NANO LETTERS
Fundamental Mechanism of Translocation across Liquidlike Membranes: Toward Control over Nanoparticle Behavior
2006 Vol. 6, No. 1 78-83
Lucian 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-dominated pore on the membrane by a nanoparticle functionalized with a peptide aggregate. Following fluctuations in the position and orientation of the nanoparticle, the peptide aggregate incorporates into the membrane and locally destabilizes it. The nucleation of a pore centered at the peptide aggregate attached to the particle is a precursor to particle translocation. The subsequent opening of the pore is assisted by adhesion of the membrane to the particle. We determine the conditions in which thermal fluctuations in the membrane shape and the pore size can induce translocation of the particle. For different system parameters quantities such as the free energy, entropy, pore size, degree of particle wrapping, and the probability of spontaneous translocation are obtained.
Introduction The interaction between fluid membranes and nanosized particles is a frequently occurring topic in nanoscience and nanotechnology. It is inevitably associated with the issues of nanoparticle toxicity1 and plays a vital role in phenomena such as drug delivery,1-3 virus entry into a cell,4,5 nanoparticle adsorption, self-assembly, and phase-transitions at fluidfluid interfaces.6,7 From this perspective, a detailed investigation of the phenomena associated with the presence of a nanoparticle near a flexible membrane or at a liquid-liquid interface comes forth as a priority. Membrane-particle adhesion can induce partial or total wrapping of a particle,8-10 a phenomenon which commonly occurs in cells. A purely energetic analysis10 predicts that a particle is completely wrapped by an essentially tensionless vesicle if its radius exceeds the critical value Rp ) (2κ/| w|)1/2, where κ is the bending modulus and w < 0 is the contact energy per unit area, whereas particles with smaller radii are completely detached. For lipid bilayers, this leads to an estimate for a critical radius of about 5 nm. If the surface * To whom correspondence may be addressed. E-mail: andriy.kovalenko@ nrc.ca. † National Institute for Nanotechnology, National Research Council of Canada. ‡ Department of Mechanical Engineering, University of Alberta. 10.1021/nl052073s CCC: $33.50 Published on Web 12/09/2005
Published 2006 by the American Chemical Society
tension σ is not negligible, the critical radius for enveloping increases (from the tensionless case) and depends on σ. Biologically meaningful phase diagrams for the degree of wrapping were constructed for this case, which are relevant for particle sizes ranging from a few to about a hundred nanometers, such as nucleocapsids of animal viruses.8 The interaction between latex spheres and unilamellar lipid vesicles was studied experimentally for particle size in the micrometer range11 comparable to the size of the vesicle. It was suggested that particle encapsulation by the vesicle is accompanied by the formation of a dynamically stabilized pore in the membrane, but the mechanism and precise conditions for it remained unknown. The presence of a nanoparticle inside a small vesicle was found to promote vesicle fission and fusion in a model system studied by Brownian dynamics simulations.12 As a crucial step in this process, aided by the particle-membrane adhesion, the nanoparticle induces the opening of a pore in the contact region between the two transient partitions of a vesicle undergoing fission/fusion, thus enabling their complete splitting/joining. On a different scale, theoretical studies on model lipid membranes found that under lateral tension, depending on the architecture of the component lipid, either rupture occurs
or the membrane is in a phase characterized by a regular array of pores.13 In another study, Monte Carlo simulations were used to investigate a related topic regarding the conditions in which small pores can coalesce and be spontaneously replaced by a single large hole and membrane rupture occurs.14 The insertion of foreign objects into a membrane15-17 effectively creates structural defects, which can alter the shape and the integrity of the membrane. Polymer chains anchored to a lipid membrane have been found to exert an entropic force on the membrane and induce shape deformations.18 Another example is the class of amphipathic peptides (featuring both hydrophobic and hydrophilic surfaces), known for their ability to adsorb and incorporate into a membrane.19-21 Recent insight into peptide-membrane complexes shows a strong tendency of the peptide to reorient and insert into the membrane with an energy gain of about 4kBT.22 It was strongly suggested that the reorientation mechanism is the basis of a generic driving force of pore formation. Once adsorbed and reoriented, peptides selfassemble and can promote the growth of pores with mixed lipid-peptide lining composition upon increasing surface tension. This could result in a selective permeabilization of a membrane under osmotic tension. Considering the above facts, a design of a peptidenanoparticle complex with novel controlled membranetranslocation functionality seems attainable. A combined action of the peptide insertion and the particle adhesion to the membrane can lead to translocation of the particle by opening a tight pore in the membrane, large enough to accommodate its size, and simultaneously sealed by it (against leakage of material from the other side of the membrane). Below we develop a statistical-mechanical approach to such a system and explore this new possible mechanism of translocation of a functionalized nanoparticle through a membrane. Model and Method. A key feature of the present theory is a rigorous statistical-mechanical account of the entropy associated with shape fluctuations of the membrane,18,23 as opposed to other studies that only account for the energy of the system. We consider not only the membrane shape that minimizes the system Hamiltonian but also fluctuations around it enabling the new translocation mechanism. Within the continuum elasticity theory, the Helfrich Hamiltonian for a two-dimensional surface h with principal curvatures c1 and c2 placed in an external field U(r) reads H({h}) )
∫ dS [σ + 2κ(c1 + c2)2 + κjc1c2 + U(r)]
(1)
where dS is the surface element at position r and κj is the Gaussian curvature modulus. The partition function of a system with this Hamiltonian is Z ) ∫ D{h} exp(-βH({h})), where D{h} represents the integral over all the possible configurations (shapes) of the surface and β ) 1/kBT is the inverse temperature. Notice the absence of a spontaneous curvature in eq 1. Many studies revolve around the idea of minimizing the Helfrich Hamiltonian, and thus they account Nano Lett., Vol. 6, No. 1, 2006
for the energetics of the system, while completely ignoring the entropy contribution to the partition function. In contrast, for a rigorous statistical mechanical analysis one needs to include not only the mean field solution (minimizing the Hamiltonian) but also a continuum spectrum of excited states. Physically this corresponds to taking into account the presence of capillary waves at the interface, which are properly built into our theory. Recently, this phenomenon was directly observed in an experiment for interfaces in colloidal suspensions.24 Theoretical studies of capillary waves based on the Ornstein-Zernike equation for the liquid-vapor interface25 revealed the fact that capillary waves are related to the presence of a term quadratic in k in the Fourier transform of the direct correlation function c(k), whereas the bending energy corresponds to a term proportional to k4 (assuming the interface fluctuations to be gentle enough so that the Hamiltonian (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 impenetrable walls and an attractive region just outside its surface, which defines28 the external field U(r) in (1). The particle is functionalized with a peptide aggregate or some other membrane destabilizing agent. Effectively, these agents are able incorporate themselves into the membrane and promote the formation and growth of pores by lowering the line tension associated with the perimeter of the pore. For simplicity, in this study we treat only planar circular pores that are small enough so that significant out-of-plane shape deviations are unlikely. The line energy of a circular pore is considered to be of the simple form26 Eline ) 2πΓ′r, where Γ′ is the three-phase line tension including the particle-edge contact energy as well as the peptide effect on the pore. In the case of a hydrophilic pore, the specific adhesion energy w′ between the pore perimeter and the particle is approximately the same as that between the membrane leaflet and the particle. If this interaction is attractive, w < 0, then the effective line tension is decreased when the nanoparticle comes in contact with the pore. However, this contribution to the line tension takes effect only as long as the pore diameter is less than or equal to the particle diameter, making the opening of an “unsealed” pore (larger than the particle) less favorable. Alternatively, the presence of the nanoparticle can favor the closing of the pore if the line tension is effectively increased by the particle-edge interaction. We consider an open, framed thermodynamic ensemble for the membrane, in which the area stretching coefficient σ′ and the projected area (to the xy-plane) are constant.27 In this ensemble, the statistical mechanics of the membrane patch is carried out by the Green’s function (probability distribution function) method in the spirit of Feynman’s path integral.28 It effectively accounts for all the membrane configurations and their contributions to the partition function in the presence of an external field. The membrane is described by its axial and azimuthal cross sections, which in turn are separately modeled as a discrete number of joint segments. Two intersecting cross sections define a vertex, and the set of all vertexes constitute the conformation of the 79
Figure 1. Successive positions of the nanoparticle and the shapes of the membrane: a, partial pore opening; b, early pore opening enhanced by adhesion with particle trapping; c, no pore opening in the absence of adhesion; d, particle translocation through the membrane and fluctuations in the shape of the membrane at successive stages, with the membrane probability density changing from F ) 0 to 1 (color map from white to black). Parameter values for each case are given in the text.
membrane. The shapes of the axial and azimuthal curves are coupled via the mean position and curvature at each vertex. We thus decompose the membrane fluctuations into the axial and azimuthal components. The boundary conditions for the probability distribution are chosen to mimic a loosely constrained membrane at the edge of the patch.28 On discretizing the axial and azimuthal profiles, the corresponding equations are solved numerically. Results and Discussion of the Proposed Translocation Mechanism. Because we work with an open, framed system,27 the stretching constant of the model σ′ is the area coefficient and not the film tension σ; these quantities coincide for an infinite surface. Also, for our discrete membrane model we adopt a “discrete” bending modulus κ′ which is generally different from the continuum bending modulus κ but becomes identical for a sufficiently fine discretization. The exact relation between the values of the discrete “intrinsic” and the measured elastic constants is directly amenable by our approach. However, it does not affect the qualitative findings below, and we leave it beyond the present investigation to focus on the essence of the translocation mechanism. We choose κ′ ) 10kBT and σ′ ) 10-3kBT/a2, which yield values of the bending modulus and the surface tension within the acceptable range for biological membranes. Here a is the half-thickness of the membrane (of the order of 1 nm) and serves as the length scale of our model. The value of the Gaussian curvature modulus was found to play a negligible role in this investigation (at least, for magnitudes not exceeding the value of κ); therefore for illustration purposes, we chose κj ) 0. It is also noteworthy that, although accounted for below, fluctuations of the 80
membrane without axial symmetry do not have an essential role in our results. This is due to the fact that they introduce additional bending and stretching in the azimuthal plane, which will increase the potential energy of that particular conformation, therefore reducing its statistical weight. The shapes of the membrane and the opening of the pore for various positions of the nanoparticle are shown in Figure 1. The particle, with the functional aggregate on top, is moved up from below at a sufficiently slow rate so that the membrane is maintained in quasi-equilibrium. Thus, the membrane is considered to exhibit equilibrium fluctuations in the “external field” created by the particle placed at a certain height zp. For each position of the particle (i.e., external field configuration) we calculate the partition function, the free energy, and the average shape of the membrane corresponding to our ensemble. We define the degree of wrapping of the particle, ξ, as the ratio of the area covered by the membrane to the total particle area. Note that each curve shown in Figure 1 is not a fixed shape of the membrane but is obtained as a set of average locations of consecutive fluctuating segments of the profile. For particular calculations we chose the following values for the system size: the particle radius Rp ) 7a, and the patch frame radius of 10Rp. Figure 1 shows three distict extents of pore formation for different sets of the contact energies. In case a, the membrane-particle contact energy per unit area is w ) -0.1kBT/a2 and the pore line tension is Γ′ ) 0.75kBT/a. A pore starts forming as soon as the particle’s top is higher than the level of the unperturbed membrane at the edge of the membrane frame (we call this the contact point). Below that height, the adhesion is Nano Lett., Vol. 6, No. 1, 2006
Figure 2. Membrane characteristics as functions of nanoparticle position: a, the free energy of the membrane A/kBT; b, the pore radius Rpore/Rp; c, the degree of wrapping ξ versus the nanoparticle position; d, the ratio of the probability of translocation to that of desorption as a function of the adhesion strength for a fixed value of Γ′ ) 0.1kBT/a; e, for the case in Figure 1d, the free energy for ∆γ ) 0 (thin solid line), for ∆γ ) 0.01kBT (thick solid line), and the entropy S/kB (short-dashed line). Long-dashed, dash-dotted, dotted, and solid lines correspond to the system in cases a-d in Figure 1, respectively.
relatively weak. The degree of wrapping grows steadily with the advancement of the particle up to zp/Rp ) 1 and levels off from that point on. In case b, the adhesion is increased to w ) -0.15kBT/a2 to single out its effect on the pore size, whereas the line tension value is the same as in (a). Clearly, the pore opening is promoted by the increase in the magnitude of the contact energy. This finding is supported by the free energy profiles in Figure 2a showing a decrease of the free energy barrier for zp > 0 as compared to case a. Another essential difference is that in case b the particle is strongly adsorbed to the membrane, a fact supported by the free energy slopes in the two cases for zp < 0 and by a degree of wrapping with a maximum corresponding to the global minimum of the free energy curve. Nano Lett., Vol. 6, No. 1, 2006
The pore expansion is aided by a local stretching of the membrane by adhesion to the particle. This finding is corroborated by the experimental observation showing that strong adsorption of a vesicle onto highly curved surfaces (in a porous substrate) increases the permeability of its membrane by locally stretching it.29 However, the main difference is that in our study the pore is formed around the peptide aggregate and is sealed by the nanoparticle, whereas in ref 29 pores can form over a stretched region of the membrane which is not in contact with the substrate. The case (c) of zero contact energy demonstrates the essential role of adhesion in pore formation. The pore exhibits virtually no growth with the onward motion of the particle (Figure 1c). The degree of wrapping remains all but constant for the whole range of particle displacement (Figure 2c). The 81
free energy exhibits an almost linear increase from the point of contact on and becomes much higher than in all other cases (Figure 2a). A successful case of translocation is illustrated in Figure 1d for parameters w ) -0.05kBT/a2 and Γ′ ) 0.1kBT/a. For a ) 1 nm the numerical value of Γ is about 4 × 10-12 N, which is a relatively low value of the line tension when compared to typical estimates for lipid bilayers, 2 × 10-11 N.30 This difference is attributable to the presence of peptides and the interaction between molecules at the pore perimeter and the nanoparticle. We assume that the particle is pushed from below against the membrane sufficiently slow so that the membrane is maintained in quasi-equilibrium. Shown as a color spectrum is the axial cross section of the probability density of membrane segments for advancing positions of the particle. This provides a good illustration of the fluctuations in the shape of the membrane. Compared to the previous cases, the pore grows steadily and at a higher rate with the advance of the particle, and the free energy profile is more favorable to translocation. The shape fluctuations monotonically increase with the onward motion of the particle and for each stage reach the greatest magnitude just outside the contact region with the particle. This observation corroborated by the entropy curve (short-dashed line in Figure 2e), indicates that in the advanced stages of pore opening a strong competition exists between membrane conformations that have a small curvature and do not adhere to the particle and those that have both strong adhesion and large curvature. The latter group of conformations eventually dominates and has a decisive influence in favor of the translocation. Within a mean-field view, translocation occurs when the average pore radius becomes equal to the particle radius. It is accompanied by an abrupt jump in the quantities shown in Figure 2 in the vicinity of the point zp/Rp ) 1.1. After reaching its maximum size, the pore closes abruptly “behind” the nanoparticle. The free energy, entropy, and the degree of wrapping drop to low values and are more or less constant. The jump corresponds to the system accessing a subspace of its configurational space with the particle above the membrane, which was inaccessible before the translocation. We also briefly checked the effects of the other parameters to compare with the expectations.10 Increasing the particle size alone causes a rise in ξ. A decrease of the surface tension promotes the proliferation of the wrapping ξ. If the adhesion is strong enough, the particle will be trapped within the membrane by becoming embedded and concomitantly sealing the pore. The minimum of the free energy profile can be identified as a trapping region, if the energy barrier around it is considerably greater than the thermal fluctuations of the particle. Its location depends on the contact energy as well as on the line tension of the pore. We emphasize that this trapping mechanism is quite different from that involving the total enveloping of the particle.10 Our results for the degree of wrapping show an important contribution of fluctuations to the shape of the membrane. Our value of the degree of wrapping is always less than 1 even when the particle radius exceeds the critical radius 82
predicted by the purely energetic analysis.10 The reasons of this distinction are the presence of a small but finite surface tension, and the fluctuations in the membrane shape. The fluctuations act against the adhesion and have an unwrapping effect, thus decreasing the area of contact with the nanoparticle. The mechanism for particle translocation (enabled by a suitable set of interaction parameters) involves as a precursor the peptide group incorporation into the membrane after the initial adsorption of the particle. Once the particle is partially wrapped, translocation becomes possible due to fluctuations in the membrane shape and in the size of the pore (see Figure 1d), by virtue of the system exploring its whole conformational space. In the case when the particle is strongly adsorbed to the membrane, spontaneous translocation can occur enabled by a sufficiently strong fluctuation in the pore size. From a dynamics point of view, the time scale of this process τtrl is inversely proportional to the probability of occurrence of a fluctuation with a pore radius greater than the sphere radius ptrl ) p(rpore > Rp), times the frequency of attempts to translocate, ftrl. If this time scale is less than the time scale for desorption, τdes ∝ p-1desf-1des, and the lifetime of the membrane, the translocation occurs spontaneously. Assuming translocation and desorption attempts to be of roughly equal frequencies, ftrl ≈ fdes, the success of translocation defined as τdes/τtrl is determined by the ratio of the translocation and desorption probabilities ptrl/pdes. It is plotted in Figure 2d as a function of w for a fixed value of Γ′ ) 0.1kBT/a. For this case the translocation occurs spontaneously for magnitudes of w greater than 0.126kBT/a2. Besides the above, there are other factors that can influence the probability of translocation, in particular, those involving a driving force to overcome the energy barrier. We list here the following scenarios: (i) the existence of a difference between the interfacial energy densities of the particle on the two different sides of the membrane (especially if the particle is charged); (ii) a ratchet mechanism associated with ligand-receptor binding to the nanoparticle; (iii) the adsorption of molecules on the protruding region and/or on the pore edge, which can apply additional lateral pressure on the pore or further decrease the line tension. Using our results for the free energy profiles (Figure 2a) as the main ingredient, a transition state theory of nanoparticle translocation through a membrane or a liquid interface can be readily formulated. The free energy profile is drastically affected toward favoring particle translocation by the existence of a difference in the interfacial energy densities of the particle at the surface of contact with the environments on the two sides of the membrane. To quantify this change we introduced a small difference ∆γ ) γ2 - γ1 of just 0.01kBT/a2 to the system with all other parameters as in case of Figure 1d. The resulting free energy profile displayed in Figure 2e as a thick solid line has a much smaller energy barrier for translocation, which is easily accessible to fluctuations. To conclude, we have explored the possibility of penetration of a membrane by a functionalized nanoparticle. By formulating an amenable statistical mechanical model, we 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 energy of the membrane. We have predicted a new possible mechanism of nanoparticle translocation or trapping, which does not involve total enveloping of the particle by the membrane. Initially, the nanoparticle functionalized with a peptide aggregate gets adsorbed onto the membrane. Following fluctuations in nanoparticle position and orientation, the peptide aggregate incorporates into the membrane, which becomes locally destabilized by it. A pore starts forming with the assistance of the adhesion of the membrane to the particle. Even in the absence of a driving engine against the membrane energy barrier, the fluctuations in the pore size eventually induce translocation of the particle for certain systems. The success of this scenario is determined by the interaction parameters accounted for in our analysis and by the time scale of the fluctuations. We assert that this new scheme of control over translocation by appropriate functionalization of the nanoparticle can find applicability in biological realm, where drug delivery in a nanoparticle scope is desired, as well as in nanochemical systems. Acknowledgment. This work is supported by the National Research Council of Canada. Supporting Information Available: Detailed information regarding the model and methods used in this study. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Mu¨ller, R. H.; Ma¨der, K.; Gohla, S. Solid lipid nanoparticles for controlled 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-Brain Barrier. 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 via endocytosis. 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. Science 2003, 299, 226-229. (7) Koltover, I.; Radler, J. O.; Safinya, C. R. Membrane mediated attraction and ordered aggregation of colloidal particles bound to giant phospholipid bilayers. Phys. ReV. Lett. 1999, 82, 1991-1994.
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