Computational Modeling of Macromolecular Systems Dr. GuanHua CHEN Department of Chemistry University of Hong Kong

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

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

Cytochrome c (involved in the ATP synthesis) heme Cytochrome c is a peripheral membrane protein involved in the long distance electron transfers 1997 Nobel Prize in Biology: ATP Synthase in Mitochondria

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 to s; 0.01 to 1 A o 2. Collective Motions to s; 0.01 to >5 A o 3. Conformational Changes to 10 3 s; 0.5 to >10 A o

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. 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

2. Roothaan Method (introduction of Basis functions)  i =  k c ki  k LCAO-MO {  k } is a set of atomic orbitals (or basis functions) 3. Hartree-Fock-Roothaan equation  j ( F ij -  i S ij ) c ji = 0 F ij  i  F  j  S ij  i  j  4. 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, 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: 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) 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 Ground State : Phys. Rev. 140, A1133 (1965) 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) A popular exchange-correlation functional E xc [  (r) ] : B3LYP

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%  

(1) Neglect or incomplete treatment of electron correlation (2) Incompleteness of the Basis set Four Sources of error in ab initio Calculation How to simulate large molecules?

Quantum Chemistry for Complex Systems

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

Linear Scaling Quantum Mechanical Methods

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 Divide-and-Conqure (DAC)

Linear Scaling Calculation for Ground State Yang, Phys. Rev. Lett Li, Nunes & Vanderbilt, Phy. Rev. B Baroni & Giannozzi, Europhys. Lett Gibson, Haydock & LaFemina, Phys. Rev. B Aoki, Phys. Rev. Lett Cortona, Phys. Rev. B Galli & Parrinello, Phys. Rev. Lett Mauri, Galli & Car, Phys. Rev. B Ordej ó n et. al., Phys. Rev. B Drabold & Sankey, Phys. Rev. Lett

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

Carbon Nanotube Chirality: (m, n) Smalley et. al., Nature (1998)

Quantum mechanical investigation of the field emission from the tips of carbon nanotubes Experimental Results J-M. Bonard et al., Phys. Rev. Lett (2002) F-N theory breaks down For strong CNT emission

Field Emission Basics Classical Model : Laplace’s Equation: Boundary Conditions: V(anode) = V a V(cathode-tube) = 0 Single nanotube model outline

Boundary conditions: V(anode) = V a V(cathode) = 0 Quantum Model Problems: 1.100,000 atoms 2.Boundary Condition: OPEN SYSTEM! 3.Number of electrons transferred to CNT

Boundary Condition Mirror image of charges

Charge distributions before & after external field (5,5)

Potential energy contour plot for SWNT (5,5) under a 14 V/μm applied field

Potential energy contour plot in the vicinity of cap under a 14 V/µm applied field Equipotential line corresponding to the Fermi energy (-4.5 eV) is presented

Potential energy distributions along the central axis of entire tube A layer of atoms is sufficient to shield most of external field!

E appl 010 V/  m14 V/  m Barrier height 4.5 eV3.0 eV2.0 eV Penetration does occur at the tip !

Effective enhancement factor  : 500 for E appl = 10 V/  m 1200 for E appl = 14 V/  m Calculated emission currents: 0.34 pA for E apply = 10 V/  m 0.20 µA for E apply = 14 V/  m Experiment [Zettl et. al., PRL 88, (2002)]: A Multi-Walled CNT: 0.40 pA for E apply = 11.7 V/  m 0.54 µA for E apply = 20.0 V/  m

ExperimentSimulation The multi-walled CNT is of same potential !!!

Linear Scaling Calculation for EXCITED STATE ? A Much More Difficult Problem !

Localized-Density-Matrix (LDM) Method  ij (0) = 0 r ij > r 0  ij = 0 r ij > r 1 Yokojima & Chen, Phys. Rev. B, 1999    Principle of the nearsightedness of equilibrium systems (Kohn, 1996) Linear-Scaling Calculation for excited states  t 

Heisenberg Equation of Motion Time-Dependent Hartree-Fock Random Phase Approximation

PPP Semiempirical Hamitonian Polyacetylene

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

Flat Panel Display

Cambridge Display Technology Weight: 15 gram Resolution: 800x236 Size: 45x37 mm Voltage: DC, 10V

Energy Intensity electron hole

Low-Lying Excited States of Light Harvesting System II in Purple Bacteria 1. “ Ng, Zhao and Chen, J. Phys. Chem. B 107, 9589 (2003) Application of O(N) method for excited states

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

B800 ring: strong 800nm B850 ring: strong 850nm 1α1α 1β1β 2α2α ~8.9Å ~9.2Å generated by VMD J1J1 J2J2 W Frenkel Exciton Model:  n J n  n +  n  n  n J nm  m +  n

Two issues: 1.Is the Frenkel exciton model a good description of the low-lying excitations in LH2? does the electron-hole pair span one B-chlorophyll at a time? values of J 1 & J 2 2.What is the energy transfer mechanism on B850? Energy transfer mechanisms: 1.Förster Incoherent hopping (Markovian) process; (small polaron) 2.Coherent exciton migration. (large polaron) The size of electron-hole pair is determined by the ratio of the n.n. coupling constant vs. the disorder in energy Static energy disorder:200 ~ 500 cm -1 Dynamic disorder:~200 cm -1 n.n. coupling << disorder: localized (Förster Incoherent hopping) n.n. coupling >> disorder: delocalized ( Coherent exciton transfer)

Calculated Parameters by others (Zerner, Fleming, Mukamel & etc.) INDO/S-CEO (a)PDA with (b)INDO/S-CIS (c) J 1 / cm J 2 / cm (a)Tretiak, S.; Chernyak, V.; Mukamel, S. J. Phys. Chem., , 2000 (b)Pullerits, T.; Sundstrom, V.; van Grondelle, R. J. Phys. Chem. 1999, 103, 2327 (c)Cory, M. G.; Zerner, M.C.; Hu, X.; Schulten, X. K.; J. Phys. Chem. B 1998, 102, 7640 Cory, M. G.; Zerner, M.C.; Hu, X.; Schulten, X. K.; J. Phys. Chem. B 1998, 102, 7640 Our task: what are J 1 & J 2 ?

Photo-excitations in Light Harvesting System II

736 atoms P3 / 700 MHz 500 MB RAM

Distorted field K= +/-  /8K=0,+/-  /4,+/-  /2, +/-3  /4     K = +/-7  /8 COS(  /2·n) & COS(7  /8·n): K = +/-3  /8, +/-5  /8 & K = , respectively k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8

CIS (Zerner et. al.) LDM

/ cm -1 J1J1 J2J2  1 1  2 2 C*rms Dimer# B Zerner Calculated parameters in Frenkel excition model (least square fitting) *transition dipole of monomer = e·A: C = cm -1 B        Doubly degenerate The B850 energies (eV) calculated by LDM

Solvation Correction J 1 ~ 445 cm -1 J 2 ~ 367 cm -1 Static disorder:200 ~ 500 cm -1 Dynamic disorder:~200 cm -1

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

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

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

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

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