Advertisement

The Role of Energy Minimization in Simulation Strategies of Biomolecular Systems

  • D. H. J. Mackay
  • A. J. Cross
  • A. T. Hagler

Abstract

It is now possible to calculate the classical energy of a complex system such as a protein as a function of its coordinates. By making many such calculations for various coordinate values, one can explore multidimensional energy surfaces. These energy surfaces are the basis for molecular dynamics and Monte Carlo studies. Another important method for exploring these energy surfaces is to find configurations for which the energy is a minimum. By this, we mean finding a point in configuration space where all of the forces on the atoms are balanced. By simply minimizing the energy of a molecule, we can identify stable conformations. Perhaps more importantly, by adding external to the molecule in the form of restraints and constraints, a wide range of modeling strategies can be developed using minimization techniques as the foundation to answer specific questions. For example, by forcing specific atoms to overlap atoms in a template structure during a molecular geometry minimization, one can answer the question, “how much energy is required for one molecule to adopt the shape of another.” In this chapter, we discuss how minimization techniques are used in a variety of molecular strategies, focusing on the use of constraints and restraints to extend the scope and utility of traditional structure minimization.

Keywords

Energy Minimization Conjugate Gradient Line Search Steep Descent Target 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abe, H., Braun, W., Noguti, T., and Go, N., 1984, Rapid calculation of firSt and second derivatives of conformational energy with respect to dihedral angles for proteins. General recurrent equations. Compo Chem. 8:239–247.CrossRefGoogle Scholar
  2. Baccanari, D. P., Daluge, S., and King, R. W., 1982. Inhibition of dihydrofolate reductase: Effect of reduced nicotinamide adenine dinucleotide phosphate on the selectivity and affinity of diaminobenzylpyrimidines. Biochemistry 21:5068–5075.PubMedCrossRefGoogle Scholar
  3. Baniak, E. L., Rivier, J. E., Struthers, R. S., Hagler, A. T., and Gierasch, L. M., 1987. Nuclear magnetic resonance analysis and conformational characterization of a cyclic decapeptide antagonist of gonadotropin-releasing hormone. Biochemistry 26: 2642–2656.PubMedCrossRefGoogle Scholar
  4. Berendsen. H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., and Haak, J. R., 1984. Molecular dynamics with coupling to an external bath, J. Chem. Phys. 81:3684–3690.CrossRefGoogle Scholar
  5. Braun, W., and Go, N., 1985. Calculation of protein conformations by proton-proton distance constraints: A new efficient algorithm. J. Mol. Biol. 186:611–626.PubMedCrossRefGoogle Scholar
  6. Brisson, A., and Unwin, P. N. T., 1985. Quaternary structure of the acetylcholine receptor, Nature 315:474–477.PubMedCrossRefGoogle Scholar
  7. Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J., Swaminathan,. S., and Karplus, M., 1983. CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Compo Chem. 4:187–217.CrossRefGoogle Scholar
  8. Brooks, C. L. III, Pettitt, B. M., and Karplus, M., 1985, Structural and energetic effects of truncating long ranged interactions in ionic and polar fluids, J. Chem. Phys. 83:5897–5908.CrossRefGoogle Scholar
  9. Chou, K. C., Nemethy, G., and Scheraga, H. A., 1983. Energetic approach to the packing of α-helices. 1. Equivalent helices, J. Phys. Chem. 87:2869–2881.CrossRefGoogle Scholar
  10. Clore, G. M., Bruenger, A. T., Karplus, M., and Gronenborn, A. M., 1986. Application of molecular dynamics with interproton distance restraints to three-dimensional protein structure determination. A model study of crambin, J. Mol. Biol. 191:523–551.PubMedCrossRefGoogle Scholar
  11. Crippen, G. M., 1977. A novel approach to calculation of conformation: distance geometry, J. Compo Phys. 24: 96–107.CrossRefGoogle Scholar
  12. Crippen, G. M., and Scheraga, H. A., 1969, Minimization of polypeptide energy. VIII. Application of the deflation technique to a dipeptide, Proc. Natl. Acad. Sci. U.S.A. 64:42–49.PubMedCrossRefGoogle Scholar
  13. Crippen, G. M., and Scheraga, H. A., 1971a, Minimization of polypeptide energy. X. A global search algorithm, Arch. Biochem. Biophys. 144:453–461.PubMedCrossRefGoogle Scholar
  14. Crippen, G. M., and Scheraga, H. A., 1971b, Minimization of polypeptide energy. XI. Method of gentlest ascent, Arch. Biochem. Biophys. 144:462–466.PubMedCrossRefGoogle Scholar
  15. Crippen, G. M., and Scheraga, H. A., 1973, Minimization of polypeptide energy. XII. Methods of partial energies and cubic subdivision, J. Comput. Phys. 12:491–497.CrossRefGoogle Scholar
  16. Dauber-Osguthorpe, P., Roberts, V. A., Osguthorpe, D. J., Wolff, J., Genest, M., and Hagler, A. T., 1988, Structure and energetics of ligand binding to proteins: E. coli dihydrofolate reductase-trimethoprim, a drug-receptor system, Proteins: Structure, Function and Genetics, 4: 31–47.CrossRefGoogle Scholar
  17. Dayringer, H. E., Tramontano, A., Sprang, S. R., and Retterick, R. J., 1986, Interactive program for visualization and modelling of proteins, nucleic acids, and small molecules, J. Mol. Graphics 6:82–87.CrossRefGoogle Scholar
  18. Eisenberg, D., 1984, Three-dimensional structure of membrane and surface proteins, Annu. Rev. Biochem. 53: 595–623.PubMedCrossRefGoogle Scholar
  19. Ermer, O., 1976, Calculation of molecular properties using force fields. Applications in organic chemistry, Structure Bonding 27:161–211.Google Scholar
  20. Fine, R. M., Wang, H., Shenkin, P. S., Yarmush, D. L., and Levinthal, C., 1986, Predicting antibody hypervariable loop conformations II: Minimization and molecular dynamics studies of MCPC603 from many randomly generated loop conformations, Proteins 1:342–362.PubMedCrossRefGoogle Scholar
  21. Fletcher, R., 1980, Practical Methods of Optimization, Volume 1, John Wiley & Sons, New York.Google Scholar
  22. Furois-Corbin, S., and Pullman, A., 1986, Theoretical study of the packing of α-helices by energy minimization: Effect of the length of the helices on the packing energy and on the optimal configuration of a pair, Chem. Phys. Lett. 123:305–310.CrossRefGoogle Scholar
  23. Gibson, K. D., and Scheraga, H. A., 1986, Predicted conformations for the immunodominant region of the circumsporozoite protein of the human malaria parasite Plasmodium falciparum, Proc. Natl. Acad. Sci. U.S.A. 83:5649–5653.PubMedCrossRefGoogle Scholar
  24. Gilson, M. K., Rashin, A., Fine, R., Honig, B., Kline, A. D., and Wüthrich, K., 1985, Secondary structure of the α-amylase polypeptide inhibitor tendamistat from Streptomyces tendae determined in solution by 1H nuclear magnetic resonance, J. Mol. Biol. 183:503–507.CrossRefGoogle Scholar
  25. Greer, J., 1985, Protein structure and function by comparative model building, Ann. N.Y. Acad. Sci. 439:44–63.CrossRefGoogle Scholar
  26. Hagler, A. T., 1985, Theoretical simulation of conformation, energetics, and dynamics of peptides, in: The Peptides, Volume 7 (V. J. Hruby and J. Meienhofer, eds.), Academic Press, New York, pp. 213–299.Google Scholar
  27. Hagler, A. T., and Moult, J., 1978, Computer simulation of the solvent structure in biological macromolecules, Nature 272:222–226.PubMedCrossRefGoogle Scholar
  28. Hagler, A. T., Lifson, S., and Dauber, P., 1979a, Consistent force field studies of intermolecular forces in hydrogen bonded crystals. 2. A benchmark for the objective comparison of alternative force fields, J. Am. Chem. Soc. 101:5122–5130.CrossRefGoogle Scholar
  29. Hagler, A. T., Dauber, P., and Lifson, S., 1979b, Consistent force field studies of intermolecular forces in hydrogen bonded crystals. 3. The C=O···H-O hydrogen bond and the analysis of the energetics and packing of carboxylic acids, J. Am. Chem. Soc. 101:5131–5141.CrossRefGoogle Scholar
  30. Hagler, A. T., Stern, P. S., Sharon, R., Becker, J. M., and Naider, F., 1979c, Computer simulation of the conformational properties of oligopeptides: Comparison of theoretical methods and analysis of experimental results, J. Am. Chem. Soc. 101:6842–6852.CrossRefGoogle Scholar
  31. Hagler, A. T., Osguthorpe, D. J., Dauber-Osguthorpe, P., and Hempel, J. C., 1985, Dynamics and conformational energetics of a peptide hormone: Vasopressin, Science 227:1309–1315.PubMedCrossRefGoogle Scholar
  32. Harvey, S. C., and McCammon, J. A., 1982, Macromolecular conformational energy minimization: An algorithm varying pseudodihedral angles, Comput. Chem. 6:173–179.CrossRefGoogle Scholar
  33. Havel, T. F., and Wuthrich, K., 1984, A distance geometry program for determining the structures of small proteins and other macromolecules from nuclear magnetic resonance measurements of intramolecular 1H-1H proximities in solution, Bull. Math. Biol. 46:673–698.Google Scholar
  34. Havel, T. F., Kuntz, I. D., and Crippen, G. M., 1983, The theory and practice of distance geometry, Bull. Math. Biol. 45:665–720.Google Scholar
  35. Hwang, J. K., and Warshel, A., 1987, Semiquantitative calculations of catalytic free energies in genetically modified enzymes, Biochemistry 26:2669–2273.PubMedCrossRefGoogle Scholar
  36. Jones, T. A., 1982, FRODO: A graphics fitting program for macromolecules, in: Computational Crystallography (D. Sayre, ed.), Clarendon Press, London, p. 303.Google Scholar
  37. Karfunkel, H. R., 1986, A fast algorithm for the interactive docking maneuver with flexible macromolecules and probes, J. Comput. Chem. 7:113–128.CrossRefGoogle Scholar
  38. Katz, L., and Levinthal, C., 1972, Interactive computer graphics and representation of complex biological structures, Annu. Rev. Biophys. Bioeng. 1:465–504.PubMedCrossRefGoogle Scholar
  39. Kirkwood, J. G., 1935, Statistical mechanics of fluid mixtures, J. Chem. Phys. 3:300–313.CrossRefGoogle Scholar
  40. Kitson, D. H., and Hagler, A. T., 1988, Theoretical studies of the structure and molecular dynamics of a peptide crystal, Biochemistry 27: 5246–5257.PubMedCrossRefGoogle Scholar
  41. Lifson, S., Hagler, A. T., and Dauber, P., 1979, Consistent force field studies of intermolecular forces in hydrogen bonded crystals. I. Carboxylic acids, amides, and the C=O···H-O hydrogen bonds, J. Am. Chem. Soc. 101:5111–5121.CrossRefGoogle Scholar
  42. Maple, J. R., Dinur, U., and Hagler, A. T., Derivation of forcefields for molecular mechanics and dynamics from ab initio energy surfaces, Proc. Natl. A cad. Sci. U.S.A. 85:5350–5354.Google Scholar
  43. Matthews, D. A., Alden, R. A., Bolin, J. T., Freer, S. T., Hamlin, R., Xuong, N., Kraut, J., Poe, M., Williams, M., and Hoogsteen, K., 1977, Dihydrofolate reductase: X-ray structure of the binary complex with methotrexate, Science 197:452–455.PubMedCrossRefGoogle Scholar
  44. Mezei, M., and Beveridge, D., 1986, Free energy simulations. Ann. N.Y. Acad.Sci. 482: 1–23.PubMedCrossRefGoogle Scholar
  45. Moult, J., and James, M. N. G., 1986, An algorithm for determining the conformation of polypeptide segments in proteins by systematic search, Proteins 1:146–163.PubMedCrossRefGoogle Scholar
  46. Nilsson, L., and Karplus, M., 1986, Empirical energy functions for energy minimization and dynamics of nucleic acids, J. Comput. Chem. 7:591–616.CrossRefGoogle Scholar
  47. Noguti, T., and Go, N., 1983, A method of rapid calculation of a second derivative matrix of conformational energy for large molecules, J. Phys. Soc. (Jpn.) 52:3685–3690.CrossRefGoogle Scholar
  48. Pattabiraman, N., Levitt, M., Ferrin, T. E., and Langridge, R., 1985, Computer graphics in real-time docking with energy calculation and minimization, J. Comput. Chem. 6:432–436.CrossRefGoogle Scholar
  49. Poe, M., Hoogsteen, K., and Matthews, D. A., 1979, Proton magnetic resonance studies on E. coli dihydrofolate reductase, J. Biol. Chem. 254:8143–8152.PubMedGoogle Scholar
  50. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., 1986, Numerical Recipes, The Art of Scientific Computing, Cambridge University Press, Cambridge.Google Scholar
  51. Purisima, E. O., and Scheraga, H. A., 1986, An approach to the multiple-minima problem by relaxing dimensionality, Proc. Natl. Acad. Sci. U.S.A. 83:2782–2786.PubMedCrossRefGoogle Scholar
  52. Quirke, N., and Jacucci, G., 1982, Energy difference functions in Monte Carlo simulations: Application to (1) the calculation of free energy of liquid nitrogen, (2) the calculation of fluctuation in Monte Carlo averages, Mol. Phys. 45:823–838.CrossRefGoogle Scholar
  53. Rivier, J., Kupryszewski, G., Varga, J., Porter, J., Rivier, C., Perrin, M., Hagler, A., Struthers, S., and Corrigan, A., Design of potent cyclic gonadotropin releasing hormone antagonists, J. Med. Chem. 31: 677–682.Google Scholar
  54. Roberts, V. A., Dauber-Osguthorpe, P., Osguthorpe, D. J., Levin, E., and Hagler, A. T., 1986, A comparison of the binding of the ligand trimethoprim to bacterial and vertebrate dihydrofolate reductases, Isr. J. Chem 27:198–210.Google Scholar
  55. Singh, U. C., Brown, F. K., Bash, P. A., and Kollman, P. A., 1987, An approach to the application of free energy perturbation methods using molecular dynamics: Applications to the transformations of methanol to ethane, oxonium to ammonium, glycine to alanine, and alanine to phenylalanine in aqueous solution and to H3O+(H3O)3 NH4 + (H2O3) in the gas phase, J. Am. Chem. Soc. 109:1607–1614.CrossRefGoogle Scholar
  56. Stem, P. S., Chorev, M., Goodman, M., and Hagler, A. T., 1983, Computer simulation of the conformational properties of retro-inverso peptides. I. Empirical force field calculations of rigid and flexible geometries of N-acetylglycine-N′-methylamide, bis(acetamido)methane, and N,N -dimethylmalonamide and their corresponding Cα-methylated analogs, Biopolymers 22:1885–1900.CrossRefGoogle Scholar
  57. Straatsma, T. P., Berendsen, H. J. C., and Postma, J. P. M., 1986, Free energy of hydrophobic hydration: A molecular dynamics study of noble gases in water, J. Chem. Phys. 85:6720–6727.CrossRefGoogle Scholar
  58. Struthers, R. S., Rivier, J., and Hagler, A. T., 1984, Design of peptide analogs: Theoretical simulation of conformation, energetics, and dynamics, in: Coriformationally Directed Drug Design: Peptides and Nucleic Acids as Templates or Targets (J. A. Vida and M. Gordon, eds.), American Chemical Society, Washington, pp. 239–261, American Chemical Society, Washington.CrossRefGoogle Scholar
  59. Tembe, B. L., and McCammon, J. A., 1984, Ligand-receptor interactions, Comput. Chem. 8:281–283.CrossRefGoogle Scholar
  60. Van Gunsteren, W. F., and Karplus, M., 1980, A method for constrained energy minimization of macromolecules, J. Comput. Chem. 1:266–274.CrossRefGoogle Scholar
  61. Vasquez, M., and Scheraga, H. A., 1985, Use of buildup and energy-minimization procedures to compute low-energy structures of the backbone of enkephalin, Biopolymers 24:1437–1447.PubMedCrossRefGoogle Scholar
  62. Warme, P. K., and Scheraga, H. A., 1974, Refinement of the x-ray structure of lysozyme by complete energy minimization, Biochemistry 13:757–767.PubMedCrossRefGoogle Scholar
  63. Warshel, A., Sussman, F., and King, G., 1986, Free energy of charges in solvated proteins: Microscopic calculations using a reversible charging process, Biochemistry 25:8368–8372.PubMedCrossRefGoogle Scholar
  64. Warwicker, J., 1986, Continuum dielectric modelling of the protein-solvent system, and calculation of the long-range electrostatic field of the enzyme phosphoglycerate mutase, J. Theor. Biol. 121:199–210.PubMedCrossRefGoogle Scholar
  65. Warwicker, J., Ollis, D., Richards, F. M., and Steitz, T. A., 1985, Electrostatic field of the large fragment of Escherichia coli DNA polymerase I, J. Mol. Biol. 186:645–649.PubMedCrossRefGoogle Scholar
  66. Weiner, S. J., Kollman, P. A., Case, D. A., Singh, U. C., Ghio, C., Alagona, G., Profeta, S., Jr., and Weiner, P., 1984, A new force field for molecular mechanical simulation of nucleic acids and proteins, J. Am. Chem. Soc. 106:765–784.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • D. H. J. Mackay
    • 1
  • A. J. Cross
    • 1
  • A. T. Hagler
    • 1
    • 2
  1. 1.Biosym Technologies, Inc.San DiegoUSA
  2. 2.Agouron InstituteLa JollaUSA

Personalised recommendations