2. Simulation Methodology

2.1. Modelling method of nanoparticle

Rigid particle is the most frequently introduced model (Figure. 2A). This coarse-grained morphology is generally mapped from an existing particle model such as the Fcc lattice structure19. The coarse-grained beads are not necessary to be connected by bonds, but the relative position between the beads is usually required to be invariable to avoid shape degradation or even disintegration during the simulation process. In order to ensure the rigidity of nanoparticles, it is usually indispensable to furnish geometric constraints to the beads by adjusting the parameters of the potential energy equation:
where the eq (1) ~ (3) is applied to bond length, bond angle, and dihedral angle, respectively. Based on different research purposes, the bead types on the particle surface can be changed or just be connected with other bead fragments to finish surface modifying. During this process, researchers can adjust the modification ratio and distribution according to actual needs to obtain a wide variety of nanoparticles. This particle modelling is very common in this field since its method is simple and can simulate some rigid inorganic particles and cell membranes. However, it cannot well reflect the mechanical behaviour of soft matter particles due to the neglect of the nanoparticle deformation. Another common nanoparticle model comes from most studies block copolymer self-assembly20-22. This type of particle contains a hydrophobic drug-carrier inside and a protective hydrophilic layer outside (Figure. 2B). This type of model conforms to the practical design of general amphiphilic polymer drug-loading nanoparticles. However, due to the flexibility of the particles, the nanoparticles will experience large deformations during the simulation process, which will lead to large energy fluctuations. Simultaneously, the insufficient cohesive energy may render nanoparticles disintegrate before the process of nanoparticles entering the cell to be observed23.
The complex components have also brought huge challenges to parameter controlling, so this modelling method is not commonly used in this field. To overcome this adversity, Pivkin24 et al. proposed a new method using a mesh model to model nanoparticles: each bead on the surface of the nanoparticle is a vertex of a two-dimensional curved triangular surface, and each vertex is connected by springs with a certain number of other vertices (Figure. 2C). The typical feature of this model is that it can adjust the elastic modulus of nanoparticles by altering many limited parameters to assist researchers in discussing the influence of nanoparticle elastic deformation on the process of cell absorption. Readers may check these references25,26for the detailed control methods parameters.
In addition to building a nanoparticle model, the modeler should also consider the driving force for the nanoparticle to enter the cell, such as ligand-receptor binding, van der Waals force, hydrophobic interaction, chemical potential gradient, etc27. One simple method is to directly apply a spring force directed to the cell membrane on the nanoparticles28. The most common method is to define ligand beads on the surface of the nanoparticle and receptor beads on the surface of the cell membrane model, and the strong interaction between receptors and ligands can be defined by adjusting the force filed parameters between the two beads29. For example, some researchers may set the repulsive parameters in the DPD force field of the two beads to approximately zero to highlight the strong attraction30. A more general method is to quantify the ligand-receptor interaction by Lennard-Jones (L-J) potential as follows31:
among which, ε represents the strength of the ligand-receptor interaction. Since the background of this type of research often involves the cancer cell membrane where the negative charge density on the surface is significantly greater than that of normal cells23, researchers should also pay attention to the calculation of the electrostatic force. The long-range electrostatic forces in molecular dynamics simulation are generally handled by the Ewald summation or particle grid Ewald method. Furthermore, there are also many well-matched electrostatic force models for specific simulation methods, such as the electrostatic force model for DPD force fields based on particle-particle-particle-mesh (PPPM) method proposed by Groot32,33. In short, the modelling of nanoparticles needs to consider various settings such as particle geometric constraints, surface modification and driving force.

2.2. Modelling method of cell membrane

The cell membrane is usually represented by the phospholipid bimolecular membrane model, involving two main aspects: the coarse-grained model of the phospholipid and the construction of bilayer topology. Figure 3A illustrates several common phospholipid coarse-grained models. The Y-shape phospholipid coarse-grained model, which contains three hydrophilic beads and two hydrophobic bead tails, is established by Groot34 and revised by Kranenburg35. Then, Shillcock and Lipowsky have proposed a λ -shape model that could better match the actual phospholipid structure36. These models emphasize the mapping of three heavy atoms into a coarse-grained bead, but this is not consistent with the conception of the MARTINI force field. Therefore, Gao37 et al. have established the H-shape model on the basis of the four-to-one scheme with more detailed bead classification according to the MARTINI force field and calculated the repulsive parameters suitable for the explicit / implicit-solvent DPD simulation38. They have also verified that this new force field could well reproduce the structure and thermodynamic properties of the bimolecular membranes37. Although the above model can more accurately reflect the real physical and chemical properties of cell membranes, the complicated bead classification and irregular topology have caused a great challenge for subsequent simulations. Therefore, more researchers have adopted more simplified models or even linear models to simulate the phospholipid molecule. Such a simplified model can greatly improve simulation efficiency, thereby broadening the scope of biomedical application.
It is inevitable that the simplified model may ignore many details of the cell membrane, so a huge amount of effort is obliged to ensure that the formed bimolecular membrane conforms to the morphological and mechanical properties of the real cell membrane. First, a weaker bond length constraint is supposed to be given to the first hydrophobic bead of the two hydrophobic tails in the lipid model to ensure that the majority of phospholipid molecules can maintain a reasonable phospholipid tail orientation39. Simultaneously, the bond angle constraint is also applied to ensure the phospholipid molecule’s rigidity31. Besides, the boundary conditions may cause the molecular relaxation, thereby destroy the continuity of the membrane. In order to reduce this effect, it is necessary to maintain the global conformation of the molecular membrane by limiting the movement in the longitudinal direction of the phospholipids near the boundary region27.
Furthermore, the N-varied DPD method is introduced to control the membrane tension, which can monitor the number of phospholipids per unit area (LNPA) in the boundary area. When LNPA is lower than the set value (the value of the tensionless membrane equals 0.64 nm), a certain number of water beads in this area will be replaced with phospholipid beads to help researchers maintain membrane tension or artificially increase external tension40. Figure 3B is a common coarse-grained phospholipid bimolecular membrane model. Generally, the rationality of the model is manifested by the membrane thickness and the phospholipid diffusion coefficient which should be close to 8 nm and 5μ m2s-1, respectively41.
There are multiple components in cell membranes and diverse types of phospholipids. Beena Rai’s group19 has used the MARTINI force field to construct a multi-component cell membrane containing different phospholipid molecules and embedded transport proteins as shown in Figure 3C. This model has been utilized to study the interaction between nanoparticles with different shapes as well as chemical properties and cell membranes. Yang42 et al. have discussed the influence of phase separation of disordered-gel phase on nanoparticle interaction by introducing unsaturated phospholipid. With the development of abundant modelling methods for nanoparticles and cell membranes, the credibility of computational simulation results has also been greatly enhanced.