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. 

Recent years have witnessed significant improvement in implicit solvents based on the Poisson-Boltzmann theory, whether in the forms of numerical solution or analytical approximation. Especially worth noting are the improvements and revisions of those implicit solvents for stable dynamics simulations. Given these technical advancements, attentions are now paid to the quality of implicit solvents as compared with the more expensive explicit solvents. The new developments in nonpolar solvents mentioned above and reviewed elsewhere will also result in more accurate simulations of biomolecules. We have also touched the new challenges facing the implicit solvents. That is how to incorporate these solvents in the emerging polarizable force fields. New challenges could also arise from the assumptions underlying all implicit solvents, as recently explored to couple electrostatic and nonelectrostatic components together. In addition, hybrid solvents could eventually become a reality for dynamics simulation even this has been proposed in the early days of computational biochemistry. It is likely that such hybrid solvents will offer the necessary accuracy, as they no longer average out the very degrees of freedom that are of interest in studies where solute/solvent coupling is crucial.

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 Poisson-Boltzmann 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.

Long-range electrostatic interactions in proteins/peptides associating to nucleic acids are reflected in the salt-dependence of the binding process. According to the oligocationic binding model, which is based on counterion condensation theory, only the cationic residues of peptides/proteins near the binding interface are assumed to affect the salt dependence in the association of peptides and proteins to nucleic acids. This model has been used to interpret and predict the binding of oligocationic chains–such as oligoarginines/lysines–to nucleic acids, and does an excellent job in these kinds of systems. This simple relationship, which is used to compare or count the number of ionic interactions in protein-nucleic acid complexes, does not hold when acidic residues, i.e. glutamate and aspartate, are incorporated in the protein matrix. Here, we report a combined molecular mechanics (by means of energy-minimization of the structure under the influence of an empirical energy function) and Poisson-Boltzmann (PB) study on the salt-dependence in binding to tRNA of two important enzymes that are involved in the seminal step of peptide formation in the ribosome: Glutamine synthetase (GluRS) and Glutaminyl synthetase (GlnRS) bound to their cognate tRNA. These two proteins are anionic and contain a significant number of acidic residues distributed over the entire protein. Some of these residues are located in the binding interface to tRNA. We computed the salt-dependence in association, SK_pred, of these enzyme-tRNA complexes using both the linear and nonlinear solution to the Poisson-Boltzmann Equation (PBE). Our findings demonstrate that the SK_pred obtained with the nonlinear PBE is in good agreement with the experimental SK_obs, while use of the linear PBE resulted in the SK_pred being anomalous. We conclude that electrostatic interactions between the binding partners in these systems are less favorable by means of charge-charge repulsion between negatively charged protein residues and phosphate oxygens in the tRNA backbone but also play a significant role in the association process of proteins to tRNA. Some unfavorable electrostatic interactions are probably compensated by hydrogen-bonds between the carboxylate group of the side chain in the interfacial acidic protein residues and the tRNA backbone. We propose that the low experimentally observed SK_obs values for both GlnRS- and GluRS-tRNA depend on the distribution and number of anionic residues that exist in these tRNA synthetases. Our computed electrostatic binding free energies were large and unfavorable due to the Coulombic and de-solvation contribution for the GlnRS-tRNA and GluRS-tRNA complexes, respectively. Thus, low SK_obs values may not reflect small contributions from the electrostatic contribution in complex-formation, as is often suggested in the literature. When charges are "turned off" in a computer-experiment, our results indicated that "turning off" acidic residues far from a phosphate group significantly influences SK_pred. If cationic residues are "turned off", less impact on SK_pred is observed with respect to the distance to the nearest phosphate-group.

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 Poisson-Boltzmann 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.

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).

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 Poisson-Boltzmann (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.

The change in some thermodynamic quantities such as Gibbs' free energy, entropy and enthalpy of the binding of two DNA strands (forming a double helix), while one is tethered to a surface and are analytically calculated. These particles are submerged in an electrolytic solution; the ionic strength of the media allows the linearized version of the Poisson-Boltzmann equation (from the theory of the double layer interaction) to properly describe the interactions [13]. There is experimental and computational evidence that an ion penetrable ellipsoid is an adequate model for the single strand and the double helix [22–25]. The analytic solution provides simple calculations useful for DNA chip design. The predicted electrostatic effects suggest the feasibility of electronic control and detection of DNA hybridization in the fast growing area of DNA recognition.

The nonlinear Poisson-Boltzmann predictions of the salt-dependent association of proteins to DNA, SK_pred, are fairly insensitive to the choice of atomic charges, radii, interior dielectric constant and treatment of the boundary between a biomolecule and the solvent. In this study we  show that the SK_pred is highly correlated with the conformational adaptability of the partners involved in the biomolecular binding process. This is demonstrated for the wild-type and mutant forms of the archaeon Pyrococcus woesi TATA-binding protein (PwTBP) in complex with DNA, on which we performed molecular mechanics energy minimizations with different protocols, and molecular dynamics simulations and then computed the SK_pred on the resulting structures. It was found that the inter-molecular non bonded force field energy between the DNA and protein correlates linearly and significantly well with the SK_pred. This correlation encompasses the wild-type and mutant variants of the PwTBP and provides us with a quick way to estimate the SK_pred from a large ensemble of structures generated with Molecular Dynamics or Monte Carlo simulations. The corresponding experimental SK_obs should also correlate with the inter-molecular non bonded force field energy between the protein and DNA, given that the underlying mechanisms in binding and salt-dependent effects are in fact the main contributors in the association of proteins/peptides to nucleic acids. We show that it is possible to fit experiments versus the inter-molecular non bonded force field energy between the protein and DNA, and use this relation to predict the SK_obs in absolute numbers. Thus, we present two novel approaches to estimate both the SKpred and the SK_obs for in silico modelled PwTBP novel mutants and even for TBPs from other organisms. This is a simple but powerful tool to suggest new experiments on the TBP-DNA type of macromolecular assemblies. We conclude by suggesting some mutants and a possible biological interpretation of how changes in solvent salinity affect the binding of proteins to DNA.