BIOS 203 Lecture 4: Ab initio molecular dynamics

Ab initio molecular dynamics"

Lee-Ping Wang"leeping @ stanford.edu "

March 4, 2013"

Outline

Review •  A wide range of simulation methods •  Schrödinger’s equation and the potential energy surface

Methods •  Molecular forces and the Hellmann-Feynman theorem •  Born-Oppenheimer and Car-Parrinello molecular dynamics •  Hybrid quantum/classical methods

Applications •  O-O bonding mechanism of cobalt water oxidation catalyst •  Simulating the Urey-Miller experiment •  Enzyme catalysis and rational enzyme design

A wide scope of computer simulations

10 ps, 100 atoms: fast chemical reactions

100 fs, 30 atoms: photochemistry

10 µs, 10k atoms: protein dynamics, drug binding

One calculation for 2-3 atoms

1 ms+, 1 million atoms: folding of large proteins, virus capsids

•  Computer simulations of molecules span a wide range of resolutions

•  More detailed theories can describe complex phenomena and offer higher accuracy

•  Less detailed theories allow for simulation of larger systems / longer timescales

•  Quantum chemistry and density function theory are used to study chemical reactions

Categories of simulations

Focus on the physics rather than the nomenclature. •  Quantum treatment of electrons or force "eld?

QM: ab initio molecular dynamics (note: nuclei may be classical) Force "eld: classical MD or molecular mechanics

•  Born-Oppenheimer approximation broken or excited states? Nonadiabatic dynamics, excited state dynamics

•  Quantum treatment of nuclei? Quantum dynamics, path integral MD

!"i2

2i# +

ZIZJ

R I !RJiI$J# !

ZI

R I ! rii,I# +

1ri ! rji$ j

#%

&''

(

)**+ ri;R I( ) = E R I( )+ ri;R I( )

The Schrödinger equation All ground-state quantum chemistry is based on the

time-independent Schrödinger’s equation.

Ψ=Ψ EΗ( ) Ψ=Ψ+ EVT

Kinetic energy

operator

Electron nuclear

attraction

Electron electron repulsion

Electron wavefunction

Electronic energy

Born-Oppenheimer approximation: when solving for the electronic wavefunction, treat nuclei as static external potentials

Nuclear repulsion

(just a “number”)

Introduction: Molecular dynamics simulation e fundamental equation of classical molecular

dynamics is Newton’s second law.

F = ! dEdR

Key approximations:

•  Born-Oppenheimer approximation; no transitions between electronic states

•  Approximate potential energy surface, either from quantum chemistry or from the force "eld

•  Classical mechanics! Nuclei are quantum particles in reality, but this is ignored.

F =ma

e force is given by the negative gradient of the potential energy.

Knowing the force allows us to accelerate the atoms in the direction of the force.

Outline



Applications •  O-O bonding mechanism of cobalt water oxidation catalyst •  Simulating the Urey-Miller experiment •  Enzyme catalysis; mechanistic investigation and rational design

Hellmann-Feynman theorem e Hellmann-Feynman theorem makes the

calculation of molecular forces possible.

dEdR

=ddR

dr!! r( )" H! r( )

=d!!

dRH! +!! dH

dR! +!!H d!

dR"

H ! = E ! ; ! ! =1

! H ! = ! E ! = E ! ! = E

dEdR

=ddR

! H !

=d!dR

H ! + !dHdR

! + ! H d!dR

= !dHdR

! +d!dR

E ! + ! E d!dR

= !dHdR

! +E d!dR

! +E !d!dR

= !dHdR

! +E ddR

! !

= !dHdR

! +E ddR1 = !

dHdR

!

=d!!

dRE! +!! dH

dR! +!!E d!

dR"

= E d!!

dR! +!! d!

dR"

#$

%

&'+!! dH

dR!(

= E ddR

!!!( )+!! dHdR

!"

= E ddR1+ !! dH

dR!" = !! dH

dR!"

H! r( ) = E! r( ); dr! !" r( )! r( ) =1

dr! !" r( ) H! r( ) = !"E!! = E !"!! = E

dEdR

= dr!! r( )" dHdR

r( )! r( )

Ab initio MD simulation protocols

Starting positions and velocities"

Generate initial wavefunction guess"

Solve approximate Schrodingerʼs

equation"

Evaluate gradient of energy"

Update positionsand velocities"

Update wavefunction coefficients"

Evaluate gradient of energy"

Update positionsand velocities"

Born-Oppenheimer MD: Most costly step is

"nding the wavefunction Car-Parrinello MD:

Save time by using “forces” on electron degrees of freedom

µd 2!i

dt2r, t( ) = ! !E

!"i" r, t( )

+ #ij! j r, t( )j$

ONIOM = MM (Entire system) + QM (Selection) – MM (Selection)

•  Many interesting problems contain quantum behavior but are too large for quantum methods

•  However, QM behavior is often con"ned to a small part of the system

•  A simple approximation: Run QM only where exciting stuff happens

ONIOM does not explicitly contain interactions between QM and MM subsystems

S. Dapprich and coworkers, JMS Theochem (1999), 462, 1.

Hybrid quantum / classical methods Hybrid methods allow us to simulate different parts

of a system using different levels of theory.

If the two subsystems are independent (i.e. wavefunction is close to an antisymmetric product), then the QM/MM Hamiltonian is simple:

P. Slavicek and T. J. Martinez, J. Chem. Phys. (2006), 124, 084107.

ETotal = EQM rQM( )+EMM rMM( )+EQM/MM rQM,rMM( )

MM Nuclei / QM Electrons!

QM Electrons / MM Electrons!

MM Nuclei / QM Nuclei! MM Electrons / QM Nuclei!

Now approximate: MM Nuclei+Electrons = partial charges !

In QM/MM, the QM and MM subsystems interact via the QM/MM Hamiltonian.

QM/MM Framework

Neglected effects:

•  Exchange antisymmetry, dispersion (still need an empirical vdW interaction between QM and MM atoms)

• Covalent bonds (leads to the link atom problem)

P. Slavicek and T. J. Martinez, J. Chem. Phys. (2006), 124, 084107.

From the QM system’s perspective, the MM system looks like a bunch of point charges.

QM/MM Framework

Outline




•  Catalyst electrodeposits in solution of Co2+ and phosphate buffer (Pi)

•  Rapid water oxidation catalysis at mild overpotential

•  Catalyst is self-healing; does not corrode, catalysis rate continues to increase

Image courtesy Y. Surendranath

Kanan and Nocera, Science (2008).

Cobalt water oxidation catalyst

•  Catalyst is amorphous at all growth conditions with no long-range order

•  X-ray absorption spectrum indicates interatomic distances of 1.90Å (Co-O) and 2.82Å (Co-Co)

•  Proposed structure: Co-oxo clusters or sheets

1.90 Å

2.86 Å 3.80 Å

1.9 Å 2.85 Å

Experimental clues: Structure

•  Our calculations use a cubic model for the catalyst

•  Hypothesis 4: O–O bonding occurs between cofacial terminal oxo groups

•  Bonding occurs with large driving force (-23 kcal/mol) and small barrier (2.3kcal/mol)

•  Orbitals and spin populations show that single bond is formed

L. P. Wang and T. Van Voorhis, J. Phys. Chem. Lett., 2011.

Direct Coupling Pathway

•  QM/MM simulations show that the O-O bond is formed spontaneously

•  Free energy pro"le calculated from snapshot histograms

•  Small (2.7 kcal/mol) free energy barrier found, similar to vacuum DFT result (no signi"cant solvent reorganization)

Free Energy Profile via QM/MM

Outline




Radical polymerization

Nanoreactor: Introduction Interesting applications include complex systems

where many mechanisms may come into play.

Urey-Miller experiment (formation of amino acids from early Earth atmosphere)

Combustion

•  GPU-accelerated quantum chemistry is potentially useful for long timescale (~1 ns) reactive molecular dynamics. •  Need improved analysis tools to extract knowledge from simulations. Early analysis of a very simple AIMD trajectory (HF / 3-21G, 120 atoms, 1500 K, 27 ps) Red = Transient species Gray = Hydrogen Blue = Acetylene Gold = Ethane Violet = Ethylene Green = Vinylidene radical

Early work

Time-dependent spherical boundary condition e square-wave spherical boundary condition leads

to high-velocity molecular collisions. •  Kinetic energy momentarily reaches equivalent of 6000-10000K •  Equivalent pressure to compress an ideal gas is 10 kbar (bottom of Earth’s crust) •  Analogous conditions in the real world: Meteorite impacts?

Analysis of Results Identi"ed chemical reactions are extracted and

re"ned at a higher level of theory.

Raw data from MD trajectory Final intrinsic reaction coordinate

•  Extract a set of atoms that undergo bond rearrangement between two stable topologies. •  Minimize energy of reactants and products, superimpose onto trajectory. •  NEB optimization of minimum energy path. •  Transition state optimization. •  Intrinsic reaction coordinate.

Urey-Miller experiment Hundreds of compounds were found, including

several amino acid analogues.

Starting reactants: CH4, H2O, CO, H2, NH3

Reaction 1: Water + aldehyde à diol (close enough) Reaction 2: Formic acid + ethylidene radical + ammonia à aldehyde + water (water cat.) Reaction 3: Water + carbon monoxide à formic acid (ammonia cat.) Reaction 4: Ethylidene radical from a big collision…

Outline




reaction coordinate

Ene

rgy

uncatalyzed transition state (TS)

ES

EP

E + S

uncatalyzed transition state (TS)

ES# TS

Catalysis through binding of the transition state

Thanks to Gert Kiss (Pande Group) for these images

reactant

TS

product

reactant

Catalysis through binding of the transition state


D-luciferin

S

N

N

S

HO COO-

H

excited Oxyluciferin

S

N

N

S

-O OS

N

N

S

-O O*

ATP / Mg2+

LIGHT

Oxyluciferin

Firefly Luciferase

active site pocket

Example: Firefly bioluminescence


Energy

S

P

I1

I2 I3

TS1

TS2

TS3

TS4

reaction coordinate

Bioluminescence is a multi-step process…

D-luciferin

S

N

N

S

HO COO-

H

excited Oxyluciferin

S

N

N

S

-O OS

N

N

S

-O O*

ATP / Mg2+

LIGHT

Oxyluciferin

light emission

oxygenation

adenylation

… that is entirely catalyzed by a single enzyme

Kemp elimination

•  Kemp eliminases are a class of computationally designed enzymes that catalyze a reaction not observed in nature •  Glutamate is used as the base in a critical deprotonation step •  How was the design carried out?

D. Röthlisberger et al., Nature 2008.

QM/MM characterization of mechanism

Kemp elimination – rational design

G. Kiss et al, upcoming cover issue of Angewandte Reviews: Enzyme Design

Enzyme design involves an advanced combination of QM and molecular mechanics, structure optimization, and molecular dynamics.

•  Start with a “theozyme” – QM calculation of transition state •  Search for PDB structures that provide a good scaffold for the the QM transition state (theozyme) •  Mutate amino acid side chains to optimize packing of QM theozyme •  Molecular dynamics to determine protein rearrangements, water molecules •  Protein is experimentally expressed, further optimization with directed evolution

Education

BIOS 203 Lecture 4: Ab initio molecular dynamics