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					                                  Literature Review
                                         For
        Mechanical Properties of Red Blood Cell Membrane

                               Yun Ge (yge@colorado.edu)

The red blood cell’s shape is encoded in the mechanical properties of the membrane [5].
Under physiological conditions, a normal red blood cell assumes a biconcave discoid
(discocyte) shape  8 m in diameter. The membrane material is a composite design based
on a fluid lipid bilayer supported by scaffolding of interconnected proteins and studded by
a superficial forest of peptidoglycans. From the electron microscopy and biochemical
analysis, a detailed model has been constructed of a triangulated network of flexible
spectirn proteins plus its complex protein linkage to the bilayer (i.e. Figure 3). There are
rich description of structure, composition, and mechanics.

For thin materials, deformation and rate of deformation are quantitated by variables that
represent a sequence of simple shape changes for local regions (elements) of the surface:
Area dilation or condensation,   A / A0 ; in-plane extension, (i.e. surface shear),
  L / L0 at constant surface density; and bending or curvature change, C  (1/ R) ,
without change in rectangular shape.[2],[3] The first-order ideal elastic (conservative)
relations for the surface-isotropic materials (i.e. figure 4):
                    K ;   ( 1 + 2 )/2
                  s   ( 2   2 ) / 2  2 s ;  s  ( 1   2 )/2   (1)
                 M  B(1/ R1  1/ R2 )
When the viscous-lipid (non-conservative) response of a bilayer is considered, the relations
can be represented by similar proportionalities between forces-moment resultants and rates
of deformation:
                                 ln(1   )
                          = a
                                      t
                                                    d
                          s  2 sVs ;      Vs  2( s ) /( 2   2 ); (2)
                                                     dt
                                  C
                         M  c
                                   t
The (  a , s , c ) are the surface viscosities for dilation, shear and bending, respectively.
These viscous properties are caused by the acyl chains which are liquid in bilayer.

Since the dilation and bending deformations are usually elastic, viscosities for these kinds
of deformations may be estimated form the elastic modulus multiplies by the relaxation
time [3]. And the shear viscosity is so large that the mechanical transition of the bilayer
from solid to liquid behavior is modeled by an ideal plastic relation. Hence, it is assumed
that the solid below a yield threshold and perfect liquid above the yield [3].

Therefore the red blood cell shape, behavior and deformability always are consistently
accounted for by a model for the elastic properties of the membrane that includes the
elasticity of the membrane skeleton in dilation and shear, and the local and non-local
resistance of the bilayer to bending [4].

Sensitive micro-pipette methods have been used to measure the relation between tension
and the projected surface area in fluid membranes of vesicles. Diacyl-lipid surfactants have
such low solubility in the aqueous environment that the membrane surface remains closed
to exchange of amphiphiles with the aqueous environment. Thus, changes in length of the
vesicle projection in the pipette represent expansions of the projected area due to reduction
of membrane undulations plus a direct dilation in area per molecule [8].
                                              PR p
The time-average membrane tension  
                                          2(1  R p / Ro )
and R p and Ro are the pipet and exterior segment radii,respectively.
The experimental procedure is to initially determine the stable projection length the tube at the
lowest suction pressure.
                       1
The areal strain   [( R p / Ro ) 2  ( R p / R0 )3 ]L / R p
                       2
                                                              kT
Area dilation as a function of tension given by                  ln(1  c A)   / K a   (3)
                                                            8 kc
kc is the elastic moduli for bending,while K ais the elasic moduli for the direct area expension.
coefficient c =1/ 2 in the plane-wave approximation;c = 1/24 in the quasispherical approach.

Examination of the results for the plot of tension versus area dilatation defines the
‘apparent’ compressibility of the bilayer: in low-tension regime (<0.5dyn/cm), the
projected area increase logarithmically with tension as shape fluctuations become
progressively restricted, and the slope of ln(tension) versus area dilation yields the elastic
bending modulus of the membrane; in the high-tension regime, the projected area crosses
over to vary linearly with tension due to expansion of area per molecule.(i.e. Figure 5 [8])
The result demonstrates that the condensed-fluid membranes behave nearly as 2-D
generalizations of linear, flexible polymers [9](i.e. Figure 6 [12]).

One former common model is simplified the cell as a hyperelastic effective continuum
shell [11]. There are only one material moludus concerned: in-plane shear modulus  and
two structure properties: bending stiffness B and membrane thickness d. Since the scalling
of bending rigidity with the thickness of the fluid membrane is nearly quadratic [7]. It is
measured that the thickness of the bilayer is ~4nm and this thick is extended to ~10nm by
the exterior glycocalyx [6]. There are only two independent parameters in this model. It is
very simple, but since it mixes the contribution of the skeleton and bilayer for the
deformation, it can’t explain the asymmetric deformation of stomatocyte-discocyte-
echinocyte sequence of the red cell.

Compared to the in-plane characteristics, little is known about how structural components
are arrayed and interact in the third dimension normal to the membrane [6].

The interaction between the bilayer might play an important role in asymmetric
deformation of cell. It is said bending elastic moduli are predicted to be proportional to the
molecular area compressibility moduli. The ratio of bending to corresponding area elastic
moduli (kc/Ka)~=d2/b, also depends on the constant b which depends on the distribution of
lateral pressure across the bilayer as derived from the moment of the stress deviation. [1],[2]
There are some scientists want to use a single parameter, the relax area difference effect
between the outer and inner leaflets of the bilayer. The foundation of the work is the
bilayer-couple hypothesis which tries to interpret on the basis of energy minimization in
response to deformation. [5]

Furthermore, a developing theory is to combine the skeleton energy contribution to the
bilayer-couple energy [4]. It seems the system favor to trading an increasing in dilation
energy for a larger decrease in shear energy. Because the red cell skeleton is attached to the
lipid bilayer, there is a constraint on the total skeleton surface area requiring that when
there is condensation of the skeleton is rejoins where shear deformation is high, dilation
must occur in regions where shear deformation is low [4]. However, this model does not
concern the special polymer chain structure of the skeleton.

The skeleton network of peripheral proteins plays an important role in determination of the
durability and deformability of dilation and shear, and it has a refinement structure of
hexagonal lattice. There is some work that focuses on using a polymer chain model to
simulation the large deformation of the cell.[9],[10]

The distance between the spectrin network and the bilayer is ~10nm. The junctional
complexes, which represent the foundation for nodes of the spectrin network, separate the
network from the lipid interface by~10nm. Simulations of spectrin network with different
numbers of segments per chain have indicated that the chains extend ~20-30nm from the
junctional complexes in to the cytophasm. Hence, a red cell membrane is expected to
span~40-50nm in thickness and, if squeezed, to exhibit a structural thickness of ~20-25nm
(i.e. Figure 1 and Figure 3).However other experiment like using the optical interferometry
method of Newton-ring interference pattern by Heinrich [6] found that the apparent material
thickness is~90nm per membrane under force~1-2pN, and ~40nm with further
impingement and stiffened on approach to~50nm under ~100pN. It means that the spectrin
network not only influent the in-plane dilation properties of the membrane, but also
influent the third dimension structure of the membrane. Hence the bending moduli might
also be changed by the structure of the skeleton that affects the thickness of the membrane.
The scientist also tried to explain the phenomena with correlation of the spectrin network
modeled as ideal polymer loops. It led to the statistical picture of spectrin as a freely
jointed chain of 20-40 segments. (i.e. 26 segments in a contour length of 200nm yield a
mean thickness for the spectrin network of ~15nm). This ideal polymer chain networks can
explain this phenomena in a certain extent.

How to model the cooperation of the skeleton and the bilayer is still a problem. The
polymer network can be directly modeled as a in-plane Langevin chain network [12],
because the protein c has polymer chain’s mechanical properties and that seems more
closer to the real nature properties than the Hookean spring. One of the exist model of the
lipid-protein interaction is the ‘mattress model’(i.e. Figure 4[2]), which is anticipate that the
acyl chains next to the integral proteins would be ‘immobilized’ because of the relative
rigidity of the proteins, leading to much larger orientational order parameters than in the
absence of proteins [2].



[1] Mohandas, Narla; Evans, Evan: Mechanical properties of the red cell membrane in
relation to molecular structures and genetic defects. Annu. Rec. Biophys. Struct. 1994.
23:787-818
[2] Bloom, Myer; Evans, Evan; Mouritsen, Oleg.: Physical properties of the fluid lipid-
bilayer component of cell membranes: a perspective. Quarterly Reviews of Biophysics 24,
3, 1991, pp. 293-397
[3] Evans, Evan; Needham, David.: Physical properties of surfactant bilayer membranes:
thermal transitions, elasticity, rigidity, cohesion and colloidal interactions. J. Phys.
Chem,1987, 91,4219-4228
[4] Svetina, Sasa; Kuzman, Drago; Waugh, Richard E.; Ziherl, Primoz; Zeks,Bostjan: The
cooperative role of membrane skeleton and bilayer in the mechanical behavior of red blood
cells. BIoelectrochemistry 62 (2004)107-113
[5] Lim, G.H.W.; Wortis, Michael; Mukhopadhyay, Ranjan: Stomatocyte-discocyte-
echinocyte sequence of the human red blood cell: Evidence for the bilayer-couple
hypothesis from membrane mechanics. Proc. Natl. Acad. Sci. U.S.A. 99(2002) 16766-
16769
[6] Heinrich, Volkmar; Ritchie, Ken; Mohandas, Narla; Evans, Evan: Elastic thickness
compressibility of the red cell membrane. Biophysical Journal Volume 81 September 2001,
pp. 1452-1463
[7] Bermudezh, H.; Hammer, D.A.; Disher, D.E.: Effect of bilayer thickness on membrane
bending rigidity. Langmuir 2004, 20.540-543
[8] Rawicz, W.; Evans, Evan: Entropy-driven tension and bending elasticity in condensed-
fluid membranes. Physical Review Letters, 23 April 1990, Volume 64, Number 17
[9] Boey, Seng K.; Boal, David H.; Disher, Dennis E.: Simulation of the erythrocyte
cytoskeleton at large deformation I. Microscopic Models. Biophysical Journal Volume 75
September 1998 1573-1583
[10] Disher, Dennis E.; Boal, David H.; Boey, Seng K.: Simulation of the erythrocyte
cytoskeleton at large deformation I. Micropipette Aspiration. Biophysical Journal Volume
75 September 1998 1584-1597
[11] Dao, M.; Lim, C.T.; Suresh, S.: Mechanics of the human red blood cell deformed by
optical tweezers. Journal of the Mechanics and Physics of Solids 51, 2003, 2259-2280
[12] Arruda, Ellen M. and Boyce, Mary: A three-dimensional constitutive model for the
large stretch behavior of rubber elastic materials. of the Mechanics and Physics of Solids,
Volume 41, No. 2. pp. 389-412,1993




                                                                                    [5]
                                                                       [2]                               [1]
Figure 2 Simple Model of interaction of integral protein and bilayer         Figure 3 spectrin network




                                           Figure 4[1],[3]
Figure 5 [8]




Figure 6 [12]

				
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