Marcia O. Fenley

Hybrid boundary element and finite difference method for solving the nonlinear Poisson–Boltzmann equation

A hybrid approach for solving the nonlinear Poisson–Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion‐exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.

A new outer boundary formulation and energy corrections for the nonlinear Poisson–Boltzmann equation

The nonlinear Poisson–Boltzmann equation (PBE) has been successfully used for the prediction of numerous electrostatic properties of highly charged biopolyelectrolytes immersed in aqueous salt solutions. While numerous numerical solvers for the 3D PBE have been developed, the formulation of the outer boundary treatments used in these methods has only been loosely addressed, especially in the nonlinear case. The de facto standard in current nonlinear PBE implementations is to either set the potential at the outer boundaries to zero or estimate it using the (linear) Debye–Hückel (DH) approximation. However, an assessment of how these outer boundary treatments affect the overall solution accuracy does not appear to have been previously made. As will be demonstrated here, both approximations can, under certain conditions, produce completely erroneous estimates of the potential and energy salt dependencies. A related concern for calculations carried out on grids of finite extent (e.g., all current finite difference and finite element implementations) is the contribution to the energy and salt dependence from the exterior region outside the computational grid. This too is shown to be significant, especially at low salt concentration where essentially all of the contributions to the excess osmotic pressure and ion stress energies originate from this exterior region. In this paper the authors introduce a new outer boundary treatment that is valid for both the linear and nonlinear PBE. The authors also formulate energy corrections to account for contributions from outside the computational domain. Finally, the authors also consider the effects of general ion exclusion layers upon biomolecular electrostatics. It is shown that while these layers tend to increase the surface electrostatic potential, under physiological salt conditions and high net charges their effect on the excess osmotic pressure term, which is a measure of the salt dependence of the total electrostatic free energy, is weak. To facilitate presentation and allow very fine resolutions and/or large computational domains to be considered, attention is restricted to the 1D spherically symmetric nonlinear PBE. Though geometrically limited, the modeling principles nevertheless extend to general PBE solvers as discussed in the Appendix . The 1D model can also be used to benchmark and validate the salt effect prediction capabilities of existing PBE solvers.

The electrostatic characteristics of G · U wobble base pairs

G·U wobble base pairs are the most common and highly conserved non-Watson–Crick base pairs in RNA. Previous surface maps imply uniformly negative electrostatic potential at the major groove of G·U wobble base pairs embedded in RNA helices, suitable for entrapment of cationic ligands. In this work, we have used a Poisson–Boltzmann approach to gain a more detailed and accurate characterization of the electrostatic profile. We found that the major groove edge of an isolated G·U wobble displays distinctly enhanced negativity compared with standard GC or AU base pairs; however, in the context of different helical motifs, the electrostatic pattern varies. G·U wobbles with distinct widening have similar major groove electrostatic potentials to their canonical counterparts, whereas those with minimal widening exhibit significantly enhanced electronegativity, ranging from 0.8 to 2.5 kT/e, depending upon structural features. We propose that the negativity at the major groove of G·U wobble base pairs is determined by the combined effect of the base atoms and the sugar-phosphate backbone, which is impacted by stacking pattern and groove width as a result of base sequence. These findings are significant in that they provide predictive power with respect to which G·U sites in RNA are most likely to bind cationic ligands.

Properties of the nucleic-acid bases in free and Watson-Crick hydrogen-bonded states: computational insights into the sequence-dependent features of double-helical DNA

The nucleic-acid bases carry structural and energetic signatures that contribute to the unique features of genetic sequences. Here, we review the connection between the chemical structure of the constituent nucleotides and the polymeric properties of DNA. The sequence-dependent accumulation of charge on the major- and minor-groove edges of the Watson–Crick base pairs, obtained from ab initio calculations, presents unique motifs for direct sequence recognition. The optimization of base interactions generates a propellering of base-pair planes of the same handedness as that found in high-resolution double-helical structures. The optimized base pairs also deform along conformational pathways, i.e., normal modes, of the same type induced by the binding of proteins. Empirical energy computations that incorporate the properties of the base pairs account satisfactorily for general features of the next level of double-helical structure, but miss key sequence-dependent differences in dimeric structure and deformability. The latter discrepancies appear to reflect factors other than intrinsic base-pair structure.

Revisiting the Association of Cationic Groove-Binding Drugs to DNA Using a Poisson-Boltzmann Approach

Proper modeling of nonspecific salt-mediated electrostatic interactions is essential to understanding the binding of charged ligands to nucleic acids. Because the linear Poisson-Boltzmann equation (PBE) and the more approximate generalized Born approach are applied routinely to nucleic acids and their interactions with charged ligands, the reliability of these methods is examined vis-à-vis an efficient nonlinear PBE method. For moderate salt concentrations, the negative derivative, SK(pred), of the electrostatic binding free energy, DeltaG(el), with respect to the logarithm of the 1:1 salt concentration, [M(+)], for 33 cationic minor groove drugs binding to AT-rich DNA sequences is shown to be consistently negative and virtually constant over the salt range considered (0.1-0.4 M NaCl). The magnitude of SK(pred) is approximately equal to the charge on the drug, as predicted by counterion condensation theory (CCT) and observed in thermodynamic binding studies. The linear PBE is shown to overestimate the magnitude of SK(pred), whereas the nonlinear PBE closely matches the experimental results. The PBE predictions of SK(pred) were not correlated with DeltaG(el) in the presence of a dielectric discontinuity, as would be expected from the CCT. Because this correlation does not hold, parameterizing the PBE predictions of DeltaG(el) against the reported experimental data is not possible. Moreover, the common practice of extracting the electrostatic and nonelectrostatic contributions to the binding of charged ligands to biopolyelectrolytes based on the simple relation between experimental SK values and the electrostatic binding free energy that is based on CCT is called into question by the results presented here. Although the rigid-docking nonlinear PB calculations provide reliable predictions of SK(pred), at least for the charged ligand-nucleic acid complexes studied here, accurate estimates of DeltaG(el) will require further development in theoretical and experimental approaches.

Recognition of the spliceosomal branch site RNA helix on the basis of surface and electrostatic features

We have investigated electrostatic and surface features of an essential region of the catalytic core of the spliceosome, the eukaryotic precursor messenger (pre-m)RNA splicing apparatus. The nucleophile for the first of two splicing reactions is the 2′-hydroxyl (OH) of the ribose of a specific adenosine within the intron. During assembly of the spliceosomes catalytic core, this adenosine is positioned by pairing with a short region of the U2 small nuclear (sn)RNA to form the pre-mRNA branch site helix. The solution structure of the spliceosomal pre-mRNA branch site [Newby,M.I. and Greenbaum,N.L. (2002) Nature Struct. Biol., 9, 958–965] showed that a phylogenetically conserved pseudouridine (ψ) residue in the segment of U2 snRNA that pairs with the intron induces a markedly different structure compared with that of its unmodified counterpart. In order to achieve a more detailed understanding of the factors that contribute to recognition of the spliceosomes branch site helix and activation of the nucleophile for the first step of pre-mRNA splicing, we have calculated surface areas and electrostatic potentials of ψ-modified and unmodified branch site duplexes. There was no significant difference between the total accessible area or ratio of total polar:nonpolar groups between modified and unmodified duplexes. However, there was substantially greater exposure of nonpolar area of the adenine base, and less exposure of the 2′-OH, in the ψ-modified structure. Electrostatic potentials computed using a hybrid boundary element and finite difference nonlinear Poisson–Boltzmann approach [Boschitsch, A.H. and Fenley, M.O. (2004) J. Comput. Chem., 25, 935–955] revealed a region of exceptionally negative potential in the major groove surrounding the 2′-OH of the branch site adenosine. These surface and electrostatic features may contribute to the overall recognition of the pre-mRNA branch site region by other components of the splicing reaction.

Monte Carlo-based linear Poisson-Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible

The prediction of salt-mediated electrostatic effects with high accuracy is highly desirable since many biological processes where biomolecules such as peptides and proteins are key players can be modulated by adjusting the salt concentration of the cellular milieu. With this goal in mind, we present a novel implicit-solvent based linear Poisson-Boltzmann (PB) solver that provides very accurate nonspecific salt-dependent electrostatic properties of biomolecular systems. To solve the linear PB equation by the Monte Carlo method, we use information from the simulation of random walks in the physical space. Due to inherent properties of the statistical simulation method, we are able to account for subtle geometric features in the biomolecular model, treat continuity and outer boundary conditions and interior point charges exactly, and compute electrostatic properties at different salt concentrations in a single PB calculation. These features of the Monte Carlo-based linear PB formulation make it possible to predict the salt-dependent electrostatic properties of biomolecules with very high accuracy. To illustrate the efficiency of our approach, we compute the salt-dependent electrostatic solvation free energies of arginine-rich RNA-binding peptides and compare these Monte Carlo-based PB predictions with computational results obtained using the more mature deterministic numerical methods.

Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation.

Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the models predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the models predictions of ΔGel.

Salt-Mediated Electrostatics in the Association of TATA Binding Proteins to DNA: A Combined Molecular Mechanics/Poisson-Boltzmann Study

The TATA-binding protein (TBP) is a key component of the archaea ternary preinitiation transcription assembly. The archaeon TBP, from the halophile/hyperthermophile organism Pyrococcus woesei, is adapted to high concentrations of salt and high-temperature environments. Although most eukaryotic TBPs are mesophilic and adapted to physiological conditions of temperature and salt, they are very similar to their halophilic counterparts in sequence and fold. However, whereas the binding affinity to DNA of halophilic TBPs increases with increasing salt concentration, the opposite is observed for mesophilic TBPs. We investigated these differences in nonspecific salt-dependent DNA-binding behavior of halophilic and mesophilic TBPs by using a combined molecular mechanics/Poisson-Boltzmann approach. Our results are qualitatively in good agreement with experimentally observed salt-dependent DNA-binding for mesophilic and halophilic TBPs, and suggest that the distribution and the total number of charged residues may be the main underlying contributor in the association process. Therefore, the difference in the salt-dependent binding behavior of mesophilic and halophilic TBPs to DNA may be due to the very unique charge and electrostatic potential distribution of these TBPs, which consequently alters the number of repulsive and attractive electrostatic interactions.

Understanding the physical basis of the salt dependence of the electrostatic binding free energy of mutated charged ligand–nucleic acid complexes

The predictions of the derivative of the electrostatic binding free energy of a biomolecular complex, ΔG(el), with respect to the logarithm of the 1:1 salt concentration, d(ΔG(el))/d(ln[NaCl]), SK, by the Poisson-Boltzmann equation, PBE, are very similar to those of the simpler Debye-Hückel equation, DHE, because the terms in the PBEs predictions of SK that depend on the details of the dielectric interface are small compared to the contributions from long-range electrostatic interactions. These facts allow one to obtain predictions of SK using a simplified charge model along with the DHE that are highly correlated with both the PBE and experimental binding data. The DHE-based model developed here, which was derived from the generalized Born model, explains the lack of correlation between SK and ΔG(el) in the presence of a dielectric discontinuity, which conflicts with the popular use of this supposed correlation to parse experimental binding free energies into electrostatic and nonelectrostatic components. Moreover, the DHE model also provides a clear justification for the correlations between SK and various empirical quantities, like the number of ion pairs, the ligand charge on the interface, the Coulomb binding free energy, and the product of the charges on the complexs components, but these correlations are weak, questioning their usefulness.