Alexander H. Boschitsch

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.

Fast adaptive multipole method for computation of electrostatic energy in simulations of polyelectrolyte DNA

This article presents a fast adaptive method for the computation of long‐range electrostatic interactions in computer simulations of polyelectrolyte DNA. Classically, the computation of electrostatic energy involves a direct summation of all pairwise in teractions due to the charged phosphate groups in the molecule. This results in an N‐body interaction problem with an asymptotic time complexity of O(N2) which is computationally very expensive and limits the number of phosphate groups that can be used in computer simulations of polyelectrolyte DNA to at most several hundred. We describe an effort to speed up computer simulations of polyelectrolyte DNA with the use of a fast adaptive hierarchical algorithm for the computation of electrostatic energy (i.e., modified Debye–Hückel energy). The asymptotic time complexity is reduced to O(N) with the implementation of the fast hierarchical algorithm on serial computers. This is achieved by grouping phosphate groups into an adaptive hierarchical data structure and computing the interactions between groups using low order multipole and Taylor series expansions expressed in Cartesian coordinates. We first examine the accuracy and speed enhancements of the fast hierarchical method in the computation of the electrostatic energy of circular DNA at zero and high salt concentrations. The fast hierarchical method is further tested in a one‐step Monte Carlo (MC) simulated annealing algorithm for closed circular supercoiled DNA. In all cases, we observe order of magnitude reductions in the computation time with negligible loss of numerical accuracy in the electrostatic energy computation.

Formulation of a new and simple nonuniform size-modified poisson–boltzmann description

The nonlinear Poisson–Boltzmann equation (PBE) governing biomolecular electrostatics neglects ion size and ion correlation effects, and recent research activity has focused on accounting for these effects to achieve better physical modeling realism. Here, attention is focused on the comparatively simpler challenge of addressing ion size effects within a continuum‐based solvent modeling framework. Prior works by Borukhov et al. (Phys. Rev. Lett. 1997, 79, 435; Electrochim. Acta 2000, 46, 221) have examined the case of uniform ion size in considerable detail. Generalizations to accommodate different species ion sizes have been performed by Li (Nonlinearity 2009, 22, 811; SIAM J. Math. Anal. 2009, 40, 2536) and Zhou et al. (Phys. Rev. E 2011, 84, 021901) using a variational principle, Chu et al. (Biophys. J. 2007, 93, 3202) using a lattice gas model, and Tresset (Phys. Rev. E 2008, 78, 061506) using a generalized Poisson–Fermi distribution. The current work provides an alternative derivation using simple statistical mechanics principles that place the ion size effects and energy distributions on a consistent statistical footing. The resulting expressions differ from the prior nonuniform ion‐size developments. However, all treatments reduce to the same form in the cases of uniform ion‐size and zero ion size (the PBE). Because of their importance to molecular modeling and salt‐dependent behavior, expressions for the salt sensitivities and ionic forces are also derived using the nonuniform ion size description. Emphasis in this article is on formulation and numerically robust evaluation; results are presented for a simple sphere and a previously considered DNA structure for comparison and validation. More extensive application to biomolecular systems is deferred to a subsequent article.

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.

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.

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.

High accuracy computation of fluid-structure interaction in transonic cascades

A coupling strategy for simulating fluid-structure interaction phenomena is formulated and applied to the prediction of flutter in transonic cascades. The flow is governed by the Euler equations and discretized using a finite volume flux-splitting scheme. The structure is modeled using an isoparametric finite element formulation. The coupling strategy successfully reconciles these two formulations at the fluid-structure interface by enforcing both kinematic and kinetic boundary conditions. In particular, the conservation laws applicable to the combined fluid-structure system are preserved across the interface. Since the primary mechanism driving aeroelastic phenomena involves energy exchange occurring at the interface, this highly accurate coupling mechanism is believed to improve the predictive capability of the scheme. The coupled equations are advanced simultaneously in time using an implicit time integration method. Results obtained using the coupling method are presented for cascade geometries operating in transonic flow.

Computation of rotor aerodynamic loads with a constant vorticity contour free wake model

The prediction of vibratory loads on both fixed and rotating wings depends critically on closely coupled interactions between wake-induced aerodynamic loading and structural deformation. Previous rotary wing aerodynamic analyses have used models lacking many important features of the actual wake structure. This paper summarizes the development of a comprehensive analysis of isolated rotors that employs an improved approach to wake simulation. The model discretizes the sheet of vorticity that trails from each blade by laying out vortex filaments along contours of constant sheet strength. In this Constant Vorticity Contour (CVC) wake model, the filaments are composed of curved vortex elements that distort freely in response to the local velocity field. This approach captures the wake of the full span of each rotor blade and includes many complex features of the vorticity field absent from earlier, more simplified models. The present paper outlines the capabilities of a new comprehensive rotor code, RotorCRAFT, that incorporates the CVC model. Results obtained with this code confirm the conclusion of previous exploratory work that the CVC approach to wake modeling is an essential prerequisite for the prediction of aerodynamic loads in forward flight.