Mesoscopic models of lipid bilayers and bilayers with embedded ...
Mesoscopic models of lipid bilayers and bilayers with embedded ... Mesoscopic models of lipid bilayers and bilayers with embedded ...
28 Surface tension in lipid bilayers the interface (xy plane), and Lz = L⊥ perpendicular to the interface (z axis), so that the system volume is V = L⊥L2 and the area of the interface A = L2 . We define a transformation of the box sizes which changes the area and the height but keeps the volume constant. Such a transformation can be written in the form L ′ = λ L (3.15) L ′ ⊥ = 1 L⊥ λ2 where λ is the parameter of the transformation. By changing λ, the above expression generates a transformation of coordinates which preserves the total volume of the system, hence no work against the external pressure is performed. The coordinate phase space has now an extra degree of freedom represented by the parameter λ. To write the partition function corresponding to this ensemble it is convenient to introduce a set of scaled coordinates s ∈ [0, 1], defined as r = (L sx, L sy, L⊥sz). (3.16) By a transformation of the box sizes with λ (equation 3.15) the coordinates of the particles rescale as r ′ = λLsx, λLsy, 1 L⊥sz . λ2 (3.17) In terms of these scaled coordinates the partition function of the system takes the expression Q = VN Λ3N dλ ds N! N exp −β U(s N ; λ) − γA(λ) . (3.18) The probability of finding a configuration with scaled positions s N and parameter λ is then given by [76] N(s N , λ) ∝ exp −β U(s N ; λ) − γA(λ) . (3.19) In a MC move an attempt of changing the parameter λ is then accepted with a probability Pacc(λ → λ ′ exp −β U(s ) = ′N ′ ′ ; λ ) − γA(λ ) exp {−β [U(sN ; λ) − γA(λ)]} (3.20) where γ is the imposed surface tension. If we choose the particular value γ = 0, then the explicit term depending on the area in equation 3.20 drops. The described scheme can be applied to impose any value of the surface tension. It is important to remark that this scheme assumes that the stress tensor is diagonal, which is true for fluid systems. Since, as we have discussed in Chapter 2, an Hamiltonian for the conservative part of the interaction energy in DPD can be defined, it is possible to imple-
3.3 Constant surface tension ensemble 29 ment the described MC scheme in combination with DPD simulations. The method combines DPD to evolve the positions of the particles and MC moves to change the shape of the simulation box [75]. A simulation done with this hybrid method consists of cycles. In each cycle we choose at random whether to perform a number of DPD steps or an attempt to change the parameter λ (i.e. the box aspect ratio) by δλ. The number of DPD steps to perform when a DPD-type move is chosen is also selected at random between one and a maximum number of steps Nmax. We have chosen Nmax = 50. The fraction of accepted box shape moves was set at 30%, and δλ is automatically adjusted at regular intervals during the simulation to match this percentage. Unless otherwise stated, these are the values used in all the simulations presented in this thesis. Validation of the constant surface tension scheme To validate the constant surface tension scheme, we performed simulations of a monolayer of amphiphilic dumb-bell surfactants at water-oil interface. The system consist of 1400 oil-like beads (o), 1400 water-like beads (w), and 200 dumb-bell surfactants built by connecting via a spring one water-like bead to one oil-like bead. The DPD repulsion parameter between beads of the same type was chosen as aoo = aww = 25, and the one between different beads as aow = 80. The overall number density of the system was ρ = 3. We considered the monolayer with two different initial values of the area, corresponding to a surface tensions of γ = 0 and γ = 4, respectively. In both systems we imposed a surface tension of γ = 2 and calculated the area and the surface tension as function of MC cycles. The results are plotted in figure 3.3. In both cases the imposed value of the surface tension was already achieved after only 1000 MC cycles, and the area of both systems converged to the same value. 100 85 A 70 55 3 γ 2 1 0 0 1000 2000 3000 4000 5000 MC cycles Figure 3.3: Area, A, and surface tension, γ, as function of MC cycles. The two sets of lines correspond to different initial conditions: area A = 53.8 and surface tension γ = 0 (black lines), area A = 101.4 and surface tension γ = 4 (gray lines). The plotted values of the surface tension are running averages of length 100.
- Page 1: Mesoscopic models of lipid bilayers
- Page 4 and 5: Promotiecommissie: Promotor: • pr
- Page 6 and 7: ii CONTENTS 5.3 Double-tail lipid b
- Page 8 and 9: 2 Introduction 1.1 The cell membran
- Page 10 and 11: 4 Introduction between the beads, a
- Page 12 and 13: 6 Introduction whether the preferre
- Page 14 and 15: 8 Simulation method for coarse-grai
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- Page 27 and 28: III Surface tension in lipid bilaye
- Page 29 and 30: 3.2 Method of calculation of surfac
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- Page 33: 3.3 Constant surface tension ensemb
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- Page 43 and 44: IV Structural characterization of l
- Page 45 and 46: 4.2 Structural quantities 39 been r
- Page 47 and 48: 4.3 Computational details 41 lipid
- Page 49 and 50: 4.4 Results and discussion 43 ρ(z)
- Page 51 and 52: 4.4 Results and discussion 45 one l
- Page 53 and 54: 4.4 Results and discussion 47 WH HT
- Page 55 and 56: 4.4 Results and discussion 49 Shill
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3.3 Constant surface tension ensemble 29<br />
ment the described MC scheme in combination <strong>with</strong> DPD simulations. The method<br />
combines DPD to evolve the positions <strong>of</strong> the particles <strong>and</strong> MC moves to change the<br />
shape <strong>of</strong> the simulation box [75]. A simulation done <strong>with</strong> this hybrid method consists<br />
<strong>of</strong> cycles. In each cycle we choose at r<strong>and</strong>om whether to perform a number<br />
<strong>of</strong> DPD steps or an attempt to change the parameter λ (i.e. the box aspect ratio) by<br />
δλ. The number <strong>of</strong> DPD steps to perform when a DPD-type move is chosen is also<br />
selected at r<strong>and</strong>om between one <strong>and</strong> a maximum number <strong>of</strong> steps Nmax. We have<br />
chosen Nmax = 50. The fraction <strong>of</strong> accepted box shape moves was set at 30%, <strong>and</strong><br />
δλ is automatically adjusted at regular intervals during the simulation to match this<br />
percentage. Unless otherwise stated, these are the values used in all the simulations<br />
presented in this thesis.<br />
Validation <strong>of</strong> the constant surface tension scheme<br />
To validate the constant surface tension scheme, we performed simulations <strong>of</strong> a monolayer<br />
<strong>of</strong> amphiphilic dumb-bell surfactants at water-oil interface. The system consist<br />
<strong>of</strong> 1400 oil-like beads (o), 1400 water-like beads (w), <strong>and</strong> 200 dumb-bell surfactants<br />
built by connecting via a spring one water-like bead to one oil-like bead. The DPD<br />
repulsion parameter between beads <strong>of</strong> the same type was chosen as aoo = aww = 25,<br />
<strong>and</strong> the one between different beads as aow = 80. The overall number density <strong>of</strong> the<br />
system was ρ = 3. We considered the monolayer <strong>with</strong> two different initial values <strong>of</strong><br />
the area, corresponding to a surface tensions <strong>of</strong> γ = 0 <strong>and</strong> γ = 4, respectively. In<br />
both systems we imposed a surface tension <strong>of</strong> γ = 2 <strong>and</strong> calculated the area <strong>and</strong> the<br />
surface tension as function <strong>of</strong> MC cycles. The results are plotted in figure 3.3. In both<br />
cases the imposed value <strong>of</strong> the surface tension was already achieved after only 1000<br />
MC cycles, <strong>and</strong> the area <strong>of</strong> both systems converged to the same value.<br />
100<br />
85<br />
A<br />
70<br />
55<br />
3<br />
γ 2<br />
1<br />
0<br />
0 1000 2000 3000 4000 5000<br />
MC cycles<br />
Figure 3.3: Area, A, <strong>and</strong> surface tension, γ, as function <strong>of</strong> MC cycles. The two sets <strong>of</strong> lines<br />
correspond to different initial conditions: area A = 53.8 <strong>and</strong> surface tension γ = 0 (black<br />
lines), area A = 101.4 <strong>and</strong> surface tension γ = 4 (gray lines). The plotted values <strong>of</strong> the surface<br />
tension are running averages <strong>of</strong> length 100.
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