This article summarizes the development of a fast boundary element method for the linear Poisson-Boltzmann equation governing biomolecular electrostatics. Unlike previous fast boundary element implementations, the present treatment accommodates finite salt concentrations thus enabling the study of biomolecular electrostatics under realistic physiological conditions. This is achieved by using multipole expansions specifically designed for the exponentially decaying Green's function of the linear Poisson-Boltzmann equation. The particular formulation adopted in the boundary element treatment directly affects the numerical conditioning and thus convergence behavior of the method. Therefore, the formulation and reasons for its choice are first presented. Next, the multipole approximation and its use in the context of a fast boundary element method are described together with the iteration method employed to extract the surface distributions. The method is then subjected to a series of computational tests involving a sphere with interior charges. The purpose of these tests is to assess accuracy and verify the anticipated computational performance trends. Finally, the salt dependence of electrostatic properties of several biomolecular systems (alanine dipeptide, barnase, barstar, and coiled coil tetramer) is examined with the method and the results are compared with finite difference Poisson-Boltzmann codes.

Watkins et al., 1996; Samarsky et al., 1998). It is also clear that a fraction of snoRNAs, such as U3, U8, and U14, depend on the same sets of conserved box C/D sequences to carry out site-specific cleavage of ribo-somal RNAs (Baserga et al. Peculis and Steitz, 1994; Samarsky and Fournier, 1998; Watkins et al., 2002). Another type of snoRNP, the box H/ACA snoRNPs, catalyzes site-specific pseudouridyla-tion of rRNA by employing similar mechanisms of sub-Summary strate recognition and enzyme organization (Ofengand and Fournier, 1998; Venema and Tollervey, 1999; Kiss, We have determined and refined a crystal structure 2002; Rozhdestvensky et al., 2003). of the initial assembly complex of archaeal box C/D Earlier biochemical studies have shown that the con-sRNPs comprising the Archaeoglobus fulgidus (AF) served box C and D sequences are the sites of protein L7Ae protein and a box C/D RNA. The box C/D RNA assembly (Kuhn et al., 2002; Szewczak et al., 2002; Wat-forms a classical kink-turn (K-turn) structure and the kins et al., 2002). The first protein that interacts with the resulting protein-RNA complex serves as a distinct box C/D sequences is a member of an RNA binding platform for recruitment of the fibrillarin-Nop5p com-protein family. This protein was identified to be 15.5 kDa plex. The cocrystal structure confirms previously pro-(Watkins et al., 2000) in human and L7Ae in Archaea posed secondary structure of the box C/D RNA that (Kuhn et al., 2002). The initial interaction established includes a protruded U, a UU mismatch, and two between the RNA binding protein and the box C/D motif sheared tandem GA base pairs. Detailed structural is required for subsequent assembly of other snoRNP or comparisons of the AF L7Ae-box C/D RNA complex sRNP proteins, including fibrillarin, the eucaryal Nop56/ with previously determined crystal structures of L7Ae 58p, or the archaeal Nop5p proteins (Watkins et al., homologs in complex with functionally distinct K-turn 2002; Cahill et al., 2002; Rashid et al., 2003; Tran et al., RNAs revealed a set of remarkably conserved princi-2003). Although the Nop56/Nop5p proteins appear to ples in protein-RNA interactions. These analyses pro-interact directly with the conserved box C/D elements vide a structural basis for interpreting the functional upon binding of the 15.5 kDa/L7Ae protein (Cahill et al., roles of the box C/D sequences in directing specific 2002; Rashid et al., 2003; Watkins et al., 2002), fibrillarin assembly of box C/D sRNPs. is the subunit that catalyzes the actual methyl transfer reaction (Niewmierzycka and Clarke, 1999; Omer et al., Introduction 2000, 2002; Wang et al., 2000). Thus the recruitment of fibrillarin in the assembly of box C/D s(no)RNPs medi-Box C/D small nucleolar ribonucleoprotein particles ated by the 15.5 kDa/L7Ae protein suggests an impor-(snoRNPs) catalyze site-specific 2-O-methylation and tant functional role for 15.5 kDa/L7Ae in constructing the processing of ribosomal RNAs (rRNAs). These reactions RNA architecture around the active site before catal...

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.