Bending stiffness of lipid bilyers. II. Spontneous curvture of monolyers Thoms Fischer To cite this version: Thoms Fischer. Bending stiffness of lipid bilyers. II. Spontneous curvture of monolyers. Journl de Physique II, EDP Sciences, 1992, 2 (3), pp.327-336. <10.1051/jp2:1992129>. <jp- 00247635> HAL Id: jp-00247635 https://hl.rchives-ouvertes.fr/jp-00247635 Submitted on 1 Jn 1992 HAL is multi-disciplinry open ccess rchive for deposit nd dissemintion of scientific reserch documents, wher y re published or not. documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.
J. Phys. II Frnce 2 (1992) 327-336 MARCH 1992, PAGE 327 stiffness of lipid bilyers. II. Spontneous curvture of Bending monolyers Abstrct. In clssicl formultion of elstic bending energy stored in bilyer, us to explin following experimentl results (I) decrese in «pprent» bending llows of pure bilyers with incresblg unsturtion of ir lipids nd (it) instbility nd stiffness se simple systems mny observtions hve not yet been explined. In this pper, new formultion of elstic bending energy density stored in bilyer Which present us to understnd s yet unexplined experimentl results. llows resistnce of bilyer to bending cn be decomposed into two contributions which distnce. single lyer bending stiffness which is lso clled intrinsic bending interlyer rises from resistnce of molecules mking up stiffness monolyer to chnge in e. g. from cylindricl to conicl. In bilyer couple bending, eir locl or globl shpe, or mixture of both occurs, depending on time scle of deformtion [il. In nture composed of single lipid species single lyer bending is locl. If more thn one lipid bilyers is present single lyer bending cn lso become globl. species shpes hve been ssigned to lipids to describe ir pcking behviour [2]. Idelized (PC) with two sturted hydrocrbon chins re ssumed to be lmost Phosphtidylcholines Hover, molecule becomes incresingly inverted cone shped with cylindricl. degree of unsturtion [3], I-e- cross sectionl re t hedgroup position is incresing Clssifiction Physics Abstrcts 87.20 87.45 Thoms M. Fischer Institut for Physiologie, Medizblische Fkultit, Rheinisch-Westfilische, Technische Hochschule, D-W-5100 Achen, Gennny (Received 7 November1991, ccepted in finl form 5 December 1991) curvtures of monolyers enter vi ir sum. Accounting for spontneous spontneous of monolyers individully leds to n essentilly different formultion which curvtures concomitnt with budding of vesicles induced by chnge in temperture. new hysteresis predicts bove criticl unsturtion of ir lipids (I) micro roughness of formultion surfce nd (it) spontneous budding of vesicles in pure bilyers. 1. Introduction. Nturl membrnes re composed of mny different kinds of molecules nd complex models re required to understnd ir mechnicl behviour. Such models use bsic informtion obtined from pure systems, e.g. lipid-bilyer vesicles of cellulr dimensions. But even in cll bilyer couple bending nd single lyer bending [ii. Bilyer couple bending rises from resistnce of two monolyers to chnge in surfce re nd from ir fixed
smller thn tht t loose end of hydrocrbon chins. This cn lso be chieved by decrese in hedgroup size. LysoPc which hs single hydrocrbon chin is tken to be cone becuse cross sectionl re of hedgroup is lrger thn tht of shped, chin. greter cone ngle of such idelized shpes greter is hydrocrbon curvture of monolyer composed of corresponding lipids. spontneous experiments with lipid vesicles use exclusively symmetric bilyers. Mechnicl curvture (in single lyer bending) of such bilyers vnishes due to opposite spontneous of monolyers within bilyer. For this reson spontneous curvture of orienttion individul monolyers hs been neglected in oreticl tretment of bilyer elsticity. this pper elstic energy density in single lyer bending of bilyer is formulted in In of rdius of curvture nd becomes relevnt when rdius corresponding independent spontneous curvture of monolyer is comprble to thickness of bilyer. to clssicl ory of thin shells is bsed on one neutrl surfce, which is defined s (energy per surfce re) of lipid bilyer is : e couple bending is not included in eqution (I), s B is intrinsic (or single lyer) bilyer stiffness of bilyer, nd $ its spontneous curvture in single lyer bending. bending Iii. now decompose bilyer in its two monolyers. y re chrcterized by indices o We I, representtive for outside or inside of closed vesicle, respectively. Ech monolyer or ssigned its own neutrl surfce. It is defined s bove whereby ssume cn bend is monolyer seprtely. isotropic tensions rising in ech monolyer from bilyer ech bending re ssumed to be linerly superposble to stresses rising from single couple ri/8r/8 ' tie index s is dropped for simplicity. It is obvious tht se re quntifies in respectively. lyer bending. For inner lyer sme formul pplies, with index o replced single by I. Here sign convention is sme for i nd it irrespective of orienttion of 328 JOURNAL DE PHYSIQUE II N 3 such wy tht it ccounts for ech monolyer seprtely. In this cse two effects re found. first is independent of spontneous curvture of monolyers nd becomes relevnt t rdii of curvture comprble to thickness of bilyer. second is 2. Anlysis. surfce within bilyer re of which does not chnge when bilyer is conceptul Clssiclly contribution (e) of locl single lyer bending to elstic energy density bent. + 2 ) II ) ri r2 Here ri nd r re principl rdii of curvture. bending energy originting from definition of f differs by fctor of 2 from one suggested by Helfrich [4], for consistency bending. distnce of two neutrl surfces from midsurfce of bilyer re lyer nd 8, respectively. If bilyer is bent rdii of curvture of two monolyers 8 hve different vlues. To distinguish quntities in new nlysis (which ccounts for individully) from quntities in clssicl nlysis (Eq. (I)), dd tilde to monolyers symbols. For energy density of outer lyer (i) obtin : where fi nd i re bending stiffness nd spontneous curvture of outer lyer,
monolyers within bilyer. From blnce of bending moments obtin spontneous curvture of bilyer s ighted sum : j j is" This is nlogous to unstretched length of prllel rrngement of springs. energy density of bilyer is given by sum : To simplify expressions set 8 8; for rtio 8/r obtin fter some lgebr : 8 nd fi fi fij2. First tret cse () ) )( i )rirj' lst two terms contin corrections which re positive nd qudrtic in 8/r. Consequently become significnt when rdii of curvture re comprble to thickness of y bilyer. now tret cse of non zero spontneous curvture of monolyers. After some We (i+i;)( + +&(i-i)(j+jj Iiv+fiil+ii- 2 2 i. Eqution (6) reduces to : i since ccording to eqution (3) : f i. f f monolyers hve opposite vlues. We set : l 2 ri r2 l. 16) io-ii ip. 18) cone shped lipids hve negtive i. A cylindricl shpe corresponds to i N 3 BENDING STIFFNESS OF LIPID BILAYERS. II 329 (3) B +B; Ii+I;. (4) i 0. elstic energies clculted under lst ssumption re chrcterized by i index v. After expnding in Tylor series nd neglecting terms with pors lrger thn 2 more lgebr obtin : &(io + 11) + ri r We consider specil cse : f nd cll corresponding energy density i + + (7) 8 i i + fili( We hve three terms in ddition to eqution (5). first two re expected from eqution (1) third is qudrtic in 8/r. This mens tht, s for (5), i is different from e only when rdii of curvture become comprble to eqution of bilyer. thickness cse of interest is symmetric bilyer, where spontneous curvtures of two From sign convention implicit in eqution (2) nd (8) it cn be pprecited tht inverted 0. Conic1lipids
i. Eqution (6) reduces to : ip + )). (9) ii+2fili(+i8( 2 l We consider two specil cses of eqution (9). First, deformtion into sphericl cp (ri r r). We cll respective energy density i nd We obtin : +3+I(+2il. (10) rc rc rc i2fi( In contrst to eqution (I) re is no vlue of r for which i becomes zero. If now i, obtin : 81i. (p ) nd inverted cone l ) shped lipids. It is obvious tht energy stored in flt configurtion is sme in both cses, provided fi is identicl. This energy corresponds to term in eqution (9) which is qudrtic in i. figure lc both bilyers re bent by exteml moments into sphericl cp with rdius In thn tht corresponding to spontneous curvture. For cse this mens energy lrger in upper lyer increses nd in lor one it decreses. If neglect stored 330 JOURNAL DE PHYSIQUE II N 3 re chrcterized by i0. energy density for symmetric bilyer is clled We hve two terms contining i. first is independent of 8 nd r. If i is uniform on surfce integrtion of this term over surfce gives constnt vlue independent of of vesicle. shpe second term is correction term tht is liner in 8. size of correction is independent of rdii of curvture. It becomes relevnt when )i 8 cnnot be neglected s compred to unity. energy density decreses for i<0 nd increses for 0. consider deformtion in sddle with ri r r nd cll respective energy density 2 2 B j + f (I I) l i rd ln eqution (11) energy density vnishes for negtive i when r( 3. Discussion. Two kinds of modifiction of clssicl energy density (Eq. (1)) hve been found by for monolyers individully. first is correction consisting of severl terms ccounting become relevnt when (8/r) becomes comprble to 1. which (1) zero elstic energy to stte of inteml stress which previls if Eqution ssigns it. This is corrected by second modifiction. It consists of two terms which cn be # i understood s follows. Figure 18 shows schemticlly symmetric bilyers in cross intuitively No exteml bending moments re cting nd hence y re flt. Figure1A shows section. two cses for spontneous curvture of monolyers, corresponding to cone difference in curvture beten two lyers obtin net increse in energy which is to squre of curvture imposed in figure1c. This increse cn be proportionl from eqution (1) with f 0. Tking into ccount difference in curvture clculted
N 3 BENDING STIFFNESS OF LIPID BILAYERS. II 331, Fig. I.-Schemticl drwing of spontneous curvtures of two types ( : f 0, fl : i 0) of monolyers (A), (symmetric) bilyer formed from se monolyers without exteml bending moments present (B), bilyer bent by exteml bendblg moments (C). For detils see text. eqution (9) which is liner in i. By n nlogous rgument find tht energy increses cse fl. in modelling equilibrium shpes by minimiztion of elstic energy first term only In provided lipid species of different intrinsic shpes re present. seprtion second term is relevnt in pure s ll s in mixed bilyers. It is lrger greter series [1] it ws noted tht rtio beten isotropic modulus (K) nd bending this in single lyer bending (B) Ws not independent of type of lipid used. stiffness A B c " %w«jp fl(((flflfl #w p (it))(i( jil/ fl(it# %dnll# beten two lyers mkes energy increse in upper lyer smller nd decrese in lor lyer lrger. This leds to reduction in energy which corresponds to term in becomes relevnt when i is not uniform on surfce. This cn occur by lterl phse difference in spontneous curvture of two monolyers. For symmetric bilyers it becomes relevnt when (i 8 cnnot be neglected s compred to unity. Recent mesurement of geometry of lipid structures in excess hydrophilic nd excess hydrophobic solvent llod for first time mesurement of i. Two vlues hve been published: nm) for DOPE monolyer nd -1/(3.78 nm) for 3/1 mixture of DOPE nd -1/(3.15 [5]. If use 1nm (one qurter of bilyer thickness) for 8 obtin DOPC )i 8 0.3. This shows tht re is n pprecible correction to energy density in bending for frequently used bilyers. experimentl vlue of i used bove ws dopted from geometry of inverted hexgonl phses in wter nd tetrdecne s solvents [5]. In bilyer re is no orgnic solvent. It cn refore be rgued tht bsolute vlue of i in bilyer nd consequently its influence is much smller thn envisged bove. re comes, hover, circumstntil evidence from mesurements on bending stiffness of lipid bilyers. In first pper of
hd roughly hlf vlue for lipids contining 6 or 8 double bonds thn lipids contining BJK one or none [6]. According to continuum mechnicl model constnt rtio ws just expected [1]. If used eqution (9) to fit experimentl results insted of eqution (1) could mke rtio fijk constnt by choosing pproprite vlues for i (i m 0.5 for lipids with multiunsturted hydrocrbon chins). depends on geometry of bilyer since curvtures enter differently in eqution B nd (9) (even if neglect terms proportionl to 8). Second, t curvtures (1) simple model for con1jgted surfce consider squre 2D lttice of ltemting As nd depressions. lttice constnt is ssumed to be smll compred with elevtions depressions. With respect to energy density, elevtions nd depressions re To mke rough estimte of wher micro roughness cn exist in sttic equilibrium, clculte wher, when strting from flt configurtion bending energy is relesed in deformtion tht is now vilble for formtion of sphericl cp. To this purpose sddle. A negtive vlue of i would indicte stble corrugted configurtion. Setting r r r get : e j 2fi &2 332 JOURNAL DE PHYSIQUE II N 3 0 for lipids with sturted nd i 8 This choice is in qulittive ggreement to trend of i With degree of unsturtion. introduction of i seems to increse number of elstic constnts, but We show tht this is not cse. In clssicl description 3 independent prmeters re necessry to describe elstic behviour of symmetric bilyers. One possible choice is K, B, nd bending stiffness in bilyer couple bending. Anor choice is K, B, nd 8 since B cn be expressed by K nd 8 [1]. In new description prmeters would be K, 8, nd B since fi is chosen to depend on K. As furr refinement fi could depend on 8 s ll. i specil cse tht time scle nd geometry of deformtion re such tht For couple bending does not contribute [ii, B s obtined from fitting to eqution ii is bilyer s n pprent bending stiffness tht describes observtion tht bilyers of lipids useful with unsturted hydrocrbon chins bend more esily thn those with sturted ones. But even in this specil cse use of eqution ii) hs two drwbcks. First, vlue of comprble in bsolute vlue to )i( new effects emerge tht remin hidden when only eqution (I) is used. Some of se effects re described in next chpter. 4. Applictions. 4.I MICRO ROUGHNBSS. -To explin ir results in bilyer dhesion Helfrich nd coworkers [7, 8] postulted corrugted bilyer surfce. To support this hyposis Helfrich tht this ws due to energy terms proportionl to 4th por in curvture [9]. proposed developed in this pper suggests tht terms proportionl to second por of ory curvture re sufficient. rdius of vesicle. This mens side from corrugtions bilyer cn be considered to be flt. In center of ech squre is sddle point. On comers re elevtions nd This mens for ech sddle point re is one elevtion eir oriented outwrd or equivlent. inwrd. dd up two contributions given by equtions (lo) nd (I I) nd cll it i l + 4+ 4 f (12) r r We tret cse of pure bilyers. f cn refore considered to be uniform on surfce. Consequently terms proportionl to I( re omitted in eqution (12). Bilyer couple
1/(4 nm) for criticl vlue for i below which minimum exists. Figure 2 shows dependence of r s function of i. Nturlly se vlues do not give ctul rdii of surfce. In rigourous clcultion minimiztion must be done for whole corrugted In ddition bending energy postulted to be ssocited with sddle deformtion surfce. E C fi 5 < Cn z UJ' E o # UJO Fig. 2. -Rdius r nd energy density i of corrugted (symmetric) bilyer s function of i, spontneous curvture of monolyers. For detils see text. According to chpter 3, fi for lipids with unsturted hydrocrbon chins corresponds to (r) from eqution (12) 2. energy density of corrugted bilyer my be n order of mgnitude smller figure sddles nd elevtions re expected to contribute much more thn regions since corrugtion strongly increses with decresing i. Adhesion of flccid bilyers to loclly flt N 3 BENDING STIFFNESS OF LIPID BILAYERS. II 333 is not expected to ply n pprecible role since men curvture verged over bending bilyer surfce does not chnge when corrugtions form. We now look for rdii (r ) tht mke i miniml. derivtive of i with respect to r vnishes t r$ 8 8/(1 + 4 i 8 ). For i 3 1/4 this eqution hs rel solutions nd < second derivtive becomes positive, indicting minimum. Assuming 8 Inm obtin [10] would hve to be included. refore r nd criticl vlue of i my hve quite different vlues in rel bilyers. io -1/2-1/3-1/( ( flm x _ j z B for lipids with sturted hydrocrbon chins. To clculte i use 1.15 x10-l erg, vlue for B of DMPC bilyers [1il. vlues re plotted in inbeten. Neverless cn see from trend tht energy density stbilizing substrtes requires dhesive energy densities lrger thn se vlues. A flttening of postulted corrugtions by isotropic tensions ws suggested by Helfrich nd coworkers [7, 8] to be responsible for ir observtion of tension dependent dhesion. dt in figure 2 suggest tht below certin vlue of / energy stbilizing corrugtions s lrge s predicted by se uthors. Hover, if two bilyers composed of sme lipids is under sme isotropic tension corrugtions in both bilyers should be identicl. re tlis should llow close pposition nd refore tension independent dhesion.
provided re is excess surfce re [6]. In second mor vesicle is stble despite n to energy re considered: first rising from single lyer bending contributions (9)) nd second from bilyer couple bending [il. In third pper of this series [10] (Eq. dditionl elstic contribution is postulted which is due to deformtion of lipid n which does not preserve ir idelized xisymmetric shpe. This contribution is molecules dughter to be lrge enough to neglect term with 8. /( term is omitted since sphere (l/) is sme E 2 lf4rfi(r-2)((-) +2i8). 8 (14) AElf+lf. (15) ssumption defines verge spontneous curvture in bilyer couple bending chnge. be (R$ +R() In budded stte locl vlues of spontneous curvture in to fm id " > 334 JOURNAL DE PHYSIQUE II N 3 4.2 VEsicuLATioN. Vesicultion implies budding of dughter vesicles from mor vesicle. Two kinds of vesicultion hve been observed. first occurs spontneously excess surfce re but budding cn be induced by n increse in temperture [12]. Both mechnisms cn be explined bsed on concepts developed bove. 4.2, I Spontneous vesicultion. mechnism is demonstrted by mens of n exmple. We begin with sphericl mor nd reduce volume slightly by mechnism tht is not for rgumenttion. We ssume tht smll dughter vesicle (with rdius importnt budds from mor which is n sphericl (with rdius R). budding process R) occurs spontneously when bending energy is lor for budded stte. Two not tken into ccount s it does not chnge qulittive conclusion of nlysis below. presented energy stored in single lyer bending is obtined by integrtion of energy density surfce. For sphericl portions, use eqution (10). We ssume dimeter of over tret pure bilyers. Smll impurities would not chnge results. contribution of ech 8 rb(1 + 2 f 8 ). (13) As to neck, ssume tht this region hs shpe of inner hlf of torus. cler (N of torus nd its thickness prllel to its xis of symmetry re ssumed to be spn For simplicity, ssume curvture to hve everywhere sme vlue s on equl. midline of torus. We n obtin n expression for energy stored in neck (lf) from eqution (11) N For simplicity ssume elstic energy stored in mor before budding to be sme s if mor ws sphericl lthough this would not llow budding to tke plce due to constnt volume restriction. energy difference due to single lyer bending before nd fter budding (AE) is n sum of two contributions As to energy due to bilyer couple bending, ssume it to be zero before volume ws reduced. This is resonble if mor vesicle ws grown t sme temperture t which experiment ws performed. If it ws not cse finl conclusion would not couple bending re different from verge vlue. Tking dvntge from fct bilyer in sphere energy density in locl nd globl bending is sme obtin : tht R R (16)
(AE) concomittnt with budding obtin R 10 Lm nd R AE i II (3. I nm). < ssumptions mde on geometry of neck region nd dimeter of dughter vesicle re rbitrry. Using correct vlues might give different number for Complete seprtion would require n energy input of l/. For numericl estimte explins observtions. 4.2.2 Vesicultion upon temperture chnge. If i is lrger thn criticl vlue no 0) chnge ir equilibrium shpe towrds n exo- or endovesicu- stte when temperture is incresed [12]. To explin this finding bilyer couple lted ws invoked nd sequences of shpes re successfully modelled [13]. budding bending wrming. propose impurities with one or severl double bonds per hydrocrbon chin to be We for instbility nd hysteresis. At criticl vlue of inteml stress suggest responsible se molecules ccumulte in neck region in process chrcterized by positive tht pper. trnsitory opening of neck upon recooling hs been explined [12] by decrese in surfce re of vesicle nd constncy in its volume. reformtion of neck cn N 3 BENDING STIFFNESS OF LIPID BILAYERS. II 335 where f nd f denote spontneous curvtures in bilyer couple bending in mor nd dughter, respectively. For increse in elstic energy due to bilyer couple bending 8 rb( (I R f) + (l R f)) (17) Here B is bending stiffness in bilyer couple bending. budding is expected when sum AE + AE becomes negtive. To mke Spontneous estimte, tke I nm for 8 nd rbitrrily 10 nm for N nd s representtive vlues rough 0.2 Lm. From verge spontneous curvture nd eqution (16) obtin f nd f. For B tke 3fi [Ii. n AE + AE becomes negtive when criticl vlue of i below which spontneous vesicultion is expected. Unfortuntely kinds of lipid for which it ws observed re not specified [6]. It cn be shown tht AE decreses when decrese R while keeping totl surfce re constnt. This explins why very smll vesicles re observed [6]. To determine vlue of R 8 term from eqution (10) would hve to be crried over to equilibrium (13). eqution It is observed [6] tht fter budding dughters remin connected to mor. insert numbers for 8 nd N s bove. For fi use 28 kt [I II nd for i criticl vlue -1/(3.I nm). We obtin -lf 260kT. This is much lrger thn rml energies nd vesicultion is expected. Hover, vesicles mde of lipids with sturted spontneous chins (i hydrocrbon itself is often ccompnied by n instbility which could not be modelled with this nlysis. recooling hysteresis ws observed in tht non budded equilibrium shpe ws Upon t temperture (T) pprecibly below one where budding occurred during ttined feedbck. decrese in totl free energy would be equl in bsolute vlue to increse in elstic energy during wrming from T to temperture where budding occurred. To clculte this energy difference would require fitting procedure which is beyond scope of this be explined by much lrger locl concentrtion of impurities thn in wrming process.
ISRAELACHVILI J. N., MITCHELL D. J. nd NINHAM B. W., Biochim. Biophys. Act 470 (1977) 185. [3] HELFRICH W., Z. Ntufiorsch. 28C (1973) 693. [4] RAND R. P., FULLER N. L., GRUNER S. M. nd PARSEGIAN V. A., Biochemistry 29 (1990) 76. [5] EVANS E. nd RAWICz W., Phys. Rev. Lett. 64 (1990) 2094. [6] HELFRICH W., Liquid Crystls 5 (1989) 1647. [9] FISCHER T. M., J. Phys. II Frnce 2 (1992) 337. [10] I] DUWE H. P. nd SACKMANN E., Physic A 163 (1990) 410. [I KAS J. nd SACKMANN E., Biophys. J. 60 (1991) 825. [12] [13] BERNDL K., KAS J., LIPOWSKY R., SACKMANN E. nd SEWERT U., Europhys. Lett. 13 (1990) 659. 336 JOURNAL DE PHYSIQUE II N 3 Acknowledgements. I thnk Dipl. Phys. J. Khs, TU Munich for discussions nd for sending me preprint of his pper nd Dip]. Ing. A. Kshefi for his help in prepring figures. References FISCHER T. M., Biophys. J. submitted. [ii ISRAELACHVILI J. N., MARCELJA S. nd HORN R. G., Qurt. Rev. Biophys. 13 (1980) 121. [2] SERVUSS R. M. nd HELFRICH W., J. Phys. Frnce 50 (1989) 809. [7] MUTz M., SERvuss R. M. nd HELFRICH W., J. Phys. Frnce 51 (1990) 2557. [8]