Adhesion, Intake, and Release of Nanoparticles by Lipid Bilayers ⇑ Sean Burgess A, Zhengjia Wang A,B, Aleksey Vishnyakov A,C, Alexander V

Journal of Colloid and Interface Science 561 (2020) 58–70

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Journal of Colloid and Interface Science

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Adhesion, intake, and release of nanoparticles by lipid bilayers ⇑ Sean Burgess a, Zhengjia Wang a,b, Aleksey Vishnyakov a,c, Alexander V. Neimark a, a Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA b Harbin Institute of Technology, 92 Xidazhi St, Nangang Qu, Haerbin Shi, Heilongjiang Sheng 150001, China c Skolkovo Institute of Science and Technology, Nobel St. 1, 121205 Moscow, Russia graphical abstract

article info abstract

Article history: Understanding the interactions between nanoparticles (NP) and lipid bilayers (LB), which constitute the Received 17 October 2019 foundations of cell membranes, is important for emerging biomedical technologies, as well as for assess- Revised 17 November 2019 ing health threats related to nanoparticle commercialization. Applying dissipative particle dynamic sim- Accepted 27 November 2019 ulations, we explore adhesion, intake, and release of hydrophobic nanoparticles by DMPC bilayers. To Available online 28 November 2019 replicate experimental conditions, we develop a novel simulation setup for modeling membranes at isotension conditions. NP-LB interactions are quantified by the free energy landscape calculated by the Keywords: ghost tweezers method. NPs are studied z of diameter 2 nm (comparable with the LB hydrophobic core), Lipid membranes 4 nm (comparable with the LB thickness) and 8 nm (exceeding the LB thickness). NPs are pre-covered by Nanoparticles Nanoparticle translocation an adsorbed lipid monolayer. It is shown that NP translocation across LB includes (1) NP intake into the Adhesion hydrophobic core via merging of the monolayer adsorbed on NP with the outer leaflet of bilayer (2) NP release via formation and rupture of a lipid junction connecting NP and LB. Both stages are associated with free energy barriers. The barrier for the intake stage increases with the NP size and becomes prohibitively high for 8 nm NP. The barriers for the release stage are significantly higher which implies that the release stage controls the translocation rate and dynamics. The release energy barrier of 4 nm NP is found smaller than those for 2 and 8 nm NPs which implies the existence of the optimal NP size for unforced trans-membrane transport. Based on the calculated free energy landscape, the dynamics of unforced transport of NP across LB is evaluated using the Fokker-Planck equation, which mimics NP

⇑ Corresponding author. E-mail address: [email protected] (A.V. Neimark). https://doi.org/10.1016/j.jcis.2019.11.106 0021-9797/Ó 2019 Elsevier Inc. All rights reserved. S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 59

diffusion along the free energy landscape with multiple attempts to reach the barrier. We found that the number of attempts required for successful translocation scales exponentially with the energy barrier. Ó 2019 Elsevier Inc. All rights reserved.

1. Introduction [30] employed thermodynamic integration to explore the free energy landscapes of NPs functionalized with hydrophilic/ Understanding of physico-chemical mechanisms of nanoparti- hydrophobic ligands upon penetration into LBs. The authors pulled cle (NP) adhesion to, encapsulation by, and translocation through the NP with a spring towards the membrane of fixed area and mea- lipid bilayers (LB) is of utmost importance for development of sured the applied force as a function of the NP distance to the cen- biomedical nanotechnologies for imaging and drug delivery [1], ter line of LB [21,30–32]. Van Lehn et al. [33] reported theoretical design of novel bio-inspired materials and devices [2,3], as well studies of embedding hydrophobic NPs functionalized with differ- as for assessing health threats related to the expansion of nanopar- ent sidechains into LBs. The free energies showed a strong depen- ticle commercialization. NP-LB interactions are affected by a vari- dence on NP size and ligand type and density. Remarkably, the free ety of physico-chemical factors such as particle size, shape, energy dependence on NP size exhibited minima at NP diameter hydrophobicity, charge density, and physisorption of lipids and between 2 and 4 nm. Free energy landscapes have also been calcu- proteins [4–6]. Silica NPs penetration into cells has been found to lated with the single-chain mean field theory [21,31,32]. be largely dependent upon particle size [7,8]. These and many It should be mentioned that in most of the simulation studies the other experimental observations related to interfacial interactions membrane sample had a constant surface area, (e.g. [15,30])whilein of NPs with lipid membranes are still poorly understood, and their practical situations (e.g. during NP transport across a cell mem- effects are hardly predictable by using classical approaches of brane), the membrane tension (and, correspondingly, the chemical interfacial and colloidal science. Consequently, a more robust sim- potential of the lipid molecules, which is directly related to the ulation technique must be applied to gain a greater understanding membrane tension) remains constant, because a cell or liposome is of these interactions. large compared to a NP. In order to maintain the membrane tension, In this work, we explore adhesion, intake, and release of the semi-isotropic pressure coupling for the simulation box resizing hydrophobic NPs by 1,2-dimyristoyl-sn-glycero-3-phosphocholine was employed in several studies, e.g. [19,29].Inourrecentwork[34] (DMPC) membranes using dissipative particle dynamics (DPD) on studies of the stability of NP loaded lipid membranes, the simulations [9,10]. DPD is a coarse-grained simulation technique required control of the membrane tension and membrane stabiliza- popularly employed for modeling soft matter, including lipid tion was achieved with a special simulation set-up that is employed membranes, due to its versatility and computational efficiency. and modified in this work for efficient modeling of NP adhesion and The feasibility of DPD modeling in simulating NP adhesion to and translocation. In this work, we study the equilibrium adhesion states engulfing by LBs has been demonstrated in many publications and the dynamics of intake and release of hydrophobic NPs by a [11–14]. In particular, DPD has been employed to mimic transloca- DMPC membrane using original simulation methods implemented tion of NPs of different shapes, sizes and ligand functionalization into the DPD computational framework. Calculations are performed [15–17]. While the aforementioned studies provided valuable for NPs of diameter 2 nm (comparable with the width of the LB qualitative insights into the process of NP translocation, to make hydrophobic core), 4 nm (comparable with the LB thickness), and NP-LB simulation results applicable to real experimental systems, 8 nm. To replicate the experimental conditions, we develop a novel the translocation dynamics must be analyzed by evaluating the simulation setup that maintains the LB at a constant tension and energy barriers of the interfacial transitions. For very small (up allows for the lipids to be freely exchanged between the membrane to 2 nm in diameter) hydrophobic NPs, the energy barriers for and solution; this is equivalent to the condition of the lipid chemical intake are small and their transport can be followed in a straight- potential constancy [34]. At this condition, a hydrophobic NP in solu- forward manner in coarse-grained simulations due to short obser- tion is coated by an equilibrium lipid monolayer (LM). We examine vation times required [18–20]. Such NPs tend to penetrate into the previously unexplored mechanisms of NP translocation relevant pro- hydrophobic core of the LB and accumulate there, as observed in cesses: merging of the lipid monolayer adhered to the hydrophobic simulations and experiments alike [21–25]. However, in most NP with the outer leaflet of LB during NP intake and formation and cases, unforced transport of NPs through LBs is too slow to be rupture of a lipid junction connecting the NP and LB during NP directly followed even in coarse-grained simulations. The mecha- release. Both these mechanisms are associated with the energy bar- nisms and dynamics of NP transport can be elucidated from the riers with depends on the NP size. To monitor, visualize and quantify free energy landscapes - the dependence of the free energy of these mechanisms, we employ the ghost tweezers (GT) method [35], NP-LB interactions on the particle position with respect to the which mimics the experiments with optical and magnetic tweezers. LB. The potential of mean force and different versions of umbrella The GT method allows us to calculate the free energy landscapes and sampling methods are commonly used to study the free energy of energy barriers associated with intake and release of NPs. Addition- NP-LB interactions [19,26–29]. In particular, Jusufi et al. [29] calcu- ally, the dynamics of unforced transport of NP across LB is evaluated lated the free energy of absorption of fullerenes by a lipid bilayer using the Fokker-Planck (FP) method [36], which mimics NP diffu- using umbrella sampling with weighted histogram analysis with sion along the calculated free energy landscapes. fullerene coordinate as the integration variable. The distance between NP and LB was controlled by an external field and the 2. Modeling methods interfacial tension of the bilayer was kept constant with anisotro- pic NPT conditions. The largest fullerene was 2.4 nm in size, twice Dissipative particle dynamics implementation, lipid and as small as the membrane thickness. All fullerenes entered the LB water models. Dissipative particle dynamics [9,10] is a coarse- without any noticeable potential barrier, and the free energy land- grained simulation method, which uses Newton’s equations of scape along the translocation trajectory was continuous. Li et al. motion governed by interparticle interaction forces, 60 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 X = ¼ v ; v = ¼ ; ¼ ðÞC þ ðÞB þ ðÞR þ ðÞD ð Þ a previously published paper and it reasonably reproduces the dri dt i d i dt fi fi Fij Fij Fij Fij 1 j–i experimental density and elasticity of DMPC membranes [34]. Rel- evant data is provided in the Supporting Information, Section 5. DPD particles, or beads, represent characteristic fragments of sys- The force-field parameters between the beads of different types tem chemical species comprised of several atoms. The force fi acting are listed in Table 1. on a bead ‘i’ include several components. We use a standard DPD The simulations are performed in an orthorhombic box of size framework with all beads having the same size and interacting by 56 40 70 Rc (36 26 45 nm). The system has a density of conservative linear short-range repulsion forces, 3 beads per unit volume (R3), giving a total of 470,400 beads. The ( c ^ velocity Verlet leapfrog integration algorithm [38] is implemented aij 1 rij =Rc rijif rij Rc FC ¼ : ð2Þ with a time step size of 0.01 in the NVT ensemble. Simulations ij > 0if rij Rc begin with a 106-step initial equilibration run, followed by eight parallel 4 105-step data runs for each NP position. Calculations Here aij represents the repulsion strength parameter, and Rc the effective bead radius which is assumed equal for all bead types. presented in the main text are performed with LAMMPS [39] run Lipid beads are connected using harmonic bonds on the Extreme Science and Engineering Discovery Environment ðÞ ðÞ ðÞ (XSEDE) [40]. Additional simulations were done using DL_Meso F B ¼K B r r br where K B is the spring constant and r is ij i ij e ij i e [41] and an in-house DPD package. The results of simulations with the equilibrium spring position. To secure the isothermal condi- the different software packages are comparable. Further details are tions, random and drag forces are implemented using the pairwise presented in Supporting Information, Section 1. A second set of Langevin thermostat, simulations was performed using a =25kT/R between lipid tails, TS c ðÞR ðÞR b ðÞD ðÞD b b ‘T’, and NP shell, ‘S’, effectively providing an extremely strong F ¼ rw rij hijðÞt rij; F ¼cw rij rijvij rij ð3aÞ ij ij attraction between tails and NP; the results are presented in Sup- porting Information, Section 1. ðÞR rij ðÞD ðÞR 2 2 w rij ¼ 1 ; w rij ¼ w rij ; r ¼ 2ckBT; ð3bÞ Nanoparticle Model. The NP is formed from beads arranged Rc into a 3D hexagonal close packed (HCP) lattice. The nearest neigh- bors are bonded together by harmonic bonds. Bond length is cho- where hijðÞt is a randomly fluctuating variable with Gaussian 3 statistics. sen to provide bead density of 3 beads/Rc. The NP is designed to The coarse-graining scheme and interaction parameters for represent an approximately spherical particle carved out of a crys- lipid and water models are adopted from the work of Groot and tal, which maintains its shape due to high bond rigidity. The inner Rabone [37]. Water is represented by water beads ‘‘W” of the effec- layers of the NP are formed by core, ‘C’, beads that strongly repel all other beads in the system (except the outer layer of the NP) to tive radius, Rc = 0.645 nm comprised of 3 water molecules. The DMPC molecule is dissected into 11 fragments of similar volume ensure that the NP is impenetrable. The shell layers of the NP con- sist of hydrophobic ‘S’ beads, which effectively attract lipid tail represented by 11 beads of the same effective radius, Rc, as the water bead, Fig. 1. The zwitterionic choline-phosphate head is rep- beads (Fig. 1). resented by two hydrophilic ‘H’ beads. The carboxyl junction is NPs of three diameters are considered in order to probe three represented by one semi-hydrophobic ‘J’ bead, and the two alipha- distinct situations: a small NP of 2 nm in diameter (NP diameter tic tails are represented by 4 ‘T’ beads each. The intra-component approximately equals the thickness of the hydrophobic core of LB; total number of all beads in NP M=44), a 4 nm NP (NP diam- self-repulsion parameter, aii = aWW = 78, is the same for all types of beads and secures the density and compressibility of water. eter comparable with the LB thickness, M = 355), and an 8 nm NP (NP diameter double the LB thickness, M = 2996). The intercomponent repulsion parameters, aij, are chosen to It is worth noting that hydrophobic NPs tend to aggregate in account for the respective beads hydrophilicity (aiW smaller of aqueous environments and adsorb various amphiphilic entities, equal aWW) or hydrophobicity (aiW larger aWW). It is worth noting that Groot & Rabone [37] modeled 1,2-dipalmitoylphosphatidylcho such as phospholipids and other biomolecules. In order to prevent line (DPPC), whereas we model DMPC, the only difference being aggregation and uncontrolled adsorption of different species on the DMPC model has one less bead per tail. The validity of using their surface in experiments, hydrophobic NPs are covered by lipid this model for a freestanding DMPC bilayer have been checked in monolayers. Characteristic example is the NP-based drug delivery

Fig. 1. Left: Coarse grained model of DMPC composed of 2 hydrophilic ‘H’ head beads, 1 ‘J’ junction bead, and 8 hydrophobic ‘T’ tail beads. Water bead ‘‘W” contains 3 water molecules. Right: Nanoparticle with surface hydrophobic beads ‘‘S” (red) and pre-adsorbed lipid monolayer. S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 61

Table 1 Parameters for repulsion and harmonic bond interactions. Bead types: H – lipid head; J – lipid junction; T – lipid tail; F – frame; R – roller bar; W – water; U – upper part of pulling bar; B – bottom part of pulling bar; S – outer shell of NP; C – inner core of NP; G – ghost NP. Note that for NP-GT bonds, MK(B) *rather than K(B)is given where M is the total number of beads in NP and depends on NP size.

Repulsion parameters aij, kT/Rc HJ T FRWUBSCG H 86.7 89.3 104 78 104 75.8 78 104 104 104 0 J 78 86.7 78 104 79.3 78 104 104 104 0 T 78 25 104 104 25 104 78* 104 0 F 7878787878781040 R 78 78 78 78 78 104 0 W 78 78 78 104 104 0 U 78 78 78 104 0 B 78 78 104 0 S 78 78 0 C 78 0 G 0 Bond parameters Bond NP-NP NP-GT H-H J-T T-T

(B) 2 K , kT/Rc 100 17.75 4 4 4 e r , Rc 0.8 0 0 0 0

using LM-coated hydrophobic NPs and nanotubes, which are stabi- chosen for all force and free energy calculations for computational lized by sonication [21,42,43]. Here, we consider a simplistic model efficiency due to substantial fluctuations of the membrane at low of the three component system consisted of DMPC LB, water, and a tensions, as we do not stabilize the membrane by applying external single hydrophobic NP coated by DMPC LM in the absence of other potentials [45]. Although this tension exceeds the range of tensions amphiphilic entities [43,21]. Therefore, at the start of the simula- of stable LBs in experiments, the simulated membranes are stable tion, an equilibrium lipid monolayer (LM) is pre-adsorbed to the up to applied tensions of ~30 mN/m [34]. To create a system that NP (Fig. 1). For the purpose of creating a LM, a configuration is cre- allows for the NP-LB interactions to be probed without compro- ated where the NP is encapsulated in the hydrophobic core of the mising the LB integrity, the system must be large enough that there LB and gradually removed. Eventually, the NP detaches with the are no periodic or edge effects, yet small enough that the simula- LM adsorbed at its surface, and the chemical potential of lipids in tion is relatively fast. For this purpose, two simulation setups are the LM approximately equals that in the LB at given surface ten- developed, as presented in Fig. 2. sion. For 4 nm NP, the average LM density of 1.8 DMPC molecules A LB is placed in a P-shaped frame that is formed from three per nm2 of NP surface, or 0.55 nm2 per lipid was determined over horizontal planks: two along the y-axis and one along the x-axis. ten trials with standard deviation of 3.7%. Noteworthy, this density The planks are made of two layers of beads: inner layer of is ~10% smaller than the experimental lipid density in the tension- hydrophobic ‘F’ beads on the inside of the P frame, and outer less freestanding DMPC LB (~0.6 nm2 per lipid [44]) and ~20% smal- layer of hydrophilic ‘R’ beads on the outside of the P frame. This ler than the simulated lipid density in the LB under 10 pN/nm construction allows the lipid bilayer to stick to the frame without tension [34]). This difference results from the fact that molecular wrapping around. On the open end, the LB is pulled either horizon- arrangement in the monolayer on the highly curved NP surface is tally in the linear set-up (Fig. 2, left) or in the rollover set-up (Fig. 2, different from that in the plain free-standing bilayer, and the effec- right) vertically down over a ‘‘roll” – a cylindrical block made of tive density of LM is lower [44]. hydrophilic ‘R’ beads. The LB is stretched by a movable plank, to System setup. One of the main goals of this work is to explore which a constant force is applied in order to maintain given LB ten- the free energy landscapes of a hydrophobic NP interacting with a sion by allowing the LB to advance/recede when it is deformed by LB under isotension conditions. A surface tension of 10 mN/m is the NP. The movable plank is constructed in the same manner as

Fig. 2. Linear and rollover simulation setups to study the free energy landscape of NP-LB interaction under isotension conditions. The inserts show the top-down projections. The LB is placed on a P-shaped static frame and stretched by a movable plank to which a constant force is applied to maintain given membrane tension. 62 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 the static planks but rotated by 90 degrees - hydrophobic side facing up and hydrophilic side facing down. To each bead in the movable plank, a constant force Fp is applied to maintain membrane surface tension, k ¼ FpMp=Lx (Fig. 2). Here Mp is the number of beads in the movable plank, and Lx is the length of that plank in the x-direction. While the linear setup is simpler, it requires larger y-dimension to prevent the movable plank from interacting with the static plank. It also requires larger x-dimension to prevent bending of the LB across the periodic image. The rollover setup Fig. 3. Diagram of the GT implementation. The twin GT particle (black), formed of allows for a smaller simulation box size since the LB is stretched the beads arranged in the same manner as the beads in the real NP, is placed at downward in the z-direction. Most of the simulations were per- given position ZGT in space. The real NP (red) is attached to the GT by the springs D ¼ formed with the rollover setup. connecting respective NP and GT beads. The average separation, Z Z ZGT, between the two is measured allowing for the force needed to keep NP at given The proposed setup, which imitates the standard experimental distance Z from the membrane to be calculated through Eq. (4b). set-up for measuring the tension of thin films, is found practically efficient for maintaining isotension conditions and keeping the membrane location fixed in the process of NP translocation. The where Z is the average normal coordinate of the NP center of mass, existence of the pre-adsorbed lipid monolayer on the NP surface Z is the normal coordinate of the center of mass of the immobile makes a principal difference compared with bared NPs. The mono- GT ðÞB ¼ ðÞB layer prevents the membrane position stabilization through the GT particle, and KGT MkGT is the overall GT spring constant acting ðÞB application of an external field (as done in Ref. [29]), since the between the real and ghost NPs. KGT is chosen to optimize the effi- lipids adsorbed on the NP mix with the membrane lipids as the ciency of calculations. The overall spring constant between the real NP passes into LB, and are not distinguishable for the external field. ðÞB 2 and ghost particle, KGT = 176 kT/Rc (1.7 N/m), is used independent of NPs prior to intake and after release contain different lipid mole- the NP size, whereas the individual spring constant varies. An image cules in their coatings. This factor makes it nearly impossible to illustrating the GT method is presented in Fig. 3. This GT setup with regulate the mutual arrangement of the NP and LB with a single multiple spring bonds between the NP and GT beads prevents NP external field, harmonic or otherwise, as it not clear to which beads rotation and reduces fluctuations of the NP position and, the calcu- the field should be applied. Another factor requiring the use of the lated GT force, respectively. proposed setup is that the membrane in our system experiences At the initial state, the GT is placed in a certain position suffi- large expansion and contraction in the processes NP intake and ciently far from the membrane, the system is equilibrated with release due to absorption and extraction of lipids coating NP. the NP fluctuating around the GT with zero mean displacement NPT simulations with semi-isotropic pressure coupling in the sys- and, respectively, zero mean force. After the system equilibration, tem with such large fluctuations of the membrane area could be the GT is placed in a new position on the z-axis (the x and y coor- extremely computationally expensive as one needs to reshape dinates are constant), and a new simulation is run, probing the the entire box to extend/contract the membrane that brings about force at the new location. This process is continued in incremental the convergence problem. The proposed setup with the membrane steps and the force experienced by NP from LB is measured in a placed on the immobile frame prevents such drastic changes in the quasi-equilibrium fashion along the translocation trajectory until box shape that would occur during the simulation in the NPT a spontaneous transition takes place (e.g. NP intake by LB). This ensemble. The rollover setup is computationally more efficient in method ensures that no force or energy minima are missed comparison with the horizontal setup, as it utilizes the vertical although it does not allow calculation of the free energy change dimension to relax the film. during spontaneous transformations. It is important to note that Force and free energy calculations using the Ghost Tweezer this is not a towing process, but rather a sequential, quasi-static, method. To calculate force and free energy landscapes, we use quasi-equilibrium, incremental process that was confirmed by the GT method developed in our earlier work [35]. The GT is imple- checking the reproducibility of the results and scanning the trajec- mented by introducing a ‘‘twin” GT particle, that is identical in the tory backwards. Exceptions are the points of spontaneous transi- bead configuration to the real NP. The ‘‘ghost” NP beads are kept in tions of the NP insertion into LB in the process of intake and NP undisturbed hexagonal order and do not interact with any system detachment from LB in the process of release. While at these points beads except for the NP beads. Harmonic spring potentials are the GT position is kept fixed, the NP experiences a finite displace- applied between each respective pair of real NP and GT beads ment across the LB boundary ‘‘jumping” from one valley of the ... (i =1, ,M). Each GT bead interacts with no other bead but the cor- energy landscape to another. These spontaneous irreversible tran- responding NP bead through a harmonic bond, effectively tether- sitions make it impossible for the NP to pass through the LB along a ing the real NP to a point in space. Thus, the NP is attached to a continuous trajectory preventing the use of standard free energy certain position by the GT force between the NP and its ghost calculation methods like umbrella sampling. The GT method image: enables the construction of the two energy landscapes, outside XM and inside the LB, which resemble two valleys separated by a ridge ¼ ðÞB ð Þ FGT kGT RNP;i RGT;i 4 that is impassible in incremental steps: a mountaineer ascending i¼0 the ridge from one valley can find a pass and climb up to the crest of the ridge and then jump down. This is illustrated in Figure S6 in where RNP;i and RGT;i represent the coordinate of bead ‘i’ within the ðÞ the Supplementary Information. The advantages of the GT method NP and the GT respectively, and k B is the spring constant of the GT compared to umbrella sampling are its simplicity and relevance to harmonic potential between the respective NP and GT beads. experimental optical tweezers method. Because the average lateral contribution to F due to the system GT From the force the GT exerts on the NP, F (Z), the Helmholtz symmetry is zero and the structures of the NP and its ghost image GT free energy, E (Z), can be obtained by thermodynamic integration are the same, the average force on the NP from the GT is equal to NP and calculation of the mechanical work needed to bring the NP ðÞ ¼ B ; ð Þ from the initial state at position Z0 to given position Z in the vicin- FGT KGT Z ZGT 4b ity of the membrane. There are two characteristic initial states S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 63 where the NP equilibrium is achieved without applying an external approaches the LB, it experiences repulsion due to the disjoining force: free state in the solvent bulk, Z0 = Zb, far from the membrane pressure in the water film squeezed between the NP coated by and encapsulated state at the center of LP, Z0 = 0. These initial LM and the outer leaflet of LB. This repulsion force is counter bal- states are used to monitor the NP intake and release, respectively. anced by a positive GT force. As the NP advances towards the LB,

In our simulation set-up, the process of NP-LB adhesion and the GT force increases up to a maximum value of 3.2 kT/Rc translocation is associated with the deformation of the membrane (2.04 1011 N), which is weaker than a characteristic force and redistribution of lipids and respective displacement of the between neighboring DPD beads. The respective free energy land- movable plank. The plank displacement causes the work done scape is calculated with Eq. (5). The LB bending on NP approach against the constant force Fp applied to the plank to maintain given causes a free energy increase with a maximum of 3.4 kT, a rela- membrane tension. To measure this work, the average position tively low barrier meaning the intake would occur rapidly in a bio- LðÞZ of the mobile plank is determined during the NP equilibration logical system. Upon achieving the free energy maximum, the system experiences a spontaneous transformation: the LM ðÞ at the position Z. L Z determines the membrane extension adsorbed on the NP surface merges with the LB, resulting in the between the static and mobile planks. The work done by the GT NP intake by the LB hydrophobic core. The membrane extends D ðÞ equals the change of the Helmholtz free energy of NP-LB, EZ, due to the pulling force applied to the mobile plank (the work plus the mechanical work performed against the plank force: made against the pulling force on the system accounted for in Z Z hi Eq. (5)). D ðÞ¼ ðÞ ðÞðÞ ð Þ EZ FGT Z dZ cLx L Z L Z0 5 The value of the free energy gained by the system in the process Z 0 of NP transfer from the solvent bulk to the point of the onset of the hi spontaneous interfacial transition constitutes the energy barrier of where LðÞZ LðÞZ0 ¼ DLÞ is the displacement of the plank corre- the NP intake. The barrier is likely due to the resistance of the dis- sponding to either contraction or extension of the membrane. Note joining pressure of the water interlayer [46,47] that hinders the LM that cL DL represents the work needed to extend/contact the mem- x fusion with the outer leaflet of LB. Note that bare fullerene NPs of brane area by L DL. In the process of intake, the membrane first x comparable size are absorbed by LB membranes without any bends and the plank displacement DL is negative, however, upon noticeable barrier along a continuous translocation trajectory incorporation of the NP and absorption of lipids in the NP coating, [29]. The statistical error in the energy barrier calculations is high, the membrane extends causing the positive plank displacement. due to the small magnitude of the barrier and large fluctuation of The maximum membrane extension is achieved at the equilibrium the plank position (the graphs of the plank position L(Z) and the position with the NP fully encapsulated at the LB center at Z =0.In 0 free energy error calculations are discussed in the Supporting the course of NP release from the fully encapsulated state at Z0 =0, Information, section 2). the membrane bends and LðÞZ progressively decreases up to the It is worth noting that during the spontaneous interfacial trans- point of spontaneous rupture of the NP-LB junction and NP detach- fer, the position of NP with respect to the LB center plane changes ment from the membrane. on the order of NP diameter upon incremental displacement of the GT that cause discontinuity of the GT force. Immediately after the 3. Results and discussion intake (position 2 in Fig. 4), the GT force acting on the NP is negative in order to retain NP from moving further into the equilibrium Calculation of the free energy landscape of NP-LB interac- state in the LB center (Z = 0). The equilibrium corresponds to a free tions 2 nm nanoparticle. Fig. 4 shows the force experienced energy minimum with zero effective force acting on the NP. It is by the NP as a function of NP position and the free energy land- taken as the initial reference point for monitoring the process of scape for a 2 nm NP near a DMPC LB. NP release. For the encapsulated configurations, the effective force At the initial reference state, the NP pre-coated with LM is on the NP increases linearly as the NP departs from the equilibrium placed far from the LB in the solvent bulk and does not feel the position up to Z < 4.5 nm due to the elasticity of the LB. Due the presence of the LB: FGT =0(Fig. 4, snapshot 1). As the NP spontaneous nature of the NP encapsulation, the free energy profile

Fig. 4. The position dependence of the GT force counterbalancing the NP-LB interaction force (top) and the free energy landscape (bottom) for the 2 nm NP. The LB snapshot in the graph background is scaled to the characteristic bilayer thickness. Blue (purple) and red (brown) circles show, respectively, the force (free energy) in the course of intake and release. Z coordinate represent the center of mass position of the NP. The panels on the right are the snapshots of characteristic conﬁgurations. (1) – initial position of NP in bulk solvent; (2) – immediately after NP intake; (3) – equilibrium state with the NP in the center of LB, initial position for the release process; (4) – NP-LB junction formation at the onset of the spontaneous detachment of the NP. 64 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70

Fig. 5. Dynamics of the formation and breakup of the NP-LB junction. The left most image is taken immediately after the GT is incrementally displaced outwards the frame plane, the second image (after 200,000 timesteps) is taken as the junction begins to form, and the third image (after 400,000 timesteps) shows the equilibrated state with a well-developed junction. As the junction forms, the LB curvature decreases causing the free energy to increase dramatically and the force to decrease as observed in Fig. 4, point 4. In the fourth image, the GT is positioned farther away from the frame plane, and the junction begins to rupture causing spontaneous detachment of the NP. for NP motion is asymmetric: the state with an encapsulated NP 2 nm NP is an absence of the cylindrical junction between the NP has a different free energy than the state with a free NP located and LB. The LB does not recede back to the frame, and the GT force and the same coordinate. As the NP moves further, the LB configu- levels out rather than decreases. As the GT moves farther, the lipid ration changes, and a cylindrical junction forms between the NP junction ruptures, the NP coated by LM separates from the LB and and the LB. The LB relaxes, moving closer to the frame plane the LB returns to its equilibrium position within the frame plane. (Figs. 4-4 and 5). The GT force magnitude decreases. The junction The maximum effective force experienced by the NP on the onset 10 formation causes the number of lipid molecules inside the LB to of spontaneous detachment is 25.0 kT/Rc (1.7 10 N), and the decrease, therefore the movable plank recedes which brings about free energy barrier is 37.2 kT (Fig. 6, snapshot 4). a negative contribution to the free energy. As a result, the free 8 nm nanoparticle. The translocation of the largest NP (8 nm in energy landscape shows a plateau (Fig. 4, bottom panel). This diameter) considered in this work follows similar stages as the behavior is reliably observed for small NPs, even with different translocation of smaller NPs, the energy barriers are higher. The parameters. The process of the NP transfer from initial encapsu- intake transition takes place much later compared to 2 nm and lated state to the point of junction formation is discussed in more 4 nm NP. It requires a substantial deformation of the LB (Fig. 8, detail in the Supporting Information, section 1. The dynamics of snapshot 2) and therefore is associated with a substantial repulsion 10 junction formation is illustrated with the characteristic snapshots force (about 34 kT/Rc or 2.2 10 N, see Fig. 8, top left) and free in Fig. 5. During the first three frames, the system evolves with energy barrier of about 40 kT (Fig. 8, bottom left). Following the time with the GT kept in the same position. The LB pulls away from intake, the qualitative snapshot is like that of smaller NPs – the the NP and creates a cylindrical junction connecting the LM to the NP sits in the center of the LB relaxed within the frame plane LB. In the fourth frame, the GT is moved once again causing the (see Fig. 9). junction to rupture. The barrier is much higher than those observed with smaller Scanning simulations are performed, varying back and forward NPs. However, it is not high enough to effectively prohibit the the GT position around the point of junction formation. No hystere- intake transitions in natural environment. The reason for the dif- sis on F(Z) dependence is found in the 5 nm < Z < 12 nm range con- ference between 8 nm and smaller particles is the increase of the firming that the junction formation is reversible from any stability of the LM adsorbed on the particle surface in its size. configuration with encapsulated NP prior to the juncture rupture The 2 nm NP is about the effective length of the hydrophobic tail and spontaneous NP detachment. The forces observed on the NP of the DMPC molecule. The NP surrounded by DMPC molecules is approach to the frame plane reasonably equal those on reproach, essentially a spherical micelle, and a micelle is generally less stable which indicates an absence of a spontaneous transition associated than an LB, has lower barrier for merging with the latter. with the junction formation. As the NP moves farther from the An LM adsorbed on 8 nm NP is denser and, most importantly, plane frame, the junction neck spontaneously ruptures, and the more uniform. It therefore reasonable that the barrier associated NP separates from the LB. The release process is now complete. with LM and LB fusion increases with the NP size, and this depen- The necessity to proceed through the junction configurations cre- dence is most pronounces when the latter is comparable with the ates a high potential barrier (75.1 kT) that the 2 nm NP must over- LB thickness or slightly exceeds it. come in order to escape the LB core. The mechanism of NP release is illustrated in Fig. 10. During 4 nm nanoparticle. The process is qualitatively similar to what the process of release, outward displacement of the encapsulated is observed for the 2 nm NP. As the NP moves towards the LB from NP deforms the membrane, which bends creating elastic resis- solvent bulk (Fig. 6, snapshot 1, FGT = 0) it experiences repulsion tance. As the NP moves further, a similar phenomenon to the due to LB deformation. The force reaches a maximum of 3.1 kT/Rc smaller NPs’ release is observed: a junction forms between the (0.47 1010 N), and free energy a maximum of 4.2 kT. As the NP and the LB (Fig. 10, images a–c). The shape of the junction GT advances further, a spontaneous intake of the NP by the LB resembles a tubular micelle with water inside the bilayer neck, (Fig. 6, snapshot 3) is observed, with a step-like drop in the force rather than a thin rope-like bridge with a single hydrophobic magnitude. NP intake and its incorporation into the LB hydropho- core observed for 2 and 4 nm NPs. As the LB stretches, the water bic core is preceded by water displacement from the NP-LM gap is squeezed out and a cylindrical junction forms (Fig. 10, images and fusion of the LM with the outer leaflet of LB (Fig. 7). The LB d–e). This junction forms a neck, which at a certain point (Fig. 10, spontaneously relaxes and the mobile bar advances. image f) ruptures. During the junction extension, the GT force The process of release of 4 nm NP is also similar to that for 2 nm becomes nearly independent on the NP position, while the free NP: as the encapsulated particle moves from the equilibrium posi- energy increases in excess of 110 kT before the junction snaps tion at the frame plane, the LB stretches (Fig. 6, snapshot 4) and the and the NP covered by LM escapes the LB. The barrier of such NP experiences a positive force increasing approximately linearly height practically prohibits unforced translocations via this with Z. (Fig. 6, top panel) An important distinction from the mechanism. S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 65

Fig. 6. The position dependence of the GT force counterbalancing the NP-LB interaction force (top) and the free energy landscape (bottom) for the 4 nm NP. The LB snapshot in the graph background is scaled to the characteristic bilayer thickness. Blue (purple) and red (brown) circles show, respectively, the force (free energy) in the course of intake and release. Z coordinate represent the center of mass position of the NP. The panels on the right are the snapshots of characteristic conﬁgurations. (1) – initial position of NP in bulk solvent; (2) – LB bends as NP approaches LB just before NP intake; (3) – equilibrium state with the NP in the center of LB, initial position for the release process; (4) – maximum stretching of LB before the NP breaks free of the LB. Note the lack of pronounced junction compared to the 2 nm NP in Fig. 5.

Fig. 7. Dynamics of the LM fusion with the LB for the 4 nm NP. The left most image represents the point immediately after the GT is moved. The second image (after 100,000 timesteps) is taken as water is displaced from the NP-LB gap and the LM begins to merge with the outer leaﬂet of the membrane. Finally, in the third image, the system is equilibrated, and the NP is completely encapsulated within the LB. The black line represents the location of the GT, and black beads represent NP core beads. The GT is in the same position relative to the frame plane in all snapshots. The same process occurs for the 2 nm NP.

Fig. 8. The position dependence of the GT force counterbalancing the NP-LB interaction force (top) and the free energy landscape (bottom) for the 8 nm NP. The LB snapshot in the graph background is scaled to the characteristic bilayer thickness. Blue (purple) and red (brown) circles show, respectively, the force (free energy) in the course of intake and release. Z coordinate represent the position of the NP center of mass. The panels on the right are the snapshots of characteristic conﬁgurations. (1) – initial position of NP in bulk solvent; (2) position just prior the spontaneous intake; - LB bends as NP approaches; (3) – equilibrium state with the NP in the center of LB, the initial position for the release process; (4) – position just prior the spontaneous detachment with maximum stretching of the LB before the junction break-up.

Dynamics of unforced NP transport through LBs. The Fokker- ers. In the absence of external forces, these barriers should be over- Planck approach. The GT simulations show that both NP intake come due to thermal ﬂuctuations that cause NP Brownian diffusion and release processes are associated with particular energy barri- that is possible within realistic experimental timescales only in 66 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70

Fig. 9. Dynamics of the LM fusion with the LB for the 8 nm NP. The left most image represents the initial position of the simulation run immediately after the GT is incrementally moved toward the membrane; NP coated with LM is separated from the membrane by a water layer. The second image (after 40,000 timesteps) is taken at the onset of water expulsion and fusion of LM with the outer leaﬂet of LB. In the third image (after 120,000 timesteps), the LM is progressively absorbed and the membrane, in order to accommodate additional lipids, begins to stretch around the NP. In the last image (after 400,000 timesteps), the system has reached equilibrium with fully encapsulated NP. The GT remains at the same position relative to the frame plane during the simulation run.

Fig. 10. Dynamics of release for the 8 nm NP. Snapshots correspond to a set of consequent GT positions. Beginning in the top left, the LB experiences extreme bending caused by the LM-coated NP. As the force increases (the GT is moved further from the frame plane) water filled tube-like junction forms (images a-c). Further along, the water is squeezed out, a cylindrical junction forms, and the LB recedes (images d-e). Junction formation is similar to 2 and 4 nm NPs, but with much stronger bending of the LB. Finally, the junction ruptures and the NP is detached from the LB. The final snapshot is a non-equilibrated configuration just at the onset of junction break-up (image f).

case of reasonably low barriers. To evaluate the timescale needed Table 2 for NP intake and release from the LB we employ the Fokker Planck Dynamics of the NP release. (FP) equation, which describes particle motion as a random walk 2nm 4nm 8nm along a free energy landscape. The FP approach has been used Energy barrier,DE (kT) 75.1 37.2 110.3 extensively in a wide range of applications, from translocation Translocation scale, L (nm) 7.40 7.71 10.28 2 dynamics [48–49], nucleation [50], and of particular interest to this Time unit, L /D (s) 2:23 10 7 4:85 10 7 1:72 10 6 ; h i 30 12 43 study, probabilistic tracking of particle locations whose positions Mean translocation time sT 2:21 10 4:90 10 5:38 10 evolve according to Langevin dynamics [51–53]. Number of escape attempts; hni 1:06 1035 4:80 1016 8:20 1048 Using the free energy landscapes, EZðÞ, obtained with the GT simulations, Eq. (5), the dynamics of NP transport is described by the FP equation given in a dimensionless form, as Table 3 hi ¼ Dynamics of NP intake. tT provided considers external diffusion with L 1 lm and @ ðÞ; @ @ ðÞ @ ðÞ; WZs ¼ EZ ðÞþ; WZs ð Þ NP respective SE diffusion coefficient. All times (except the last) are dimensionless. @ @ @ WZs @ 6 s Z Z Z 2nm 4nm 8nm Energy barrier,DE (kT) 3.4 4.2 39.6 Here, WZðÞ; s is the probability distribution function representing Translocation scale, L (nm) 1.86 1.58 6.73 the density of the probability of NP location at distance Z at time Time unit, L2/D (s) : 8 : 8 : 7 1 41 10 2 03 10 7 40 10 3 3 5 s. Dimensionless distance Z is reduced by the distanceL from the Single attempt return time s ; : : : R 1 4 53 10 1 84 10 2 63 10 free energy landscape minimum to its maximum at the point of : : 2 Single attempt escape time sT;1 0 187 0 168 4:06 10 spontaneous interfacial transition. For the NP-LB systems consid- Number of escape attempts; hin 430 1710 1:15 1018 ; hi : : 16 ered here, L varies from 2 to 10 nm, see below Tables 2 and 3. Mean translocation time tT (s) 1 6140 1:89 10 The dimensionless time, s ¼ tL2=D, is reduced by the characteristic diffusion time over distance L determined by an effective diffusion coefficient D. The effective diffusion coefficient, D, in the FP equation characterizes NP mobility and hydrodynamic resistance to its equals the NP radius augmented by the effective thickness of the motion. In a bulk solvent, it can be estimated through the Stokes- LM covering the NP). For NPs of 2, 4, and 8 nm, the SE relationship 10 10 11 2 Einstein (SE) relationship, as D ¼ kT=6pgRP, where g is the viscosity gives D ¼ 2:45 10 ; 1:23 10 ; and 6:13 10 m =s respec- of water and RP is the hydrodynamic radius of a LM-coated NP (RP S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 67

ðÞ¼j ðÞ¼j ð Þ tively. The effective diffusivity of the NP interacting with the LB is pT s J Z¼1 and pR s J Z¼0 9b difficult to estimate. Moreover, a NP’s mobility outside (for intake Here, we denote the probabilities of successful translocation p ðÞs processes) and inside (for release processes) may differ. For exam- T and return p ðÞs in small letters to distinguish them from the prob- ple, the release of an encapsulated NP involves LB bending which R ability P ðÞs , associated with multiple attempts of translocation in increases hydrodynamic resistance to the NP translocation. It may T the symmetric potential well. The average times of successful be agreed that the SE equation gives the upper estimate for D. For the FP model numerical calculations, the free energy landscapes sT;1 and unsuccessful sR;1 translocation attempts are given by Z Z determined in the simulation by the GT method are approximated 1 1 ¼ ðÞ = ðÞ ð Þ with 4th ordered polynomials. Exact equations are presented in sT;1=R;1 spT=R s ds pT=R s ds 10b 0 0 the Supporting Information, section 5. 00 Dynamics of NP release. The FP Eq. (6) requires initial and Subscript ‘‘1 stands for the single translocation attempt. The boundary conditions which reflect the physics of the process under normalization constants in Eq. (10b) equal the probabilities of consideration. In the case of release of an encapsulated NP from the successful pT and unsuccessfulR pR translocationR within one translo- ¼ 1 ðÞ; ¼ 1 ðÞ; þ ¼ LB, we deal with a symmetric system. At time zero, the NP is placed cation attempt, pT 0 pT s pR 0 p s pT pR 1. The proba- in the equilibrium position at the center of the LB and can diffuse bility of a successful translocation attempt, pT, allows one to inside the free energy potential well until it achieves the edge of calculate the probability, pT;n, that the translocation occurs during the potential barrier on either side of the LB at Z ¼1 and is the n-th translocation attempt. This means that the first (n 1) released. In this case, the initial condition is represented by d- attempts failed and the NP returned to the initial state at Z ¼ 0, function at Z ¼ 0. The system has absorbing boundary conditions: but the n-th attempt is successful. Thus, since the translocation attempts are statistically independent, WZðÞ¼; s dðÞZ0 at s ¼ 0 ð7Þ ¼ ðÞ n1 ð Þ pT;n pT 1 pT 12 WZðÞ¼; s 0atZ ¼1 ð8aÞ The mean number of attempts needed for successful Pairing Eqs. (6), (7) and (8a) with the free energy landscape EZðÞ translocation, obtained by the GT simulation, Eq. (5), the particle location proba- X1 X1 bility distribution function, WZðÞ; s , is found. The inner term in hi¼n np ¼ p nðÞ1 p n 1 ¼ 1=p ð13Þ Eq. (6), represents the probability flux, JðÞ¼s ½@EZðÞ=@ZWðÞþ Z; s T;n T T T 1 1 @WZðÞ; s =@Z, allowing Eq. (6) to be rewritten as @WZðÞ; s =@s ¼ hi @J=@Z. The translocation probability, defined as the probability of The mean time of translocation, sT , is proportional to the mean number of unsuccessful attempts, hin 1, multiplied by approaching the system boundaries at given time s, PTðÞs ,is equaled to the sum of the probability fluxes at the boundaries, the mean time of returnsR;1 plus the mean time of one successful P ðÞ¼s Jj Jj ð9aÞ translocation event sT;1 , T Z¼1 Z¼1 hi¼sT sR;1 ðÞþhin 1 sT;1 sR;1 hin ð14Þ The probability distribution PTðÞs is normalized as in the limit of infiniteR observation time the NP is released with the probability of 1 This equation gives the mean translocation time in dimension- one, P ðÞs ds ¼ 1. The mean time of translocation, his , repre- 0 T less units reduced to L2=D . The approximation in Eq. (14) holds sents the first moment of this distribution, Z 1 when hin 1 and sR;1 hin >> sT;1 which is always the case in hisT ¼ sPTðÞs ds: ð10aÞ the systems studied. 0 A characteristic example of calculation results for the 4 nm NP Alternatively, the mean first passage time can be calculated release obtained by solution of the FP equation with symmetric directly using the Kramers method [49] as is done by Su et al. and asymmetric boundary conditions (8a) and (8b) is given in [24,54] Fig. 11. The left graph represents the probability distribution, Z Z ðÞ 1 Z PT s , of the release time defined as the first passage time of NP dif- 0 0 hi¼sT dZ exp ðÞEZðÞ=kT dZ exp EZ =kT ð11Þ fusion within the symmetric energy landscape. Since this process 0 0 consists of multiple cycles of NP return to the initial equilibrium hi In the process of a random walk along the symmetric free state at Z = 0, the mean time of translocation sT is very large, of 13 energy landscape, the NP makes multiple attempts to achieve the the order of 10 in dimensionless units. The right graph represents ðÞ barrier and escape, returning to the initial equilibrium state after the probability distributions pT=R s of the successful release and each attempt. As such, the dynamics of release may be considered return time during a single translocation attempt. This process is differently. The FP Eq. (6) with absorbing boundary conditions at defined within the asymmetric energy landscape with the bound- Z ¼ 0 and Z ¼ 1, ary condition (8b). This process has a very short time scale because it describes the escape and return time during a single transloca- WZðÞ¼; s 0atZ ¼ 0; 1 ð8bÞ tion attempt, of the order of 10 4 for the return and 10 1 for describes one translocation attempt which may be either success- release. The probability of release is so small that it requires on ful, when the NP achieves the barrier atZ ¼ 1 and escapes, or unsuc- average hin 5 1016 attempts for successful translocation. As cessful when the NP returns to the initial state at Z ¼ 0. Note that in expected, the mean time of translocation hisT estimated via Eq. this case, the initial d-function condition must be shifted from (14), that is proportional to hin , reasonably agrees with hisT found Z ¼ 0, by a small increment, DZ, equal to the finite difference for via the symmetric case solution. Both estimates agree well with hi 12 integration WZðÞ¼; 0 dðÞZ0 DZ . DZ ranges from 0.01 to 0.0001 the Kramers relationship that gives sT 5 10 . Detailed com- depending on the system. Smaller DZ is selected for higher energy parison of the translocation times calculated by the three methods barrier systems to increase precision. is provided in the Supporting Information, section 4. It is worth ðÞ The probabilities of successful translocation into the LB, pT s , noting, that solution of the FP equation with asymmetric boundary ðÞ and return to the bulk, pR s , within given time s are given by conditions is more computationally efficient, especially for the sys- the respective probability fluxes: tems with large energy barriers, than with the symmetric boundary conditions due to the significant timescale difference. Also, it 68 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70

Fig. 11. Distributions of the release and return times for the 4 nm NP trapped in the LB core. Time is given is dimensionless units. Left: The probability distribution, PTðÞs ,of the translocation time from the solution of the FP equation with symmetric boundary conditions (7a). Insert shows the mean translocation time indicated by the broken line. ðÞ Right: The probability distributions pT=R s of the successful escape and return time during a single translocation attempt from the solution of FP equation with asymmetric boundary conditions (7b). Note the difference in the time scales. Insert shows the mean times of release sT;1 and return sR;1 during one translocation attempt, the mean number of attempts needed for achieving the successful escape hin , and the mean translocation time hisT (13). Note that despite of the huge difference in the timescales, the mean translocation time hisT estimated by the two methods are in reasonable agreement.

contains a more detailed description of the physics of translocation attempt cycles, the mean time of translocation hitT is given tion. At the same time, the Kramers relationship (10) provides a by the similar equation as Eq. (14) for the mean time of release, direct easy method for calculating the mean translocation time. his ¼ s ; þ s ðÞþhin 1 s ; s hin ð15Þ Results of calculations are summarized Table 2. T R 1 ext T 1 ext The translocation probability and characteristic time is mainly Here s ¼ t = L2=D is the dimensionless mean time of external determined by the height of the energy barrier. Interestingly, that ext ext the energy barrier for release is not monotonic – the 4 nm NP has diffusion. The approximate equality holds for all cases considered, the lowest energy barrier for release. To overcome this barrier is since the time of external diffusion significantly exceeds the time 16 hi takes about 510 attempts, yet it’s feasible to escape with a rea- of return, sext sR;1 , the number of attempts n 1, and the sonable physiological time (this estimate is made assuming the SE time of the single successful translocation attempt makes insignifi- diffusion coefficient). For 2 and 8 nm NPs the energy barriers are cant contribution compared with the time of multiple return cycles. too high, and the release probability is so small, that it would In the real units of time, Eq. (15), converts into require an unrealistic number of attempts. The high energy barri- h i¼½ h ihðÞþhi i 2= þ ðÞh i h iðÞ ers are explained by the mechanism of interfacial NP transfer tT sR;1 n 1 sT;1 L D text n 1 text n 16 related to the membrane bending, formation and rupture of lipid The results of simulations are summarized in Table 3. The junctions, as discussed above in Section 3.1. The 2 nm NP nicely fits energy barriers of intake are substantially lower than those of the LB hydrophobic core and thus is strongly retained. Diffusion of release. The energy barriers of several kT for small NPs do not rep- 8 nm NP release from the membrane requires substantial LB defor- resent any hindrance for translocation, as the required number of mation that is restricted by the condition of constant tension. attempts does not exceed several thousand. Several thousand Dynamics of NP intake. NP transport in the process of intake is attempts translate into the timescale of intake on the order of sec- physically different, as the system is asymmetric – the successful onds, assuming that the characteristic scale of external diffusion is translocation occurs only in one direction, towards the LB. In order in microns. to translocate inside the membrane, the NP has to first diffuse from The energy barrier, DE, and respectively the number of required the bulk toward the membrane and then to attempt to overcome translocation attempts, hin , dramatically increases with the NP the energy barrier by diffusion along the free energy landscape. size, scaling exponentially with the energy barrier, hin eDE=kT ,as This attempt may be either successful or unsuccessful. If the shown in Fig. 12. For the 8 nm NP, the energy barrier amounts translocation attempt is unsuccessful, the NP returns to the bulk DE 40 kT and the respective number of required translocation and this cycle of external diffusion and translocation attempts attempts is hin 1018. At these conditions, the characteristic time are repeated until the NP successfully reaches the edge to the energy barrier and translocates into the membrane. The characteristic time of the external diffusion stage depends on the environment outside the membrane and can be estimated as ¼ 2 = text Lext D, where Lext is the characteristic length of the external diffusion. One may expect that depending on the environment,

Lext varies on the order from microns to millimeters. Therefore, the Stokes-Einstein equation for NP diffusion provides an estimate 3 3 for text ranging from 10 to 10 sec for NPs of different size. The FP Eq. (6) with absorbing boundary conditions (8b) describes the dynamics of a single translocation attempt. The probabilities and mean times of successful and failed translocation attempts, and the mean number of attempts needed for the successful translocations are given by Eqs. (10)–(12). Accounting for the mean time of external diffusion during each of the transloca- Fig. 12. Scaling of lnhi n as a function of the energy barrier, DE. S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70 69

of intake, hitT , is determined by the time of external diffusion text, It follows from our analysis that there is an optimal size for the Eq. (16), which, by any conservative estimates, exceeds millisec- unforced trans-membrane transfer of hydrophobic NPs by the onds. That leads to absolutely unrealistic probability of unforced encapsulation-release mechanism. While 4 nm NPs may swiftly encapsulation. penetrate the membrane core and then escape within reasonable timeframe, 2 nm NPs are captured by the hydrophobic core and strongly retained, and 8 nm NPs experience prohibitively high- 4. Conclusions energy barrier preventing their intake. Moreover, if 8 nm NP is encapsulated by the membrane, it is strongly retained, and its Using DPD simulations, we investigated the interactions release is restricted by prohibitively high barrier. Noteworthy, that between LB membranes at isotension conditions and hydrophobic the mechanism of encapsulation in the membrane interior we con- NPs covered by an equilibrium LM. A novel simulation setup was sidered is different from the mechanism of NP wrapping by the developed, permitting us to probe the force of NP-LB interaction membrane, which is possible only if the NP size exceeds a certain and to calculate the free energy landscapes of NP intake and threshold [55–57]. release. The simulations were performed for particles of 2 (compa- It is worth noting that here are multiple factors affecting trans- rable with the LB hydrophobic core thickness), 4 (comparable with membrane transport that are not present in our simplistic coarse- the LB thickness), and 8 nm (exceeding the LB thickness), interact- grained models of the lipid membrane and its environment and ing with a DMPC bilayer held at isotension conditions. We reveal should be addressed in the further studies. First, we consider an the mechanisms of NP intake and release, which are associated ideally homogeneous LB without any defects and inclusions, which with irreversible spontaneous interfacial transitions. In the course may facilitate NP translocation. Second, we do not consider any of intake, the hydrophobic NP must overcome the resistance of the additives in the solvent. Third, the NP is round, smooth and chem- water interlayer between particle and membrane, which prevents ically uniform without any ligands and hydrophilic entities. It is fusion of the LM coating and the outer leaflet of the membrane. As expected that non-spherical NPs with non-uniform hydrophobicity a result, the membrane bends, counter-balancing the disjoining may translocate easier. Fourth, our estimates are done for one pressure in the water interlayer until the latter becomes unstable nanoparticle ignoring cooperative effects, like accumulation of and ruptures allowing for rapid absorption of monolayer lipids small NPs inside the LB. Finally, our DPD model of water does by the membrane and spontaneous encapsulation of NP in the LB not account for the long-range effects of the disjoining pressure exterior. The process of release of encapsulated NPs involves mem- in the thin water layers between NPs and membranes [46,47]. brane deformation and formation of a lipid junction retaining the These long-range interactions may prevent LB-LM fusion and hin- NP, which extends and eventually breaks; upon the junction der NP intake. In order to evaluate NP translocation through real break-up, the LM-coated NP spontaneously detaches. cell membranes, this list must be expanded to account for their The energetics of the intake and release processes was charac- multicomponent nature, presence of cholesterol and proteins, terized by the respective free energy landscapes calculated with inherent morphological defects (e.g. gel islands and rafts), and the GT method. Noteworthy, due to the spontaneous interfacial electrostatic interactions involving charged membrane compo- transitions, no continuing translocation trajectory exists. The free nents and solvent ions, etc. Further studies should also address energy landscapes of intake and release terminate at the respective the cooperative effects of nanoparticle interactions outside and points of spontaneous transitions and do not merge. This behavior inside the membrane to explore their possible aggregation. of lipid pre-coated hydrophobic NPs is distinct from that of bare Despite the aforementioned reservations, the proposed model NPs with continuous translocation trajectories.[29] The interfacial to study the NP-LB interactions sheds lights on the specifics of transitions during intake and release are associated with free the mechanisms and dynamics of NP adhesion, intake and release, energy barriers, which depend on the particle size. The dynamics provides instructive quantitative estimates, and lays down a foun- of these transitions was studied by the Fokker-Planck equation dation for exploring more complex interfacial phenomena in bio- mimicking the NP Brownian motion along the calculated free logical environments. The conclusion about the existence of an energy landscape. We presented a novel modification of the FP optimal NP size for unforced translocation through the membrane approach, which accounts for the external diffusion of NP during may have practical implications for the choice of NPs for intracel- the intake attempts. This factor has not been considered in prior lular drug delivery and imaging. works and, as shown here, it significantly affects the mean translocation time. This FP method allowed us to establish the relationships for the probabilities of intake and release transition and 5. Notes characteristic time needed for the successful translocation. We found a linear scaling between the translocation probability and The authors declare no competing financial interest the energy barrier. While the energy barrier and respectively the rate of intake monotonically increase with the NP size, the rate of release does not scale exhibits a pronounced maximum for 4 nm NPs. The 6. Author contribution statement smallest (2 nm) and the largest (8 nm) particles have significantly higher energy barriers than the 4 nm particle. At the same time, the S.B. performed DPD and FP calculations. A.V. and A.V.N. barriers of intake for small (2 and 4 nm) particles are negligible so designed the simulation methodology. ZW performed DPD simula- that such particles would be eagerly encapsulated by lipid mem- tions on the initial stage of the project. S.B., A.V. and A.V.N. ana- branes and retained due the high release energy barriers. This con- lyzed and summarized data and prepared the manuscript. clusion is consistent with experimental observations from the literature that hydrophobic NPs comparable in size to the LB thickness remain inside the membrane without a chance to escape Declaration of Competing Interest within reasonable time limits [21–24]. For 8 nm NP, the energy barriers for both intake and release are significantly higher com- The authors declare that they have no known competing finan- pared to those of 2 and 4 nm NPs, that makes unforced trans- cial interests or personal relationships that could have appeared membrane transfer of NP larger than ~8 nm hardly probable. to influence the work reported in this paper. 70 S. Burgess et al. / Journal of Colloid and Interface Science 561 (2020) 58–70

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