There	
  are	
  several	
  topics	
  from	
  BFCH1	
  (lectures:	
  Structure	
  
of	
  biopolymers	
  (2),	
  Canonical	
  ensemble)	
  that	
  are	
  expected	
  	
  
that	
  students	
  know:	
  
-­‐  canonical	
  ensemble	
  
-­‐  Boltzmann	
  probability	
  
-­‐  free	
  energy,	
  enthalpy,	
  entropy	
  	
  
-­‐  free	
  energy	
  cycle	
  	
  
	
  
For	
  which	
  problems	
  are	
  simula1ons	
  useful	
  ?	
  
Simulation can replace or complement the experiment:
1. Experiment is impossible Inside of stars
Weather forecast
2. Experiment is too dangerous Flight simulation
Explosion simulation
3. Experiment is expensive High pressure simulation
Windchannel simulation
4. Experiment is blind Some properties cannot be
observed on very short timescales
and very small space-
scales
5. Fantasy It is possible to simulate
unrealistic phenomena
Simula1ons	
  can	
  complement	
  the	
  experiment:	
  
	
  
	
  
	
  
•	
  SimulaGon	
  explains	
  experiments	
   	
   	
  Proper&es	
  of	
  water	
  
	
   	
   	
   	
   	
   	
   	
   	
   	
  Folding	
  of	
  protein	
  molecules	
  
	
  
	
  
	
  
	
  
	
  
	
  
	
  
•	
  SimulaGon	
  suggests	
   	
   	
   	
   	
   	
  Design	
  of	
  drugs,	
  enzymes	
  
	
   	
  new	
  experiments	
   	
   	
   	
   	
  Stock	
  market	
  prices	
  
	
  
	
  
	
  
less	
  experiments	
  
beHer	
  chance	
  of	
  success	
  
knowledge	
  
new	
  ideas	
  
For	
  which	
  problems	
  are	
  simula1ons	
  useful	
  ?	
  
Hritz,	
  J.;	
  de	
  Ruiter,	
  A.;	
  Oostenbrink,	
  C.	
  Impact	
  of	
  plasGcity	
  and	
  flexibility	
  on	
  docking	
  
results	
  for	
  Cytochrome	
  P450	
  2D6:	
  a	
  combined	
  approach	
  of	
  molecular	
  dynamics	
  and	
  
ligand	
  docking.	
  J.	
  Med.	
  Chem.	
  2008,	
  51,	
  7469-­‐7477	
  
Simulation and experiment are complementing methods
to study different aspects of nature
experiment simulation
Typical space / time scales
size : 10-3 meter 10-7 meter
time : 103 seconds 10-3 seconds
Resolution*
size : 1023 molecules 1 molecule
time : 1 second 10-15 seconds
*: Single molecules / 10-15 seconds possible
(but not both in the liquid phase)
(restricted)	
   (unrestricted)	
  
Molecular	
  simulaGon	
  and	
  experiment	
  
How would you calculate movement
of planets in our solar system?
Choose	
  relevant	
  degrees	
  of	
  freedom:	
  elementary	
  par1cles	
  
.	
  .	
  .	
  .	
  .	
  .	
  
atomic	
  nuclei	
  	
  
	
  	
  	
  	
  	
  	
  	
  	
  	
  +	
  electrons	
  
	
  
	
  
quantummechanics	
  
	
  
	
  
	
  
electrosta&cs	
  
all	
  atoms	
  
(excluding	
  solvent)	
  
	
  
	
  
classical	
  
mechanics	
  
	
  
	
  
	
  
Force	
  Field	
  
(including	
  solvent)	
  
monomers	
  
	
  
	
  
	
  
classical	
  
mechanics	
  
	
  
	
  
	
  
Force	
  Field	
  
(sta&s&c)	
  
Par1cles:	
  
Descrip1on:	
  
all	
  atoms	
  
	
  
	
  
	
  
classical	
  
mechanics	
  
	
  
	
  
	
  
Force	
  Field	
  
(atomis&c)	
  
Interac1ons:	
  
Broader	
  applicability	
  
Less	
  model	
  parameters	
  	
  
Physical	
  parameters	
  
More	
  expensive	
  
Restricted	
  applicability	
  
More	
  model	
  parameters	
  
Empirical	
  parameters	
  
Less	
  expensive	
  
=	
  
Interac1ons	
  in	
  atomic	
  simula1es	
  :	
  Force	
  Field	
  
	
   	
   	
  	
  	
  	
  	
  	
  	
   	
  physico-­‐chemical	
  knowledge	
  
RotaGon	
  around	
  
bond	
  
Planar	
  
atomgroups	
  
van	
  der	
  Waals	
  
interacGons	
  
ElectrostaGc	
  
interacGons	
  
-­‐	
  
+	
  
-­‐	
  
-­‐	
  
	
  	
  Bond	
  stretching	
  
non-­‐bonded	
  	
  
interac3ons	
  
bonded	
  	
  
interac3ons	
  
Angle	
  bending	
  
InteracGng	
  ParGcles	
  
( )
i<j
. .
12 6
4
ij ijN
i
v d Waal
pairs ij ij
s
j
r
V
r
r
σ σ
ε
# $% & % &
' (= −* + * +* + * +' (, - , -. /
∑

( )
0i<j
1
4
i jN
rpair
Coulom
s j
b
ir
q q
V r
πε ε
= ∑

Physical Terms
( ) ( ) 0
i
21
2
N b Nbond
i ii
bonds
V r K r bb! "= −$ %∑
 
( ) ( ) 0
i
21
2
N a Nangl
i ii
angle
e
s
V r K rθ θ" #= −% &∑
 
( ) ( )
i
cos i
N Nt
torsi
orsion
i
on
i iV r K m rϕ
ϕ δ# $= +% &∑
  ( ) N-body polarization energypol N
V r =

( ) external fields energyext N
V r =

Special Interaction Terms examples
restraints on the system:
-  from experimental data
-  to bias the sampling
Bond	
  stretch	
  term	
  
•  The	
  bond	
  stretch	
  term	
  is	
  almost	
  always	
  
approximated	
  with	
  a	
  harmonic	
  potenGal:	
  
	
  	
  
	
  	
   	
   	
  U(r)	
  =	
  ½	
  k	
  (r	
  –	
  r0)2	
  
Torsion	
  term	
  
ethane:	
  sp3-­‐sp3	
  bond	
  
CH3-­‐CH3	
  
periodicity	
  =	
  3	
  
phase	
  	
  =	
  0	
  degrees	
  
	
  
ethene:	
  sp2-­‐sp2	
  bond	
  
CH2=CH2	
  
periodicity	
  =	
  2	
  
phase	
  	
  =	
  0	
  degrees	
  
Nonbonded	
  interacGons	
  
•  All	
  atoms	
  see	
  each	
  other	
  through	
  space	
  
	
  	
  ~	
  ½	
  N(N	
  –	
  1)	
  	
  	
  	
  interacGons	
  
•  Bound	
  atoms	
  (1,2	
  neighbours)	
  and	
  their	
  neighbours	
  (1,3	
  
neighbours)	
  are	
  ocen	
  excluded	
  because	
  the	
  bonded	
  
interacGons	
  already	
  take	
  these	
  into	
  account	
  
•  SomeGmes	
  1,4	
  neighbours	
  are	
  treated	
  differently	
  to	
  keep	
  
the	
  proper	
  torsional	
  profile	
  
•  Ocen	
  we	
  work	
  with	
  a	
  cutoff	
  for	
  the	
  nonbonded	
  interacGon	
  
	
  
Other	
  interacGons	
  
•  Some	
  force	
  fields	
  show	
  alternaGve	
  interacGons	
  
–  Specific	
  hydrogen	
  bond	
  interacGons	
  
usually	
  a	
  balance	
  between	
  vdw	
  and	
  electrostaGcs	
  
–  Crossterms,	
  e.g.	
  Energy	
  as	
  funcGon	
  of	
  bond	
  length	
  
and	
  angle:	
  U(r,θ)	
  
needed	
  to	
  reproduce	
  vibraGonal	
  spectra	
  
–  …	
  
Example	
  of	
  force	
  field	
  parameters	
  A
B
PotenGal	
  Energy	
  Surface	
  (PES)	
  
Levinthal’s	
  Paradox	
  
	
  
HeurisGc	
  energy	
  minimizaGon	
  	
  
techniques:	
  
-­‐  Simulated	
  annealing	
  
-­‐  GeneGc	
  algorithm	
  
-­‐  InterpretaGon	
  of	
  EM	
  
PotenGal	
  Energy	
  Surface	
  (PES)	
  
Defini1on	
  of	
  a	
  model	
  for	
  molecular	
  simula1on	
  
MOLECULAR	
  
MODEL	
  
Degrees	
  of	
  freedom:	
  
atoms	
  are	
  the	
  
elementary	
  par3cles	
  
Forces	
  or	
  
interac3ons	
  
between	
  atoms	
   Boundary	
  condi3ons	
  
Methods	
  to	
  generate	
  	
  
configura3ons	
  of	
  
atoms:	
  Newton	
  	
  	
  
system	
  
temperature	
  
pressure	
  
Force	
  Field	
  =	
  
physico-­‐chemical	
  
knowledge	
  
Every	
  molecule	
  consists	
  of	
  atoms	
  that	
  are	
  very	
  strongly	
  aHached	
  
Situa3on	
  at	
  3me	
  t+Δt	
  
Classical	
  dynamics	
  
Situa3on	
  at	
  3me	
  t	
  
	
  
	
  
	
  
Force	
  is	
  determined	
  by	
  rela3ve	
  posi3ons	
  
	
  
accelera3on	
  =	
  force	
  /	
  mass	
  	
  	
  	
  	
  
Δ	
  velocity	
  =	
  accelera3on	
  ×	
  Δ	
  t	
  
Δ	
  posi3on	
  =	
  velocity	
  ×	
  Δ	
  t	
  
	
  force	
  
velocity	
  
posi3on	
  
Determinism	
  …	
  
Sir	
  Isaac	
  Newton	
  
1642	
  -­‐1727	
  
Genera1ng	
  configura1ons	
  in	
  atomic	
  	
  
simula1ons:	
  molecular	
  dynamics	
  
new	
  posiGons	
  
Time	
  t	
  
Time	
  (t+Δt)	
  
posiGons	
  
velociGes	
  
forces	
  
new	
  velociGes	
  
...	
  comparable	
  to	
  shooGng	
  a	
  movie	
  
of	
  a	
  molecular	
  system...	
  
Δt	
  ≈	
  10-­‐15	
  seconds	
  
Monte-­‐Carlo:	
  AlternaGve	
  methodology	
  for	
  generaGng	
  canonical	
  
Hritz,	
  J.;	
  de	
  Ruiter,	
  A.;	
  Oostenbrink,	
  C.	
  Impact	
  of	
  plasGcity	
  and	
  flexibility	
  on	
  docking	
  
results	
  for	
  Cytochrome	
  P450	
  2D6:	
  a	
  combined	
  approach	
  of	
  molecular	
  dynamics	
  and	
  
ligand	
  docking.	
  J.	
  Med.	
  Chem.	
  2008,	
  51,	
  7469-­‐7477	
  
Monte-­‐Carlo:	
  AlternaGve	
  methodology	
  for	
  generaGng	
  	
  
canonical	
  structural	
  ensemble	
  
	
  
Metropolis	
  criterion	
  
-­‐probability	
  of	
  moving	
  from	
  state	
  1	
  (having	
  energy	
  E1)	
  to	
  	
  
	
  the	
  state	
  2	
  	
  (having	
  energy	
  E2)	
  is:	
  
	
   	
   	
   	
   	
  	
  
	
  
p12
acc
= min[1,e
−
E2−E1
kT
]
X	
  =	
  H,	
  F,	
  Cl,	
  Br,	
  CH3	
  	
  
Enhanced	
  sampling	
  by	
  replica	
  exchange	
  	
  
molecular	
  dynamics	
  (REMD)	
  	
  
Hritz	
  J.,	
  Oostenbrink	
  C.	
  	
  
Hamiltonian	
  replica	
  exchange	
  molecular	
  dynamics	
  using	
  soc-­‐core	
  
interacGons.	
  J.	
  Chem.	
  Phys.	
  2008,	
  128,	
  144121	
  
Hritz	
  J,	
  Oostenbrink	
  C.	
  	
  	
  
OpGmizaGon	
  of	
  Replica	
  Exchange	
  Molecular	
  Dynamics	
  by	
  Fast	
  Mimicking.	
  J.	
  
Chem.	
  Phys.	
  2007,	
  127,	
  204104	
  
Replica	
  exchange	
  probabiliGes	
  
β= −( ) exp( ( , ))W x H r p probability of state x
(Boltzmann factor)
( )( ) exp ,
M
REM i i i
i
W X Hβ
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
∑ r p
probability of global state X
(product of single state x
probabilities)
detailed balance condition on the transition probability w(X→X’):
( ) ( ') ( ') ( ' )REM REMW X w X X W X w X X→ = →
Metropolis criterion for the exchange probability:
( )
( ) ( ) ( )( )
1 for 0( ')
( ')
exp for 0( ' )
i j j i
w X X
p X X
w X X
U Uβ β
Δ ≤⎧→
→ = = ⎨
−Δ Δ >→ ⎩
Δ = − −r r 1/ Bk Tβ =
Hansmann,	
  U.H.E.;	
  Chem.	
  Phys.	
  LeH.	
  1997,	
  281,	
  140-­‐150.	
  
	
  
Temperature	
  REMD	
  
ΔT	
  ~	
  (square	
  root	
  of	
  degrees	
  of	
  freedom)-­‐1	
  
Is	
  there	
  alternaGve	
  way	
  how	
  to	
  get	
  from	
  one	
  
side	
  of	
  barrier	
  to	
  other	
  one?	
  
Hamiltonian	
  replica	
  exchange	
  
(Fukunishi,	
  H.;	
  Watanabe,	
  O.;	
  Takada,	
  S.;	
  J.	
  Chem.	
  Phys.	
  2002,	
  113,	
  6042-­‐6051.)	
  
	
  
IDEA: instead of changing a temperature,
rather alter specific interactions
( , ) ( ) ( )i i i i i iH K U= +r p p r
( ) ( )( ) ( ) ( )( )i i j i i j j j j iU U U Uβ βΔ = − − −r r r r
Soc	
  core	
  interacGons	
  
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the red x still appears, you may have to delete the image and then insert it again.
2
2
2
6
12
666
12
;
1
4
)(
;
1
)(
λα
πε
λα
α
α
el
ij
ji
ij
el
vdw
ijij
ij
vdw
B
rB
qq
rE
C
C
A
rA
C
rA
C
rE
=
+
=
=
+%
%
&
'
(
(
)
*
−
+
=
Where	
  else	
  to	
  get	
  crazy	
  ideas	
  J?	
  
Real	
  REMD	
  of	
  GTP	
  
REMD	
  costs:	
  6x5ns	
  =	
  30ns	
  
	
  	
  	
  	
  	
  	
  	
  	
  anGàsyn	
  transiGons:16	
  
	
  	
  	
  	
  	
  	
  	
  	
  synàanG	
  transiGons:16	
  	
  
MD	
  costs:	
  2x20ns	
  =	
  40ns	
  
	
  	
  	
  	
  	
  	
  	
  	
  anGàsyn	
  transiGons:	
  0	
  
	
  	
  	
  	
  	
  	
  	
  	
  synàanG	
  transiGons:	
  1	
  	
  
RelaxaGon	
  
PotenGal	
  of	
  mean	
  force	
  
GTP
antisynG −Δ GTP
antisynG −Δ
GTP
antisynG −Δ
(REMD)	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  =	
  7.6	
  ±	
  0.3	
  kJ.mol-­‐1	
  	
  	
  	
  	
  	
  (REMD)	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  =	
  -­‐6.8	
  ±	
  0.9	
  kJ.mol-­‐1	
  
	
  	
  	
  	
  	
  	
  	
  (TI)	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  =	
  7.7	
  ±	
  1.2	
  kJ.mol-­‐1	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  (TI)	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  =	
  -­‐5.8	
  ±	
  1.6	
  kJ.mol-­‐1	
  
	
  
GTP
antisynG −Δ
GTP
antisynG −Δ
ΔGsyn−anti
8−Br−GTP
ΔGsyn−anti
8−Br−GTP
Molecular	
  dynamics	
  is	
  more	
  than	
  
super-­‐microscope!	
  
ENERGY!	
  
Efficient	
  free	
  energy	
  calcula1ons	
  for	
  compounds	
  
with	
  high	
  intramolecular	
  energy	
  barriers	
  
Hritz,	
  J.;	
  Lappchen	
  T.;	
  Oostenbrink,	
  C.	
  	
  
CalculaGons	
  of	
  binding	
  affinity	
  between	
  C8-­‐subsGtuted	
  GTP	
  analogs	
  	
  
and	
  the	
  bacterial	
  cell-­‐division	
  protein	
  FtsZ.	
  	
  
Eur.Biophys.	
  J..	
  2010,	
  39,	
  1573-­‐1580	
  
	
  	
  
Hritz,	
  J.;	
  Oostenbrink,	
  C.	
  	
  
Efficient	
  free	
  energy	
  calculaGons	
  for	
  compounds	
  with	
  mulGple	
  stable	
  conformaGons	
  
separated	
  by	
  high	
  energy	
  barriers.	
  	
  
J.	
  Phys.	
  Chem.	
  B	
  2009,	
  113,	
  12711-­‐12720	
  
Thermodynamic	
  cycle	
  
Cl Cl
OH OH
ΔGbind(1)	
  
ΔGbind(2)	
  
ΔG21(bound)	
  ΔG21(free)	
  
1	
  
2	
  
( ) ( )
( ) ( )
bind bind bindG G G
G free G bound
ΔΔ = Δ − Δ
= Δ − Δ21 21
2 1
One-step perturbation
( )/
ln BA BE
A
k T
B
E
B B
G k T e− −
Δ = −
,BAG Gλ λ δλ
λ
+Δ = Δ∑
A ARB RBG G GΔ = Δ −Δ
A	
  
B	
  
C	
  
D	
  
E	
  
…	
  
R	
  
X	
  =	
  H,	
  F,	
  Cl,	
  Br,	
  CH3	
  	
  
( )
S
Tk
HH
BSRSR
B
SR
eTkGGG
),(),(
ln
pqpq −
−
−=−=Δ
( )
( )
∑
−
−
−
−
=
i
Tk
HH
Tk
HH
R
i
B
iiSiiR
B
iiSiiR
e
e
p ),(),(
),(),(
pqpq
pqpq
Enhanced	
  sampling	
  one	
  step	
  perturbaGon	
  method	
  (ES-­‐OS)	
  
Crystal	
  structures	
  of	
  FtsZ	
  from	
  Aquifex	
  aeolicus	
  
(Oliva	
  et	
  al.	
  J.	
  Mol.	
  Biology	
  (2007);	
  Lappchen	
  et	
  al.	
  Chemistry	
  &	
  Biology	
  (2008))	
  	
  
Figures	
  taken	
  from	
  Lappchen	
  et	
  al.	
  Chemistry	
  &	
  Biology	
  (2008)	
  
	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
Thermodynamic	
  cycle	
  
Cl Cl
OH OH
ΔGbind(1)	
  
ΔGbind(2)	
  
ΔG21(bound)	
  ΔG21(free)	
  
1	
  
2	
  
( ) ( )
( ) ( )
bind bind bindG G G
G free G bound
ΔΔ = Δ − Δ
= Δ − Δ21 21
2 1
The	
  comparison	
  of	
  relaGve	
  binding	
  affiniGes	
  obtained	
  from	
  
computaGonal	
  (Hritz,	
  J.	
  et	
  al.	
  Eur.Biophys.	
  J.	
  (2010))	
  simulaGons	
  
and	
  experimental	
  assays	
  (Lappchen	
  et	
  al.	
  Chemistry	
  &	
  Biology	
  
(2008)).	
  
Conclusion	
  
•  VisualizaGon	
  of	
  biomolecular	
  structures,	
  their	
  electrostaGcs	
  
potenGal,	
  etc.	
  can	
  provide	
  very	
  useful	
  informaGon	
  
•  Dynamical	
  features	
  can	
  be	
  obtained	
  from	
  molecular	
  dynamics	
  
simulaGons.	
  Be	
  aware	
  about	
  a	
  lot	
  of	
  used	
  approximaGons,	
  
simplificaGons	
  and	
  restricted	
  Gme	
  length	
  of	
  simulaGon.	
  	
  
•  PotenGal	
  of	
  molecular	
  dynamics	
  can	
  be	
  enhanced	
  sampling	
  
methods	
  such	
  as	
  replica	
  exchange	
  molecular	
  dynamics:	
  T-­‐REMD	
  
or	
  H-­‐REMD.	
  	
  
•  Free	
  energy	
  calculaGons	
  allows	
  for	
  calculaGng	
  the	
  binding	
  
affiniGes,	
  solvaGon	
  free	
  energies,	
  lipophiliciGes,	
  etc.