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Dielectric Boundary Force in Biomolecular Solvation
Bo Li
Department of Mathematics and Center for Theoretical Biological Physics
UC San Diego
Funding: NSF, NIH, and CTBP
The 52nd Meeting of the Society for Natural Philosophy Rio de Janeiro, Brazil, October 22 - 24, 2014
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protein folding molecular recognition
solvation
conformational change water
water
solute solute
solute
water
receptor ligand
binding
€
ΔG = ?
Solvation
solvent
solute
solvent
solute
Biomolecular Modeling: Explicit vs. Implicit
MD simulations
Statistical mechanics
mi!!ri = −∇riV (r1,…, rN )
A =1Z
A(p, r)∫∫ e−βH ( p,r )dpdr = Atime
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! Solute-solvent interfacial property
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γ 0
R
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γ 0 = 73mJ /m2
€
γ = γ 0 1− 2τH( )
Curvature effect
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τ = 0.9
Huang et al., JPCB, 2001.
A!
Solute-solvent interface: vapor-liquid interface Widom 1969, Weeks 1977, Chandler 2010.
Solute
Water
€
ULJ (r) = 4ε σr( )12 − σ
r( )6[ ]σ rO
€
−ε
The Lennard-Jones (LJ) potential ! Excluded volume and van der Waals (vdW) dispersion
! Electrostatic interactions
€
∇ ⋅εε0∇ψ = −ρPoisson’s equation: solvent
solute
€
ε =1
€
ε = 80
ρ = ρ f + ρi
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Dielectric boundary
Hasted, Ritson, & Collie, JCP, 1948.
Surface energy
PB/GB calcula1ons
Commonly used, surface based, implicit-solvent models
solvent accessible surface (SAS)
probing ball
vdW surface
solvent excluded surface (SES)
Possible issues
! Hydrophobic cavities ! Curvature correction ! Decoupling of polar and
nonpolar contributions
5 Koishi et al., PRL, 2004. Sotomayor et al., Biophys. J., 2007.
PB = Poisson-Boltzmann GB = Generalized Born
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1. The Poisson-Boltzmann (PB) Theory 2. Dielectric Boundary Force 3. Stability of a Cylindrical Dielectric Boundary
Dzubiella, Swanson, & McCammon: PRL, 2006; JCP, 2006.
Free-energy functional
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r i
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Ωm
Γ
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Qi
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Ωw
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c j∞,
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q j , wρ
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G[Γ] = Pvol(Ωm ) + γ 0 (1− 2τH)dSΓ
∫+ρw ULJ ,i
i∑
Ωw
∫ (| !r − "ri |)dV
Variational Implicit-Solvent Model (VISM)
+Gelec[Γ]
BphC
p53/MDM2
This talk focuses on Two paraffin plates
Consider an ionic solution ! : local ionic concentrations ! dielectric coefficient ! fixed charge density
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∇⋅εε0∇ψ −B '(ψ) = −ρ f
€
B(ψ) = β−1 c j∞ e−βq jψ −1( )j=1
M∑
G =12∫ ρψdV = −
εε02|∇ψ |2 +ρ fψ −B(ψ)
$
%&'
()∫ dV
1. The Poisson-Boltzmann (PB) Theory
Poisson equation
Boltzmann distributions Charge density
∇⋅εε0∇ψ = −ρ
ρ = ρ f + ρi = ρ f + qicii=1
M∑
ci = ci∞e−βqiψ
! Linearized PBE ! Sinh PBE
∇⋅εε0∇ψ −κ2ψ = −ρ f
∇⋅εε0∇ψ − 2c∞ sinh(βψ) = −ρ f
ci = ci (x) (i =1,...,M )ε :ρ f :
PBE
PB free energy
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Theorem (Li, Cheng, & Zhang, SIAP 2011) has a unique minimizer, bouded in and uniformly with respect to It is the unique solution to the PBE.
I[•]
Proof. ! Existence and uniqueness of a minimizer in by direct
methods in the calculus of variations and the convexity of . ! Uniform bound by comparison. ! Regularity theory and routine calculations. Q.E.D.
€
H1
€
L∞
∇⋅εε0∇ψ −B '(ψ) = −ρ f
B(ψ) = β −1 ci∞ e−βqiψ −1( )i=1
M∑
I[ψ]= εε02|∇ψ |2 −ρ fψ +B(ψ)
#
$%&
'(∫ dV
PBE
Charge neutrality o s
B
Define
−B '(0) = ci∞qi = 0i=1
M∑
Hg1(Ω) = {φ ∈ H1(Ω) :φ = g on ∂Ω}
ε ∈ [εmin,εmax ].
Hg1(Ω)
L∞(Ω)I[•]
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Electrostatic Free-energy functional F[c]= 1
2ρψ +β −1 ci ln(Λ
3ci )i=1
M
∑ − µicii=1
M
∑$
%&
'
()∫ dV
ρ = ρ f + qicii
M∑
€
∇ ⋅εε0∇ψ = −ρ
€
δiF[c] = 0 ci = ci∞e−βqiψ
O s
slns
Theorem (B.L. SIMA 2009). ! has a unique minimizer . ! There exist such that for all ! All satisfy the Bolzmann distributions. ! The corresponding is the unique solution to PBE.
F[c] c = (c1,…,cM )θ1,θ2 > 0 θ1 ≤ ci ≤θ2 i =1,…,M.
ψ
(+ B.C.)
ci
Proof. ! Existence and uniqueness of a minimizer by direct methods, using
convexity of and the superlinear growth of ! Uniform bounds for equilibrium concentrations by Lemma below. ! Regularity theory and routine calculations. Q.E.D.
F[c] s! s ln s.
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€
r i
€
Ωm
Γ
€
Qi
€
Ωw
€
c j∞,
€
q j , wρ€
εm =1
€
εw = 80
The PB theory applied to molecular solvation
! Dielectric coefficient ! Fixed charge density ! All
ε = εΓ =εm in Ωm
εw in Ωw
ρ f = Qiδrii=1
N∑
ci = 0 (i =1,...,M ) in Ωm.
∇⋅εε0∇ψ − χwB '(ψ) = − Qiδrii=1
N∑
€
B(ψ) = β−1 c j∞ e−βq jψ −1( )j=1
M∑
G =12
Qiψreac (!ri )−
12
B '(ψ)ψreac dVΩw∫i=1
N∑
PBE
PB free energy
Reaction field
Reference potential
ψreac =ψ −ψref
ψref (!r ) = Qi
4πεmε0 |!r − !ri |i=1
N∑
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A shape derivative approach Perturbation defined by
€
V :R3 → R3 :
€
˙ x = V (x)
€
x(0)= X{
€
x = x(X,t) = Tt (X)
€
Γt PBE:
€
ψt
€
Gelec[Γt ]
€
δΓGelec[Γ] =ddt$
% &
'
( ) t= 0
Gelec[Γt ]
2. Dielectric Boundary Force (DBF):
€
Fn = −δΓGelec[Γ]
€
r i
€
Ωm
Γ
€
Qi
€
Ωw
€
c j∞,
€
q j , wρ€
εm =1
€
εw = 80
Structure Theorem
Shape derivative
∇⋅εΓε0∇ψ − χwB '(ψ) = −ρ f
Gelec[Γ]= −εΓε02|∇ψ |2 +ρ fψ − χwB(ψ)
$
%&'
()∫ dV
PBE
PB free energy
n
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Che, Dzubiella, Li, & McCammon, JPCB, 2008. Li, Cheng, & Zhang, SIAP, 2011. Luo et al., PCCP 2012 & JCP 2013.
δΓGelec[Γ]=ε02
1εm
−1εw
#
$%
&
'( |εΓ∂nψ |
2 +ε02εw −εm( ) (I − n⊗ n)∇ψ 2
+B(ψ).
Theorem (Li, Cheng, & Zhang, SIAP, 2011). Let point from to . Then
n Ωm Ωw
Consequence: Since the force
Chu, Molecular Forces, based on Debye’s lectures, Wiley, 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.”
εw > εm, −δΓGelec[Γ]> 0.
∇⋅εΓε0∇ψ − χwB '(ψ) = −ρ f
Gelec[Γ]= −εΓε02|∇ψ |2 +ρ fψ − χwB(ψ)
$
%&'
()∫ dV
PBE
PB free energy
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3. Stability of a Cylindrical Dielectric Boundary Cheng, Li, White, & Zhou, SIAP, 2013.
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25 Yin, Hummer, Rasaiah, JACS 2007. Yin, Feng, Clore, Hummer, Rasaiah, JPCB 2010.
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µw∇2u−∇pw − nw∇Uext +∇⋅Σ = 0 in
€
Ωw (t)
€
∇ ⋅ u = 0 in
€
Ωw (t)
pm,i (t)Ωm,i (t) = NikBT =Cm
at
€
Γ(t)
Fluctuating solvent fluid:
€
∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f
€
r i
€
Ωm
Γ
€
Qi
€
Ωw
Solvent Fluid Dielectric Boundary Model
Σij (x, t)Σkl (x ', t ') = 2µwkBTδ(x − x ')δ(t − t ')(δikδ jl +δilδ jk )
Interface motion Vn = u ⋅n
Electrostatics
Force balance
− fele =ε02
1εm
−1εw
"
#$
%
&' |ε∂nψ |
2 +ε02εw −εm( ) (I − n⊗ n)∇ψ 2
+B(ψ)
2µwD(u)n+ (pm − pw − 2γ0H + nwUvdW + fele )n = 0
M. White, Ph.D. thesis, UCSD, 2013. Li, Sun, and Zhou, SIAP, 2014 (submitted).
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Dispersion relation
Li, Sun, and Zhou, SIAP, 2014 (submitted)
Stability of a cylindrical dielectric boundary: Effect of geometry, electrostatics, and hydrodynamics
Viscosity slows down the decay of perturbations.
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5e+4 6e+4 7e+4 8e+45
7
9
11
R0
( A)
ρ0 (e·A−1)1e+4 3e+4 5e+4 6e+45
7
9
11
R0
( A)
ρ0 (e·A−1)
Steady-state radius vs. a charge parameter: multiple hydration states
0 1 2 3−2e+4
−1e+4
0
5e+3
ω
(ps−1)
l0=2.5e+4, R0= 8.03
l0=4e+4, R0= 6.88
l0=5e+4, R0= 6.50
l0=5.5e+4, R0= 6.50
0 1 2 30
100
200
k (A−1)
l0=2.5e+4, R0= 8.96
l0=4e+4, R0= 9.61
l0=5e+4, R0= 9.80
0 1 2 3−100
−50
0
l0=2e+4, R0= 9.99
l0=2.5e+4, R0= 9.98
l0=4e+4, R0= 9.96
l0=5e+4, R0= 9.91
0 2e+4 4e+4
−2e+4
0
0 2e+4 4e+4
−2e+4
0
0 2e+4 4e+4
−2e+4
0
Dispersion relations
0 1 2 3
−600
−300
0
300
k (A−1)
ω(ps−1)
R0= 8.03R0= 8.96R0= 9.98
0 1 2 30
0.5
1ωhyd(k )
0 1 2 3−100
−50
0
ωvdW(k )
0 1 2 3−2
0
2
ωcurv (k )
0 1 2 30
200
400ωele (k )
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0 500 1000 1500−1e+3
−600
0
400
k (A−1)
ω(ps−1)
R0= 8.03R0= 8.96R0= 9.98
0 500 1000 15000
500ωhyd(k )
0 500 1000 1500−100
−50
0
ωvdW(k )
500 1000
−2e+50
2e+5 ωsurf (k )
0 500 1000 15000
1e+5
2e+5ωele (k )
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! PB theory: analysis and application to molecular solvation. Ionic
size effects? Ion-ion correlation? ! The dielectric boundary force always points to charged molecules,
regardless of charge asymmetry and other microscopic details. ! Electrostatics contributes to instability. Viscosity slows down the
stabilization. ! Prove the well-posedness of proposed variational solvation model. ! Calculate the second variations to study the conformational stability
of a biomolecular system, in particular the dry-wet transition. ! Modeling and analysis of dielectric boundary fluctuations. ! Develop hybrid solute MD and solvent fluid dielectric boundary
models, bridging the time scale.
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Concluding Remarks
Thank you!
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