The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of $1.5×10−6 N/m$ over a $1–2 μm$ range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus $B≈1×10−19 Nm$ and an upper limit to the in-plane shear modulus $H<2×10−6 N/m$. The partial agreement of these results with other published values may serve to highlight the dependence of the cell’s resistance to deformation on the scale and geometry of the deformation.

1.
Lenormand
,
G.
,
Hénon
,
S.
,
Richert
,
A.
,
Simon
,
J.
, and
Gallet
,
F.
, 2001, “
Direct Measurement of the Area Expansion and Shear Moduli of the Human Red Blood Cell Membrane Cytoskeleton
,”
Biophys. J.
0006-3495,
81
, pp.
43
56
.
2.
Waugh
,
R. E.
, and
Evans
,
E. A.
, 1979, “
Thermoelasticity of Red Blood Cell Membranes
,”
Biophys. J.
0006-3495,
26
, pp.
115
131
.
3.
Brochard
,
F.
, and
Lennon
,
J. F.
, 1975, “
Frequency Spectrum of the Flicker Phenomenon in Erythrocytes
,”
J. Phys. (France)
0302-0738,
11
, pp.
1035
1047
.
4.
Zilker
,
A.
,
Engelhardt
,
H.
, and
Sackmann
,
E.
, 1987, “
Dynamic Reflection Interference Contrast Microscopy: A New Method to Study Surface Excitations of Cells and to Measure Membrane Bending Elastic Moduli
,”
J. Phys. II
1155-4312,
48
, pp.
2139
2151
.
5.
Lee
,
J. C.
, and
Discher
,
D. E.
, 2001, “
Deformation-Enhanced Fluctuations in the Red Cell Skeleton With Theoretical Relations to Elasticity, Connectivity, and Spectri Unfolding
,”
Biophys. J.
0006-3495,
81
, pp.
3178
3192
.
6.
Strey
,
H.
,
Peterson
,
M.
, and
Sackmann
,
E.
, 1995, “
Measurement of Erythrocyte Membrane Elasticity by Flicker Eigenmode Decomposition
,”
Biophys. J.
0006-3495,
69
, pp.
478
488
.
7.
Yoon
,
Y. -Z.
,
Hong
,
H.
,
Brown
,
A.
,
Kim
,
D. C.
,
Kang
,
D. J.
,
Lew
,
V. L.
, and
Cicuta
,
P.
, 2009, “
Flickering Analysis of Erythrocyte Mechanical Properties: Dependence on Oxygenation Level, Cell Shape, and Hydration Level
,”
Biophys. J.
0006-3495,
97
, pp.
1606
1615
.
8.
Marcelli
,
G.
,
Parker
,
K. H.
, and
Winlove
,
C. P.
, 2005, “
Thermal Fluctuations of Red Blood Cell Membrane via a Constant-Area Particle-Dynamics Model
,”
Biophys. J.
0006-3495,
89
, pp.
2473
2480
.
9.
Lim
,
G.
,
Wortis
,
M.
, and
,
R.
, 2002, “
Stomatocyte-Discocyte-Echinocyte Sequence of the Human Red Blood Cell: Evidence for the Bilayer-Couple Hypothesis From Membrane Mechanics
,”
0027-8424,
99
, pp.
16766
16769
.
10.
Hénon
,
S.
,
Lenormand
,
G.
,
Richert
,
A.
, and
Gallet
,
F.
, 1999, “
A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers
,”
Biophys. J.
0006-3495,
76
, pp.
1145
1151
.
11.
Dao
,
M.
,
Lim
,
C. T.
, and
Suresh
,
S.
, 2003, “
Mechanics of the Human Red Blood Cell Deformed by Optical Tweezers
,”
J. Mech. Phys. Solids
0022-5096,
51
, pp.
2259
2280
.
12.
Sleep
,
J.
,
Wilson
,
D.
,
Simmons
,
R.
, and
Gratzer
,
W.
, 1999, “
Elasticity of the Red Cell Membrane and Its Relation to Hemolytic Disorders: An Optical Tweezers Study
,”
Biophys. J.
0006-3495,
77
, pp.
3085
3095
.
13.
Parker
,
K. H.
, and
Winlove
,
C. P.
, 1999, “
The Deformation of Spherical Vesicles With Permeable, Constant-Area Membranes: Application to the Red Blood Cell
,”
Biophys. J.
0006-3495,
77
, pp.
3096
3107
.
14.
Pamplona
,
D. C.
, and
,
C. R.
, 1993, “
The Mechanics of Axially Symmetric Liposomes
,”
ASME J. Biomech. Eng.
0148-0731,
115
, pp.
149
159
.
15.
Guck
,
J.
,
Ananthakrishnan
,
R.
,
Mahmood
,
H.
,
Moon
,
T. J.
,
Cunningham
,
C. C.
, and
Käs
,
J.
, 2001, “
The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells
,”
Biophys. J.
0006-3495,
81
, pp.
767
784
.
16.
Neuman
,
K. C.
, and
Block
,
S. M.
, 2004, “
Optical Trapping
,”
Rev. Sci. Instrum.
0034-6748,
75
, pp.
2787
2809
.
17.
Heinrich
,
V.
,
Wong
,
W. P.
,
Halvorsen
,
K.
, and
Evans
,
E.
, 2008, “
Imaging Biomolecular Interactions by Fast Three-Dimensional Tracking of Laser-Confined Carrier Particles
,”
Langmuir
0743-7463,
24
, pp.
1194
1203
.
18.
Neuman
,
K. C.
,
Abbondanzieri
,
E. A.
, and
Block
,
S. M.
, 2005, “
Measurement of the Effective Focal Shift in an Optical Trap
,”
Opt. Lett.
0146-9592,
30
, pp.
1318
1320
.
19.
Heinrich
,
V.
,
Ritchie
,
K.
,
Mohandas
,
N.
, and
Evans
,
E.
, 2001, “
Elastic Thickness Compressibility of the Red Cell Membrane
,”
Biophys. J.
0006-3495,
81
, pp.
1452
1463
.
20.
Evans
,
E. A.
, 1983, “
Bending Elastic Modulus of Red Blood Cell Membrane Derived From Buckling Instability in Micropipet Aspiration Tests
,”
Biophys. J.
0006-3495,
43
, pp.
27
30
.
21.
Scheffer
,
L.
,
Bitler
,
A.
,
Ben-Jacob
,
E.
, and
Korenstein
,
R.
, 2001, “
Atomic Force Pulling: Probing the Local Elasticity of the Cell Membrane
,”
Eur. Biophys. J.
0175-7571,
30
, pp.
83
90
.
22.
Waugh
,
R. E.
,
Song
,
J.
,
Svetina
,
S.
, and
Žekš
,
B.
, 1992, “
Local and Nonlocal Curvature Elasticity in Bilayer Membranes by Tether Formation From Lecithin Vesicles
,”
Biophys. J.
0006-3495,
61
, pp.
974
982
.
23.
Peterson
,
M. A.
, 1992, “
Linear Response of the Human Erythrocyte to Mechanical Stress
,”
Phys. Rev. A
1050-2947,
45
, pp.
4116
4131
.
24.
Mirijanian
,
D. T.
, and
Voth
,
G. A.
, 2008, “
Unique Elastic Properties of the Spectrin Tetramer as Revealed by Multiscale Coarse-Grained Modeling
,”
0027-8424,
105
, pp.
1204
1208
.
25.
Flügge
,
W.
, 1973,
Stresses in Shells
,
Springer
,
New York
.
26.
Schumaker
,
L. L.
, 1981,
Spline Function: Basic Theory
,
Wiley
,
New York
.
27.
Mesquita
,
L. G.
,
Agero
,
U.
, and
Mesquita
,
O. N.
, 2006, “
Defocusing Microscopy: An Approach for Red Blood Cell Optics
,”
Appl. Phys. Lett.
0003-6951,
88
, p.
133901
.
28.
Engelhardt
,
H.
, and
Sackmann
,
E.
, 1988, “
On the Measurement of Shear Elastic Moduli and Viscosities of Erythrocyte Plasma Membranes by Transient Deformation in High Frequency Electric Fields
,”
Biophys. J.
0006-3495,
54
, pp.
495
508
.