Poisson-Boltzmann equation
Encyclopedia
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):
or (in SI units or mks
):
where
is the divergence
operator,
represents the position-dependent dielectric, 
represents the gradient
of the electrostatic potential,
represents the charge density of the solute, 
represents the concentration of the ion i at a distance of infinity from the solute, 
is the charge of the ion, q is the charge of a proton, 
is the Boltzmann constant, T is the temperature
, and
is a factor for the position-dependent accessibility of position r to the ions in solution. If the potential is not large compared to kT, the equation can be linearized
to be solved more efficiently, leading to the Debye–Hückel equation.
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
that describes electrostatic interactions between molecules in ionic solutions. It is the mathematical base for the Gouy–Chapman double layer (interfacial)
Double layer (interfacial)
A double layer is a structure that appears on the surface of an object when it is placed into a liquid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The DL refers to two parallel layers of charge surrounding the object...
theory; first proposed by Gouy
Louis Georges Gouy
Louis Georges Gouy was a French physicist who was born at Vals-les-Bains, Ardèche in 1854 and died January 27 1926. He is the namesake of the Gouy balance, the Gouy-Chapman electric double layer model and the ....
in 1910 and complemented by Chapman in 1913. The equation is important in the fields of molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
and biophysics
Biophysics
Biophysics is an interdisciplinary science that uses the methods of physical science to study biological systems. Studies included under the branches of biophysics span all levels of biological organization, from the molecular scale to whole organisms and ecosystems...
because it can be used in modeling implicit solvation
Implicit solvation
Implicit solvation is a method of representing solvent as a continuous medium instead of individual “explicit” solvent molecules most often used in molecular dynamics simulations and in other applications of molecular mechanics...
, an approximation of the effects of solvent
Solvent
A solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...
on the structures and interactions of proteins, DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...
, RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....
, and other molecules in solutions of different ionic strength
Ionic strength
The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such as the dissociation or the solubility of different salts...
. It is often difficult to solve the Poisson–Boltzmann equation for complex systems, but several computer programs have been created to solve it numerically
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
.
The equation can be written as (in cgs):
or (in SI units or mks
Mks system of units
The MKS system of units is a physical system of units that expresses any given measurement using fundamental units of the metre, kilogram, and/or second ....
):
where
is the divergenceDivergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
operator,
represents the position-dependent dielectric, represents the gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of the electrostatic potential,
represents the charge density of the solute, represents the concentration of the ion i at a distance of infinity from the solute, is the charge of the ion, q is the charge of a proton, is the Boltzmann constant, T is the temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, and
is a factor for the position-dependent accessibility of position r to the ions in solution. If the potential is not large compared to kT, the equation can be linearizedLinearization
In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
to be solved more efficiently, leading to the Debye–Hückel equation.
See also
- DelPhiDelPhi (software)DelPhi is a scientific application which calculates electrostatic potential and energies in systems of biomolecules and geometrical objects. One of the main problems in modeling the electrostatic potential of biological macromolecules is that they exist in water at a given ionic strength and that...
: a canonical finite difference Poisson–Boltzmann solver for protein, now distributed as free software
External links
- Adaptive Poisson–Boltzmann Solver
- Zap - A Poisson–Boltzmann electrostatics solver.
- MIBPB Matched Interface & Boundary based Poisson–Boltzmann solver
- CHARMM-GUI: PBEQ Solver