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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
⎛
⎝ ⎜
⎞
⎠ ⎟