Univerza v Ljubljani Fakulteta za elektrotehniko Miha Fošnaric Vpliv

Transcription

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
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
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
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
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
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
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
−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
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
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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
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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
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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.
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Vol. 6. No. 2. 2001
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6. Sheetz, M.P., and Singer, S.J. Biological membranes as bilayer couples. A
molecular mechanism of drug-erythrocyte interactions. Proc. Natl. Acad. Sci.
71 (1974) 4457-4461.
7. Sheetz, M.P., and Singer, S.J. Equilibrium and kinetic effects of drugs on the
shape of human erythrocytes. J. Cell. Biol. 70 (1976) 247-251.
8. Evans, E. Bending resistance and chemically induced moments in
membrane bilayer. Biophys. J. 14 (1974) 923-931.
9. Iglic, A., Kralj-Iglic, V. and Hägerstrand, H. Amphiphile induced
echinocyte-spheroechinocyte transformation of red blood cell shape. Eur.
Biophys. J. 27 (1998) 335-339.
10. Deuticke, B., R. Grebe, and C.W.M. Haest. Action of drugs on the erythrocyte
membrane. In: Blood Cell Biochemistry. Erythroid cells. (Harris, J.R., Ed.)
Plenum Publishing Corporation, New York. 1990, 475-529.
11. Le Maire, M., Moller, J.V. and Champeil, P. Binding of a nonionic
detergent to membranes: Flip-flop rate and location on the bilayer.
Biochemistry 26 (1987) 4803-4810.
12. Kragh-Hansen, U., le Maire, M, Moller, J.V. The mechanism of detergent
solubilization of liposomes and protein-containing membranes. Biophys. J.
75 (1998) 2932-2946.
13. Bobrowska-Hägerstrand, M., Kralj-Iglic, V., Iglic, A., Bialkowska, K.,
Isomaa, B. and Hägerstrand, H. Toroidal membrane endovesicles induced
by polyethyleneglycol dodecylether in human erythrocytes. Biophys. J. 77
(1999) 3356-3362.
14. Liljekvist, P and Kronberg, B. Comparing Decyl-beta-maltoside and
Octaethyleneglycol Mono n-Decyl Ether in Mixed Micelles with Dodecyl
Benzenesulfonate. 1. Formation of micelles. J. Colloid Interface Sci. 222
(2000) 159-164.
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15. Kikuchi, Y., Takahashi, N., Suzuki, T. and Sawada, K. Liquid-liquid
extraction of alkali metal ions with non-ionic surfactant having a
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the potentiometric response of ion-selective polymeric membrane
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17. Malinowska, E., Manzoni, A. and Meyerhoff, M.E. Potentiometric
response of magnesium-selective membrane electrode in the presence of
nonionic surfactants. Anal. Chim. Acta 382 (1999) 265-275.
18. Israelachvili, J.N. Intermolecular and Surface Forces, 2nd ed., Academic
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19. 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
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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.
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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.
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Hägerstrand, H. Torocyte shapes of red blood cell daughter vesicles.
Bioelectrochemistry 52 (2000) 203-211.
F
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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
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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
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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)
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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
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228
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
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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)
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230
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.
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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
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232
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
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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
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234
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
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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
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236
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
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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.
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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: ales.iglic@fe.uni-lj.si (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
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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
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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].
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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].
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C
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*.
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248
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,
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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-
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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.
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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
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252
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.
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F
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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: ales.iglic@fe.uni-lj.si (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
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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)
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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)
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154
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
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D
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].
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F
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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: damijan@svarun.fe.uni-lj.si (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
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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).
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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.
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208
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.
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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
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210
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
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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
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212
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
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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
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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
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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
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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)
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J. Phys. Chem. 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: may@lily.molebio.uni-jena.de.
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
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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
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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
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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
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F
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
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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
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
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
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. Kandušer for
stimulating discussions and to MESS of the Republic of Slovenia
for financial support. S.M. thanks TMWFK for support.
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Univerza v Ljubljani Fakulteta za elektrotehniko Miha Fošnaric Vpliv

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