Computational biochemistry ferenc Bogár György Ferency
Download 2.94 Mb.
|
- Navigate this page:
- Non-bonded interactions
- Polarizability
- Chapter 3. Electrostatics in Molecules
- What is described here
Hooke’s law: It describe the deformations in bonds and angles by a harmonic oscillator equation. Force field: The force field includes the functions and the parameters of the functions to describe the deformation of the molecules. Bond deformations: The bond deformations are described as a harmonic function by the Hook’s law. Angle deformation: The angle deformations are described as a harmonic function. Non-bonded interactions: Electrostatic (multipole) interactions and van der Waals interactions. Coulomb function or multipole interaction functions are used int he previous, 12-6, 10-6 Lennard-Jones or Buckingham potential is applied for the calculations. Point charges: Most of the traditional force fields apply point charges. Atoms are not points, that is why RESP or CHELP method is used to estimate the point charges. Polarizability: Most of the atoms have polarizability. On the effect of charges induced charges can be formed. Multipole interactions: Interactions of point-charges-point charges, dipole-point charges, dipole-dipole, point charges-quadrupole, dipole-quadrupole, etc. interaction are multipole interactions. Chapter 3. Electrostatics in Molecules (Tamás Körtvélyesi)
1. Introduction The atoms in molecules have partial charges, which direct the association of these molecules and the solvation of the molecules in different solvents. The counter ions (ionic strength) in the solution have also effect on the interactions. The solution of PBE and/or TKE make possible to obtain the solvation free energy, the association energy with ligands and the association of peptides/proteins with metal (gold, silver, etc.) surfaces in the nano scale with modeling of the field of solvents.The method is simpler than the all atom models with explicit water molecules (see Chapter 4), but it does not contain some real effects in the solution (e.g. viscosity, the real interactions between the solvent molecules and the solute, etc.). The method is suitable for calculating the potential around an extended structure, too. The solution includes grid calculations with fixed grid space, grid space for refinement or multigrid space. 2. Coulomb Equation The electrostatic pair potential (φi(r)) around a point charge (qi) in a homogeneous medium with an ε effective dielectric constant (ε = εr εo,where εr is the relative dielectric constant and εo is the dielectric constant in vacuo) is
where εo = 8.854 x 10-12 C2 N-1 m-2. The location of the charge i is ri. The sign is negative at negative charge and positive at positive charge. The electrostatic potential in a system with N point charges
The total electrostatic interactions in a protein [1] used the pairwise Coulomb’s law in a system with homogeneous medium consisting of N point charges can be written as
ΔGel is the electrostatic interaction energy (free energy) at room temperature relative to the energy between the point charges at infinite distances, εr is the relative dielectric constant. The sign is negative at the interaction of different charges (attractive) and positive at the same point charges (repulsive interactions) [2]. The interaction energies calculated by the Coulomb’s law are pairwise interaction energies without considering the many-body interactions (see e.g. Axelrod-Teller’s formula [3]). These formulas are not totally valid for extended structures with charges.. 3. Poisson Equation The electrostatic potential φ(r) in vacuo for an extended structure with arbitrary shape can be given by the Poisson Eq. 3.4.
where ρ(r) is the charge density as a function of position (r is the Cartesian coordinates of a point in space, εo is the effective dielectric constant in vacuo. Nabla is defined by Eq. 3.5 as a vector operator
and
In spherical coordinates the Laplacian operator is given by Eq. 3.7
and in cylindrical coordinates the operator is
The electrostatic potential φ(r) in a uniform dielectric medium with a relative dielectric constant εr (ε = εo εr)
where ρ(r) is the charge density and ε = εo εr is the dielectric constant in a uniform media. The electrostatic potential φ(r) is
The integral is over the space. The electric field is given by Eq. 3.11 [4]
The Poisson equation becomes in a uniform media
If the effective dielectric constant, ε depends on the coordinates (ε(r) in a non-uniform media), Eq. 3.9 equation is modified to Eq. 3.13 as the general form of the Poisson equation.:
The solute of the equation is with a low relative dielectric constant of εr of about 2 to 4 (e.g. in protein, where protein can be considered as a dielectric) Eq. 3.13 can be used. In some cases εr was suggested to be of about 20. We can not say anything exactly on εr, the structure of proteins are different and that is why εr, can be also different. These values are only approximations. In a water solution εr is ca. 80 [5]. The relative dielectric constants are handled in water (protein in water) as it can be seen in Figure 3.1. The analytical solution is available only for spherical, cylindrical or planar structures . The spherical model of an ion can be seen in Figure 3.2. The relative dielectric constant of the media is εr. Figure 3.1. A shematic figure on the protein in water with the εr in the solvent and in the solute. The spherical cylindtical model of an ion can be seen in Figure 3.2. 
Figure 3.2. The spherical model of the ions. The analytical solution of the model for sphericaln cylindrical, or planar symmetry is given in Eq. 3.14 - 3.15.
The random positions of the ions around the large molecules can be described by the Boltzmann distribution.
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:
|
