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 xn   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 vi v0i t tt2mi 2mit  t
xii 2t 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  cosnφ  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  
Cij12
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   Ex 
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  Ee
• 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   2me    
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  12 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 rn1  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 rn1  m rn 
- molecular orbitals
 i r    c ji  ao
j r 
j
- atomic orbitals (basisset)
• optimization of MO coefficents cji
- variation principle
* ˆ

 e He 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
A1
- 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  m1 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   rij1   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   Eelel  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  Ee
- 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
A1
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