stalk2 apposed
5.5. GRAND POTENTIAL OF THE STALK 93
density 6 5 4 3 2 1 0 -1 -2 -3 -4 -5
x
r
Figure 5.16: 2D contour plot of the difference between the molecular density distribution of hydrophobic A-species in the stalk morphology at NA=28, radially averaged around the central axis of the stalk, in the grand canonical ensemble atµ−µ0=0.100kBT and the one at µ−µ0=−0.100kBT (see also Fig. 5.5). The graph shows the molecular density distribution of tails atµ−µ0=0.100kBT minus the distribution atµ−µ0=−0.100kBT . r and x denote the radial distance from the central axis and the coordinate along the membrane normal respectively. This result indicates that, in the vicinity of the stalk, the materials mostly increase on the stalk itself with chemical potential. On the other hand, outside this vicinity, i.e. in bilayers, the number of molecules homogeneously increases.
data indicate the similar dependence on the chemical potential to the results at NA =28. We also note that the excess number of amphiphiles in the stalk is larger than the result at NA =28 and that the spontaneous stalk formation is observed in the system at NA =29.
-0.10 0.00 0.10 (µ-µo)/kBT
58.0 60.0 62.0 64.0
<np>R e 2 /(2)L yL z
stalk bilayers single bilayer
-0.10 0.00 0.10 80 90 100 110 120
∆<np>
molecules in stalk stalk bilayers
Figure 5.17: Simulation resutls at NA =29 in the grand canonical ensemble. Number of molecules per area in a single bilayer (diamonds), two-apposed bilayers (squares), and the stalk-morphology (circles) as a function of the chemical potential referred to the chemical potential,µ0, of the tensionless state. The inset depicts the excess number of molecules of the stalk,∆hnpi=hnpistalk− hnpibilayersas a function ofµ−µ0.
-5.0 0.0 5.0 10.0
σRe2/kBT -20
-10 0
∆ΦG /kBT
excess free
-5.0 0.0 5.0 10.0
Re2/kBT
-0.20 -0.10 0.00 0.10
(µ-µ o)/k BT
chemical potential energy of stalk
Figure 5.18: Excess grand canonical potential,∆ΦG(circles), of the stalk as a function of the membrane tension,σ, at NA =29. On the right hand side, we show the dependence of the chemical potential,µ(dashed line), on the membrane tension,σ.
Chapter 6
Conclusions
We have devised a general computational strategy for computing free energies of self-assembling systems [1, 36]. It has been employed to determine the excess free energy of a stalk that bridges a pair of apposed bilayer membranes, using a ther-modynamic integration scheme based on external fields. The technique relies on reversibly transforming one self-assembled structure into another by substituting the non-bonded interactions by external, ordering fields. To ensure reversibility, these external, ordering fields have to be chosen as to generate the structure of the self-assembled system in a system, where the non-bonded interactions have been turned off, i.e., an ideal gas. Applications to dense polymer system study-ing fluctuation-induced, first-order transition between a disordered and a lamellar phases in diblock copolymers [36] and the free energies of grain boundariew [37]
demonstrated that the external field can be estimated from mean field theory. For the case of the strongly fluctuating bilayer systems considered in the main part of this work the mean field approximation fails and we have devised a numerical strategy for calculating the external fields.
Along the thermodynamic integration paths, the excess Helmholtz free energy of the stalk has accurately been calculated with the use of expanded ensembles sim-ulation. This method quantitatively verifies the absence of the first order transition along the paths.
The simulations have been performed using a solvent-free coarse-grained model which reduces the computational time and facilitates the Monte Carlo simulations due to the reduced number of the degrees of freedom and the soft interaction. This allows, for example, for a very accurate measurment of chemical potential required for the simulations of lipid bilayers in the grand canonical ensemble. The Widom’s insertion and deletion schemes that we used would not be efficient in a system with hard interactions. However the proposed TDI approach can be used in context of a broad scope of different models (e.g. Lennard-Jones potentials) and simulation techniques such as DPD, conventional molecular dynamics, and single-chain-in-mean-field-simulations [17]. For example we have utilized this technique within the framework of a Flory-Huggins type density functional to compute the free
en-95
ergy of T-junctions and the free energy cost of the surface reconstruction of the soft morphologies in thin films of lamella forming diblock copolymers assembled on patterned substrates.
Focusing on the simulations of bilayers, our modeling approach provided in-formation on the stalk structure which consists of at most one hundred amphiphilic molecules, a few % of the total molecules in our system. This in combination with the fact that the free energy differences between different morphololgies in soft-matter systems typically are small, the accurate measurement of the free en-ergy is a computational challenge. In this work, we have utilized a combination of sophisticated simulation techniques (e.g. expanded ensemble and reweighting methods [1, 65, 66]) in conjunction with thermodynamic integration which are well suitable for parallel computing [37]. We obtained an accuracy of 4kBT in the free energy calculation, which corresponds to the relative accuracy of 10−5, within our computational resources.
Once the excess Helmholtz free energy of the stalk,∆F, is determined in the canonical ensemble, the dependence of the excess free energies in the canonical and grand canonical ensemble,∆F and∆ΦG, on the molecular architecture and the chemical potentialµcan be obtained with a relatively low computational cost. The dependence of the excess free energy∆ΦGon the chemical potential can be utilized to extract the dependence of ∆ΦG on the membrane tension, σ; an information which can provide a link to experiments.
We have determined the excess free energy of a stalk connecting two tension-less bilayer membranes to be∆ΦG = 16.2kBT with NA = 28. This free energy is lower than earlier estimates based on the phenomenological theory, whereas it is consistent with the results in self-consistent field calculations [30]. This result is particularly notable owing to significant differences in the microscopic struc-tures between our solvent-free model and the model of the self-consistent field calculations, e.g., we use an implicit solvent while the self-consistent field model represents the solvent by homopolymers. This finding suggests that the excess free energy of the stalk is not very sensitive to the specific interactions of the model and that the results for our simple coarse-grained model are also relevant for synthetic or biological bilayer membranes.
When the membrane tension increases, we observe that the number of molecules, of which the stalk is composed, increases and, in turn, the excess free energy,∆ΦG
also increases slightly. This finding differs from the results of self-consistent field calculations [30], which observe that the free energy of the stalk is almost indepen-dent from the membrane tension or decreases withσ. In the vicinity of the stalk, the distribution of the molecular density changes mostly in the stalk itself with the tensionσ. This accounts for the high stability of the stalk at lowσ. We observe that the stalk becomes unstable for more symmetric molecules in agreement with self-consistent field calculations [30]. We also observe that in lipid mixtures, com-prised of two species with different molecular asymmetries, the more asymmetric species segregates to the stalk.
In this work we have focused on the study of the free energy of the stalk
inter-97 mediate without considering its further evolution into a fusion pore. In this scope, it would be interesting to investigate potential pathways which the fusion process can follow after the stalk structure is formed. This can be performed via Monte Carlo simulations utilizing moves that mimic a realistic chain dynamics. In case some specific fusion intermediate structures are indentified rluving the simulations, a TDI technique similar to the one developed in this work could be used to estimate the free energy of these fusion intermediates and exploring the observed pathway.
An additional topic that we would like to address in the future refers to investi-gating further the influence of the chain architecture on the stalk stability and the kinetics of its formation. For instance, it has been argued [67] that the double tail structure, a common feature of many lipid molecules, can play an important roll during stalk nucleation.