Atomic-Detail Computer Simulation

Model System

Molecular Mechanics Potential

ji ij

ji

ji ij

ij

ij

ijij

impropersdihedrals

N

n

n

anglesbondsb

Dr

qq

rr

KnK

kbbkV

,,

612

20

1

20

20

4

cos1

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

2/8MM Energy Function

l

r

qi qj

ij

jielec r

qqV

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.

ij

elecij r

VF

20 )( llkV ll

kl = force constantlo=equilibrium value

first approximation

- a molecule will tend to minimize its potential energy.

20 )( kV

nkV 1

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

6min12min

ij

ij

ij

ijijvdw r

R

r

RV

2/8MM Energy Function

l

r

qi qj

Molecular Mechanics Force Field

bonds angles dihedrals impropers

bbonded knkkbbkE 20

20

20 )(])cos[1()()(

bondednonbonded EERV )(

ij

ji

jiji ij

ij

ij

ijijbondednon r

qq

rrE

,,

612

4

14

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

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.)

(k 2)

Determining Force Constants

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

Ne2Ne

Number of Electrons (N)

3 Mio years

1 year

1 month

12 hours

Size30 100 00010 0001 00010010

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

Approximate the complex energy landscape by harmonic potentials

Force Constant Matrix: Hessianji

ij rr

VH

)(2 r

Normal Modes

at the energy minimum

vibrational frequencies energy

eigenvectors internal motions

Water

Normal Modes

MM

QM

Automated Frequency Matching Method for Parameter Development*

• Fitting the molecular mechanics potential (CHARMM):

• vibrational frequencies

• eigenvector projections

From quantum chemical calculations

* A.C. Vaiana et al., J.Comput.Chem., 24: 632, 2003

• Frequencies AND the sets of eigenvectors should coincide

NWChem - DFT (B3LYP)

Automated Frequency Matching (2)

• Refinement of parameter set: Monte Carlo Algorithm

• Optimizations performed separately for bond, angle, torsion and improper constants

• VDW parameters were not optimized

1) Project the CHARMM eigenvectors onto the reference NWChem

CHARMM eigenvectors:

NWChem eigenvectors:

C

N

max):(max Nj

Cij jv

ijNj

Ci

2) Minimize Merit Function:

3) Results are iteratively refined to fit the results of the quantum chemical normal mode calculations

63

2max2 )(N

ji vvY

Ideal case: maxji vv

Projection:

Frequency correspondingto max. projection:

Starting parameters

Compare MM and QM NMA results

Calculate Y2

Y2new Y2

old

Run NMA in CHARMM

Keep old parameters

N

Keep newparameters

Check forconverg.

Change Parameters

Y

N

YSTOP

• Convergence criterion:2.500 steps of constant Y2

63

2max2 )(N

ji vvY

Results

Root Mean Square Deviation:

163

2max

98.3973

cmN

vvN

ji

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

•Basis Set: 6-31G*

•Method: CHELPG

• 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

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)).

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)

Experimental. Theoretical.

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

H

O C3

C2

C2

C3OH

cyclohexanol

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

Rotational Barrier of H – O – C3 – C2

(Kept fixed during optimization)

Example of a torsional potential.Potential energy profile for rotation of the two ringsof biphenyl around the central bond.

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

Mean Rmsd = 0.973

Mean Rmsd = 0.617

Mean Rmsd = 0.195 Mean Rmsd = 0.069

Rmsd calculated over the whole trajectory including all atoms

Rmsd calculated over the whole trajectory including atoms with B factors < 10 Å2

RMSD Calculations

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

Application:

Cholesterol in Biomembrane Simulations

Structural Analysis

Dynamical Analysis

• organization in membrane

• interactions with lipids

• H bonding

• motion of cholesterol

• influence on lipid dynamics

• diffusion