presence of both a minimum and clear oscillations in the frequency dependence of the translocation time of a polymer described as a unidimensional Rouse chain driven by a spatially localized oscillating linear potential. The observed oscillations of the mean translocation time arise from the synchronization between the very mean translocation time and the period of the external force. We have checked the robustness of the frequency value for the minimum translocation time by changing the damping parameter, finding a very simple relationship between this frequency and the correspondent translocation time. The translocation time as a function of the polymer length has been also evaluated, finding a precise L2 scaling. Furthermore, the role played by the thermal fluctuations described as a gaussian uncorrelated noise has been also investigated, and the analogies with the resonant activation phenomenon are commented.
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Affiliation(s)
- Alessandro Fiasconaro
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain.
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Kauttonen J, Merikoski J. Characteristics of the polymer transport in ratchet systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041112. [PMID: 20481682 DOI: 10.1103/physreve.81.041112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2010] [Revised: 03/25/2010] [Indexed: 05/29/2023]
Abstract
Molecules with complex internal structure in time-dependent periodic potentials are studied by using short Rubinstein-Duke model polymers as an example. We extend our earlier work on transport in stochastically varying potentials to cover also deterministic potential switching mechanisms, energetic efficiency, and nonuniform charge distributions. We also use currents in the nonequilibrium steady state to identify the dominating mechanisms that lead to polymer transportation and analyze the evolution of the macroscopic state (e.g., total and head-to-head lengths) of the polymers. Several numerical methods are used to solve the master equations and nonlinear optimization problems. The dominating transport mechanisms are found via graph optimization methods. The results show that small changes in the molecule structure and the environment variables can lead to large increases of the drift. The drift and the coherence can be amplified by using deterministic flashing potentials and customized polymer charge distributions. Identifying the dominating transport mechanism by graph analysis tools is found to give insight in how the molecule is transported by the ratchet effect.
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Affiliation(s)
- Janne Kauttonen
- Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland.
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Woo HJ. Relaxation dynamics near nonequilibrium stationary states in Brownian ratchets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:021101. [PMID: 19391700 DOI: 10.1103/physreve.79.021101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2008] [Revised: 11/23/2008] [Indexed: 05/27/2023]
Abstract
A comprehensive study of the static and dynamical properties of a representative stochastic model of Brownian ratchet effects for molecular motors is reported. The model describes Brownian motions on two periodic potentials under static and time-dependent forces, where there are two distinct locations of chemical reactions coupling the levels with reversible rates within a period. Complete stationary properties have been obtained analytically for arbitrary potentials under external force. Dynamical relaxation properties near nonequilibrium stationary states were examined by considering the response function of velocity upon time-dependent external force, expressed in terms of the conditional probability density of the model. The latter is fully calculated using a systematic numerical method using matrix diagonalization, which is easily generalized to more complicated models for studying both static and dynamical properties. The behavior of the time-dependent response examined for model potentials suggests that the characteristic relaxation time near stationary states generally decreases linearly with respect to increasing velocity as one goes away from equilibrium via an increase in chemical potential of fuel species, a prediction testable in single molecule experiments.
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Affiliation(s)
- Hyung-June Woo
- Department of Chemistry, University of Nevada, Reno, Nevada 89557, USA
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Evstigneev M, von Gehlen S, Reimann P. Interaction-controlled Brownian motion in a tilted periodic potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:011116. [PMID: 19257010 DOI: 10.1103/physreve.79.011116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2008] [Indexed: 05/27/2023]
Abstract
The drift and diffusion of a few interacting, overdamped Brownian particles in a tilted periodic potential are studied analytically and numerically. Both quantities exhibit a complex multipeaked structure as a function of the equilibrium interparticle separation. Upon variation of the interaction strength, both drift and diffusion may exhibit a nonmonotonic, resonancelike behavior.
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Kenward M, Slater GW. Polymer deformation in Brownian ratchets: theory and molecular dynamics simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:051806. [PMID: 19113148 DOI: 10.1103/physreve.78.051806] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2008] [Indexed: 05/27/2023]
Abstract
We examine polymers in the presence of an applied asymmetric sawtooth (ratchet) potential which is periodically switched on and off, using molecular dynamics (MD) simulations with an explicit Lennard-Jones solvent. We show that the distribution of the center of mass for a polymer in a ratchet is relatively wide for potential well depths U0 on the order of several kBT. The application of the ratchet potential also deforms the polymer chains. With increasing U0 the Flory exponent varies from that for a free three-dimensional (3D) chain, nu=35 (U0=0), to that corresponding to a 2D compressed (pancake-shaped) polymer with a value of nu=34 for moderate U0. This has the added effect of decreasing a polymer's diffusion coefficient from its 3D value D3D to that of a pancaked-shaped polymer moving parallel to its minor axis D2D. The result is that a polymer then has a time-dependent diffusion coefficient D(t) during the ratchet off time. We further show that this suggests a different method to operate a ratchet, where the off time of the ratchet, toff, is defined in terms of the relaxation time of the polymer, tauR. We also derive a modified version of the Bader ratchet model [Bader, Proc. Natl. Acad. Sci. U.S.A. 96, 13165 (1999)] which accounts for this deformation and we present a simple expression to describe the time dependent diffusion coefficient D(t). Using this model we then illustrate that polymer deformation can be used to modulate polymer migration in a ratchet potential.
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Affiliation(s)
- Martin Kenward
- Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455, USA.
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Symmetry based mechanism for hand-over-hand molecular motors. Biosystems 2008; 93:8-15. [DOI: 10.1016/j.biosystems.2008.04.005] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2008] [Revised: 04/10/2008] [Accepted: 04/17/2008] [Indexed: 11/23/2022]
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Kauttonen J, Merikoski J, Pulkkinen O. Polymer dynamics in time-dependent periodic potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:061131. [PMID: 18643241 DOI: 10.1103/physreve.77.061131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2008] [Indexed: 05/26/2023]
Abstract
The dynamics of a discrete polymer in time-dependent external potentials is studied with the master equation approach. We consider both stochastic and deterministic switching mechanisms for the potential states and give the essential equations for computing the stationary-state properties of molecules with internal structure in time-dependent periodic potentials on a lattice. As an example, we consider standard and modified Rubinstein-Duke polymers and calculate their mean drift and effective diffusion coefficient in the two-state nonsymmetric flashing potential and symmetric traveling potential. Rich nonlinear behavior of these observables is found. By varying the polymer length, we find current inversions caused by the rebound effect that is only present for molecules with internal structure. These results depend strongly on the polymer type. We also notice increased transport coherence for longer polymers.
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Affiliation(s)
- Janne Kauttonen
- Department of Physics, University of Jyväskylä, PO Box 35, Jyväskylä, Finland.
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von Gehlen S, Evstigneev M, Reimann P. Dynamics of a dimer in a symmetric potential: ratchet effect generated by an internal degree of freedom. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031136. [PMID: 18517358 DOI: 10.1103/physreve.77.031136] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2007] [Indexed: 05/26/2023]
Abstract
The one-dimensional dynamics of a dimer consisting of two harmonically coupled components is considered. The mutual distance between the dimer components plays the role of an internal degree of freedom. Both components are in contact with the same heat bath and are coupled to a spatially periodic, symmetric potential, whose amplitude is modulated periodically in time and whose coupling strength is different for the two components. In the absence of any external bias, a ratchet effect (directed transport) arises generically unless the mutual coupling of the dimer components tends to zero or infinity. In other words, the ratchet effect is generated by the internal degree of freedom. An accurate analytical approximation for the dimer's velocity and diffusion coefficient is obtained. The velocity of the system is maximized by adding an optimal amount of noise and by tuning the driving frequency to an optimal value. Furthermore, there exists an optimal coupling strength at which the velocity is the largest.
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