Computer Modeling Dr. GuanHua CHEN Department of Chemistry University of Hong Kong

Mulliken,1966Fukui, 1981Hoffmann, 1981 Pople, 1998Kohn, 1998 Nobel Prizes for Computational Chemsitry

Computational Chemistry Quantum Chemistry Schr Ö dinger Equation H  = E  Molecular Mechanics F = Ma F : Force Field Bioinformatics

Computational Chemistry Industry CompanySoftware Gaussian Inc.Gaussian 94, Gaussian 98 Schrödinger Inc.Jaguar WavefunctionSpartanQ-Chem AccelrysInsightII, Cerius 2 HyperCubeHyperChem Informatix Celera Genomics Applications: material discovery, drug design & research R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billion Bioinformatics: Total Sales in 2001 US$ 225 million Project Sales in 2006US$ 1.7 billion

Vitamin C C60 Cytochrome c heme OH + D 2 --> HOD + D energy

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  Schr Ö dinger Equation Hamiltonian H =   (  h 2 /2m      h 2 /2m e )  i  i 2  i     e 2 /r i   +     Z  Z  e   r    i  j  e 2 /r ij Wavefunction Energy One-electron terms:   (  h 2 /2m      h 2 /2m e )  i  i 2  i     e 2 /r i  Two-electron term:   i  j  e 2 /r ij

1. Many-Body Wave Function is approximated by Single Slater Determinant 2. Hartree-Fock Equation F  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 Orbitals

3. Roothaan Method (introduction of Basis functions)  i =  k c ki  k LCAO-MO {  k } is a set of atomic orbitals (or basis functions) 4. Hartree-Fock-Roothaan equation  j ( F ij -  i S ij ) c ji = 0 F ij  i  F  j  S ij  i  j  5. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

Graphic Representation of Hartree-Fock Solution 0 eV Ionization Energy Electron Affinity

Basis Set  i =  p c ip  p {  k } is a set of atomic orbitals (or basis functions) STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G**  complexity & accuracy # HF/6-31G(d) Route section water energy Title 0 1 Molecule Specification O (in Cartesian coordinates H H A Gaussian Input File for H 2 O

Gaussian type functions g ijk = N x i y j z k exp(-  r 2 ) (primitive Gaussian function)  p =  u d up g u (contracted Gaussian-type function, CGTF) u = {ijk}p = {nlm}

STO-3G Basis Set

3-21G Basis Set

6-31G Basis Set

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

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 = H 0 + H’ H 0  n (0) = E n (0)  n (0)  n (0) is an eigenstate for unperturbed system H’ is small compared with H 0 Perturbation Theory

Moller-Plesset (MP) Perturbation Theory The MP unperturbed Hamiltonian H 0 H 0 =  m F(m) where F(m) is the Fock operator for electron m. And thus, the perturbation H ’ H ’ = H - H 0 Therefore, the unperturbed wave function is simply the Hartree-Fock wave function . Ab initio methods: MP2, MP3, MP4

 = e T  (0)  (0) : Hartree-Fock ground state wave function  : Ground state wave function T = T 1 + T 2 + T 3 + T 4 + T 5 + … T n : n electron excitation operator Coupled-Cluster Method = T1T1

 CCD = e T 2  (0)  (0) : Hartree-Fock ground state wave function  CCD : Ground state wave function T 2 : two electron excitation operator Coupled-Cluster Doubles (CCD) Method = T2T2

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) E 0 =  h 2 /2m e )  i    dr   e 2  (r) / r 1    dr 1 dr 2 e 2 /r 12 + E xc [  (r) ] Kohn-Sham Equation: F KS  i =  i  i F KS   h 2 /2m e )  i  i 2     e 2 / r 1   j  J j + V xc V xc   E xc [  (r) ] /  (r)

Ground State Excited State CPU Time Correlation Geometry Size Consistent (CHNH,6-31G*) HFSCF   1 0 OK  DFT   ~1   CIS   <10 OK  CISD   %   (20 electrons) CISDTQ   very large 98-99%   MP2   %   (DZ+P) MP4   5.8 >90%   CCD   large >90%   CCSDT   very large ~100%  

Reactant Product Transition State: one negative frequency Reaction Coordinate Search for Transition State GG k  e -  G/RT

#b3lyp/6-31G opt=qst2 test the first is the reactant internal coordinate 0 1 O H 1 oh1 H 1 oh1 2 ohh1 oh ohh The second is the product internal coordinate 0 1 O H 1 oh2 H 1 oh3 2 ohh2 oh2 0.9 oh ohh Gaussian Input File for Transition State Calculation

Semiempirical Molecular Orbital Calculation Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman) Independent electron approximation Schrodinger equation for electron i H val =  i H eff (i) H eff (i) = -(h 2 /2m)  i 2 + V eff (i) H eff (i)  i =  i  i

LCAO-MO:  i =  r c ri  r  s ( H eff rs -  i S rs ) c si = 0 H eff rs  r  H eff  s  S rs  r  s  Parametrization: H eff rr  r  H eff  r   minus the valence-state ionization  potential (VISP)

Atomic Orbital Energy VISP e 5 -e e 4 -e e 3 -e e 2 -e e 1 -e 1 H eff rs = ½ K (H eff rr + H eff ss ) S rs K:1  3

CNDO, INDO, NDDO (Pople and co-workers) Hamiltonian with effective potentials H val =  i [ -(h 2 /2m)  i 2 + V eff (i) ] +  i  j>i e 2 / r ij two-electron integral: (rs|tu) = 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 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.

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

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).

(1) Neglect or incomplete treatment of electron correlation (2) Incompleteness of the Basis set (3) Relativistic effects (4) Deviation from the Born-Oppenheimer approximation Four Sources of error in ab initio Calculation

Quantum Chemistry for Complex Systems

Quantum Mechanics / Molecular Mechanics (QM/MM) Method Combining quantum mechanics and molecular mechanics methods: QM MM

Hamiltonian of entire system: H = H QM +H MM +H QM/MM Energy of entire system: E = E QM ( QM ) + E MM ( MM ) + E QM/MM ( QM/MM ) E QM/MM ( QM/MM ) = E elec ( QM/MM ) + E vdw ( MM ) + E MM-bond ( MM ) E QM ( QM ) + E elec ( QM/MM ) = H eff = - 1/2  i  i 2 +  ij 1/r ij -    i Z  /r i  -    i q  /r i  +    i V v-b (r i ) +     Z  Z  /r  +     Z  q  /r  QM MM

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 Divide-and-Conqure (DAC)

Superoxide Dismutase (4380 atoms) York, Lee & Yang, JACS, 1996  Strain, Scuseria & Frisch, Science (1996): LSDA / 3-21G DFT calculation on 1026 atom RNA Fragment

Liang, Yokojima & Chen, JPC, 2000 Linear Scaling Calculation for Excited State

LDM-TDDFT: C n H 2n+2 Fast Multiple Method

LODESTAR: Software Package for Complex Systems Characteristics ： O(N) Divide-and-Conquer O(N) TDHF (ab initio & semiemptical) O(N) TDDFT CNDO/S-, PM3-, AM1-, INDO/S-, & TDDFT-LDM Light Harvesting System Nonlinear Optical

Photo-excitations in Light Harvesting System II generated by VMD strong absorption: ~800 nm generated by VMD

Carbon Nanotube

Quantum mechanical investigation of the field emission from the tips of carbon nanotubes Zettl, PRL 2001 Zheng, Chen, Li, Deng & Xu, Phys. Rev. Lett. 2004

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

Bond Stretching Potential E b = 1/2 k b (  l) 2 where, k b : stretch force constant  l : difference between equilibrium & actual bond length Two-body interaction

Bond Angle Deformation Potential E a = 1/2 k a (  ) 2 where, k a : angle force constant   : difference between equilibrium & actual bond angle  Three-body interaction

Periodic Torsional Barrier Potential E t = (V/2) (1+ cosn  ) where, V : rotational barrier  : torsion angle n : rotational degeneracy Four-body interaction

Non-bonding interaction van der Waals interaction for pairs of non-bonded atoms Coulomb potential for all pairs of charged atoms

Force Field Types MM2Molecules AMBERPolymers CHAMMPolymers BIOPolymers OPLSSolvent Effects

MM2 Force Field

CHAMM FORCE FIELD FILE

/Ao/Ao /(kcal/mol)

/(kcal/mol/A o2 ) /Ao/Ao

/(kcal/mol/rad 2 ) /deg

/(kcal/mol)/degn

AMBER FORCE FIELD

OPLS Force Field

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) (s 1 +2s 2 +2s 3 +s 4 )  t +O(  t 5 ) s 1 = dx/dt s 2 = dx/dt [w/ t=t+  t/2, x = x(t)+s 1  t/2] s 3 = dx/dt [w/ t=t+  t/2, x = x(t)+s 2  t/2] s 4 = dx/dt [w/ t=t+  t, x = x(t)+s 3  t] Very accurate but slow!

Algorithms for Molecular Dynamics Verlet Algorithm: x(t+  t) = x(t) + (dx/dt)  t + (1/2) d 2 x/dt 2  t x(t -  t) = x(t) - (dx/dt)  t + (1/2) d 2 x/dt 2  t x(t+  t) = 2x(t) - x(t -  t) + d 2 x/dt 2  t 2 + O(  t 4 ) Efficient & Commonly Used!

Goddard, Caltech Multiple Scale Simulation

Large Gear Drives Small Gear G. Hong et. al., 1999

Nano-oscillators Zhao, Ma, Chen & Jiang, Phys. Rev. Lett Nanoscopic Electromechanical Device (NEMS)

Computer-Aided Drug Design GENOMICS Human Genome Project

Computer-aided drug design Chemical Synthesis Screening using in vitro assay Animal Tests Clinical Trials

ALDOSE REDUCTASE Diabetes Diabetic Complications Glucose Sorbitol

Design of Aldose Reductase Inhibitors Aldose Reductase Inhibitor

Database for Functional Groups Descriptors: Electron negativity Volume

Possible drug leads: ~ 350 compounds

TYR48LYS77 HIS110 TRP111 PHE122 TYP219 TRP20 CYS298 LEU300 NADPH TRP79 VAL47 Aldose Reductase Active Site Structure Cerius2 LigandFit

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

Docking of aldose reductase inhibitor Cerius2 LigandFit Aldose reducatse (4R)-6’-fluoro-7’-hydroxyl-8’-bromo-3’-methylspiro- [imidazoli-dine-4,4’(1’H)-quinazoline]-2,2’,5(3’H)-trione Inhibitor Hu & Chen, 2003

Interaction energy between ligand and protein Quantum Mechanics/Molecular Mechanics (QM / MM) Hu & Chen, 2003

a:Inhibitor concentration of inhibit Aldose Reductase; b: the percents of lower sciatic nerve sorbitol levels c: interaction with AR in Fig. 4

Our Design Strategy QSAR determination & prediction (Neural Network) Docking (Cerius2) QM / MM (binding energy) ?

SARS 3CL Protease “Identification of novel small molecule inhibitors of severe acute respiratory syndrome associated coronavirus by chemical genetics”, Richard Y. Kao, Wayne H.W. Tsui, Terri S. W. Lee, Julian A. Tanner, Rory M. Watt, Jian-Dong Huang, Lihong Hu, Guanhua Chen, Zhiwei Chen, linqi Zhang, Tien He, Kwok-Hung Chan, Herman Tse, Amanda P. C. To, Louisa W. Y. Ng, Bonnie C. W. Wong, Hoi-Wah Tsoi, Dan Yang, David D. Ho, Kwok-Yung Yuen, Chemistry & Biology 11, 1293 (2004). A B Inhibitor site Complex with hexapeptidyl CMK inhibitor

New ligand candidates for SARS 3Cl-Protease generated by a known compound AG7088 AG7088 Anand, et al, Science, 300, 1763 (2003) Our prediction

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 LDM (localized-density-matrix) Neural Network for QSAR Monte Carlo & Molecular Dynamics