Thermal Tunneling of Homopolymers through Amphiphilic Membranes

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Thermal Tunneling of Homopolymers through Amphiphilic Membranes Marco Werner,*,†,‡ Jasper Bathmann,§ Vladimir A. Baulin,† and Jens-Uwe Sommer‡,§ †

Universitat Rovira i Virgili, Departament d’Enginyeria Quı ́mica, Av. Paisos Catalans 26, 43007 Tarragona, Spain Leibniz-Institut für Polymerforschung Dresden e.V., Hohe Straße. 6, 01069 Dresden, Germany § Technische Universität Dresden, Institute of Theoretical Physics, Zellescher Weg 17, 01069 Dresden, Germany ‡

S Supporting Information *

ABSTRACT: We propose a theory to predict the passive translocation of flexible polymers through amphiphilic membranes. By using a generic model for the potential felt by a monomer across the membrane we calculate the free energy profile for homopolymers as a function of their hydrophobicity. Our model explains the translocation window and the translocation rates as a function of chain hydrophobicity in quantitative agreement with simulation results. The potential model leads to a new adsorption transition where chains switch from a one-sided bound adsorbed state into a bridging state through the membrane core by increasing the hydrophobicity beyond a critical value. We demonstrate that the hydrophobicity leading to the fastest translocation coincides with the solution for the critical point of adsorption in the limit of long chains.

S

dynamic Monte Carlo (MC) simulation results including all details of coarse-grained lipids and solvent.8,9 The passive transport of homogeneous small molecules through a membrane can be understood as a one-dimensional diffusion process. For nonionic molecules one finds a correlation between the hydrophobicity of the object and its permeability.12−14 On the other hand, with increasing hydrophobicity, the trapping within the membrane’s core may delay the passage through the membrane.15 As a consequence, mean first passage times through the membrane are expected to have a minimum near a point of hydrophobicity characterized by a vanishing free energy of transfer from solvent into a membrane’s core. Indeed, in coarse-grained molecular simulation results for nanoparticles16 as well as for polymers8,9,17 one finds a minimum mean first escape time close to this balanced point. Different degrees of adsorption at the membrane−solvent interfaces as well as additional degrees of freedom of a molecule18 however complicate the situation. The theoretical framework provided in this work takes into account both surface adsorption and conformational entropy of homopolymers on the basis of a proper abstraction of the monomer−membrane interactions. Let us consider the free energy of transfer of a single monomer from solvent into the core of the membrane. It shall be controlled by the selectivity, Δχ, of the monomer with respect to both environments. Upon insertion of the monomer, it enforces a small change of the self-assembled arrangement, which is expressed as an additional barrier, μins, against

elf-assembled bilayer membranes form the essential diffusion barrier between cells, cell compartments, and intercellular spaces. To surmount this barrier, nanoparticles and polymers are investigated as delivery vectors for small biochemical molecules, gene transfection, and cell reprogramming. Cell-penetrating peptides (CPPs), for instance, are natural polymer molecules that translocate through cell membranes1 and form part of transcription factors.2 Suggested mechanisms of CPP translocation include active endocytotic processes,3 cooperative effects such as specific self-assembly into pores4,5 and inverted micelles,1 and complexation of charged amphiphilic sequences with counterions.6 It was also demonstrated that flexible synthetic polymers7 translocate into mammalian cells in vitro without membrane disruption by a passive process in agreement with recent simulation studies.8,9 Using the analogy between quantum particle paths and single polymer conformations proposed by Edwards10,11 we predict the frequency of passive translocation of homopolymers through amphiphilic membranes induced by thermal energy only. In particular, we describe monomer−membrane interactions by means of a generic potential across the membrane and employ the Edwards equation,11 in order to access the free energy profile of a polymer. We show that polymer translocation is promoted close to a point of hydrophobicity, where the chain path has the highest probability to be found in a “tunneled” state, and a residual potential barrier is surmounted by thermal activation. At this point the polymer is compatible neither with solvent nor with the bilayer’s core and assumes a globular state where only a fraction of monomers feels the external potential. By taking into account this fact by a screening factor, we obtain excellent quantitative agreement when comparing predicted translocation times with earlier © XXXX American Chemical Society

Received: December 31, 2016 Accepted: February 16, 2017

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DOI: 10.1021/acsmacrolett.6b00980 ACS Macro Lett. 2017, 6, 247−251

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ACS Macro Letters

Figure 1(a) illustrates a potential model fitting to all of the above requirements. By using the Heaviside functions Θ(z) we compose the monomer potential as

monomer insertion independent from selectivity. For a fluid membrane we expect μins to be of the order of kBT,17 where kB is the Boltzmann constant and T is the absolute temperature. The total free energy of transfer then is Δχ + μins. The selectivity we formulate as Δχ = Z ϵ0(1 − 2H )

V (z ) = (Δχ + μins )Θ(h − |z|) + Γ αΔχ Θ(|z| − h)Θ(h + b − |z|)

(1)

(2)

On the lhs, we introduce a factor Γ in order to respect a conformation-dependent screening of the potential as described further below. The first term on the rhs represents the core of thickness 2h where the total free energy of transfer Δχ + μins is used. The second term in eq 2 implements two membrane− solvent interface regions of the size b representing the typical monomer size. The factor α denotes the fraction of monomer− membrane contacts at the interfaces that can be established without entering the core. We set this factor to α = 1/2 assuming that a monomer faces half of membrane contacts. We emphasize that this model represents the main features of small solute-free energy profiles both on coarse-grained scale17 and on atomistic scale.19 Specific interactions with lipid heads as observed in ref 19 are not included but can be taken into account if necessary to fine-tune the model. Polymers in the regime of interest, H ≳ 1/2, are compatible neither with the membrane core nor with solvent but found in a globular state.8,9 In this case, the fraction of monomers exposed to the external potential decreases with the increasing chain length, N, as ∝N−1/3 assuming an ideal spherical globule surface, and we set Γ = N−1/3. For the interesting range of hydrophobicity 0.6 ≤ H ≤ 0.7 polymer globules are far from Θconditions8,9,20 both in solvent and in the bilayer’s core, and the screened potential is too weak to break up the globule state. Nevertheless, a statistical description of polymer conformations is essential: a polymer globule does not have an ideal spherical surface but possesses shape and density fluctuations at its surface. Inspection of radial globule density distributions reveals that the total diameter of the diffuse surface layer can be of the order of the membrane thickness.20,21 Length scales are not sufficiently separated to treat the polymer either as an inert sphere or as a random coil only. We note further that the interface of the globule exposes sections of fluctuating strands that obey nearly Gaussian statistics. The typical length of such a strand is given by the size of the globule and thus is of the order of N2/3. In order to describe a polymer within the potential eq 2 statistically, we use the Edwards equation11

where Z denotes the mean number of contacts between a monomer and the surrounding molecules and ϵ0 is a repulsive energy per contact between monomers. We introduce a hydrophobicity parameter, H, in order to control the monomer selectivity in eq 1 between H = 0 (hydrophilic) and H = 1 (hydrophobic). It is important to note that for slightly hydrophobic monomers, H ≳ 1/2, the membrane−solvent interfaces become attractive since here the monomer may avoid solvent contacts without the penalty of μins for full entering into the core.

Figure 1. (a) Illustration for the monomer potential V(z) (eq 2) as a function of distance z from the bilayer’s midplane. The dashed circle shows the scale of a monomer size, and the thick line illustrates a polymer. (b) Sketches of the shape of V(z) at various values of hydrophobicity, H (eq 1).

Figure 2. (a) Free energy F(z) for chain length N = 64 as a function of the distance from the bilayer’s midplane. We compare results based on gC (eq S1) and (b) corresponding simulation results “MC”.8,9 Dashed lines show the position of bilayer−solvent interfaces. 248


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first arrival at point z+, after it started to diffuse from a wall located at z−. In Figure 3 we show the frequency of polymer translocation, TR ≡ 1/τ, on the basis of the inverse mean first escape time.

⎡∂ ⎤ b2 ∂ 2 + K G ( z , z ′ , s ) = 0 with K = − + β V (z ) ⎢⎣ ⎥⎦ ∂s 2 ∂z 2 (3)

to find the propagator G = G(z, z′, s) between two points z and z′ of a chain path of contour length s. In eq 3, b2 is the onedimensional mean-squared bond length, and β = 1/(kBT) is the inverse temperature. We emphasize that the screening constant Γ depends on the total chain length N only, and the potential V(z) is N-dependent but not contour length s-dependent. We thereby assume that the mean potential felt by each monomer is most significant for the partition sum. Integration of eq 3 with respect to s and the spectral theorem allows for writing the solution as a series ∞

G (z , z′ , s ) =

∑ e−λ sϕk(z)ϕk(z′) k

k=0

(4)

with eigenfunctions ϕk and eigenvalues λk fulfilling Kϕk = λkϕk. Due to Γ = Γ(N) in eq 2 the eigenvalue problem has to be solved for each chain length independently, and intrachain correlations described by G(z, z′, s) become a function of N. We solve the eigenvalue problem in a discretized form22 on an interval z ∈ [−L, L], where we denote Δz as the grid size and chose Δz = b/4 with b as described below. The propagator G(z, z′, s) holds all essential information for calculating the probability distribution of the center of mass, gC(z) (see Supporting Information). The free energy landscape of the polymer is given by F(z) = −kBT log[gC(z)/g0]

Figure 3. Predicted frequency of translocations, TR = 1/τ (eq 6), as functions of homopolymer hydrophobicity H for various chain lengths N (continuous lines, labeled by approximate maximum levels) in comparison to simulation results (points) given as the number of translocation events per 107MC steps.8,9 For the mapping of time scales we use a diffusion constant of D = 1/(N × 102MCS) in eq 6 as obtained earlier for single chains in explicit solvent20 and apply a factor of 0.58 for TR in order to overlay the graphs. Error bars reflect the standard deviation assuming that events follow the Poisson statistics. We indicated the points of balanced hydrophobicity, H0 (eq 7), and critical adsorption threshold, Hc (eq 9), for N → ∞.

There, we also include simulation results on the number of polymer translocations detected during MC simulations.8,9 Excellent agreement is found for all chain lengths using the diffusion constant D ∝ 1/N of the chain20 and rescaling the time by a factor of order unity. The results demonstrate further that introduction of Γ ∝ N−1/3 is sufficient to reproduce the chain-length dependence as determined by the globular conformations. In MC simulations we have shown a correlation between a maximum of polymer translocation frequency and the value of H, where the polymer is at the verge of being adsorbed at the membrane.8 In order to understand this result, we have to recognize that interface adsorption partially suppresses polymer translocation, and therefore the minimum escape time is found not at the compensation point, Δχ + μins = 0

(5)

where we normalize gC(z) by its bulk value, g0. Figure 2 shows the polymer free energy profiles for various values of hydrophobicity in comparison with MC simulation results for N = 64.8,9 For both simulation results and Edwards model we see the lipid bilayer acting either as a potential barrier (smallest values of H) or as a potential trap (largest values of H) (see Figure 1(b)). In order to quantitatively compare the analytical model to MC simulation results, we set ϵ0 = 0.8kBT as defined in the MC model,8,9 while b = 1.49 and h = 6.9 are based on simulation results (Table S1). The remaining parameters, Z = 4.1 and μins = 1.17kBT, have been gauged such that the approximate values of the free energy of transfer, ΔF = F(0)−F(z → ∞), for H = 0.62 and H = 0.74 in Figure 2(b) match with the theoretical lines (Figure 2(a)). When comparing Figures 2(a) and (b) we see that beside the gauge of Z and μins no curve fitting is needed for a good agreement with molecular simulation results. Despite the idealized shape of the potential (Figure 1(a)) both Z and μins show a remarkably good consistency with the corresponding MC results (Table S1). Note that all results in this work with various values of hydrophobicity and chain length are represented with the same set of parameters sumarized in Table S1 resulting in the set of curves presented in Figure 2. The free energy profile (eq 5) forms the basis to calculate the mean first escape time23 τ=

1 D

∫z

z+ −

dz e F(z)/(kBT )

∫z

z −

dz′e−F(z′)/(kBT )

H0 = 1/2 + μins /(2Z ϵ0) ≈ 0.6784

(7)

but at slightly lower values of H, where the interface adsorption is partially compensated by a slight repulsion from the membrane’s core (Figure 1). Now the question remains, is the maximum translocation frequency still found exactly at the adsorption threshold? In Figure 4 we compare the points, Hm, of minimum escape time τ (maximum TR) to the critical adsorption threshold, Hc, in the limit of long polymers, Nb2 ≫ h2. We define the critical adsorption threshold, Hc, as the point where a bound ground state (eigenvalue λ0 < 0) appears first with increasing hydrophobicity. When using a more abstract version of the potential V(z) (eq 2), where the surface term is replaced by a δdistribution of the same integral strength

(6)

from the interval [z−,z+], where z− and z+ are reflecting and absorbing boundaries, and D is the polymer’s diffusion constant. The time τ is needed on average for the polymer’s

αΔχ Θ(|z| − h)Θ(h + b − |z|) → bαΔχδ(|z| − h)

(8)

24

the critical adsorption threshold, Hc, is determined by 249


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the membrane surface, hydrophobic attraction, and the free energy effort for insertion of monomers into the membrane core. We showed that it is possible to take into account the globular state via a screening factor for the potential strength. This leads to good quantitative agreement even for polymer translocation time with simulation results. In the limit of long chains the mean first escape time through the membrane becomes minimal, if the polymer hydrophobicity is slightly below the compensation point of balanced hydrophobicity, where the membrane core would be energetically transparent for chain monomers (Figure 1(b), center). Here, the remaining repulsion barrier of the core together with the two potential traps at the membranes surface define a novel adsorption problem. Above a critical hydrophobicity, Hc, long chains are dominated by conformational states bridging the membrane, while below Hc conformations are dominated by a half-space restriction and adsorption at one surface of the membrane. For even lower hydrophobicities only nonbound states persist. We demonstrate that the point of minimum escape time, Hm, is found above the adsorption transition, Hc, for small chain lengths. The upper bound of Hm is set by the point, Ht, where the probability to observe both chain ends on opposite sides becomes extremal. All three characteristic values, Hc, Hm, and Ht, coincide in the limit N → ∞. Our results open a perspective to the systematic design of membrane-active polymers with respect to their passive transport through membranes, where both the monomer hydrophobicity as well as the interface adsorption properties are subject to optimization. Promising candidates of translocating polymers in practice are short-block amphiphilic copolymers, which in the limit of small block sizes resemble homopolymers on a coarse-grained level.

Figure 4. Critical hydrophobicity Hc defined by eq 9 (continuous line), points Hm of minimum escape time τ (eq 6), and points Ht of maximum “tunneling” probability (eq 10) as functions of inverse chain length, 1/N. The balanced point H0 is labeled according to eq 7.

V1 +

V 0 tanh( V 0 h) = 0

(9)

with V0 = Γ(Δχ(H) + μins) and V1 = ΓbαΔχ(H). In Figure 4, the corresponding critical hydrophobicity Hc is shown, which is a function of chain length due to the Ndependent factor Γ. Since the ground state dominates the propagator in eq 4 for N → ∞, the precise catching up of Hc with the minimum escape time positions, Hm, demonstrates that minimum escape times are found at the adsorption transition point in this limit. Curves and data points in Figure 4 eventually meet within an interval H = 0.661 ± 0.002. This value is close to the projected critical adsorption threshold, H = 0.667 ± 0.005, based on the corresponding MC simulations.8,9 Vice versa, the symmetry of the ground state, ϕ0(z) = ϕ0(−z), gets increasingly pronounced in the propagator G(z, z′, N) close to the critical value H = Hc in the limit of long chains. This can be seen by identifying the values of hydrophobicity, where the propagator G(z, z′, N) becomes maximally symmetric with respect to the membrane’s midplane. In particular, we calculate the probability − h − 2b

pt =

∫−L

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* Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00980. Calculation of the center of mass distribution gC(z) (Section SA) as well as a summary of parameters used (Section SB, Table S1) (PDF)

dz′G(h + 2b , z′, N )

■

+L

∫−L dz′G(h + 2b , z′, N )

ASSOCIATED CONTENT

S

(10)

AUTHOR INFORMATION

Corresponding Author

to observe the second chain end on the opposite solvent side if the first chain end is fixed near the interface at z = h + 2b. In Figure 4 we show the points of hydrophobicity, Ht, where pt has its maximum with respect to H characterizing a maximum bridging probability through the membrane. According to Figure 4, Ht approaches the point of critical adsorption threshold in the limit N → ∞. Hence, in this limit the translocation time of the chain as a whole becomes minimal, when the propagator between the two chain ends maximally promotes a tunneled state of the chain contour between the two solvents sides. For short chains, however, minimum translocation times are found between the adsorption transition and a point of maximum symmetry, Hc < Hm < Ht. At the lower bound, Hc, a residual barrier in the membrane’s core suppresses translocation, while at the upper bound, Ht, interface adsorption hinders the polymer from escaping the membrane. To summarize, we demonstrate that homopolymer translocation and adsorption at self-assembled amphiphilic membranes can be understood quantitatively by solving the Edwards equation in a potential which takes into account adsorption at

*E-mail: [email protected]. ORCID

Marco Werner: 0000-0001-5433-8443 Jens-Uwe Sommer: 0000-0001-8239-3570 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Toni Müller and Hauke Rabbel for fruitful discussions. MW, VAB, and JUS gratefully acknowledge the European Union’s funding of the Initial Training Network SNAL (grant agreement no. 608184) under the 7th Framework Programme.

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Thermal Tunneling of Homopolymers through Amphiphilic Membranes

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