Tokyo ISSP/SOFT2010 Workshop Dynamical curvature instability controlled by inter-monolayer friction, causing tubule ejection in membranes Jean-Baptiste Fournier Laboratory « Matière et Systèmes Complexes » (MSC), University Paris Diderot & CNRS, France. M. I. Angelova, A.-F. Bitbol, N. Khalifat, L. Peliti, N. Puff
Tokyo ISSP/SOFT2010 Workshop Question : Instantaneous local modification of the lipids of one of the two monolayers: what happens? e.g., local pH variation. M. I. Angelova, N. Puff et al.
Tokyo ISSP/SOFT2010 Workshop Outline 1. Review of the elastic and dynamical models of membranes and monolayers. 2. Experiment by M. I. Angelova, N. Puff et al. 3. Theory of the curvature instability caused by a local modification of the lipids of one of the monolayers 4. Comparison with the pH-micropipette experiment of M. I. Angelova, N. Puff et al. 5. Non-linear development : tubule ejection
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Helfrich model P. Canham (1970), W. Helfrich (1973) � F = dA f f = σ 0 + κ 2 c 2 − κ c b 0 c . c 2 c 1 free-energy density depends on the c = c 1 + c 2 ✤ Bilayer structure neglected ✤ Gaussian term discarded (Gauss Bonnet) c 1 c 2 ✤ sets the area constraint (hides lipid density) σ 0 ✤ spontaneous curvature of the bilayer (if asymmetric) c b 0
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Area Difference Elasticity (ADE) model S. Svetina & B. Ž ek š (1989) – U. Seifert, I. Miao, H.-G. Döbereiner & M. Wortis (1991) ✤ Lipid density is involved, but in a global manner Preferred (relaxed) area A + ✤ related to the ∆ A 0 integrated curvature A − 0 ∆ A 0 = A + 0 − A − ∆ A 0 A � � σ 0 + κ k � 4 A (∆ A − ∆ A 0 ) 2 2 c 2 − κc b F = dA + 0 c Cost to deviate from A = ( A + Fixes 0 + A − 0 ) / 2 ∆ A = ∆ A 0
72 EUROPHYSICS LElTERS on which lateral redistribute laterally? Obviously, the answer depends on the time scale 7 ; ' redistribution within the monolayers can take place. The driving force for this redistribution modulus k, is the monolayer elasticity, characterized by an elastic area compressibility while the main dissipative mechanism is intermonolayer friction with a phenomenological friction coefficient b. Dimensional analysis then yields y2 - kq2/b, with the q2 arising from the fact that densities are conserved quantities. Comparing the two time scales, one finds that for long wavelengths, q < < qklbic, bending fluctuations occur at relaxed lipid monolayer densities, while at shorter wavelengths, q > > qk/bK, the effective bending rigidity increases, since the lipid molecules cannot redistribute themselves quickly enough [5]. Therefore, bending fluctuations and fluctuations in the lipid density of the two monolayers are dynamically coupled, giving rise to an interesting dispersion relation which is characterized by a mixing between two viscous modes. We start the derivation of the dispersion relations by introducing two densities I$* and $* #* for the upper (+) and lower (-) monolayers (see fig. 1). describes the density of lipids at the neutral surface of each monolayer. When the membrane is curved, the densities $* projected onto the midsurface of the bihyer will differ from the densities I$* on the neutral surfaces of the monolayers. To lowest order i dH these two densities are related by n where H is the mean curvature $* = 2 2dH), of the bilayer and d the distance between $* (1 the midsurface of the bilayer and the neutral surface of a monolayer. The elastic energy - density of each monolayer is given by (k/2)(I$*/& = (k/2)(p* 2dHI2, where 1)2 k pf = ($*/#o - is the scaled deviation of the projected density from its equilibrium value $o 1) for a flat membrane. Thus, the continuum free energy, F, for the entire membrane reads I (z I Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics + k [(p' + + (p- - 2dEO2] . F = dA E (2H)2 2dH)2 ( 1 ) 2 The first term arises from the bending energy of each monolayer, with the usual bilayer Bilayer curvature — density elasticity bending rigidity K. (We have implicitly assumed that the monolayers are symmetric and have < < d -'.) As written, F spontaneous curvature C g " ) is a functional of the membrane shape and the two densities p*. U. Seifert & S. A. Langer (1993) We are interested only in the small displacements of a nearly planar membrane. Letting the planar membrane lie in the (z,y)-plane, we describe its fluctuations in the Monge ρ = density on midsurface − 1 neutral surface equilibrium density MID-SURFACE e � κ � 2 �� � 2 c 2 + k � 2 + �� ρ + + ec � F = dA ρ − − ec 2 1. - F i g . Schematic geometry of a bilayer membrane. The circles with squiggly tails represent the lipid the neutral surfaces of the monolayers, on which the densities #* molecules. The dashed lines are are defined. The dark solid line i the midsurface of the bilayer, on which the projected densities $' and the s ✤ On the «neutral surface» density and curvature are independent scaled projected densities p' are defined. variables (decoupled). ✤ Not if they are defined on the «mid-surface» ✤ Inter-monolayer friction and membrane bending : MID-SURFACE E. Evans & Y. Yeung (1992)
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Bilayer curvature — density elasticity A.-F. Bitbol, L. Peliti & J.-B. Fournier (2010) All quantities defined on the MID-SURFACE f ( r, H, K ) = A 0 + A 1 H + A 2 ( r + H ) 2 + A 3 H 2 : reference density ρ 0 � = ρ eq + A 4 K + O ( ǫ 3 ) r = ρ − ρ 0 = O ( � ) , ρ 0 ✤ No term : total number of ∝ r H = ( c 1 + c 2 ) e = O ( � ) , lipids fixed K = c 1 c 2 e 2 = O � 2 � � ✤ Coupling term : -> sets , rH e ✤ All terms ( ) depend on ρ 0 σ 0 4 c 2 ± κ c 0 2 c + k f ± = σ 0 2 + κ r ± ± ec � 2 � 2 ula, is the stretching elastic constant of a monola NB. Minimizing with respect to the densities -> ADE
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Monolayer stress tensor d � f = Σ · � m d ℓ A.-F. Bitbol, L. Peliti & J.-B. Fournier (2010) ✤ Projected quantities (Monge) d� f h ( x, y ) ¯ ρ ( x, y ) Two-component membrane y � d 2 r ¯ F = f (¯ ρ , φ , h i , h ij ) x Ω Ordinary isotropic fluid term = pressure (or tension) � ∂ ¯ ρ∂ ¯ ∂ ¯ � f � f f � ¯ Σ ij = f − ¯ h i δ ij − − ∂ k ∂ ¯ ∂ h j ∂ h kj ρ ∂ ¯ f h ki , − Other terms due to the ∂ h kj curved layer structure Σ zj = ∂ ¯ ∂ ¯ f f , − ∂ k ∂ h j ∂ h kj i, j ∈ { x, y }
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Monolayer stress tensor d � f = Σ · � m d ℓ A.-F. Bitbol, L. Peliti & J.-B. Fournier (2010) d� f 4 c 2 ± κ c 0 2 c + k f ± = σ 0 2 + κ r ± ± ec � 2 � 2 ula, is the stretching elastic constant of a monola r = ρ − ρ 0 ρ 0 y c = ∇ 2 h + O ( ǫ 2 ) x Stress tensor normal components zj = σ 0 2 h j − κ Σ + 2 ∂ j ∇ 2 h − ke ∂ j r + e ∇ 2 h + O ( ǫ 2 ) � � From Helfrich Force density zj = σ 0 2 ∇ 2 h − κ p + z = ∂ j Σ + 2 ∇ 4 h − ke ∇ 2 � r + e ∇ 2 h + O ( ǫ 2 ) �
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Monolayer stress tensor d � f = Σ · � m d ℓ A.-F. Bitbol, L. Peliti & J.-B. Fournier (2010) d� f 4 c 2 ± κ c 0 2 c + k f ± = σ 0 2 + κ r ± ± ec � 2 � 2 ula, is the stretching elastic constant of a monola r = ρ − ρ 0 ρ 0 y c = ∇ 2 h + O ( ǫ 2 ) x Stress tensor tangential components � σ 0 − κc 0 δ ij + κc 0 � Σ + r + e ∇ 2 h 2 ∇ 2 h 2 h ij + O ( ǫ 2 ) � � ij = 2 − k tension of the flat membrane with r=0. Force density p + i = ∂ j Σ + r + e ∇ 2 h + O ( ǫ 2 ) � � ij = − k ∂ i
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics Dynamics of structureless membranes zj = σ 0 2 ∇ 2 h − κ p + z = ∂ j Σ + 2 ∇ 4 h − ke ∇ 2 � r + e ∇ 2 h + O ( ǫ 2 ) � h ( x, y, t ) ✤ Relaxation time p z ( q ) = − ( σ 0 q 2 + κq 4 ) h q 4 η τ R = σ 0 q + κq 3 ✤ Valid if at σ 0 ≈ 0 intermediate lengthscales (inter-monolayer friction). σ = 10 − 9 J / m 2 → [10 µ m , 3000 µ m] σ = 10 − 8 J / m 2 → [10 µ m , 300 µ m] σ = 10 − 7 J / m 2 → never valid T zz ( q ) = − 4 ηq dh q dt τ R ≃ 10 s at λ = 150 µ m
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics In-plane dynamics in a flat membrane (symmetric mode) p + i = ∂ j Σ + r + e ∇ 2 h + O ( ǫ 2 ) � � ij = − k ∂ i ✤ Relaxation of a SYMMETRIC − 2 η qv q density modulation in a FLAT MEMBRANE − η 2 q 2 v q − ikqr q R = η 2 + 2 η/q τ s k ✤ Crossover in the range µ m τ s R ≈ 10 ns dr q dt + iqv q = 0
Tokyo ISSP/SOFT2010 Workshop 1. Review of elasticity & Dynamics In-plane dynamics in a flat membrane (anti-symmetric mode) ij = − k ∂ i ( r + + e ∇ 2 h ) + O ( ǫ 2 ) p + i = ∂ j Σ + ✤ Relaxation of an ANTI-SYMMETRIC − 2 η qv + density modulation in a FLAT MEMBRANE q − η 2 q 2 v + − ikqr + q q − b ( v + q ) q − v − R = η 2 + 2 η /q + 2 b/q 2 τ a k τ a R ≈ 10 s at λ = 150 µ m dr ± dt + iqv ± q q = 0
Recommend
More recommend
