Macroscopic Electrostatic Force in Molecular Conformational Dynamics

Macroscopic Electrostatic Force in Molecular

Conformational Dynamics

Bo Li

Department of Mathematics University of California, San Diego

Funding: NSF and NIH

Workshop on Mathematical Modeling and Computation in Biophysics

Shanghai Jiao Tong University July 31–August 2, 2021

Biomolecular Solvation

Electrostatic Interactions

• Solute partial charges

• Mobile ions in water

• Solvent polorization

• Dielectric response

Coulomb’s law F₂₁= − q₁q₂

4πεr²r₂₁ Poisson’s equation

∇ · ε∇ψ = −ρ

2

q₁ r q

O ε R

m ε

w

Q Born’s model (1920): W = − Q² 8πR

 1 ε_m − 1

ε_w



Hasted, Ritson, & Collie, JCP 1948)

Variational Implicit-Solvent Model (VISM)

(Dzubiella, Swanson, & McCammon, PRL & JCP 2006) Solvation free-energy functional

G [Γ] = surface energy + solute-solvent interaction + electrostatics

= γ0Area (Γ) + ρw Z

Ωw

U_vdWdV + G_ele[Γ]

n boundary dielectric

x Q_i Ω_w Ω_m i

ε =80 ε =1

w m

Γ

ε_Γ =

(εm in Ωm εw in Ωw

U_vdW(x ) =

N

X

i =1

U_LJ^{(i )}(|x − x_i|)

Poisson–Boltzmann (PB) free energy and PBE G_ele[Γ] =

Z

Ω

h

−ε_Γ

2 |∇ψ|²+ f ψ − χwB(ψ) i

dV

∇ · ε_Γ∇ψ − χ_wB⁰(ψ) = −f B(ψ) = β⁻¹XM

j =1c_j^∞

e^−βqjψ− 1 Boundary force: −δΓG [Γ] = −2γ0H + ρwU_vdW− δ_ΓG_ele[Γ]

Dielectric boundary force (DBF): −δ_ΓG_ele[Γ]

∇ · ε_Γ∇ψ − χ_wB⁰(ψ) = −f G_ele[Γ] =

Z

Ω

h−ε_Γ

2|∇ψ|²+ f ψ − χwB(ψ)i dV Define the DBF

n boundary dielectric

x Q_i Ω_w Ω_m i

ε =80 ε =1

w m

Γ

Γ =⇒ Γt =⇒ ψ_Γt =⇒ G_ele[Γt] =⇒ δ_ΓG_ele[Γ] := d dt

   t=0

G_ele[Γt] For any V ∈ C_c^∞(R³, R³), define x : [0, ∞) × R³ → R³ by

˙

x = V (x ) and x (0, X ) = X . Set T_t(X ) = x (t, X ) and Γ_t = T_t(Γ).

δ_Γ,VG_ele[Γ] := lim

t→0

G_ele[Γ_t] − G_ele[Γ]

t =

Z

Γ

(V · n) h_ΓdS Definition. δ_ΓG_ele[Γ] = h_Γ: Γ → Γ.

Theorem(Li, Cheng, & Zhang, SIAP 2011; Li, Zhang, & Zhou, J.

Nonlinear Sci. 2021 (under review).) The DBF is given by

− δ_ΓG_ele[Γ]

= −1 2

 1 ε_m − 1

ε_w



|ε_Γ∇ψ · n|²

−1

2(ε_w− ε_m) |(I − n ⊗ n)∇ψ|²− B(ψ)

< 0.

n

boundary dielectric

x Q_i Ω_w Ω_m i

ε =80 ε =1

w m

Γ

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.”

4 6 8 10 12 14 16 18 20

−140

−120

−100

−80

−60

−40

−20 0 20 40

d (˚A) Gtotpmf(kB T)

PS Tight PS Loose PB Tight PB Loose

Wang et al. JCTC 2012, Guo et al. JCTC 2014, Zhou et al. JCTC 2014.

Stability of a Cylindrical Dielectric Boundary (Cheng, Li, White, & Zhou. SIAP 2013)

F [Γ] = γ0Area (Γ) + Z

Ω

1 2f ψ_ΓdV (∇ · ε_Γ∇ψ_Γ= −f in Ω

ψΓ= 0 on r = R∞

8R

Γ: r=u(z)

L O

x

y

z Ω_

Ω₊

ε_Γ=

(ε− in Ω−

ε+ in Ω+

Assume: ε−> ε+.

• Water molecules deep in a protein.

• Competition: surface energy vs. electrostatic energy.

• Stability of an equilibrium dielectric boundary.

Steepest descent: Vn= −δ_ΓF [Γ]





































 ut= γ0

 uzz

1 + u_z² −1 u



−1 2

 1 ε−

− 1 ε+

 ε_Γ(t)(ψ_r − u_zψ_z)2

p1 + u_z²

−1

2(ε+− ε−)(uzψr + ψz)²

p1 + u_z² ∀(z, t) ∈ (−∞, ∞) × (0, T ] u(z, t) is L-periodic in z for each t ∈ [0, T ]

u(z, 0) is given for all z ∈ (−∞, ∞)

∇ · ε_Γ(t)∇ψ = −f in Ω

ψ(r , z, t) is L-periodic in z for each (r , t) ∈ [0, R∞) × [0, T ] ψ(R∞, z, t) = 0 ∀(z, t) ∈ (−∞, ∞) × [0, T ]

Step 1. Steady-state solutions. u = u₀ and ψ = ψ₀(r ).

u₀ = 1 η₀

Z u0

0

sf (s) ds

2

ψ₀(r ) =











− 1 ε−

Z r 0

 1 s

Z s 0

τ f (τ ) d τ



ds + C₂ if r < u₀

− 1 ε+

Z r u0

 1 s

Z s u0

τ f (τ ) d τ



ds + C3log r + C4 if r > u0



















C2− C₃log u0− C₄= 1 ε−

Z u0

0

 1 s

Z s 0

τ f (τ ) d τ

 ds C₃ = − 1

ε₊ Z u0

0

sf (s) ds C3log R∞+ C4= 1

ε₊ Z R∞

u0

 1 s

Z s u0

τ f (τ ) d τ

 ds

η = 2γ₀ 1 ε₊ − 1

ε−

−1

> 0 and ηR∞<

Z R∞

0

sf (s) ds

²

Step 2. Linearization.

u = u(z, t, δ) = u₀+ δu₁(z, t) + · · · ,

ψ = ψ(r , z, t, δ) = ψ0(r ) + δψ1(r , z, t) + · · · .



































∂tu1 = γ0∂²_zu1+ γ₀

u₀² − 1 ε−

− 1 ε+



ε²₊ψ₀⁰(u₀⁺)ψ₀⁰⁰(u₀⁺)

 u1

− 1 ε−

− 1 ε₊



ε²₊ψ⁰₀(u₀⁺)∂rψ1(u₀⁺, z, t) ∀z, t,

∆ψ1 = 0 if 0 < r < u0,

∆ψ₁ = 0 if u₀ < r < R∞,

ψ₁(u₀⁺, z, t) − ψ₁(u₀⁻, z, t) = −u₁(z, t)ψ₀⁰(u₀⁺) − ψ₀⁰(u₀⁻)

∀z, t, ε−∂_rψ₁(u₀⁻, z, t) = ε₊∂_rψ₁(u₀⁺, z, t) ∀z, t,

ψ1(R∞, z, t) = 0 ∀z, t.

Step 3. Dispersion relations.

Assume

u1(z, t) = Ae^ωte^ikz with k = 2πk⁰/L, k⁰ ∈ Z, ψ₁(r , z, t) = u₁(z, t)φ_k(r ).

Then the dispersion relation ω = ω(k) is given by ω(k) = −γ0k²+γ0

u₀²− 1 ε−

− 1 ε₊



ε²₊ψ₀⁰(u₀⁺)ψ⁰⁰₀(u⁺₀) + φ⁰_k(u₀⁺) ,























φ⁰⁰_k(r ) + 1

rφ⁰_k(r ) − k²φ_k(r ) = 0 if 0 < r < u₀, φ⁰⁰_k(r ) + 1

rφ⁰_k(r ) − k²φ_k(r ) = 0 if u₀< r < R∞, φk(u₀⁺) − φk(u⁻₀) = −ψ₀⁰(u₀⁺) − ψ₀⁰(u₀⁻) , ε−φ⁰_k(u⁻₀) = ε₊φ⁰_k(u₀⁺),

φ_k(R∞) = 0.

The modified Bessel differential equation

x²y⁰⁰(x ) + xy⁰(x ) − x²y (x ) = 0.

The modified Bessel functions I0(x ) =

∞

X

s=0

1 (s!)²

x 2

2s

and K0(x ) = Z ∞

0

cos(xs)

√

1 + s²ds.

φk(r ) =











µε₊ψ₀⁰(u₀⁺) − ψ₀⁰(u₀⁻) I₀(kr )

K₀(kR∞)I₀⁰(ku₀) − I₀(kR∞)K₀⁰(ku₀)

if 0 < r < u₀, µε−ψ₀⁰(u₀⁺) − ψ₀⁰(u₀⁻) I₀⁰(ku₀)

[K0(kR∞)I0(kr ) − I0(kR∞)K0(kr )] if u0 < r < R∞, 1

µ = ε−I1(ku0) [I0(kR∞)K0(ku0) − I0(ku0)K0(kR∞)]

+ ε₊I₀(ku₀) [I₁(ku₀)K₀(kR∞) + I₀(kR∞)K₁(ku₀)] .

1 2 3 4 5 k

-1.5 -1.0 -0.5 0.5

Ω₂ Ω₁ Ω

ω(k) = ω₁(k) + ω₂(k) ω1(k) = − 1

ε−

− 1 ε₊



ε²₊ψ₀⁰(u₀⁺)ψ₀⁰⁰(u₀⁺) + φ⁰_k(u₀⁺) ω2(k) = −γ0k²+γ₀

u²₀

Conclusion: linearly stable if and only if k > kc.

Possible implication in biology: Water molecules inside proteins are unstable—no hydrogen network.

(Rasaiah et al. JACS 2007 & JPCB 2010)

Dynamic Implicit-Solvent Model (DISM)

Li, Sun, & Zhou, SIAP 2015,

Sun, Zhou, Cheng, & Li, J. Comput. Phys. 2018, Fan, Li, & White, SIAP 2021.

n boundary dielectric

x Q_i Ω_w Ω_m i

ε =80 ε =1

w m

Γ

Interface motion V_n= u · n on Γ(t), Incompressibility ∇ · u = 0 in Ωw(t),

Stochastic Stokes eq. µw∇²u = ∇Pw(t) + ∇ · Σ in Ωw(t), Ideal-gas law P_m^{(i )}(t) = N_i(t) k_BT

Vol(Ω^{(i )}_m(t))

in Ω^{(i )}_m(t) (i = 1, . . . , m(t)), PBE ∇ · ε_Γ(t)∇ψ − χ_w(t)B⁰(ψ) = −XN

j =1Qjδxj in Ω, Force balance 2µ_wD(u)n − δ_ΓG [Γ]n = 0 on Γ(t).

Σ: Landau–Lifshitz stochastic stress tensor

hΣ_ij(x, t)Σ_{k`</sub>(x<sup>0</sup>, t<sup>0</sup>)i = 2 µ<sub>w</sub>k<sub>B</sub>T δ(x − x<sup>0</sup>)δ(t − t<sup>0</sup>) (δ<sub>ik</sub>δ<sub>j `}+ δ_{i `}δ_jk) .

Stability of a Cylindrical Dielectric Boundary, Again

(Li, Sun, & Zhou, SIAP 2015)

ω(k) = ω_surf(k) + ω_vdW(k) + ω_ele(k) ω_hyd

= − γ0

2µ_wk + O(1) as k → ∞.

ω_surf(k) = γ₀

 1 R₀² − k²

 , ω_vdW(k) = nwU_vdW⁰ (R0), ω_ele(k) = (εw− ε_m)²

ε_wε_m(ε_w+ ε_m)

 1 R₀

Z R0

0

sρ(s)ds

²

k + O(1), ω_hyd(k) = 2µwk + O(1).

Numerical Simulations of a Two-Plate System (Sun, Zhou, Cheng, & Li, J. Comput. Phys. 2018)

0 5 10 15 20 25 30 35 40 45

3100 3200 3300 3400 3500 3600 3700 3800

=0.12

=0.25

=0.5

The area of the solute-solvent interface vs. the rescaled simulation time for three different values of solvent viscosity µ = µw.

A Generalized Rayleigh–Plesset Equation for an Ion

(Fan, Li, & White, SIAP 2021)

O ε R

m ε

w Q

4µwR˙

R = F (R) + ξ (ξ : white noise) hξ(t)ξ(t⁰)i = 4

3µwk_BT δ(t − t⁰) F (R) = Pm(R) − P∞− 2γ₀ 1

R − τ R²



+ nwU_LJ(R) + f_ele(R) Pm(R) = 3kBT

4πR³ f_ele(R) = Q²

32π²

 1 ε_w − 1

ε_m

 1

R⁴ − κ² ε_w(1 + κR)²R²



2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

−1

−0.5 0 0.5 1 1.5 2

R (˚A) Surface forces (kBT/˚A2 )

Scaled LJ interactions Pressure difference Surface tension Electrostatic interactions Total

The total surface force density F = F (R).

2 3 4 5 6 7 8 Distance

0 0.25 0.5 0.75 1 1.25 1.5 1.75

Radial distribution

Radial distribution Peak Half peak Bulk

2 3 4 5 6 7 8

Distance 0

0.5 1 1.5 2 2.5

Radial distribution

1 2 3 4 5 6 7 8

Distance 0

0.5 1 1.5 2 2.5 3

Radial distribution

1 2 3 4 5 6 7 8

Distance 0

1 2 3 4 5 6 7

Radial distribution

The radius of an ion (Fan, Li, & White, SIAP 2021)

Q (e) Peak Half-Peak Bulk First nonzero RP

1 3.32 3.00 3.03 2.48 2.80

−1 2.04 1.90 1.86 1.56 2.80

2 2.96 2.83 2.81 2.32 2.46

−2 1.86 1.74 1.67 1.46 2.46

(Unit of length: ˚A.)

• With the same |Q|, the MD radius of a cation is larger than that of an anion But the RP radius remains the same. Charge asymmetry!

• The RP radius approximates well the averaged peak radius over those for the two ions with the same |Q|.

• For a cation, the RP radius is close to the average of the first nonzero and bulk radii.

• For an anion, an effective radius is the RP radius minus 0.5 ˚A.

Conclusions and Discussions

• The PB based DBF is crucial in qualitative descriptions of conformational dynamics of charged molecules.

• Dielectric boundary is sensitive in calculating the electrostatic energy and hence the DBF.

• Known problems of continuum electrostatics: charge asymmetry, ionic size effect, charge-charge correlations, etc.

Current Work

• Fast numerical method for minimizing the surface energy and electrostatic energy, using the Legendre transform.

• A sharp boundary is often too “rigid”, leading to high barriers in conformational changes. Relaxation?

• Including the solvent polarization.

• Coupling continuum and atomistic descriptions.

Thank You!