Multivariate Minimization in Computational Chemistry

Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 21)


Optimization is a fundamental component of molecular modeling. The determination of a low-energy conformation for a given force field can be the final objective of the computation. It can also serve as a starting point for subsequent calculations, such as molecular dynamics simulations or normal-mode analyses.


Conjugate Gradient Line Search Conjugate Gradient Method Descent Direction Potential Energy Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Adams and J. L. Nazareth, editors. Linear and Nonlinear Conjugate Gradient- Related Methods. SIAM, Philadelphia, PA, 1996.Google Scholar
  2. 106.
    D. Beard and T. Schlick. Inertial stochastic dynamics: I. long-timestep methods for langevin dynamics. J. Chem. Phys., 112:7313–7322, 2000.Google Scholar
  3. 153.
    J. M. Borwein and A. S. Lewis. Convex Analysis and Nonlinear Optimization. heory and Examples, volume 3 of Canadian Mathematical Society (CMS) Books in Mathematics. Springer-Verlag, New York, NY, 2000.Google Scholar
  4. 192.
    R. H. Byrd, P. Lu, and J. Nocedal. A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Stat. Comput., 16:1190–1208, 1995.MathSciNetMATHCrossRefGoogle Scholar
  5. 193.
    R. H. Byrd, J. Nocedal, and R. B. Schnabel. Representations of quasi-Newton matrices and their use in limited memory methods. Math. Prog., 63:129–156, 1994.MathSciNetMATHCrossRefGoogle Scholar
  6. 194.
    R. H. Byrd, J. Nocedal, and C. Zhu. Towards a discrete Newton method with mem- ory for large-scale optimization. In G. Di Pillo and F. Giannessi, editors, Nonlinear Optimization and Applications. Plenum, 1996.Google Scholar
  7. 196.
    F. Calvo. Non-genetic global optimization methods in molecular science: An overview. Comp. Mat. Sci., 45:8–15, 2009.CrossRefGoogle Scholar
  8. 261.
    A. R. Conn, N. I. M. Gould, and Ph. L. Toint. LANCELOT: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A), volume 17 of Springer Series in Computational Mathematics. Springer-Verlag, New York, NY, 1992.Google Scholar
  9. 295.
    R. S. Dembo and T. Steihaug. Truncated-Newton algorithms for large-scale unconstrained optimization. Math. Prog., 26:190–212, 1983.MathSciNetMATHCrossRefGoogle Scholar
  10. 297.
    J. E. Dennis, Jr. and R. B. Schnabel. Numerical Methods for Unconstrained Op- timization and Nonlinear Equations. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. (Reprinted by SIAM, 1996).Google Scholar
  11. 302.
    P. Derreumaux, G. Zhang, B. Brooks, and T. Schlick. A truncated-Newton method adapted for CHARMM and biomolecular applications. J. Comput. Chem., 15:532–552, 1994.CrossRefGoogle Scholar
  12. 407.
    D. Flatters and R. Lavery. Sequence-dependent dynamics of TATA-box binding sites. Biophys. J., 75:372–381, 1998.CrossRefGoogle Scholar
  13. 411.
    J. Flori´an, M. F. Goodman, and A. Warshel. Computer simulations of protein functions: searching for the molecular origin of the replication fidelity of DNA polymerases. Proc. Natl. Acad. Sci. USA, 102:6819–6824, 2005.Google Scholar
  14. 417.
    N. Foloppe and A. D. MacKerell, Jr. All-atom empirical force field for nucleic acids: I. Parameter optimization based on small molecule and condensed phased macromolecular target data. J. Comput. Chem., 21:86–104, 2000.Google Scholar
  15. 456.
    W. W. Gibbs. Evolution in a bottle: Synthetic life oozes closer to reality. self- replicating RNAs advance science another step toward artificial life. Sci. Amer., 300:18–21, 2009.Google Scholar
  16. 457.
    K. B. Gibson and H. A. Scheraga. Decisions in force field development: Reply to Kollman and Dill. J. Biomol. Struct. Dyn., 8:1109–1111, 1991.CrossRefGoogle Scholar
  17. 458.
    J. C. Gilbert and C. Lemarechal. Some numerical experiments with variable- storage quasi-Newton algorithms. Math. Prog. B, 45:407–435, 1989.MathSciNetMATHCrossRefGoogle Scholar
  18. 459.
    J. C. Gilbert and J. Nocedal. Global convergence properties of conjugate gradient methods for optimization. Technical Report 1268, Institut National de Recherche en Informatique et en Automatique, January 1991.Google Scholar
  19. 467.
    V. Gogonea, D. Su´arex, A. van der Vaart, and K. M. Merz, Jr. New developments in applying quantum mechanics to proteins. Curr. Opin. Struct. Biol., 11:217–223, 2001.Google Scholar
  20. 482.
    L. Greengard and V. Rokhlin. On the evaluation of electrostatic interactions in molecular modeling. Chemica Scripta, 29A:139–144, 1989.Google Scholar
  21. 490.
    F. Guo, A. R. Gooding, and T. R. Cech. Structure of the tetrahymena ribozyme: base triple sandwich and metal ion at the active site. Mol Cell., 16:351–362, 2004.Google Scholar
  22. 549.
    T. L. Hill. An Introduction to Statistical Thermodynamics. Dover, New York, NY, 1986.Google Scholar
  23. 550.
    B. E. Hingerty, R. H. Ritchie, T. L. Ferrell, and J. E. Turner. Dielectric effects in biopolymers: The theory of ionic saturation revisited. Biopolymers, 24:427–439, 1985.CrossRefGoogle Scholar
  24. 773.
    J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, Jr., and C. Bustamante. Reversible unfolding of single RNA molecules by mechanical force. Science, 292:733–737, 2001.CrossRefGoogle Scholar
  25. 774.
    C. Lipinski and A. Hopkins. Navigating chemical space for biology and medicine. ature, 432:855–861, 2004.Google Scholar
  26. 789.
    Y. Lu and J. Liu. Functional DNA nanotechnology: Emerging applications of DNAzymes and aptamers. Curr. Opin. Biotech., 17:580–588, 2006.CrossRefGoogle Scholar
  27. 876.
    G. T. Montelione and S. Anderson. Structural genomics: Keystone for a human proteome project. Nature Struc. Biol., 6:11–12, 1999.CrossRefGoogle Scholar
  28. 877.
    P. B. Moore. Structural motifs in RNA. Ann. Rev. Biochem., 68:287–300, 1999.CrossRefGoogle Scholar
  29. 891.
    E. W. Myers, G. G. Sutton, H. O. Smith, M. D. Adams, and J. C. Venter. On the sequencing and assembly of the human genome. Proc. Natl. Acad. Sci. USA, 99:4145–4146, 2002.CrossRefGoogle Scholar
  30. 914.
    P. Nissen, J. Hansen, N. Ban, P. B. Moore, and T. A. Steitz. The structural basis of ribosome activity in peptide bond synthesis. Science, 289:920–930, 2000.CrossRefGoogle Scholar
  31. 915.
    P. Nissen, J. A. Ippolito, N. Ban, P. B. Moore, and T. A. Steitz. RNA tertiary inter- actions in the large ribosomal subunit: The A-minor motif. Proc. Natl. Acad. Sci. SA, 98:4899–4903, 2001.CrossRefGoogle Scholar
  32. 916.
    K. T. No, S. G. Kim, K.-H. Cho, and H. A. Scheraga. Description of hydration free energy density as a function of molecular physical properties. Biophys. Chem., 78:127–145, 1999.CrossRefGoogle Scholar
  33. 917.
    M. E. M. Noble, J. A. Endicott, and L. N. Johnson. Protein kinase inhibitors: Insights into drug design from structure. Science, 303:1800–1805, 2004.CrossRefGoogle Scholar
  34. 918.
    J. Nocedal. Updating quasi-Newton matrices with limited storage. Mathematics of Computation, 35:773–782, 1980.MathSciNetMATHCrossRefGoogle Scholar
  35. 954.
    C. Oostenbrink, A. Villa, A. E. Mark, and W. F. Van Gunsteren. A biomolecular force field based on the free enthalpy of hydration and solvation: The GROMOS force-field parameter sets 53A5 and 53A6. J. Comput. Chem., 25:1656–1676, 2004.CrossRefGoogle Scholar
  36. 958.
    M. L. Overton. Numerical Computing with IEEE Floating Point Arithmetic. SIAM, Philadelphia, PA, 2001.Google Scholar
  37. 997.
    G. A. Petsko. An idea whose time has gone. Genome Biol., 8:107, 2007.CrossRefGoogle Scholar
  38. 1000.
    J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, L. Kalé, and K. Schulten. Scalable molecular dynamics with NAMD. . Comput. Chem., 26:1781–1802, 2005.Google Scholar
  39. 1012.
    G. E. Plum and K. J. Breslauer. Calorimetry of proteins and nucleic acids. Curr. pin. Struct. Biol., 5:682–690, 1995.CrossRefGoogle Scholar
  40. 1104.
    C. A. Schiffer, J. W. Caldwell, P. A. Kollman, and R. M. Stroud. Protein structure prediction with a combined solvation free energy-molecular mechanics force field. ol. Sim., 10:121–149, 1993.Google Scholar
  41. 1108.
    T. Schlick. A recipe for evaluating and differentiating cos φ expressions. J. Com- put. Chem., 10:951–956, 1989.CrossRefGoogle Scholar
  42. 1116.
    T. Schlick. Engineering teams up with computer-simulation and visualization tools to probe biomolecular mechanisms. Biophys. J., 85:1, 2003.CrossRefGoogle Scholar
  43. 1117.
    T. Schlick. The critical collaboration between art and science: Applying an ex- periment on a bird in an air pump to the ramifications of genomics on society. eonardo, 38:323–329, 2005.Google Scholar
  44. 1121.
    T. Schlick. Molecular-dynamics based approaches for enhanced sampling of long-time, large-scale conformational changes in biomolecules. F1000 Biol. Rep., 1:51, 2009.Google Scholar
  45. 1122.
    T. Schlick. Monte Carlo, harmonic approximation, and coarse-graining approaches for enhanced sampling of biomolecular structure. F1000 Biol. Rep., 1:48, 2009.Google Scholar
  46. 1130.
    T. Schlick, B. Li, and W. K. Olson. The influence of salt on DNA energetics and dynamics. Biophys. J., 67:2146–2166, 1994.CrossRefGoogle Scholar
  47. 1131.
    T. Schlick, M. Mandziuk, R.D. Skeel, and K. Srinivas. Nonlinear resonance artifacts in molecular dynamics simulations. J. Comput. Phys., 139:1–29, 1998.MathSciNetCrossRefGoogle Scholar
  48. 1137.
    T. Schlick and C. S. Peskin. Can classical equations simulate quantum-mechanical behavior? A molecular dynamics investigation of a diatomic molecule with a Morse potential. Comm. Pure App. Math., 42:1141–1163, 1989.MathSciNetMATHCrossRefGoogle Scholar
  49. 1168.
    eferences [1163] H. M. Senn and W. Thiel. QM/MM methods for biological systems. Top. Curr. hem., 268:173–290, 2007.Google Scholar
  50. 1358.
    J. D.Watson. The Double Helix. A Personal Account of the Discovery of the Struc- ture of DNA. Norton Critical Edition G.S. Stent (Editor), Norton & Company, New York, NY, 1980.Google Scholar
  51. 1390.
    P. G.Wolynes. Folding funnels and energy landscapes of larger proteins within the capillarity approximation. Proc. Natl. Acad. Sci. USA, 94:6170–6175, 1997.CrossRefGoogle Scholar
  52. 1397.
    B. Wu, P. Dr¨oge, and C. A. Davey. Site selectivity of platinum anticancer therapeutics. Nat. Chem. Biol., 4:110–112, 2008.Google Scholar
  53. 1398.
    X. Wu and S. Wang. Self-guided molecular dynamics simulation for efficient conformational search. J. Phys. Chem. B, 102:7238–7250, 1998.CrossRefGoogle Scholar
  54. 1400.
    X.Wu, S.Wang, and B. R. Brooks. Direct observation of the folding and unfolding of β-hairpin in explicit water through computer simulation. J. Amer. Chem. Soc., 124:5282–5283, 2002.CrossRefGoogle Scholar
  55. 1424.
    M. A. Young, B. Jayaram, and D. L. Beveridge. Intrusion of counterions into the spine of hydration in the minor groove of B-DNA: Fractional occupancy of electronegative pockets. J. Amer. Chem. Soc., 119:59–69, 1997.CrossRefGoogle Scholar
  56. 1438.
    A. H. Zewail. Physical biology: 4D visualization of complexity. In A. H. Zewail, editor, Physical Biology: From Atoms to Medicine, pages 23–50. Imperial College Press, London, UK, 2008.Google Scholar
  57. 1450.
    Q. Zhao and W. Yang. Analytical energy gradients and geometry optimization in the divide-and-conquer method for large molecules. J. Chem. Phys., 102:9598– 9603, 1995.CrossRefGoogle Scholar
  58. 1451.
    Y. Zhao, B. L. Kormos, D. L. Beveridge, and A. M. Baranger. Molecular dynamics simulations studies of a protein-RNA complex with a selectively modified binding interface. Biopolymers, 81:256–269, 2006.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Courant Institute of Mathematical Sciences and Department of ChemistryNew York UniversityNew YorkUSA

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