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Ping-ping Xia, Yue Shan, Lin-li He, Yong-yun Ji, Xiang-hong Wang, Shi-ben Li. Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes[J]. Chinese Journal of Chemical Physics , 2020, 33(4): 468-476. doi: 10.1063/1674-0068/cjcp1910174
Citation: Ping-ping Xia, Yue Shan, Lin-li He, Yong-yun Ji, Xiang-hong Wang, Shi-ben Li. Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes[J]. Chinese Journal of Chemical Physics , 2020, 33(4): 468-476. doi: 10.1063/1674-0068/cjcp1910174

Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes

doi: 10.1063/1674-0068/cjcp1910174
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  • Corresponding author: Shi-ben Li. E-mail:shibenli@wzu.edu.cn
  • Received Date: 2019-10-06
  • Accepted Date: 2020-03-08
  • Publish Date: 2020-08-27
  • Multinanoparticles interacting with the phospholipid membranes in solution were studied by dissipative particle dynamics simulation. The selected nanoparticles have spherical or cylindrical shapes, and they have various initial velocities in the dynamical processes. Several translocation modes are defined according to their characteristics in the dynamical processes, in which the phase diagrams are constructed based on the interaction strengths between the particles and membranes and the initial velocities of particles. Furthermore, several parameters, such as the system energy and radius of gyration, are investigated in the dynamical processes for the various translocation modes. Results elucidate the effects of multiparticles interacting with the membranes in the biological processes.
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Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes

doi: 10.1063/1674-0068/cjcp1910174

Abstract: Multinanoparticles interacting with the phospholipid membranes in solution were studied by dissipative particle dynamics simulation. The selected nanoparticles have spherical or cylindrical shapes, and they have various initial velocities in the dynamical processes. Several translocation modes are defined according to their characteristics in the dynamical processes, in which the phase diagrams are constructed based on the interaction strengths between the particles and membranes and the initial velocities of particles. Furthermore, several parameters, such as the system energy and radius of gyration, are investigated in the dynamical processes for the various translocation modes. Results elucidate the effects of multiparticles interacting with the membranes in the biological processes.

Ping-ping Xia, Yue Shan, Lin-li He, Yong-yun Ji, Xiang-hong Wang, Shi-ben Li. Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes[J]. Chinese Journal of Chemical Physics , 2020, 33(4): 468-476. doi: 10.1063/1674-0068/cjcp1910174
Citation: Ping-ping Xia, Yue Shan, Lin-li He, Yong-yun Ji, Xiang-hong Wang, Shi-ben Li. Multinanoparticle Translocations in Phospholipid Membranes: Translocation Modes and Dynamic Processes[J]. Chinese Journal of Chemical Physics , 2020, 33(4): 468-476. doi: 10.1063/1674-0068/cjcp1910174
  • With the development of nanotechnology, nanoparticles (NPs) of various shapes and sizes have been extensively used in biology applications [1-3]. For example, NP as a carrier can bring desired substances, such as proteins and drug molecules, by penetrating the membranes into the intracellular place, thereby providing broad application prospects in drug delivery [4-7]. These potential applications require the efficient control of the interaction between the NPs and cell membranes. Thus, clear understanding of the interaction between the NPs and cell membranes is very important in the biological technique [8-10].

    Considerable experiments and computer simulations have contributed to the interactions between the NPs and membranes during the last few decades [11]. The latest experimental data show that the complexity of the diffusion dynamics of particle-membrane interaction is beyond the scope of classical Brownian dynamics. Particles exhibit a variety of diffusion motions from anomalies to classical diffusion on a wide range of time scales, which has prompted researchers to propose different models [12, 13]. In general, the two basic ways that the NPs transfer through a membrane are endocytosis and direct penetration [14, 15]. Endocytosis, also known as endocytosis or endocytosis [16, 17], is the process of transporting extracellular materials into cells through the deforming movement of lipid membranes. As reported, the NP characteristics, such as size, shape, and surface decoration, can significantly affect their interaction with the cell membranes. As observed in experiments, the simulations predicted that the large particles between tens and hundreds of nanometers in size can be engulfed by the membranes [18-20]. Larger-sized NPs are easier to enter cells through endocytosis. Specifically, this translocation can be divided into three sub-processes [21, 22]. The first sub-process is the particle attachment to the membrane. The second is the enclosing of the particles by the membrane. Finally, the separation occurs, that is, the particles are separated from the membrane. Meanwhile, the NPs of various sizes can also be wrapped in different ways, in which the NPs are enclosed within the cell membranes [19, 23, 24]. The experimental and theoretical studies and the computer simulations have shown that the wrapped pathway is related to the sizes of NPs [14, 25]. For smaller NPs, the bending energy required to overcome the endocytosis process increases significantly, making endocytosis no longer very effective at this time. In this case, the direct penetration method becomes very effective [14, 26, 27]. In addition, some charged NPs may induce the formation of nano-scale holes on the cell membrane surface, so that they or other NPs can enter the cells from these nanopores [28, 29]. But reaching some holes can often cause damage to cells, and even cause them to die [30]. Conversely, the membrane deformation induced by NPs is also determined by the interaction strength between them, in which the NPs' shape and size play an important role during the wrapping processes [31-33].

    Another factor that affects the interactions between the NPs and membranes is the shape of NPs, this condition has been extensively investigated in the experiments and simulations [34-38]. For example, the NPs with the ellipsoidal cross-section, e.g., the nanorods, cubes, nuts, and hexagonal columns, have different rates in the penetrating processes [39]. Furthermore, the simulation predicted that a sharp shape provides easy penetration, consistent with the experimental results [40]. The physicochemical properties, for example whether NPs carry electricity or not, also limit the penetration of the NPs [41-43]. However, less attention was given to the effect of velocity and interaction strength of NPs between the membranes on the translocation processes. In the present work, we focus on the influence of NP velocity and the interaction strength on the translocation processes and construct the phase diagrams of translocation modes based on these two parameters, in which the dynamical processes are discussed to provide a further understanding.

  • We used the dissipative particle dynamics (DPD) method to simulate the dynamical processes of the system in the present work. In the DPD method, a group of atoms are coarse-grained to be a single bead, in which these DPD beads interact with each other [22, 44, 45]. Typically, the dynamics of the $i$-th DPD bead with the mass of $m_i$ is controlled by Newton's equation of motion, i.e., $\textrm{d}{\bm v_i}/\textrm{d}t$= ${\textbf{F}_i}/{{\textbf{m}_i}}$, where ${\bm v_i}$ is the velocity, and $ \textbf{F}_i$ is a force acting on the $i$-th bead. The force appears in a pairwise manner, which includes three contributions, namely, a conservation force $\textbf{F}_{ij}^\textrm{C}$, a dissipative force $\textbf{F}_{ij}^\textrm{D}$, and a random force $\textbf{F}_{ij}^\textrm{R}$. Thus, the total force acting on the $i$-th bead obeys the following equation:

    where the sum of the forces acting on the bead or particle depends on its cutoff radius, $r_\textrm{c}$. In other words, the total force can be specifically expressed as follows:

    where ${a_{ij}}$ is the maximum repulsive force between the bead $i$ and bead $j$; $r_{ij}$ is the distance between them; ${\bf{\hat r}_{ij}}$ is the unit vector, and ${\bf{\hat r}}_{ij}$=${{\bf{r}}_{ij}}{\rm{ = }}{{\bf{r}}_{ij}}/\left| {{{\bf{r}}_{ij}}} \right|.{{\bf{v}}_{ij}}$ is the difference of the velocity between beads $i$ and $j$.${\zeta _{ij}}$ denotes a random number with zero mean and variance 1, and $\gamma$ and $\sigma$ are parameters coupled by ${\sigma^2} $= $2\gamma {k_\textrm{B}}T$, where $k_\textrm{B}$ is the Boltzmann constant. The weighting function $w(r_{ij})$ can be expressed as follows:

    The standard values $\sigma$=3.0 and $\gamma$=4.5 are used in the present simulation similar to the previous simulation [22]. In the DPD simulations, the length, timescales, and mass are all in unity.

  • The simulation system consists of water, phospholipid bilayer membrane, and NPs or cylindrical NPs. For simplicity, each phospholipid membrane consists of a hydrophilic head chain and two hydrophobic tails, which have been widely used in previous studies [20, 22, 46]. The phospholipid molecules are represented by the H$_5$(T$_9$)$_2$ coarse-grained model, as shown in FIG. 1. The hydrophilic beads (H) and hydrophobic tails (T) are shown as red and deep blue beads, where $N_{\textrm{HB}}$ and $N_{\textrm{TB}}$ are used to denote the number of head beads and tail beads, respectively. Light yellow beads and sticks represent NPs (P) and cylindrical NPs, respectively. The number of beads in the head chain is five, and the number of beads in each tail chain is nine. Meanwhile, the rigid cylindrical NPs are made of eight beads. The head chain is rigid, and the tail chain is semi-flexible. For the phospholipid molecule, an extra elastic harmonic force is defined between the adjacent particles in the chain to hold the beads together, given by

    Figure 1.  The schematic diagram for phospholipid molecule with one head group and two tail group. (a) Spherical and (c) cylindrical nanoparticles interacting with the membranes are shown. (b) The corresponding coarse-grained model for the phospholipid molecules is shown. The red and blue colors represent the head and tail beads, and nanoparticles are represented in yellow colors.

    where $k_\textrm{s}$ and $r_\textrm{s}$ are the spring constant and equilibrium bond length, respectively. To construct the coarse-grained models of phospholipid molecule, we use spring constant $k_\textrm{s}$=100.0 and an equilibrium bond length $r_\textrm{s}$=0.7$r_\textrm{c}$ in the present simulation, similar to the previous studies [22, 47]. Additional forces were generated by harmonic constraints to indicate bending resistance on two consecutive bonds of the phospholipid chain.

    where $k_\theta$ is the bending constant, $\theta$ is the inclination angle, and $\theta_0$ is equilibrium angle. We use $k_\theta$=6 and $\theta$=180$^{\circ}$ to represent the bending constant and equilibrium angle of three consecutive tail beads or three consecutive head beads on each phospholipid molecule chain. For the last head bead and the two top beads, $k_\theta$=3 and $\theta$=120$^{\circ}$. For the last two consecutive beads and the first bead, we take $k_\theta$=4.5 and $\theta$=120$^{\circ}$, as shown in FIG. 1.

  • The simulations are carried out in the cube box with the size of 30$r_\textrm{c}$$\times$30$r_\textrm{c}$$\times$30$r_\textrm{c}$, where the periodic boundary conditions are used [22]. In the DPD method, the simplified units are generally used for convenience. The cutoff radius $r_\textrm{c}$ determines the unit of length. The mass of the particle mass $m$ defines the unit of mass, and the energy $k_\textrm{B}T$ defines the unit of energy. Additionally, ${r_c}$=${\left( {\rho {V_{\rm{b}}}} \right)^{1/3}}$, where $V_\textrm{b}$ and $\rho$ are, respectively, the volume of one DPD bead and the particle density. In the water-like mixture systems, a cube DPD particle volume approximately can generally be assumed. Here, we take $\rho$=3, and find the cutoff radius by $r_\textrm{c}$=1.0 nm [48]. The modified velocity-Verlet algorithm is applied to the time integral of motion equation with a time step $\Delta t$=0.01$\tau$ [44]. In the present simulation, $\tau$ is the natural unit of time with the following form:

    where $m$ is the unit of bead mass. The value of natural unit of time for the DPD simulation can be determined so that the numerical value corresponds to the previous experimental value due to the diffusion constant of the lipid in the plane in the experiment [49, 50]. All the simulations are performed using LAMMPS [51]. In the simulations, to obtain good simulation results, the entire dynamic process went through 300, 000 DPD time steps. The system contains 81, 000 particles, with a particle density of approximately 3. The water beads are not shown in the following section for clarity. During the simulation, the repulsive interaction parameter between the two identical molecules of beads within the NPs, i.e. water and membrane, has been chosen as ${a_{ij}}$=25. In addition, the two types of microspheres of different types are fixed to be ${a_{{\rm{HT}}}}$=${a_{{\rm{TW}}}}$=${a_{{\rm{WP}}}}$=100 and ${a_{{\rm{HW}}}}$=25 (W stands for water bead). The repulsive interactions between the NPs and the phospholipid head and tail chains are denoted as ${a_{\textrm{HP}}}$, ${a_{\textrm{TP}}}$, respectively. NPs with spherical and cylindrical shapes are shown as yellow beads (FIG. 1) with parameters $\left({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}} \right)$=(0, 0), (25, 25), (50, 50), (75, 75), and (100, 100) [22, 52]. In the simulations, both the numbers of NPs with spherical and cylindrical shapes are chosen to be 25. We repeated each of the systems studied by changing the interaction parameters, $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$, and the velocity of the NPs, $v$.

  • Multiple factors, including the particle shape, surface properties, concentration, velocity, and interaction parameters $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$, have complicated effects on modulating the membrane translocation of NPs. Particle shape, surface properties, and concentration have been studied in previous experiments [14, 36]. In the present work, we are interested in the effect of $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) and the particle velocity $v$ on the particle penetration membranes.

    In this sub-section, we focus on the effect of interaction parameters $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) and the particle velocity $v$ on the translocation modes. By careful examination, several translocation models are sorted out for the NPs with spherical and cylindrical shapes, respectively, as shown in FIG. 2 and FIG. 4. Then, these modes are arranged based on $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) and $v$, constructing into the phase diagrams, as shown in FIG. 3 and FIG. 5, respectively.

    Figure 2.  Four typical translocation processes of SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, SP-Ⅳ. The nanoparticles are spherical, with the number of 25, while the lipid membranes have the parameters of $N_{\textrm{HB}}$=5 and $N_{\textrm{TB}}$=9. The red and blue colors represent the head and tail beads, and nanoparticles are represented in yellow colors.

    Figure 3.  The phase diagrams for the four typical translocation modes of SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, SP-Ⅳ, which is arranged by the interaction parameter $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$ and velocity $v$. The nanoparticles are spherical with the number of 25.

    Figure 4.  Four typical translocation processes of CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ. The nanoparticles are cylindrical, with the number of 25, while the lipid membranes have the parameters of $N_{\textrm{HB}}$=5 and $N_{\textrm{TB}}$=9. The red and blue colors represent the head and tail beads, and nanoparticles are represented in yellow colors.

    Figure 5.  The phase diagrams for the four typical translocation modes of CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ, which is arranged by the interaction parameter $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$ and velocity $v$. The nanoparticles are spherical with the number of 25.

    For the spherical NPs interacting with the membranes, four translocation modes are picked out by examining various $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) and $v$, as shown in FIG. 2, in which the mode symbols and translocation times are also listed. The first mode is defined as SP-Ⅰ. A typical example is shown in FIG. 2(a), in which the spherical particles are completely embedded into the membranes at the final stage. In the SP-Ⅰ mode, the particles are distributed randomly in the beginning and then moved to the membranes. By attracting from the membranes and the initial velocity of NPs, the NPs translocate through the membranes. This phenomenon is the second stage in the SP-Ⅰ mode. However, the NPs are adsorbed by the membranes, pulled back, and finally embedded into the membranes. In SP-Ⅰ, the membranes are not damaged in the stages, thereby exhibiting a good elasticity; the NPs are aggregated, as extensively reported in the previous simulations [15, 23, 53]. For the second mode, SP-Ⅱ, as shown in FIG. 2(b), the membranes are distorted severely. In SP-Ⅱ, the NPs are distributed randomly in the beginning and then are absorbed into the membranes in a disperse manner. In this way, the membranes are distorted, and subsequently, the NPs gather together and penetrate through the membranes. At the final stage, the NPs are attracted back to the membrane, similar to the SP-Ⅰ mode. This phenomenon is also consistent with previously reported simulations and experiments [45, 53].

    A third mode, SP-Ⅲ mode, is shown in FIG. 2(c), in which the NPs come together and pass through the membrane. NPs are randomly dispersed at the beginning, then absorbed by the membrane in a dispersed form until the NPs gather together, and shifted through the membrane in the form of clusters. In this case, the translocation of the NPs on the membrane is similar to that of SP-Ⅱ, and it produces a strong disturbance to the membrane. The main difference of SP-Ⅲ from the SP-Ⅰ and SP-Ⅱ is that the NPs are not trapped in the membrane. During this period, the membrane is disturbed obviously, as mentioned in previous experiments [54, 55]. Interestingly, in the SP-Ⅳ shown in FIG. 2(d), NPs are randomly distributed on one side of the membrane, and then, the NPs easily shift with the membrane in a dispersed form. In the end, NPs are distributed randomly on both sides of the membrane. In this case, the translocation of the NPs on the membrane creates a strong disturbance to the membrane and even breaks. These observations clearly show that the change in NP velocities and interaction parameters $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) may make the NP membrane penetration behavior completely different, thereby further confirming the complexity of the interaction of multiple NPs with lipid membranes in the previous studies [54, 56, 57]. For the translocation time of four SP modes, as shown in FIG. 2, we state that the processes have various times to reach the stable states, i.e., 450$\tau$, 1020$\tau$, 1560$\tau$, and 2040$\tau$ for the SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, and SP-Ⅳ, respectively. These data indicate that the system most easily reaches the stable state in the SP-Ⅰ mode. Besides, in order to balance the translocation efficiency of NPs and maintain the integrity of the phospholipid membrane [56], SP-Ⅰ is the best mode compared to other modes.

    We investigate the interaction parameters $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$ by fixing other parameters. As shown in FIG. 3, the head-bead interaction parameter ${a_{{\rm{HP}}}} $ and the tail-chain bead ${a_{{\rm{TP}}}}$ of the nanoparticle and the lipid chain are varied in the range of 0-100. Each step is increased by 25. The initial velocity of the nanoparticle is set to 1-10, in units of $\left( {{r_\textit{c}}{\tau ^{ - 1}}, \times {{10}^{ - 1}}} \right)$, to observe their effect on the phase behavior. The SP-Ⅳ mode occurs when the NP speed or the interaction parameters are relatively large, that is, the NPs can easily pass through the membrane and cause a large change in the membrane. This phenomenon indicates that high velocity will benefit the particle penetration [14]. SP-Ⅰ mode is observed when the velocity is small, and the interaction parameters are large. In SP-Ⅰ, the particles come together and interact with the membrane. Single NP with relatively small size cannot be fully wrapped by the cell membrane, prohibiting its uptake. One feasible way is cooperative entry [59]. This observation is similar to the results obtained in previous experiments [53, 54]. We also observe that SP-Ⅲ mode occurs when the interaction parameter is zero. The increase in velocity at this time does not affect the penetration of the NPs. This result may be attributed to the "softness" of the lipid membrane. The deformation of the bilayer produced by the interaction between NPs and lipid bilayer minimizes the influence of different impulses of particles [14]. SP-Ⅱ occurs at a chamfer, i.e., SP-Ⅱ is between the SP-Ⅰ, SP-Ⅱ, and SP-Ⅳ, in which the velocity is medium or has less interaction.

    Subsequently, we investigate the physical translocation processes of cylindrical NPs across the membranes. For the cylindrical NPs interacting with the membranes, four translocation modes are picked out by examining various $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) and $v$, as shown in FIG. 4. The first mode is defined as CP-Ⅰ. A typical example is shown in FIG. 4(a), in which the cylindrical particles do not physically move across the membrane at the final stage. The cylindrical NPs start to randomly distribute, but they do not cross the membrane over time. In the CP-Ⅰ mode, the membrane remains nearly unchanged during the whole process, because the membranes have good physical elasticity, and the cylindrical NPs are only attached to the membrane [14, 60]. Similar results have been found in other previous studies [61, 62]. In the CP-Ⅰ mode, the system quickly reaches the stable state, with time of 480$\tau$. For the second mode, CP-Ⅱ, as shown in FIG. 4(b), the cylindrical NPs are sucked into the membrane and finally embedded in the membrane, with a time of 540$\tau$. First, the contact between the cylindrical NPs and the membrane causes a strong deformation. Then, through the attraction of the membrane to the cylindrical NPs and the initial velocity of the cylindrical NPs, the cylindrical NPs are slowly pulled back into the membrane until they are embedded. In addition, the membrane slowly recovers its elasticity. In the third mode, CP-Ⅲ, some cylindrical NPs shift with the membrane, as shown in FIG. 4(c). The cylindrical NPs are randomly distributed, shifting through the membrane under the influence of membrane attraction and the initial velocity of the cylindrical NPs. In the first stage, the cylindrical NPs penetrate the membrane and cause the rupture of the membrane due to the large initial velocity of the cylindrical NPs. Gradually, under the action of the attraction of the membrane, the cylindrical NPs are randomly distributed on both sides of the membrane, and the membrane also regained elasticity. In CP-Ⅲ mode, the stable time is 720$\tau$. This result is similar to the deformation of the CP-Ⅱ mode membrane. Finally, in the fourth mode, CP-Ⅳ, the cylindrical NPs penetrate the membrane and break it, as shown in FIG. 4(d). First, the cylindrical NPs are randomly distributed near the membrane, and then, the cylindrical NP dispersion shifts with the membrane. Finally, cylindrical NPs are randomly distributed on both sides of the membrane. Among the four modes of CP obtained from the simulation, to maintain the balance between the nanoparticle's equilibrium shift efficiency and cytotoxicity, we can conclude that the mode of CP-Ⅱ is more meaningful.

    To clearly study the influence of initial velocity and interaction parameters of cylindrical NPs on the interaction with the membrane, we reduce the complex parameters to subspaces containing only velocity, interaction parameters (${{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$) by fixing other parameters. As shown in FIG. 5, we vary the head-bead interaction parameter ${a_{{\rm{HP}}}}$ and the tail-chain bead ${a_{{\rm{TP}}}}$ of the cylindrical NPs and the lipid chain in the range 0$ \le$${a_{{\rm{HP}}}}$$ \le$100, 0$\le$$ {a_{{\rm{TP}}}} $$\le$100, respectively. Each step is increased by 25. The initial velocity of the cylindrical NPs is varied within $v$=0.2-2.0 in the simulation in step $\Delta v $=0.2(${r_\textrm{c}}{\tau ^{ - 1}}$, $\times$$ {{10}^{ - 1}}$) to observe its effect on phase behavior. We observe that the second mode, CP-Ⅱ, occurs in the case of small velocity of cylindrical NPs. In this case, cylindrical NPs are embedded in the membrane, thereby indicating that the interaction parameters do not play a dominant role in small velocity. We also observe that CP-Ⅳ occurs in high velocity, when the interaction parameters have minimal influence on it, like CP-Ⅱ. Then, CP-Ⅰ occurs in small velocity and large interaction parameters, thereby indicating the difficulty of physical shifting of cylindrical NPs with the membrane when the velocity is small and the interaction parameters are large. Similar results have been reported previously [60]. Finally, CP-Ⅲ occurs when the velocity is appropriate, and the interaction parameters are large. Some cylindrical NPs shift with the membrane in scattered forms. By comparison, cylindrical NPs move much more slowly than NPs with membranes. Recently, synthetic nonspherical NPs have shown significantly improved biological properties over their spherical counterparts [61, 63].

  • In this subsection, we investigate the dynamic processes for the various translocation modes. We consider the system energies and chain conformations for the translocation modes, as shown in FIGs. 6-8. First, we consider the system energy $\langle E_{\textrm{tot}}/k_\textrm{B}T\rangle$ for the various SP and CP modes in the whole processes and show the functions of system energy with time step in FIG. 6. FIG. 6(a) describes the energy variation trends for the four SP modes, in which the parameters are chosen to be $v$=1.0 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(0, 0), $v$=4.0 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}}$)=(75, 75), $v$=4.0 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(0, 0), and $v$=7.0 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(75, 75) for SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, and SP-Ⅳ, respectively. As shown in FIG. 6(a), the energies for the SP-Ⅱ, SP-Ⅲ, and SP-Ⅳ show increasing trends and finally stabilize. However, the energy of the SP-Ⅰ basically does not change with the time step and tends to be stable. In the SP-Ⅰ, the dynamic interaction between the NPs and the membrane is not violent, and the membrane remains completely elastic. SP-Ⅱ exhibits more obvious variance than those in SP-Ⅰ because the NPs move fast. The same cases are observed in the SP-Ⅲ and SP-Ⅳ. Specifically, SP-Ⅳ has the greatest energy variation because the NPs are fast and the interaction parameters are large, thereby causing the complete breakage of the membrane. Then, we investigate the energy variances for the four CP modes to illustrate their trends, as shown in FIG. 6(b), in which the parameters are chosen to be $v$=0.2 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(100, 100), $v$=0.6 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(0, 0), $v$=0.8 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(75, 75), $v$=1.4 and $({{a_{{\rm{HP}}}}, {a_{{\rm{TP}}}}})$=(0, 0) for CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ, respectively. The data clearly show the stable $\langle E_{\textrm{tot}}/k_\textrm{B}T\rangle$=6.05, 6.15, 6.35, and 6.95 for these four modes, respectively. The system energy of the four modes increases with the time step. First, it gradually increases and then stabilizes. The energy also gradually increases and gradually reaches stability because of the unstable initial interaction between the cylindrical NPs and the membrane. Correspondingly, the four modes take different time to reach the stable states. Specifically, CP-Ⅰ maintains the energy increase stage of approximately 240$\tau$ and then remains stable, whereas CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ modes have time of 245$\tau$, 340$\tau$, and 480$\tau$, respectively. These results are consistent with FIG. 4. These data indicate that the velocity and interaction parameters of the cylindrical NPs affect the dynamic process and energy required to achieve a stable energy. In these modes, the energy change of the system is similar to the previous result [64, 65].

    Figure 6.  The system energy as functions of time steps for various (a) SP and (b) CP modes.

    Figure 7.  The gyration radii $\langle R_\textrm{g}\rangle$ of various SP modes of SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, and SP-Ⅳ during the dynamic process.

    Figure 8.  The gyration radii $\langle R_\textrm{g}\rangle$ of various CP modes of CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ during the dynamic process.

    Although both the system energies of SP and CP modes converge to stable values, the conformations of phospholipid molecules have periodical changes in these stages. Then, we investigate the chain conformations of phospholipid molecules in dynamical processes and plot their variances in FIGs. 7 and 8, in which the time begins at the stable stage. FIG. 7 shows the dynamic processes of gyration radius $\langle R_\textrm{g}\rangle$ for phospholipid molecules in different SP modes, in which the parameters of SP-Ⅰ, SP-Ⅱ, SP-Ⅲ, and SP-Ⅳ are the same as those in FIG. 6(a). As FIG. 7 shows, the gyration radii in SP-Ⅰ and SP-Ⅱ modes have changing periods in the stable stages of $T_{\textrm{SP-Ⅰ}}$=35$\tau$ and $T_{\textrm{SP-Ⅱ}}$=12$\tau$, respectively. The time is different due to the interaction between the NPs and membranes and the movement of the NPs. Similarly, the gyration radius of phospholipid molecules in the SP-Ⅲ and SP-Ⅳ modes has similar change trends but has different time, i.e., SP-Ⅲ and SP-Ⅳ period time are $T_{\textrm{SP-Ⅲ}}$=11$\tau$ and $T_{\textrm{SP-Ⅳ}}$=7$\tau$, respectively. In general, the gyration radii of the four modes gradually increase, then decrease, and finally stabilize. These results suggest that NPs interact with the membrane and then pass through. The periods are different because of the different velocities of NPs and different interaction parameters [66]. When we change the interaction parameters between the membrane and NP, different modes are found. Given that the interaction parameters between the NPs and membrane are very large, the membrane wrapping of NPs has a slow rate and limited extent, as observed in our previous work [67].

    FIG. 8 shows the gyration radii of phospholipid molecules in different CP dynamic processes, in which the CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ parameters are the same as those in FIG. 6(b). A shown in FIG. 8, the gyration radii variation of phospholipid molecules in CP-Ⅰ, CP-Ⅱ, CP-Ⅲ, and CP-Ⅳ modes with time have a changing time of $T_{\textrm{CP-Ⅰ}}$=130$\tau$, $T_{\textrm{CP-Ⅱ}}$=41$\tau$, $T_{\textrm{CP-Ⅲ}}$=38$\tau$, $T_{\textrm{CP-Ⅳ}}$=22$\tau$, respectively. CP-Ⅰ takes the longest changing time. The modified diffusion model proposed by Zhu et al. provides dynamic analysis of the components of cells and other microscopic objects for our time-resolution measurements [13]. Meanwhile, obstacles existing in the process of dynamic energy cannot be easily overcome, and the membrane can bend. As a result, a long time of fluctuation is required. In other words, the cylindrical NPs of different speeds and different interaction parameters lead to different period time [66]. Furthermore, our simulation results indicate that the four modes show different cylindrical NP velocities and interaction parameters depending on the dynamic process. These observations indicate that NP adhesion has an important effect on endocytosis [19].

  • The translocations of spherical and cylindrical NPs in the phospholipid membranes are studied using the DPD method. By changing the initial velocity and interaction parameters of NPs, we observe a variety of translocation mode for spherical and cylindrical NPs. When the velocity of the NPs or the interaction parameter is small, the NPs have difficulty passing through the phospholipid membrane and are eventually embedded in the membrane. When the velocity or interaction parameter is relatively large, the spherical and cylindrical NPs can penetrate the lipid membrane relatively easily, thereby causing large membrane deformation and even cracking of the lipid membrane. However, for cylindrical NPs, penetrating the membranes is difficult, when the velocity is small and the interaction force is relatively large. Our results suggest that the increasing interactions may significantly affect the translocation and result in different membrane granule superstructures. This outcome may not be achieved by using moderate interaction intensities. Among the four modes formed by the interaction between spherical NPs and cylindrical NPs and the membrane, the total energy of the system, the time to reach equilibrium, and the radius of rotation of the phospholipid membrane are different, which may be due to the different shapes of the NPs. In comparison, cylindrical NPs are more likely to interact with the membrane.

    We further study the dynamic processes of the interaction between NPs and the membrane. The total energy and the gyratory radius of the system differ between SP and CP modes. In comparison, the total energy of SP mode is larger than that of the CP mode. In addition, the gyratory radius has similar changes in the stable processes. The translocation becomes complicated with increasing interaction strength between the membrane and the particle. With increasing interaction strength, less time is needed for particle endocytosis. Large interaction strength enhances the ability of phospholipid diffusion. In comparison, the dynamics of spherical NPs are more complex than that of cylindrical NPs. Our results deepen our knowledge on the effects of multiparticles interacting with the membranes in the biological processes and provide some hints for designing NPs as stimulus materials in an experiment.

  • This work was supported by the National Natural Science Foundation of China (No.21973070, No.21474076, No.21674082, and No.11875205)

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