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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
Wsol
+
+
Sol
ch
W
W
Vac
ch
Wnp
Wsol  Welec  Wnp  W
sol
ch
W
vac
ch
  W
vdW
 Wcav 
A Poisson-Boltzmann view of Electrostatics
Elementary Electrostatics in vacuo
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:
q
 E  dA  
Notes:
0
(X )
div ( E ( X )) 
0
- for a point charge q at position X0, (X)=qd(X-X0)
- Coulomb’s law for a charge can be retrieved from Gauss’s law
Elementary Electrostatics in vacuo
Poisson equation:


div E 
0



div grad             
0
2
Laplace equation:
 0
2
(charge density = 0)
Uniform Dielectric Medium
Physical basis of dielectric screening
An atom or molecule in an externally imposed electric field develops a non
zero net dipole moment:
-
+
(The magnitude of a dipole is a measure of charge separation)
The field generated by these induced dipoles runs against the inducing
field
the overall field is weakened (Screening effect)
The negative
charge is
screened by
a shell of positive
charges.
Uniform Dielectric Medium
Polarization:
The dipole moment per unit volume is a vector field known as
the polarization vector P(X).
In many materials:
P( X )   E ( X ) 
 1
E( X )
4
 is the electric susceptibility, and  is the electric permittivity, or dielectric constant
The field from a uniform dipole density is -4P, therefore the total field is
E  E applied  4 P
E
E applied

Uniform Dielectric Medium
Modified Poisson equation:



div grad        
 0
2
Energies are scaled by the same factor. For two charges:
U
q1q2
40r
System with dielectric boundaries
The dielectric is no more uniform:  varies, the Poisson equation becomes:
 X 
div   X grad  ( X )       X  ( X )  
0


If we can solve this equation, we have the potential, from which we can derive
most 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
(X) is the density of charges. For a biological system, it includes the charges
of the “solute” (biomolecules), and the charges of free ions in the solvent:
 ( X )   solute( X )  ions ( X )
The ions distribute themselves in the solvent according to the electrostatic
potential (Debye-Huckel theory):
ni ( X )
e
0
ni
 qi ( X )
kT
ni : number of ions of type i per unit volume
qi : charge on type i ion
N
ions ( X )   qi ni ( X )
i 1
The potential  is itself influenced by the redistribution of ion charges, so the
potential and concentrations must be solved for self consistency!
The Poisson Boltzmann Equation
 X  1
0
    X  ( X )  
  qi ni e
0
 0 i 1
N

qi ( X )
kT
Linearized form:
 X 
    X  ( X )  
  ( X ) 2 ( X ) ( X )
0
N
1
2
2
0 2
 
ni qi 
I

 0kT i 1
 0kT
I: ionic strength
Solving the Poisson Boltzmann Equation
• 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
Linear Poisson Boltzmann equation:
Numerical solution
• Space discretized into a
cubic lattice.
• Charges and potentials are
defined on grid points.
w
• Dielectric defined on grid lines
• Condition at each grid point:
P
j : indices of the six direct neighbors of i
6
i 
 ij j 
j 1
6

j 1
ij
qi
 0h
  h
2 2
ij ij
Solve as a large system of linear
equations
Meshes
•
•
Unstructured mesh have advantages over structured mesh
on boundary conformity and adaptivity
Smooth surface models for molecules are necessary for
unstructured mesh generation
Molecular Surface
Disadvantages
• Lack of smoothness
• Cannot be meshed with good quality
An example of the self-intersection of molecular surface
Molecular Skin
• 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
Skin Decomposition
Sphere patches
card(X) =1, 4
Hyperboloid patches
card(X) =2, 3
Building a skin mesh
Sample points
Join the points to form
a mesh of triangles
Building a skin mesh
A 2D illustration of skin surface meshing algorithm
Building a skin mesh
Full Delaunay of sampling points
Restricted Delaunay defining
the 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
Generalized Gauss Equation:
(X)
div(E(X)  P(X)) 
0
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
Generalized Poisson-Boltzmann Langevin Equation


2 2
2ion sinh ezr   po dip F1 (u) r 

 r    f r   

4
a 3 Dr 
a 3 Dr 





 4 po4 dip F1' (u)r   r    r 
a Dr u
3
2 2 po2 ion dip F1 (u) r  ez sinh ezr 
2

a 3 Dr 
2
2

2
Dr   1 2ion cosezr  dip

sinh po r 
po r 
1  sinh( u)  1 ucosh( u)  sinh( u) 
F1u 

 

2




u u
u
u
u

and


a 3 Dr 
with


 4 po4 2dip F1(u) r   r    r 
u  p0 E 
p0 E
kB T
