Computational biochemistry ferenc Bogár György Ferency
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- 5. Poisson-Boltzmann Equation (PBE)
- 5.1. Linearized Poisson-Boltzman Equation (LPBE)
| 4. Boltzmann Distribution
The charge density (ρion(r)) is the sum of the charges in solutes and the mobile ions (Na+, Cl-, K+, Ca2+, etc.) in the solvent. The mobile ions in the solvent are handled to be uniform and the the Boltzmann distribution is used for the ion distribution:
where e is the charge of the electron, zi and ci are the charge number and the concentration of the ions i in the bulk solution, respectively [1,4]. kB is the Boltzmann constant. m is the number of the mobile ion species in the solution. The charge density of the solute can be determined as charges with fixed positions („source charges”) (see Lit.[5]):
δ(x) is the delta function. δ(x-y)=0 if x≠y and δ(x-y)=1 if x=y. M is the number of charges on biomolecules. In a one-to-one electrolyte (one positive and one negative ion, e.g. Na+ and Cl- ions) the Eq. 3.17 is simplified to
κ’(r) is the modified Debye-Hückel parameter is defined by Eq. 3.19. κ is the Debye-Hückel inverse length.
where NA is the Avogadro number, e is the electric charge, kB is the Boltzmann constant, T is the temperature, I is the ionic strength of the bulk solution
c(r) is the concentration of the ions in the molecules with the fixed charges [6]. ci and zi are the concentration and the charge of the ions in the bulky solution, respectively. These expressions make possible to find the dependence between electrostatic potential (φ(r)) and the the charge density (ρ(r)). 5. Poisson-Boltzmann Equation (PBE) The nonlinear form of the PBE [5-10] (NLBE) is a second order nonlinear elliptic partial differential equation . Analytical solution is available only for spheres and cylinders. For biomolecules, as proteins, DNA only numerical solution is possible. The charge density („source charges”) of the solute is given by the Eq. 3.21.
It means that the electrostatic interactions between charges in biological systems depends on the ionic strength and the pH of the medium, too (see in Chapter 6). The first theoretical studies were published almost a century ago. The analytic solution of Eq. 3.21. is available only for simple geometric objects. For complex systems it is possible to solve it by iterative finite difference methods (see later). 5.1. Linearized Poisson-Boltzman Equation (LPBE)
Considering sinh φ(r) ~ φ(r) (see Eq. 3.21), the linearized form of PBE (LPBE) can be obtained [11],
Both the PBE and LPBE equations are determined by εr(r), c(r) and the positions of the atoms in molecules (q) [5]. The model of an ion pair can be seen on Figure 3.3. 
Figure 3.3. The model of the electrostatic calculation in an extended structure (e.g. in a protein) One model can be described [6]:
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