Theoretical analysis of the formation of membrane microtubes on axially strained vesicles

Transcription

1 PHYSICAL REVIEW E VOLUME 55, NUMBER 5 MAY 1997 Theoetical analysis of the fomation of membane micotubes on axially stained vesicles Bojan Božič, 1 Saša Svetina, 1,2 and Boštjan Žekš 1,2 1 Institute of Biophysics, Medical Faculty, Lipičeva 2, SI-1105 Ljubljana, Slovenia 2 J. Stefan Institute, Jamova 39, SI-1111 Ljubljana, Slovenia Received 29 May 1996; evised manuscipt eceived 10 Decembe 1996 The fomation of membane micotubes tethes was analyzed by a theoetical study of the shape changes of an axisymmetical phospholipid vesicle caused by a pulling axial foce applied at the vesicle poles. Equilibium vesicle shapes wee obtained by vaiationally seeking the minimum of the sum of membane local and nonlocal bending enegies at constant vesicle volume, membane aea, and the distance between the vesicle poles. The effect of axial foce on vesicle shapes was studied by examining the shape behavio of polate axisymmetical vesicles with equatoial mio symmety. Fo a vesicle with a given elative volume, the esulting shapes eside within a given egion of the phase diagam fo this vesicle as a function of the distance between vesicle poles and the elative diffeence between the aeas of the membane layes. The uppe limit of this egion was obtained by a vaiational pocedue fo the detemination of vesicle shapes that coespond, at given vesicle volume, membane aea, and diffeence between the aeas of membane layes, to the maximum distance between the vesicle poles. It was shown that fo finite values of the atio between the nonlocal and local bending moduli, at high enough axial foce the vesicle shape exhibits an elongated tubula ending at each pole. The equation fo the adius of such a tubula ending obtained by the igoous teatment pesented matches the equation that has peviously been used as an appoximation in analyses of tethe fomation methods fo the detemination of membane bending moduli. Futhemoe, it is pedicted that below a cetain citical value of the atio between the two bending moduli that depends on the vesicle volume, the shape chaacteized by the tubula endings is attained, with continuously inceasing the axial foce, by a discontinuous shape tansfomation. S X PACS numbe s : Bt, y, Lj I. INTRODUCTION Membane micotubes, known also as tethes, ae elatively long cylindical extensions of membanes of vesicles and cells. They ae fequently obseved as intacellula micotubule-associated membaneous tubula oganelles 1. A tubulovesicula netwok can be fomed by membane associated motos moving upon micotubules 2, o a tethe can be extacted fom neuonal gowth cones by applying an extenal foce 3. Extenal foces wee also shown to cause the fomation of tethes fom membanes of ed blood cells when these, afte being adheed to a glass slide, ae subjected to an oveflow of liquid. Tethes pulled out of a ed blood cell membane do not contain a membane skeleton and ae deficient in at least some integal membane poteins 5. Tethes wee also extacted fom phospholipid vesicles 6, which indicates that thei fomation could be undestood on the basis of the popeties of phospholipid membanes. The poblem of tethe extaction fom vesicles can be viewed as a specific case of the poblem of the fomation of the vesicle shape. The above examples suggest that we ae dealing with the shape of the vesicle when it is stained by an extenal foce acting at a point. The shapes of phospholipid vesicles ae govened by the elastic popeties of the closed phospholipid membane 7,8, and in this espect tethes wee found to epesent a suitable expeimental system fo studying membane mateial popeties 9. They can be conveniently used fo the detemination of both local and nonlocal membane bending constants, paticulaly since, because of thei small adius, the effects of bending ae much moe ponounced in tethes than in ounded vesicles 10. Tethes wee ecently used also fo the detemination of intelaye fiction 11. The aim of this wok is to contibute by a igoous theoetical teatment to the geneal undestanding of the way in which the application of a point foce on a vesicle affects its shape, with paticula emphasis on evealing the conditions when such a foce causes the fomation of a tethe, i.e., a membane micotube. Since the usual basis fo analyzing tethe behavio in the detemination of membane viscoelastic constants is to estimate the elastic enegy of the vesicle involving the tethe by descibing the system by a simple geometical model fo which the enegies of diffeent pats of the vesicle can be easily calculated 12, the pesent igoous theoetical teatment also epesents the necessay basis fo justification of these appoximate models and fo defining the limits of thei possible applications. Moeove, the teatment of vesicles unde the influence of extenal foces in geneal epesents a non-tivial genealization of the appoaches that wee developed in the past fo the detemination of the shapes of feely suspended vesicles 13. The case of a vesicle stained by an axial foce was hitheto not consideed in detail. It has been shown 1 that the Eule equations that ae obtained by the applying the vaiational pinciple to the poblem of the shape of a vesicle stained by an axial foce ae identical to the Eule equations deived fo an axisymmetical vesicle 15 fom the geneal shape equation 16. Possible solutions of the geneal shape equation and thei chaacteization epesent a cuently developing eseach inteest 17. In this espect the pesent wok contibutes in descibing some examples of these solutions fo axisymmetical vesicles of spheical topology. The analysis X/97/55 5 /583 9 /$ The Ameican Physical Society

2 55 THEORETICAL ANALYSIS OF THE FORMATION OF pesented also foms the pope basis fo an undestanding of the fomation of vesicle shapes that wee obseved in expeiments on vesicles containing micotubule assemblies elongating along the vesicle axis and thus exeting foce on its poles 18. In this aticle the elevant contibutions to the membane elastic enegy ae pesented fist. Then the theoetical pocedue is descibed fo the detemination of axisymmetical shapes of bilaye vesicles unde the effect of the axial foce, with special emphasis on the behavio of the vesicle at its poles. It is then shown sepaately how to detemine vesicle shapes with maximal length at a given vesicle volume, membane aea, and diffeence between the aeas of membane layes. The theoy is used to analyze the behavio of the class of axisymmetical polate cigalike shapes involving equatoial mio symmety. II. DETERMINATION OF STATIONARY SHAPES OF AXIALLY STRAINED VESICLES It is commonly believed that the shape of a vesicle o a simple cell such as an eythocyte is detemined by the minimum of the elastic enegy of the membane. We analyze the shape of a phospholipid vesicle unde the assumption that the membane aea (A) does not change. Thus the elastic enegy of a closed symmetical bilaye composed of layes of the same composition is the sum of only two tems (W RE W b ) 12, i.e., a the elative expansivity tem k W RE 2Ah 2 A A 0 2, whee k is the nonlocal bending modulus, A is the diffeence between the aeas of the oute and the inne monolayes in the defomed state and is equal to h (c 1 c 2 )da, with c 1 and c 2 the pincipal cuvatues and h the distance between the neutal sufaces of the oute and the inne monolaye, and A 0 is the coesponding equilibium aea diffeence, and b the bending enegy tem W b 1 2 k c c 1 c 2 2 da, 1 2 whee k c is the bending modulus. The spontaneous cuvatue in the bending enegy tem 19 was taken to be zeo because we ae consideing a symmetical bilaye membane. Also, we only conside vesicles with spheical topology and theefoe the contibution of the Gaussian bending tem to the bending enegy is constant and is thus omitted. The minima of elastic enegy coespond to stationay shapes of the vesicle, so the poblem is to find the exteme values of the membane elastic enegy. The shape of a flaccid vesicle is obtained fom the minimum of W RE W b, whee the membane aea is fixed. Duing the minimization pocedue we also take into account that the volume of the vesicle (V) is fixed because of the incompessibility of wate and the low wate tansmembane tanspot. The constaints in volume and aea can be incopoated in the enegy minimization by intoducing the Lagange multiplies and, which epesent the pessue diffeence acoss the membane and the lateal tension. We wish to detemine the axisymmetical shape of a vesicle that is stained between two point foces acting in opposite diections at the vesicle poles. So the additional constaint that is impotant fo the shape of the vesicle is the distance between the poles (Z 0 ). This distance can be kept constant by intoducing the Lagange multiplie f, which epesents the axial foce. Thus the shape equation fo the vesicle is obtained by minimizing the functional G W RE W b V A fz 0. Because of ecently debated contovesies egading the deivation of the shape equation fo an axisymmetic vesicle cf. 1,15,17,20, the pocedue fo obtaining it fom Eq. 3 is outlined in the following in some detail. The expession fo the membane bending enegy Eq. 2 fo a given vesicle shape does not depend on the vesicle size. Due to this scale invaiance, in the fothcoming minimization analysis the unit of length is chosen in such a way 2 that the elative membane aea equals unity (a A/ R 0 1); thus R 0 A/ is the adius of the sphee with the membane aea A. Then the elative vesicle volume is defined as v V/ 3 R 3 0, the elative diffeence between the aeas of the two membane monolayes is defined as a A/8 hr 0, whee 8 hr 0 is the elative diffeence between the aeas of the two membane monolayes fo the sphee, and z 0 Z 0 /R 0 is the distance between the poles in elative units. It is also appopiate to measue the elative expansivity tem, the membane bending enegy, and the functional G elative to the bending enegy of the sphee (8 k c ): w RE W RE /8 k c, w b W b /8 k c, and g G/8 k c. Thus w RE k k c a a 0 2, whee a 0 A 0 /8 hr 0 is the equilibium elative aea diffeence. The functional Eq. 3 in the dimensionless fom then eads g w RE w b Mv La Fz 0, whee the new Lagange multiplies M, L, and F ae elated to,, and f M R 3 0, L R 2 0, F R 0 f. 6k c 2k c 8 k c It is convenient to minimize the functional g sepaately with espect to a and fo a given a with espect to the vesicle shape. The minimization of functional g with espect to a is pefomed as follows. At equilibium the patial deivative of the functional g with espect to the elative aea diffeence equals zeo. This equiement leads to w b w RE a eq w b a eq dw RE 0, d a eq whee the fact that w RE depends only on a is used. By consideing the equation fo the elative expansivity enegy Eq. one immediately obtains the equation

3 5836 BOJAN BOŽIČ, SAŠA SVETINA, AND BOŠTJAN ŽEKŠ 55 w b a eq 2 k a a k 0, c which epesents the condition fo equilibium. This equilibium condition Eq. 8 is the same as fo the case of no axial foce 21,22. The vaiation of the functional g with espect to the vesicle shape is pefomed by defining as a new vaiable N, the patial deivative of the elative bending enegy with espect to the elative aea diffeence (N w b / a eq ). Fom Eq. 7 it follows that N epesents the elative lateal tension between the monolayes: 8 N dw RE. 9 d a eq Because the elative expansivity tem (w RE ) depends only on the elative aea diffeence ( a), the abitay vaiation of the elative expansivity tem also depends only on the vaiation of the elative aea diffeence ( w RE adw RE / d a eq ). Afte using Eq. 9 the vaiation of the elative expansivity tem ( w RE ) eads N a. An axisymmetical suface can be conveniently paametized in elative units by the coodinates (s) and z(s) 20, whee is the distance between the symmety axis and a cetain point on the contou, z is the position of this point along the symmety axis, and s is the aclength along the contou. The angle of the contou (s) is defined though the equation tan dz/d, so the coodinates and z depend on the angle though the equations ṙ cos and ż sin, whee the ovedot denotes the deivative with espect to the aclength s. The angle and coodinate ae taken as two independent vaiables and the estiction fo the geometical elation between them is consideed by a new Lagange multiplie (s), which epesents the component of the tansvese shea foce 11 in the adial diection. The vaiation of the functional y can then be expessed fo an axisymmetical vesicle as s* g L ds, 10 0 whee L is the Lagange function L 8 sin 2 M 32 sin F sin ṙ cos L 2 N sin 11 and s* is the length of the contou. The bending enegy tem in the functional Eq. 10 w b 8 1 s* 0 (sin / )2 ds] includes the pincipal cuvatues along the paallels (sin /) and the meidians ( ). The geometical paametes of the vesicle ae given by integals: v 3 s* 0 2 sin ds is the elative volume of the vesicle, a 2 1 s* 0 dsis the elative aea of the vesicle, a 1 s* 0 (sin / )ds is the elative aea diffeence, and z 0 s* 0 sin ds is the distance between the poles, espectively. Because the vaiation of the functional Eq. 10 with espect to all independent vaiables along the contou has to vanish ( g 0), one obtains diffeential equations ṙ sin ṙ 3M 2 cos F cos sin, 12 and conditions whee sin2 2 H 8 2 sin2 2 3M sin L 2 2 ṙ cos, H s 0 s* 0, N, sin N 0 s* 0, 16 cos F sin 0 s* 0, 3M2 sin L N sin is the Hamiltonian function. The contou of a vesicle is obtained by solving Eqs Equations epesent the bounday conditions that have to be fulfilled at the beginning (s 0) and at the end (s s*) of the contou 20. Because the length (s*) is not fixed ( s s * 0) and H is constant, Eq. 15 shows that H 0. Equation 16 shows that 0 on the axis. This means that the vesicles ae smooth at the poles. Because the coodinate equals zeo at the beginning and at the end of the contou (0) (s*) 0], Eq. 17 is automatically fulfilled on the axis. In ode to solve the diffeential equations the behavio of the contou, i.e., the dependence of the angle on close to the symmety axis, has to be investigated. Fo this it is convenient to eliminate and s fom the diffeential equations 20. This is done by fist ewiting Eq. 12 as (,,,) and then inseting the expession obtained fo into expession 18 fo H 0, which gives the equation of the contou in the fom (,,). Then the aclength s is eliminated by consideing Eq. 1, and the shape equation appeas in the fom cos 2 sin 2 cos 2 sin L sin 2N sin2 2 sin2 2 2 cos 2 8F 1 6M cos 2 sin 2, 19 whee the pime denotes the deivative with espect to the coodinate. The same equation fo N 0 was pesented by

4 55 THEORETICAL ANALYSIS OF THE FORMATION OF Zheng and Liu 23, who showed that Eq. 19 is also the fist integal of the geneal shape equation 16 fo the axisymmetical case 15 with F an integation constant. In the limit 0 the solution of Eq. 19 has the fom 1 2F ln B, 20 whee B is a constant. In the absence of the axial foce the constant B epesents the value of the two pincipal cuvatues on the vesicle poles 8. In the pocedue fo obtaining the vesicle shape the values of M, L, N, F, and B ae found to fulfill the conditions of the chosen v, a, a 0, and z 0 and to fulfill the condition that the tansvese shea foce at the equato equals zeo due to the mio symmety of the vesicle shape. III. LIMITING SHAPES The question of the maximal length of a vesicle with given volume, aea, and aea diffeence may be posed. In dimensionless epesentation, this means that we ae looking fo the shape with the extemal distance between the poles (z 0 ) unde the conditions that the elative aea (a) equals one and that the elative volume (v) and the elative aea diffeence ( a) ae fixed. The maximal distance coesponds to an infinitely lage axial foce. We thus study the dimensionless functional g z 0 M v L a Ñ a, whee the Lagange multiplies M,L,Ñ ae M dz 0 dv, L dz 0 da, Ñ dz 0 d a The functional g can be expessed fo an axisymmetical vesicle as s* g L ds, 23 0 whee L is the Lagange function L sin M 32 sin L sin 2 Ñ ṙ cos. 2 The equiement fo the geometical elation between in is consideed by the Lagange multiplie. Because the vaiation of the functional Eq. 23 at equilibium has to be zeo, we obtain the equations cos 3 M 2 cos sin 0, 25 3M sin L Ñ 0, ṙ cos. 27 and the bounday conditions H s s* 0 0, (Ñ/) 0 s* 0, and s* 0 0, whee H sin 3M 2 sin L 2 Ñ sin cos 28 is the Hamiltonian function fo this case. Because L does not explicitly depend on the aclength s, the Hamiltonian function is constant. Since the vaiation of g with espect to vaiation of the contou length at the two end points must vanish, one obtains the condition H (s*) 0. Because the Hamiltonian function is constant along the contou, it follows that H 0. To simplify the numeical calculations, we can obtain an equation in a fom without the Lagange multiplie. Fist Eq. 25 is ewitten as (,). This expession fo gives, togethe with H 0 Eq. 28, 3M 2 2L sin Ñ sin The contou of a limiting vesicle is obtained by solving Eq. 29. On the poles, whee 0, one obtains sin 2 0 Ñ, 30 whee 0 is the angle of the contou on the symmety axis. Equation 30 shows that the limiting vesicle shapes ae not smooth at the poles and Ñ must be geate than o equal to because thee ae no solutions that begin on the axis if Ñ is lowe than. Equation 30 also shows that the contou of the limiting vesicle shape begins with the same angle 0 on both poles. It can easily be seen fom Eq. 25 that is zeo on the equato, as it has to be because of the mio symmety of the limiting vesicle shapes. IV. RESULTS The esults pesented ae esticted to vesicles belonging to a class of axisymmetical polate cigalike shapes involving equatoial mio symmety. A systematic desciption of vesicle shapes unde the effect of an axial foce fo othe shape classes will be pesented sepaately. An incease in axial foce in geneal causes an elongation of the vesicle and a change of the elative membane aea diffeence. Thus it is pactical to epesent vesicles of a given elative volume in a two-dimensional phase diagam as a function of the distance between the poles and the elative aea diffeence z 0 - a phase diagam. In Fig. 1 the egion of the z 0 - a phase diagam is shown fo a vesicle with a elative volume of 0.95 in the ange of elative aea diffeences whee axisymmetical polate shapes with equatoial mio symmety exist in the absence of the axial foce. Each point fom this egion is chaacteized by the coesponding Lagange multiplies M,L,F and the elative lateal tension between the monolayes (N). The egion is bounded fom below by the cuve denoted by P epesenting the lowest bending enegy shapes of the pescibed symmety when the axial foce is equal to zeo. These ciga class shapes exist within the inteval of values of the aea diffeence ( a), whee the shape with the smallest aea diffeence is composed of a cylinde with two hemispheical caps shape g in

5 5838 BOJAN BOŽIČ, SAŠA SVETINA, AND BOŠTJAN ŽEKŠ 55 FIG. 1. The z 0 - a phase diagam fo polate axisymmetical shapes with equatoial mio symmety stained by an axial foce fo a vesicle with the elative volume v The cuve designated by P shows the dependence of the distance between the poles in elative units (z 0 ) on the elative aea diffeence ( a) fo the polate vesicles of the ciga class in the absence of the foce. Some epesentatives of vesicles denoted by g, h, i, and j that belong to this cuve ae depicted in Fig. 2. The cuve designated by M shows the dependence of the maximal distance between the poles on the elative aea diffeence fo the polate vesicles. Some limiting shapes denoted by g, a, b, and c fom this cuve ae depicted in Fig. 2. The dotted line is at a b Cuves designated by Q 0, Q, Q 0, and Q 00 show the distance between the poles as a function of the elative aea diffeence fo the vesicles with the atio between the nonlocal bending modulus and the bending modulus k /k c 0,, 0, and 00. The cuves Q 0, Q, Q 0, and Q 00 ae obtained by solving Eqs and 8 fo diffeent values of the axial foce whee the equilibium aea diffeence ( a 0 ) is They begin on cuve P shape h, whee a a 0, and they end whee numeical poblems appea. Some examples of the shapes that ae on the cuve designated by Q denoted by d, e, and f ae depicted in Fig. 2. Fig. 2 and the shape with the lagest aea diffeence is the combination of one lage and two small sphees shape j. Two intemediate zeo foce shapes ae also shown in Fig. 2 shapes h and i. The shape denoted by h coesponds to the absolute minimum of bending enegy. The paametes of these shapes ae given in Table I. The cuve M in Fig. 1 epesents the shapes coesponding to an infinite foce. Fo v 0.95 the infinite foce shape at a coincides with the shape fo F 0 shape g, as this shape cannot be defomed. At highe a values the angle of the contou on the symmety axis ( 0 ) inceases shape a, until at a b it eaches the value /2 shape b whee Ñ equals Table II. The limiting shapes at highe a values diffe fom the shape at a b by having on the axis on both sides an infinitesimally thin cylinde of length 2( a a b ) e.g., shape c. Namely, if Ñ, the solution of Eq. 29 fo the infinitely thin cylinde is 0, The infinitely thin cylinde has no volume and no aea. It contibutes only to the elative aea diffeence and to the FIG. 2. Chaacteistic examples of axisymmetical shapes with equatoial mio symmety at elative volume v The position of each of these shapes in the z 0 - a phase diagam is indicated in Fig. 1. The vesicle otational symmety axis is in the vetical diection. The shapes denoted by g, h, i, and j coespond to polate vesicles of the ciga class in the absence of the foce. The shapes denoted by d, e, and f ae epesentatives of the vesicles that lie in Fig. 1 on the cuve designated by Q. The contous of the vesicles denoted by d, e, f, g, h, i, and j ae obtained by solving the system of diffeential equations Eqs The paametes of these vesicles ae given in Table I. The contous of the vesicles denoted by a, b, and c ae obtained by solving Eq. 29 and the vesicles with these shapes ae chaacteized in Table II. The shapes b, e, and i ae at a b length of the limiting vesicle shape whee dz 0 /d a. The esult pesented can be consideed as the uppe limit valid exactly fo h/r 0 0 fo the distance between the poles (z 0 ) at a given elative aea diffeence ( a) 2. The paametes fo diffeent limiting shapes ae given in Table II. The limiting shapes cuve M of vesicles epesent the limit fo an infinitely lage axial foce applied on the polate vesicles cuve P by keeping the elative aea diffeence constant (k /k c ). As an example of the effect of the axial foce one can note the shapes along the dotted line in Fig. 1 shapes i, e, and b, which have the same aea diffeence. It can be seen that the distance between the poles fo these shapes inceases on inceasing the elative axial foce. Shape e is simila to shape b except close to the poles. Fo phospholipid vesicles the estimated atio between the nonlocal and local bending moduli is finite 10 and theefoe it is of inteest to follow the vesicle shape changes by taking into consideation that the elastic enegy is the sum of the

6 55 THEORETICAL ANALYSIS OF THE FORMATION OF TABLE I. Popeties of the vesicles that ae pesented in Fig. 2, whee a is the elative aea diffeence between the two membane monolayes, z 0 is the distance between the poles in elative units, e is the elative adius of the vesicle on the equato, B is the paamete that denotes the behavio of the contou close to the pole Eq. 20, M is the elative pessue diffeence acoss the membane, L is the elative lateal tension of the membane, N is the elative lateal tension between the monolayes, and F is the axial foce in elative units. Shape a z 0 e B M L N F d e f g h i j local and nonlocal bending tems Eqs. 2 and 1. Asan example we look fo the effect of the axial foce on the vesicle with the equilibium elative aea diffeence ( a 0 ), coesponding to the shape h in Fig. 1, i.e., the shape with the absolute bending enegy minimum in the case of a zeo axial foce. Fo this case the elative aea diffeence equals the equilibium elative aea diffeence. Figue 1 shows the cuves Q 0, Q, Q 0, and Q 00 in the z 0 - a phase diagam of shape changes due to the incease of axial foce fo fou diffeent values of the atio k /k c 0,, 0, and 00. Some examples of the coesponding shapes that lie on the cuve fo k /k c ae also shown in Fig. 2 shapes h, d, e, and f. The tubula endings at the poles of shapes e and f show the fomation of tethes. In Fig. 3 the dependence of the distance between the poles on the applied axial foce when staining the vesicle with the initial shape h is pesented. It can be noted that at smalle values of the atio k /k c thee is a steep incease in the distance between the poles in the egion of the shape changes whee the vesicle begins to fom the tethe i.e., fom shape d to shape e fo k /k c. Fo the elative volume 0.95 at values whee the atio k /k c is smalle than the citical atio (k /k c ) c 1.89, a discontinuous tansition into tethe confomation is pedicted. The value of the citical axial foce in elative units (F c ), which coesponds to this citical atio, is equal to.7. The value of the citical atio between nonlocal and local bending moduli (k /k c ) c and the value of the coesponding citical axial foce (F c ) depend on the elative volume Fig.. It may be seen fom Fig. a that at highe elative volumes a discontinuous shape tansition occus at highe values of the atio k /k c. It can also be seen Fig. b that the citical axial foce (F c ) inceases with the elative volume. The dependence of (k /k c ) c on v shows that thee is no discontinuous shape tansition below a cetain elative volume. The Lagange multiplies that epesent the pessue diffeence acoss the membane, the lateal tension, the elative lateal tension between the monolayes, and the axial foce steeply incease when a tethe is elongated. If the Lagange multiplies ae sufficiently lage, an estimate of the elative adius of the tethe ( t ) could be given, because on the section of the tethe whee /2 the fist deivative of the angle with espect to the aclength s is almost zeo ( 0). One obtains fom Eq. 18 and fom Eq M 2 t L t 8 t 2 N F M t L 8 t Afte eliminating L fom Eqs. 32 and 33 we have 3M t 3 1 N t F t Fo the shape denoted by e the poduct 3M t 3 is equal to 0.039, the poduct N t is equal to 0.06, and the poduct F t is equal to The poduct M t 3 deceases with inceasing elative axial foce fo the vesicle with the cetain atio k /k c. Because fo the elongated tethe the poduct M t 3 is much smalle than N t F t 1, afte using Eq. 8 the adius of the tethe (R t R 0 t ) can be appoximated in dimensional fom by the equation TABLE II. Popeties of the limiting vesicles that ae pesented in Fig. 2, whee a is the elative aea diffeence between the two membane monolayes, z 0 is the distance between the poles in elative units, e is the elative adius of the vesicle on the equato, and 0 is the angle of the contou on the poles. The Lagange multiplies M,L,Ñ ae defined though Eqs. 22. Shape a z 0 e 0 M L Ñ a b / c / g

7 580 BOJAN BOŽIČ, SAŠA SVETINA, AND BOŠTJAN ŽEKŠ 55 2 k c R t 2 k A A 0 f 0. Ah 35 This equation pedicts the same adius fo the tethe as the equation fo the tethe equilibium of the simple geometical model Eq. 21 of Ref. 12. V. DISCUSSION FIG. 3. Distance between the poles in elative units (z 0 )asa function of the elative axial foce (F) whee the cuves designated by Q 0, Q, Q 0, Q 00, and Q show the dependences fo five diffeent values of the atio between the nonlocal and local bending moduli k /k c 0,, 0, 00, and. The equilibium elative aea diffeence of the two membane monolayes ( a 0 ) is The cuves fo k /k c 0,, and 0 ae the most inclined close to z 0 3.2, whee the tethe appeas. The lettes h, d, e, and f indicate the positions of the vesicle shapes that ae depicted in Fig. 2. Fo k /k c cuve designated by Q the maximal distance between the poles exists at z dotted line. FIG.. a Dependence of the citical atio between the elastic constants (k /k c ) c on the elative vesicle volume (v). Thee is a discontinuous tansition in vesicle shape fo the vesicles that lie below the cuve. The cuve ends at elative volume 0.966, whee numeical poblems appea. b Coesponding dependence of the citical axial foce (F c ). The vaiety of vesicle shapes obtained unde the conditions of the applied axial foce is discussed fist in elation to the vaiety of shapes of feely suspended flaccid phospholipid vesicles o stuctually elated cells. A lage vaiety of shapes within diffeent symmety classes has aleady been found in the latte case 7,8,25. Unde the influence of the extenal axial foce the shape vaiety geatly inceases. A demonstation of this is the phase diagam fo the class of polate axisymmetical shapes exhibiting equatoial mio symmety, as a function of distance between the vesicle poles and the aea diffeence pesented in Fig. 1. In this phase diagam the shapes fo the case of zeo foce ae points on a cuve cuve P, wheeas the shapes unde applied axial foces coespond to the points within a cetain aea of the z 0 - a phase diagam bounded on two sides by the cuves P and M. In both cases the shapes ae solutions of the genealized shape equation 16 ; howeve, these solutions diffe in thei behavio at the poles. In the case of a nonzeo foce the pincipal cuvatues at the poles depend logaithmically on the distance fom the axis see 1 and Eq. 20, wheeas in the case of a feely suspended vesicle the shape behavio in the poles is nomal, giving ise to the equiement that the two pincipal cuvatues ae always finite and equal 8. The diffeent extent of the shape vaiety can thus be diectly elated to the diffeent estictions in the bounday condition on the axis. By following up the shape tansfomations unde the effect of the axial foce it is possible to envisage why tethes ae fomed. The vesicle shapes unde the effect of the axial foce ae govened, on the one hand, by the tendency of the system to be as elongated as possible and, on the othe hand, by the opposing tendencies due to the constaints on the constant vesicle aea and volume. The dependence of the distance between the poles on the axial foce detemined fo a easonable value of the atio between the bending moduli cuve Q in Fig. 3 eveals two egimes with egad to the esponse of the vesicle to the axial foce. At elatively low values of the foce befoe eaching shape d a vesicle can adjust to the stain by changing its shape ove its whole suface, wheeas at lage foces afte eaching shape e the vesicle can futhe adapt only by foming tethes, by which the distance between the poles can incease the most by taking into the tethes the minimum possible amount of the vesicle inteio and membane aea. A tethe is an almost cylindical section of the vesicle whee the distance between the membane and the symmety axis is pactically constant. Afte a tethelike confomation is established, a futhe incease of axial foce causes elongation of the tethes and a decease in thei adii Eq. 3. The longe the tethes, the moe membane mateial than wate dawn fom the main vesicle body to such thin tethes. Thus, fo a lage axial foce

8 55 THEORETICAL ANALYSIS OF THE FORMATION OF FIG. 5. Examples of the limiting shapes b fo the diffeent elative volumes v 0.91, 0.95, 0.99, and These shapes coespond to the limiting vesicle shapes of the main vesicle body at infinitely lage axial foce. The otational symmety axis is in the vetical diection. The contous of the main vesicle bodies ae obtained by solving Eq. 29 fo Ñ. the pessue diffeence acoss the membane, the lateal tension, and the elative lateal tension between the monolayes incease vey much. Because the Lagange multiplies F, M, L, and N steeply incease, the solutions of the diffeential equation fo the vesicle shapes Eq. 19 in the limit F,M,L,N coespond to solutions of the diffeential equation fo the limiting vesicle shapes Eq. 29, and the shape of the vesicle can be detemined by integating this equation. Consequently, the shape of the main body of a vesicle is expected to become simila to the limiting shape b Figs. 1 and 2 thoughout the whole egion of space whee tethes ae fomed. This notion can be visualized by compaing shapes e and b in Fig. 2. Howeve, at lage tethe lengths, the tethe faction of the membane aea may become significant elative to the total membane aea, meaning that effectively the shape of the main body would tend to assume the shape b coesponding to a highe elative volume. The limiting shapes b fo diffeent elative volumes ae shown in Fig. 5. It can be visualized that at high enough tethe lengths the shape of the vesicle main body would attain a spheical aspect. The adius of the vesicle on the equato inceases on inceasing the axial foce cf. shapes e and f in Table II. At an infinitely lage axial foce the adius of the main vesicle body on the equato is equal to the adius on the equato of the coesponding shape b. Fo a lage axial foce the elative aea diffeence is popotional to the distance between the poles since the contibution of the membane on the tethe section to the aea diffeence depends only on the tethe length. Because at lage axial foces thee ae only slight changes in the shape of the main vesicle body, also the membane aea diffeence of the main vesicle body and the length of the main vesicle body ae almost constant. Thus, when the distance between the poles inceases by the lengthening of the tethes, the total aea diffeence is popotional to the distance between the poles. The deivative of the elative aea diffeence with espect to distance between the poles in elative units equals 1. The value of this deivative, which is the same as fo the limiting vesicle shape with infinitely thin tethes cuve M in Fig. 1 fo a a b, does not depend on the atio k /k c. This means that at lage axial foces the lines that epesent the dependence of the distance between the poles on the elative aea diffeence fo diffeent atios k /k c ae paallel to the line of the coesponding dependence fo the limiting shape. The highe the atio k /k c, the close the line fo the dependence of the distance between the poles on the elative aea diffeence to the line fo the coesponding dependence of the limiting shape. Fo any finite atio k /k c the limiting shape with infinitely thin tethes is neve eached because the bending enegy of such tethes is infinitely lage. When tethe confomation is established, vesicle lengthening is essentially esisted by the contibutions to the elastic enegy of the tethe sections of the vesicle. This was a basic assumption in the appoximate analysis of the tethe pulling expeiment 12,10,11, whee the shape of the tethe was appoximated by a cylinde. The pesent analysis shows that the magnitudes of the axial foces ae elated to the tethe adius in the same manne Eq. 35 as was aleady pedicted on the basis of simple tethe models 12. This esult thus justifies the use of simple geometical tethe models in the analysis of equilibium tethe expeiments. In this wok the question of the stability of the calculated shapes of the teated shape class was not addessed systematically. It was tacitly assumed that the shapes calculated at given elative volume, elative aea diffeence, and distance between the poles ae the lowest-enegy shapes of the teated symmety, which is a genealization of the case of zeo extenal foce. The poblem of stability is a elevant poblem in view of the fact that in the case of zeo foce the ciga class shapes fo a values lage than ae unstable, having lage enegies than the shapes with no equatoial mio symmety 25,26. Howeve, at least some of the shapes with mio equatoial symmety ae also elevant at nonzeo foce, which is substantiated by obsevations of axially stained vesicles 18. It is to be pointed out that the pesent analysis evealed a diffeent type of instability within the teated class, i.e., the egions of instability at smalle atios of k /k c, as evidenced by the esult pesented in Fig. 3. The mateial constant that essentially affects the behavio of axially stained phospholipid vesicles appeas to be the atio between the nonlocal and local bending constants k /k c. The atio k /k c depends on the sot of lipid and on the numbe of layes in the membane. The atio k /k c obtained fo a mixtue of 1-steaoyl-2- oleoyl-phosphatidylcholine and 1-palmitoyl-2-oleoylphosphatidylseine was appoximately This value applies to unilamella membanes. Fo multilamella phospholipid vesicles the atio k /k c can be consideably lage than fo a bilaye 27. At sufficiently low values of the atio k /k c a definite change into the tethe egime occus in a small inteval of foces Fig. 3. It is of paticula inteest that thee is a discontinuous tansition of the shape fom the petethe to the tethe confomation below the citical atio of k /k c, which is of the same ode of magnitude as the measued value 10. The citical atio k /k c is smalle fo smalle elative volumes Fig. meaning that moe flaccid vesicles can

9 582 BOJAN BOŽIČ, SAŠA SVETINA, AND BOŠTJAN ŽEKŠ 55 adapt to the foce by shape changes ove thei whole suface moe easily. The citical atio k /k c inceases with inceasing elative volume of the vesicle, but also the coesponding elative foces at which the discontinuous tansitions occu ae lage. The pedicted dependence of the citical atio k /k c on elative volume could povide a sensitive method fo the detemination of this mateial paamete, consisting of measuing the dependence of the vesicle length on the foce at diffeent elative vesicle volumes. Fo instance, fom the esults pesented in Fig. fo the atio k /k c, the discontinuous tansition of the vesicle shape is expected at elative volumes lage than The elative volume at which the discontinuities in this dependence appea would povide fo the atio k /k c, wheeas fom the coesponding foce one could detemine k c. The esults pesented may have elevance in some cellula pocesses. The foce exeted by a single kinesin molecule is appoximately 5 pn 28. The foces needed fo pulling the tethe ae of the same ode of magnitude, which indicates that the fomation of tubula cellula systems may actually be the natual consequence of the foces exeted by cytoskeletal systems. An estimation fo the minimal foce needed fo fomation of a tethe can be given fom the value of the poduct F t fo the shape denoted by e Fig. 2, whee the micotubes appea. Because fo this shape the value fo the poduct F t is appoximately 1, the foce needed fo tethe fomation can be given by the equation in dimensional fom f 2 k c /R t. Fo k c J the adius of the tethe, fo foces ( f ) between 25 and 5 pn, anges between 30 and 150 nm. 1 N. Benlimame, D. Simad, and I. R. Nabi, J. Cell Biol. 129, S. L. Daboa and M. P. Sheetz, Cell 5, ; R. D. Vale and H. Hotani, J. Cell Biol. 107, J. Dai and M. P. Sheetz, Biophys. J. 68, R. M. Hochmuth, N. Mohandas, and P. L. Blackshea, J., Biophys. J. 13, R. E. Waugh and R. G. Bauseman, Ann. Biomed. Eng. 23, R. E. Waugh, Biophys. J. 38, H. J. Deuling and W. Helfich, J. Phys. Pais 37, S. Svetina and B. Žekš, Eu. Biophys. J. 17, L. Bo and R. E. Waugh, Biophys. J. 55, R. E. Waugh, J. Song, S. Svetina, and B. Žekš, Biophys. J. 61, E. Evans and A. Yeung, Chem. Phys. Lipids 73, B. Božič, S. Svetina, B. Žekš, and R. E. Waugh, Biophys. J. 61, S. Svetina and B. Žekš, innonmedical Applications of Liposomes, edited by D. D. Lasic and Y. Baenholz CRC, Boca Raton, FL, 1996, p R. Podgonik, S. Svetina, and B. Žekš, Phys. Rev. E 51, Hu Jian-Guo and Ou-Yang Zhong-Can, Phys. Rev. E 7, Ou-Yang Zhong-Can and W. Helfich, Phys. Rev. A 39, H. Naito and M. Okuda, Phys. Rev. E 8, ; H. Naito, M. Okuda, and Ou-Yang Zhong-Can, Phys. Rev. Lett. 7, H. Hotani and H. Miyamoto, Adv. Biophys. 26, ; M. Elbaum, D. K. Fygenson, and A. Libchabe, Phys. Rev. Lett. 76, W. Helfich, Z. Natufosch. Teil C 28, F. Jüliche and U. Seifet, Phys. Rev. E 9, V. Heinich, S. Svetina, and B. Žekš, Phys. Rev. E 8, M. Jaić, U. Seifet, W. Wintz, and M. Wotis, Phys. Rev. E 52, Wei-Mou Zheng and Jixing Liu, Phys. Rev. E 8, The length of the infinitely thin cylinde also epesents a good estimate fo thin enough membane tubes. Fo instance, the distances between the poles in elative units (z 0 ) obtained fo a vesicle with R 0 10 m, h 2.5 nm, v 0.95, and a cylinde of adius 10h diffe fom distances z 0 obtained fo the vesicle with an infinitely thin cylinde by less than of thei values in the ange of a between a b and the elative aea diffeence of the shape c U. Seifet, K. Bendl, and R. Lipowsky, Phys. Rev. A, S. Svetina and B. Žekš, J. Theo. Biol. 16, S. Svetina and B. Žekš, Eu. Biophys. J. 21, K. Svoboda and S. M. Block, Cell 77,