Symmetries and Solutions of the Membrane Shape Equation Vladimir - PowerPoint PPT Presentation

Symmetries and Solutions of the Membrane Shape Equation Vladimir Pulov 1 Mariana Hadjilazova, 2 Ivailo Mladenov 2 1 Department of Physics, Technical University of Varna 2 Institute of Biophysics, Bulgarian Academy of Science Geometry, Integrability and Quantization, June 8-13, 2012

Outline 1. Closed Biomembranes – Vesicles Molecule Bilayers Equilibrium Shapes Membrane Shape Equation 2. Membrane Shape Equation Exact Analytic Solutions Mong´ e Representation Conformal Metric Representation 3. Group Analysis of the Membrane Shape Equation Symmetries Symmetry Reduction Group-Invariant Solutions

Closed Biomembranes – Vesicles Molecule Bilayers Lipid Vesicles Formation • In aqueous solution, amphiphilic molecules (e.g., phospholipids) may form bilayers, the hydrophilic heads of these molecules being located in both outer sides of the bilayer, which are in contact with the liquid, while their hydrophobic tails remain at the interior. • A bilayer may form a closed membrane – vesicle. Vesicles constitute a well-defined and sufficiently simple model system for studying basic physical properties of the more complex cell biomembranes.

hydrophilic heads hydrophilic heads hydrophobic tails hydrophobic tails bilayer bilayer aqueous solution aqueous solution hydrophilic heads hydrophilic heads

Closed Biomembranes – Vesicles Equilibrium Shapes Spontaneous Curvature Model (Helfrich, 1973) The equilibrium shapes of lipid vesicles are determined by the extremals of the Helfrich’s functional ∫ ∫ F = F c + λ dS + p dV S F c = k c h ) 2 dS + k G ∫ ∫ S (2 H − I – curvature free energy S KdS 2 k c , k G – bending and Gaussian rigidities λ – tensile stress p – osmotic pressure H , K – mean and Gaussian curvatures I h – Helfrich’s spontaneous curvature

Closed Biomembranes – Vesicles Equilibrium Shapes Membrane Shape Equation h )( H 2 + I p 2 H − K ) − λ h ∆ H + (2 H − I k c H + 2 k c = 0 • is Euler-Lagrange equation of the Helfrich’s functional F • derived by Ou-Yang and Helfrich (1989) • describes the equilibrium shapes of lipid vesicles • λ, p (stress and pressure) – Lagrangian multipliers ∆ – Laplace-Beltrami operator H , K , I h – mean, Gaussian and spontaneous curvatures k c – curvature bending rigidity

Membrane Shape Equation Exact Analytic Solutions Equilibrium Vesicle Shapes I • Spheres and Circular Cylinders Ou-Yang and Helfrich, 1989 • Clifford tori Ou-Yang, 1990, 1993; Hu and Ou-Yang, 1993 • Delaunnay Surfaces Naito, Okuda and Ou-Yang, 1995; Mladenov, 2002 • Circular Biconcave Discoids Naito, Okuda and Ou-Yang, 1993, 1996

Membrane Shape Equation Exact Analytic Solutions Equilibrium Vesicle Shapes II • Nodoidlike and Unduloidlike Shapes Naito, Okuda and Ou-Yang, 1995 • Willmore and Constant Squared Mean Curvature Surfaces Willmore, 1993; Konopelchenko, 1997; Vassilev and Mladenov, 2004 • Generalized Cylindrical Surfaces Ou-Yang, Liu and Xie, 1999; Vassilev, Djondjorov and Mladenov, 2008

Membrane Shape Equation Mong´ e Representation • S : x 3 = w ( x 1 , x 2 ) – Mong´ e representation of S • ( x 1 , x 2 , x 3 ) Cartesian coordinates of R 3 – • w ( x 1 , x 2 ) e gauge of S immersed in R 3 – Mong´ ∂ k w • w α 1 α 2 ...α k = ∂ x α 1 ...∂ x α k , k = 1 , 2 , . . . • ( g αβ ) – first fundamental tensor (contravariant components) • g = det( g αβ ) Fourth-Order PDE 1 2 g − 1 / 2 g αβ g µν w αβµν + Φ( x 1 , x 2 , w , w 1 , . . . , w 222 ) = 0 Φ( x 1 , x 2 , w , w 1 , . . . , w 222 ) – third-order differential equation

Membrane Shape Equation Conformal Metric Representation Conformal Metric (Konopelchenko, 1997) • ds 2 = 4 q 2 ϕ 2 ( dx 2 + dy 2 ) – conformal metric ( ϑ ω ) – second • ( b αβ ) = 8 q 2 ϕ (1 + I ω h ϕ ) − ϑ fundamental tensor • q ( x , y ), ϕ ( x , y ), ϑ ( x , y ), ω ( x , y ) – unknown functions Gauss-Codazzi-Mainardi Equations (Γ 2 12 ) x − (Γ 2 11 ) y + Γ 1 12 Γ 2 11 + Γ 2 12 Γ 2 12 − Γ 2 11 Γ 2 22 − Γ 1 11 Γ 2 12 = − g 11 K ( b 11 ) y − ( b 12 ) x − b 11 Γ 1 12 − b 12 (Γ 1 12 − Γ 1 11 ) + b 22 Γ 2 11 = 0 ( b 12 ) y − ( b 22 ) x − b 11 Γ 1 22 − b 12 (Γ 2 22 − Γ 1 12 ) + b 22 Γ 2 12 = 0 • Γ σ – Christoffel symbols (depend on q , q x , q y , ϕ, ϕ x , ϕ y ) αβ

Membrane Shape Equation Conformal Metric Representation System of Second-Order PDEs (De Matteis, 2002) q 2 ( ϕ xx + ϕ yy ) + 2 q ϕ ( q xx + q yy ) − 2 ϕ ( q 2 x + q 2 y )+ + q 4 (8 ϕ + α 2 ϕ 2 + α 3 ϕ 3 + α 4 ϕ 4 ) = 0 ϑ y − ω x − (8 + α 2 3 ϕ )( ϕ q y + q ϕ y ) = 0 ω y + ϑ x − α 2 3 q ϕ ( ϕ q x + q ϕ y ) + 8 q ϕ q x = 0 4 q ϕ 2 ( q xx + q yy )+4 ϕ q 2 ( ϕ xx + ϕ yy ) − 4 ϕ 2 ( q 2 x + q 2 y ) − 4 q 2 ( ϕ 2 x + ϕ 2 y ) − − ω 2 − ϑ 2 + (8 + α 2 3 ϕ ) q 2 ϕθ = 0 • four equations • four unknown functions: q ( x , y ) , ϕ ( x , y ) , ϑ ( x , y ) , ω ( x , y ) • conformal coordinates: ( x , y )

Group Analysis Symmetries of the Membrane Shape Equation Symmetry Algebra (De Matteis, 2002) • Case I: ( α 2 , α 3 , α 4 ) ̸ = (0 , 0 , 0) Special Conformal Transformations (ˆ L I = ˆ L c ) V c ( ξ ) = ξ∂ z + ξ z [ − ( ϑ − i ω ) ∂ ϑ − − ( ω + i ϑ − 4 iq 2 ϕ − i α 2 6 q 2 ϕ 2 ) ∂ ω − q ] 2 ∂ q + c.c. ξ = ξ 1 + i ξ 2 – complex notation z = x + iy , ξ 1 , ξ 2 – arbitrary real harmonic functions of z ξ 1 y = − ξ 2 x , ξ 1 x = ξ 2 y – Cauchy-Riemann conditions

Group Analysis Symmetries of the Membrane Shape Equation Symmetry Algebra (De Matteis, 2002) • Case II: ( α 2 , α 3 , α 4 ) = (0 , 0 , 0) Conformal Transformations and Dilatations (ˆ L II = ˆ L c ⊕ ˆ L d ) V c ( ξ ) = ξ∂ z + ξ z [ − ( ϑ − i ω ) ∂ ϑ − ( ω + i ϑ − 4 iq 2 ϕ ) ∂ ω − q ] 2 ∂ q + c.c. V d = ϑ∂ ϑ + ω∂ ω + ϕ∂ ϕ ξ = ξ 1 + i ξ 2 – complex notation z = x + iy , ξ 1 , ξ 2 – arbitrary real harmonic functions of z ξ 1 y = − ξ 2 x , ξ 1 x = ξ 2 y – Cauchy-Riemann conditions

Group Analysis Symmetry Reduction of the Membrane Shape Equation Reduced System of Second-Order ODEs for Solutions Invariant under the Subgroup Generated by V c (1 / 2) = ∂ x ) 2 q 2 d 2 ϕ dy 2 + 2 q ϕ d 2 q ( + q 4 (8 ϕ + α 2 ϕ 2 + α 3 ϕ 3 + α 4 ϕ 4 ) = 0 dq dy 2 − 2 ϕ dy dy − (8 + α 2 3 ϕ ) q ( ϕ dq dy + q d ϕ d ϑ dy ) = 0 ) 2 ) 2 dy 2 − 4 ϕ 2 ( − 4 q 2 ( 4 q ϕ 2 d 2 q dy 2 + 4 ϕ q 2 d 2 ϕ dq d ϕ − α 2 5 − ϑ 2 + dy dy +(8 + α 2 3 ϕ ) q 2 ϕϑ = 0 • three equations • three unknown functions: q ( y ) , ϕ ( y ) , ϑ ( y ); ω ( y ) ≡ const • three phenomenological constants: α 2 , α 3 , α 4 ; α 5 = ω

Group Analysis Group-Invariant Solutions of the Membrane Shape Equation Classification of the Group-Invariant Solutions of the Membrane Shape Equation (De Matteis, 2002) • All one-parameter subgroups of the general symmetry group of the membrane shape equation (in the above conformal metric presentation) are equivalent through the adjoint representation of the symmetry group on its Lie algebra. • Any solution invariant under one-parameter subgroup of the general symmetry group of the membrane shape equation can be obtained by applying a symmetry group transformation to some solution invariant under the one-parameter symmetry subgroup generated by V c (1 / 2) = ∂ x .

Group Analysis Group-Invariant Solutions Vesicle Shapes Derived from Solutions Invariant Under the Translation Symmetry Subgroup ( x , y , q , ϕ, ϑ, ω ) �→ ( x + ε, y , q , ϕ, ϑ, ω ) , ε ∈ R (De Matteis, 2002) • Sphere (for H = const) • Delaunay’s Surfaces (for H = I h ) • Toroidal Surfaces (for q = const) • Circular Biconcave Discoid (for q = ρ ( ϕ ) 2 ϕ , ρϕ ρ = − c ϕ 2 )

Figure : The open parts of the Delaunay surfaces - cylinder, sphere, catenoid, unduloid and nodoid.

Group Analysis Group-Invariant Solutions Sphere Obtained from Group-Invariant Solution for H = H 0 = const ̸ = I h Cartesian Coordinates x 1 = − R sin x ′ cosh y ′ , x 2 = − R cos x ′ cosh y ′ , x 3 = − R tanh y ′ Metric 0 cosh 2 y ′ ( dx ′ 2 + dy ′ 2 ) ds 2 = 1 H 2 Second Fundamental Form H 0 cosh 2 y ′ ( dx ′ 2 + dy ′ 2 ) 1 Ω = ( x ′ , y ′ ) = δ 0 ( x , y ) – coordinate scaling R = 1 / H 0 – radius of the sphere δ 0 = const; h – spontaneous curvature I

Group Analysis Group-Invariant Solutions Delaunay Surfaces Obtained from Group-Invariant Solution for H = I h h 3 ϑ 0 < 0 Nodoids obtained for I h 3 ϑ 0 < 1 Unduloids obtained for 0 < I Metric ds 2 = p 2 d Φ 2 − 4 p 2 h 2 − 4 dp 2 h 2 p 4 +2 ϑ 0 I h 3 − I 4 I Second Fundamental Form h p 2 + 1 h 2 ) d Φ 2 − h p 2 − ϑ 0 I h 2 4 I h 2 − 4 dp 2 Ω = ( I 4 ϑ 0 I h 2 p 4 +2 ϑ 0 I h 3 + I 4 I √ √ ( 1 − σ 2 sn(2 ) p = 2 y , σ ) , Φ = 2 x / I – ( x , y ) → (Φ , p ) r h √ r = C ( I h ) / 4 , σ = 2 C ( I h ) / r coordinate change √ h 3 + 4 h 6 − I h 4 / 4 − 4 I C ( I h ) = I h – spontaneous curvature I