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8 Simulation method for coarse-grained lipids In this Chapter we describe the Dissipative Particle Dynamics (DPD) simulation technique that we have applied to the study of biological membranes. In addition, we introduce the mesoscopic model that we use to represent coarse-grained (CG) lipid molecules. In this CG model that we propose, the lipids are represented by soft spheres interacting by purely repulsive forces, and are constructed by connecting beads via harmonic springs. We also demonstrate that DPD in combination with these mesoscopic models gives us a sufficiently large gain in CPU-time such that we can form the bilayer by self-assembly. 2.1 Dissipative Particle Dynamics Dissipative Particle Dynamics was introduced by Hoogerbrugge and Koelman [26] as a particle based method to simulate complex fluids with the correct hydrodynamics on mesoscopic time and length scales. DPD describes the motion of particles, or beads, which represent the center of mass of a fluid ’droplet’, i.e. each DPD particle is a momentum carrying group of molecules. All the degrees of freedom smaller than a bead radius are assumed to have been integrated out, resulting in a coarse-grain representation of fluid elements and in the possibility to adopt soft interactions between the beads. DPD resembles Molecular Dynamics (MD) in that the particles move in continuous space and discrete time step, according to Newton’s laws, but, due to the soft interactions, larger time and length scales are accessible, compared to standard MD. This allows for the study of physical behavior on time scales many orders of magnitude greater than possible with MD. In addition to the soft-repulsive (conservative) interaction, DPD models include two other forces: a dissipative force that slows the particles down and removes energy, which can be seen as a viscous resistance, and a stochastic force, which, on average, adds energy to the system and accounts for the degrees of freedom which have been removed by the coarse-graining process. These two forces act together as a thermostat for the system. Each of the three DPD forces is pairwise additive, conserves momentum, and acts along the line joining two particles. One of the attractive features of DPD is the easy way in which complex fluids can be constructed by introducing additional conservative interactions between the particles, like harmonic potentials to connect particles to build a “bead-and-spring” representation of polymer chains. DPD has been used to study phase separation and domain growth [31–33] of binary immiscible fluids by assuming two types of particles which repel each other with a strength larger than the repulsion between equal particles. The kinetics of microphase separation of diblock copolymer has also been investigated with DPD [34,35] showing the critical role of hydrodynamic forces. Another successful application of DPD is the modeling of dilute polymer solutions [36, 37] and the effect of sol-
2.1 Dissipative Particle Dynamics 9 vent quality on the static and rheological properties of polymer solutions has been investigated [38]. The DPD method applied to the study of polymer melts [39] found results consistent with the Rouse-Zimm model. Other applications of DPD are to simulation of suspensions of colloidal particles [40–42], polymer mixtures [28,36,43], tethered polymers in shear flow [44], amphiphilic mesophases [45], and polymersurfactant aggregation [46]. For an exhaustive review of the DPD technique and its applications we refer the reader to refs. [47] and [48]. The simulation technique The total force acting on a DPD particle i is expressed as a summation over all the other particles, j, of three forces [26, 28] of the pairwise-additive type: fi = (F C ij + F D ij + F R ij). (2.1) j=i The first term in the above equation refers to a conservative force, the second to a dissipative force and the third to a random force. The conservative force has two contributions, one describing non bonded interactions and the other describing the interactions used to tie the beads together in the building of a polymer like molecule (see equation 2.13). Most DPD studies express the former contribution as a soft repulsive force in the form F C ij = aij(1 − rij/Rc)^rij (rij < Rc) 0 (rij ≥ Rc) (2.2) where the coefficient aij > 0 represents the maximum repulsion strength, rij = ri−rj is the distance between particles i and j (rij = |rij|), and Rc is a cutoff radius which gives the range of the interactions. The dissipative and random forces are expressed as: F D ij = −ηw D (rij)(^rij · vij)^rij F R ij = σw R (rij)ζij^rij (2.3) where vij = vi − vj is the velocity difference between particles i and j, η is a friction coefficient and σ is the noise amplitude. ζij is a Gaussian (or uniform) distributed random number, independent for each pair of particles, with zero mean and unit variance. The requirement ζij = ζji enforces momentum conservation. Español and Warren [27] demonstrated that, in the limit of infinitesimal timestep, the system satisfies detailed balance and achieves a well-defined equilibrium state, the Gibbs Canonical NVT ensemble, with a well defined temperature, derived from a fluctuation dissipation theorem, if the weight functions and coefficients of the
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 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 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
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58 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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