Dielectric Boundary Forces in Variational Implicit-Solvent Modeling of Biomolecules

Transcript

1 Dielectric Boundary Forces in Variational Implicit-Solvent Modeling of Biomolecules Bo Li Department of Mathematics and NSF Center for Theoretical Biological Physics UC San Diego Collaborators: Hsiao-Bing Cheng, Li-Tien Cheng, Xiaoliang Cheng, and Zhengfang Zhang Funding: NIH, NSF, and Zhejiang Univ. Lu Foundation ICMCEC, Chinese Academy of Sciences, Beijing June 17, 2011

2 Outline 1. Introduction 2. The Poisson Boltzmann Theory 3. The Coulomb-Field Approximation 4. The Yukawa-Field Approximation 5. Conclusions and Discussions

3 1. Introduction

4 Biomolecular Interactions Variational Implicit-Solvent Model (VISM) (Dzubiella, Swanson, & McCammon, 2006) min G[Γ] = Equilibrium structures Equilibrium dielectric boundary (or solute-solvent interface) Minimum solvation free energy n dielectric boundary Ω w Ω m ε m=1 ε =80 w Γ x Q i i

5 n dielectric boundary Ω w Ω m ε m=1 ε =80 w Γ x Q i i Free-energy functional G[Γ] = P Vol (Ω m ) + γ 0 Γ (1 2τH)dS + ρ w U (i) i LJ ( x x i )dv x Ω w + G ele [Γ] (electrostatic free energy) Level-set simulations of BphC. Left: no charges. Right: with charges.

6 Charges Point charges: Q i at x i Mobile ions: valence Z j, volume V j, bulk c j, temperature T Dielectric coefficient { εm ε 0 in Ω m ε Γ = ε w ε 0 in Ω w Continuum electrostatics n dielectric boundary Ω w Ω m ε m=1 ε =80 w Γ x Q i i dielectric boundary Γ = potential ψ Γ = free energy G ele [Γ] (Normal component of) Dielectric boundary force (DBF) F n = δ Γ G ele [Γ]

7 Mathematical definition of δ Γ G ele [Γ] : Shape derivatives Let V C c (R 3, R 3 ). Define x : [0, ) R 3 R 3 by {ẋ = V(x) for t > 0, x(0, X) = X. Denote T t (X) = x(t, X). Then T t (X) = X + tv(x) + O(t 2 ) for small t > 0. Define δ Γ,V G ele [Γ] = d G ele [Γ t ] G ele [Γ] dt G ele [Γ t ] = lim. t=0 t 0 t Structure Theorem. There exists w : Γ R such that δ Γ,V G ele [Γ] = Γ w(x)[v(x) n(x)]ds X V C c (R 3, R 3 ). Shape derivative δ Γ G elel [Γ](X) = w(x) X Γ

8 Basic properties Let J t (X) = det T t (X). Then dj t dt = J t( V) T t. Let A(t) = J t ( T t ) 1 ( T t ) T. Then If u L 2 (Ω) then If u H 1 (Ω) then A (t) = [ (( V) T t ) ( T t ) 1 (( V) T t ) T t ( T t ) 1 (( V) T t ) T ( T t ) ] A(t). lim u T t = u and lim u Tt 1 = u in L 2 (Ω). t 0 t 0 (u T 1 t For any u H 1 (Ω) and t 0, ) = ( T 1 t ) T ( u Tt 1 ), (u T t ) = ( T t ) T ( u T t ). d dt (u T t) = ( u V) T t.

9 2. The Poisson Boltzmann Theory

10 The (generalized) Poisson Boltzmann equation (PBE) ε Γ ψ χ w B (ψ) = f Continuum electrostatics Poisson s equation: ε Γ ψ = ρ Charge density: ρ = f + χ w ρ i Boltzmann distribution: ρ i = B (ψ) (χ w = 1 in Ω w and χ w = 0 in Ω m.) Examples of B = B(ψ) Nonlinear PBE without size effect β 1 ( j c j e βz j eψ 1 ) Linearized PBE 1 2 κ2 ψ 2 n dielectric boundary Ω w Nonlinear PBE ( with size effect (βv) 1 log 1 + v ) j c j e βz o jeψ Ω m B ε m=1 ε =80 w Γ x Q i i ψ

11 The (generalized) Poisson Boltzmann equation (PBE) ε Γ ψ χ w B (ψ) = f Electrostatic free energy: G[Γ] = G ele [Γ] G[Γ] = Ω [ ε ] Γ 2 ψ 2 + f ψ χ w B(ψ) dv The region Ω is the union of Ω m, Ω w, and Γ. The integral as a functional of ψ is concave. The PBE is the Euler Lagrange equation of the functional.

12 Notations Hg(Ω) 1 = {φ H 1 (Ω) : φ = g on Ω} [ G[Γ, φ] = ε ] Γ 2 φ 2 + f φ χ w B(φ) dv Ω Theorem. G[Γ, ] : H 1 g(ω) R has a unique maximizer ψ 0 : Γ-uniformly bounded in H 1 (Ω) and L (Ω), and the unique solution to the PBE. Proof. Step 1. Existence and uniqueness by the direct method, using the concavity. Step 2. Key: The L -bound. Let λ > 0 and define λ if ψ 0 (X) < λ, ψ λ (X) = ψ 0 (X) if ψ 0 (X) λ, λ if ψ 0 (X) > λ. G[Γ, ψ 0 ] G[Γ, ψ λ ], ψ λ ψ 0, the properties of B, and the uniqueness of maximizer = ψ 0 = ψ λ for large λ. Step 3. Regularity theory and routine calculations. Q.E.D.

13 Electrostatic free energy: G[Γ] = max φ H 1 g (Ω) G[Γ, φ] Theorem. Assume n points from Ω m to Ω w and f H 1 (Ω). Then δ Γ G[Γ] = ε w 2 ψ+ 0 2 ε m 2 ψ 0 2 ε w ψ 0 + n 2 + ε m ψ0 n 2 + B(ψ 0 ) = 1 ( 1 1 ) ε Γ ψ 0 n 2 + ε w ε m (I n n) ψ B(ψ 0 ). 2 ε m ε w 2 Consequence: Since ε w > ε m, the force F n = δ Γ G[Γ] < 0. B. Chu, Molecular Forces Based on the Baker Lectures of Peter J. W. Debye, John Wiley & Sons, 1967: 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.

14 Proof of Theorem. Let V C c (R 3, R 3 ) be local, Γ 0 = Γ, and G[Γ t ] = G[Γ t, ψ t ] = max φ H 1 g (Ω) G[Γ t, φ]. Hence ψ t is the solution to the PBE corresponding to Γ t. Denote z(t, φ) = G[Γ t, φ Tt 1 ]. We have G[Γ t ] = max z(t, φ). φ Hg(Ω) 1 Step 1. Easy to verify for 0 < t 1 that z(t, ψ 0 ) z(0, ψ 0 ) t Hence G[Γ t] G[Γ] t t z(ξ, ψ 0 ) G[Γ t] G[Γ] t z(t, ψ t T t ) z(0, ψ t T t ). t t z(η, ψ t T t ), ξ, η [0, t].

15 Step 2. Direct calculations lead to [ t z(t, φ) = ε Γ 2 A (t) φ φ + (( (fv)) T t )φj t Ω χ w B(φ)(( V) T t )J t ]dv. Replacing t by η and φ by ψ t T t, respectively, we obtain lim tz(η, ψ t T t ) = t z(0, ψ 0 ) t 0 and hence provided that δ Γ,V G[Γ] = t z(0, ψ 0 ), lim ψ t T t ψ 0 H t 0 1 (Ω) = 0.

16 Step 3. The limit follows from: lim ψ t T t ψ 0 H t 0 1 (Ω) = 0 Weak form of the Euler Lagrange equation for the maximization of z(t, ) by ψ t T t for t > 0 and by ψ 0 for t = 0, respectively; Subtract one from the other; Use the properties of T t (X) and the convexity of B. Step 4. We now have δ Γ,V G[Γ] = t z(0, ψ 0 ). Direct calculations complete the proof. Q.E.D.

17 3. The Coulomb-Field Approximation

18 ε =1 m Ω m Ω m ε m=1 ε =80 Ω w w Γ Electrostatic free energy 1 G ele [Γ] = Ω 2 E 1 sol D sol dv Ω 2 E vac D vac dv Electric field and displacement: E = ψ and D = εε 0 E. The charge density ρ = i Q iδ xi ε m ε 0 ψ vac = ρ = ψ vac (x) = Q i i 4πε 0 ε m x x i ε 0 ε Γ ψ sol = ρ = ψ sol =? Poisson s equation = D sol = D vac

19 The Coulomb-field approximation: D sol D vac 1 G ele [Γ] = Ω 2 E 1 sol D sol dv Ω 2 E vac D vac dv = 1 1 D sol 2 dv 1 1 D vac 2 dv 2 Ω ε 0 ε Γ 2 Ω ε 0 ε m 1 1 D vac 2 dv 1 1 D vac 2 dv 2 Ω ε 0 ε Γ 2 Ω ε 0 ε m = 1 1 ε 0 ε m ψ vac 2 dv 1 1 ε 0 ε m ψ vac 2 dv 2 Ω ε 0 ε Γ 2 Ω ε 0 ε m = 1 1 ε 0 ε m ψ vac 2 dv 1 1 ε 0 ε m ψ vac 2 dv 2 Ω w ε 0 ε w 2 Ω w ε 0 ε m = 1 ( 1 32π 2 1 ) Q i (x x i ) 2 ε 0 ε m ε w x x i 3 dv. Ω w i

20 G ele [Γ] = 1 ( 1 32π 2 1 ) ε 0 ε m ε w Ω w Exact for a single-particle, spherical solute! i Q i (x x i ) x x i 3 2 dv Born s calculation (1920) G ele [Γ] = 1 2 Q ψ reac(o) = 1 2 Q (ψ sol ψ vac )(O) Q O R εm ε w ψ reac (r) = ( 1 1 ε m ε w ( 1 1 ε m ε w ) Q 4πε 0 R ) Q 4πε 0 r if r < R if r > R Q2 G ele [Γ] = 32π 2 ε 0 R ( 1 ε m 1 ε w ψ ) reac O R r

21 G ele [Γ] = 1 ( 1 32π 2 ε π 2 ε 0 1 ε m ε w ( 1 1 ε m ε w ) i Q 2 i Ω w ) Q i Q j i<j dv x x i 4 (x x i ) (x x j ) Ω w x x i 3 x x j 3 dv Generalized Born models (Still, Tempczyk, Hawley, & Hendrickson, 1990) G elec = 1 ( 1 32π 2 1 ) Qi 2 1 ( 1 ε 0 ε m ε w R i 16π 2 1 ) Q i Q j ε 0 ε m ε w f ij i i<j Generalized Born radii R i : R 1 dv i = Ω w x x i 4 Interpolation: f ij = x i x j 2 + R i R j exp ( x i x j 2 4R i R j )

22 G ele [Γ] = 1 ( 1 32π 2 1 ) ε 0 ε m ε w Ω w i Q i (x x i ) x x i 3 Theorem. Assume the normal n points from Ω m to Ω w. Then ( 1 1 δ Γ G ele [Γ] = 32π 2 1 ) Q i (x x i ) 2 ε 0 ε m ε w x x i 3 x Γ. i Proof. Let V Cc (R 3, R 3 ) with V(x i ) = 0 for all i. Then G ele [Γ t ] = w(x)dv = w(t t (X))J t (X)dV. T t(ω w) Ω w d dt G ele [Γ t ] = [ w T t(x)j t (X) + w(x)j t(x)] dv t=0 Ω w t=0 = [ w V(X) + w(x) V(X)] dv Ω w = (wv)dv = w(v n)ds. Q.E.D. Ω w Γ 2 dv

23 4. The Yukawa-Field Approximation

24 Definition. The Yukawa potential Y µ for µ > 0 is Y µ (x) = 1 4πr e µr (r = x ). It is the fundamental solution to + µ 2, i.e., ( + µ 2 )Y µ = δ and Y µ ( ) = 0. A property: R 3 Y µ dv = 1 µ 2.

25 ε =1 m Ω m Ω m ε m=1 ε =80 κ w Ω w Γ Electrostatic free energy 1 G ele [Γ] = 2 E sol D sol dv Poisson s equation: ε m ε 0 ψ vac = i Q iδ xi Ω Ω 1 2 E vac D vac dv = ψ vac (x) = i Q i 4πε 0 ε m x x i The Debye Hückel (or linearized Poisson Boltzmann) equation: ε 0 ε Γ ψ sol χ w ε w κ 2 ψ sol = i Q iδ xi = ψ sol =?

26 Definition. A Yukawa-field approximation is D vac (x) = Q i (x x i ) i 4π x x i D sol (x) 3 if x Ω m, x x i i, i f i(x, κ, Γ) Q i(x x i ) 4π x x i 3 if x Ω w. The electrostatic solvation free energy with the Yukawa-field approximation G ele [Γ] = 1 32π 2 ε 0 Ω w 1 ε w 1 ε m i f i (x, κ, Γ) Q i(x x i ) x x i 3 i Q i (x x i ) x x i 3 2 dv 2

27 Conditions on all f i (, κ, Γ) : Ω w R: (1) κ = 0 = f i (, κ, Γ) = 1 for all i; (2) f i (x, κ, Γ) Q i(x x i ) 4π x x i 3 e κr r as r = x ; (3) Exact for a spherical solute. Ω m ε m=1 ε =80 κ w Ω w Γ x Final formulas f i (x, κ, Γ) = 1 + κ x x i 1 + κli m (x) e κlw i (x) li m (x) = [x i, x] Ω m li w (x) = [x i, x] Ω w xi Γ Ωm Ω w

28 f i (x, κ, Γ) = 1 + κ x x i 1 + κli m (x) e κlw i (x) Exact for a single-particle, spherical solute! Q 4πε w ε 0 R(1 + κr) + Q ( 1 4πε m ε 0 x 1 R ψ sol (x) = Q e κ( x R) 4πε w ε 0 (1 + κr) x ) if x < R, if x > R. D sol (x) = Qx 4π x 3 if x < R, 1 + κ x 1 + κr e κ( x R) Qx 4π x 3 if x > R. Q O R εm ε w

29 f i (x, κ, Γ) = 1 + κ x x i 1 + κli m (x) e κlw i (x) Asymptotic analysis with κ 1 for a model system D sol (x) Q(1 + κ x )e κ(r R 3+R 2 R 1 ) 4π(1 + κ(r 3 R 2 + R 1 )) x 3x x Ω w O Q R R 3 r 1 R 2

30 Assume a total of N solute particles x i. The electrostatic free energy with the Yukawa-field approximation is 1 G ele [Γ] = 1 N 32π 2 f ε 0 Ω w ε w i (x, κ, Γ) Q 2 i(x x i ) x x i 3 i=1 1 2 N Q i (x x i ) ε m x x i 3 dv i=1 [ 1 = F(x, l1 w (x),...,ln w ε (x)) 1 ] C(x) dv w ε m Ω w Theorem. Assume n points from Ω m to Ω w. Denote L w i (x) = {x i + s(x x i ) : 1s < } Ω w for x Ω w. Then [ 1 δ Γ G ele [Γ](x) = F(x, l1 w (x),...,ln w ε (x)) 1 ] C(x) w ε m N i=1 1 x x i 2 L w i (x) y x i 2 i F(y)dl y x Γ.

31 First Proof. Partition of unity. Local polar coordinates. Apply a generalized version of Leibniz formula: d dy b(y) a Second Proof. f (x, y)dx = Local perturbation. Level-set representation. Co-area formula {φ>t} Q.E.D. u dx = t ( b(y) a {φ=s} y f (x, y)dx + d b(y)f (b(y), y). dy ) u φ ds ds. Ω xi Γ P Λ B(z,d) Ω + 0 x

32 5. Conclusions and Discussions

33 Summary Mathematical notion and tool: shape derivatives. Definition and formulas for the dielectric boundary force. The Poisson Boltzmann theory The Coulomb-field approximation The Yukawa-field approximation The dielectric boundary force always pushes charged solutes. Proof of existence and uniqueness of solution to the PBE.

34 Current work: Incorporate the dielectric boundary force into the level-set variational implicit-solvent model. The dielectric boundary force is part of the normal velocity for the level-set relaxation. Numerical implementation The Poisson Boltzmann theory: Need a highly accurate and efficient numerical method. The Coulomb-field approximation: Simple implementation, very efficient but less accurate. The Yukawa-field approximation: Difficult to implement.

35 Include ionic size effects with different ionic sizes and valences. No explicit PBE type of equation for non-uniform ion sizes. Constrained optimization method. Discovery: the valence-to-volume ratio of ions is the key parameter in the stratification of multivalent counterions near a charged surface. Concentration of counterion (M) (a) Distance to a charged surface From S. Zhou, Z. Wang, and B. Li, Phys. Rev. E, 2011 (in press)

36 Main References H.-B. Cheng, L-T. Cheng, and B. Li, Yukawa-field approximation of electrostatic free energy and dielectric boundary force, 2011 (submitted). B. Li, X. Cheng, and Z. Zhang, Dielectric boundary force in molecular solvation with the Poisson Boltzmann free energy: A shape derivative approach, 2011 (submitted). Other Closely Related References B. Li, Minimization of electrostatic free energy and the Poisson Boltzmann equation for molecular solvation with implicit solvent, SIAM J. Math. Anal., 40, , B. Li, Continuum electrostatics for ionic solutions with nonuniform ionic sizes, Nonlinearity, 22, , S. Zhou, Z. Wang, and B. Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 2011 (in press).

37 Thank you!