Dielectric Boundary Forces in Variational Implicit-Solvent...
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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
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Outline
1. Introduction
2. The Poisson–Boltzmann Theory
3. The Coulomb-Field Approximation
4. The Yukawa-Field Approximation
5. Conclusions and Discussions
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1. Introduction
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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
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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.
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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]
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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
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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 .
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2. The Poisson–Boltzmann Theory
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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
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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.
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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.
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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.”
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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].
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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.
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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.
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3. The Coulomb-Field Approximation
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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
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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 .
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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
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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
)
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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.
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4. The Yukawa-Field Approximation
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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.
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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 = ?
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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
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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
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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
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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
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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.
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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
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5. Conclusions and Discussions
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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.
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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.
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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).
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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).
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Thank you!