Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negat...Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 〈 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration 〉 0.15 tool/1 in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.展开更多
In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions....In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Compared with the former result obtained by Duan and Liu in [12] for the two species model, we do not ask the initial perturbation to satisfy the neutral condition and our result covers all physical collision kernels for the full range of intermolecular repulsive potentials.展开更多
Nucleic acids are negatively charged biomolecules, and metal ions in solutions are important to their folding structures and thermodynamics, especially multivalent ions. However, it has been suggested that the binding...Nucleic acids are negatively charged biomolecules, and metal ions in solutions are important to their folding structures and thermodynamics, especially multivalent ions. However, it has been suggested that the binding of multivalent ions to nucleic acids cannot be quantitatively described by the well-established Poisson-Boltzmann （PB） theory. In this work, we made extensive calculations of ion distributions around various RNA-like macroions in divalent and trivalent salt solutions by PB theory and Monte Carlo （MC） simulations. Our calculations show that PB theory appears to underestimate multi- valent ion distributions around RNA-like macroions while can reliably predict monovalent ion distributions. Our extensive comparisons between PB theory and MC simulations indicate that when an RNA-like macroion gets ion neutralization be- yond a ＂critical＂ value, the multivalent ion distribution around that macroion can be approximately described by PB theory. Furthermore, an empirical formula was obtained to approximately quantify the critical ion neutralization for various RNA- like macroions in multivalent salt solutions, and this empirical formula was shown to work well for various real nucleic acids including RNAs and DNAs.展开更多
With the help of the method of separation of variables and the Debye-Hüchel approximation, the Poisson-Boltzmann equation that describes the distribution of the potential in the electrical double layer of a cylin...With the help of the method of separation of variables and the Debye-Hüchel approximation, the Poisson-Boltzmann equation that describes the distribution of the potential in the electrical double layer of a cylindrical particle with a limited length has been firstly solved under a very low potential condition. Then with the help of the functional analysis theory this equation has been further analytically solved under general potential conditions and consequently, the corresponding surface charge densities have been obtained. Both the potential and the surface charge densities cointide with those results obtained from the Debye-Hüchel approximation when the very low potential of zeψ〈〈kT is introduced.展开更多
We consider the problem of measuring the electric charge of nanoparticles immersed in a fluid electrolyte. We develop a mathematical framework based on the solution of the nonlinear Poisson-Boltzmann equation to obtai...We consider the problem of measuring the electric charge of nanoparticles immersed in a fluid electrolyte. We develop a mathematical framework based on the solution of the nonlinear Poisson-Boltzmann equation to obtain interaction forces between nanoparticles immersed in a fluid electrolyte and an Atomic Force Microscopy micro spherical probe. This force-separation information is shown explicitly to depend on the charge of the nanoparticle.? This method overcomes the statistical nature of extant methods and renders a charge value for an individual single nanoparticle.展开更多
The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’...The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions,where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation.Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function,building upon our previous result of two dimensions.This approach yields an algorithm with a complexity of O(N log N)by strategically leveraging the locality and low-rank characteristics of the corresponding operators.Additionally,the theoretical O(N)complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction.Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.展开更多
Proteins perform various biological functions in the cell by interacting and binding to other proteins,DNA,or other small molecules.These interactions occur in cellular compartments with different salt concentrations,...Proteins perform various biological functions in the cell by interacting and binding to other proteins,DNA,or other small molecules.These interactions occur in cellular compartments with different salt concentrations,which may also vary under different physiological conditions.The goal of this study is to investigate the effect of salt concentration on the electrostatic component of the binding free energy(hereafter,salt effect)based on a large set of 1482 protein-protein complexes,a task that has never been done before.Since the proteins are irregularly shaped objects,the calculations have been carried out by a means of finite-difference algorithm that solves PoissonBoltzmann equation(PB)numerically.We performed simulations using both linear and non-linear PB equations and found that non-linearity,in general,does not have significant contribution into salt effect when the net charges of the protein monomers are of different polarity and are less than five electron units.However,for complexes made of monomers carrying large net charges non-linearity is an important factor,especially for homo-complexes which are made of identical units carrying the same net charge.A parameter reflecting the net charge of the monomers is proposed and used as a flag distinguishing between cases which should be treated with non-linear Poisson-Boltzmann equation and cases where linear PB produces sound results.It was also shown that the magnitude of the salt effect is not correlated with macroscopic parameters(such as net charge of the monomers,corresponding complexes,surface and number of interfacial residues)but rather is a complex phenomenon that depends on the shape and charge distribution of the molecules.展开更多
In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlin...In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.展开更多
In this paper we establish the convergence of the Vlasov-Poisson-Boltzmann system to the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously proved for time intervals on...In this paper we establish the convergence of the Vlasov-Poisson-Boltzmann system to the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously proved for time intervals on which the smooth solution of the Euler equations of the incompressible fluid exists. The proof relies on the relative-entropy method.展开更多
The applicability of the Poisson-Boltzmann model for micro-and nanoscale electroosmotic flows is a very important theoretical and engineering problem.In this contribution we investigate this problem at two aspects:fir...The applicability of the Poisson-Boltzmann model for micro-and nanoscale electroosmotic flows is a very important theoretical and engineering problem.In this contribution we investigate this problem at two aspects:first the high ionic concentration effect on the Boltzmann distribution assumption in the diffusion layer is studied by comparisons with the molecular dynamics(MD)simulation results;then the electrical double layer(EDL)interaction effect caused by low ionic concentrations in small channels is discussed by comparing with the dynamic model described by the coupled Poisson-Nernst-Planck(PNP)and Navier-Stokes(NS)equations.The results show that the Poisson-Boltzmann(PB)model is applicable in a very wide range:(i)the PB model can still provide good predictions of the ions density profiles up to a very high ionic concentration(∼1 M)in the diffusion layer;(ii)the PB model predicts the net charge density accurately as long as the EDL thickness is smaller than the channel width and then overrates the net charge density profile as the EDL thickness increasing,and the predicted electric potential profile is still very accurate up to a very strong EDL interaction(λ/W∼10).展开更多
Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics.Recent developments in boundary element methods,interface methods,ad...Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics.Recent developments in boundary element methods,interface methods,adaptive methods,finite element methods,and other approaches for the Poisson-Boltzmann equation as well as related mesh generation techniques are reviewed.We also discussed the challenging problems and possible future work,in particular,for the aim of biophysical applications.展开更多
Stochastic walk-on-spheres(WOS)algorithms for solving the linearized Poisson-Boltzmann equation(LPBE)provide several attractive features not available in traditional deterministic solvers:Gaussian error bars can be co...Stochastic walk-on-spheres(WOS)algorithms for solving the linearized Poisson-Boltzmann equation(LPBE)provide several attractive features not available in traditional deterministic solvers:Gaussian error bars can be computed easily,the algorithm is readily parallelized and requires minimal memory and multiple solvent environments can be accounted for by reweighting trajectories.However,previouslyreported computational times of these Monte Carlo methods were not competitive with existing deterministic numerical methods.The present paper demonstrates a series of numerical optimizations that collectively make the computational time of these Monte Carlo LPBE solvers competitive with deterministic methods.The optimization techniques used are to ensure that each atom’s contribution to the variance of the electrostatic solvation free energy is the same,to optimize the bias-generating parameters in the algorithm and to use an epsilon-approximate rather than exact nearest-neighbor search when determining the size of the next step in the Brownian motion when outside the molecule.展开更多
We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous probl...We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.展开更多
The structure and function of BNS(bionanosystems)such as macromolecules,viruses and ribosomes are strongly affected by electrostatic interactions.Yet their supra-million atom size makes them difficult to simulate via ...The structure and function of BNS(bionanosystems)such as macromolecules,viruses and ribosomes are strongly affected by electrostatic interactions.Yet their supra-million atom size makes them difficult to simulate via a straightforward PoissonBoltzmann(PB)approach.Here we explore a multiscale approach that results in a coarse-grained PB equation that follows rigorously from the all-atom PB equation.The derivation of the coarse-grained equation follows from an ansatz on the dependence of the electrical potential in two distinct ways,i.e.one reflecting atomic-scale variations and the other capturing nanometer-scale features.With this ansatz and a series expansion of the potential in a length-scale ratio,the coarse-grained PB equation is obtained.This multiscale methodology and an efficient computational methodology provide a way to efficiently simulate BNS electrostatics with atomic-scale resolution for the first time,avoiding the need for excessive supercomputer resources.The coarse-grained PB equation contains a tensorial dielectric constant that mediates the channeling of the electric field along macromolecules in an aqueous medium.The multiscale approach and novel salinity connections to the PB equation presented here should enhance the accuracy and wider applicability of PB modeling。展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.40971146 and 40740420660the National Basic Research Program of China under Grant No.2010CB134511Scientific and Technological Innovation Foundation of Southwest University for Graduates under Grant No.kb2010013
文摘Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 〈 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration 〉 0.15 tool/1 in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
基金supported by the Fundamental Research Funds for the Central Universitiessupported by a grant from the National Science Foundation of China under contract 11501556+1 种基金supported by a grant from the National Natural Science Foundation under contract 11501187supported by three grants from the National Natural Science Foundation of China under contracts 10925103,11271160,and 11261160485
文摘In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Compared with the former result obtained by Duan and Liu in [12] for the two species model, we do not ask the initial perturbation to satisfy the neutral condition and our result covers all physical collision kernels for the full range of intermolecular repulsive potentials.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11374234,11575128,11774272,and 11647312)
文摘Nucleic acids are negatively charged biomolecules, and metal ions in solutions are important to their folding structures and thermodynamics, especially multivalent ions. However, it has been suggested that the binding of multivalent ions to nucleic acids cannot be quantitatively described by the well-established Poisson-Boltzmann （PB） theory. In this work, we made extensive calculations of ion distributions around various RNA-like macroions in divalent and trivalent salt solutions by PB theory and Monte Carlo （MC） simulations. Our calculations show that PB theory appears to underestimate multi- valent ion distributions around RNA-like macroions while can reliably predict monovalent ion distributions. Our extensive comparisons between PB theory and MC simulations indicate that when an RNA-like macroion gets ion neutralization be- yond a ＂critical＂ value, the multivalent ion distribution around that macroion can be approximately described by PB theory. Furthermore, an empirical formula was obtained to approximately quantify the critical ion neutralization for various RNA- like macroions in multivalent salt solutions, and this empirical formula was shown to work well for various real nucleic acids including RNAs and DNAs.
基金Supported by the National Natural Science Foundation of China(No.20473034) the Taihu Scholar Foundation of SouthernYangtze University(2003).
文摘With the help of the method of separation of variables and the Debye-Hüchel approximation, the Poisson-Boltzmann equation that describes the distribution of the potential in the electrical double layer of a cylindrical particle with a limited length has been firstly solved under a very low potential condition. Then with the help of the functional analysis theory this equation has been further analytically solved under general potential conditions and consequently, the corresponding surface charge densities have been obtained. Both the potential and the surface charge densities cointide with those results obtained from the Debye-Hüchel approximation when the very low potential of zeψ〈〈kT is introduced.
文摘We consider the problem of measuring the electric charge of nanoparticles immersed in a fluid electrolyte. We develop a mathematical framework based on the solution of the nonlinear Poisson-Boltzmann equation to obtain interaction forces between nanoparticles immersed in a fluid electrolyte and an Atomic Force Microscopy micro spherical probe. This force-separation information is shown explicitly to depend on the charge of the nanoparticle.? This method overcomes the statistical nature of extant methods and renders a charge value for an individual single nanoparticle.
基金support from the National Natural Science Foundation of China(Grant Nos.12071288 and 12325113)the Science and Technology Commission of Shanghai Municipality of China(Grant No.21JC1403700)+1 种基金Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25010403)support of the US National Science Foundation under awards DMS-2244988 and DMS-2206333.
文摘The modified Poisson-Boltzmann(MPB)equations are often used to describe the equilibrium particle distribution of ionic systems.In this paper,we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions,where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation.Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function,building upon our previous result of two dimensions.This approach yields an algorithm with a complexity of O(N log N)by strategically leveraging the locality and low-rank characteristics of the corresponding operators.Additionally,the theoretical O(N)complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction.Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.
文摘Proteins perform various biological functions in the cell by interacting and binding to other proteins,DNA,or other small molecules.These interactions occur in cellular compartments with different salt concentrations,which may also vary under different physiological conditions.The goal of this study is to investigate the effect of salt concentration on the electrostatic component of the binding free energy(hereafter,salt effect)based on a large set of 1482 protein-protein complexes,a task that has never been done before.Since the proteins are irregularly shaped objects,the calculations have been carried out by a means of finite-difference algorithm that solves PoissonBoltzmann equation(PB)numerically.We performed simulations using both linear and non-linear PB equations and found that non-linearity,in general,does not have significant contribution into salt effect when the net charges of the protein monomers are of different polarity and are less than five electron units.However,for complexes made of monomers carrying large net charges non-linearity is an important factor,especially for homo-complexes which are made of identical units carrying the same net charge.A parameter reflecting the net charge of the monomers is proposed and used as a flag distinguishing between cases which should be treated with non-linear Poisson-Boltzmann equation and cases where linear PB produces sound results.It was also shown that the magnitude of the salt effect is not correlated with macroscopic parameters(such as net charge of the monomers,corresponding complexes,surface and number of interfacial residues)but rather is a complex phenomenon that depends on the shape and charge distribution of the molecules.
基金The research of the first author is supported by the Hong Kong Baptist University. The research of the second author is partially supported by a USA-AR0 grant 43751-MA and USA- NFS grants DMS0201094 and DMS-0412654. The third author is partially supported by CERG Grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, and an NSAF Grant （#10476032） of National Science Foundation of Chian.
文摘In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.
基金the Special Funds of State Major Basic Research Projects(Grant 1999075107)The Grant of NSAF(No.10276036)+4 种基金NSFC(Grant 10431060)Tianyuan Youth Funds of China(Grant 10426030)NSFC(Grant 10501047)Nanjing University Talent Development FoundationNSFC(Grant 10471009)
文摘In this paper we establish the convergence of the Vlasov-Poisson-Boltzmann system to the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously proved for time intervals on which the smooth solution of the Euler equations of the incompressible fluid exists. The proof relies on the relative-entropy method.
文摘The applicability of the Poisson-Boltzmann model for micro-and nanoscale electroosmotic flows is a very important theoretical and engineering problem.In this contribution we investigate this problem at two aspects:first the high ionic concentration effect on the Boltzmann distribution assumption in the diffusion layer is studied by comparisons with the molecular dynamics(MD)simulation results;then the electrical double layer(EDL)interaction effect caused by low ionic concentrations in small channels is discussed by comparing with the dynamic model described by the coupled Poisson-Nernst-Planck(PNP)and Navier-Stokes(NS)equations.The results show that the Poisson-Boltzmann(PB)model is applicable in a very wide range:(i)the PB model can still provide good predictions of the ions density profiles up to a very high ionic concentration(∼1 M)in the diffusion layer;(ii)the PB model predicts the net charge density accurately as long as the EDL thickness is smaller than the channel width and then overrates the net charge density profile as the EDL thickness increasing,and the predicted electric potential profile is still very accurate up to a very strong EDL interaction(λ/W∼10).
基金the NIH,NSF,the Howard Hughes Medical Institute,National Biomedical Computing Resource,the NSF Center for Theoretical Biological Physics,SDSC,the W.M.Keck Foundation,and Accelrys,Inc.Michael Holst was supported in part by NSF Awards 0411723,0511766,and 0225630,and DOE Awards DEFG02-05ER25707 and DE-FG02-04ER25620.
文摘Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics.Recent developments in boundary element methods,interface methods,adaptive methods,finite element methods,and other approaches for the Poisson-Boltzmann equation as well as related mesh generation techniques are reviewed.We also discussed the challenging problems and possible future work,in particular,for the aim of biophysical applications.
基金the support of NIH grant 5R44GM073391-03(PI:Dr.Alexander H.Boschtisch).
文摘Stochastic walk-on-spheres(WOS)algorithms for solving the linearized Poisson-Boltzmann equation(LPBE)provide several attractive features not available in traditional deterministic solvers:Gaussian error bars can be computed easily,the algorithm is readily parallelized and requires minimal memory and multiple solvent environments can be accounted for by reweighting trajectories.However,previouslyreported computational times of these Monte Carlo methods were not competitive with existing deterministic numerical methods.The present paper demonstrates a series of numerical optimizations that collectively make the computational time of these Monte Carlo LPBE solvers competitive with deterministic methods.The optimization techniques used are to ensure that each atom’s contribution to the variance of the electrostatic solvation free energy is the same,to optimize the bias-generating parameters in the algorithm and to use an epsilon-approximate rather than exact nearest-neighbor search when determining the size of the next step in the Brownian motion when outside the molecule.
基金supported in part by NSF Awards 0715146,0821816,0915220 and 0822283(CTBP)NIHAward P41RR08605-16(NBCR),DOD/DTRA Award HDTRA-09-1-0036+1 种基金CTBP,NBCR,NSF and NIHsupported in part by NIH,NSF,HHMI,CTBP and NBCR.The third,fourth and fifth authors were supported in part by NSF Award 0715146,CTBP,NBCR and HHMI.
文摘We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
文摘The structure and function of BNS(bionanosystems)such as macromolecules,viruses and ribosomes are strongly affected by electrostatic interactions.Yet their supra-million atom size makes them difficult to simulate via a straightforward PoissonBoltzmann(PB)approach.Here we explore a multiscale approach that results in a coarse-grained PB equation that follows rigorously from the all-atom PB equation.The derivation of the coarse-grained equation follows from an ansatz on the dependence of the electrical potential in two distinct ways,i.e.one reflecting atomic-scale variations and the other capturing nanometer-scale features.With this ansatz and a series expansion of the potential in a length-scale ratio,the coarse-grained PB equation is obtained.This multiscale methodology and an efficient computational methodology provide a way to efficiently simulate BNS electrostatics with atomic-scale resolution for the first time,avoiding the need for excessive supercomputer resources.The coarse-grained PB equation contains a tensorial dielectric constant that mediates the channeling of the electric field along macromolecules in an aqueous medium.The multiscale approach and novel salinity connections to the PB equation presented here should enhance the accuracy and wider applicability of PB modeling。
