1. Computational Chemistry Molecular Mechanics/Dynamics F = Ma Quantum Chemistry Schr Ö dinger Equation H  = E 

2. Simulation of a pair of polypeptides Duration: 100 ps. Time step: 1 ps (Ng, Yokojima & Chen, 2000)

3. Large Gear Drives Small Gear G. Hong et. al., 1999

4. Molecular Mechanics Force Field Bond Stretching Term Bond Angle Term Torsional Term Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction

5. Bond Stretching Potential E b = 1/2 k b (  l) 2 where, k b : stretch force constant  l : difference between equilibrium & actual bond length Two-body interaction

6. Bond Angle Deformation Potential E a = 1/2 k a (  ) 2 where, k a : angle force constant   : difference between equilibrium & actual bond angle  Three-body interaction

7. Periodic Torsional Barrier Potential E t = (V/2) (1+ cosn  ) where, V : rotational barrier  : torsion angle n : rotational degeneracy Four-body interaction

8. Non-bonding interaction van der Waals interaction for pairs of non-bonded atoms Coulomb potential for all pairs of charged atoms

9. VDW potential of CHARMM ε i and ε j are constants characteristic of the strengths of the van der Waals interactions of the two atoms, R min,i and R min,j are constants characteristic of the radii of the two atoms

10. MM Force Field Types MM2Small molecules AMBERPolymers CHAMMPolymers BIOPolymers OPLSSolvent Effects

11. CHAMM FORCE FIELD FILE

12. atom 1 1 HA "Nonpolar Hydrogen" atom 2 2 HP "Aromatic Hydrogen" atom 3 3 H "Peptide Amide HN" atom 4 4 HB "Peptide HCA" atom 5 4 HB "N-Terminal HCA" atom 6 5 HC "N-Terminal Hydrogen" atom 7 5 HC "N-Terminal PRO HN" atom 8 3 H "Hydroxyl Hydrogen" atom 9 3 H "TRP Indole HE1" atom 10 3 H "HIS+ Ring NH" atom 11 3 H "HISDE Ring NH" atom 12 6 HR1 "HIS+ HD2/HISDE HE1" atom 13 7 HR2 "HIS+ HE1" H-HH-CH2-CH3 H H H H H HO- H + H + H H H H

13. atom 20 10 C "Peptide Carbonyl" atom 21 11 CA "Aromatic Carbon" atom 22 12 CC "C-Term Carboxylate" atom 23 13 CT1 "Peptide Alpha Carbon" atom 24 13 CT1 "N-Term Alpha Carbon" atom 25 13 CT1 "Methine Carbon" atom 26 14 CT2 "Methylene Carbon" atom 27 15 CT3 "Methyl Carbon" atom 28 14 CT2 "GLY Alpha Carbon" atom 29 14 CT2 "N-Terminal GLY CA" atom 30 16 CP1 "PRO CA Carbon"

14. /A o /(kcal/mol)

15. /(kcal/mol/A o2 ) /Ao/Ao

16. /(kcal/mol/rad 2 ) /deg

17. /(kcal/mol)/deg

18. Algorithms for Molecular Dynamics Runge-Kutta methods: x(t+  t) = x(t) + (dx/dt)  t Fourth-order Runge-Kutta x(t+  t) = x(t) + (1/6) (s 1 +2s 2 +2s 3 +s 4 )  t +O(  t 5 ) s 1 = dx/dt s 2 = dx/dt [w/ t=t+  t/2, x = x(t)+s 1  t/2] s 3 = dx/dt [w/ t=t+  t/2, x = x(t)+s 2  t/2] s 4 = dx/dt [w/ t=t+  t, x = x(t)+s 3  t] Very accurate but slow!

19. Algorithms for Molecular Dynamics Verlet Algorithm: x(t+  t) = x(t) + (dx/dt)  t + (1/2) d 2 x/dt 2  t 2 +... x(t -  t) = x(t) - (dx/dt)  t + (1/2) d 2 x/dt 2  t 2 -... x(t+  t) = 2x(t) - x(t -  t) + d 2 x/dt 2  t 2 + O(  t 4 ) Efficient & Commonly Used!