Computational Modeling of Macromolecular Systems

Dr. GuanHua CHEN

Department of Chemistry

University of Hong Kong

Computational Chemistry

• Quantum Chemistry

SchrÖdinger Equation

H = E• Molecular Mechanics

F = Ma

F : Force Field

Computational Chemistry Industry

Company Software

Gaussian Inc. Gaussian 94, Gaussian 98Schrödinger Inc. Jaguar Wavefunction SpartanQ-Chem Q-ChemMolecular Simulation Inc. (MSI) InsightII, Cerius2, modelerHyperCube HyperChem

Applications: material discovery, drug design & research

R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billionSales of Scientific Computing in 2000: > US$ 200 million

Cytochrome c (involved in the ATP synthesis)

heme

Cytochrome c is a peripheral membrane protein involved in the long distance electron transfers

1997 Nobel Prizein Biology:

ATP Synthase inMitochondria

Simulation of a pair of polypeptides

Duration: 100 ps. Time step: 1 ps (Ng, Yokojima & Chen, 2000)

Protein Dynamics

Theoretician leaded the way ! (Karplus at Harvard U.)

1. Atomic Fluctuations 10-15 to 10-11 s; 0.01 to 1 Ao

2. Collective Motions

10-12 to 10-3 s; 0.01 to >5 Ao

3. Conformational Changes10-9 to 103 s; 0.5 to >10 Ao

Scanning Tunneling Microscope

Manipulating Atoms by Hand

Large Gear Drives Small Gear

G. Hong et. al., 1999

Calculated Electron distribution at equator

The electron density around the vitamin C molecule. The colors show the electrostatic potential with the negative areas shaded in red and the positive in blue.

Vitamin C

Molecular Mechanics (MM) Method

F = MaF : Force Field

Molecular Mechanics Force Field

• Bond Stretching Term

• Bond Angle Term

• Torsional Term

• Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction

Bond Stretching PotentialEb = 1/2 kb (l)2

where, kb : stretch force constantl : difference between equilibrium & actual bond length

Two-body interaction

Bond Angle Deformation PotentialEa = 1/2 ka ()2

where, ka : angle force constant

: difference between equilibrium & actual bond angle

Three-body interaction

Periodic Torsional Barrier PotentialEt = (V/2) (1+ cosn )where, V : rotational barrier

: torsion angle n : rotational degeneracy

Four-body interaction

Non-bonding interaction

van der Waals interactionfor pairs of non-bonded atoms

Coulomb potential

for all pairs of charged atoms

MM Force Field Types

• MM2 Small molecules

• AMBER Polymers

• CHAMM Polymers

• BIO Polymers

• OPLS Solvent Effects

######################################################## ## ## ## TINKER Atom Class Numbers to CHARMM22 Atom Names ## ## ## ## 1 HA 11 CA 21 CY 31 NR3 ## ## 2 HP 12 CC 22 CPT 32 NY ## ## 3 H 13 CT1 23 CT 33 NC2 ## ## 4 HB 14 CT2 24 NH1 34 O ## ## 5 HC 15 CT3 25 NH2 35 OH1 ## ## 6 HR1 16 CP1 26 NH3 36 OC ## ## 7 HR2 17 CP2 27 N 37 S ## ## 8 HR3 18 CP3 28 NP 38 SM ## ## 9 HS 19 CH1 29 NR1 ## ## 10 C 20 CH2 30 NR2 ## ## ## ########################################################

CHAMM FORCE FIELD FILE

atom 1 1 HA "Nonpolar Hydrogen" 1 1.0081atom 2 2 HP "Aromatic Hydrogen" 1 1.0081atom 3 3 H "Peptide Amide HN" 1 1.0081atom 4 4 HB "Peptide HCA" 1 1.0081atom 5 4 HB "N-Terminal HCA" 1 1.0081atom 6 5 HC "N-Terminal Hydrogen" 1 1.0081atom 7 5 HC "N-Terminal PRO HN" 1 1.0081atom 8 3 H "Hydroxyl Hydrogen" 1 1.0081atom 9 3 H "TRP Indole HE1" 1 1.0081atom 10 3 H "HIS+ Ring NH" 1 1.0081atom 11 3 H "HISDE Ring NH" 1 1.0081atom 12 6 HR1 "HIS+ HD2/HISDE HE1" 1 1.0081

################################ ## ## ## Van der Waals Parameters ## ## ## ################################

vdw 1 1.3200 -0.0220vdw 2 1.3582 -0.0300vdw 3 0.2245 -0.0460vdw 4 1.3200 -0.0220vdw 5 0.2245 -0.0460vdw 6 0.9000 -0.0460vdw 7 0.7000 -0.0460vdw 8 1.4680 -0.0078vdw 9 0.4500 -0.1000vdw 10 2.0000 -0.1100

/Ao /(kcal/mol)

################################## ## ## ## Bond Stretching Parameters ## ## ## ##################################

bond 1 10 330.00 1.1000bond 1 11 340.00 1.0830bond 1 12 317.13 1.1000bond 1 13 309.00 1.1110bond 1 14 309.00 1.1110bond 1 15 322.00 1.1110bond 1 17 309.00 1.1110bond 1 18 309.00 1.1110bond 1 21 330.00 1.0800

/(kcal/mol/Ao2) /Ao

################################ ## ## ## Angle Bending Parameters ## ## ## ################################

angle 3 10 34 50.00 121.70angle 13 10 24 80.00 116.50angle 13 10 27 20.00 112.50angle 13 10 34 80.00 121.00angle 14 10 24 80.00 116.50angle 14 10 27 20.00 112.50angle 14 10 34 80.00 121.00angle 15 10 24 80.00 116.50angle 15 10 27 20.00 112.50angle 15 10 34 80.00 121.00angle 16 10 24 80.00 116.50angle 16 10 27 20.00 112.50

/(kcal/mol/rad2) /deg

############################ ## ## ## Torsional Parameters ## ## ## ############################torsion 1 11 11 1 2.500 180.0 2torsion 1 11 11 11 3.500 180.0 2torsion 1 11 11 22 3.500 180.0 2torsion 2 11 11 2 2.400 180.0 2torsion 2 11 11 11 4.200 180.0 2torsion 2 11 11 14 4.200 180.0 2torsion 2 11 11 15 4.200 180.0 2torsion 2 11 11 22 3.000 180.0 2torsion 2 11 11 35 4.200 180.0 2torsion 2 11 11 36 4.200 180.0 2torsion 11 11 11 11 3.100 180.0 2torsion 11 11 11 14 3.100 180.0 2torsion 11 11 11 15 3.100 180.0 2torsion 11 11 11 22 3.100 180.0 2torsion 11 11 11 35 3.100 180.0 2torsion 11 11 11 36 3.100 180.0 2

/(kcal/mol) /deg

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) (s1+2s2+2s3+s4) t +O(t5) s1 = dx/dt s2 = dx/dt [w/ t=t+t/2, x = x(t)+s1t/2] s3 = dx/dt [w/ t=t+t/2, x = x(t)+s2t/2] s4 = dx/dt [w/ t=t+t, x = x(t)+s3 t]

Very accurate but slow!

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!

Calculated Properties

• Structure, Geometry

• Energy & Stability

• Mechanic Properties: Young’s Modulus

• Vibration Frequency & Mode

Crystal Structure of C60 solid

Crystal Structure of K3C60

Vibration Spectrum of K3C60

GH Chen, Ph.D. Thesis, Caltech (1992)

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

H E

SchrÖdinger Equation

HamiltonianH = (h2/2mh2/2me)ii

2 + ZZeri e2/ri

ije2/rij

Wavefunction

Energy

f(1)+ J2(1) K2(1)1(1)11(1)f(2)+ J1(2) K1(2)2(2)22(2)

F(1) f(1)+ J2(1) K2(1) Fock operator for 1F(2) f(2)+ J1(2) K1(2) Fock operator for 2

Hartree-Fock Equation:

Fock Operator:

+e-

e-

f(1) h2/2me)12 N ZNr1N

one-electron term if no Coulomb interactionJ2(1) dr2

e2/r122Ave. Coulomb potential on electron 1 from 2 K2(1) 2 dr2

*e2/r12 Ave. exchange potential on electron 1 from 2f(2) h2/2me)2

2 N ZNr2NJ1(2) dr1

e2/r121K1(2) 1 dr1

*e2/r12 Average Hamiltonian for electron 1 F(1) f(1)+ J2(1) K2(1)

Average Hamiltonian for electron 2 F(2) f(2)+ J1(2) K1(2)

1. Many-Body Wave Function is approximated by Single Slater Determinant

2. Hartree-Fock EquationF i = i i

  F Fock operator

i the i-th Hartree-Fock orbital

i the energy of the i-th Hartree-Fock orbital

Hartree-Fock Method

3. Roothaan Method (introduction of Basis functions)i = k cki k LCAO-MO

  {k } is a set of atomic orbitals (or basis functions)

4. Hartree-Fock-Roothaan equation j ( Fij - i Sij ) cji = 0

  Fij iF j Sij ij

5. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

Graphic Representation of Hartree-Fock Solution

0 eV

IonizationEnergy

ElectronAffinity

The energy required to remove an electron from aclosed-shell atom or molecules is well approximatedby minus the orbital energy of the AO or MO fromwhich the electron is removed.

Koopman’s Theorem

Slater-type orbitals (STO)  nlm = N rn-1exp(r/a0) Ylm(,)

 the orbitalexponent

Gaussian type functions (GTF)gijk = N xi yj zk exp(-r2)

(primitive Gaussian function)p = u dup gu

(contracted Gaussian-type function, CGTF)u = {ijk} p = {nlm}

Basis Set i = p cip p

Basis set of GTFs STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G**------------------------------------------------------------------------------------- complexity & accuracy

Minimal basis set: one STO for each atomic orbital (AO)

STO-3G: 3 GTFs for each atomic orbital3-21G: 3 GTFs for each inner shell AO 2 CGTFs (w/ 2 & 1 GTFs) for each valence AO 6-31G: 6 GTFs for each inner shell AO 2 CGTFs (w/ 3 & 1 GTFs) for each valence AO 6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows

and a set of p functions to hydrogen Polarization Function

Diffuse Basis Sets:For excited states and in anions where electronic densityis more spread out, additional basis functions are needed.

Diffuse functions to 6-31G basis set as follows: 6-31G* - adds a set of diffuse s & p orbitals to atoms in 1st & 2nd rows (Li - Cl). 6-31G** - adds a set of diffuse s and p orbitals to atoms in 1st & 2nd rows (Li- Cl) and a set of diffuse s functions to H Diffuse functions + polarisation functions:6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets.

Double-zeta (DZ) basis set: two STO for each AO

6-31G for a carbon atom: (10s12p) [3s6p]

1s 2s 2pi (i=x,y,z)

6GTFs 3GTFs 1GTF 3GTFs 1GTF

1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)

Electron Correlation: avoiding each other

Two reasons of the instantaneous correlation:(1) Pauli Exclusion Principle (HF includes the effect)(2) Coulomb repulsion (not included in the HF)

Beyond the Hartree-FockConfiguration Interaction (CI)*Perturbation theory*Coupled Cluster MethodDensity functional theory

Configuration Interaction (CI)

+

+ …

Single Electron Excitation or Singly Excited

Double Electrons Excitation or Doubly Excited

Singly Excited Configuration Interaction (CIS): Changes only the excited states

+

Doubly Excited CI (CID):Changes ground & excited states

+

Singly & Doubly Excited CI (CISD):Most Used CI Method

Full CI (FCI):Changes ground & excited states

++

+ ...

H = H0 + H’H0n

(0) = En(0)n

(0)

n(0) is an eigenstate for unperturbed system

H’ is small compared with H0

Perturbation Theory

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

= eT(0)

(0): Hartree-Fock ground state wave function: Ground state wave functionT = T1 + T2 + T3 + T4 + T5 + …Tn : n electron excitation operator

Coupled-Cluster Method

=T1

CCD = eT2(0)

(0): Hartree-Fock ground state wave functionCCD: Ground state wave functionT2 : two electron excitation operator

Coupled-Cluster Doubles (CCD) Method

=T2

Complete Active Space SCF (CASSCF)

Active space

All possible configurations

Density-Functional Theory (DFT)Hohenberg-Kohn Theorem: 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 <i |i

2 |i > dr e2(r) /

r1 dr1 dr2 e2/r12 + Exc[(r)]

Kohn-Sham Equation: FKS i = i i

FKS h2/2me)ii2 e2 / r1jJj + Vxc

Vxc Exc[(r)] / (r)

Semiempirical Molecular Orbital Calculation

Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman)

Independent electron approximation

Schrodinger equation for electron i 

Hval = i Heff(i)

Heff(i) = -(h2/2m) i2 + Veff(i)

Heff(i) i = i i

LCAO-MO: i = r cri r

  s ( Heff

rs - i Srs ) csi = 0

  Heffrs rHeff s Srs

rs Parametrization: Heff

rr rHeff r minus the valence-state ionization potential (VISP)

Atomic Orbital Energy VISP--------------- e5 -e5

--------------- e4 -e4

--------------- e3 -e3

--------------- e2 -e2

--------------- e1 -e1

 Heff

rs = ½ K (Heffrr + Heff

ss) Srs K:

13

CNDO, INDO, NDDO(Pople and co-workers)

Hamiltonian with effective potentialsHval = i [ -(h

2/2m) i2 + Veff(i) ] + ij>i e

2 / rij

two-electron integral:(rs|tu) = <r(1) t(2)| 1/r12 | s(1) u(2)>

 CNDO: complete neglect of differential overlap (rs|tu) = rs tu (rr|tt) rs tu rt

INDO: intermediate neglect of differential overlap(rs|tu) = 0 when r, s, t and u are not on the same atom.

NDDO: neglect of diatomic differential overlap(rs|tu) = 0 if r and s (or t and u) are not on the same atom.

CNDO, INDO are parametrized so that the overallresults fit well with the results of minimal basis abinitio Hartree-Fock calculation.

CNDO/S, INDO/S are parametrized to predict optical spectra.

MINDO, MNDO, AM1, PM3(Dewar and co-workers, University of Texas, Austin) MINDO: modified INDOMNDO: modified neglect of diatomic overlap AM1: Austin Model 1PM3: MNDO parametric method 3 *based on INDO & NDDO *reproduce the binding energy

Relativistic Effects

Speed of 1s electron: Zc / 137

Heavy elements have large Z, thus relativistic effects areimportant.

Dirac Equation:Relativistic Hartree-Fock w/ Dirac-Fock operator; orRelativistic Kohn-Sham calculation; orRelativistic effective core potential (ECP).

Ground State: ab initio Hartree-Fock calculation

Computational Time: protein w/ 10,000 atoms

ab initio Hartree-Fock ground state calculation:

~20,000 years on CRAY YMP

In 2010: ~24 months on 100 processor machine

One Problem: Transitor with a few atoms

Current Computer Technology will fail !

Quantum Chemist’s Solution

Linear-Scaling Method: O(N)

Computational time scales linearly with system size

Time

Size

Linear Scaling Calculation for Ground State

W. Yang, Phys. Rev. Lett. 1991

Divide-and-Conqure (DAC)

Density-Matrix Minimization (DMM) Method

Li, Nunes and Vanderbilt, Phy. Rev. B. 1993

Minimize the Energy or the Grand Potential:

= Tr [ (32 - 23) (H-I) ]

Orbital Minimization (OM) Method

Mauri (1993), Ordejon (1993), Galii (1994), Kim (1995)

Minimize the Energy or the Grand Potential:

= 2 nij cni (H-I)ij cn

j - nmij cn

i (H-I)ij cmj l cn

l cml

Fermi Operator Expansion (FOE) Method

Goedecker & Colombo (1994)

Expand Density Matrix in Chebyshev Polynomial:

(H) = c0I + c1H + c2H2 + … = c0I / 2 + cjTj(H) + …

T0(H) = IT1(H) = H

Tj+1 (H) = 2HTj(H) - Tj-1(H)

Superoxide Dismutase (4380 atoms)

York, Lee & Yang, JACS, 1996

Linear Scaling First Principle Method

Two-electron integrals :

Vabcd = abe2 / r12 dc

Coulomb Integrals: Fast Multiple Method (FMM)

Exchange-Correlation (XC):Use of Locality

Strain, Scuseria & Frisch, Science (1996):LSDA / 3-21G DFT calculation on 1026 atom RNA Fragment

Linear Scaling Calculation for Ground State

Yang, Phys. Rev. Lett. 1991Li, Nunes & Vanderbilt, Phy. Rev. B. 1993Baroni & Giannozzi, Europhys. Lett. 1992. Gibson, Haydock & LaFemina, Phys. Rev. B 1993.Aoki, Phys. Rev. Lett. 1993.Cortona, Phys. Rev. B 1991.Galli & Parrinello, Phys. Rev. Lett. 1992.Mauri, Galli & Car, Phys. Rev. B 1993.Ordejón et. al., Phys. Rev. B 1993.Drabold & Sankey, Phys. Rev. Lett. 1993.

Linear Scaling Calculation for EXCITED STATE ?

A Much More Difficult Problem !

Localized-Density-Matrix (LDM) Method

ij(0) = 0 rij > r0

ij = 0 rij > r1Yokojima & Chen, Phys. Rev. B, 1999

Principle of the nearsightedness of equilibrium systems (Kohn, 1996)

Linear-Scaling Calculation for excited states

t

,Hi

Heisenberg Equation of Motion

Time-Dependent Hartree-Fock Random Phase Approximation

PPP Semiempirical Hamitonian

Polyacetylene

1

2

3

4

5

6

7

8

9

10

11

12

N-3

N-2

N-1

N

...

CH CH2N

extcckeluH HHHH ˆˆˆˆ

Liang, Yokojima & Chen, JPC, 2000

0 5000 10000 15000 200000

10,000

20,000

30,000

40,000

LDM

=50

0=20

LDM

=80

c=30

HF

CP

U T

ime

(s)

Number of Atoms

0 200 400 600 8000

1000

2000

3000

LDM

=50

c=20

LDM

=80

c=30

HF

CP

U T

ime

(s)

Number of Atoms

Yokojima, Zhou & Chen, Chem. Phys. Lett., 1999

Liang, Yokojima & Chen, JPC, 2000

Flat Panel Display

Cambridge Display Technology

Weight: 15 gramResolution: 800x236Size: 45x37 mmVoltage: DC, 10V

Energy

Inte

nsi

ty

electron

hole

Carbon Nanotube

Liang, Wang, Yokojima & Chen, JACS (2000)

Surprising!DFT: no or very small gap

Absorption Spectra of (9,0) SWNTs

Smallest SWNT: 0.4 nm in diameter

Wang, Tang & etc., Nature (2000)

Three possibilities:

(4,2), (3,3) & (5,0) SWNTs

Tang et. al, 2000

Absorption of SWNTs (4,2), (3,3) & (5,0)

C332H12

C420H12

C330

Liang, & Chen (2001)

Quantum Mechanics / Molecular Mechanics (QM/MM) Method

Combining quantum mechanics and molecular mechanics methods:

QM

MM

GENOMICSHuman Genome Project

Design of Aldose Reductase Inhibitors

Aldose Reductase

Goddard, CaltechGoddard, Caltech