Computer Modeling

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Computer Modeling

• Dr. GuanHua CHEN
• Department of Chemistry
• University of Hong Kong
• http//yangtze.hku.hk/lecture/comput06-07.ppt

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Computational Chemistry

• Quantum Chemistry
• SchrÖdinger Equation
• H? E?
• Molecular Mechanics
• F Ma
• F Force Field
• Bioinformatics

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Computational Chemistry Industry
Company
Software
Gaussian Inc. Gaussian 94, Gaussian
98 Schrödinger Inc. Jaguar Wavefunction Spart
an Q-Chem Q-Chem Accelrys InsightII,
Cerius2 HyperCube HyperChem Informatix Celera
Genomics
Applications material discovery, drug design
research
RD in Chemical Pharmaceutical industries in
2000 US 80 billion Bioinformatics Total Sales
in 2001 US 225 million Project Sales in
2006 US 1.7 billion
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Vitamin C
C60
energy
heme
OH D2 --gt HOD D
Cytochrome c
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Quantum Chemistry Methods

• Ab initio Molecular Orbital Methods
• Hartree-Fock, Configurationa Interaction (CI)
• MP Perturbation, Coupled-Cluster, CASSCF
• Density Functional Theory
• Semiempirical Molecular Orbital Methods
• Huckel, PPP, CNDO, INDO, MNDO, AM1
• PM3, CNDO/S, INDO/S

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SchrÖdinger Equation
H y E y
Wavefunction
Hamiltonian H ??(-h2/2ma)??2 - (h2/2me)?i?i2
- ?i ?? Zae2/ria ( ???? ZaZbe2/rab ) ?i ?j
e2/rij
Energy
One-electron terms ??(-h2/2ma)??2 -
(h2/2me)?i?i2 - ?i ?? Zae2/ria Two-electron
term ?i ?j e2/rij
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Hartree-Fock Method
Orbitals
1. Hartree-Fock Equation F fi ei fi   F Fock
operator fi the i-th Hartree-Fock orbital ei
the energy of the i-th Hartree-Fock orbital
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2. Roothaan Method (introduction of Basis
functions) fi ?k cki yk LCAO-MO  
yk is a set of atomic orbitals (or basis
functions) 3. Hartree-Fock-Roothaan equation
?j ( Fij - ei Sij ) cji 0   Fij ? lt ?iF
?j gt Sij ? lt ?i ?j gt 4. Solve the
Hartree-Fock-Roothaan equation
self-consistently (HFSCF)
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Graphic Representation of Hartree-Fock Solution
0 eV
Electron Affinity
Ionization Energy
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A Gaussian Input File for H2O
HF/6-31G(d)
Route section water energy
Title 0 1
Molecule
Specification O -0.464 0.177 0.0
(in Cartesian coordinates H
-0.464 1.137 0.0 H 0.441 -0.143 0.0
Basis Set ?i ?p cip ?p yk is a set of
atomic orbitals (or basis functions) STO-3G,
3-21G, 4-31G, 6-31G, 6-31G, 6-31G -------------
--------------------------------------------------
----------------------?
complexity accuracy
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Gaussian type functions gijk N xi yj zk
exp(-ar2) (primitive Gaussian function) yp ?u
dup gu (contracted Gaussian-type function,
CGTF) u ijk p nlm
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Electron Correlation avoiding each other
The reason of the instantaneous
correlation Coulomb repulsion (not included in
the HF) Beyond the Hartree-Fock Configuration
Interaction (CI) Perturbation theory Coupled
Cluster Method Density functional theory
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Configuration Interaction (CI)


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Single Electron Excitation or Singly Excited
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Double Electrons Excitation or Doubly Excited
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Singly Excited Configuration Interaction (CIS)
Changes only the excited states

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Doubly Excited CI (CID) Changes ground
excited states

Singly Doubly Excited CI (CISD) Most Used
CI Method
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Full CI (FCI) Changes ground excited states


...
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Perturbation Theory
H H0 H H0yn(0) En(0) yn(0) yn(0) is an
eigenstate for unperturbed system H is small
compared with H0
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Moller-Plesset (MP) Perturbation Theory The MP
unperturbed Hamiltonian H0 H0 ?m
F(m) where F(m) is the Fock operator for
electron m. And thus, the perturbation H
  H H - H0   Therefore, the unperturbed
wave function is simply the Hartree-Fock wave
function ?.   Ab initio methods MP2, MP3, MP4
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Coupled-Cluster Method
y eT y(0) y(0) Hartree-Fock ground state wave
function y Ground state wave function T T1
T2 T3 T4 T5 Tn n electron
excitation operator
T1

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Coupled-Cluster Doubles (CCD) Method
yCCD eT2 y(0) y(0) Hartree-Fock ground state
wave function yCCD Ground state wave
function T2 two electron excitation operator
T2

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Complete Active Space SCF (CASSCF)
Active space
All possible configurations
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Density-Functional Theory (DFT)
Hohenberg-Kohn Theorem Phys. Rev. 136, B864
(1964) The ground state electronic density
?(r) determines uniquely all possible
properties of an electronic system ?(r) ?
Properties P (e.g. conductance), i.e. P ?
P?(r) Density-Functional Theory (DFT) E0 -
(h2/2me)?i lt?i ?i2 ?i gt- ?? ? dr Zae2?(r) / r1a
(1/2) ? ? dr1 dr2 e2/r12
Exc?(r) Kohn-Sham Equation Ground State Phys.
Rev. 140, A1133 (1965) FKS yi ei yi FKS ? -
(h2/2me)?i?i2 - ?? Zae2 / r1a ?j Jj Vxc
Vxc ? dExc?(r) / d?(r) A popular
exchange-correlation functional Exc?(r) B3LYP
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Hu, Wang, Wong Chen, J. Chem. Phys. (Comm)
(2003)
B3LYP/6-311G(d,p)
B3LYP/6-311G(3df,2p)
RMS21.4 kcal/mol
RMS12.0 kcal/mol
RMS3.1 kcal/mol
RMS3.3 kcal/mol
B3LYP/6-311G(d,p)-NEURON B3LYP/6-311G(d,p)-NEU
RON same accuracy
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Time-Dependent Density-Functional Theory (TDDFT)
Runge-Gross Extension Phys. Rev. Lett. 52, 997
(1984) Time-dependent system ?(r,t) ?
Properties P (e.g. absorption) TDDFT
equation exact for excited states
Isolated system
Open system
Density-Functional Theory for Open System ???
Further Extension X. Zheng, F. Wang G.H. Chen
(2005) Generalized TDDFT equation exact
for open systems
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Ground State Excited State CPU Time
Correlation Geometry Size Consistent

(CHNH,6-31G) HFSCF ?
? 1 0
OK ? DFT
? ?
1 ?
? CIS ?
? lt10
OK ? CISD
? ?
17 80-90 ?
?
(20
electrons) CISDTQ ? ?
very large 98-99 ?
? MP2 ?
? 1.5
85-95 ? ?

(DZP) MP4
? ?
5.8 gt90 ?
? CCD ? ?
large gt90
? ? CCSDT ?
? very large
100 ? ?
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Search for Transition State
Transition State one negative frequency
k ? ? e-DG/RT
DG
Reactant
Product
Reaction Coordinate
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Gaussian Input File for Transition State
Calculation
b3lyp/6-31G optqst2 testthe first is the
reactant internal coordinate0 1OH 1 oh1 H
1 oh1 2 ohh1oh1 0.90ohh1 104.5The second
is the product internal coordinate0 1OH 1
oh2H 1 oh3 2 ohh2oh2 0.9oh3 10.0ohh2
160.0
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Semiempirical Molecular Orbital Calculation
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LCAO-MO fi ?r cri yr   ?s ( Heffrs - ei
Srs ) csi 0   Heffrs ? lt ?r Heff ?s
gt Srs ? lt ?r ?s gt

• Parametrization
• Heffrr ? lt ?r Heff ?r gt
• minus the valence-state
  ionization
• potential (VISP)

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Atomic Orbital Energy
VISP --------------- e5 -e5 --------------- e
4 -e4 --------------- e3 -e3 --------------
- e2 -e2 --------------- e1 -e1   Heffrs
½ K (Heffrr Heffss) Srs K 1?3
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CNDO, INDO, NDDO (Pople and co-workers) Hamiltoni
an with effective potentials Hval ?i -(h2/2m)
?i2 Veff(i) ?i?jgti e2 / rij
two-electron integral (rstu) lt?r(1) ?t(2)
1/r12 ?s(1) ?u(2)gt   CNDO complete neglect of
differential overlap (rstu) ?rs ?tu (rrtt) ?
?rs ?tu ?rt
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INDO intermediate neglect of differential
overlap (rstu) 0 when r, s, t and u are not on
the same atom. NDDO neglect of diatomic
differential overlap (rstu) 0 if r and s (or t
and u) are not on the same atom. CNDO, INDO are
parametrized so that the overall results fit well
with the results of minimal basis ab initio
Hartree-Fock calculation. CNDO/S, INDO/S are
parametrized to predict optical spectra.
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MINDO, MNDO, AM1, PM3 (Dewar and co-workers,
University of Texas, Austin)   MINDO modified
INDO MNDO modified neglect of diatomic overlap
AM1 Austin Model 1 PM3 MNDO parametric method
3   based on INDO NDDO reproduce the binding
energy
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Relativistic Effects
Speed of 1s electron Zc / 137 Heavy elements
have large Z, thus relativistic effects
are important. Dirac Equation Relativistic
Hartree-Fock w/ Dirac-Fock operator
or Relativistic Kohn-Sham calculation
or Relativistic effective core potential (ECP).
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Four Sources of error in ab initio Calculation
(1) Neglect or incomplete treatment of electron
correlation (2) Incompleteness of the Basis
set (3) Relativistic effects (4) Deviation from
the Born-Oppenheimer approximation
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Quantum Chemistry for Complex Systems
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Quantum Mechanics / Molecular Mechanics (QM/MM)
Method
Combining quantum mechanics and molecular
mechanics methods
QM
MM
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Hamiltonian of entire system H HQM HMM
HQM/MM Energy of entire system E EQM(QM)
EMM(MM) EQM/MM(QM/MM) EQM/MM(QM/MM)
Eelec(QM/MM) Evdw(MM) EMM-bond(MM) EQM(QM)
Eelec(QM/MM) lt? Heff ?gt Heff -1/2 ?i?i2
?ij 1/rij - ?a?i Za/ria - ?b?i qb/rib
?g?i Vv-b(ri) ?a?d Za Zd/rad ?b?a Zaqb/rba

QM
MM
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Quantum Chemists Solution
Linear-Scaling Method O(N) Computational time
scales linearly with system size
Time
Size
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Linear Scaling Calculation for Ground State
Divide-and-Conqure (DAC)
W. Yang, Phys. Rev. Lett. 1991
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York, Lee Yang, JACS, 1996
Superoxide Dismutase (4380 atoms)
AM1
Strain, Scuseria Frisch, Science (1996) LSDA /
3-21G DFT calculation on 1026 atom RNA Fragment
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Linear Scaling Calculation for Excited State
Liang, Yokojima Chen, JPC, 2000
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Fast Multiple Method
LDM-TDDFT CnH2n2
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LODESTAR Software Package for Complex Systems
Characteristics O(N) Divide-and-Conquer O(N)
TDHF (ab initio semiemptical) O(N) TDDFT
Light Harvesting System
Nonlinear Optical
CNDO/S-, PM3-, AM1-, INDO/S-, TDDFT-LDM
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Photo-excitations in Light Harvesting System II
strong absorption 800 nm
generated by VMD
generated by VMD
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Carbon Nanotube
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Quantum mechanical investigation of the field
emission from the tips of carbon nanotubes
Zheng, Chen, Li, Deng Xu, Phys. Rev. Lett. 2004
Zettl, PRL 2001
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Molecular Mechanics Force Field

• Bond Stretching Term
• Bond Angle Term
• Torsional Term
• Electrostatic Term
• van der Waals interaction

Molecular Mechanics F Ma F Force Field
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Bond Stretching Potential Eb 1/2 kb
(Dl)2 where, kb stretch force
constant Dl difference between equilibrium
actual bond length
Two-body interaction
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Bond Angle Deformation Potential Ea 1/2 ka
(D?)2 where, ka angle force
constant D? difference between equilibrium
actual bond angle
?
Three-body interaction
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Periodic Torsional Barrier Potential Et
(V/2) (1 cosn? ) where, V rotational
barrier t torsion angle n
rotational degeneracy
Four-body interaction
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Non-bonding interaction van der Waals
interaction for pairs of non-bonded atoms
Coulomb potential for all pairs of charged
atoms
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Force Field Types

• MM2 Molecules
• AMBER Polymers
• CHAMM Polymers
• BIO Polymers
• OPLS Solvent Effects

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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) (s12s22s3s4) ?t O(?t5) s1
dx/dt s2 dx/dt w/ tt?t/2, x
x(t)s1?t/2 s3 dx/dt w/ tt?t/2, x
x(t)s2?t/2 s4 dx/dt w/ tt?t, x
x(t)s3 ?t
Very accurate but slow!
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Algorithms for Molecular Dynamics
Verlet Algorithm x(t?t) x(t) (dx/dt) ?t
(1/2) d2x/dt2 ?t2 ... x(t -?t) x(t) -
(dx/dt) ?t (1/2) d2x/dt2 ?t2 -
... x(t?t) 2x(t) - x(t -?t) d2x/dt2
?t2 O(?t4)
Efficient Commonly Used!
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Multiple Scale Simulation
Goddard, Caltech
Goddard, Caltech
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Large Gear Drives Small Gear
G. Hong et. al., 1999
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Nano-oscillators
Nanoscopic Electromechanical Device (NEMS)
Zhao, Ma, Chen Jiang, Phys. Rev. Lett. 2003
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Computer-Aided Drug Design
Human Genome Project
GENOMICS
Drug Discovery
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Computer-aided drug design
Chemical Synthesis
Screening using in vitro assay
Animal Tests
Clinical Trials
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ALDOSE REDUCTASE
Diabetic Complications
Diabetes
Sorbitol
Glucose
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Inhibitor
Aldose Reductase
Design of Aldose Reductase Inhibitors
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Descriptors Electron negativity Volume
Database for Functional Groups
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Possible drug leads 350 compounds
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Aldose Reductase Active Site Structure
Cerius2 LigandFit
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To further confirm the AR-ARI binding, We perform
QM/MM calculations on drug leads. CHARMM 5'-OH,
6'-F, 7'-OH
Binding energy is found to be 45 kcal / mol
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Docking of aldose reductase inhibitor
Aldose reducatse
Inhibitor
(4R)-6-fluoro-7-hydroxyl-8-bromo-3-methylspiro
- imidazoli-dine-4,4(1H)-quinazoline-2,2,5(3
H)-trione
Cerius2 LigandFit
Hu Chen, 2003
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Interaction energy between ligand and protein
Quantum Mechanics/Molecular Mechanics
(QM / MM)
Hu Chen, 2003
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aInhibitor concentration of inhibit Aldose
Reductase b the percents of lower sciatic nerve
sorbitol levels c interaction with AR in Fig. 4
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Our Design Strategy
QSAR determination prediction (Neural Network)
Docking (Cerius2)
QM / MM (binding energy)
?
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Software in Department 1. Gaussian 2. Insight
II CHARMm molecular dynamics
simulation, QM/MM Profiles-3D Predicting
protein structure from sequences SeqFold
Functional Genomics, functional
identification of protein w/ sequence and
structure comparison NMR Refine Structure
determination w/ NMR data 3. Games 4.
HyperChem 5. AutoDock (docking) 6. MacroModel 6.
In-House Developed Software LODERSTAR Neural
Network for QSAR Monte Carlo Molecular
Dynamics
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HYPERCHEM Exercise Part A Study the electronic
structure and vibrational spectrum of formaldehyde
Step 1 Build up the structure of the
formaldehyde. 1.    Run HYPERCHEM software in the
start menu. 2.  Double click the drawing tool to
open the elements table dialogue box and select
carbon atom. Close the element table. (Drawing
tool) 3.    L-click the cursor on the workspace.
A carbon atom will appear. (Make sure drawing
tool is selected. R-click on the atom if you want
to delete it) 4.    Repeat (2) and choose oxygen
instead of carbon. Move the cursor to the carbon
centre and drag the mouse from the carbon centre
to an empty workspace. (A single bond is created
between carbon atom and oxygen atom.) 5.   
L-click the bond between carbon and oxygen to
create a double bond. 6.    L-click on Build in
the menu bar and switch on add H model build
(i.e. make sure a tick appeared on the left of
this function.). Step 2 Optimize the structure
using RHF and 6-31G basis set. 7.    L-Click on
Setup in the menu bar and L-click ab
Initio L-Click on 6-31G then, L-Click on
Options button Select RHF, set Charge to 0 and
Multiplicity to 1 (default for charge
0) L-Click OK buttons after modifications were
done. 8.    L-Click on Compute in the menu bar
and select Geometry Optimization Select
Polak-Ribiere and set RMS gradient to 0.05 and
max cycles to 60 L-Click OK button (The
calculation will be started. Repeat the step till
ConvYES appears in the status line.). Record
the energy appeared in the status line 9.   
L-Click on Compute in the menu bar and select
Orbitals. Record energy levels and point groups
of required molecular orbitals (MO) (Optional
You can draw the contour plot of the selected
orbital and visualize the shape of the
orbital.) 10. L-Click on Compute in the menu bar
and select Vibrations. 11. L-Click on Compute in
the menu bar and select Vibrational
Spectrum. Record the frequencies of different
vibrational modes and their corresponding
oscillator strengths. (Optional You can turn on
animate vibrations, select any vibrational modes,
and L-Click on OK button. The molecule begins to
vibrate. To suspend the animation, L-Click on
Cancel button.)
Formaldehyde
Procedures

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Part B Molecular Dynamics of
Tetrapeptide 1.    L-click Databases on the menu
bar. Choose Amino Acids. 2.    Select Beta
sheet. 3.    L-click Ala, Tyr, Asp and Gly to
create tetrapeptide Ala-Tyr-Asp-Gly. 4.   
L-click on rotate-out-of-plane tool and use it to
rotate the molecule to a proper angle for
observation and measurements. (Rotate-out-of-plane
tool) 5.    L-Click on Setup in the menu bar
and L-click Molecular Mechanics L-Click on
MM L-Click OK buttons after modifications were
done. 6.    L-Click on Compute in the menu bar
and select Geometry Optimization 7.    Record
the total energies. 8.    L-Click on Compute in
the menu bar and L-click Molecular Dynamics Run
molecular dynamics at 0K and 300K with constant
temperature. Simulation Time 1ps 9.    Record
the total energies.   Part C Molecular
Dynamics of Ribosomal Protein Procedures 10.    
Use a web-browser and Go to http//yangtze.hku.hk/
lecture_notes.htm. 11.     R-click the title
labeled Download molecule and save it in a
folder in your local disk (C\). 12.     L-click
on File in the menu bar and select open to load
in the molecule. (You should notify that this
file has extension filename .ENT and is in PDB
format.) 13.     L-click on rotate-out-of-plane
tool and use it to rotate the molecule to a
proper angle for observation and
measurements. (Rotate-out-of-plane tool) 14.    
L-Click on Setup in the menu bar and L-click
Molecular Mechanics L-Click on MM L-Click OK
buttons after modifications were done. 15.    
L-Click on Compute in the menu bar and L-click
Molecular Dynamics Run molecular dynamics at
300K with constant temperature. Simulation Time
1ps 16.     Record the total energy.