This paper is the rst of two papers on the adaptive multilevel nite element treatment of the nonlinear poissonboltzmann equation pbe, a nonlinear elliptic equation arising in biomolecular modeling. The poissonboltzmann equation is widely used for modeling the electrostatics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. The poissonboltzmann equation pbe is one of the most popular implicit solvent models which describes the solvent in a continuum model through the boltzmann distribution. The validity of the poissonboltzmann pb equation is reconsidered on the basis of functional expansion techniques supplemented by the mean spherical approximation.
If the charge density is zero, then laplaces equation results. The journal of physical chemistry b 2019, 123 5, 957973. A quasilinear poisson boltzmann model based on a simple experimental result two numerical methods to solve the nonlinear equation 3d simulations in applications of electrostatic analysis for biomolecules future work. Nonlinear poissonboltzmannequation for a zzelectrolyte with concentration c0 the poissonboltzmann equation has an explicit solution.
Minimizers and bounds i pb does not predict likecharge attraction i references. A solution of the twodimensional modi ed poissonboltzmann. Ren university of waterloo, waterloo, ontario n2l 3g1, canada doi. Cylindrical poisson boltzmann equation, tanh method, ricatti functions. Fragment molecular orbital calculations with implicit solvent based on the poissonboltzmann equation. The poissonboltzmann equation is a nonlinear partial di. In this work we consider these solutions for the membrane of. The results have been compared with those obtained fromthe linearized equation. Because of the spherical symmetry of simple ions, the various functions of x above are in fact. A solution of the cylindrical poissonboltzmann equation. Eliminating by substitution, we have a form of the poisson equation. The proposed pbnp model can be used for the prediction of ion transport of membrane proteins subject to applied external electric fields.
Sep 30, 2003 chapter 4 is a thorough discussion of the theoretical underpinnings of the widely. This work presents the solution of the electrostatic potential for the cylindrical poissonboltzmann equation, 23, applying the tanh method. Parallelized successive over relaxation sor method and its. The tutorial is divided into four parts, the first of which is a brief history of the pb equation and its derivation. Poissonboltzmann equation an overview sciencedirect topics. The poissonboltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. If the charge density follows a boltzmann distribution, then the poissonboltzmann equation results.
Solving poisson s equation for the potential requires knowing the charge density distribution. It cannot be used as a local minimization principle. As a result, we obtain coupled poissonboltzmannnernstplanck equations from the variational principle. Numerical solution of nonlinear poisson boltzmann equation 1. Greens function method, pka calculation, and poissonboltzmann equation jingzhen hu 1 introduction this project is motivated by interest in computing the acid dissociation rate pka at an amino acid titration site. The galerkin method is a viable approach, due its relative simplicity, strong theoretical base, and proven results in practice. The default choice for the dielectric boundary has been the molecular surface, but the use of.
To simplify the poissonboltzmann equation, gc theory makes a few assumptions. The galerkin method is a viable approach, due its relative simplicity, strong theoretical base, and proven results in. If the charge density follows a boltzmann distribution, then the poisson boltzmann equation results. For this particular case, our numerical solution of the poissonboltzmann equation can be compared to the analytical onedimensional gouychapman solution for a monovalent and symmetric salt. This way, the width of the saturated layer depends also on the surface charge and is not limited a priori to only one counterion layer 41. In an ideal situation, this is a sharp boundary located at z 0 which limits the ionic solution to the half space z0. A multigrid method is presented for the numerical solution of the linearized poissonboltzmann equation arising in molecular biophysics. Modi ed poisson boltzman equation, tanh method 1 introduction when in a electrolyte solution is incorporated the nite size of particles, an entropic e ect, called excluded volume, we get the modi ed poisson boltzmann equation 14. In an ideal situation, this is a sharp boundary located. Jul 28, 2008 the validity of the poissonboltzmann pb equation is reconsidered on the basis of functional expansion techniques supplemented by the mean spherical approximation. The poissonboltzmann equation 61 is derived from two components. A weighted adaptive leastsquares finite element method. The poisson boltzmann equation pbe is one of the most popular implicit solvent models which describes the solvent in a continuum model through the boltzmann distribution.
In the search to nd new solutions to nonlinear partial di er. A weighted adaptive leastsquares finite element method for. The pbe solves the electrostatic potential in the entire domain which comprises both the molecule and the solvent. To motivate the work, we provide a thorough discussion of the poissonboltzmann equation, including derivation from a few basic assumptions, discussions of special case solutions, as well as common analytical approximation techniques. Poissonboltzmann equation the poissonboltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in. A minimizing principle for the poissonboltzmann equation. Multigrid solution of the poissonboltzmann equation.
Normally, the differences between these protonation states are modeled by changing the charges on a few atoms. A quasilinear poissonboltzmann model based on a simple experimental result two numerical methods to solve the nonlinear equation 3d simulations in applications of electrostatic analysis for biomolecules future work. In southern methodist university i worked under the instruction of professor weihua geng on the topic of computing pka. Fragment molecular orbital calculations with implicit solvent.
To simplify the poisson boltzmann equation, gc theory makes a few assumptions. Analytical solutions of the poissonboltzmann equation within. The spatial dependence of gas properties is sufficiently slow distribution function is constant over the interaction region 4. In this paper the cylindrical poissonboltzmann equation in reduced coordinates is transformed into an algebraically nonlinear second order ordinary differential equation, which is a particular case of painleves third equation. The relation between potential and charge density of the diffuse layer is given by m d d 0 12,0 0 0 0 8 sinh. Greens function method, pka calculation, and poisson. I the potential is the unique solution of the boundaryvalue problem of the implicit poissonboltzmann equation r 0r b0. In mathematics, poissons equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. A multigrid method is presented for the numerical solution of the linearized poisson boltzmann equation arising in molecular biophysics. Numerical study of the poissonboltzmann equation for. Chapter 2 poissons equation university of cambridge. The poisson boltzmann problem in spherical symmetry has been considered using the distribution of a selfconsistent potential around a charged grain in a thermal collisional plasma as an example.
The equation is discretized with the nite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for twodimensional interface. From this potential, further information can be obtained. Poissonboltzmann equation is derived where the steric forces lead to saturation of the ion density at high potentials. Wuanalytical solutions of nonlinear poisson boltzmann equation for colloidal particles immersed in a general electrolyte solution by homotopy perturbation technique colloid polym. The poissonboltzmann equation is widely used to treat this electrostatic effect in an ionic solution. We find that this model greatly amplifies the steric effects predicted by the usual modified poissonboltzmann equation, which imposes only a restriction on. Poissonboltzmann description of the electrical double layer. The poissonboltzmann equation and its application to.
The poisson boltzmann equation describes the electrostatic potential of a biomolecular system in a. In this work, a simple mixed discretecontinuum model is considered and boundary element method is used to solve for the solution. Keywords boundary element method, biomolecular electrostatics, poissonboltzmann equation. In mathematics, poisson s equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. This work presents the solution of the electrostatic potential for the cylindrical poisson boltzmann equation, 23, applying the tanh method. Cylindrical poissonboltzmann equation, tanh method, ricatti functions. In the application of greatest interest a strong coulomb potential originating in an external source, such as a polyelectrolyte molecule, acts on a salt solution of small mobile. This is the eulerlagrange equation of the convex functional j. Poissonboltzmann lpb equation, 22xx x1, 6 fkf er f wherek1. May 21, 2011 as a result, we obtain coupled poissonboltzmannnernstplanck equations from the variational principle. Poissonboltzmann equation for microfluidic transport phenomena with statistical thermodynamics approach p.
Collisions can be thought of as being instantaneous. For low electrostatic potentials less than 25 mv, the pb equation can be linearized and yields the debyehuckel theory 3. Solving poissons equation for the potential requires knowing the charge density distribution. Abstract theunidimensional poissonboltzmann equation for a 1. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution.
The poisson boltzmann equation is widely used for modeling the electrostatics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. Combining the poisson equation and the boltzmann distribution applied to ion concentrations yields the poisson boltzmann equation. Combining the poisson equation and the boltzmann distribution applied to ion concentrations yields the poissonboltzmann equation. The electrical double layer is examined using a generalized poissonboltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. The poissonboltzmann equation article pdf available in european journal of physics 395 march 2018 with 88 reads how we measure reads. Nonlinear poisson boltzmann equation for a zzelectrolyte with concentration c0 the poisson boltzmann equation has an explicit solution. This model is further reduced to the poissonboltzmann equation when the external voltage is. The poissonboltzmann equation i background i the pb equation. However, the functional has the defect of being nonconvex. With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of poissonboltzmann. The poissonboltzmann equation describes the electrostatic potential of a biomolecular system in a.
The poissonboltzmann problem in spherical symmetry has been considered using the distribution of a selfconsistent potential around a charged grain in. Numerical solution of nonlinear poisson boltzmann equation. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. Fast solution of the linearized poissonboltzmann equation. Pdf nonlinear poissonboltzmann equation in spherical symmetry. In the linear case, we implement these algorithms on real size biomolecules, the geometrical complexity being handled thanks to e cient computational geometry algorithms. The only singularities of solutions to this equation are movable poles of second order. Depends only on the electrostatic energy, permittivity is a constant given by the bulk value. Wuanalytical solutions of nonlinear poissonboltzmann equation for colloidal particles immersed in a general electrolyte solution by homotopy perturbation technique colloid polym. Adsorption of large ions from an electrolyte solution. Chuan li epadel spring 2017 section meeting kutztown university april 1, 2017.
We find that this model greatly amplifies the steric effects predicted by the usual modified poissonboltzmann equation, which. Modi ed poissonboltzman equation, tanh method 1 introduction when in a electrolyte solution is incorporated the nite size of particles, an entropic e ect, called excluded volume, we get the modi ed poissonboltzmann equation 14. We solve the poissonboltzmann equation for a monovalent salt, i. Some examples i existence, uniqueness, and uniform bound i freeenergy functional. Chapter 4 is a thorough discussion of the theoretical underpinnings of the widely.
T5534 with the advent of microfluidics and labonchip systems, dna and protein separation technologies are being. With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of poisson boltzmann. The density is sufficiently low so that only binary collisions need be considered 2. It is shownthat in physiological conditions the difference maybegreater than 10%. A charged spherical particle at various distances from a charged cylindrical pore in a planar surface, journal of colloid interface science 187 1997 z. Poisson boltzmann equation for microfluidic transport.
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