Langmuir 1994,10, 4219-4224
4219
Polymer Stretching and Membrane Deformation in Tethers of Partially Polymerized Bilayer M. M. Kozlov* and W. Helfrich Fachbereich Physik, Freie Universitat Berlin, Arnimallee 14, 14195 Berlin, Germany Received May 2, 1994. In Final Form: August 16, 1994@ We consider tethers pulled from vesicles that are made of a membrane embedding polymers, e.g. a partially polymerized bilayer at a low degree of polymerization. Assuming that straight polymers produce a ridge in the stressed membrane, we deal with two effects, the alignment of polymers parallel to the axis of the tether and the deformation of the initially circular cross section of the tether by the aligned polymers. Studying the cases of two or more parallel polymers and of a single polymer, we calculate the free energy and the shape of the tether. Three or more polymers can collapse the tether, depending on their ridge angle. An estimate ofthe energiesinvolved suggeststhat the stretching of polymersby tethers is practicable.
Introduction
All existing theories of random polymers floating in membranes consider only curvature and bending elasticUnfolded polymer chains floating in fluid bilayers give ity. However, in many cases of practical interest the A direct way to rise to novel physical membrane is subjected to lateral tension. Lateral tension produce such systems is in situ polymerization of unsatacts in osmotic lysis" and fusion1*J9ofbiological cells and urated amphiphiles by W Depending on membranes. Weak lateral tensions are one of the factors the material, this can result in cross polymerization or, controlling the shape of vesicles20 and induce mutual in other cases, linear polymerization that joins up to lo4 adhesion in the case of lipid membranes.21,22However, it monomers.'j Other possibilities are adsorption or anchoris also possible to generate tension by micromanipulation. ing of hydrophylic polymer chains on the membrane For instance, one can hold a vesicle with a pipette, thus Experiments have shown that polymerizacontrolling lateral tension, and pull from the other side tion roughens the bilayer and, thus, reduces the size of ofthe vesicle a very long tube, called a tether, whose radius giant vesicles.2-6 The partially cross-linked bilayers of is much smaller than its length.23-2s particular lipids were found to undergo a reversible phase In the present work we consider a stressed cylindrical transition from smooth to rough, called a wrinkling membrane with linear polymers embedded in it. The Long linear polymers, generated in situs or polymerized molecules are assumed to impose a local multiply a n ~ h o r e dwere , ~ shown to give rise to bulging curvature on the bilayer so that a straight polymer and budding of vesicle membranes. Evidently, the produces along its length a sharp bend. As a result, polymers produce spontaneous curvature in initially polymer alignment and membrane shape interact with symmetric bilayers. each other in a tether. We consider in the following a case There are also theoretical studies analyzing possible of strong interaction in which the polymers become effects of polymers on membrane shape and mechanical stretched along the cylinder axis, while the cylinder cross properties caused by polymers. The wrinkling transition section becomes noncircular and decreases in circumferhas been treated in terms of quenched local c ~ r v a t u r e s . ~ ~ - ence. ~ ~ Contours and energies of tubes containing straight Detailed calculations have been performed to deal with polymers will be calculated. In addition, we will argue the effect ofweakly adsorbed polymers on bending stiffness that the presence of more than two polymers in a tube and spontaneous curvature of a membrane." Recently, may lead to the collapse of the tube by allowing its we examined how polymers behaving like self-avoiding circumference to shrink to zero. walks in a two-dimensional medium modify the modulus The possible alignment of polymer chains in tethers of Gaussian curvature15 as well as the bending rigidity could permit the microscopic detection of single polymers and roughness16 of fluid membranes. and may turn out to be interesting for practical applications. @
Abstract published inAdvanceACSAbstracts, October 1,1994.
(1)Ringsdorf, H.; Schlarb, B.; Venzmer, J.Angew. Chem. 1988,100, ..111. (2)Gaub, H.; Sackmann, E.; Biischl, R.; Ringsdorf, H. Biophys. J. 1984,45,725. (3)Gaub, H.; Biischl, R.; Ringsdorf, H.; Sackmann, E. Chem. Phys. Lip. 1985,37,19. (4) Mutz. M.: Bensimon, D.; Brienne, M. J . Phys. Rev. Lett. 1991,67, 923. (5)Dvolaitzky, M.; Guedeau-Boudeville, M. A.; Leger, L. Langmuir 1992,8,2595. (6)Sackmann, E.; Eggl, P.; Fahn, C.; Bader, H.; Ringsdorf, H.; Schollmeier, M. Ber. Bensen-Ges. Phys. Chem. 1986,89,1198.Meier, H.; Sprenger, I.; B h n a n n , M.; Sackmann, E. Preprint. (7)Simon, J.; Kiihner, M.; Ringsdorf, H.; Sackmann, E. Preprint. (8)Ji, H.; Hone, D. Macromolecules 1988,21,2600. (9)de Gennes, P.G. Adv. Colloid Interface Sci. 1987,27,189. (10)de Gennes, P.G. J. Phys. Chem. 1990,94,8407. (11)Brooks, J . T.;Marques, C. M.; Cates, M. E. J.Phys. ZZFr. 1991, 1, 673. (12)Nelson, D. R.; Radzihovsky, L. Europhys. Lett. 1991,16,79. (13)Radzihovsky, L.;Nelson, D. R. Phys. Rev. A 1991,44,3525. (14)Radzihovsky, L.;Le Doussal, P. J . Phys. Z Fr. 1992,2.599. (15)Kozlov, M.M.;Helfrich, W. Langmuir 1993,9,2761.
Description of the System We consider a tether formed by locally applying a pulling force to the membrane of a large vesicle (Figure 1). The (16)Helfrich, W.; Kozlov, M. M. J . Phys. IZ 1994,4,1427. (17)Kozlov, M. M.; Markin, V. S. J. Theor. Biol. 1984,169,17. (18)Finkelstein, A.;Zimmerberg, - J.;Cohen, F. S.Annu.Rev. Physiol. 1986,48,163. (19)Kozlov, M. M.;Leikin, S. L.; Chemomordik, L. V.; Markin, V. S.; Chizmadzhev, Yu. A. Eur. J. Biophys. 1989,17,121. (20)Kozlov, M. M.; Markin, V. S. J.Phys. ZZ Fr. 1991,1, 805. (21)Helfrich, W.; Servuss, R. M. Zl N w u o Cimento D 1984,3,137151. Servuss, R. M.; Helfrich, W. In Physics of Complex and SupermolecularFluids; Safran, S. A. Clark, N. A., Eds.; John Wiley and Sons: New York, 1987;p 85. Servuss, R. M.; Helfrich, W. J.Phys. Fr 1989, .m - - , 80s. ---. (22)Evans, E. Langmuir 1991, 7, 1900. (23)Hochmuth, R. M.; Evans, E. A. Biophys. J.1982,39,71-81and 83-89. (24)Waugh, R. E.Biophys. J. 1982,38,19-27 and 29-37. (25)Song, J . ; Waugh, R. E. J. Biomech. Engr. 1990,112,235.
0743-746319412410-4219$04.50/0 0 1994 American Chemical Society
Kozlov and Helfrich
4220 Langmuir, Vol. 10, No. 11, 1994
Figure 1. Large vesicle with a tether in a cross section along the tether. membrane is cylindrical in the tether of constant length, while it is practically flat around the vesicle. Since there is a free exchange of bilayer between tether and vesicle, the vesicle can be regarded as a reservoir of membrane area. The bending elasticity of the membrane is characterized by the bendingrigidity K . The ~ ~ whole system is stressed, the lateral tension in the flat membrane of the reservoir being yo. In a uniform tether, one of the two principal curvatures of the membrane is identical to zero so that the other makes up the total curvature J. The geometry of such a tether is completely determined by the function J(s),s being the arc length along the contour of a normal section of the tube. The Gaussian curvature vanishes, and the spontaneous curvature is taken to be zero. The membrane is assumed to contain random polymer chains for which it is a two-dimensional medium. Let us consider the simplest case, a partially polymerized lipid membrane representing a dilute solution of polymer chains. The polymerization links rows of molecules by chemical bonds and can thereby change the effectiveshape of the lipid molecules involved. We assume a straight polymer to produce a sharp bend in the membrane. Since the membrane is stretched, this will result in the formation of a ridge (or furrow) of a height (or depth) which depends on the ridge angle ~ 0 the , lateral tension, and the bending rigidity. Any bending elasticity of the polymer, which may tend to warp it, is disregarded. The ridge angle V O is taken to be independent of lateral tension and curvature of the surrounding membrane, which is legitimate as the bend at the ridge is highly localized. A polymer can be either in the reservoir or in the tether. The tether will attract polymers which reduce its bending energy, i.e. those with the ridge on the outside of the tube. We will consider only this type of polymer and assume them to align parallel to the tube axis. Although the stretched conformation of the polymer is entropically unfavorable, the assumption will be justified by an estimate of the elastic energy of the membrane. To keep the tether uniform, polymers in it will be thought to extend over its whole length.
Shape Equation for Tethers Both in the tether and in the reservoir the polymer perturbs the initial shape of the membrane, resulting in a change of the membrane elastic energy. The energy change per unit length of straight polymer will be denoted as Aft and Afr for the tether and the reservoir, respectively, and referred to for simplicity as the polymer energy. The net change of the energy when the polymer passes from the reservoir to the tether is Af= Aft - Afr. We consider below the polymer energy in the tether Aft, since, as shown in Appendix A, the polymer energy in the reservoir Afr can be neglected, unless the ridge angle is very large, y j o 2 1. Provided that the membrane is in lateral equilibrium and practically unstretchable, its free energy per unit area (26)Helfrich, W.2.Naturforsch. 1973,C28,693.
Figure 2. Membrane ridge produced in a stretched flat membrane by a straight polymer with ridge angle VO. in the tether may be expressed by27
The first term in (1) is the lateral tension of the flat membrane in the reservoir. The second term is the energy of bending per unit area. The quantity y also represents the total lateral tension ofthe curved membrane. In order to calculate Aft, we have to find the membrane curvature J around the tether and then to integrate the energy density (1)over the surface of the tether. This gives us the total energy needed to pull the tether from a reservoir of constant lateral tension yo. The membrane curvature J can be determined from It the known shape equations of fluid membrane~.l~,~* depends only on the arc length s measured along the contour of the membrane cross section (Figure 2). The equilibrium equation then takes the simple form
We have omitted here any pressure inside the tether, which is permissible as long as the radius of the tether is much smaller than that of the sphere from which it is pulled. The tangent angle to the membrane contour (Figure 2) is related to the curvature by
(3) The solutions v ( s ) have to satisfy boundary conditions imposed by the ridge angle of the polymer. If there are n parallel polymers in the tether, its contour may be anticipated to have n-fold rotational symmetry in the state of minimal energy. The shape of the membrane contour in Cartesian coordinates is given by
Tether with T w o or More Polymers The energy of the tether can be reduced dramatically, sometimes down to zero, if there are two or more polymers in the tether. Moreover, this case is the simplest to analyze. To calculate the polymer energy Aft, let us first consider a tether without polymers. We know that it has a circular cross section and uniform membrane curvature which, because of (21, is
J=
JYXO 2-
(27)Helfrich, W.;Kozlov, M. M. J. Phys. ZZ Fr. 1993,3,287. (28)Ou-Yang, Zhong-can; Helfrich, W. Phys. Reu. A 1989,39,5280.
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Tethers of Partially Polymerized Bilayer
To choose between these possibilities, we recall that each polymer bends the membrane by its ridge angle VO. It
1
follows from Figure 3a that the positive curvature (9a) will apply to small angles $Jo < 2x13. The negative curvature (9b) (Figure 3b) holds for large angles, 2n/3 < $Jo < n. The case of zero curvature (9c)applies to only one particular value of the ridge angle, OJ!,I = 2x13, and represents equilibrium of membrane shape only, but not of the tether as a whole (Figure 312). The circumference of the tether depends on the ridge angle. For three polymers it equals
if the curvature is positive (gal, and
if it is negative (9b). Because of $Jo < n,the circumference
L in the presence of polymers is always smaller than (7)) the circumference of the undeformed, circular tether. The energy per unit area (1)of a membrane with positive (9a)and negative (9b)curvatures is equal to that of circular tether (6), but the circumference of the tether is reduced. Therefore, the free energy of the tether is decreased by the polymers. Multiplying (6)by the circumference (lo), subtracting the energy (8)of the unperturbed tether, and dividing by the number of polymers, we get the polymer energy Aft. For t,bo < 2n13, i.e. positive membrane curvature, it is Figure 3. Cross section of a tether with three polymers for different values of the ridge angle YO. (a, Top) Positive curvature: (1)vo= 0, (2) vo= 0.5, (3) l//o = 1, (4)V O= 1.5. (b, Middle)Negative curvature: (1)V O= 2.5, (2)V O= n. (c, Bottom) Zero curvature: disequilibrium, V O= 2nI3.
Substitution of (5) in (1)gives for the energy per unit area Y = 2Yo (6) Multiplying (6) by the circumference of the tether,
L, = f i nJr
Yo
(7)
we obtain the energy per unit length of cylinder
Now we consider a tether with more than one polymer. For simplicity, we take the case of three polymers, which will be easily generalized. We expect the spacings of the polymers along the contour to be equal and the curvature of the membrane to be uniform, as illustrated in Figure 3a-c. According to (21, the curvature can have one of the three following values:
J=
JYKO J: 2-
I
J=-
2-
or
J = O
(9c)
(11) For $Jo > 2n13, i.e. negative curvature, we have = -$To
(F-
w0)
(12)
The tether is out of equilibrium for any finite width in the special case $Jo = 2nl3 with zero curvature. This is because its energy can be reduced continuously to zero by decreasing the spacing of the polymers at fixed free energy per unit area of membrane, which according to (1)is y = yo. Finally, all the lipid material has gone to the reservoir, and only the three polymers form a very thin residual utether)’. The polymer energy in the final state is simply the negative of the energy per unit length of the unperturbed tether (8) divided by the number of polymers
(13)
ut
The dependence of the energy on the ridge angle WO is illustrated in Figure 4a, where the minimal energy corresponds to the collapsed tether. Squeezing the water out of the collapsing tether will require pressures which can be produced by a slight deformation of the membrane from its flat state. In the present work we do not analyze the dynamics. The particular case of three polymers can be generalized to allow for an arbitrary number M > 2 of polymers stretched along the tether axis. The equally spaced polymers separate regions ofuniform membrane curvature between them. This curvature is positive and is given by (9a), if the ridge angle is small, $JO < 2nIM. For large ridge angles, 2nlM < vo < n,the curvature is negative
4222 Langmuir, Vol. 10,No. 11, 1994
n
-'
-1.2
0.5
1
1.5
2
Kozlov and Helfrich 2.5
3
*o
1
Figure4. Energy per unit length of one polymer vs ridge angle ~ 0 (1) : The case of three polymers; (2) The case of a single polymer.
and is equal to (9b). The special case of zero curvature leading to the collapse of the tether corresponds to the ridge angle ~0 = 2dM. The polymer energies are obtained for vo< 2 d M from the unchanged (11)and for voI2 d M from (12) and (13) upon replacing 3 by M. Interestingly, the negative polymer energy in the tether is proportional to the first power of the ridge angle voso that even small ridge angles make a difference. Moreover, Hence, the the polymer energy is proportional to aligning effect of the tether on the polymer increases with the ridge angle and with lateral tension and bending rigidity. The polymer energy (11)multiplied by the length of one monomer f; can be used to assess if a polymer chain will be aligned parallel to the tether. Assuming yo = lov3 J/m2 (a typical lateral tension in tether^,^^,^^ K = 213 x J , 6 = 1 nm (the width of a lipid molecule in the plane of the membrane), and lye = 0.33 (= 19"))we find the energy per one monomer to be about -3kT. This energy is more than sufficient if kT or less is needed per monomer to align the polymer in the tether. We have used in the estimation a rather high value of the ridge angle qo. On the other hand, the polymer persistence length may actually be much larger than 1 nm because of backbone elasticity or an effective stiffness resulting from the membrane ridge. Preliminary calculations indicate that the latter effect can raise the persistence length to 10 nm,29which would result in AfiE = -30kT.
Figure 6. Cross section of a tether with a single polymer for different values of ridge angle VO: (1)~0 = 0.001, (2) ~0 = 1, (3)~0 = 1.72 (zero curvature of membrane at the position of polymer), and (4) ~0 = 3. case of relatively small ridge angles, i.e. for
Yo
lll/oK.
Tether with a Single Polymer If there is only one polymer in the membrane of the tether, the shape of the cross section is no longer composed of circular sections. As the membrane curvature is nonuniform, we have to solve (2) to obtain J(s). The first integration of (2) gives a first-order differential equation for the curvature,
where Jo is the unknown curvature of the membrane opposite to the location of polymer (Figure 5). The value of Jo depends on the ridge angle V Oand can be found from boundary conditions upon solving (14).For polymers with zero ridge angle the curvature JOis, of course, equal to the curvature on the undeformed cylindrical tether (5). As increases, JOturns out to increase also. The function J(s)can be obtained from (14) my means of elliptic integrals.30 The form of the solution of (14) depends on the value of J i as compared with 4 y d ~ .We present here the solution for the most practically useful (29)Kozlov, M. M.; Helfrich, W. Phys. Rev. E , submitted for publication.
454-
K
The solutions for larger values of Jo and, thus, larger ridge angles are given in Appendix B. Using standard notation and the abbreviation 7 = 4 ~ 4 -~ 1,4we obtain for J ( s ) the implicit equation
1 F ( q , k )= -do 2
(16)
where
q = arcsin and
are the parameters in the incomplete elliptic integral of the first kind F ( ~ I , ~ ) . ~ O The energy ofthe polymer, as determined by integration of (1)over the circumference of the cross section, is equal to
Upon calculating V(J) from (31, one can derive x ( s ) and y(s) from (4). If the tether is placed as in Figure 5, the result is
y = 2 p J (1 - J a ) Yo (1- ?1)
(21)
where E(q,m)is an incomplete elliptic integral of the second kind.30 (30)Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic Press: New York, 1965.
Tethers of Partially Polymerized Bilayer
Langmuir, Vol. 10, No. 11, 1994 4223
The unknown constant JO entering into (16)-(21) is determined for each value of the ridge angle WO by the equation
2E(n - :,Ji-llj = (1
+ v)F(n - "",fij 4
(22)
which expresses the condition that the contour of the cross section be closed at the position of the polymer. Solving (22) numerically and taking into account (151, we find that (16)-(22) are valid for the following range of ridge angles: 0 IqoI1.17
(23)
For small values of the ridge angle, q o 1 polymers, each of them reduces the energy of the tether by an amount proportional to qo before Mqo reaches 2n. Since the energy reduction is less than that for the first polymer, it is the more for the second.
8
Concluding Remarks The present paper tackles the problem of how random membrane polymers and the shape of bilayer tethers influence each other. Polymers embedded in a stretched membrane can produce in it a ridgelike deformation.This means that a straight polymer can change the shape of the cross section of a tether. If the change involves a sufficient reduction of the free energy of the tether, it will in turn stabilize its stretched configuration. The energy reduction per polymer is larger for two and more polymers than for a single one. In special conditions the right number of polymers can cause the collapse of the tether, i.e. shrinkage to (theoretically) zero radius. We analyzed the problem in the simplest possible terms, assuming the stretched polymer to be a straight line parallel to the cylinder axis. The real conformation of a chain cannot be exactly straight, since this would cost an infinite amount of free energy resulting from the suppression of alignment fluctuations. However, the above estimate suggests that the deformation of the tether can release sufficient energy to keep the polymer chain reasonably straight. We did not consider the possibility that a polymer forms a spiral on the tether surface. The exact calculation of membrane shape and energy is difficult to perform analytically in this case. However, it is clear that the Gaussian curvature and the correspondingelastic modulus E come into play as a helical ridge of the membrane surface produces a non-zero Gaussian curvature, whose energy
has to be taken into account. Simple considerations show that a positive modulus of Gaussian curvature is required for the formation of a spiral. A full theory of spiraling polymers has to include the bending elasticity not only of the membrane ridge but also of the polymer backbone. We plan to come back to the possibility of spirals in future work. As a final remark, let us point out that our theory can be checked experimentally by investigating tethers pulled from polymerized membranes at a low enough degree of polymerization. Any thinning of the tether can be compared with our prediction. We considered only the case of a tether with constant length exchanging material with an external reservoir. However,the results are easily extended to the situation where the tether has a constant membrane area but can change its length under the action of a k e d external pulling force. The accompanying change of the lateral tension poses no problems as the tether contours at a given yjo are scale invariant, all lengths being A decrease of the circumference proportional to gives rise t o an elongation of the tether, which could be measured directly in the experiment.26
6.
Acknowledgment. We are grateful to the Deutsche Forschungsgemeinschaftfor supporting this work through Grant He 952/15-1 and Sonderforschungsbereich312 for supporting this work. Appendix A Let us calculate the energy Afr and the shape of the ridge associated with a straight polymer in the flat membrane ofthe reservoir. The curvature J derived from (2) is equal to
(-41)
6
where 6 = is the characteristic length of the decay of the ridgelike deformation of the membrane produced by the polymer. The energy of the polymer Afr as obtained by inserting (All in (1)and integrating over the arc length s on both sides of the polymer is given by
In the practically important case of small ridge angles, qo
