Computational biochemistry ferenc Bogár György Ferency
Download 2.94 Mb.
|
- Navigate this page:
- 5.2. Tanford-Kirkwood Equation (TKE)
- 6. Molecular Surface and Volume
- 7. Numerical Solution of non-linear Poisson-Boltzmann Equation (NPBE), Linear Poisson-Boltzmann Equation (LPBE) and Tanford-Kirkwood Equation (TKE)
Region 1: Inside the molecule. εr(r) is 2 to 20., c(r) is set to 0 (no ions) and qi positioned by the atomic coordinates. Region 2: Stern region. εr(r) is set to the value in bulky solvent (εr = 80). It is supposed it is a region without ions. c(r) is 0 (no ions). The source charge density is zero. Region 3: In bulk solvent. In the bulk solvent the relative dielectric constant is 80. c is 1. The source charge density is zero. The classical PBE does not include the possible difference in the size of ions. A modified PBE was developed which considers this difference („size modified PB (SMPB) equation”) [12]. The orientation and strong dipolar moments of water molecules is described by the „dipolar Poisson-Boltzmann (DPB)” model [13]. The hydration forces, ionic associations and short range hydrophobic effects are calculated by the combination of SMPB and DPB methods [6]. 5.2. Tanford-Kirkwood Equation (TKE) The Tanford-Kirkwood equation [17-20] is a separeted partial differential equiation of PBE in two media. A model for proteins with the ε1 relative dielectric constant in the molecule and ε2 in the bulky solvent can be seen in Figure 3.4. 
Figure 3.4. The model of the proteins in solution with counter ions The TK equation can be seen in Eq. 3.27 and 3_28.
κ2 is given by Eq. 3.19. κ2~ βI. 6. Molecular Surface and Volume In the calculation the surface and/or volume of the molecule is necessarry. The different surface and volume types are depicted on Figure 3.5. The radius of the probe sphere is 1 to 1.4 Ǻ in water as solvent. The difference between the van der Waals surface, molecular surface and the solvent accessible surface (SAS) can be seen. The no-entrant surface is the part of the surface where the probe sphere does not reach the atoms. The fast calculation of the surface and volume is basic in the calculations. The grid calculation and the potential on the (van der Waals) surface is calculated by sophisticated methods [1]. The grid geometry and the interpolations of the potentials between the grid points (2D or 3D grids – trigonal, tetrahedral, tetraeder, …) to smooth the difference in dielectric constants) are also very important (see e.g. Lit. [6]). The grid spacing for calculating solvation energy is 0.2-0.3 Ǻ in UHBD. Int he active centre the grid spacing can be refined. 
Figure 3.5. The molecular surface, van der Waals surface and the surface accessible surface (SAS). The probe sphere is the model of solvent. 7. Numerical Solution of non-linear Poisson-Boltzmann Equation (NPBE), Linear Poisson-Boltzmann Equation (LPBE) and Tanford-Kirkwood Equation (TKE) The application of the results of PBE and LPBE is important to know (i) the electrostatic potential ont he surface of a biomolecule, (ii) the electrostatic potential outside the molecule, (iii) calculation of the free energy of a biomolecule and (iv) calculation of the electrostatic field to give the mean forces [18]. The analytical solution of PBE for real molecules as proteins, DNA, PNA, etc. are not available. Only numerical methods can give solutions. Several program packages for the solution of PBE were developed: DELPHI [19], UHBD (University of Houston Brownian Dynamics) [20], APBS (Adaptive Poisson-Boltzmann Solver) [21], MEAD [22], ZAP [23]. Two main methods are developed: (i) surface based methods and (ii) volume based methods [5]. 
Figure 3.6. Grids for the solution of PBE with the +1 charge probe (PDB Id.: 1yet without ligand: geldanomycin and without structural water) Solution of PBE on a surface mesh (BEM). The molecules („interior”) and the space around the molecule is handled separately. The electrostatics of the interior is solved by the Poisson equation. The outside part is solved by the PB equation. The molecular surface is by the method of polygonal approach, triangular mesh (the discretization is in 2D). The interface between the two regions is handled by a continuous displacement field. The BEM method is faster than the volumetric methods. That is why the two methods are combined. FD and FEM approaches in the volumetric mesh. The discretization is in 3D space. The application of the results of PBE and LPBE is important to know (i) the electrostatic potential on the surface of a biomolecule, (ii) the electrostatic potential outside the molecule, (iii) calculation of the free energy of a biomolecule and (iv) calculation of the electrostatic field to give the potential energy mean forces (PMF) [18], which describes how the free energy changes along a coordinate. The electrostatics of the protein in the solvent is calculated by an iterative finite-difference approach with mapping on a cubic lattice with parameters ρ(r), κ(r) and ε(r). DelPhi [19]: With the Cartesian coordinates it calculates the electrostatic potential from the known geometry and the known charge distribution by using finite difference method in the solution of LPBE and full NPBE for proteins with arbitrary shape and charge considering the ionic strength. It considers ionic strength of the media. It can be used for extremly highly dimensions. Directory: tartalom -> tamop412A tamop412A -> Agricultural Economics II. Popp, József Agricultural Economics II tamop412A -> Ethology practical Vilmos Altbäcker Márta Gácsi András Kosztolányi Ákos Pogány Gabriella Lakatos tamop412A -> Operating Systems Lecture Notes Attila Dr. Adamkó Operating Systems Lecture Notes tamop412A -> Agroinformatics Herdon, Miklós Agroinformatics tamop412A -> Paternoster of Programmers Reloaded tamop412A -> Lőrincz, András Mészáros, Tamás Pataki, Béla Embedded Intelligent Systems tamop412A -> This course is realized as a part of the TÁmop 1 A/1-11/1-2011-0038 project tamop412A -> Fundamentals of geology I. (lithosphere) 1 1. The formation of the Earth 1 Download 2.94 Mb. Share with your friends:
|
