Dielectric Boundary Forces in VariationalImplicit-Solvent Modeling of Biomolecules

Bo LiDepartment of Mathematics and

NSF Center for Theoretical Biological PhysicsUC San Diego

Collaborators: Hsiao-Bing Cheng, Li-Tien Cheng, XiaoliangCheng, and Zhengfang Zhang

Funding: NIH, NSF, and Zhejiang Univ. Lu Foundation

ICMCEC, Chinese Academy of Sciences, Beijing

June 17, 2011

Outline

1. Introduction

2. The Poisson–Boltzmann Theory

3. The Coulomb-Field Approximation

4. The Yukawa-Field Approximation

5. Conclusions and Discussions

1. Introduction

Biomolecular Interactions

Variational Implicit-Solvent Model (VISM)

(Dzubiella, Swanson, & McCammon, 2006)

minG [Γ] =⇒ Equilibrium structures

Equilibrium dielectric boundary(or solute-solvent interface)

Minimum solvation free energy

n

boundarydielectric

xQi

iΩw Ωm

ε =80ε =1

w

m

Γ

n

boundarydielectric

xQi

iΩw Ωm

ε =80ε =1

w

m

ΓFree-energy functional

G [Γ] = P Vol (Ωm) + γ0

∫

Γ(1 − 2τH) dS

+ ρw

∑

i

∫

Ωw

U(i)LJ (|x − xi |) dVx

+ Gele [Γ] (electrostatic free energy)

Level-set simulations of BphC. Left: no charges. Right: with charges.

Charges

Point charges: Qi at xi

Mobile ions: valence Zj , volume Vj , bulk c∞j , temperature T

Dielectric coefficient

εΓ =

εmε0 in Ωm

εwε0 in Ωw

n

boundarydielectric

xQi

iΩw Ωm

ε =80ε =1

w

m

Γ

Continuum electrostatics

dielectric boundary Γ =⇒ potential ψΓ =⇒ free energy Gele [Γ]

(Normal component of) Dielectric boundary force (DBF)

Fn = −δΓGele [Γ]

Mathematical definition of δΓGele [Γ] : Shape derivatives

Let V ∈ C∞c (R3, R3). Define x : [0,∞) × R

3 → R3 by

x = V (x) for t > 0,

x(0, X ) = X .

Denote Tt(X ) = x(t, X ). Then

Tt(X ) = X + tV (X ) + O(t2) for small t > 0.

Define

δΓ,V Gele [Γ] =d

dt

∣

∣

∣

∣

t=0

Gele [Γt ] = limt→0

Gele [Γt ] − Gele [Γ]

t.

Structure Theorem. There exists w : Γ → R such that

δΓ,V Gele [Γ] =

∫

Γw(X )[V (X ) · n(X )] dSX ∀V ∈ C∞

c (R3, R3).

Shape derivative δΓGelel [Γ](X ) = w(X ) ∀X ∈ Γ

Basic properties

Let Jt(X ) = det∇Tt(X ). Then

dJt

dt= Jt(∇ · V ) Tt .

Let A(t) = Jt (∇Tt)−1

(∇Tt)−T

. Then

A′(t) =[

((∇ · V ) Tt) − (∇Tt)−1((∇V ) Tt)∇Tt

− (∇Tt)−1((∇V ) Tt)

T (∇Tt)]

A(t).

If u ∈ L2(Ω) then

limt→0

u Tt = u and limt→0

u T−1t = u in L2(Ω).

If u ∈ H1(Ω) then

∇(u T−1t ) = (∇T−1

t )T(

∇u T−1t

)

,

∇(u Tt) = (∇Tt)T (∇u Tt) .

For any u ∈ H1(Ω) and t ≥ 0,

d

dt(u Tt) = (∇u · V ) Tt .

2. The Poisson–Boltzmann Theory

The (generalized) Poisson–Boltzmann equation (PBE)

∇ · εΓ∇ψ − χwB ′(ψ) = −f

Continuum electrostatics

Poisson’s equation: ∇ · εΓ∇ψ = −ρ

Charge density: ρ = f + χwρi

Boltzmann distribution: ρi = −B ′(ψ)(χw = 1 in Ωw and χw = 0 in Ωm.)

n

boundarydielectric

xQi

iΩw Ωm

ε =80ε =1

w

m

Γ

Examples of B = B(ψ)

Nonlinear PBE without size effect

β−1∑

j c∞j(

e−βZjeψ − 1)

Linearized PBE12κ2ψ2

Nonlinear PBE with size effect

−(βv)−1 log(

1 + v∑

j c∞j e−βZjeψ) o ψ

B

The (generalized) Poisson–Boltzmann equation (PBE)

∇ · εΓ∇ψ − χwB ′(ψ) = −f

Electrostatic free energy: G [Γ] = Gele [Γ]

G [Γ] =

∫

Ω

[

−εΓ

2|∇ψ|2 + f ψ − χwB(ψ)

]

dV

The region Ω is the union of Ωm, Ωw , and Γ.

The integral as a functional of ψ is concave.

The PBE is the Euler–Lagrange equation of the functional.

Notations H1g (Ω) = φ ∈ H1(Ω) : φ = g on ∂Ω

G [Γ, φ] =

∫

Ω

[

−εΓ

2|∇φ|2 + f φ − χwB(φ)

]

dV

Theorem. G [Γ, ·] : H1g (Ω) → R has a unique maximizer ψ0 :

Γ-uniformly bounded in H1(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

ψλ(X ) =

− λ if ψ0(X ) < −λ,

ψ0(X ) if |ψ0(X )| ≤ λ,

λ if ψ0(X ) > λ.

G [Γ, ψ0] ≥ G [Γ, ψλ], |∇ψλ| ≤ |∇ψ0|, the properties of B, and theuniqueness of maximizer =⇒ ψ0 = ψλ for large λ.

Step 3. Regularity theory and routine calculations. Q.E.D.

Electrostatic free energy: G [Γ] = maxφ∈H1g (Ω) G [Γ, φ]

Theorem. Assume n points from Ωm to Ωw and f ∈ H1(Ω). Then

δΓG [Γ]

=εw

2|∇ψ+

0 |2 −

εm

2|∇ψ−

0 |2 − εw |∇ψ+

0 · n|2 + εm|∇ψ−0 · n|2 + B(ψ0)

=1

2

(

1

εm

−1

εw

)

|εΓ∇ψ0 · n|2 +

εw − εm

2|(I − n ⊗ n)∇ψ0|

2 + B(ψ0).

Consequence: Since εw > εm, the force Fn = −δΓG [Γ] < 0.

B. Chu, Molecular Forces Based on the Baker Lectures of PeterJ. W. Debye, John Wiley & Sons, 1967:

“Under the combined influence of electric field generated by solutecharges and their polarization in the surrounding medium which iselectrostatic neutral, an additional potential energy emerges anddrives the surrounding molecules to the solutes.”

Proof of Theorem. Let V ∈ C∞c (R3, R3) be local, Γ0 = Γ, and

G [Γt ] = G [Γt , ψt ] = maxφ∈H1

g (Ω)G [Γt , φ].

Hence ψt is the solution to the PBE corresponding to Γt . Denotez(t, φ) = G [Γt , φ T−1

t ]. We have

G [Γt ] = maxφ∈H1

g (Ω)z(t, φ).

Step 1. Easy to verify for 0 < t ≪ 1 that

z(t, ψ0) − z(0, ψ0)

t≤

G [Γt ] − G [Γ]

t≤

z(t, ψt Tt) − z(0, ψt Tt)

t.

Hence

∂tz(ξ, ψ0) ≤G [Γt ] − G [Γ]

t≤ ∂tz(η, ψt Tt), ξ, η ∈ [0, t].

Step 2. Direct calculations lead to

∂tz(t, φ) =

∫

Ω

[

−εΓ

2A′(t)∇φ · ∇φ + ((∇ · (fV )) Tt)φJt

− χwB(φ)((∇ · V ) Tt)Jt

]

dV .

Replacing t by η and φ by ψt Tt , respectively, we obtain

limt→0

∂tz(η, ψt Tt) = ∂tz(0, ψ0)

and henceδΓ,V G [Γ] = ∂tz(0, ψ0),

provided thatlimt→0

‖ψt Tt − ψ0‖H1(Ω) = 0.

Step 3. The limit

limt→0

‖ψt Tt − ψ0‖H1(Ω) = 0

follows from:

Weak form of the Euler–Lagrange equation for themaximization of z(t, ·) by ψt Tt for t > 0 and by ψ0 fort = 0, respectively;

Subtract one from the other;

Use the properties of Tt(X ) and the convexity of B.

Step 4. We now have

δΓ,V G [Γ] = ∂tz(0, ψ0).

Direct calculations complete the proof. Q.E.D.

3. The Coulomb-Field Approximation

m

Ωm

ε =1

wΩm

ε =80ε =1

w

m

Γ

Ω

Electrostatic free energy

Gele [Γ] =

∫

Ω

1

2Esol · Dsol dV −

∫

Ω

1

2Evac · Dvac dV

Electric field and displacement: E = −∇ψ and D = εε0E .

The charge density ρ = −∑

i Qiδxi

∇ · εmε0∇ψvac = −ρ =⇒ ψvac(x) =∑

i

Qi

4πε0εm|x − xi |

∇ · ε0εΓ∇ψsol = −ρ =⇒ ψsol = ?

Poisson’s equation =⇒ ∇ · Dsol = ∇ · Dvac

The Coulomb-field approximation: Dsol ≈ Dvac

Gele [Γ] =

∫

Ω

1

2Esol · Dsol dV −

∫

Ω

1

2Evac · Dvac dV

=1

2

∫

Ω

1

ε0εΓ|Dsol |

2dV −1

2

∫

Ω

1

ε0εm

|Dvac |2dV

≈1

2

∫

Ω

1

ε0εΓ|Dvac |

2dV −1

2

∫

Ω

1

ε0εm

|Dvac |2dV

=1

2

∫

Ω

1

ε0εΓ|ε0εm∇ψvac |

2dV −1

2

∫

Ω

1

ε0εm

|ε0εm∇ψvac |2dV

=1

2

∫

Ωw

1

ε0εw

|ε0εm∇ψvac |2dV −

1

2

∫

Ωw

1

ε0εm

|ε0εm∇ψvac |2dV

= −1

32π2ε0

(

1

εm

−1

εw

)∫

Ωw

∣

∣

∣

∣

∣

∑

i

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

dV .

Gele [Γ] = −1

32π2ε0

(

1

εm

−1

εw

)∫

Ωw

∣

∣

∣

∣

∣

∑

i

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

dV

Exact for a single-particle, spherical solute!

Born’s calculation (1920)

Gele [Γ] =1

2Q ψreac(O) =

1

2Q (ψsol − ψvac) (O)

O

ε

R

m εw

Q

ψreac(r) =

−

(

1

εm

−1

εw

)

Q

4πε0Rif r < R

−

(

1

εm

−1

εw

)

Q

4πε0rif r > R

Gele [Γ] = −Q2

32π2ε0R

(

1

εm

−1

εw

)

reacr

ψR

O

Gele [Γ] = −1

32π2ε0

(

1

εm

−1

εw

)

∑

i

Q2i

∫

Ωw

dV

|x − xi |4

−1

16π2ε0

(

1

εm

−1

εw

)

∑

i<j

QiQj

∫

Ωw

(x − xi ) · (x − xj)

|x − xi |3|x − xj |3dV

Generalized Born models (Still, Tempczyk, Hawley, &Hendrickson, 1990)

Gelec = −1

32π2ε0

(

1

εm

−1

εw

)

∑

i

Q2i

Ri

−1

16π2ε0

(

1

εm

−1

εw

)

∑

i<j

QiQj

fij

Generalized Born radii Ri : R−1i =

∫

Ωw

dV

|x − xi |4

Interpolation: fij =

√

|xi − xj |2 + RiRj exp(

−|xi−xj |2

4RiRj

)

Gele [Γ] = −1

32π2ε0

(

1

εm

−1

εw

)∫

Ωw

∣

∣

∣

∣

∣

∑

i

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

dV

Theorem. Assume the normal n points from Ωm to Ωw . Then

δΓGele [Γ] =1

32π2ε0

(

1

εm

−1

εw

)

∣

∣

∣

∣

∣

∑

i

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

∀x ∈ Γ.

Proof. Let V ∈ C∞

c (R3, R3) with V (xi ) = 0 for all i . Then

Gele [Γt ] =

∫

Tt(Ωw )

w(x) dV =

∫

Ωw

w(Tt(X ))Jt(X ) dV .

d

dt

∣

∣

∣

∣

t=0

Gele [Γt ] =

∫

Ωw

[∇w · T ′

t (X )Jt(X ) + w(X )J ′

t(X )]

∣

∣

∣

∣

t=0

dV

=

∫

Ωw

[∇w · V (X ) + w(X )∇ · V (X )] dV

=

∫

Ωw

∇ · (wV )dV = −

∫

Γ

w(V · n) dS . Q.E.D.

4. The Yukawa-Field Approximation

Definition. The Yukawa potential Yµ for µ > 0 is

Yµ(x) =1

4πre−µr (r = |x |).

It is the fundamental solution to −∆ + µ2, i.e.,

(−∆ + µ2)Yµ = δ and Yµ(∞) = 0.

A property:∫

R3

Yµ dV =1

µ2.

m

Ωm

ε =1

κΩm

ε =80ε =1

w

m

Γ

Ωw

Electrostatic free energy

Gele [Γ] =

∫

Ω

1

2Esol · Dsol dV −

∫

Ω

1

2Evac · Dvac dV

Poisson’s equation:

∇ · εmε0∇ψvac = −∑

iQiδxi

=⇒ ψvac(x) =∑

i

Qi

4πε0εm|x − xi |

The Debye–Huckel (or linearized Poisson–Boltzmann) equation:

∇ · ε0εΓ∇ψsol − χwεwκ2ψsol = −∑

iQiδxi

=⇒ ψsol = ?

Definition. A Yukawa-field approximation is

Dsol(x) ≈

Dvac(x) =∑

i

Qi (x − xi )

4π|x − xi |3if x ∈ Ωm, x 6= xi ∀i ,

∑

ifi (x , κ, Γ)

Qi (x − xi )

4π|x − xi |3if x ∈ Ωw .

The electrostatic solvation free energy with the Yukawa-fieldapproximation

Gele [Γ] =1

32π2ε0

∫

Ωw

1

εw

∣

∣

∣

∣

∣

∑

i

fi (x , κ, Γ)Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

−1

εm

∣

∣

∣

∣

∣

∑

i

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

dV

Conditions on all fi (·, κ, Γ) : Ωw → R:

(1) κ = 0 =⇒ fi (·, κ, Γ) = 1 for all i ;

(2) fi (x , κ, Γ)Qi (x − xi )

4π|x − xi |3∼

e−κr

ras r = |x | → ∞;

(3) Exact for a spherical solute. κΩm

ε =80ε =1

w

m

Γ

Ωw

Final formulas

fi (x , κ, Γ) =1 + κ|x − xi |

1 + κlmi (x)e−κlw

i(x)

lmi (x) = |[xi , x ] ∩ Ωm|

lwi (x) = |[xi , x ] ∩ Ωw |

m

x

x

Ω

Γ

i

Ωw

fi (x , κ, Γ) =1 + κ|x − xi |

1 + κlmi (x)e−κlw

i(x)

Exact for a single-particle, spherical solute!

ψsol(x) =

Q

4πεwε0R(1 + κR)+

Q

4πεmε0

(

1

|x |−

1

R

)

if |x | < R,

Q

4πεwε0(1 + κR)

e−κ(|x |−R)

|x |if |x | > R.

Dsol(x) =

Qx

4π|x |3if |x | < R,

1 + κ|x |

1 + κRe−κ(|x |−R) Qx

4π|x |3if |x | > R.

O

ε

R

m εw

Q

fi (x , κ, Γ) =1 + κ|x − xi |

1 + κlmi (x)e−κlw

i(x)

Asymptotic analysis with κ ≪ 1 for a model system

Dsol(x) ≈Q(1 + κ|x |)e−κ(r−R3+R2−R1)

4π(1 + κ(R3 − R2 + R1))|x |3x ∀x ∈ Ωw

3 rOQ

R R1 R2

Assume a total of N solute particles xi . The electrostatic freeenergy with the Yukawa-field approximation is

Gele [Γ] =1

32π2ε0

∫

Ωw

1

εw

∣

∣

∣

∣

∣

N∑

i=1

fi (x , κ, Γ)Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

−1

εm

∣

∣

∣

∣

∣

N∑

i=1

Qi (x − xi )

|x − xi |3

∣

∣

∣

∣

∣

2

dV

=

∫

Ωw

[

1

εw

F (x , lw1 (x), . . . , lwN (x)) −1

εm

C (x)

]

dV

Theorem. Assume n points from Ωm to Ωw . DenoteLw

i (x) = xi + s(x − xi ) : 1s < ∞ ∩ Ωw for x ∈ Ωw . Then

δΓGele [Γ](x) = −

[

1

εw

F (x , lw1 (x), . . . , lwN (x)) −1

εm

C (x)

]

−N

∑

i=1

1

|x − xi |2

∫

Lwi(x)

|y − xi |2∂iF (y) dly ∀x ∈ Γ.

First Proof.

Partition of unity.

Local polar coordinates.

Apply a generalized version of Leibniz formula:

d

dy

∫ b(y)

a

f (x , y) dx =

∫ b(y)

a

∂y f (x , y) dx +d

dyb(y)f (b(y), y).

Second Proof.

Local perturbation.

Level-set representation.

Co-area formula

∫

φ>tu dx =

∫ ∞

t

(

∫

φ=s

u

|∇φ|dS

)

ds.

Q.E.D.

0

x

x

Γ

i

+Ω

PΩ− B(z,d)

Λ

5. Conclusions and Discussions

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.

Current work: Incorporate the dielectric boundary force into thelevel-set variational implicit-solvent model.

The dielectric boundary force is part of the normal velocity for thelevel-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.

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 keyparameter in the stratification of multivalent counterions neara charged surface.

5 10 15 20 250

5

10

15

Distance to a charged surface

Con

cent

ratio

n of

cou

nter

ion

(M)

+3+2+1

(a)

From S. Zhou, Z. Wang, and B. Li, Phys. Rev. E, 2011 (in press).

Main References

H.-B. Cheng, L-T. Cheng, and B. Li, Yukawa-fieldapproximation of electrostatic free energy and dielectricboundary force, 2011 (submitted).

B. Li, X. Cheng, and Z. Zhang, Dielectric boundary force inmolecular 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 thePoisson–Boltzmann equation for molecular solvation withimplicit solvent, SIAM J. Math. Anal., 40, 2536–2566, 2009.

B. Li, Continuum electrostatics for ionic solutions withnonuniform ionic sizes, Nonlinearity, 22, 811–833, 2009.

S. Zhou, Z. Wang, and B. Li, Mean-field description of ionicsize effects with non-uniform ionic sizes: A numericalapproach, Phys. Rev. E, 2011 (in press).

Thank you!