Dielectric Boundary in Biomolecular Solvation Bo Li Department of Mathematics and Center for Theoretical Biological Physics UC San Diego

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1 Dielectric Boundary in Biomolecular Solvation Bo Li Department of Mathematics and Center for Theoretical Biological Physics UC San Diego International Conference on Free Boundary Problems Newton Institute, Cambridge, UK June 23-27, 2014

2 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 2

3 Variational Implicit-Solvent Model (VISM) Dzubiella, Swanson, & McCammon: PRL, 2006; JCP, Free-energy functional G[Γ] = Pvol(Ωm ) + γ 0 (1 2τH)dS Γ Ωw Ωm! " +ρ w U LJ,i ( r ri )dv +Gelec [Γ] Ωw i Γ ri Qi c j, q j, ρw Two paraffin plates This talk focuses on: BphC p53/mdm2 The Poisson-Boltzmann (PB) Theory and Dielectric Boundary Force; Stability of a Cylindrical Solute-Solvent Interface..3

4 Main Collaborators Li-Tien Cheng (UCSD) Zhongming Wang (Florida International Univ.) Shenggao Zhou (UCSD) Michael White (Univ. of Minnesota) Hui Sun (UCSD) Zhengfang Zhang (Zhejiang Univ.) J. Andrew McCammon (UCSD) Joachim Dzubiella (Humboldt Univ. Berlin) Piotr Setny (Munich & Warsaw) Jianwei Che (GNF) Zuojun Guo (GNF) Yang Xie (Georgia Tech) Funding: NIH, NSF, and CTBP 4

5 The Poisson-Boltzmann (PB) Theory εε 0 ψ B'(ψ) = ρ f $ G elec [Γ] = εε 0 2 ψ ' 2 +ρ f ψ B(ψ) % & ( ) dv B(ψ) = β 1 M j=1 Poisson equation Charge density ( ) c j e βq jψ 1 Boltzmann distributions Γ ε m =1 ε w = 80 r Q i i Ω m c j, q j, εε 0 ψ = ρ N M ρ = ρ f + ρ i = Q i δ i=1 ri + q j c i=1 j c j = c j e βq jψ Ω w ρw! Linearized PBE:! Sinh PBE G elec [Γ, ] εε 0 ψ κ 2 ψ = ρ f εε 0 ψ 2c sinh(βψ) = ρ f Theorem. has a unique maximizer, uniformly bouded in 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. Li,, SIMA, 2009 & 2011; Nonlinearity, 2009; Li, Cheng, & Zhang, SIAP, H 1 5

6 Electrostatic Free-energy functional F[c] = $ M M 1 ' 1 & ρψ + β c i ln(λ 3 c i ) µ i c i ) dv % 2 ( ρ = ρ f + M i q i c i i=1 εε 0 ψ = ρ δ i F[c] = 0 c j = c j e βq jψ O i=1 slns s Theorem (B.L. SIMA 2009) has a unique minimizer satisfying the F[c] c = (c 1,, c M ) Boltzmann distributions. θ 1,θ 2 > 0 θ 1 c j θ 2 j =1,, M. There exist such that for all The corresponding ψ is the unique solution to PBE. 6

7 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 7

8 ) 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 2 & 1 ( 1 ' ε m ε w ) + ε n ψ 2 + ε 0 ( * 2 ε ε w m)(i n n) ψ 2 + B(ψ). Li, Cheng, & Zhang, SIAP, 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. 8

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12 Surface energy vs. electrostatic energy Cheng, Li, White, & Zhou, SIAP,

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21 Solvent Fluid Dynamics and Fluctuations Interface motion V n = u n Fluctuating solvent fluid: µ w 2 u p w n w U ext + Σ = 0 in Ω w (t) u = 0 in Ω w (t) p m,i (t) Ω m,i (t) = N i k B T εε 0 ψ χ w B'(ψ) = ρ f Γ r i Ω m Σ ij (x,t)σ kl (x',t') = 2µ w k B Tδ(x x')δ(t t ')(δ ik δ jl +δ il δ jk ) Electrostatics " 1 1 ε m f ele = ε 0 $ 2 # Force balance ε w % ' ε n ψ 2 + ε 0 & 2 ε w ε m 2µ w D(u)n + (p m p w 2γ 0 H + n w U vdw + f ele )n = 0 at Q i Γ(t) Ω w ( ) (I n n) ψ 2 + B(ψ) 21

22 Stability of a cylindrical solute-solvent interface Li, Sun, and Zhou, SIAP, 2014 (submitted) 22

23 R 0 = R 0 = 8.96 R 0 = 9.98 ω ωhyd (k ) ω ele (k ) ω vdw (k ) ω curv (k ) k

24 400 2e+5 1e+5 ω ele (k ) R 0 = 8.03 R 0 = 8.96 R 0 = ω curv (k ) ω 500 ω hyd (k ) ω vdw (k ) e k 24

25 A generalized Rayleigh-Plesset equation for a charged sphere dr = R # p 0 (R) p 2γ 0 4µ w R + n U(R)+ f & % w ele (R)+ dη( $ ' η(t)η(t ') = 4 3 k BTδ(t t ') p 0 (R) = 3k BT 4π R 3 Q O R(t)! m! w r U(r) =U vdw (r)+u ext (r) = 4ε " σ ( r ) 12 ( r σ ) 6 $ # % +U ext(r) Q 2 (" 1 f ele (R) = $ 1 % ' 1 32π 2 ε 0 # ε w ε m & R κ * )* 4 ε w ( 1+κR) 2 R 2,- 25

26 Concluding remarks! Dielectric boundary force: pointing to charged molecules.! Electrostatics and hydrodynamics: instability.! Mathematical analysis! Modeling: solute-solvent interfacial fluctuations.! Hybrid approach: solute MD + implicit solvent.! Application: protein-ligand binding, rational drug design. Green: molecular surface. Red: VISM surface. 26

27 Thank you! 27