A coupled (hybrid) potential QM/MM/MD simulations in Amber Dr. Vladislav Vasilyev Supercomputer...

A coupled (hybrid) potential A coupled (hybrid) potential QM/MM/MD simulations in AmberQM/MM/MD simulations in Amber

Dr. Vladislav Vasilyev

Supercomputer Facility

The Australian National University

Hybrids are popular from the Hybrids are popular from the ancient timeancient time

Why Do We Need a Hybrid QM/MM Why Do We Need a Hybrid QM/MM Approach?Approach?

Quantum chemical methods are generally applicable and allow the calculation of ground and excited state properties (molecular energies and structures, energies and structures of transition states, atomic charges, reaction pathways etc.)

Molecular Mechanical methods are restricted to the classes of molecule it have been designed for and their success strongly depends on the careful calibration of a large number of parameters.

Why Do We Need a Hybrid QM/MM Why Do We Need a Hybrid QM/MM Approach?Approach?

CPU Time Memory

Method Seconds Time units KB Memory units

Quantum chemical*

273.00 1820 4889 85

Molecular Mechanical

0.15 1 58 1

The main bottleneck of quantum chemical methods is that they are CPU and memory hungry.

For example, for small peptide of 126 atoms one energy evaluation requires:

*Semi-empirical PM3 method

In general, CPU and memory requirements:

Molecular Mechanical methods ~ N2

Semiempirical Quantum Chemical methods

~ N2

Ab initio Quantum Chemical methods ~ N4

A Hybrid QM/MM ApproachA Hybrid QM/MM ApproachThe development of hybrid QM/MM approaches is guided by the general idea that large chemical systems may be partitioned into an electronically important region which requires a quantum chemical treatment and a remainder which only acts in a perturbative fashion and thus admits a classical description.

The Simplest Hybrid QM/MM ModelThe Simplest Hybrid QM/MM Model Hamiltonian for the molecular system in the Born-Oppenheimer approximation:

esChExternalofEffect

nuclei

i

esch

k ik

kielectrons

i

esch

k ik

knuclei

ij ij

jinuclei

i

nuclei

j

electrons

ij ij

electrons

iij

jelectrons

i

electrons

i

nuclei

ij ij

jinuclei

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nuclei

j

electrons

ij ij

electrons

iij

jelectrons

i

electrons

i

R

QZ

R

Q

R

ZZ

rR

ZH

R

ZZ

rR

ZH

arg

argarg2

2

1

2

1

1

2

1

The main drawbacks of this simple QM/MM model are: it is impossible to optimize the position of the QM part relative to the external charges because QM nuclei will collapse on the negatively charged external charges. some MM atoms possess no charge and so would be invisible to the QM atoms the van der Waals terms on the MM atoms often provide the only difference in the interactions of one atom type versus another, i.e. chloride and bromide ions both have unit negative charge and only differ in their van der Waals terms.

“Standard” QM hamiltonian

A Hybrid QM/MM ModelA Hybrid QM/MM Model So, it is quite reasonable to attribute the van der Waals parameters (as it is in the MM method) to every QM atom and the Hamiltonian describing the interaction between the QM and MM atoms can have a form:

The van der Waals term models also electronic repulsion and dispersion interactions, which do not exist between QM and MM atoms because MM atoms possess no explicit electrons.

A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103(1976), 227-49

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A

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QZ

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The Hybrid QM/MM ModelThe Hybrid QM/MM Model Now we can construct a “real” hybrid QM/MM Hamiltonian:

A “standard” MM force field can be used to determine the MM energy. For example, AMBER-like force field has a form:

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MMMMQMQM HHHH ˆˆˆˆ/

dihedrals

anglesbondsb

bondsH ij

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Choice of QM methodChoice of QM method... is a compromise between computational efficiency and practicality and the desired chemical accuracy.

The main advantage of semi-empirical QM methods is that their computational efficiency is orders of magnitude greater than either the density functional or ab initio methods

OP

O O

O

NOO

O

F F

F

NOO

Calculation times (in time units)

1800

1

36228

1

RHF/6-31G*

PM3

QM Methods in Amber 9QM Methods in Amber 9

• Available semi- empirical Hamiltonians are PM3, AM1, MNDO, PDDG/PM3, PDDG/MNDO.

• They can be used for gas phase, generalized Born and PME periodic simulations.

QM Methods in Amber 9QM Methods in Amber 9

• Support is also available, on a functionally limited basis:

1. The Density Functional Theory-based-tight-binding (DFTB) Hamiltonian

2. The Self-Consistent-Charge version, SCC-DFTB

The DFTB/SCC-DFTB implementation does not currently support generalized Born, PME or Ewald calculations,

The elements supported by QM The elements supported by QM methods in Amber 9methods in Amber 9

• MNDO: H, Li, Be, B, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, Sn, I, Hg, Pb

• AM1: H, C, N, O, F, Al, Si, P, S, Cl, Zn, Ge, Br, I, Hg

• PM3: H, Be, C, N, O, F, Mg, Al, Si, P, S, Cl, Zn, Ga, Ge, As, Se, Br, Cd, In, Sn, Sb, Te, I, Hg, Tl, Pb, Bi

• PDDG/PM3: H, C, N, O, F, Si, P, S, Cl, Br, I• PDDG/MNDO: H, C, N, O, F, Cl, Br, I• PM3CARB1: H, C, O• DFTB/SCC-DFTB: H, C, N, O, S, Zn

Calibration of the QM/MM potentialCalibration of the QM/MM potential

• Crucial aspect is how the interaction between QM and MM parts is determined.

• In choosing the appropriate form, it is required that the balance between attractive and repulsive forces must be preserved and the QM/MM interactions must be of the correct magnitude with respect to the separate QM and MM contributions

Calibration of the QM/MM potential:Calibration of the QM/MM potential:ParameterizationsParameterizations

• 1) Modification of the one-electron terms arising from interaction of the electron cloud of the QM fragment with the point charge of an MM atom.

• 2) By varying the radii in the van der Waals terms.

• 3) By varying 1)+2)

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j ij

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(1) (2)

Calibration of the QM/MM potentialCalibration of the QM/MM potential

• 1) By hand, to find the optimum values of the parameters by calculating interaction curves for charge/ion systems and comparing them with the MP2/6-311++G** ab initio results.

M.J. Field, P.A. Bash, M. Karplus, J.Comp.Chem., 11(1990), 700-733.

• 2) Fitting calculated H-bond energies to experimental data on ion-molecular complexes in the gas phase.

V.V. Vasilyev, A.A. Bliznyuk, A.A. Voityuk, Int.J.Quant.Chem. 44(1992), 897-930.

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Calibration of the QM/MM potential:Calibration of the QM/MM potential:

• 3) Optimizing van der Waals parameters on QM atoms to reproduce the 6-31G(d) interaction energies for H-bonded complexes in the gas phase.

P.A. Bash, L. Lawrence, A.D. MacKerell, Jr., D. Levine, P. Hallstrom, PNAS USA, 93(1996), 3698-703.

• 4) Optimizing van der Waals parameters on QM atoms to reproduce the MP2/6-31G(dp) interaction energies for H-bonded complexes in the gas phase.

J. Gao // Toward a molecular Orbital Derived Empirical Potential for Liquid Simulations // J.Phys.Chem. B 101(1997), 657-63

• 5) By varying the radii in the van der Waals terms to reproduce experimental free energies of solvation using MD simulations.

P.L. Cummins, J.E. Gready, J.Comp.Chem., 18(1997), 1496-512.

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Dividing Covalent Bonds across the Dividing Covalent Bonds across the QM and MM RegionsQM and MM Regions

In many simulations it is necessary to have the QM/MM boundary cut covalent bonds, and a number of additional approximations have to be made.


A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103 (1976), 227-249

V. Thery, D. Rinaldi, J.-L. Rivail, B. Maigret, G.G. Ferenczy, J.Comp.Chem. 15 (1995), 269

Using a hybrid orbital on the frontier MM atom

O

O

QM Region

MM Region

Frontier MM Atom

Frontier QM Atom

Frontier MM Atom


“Link” atoms are used to gracefully cap the electron density.

This approach is used in Amber

Using “link” atoms

Implementation of “link” atom Implementation of “link” atom Approach in Amber 9Approach in Amber 9

The link atom is placed along the bond vector joining the QM and MM atom

The default link atom type is hydrogen

It interacts with MM region only electrostatically (no VDW term).

WdV interaction between QM and MM atoms which form 1-2 and 1-3 “bonded” pairs is not calculated.

Bond stretching, angle bending, and torsion interactions between QM and MM regions are calculated as those in MM if 1-2, 1-2-3, or 1-2-3-4 terms contain at least one MM atom

Example of Application of the Example of Application of the QM/MM Method to the Enzyme QM/MM Method to the Enzyme

CatalysisCatalysis

Tetrahedral Intermediate Formation in the Tetrahedral Intermediate Formation in the Acylation Step of AcetylcholinesterasesAcylation Step of Acetylcholinesterases

• Acetylcholinesterases (AChE) are the serine protease enzymes which hydrolyze the neurotransmitter acetylcholine (Ach)

CH3COOCH2CH2N+(CH3)3 + AchE CH3CO-AchE + HOCH2CH2N+(CH3)3 CH3COO- + H+ + AChE

and which function at a rate approaching that of a diffusion-controlled reaction.

• Remark: Human Cathepsin G is also a serine protease

enzyme

V.V. Vasilyev, J.Mol.Struct. (Theochem), 304(1994), 129.

Steps along the reaction pathway of serine Steps along the reaction pathway of serine protease catalyzed bond cleavageprotease catalyzed bond cleavage

H N N H

C

ONC

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R1

H

R2

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R

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O

R3

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Asp-102His-57

Ser-195

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CO

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Ser-195

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CSer-195

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C

O

C

H

R2

N

R

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O N N H

Acyl-Enzyme

Tetrahedral Intermediate

O

H

H

Rate limiting step of the reaction

“Catalytic triad”

Partition into the QM and MM PartsPartition into the QM and MM Parts

H N N H

C

ONC

H

R1

H

R2

N

R

C

O

R3

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Ser-195

O

CO

C

-+

C

O

NC

H

R1

H

R2

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R

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Ser-195

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CSer-195

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C

O

C

H

R2

N

R

C

O

R3

O N N H

Acyl-Enzyme

Tetrahedral Intermediate

O

H

H

QM Region (PM3) – CH3OH (as Ser-200), methylimidazole (as His-440), CH3CH2COOH (as Glu-327), and CH3COOCH3 (as a substrate)

MM Region – 5161 enzyme atoms

Activation of the Serine ResidueActivation of the Serine Residue

O H NN O NNH-

+

Critical Step of the Reaction is activation of the Serine residue

In Gas Phase: PA(CH3O-) = ~ -350 kcal/molePA(Imidazole) = ~ -220 kcal/mole

where PA is a Proton Affinity

Activation of the Serine ResidueActivation of the Serine ResidueA proton transfer from the Serine to Histidine residue is very unfavorable in the gas phase (~34.6 kcal/mol).

A proton transfer is less unfavorable in the enzyme (energy barrier is about 21.5 kcal/mol).

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10

20

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Ser(0) Ser(-)

In G as Phase

In Enzym e

Energetics of the proton transferfrom Ser to His in the enzyme and in the gas phase

The energy of the first point, Ser(0)-His(0)-Glu(-), is taken as zero

Activation of the Serine ResidueActivation of the Serine Residue

Enzyme environment creates electrostatic potential which favours the proton transfer from the Serine to Histidine residue.

0

10

20

30

40

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Ser(0) Ser(-)

In G as Phase

In Enzym e

Energetics of the proton transferfrom Ser to His in the enzyme and in the gas phase

The energy of the first point, Ser(0)-His(0)-Glu(-), is taken as zero

O H NN

ElectrostaticPotential (kcal/mol)

-0.8

-28.4

-54.3

Tetrahedral Intermediate Formation in the Tetrahedral Intermediate Formation in the Acylation Step of AcetylcholinesterasesAcylation Step of Acetylcholinesterases

1.00 2.00 3.00 4.00Serine-Substrate D istance, Angstrom

-40

-20

0

20

40

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kca

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l

T I

Reaction path

In Enzym e

In G as Phase

Incorporation of enzyme environment in simulation changes drastically the flow of the reaction

A decrease in the activation energy of TI formation in the enzyme environments versus the uncatalysed gas-phase reaction is about 27 kcal/mol

(experimental estimation of the reduction in activation energy for the whole enzyme reaction versus uncatalysed neutral, basic, and acidic hydrolysis of AchE is 14, 11, and 18 kcal/mol, respectively.)

Hints for running QM/MM calculationsHints for running QM/MM calculationsChoosing the QM regionChoosing the QM region

• One might want to have as large a QM region as possible

• There are no good universal rules here

• However, having more than 80-100 atoms in the QM region will lead to simulations that are very expensive.


• for many features of conformational analysis, a good MM force field may be better than a semi-empirical or DFTB quantum description.

Hints for running QM/MM calculationsHints for running QM/MM calculationsParallel SimulationsParallel Simulations

• At present all parts of the QM simulation are parallel except the density matrix build and the matrix diagonalisation.

• For small QM systems these two operations do not take a large percentage of time and so acceptable scaling can be seen to around 8 cpus.

• However, for large QM systems the matrix diagonalization time will dominate and so the scaling will not be as good.

Amber 9 QM/MM ExampleAmber 9 QM/MM Example

ResumeResume

• Amber 9 features new and significantly improved QM/MM support

• The QM/MM facility supports gas phase, implicit solvent (GB) and periodic boundary (PME) simulations

• Compared to earlier versions, the QM/MM implementation offers improved accuracy, energy conservation, and performance.

A coupled (hybrid) potential QM/MM/MD simulations in Amber Dr. Vladislav Vasilyev Supercomputer...

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