Reduction Potential Calculations

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Introduction
The standard reduction potential Eo for the reduction of a molecule A,

$$A+e^- -> A^-$$

is related to the standard free energy of reduction &Delta;G&deg; through the Nernst equation,

$$-nFE^{\circ} = {\Delta}G^{\circ} $$

where n is the number of electrons transferred and F is Faraday’s constant. &Delta;G&deg; can be further decomposed as

$${\Delta}G^{\circ} = {\Delta}G_{in} + {\Delta}G_{out}  + {\Delta}G_{SHE}  $$

where &Delta;Gin is the inner sphere contribution, &Delta;Gout is the outer sphere contribution, and &Delta;GSHE/F = 4.43 V is the absolute electrode potential for the standard hydrogen electrode (SHE). In a protein, the inner sphere can be defined as the redox site only, i.e., the metal plus its direct ligands, while the outer sphere includes the rest of the protein and solvent.

&Delta;G&deg; can be calculated from a thermodynamic cycle for the reduction of a protein (Figure 1). &Delta;Gin can be calculated as the change in energy of the redox site in the oxidized versus reduced states. Quantum mechanical electronic structure calculations are required for calculations of the inner sphere energies and density functional theory (DFT) is generally accepted as an accurate and efficient method for treating transition metal complexes. Niu and Ichiye have determined ΔGin using broken symmetry (BS) DFT calculations for analogs of iron-sulfur protein redox sites in the gas phase. The calculated adiabatic detachment energy (ADE) was calibrated to the measured value for the same analogs, also in gas phase, by electrospray ionization-photoelectron spectroscopy (EI-PES). The best agreement between the calculation and experiment was obtained using B3LYP and double-zeta basis sets. Additionally, these DFT calculations confirmed that &Delta;Gin for the Fe-S clusters is relatively independent of environment, although the ligand dihedral conformation may tune &Delta;Gin by ~ 200 mV.

This allows ΔGout to be calculated as

$${\Delta}G_{out} = {\Delta}G_{sol} (A^{n-1}) - {\Delta}G_{sol}(A^{n}) $$

where ΔGsol(An) is the solvation energy for a redox site A with a net charge n. Here, the protein is considered as part of the solvent, so ΔGsol is calculated as the change in free energy between the redox site surrounded by protein and solvent and the redox site in vacuum. The redox site and protein are modeled atomistically with each atom given a radius and partial charge. X-ray crystal structures from the Protein Data Bank22 provide the atomic coordinates for the protein and redox site. The redox site is assigned a dielectric value of 1 (εc). The protein is assigned a dielectric (εp) between 1 and 10 to account for the dielectric relaxation.23 The solvent is modeled implicitly by a dielectric (εs) of 78, this is the dielectric constant of water (measured experimentally). From this model, the solvation energy, ΔGsol, is calculated as the change in free energy between the redox site partial charges, protein partial charges, εc, εp, and εs and the redox site partial charges in a vacuum. The change in free energy of the protein upon oxidation, ΔGout, can be calculated as the change ΔGsol between the reduced and oxidized redox site.

Introduction
Continuum electrostatics models parts of the system as a dielectric and other parts atomistically.

Solvation Free Energy
The solvation free energy is the energy required to surround a system in water relative to the system in vacuum. In continuum electrostatic calculations, the water is approximated as a dielectric.

Solvation Energy of a Proteins
For a protein, the protein is modeled atomistically, with each atom having a radius and charge. The radii are used to determine the solvent accessible surface area (the region at which protein volume transitions to water). The solvation free energy is then the energy to dunk the atomic partial charges in the water. Two calculations are done, one for charges in vacuum and one for charges in a dielectric of water. The difference in energy is the solvation free energy of the protein.

Binding Energy
The binding free energy of a molecule to a protein can also be calculated using continuum electrostatics. The binding free energy is the change in solvation energy between the protein and ligand separately and the solvation energy of the protein and ligand complex.

Poisson-Boltzmann Equation
Poisson-Boltzmann equation is a method for calculating the solvation energy of a system.

Calculation Parameters
Finite-difference methods for solving the Poisson-Boltzmann equation involve extrapolating the system to a grid and solving for the discretized system.

Atomic Partial Charges and Radii
Each atom is given a partial atomic charge and a radii. The sum of the partial atomic charges for a molecule should sum to a net integer. If not, you did something wrong.

Dielectric Constants
The polarizability of a system is modeled by dielectric constant. The dielectric constant for water is 78 or 80 and the dielectric constant for a protein is anywhere between 2 and 20. This is pretty controversial.

Grid Definition
The grid is defined as a set of points in the x, y, and z direction. One can define the grid by defining two the these three parameters: number of grid point in each of the x, y, and z direction, grid spacing between grid points in each direction, and length of the calculation grid. The center of the grid must also be defined.

Charge Extrapolation to Grid
There are several methods in which charges can be placed onto the grid.

PBEQ stream apbs_radii.str APBS mgmanual lpbe rdiel dimx 40.0 dimy 40.0 dimz 40.0 - grdx 0.4 grdy 0.4 grdz 0.4 - cmet 0  cntx 0.0 cnty 0.0 cntz 0.0 - bcfl 2 pdie 2 sdie 78 srfm 1 chgm 1 - sdens 10 srad 1.4 swin 0.3 temp 289.15 - calcene 1 sele all end END