Therefore it was only used in a few simulation experiments to determine the model’s phase diagram, while most of the simulations employed the second model, the phan-tom solvent beads. Therefore, the bilayer reference model uses the phanphan-tom solvent beads.
5.4.1. Surface potential solvent environment model
In the first solvent environment model, the bilayer is constrained by two parallel planes. The lower plane is thex-y-plane itself, the upper plane is shifted byzupper≥0.
The tail beads of the bilayer are confined between the planes by the surface potential VST (equation 5.6), while the head beads are forced to stay above the upper plane respectively below the lower plane byVSH(equation 5.7). The parameters have been chosen arbitrarily to be= 10,r0= 0 and∆rmax= 0.5).
VST(r) =
VFENE(z) ,if z <0 VFENE(z−zupper) ,if z > zupper
0 ,otherwise
(5.6)
VSH(z) =
VFENE(z) ,if 0< z < 12zupper
VFENE(z−zupper) ,if 12zupper< z < zupper
0 ,otherwise
(5.7) This solvent environment model is very simple to implement and is also very effi-cient when it comes to computing. Unfortunately, the bilayer is not flexible in this model, i.e. it can not undulate or deform on longer length scales. Therefore, it is useless for the simulation of phenomena that involve any membrane deformations, such as undulations or hydrophobic mismatch effects of membrane integral proteins.
As these phenomena are the main goal addressed by this work, the surface potential model was used only during the first simulation studies performed for this work and is seen as a variant of the bilayer reference model (see section 7.3 on page 84).
Note, that the upper plane position zupper can drop to 0, as the soft potentials associated with the planes allow for beads to permeate up to a certain depth into the disallowed regions. When the volume of the system is defined to beV =LxLyzupper, this would actually allow the volume to drop to0.
5.4.2. Phantom solvent beads
The phantom solvent bead model was developed to retain the full membrane flexibil-ity, while still adding only a small computational overhead. In this model, the solvent is represented by explicit solvent beads. These beads behave exactly like additional, non-bonded head beads, i.e. they have a purely repulsive soft-core interaction with the lipid beads. What is special about these beads, however, is that they do not interact with each other.
The phantom solvent beads have two effects on the lipid bilayer. The main effect of the solvent is, that it mediates the external pressure onto the bilayer by means
5.4. Solvent environment model
of an excluded volume interaction: the phantom solvent probes the lipid-bead free volume, gouverned by the ideal gas law, thus excerting pressure onto the bilayer, which keeps the bilayer together and gives the system a preference for well-packed lipid configurations. This is an advantage over implicit solvent models, where external pressure can not be applied.
Another, more subtle effect of the phantom solvent is, that it creates an attractive depletion interaction between the lipid beads at the interface between the phantom solvent and the the lipid, yielding in a surface tension. As in the most cases, only head beads can be found at the interface, this results in an effective attractive interaction between the head beads. The strength of the interaction is controlled by the external pressure, while the range of the interaction is gouverned by the size of the solvent beads.
Note, that the depletion interaction only affects the beads at the solvent–lipid in-terface, i.e. the head beads, while the effect of the external pressure to minimise the system’s volume in general affects the whole bilayer. Therefore, the relation between the head group attraction and the tail group attraction can be finely tuned by the choice of the pressure. This influences the general phase behaviour of the model, and the ripple phase Pβ0 and interdigitated phaseLβI in particular (see chapter 7 on page 73).
Using the phantom solvent in a computer simulation of lipids has a number of advantages. First of all, the model is still computationally very efficient: phantom solvent beads that are far from the bilayer do not have any interaction partner. In a Monte-Carlo simulation together with the cell-lists algorithm, this means that the energy computation for these beads can be skipped. Only those solvent beads that are actually close to the bilayer significantly contribute to the computing time. In fact, simple considerations show, that the efficiency of the phantom solvent environment model at low to intermediate pressures is comparable to the efficiency of the so-called solvent-free models that use an implicit solvent, as all of these models require longer ranged potentials.
In contrast to the surface potential model, the phantom solvent model does not put any constraints on the shape of the bilayer: the bilayer is fully flexible and can undulate or bulge. However, the surface tension induced by the depletion interaction between the interfacial beads leads to a preference for assemblies of the lipids.
Compared to a model of explicit, interacting solvent beads, the phantom solvent has the advantage, that the phantom solvent can not develop an internal structure. This rules out any artefacts in the bilayer behaviour caused by the solvent. Furthermore, only a relatively thin layer of solvent is required to screen the bilayer from interacting with either the system’s wall or the bilayer’s periodic image, depending on the type of boundary conditions used. If a thicker layer of phantom solvent beads is used in a constant volume simulation, the pressure of the system can be measured by measuring the solvent density far from the bilayer, as the ideal gas equation of state holds for the phantom solvent beads there.
It is also possible to use the phantom solvent environment model in a molecular dynamics simulation. In that case, it should even be possible to use a dissipative
particle dynamics (DPD) thermostat to study the solvent’s hydrodynamics.
To summarise, the phantom solvent model is a simple way to excert external pres-sure onto the bilayer without putting any constraints onto the bilayers flexibility, with-out the disadvantages of solvent artefacts and with a computational effort comparable to that of implicit solvent models.