Atomic-Detail Computer Simulation Model System Molecular Mechanics Potential

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

Lysozyme in explicit water

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

2/8 MM Energy Function q l r qi qj

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.

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

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

2/8 MM Energy Function q l r qi qj

Molecular Mechanics Force Field CHARMM Energy Function:

Interaction Energy of Two Peptide Groups

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). B 55, 104 -113).

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

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, 12956 -12957. )

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.

2 x 1020 years b. R time ~ N 6 3 Mio years 1 year 1 month Ne 12 hours 10 Ne 2 30 100 1 000 Number of Electrons (N) 10 000 Size 100 000

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

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

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

Automated Frequency Matching Method for Parameter Development* • Fitting the molecular mechanics potential (CHARMM): From quantum • vibrational frequencies chemical calculations • eigenvector projections NWChem - DFT (B 3 LYP) • Frequencies AND the sets of eigenvectors should coincide * A. C. Vaiana et al. , J. Comput. Chem. , 24: 632, 2003

Automated Frequency Matching (2) 1) Project the CHARMM eigenvectors onto the reference NWChem CHARMM eigenvectors: Projection: NWChem eigenvectors: 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

Starting parameters • Convergence criterion: 2. 500 steps of constant Y 2 Run NMA in CHARMM Change Parameters Compare MM and QM NMA results N Check for converg. Calculate Y 2 new Y 2 old N Keep old parameters Y Keep new parameters Y STOP

Results 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

Calculating the Point Charges

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 • Basis Set: 6 -31 G* • Method: CHELPG • grouping in subsets of atoms constrained to have zero charge

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. 22 436 (2001)).

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

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 | 8742 -8747)

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), 10225 -10235, 2002. )

Rotational Barrier O H cyclohexanol C 2 C 3

Rotational Barrier of H – O – C 3 – C 2 dihedral k n CTL 2 CTL 1 OHL HOL 0. 23 3 0. 00 HAL 1 CTL 1 OHL HOL 1. 3 180. 00 1 (Kept fixed during optimization)

Crystal Simulation • Crystal Symmetry: P 1 • 2 ns MD simulation of single cholesterol molecule to ensure that stereochemistry is preserved • 2 ns MD of crystal • Calculation of RMSD … The experimental unit cell Superposition of the experimental and the CHARMM minimized structures for an individual cholesterol molecule

RMSD Calculations Mean Rmsd = 0. 617 Mean Rmsd = 0. 973 Rmsd calculated over the whole trajectory including all atoms Mean Rmsd = 0. 195 Rmsd comparing 1 averaged cholesterol molecule (from the crystal structure) with the averaged cholesterol from trajectory Rmsd calculated over the whole trajectory including atoms with B factors < 10 Å2 Mean Rmsd = 0. 069 Rmsd comparing 1 averaged cholesterol molecule (from the crystal structure) with the averaged cholesterol from trajectory, incl. only atoms with B factors < 10 Å2

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