Variational Implicit Solvation: Empowering Mathematics and Computation to Understand Biological Building Blocks

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1 Variational Implicit Solvation: Empowering Mathematics and Computation to Understand Biological Building Blocks Bo Li Department of Mathematics and NSF Center for Theoretical Biological Physics UC San Diego Funding: NIH, NSF, DOE, CTBP UC Irvine, October 25, 2012

2 MBB (Math & Biochem-Biophys) group Li-Tien Cheng (UCSD) Zhongming Wang (Florida Intern l Univ.) Tony Kwan (UCSD) Yanxiang Zhao (UCSD) Shenggao Zhou (Zhejiang Univ. & UCSD) Tim Banham (UCSD) Maryann Hohn (UCSD) Jiayi Wen (UCSD) Michael White (UCSD) Yang Xie (Georgia Tech) Hsiao-Bing Cheng (UCLA) Rishu Saxena (UCSD/MSU) Collaborators J. Andrew McCammon (UCSD) Joachim Dzubiella (Humboldt Univ.) Piotr Setny (Munich & Warsaw) Jianwei Che (GNF) Zuojun Guo (GNF) Xiaoliang Cheng (Zhejiang Univ.) Zhengfang Zhang (Zhejiang Univ.) Zhenli Xu (Shanghai Jiaotong Univ.) 2

3 OUTLINE 1. Biomolecules: What and Why? 2. Variational Implicit-Solvent Models 3. Dielectric Boundary Force 3.1 The Poisson-Boltzmann Theory 3.2 The Coulomb-Field and Yukawa-Field Approximations 4. Computation by the Level-Set Method 5. Move Forward: Solvent Fluctuations 6. Conclusions 3

4 1. Biomolecules: What and Why? 4

5 Biomolecules 5

6 Protein structures We have more than 100,000 proteins in our bodies. Each protein is produced from a set of only 20 building blocks. Protein functions Antibodies, enzymes, contractile, structural, storage, transport, etc. Wiki To function, proteins fold into three-dimensional compact structures. 6

7 Protein Folding Misfolding diseases Alzheimer s, Parkinson s, etc. Levinthal s paradox If a protein with 100 amino acids can try out 10^13 configurations per second, then it would take 10^27 years to sample all the configurations. But proteins fold in seconds. Free-energy landscape Averagely, more than 10^100 local minima for a protein with 100 amino acids each of which has 10 configurations. 7

8 Solvation water solvation water conformational change solute solute ΔG =? solute binding water receptor ligand protein folding molecular recognition 8

9 2. Variational Implicit-Solvent Models 9

10 Explicit vs. Implicit solute solvent solute solvent Molecular dynamics (MD) simulations Statistical mechanics 10

11 What to model with an implicit solvent?! Solute-solvent interfacial property γ 0 R Curvature effect Symbols: MD. ( ) γ = γ 0 1 2τH γ 0 = 73mJ /m 2 τ : τ = 0.9 A H : the Tolman length mean curvature Huang et al., JCPB,

12 ! Excluded volume and van der Waals dispersion Solute The Lennard-Jones (LJ) potential U LJ (r) = 4ε σ r [( ) 12 ( σ ) 6 ] r Fermi repulsion vdw attraction Water O σ ε! Electrostatic interactions Poisson s equation εε 0 ψ = ρ solute solvent ε =1 ε = 80 r 12

13 Commonly used implicit-solvent models Surface energy PB/GB calcula1ons PB = Poisson-Boltzmann GB = Generalized Born solvent excluded surface (SES) probing ball vdw surface Possible issues! Hydrophobic cavities! Curvature correction solvent accessible surface (SAS)! Decoupling of polar and nonpolar contributions 13

14 Koishi et al., PRL, Liu et al., Nature, Sotomayor et al., Biophys. J

15 Variational Implicit-Solvent Model (VISM) Dzubiella, Swanson, & McCammon: Phys. Rev. Lett. 96, (2006) Free-energy functional G[Γ] = Pvol(Ω m ) + γ 0 (1 2τH)dS +ρ w U LJ,i ( r r i )dv + G elec [Γ] τ : Ω w i Γ J. Chem. Phys. 124, (2006) Γ the Tolman length, a fitting parameter G elec [Γ] : electrostatic free energy! The Poisson-Boltzmann (PB) theory r Q i i Ω m c j, q j,! The Coulomb-field or Yukawa-field approximation Ω w ρw 15

16 Geometrical part: Hadwiger s Theorem Pvol(Ω m )+γ 0 area(γ) 2γ 0 τ H ds +c K 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,! conditionally continuous: U, U C, U U F( U ) F( U ), Γ ( ) j j j then F( U) avol( U) + barea( U ) + c HdS + d KdS U M. = U Application to nonpolar solvation Roth, Harano, & Kinoshita, PRL, Harano, Roth, & Kinoshita, Chem. Phys. Lett., U Γ 16

17 Coupling solute molecular mechanics with implicit solvent V[ r 1,..., r N ] = i, j i, j,k,l i, j W bond + W torsion ( r + W Coulomb ( r i, r j ) + i, j,k ( r i,q i ; r j,q j ) An effective total Hamiltonian W bend ( r i, r j, r k ) H[Γ; r 1,..., r N ] = V[ r 1,..., r N ] + G[Γ; r 1,..., r N ] minh[γ; r 1,..., r N ] i, r j, r k, r l ) + i, j W LJ Γ ( r i, r j ) r Q i i Ω m Ω w Equilibrium conformations Cheng,..., Li, JCTC,

18 3. Dielectric Boundary Force 18

19 Dielectric boundary force (DBF): F n = δ Γ G elec [Γ] A shape derivative approach Perturbation defined by V : R 3 R 3 : { x = V (x) x = x(x,t) = T t (X) x(0)= X Γ t PBE: ψ t G elec [Γ t ] δ Γ G elec [Γ] = d % dt Structure Theorem Γ ε m =1 ε w = 80 r Q i i Ω m c j, q j, $ & ' ) ( t= 0 Ω w ρw G elec [Γ t ] Shape derivative 19

20 4.1 The Poisson-Boltzmann Theory εε 0 ψ χ w B'(ψ) = ρ f ) G elec [Γ] = εε 0 2 ψ, 2 +ρ f ψ χ w B(ψ) * + -. dv B(ψ) = β 1 M j=1 ( ) c j e βq jψ 1 Γ ε m =1 ε w = 80 r Q i i Ω m c j, q j, Ω w Theorem. G elec [Γ, ] has a unique maximizer, uniformly bouded in H 1 and L. It is the unique solution to the PBE. Proof. Direct methods in the calculus of variations. Uniform bounds by comparison. Regularity theory and routine calculations. Q.E.D. ρw Li,, SIMA, 2009 & 2011; Nonlinearity, 2009; Li, Cheng, & Zhang, SIAP,

21 ) G elec [Γ] = εε 0 2 ψ, 2 +ρ f ψ χ w B(ψ) * + -. dv εε 0 ψ χ w B'(ψ) = ρ f Theorem. Let n point from Ω m to Ω w. Then δ Γ G elec [Γ] = ε & 0 1 ( 1 ) + ε n ψ 2 + ε 0 ( 2 ' ε m * 2 ε ε w m)(i n n) ψ 2 + B(ψ). ε w Li, Cheng, & Zhang, SIAP, Luo, Private communications. Cai, Ye, & Luo, PCCP, Consequence: Since ε w > ε m, the force δ Γ G elec [Γ] > 0. Chu, Molecular Forces, based on Debye s lectures, Wiley, Under the combined influence of electric field generated by solute charges and their polarization in the surrounding medium which is electrostatic neutral, an additional potential energy emerges and drives the surrounding molecules to the solutes. 21

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25 4.2 The Coulomb-Field and Yukawa-Field Approximations Electric field: Electric displacement: Electrostatic free energy: x i Q i " " m # m $ G E = ψ D = εε 0 E G elec [Γ] = x i Q i " m # m The Coulomb-field approximation (CFA): The Yukawa-field approximation (CFA): & 1 D 2 E 2 dv 2 % &! " w # w! D 2 D 2 1 D 1 E 1 dv 2 D 1 (κ = 0) D 1 (κ > 0) No need to solve partial differential equations. 25

26 The Colulomb-field approximation (CFA) G elec [Γ] = 1 # 1 % 1 32π 2 ε 0 $ ε w ε m & ( ' Ωw N i=1 Q i ( r r i ) r r i 3 2 dv δ Γ G elec [Γ]( r) = The Yukawa-field approximation (YFA) 1 G elec [Γ] = ( N * 1 f 32π 2 i ( r,κ,γ) Q ( i r r ε 0 ε w r r Ω 3 w * i=1 ) i f i ( r,κ,γ) = 1+ κ r r i 1+ κ r i P i ( r exp( κ( r P i ( r )) ) δ Γ G elec [Γ] : Γ R 1 # 1 % 1 32π 2 ε 0 $ ε w ε m Too complicated! Cheng, Cheng, & Li, Nonlinearity, & ( ' N i=1 Q i ( r r i ) r r i 3 i ) N Q i ( r r ε m r r x i=1 " (x) 2 i ) 3 i + - -,! m x i p i dv! w 26

27 4. Computation by the Level- Set Method 27

28 The Level-Set Method! Interface motion V = V ( n n r,t) for! Level-set representation r Γ(t) Γ(t) = { r Ω :ϕ( r,t) = 0}! The level-set equation ϕ + V ϕ = t n 0 Topological changes z = 0 n r Γ(t) z = ϕ( r,t) Γ(t) 28

29 Application to variational solvation ϕ + V ϕ = t n 0 Relaxation dr i dt = r i H[Γ; r 1,..., r N ] = r i V[ r 1,..., r N ] r i G[Γ] V n = δ δ Γ G[Γ]( r Γ H[Γ;, r 1,..., r N ] = δ ) = P + 2γ 0 [H( r Γ G[Γ] ) τk( r )] ρ w U( r ) + δ Γ G elec [Γ] JCP, 2007, 2009; JCTC, 2009, 2012; PRL, 2009; J. Comput. Phys.,

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36 2 Two xenon atoms 1 w(d)/k B T W(d)/k B T d/ Å PMF: Level-set (circles) vs. MD (solid line). MD: Paschek, JCP,

37 Two paraffin plates PMF: Level-set (circles) vs. MD (line). MD: Koishi et al. PRL, 2004; JCP,

38 A hydrophobic receptor-ligand system "&$ 0( /:5442)(3)4.7) 0( ()245*+)2)(3)4.7) "&! PMF "&" "&# "#% "#$ wall-particle distance "#!!!!" # "! $ % &# 38

39 A benzene molecule 39

40 BphC 40

41 The p53/mdm2 complex (PDB code: 1YCR) 41

42 Molecular surface (green) vs. VISM loose (red) and VISM tight initials (blue) at 12 A. 42

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45 5. Move Forward: Solvent Fluctuations 45

46 General description Du ρ w Dt µ 2 u + p w = f + η u = 0 in Ω w (t) p m (t)ω m (t) = K(T) εε 0 ψ χ w B'(ψ) = ρ f in Ω w (t) (p m p w )n + 2µD(u)n = (γ 0 H f ele )n at Γ(t) Γ ε m =1 ε w = 80 r Q i i Ω m c j, q j, Ω w ρw Assumptions! Small inertia:! Body force: Du Dt 0 f = ρ w U, U( r ) = i U LJ,i ( r r i ) 46

47 A charged sphere! Linearized PBE! Fluctuations with decay Q O R(t)! m! w A generalized Rayleigh-Plesset equation dr dt = F(R) + η R 0 4µα e αr W t F(R) = R % 4µ U LJ (R) + K(T) 2γ 0 ' R 3 R p + f & ele Q 2,% 1 f ele = ' 1 ( * 1 32π 2 ε 0 & ε w ε m ) R κ ε w 1+ κr The Euler-Maruyama method ( * ) ( ) 2 R 2 / 1 01 r R n +1 = R n + F(R n )Δt + η 0R n 4µα e αr n ΔW n ΔW n : iid Gaussians with mean 0 variance Δt 47

48 The force F(R) vs. R The potential U(R) 48

49 R = R(t) Probability density of R 49

50 6. Conclusions 50

51 Achivement! Level-set VISM with solute molecular mechanics; freeenergy functional; hydrophobic cavities, charge effects, multiple states, etc.! Effective DBF: PB theory, CFA and YFA.! Initial work on the solvent dynamics with fluctuations. Issues! Efficiency: mimutes to hours.! Parameters: similar to that for MD force fields.! More details: charge asymmetry, hydration shells, etc.! Coarse graining, coupling with other models. 51

52 Current and future work! Level-set VISM coupled with the full PBE.! Molecular recognition + drug design: host-guest systems.! Solvent dynamics: hydrodynamics + fluctuation.! Brownian dynamics coupled with continuum diffusion.! Fast algorithms, GPU computing, software development.! Multiscale approach: solute MD + solvent fluid motion.! Mathematics and statistical mechanics of VISM. 52

53 Roles of mathematics and computation! Many mathematical concepts and methods are used: differential geometry, PDE, stochastic processes, numerical PDE, numerical optimization, etc.! More is needed: geometrical flows for protein folding; stochastic methods for hydrodynamic interactions; topological methods for DNA and RNA structures; etc.! Computation is essential: real biomolecular systems are very complicated and the mathematical problems cannot be solved analytically.! Collaboration between mathematics and biological sciences is crucial. 53

54 Thank you! 54