Abstract

Pure lipid bilayers are frequently used to mimic membranes that enclose living cells. However, real biological membranes are highly heterogeneous and have a complex structure. The so-called Helfrich Hamiltonian is frequently used to characterize the mechanical behavior of such membranes. Thermal fluctuations and, in general, statistical mechanics are used to explain a variety of cellular behaviors, but are very difficult to carry out in the case heterogeneous membranes. We propose to use a homogenized Hamiltonian that accounts for the presence of proteins to simplify the statistical mechanics analysis of realistic biological membranes. We recognize that (i) the effective Hamiltonian structure itself may be different from what is used for a homogeneous lipid bilayer and (ii) experimental evidence indicates that rigid proteins may introduce both stiffening and softening in the membrane. We consider generalized boundary conditions at the protein–lipid interface within the Helfrich Hamiltonian as a simple route to capture the protein membrane specificity and to account for both softening and stiffening due to rigid proteins. We postulate that real biological membranes require an effective elastic energy form that is far more complex than what is conventionally used and also propose to add a nonlocal elastic energy functional. The new augmented Helfrich Hamiltonian, in a mean-field setting, accounts for the presence of proteins by capturing their short- and long-range effects. Finally, by using the developed effective field theory, we present statistical mechanics results that illustrate the effect of proteins on the interaction between fluctuating membranes.

References

1.
Cooper
,
G. M.
,
Hausman
,
R. E.
, and
Hausman
,
R. E.
,
2007
,
The Cell: A Molecular Approach
, Vol.
4
,
ASM Press
,
Washington, DC
.
2.
Helfrich
,
W. S. R. M.
, and
Servuss
,
R.-M.
,
1984
, “
Undulations, Steric Interaction and Cohesion of Fluid Membranes
,”
Il Nuovo Cimento D
,
3
(
1
), pp.
137
151
.
3.
Janke
,
W.
, and
Kleinert
,
H.
,
1986
, “
Fluctuation Pressure of Membrane Between Walls
,”
Phys. Lett. A.
,
117
(
7
), pp.
353
357
.
4.
Freund
,
L. B.
,
2013
, “
Entropic Pressure Between Biomembranes in a Periodic Stack Due to Thermal Fluctuations
,”
Proc. Natl. Acad. Sci. USA
,
110
(
6
), pp.
2047
2051
.
5.
Sharma
,
P.
,
2013
, “
Entropic Force Between Membranes Reexamined
,”
Proc. Natl. Acad. Sci. USA
,
110
(
6
), pp.
1976
1977
.
6.
Lipowsky
,
R.
, and
Seifert
,
U.
,
1991
, “
Adhesion of Vesicles and Membranes
,”
Mol. Cryst. Liq. Cryst.
,
202
(
1
), pp.
17
25
.
7.
Evans
,
E.
,
1991
, “
Entropy-driven Tension in Vesicle Membranes and Unbinding of Adherent Vesicles
,”
Langmuir
,
7
(
9
), pp.
1900
1908
.
8.
Rao
,
M.
,
2001
, “
Active Fusion and Fission Processes on a Fluid Membrane
,”
Phys. Rev. Lett.
,
87
(
12
), p.
128101
.
9.
Shillcock
,
J. C.
, and
Lipowsky
,
R.
,
2006
, “
The Computational Route From Bilayer Membranes to Vesicle Fusion
,”
J. Phys.: Condens. Matter.
,
18
(
28
), p.
S1191
.
10.
Deserno
,
M.
,
2005
, “
The Influence of Thermal Fluctuations on the Bending Rigidity of Fluid Membranes
,”
Max-Planck-Institut für Polymerforschung, Ackermannweg
,
10
, p.
55128
.
11.
Hanlumyuang
,
Y.
,
Liu
,
L.
, and
Sharma
,
P.
,
2014
, “
Revisiting the Entropic Force Between Fluctuating Biological Membranes
,”
J. Mech. Phys. Solids.
,
63
, pp.
179
186
.
12.
Mozaffari
,
K.
,
Ahmadpoor
,
F.
, and
Sharma
,
P.
,
2021
, “
Flexoelectricity and the Entropic Force Between Fluctuating Fluid Membranes
,”
Math. Mech. Solids
,
26
(
12
), pp.
1760
1778
.
13.
Swain
,
P. S.
, and
Andelman
,
D.
,
1999
, “
The Influence of Substrate Structure on Membrane Adhesion
,”
Langmuir
,
15
(
26
), pp.
8902
8914
.
14.
Helfrich
,
W.
,
1973
, “
Elastic Properties of Lipid Bilayers: Theory and Possible Experiments
,”
Zeitschrift für Naturforschung c
,
28
(
11–12
), pp.
693
703
.
15.
Steigmann
,
D.
,
1999
, “
Fluid Films With Curvature Elasticity
,”
Arch. Rat. Mech. Anal.
,
150
(
2
), pp.
127
152
.
16.
Maleki
,
M.
,
Seguin
,
B.
, and
Fried
,
E.
,
2013
, “
Kinematics, Material Symmetry, and Energy Densities for Lipid Bilayers With Spontaneous Curvature
,”
Biomech. Model. Mechanobiol.
,
12
(
5
), pp.
997
1017
.
17.
Deserno
,
M.
,
2015
, “
Fluid Lipid Membranes: From Differential Geometry to Curvature Stresses
,”
Chem. Phys. Lipids.
,
185
, pp.
11
45
.
18.
Berdičevskij
,
V. L.
,
2009
,
Variational Principles of Continuum Mechanics: Applications
,
Springer
,
Berlin/Heidelberg
.
19.
Parton
,
V. Z.
, and
Kudriavtsev
,
B. A.
,
1993
,
Engineering Mechanics of Composite Structures
,
CRC Press, Boca Raton, FL
.
20.
Mindlin
,
R. D.
,
1963
, “
Microstructure in Linear Elasticity
,”
Columbia University: New York Depatment of Civil Engineering and Engineering Mechanics
, Technical Report No. 50.
21.
Bacca
,
M.
,
Bigoni
,
D.
,
Dal Corso
,
F.
, and
Veber
,
D.
,
2013
, “
Mindlin Second-Gradient Elastic Properties From Dilute Two-Phase Cauchy-Elastic Composites. Part I: Closed Form Expression for the Effective Higher-Order Constitutive Tensor
,”
Int. J. Solids. Struct.
,
50
(
24
), pp.
4010
4019
.
22.
Bacca
,
M.
,
Bigoni
,
D.
,
Dal Corso
,
F.
, and
Veber
,
D.
,
2013
, “
Mindlin Second-Gradient Elastic Properties From Dilute Two-Phase Cauchy-Elastic Composites Part II: Higher-Order Constitutive Properties and Application Cases
,”
Int. J. Solids. Struct.
,
50
(
24
), pp.
4020
4029
.
23.
Sharma
,
P.
, and
Dasgupta
,
A.
,
2002
, “
Average Elastic Fields and Scale-Dependent Overall Properties of Heterogeneous Micropolar Materials Containing Spherical and Cylindrical Inhomogeneities
,”
Phys. Rev. B
,
66
(
22
), p.
224110
.
24.
Sharma
,
P
,
2004
, “
Size-dependent Elastic Fields of Embedded Inclusions in Isotropic Chiral Solids
,”
Int. J. Solids. Struct.
,
41
(
22–23
), pp.
6317
6333
.
25.
Kozlov
,
M. M.
,
1992
, “
Energy of Nonhomogeneous Bending of Surfactant Monolayer. Persistence Length
,”
Langmuir
,
8
(
6
), pp.
1541
1547
.
26.
Kralj-Iglič
,
V.
,
Heinrich
,
V.
,
Svetina
,
S.
, and
Žekš
,
B.
,
1999
, “
Free Energy of Closed Membrane With Anisotropic Inclusions
,”
,
10
(
1
), pp.
5
8
.
27.
Vyas
,
P.
,
Sunil Kumar
,
P. B.
, and
Lal Das
,
S.
,
2022
, “
Sorting of Proteins With Shape and Curvature Anisotropy on a Lipid Bilayer Tube
,”
Soft. Matter.
,
18
(
8
), pp.
1653
1665
.
28.
Dimova
,
R.
,
2014
, “
Recent Developments in the Field of Bending Rigidity Measurements on Membranes
,”
Adv. Colloid. Interface. Sci.
,
208
, pp.
225
234
.
29.
Bouvrais
,
H.
,
Méléard
,
P.
,
Pott
,
T.
,
Jensen
,
K. J.
,
Brask
,
J.
, and
Ipsen
,
J. H.
,
2008
, “
Softening of Popc Membranes by Magainin
,”
Biophys. Chem.
,
137
(
1
), pp.
7
12
.
30.
Vitkova
,
V.
,
Méléard
,
P.
,
Pott
,
T.
, and
Bivas
,
I.
,
2006
, “
Alamethicin Influence on the Membrane Bending Elasticity
,”
Eur. Biophys. J.
,
35
(
3
), pp.
281
286
.
31.
Häckl
,
W.
,
Seifert
,
U.
, and
Sackmann
,
E.
,
1997
, “
Effects of Fully and Partially Solubilized Amphiphiles on Bilayer Bending Stiffness and Temperature Dependence of the Effective Tension of Giant Vesicles
,”
J. de Physique II
,
7
(
8
), pp.
1141
1157
.
32.
Ratanabanangkoon
,
P.
,
Gropper
,
M.
,
Merkel
,
R.
,
Sackmann
,
E.
, and
Gast
,
A. P.
,
2003
, “
Mechanics of Streptavidin-Coated Giant Lipid Bilayer Vesicles: A Micropipet Study
,”
Langmuir
,
19
(
4
), pp.
1054
1062
.
33.
Shchelokovskyy
,
P.
,
Tristram-Nagle
,
S.
, and
Dimova
,
R.
,
2011
, “
Effect of the HIV-1 Fusion Peptide on the Mechanical Properties and Leaflet Coupling of Lipid Bilayers
,”
New. J. Phys.
,
13
(
2
), p.
025004
.
34.
Tristram-Nagle
,
S.
,
Chan
,
R.
,
Kooijman
,
E.
,
Uppamoochikkal
,
P.
,
Qiang
,
W.
,
Weliky
,
D. P.
, and
Nagle
,
J. F.
,
2010
, “
Hiv Fusion Peptide Penetrates, Disorders, and Softens T-Cell Membrane Mimics
,”
J. Mol. Biol.
,
402
(
1
), pp.
139
153
.
35.
Pan
,
J.
,
Tieleman
,
D. P.
,
Nagle
,
J. F.
,
Kučerka
,
N.
, and
Tristram-Nagle
,
S.
,
2009
, “
Alamethicin in Lipid Bilayers: Combined Use of X-ray Scattering and Md Simulations
,”
Biochimica et Biophysica Acta (BBA)-Biomembranes
,
1788
(
6
), pp.
1387
1397
.
36.
Pabst
,
G.
,
Danner
,
S.
,
Podgornik
,
R.
, and
Katsaras
,
J.
,
2007
, “
Entropy-Driven Softening of Fluid Lipid Bilayers by Alamethicin
,”
Langmuir
,
23
(
23
), pp.
11705
11711
.
37.
Pott
,
T.
,
Gerbeaud
,
C.
,
Barbier
,
N.
, and
Méléard
,
P.
,
2015
, “
Melittin Modifies Bending Elasticity in an Unexpected Way
,”
Chem. Phys. Lipids.
,
185
, pp.
99
108
.
38.
Mills
,
J. P.
,
Diez-Silva
,
M.
,
Quinn
,
D. J.
,
Dao
,
M.
,
Lang
,
M. J.
,
Tan
,
K. S. W.
,
Lim
,
C. T.
,
Milon
,
G.
,
David
,
P. H.
,
Mercereau-Puijalon
,
O.
, and
Bonnefoy
,
S.
,
2007
, “
Effect of Plasmodial Resa Protein on Deformability of Human Red Blood Cells Harboring Plasmodium Falciparum
,”
Proc. Natl. Acad. Sci. USA
,
104
(
22
), pp.
9213
9217
.
39.
Agrawal
,
H.
,
Liu
,
L.
, and
Sharma
,
P.
,
2016
, “
Revisiting the Curvature-Mediated Interactions Between Proteins in Biological Membranes
,”
Soft. Matter.
,
12
(
43
), pp.
8907
8918
.
40.
Agrawal
,
H.
,
Zelisko
,
M.
,
Liu
,
L.
, and
Sharma
,
P.
,
2016
, “
Rigid Proteins and Softening of Biological Membranes—With Application to HIV-Induced Cell Membrane Softening
,”
Sci. Rep.
,
6
(
1
), pp.
1
12
.
41.
Gurtin
,
M. E.
,
Weissmüller
,
J.
, and
Larche
,
F.
,
1998
, “
A General Theory of Curved Deformable Interfaces in Solids at Equilibrium
,”
Phil. Mag. A
,
78
(
5
), pp.
1093
1109
.
42.
Gurtin
,
M. E.
, and
Ian Murdoch
,
A.
,
1975
, “
A Continuum Theory of Elastic Material Surfaces
,”
Arch. Rat. Mech. Anal.
,
57
(
4
), pp.
291
323
.
43.
Gurtin
,
M. E.
, and
Ian Murdoch
,
A.
,
1978
, “
Surface Stress in Solids
,”
Int. J. Solids. Struct.
,
14
(
6
), pp.
431
440
.
44.
Mozaffari
,
K.
,
Yang
,
S.
, and
Sharma
,
P.
,
2020
,
Surface Energy and Nanoscale Mechanics
,
A.
Wanda
and
Y.
Sidney
, eds.,
Springer International Publishing
, pp.
1949
1974
.
45.
Krichen
,
S.
,
Liu
,
L.
, and
Sharma
,
P.
,
2019
, “
Liquid Inclusions in Soft Materials: Capillary Effect, Mechanical Stiffening and Enhanced Electromechanical Response
,”
J. Mech. Phys. Solids.
,
127
, pp.
332
357
.
46.
Nelson
,
P.
,
2004
,
Biological Physics
,
WH Freeman
,
New York
.
47.
Phillips
,
R.
,
Kondev
,
J.
,
Theriot
,
J.
,
Garcia
,
H. G.
, and
Orme
,
N.
,
2012
,
Physical Biology of the Cell
,
Garland Science
,
New York
.
48.
Seifert
,
U.
,
1997
, “
Configurations of Fluid Membranes and Vesicles
,”
Adv. Phys.
,
46
(
1
), pp.
13
137
.
49.
Safran
,
S. A.
,
2018
,
Statistical Thermodynamics of Surfaces, Interfaces, and Membranes
,
CRC Press
,
Boca Raton, FL
.
50.
Ahmadpoor
,
F.
, and
Sharma
,
P.
,
2016
, “
Thermal Fluctuations of Vesicles and Nonlinear Curvature Elasticity—Implications for Size-Dependent Renormalized Bending Rigidity and Vesicle Size Distribution
,”
Soft. Matter.
,
12
(
9
), pp.
2523
2536
.
51.
Ahmadpoor
,
F.
, and
Sharma
,
P.
,
2017
, “
A Perspective on the Statistical Mechanics of 2D Materials
,”
Extreme Mech. Lett.
,
14
, pp.
38
43
.
52.
Ahmadpoor
,
F.
,
Wang
,
P.
,
Huang
,
R.
, and
Sharma
,
P.
,
2017
, “
Thermal Fluctuations and Effective Bending Stiffness of Elastic Thin Sheets and Graphene: A Nonlinear Analysis
,”
J. Mech. Phys. Solids.
,
107
, pp.
294
319
.
53.
Lipowsky
,
R.
,
1991
, “
The Conformation of Membranes
,”
Nature
,
349
(
6309
), pp.
475
481
.
54.
Nelson
,
D.
,
Piran
,
T.
, and
Weinberg
,
S.
,
2004
,
Statistical Mechanics of Membranes and Surfaces
,
World Scientific, Singapore
.
55.
Podgornik
,
R.
, and
Parsegian
,
V. A.
,
1992
, “
Thermal-Mechanical Fluctuations of Fluid Membranes in Confined Geometries: The Case of Soft Confinement
,”
Langmuir
,
8
(
2
), pp.
557
562
.
56.
Helfrich
,
W.
,
1978
, “
Steric Interaction of Fluid Membranes in Multilayer Systems
,”
Zeitschrift für Naturforschung A
,
33
(
3
), pp.
305
315
.
57.
Farago
,
O.
,
2008
, “
Membrane Fluctuations Near a Plane Rigid Surface
,”
Phys. Rev. E
,
78
(
5
), p.
051919
.
58.
Janke
,
W.
, and
Kleinert
,
H.
,
1987
, “
Fluctuation Pressure of a Stack of Membranes
,”
Phys. Rev. Lett.
,
58
(
2
), p.
144
.
59.
Lu
,
B. S.
, and
Podgornik
,
R.
,
2015
, “
Effective Interactions Between Fluid Membranes
,”
Phys. Rev. E
,
92
(
2
), p.
022112
.
60.
Liang
,
X.
, and
Purohit
,
P. K.
,
2016
, “
A Fluctuating Elastic Plate and a Cell Model for Lipid Membranes
,”
J. Mech. Phys. Solids.
,
90
, pp.
29
44
.
61.
Schneider
,
M. B.
,
Jenkins
,
J. T.
, and
Webb
,
W. W.
,
1984
, “
Thermal Fluctuations of Large Quasi-Spherical Bimolecular Phospholipid Vesicles
,”
J. de Physique
,
45
(
9
), pp.
1457
1472
.
62.
Bachmann
,
M.
,
Kleinert
,
H.
, and
Pelster
,
A.
,
1999
, “
Strong-Coupling Calculation of Fluctuation Pressure of a Membrane Between Walls
,”
Phys. Lett. A.
,
261
(
3–4
), pp.
127
133
.
63.
Gompper
,
G.
, and
Kroll
,
D. M.
,
1989
, “
Steric Interactions in Multimembrane Systems: A Monte Carlo Study
,”
EPL (Europhysics Letters)
,
9
(
1
), pp.
59
.
64.
Martin
,
H. S.
,
Podolsky
,
K. A.
, and
Devaraj
,
N. K.
,
2021
, “
Probing the Role of Chirality in Phospholipid Membranes
,”
ChemBioChem
,
22
(
22
), pp.
3148
3157
.
65.
Torbati
,
M.
,
Mozaffari
,
K.
,
Liu
,
L.
, and
Sharma
,
P.
,
2022
, “
Coupling of Mechanical Deformation and Electromagnetic Fields in Biological Cells
,”
Rev. Mod. Phys.
,
94
, p.
025003
.
You do not currently have access to this content.