qm/mm background
DESCRIPTION
First lecture in QM/MM courseTRANSCRIPT
QM/MM methods applied to reactionmechanisms in enzymesRequired 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 slidesBlog post* summarizing in-class discussion
Required for extra credit (2.5 ECTS):Proposal describing improvement to QM/MM
*http://proteinsandwavefunctions.blogspot.com/
1Monday, January 31, 2011
2
QM/MM methods applied to reactionmechanisms in enzymes
Week 1 (Feb 3rd): Jan - QM/MM BackgroundWeek 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 ?
Intro + 7 papers in 8 weeks6 students: Casper, Anders, Martin, Kasper, Eric, Janus
2Monday, January 31, 2011
Measured: rate [P]/sRate => kcat
310.1126/science.1088172
3Monday, January 31, 2011
kcat ⇒ ΔGact0
kcat is converted to free energyvia transition state theory
Most QM/MM studies assumeΔGextra ≈ 0
410.1126/science.1088172
4Monday, January 31, 2011
ΔGTS ,0 = GTS −GES
The activation free energy
GX = −RT ln e−GiX /RT
i
conformations
∑⎛⎝⎜
⎞⎠⎟
= G0X − RT ln e− Gi
X −G0X( )/RT
i
conformations
∑⎛⎝⎜
⎞⎠⎟
Some QM/MM studies assume
GX ≈ G0X
(this also assumes the lowest energy conf has been found)
0 is the conformation with lowest G
5
5Monday, January 31, 2011
The free energy change has an electronic and vibrational contribution
GX ≈ EeleX +Gvib
X
6
6Monday, January 31, 2011
Challenges for QM/MM studies
Computing Eele and Gvib
Finding the TS
Eele ≈ EQM + EMM + EQM /MM + Eboundary
7image: 10.1080/014423509034954177Monday, January 31, 2011
Eele ≈ EQM + EMM + EQM /MM + Eboundary
Computing the “electronic” QM/MM energy
EQM = Ψ H Ψ + ZIZJRIJ−1
J > I∑
I∑
EMM = ki ri − ri,e( )2i
bonds
∑ + ki θi −θi,e( )2i
angles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
somedihedrals
∑
+ −AiAj
rij6 +
BiBj
rij12 +
qiqjrij
⎛
⎝⎜⎞
⎠⎟j>i
MMatoms
∑i
MMatoms
∑
8
8Monday, January 31, 2011
EQM /MM = Ψqirii
MMatoms
∑ Ψ +ZIqjrIjj
MMatoms
∑I
QMatoms
∑ + −AIAj
rIj6 +
BIBj
rIj12
⎛
⎝⎜⎞
⎠⎟j
MMatoms
∑I
QMatoms
∑
AI and BI may need to be re-adjusted
What are AI and BI for atoms in a TS?
Notice that is polarized by qi’s(this is called electrostatic embedding)
Ψ
Computing the “electronic” QM/MM energy
9
9Monday, January 31, 2011
The QM/MM covalent boundary
1010.1080/01442350903495417
requires special consideration because anMM atom does not help satisfy QM valence
most popular(easiest to implement)
10Monday, January 31, 2011
image and text:10.1021/jp9924124 11
Eboundary = ki ri − ri,e( )2 + ki θi −θi,e( )2i
angles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
somedihedrals
∑
H
The link atom methodBoundary constraints
11Monday, January 31, 2011
QM MM
density
EQM /MM = Ψqirii
MMatoms
∑ Ψ + ...
The link atom methodBoundary charge adjustment
charges close to density cause over-polarization
Solutions
All q’s in residue areset to 0
Closest q’s set to 0remaining q’s rescaled
Closest q’s representedby Gaussian functions
(Deleting 1-e- integralsinvolving link atom,
large errors for ab initio)image: 10.1021/jp0743469 12
12Monday, January 31, 2011
The Localized-SCF methodThe density localized molecular orbital
of the boundary bond is kept frozen during the SCF
image: 10.1021/jp000887l text: 10.1016/S0009-2614(00)00289-X 13
Eboundary = ki ri − ri,e( )2 + ki θi −θi,e( )2i
angles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
somedihedrals
∑
13Monday, January 31, 2011
The Generalized Hybrid Orbital method
vs
1410.1080/01442350903495417
frozenorbital
14Monday, January 31, 2011
QM/MM = QM program + MM program
15
Eele = Ψ H +qirii
MMatoms
∑ Ψ + ZIZJRIJ−1
J > I∑
I∑ +
ZIqjrIjj
MMatoms
∑I
QMatoms
∑
+ −AIAj
rIj6 +
BIBj
rIj12
⎛
⎝⎜⎞
⎠⎟j
MMatoms
∑I
QMatoms
∑ + −AiAj
rij6 +
BiBj
rij12 +
qiqjrij
⎛
⎝⎜⎞
⎠⎟j>i
MMatoms
∑i
MMatoms
∑
+ ki ri − ri,e( )2i
bonds
∑ + ki θi −θi,e( )2i
angles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
somedihedrals
∑
+ ki ri − ri,e( )2i
boundarybonds
∑ + ki θi −θi,e( )2i
boundaryangles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
boundarydihedrals
∑
15Monday, January 31, 2011
(some MM programs have semiempirical QM in them)
Eele = EQM + EQM/mm + Eqm,MM + EMM
QM/MM = QM program + MM program
GAMESS, GAUSSIAN, Turbomole, Molpro, ...
Chemshell, QoMMMa, COMQUM
AMBER, CHARMM, GROMACS, ....
The interface programs also often performgeometry optimizations after collecting
gradient terms from both programs
16Monday, January 31, 2011
EQM + EQM /mm = Ψ H +qirii
MMatoms
∑ Ψ + ZIZJRIJ−1
J > I∑
I∑ +
ZIqjrIjj
MMatoms
∑I
QMatoms
∑
Eqm /MM = −AIAj
rIj6 +
BIBj
rIj12
⎛
⎝⎜⎞
⎠⎟j
MMatoms
∑I
QMatoms
∑ + ki ri − ri,e( )2i
boundarybonds
∑ + ki θi −θi,e( )2i
boundaryangles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
boundarydihedrals
∑
EMM = ki ri − ri,e( )2i
bonds
∑ + ki θi −θi,e( )2i
angles
∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i
somedihedrals
∑ + −AiAj
rij6 +
BiBj
rij12 +
qiqjrij
⎛
⎝⎜⎞
⎠⎟j>i
MMatoms
∑i
MMatoms
∑
gx,QM /mm =∂ EQM + EQM /mm( )
∂xQM
gx,qm /MM =∂Eqm /MM
∂xQM
gx,MM =∂EMM
∂xMM
gQM ,mm + gqm,MM
gMM
QM/MM = QM program + MM program
17Monday, January 31, 2011
Workflowprotein structure form PDB
repair, add hydrogens, determine protonation statebuild in substrate
MM minimize, MD?Define QM region => boundary
coord + charges fed into QM programCompute EQM/mm + g for QM atoms
coord + vdW param for substrate fed into MM programspecial MM parameters for boundary?
Compute Eqm/MM + EMM + g for all atomsAdd g’s compute new coord
QM/MM = QM program + MM program
18Monday, January 31, 2011
Eele ≈ EQM + EMM + EQM /MM + Eboundary
Hij =∂2Eele
∂xi∂yjk = LtHL
ν i =ki2π
Gvib = −RT ln e−hν /2kT
1− e−hν /2kT⎛⎝⎜
⎞⎠⎟
Computing the QM/MM Gvib
too time consuming for larger systems
matrix diagonalizationscales as N3
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Computing the QM/MM Gvib
Solutions
2. Compute Gvib for model reaction(not good approximation of )
ΔGvib ≈ 0
20
ZPE ≈
1.5 ν kcal/mol1000 cm−1
i.e. breaking a covalent bond contributes roughly 3-4 kcal/mol to ΔHvib
1.
ΔSvib
20Monday, January 31, 2011
Finding the TS
Conventional TS finding algorithmsuse the Hessian Hqn+1 = qn −Hn
−1gn
Common solution:adiabatic mapping
21 text: 10.1080/0144235090349541721Monday, January 31, 2011
GX = −RT ln e−GiX /RT
i
conformations
∑⎛⎝⎜
⎞⎠⎟
= G0X − RT ln e− Gi
X −G0X( )/RT
i
conformations
∑⎛⎝⎜
⎞⎠⎟
≈ GrefX − RT 1
Ne− E (τ )−Eref( )/RT
τ =1
N
∑⎡⎣⎢
⎤⎦⎥
Dynamic Effects via MD
E(t)’s are energies along an MD trajectory
22Monday, January 31, 2011