a quasi-linear poisson-boltzmann equation · 2015-07-29 · a quasi-linear poisson-boltzmann...

A quasi-linear Poisson-Boltzmann equationModeling, computation and biological application

Duan Chen

University of North Carolina at Charlotte

IMA Hot Topics Workshop, July 23, 2015

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Cancers and molecules

PLoS ONE. 2014;9(10):e107511.

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Structure matters

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Environment matters

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implicit solvent model

Poisson-Boltzmann equation

−∇ · (ε(r)∇φ) + κ2 sinh

(φe

kBT

)= 4πρfe,

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Improvement of PB or PNP equations

Finite size or steric effectsC. Liu, B. Eisenberg, T.C. Lin, W. Liu, B. Lu, Y. Zhou, B. Li,et. al

Non-local interactionsD. Xie, J.L. Liu, et. al

Generalized correlations of ion species and environments.D. Chen and G. Wei, et. al;

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Ion-concentration dependent dielectric constant

Some experimental results:

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Some explanation

Na+$

E$

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Objectives

Based on the simple assumptions:

what kind of new equation we can derive? Analysis?

Any difficulties in numerical simulations?

Applications in molecular biology?

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Mathematical modeling

A slight change of the total free energy:

G[φ, n1, ..., nNc ] =

∫Ω

kBT Nc∑j=1

nj lnnjn0j

− ε(n1, ..., nNc)

8π|∇φ|2 + φρ

dr+

∫ΩkBT

Nc∑j=1

(n0j − nj)dr. (1)

ρ = ρfe+

Nc∑j=1

njqj .

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Mathematical modeling

Variation of the free energy:

δGTotal[φ, n1, ..., nNc ]

δφ= 0⇒ −∇·(ε(n1, ..., nNc)∇φ) = 4πρfe+4π

Nc∑j=1

njqj ,

δGTotal[φ, n1, ..., nNc ]

δnj= µj ⇒ µj = kBT ln

njn0j

+ qjφ−δε

δnj

|∇φ|2

8π,

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A simple case

We consider:

Equilibrium state, i.e., ∇µj = 0

1:1 electrolyte, i.e., Nc = 2 and q1 = −q2 = q

linear dependence, i.e., ε(p, n) = ε− β(p+ n).

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A quasi-linear Poisson-Boltzmann equation

−∇ · (ε(r)∇φ) + 8πn0qλ sinh

(φq

kBT

)= 4πρfq, (2)

whereλ = e−β|∇φ|

2/8πkBT , (3)

and

ε(r) = ε− λβn0 cosh

(φq

kBT

). (4)

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Computational method

Non-dimensionlization: u =φe

kBT, L =

L

A, n0 =

n0M

, β = βM .

We arrive at

−∇ ·[(ε− Isβe−c1β|∇u|

2

coshu)∇u]

+ c2Ise−c1β|∇u|2 sinhu = c3ρf , (5)

Constants c1 =1027kBTA

8πe2NA≈ 0.12, c2 =

8πe2NA

1027kBTA≈ 8.44, and

c3 =4πe2

kBTA≈ 7046

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Iteration methods

Discretized system:A(U)U +N(U) = f , (6)

Full Newton’s method.

J(U) =∂

∂U[A(U)U +N(U)] = A(U) +

∂

∂UA(U)U +

∂

∂UN(U),

Fixed-point-Newton’s method

−∇ ·[(ε− Isβe−β|∇u|

2coshu)∇u∗

]+ Ise

−β|∇u|2 sinhu∗ = f.

Then

J(U,U∗) =∂

∂U∗[A(U)U∗] +

∂

∂U∗[N(U∗)].

Mol. Based Math. Biol. 2014; 2:107127

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Computational efficiency and accuracy

0 10 20

−4

−2

0

Iteration steps

Itera

tio

n e

rro

r (l

og

10)

β=20

β=12

0 200 400−5

−4

−3

−2

−1

0

Iteration steps

Itera

tio

n e

rro

r (l

og

10)

β=20

β=12

0 20 40

−4

−2

0

Iteration steps

Itera

tio

n e

rro

r (l

og

10)

Newton

Fixed−Newton

(a) (b) (c)

Figure 1 : Computational efficiency of the Newton’s method and thefixed-point-Newton’s method. (a): Newton’s method; (b)Fixed-point-Newton’s method. For (a) and (b), Is = 0.2, g = −10. (c):comparison of the two method with Is = 0.2, β = 12 but g = −2.

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Computational efficiency and accuracy

Newton’s method F-N Method Relative differenceError Order Error Order

5.25e-5 5.26e-5 0.07%1.29e-5 2.0 1.299e-5 2.0 0.07%3.09e-6 2.0 3.09e-6 2.0 0.07%6.18e-7 2.3 6.19e-7 2.3 0.07%

Table 1 : Convergence rates of the Newton’s method (first two columns) andthe fixed-point-Newton’s (F-N)method (the third and fourth columns). Therelative differences of the solutions from the two methods are in the lastcolumn.

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Comparison with Poisson-Boltzmann equation

0 5 10−2

−1.5

−1

−0.5

Distance from the boundary (A)

Ele

ctr

os

tati

cs

(k

BT

/e)

PBE

QPBE

0 2 4 6 8 10−10

−8

−6

−4

−2

0

Distance from the boundary (A)

Ele

ctr

os

tati

cs

(k

BT

/e)

PBE

QPBE

(a) Is = 0.2, β = 12 (b)

0 2 4 6 8 100

0.5

1

1.5

Distance from the boundary (A)

Ion

ic c

on

ce

ntr

ati

on

(M)

PBE

QPBE

0 2 4 6 8 101.5

2

2.5

3

3.5

4

Distance from the boundary (A)

Ion

ic c

on

ce

ntr

ati

on

(M)

QPBE

0 0.5 10

0.5

1

1.5

2x 10

4

PBE

(c) (d)Duan Chen (UNCC) IMA workshop July 23, 2015 18 / 25

Comparison with Poisson-Boltzmann equation

0 2 4 6 8 10−10

−8

−6

−4

−2

0

Distance from the boundary (A)

Ele

ctr

os

tati

cs

(k

BT

/e)

β=0

β=10

β=20

0 2 4 6 8 10−10

−8

−6

−4

−2

0

Distance from the boundary (A)

Ele

ctr

os

tati

cs

(k

BT

/e)

β=0

β=10

β=20

(a)Is = 0.2 (b)Is = 1.5

0 2 4 6 8 101

2

3

4

Distance from the boundary (A)

Ion

ic c

on

ce

ntr

ati

on

(M)

β=10

β=20

0 2 4 6 8 102

2.5

3

3.5

4

Distance from the boundary (A)

Ion

ic c

on

ce

ntr

ati

on

(M)

β=10

β=20

(c) (d)Duan Chen (UNCC) IMA workshop July 23, 2015 19 / 25

Applications

ε(r) =

ε+, r ∈ Ω+,

ε− − Isβe−c1β|∇u|2

coshu, r ∈ Ω−,

(7)

ρf =

Na∑i=1

ziδ(r− ri), (8)

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Regularization

u(r) = u(r) + u∗(r) + u0(r), (9)

−∇ · (ε(r)∇u(r)) + c2Ise−c1β|∇u|2 sinh u = 0 (10)

[u]Γ = 0 (11)

[ε∇u · ~n]|Γ = −[ε∇(u∗ + u0) · ~n]|Γ (12)

where

u∗(r) = c3

Na∑i=1

ziε+|r− ri|

(13)

and u0(r) is a harmonic function on Ω+ and

u0(r) = −u∗(r), ∀r ∈ ∂Ω+. (14)

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Application I: Electrostatic solvation energy

∆Gelec =1

2

Na∑i=1

zi[u0(ri) + u(ri)].

Protein ID PBE QPBE Difference

1AJJ -1337.34 -1334.73 2.611AJK -1322.24 -1319.95 2.291AL1 -692.08 -690.5 1.581BBL -1368.0 -1355.0 13.01BOR –1122.83 -1118.50 4.331BPI -1825.73 -1821.21 4.521CBN -494.63 -493.48 1.151FCA -1486.81 -1475.06 11.751FXD -3643.70 -3640.71 2.291HPT -1231.29 -1225.98 5.311PTQ -1237.31 -1235.02 2.291R69 -1553.86 -1552.00 1.861UXC -1600.44 -1595.01 5.431VII -1210.59 -1204.20 6.391YSN -1480.50 -1477.51 2.992ERL -115.06 -1114.28 0.782PDE -1047.46 -1045.88 1.58

Table 2 : Comparison of electrostatic solvation energy (unit: kcal/mol)calculated by the PBE and the QPBE for a set of proteins. Is = 0.2 andβ = 12.

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Application II: Electrostatic analysis of a DNA segment

(a) Is = 0.2, β = 12 (b) Is = 1.5, β = 12

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Application II: Electrostatic analysis of a DNA segment

(c) Is = 0.2, β = 20 (d) Is = 1.5, β = 20

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Conclusion

A quasi-linear Poisson-Boltzmann model based on a simpleexperimental result

Two numerical methods to solve the nonlinear equation

3D simulations in applications of electrostatic analysis forbiomolecules

Future work: modified PNP equation for ion channels.

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