Coupling the Level-Set Method with Variational Implicit Solvent Modeling of Molecular Solvation

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1 Coupling the Level-Set Method with Variational Implicit Solvent Modeling of Molecular Solvation Bo Li Math Dept & CTBP, UCSD Li-Tien Cheng (Math, UCSD) Zhongming Wang (Math & Biochem, UCSD) Yang Xie (MAE, UCSD) Joachim Dzubiella (Phys., Tech. Univ. Munich) Jianwei Che (Genomics Inst. of Novartis Res. Found.) J. Andrew McCammon (Biochem & CTBP, UCSD) Supported by NSF, DOE, DFG, Sloan, NIH, HHMI, CTBP 1

2 References 1. J. Dzubiella, J.M.J. Swanson, & J.A. McCammon, Phys. Rev. Lett., 96, , J. Dzubiella, J.M.J. Swanson, & J.A. McCammon, J. Chem. Phys., 124, , L.-T. Cheng, J. Dzubiella, J.A. McCammon, & B. Li, J. Chem. Phys., 127, , J. Che, J.Dzubiella, B. Li, & J.A. McCammon, J. Phys. Chem. B, 112, 3058, B. Li, SIAM J. Math. Anal., 2008 (revision submitted). 6. L.-T. Cheng, Y. Xie, J. Dzubiella, J.A. McCammon, J. Che, & B. Li, J. Chem. Theory Comput., 2008 (submitted). 2

3 OUTLINE 1. Introduction 2. Varitaional implicit solvent models 3. The level-set method 4. Results of level-set calculations 5. Conclusions and outlook 3

4 1. Introduction 4

5 Molecules Bond, angle, torsion Polarity, electrostatics Complex structure Bohr s model Ball and stick model ( Like-charge attraction of (-) actin protein rods mediated by (+) barium ions. Wong et al., UIUC. 5

6 Biomolecules (proteins, nucleic acids, lipids,...) U. Conn Kyushu Inst. of Tech., Japan UCSB UCSB 6

7 solvent Solvation solute Attraction and association of molecules of a solvent (e.g., water) with molecules or ions of a solute Molecular surface: geometry Solvent and solute: polar and nonpolar determined by dielectrics (water 80, ethanol 24, hexane 2) Stable solvent-solute interaction: small and large scales, hydrophobicity, etc. Energy landscape structure, function, dynamics, etc. 7

8 Implicit vs. explicit Explicit solvent: treat solvent atoms individually (e.g. MD) First principle Accurate Small systems Statistics solvent solute Implicit solvent: coarse-grained solvent Approximation Efficient Large systems Thermodynamics solute solvent 8

molecular mechanics Numerical methods the level-set method D.

9 This work Solvation free energy and potential of mean forces Molecular structures and interactions Implicit solvent and molecular mechanics Numerical methods the level-set method D. Cooper, U. Virginia Water inside a protein. Imai et al., J. Am. Chem. Soc. 9

10 2. Variational Implicit Solvent Models 10

11 Molecular surface approach molecular surface: SAS, SES SAS = level-set of distance function surface area + Poisson-Boltzmann free energy 11

12 Hydrophobic effect 12

13 A variational implicit solvent model (Dzubiella, Swanson, & McCammon, 2006) solvent solute Experimentally observed solute-solvent interfaces are those surfaces that minimize a solvation free energy functional. The free energy functional should couple together all kinds of different interactions (polar, nonpolar, dispersive, etc.). 13

14 Γ x q i i solute Ω solvent Ω Γ x i q i ρ q j j ρ w solute region solute-solvent interface (dielectric boundary) center of the i-th solute atom fixed point charge at the i-th solute particle bulk density of the j-th ion species bulk charge of the j-th ion species solvent density 14

15 A free energy functional G[ Γ] = G [ Γ] + G [ Γ] + G [ Γ] geom vdw elec Γ x q i i solute Ω solvent G geom [ Γ] Pvol(Ω): P= = γds : Γ γ = Pvol( Ω) + γds Γ Creation of a cavity in the solvent pressure difference Molecule rearrangement near the interface surface energy density ( 1+ τh ) γ ( τh ) γ = γ 1 / 0 0 (Tolman 1949) a fitting prarameter H = mean curvature 15

16 Geometrical part of the free energy (K: Gaussian curvature) Pvol ( 0 Ω ) + γ area( Γ) + c Hadwiger s Theorem H Γ HdS ( ) + c KdS Let C = the set of all convex bodies, M = the set of finite union of convex bodies. If F : M R is rotational and translational invariant, additive: F( U V ) = F( U ) + F( V ) F( U V ) U, V M, then conditionally continuous: U, U C, U U F( U ) F( U ), j j F( U) avolu ( ) + barea( U) + c HdS+ d KdS U M. = j U K Γ U 16

17 Ω G vdw[ Γ] = ρw U( x x c i ) dv i Non-electrostatic, van der Waals type, solute-solvent interaction governed by a pair-wise interaction potential ρ w = U = U ( r) = 4ε [( σ) ( σ ) 6 ] LJ 0 U LJ solvent density r 12 r U Lennard-Jones potential O σ r ε 0 17

18 G elec [ Γ] = Electrostatic free energy ψ = ψ = The Poisson-Boltzmann equation ( ε ψ ) + 4πχ c ρ q exp( q ψ / k T ) = 4π q δ ( x x ) χ ref Ω c ε = 1 2 electrostatic potential = i + k B reference electrostatic potential Ω j j characteristic function of dielectrics = j ( ε ψ ) = 4π q δ( x x ) solute q T i ( ψ ψ j ρ ref j Ω ref c )( x ( q jψ / kbt e 1)dx { i ) ε ε i Ω i solute solvent ε ( x) 8π j ψ i B 2 Ω c dx in solute region in solvent region i i Ω i c Ω 18

19 Remarks { Pvol( Ω) + γ area( )} min 0 Γ The Laplace-Young relation: P( = P P ) = 2γ H l s 0 G elec min U LJ ( x x1 ) dv Γ Ω c Ω = B( x 1, σ ) Extremize ε( x) q x dx k T ( q j kbt [ Γ] = 1 2 / 2 i( ψ ψ ref )( i) ψ + B ρj e ψ 1)dx c Ω 8π Ω i U LJ ε 0 The Poisson-Boltzmann equation ( ε ψ ) + 4πχ c ρ q exp( q ψ / k T ) = 4π q δ ( x x ) Ω j j j j O B σ j i i i r 19

20 From variational implicit solvent models to SAS/SES types of implicit solvent models { } area ( Γ) + U ( x x ) dv Ω c min γ 0 1 Molecular surface (van der Waals, SAS, SES, etc.) Potentials: hard sphere, square well, etc. Accuracy vs. efficiency 20

21 21 Mechanical energy of solute atoms ), ( ), ( ],..., [,, 1 j i j i bond j i j i LJ N x x W x x W x x V r r r r r r = + ),,, ( ),, (,,,,, l k j l k j i i torsion k j i k j i bend x x x x W x x x W r r r r r r + + An effective total Hamiltonian ],,..., ; [ ],..., [ ],..., ; [ N N N x x G x x V x x H r r r r r r Γ + = Γ ],..., ; [ min 1 N x x H r r Γ Equilibrium conformations

22 3. The Level-Set Method 22

23 The level-set method Interface motion V = n Vn ( x, t) for x Γ(t) Level-set representation Γ( t) = { x Ω : ϕ( x, t) = The level-set equation n x Γ(t) z = ϕ( x, t) 0} z = 0 Γ (t) ϕt + Vn ϕ = 0 [ ϕ( x( t), t) = 0 ϕt + ϕ x& = 0 ] ϕ ϕ x & = x& ϕ = ( n x& ) ϕ = ( ) ϕ V n ϕ 23

24 Examples of normal velocity Geometrically based motion Motion by mean curvature V n = H Motion by the surface Laplacian of mean curvature u t Δu =0 in Ω Ω+ u = H on Γ u n V n V = Δ H s External field = 0 = n [ u ] n on on Γ Ω Γ n n Ω Ω + Ω 24

25 Level-set formulas of geometrical quantities Normal n = ϕ ϕ Mean curvature 1 H = n 2 Gaussian curvature K = n adj ( He( ϕ )) n 25

26 Topological changes Merging Break-up Disappearing Nucleation? Accuracy issues Interface approximation Conservation of mass Rigorous analysis 26

27 Application to solvation of nonpolar molecules First variation δg[ Γ] = P + 2γ 0[ H ( x) τk ( x)] ρw U ( x xi i δ dv =1 Ω ds = 2H δ Γ ) δ HdS = K Γ Normal velocity of the solute-solvent interface V n = δg[γ ] Relaxation r of solute atoms dxi r r = r x H[ Γ; x1, K, xn ], i = 1,..., N, i dt 27

28 Algorithm Step 1. Input parameters and initialize level-set function Step 2. Calculate the normal and curvatures Step 3. Calculate and extend the normal velocity Step 4. Solve the level-set equation Step 5. Reinitialize the level-set function Step 6. Solve ODEs for the motion of solute particles t := t + Δt Step 7. Set and go to Step 2 28

29 New level-set techniques Regularization of degenerated geometrically based motion of interface. Calculation of the Lennard-Jones potential using the largest ball in the solute region centered at a fixed solute particle. A two grid method for calculating the interaction energy with a Lennard-Jones potential. 29

30 4. Results of Level-Set Calculations 30

31 Numerical calculations with fixed solute atoms Units: Energy = k B T, Length = Example 1. Two xenon atoms P = 0, γ 0 = 0.174, δ = 1 ρ = 0.033, σ = 3.57, ε 0 = w Example 2. Two paraffin plates P = 0, γ 0 = 0.174, δ = 0. 9 ρ = 0.033, σ = 3.538, ε 0 = w Example 3. Two alkane helical chains P = 0, γ 0 = 0.176, δ = 1. 3 ρ = 0.033, σ = 3.538, ε 0 = w Example 4. A C60 molecule buckyball Example 5. A large molecule of about 1,000 atoms. A o 31

32 Example 1. Two xenon atoms 32

33 2 1 w(d)/k B T W(d)/k B T d/ Å Comparison of PMF by the level-set (circles) and by molecular dynamics (solid line) calculations. 33

34 34

35 Example 2. Two paraffin plates 35

36 Comparison of the level-set and molecular dynamics calculations (Koishi et al., 2004 and 2005) for the two paraffin plates. 36

37 Example 3. Two helical alkane chains 37

38 A Richards-Lee type surface Level-set calculation 38

39 Mean curvature field 39

40 Example 4. A C60 molecule the buckyball C60.mpg 40

41 Mean curvature field 41

42 Example 5. A biomolecule of 800 atoms 42

43 Numerical calculations with moving solute atoms (see also movies) 43

44 A benzene molecule 44

45 Free Energy Computational Step 45

46 An ethane molecule 46

47 Free Energy Computational Step 47

48 Tolman length 48 Free energy (surface energy + solute water interaction)

49 Hydrophobic pocket 49

50 12 10 water density profile LS τ=0.8: x=r, y=0, φ=0 LS τ=1.0: x=r, y=0, φ=0 LS τ=1.2: x=r, y=0, φ=0 LS τ=1.4: x=r, y=0, φ=0 MD ρ=0.25 MD ρ=0.5 MD ρ= r z 50

51 5. Conclusions and Outlook 51

52 Accomplishments Examined and improved a class of variational implicit solvent models, coupled with molecular mechanics Developed a level-set method for solvation of nonpolar molecules. Level-set calculations captured dewetting, local minima, etc., and agree well with MD simulations. Derived the variation of electrostatic free energy with respect to the location change of the solutesolvent interface. 52

53 Current and future work Revisit the surface tension and Tolman correction Coupling the Poisson-Boltzmann calculations with the level-set relaxation Improvement of generalized Born models Dynamics Multiscale modeling and simulation 53

Level-Set Variational Solvation Coupling Solute Molecular Mechanics with Continuum Solvent

Level-Set Variational Solvation Coupling Solute Molecular Mechanics with Continuum Solvent Bo Li Department of Mathematics and Center for Theoretical Biological Physics (CTBP) University of California,