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Computational Modeling of Macromolecular Systems

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

  2. Computational Chemistry • Quantum Chemistry SchrÖdinger Equation H = E • Molecular Mechanics F = Ma F : Force Field

  3. Computational Chemistry Industry Company Software Gaussian Inc. Gaussian 94, Gaussian 98 Schrödinger Inc. Jaguar Wavefunction Spartan Q-Chem Q-Chem Accelrys InsightII, Cerius2 HyperCube HyperChem 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 2006 US$ 1.7 billion

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

  5. Simulation of a pair of polypeptides Duration: 100 ps. Time step: 1 ps (Ng, Yokojima & Chen, 2000)

  6. Protein Dynamics 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 Changes 10-9 to 103 s; 0.5 to >10 Ao Theoretician leaded the way ! (Karplus at Harvard U.)

  7. 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

  8. SchrÖdinger Equation Hy = Ey Wavefunction Hamiltonian H = (-h2/2ma)2 - (h2/2me)ii2 - i Zae2/ria (+  ZaZbe2/rab ) + ije2/rij Energy One-electron terms: (-h2/2ma)2 - (h2/2me)ii2 - i Zae2/ria Two-electron term: ije2/rij

  9. Hartree-Fock Method Orbitals 1. Hartree-Fock Equation Ffi = ei fi FFock operator fi the i-th Hartree-Fock orbital ei the energy of the i-th Hartree-Fock orbital

  10. 2. Roothaan Method (introduction of Basis functions) fi= k ckiyk LCAO-MO { yk }is a set of atomic orbitals (or basis functions) 3. Hartree-Fock-Roothaan equation j ( Fij - ei Sij ) cji = 0 Fij  < i|F | j > Sij  < i| j > 4. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

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

  12. 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

  13. Gaussian type functions gijk = N xi yj zk exp(-ar2) (primitive Gaussian function) yp = u dupgu (contracted Gaussian-type function, CGTF) u = {ijk} p = {nlm}

  14. STO-3G Basis Set

  15. 3-21G Basis Set

  16. 6-31G Basis Set

  17. 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

  18. Configuration Interaction (CI) + + …

  19. Single Electron Excitation or Singly Excited

  20. Double Electrons Excitation or Doubly Excited

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

  22. Doubly Excited CI (CID): Changes ground & excited states + Singly & Doubly Excited CI (CISD): Most Used CI Method

  23. Full CI (FCI): Changes ground & excited states + + + ...

  24. 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

  25. Moller-Plesset (MP) Perturbation Theory The MP unperturbed Hamiltonian H0 H0 = mF(m) whereF(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, MP4

  26. 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 =

  27. 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 =

  28. Complete Active Space SCF (CASSCF) Active space All possible configurations

  29. 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 <i |i2|i >-  drZae2(r) /r1a + (1/2)   dr1 dr2e2/r12 + Exc[(r)] Kohn-Sham EquationGround State: Phys. Rev. 140, A1133 (1965) FKSyi = ei yi FKS- (h2/2me)ii2-  Zae2 /r1a + jJj + Vxc Vxc dExc[(r)] / d(r) A popular exchange-correlation functional Exc[(r)]: B3LYP

  30. Ground State Excited State CPU Time Correlation Geometry Size Consistent (CHNH,6-31G*) HFSCF   1 0 OK  DFT   ~1   CIS   <10 OK  CISD   17 80-90%   (20 electrons) CISDTQ   very large 98-99%   MP2   1.5 85-95%   (DZ+P) MP4   5.8 >90%   CCD   large >90%   CCSDT   very large ~100%  

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

  32. Quantum Chemistry for Complex Systems

  33. Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman) Independent electron approximation Schrodinger equation for electron i Hval = iHeff(i) Heff(i) = -(h2/2m) i2 + Veff(i) Heff(i) i = i i Semiempirical Molecular Orbital Calculation

  34. LCAO-MO: fi= r criyr s (Heffrs- eiSrs ) csi = 0 Heffrs < r|Heff| s >Srs< r| s > • Parametrization: • Heffrr < r|Heff| r > • = minus the valence-state ionization • potential (VISP)

  35. Atomic Orbital Energy VISP --------------- e5 -e5 --------------- e4 -e4 --------------- e3 -e3 --------------- e2 -e2 --------------- e1 -e1 Heffrs = ½ K(Heffrr + Heffss) SrsK: 13

  36. CNDO, INDO, NDDO (Pople and co-workers) Hamiltonian with effective potentials Hval = i [ -(h2/2m) i2 + Veff(i) ] + ij>i e2 / 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

  37. 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, INDOare 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.

  38. 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

  39. Linear Scaling Quantum Mechanical Methods

  40. Quantum Mechanical Simulation of Nano-size Systems Ground State:ab initio Hartree-Fock calculation

  41. Computational Time: protein w/ 10,000 atoms ab initio Hartree-Fock ground state calculation: ~20,000 years on CRAY YMP

  42. In 2010: ~24 months on 100 processor machine One Problem: Transitor with a few atoms Current Computer Technology will fail !

  43. Quantum Chemist’s Solution Linear-Scaling Method: O(N) Computational time scales linearly with system size Time Size

  44. Linear Scaling Calculation for Ground State Divide-and-Conqure (DAC) W. Yang, Phys. Rev. Lett. 1991

  45. Linear Scaling Calculation for Ground State Yang, Phys. Rev. Lett. 1991 Li, Nunes & Vanderbilt, Phy. Rev. B.1993 Baroni & 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.

  46. 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

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

  48. Quantum mechanical investigation of the field emission from the tips of carbon nanotubes Experimental Results F-N theory breaks down For strong CNT emission J-M. Bonard et al., Phys. Rev. Lett. 89 19 (2002)