We present a complete analysis for the deformation profiles of lipid membranes induced by their interactions with solid elliptical cylinder substrates (e.g., proteins). The theoretical framework for the mechanics of lipid membranes is described in terms of the classical Helfrich model, and the resulting shape equation is formulated in general curvilinear coordinates to accommodate the elliptical shape of the contour surrounding the contact area. Admissible boundary conditions for the contact region are taken from the existing literature but reformulated and adapted to the current framework. A complete semi-analytic solution (in terms of Mathieu functions) is obtained within the limitation of superposed incremental deformations and the Monge representation in the deformed configuration functions. The results predict smooth morphological transitions over the domain of interest when a lipid membrane interacts with a rigid substrate through an elliptical contact region.

References

References
1.
Gorter
,
E.
, and
Grendel
,
F.
,
1925
, “
On Bimolecular Layers of Lipoids on the Chromocytes of the Blood
,”
J. Exp. Med.
,
41
(
4
), pp.
439
443
.
2.
Robertson
,
J. D.
,
1959
, “
The Ultrastructure of Cell Membranes and Their Derivatives
,”
Biochem. Soc. Symp.
,
16
, pp.
3
43
.
3.
Chernomrdik
,
L.
, and
Kozlov
,
M.
,
2008
, “
Mechanics of Membrane Fusion
,”
Nat. Struct Mol. Biol.
,
15
(
7
), pp.
675
683
.
4.
Lenz
,
M.
,
Morlot
,
S.
, and
Roux
,
A.
,
2009
, “
Mechanical Requirements for Membrane Fission: Common Facts From Various Examples
,”
FEBS Lett.
,
583
(
23
), pp.
3839
3846
.
5.
Razani
,
B.
, and
Lisanti
,
M. P.
,
2001
, “
Caveolins and Caveolae: Molecular and Functional Relationships
,”
Exp. Cell Res.
,
271
(
1
), pp.
36
44
.
6.
Nigel
,
U.
,
2005
, “
Refined Structure of the Nicotinic Acetyl-Choline Receptor at 4a Resolution
,”
J. Mol. Biol.
,
346
(
4
), pp.
967
989
.
7.
Bruinsma
,
R.
, and
Pincus
,
P.
,
1996
, “
Protein Aggregation in Membranes
,”
Curr. Opin. Solid State Mater. Sci.
,
1
(
3
), pp.
401
406
.
8.
Benedict
,
J. R.
,
Gregoria
,
I.
,
Vagelis
,
A. H.
,
Martin
,
M.
,
Kurt
,
K.
, and
Markus
,
D.
,
2007
, “
Aggregation and Vesiculation of Membrane Proteins by Curvature-Mediated Interactions
,”
Nature
,
447
(
7143
), pp.
461
464
.
9.
Agrawal
,
A.
, and
Steigmann
,
D. J.
,
2009
, “
Boundary-Value Problems in the Theory of Lipid Membranes
,”
Continuum Mech. Thermodyn.
,
21
(
1
), pp.
57
82
.
10.
Agrawal
,
A.
, and
Steigmann
,
D. J.
,
2009
, “
Modelling Protein-Mediated Morphology in Biomembranes
,”
Biomech. Model. Mechanobiol.
,
8
(
5
), pp.
371
379
.
11.
Ross
,
R.
, and
Virga
,
E. G.
,
1999
, “
Adhesive Borders of Lipid Membranes
,”
Proc. R. Soc. London A
,
455
(
1992
), pp.
4145
4168
.
12.
Deserno
,
M.
,
Muller
,
M. M.
, and
Guven
,
J.
,
2007
, “
Contact Lines for Fluid Surface Adhesion
,”
Phys. Rev. E
,
76
(
1
), p.
011605
.
13.
Tero
,
R.
,
2012
, “
Substrate Effects on the Foundation Process, Structure and Physicochemical Properties of Supported Lipid Bilayers
,”
Materials
,
5
(
12
), pp.
2658
2680
.
14.
Skar-Gislinge
,
N.
,
Simonsen
,
J. B.
,
Mortensen
,
K.
,
Feindenhans'l
,
R.
,
Sligar
,
S. G.
,
Mø ller
,
B. L.
,
Bjønholm
,
T.
, and
Arleth
,
L.
,
2010
, “
Elliptical Structure of Phospholipid Bilayer Nanodiscs Encapsulated by Scaffold Proteins: Casting the Roles of the Lipids and the Protein
,”
J. Am. Chem. Soc.
,
132
(
39
), pp.
13713
13722
.
15.
Belay
,
T.
,
Kim
,
C. I.
, and
Schiavone
,
P.
,
2015
, “
Analytical Solution of Lipid Membrane Morphology Subjected to Boundary Forces on the Edges of Rectangular Membrane
,”
Continuum Mech. Thermodyn.
(in press).
16.
Helfrich
,
W.
,
1973
, “
Elastic Properties of Lipid Bilayers: Theory and Possible Experiments
,”
Z. Naturforsch.
,
28
(
11
), pp.
693
703
.
17.
Rangamani
,
P.
,
Agrawal
,
A.
,
Mandadapu
,
K. K.
,
Oster
,
G.
, and
Steigmann
,
D. J.
,
2012
, “
Interaction Between Surface Shape and Intra-Surface Viscous Flow on Lipid Membrane
,”
Biomech. Model. Mechanobiol.
,
12
(
4
), pp.
833
845
.
18.
Kim
,
C. I.
, and
Steigmann
,
D. J.
,
2014
, “
Distension-Induced Gradient Capillarity in Lipid Membranes
,”
Continuum Mech. Thermodyn.
,
27
(
4
), pp.
609
621
.
19.
Steigmann
,
D. J.
,
1999
, “
Fluid Films With Curvature Elasticity
,”
Arch. Ration. Mech. Anal.
150
(
2
), pp.
127
152
.
20.
Steigmann
,
D. J.
,
1999
, “
On the Relationship Between the Cosserat and Kirchhoff-Love Theories of Elastic Shells
,”
Math. Mech. Solids
,
4
(
3
), pp.
275
288
.
21.
Landau
,
L. D.
, and
Lifschitz
,
E. M.
,
1947
, “
Theory of Elasticity
,”
Course of Theoretical Physics
,
3rd ed.
, Vol.
7
,
Pergamon
,
Oxford, UK
.
22.
Virga
,
E. G.
,
1947
,
Variational Theories for Liquid Crystals
,
Chapman & Hall
,
London, UK
.
23.
Moon
,
P.
, and
Spencer
,
D. E.
,
1971
,
Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions
,
2nd ed.
,
Springer
,
New York.
24.
McLachlan
,
N. W.
,
1947
,
Theory and Application of Mathieu Functions
,
Oxford
,
UK
.
25.
Sato
,
K.
,
2006
, “
Bending of an Elliptical Plate on Elastic Foundation and Under the Combined Action of Lateral Load and In-Plane Force
,”
III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering
,
Springer
,
Dordrecht
, pp.
49
.
26.
Gutiérrez-Vega
,
J.
,
Chávez-Cedra
,
S.
, and
Rodríguez-Dagnino
,
R.
,
1999
, “
Free Oscillations in an Elliptical Membrane
,”
Rev. Mex. Fis.
,
45
(
6
), pp.
613
622
.
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