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

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10 Simulation method for coarse-grained lipids dissipative and the random force satisfy the relations: w D (r) = [w R (r)] 2 σ 2 = 2ηkBT. (2.4) This is because the time evolution of the distribution function of the system f(r, p, t), which specifies the probability of finding the system at time t in a state where the particles have positions r={ri} and momenta p={pi}, is governed by the Fokker-Planck equation ∂f ∂t = LC f + L D f, (2.5) where L C and L D are the evolution operators. L C is the Liouville operator of the Hamiltonian system interacting with conservative forces, and L D contains the dissipative and the noise terms. If only L C is present, the system is Hamiltonian with equilibrium distribution, feq, the Gibbs distribution. In order to have the Gibbs distribution also when the random and the dissipative forces are present, the corresponding evolution operator should satisfy L D feq = 0. The relations 2.4 specify this condition. The existence of a Hamiltonian implies that only the conservative part of the force (or better, the potential UC related to it: F C = −∇UC), determines the equilibrium averages of the system observables. Furthermore, as shown by Willemsen et al. in [49], it implies that DPD can be combined with Monte-Carlo (MC) methods. We will use this property in section 3.3 of Chapter 3, where we describe an hybrid DPD-MC scheme to ensure that the simulations are performed at constant surface tension. All the forces assume the same functional dependence on the interparticle distance rij, as the conservative force F C ij does, if the weight function wR (r) has the following functional form w R (r) = (1 − r/Rc) (r < Rc) 0 (r ≥ Rc) (2.6) and w D (r) follows from equation 2.4. The Newton’s equations of motion are integrated using a modified version of the velocity-Verlet algorithm [28]: ri(t + ∆t) = ri(t) + ∆tvi(t) + 1 2 (∆t)2 fi(t) ˜vi(t + ∆t) = vi(t) + λ∆tfi(t) (2.7) fi(t + ∆t) = fi[r(t + ∆t), ˜v(t + ∆t)] vi(t + ∆t) = vi(t) + 1 2 ∆t [fi(t) + fi(t + ∆t)]. This integration algorithm becomes the original velocity-Verlet algorithm when the
2.1 Dissipative Particle Dynamics 11 force is independent of velocity and λ = 0.5. Since in DPD the force depends on the velocity, the iterative scheme of equation 2.7 is used, where ˜v is a prediction for the velocity. In practice λ is an empirical parameter which takes into account the fact that the underlying equations of DPD are stochastic differential equations [28]. Following Groot and Warren in [28] we choose the values λ = 0.65 and σ = 3 which are a compromise between fast simulations (large timestep ∆t) and stability of the system without large deviations from the imposed temperature. Reduced and physical units Usually within the DPD approach one makes use of reduced units for the mass, length and energy. [24, 28]. The unit of length is defined by the cut-off radius Rc, the unit of mass by the mass m of a DPD bead (where all the beads in the system have equal mass), and the unit of energy by kBT. From these, the unit of time τ follows as τ = Rc m/kBT. (2.8) In the following, unless otherwise stated, all the reported quantities will be expressed in these reduced units. An important—and non trivial—aspect in mesoscopic simulations is the mapping of the above mentioned units onto physical units. The level of coarse-graining, i.e. the number Nm of molecules represented by a DPD bead, can be seen as the renormalization factor for this mapping [24]. As an example we describe below the procedure to derive the physical unit of length. If a DPD bead corresponds to Nm water molecules, then a cube of volume R 3 c represents ρNm water molecules, where ρ is the number density, i.e. the number of DPD beads per cubic Rc. Considering that a water molecule has approximately a volume of 30 ˚A 3 , we then have R 3 c = 30ρNm [˚A 3 ]. (2.9) Taking a bead density of ρ = 3 [28], from equation 2.9 we have Rc = 4.4814(Nm) 1/3 [˚A]. (2.10) For instance, if Nm = 1 then Rc = 4.4814 ˚A, and if Nm = 3 then Rc = 6.4633 ˚A. Different mapping criteria have been used to derive the physical unit of time, all based on the mapping of diffusion constants for the system components. For example, Groot and Rabone [24] used a mapping based on the comparison of the experimental value of the self-diffusion constant of water and the corresponding value computed in DPD simulations, while Groot [46] used the diffusion constant of a surfactant micelle. In both references a value of the integration timestep (see equation 2.7) of ∆t = 0.06τ was taken, which gives ∆t ≈5ps or ∆t ≈25ps depending on which diffusion constant is considered. Both values show that DPD simulations allow for
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 18 and 19: 12 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
Page 64 and 65: 58 Phase behavior of coarse-grained
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60 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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