The Geometry of Biomolecular Solvation

2. Electrostatics

Patrice Koehl

Computer Science and Genome Center

http://www.cs.ucdavis.edu/~koehl/

++

Solvation Free Energy

Wnp

Wsol

VacchW−

SolchW

( ) ( )cavvdWvac

chsol

chnpelecsol WWWWWWW ++−=+=

A Poisson-Boltzmann view of Electrostatics

Elementary Electrostatics in vacuo

∫ =•0ε

qdAE

0

)())((

ερ X

X =Ediv

Gauss’s law:

The electric flux out of any closed surface is proportional to the total charge enclosed within the surface.

Integral form: Differential form:

Notes:- for a point charge q at position X0, ρ(X)=q(X-X0)

- Coulomb’s law for a charge can be retrieved from Gauss’s law

Elementary Electrostatics in vacuo

( )

( )( ) ( ) ( )0

2

0

ε

ρφφφ

ε

ρ

−=∇=∇•∇=

=

graddiv

Ediv

Poisson equation:

Laplace equation:

02 =∇φ (charge density = 0)

+-

Uniform Dielectric MediumPhysical basis of dielectric screening

An atom or molecule in an externally imposed electric field develops a nonzero net dipole moment:

(The magnitude of a dipole is a measure of charge separation)

The field generated by these induced dipoles runs against the inducingfield the overall field is weakened (Screening effect)

The negativecharge is screened bya shell of positivecharges.

Uniform Dielectric MediumPolarization:

The dipole moment per unit volume is a vector field known asthe polarization vector P(X).

In many materials: )(4

1)()( XEXEXP

πεχ −

==

χ is the electric susceptibility, and ε is the electric permittivity, or dielectric constant

The field from a uniform dipole density is -4πP, therefore the total field is

ε

π

applied

applied

EE

PEE

=

−= 4

Uniform Dielectric Medium

Modified Poisson equation:

( )( ) ( )εερφφ0

2 −=∇=graddiv

Energies are scaled by the same factor. For two charges:

r

qqU

επε0

21

4=

System with dielectric boundaries

The dielectric is no more uniform: ε varies, the Poisson equation becomes:

( ) ( )( ) ( ) ( )0

)()(ε

ρφεφε

XXXXgradXdiv −=∇•∇=

If we can solve this equation, we have the potential, from which we can derivemost electrostatics properties of the system (Electric field, energy, free energy…)

BUT

This equation is difficult to solve for a system like a macromolecule!!

The Poisson Boltzmann Equation

ρ(X) is the density of charges. For a biological system, it includes the chargesof the “solute” (biomolecules), and the charges of free ions in the solvent:

The ions distribute themselves in the solvent according to the electrostatic potential (Debye-Huckel theory):

∑=

=N

iiiions XnqX

1

)()(ρ €

ni : number of ions of type i per unit volume

qi : charge on type i ionkT

Xq

i

ii

en

Xn )(

0

)( φ−

=

)()()( XXX ionssolute ρρρ +=

The potential is itself influenced by the redistribution of ion charges, so thepotential and concentrations must be solved for self consistency!

( ) ( ) ∑=

−−−=∇•∇

N

i

kT

Xq

ii

i

enqX

XX1

)(0

00

1)(

φ

εε

ρφε

The Poisson Boltzmann Equation

Linearized form:

( ) ( )

IkT

qnkT

XXXX

XX

N

iii εεεε

κ

φκεε

ρφε

01

20

0

2

2

0

21

)()()()(

==

−−=∇•∇

∑=

I: ionic strength

• Analytical solution

• Only available for a few special simplification of the molecular shape and charge distribution

• Numerical Solution

• Mesh generation -- Decompose the physical domain to small elements;• Approximate the solution with the potential value at the sampled mesh

vertices -- Solve a linear system formed by numerical methods like finite difference and finite element method

• Mesh size and quality determine the speed and accuracy of the approximation

Solving the Poisson Boltzmann Equation

Linear Poisson Boltzmann equation:Numerical solution

εP

εw

• Space discretized into a cubic lattice.

• Charges and potentials are defined on grid points.

• Dielectric defined on grid lines

• Condition at each grid point:

∑

∑

=

=

+

+= 6

1

22

0

6

1

jijijij

i

jjij

i

h

hq

κεε

εφε

φ

j : indices of the six direct neighbors of i

Solve as a large system of linearequations

• Unstructured mesh have advantages over structured mesh on boundary conformity and adaptivity

• Smooth surface models for molecules are necessary for unstructured mesh generation

Meshes

Disadvantages • Lack of smoothness• Cannot be meshed with good quality

An example of the self-intersection of molecular surface

Molecular Surface

• The molecular skin is similar to the molecular surface but uses hyperboloids blend between the spheres representing the atoms

• It is a smooth surface, free of intersection

Comparison between the molecular surface model and the skin model for a protein

Molecular Skin

• The molecular skin surface is the boundary of the union of an infinite family of balls

Molecular Skin

Skin Decomposition

Sphere patches Hyperboloid patches

card(X) =1, 4 card(X) =2, 3

Building a skin mesh

Sample pointsJoin the points to form a mesh of triangles

A 2D illustration of skin surface meshing algorithm

Building a skin mesh

Building a skin mesh

Full Delaunay of sampling points Restricted Delaunay definingthe skin surface mesh

Mesh Quality

Mesh Quality

Triangle quality distribution

Example

Skin mesh

Volumetric mesh

Problems with Poisson Boltzmann

• Dimensionless ions

• No interactions between ions

• Uniform solvent concentration

• Polarization is a linear response to E, with constant proportion

• No interactions between solvent and ions

Modified Poisson Boltzmann Equations

€

div(E(X ) +r P (X)) =

ρ(X )

ε0

Generalized Gauss Equation:

Classically, P is set proportional to E.

A better model is to assume a density of dipoles, with constant module po

Also assume that both ions and dipoles have a fixed size a

with

Generalized Poisson-Boltzmann Langevin Equation

and

€

u = βp0

r E =

p0

r E

kBT

€

β4π

r ∇ • ε

r ∇Φ

r r ( )( ) + βρ f

r r ( ) = −

2βλ ion sinh βezΦr r ( )( )

a3D Φr r ( )( )

+β 2 po

2λ dipF1(u)r

∇ •r

∇Φr r ( )( )

a3D Φr r ( )( )

+β 4 po

4λ dipF1'(u)

r ∇Φ

r r ( ) •

r ∇Φ

r r ( ) •

r ∇( )

r ∇Φ

r r ( )

a3D Φr r ( )( )u

−2β 2 po

2λ ionλ dipF1(u)r

∇Φr r ( )

2βezsinh βezΦ

r r ( )( )

a3D Φr r ( )( )

2

−β 4 po

4λ dip2 F1(u)( )

2 r ∇Φ

r r ( ) •

r ∇Φ

r r ( ) •

r ∇( )

r ∇Φ

r r ( )

a3D Φr r ( )( )

2

€

D Φr r ( )( ) =1+ 2λ ion cos βezΦ

r r ( )( ) + λ dip

sinh βpo

r ∇Φ

r r ( )( )

βpo

r ∇Φ

r r ( )

€

F1 u( ) =1

u

∂

∂u

sinh(u)

u

⎛

⎝ ⎜

⎞

⎠ ⎟=

1

u

ucosh(u) − sinh(u)

u2

⎛

⎝ ⎜

⎞

⎠ ⎟