The Poisson–Boltzmann equation is a differential equation that describes electrostatic interactions between molecules in ionic solutions. It is the mathematical base for the Gouy–Chapman double layer (interfacial) theory; first proposed by Gouy in 1910 and complemented by Chapman in 1913. The equation is important in the fields of molecular dynamics and biophysics because it can be used in modeling implicit solvation, an approximation of the effects of solvent on the structures and interactions of proteins, DNA, RNA, and other molecules in solutions of different ionic strength. It is often difficult to solve the Poisson–Boltzmann equation for complex systems, but several computer programs have been created to solve it numerically.
The equation can be written as (in cgs):
- is the divergence operator,
- represents the position-dependent dielectric constant,
- represents the gradient of the electrostatic potential,
- represents the charge density of the solute,
- represents the concentration of the ion at a distance of infinity from the solute,
- is the charge of the ion, is the charge of a proton,
- is the Boltzmann constant,
- is the temperature,
- is a factor for the position-dependent accessibility of position to the ions in solution.
- DelPhi: a canonical finite difference Poisson–Boltzmann solver for proteins, now distributed as free software.
- Fogolari F, Brigo A, Molinari H. (2002). The Poisson–Boltzmann equation for biomolecular electrostatics: a tool for structural biology. J Mol Recognit 15(6):377–392. (See this paper for derivation.)
- G.L. Gouy, j. de phys 9, 457 (1910)
- D.L. Chapman, Philos. Mag. 25, 475 (1913)
|Wikiversity has learning materials about Poisson–Boltzmann profile for an ion channel|
- Adaptive Poisson–Boltzmann Solver - A free, open-source Poisson-Boltzmann electrostatics and biomolecular solvation software package.
- Zap - A Poisson–Boltzmann electrostatics solver.
- MIBPB Matched Interface & Boundary based Poisson–Boltzmann solver
- CHARMM-GUI: PBEQ Solver
- AFMPB Adaptive Fast Multipole Poisson–Boltzmann Solver, free and open-source.