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40 Structural characterization of lipid bilayers the same as in equation 4.1, but the angle φ is now defined as the angle between the orientation of the vector along two beads in the chain and the normal to the bilayer plane: cos φ = rij · ^n rij = zij rij (4.2) where ^n is a unit vector normal to the bilayer, and rij = ri − rj is the vector between beads i and j (rij = |rij|). The order parameter has value 1 if this vector is on average parallel to the bilayer normal, 0 if the orientation is random, and −0.5 if the bond is on average parallel to the bilayer plane. With this definition of the angle φ, we can compute the order parameter for a vector between any two beads in the lipid. In particular we are interested in characterizing the overall order of the chains and the local order. For the first quantity we define the indexes of the vector rij in equation 4.2 as: i = tn and j = t1, where tn is the last bead in the lipid tail and t1 is the first one. We call S tail the corresponding order parameter. For the local order we define i = tm+1 and j = tm with the index m increasing going toward the tail end, and call the corresponding order parameter Sm. If m is taken progressively from the headgroup to the tail-end of the molecule, a plot of the corresponding order parameters, Sm, gives an indication of the persistence of order from the interfacial region to the bilayer core. Area per lipid and bilayer thickness The area per lipid can be experimentally estimated from X-ray or neutron scattering [84, 85], or from the just described lipid order parameter profiles [86]. We compute the area per lipid, AL, by dividing the total bilayer projected area by half the number of lipids in the bilayer, since we find that, on average, there is an equal number of lipids in each monolayer. Experimentally the bilayer (total) thickness is computed from the peak-to-peak headgroup distance measured by X-ray diffraction. In simulations the same approach can also be used, and the thickness can be computed as the distance between the peaks in the density profile. To compute the thickness of the bilayer hydrophobic core, Dc, we consider the average distance along the bilayer normal (which we assume to be the z-axis) between the tail bead (or beads in case of double-tail lipids) connected to the headgroup of the lipids in one monolayer and the ones in the opposite monolayer: where zi t1 (i = 1, 2). Dc = z 1 t1 − z 2 t1 (4.3) is the average z position of the first tail beads of the lipids in monolayer i It is also useful to consider the lipid end-to-end distance, Lee, defined as the distance between the positions of the first bead(s), rt1 , and the last bead(s) rtn , of the
4.3 Computational details 41 lipid tail(s): Lee = 〈|rtn − rt1 |〉 (4.4) and its projection, L n ee, onto the normal to the bilayer plane: L n ee = 〈|ztn − zt1 |〉 (4.5) where the bilayer is taken parallel to the xy-plane. If the bilayer consists of two opposing monolayers which are in contact by the terminal carbons in the tails, the area per lipid and bilayer thickness are related by VL = ALDc/2, where VL is the volume of one lipid; and the thickness and the (projected) end-to-end distance are related by Dc = 2L n ee. However, if the two monolayers are interdigitated, the above relations do not hold [86]. For example, for partially interdigitated bilayers it will be Dc < 2L n ee and for fully interdigitated bilayers Dc = L n ee. To investigate the presence of an interdigitated phase we define a measure for the extent of interpenetration of the hydrophobic cores (tails) of the lipids on opposite sides of the bilayer by defining the chain overlap D overlap, as Doverlap = 2Lnee − Dc Ln . (4.6) ee where Dc and L n ee are defined in equations 4.3 and 4.4, respectively. 4.3 Computational details We first study the structural properties of single-tail lipids, in which we vary the length of the hydrophobic tail, the chain stiffness, and the headgroup interaction parameter. All the studied bilayers consist of 400 lipids, and approximately 5000 water beads, with a total bead density of ρ = 3. The non-bonded interactions between the beads are represented by the soft repulsion of equation 2.2, with the parameter set derived by Groot in [46] and reported in table 4.3. The reduced temperature was T ∗ = 1, and at this temperature all the considered bilayers are in the fluid phase. All the bilayers aij w h t w 25 15 80 h 15 35 (15) 80 t 80 80 25 Table 4.1: Repulsion parameters aij (see equation 2.2) used in our simulations. Water beads are indicated as w, hydrophilic head beads as h and hydrophobic tail beads as t. The parameters are in units of kBT. The value in parenthesis corresponds to a repulsion parameter between the headgroups which results in a non interdigitated bilayer (see text).
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
Page 16 and 17: 10 Simulation method for coarse-gra
Page 27 and 28: III Surface tension in lipid bilaye
Page 29 and 30: 3.2 Method of calculation of surfac
Page 31 and 32: 3.2 Method of calculation of surfac
Page 33 and 34: 3.3 Constant surface tension ensemb
Page 35 and 36: 3.3 Constant surface tension ensemb
Page 37 and 38: 3.4 Surface tension in lipid bilaye
Page 43 and 44: IV Structural characterization of l
Page 45: 4.2 Structural quantities 39 been r
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
Page 57 and 58: 4.4 Results and discussion 51 chain
Page 59 and 60: 4.4 Results and discussion 53 4.4.4
Page 61: 4.4 Results and discussion 55 headg
Page 64 and 65: 58 Phase behavior of coarse-grained
Page 85 and 86: VI Interaction of small molecules w
Page 87 and 88: 6.2 Computational details 81 For re
Page 89 and 90: 6.3 Results and Discussion 83 withi
Page 91 and 92: 6.3 Results and Discussion 85 ρ(z)
Page 93 and 94: 6.3 Results and Discussion 87 ρ(z)
Page 95 and 96: 6.3 Results and Discussion 89 S m 0
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6.3 Results and Discussion 91 π(z)
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6.3 Results and Discussion 93 the l
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96 Mesoscopic model for lipid bilay
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98 Mesoscopic model for lipid bilay
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100 Mesoscopic model for lipid bila
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References [1] Tanford, C. Science
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7.4 Conclusion 121 [68] Ono, S.; Ko
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7.4 Conclusion 123 [139] Lee, A. Bi
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Summary Biological membranes, as th
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agreement we find gives us confiden
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Samenvatting Biologische membranen,
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de lage temperatuur gel fase of Lβ
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ied dat het dichtst bij het hydrofo
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This thesis is based on the followi
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