Elasticities and stabilities: lipid membranes vs cell membranes

0.17 MB·English

by  Z. C. Tu

Elasticities and stabilities: lipid membranes vs
5
cell membranes
0
0
2
Z. C. Tu1, R. An2, and Z. C. Ou-Yang1
n
1Institute of Theoretical Physics, Chinese Academy of Sciences
a
J P. O. Box 2735 Beijing 100080, China

3 2Department of Mathematics

Hong Kong University of Science and Technology
]
T
Clear Water Bay, Kowloon, Hong Kong
O
.
o
i ABSTRACT
b
-
q
A cell membrane can be simply regarded as composite material consisting of
[
lipidbilayer,membranecytoskeletonbeneathlipidbilayer,andproteinsembed-

1 dedinlipidbilayerandlinkedwithmembranecytoskeletonifoneonlyconcerns
v
its mechanical properties. In this Chapter, above all, the authors give a brief
1
introduction to some important work on mechanical properties of lipid bilay-
0
0 ers following Helfrich’s seminal work on spontaneous curvature energy of lipid
1 bilayers. Next, the entropy of a polymer confined in a curved surface and the
0 free energy of membrane cytoskeleton are obtained by scaling analysis. It is
5 foundthatthe freeenergyofcellmembraneshasthe formofthe in-planestrain
0
energy plus Helfrich’s curvature energy. The equations to describe equilibrium
/
o shapesandin-planestrainsofcellmembranesbyosmoticpressuresareobtained
i
b by taking the first order variation of the total free energy containing the elas-
- tic free energy, the surface tension energy and the term induced by osmotic
q
pressure. The stability of spherical cell membrane is discussed and the critical
:
v pressure is found to be much larger than that of spherical lipid bilayer without
i
X membranecytoskeleton. Lastly,theauthorstrytoextendthepresentstaticme-
chanical model of cell membranes to the cell structure dynamics by proposing
r
a a group of coupling equations involving tensegrity architecture of cytoskeleton,
fluid dynamics of cytoplasm and elasticities of cell membranes.
INTRODUCTION
Cells are the basic elements of living organisms, such as plants and animals.
Viewedfromstructure,aneukaryoticcellconsistsofcellmembrane,cytoplasm,
cytoskeleton, nucleus and so on. Cell membrane defines the boundary between
the living cell and its environment, including extracellular matrix and liquid
1
surroundings. Cytoplasm is a kind of viscous fluid. Cytoskeleton is a three-
dimensional structure composed of three classes of fibers [1]: microtubules (20
nm in diameter), elements built of polymers of the protein tubulin; microfila-
ments(7nmindiameter),builtofthe proteinactin;andintermediatefilaments
(10 nm in diameter), built of one or more rod-like protein subunits. Nucleus of
the eukaryotic cell is enclosed by membrane.
Membranes consist of lipids, proteins and carbohydrates etc. Lipids and
proteinsaredominantcomponentsofmembranes. One ofthe principaltypesof
lipids in membranes is phospholipid. A phospholipid molecule has a polar hy-
drophilic headgroupandtwo hydrophobic hydrocarbontails. In physicalpoint
of view, a lipid molecule can be simply regardedas an amphipathic rod. When
a quantity of lipid molecules disperse in water, they will assemble themselves
into a bilayer in which the hydrophilic heads shield the hydrophobic tails from
the water surroundings because of the hydrophobic forces.
There are many simplified models for cell membranes in history [2]. Among
them, the widely accepted one is the fluid mosaic model proposed by Singer
and Nicolson in 1972 [3]. In their model, cell membrane is consideredas a lipid
bilayer where the lipid molecules can move freely in the membrane surface like
fluid, while the proteins are embedded in the lipid bilayer. Some proteins are
called integral membrane proteins because they traverse entirely in the lipid
bilayer and play the role of information and matter communications with both
theinsideofthecellanditsouterenvironment. Theothersarecalledperipheral
membrane proteins because they are partially embedded in the bilayer and
accomplish the other biological functions. Beneath the lipid membrane, the
membrane cytoskeleton, a network of proteins, links with the proteins in lipid
membrane.
Thefirststeptostudytheelasticityofcellmembraneistostudylipidbilayer.
Usually, the thickness of lipid bilayer is about 4 nanometers which is much less
thanthescaleofthewholelipidbilayer(aboutseveralmicrometers). Therefore,
it is reasonable to describe the lipid bilayer by a mathematical surface. In
1973,Helfrich[4]recognizedthat the lipid bilayerwasjust like a nematic liquid
crystal at room temperature. Based on the elastic theory of liquid crystal [5],
he proposed the curvature energy per unit area of the bilayer
f =(k /2)(2H+c )2, (1)
c c 0
where k is an elastic constant; and H, c are mean curvature and spontaneous
c 0
curvature of the membrane surface, respectively. In (1), Gaussian curvature K
is not written explicitly because only close bilayer is discussed in this chapter.
We can safely ignore the thermodynamic fluctuation of the curved bilayer at
the room temperature because of k 10 19J k T [6,7], where k is the
c − B B
≈ ≫
Boltzmann factor and T the room temperature. Based on Helfrich’s curvature
energy, we can express the free energy of a closed bilayer under the osmotic
pressure ∆p (the outer pressure minus the inner one) as:
= (f +µ)dA+∆p dV, (2)
H c
F
Z Z
2
where dA is the area element and V the volume enclosed by the closed bilayer.
µ is the surface tension of the bilayer. Based on above free energy, many re-
searchers studied the shapes of bilayers [8–10]. Especially, by taking the first
order variation of above free energy and doing the complicated calculations of
tensors, Ou-Yang et al. derived an equation to describe the equilibrium shape
of the bilayer as [11,12]:
∆p 2µH+k (2H +c )(2H2 c H 2K)+k 2(2H)=0. (3)
c 0 0 c
− − − ∇
This equation is now called the shape equation of closed membranes or gen-
eralized Laplace equation. They also obtained that the threshold pressure for
instability of spherical bilayer was ∆p k /R3, where R being the radius of
c c
∼
spherical bilayer.
Usingtheshapeequation(3)ofclosedbilayers,Ou-Yangpredictedthatthere
wasalipidtoruswiththeratiooftworadiibeingexactly√2[13]. Hisprediction
wassoonconfirmedbytheexperiments[14–16]. Otherwise,researchersobtained
ananalyticalsolutionto Eq.(3)whichexplainedthe classicalproblem[17]—the
biconcave discoidal shape of normal red cells [18].
Recently, TuandOu-Yanghaveproposeda mathematicalschemeto discuss
the elasticities and stabilities of cell membranes with membrane cytoskeleton
andfoundthatthemembranecytoskeletonenhancesthestabilitiesofcellmem-
branes [19]. But they have omitted the effect of in-plane modes of membranes
on the stabilities. In this chapter, we will first retrospect to elasticities and
stabilities of lipid bilayers following Helfrich’s seminal work on spontaneous
curvature energy of lipid bilayers. Next, we will fully discuss the entropic elas-
ticity of membrane cytoskeleton,as well as the elasticities and stabilities of cell
membranes with membrane cytoskeleton. Last, we will expatiate on how to
construct the framework of cell structure dynamics involving tensegrity archi-
tecture of cell cytoskeleton, fluid dynamics of cytoplasm and elasticities of cell
membranes (with membrane cytoskeleton).
LIPID MEMBRANE
In this section, we will recur to some main results on elasticities and stabilities
oflipidbilayersby adoptingthe mathematicalschemeproposedinRef.[19–21].
We use a smooth and closed surface M in 3-dimensionalEuclid space E3 to
represent a membrane. As shown in Fig. 1, we can construct a right-hand or-
thonormalsystem e ,e ,e atanypointrinthesurfaceandcall r;e ,e ,e
1 2 3 1 2 3
{ } { }
a moving frame. The differential of the frame is denoted by
dr=ω e +ω e ,
1 1 2 2 (4)
de =ω e (i=1,2,3),
i ij j
(cid:26)
whereω ,ω andω = ω (i,j =1,2,3)are1-forms. Thestructureequations
1 2 ij ji
−
of the surface are
dω = ω ω ; (5)
1 12 2
∧
3
dω = ω ω ; (6)
2 21 1
∧
ω = aω +bω , ω =bω +cω ; (7)
13 1 2 23 1 2
dω = ω ω (i,j =1,2,3). (8)
ij ik kj
∧
Readers should notice that the operator “d” is an exterior differential operator
[19]in this chapter. The areaelement, mean curvature and Gaussiancurvature
are respectively expressed as [22]:
dA=ω ω , (9)
1 2
∧
H =(a+c)/2, (10)
K =ac b2. (11)
−
e
3
r
e
e 2
1
M
Figure1: SmoothandorientablesurfaceM. we canconstructaright-handor-
thonormalsystem e ,e ,e atanypointrinthesurfaceandcall r;e ,e ,e
1 2 3 1 2 3
{ } { }
a moving frame.
If M undergoes an infinitesimal deformation such that every point r of M
has a displacement δr, we obtain a new surface M = r r = r+δr . δr is
′ ′ ′
{ | }
called the variation of surface M and can be expressed as
δr=δ r+δ r+δ r, (12)
1 2 3
δ r=Ω e (i=1,2,3), (13)
i i i
where the repeated subindexes do not represent Einstein summation. Due to
the deformation of M, e ,e ,e also change. We denote the change as
1 2 3
δ e =Ω e , Ω = Ω . (14)
l i lij j lij lji
−
Using the commutativity between δ (i = 1,2,3) and d, we obtain the fun-
i
damentally variational identities of the move frame [19]:
δ ω =dΩ ω Ω , (15)
1 1 1 2 121
−
4
δ ω =Ω ω ω Ω , (16)
1 2 1 12 1 112
−
Ω =aΩ , Ω =bΩ ; (17)
113 1 123 1
δ ω =Ω ω ω Ω , (18)
2 1 2 21 2 221
−
δ ω =dΩ ω Ω , (19)
2 2 2 1 212
−
Ω =bΩ , Ω =cΩ ; (20)
213 2 223 2
δ ω =Ω ω ω Ω , (21)
3 1 3 31 2 321
−
δ ω =Ω ω ω Ω , (22)
3 2 3 32 1 312
−
dΩ =Ω ω +Ω ω ; (23)
3 313 1 323 2
δ ω =dΩ +Ω ω ω Ω . (24)
l ij lij lik kj ik lkj
−
Now we can take the first order variation of functional (2) by considering
above variational identities (15)–(24) and Stokes theorem. It is not hard to
obtain [19]:
δ = [∆p 2µH + k (2H +c )(2H2 c H 2K)
H c 0 0
F − − −
Z
+ k 2(2H)]Ω dA. (25)
c 3
∇
Thus the Euler-Lagrange equation corresponding to the functional reduces
H
F
to equation (3) because Ω is an arbitrary function. Equation(3) describes the
3
equilibrium shapes of lipid membranes.
Similarly, we can obtain the second order variation of functional (2):
δ2 = Ω 2Ω k (14H2+2Hc 4K c2/2) µ dA+k ( 2Ω )2dA
FH 3∇ 3 c 0− − 0 − c ∇ 3
Z Z
(cid:2) (cid:3)
+ k (2H +c ) (2HΩ ) Ω 2 Ω ˜Ω 4Ω ˜Ω dA
c 0 3 3 3 3 3 3
∇ ·∇ − ∇ ·∇ − ∇·∇
Z h i
+ Ω2 k (16H4 20H2K+4K2+Kc2)+2Kµ 2H∆p dA. (26)
3 c − 0 −
Z
(cid:2) (cid:3)
Thedetailedexpressionsofoperators , ¯, ˜, 2, ¯, ˜ inaboveequation
∇ ∇ ∇ ∇ ∇·∇ ∇·∇
can be found in Appendix D of Ref. [19].
Now,wediscussmechanicalstabilityofsphericallipidmembranewithradius
R. In this case, we have H = 1/R and K = 1/R2. Substituting them into
−
equation (3), we arrive at
∆pR2+2µR k c (2 c R)=0. (27)
c 0 0
− −
Using r = R(sinθcosφ,sinθsinφ,cosθ) to represent the point in the spherical
membrane,wehave ˜ = (1/R) , ˜ = (1/R) 2and 2 = 1 ∂ sinθ ∂ +
∇ − ∇ ∇·∇ − ∇ ∇ R2sinθ∂θ ∂θ
1 ∂2 . Under the condition (27), equation (26) is transformed into
R2sin2θ∂φ2 (cid:0) (cid:1)
π 2π
δ2 = (2c k /R+∆pR) sinθdθ dφΩ2
FH 0 c 3
Z0 Z0
5
π 2π
+ (k c R+2k +∆pR3/2) sinθdθ dφΩ 2Ω
c 0 c 3 3
∇
Z0 Z0
π 2π
+ k R2 sinθdθ dφ( 2Ω )2. (28)
c 3
∇
Z0 Z0
Expand Ω by the spherical harmonic functions [23]:
3
m=l
∞
Ω = a Y (θ,φ), a =( 1)ma . (29)
3 lm lm ∗lm − l,−m
l=0m= l
X X−
Ifconsidering∇2Ylm =−l(l+1)Ylm/R2and 0πsinθdθ 02πdφYl∗mYl′m′ =δmm′δll′,
we transform equation (28) into
R R
δ2 =(R/2) a 2[l(l+1) 2] 2k /R3[l(l+1) c R] ∆p . (30)
H lm c 0
F | | − { − − }
l,m
X
Obviously, δ2 is a positive definite form if ∆p<p (2k /R3)[l(l+1)
H l c
F ≡ −
c R] (l=2,3, ). Therefore, we can take the critical pressure as
0
···
∆p =min p =p =(2k /R3)(6 c R). (31)
c l 2 c 0
{ } −
If ∆p > ∆p , the spherical bilayer will be instable and inclined to transform
c
into the biconcave discoid shape.
Inthissection,wehavediscussedtheelasticityofandstabilityoflipidbilayer
under the pressure. But lipid bilayer is oversimplified model of cell membrane
whichcontainsthemembranecytoskeleton. Thatis,forcellmembrane,wemust
take into account the contribution of membrane cytoskeleton.
MEMBRANE CYTOSKELETON
In this section, we will discuss the contribution of membrane cytoskeleton to
the free energy of cell membrane. The membrane cytoskeleton is cross-linking
chain-like protein structure which can be thought of as a polymer membrane.
Now we will derive its free energy by analogywith the polymer membrane [24].
We take de Gennes’ convention [25] in the this section: the entropy S is a
dimensionlessquantityandBoltzmannfactork isimplicatedintemperatureT.
B
If we regard the protein in membrane cytoskeleton as Gaussian chains [34], its
rootofmeansquareend-to-enddistanceisR √Nb ,whereb isthesegment
0 0 0
∼
length of protein and N 1 is the number of segments. Assume that the
≫
principal radii of the membrane are much larger than R . If we denote the in-
0
planestraintensorby ǫwhichisassumedtobe asmallquantity,the entropyof
theproteinchainmustbethefunctionof2HR ,KR2,2J andQbecauseentropy
0 0
isa dimensionlessinvariableunder the transformationofcoordinates,whereH,
K, J = trǫ and Q = trǫ are the mean curvature, the Gaussian curvature, the
trace of strain tensor and the determinant of strain tensor, respectively. Thus
6
wecanexpanditasS A (2HR )+A (2HR )2+A KR2+B (2J)2+B Qup
∼ 1 0 2 0 3 0 2 3
to the second order terms, where A ,A ,A ,B ,B are constants. There is no
1 2 3 2 3
firstordertermof2J intheexpressionoftheentropybecauseweexpectthat ǫ
playsthesameroleasǫintheentropy. Itisusefultowritetheentropyinanoth−er
equivalentformS A R2(2H+c )2+A R2K+B (2J)2+B Q, where c is a
∼ 2 0 0 3 0 2 3 0
constantcalledspontaneouscurvatureandweexpect c R 1. Consequently,
0 0
| |≪
the elastic free energy per area of a membrane consisting of protein chains has
the following form
k k
f = (Mh)TS = d[(2J)2 νQ]+ c′(2H +c )2 k¯ K, (32)
0 ′
− 2 − 2 −
where h is thickness of the membrane, M the number of protein chains per
volume,and k , ν, k , k¯ are unknownuniversalconstants. Here we neglect the
d c′ ′
entanglement of proteins. We will show k =4MhT and ν =1 as follows.
d
Letusconsideranidealcase–theplanarmembranewiththehomogenousin-
planestrains. Inthiscase,H,K andc arevanishingforplanarmembranewith
0
symmetry between its two sides. On the one hand, equation (32) is simplified
as
k
f = d[(2J)2 νQ]. (33)
2 −
For homogenousstain ǫ, we can expressit by its components ǫ , ǫ andǫ =
11 22 12
ǫ = 0 in some orthonormal coordinate system so that 2J = ǫ +ǫ and
21 11 22
Q=ǫ ǫ .
11 22
On the other hand, this structure can be compared with the structure of
rubber. Intermsoftheelasticitytheoryofrubber[26],thedeformationenergyof
aplanarrubberperareacanbeexpressedasf =(MhT/2)[λ2+λ2+1/(λ2λ2)
r 1 2 1 2 −
3], where λ = 1+ǫ and λ = 1+ǫ are extensions. For small strains, it
1 11 2 22
is expanded to the lowest order terms as f 2MhT(ǫ2 + ǫ ǫ + ǫ2 ) =
r ∼ 11 11 22 22
2MhT[(2J)2 Q]. Thus we can obtain k =4MhT and ν =1 by comparing it
d
−
with equation (33).
In this section we do not discuss the mechanical property of membrane cy-
toskeletonalone. We will discuss it with the cell membrane in the next section.
CELL MEMBRANE WITH MEMBRANE CY-
TOSKELETON
The free energy of cell membrane with membrane cytoskeleton is taken as the
sum of the free energies of lipid bilayer and membrane cytoskeleton. Thus we
write the free energy of cell membrane under the osmotic pressure ∆p as:
= (k /2)[(2J)2 Q]dA+ [(k¯ /2)(2H +c¯ )2+µ]dA+∆p dV, (34)
d c 0
F −
Z Z Z
wherek¯ =k +k ,c¯ =k c /k¯ ,andthetermrelatedtotheGaussiancurvature
c c c′ 0 c 0 c
disappearsbecause its integration KdAis anunimportant constantso that it
is omitted.
R
7
Before taking the first order variation of the free energy (34), we must in-
troduce the strain analysis expressed by the notation of differential forms [19].
If a point r in a surface undergoing a displacement u to arrive at point r,
0
we have du=dr dr and naturally δ du=δ dr (i=1,2,3).
0 i i
−
Ifdenotedr=ω e +ω e anddu=U ω +U ω with U 1, U 1,
1 1 2 2 1 1 2 2 1 2
| |≪ | |≪
we can define the in-plane strains [27]:
du e
ε = · 1 U e , (35)
11 dr ≈ 1· 1
(cid:20) | 0| (cid:21)ω2=0
du e
ε = · 2 U e , (36)
22 dr ≈ 2· 2
(cid:20) | 0| (cid:21)ω1=0
1 du e du e 1
ε = · 2 + · 1 (U e +U e ).(37)
12 2"(cid:18) |dr0| (cid:19)ω2=0 (cid:18) |dr0| (cid:19)ω1=0#≈ 2 1· 2 2· 1
Usingδ du=δ drandthedefinitionsofstrains(35)–(37),wecanobtainthe
i i
leading terms of variational relations:
δ ε ω ω = δ ω ω , (38)
i 11 1 2 i 1 2
∧ ∧
1
δ ε ω ω = [ω δ ω +δ ω ω ], (39)
i 12 1 2 1 i 1 i 2 2
∧ 2 ∧ ∧
δ ε ω ω = ω δ ω . (40)
i 22 1 2 1 i 2
∧ ∧
From equations (15)–(24) and (38)–(40), we have [19]:
ε dω ε dω d(ε ω +ε ω )
11 2 12 1 12 1 22 2
δ = k [ d(2J) ω − + ]Ω ,(41)
1 d 2 1
F − ∧ − 2 2
Z
ε dω ε dω d(ε ω +ε ω )
12 2 22 1 11 1 12 2
δ = k [d(2J) ω − ]Ω , (42)
2 d 1 2
F ∧ − 2 − 2
Z
δ = [k¯ (2H +c¯ )(2H2 c¯ H 2K)+k¯ 2(2H)
3 c 0 0 c
F − − ∇
Z
k
d
+∆p 2H(µ+k J) (aε +2bε +cε )]Ω dA. (43)
d 11 12 22 3
− − 2
Thus the Euler-Lagrangeequations corresponding to the functional (34) are
1 1
k [ d(2J) ω (ε dω ε dω )+ d(ε ω +ε ω )]=0, (44)
d 2 11 2 12 1 12 1 22 2
− ∧ − 2 − 2
1 1
k [d(2J) ω (ε dω ε dω ) d(ε ω +ε ω )]=0, (45)
d 1 12 2 22 1 11 1 12 2
∧ − 2 − − 2
∆p 2H(µ+k J)+k¯ (2H +c¯ )(2H2 c¯ H 2K)+k¯ 2(2H)
d c 0 0 c
− − − ∇
k
d
(aε +2bε +cε )=0. (46)
11 12 22
− 2
Equations (44) and (45) are called the in-plane strain equations because they
describethe in-planestrainsofcellmembraneunderthe pressure∆p. Equation
8
(46) is called the shape equation because it describes the equilibrium shape of
cell membrane under the pressure ∆p.
If we take a local orthonormal coordinates (u1,u2) such that the first and
second fundamental forms of the surface are I = g (du1)2 + g (du2)2 and
11 22
II =L (du1)2+2L du1du2+L (du2)2, respectively, above equations (44)–
11 12 22
(46) can be expressed as:
∂√g22 ∂ ∂ǫ12 ∂√g11
k (ǫ ǫ ) √g (2ǫ +ǫ ) √g 2ǫ = 0,
d 22− 11 ∂u1 − 22∂u1 11 22 − 11∂u2 − 12 ∂u2
(cid:20) (cid:21)
∂√g11 ∂ ∂ǫ12 ∂√g22
k (ǫ ǫ ) √g (ǫ +2ǫ ) √g 2ǫ = 0,
d 11− 22 ∂u2 − 11∂u2 11 22 − 22∂u1 − 12 ∂u1
(cid:20) (cid:21)
∆p 2(µ+k J)H +k¯ (2H +c¯ )(2H2 c¯ H 2K)+k¯ 2(2H)
d c 0 0 c
− − − ∇
k
d
[aǫ +2bǫ +cǫ ] = 0.
11 12 22
− 2
Obviously, if k =0, then equations (44) and (45) are two identities. More-
d
over equation (46) degenerates into shape equation (3) of closed lipid bilayers
inthis case. Generally speaking,it is difficult to findthe analyticalsolutionsto
equations (44)–(46). But we can verify that ǫ =ǫ =ε (a constant), ǫ =0
11 22 12
can satisfy (44)–(46) for a spherical membrane with radius R if the following
equation is valid:
∆pR2+(2µ+3k ε)R+k¯ c¯ (c¯ R 2)=0. (47)
d c 0 0
−
Now we discuss the effect of membrane cytoskeleton on the stability of cell
membrane. That is, we will calculate the second order variation of functional
(34). Additionally, we only consider the spherical membrane with uniform in-
plane strains whose radius satisfies equation (47) for simplicity.
In Ref. [19], only the term δ2 related to the out-plane mode is calculated.
3F
Here we also consider the contribution of in-plane modes. Due to the notation
ofexteriordifferentialandHodgestar ,Ω andΩ canbeexpressedasΩ ω +
1 2 1 1
∗
Ω ω = dΩ+ dχ by two scalar functions Ω and χ. Using equations (15)–(24)
2 2
∗
and (38)–(40), we can calculate δ2 , δ2 , δ2 , δ δ , δ δ , and δ δ from
1F 2F 3F 1 2F 1 3F 2 3F
equations (41)–(43) and (47). Eventually, we arrive at
δ2 =δ2 +δ2 +δ2 +2δ δ +2δ δ +2δ δ G +G , (48)
F 1F 2F 3F 1 2F 1 3F 2 3F ≡ 1 2
where
G = Ω2 3k /R2+(2k¯ c¯ /R3)+∆p/R dA
1 3{ d c 0 }
Z
+ Ω 2Ω k¯ c¯ /R+2k¯ /R2+∆pR/2 dA+ k¯ ( 2Ω )2dA
3 3 c 0 c c 3
∇ { } ∇
Z Z
+3kd Ω 2ΩdA+k 2Ω 2dA+ kd Ω 2ΩdA, (49)
R 3∇ d ∇ 2R2 ∇
Z Z Z
k k (cid:0) (cid:1)
G = d ( 2χ)2dA+ d χ 2χdA. (50)
2 4 ∇ 2R2 ∇
Z Z
9
If we take κ=k¯ /2, K =3k /2, µ=k /2, w =Ω and Ψ=Ω in equations (6)
c d d 3
and(7)ofZhanget al.’spaper[28],thenG andG correspondtoF [w,Ψ]and
1 2 1
F [χ] in that paper under the conditions of ∆p = 0 and c¯ R = 2. Obviously,
2 0
there is no coupling between modes χ and Ω,Ω ; but there is coupling
3
{ } { }
between in-plane mode Ω and out-of-plane mode Ω . We will show that
3
{ } { }
in-plane modes have quantitive effect on the stability of the cell membrane
although they can not qualitatively modify the results of Ref. [19].
Because G is obviously positive definite, we merely need to discuss G . Ω
2 1 3
andΩintheexpressionofG canbe expandedbysphericalharmonicfunctions
1
[23] as Ω3 = ∞l=0 mm==l lalmYlm(θ,φ) and Ω = ∞l=0 mm==l lblmYlm(θ,φ)
with a =( 1)ma an−d b =( 1)mb . It follows that −
∗lm −P l,−Pm ∗lm − l,−m P P
l
G = ∞ 2a 2 3k +[l(l+1) 2][l(l+1)k¯ /R2 k¯ c¯ /R ∆pR/2]
1 lm d c c 0
| | { − − − }
l=0m=0
X X
l
∞ 3kd
− R l(l+1)(a∗lmblm+almb∗lm)
l=0m=0
X X
l
+ ∞ kd 2l2(l+1)2 l(l+1) b 2. (51)
R2 − | lm|
l=0m=0
X X (cid:2) (cid:3)
We find that if ∆p <pl = [2l(l+31k)d 1]R + 2k¯c[l(l+R13)−c¯0R] (l =2,3,···), then
G ispositivedefinite,i.e.,themembr−aneisstable. Wemusttaketheminimum
1
of p to obtain the critical pressure:
l
∆p =min p = 131kRd + 2k¯c[6R−3c¯0R] < k¯c[23R−32c¯0R], (3kdR2 <121k¯c) .
c { l} ( 2√R3k2dk¯c + Rk¯c3(1−2c¯0R), (3kdR2 >121k¯c)
(52)
But if we do not consider the in-plane mode Ω , we will obtain the critical
{ }
pressure [19]:
∆p = 32kRd + 2kc(6R−3c0R) < 2kc(1R0−3c0R), (3kdR2 <16kc) . (53)
c ( 4√3Rk2dkc + 2Rk3c(2−c0R), (3kdR2 >16kc)
Comparingequation(52)with(53), wefindthatin-plane modeshavequan-
titive effect on the stability of the cell membrane although they can not quali-
tatively modify the result without considering them.
On the one hand, equation (52) includes the classical results for stability
of elastic shells. For classic shell, the critical pressure for spherical shell is
∆p Yh2/R2 [29,30], where Y is the Young’s modulus of the shell. If taking
c
c¯ =∼0, k Yh, k¯ Yh3 and R h, our equation (52) also gives ∆p
0 d c c
∼ ∼ ≫ ∼
Yh2/R2. On the other hand, equation (52) includes the critical pressure for
stabilityoflipidmembranes. Forsphericallipidmembranes,thecriticalpressure
is ∆p k /R3 [11], which is the natural result of equation (52) only if let
c c
∼
k = 0 in it. Otherwise, if we take the typical parameters of cell membranes
d
10