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

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

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.

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

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). B55, 104-113).

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

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.

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

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

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

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

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

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

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

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

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

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)).

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)).

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 f C2 O C3 H cyclohexanol

Rotational Barrier of H – O – C3 – C2 dihedral k n  CTL2 CTL1 OHL HOL 0.23 3 0.00 HAL1 CTL1 OHL HOL 0.23 3 0.00 HAL1 CTL1 OHL HOL 1.3 1 180.00 (Kept fixed during optimization)

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

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

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