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22
Continuum electrostatics for ionic solutions with nonuniform ionic sizes
 Nonlinearity
"... This work concerns electrostatic properties of an ionic solution with multiple ionic species of possibly different ionic sizes. Such properties are described by the minimization of an electrostatic freeenergy functional of ionic concentrations. Bounds are obtained for ionic concentrations with low ..."
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This work concerns electrostatic properties of an ionic solution with multiple ionic species of possibly different ionic sizes. Such properties are described by the minimization of an electrostatic freeenergy functional of ionic concentrations. Bounds are obtained for ionic concentrations with low electrostatic free energies. Such bounds are used to show that there exists a unique set of equilibrium ionic concentrations that minimizes the freeenergy functional. The equilibrium ionic concentrations are found to depend sorely on the equilibrium electrostatic potential, resembling the classical Boltzmann distributions that relate the equilibrium ionic concentrations to the equilibrium electrostatic potential. Unless all the ionic and and solvent molecular sizes are assumed to be the same, explicit formulas of such dependence are, however, not available in general. It is nevertheless proved that in equilibrium the ionic charge density is a decreasing function of the electrostatic potential. This determines a variational principle with a convex functional for the electrostatic potential. Mathematical Subject Classification: 35J20, 35J60, 49S05, 81V55, 92E99. 1
MeanField Description of Ionic Size Effects with NonUniform Ionic Sizes: A Numerical Approach
, 2011
"... Ionic size effects are significant in many biological systems. Meanfield descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic p ..."
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Cited by 14 (9 self)
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Ionic size effects are significant in many biological systems. Meanfield descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic potential, i.e., there is no explicit, Boltzmann type distributions. This work begins with a variational formulation of the continuum electrostatics of an ionic solution with such nonuniform ionic sizes as well as multiple ionic valences. An augmented Lagrange multiplier method is then developed and implemented to numerically solve the underlying constrained optimization problem. The method is shown to be accurate and efficient, and is applied to ionic systems with nonuniform ionic sizes such as the sodium chloride solution. Extensive numerical tests demonstrate that the meanfield model and numerical method capture qualitatively some significant ionic size effects, particularly those for multivalent ionic solutions, such as the stratification of multivalent counterions near a charged surface. The ionic valencetovolume ratio is found to
Predicting ion binding properties for RNA tertiary structures
 Biophys. J. 2010
"... dic cre d o mp erti s t str ang (2–17). One of the difficulties comes from the fact that of the meanfield distribution) of the ions become important. acid structures with finite size in a solution of finite ion helix assembly/bending (38–41). However, the model is coarsegrained model (for helices) ..."
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Cited by 8 (5 self)
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dic cre d o mp erti s t str ang (2–17). One of the difficulties comes from the fact that of the meanfield distribution) of the ions become important. acid structures with finite size in a solution of finite ion helix assembly/bending (38–41). However, the model is coarsegrained model (for helices) cannot treat RNA tertiary folds. Therefore, we need a model that can handle the ion*Correspondence:
Competitive Adsorption and Ordered Packing of Counterions near Highly Charged Surfaces: From MeanField Theory to Monte Carlo Simulations
, 2012
"... Competitive adsorption of counterions of multiple species to charged surfaces is studied by a sizeeffect included meanfield theory and Monte Carlo (MC) simulations. The meanfield electrostatic freeenergy functional of ionic concentrations, constrained by Poisson’s equation, is numerically minimi ..."
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Competitive adsorption of counterions of multiple species to charged surfaces is studied by a sizeeffect included meanfield theory and Monte Carlo (MC) simulations. The meanfield electrostatic freeenergy functional of ionic concentrations, constrained by Poisson’s equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both of the meanfield theory and MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valencetovolume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial
Simulations of Cyclic Voltammetry for Electric Double Layers in Asymmetric Electrolytes: A Generalized Modified Poisson−Nernst−Planck Model
"... ABSTRACT: This paper presents a generalized modified Poisson−Nernst−Planck (MPNP) model derived from first principles based on excess chemical potential and Langmuir activity coefficient to simulate electric doublelayer dynamics in asymmetric electrolytes. The model accounts simultaneously for (1) ..."
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Cited by 5 (2 self)
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ABSTRACT: This paper presents a generalized modified Poisson−Nernst−Planck (MPNP) model derived from first principles based on excess chemical potential and Langmuir activity coefficient to simulate electric doublelayer dynamics in asymmetric electrolytes. The model accounts simultaneously for (1) asymmetric electrolytes with (2) multiple ion species, (3) finite ion sizes, and (4) Stern and diffuse layers along with Ohmic potential drop in the electrode. It was used to simulate cyclic voltammetry (CV) measurements for binary asymmetric electrolytes. The results demonstrated that the current density increased significantly with decreasing ion diameter and/or increasing valency zi  of either ion species. By contrast, the ion diffusion coefficients affected the CV curves and capacitance only at large scan rates. Dimensional analysis was also performed, and 11 dimensionless numbers were identified to govern the CV measurements of the electric double layer in binary asymmetric electrolytes between two identical planar electrodes of finite thickness. A selfsimilar behavior was identified for the electric doublelayer integral capacitance estimated from CV measurement simulations. Two regimes were identified by comparing the half cycle period τCV and the “RC time scale ” τRC corresponding to the characteristic time of ions ’ electrodiffusion. For τRC ≪ τCV, quasiequilibrium conditions prevailed and the capacitance was diffusionindependent while for τRC ≫ τCV, the capacitance was diffusionlimited. The effect of the electrode was captured by the dimensionless electrode electrical conductivity representing the ratio of characteristic times associated with charge transport in the electrolyte and that in the electrode. The model developed here will be useful for simulating and designing various practical electrochemical, colloidal, and biological systems for a wide range of applications. 1.
Saltdependent folding energy landscape of RNA threeway junction
 Biophys. J. 98:111–120
, 2010
"... ABSTRACT RNAs are highly negatively charged chain molecu ly d utio pre nfo ts h s a ore abi e d ct, lizin e C an loops/junctions. In response to the cofactor (ligand, protein, the unfolded threeway junction system can adopt an cannot completely predict the experimental results (22). aqueous solutio ..."
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Cited by 5 (5 self)
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ABSTRACT RNAs are highly negatively charged chain molecu ly d utio pre nfo ts h s a ore abi e d ct, lizin e C an loops/junctions. In response to the cofactor (ligand, protein, the unfolded threeway junction system can adopt an cannot completely predict the experimental results (22). aqueous solution containing multivalent ions. The predicted slopes of log(ko/kf) are compared with the experimental data,
Chromatin ionic atmosphere analyzed by a mesoscale electrostatic approach
, 2010
"... ABSTRACT Characterizing the ionic distribution around chromatin is important for understanding the electrostatic forces governing chromatin structure and function. Here we develop an electrostatic model to handle multivalent ions and compute the ionic distribution around a mesoscale chromatin model ..."
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ABSTRACT Characterizing the ionic distribution around chromatin is important for understanding the electrostatic forces governing chromatin structure and function. Here we develop an electrostatic model to handle multivalent ions and compute the ionic distribution around a mesoscale chromatin model as a function of conformation, number of nucleosome cores, and ionic strength and species using PoissonBoltzmann theory. This approach enables us to visualize and measure the complex patterns of counterion condensation around chromatin by examining ionic densities, free energies, shielding charges, and correlations of shielding charges around the nucleosome core and various oligonucleosome conformations. We show that: counterions, especially divalent cations, predominantly condense around the nucleosomal and linker DNA, unburied regions of histone tails, and exposed chromatin surfaces; ionic screening is sensitively influenced by local and global conformations, with a wide ranging net nucleosome core screening charge (56–100e); and screening charge correlations reveal conformational flexibility and interactions among chromatin subunits, especially between the histone tails and parental nucleosome cores. These results provide complementary and detailed views of ionic effects on chromatin structure for modest computational resources. The electrostatic model developed here is applicable to other coarsegrained macromolecular complexes.
Ionic size effects: generalized Boltzmann distributions, counterion stratification and modified Debye length
 Nonlinearity
, 2013
"... Abstract Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Both recent studies using a variational meanfield approach ..."
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Abstract Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Both recent studies using a variational meanfield approach that includes ionic size effects and Monte Carlo simulations suggest that counterion stratification is determined by the ionic valencetovolume ratios. Central in the meanfield approach is a freeenergy functional of ionic concentrations in which the ionic size effects are included through the entropic effect of solvent molecules. The corresponding equilibrium conditions define the generalized Boltzmann distributions relating the ionic concentrations to the electrostatic potential. This paper presents a detailed analysis and numerical calculations for such a freeenergy functional to understand the dependence of the ionic charge density on the electrostatic potential through the generalized Boltzmann distributions, the role of ionic valencetovolume ratios in the counterion stratification and the modification of Debye length due to the effect of ionic sizes. Mathematics Subject Classification: 35J20, 35J60, 35Q99, 82B21, 92C05
Ion Counting from ExplicitSolvent Simulations and 3DRISM
, 2013
"... ABSTRACT The ionic atmosphere around nucleic acids remain (IC) experiments provide a quantitative measure of the ionic atmo e re fere fertin d 3 the r a rela tes ntr ion ght NMR relaxation rates have also been employed (7,8). There shown that this phenomenon is the result of the attempt than simila ..."
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ABSTRACT The ionic atmosphere around nucleic acids remain (IC) experiments provide a quantitative measure of the ionic atmo e re fere fertin d 3 the r a rela tes ntr ion ght NMR relaxation rates have also been employed (7,8). There shown that this phenomenon is the result of the attempt than similar calculations using NLPB or IET. NLPB calcudimensional reference interaction site model (3DRISM)
The Geometry Behind Numerical Solvers of the Poisson Boltzmann Equation
"... Abstract. Electrostatics interactions play a major role in the stabilization of biomolecules: as such, they remain a major focus of theoretical and computational studies in biophysics. Electrostatics in solution is strongly dependent on the nature of the solvent and on the ions it contains. While me ..."
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Abstract. Electrostatics interactions play a major role in the stabilization of biomolecules: as such, they remain a major focus of theoretical and computational studies in biophysics. Electrostatics in solution is strongly dependent on the nature of the solvent and on the ions it contains. While methods that treat the solvent and ions explicitly provide an accurate estimate of these interactions, they are usually computationally too demanding to study large macromolecular systems. Implicit solvent methods provide a viable alternative, especially those based on Poisson theory. The PoissonBoltzmann equation (PBE) treats the system in a mean field approximation, providing reasonable estimates of electrostatics interactions in a solvent treated as continuum. In the first part of this paper, we review the theory behind the PBE, including recent improvement in which ions size and dipolar features of solvent molecules are taken into account explicitly. The PBE is a non linear second order differential equation with discontinuous coefficients, for which no analytical solution is available for large molecular systems. Many numerical solvers have been developed that solve a discretized version of the PBE on a mesh, either using finite difference, finite element, or boundary element methods. The accuracy of the solutions provided by these solvers highly depend on the geometry of their underlying meshes, as well as on the method used to embed the physical system on the mesh. In the second part of the paper, we describe a new geometric approach for generating unstructured tetrahedral meshes as well as simplifications of these meshes that are well fitted for solving the PBE equation using multigrid approaches.