The Geometry of Biomolecular Solvation 2. Electrostatics

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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 Poisson-Boltzmann 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/

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:

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

Elementary Electrostatics in vacuo

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 (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!

The Poisson Boltzmann Equation

Linearized form:

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.
• 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

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

• 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

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

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:

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