Continuum Methods 1. John Lowengrub Dept Math UC Irvine

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1 Continuum Methods 1 John Lowengrub Dept Math UC Irvine

2 Membranes, Cells, Tissues complex micro-structured soft matter Micro scale Cell: ~10 micron Sub cell (genes, large proteins): nanometer Macro scale Tissues: billion to 1000 billion cells or 1 10 centimeter Processes at cell scale Signals at sub-cell scale cell aggregation, organ development

3 Outline Give examples of continuum methods applied to various problems in the four workshop areas: Membrane and protein science Microfluidics Angiogenesis and neovascularization Systems biology Today: Biomembranes (1 st talk) Microfluidics (2 nd talk) Focus: Different modeling approaches, advantages/disadvantages

4 Biomembranes main structural component complex architecture of biological systems Form interface between cell, and organelle structures with microenvironment Lipid bilayer containing cholesterol, proteins Several nm thick, surface area can be several mm Can be highly mobile and fluid-like McMahon, Gallop. Nature 2005 Membrane morphology changes during cellular movement, division and vesicle trafficking. Active role.

5 Biomembranes Contd. Membrane subdomains with particular curvature may have precise biological properties and function Mechanisms to control membrane morphologies: McMahon, Gallop. Nature 2005

6 Mathematical Modeling Bending energy/spontaneous curvature model: ( ) 2 = + H 0 + G Σ Σ E τ b H H b Gd surface energy normal bending stiffness spontaneous curvature Gaussian bending stiffness H = (1/ R + 1/ R ) / 2 Mean curvature 1 2 G = 1/( RR ) Gaussian curvature 1 2 Lipowsky, Nature 1991.

7 Constraints Area constraint A= dσ= A 0 Σ (exchange of lipids with surrounding microenvironment is typically very slow) Volume constraint Vol( Σ ) = 1/3 x n dσ= V Σ (limited osmosis, can be controlled however) Morphology: Minimization of E subject to above constraints. 0

8 Related Problems Willmore flow Surface diffusion in materials Image processing

9 Willmore flow E 2 = H dσ Σ Normal velocity: δ E V = µ = = ΣH 2H H K δσ ( 2 ) High order, nonlinear equation on moving boundary. Germain, 1810 Willmore, Riemannian Geometry 1993.

10 n Surface diffusion in materials E = τ ( n, H) dσ Σ Normal vector (reflects crystalline anistropy) Example (2D): τ 2 δ, = γ( θ) + 2 ( n H) DiCarlo, Gurtin, Podio-Duidugli, SIAM J. Appl. Math. (1992);Gurtin, Jabbour, Arch. Rat. Mech. Anal. (2002); Spencer, Phys. Rev. E (2004) Then, chemical potential given by (2D) δ E µ = = γ θ + γ θ + δ + δσ Normal velocity: H ( ( ) ''( )) H 2 ( H 3 ) ss H V = µ ss 2 6 th order system!

11 Wulff Shapes for 4-fold Anisotropy Wulff Shapes: increasing (given enclosed area) increasing Nonphysical ears form for

12 Willmore regularization blow-up Willmore (α=1)

13 Image processing Xu, Pan, C.A.D Clarenz, et al, C. A. Geom. Design 2004

14 Difficulties Numerical stiffness: V Σ H t s 4 Surface diffusion V = Σ H Computational Methods with Willmore regularization V 2 Σ H Sharp interface/front-tracking Level-set methods/front capturing Phase-field methods/front capturing

15 Sharp interface methods To overcome stiffness: Implicit discretizations, marker-point redistribution Surface diffusion Axisymmetric Willmore t Bansch, Morin, Nochetto JCP 2005 Small scale decomposition (2D, Axisymmetric). Hou, Lowengrub, Shelley JCP 1994 Mayer, Simonett. 2001

16 SSD for axisymmetric surfaces Reformulation: and Total curvature 2H s = θ + sinθ r Willmore flow/ Surface diffusion: V θ sss Dominant term at small spatial scales

17 Special choice of Tangential velocity Marker-points equally spaced in arclength: 1 α 0 α' α s = s d ' = L( t) α T( α, t) = T(0, t) Vθ dα' + α Vθα' dα' 1 α' 0 0 Linear, constant coefficient equation at leading order: 1 θt = θ (, ) 4 ssss + N α t Lt () Easy to apply implicit time integration algorithms

18 Extended form Deviations from equal-arclength may arise. Can overcome by requiring instead: t sα L() t = sα L() t ( ) ( ) This makes s Lt () α = a stable manifold. α T( α, t) = T(0, t) Vθ dα' + α Vθ dα' s dα' + αl( t) 1 α' α' α' Many other choices of tangential velocity possible. (e.g., cluster points in regions of high curvature, etc.) α additional terms

19 3D Bansch, Morin, Nochetto JCP Finite element approach. Update: Solve: By: Evaluated on current (time n) surface Using C0 elements. Extension to Willmore flow Burger, Voigt et al (2006)

20 3D Surface diffusion Mesh adaptivity using a posteriori error estimates Implementation using Albert Schmidt, Sieberg, Acta Math Leads to topology change Bansch, Morin, Nochetto JCP 2005.

21 Level-set methods Osher, Sethian 1988 Interface capturing. Σ () t = x φ( x,) t = 0 { } Interface moves with speed V: φ + V φ = 0 t ext Willmore flow. Droske, Rumpf IFB 2004 where V ext is an extension of V off Σ weighted mean curvature Semi-implicit discretization

22 Level-set results (sample) Willmore flow Droske, Rumpf IFB 2004 Anisotropic surface diffusion with Willmore regularization Burger, JCP 2005.

23 Phase-field methods c is a smooth order parameter c = 0.5 (Γ) y x y x y=const cross-section

24 Why Phase-field? Allows the easy capture of interface dynamics. Avoid explicit tracking. Easy to add more physics (e.g., elasticity, multiple phases). Downside: introduce finite thickness. diffuse interface

25 Phase-field Willmore Problem Du et al. showed rigorous asymptotic convergence ( ) of solutions of to solutions of the classical Willmore problem Qiang Du et al, A phase field formulation of the Willmore problem, Nonlinearity (2005).

26 Evolution equations Gradient flow approach: δ E W, ε 1 ( 2 = µ = νf''( c) ε ν), 3 δc ε 2 ν = F'( c) ε c Phase field equation: 1 ( 2 c ) t = µ = νf''( c) ε ν, 3 ε 2 ν = F'( c) ε c 4 th order nonlinear equation. (non conserved)

27 3D Anisotropic Diffuse Interface Model order parameter finite interfacial thickness corner rounding double well energy density anisotropy asymptotically like for α=1 (α=0) Wise, Lowengrub, Kim, Thornton, Voorhees, Johnson, Appl. Phys. Lett. (2005) (α=1) Ratz, Ribalta, Voigt, J. Comput. Phys. (2005) (α=0,1) Kim, Wise, Lowengrub (in preparation)

28 Evolution: 6 th -order Cahn-Hilliard Eqn mass conservation: M = 1, bulk diff.; M = c(1-c), surface diff. diagonal anisotropy matrix linear biharmonic for α = 0 Asymptotic convergence to the sharp interface surface diffusion model?

29 Discretization of Cahn-Hilliard Eq.: Crank-Nicholson α=0 case center-differencing (compact stencils) Solve nonlinear elliptic problem using adaptive full approximation scheme (AFAS) for each time step n.

30 Linearizations local Picard-linearization: GS local Newton-linearization: GS local Picard-linearization: Vcycle anisotropy is fully implicit!

31 Details Mild solvability time-step restriction (being remedied) No time-step restriction owing to anisotropy Tests indicate 2nd order accuracy method (c, a posteriori in l 2 ) J.S. Kim, K. Kang, and J.S. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., (2004) Wise et al., Appl. Phys. Lett. (2005) (QD self assembly) Kim, Wise, Lowengrub, Adaptive Method for Strong Anisotropy (in preparation)

32 Isotropic Spinodal Decomposition (a=0, δ=0) base level = levels of refinement effective e Number of mesh points Uniform mesh Adaptive mesh time 30% volume fraction 1/6 cost (long times)

33 2D Mesh: 2 Levels of Refinement Kim, Wise, Lowengrub, (in preparation).

34 3D Adaptive Computations interface thickness corner rounding ε = 1.8e-02, δ = 5.0e-04, a = fold anisotropy h 2 ~ ε base grid 32 3, 2 levels, h 2 = 3.2/128 surface diffusion, M = c(1-c) c = 0.5 isosurfaces

35 Anisotropic surface energy + Willmore regularization Evolution to Wulff shape

36 Adaptive Mesh

37 Phase-field formulation of membrane problem Du, Liu, Wang JCP 2004 and Bending energy volume Surface area Gradient flow: λ1, λ2 Lagrange multipliers for volume and area conservation.

38 Numerical methods Implicit time discretization, Discrete energy law, Periodic BC, pseudo-spectral methods Evolution to a discocyte Evolution to a torus Du, Liu, Wang JCP 2006

39 Further directions Effects of fluid flow Du et al. for single-component membranes (to appear) Lowengrub et al for multicomponent membranes (in progress) Multicomponent membranes More than one lipid component Baumgart, Hess, Webb Nature 2003 Du et al., phase-field models Lowengrub, Voigt et al., sharp interface models