Hybrid Quantum and Classical Methods for Computing Kinetic Isotope Effects of Chemical Reactions in Solutions and in Enzymes

  • Jiali Gao
  • Dan T. Major
  • Yao Fan
  • Yen-lin Lin
  • Shuhua Ma
  • Kin-Yiu Wong
Part of the Methods Molecular Biology™ book series (MIMB, volume 443)


A method for incorporating quantum mechanics into enzyme kinetics modeling is presented. Three aspects are emphasized: 1) combined quantum mechanical and molecular mechanical methods are used to represent the potential energy surface for modeling bond forming and breaking processes, 2) instantaneous normal mode analyses are used to incorporate quantum vibrational free energies to the classical potential of mean force, and 3) multidimensional tunneling methods are used to estimate quantum effects on the reaction coordinate motion. Centroid path integral simulations are described to make quantum corrections to the classical potential of mean force. In this method, the nuclear quantum vibrational and tunneling contributions are not separable. An integrated centroid path integral—free energy perturbation and umbrella sampling (PI-FEP/UM) method along with a bisection sampling procedure was summarized, which provides an accurate, easily convergent method for computing kinetic isotope effects for chemical reactions in solution and in enzymes. In the ensemble-averaged variational transition state theory with multidimensional tunneling (EA-VTST/MT), these three aspects of quantum mechanical effects can be individually treated, providing useful insights into the mechanism of enzymatic reactions. These methods are illustrated by applications to a model process in the gas phase, the decarboxy-lation reaction of N-methyl picolinate in water, and the proton abstraction and reprotonation process catalyzed by alanine racemase. These examples show that the incorporation of quantum mechanical effects is essential for enzyme kinetics simulations.


Combined QM/MM Dual-level potential Enzyme kinetics Kinetic isotope effects Path integral simulations PI-FEP/UM Solvent effects 



Acknowledgements This work was supported in part by the National Institutes of Health and by the Army Research Laboratory through the Army High-Performance Computing Research Center (AHPCRC) under the auspices of Army Research Laboratory DAAD 19-01-2-0014. JG is grateful to Professor Modesto Orozco for hospitality during his sabbatical leave at the Barcelona Super-computing Center, and acknowledges support from the Ministerio de Educación y Ciencia, Spain.


  1. 1.
    1. Sumner, J. B. (1926) “THE ISOLATION AND CRYSTALLIZATION OF THE ENZYME UREASE. PRELIMINARY PAPER,” J. Biol. Chem. 69, 435–441.Google Scholar
  2. 2.
    2. Pauling, L. (1946) “Molecular architecture and biological reactions,” Chem. Eng. News 24, 1375.Google Scholar
  3. 3.
    3. Schowen, R. L. (1978) in Transition States of Biochemical Processes (Gandour, R. D., and Schowen, R. L., Eds.) pp 77–114, Plenum Press, New York.Google Scholar
  4. 4.
    4. Gao, J., Ma, S., Major, D. T., Nam, K., Pu, J., and Truhlar, D. G. (2006) “Mechanisms and free energies of enzymatic reactions,” Chem. Rev. 106, 3188–3209.PubMedGoogle Scholar
  5. 5.
    5. Garcia-Viloca, M., Gao, J., Karplus, M., and Truhlar, D. G. (2004) “How enzymes work: Analysis by modern rate theory and computer simulations,” Science 303, 186–195.PubMedGoogle Scholar
  6. 6.
    6. Villa, J. and Warshel, A. (2001) “Energetics and dynamics of enzymatic reactions,” J. Phys. Chem. B 105, 7887–7907.Google Scholar
  7. 7.
    7. Gao, J. and Truhlar, D. G. (2002) “Quantum mechanical methods for enzyme kinetics,” Ann. Rev. Phys. Chem. 53, 467–505.Google Scholar
  8. 8.
    8. Pu, J., Gao, J., and Truhlar, D. G. (2006) “Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions,” Chem. Rev. 106, 3140–3169.PubMedGoogle Scholar
  9. 9.
    9. Wolfenden, R. and Snider, M. J. (2001) “The depth of chemical time and the power of enzymes as catalysts,” Acc. Chem. Res. 34, 938–945.PubMedGoogle Scholar
  10. 10.
    10. Snider, M. G., Temple, B. S., and Wolfenden, R. (2004) “The path to the transition state in enzyme reactions: A survey of catalytic efficiencies,” J. Phys. Org. Chem. 17, 586–591.Google Scholar
  11. 11.
    11. Truhlar, D. G., Garrett, B. C., and Klippenstein, S. J. (1996) “Current status of transition-state theory,” J. Phys. Chem. 100, 12771–12800.Google Scholar
  12. 12.
    12. Fernandez-Ramos, A., Miller, J. A., Klippenstein, S. J., and Truhlar, D. G. (2006) “Modeling the kinetics of bimolecular reactions,” Chem. Rev. 106, 4518–4584.PubMedGoogle Scholar
  13. 13.
    13. Bennett, C. H. (1977) “Molecular dynamics and transition state theory: the simulation of infrequent events,” ACS Symp. Ser. 46, 63–97.Google Scholar
  14. 14.
    14. Chandler, D. (1978) “Statistical mechanics of isomerization dynamics in liquids and the transition state approximation,” J. Chem. Phys. 68, 2959–2970.Google Scholar
  15. 15.
    15. Neria, E. and Karplus, M. (1997) “Molecular dynamics of an enzyme reaction: proton transfer in TIM,” Chem. Phys. Lett. 267, 26–30.Google Scholar
  16. 16.
    16. Nam, K., Prat-Resina, X., Garcia-Viloca, M., Devi-Kesavan, L. S., and Gao, J. (2004) “Dynamics of an enzymatic substitution reaction in haloalkane dehalogenase,” J. Am. Chem. Soc. 126, 1369–1376.PubMedGoogle Scholar
  17. 17.
    17. Voth, G. A. and Hochstrasser, R. M. (1996) “Transition state dynamics and relaxation processes in solutions: a frontier of physical chemistry,” J. Phys. Chem. 100, 13034–13049.Google Scholar
  18. 18.
    18. Alhambra, C., Corchado, J., Sanchez, M. L., Garcia-Viloca, M., Gao, J., and Truhlar, D. G. (2001) “Canonical variational theory for enzyme kinetics with the protein mean force and multidimensional quantum mechanical tunneling dynamics. Theory and application to liver alcohol dehydrogenase,” J. Phys. Chem. B 105, 11326–11340.Google Scholar
  19. 19.
    19. Valleau, J. P. and Torrie, G. M. (1977) in Modern Theoretical Chemistry (Berne, B. J., Ed.) pp 169–194, Plenum, New York.Google Scholar
  20. 20.
    20. Jorgensen, W. L. (1989) “Free energy calculations: a breakthrough for modeling organic chemistry in solution,” Acc. Chem. Res. 22, 184–189.Google Scholar
  21. 21.
    21. Hwang, J. K., King, G., Creighton, S., and Warshel, A. (1988) “Simulation of free energy relationships and dynamics of SN2 reactions in aqueous solution,” J. Am. Chem. Soc. 110, 5297–5311.Google Scholar
  22. 22.
    22. Gertner, B. J., Bergsma, J. P., Wilson, K. R., Lee, S., and Hynes, J. T. (1987) “Nonadiabatic solvation model for SN2 reactions in polar solvents,” J. Chem. Phys. 86, 1377–1386.Google Scholar
  23. 23.
    23. Gao, J. (1996) “Hybrid quantum mechanical/molecular mechanical simulations: an alternative avenue to solvent effects in organic chemistry,” Acc. Chem. Res. 29, 298–305.Google Scholar
  24. 24.
    24. Muller, R. P. and Warshel, A. (1995) “Ab initio calculations of free energy barriers for chemical reactions in solution,” J. Phys. Chem. 99, 17516–17524.Google Scholar
  25. 25.
    25. Mo, Y. and Gao, J. (2000) “Ab initio QM/MM simulations with a molecular orbital-valence bond (MOVB) method: application to an SN2 reaction in water,” J. Comput. Chem. 21, 1458–1469.Google Scholar
  26. 26.
    26. Truhlar, D. G. and Garrett, B. C.(2000) “Multidimensional transition state theory and the validity of Grote-Hynes theory,” J. Phys. Chem. B 104, 1069–1072.Google Scholar
  27. 27.
    27. Truhlar, D. G., Gao, J., Alhambra, C., Garcia-Viloca, M., Corchado, J., Sanchez, M. L., and Villa, J. (2002) “The incorporation of quantum effects in enzyme kinetics modeling,” Acc. Chem. Res. 35, 341–349.PubMedGoogle Scholar
  28. 28.
    28. MacKerell, A. D., Jr., Bashford, D., Bellott, M., Dunbrack, R. L., Evanseck, J. D., Field, M. J., Fischer, S., Gao, J., Guo, H., Ha, S., Joseph-McCarthy, D., Kuchnir, L., Kuczera, K., Lau, F. T. K., Mattos, C., Michnick, S., Ngo, T., Nguyen, D. T., Prodhom, B., Reiher, W. E., III, Roux, B., Schlenkrich, M., Smith, J. C., Stote, R., Straub, J., Watanabe, M., Wiorkiewicz-Kuczera, J., Yin, D., and Karplus, M. (1998) “All-atom empirical potential for molecular modeling and dynamics studies of proteins,” J. Phys. Chem. B 102, 3586–3616.Google Scholar
  29. 29.
    29. Ponder, J. W. and Case, D. A. (2003) “Advances in Protein Chemisty”, edited by V. Daggett, QM/MM and related approaches, Adv. Protein Chem. 66, 27–85.PubMedGoogle Scholar
  30. 30.
    30. Gao, J. and Xia, X. (1992) “A prior evaluation of aqueous polarization effects through Monte Carlo QM-MM simulations,” Science 258, 631–635.PubMedGoogle Scholar
  31. 31.
    31. Gao, J. (1995) in Rev. Comput. Chem. (Lipkowitz, K. B., and Boyd, D. B., Eds.) Methods and applications of combined QM/MM methods. pp 119–185, VCH, New York.Google Scholar
  32. 32.
    32. Chandrasekhar, J., Smith, S. F., and Jorgensen, W. L. (1984) “SN2 reaction profiles in the gas phase and aqueous solution,” J. Am. Chem. Soc. 106, 3049–3050.Google Scholar
  33. 33.
    33. Gao, J. (1991) “A priori computation of a solvent-enhanced SN2 reaction profile in water: the Menshutkin reaction,” J. Am. Chem. Soc. 113, 7796–7797.Google Scholar
  34. 34.
    34. Donini, O., Darden, T., and Kollman, P. A. (2000) “QM-FE calculations of aliphatic hydrogen abstraction in citrate synthase and in solution: reproduction of the effect of enzyme catalysis and demonstration that an enolate rather than an enol is formed,” J. Am. Chem. Soc. 122, 12270–12280.Google Scholar
  35. 35.
    35. Zhang, Y. K., Liu, H. Y., and Yang, W. T. (2000) “Free energy calculation on enzyme reactions with an efficient iterative procedure to determine minimum energy paths on a combined ab initio QM/MM potential energy surface,” J. Chem. Phys. 112, 3483–3492.Google Scholar
  36. 36.
    36. Liu, H., Lu, Z., Cisneros, G. A., and Yang, W. (2004) “Parallel iterative reaction path optimization in ab initio quantum mechanical/molecular mechanical modeling of enzyme reactions,” J. Chem. Phys. 121, 697–706.PubMedGoogle Scholar
  37. 37.
    37. Warshel, A. and Weiss, R. M. (1980) “An empirical valence bond approach for comparing reactions in solutions and in enzymes,” J. Am. Chem. Soc. 102, 6218–6226.Google Scholar
  38. 38.
    38. Warshel, A. (1991) Computer Modeling of Chemical Reactions in Enzymes and Solutions, Wiley, New York.Google Scholar
  39. 39.
    39. Aaqvist, J., Fothergill, M., and Warshel, A. (1993) “Computer simulation of the carbon dioxide/bicarbonate interconversion step in human carbonic anhydrase I,” J. Am. Chem. Soc. 115, 631–635.Google Scholar
  40. 40.
    40. Kim, Y., Corchado, J. C., Villa, J., Xing, J., and Truhlar, D. G. (2000) “Multiconfiguration molecular mechanics algorithm for potential energy surfaces of chemical reactions,” J. Chem. Phys. 112, 2718–2735.Google Scholar
  41. 41.
    41. Olsson, M. H. M. and Warshel, A. (2004) “Solute solvent dynamics and energetics in enzyme catalysis: the SN2 reaction of dehalogenase as a general benchmark,” J. Am. Chem. Soc. 126, 15167–15179.PubMedGoogle Scholar
  42. 42.
    42. Warshel, A. (2003) “Computer simulations of enzyme catalysis: Methods, progress, and insights,” Ann. Rev. Biophys. Biomol. Struct. 32, 425–443.Google Scholar
  43. 43.
    43. Aqvist, J. and Warshel, A. (1993) “Simulation of enzyme reactions using valence bond force fields and other hybrid quantum/classical approaches,” Chem. Rev. 93, 2523–2544.Google Scholar
  44. 44.
    44. Warshel, A., Sharma, P. K., Kato, M., Xiang, Y., Liu, H., and Olsson, M. H. M. (2006) “Electrostatic basis for enzyme catalysis,” Chem. Rev. 106, 3210–3235.PubMedGoogle Scholar
  45. 45.
    45. Mo, Y., Zhang, Y., and Gao, J. (1999) “A simple electrostatic model for trisilylamine: theoretical examinations of the n?sigma. Negative hyperconjugation, p.pi.→d.pi. bonding, and stereoelectronic interaction,” J. Am. Chem. Soc. 121, 5737–5742.Google Scholar
  46. 46.
    46. Mo, Y. and Gao, J. (2000) “An ab initio molecular orbital-valence bond (MOVB) method for simulating chemical reactions in solution,” J. Phys. Chem. A 104, 3012–3020.Google Scholar
  47. 47.
    47. Gao, J. and Mo, Y. (2000) “Simulation of chemical reactions in solution using an ab initio molecular orbital-valence bond model,” Prog. Theor. Chem. Phys. 5, 247–268.Google Scholar
  48. 48.
    48. Gao, J., Garcia-Viloca, M., Poulsen, T. D., and Mo, Y. (2003) “Solvent effects, reaction coordinates, and reorganization energies on nucleophilic substitution reactions in aqueous solution,” Adv. Phys. Org. Chem. 38, 161–181.Google Scholar
  49. 49.
    49. Albu, T. V., Corchado, J. C., and Truhlar, D. G. (2001) “Molecular mechanics for chemical reactions: a standard strategy for using multiconfiguration molecular mechanics for variational transition state theory with optimized multidimensional tunneling,” J. Phys. Chem. A 105, 8465–8487.Google Scholar
  50. 50.
    50. Liu, H., Elstner, M., Kaxiras, E., Fraunheim, T., Hermans, J., and Yang, W. (2001) “Quantum mechanics simulation of protein dynamics on long timescale,” Proteins: Struct., Funct. Gen. 44, 484–489.Google Scholar
  51. 51.
    51. Monard, G. and Merz, K. M., Jr. (1999) “Combined quantum mechanical/molecular mechanical methodologies applied to biomolecular systems,” Acc. Chem. Res. 32, 904–911.Google Scholar
  52. 52.
    52. Car, R. and Parrinello, M. (1985) “Unified approach for molecular dynamics and density-functional theory,” Phys. Rev. Lett. 55, 2471–2474.PubMedGoogle Scholar
  53. 53.
    53. Tuckerman, M. E., Laasonen, K., Sprik, M., and Parrinello, M. (1994) “Ab initio simulations of water and water ions,” J. Phys.: Conden. Matter 6, A93–A100.Google Scholar
  54. 54.
    54. Sprik, M., Hutter, J., and Parrinello, M. (1996) “Ab initio molecular dynamics simulation of liquid water: comparison of three gradient-corrected density functionals,” J. Chem. Phys. 105, 1142–1152.Google Scholar
  55. 55.
    55. Rothlisberger, U., Carloni, P., Doclo, K., and Parrinello, M. (2000) “A comparative study of galactose oxidase and active site analogs based on QM/MM Car-Parrinello simulations,” J Biol Inorg Chem 5, 236–250.PubMedGoogle Scholar
  56. 56.
    56. Rohrig, U. F., Guidoni, L., and Rothlisberger, U. (2005) Chem Phys Chem 6, 1836.PubMedGoogle Scholar
  57. 57.
    57. York, D. M., Lee, T.-S., and Yang, W. (1998) “Quantum mechanical treatment of biological macromolecules in solution using linear-scaling electronic structure methods,” Phys. Rev. Lett. 80, 5011–5014.Google Scholar
  58. 58.
    58. Titmuss, S. J., Cummins, P. L., Rendell, A. P., Bliznyuk, A. A., and Gready, J. E. (2002) “Comparison of linear-scaling semiempirical methods and combined quantum mechanical/molecular mechanical methods for enzymic reactions. II. An energy decomposition analysis,” J. Comput. Chem. 23, 1314–1322.PubMedGoogle Scholar
  59. 59.
    59. Van der Vaart, A. and Merz, K. M., Jr. (1999) “The role of polarization and charge transfer in the solvation of biomolecules,” J. Am. Chem. Soc. 121, 9182–9190.Google Scholar
  60. 60.
    60. Stewart, J. J. P. (1996) “Application of localized molecular orbitals to the solution of semi-empirical self-consistent field equations,” Int. J. Quantum Chem. 58, 133–146.Google Scholar
  61. 61.
    61. Gao, J. and Thompson, M. A. (1998) Combined Quantum Mechanical and Molecular Mechanical Methods, Vol. 712, American Chemical Society, Washington, DC.Google Scholar
  62. 62.
    62. Warshel, A. and Levitt, M. (1976) “Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme,” J. Mol. Biol. 103, 227–249.PubMedGoogle Scholar
  63. 63.
    63. Singh, U. C. and Kollman, P. A. (1986) “A combined ab initio quantum mechanical and molecular mechanical method for carrying out simulations on complex molecular systems: applications to the CH3Cl + Cl-exchange reaction and gas phase protonation of polyenes.,” J. Comput. Chem. 7, 718–730.Google Scholar
  64. 64.
    64. Tapia, O., Lluch, J. M., Cardenas, R., and Andres, J. (1989) “Theoretical study of solvation effects on chemical reactions. A combined quantum chemical/Monte Carlo study of the Meyer-Schuster reaction mechanism in water,” J. Am. Chem. Soc. 111, 829–835.Google Scholar
  65. 65.
    65. Field, M. J., Bash, P., A., and Karplus, M. (1990) “A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations,” J. Comput. Chem. 11, 700–733.Google Scholar
  66. 66.
    66. Gao, J. (1992) “Absolute free energy of solvation from Monte Carlo simulations using combined quantum and molecular mechanical potentials,” J. Phys. Chem. 96, 537–540.Google Scholar
  67. 67.
    67. Stanton, R. V., Hartsough, D. S., and Merz, K. M., Jr. (1993) “Calculation of solvation free energies using a density functional/molecular dynamics coupled potential,” J. Phys. Chem. 97, 11868–11870.Google Scholar
  68. 68.
    68. Freindorf, M. and Gao, J. (1996) “Optimization of the Lennard-Jones parameters for a combined ab initio quantum mechanical and molecular mechanical potential using the 3-21G basis set.,” J. Comput. Chem. 17, 386–395.Google Scholar
  69. 69.
    69. Hillier, I. H. (1999) “Chemical reactivity studied by hybrid QM/MM methods,” Theochem 463,45–52.Google Scholar
  70. 70.
    70. Morokuma, K. (2002) Phil. Trans. Roy. Soc. London, Ser. A 360, 1149.Google Scholar
  71. 71.
    71. Warshel, A. and Karplus, M. (1972) “Calculation of ground and excited state potential surfaces of conjugated molecules. I. Formulation and parametrization,” J. Amer. Chem. Soc. 94, 5612–5625.Google Scholar
  72. 72.
    72. Bash, P. A., Field, M. J., and Karplus, M. (1987) “Free energy perturbation method for chemical reactions in the condensed phase: a dynamic approach based on a combined quantum and molecular mechanics potential,” J. Am. Chem. Soc. 109, 8092–8094.Google Scholar
  73. 73.
    73. Gao, J. (1995) “An automated procedure for simulating chemical reactions in solution. Application to the decarboxylation of 3-carboxybenzisoxazole in water,” J. Am. Chem. Soc. 117, 8600–8607.Google Scholar
  74. 74.
    74. Wu, N., Mo, Y., Gao, J., and Pai, E. F. (2000) “Electrostatic stress in catalysis: structure and mechanism of the enzyme orotidine monophosphate decarboxylase,” Proc. Natl. Acad. Sci. U.S.A. 97, 2017–2022.PubMedGoogle Scholar
  75. 75.
    75. Gao, J. (1994) “Computation of intermolecular interactions with a combined quantum mechanical and classical approach,” ACS Symp. Ser. 569, 8–21.Google Scholar
  76. 76.
    76. Dewar, M. J. S., Zoebisch, E. G., Healy, E. F, and Stewart, J. J. P. (1985) “Development and use of quantum mechanical molecular models. 76. AM1: a new general purpose quantum mechanical molecular model,” J. Am. Chem. Soc. 107, 3902–3909.Google Scholar
  77. 77.
    77. Stewart, J. J. P. (1989) “Optimization of parameters for semiempirical methods I. Method,” J. Comp. Chem. 10, 209–220.Google Scholar
  78. 78.
    78. Orozco, M., Luque, F. J., Habibollahzadeh, D., and Gao, J. (1995) “The polarization contribution to the free energy of hydration. [Erratum to document cited in CA122:299891],” J. Chem. Phys. 103, 9112.Google Scholar
  79. 79.
    79. Major, D. T., York, D. M., and Gao, J. (2005) “Solvent polarization and kinetic isotope effects in nitroethane deprotonation and implications to the nitroalkane oxidase reaction,” J. Am. Chem. Soc. 127, 16374–16375.PubMedGoogle Scholar
  80. 80.
    80. Gao, J. (1994) “Monte Carlo quantum mechanical-configuration interaction and molecular mechanics simulation of solvent effects on the n → pi.* blue shift of acetone,” J. Am. Chem. Soc. 116, 9324–9328.Google Scholar
  81. 81.
    81. Gao, J. and Byun, K. (1997) “Solvent effects on the n → pi* transition of pyrimidine in aqueous solution,” Theor. Chem. Acc. 96, 151–156.Google Scholar
  82. 82.
    82. Garcia-Viloca, M., Truhlar, D. G, and Gao, J. (2003) “Importance of substrate and cofactor polarization in the active site of dihydrofolate reductase,” J. Mol. Biol. 327, 549–560.PubMedGoogle Scholar
  83. 83.
    83. Byun, K., Mo, Y., and Gao, J. (2001) “New insight on the origin of the unusual acidity of Meldrum's acid from ab initio and combined QM/MM simulation study,” J. Am. Chem. Soc. 123, 3974–3979.PubMedGoogle Scholar
  84. 84.
    84. Maseras, F. and Morokuma, K. (1995) “IMOMM: a new integrated ab initio + molecular mechanics geometry optimization scheme of equilibrium structures and transition states,” J. Comput. Chem. 16, 1170–1179.Google Scholar
  85. 85.
    85. Mulholland, A. J. (2001) “The QM/MM approach to enzymatic reactions,” Theor. Comput. Chem. 9, 597–653.Google Scholar
  86. 86.
    86. Zwanzig, R. (1954) “High-temperature equation of state by a perturbation method. I. Non-polar gases,” J. Chem. Phys. 22, 1420–1426.Google Scholar
  87. 87.
    87. Jorgensen, W. L. and Ravimohan, C. (1985) “Monte Carlo simulation of differences in free energies of hydration,” J. Chem. Phys. 83, 3050–3054.Google Scholar
  88. 88.
    88. Kollman, P. (1993) “Free energy calculations: Applications to chemical and biochemical phenomena,” Chem. Rev. 93, 2395–2417.Google Scholar
  89. 89.
    89. Simonson, T. (2001) in Computational Biochemistry and Biophysics (Becker, O. M., MacKerell, A. D., Jr., Roux, B., and Watanabe, M., Eds.) pp. 169–197. Dekker, New York.Google Scholar
  90. 90.
    90. Espinosa-Garcia, J., Corchado, J. C., and Truhlar, D. G. (1997) “importance of quantum effects for C-H bond activation reactions,” J. Am. Chem. Soc. 119, 9891–9896.Google Scholar
  91. 91.
    91. Alhambra, C., Gao, J., Corchado, J. C., Villa, J., and Truhlar, D. G. (1999) “Quantum mechanical dynamical effects in an enzyme-catalyzed proton transfer reaction,” J. Am. Chem. Soc. 121, 2253–2258.Google Scholar
  92. 92.
    92. Billeter, S. R., Webb, S. P., Iordanov, T., Agarwal, P. K., and Hammes-Schiffer, S. (2001) “Hybrid approach for including electronic and nuclear quantum effects in molecular dynamics simulations of hydrogen transfer reactions in enzymes,” J. Chem. Phys. 114, 6925–6936.Google Scholar
  93. 93.
    93. Garcia-Viloca, M., Alhambra, C., Truhlar, D. G., and Gao, J. (2001) “Inclusion of quantum-mechanical vibrational energy in reactive potentials of mean force,” J. Chem. Phys. 114, 9953–9958.Google Scholar
  94. 94.
    94. Hammes-Schiffer, S. (1998) “Mixed quantum/classical dynamics of hydrogen transfer reactions,” J. Phys. Chem. A 102, 10443–10454.Google Scholar
  95. 95.
    95. Webb, S. P. and Hammes-Schiffer, S. (2000) “Fourier grid Hamiltonian multiconfigurational self-consistent-field: A method to calculate multidimensional hydrogen vibrational wave-functions,” J. Chem. Phys. 113, 5214–5227.Google Scholar
  96. 96.
    96. Alhambra, C., Corchado, J., Sanchez, M. L., Gao, J., and Truhlar, D. G. (2000) “Quantum dynamics of hydride transfer in enzyme catalysis,” J. Am. Chem. Soc. 122, 8197–8203.Google Scholar
  97. 97.
    97. Hwang, J.-K. and Warshel, A. (1996) “How important are quantum mechanical nuclear motions in enzyme catalysis?,” J. Am. Chem. Soc. 118, 11745–11751.Google Scholar
  98. 98.
    98. Liu, Y. P., Lynch, G. C., Truong, T. N., Lu, D. H., Truhlar, D. G., and Garrett, B. C. (1993) “Molecular modeling of the kinetic isotope effect for the [1,5]-sigmatropic rearrangement of cis-1,3-pentadiene,” J. Am. Chem. Soc. 115, 2408–2415.Google Scholar
  99. 99.
    99. Liu, Y. P., Lu, D. H., Gonzalez-Lafont, A., Truhlar, D. G., and Garrett, B. C. (1993) “Direct dynamics calculation of the kinetic isotope effect for an organic hydrogen-transfer reaction, including corner-cutting tunneling in 21 dimensions,” J. Am. Chem. Soc. 115, 7806–7817.Google Scholar
  100. 100.
    100. Garrett, B. C., Truhlar, D. G., Wagner, A. F., and Dunning, T. H., Jr. (1983) “Variational transition state theory and tunneling for a heavy-light-heavy reaction using an ab initio potential energy surface. Atomic chlorine-37 + hydrogen chloride [H(D)35Cl] .fwdarw. hydrogen chloride [H(D)37Cl] + atomic chlorine-35,” J. Chem. Phys. 78, 4400–4413.Google Scholar
  101. 101.
    101. Fernandez-Ramos, A. and Truhlar, D. G. (2001) “Improved algorithm for corner-cutting tunneling calculations,” J. Chem. Phys. 114, 1491–1496.Google Scholar
  102. 102.
    102. Feynman, R. P. and Hibbs, A. R. (1965) Quantum Mechanics and Path Integrals, McGraw-Hill, New York.Google Scholar
  103. 103.
    103. Gillan, M. J. (1988) “The quantum simulation of hydrogen in metals,” Phil. Mag. A 58, 257–283.Google Scholar
  104. 104.
    104. Voth, G. A., Chandler, D., and Miller, W. H. (1989) “Rigorous formulation of quantum transition state theory and its dynamical corrections,” J. Chem. Phys. 91, 7749–7760.Google Scholar
  105. 105.
    105. Messina, M., Schenter, G. K., and Garrett, B. C. (1993) “Centroid-density, quantum rate theory: variational optimization of the dividing surface,” J. Chem. Phys. 98, 8525–8536.Google Scholar
  106. 106.
    106. Cao, J. and Voth, G. A. (1994) “The formulation of quantum statistical mechanics based on the Feynman path centroid density. V. Quantum instantaneous normal mode theory of liquids,” J. Chem. Phys. 101, 6184–6192.Google Scholar
  107. 107.
    107. Hwang, J. K. and Warshel, A. (1993) “A quantized classical path approach for calculations of quantum mechanical rate constants,” J. Phys. Chem. 97, 10053–10058.Google Scholar
  108. 108.
    108. Thomas, A., Jourand, D., Bret, C, Amara, P., and Field, M. J. (1999) “Is there a covalent intermediate in the viral neuraminidase reaction? A hybrid potential free-energy study,” J. Am. Chem. Soc. 121, 9693–9702.Google Scholar
  109. 109.
    109. Makarov, D. E. and Topaler, M. (1995) “Quantum transition-state theory below the crossover temperature,” Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 52, 178–188.Google Scholar
  110. 110.
    110. Messina, M., Schenter, G. K., and Garrett, B. C. (1995) “A variational centroid density procedure for the calculation of transmission coefficients for asymmetric barriers at low temperature,” J. Chem. Phys. 103, 3430–3435.Google Scholar
  111. 111.
    111. Mills, G., Schenter, G. K., Makarov, D. E., and Jonsson, H. (1997) “Generalized path integral based quantum transition state theory,” Chem. Phys. Lett. 278, 91–96.Google Scholar
  112. 112.
    112. Jang, S. and Voth, G A. (2000) “A relationship between centroid dynamics and path integral quantum transition state theory,” J. Chem. Phys. 112, 8747–8757. Erratum: 2001. 114, 1944.Google Scholar
  113. 113.
    113. Feierberg, I., Luzhkov, V., and Aqvist, J. (2000) “Computer simulation of primary kinetic isotope effects in the proposed rate-limiting step of the glyoxalase I catalyzed reaction,” J. Biol. Chem. 275, 22657–22662.PubMedGoogle Scholar
  114. 114.
    114. Sprik, M., Klein, M. L., and Chandler, D. (1985) “Staging: a sampling technique for the Monte Carlo evaluation of path integrals,” Phys. Rev. B 31,4234–4244.Google Scholar
  115. 115.
    115. Major, D. T. and Gao, J. (2005) “Implementation of the bisection sampling method in path integral simulations,” J. Mol. Graph. Model. 24, 121–127.PubMedGoogle Scholar
  116. 116.
    116. Major, D. T., Garcia-Viloca, M., and Gao, J. (2006) “Path integral simulations of proton transfer reactions in aqueous solution using combined QM/MM potentials,” J. Chem. Theory Comput. 2, 236–245.Google Scholar
  117. 117.
    117. Ceperley, D. M. (1995) “Path integrals in the theory of condensed helium,” Rev. Mod. Phys. 67, 279–355.Google Scholar
  118. 118.
    118. Major, D. T., Nam, K., and Gao, J. (2006) “Transition state stabilization and a-amino carbon acidity in alanine racemase,” J. Am. Chem. Soc. 128, 8114–8115.PubMedGoogle Scholar
  119. 119.
    119. Major, D. T. and Gao, J. (2007) “An integrated path intergral and free-energy perturbation-umbrella sampling method for computing kinetic isotope effects of chemical reactions in solution and in enzymes,” J. Chem. Theory Comput. 3, 949–960.Google Scholar
  120. 120.
    120. Ramirez, R. (1997) “Dynamics of quantum particles of by path-integral centroid simulations: The symmetric Eckart barrier,” J. Chem. Phys. 107, 3550–3557.Google Scholar
  121. 121.
    121. Johnston, H. S. (1966) Gas Phase Reaction Rate Theory, Ronald Press, New York.Google Scholar
  122. 122.
    122. Shavitt, I. (1959) “Calculation of the rates of the ortho-para conversions and isotope exchanges in hydrogen,” J. Chem. Phys. 31, 1359.Google Scholar
  123. 123.
    123. Rishavy, M. A. and Cleland, W. W. (2000) “Determination of the mechanism of orotidine 5′-monophosphate decarboxylase by isotope effects,” Biochemistry 39, 4569–4574.PubMedGoogle Scholar
  124. 124.
    124. Nam, K., Gao, J., and York, D. M. (2005) “An efficient linear-scaling ewald method for long-range electrostatic interactions in combined QM/MM calculations,” J. Chem. Theory Comput. 1, 2–13.Google Scholar
  125. 125.
    125. Toney, M. D. (2005) “Reaction specificity in pyridoxal phosphate enzymes,” Arch. Biochem. Biophys. 433, 279–283.PubMedGoogle Scholar
  126. 126.
    126. Ondrechen, M. J., Briggs, J. M., and McCammon, J. A. (2001) “A model for enzyme-substrate interaction in alanine racemase,” J. Am. Chem. Soc. 123, 2830–2834.PubMedGoogle Scholar
  127. 127.
    127. Major, D. T. and Gao, J. (2006) “A combined quantum mechanical and molecular mechanical study of the reaction mechanism and a-amino acidity in alanine racemase,” J. Am. Chem. Soc. 128, 16345–16357.PubMedGoogle Scholar
  128. 128.
    128. Spies, M. A., Woodward, J. J., Watnik, M. R., and Toney, M. D. (2004) “Alanine racemase free energy profiles from global analyses of progress curves,” J. Am. Chem. Soc. 126, 7464–7475.PubMedGoogle Scholar

Copyright information

© Humana Press, a part of Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Jiali Gao
    • 1
  • Dan T. Major
    • 1
  • Yao Fan
    • 1
  • Yen-lin Lin
    • 1
  • Shuhua Ma
    • 1
  • Kin-Yiu Wong
    • 1
  1. 1.Department of Chemistry and Supercomputer InstituteUniversity of MinnesotaMinneapolisMN

Personalised recommendations