1 / 98

Computational Modeling of Macromolecular Systems

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. Company.

pisces
Download Presentation

Computational Modeling of Macromolecular Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 Molecular Simulation Inc. (MSI) InsightII, Cerius2, modeler HyperCube HyperChem Applications: material discovery, drug design & research R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billion Sales of Scientific Computing in 2000: > US$ 200 million

  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. Nanotechnology Scanning Tunneling Microscope Manipulating Atoms by Hand

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

  9. Calculated Electron distribution at equator

  10. Vitamin C 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.

  11. Molecular Mechanics (MM) Method F = Ma F : Force Field

  12. Molecular Mechanics Force Field • Bond Stretching Term • Bond Angle Term • Torsional Term • Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction

  13. Bond Stretching Potential Eb = 1/2 kb (Dl)2 where, kb : stretch force constant Dl : difference between equilibrium & actual bond length Two-body interaction

  14. Bond Angle Deformation Potential Ea = 1/2 ka (D)2 where, ka : angle force constant D : difference between equilibrium & actual bond angle  Three-body interaction

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

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

  17. MM Force Field Types • MM2 Small molecules • AMBER Polymers • CHAMM Polymers • BIO Polymers • OPLS Solvent Effects

  18. CHAMM FORCE FIELD FILE

  19. /(kcal/mol) /Ao

  20. /(kcal/mol/Ao2) /Ao

  21. /deg /(kcal/mol/rad2)

  22. /(kcal/mol) /deg

  23. 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)+s1t/2] s3 = dx/dt [w/ t=t+t/2, x = x(t)+s2t/2] s4 = dx/dt [w/ t=t+t, x = x(t)+s3t] Very accurate but slow!

  24. 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!

  25. Calculated Properties • Structure, Geometry • Energy & Stability • Mechanic Properties: Young’s Modulus • Vibration Frequency & Mode

  26. Crystal Structure of C60 solid Crystal Structure of K3C60

  27. Vibration Spectrum of K3C60 GH Chen, Ph.D. Thesis, Caltech (1992)

  28. 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

  29. SchrÖdinger Equation Hy = Ey Wavefunction Hamiltonian H = (-h2/2ma)2 - (h2/2me)ii2 +  ZaZbe2/rab - i Zae2/ria + ije2/rij Energy

  30. Hartree-Fock Equation: [ f(1)+ J2(1) -K2(1)] f1(1) = e1 f1(1) [ f(2)+ J1(2) -K1(2)] f2(2) = e2 f2(2) Fock Operator: F(1) f(1)+ J2(1) -K2(1) Fock operator for 1 F(2) f(2)+ J1(2) -K1(2) Fock operator for 2 e- + e-

  31. f(1) -(h2/2me)12 -N ZN/r1N one-electron term if no Coulomb interaction J2(1) dr2 f2*(2)e2/r12 f2(2) Ave. Coulomb potential on electron 1 from 2 K2(1) f1(1)  f2(1)  dr2 f2*(2) e2/r12 f1(2) Ave. exchange potential on electron 1 from 2 f(2) -(h2/2me)22 -N ZN/r2N J1(2) dr1 f1*(1)e2/r12 f1(1) K1(2) q(2)  f1(1)  dr1 f1*(1) e2/r12 q(1) 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)

  32. Hartree-Fock Method 1. Many-Body Wave Function is approximated by Single Slater Determinant 2. 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

  33. 3. Roothaan Method (introduction of Basis functions) fi= k ckiyk LCAO-MO { yk }is a set of atomic orbitals (or basis functions) 4. Hartree-Fock-Roothaan equation j ( Fij - ei Sij ) cji = 0 Fij  < i|F | j > Sij  < i| j > 5. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

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

  35. Koopman’s Theorem The energy required to remove an electron from a closed-shell atom or molecules is well approximated by minus the orbital energy e of the AO or MO from which the electron is removed.

  36. Slater-type orbitals (STO) nlm = Nrn-1exp(-r/a0) Ylm(,) x the orbitalexponent Basis Set i = p cip p Gaussian type functions (GTF) gijk = N xi yj zk exp(-ar2) (primitive Gaussian function) p = u dupgu (contracted Gaussian-type function, CGTF) u = {ijk} p = {nlm}

  37. 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 orbital 3-21G: 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 rows 6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen Polarization Function

  38. Diffuse Basis Sets: For excited states and in anions where electronic density is 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

  39. 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)

  40. 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-Fock Configuration Interaction (CI)* Perturbation theory* Coupled Cluster Method Density functional theory

  41. Configuration Interaction (CI) + + …

  42. Single Electron Excitation or Singly Excited

  43. Double Electrons Excitation or Doubly Excited

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

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

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

  47. 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

  48. 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, MP3, MP4

  49. 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 =

More Related