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.