2. Atomic-Detail Computer Simulation
Model System
Molecular Mechanics Potential
Energy Surface 
Exploration by Simulation..

3. Lysozyme in explicit water

4. Model System
set of atoms
explicit/implicit solvent
periodic boundary conditions
Potential Function
empirical
chemically intuitive
quick to calculate
Tradeoff: simplicity (timescale) versus accuracy

5. MM Energy Function
2/8 l q f r q i j

6. Electrostatic interaction potential energy between two like-charged atoms.
A particular value of rij specifies the configuration of the system. In the above case one coordinate (degree of freedom) suffices to define the configuration of the system.

7. first approximation
- a molecule will tend to minimize its potential energy.
kl = force constant
lo=equilibrium value

9. Each different potential energy minimum defines
a separate conformation of the molecule.

11. MM Energy Function
2/8 l q f r q i j

12. Molecular Mechanics Force Field
CHARMM Energy Function:

13. Interaction Energy of Two Peptide Groups

14. Crystal structure of L-Leu-L-Val methanol solvate showing methanol-peptide group hydrogen bonding. (From C. H. Görbitz and E. Torgersen Acta Cryst. (1999). B55, ).

15. Determining Parameters
experimental data
ab initio results
X-ray and neutron scattering crystal structures
vibrational frequencies (IR-Raman)
NMR measurements
crystal lattice constants
Hessian matrix elements  normal modes
forces
energy barriers
electrostatic potential

16. Determining Force Constants
(k  2)
Infrared spectrum of arginine.
The frequency is given in wavenumbers.
(From Chapo, C. J.; Paul, J. B.; Provencal, R. A.;
Roth, K.; Saykally, R. J. J. Am. Chem. Soc. 1998, 120, )

17. Basics of Quantum Chemistry.
Schrödinger equation:
H=E
where E is the energy of the system,
H is the Hamiltonian operator,
H=T+V.
V=Vnn+Vne+Vee.
Born-Oppenheimer Approximation Potential Energy Surface.

18. Number of Electrons (N)
2 x 1020 years
Ne2 Ne
Number of Electrons (N)
3 Mio years
1 year
1 month
12 hours
Size 30
10 000
1 000
100 10
time ~ N6 bR

19. Quantum-chemically optimized structure of a fluorescent probe: Rhodamine 6G.

20. Case Study: Cholesterol
Regulates:
membrane fluidity
membrane permeability
lateral mobility of proteins
Cholesterol (~ 40%)
in plasma membrane

21. Normal Mode Analysis  MM QM Force Constant Matrix: Hessian
Approximate the complex energy landscape by harmonic potentials MM QM
Force Constant Matrix: Hessian
at the energy minimum
vibrational frequencies  energy 
Normal Modes
eigenvectors  internal motions
Water
Normal Modes

22. Automated Frequency Matching Method for Parameter Development*
Fitting the molecular mechanics potential (CHARMM):
vibrational frequencies
eigenvector projections
From quantum chemical calculations
NWChem - DFT (B3LYP)
Frequencies AND the sets of eigenvectors should coincide
* A.C. Vaiana et al., J.Comput.Chem., 24: 632, 2003

23. Automated Frequency Matching (2)
1) Project the CHARMM eigenvectors onto the reference NWChem
CHARMM eigenvectors:
NWChem eigenvectors:
Projection:
Frequency corresponding
to max. projection:
Ideal case:
2) Minimize Merit Function:
3) Results are iteratively refined to fit the results of the quantum chemical normal mode calculations
Refinement of parameter set: Monte Carlo Algorithm
Optimizations performed separately for bond, angle, torsion and improper constants
VDW parameters were not optimized

24. Convergence criterion: 2.500 steps of constant Y2
Starting parameters
Convergence criterion:
2.500 steps of constant Y2
Run NMA in
CHARMM
Change
Parameters
Compare MM and
QM NMA results N
Check for
converg. Y
STOP
Calculate Y2
Y2new  Y2old Y
Keep new
parameters N
Keep old
parameters

25. Results overall agreement of CHARMM and quantum chemical normal modes
Root Mean Square Deviation:
Fig. The line is the ideal case of perfectly matched frequencies
and eigenvector projections ; points refer to optimized parameters
overall agreement of CHARMM and quantum chemical normal modes
biologically relevant modes (low frequencies) are well reproduced

26. Calculating the Point Charges

27. Calculating the Point Charges
not within atom radius - unrealistic charge
not too far away from the molecule
 calculate the potential on a grid
Constraints:
sum of the charges equal to zero
grouping in subsets of atoms constrained to have zero charge
Basis Set: 6-31G*
Method: CHELPG

28. The electrostatic potential (r) at a point r is defined
as the work done to bring a unit positive charge
from infinity to the point.
The electrostatic interaction energy between a
point charge q located at r and the molecule
equals q(r).
Electrostatic potential
mapped onto the electron density surface
for 2-bromo-2-chloro-1,1,1-trifluoroethane
(halothane). (From: Pei Tang, Igor Zubryzcki,
Yan Xu J comp chem (2001)).

29. Quantum Chemistry X-Ray
Electron density in the peptide bond plane of DL-alanyl-methionine
(from Guillot et al Acta Cryst B 57(4) 567 (2001)).

30. Experimental. Theoretical.
Electrostatic potential generated by the NADP+ cofactor in the
plane of the nicotinamide ring an aldose reductase complex.
Blue, positive; red, negative; black dotted line, zero level.
(From Nicolas Muzet , Benoît Guillot, Christian Jelsch, Eduardo Howard
and Claude Lecomte PNAS 2003 | vol. 100 | no. 15 | )

31. Transition state structure for the catalytic mechanism of a Tyrosine Phosphatase
calculated using Density Functional Theory (From Dilipkumar Asthagiri, Valerie Dillet,
Tiqing Liu, Louis Noodleman, Robert L. Van Etten, and Donald Bashford J. Am. Chem. Soc.,
124 (34), , 2002.)

32. Rotational Barrier f C2 O C3 H
cyclohexanol

33. Rotational Barrier of H – O – C3 – C2
dihedral k n 
CTL2 CTL1 OHL HOL
HAL1 CTL1 OHL HOL
HAL1 CTL1 OHL HOL
(Kept fixed during optimization)

35. Crystal Simulation Crystal Symmetry: P1
2ns MD simulation of single cholesterol molecule to ensure that stereochemistry is preserved
2ns MD of crystal
Calculation of RMSD …
Superposition of the experimental and the CHARMM minimized structures for an individual cholesterol molecule
The experimental unit cell

36. RMSD Calculations Mean Rmsd = 0.617 Mean Rmsd = 0.973
Rmsd calculated over the whole trajectory including all atoms
Rmsd calculated over the whole trajectory including atoms with B factors < 10 Å2
Mean Rmsd = 0.195
Mean Rmsd = 0.069
Rmsd comparing 1 averaged cholesterol molecule (from the crystal structure) with the averaged cholesterol from trajectory
Rmsd comparing 1 averaged cholesterol molecule (from the crystal structure) with the averaged cholesterol from trajectory, incl. only atoms with B factors < 10 Å2

38. Cholesterol in Biomembrane Simulations
Application:
Cholesterol in Biomembrane Simulations
Structural Analysis
organization in membrane
interactions with lipids
H bonding
Dynamical Analysis
motion of cholesterol
influence on lipid dynamics
diffusion