Univerza v Ljubljani Fakulteta za elektrotehniko Miha Fošnaric Vpliv
Transcription
Univerza v Ljubljani Fakulteta za elektrotehniko Miha Fošnaric Vpliv
Univerza v Ljubljani Fakulteta za elektrotehniko Miha Fošnarič Vpliv anizotropnih in električnih lastnosti membrane na stabilnost membranskih mikro in nano struktur DOKTORSKA DISERTACIJA Mentor: prof. dr. Aleš Iglič Somentorica: prof. dr. Veronika Kralj-Iglič Ljubljana, 2004 Zahvala Veroniki Kralj–Iglič in Alešu Igliču. In seveda staršem. 4 KAZALO 1. Uvod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. Teoretični model anizotropne membrane . . . . . . . . . . . . . . . 17 2.1 Energija inkluzije v membranskem kontinuumu . . . . . . . . . 17 2.2 Anizotropne inkluzije v približku majhnih koncentracij . . . . 21 2.3 Membrana sestavljena iz ene vrste anizotropnih gradnikov . . 23 3. Elektrostatika membrane . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Električni potencial . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Elektrostatska prosta energija . . . . . . . . . . . . . . . . . . 31 4. Stabilizacija por v naelektreni lipidni dvojni plasti . . . . . . . . . . 33 4.1 Uvod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Teorija . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Rezultati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5. Stabilnost torocitnih oblik membranskih struktur . . . . . . . . . . 39 6. Stabilnost in sesedanje anorganskih mikro in nano cevk . . . . . . . 43 7. Sklep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 8. Pomembnejši originalni prispevki . . . . . . . . . . . . . . . . . . . 49 6 Kazalo Dodatki: originalne publikacije 57 A. H. Hägerstrand, J. Bobacka, M. Bobrowska-Hägerstrand, V. KraljIglič, M. Fošnarič in A. Iglič: Oxyethylene chain-cation complexation: Nonionic polyoxyethylene detergents attain a positive charge and demonstrate electrostatic head group interactions. Cell. Mol. Biol. Lett. (2001) 6: 161–165. . . . . . . . . . . . . . . . . . . . . . . . . 59 B. M. Fošnarič, A. Iglič in V. Kralj-Iglič: Influence of Anisotropic Membrane Properties on the Shape of the Membrane. V: Geometry, Integrability an Quantization III (I. M. Mladenov in G. L. Naber, ur.) Coral Press, Sofia, 2002, 224–237. . . . . . . . . . . . . . . . . . . . 65 C. M. Fošnarič, M. Nemec, V. Kralj-Iglič, H. Hägerstrand, M. Schara in A. Iglič: Possible role of anisotropic membrane inclusions in stability of torocyte red blood cell daughter vesicles. Colloid. Surface. B (2002) 26: 243–253. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 D. V. Kralj-Iglič, M. Remškar, G. Vidmar, M. Fošnarič in A. Iglič: Deviatoric elasticity as a possible physical mechanism explaining collapse of inorganic micro and nanotubes. Phys. Lett. A (2002) 296: 151–155. 91 E. M. Kandušer, M. Fošnarič, M. Šentjurc, V. Kralj-Iglič, H. Hägerstrand, A. Iglič in D. Miklavčič: Effect of surfactant polyoxyethylene glycol (C12 E8 ) on electroporation of cell line DC3F. Colloid. Surface. A (2003) 214: 205–217. . . . . . . . . . . . . . . . . . . . . . . . . . 97 F. M. Fošnarič, V. Kralj-Iglič, K. Bohinc, A. Iglič in S. May: Stabilization of pores in lipid bilayers by anisotropic inclusions, J. Phys. Chem. B (2003) 107: 12519–12526. . . . . . . . . . . . . . . . . . . . . . . 111 POVZETEK V doktorskem delu predstavimo teoretični model membrane, ki jo sestavljajo v splošnem anizotropni gradniki. Kot membranski gradnik obravnavamo molekulo ali pa skupek molekul v membrani, ki ga v izbranem teoretičnem modelu popišemo kot celoto. Za anizotropen gradnik je značilno, da njegova rotacijska stanja okoli normale na površino membrane v splošnem niso energijsko enakovredna. V predstavljenem delu poskušamo z anizotropnimi lastnostmi membranskih gradnikov pojasniti nekatere pojave v organskih in anorganskih mikro in nano strukturah, kot so stabilnost por v membranski dvojni plasti, stabilnost torocitnih oblik membranskih mehurčkov ter sesedanje anorganskih mikro in nano cevk. Vpeljemo teoretični model, ki pojasni povečano stabilnost por v lipidni dvojni plasti membrane kot posledico lateralne preporazdelitve anizotropnih membranskih gradnikov. Prosto energijo membrane obravnavamo kot vsoto energije roba pore, proste energije anizotropnih membranskih gradnikov in elektrostatske proste energije naelektrene membrane. Pri slednji upoštevamo, da membrano obdaja elektrolitska raztopina. Zaradi povečane energije roba je energijsko neugodno, da membrana tvori poro. Po drugi strani pa nastanek por v membrani zmanjšuje elektrostatsko prosto energijo membrane. Zaradi nastanka pore so namreč nekateri enako naelektreni gradniki membrane med seboj oddaljeni bolj kot v nepretrgani membrani. Zato elektrostatski prispevek k prosti energiji membrane favorizira čim večjo poro. Od naelektrenosti membrane in od ionske jakosti okolne raztopine je odvisno, ali bo prevladal pozitivni prispevek energije roba ali pa negativni prispevek elektrostatske proste energije. V prvem primeru se nastala pora zapre, v drugem primeru 8 Povzetek pa radij pore zelo naraste in s tem uniči membrano. Stabilne pore tako ne moremo razložiti z minimizacijo vsote energije roba in elektrostatske proste energije membrane. S predstavljenim modelom pokažemo, da se lahko primerno izbrani anizotropni gradniki naberejo na robu pore in z njega izrinejo izotropne lipidne molekule. Na ta način zmanjšajo energijo roba pore in elektrostatska energija lahko močno zniža prosto energijo membrane. Le-ta ima tako globok minimum pri končnem radiju pore, ki predstavlja stabilno ravnovesno stanje sistema. V nadaljevanju pokažemo, da lahko anizotropija membranskih gradnikov pojasni tudi stabilnost opaženih torocitnih oblik membranskih struktur. Torocit je mehurček, ki ima ploščat, tanek osrednji del ter odebeljen, toroiden zunanji del. V osrednjem delu mehurčka sosednji membrani nista v stiku. Stabilnosti opisanih mehurčkov ni mogoče razložiti z doslej znanimi izotropnimi modeli membrane. V doktorskem delu pokažemo, da se anizotropni membranski gradniki naberejo na odebeljenih robovih torocitnega mehurčka in tako stablizirajo značilno torocitno obliko. Na koncu predstavljeni koncept anizotropije tankih struktur uporabimo pri teoretični razlagi sesedanja anorganskih mikro in nano cevk. Predstavimo preprost model, ki vsebuje anizotropijo večplastne stene cevke. Pokažemo, da je lahko cevka pri majhnih debelinah stene cevke stabilna, pri večjih debelinah stene pa se sesede in splošči v trak. V doktorskem delu torej pokažemo, da imajo lahko anizotropne lastnosti tankih membranskih struktur pomemben vpliv na obliko membrane. Anizotropne lastnosti pridejo še posebej do izraza v primerih, kjer ima del membrane močno anizotropno geometrijo in velike ukrivljenosti. ABSTRACT In Doctoral thesis we present a theoretical model of a membrane composed of constituents that are in general anisotropic. As a membrane constituent we consider a molecule or a complex of molecules in the membrane that is in a given theoretical framework treated as a single membrane building block. Different rotational states of an anisotropic constituent around the axis normal to the membrane are in general not energetically equivalent. In presented work we use anisotropic properties of the membrane constituents to explain phenomena observed in organic and inorganic micro and nano-scale structures, like stability of pores in membrane bilayers, stability of torocyte vesicles and collapse of inorganic micro and nanotubes. We introduce a theoretical model explaining the increased stability of pores in membrane’s lipid bilayer as a consequence of lateral redistribution of anisotropic membrane constituents. We consider the free energy of the membrane as the sum of three contributions: energy of the edge of the pore, free energy of the anisotropic membrane constituents and electrostatic free energy of the electrically charged membrane immersed in an ionic solution. Due to increased energy of the edge of the pore it is energetically unfavorable for a membrane to form pores. On the other hand the creation of pores in the membrane lowers the electrostatic free energy of the membrane. In the membrane containing a pore, some of the membrane constituents with the same sign of charge are more apart than in the pore-free membrane. Therefore the electrostatic contribution to the free energy of the membrane favors large pores. It depends on the charge of the membrane and on the ionic strength of the surrounding solution, which one prevails: the positive contribution of the energy of the edge of the pore or the negative contribution 10 Abstract of the electrostatic free energy. In the first case the pore closes and in the second case the radius of the pore grows very large, thereby destroying the membrane. Stable holes therefore cannot be explained by minimizing the sum of the energy of the edge of the pore and the electrostatic free energy of the membrane. With the proposed model we show that properly chosen anisotropic membrane constituents accumulate on the edge of the pore, replacing some of the isotropic lipid molecules. Because of that, the energy of the edge is lowered. The electrostatic free energy can therefore significantly lower the total free energy of the membrane. This results in a deep minimum at some finite radius of the pore, corresponding to a stable equilibrium state of the system. Furthermore, we show that anisotropy of membrane constituents can explain the stability of the observed torocyte-like membrane structures. Torocyte is a vesicle having a thin plate-like central region and a toroidal periphery. In the flat central part of the vesicle the adjacent membranes are not in close contact. Stability of the described vesicles cannot be explained by the known isotropic models of the membrane. In the thesis we show that anisotropic constituents can accumulate in the toroidally shaped periphery of the vesicle, stabilizing the characteristic torocyte shape. The concept of the anisotropy of thin membrane structures shown in the end is used as a possible explanation for the collapse of inorganic micro and nanotubes. We present a simple model containing the anisotropy of the multilayered wall of the tube. We show that the tube can be stable at smaller thicknesses of the tube’s wall but collapses into a thin ribbon at larger thicknesses of the wall. In Doctoral thesis we therefore show that anisotropic properties of thin membrane structures can have important influence on the shape of the membrane. Anisotropic properties are especially important in cases where part of the membrane has largely anisotropic geometry and large curvatures. 1. UVOD Biološke celice obdaja biološka membrana, ki ločuje notranjost celice od okolja in opravlja še številne druge funkcije. Preko membrane poteka selektiven transport snovi iz celice in v njo. Celica mora pogosto znatno spremeniti svojo obliko zaradi vplivov okolja, ne da bi pri tem membrana utrpela poškodbe. Vse to kaže na pomembno vlogo in kompleksno zgradbo bioloških membran. Biološka membrana je sestavljena iz dvojne plasti fosfolipidnih molekul, v katero so vgrajene različne molekule, kot so integralni membranski proteini, peptidi, oglikovi hidrati (slika 1.1) [1]. Slika 1.1: Shema biološke membrane [2]. 12 1. Uvod Posamezno fosfolipidno molekulo v dvojni lipidni plasti sestavljajo polarna glava (ki ima lahko pozitiven ali negativen naboj ali pa je električno nevtralna) in dva nevtralna repa (slika 1.2) [1]. V vodni raztopini se fosfolipidne molekule ponavadi organizirajo v sisteme, v katerih so nevtralni repi skriti pred vodnimi molekulami, multipolne glave pa mejijo na okolno raztopino [3]. Pravimo, da imajo fosfolipidne molekule hidrofobne repe in hidrofilno glavo. Molekule, ki so sestavljene iz hidrofobnega in hidrofilnega dela, imenujemo tudi amfifilne molekule. Slika 1.2: Shema fosfolipidne molekule fosfatidilholina [2]. Prikazana je razdelitev v osnovne skupine (levo), kemijska zgradba (na sredini) in prostorska zapolnitev (desno). Dejavnikov, ki vplivajo na hidrofobno in hidrofilno naravo fosfolipidnih molekul, je veliko. Statistična mehanika nam pove, da se poskuša sistem amfifilnih molekul in vodne raztopine v ravnovesju organizirati tako, da je njegova prosta energija (F ) čim manjša [4, 5]. Iz zveze ∆F = ∆Wn − T ∆S vidimo, da lahko sistem pri konstantni temperaturi (T ) spreminja prosto 13 energijo preko notranje energije (Wn ) ali preko entropije (S). Na spremembo notranje energije sistema (∆Wn ) vpliva na primer elektrostatska interakcija med dipolnimi momenti vodnih molekul v raztopini in multipolnimi ali naelektrenimi glavami fosfolipidnih molekul. Kot entropijski prispevek k prosti energiji pa navedimo urejanje vodnih molekul v raztopini zaradi prisotnosti fosfolipidnih molekul. Če voda obliva repe fosfolipidnih molekul, se molekule vode okoli repov namreč uredijo tako, da se entropija (S) sistema vodnih molekul zmanjša [3]. Z drugimi besedami, prisotnost repov vodne molekule prisili k večji urejenosti, kot jo imajo v sami vodi, kar je energijsko neugodno. Fosfolipidne molekule se torej lahko zaradi svoje amfifilne narave organizirajo v dvojno lipidno plast, kjer so hidrofobni repi skriti v notranjosti plasti, hidrofilne glave pa so v stiku z zunanjo raztopino (slika 1.1). Takšna dvojna lipidna plast je debela 4-5 nm [6, 7] in je tanka v primerjavi z lateralnimi razsežnostmi membrane. Najmanše biološke celice imajo namreč premer približno 0.3 µm, medtem ko je lahko premer največjih bioloških celic tudi več kot 100 µm [7]. V zadnjih tridesetih letih je bilo razvitih več fizikalnih modelov, ki popisujejo biološke membrane kot dvodimenzionalno ploskev ali kot več zelo tankih plasti, ki so tesno naložene druga na drugo. Že leta 1970 je Canham študiral obliko eritrocita pri fizioloških pogojih v raztopini [8]. Eritrociti sesalcev so celice brez jedra in notranje strukture. Tako obliko eritrocita, pri dani prostornini, določajo lastnosti njegove membrane. Canham je membrano obravnaval kot homogeno elastično lupino z zaključeno površino. Predpostavil je, da ravnovesna oblika eritrocita, pri določeni površini membrane in določeni prostornini celice, ustreza minimumu upogibne energije njegove membrane. Za upogibno energijo tanke elastične membrane, homogene in izotropne v vseh treh razsežnostih, lahko zapišemo izraz [9] Z Z kc 2 Wb = (C1 + C2 ) dA + kG C1 C2 dA, (1.1) 2 A A kjer sta kc in kG elastični konstanti, C1 in C2 glavni ukrivljenosti membrane v izbrani točki, dA je element površine membrane, integrala pa tečeta po 14 1. Uvod celotni površini membrane (A). Drugi člen v izrazu (1.1) je pri nespremenjeni topologiji zaključenih ploskev konstanten (Gauss-Bonetov teorem) in ga pri minimizaciji upogibne energije lahko izpustimo. Ravnovesno obliko membrane dobimo tako, da rešimo variacijski problem z vezmi Wb = min, (1.2) Z dA = A, (1.3) dV = V. (1.4) A Z V Canham je približno rešitev problema (1.2) dobil z uporabo parametričnega modela za obliko celice. Predpostavil je, da je oblika eritrocita osnosimetrična in da obris celice v ravnini, ki vsebuje os simetrije, opiše triparametrična funkcija [8]. Tako je ob izbranih pogojih izračunal vrednosti parametrov funkcije in dobil ravnovesne oblike, ki se lepo ujemajo z opaženimi oblikami eritrocita. Helfrich je leta 1973 [10] upošteval tudi spontano ukrivljenost membrane (C0 ) in zapisal izraz za upogibno energijo v obliki: Z Z kc 2 Wbc = (C1 + C2 − C0 ) dA + kG C1 C2 dA. 2 A A (1.5) Kasneje so [11, 12, 13, 14] izrazu za upogibno energijo membrane dodali nov kvadratni člen, sorazmeren (∆A)2 , ki upošteva lateralno raztegovanje zunanje in notranje plasti lipidne dvojne plasti. ∆A je razlika med površinama nevtralnih ravnin zunanje in notranje plasti (nevtralna ravnina plasti imenujemo ploskev, ki ob upogibu plasti ohrani konstantno površino) in je podana z enačbo ∆A = h Z (C1 + C2 )dA, (1.6) A kjer je h razdalja med nevtralnima ravninama zunanje in notranje plasti. Z zgoraj opisanimi modeli in nekaterimi kasnejšimi dopolnitvami [15, 16, 17, 18], je bilo z minimizacijo elastične energije membrane uspešno pojasnjenih več opaženih oblik celic brez notranje strukture in oblik lipidnih mehurčkov. 15 Omenjeni modeli obravnavajo membrano kot kontinuum fosfolipidnih molekul, izotropen v ravnini membrane. Vendar pa se v naravi pojavljajo tudi oblike celic in lipidnih mehurčkov, ki jih z izotropnimi modeli ne moremo pojasniti [19, 20]. To nas napelje na misel, da v nekaterih primerih pridejo do izraza anizotropne lastnosti membrane. Izotropna inkluzija 90o 90o Anizotropna inkluzija 90o 90o Slika 1.3: Shema izotropne in anizotropne lipidne molekule. Pogosto je inkluzija skupek molekul, ki ga v izbranem modelu obravnavamo kot celoto. Vzrok za anizotropijo membrane so lahko same lipidne molekule, ki že zaradi dveh repov nimajo rotacijske simetrije glede na os normalno na površino membrane. Lahko pa k anizotropnim lastnostim membrane prispevajo druge molekule, ki so vgrajene v dvojno plast lipidnih molekul. Molekulo, vgrajeno v dvojno lipidno plast, skupek lipidov ali pa kombinacijo obojega, ki se v izbranem fizikalnem modelu obnaša kot celota, imenujemo s skupno besedo inkluzija (slika 1.3). V doktorskem delu bomo predstavili teoretični model, ki upošteva anizotropne lastnosti membrane (2. in 3. poglavje in dodatka A in B). Pokazali 16 1. Uvod bomo, da lahko anizotropne inkluzije stabilizirajo pore (luknje) nastale v dvojni lipidni plasti (4. poglavje in dodatek F). Omenjena ugotovitev ima lahko praktičen pomen pri izboljšavi metode elektroporacije, ki se v praksi že uporablja v biologiji in medicini. Pri tej metodi z zunanjim električnim poljem ustvarjajo pore v membrani in s tem povečujejo njeno prepustnost [dodatek E]. V nadaljevanju bomo pokazali, da anizotropne inkluzije v dvojni lipidni plasti lahko stabilizirajo opažene torocitne oblike bioloških membranskih struktur [5. poglavje in dodatek C]. Na koncu bomo podali tudi možno razlago za sesedanje organskih in anorganskih mikro in nano cevk [6. poglavje in dodatek D]. 2. TEORETIČNI MODEL ANIZOTROPNE MEMBRANE 2.1 Energija inkluzije v membranskem kontinuumu V dvojni lipidni plasti se plasti tesno prilegata druga k drugi zaradi hidrofobnega efekta. Plasti lahko skoraj prosto drsita ena ob drugi. V naših modelih bomo membrano celice obravnavali kot sestavljeno iz dveh lipidnih plasti, v katero so potopljene membranske inkluzije, na primer proteini. Predpostavili bomo, da sta plasti povsod v stiku. Izpeljali bomo prosto energijo membranske plasti za nekatere posebne primere. Tudi če bomo obravnavali samo posamezno membransko plast, bomo zaradi enostavnosti običajno uporabljali kar izraz membrana. Eksplicitno bomo zanemarili vse interakcije in izmenjavo molekul med membranskima plastema. V resnici membranski plasti seveda interagirata in izmenjujeta molekule. Poleg tega se nekatere molekule ne nahajajo samo v eni lipidni plasti ampak v obeh. Posamezno membransko plast obravnavajmo kot dvodimenzionalno ploskev. Potem je oblika membranske plasti v poljubni točki na plasti določena z glavnima ukrivljenostima membrane, C1 in C2 , v tej točki (slika 2.1). Zamislimo si poljuben, zelo majhen delček membrane, in ga imenujmo inkluzija. Inkluzija naj ima neko lastno obliko, ki jo popišemo z lastnima glavnima ukrivljenostima inkluzije, C1m in C2m . Včasih lastni obliki inkluzije rečemo tudi efektivna oblika inkluzije. Lastna oblika inkluzije namreč popisuje obliko membrane, ki bi inkluziji najbolj ustrezala in je posledica same inkluzije in njene interakcije z okolno membrano. Dopustimo možnost, da je inkluzija lahko anizotropna, torej da je lahko C1m 6= C2m . V splošnem se seveda oblika membrane na mestu inkluzije razlikuje od lastne oblike inkluzije. Inkluzija je vrtljiva okoli svoje osi normalne na mem- 18 2. Teoretični model anizotropne membrane R2 = 1/C2 R1 = 1/C1 Slika 2.1: Shema glavnih ukrivljenosti membrane. Ukrivljenost (C) je recipročna vrednost krivinskega radija (R). brano in gibljiva po membranski plasti. Lateralna gostota inkluzij torej v splošnem ni enaka povsod na membranski plasti. Izhodišče pravokotnega koordinatnega sistema postavimo na mesto izbrane inkluzije, njegovi osi pa naj sovpadata s smerema glavnih ukrivljenosti membranske plasti. V tem sistemu ima tenzor ukrivljenosti membrane diagonalno obliko, C= " C1 0 0 C2 # . (2.1) Zapišimo še tenzor lastne ukrivljenosti inkluzije v diagonalni obliki: Cm = " C1m 0 0 C2m # . (2.2) Ker je inkluzija anizotropna, v splošnem ni vseeno, kako je inkluzija zavrtena okoli svoje osi normalne na membrano. Naj bo ω kot zasuka lastnega sistema tenzorja lastne ukrivljenosti inkluzije glede na lasten sistem tenzorja ukrivljenosti membrane. Vpeljemo tenzor M [dodatka B in D], ki podaja neujemanje med lastno obliko inkluzije in obliko membrane na mestu inkluzije, M = R Cm R−1 − C, (2.3) 2.1. Energija inkluzije v membranskem kontinuumu 19 kjer je R rotacijska matrika, R= " cos ω − sin ω sin ω cos ω # . (2.4) Vpeljimo energijo inkluzije kot energijo, ki je potrebna, da obliko membrane na mestu inkluzije prilagodimo lastni obliki inkluzije. Torej čim večje je neujemanje med lastno obliko inkluzije in obliko membrane na mestu inkluzije, tem večja je energija inkluzije. Zavedati se moramo, da smo zapletene interakcije med inkluzijo in okolno membrano (elektrostatske multipolne interakcije, sterične interakcije, . . . ) pospravili v sorazmerno preprost geometrijski model. Da dobimo izraz za energijo inkluzije, uporabimo Landauov razvoj in energijo sistema razvijemo kot potenčno vrsto po primerno izbranem parametru urejenosti. Pri tem upoštevamo samo tiste člene vrste, ki so v skadu s simetrijskimi lastnostmi sistema. Za parameter urejenosti izberemo tenzor M in energijo inkluzije razvijemo v potenčno vrsto po komponentah tenzorja M. Energija mora biti seveda invariantna na rotacijo koordinatnega sistema. Iz linearne algebre vemo, da sta invarianti tenzorja njegova sled (Tr) in njegova determinanta (Det). Torej moramo člene potenčne vrste tvoriti kot kombinacijo sledi in determinante tenzorja M. Kot edini kandidat za linearni člen ostane TrM, ki pa ni invarianten na transformacijo M → −M in ga izpustimo. Za kvadratni člen lahko izberemo dve invarianti: kvadrat sledi in determinanto. Če se ustavimo pri kvadratnih členih, lahko energijo inkluzije aproksimiramo z izrazom [dodatka B in D] E= K (TrM)2 + K̄ DetM, 2 (2.5) kjer sta K in K̄ konstanti. Če uporabimo enačbe (2.1)-(2.5), lahko energijo inkluzije zapišemo v obliki [21] 2 E = (2K + K̄)(H − Hm )2 − K̄(D 2 − 2DDm cos (2ω) + Dm ), kjer sta 1 1 H = (C1 + C2 ), Hm = (C1m + C2m ) 2 2 (2.6) (2.7) 20 2. Teoretični model anizotropne membrane povprečna ukrivljenost membrane in povprečna lastna ukrivljenost inkluzije ter 1 1 (2.8) D = (C1 − C2 ), Dm = (C1m − C2m ) 2 2 deviator ukrivljenosti membrane in deviator lastne ukrivljenosti inkluzije. Deviator ukrivljenosti je nenegativna količina. Iz definicije glavnih ukri- vljenosti namreč velja, da je C1 ≥ C2 in C1m ≥ C2m . Deviator ukrivljenosti je prav tako kot povprečna ukrivljenost invarianta tenzorja ukrivljenosti, saj ga lahko izrazimo z determinanto in sledjo tenzorja ukrivljenosti, p p D = (TrC/2)2 − DetC = H 2 − C1 C2 . (2.9) Naj le omenimo, da se zaradi zgodovinskih razlogov [21] izraz (2.6) včasih zapiše s konstantama ξ in ξ ∗, ki sta s konstantama K in K̄ v preprostih zvezah, ξ = 4K + 2K̄ in ξ ⋆ = −4K − 6K̄. Izpeljali smo torej energijo inkluzije za primer, ko je lasten sistem lastne ukrivljenosti inkluzije za kot ω zavrten glede na lasten sistem ukrivljenosti membrane. Običajno pa se v bioloških membranah in liposomih inkluzija lahko prosto vrti okrog svoje osi normalne na membrano. Ponavadi je časovna skala orientacijskih sprememb inkluzije v membrani majhna v primerjavi s spremembami oblike membrane. Torej je smiselno uporabiti statističnomehansko povprečje po kotu ω. Fazni integral posamezne inkluzije (q), ki se lahko vrti okrog svoje osi normalne na membrano, je [22, 4, 5] Z 2π 1 E(ω) q= exp (− ) dω, ω0 0 kT (2.10) kjer je ω0 kvant kota (normalizacijska konstanta, ki jo pridelamo pri prehodu iz vsote v integral), k Boltzmannova konstanta in T absolutna temperatura. V faznem integralu lahko ločimo prispevek orientacijskih stanj qorient od prispevka ostalih stanj qc [21], q = qc qorient , 2K + K̄ K̄ 2 2 2 qc = exp − (H − Hm ) + (D + Dm ) , kT kT (2.11) (2.12) 2.2. Anizotropne inkluzije v približku majhnih koncentracij qorient 1 = ω0 Z 2π 0 2K̄ exp − DDm cos (2ω) dω. kT 21 (2.13) Integracija po kotu ω nam da qorient = 1 2K̄ I0 ( DDm ), ω0 kT (2.14) kjer je I0 modificirana Besselova funkcija. (1) Prosto energijo inkluzije dobimo iz enačbe Fi = −kT ln q (indeks (1) označuje, da gre za prosto energijo ene inkluzije) [4, 5]. Tako lahko zapišemo: 2K̄ (1) 2 2 2 DDm ) . (2.15) Fi = (2K + K̄)(H − Hm ) − K̄(D + Dm ) − kT ln I0 ( kT V zgornjem izrazu smo izpustili konstantni člen kT ln ω0 . 2.2 Anizotropne inkluzije v približku majhnih koncentracij Statistično-mehanski opis anizotropnih inkluzij v približku majhnih koncentracij Prosto energijo posamezne inkluzije smo torej zapisali z dvema invariantama tenzorja ukrivljenosti, s povprečno ukrivljenostjo H in z deviatorjem ukrivljenosti D. Zanima pa nas, kolikšna je prosta energija vseh inkluzij v membranski plasti. Oglejmo si enostaven primer, kjer membransko plast obravnavamo kot kontinuum, v katerem so porazdeljene inkluzije. Predpostavimo, da je lateralna koncentracija inkluzij majhna povsod v membrani. Inkluzije se lahko prosto gibljejo po membrani in se vrtijo okoli svoje osi normalne na membrano. Direktne interakcije med inkluzijami ne upoštevamo eksplicitno. Membransko plast površine A v mislih razdelimo na majhne koščke. Ti naj bodo dovolj majhni, da lahko ukrivljenost membrane vzamemo za konstanto po celotnem koščku in vendar dovolj veliki, da vsebujejo primerno število inkluzij za statistično-mehansko obravnavo. Izbrani košček obravnavamo kot sistem z dano površino Ap , dano ukrivljenostjo in dano temperaturo T ter z danim številom inkluzij M. Sistem je v termodinamičnem ravnovesju 22 2. Teoretični model anizotropne membrane in zanj velja kanonična porazdelitev. Predpostavimo še, da so inkluzije nerazločljive. Potem kanonični fazni integral za izbrani delček membranske plasti zapišemo kot Q = q M /M!, kjer je q fazni integral posamezne inkluzije, podan z enačbami (2.11), (2.12) in (2.14). Če poznamo fazni integral Q, lahko izračunamo prosto energijo inkluzij v izbranem delčku membranske plasti iz zveze F p = −kT ln Q [4, 5]. Sedaj uporabimo Stirlingovo formulo ln M! = M ln M − M, ki velja v limiti M → ∞. Potem lahko površinsko gostoto proste energije inkluzij zapišemo v obliki Fp 2K̄ = −kT m ln qc I0 ( DDm ) + kT (m ln m − m) + kT m ln(ω0 Ap ), (2.16) Ap kT kjer je m = M/Ap površinska gostota inkluzij. Da dobimo izraz za prosto energijo vseh inkluzij v membranski plasti, Fi , moramo sešteti prispevke vseh delčkov membrane. Zgornji izraz torej integriramo po celotni površini membranske plasti A, Z Fp Fi = dA. p A A (2.17) Ne vemo še, kako se inkluzije v termodinamičnem razvnovesju porazdelijo po membranski plasti. Zanima nas torej izraz za površinsko gostoto inkluzij, m. Prosta energija sistema, ki je pri konstantni temperaturi in ne prejema ali oddaja dela, doseže v termodinamičnem ravnovesju svoj minimum. Ali zapisano drugače, δFi = 0. Upoštevajmo še, da je število vseh inkluzij v membranski plasti (N) konstantno, Z m dA = N. (2.18) A Zgoraj definiran izoperimetrični problem lahko prevedemo v navaden variaR R cijski problem, če sestavimo funkcional Fi + λm A m dA = A L(m) dA, kjer je λm Lagrangeov multiplikator in L(m) Lagrangeova funkcija, 2K̄ L(m) = −kT m ln qc I0 ( DDm ) + kT (m ln m − m) + λ m. kT (2.19) V zgornjem izrazu je λ nov Lagrangeov multiplikator, λ = λm + kT ln(ω0 Ap ). 2.3. Membrana sestavljena iz ene vrste anizotropnih gradnikov ∂L ∂m Euler-Lagrangeova diferencialna enačba je torej 23 = 0. Ko odvajamo (2.19) po m in upoštevamo (2.18), dobimo za porazdelitev inkluzij v membranski plasti Boltzmannovo porazdelitveno funkcijo z dodatnim faktorjem, ki je modificirana Besselova funkcija I0 : K̄ qc I0 ( 2kT DDm ) m = n 1 A q I ( 2K̄ DDm ) A c 0 kT R dA . (2.20) Faktor qc je definiran z (2.12), n pa je površinska gostota inkluzij pri enakomerni porazdelitvi, n = N/A. Sedaj izraz za površinsko gostoto inkluzij v termodinamičnem ravnovesju (2.20) vstavimo v izraz (2.16) in integriramo po celotni površini membranske plasti. Izraz za prosto energijo inkluzij v membranski plasti zapišemo v obliki [21] 1 Fi = −kT N ln A 2K̄ qc I0 ( DDm ) dA , kT A Z (2.21) kjer smo izpustili konstantni člen kT N(ln(ω0 NAp /A) − 1). 2.3 Membrana sestavljena iz ene vrste anizotropnih gradnikov Poglejmo sedaj še primer, kjer je membranska plast sestavljena iz ene vrste gradnikov (ponavadi lipidnih molekul). Vsak gradnik membrane obravnavajmo kot inkluzijo. Izpeljava nas pripelje do že znanega izraza za upogibno energijo izotropne membrane (1.5), ki pa mu je treba prišteti še dodaten člen zaradi anizotropije (glej dodatek B in [20]). Inkluzije so sedaj seveda enakomerno porazdeljene po membranski plasti. Prosto energijo membranske plasti (Fm ), torej prosto energijo vseh inkluzij v membrani, dobimo z integracijo Fm = n Z A (1) Fi dA, (2.22) kjer je n površinska gostota inkluzij pri enakomerni porazdelitvi, n = N/A, (1) in Fi prosta energija ene inkluzije, podana z enačbo (2.15). 24 2. Teoretični model anizotropne membrane Zgornji integral lahko zapišemo kot Z 2K̄ Fm = Wb − nkT ln I0 ( DDm ) dA, 2kT A (2.23) kjer je drugi člen posledica anizotropije molekul. Prvi člen pa lahko, če uporabimo zvezo (2.9), zapišemo v obliki Z Z nK 2 (2H − C0 ) dA + nK̄ C1 C2 dA, Wb = 2 A A (2.24) kjer je C0 spontana ukrivljenost membrane, C0 = (2K + K̄)Hm /K. V izrazu (2.23) smo izpustili konstantna člena nKC1m C2m ter n(2K + (2K + 2 K̄)/2K)Hm . Izraz za Wb je že znan izraz za upogibno energijo izotropne membrane [10] in primerjava enačbe (2.24) z enačbo (1.5) nam da zvezi med konstantami: kc = nK in kG = nK̄. S pomočjo zgoraj opisanega modela je razložena stabilnost tubastih izrastkov, opaženih pri nekaterih liposomih [20, 23]. Omenimo še primer, ko se gradniki membrane ne morejo prosto vrteti okrog svoje osi normalne na membrano [dodatek D]. Takrat moramo v izrazu za prosto energijo membranske plasti (2.22), namesto statistično-mehanskega povrečja po kotu ω, vstaviti izraz za energijo majhnega delčka membrane (inkluzije), ki je zavrten za nek določen kot ω. Namesto izraza za prosto (1) energijo inkluzije, Fi , torej uporabimo enačbo (2.6). 3. ELEKTROSTATIKA MEMBRANE V prejšnjem poglavju smo govorili o energiji membrane, ki je posledica interakcij med gradniki membrane. Pri tem smo upoštevali samo interakcije kratkega dosega (reda razdalje med gradniki), na primer elektrostatske interakcije med multipoli in sterične interakcije [3]. Vse to smo za naš namen pospravili v enostaven geometrijski model. Vendar pa biološke membrane pogosto niso električno nevtralne. V tem primeru imamo opravka tudi z elektrostatskimi interakcijami med membrano in elektrolitsko raztopino, ki membrano obdaja. V tem poglavju bomo membrano obravnavali kot tanko, ravno in neskončno razsežno ploščo z okroglo odprtino, ki je v stiku z elektrolitsko raztopino. Predpostavili bomo tudi, da je plošča naelektrena, pri čemer je električni naboj enakomerno porazdeljen po plošči. Izpeljali bomo elektrostatski prispevek okrogle odprtine k prosti energiji naelektrene membrane. V ravnini z = 0 si zamislimo ploskev, ki zavzema površino A. Po tej ploskvi je enakomerno razmazan električni naboj s površinsko gostoto σ. Ploskev je na eni strani v stiku z elektrolitsko raztopino. Ker je ploskev naelektrena, privlači ione nasprotnega predznaka (pravimo jim protiioni) in odbija ione istega predznaka (pravimo jim koioni), pri čemer ureditvi ionov nasprotuje termično gibanje delcev v raztopini. Razen privlačnih elektrostatskih interakcij med naelektreno ploskvijo in protiioni in odbojnih interacij med nabito ploskvijo in koioni delujejo v raztopini še druge interakcije, kot na primer elektrostatske interakcije med ioni, van der Waalsove interakcije in sterične interakcije [3, 24]. V ravnovesnem stanju se protiioni v povprečju naberejo v bližini naelektrene ploskve s površinsko gostoto naboja σ, koionov pa je v tem območju 26 3. Elektrostatika membrane Naelektrena Naelektrena difuzna plast ploskev - + + + + + + + + + + - + + + + + + + + + + + + + + + + + + + + + + - + - + + - - - + + + + - 0 - - + + - + + - - z Slika 3.1: Shema električne dvojne plasti, sestavljene iz negativno naelektrene plasti pri z = 0, in v povprečju pozitivno naelektrene difuzne plasti. Kationi (+) se v povprečju naberejo v bližini naelektrene ploskve, anionov (−) pa je v tem območju v povprečju manj. Daleč stran od naelektrene plošče je kationov toliko kot anionov. v povprečju manj. Nastane električna dvojna plast, ki jo sestavljata naelektrena plast pri z = 0, in v povprečju nasprotno naelektrena difuzna plast, ki se razteza v raztopino [25, 26, 27] (slika 3.1). Ker naelektrena ploskev pritegne v svojo bližino protiione, je njen vpliv na izbran protiion globlje v raztopini zaradi senčenja zmanjšan. Pravimo, da oblak protiionov, ki nastane v bližini nabite ploskve, zasenči naboj na ploskvi. Vpliv ploskve na naelektrene delce pada z razdaljo od ploskve in na dovolj veliki razdalji postane zanemarljiv [27, 3]. Električno dvojno plast opišemo z makroskopskimi količinami, kot so električni potencial φ, koncentracije vseh vrst ionov, dielektričnost raztopine ε in temperatura T . Namen naslednjega podpoglavja je opisati vpliv okrogle odprtine v naelektreni ploskvi na krajevno odvisnost električnega potenciala v ravni električni dvojni plasti [28]. Takšen primer nam bo kasneje prišel prav v poglavju 4, kjer bomo obravnavali poro v naelektreni lipidni dvojni plasti. 3.1. Električni potencial 27 3.1 Električni potencial v ravni električni dvojni plasti z okroglo odprtino Površina naelektrene ploskve v stiku z elektrolitsko raztopino naj bo dovolj velika, da lahko vplive robov zanemarimo. Raztopina zavzema prostornino, ki je omejena z naelektreno ploskvijo v ravnini z = 0, v smeri pozitivne osi z pa se razteza od z = 0 do z = d. Vzamemo, da je razdalja d, do katere se razteza raztopina, tako velika, da je tam vpliv nabite ploskve zanemarljiv. Električno dvojno plast torej obravnavamo kot neskončno razsežno v ravnini z = 0 in neskončno razsežno v smeri pozitivne osi z. Omejimo se na posebno preprost sistem, v katerem elektrolitsko raztopino sestavljajo enovalentni kationi in enovalentni anioni. Če je ploskev pri z = 0 negativno nabita (σ < 0), privlači katione in odbija anione. Kationi se naberejo v bližini z = 0, anionov je pa tam manj kot daleč stran od plošče. Daleč stran od ravnine z = 0 je kationov toliko kot anionov (slika 3.1). Zanima nas potencial povprečnega električnega polja (φ), v odvisnosti od oddaljenosti od ravnine z = 0. Rešujemo Poisson-Boltzmannovo enačbo [27, 3, 24]: ∇2 φ = − ρ εε0 (3.1) kjer je volumska gostota naboja v elektrolitski raztopini porazdeljena v skladu z Boltzmannovo porazdelitvijo : ρ = e0 nd e−e0 φ/kT − e0 nd ee0 φ/kT , (3.2) nd je številska gostota ionov obeh vrst daleč stran od naelektrene plošče, k Boltzmannova konstanta, e0 osnovni naboj, ε0 dielektrična konstanta in T absolutna temperatura. Enačbi (3.1) in (3.2) lahko zapišemo tudi v obliki : 2e0 cd NA e0 φ ∇ φ= , sinh ε0 ε kT 2 (3.3) kjer je cd = nd /NA koncentracija obeh vrst ionov daleč stran od naelektrene plošče in NA Avogadrovo število. 28 3. Elektrostatika membrane V nadaljevanju se zaradi lažjega računanja omejimo na primer, ko je razmerje e0 φ kT tako majhno, da lahko desno stran Poisson-Boltzmannove enačbe (3.3) razvijemo v vrsto (obdržimo samo člene prvega reda), ki tako preide v obliko: ∇2 φ = κ2 φ, (3.4) s (3.5) kjer je κ= 2cd NA e20 . εε0 kT Enačbi (3.4) pravimo linearizirana Poisson-Boltzmannova enačba, parametru 1/κ pa Debyeva debelina električne dvojne plasti. Rešitev linearizirane Poisson-Boltzmannove enačbe za primer neskončno razsežne naelektrene plošče brez luknje (φ∞ ), ki zadošča robnim pogojem, φ(z → ∞) = 0, (3.6) dφ σ (z = 0) = − , dz ǫǫ0 (3.7) lahko zapišemo v obliki [27]: φ∞ (z) = σ −κz e . ǫǫ0 κ (3.8) Krajevno odvisnost električnega potenciala ravne, neskončno razsežne naelektrene plošče s površinsko gostoto naboja σ in luknjo radija R0 zapišemo kot razliko med potencialom neskončne ravne plošče (φ∞ ) in potencialom krožne plošče (φp ) z radijem R0 (obe s površinsko gostoto naboja σ ter v stiku z enako elektrolitsko raztopino) [dodatek F]: φ = φ∞ − φp . (3.9) Krajevno odvisnost električnega potenciala ravne krožne plošče s površinsko gostoto naboja σ in radija R0 izračunamo v cilindričnih koordinatah. 3.1. Električni potencial 29 Linearizirano Poisson-Boltzmannovo enačbo v cilindričnih koordinatah za osnosimetrični primer zapišemo v obliki : ∂ 2 φp 1 ∂ ∂φp (r )+ = κ2 φp , r ∂r ∂r ∂z 2 (3.10) kjer postavimo izhodišče koordinatnega sistema v središče krožne plošče, os z pa kaže v smeri normale na naelektreno ploščo. Rešitev enačbe (3.10) iščemo z nastavkom φp (r, z) = R(r)Z(z), (3.11) kjer je R(r) samo funkcija r in Z(z) samo funkcija z. Tako dobimo: 1 1 ∂ ∂R 1 ∂2Z (r )+ = κ2 . R r ∂r ∂r Z ∂z 2 (3.12) Ker je prvi člen v enačbi (3.12) odvisen samo od koordinate r, drugi člen pa samo od koordinate z, je njuna vsota vedno enaka konstanti κ2 le v primeru, da sta oba člena konstantna: 1 ∂2Z = κ2 + k 2 , Z ∂z 2 (3.13) 1 1 ∂ ∂R (r ) = −k 2 . R r ∂r ∂r (3.14) Splošna rešitev enačbe (3.13) ima obliko: Z = Ce− √ κ2 +k 2 z √ + C1 e κ2 +k 2 z . (3.15) Ob upoštevanju robnega pogoja φ = 0 v limiti z → ∞ postavimo v enačbi (3.15) konstanto C1 = 0: Z = Ce− √ κ2 +k 2 z . (3.16) Enačbo (3.14) zapišemo v obliki: r2 ∂2R ∂R +r + k 2 r 2 R = 0. 2 ∂r ∂r (3.17) 30 3. Elektrostatika membrane Regularna rešitev diferencialne enačbe (3.17) je Besslova funkcija ničtega reda [29] R = CJ0 (kr). (3.18) Rešitev enačbe (3.10) v obliki φp (r, z) = R(r)Z(z) (en.3.11) je tako: φp (r, z) = C(k) J0 (kr)e− √ κ2 +k 2 z . (3.19) Splošna rešitev enačbe (3.10) pa je: Z ∞ √ 2 2 φp (r, z) = dk C(k) J0 (kr)e− κ +k z . (3.20) 0 V nadaljevanju enačbo (3.20) odvajamo po koordinati z: Z ∞ √ √ ∂φp (r, z, ) 2 2 = dk (− κ2 + k 2 )C(k) J0 (kr)e− κ +k z , ∂z 0 potem pa postavimo z = 0. Tako dobimo: Z ∞ √ ∂φp (r, z) |z=0 = dk (− κ2 + k 2 )C(k) J0 (kr). ∂z 0 (3.21) (3.22) Z uporabo Hanklove transformacije [29] iz enačbe (3.22) izračunamo koeficiente C(k): k C(k) = − √ κ2 + k 2 Z ∞ dr rJ0 (kr) 0 ∂φp (r, z) |z=0. ∂z (3.23) Integral v enačbi (3.23) razdelimo na dva dela: k C(k) = − √ κ2 + k 2 + Z ∞ R0 Z R0 dr rJ0 (kr) 0 ∂φp (r, z) |z=0+ ∂z ∂φp (r, z) |z=0 , dr rJ0 (kr) ∂z (3.24) in upoštevamo robna pogoja: ∂φp σ (z = 0) = − ∂z ǫǫ0 , r < R0 , (3.25) 3.2. Elektrostatska prosta energija ∂φp (z = 0) = 0 , ∂z r ≥ R0 . 31 (3.26) Tako dobimo σ √ C(k) = kǫǫ0 κ2 + k 2 Z R0 d(kr) krJ0 (kr). (3.27) 0 Ob upoštevanju zveze [29], kR0 J1 (kR0 ) = Z R0 d(kr)krJ0 (kr), (3.28) 0 iz enačbe (3.27) sledi C(k) = σR0 J1 (kR0 ) √ . ǫǫ0 κ2 + k 2 (3.29) Dobljeni izraz za koeficiente C(k) vstavimo v enačbo (3.20), tako dobimo: Z σR0 ∞ J0 (kr)J1 (kR0 ) −√κ2 +k2 z dk √ . (3.30) e φp (r, z) = ǫǫ0 0 κ2 + k 2 Iz enačb (3.8), (3.9) in (3.30) pa sledi izraz za krajevno odvisnost električnega potenciala neskončno razsežne naelektrene plošče v stiku z elektrolitsko raztopino [30]: σ −κz σR0 φ(r, z) = e − ǫǫ0 κ ǫǫ0 Z ∞ dk 0 J0 (kr)J1 (kR0 ) −√κ2 +k2 z √ e , κ2 + k 2 (3.31) kjer je R0 radij krožne luknje v plošči. 3.2 Elektrostatska prosta energija električne dvojne plasti z okroglo odprtino V prejšnjem podpoglavju smo izpeljali izraz za krajevno odvisnost električnega potenciala neskončno razsežne naelektrene plošče z okroglo odprtino, ki je v stiku z elektrolitsko raztopino. Sedaj bi radi za ta sistem zapisali še elektrostatsko prosto energijo. Izraz za izračun proste energije lahko zapišemo po naslednjem kratkem razmisleku [27]. Predstavljajmo si ploščo, ki je v stiku z elektrolitsko raztopino in je v začetku električno nevtralna. Nato na delček plošče izotermno 32 3. Elektrostatika membrane in reverzibilno nanesemo električni naboj do končne gostote električnega naboja σ. Pri tem predpostavimo, da je izbrani delček plošče sicer popolnoma izoliran od okolice. Ker smo nanesli naboj izotermno in reverzibilno, je sprememba proste energije izbranega delčka plošče kar enaka dovedenemu delu zaradi dodajanja električnega naboja. To spremembo proste energije delimo s površino obravnavanega delčka plošče in dobimo površinsko gostoto elektrostatske proste energije (uel ) [27]: Z σ uel = φσ′ (z = 0) dσ ′ , (3.32) 0 kjer je φσ′ (z = 0) električni potencial na površini plošče pri površinski gostoti električnega naboja σ ′ . V našem primeru, ko imamo opravka z linearizirano Poisson-Boltzmannovo enačbo (3.4), je potencial na površini plošče, φ(z = 0), linearno odvisen od površinske gostote naboja σ [31, 32]. To je razvidno tudi iz enačbe (3.31). Rσ Tako je integral v enačbi (3.32) sorazmeren z 0 σ ′ dσ ′ = 21 σ 2 in površinsko gostoto elektrostatske proste energije lahko zapišemo kot uel = 1 φ(z = 0) σ. 2 (3.33) Skupno elektrostatsko prosto energijo dobimo z integracijo izraza (3.33) po celotni nabiti površini. V našem primeru naj bo tanka ravna plošča (oziroma membrana) z obeh strani v stiku z elektrolitsko raztopino. Na obeh straneh plošče naj bo enakomerno porazdeljen površinski naboj σ, razen seveda v okrogli odprtini z radijem R0 . Skupna elektrostatska prosta energija je potem Z ∞ Uel,tot = 2πrσφ(r, z = 0) dr. (3.34) R0 V zgornjo enačbo vstavimo izraz (3.31) za elektrostatski potencial pri R z = 0, uporabimo zvezo xJ0 (x) dx = xJ1 (x) in po krajšem računu dobimo [30] πσ 2 R02 2πσ 2 Uel = − + ǫǫ0 κ ǫǫ0 Z ∞ 0 J (x)2 p 1 dx. x2 + κ2 R02 (3.35) 4. STABILIZACIJA POR V NAELEKTRENI LIPIDNI DVOJNI PLASTI Z ANIZOTROPNIMI INKLUZIJAMI 4.1 Uvod V bioloških celicah poteka izmenjava snovi preko celične membrane. Eden izmed možnih načinov za prehod snovi preko membrane je nastanek por v lipidni dvojni plasti membrane. Pore v membrani so opazili na primer pri eritrocitih [33, 34, 35], kjer je velikost pore odvisna od koncentracije ionov v okolni raztopini [34]. Nastanek por lahko povzroči tudi električno polje. Metodo, pri kateri uporabljajo električno polje za nastanek por v membrani, imenujemo elektroporacija in je razširjena metoda v medicini in biologiji [36, 37, 38, 39, 40, 41]. Nenazadnje nastanek por igra pomembno vlogo tudi pri funkciji mnogih protimikrobnih peptidov v membrani [42]. V tem poglavju bomo predstavili teoretični model [dodatka F in E], s katerim poskušamo razložiti stabilizacijo por v lipidni dvojni plasti z anizotropnimi inkluzijami. 4.2 Teorija Obravnavamo ravno membrano s poro (odprtino), ki naj ima rotacijsko simetrijo okoli osi, ki gre skozi središče pore in je pravokotna na ravnino membrane. Membrana naj bo sestavljena iz izotropnega lipidnega dvosloja, v katerega so vgrajene tudi anizotropne inkluzije. Iz slike 4.1 je razvidno, da je edini prosti parameter, ki določa geometrijo sistema, radij pore. Zanima nas ravnovesno stanje sistema, torej radij pore (r), pri katerem prosta energija sistema doseže minimum. Predpostavimo, da je prosta ener- 34 4. Stabilizacija por v naelektreni lipidni dvojni plasti z 2Ri b r x Slika 4.1: Shematski prikaz ravne dvojne lipidne plasti s poro v središču [dodatek F]. Slika prikazuje prečni prerez v ravnini x-z. Os rotacijske simetrije sovpada z osjo z. Na levi je shematsko prikazana ureditev lipidnih molekul. Polarne glave lipidnih molekul so prikazane s polnimi krogi. Puščica kaže na membransko inkluzijo, ki je prav tako prikazana shematsko. Zaradi nazornosti je prikazana le ena membranska inkluzija. gija pore vsota treh prispevkov [dodatek F]: F = Wrob + Uel + F̃i , (4.1) kjer je Wrob energija roba pore zaradi površinske napetosti lipidnega dvosloja, Uel je elektrostatska prosta energija zaradi električnega naboja membrane in F̃i je prosta energija inkluzij v membranski plasti. Energijski prispevek Wrob je posledica reorganizacije lipidnih molekul na robu pore (slika 4.1). Za lipidni dvosloj brez anizotropnih inkluzij je energijski prispevek roba kar 2πΛr, kjer je r radij krožne odprtine, Λ pa je energija na enoto dolžine roba pore (površinska napetost lipidnega dvosloja). Ker so v lipidnem dvosloju prisotne tudi membranske inkluzije, le-te zamenjajo (izrinejo) nekatere izmed lipidnih molekul. Izrinjene lipidne molekule tako ne prispevajo več k energiji roba (Wrob ). To upoštevamo v formuli za energijski prispevek in zapišemo [dodatek F]: Wrob = 2Λ (πr − NP Ri ) , (4.2) kjer je NP število inkluzij v robu pore in 2Ri premer prečnega preseka inkluzije (slika 4.1). 4.3. Rezultati 35 Za elektrostatski prispevek k prosti energiji uporabimo kar izraz za tanko, neskončno in enakomerno nabito ploščo s krožno odprtino, ki je v stiku z elektrolitsko raztopino. Za Uel lahko torej zapišemo izraz, ki smo ga izpeljali v poglavju 3 (enačba (3.35)): πσ 2 r 2 2πσ 2 + Uel = − ǫǫ0 κ ǫǫ0 Z 0 ∞ √ J1 (x)2 dx. x2 + κ2 r 2 (4.3) Izraz za prosto energijo inkluzij prav tako že poznamo iz poglavja 2.2. Za referenčno stanje izberemo ravno membrano brez pore, torej F̃i = Fi − Fi (H = D = 0). (4.4) Vstavimo v zgornji izraz enačbo (2.21) in odštejemo referenčno stanje, pa dobimo 1 F̃i = −kT N ln A kjer je 2K̄ qc I0 ( DDm ) dA , kT A Z K̄ 2 2K + K̄ 2 qc = exp − (H − 2HHm ) + D . kT kT (4.5) (4.6) Integracija v enačbi (4.5) sicer teče po celotni površini membrane, vendar je od nič različna samo integracija po robu pore, kjer membrana ni ravna. Ker je površina ukrivljenega roba pore (Ap ) veliko manjša od celotne površine membrane, lahko logaritem v enačbi (4.5) razvijemo in prosto energijo inkluzij zapišemo v obliki F̃i = nAP − NP , kT kjer je število inkluzij v robu pore Z 2K̄ DDm dAP NP = n qc I0 kT (4.7) (4.8) AP in n površinska gostota inkluzij pri enakomerni porazdelitvi, n = N/A. 4.3 Rezultati Izberimo za debelino lipidne plasti b = 2.5 nm, za površinsko napetost Λ = 10−11 J/m in za površinsko gostoto električnega naboja σ = −0.05 As/m2 = 36 4. Stabilizacija por v naelektreni lipidni dvojni plasti −e0 /3.2 nm2 . Če upoštevamo, da je površina membrane, ki jo zavzema ena lipidna molekula 0.6-0.8 nm2 , izbrana vrednost za σ pomeni, da je približno vsaka četrta lipidna molekula električno nevtralna. Tri četrtine lipidnih molekul pa so monovalentno naelektrene (imajo torej osnovni naboj −e0 ). Takšno razmerje je v bioloških membranah pogosto. 10 0 F kT −10 −20 −30 .. . .... .... (a) ..... ... (b) .... ... . .... ............ .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . .... .. ................ ............... .... .... .... .... .... .............................. .... .... .................... ........... .... ................ ...... .............. .... ... ........................... . . . ......... . . . . .... .... ...... .. .......... .......... ....................... .... .... ......... ................... ... .. .... .... . . . . . .... .... ..... .... .... ..... ..... ... ............... ... . .... . ... ... ... .. ... ... ... ... .... .. ... . .. .. .. .. ... .. ... .. .... .. . . . . ... ... ... ... ... ... P ... ... ... ... ... ... ... ... ... ... .. .. . . ..... . .. .. . . . ...... . ... .......... ... ............... ... ... .. .......... ... ... ... ...................... ......... ... ... .. ... . ... .... ... ... ...... (d) (c) (e) 6 4 2 0 (e) N (d) 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r/b Slika 4.2: Prosta energija pore (F ) kot funkcija radija pore (r) [dodatek F]. Črtkane črte ustrezajo membrani brez inkluzij, s površinsko gostoto naboja σ = −0.05 As/m2 ter z Debyevo dolžino ld = 2.6 nm (a), ld = 2.8 nm (b) in ld = 3.0 nm (c). Povezane črte ustrezajo membrani z anizotropnimi inkluzijami (z lastnostmi K = 98 kT /nm2 , K̄ = −2K/3, C1m = −C2m = 1/b) in s površinsko gostoto naboja σ = −0.05 As/m2 ter Debyevo dolžino ld = 2.8nm. Ravnovesna površinska gostota inkluzij je n = 1/70000nm2 (d) in n = 1/14000nm2 (e). Notranja slika prikazuje število inkluzij v robu pore (NP ), kot funkcijo r/b (za krivulji d in e). Debelina lipidne plasti je b = 2.5 nm. Slika 4.2 prikazuje, kako anizotropne inkluzije vplivajo na stabilnost in velikost pore v membrani. Če v membrani ni anizotropnih inkluzij, je prosta energija samo vsota energije roba in elektrostatskega prispevka. Prvi poskuša poro zapreti, saj je rob energijsko neugoden, drugemu pa bi ustrezala 4.3. Rezultati 37 čim večja pora. S tekmovanjem med obema energijskima prispevkoma ne moremo dobiti minimuma proste energije, ki bi lahko ustrezal ravnovesnemu stanju sistema. To sta ugotovila že Betterton in Brenner [30]. Z dodajanjem anizotropnih inkluzij, ki jim geometrija roba pore ustreza, pa lahko nastane globok minimum proste energije pri določenem radiju pore. Anizotropne inkluzije se namreč naberejo v robu pore in iz njega izrinejo del izotropnih lipidnih molekul. Tako se pozitivni prispevek energije roba pore zmanjša in elektrostatska energija precej zniža skupno prosto energijo pore. Vidimo torej, da je z vgrajevanjem primernih molekul v biološke membrane mogoče povečati stabilnost membranskih por. Na to kažejo tudi eksperimentalni rezultati. Izkaže se [dodatek E], da v ravno lipidno dvojno plast dodane molekule oktaetilenglikol-dodeciletra (C12 E8 ) znižajo električno napetost potrebno za ireverzibilno elektroporacijo. Drugi poskusi (glej dodatek A in [43]) nakazujejo, da amfifilne molekule C12 E8 v membrani tvorijo anizotropne inkluzije [dodatek C]. Oboje se sklada z napovedmi zgoraj opisanega teoretičnega modela. Omeniti je potrebno, da v tem poglavju opisani teoretični model ne popisuje nastanka, ampak zgolj stabilnost por v lipidni dvojni plasti. 38 4. Stabilizacija por v naelektreni lipidni dvojni plasti 5. STABILNOST TOROCITNIH OBLIK MEMBRANSKIH STRUKTUR V membrano eritrocita vezane amfifilne molekule oktaetilenglikoldodeciletra (C12 E8 ) interagirajo s sosednjimi lipidnimi molekulami in tvorijo z njimi komplekse, ki jih imenujemo inkluzije C12 E8 . Vezava molekul C12 E8 pretežno v notranjo plast membrane eritrocita povzroči gubanje membrane navznoter [dodatek A] (slika 5.1). Proces se lahko nadaljuje v nastanek notranjih hčerinskih mehurčkov (endovesiklov), ki imajo značilno torocitno obliko [43] (slika 5.2). Takšni hčerinski mehurčki imajo sploščen, tanek osrednji del ter odebeljen, torusu podoben zunanji del. Kljub temu da je debelina tankega osrednjega dela mehurčka le nekaj debelin membrane, pa membrani nista v stiku. Slika 5.1: Slika eritrocita [44], ki se mu membrana naguba navznoter (stomatocit) zaradi vezave amfifilnih molekul C12 E8 v pretežno v notranjo plast membrane eritrocita. Podobne torocitne oblike najdemo tudi v drugih bioloških sistemih. Toro- 40 5. Stabilnost torocitnih oblik membranskih struktur 1ìm Pogled s strani 100 nm Preèni prerez Slika 5.2: Levo: Slika torocita [43], posneta s transmisijskim elektronskim mikroskopom - prečni presek. Lepo se razloči tanek osrednji del in odebeljen zunanji del mehurčka. V tankem osrednjem delu sosednji membrani nista v stiku. Desno: Shema torocita. citne oblike so na primer mehurčki, ki sestavljajo Golgijev aparat v bioloških celicah. Predstavili bomo teoretični model, ki predstavlja možno razlago za stabilnost torocitnih bioloških membranskih struktur [dodatek C]. Stabilne oblike mehurčkov iščemo z minimizacijo proste energije membrane mehurčka, pri dani prostornini mehurčka in dani površini njegove membrane (glej poglavje 1). Prosto energijo membrane zapišemo kot vsoto F = Wb + Fzun + Fnot . (5.1) Pri tem je Wb upogibna energija izotropne membrane brez inkluzij. Zanjo uporabimo izraz (1.1) iz poglavja 1. Prispevka Fzun in Fnot sta prosti energiji membranskih inkluzij C12 E8 v zunanji in notranji plasti membrane mehurčka. Zanju uporabimo enačbo (2.21) iz poglavja 2.2, Z 1 2K̄ ∓ Fzun/not = −kT N ln q I0 ( DDm ) dA , A A c kT (5.2) pri čemer moramo upoštevati, da se v notranji plasti predznaki ukrivljenosti membrane spremenijo. V izrazu za qc (2.12), 2K + K̄ K̄ ∓ 2 2 2 qc = exp − (H ∓ Hm ) + (D + Dm ) , kT kT (5.3) je torej predznak pred Hm negativen za inkluzije v zunanji plasti in pozitiven za inkluzije v notranji plasti. Razliko v ukrivljenostih zunanje in notranje plasti zaradi končne debeline membrane zanemarimo. 41 Omenili smo že, da se pri nastanku torocitnih mehurčkov zaradi vezave molekul C12 E8 v membrano eritrocita, te vežejo pretežno v notranjo plast membrane. Ko iz invaginacije nastane hčerinski mehurček v eritrocitu, je torej večina molekul C12 E8 v njegovi zunanji plasti, saj notranja plast v membrani eritrocita postane zunanja plast v membrani endovesikla. Slika 5.3 kaže izračunano [dodatek C] stabilno torocitno obliko mehurčka, pri čemer je 80 % anizotropnih membranskih inkluzij v zunanji in 20 % v notranji plasti. Na sliki je prikazana tudi porazdelitev anizotropnih inkluzij v zunanji in notranji plasti. Izkaže se, da lahko anizotropne inkluzije s primerno izbrano efektivno obliko stabilizirajo torocitne oblike membranskih struktur. S slike je razvidno, da se v zunanji plasti anizotropne inkluzije porazdelijo večinoma po odebeljenem robu mehurčka. Tam jim, zaradi njihove efektivne oblike, geometrija membrane najbolj ustreza. Inkluzije v notranji plasti pa čutijo membrano z nasprotno ukrivljenostjo kot inkluzije v zunanji plasti, zato jim geometrija membrane na robu mehurčka ne ustreza. Na sliki je tudi vidno, da v sploščenem osrednjem delu mehurčka sosednji membrani nista v stiku, kar kažejo tudi eksperimenti [43] (slika 5.2). Takšnih oblik z minimizacijo upogibne energije izotropne membrane ne moremo dobiti [dodatek C]. Slika 5.3: Izračunana stabilna torocitna oblika mehurčka in porazdelitev anizotropnih inkluzij v zunanji (črtkana črta) in notranji (točkasta črta) plasti membrane mehurčka, pri čemer je 80 % inkluzij v zunanji in 20 % v notranji plasti. Vrednosti parametrov so: C1m = 34/R0 , C2m = 0, N kT /(8πkc ) = 100 in 2πK/(R02 kT ) = 0.01, pri čemer je 4πR02 = A. 42 5. Stabilnost torocitnih oblik membranskih struktur Omeniti je treba, da teoretični model predstavljen v tem poglavju velja le v primeru, ko je inkluzij povsod v membrani veliko manj kot izotropnih lipidnih molekul in so interakcije med inkluzijami in okolno membrano veliko večje kot med samimi lipidnimi molekulami. V nasprotnem primeru vezava inkluzij v membrano neposredno vpliva tudi na upogibno energijo membrane (Wb ), saj inkluzije iz membrane izrinejo nekaj lipidnih molekul. Tega pa v izračunih nismo upoštevali. Poudariti je treba še, da smo v zgoraj predstavljenem modelu zanemarili direktne interakcije med inkluzijami in efekt nasičenja inkluzij zaradi njihove končne velikosti. 6. STABILNOST IN SESEDANJE ANORGANSKIH MIKRO IN NANO CEVK V zadnjih dveh desetletjih so mikro in nano cevke predmet številnih raziskav: sintetizirane so bile nano cevke iz ogljika [45] in drugih anorganskih snovi (glej [46] in dodatek D), veliko stabilnih mikro in nano cevk pa so opazili tudi v organskih sistemih [20, 19]. Pri sintetiziranju mikro in nano cevk iz molibdenovega disulfida (MoS2 ), so opazili zanimiv pojav sesedanja cevk [47]. Mikro in nano cevke iz MoS2 so sestavljene iz več plasti trosloja S–Mo–S. Večinoma so cevke cilindrične oblike (slika 6.1A), opazili pa so tudi nekatere sploščene (sesedene) oblike cevk iz MoS2 (slika 6.1B). Čeprav so cevke iz MoS2 zelo občutljive na radialne sile, pa sesedanje najverjetneje ni posledica zunanjih sil [47]. Sesedanje cevk se namreč pojavi, ko pri pripravi vzorca cevka zaradi ovire ne more več rasti v dolžino. Takrat se trosloji MoS2 nalagajo eden na drugega in stena cevke se debeli. V tem delu predstavimo teoretični model, ki podaja možno razlago za sesedanje mikro in nano cevk [dodatek D]. Zanimajo nas samo osnovni fizikalni principi, zato zaradi enostavnosti najprej obravnavajmo le en trosloj MoS2 zvit v cevko. Cevka naj bo dovolj dolga, da lahko vplive robov zanemarimo. Trosloj si predstavljamo kot tanko elastično ploščo, sestavljeno iz anizotropnih gradnikov. Spomnimo se poglavja 2.3, ki govori o membrani sestavljeni iz ene vrste anizotropnih gradnikov. Za prosto energijo trosloja MoS2 lahko uporabimo izraz (2.22). Predpostavimo, da se gradniki plasti ne morejo prosto vrteti okrog svoje osi normalne na plast. Tako namesto statistično-mehanskega povrečja po kotu ω, ki nastopa v izrazu (2.22), uporabimo izraz za energijo (1) gradnika, ki je zavrten za nek dani kot ω. Namesto izraza za Fi torej v 44 6. Stabilnost in sesedanje anorganskih mikro in nano cevk Slika 6.1: A: Nano cevka, sestavljena iz sedmih plasti trosloja MoS2 . Temne črte predstavljajo plasti atomov molibdena. B: Sesedena mikro cevka iz MoS2 . Debelina cevke je približno 70 nm. Obe sliki sta posneti s transmisijskim elektronskim mikroskopom, vendar sta skali na slikah različni [dodatek D]. (2.22) vstavimo enačbo (2.6). Za prosto energijo trosloja MoS2 tako velja Z Fm = n (2K + K̄)(H − Hm )2 − A 2 − K̄(D 2 − 2DDm cos (2ω) + Dm ) dA, (6.1) kjer je n = N/A površinska gostota gradnikov trosloja MoS2 . Še enkrat le omenimo, da za izotropne gradnike (Dm = 0) enačba (6.1) preide v znan izraz za upogibno energijo izotropne tanke plošče [10]: Z Z kc 2 Fm (Dm = 0) = (2H − C0 ) dA + kG C1 C2 dA, 2 A A (6.2) kjer je C0 = (2K + K̄)Hm /K spontana ukrivljenost plošče, kc = nK in kG = nK̄. 45 Slika 6.2: Skica gradnika v tanki plasti, ki se popolnoma prilega lastni obliki gradnika. Prikazan je izotropen gradnik (Dm = 0) s povprečno lastno ukrivljenostjo Hm = 0 (A) ter anizotropni gradniki (Dm > 0) sedlaste lastne oblike s Hm = 0 (B), cilindričnine lastne oblike s Hm > 0 (C) in cilindrične lastne oblike s Hm < 0 (D). Ravnovesna oblika cevke ustreza stanju z najmanjšo prosto energijo. Pri računanju proste energije trosloja MoS2 uporabimo enačbo (6.1). Pri tem predpostavimo da imajo gradniki trosloja MoS2 cilindrično lastno obliko (|Hm | = Dm > 0; slika 6.2). Privzamemo tudi, da so gradniki glede na cevko zavrteni za kot ω = 0, tako da lastni sistem ukrivljenosti cevke na mestu gradnika sovpada z lastnim sistemom lastne ukrivljenosti gradnika (poglavje 2.1). Obliko preseka nesesedene cevke podamo s krožnico, medtem ko obliko preseka sesedene cevke popišemo z dvoparametrično modelsko funkcijo [dodatek D]. Izkaže se, da je od premera cevke odvisno, katero stanje cevke je energijsko ugodnejše: sesedeno ali nesesedeno. Pri manjših premerih cevke ima nesesedena cevka manjšo prosto energijo kot sesedena, pri večjih premerih pa je energijsko ugodnejše sesedeno stanje (slika 6.3). Tako nam opisani model plasti cevke iz anizotropnih gradnikov ponuja razlago za stabilnost cevke pri manjših premerih in vzrok za sesedanje cevke pri večjih premerih. Pri rasti cevke v širino se nalagajo trosloji MoS2 eden na drugega in premer zaporednih nanešenih slojev raste. Pri dovolj velikem številu slojev cilindrična oblika cevke ni več energijsko najugodnejša rešitev in 46 6. Stabilnost in sesedanje anorganskih mikro in nano cevk R0 Dm Slika 6.3: Fazni diagram ravnovesnih oblik trosloja MoS2 [dodatek D]. R0 je brezdimenzijski radij preseka nesesedene cevke in Dm je brezdimenzijski deviator lastne ukrivljenosti anizotropnih gradnikov (torej mera za anizotropijo gradnikov plasti). Shematsko sta prikazana tudi prečna preseka cevke v sesedeni fazi (desno) in nesesedeni fazi (levo). Vrednosti ostalih parametrov so: Hm = Dm in ω = 0. cevka se sesede. Opisan model lahko nakaže le začetni vzrok sesedanja ne pa celotnega sesedanja cevke v opaženo sploščeno obliko (slika 6.1B). Verjetno igrajo v zadnji fazi sesedanja pomembno vlogo privlačne van der Waalsove sile med stenami cevke. 7. SKLEP V doktorskem delu smo predstavili teoretični model membrane, ki upošteva njene anizotropne lastnosti. Izkaže se, da lahko z anizotropnim modelom membrane popišemo nekatere opažene stabilne oblike membranskih struktur in lastnosti membrane, ki jih izotropni modeli membrane ne morejo. Anizotropija membranskih gradnikov pride do izraza predvsem pri membranskih mikro in nano strukturah z izrazito anizotropno geometrijo. Kot primere smo prikazali organske in anorganske mikro in nano cevke, torocitne oblike mehurčkov membrane eritrocita in pore v membranski dvojni plasti. Anizotropija v modele membranskih struktur vstopa na različne načine. Pri anorganskih mikro in nano cevkah je anizotropija vgrajena v gradnike cevke. Le-ti niso gibljivi, saj jim zgradba cevke tega ne dopušča. V primerih por v lipidni dvojni plasti membrane in torocitnih mehurčkov membrane eritrocita pa smo predpostavili, da so anizotropni gradniki inkluzije, ki se lahko gibljejo v ravnini membrane in vrtijo okoli svoje osi. Za ta primera smo pokazali, da se anizotropne inkluzije s primerno efektivno obliko porazdelijo v dele membrane z anizotropno geometrijo (na primer v rob pore ali v odebeljen robni del torocita) in tako pripomorejo k stabilizaciji opaženih oblik. Potrebno je poudariti, da so predstavljeni teoretični modeli membrane le približni. Kompleksne interakcije med gradniki membrane so skrite v sorazmerno majhnem številu modelskih parametrov. Tako ne moremo pričakovati, da nam bo takšen fizikalni model podal dejansko mikroskopsko sliko sistema. Naš namen je bil s sorazmerno enostavnim modelom opisati vplive različnih gradnikov membrane na lastnosti membrane. Upamo, da bomo z rezultati naših modelov pripomogli k boljšemu razumevanju zanimivih pojavov in metod na področju membranskih mikro in nano struktur, pri katerih anizotropne 48 7. Sklep lastnosti membrane gotovo igrajo pomembno vlogo. 8. POMEMBNEJŠI ORIGINALNI PRISPEVKI • teoretični model membrane, ki upošteva anizotropne lastnosti njenih gradnikov, • teoretični model stabilizacije vodne pore v lipidni dvojni plasti membrane z anizotropnimi inkluzijami, • stabilizacija torocitnih oblik mehurčka membrane eritrocita vsled ne- homogene lateralne porazdelitve v membrano vgrajene anizotropne inkluzije, • razlaga sesedanja anorganskih mikro in nano cevk iz MoS2 , ob upoštevanju anizotropnih gradnikov 50 8. Pomembnejši originalni prispevki LITERATURA [1] L. Vodovnik, D. Miklavčič in T. Kotnik, Biološki sistemi, Založba FE in FRI, Ljubljana, 1998. 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Lett., 5, str. 423–426, 1998. 56 Literatura DODATKI: ORIGINALNE PUBLIKACIJE Priloga A CELLULAR & MOLECULAR BIOLOGY LETTERS Volume 6, (2001) pp 161 – 165 Received 30 March 2001 Accepted 11 May 2001 Short Communication OXYETHYLENE CHAIN-CATION COMPLEXATION; NONIONIC POLYOXYETHYLENE DETERGENTS ATTAIN A POSITIVE CHARGE AND DEMONSTRATE ELECTROSTATIC HEAD GROUP INTERACTIONS HENRY HÄGERSTRAND 1 , JOHAN BOBACKA2 , MALGORZATA BOBROWSKA-HÄGERSTRAND1 , VERONIKA KRALJ-IGLIC3 , MIHA FOŠNARIC4 and ALEŠ IGLIC 4 1 Department of Biology and 2 Laboratory of Analytical Chemistry, Åbo Akademi University, FIN-20520, Åbo/Turku, Finland, 3 Institute of Biophysics, Medical Faculty, University of Ljubljana and Clinical Centre Ljubljana and 4 Laboratory of Applied Physics, Faculty of Electrical Engineering, University of Ljubljana, SI1000, Ljubljana, Slovenia Abstract: We report literature data indicating that the polyoxyethylene chain of polyoxyethylene detergents attracts cations via dipole-ion interactions thereby attaining a positive charge character. This implies that nonionic polyoxyethylene detergents like Triton X-100 and C12 E8 may interact electrostatically with phospholipid head groups. We describe how a positive charge character of Triton X-100 and C12 E8 can explain their hitherto mysterious stomatocytogenic shape altering effect in human erythrocytes. Key Words: Erythrocyte Membrane, Shape Change, Surfactant, CmEn , Triton X100, PEG, Polyoxyethylene INTRODUCTION The mammalian erythrocyte has been frequently used as an experimental model to study plasma membrane dynamics. Manipulations of erythrocyte shape (membrane inward and outward bending) give information about membrane physical properties which are important in different physiological processes such as endocytosis and exocytosis. Detergents (water-soluble amphiphiles) have been shown to induce either spiculated (echinocytic) or invaginated Priloga A 162 CELL. BIOL. MOL. LETT. Vol. 6. No. 2. 2001 (stomatocytic) shapes in human erythrocytes [1-5]. These shape transformations are thought to depend mainly on the distribution of the detergent in the bilayer, i.e. whether the detergent is predominantly incorporated into the outer or inner membrane leaflet, thereby expanding this leaflet relative to the other [6-9]. The distribution of charged detergents, in case they can translocate to the inner bilayer leaflet (flip), is thought to depend mainly on electrostatic interactions (attraction/repulsion) between the detergent and the negatively charged phospholipids (mainly phosphatidylserine) in the inner leaflet. At equilibrium the anionic detergents are regarded to preferentially stay in the outer monolayer, thereby being echinocytogenic, while cationic ones are thought to be trapped mainly in the inner membrane leaflet, thereby being stomatocytogenic. Although this model qualitatively explains a majority of observed shape alterations induced by a variety of detergents [1-5,9], it has been thought that it cannot account for the stomatocytic effect of certain nonionic detergents, e.g. C12E8 and Triton X-100, for which there should be no a priori reason to assume an asymmetrical membrane insertion on electrostatic grounds [3,10]. Thus, although detergents like C12 E8 must be expected to easily flip in the erythrocyte membrane [see 11,12], the reason for the stomatocytogenic effect of nonionic amphiphiles like C12E8 and Triton X-100, as well as their transbilayer distribution (at equilibrium), has been regarded as unclear [10,13]. RESULTS AND DISCUSSION Recent results indicate that C10E8 shows a weak positive charge character [14]. This observation is in line with results showing that nonionic surfactants containing polyoxyethylene (or similar) units bind cations (monovalent as well as divalent) resulting in positively charged complexes [15-17 and references therein]. The detergent-cation complex formation apparently occurs due to dipole-ion interactions [18,19]. The polyoxyethylene chain complex cations via their co-ordination to oxygen atoms resulting in a helical structure where the cation is located inside the helix formed by the polyoxyethylene chain [20,21]. Analogously to crown ethers, several oxygen atoms in the polyoxyethylene chain interact with one cation and therefore the strength of the attraction depends on the length of the polyoxyethylene chain [20,22,23]. Similar characteristics may be expected for polyethylene glycol (PEG) [20]. The results presented above may explain the stomatocytogenic effect of polyoxyethylene detergents like C12E8 and Triton X-100. The detergents may bind cations in the buffer solution. A positive charge of the polyoxyethylene detergent-cation complex should lead to its electrostatic attraction to negatively charged phospholipid head groups in the inner membrane leaflet and thereby to stomatocytosis in line with the bilayer couple hypothesis. Thus, it seems that oppositely to what was previously believed [3,24], polyoxyethylene detergents like C12 E8 and Triton X-100 show electrostatic head group interactions with Priloga A CELLULAR & MOLECULAR BIOLOGY LETTERS 163 phospholipids. In accordance with the results presented above, recent results from our laboratory show that the C12 En oxyethylene chain must be at least five units long (n 5) for stomatocytosis to occur. A shorter oxyethylene chain may not bind the cation strongly enough. It can not be excluded that non-zero dipole moments of the polyoxyethylene chain additionally interact directly with phospholipid head groups. There are some experimental results from our laboratory which indirectly support the assumption that C12 E8 and Triton X-100 are predominantly accumulated in the inner erythrocyte membrane leaflet. Namely, erythrocytes treated with echinocytogenic amphiphiles (e.g. dodecylmaltoside, dodecylzwittergent and sodium dodecylsulphate) attained a stomatocytic shape upon extraction (washing) with bovine serum albumin containing buffer, thereby indicating a washing away of the echinocytogenic detergent from the outer membrane leaflet. However, erythrocytes treated with stomatocytogenic amphiphiles like C12E8 , Triton X-100 and chlorpromazine remained stomatocytic upon washing, indicating a location of these detergents in the inner membrane leaflet from where they cannot easily be washed away [see 3]. Polyoxyethylene detergents show some specific properties among detergents. We have shown that C12 E8 and Triton X-100 propagate transmembrane phospholipid movements exceptionally strongly [25], an observation subsequently confirmed by Pantaler et al. [26]. Furthermore, C12 E8 induces unique torocyte-shaped endovesicles in human erythrocytes [13]. It was suggested that a specific co-operative interaction of membrane intercalated C12 E8 with adjacent phospholipids leads to the formation of C12 E8 /phospholipid complexes. The properties of such complexes, i.e. their orientational ordering in the regions with a nonzero membrane curvature deviator, may favour the formation of torocyte endovesicles, characterised by a low average mean membrane curvature and a high average curvature deviator [13,27]. Interrelated molecular properties like a cationic charge, a relatively high lipophilicity and an appropriate effective molecular shape may be important for the above described membrane effects. To conclude, the ability of polyoxyethylene chains to complex cations and attain a positive charge explains the for long mysterious stomatocytogenic effect of nonionic detergents like C12 E8 and Triton X-100. Acknowledgements. We are indebted to the Research Institute at the Åbo Akademi University, TEKES, the Academy of Finland and the Ministry of Education and Science of Republic of Slovenia, for their economical support, and M. Kaè for useful discussion. Priloga A 164 CELL. BIOL. MOL. LETT. Vol. 6. No. 2. 2001 REFERENCES 1. Deuticke, B. Transformation and restoration of biconcave shape of human erythrocytes induced by amphiphilic agents and changes of ion environment. Biochim. Biophys. Acta. 163 (1968) 494-500. 2. Fujii, T., T. Sato, A. Tamura, M. Wakatsuki, and Y. Kanaho. Shape changes of human erythrocytes induced by various amphiphatic drugs acting on the membrane of the intact cell. Biochem. Pharmacol. 28 (1979) 613-620. 3. Isomaa, B., H. Hägerstrand, and G. Paatero.. 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Markuzina, N.N., Mokrov, S.B., Stefanova, O.k., Sementsov, S.N., Volkov, Yu. M. and Ranieva, E.A. Electrode properties of film membranes containing alkoxylated alkylophenols as nonionic surfactants. Russ. J. Appl. Chem. 66 (1993) 1765-1769. 20. Yanagida, S., Takahashi, K. and Okahara, M. Metal-ion complexation of noncyclic polyoxyethylene derivataives. Bull. Chem. Soc. Jpn. 50 (1977) 1386-1390. 21. Adams, M.D., Wade, P.W. and Hancock, R.D. Extraction of aurocyanide ion-pairs by polyoxyethylene extractants. Talanta 37 (1990) 875-883. 22. Liu, K-J. Nuclear magnetic resonance studies of polymer solutions. V. Cooperative effects in the ion-dipole interaction between potassium iodide and poly(ethylene oxide). Macromolecules 1 (1968), 308-311. 23. Sakai, Y., Ono, K., Hidaka, T., Takagi, M., and Cattrall, R.W. Extraction of alkali metal ions and tetraalkylammonium ions with ionic surfactants containing a polyoxyethylene chain. Bull. Chem. Soc. 73 (2000) 11651169. 24. Thurmond R.L., Otten, D., Brown, M.F. and Beyer, K. Structure and packing of phosphatidylcholines in lamellar and hexagonal liquidcrystalline mixtures with a nonionic detergent: a wide-line deuterium and phosphorus-31 study. J. Phys. Chem. 98 (1994) 972-983. 25. Hägerstrand, H., Holmström, T.H., Bobrowska-Hägerstrand, M., Eriksson, J.E. and Isomaa, B. Amphiphile -induced phosphatidylserine exposure in human erythrocytes. Mol. Membr. Biol. 15 (1998) 89-95. 26. Pantaler, E., Kamp, D. and Haest, C.W. Acceleration of phospholipid flipflop in the erythrocyte membrane by detergents differing in polar head group and alkyl chain length. Biochim. Biophys . Acta 1509 (2000) 397408. 27. Iglic , A., Kralj-Iglic , V., Bozic , B., Bobrowska-Hägerstrand, M. and Hägerstrand, H. Torocyte shapes of red blood cell daughter vesicles. Bioelectrochemistry 52 (2000) 203-211. F Priloga B Third International Conference on Geometry, Integrability an Quantization June 14-23, 2001, Varna, Bulgaria Ivaı̈lo. M. Mladenov in Gregory L. Naber, Editors Coral Press, Sofia 2001, pp 224–237 INFLUENCE OF ANISOTROPIC MEMBRANE PROPERTIES ON THE SHAPE OF THE MEMBRANE MIHA FOŠNARIČ1 , ALEŠ IGLIČ1 and VERONIKA KRALJ-IGLIČ2 1 2 Laboratory of Applied Physics, Faculty of Electrical Engineering Tržaška 25, SI-1000 Ljubljana, Slovenia Institute of Biophysics, Faculty of Medicine Lipičeva 2, SI-1000 Ljubljana, Slovenia Abstract Membranous structures that are composed of particles with inplane C2 group symmetry are studied. Starting from a single-constituent energy it is derived that the relevant invariants for description of such systems are the mean curvature and the curvature deviator; the energy of the system can be expressed in a simple and transparent form by these two invariants while their average values span the phase diagram of shapes that can be attained by the membrane enclosed structures. A variational problem for axisymmetric shapes is stated where the shapes with extreme average mean curvature and extreme average curvature deviator at relevant constraints are sought for. It is shown that, at fixed membrane area and at fixed enclosed volume, the solutions of the variational problem correspond to spherical, cylindrical and toroidal shapes. These solutions form lines in the phase diagram of possible shapes that separate classes of shapes with different symmetry properties. Physically, these lines represent limits of trajectories formed by processes that cause change of the average mean curvature and/or average curvature deviator, such as intercalation of particles into the membrane. Correspondence with some experiments involving toroidal structures of erythrocyte membrane, induced by exogenously added amphiphilic molecules, is considered. 224 Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 1 225 Introduction The membrane is considered as a system with one of its extensions much smaller than the other two. Therefore the shape of the membranous structure resembles a two-dimensional surface embedded into a three-dimensional space R3 . The two-dimensional surface in R 3 can be described in an elegant and easy-to-visualize way by a local 2x2 curvature tensor. However, within two dimensions the curvature cannot be directly perceived. In physical description, the membrane is composed of particles that act one upon another. The interaction between a chosen particle and its surrounding depends on the properties of the membrane constituents and their mutual configuration within the membrane. So, the particle directly perceives the local membrane curvature. In other words, the energy of the particle depends on the local membrane curvature. In describing the shape of the membranous structure as a shape of the two-dimensional surface in R 3 , we should have in mind that the membrane is really a three dimensional structure. In our description the ”third dimension” is accounted for in the free energy of the system. Knowing the curvature tensor in every point of the membrane and assuming that the system attains the shape of minimal free energy at given constraints, it is convenient to chose the invariants of the curvature tensor by which the free energy can be expressed in a simple and transparent way. In this work, we determine the relevant invariants of the curvature tensor by the single-inclusion energy that forms an origin for the derivation of the free energy of the membrane. We study the shapes of the membranous structures within the frame of these invariants. 2 Single-inclusion energy Any constituent of the membrane can be considered as an inclusion that is acted upon by a curvature field of the membrane. It is taken that the inclusions have a C2 group symmetry with respect to the axis normal to the membrane surface and can therefore exhibit orientational effects. Let us imagine that there exists a shape which would completely fit the inclusion. This shape is referred to as the membrane shape intrinsic to the inclusion. In general, the local membrane shape differs from the shape intrinsic to the inclusion. The origin of the coordinate system is taken at the site of the inclusion. The membrane shape at this site is described by the diagonalized curvature Priloga B 226 M. Fošnarič, A. Iglič and V. Kralj-Iglič tensor C, C= C1 0 0 C2 , (1) while the membrane shape intrinsic to the inclusion is described by the diagonalized curvature tensor C m 0 C1m Cm = . (2) 0 C2m The principal directions of the membrane are in general different from the principal directions intrinsic to the inclusion, the systems being mutually rotated for an angle ω. We introduce the mismatch tensor M, M = R Cm R−1 − C where R is the rotation matrix, cos ω − sin ω R= . sin ω cos ω (3) (4) The single-inclusion energy is defined as the energy that is spent by adjusting the inclusion into the membrane. Since energy is a scalar quantity, it is determined by the terms composed of invariants of the mismatch tensor M, i.e. the trace and the determinant of the tensor. The energy is approximated by the expansion of these invariants up to the second order in the elements of M, K (5) E = (TrM)2 + K̄ DetM, 2 where K and K̄ are constants. Using the expressions (1)–(5) yields the expression for the single-inclusion energy ξ + ξ 2 ξ 2 (Ĉ − 2Ĉ Ĉm cos 2ω + Ĉm ), E = (H − Hm )2 + 2 4 (6) where 1 1 (C1 + C2 ), Hm = (C1m + C2m ) 2 2 are the respective mean curvatures, while H= Ĉ = 1 1 (C1 − C2 ), Ĉm = (C1m − C2m ). 2 2 The constants used in Eq.(6) are ξ = 4K + 2K̄ and ξ = −4K − 6K̄. (7) (8) Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 3 227 Free energy of the inclusions The partition function q of a single inclusion is [5] 2π E(ω) 1 exp(− q= )dω, ω0 0 kT (9) with ω0 an arbitrary angle quantum and k the Boltzmann constant. In the partition function of the inclusion the contribution of the orientational states qorient is distinguished from the contribution of the other states q c , q = qc qorient , ξ ξ + ξ 2 2 2 qc = exp − (H − Hm ) − (Ĉ + Ĉm ) , (10) 2kT 4kT 2π 1 (ξ + ξ )Ĉ Ĉm cos 2ω exp dω. (11) qorient = ω0 0 2kT The integration over ω yields qorient = 1 (ξ + ξ )Ĉ Ĉm ), I0 ( ω0 2kT (12) where I0 is the modified Bessel function. The values of Ĉ and Ĉm in the expressions (10) and (12) can be replaced by its absolute values D = | Ĉ| and Dm = |Ĉm |, where D and Dm are the respective curvature deviators. The curvature deviator is also an invariant of the curvature tensor as it can be expressed by the trace Tr C = 2H and the determinant Det C = C 1 C2 , (13) D = (Tr C /2)2 − Det C = H 2 − C1 C2 . The free energy of the inclusion is then obtained by the expression Fi = −kT ln q, ξ ξ + ξ 2 (ξ + ξ )DDm 2 2 (D + Dm ) − kT ln I0 ( ) . (14) Fi = (H − Hm ) + 2 4 2kT Therefore we can say that the free energy of the inclusion is in a simple and transparent way expressed by the two invariants of the membrane curvature tensor: the membrane mean curvature H and the curvature deviator D. It is imagined that the membrane is divided into patches that are so small that the curvature is constant over the patch, however, they are large enough to contain a sufficiently large number of inclusions, so that they can Priloga B 228 M. Fošnarič, A. Iglič and V. Kralj-Iglič be treated by statistical methods. A chosen patch can then be treated as a system with well defined field C, given area A p , given number of inclusions M and temperature T and can therefore be subject to a local thermodynamic equilibrium. To describe the local thermodynamic equilibrium we chose canonical statistics [7]. We consider two simple cases. In the first case the inclusions are described as an ”ideal gas” in the membrane continuum; the area density of the inclusions depends on the local membrane curvature. In the second case each membrane constituent is considered as an inclusion. The area density of the inclusions is taken to be uniform. Both cases can be described within a lattice model. In a chosen small patch of the membrane there is a fixed number of equal lattice sites. In the ideal gas model, only few of these sites are occupied by the inclusions while in the model describing the membrane composed of a single species of the molecules, all sites are occupied. 3.1 Lattice gas model for inclusions embedded in the membrane continuum When the inclusions are treated as an ideal gas embedded in the membrane continuum, the canonical partition function of the inclusions in the small patch of the membrane is Q = q M /M !, where q is the partition function of the inclusion and M is the number of the inclusions in the patch. Knowing the canonical partition function of the patch Q, we obtain the Helmholtz free energy of the patch, F p = −kT lnQ. The Stirling approximation is used and the area density of the number of inclusions m = M/A p is introduced. This gives for the area density of the free energy ξ + ξ Fp = −kT mln qc I0 ( DDm ) + kT (m ln m − m). Ap 2kT (15) To obtain the free energy of the inclusions of the whole layer F o the contributions of all the patches are summed, i.e. the integration over the layer p area A is performed Fo = FAp dA, where dA is the area element. The explicit dependence of the area density m on the position can be determined by the condition for the free energy of all the membrane inclusions to be at its minimum in the thermodynamic equilibrium δF o = 0. It is taken into account that the total number of inclusions M T in the layer is fixed, m dA = MT (16) A Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 229 and that the area of the membrane A is fixed. The above isoperimetric problem is reduced to the ordinary variational problem by constructing a functional Fo + λm A m dA = A L(m) dA, where ξ + ξ DDm ) + kT (m ln m − m) + λm m L(m) = −kT m ln qc I0 ( 2kT (17) and λm is the Lagrange multiplier. The variation is performed by solving ∂L = 0. Deriving (17) with respect to m and taking into the Euler equation ∂m account (16) gives the Boltzmann distribution function modulated by the modified Bessel function I0 m = mu 1 A qc I0 ( ξ+ξ 2kT DDm ) qc I0 ( ξ+ξ 2kT DDm )dA , (18) where qc is given by (11) and mu is defined by mu A = MT . To obtain the equilibrium free energy of the layer the equilibrium area density (18) is inserted into the expression (15) and integrated over the area A. Rearranging the terms yields [10] 1 Fo = −kT MT ln A 3.2 ξ + ξ q c I0 ( DDm )dA . 2kT (19) Lattice model for membrane composed of single species of molecules In this case we are describing a membrane that is composed of a single species of molecules. Each molecule is considered as an inclusion. The energy of the interaction between the inclusions and the membrane continuum F m is obtained by multiplying the free energy of a single inclusion (Eq.(14)) by the area density of the inclusions m u = MT /A and integrating over the membrane area: (20) Fm = mu Fi dA, ξmu Fm = 2 ξ + ξ 2 2 (H − Hm )2 dA + ) (D + Dm 4 (ξ + ξ )DDm ) dA. −kT mu ln I0 ( 2kT (21) Priloga B 230 M. Fošnarič, A. Iglič and V. Kralj-Iglič Expressing the second term in the above expression by Eq.(13) and omitting the constant terms yields 3ξ + ξ ξ + ξ 2 (22) Fm = mu (H − Hm ) dA − mu C1 C2 dA 4 4 (ξ + ξ )DDm ) dA. −kT mu ln I0 ( 2kT The first two terms recover the energy of isotropic bending [6], kc 2 (2H − C0 ) dA + κG C1 C2 dA, Wb = 2 (23) where C0 is the membrane spontaneous curvature while k c and κG are respectively, the membrane splay modulus and the membrane saddle splay modulus, kc = (3ξ + ξ )mu /8 and κG = −(ξ + ξ )mu /4. The above procedure, leading to Eq.(22), links the statistical mechanical derivation description to the continuum elastomechanics. It follows from Eq.(22) that the saddle splay modulus is negative for the one-component membrane. Also, we get an additional term (third term in Eq.(22)) that originates in orientational ordering of the membrane constituents. Therefore, we can say that C2 symmetry of the membrane constituents yields the membrane bending energy but gives also a new term that is called the deviatoric splay. It can be seen from Eq.(21) that the free energy of the membrane is in a simple and transparent way expressed by the two invariants of the local curvature tensor: by the mean curvature and the curvature deviator. We therefore chose these invariants as fundamental invariants for description of the membrane with in-plane orientational order, rather than the mean and the Gaussian curvatures that are considered as the fundamental invariants for description of isotropic continuum. 3.3 Determination of the equilibrium shape The above expressions for the free energy of the inclusions are subject to local thermodynamic equilibrium and global thermodynamic equilibrium with respect to the distribution functions. However, the equilibrium shape, i.e. the principal membrane curvatures at each point of the membrane, are at this point not known. In order to determine the equilibrium shape the membrane free energy should be minimized also with respect to the shape at the relevant constraints. Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 231 To find the minimum of the membrane free energy F at the given area of the membrane A, and at the given volume enclosed by the bilayer membrane V , we construct a functional G G = F − λA · dA − A − λV · dV − V , (24) where λA and ΛV are the Lagrange multipliers, and dV is the volume element. The equilibrium shape is then obtained by solving the variational problem δG = 0. It would be preferred if this variational problem were rigorously solved. The rigorous solution would in general be given in a numerical form. However, to grasp the main characteristics of the behavior of the system, approximative solutions may be used. The limiting cases are studied, the analysis is restricted to certain classes of shapes and the probe functions with adjustable parameters are used. Let us consider a special case of a one-component membrane where H m and/or Dm are much larger than any H or D in the membrane. Also, let (ξ + ξ )DDm /2kT ≥ 1 so that we can expand the modified Bessel function for large arguments (ln[I0 (x)] x). Considering the above assumptions, it follows from Eq.(21) that the membrane free energy is up to a constant equal to 1 Fm = −ξHm H − (ξ + ξ )Dm D, 2 (25) 1 1 HdA, D = DdA, (26) H = A A H is the average mean curvature and D is the average curvature deviator. We can see from Eq.(25) that the shape of the minimal free energy would in this special case have an extreme average mean curvature (the nature of this extreme depending on the sign of H m ) and/or a maximal average curvature deviator (since deviator is always positive). The shapes of the extreme average invariants of the curvature tensor are therefore distinguished shapes in the set of possible shapes. where 4 Shapes of extreme averages of curvature tensor invariants In order to obtain the shapes of the membrane of an extreme average mean curvature < H > and the shapes of an extreme average curvature deviator < D > at a given area of the membrane surface A and a given volume Priloga B 232 M. Fošnarič, A. Iglič and V. Kralj-Iglič enclosed by the membrane V , the variational problems are stated by constructing the respective functionals GH =< H > − λA · ( dA − A) + λV · ( dV − V ), (27) GD =< D > + λA · ( dA − A) − λV · ( dV − V ), (28) where λA and λV are the Lagrange multipliers while H and D are given by Eqs.(26). The analysis is restricted to axisymmetric shapes. It is chosen that the symmetry axis of the body coincides with the x axis. The shape is given by the rotation of the function y(x) around the x axis. In this case the principal curvatures are expressed by y(x) and its derivatives with respect to x; y = ∂y/∂x and y = ∂ 2 y/∂x2 , as C1 = ± y −1 (1 + y 2 )−1/2 and C2 = ∓ y (1 + y 2 )−3/2 . The area element is dA = 2π(1 + y 2 )1/2 y dx and the volume element is dV = ±πy 2 dx. By ± it is taken into account that the function y(x) may be multiple valued. The sign may change in the points where y → ∞. Inserting the above expressions for C 1 , C2 , dA and dV into Eqs.(27) and (28) and rearranging, the functionals become GH = with gH (x, y, y , y ) dx, GD = gD (x, y, y , y ) dx, (29) yy − λA y gH (x, y, y , y ) = ±1 ∓ 1 + y2 1 1 + y 2 ∓ λV y 2 , 2 (30) yy − λA y 1 + y2 1 1 + y 2 ∓ λV y 2 . 2 (31) gD (x, y, y , y ) = ∓1 ∓ The variations δGH = 0 and δGH = 0 are performed by solving the corresponding Poisson - Euler equations d ∂gi − ∂y dx ∂gi ∂y d2 + 2 dx ∂gi ∂y = 0, i = H, D. (32) As the first terms on the right side of Eqs. (30) and (31), that originate from C1 , are constant, this curvature does not influence the solution of the variational problem. Further, by the particular choice of the sign before the Lagrange multipliers in Eqs. (27) and (28) we can express both variational Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 233 problems with a single Poisson-Euler equation. Obtaining the necessary differentiations, this Poisson - Euler equation is yy 1 2y − − yλV = 0, i = H, D, (33) δi 2 + λA 1 + y 2 ( 1 + y 2 )3 (1 + y 2 ) where δH = ±1 while the choice of the sign of δD depends also on the sign of (C1 − C2 ). It follows from the above that the solutions for the extremes of the average invariants of the curvature tensor are equal. The nature of the obtained extreme may however be different. So it is possible that some solution corresponds to the maximal average mean curvature and the maximal average curvature deviator. Some other solution may correspond to the minimum of the average mean curvature and the maximum of the average curvature deviator etc. We will consider some simple analytic solutions of Eq.(33). The ansatz y = λA /λV fulfills Eq.(33) [8, 11]. This solution represents a cylinder of the solution of Eq.(33) is given by a radius rcyl = λA /λV . Another analytical 2 − x2 where (0, y ) is the center of circle of the radius rcir , y = y0 ± rcir 0 the circle. If y0 = 0 the ansatzfulfills Eq.(33) for two different radii [8], (rcir )1,2 = 2/ λA ± λ2A − 2λV , representing spheres with two different radii. If y0 = 0, the circle is the solution of Eq.(33) only when the Lagrange multipliers are interdependent, λ2A = 2|λV |. For rcir < y0 , the solution represents a torus of the thickness 2r cir and radius y0 , and a torocyte [9]. As a sum of the solutions of the differential equation within each of the above categories is also a solution of the same equation at the chosen constraints, different combinations of shapes within the corresponding category are possible, provided that the combined shape fulfills the constraints [2]. If the area A and the volume V are fixed, two independent parameters can be determined from the respective constraints. The cylinder, the two spheres, the torus, the cylinder ended by the hemispheres of the cylinder radius, and the sequence of a fixed number of beads are some of the possible shapes that are characterized by exactly two parameters each. The two constraints determine the shape completely so that these shapes are the shapes of the extreme H and of the extreme D. The equilibrium shape can be characterized by a volume to area ratio defined as the isoperimetric quotient IQ = 36πV 2 /A3 and both average invariants of the curvature tensor, and can be presented in a (H, D, IQ) phase diagram. The shapes of the extreme average invariants of the curvature tensor form curves in this phase diagram (Fig.1). These lines in turn Priloga B 234 M. Fošnarič, A. Iglič and V. Kralj-Iglič Figure 1: The (h, d, IQ) phasediagram, where dimensionless2 quantities A/4π H, d = A/4π D and IQ = 36πV /A3 . The are: h = lines pertaining to three sets of limit shapes are depicted: the set of shapes composed of two spheres, the set of tori and the set of shapes composed of a cylinder ended by two hemispheres. The corresponding projections to the d = 0 and to the h = 0 planes are also shown. form limits of the trajectories that correspond to the processes with changing average curvature invariants. 5 Some examples outlining the deviatoric splay Intercalation of specific amphiphilic molecules into the red blood cell membrane can induce membrane invaginations or evaginations which can finally close, forming an endovesicle or exovesicle, respectively. Spherical, cylindrical and torocyte shaped vesicles were observed [1, 11]. In this work we will briefly describe endovesicles with a toroid-like periphery and a large flat and thin central region, called torocytes (see Fig.2). These vesicles were observed after incubation of erythrocyte suspension with nonionic surfactant octaethyleneglycol dodecilether (C12E8). Let us consider the membrane of a torocyte as a continuum with anisotropic C12E8 inclusions (see section 3.1). The inclusions consist of a single C12E8 molecule and of the surrounding membrane constituents that are significantly distorted due to the presence of the embedded C12E8 molecule. It was shown [4] that the equilibrium shapes that resemble the observed torocytes can be obtained by considering that the C12E8 inclusions are anisotropic (Fig.2). The second example also involves C12E8 molecules intercalated into the Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 235 Figure 2: The image on the left is a transmission electron micrograph of the torocyte vesicle of the human erythrocyte membrane [1, 9]. The vesiculation was induced by incubating the erythrocyte suspension with C12E8 [1, 9]. The figure on the right shows the cross-section of the calculated equilibrium vesicle shape obtained by minimization of the membrane free energy where contribution of the C12E8 induced inclusions was taken into account [4]. The distribution of the inclusions over the outer membrane layer (dashed line) and over the inner layer (dotted line) is also depicted. It was taken that 80% of the inclusions are in the outer layer and 20% in the inner layer. It was also MT kT /8πkc = 100 and ξ/kT R02 = 1, C1m = −2.6/R0 , taken that ξ = ξ ∗ , C2m = 0 and R0 = A/4π [4]. lipid bilayer. Large molecules can pass through the cell membrane trough transient pores. Artificially, the formation of pores in the membrane can be achieved by applying an electric field across the membrane. This phenomenon is known as electroporation [12]. Some recent experiments [13] indicate that C12E8 molecules in lipid bilayers make transient pores, that are created by electroporation, more stable. It was shown theoretically [3] that anisotropic inclusions may stabilize circular pore in a flat membrane segment, where the edge of the pore was described as a part of a torus (see Fig.3). 6 Conclusions In this work a single-particle energy was expressed by the invariants of the curvature tensor of two-dimensional surface. Therefrom the membrane free energy was derived. Usually, it is considered that the fundamental invariants for the description of two-dimensional surface are the mean curvature and the Gaussian curvature. Our results indicate that for determination of Priloga B 236 M. Fošnarič, A. Iglič and V. Kralj-Iglič Figure 3: The diagram on the right shows the relative membrane free energy f = F/8πkc as a function of the relative radius of the pore R (see below). A flat circular segment of the double-layered membrane is considered [3]. The circular pore is in the center of the segment and the edge of the pore forms the inner half of a torus with larger relative radius R and smaller relative radius r. The unit of length is the radius of the circular membrane segment without a pore l0 = A/2π. The membrane free energy reaches minimum at Rmin = 0.029 (for lipid bilayer of the thickness 2r ≈ 5 nm this gives Rmin ≈ 10 nm). The drawing on the left represents the top view of the corresponding membrane segment with a pore of the radius R min . The values of the parameters are C1m = −50/l0 , C2m = 100/l0 , ξ = ξ ∗ , MT kT /8πkc = 10, ξ/kT l02 = 0.01 and r = 0.01. the equilibrium shape this is not always appropriate. It is suggested that when the membrane is composed of particles that have an in-plane C 2 group symmetry, the relevant invariants are the mean curvature and the curvature deviator. If we assume that the reference state of the membrane is isotropic, which would be true for the membrane composed solely of particles that are axisymmetric with respect to the axis normal to the membrane, the use of either set of invariants is equivalent, i.e. the equilibrium shapes would not depend on the choice of the set of the invariants. However, for the membrane containing particles that may be orientationally ordered, the choice of the set of the invariants is important. The presented results could probably be generalized by considering three-, four- or even n-dimensional differentiable manifold. A possibility should be considered, that the reference state of the system is not isotropic. Acknowledgements: The authors would like to thank A. Čadež for fruitful and Priloga B Influence of Anisotropic Membrane Properties on the Shape of the Membrane 237 stimulating discussion. References [1] Bobrowska-Hägerstrand M., Kralj-Iglič V., Iglič A., Bialkowska K., Isomaa B., Hägerstrand H., Torocyte membrane endovesicles induced by octaethyleneglycol dodecylether in human erythrocytes, Biophys. J. 77 (1999) 3356-3362. [2] Elsgolc L. E., Calculus of Variations, Pergamon Press, Oxford, 1961. [3] Fošnarič M., Kralj-Iglič V., Hägerstrand H., Iglič A., On stability of circular hole in membrane bilayer, Cell. Mol. Biol. Lett. 6 (2001) 167-171. [4] Fošnarič M., Nemec M., Kralj-Iglič V., Hägerstrand H., Schara M., Iglič A., Possible role of anisotropic membrane inclusions in stability of torocyte red blood cell daughter vesicle, Coll. Surf. B (in print). [5] Fournier J. B., Nontopological Saddle-Splay and Curvature Instabilities from Anisotropic Membrane Inclusions, Phys. Rev. Lett. 76 (1996) 4436–4439. [6] Helfrich W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturförsch 28c (1973) 693–703. [7] Hill T. R., An Introduction to Statistical Thermodynamics, Dover Publications, New York, 1986. [8] Iglič A., Kralj-Iglič V., Majhenc J., Cylindrical shapes of closed lipid bilayer structures correspond to an extreme area difference between the two monolayers of the bilayer, J. Biomechanics 32 (1999) 1343–1347. [9] Iglič A., Kralj-Iglič V., Božič B., Bobrowska-Hägerstrand M., Isomaa B., Hägerstrand H., Torocyte shapes of red blood cell daughter vesicles, Bioelectrochemistry 52 (2000) 203-211. [10] Kralj-Iglič V., Heinrich V., Svetina S., Žekš B., Free energy of closed membrane with anisotropic inclusions, Eur. Phys. J. B 10 (1999) 5–8. [11] Kralj-Iglič V., Iglič A., Hägerstrand H., Peterlin P., Stable tubular microexovesicles of the erythrocyte membrane induced by dimeric amphiphiles, Phys. Rev. E 61 (2000) 4230–4234. [12] Neuman E., Sowers A. E., Jordan C. A., Electroporation and Electrofusion in Cell Biology, Plenum Press, New York and London, 1989. [13] Troiano G., Stebe K., Sharma V., Tung L., The electroporation of artificial planar POPC bilayers and the effects of C12E8 on its properties, Biophys. J. 74 (1998) 880-888. Priloga C Colloids and Surfaces B: Biointerfaces 26 (2002) 243 – 253 www.elsevier.com/locate/colsurfb Possible role of anisotropic membrane inclusions in stability of torocyte red blood cell daughter vesicles Miha Fošnarič a, Marjana Nemec b, Veronika Kralj-Iglič c, Henry Hägerstrand d, Milan Schara b, Aleš Iglič a,* a Laboratory of Applied Physics, Faculty of Electrical Engineering, Uni6ersity of Ljubljana, Tržaška 25, SI-1000 Ljubljana, Slo6enia b J. Stefan Institute, Ljubljana, Slo6enia c Institute of Biophysics, Faculty of Medicine, Uni6ersity of Ljubljana, Ljubljana, Slo6enia d Department of Biology, A, bo Akademi Uni6ersity, A, bo/Turku, Finland Received 23 March 2001; received in revised form 15 October 2001; accepted 2 January 2002 Abstract The stability of torocyte red blood cell daughter endovesicles induced by octaethylene-glycol dodecylether (C12E8) was studied theoretically. In addition, the effects of C12E8 and tetraethylene-glycol dodecylether (C12E4) on physical properties of the red blood cell membrane were studied experimentally, using the electron spin resonance (ESR) technique. In the theoretical part, it was assumed that the stable vesicle shape corresponds to the minimum of its membrane free energy, which is the sum of the membrane bending energy and the contribution of the C12E8-induced membrane inclusions. We found that the torocytic vesicle shape may be stable due to quadrupolar ordering of the C12E8 anisotropic inclusions that are embedded in the vesicle membrane. It was also shown how a preference of the membrane inclusions for a specific membrane curvature might lead to their non-homogeneous lateral distribution. In the experimental part, it was shown that C12E4 drastically changes the proportions of the membrane lipid domains (characterized by different ‘fluidity’), while C12E8 induces much smaller changes in the proportions of the domains. A possible relation between the difference in the effects of C12E8 and C12E4 on the membrane lipid domains, and their distribution between the membrane leaflets, is discussed. © 2002 Published by Elsevier Science B.V. Keywords: Red blood cell; C12E8; Non-ionic detergent; Vesiculation; Torocyte shape; Orientational ordering; Electron spin resonance 1. Introduction We have recently reported [1] that the nonionic surfactant octaethylene-glycol dodecylether * Corresponding author. Tel.: +386-1-4250-278; fax: +3861-4264-630. E-mail address: [email protected] (A. Iglič). (C12E8) (Fig. 1) may induce in erythrocytes stable torocyte endovesicles having a thin plate-like central region and a toroidal periphery (Fig. 2). It was suggested that the torocyte endovesicle originates from a large stomatocytic invagination of the erythrocyte membrane, which looses volume and finally forms a toroidal endovesicle. Since intercalation of C12E8 into the membrane induces 0927-7765/02/$ - see front matter © 2002 Published by Elsevier Science B.V. PII: S 0 9 2 7 - 7 7 6 5 ( 0 2 ) 0 0 0 1 6 - 4 Priloga C 244 M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 Fig. 1. Schematic illustrations of the chemical structure of octaethylene-glycol dodecylether (C12E8) (A) and of tetraethylene-glycol dodecylether (C12E4) (B). stomatocytosis (inward membrane bending) and endovesiculation, it was assumed that C12E8 should be located mostly in the inner leaflet of the erythrocyte membrane, i.e. in the outer leaflet of the torocyte endovesicle membrane [1,2]. Although the phase diagram of the stable shapes of the vesicles and the cells with no internal structure has been extensively studied in the past [3 – 5], aside from a few works [6,2] the torocyte and codocyte shape classes were not given attention. Within the standard bending Fig. 2. Transmission electron micrograph of torocyte endovesicles of human red blood cells incubated with C12E8 (adapted from Bobrowska-Hägerstrand et al. [1]). elasticity models of the bilayer membrane [7,8,4], the calculated torocyte vesicle shapes, corresponding to the minimal bending energy, have a thin central region where the membranes on both sides of the vesicle are in close contact, i.e. the resultant forces on both membranes in contact are balanced [2,6]. However, as it was shown by the confocal laser scanning microscopy [1] and as it can also be seen by the transmission electron microscopy (Fig. 2), the adjacent membranes in the flat central region of the torocyte are separated by a certain distance indicating that the stability of the observed torocyte shapes of the erythrocyte endovesicles can not be explained by the standard bending elasticity model. Therefore it is of interest to understand which additional mechanisms (beside the minimization of the membrane bending energy) might take place in the shape determination of the C12E8 induced torocyte endovesicles. Three partly complementary mechanisms were suggested in order to explain the formation and stability of the observed torocyte endovesicles [1,2]. The first is a preferential intercalation of the C12E8 molecules into the inner membrane layer, resulting in a membrane invagination that may finally close, forming an inside-out endovesicle. The second mechanism is a preference of the C12E8 induced membrane inclusions (dynamic co-operative units composed of the embedded C12E8 molecule and adjacent membrane constituents that are significantly distorted due to the presence of the embedded C12E8 molecule) for zero or slightly negative [1,2] local mean curvature. Such inclusions would induce the vesicle shape with large regions of zero or slightly negative membrane mean curvatures. The third mechanism is the orientational ordering of anisotropic C12E8 membrane inclusions in the regions of nonzero local membrane curvature deviator [1,2]. The aim of the present work is to investigate the role of these mechanisms in explaining the origin and stability of the torocyte endovesicle shape. The paper is organized as follows. Materials and methods used in ESR experiments and the Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 corresponding methods for interpretation of the measured results of ESR experiments are described in Section 2. The expressions for the free energy of the membrane inclusions and for their lateral distribution are derived in Section 3. In Section 4.1, the mathematical model that is used for the theoretical study of the stable torocyte endovesicle shapes is described and the predictions of the model are given. The experimental results from the ESR measurements are reported in Section 4.2. In Section 5, the conclusions are drawn from both experimental and theoretical results. The possible origins of anisotropy of the C12E8-induced inclusions and of the asymmetric distribution of the C12E8 molecules between both membrane leaflets are discussed. The interdependence between the stable vesicle shape and the non-homogeneous lateral distribution of C12E8-induced inclusions is also discussed. 2. Materials and methods 2.1. Preparation of erythrocytes and spin labelling of erythrocytes The cell pellet, 0.2 ml, was dispersed in 6 ml phosphate buffered saline (PBS), transferred into the glass test tube containing 3.0 nmol of the spin probe methyl ester of 5 dioxyl palmitate (MeFASL(10,3)) [9] and deposited on the walls. A total of 13.2 ml of PBS for the control or 13.2 ml of PBS containing dissolved Cm En was added and all together exposed for 10 min. The sample was centrifuged for 4 min and the pellet was transferred into a glass capillary (inner diameter 1 mm). 2.2. Measurement and simulation of membrane spectra The electron spin resonance (ESR) spectra were recorded on the Bruker ESP 300 X-band spectrometer at 37 °C. The spectra were recorded at 9.6 GHz microwave frequency and 20 mW power. The modulation frequency and amplitude were 100 kHz and 2.0 gauss. The magnetic scan range was 100 gauss and the scan time was 168 s. The 245 molar ratio between the membrane spin probe and the phospholipids was about 1/100. The measured spectra of the spin probe in the erythrocyte membrane are superimpositions of spectra of the spin probes dissolved in particular coexisting lateral membrane domains [9,10]. Therefore, the experimental spectra are fitted by the calculated spectra. Three lateral domain types have been assumed. For each type of spectra the corresponding parameters of the spin Hamiltonian function have been selected. The averaged interaction tensor components have been calculated for the properly chosen characteristics of the particular domain, i.e. the molecular ordering of the lipid acyl chains, the rotational correlation times and the polarity corrections due to the alterations of the electronic structure induced by the polarity of the nitroxide environment. The orientation order parameter S describes the alignment of the lipid hydrocarbon chains in the membrane, i.e. a stronger ordering of molecules can be described by higher values of S. Herewith the hyperfine splitting of lines in the spectrum can be directly modified. On the other hand, the line widths depend on the rotational correlation time ~, which reflects the rate of molecular angular deflections of the fluctuating molecular segment, bearing the nitroxide in conformity with the mobility of the neighbouring lipid molecules. The line widths of the spectra are additionally modified by the inherent unresolved hyperfine splitting of the paramagnetic nuclei of the near hydrogen atoms. The polarity corrections due to the membrane regions influence the g an A tensors [11], describing the already mentioned shifts and hyperfine splitting of the spectral lines via the displacement of the unpaired electrons of the nitroxide, that perturbs the wave functions describing the electronic structure of the spin probe. The optimization of the typical coexisting lateral domains in the proportion set by the population weight parameters and the corresponding line shape parameters was performed for the evaluation of the particular domain’s spectra. The program EPR SIM 4.0 was used to solve the described inverse optimization problem. The details are described in Ref. [11]. Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 246 qc = exp 3. Theory 3.1. Free energy of the membrane inclusions qorient = The anisotropic membrane inclusion in the curvature field of the membrane is considered. It is taken that the inclusion has a C2 group symmetry with respect to the axis normal to the membrane surface. Let us imagine that there exists a membrane shape that would perfectly fit the inclusion. This shape is referred to as the membrane shape intrinsic to the inclusion. In general, the local membrane shape at the site of the inclusion differs from the shape intrinsic to the inclusion. The energy of the inclusion is defined as the energy that is spent by adjusting the inclusion into the membrane. Let C1 and C2 be the membrane principal curvatures at the site of the inclusion and C1m and C2m the principal curvatures of the shape intrinsic to the inclusion [12]. The energy of a single inclusion can be approximated by the expression [13]: x E = (C − Cm)2 2 + x+ x* 2 (C. − 2C. C. m cos(2 ) + C. 2m) 4 (1) where x and x* are the constants representing the strength of the interaction between the inclusion and the surrounding membrane [13,12], C= (C1 + C2)/2, C. = (C1 − C2)/2, C( m = (C1m + C2m )/2, C. m = (C1m − C2m )/2 and is the angle between the principle directions of the local membrane shape and the corresponding principle directions of the shape intrinsic to the inclusion. The partition function q of the single inclusion is [14] q= 1 0 & 2y 0 exp − E( ) d kT (2) where 0 is an arbitrary angle quantum, k is the Boltzmann constant and T is the temperature. In the partition function of the inclusion the contribution of the orientational states qorient is distinguished from the contribution of the other states qc, q = qcqorient, n −x x+ x* 2 (C( − C( m )2 − (C. + C. 2m) 2kT 4kT 1 0 & 2y exp 0 (3) n x+ x* C. C. m cos(2 ) d 2kT (4) The integration over yields qorient = 1 x+ x* C. C. m I 0 0 2kT n (5) where I0 is the modified Bessel function. The free energy of the single inclusion is then obtained by the expression F1 = −kT ln q, x x+ x* 2 F1 = (C − Cm )2 + (C. + C. 2m) 2 4 − kT ln I0 x+ x* C. C. m 2kT n (6) 3.2. Lattice gas model for inclusions embedded in membrane continuum To obtain the energy contribution of all the inclusions in a membrane layer it is imagined that the membrane layer is divided into patches. The patches are small enough, so that the curvature can be taken as constant over the patch, however they contain enough molecules to be treated by statistical mechanics. The chosen patch can then be considered as a system with well defined principal curvatures C1 and C2, given area Ap, given number of inclusions Np and given temperature T and is considered to be in local thermodynamic equilibrium. The inclusions are treated as a two dimensional ideal gas [13,12]. The canonical partition function of the inclusions in a small patch of the membrane layer is Q = q Np/Np! [12,13], where q is the partition function of the inclusion and Np is the number of the inclusions in the patch. Knowing the canonical partition function Q, we can obtain the free energy of the patch, Fp = −kT ln Q. The Stirling approximation is used and the area density of the number of inclusions n = Np/Ap is introduced. This gives for the area density of the free energy [13]. Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 Fp x+ x* C. C. m = − nkT ln qcI0 2kT Ap n + kT(n ln n − n) & 1 A FL = −NkT ln qcI0 A 247 n x+ x* C. C. m dA 2kT (7) (11) 4. Results To obtain the free energy of the inclusions of the whole membrane layer FL the contributions of all the patches are summed, i.e. the integration over the layer area A is performed FL = A (Fp/ Ap) dA, where dA is the area element. The explicit dependence of the area density n on the position can be determined by the condition for the free energy of all the inclusions to be at its minimum in the thermodynamic equilibrium, lFL = 0. It is taken into account that the total number of inclusions N in the layer is fixed, & n dA = N (8) A and that the area of the layer A is fixed. The above isoperimetric problem is reduced to the ordinary variational problem by constructing a functional FL + u A n dA= A L(n) dA, where L(n)= − nkT ln qcI0 x+ x* C. C. m 2kT + kT(n ln n − n) + un n (9) and u is the Lagrange multiplier. The variation is performed by solving the Euler equation (L/(n = 0. Deriving Eq. (9) with respect to n and taking into account Eq. (8) gives the Boltzmann distribution function modulated by the modified Bessel function I0 & x+ x* C. C. m qcI0 2kT n = n̄ 1 x+ x* C. C. m dA qI A A c 0 2kT 4.1. Theoretical predictions To obtain the equilibrium shape of the vesicle at given area (A) and volume (V), we should minimize the membrane free energy consisting of the contribution of the Canham–Helfrich bending energy and the contribution of the C12E8 induced membrane inclusions in the outer and in the inner membrane bilayer leaflet (Fo and Fi, respectively), 1 F= kc 2 where qc is given by Eq. (3) and n̄ is defined as n̄= N/A. To obtain the equilibrium free energy of the inclusions in the membrane layer the equilibrium density (10) is inserted into the expression (7) and integrated over the area A. Rearranging the terms yields [13] (2C( − C0)2 dA + Fi + Fo (12) A where kc is the local bending modulus and C0 is the spontaneous curvature. In this work it is taken that C0 = 0. The first term in Eq. (12) represents the local bending energy [8]. For the sake of simplicity the non-local bending energy [15–17,4,18] is not considered in Eq. (12). Including the non-local bending energy would not affect the set of possible shapes obtained by the minimization procedure [4,19], since the non-local bending energy does not depend on the details of the shape [20,17]. The contribution of the C12E8 inclusions in the j-th leaflet of the membrane bilayer to the membrane free energy is (see Eq. (11)) & Fj = −Nj kT ln 1 A qcj I0 A n x+ x* C. C. m dA 2kT (13) with (10) & qcj = exp x+ x* 2 −x (C. + C. 2m) (ljC( − C( m )2 − 4kT 2kT n (14) where j = o, i. The index o denotes the outer leaflet while the index i denotes the inner leaflet of the membrane bilayer, li = −1 and lo = 1. N is the total number of the C12E8 inclusions in the j-th leaflet of the vesicle membrane. For the sake of simplicity it is taken in this work that x= x*. Priloga C 248 M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 Fig. 3. Schematic figure illustrating different intrinsic shapes of the membrane inclusions characterized by the two intrinsic principal curvatures C1m and C2m. Shading marks the hydrophilic surface of the inclusion. Three characteristic intrinsic shapes of inclusions are shown in the figure: C1m = C2m = 0 (A), C1m \ 0 and C2m B 0 (B), C1m B 0 and C2m = 0 (C). The corresponding most favourable membrane surfaces are also shown. The integrations in Eqs. (12) and (13) are performed over the entire membrane area A. In the expression for Fj (Eq. (13)), the quantities C( m = (C1m + C2m )/2 and C. m = (C1m − C2m )/2 contain the information about the effective shape of the inclusions. Here, C1m and C2m are the two intrinsic principal curvatures of the inclusion (Fig. 3). The inclusions are called isotropic if C1m = C2m [20] and anisotropic if C1m " C2m [14,12,13] (Fig. 3B and C). The in-plane rotational ordering of the anisotropic inclusions in the curvature field of the membrane [2] may be strongly coupled to the lateral area density of the membrane inclusions and to the shape of the membrane [12,13,21,14]. In the following, the analysis is restricted to axisymmetric vesicle shapes where the symmetry axis of the vesicle coincides with the y axis, so that the shape is given by the rotation of the function y(x) around the y axis (Fig. 4). In this case the principal curvatures are expressed by y(x) and its derivatives with respect to x as follows, Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 C1 = − y¦(1+ y%2) − 3/2 (15) C2 = − y%x − 1(1 + y%2) − 1/2 (16) where y% = dy/dx and y¦= d y/dx . 2 2 The axisymmetric vesicle shape is parametrized by a function of the form y(x)= h+ k mx m i 2 − x 2 1 + k mx m (17) including four free parameters (h, i, k, m). In seeking for the minimum of the membrane free energy F, the constraints of fixed vesicle area, A= 4y & i x 1 + y%2 dx (18) 0 & and volume, V = 4y i xy dx (19) 0 are taken into account. Dimensionless quantities are introduced (see Appendix A). In the minimization procedure, the parameters h and i, as functions of the parameters k and m, are determined numerically from the constraints for the vesicle volume (Eq. (19)) and area (Eq. (18)). The remaining parameters k and m are then determined numerically by the mini- Fig. 4. The calculated equilibrium vesicle shape as a function of the increasing intrinsic principal curvature C1m for C2m = 0, 6= 0.2, s = 1, mo = 100 and mi = 0. The values of C1m are: 0 (a), − 1 (b), − 2 (c), − 2.5 (d) and − 3 (e). The corresponding area density of the membrane inclusions in the outer leaflet (no) is also shown (broken lines). 249 mization of the relative membrane free energy f = F/8ykc (see Appendix A). The integrals in Eqs. (A1), (18) and (19) are calculated numerically. Fig. 4 shows the dependence of the calculated equilibrium vesicle shape on the intrinsic principal curvature of the membrane inclusion C1m, for C2m = 0. The relative volume is 6= 0.2 and the values of the parameters are s= 1, mo = 100 and mi = 0 (where mj 8 Nj T/kc and s8 x/T, see Appendix A). The condition mi = 0 means that the inclusions are distributed only in the outer leaflet of the membrane bilayer. To illustrate the interdependence between the intrinsic effective shape of the membrane inclusions (determined by C1m and C2m ) and the equilibrium vesicle shape (determined by C1 and C2), the area density of the membrane inclusions nj (x) in the j-th membrane leaflet is also calculated (see Eq. (A6)). The area density of the inclusions in the outer layer no(x) is shown in Fig. 4 (the value of ni (x) is zero for all x, since mi = 0). It can be seen (Fig. 4a) that the membrane inclusions with zero intrinsic principal curvatures (C1m = C2m = 0) favour an oblate shelllike vesicle shape. In this case the vesicle has a large central region of small membrane local mean curvature C( = (C1 + C2)/2, where the area density of the inclusions no(x) is nearly constant. On the other hand the area density no(x) of inclusions with zero intrinsic curvatures is very small at the edge of the vesicle, where the membrane local mean curvature C( is large and positive. The central part of the vesicle becomes increasingly thinner and more plate-like while the thickness of the periphery increases with increasing C1m. At C1m = −3 (Fig. 4e), the vesicle has a thin plate-like central region and a toroidal periphery and resembles the observed torocytes. On the basis of the presented results it can be concluded that the calculated equilibrium vesicle shape approaches the shape of the torocyte with increasing the anisotropy of the membrane inclusions that are embedded in the vesicle membrane. The membrane inclusions with zero intrinsic curvatures (C1m = C2m = 0) do not favour the torocyte vesicle shape (Fig. 4a). For the isotropic conical (C1m = C2m \ 0) and inverted conical (C1m = C2m B 0) membrane inclu- Priloga C 250 M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 4.2. Experimental results Fig. 5. The calculated equilibrium vesicle shape as a function of the increasing intrinsic principal curvature C1m for C2m = 0, 6= 0.2, s = 1, mo = 80 and mi = 20. The values of C1m are: 0 (a), − 2.5 (b) and − 2.6 (c). The corresponding area densities of the membrane inclusions in the outer (broken lines) and inner leaflet (dotted lines) of the membrane bilayer are also shown. sions, the calculated stable torocyte vesicle shapes were not found, neither for the inclusions distributed in the outer leaflet (mo " 0, mi = 0) nor for the inclusions distributed in both leaflets of the membrane bilayer (mo " 0, mi " 0). For comparison, Fig. 5 shows the dependence of the calculated equilibrium vesicle shape on the intrinsic principal curvature of the membrane inclusion C1m for the case where the inclusions are distributed in both leaflets of the membrane bilayer, e.g. mo = 80 and mi = 20. The values of the other parameters are: C2m = 0, 6= 0.2 and s = 1. It can be seen in Fig. 5 that the influence of the intrinsic effective shape of the inclusions on the equilibrium vesicle shape is essentially the same as in Fig. 4. The area density of the inclusions in the outer leaflet no(x) (broken lines) differs from the area density of the inclusions in the inner leaflet ni(x) (dotted line). The lateral membrane heterogeneity in terms of co-existing lateral lipid domains with different molecular composition and distinct physical properties has been intensively studied in biological membranes in the last years [22,23,9,10]. In this work, the coexistence of the lateral lipid domains, characterized by different order parameters and rotational correlation times [9,10] was studied using the ESR technique. The ESR experiment shows that the Cm En molecules interact with the erythrocyte membrane. We anticipate that the hydrophobic tail of Cm En incorporates into the hydrophobic portion of the membrane bilayers. The experimental ESR spectra of the control erythrocytes, as well as the spectra of Cm En treated erythrocyte samples have been decomposed into three domain types, where the type I pertains to the most disordered fluid domain, and type III to the most ordered domain. The population proportions of the membrane domains, i.e. the relative weight factors are given in Fig. 6. They have been evaluated using the program EPR SIM 4.0 [11], by which the experimental spectra have been fitted with the calculated spectra for the considering domains. The parameters used are the optimized values of the molecular orientation ordering and molecular dynamics. The obtained parameters and the polarity corrections are shown in Table 1. Fig. 6. The lateral domain population of erythrocyte membrane for the control and C12E8 treated samples. The spin probe methyl ester of 5 dioxyl palmitate MeFASL(10,3) was used for EPR measurement. The bars indicate the S.D., referring to five independent experiments. The parameters of domains are given in Table 1. Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 Table 1 Values of the parameters obtained from the simulation of the experimental ESR spectra for the three types of domains Parameters Order parameter S Rotational correlation time ~ (ns) Additional relax. width (G) Polarity corrections on A Polarity corrections on g Domains I II III 0.13 1.1 0.38 0.6 0.73 1.0 0.7 0.95 1.00015 1.3 0.99 1.00015 2.7 0.99 1.0001 Fig. 6 shows that the parameters of the simulation of the experimental ESR spectra of the samples spin-labelled with MeFASL(10,3) changed most strikingly upon binding of C12E4 in the erythrocyte membrane. The effect of C12E8 is smaller. The portion of the most fluid membrane domain I relative to the control membrane is increased in the samples containing C12E8 and also in the samples containing C12E4. However, in the case of C12E4 the increase of the membrane fluidity is drastic. 5. Discussion In this work the role of C12E8-induced membrane inclusions in the formation and stability of the torocyte endovesicles was studied theoretically. The equilibrium shape of the vesicle was determined by the minimization of the membrane free energy. A strong coupling between the calculated equilibrium vesicle shape and the lateral distribution of C12E8-induced inclusions was observed. It was shown that the calculated equilibrium vesicle shapes are torocytic only if the membrane inclusions embedded in the vesicle membrane are anisotropic. In contrast, the isotropic membrane inclusions do not favour the torocyte vesicle shapes. A possible source of anisotropy of the C12E8 membrane inclusions, favouring the torocyte formation, may be the large head group of C12E8 molecule. Another possible reason for the anisotropy of the C12E8-induced inclusions is that the acyl chains of the phospho- 251 lipid molecules in the neighbourhood of C12E8 move apart sideways (local phase transition) [24– 26]. Based on the described properties of C12E8 molecules and on the C12E8-phospholipid interactions [27] it has been anticipated that the effective shape of the C12E8-phospholipid complexes (inclusions) may be anisotropic (C1m " C2m ) [1,2]. In addition, it is known that C12E8 molecules may also interact with membrane proteins and form C12E8-protein complexes (inclusions) [28]. In general, the effective shape of the C12E8-protein membrane inclusion may also be anisotropic. Therefore the anisotropic C12E8-induced membrane inclusions may be C12E8-phospholipid complexes and/or C12E8-protein complexes. However in our calculations, the inclusions have been, for the sake of simplicity, considered as equal. In line with the theoretical predictions given in this work, it was recently suggested that the C12E8 molecules might also stabilize the membrane pores induced by the electroporation [29]. This could be the consequence of an increased area density of the anisotropic C12E8-induced inclusions at the edge of the pore, favouring highly anisotropic membrane shape at the pore edge. The effect of C12E8 on the physical properties of the erythrocyte membrane was compared to the effect of C12E4. The C12E8 molecule can be distinguished from the analogous C12E4 molecule as it has a larger hydrophilic head (Fig. 1). The spin probe MeFASL(10,3) [9] may report about the effects of the Cm En molecules on the lateral domain distribution (domains I, II and III) at the level of the incorporated nitroxide. The used spin probe is thought to distribute evenly between the inner and the outer leaflet. Fig. 6 shows that C12E4 drastically changes the proportions of the membrane lipid domains relative to the control membrane, while C12E8 induces much smaller changes in the proportions of the domains. This may be partially due to the larger hydrophilic polyethylene head group of the C12E8 molecule that may not be pulled so deep into the membrane as the head group of the C12E4 molecule. Therefore the bound C12E8 induces a different perturbation at the level of the incorporated nitroxide than C12E4. Also, specific interactions between the spin Priloga C 252 M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 probe and the Cm En molecules cannot be excluded. In addition, the difference between the effects of C12E8 and C12E4 on the ESR spectra may be related to the preferential distribution of C12E8 and C12E4 between the two membrane leaflets. Since C12E8 induces stomatocytic shapes it should be predominantly accumulated in the inner membrane leaflet, while C12E4, which does not markedly affect erythrocyte shape, should be rather evenly distributed between both membrane leaflets [30,31]. A recent report [32] suggests that C12E8 is accumulated in the inner membrane leaflet because the oxyethylene chain binds cations, thereby giving the molecule a positive charge character. The positively charged C12E8cation complexes would then be attracted to the negatively charged phospholipids head groups in the inner membrane leaflet. It is also possible that C12E8 is accumulated in the inner membrane leaflet due to polarization interactions [33] directly with the phospholipid head groups in the inner leaflet or with the membrane skeleton. The presence of Cm En molecules in both leaflets requires that Cm En molecules can be exchanged between the leaflets. The results from the previous studies indicate a rapid C12E8 transport across the cell membrane [34,35]. The transport of C12E8 from the outer to the inner leaflet of the erythrocyte membrane is also in accordance with the stomatocytogenic effect of C12E8, i.e. the predominant binding of C12E8 in the inner leaflet of the erythrocyte membrane. Acknowledgements We are indebted to the Research Institute at the A, bo Akademi University and to the Ministry of Education, Science and Sport of the Republic of Slovenia for their financial support. Appendix A. Introduction of dimensionless quantities The unit of length is chosen to be the radius of a spherical vesicle R0 = (A/4y)1/2 that has the same area A as the vesicle under consideration. The variables and parameters are redefined as follows: x/R0 x, y/R0 y, C1R0 C1, C2R0 C2, C1mR0 C1m, C2mR0 C2m, C( R0 C( , C. R0 C. , C( mR0 C( m, C. mR0 C. m, i/R0 i, k/R0 k. The volume and the area are normalized relative to the corresponding values of the spherical vesicle with radius R0. The relative vesicle volume is 6= (36yV 2/A 3)1/2 and the relative vesicles area is a = A/4yR 20 = 1. The relative membrane free energy f= F/8ykc can be written in terms of normalized quantities as f= & C( 2 da − moo − mii (A1) a where j = o, i, & qcj I0(ljsC. C. m ) da j = ln n a s qcj = exp − ((ljC( − C( m )2 + C. 2 + C. 2m) 2 n (A2) (A3) mj = Nj kT/8ykc (A4) s= x/R 20kT (A5) The expression for the area density of the membrane inclusions nj (x) in the j-th membrane leaflet is normalized as follows (see Eq. (10)): nj (x) = n̄j & qcj I0(sC. C. m ) (A6) qcj I0(sC. C. m ) da a where n̄j = Nj /A is the uniform area density. It was estimated for R0 $ 10 mm that the interaction constant s might be of the order of 10 − 3 [13]. In our case R0 5 1 mm, therefore s may be of the order of 10 − 1 or larger. By taking into account that the area per C12E8 molecule is around 1 nm2 [36], the area of the torocyte vesicle is A$ 5 mm2 (Fig. 2), kT$ 5 × 10 − 21 J, kc $ 10 − 19 [18] and that the C12E8 molecules occupy around 1% of the leaflet area, we can estimate that mj $ 100. References [1] M. Bobrowska-Hägerstrand, V. Kralj-Iglič, A. Iglič, K. Bialkowska, B. Isomaa, H. Hägerstrand, Biophys. J. 77 (1999) 3356. Priloga C M. Fošnarič et al. / Colloids and Surfaces B: Biointerfaces 26 (2002) 243–253 [2] A. Iglič, V. Kralj-Iglič, B. Božič, M. Bobrowska-Hägerstrand, B. Isomaa, H. Hägerstrand, Bioelectrochemistry 52 (2000) 203–211. [3] E. Sackmann, FEBS Lett. 346 (1994) 3. [4] U. Seifert, Adv. Phys. 46 (1997) 13. [5] K.H. Parker, C.P. Winlove, Biophys. J. 77 (1999) 3096. [6] H.J. Deuling, W. Helfrich, J. Phys. (France) 37 (1976) 1335. [7] P.B. Canham, J. Theoret. Biol. 26 (1970) 61. [8] W. Helfrich, Z. Naturforsch. 28c (1973) 693. [9] J. Svetek, B. Kirn, B. Vilhar, M. Schara, Physiol. Plant. 105 (1999) 499. [10] M. Zuvic-Butorac, P. Müller, T. Pomorski, J. Libera, A. Herrmann, M. Schara, Eur. Biophys. J. 28 (1999) 302. [11] J. S& trancar, M. S& entjurc, M. Schara, J. Magn. Reson. 142 (2000) 254. [12] V. Kralj-Iglič, S. Svetina, B. Z& ekš, Eur. Biophys. J. 24 (1996) 311. [13] V. Kralj-Iglič, V. Heinrich, S. Svetina, B. Z& ekš, Eur. Phys. J. B 10 (1999) 5. [14] J.B. Fournier, Phys. Rev. Lett. 76 (1996) 4436. [15] E.A. Evans, Biophys. J. 16 (1974) 13. [16] E. Evans, Biophys. J. 30 (1980) 265. [17] S. Svetina, A. Iglič, B. Zekš, Ann. N.Y. Acad. Sci. 710 (1994) 179. [18] W.C. Hwang, R.A. Waugh, Biophys. J. 72 (1997) 2669. [19] A. Iglič, V. Kralj-Iglič, H. Hägerstrand, Eur. Biophys. J. 27 (1998) 335. [20] V. Kralj-Iglič, A. Iglič, M. Bobrowska-Hägerstrand, H. Hägerstrand, Coll. Surf. A 179 (2001) 57. 253 [21] V. Kralj-Iglič, A. Iglič, H. Hägerstrand, P. Peterlin, Phys. Rev. E 61 (2000) 4230. [22] M. Edidin, Curr. Opin. Cell. Biol. 7 (1997) 528. [23] K. Simons, E. Ikonen, Nature (1997) 569. [24] R.L. Thurmond, D. Otten, M.F. Brown, K. Beyer, J. Phys. Chem. 98 (1994) 972. [25] D. Otten, L. Lobbecke, K. Beyer, Biophys. J. 68 (1995) 584. [26] H. Heerklotz, H. Binder, G. Lantzsch, G. Klose, A. Blume, J. Phys. Chem. (1997) 639. [27] H.H. Heerklotz, H. Binder, H. Schmiedel, J. Phys. Chem. B 102 (1998) 5363. [28] J.V. Møller, M. le Maire, J. Biol. Chem. 268 (1993) 18659. [29] M. Fošnarič, V. Kralj-Iglič, H. Hägerstrand, A. Iglič, Cell. Mol. Biol. Lett. 6 (2001) 171. [30] M.P. Sheetz, S.J. Singer, Proc. Natl. Acad. Sci. 71 (1974) 4457. [31] M.P. Sheetz, S.J. Singer, J. Cell. Biol. 70 (1976) 247. [32] H. Hägerstrand, J. Bobacka, M. Bobrowska- Hägerstrand, V. Kralj-Iglič, M. Fošnarič, A. Iglič, Cell. Mol. Biol. Lett. 6 (2001) 161. [33] J. Israelachvili, Intermolecular and Surface Forces, Academic Press, New York, 1997. [34] U. Kragh-Hansen, M. le Maire, J.V. Moller, Biophys. J. 75 (1998) 2932. [35] M. le Maire, J.V. Moller, P. Champeil, Biochemistry 26 (1987) 4803. [36] G. Lantzsch, H. Binder, H. Heerklotz, M. Wendling, G. Krose, Biophys. Chem. 58 (1996) 289. F Priloga D 15 April 2002 Physics Letters A 296 (2002) 151–155 www.elsevier.com/locate/pla Deviatoric elasticity as a possible physical mechanism explaining collapse of inorganic micro and nanotubes V. Kralj-Iglič a , M. Remškar b , G. Vidmar c , M. Fošnarič c , A. Iglič c,∗ a Institute of Biophysics, Faculty of Medicine, Lipičeva 2, SI-1000 Ljubljana, Slovenia b Jožef Stefan Institute, Jamova 39, SI-1111 Ljubljana, Slovenia c Laboratory of Applied Physics, Faculty of Electrical Engineering, Tržaška 2, SI-1000 Ljubljana, Slovenia Received 19 November 2001; received in revised form 18 February 2002; accepted 5 March 2002 Communicated by L.J. Sham Abstract A mechanism is proposed that explains collapse of the multishell inorganic micro and nanotubes. A single shell is considered as a thin elastic plate with anisotropic properties. The derived elastic energy is expressed by the mean curvature and the curvature deviator. If the tube perimeter exceeds a certain threshold, the collapsed shape corresponds to the absolute minimum of the elastic energy. © 2002 Elsevier Science B.V. All rights reserved. PACS: 81.07.De; 82.70.Uv Keywords: Microtubes; Nanotubes; Deviatoric elasticity; Collapse In the last years, the interest in inorganic micro and nanotubes has been increasing. The carbon nanotubes [1] are being extensively studied [2]. Also, the nanotubes composed of other materials have been synthesized and explored [3–5]. In synthesizing MoS2 micro and nanotubes, an interesting phenomenon, that we refer to as a collapse, was noted [6]; usually, MoS2 micro and nanotubes are hollow cylinders composed of many S–Mo–S trilayers (Fig. 1A), however some stable flattened (i.e., collapsed) multitrilayer structures also appear (Fig. 1B). Although the MoS2 tubes are very soft against radial forces, it seems that the col- * Corresponding author. E-mail address: [email protected] (A. Iglič). lapse is not caused by mechanical manipulation during the sample preparation [6]. The collapse could rather be triggered by an obstacle that would affect the tube growth [6] so that the tube becomes thicker due to increasing number of trilayers. In this Letter we propose that deviatoric elasticity that originates in the anisotropic properties of the layered structure is a possible physical mechanism that can explain the collapse of cylindrical MoS2 micro and nanotubes and the stability of the collapsed tubes (Fig. 1). As we are interested in the general principles of the phenomenon we consider a single S–Mo–S trilayer closed into a tube. We take that the tube is very long so that the end effects can be neglected. A single trilayer is treated as a thin elastic plate. Its shape is described by the equations of differential geometry of a two- 0375-9601/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 2 6 5 - 7 Priloga D 152 V. Kralj-Iglič et al. / Physics Letters A 296 (2002) 151–155 and Cm = C1m 0 0 . C2m (2) The principal systems of these two tensors are in general rotated for an angle ω with respect to each other. The mismatch between the actual local continuum curvature and the intrinsic continuum curvature in the absence of the external forces is characterized by the tensor M: M = R C m R −1 − C, Fig. 1. A: High-resolution transmission electron micrograph of the MoS2 nanotube. Dark fringes correspond to molybdenum atomic layers. The nanotube consists of seven MoS2 layers. B: Transmission electron micrograph of the collapsed MoS2 microtube. The thickness of the tube estimated from the ribbon turn is approximately 70 nm. dimensional surface. However, the three-dimensional structure of the trilayer as well as its interaction with the adjacent trilayers is taken into account by the appropriate choice of energy. The continuum approach is used where it is taken into account that the trilayer is in general anisotropic within the trilayer plane. It is considered that the elastic energy of a chosen very small element of the trilayer is in the absence of the external forces equal to zero at the characteristic principal curvatures C1m and C2m . We call these principal curvatures the intrinsic principal curvatures. If a given shape had such principal curvatures in all its points, the elastic energy of such shape would be zero. We define the elastic energy per unit area of a very small element of the thin plate with area dA as the energy of mismatch between the actual local continuum curvature of this element and the intrinsic continuum curvature. The shape of both continuums are described by the curvature tensors C and C m , respectively. The tensor C describes the actual curvature while the tensor C m describes the intrinsic curvature, i.e., the curvature which would be energetically the most favorable. In the respective principal systems the curvature tensors include only the diagonal elements C1 0 C= (1) , 0 C2 where R is the rotation matrix, cos ω − sin ω . R= sin ω cos ω (3) (4) The small patch of the membrane should overcome this mismatch in order to fit into its place in the actual membrane. This is reflected in the energy that is needed for such deformation. The elastic energy per unit area dE/dA is a scalar quantity. Therefore each term in the expansion of dE/dA must also be a scalar [7], i.e., an invariant with respect to the rotation of the local coordinate system. In this Letter, the elastic energy density dE/dA is approximated by an expansion in powers of the invariants of the tensor M up to the second order in the components of M. The trace and the determinant of the tensor are taken as the set of invariants, dE = K1 (Tr M )2 + K2 Det M, dA (5) where K1 and K2 are constants. Taking into account Eqs. (1)–(4), the energy density dE/dA can be written as dE ξ = (H − Hm )2 dA 2 ξ + ξ 2 2 , + Ĉ − 2Ĉ Ĉm cos(2ω) + Ĉm 4 (6) where 1 H = (C1 + C2 ), 2 (7) is the mean curvature, 1 Ĉ = (C1 − C2 ), 2 (8) Priloga D V. Kralj-Iglič et al. / Physics Letters A 296 (2002) 151–155 is the curvature deviator, 1 Hm = (C1m + C2m ), 2 is the mean curvature of the intrinsic shape, (9) 1 Ĉm = (C1m − C2m ), (10) 2 is the curvature deviator of the intrinsic shape, ξ = 8K1 + 2K2 and ξ = −8K1 − 6K2 . It can be seen from Eq. (6) that the area density of the energy of the tube is characterized by two constants ξ and ξ and three parameters ω, Hm and Ĉm . The energy density (6) can be obtained also by integrating the square of the difference between the curvatures of the normal cuts of the intrinsic shape and the actual shape over all possible normal cuts through the selected point [8]. If the trilayer were isotropic (Ĉm = 0) Eq. (6) can be written in the form of the area density of the energy of isotropic bending [7,9] dEb kc = (2H − C0 )2 + kG K, dA 2 where C0 is the spontaneous curvature, K = C1 C2 = H 2 − Ĉ 2 , (11) behavior of the system and not in the details of the shape we describe the contour by a variational ansatz with a sufficient number of parameters. We consider only the shapes with a constant cross-section along the longitudinal (ζ ) axis. This cross-section lies in the (χ, ψ) plane and is given by the variational ansatz (cχ)2 b2 − χ 2 , ψ(χ) = ± a + (13) 1 + (cχ)2 where a, b and c are parameters and χ ∈ [−b, +b]. The sign + pertains to the contour above the χ axis and the sign − pertains to the contour below the χ axis. By taking into account the definition of the principal curvatures C1 and C2 [10], the mean curvature H and the curvature deviator Ĉ can be expressed as H =− ψ , 2(1 + ψ 2 )3/2 is the Gaussian curvature, while kc and kG are the splay modulus and the saddle-splay modulus, respectively. The microscopic constants ξ , ξ , Hm and Ĉm are connected to the elastic constants kc , kG and C0 . By comparing the expressions (6) and (11) we get the relations kc = (3ξ + ξ )/8, kG = −(ξ + ξ )/4 and C0 = 4ξ Hm /(3ξ + ξ ). It can be seen from Eq. (6) that the area density of the energy of an anisotropic thin plate can be expressed in a simple way by two invariants of the curvature tensor: the mean curvature and the curvature deviator. For the trilayer with intrinsically anisotropic properties this set of invariants is favored over a set composed of the mean curvature and the Gaussian curvature that are usually considered as the fundamental invariants for description of the twodimensional surfaces. The main difference between the expressions (6) and (11) derives from the choice of the reference state of the system. For isotropic material, however, the use of both sets of invariants is equivalent. The equilibrium shape is the shape with the minimal elastic energy. As we are interested in the general (14) and Ĉ = |H |, ψ (12) 153 (15) ψ where = dψ/dχ and = The infinitesimal area element is dA = 1 + ψ 2 dχ dζ . The mean curvature defined by Eq. (14) is positive for the convex regions (such as parts of sphere or cylinder) and automatically negative for the concave regions. The normal direction to the surface is outwards for all points. The curvature deviator (Eq. (15)) is always positive. In the following, dimensionless quantities are used. The elastic energy of the tube is divided by ξ/2 and calculated per unit of the normalized length L to yield a dimensionless quantity dE/dζ , 2 dE dE = dA. (16) dζ ξL dA The minimum of the elastic energy is sought at constant dimensionless contour perimeter 2π (17) 1 + ψ 2 dχ = 2πR0 , d 2 ψ/dχ 2 . where R0 is the dimensionless radius of the cylindrical tube with the circular contour. In the minimization procedure the parameter c as the function of the parameters a and b is determined numerically from the constraint (17). The parameters a and b are then determined by the minimization of dE/dζ . The integrals in Eqs. (16) and (17) Priloga D 154 V. Kralj-Iglič et al. / Physics Letters A 296 (2002) 151–155 Fig. 2. The calculated cross-sections of the equilibrium shapes for increasing relative perimeter of the tube; R0 : 1.0 (1), 1.5 (2), 2.0 (3) and 2.5 (4) at Hm = Ĉm = 1 and ω = 0. The cross-section of the tube is circular for R0 < 1.87 while the tube is in the collapsed state for larger R0 . are calculated numerically. The material properties of the tube are described by the intrinsic mean curvature Hm and intrinsic curvature deviator Ĉm . As the tube at the beginning grows into a cylindrical shape, from a cylindrical shape of the S–Mo–S we assume that the intrinsic shape is a cylinder with Hm = Ĉm > 0. For simplicity, it is taken that ω = 0. In general, for anisotropic thin closed plates (i.e., for Ĉm = 0) the energy per unit of normalized length dE/dζ has two minima with respect to a and b. One minimum corresponds to the cylindrical tube with the circular contour, while the second minimum corresponds to the collapsed tube. At smaller values of R0 the minimum of dE/dζ corresponding to the cylindrical tube is the global minimum of dE/dζ . However, with increasing R0 , at a certain threshold, the minimum of dE/dζ corresponding to the collapsed tube, becomes the global minimum of dE/dζ . Fig. 2 shows the cross section of the calculated equilibrium shapes of the material with anisotropic properties (Hm = Ĉm = 1). For simplicity, it is taken that ξ = ξ . The contour length R0 is increased from the top to the bottom. The tube cross-section is circular at smaller values of R0 while it is in the collapsed state above some threshold value of R0 . In contrast, for isotropic thin plates (Ĉm = 0 and Hm 0) the calculated equilibrium state of the tube is cylindrical for all values of R0 . A nonzero intrinsic curvature deviator is therefore prerequisite for the initiation of the collapse of the cylindrical tube with a large contour length. Fig. 3. A (Ĉm , R0 ) phase diagram of equilibrium shapes of the tube with constant cross-section along the longitudinal axis; Hm = Ĉm , ω = 0. Based on these theoretical results we suggest that the observed collapse of the cylindrical MoS2 micro and nanotubes that occurs during growth of the tube into a multitrilayer structure [6] is spontaneous, in order to keep the elastic energy of the tube as low as possible. Namely, during the growth of the tube into the multilayer structure the perimeter of the layers increases. For the outer layers the collapsed state becomes energetically more favourable. When this effect becomes large enough to render the collapsed state of the whole tube energetically the most favourable, the collapsed state becomes the stable state of the tube. These suggestions are in accordance with observations [6]. Fig. 3 shows the (Ĉm , R0 ) phase diagram exhibiting the regions corresponding to the stable shapes of the single-trilayer tube with constant closed crosssection along its longitudinal axis. The phase diagram shows two different regions of shapes: the region of cylindrical tubes with circular cross-section and the region of collapsed tubes. The critical value of R0 where the collapse of the tube occurs, decreases with increasing intrinsics curvature deviator Ĉm (Fig. 3). The variational problem regarding the shape with minimal energy can be expressed by the differential equation [13]. In this Letter we have used a variational ansatz as we were interested in general behavior of the system and not in the details of the shape. However, the model can be in the future upgraded also in this direction. The collapsed shapes were observed also in carbon nanotubes [11]. It was suggested that the collapse of the carbon nanotube is initiated by some external Priloga D V. Kralj-Iglič et al. / Physics Letters A 296 (2002) 151–155 mechanical force while the collapsed structure is kept stable by the van der Waals attractive forces between the nanotube walls [11]. We think that the van der Waals forces are important also in stabilizing the collapsed shapes of MoS2 micro and nanotubes, however, based on the results presented in this Letter we argue that some other mechanism such as the intrinsic anisotropy of the trilayer (described by the parameters Hm and Ĉm ) is necessary to trigger the collapse. The intrinsic anisotropy of the trilayer may be a consequence of the interaction between the trilayers. A perfect match of the two adjacent trilayers cannot be obtained as the curvature of the adjacent layers is different. While in the direction of the tube axis the distance between the atoms may stay the same, the differences in the interatomic distances between individual layers are necessarily present along the tube circumference. In order to yield the most favorable match, defects in the structure may appear [12]. We may say that in this Letter the trilayer is described as a thin elastic plate with uniformly distributed anisotropic defects. In this Letter we start with the assumption that the initial equilibrium shape of the S–Mo–S trilayer is a cylinder (Hm = Ĉm > 0) with certain orientation of the atomic lattice with respect to the geometrical axes of the cylinder (ω = 0) and propose an explanation for the collapse of this structure. At the present stage of the knowledge on the process of the formation of the MoS2 nano and microtubes we cannot say why the S–Mo–S trilayer initially attains the cylindrical shape. Stable nano and microtubes have been found also in organic systems such as in surfactant systems [14], protein systems [15], in phospholipid membranes [16, 17] and in cell membranes [18]. The deviatoric elasticity provides an explanation for the stability of the cylindrical phospholipid nano and microtubes [17] and of the tethers connecting a vesicle and a mother cell [18]. In these systems the anisotropic properties of the membrane were explained by orientational ordering of the membrane constituents [8,17–19]. References [1] S. Iijima, Nature 354 (1991) 56; 155 B.I. Yakobson, C.J. Brabec, J. Bernholc, Phys. Rev. Lett. 76 (1996) 2511. [2] W. Marx, M. Wanitschek, H. Schier, Condens. Matter News 7 (1999) 3. [3] N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Science 269 (1995) 966. [4] H. Nakamura, Y. Matsui, J. Am. Chem. Soc. 117 (1995) 2651; P. Hoyer, Langmuir 12 (1996) 141; M. Remškar, Z. Škraba, R. Sanjines, F. Levy, Appl. Phys. Lett. 74 (1999) 3633; W.Q. Han, S.S. Fan, Q.Q. Li, Y.D. Hu, Science 277 (1997) 1287; L. Guo, Z. Wu, T. Liu, W. Wang, H. Zhu, Chem. Phys. Lett. 318 (2000) 49; G. Seifert, H. Terrones, M. Terrones, T. Frauenheim, Solid State Commun. 15 (2000) 635. [5] M. Remškar, A. Mrzel, Z. Škraba, A. Jesih, M. Čeh, J. Demšar, P. Stadelmann, F. Levy, D. Mihailovič, Science 292 (2001) 479. [6] M. Remškar, Z. Škraba, F. Cleton, R. Sanjines, F. Levy, Surf. Rev. Lett. 5 (1998) 423. [7] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, 3rd edn., Butterworth-Heinemann, Oxford, 1997. [8] V. Kralj-Iglič, V. Heinrich, S. Svetina, B. Žekš, Eur. Phys. J. B 10 (1999) 5. [9] W. Helfrich, Z. Naturforsch. 28c (1973) 693. [10] M. Vygodsky, Mathematical Handbook—Higher Mathematics, Mir Publishers, Moscow, 1975. [11] N.G. Chopra, L.X. Benedict, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Nature 377 (1995) 135. [12] D.J. Srolovitz, S.A. Safran, M. Homyonfer, R. Tenne, Phys. Rev. Lett. 74 (1995) 1779. [13] H.J. Deuling, W. Helfrich, J. Phys. Paris 37 (1976) 1335. [14] J.M. Schnur, Science 262 (1985) 1669; S. Chiruvolu, H.E. Warriner, E. Naranjo, K. Kraiser, S.H.J. Idziak, J. Radler, R.J. Plano, J.A. Zasadzinsky, C.R. Safinya, Science 266 (1994) 1222; N. Shahidzadeh, D. Bonn, O. Aguerre-Chariol, J. Meunier, Phys. Rev. Lett. 81 (1998) 4268; R. Oda, I. Huc, M. Schmutz, S.J. Candau, F.C. MacKintosh, Nature 399 (1999) 566. [15] A.A. Boulbitch, Phys. Rev. E 56 (1997) 3395. [16] L. Mathivet, S. Cribier, P.F. Devaux, Biophys. J. 70 (1996) 1112; V. Kralj-Iglič, G. Gomišček, J. Majhenc, V. Arrigler, S. Svetina, Colloids Surf. A 181 (2001) 315. [17] V. Kralj-Iglič, A. Iglič, G. Gomišček, F. Sevšek, V. Arrigler, H. Hägerstrand, J. Phys. A 35 (2002) 1533. [18] V. Kralj-Iglič, A. Iglič, H. Hägerstrand, P. Peterlin, Phys. Rev. E 61 (2000) 4230; V. Kralj-Iglič, A. Iglič, H. Hägerstrand, M. BobrowskaHägerstrand, Colloids Surf. A 179 (2001) 57. [19] J.B. Fournier, Phys. Rev. Lett. 76 (1996) 4436. F Priloga E Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 www.elsevier.com/locate/colsurfa Effect of surfactant polyoxyethylene glycol (C12E8) on electroporation of cell line DC3F Maša Kandušer a, Miha Fošnarič a, Marjeta Šentjurc b, Veronika Kralj-Iglič c, Henry Hägerstrand d, Aleš Iglič a, Damijan Miklavčič a,* a Faculty of Electrical Engineering, University of Ljubljana, SI-1000, Ljubljana, Tržaška 25, SI-1000 Ljubljana, Slovenia b J. Stefan Institute, SI-1000, Ljubljana, Slovenia c Faculty of Medicine, University of Ljubljana, SI-1000, Ljubljana, Slovenia d Department of Biology, Åbo Akademi University, FIN-20520, Abo/Turku, Finland Received 28 October 2001; accepted 20 August 2002 Abstract Surfactant polyoxyethylene glycol (C12E8) decreases the voltage required for irreversible electroporation in planar lipid bilayers. In our study the effect of non-cytotoxic concentration of C12E8 on cell membrane reversible and irreversible electroporation voltage was investigated in DC3F cell line. Cell suspension was exposed to a train of 8 electric pulses of 100 ms duration, repetition frequency 1 Hz and amplitudes from 0 to 400 V at electrode distance 2 mm. The effect of C12E8 on the reversible and irreversible electroporation was investigated. We found that C12E8 decreases the voltage necessary for irreversible electroporation but has no effect on reversible electropermeabilization. Cell membrane fluidity measured by electron paramagnetic resonance spectrometry, using the spin probe methylester of 5doxyl palmitate was not significantly changed due to the addition of C12E8. Based on this we conclude that the main reason for the observed effect were not the changes in the membrane fluidity. As an alternative explanation we suggest that C12E8 induced anisotropic membrane inclusions may stabilize the hydrophilic pore, by accumulating on a toroidally shaped edge of the pore and attaining favorable orientation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Polyoxyethylene glycol; C12E8; Electroporation; Membrane fluidity; DC3F; Anisotropic membrane inclusions 1. Introduction The cell membrane represents a semi-permeable barrier between the cell interior and its surround- * Corresponding author. Tel.: /386-1-4768456; fax: /3861-4264658 E-mail address: [email protected] (D. Miklavčič). ings. The application of high intensity electric pulses of short duration causes permeabilization of cell membrane termed electropermeabilization or electroporation [1,2]. Electroporation is widely used in biotechnology for gene transfer and has a good prospect to be used in gene therapy [3]. In clinical oncology electroporation is already used in combination with chemotherapy as a method termed electrochemotherapy [4 /6]. 0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 4 1 0 - 7 Priloga E 206 M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 There are different theoretical models that describe mechanisms of electropermeabilization of cell membrane. Among them the electroporation model is the most widely accepted and accounts for the key electropermeabilization phenomena [7]. According to this model in the presence of an external electric field, hydrophilic pores are formed in the cell membrane [7,8]. The number of hydrophilic pores formed due to exposure to electric pulses is depending on the voltage applied, and is thus a continuous process. Increased membrane permeability however is a threshold phenomenon since the increased flow is observed once the number of hydrophilic pores per membrane area is sufficiently high [9]. Namely, the external electric field must reach a critical value to induce a threshold transmembrane potential that leads to increased membrane permeability [2]. We use the term ‘reversible electropermeabilization’ in those cases in our study when we observe membrane permeabilization and the term ‘irreversible electroporation’ when we observe cell death due to membrane rupture. Both phenomena are described in the existing literature on electroporation; reversible (when cell membrane reseals and cell survives) or irreversible (when cell membrane cannot reseal and cell dies) [10 /15]. The electroporation depends on different parameters among which the parameters of external electric field (electric field strength, duration and number of pulses) have been studied extensively [14,16/24]. The choice of these parameters determines whether the electroporation is reversible or irreversible in nature. Besides parameters of electric field, other parameters have been shown to affect electroporation; among those, the physical properties of the cell membrane, such as membrane fluidity. The study of the cell culture of Chinese hamster ovary cells treated with ethanol and lysolecitin, which incorporate into the cell membrane and change the membrane fluidity, shows that electroporation behavior of treated cells is affected [13]. Nevertheless, little is known about the effect of surfactants on cell membrane electroporation. Surfactants are a group of amphiphilic substances that incorporate in the cell membrane and change its properties [25,26]. The polyoxyethylene glycol surfactant C12E8 is widely used as a solubilizer of membrane proteins. Its critical micellar concentration is 0.09 mM [27]. The results of previous studies indicate a rapid flip /flop of C12E8 across the lipid and cell membranes [28 /30]. Effects of C12E8 on the cell membrane depend on its concentration. At sub-solubilizing concentration, C12E8 perturbs membrane structure and function and changes its physical properties [27,31,32]. The C12E8 molecules, while bound in the membrane bilayer, may form C12E8 /phospholipid [32,33] and C12E8 /protein [34] complexes that we call inclusions [33,35,36]. It was shown that in POPC planar lipid bilayers C12E8 significantly lowers the voltage required for irreversible electroporation [37]. The aim of the present study was to test the effect of a non-cytotoxic concentration of C12E8 on reversible and irreversible electroporation of cells in vitro and to investigate the possible mechanisms responsible for these phenomena. Therefore, a non-cytotoxic concentration of C12E8 was chosen which did not change the membrane fluidity. The possible mechanism of stabilization of the hydrophilic pores by C12E8 molecules was studied theoretically. 2. Experiments 2.1. Materials and methods 2.1.1. Cells Transformed Chinese hamster lung fibroblast cells, DC3F, were used. Cells were grown in Eagle’s minimum essential medium with Earles salts, 2 mM L-glutamine (Sigma-Aldrich Chemie GmbH, Deisenhofen, Germany), sodium bicarbonate (Braum, Melsungen, Germany), benzyl penicillin (Crystacillin, Pliva d.d., Zagreb, Croatia) and gentamicin sulphate (Lek d.d., Ljubljana, Slovenia), supplemented with 10% fetal bovine serum (Sigma-Aldrich Chemie GmbH). The cells were maintained at 37 8C in a humidified atmosphere that contained 5% CO2 for 3/4 days to obtain confluent culture from which the cell suspension was prepared with 0.05% trypsin solution containing 0.02% EDTA (Sigma-Aldrich Chemie GmbH). Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 2.1.2. Citotoxicity of C12E8 on cell culture To determine cytotoxicity of C12E8 (Fluka, Sigma-Aldrich Chemie GmbH) the cell suspension was incubated with a sub-solubilization concentration (0.5, 5 and 13 mg ml1) of C12E8 in Spinner modification of Eagle’s minimum essential medium (Life Technologies Ltd, Paisley, UK) for 1 h at 378 (treated cells) while untreated cells were incubated with Spinner modification of Eagle’s minimum essential medium for the same time at the same temperature. After incubation the cells were diluted in Eagle’s minimum essential medium and plated in the same medium supplemented with 10% fetal bovine serum in concentration 200 cells per Petri dish for clonogenic test. Colonies grown for 5 days were fixed with methanol (Merck KGaA, Darmstadt, Germany) and stained with crystal violet (Sigma-Aldrich Chemie GmbH). Colonies were counted and results normalized to untreated cells and expressed as a percentage of survival. Results of four independent experiments were pooled and presented as mean9/S.E. Differences between untreated and C12E8 treated cells were tested by Student t-test. For further experiments, a ten times lower concentration than the one that was determined as non-cytotoxic was used (i.e. 0.05 mg ml1 C12E8). 2.1.3. Cell membrane fluidity measurements The cells in suspension were incubated for 45 min at 37 8C; 0.05 mg ml 1 C12E8 was added to treated cells, while the medium in which C12E8 was dissolved was added to untreated cells. The membrane fluidity was measured by electron paramagnetic resonance method (EPR), using the spin probe methylester of 5-doxyl palmitate (MeFASL (10,3)), which is lipophilic and therefore incorporates primarily into the membrane lipid bilayer [38]. Sixty milliliters of 0.1 mM MeFASL (10.3) in ethanol solution was placed into the glass tube, then ethanol was evaporated by rotavapor. 1 ml of cell suspension that contained 20/106 cells was added to the MeFASL film formed on the wall of the glass tube, and incubated for 15 min while shaking. After that, the cell suspension was centrifuged and the pellet was placed in glass capillary for EPR measurements. From the EPR spectra at 4 8C, the maximal hyperfine splitting 207 constant 2AII that reflects the order parameter was measured. At 37 8C, the empirical rotational correlation time (tc) that reflects rotation of the low and middle field amplitudes, was calculated pffiffiffiffiffiffiffiffiffiffiffiffiffiffi using the following equation: tc KDH0 h1 =h0 (Fig. 1). Both parameters reflect ordering and dynamics of phospholipids in the membrane bilayer and are the measure of membrane fluidity [39]. Each measurement was repeated three times in three independent experiments. Differences between untreated and C12E8 treated cells were tested by the Student t-test. 2.1.4. Electroporation To determine the effect of non-cytotoxic concentration of C12E8 (that does not affect membrane fluidity) on reversible electropermeabilization, the cell suspension was prepared in Spinner modification of Eagle’s mini- Fig. 1. Typical EPR spectra of DC3F cells at 4 8C (A) and at 37 8C (B). At 4 8C the maximal hyperfine splitting constant 2AII was measured, and at 37 8C the empirical rotational correlation time (tc) was calculated from h0, DH0 and h 1. Priloga E 208 M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 mum essential medium. Before electroporation was performed, bleomycin was added to the cell suspension in final concentration of 5 nM. At this concentration, bleomycin is non-cytotoxic as the cell membrane is unpermeant for this substance [40]. When the electroporation takes place bleomycin gains access to the cell interior and provokes cell death [40]. In this way reversible electropermeabilization is detected. In all experiments, cell suspension was incubated with C12E8 as described above for the cell membrane fluidity measurements. A 50 ml droplet of cell suspension that contained 106 cells was placed between stainless steel electrodes, which were 2 mm apart. The train of 8 square pulses, 100 ms of duration and of the repetition frequency 1 Hz was generated (we used a prototype electroporator developed in our laboratory at the University of Ljubljana, Faculty of Electrical Engineering, Slovenia [41]). The voltage from 0 (control) to 400 V in 40 V steps was applied. Cells were then incubated at room temperature for 30 min to allow cell membrane resealing. After that the uptake of bleomycin was determined by clonogenic test as already described above for citotoxicity experiments. Cell colonies were counted for untreated and C12E8 treated cells and results were normalized to the corresponding results of the control, i.e. cells that were not exposed to electric pulses. The percentage of colonies was subtracted from 100 percent to obtain the percentage of permeabilized cells. Results of five independent experiments were pooled and presented as a mean value9/S.E. Differences between the untreated and the C12E8 treated cells were tested by the Student t -test. To determine the effect of the non-cytotoxic concentration of C12E8 on irreversible electroporation, the protocol that is described above was used without the addition of bleomycin. To gain more information about the effect of C12E8 on irreversible electroporation, additional experiments were performed. The cell suspension was electroporated using the same protocol as described above while C12E8 was added immediately after the application of the train of 8 pulses when the cell membrane is still permeable for small molecules like C12E8. Cell survival was determined by the clonogenic test as described above. The results of three independent experiments were pooled and presented as a mean value9/S.E. Differences between the untreated and the C12E8 treated cells were tested by Student t -test. The results of the permeabilization and the cell survival with respect to the applied electric field (i.e. voltage applied on the electrodes) are reported. The nominal electric field for the geometry and dimensions of the electrodes used in our experiments can be estimated as the voltage applied to the electrodes divided by the distance between the electrodes. 2.2. Experimental results 2.2.1. The cytotoxic effect of C12E8 on the cell culture and cell membrane fluidity The cytotoxic effect of different concentrations of C12E8 was tested to establish the non-cytotoxic concentration. The results show that sub-solubilizing concentrations of C12E8 (lower than 0.5 mg ml1), 60 min incubation at 37 8C are noncytotoxic for the DC3F cell line (Table 1). The maximal hyperfine splitting 2AII of the EPR spectra, which reflects the order parameter of the membrane lipids and the empirical rotational correlation time tc were measured (Fig. 1). No differences were found by Student t -test between the C12E8 treated and the untreated cells (Table 2). From these results, we conclude that C12E8 at noncytotoxic concentration of 0.05 mg ml1 affects neither the order parameter (packing of phospholipids) nor the dynamics of the motion of lipids in the bilayer of the DC3F cell line due to the low molar ratio of C12E8 to membrane phospholipids. Table 1 Cytotoxic effect of different concentrations of C12E8 mg ml 1 on cell line DC3F C12E8 mg ml 1 Survival (% of control) t -test 0 0.5 5 13 1009/0 929/3 499/9 209/8 P/0.06 P/0.03 P/ B/0.001 Cell suspension was incubated for 60 min at 37 8C. Values are means of 4 experiments9/S.E. Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 Table 2 Order parameter and rotational correlation time in DC3F cell line after 45 min incubation with SMEM control and with C12E8 at concentration 0.05 mg ml 1 treated cells Order parameter 2AII (mT) Rotational correlation time tc (ns) Control C12E8 ttest 5.819/0.4 2.079/ 0.02 5.719/1.4 2.219/ 0.14 NS NS Values are means of 3 experiments9/STD. 2.2.2. Electroporation The electroporation experiments revealed that C12E8 does not affect the reversible electroporation, however, it significantly increases the irreversible electroporation i.e. the irreversible electroporation occurs at lower applied voltages. The differences between the C12E8 treated cells and the control was most pronounced when the cell suspension was exposed to the train of 8 pulses and voltages 160V (P /0.037) and 200V (P / 0.022), as shown by Student t -test (Fig. 2). The addition of C12E8 caused the cell death at the same voltage at which the reversible electroporation takes place. This can be explained by a pore Fig. 2. The effect of C12E8 on reversible and irreversible electroporation measured by bleomycin uptake and cell survival, respectively on cell line DC3F. The train of 8 pulses, 100 ms, and repetition frequency 1 Hz was applied. The voltage of each train of pulses applied to 2 mm stainless still electrodes was 0, 80, 120, 160, 200, 240, 280, 320, 360 and 400 V. Cell suspension was incubated with 0.05 mg ml 1 C12E8 for 45 min (C12E8) or with electroporation medium (control). Values are means of 5 experiments9/S.E. 209 stabilization effect of C12E8 as explained in detail in our theoretical considerations. We presume that hydrophilic pores are formed at the voltage at which reversible electroporation takes place and that these pores are prerequisite for the bleomycin access to the cell interior that causes cell death. On the other hand cell death that is a consequence of irreversible electroporation is caused by electric field itself that provokes irreversible changes in the cell membrane. In control cells, which were not treated with C12E8 so that the pore stabilization did not occur, 50% of the cells survived the application of pulses of the amplitude of 250 V. In these cells, a resealing of the cell membrane took place while in the C12E8 treated cells no cell survived the application of pulses of the amplitude of 250 V, as the resealing was prevented by C12E8 (Fig. 2). In control cells, we observe 50% of the permeabilization as determined by bleomycin uptake at 160 V. At the same voltage, in the C12E8 treated cells, we observe 50% of the permeabilization and also only 50% of cell survival after treatment with electric field (Fig. 2). This shows that the irreversible electroporation of the C12E8 treated cells is shifted to the same voltage at which reversible electroporation occurs. In other words, electropermeabilization in the presence of C12E8 becomes irreversible as soon as it occurs. These results lead us to the conclusion that stabilization of the hydrophilic pores is responsible for the observed behavior as described latter in the theoretical part of this work. To confirm our conclusion we performed additional experiments. From the described experiments we could not distinguish between the pore stabilization effect of C12E8 and the possibility that C12E8 could be toxic when it has access to cell interior. Therefore, in these additional experiments the C12E8 was added immediately after the application of the train of 8 pulses. We showed that C12E8 was not cytotoxic when it gained access to the cell interior (Fig. 3), as after electroporation the cell membrane remains permeable for small molecules such as C12E8. From these results we concluded that the cell death observed in the previous experiment (Fig. 2) had to be caused by the effect of C12E8 on the pore stabilization and Priloga E 210 M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 Fig. 3. The effect of C12E8 added immediately after application of electric pulses on irreversible electroporation measured by cell survival on cell line DC3F. The train of 8 pulses, 100 ms, and repetition frequency 1 Hz was applied. The voltage of each train of pulses applied to 2 mm stainless still electrodes was 0, 80, 120, 160, 200, 240, 280, 320, 360 and 400 V. 0.05 mg ml 1 C12E8 (C12E8) or electroporation medium (control) was added to electroporated cells. Values are means of 3 experiments9/ S.E. that C12E8 has to be incorporated in the cell membrane at pulse application. 3. Theoretical considerations In this section, we present the theory that describes the formation of hydrophilic pores, which are responsible for increased membrane permeability [7]. Within this theory, a possible role of C12E8 membrane inclusions in the increased stability of hydrophilic pores [9,42] is described. A circular segment of a planar lipid bilayer with a hydrophilic circular pore in the center is studied. The formation of the hydrophilic pore in a lipid bilayer implies the existence of a bilayer edge [28,43]. In the process of pore formation the membrane constituents attain a configuration where the polar parts are used to shield the hydrophobic parts from the water (Fig. 4). For the sake of simplicity we assume a molecular arrangement, where both membrane layers at the edge of the pore bend towards each other [28,9], forming an inner half of the torus (Fig. 4). We consider a flat circular bilayer membrane segment of a radius l and of a thickness 2r , having Fig. 4. Cross section of the geometrical model of the circular membrane bilayer segment with a toroidal pore in the center. The whole segment and the pore are assumed to have rotational symmetry around the y -axis. R is the radius of the pore and 2r is the distance between the surfaces of both layers at the flat part of the segment. Principal membrane curvatures C1 and C2 are zero in the region R B/x B/l , where the membrane is flat. In the region R/r B/x B/R one of the principal curvatures is C1 /1/r , and the other is C2 /(x/R )/rx . a toroidal pore of a radius R at its center. Due to the rotational symmetry of the segment around the y-axis, the orientation of the x -axis in the y /0 plane is arbitrary. Outside the outer border of the circular bilayer segment (i.e. for x /l) the membrane bilayer remains flat, however these parts are not considered in our calculations (periodical boundary conditions are imposed at x/l). We describe the shape of the membrane with two principal curvatures, C1 and C2, given at each point on the membrane. Both principal membrane curvatures are zero in the region R B/x B/l, where the membrane is flat. In the region of the inner half of the torus (R/r B/x B/R ) one of the principal membrane curvatures is constant, C1 /1/r, while the other is C2 /(x/R )/rx (Appendix A). It was recently suggested [37] that C12E8 lowers the voltage required for irreversible electroporation of planar lipid bilayers. This increased susceptibility of lipid bilayers to electroporation could be related to the specific effect of the C12E8 molecules that changes the macroscopic physical properties of the membrane in a way that the formation of the pores becomes energetically more favorable. It was also proposed that C12E8 could be cooperatively bound in the region of the pore edge, which may stabilize the pore shape [30]. This scheme assumes an increased local area density of C12E8 in the region of the pore edge and rapid transport of C12E8 across the membrane [30]. The last assumption is in accordance with some recent Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 experimental results of the same authors [29] and also with the results [28,29]. It is known that C12E8 molecules may interact with membrane proteins and form C12E8 /protein complexes (inclusions) [34]. Therefore C12E8 may solubilize membrane proteins and change the membrane protein activity [34]. It has also been shown recently that the cooperative interaction of C12E8 with larger number of neighboring lipid molecules [32] may lead to the formation of the C12E8 /lipid membrane inclusions (clusters/rafts) [33,36]. Recent experimental results show that the effective shape of the lipid molecules interacting with the C12E8 molecule is changed because the acyl chains of the lipid molecules are moved apart sideways [31,32]. Therefore, the effective intrinsic shape of the C12E8 /lipid inclusion may be in general anisotropic [33,36] (see also Fig. 5). Also, the effective intrinsic shape of the detergent/ protein membrane inclusion may be in general 211 anisotropic [35,50,51,44,45]. In the present work the possible role of the effective intrinsic shape of the complexes formed by C12E8 molecules and the neighboring membrane constituents in the stability of the membrane hydrophilic pore is studied theoretically. In order to obtain the equilibrium shape of the hydrophilic pore in the planar membrane bilayer segment we are looking for the minimum of the membrane free energy (F ) as the sum of the pore edge energy (Wb) and the contribution of the C12E8 membrane inclusions (Fi), F Wb Fi : (1) The edge energy (line tension) of the pore [9,46] is approximated by the (monolayer) bending energy [47,48] of the pore (Wb): 1 Wb kc 2 g 4C̄ dA 2 k g C C dA; 1 2 G A 1 2 (2) A where kc is the local bending modulus, kG is the Gaussian bending modulus, C̄ (C1 C2 )=2; A is the area of the membrane segment and dA is the infinitesimal membrane area element. For the Gaussian bending modulus it is taken that kG / /2kc [49,52]. For the sake of simplicity, the spontaneous curvature was not included in Eq. (2). The C12E8 membrane inclusions may be C12E8/ lipid complexes and/or C12E8 /protein complexes. However, in this work, all the inclusions are for the sake of simplicity considered as equal. In general, the effective intrinsic shape of the inclusions is considered as anisotropic. The contribution of the C12E8 induced membrane inclusions to the membrane energy can be written in the form [51,36,44]: 1 j j+ Fi NkTln qI0 (3) Ĉ Ĉ m dA ; A 2kT g A Fig. 5. Schematic figure illustrating different intrinsic shapes of the membrane inclusions characterized by the two principal curvatures C1m and C2m. Shading marks the hydrophilic surface of the inclusion. The characteristic intrinsic shapes of inclusions are shown in the figure: C1m /C2m /0 (A), C1m /0, C2m /0 (B) and C1m /0, C2m B/0 (C). The corresponding most favorable membrane surfaces are also shown. where q is defined as j j j+ (C̄ C̄ m )2 qexp 2kT 4kT (Ĉ 2 Ĉ 2m ) ; (4) j and j * are the constants representing the Priloga E 212 M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 strength of the interaction between the inclusion and the membrane continuum, Ĉ /(C1/C2)/2, C̄ m (C1m C2m )=2; Ĉm /(C1m/C2m)/2, where C1m and C2m are the principal curvatures of the intrinsic shape (Fig. 5) of the inclusion, N is the total number of the inclusions in the membrane bilayer segment, I0 is the modified Bessel function of the first kind, k is the Boltzmann constant and T is the temperature. It is taken for simplicity that j /j * [51]. Due to the rapid C12E8 transport across the membrane [28 /30] we assumed that C12E8 inclusions are equally distributed between both membrane layers in the membrane region around the pore (the trans-bilayer transport of C12E8 molecules can be partially carried out also due to lateral flow of C12E8 through the membrane pores). For fixed values of parameters r, o , k and for fixed C1m and C2m, the membrane free energy is a function only of the radius of the pore R , i.e. F (R )/Wb(R )/Fi(R ), and we can search for the value (R /Rmin), which gives the minimal possible relative membrane free energy: Fmin /F (Rmin). The area A of the membrane segment must remain constant throughout the minimization, so when the size of the hydrophilic pore is changed, the radius of the segment (l) must change accordingly. The pore edge (bending) energy Wb is a monotonically increasing function of the radius of the pore R (Appendix A). Therefore Wb(R) does not have a minimum for R larger than r, which is the smallest possible radius of the pore in the described model. The membrane bending energy is minimal when the membrane surface is completely flat and without pores. Any anomaly in the flatness of the membrane surface increases its bending energy. The relative free energy of the membrane inclusions Fi/8pkc and the relative total membrane free energy Fi/8pkc /(Wb/Fi)/8pkc as functions of the radius of the pore R are shown in Fig. 6 for three different intrinsic shapes of inclusions. Fig. 6(A) shows the case of a conical isotropic intrinsic shape of the inclusions (C1m /C2m). It can be seen, that for isotropic inclusions the energy Fi decreases monotonically with increasing R. Summation of Wb and Fi gives us the total membrane free energy F, which can have a minimum for a particular radius of the pore R /Rmin. The number of inclusions N enters the expression (3) as the multiplication factor. Since Fi(R ) is a monotonically decreasing function for isotropic inclusions and Wb(R ) is monotonically increasing, we can always find such N , that the sum F /Wb/Fi will have a minimum. However, the position and the existence of such minimum depend very strongly on the value of N . Also, such pores with isotropic inclusions are unrealistically large (:/100 nm). A much stronger argument for the stability of the pore would be a minimum in the function Fi(R ) itself. This would also indicate that inclusions favor a particular size of the pore. The local minimum actually appears in the function Fi(R ) if the shape of the inclusions is sufficiently anisotropic, with C1m B/0 and C2m /0. For illustration, Fig. 7 shows the calculated radius of the stable pore (Rmin) as the function of C1m. With the constant value C2m /1/r, we have isotropic inclusions at C1m /1/r (indicated as point A in Fig. 7). With decreasing value of C1m, the shape of the inclusions becomes more and more anisotropic. When the intrinsic shape of the inclusions becomes anisotropic enough, i.e. when jĈm j is large enough, inclusions start to favor the toroidal shape of the edge of the pore and the local minimum appears in the function Fi(R ). Fig. 6(B) corresponds to the point B in Fig. 7, where the local minimum just begins to appear. In Fig. 6(C) it can be seen, that inclusions with C1m //0.5/r and C2m /1/r strongly stabilize the pore of a radius Rmin /2.9r . Based on the presented results it can be concluded that anisotropic inclusions can significantly contribute to stabilization of hydrophilic pores in bilayer membranes. The radii of such pores are of the order of the membrane thickness. On the other hand, the pores with isotropic inclusions are very weakly stabilized and are unrealistically large. 4. Discussion The question addressed by the present study was how non-cytotoxic concentrations of surfactant C12E8 affect the electroporation behavior of the cell line DC3F. The main finding is that incubation of cell suspension with C12E8 at non-cytotoxic Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 213 Fig. 6. The relative free energy of the inclusions (fi /Fi/8pkc) and the relative membrane total free energy (f/F /8pkc) as functions of the relative radius of the pore (R /r ) calculated for conical isotropic intrinsic shape of inclusions with C1m /C2m /1/r (A) and for two different anisotropic intrinsic shape of the inclusions: C1m //0.3/r , C2m /1/r (B) and C1m //0.5/r , C2m /1/r (C). The membrane segment free energy reaches minimum at: (A) Rmin /42r , (B) Rmin /5.2r and (C) Rmin /2.9r . The corresponding membrane segments with a pore with radius Rmin are also shown in the bottom. The outer circles represent the outer borders of the membrane segments and the inner circles represent the edges of the pores in the center. The values of parameters are: o /10, k/100, A /4pr2 /104 (the values are within the previously estimated range [51]). concentration of 0.05 mg ml 1 (which does not affect membrane fluidity) significantly decreases voltage for irreversible electroporation when the train of 8 pulses at frequency 1 Hz is applied. We propose that pore stabilization due to the incorporation of C12E8 into the cell membrane is the physical mechanism that may explain these experimental observations. Incubation of cell suspension with non-cytotoxic concentration of 0.05 mg ml1 (i.e. 0.09 mM) C12E8 for 45 min does not affect cell membrane fluidity (Table 1) however, it reduces irreversible electroporation for 63% compared with untreated cells (Fig. 2). These results are in agreement with previous results obtained on planar lipid bilayers [37]. On planar lipid bilayers, the electroporation with 100 ms pulses and simultaneous addition of 1 mM C12E8 reduced irreversible electroporation (measured as a voltage at which the membrane rupture takes place) for 69% with respect to untreated POPC membranes. Our results indicate that the reason for this effect could not be the changes in the membrane fluidity caused by incorporation of C12E8 into the cell membrane since C12E8 in concentration of 0.09 mM does not cause significant changes in membrane fluidity Priloga E 214 M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 Fig. 7. The relative equilibrium radius of the pore Rmin as the function of C1m, for C2m /1/r . The points A, B and C denote the values of Rmin for the corresponding Fig. 6(A /C). The values of parameters o , k and A are the same as given in Fig. 6. (Table 2). Although C12E8 increase membrane fluidity in POPC membranes [31,32], the molar ratio of phospholipids to C12E8 in our study was one order of magnitude lower than in published studies that report this effect of C12E8. The possibility that the effect of C12E8 on irreversible electroporation was the consequence of cytotoxic effect C12E8 when it has access to cell interior was disproved in the experiment when C12E8 was added immediately after pulse application (Fig. 3). In accordance, we proposed a mechanism that may explain our experimental results. Our explication is based on the findings obtained in planar lipid bilayers where it was hypothesized that in voltage induced metastable conductive pores hypothetical contaminants gradually replace the background lipids at the pore edge to lower its energy [53]. The effect of C12E8 on irreversible electroporation can be explained similarly. As shown by our theoretical study, C12E8 may stabilize pores by incorporation into the hydrophilic pores thereby preventing resealing of the cell membrane after electroporation and consequently causing cell death. This agrees with our experimental results (Fig. 2). In our theoretical model, a hydrophilic pore was considered as a toroidal pore in a center of a circular flat membrane segment. For a lipid bilayer with thickness 2r $/5 nm, the radius of the pore for isotropic inclusions is in the range of 100 nm (Fig. 6(A)). Although the minimum in the total membrane free energy was found for isotropic inclusions, the corresponding pores are unrealistically large. On the other hand, if the inclusions have an anisotropic intrinsic shape, the energy contribution of C12E8 inclusions itself can have a minimum for the specific size of the pore. For anisotropic inclusions, the radius of the pore is in the range of the thickness of the bilayer (Fig. 6(C)). It must be emphasized that the proposed expression for the continuum bending energy that was used for the pore edge energy (Eq. (2)) was originally derived for small principal curvatures [48]. In our case, at least one principal curvature is very large. Therefore the correct expression for the pore bending energy should in general include also higher terms than quadratic. However, in our theoretical model the major effect that stabilizes the pore originates from the energy of the membrane inclusions (Fi(R )), so the inaccuracy of the expression (2) for the edge energy of the pore does not have a great significance for the interpretation of our main theoretical predictions. In the described theoretical approach the area density of the membrane inclusion is in general non-homogeneous. Namely, the local area density of membrane inclusions in the region of the pore edge could be very high although the area density of inclusions far from the pore is very small. However, since the inclusions are treated as dimensionless (Eq. (3)) [51] and the excluded volume effect [54] is not taken into account, the local density of the inclusions at the pore edge may become unrealistically high if the inclusions favor the local shape of the membrane at the edge of the pore. Nevertheless, our purpose was only to describe the basic principles of the possible physical mechanism that determines the increased stability of pores in membrane bilayers when C12E8 molecules are incorporated in the membrane and to explain our experimental results (Fig. 2). Therefore, for simplicity the excluded volume effect was neglected. Although our theoretical model may offer a possible qualitative explanation of the influence of the C12E8 molecules on the stability of hydrophilic pores in membrane bilayer, the uncertainties or lack in experimentally measured model parameters prevent us from precise numerical calculations of Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 215 the stable size of the hydrophilic pore. In addition, we have neglected some other important contributions to the free energy of the pore [42,43] such as non-local bending energy also called area-difference-elasticity energy [55,56], surface pressure [9,57] and electrostatic energy of the pore [46]. In conclusion, we have shown that the detergent C12E8 affects irreversible electroporation of the cell line DC3F in a similar way as it was previously demonstrated in electroporation of planar lipid bilayer as a membrane model [37]. Our theoretical considerations indicate that C12E8-lipid and C12E8-protein induced anisotropic membrane inclusions [33,36] may stabilize a hydrophilic pore that is formed during the pulse application in the membrane. Recently the existence of hydrophylic pores in the lipid mebranes that can become stable under electrocompressive stress was proposed [58]. Our results could therefore be considered as a circumstantial evidence for the existence of hydrophilic pores that become stabilized by anisotropic membrane inclusions that prevent membrane resealing and consecutively transform the reversible electroporation into the irreversible electroporation. F is normalized relative to the energy 8pkc, f/F / 8pkc /wb/fi, where wb /Wb/8pkc and fi /Fi/ 8pkc. In the described geometrical model of the membrane bilayer segment with a circular pore in the center (Fig. 4), both membrane layers are flat in the region R B/x B/l and bend towards each other in the region R/r B/x B/R , forming an inner half of the torus with the larger radius R and the smaller radius r. The principal membrane curvatures of the membrane can be derived from the expressions for principal curvatures of a surface obtained by the function y(x ) rotated around the y -axis, C1 //y ƒ(1/y ?2) 3/2 and C2 // y?x1(1/y?2) 1/2. The principal curvatures C1 and C2 are zero in the region R B/x B/l, where the membrane is flat (Fig. 4). In the region of the pore edge (R/r B/x B/R ), the function y (x ) obeys the equation (x/R)2/y2 /r2, therefore C1 /1/r and C2 /(x/R )/rx. Knowing C1 and C2, integration in the expression for the pore edge energy (Eq. (2)) can be performed. The relative pore edge energy can be then written as Acknowledgements Expansion of wb into the power series [59] of R yields wb p8(R0:5R1 O(R2 )): For an wb arcsin (1=R) p=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2=R)2 1 R2 2 (A1) This research was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia through various grants. Authors wish to express their thanks to Zorka Stolič from J. Stefan Institute, Ljubljana, Slovenia, for technical assistance and Leslie Tung and Kate Stebe from Johns Jopkins University, Baltimore, USA, for stimulating our experimental work. open pore (R /1) the pore edge energy wb(R ) is a monotonically increasing function of R . In our range of interest, wb(R ) only slightly deviates from the linear function. To obtain the expression for the free energy of the inclusions, the parameters o /NkT /8pkc and k/j /kTr2 are introduced and integration in Eq. (3) is performed. For the relative free energy of the inclusions we can write the expression: Appendix A fi oln (2apR) In this work dimensionless quantities are introduced. The effective thickness of a lipid monolayer r is chosen for the unit of length (Fig. 4). The variables and parameters are redefined as follows: x /r 0/x , r/r 0/1, R /r 0/R , l/r 0/l, C1r 0/C1, C2r 0/ C2, C1mr 0/C1m, C2mr 0/C2m, C̄r 0 C̄; Ĉr 0/Ĉ , C̄ m r 0 C̄ m ; Ĉm r 0/Ĉm . The membrane free energy k exp (C̄ 2m Ĉ 2m ) 2Ĩ 2 (A2) where a /A /4pr2 is the relative area of a flat membrane segment without a pore and Ĩ is: Priloga E M. Kandušer et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 214 (2003) 205 /217 216 R g x q̃I0 (kĈ Ĉ m ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx Ĩ 1 (R x)2 R1 with q̃ defined as k 2 2 2 q̃ exp ((C̄ C̄ m ) Ĉ Ĉ m ) : 2 (A3) (A4) References [1] D.C. Chang, B.M. Chassy, J.A. Saunders, A.E. Sower (Eds.), Guide to Electroporation and Electrofusion, Academic Press, New York, 1992. [2] E. Neumann, A.E. Sowers, C.A. Jordan (Eds.), Electroporation and Electrofusion in Cell Biology, Plenum Press, New York and London, 1989. [3] M.F. Bureau, J. Gehl, V. Deleuze, L.M. Mir, D. Scherman, Biochim. Biophys. Acta 1474 (2000) 353. [4] G. Serša, B. .Štabuc, M. Čemazar, B. Jančar, D. Miklavčič, Z. Rudolf, Eur. J. Canc. 34 (1998) 1213. [5] R. Heller, R. Gilbert, M.J. Jaroszeski, Adv. 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Champeil, Biochemistry 26 (1987) 4803. [29] U. Kragh-Hansen, M. le Maire, J.V. Møller, Biophys.J. 75 (1998) 2932. [30] H. Hägerstrand, J. Bobacka, M. Bobrowska- Hägerstrand, V. Kralj-Iglič, M. Fošnaric, A. Iglič, Cell Mol. Biol. Lett. 6 (2001) 161. [31] R.L. Thurmond, M.F. Otten, Brown, K. Beyer, J. Phys. Chem. 98 (1994) 972. [32] H.H. Heerklotz, H. Binder, H. Schmiedel, J. Phys. Chem. 102 (1998) 5363. [33] M. Bobrowska-Hagerstrand, V. Kralj-Iglič, A. Iglič, K. Bialkowska, B. Isomaa, H. Hägerstrand, Biophys. J. 77 (1999) 3356. [34] J.V. Møller, M. le Maire, J. Biol. Chem. 268 (1993) 18659. [35] J.B. Fournier, Phys. Rev. Lett. 76 (1996) 4436. [36] A. Iglič, V. Kralj-Iglič, B. Bozič, M. Bobrowska-Hägerstand, B. Isomaa, H. Hägerstrand, Bioelectrochemistry 52 (2000) 203. [37] G. Troiano, K. Stebe, V. Sharma, L. Tung, Biophys. J. 75 (1998) 880. [38] M. .Šentjurc, M. Zorec, M. Čemazar, M. Auersperg, G. Serša, Cancer Letts. 130 (1998) 183. [39] D. Marsh, in: E. 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B 2003, 107, 12519-12526 12519 Stabilization of Pores in Lipid Bilayers by Anisotropic Inclusions Miha Fošnarič,† Veronika Kralj-Iglič,‡ Klemen Bohinc,† Aleš Iglič,† and Sylvio May*,§ Laboratory of Applied Physics, Faculty of Electrical Engineering, UniVersity of Ljubljana, Tržaška 25, SI-1000 Ljubljana, SloVenia, Institute of Biophysics, Faculty of Medicine, UniVersity of Ljubljana, LipičeVa 2, SI-1000 Ljubljana, SloVenia, and Institute of Molecular Biology, Friedrich-Schiller-UniVersity, Winzerlaer Strasse 10, Jena 07745, Germany ReceiVed: April 17, 2003; In Final Form: September 5, 2003 Pores in lipid bilayers are usually not stable; they shrink because of the highly unfavorable line tension of the pore rim. Even in the presence of charged lipids or certain additives such as detergents or isotropic membrane inclusions, membrane pores are generally not expected to be energetically stabilized. We present a theoretical model that predicts the existence of stable pores in a lipid membrane, induced by the presence of anisotropic inclusions. Our model is based on a phenomenological free energy expression that involves three contributions: the energy associated with the line tension of the pore in the absence of inclusions, the electrostatic energy of the pore for charged membranes, and the interaction energy between the inclusions and the host membrane. We show that the optimal pore size is governed by the shape of the anisotropic inclusions: saddle-like inclusions favor small pores, whereas more wedgelike inclusions give rise to larger pore sizes. We discuss possible applications of our model and use it to explain the observed dependency of the pore radius in the membrane of red blood cell ghosts on the ionic strength of the surrounding solution. Introduction Biological cells exchange material with the surrounding environment through the cell membrane. One of the mechanisms for transmembrane transport involves the presence of pores in the lipid bilayer, through which a substantial flow of material can take place. For example, pores were observed in red blood cell ghosts,1-3 where the pore size depends on the ionic strength of the surrounding fluid.2 The formation of pores in the membrane can also be induced by applying an AC electric field across the membrane.4 This phenomenon is known as electroporation and has become widely used in medicine and biology.5-7 Finally, the formation of pores plays an important role in the action of many antimicrobial peptides.8 A number of theoretical studies have been made to understand the physical basis of electroporation9,10 and peptide-induced pore formation.11-14 However, the mechanisms responsible for the energetics and stability of membrane pores are still obscure and require further clarification. The formation of a pore in a lipid bilayer implies the existence of a bilayer edge. It is likely15-17 that in the process of pore formation the lipid molecules near the edge of the pore rearrange themselves in such a way that their polar headgroups shield the hydrocarbon tails from water (Figure 1). Modified molecular packing of the phospholipid molecules at the bilayer edge causes the membranes to have high line tension, Λ, that is, high excess energy per unit length of the exposed edge, making pores energetically unfavorable. In fact, even if the membrane is subject to a lateral tension, holes in membranes are not stable: they either shrink, or above a critical size, they grow. On the other hand, there are various examples in which membrane pores * Corresponding author. E-mail: [email protected]. Phone: ++49-3641-657582. Fax: ++49-3641-657520. † Faculty of Electrical Engineering, University of Ljubljana. ‡ Faculty of Medicine, University of Ljubljana. § Friedrich-Schiller-University. live long enough to be observed experimentally.2,18-20 The question arises what mechanisms could be responsible for the stabilization of pores against immediate and spontaneous closure or widening. One such mechanism has recently been suggested by Betterton and Brenner.21 It applies to charged membranes and is based on competition between line tension and electrostatic repulsion between the opposed membrane rims within a pore. An analysis based on linearized Poisson-Boltzmann theory showed for certain combinations of membrane charge density, σ, line tension, Λ, and Debye length, ld, that holes become energetically stabilized. However, for common lipid membranes, the depth of the minimum is so shallowsbelow kT where k is Boltzmann’s constant and T the absolute temperaturesthat additional stabilizing effects are required to explain the existence of experimentally observed pores.21 In the present work, we suggest and analyze a different explanation for the stabilization of pores in fluid membranes, the presence of anisotropic inclusions. Inclusions are rigid, membrane-inserted bodies that appear in biological or model membranes such as (often transmembrane) proteins or peptides, detergents, or sterols. If not all in-plane orientations of the inclusion are energetically equivalent, then the inclusion is referred to as anisotropic. Anisotropic inclusions are candidates for the formation of membrane pores because the pore rim provides a lipid packing geometry with which anisotropic inclusions can favorably interact.22-27 Of particular interest in this respect are certain antimicrobial peptides. These peptides are positively charged and amphipathic, often exhibiting their lytic activity through the cooperative formation of membrane pores.8,19 Most importantly in connection with the present work, antimicrobial peptides are typically elongated in shape, which renders their interaction with curved membranes highly anisotropic. Examples of anisotropic inclusions also include various lipids28 (certain cationic lipids,29 glycolipids, or lipoproteins30), 10.1021/jp035035a CCC: $25.00 © 2003 American Chemical Society Published on Web 10/16/2003 Priloga F 12520 J. Phys. Chem. B, Vol. 107, No. 45, 2003 Fošnarič et al. detergents,31 gemini or detergents with a large and anisotropic headgroup.32 The abundance of proteins in biological membranes has motivated numerous theoretical studies on membrane-inclusion interactions; see, for example, the reviews by Gil et al.33 and Goulian34 and references therein. Among the various lines of research, some focus has recently been put on anisotropic inclusions. These inclusions are nonaxisymmetric but, for simplicity, are usually considered to still have quadrupolar symmetry. One principal question concerns the lateral organization of anisotropic inclusions and the corresponding response of the membrane shape. This question was addressed recently on two different levels of approximation. The first approach considers indiVidual inclusions. Here, an angular matching condition between a given inclusion and the host membrane is imposed. For example, a single isotropic inclusion induces a catenoid-like membrane shape (for which the mean curvature vanishes at each given point). Interference of the inclusion-induced membrane perturbations gives rise to membrane-mediated interaction between inclusions. For two isotropic inclusions, this interaction is known to be repulsive, at least in the low-temperature limit.35 However, more than two inclusions (isotropic or anisotropic) cause nontrivial manybody effects that induce complex spatial patterns of the inclusion arrangement.36-38 There is a second, mean-field level approach27 which we adopt in the present work. Here, a given small membrane patch contains an ensemble of inclusions. The inclusions do not individually deform the membrane but energetically couple to the shape of the membrane patch. Note that the shape of the membrane patch is prescribed (but may later be optimized). The coupling between the inclusions and the membrane results from a mismatch of the given membrane curvatures and the preferred (“spontaneous”) curvatures of the inclusions. Thus, the membrane curvatures act as a mean-field that must self-consistently be determined so as to minimize the overall free energy, yielding the local inclusion density everywhere on the membrane and the corresponding optimal membrane shape. We will show that anisotropic membrane inclusions are candidates for the stabilization of pores in lipid bilayers. To this end, we analyze the energetics of a single membrane pore in a binary lipid membrane, consisting of (charged) lipids and anisotropic inclusions. The free energy of the inclusion-doped membrane contains the line tension contribution due to the rearrangement of lipids within the pore region, the interaction energy between the anisotropic inclusions and the membrane, and the electrostatic energy of the charged lipids. The latter is taken into account to allow a prediction of how the pore size depends on the salt concentration. Theoretical Model We consider a lipid membrane that contains a single pore. For our purpose, it is most convenient to assume a perfectly planar membrane and a pore of circular shape, say of aperture radius r. We locate a Cartesian coordinate system at the pore center with the axis of rotational symmetry (the z-axis) pointing normal to the bilayer midplane. The presence of the membrane pore is likely to imply some structural rearrangement of the lipids at the bilayer rim. This reorganization is driven by the unfavorable interaction of the lipid tails when exposed directly to the aqueous environment. Even though experimentally obtained evidence is currently not available, it seems a reasonable approximation to assume (and we base our present work on this notion) that the lipids within the rim assemble into a Figure 1. A planar lipid bilayer with a pore in the center. The figure shows the cross-section in the x-z plane. Rotational symmetry around the z-axis is indicated. On the left side, the packing of the lipid molecules is shown schematically. The headgroups of lipid molecules are represented by filled circles. The arrow denotes the membrane inclusion, which is shown schematically. semitoroidal cap to shield the hydrocarbon chains from contact with the aqueous environment. The height of the cap fits the bilayer thickness 2b, implying a radius, b, of its circular crosssectional shape. The pore geometry is schematically shown in Figure 1. A parametrization of the semitoroidal cap is given by x ) b cos φ (r/b + 1 + cos θ), y ) b sin φ (r/b + 1 + cos θ), and z ) b sin θ with 0 e φ e 2π and π/2 e θ e 3π/2. The principal curvatures of the cap are then C1 ) 1 b C2 ) cos θ r + b(1 + cos θ) (1) and the area element dAP ) b[r + b(1 + cos θ)] dφ dθ. The local geometry within the rim is saddle-like everywhere and most pronounced at θ ) π where C1/C2 ) -r/b. Note again that the semitoroidal shape of the rim is an assumption; alternative choices could be considered but are not expected to alter the conclusions of the present work. Our objective is to analyze the influence of anisotropic inclusions on the energetics of a membrane pore. As is wellknown, lipid membranes are two-dimensional fluids that allow inserted inclusions to redistribute laterally. Consequently, inclusions accumulate at energetically beneficial membrane regions or even induce their formation. The creation of a membrane pore is a drastic example for an inclusion-induced reorganization of a lipid membrane. Obviously, if inclusions induce stable pores, they must be able to affect substantially the pore free energy F as a function of the pore radius r. To obtain the equilibrium size of the pore, the overall free energy, F, of the pore is minimized. We assume that F is the sum of three contributions: F ) Wedge + Uel + Fi (2) where Wedge is the energy due to the line tension of a lipid bilayer without the inclusions, Uel is the electrostatic energy of the charged lipids, and Fi is the energy due to the interactions between the membrane inclusions and the host membrane. We note that F and all its contributions are excess free energies, measured with respect to a planar, pore-free membrane. We also note that our work does not involve any additional constraints of the membrane area, A. Hence, we work at vanishing lateral tension, as should be appropriate for most bilayers. Line Tension of a Lipid Bilayer. The modified molecular packing of the lipid molecules at the edge of the pore (see Figure 1) entails an energy cost Wedge. For an inclusion-free membrane, this energy cost is given by Wedge ) 2πΛr, where r is the radius of the circular membrane pore and Λ is the line tension (i.e., energy per unit length of the exposed edge) of the lipid bilayer. One can easily obtain a rough estimate for Λ on the basis of the elastic energy required to bend a lipid monolayer into a semicylindrical micellar cap. Adopting the usual quadratic Priloga F Stabilization of Pores in Lipid Bilayers J. Phys. Chem. B, Vol. 107, No. 45, 2003 12521 Helfrich,39 curvature expansion for the free energy according to one finds16 Λ ) πkc/(2b) where kc is the lipid layer’s bending rigidity. Typically for a single lipid layer, b ) 2.5 nm and kc ) 10kT, implying Λ ≈ 6kT/nm ≈ 2 × 10-11 J/m (at room temperature). Indeed, this order of magnitude corresponds to experimental results for the line tension of lipid bilayers.18,40,41 It is worth noting that the semicylindrical micellar cap has large curvatures, C1 ) 1/b and |C2| e 1/r, for which the quadratic curvature expansion might completely fail.18 The fact that it does not fail indicates the robustness of the membrane elasticity approach; it is often used successfully to model experimentally observed structural reorganization of lipid assemblies, even if that involves large changes in curvature.42,43 If rigid membrane inclusions are present within the membrane pore, they replace some of the lipids, depending on their lateral extensions. The replaced lipids no longer contribute to the line tension Wedge. We shall account approximately for this reduction in line tension by writing Wedge ) 2Λ(πr - NPRi) (3) where NP denotes the number of inclusions within the membrane rim and 2Ri is the lateral extension of the cross-sectional shape of the inclusions. Steric interactions limit the number of inclusions within the membrane rim; NP e Nmax ) πr/Ri. Thus P Wedge g 0, and the line tension always provides a tendency for the pore to shrink. Electrostatic Energy of a Pore. The calculation of the electrostatic energy of a membrane pore follows Betterton and Brenner21 who derived an expression valid for a very thin membrane (b f 0) in linearized Poisson-Boltzmann (PB) theory. It is well-known that linearized PB theory greatly overestimates the electrostatic free energies for lipid membranes. Yet, solutions within nonlinear Poisson-Boltzmann theory (and for more realistic choices of b) are numerically demanding. Hence, to keep our model tractable, we adopt the result of linear PB theory where the equation ∇2φ ) κd2φ determines the (dimensionless) electrostatic potential, φ, at a given Debye length ld ) κd-1 ) w0kT/(2n0NAe02). Here w is the dielectric constant of the aqueous solution, 0 is the permittivity of free space, n0 is the ionic strength of the surrounding solution (i.e., bulk salt concentration, assuming a 1:1 salt such as NaCl), NA is Avogadro’s number, and e0 is the unit charge. The solution of the linear PB equation is written as the difference, φ(x,z) ) φ∞(z) - φ0(x,z), between the electrostatic potential of a flat infinite pore-free membrane (φ∞) and the electrostatic potential of the circular flat membrane segment with radius r (φ0), both having constant surface charge density, σ. The value of φ∞ is44 φ∞(z) ) (σ/(w0κd)) exp(-κdz). The expressions for φ0 can be obtained using the Hankel transformation and taking into account the boundary conditions φ(zf∞) ) 0 and ∂zφ(z)0) ) 0 for x > r, ∂zφ(z)0) ) -σ/(w0) for x < r. Then the electric potential φ(x,z) can be written as σr σ e-κdz φ(x,z) ) + w0κd w 0 J (kx)J (kr) ∫0∞ 0 2 1 2 κd + k e-z(κd +k ) dk 2 2 1/2 (4) where J0 and J1 are Bessel functions. Using eq 4 for the electric potential, we can derive the electrostatic free energy via a charging process:44,45 Uel,tot ) 2π ∫0∞σ(x)φ(z)0)x dx (5) Equation 5 can be further processed analytically. By subtracting the electrostatic energy of the charged pore-free membrane, one obtains an explicit expression for the excess electrostatic energy of the pore: Uel ) - πσ2r2 2πσ2r3 + w0κd w0 J1(x)2 ∫0∞ xx2 + κd2r2 dx (6) This is the result of Betterton and Brenner,21 which we shall use in our present work. Free Energy of Inclusions. Membrane inclusions are embedded within the host membrane, and the inclusion-membrane interactions are mainly governed by the hydrophobic effect. To describe the corresponding free energy, Fi, we use a phenomenological model26 in which the mismatch between the effective intrinsic shape of the inclusions and the actual shape of the membrane at the site of the inclusions causes an interaction energy. The actual shape of the membrane at the site of the inclusion can be described by the diagonalized curvature tensor C, C) [ C1 0 0 C2 ] (7) where C1 and C2 are the two principal curvatures. Similarly, the intrinsic shape of a given inclusion can be described by the diagonalized curvature tensor Cm, Cm ) [ C1m 0 C2m 0 ] (8) where C1m and C2m are the two intrinsic principal curvatures of the inclusion. In general, inclusions are anisotropic,22,23,27,28,31 which means that C1m * C2m. The principal directions of the tensor C deviate in general from the principal directions of the tensor Cm; say, a certain angle ω quantifies this mutual rotation. The single-inclusion energy (Ei) can then be expressed in terms of the two invariants (trace and determinant) of the mismatch tensor M ) RCmR-1 - C where R is the rotation matrix, R) [ cos ω -sin ω sin ω cos ω ] (9) Terms up to second order in the elements of the tensor M are taken into account: K h det M Ei ) (tr M)2 + K 2 (10) where K and K h are the interaction constants between the inclusion and the surrounding membrane. Using eqs 7-10, we can write the single-inclusion energy (Ei) in the form27 h )(H - Hm)2 - K h [D2 - 2DDm cos(2ω) + Dm2] Ei ) (2K + K (11) Quantities H ) (C1 + C2)/2 and Hm ) (C1m + C2m)/2 are the respective mean curvatures, while D ) (C1 - C2)/2 and Dm ) (C1m - C2m)/2 are the curvature deviators. Curvature deviator Dm describes the intrinsic anisotropy of the single membrane inclusion.26,28 The time scale for orientational changes of the anisotropic inclusions is usually small compared to shape changes of the lipid bilayer. It is therefore reasonable to employ an orientational averaging of the inclusions according to the rules of statistical Priloga F 12522 J. Phys. Chem. B, Vol. 107, No. 45, 2003 Fošnarič et al. mechanics. To this end, we take into account that the inclusions can rotate around the axis defined by normal to the membrane at the site of the inclusion. Then the partition function, q, of a single inclusion is22,28 q) 1 ω0 ( ∫02π exp - ) Ei(ω) dω kT (12) where ω0 is an arbitrary angle quantum. Inclusions can also move laterally over the phospholipid layer, so they can distribute laterally over the membrane in the way that is energetically the most favorable.23,27 The lateral distribution of the inclusions in a membrane of overall area A is in general nonuniform. Treating inclusions as pointlike, independent, and indistinguishable, we can derive the expression for the contribution of the inclusions to the membrane free energy on the basis of eqs 11 and 12:27 [ ∫ q I (2KkTh DD ) dA] Fi 1 ) -N ln kT A A c 0 (13) m where N is the total number of inclusions in the membrane, qc is defined as ( qc ) exp - 2K + K h 2 K h (H - 2HHm) + D2 kT kT ) (14) and I0 is the modified Bessel function. The integration in eq 13 is performed over the whole area, A, of the membrane, including the pore region of area AP and the two flat monolayers that constitute the planar bilayer part. Recall that Fi in eq 13 (together with eq 14) is an excess free energy with respect to the pore-free planar membrane, as was defined in eq 2. Indeed, it is Fi(H)D)0) ) 0 and hence only those inclusions that are located within the pore rim (but not those in the planar membrane) contribute to Fi. Because the overall area A is assumed to be large, we can expand Fi with respect to A, yielding Fi )n kT h DDm)] dAP ∫A [1 - qcI0(2K kT (15) P where n ) N/A is the area density of the inclusions in the membrane and where the integration extends only over the area, AP, of the membrane rim. The influence of the inclusion’s h /(kT)) anisotropy is contained in the Bessel function I0(2DDmK (the coefficient qc is independent of Dm). Because I0 g 1, we see from eq 15 that the anisotropy of inclusions always tends to lower Fi. Yet, whether inclusions eventually lower or increase F depends on all inclusion properties, on the geometry (Dm and Hm) and on the interaction constants (K and K h ). We also note that the number of inclusions contained within the membrane rim is given by NP ) n h DDm) dAP ∫A qcI0(2K kT (16) P Combination of eqs 15 and 16 yields Fi ) nAP - NP kT Figure 2. Cross sections through a lipid layer containing a single cylindrical inclusion (black square). Some lipids are shown schematically. In the planar layer (a), all lipids pack, on average, into a cylindrical shape (schematized by the shaded rectangle). Bending the monolayer (b) induces a splay deformation of the lipids. Because the inclusion is rigid, it cannot participate in the splay deformation, thus inducing an extra (excess) splay of the lipids in its vicinity. (17) Of course, if no energetic preference exists for the inclusions to partition into the membrane rim (qcI0 ) 1), then eq 16 predicts NP/AP ) n and thus Fi ) 0. On the other hand, if the density of the inclusions within the pore region greatly exceeds the bulk density, then nAP , NP and Fi ≈ -NPkT. Hence, each inclusion that enters the membrane pore in excess to the bulk density contributes 1kT to the inclusion free energy. The inclusion size determines the maximal number, Nmax P , that can enter the pore rim. For rather large inclusions Ri ≈ b and small pores r ≈ b, is of the order of a very few inclusions. we expect that Nmax P That seems to indicate that for small pores Fi is not able to decrease F by more than a few kT. Below we show that nevertheless, for charged membranes, anisotropic inclusions can dramatically reduce F (substantially more than -NPkT). Estimation of the Constants. We discuss a simple generic, molecular-level model for the interaction between a single anisotropic inclusion and a lipid bilayer of principal curvatures C1 and C2 into which the inclusion is embedded. The model allows us to estimate the phenomenological constants, K, K h, Hm, and Dm, in terms of the inclusion shape and the elastic properties of the lipid bilayer. Bending a lipid layer implies a change in the average molecular shape of each individual lipid: a splay (or saddlesplay) deformation is imposed by the membrane curvature. Membrane-embedded inclusions cannot participate in the curvature-induced splay deformation because of their stiffness. That is, some lipids in the vicinity of the inclusion have to compensate for the stiffness of the inclusion by adopting an additional (“excess”) splay, beyond that of the lipids far from the inclusion. Figure 2 provides a schematic illustration of this mechanism for a cylindrical inclusion of radius Ri. The energy associated with the excess splay determines the inclusion energy Fi. It depends on both the inclusion’s shape and size and the elastic properties of the lipid layer. The inclusion’s shape can conveniently be characterized by a circular cross section (of radius Ri) and a modulated “cone angle” θi(φ) ) θ h i + ∆θi cos(2φ) along its circumference with a corresponding azimuthal angle φ. The average “coneness” of such an inclusion is θ h i, and the deviation along the inclusion’s circumference is ∆θi. Clearly, for ∆θi ) 0, the inclusion is isotropic, and for ∆θi ) θ h i ) 0, the inclusion is cylindrical. The elastic properties of a lipid layer can be characterized by the bending constant kc and the area stretching modulus Kc. From membrane elasticity theory, it is well-known46,47 that the decay length of the perturbation induced by a single inclusion is ξ ) 2b(kc/(Kcb2))1/4. If the inclusion radius is not much smaller than ξ (that is, for Ri J ξ), one can roughly estimate Priloga F Stabilization of Pores in Lipid Bilayers J. Phys. Chem. B, Vol. 107, No. 45, 2003 12523 the interaction constants 3 3 Ri K ≈ πkc , 2 ξ 2 K h ≈- K 3 (18) as well as Hm ) θ hi , Ri Dm ) ∆θi Ri (19) The derivation of eqs 18 and 19 will be given elsewhere. Note that the interaction constants K and K h depend not only on the monolayer’s bending stiffness kc but also on the area stretching modulus Kc (through ξ). The presence of Kc implies that local changes in the thickness of the monolayer leaflets are involved in the spatial relaxation of the inclusion-induced perturbation. This is not obvious because we do not impose any thickness mismatch between the inclusion and the membrane. However, as Figure 2 illustrates, insertion of the inclusion imposes an angular mismatch which spatially relaxes with the same decay length as a thickness mismatch.47-49 The relaxation involves a compromise between splay and dilation of the lipid chains. Typically for a lipid layer ξ ) 1 nm, kc ) 10kT, and thus K ≈ 50Ri3kT/nm and K h ≈ - 33Ri3kT/nm. The interaction constants increase strongly with the inclusion radius Ri. Because the validity of eq 18 requires Ri J 1 nm (below we shall use Ri ) b/2 ) 1.25 nm), our present estimate will necessarily predict a strong membrane-inclusion interaction. To this end, we note that eq 11 with Dm ) 0 has the same structure as the Helfrich bending energy for isotropic membranes. The corresponding interaction constant, K, (and similarly for K h ) for a lipid membrane could thus be identified28 with K ) kca0, where kc ≈ 10kT is the bending constant of an ordinary lipid monolayer (that is part of a bilayer membrane) and a0 ) 0.6-0.8 nm2 is the cross sectional area per lipid. Thus kca0 ≈ 7kT nm2, which is at least an order of magnitude smaller than the interaction constant K in eq 18. Hence, sufficiently large, rigid membrane inclusions are generally expected to partition strongly into “appropriately curved” membrane regions. We note that partitioning rigid inclusions into the rim of a membrane poresbesides causing an extra (excess) splay, see abovesalso replaces some structurally perturbed lipids. These lipids no longer contribute to the energy of the pore. The corresponding energy gain is not contained in Fi because we have taken it into account already in Wedge (see eq 3). Results and Discussion All of the following results are derived for a fixed thickness, b ) 2.5 nm, of the lipid layer, for a line tension of Λ ) 10-11 J/m, and for a surface charge density σ ) - 0.05 A s/m2 ) -e0/3.2 nm2 of the lipid layer. Taking into account a typical cross-sectional area per lipid of a0 ) 0.6-0.8 nm2, the value for σ would correspond roughly to a 1:4 mixture of (monovalently) charged and uncharged lipids. This is not an unusual situation in (biological and model) membranes. Inclusion-Free Membrane. Let us start with an inclusionfree membrane. This case has recently been analyzed by Betterton and Brenner.21 The free energy consists only of the line tension contribution (see eq 3) and the electrostatic free energy (see eq 6); the former favors shrinking and the latter growing of a membrane pore. The parameter that governs the resulting behavior is the Debye length, ld. For small ld, pores close; for large ld, pores grow. Betterton and Brenner have found that for intermediate ld a local minimum in F(r) exists that may Figure 3. The pore free energy, F, as a function of the pore size, r. The dashed lines correspond to a charged inclusion-free membrane of charge density σ ) -0.05 A s/m2 with Debye length ld ) (a) 2.6, (b) 2.8, and (c) 3.0 nm. The solid lines describe the effect of adding anisotropic inclusions (characterized by K ) 98kT nm2, K h ) -2K/3, C1m ) -C2m ) 1/b) to the charged membrane with σ ) -0.05 A s/m2 and ld ) 2.8 nm. The inclusion concentrations are (d) n ) 1/70 000 nm2 and (e) n ) 1/14 000 nm2. The inset shows the corresponding numbers NP as function of b/r for curves d and e. give rise to stable pores. As an illustration, we plot F(r) in Figure 3 for three different choices of ld, namely, ld ) (a) 2.6, (b) 2.8, and (c) 3.0 nm; a local minimum in F(r) is present only in curve b. Figure 3 exemplifies a general finding: the local minimum of F(r) is very shallowsbelow kTsand appears in a very narrow region of ld. Hence, pores in lipid membranes cannot be stabilized solely by electrostatic interactions. Whether the local minimum in F(r) will be preserved in a full nonlinear PB treatment is not known to us. Saddle-Shaped Inclusion. Let us now add anisotropic inclusions to the charged membrane. As argued above, we shall employ the interaction constants K ≈ 50Ri3kT/nm and K h ) -2K/ 3. The inclusion radius Ri should be larger than ξ ≈ 1 nm; we shall use Ri ) b/2 ) 1.25 nm. To illustrate the effect of the inclusion’s anisotropy, we chose a perfectly saddle-like inclusion with C1m ) -C2m ) 1/b. Figure 3 shows F(r) for two examples of different area density, n, of the inclusions: n ) 1/70 000 nm2 (curve d), and n ) 1/14 000 nm2 (curve e). We clearly see the pronounced ability of the inclusions to lower and deepen the local minimum of F(r). Because of their favorable interaction, the anisotropic inclusions tend to accumulate within the membrane rim. The corresponding number, NP, of inclusions in this region is given in eq 16. This number is plotted in the inset of Figure 3 for n ) 1/70 000 nm2 (curve d) and n ) 1/14 000 nm2 (curve e). Recall that the statistical mechanical approach leading to eq 15 is based on pointlike particles. Within this approach, the inclusion size will not limit entry into the pore rim. Yet, for a realistic (that is, nonvanishing) size (recall our choice Ri ) b/2), NP should stay sufficiently small to ensure steric fitting of the inclusions into the pore. In fact, at r ) b, there is full coverage for NP ) Nmax ) πb/Ri ) 2π. More P precisely, the interactions of the inclusions with the membrane rim should dominate over direct inclusion-inclusion interactions. For the examples shown in Figure 3, the inclusion number, NP, is not prohibitively high. But above n ) 1/14 000 nm2, our present approach (with the present choice of interaction parameters, particularly with C1m ) -C2m ) 1/b), is no longer applicable. The minimum for n ) 1/14 000 nm2 (curve e in Figure 3) is roughly F ) - 30kT. It arises from the presence of inclusions in the pore region. On the other hand, eq 17 predicts that the inclusion energy Fi ≈ -NPkT. The inset of Figure 3 shows that NP e 6. Hence, the deep minimum of F does not arise solely from the inclusion contribution Fi. It is also the electrostatic Priloga F 12524 J. Phys. Chem. B, Vol. 107, No. 45, 2003 Fošnarič et al. Figure 4. The pore free energy, F, as a function of the pore size, r, for differently shaped anisotropic inclusions: C2m/C1m ) (a) -1, (b) -0.8, (c) -0.6, and (d) 0. In all cases, the membrane is charged (σ ) -0.05 A s/m2, ld ) 2.8 nm), and it is C1m ) 1/b and n ) 1/14 000 nm2. The inset shows the position of the local minima, ropt, as a function of C2m/C1m (solid line). The dashed line in the inset shows -C2m/C1m ) b/ropt. energy, Uel, which lowers F. To explain the mechanism, we recall that in the inclusion-free membrane the electrostatic energy and the line tension nearly balance each other. If inclusions enter the pore region, they reduce the line tension (see eq 3). As a result, Uel is no longer counterbalanced by Wedge and thus strongly lowers F. It can be therefore concluded that also in the case of the charged membranes the deepness of the minimum of F is mainly determined by the inclusions, directly because of their energy contribution Fi and indirectly because of their influence on Uel + Wedge. Influence of Inclusion Shape. Deepening of the minimum in F(r) occurs in Figure 3 at r ≈ b [see curves d and e]. This reflects our choice of the inclusion geometry, C1m ) -C2m ) 1/b. In fact, for r ) b, the principal curvatures of the rim at θ ) π (see eq 1) are C1 ) -C2 ) 1/b, coinciding with the inclusion’s preference. This observation suggests the possibility of increasing the optimal size of the pore by altering the intrinsic curvatures of the inclusion from a saddle-like (C2m/C1m ≈ -1) toward a more wedgelike shape (C2m/C1m ≈ 0). The smaller the magnitude of |C2m/C1m|, the larger should be the preferred pore size. With regard to the principal curvatures at the waist of the rim, C1 ) 1/b and C2 ) -1/r, one would expect the optimal pore size, ropt, to be determined by the relation - C2m C2 b )) C1m C1 ropt (20) In Figure 4, we add anisotropic inclusions of area density n ) 1/14 000 nm2 to a charged membrane (σ ) - 0.05 A s/m2, ld ) 2.8 nm); the shape of the inclusions is characterized by C1m ) 1/b and (a) C2m/C1m ) -1, (b) C2m/C1m ) -0.8, (c) C2m/ C1m ) -0.6, and (d) C2m/C1m ) 0. Indeed, the local minimum of F(r) shifts to larger pore sizes as the inclusions become more wedge-shaped [compare the position of the local minimum of curves a-c]. The solid line in the inset of Figure 4 shows how the optimal pore radius ropt changes with C2m/C1m. The broken line in the inset displays the prediction according to eq 20. Clearly, the optimal pore size, ropt, is even larger than what would be expected from the inclusion’s geometry. The reason can be found in the presence of the inclusion reservoir in the bulk membrane. Increasing the pore size allows incorporation of more inclusions that interact only somewhat less favorably with the membrane rim. Our calculations also show that below a critical ratio of |C2m/ C1m| the local minimum in F(r) vanishes (in Figure 4, we find |C2m/C1m| < 0.4), and the pore grows. Again, the inclusion Figure 5. The optimal pore size r as a function of the ionic strength of the surrounding aqueous medium. The charge density of the membrane is σ ) -0.05 A s/m2, the area density of the inclusions is n ) 1/2000 nm2, and the inclusion’s preferred curvatures are C1m ) 1/b and C2m/C1m ) -0.4. Experimental values2 are also shown (b). The inset shows the actual number of inclusions, NP, residing in the pore of optimal size, ropt (solid line), and the maximal number, Nmax ) P πropt/Ri (broken line). reservoir in the membrane favors additional partitioning into the membrane rim, which increases the pore size. Below a critical ratio of |C2m/C1m|, this process never stops. We also note that for isotropic inclusions (where C1m ) C2m), we do not find energetically stabilized pores. Even more generally, if C1m and C2m have the same sign, stable pores cannot be predicted. Stabilization of the pore derives from the matching of the rim geometry with the inclusion’s preference. The rim provides a saddle-like geometry (that is, different signs of C1 and C2); consequently, a saddle-like inclusion geometry (that is, different signs of C1m and C2m) is needed to stabilize a pore. Salt Concentration and Pore Size. In all of the examples presented so far, we added anisotropic inclusions to charged membranes with a specifically selected Debye length, ld ) 2.8 nm. We recall from Figure 3 that this was the choice for which an inclusion-free membrane already exhibits a (shallow) minimum in F(r). The question arises whether pores can also be stabilized for uncharged membranes or, more generally, under varying electrostatic conditions. In this respect, it is interesting to note the experimental observation of stable pores in red blood ghosts for which data exist2 on the optimal pore radius, ropt(n0), as a function of the salt concentration, n0. Our present theoretical approach is able to reproduce the experimental data as shown in Figure 5. We used a charge density of σ ) -0.05 A s/m2, an area density of the inclusions of n ) 1/2000 nm2, and an inclusion geometry characterized by C1m ) 1/b and C2m/C1m ) -0.4. The inset of Figure 5 shows the corresponding number of inclusions (solid line), as well as the maximal number Nmax P ) πropt/Ri (broken line) at which the inclusions would sterically indicates the occupy the entire rim. The observation NP < Nmax P applicability of our approach for the selected area density (n ) 1/2000 nm2). Of course, the number of approximations in our approach may still render the good agreement in Figure 5 fortuitous. What adds to this uncertainty is the complexity of the red blood cell membrane. In particular, the attached cytoskeleton can be expected to affect the membrane pore energetics. Therefore, Figure 5 should be understood as an illustration of the principal ability of anisotropic membrane inclusions to stabilize membrane pores, even under changing electrostatic conditions. In our present investigation, we included electrostatic interactions to allow a prediction of how the pore size depends on salt content. Note that it is generally the anisotropy of the inclusions but not electrostatic interactions that stabilizes pores. Thus, for Priloga F Stabilization of Pores in Lipid Bilayers uncharged membranes or in the limit of high salt content, qualitatively similar considerations as those presented above account for pore stability. However, the minimum of the overall free energy F would be less deep compared to the case of charged membranes. Isotropic vs Anisotropic Inclusions. The influence that admixed inclusions have on the energetics (particularly the line tension) of membrane pores is often interpreted in terms of altering the elastic properties of the membrane. For example, the addition of cosurfactants typically reduces the bending stiffness of a membrane.50 Even more important for membrane pores, the presence of conelike (and analogously, inverted conelike) inclusions can induce a shift in the spontaneous curvature. This shift can be translated into a change in line tension, which provides a common basis for analyzing the energetics of a membrane pore.18 Our present approach contains this scenario as a special case, namely, if the inclusions are isotropic (Dm ) 0). In fact, it is the spontaneous mean curvature, Hm, that plays the role of the spontaneous curvature. Beyond the effect of conelike and inverted conelike inclusions, our present approach also allows us to analyze other inclusion shapes, such as wedgelike or saddle-like inclusions. These inclusions can be characterized by an appropriate combination of Hm and Dm. In the following, we shortly discuss a few examples in which we think that the anisotropy of admixed inclusions could be particularly relevant to the pore energetics. Electroporation is a method of artificial formation of pores in biological membranes by applying an electric field across the membrane. A problem in the electroporation of living tissue is that it often causes irreversible damage to the exposed cells and tissue.6 Increasing the amplitude of the electric field in electroporation diminishes cell survival rates.7 On the other hand, if the applied electric field is too low, stable pores are not formed. A way to improve the efficacy of electroporation is chemical modification of the membranes. It has been reported recently51,52 that adding the nonionic surfactant octaethyleneglycol dodecyl ether (C12E8) to the outer solution of the phospholipid membrane or the cell membrane causes a decrease in the threshold for irreversible electroporation. In other words, C12E8 molecules make transient pores in a membrane more stable. C12E8 molecules were recently suggested to act as anisotropic inclusions in bilayer membranes.26 Our theoretical model could add to the understanding of pore energetics as recently investigated by Karatekin et al.18 For example, these authors measured a dramatic increase of the transient pore lifetime induced by the detergent Tween 20, which has an anisotropic polar headgroup. The importance of the anisotropy of such polar heads of the detergents for the stability of anisotropic membrane structures has been indicated recently. It has been shown that a single-chain detergent with an anisotropic dimeric polar head (dodecyl D-maltoside) may induce tubular nanovesicles32 in a way similar to those induced by strongly anisotropic dimeric detergents.31 Our approach could also add to the understanding of pore formation induced by certain antimicrobial peptides.8,19 These peptides are often amphipathic, partially penetrating the host membrane. In addition to that, they have a pronounced elongated shape, which arises from their R-helical backbone structure and, apparently, renders them highly anisotropic. Some of these peptides are believed to cooperatively self-assemble into membrane pores. Thus, they not only facilitate pore formation, but they actiVely induce it. Despite their importance, there are currently few theoretical investigations about the energetics of peptide-induced pore formation.11,12,14,53 Our model provides J. Phys. Chem. B, Vol. 107, No. 45, 2003 12525 perhaps the most simple way to capture the underlying physics of peptide-induced pore formation in lipid membranes. Discussion of Approximations. We analyzed the energetics of a single membrane pore on the basis of a simple, physically transparent model. It involves a number of approximations that we discuss in the following. We adopted a phenomenological expression for the membrane-inclusion interaction energy. This expression is valid on a mean-field level. The “mean field” is provided by the local membrane curvatures, C1 and C2, that adjust to minimize the system’s free energy. This approach is, similar to the Helfrich bending energy, valid if the local curvatures, C1 and C2, do not change too drastically. On the other hand, the membrane rim provides local curvatures that differ greatly from those of the planar membrane (the latter, in fact, vanish). Hence, formation of a pore in a planar membrane necessarily involves large curvature changes. Yet, as we have seen, the Helfrich bending energy describes the line tension of an inclusion-free membrane; on the same ground, we are confident about the applicability of the inclusion free energy. The phenomenological expression for the inclusion free energy, Fi, contains four interaction constants (K, K h , Hm, and Dm). Two of them, Hm and Dm, characterize the shape of the anisotropic inclusion. If the lipid bilayer is required to exactly match the angular shape of individual inclusions, then no further interaction constants are needed. In this case, the angular matching appears as a boundary condition for an appropriate differential equation.35,36,38,54 In our present approach, an ensemble of inclusions interacts (in a mean-field fashion) with a membrane patch of prescribed principal curvatures (the curvatures may afterward be optimized). In this case, there appear two additional interaction constants, K and K h . These constants account for the energy to insert anisotropic inclusions into a bilayer patch of fixed principal curvatures. This process is supposed to locally perturb the two monolayer leaflets of the bilayer. One can thus roughly obtain the interaction constants, K and K h , by estimating the microscopic (short-range) interaction between the perturbed monolayers and the inclusion. To this end, we used membrane elasticity theory,46 which involves a spatial decay length, ξ ≈ 1 nm, of the inclusion-induced perturbation. Note that the discreteness of the lipids should not be neglected at these length scales, yet membrane elasticity theory actually does neglect it. Still, this approach is commonly used to estimate membrane-inclusion interactions and, where possible, gives good agreement with experimental observations.55,56 There are structural approximations concerning the membrane pore. Its shape is assumed to be circular, covered by a semitoroidal rim. These assumptions seem to us the most reasonable ones. Still, there could be, say, an inclusion-induced change in the cross-sectional shape of the membrane rim. In fact, we performed additional calculations in which we allowed for a semiellipsoidal shape of the membrane rim. The free energy was then minimized with respect to the corresponding aspect ratio. With this additional degree of freedom, we found qualitatively the same results as with the semitoroidal rim. The statistical mechanical approach to derive the inclusion free energy Fi in eq 13 assumes noninteracting, pointlike particles. Even though we ensured that the number of inclusions in the pore, NP, never exceeds the sterically possible maximal number, Nmax P , we cannot exclude direct inclusion-inclusion interactions within the pore. In fact, such interactions are important in nearly all realistic situations. The average distance between neighboring inclusions in membrane pores is generally Priloga F 12526 J. Phys. Chem. B, Vol. 107, No. 45, 2003 of the order of molecular dimensions. Hence, direct interactions matter. Yet, these interactions are specific, depending on molecular details. Our approach is of generic nature; specific interactions add to the mechanisms specified in our work. Finally, we employed linearized PB theory. Calculations within nonlinear PB theory with regard to the semitoroidal shape of the membrane rim and the local demixing between charged and uncharged lipids are much more demanding but are currently being carried out. None of the approximations employed can detract from our principal conclusion: anisotropic membrane inclusions are candidates for the energetic stabilization of membrane pores. Conclusions Our theoretical approach adds three aspects to the analysis of pores in lipid membranes. First, the modification of the elastic properties of the membranes in the presence of inclusions is taken into account, as is reflected in the calculation of the membrane-inclusion interaction constants. Second, we allow for anisotropy of the inclusions, which enables us to consider various inclusion shapes, conelike, inverted conelike, wedgelike, and saddle-like inclusions. Third, the lateral density of the inclusions is not kept constant. Instead, we calculate the pore energetics for a fixed chemical potential of the inclusions. The last point is especially important in studies where the admixed compounds are predominantly localized in the region of the pore edges, such as the detergent sodium cholate18 or the protein talin.57 Our model is simple and approximate, but it provides a lucid and reproducible framework to analyze pore formation in lipid membranes. Acknowledgment. We are indebted to A. Ben-Shaul, S. Bezrukov, D. Miklavčič, H. Hägerstrand, and M. 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