Variational Implicit Solvation of Biomolecules: From Theory to Numerical Computations

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1 Variational Implicit Solvation of Biomolecules: From Theory to Numerical Computations Bo Li Department of Mathematics and Center for Theoretical Biological Physics UC San Diego CECAM Workshop: New Perspectives in Liquid State Theories for Complex Molecular Systems Institut Henri Poincare, Paris, June 20-22, 2013

2 OUTLINE 1. Introduction 2. Variational Implicit-Solvent Models 3. The Level-Set Method 4. Test and Applications 5. Conclusions 2

3 Biomolecular interactions Solvation solute solvation ΔG =? water solute 1. Introduction Fundamental in biological structures and functions Molecular mechanical, charge-charge, and van der Waals interactions Water is essential Fluctuations, multiple scales, complex energy landscapes, and explosive data Applications: protein folding, molecular recognition, etc. water conformational change binding water solute receptor ligand protein folding molecular recognition 3

4 Biomolecular Modeling solute solvent solute solvent explicit solvent vs. implicit solvent Solvent molecules are treated explicitly as in molecular dynamics (MD) simulations. First principle and atomic resolution Sampled statistical information Relatively small systems and long computation: less efficient Solvent molecules are treated implicitly. Mean-field descriptions: efficient Large systems Direct thermodynamic data Coupling with fluid motion, fluctuations, etc. Statistical mechanics of implicit-solvent modeling X = solute deg. of freedom βu (X,Y ) Probability P(X,Y ) = P 0 e Y = solvent deg. of freedom P(X) = P(X,Y )dy = P 0 e total interaction potential W (X) : potential of mean force U(X,Y ) =U m (X)+U w (Y )+U mw (X,Y ) βw (X ) 4

5 What to model with an implicit solvent? Solute-solvent interfacial energy symbol: MD solid line: γ = γ 0 1 2τH ( ) γ 0 Curvature effect ( H is mean curvature) Excluded volume and solute-solvent van der Waals (vdw) interactions Solute The Lennard-Jones (LJ) potential U LJ (r) = 4ε σ r [( ) 12 ( σ r ) 6 ] Fermi repulsion Water Electrostatic interactions Poisson s equation εε 0 ψ = ρ vdw attraction solvent solute ε =1 ε = 80 dielectric coefficient 5

6 Commonly used surface-type implicit-solvent models Possible issues PB = Poisson-Boltzmann GB = Generalized Born No curvature correction Unable to describe hydrophobic cavities Decoupling of polar and nonpolar contributions This work Surface energy PB/GB calcula1ons solvent excluded surface (SES) probing ball vdw surface solvent accessible surface (SAS) Develop a robust implicit-solvent model for biomolecules interactions. Design and implement highly efficient and accurate computational methods. Apply to molecular recognition, protein-protein interaction, protein folding, and many other processes. 6

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

8 2. Variational Implicit-Solvent Model (VISM) Dzubiella, Swanson, & McCammon (PRL 2006; JCP 2006) Free-energy minimization determines stable equilibrium conformations. Different interactions are coupled in the free-energy functional. Free-energy functional volumetric and surface energies G[Γ] = Pvol(Ω m ) + γ 0 (1 2τH)dS Γ +ρ w U LJ,i ( r r i )dv + G elec [Γ] Ω w solute-solvent vdw interaction i electrostatic free energy dielectric boundary Γ r Q i i Ω m c j, q j, Ω w ρw P : pressure difference between inside/outside solutes : surface tension for flat solute-solvent interface : coefficient of curvature expansion : bulk solvent density γ 0 τ ρ w 8

9 Electrostatic free energy and dielectric boundary force The Poisson-Boltzmann (PB) theory B(ψ) = β 1 M j=1 εε 0 ψ χ w B'(ψ) = Q i δ ri i ) G elec [Γ] = εε 0 2 ψ, 2 +ρ f ψ χ w B(ψ) * + -. dv ( ) c j e βq jψ 1 Γ dielectric coefficient ε m =1 ε w = 80 r Q i i Ω m c j, q j, The (normal component of ) dielectric boundary force F n = δ Γ G elec [Γ] n Theorem. If is the unit normal at the dielectric boundary Γ pointing from the solute region to solvent region then δ Γ G elec [Γ] = ε 0 2 # 1 % 1 $ ε m ε w Ω m ε w > ε m, & ( ε n ψ 2 + ε 0 ' 2 ε ε w m Ω w Ω w ( ) (I n n) ψ 2 + B(ψ). Corollary. Since the force δ Γ G elec [Γ] < 0 in the direction. The electrostatic force always points from the solvent to the solute region. n ρw Che, Dzubiella, Li, & McCammon, JPCB, Li,Cheng, & Zhang, SIAM J. Applied Math,

10 The Coulomb-Field Approximations (CFA) Electric field E and electric displacement D : Electrostatic free energy (Born cycle): G elec [Γ] = D = εε 0E 1 D 2 E 2 dv 2 1 D 1 E 1 dv 2 x i Q i & " " m # m $ G x i Q i " m # m % &! " w # w! State 1: before immersion The Coulomb field Electrostatic energy D 1 = N i=1 Dielectric boundary force Q i ( r r i ) 4π r r i 3 G elec [Γ] = No need to solve partial differential equations. State 2: after immersion, creating a dielectric boundary. The CFA: D 2 1 # 1 % 1 32π 2 ε 0 $ ε w ε m δ Γ G elec [Γ]( r) 1 # 1 = 1 & % ( 32π 2 ε 0 $ ε w ' & ( ' Ωw ε m N i=1 N i=1 Q i ( r r i ) r r i 3 Q i ( r r i ) r r i dv D 1 Cheng, Cheng, & Li, Nonlinearity, Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC,

11 Coupling solute molecular mechanics with VISM Force field of solute mechanical interactions V[ r 1,..., r N ] = H[Γ; r 1,..., i, j W bond ( r i, r j ) + + W LJ ( r i, r j ) i, j minh[γ; r 1,..., r N ] i, j,k W bend + W Coulomb An effective total Hamiltonian i, j ( r i, r j, r k ) ( r i,q i ; r j,q j ) r N ] = V[ r 1,..., r N ] + G[Γ; r 1,..., r N ] + W torsion ( r i, r j, r k, r l ) i, j,k,l equilibrium conformations Cheng, Xie, Dzubiella, McCammon, Che, & Li, JCTC,

12 Interface motion by the normal velocity V n = V n ( r,t) for r Γ(t) Level-set representation Γ(t) = { r Ω :ϕ( r,t) = 0} The level-set equation ϕ + V ϕ = t n 3. The Level-Set Method 0 Easy handle of topological changes Level-set formulas of geometrical quantities Unit normal Mean curvature Gaussian curvature Surface integral Volume integral n = ϕ/ ϕ H = n / 2 z = 0 K = n adj(he(ϕ)) n Γ Ω f ( r )ds = f ( r )dv = R 3 R 3 n r f ( r )δ(ϕ)dv Γ(t) z = ϕ( r,t) f ( r )[1 H(ϕ)]dV Γ(t) 12

13 Application to variational solvation Relaxation ϕ + V ϕ = t n 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 Γ H[Γ;, r 1,..., r N ] = δ Γ G[Γ] 0 δ Γ G[Γ]( r) = P + 2γ 0 [H( r) τ K( r)] ρ w i U LJ,i ( r r i )+δ Γ G elec [Γ] Initial surfaces: tight wraps, loose wraps, or their combinations 13

14 Discretization of the level-set equation ϕt + Vn ϕ = 0 V n = P 2γ 0 [H( r ) τk( r )]+ ρ w U( r ) Special case: Semi-implicit τ = 0 ϕ t = 2γ 0 Δϕ + N( ϕ, 2 ϕ) Central differencing + FFT or Cholesky decomposition General case: Forward Euler τ > 0 ϕ k +1 (x) ϕ k (x) = ΔtV n k (x) ϕ k (x) Decomposition Central differencing Upwinding ϕ t = A + B A = 2γ 0 [H( r ) τk( r )] ϕ B = [P ρ w U( r )] ϕ Cheng, Li, & Wang, J. Comput. Phys.,

15 4. Test and Applications 2 1 w(d)/k B T W(d)/k B T d/ Å two xenon atoms two paraffin plates Potential of mean forces (PMF): Level-set (circles) vs. MD (solid line). Cheng, Dzubiella, McCammon, & Li, JCP,

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18 Effect of charges to hydration Two charged paraffin plates Plate-plate separation d = 10 A. Left: no charges. Middle: charges (0.2 e, 0.2 e). Right: charges (0.2 e, -0.2 e). Color code represents mean curvature. Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, VISM surfaces vs. other fixed surfaces Molecular surface (green) vs. VISM loose (red) and VISM tight initials (blue) at 12 A. The p53/mdm2 complex (PDB code: 1YCR) Guo, Li, Dzubiella, Cheng, McCammon, Che, JCTC,

19 BphC: a two-domain protein Upper row: uncharged. Lower row: charged. The domain separations are: 8 (left), 14 (middle), and 16 (right) A. Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC,

20 A receptor-ligand system p53/mdm2 "&$ "&! 0( /:5442)(3)4.7) 0( ()245*+)2)(3)4.7) uncharged charged "&" "&# PMF "#% A host-guest system CB[7]-B2 "#$ particle-wall distance "#!!!!" # "! $ % &# Cheng, Wang, Setny,Dzubiella, Li, & McCammon, JCP, Setny, Wang, Cheng, Li, McCammon, & Dzubiella, PRL, Zhou, Rogers, de Oliveira, Baron, Cheng, Dzubiella, Li, & McCammon,

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23 Summary VISM 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] Γ " Coupling solute molecular mechanics " Effective dielectric boundary force " The Coulomb-field approximation of electrostatic energy The level-set method: algorithm and coding Numerical computations 5. Conclusions " Estimates of solvation free energy " Hydrophobic cavities and multiple states of hydration " Charge effect Ω w i 23

24 Issues " Parameters: MD force fields; fit-parameters " Efficiency: minutes to hours " Molecular details: charge asymmetry, hydration shells, etc. " Entropy calculations SAS/SES vs. VISM vs. MD simulations Application of VISM " Estimate solvation free energy " Describe equilibrium conformations 24

25 Current and future work " Level-set VISM coupled with the full PBE " Application: host-guest systems, protein-protein interactions " Fast algorithms, GPU computing, software development " Phase-field VISM implementation " Solvent dynamics: hydrodynamics + fluctuation " Multiscale approach: solute MD + solvent fluid motion " Mathematics and statistical mechanics of VISM 25

26 Acknowledgment Main Collaborators Joachim Dzubiella (Humboldt Univ.) J. Andrew McCammon (UCSD) Li-Tien Cheng (UCSD) Jianwei Che (GNF) Zhongming Wang (Florida Intern l Univ.) Piotr Setny (Munich & Warsaw) Zuojun Guo (GNF) Funding: NSF, DOE, NIH, CTBP 26

27 Thank you! 27

28 References [1] Dzubiella, Swanson, McCammon, PRL, 96, , [2] Dzubiella, Swanson, McCammon, JCP, 124, , [3] Cheng, Dzubiella, McCammon, & Li, JCP, 127, , [4] Che, Dzubiella, Li, and McCammon, JPC-B, 112, [5] Cheng, Xie, Dzubiella, McCammon, Che, & Li, JCTC, 5, 257, [6] Cheng, Wang, Setny, Dzubiella, Li, & McCammon, JPC, 113, , [7] Setny, Wang, Cheng, Li, McCammon, & Dzubliella, PRL, 103, , [8] Cheng, Li, & Wang, J. Comput. Phys., 229, 8497, [9] Cheng, Cheng, & Li, Nonlinearity, 24, 3215, [10] Li, Cheng, & Zhang, SIAM J. Applied Math, 71, 2093, [11] Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, 8, 386, [12] Cheng, Li, White, & Zhou, SIAM J. Applied Math, 73, 594, [13] Guo, Li, Dzubiella, Cheng, McCammon, & Che, JCTC, 9, 1778, [14] Zhao, Kwan, Che, Li, & McCammon, JCP, 2013 (accepted). [15] Zhou, Rogers, Oliveira, Baron, Cheng, Dzubiella, Li, & McCammon, JCTC, 2013 (submitted) 28