Computational Modeling of Macromolecular Systems Dr. GuanHua CHEN Department of Chemistry University of Hong Kong
Jan 13, 2016
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