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

Transcription

1 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, San Diego (UCSD), USA

2 Collaborators Jianwei Che (Genomics Inst. of Novartis Res. Found) Li-Tien Cheng (Math, UCSD) Joachim Dzubiella (Phys., Tech. Univ. Munich) J. Andrew McCammon (Biochem & CTBP, UCSD) Piotr Setny (Biophysics, Univ. of Warsaw) Zhongming Wang (Math & Biochem, UCSD) Yang Xie (ME, Georgia Tech) Support NSF, DOE, DFG, Sloan, NIH, HHMI, CTBP

3 OUTLINE 1. Introduction 2. A variational model of solvation 3. The level-set method 4. Numerical results 5. Electrostatic free energy 6. Conclusions 3

4 1. Introduction 4

5 water solvation water conformational change solute solute G =? solute binding water Biological prcess and free energy receptor ligand solvent solvent solute solute Explicit solvent vs. implicit solvent 5

6 Established implicit-solvent models Get data of biomolecules. Generate solute-solvent interface. Calculate surface energy. Calculate the electrostatic free energy using PB/GB with the surface as dielectric boundary. G = G np + G p ( : Surface area) 6

7 Example 1. Capillary evaporation in hydrophobic confinement. D L Koishi et al., Phys. Rev. Lett., 93, ,

8 Example 2. A receptorligand (pocket wallmethane atom) system. Setny, J. Chem. Phys., 127, , MD: weakly solvated pocket, strong hydrophobic attraction. SASA/MSA: Onset of attraction is wrong by 2-4 Angstroms! 8

9 Example 3. Evaporation in proteins. MD simulations of the melittin protein tetramer Water in hydrophobic core Stable nanobubble Liu et al., Nature, 437, 159, More MD simulations Electrostatics Curvature Giovambattista et al., PNAS, 105, 2274,

10 Possible issues of fixed-surface models Hydrophobic cavities Curvature correction Decoupling of polar and nonpolar contributions Strong curvature effects at small scales R 0 = 73mJ /m 2 = 0.9 Symbols: MD, SPC/E water, P=1bar, T=300K. = 0 1 2H : H : ( ) the Tolman length mean curvature Huang, Geissler, & Chandler, J. Phys. Chem. B, 105, 6704,

11 2. A Variational Model of Solvation 11

12 A variational implicit-solvent model (VISM) Dzubiella, Swanson, & McCammon, Phys. Rev. Lett., 96, , Dzubiella, Swanson, & McCammon, J. Chem. Phys., 124, , Guiding principles solute solvent Solvation structure = Solute atomic positions + Solute-solvent interface. Free-energy minimization determines solute-solvent interfaces. Free energy couples different interactions: polar, nonpolar, dispersive, etc. 12

13 A free-energy functional r i m Q i w G geom [] = Pvol() + ( r )ds c j, q j, ( r )ds : Creation of a cavity in the solvent Liquid-vapor pressure difference Molecular rearrangement near the interface = ( r ): ( r ) = 0 1 2H( r ) : H = H( r ): Surface tension [ ] (Scaled Particle Theory) 0 : the (planar) surface tension the Tolman length, a fitting parameter mean curvature 13

14 G geom [] = Pvol() + 0 area() 2 0 HdS +c K KdS Hadwiger s Theorem ( ) Let C = the set of all convex bodies, M = the set of finite union of convex bodies. If is rotationally and translationally invariant, additive: then conditionally continuous: Application to nonpolar solvation Roth, Harano, & Kinoshita, Phys. Rev. Lett., 97, , Harano, Roth, & Kinoshita, Chem. Phys. Lett., 432, 275,

15 G vdw [] = w w U( r )dv r i m Q i w solute-solvent van der Waals interaction U( r ) = i U i ( r r i ) U i (r) = U LJ,i (r) = 4 i ( i ) 12 r i r ( ) 6 G elec [] - Electrostatic free energy O c j, q j, The Poisson-Boltzmann (PB) theory The generalized Born (GB) model 15

16 Coupling solute molecular mechanics with implicit solvent Molecular mechanical interactions of solute atoms V[ r 1,..., r N ] = i, j W bond i, j,k,l i, j ( r i, r j ) + + W torsion ( r + W Coulomb An effective total Hamiltonian i, j,k W bend ( r i,q i ; r j,q j ) ( r i, r j, r k ) H[; r 1,..., r N ] = V[ r 1,..., r N ] + G[; r 1,..., r N ], minh[; r 1,..., r N ] i, r j, r k, r l ) + i, j W LJ ( r i, r j ) Equilibrium conformations 16

17 3. The Level-Set Method 17

18 The level-set method Interface motion V n = V n ( r,t) for Level-set representation r (t) (t) = { r :( r,t) = 0} r z = ( r,t) The level-set equation ( r (t),t) = 0 t + r t = 0 r t = ( r t ) = ( n r t ) = V n 18

19 Examples of normal velocity Geometrically based motion Motion by mean curvature V n = H Motion by the surface Laplacian of mean curvature V n = s H External field u = H in on on on n n 19

20 Level-set formulas of geometrical quantities Unit normal Mean curvature Gaussian curvature n = H = 1 2 n K = n adj(he()) n Surface integral f ( r )ds = R 3 f ( r )()dv Volume integral f ( r )dv = R 3 f ( r )[1 H()]dV 20

21 Topological changes Merging Break-up Disappearing Nucleation? Accuracy issues Interface approximation Conservation of mass Rigorous analysis 21

22 Application to variational solvation Cheng, Dzubiella, McCammon, & Li, J. Chem. Phys. 127, , Cheng, Xie, Dzubiella, McCammon, Che, & Li, J. Chem. Theory Comput., 5, 257, Cheng, Wang, Setny, Dzubiella, Li, & McCammon, J. Chem. Phys., Relaxation dr i dt = r i H[; r 1,..., r N ] = r i V[ r 1,..., r N ] r i G[] V n = H[;, r 1,..., r N ] = G[] G[]( r ) = P [H( r ) K( r )] w U( r ) + G elec [] dv =1 ds = 2H HdS = K 22

23 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 Step 7. Set and go to Step 2 23

24 New level-set techniques Pre-computation of the potential Numerical regularization Fast numerical integration Local level-set method Efficiency 4,000 solute atoms, 50x50x50 grid size, a good initial guess 5 minutes 4,000 solute atoms, high resolution, a bad initial guess about 2 4 hours Dynamics: a different situation 24

25 4. Numerical Results 25

26 Comparison of PMF by the level-set (circles) and MD (solid line) calculations. Paschek, J. Chem. Phys., 120, 6674,

27 Comparison of the level-set and MD calculations for the two paraffin plates. MD: Koishi et al. Phys. Rev. Lett., 93, , 2004; J. Chem. Phys., 123, ,

28 Two helical alkanes ( ~30 atoms) Mean curvature Parameters: =1.2, =

29 Solvation of C60 fullerene (nonpolar) Mean curvature Solvation free energy from MD Best fit Tolman length =1.2 Side note: enthalpy-entropy compensation in solvation: Solvation free energy is a difference of big numbers: Solvation entropy Solvation enthalpy A big problem for solvation free-energy calculations! 29

30 A Hydrophobic receptor-ligand system Each wall consists of 4,242 atoms. System setup for the levelset VISM calculation. 30

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33 Free energy vs. the distance between ligand and wall: a bimodal behavior. 33

34 A model system of 4 atoms Left: initial positions. Right: final positions. 34

35 A benzene molecule 35

36 A two-particle system: the surface motion influences the particle motion 36

37 5. Electrostatic Free Energy 37

38 The Coulomb-field approximation G elec [] = w m w G elec []( r 1 1 ) = w Q i ( r r 2 i ) r r 3 dv i Q i ( r 2 N r i ) r r 3 R m i=1 i Q A Single charged particle 12 6 G(R) = 4(R 2 2R) +16 w 9 Q R 3R m N i=1 w Level-set VISM vs. exact solution 38

39 The Poisson-Boltzmann (PB) theory Electrostatic free energy G elec [] = ( r ) 8 ( r ) 2 + f ( r )( r ) 1 w electrostatic potential PBE: ( r ) = f w m w in solute region ( r )( r ) w in solvent region m w = fixed charges of molecular atoms = characteristic function of w c j q j e q j( j j c j (e q j( r ) 1) dv r ) = 4 f ( r ) 39

40 Effective electrostatic surface force G elec []( r ) = m s ( r )( r ) 2 1 j c j (e q j( r ) Charge neutrality, convexity, and Jensen s inequality G elec [] >0 Force attractive to solutes! See: B. Chu, Molecular Forces, Wiley, Lemma 1) (,z u )vdv = (u m u w )v(z) 40

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47 6. Conclusions 47

48 Variational implicit-solvent model Coupling polar and nonpolar interactions Capturing hydrophobic cavities Curvature correction Extension Coupling with molecular mechanics Electrosttic surface forces A level-set method for variational solvation Capturing hydrophobic cavities New level-set techniques 48

49 Poisson-Boltzmann theory Mathematical analysis: bounds Extension to include the excluded volume effect Further development Coupling the PB and level-set calculations Stochastic level-set VISM Solvent dynamics: Rayleigh-Plesset equation Multiscale modeling and simulation Mathematical problems Derivation of the free-energy functional Constrained motion by mean curvature 49

50 References 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, L.-T. Cheng, Y. Xie, J. Dzubiella, J.A. McCammon, J. Che, & B. Li, J. Chem. Theory Comput., 5, 257, B. Li, SIAM J. Math. Anal., 40, 2536, B. Li, Nonlinearity, 19, 2581, L.-T. Cheng, Z. Wang, P. Setny, J. Dzubiella, B. Li, & J.A. McCammon, J. Chem. Phys., 2009 (accepted). P. Setny, Z. Wang, L-T. Cheng, B. Li, J. A. McCammon, & J. Dzubliella, 2009 (submitted). 50

51 Thank You! 51