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Wang H, Qian T, Xu X. Onsager's variational principle in active soft matter. SOFT MATTER 2021; 17:3634-3653. [PMID: 33480912 DOI: 10.1039/d0sm02076a] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.
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Affiliation(s)
- Haiqin Wang
- Technion - Israel Institute of Technology, Haifa, 32000, Israel
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Al-Izzi SC, Sens P, Turner MS, Komura S. Dynamics of passive and active membrane tubes. SOFT MATTER 2020; 16:9319-9330. [PMID: 32935733 DOI: 10.1039/d0sm01290d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Utilising Onsager's variational formulation, we derive dynamical equations for the relaxation of a fluid membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across the membrane and variations in surface tension due to membrane incompressibility. We compute the relaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making new predictions for higher order (azimuthal) m-modes. We analyse the long and short wavelength limits of these modes by making use of various asymptotic arguments. We incorporate stochastic terms to our dynamical equations suitable to describe both passive thermal forces and non-equilibrium active forces. We derive expressions for the fluctuation amplitudes, an effective temperature associated with active fluctuations, and the power spectral density for both the thermal and active fluctuations. We discuss an experimental assay that might enable measurement of these fluctuations to infer the properties of the active noise. Finally we discuss our results in the context of active membranes more generally and give an overview of some open questions in the field.
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Affiliation(s)
- Sami C Al-Izzi
- School of Physics & EMBL-Australia node in Single Molecule Science, University of New South Wales, Sydney, Australia and Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK and Institut Curie, PSL Research University, CNRS, Physical Chemistry Curie, F-75005, Paris, France and Sorbonne Université, CNRS, UMR 168, F-75005, Paris, France
| | - Pierre Sens
- Institut Curie, PSL Research University, CNRS, Physical Chemistry Curie, F-75005, Paris, France and Sorbonne Université, CNRS, UMR 168, F-75005, Paris, France
| | - Matthew S Turner
- Department of Physics & Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK and Department of Chemical Engineering, University of Kyoto, Kyoto 615-8510, Japan
| | - Shigeyuki Komura
- Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan.
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Bivas I, Tonchev NS. Membrane stretching elasticity and thermal shape fluctuations of nearly spherical lipid vesicles. Phys Rev E 2019; 100:022416. [PMID: 31574724 DOI: 10.1103/physreve.100.022416] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2018] [Indexed: 11/07/2022]
Abstract
One of the most widely used methods for determination of the bending elasticity modulus of model lipid membranes is the analysis of the shape fluctuations of nearly spherical lipid vesicles. The theoretical basis of this analysis is given by Milner and Safran [Phys. Rev. A 36, 4371 (1987)0556-279110.1103/PhysRevA.36.4371]. In their theory the stretching effects are not considered. In the present study we generalized their approach including the stretching effects deduced after application of the statistical mechanics to vesicles.
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Affiliation(s)
- Isak Bivas
- Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee blvd., Sofia 1784, Bulgaria
| | - Nicholay S Tonchev
- Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee blvd., Sofia 1784, Bulgaria
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Yoshioka J, Fukao K. Horizontal transportation of a Maltese cross pattern in nematic liquid crystalline droplets under a temperature gradient. Phys Rev E 2019; 99:022702. [PMID: 30934222 DOI: 10.1103/physreve.99.022702] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2018] [Indexed: 11/07/2022]
Abstract
Flow and director fields strongly couple with each other in liquid crystalline systems, and herein we discuss the coupling effect in cylindrical and spherical-cap droplets formed by nematic liquid crystal. Applying a temperature gradient to droplets dispersed in a liquid solvent, we observed a crosslike texture in the droplets moved toward the high-temperature side, indicating that the director field was deformed from equilibrium. Additionally, measurement of the flow field revealed that a convective flow was induced in the droplets under temperature gradient. These results suggested that the director deformation in the droplet was induced by convection. By designing a simplified model based on this, we theoretically analyzed the above phenomenon based on Onsager's variational principle. The results show that the phenomenon was well described by a balance of surface energy gradient with viscous and elastic forces.
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Affiliation(s)
- Jun Yoshioka
- Department of Physical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, Shiga 525-0058, Japan
| | - Koji Fukao
- Department of Physical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, Shiga 525-0058, Japan
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Oya Y, Kawakatsu T. Onsager's variational principle for the dynamics of a vesicle in a Poiseuille flow. J Chem Phys 2018; 148:114905. [PMID: 29566523 DOI: 10.1063/1.4999049] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We propose a systematic formulation of the migration behaviors of a vesicle in a Poiseuille flow based on Onsager's variational principle, which can be used to determine the most stable steady state. Our model is described by a combination of the phase field theory for the vesicle and the hydrodynamics for the flow field. The dynamics is governed by the bending elastic energy and the dissipation functional, the latter being composed of viscous dissipation of the flow field, dissipation of the bending energy of the vesicle, and the friction between the vesicle and the flow field. We performed a series of simulations on 2-dimensional systems by changing the bending elasticity of the membrane and observed 3 types of steady states, i.e., those with slipper shape, bullet shape, and snaking motion, and a quasi-steady state with zig-zag motion. We show that the transitions among these steady states can be quantitatively explained by evaluating the dissipation functional, which is determined by the competition between the friction on the vesicle surface and the viscous dissipation in the bulk flow.
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Affiliation(s)
- Yutaka Oya
- Department of Aerospace Engineering, Tohoku University, Sendai 980-8579, Japan
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Okamoto R, Komura S, Fournier JB. Dynamics of a bilayer membrane coupled to a two-dimensional cytoskeleton: Scale transfers of membrane deformations. Phys Rev E 2018; 96:012416. [PMID: 29347262 DOI: 10.1103/physreve.96.012416] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2017] [Indexed: 11/07/2022]
Abstract
We theoretically investigate the dynamics of a floating lipid bilayer membrane coupled with a two-dimensional cytoskeleton network, taking into account explicitly the intermonolayer friction, the discrete lattice structure of the cytoskeleton, and its prestress. The lattice structure breaks lateral continuous translational symmetry and couples Fourier modes with different wave vectors. It is shown that within a short time interval a long-wavelength deformation excites a collection of modes with wavelengths shorter than the lattice spacing. These modes relax slowly with a common renormalized rate originating from the long-wavelength mode. As a result, and because of the prestress, the slowest relaxation is governed by the intermonolayer friction. Conversely, and most interestingly, forces applied at the scale of the cytoskeleton for a sufficiently long time can cooperatively excite large-scale modes.
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Affiliation(s)
- Ryuichi Okamoto
- Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Shigeyuki Komura
- Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Jean-Baptiste Fournier
- Université Paris Diderot, Sorbonne Paris Cité, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205 Paris, France
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Fujiwara K, Yanagisawa M. Liposomal internal viscosity affects the fate of membrane deformation induced by hypertonic treatment. SOFT MATTER 2017; 13:9192-9198. [PMID: 29184957 DOI: 10.1039/c7sm01421j] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Artificial lipid membranes have been utilized to understand the physical mechanisms of the deformation patterns of live cells. However, typical artificial membrane systems contain only dilute components compared to those in the cytoplasm of live cells. By using giant unilamellar liposomes containing dense protein solutions similar to those in live cells, we here reveal that viscosity derived from internal crowding affects the deformation patterns of lipid membranes. After hypertonic treatment, liposome deformation patterns transitioned from budding to tubing when the initial internal macromolecular concentrations were increased. Remarkably, instead of observing different transition concentrations between two species of macromolecules, the viscosity at the transition concentration was found to be similar. Further analyses clearly demonstrated that the internal viscosity affects the deformation patterns of lipid membranes induced by hypertonic treatment. These results indicate that the viscosity of the cytoplasm is a key factor in determining cell deformation, and suggest the association of a process involving dynamic instability, such as a viscous fingering phenomenon, during the determination of deformation patterns by hypertonic treatment.
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Affiliation(s)
- Kei Fujiwara
- Department of Biosciences and Informatics, Faculty of Science and Technology Keio University, Yokohama 223-8522, Japan.
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