qm/mm background

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

http://proteinsandwavefunctions.blogspot.com

http://proteinsandwavefunctions.blogspot.com

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


Measured: rate [P]/sRate => kcat

310.1126/science.1088172


kcat ⇒ ΔGact0

kcat is converted to free energyvia transition state theory

Most QM/MM studies assumeΔGextra ≈ 0

410.1126/science.1088172


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


The free energy change has an electronic and vibrational contribution

GX ≈ EeleX +Gvib

X

6


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


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


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


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)


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


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


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

∑


http://dx.doi.org/10.1016/S0009-2614%2800%2900289-X

http://dx.doi.org/10.1016/S0009-2614%2800%2900289-X

The Generalized Hybrid Orbital method

vs

1410.1080/01442350903495417

frozenorbital


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


angles

∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i

somedihedrals

∑

+ ki ri − ri,e( )2i

boundarybonds


boundaryangles

∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i

boundarydihedrals

∑


(some MM programs have semiempirical QM in them)

Eele = EQM + EQM/mm + Eqm,MM + EMM


GAMESS, GAUSSIAN, Turbomole, Molpro, ...

Chemshell, QoMMMa, COMQUM

AMBER, CHARMM, GROMACS, ....

The interface programs also often performgeometry optimizations after collecting

gradient terms from both programs


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


boundaryangles

∑ + Vi 1± cos niφi( )⎡⎣ ⎤⎦i

boundarydihedrals

∑

EMM = ki ri − ri,e( )2i

bonds


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



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




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

19


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


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


qm/mm background

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