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Combined QM

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Combined QM

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    1. Combined QM/MM studies of enzymes 5.1.325.1.32

    2. QM/MM approach: General overview Border region: hydrogen link atoms L charge shift for q(M1) 5.1.245.1.24

    3. ChemShell: A modular QM/MM package 5.2.215.2.21

    4. Exploring potential surfaces of complex systems 5.2.525.2.52

    5. PHBH : p-hydroxybenzoate hydroxylase 5.7.215.7.21

    6. Aromatic hydroxylation of p-hydroxybenzoate 5.7.225.7.22

    7. PHBH : General setup 5.7.245.7.24

    8. PHBH : Motion in transition state (40 ps snapshot) 5.7.275.7.27

    9. PHBH: Active site 5.7.375.7.37

    10. PHBH : Comparing different snapshots 5.7.305.7.30

    11. Thermodynamic integration 5.7.325.7.32

    12. 5.7.48 new version of MD movie, July 2005, zu dieser Folie gehört ein avi-film PHBH_10fps5.7.48 new version of MD movie, July 2005, zu dieser Folie gehört ein avi-film PHBH_10fps

    13. PHBH : Role of Pro293 5.7.385.7.38

    14. PHBH : DE versus DA for different snapshots 5.7.605.7.60

    15. Umbrella sampling: biased MD MD with a restraint (bias) on the reaction coordinate: Windows with different ?i are sampled The biased distribution of ?, Pib(?), is sampled 5.12.45.12.4

    16. Umbrella sampling: unbiasing Unbiasing provides the free energy for each window Unknown constant Fi for each window Combination of the windows (weighted average and estimation of Fi): Weighted histogram analysis method (WHAM), or Umbrella integration 5.12.55.12.5

    17. Umbrella integration: method Conceptual combination of thermodynamic integration and umbrella sampling Analysis of umbrella sampling data Calculate the mean force for each window: Combine the windows by a weighted sum: Integrate to obtain A(?) 5.12.65.12.6

    18. Umbrella integration: weighted average Reaction coordinate is divided into bins of uniform width The unbiased mean forces of the windows are averaged on the grid provided by the bins Weight: with Ni being the number of MD steps for window i Numerical integration yields DA 5.12.75.12.7

    19. Umbrella integration: normal distribution Full distribution Pib(?): Normal distribution of Pib(?) through: Truncation of Ai(?) after the quadratic term in ? Truncation of a cumulant expansion of Pib(?) Results depend only on the mean and the variance for each window 5.12.85.12.8

    20. Power series truncation Noise reduction: Linear and quadratic contributions contain relevant information Higher terms (residuum) predominantly contain noise The central region contributes mainly (> 50 %) to A(?) 5.12.95.12.9

    21. Umbrella integration: analytic potential 2-dimensional function Barrier between two minima Monte Carlo sampling, T = 300 K 20,000 steps in each of 40 windows Results: Umbrella integration converges with the number of bins. Errors in barrier heights: 0.097 and -0.136 kJ/mol. With better sampling (80 windows, 80,000 steps): -0.013 and -0.035 kJ/mol WHAM does not converge with the number of bins. Errors with 4500 bins: 1.040 and 0.831 kJ/mol 5.12.105.12.10

    22. Application: PHBH - results Snapshot after 40 ps Molecular dynamics, T=300K 8000 steps in each of 38 windows Tests for equilibration of and (sib)2 Activation barrier: Umbrella integration: 101.5 kJ/mol WHAM: 100.1 – 102.3 kJ/mol, depending on the number of bins Thermodynamic integration: 101±2 kJ/mol 5.12.215.12.21

    23. Error analysis: summary Data collection in each window Combining the windows Integration Confidence interval (95%): This estimate only covers the statistical error, not the systematic error. 5.12.165.12.16

    24. Umbrella integration: summary Combines thermodynamic integration and umbrella sampling Advantages over WHAM analysis: Enables control of the equilibration of the system Analysis independent of the bin width Error bar estimate is available Advantages over thermodynamic integration: Metric tensor correction is avoided Easier implementation of new types of reaction coordinates 5.12.225.12.22

    25. QM/MM free-energy perturbation (FEP) Reaction profile: Full QM/MM calculations QM and MM atoms optimized 5.12.32 mdfreeze2.avi gehört dazu 5.12.32 mdfreeze2.avi gehört dazu

    26. FEP applied to PHBH Good agreement between FEP and termodynamic integration 5.12.365.12.36

    27. PHBH : B3LYP/GROMOS results (TZVP basis) 5.7.505.7.50

    28. SP LMP2/GROMOS barriers (TZ basis) 5.7.525.7.52

    29. SP LMP2/GROMOS and LCCSD(T0)/GROMOS barriers (TZ basis) 5.7.535.7.53

    30. PHBH: Comparison of barriers 5.7.615.7.61

    31. PHBH and CM: Comparison of barriers 5.7.595.7.59

    32. How to introduce classical explicit polarisation 5.17.21 How to include polarisability explicitly5.17.21 How to include polarisability explicitly

    33. COS Model: Overview Charge-on-spring (COS) model: Virtual site with qv attached to polarizable center adapts position to electric field Ei ? induced dipole ?ind,i: Positions of charges-on-spring and electric field components depend on each other ? iterative scheme employed 2-3 iterations per step ? MD 3-4 times more expensive 5.17.1 Additional point-charge (?) is added to the polarizable atoms which adapts its position to the electric field to minimize the total electrostatic energy in the system.5.17.1 Additional point-charge (?) is added to the polarizable atoms which adapts its position to the electric field to minimize the total electrostatic energy in the system.

    34. Solvent effects on an SN2 reaction 5.17.235.17.23

    35. Comparison of (free) energy profiles 5.17.295.17.29

    36. Acknowledgement Ahmet Altun Iris Antes Dirk Bakowies Salomon Billeter Marco Bocola Johannes Kästner Hai Lin Nikolaj Otte Jan Schöneboom Hans Martin Senn Frank Terstegen Stephan Thiel Alexander Turner Tell Tuttle Dongqi Wang Jingjing Zheng Richard Catlow Shimrit Cohen Karl-Erich Jaeger Christian Lennartz Frank Neese David O‘Hagan Manfred Reetz Ansgar Schäfer Sason Shaik Paul Sherwood Wilfred van Gunsteren Hans-Joachim Werner 5.1.265.1.26

    37. EndeEnde

    38. PHBH : QM/MM approach 5.7.265.7.26

    39. PHBH : Comparison of QM/MM and full QM results 5.7.425.7.42

    40. PHBH: References to QM/MM studies 5.7.435.7.43

    41. PHBH : Optimized B3LYP/GROMOS structures (TZVP basis) 5.7.495.7.49

    42. Free energy changes from simulations: overview Fixed constraint Thermodynamic integration: sampling the mean force Continuously changing constraint Slow growth: sampling the mean force Fast growth: fast changing constraint, exponential average of the energy change Restraint (bias) Umbrella sampling: sampling the distribution of the reaction coordinate Free-energy perturbation: instantaneous changes, exponential average of the energy change And other methods … 5.12.25.12.2

    43. Thermodynamic integration The reaction is split into windows with different ? The force on the constrained ? is sampled Mean force ˜ force of constraint Fc Numerical integration along ? yields ?A: Metric tensor correction: accounts for constraint on the momentum canonically conjugated to ? 5.12.35.12.3

    44. Error analysis: strategy Determine error bars for the mean and the variance (sib)2 in each window. Apply error propagation in each step of umbrella integration (data collection, combination of windows, integration) to calculate the sampling error. Use the insight gained to choose the parameters of umbrella simulations in an optimum manner. Test approximate expressions for the statistical error against exact results available for an analytical example potential. 5.12.115.12.11

    45. Tests for equilibration MD trajectories are correlated De-correlation through coarse-graining Tests for Lack of trend in the mean Lack of trend in the variance Normality Lack of correlation Tests provide well-defined error bars for the mean and the variance 5.12.125.12.12

    46. Error analysis: data collection Statistical tests provide variances for and (sib)2 in each window: Error propagation leads to 5.12.13 new. Figure: relevant region between the vertical lines5.12.13 new. Figure: relevant region between the vertical lines

    47. Error analysis: combining the windows Variations in the weights pi can be neglected. 5.12.14 5.12.14

    48. Error analysis: integration var(?A/??) is defined on the bin-grid (width: h) Integration from ?a to ?b according to Simpson’s rule Taking into account the correlation between the bins: Bins are correlated if influenced by the same window. An approximation of the covariance leads to: sb: average of sib (width of window) over the integration range 5.12.155.12.15

    49. Choice of umbrella potential bias: K = 2? recommended (? is the maximum curvature of A(?)) Global histogram: enough, but not too much overlap 5.12.175.12.17

    50. Number and range of windows Overlap between the windows not required in UI, but advantageous to reduce the sampling error. Distance between the window centers should be . Stronger bias (larger K) requires more windows. 5.12.185.12.18

    51. FEP formalism States A and B are part of the reaction profile Perturbation: Energy of state A (EA) is calculated QM-atoms are perturbed: moved to their places in state B, MM-atoms remain Perturbed energy EB is calculated Sampling: QM part frozen ?E=EB-EA is sampled at state A 5.12.335.12.33

    52. Hysteresis in the optimization: PHBH The change between two hydrogen-bond patterns leads to a hysteresis in the energies of the optimized structures. Significant structural changes challenge FEP sampling: intermediate states required! 5.12.345.12.34

    53. FEP energy contributions (kJ/mol) in PHBH 5.12.375.12.37

    54. Sampled free-energy changes in PHBH 5.12.355.12.35

    55. Double iterative scheme for QM/MM-pol 5.17.285.17.28

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