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Majumdar SN, Mori F, Vivo P. Nonlinear-cost random walk: Exact statistics of the distance covered for fixed budget. Phys Rev E 2023; 108:064122. [PMID: 38243551 DOI: 10.1103/physreve.108.064122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2023] [Accepted: 11/22/2023] [Indexed: 01/21/2024]
Abstract
We consider the nonlinear-cost random-walk model in discrete time introduced in Phys. Rev. Lett. 130, 237102 (2023)10.1103/PhysRevLett.130.237102, where a fee is charged for each jump of the walker. The nonlinear cost function is such that slow or short jumps incur a flat fee, while for fast or long jumps the cost is proportional to the distance covered. In this paper we compute analytically the average and variance of the distance covered in n steps when the total budget C is fixed, as well as the statistics of the number of long or short jumps in a trajectory of length n, for the exponential jump distribution. These observables exhibit a very rich and nonmonotonic scaling behavior as a function of the variable C/n, which is traced back to the makeup of a typical trajectory in terms of long or short jumps, and the resulting entropy thereof. As a by-product, we compute the asymptotic behavior of ratios of Kummer hypergeometric functions when both the first and last arguments are large. All our analytical results are corroborated by numerical simulations.
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Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, and Université Paris-Saclay, 91405 Orsay, France
| | - Francesco Mori
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
| | - Pierpaolo Vivo
- Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
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2
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Sáinz-Agost A, Falo F, Fiasconaro A. Polymer translocation driven by longitudinal and transversal time-dependent end-pulling forces. Phys Rev E 2023; 108:034501. [PMID: 37849105 DOI: 10.1103/physreve.108.034501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2023] [Accepted: 08/07/2023] [Indexed: 10/19/2023]
Abstract
In this article, we simulate the translocation of a semiflexible homopolymer through an extended pore, driven by both a constant and a time-dependent end-pulled force, employing a model introduced in previous studies. The time dependence is simplistically modeled as a cosine function, and we distinguish between two scenarios for the driving--longitudinal force and transversal force-depending on the relative orientation of the force, parallel or perpendicular, respectively, with respect to the pore axis. Besides some key differences between the two drivings, the mean translocation times present a large minimum region as a function of the frequency of the force that is typical of the resonant activation effect. The presence of the minimum is independent on the elastic characteristics of the polymeric chains and reveals a linear relation between the optimum mean translocation time and the corresponding period of the driving. The mean translocation times show different scaling exponents with the polymer length for different flexibilities. Lastly, we derive an analytical expression of the mean translocation time for low driving frequency, which clearly agrees with the simulations.
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Affiliation(s)
- A Sáinz-Agost
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, Zaragoza 50009, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Zaragoza 50018, Spain
| | - F Falo
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, Zaragoza 50009, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Zaragoza 50018, Spain
| | - A Fiasconaro
- Departamento de Física de la Materia Condensada, Universidad de Zaragoza, Zaragoza 50009, Spain
- Instituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Zaragoza 50018, Spain
- Istituto di Biofisica, Consiglio Nazionale delle Ricerche, Palermo 90146, Italy
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Majumdar SN, Mori F, Vivo P. Cost of Diffusion: Nonlinearity and Giant Fluctuations. PHYSICAL REVIEW LETTERS 2023; 130:237102. [PMID: 37354426 DOI: 10.1103/physrevlett.130.237102] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2023] [Revised: 04/14/2023] [Accepted: 05/15/2023] [Indexed: 06/26/2023]
Abstract
We introduce a simple model of diffusive jump process where a fee is charged for each jump. The nonlinear cost function is such that slow jumps incur a flat fee, while for fast jumps the cost is proportional to the velocity of the jump. The model-inspired by the way taxi meters work-exhibits a very rich behavior. The cost for trajectories of equal length and equal duration exhibits giant fluctuations at a critical value of the scaled distance traveled. Furthermore, the full distribution of the cost until the target is reached exhibits an interesting "freezing" transition in the large-deviation regime. All the analytical results are corroborated by numerical simulations. Our results also apply to elastic systems near the depinning transition, when driven by a random force.
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Affiliation(s)
- Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Francesco Mori
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3RH, United Kingdom
| | - Pierpaolo Vivo
- Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom
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Singh SL, Chauhan K, Bharadwaj AS, Kishore V, Laux P, Luch A, Singh AV. Polymer Translocation and Nanopore Sequencing: A Review of Advances and Challenges. Int J Mol Sci 2023; 24:6153. [PMID: 37047125 PMCID: PMC10094227 DOI: 10.3390/ijms24076153] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2022] [Revised: 02/01/2023] [Accepted: 02/28/2023] [Indexed: 03/31/2023] Open
Abstract
Various biological processes involve the translocation of macromolecules across nanopores; these pores are basically protein channels embedded in membranes. Understanding the mechanism of translocation is crucial to a range of technological applications, including DNA sequencing, single molecule detection, and controlled drug delivery. In this spirit, numerous efforts have been made to develop polymer translocation-based sequencing devices, these efforts include findings and insights from theoretical modeling, simulations, and experimental studies. As much as the past and ongoing studies have added to the knowledge, the practical realization of low-cost, high-throughput sequencing devices, however, has still not been realized. There are challenges, the foremost of which is controlling the speed of translocation at the single monomer level, which remain to be addressed in order to use polymer translocation-based methods for sensing applications. In this article, we review the recent studies aimed at developing control over the dynamics of polymer translocation through nanopores.
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Affiliation(s)
- Swarn Lata Singh
- Department of Physics, Mahila Mahavidyalaya (MMV), Banaras Hindu University, Varanasi 221005, UP, India
| | - Keerti Chauhan
- Department of Physics, Banaras Hindu University, Varanasi 221005, UP, India
| | - Atul S. Bharadwaj
- Department of Physics, CMP Degree College, University of Allahabad, Prayagraj 211002, UP, India
| | - Vimal Kishore
- Department of Physics, Banaras Hindu University, Varanasi 221005, UP, India
| | - Peter Laux
- Department of Chemical and Product Safety, German Federal Institute of Risk Assessment (BfR) Maxdohrnstrasse 8-10, 10589 Berlin, Germany
| | - Andreas Luch
- Department of Chemical and Product Safety, German Federal Institute of Risk Assessment (BfR) Maxdohrnstrasse 8-10, 10589 Berlin, Germany
| | - Ajay Vikram Singh
- Department of Chemical and Product Safety, German Federal Institute of Risk Assessment (BfR) Maxdohrnstrasse 8-10, 10589 Berlin, Germany
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Paun MA, Paun VA, Paun VP. Time Estimation of Polymer Translocation through Nano-Membrane. Polymers (Basel) 2022; 14:2090. [PMID: 35631973 PMCID: PMC9147610 DOI: 10.3390/polym14102090] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Revised: 05/15/2022] [Accepted: 05/17/2022] [Indexed: 12/10/2022] Open
Abstract
In this paper, the charged polymer escapement phenomenon, via a little hole of nano-metric dimensions arranged in a constitutive biological membrane, is studied. We will present the case of the transport process of an ideal polymer in a 3-dimensional extended region separated by a fine boundary named membrane in a free energy barrier attendance. Additionally, the general translocation time formula, respectively, the transition time from the cis area to the trans area, is presented. The model for estimation of the likelihood, designated by P(x, t), as a macromolecular chain of lengthiness equal to x, to be able to pass by the nanopore in escape period t, was optimized. The longest-lasting likely escape time found with this model is indicated to be tp = 330 μs. Thus, the results obtained with the described formula are in good agreement with those announced in the specialized literature.
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Affiliation(s)
- Maria-Alexandra Paun
- School of Engineering, Swiss Federal Institute of Technology (EPFL), 1015 Lausanne, Switzerland;
- Division Radio Monitoring and Equipment, Section Market Access and Conformity, Federal Office of Communications OFCOM, 2501 Bienne, Switzerland
| | | | - Viorel-Puiu Paun
- Department of Physics, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
- Academy of Romanian Scientists, 050094 Bucharest, Romania
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Paun MA, Paun VA, Paun VP. Polymer Translocation through Nanometer Pores. Polymers (Basel) 2022; 14:polym14061166. [PMID: 35335497 PMCID: PMC8950326 DOI: 10.3390/polym14061166] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2022] [Revised: 02/25/2022] [Accepted: 03/12/2022] [Indexed: 11/16/2022] Open
Abstract
In this paper the loaded polymer transport and its escape via a nanometer size aperture, virtually by nanomembrane, the polymer being moved by an exterior electrostatic field, has been studied. Assuming a linear dependency of the friction coefficient on the number of segments m and a parabolic behavior for the open-free (Gibbs) energy, in attendance of a present electrical potential across nanopore, an explicit flux formula for the polymers passed over a dimensional restricted pore, was derived. In addition, the linear polymers transport through a nanometer-sized pore under the action of a constant force is presented. The important mechanical effects of superimposed steady force and the monomers number of macromolecule chain on the polymer translocation process by nanomembranes, in a 2D diffusion model, have been demonstrated. The escape time by a three-dimensional graph as a function of the electric field intensity and monomers number of polymer was represented.
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Affiliation(s)
- Maria-Alexandra Paun
- School of Engineering, Swiss Federal Institute of Technology (EPFL), 1015 Lausanne, Switzerland;
- Division Radio Monitoring and Equipment, Section Market Access and Conformity, Federal Office of Communications OFCOM, Federal Department of the Environment, Transport, Energy and Communications DETEC, Rue de l’Avenir 44, 2501 Bienne, Switzerland
| | | | - Viorel-Puiu Paun
- Department of Physics, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
- Academy of Romanian Scientists, 050094 Bucharest, Romania
- Correspondence: or
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Chen X, Chen J, Zhuo BY, Yang X, Luo MB. Simulation study for the pulling translocation of a polymer globule. Polym J 2021. [DOI: 10.1038/s41428-021-00502-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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8
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Hsiao PY. Translocation of a Polyelectrolyte through a Nanopore in the Presence of Trivalent Counterions: A Comparison with the Cases in Monovalent and Divalent Salt Solutions. ACS OMEGA 2020; 5:19805-19819. [PMID: 32803076 PMCID: PMC7424739 DOI: 10.1021/acsomega.0c02647] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2020] [Accepted: 07/10/2020] [Indexed: 05/08/2023]
Abstract
A polyelectrolyte threading through a nanopore in a trivalent salt solution is investigated by means of molecular dynamics simulations under a reflective wall boundary. By varying the chain length N and the strength E of the driving electric field applied inside the pore, the translocation time is carefully calculated to get rid of the bouncing effect because of the boundary. The results are analyzed under the scaling form ⟨τ⟩ ∼ N α E -δ and four driving force regimes; namely, the unbiased, the weakly driven, the strongly driven trumpet, and the strongly driven isoflux regime, are distinguished. The exponents are calculated in each regime and compared with the cases in the monovalent and divalent salt solutions. Owing to strong condensation of counter ions, the changes of the exponents in the force regimes are found to be nontrivial. A large increase in translocation time can be, however, achieved as the driving field is weak. The variations of the chain size, the ion condensation, and the effective chain charge show that the process is proceeded in a quasi-equilibrium way in the unbiased regime and deviated to exhibit strong nonequilibrium characteristics as E increases. Several astonishing scaling behaviors of the waiting time function, the translocation velocity, and the diffusion properties are discovered in the study. The results provide deep insights into the phenomena of polyelectrolyte translocation in various salt solutions at different driving forces.
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Affiliation(s)
- Pai-Yi Hsiao
- Department
of Engineering and System Science, National
Tsing Hua University, Hsinchu, Taiwan 30013, R. O. C
- Institute
of Nuclear Engineering and Science, National
Tsing Hua University, Hsinchu, Taiwan 30013, R. O. C
- ,
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Nagarajan K, Chen SB. Polyelectrolyte Translocation through a Corrugated Nanopore. MACROMOL THEOR SIMUL 2020. [DOI: 10.1002/mats.202000042] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
Affiliation(s)
- Karthik Nagarajan
- Department of Chemical & Biomolecular EngineeringNational University of Singapore Singapore 117585 Singapore
| | - Shing Bor Chen
- Department of Chemical & Biomolecular EngineeringNational University of Singapore Singapore 117585 Singapore
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Abstract
We employ a three-dimensional molecular dynamics to simulate a driven polymer translocation through a nanopore by applying an external force, for four pore diameters and two external forces. To see the polymer and pore interaction effects on translocation time, we studied nine interaction energies. Moreover, to better understand the simulation results, we investigate polymer center of mass, shape factor and the monomer spatial distribution through the translocation process. Our results reveal that increasing the polymer-pore interaction energy is accompanied by an increase in the translocation time and decrease in the process rate. Furthermore, for pores with greater diameter, the translocation becomes faster. The shape analysis of the polymer indicates that the polymer shape is highly sensitive to the interaction energy. In great interactions, the monomers come close to the pore from both sides. As a result, the translocation becomes fast at first and slows down at last. Overall, it can be concluded that the external force does not play a major role in the shape and distribution of translocated monomers. However, the interaction energy between monomer and nanopore has a major effect especially on the distribution of translocated monomers on the trans side.
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Abstract
Although nanopores have shown tremendous promise for use in DNA sequencing, the rate of translocation through most pores studied previously is too rapid for the genetic information to be read accurately. In this study, dissipative particle dynamics simulations were employed to investigate the feasibility of using tortuous nanopores to control the rate of polyelectrolyte translocation. Unlike many previous studies, our simulation method incorporates the effects of hydrodynamic and electrostatic interactions and the spatial variation of electric field strength. The average translocation time, ⟨τ⟩, increases with the pore length and tortuosity but decreases as the pore width increases. For the longest pore investigated, the introduction of tortuosity results in ⟨τ⟩ increasing by as much as 187% as compared to a straight pore. The temporal variation of bond tension indicates that slower translocation in tortuous nanopores is caused by inhibition of tension propagation.
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Affiliation(s)
- Karthik Nagarajan
- Department of Chemical and Biomolecular Engineering , National University of Singapore , 117585 , Singapore
| | - Shing Bor Chen
- Department of Chemical and Biomolecular Engineering , National University of Singapore , 117585 , Singapore
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12
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Hsiao PY. Translocation of Charged Polymers through a Nanopore in Monovalent and Divalent Salt Solutions: A Scaling Study Exploring over the Entire Driving Force Regimes. Polymers (Basel) 2018; 10:E1229. [PMID: 30961154 PMCID: PMC6290626 DOI: 10.3390/polym10111229] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2018] [Revised: 10/31/2018] [Accepted: 11/02/2018] [Indexed: 11/16/2022] Open
Abstract
Langevin dynamics simulations are performed to study polyelectrolytes driven through a nanopore in monovalent and divalent salt solutions. The driving electric field E is applied inside the pore, and the strength is varied to cover the four characteristic force regimes depicted by a rederived scaling theory, namely the unbiased (UB) regime, the weakly-driven (WD) regime, the strongly-driven trumpet (SD(T)) regime and the strongly-driven isoflux (SD(I)) regime. By changing the chain length N, the mean translocation time is studied under the scaling form 〈 τ 〉 ∼ N α E - δ . The exponents α and δ are calculated in each force regime for the two studied salt cases. Both of them are found to vary with E and N and, hence, are not universal in the parameter's space. We further investigate the diffusion behavior of translocation. The subdiffusion exponent γ p is extracted. The three essential exponents ν s , q, z p are then obtained from the simulations. Together with γ p , the validness of the scaling theory is verified. Through a comparison with experiments, the location of a usual experimental condition on the scaling plot is pinpointed.
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Affiliation(s)
- Pai-Yi Hsiao
- Department of Engineering and System Science, National Tsing Hua University, Hsinchu 30013, Taiwan.
- Institute of Nuclear Engineering and Science, National Tsing Hua University, Hsinchu 30013, Taiwan.
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de Haan HW, Sean D, Slater GW. Reducing the variance in the translocation times by prestretching the polymer. Phys Rev E 2018; 98:022501. [PMID: 30253469 DOI: 10.1103/physreve.98.022501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2018] [Indexed: 11/07/2022]
Abstract
Langevin dynamics simulations of polymer translocation are performed where the polymer is stretched via two opposing forces applied on the first and last monomer before and during translocation. In this setup, polymer translocation is achieved by imposing a bias between the two pulling forces such that there is net displacement towards the trans side. Under the influence of stretching forces, the elongated polymer ensemble contains less variations in conformations compared to an unstretched ensemble. Simulations demonstrate that this reduced spread in initial conformations yields a reduced variation in translocation times relative to the mean translocation time. This effect is explored for different ratios of the amplitude of thermal fluctuations to driving forces to control for the relative influence of the thermal path sampled by the polymer. Since the variance in translocation times is due to contributions coming from sampling both thermal noise and initial conformations, our simulations offer independent control over the two main sources of noise and allow us to shed light on how they both contribute to translocation dynamics. Simulation parameter space corresponding to experimentally relevant conditions is highlighted and shown to correspond to a significant decrease in the spread of translocation times, thus indicating that stretching DNA prior to translocation could assist nanopore-based sequencing and sizing applications.
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Affiliation(s)
- Hendrick W de Haan
- Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada, L1H 7K4
| | - David Sean
- Physics Department, University of Ottawa, Ottawa, Ontario, Canada, K1N 6N5.,Institut für Computerphysik, Universität Stuttgart, 70569 Stuttgart, Germany
| | - Gary W Slater
- Physics Department, University of Ottawa, Ottawa, Ontario, Canada, K1N 6N5
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Haridasan N, Kannam SK, Mogurampelly S, Sathian SP. Translational mobilities of proteins in nanochannels: A coarse-grained molecular dynamics study. Phys Rev E 2018; 97:062415. [PMID: 30011556 DOI: 10.1103/physreve.97.062415] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Indexed: 05/18/2023]
Abstract
We investigated the translation of a protein through model nanopores using coarse-grained (CG) nonequilibrium molecular dynamics (NEMD) simulations and compared the mobilities with those obtained from previous coarse-grained equilibrium molecular dynamics model. We considered the effects of nanopore confinement and external force on the translation of streptavidin through nanopores of dimensions representative of experiments. As the nanopore radius approaches the protein hydrodynamic radius, r_{h}/r_{p}→1 (where r_{h} is the hydrodynamic radius of protein and r_{p} is the pore radius), the translation times are observed to increase by two orders of magnitude. The translation times are found to be in good agreement with the one-dimensional biased diffusion model. The results presented in this paper provide useful insights on nanopore designs intended to control the motion of biomolecules.
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Affiliation(s)
- Navaneeth Haridasan
- Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Sridhar Kumar Kannam
- Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
- School of Sciences, RMIT University, Melbourne, Victoria 3001, Australia
| | - Santosh Mogurampelly
- Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania 19122, USA
| | - Sarith P Sathian
- Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
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