Fundamental Mechanism ofTranslocation across LiquidlikeMembranes: Toward Control overNanoparticle BehaviorLucian Livadaru † and Andriy Kovalenko* ,†,‡

National Institute for Nanotechnology, National Research Council of Canada,9107-116 St., W6-010, ECERF Building, Edmonton, Alberta T6G 2V4, Canada,and Department of Mechanical Engineering, UniVersity of Alberta, Edmonton,Alberta, Canada

Received October 19, 2005; Revised Manuscript Received November 9, 2005

ABSTRACT

We envision and theoretically investigate a novel behavior of a functionalized nanoparticle designed to translocate through a liquidlike membrane .We develop a statistical −mechanical approach to such a system. We predict a new mechanism for the opening of a circular energy-dominatedpore on the membrane by a nanoparticle functionalized with a peptide aggregate. Following fluctuations in the position and orientation of thenanoparticle, the peptide aggregate incorporates into the membrane and locally destabilizes it. The nucleation of a pore centered at thepeptide aggregate attached to the particle is a precursor to particle translocation. The subsequent opening of the pore is assisted by adhesionof the membrane to the particle. We determine the conditions in which thermal fluctuations in the membrane shape and the pore size caninduce translocation of the particle. For different system parameters quantities such as the free energy, entropy, pore size, degree of particlewrapping, and the probability of spontaneous translocation are obtained.

Introduction

The interaction between fluid membranes and nanosizedparticles is a frequently occurring topic in nanoscience andnanotechnology. It is inevitably associated with the issuesof nanoparticle toxicity1 and plays a vital role in phenomenasuch as drug delivery,1-3 virus entry into a cell,4,5 nanoparticleadsorption, self-assembly, and phase-transitions at fluid-fluid interfaces.6,7 From this perspective, a detailed investiga-tion of the phenomena associated with the presence of ananoparticle near a flexible membrane or at a liquid-liquidinterface comes forth as a priority.

Membrane-particle adhesion can induce partial or totalwrapping of a particle,8-10 a phenomenon which commonlyoccurs in cells. A purely energetic analysis10 predicts that aparticle is completely wrapped by an essentially tensionlessvesicle if its radius exceeds the critical valueRp ) (2κ/| w|)1/2,whereκ is the bending modulus andw < 0 is the contactenergy per unit area, whereas particles with smaller radiiare completely detached. For lipid bilayers, this leads to anestimate for a critical radius of about 5 nm. If the surface

tensionσ is not negligible, the critical radius for envelopingincreases (from the tensionless case) and depends onσ.Biologically meaningful phase diagrams for the degree ofwrapping were constructed for this case, which are relevantfor particle sizes ranging from a few to about a hundrednanometers, such as nucleocapsids of animal viruses.8

The interaction between latex spheres and unilamellar lipidvesicles was studied experimentally for particle size in themicrometer range11 comparable to the size of the vesicle. Itwas suggested that particle encapsulation by the vesicle isaccompanied by the formation of a dynamically stabilizedpore in the membrane, but the mechanism and preciseconditions for it remained unknown.

The presence of a nanoparticle inside a small vesicle wasfound to promote vesicle fission and fusion in a model systemstudied by Brownian dynamics simulations.12 As a crucialstep in this process, aided by the particle-membraneadhesion, the nanoparticle induces the opening of a pore inthe contact region between the two transient partitions of avesicle undergoing fission/fusion, thus enabling their com-plete splitting/joining.

On a different scale, theoretical studies on model lipidmembranes found that under lateral tension, depending onthe architecture of the component lipid, either rupture occurs

* To whom correspondence may be addressed. E-mail: andriy.kovalenko@nrc.ca.

† National Institute for Nanotechnology, National Research Council ofCanada.

‡ Department of Mechanical Engineering, University of Alberta.

NANOLETTERS

2006Vol. 6, No. 1

78-83

10.1021/nl052073s CCC: $33.50 Published 2006 by the American Chemical SocietyPublished on Web 12/09/2005

or the membrane is in a phase characterized by a regulararray of pores.13 In another study, Monte Carlo simulationswere used to investigate a related topic regarding theconditions in which small pores can coalesce and bespontaneously replaced by a single large hole and membranerupture occurs.14

The insertion of foreign objects into a membrane15-17

effectively creates structural defects, which can alter theshape and the integrity of the membrane. Polymer chainsanchored to a lipid membrane have been found to exert anentropic force on the membrane and induce shape deforma-tions.18 Another example is the class of amphipathic peptides(featuring both hydrophobic and hydrophilic surfaces),known for their ability to adsorb and incorporate into amembrane.19-21 Recent insight into peptide-membranecomplexes shows a strong tendency of the peptide to reorientand insert into the membrane with an energy gain of about4kBT.22 It was strongly suggested that the reorientationmechanism is the basis of a generic driving force of poreformation. Once adsorbed and reoriented, peptides self-assemble and can promote the growth of pores with mixedlipid-peptide lining composition upon increasing surfacetension. This could result in a selective permeabilization ofa membrane under osmotic tension.

Considering the above facts, a design of a peptide-nanoparticle complex with novel controlled membrane-translocation functionality seems attainable. A combinedaction of the peptide insertion and the particle adhesion tothe membrane can lead to translocation of the particle byopening a tight pore in the membrane, large enough toaccommodate its size, and simultaneously sealed by it(against leakage of material from the other side of themembrane). Below we develop a statistical-mechanicalapproach to such a system and explore this new possiblemechanism of translocation of a functionalized nanoparticlethrough a membrane.

Model and Method. A key feature of the present theoryis a rigorous statistical-mechanical account of the entropyassociated with shape fluctuations of the membrane,18,23 asopposed to other studies that only account for the energy ofthe system. We consider not only the membrane shape thatminimizes the system Hamiltonian but also fluctuationsaround it enabling the new translocation mechanism.

Within the continuum elasticity theory, the HelfrichHamiltonian for a two-dimensional surfaceh with principalcurvaturesc1 andc2 placed in an external fieldU(r ) reads

where dS is the surface element at positionr andκj is theGaussian curvature modulus. The partition function of asystem with this Hamiltonian isZ ) ∫ D{h} exp(-âH({h})),where D{h} represents the integral over all the possibleconfigurations (shapes) of the surface andâ ) 1/kBT is theinverse temperature. Notice the absence of a spontaneouscurvature in eq 1. Many studies revolve around the idea ofminimizing the Helfrich Hamiltonian, and thus they account

for the energetics of the system, while completely ignoringthe entropy contribution to the partition function. In contrast,for a rigorous statistical mechanical analysis one needs toinclude not only the mean field solution (minimizing theHamiltonian) but also a continuum spectrum of excited states.Physically this corresponds to taking into account thepresence of capillary waves at the interface, which areproperly built into our theory. Recently, this phenomenonwas directly observed in an experiment for interfaces incolloidal suspensions.24

Theoretical studies of capillary waves based on theOrnstein-Zernike equation for the liquid-vapor interface25

revealed the fact that capillary waves are related to thepresence of a term quadratic ink in the Fourier transform ofthe direct correlation functionc(k), whereas the bendingenergy corresponds to a term proportional tok4 (assumingthe interface fluctuations to be gentle enough so that theHamiltonian (1) can be expressed as∫ dx dy [(σ/2)(∇h)2 +(κ/2)(∇2h)2)], in the absence of external fields).

We consider a spherical nanoparticle with impenetrablewalls and an attractive region just outside its surface, whichdefines28 the external fieldU(r ) in (1). The particle isfunctionalized with a peptide aggregate or some othermembrane destabilizing agent. Effectively, these agents areable incorporate themselves into the membrane and promotethe formation and growth of pores by lowering the linetension associated with the perimeter of the pore. Forsimplicity, in this study we treat only planar circular poresthat are small enough so that significant out-of-plane shapedeviations are unlikely. The line energy of a circular pore isconsidered to be of the simple form26 Eline ) 2πΓ′r, whereΓ′ is the three-phase line tension including the particle-edgecontact energy as well as the peptide effect on the pore. Inthe case of a hydrophilic pore, the specific adhesion energyw′ between the pore perimeter and the particle is ap-proximately the same as that between the membrane leafletand the particle. If this interaction is attractive,w < 0, thenthe effective line tension is decreased when the nanoparticlecomes in contact with the pore. However, this contributionto the line tension takes effect only as long as the porediameter is less than or equal to the particle diameter, makingthe opening of an “unsealed” pore (larger than the particle)less favorable. Alternatively, the presence of the nanoparticlecan favor the closing of the pore if the line tension iseffectively increased by the particle-edge interaction.

We consider an open, framed thermodynamic ensemblefor the membrane, in which the area stretching coefficientσ′ and the projected area (to thexy-plane) are constant.27 Inthis ensemble, the statistical mechanics of the membranepatch is carried out by the Green’s function (probabilitydistribution function) method in the spirit of Feynman’s pathintegral.28 It effectively accounts for all the membraneconfigurations and their contributions to the partition functionin the presence of an external field. The membrane isdescribed by its axial and azimuthal cross sections, whichin turn are separately modeled as a discrete number of jointsegments. Two intersecting cross sections define a vertex,and the set of all vertexes constitute the conformation of the

H({h}) ) ∫ dS [σ + κ

2(c1 + c2)

2 + κjc1c2 + U(r )] (1)

Nano Lett., Vol. 6, No. 1, 2006 79

membrane. The shapes of the axial and azimuthal curvesare coupled via the mean position and curvature at eachvertex. We thus decompose the membrane fluctuations intothe axial and azimuthal components. The boundary condi-tions for the probability distribution are chosen to mimic aloosely constrained membrane at the edge of the patch.28

On discretizing the axial and azimuthal profiles, the corre-sponding equations are solved numerically.

Results and Discussion of the Proposed TranslocationMechanism. Because we work with an open, framedsystem,27 the stretching constant of the modelσ′ is the areacoefficient and not the film tensionσ; these quantitiescoincide for an infinite surface. Also, for our discretemembrane model we adopt a “discrete” bending modulusκ′which is generally different from the continuum bendingmodulus κ but becomes identical for a sufficiently finediscretization. The exact relation between the values of thediscrete “intrinsic” and the measured elastic constants isdirectly amenable by our approach. However, it does notaffect the qualitative findings below, and we leave it beyondthe present investigation to focus on the essence of thetranslocation mechanism. We chooseκ′ ) 10kBT andσ′ )10-3kBT/a2, which yield values of the bending modulus andthe surface tension within the acceptable range for biologicalmembranes. Herea is the half-thickness of the membrane(of the order of 1 nm) and serves as the length scale of ourmodel. The value of the Gaussian curvature modulus wasfound to play a negligible role in this investigation (at least,for magnitudes not exceeding the value ofκ); therefore forillustration purposes, we choseκj ) 0. It is also noteworthythat, although accounted for below, fluctuations of the

membrane without axial symmetry do not have an essentialrole in our results. This is due to the fact that they introduceadditional bending and stretching in the azimuthal plane,which will increase the potential energy of that particularconformation, therefore reducing its statistical weight.

The shapes of the membrane and the opening of the porefor various positions of the nanoparticle are shown in Figure1. The particle, with the functional aggregate on top, ismoved up from below at a sufficiently slow rate so that themembrane is maintained in quasi-equilibrium. Thus, themembrane is considered to exhibit equilibrium fluctuationsin the “external field” created by the particle placed at acertain heightzp. For each position of the particle (i.e.,external field configuration) we calculate the partitionfunction, the free energy, and the average shape of themembrane corresponding to our ensemble. We define thedegree of wrapping of the particle,ê, as the ratio of the areacovered by the membrane to the total particle area. Note thateach curve shown in Figure 1 is not a fixed shape of themembrane but is obtained as a set of average locations ofconsecutive fluctuating segments of the profile.

For particular calculations we chose the following valuesfor the system size: the particle radiusRp ) 7a, and thepatch frame radius of 10Rp. Figure 1 shows three distictextents of pore formation for different sets of the contactenergies. In case a, the membrane-particle contact energyper unit area isw ) -0.1kBT/a2 and the pore line tension isΓ′ ) 0.75kBT/a. A pore starts forming as soon as theparticle’s top is higher than the level of the unperturbedmembrane at the edge of the membrane frame (we call thisthe contact point). Below that height, the adhesion is

Figure 1. Successive positions of the nanoparticle and the shapes of the membrane: a, partial pore opening; b, early pore opening enhancedby adhesion with particle trapping; c, no pore opening in the absence of adhesion; d, particle translocation through the membrane andfluctuations in the shape of the membrane at successive stages, with the membrane probability density changing fromF ) 0 to 1 (color mapfrom white to black). Parameter values for each case are given in the text.

80 Nano Lett., Vol. 6, No. 1, 2006

relatively weak. The degree of wrapping grows steadily withthe advancement of the particle up tozp/Rp ) 1 and levelsoff from that point on.

In case b, the adhesion is increased tow ) -0.15kBT/a2

to single out its effect on the pore size, whereas the linetension value is the same as in (a). Clearly, the pore openingis promoted by the increase in the magnitude of the contactenergy. This finding is supported by the free energy profilesin Figure 2a showing a decrease of the free energy barrierfor zp > 0 as compared to case a. Another essential differenceis that in case b the particle is strongly adsorbed to themembrane, a fact supported by the free energy slopes in thetwo cases forzp < 0 and by a degree of wrapping with amaximum corresponding to the global minimum of the freeenergy curve.

The pore expansion is aided by a local stretching of themembrane by adhesion to the particle. This finding iscorroborated by the experimental observation showing thatstrong adsorption of a vesicle onto highly curved surfaces(in a porous substrate) increases the permeability of itsmembrane by locally stretching it.29 However, the maindifference is that in our study the pore is formed around thepeptide aggregate and is sealed by the nanoparticle, whereasin ref 29 pores can form over a stretched region of themembrane which is not in contact with the substrate.

The case (c) of zero contact energy demonstrates theessential role of adhesion in pore formation. The pore exhibitsvirtually no growth with the onward motion of the particle(Figure 1c). The degree of wrapping remains all but constantfor the whole range of particle displacement (Figure 2c). The

Figure 2. Membrane characteristics as functions of nanoparticle position: a, the free energy of the membraneA/kBT; b, the pore radiusRpore/Rp; c, the degree of wrappingê versus the nanoparticle position; d, the ratio of the probability of translocation to that of desorptionas a function of the adhesion strength for a fixed value ofΓ′ ) 0.1kBT/a; e, for the case in Figure 1d, the free energy for∆γ ) 0 (thin solidline), for ∆γ ) 0.01kBT (thick solid line), and the entropyS/kB (short-dashed line). Long-dashed, dash-dotted, dotted, and solid linescorrespond to the system in cases a-d in Figure 1, respectively.

Nano Lett., Vol. 6, No. 1, 2006 81

free energy exhibits an almost linear increase from the pointof contact on and becomes much higher than in all othercases (Figure 2a).

A successful case of translocation is illustrated in Figure1d for parametersw ) -0.05kBT/a2 andΓ′ ) 0.1kBT/a. Fora ) 1 nm the numerical value ofΓ is about 4× 10-12 N,which is a relatively low value of the line tension whencompared to typical estimates for lipid bilayers, 2× 10-11

N.30 This difference is attributable to the presence of peptidesand the interaction between molecules at the pore perimeterand the nanoparticle. We assume that the particle is pushedfrom below against the membrane sufficiently slow so thatthe membrane is maintained in quasi-equilibrium. Shown asa color spectrum is the axial cross section of the probabilitydensity of membrane segments for advancing positions ofthe particle. This provides a good illustration of the fluctua-tions in the shape of the membrane. Compared to theprevious cases, the pore grows steadily and at a higher ratewith the advance of the particle, and the free energy profileis more favorable to translocation. The shape fluctuationsmonotonically increase with the onward motion of theparticle and for each stage reach the greatest magnitude justoutside the contact region with the particle. This observationcorroborated by the entropy curve (short-dashed line inFigure 2e), indicates that in the advanced stages of poreopening a strong competition exists between membraneconformations that have a small curvature and do not adhereto the particle and those that have both strong adhesion andlarge curvature. The latter group of conformations eventuallydominates and has a decisive influence in favor of thetranslocation.

Within a mean-field view, translocation occurs when theaverage pore radius becomes equal to the particle radius. Itis accompanied by an abrupt jump in the quantities shownin Figure 2 in the vicinity of the pointzp/Rp ) 1.1. Afterreaching its maximum size, the pore closes abruptly “behind”the nanoparticle. The free energy, entropy, and the degreeof wrapping drop to low values and are more or less constant.The jump corresponds to the system accessing a subspaceof its configurational space with the particle above themembrane, which was inaccessible before the translocation.

We also briefly checked the effects of the other parametersto compare with the expectations.10 Increasing the particlesize alone causes a rise inê. A decrease of the surface tensionpromotes the proliferation of the wrappingê.

If the adhesion is strong enough, the particle will betrapped within the membrane by becoming embedded andconcomitantly sealing the pore. The minimum of the freeenergy profile can be identified as a trapping region, if theenergy barrier around it is considerably greater than thethermal fluctuations of the particle. Its location depends onthe contact energy as well as on the line tension of the pore.We emphasize that this trapping mechanism is quite differentfrom that involving the total enveloping of the particle.10

Our results for the degree of wrapping show an importantcontribution of fluctuations to the shape of the membrane.Our value of the degree of wrapping is always less than 1even when the particle radius exceeds the critical radius

predicted by the purely energetic analysis.10 The reasons ofthis distinction are the presence of a small but finite surfacetension, and the fluctuations in the membrane shape. Thefluctuations act against the adhesion and have an unwrappingeffect, thus decreasing the area of contact with the nano-particle.

The mechanism for particle translocation (enabled by asuitable set of interaction parameters) involves as a precursorthe peptide group incorporation into the membrane after theinitial adsorption of the particle. Once the particle is partiallywrapped, translocation becomes possible due to fluctuationsin the membrane shape and in the size of the pore (see Figure1d), by virtue of the system exploring its whole conforma-tional space.

In the case when the particle is strongly adsorbed to themembrane, spontaneous translocation can occur enabled bya sufficiently strong fluctuation in the pore size. From adynamics point of view, the time scale of this processτtrl isinversely proportional to the probability of occurrence of afluctuation with a pore radius greater than the sphere radiusptrl ) p(rpore > Rp), times the frequency of attempts totranslocate,ftrl. If this time scale is less than the time scalefor desorption,τdes ∝ p-1

desf-1des, and the lifetime of the

membrane, the translocation occurs spontaneously. Assumingtranslocation and desorption attempts to be of roughly equalfrequencies,ftrl ≈ fdes, the success of translocation definedasτdes/τtrl is determined by the ratio of the translocation anddesorption probabilitiesptrl/pdes. It is plotted in Figure 2d asa function ofw for a fixed value ofΓ′ ) 0.1kBT/a. For thiscase the translocation occurs spontaneously for magnitudesof w greater than 0.126kBT/a2.

Besides the above, there are other factors that can influencethe probability of translocation, in particular, those involvinga driving force to overcome the energy barrier. We list herethe following scenarios: (i) the existence of a differencebetween the interfacial energy densities of the particle onthe two different sides of the membrane (especially if theparticle is charged); (ii) a ratchet mechanism associated withligand-receptor binding to the nanoparticle; (iii) the adsorp-tion of molecules on the protruding region and/or on the poreedge, which can apply additional lateral pressure on the poreor further decrease the line tension. Using our results forthe free energy profiles (Figure 2a) as the main ingredient,a transition state theory of nanoparticle translocation througha membrane or a liquid interface can be readily formulated.

The free energy profile is drastically affected towardfavoring particle translocation by the existence of a differencein the interfacial energy densities of the particle at the surfaceof contact with the environments on the two sides of themembrane. To quantify this change we introduced a smalldifference∆γ ) γ2 - γ1 of just 0.01kBT/a2 to the systemwith all other parameters as in case of Figure 1d. Theresulting free energy profile displayed in Figure 2e as a thicksolid line has a much smaller energy barrier for translocation,which is easily accessible to fluctuations.

To conclude, we have explored the possibility of penetra-tion of a membrane by a functionalized nanoparticle. Byformulating an amenable statistical mechanical model, we

82 Nano Lett., Vol. 6, No. 1, 2006

have determined the free energy profiles for translocation,properly accounting for the entropy as well as for the energyof the membrane. We have predicted a new possiblemechanism of nanoparticle translocation or trapping, whichdoes not involve total enveloping of the particle by themembrane. Initially, the nanoparticle functionalized with apeptide aggregate gets adsorbed onto the membrane. Fol-lowing fluctuations in nanoparticle position and orientation,the peptide aggregate incorporates into the membrane, whichbecomes locally destabilized by it. A pore starts forming withthe assistance of the adhesion of the membrane to the particle.Even in the absence of a driving engine against the membraneenergy barrier, the fluctuations in the pore size eventuallyinduce translocation of the particle for certain systems. Thesuccess of this scenario is determined by the interactionparameters accounted for in our analysis and by the timescale of the fluctuations. We assert that this new scheme ofcontrol over translocation by appropriate functionalizationof the nanoparticle can find applicability in biological realm,where drug delivery in a nanoparticle scope is desired, aswell as in nanochemical systems.

Acknowledgment. This work is supported by the Na-tional Research Council of Canada.

Supporting Information Available: Detailed informationregarding the model and methods used in this study. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.

References

(1) Muller, R. H.; Mader, K.; Gohla, S. Solid lipid nanoparticles forcontrolled drug deliverysa review of the state of the art.Eur. J.Pharm. Biopharm.2000, 50, 161-177.

(2) Lockman, P. R.; Mumper, R. J.; Khan, M. A.; Allen, D. D.Nanoparticle Technology for Drug Delivery Across the Blood-BrainBarrier.Drug DeV. Ind. Pharm.2002, 28, 1-13.

(3) Barratt, G.; Colloidal drug carriers: achievements and perspectives.Cell. Mol. Life Sci.2003, 60, 21-37.

(4) Sieczkarski, S. B.; Whittaker, G. R. Dissecting virus entry viaendocytosis.J. Gen. Virol.2002, 83, 1535-1545.

(5) Carrasco, L. Entry of animal viruses and macromolecules into cells.FEBS Lett.1994, 350, 151-154.

(6) Lin, Y.; Skaff, H.; Emrick, T.; Dinsmore, A. D.; Russell, T. P.Nanoparticle assembly and transport at liquid-liquid interface.Science2003, 299, 226-229.

(7) Koltover, I.; Radler, J. O.; Safinya, C. R. Membrane mediatedattraction and ordered aggregation of colloidal particles bound to giantphospholipid bilayers.Phys. ReV. Lett. 1999, 82, 1991-1994.

(8) Deserno, M. Elastic deformation of a fluid membrane upon colloidbinding.Phys. ReV. E 2004, 69, 031903 (1-11).

(9) Deserno, M.; Gelbart, W. M. Adhesion and wrapping in colloid-vesicle complexes.J. Phys. Chem. B2002, 106, 5543-5552.

(10) Lipowsky, R.; Do¨bereiner, H.-G. Vesicles in contact with nanopar-ticles and colloids.Europhys. Lett.1998, 43, 219-225.

(11) Dietrich, C.; Angelova, M.; Pouligny, B. Adhesion of latex spheresto giant phosholipid vesicles: statics and dynamics.J. Phys. II1997,7, 1651-1682.

(12) Noguchi, H.; Takasu, M. Adhesion of nanoparticles to vesicles: aBrownian dynamics simulation.Biophys. J. 2002, 83, 299-308.

(13) Netz, R. R.; Schick, M. Pore formation and rupture in fluid bilayers.Phys. ReV. E 1996, 53, 3875-3885.

(14) Shillcock, J. C.; Seifert, U. Thermally induced proliferation of poresin a model fluid membrane.Biophys. J.1998, 74, 1754-1766.

(15) Partenskii, M. B.; Jordan, P. C. Membrane deformation and the elasticenergy of insertion: perturbation of membrane elastic constants dueto peptide insertion.J. Chem. Phys.2002, 117, 10768-10776.

(16) Maddox, M. W.; Longo, M. L. A Monte Carlo study of peptideinsertion into lipid bilayers: equilibrium conformations and insertionmechanism.Biophys. J.2002, 82, 244-263.

(17) Kessel, A.; Shental-Bechor, D.; Haliloglu, T.; Ben-Tal, N. Interactionsof hydrophobic peptides with lipid bilayers: Monte Carlo simulationswith M2δ. Biophys. J.2003, 85, 3431-3444.

(18) Breidenich, M.; Netz R. R.; Lipowsky, R. The shape of polymer-decorated membranes.Europhys. Lett.2000, 49, 431-437.

(19) Polozov, I. V.; Anantharamaiah, G. M.; Segrest, J. P.; Epand, R. M.Osmotically induced membrane tension modulates membrane per-meabilization by class L amphipatic helical peptides: nucleationmodel of defect formation.Biophys. J.2001, 81, 949-959.

(20) Nir, S.; Nieva, J. L. Interactions of peptides with liposomes: poreformation and fusion.Prog. Lipid Res.2000, 39, 181-206.

(21) Shai, Y. Mechanism of the binding, insertion and destabilization ofphospholipid bilayer membranes byR-helical antimicrobial and cellnonselective membrane-lytic peptides.Biochim. Biophys. Acta1999,1462, 55-70.

(22) Zemel, A.; Ben-Shaul A.; May, S. Perturbation of a lipid membraneby amphipathic peptides and its role in pore formation.Eur. Biophys.J 2005, 34, 230-242.

(23) Netz, R. R. Inclusions in fluctuating membranes: Exact results.J.Phys. I1997, 7, 833-852.

(24) Aarts, D. G. A. L.; Schmidt, M.; Lekkerkerker, H. N. W. Direct visualobservation of thermal capillary waves.Science2004, 304, 847-850.

(25) Stecki, J. Total and direct correlations in the liquid-vapor interfaceand the capillary-wave theory.J. Chem. Phys.1997, 107, 7967-7974.

(26) Farago, O.; Santagelo, C. Pore formation in fluctuating membranes.J. Chem. Phys.2005, 122, 044901(1-9).

(27) David, F.; Leibler, S. Vanishing tension of fluctuating membranes.J. Phys. II1991, 1, 959-976.

(28) See the Supporting Information.(29) Guedeau-Boudeville, M. A.; Jullien, L.; di Meglio, J.-M. Drug

delivery: piercing vesicles by their adsorption onto a porous medium.Proc. Natl. Acad. Sci. U.S.A.1995, 92, 9590-9592.

(30) Moroz, J. D.; Nelson, P. Dynamically stabilized pores in bilayermembranes.Biophys. J.1997, 72, 2211-2216.

NL052073S

Nano Lett., Vol. 6, No. 1, 2006 83