minimal models for lipid membranes: local modifications around fusion objects

Transcription

1 minimal models for lipid membranes: local modifications around fusion objects Giovanni Marelli Georg August Universität, Göttingen January 21, 2013 PhD defense

2 collective phenomena structures and morphologies lipid bilayer minimal models for membranes motivations 2 / 22

3 collective phenomena structures and morphologies lipid bilayer stalk pore fusion objects transmembrane protein minimal models for membranes motivations 2 / 22

4 membrane fusion how does membrane fusion work? viral fusion K. K. Lee EMBO J. 29, 1299 (2010) minimal models for membranes motivations 3 / 22

5 membrane fusion viral fusion how does membrane fusion work? possible (heuristic) explanations? K. K. Lee EMBO J. 29, 1299 (2010) minimal models for membranes motivations 3 / 22

6 membrane fusion how does membrane fusion work? artificial pore possible (heuristic) explanations? capture morphology change and dynamics isolate metastable states minimal models for membranes motivations 3 / 22

7 membrane fusion how does membrane fusion work? artificial pore under lateral tension, low hydration identify the fusion pathways capture morphology change and dynamics isolate metastable states minimal models for membranes estimate energies and times role of the peptides motivations 3 / 22

8 contents minimal models for collective phenomena in lipid membranes lipids as diblocks atomistic martini diblock coarse-grained solvent-free soft large time scales, many lipids computational efficiency sampling from thermal fluctuations minimal models for membranes motivations 4 / 22

9 contents minimal models for collective phenomena in lipid membranes lipids as diblocks atomistic martini diblock coarse-grained solvent-free soft large time scales, many lipids computational efficiency sampling from thermal fluctuations membranes as continuum sheets minimal models for membranes motivations 4 / 22

10 contents minimal models for collective phenomena in lipid membranes lipids as diblocks atomistic martini diblock coarse-grained solvent-free soft large time scales, many lipids computational efficiency sampling from thermal fluctuations membranes as continuum sheets mechanical properties of unperturbed membranes: time and length scales energetic contributions local modifications by fusion objects thickness and density profiles in simulations and numerical calculations superposition of effects lipid mediated interactions the peptides stabilize the pore the peptide lowers the line tension of the pore minimal models for membranes motivations 4 / 22

11 overview of the thesis planar membranes reference values protein/lipid interactions superposition of effects stalk morphologies universality of structures in models and experiments minimal models for membranes hydrophobic stability conditions for stability pore/protein interactions superposition of thickness profiles motivations 5 / 22

12 model solvent-free, coarse-grained model bonded (harmonic spring, bending): U sp k B T = P N b 1 k sp i=1 2 (r i r i+1 ) 2 U b k B T = P N b 1 i=2 k b (1 cos θ i±1 ) non bonded (density functional): H nb [ρ] k B T = R «d 3 r v 2 Re 3 AA 2 ρ 2 A + v AAA 3 ρ 3 3 A corresponding eq of state: PRe 3 k B T = ρ + v AA 2 ρ 2 A v AAAρ 3 A (fluid/vapor phase coexistance) the virial coefficients are set by ρ coex and the kn b : v AA = 2 kn b +3 v ρ coex AAA = 3 kn b +2 2 ρ 2 coex mixed virial coefficient χn MF = ρ coex `vab 1 2 (v AA + v BB ) M. Hömberg and M. Müller, J. Chem. Phys. 132, (2010) minimal models for membranes model 6 / 22

13 model solvent-free, coarse-grained model bonded (harmonic spring, bending): U sp k B T = P N b 1 k sp i=1 2 (r i r i+1 ) 2 U b k B T = P N b 1 i=2 k b (1 cos θ i±1 ) non bonded (density functional): H nb [ρ] k B T = R «d 3 r v 2 Re 3 AA 2 ρ 2 A + v AAA 3 ρ 3 3 A corresponding eq of state: PRe 3 k B T = ρ + v AA 2 ρ 2 A v AAAρ 3 A (fluid/vapor phase coexistance) the virial coefficients are set by ρ coex and the kn b : v AA = 2 kn b +3 v ρ coex AAA = 3 kn b +2 2 ρ 2 coex mixed virial coefficient χn MF = ρ coex `vab 1 2 (v AA + v BB ) M. Hömberg and M. Müller, J. Chem. Phys. 132, (2010) minimal models for membranes model 6 / 22

14 model solvent-free, coarse-grained model bonded (harmonic spring, bending): U sp k B T = P N b 1 k sp i=1 2 (r i r i+1 ) 2 U b k B T = P N b 1 i=2 k b (1 cos θ i±1 ) non bonded (density functional): H nb [ρ] k B T = R «d 3 r v 2 Re 3 AA 2 ρ 2 A + v AAA 3 ρ 3 3 A corresponding eq of state: PRe 3 k B T = ρ + v AA 2 ρ 2 A v AAAρ 3 A (fluid/vapor phase coexistance) the virial coefficients are set by ρ coex and the kn b : v AA = 2 kn b +3 v ρ coex AAA = 3 kn b +2 2 ρ 2 coex mixed virial coefficient χn MF = ρ coex `vab 1 2 (v AA + v BB ) M. Hömberg and M. Müller, J. Chem. Phys. 132, (2010) minimal models for membranes model 6 / 22

15 properties of planar membranes planar lamellar membrane time scale length scale coupling moduli fluctuation spectrum minimal models for membranes planar membranes 7 / 22

16 time and length scale diffusivity D` 2E x(t) x(0) D = lim t 4t timestep: 1[ t s] 8[ps] (martini 0.04[ps], atomistic 4[fs]) K. Weiß and J. Enderlein, ChemPhysChem 13, 990 (2012) varying kn b minimal models for membranes planar membranes 8 / 22

17 time and length scale diffusivity D` 2E x(t) x(0) D = lim t 4t timestep: 1[ t s] 8[ps] (martini 0.04[ps], atomistic 4[fs]) K. Weiß and J. Enderlein, ChemPhysChem 13, 990 (2012) varying kn b thickness d = [ L] 1.08[nm] N. Kučerka et al. Biophysica Acta 1808, 2761 (2011) minimal models for membranes planar membranes 8 / 22

18 quantities fluctuations Z F s = dxdy ks (s(x, y) s) 2 2 s 2 harmonic approximation we sample from thermal fluctuations s fluctuating quantity, k s coupling constant, L patch area. minimal models for membranes planar membranes 9 / 22

19 quantities fluctuations Z F s = dxdy ks 2 harmonic approximation we sample from thermal fluctuations (s(x, y) s) 2 s 2 1 standard deviation = lim k s L L 2 (s s) 2 L s 2 minimal models for membranes planar membranes 9 / 22

20 quantities fluctuations Z F s = dxdy ks 2 harmonic approximation we sample from thermal fluctuations (s(x, y) s) 2 s 2 1 standard deviation = lim k s L L 2 (s s) 2 L s 2 σ = ρd thickness `d d 2 k el = 22.7[k B T / L 2 ] lipids/area (σ σ) 2 k com = 38.7[k B T / L 2 ] density (ρ ρ) 2 k melt = 34.6[k B T / L 2 ] σ 2 σ 2 = ρ2 ρ 2 + d2 d 2 1 = k com k melt k el minimal models for membranes planar membranes 9 / 22

21 quantities fluctuations we sample from thermal fluctuations Z F s = dxdy ks (s(x, y) s) 2 1 L 2 (s s) 2 L 2 s 2 = lim k s L s 2 harmonic approximation standard deviation compressibility σ = ρd thickness `d d 2 k el = 22.7[k B T / L 2 ] lipids/area (σ σ) 2 k com = 38.7[k B T / L 2 ] density (ρ ρ) 2 k melt = 34.6[k B T / L 2 ] σ 2 σ 2 = ρ2 ρ 2 + d2 d 2 1 = k com k melt k el minimal models for membranes planar membranes 9 / 22

22 energetic contributions bending (height), peristaltic (thickness), protrusion (lipid separation) F = 1 R 2 dxdy k 1 ben + 1 (d d) + k 2 (ρ ρ) R 1 R 2 el d 2 + k 2 melt ρ 2 Helfrich model bending: k ben bending rigidity peristaltic: k el elastic coupling c 0 spontaneous curvature a area/lipids ζ := c 0 a a c 0 protrusion: k λ protrusion rigidity γ λ microscopic surface tension minimal models for membranes planar membranes 10 / 22

23 energetic contributions bending (height), peristaltic (thickness), protrusion (lipid separation) F ben = 1 2 R dxdy kben ( 2 h(x, y)) 2 Monge representation height power spectrum h 2 (q) L 2 = 1 k ben q 4 bending: k ben bending rigidity 14.3[k B T ] peristaltic: k el elastic coupling c 0 spontaneous curvature a area/lipids ζ := c 0 a a c 0 protrusion: k λ protrusion rigidity γ λ microscopic surface tension minimal models for membranes planar membranes 10 / 22

24 energetic contributions F ben = 1 2 bending (height), peristaltic (thickness), protrusion (lipid separation) R dxdy kben ( 2 h(x, y)) 2 F per = 1 2 R dxdy kel d 2 (x,y) d 2 thickness height h 2 (q) L 2 = 1 k ben q 4 d 2 (q) L 2 = 1 k el / d 2 power spectrum bending: k ben bending rigidity 14.3[k B T ] peristaltic: k el elastic coupling 1.74/ d 2 [k B T / L 4 ] c 0 spontaneous curvature a area/lipids ζ := c 0 a a c 0 protrusion: k λ protrusion rigidity γ λ microscopic surface tension minimal models for membranes planar membranes 10 / 22

25 energetic contributions bending (height), peristaltic (thickness), protrusion (lipid separation) thickness height h 2 (q) L 2 = 1 d 2 (q) L 2 = 1 k ben q 4 + 2(k λ +γ λ q 2 ) 1 1 k ben q 4 4ζk ben q 2 +k el / d 2 + 2(k λ +γ λ q 2 ) Branningan and Brown, Biophys. J. 92, 864 (2007) power spectrum bending: k ben bending rigidity 12.5[k B T ] peristaltic: k el elastic coupling 1.96/ d 2 [k B T / L 4 ] c 0 spontaneous curvature a area/lipids ζ := c 0 a a c 0 = 0.14/ d[ L 2 ] protrusion: k λ protrusion rigidity γ λ microscopic surface tension we extract: k ben, ζ, k el minimal models for membranes planar membranes 10 / 22

26 lipid/protein interactions hydrophobic inclusion radial thickness profile weakening of the membrane continuum model superposition of effects pore stabilisation minimal models for membranes lipids/peptide 11 / 22

27 thickness profile contact angle and thinning cylindrical transmembrane protein cluster of connected monomers radius 1[ L] hydrophobic mismatch 1.3 N b /A = 38[ L 2 ] θ c = 31 o minimal models for membranes lipids/peptide 12 / 22

28 thickness profile contact angle and thinning cylindrical transmembrane protein cluster of connected monomers radius 1[ L] hydrophobic mismatch 1.3 N b /A = 38[ L 2 ] θ c = 31 o radial thickness profiles minimal models for membranes lipids/peptide 12 / 22

29 lipid frustration bond length minimal models for membranes lipids/peptide 13 / 22

30 lipid frustration oil partitions close to the peptide bond length density of the oil oil molecule: dodecane hydrocarbon tail the oil loses its translational entropy acts as a sensor to partion where most of the stress is minimal models for membranes lipids/peptide 13 / 22

31 lipid frustration oil partitions close to the peptide bond length density of the oil point to point hydrophobic density difference with and without oil oil molecule: dodecane hydrocarbon tail the oil loses its translational entropy acts as a sensor to partion where most of the stress is minimal models for membranes lipids/peptide 13 / 22

32 discrete description mean separation rescaled to the bulk value thickness from the polar/apolar density intersection Delaunay triangulation thickness mean area minimal models for membranes lipids/peptide 14 / 22

33 discrete description mean separation rescaled to the bulk value thickness from the polar/apolar density intersection bead number normalization upper and lower monolayers Delaunay triangulation thickness mean area minimal models for membranes lipids/peptide 14 / 22

34 weakening of the membrane lowering the line tension of the pore classical nucleation theory G = G0 + 2πRp λ πrp2 Pl line tension of the pore in NVT ensemble Pl = λ Rp minimal models for membranes lipids/peptide 15 / 22

35 weakening of the membrane lowering the line tension of the pore classical nucleation theory G = G0 + 2πRp λ πrp2 Pl line tension of the pore in NVT ensemble Pl = λ Rp the peptide lowers the line tension of the pore pore s radius correction N Pl = Rλp 1 αpep Rpep p minimal models for membranes lipids/peptide 15 / 22

36 continuum description F = R dxdy k ben d s(x, y) := d(x, y) d k ben 4 c + 4k ben ζ d elastic model 2 ( 2 d s) 2 + 4k ben c 0 2 d s + 2 k ben ζ d d s 2 d s + k el 2 d 2 ds 2 Euler Lagrange equations: 2 c + k el d 2 d s(x, y) = 0 2 c := 2 x 0 0 y 2 δf = 0 δd s «r/ξ sin(kr) fitting equation: Ae r ξ = 2.2[ L] k = 0.88[ L 1 ] sketch B. West, F. L. H. Brown and F. Schmid, Biophys. J. 96, 101 (2009) minimal models for membranes lipids/peptide 16 / 22

37 continuum description F = R dxdy k ben d s(x, y) := d(x, y) d k ben 4 c + 4k ben ζ d elastic model 2 ( 2 d s) 2 + 4k ben c 0 2 d s + 2 k ben ζ d d s 2 d s + k el 2 d 2 ds 2 Euler Lagrange equations: 2 c + k el d 2 d s(x, y) = 0 2 c := 2 x 0 0 y 2 δf = 0 δd s «r/ξ sin(kr) fitting equation: Ae r ξ = 2.2[ L] k = 0.88[ L 1 ] sketch B. West, F. L. H. Brown and F. Schmid, Biophys. J. 96, 101 (2009) minimal models for membranes lipids/peptide 16 / 22

38 continuum description F = R dxdy k ben d s(x, y) := d(x, y) d k ben 4 r + 4k ben ζ d elastic model 2 ( 2 d s) 2 + 4k ben c 0 2 d s + 2 k ben ζ d d s 2 d s + k el 2 d 2 ds 2 Euler Lagrange equations: 2 r + k el d 2 d s(r) = 0 2 r := 1 r (r r ) r δf δd s = 0 r/ξ sin(kr) fitting equation: Ae r ξ = 2.2[ L] k = 0.88[ L 1 ] sketch B. West, F. L. H. Brown and F. Schmid, Biophys. J. 96, 101 (2009) minimal models for membranes lipids/peptide 16 / 22

39 continuum description F = R dxdy k ben d s(x, y) := d(x, y) d k ben 4 r + 4k ben ζ d elastic model 2 ( 2 d s) 2 + 4k ben c 0 2 d s + 2 k ben ζ d d s 2 d s + k el 2 d 2 ds 2 Euler Lagrange equations: 2 r + k el d 2 d s(r) = 0 2 r := 1 r (r r ) r δf δd s = 0 r/ξ sin(kr) fitting equation: Ae r ξ = 2.2[ L] k = 0.88[ L 1 ] sketch B. West, F. L. H. Brown and F. Schmid, Biophys. J. 96, 101 (2009) minimal models for membranes lipids/peptide 16 / 22

40 fine exploration thickness characteristics the solution of the continuum model is times faster than simulations maximum thinning d max minimum distance r min minimal models for membranes lipids/peptide 17 / 22

41 superposition of effects two inclusions rmin II : minimum splits in two dmax II : superposition effect is stronger clusterisation of gramicidins: (hydrophobic matching) T. Harroun et al Biophys. J. 76, 937 (1999) minimal models for membranes lipids/peptide 18 / 22

42 superposition of effects two inclusions rmin II : minimum splits in two dmax II : superposition effect is stronger clusterisation of gramicidins: (hydrophobic matching) T. Harroun et al Biophys. J. 76, 937 (1999) minimal models for membranes lipids/peptide 18 / 22

43 many peptides thickness (simulations) minimal models for membranes thickness (calculations) lipids/peptide 19 / 22

44 many peptides thickness (simulations) minimal models for membranes thickness (calculations) lipids/peptide 19 / 22

45 many peptides thickness (simulations) minimal models for membranes thickness (calculations) lipids/peptide 19 / 22

46 summary unperturbed membranes sampling from thermal fluctuations time and length scale energetic contributions power spectrum material properties parametrisation of the continuum model minimal models for membranes conclusions 20 / 22

47 summary unperturbed membranes sampling from thermal fluctuations time and length scale energetic contributions power spectrum material properties parametrisation of the continuum model lipid/peptide thickness profile undershoot below the bulk value good agreement between simulations and calculations superposition of effects the membrane gets thinner pore stabilisation minimal models for membranes conclusions 20 / 22

48 summary unperturbed membranes sampling from thermal fluctuations time and length scale energetic contributions power spectrum material properties parametrisation of the continuum model lipid/peptide thickness profile undershoot below the bulk value good agreement between simulations and calculations superposition of effects the membrane gets thinner pore stabilisation oil partition oil partitions at the interface with the peptide (pore s rim and stalk s ends) (relaxes the lipid frustration) minimal models for membranes conclusions 20 / 22

49 summary unperturbed membranes sampling from thermal fluctuations time and length scale energetic contributions power spectrum material properties parametrisation of the continuum model oil partition oil partitions at the interface with the peptide (pore s rim and stalk s ends) (relaxes the lipid frustration) lipid/peptide thickness profile undershoot below the bulk value good agreement between simulations and calculations superposition of effects the membrane gets thinner pore stabilisation fusion pathways metastable states identify them (evolution change by lipid composition) minimal models for membranes conclusions 20 / 22

50 summary unperturbed membranes sampling from thermal fluctuations time and length scale energetic contributions power spectrum material properties parametrisation of the continuum model oil partition oil partitions at the interface with the peptide (pore s rim and stalk s ends) (relaxes the lipid frustration) lipid/peptide thickness profile undershoot below the bulk value good agreement between simulations and calculations superposition of effects the membrane gets thinner pore stabilisation fusion pathways metastable states identify them (evolution change by lipid composition) minimal models fast and fine exploration of phenomena tuned by more detailed models confirm the multiscale approach (universality of stalk s morphologies) (bilayer repulsion) minimal models for membranes conclusions 20 / 22

51 future developement lipid/peptide dynamics study the peptide diffusion by changing the peptide description study the lipid diffusivity depending on the radial distance from the peptide minimal models for membranes conclusions 21 / 22

52 future developement lipid/peptide dynamics study the peptide diffusion by changing the peptide description study the lipid diffusivity depending on the radial distance from the peptide stalk/pore/peptides competing line tensions interactions between a boundle of peptide and a stalk minimal models for membranes conclusions 21 / 22

53 acknowledgments thank you for your attention acknowledgments Marcus Müller Yuliya Smirnova Jelger Risselada Martin Hömberg Kostas Daoulas Ulrich Welling foundings: SFB project minimal models for membranes appendix 22 / 22

54 Dissipative Particle Dynamics thermostat simulation technique: MD + MDPD thermostat characteristic: the total force is the sum of the conservative, random and dissipative forces F tot = F c + F r + F d the forces are pairwise F tot i = P j Fc ij + Fr ij + Fd ij the forces are soften via a weighting function F ij = F ij w(r ij ) features of the multibody DPD: local density dependance pragmatic extension of the classical DPD that allows one to prescribe the thermodynamic behavior of a system ensembles NPT NVT NP t T tensionless membrane. P. Español and P. B. Warren Europhys. Lett (1995) S. Y. Trominov et al. J. Chem. Phys (2002) minimal models for membranes appendix 23 / 22

55 Dissipative Particle Dynamics thermostat simulation technique: MD + MDPD thermostat characteristic: the total force is the sum of the conservative, random and dissipative forces F tot = F c + F r + F d the forces are pairwise F tot i = P j Fc ij + Fr ij + Fd ij the forces are soften via a weighting function F ij = F ij w(r ij ) features of the multibody DPD: local density dependance pragmatic extension of the classical DPD that allows one to prescribe the thermodynamic behavior of a system ensembles NPT NVT NP t T tensionless membrane. P. Español and P. B. Warren Europhys. Lett (1995) S. Y. Trominov et al. J. Chem. Phys (2002) minimal models for membranes appendix 23 / 22

56 oil relaxation P αβ (r, z) = k B T ρ(r, z) + k B T ρ(r, z) + 1 V normal component Pz l (r, z) = P zz (r, z) P θθ(r,z )+P rr (r,z ) 2 no oil with oil (r,z P ) ij du ij dr x α ij x β ij x radial component P l r (r, z) = P rr (r, z) P θθ(r,z )+P zz (r,z ) 2 identification of the interface position of the relaxation by the oil minimal models for membranes appendix 24 / 22

57 oil relaxation P αβ (r, z) = k B T ρ(r, z) + k B T ρ(r, z) + 1 V normal component Pz l (r, z) = P zz (r, z) P θθ(r,z )+P rr (r,z ) 2 no oil with oil (r,z P ) ij du ij dr x α ij x β ij x radial component P l r (r, z) = P rr (r, z) P θθ(r,z )+P zz (r,z ) 2 identification of the interface position of the relaxation by the oil minimal models for membranes appendix 24 / 22

58 oil relaxation P αβ (r, z) = k B T ρ(r, z) + k B T ρ(r, z) + 1 V normal component Pz l (r, z) = P zz (r, z) P θθ(r,z )+P rr (r,z ) 2 no oil with oil (r,z P ) ij du ij dr x α ij x β ij x lateral pressure difference radial component P l r (r, z) = P rr (r, z) P θθ(r,z )+P zz (r,z ) 2 identification of the interface position of the relaxation by the oil minimal models for membranes appendix 24 / 22

59 monolayer spontaneous curvature calculation of the spontaneous curvature lateral pressure P l zz (r, z) = Pzz (r, z) Pxx (r, z) + P yy (r, z) 2 spontaneous curvature c 0 = 0.36[ L 1 ] Z L k ben c 0 = dz zpzz l (z) 0 «normal lateral pressure profile minimal models for membranes appendix 25 / 22

60 elastic model F = Z dxdy k ben 2( 2 h) 2 + k λ (z z 2 ) + γ λ `( z + ) 2 + ( z ) 2 + h z + /2 + d z ) + kcom 2 d 2 + k ben c 0 2 d + k ben ζd d 2 d + k ben 2 ( 2 d) 2 comparison with the solution of the differential equation in 2d finite elements calculation (surface evolver) finite differences calculation minimal models for membranes appendix 26 / 22

61 slab mean square displacement mean square displacement in radial slabs minimal models for membranes appendix 27 / 22

62 superposition of effects/two inclusions density plot around the inclusions division in sectors by angles minimum distance and depth the membrane is thinner between the two inclusions the position of the minimum shifts minimal models for membranes appendix 28 / 22

63 stiffness of a circular stalk we search iteratively a torus surrounding the stalk with: we count the beads inside the torus the smallest lateral radius the largest normal radius the most contact with heads the less contact with tails the center of mass of the beads is the torus position the radius of the torus is r t = A/2π, A the area occupied by the beads radial evolution diffusivity D stalk = D lipid /3 minimal models for membranes appendix 29 / 22

64 stiffness and density profile exp data PhD thesis of S. Aeffner radial density profiles - model/scale dependence Martini force field, dft, implicit solvent explicit solvent flexible chains, explicit solvent d t /d s = d t /d b = 2 d s/d b = d t /d s = 2 d t /d b = 2.5 d s /d b = 1.2 d t /d s = 2.1 d t /d b = 2.1 d s /d b = 1.0 d t /d s = 2.9 d t /d b = 2.2 d s /d b = 0.77 stiffness dft Martini minimal models for membranes appendix 30 / 22

65 oil contribution addition of small oil chains in the bilayers dft 10% oil partitioning density profile tails density difference Martini 15% minimal models for membranes appendix 31 / 22