Free Energy of Translocating an Arginine-Rich Cell-Penetrating Peptide across a Lipid Bilayer Suggests Pore Formation

Abstract

The molecular mechanism and energetics of the translocation of arginine-rich, cell-penetrating peptides through membranes are still under debate. One possible mechanism involves the formation of a water pore in the membrane such that the hydrophilic residues of the peptide are solvated throughout the translocating process. In this work, employing two different order parameters, we calculate the free energies of translocating a cyclic Arg₉ peptide into a lipid bilayer along one path that involves a water-pore formation and another path that does not form a separate pore. The free-energy barrier of translocating the peptide along a pore path is 80 kJ/mol lower than along a pore-free path. This suggests that the peptide translocation is more likely associated with a water-pore formation.

Introduction

Cell-penetrating peptides (CPPs) are short sequences of amino acids (<30) that are capable of introducing cargoes into living cells. The cargoes carried with them vary over a wide range of molecular sizes, such as proteins, oligonucleotides, and even 200-nm-large liposomes. Most CPPs are noncytotoxic and penetrate into cells rapidly. These properties also make them promising drug-delivery agents (1–3). Although >100 CPPs have been identified, their translocating mechanisms have remained elusive. Conflicting results have been reported. Some experiments seem to indicate an energy-dependent or endocytotic pathway, and others an energy-independent pathway (4–8). There is also a possibility that, considering the physico-chemical diversity of the known CPPs, different mechanisms of the internalization occur for different CPPs or CPP with different cargoes (9–12).

Arginine-rich peptides are a subclass of CPPs that penetrate cells efficiently. There is evidence that the uptake is free of endocytosis and may be energy-independent (8,13). Even when the uptake initially follows an endocytotic pathway, CPPs are still able to breach the endosome membrane to reach the cell nucleus (14). Furthermore, experiments have shown that arginine-rich CPPs can penetrate across artificial membranes such as giant unilamellar vesicles (GUVs) (15–17) and black lipid bilayers (18) by direct peptide-lipid interactions. These membrane mimics are free of any energetic, receptor-driven pathways such as endocytosis. Various models have been proposed to explain the translocating mechanism of CPPs; however, a central question that has to be answered by any candidate mechanism is how the highly hydrophilic peptides are able to cross the hydrophobic barrier imposed by the cell or endocytotic membrane. A water-pore-assisted translocation mechanism has been suggested by our group recently, showing that these peptides may be able to nucleate a pore in membrane and diffuse across the membrane along the pore (19). It highlights the important interaction between arginine and lipid phosphates that distorts the membrane structure and initiates the pore formation, which is supported by recent experiments (16,18,20–22).

It is essential to understand the water-pore model from a thermodynamic perspective. In this work, we calculate the free energy of translocating a cyclic Arg₉ (cR₉) from bulk water into a lipid bilayer using molecular-dynamics umbrella sampling simulations. We choose the cR₉ peptide for the calculations because, recently, it has been reported that cyclic CPPs penetrate the cell membrane more efficiently than their linear counterparts (23). The free energy is usually described as a function of an order parameter (OP) which describes the position of the peptide relative to the bilayer center. Conventionally, umbrella sampling restraints are applied along the same OP; however, we find that this is problematic for our system, which can be illustrated with the aid of Fig. 1. In Fig. 1, the reaction coordinate presents the distance between the peptide and bilayer center. The orthogonal degrees of freedom (DOFs) might have multiple basins and the free-energy barriers prevent fast hopping between basins in simulation timescale (usually nanosecond). Those orthogonal slow-relaxing DOFs (sDOFs) could be associated with peptide rotation and bilayer deformation (24). When we perform umbrella sampling using initial configurations generated through successively pulling, the calculated free-energy profile is actually a local profile along a certain path. One way to solve this problem is by employing multidimensional umbrella samplings and applying umbrella potential along each sDOF. This is barely possible for our system because many sDOFs are unknown a priori.

Figure 1.

Schematic illustration of free-energy sampling with barriers in orthogonal degrees of freedom (DOFs). The reaction coordinate represents the distance between the peptide and bilayer center. S is the starting state, and A, B, and C are the possible paths for the translocation. The free-energy sampling will very likely be trapped in a local path because of the barriers in orthogonal DOFs.

In this work, we take a different approach. Understanding the difficulty in obtaining the full free-energy profile of translocation, we aim to compare the free-energy profiles along different paths. By running umbrella simulations employing two different OPs, we constructed the free-energy profiles of peptide translocation along a path involving a water-pore formation and a path which does not separately form a pore. We found the free-energy barrier along a path involving a water pore is 80 kJ/mol lower than along a pore-free path. Because the energy-independent mechanism for CPP translocation has been shown experimentally (16,18,22), our work indicates that the energy-independent translocation should be accompanied by the formation of a water pore, as suggested by previous studies (19). Here we provide the free-energy estimates for the process in neutral 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) lipid bilayers.

Materials and Methods

System setup

We use umbrella simulations to study the free energy of transferring a cR₉ into DOPC bilayer under different peptide concentrations. Systems with three different peptide/lipid ratios (P/L), 1:128, 1:32, and 1:14, were constructed. cR₉s were initially placed near one side of the bilayer. Because of the periodic boundary condition, peptides could bind either to the proximal or the image of the distal bilayer leaflet. The interaction between the peptides and the periodic image of the distal leaflet was reduced by solvating the system with a large amount of water (water/lipid > 80). For systems with P/L of 1:32, a base system with 1 peptide, 32 lipids is constructed initially. The base system was equilibrated and then duplicated four times on a 2×2 grid perpendicular to the bilayer-normal. This results in a maximal lateral separation between peptides. Systems with P/L of 1:14 were built similarly. Cl⁻ ions are added to explicitly satisfy electroneutrality. Excess salt was not added to the system for reason of simplicity, because cations might compete with the arginines for zwitterionic lipids carboxyl groups and result in longer equilibration times (25,26). We assume that the arginine side chains remain charged, which is consistent with a recent potential-of-mean-force calculation by MacCallum et al. (27) and observations in protein systems (28). We define the Z axis as the bilayer-normal and refer the values of Z > 0 nm and Z < 0 nm as upper and lower monolayers. The peptide was transferred along the Z direction through the bilayer. More details are provided in the Supporting Material.

Umbrella sampling calculations

Five sets of umbrella sampling simulations were performed to determine the free energy to transfer an Arg in cR₉ into the DOPC bilayer along different paths with increasing peptide concentration. We use the Cz atom of the Arg as reference. Cz atom is the fourth carbon in the Arg side chain, which is the center of charge of the guanidinium group. Because every Arg in the cR₉ is identical, it does not matter which Arg (and Cz atom) we choose as reference. Details of five simulations, labeled A–E, are given in Table 1. In Simulation A, the vertical distance (in Z direction) between the restrained Cz atom and the center-of-mass (COM) of lipid bilayer was used as the order parameter to restrain the system. In simulations B–E, a cylinder geometry, implemented in GROMACS, was used to define the order parameter (OP). We refer to the two OPs as the position and cylinder OPs, respectively. A detailed description of the cylinder OP is in the Supporting Material.

Table 1.

Systems studied by molecular-dynamics simulations

Simulation	Lipid	cR9	Water	Ions, Cl−	Time, ns	Windows	OP, range	Detail
A	128	1	11,182	9	80–240	38∗	P: Cz-Com,† 0–3.6 nm	No pore
B	128	1	11,182	9	80–240	38	C: Cz-P8s, 0–3.7 nm	Pore
C	128	1	11,182	9	160	15	C: Cz-P8s, 1.2 nm‡	Different radius
D	128	4	12,112	36	80–160	31	C: Cz-P8s, 0–3.0 nm	4 cR9s
E	126	9	12,600	81	90	20	C: Cz-P8s, 1.0–2.9 nm	9 cR9s

^∗One more window was added at a restrained distance of 1.15 nm to ensure enough histogram overlapping between windows restrained at 1.1 and 1.2 nm.

^†P/C: A–B stands for position/cylinder distance between groups A and B.

^‡Each window was restrained at cylinder distance of 1.2 nm, with r₁ changing from 0.5 to 1.9 nm.

In simulation A, the peptide translocates along a path in absence of a water pore. Simulations B and C combined translocate the peptide along a path involving a water pore. Simulations D and E translocate a peptide along a water-pore path under different peptide concentrations. For simulations D and E, though multiple peptides are bound to one bilayer leaflet, the umbrella restraint is only applied to one Arg of one peptide. All other peptides are unconstrained. Different arrangements of the free peptides were not sampled. When the cylinder OP is used, the outer radius r₀ was always chosen 0.5-nm larger than the inner radius r₁. The region between the two radii serves as a buffered zone to avoid abrupt changes in the OP. In simulations B, D, and E, r₁ was chosen as 0.5 nm and the cylinder OP changed among umbrella windows. In simulation C, r₁ increased from 0.5 nm to 1.9 nm among umbrella windows but we fixed the cylinder OP at 1.2 nm for every window.

Unless otherwise specified, each umbrella window was separated by 0.1 nm in the Z direction, with a harmonic bias potential with spring constant 3000 kJ/mol/nm². Initial configurations in each umbrella window for every set of simulations were generated from corresponding pulling simulations with a force constant of 3000 kJ/mol/nm² at a rate of 0.1 nm/ns.

Free-energy calculations

The weighted histogram analysis method (WHAM) was used to remove biases in the umbrella sampling data (29). Following Roux’s work (30), free energies parameters {f_i} of each umbrella window can be solved from

$$e^{-\beta f_k}=\int d\eta \sum _{i=1}^N\frac{n_i{e}^{-\beta W_k\left(\eta \right)}}{\sum _{j=1}^N{n}_j{e}^{-\beta \left[W_j\left(\eta \right)-f_j\right]}}{\rho }_i^{\left(b\right)}\left(\eta \right),$$ (1)

where N is the number of umbrella windows, the value η is the OP, and the values n_i, W_i(η), and ρ_i^(b)(η) are the number of sampling, biased-potential, and biased-probability-distribution functions of the $i^{th}$ window. Because {f_i} is involved in both sides of Eq. 1, it has to be solved iteratively. All {f_i} values were initialized to 0 and plugged into the right-hand side of Eq. 1 to calculate a new set of {f_i} values from the left-hand side of the equation. This was done iteratively until the deviation of each f_i value between the old and new sets is smaller than ${10}^{-3}$ kJ/mol.

Once the f_i values are known, it is straightforward to assign a correct probability to every sampled configuration obtained from umbrella simulations, using

$${\rho }_0\left(\mathbf{R}\right)=\sum _{i=1}^N\sum _{l=1}^{n_i}\frac{\delta \left(\mathbf{R}-{\mathbf{R}}_{i,l}\right)}{\sum _{j=1}^N{n}_j{e}^{-\beta \left[W_j\left(\mathbf{R}\right)-f_j\right]}},$$ (2)

where ρ₀(R) is the unbiased probability of configuration R obtained from umbrella simulations. The free energy can then be projected along any OP using

$$W\left(\xi \right)=-k_{\text{B}}T\text{ln}\int d\mathbf{R}\delta \left({\xi }^{\prime }\left(\mathbf{R}\right)-\xi \right){\rho }_0\left(\mathbf{R}\right),$$ (3)

where ξ could be an OP different from η along which the umbrella potentials apply.

The above method works well for simulations A, B, D, and E; however, for simulation C, a second version of WHAM was used to obtain {f_i}. This version was also described in Souaille and Roux’s work (30) as

$$e^{-\beta f_k}=\sum _{i=1}^N\sum _{l=1}^{n_i}\frac{e^{-\beta W_k\left({\mathbf{R}}_{i,l}\right)}}{\sum _{j=1}^N{n}_j{e}^{-\beta \left[W_j\left({\mathbf{R}}_{i,l}\right)-f_j\right]}},$$ (4)

where R_i,l is the l^th configuration in the i^th umbrella window. This was also solved iteratively until the deviation of each f_i between the old and new set is smaller than 10⁻³ kJ/mol. We adopted this version because the cylinder radii are different for each umbrella in simulation C. We modified an existing WHAM code to suit the above calculations (31).

The standard errors were estimated from the potentials of mean force (PMFs) calculated from different time blocks in the production run. In each umbrella window, the value of OP was monitored as a function of time and the production run was defined when the OP distribution ceased drifting. Details of the production runs and block size for each simulation are provided in the Supporting Material. In all calculations, we ensure that the block size is significantly larger than the correlation time in each umbrella window.

Results

Translocation along a pore-free path

Simulation A was conducted by applying umbrella potentials along the position OP. An Arg (Cz) of cR₉ was transferred from water into the bilayer center. The corresponding PMF is shown as a blue line in Fig. 2 A. We shifted the PMF minimum to 0. The PMF decreases in the region between 2.5 nm and 3.5 nm due to the favorable binding of the peptide to the bilayer amphiphilic region. It then increases continuously to ∼200 kJ/mol because of the cost of dehydrating charged residues in bilayer hydrophobic core. Fig. 2 E is a snapshot of the starting configuration of the system when the peptide bound to the bilayer. Fig. 2 B shows the restrained Arg at the bilayer center. During the whole simulation process, no water pore was formed spontaneously, and a water defect on one side of the bilayer was formed.

Figure 2.

Free-energy profile (or potential of mean force, PMF) to transfer an Arg of cR₉ from water into the center of DOPC bilayer. (A) PMF as a function of the distance between the Arg and the center-of-mass (COM) of bilayer along z axis. (Blue line) PMF calculated from simulation A associated with a translocation path without a water pore. (Red and green lines) Calculated from simulations B and C, respectively, associated with a translocation path involving a water pore. (B–E) Snapshots of the system at different translocation states. (B and C) Arginine at the center of the bilayer with a water pore absent and present, correspondingly. (D) Arginine at the depth at which the water pore started to form. (E) The starting configuration of simulation A where the cR₉ bound to the bilayer. The free-energy cost calculated along a path with a water pore is significantly lower than along a path without a water pore, suggesting that the peptide translocation involving a water pore is energetically more favorable.

Translocation along a pore path

Previous studies have suggested a mechanism of translocating CPPs through a lipid bilayer involving the formation of a water pore (18,19). However, in simulation A, we did not observe the spontaneous formation of a water pore. We reason that the pore-forming DOF might belong to the slow-relaxing DOFs and there is a activation energy required for the process. Thus, we use a different OP to direct the system to a translocating path involving a pore.

In simulation B, we apply biased potentials on the cylinder OP to force the bilayer to form a water pore. Fig. 3 A shows the PMF of transferring an Arginine from the binding position into the bilayer along the cylinder OP. The cylinder OP restraint restricted the distance between pulled Arg and the phosphate atoms contained in the lower leaflets, within the cylinder boundary. Shortening the distance resulted in pulling both the Arg and phosphate atoms within the cylinder boundary inside the bilayer, therefore creating a pore.

Figure 3.

(A–C) Free-energy profile of transferring an Arg of cR₉ from a binding position into the bilayer through umbrella sampling using the cylinder order parameter (cylinder OP). The total number of peptide(s) in the system are one, four, and nine, respectively, which corresponds to simulations B, D, and E. The z axis represents the cylinder distance between the restrained arginine and the phosphates in the distal monolayer. (Curves in different color in each plot) PMFs calculated from different time blocks. Block sizes of simulation A varies from 20 ns to 40 ns, depending on the correlation time of each window. Simulations B and C use 20 ns as block size for every window. The kink in each free-energy profile (pointed by an arrow) indicates the position where a water pore is formed.

The kink in the PMF shown in Fig. 3 A marks the starting position at which a water pore was formed. A configuration chosen from the umbrella window constrained at the kink position is shown in Fig. 4 A (with water) and Fig. 4 B (without water). In Fig. 4 B, we observe that a lipid could flip deep inside the bilayer to interact with the peptide. When a lipid from the lower leaflet flipped, a water defect extended from the lower leaflet began to touch the upper-layer water defect, mimicking the rudiments of a water pore. The discontinuity of the free-energy profile at 1.8 nm in Fig. 3 A marks the intersection between the two free-energy surfaces. After a water pore was formed, transferring the Arginine further into the bilayer was no longer the same as transferring through the bilayer hydrophobic core. We observe that the PMF flattened in the region between 1.5 nm and 1.8 nm.

Figure 4.

Snapshots of the peptide at different stages along a translocation path involving a water pore. (A) The restrained arginine is at the position where a water pore starts to form. (C) The restrained arginine is at the center of the bilayer. Snapshot B is the same as A, and snapshot D is the same as C, but without displaying the water molecules. Phosphates (gold and orange) from the upper and lower leaflets, respectively. (Yellow circles in B and D) Restrained arginines. In snapshot D, an Arg side chain not restrained by the umbrella potential passively diffused along the pore, indicating that transferring an arginine in a hydrated environment requires little energy.

To compare with the PMF calculated from simulation A, we projected the data in simulation B onto the position OP. Equation 2 is used to assign an unbiased probability to each datum in simulation B and Eq. 3 is used construct a PMF along the position OP. The projected PMF is plotted as the red line in Fig. 2 A. A water pore formed when the restrained Arg was 0.5 nm away from the bilayer center. The projected PMF with position OP < 0.5 nm is not shown in Fig. 2 A, because we find that simulation C describes a better way (requires less energy) to insert the peptide after a water pore is formed. Because in simulation B, the initial position of the restrained Arg was 1.6 nm away from the bilayer center, we aligned the starting point of the red line in Fig. 2 A with the value of blue line at the same depth.

We hypothesized that the energy cost to transfer the Arg further into the bilayer center through a hydrated environment would be small. Simulation C transfers the restrained Arg for the extra 0.5 nm after the pore was formed. In simulation C, the cylinder OP was fixed at 1.2 nm (pore formed at such distance in simulation B) for every umbrella window, but the radius (r₁) of the cylinder was gradually increased from 0.5 nm to 1.9 nm, at a 0.1-nm interval. The value r₀ increases accordingly because r₀ is always set 0.5 nm larger than r₁. The initial configuration for the umbrella window with r₁ = 0.6 nm in simulation C was taken from simulation B (r₁ = 0.5 nm) with the cylinder distance restrained at 1.2 nm. This configuration was then equilibrated for 10 ns and the last configuration was used as the initial configuration for the neighboring window with a larger radius. Initial configurations in other windows were generated in a similar fashion. When we increased the radius of the cylinder, while maintaining the cylinder OP distance, the system could respond in two ways. It could either pull more phosphates from the lower monolayer inside the bilayer or it could insert the restrained Arg more deeply. The simulation shows that the size of the water pore did not increase proportionally to the cylinder radius but instead the Arg was transferred further across the bilayer. This suggests that the second way to maintain the restraint helps sample lower energy configurations that satisfy the intended restraint.

Applying Eqs. 2–4, we could unbias the data in simulation C and project the PMF along the position OP which is shown as the green curve in Fig. 2. The green curve increases at a significantly slower rate than the blue curve and shows that transferring the Arg in a hydrated environment requires much less energy. The total free-energy barrier to transfer an Arg in cR₉ into the center of the bilayer, involving a water pore, is ∼120 kJ/mol, which is 80 kJ/mol lower than the barrier in absence of a water pore. This indicates that although forming a water pore in the bilayer costs energy, this cost will finally be overcome by the energy required to dehydrate the charged arginines of the peptide in the bilayer.

A snapshot of the system with the restrained Arg lying at the center of the bilayer is shown in Fig. 4 C (with water) and Fig. 4 D (without water). From Fig. 4 D, we observe that there are actually two arginines at the bilayer center, though only one is restrained. This is an interesting result indicating that after the water pore forms, the charged arginines could diffuse across the bilayer without much energetic cost, which supports our initial hypothesis and agrees with a recent work by MacCallum et al. (32).

To test the stability of the water pore, a configuration similar to Fig. 4 D was chosen and simulated for 250 ns without any restraints. The water defect persisted through the whole simulation. The amount of water inside the membrane fluctuated but the water pore never closed. We observe that as long as there are phosphate groups coordinated with the Arg-residue side chains, water could easily penetrate inside the bilayers. We hypothesize that the peptide could further diffuse along the pore and complete the penetration. This, however, was not observed in this work, as the process timescale might exceed our computational capability.

Effect of different peptide concentrations

The effect of peptide concentrations on the PMF of translocating an Arg of cR₉ is studied. Simulations B–E were simulated for system with one, four, and nine peptides, respectively, which corresponds to 1:128, 1:32, and 1:14 P/L. The cylinder OP was used because previous sections have shown that the translocation path involving a water pore is energetically more favorable. PMFs along the cylinder OP are shown in Fig. 3. In each subplot, the kink indicates the depth at which a water pore is formed. The free energy to form a pore rises from ∼67 kJ/mol to ∼90 kJ/mol with increasing peptide concentration. In each concentration, PMFs calculated using different time blocks are displayed, and the overlap between them suggests the convergence of free-energy calculation.

Discussion

PMF calculations

In this work, we use a nonconventional way to calculate the free energy to translocate a cR₉ into lipid bilayers. Understanding the complexity of the system for free-energy calculation (due to slow-relaxing orthogonal DOFs), we aim to compare free-energy profiles along different paths rather than calculating the full free-energy profile. The local equilibrium at points along a path can be achieved within hundred-nanosecond simulations and the corresponding PMFs have small statistical error. If there are multiple paths for the translocation (as illustrated in Fig. 1) and assuming similar diffusion coefficient along the paths, we argue that the path that has the lowest barrier should dominate the translocation. The means that the free energy calculated along a single path should serve as an upper boundary of the true free energy. Nevertheless, by comparing the free energy along paths with and without a pore, we can claim that the peptide translocation will most likely involve the formation of a water pore.

From a thermodynamic point of view, the PMF calculated from simulation A (without a water pore) does not correspond to an equilibrated low free-energy path. Although it has a small statistical error, it has a large systematic error. Eventually a water pore should form in simulation A when the arginine is restrained in the bilayer center. However, the free-energy barrier in the orthogonal DOFs prevents it from happening in the hundred-nanosecond simulation timescale. The quantitative value of the barrier in the orthogonal DOFs is not known from this work.

Our choice to use a single arginine in cR₉ as OP seems unconventional, compared to the COM of the peptide. Admittedly, when the restrained arginine is in the bilayer center, the COM of the peptide is still above the bilayer center. If we identify the transition state as the peptide’s COM in the bilayer center, transferring an arginine into the bilayer center is only part of the translocation process. Thus, the barrier for translocating an arginine should be no higher than the barrier for translocating the COM. However, considering further transferring the COM of the peptide into the bilayer center along a pore-free path should cost more than along a pore path. Therefore, we are confident that our argument that the free-energy barrier along a pore path is lower than a pore-free path will still hold even if we use the peptide’s COM as OP. The choice of a single arginine as OP may introduce a degeneracy of the peptide position with the whole peptide attached to the distal leaflet while keeping the Arg position the same. However, such rotation is highly unlikely and therefore unnecessary to sample.

Surprisingly, the free energy increases when the peptide concentration increases, which seems to be in contrast with experiments where increasing peptide concentration increases the CPPs’ uptake. This could be due to several effects. The most likely effect is the small size of our system. To form pores, the bilayers should bend inwards. The smaller the system, the more the curvature of the bilayer is required to bend. When the number of peptides attached to the bilayers increases, they tend to flatten the bilayer because they are positively charged and repel each other. As a result, this increases the rigidity of the bilayers and makes the pore harder to form. Larger bilayers will include longer wave-length undulation modes and thus may alleviate this situation (33). Furthermore, we only tried a single starting configuration where peptides are maximally separated. The diffusion of peptides on the bilayers surface is another slow DOF. Among peptides, an optimal uptake configuration may exist in which they can interact cooperatively to distort the membrane and lower the free energy to form pores. However, such possibility is not explored in this work.

Characterization of the water pore

The radius of the water pore is defined as the two-dimensional radius of gyration (in x-y plane) of water molecules that are within a 1-nm-thick slab centered at the bilayer center. For systems with one, four, and nine peptides, the pore radii are 0.62 ± 0.07, 0.52 ± 0.07, and 0.58 ± 0.17 nm, respectively. The average pore radius is similar in different peptide concentrations but the fluctuation of radius is larger in the nine-peptide system. The two-dimensional radius of gyration of the peptide is 0.64 ± 0.01 nm, which is similar to the radius of the pore. Thus, we conclude that in all systems, the pore is sufficiently large for the translocation.

The translocation of a charged arginine into lipid bilayers has been discussed extensively in the literature (27,32,34–36). Water pores are only observed when we have translocation in bilayers with short-tailed lipids such as 1,2-Didecanoyl-sn-glycero-3-phosphocholine and DLPC. For DMPC and dipalmitoylphosphatidlycholine (DPPC), water defects are observed (36). Here we want to emphasize the subtle difference between a water defect and a water pore. A water pore is a connected water channel opened from both bilayer leaflets and a water defect is a water perturbation that resides only in one leaflet but does not extend through the whole bilayer. A water defect is usually sufficient to keep a single arginine hydrated; however, in the case of peptide translocation, a water pore is required. For DPPC bilayers, Wohlert et al. (37) found that the free energy of forming a small pore is in the range of 75–100 kJ/mol. Tieleman and Marrink (38) found that the free energy of a lipid flip-flop in DPPC bilayer is ∼80 kJ/mol. Because the flip-flop in their work formed a water pore, the energetics required for a flip-flop can also be understood as the energetics to form a water pore. For the DOPC bilayer, no water pore is observed in lipid flip-flop and the flip-flop energy is ∼90 kJ/mol (39). Therefore, the free energy to form a water pore in a DOPC bilayer should be higher than 90 kJ/mol. In our work, the water-pore formation is convoluted with the peptide translocation and the total energy of these two processes is ∼100 kJ/mol, which is of the same magnitude as previous works.

Translocation rate

We estimate the translocation rate by using the pore-formation rate-estimate from Tieleman and Marrink (38). The detailed balance condition of a reversible process is

$$\frac{k_f}{k_b}=exp\left(\frac{-\Delta G}{kT}\right),$$

where k_f and k_b are the forward and backward reaction rates, respectively. Previous simulations found a pore lifetime is 10–100 ns (40,41). Assuming the peptide residence time inside the pore is similar to the pore lifetime, we estimate the peptide dissipation rate k_b ∼ 10⁸ s⁻¹. For our one-peptide system, ΔG = 120 kJ/mol along a pore path, which gives the translocation rate k_f ∼ 10⁻¹¹ s⁻¹. This is the translocation rate per peptide. Using the area per lipid as 64 A², and peptide/lipids per leaflet ratio as 1:64, we can calculate the translocation rate per unit area as k_f ∼ 10⁻¹¹/(64^∗64 A² s). For a liposome with a radius 1 μm, this gives the translocation timescale upper estimate of ∼36 h, which is longer than the timescale observed in experiments.

However, we want to emphasize that the translocation rate estimated from free energy is very inaccurate. For example, if the free energy is reduced by 5 kT (≈10 kJ/mol), the translocation time will be reduced by a factor of 100, which will shorten our previous estimation from hours to minutes. On the other hand, the significant difference between free energies along two different paths can provide us enough confidence that the translocation along a water-pore path should be more favorable than a path without pore. Given that experiments have found that CPPs can penetrate through model membranes such as GUVs (15–17), which are free of endocytosis, we claim that if the CPPs go through those membranes, they should go through with a pore.

Due to the complexity of cellular environment, it is inappropriate to use this work as evidence to use against other translocation mechanisms, such as micropinocytosis (42). Admittedly, the previous work might overestimate the rate of translocation, mainly due to the absence of counterions (19). The lack of local screening of excess charges may have produced a large local electric field which induces electroporation. Once the cost of forming a pore is eliminated from the free energy, the peptide can translocate in a shorter timescale. One factor missing in our model is the effect of different lipids components. A recent experiment suggests that including PE or PG lipids in PC GUVs facilitates the CPP uptakes (16), thus we reason that PE or PG lipids may further reduce the free energy of translocation. It will be interesting to extend our work on those mixed lipid bilayers in the future.

Conclusions

In this work, umbrella sampling simulations were used to estimate the free-energy profile of translocation of a cR₉ into a lipid bilayer. The free-energy calculation in such a system is very challenging due to the orthogonal slow-relaxing DOFs and many of them are unknown a priori. The fully converged free-energy profile should correctly sample the orthogonal DOFs, but the free-energy barriers in the orthogonal DOFs make such sampling impossible in our simulation timescale. Depending on the initial configurations and OPs used to restrain the system, etc., umbrella simulations can fall into different paths in the energy landscape, and the calculated free energy should be viewed as the free energy along a certain path. Although the full free-energy profile is hard to obtain, the comparison between free energies along different paths can still shed insight on the translocation mechanism. Comparisons can be made by projecting the free-energy profiles of different paths onto the same OP.

Here we focus on the effect of water-pore formation. Employing two different OPs, we guided the peptide to translocate along two different paths—one that involves a water-pore formation and another that does not. We found that the free-energy barrier along the translocation path involving a water pore is 80 kJ/mol lower than the barrier along the path without a pore. Once the water pore is formed, transferring the peptide along the pore costs little energy. This indicates the energy-independent translocation mechanism of CPPs should involve the formation of a water pore. Because the water-pore formation is only one of the many slow-relaxing DOFs, exploring other slow-relaxing DOFs should further lower the free energy. Although our result cannot be used to eliminate other possible translocation mechanisms in cellular membranes, it provides thermodynamic evidence for the pore-assisted translocation model. The peptide translocation process is dominated by several time-limiting DOFs in the system and the calculation of the full free-energy profile of translocation remains challenging. However, we emphasize the importance of finding a good OP that can sample low energy configurations and that biological relevant information can be extracted even without obtaining the full free-energy profile. We expect our work will open new ways to tackle systems with multiple slow-relaxing DOFs.