QM/MM Background

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First lecture in QM/MM course

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QM/MM Background

  1. 1. QM/MM methods applied to reaction mechanisms in enzymes Required for credit (7.5 ECTS): Present the method used in one of the papers on the list Appreciated: PDF file of slides before presentation (on web site) Links to papers you used to prepare slides Blog post* summarizing in-class discussion Required for extra credit (2.5 ECTS): Proposal describing improvement to QM/MM *http://proteinsandwavefunctions.blogspot.com/Monday, January 31, 2011 1
  2. 2. QM/MM methods applied to reaction mechanisms in enzymes Intro + 7 papers in 8 weeks 6 students: Casper, Anders, Martin, Kasper, Eric, Janus Week 1 (Feb 3rd): Jan - QM/MM Background Week 2 (Feb 10) Jan - Yang paper Week 3 (Feb 17) ? - Paper ? Week 4 (Feb 24) ? - Paper ? Week 5 (March 10) ? - Paper ? Week 6 (March 17) ? - Paper ? Week 7 (March 24) ? - Paper ? Week 8 (March 31) ? - Paper ? 2Monday, January 31, 2011 2
  3. 3. Measured: rate [P]/s Rate => kcat 10.1126/science.1088172 3Monday, January 31, 2011 3
  4. 4. kcat is converted to free energy via transition state theory kcat ⇒ ΔG 0 act Most QM/MM studies assume ΔGextra ≈ 0 10.1126/science.1088172 4Monday, January 31, 2011 4
  5. 5. The activation free energy ΔG TS,0 =G −G TS ES ⎛ conformations ⎞ G = −RT ln ⎜ X ⎝ ∑ e −GiX / RT ⎟ ⎠ i ⎛ conformations ( ) ⎞ ∑ − GiX −G0 / RT X = G − RT ln ⎜ X e ⎟ ⎝ 0 i ⎠ 0 is the conformation with lowest G Some QM/MM studies assume G ≈G X X 0 (this also assumes the lowest energy conf has been found) 5Monday, January 31, 2011 5
  6. 6. The free energy change has an electronic and vibrational contribution G ≈ E +G X X ele X vib 6Monday, January 31, 2011 6
  7. 7. Challenges for QM/MM studies Computing Eele and Gvib Finding the TS Eele ≈ EQM + EMM + EQM / MM + Eboundary 7 image: 10.1080/01442350903495417Monday, January 31, 2011 7
  8. 8. Computing the “electronic” QM/MM energy Eele ≈ EQM + EMM + EQM / MM + Eboundary EQM = Ψ H Ψ + ∑ ∑ Z I Z J RIJ ˆ −1 I J >I some bonds angles dihedrals ∑ k (r − r ) ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 EMM = i i i,e + i i ⎡ ⎤ i i i MM MM atoms atoms ⎛ Ai A j Bi B j qi q j ⎞ +∑ ∑ ⎝ − r 6 + r12 + r ⎟ ⎜ i j >i ij ij ij ⎠ 8Monday, January 31, 2011 8
  9. 9. Computing the “electronic” QM/MM energy MM QM MM QM MM atoms qi atoms atoms ZI qj ⎛ AI A j BI B j ⎞ atoms atoms EQM / MM = Ψ ∑ ri Ψ + ∑∑ rIj + ∑ ∑ ⎝ − r 6 + r12 ⎠ ⎜ ⎟ i I j I j Ij Ij AI and BI may need to be re-adjusted What are AI and BI for atoms in a TS? Notice that Ψ is polarized by qi’s (this is called electrostatic embedding) 9Monday, January 31, 2011 9
  10. 10. The QM/MM covalent boundary requires special consideration because an MM atom does not help satisfy QM valence most popular (easiest to implement) 10.1080/01442350903495417 10Monday, January 31, 2011 10
  11. 11. The link atom method Boundary constraints H some angles dihedrals Eboundary = ki ( ri − ri,e ) + ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 i i ⎡ ⎤ i i image and text:10.1021/jp9924124 11Monday, January 31, 2011 11
  12. 12. The link atom method Boundary charge adjustment charges close to density cause over-polarization Solutions All q’s in residue are density set to 0 Closest q’s set to 0 remaining q’s rescaled QM MM Closest q’s represented by Gaussian functions MM atoms qi EQM / MM = Ψ ∑ ri Ψ + ... (Deleting 1-e- integrals i involving link atom, image: 10.1021/jp0743469 12 large errors for ab initio)Monday, January 31, 2011 12
  13. 13. The Localized-SCF method The density localized molecular orbital of the boundary bond is kept frozen during the SCF some angles dihedrals Eboundary = ki ( ri − ri,e ) + ∑ ki (θ i − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 ⎡ ⎤ i i image: 10.1021/jp000887l text: 10.1016/S0009-2614(00)00289-X  13Monday, January 31, 2011 13
  14. 14. The Generalized Hybrid Orbital method frozen orbital vs 10.1080/01442350903495417 14Monday, January 31, 2011 14
  15. 15. QM/MM = QM program + MM program MM QM MM atoms atoms atoms qi ZI qj ˆ Eele = Ψ H + ∑ ri Ψ + ∑ ∑ Z I Z J RIJ + −1 ∑∑ rIj i I J >I I j QM MM MM MM atoms atoms ⎛ AI A j BI B j ⎞ atoms atoms ⎛ Ai A j Bi B j qi q j ⎞ +∑ ∑ ⎝ − r 6 + r12 ⎠ + ⎜ ⎟ ∑ ∑ ⎝ − r 6 + r12 + r ⎟ ⎜ I j Ij Ij i j >i ij ij ij ⎠ some bonds angles dihedrals + ∑ ki ( ri − ri,e ) + ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 i i ⎡ ⎤ i i i boundary boundary boundary bonds angles dihedrals ∑ k (r − r ) ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 + i i i,e + i i ⎡ ⎤ i i i 15Monday, January 31, 2011 15
  16. 16. QM/MM = QM program + MM program Eele = EQM + EQM/mm + Eqm,MM + EMM GAMESS, GAUSSIAN, Turbomole, Molpro, ... Chemshell, QoMMMa, COMQUM AMBER, CHARMM, GROMACS, .... (some MM programs have semiempirical QM in them) The interface programs also often perform geometry optimizations after collecting gradient terms from both programsMonday, January 31, 2011 16
  17. 17. QM/MM = QM program + MM program MM QM MM atoms atoms atoms qi ZI qj EQM + EQM / mm ˆ = ΨH+ ∑ ri Ψ + ∑ ∑ Z I Z J RIJ + −1 ∑∑ rIj i I J >I I j QM MM boundary boundary boundary ⎛ AI A j BI B j ⎞ atoms atoms bonds angles dihedrals ∑ ∑ ⎝ − r 6 + r12 ⎠ + ∑ k (r − r ) ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ 2 2 Eqm / MM = ⎜ ⎟ i i i,e + i i ⎡ ⎤ I j Ij Ij i i i some MM MM bonds angles dihedrals atoms atoms ⎛ Ai A j Bi B j qi q j ⎞ ∑ k (r − r ) ∑ k (θ − θ i,e ) + ∑ Vi ⎣1 ± cos ( niφi ) ⎦ + ∑ ∑ ⎝ − r 6 + r12 + r ⎟ 2 2 EMM = i i i,e + i i ⎡ ⎤ ⎜ i i i i j >i ij ij ij ⎠ gQM , mm + g qm, MM gx,QM / mm = ( ∂ EQM + EQM / mm ) ∂xQM ∂Eqm / MM g MM gx,qm / MM = ∂xQM ∂EMM gx, MM = ∂x MMMonday, January 31, 2011 17
  18. 18. QM/MM = QM program + MM program Workflow protein structure form PDB repair, add hydrogens, determine protonation state build in substrate MM minimize, MD? Define QM region => boundary coord + charges fed into QM program Compute EQM/mm + g for QM atoms coord + vdW param for substrate fed into MM program special MM parameters for boundary? Compute Eqm/MM + EMM + g for all atoms Add g’s compute new coordMonday, January 31, 2011 18
  19. 19. Computing the QM/MM Gvib Eele ≈ EQM + EMM + EQM / MM + Eboundary ∂ 2 Eele H ij = ∂xi ∂y j too time matrix diagonalization consuming for k = L HL t scales as N3 larger systems ki νi = 2π ⎛ e− hν /2 kT ⎞ Gvib = −RT ln ⎜ − hν /2 kT ⎟ ⎝1− e ⎠ 19Monday, January 31, 2011 19
  20. 20. Computing the QM/MM Gvib Solutions 1. ΔGvib ≈ 0 1.5ν kcal/mol  ZPE ≈ −1 1000 cm i.e. breaking a covalent bond contributes roughly 3-4 kcal/mol to ΔH vib 2. Compute Gvib for model reaction (not good approximation of ΔSvib) 20Monday, January 31, 2011 20
  21. 21. Finding the TS Conventional TS finding algorithms use the Hessian H −1 q n +1 = q n − H g n n Common solution: adiabatic mapping 21 text: 10.1080/01442350903495417Monday, January 31, 2011 21
  22. 22. Dynamic Effects via MD ⎛ conformations ⎞ G = −RT ln ⎜ X ⎝ ∑ e −GiX / RT ⎟ ⎠ i ⎛ conformations ( ) ⎞ ∑ − GiX −G0 / RT X = G − RT ln ⎜ X e ⎟ ⎝ 0 i ⎠ ⎡ 1 N − ( E (τ )− Eref ) / RT ⎤ ≈G X ref − RT ⎢ ∑ e ⎥ ⎣ N τ =1 ⎦ E(t)’s are energies along an MD trajectoryMonday, January 31, 2011 22

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