## Thursday, July 27, 2006

### Electrostatic interactions

The high dielectric coefficient of water, together with the tendency of diffusible ions to move toward biopolymer charges of opposite sign, reduces the effective interactions amonr the biopolymer charges (solvent-screened interactions).

Electrostatic-steering effects can lead to increases in the rate constant of association by two orders of magnitude.

The thermodynamics and kinetics of protein-protein association and larger-scale supramolecular assembly can be analyzed and predicted in many cases with the aid os electrostatic calculations, supplemented in the kinetics area by simulation of the diffusional motion of proteins.

Poisson-Boltzmann Theory

Canonical expression:

$- \nabla \cdot \epsilon (\mathbf{x}) \nabla \varphi(\mathbf{x}) = \rho (\mathbf{x})$

dielectric coefficient (2-20 for molecules inside the solute and smooth evolution to 80 in the solvent ), electrostatic potential and charge distribution (a collection of Dirac delta functions that model the $N_f$ atomic partial charges of the solute.

Poisson-Boltzmann equation (PBE): a variant where mobile counterion charges are introduced to the charge distribution in a mean field fashion: $\rho(x) = {\rho_f } (x) + {\rho_m} (x)$. There exists a linearized PBE.

Energies

The free energy is a function of the electrostatic potential as well as the atomic positions, charges, and radii. The calculated energies (protonation, binding and solvation energies) are combined and converted to a $pK_a$ value.

Forces

The dynamic trajectory of a solute is calculated without the inclusion of the numerous explicit solvent molecules required for traditional MD simulations. The solvent effects are modeled by stochastic forces applied to the biomelecule.

Numerical solutions of the PBE

Software: Delphi, APBS, MEAD, UHBD, MAcroDox, AMBER, CHARMM.

Finite Difference Discretization: cartesian meshes, but it does not provide a way to locally increase the accuracy of the solution in a specific region without increasing the number of unknowns across the entire grid.

Adaptive Finite Element Discretization: offer the ability to place computational effort in specific regions of the problem domain.

Multilevel solvers: iterative method until the desired accuracy is reached by projecting the discretized system onto meshes (or grids) at multiple resolutions.

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