The Geometry of Biomolecular Solvation 2. Electrostatics Patrice Koehl Computer Science and Genome Center

+ + Solvation Free Energy W np W sol

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: Notes: - for a point charge q at position X 0,  (X)=q  (X-X 0 ) - 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:  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

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:  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  (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

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 PP ww Space discretized into a cubic lattice. Charges and potentials are defined on grid points. Dielectric defined on grid lines Condition at each grid point: 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 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 Mixed complex Computing the skin

Skin Decomposition Sphere patches Hyperboloid patches card(X) =1, 4card(X) =2, 3

Building a skin mesh Sample points Join the points to form a mesh of triangles

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

Delaunay Refinement Insert the circumcenter of the skinny tetrahedron iteratively Volumetric Meshing

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 p o Also assume that both ions and dipoles have a fixed size a

with Generalized Poisson-Boltzmann Langevin Equation and