Level-Set Variational Implicit-Solvent Modeling of Biomolecular Solvation

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1 Level-Set Variational Implicit-Solvent Modeling of Biomolecular Solvation Bo Li Department of Mathematics and Quantitative Biology Graduate Program UC San Diego The 7 th International Congress of Chinese Mathematicians Beijing, China August 6 11, 2016

2 Main collaborators Li-Tien Cheng (UCSD) Zhongming Wang (Florida Intern l Univ.) Shenggao Zhou (Soochow Univ.) Yanxiang Zhao (George Washington Univ.) Hui Sun (UCSD) J. Andrew McCammon (UCSD) Joachim Dzubiella (Humboldt Univ., Berlin) Piotr Setny (Munich & Warsaw) Jianwei Che (GNF) Zuojun Guo (GNF) Funding: NIH and NSF 2

3 water Solvation solvation water conformational change solute solute ΔG =? solute binding water protein folding molecular recognition receptor ligand Biomolecular Modeling: Explicit vs. Implicit solvent solvent solute solute Statistical mechanics MD simulations m i!! r i = ri V(r 1,, r N ) A = 1 Z A(p,r) e βh ( p,r) dpdr = A time 3

4 Dielectric boundary Hasted, Ritson, & Collie, JCP,

5 Commonly used, surface based, implicit-solvent models PB = Poisson-Boltzmann GB = Generalized Born Surface energy PB/GB calcula1ons solvent excluded surface (SES) probing ball vdw surface solvent accessible surface (SAS) Koishi et al., PRL,

6 OUTLINE 1. Free-Energy Functional 2. Dielectric Boundary Force 3. The Level-Set Computation 4. Interfacial Fluctuations 5. Conclusions 6

7 1. Free-Energy Functional Dzubiella, Swanson, & McCammon, PRL, JCP, G[Γ] = Pvol(Ω m ) + γ 0 τ : PBE PB free energy Γ εε 0 ψ B'(ψ) = ρ f G elec (1 2τH)dS +ρ w U LJ,i ( r r i )dv + G elec [Γ] Ω w i the Tolman length, a fitting parameter B(ψ) = β 1 ρ f = Q i δ! i ri M c ( j e βq jψ 1) j=1 Γ [ Γ] = εε 0 2 ψ 2 +ρ f ψ B(ψ) dv ε m =1 ε w = 80 r Q i i Ω m ε = ε Γ = c j, q j, ε m ε w Ω w ρw in Ω m in Ω w! Linearized PBE! Sinh PBE εε 0 ψ κ 2 ψ = ρ f εε 0 ψ 2c sinh(βψ) = ρ f 7

8 Coupling solute molecular mechanics with implicit solvent V[ r 1,..., r N ] = W bond ( r i, r j ) + W bend ( r i, r j, r k ) + W torsion ( r! i, r! j, r! k, r! l ) i, j + W LJ ( r! i, r! j ) i, j i, j,k + W Coulomb An effective total Hamiltonian H[Γ; r 1,..., r N ] = V[ r 1,..., r N ] + G[Γ; r 1,..., r N ] minh[γ; r 1,..., r N ] Generalized Langevin equations i, j i, j,k,l ( r i,q i ; r j,q j ) Γ r Equilibrium conformations Q i i Ω m Ω w d r! i dt = M i! ri H[Γ; r! 1,..., r! N ]+η i, i =1,..., N V n = δ Γ H[Γ;,! r 1,...,! r N ]+η Γ Cheng,Xie, Dzubiella, McCammon, Che, & Li, JCTC 2009, Zhou, Sun, Cheng, Li, & McCammon, JCP

9 PBE Charge neutrality Define εε 0 ψ B'(ψ) = ρ f M ( ) B(ψ) = β 1 c i e βq iψ 1 i=1 M B'(0) = c i q i = 0 i=1 # εε I[ψ] = 0 2 ψ & 2 ρ f ψ + B(ψ) $ % ' ( dv H 1 g (Ω) = {φ H 1 (Ω) :φ = g on Ω} B o Theorem (Li, Cheng, & Zhang, SIAP 2011) I[ ] has a unique minimizer, bouded in and uniformly with respect to It is the H 1 L unique solution to the PBE. ε [ε min,ε max ]. Proof.! Existence and uniqueness of a minimizer in H 1 g (Ω) by direct methods in the calculus of variations and the convexity of I[ ].! Uniform L (Ω) bound by comparison.! Regularity theory and routine calculations. Q.E.D. 9

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12 2. Dielectric Boundary Force F n = δ Γ G elec [Γ] PB free energy PBE 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 $ G elec [Γ] = ε Γε 0 2 ψ ' 2 +ρ f ψ χ w B(ψ) % & ( ) dv ε Γ ε 0 ψ χ w B'(ψ) = ρ f Γ n 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 12

13 PB free energy PBE $ G elec [Γ] = ε Γε 0 2 ψ ' 2 +ρ f ψ χ w B(ψ) % & ( ) dv ε Γ ε 0 ψ χ w B'(ψ) = ρ f Theorem (Li, Cheng, & Zhang, SIAP 2011). Let n point from Ω m to Ω w. Then δ Γ G elec [Γ] = ε # 0 1 % 1 & ( ε Γ n ψ 2 + ε 0 2 $ ε m ' 2 ε w ε m ε w ( ) (I n n) ψ 2 + B(ψ). Che, Dzubiella, Li, & McCammon, JPCB, Luo et al., PCCP 2012 & JCP 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. 13

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17 3. The Level-Set Computation! Interface motion V n = V n ( r,t) r for Γ(t)! The level-set representation Γ(t) = { r Ω :ϕ( r,t) = 0}! The level-set equation [ t n 0 ϕ + V ϕ =! Geometrical quantities z = 0 n r Γ(t) z = ϕ( r,t) Γ(t) ϕ( r! (t),t) = 0 ϕ t + ϕ r! t = 0 ϕ! r t =! n = ϕ ϕ H = 1 2!! Surface and volume integrals f ( r! )ds = f ( r! )δ(ϕ)dv Γ R 3 ( ϕ r! ) ϕ t ϕ = (! n r! t ) ϕ = V n ϕ n K = n! adj(he(ϕ)) n! f ( r! )dv = f ( r! )[1 H(ϕ)]dV R 3 Ω ] Topological changes 17

18 Application to variational solvation Relaxation, or gradient flow Free Energy Computational Step ϕ d t + Vn ϕ = r i dt = r i H[Γ; 0 r 1,..., r N ] = V n = δ Γ δ Γ G[Γ]( r H[Γ;, r r 1,..., r i V[ r 1,..., r N ] r i G[Γ] N ] = ) = P + 2γ 0 [H( r δ Γ G[Γ] ) τk( r )] ρ w U( r ) + δ Γ G elec [Γ] JCP 2007, 2009, & 2016; JCTC 2009, PRL J. Comput. Phys

19 Cheng, Li, & Wang, J. Comput. Phys

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22 Estimation of Solvation Free Energies Two xenon atoms Two paraffin plates 2 1 w(d)/k B T W(d)/k B T d/ Å PMF: Level-set (circles) vs. MD (solid line). MD: Paschek, JCP Koishi et al. PRL 2004; JCP Cheng, Dzubiella, McCammon, & Li, JCP

23 Dry and Wet States Setny, Wang, et al. PRL, JCP A receptor-ligand system Tight initial Loose initial The p53/mdm2 complex Guo, Che, et al., JCTC F(d) d/å MD PMF LI TI LS PMF MS (green) vs. VISM loose (red) and VISM tight initials (blue). 23

24 Charge Effect Two charged paraffin plates Wang, Cheng, et al. JCTC Zhou, Cheng, et al. JCTC BphC Plate-plate separation d = 10! A. Left: no charges. Middle: partial charges (0.2 e, 0.2 e). Right: partial charges (0.2 e, -0.2 e). Two (oppositely) charged plates Hua, Zangi, & Berne, JPCC 2009 Stepwise cavitation 24

25 VISM energy map correlates water density in MDM2 binding pocket MD Low water density high water density VISM Low energy density high energy density Guo, Li, Dzubiella, Cheng, McCammon, & Che, JCTC 2014.

26 4. Interfacial Fluctuations Zhou, Hui, Cheng, Dzubiella, Li, & McCammon, JCP G[Γ] = γ 0 Area(Γ)+ ρ w The level-set eq. Ω w i U LJ,i (! x! x i )dv Γ! x i Ωm Ω w Interfacial noise : components of standard M-dim Wiener process A stochastic level-set eq. (Stratonovich) 26

27 A stochastic level-set eq. (Ito) 27

28 Numerical Methods! Central differencing for the parabolic part! CFL condition! Fifth-order WENO for the hyperbolic part! Level-set re-initialization! Euler-Maruyama scheme! Simulated annealing! FFT to filter out high frequencies and ifft to keep smooth noises 28

29 Dewetting process 29

30 Energy barrier StoLSM Fitted quadratic curve Energy barrier (kbt ) Distance between two plates (Å) 30

31 5. Conclusions! A variation model for molecular solvation. Continuum electrostatics. Mathematical analysis of PB theory. Dielectric boundary force.! The level-set method: detailed algorithms and coding.! The level-set variational implicit-solvent model: dry and wet states; charge effects; free energy estimates; etc.! A stochastic level-set variational approach to dewetting process.! GPU implementation.! Hybrid model: solute molecular mechanical interactions and implicit-solvent modeling; bridging time scales.! Fluctuating solvent flow. 31

32 Thank you! 32