Dielectric Boundary Force in Biomolecular Solvation

Bo Li

Department of Mathematics and Quantitative Biology Graduate Program

UC San Diego

Funding: NSF and NIH

IMA Workshop 22nd Century Mathematics and Mechanics: Seven Decades

and Counting … Eugene, Oregon, October 23 – 25, 2015

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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

€

γ 0 = 73mJ /m2

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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 effect and van der Waals (vdW) dispersion

!   Electrostatic interactions

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∇ ⋅εε0∇ψ = −ρPoisson’s equation: solvent

solute

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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 4. Implications to Molecular Recognition

Dzubiella, Swanson, & McCammon, PRL, 2006; JCP, 2006.

Free-energy functional

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r i

€

Ωm

Γ

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Qi

€

Ωw

€

c j∞,

€

q j , wρ

€

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: Two paraffin plates

Cheng, Wang, Zhou, Guo, Setny, Che, Dzubiella, Li, & McCammon, JCP, JPCB, JCTC, PRL, J. Comput. Phys., J. Comput. Chem., SIMA, SIAP, Nonlinearity, 2007-2015.

Hadwiger’s Theorem

Pvol(Ωm )+γ0area(Γ)− 2γ0τ H dSΓ∫ +cK K

Γ∫ dS( )

Let C = the set of all convex bodies, M = the set of finite union of convex bodies. If is

!   rotational and translational invariant, !   additive:

!   conditionally continuous: ),()(,, UFUFUUCUU jjj →⇒→∈

RMF →:

,,)()()()( MVUVUFVFUFVUF ∈∀∩−+=∪

.)()()( MUKdSdHdScUbAreaUaVolUFUU

∈∀++∂+= ∫∫ ∂∂

then

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Application to solvation of nonpolar molecules Roth, Harano, & Kinoshita, PRL, 2006. Harano, Roth, & Kinoshita, Chem. Phys. Lett., 2006.

Geometrical part:

Consider an ionic solution !   : local ionic concentrations !   dielectric coefficient !   fixed charge density

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∇⋅εε0∇ψ −B '(ψ) = −ρ f

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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, bounded 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∑

Gelec[Γ]=12

Qiψreac (!ri )−

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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)

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x(0)= X{

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x = x(X,t) = Tt (X)

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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 we have The dielectric boundary force always points to solutes!

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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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, 2015. Sun et al., JSC, 2015. Luo et al., Chem. Phys. Lett., 2014.

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Dispersion relation

Li, Sun, and Zhou, SIAP, 2015.

Stability of a cylindrical dielectric boundary: Effect of geometry, electrostatics, and hydrodynamics

Viscosity slows down the decay of perturbations.

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4. Implications to Molecular Recognition !   Water molecules inside a protein are unhappy – broken bounds.

(Yin, Hummer, & Rasaiah, JACS, 2007. Yin, Feng, Clore, Hummer, & Rasaiah, JPCB, 2010, Baron & McCammon, Annu. Rev. Phhy. Chem., 2013.)

!   Instability and fluctuations lead to drying near a pocket on protein surface, allowing a diffusional ligand to bind to the protein.

!   Dielectric boundary force in the transient complex theory for the binding kinetics of two molecules in water: electrostatic steering.

J. A. McCammon, PNAS, 2009.