DNA

RNA

proteins

organs

tissues

Living systems

lipids

cell

Structural Analysis of Biological Systems

Joel IretaFritz-Haber-Institut der Max-Planck-Gesellschaft

Biological Macromolecules

LipidsCarbohydrates

P P P P PA B C D ENucleic Acids Proteins

Can DFT help us to find conformation-stability-activity relationships ?

Protein Activity

Native conformation random coil

unfolding(denaturation)

folding

Conformation

Stability

Activity

Covalent bondsweak interactions

TemperaturePressureEnvironment (solvent)

Charge transfer processesMechanical response

eletrons

protons

dynamics

Outline

α-helixsecondarystructure

Hydrogen bonds

Mechanical responsestability

Weak interactions

Covalent bondsDFT accuracy conformation

activityThe role and Perspective of Ab Initio Molecular Dynamics in Study of Biological SystemsP. Carloni, U. Rothlisberger and M. ParrinelloAcc. Chem. Res., 35, 455 (2002)

Weak Interactions

R

E

Repulsion of the electronic shells

Attraction due tooscillations of thecharge density(dispersion)

RX Y Interaction between

neutral molecules

δ+ δ-

electrostatics

+

DFT can not describe van der Waals interactions !

Rqq

RB

RAE yx+−= 612

van der Waals(isotropic interaction)

< 2 kcal/mol(0.084 eV)

Weak Interactions

hydrogenbonding interaction

a) strong > 10 kcal/mol ( 0.43 eV)

b) moderate 3 kcal/mol (0.13 eV) to 10 kcal/mol (0.43)

c) weak < 3 kcal/mol (0.13 eV)D = donor atom A = acceptor atom

rhb

r2

θσ

BD

H

Aα

δ+

δ- µ1

µ2

Hydrogen bonds are predominantlyelectrostatic interactions. However...

-3

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8 10

r2(Å)

E (K

cal/m

ol)

Hydrogen bonds are directional : σ usually ranges from 140° to 180°

Hydrogen bonds are cooperative: they strongly interact each other modifying its bond strength

For small r2 multipole expansion ofthe electrostatic interaction doesnot converge properly

Full electrostaticinteraction energy

R-3

R-4

R-5

R-6

S Scheiner, Hydrogen bonding a theoretical perspectiveOxford University Press (1997)

Techniques accounting for the electroniccorrelation are needed for an accurate description of the hydrogen bonds

Dispersion energies contributes significantly to the Hydrogen bond energy

( ) ( ) ( ) ( )( )( ) Bluer

Yellowrrrrr BAAB

;0;0

<∆>∆

−−=∆

ρρ

ρρρρ

H

Attractive part : electrostaticinduction an dispersion energies (charge transfer ?)

Repulsion part: electronic exchange interaction

Hydrogen Bond Nature O

H

N

Projection of the electrostatic potential on a charge density isosurface.System: alanine peptide dimers forming a hydrogen bond

O N

Hydrogen Bond Nature

Water dimer

-6

-5

-4

-3

-2

-1

0HF MP2 CCSD CCSD(T)

Ener

gy (K

cal/m

ol)

At least MP2 is needed to accuratelydescribe the hydrogen bond interactionJ. E. del Bene, Hydrogen Bonds. Encyclopedia

of Computational Chemistry Vol. 2. Schleyer, D. Ed. in Chief.(John Wiley, Chichester U. K. 1998).

ABBA →+

BAABbindinghb EEEEE −−==

LDA or GGA?

hbinitioabbest

hbDFT EEE __−=∆ Hartree-Fock plus

configuration interactionor coupled-cluster

-10123456789

10

∆E

(kca

l/mol

)

Error PBEError LDA

(HF)

2

(HC

l) 2

(H2O

) 2

(OC

)(HF)

(ClH

)(NH

3 )

(FH

)(NH

3 )

(H2O

)(NH

3 )

(CO

)(HF)

GGA is needed !

Tuma et. al

C. Tuma, D. Boese, N. C. Handy Phys. Chem. Chem. Phys. 1, 3939 (1999)

Accuracy of DFT Plane-wave Pseudopotential Method for the Description of Hydrogen Bonds

hbinitioabbest

hbDFT EEE __−=∆ CCSD(T)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

∆E

(Kca

l/mol

)

The error bar is less than 1 kcal/mol (0.042 eV)

0

1

2

3

4

5

6

fm2 fm5 dmf1 dmf2 dmf3 dmf4 nma1 nma2 nma3

E (K

cal/m

ol)

PBE Accuracy to Describe Hydrogen Bonded Systems:Dependence on the Bond Directionality

hbinitioabbest

hbDFT EEE __−=∆ MP2

X-HO=C

r1

σ hydrogen bonds are highly bent.i.e. σ < 130°

PBE Accuracy to Describe Hydrogen Bond Strength

O

O

N

NC

C

C

C

N

N

O

O

N N

O

O

C

C

C

C C

CC

C

O

O N

N

C

C

C

C

C

N

N

O

O

Formamidedimers

N-Methyl acetamidedimers

PBE accuracy:1 kcal/mol per hydrogen bondwith respect to Møller-Plesset (MP2) level of theory if the hydrogen bonds are close to linearityi.e. σ = (130° , 180°)

X-HO=C

r1

σ

-7.2 kcal/mol (MP2)-7.2 kcal/mol (PBE)

-2.5 kcal/mol (MP2)-1.8 kcal/mol (PBE)

-7.3 kcal/mol (MP2)-6.8 kcal/mol (PBE)

-5.4 kcal/mol (MP2)-4.3 kcal/mol (PBE)

-8.6 kcal/mol (MP2)-7.6 kcal/mol (PBE)

MP2 results: R. Vargas et al J. Phys. Chem. A 105, 4963, 2001.

PBE Accuracy to Describe Hydrogen Bond Strength

C

C

CCC

C

N NO

O

C

C

CC

C

CN

N

O

O

C

C

C

C

C

C

N

N

O

O

C C

C

C

C

C

N

N

O

O

N-Methyl acetamidedimers

N, N-dimethyl formamidedimers

X-HO=C

r1

σ

PBE accuracy:1.5 kcal/mol per hydrogen bondwith respect to Møller-Plesset (MP2) level of theory if the hydrogen bonds are benti.e. σ < 130°

-4.1 kcal/mol (MP2)-2.7 kcal/mol (PBE)

-4.8 kcal/mol (MP2)-3.5 kcal/mol (PBE)

-2.1 kcal/mol (MP2)-0.9 kcal/mol (PBE)

-2.2 kcal/mol (MP2)-1.0 kcal/mol (PBE)

MP2 results: R. Vargas et al J. Am. Chem Soc 122, 4750, 2000.R. Vargas et al J. Phys. Chem. A 105, 4963, 2001.

PBE Accuracy to Describe Hydrogen Bond Cooperativity

Å

Method hb dimer(kcal/mol)

hb Infinite chain(kcal/mol)

∆hb, cooperativity(kcal/mol)

MP2-tz(2df,2pd)

-5.08 -8.14 -3.06

BLYP-tz(2df,2pd)

-4.39 -7.66 -3.27

BLYP-lno -5.13 -7.42 -2.29

PBE-lno -5.91 -8.55 -2.64

PBE-pw -5.7 -8.3 -2.60

1

1

1. S. Suhai, J. Phys. Chem. 100, 3950 (1996)

Method r2 dimer ( ) r2 infinite chain ( ) ∆r2 ( )

MP2-tz(2df,2pd)

3.0012 2.8784 -0.1228

BLYP-tz(2df,2pd)

3.0450 2.8516 -0.1934

BLYP-lno 3.05 2.87 -0.18

PBE-lno 3.01 2.83 -0.18

PBE-pw 2.99 2.82 -0.17

Å Å

1

1

PBE accuracy: 0.05 Å in distance changes0.5 kcal/mol in hb strength changes

O

O

N

N

N

N

O

O

C

C

Formamidedimer

Formamideinfinitechain

Unit cell

r2

-2

-1

0

1

2

3

LDA 50 Ryd BP 50 Ry PBE 50 Ry PBE 70 Ry

C-C

N-C

C-O

C=O

O-H

C-H

N-H

<NCC

<CCO

<CC=O

<COH

%Error

covalent bonds are well describedDFT-PBE gives errors smaller than 1% !

Structural parameters of an isolated glycine molecule calculated with different functionals.•Compared against HF/CISD1

C

H

C

HH

O

ON

1. C.-H. Hu, M. Shen and H. F. Shaefer III, J. Am. Chem. Soc. 115, 2923 (1993).

Accuracy of DFT for Hydrogen Bonded Systems

Protein Structure

secondary structure(β-sheet)

The Peptide Bond

C HC

ON

C

Rn

Rn-1 The peptide bond has a partial doublebond character

Peptide group characteristics

Planar

RigidPeptide group

ϕ

ψ

Ramachandran-Diagramm

C HC

ON

C

Rn

Rn-1

Secondary Structure of proteins

The α-helix conformation is the most common secondary structure

α-Helixβ-Sheet

ϕ

ψ

Helix Stability

Several factors are responsible for the α-helix stability

Hydrogenbonds

α-helix

α-helix is a prominent secondary structure in protein conformation

CappingR1

R2Capping

Hel

ix d

ipol

e

q-

q+

Solvent

Hydrogen bonds are consideredone of the main interactionsstabilizing the α-helix structure

+

-Hydrogen bonds are cooperativeThe strength of an hb is increased by its interaction with another hb

Open questions:

How large is the hydrogen bond strength in an α-helix?

How large is the hydrogen bond cooperativity in an α-helix?

Helixaxis

Zr

r

θ

LCarbon

Oxygen

α− Carbon

Nitrogen

α− Carbon

Model

Unit cell5.0 Å

5.0 Å •11 Peptide units• 3 turns•110 atoms/cell• Γ Point for sampling

Brillouing zone

Nmo360=θ

M turns per unit cell

N peptide units per unitcell

o57.99exp =θo

el 2.98mod =θ

zyxn nZeenrenrR ++= )sin()cos( θθ

One dimensionalcrystal

Unitcell

No ending effects

H

NC

O R

R

ψ

φ

ω

Parameters Calculated Experimental

hb 1.950 Å ± 0.005 2.06 Å ± 0.16

NO 2.950 Å ± 0.005 2.99 Å ± 0.14

NHO 163.6° ± 0.3 155° ± 11

HOC 147.3° ± 0.5 147° ± 9

φ -63.5° ± 0.5 -63.8° ± 6.6

ψ -43.0° ± 0.5 -41.0° ± 7.2

ω

Pitch

177.4° ± 0.7

5.48 Å

180° ± 5

5.4 Å

Equilibrium structure of the helix

Good agreement between calculated and experimental parameters!

NO

<HOC

<NHO

hb Pitch

α-Helix Geometry

Hydrogen Bond Strength in a α-helix

Problem: back bone is not taken into account !

N C

O H hb

molecule : • formamide [1]

MP2 and DFT calc.60-70% cooperativity in an infinite array

• N-methylacetamide [2]cluster with five molecules HF calc.38-42% cooperativity

1. S. Suhai, J. Phys. Chem. 100, 3950 (1996) 2. R. Ludwid, F. Weinhold, T. C. Farrar, J. Chem. Phys. 107, 499 (1997).

-P-P-P-P-

hb

1 4Back bone

How to extract the hb strength?

α-helix conformation

Previous studies: molecular cluster approach:

α-helix withouthb

Ehb = Hydrogen bond energy

Hydrogen Bond Strength

onconformatiE

Stability

Fully extended structure(FES)

α-helixµ = Energy per peptide unit

onconformatiFEShb EE −−= ∞∞∞ µµα infinite chain

onconformatiFESNNN

hb EEEHE −−−=∆= ∞− µααα1

finite chain

21211 RPRPRPR NN →+−

∞− −−≈ FESNN

onconformati EEE µαα1 N=3 ( α-helices )

N=2 ( 310-helices )

Hydrogen Bond StrengthSystem Econformational Ehb

(first turn, i—i+3)

Ehb

(infinite chain)

∆Ehb

(cooperativity)

Polyalanine 5.9 kcal/mol -3.5 kcal/mol -8.6 kcal/mol -5.1 kcal/mol

Polyglycine 7.2 kcal/mol -4.1 kcal/mol -9.9 kcal/mol -5.8 kcal/molα-helixhbs (i,i+3)

310-helixhbs (i,i+2)

System Econformational Ehb

(first turn, i—i+2)

Ehb

(infinite chain)

∆Ehb

(cooperativity)

Polyalanine 5.8 kcal/mol -4.4 kcal/mol -8.0 kcal/mol -3.6 kcal/mol

-5.9 kcal/mol polyalanine α-helix

-5.9 kcal/mol polyglycine α-helix

Hydrogen bond strength as calculated in a cluster approach

1

4

The back bone significantly affects the strength of neighboring hb’s Without back bone the hb energy is larger by 50 %

-50

-40

-30

-20

-10

0

10

20

30

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Number of peptide units

Stab

ility

res

pect

FES

(K

cal/mol)

α-helix glycine

α-helix alanine

310−helix alanine

The Importance of Cooperativity

stabilization energyelastic energy

( )ANENE hbonconformati −−< ∞ A = 3 for α-helixA = 2 for 310-helix

J. Ireta, J. Neugebaure, M. Scheffler, A. Rojo, M. Galván J. Phys. Chem. B, 107, 1432 (2003)

1

2

3

4

5

6

7

Helix axis

After the second turn the hydrogen bond strength increases smoothly

10

The hydrogen bond strength difference between long finite chains and the infinite one is due to the large electric field at the ends of the finite chains

9

8

-5

-4

-3

-2

-1

0

2 4 6 8 10 12 14 16 18 20Number of peptide uni ts

∆Eh

b (k

cal/m

ol)

cooperativityPolyGlyPolyAla

First turn

second turn

third turn

+ -

Electrostatic potential

Helix axis

Ending Effects

-10

-9

-8

-7

-6

-5

-4

-3

2 4 6 8 10 12 14 16 18 20

Number of peptide un its

Ehb

(kca

l/mol

)

PolyGlyPolyAla

-5.4 kcal/mol, N=7

Ehb, ∞

Ehb , ∞ ~ 1 kcal/mol

How does the peptide bond respond to strain ?

Open questionsHow does the helix structure responds to tensileor compressive loads?

How do the hydrogen bonds respond to tensileor compressive loads? Random coil

Denaturation( unfolding )

Experiment:

Proteins denaturates when uniaxial compression above 3 GPa is applied (fast ultra shock waves experiments)

α-helix unfolds under tensile load (atomic force microscope experiments)

The Resonant Model

The hydrogen bonds shifts the equilibrium towards the zwitterion state

C HC

ON

C

Rn

Rn-1O

H

hb

hb

Hydrogen bond effect on the peptide group structure

Single bond

double bond(zwitterion)

R1 R1

C N

O

HCα

Cα

C N

O

HCα

Cα

R2 R2

-

+

Singlebond

doublebond

Singlebond state

Doublebond state(zwitterion)

Effect of the Secondary Structure on the Peptide Bond

α-helix

C HC

ON

C

Rn

Rn-1O

H

hb

hb- 0.017 Å

0.012 Å

0.019 Å

-8.6 kcal/mol per hbmonomer

Changes in the peptide bond are modestif they are compared with changes in othersystems with hbs of similar strength

Peptide Bond Response to Strain

Unit cell

compression

peptide bond is compressed by -0.006 ÅN-H bond is elongated by 0.005 Å C=O bond is elongated by 0.002 Å

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

strain

Cov

alen

t bo

nds

dist

ortio

n (Å

)

Peptide bond

C=O

N-H

Peptide Bond Behavior

- 0.017 Å

8.6 kcal/molper hb N-H stretch 3314 cm-1

hb effect

doublebond

Single bond

strain effect

- 0.023 Å

9.0 kcal/mol per hb

N-H stretch 3215 cm-1

low strain

0.029 Å

no hb

N-H stretch 3514 cm-1

high strain

Hydrogenbonds

α-helix

Hydrogen Bond Response to Strain

1.51.71.92.12.32.52.72.93.1

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05strain

hbbo

nd

dist

ance

(Å

) 8.6 kcal/mol

9.0 kcal/mol

At high strain the hydrogen bonds are broken

N-H stretch 3314 cm-1

N-H stretch 3215 cm-1

Carbon Pyramidalization

HN

C

CPyramidalization

C

O

d

θ

0123456789

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

strain

III

θ (º)

At high strain carbon pyramidalizes

-40

160

360

560

760

960

1160

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

s

F (p

N)

-0.5

1.5

3.5

5.5

7.5

9.5

11.5

13.5

P (G

Pa)

III

Strain Induced First Order Phase Transition

Cα

Cα

N C O

Planar peptide unithydrogen bond strength ~ 9 kcal/mol zwitterionic like state

CO

CαCαN

Highly distorted peptide unitBroken hydrogen bondssingle bond like state

Backbone Response to Strain

C HC

ON

C

Rn

Rn-1

ϕψ

Dihedral angles

-180

-120

-60

0

60

120

180

-180 -120 -60 0 60 120 180

φ

ψ

α-helix

310-helix

β-sheet

Phase II is out of the helical regionin a Ramachandran diagram

Ramachandram diagram

Phase II

-45

-30

-15

0

15

30

45

60

75

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

s

III

tors

iona

lang

les d

evia

tion

(º) ψ

ω

φ

ConclusionsThe DFT plane-wave pseudopotential method is areliable tool to study biological systems

PBE describes the hb strength and cooperativitywithin an error bar of 1 kcal/mol

Cooperative effects within an infinite α-helix strengthen the hb by a factor of two

Compressive strain stabilizes the zswitterionicform of a peptide unit

At high compressive strain helices undergoa first order phase transition

The interplay between hydrogen bond strengthand carbonyl pyramidalization drives the phase transition

-30 -20 -10 0Energy (eV)

0.0

0.09

0.15

Strain induces a qualitative change in the electronic charge density at the carbonyl bond: (sp2⇒sp3 like hybridization)!

Carbonyl bond

Electronic structure response to compression

Densityof states

Strain

0.0 0.09 0.15Strain

-100

-50

0

50

100

150

200

250

300

350

400

-0.4 -0.2 0 0.2 0.4 0.6

s

Fza (pN)

Under tensionalso a phase transitionis observed

Helix stability

hbstrainFES EEStability −=−= µµα

-4

-2

0

2

4

6

8

10

12

14

-0.4 -0.2 0 0.2 0.4 0.6

s

stability (kcal/mol)

Stabilityregion

nativeconformationphase 1

phase IIIphase

II