Proc Natl Acad Sci U S A 98(18): 10037-10041

Electrostatics of nanosystems: Application to microtubules and the ribosome

Departments of ^{Chemistry and Biochemistry,}^{Mathematics, and}^{Pharmacology, and}^{Howard Hughes Medical Institute, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093; and}^{Department of Biomedical Engineering, Washington University, One Brookings Drive, St. Louis, MO 63130-4899}

^{To whom reprint requests should be addressed at: University of California at San Diego, 9500 Gilman Drive, Mail Code 0365, La Jolla, CA 92093-0365. E-mail:}ude.dscu.nommaccm@rekabn.

Communicated by Peter G. Wolynes, University of California at San Diego, La Jolla, CA

Received 2001 May 8; Accepted 2001 Jul 5.

Abstract

Evaluation of the electrostatic properties of biomolecules has become a standard practice in molecular biophysics. Foremost among the models used to elucidate the electrostatic potential is the Poisson-Boltzmann equation; however, existing methods for solving this equation have limited the scope of accurate electrostatic calculations to relatively small biomolecular systems. Here we present the application of numerical methods to enable the trivially parallel solution of the Poisson-Boltzmann equation for supramolecular structures that are orders of magnitude larger in size. As a demonstration of this methodology, electrostatic potentials have been calculated for large microtubule and ribosome structures. The results point to the likely role of electrostatics in a variety of activities of these structures.

Abstract

The importance of electrostatic modeling to biophysics is well established; electrostatics have been shown to influence various aspects of nearly all biochemical reactions. Advances in NMR, x-ray, and cryo-electron microscopy techniques for structure elucidation have drastically increased the size and number of biomolecules and molecular complexes for which coordinates are available. However, although the biophysical community continues to examine macromolecular systems of increasing scale, the computational evaluation of electrostatic properties for these systems is limited by methodology that can handle only relatively small systems, typically consisting of fewer than 50,000 atoms. Despite these limitations, such computational methods have been immensely useful in analyses of the stability, dynamics, and association of proteins, nucleic acids, and their ligands (1–3). Here we describe algorithms that open the way to similar analyses of much larger subcellular structures.

One of the most widespread models for the evaluation of electrostatic properties is the Poisson-Boltzmann equation (PBE) (4, 5)

a second-order nonlinear elliptic partial differential equation that relates the electrostatic potential (φ) to the dielectric properties of the solute and solvent (ɛ), the ionic strength of the solution and the accessibility of ions to the solute interior ( $\bar{κ}$ ^{), and the distribution of solute atomic partial charges (}f). To expedite solution of the equation, this nonlinear PBE is often approximated by the linearized PBE (LPBE) by assuming sinhφ(x) ≈ φ(x). Several numerical techniques have been used to solve the nonlinear PBE and LPBE, including boundary element (6–8), finite element (9–11), and finite difference (12–14) algorithms. However, despite the variety of solution methods, none of these techniques has been satisfactorily applied to large molecular structures at the scales currently accessible to modern biophysical methods. To accommodate arbitrarily large biomolecules, algorithms for solving the PBE must be both efficient and amenable to implementation on a parallel platform in a scalable fashion, requirements that current methods have been unable to satisfy. Although boundary element LPBE solvers provide an efficient representation of the problem domain, they are not useful for the nonlinear problem and have not been applied to the PBE on parallel platforms. Similarly, adaptive finite element methods have shown some success in parallel evaluation of both the LPBE and nonlinear PBE (15), but limitations in current solver technology and difficulty with efficient representation of the biomolecular data prohibits their practical application to large biomolecular systems. Finally, unlike the boundary and finite element techniques, finite difference methods have the advantage of very efficient multilevel solvers (12, 16) and applicability to both the linear and nonlinear forms of the PBE; however, existing parallel finite difference algorithms often require costly interprocessor communication that limits both the nature and scale of their execution on parallel platforms (17–21) [see especially Van de Velde (19) for reviews of the various methods].

Acknowledgments

N.A.B. thanks A. Elcock and C. Wong for helpful discussions on the PBE and A. Majumdar and G. Chukkapalli for assistance with technical issues on the Blue Horizon supercomputer. N.A.B. was supported by predoctoral fellowships from the Howard Hughes Medical Institute and the Burroughs-Wellcome La Jolla Interfaces in Science program. This work was supported, in part, by an IBM/American Chemical Society award to N.A.B., providing time on the Minnesota Supercomputing Institute IBM SP2; by grants to J.A.M. from the National Institutes of Health, National Science Foundation, and NPACI/San Diego Supercompter Center; by National Science Foundation CAREER Award 9875856 to M.J.H.; and by National Science Foundation Grant MCB-0078322 to S.J. Additional support has been provided by the W. M. Keck Foundation and the National Biomedical Computation Resource.

Acknowledgments

Abbreviations

PBE	Poisson-Boltzmann equation
LPBE	linearized PBE
NPACI	National Partnership for Advanced Computational Infrastructure

Abbreviations

Footnotes

^{For example, let}v(P) = h_x(P)h_y(P)h_z(P) be the volume of a grid element in a P-processor calculation with 100f% overlap on a n_x × n_y × n_z mesh with spacings h_x(P),h_y(P),h_z(P) over a problem domain of volume V. One measure of the mesh resolution is the inverse element volume, which behaves as v(P)^{∼(1 − 2}f)^P(n_x − 1)(n_y − 1)(n_x− 1)/V, which is linear in P.

^{The 50S Protein Data Bank coordinate file (}1FFK) (27) contained only Cα coordinates for the protein constituents. Therefore, each protein residue was simply represented by its Cα atom, which was assigned the total charge of the residue and a radius of 4.0 Å.

Footnotes

References

1. Elcock A H, Sept D, McCammon J A. J Phys Chem B. 2001;105:1504–1518.[PubMed]
2. Elcock A H, Gabdoulline R R, Wade R C, McCammon J A. J Mol Biol. 1999;291:149–162.[PubMed]
3. Sept D, Elcock A H, McCammon J A. J Mol Biol. 1999;294:1181–1189.[PubMed]
4. Davis M E, McCammon J A. Chem Rev. 1990;94:7684–7692.[PubMed]
5. Honig B, Nicholls A. Science. 1995;268:1144–1149.[PubMed]
6. Zahaur R J, Morgan R S. J Comput Chem. 1988;9:171–187.[PubMed]
7. Zhou H-X. Biophys J. 1993;65:955–963.
8. Vorobjev Y N, Scheraga H A. J Comput Chem. 1997;18:569–583.[PubMed]
9. Holst M J, Baker N A, Wang F. J Comput Chem. 2000;21:1319–1342.[PubMed]
10. Baker N A, Holst M J, Wang F. J Comput Chem. 2000;21:1343–1352.[PubMed]
11. Cortis C M, Friesner R A. J Comput Chem. 1997;18:1591–1608.[PubMed]
12. Holst M, Saied F. J Comput Chem. 1995;16:337–364.[PubMed]
13. Davis M E, McCammon J A. J Comput Chem. 1989;10:386–391.[PubMed]
14. Gilson M K, Sharp K A, Honig B H. J Comput Chem. 1987;9:327–335.[PubMed]
15. Baker N A, Sept D, Holst M J, McCammon J A. IBM J Res Dev. 2001;45:427–438.[PubMed]
16. Holst M, Saied F In: Multigrid and Domain Decomposition Methods for Electrostatics Problems. Keyes D E, Xu J, editors. Providence, RI: Am. Math. Soc.; 1995. [PubMed][Google Scholar]
17. Schwarz H A. J für die reine und angewandte Mathematik. 1869;70:105–120.[PubMed]
18. Bjørstad P E, Widlund O B. Soc Ind Appl Math J Num Anal. 1986;23:1097–1120.[PubMed]
19. Van de Velde E F Concurrent Scientific Computing. New York: Springer; 1994. [PubMed][Google Scholar]
20. Ilin A, Bagheri B, Scott L R, Briggs J M, McCammon J A Parallelization of Poisson-Boltzmann and Brownian Dynamics Calculations. Washington, DC: Am. Chem. Soc.; 1995. pp. 170–185. [PubMed][Google Scholar]
21. Tuminaro R S, Womble D E Analysis of the Multigrid FMV Cycle on Large-Scale Parallel Machines. Albuquerque, NM: Sandia National Laboratories; 1992. , Technical Report SAND91–2199J. [PubMed][Google Scholar]
22. Bank R, Holst M. Soc Ind Appl Math J Sci Comput. 2000;22:1411–1443.[PubMed]
23. Holst, M(2001) Adv. Comput. Math., in press.
24. Nogales E, Whittaker M, Milligan R A, Downing K H. Cell. 1999;96:79–88.[PubMed]
25. Walker R A, O'Brien E T, Pryer N K, Soboeiro M F, Voter W A, Erikson H P, Salmon E D. J Cell Biol. 1988;107:1437–1448.
26. Giannakakou P, Sackett D L, Kang Y-K, Zhan Z R, Buters J T M, Fojo T, Poruchynsky M S. J Biol Chem. 1997;272:17118–17125.[PubMed]
27. Ban N, Nissen P, Hansen J, Moore P B, Steitz T A. Science. 2000;289:905–920.[PubMed]
28. Schluenzen F, Tocilj A, Zarivach R, Harms J, Gluehmann M, Janell D, Bashan A, Bartels H, Agmon I, Franceschi F, Yonath A. Cell. 2000;102:615–623.[PubMed]
29. Wimberly B T, Brodersen D E, Clemons W M, Morgan-Warren R J, Carter A P, Vonrhein C, Hartsch T, Ramakrishnan V. Nature (London) 2000;407:327–339.[PubMed]
30. Cate J H, Yuspov M M, Yusupova G Z, Earnest T N, Noller H F. Science. 1999;285:2095–2104.[PubMed]
31. Culver G M, Cate J H, Yusupova G Z, Yusupov M M, Noller H F. Science. 1999;285:2133–2135.[PubMed]
32. Klein H A, Ochoa S. J Biol Chem. 1972;247:8122–8128.[PubMed]
33. Boublik M, Wydro R M, Hellmann W, Jenkins F. J Supramol Struct. 1979;10:397–404.[PubMed]
34. Carter A P, Clemons W M, Brodersen D E, Morgan-Warren R J, Wimberly B T, Ramakrishnan V. Nature (London) 2000;407:340–348.[PubMed]
35. Nissen P, Hansen J, Ban N, Moore P B, Steitz T A. Science. 2000;289:920–930.[PubMed]
36. Samaha R R, Green R, Noller H F. Nature (London) 1995;377:309–314.[PubMed]