The Geometry ofBiomolecular Solvation
2. Electrostatics
Patrice KoehlComputer Science and Genome Centerhttp://www.cs.ucdavis.edu/~koehl/
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Solvation Free Energy
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A Poisson-Boltzmann view of Electrostatics
Elementary Electrostatics in vacuo
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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, ρ(X)=qδ(X-X0)
- Coulomb’s law for a charge can be retrieved from Gauss’s law
Elementary Electrostatics in vacuo
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Poisson equation:
Laplace equation:
02=! " (charge density = 0)
+-
Uniform Dielectric MediumPhysical basis of dielectric screening
An atom or molecule in an externally imposed electric field develops a nonzero net dipole moment:
(The magnitude of a dipole is a measure of charge separation)
The field generated by these induced dipoles runs against the inducingfield the overall field is weakened (Screening effect)
The negativecharge is screened bya shell of positivecharges.
Uniform Dielectric MediumPolarization:
The dipole moment per unit volume is a vector field known asthe polarization vector P(X).
In many materials: )(4
1)()( XEXEXP
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χ is the electric susceptibility, and ε is the electric permittivity, or dielectric constant
The field from a uniform dipole density is -4πP, therefore the total field is
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Uniform Dielectric Medium
Modified Poisson equation:
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Energies are scaled by the same factor. For two charges:
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System with dielectric boundaries
The dielectric is no more uniform: ε varies, the Poisson equation becomes:
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If we can solve this equation, we have the potential, from which we can derivemost 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
ρ(X) is the density of charges. For a biological system, it includes the chargesof 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):
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ni : number of ions of type i per unit volume
qi : charge on type i ionkT
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The potential φ is itself influenced by the redistribution of ion charges, so thepotential and concentrations must be solved for self consistency!
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The Poisson Boltzmann Equation
Linearized form:
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I: ionic strength
• Analytical solution
• Only available for a few special simplification of the molecularshape 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 finitedifference and finite element method
• Mesh size and quality determine the speed and accuracy of theapproximation
Solving the Poisson Boltzmann Equation
Linear Poisson Boltzmann equation:Numerical solution
εP
εw
• Space discretized into a cubic lattice.
• Charges and potentials are defined on grid points.
• Dielectric defined on grid lines
• Condition at each grid point:
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j : indices of the six direct neighbors of i
Solve as a large system of linearequations
• Unstructured mesh have advantages over structured meshon boundary conformity and adaptivity
• Smooth surface models for molecules are necessary forunstructured 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 molecularsurface but uses hyperboloids blend betweenthe 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 boundaryof 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, 4 card(X) =2, 3
Building a skin mesh
Sample pointsJoin the points to forma mesh of triangles
A 2D illustration of skin surface meshing algorithm
Building a skin mesh
Building a skin mesh
Full Delaunay of sampling points Restricted Delaunay definingthe skin surface mesh
Mesh Quality
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
!
div(E(X ) +r P (X)) =
"(X )
#0
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
with
Generalized Poisson-Boltzmann Langevin Equation
and
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r E =
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r E
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