Mesoscopic models of lipid bilayers and bilayers with embedded ...

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22 Surface tension in lipid bilayers 3.1 Introduction Lipid bilayers are self-assembled structures, which are not constrained by the total area, and hence will adopt a conformation that will have the lowest free energy. Since the thermodynamic definition of surface tension is the derivative of the free energy with respect to the area of the interface [53], for a unconstrained bilayer the free energy minimum will be a tensionless state [54]. Experimental results on unilamellar vesicles have also indicated that bilayers are in a stress free state [55]. In molecular simulation, for both self-assembled and pre-assembled membranes, a fixed number of lipid molecules and a fixed area are usually combined with periodic boundary conditions. The periodic boundary conditions correspond to an infinitely large membrane, but the fixed size of the simulation box, and the fixed number of lipids at the interface, impose a constraint on the bilayer area which results in a finite surface tension. Although the constraint on the fixed area can be released by performing simulations of membranes at constant pressure or constant surface tension [56, 57], it is an important — and still open — question which value of the surface tension should be used in simulations to reproduce the state and the area per lipid of a real membrane. In their molecular dynamics simulations Feller and Pastor [58,59] observed that a tensionless state did not reproduce the experimental value of the area per lipid. They explained this result by considering that, since the typical undulations and and outof-plane fluctuations of a macroscopic membrane do not develop in a small patch of a membrane (theirs was composed of 36 lipids for monolayer), a positive surface tension (stretching) must be imposed in order to compensate for the suppressed undulations, and hence to recover the experimental values of the area per lipid. Recently, Marrink and Mark [60] investigated the system size dependence of the surface tension in membrane patches ranging from 200 to 1800 lipids, simulated for times up to 40ns. Their calculations show that, in a stressed membrane, the surface tension is size dependent, i.e. it drops if the system size is increased (at fixed area per lipid), which is in agreement with the results by Feller and Pastor. On the other hand, their results show that at zero stress simulation conditions the equilibrium does not depend on the system size. Marrink and Mark then concluded that simulations at zero surface tension correctly reproduce the experimental surface areas for a stress free membrane. Simulations at constant surface tension have been introduced by Chiu et al. in [56]. A constant surface tension ensemble (NVγ) has been considered in literature and the corresponding equations of motion for Molecular Dynamics simulations have been derived [57], and applied to the simulation of phospholipid bilayers [59, 61–65]. Alternatively, to ensure a tensionless state, Goetz and Lipowsky [21] performed several simulations to determine the area per lipid that gives a state of zero tension.
3.2 Method of calculation of surface tension 23 In our simulations we use a different approach, based on a Monte Carlo (MC) scheme, to simulate a membrane at a given state of tension, of which the tensionless state is a particular case. Similar to constant pressure simulation, we impose a given value for the surface tension and from the simulation we obtain the average area per molecule. Such method has the advantage that we do not have to perform several – relatively expensive – simulations to locate the area of zero tension. Moreover, by releasing the constraint of a priori chosen area and allowing dynamic fluctuations of the bilayer area, the system is able to explore the phase space and assume the configurational structure that corresponds to the free energy minimum at the given thermodynamic conditions. One of the implications of this extra degree of freedom is that it allows to observe directly phase transitions in which the area per lipid changes. We will exploit this advantage in Chapter 5 where we study the phase behavior of lipid bilayers. In this Chapter we first introduce the definition of surface tension and its calculation method in computer simulations. We then describe the scheme we use to impose a given value of the surface tension and we validate the method by applying it to a monolayer of amphiphilic dumb-bells at oil-water interface. We then utilize the constant surface tension scheme in simulations of lipid bilayers to study the dependence of the area per lipid on the system size, and on the value of the applied surface tension. 3.2 Method of calculation of surface tension In homogeneous systems at equilibrium the pressure is constant and equal at each point in space, while for an inhomogeneous system the pressure is a tensor P(r) that depends on the spatial direction and on the position r where it is calculated. Here we follow the discussion presented in [53] and [66] to derive the properties of the pressure tensor. Consider a system of two immiscible liquids forming a planar interface normal to the z direction. In equilibrium, mechanical stability requires that the gradient of the pressure tensor is zero everywhere ∇ · P = 0. (3.1) Shear forces are also zero and the non diagonal components of P vanish; also, because of planar symmetry, the components of the pressure tensor parallel to the interface should be identical. The pressure tensor is then diagonal ⎛ P = ⎝ Pxx 0 0 0 Pyy 0 0 0 Pzz ⎞ ⎠ , (3.2)
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: III Surface tension in lipid bilaye
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 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
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
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72 Phase behavior of coarse-grained
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VI Interaction of small molecules w
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6.2 Computational details 81 For re
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6.3 Results and Discussion 83 withi
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6.3 Results and Discussion 85 ρ(z)
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6.3 Results and Discussion 87 ρ(z)
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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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