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A.Sokirko1*, V.Pastushenko2*, S.Svetina3 and B. ek 3
1Dept.of Chemistry, Trent University, Peterborough K9J 7B8 Ont., Canada
2Institut für Biophysik, Johannes-Kepler-Universitat, Altenbergerstr. 69,
A-4040 Linz-Auhof, Austria*
3Institute of Biophysics, Medical Faculty and J.Stefan Institute, University of Ljubljana, Lipi eva 2, 61105 Ljubljana, Slovenia
Abstract
The approximate dependency of the vesicle shape on external field strength, bending elasticity modulus and the vesicle size is established by modelling the behaviour of an ellipsoidal vesicle in an electric field (Eo). It is shown that vesicle elongates with increasing field strength. It is suggested that the obtained dependency may be used in experimental determination of the bending elasticity modulus. Kinetics of vesicle deformation has been studied under assumption that the volume of the vesicle changes starting from initial value Vo due to water transport. For weak fields the characteristic time of vesicle deformation t is tªV E , whereas for strong fields it is tªV .
Introduction
Application of an electrical field causes deformation of a lipid vesicle or a cell. Deformation of a lipid vesicle as well as of cells with no internal structure (such as erythrocytes), depends on the elastic properties of their membranes. Therefore studies of the dependency of a vesicle or a cell deformation on strength of the electric field make possible determination of elastic parameters of these systems. Several papers were already devoted to investigation of a phospholipid vesicle or cell shape in electrical field. The shapes close to spherical ones were investigated theoretically in the case of constant [1] and alternating [2] electrical field. These studies were carried out assuming that the area of the cell membrane is constant and the analysis was performed for small deformations. Small changes of the membrane area were taken into account in Ref. [3], where the case of alternating electric field was analysed for initially spherical cell.
The strict formalism describing the vesicle shape should be based on a condition of the local equilibrium of each surface element being subject to the action of electrical and mechanical forces. The exact solution of this problem can be obtained by numerical iterative procedure [4]. Such a procedure is cumbersome and therefore it is convenient to look for an adequate simple but approximate solution of the problem. The aim of this paper is, firstly, theoretical analysis of a large vesicle deformation in an electric field by taking into account the bending elastic properties of the membrane and parametrising the shape in such a way that it is possible to determine electrical forces analytically. For this reason the shape of the vesicle is taken to be that of a rotation ellipsoid. The shape obtained by a small deformation of a sphere always belongs to this set and it is believed that at least for those volumes that do not deviate too much from the volume of a sphere they still represent a good approximation. The second aim of this paper is to elucidate the corresponding dynamic behaviour of the system by including possible water transport through the membrane.
Spheroidal static vesicle deformation
The vesicle we are dealing with should have an axisymmetric shape and a centre of symmetry as well. In order to calculate electrical forces it is necessary to find electrical field outside the vesicle. This problem can be solved analytically only for a very limited set of surfaces. We shall simplify the problem significantly assuming that the shape of our vesicle belongs to that set. We choose the family of extended rotational ellipsoids (spheroids) with constant membrane area.
Let us introduce some notations. We shall be using cylindrical coordinates y, j, x with the symmetry axis x oriented along the vector of external field strength Eo. The elongated spheroid (i.e. ellipsoid, the body obtained by revolution of an ellipse) is then described by the equation
y = b (1)
Here a and b are the parameters of the ellipsoid, a > b. Due to the condition of constant membrane area S, only one of them is independent [5]:
S = 4p =2p a2
Here e is the excentricity of the ellipsoid,
e2 = 1 - b2/a2 (3)
In what follows we shall use the excentricity e as an independent parameter, considering a and b as functions of e with the fixed parameter S. Once the membrane area is constant, excentricity is the only parameter determining the volume of the ellipsoid:
V = 4/3 p a3
We shall first treat the mechanical aspects of the problem and then the electrical ones. The elastic energy of a flaccid vesicle is given by its membrane bending energy:
Wb = H2dS (5)
where H is the surface curvature, defined by its principal curvature radii Rm (meridians) and Rp (parallels),
H = 1/Rm + 1/Rp, (6)
and the constant kc is the modulus of bending elasticity.
The expressions for the radius Rm of curvature of the ellipse (cf. Eq. 1) generating the ellipsoid is given by [5]:
Rm = 3/2 -1
or
Rm= a (7)
According to the Meneut´s theorem for symmetrical surfaces Rp= y/cosq, where q is the angle between the plane x=0 and the normal to the surface in the point of interest. Because tgq = dy/dx , the radius Rp is expressed as follows:
Rp= a (8)
The element dS of the surface of rotation can be expressed as
dS = y dx dj (9)
We can substitute (6)-(9) into (5) and obtain the expression for the membrane bending energy:
Wb = 2 p kc (10)
The dependency of Wb on e is shown in Fig.1 (curve 1). Differentiation of Eq. (10) with respect to e gives the variation of the bending elastic energy, corresponding to the variation of excentricity, de:
dWb = 8p kc de (11)
The electrical forces involved arise as the consequence of putting the vesicle into a static field Eo. Here vesicle is surrounded by the conducting medium and its inner content is also conducting. It is assumed that the membrane is non conducting. Consequently the field exists only outside the vesicle. Because the current density is proportional to the electric field strength (no concentration gradient), the problem is reduced to solving the Laplace equation for the electrical potential f, outside the vesicle
Df = 0 (12)
At an infinite distance from the vesicle we have:
-—f = Eo (13)
where Eo is the external field strength. At the vesicle surface the normal component of the field strength must be zero:
(En) = 0 (14)
Here n is an externally oriented unit vector, normal to the surface. Equations (13) and (14) define necessary boundary conditions.
Let us consider variation in electrical free energy of the system, dWel, defined as the work of the external forces which compensate the electrical forces for an infinitely small displacement of the elements of the surface, du:
dWel = - T du dS (15)
Here T is the electrical force per unit of the surface area. It is equal to the tensor product of the Maxwell´s stress tensor and the normal to the surface, n [6]:
T = e eo
The first term here, according to Eq.(14), is zero, and as T is normal to the vesicle surface, only normal components of u contribute to dWel, i.e.
dWel = - T dundS (17)
Solution of the problem (12)-(14) is well known and may be found in [6,7]. Transformation to the spheroidal coordinates is done in accordance to the formulas:
x = c l m
y = c (18)
where ≥1, 0£m £OQ1 and c= is a half distance between the ellipsoid foci. The Laplace equation (12) in spheroidal coordinate system takes the form:
+ = 0 (19)
and the boundary conditions (13) and (14) become
f Æ - Eo c l m at lÆ∞ (20)
= 0 at l= 1/e (21)
The solution of the problem (19)-(21) is
f = - Eo c m (22)
where Q(l) is the first Legendre function of the second kind,
Q(l) = ln - 1 (23)
and Q´ is the derivative of Q(l) with respect to its argument. The value of the electric field E on the surface is:
E = - = - (24)
where h is Lamé coefficient [5]:
h = c (25)
The value of the electrical field at the vesicle surface is expressed as:
E 2= (26)
Here f stands for the depolarisation coefficient
f = (27)
Let us consider now in detail the surface movement due to changing excentricity. A small normal displacement of the vesicle surface, dun corresponding to a small change in excentricity, de, is
dun = -1/2 de(28)
In terms of Eqs. /1) and (3) we obtain:
= (1 - e2) - (a2 - x2) 2e (29)
The derivatives d(a2)/de can be found by differentiation of (2):
= (30)
Because of the identity dy/de =(2y)-1d(y2)/de equation (28) allows to obtain dun after substitution of (1), (2) and (29), (30).
Then after integration in Eq. (17), one obtains:
dWel = peeo E a3 (e-2 - 1) (1 - f)-2 I de (31)
where I is a dimensionless quantity depending on e:
I = (e-2-1) ln + a-2 - - (32)
In order to find the final expression for Wel we have to integrate (31) with respect to e:
Wel = = (33)
Combining results (11) and (31), one may calculate the variation in the free energy of the system, or the total work performed by external forces on the system, dW, at a small variation of the excentricity de:
dW = dWel + dWb (34)
Fig.1 shows the dependency of W and its components Wel and Wb on excentricity e.
It is possible to introduce the generalised force F, acting along the "e" axis,
F = - (35)
Solution of the equation F=0 will correspond to an equilibrium shape of the vesicle. The dependency F(e) is shown in Fig.2. As F=0 at e=0, the point e=0 is also an equilibrium point although a nonstable one. The stable equilibrium of the system is established at a certain value of excentricity which will be denoted by e*.
The parameters Eo, S and kc enter the equation F=0 in a single dimensionless combination
E = Eo
One may consider the quantity E as a dimensionless electrical field strength. Fig. 3 shows the dependencies of the equilibrium excentricity e*, the ratio b/a and the normalised vesicle volume V/Vo on the electrical field strength E . Here Vo= S 3/2/6
Kinetics of the vesicle deformation
Whenever condition F=0 is not satisfied, water flow through the membrane is expected. The corresponding vesicle volume change may be expressed as
= - kS p (37)
where kS is the water permeability of the membrane and p is a pressure difference.
Assume that the field was applied to a spherical vesicle at the time t=0. As the electrical charges redistribute much faster than the vesicle changes its shape, we may state that there is a generalised force F acting on the vesicle, moving it from the initial state to e*. The work dW is equal to the change in the internal energy of the vesicle, - pdV:
dW = - p dV (38)
The value of the pressure difference p may be calculated from (36) and the definition of the generalised force. Thus one obtains
p = -F de/dV (39)
Taking the derivative of the vesicle volume (4) one obtains
= pa3 (40)
Then, integrating Eq. (37), we get
t = - = - (41)
This is an implicit expression for the excentricity e, and therefore for the vesicle volume V, as functions of time t (see Fig.4).
Close enough to the equilibrium point the process of the vesicle volume change becomes an exponential one with the characteristic time of the exponential relaxation t :
t = e=e* (42)
The right part of Eq.(42) can be rewritten as a product of to, having the dimension of time, and dimemsionless combination:
t = to e=e* (43)
where v is relative volume (V/Vo) and the proportionality factor to is
t0 = (44)
The relaxation of vesicle volume and its exponential approximation are shown in Fig.4
Discussion
We have suggested an approximate method of calculation of lipid vesicle shapes in an external homogeneous constant electrical field valid for relatively large vesicle deformations. Elastic properties of the vesicle are described by bending elasticity modulus. The vesicle shape is approximated by a rotational ellipsoid. All the dimensional parameters of the problem are combined into a single dimensionless parameter E (Eq. 36).
It can be proven by asymptotic analysis that for a long enough ellipsoid the elastic forces are always greater than the electrical forces. Therefore the equilibrium solution exists for an arbitrary value of the electrical field. Simple analytical results may be obtained from the general equation F=0 for the extreme cases of weak and strong electrical fields. In the case of the weak electrical field one gets
e* = E (45)
which coincides with the results of [1] and [2]. For the strong fields we have
e* = 1 - p10/3 E -8/3 (46)
These simple formulae give results which deviate from exact solution of equation F=0 less than 20 % at e* < 0.5 and e* > 0.9, respectively (See Fig.3). Therefore they may be useful within the indicated limits for comparison with the experiment.
We shall indicate here the main practical application of the results obtained. Given the value of the experimentally measured excentricity, one may solve the equation F=0 and find the value of parameter E. This enables one to estimate the bending elasticity modulus of the membrane.
Let us discuss now the kinetics of the vesicle deformation. As it has been shown above, the kinetics of the deformation is determined by the water permeability of the membrane. The characteristic time of this process, t, has been calculated depending on parameter E (Fig. 5a).
For weak fields t can be expressed as:
t /t0 = E 4 (47)
Comparison of this result with the exact one is shown in Fig.5. It is a noteworthy in the behaviour of t (E ), which is expressed in its nonmonotoneity. The most interesting is increasing of t with E, which is not clear at a glance. This may be explained, however, if we note that the result (47) is valid for small E and therefore for small variations of the vesicle volume. On the contrary the volume change was expected to be comparable with the volume of the vesicle. The obtained result is important, because for nearly spherical vesicles the volume changes very slowly (V/Vo ª1- e4/15), although the dependence of the volume change on the excentricity is very strong.
Thus, the factor (dV/de)2, appearing in (42), is proportional to the sixth power of e, and analogous t increases with excentricity (and consequently with the electrical field). As it is clear from (47), the characteristic time t is strongly dependent on the vesicle size. Thus, for the field strength Eo = 106 Vm-1 time t changes from 10 ms for vesicles with radius 0.1 mm up to some months for vesicles with the radius of some microns.
In the case of high field strength there is another asymptotic expression for t :
t /t0 = 32 (3p)1/3 E -8/3 (48)
This dependency is also illustrated in Fig.5b. According to (48), t in this limit does not depend on the initial volume Vo.
In conclusion we should like to note that the results obtained are also valid for alternating electrical fields if the frequency is not too high, so that one may neglect the current flow across the vesicle.
Conclusions
The dependency of the vesicle shape on the external field strength, bending elasticity modulus and the vesicle size is determined. It is shown that the vesicle is elongated with increasing field strength. Equilibrium excentricity is proportional to an electrical field strength if the latter is weak. Alternatively the axis ratio b/a is inversely proportional to the electrical field in the power 4/3 , if the latter is strong enough.
Kinetics of the vesicle deformation has been studied under the assumption that the volume of the vesicle changes due to the water permeation through the membrane. The behaviour of the characteristic time of the vesicle deformation is essentially different in weak and strong electrical fields: for strong fields it does not depend on the size of the vesicle, whereas for weak fields it is proportional to the volume raised to power 10/3. For weak fields this time is proportional to 4-th power of the external field strength, whereas for strong fields it is inversely proportional to the field strength raised to the power of 8/3.
The results may be used for experimental determination of the bending elasticity modulus.
References
1. Helfrich W., Z. Naturforsch. C29 (1974) 182.
2. Winterhalter M. and W.Helfrich, J.Coll.Int.Sci. 122 (1988) 583-586.
3. Bryant G. and J. Wolfe, J.Membr.Biol. 96 (1987) 129-139.
4. ek B., Svetina S. and Pastushenko V. Biol. Membrany 8,429 (1991).
5. Korn G. and Korn T., Mathematical Handbook. , Ch.6, McGraw-Hill Book Company, inc. 1961.
6. Landau L.D. and Lifshiz E.M. Electrodynamics of continuous media, 2nd ed.Pergamon, Oxford, 1984.
7.Lamb H., Hydrodynamics, 6th ed., Dover publ., New York, 1945.
Captions to the figures
Fig. 1. The dependency of the elastic bending energy, Wb (1), the electrical energy, Wel (2), and the total energy of the vesicle W (3) on the excentricity e. All energies are given relative to the bending energy of a sphere, 8pkc. Calculations of electrical energy made for E =2.67.
Fig. 2. The dependency of the generalised force F (relative to the bending energy of a sphere, 8pkc) on the excentricity e. The equilibrium exentricity is denoted by e*. The dimensionless electrical field strength is E =2.67.
Fig. 3. The dependency of the equilibrium excentricity, e* (1), the relative equilibrium vesicle volume, V/V0 (2), and the equilibrium semiaxes ratio, (b/a)* (3) on the external dimensionless electrical field strength E . The dashed lines correspond to asymptotic expressions (45) and (46).
Fig. 4. Dependency of the vesicle volume on time (t/to). The initial volume is the volume of the sphere. The final relative volume is V/V0=0.835 which corresponds to the dimensionless electrical field E =5. Broken line shows the exponential relaxation.
Fig. 5(a). Dependency of dimensionless characteristic relaxation time (t/t0) on dimensionless electric field.
(b) The same curve plotted in the logarithmic scale. The straight lines are asymptotic dependencies, obtained from Eqs. (47) and (48).
List of symbols
a large semiaxis of ellipsoid, m
b small semiaxis of ellipsoid, m
c half distance between the ellipsoid foci
e excentricity of ellipsoid
e* equilibrium excentricity of ellipsoid
E field strength, V m-1
Eo external field strength, V m-1
E dimensionless electrical field strength
f depolarisation coefficient
Fgeneralised force, J
h Lamé coefficient, m
H surface curvature, m-1
I dimensionless parameter
kc modulus of bending elasticity, J
n externally oriented unit vector, normal to the surface, m
p pressure difference, Pa
Q the first Legendre function of the second kind
Rm meridians principal curvature radius, m
Rp parallels principal curvature radius, m
S membrane area, m2
t time, s
T electrical force per unit of the surface area, J m-3
un normal component of du, m
V volume of vesicle , m3
V o initial volume of vesicle , m3
W free energy of vesicle, J
Wb elastic energy of vesicle, J
Wel electrical energy of the system, J
x axial coordinate, m
y radial coordinate, m
duinfinitely small displacement of the elements of the surface, m
D Laplace operator
e coefficient of permittivity
eo permittivity of free space, 8.8542 ¥10-16 F m-1
kwater permeability coefficient, J-1 m4 s-1
l spheroidal coordinate
m spheroidal coordinate
q angle between the plane x=0 and the normal to the surface
t characteristic time of the exponential relaxation, s
to proportionality factor for t, s
f electrical potential, V
j angular coordinate
— operator of divergence
