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SCCDFTB as a bridge between MM and high-level QM. Jan Hermans University of North Carolina 1

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1. SCCDFTB better than MM a.Examples Simulation of crambin (Haiyan Liu) Simulation of “dipeptides” (Hao Hu) b. But why? Concerted changes of geometry in N-methyl acetamide Hydrogen bonding between two N-methyl acetamide molecules More flexible 2. Develop and test MM force fields From QM to MM via SCCDFTB 2

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Simulation of crambin (Haiyan Liu; 2001) Liu, HY, Elstner, M, Kaxiras, E, Frauenheim, T, Hermans, J, & Yang, W. Quantum mechanics simulation of protein dynamics on long time scale. Proteins, 44: 484-489, 2001. Improved agreement of backbone geometry in folded state From QM to MM via SCCDFTB Simulation of “dipeptides” (Hao Hu; 2002) Hu, H, Elstner, M., Hermans, J. Comparison of a QM/MM force field and molecular mechanics force fields in simulations of alanine and glycine "dipeptides" (Ace-Ala-Nme and Ace-Gly-Nme) in water in relation to the problem of how to model the unfolded peptide backbone in solution. Proteins, 50, 451-463 (2003). Improved agreement of backbone geometry in solution 3

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amber, charmm, gromos, opls-aa vs. each other and vs. SCCDFTB SCCDFTB Ace-Ala-Nme in explicit water Hao Hu, 2002 4

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Why better accuracy with SCCDFTB? SCCDFTB reproduces concerted changes of geometry charge fluctuations hydrogen bond geometry example: N-methyl acetamide 5

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6 Concerted changes of geometry in N-methyl acetamide, CH 3 -NH-CO-CH 3 Recipe: 1. Twist about NH-CO bond 2. Minimize the energy (with SCCDFTB) H-N-C C-N-CA2 H-N-CA2 tetrahedral planar

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7 Fluctuation of charge in N-methyl acetamide Fluctuations of charges and geometry are coupled atom:CONHN 180 º (energy minimum) 0.4911-0.5082-0.25040.1879 = 90 º (saddle point) 0.5255-0.4257-0.33430.1749

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8 Non-spherical electron distribution: C=O interacts with H-N Non-linear N-H … O=C hydrogen bonds NHO prefers 180º HOC likes 130º Cf. Side chain hydrogen bonds in proteins and by ab initio QM: Morozov, Kortemme, Baker

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SCCDFTB MM force field 9 SCCDFTB favors bent arrangement Simple Point Charge model of MM favors linear structures Distribution of COH in dimers of N-methyl acetamide. Hermans, J. Hydrogen bonds in molecular mechanics force fields. Adv. Protein Chem. 72, 105-119, 2006.

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1. Correlation of DFT (B3LYP/631G*) and SCCDFTB energies 10 But … SCCDFTB is too flexible: 1000 conformations from 1 ns MD simulation with SCCDFTB

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2. Energy profile for internal rotation in butane 11 SCCDFTB is too flexible: DFT B3LYP/631G*: eclipsed: E =±120 = 3.35 gauche: E = ±60 = 0.83 cis: E =0 = 5.69 SCCDFTB: eclipsed: E =±120 = 2.57 gauche: E = ±60 = 0.45 cis: E =0 = 3.80 (relative to trans, = 180) MP2: eclipsed: E =±120 = 3.31 gauche: E = ±60 = 0.62 cis: E =0 = 5.51

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End of part 1

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Molecular mechanics energy function: how to improve it? 1. How precise is this expansion? 2. How accurate is this model? 3. How accurate are the implementations (amber, charmm, … 13 intramolecular non-bonded

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Assume a high-level QM method as “REALITY”: DFT (B3LYP/631G*) Try to reproduce its energy. (can always choose a higher level of QM.)

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The slope is very close to 1 The RMS deviation is 0.07 kcal/mol (mean E pot = 3) 15 * By minimizing the RMS deviation Recipe STEP 1: 1. Simulate (1 ns with SCCDFTB) 2. Save 1000 conformations Example: methane, CH 4 Recipe STEP 2: 3. Compute E pot with B3LYP/631G* 4. Fit* a new MM forcefield 5. Compute E pot with the new MM force field

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What are the most important energy parameters for methane? Parametervalue rmsd 10 2 K l, C-H 3531.4361.6 2 K , H-C-H 33.20.2220.26 3 K l, C-H -8030.15726 3 K , H-C-H -7.80.1530.55 K l,l, C-H, C-H -22.80.1520.77 2 K d,H·H 20.50.0660.69 rms residual Standard quadratic MM terms include these terms (not needed in simulations with fixed bond lengths) not very useful 16 precision

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Systems studied to date (manuscript): “rigid” molecules methane, benzene, water molecules with internal rotation ethane, propane, butane, methyl-benzene Non-bonded interactions methane…methane, ethane…ethane water…methane, water…water Some results and some conclusions …. 17

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Geometric parameters agree well. Transferability between related molecules Compared with “standard” force fields LESSONS LEARNED: 18

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Nonbonded interactions 19 with independent values according to mean ESP charges with charge neutrality with one fixed value System C free C ESP C neutral C fix 1 H 2 O, H 2 O 1.004200 ‑ 105 48.1 1.3360.65(215) (-108) (54) 1.2100.2298 -149 75.5 1.004(215) -102 46.5 CH 4, CH 4 0.166190 -49 12.7 0.171-0.1(95) (-23.75) (5.94) Lowest From ESP charges Results of different fits for Coulomb interactions acceptable =

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Coulomb interactions:(we skipped a slide) (Water: Fixed Point charges based on ESP inadequate) Methane and ethane: ESP charges can be used Parametermethane dimer (1) methane dimer (2) ethane dimer 12 B C,C 1,200,000 1,110,000 12 B C,H 60,00062,00052,000 12 B H,H 1,100700840 Methane and ethane: Lennard-Jones repulsive parameters Conclusion: Nice agreement

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Geometric parameters agree well. Fixed point charge (FPC) model for Coulomb energy is poor for water … water and water … methane LESSONS LEARNED: 21

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Geometric parameters agree well. Fixed point charge (FPC) model for Coulomb energy is poor for water … water and water … methane Intermolecular parameters for methane and ethane are similar (and FPC model is OK). LESSONS: 22 LESSONS LEARNED:

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Geometric parameters agree well. Fixed point charge (FPC) model for Coulomb energy is poor for water … water and water … methane Intermolecular parameters for methane and ethane are similar (and FPC model is OK). Exponent of L-J repulsive term = 12 is good. LESSONS: 23 LESSONS LEARNED:

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Butane: “intrinsic” torsion term non-bonded interactions (1/r 12 and 1/r) 1-4 C,C1-5 and 1-4 C,H 1-6, 1-5, 1-4 H,H * In the SCCDFTB simulation forced 360º rotation about C 2 -C 3, = 14 kcal/mol * Fit several MM models: A0* has 38 parameters, = 0.441 A5 has 12 parameters, = 0.598 24 CC HH

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Butane: Fit for model A5 25

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Butane: 26 Critical tests: * Re-calculate DFT (B3LYP/631G*) energies * Compare energies at minima and barriers DFT vs. A5 (and 2 others) * Simulate butane with A5 force field (and 2 others) Calculate PMF for torsion about C 2 -C 3

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red curve = MM energy black dots = DFT energy black curve = PMF DFT energy is systematically high 27 Simulation with A5 force field

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Slope of best fit is 1.04 28

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modelnpnp E = 120 E = 60 E =0 A = 120 A = 60 A =0 slopermsd A0h323.880.765.813.870.866.081.020.700 A1233.850.725.833.890.866.171.020.696 A5123.710.675.633.650.805.911.040.734 DFT3.350.835.69 With more parameters (n p ) in the MM force field: The slope goes down to 1.02 The PMF becomes a little bit sharper Energies and free energies at minima and maxima (relative to minimum at = 180º) Slope and rmsd of correlation between DFT and MM energies 29

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Geometric parameters agree well. Fixed point charge (FPC) model for Coulomb energy is poor for water … water and water … methane Intermolecular parameters for methane and ethane are similar and FPC model is OK. Exponent of L-J repulsive term = 12 is good. Torsion in ethane, propane, butane: omit terms in 1/r “messy” set of 1-4, 1-5 and 1-6 repulsive terms LESSONS: 30 LESSONS LEARNED:

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Why is SCCDFTB important in this project: (1)Fast to run (2)Easy to set up (need only coordinates) (3)Equilibrium geometry agrees well with DFT (4)Slightly more flexible: do not miss anything

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Thanks to Weitao Yang Hao Hu (coauthor of paper) Future work: I hope so 32

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