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Zabala-Ferrera O, Liu P, Beltramo PJ. Determining the Bending Rigidity of Free-Standing Planar Phospholipid Bilayers. MEMBRANES 2023; 13:129. [PMID: 36837632 PMCID: PMC9959114 DOI: 10.3390/membranes13020129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/30/2022] [Revised: 01/14/2023] [Accepted: 01/17/2023] [Indexed: 06/18/2023]
Abstract
We describe a method to determine membrane bending rigidity from capacitance measurements on large area, free-standing, planar, biomembranes. The bending rigidity of lipid membranes is an important biological mechanical property that is commonly optically measured in vesicles, but difficult to quantify in a planar, unsupported system. To accomplish this, we simultaneously image and apply an electric potential to free-standing, millimeter area, planar lipid bilayers composed of DOPC and DOPG phospholipids to measure the membrane Young's (elasticity) modulus. The bilayer is then modeled as two adjacent thin elastic films to calculate bending rigidity from the electromechanical response of the membrane to the applied field. Using DOPC, we show that bending rigidities determined by this approach are in good agreement with the existing work using neutron spin echo on vesicles, atomic force spectroscopy on supported lipid bilayers, and micropipette aspiration of giant unilamellar vesicles. We study the effect of asymmetric calcium concentration on symmetric DOPC and DOPG membranes and quantify the resulting changes in bending rigidity. This platform offers the ability to create planar bilayers of controlled lipid composition and aqueous ionic environment, with the ability to asymmetrically alter both. We aim to leverage this high degree of compositional and environmental control, along with the capacity to measure physical properties, in the study of various biological processes in the future.
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2
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Pinigin KV. Determination of Elastic Parameters of Lipid Membranes with Molecular Dynamics: A Review of Approaches and Theoretical Aspects. MEMBRANES 2022; 12:membranes12111149. [PMID: 36422141 PMCID: PMC9692374 DOI: 10.3390/membranes12111149] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2022] [Revised: 11/07/2022] [Accepted: 11/10/2022] [Indexed: 05/12/2023]
Abstract
Lipid membranes are abundant in living organisms, where they constitute a surrounding shell for cells and their organelles. There are many circumstances in which the deformations of lipid membranes are involved in living cells: fusion and fission, membrane-mediated interaction between membrane inclusions, lipid-protein interaction, formation of pores, etc. In all of these cases, elastic parameters of lipid membranes are important for the description of membrane deformations, as these parameters determine energy barriers and characteristic times of membrane-involved phenomena. Since the development of molecular dynamics (MD), a variety of in silico methods have been proposed for the determination of elastic parameters of simulated lipid membranes. These MD methods allow for the consideration of details unattainable in experimental techniques and represent a distinct scientific field, which is rapidly developing. This work provides a review of these MD approaches with a focus on theoretical aspects. Two main challenges are identified: (i) the ambiguity in the transition from the continuum description of elastic theories to the discrete representation of MD simulations, and (ii) the determination of intrinsic elastic parameters of lipid mixtures, which is complicated due to the composition-curvature coupling effect.
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Affiliation(s)
- Konstantin V Pinigin
- A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31/4 Leninskiy Prospekt, 119071 Moscow, Russia
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3
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Hénin J, Lopes LJS, Fiorin G. Human Learning for Molecular Simulations: The Collective Variables Dashboard in VMD. J Chem Theory Comput 2022; 18:1945-1956. [PMID: 35143194 DOI: 10.1021/acs.jctc.1c01081] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The Collective Variables Dashboard is a software tool for real-time, seamless exploration of molecular structures and trajectories in a customizable space of collective variables. The Dashboard arises from the integration of the Collective Variables Module (also known as Colvars) with the visualization software VMD, augmented with a fully discoverable graphical interface offering interactive workflows for the design and analysis of collective variables. Typical use cases include a priori design of collective variables for enhanced sampling and free energy simulations as well as analysis of any type of simulation or collection of structures in a collective variable space. A combination of those cases commonly occurs when preliminary simulations, biased or unbiased, reveal that an optimized set of collective variables is necessary to improve sampling in further simulations. Then the Dashboard provides an efficient way to intuitively explore the space of likely collective variables, validate them on existing data, and use the resulting collective variable definitions directly in further biased simulations using the Collective Variables Module. Visualization of biasing energies and forces is proposed to help analyze or plan biased simulations. We illustrate the use of the Dashboard on two applications: discovering coordinates to describe ligand unbinding from a protein binding site and designing volume-based variables to bias the hydration of a transmembrane pore.
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Affiliation(s)
- Jérôme Hénin
- Laboratoire de Biochimie Théorique UPR 9080, CNRS, Université de Paris, 75005 Paris, France.,Institut de Biologie Physico-Chimique-Fondation Edmond de Rothschild, PSL Research University, 75005 Paris, France
| | - Laura J S Lopes
- Theoretical Division T-1, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States
| | - Giacomo Fiorin
- National Institute of Neurological Disorders and Stroke (NINDS) and National Heart, Lung and Blood Institute (NHLBI), Bethesda, Maryland 20892, United States
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4
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Smirnova YG, Müller M. How does curvature affect the free-energy barrier of stalk formation? Small vesicles vs apposing, planar membranes. EUROPEAN BIOPHYSICS JOURNAL : EBJ 2021; 50:253-264. [PMID: 33547940 PMCID: PMC8071802 DOI: 10.1007/s00249-020-01494-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 12/08/2020] [Accepted: 12/31/2020] [Indexed: 11/26/2022]
Abstract
Using molecular simulations of POPC lipids in conjunction with the calculation of the Minimum Free-Energy Path (MFEP), we study the effect of strong membrane curvature on the formation of the first fusion intermediate-the stalk between a vesicle and its periodic image. We find that the thermodynamic stability of this hourglass-shaped, hydrophobic connection between two vesicles is largely increased by the strong curvature of small vesicles, whereas the intrinsic barrier to form a stalk, i.e., associated with dimple formation and lipid tails protrusions, is similar to the case of two, apposing, planar membranes. A significant reduction of the barrier of stalk formation, however, stems from the lower dehydration free energy that is required to bring highly curved vesicle into a distance, at which stalk formation may occur, compared to the case of apposing, planar membranes.
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Affiliation(s)
- Y G Smirnova
- Institute for Theoretical Physics, Georg-August University, 37077, Göttingen, Germany.
| | - M Müller
- Institute for Theoretical Physics, Georg-August University, 37077, Göttingen, Germany
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5
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Risselada HJ, Grubmüller H. How proteins open fusion pores: insights from molecular simulations. EUROPEAN BIOPHYSICS JOURNAL : EBJ 2021; 50:279-293. [PMID: 33340336 PMCID: PMC8071795 DOI: 10.1007/s00249-020-01484-3] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Revised: 11/16/2020] [Accepted: 11/24/2020] [Indexed: 02/06/2023]
Abstract
Fusion proteins can play a versatile and involved role during all stages of the fusion reaction. Their roles go far beyond forcing the opposing membranes into close proximity to drive stalk formation and fusion. Molecular simulations have played a central role in providing a molecular understanding of how fusion proteins actively overcome the free energy barriers of the fusion reaction up to the expansion of the fusion pore. Unexpectedly, molecular simulations have revealed a preference of the biological fusion reaction to proceed through asymmetric pathways resulting in the formation of, e.g., a stalk-hole complex, rim-pore, or vertex pore. Force-field based molecular simulations are now able to directly resolve the minimum free-energy path in protein-mediated fusion as well as quantifying the free energies of formed reaction intermediates. Ongoing developments in Graphics Processing Units (GPUs), free energy calculations, and coarse-grained force-fields will soon gain additional insights into the diverse roles of fusion proteins.
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Affiliation(s)
- H. Jelger Risselada
- Department of Theoretical Physics, Georg-August University of Göttingen, Göttingen, Germany
- Leiden University, Leiden Institute of Chemistry (LIC), Leiden, The Netherlands
| | - Helmut Grubmüller
- Max Planck Institute for Biophysical Chemistry, Theoretical and Computational Biophysics Department, Göttingen, Germany
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6
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Endter LJ, Smirnova Y, Risselada HJ. Density Field Thermodynamic Integration (DFTI): A "Soft" Approach to Calculate the Free Energy of Surfactant Self-Assemblies. J Phys Chem B 2020; 124:6775-6785. [PMID: 32631061 DOI: 10.1021/acs.jpcb.0c03982] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Thermodynamic integration is one of the most established methods to quantify excess free energies between different metastable states. Excess intermolecular interactions in surfactant assemblies are on the scale of the energy of thermal fluctuations. Therefore, these materials can be deformed and topologically altered via relatively small mechanical stresses. It is thus intuitive to design reaction paths and associated order parameters that exploit the "soft" nature of these materials to mechanically rather than alchemically morph surfactant assemblies from state to state. Here, we propose a novel method coined "density field thermodynamic integration" (DFTI) that adopts the universality and transferability of alchemical methods while simultaneously exploiting the soft excess interactions between surfactant molecules. DFTI was designed for a rapid quantification of the free energy differences between different metastable structures in soft fluid materials. The DFTI method uses an external field coupled to the local density to mechanically morph the system between metastable states of interest. Here, we explored the capability of the DFTI method to swiftly and accurately calculate free energy differences between states. To this aim, we studied two different coarse-grained lipidic surfactant systems: (i) a fusion stalk and (ii) a worm-like micelle. Our results illustrate that DFTI can provide an efficient, versatile, and rather reliable method to calculate the free energy differences between surfactant assemblies.
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Affiliation(s)
- Laura Josefine Endter
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
| | - Yuliya Smirnova
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
| | - Herre Jelger Risselada
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany.,Leiden Institute of Chemistry (LIC), University of Leiden, 2311 Leiden,The Netherlands.,Chemical Deptartment, Leibniz Institute of Surface Modifications, 04318 Leipzig, Germany
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7
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Bouvier B. Curvature as a Collective Coordinate in Enhanced Sampling Membrane Simulations. J Chem Theory Comput 2019; 15:6551-6561. [DOI: 10.1021/acs.jctc.9b00716] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Affiliation(s)
- Benjamin Bouvier
- Laboratoire de Glycochimie, des Antimicrobiens et des Agroressources, CNRS UMR7378/Université de Picardie Jules Verne, 10, rue Baudelocque, 80039 Amiens Cedex, France
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8
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Fiorin G, Marinelli F, Faraldo-Gómez JD. Direct Derivation of Free Energies of Membrane Deformation and Other Solvent Density Variations From Enhanced Sampling Molecular Dynamics. J Comput Chem 2019; 41:449-459. [PMID: 31602694 DOI: 10.1002/jcc.26075] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2019] [Revised: 08/04/2019] [Accepted: 08/28/2019] [Indexed: 12/20/2022]
Abstract
We report a methodology to calculate the free energy of a shape transformation in a lipid membrane directly from a molecular dynamics simulation. The bilayer need not be homogeneous or symmetric and can be atomically detailed or coarse grained. The method is based on a collective variable that quantifies the similarity between the membrane and a set of predefined density distributions. Enhanced sampling of this "Multi-Map" variable re-shapes the bilayer and permits the derivation of the corresponding potential of mean force. Calculated energies thus reflect the dynamic interplay of atoms and molecules, rather than postulated effects. Evaluation of deformations of different shape, amplitude, and range demonstrates that the macroscopic bending modulus assumed by the Helfrich-Canham model is increasingly unsuitable below the 100-Å scale. In this range of major biological significance, direct free-energy calculations reveal a much greater plasticity. We also quantify the stiffening effect of cholesterol on bilayers of different composition and compare with experiments. Lastly, we illustrate how this approach facilitates analysis of other solvent reorganization processes, such as hydrophobic hydration. Published 2019. This article is a U.S. Government work and is in the public domain in the USA.
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Affiliation(s)
- Giacomo Fiorin
- Theoretical Molecular Biophysics Laboratory, National Heart, Lung and Blood Institute, National Institutes of Health, 10 Center Drive, Bethesda, Maryland, 20814
| | - Fabrizio Marinelli
- Theoretical Molecular Biophysics Laboratory, National Heart, Lung and Blood Institute, National Institutes of Health, 10 Center Drive, Bethesda, Maryland, 20814
| | - José D Faraldo-Gómez
- Theoretical Molecular Biophysics Laboratory, National Heart, Lung and Blood Institute, National Institutes of Health, 10 Center Drive, Bethesda, Maryland, 20814
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9
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Smirnova YG, Risselada HJ, Müller M. Thermodynamically reversible paths of the first fusion intermediate reveal an important role for membrane anchors of fusion proteins. Proc Natl Acad Sci U S A 2019; 116:2571-2576. [PMID: 30700547 PMCID: PMC6377489 DOI: 10.1073/pnas.1818200116] [Citation(s) in RCA: 46] [Impact Index Per Article: 9.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Abstract
Biological membrane fusion proceeds via an essential topological transition of the two membranes involved. Known players such as certain lipid species and fusion proteins are generally believed to alter the free energy and thus the rate of the fusion reaction. Quantifying these effects by theory poses a major challenge since the essential reaction intermediates are collective, diffusive and of a molecular length scale. We conducted molecular dynamics simulations in conjunction with a state-of-the-art string method to resolve the minimum free-energy path of the first fusion intermediate state, the so-called stalk. We demonstrate that the isolated transmembrane domains (TMDs) of fusion proteins such as SNARE molecules drastically lower the free energy of both the stalk barrier and metastable stalk, which is not trivially explained by molecular shape arguments. We relate this effect to the local thinning of the membrane (negative hydrophobic mismatch) imposed by the TMDs which favors the nearby presence of the highly bent stalk structure or prestalk dimple. The distance between the membranes is the most crucial determinant of the free energy of the stalk, whereas the free-energy barrier changes only slightly. Surprisingly, fusion enhancing lipids, i.e., lipids with a negative spontaneous curvature, such as PE lipids have little effect on the free energy of the stalk barrier, likely because of its single molecular nature. In contrast, the lipid shape plays a crucial role in overcoming the hydration repulsion between two membranes and thus rather lowers the total work required to form a stalk.
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Affiliation(s)
- Yuliya G Smirnova
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany;
| | - Herre Jelger Risselada
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
- Leiden Institute of Chemistry, Leiden University, 2333 CC Leiden, The Netherlands
| | - Marcus Müller
- Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany
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10
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Sun DW, Müller M. Numerical algorithms for solving self-consistent field theory reversely for block copolymer systems. J Chem Phys 2018; 149:214104. [PMID: 30525732 DOI: 10.1063/1.5063302] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
Abstract
Besides dictating the equilibrium phase diagram, the rugged free-energy landscape of AB block copolymers gives rise to a multitude of non-equilibrium phenomena. Self-consistent field theory (SCFT) can be employed to calculate the mean-field free energy, F [ ϕ A t a r g e t ] , of a non-equilibrium unstable state that is characterized by a given spatial density distribution, ϕ A t a r g e t , in the incompressible system. Such a free-energy functional is the basis of describing the structure formation by dynamic SCFT techniques or the identification of minimum free-energy paths via the string method. The crucial step consists in computing the external potential fields that generate the given density distribution in the corresponding system of non-interacting copolymers, i.e., the potential-to-density relation employed in equilibrium SCFT calculations has to be inverted (reverse SCFT calculation). We describe, generalize, and evaluate the computational efficiency of two different numerical algorithms for this reverse SCFT calculation-the Debye-function algorithm based on the structure factor and the field-theoretic umbrella-potential (FUP) algorithm. In contrast to the Debye-function algorithm, the FUP algorithm only yields the exact mean-field values of the given target densities in the limit of a strong umbrella potential, and we devise a two-step variant of the FUP algorithm that significantly mitigates this issue. For Gaussian copolymers, the Debye-function algorithm is more efficient for highly unstable states that are far away from the equilibrium, whereas the improved FUP algorithm outperforms the Debye-function algorithm closer to metastable states and is easily transferred to more complex molecular architectures.
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Affiliation(s)
- De-Wen Sun
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Marcus Müller
- Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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11
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Berthault A, Werner M, Baulin VA. Bridging molecular simulation models and elastic theories for amphiphilic membranes. J Chem Phys 2018; 149:014902. [PMID: 29981558 DOI: 10.1063/1.5027895] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The Single Chain Mean Field theory is used to link coarse-grained models of amphiphilic molecules with analytical models for membrane elasticity, where phenomenological parameters are deduced from explicit molecular models and force fields. We estimate the elastic constants based on the free energy of the amphiphilic bilayer in planar and cylindrical geometries on the example of four amphiphilic molecules that differ in length and stiffness. We study how these variations affect the equilibrium bilayer structure, the equilibrium free energy, and the elastic constants. Bending rigidities are obtained within the typical range of experimental values for phospholipid membranes in a liquid state.
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Affiliation(s)
- Adrien Berthault
- Department d'Enginyeria Quimica, Universitat Rovira i Virgili, Ave. dels Paisos Catalans 26, 43007 Tarragona, Spain
| | - Marco Werner
- Department d'Enginyeria Quimica, Universitat Rovira i Virgili, Ave. dels Paisos Catalans 26, 43007 Tarragona, Spain
| | - Vladimir A Baulin
- Department d'Enginyeria Quimica, Universitat Rovira i Virgili, Ave. dels Paisos Catalans 26, 43007 Tarragona, Spain
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12
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Masone D, Uhart M, Bustos DM. Bending Lipid Bilayers: A Closed-Form Collective Variable for Effective Free-Energy Landscapes in Quantitative Biology. J Chem Theory Comput 2018; 14:2240-2245. [PMID: 29506389 DOI: 10.1021/acs.jctc.8b00012] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
Abstract
Curvature-related processes are of major importance during protein-membrane interactions. The illusive simplicity of membrane reshaping masks a complex molecular process crucial for a wide range of biological functions like fusion, endo- and exocytosis, cell division, cytokinesis, and autophagy. To date, no functional expression of a reaction coordinate capable of biasing molecular dynamics simulations to produce membrane curvature has been reported. This represents a major drawback given that the adequate identification of proper collective variables to enhance sampling is fundamental for restrained dynamics techniques. In this work, we present a closed-form equation of a collective variable that induces bending in lipid bilayers in a controlled manner, allowing for straightforward calculation of free energy landscapes of important curvature-related events, using standard methods such as umbrella sampling and metadynamics. As a direct application of the collective variable, we calculate the bending free energies of a ternary lipid bilayer in the presence and the absence of a Bin/Amphiphysin/Rvs domain with an N-terminal amphipathic helix (N-BAR), a well-known peripheral membrane protein that induces curvature.
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13
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Bubnis G, Risselada HJ, Grubmüller H. Exploiting Lipid Permutation Symmetry to Compute Membrane Remodeling Free Energies. PHYSICAL REVIEW LETTERS 2016; 117:188102. [PMID: 27834997 DOI: 10.1103/physrevlett.117.188102] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2016] [Indexed: 05/26/2023]
Abstract
A complete physical description of membrane remodeling processes, such as fusion or fission, requires knowledge of the underlying free energy landscapes, particularly in barrier regions involving collective shape changes, topological transitions, and high curvature, where Canham-Helfrich (CH) continuum descriptions may fail. To calculate these free energies using atomistic simulations, one must address not only the sampling problem due to high free energy barriers, but also an orthogonal sampling problem of combinatorial complexity stemming from the permutation symmetry of identical lipids. Here, we solve the combinatorial problem with a permutation reduction scheme to map a structural ensemble into a compact, nondegenerate subregion of configuration space, thereby permitting straightforward free energy calculations via umbrella sampling. We applied this approach, using a coarse-grained lipid model, to test the CH description of bending and found sharp increases in the bending modulus for curvature radii below 10 nm. These deviations suggest that an anharmonic bending term may be required for CH models to give quantitative energetics of highly curved states.
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Affiliation(s)
- Greg Bubnis
- Department of Theoretical and Computational Biophysics, Max-Planck-Institute for Biophysical Chemistry, Göttingen 37077, Germany
| | - Herre Jelger Risselada
- Department of Theoretical and Computational Biophysics, Max-Planck-Institute for Biophysical Chemistry, Göttingen 37077, Germany
- Chemistry Department, Leibniz Institute of Surface Modification, Leipzig 04318, Germany
- Deptartment of Theoretical Physics, Georg-August University Göttingen, Göttingen 37077, Germany
| | - Helmut Grubmüller
- Department of Theoretical and Computational Biophysics, Max-Planck-Institute for Biophysical Chemistry, Göttingen 37077, Germany
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