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Zhu Q, Tree DR. Simulations of morphology control of self‐assembled amphiphilic surfactants. JOURNAL OF POLYMER SCIENCE 2023. [DOI: 10.1002/pol.20220771] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/19/2023]
Affiliation(s)
- Qinyu Zhu
- Department of Chemical Engineering Brigham Young University Provo Utah USA
| | - Douglas R. Tree
- Department of Chemical Engineering Brigham Young University Provo Utah USA
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2
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Zhang T, Wolgemuth CW. A general computational framework for the dynamics of single- and multi-phase vesicles and membranes. JOURNAL OF COMPUTATIONAL PHYSICS 2022; 450:110815. [PMID: 35355617 PMCID: PMC8959479 DOI: 10.1016/j.jcp.2021.110815] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
The dynamics of thin, membrane-like structures are ubiquitous in nature. They play especially important roles in cell biology. Cell membranes separate the inside of a cell from the outside, and vesicles compartmentalize proteins into functional microregions, such as the lysosome. Proteins and/or lipid molecules also aggregate and deform membranes to carry out cellular functions. For example, some viral particles can induce the membrane to invaginate and form an endocytic vesicle that pulls the virus into the cell. While the physics of membranes has been extensively studied since the pioneering work of Helfrich in the 1970's, simulating the dynamics of large scale deformations remains challenging, especially for cases where the membrane composition is spatially heterogeneous. Here, we develop a general computational framework to simulate the overdamped dynamics of membranes and vesicles. We start by considering a membrane with an energy that is a generalized functional of the shape invariants and also includes line discontinuities that arise due to phase boundaries. Using this energy, we derive the internal restoring forces and construct a level set-based algorithm that can stably simulate the large-scale dynamics of these generalized membranes, including scenarios that lead to membrane fission. This method is applied to solve for shapes of single-phase vesicles using a range of reduced volumes, reduced area differences, and preferred curvatures. Our results match well the experimentally measured shapes of corresponding vesicles. The method is then applied to explore the dynamics of multiphase vesicles, predicting equilibrium shapes and conditions that lead to fission near phase boundaries.
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Affiliation(s)
- Tiankui Zhang
- Department of Physics, University of Arizona, Tucson, AZ 85721
| | - Charles W Wolgemuth
- Department of Physics, University of Arizona, Tucson, AZ 85721
- Department of Molecular and Cellular Biology, University of Arizona, Tucson, AZ 85721
- Johns Hopkins Physical Sciences-Onocology Center, Johns Hopkins University, Baltimore, MD 21218
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Saito N, Sawai S. Three-dimensional morphodynamic simulations of macropinocytic cups. iScience 2021; 24:103087. [PMID: 34755081 PMCID: PMC8560551 DOI: 10.1016/j.isci.2021.103087] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Revised: 08/13/2021] [Accepted: 09/01/2021] [Indexed: 12/02/2022] Open
Abstract
Macropinocytosis refers to the non-specific uptake of extracellular fluid, which plays ubiquitous roles in cell growth, immune surveillance, and virus entry. Despite its widespread occurrence, it remains unclear how its initial cup-shaped plasma membrane extensions form without any external solid support, as opposed to the process of particle uptake during phagocytosis. Here, by developing a computational framework that describes the coupling between the bistable reaction-diffusion processes of active signaling patches and membrane deformation, we demonstrated that the protrusive force localized to the edge of the patches can give rise to a self-enclosing cup structure, without further assumptions of local bending or contraction. Efficient uptake requires a balance among the patch size, magnitude of protrusive force, and cortical tension. Furthermore, our model exhibits cyclic cup formation, coexistence of multiple cups, and cup-splitting, indicating that these complex morphologies self-organize via a common mutually-dependent process of reaction-diffusion and membrane deformation.
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Affiliation(s)
- Nen Saito
- Exploratory Research Center on Life and Living Systems, National Institutes of Natural Sciences, 5-1 Higashiyama, Myodaiji-cho, Okazaki, Aichi 444-8787, Japan
| | - Satoshi Sawai
- Department of Basic Science, University of Tokyo, Meguro-ku, Tokyo 153-8902, Japan
- Research Center for Complex Systems Biology, Graduate School of Arts and Sciences, University of Tokyo, Meguro-ku, Tokyo 153-8902, Japan
- Department of Biological Sciences, Graduate School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
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Rojas Molina R, Liese S, Alimohamadi H, Rangamani P, Carlson A. Diffuso-kinetic membrane budding dynamics. SOFT MATTER 2020; 16:10889-10899. [PMID: 33125025 DOI: 10.1039/d0sm01028f] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
A wide range of proteins are known to create shape transformations of biological membranes, where the remodelling is a coupling between the energetic costs from deforming the membrane, the recruitment of proteins that induce a local spontaneous curvature C0 and the diffusion of proteins along the membrane. We propose a minimal mathematical model that accounts for these processes to describe the diffuso-kinetic dynamics of membrane budding processes. By deploying numerical simulations we map out the membrane shapes, the time for vesicle formation and the vesicle size as a function of the dimensionless kinetic recruitment parameter K1 and the proteins sensitivity to mean curvature. We derive a time for scission that follows a power law ∼K1-2/3, a consequence of the interplay between the spreading of proteins by diffusion and the kinetic-limited increase of the protein density on the membrane. We also find a scaling law for the vesicle size ∼1/([small sigma, Greek, macron]avC0), with [small sigma, Greek, macron]av the average protein density in the vesicle, which is confirmed in the numerical simulations. Rescaling all the membrane profiles at the time of vesicle formation highlights that the membrane adopts a self-similar shape.
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Affiliation(s)
- Rossana Rojas Molina
- Mechanics Division, Department of Mathematics, University of Oslo, 0316 Oslo, Norway.
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Omar YAD, Sahu A, Sauer RA, Mandadapu KK. Nonaxisymmetric Shapes of Biological Membranes from Locally Induced Curvature. Biophys J 2020; 119:1065-1077. [PMID: 32860742 DOI: 10.1016/j.bpj.2020.07.021] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Revised: 07/07/2020] [Accepted: 07/15/2020] [Indexed: 01/24/2023] Open
Abstract
In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved morphologies. Previous models to study membrane morphologies resulting from locally induced curvature often only consider the possibility of axisymmetric shapes-an indeed unphysical constraint. Past studies predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction to axisymmetry to study all possible membrane morphologies. Only if the resting tension of the membrane is low, we reproduce axisymmetric membrane morphologies. When the resting tension is moderate to high, we show that 1) axisymmetric membrane pits are unstable and 2) nonaxisymmetric ridge-shaped structures are energetically favorable. Furthermore, we find the interplay between intramembrane viscous flow and the rate of induced curvature affects the membrane's ability to transition into nonaxisymmetric ridges and axisymmetric buds. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, whereas nonaxisymmetric ridges are favored when the curvature is slowly increased. Our results hold relevant implications for biological processes such as endocytosis and physical phenomena like phase separation in lipid bilayers.
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Affiliation(s)
- Yannick A D Omar
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California.
| | - Amaresh Sahu
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California.
| | - Roger A Sauer
- Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University, Aachen, Germany.
| | - Kranthi K Mandadapu
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California; Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California.
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Liu K, Lowengrub J, Allard J. Efficient simulation of thermally fluctuating biopolymers immersed in fluids on 1-micron, 1-second scales. JOURNAL OF COMPUTATIONAL PHYSICS 2019; 386:248-263. [PMID: 31787778 PMCID: PMC6884323 DOI: 10.1016/j.jcp.2018.12.039] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The combination of fluid-structure interactions with stochasticity, due to thermal fluctuations, remains a challenging problem in computational fluid dynamics. We develop an efficient scheme based on the stochastic immersed boundary method, Stokeslets, and multiple timestepping. We test our method for spherical particles and filaments under purely thermal and deterministic forces and find good agreement with theoretical predictions for Brownian Motion of a particle and equilibrium thermal undulations of a semi-flexible filament. As an initial application, we simulate bio-filaments with the properties of F-actin. We specifically study the average time for two nearby parallel filaments to bundle together. Interestingly, we find a two-fold acceleration in this time between simulations that account for long-range hydrodynamics compared to those that do not, suggesting that our method will reveal significant hydrodynamic effects in biological phenomena.
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Affiliation(s)
- Kai Liu
- Department of Mathematics, University of California at Irvine
| | - John Lowengrub
- Department of Mathematics, University of California at Irvine
- Center for Complex Biological Systems, University of California at Irvine
- Department of Biomedical Engineering, University of California at Irvine
| | - Jun Allard
- Department of Mathematics, University of California at Irvine
- Center for Complex Biological Systems, University of California at Irvine
- Department of Physics, University of California at Irvine
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Liu K, Chu B, Newby J, Read EL, Lowengrub J, Allard J. Hydrodynamics of transient cell-cell contact: The role of membrane permeability and active protrusion length. PLoS Comput Biol 2019; 15:e1006352. [PMID: 31022168 PMCID: PMC6504115 DOI: 10.1371/journal.pcbi.1006352] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2018] [Revised: 05/07/2019] [Accepted: 02/01/2019] [Indexed: 11/19/2022] Open
Abstract
In many biological settings, two or more cells come into physical contact to form a cell-cell interface. In some cases, the cell-cell contact must be transient, forming on timescales of seconds. One example is offered by the T cell, an immune cell which must attach to the surface of other cells in order to decipher information about disease. The aspect ratio of these interfaces (tens of nanometers thick and tens of micrometers in diameter) puts them into the thin-layer limit, or "lubrication limit", of fluid dynamics. A key question is how the receptors and ligands on opposing cells come into contact. What are the relative roles of thermal undulations of the plasma membrane and deterministic forces from active filopodia? We use a computational fluid dynamics algorithm capable of simulating 10-nanometer-scale fluid-structure interactions with thermal fluctuations up to seconds- and microns-scales. We use this to simulate two opposing membranes, variously including thermal fluctuations, active forces, and membrane permeability. In some regimes dominated by thermal fluctuations, proximity is a rare event, which we capture by computing mean first-passage times using a Weighted Ensemble rare-event computational method. Our results demonstrate a parameter regime in which the time it takes for an active force to drive local contact actually increases if the cells are being held closer together (e.g., by nonspecific adhesion), a phenomenon we attribute to the thin-layer effect. This leads to an optimal initial cell-cell separation for fastest receptor-ligand binding, which could have relevance for the role of cellular protrusions like microvilli. We reproduce a previous experimental observation that fluctuation spatial scales are largely unaffected, but timescales are dramatically slowed, by the thin-layer effect. We also find that membrane permeability would need to be above physiological levels to abrogate the thin-layer effect.
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Affiliation(s)
- Kai Liu
- Department of Mathematics, University of California Irvine, Irvine, California, United States of America
- Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China
| | - Brian Chu
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, California, United States of America
| | - Jay Newby
- Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada
| | - Elizabeth L. Read
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, California, United States of America
- Center for Complex Biological Systems, University of California Irvine, Irvine, California, United States of America
| | - John Lowengrub
- Department of Mathematics, University of California Irvine, Irvine, California, United States of America
- Center for Complex Biological Systems, University of California Irvine, Irvine, California, United States of America
| | - Jun Allard
- Department of Mathematics, University of California Irvine, Irvine, California, United States of America
- Center for Complex Biological Systems, University of California Irvine, Irvine, California, United States of America
- Department of Physics and Astronomy, University of California Irvine, Irvine, California, United States of America
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Multerer MD, Wittwer LD, Stopka A, Barac D, Lang C, Iber D. Simulation of Morphogen and Tissue Dynamics. Methods Mol Biol 2018; 1863:223-250. [PMID: 30324601 DOI: 10.1007/978-1-4939-8772-6_13] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modeling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter focuses on the mathematical description and numerical simulation of developmental processes. In particular, we discuss the mathematical representation of morphogen and tissue dynamics on static and growing domains, as well as the corresponding tissue mechanics. In addition, we give an overview of numerical methods that are routinely used to solve the resulting systems of partial differential equations. These include the finite element method and the Lattice Boltzmann method for the discretization as well as the arbitrary Lagrangian-Eulerian method and the Diffuse-Domain method to numerically treat deforming domains.
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Affiliation(s)
- Michael D Multerer
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
| | - Lucas D Wittwer
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
| | - Anna Stopka
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
| | - Diana Barac
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
| | - Christine Lang
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland
| | - Dagmar Iber
- Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland.
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Guckenberger A, Gekle S. Theory and algorithms to compute Helfrich bending forces: a review. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:203001. [PMID: 28240220 DOI: 10.1088/1361-648x/aa6313] [Citation(s) in RCA: 36] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.
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Affiliation(s)
- Achim Guckenberger
- Biofluid Simulation and Modeling, Fachbereich Physik, Universität Bayreuth, Germany
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Liu K, Marple GR, Allard J, Li S, Veerapaneni S, Lowengrub J. Dynamics of a multicomponent vesicle in shear flow. SOFT MATTER 2017; 13:3521-3531. [PMID: 28440378 PMCID: PMC5505236 DOI: 10.1039/c6sm02452a] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We study the fully nonlinear, nonlocal dynamics of two-dimensional multicomponent vesicles in a shear flow with matched viscosity of the inner and outer fluids. Using a nonstiff, pseudo-spectral boundary integral method, we investigate dynamical patterns induced by inhomogeneous bending for a two phase system. Numerical results reveal that there exist novel phase-treading and tumbling mechanisms that cannot be observed for a homogeneous vesicle. In particular, unlike the well-known steady tank-treading dynamics characterized by a fixed inclination angle, here the phase-treading mechanism leads to unsteady periodic dynamics with an oscillatory inclination angle. When the average phase concentration is around 1/2, we observe tumbling dynamics even for very low shear rate, and the excess length required for tumbling is significantly smaller than the value for the single phase case. We summarize our results in phase diagrams in terms of the excess length, shear rate, and concentration of the soft phase. These findings go beyond the well known dynamical regimes of a homogeneous vesicle and highlight the level of complexity of vesicle dynamics in a fluid due to heterogeneous material properties.
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Affiliation(s)
- Kai Liu
- Department of Mathematics, University of California at Irvine, Irvine, CA, USA.
| | - Gary R Marple
- Department of Mathematics, University of Michigan, Ann Arbor, MI, USA.
| | - Jun Allard
- Department of Mathematics, University of California at Irvine, Irvine, CA, USA. and Department of Physics, University of California at Irvine, Irvine, CA, USA and Center for Complex Biological Systems, University of California at Irvine, Irvine, CA, USA
| | - Shuwang Li
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA.
| | | | - John Lowengrub
- Department of Mathematics, University of California at Irvine, Irvine, CA, USA. and Center for Complex Biological Systems, University of California at Irvine, Irvine, CA, USA and Department of Biomedical Engineering, University of California at Irvine, Irvine, CA, USA
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