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Interface mechanics: the shape of membrane cylinder
M. Kozlov, V. Markin
To cite this version:
M. Kozlov, V. Markin. Interface mechanics: the shape of membrane cylinder. Journal de Physique II, EDP Sciences, 1991, 1 (7), pp.805-820. �10.1051/jp2:1991111�. �jpa-00247557�
J Phys II France 1 (1991) 805.820 JUILLET1991, PAGE 805
Classification Phj,sics Absiracis
8? 65D 87 20C
Interface mechanics : the shape of membrane cylinder
M M. Kozlov (1. 2) and v S. Markm (2)
(f) Freie Universitat Berlin~ FB Physik~ Institut fur Theoretische Physik, WE 2~ Amimallee 14~
1000 Berlin 33, Germany
(2) A N Frumkm Institute of Electrochemistry, US S R. Academy of Science, 31 Lenmsky
Prosp Moscow, 117071, U S S R
(Received 3 January J99J, accepied m final form J March J99J)
Abstract. We determined the shape of membrane cylinder adherent to a flat support and
derived the relationship between the energy of membrane adhesion and the geometncal
charactenstics of membrane contour The obtained theoretical results allow an altemative
method of adhesion energj measurement in addition to the method which includes the
measurement of contact angle Two different cases are analytically analysed zero transmembrane
pressure differential at which the shape of the membrane is determined entirely by the bending effects, second, high transmembrane pressure differential whose action surpasse the effect of bending elasticity In the first case the adhesion energy is expressed only through the lengths of adsorbed and free parts of membrane contour which can be readly measured by optical methods In second case the expression for adhesion energy includes the additional geometrical
characteristic the curvature ofmembrane contour in the pole which can be also calculated from the results of optical expenments The distribution of force factors along the membrane is
analysed
Introduction,
In the recent decade, the attention of researchers has been drawn to the mechanical
properties of lipid vesicles and multilayers, formed by lipid bilayers, as well as biological
membranes The mechanical properties of membranes are manifested in adhesion Mutual adhesion of vesicles and their interaction with supports have been widely studied both
experimentally and theoretically [1.7] Adhesion occurs due to attraction between bilayers
but is prevented by the mechanical resistance of the bilayer to deformation Thus, energy and force characteristics of adhesion reflect, on the one hand, mechanical properties of the
bilayers and, on the other, parameters of electrostatic, van der Waals and hydration forces [8.
12]
At present there are two fruitful developments of expenmental studies of lipid vesicle-
vesicle adhesion. Evans and coworkers have developed a method of micromechanical
experiments that yield information of the forces between the mutually adherent bilayers 11, 2, 4-6] The expenments are carried out using two vesicles partially sucked into a m~cropipette,
as a result of which the vesicle membranes are under controlled tension. Being brought into contact, the vesicles mutually adhere, their shape changes and the equihbnum area of the
established contact is expenmentally measured. Change in the pressure sucking one of the vesicles into the micropipette leads to that of the contact area. Numencal computation of such experiments, based on the theoretical model, yields elastic stresses in bilayers and their adhesion energy [1-2, 4-6]
A series of works by Helfnch and coauthors [7, 13] studied mutual adhesion of lipid bilayers
in multilamellar systems The basic expenmental method in those investigations was light
microscopy The authors succeeded in observing the fine details of changes in the membrane
shape due to adhesion. The theoretical analysis of membrane in such systems also yields informatJon on the adhesion energy of the lipid bilayers and their mechanical properties.
Extensive expenmental material on lipid vesicle adhesion, accumulated due to the above investigations, necessitates, in our view, the development of theoretical models to describe the process The works [4, 5, 7] theoretically consider the shape of the membrane, part of
which adheres to the support and the free part is under extemal tension. At the same time, the transmembrane pressure differential is supposed to be zero. This theoretical model does not quite correspond to the expenmental conditions described by the above authors. Both in the micromechanical expenments by Evans and coworkers with two micropipettes and in the
multilamellar systems studied by Helfnch and coauthors, the tension and transmembrane pressure differential do not exist separately. Therefore, description of the membrane shape
requires a more general pattern to be considered that would take into account the
interrelation of pressure and membrane tension as well as other force factors (bending
moments and transverse sheanng forces) In all theoretical works the contact angle between
the membrane and the support is calculated. The adhesion energy is related theoretically to the value of contact angle. The analysis of expenmental works showed us that the optical
measurements of membrane adhesion don't give in every case the possibility to determine
accurately the contact angle. In the same time it seems to be much more easy to measure such charactenstics of membrane contour as the length of adherent part, the length of free part and probably the curvature of membrane contour in the pole of vesicle
The purpose of our present work is to express the energy of the membrane adhesion in
terms of the geometrical charactenstics of membrane contour which seems facile to measure
by optical methods. Such expression will give the alternative method of adhesion energy determination m addition to the well known method based on the measurement of contact
angle
Statement of the problem and basic equations.
Let us consider a lipid membrane cylinder adherent to a flat support (Fig I). The membrane
is divided into the adherent and free segments The adherent segment wJth area
A~ sticks to the support and is flat The free segment of area Ai is bent and does not interact with the support. The shape of the membrane cylinder is determined by that of the cross-
section perpendicular to the cylinder element. In cross-section plane of the membrane
cylinder we shall select axis z normal to the flat segment and located such that the shape of the membrane cylinder cross-section contour is symmetncal relative to this axis (Fig. i) Let's
also assume that the cylinder is stable along its axis.
Adhesion of the membrane cylinder to the support is due to the interaction between the surfaces. The work of interacting forces dunng the contact between the membrane and the support in a unit area without the change of membrane geometry is referred to as adhesion
N 7 INTERFACE MECHANICS 807
?~J
5~ 70
x
Fig I. The cross-section of a membrane cylinder adherent to a flat support (designations)
energy The purpose of calculation below is to find the shape of the membrane cylinder adherent to a flat support and the relation between the shape of the adherent membrane and the value of adhesion energy
Equation for the local equilibrium of a membrane Let us consider the contour of the membrane cylinder cross-section (Fig I). Position of the point is charactenzed by the length
of arc s counted off along the contour. The point of transition from the adherent segment to the free part shall be taken as the reference (s
= 0). The coordinate of the membrane pole shall be designated as si The length of the adherent part of the contour is equal
s~, the length of free part is equal si. The shape of the contour wJll be described by the function
~b(s), where ~b is the angle between the tangent to the contour and the honzontal axis (Fig I) The angle is related to the curvature by the ratio d~b/ds
= J. At the vesicle pole
~b = ar and dJ/ds
= 0. Using the angle ~b, the shape of the membrane cylinder can be
represented in Cartesian coordinates with axis x perpendicular to axis z ThJs representation has the form
s s
z= o dssin~b, x=s~+ o dscos~b. (I)
The shape of membrane cylinder contour is determ~ned by the equations for local equihbnum of the membrane The equation of equihbnum relative to the normal displace-
ments, referred to as a generalized Laplace equation, can be written as [14]
yJ Cj J~ ~~
=
AP (2)
where y( ~ the Gibbs surface tension, Cj[N] is the bending moment, Q( ~ is the
m m
transverse sheanng force, AP is the transmembrane pressure differential [7, 13, 14]. We will
use below also another force factor named zero moment Co(~j which is equal
m
Co = y JCj. Zero moment Co is equivalent to generahsed tension defined and used by
Evans and Skalak [15].
The equation for lateral equihbnum appears to be [16]
~-Cj~=0. (3)
The equation of the equilibrium of a surface segment relative to rotation is
dcj
Q " ~ (4)
We shall assume that the spontaneous curvature of the membrane equal zero and the
bending moment Cj is linked w~th the membrane curvature by a common linear relation
Cj
= EjjJ, where Ejj[Nm is the membrane bending modulus. Two boundary conditions for equations (2)-(4) are a consequence of the symmetry of the cross-section shape relative to axis z First, in the membrane cylinder pole angle ~b
= ar Second, the transverse sheanng force in the membrane cylinder pole is equal to zero, Q
= 0 The third boundary condition is imposed
in the point of transition of the flat segment of the membrane to the bent part,
s = 0. Due to the membrane contour having no sharp points, in the point s = 0 the angle
~b =
0
Energy of membrane adhesion, Equilibrium between the membrane cylinder and the support. Vfhen the membrane element of area dA is brought into contact with the support, the adhesion forces do the work described by expression
dW~
= g~ dA (5)
The value g~ is reffered to as adhesion energy and charactenzes only the interaction between the membrane and the support.
Let us consider a small increment of area dA of the adherent membrane cylinder segment.
In this case the eliment of the free segment of are dA, neighbounng the point
s = 0 (Fig. I) is brought into contact with the support and takes on a flat shape. This process is
associated with the work done by the forces of membrane-support interaction as well as with
changes in the free energy of the membrane itself when it deforms At the membrane
cylinder-support equihbnum the total change in the free energy of the system including that m the free energy of the membrane cylinder and the work of interacting forces shall go to zero.
Change in membrane cylinder free energy during the deformation with constant membrane
area conform to the work of bending moment and is obtained using the expression dF
= C A dJ (6)
Calculation of the energy change of the membrane element becoming flat and that of the free part of the membrane cylinder leads to the total change in the free energy of the
membrane
oie
dF
= Jod C jo dA (7)
o
where Cio and J~ are the bend~ng moment and curvature of the membrane in the point of transitJon from the adherent to free segment (s
= 0).
bf 7 INTERFACE MECHANICS 809
Based on expressions (6)-(7) we obtain that, in the membrane cylinder-support equihbnum,
the bending moment and the curvature of the membrane are related as
cjo
Jo dcjo
= g~. (8)
o
Results.
Equations deduced in the previous chapter make it possible to find the shape and distnbutJon of force factors in the membrane cylinder and relate all thJs charactenstics with the
membrane-support adhesion energy.
Criteria of adhesion and equations for the shape.
Adhesion feasibility criterion Within the framework of the above-assumed relationshJp between the bending moment and the curvature, Cjo
= EjjJo, the integration of equation (8) leads to the relation between the adhesion energy g~ and the curvature of the membrane
contour in the point s
=
0
ga "
EJJJi (9)
Analysis of expression (9) yields two important results Similar results have been obtained
in [17, 4].
First, using (9) one succeeds in establishing the adhesion criterion, i.e., determ~ne the
minimum energy g~ at which adhesion becomes feasible. Second, if cntenon is satisfied and
adhesion occurs, expression (9) allows one to determJne the membrane curvature in the point
of transition from the adherent segment to the free part
Let us consider the adhesion cntenon The energy of adhesion g~ is determined only by the
nature of membrane-support interactJon and does not depend on membrane geometry. Let us
assume that the initial curvature of the membrane cylinder prior to its spreading on the
Ejj /$
support is /~. If the value of the adhesion energy is small, g~
~ ,
adhesion would not 2
occur : the work required for bending the membrane dunng the spreading in this case exceeds
the energy advantage due to adhesion. Adhesion becomes feasible if the work of attracting
forces between the membrane and the support is sufficiently neat and compensates for the Ejj /$
increase of the elastic energy at the spreading, g~
> 2
Let us assume that the adhesion is possible Then from expression (9) it follows that the
curvature Jo of the free part of the membrane in the point of transition between the adherent
and free parts is determined by only adhesion energy and does not depend on the other characteristics of the system : membrane cylinder size and transmembrane pressure differen- tial
Equation for contour shape. Within the framework of the above-assumed relationship between the bending moment and the curvature, equation (4) is represented as Q
=
Ejj~~
ds
Considering this relationship, from equations (2).(3) we have
Y EJJJ~j = 0 (io)
where the Gibssian surface tension ~s expressed as
y =
~~
+ Ejj ~~~+ EjjJ~ (I I)
J J ds
Equation (10) can yield the shape of the membrane cylinder. It should be noted that equation (10) is equivalent to results of Evans (1985) expressed in terms of generahsed
tension (zero moment) Co. We failed to obtain the analytical solution of this equation for the general case. Below we obtain solution for the limiting case in which the surface tension
significantly exceeds the product of the bending moment by curvature, y » Cj J, as well as for the case where the basic contnbutor to the surface tension is the bending moment,
y = Cj J.
Vesicle with large pressure differential.
Let us consider the membrane cylinder with large pressure d~fferential, AP » EjjJ~, at which
the surface tension significantly exceeds the product of the bending moment by the curvature In the equihbnum equatJons one can neglect the terms proportional to the bending moment as compared with those related to the surface tension and the bending moment denvative
From equations (10)-(11) we obtain the approximate equation
~~
= A
j J ~~
l2)
~
E jj
According to expression (2) in the case of great zero moment, Cow Cj J, the integration Co
constant Aj can be expressed in terms of the membrane zero moment, Aj
= -.
In the case
EJJ
considered the zero moment is constant (to the accuracy of the ratio ~ along the
Cj J
entire surface of the membrane The final expressions for the curvature J of the membrane
and angle ~b depending on the arc length s have the following form (Appendix A).
AP Ap
"
Co ~~ (si s)
~ ~° ~
f sinh ~~
~~~~
~ ~~~~
£
j~ AP ~
~b =
(
s + 2
~°
cosh ~~
)~ ~~ sinh ( (14)
o s~
~~~~
f
Taking (9) into account, expression (13) relates the value of the constant zero moment of the membrane wJth adhesion energy
2 g jj2 W
AP ~j
EJ~
i ~ /° coth ~~
f (15)
Figure 2 shows the membrane cylinder shape found by numerical integration using
N 7 INTERFACE MECHANICS 811
o_7 21's,
0.6
o,s
0. 4
0.3
0.2 /
o.
~,/c
0.0 ~
0.0 0,1 0.2 0.3 0.4 0.5 0.6 0.7
~
Fig 2 -The cross-section shape of a membrane cylinder adherent to a flat support at high
transmembrane pressure differential. Values of the parameters determined in the text are equal to
APsi sj
~ = 2 5, = 10. Dashes indicate the initial shape of the non-deformed membrane cylinder before
o f
adhesion
Ap si
expression (14) at the parameter values si = 2.5
= 10 The figure also shows the iniyal
Co f
non-deformed membrane cylinder shape prior to adhesion. As a result of adhesion the membrane curvature in the pole is seen to diminish as compared with the initial value At the
same time, the curvature in the point of transition from the adherent part proves to be larger
than that of the non-deformed membrane cylinder The membrane force factors are read~ly expressed from equations (13)-(14) in appendix B.
Experimentally measurable values We obtained the expressions for the shape of the membrane cylinder cross-section and distnbutJon of the force factors along the membrane
that include the zero moment Co and the transmembrane pressure differential AP.
Experimental conditJons, as a rule, do not make it possible to measure these values. At the
same time, the geometrical charactenstics are readily measurable the length of the contour of the membrane cylinder free segment si, the length of the contour of the adherent segment
s~, the curvature of the membrane in the pole J~. Based on thJs geometncal characteristics,
one can calculate the energy of adhesion. It is sufficient to use expression (14) for the membrane cylinder shape and the relationship
lsr
s~ = cos ~b ds. (16)
o
The problem can be solved numerically The case of a small area of the adherent segment, s~ «si, fives the analytical results
Co = ar
~
~( ~~ (l 7)
si W Jw si)
AP = ar
~ ~~ ~~~°'
(l8) si ar J~ si)
Besides, expressions (17) and (18) relate the measured geometrical charactenstics of the membrane cylinder contour and the adhesion energy
l~'i ~~ ~°~ +
~" ii~~ ~~ " ~
C°th ~~«~~ii
~~~
~~~
~,~~
This relationship allows one to determine the adhesion energy by the data of optical
measurements of the membrane cylinder shape s~, si and J~.
We got two expressions (9) and (19) relating the adhesion energy to the geometry of
membrane contour Equation (9) is more general and contains only the curvature
Jo in the point of transition of the flat segment of the membrane to the bent part. Equation (19) is approximate and contains the curvature of membrane contour in the pole. In the same time equation (19) can be more useful for the analysis of experimental results. If the region of
contact between flat support and membrane is not resolved expenmentally with a great
accuracy, the determination of the curvature Jo can be much more difficult than the
measurement of charactenstics of free part of membrane contour s~, si and J~
Vesicle without pressure differential.
Let us consider a membrane cylinder with zero transmembrane pressure differential
AP
= 0 In this case the shape of the membrane cylinder is determined by the bending
moment and the transverse sheanng force We shall consider the pressure differential to equal
zero. The shape of the adherent membrane cylinder in the absence of the transmembrane
pressure differential is expressed (Appendix C) as
J) J~ 1'2 J$ '~2 j
~
~~~~~
Jl JS ' 1 ~ i ~~° ~~~~
where F(~, k is the incomplete elliptic integral of the first kind [18] Considering (20) we
obtain the relation between the membrane curvature in the point of transition from the adherent part to the free part Jo and in the pole J~ :
j2 ij2
K (
= si Jo (21)
~o ~
where K[k]
=
F I, k is the complete elliptic integral. Equations (20-(21) accurately
2
determine the membrane cylinder shape at the zero value of the pressure differential Based
on these equations, one can describe the shape of the membrane cylinder in the elementary
functions for the limiting case of small deformation, ~~
~
~°'~
« l
o
