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

Symmetries and Solutions

• f the Membrane Shape Equation

Vladimir Pulov1 Mariana Hadjilazova,2 Ivailo Mladenov2

1Department of Physics, Technical University of Varna 2Institute 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.

bilayer bilayer

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

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 = Fc + λ ∫

S

dS + p ∫ dV Fc = kc

2

∫

S(2H − I

h)2dS + kG ∫

S KdS

– curvature free energy kc, kG – 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 + (2H − I h)(H2 + I

h 2 H − K) − λ kc H + p 2kc = 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 kc – 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 : x3 = w(x1, x2)

– Mong´ e representation of S

• (x1, x2, x3)

– Cartesian coordinates of R3

• w(x1, x2)

– Mong´ e gauge of S immersed in R3

• wα1α2...αk =

∂kw ∂xα1...∂xαk ,

k = 1, 2, . . .

• (gαβ) – first fundamental tensor (contravariant components)
• g = det(gαβ)

Fourth-Order PDE

1 2g−1/2gαβgµνwαβµν + Φ(x1, x2, w, w1, . . . , w222) = 0

Φ(x1, x2, w, w1, . . . , w222) – third-order differential equation

Membrane Shape Equation

Conformal Metric Representation

Conformal Metric (Konopelchenko, 1997)

• ds2 = 4q2ϕ2(dx2 + dy2)

– conformal metric

• (bαβ) =

( ϑ ω ω 8q2ϕ(1 + I hϕ) − ϑ ) – second fundamental tensor

• q(x, y), ϕ(x, y), ϑ(x, y), ω(x, y) – unknown functions

Gauss-Codazzi-Mainardi Equations (Γ2

12)x − (Γ2 11)y + Γ1 12Γ2 11 + Γ2 12Γ2 12 − Γ2 11Γ2 22 − Γ1 11Γ2 12 = −g11K

(b11)y − (b12)x − b11Γ1

12 − b12(Γ1 12 − Γ1 11) + b22Γ2 11 = 0

(b12)y − (b22)x − b11Γ1

22 − b12(Γ2 22 − Γ1 12) + b22Γ2 12 = 0

• Γσ

αβ

– Christoffel symbols (depend on q, qx, qy, ϕ, ϕx, ϕy)

Membrane Shape Equation

Conformal Metric Representation

System of Second-Order PDEs (De Matteis, 2002) q2(ϕxx + ϕyy) + 2qϕ(qxx + qyy) − 2ϕ(q2

x + q2 y)+

+q4(8ϕ + α2ϕ2 + α3ϕ3 + α4ϕ4) = 0 ϑy − ωx − (8 + α2

3 ϕ)(ϕqy + qϕy) = 0

ωy + ϑx − α2

3 qϕ(ϕqx + qϕy) + 8qϕqx = 0

4qϕ2(qxx +qyy)+4ϕq2(ϕxx +ϕyy)−4ϕ2(q2

x +q2 y)−4q2(ϕ2 x +ϕ2 y)−

−ω2 − ϑ2 + (8 + α2

3 ϕ)q2ϕθ = 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: (α2, α3, α4) ̸= (0, 0, 0)

Special Conformal Transformations (ˆ LI = ˆ Lc) Vc(ξ) = ξ∂z + ξz [−(ϑ − iω)∂ϑ − −(ω + iϑ − 4iq2ϕ − i α2

6 q2ϕ2)∂ω − q 2∂q

] + c.c. z = x + iy, ξ = ξ1 + iξ2 – complex notation ξ1, ξ2 – arbitrary real harmonic functions of z ξ1

y = −ξ2 x, ξ1 x = ξ2 y – Cauchy-Riemann conditions

Group Analysis

Symmetries of the Membrane Shape Equation

Symmetry Algebra (De Matteis, 2002)

• Case II: (α2, α3, α4) = (0, 0, 0)

Conformal Transformations and Dilatations (ˆ LII = ˆ Lc ⊕ ˆ Ld) Vc(ξ) = ξ∂z + ξz [−(ϑ − iω)∂ϑ− (ω + iϑ − 4iq2ϕ)∂ω − q

2∂q

] + c.c. Vd = ϑ∂ϑ + ω∂ω + ϕ∂ϕ z = x + iy, ξ = ξ1 + iξ2 – complex notation ξ1, ξ2 – arbitrary real harmonic functions of z ξ1

y = −ξ2 x, ξ1 x = ξ2 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 Vc(1/2) = ∂x q2 d2ϕ

dy2 + 2qϕ d2q dy2 − 2ϕ

(

dq dy

)2 + q4(8ϕ + α2ϕ2 + α3ϕ3 + α4ϕ4) = 0

dϑ dy − (8 + α2 3 ϕ)q(ϕ dq dy + q dϕ dy ) = 0

4qϕ2 d2q

dy2 + 4ϕq2 d2ϕ dy2 − 4ϕ2 ( dq dy

)2 − 4q2 (

dϕ dy

)2 − α2

5 − ϑ2+

+(8 + α2

3 ϕ)q2ϕϑ = 0

• three equations
• three unknown functions: q(y), ϕ(y), ϑ(y); ω(y) ≡ const
• three phenomenological constants: α2, α3, α4; α5 = ω

Group Analysis

Group-Invariant Solutions of the Membrane Shape Equation

Classification of the Group-Invariant Solutions

• f the Membrane Shape Equation

(De Matteis, 2002)

• All one-parameter subgroups of the general symmetry group
• f 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 Vc(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)

2ϕ , ρϕρ = −cϕ2)

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 = H0 = const ̸= I h Cartesian Coordinates x1 = − R sin x′

cosh y′ , x2 = − R cos x′ cosh y′ , x3 = −R tanh y′

Metric ds2 =

1 H2

0 cosh2 y′ (dx′2 + dy′2)

Second Fundamental Form Ω =

1 H0 cosh2 y′ (dx′2 + dy′2)

(x′, y′) = δ0(x, y) – coordinate scaling R = 1/H0 – radius of the sphere δ0 = const; I h – spontaneous curvature

Group Analysis

Group-Invariant Solutions

Delaunay Surfaces Obtained from Group-Invariant Solution for H = I h Nodoids obtained for I h3ϑ0 < 0 Unduloids obtained for 0 < I h3ϑ0 < 1 Metric ds2 = p2dΦ2 −

4p2 4I h2p4+2ϑ0I h3−I h2−4dp2

Second Fundamental Form Ω = (I hp2 + 1

4ϑ0I

h2)dΦ2 −

4I hp2−ϑ0I h2 4I h2p4+2ϑ0I h3+I h2−4dp2

p = √ r ( 1 − σ2sn(2 √ 2y, σ ) ), Φ = 2x/I h – (x, y) → (Φ, p) r = C(I h)/4, σ = √ 2C(I h)/r coordinate change C(I h) = √ I h6 − I h4/4 − 4I h3 + 4 I h – spontaneous curvature

Group Analysis

Group-Invariant Solutions

Clifford Torus Obtained from Group-Invariant Solution for q = q0 = const Detected Experimentally by Mutz and Bensimon (1991) Cartesian Coordinates [ (x1)2 + (x2)2 + (x3)2 +

1 I h2

]2 =

8 I h2

[ (x1)2 + (x2)2] Metric ds2 = ρ2dθ2 −

dρ2 1+2 √ 2I hρ+I h2ρ2

Second Fundamental Form Ω = (I hρ2 + √ 2ρ)dθ2 −

I hdρ2 1+2 √ 2I hρ+I h2ρ2

θ – rotation angle through the x3 axis ρ – radius; I h – spontaneous curvature

Group Analysis

Group-Invariant Solutions

Circular Biconcave Discoid Obtained from Group-Invariant Solution for q = q(ϕ) = ρ(ϕ)/2ϕ Shape of the Red Blood Cells Cartesian Coordinates x1 = cos x, x2 = sin x, x3 = ∫ tan ψ(ρ)dρ + z0 Metric ds2 = ρ2dx2 +

ρ2 f (ρ)dρ2

Second Fundamental Form Ω = 2I hρ2 ln ( ρ

ρ0 )dx2 + 2I

hρ2 ( ln ( ρ

ρ0 ) + 1

)

1 f (ρ)dρ2

x – angular variable; ρ – radius; ρ0, f0, z0 = const (1/ϕ2)(dϕ/dρ) = −2I h/ρ – definition of ρ(ϕ) ψ – angle between the tangent to the contour and the radius f (ρ) = ρ2 ( f0 − 4I h2ρ2 ln2 ( ρ

ρ0

) ; sec2 ψ(ρ) = ρ2/f (ρ)

References

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and Possible Experiments, Z. Naturforsch 28c 693-703.

• Hu J.-G. and Ou-Yang Z.-C. (1993) Shape Equations of the

Axisymmetric Vesicles, Phys.Rev. E 47 461-467.

• Konopelchenko B. (1997) On Solutions of the Shape Equation

for Membranes and Strings, Phys. Lett. B 414 58-64.

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