Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation

Transcription

1 Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation Bo Li Dept. of Math & NSF Center for Theoretical Biological Physics UC San Diego, USA Collaborators: Li-Tien Cheng and Zhongming Wang $ $ $ NSF, DOE, and NIH $ $ $ Workshop on Numerical Methods for PDEs Sun Yat-Sen University, China, 7/28 8/1, 2010

2 Outline 1. Free-Energy Functional and Level-Set Equation 2. Discretization and Algorithm 3. A Technique of Numerical Integration 4. Convergence Test 5. Application to Molecular Solvation 6. Conclusions

3 1. Free-Energy Functional and Level-Set Equation

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5 A variational implicit-solvent model (Dzubiella Swanson McCammon 1996).. n... Ω.. Ω in ex xn. Γ x 1. x i U LJ r G[Γ] = P 0 Vol (Ω in ) + γ 0 Area (Γ) 2γ 0 τ U(x) = N i=1 U (i) LJ ( x x i ) = i=1 Γ N [ (σi 4ǫ i r H ds + ρ 0 Ω ex UdV ) 12 ( σi ) ] 6. r

6 Hadwiger s Theorem Let C = { all convex compact subsets of R 3 }, M = { finite unions of members of C }, F : M R : translationally and rotationally invariant. Assume F(U V) = F(U) + F(V) F(U V) U, V M, F(U k ) F(U) if U, U k C (k 1) and U k U. Then there exist a 1,...,a 4 R such that for all U M F(U) = a 1 Vol (U) + a 2 Area ( U) + a 3 H ds + a 4 U U K ds.

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8 Level-set representation of a moving surface Γ(t) = {x : φ(x, t) = 0}. Level-set equation t φ + v n φ = 0. Unit normal and normal velocity n = φ φ and v n = dx dt n. Mean curvature and Gaussian curvature H = 1 2 (κ 1 + κ 2 ) = 1 2 n and K = κ 1κ 2 = n adj ( 2 φ ) n.

9 Level-set optimization method t φ + v n φ = 0, v n = δ Γ G[Γ] = P 0 2γ 0 (H τk) + ρ 0 U. Recall G[Γ] = P 0 Vol (Ω in ) + γ 0 Area (Γ) 2γ 0 τ A steepest descent method [ d dx(t) dt G[Γ(t)] = δ Γ G[Γ(t)] dt Γ(t) Γ H ds + ρ 0 Ω ex UdV. ] n ds = [v n (x)] 2 ds 0. Γ(t)

10 2. Discretization and Algorithm

11 Time discretization by forward Euler s method φ (k+1) (x) φ (k) (x) t k = v n (k) (x) φ (k) (x). Spatial discretization: Decomposition of normal velocity t φ = v n φ = A + B, parabolic part A = 2γ 0 (H τk) φ, hyperbolic part B = (P 0 ρ 0 U) φ. Central differencing with parameter correction for A = A(φ). Upwinding schemes for B = B(φ). Time step t = 0.5h max x [Trace (C(x))/h + P 0 + 2ρ 0 U(x) ].

12 Introduce P = I φ φ φ 2 : λ = 0, 1, 1, u 1 = φ, u 2, u 3 φ, Π = 1 φ P ( 2 φ ) P : λ = 0, κ 1, κ 2, u 1 = φ, u 2, u 3 φ. Then H = 1 2 Trace Π and K = 1 2 [ (Trace Π) 2 Trace ( Π 2)]. A(φ) = 2γ 0 [H(φ) τk(φ)] φ = γ 0 C(φ) : 2 φ, where a C(φ) = [1 2τH(φ)] P(φ) + τπ(φ) = Q 1 0 a 2 0 Q has eigenvalues a 1 = 1 τκ 1 and a 2 = 1 τκ 2.

13 Linearization of the parabolic part A = A(φ) δa(φ)(ψ) = d dε A(φ + εψ) = γ 0 D(φ) : 2 ψ + E( φ, 2 φ, ψ), ε=0 D(φ) = [1 4τH(φ)] P(φ) + 2τΠ(φ) : λ = 0, 1 2τκ 1, 1 2τκ 2. Recall linearized parabolicity 1 2τκ 1 > 0, 1 2τκ 2 > 0 a A(φ) = γ 0 C(φ) : 2 φ = Q 1 0 a 2 0 Q : 2 φ Eigenvalues of C(φ) : a 1 = 1 τκ 1 and a 2 = 1 τκ 2. Parameter correction to enforce the parabolicity For i = 1, 2: if a i < 0.5 then re-set a i = 0.5.

14 Algorithm (1) Initialization: Input all parameters; Fix a computational box and discretize it with a uniform grid; Compute P0 and ρ 0 U(x) at all grid points except x 1,...,x N ; Generate an initial surface Γ = {x : φ(x) = 0}: a loose wrap (e.g., a large ball) or a tight-wrap, e.g., (2) Local level-set method: φ(x) = min 1 i N ( x x i σ i ). Choose a narrow band around the surface with width grid points. At each grid point in the band, compute φ, 2 φ, H, and K using centered difference schemes. At grid points near the band boundary, use a lower-order scheme.

15 Algorithm (continued) (3) Compute the free energy using f dv = (1 Heav (φ))dv, Ω in R 3 g ds = g φ δ(φ)dv. R 3 Γ (4) Calculate and extend v n φ = A + B. Parameter correction to enforce the parabolicity. Fast sweep to extend P0 and ρ 0 U. A WENO scheme to discretize the extended P0 and ρ 0 U. (5) Calculate the time step t.

16 (6) Reinitialize the level-set function φ by solving φ t + sign(φ 0 )( φ 1) = 0 with sign(φ 0 ) = φ 0 φ 20 + h. About 12 iterations in time. fourth-order Runge-Kutta scheme for time. fifth-order WENO for space. (7) Check the convergence. If not, locate the interface Γ and go to Step (2).

17 3. A Technique of Numerical Integration

18 . x 0 D Ω \ D R 3 \D 1 x x 0 k dv = Ω\D I 1 = Ω\D Ω 1 x x 0 k dv+ 1 x x 0 k dv = Approximate χ Ω\D by a smooth function. Ω R 3 \Ω χ Ω\D (x) x x 0 k dv 1 x x 0 k dv = I 1+I 2 The composite trapezoidal rule for one-dimensional integrals. A second-order accurate scheme.

19 Q x 0 Q 2 x Q 2 ϕ 2 Q 2 x axis Q 1 I 2 = R 3 \Ω Q 1 x x 0 k dv 2 ϕ1 The integral I 2 does not depend on D. Decompose R 3 \ Ω into six regions of two types.

20 Q x 0 Q 2 x Q 2 ϕ 2 Q 2 x axis ˆQ 1 = I 2,1 = Q 1 { (r, θ, z) : 0 < r < ˆQ 1 1 x x 0 k dv = Q 2 ϕ1 β } cos θ, θ 1 < θ < θ 2, L < z < θ2 Integrated analytically in r and z. L θ 1 β cos θ 0 r (r 2 + z 2 dr dθ dz ) k/2 The composite Simpson s rule for the integral in θ.

21 Q x 0 Q 2 x Q 2 ϕ 2 Q 2 x axis Q 1 Q 2 ϕ1 { } α Q 2 = (ρ, ϕ, ψ) : cos ϕ sinψ < ρ <, ϕ 1 < ϕ < ϕ 2, 0 < ψ < π 1 π ϕ2 I 2,2 = Q 2 x x 0 k dv = sinψ dρ dϕ dψ ρk 2 0 ϕ 1 α cos ϕ sin ψ Calculated analytically.

22 4. Convergence Test

23 Γ O R A one-particle system G(R) = 4 ( ) σ 3 P 0πR 3 + 4γ 0 πr γ 0 τπr + 16πρ 0 ǫ 9R 9 σ6 3R 3.

24 6 5.5 LS Exact The total free energy Steps The decay of the free energy with respect to level-set iteration steps.

25 LS Exact Log 2 of the total free energy log 2 1/h The free energy vs. step size in logarithmic scale.

26 5. Application to Molecular Solvation

27 A two-atom system. Cross sections of the free-energy minimizing surfaces at distance d = 4Å (upper left), d = 6Å (upper right), d = 8Å (lower left), and d = 10Å (lower right).

28 Tight initial with distance of 6.6Å Final Loose initial with distance of 6.6Å Tight initial with distance of 7Å Loose initial with distance of 7Å Final Final Final A two-plate system. Cross sections of initial and final surfaces at two different separations d = 6.6Å (left) and d = 7Å (right).

29 300 Dry Wet 350 The total free energy Distance of two plates A two-plate system (continued). The free energy vs. the plate separation distance, showing a hysteresis loop.

30 Topological changes in a system of 5 atoms.

31 A helical polymer. Two different initial surfaces and their corresponding final surfaces.

32 The free-energy minimizing surface of an artificial molecule.

33 6. Conclusions

34 Developed and tested a level-set optimization method Discretization: centered differencing for curvatures, upwinding for lower-order hyperbolic terms Parameter correction: mathematical regularization and physical modeling Tested second-order convergence rate Application to solvation of simple, nonpolar molecules Initial surfaces determine final, free-energy minimizing surfaces Efficiency vs. accuracy.

35 Current work Improve the accuracy lost in the difference of large geometrical and LJ parts of the free energy Speed up the computation Convergence analysis Add the surface integral of the Gaussian curvature Include electrostatics Monte Carlo level-set method for searching local minima Application to large biomolecular systems Reference L.-T. Cheng, B. Li, and Z. Wang, Level-set minimization of potential controlled Hadwiger valuations for molecular solvation, J. Comp. Phys., 2010 (accepted).

36 Thank you!