17.Gerrit_Hybrid_QM_MM
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Transcript 17.Gerrit_Hybrid_QM_MM
mixed quantum-classical
molecular dynamics simulations
of biomolecular systems
concepts, machinery & applications
Gerrit Groenhof
dept. of biophysical chemistry
University of Groningen
Nijenborg 4, 9747 AG Groningen
The Netherlands
biomolecular simulation
• biomolecules
- proteins, DNA, lipid membranes, …
- biochemistry, biology, farmacy, medicine, …
• physical composition of biomolecules
- molecules are composed of atoms
- atoms are composed of electrons and nuclei
• laws of physics
- interaction
- motion
• computing properties of biomolecules
- static: energies, structures, spectra, …
- dynamic: trajectories, …
molecular simulation
• standard molecular dynamics
- forcefield
- single overall connectivity: no chemical reactions
- single electronic state: no photo-chemical reactions
• example
- aquaporin-1 mechanism
B. de Groot & H. Grubmüller
Science 294: 2353-2357 (2001)
molecular simulation
• QM/MM molecular dynamics
- combination of quantum mechanics and forcefield
- connectivity varies: chemical reactions
- electronic state varies: photo-chemical reactions
• examples
- Diels-Alder reaction
cycloaddition of ethene
and butadiene in cyclohexane (not shown)
molecular simulation
• QM/MM molecular dynamics
- combination of quantum mechanics and forcefield
- connectivity varies: chemical reactions
- electronic state varies: photo-chemical reactions
• examples
- photo-isomerization
QM/MM simulation of Photoactive yellow protein
J. Amer. Chem. Soc. 126:
4228-4233 (2004)
molecular simulation
• concepts & machinery
- molecular dynamics (MD)
(5m)
- molecular mechanics forcefield (MM)
(5m)
- molecular quantum mechanics (QM)
(60m)
- mixed quantum/classical mechanics (QM/MM) (30m)
- geometry optimization
(10m)
• applications
- Photoactive Yellow Protein
- Diels-Alderase enzyme (you!)
(45m)
(3h)
molecular dynamics
• nuclei are classical particles
- Newton’s equation of motion
Fn mn xn RnV x1 , x2 ,, xN
- numerically integrate equations of motion
2
1 2
xn t xn t0 xn t0 t t0 2 xn t t0
• potential energy and forces
- molecular mechanics
V x1 , x2 ,, xN vk x; pk
k
- quantum mechanics
V x1 , x2 ,, xN e Hˆ x1 , x2 ,, xN e
molecular dynamics
• numerically integrate eoms of atoms
Fi 0 Fi t 2
2
t vi v0i t tt2mi 2mit t
xii 2t t xxi i0
molecular mechanics forcefield
• approximation for energy V
V x1 , x2 ,, xN vk x1 ,, xn ; pk
k
- analytical lower dimensional functions (n << N)
bonded interactions
vb r 12 kb r r0
2
va θ 12 ka θ θ0
2
vd φ kd 1 cosnφ φ0
- empirical parameters (pk)
thermodynamic data & QM calculations
molecular mechanics forcefield
• approximation for energy V
V x1 , x2 ,, xN vk x1 ,, xn ; pk
k
- analytical lower dimensional functions (n << N)
non-bonded interactions
vlj rij
Cij12
rij12
Cij 6
rij6
vc rij
qi q j
4 πε0 rij
- empirical parameters (pk)
thermodynamic data & QM calculations
molecular mechanics forcefield
• bonded interactions:
bonds , angles & torsions
molecular mechanics forcefield
• non-bonded interactions:
Lennard-Jones & Coulomb
molecular mechanics forcefield
• popular forcefields
- CHARMM, OPLS, GROMOS, AMBER, …
• advantages
- fast
large systems: proteins, DNA, membranes, vesicles
• disadvantages
- limited validity
only valid inside harmonic regime
no bond breaking/formation
- limited transferrability
new molecules need new parametrization
fundamental quantum mechanics
• subatomic particles
Louis de Broglie
- wave character
Erwin Schrödinger
electron diffraction
Werner Heisenberg
Paul Dirac
- energy quantization
Max Born
• wavefunction
Albert Einstein
- Schrödinger wave equation
i dtd x, t Hˆ x, t
time-dependent
- Hamilton operator
2
ˆ
H 2m
d2
dx2
kinetic
V x
potential
many others
Hˆ x Ex
time-independent
molecular quantum mechanics
• solving electronic Schrödinger equation
- Born-Oppenheimer approximation
electronic and nuclear motion decoupled
- electrons move in field of fixed nuclei
Hˆ e Ee
• electronic hamiltonian
ne
ne
N
i
i
A
ne
N
e2Z I Z J
e ZI
2
e
1
1
ˆ
H R1 ,, RN 2 me i 4 0riA 2 4 0rij 2 4 0 RAB
2
kinetic
2
elec-nucl
2
i, j
elec-elec
• forces on classical nuclei
FM RM e Hˆ R1 , R2 ,, RN e
A, B
nucl-nucl
molecular quantum mechanics
• applications for molecular modeling
- electron density (charge distribution)
ρ e r e r e* r e r
2
molecular quantum mechanics
• applications for molecular modeling
- reaction pathways
Diels-Alder cyclo-addition mechanism
molecular quantum mechanics
• Hartree approximation to wavefunction
- product of one electron functions
1 r1 2 r2 n rn
e r1 , r2 ,, rn
- hamiltonian with
without
electron-electron
electron-electron
term
term
ne
ne
N
Hˆ 2me
2
2
i
i
i
kinetic
e2Z I
4 0 riI
I
ne
1
2
e2
4 0 rij
i, j
elec-nucl
elec-elec
- mean field approximation
electron i in average static field of other electrons
ne
j
e2
rij
e
2
,h , j ,k , r
r
dr
- iterative solution (self consistent field)
molecular quantum mechanics
• Hartree approximation
- illustration of mean-field approach
electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1
molecular quantum mechanics
• Hartree approximation
- illustration of mean-field approach
electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1
molecular quantum mechanics
• Hartree approximation
- illustration of mean-field approach
electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1
molecular quantum mechanics
• Pauli principle
- electrons are fermions (spin ½ particles)
- electron wavefunction is anti-symmetric
e , ri , rj , e , rj , ri ,
- no two electrons can occupy same state
• Hartree approximation
- product of one electron functions: e 12 n
- not anti-symmetric: e , ri , rj , e , rj , ri ,
• Hartree-Fock approximation
- anti-symmetric combination of Hartee products
molecular quantum mechanics
• anti-symmetric sum of Hartree products
- e.g. product of two one electron functions
Hartree approximation:
eH r1 , r2 1 r1 2 r2
Fock (Slater) correction: eF r1 , r2 1 r1 2 r2 1 r2 2 r1
- anti-symmetric
eF r2 , r1 eF r1 , r2
1 r2 2 r1 1 r1 2 r2 1 r1 2 r2 1 r2 2 r1
- no effect on wavefunction’s properties
energy, density, …
molecular quantum mechanics
• Hartree-Fock approximation
- anti-symm. product of one electron wavefunctions
e r 1, r2 ,, rn
a r1 b r1 z r1
a r2 b r2 z r2
a rn b rn z rn
- Slater determinant
e r1, r2 ,, rn a r1 b r2 z rn
molecular quantum mechanics
• one electron wavefunctions
- spatial & spin part
i x, y, z, s i x, y, z σ s
- Ĥ does not operate on s, only on x,y,z
- s(s) is a spinlabel
s 12 , 12
s 12
s 12
- spatial part (x,y,z) is a molecular orbital
max. two electrons (Pauli principle)
i x, y, z, 12 i x, y, z
j x, y, z, 12 i x, y, z
- Slater determinant with molecular orbitals
e r1, r2 ,, rn a r1 a r2 m rn1 m rn
molecular quantum mechanics
• molecular orbitals
- linear combination of atomic orbitals
i r c ji ao
j r
j
- e.g. H2 1 r 1 r 2 r ; 2 r 1 r 2 r
e r1, r2 1 r1 1 r2
molecular quantum mechanics
• atomic orbitals
- combination of simple spatial functions
Slater-type orbitals:
gaussian-type orbitals:
r ce
r
r ce
r
2
- mimic atomic s,p,d,… orbitals
- basisset: STO-3G, 3-21G, …, 6-31G*, …
1s r d i ,1s 8
3
i 1
3
i ,1s
2 s r d i , 2 s 8
3
i 1
3
i , 2 sp
3
1
3
4
e
1
4
i ,1 s r 2
e
i , 2 sp r 2
2 p r d i , 2 p 128 i5, 2 sp 3 xe
3
x
i 1
1
4
i , 2 sp r 2
molecular quantum mechanics
• restricted Hartree-Fock wavefunction
- Slater determinant
e r1, r2 ,, rn a r1 a r2 m rn1 m rn
- molecular orbitals
i r c ji ao
j r
j
- atomic orbitals (basisset)
• optimization of MO coefficents cji
- variation principle
* ˆ
e He d E0
- find cji that minimize the energy (just 3 slides)
molecular quantum mechanics
• Hartree-Fock equations
- minimization problem
E Hˆ e d 0
*
e
Hˆ hi 12 4 πe 0rij
2
i
M
hi 12 ZriAA
2
i
ij
A1
- HF equation for single moleclar orbital (meanfield)
h
2
J
K
i s ( r1 ) s s r1
1 i
i
- nonlinear set of equations
coulomb operator
J i s (r1 ) i* r2 4 πe r i r2 s (r1 )
2
0 12
K i s (r1 ) r2
*
i
e2
4 π 0 r12
s r2 i (r1 )
- total electronic energy
E 2 s 2 J is K is
s
i ,s
exchange operator
(1/3)
molecular quantum mechanics
• Roothaan-Hall equations
- HF equation for molecular orbitals
f1 h1 2 J u r1 K u r1
f1 a (r1 ) a a r1
u
- expressed in atomic orbitals
f1 c ja j (r1 ) a c ja j (r1 )
j
j
- multiply by atomic orbital i* and integrate
c
ja
j
*
i
(r1 ) f1 j (r1 )dr1 a c ja i* (r1 ) j (r1 )dr1
j
Fij i* (r1 ) f1 j (r1 )dr1
Sij i* (r1 ) j (r1 )dr1
- matrix equation
Fc Scε
- solution ({cja} and {a}) if
F aS 0
(2/3)
molecular quantum mechanics
• self consistent field procedure
- iterate until energy no longer changes (converged)
e.g. Gaussian SCF output:
Closed shell SCF:
Cycle
1 Pass 1 IDiag
E= -2929.02815281902
1:
Cycle
2 Pass 1 IDiag 1:
E= -2929.07991917607
Delta-E=
-0.051766357053 Rises=F Damp=T
Cycle
3 Pass 1 IDiag 1:
E= -2929.13887276782
Delta-E=
-0.058953591741 Rises=F Damp=F
...skipping...
Cycle 12 Pass 1 IDiag 1:
E= -2929.14125348456
Delta-E=
-0.000000000195 Rises=F Damp=F
Cycle 13 Pass 1 IDiag 1:
E= -2929.14125348457
Delta-E=
-0.000000000008 Rises=F Damp=F
Cycle 14 Pass 1 IDiag 1:
E= -2929.14125348456
Delta-E=
0.000000000012 Rises=F Damp=F
SCF Done:
E(RHF) = -2929.14125348
Convg =
0.9587D-08
S**2
=
0.0000
A.U. after
14 cycles
-V/T = 1.9993
(3/3)
molecular quantum mechanics
• Hartree-Fock based methods
- Hartree Fock wavefunction as starting point
no electron correlation
- MCSCF (CI, CASSCF)
- perturbation theory (MP2, MP4, CASPT2)
- high demand on computational resources
small to medium-size molecules in gas phase
• alternative methods
- semi-empirical methods
- density functional theory methods
molecular quantum mechanics
• limitations of HF wavefunction
- no electron correlation
dynamic: electronic motion is correlated
static: electrons avoid each other
• improving HF wavefunction
- multi-configuration self-consistent field (mcscf)
e C0 a r1 a r2 m rn 1 m rn C1 a r1 a r2 m rn 1 m1 rn
C2 a r1 a r2 m rn 1 m 2 rn
e C0 HF Ci i
i 1
single, double, triple, quadruple, quintuple, … excitations
resolves (part of) static correlation
molecular quantum mechanics
• multi-configuration self-consisitent field
e C0 HF Ci i
i 1
- size of sum
molecular quantum mechanics
• limitations of HF wavefunction
- no electron correlation
dynamic: electronic motion is correlated
static: electrons avoid each other
• improving HF wavefunction
- perturbation theory
Hˆ Hˆ 0 Hˆ pert
HF 1pert 2pert
Hˆ 0 f i hi viHF
1pert ai iHF 2pert bi iHF
i
i
Hˆ pert rij1 viHF
i j
i
i
i
- Møller-Plesset (MP): MP2, MP4, CASPT2, …
molecular quantum mechanics
• semi-empirical methods
- Roothaan-Hall equations
Fc Scε
Fij i* (r1 ) f1 j (r1 )dr1
Sij i* (r1 ) j (r1 )dr1
f1 h1 2 J u r1 K u r1
u
- zero differential overlap
Sij ij
- empirical parameters in Fij
fitted to thermochemical data
CNDO, INDO, NDDO, MINDO, MNDO, AM1, PM3
molecular quantum mechanics
• density functional theory
- Hohenberg-Kohn Theorem (1964)
electron density defines all ground-state properties
- Kohn-Sham equation (1965)
E e r Ekin e r Eelel e r Exc e r Vnucl r e r dr
- Kohn-Sham orbitals
n
e r i r
i 1
2
i r c ji ao
j r
j
- exchange-correlation functional Exc[e(r)]
- find cji that minimize the energy functional E[e(r)]
- self-consistent Roothaan-Hall equations
molecular quantum mechanics
• summary
- solving electronic Schrödinger equation
Hˆ e Ee
- computational techniques
Hartree-Fock and beyond (RHF, UHF, CASSCF, MP2,…)
semi-empirical methods
(INDO, AM1, PM3, …)
density functional theory
(Becke, BP87, B3LYP, …)
- forces on nuclei
FN RN e Hˆ e
- more accurate than any forcefield
bond breaking/formation
excited states, transitions between electronic states
molecular quantum mechanics
• high demand on computational recources
small to medium sized gas-phase systems
mixed quantum/classical methods
• reaction in condensed phase
- reactions in solution
- enzymatic conversions
• subdivision of the total system
- reactive center (QM)
- environment (MM)
• QM/MM hybrid model
- compromise between speed and accuracy
- realistic chemistry in realistic system
QM/MM hybrid model
• QM subsystem embedded in MM system
A. Warshel & M. Levitt. J. Mol. Biol. 103: 227-249 (1976)
QM/MM hybrid model
• application for molecular modeling
- catalytic Diels-Alderase antibody
J. Xu et al. Science 286: 2345-2348 (1999) (experimental)
http://md.chem.rug.nl/~groenhof/EMBO2004/html/tutorial.html
QM/MM hybrid model
• interactions in QM subsystem
- QM hamiltonian
• interactions in the MM subsystem
- forcefield
• interactions between QM and MM subsystems
- QM/MM interface
- forcefield
bonded and dispersion interactions
- QM hamiltonian
electrostatic interactions
QM/MM hybrid model
• QM/MM bonded interactions
bonds , angles & torsions
QM/MM hybrid model
• QM/MM dispersion interactions
Lennard-Jones
QM/MM hybrid model
• QM/MM boundary
link atom, frozen orbital
QM/MM hybrid model
• QM/MM electrostatic interactions
ne N MM
N QM N MM
e Qj
e 2 Z I QJ
QM / MM
QM
ˆ
ˆ
point charges: H electronic H elec 4 0riJ 4 0 RIJ
i
J
2
I
J
QM/MM hybrid model
• Roothaan-Hall equations
- HF equation for molecular orbitals
f1 a (r1 ) a a r1
f1 h1 2 J u r1 K u r1
u
- QM subsystem in cloud of pointcharges
M
N MM
A1
K 1
hi 12 i2 ZriAA QriKK
- polarization of QM subsystem
• forcefield terms
- QM/MM (bonds, angles, torsions & LJ)
- MM
QM/MM hybrid model
QM/MM hybrid model
• electrostatic QM/MM interaction
- QM subsystem in cloud of pointcharges
ne N MM
QM / MM
QM
Hˆ electronic
Hˆ elec
i
core
e 2Q j
4 0 riJ
J
elec-MMatom
N QM N MM
I
e 2 Z I QJ
4 0 RIJ
J
nucl-MMatom
- polarization of QM subsystem
• problems & inconsistencies
- no polarization of MM subsystem
implicitly incorporated in LJ and atomic charges
- pointcharges of MM atoms
forcefield dependent
alternative QM/MM interface
• ONIOM
F. Maseras & K. Morokuma, J. Comp. Chem. 16, 1170 (1995)
• two layer ONIOM energy
EQM/MM E
high
QM
E
low
total
E
low
QM
alternative QM/MM interface
• multilayer ONIOM
QM/QM/.../…/MM
geometry optimization
• potential energy surface
V x1 , x2 ,, xN
F x1 , x2 ,, xN V x1 , x2 ,, xN
• energy & forces
- MM (forcefield)
- QM (HF, DFT, …)
- QM/MM
-…
geometry optimization
• stationary points
V x1 , x2 ,, xN 0
• minima on PES
- reactants
- products
• saddle-points
- transition states
• reaction mechanism
k exp
VTST
k bT
reactants → products
geometry optimization
• stationary points
V x1 , x2 ,, xN 0
• Hess matrix (Hessian)
- matrix of second derivatives
H ij
2V
xi x j
xi x1 , y1 , z1 , x2, y2 , z2 ,, xN , y N , z N
• minima on potential energy surface
- Hessian has only positive eigenvalues
• saddle-points on potential energy surface
- Hessian has one negative eigenvalue
geometry optimization
• locating minima
- general procedure
follow the gradient downhill
• locating saddle points
- optimization with constraints
one eigenvalue of Hessian is negative
- good guess TS geometry (intuition & experience)
- linear transit
reaction coordinate
interpolation between reactant and product geometries
- always check the eigenvalues of Hessian!!
geometry optimization
• linear transit calculation
- reaction coordinate (experience & intuition)
e.g. Diels-Alder cycloaddition
- constrain/restrain reaction coordinate
- minimize/sample all other degrees of freedom
vreact or even Greact (potential of mean force)
geometry optimization
• linear transit calculation
- result for the Diels-Alder cycloaddition
end of part I
QM/MM
concepts & machinery
coming up part II
QM/MM
applications