Biomolecular Modelling: Goals, Problems, Perspectives
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Transcript of Biomolecular Modelling: Goals, Problems, Perspectives
Biomolecular Modelling: Goals, Problems, Perspectives
1. Goal
simulate/predict processes such as1. polypeptide folding thermodynamic 2. biomolecular association equilibria governed3. partitioning between solvents by weak (nonbonded)4. membrane/micelle formation forces
common characteristics:- degrees of freedom: atomic (solute + solvent) hamiltonian
or - equations of motion: classical dynamics force field- governing theory:statistical mechanics entropy
Processes: Thermodynamic Equilibria
Folding Micelle Formation
Complexation Partitioning
folded/native denatured micelle mixture
bound unbound in membrane in water in mixtures
Methods to Compute Free Energy
Classical Statistical Mechanics:
Free Energy:
Free Energy Differences:
- between two systems: and
- depending on a parameter:
- along a (phase space) coordinate:
2
1 21
N particles
Hamiltonian:
kinetic potential
, ,..., ,2
energy
Ni
Ni i
pV r r r H p r
m
3 1
one number integrand everywhere
(
>
)ln exp
,( , , ) (
0
!)N p rF dp
HN V T kT h N d
kTr
( , )A pH r
( , )B pH r
( , ; )pH r
( )R r
13' ' exp( , )
( , , ; ) ln ! ( )NR R R
HN V T kT h N
kR
T
p rF r dpdr
Methods to Compute Free Energy
• Counting of Configurations:
one simulation, but sufficient events sampled?
• Thermodynamic Integration
many simulations: ensemble average <…> for each valuethen numerical integration
• Perturbation Formula
one simulation, sufficient overlap?
0
0
conf.bou ( ) (
ndbound unbound
'
')= ln
cunbound onf. R R
R R
R
NRkTF
RF
NR
( ) ( )ln exp B A
B A
A
FH
kTT
FH
k
( , ; )B
B A
A
F Fp r
dH
Free Energy Difference via Thermodynamic Integration
( , , )B
B A
A
FH p r
F F d
- Accurate: sufficient sampling <…> sufficient-points i
- many (10 – 100) separate simulations- for each new pair of states A and B a new set of simlulations is required- for each
the state is unphysical
Very time consuming
or i A B F
i BA
state A state B
H
Free Energy Calculations
One-step perturbation technique and efficient sampling of relevant configurations
Thermodynamic Integration
N inhibitors:
12 1
M
jj j
HG G
unbound bound
1E I
NE I
2E I
1EI
2EI
NEI
2 M N simulations
10 10 200
Free Energy Calculations
One-step Perturbation
ln expBB
RR
R
ii
H HG G k T
k T
RE I
1I
2I
NI
REI
1EI
2EI
NEI
2 simulations ofan unphysical state which is chosen to optimise sampling for entire set of N inhibitors
Idea: use soft-core atoms for each site where the inhibitors possess different (or no) atoms
The reference state simulation (R) should produce an ensemble that contains low-energy configurations for all of the Hamiltonians (inhibitors)
H1, H2, … ,HN
conformational space
A
A
A
B
B
B
R
B’
C
D
E
0 1 2 3 4 5 6 7 8
ln expBB
RR
R
ii
H HG G k T
k T
H2O ProteinA
B
C
Unphysical Reference Ligand R
GAbind
GBbind
GARH2O
GBRH2O
GAB = GBbind – GA
bind = GARH2O – GAR
protein – GBRH2O + GBR
protein
Unphysical Reference Ligand R
A
B
C
……
N
N
O
NH2
Y1 (C)
N
NN
N NH2
U1 (A)
N
NN
NH2
U8
N
NH
O
O
Y2 (T)
N
NH
O
OF
Y3
N
NH
O
OBr
Y4
N
N
O
NH2
Y5
N
N
NH2
O
Y6
N
N
O
NH2Br
Y7
N
N
NH2
OBr
Y8
N
N
NH2
O
Y9
F
F
Y10
N
NHN
N O
NH2
U2 (G)
N
NN
N NH2
NH2
U3N
NN
NH
NH2O
U4
N
NN
NH
OO
NH2
U5N
NHN
N O
U6N
NN
N
U7
N
NHN
O
NH2
U9
N
NN
NNH2
U10
N
U11
N
N
U12
N
NHN
NNH2O
U13
C6
C1 C2
C3
C4C5
N4 H42
H41
H3
H22
H21RIBOSE
H6
CM5
N2
C2 C3
C6
C1
C7
C5
C4N9
C8
H1
N6
H61
H62
H21
H22
H3
RIBOSE
N8
H7
H82
H81
N2
SPYR SPUR
Free energies of base insertion, stacking, pairing in DNA
(CGCGAXYTCGCG)
2.0 ns 3.4 ns 2.0 ns 2.0 ns
Five MD simulations to obtain free energies of base insertion, stacking, pairing
Double helix d(CGCGAXYTCGCG)2 in water
Free energy of insertion and stacking for particular pairs of central bases
A B C
A, G, C, T A, U13, C, T A, G, Y9, T
Stacking of adjacent central bases
• 10x13x10x13 – 1 = 15899 values (in fact we did 1024)
• U1-2 – Y1-2 and U1-13 – Y1-10, and vice versa (520 free energies)
• Decompose the double free energies of pairing into single free energies of pairing
A G 3 4 5 6 7 8 9 10 11 12 13
C 66 25 52 77 38 46 54 59 32 52 42 40 39
T 52 85 48 65 91 88 51 48 92 38 59 67 89
3 44 69 41 54 80 76 42 32 71 24 47 51 78
4 55 99 48 65 97 92 54 48 97 39 63 73 93
5 58 20 49 80 42 43 53 60 30 56 41 46 39
6 36 84 33 48 76 70 38 33 73 22 37 60 79
7 56 14 48 78 38 38 50 58 25 54 40 44 36
8 41 90 37 52 79 71 40 37 76 25 39 61 81
9 30 47 32 27 53 57 38 25 42 14 33 42 60
10 66 96 63 82 105 97 62 60 96 52 59 79 96
110-120
100-110
90-100
80-90
70-80
60-70
50-60
40-50
30-40
20-30
10-20
kJ/mol
purine
pyrim
idin
e
Free energies of double base pairing in (CGCGAXYTCGCG)2
F
RIBOSE
H
F
N
N
N
N
O
RIBOSE
N
H
O
H
H
H
U5Y10
N
O4
RIBOSE
H
H
H
H
N3
N2
N
N1N
N
N6
RIBOSE
H
H
H
H
U10Y9
N
N4
RIBOSE
H
Br
O2
H
H
N3
N
N1
N
N
O6
RIBOSE
H
H
H
H
N2
Y7 U2 (G)
N
O4
RIBOSE
H
Br
O2
HN3
N
N1
N
N
N6
RIBOSE
H
H
H
H
O
N
O4
RIBOSE
H
Br
O2
HN3
N
N1
N
N
O6
RIBOSE
H
H
H
H
N2
U2 (G)U4Y4 Y4
14 kJ/mol 14 kJ/mol
99 kJ/mol
105 kJ/mol
65 kJ/mol
(CGCGAXYTCGCG)
2.0 ns 3.4 ns 2.0 ns 2.0 ns
Five MD simulations to obtain free energies of base insertion, stacking, pairing
Free Enthalpy of Solvation by Thermodynamic Integration
Make Hamiltonian (Interaction) dependent on a coupling parameter
2
1 21
( , , ) ( , ,... , )2
PotentialKinet
Ni
N
ic
i i
pH p r U r r r
m
( , ) ( ) ( , ) ( )uu uv vvU r U r U r U r
solute-soluteassume = 0(for simplicity)
solute-solventsmall
solvent-solventvery large
=0 no solute-solvent interaction (solute in gas phase)=1 full solute-solvent interaction (solute in solution)
(Free) Enthalpy and Entropy of Solvation
1
0
1 0
1
0
( )
( 1) ( 0)
1 ( ) ( )
uvS
S uv uv
vv uv vvB
uv
d
pV pV
UG
dk
H U
U U UT
U
U
difficult to calculate due to Uvv
same term
1
0
1
0
1( ) ( )
1( ) ( ) ( )
S vv uv vv uv
uv uv uv uv
B
B
pV pV dk T
dk
T
T
S U U U U
U U U U
assumed: only solute-solvent interaction Uuv() depends on
solvent-solvent term Uvv does not
(Free) Enthalpy and Entropy of Solvation
1 0
1
0
( 1) ( 0)
1 ( ) ( ) ( ) ( )
S S S
uv uv
uv u vB
v uv u dk
G H T S
U U
U U UT
U
Uvv terms are absent computable
Calculate instead of HS and TSS:
both computable
1 0
1
0
( 1) ( 0)
1( ) ( ) ( ) ( )
uv uv uv
uv uv uv u vB
v u dk
U U U
T S U UT
U U
yield insight into enthalpic and entropic effects
(Free) Enthalpy and Entropy of SolvationNico van der Vegt
reference: J. Phys. Chem. B. (2004)
mole fraction
Solvation of Methane in Na+Cl- Solutions
methane solvation in salt
U*uv triangles
TS*uv squares
relative to neat water
UG T S
concentration
Na+Cl- free enthalpy
energy (enthalpy)
entropy
Entropy disfavours solvation increasingly with salt concentration (non-linear).
Solvation of Methane in Acetone Solution
UG T S
methane solvation in acetone
U*uv triangles
TS*uv squares
relative to neat water: SPC water SPC/E water
free enthalpy
entropy
energy (enthalpy)
Entropy favours solvation.mole fraction
Solvation of Methane in Dimethylsulfoxide (DMSO) Solutions
free enthalpy
entropy
energy(enthalpy)
Energy favours solvation (non-linearly).
mole fraction mole fraction
reference: J. Chem. Phys. B. (2004)
UG T S
GS Uuv TSuv Relative to Solvation in Pure Water
enthalpyrelative and absolute contributions do vary
entropy
dominant counteracts enthalpy enthalpy and entropy
co-act counteractchanges sign
mole fraction different models
relative values of Uuv, TSuv change, Gs not so much
Computer-aided Chemistry: ETH Zuerich
Molecular Simulation Package
GROMOS = Groningen Molecular Simulation + GROMOS Force FieldGenerally available: http://www.igc.ethz.ch/gromos
Research Topics
• searching conformational space• force field development
– atomic– polarization– long range Coulomb
• techniques to compute free energy
• 3D structure determination– NMR data– X-ray data
• quantum MD: reactions
• solvent mixtures, partitioning• interpretation exp. data• applications
– proteins, sugar, DNA, RNA, lipids, membranes, polymers
– protein folding, stability– ligand binding– enzyme reactions
Computer-aided Chemistry: ETH Zuerich
Group members
Dirk Bakowies
Indira Chandrasekhar
David Kony
Merijn Schenk
(Alex de Vries)
(Thereza Soares)
(Nico van der Vegt)
(Christine Peter)
Alice Glaettli
Yu Haibo
Chris Oostenbrink
Peter Gee
Markus Christen
Riccardo Baron
Daniel Trzesniak
Daan Geerke
Bojan Zagrovic