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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. W sol. +. +. W np. A PoissonBoltzmann view of Electrostatics. Elementary Electrostatics in vacuo. Gauss’s law:
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The Geometry of Biomolecular Solvation2. Electrostatics
Patrice Koehl
Computer Science and Genome Center
http://www.cs.ucdavis.edu/~koehl/
Solvation Free Energy
Wsol
+
+
Wnp
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, r(X)=qd(XX0)
 Coulomb’s law for a charge can be retrieved from Gauss’s law
Poisson equation:
Laplace equation:
(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:
c is the electric susceptibility, and e is the electric permittivity, or dielectric constant
The field from a uniform dipole density is 4pP, therefore the total field is
Uniform Dielectric Medium
Modified Poisson equation:
Energies are scaled by the same factor. For two charges:
System with dielectric boundaries
The dielectric is no more uniform: e varies, the Poisson equation becomes:
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
r(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:
The ions distribute themselves in the solvent according to the electrostatic
potential (DebyeHuckel theory):
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
Linearized form:
I: ionic strength
Solving the Poisson Boltzmann Equation
Linear Poisson Boltzmann equation:
Numerical solution
ew
eP
j : indices of the six direct neighbors of i
Solve as a large system of linear
equations
Meshes
Molecular Surface
Disadvantages
An example of the selfintersection of molecular surface
Molecular Skin
Comparison between the molecular surface model and the skin model for a protein
Molecular Skin
Skin Decomposition
Hyperboloid patches
Sphere patches
card(X) =1, 4
card(X) =2, 3
Building a skin mesh
Join the points to form a mesh of triangles
Sample points
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
Modified Poisson Boltzmann Equations
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
Generalized PoissonBoltzmann Langevin Equation
with
and