1 / 27

# The Geometry of Biomolecular Solvation 2. Electrostatics - PowerPoint PPT Presentation

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 Poisson-Boltzmann view of Electrostatics. Elementary Electrostatics in vacuo. Gauss’s law:

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

## PowerPoint Slideshow about 'The Geometry of Biomolecular Solvation 2. Electrostatics' - bliss

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### The Geometry of Biomolecular Solvation2. Electrostatics

Patrice Koehl

Computer Science and Genome Center

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

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(X-X0)

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

Poisson equation:

Laplace equation:

(charge density = 0)

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.

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

Modified Poisson equation:

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

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!!

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 (Debye-Huckel theory):

The potential  is itself influenced by the redistribution of ion charges, so the

potential and concentrations must be solved for self consistency!

Linearized form:

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

Numerical solution

• Space discretized into a

• cubic lattice.

• Charges and potentials are

• defined on grid points.

• Dielectric defined on grid lines

• Condition at each grid point:

ew

eP

j : indices of the six direct neighbors of i

Solve as a large system of linear

equations

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

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

Disadvantages

• Lack of smoothness

• Cannot be meshed with good quality

An example of the self-intersection of 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

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

Hyperboloid patches

Sphere patches

card(X) =1, 4

card(X) =2, 3

Join the points to form a mesh of triangles

Sample points

A 2D illustration of skin surface meshing algorithm

Full Delaunay of sampling points

Restricted Delaunay defining

the skin surface mesh

Triangle quality distribution

Skin mesh

Volumetric mesh

• 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

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