Translocation and Induced Permeability of Random Amphiphilic

Article pubs.acs.org/Biomac

Translocation and Induced Permeability of Random Amphiphilic Copolymers Interacting with Lipid Bilayer Membranes Marco Werner† and Jens-Uwe Sommer* Leibniz-Institut für Polymerforschung Dresden, Hohe Straße 6, 01069 Dresden, Germany Technische Universität Dresden, Institute of Theoretical Physics, 01069 Dresden, Germany ABSTRACT: We investigate adsorption and passive translocation of random amphiphilic copolymers interacting with a self-assembled lipid bilayer membrane. By using the bond fluctuation model with explicit solvent, we consider random copolymers under variation of the fraction, H̅ , of hydrophobic sites and chain length. Our results indicate a point of balanced hydrophobicity, where a slight excess of hydrophobic monomers compensates an additional insertion barrier due to the self-organized packing of the bilayer. Close to balanced hydrophobicity, we observe translocation events of shorter polymers through the membrane. Compared to homopolymers, surface localization of amphiphilic polymers is considerably increased due to the polar nature of the molecules with respect to the amphiphilic environment, and translocations are suppressed for longer chains. Close to balanced hydrophobicity, the polymer induces dynamic and static perturbations in the bilayer, and permeability with respect to solvent is significantly increased around the copolymer. We discuss how to design membrane-active copolymers with a desired emphasis on either translocation or permeabilization based on a systematic sequence analysis. Our results indicate that alternating copolymers with an optimal block size smaller than the lipid size maximize perturbation of the bilayer, whereas for passive translocation, the limit of small block size or homopolymers with balanced hydrophobicity are most relevant.

1. INTRODUCTION Translocation of potential drug cargoes or gene vectors into living cells as well as controlled permeabilization of plasmamembranes for other solutes are of enormous importance for therapeutic research and cell biology. The underlying biological, biochemical, and physical mechanisms are not yet fully understood and are controversially disputed.1,2 It is wellknown that for hydrophilic molecules the membrane acts as barrier, whereas hydrophobic molecules are trapped.3 Translocation rates should be at maximum for a flat free energy profile along the membrane normal. It has been shown that this simple physical argument is in agreement with a permeability ranking for smaller solutes such as β-blocker drugs and steroid hormones4 by using molecular dynamics simulations. Besides well-known biologically active processes such as endocytosis, which can explain the uptake of substances into the cell, there is growing experimental evidence that amphiphilic polymers may translocate through lipid bilayer membranes passively. In particular, in vivo experiments by Goda et al. demonstrate translocation of polyphospholipids without energy consumption into mammalian cells.5 Passive diffusion has also been shown for poly(ethylene glycol) (PEG)-based polymeric surfactants through model lipid membranes by Mathot and co-workers.6 Our recent simulation studies7−9 and a direct application of Kramer’s theorem to the potential of mean force8 indicate that also translocations of larger macromolecules would be triggered by an overall balanced © XXXX American Chemical Society

hydrophobicity between bilayer core and the solvent phase, and specific interactions between charges and lipid head groups are not necessary. Indeed, experiments10,11 give rise to a headgroup independent translocation mechanism even for highly charged molecules such as polyarginines: Sakai et al. emphasized that translocation of polyarginin is triggered by its complexation with specific hydrophobic counterions. The enhancement of passive transport by highly charged amino-acids such as arginine may play an important role also for cell-penetrating peptides (CPP), for instance, based on homeo-domains, which are evolutionary highly conserved in natural transcription factors.2 The translocation and membrane permeabilization mechanisms of CPPs are still not clarified and contradict to the expected mechanism of charge-tail repulsion.12 Apart from direct translocation of the polymers through the membrane, polymers are also reported to mediate dynamic and static perturbation of the bilayer structure. This enables applications for permeabilization of membranes for other solutes6,13 such as small sugars14 in order to enhance cryopreservation of red blood cells.14 Amphiphilic diblock copolymers have been found to facilitate the transport of DNA through model membranes15 by a possible cooperative mechanism, which can be the self-assembly into transient Received: August 26, 2014 Revised: November 28, 2014

A

DOI: 10.1021/bm501266x Biomacromolecules XXXX, XXX, XXX−XXX

Article

Biomacromolecules pores.15 Likewise, amphiphilic triblock copolymers based on PEO−PPO−PEO (poloxamers) have been shown to facilitate passive transport of larger solutes through model membranes.16,17 Therefore, they are promising candidates for drug- and gene delivery systems,18 for instance, into tumor cells.19 Also amphiphilic polymers on a different chemical basis such as polyacrylates (amphipols) with random amphiphilic sequence have been observed to permeabilize vesicle membranes for small sugar molecules.20,1 It has been demonstrated by Song et al. that permeabilization of bacterial plasma membranes and antimicrobial activity of synthetic peptide mimics is particularly efficient for alternating short hydrophobic and cationic blocks.22 Our present work demonstrates the role of balanced hydrophobicity and the influence of amphiphilicity on the membrane activity of random copolymers that mimic most of the investigated cell penetrating peptides and polylipids. Our results confirm the universality of the role of balanced hydrophobicity also for amphiphilic polymers in line with earlier studies,4,7−9 although surface effects are considerably increased with amphiphilicity and reduce translocation. Furthermore, our results indicate that one possible pathway of permeabilization are spontaneous reorganizations of the polymer within the membrane forming transient pore-like structures with short lifetime. This may explain the observed time lapse between adsorption and translocation of CPPs representing short random copolymers, before they selforganize into such structures.12 A balanced hydrophobicity corresponds to the point of equal partition of the molecule in solvent as compared to the bilayer core, and a vanishing free energy difference ΔFz=0 = 0 comparing the bilayer core (z = 0) with the solvent. In the case of homopolymers where all monomers have an intermediate hydrophobicity, one can show that adsorption effects at the bilayer−solvent interface are weak close to the point of balanced hydrophobicity. As a consequence, very long homopolymer chains can also passively translocate. In this study, we will show an important difference for amphiphilic polymers as compared to homopolymers, which is caused by their much stronger localization at the bilayer interfaces. These localization effects can obscure the balanced hydrophobicity condition for longer chains such that the membrane always acts as a trap and translocations are suppressed. On the other hand, membrane-trapped copolymers at balanced hydrophobicity are more effective to induce transport for smaller molecules through the membrane. Throughout the paper we will analyze how amphiphilicity inhibits translocation and modifies surface activity of a random copolymer as compared to homopolymers. Furthermore, we analyze translocation as well as permeabilization of the membrane as a function of specific features of the sequence of a copolymer. Our results will point forward to a more systematic study of multiblock copolymer sequences to achieve controlled membrane-activity of macromolecules under variation of hydrophilic−lipophilic balance (HLB) as well as blockiness of the sequence. Translocation of smaller solutes3,4,23,24 as well as macromolecules such as nanoparticles,9,25−27 dendrimers,28 or polymers through phospholipid bilayer and their perturbations of the bilayer have been subject to computer simulation studies based on molecular dynamics as well as Monte Carlo simulations.7−9 All-atom molecular dynamics29,30 provide a good agreement of time scales and local relaxation mechanisms with experimental result for specific small permeants such as

benzene31 or water.3 However, with reasonable computational effort, only coarse grained methods32−34 may reach time scales that allow one to directly observe translocation events for small molecules such as water23 or even larger objects such as polymer chains, or to study shape-induced endocytotic uptake of nanoparticles27 systematically by means of continuum membrane models. In this study we use a central processing unit (CPU)- and a graphics processing unit (GPU) version35 of the bond fluctuation model with explicit solvent, which has been applied to the problem of translocation of homopolymers as well as nanoparticles in earlier studies.7−9,36 We study the interaction of single random copolymers with lipid bilayer membranes. A better understanding of the interactions of a membrane with single chains can be the starting point to investigate multichain effects as well, where micellization between the polymer or the crowding of adsorptive interfaces may come into play depending on polymer concentration.37,38 The rest of this paper is organized as follows. In section 2 we briefly summarize the coarse grained simulation method as introduced earlier. In sections 3.1 and 3.2 we present simulation results of polymer adsorption at and translocation through the model membrane, respectively. In section 3.3 we discuss polymer induced permeability for solvent and give an outline in section 3.4 to optimize the copolymer sequence with respect to induced perturbation and polymer translocation. In section 4 we summarize our results.

2. SIMULATION METHOD We use the bond fluctuation model39,40 (BFM) with an explicit solvent7,36 to simulate lipid bilayer membranes interacting with heteropolymers such as random A/B copolymers (RCP). The simulation model represents a coarse-grained view on flexible polymers where the monomer unit of the BFM corresponds to a few chemical monomers. The level of coarse graining can be estimated by mapping the statistical segment length (Kuhn segment) of polyethylene to the statistical segment in the BFM, which leads to approximately 5 CH2 units.41 We note that coarse-grained models such as the BFM, coarse-grained molecular dynamics or dissipative particle dynamics aim to guide our understanding of universal aspects of polymer-based systems. In the present case, it is the interplay between hydrophilic and hydrophobic effects which causes the selfoganization of all lipid membranes, which is in the focus of our study. Therefore, our results can be potentially mapped to various lipid membranes and copolymers by appropriately choosing the interaction/solubility parameters. Figure 1 shows the connectivity of the molecules used in our BFM simulations. Coarse-grained phospholipids are formed by head (h) groups of length nh = 3 and two tails (t) of length nt = 5. Furthermore, the simulation box is occupied by a single

Figure 1. Schematic representation of the interaction model for lipids (h,t), solvent (s) and random copolymers (A,B). B


Article

Biomacromolecules random copolymer chain (A and B) and filled up by solvent monomers to achieve a total lattice occupancy of 0.5 corresponding to a dense state in the BFM. The hydrophobic effect is mediated by a short-range repulsive interaction between hydrophilic (h and s) and hydrophobic monomers (t) according to an interaction constant ε0 = 0.8kBT. The interaction potential42 is described in more detail further below (eq 3), see also our previous work.7−9,36 If not stated otherwise, the two monomer species A and B forming the copolymer are identical to (h, s) and t, respectively. We define the mean hydrophobicity, H̅ , of the polymer as the number fraction of hydrophobic monomers, H̅ =

NB N = B NB + NA N

8 and Figure 5c). Typical simulation snapshots for various values of H̅ are shown in the upper part of Figure 3. The repulsive interaction energy between monomers of species i and j is defined as42

Ui , j( r ⃗) = εi , jc( r ⃗),

(3)

where r ⃗ is the distance vector between two monomers of species i and j, and P± defines the set of all possible permutations and sign combinations. According to eq 3, each occupation in the lattice by a monomer j on the 24 next nearest neighbor sites of a monomer i contributes εi,j to the total interaction energy between both. The hydrophobic interaction of lipids and solvent without polymer is defined as

(1)

with N giving the degree of polymerization. The degree of hydrophobicity of nonionic amphiphiles such as poloxamers and other surfactants is often quantified in terms of the value of hydrophilic/lipophilic balance (HLB).43 The definition of HLB43 and the measure H̅ in eq 1 are related by

HLB = 20(1 − H̅ )

⎧ 4 for r ⃗ ∈ P±(2, 0, 0) ⎪ ⎪ 2 for r ⃗ ∈ P±(2, 1, 0) with c( r ⃗) = ⎨ ⎪1 for r ⃗ ∈ P±(2, 1, 1) ⎪ ⎩ 0 otherwise

ε0 = εt,h = εt,s = 0.8kBT

(2)

and

εh,s = 0

(4)

According to eq 4, s-monomers are identical to h-monomers in terms of interaction constants in our model. The coupling of monomers of an additional component X to lipids and solvent is adjusted by defining additional interaction constants εX,j, where

Simulations have been performed under variation of 0 ≤ H̅ ≤ 1. The corresponding number NB has been predefined for the otherwise random sequence. The simulation box has been chosen with the dimension (64a)3 for chain length N ≤ 128 and (128 × 128 × 64···128)a3 for chain lengths N > 128 with periodic boundary conditions and a denoting the lattice constant of the simple cubic lattice. Simulations involving the larger box sizes have been performed by using a GPU version of the bond fluctuation model35 with explicit solvent.7,8 The projected area per lipid in the single leaflet has been chosen as 27.3a2 as in the previous studies,7−9 and lipids are preordered in the simulation box to form a stable bilayer membrane parallel to the x,y plane spread over the periodic boundaries. We simulated in the order of 40 random A/B sequences per combination of N and NB for polymers of length N ≤ 128, and 20 random sequences for polymers of length N > 128. The sequence of A and B has been generated for each polymer by randomly choosing NB from N monomers to be labeled as Btype monomers, while all other monomers belong to species A. N This procedure randomly choses one sequence out of N B possible combinations with fixed mean hydrophobicity, H̅ , in the manner of drawing without returning from a pool of NA and NB monomers of each species. This imprints an anticorrelation between the single monomer species and the frequence of the same species in the rest of the chain. Independent labeling of monomers (drawing with returning) would, on the other hand, introduce a scattering around a given expectation value ⟨H̅ ⟩ and smear out simulation results according to a chain-length dependent variance, ⟨ΔH̅ 2⟩ ∝ N−1. Polymers are initially placed narrow to the bilayer. Each individual simulation has been equilibrated for 107 MCS and analyzed for another 5 × 107 MCS. This corresponds to the diffusion time of polymer of length N = 64 through a box of size 64. Longer chains would not show translocation within the accessible simulation time. Instead, for intermediate and strong hydrophobicity, H̅ > 0.4, which are the focus of this work, polymers are localized close to the membrane. As a relevant time scale for reorganization of the polymer between the two leaflets, we consider the rate of lipid flip-flops, which is on the order of ∼1/(107 MCS) per lipid (ref

εX,h = εX,s = HXε0

and

εX,t = (1 − HX)ε0

(5)

and we call HX the monomer hydrophobicity for species X. Furthermore, two monomers of species X and Y repel each other according to their absolute difference of monomer hydrophobicity εX,Y = |HX − HY|ε0

Simulation results of random copolymers are obtained for hydrophobic B monomers with HB = 1 monomers and hydrophilic A monomers with HA = 0. Simulation results for the largest chain length, N = 256, have been obtained by using a parallel version of the BFM35 for GPUs with a direct mapping of monomers to parallel threads. Local properties may slightly change upon parallelization35 due to different move−move correlations as compared to the serial BFM procedure (CPU), while universal properties are expected to be unchanged. In order to nevertheless directly show simulation results from CPU and GPU together in Figure 5, we reduced this effect by diluting the number of active monomers per cycle. Monomers have been divided into four groups, where in each cycle all monomers of a randomly selected group have been attempted to move simultaneously. By this choice, bondsharing conflicts35 do not have to be taken into account even for the three-armed lipids as illustrated in Figure 2b, while lattice collisions still have to be ruled out involving atomic operations. As demonstrated by Jentzsch et al.44 even lattice conflicts can be avoided by, for instance, choosing a different lattice geometry. The bilayer density profiles in Figure 2a have a maximal absolute difference of lattice density in the order of Δϕ ≈ 0.004 (≈ 0.8%) between GPU and CPU results. For box sizes of 128 lattice units in the z-direction, the total lattice occupancy has been set to ϕ = 0.49574 to compensate for density deviations along the z-axis (see Figure 2a) with the bulk solvent density as a reference.

( )

C


Article

Biomacromolecules

Figure 2. (a) Comparison of total, lipid tail, solvent, and lipid head density profiles (listed in the order from the highest to lowest maximum value in the plot) between serial and GPU-version of the BFM model for various simulation box geometries. (b) Using the example of lipids the division of monomers into groups of simultaneously updated monomers on GPU is shown, and group 2 is marked as being selected.

Figure 3. Center part: Schematic representation of a random copolymer close to the bilayer with low (left) and high (right) fraction of hydrophobic sites, H̅ . In the top part we show corresponding simulation snapshots for chain length N = 128. In the bottom part we illustrate simplified effective potentials for blocks of B monomers (left) and A monomers (right) and both close to the balanced condition (center). The size of excess blobs under the balanced condition is denotes as g.

3. RESULTS AND DISCUSSION 3.1. Polymer Adsorption at the Bilayer−Solvent Interface. As shown in a previous work,7,8 the given configuration of lipids and solvent provides a stable selfassembled bilayer membrane. The bilayer−solvent interface is selective for the two monomer species A and B. To illustrate this, the free energy profiles, so-called potential of mean force (PMF), FA/B(z), for individual A(B) monomers are shown in Figure 4. They have been obtained by sampling the density profiles, ρA/B(z) ∝ exp(−FA/B(z)/(kBT)), of unconnected A(B) monomers. The bilayer core as formed by lipid tails (t) shows an attractive potential for B-type monomers, whereas for A-type monomers, the bilayer core acts as a potential barrier. There is a difference between the depth of the trap (FB) and the height of the barrier (FA) (see inset of Figure 4). This asymmetry of the interface reflects an additional insertion barrier μins induced by the self-organized packing of the bilayer and has been observed also for homopolymers (HP) in ref 7. Driven by the selectivity of the bilayer−solvent interface, a random A/B copolymer may reorganize its conformation close to the interface to increase the number of monomers located in the preferred environment. Excursions are formed on the two sides of a selective interface according to their random excess of A or B species (excess blobs) . For random copolymers at an ideal interface with a weak selectivity χ, one may estimate the blob size, g, by balancing the energy gain of localization with the thermal energy kBT ∼ √gχ, if √g denotes a typical random excess.45 As a consequence, the polymer chain of length N has an energy gain in the order of Fads ∼ kBTN/g ∼ Nχ2, if it is localized around the interface.46,47 The monomer PMFs as shown in Figure 4 do not correspond to an ideal sharp interface. Instead, the interface appears smooth on the scale of the membrane thickness, d ≈ 14a. Furthermore, the bilayer is subject to thermal fluctuations and may be deformed by the presence of a polymer. Instead of excess blobs at a sharp interface, reorganization of the copolymers is controlled by equilibrated forces between Aand B-rich parts of the chain in a broader interface zone. As a reference for the location of the smooth interface, we therefore

Figure 4. Potential of mean force for unconnected monomers as a function of distance from the bilayer center of mass for various values of monomer hydrophobicity H according to eq 5. The values of H = 0 = HA and H = 1 = HB correspond to A- and B-type monomers of the random copolymers throughout the text, respectively. The vertical line at z = 6.90 in the inset denotes the point of compensated forces between A- and B monomers. The corresponding PMFs can be approximated by FA ≈ 5.14 exp[−z2/26] and FB ≈ 0.029z2 − 3.34, respectively, as shown by the continuous lines. Simulation results for F in the main plot (in the inset) have been normalized such that F = 0 corresponds to bulk (local minima).

sketched the point of balanced forces ∂FA/∂z = −∂FB/∂z between A and B monomers in the inset of Figure 4. Polymer adsorption at interfaces can be characterized by the fraction of monomers located directly at the interface.47−49 As a D


Article

Biomacromolecules sharp interface cannot be clearly defined for the bilayer, let us propose an alternative order parameter of RCP localization at the bilayer−solvent interface. Here, we count the number of pairs of A and B monomers, where both are located in the preferred environment. We considered a monomer to be located in the preferred environment, if the number of contacts with surrounding monomers (according to eq 3) is dominated by nonrepulsive species. Contacts of monomers of the same species are not taken into account for the comparison. Let us call MA and MB the numbers of A and B monomers in the preferred environment. Then M = min(MB,MA) is the number of localized AB-pairs, and we define the fraction of localized monomers,

m = 2M /N

(6)

as an order parameter of polymer localization around the bilayer−solvent interface.47 In Figure 5a we show the simulation results for m as a function of H̅ . For values of H̅ close to 0.6, we observe a peak of the order parameter. This lets us to define the point of maximum interface localization as HL ≡ 0.6. As we can see from Figure 6, HL is close to the point, where the mean size of excursions around the bilayer-solvent interface is balanced with respect to both environments. Deviation from H̅ = HL , reduces the localization of the polymer by a bias of loop sizes in favor of either the bilayer core (H̅ > HL) or the solvent phase (H̅ < HL). In this sense, the point of maximum interface localization is controlled by a balanced compatibility of the polymer with respect to bilayer core and solvent. This becomes clear, when we compare the results for HL with the single monomer potentials as shown in Figure 4. Let us rewrite the free energy differences in the bilayer core (z = 0) as compared to the bulk (z → ∞) as ΔFA,0 = χ + μins and ΔFB,0 = −(χ − μins) for A(B)type monomers respectively, where we introduce χ as the mean selectivity of the membrane. Assuming linear superposition of single monomer contributions, we may write the balanced condition as (1 − H̅ )ΔFA,0 + H̅ ΔFB,0 = 0, revealing χ (1 − 2H̅ ) + μins = 0

Figure 5. (a) Interface localization order parameter, m, corresponding to eq 6, (b) rate of polymer translocation, (c) rate of lipid flip-flops, (d) lipid orientational order, and (e) polymer induced permeability of solvent as a function of fraction of hydrophobic sites, H̅ , of a random copolymer of length N as given in the legend. Translocation rates in panel b are multiplied by chain length, N, and shown for N = 16 and 32 as a function of H̅ as well as for fixed hydrophobicity H̅ = 0.6 (rounded to the next integer, NB) for chain lengths N = 24, 40, and 48 from top to bottom, corresponding to Figure 7. Results in panels c and d are taken within a projected distance of 20 lattice units from the polymer center of mass.

(7)

The empirical estimates are χ = 4.24 kBT and μins = 0.895 kBT based on PMFs as shown in Figure 4 and taken at z = 0. The balance condition (eq 7) gives H̅ = (1/2) + (μins/(2χ)) ≈ 0.606. This is in very good agreement with HL = 0.6 as obtained from the peak in the order parameter m. Let us discuss the order parameter in the limits of H̅ → 0 and H̅ → 1. Hydrophilic polymers, H̅ = 0, are exclusively found in the solvent phase, whereas hydrophobic polymers, H̅ = 1, are confined in the bilayer core. In accordance, the order parameter vanishes in both limiting cases per definition, since the largest possible values of m are given by the respective minority species (see eq 6). The different characteristics of the observed order parameter as a function of H̅ on both sides of the balanced point reflect different adsorption scenarios at the bilayer− solvent interface. The order parameter as a function of H̅ for both limiting cases can be described by considering the adsorption of small blocks of the respective minority species at practically impenetrable bilayer−solvent interfaces (see Figure 3). In the case of H̅ → 0 already the attractive interaction of a small fraction of B monomers may compensate translational entropy of the whole chain and hold the polymer in the vicinity of the membrane due to the attractive energy of ∼kBT per Btype monomer (see Figure 4). In fact, we observe a significant increase of m for values of H̅ ≳ 0.2. Here, one may consider the

polymer as being composed of loops in the solvent phase, which are divided by adsorbed B blocks. For the opposite limit, H̅ → 1, the order parameter m is controlled by the adsorption of small A blocks from inside the bilayer core at the interfaces. Confinement of the polymer in the bilayer core changes the situation as compared to H̅ → 0 by appearance of the bilayer thickness as a new relevant length scale. In the case of H̅ → 1, this becomes particularly relevant when the typical loop size becomes larger than the bilayer thickness. Then, creation of an additional loop by adsorption of an A block at the interface does only marginally reduce conformational entropy by a weak perturbation of a confinement blob. We observe an almost linear decrease of the order parameter for H̅ → 1, indicating that a dominating fraction of A monomers is located in the solvent phase. E


Article

Biomacromolecules

shortly after the event. We detected events during Δt = 5 × 107 MCS for each random copolymer and calculated a mean number of translocation events K per simulation time for a given parameter set, (N,NB). The rate K gives an estimate for the mean translocation rate, T, averaged over infinite simulation time and the whole set of sequence combinations. To estimate the actual translocation rate, T, the limitation of the time interval Δt per individual simulation run has to be taken into account. To detect a translocation, the simulation time window has to be equal or larger than the travel time, Δttravel, of the polymer between the two separated states on the opposing sides of the membrane. According to our simulation data, the travel time of individual events was typically smaller than onehalf of the simulation window Δttravel < Δt/2, indicating that the relevant part of the distribution with respect to Δttravel is taken into account. Here, we take into account Δttravel as a dead time and apply a first-order correction for the estimate of T,

Figure 6. Bias of mean loop weight in favor for the bilayer core environment as a function of mean hydrophobicity H̅ . We count the number Nloop,(h,s) and Nloop,t of monomers, which belong to excursions either in the hydrophilic (h, s) environment or the hydrophobic (t) environment (bilayer core), respectively. If the loop identity of a monomer is not clearly defined due to a lack of a dominating (h, s) or t neighbor contacts, we attribute the monomer with the loop identity of the last monomer along the chain. This way, every monomer is attributed to an environment, N = Nloop,(h,s) + Nloop,t , in contrast to MA and MB as defined in the text.

T≈K

N (2z±)2 2D1

(9)

Simulation results for the translocation rate, T, per polymer and time are presented in Figure 5b, where the error bars take into account the total number of translocations as independent and equivalent events by means of a single standard deviation. In Figure 5b we also include simulation results for chain length N = 16. For the shortest chain length simulated, N = 16, we see the largest translocation rate close to the point of balanced hydrophobicity, at H̅ ≈ 0.55. This indicates that the balance condition as given by eq 7 controls the point of maximum translocation for shorter chains, resembling our results for homopolymers.7,8 Note that we use the intrinsic time as given in eq 8 to rescale the translocation rate in Figure 5b by multiplication with N. Here, the rescaling does not lead to an overlap of the curves at H̅ = HL as was the case for homopolymers.8 Instead, for the random copolymers we observe a left shift of the peak position toward smaller hydrophobicity, H̅ < HL as well as a strong suppression of translocation processes with increasing chain length. We did not detect any event for polymers with N ≥ 64 at H = HL during a total simulation time of 1010 MCS for chain lengths N = 64 and N = 128. We notice an essential difference for random copolymers as compared to homopolymers. For the segments of a homopolymer one can find a weak attraction at the interfaces, which is induced by the interface asymmetry, μins, as argued earlier.7 Considering ideal homopolymer chains in ground state dominance51 one can show that there exists a critical adsorption transition, HAds, which is always HAds > 1/2, since only then can the surface become attractive. If μins is not too large, the adsorption transition is close to the point of balanced solubility. Vice versa, at the point of balanced solubility, the homopolymer is only at the verge of getting localized close to the membrane, and the membrane becomes energetically transparent. This has been confirmed previously by computer simulations.7,8 In contrast, we observe that random copolymers get localized close to the membrane already for values of H̅ ≈ 0.3 due to the formation of excess blobs around the interface (see Figure 3). For values of H̅ beyond H̅ > 0.3, the binding to the membrane increases, and there is no state of an effective transparent membrane for any H̅ .

3.2. Polymer Translocation. Our recent results for homopolymers show that the point of balanced hydrophobicity plays a key role for passive translocation of polymers through the membrane.8 To detect a translocation event, we define two thresholds z± = ± 20a for the distance of the center of mass of the random copolymer from the bilayer core. The distance between the two thresholds z± defines a time scale given by the diffusivity of the chain: τ∼

Δt Δt − Δt travel

(8)

where D1 is the single monomer diffusion coefficient. As demonstrated previously,36 the friction for diffusive motion in the model grows linearly with N as known for Rouse-like dynamics. The time scale τ in eq 8 corresponds to the translocation time between the thresholds, z±, if there would be no membrane in between. Note that we keep z± fixed and independent of chain length, and the time scale, τ ∝ N, does not correspond to a Rouse time (∝N2), as it would be relevant for translocation through a pore.50 To avoid the extra Ndependence allows us to observe translocation events also for longer chains within the limited simulation time. To be sure, however, that the chain is delocalized from the bilayer, we apply an extra criterion based on pairwise monomer contacts between polymer and membrane. The polymer was considered as being in contact with the membrane, if its center of mass has a distance smaller than |z±| from the membrane core or monomers of the polymer during the last 500 MCS have been in contact with lipid tail monomers (according to eq 3) on the basis of a time resolution of 100 MCS. The polymer was considered to be separated from the membrane if there was no contact and the absolute distance of the center of mass was larger than 20a. A translocation event is detected if the polymer changed its state in the order separated → in contact → separated, and after the process it has been found on the opposite side. Note that the criterion for separation does not exclude that polymer and membrane get in contact again F


Article

Biomacromolecules

the scope of this Article, T and TArr , refer to averages with respect to the ensemble of random sequences, although reorganization of the polymer and translocation rates depend on particular features of the A/B-sequence of a given polymer beside the fraction of hydrophilic and hydrophobic monomers. It is interesting to compare polymers of the same fraction H̅ but different distributions of B-monomers along the backbone with respect to their translocation rates. In Figure 7 we include simulation results for multiblock copolymers consisting in short blocks as an example, which are part of ongoing research to be published in the future. As mentioned in ref 8, amphiphilic polymers of small blocks on the monomer scale seem to resemble the passive transport behavior of homopolymers through a quasi-transparent membrane. As shown in Figure 7, those polymers obay translocation rates, which stay in the same order of magnitude in relation to the inverse diffusion time τ−1 (see eq 8) for the given chain lengths and thresholds z± applied. Note that the fraction of B-type monomers is H̅ ≈ 0.6 for all polymers in Figure 7, and translocation rates have been obtained following the same procedure. 3.3. Polymer-Induced Permeability. The explicit solvent model allows to directly analyze polymer-induced perturbations of the bilayers integrity as a diffusion barrier for small solutes. We used the procedure as documented previously7,8 to count translocation events of individual solvent molecules per time. A corresponding permeability, Ps, of the membrane with respect to solvent is then defined as the number of translocation events, Ks, per time and projected area. Figure 5e shows simulation results for the relative polymer induced change of permeability, ΔP = Ps/Ps,0 − 1, within a projected distance of 32 lattice units from the polymer center of mass, where Ps,0 corresponds to the unperturbed membrane. Permeability for solvent molecules is significantly enhanced close to the point of balanced hydrophobicity, HL (see Figure 5e). This resembles our previous result for homopolymers7,8 as well as for homogeneous and amphiphilic nanoparticles.9 As shown in Figure 5c,d, the adsorbed polymer also increases the flip-flop rate of individual lipids in its vicinity along with reduction of lipid orientation order (see also ref 8) in a broad range of polymer composition 0.3 ≤ H̅ ≤ 0.8. It is worth noting that permeability is increased in almost the same interval of H̅ . Furthermore, the representation of ΔPs multiplied by 1/N in Figure 5e shows a superlinear increase of ΔPs as a function of chain length, N. This indicates that distant parts along the chain contour may contribute to Ps cooperatively, instead of perturbing the membrane independently from each other. One possible cooperative effect is the formation of transient channel-like structures as shown in the snapshot, Figure 8. We observed peaks of translocation frequency as a function of time of a typical width of 105 to 106 MCS coinciding with the spontaneous formation and lifetime of channel-like conformations as shown in Figure 8. Such spontaneous pores of short lifetime are within the range of spontaneous fluctuations around the bilayer−solvent interface. If the interface is locally crowded by the polarized copolymer, fluctuations start to involve conformations, where additional interface area is created, as argued in ref 1. Our observation seems to confirm experimental evidence for random copolymer (amphipol)induced transient pores in vesicles by Vial et al.1,20,21 In Figure 9a we present a histogram of solvent translocation events with respect to the position of a random copolymer for the case H̅ = HL. It is interesting to compare this distribution with the corresponding result for homopolymers8 (see Figure

In order to explain the low translocation rates for copolymer chains, we consider the chain as being organized in terms of excess blobs,46 which bind on average g monomers per blob at the bilayer-solvent interfaces with a free energy of kBT, see Figure 3. The free energy of adsorption of the chain then is Fads ≈ kBTN/(2g), where a factor of 2 in the denominator accounts for the fact that one blob is always on the “right” side of the solvent; the actual size of one “anchor” is 2g. Now, considering the process of desorption from the interface as the rate limiting process for translocation and assuming that reorganization with respect to the two leaflets is much faster, one may write the translocation rate in the form of an Arrhenius law, TArr ∝ 1/(2τ ) exp[−N /(2g )]

(10)

which is given by the probalility exp(−Fads/(kBT)) to find the polymer in a separated state from the bilayer. We use the intrinsic diffusion rate with respect to the nearby thresholds, z± (eq 8), as an approximate rate constant within the simple ansatz in eq 10. Only half of the desorption events correspond to translocation, while the other half corresponds to reflection, which is respected by a prefactor of 1/2 in eq 10. In Figure 7 we plot TArr in comparison to simulation results for HL = 0.6, where additional chain lengths between N = 16 and N = 128 have been added. Simulation results as shown in Figure 7 are in good agreement with the ad-hoc result eq 10 up

Figure 7. Rescaled translocation rates of random copolymers (closed circles, see eq 9) in units 1/(107 MCS) as a function of chain length, N, and balanced hydrophobicity, H̅ = 0.6, as compared to predictions by eq 10 as indicated by dashed lines by using two different monomer diffusion coefficients D1 = 1.05/102 MCS and D1 = 0.65/102 MCS and g = 4.9. The open circle for N = 16 is a linear interpolation between the results for H̅ = 0.5675 and H̅ = 0.625 (see Figure 5b). Filled squares show results for alternating block copolymers of B-block size nB = 2, for which an example sequence for N = 17 is given below.

to a prefactor of ≈3/2−3, if we use a blob size of g = 4.9 and the proper diffusion constants as described in the Appendix. Besides the uncertainty of the rate constant, one would expect the simulation result to be smaller than the prediction, because the potential profile in combination with the random sequence adds some roughness to the free energy landscape along the center of mass reaction coordinate, which effectively reduces the center of mass diffusivity locally.52 However, although eq 10 seems to describe the essential physics behind the observed rates for H̅ = 0.6 as a function of N, one has to keep in mind that the blob picture refers to a weak seggregation limit and ideal interfaces. Its applicability in the case of selectivities ≳ kBT, but rather smooth potentials (Figure 4), remains an open question. Furthermore, the measures taken into account within G


Article

Biomacromolecules

marginal perturbations of the bilayer originated by the size mismatch. Furthermore, much larger B-blocks, nB ≫ nt , are confined in the bilayer core, where they may even increase the hydrophobic barrier for hydrophilic permeants as observed earlier for hydrophobic homopolymers.7,8 Block sizes smaller than the lipid size destabilize the interface, since they are subject to larger thermal fluctuations with respect to the interface as compared to lipids due to their smaller polarization energy. To test the effect of block size as discussed above, we performed independent simulation runs for 200 random sequences for each chain length N ∈ {32,64,128} at the point of balanced hydrophobicity, H̅ = HL. For each individual polymer with random sequence, we estimated ΔPs and calculated the fraction of B monomers of the chain, which are part of a B-block smaller than the lipid tail size, nt. In Figure 10 we show the induced permeability ΔPs as a function of the

Figure 8. Simulation snapshot of a random copolymer (N = 256) inserted across the membrane and inducing a transient pore. Lipid heads (h) are dark gray, lipid tails (t) are light gray, A-type monomers are dark blue, B-type monomers are red, solvent monomers (s) are light blue. (a) and (b) show the same configuration with solvent (a) and without solvent (b).

Figure 9. Simulation results for the relative change of permeability Ps in the presence of a random copolymer (a) with H̅ = 0.6 and a homopolymer (b) with H = 0.68 as a function of the distance (x,y) from the polymer center of mass, which is projected into the bilayer midplane. Results are shown for chain length N = 128 and relative to the result without polymer, Ps,0. Simulation results for (b) are taken from the authors’ previous work.8

9b). Random copolymers have a disordered structure and randomly induce perturbations in the bilayer correlated with their monomer density projected on the membrane. Peak solvent permeability for random copolymers is found at its projected center of mass. In contrast, homopolymers with balanced hydrophobicity (H = 0.68) on the monomer level form globular structures, since the bilayer core as well as the solvent act as poor environments for the whole chain. The globule itself blocks translocation events close to its center of mass, while the bilayer is perturbed in the vicinity of the globule and a transient pore is formed around the object itself. This leads to a ring-like distribution of solvent translocation. 3.4. A Few Considerations about the Role of Short Amphilic Sequences. Returning to Figure 5e, let us note that simulation results for ΔPs are averaged over 30 to 40 random sequences, disregarding the fact that each polymer may contribute differently to ΔPs depending on its particular quenched randomness. In the following, we analyze the sequence-systematic contributions to the scattering of ΔPs. Here, we analyzed in particular the role of the size of consecutive blocks of one species in the sequence. We expect that the induced permeability of individual polymers should depend on the blockiness of the sequence. Polarization energy of an isolated pair of AB blocks of length 2n (assuming equal size, nA = nB = n, of the two blocks for the moment) is equivalent to their segregation strength, ∼χn. Therefore, with increasing size of A(B)-blocks, the polymer tends to stabilize the interface instead of perturbing it. In particular, if the blocks are larger than the typical size of a lipid, 2n ≫ nh + 2nt , they correspond to stronger surfactants than the lipids forming the interface. In this case we expect only

Figure 10. Relative permeability change ΔPs per 100 chain monomers induced by random copolymers of length N = 32 (•), N = 64 (■), and N = 128 (▼) as a function of the fraction of B-type monomers in blocks smaller than lipid tail size. Small filled symbols correspond to simulation results for ΔPs of individual polymers with a randomly drawn sequence. Large open symbols with error bars show averaged fractions of B in small blocks (abscissa) over all individuals that fall into bins with respect to ΔPs of width 0.08 in axis units. Boundaries of these bins correspond to horizontal grid lines. The bin centered around 0.12 is marked by a blue frame as an example. The error bars correspond to the error for the estimate of the expectation value, ± σ/ (nbin − 1)1/2, based on the standard deviation, σ, where nbin is the number of individuals in the bin. Small open circles label those individual polymers of N = 32, for which we detected translocation events of the whole chain (compare Figure 5b and Figure 7).

fraction of B-type monomers in small blocks for all individual test sequences as well as binned averages with respect to intervals of ΔPs. The results in Figure 10 show an increase of polymer-induced permeability with increasing fraction of small B-blocks for all chain lengths, indeed, as expected by the above argument. We conclude that blocks of the sequence smaller than the lipid size lead to significantly larger contributions per monomer to the signal observed in Figure 5e at H̅ ≈ HL than larger blocks. A particularly interesting observation considering chain length N = 32 in Figure 10 is that translocation of the polymer itself was almost exclusively possible for sequences containing only small blocks fulfilling nB < nt. In conjunction with Figure 7, our results qualitatively reproduce the observations by Corsi et H


Article

Biomacromolecules

permeability shows a weaker dependence, ΔP ∝ N2/3 7,8 related to the surface of a collapsed homopolymer-globule. The amphiphilic copolymers studied have a random sequence. We have observed a correlation between Ps and the content of small blocks of hydrophobic species. This leads to the hypothesis that alternating copolymers with blockiness on a scale smaller than the lipid size maximize polymer induced perturbations in the bilayer as well as polymer translocation. A closer look into polymer-induced permeability for solvent molecules as a function of AB-sequence reveals a possible strategy to design membrane active polymers for various purposes. For more efficient translocation of the polymer, the mean logarithmic partition coefficient of monomers comparing the aliphatic environment and solvent should be small (corresponding to H ≈ HL) and as homogeneously distributed as possible throughout the sequence. Simulation results suggest the balanced point to be found for HLB-values ( eq 2) below 10, which reflects the asymmetry with respect to water and lipid environments due to the self-assembled packing of lipids. To induce passive transport for small solutes, we propose amphiphilic multiblock copolymers with an approximate HLB value below 10 and larger than 5, where the lower limit, however, may vary with the mentioned asymmetry. Blockiness on smaller scales enhances fluctuations around the bilayer− solvent interface and polymer-induced perturbations, but also reduces the escape barrier for the polymer itself. We note that experimental evidence for RCP translocation mammalian plasma membranes in both directions was found for polyphospholipids such as rhoPMB305 as introduced by Ishihara et al.57 Our estimate based on the atomic weights of the constituents of rhoPMB30 results in an HLB value of ≈9, in agreement with the above proposal. The polymers used by Goda et al.5 can be considered as good test candidates for our coarse grained simulation results in the sense that their phospholipid building blocksalthough they are rather large are chemically compatible with the phospholipid bilayer as is the case for BFM monomers by construction. However, the flourescence signals found by Goda et al.5 indicated a lower specificity of passive cellular uptake with respect to the amphiphilic ratio (HLB values) than would be expected from our simulation results. Interesting future experiments would be an attempt to verify passive transport of polyphospholipids through model membranes. Furthermore, we suggest testing for translocation of alternating copolymers with short PEO/PPO building blocks. This would be particularly interesting in comparison with the often used poloxamers built of three larger blocks. For polaxamers, up to now, there is no evidence for passive translocation, which may be the result of the large block lengths used. The point of balanced solubility as well as interface polarization can be properly taken into account when designing transport systems by conjugation with a cargo. For direct penetration through the membrane, from our point of view, the perfect conjugate of a cargo compensates for two of its properties: interface polarization in terms of solubility, and oil/ water partition coefficients deviating from unity. Doxorubicin (DOX), for instance, is known to be lipophilic58 and to localize at bilayer−water interfaces. The transport rate by passive diffusion could be enhanced by covering the bare hydrocarbon atoms with some weakly hydrophilic conjugate to reduce the average hydrophobicity. This would, at the same time, reduce the polarity of the molecule with respect to the interfaces.

al.53 regarding the role of small block sizes for passive translocation of multiblock copolymers through liquid interfaces by using scaling theory as well as Monte Carlo simulations. Furthermore, Gkeka et al.25 demonstrated by using molecular dynamics simulations that also translocation of amphiphilic nanoparticles through membranes is facilitated with enhanced homogeneity of the distribution of amphiphilic sites. Clearly, patterns matching the scale of the selective environment show stronger localization within the environment than disordered patterns or patterns on a smaller scale. This applies to nanoparticles54 as well as to polymers with respect to lipid bilayer membranes or, for instance, adsorbed on randomly selective surfaces.55,56 The difference for polymers is that their flexibility allows them to self-organize in a way to match the environmental pattern by spatial redistribution of amphiphilic sites. Depending on the block sizes found in the polymer sequence, reorganization costs an extra entropic contribution for each block needed to provide the match. Therefore, smaller blocks exert larger fluctuations with respect to the external order. The point of self-reorganization was also addressed by bringing the flexibility of reorganization to nanoparticles by van Lehn et al.54 Our results in Figure 10 motivate a more systematic study of polymer translocation and permeabilization by amphiphilic polymers in relation to their specific sequence. A more detailed examination of the sequence dependence, however, is the subject of further investigations to be communicated in the future.

4. CONCLUSIONS The interaction of amphiphilic polymers with self-assembled lipid bilayer membranes has been studied using lattice-based Monte Carlo simulations under variation of the fraction H̅ of lipophilic monomers and degree of polymerization. In accordance with earlier simulation studies regarding homopolymers8 and nanoparticles,9 for RCPs we identify a point of balanced hydrophobicity, HL ≈ 0.6, where the partition of the polymer on both sides of the bilayer−solvent interface is symmetric. This point is further related to a maximum of the order parameter of interface localization, a maximum polymerinduced perturbation such as increased lipid flip-flop rates, and reduced orientational order of the lipids. In marked difference to homopolymers of balanced hydrophobicity amphiphilic copolymers display strongly adsorbing conformations based on formation of excess blobs. Thus, the interfaces of the lipid bilayer “polarize” the copolymer, which organizes in loops with an excess of hydrophilic or hydrophobic monomers, respectively. As a consequence, a hydrophobicity below H̅ < 1/2 is already sufficient to localize the chain close to the bilayer. Enhanced surface localization effects for amphiphilic polymers as compared to homopolymers lead to a complete inhibition of polymer translocation for longer chains. On the other hand, amphiphilic polymers increase permeability for small solutes more effectively as compared to homopolymers with balanced hydrophobicity for two reasons. First, surface localization increases the fraction of time in which the copolymer is located close to the membrane.7,8 Second, membrane activity in terms of ΔPs is expected to increase linearly with chain length, N, if the chain localized at the bilayer in terms of independent excess blobs. Our simulation results even indicate a superlinear relation for Ps(N) within the range of chain lengths chosen. In contrast, homopolymer-induced I


Article

Biomacromolecules

(8) Werner, M.; Sommer, J.-U.; Baulin, V. A. Soft Matter 2012, 8, 11714−11722. (9) Pogodin, S.; Werner, M.; Sommer, J.-U.; Baulin, V. A. ACS Nano 2012, 6, 10555−10561. (10) Sakai, N.; Matile, S. J. Am. Chem. Soc. 2003, 125, 14348−14356. (11) Sakai, N.; Futaki, S.; Matile, S. Soft Matter 2006, 2, 636−641. (12) Wheaten, S. A.; Ablan, F. D. O.; Spaller, B. L.; Trieu, J. M.; Almeida, P. F. J. Am. Chem. Soc. 2013, 135, 16517−16525. (13) Xia, W. J.; Onyuksel, H. Pharm. Res. 2000, 17, 612−618. (14) Lynch, A. L.; Slater, N. K. H. Cryobiology 2011, 63, 26−31. (15) Huin, C.; Gall, T. L.; Barteau, B.; Pitard, B.; Montier, T.; Lehn, P.; Cheradame, H.; Guégan, P. J. Gene Med. 2011, 13, 538−548. (16) Erukova, V. Y.; Krylova, O. O.; Antonenko, Y. N.; MelikNubarov, N. S. Biochim. Biophys. Acta, Biomembr. 2000, 1468, 73−86. (17) Demina, T.; Grozdova, I.; Krylova, O.; Zhirnov, A.; Istratov, V.; Frey, H.; Kautz, H.; Melik-Nubarov, N. Biochemistry 2005, 44, 4042− 4054. (18) Fan, W.; Wu, X.; Ding, B.; Gao, J.; Cai, Z.; Zhang, W.; Yin, D.; Wang, X.; Zhu, Q.; Liu, J.; Ding, X.; Gao, S. Int. J. Nanomed. 2012, 7, 1127−1138. (19) Valeeva, Y. K.; Dorodnykh, T. Y.; Alexandrova, N. A.; Zubin, E. M.; Kachatova, A. V.; Volkov, E. M.; Gottikh, M. B.; Melik-Nubarov, N. S. J. Drug Delivery Sci. Technol. 2006, 16, 245−251. (20) Vial, F.; Oukhaled, A. G.; Auvray, L.; Tribet, C. Soft Matter 2007, 3, 75−78. (21) Vial, F.; Cousin, F.; Bouteiller, L.; Tribet, C. Langmuir 2009, 25, 7506−7513. (22) Song, A.; Walker, S. G.; Parker, K. A.; Sampson, N. S. ACS Chem. Biol. 2011, 6, 590−599. (23) Orsi, M.; Haubertin, D. Y.; Sanderson, W. E.; Essex, J. W. J. Phys. Chem. B 2008, 112, 802−815. (24) Cardenas, A. E.; Jas, G. S.; DeLeon, K. Y.; Hegefeld, W. A.; Kuczera, K.; Elber, R. J. Phys. Chem. B 2012, 116, 2739−2750. (25) Gkeka, P.; Sarkisov, L.; Angelikopoulos, P. J. Phys. Chem. Lett. 2013, 4, 1907−1912. (26) Rocha, E. L. d.; Caramori, G. F.; Rambo, C. R. Phys. Chem. Chem. Phys. 2013, 15, 2282−2290. (27) Dasgupta, S.; Auth, T.; Gompper, G. Nano Lett. 2014, 14, 687− 693. (28) Lee, H.; Larson, R. G. J. Phys. Chem. B 2006, 110, 18204− 18211. (29) Egberts, E.; Berendsen, H. J. Chem. Phys. 1988, 89, 3718−3732. (30) Damodaran, K. V.; Merz, K. M.; Gaber, B. P. Biochemistry 1992, 31, 7656−7664. (31) Bassolino-Klimas, D.; Alper, H. E.; Stouch, T. R. Biochemistry 1993, 32, 12624−12637. (32) Müller, M.; Schick, M. J. Chem. Phys. 1996, 105, 8282−8292. (33) Cooke, I. R.; Kremer, K.; Deserno, M. Phys. Rev. E 2005, 72, 011506 (4p.). (34) Müller, M.; Katsov, K.; Schick, M. Phys. Rep. 2006, 434, 113− 176. (35) Nedelcu, S.; Werner, M.; Lang, M.; Sommer, J.-U. J. Comput. Phys. 2012, 231, 2811−2824. (36) Jentzsch, C.; Werner, M.; Sommer, J.-U. J. Chem. Phys. 2013, 138, 094902−094902−7. (37) Kłos, J. S.; Romeis, D.; Sommer, J.-U. J. Chem. Phys. 2010, 132, 024907 (7p.). (38) Gazuz, I.; Sommer, J.-U. Soft Matter 2014, 10, 7247. (39) Carmesin, I.; Kremer, K. Macromolecules 1988, 21, 2819−2823. (40) Deutsch, H. P.; Binder, K. J. Chem. Phys. 1991, 94, 2294−2304. (41) Tries, V.; Paul, W.; Baschnagel, J.; Binder, K. J. Chem. Phys. 1997, 106, 738−748. (42) Hoffmann, A.; Sommer, J.-U.; Blumen, A. J. Chem. Phys. 1997, 106, 6709−6721. (43) Griffin, W. C. J. Soc. Cosmet. Chem. 1954, 5, 249−256. (44) Jentzsch, C.; Sommer, J.-U. J. Chem. Phys. 2014, 141, 104908 (10p.). (45) Garel, T.; Huse, D. A.; Leibler, S.; Orland, H. Europhys. Lett. 1989, 8, 9−13.

We conclude by noting that polymer physics identifies an intrinsic relation between passive translocation of copolymers and the pore formation due to hydrophobic balance. Design of the sequence with respect to blockiness is already sufficient to split in “pore-formers” and “translocaters”. This could give insight into the evolution of membrane proteins in an earlier stage, which might have been controlled by disordered polypeptides and shows that elementary functions such as translocation and pore formation do not necessary require unique protein structures.

■

APPENDIX We use eq 10 to estimate the blob size by taking into account our simulation results (Figure 7) for two chain lengths, N1 = 24 and N2 = 48, in relation to the corresponding translocation rates T1 and T2: N2 − N1 2g = log[N1T1/(N2T2)] (11) For HL = 0.6, we obtain g ≈ 4.9. This result should not be interpreted in a way that the chain forms loops around the bilayer−solvent interface precisely of size g = 4.9, but translocation rates indicate a typical potential trap for the polymer close to the membrane with a depth of ∼ kBT per 10 monomers. Nevertheless, we compared the value of g with a direct estimation of loop sizes by counting the mean number of changes of the environment by following the chain contour (compare Figure 6). The latter procedure gives an estimate of g ≈ 3.5 close to the point H̅ = HL. Both estimates suggest a typical blob size g on the scale of a lipid tail or somewhat smaller. In order to determine the intrinsic time scale τ appearing in eq 10, we use the definition in eq 8, the two thresholds, z± = ± 20, and the monomer diffusion constants, D1, in poor solvent (ε = 0.4kBT), D1 = 0.65a2/(102 MCS), and in good solvent (ε = 0), D1 = 1.05a2/(102 MCS), as obtained in an earlier work.36

■

AUTHOR INFORMATION

Corresponding Authors †

E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the Leibniz-Society for financial support (SAWproject). The authors gratefully acknowledge technical support and computing time from the ZIH, Technical University Dresden. M.W. thanks Hauke Rabbel, Dirk Romeis, Ron Dockhorn, and Hsiao-Ping Hsu for fruitful conversations.

■

REFERENCES

(1) Tribet, C.; Vial, F. Soft Matter 2008, 4, 68−81. (2) Prochiantz, A. Adv. Drug Delivery Rev. 2008, 60, 448−451. (3) Marrink, S. J.; Berendsen, H. J. C. J. Phys. Chem. 1996, 100, 16729−16738. (4) Orsi, M.; Essex, J. W. Soft Matter 2010, 6, 3797−3808. (5) Goda, T.; Goto, Y.; Ishihara, K. Biomaterials 2010, 31, 2380− 2387. (6) Mathot, F.; Schanck, A.; Van Bambeke, F.; Ariën, A.; Noppe, M.; Brewster, M.; Préat, V. J. Controlled Release 2007, 120, 79−87. (7) Sommer, J.-U.; Werner, M.; Baulin, V. A. Europhys. Lett. 2012, 98, 18003 (8p.). J


Article

Biomacromolecules (46) Sommer, J.-U.; Daoud, M. Europhys. Lett. 1995, 32, 407−412. (47) Sommer, J.-U.; Peng, G.; Blumen, A. J. Chem. Phys. 1996, 105, 8376−8384. (48) Eisenriegler, E.; Kremer, K.; Binder, K. J. Chem. Phys. 1982, 77, 6296−6320. (49) Kłos, J. S.; Sommer, J.-U. J. Chem. Phys. 2008, 128, 164908− 164908 (8p.). (50) Sung, W.; Park, P. J. Phys. Rev. Lett. 1996, 77, 783−786. (51) Müller, T., Bachelor Thesis, Technische Universität Dresden, 2013. (52) Hänggi, P. Rev. Mod. Phys. 1990, 62, 251−341. (53) Corsi, A.; Milchev, A.; Rostiashvili, V. G.; Vilgis, T. A. Macromolecules 2006, 39, 7115−7124. (54) Lehn, R. C. V.; Alexander-Katz, A. Soft Matter 2011, 7, 11392− 11404. (55) Jayaraman, A.; Hall, C. K.; Genzer, J. J. Chem. Phys. 2005, 123, 124702 (11p.). (56) Jayaraman, A.; Hall, C. K.; Genzer, J. Phys. Rev. Lett. 2005, 94, 078103 (4p.). (57) Ishihara, K.; Iwasaki, Y.; Nakabayashi, N. Polym. J. (Singapore) 1999, 31, 1231−1236. (58) Harrigan, P. R.; Wong, K. F.; Redelmeier, T. E.; Wheeler, J. J.; Cullis, P. R. Biochim. Biophys. Acta, Biomembr. 1993, 1149, 329−338.

K


Translocation and Induced Permeability of Random Amphiphilic

Recommend Documents