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