The Buckling of Spherical Liposomes

[+] Author and Article Information
D. C. Pamplona

Department of Civil Engineering,  PUC-RJ, Rio de Janeiro CEP 22453, Brasil

J. A. Greenwood, C. R. Calladine

Department of Engineering,  University of Cambridge, Cambridge CB2 1PZ, U.K.

J Biomech Eng 127(7), 1062-1069 (Jun 05, 2005) (8 pages) doi:10.1115/1.2073527 History: Received September 09, 2004; Revised June 05, 2005

In the classical “first approximation” theory of thin-shell structures, the constitutive relations for a generic shell element—i.e. the elastic relations between the bending moments and membrane stresses and the corresponding changes in curvature and strain, respectively—are written as if an element of the shell is flat, although in reality it is curved. In this theory it is believed that discrepancies on account of the use of “flat” constitutive relations will be negligible provided the ratio shell-radius∕thickness is of sufficiently large order. In the study of drawing of narrow, cylindrical “tethers” from liposomes it has been known for many years that it is necessary to use instead a constitutive law which explicitly describes a curved element in order to make sense of the mechanics; and indeed such tethers are generally of “thick-walled” proportions. In this paper we show that the proper constitutive relations for a curved element must also be used in the study, by means of shell equations, of the buckling of initially spherical thin-walled giant liposomes under exterior pressure: these involve the inclusion of what we call the “Mκ” terms, which are not present in the standard “first-approximation” theory. We obtain analytical expressions for both the bifurcation buckling pressure and the slope of the post-buckling path, in terms of the dimensions and elastic constants of the lipid bi-layer, and also the initial state of bending moment in the vesicle. We explain physically how the initial bending moment can affect the bifurcation pressure, whereas it cannot in “first-approximation” theory. We use these results to map the conditions under which the vesicle buckles into an oblate, as distinct from a prolate (“rugby-ball”) shape. Some of our results were obtained long ago by the use of energy methods; but our aim here has been to identify precisely what is lacking in “first-approximation” theory in relation to liposomes, and so to put the “shell equations” approach onto a firm footing in mechanics.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

A doubly-curved small element of lipid bi-layer, acted upon by in-plane membrane stress resultants Ns, Nθ and isotropic bending-stress resultants M—all of which are dimensionless. The principal radii of curvature ρs, ρθ are equal to 1∕κs, 1∕κθ, respectively; subscripts s, θ refer to the meridional, circumferential principal directions, respectively.

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Figure 2

Diagrams showing the meridian of a liposome, between pole and equator: (a) In its original, spherical form and (b) in its current, deformed conformation. Length co-ordinates are all normalized with respect to the radius of the sphere.

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Figure 3

Portion of the meridian in the current conformation, showing the (dimensionless) normal pressure loading p and the positive senses of stress-resultants Ns, Nθ, Q, and M

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Figure 4

(a) Complete perturbed meridian of a liposome of the form ϕ=s+bsin2s. (b) Schematic distribution of Ns, Nθ from pole to equator in the case C=0, κo=0.6. The equatorial value of Ns and the mean value of Nθ are marked by heavy points: These play a significant role in determining pcr. (c) As (b), but with C=0, κo=1; for which case Ns=Nθ.

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Figure 5

(a) Plot of p against κ* (apex curvature), showing the bifurcation point (at pcr) and the sloping secondary equilibrium path in its vicinity. In this case the liposome would buckle into the prolate spheroidal mode. (b) Plot of C (elastic membrane shear constant) against κo (initial curvature), showing regions of prolate and oblate buckling, according to inequality 48. The marked points on the axes come from other studies: See the text.



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