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Article

Mechanics of Curved Plasma Membrane Vesicles: Resting Shapes, Membrane Curvature, and In-Plane Shear Elasticity

[+] Author and Article Information
Tadashi Kosawada1

 Department of Mechanical Systems Engineering Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan

Kohji Inoue

 Department of Mechanical Systems Engineering Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan

Geert W. Schmid-Schönbein

 Department of Bioengineering, University of California, San Diego La Jolla, CA 92093-0412

1

Corresponding author.

J Biomech Eng 127(2), 229-236 (Sep 02, 2004) (8 pages) doi:10.1115/1.1865197 History: Received March 11, 2004; Revised August 09, 2004; Accepted September 02, 2004

Highly curved cell membrane structures, such as plasmalemmal vesicles (caveolae) and clathrin-coated pits, facilitate many cell functions, including the clustering of membrane receptors and transport of specific extracellular macromolecules by endothelial cells. These structures are subject to large mechanical deformations when the plasma membrane is stretched and subject to a change of its curvature. To enhance our understanding of plasmalemmal vesicles we need to improve the understanding of the mechanics in regions of high membrane curvatures. We examine here, theoretically, the shapes of plasmalemmal vesicles assuming that they consist of three membrane domains: an inner domain with high curvature, an outer domain with moderate curvature, and an outermost flat domain, all in the unstressed state. We assume the membrane properties are the same in these domains with membrane bending elasticity as well as in-plane shear elasticity. Special emphasis is placed on the effects of membrane curvature and in-plane shear elasticity on the mechanics of vesicle during unfolding by application of membrane tension. The vesicle shapes were computed by minimization of bending and in-plane shear strain energy. Mechanically stable vesicles were identified with characteristic membrane necks. Upon stretch of the membrane, the vesicle necks disappeared relatively abruptly leading to membrane shapes that consist of curved indentations. While the resting shape of vesicles is predominantly affected by the membrane spontaneous curvatures, the membrane shear elasticity (for a range of values recorded in the red cell membrane) makes a significant contribution as the vesicle is subject to stretch and unfolding. The membrane tension required to unfold the vesicle is sensitive with respect to its shape, especially as the vesicle becomes fully unfolded and approaches a relative flat shape.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) A trace from electron micrograph of a cross section of a small blood capillary comprised of a single endothelial cell conjoined to itself (Fig. 1 in (17)). (b) A magnified view of the typical luminal surface where one plasmalemmal vesicle is surrounded by curved plasma membrane. (c) A typical assumed model provided for theoretical analysis.

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Figure 2

Coordinate system and parameters defined on the unstressed initial cross section of an axisymmetric vesicle system composed with three membrane domains; an inner spherical domain, an outer partially spherical domain, and an outermost flat circular domain.

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Figure 3

Control structures of Runge-Kutta numerical integration with respect to A (total surface area) for the governing Eqs. 7,8,9 in 0≦A≦At

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Figure 4

Two-dimensional membrane contours with upper outer perimeter rt set around 4.4rI in the case of spontaneous radii ratio n(=rO∕rI)=6, 10, 100, ∞, and shear modulus μ=0.66×10−2dyn∕cm. The position of the upper vesicle perimeter is intentionally aligned along the x-axis (z=0). The figure shows the right side of each cross section, and the initial shape is also shown by the broken line.

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Figure 6

Relationships between nondimensional opening radius and strain energy in the case of spontaneous radii ratio n(=rO∕rI)=6, 10, 100, ∞, and shear modulus μ=0.66×10−2dyn∕cm. Numbers along with n=10 correspond to the configuration numbers in Fig. 5.

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Figure 5

(Color). Three-dimensional membrane contours of the computed equilibrium configurations in the case of spontaneous radii ratio n=10 and shear modulus μ=0.66×10−2dyn∕cm. They were drawn by using Mathematica (20). The meshes in the figure were automatically generated by Mathematica so as to show three-dimensional images effectively.

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Figure 7

Relationships between nondimensional opening radius and radial tension acting at rt in the case of spontaneous radii ratio n(=rO∕rI)=6, 10, 100, ∞, and shear modulus μ=0.66×10−2dyn∕cm. Numbers along with n=10 correspond to the configuration numbers in Figs.  56.

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Figure 8

Two-dimensional membrane contours of equilibrium shape of the vesicle system with various opening radii rt in the case of shear modulus μ=(0,0.33,0.66)×10−2dyn∕cm and spontaneous radii ratio n=6. The black circle on the line denotes the boundary between the outer membrane domains (dotted line) and the inner vesicle domain (solid line).

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Figure 9

Magnified two-dimensional membrane contours of equilibrium shape of the vesicle system selected from Fig. 8 where the neck portion suddenly disappears while unfolding the vesicle. Significant effects of the shear modulus μ on the equilibrium shapes are observed.

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Figure 10

Comparison between the computed shapes of the vesicle [in case of shear modulus μ=(0,0.33,0.66)×10−2dyn∕cm and spontaneous radii ratio n=30] and a trace picked out from electron micrograph published by Palade and Bruns ((21), Fig. 18) in capillary endothelium of the rat tongue.

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