Effect of thermal undulations on the bending elasticity and spontaneous curvature of fluid membranes

Pinnow, Henryk A.; Helfrich, W.

doi:10.1007/s101890070028

Cited by 29 publications

(45 citation statements)

References 15 publications

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“…This is not quite as clear for the membrane as for the polymer since a cap representing local mean curvature is accompanied by a brim of saddle curvatures which, however, are fully slaved. In calculations of the effective bending rigidity it has been shown that normal displacement is insufficient as statistical measure if membrane or polymer are subjected to a background curvature [10]. In the present article normal displacement fails as measure (or is at least impractical) for other reasons.…”

Section: Discussionmentioning

confidence: 66%

“…It is automatically used in Monte Carlo simulations. For the nearly flat membrane without background curvature, one-dimensional normal displacement (Du) is good enough and results, in fact, in the same configurational statistics as mean curvature (DJ), as was discussed in detail elsewhere [10]. Like bend in the case of polymers, mean curvature may be called the natural measure because it is identical to the relevant strain, i.e.…”

Section: Discussionmentioning

confidence: 86%

“…It was shown before that mean curvature is a correct measure of integration for fluctuating fluid membranes, while the use of normal displacement can lead to wrong results, e.g. when the membrane fluctuates in the presence of a background curvature [9,10]. The coincidence of the interaction energies derived by the two (or three) methods may be regarded as a further confirmation of the validity and usefulness of the curvature measure.…”

Section: Introductionmentioning

confidence: 89%

“…The fluctuating mean curvature of each local mode is confined to a molecular (or similar uniform) area which for simplicity is taken to be circular. A single such mode resembles a hat, consisting of a cap with spherical curvature and a brim forming a catenoidal minimal surface [10][11][12]. Not only the normal displacements but also the mean curvatures of sinusoidal waves and hat modes are superimposable whenever the usual approximation of the nearly flat membrane applies, which will be assumed throughout.…”

Section: Fluctuation Mode Entropiesmentioning

confidence: 98%

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Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

Helfrich

Weikl

2001

The European Physical Journal E

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The fluctuation-induced attractive interaction of rigid flat objects embedded in a fluid membrane is calculated for a pair of parallel strips and a pair of equal circular disks. Assuming flat boundary conditions, we derive the interaction from the entropy of the suppressed boundary angle fluctuation modes. Each mode entropy is computed in two ways: from the boundary angles themselves and from the meancurvature mode functions. A formula for the entropy loss of suppressing one or more mean-curvature modes is developed and applied. For the pair of disks we recover the result of Goulian et al. and Golestanian et al. in a direct manner, avoiding any mappings by Hubbard-Stratonovitch transformations. The mode-by-mode agreement of the two computed entropies in both systems confirms an earlier claim that mean curvature is the natural measure of integration for fluid membranes. PACS. 82.70.Uv Surfactants, micellar solutions, vesicles, lamellae, amphiphilic systems (hydrophilic and hydrophobic interactions) -87.16.Dg Membranes, bilayers, and vesicles -34.20.-b Interatomic and intermolecular potentials and forces, potential energy surfaces for collisions

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Section: Discussionmentioning

confidence: 66%

Section: Discussionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 89%

Section: Fluctuation Mode Entropiesmentioning

confidence: 98%

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Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

Helfrich

Weikl

2001

The European Physical Journal E

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“…Conflicting values for the prefactor—α = 1 [51, 56, 57], α = 3 [52–54] and α = −1 [58, 59]—pointing to both thermal softening (α = 1 and 3) and stiffening (α = −1) have been reported. In spite of these contradictions, it should be remembered that we measure κ R from experiments and molecular simulations, whereas we impose κ in mesoscale/continuum simulations based on the Helfrich energy functional.…”

Section: Thermodynamics-based Models For Membranesmentioning

confidence: 99%

Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins

2014

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Biological membranes constitute boundaries of cells and cell organelles. These membranes are soft fluid interfaces whose thermodynamic states are dictated by bending moduli, induced curvature fields, and thermal fluctuations. Recently, there has been a flood of experimental evidence highlighting active roles for these structures in many cellular processes ranging from trafficking of cargo to cell motility. It is believed that the local membrane curvature, which is continuously altered due to its interactions with myriad proteins and other macromolecules attached to its surface, holds the key to the emergent functionality in these cellular processes. Mechanisms at the atomic scale are dictated by protein-lipid interaction strength, lipid composition, lipid distribution in the vicinity of the protein, shape and amino acid composition of the protein, and its amino acid contents. The specificity of molecular interactions together with the cooperativity of multiple proteins induce and stabilize complex membrane shapes at the mesoscale. These shapes span a wide spectrum ranging from the spherical plasma membrane to the complex cisternae of the Golgi apparatus. Mapping the relation between the protein-induced deformations at the molecular scale and the resulting mesoscale morphologies is key to bridging cellular experiments across the various length scales. In this review, we focus on the theoretical and computational methods used to understand the phenomenology underlying protein-driven membrane remodeling. Interactions at the molecular scale can be computationally probed by all atom and coarse grained molecular dynamics (MD, CGMD), as well as dissipative particle dynamics (DPD) simulations, which we only describe in passing. We choose to focus on several continuum approaches extending the Canham - Helfrich elastic energy model for membranes to include the effect of curvature-inducing proteins and explore the conformational phase space of such systems. In this description, the protein is expressed in the form of a spontaneous curvature field. The approaches include field theoretical methods limited to the small deformation regime, triangulated surfaces and particle-based computational models to investigate the large-deformation regimes observed in the natural state of many biological membranes. Applications of these methods to understand the properties of biological membranes in homogeneous and inhomogeneous environments of proteins, whose underlying curvature fields are either isotropic or anisotropic, are discussed. The diversity in the curvature fields elicits a rich variety of morphological states, including tubes, discs, branched tubes, and caveola. Mapping the thermodynamic stability of these states as a function of tuning parameters such as concentration and strength of curvature induction of the proteins is discussed. The relative stabilities of these self-organized shapes are examined through free-energy calculations. The suite of methods discussed here can be tailored to applications in specific cellular set...

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Liquid Bilayer and its Simulation

Stecki

2010

Advances in Chemical Physics

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Effect of thermal undulations on the bending elasticity and spontaneous curvature of fluid membranes

Cited by 29 publications

References 15 publications

Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes

Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins

Liquid Bilayer and its Simulation

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