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Infinite Polypeptides: An Approach to
Study the Secondary Structure of Proteins
Joel Ireta
Fritz-Haber-Institut der Max-Planck-Gesellschaft
Berlin
Multiscale Modeling of Proteins
coarsening
Atomistic model
Reduced models
The structure results from a subtle interplay between covalent bonds and noncovalent interactions (hydrogen bonding and van der Waals forces)
We aim to get insight into the underlying physics that govern the biological
processes by properly taking into account the non-covalent interactions in
atomistic and coarse-grain modeling of biomolecules
Noncovalent Interactions
The magnitude of non-covalent interactions is difficult to
quantify and the extent of their effect on the structure
and stability of proteins remains unclear
Outline
Hydrogen bonding
The accuracy of Density Functional Theory (DFT) to
describe hydrogen bonding
Cooperativity of hydrogen bonding in finite and infinite
chains
Infinite polypeptides: models to study the role of hydrogen
bonding in the stability of the secondary structure of
proteins
Comparison of DFT results with force fields
Hydrogen Bond Nature
H
D
Attractive part :
electrostatic
induction and charge transfer ?
rhb
q
A
B
Repulsion part:
electronic exchange interaction
O
O
N
N
H
 r    AB r    A r    B r 
 r   0;Yellow
 r   0; Blue
H
Projection of the electrostatic potential
on a charge density isosurface.
System: alanine peptide dimers forming
a hydrogen bond
Techniques accounting for the electronic
correlation are needed for an accurate
description of the hydrogen bonds
Density Functional Theory
E  f  r 
Energy is a functional of the electronic density, (r)
E  T  r   EH  r   Eenucl  r   Exc  r   Enuclnucl R 
T  r 
kinetic energy of non-interacting electrons
EH  r 
classical electron-electron interaction
Enuclnucl R 
nuclei-nuclei interaction
Exc  r 
Eenucl  r 
exchange-correlation energy
(non-classical electron-electron
interaction)
Electron-nucleus interaction
Pseudopotential
approximation
LDA
Exc  f  r 
GGA Exc  f  r ,  r 
only valence electrons
are treated explicitly
core electrons are included
by using a pseudopotential
DFT Accuracy and the Hydrogen Bond
Directionality
r
H
hb
q
D
H
A
H
C
B
H
H
1.6
PBE error per hb (kcal/mol)
H
C
C
1.4
N
H
1.2
1
O
H
N-Methyl
Acetamide
0.8
0.6
2
0.4
0.2
H
O
0
110
120
130
150
q (deg)
H
H
140
C
160
170
N
180
C
H
H
formamide
H
2
H
N
H
C
H
H
C
O
N-N dimethyl
formamide
With increasing deviation from a linear arrangement
of the hydrogen bonds, the accuracy of the DFT-PBE
2 decreases.
J.Ireta, J. Neugebauer, M. Scheffler J. Phys. Chem A, 108, 5692 (2004)
Hydrogen Bonds are Cooperative
+
N 1
Ehb
 ETN  ETN 1  ET1
Formamide chain
+
+
-4
E (kcal/mol)
avg
Ehb
ETN  nET1

n 1
-6

hb
E 
-8
-10
0
5
10
15
Formamide units
20
ET
nunitcell
 nunitcellET1
An infinite network of hbs
strengthens each individual
bond by more than a factor
of two
Ending Effects
H
N
rhb (Å)
O
2
C
n=2
1.95
n=3
1.9
n=4 n=5
n=6 n=7
1.85
1.8
hbs in the chain
1.75
0
2
4
6
Electrostatic Potential
8
10
Helix Stability
Open questions:
Capping
R1
Solvent
Is the helix conformation
intrinsically stable?
q-
Is there a free energy
minimum corresponding
to an isolate helical
conformation?
+
Hydrogen
bonds
a-helix
Are the hydrogen bonds strong
enough to stabilize the helical
conformation?
q+
R2
Capping
Side
group
R1
N
C
R2
C
O
Glycine
Does not
form
helices
Why do different amino acids
have
different propensity to form
helices ?
Side
group
C
N
C
R2
R1
C
O
Alanine
the highest
propensity to
form helices
Helix-Coil Transition
Random coil
Temperature
Solvent
Pressure
helix
Denaturation
( unfolding )
The formation of a helix can be divided in two steps:
4
5
3
1. helix nucleation:
4
3
2. helix propagation:
1
2
Experimental observations:
Helix formation may not be a two-state process
1
2
Model
One-dimensional
crystal
Unit
cell
Rn  r cos(qn )ex  r sin( qn )e y  nZez
Peptide
unit
Stability
r
z
hb
q
Reference system:
Fully extended structure
(FES)
Stability
E  E (q , z )  EFES
per peptide unit
Unit
cell
Potential Energy Surface at 0 K
E  E (q , z )  EFES
left
handed
helices
folded
conformations
right
handed
helices
6
E
(kcal/mol)
3
0.5
0
1.0
1.5
folding
Extended
conformations
2.0
Z (Å)
2.5
3.0
unfolding
3.5
4.0
-3
60
90
120
150
right handed
180
210 q (degrees)
240
270
left handed
300
Minimum Energy Pathway
310helix
p-helix
Stability
(kcal/mol) (i, i + 4)
27conformation
(i, i + 1)
(i, i + 2)
a-helix
4
(i, i + 3)
2
0
left
handed
-2
fully
extended
structure
right
handed
-4
0.5
1
1.5
2
Z (Å)
2.5
3
3.5
4
a-Helix Geometry
Equilibrium structure of polyalanine in a-helix conformation
y
O
w
R
C
<HOC
N
R
H
f
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
f
-63.5° ± 0.5
-63.8° ± 6.6
y
-43.0° ± 0.5
-41.0° ± 7.2
w
177.4° ± 0.7
180° ± 5
Pitch
5.48 Å
5.4 Å
NO
hb
<NHO
Good agreement between calculated and experimental parameters!
Pitch
Trajectory
q (deg)
190
p-helix
a-helix
27helix
310helix
170
alanine
150
glycine
130
110
Fold
90
Unfold
z (Å)
70
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Structural transitions occur in approximately two steps:
1) mainly a change in the length
2) delayed adjustment of the twist
2.8
3
Hydrogen Bond Strength
a-helix without
hb
Fully extended structure
(FES)
Econformation
Ehb = Hydrogen bond energy
Stability

= Energy per peptide unit
a-helix
R1PN 1R2  P  R1PN R2

Econformation  EaN  EaN 1  FES
N=3 ( a-helices )
N=2 ( 310-helices )

Ehb  HaN  EaN  EaN 1  FES
 Econformation
finite chain


Ehb
 a  FES
 Econformation
infinite chain
Hydrogen Bond Strength in Infinite Helices
with Different ( L, q ) Parameters
q
Z
Number
of hbs
per PU
Ehb
Ground state
(kcal/mol)
1.17
80.0
1
-10.4 p
1.32
83.1
2
-3.9 (ts1)
1.50
98.2
1
-8.6 a
1.71
102.9
2
-3.3 (ts2)
1.95
120.0
1
-7.7 (310)
E (kcal/mol)
-2
p
a
-3
1
1.2 1.4 1.6 1.8
310
2
H
N
N PU i+n-1
H
hb
O
Bifurcated hbs
hb
PU i+n
H
N
The helix with the strongest
hbs is not the lowest energy
structure
ts2
-1
PU i+n
PU i
PU i
ts1
hb
O
Transition state (ts)
C
1
0
C
Z (Å)
2.2 2.4
J. Ireta, J. Neugebauer, M. Scheffler, A. Rojo,
M. Galvan J. Am. Chem. Soc. in press
Hydrogen Bond Cooperativity in a-Helix
(kcal/mol)
a-helix
hbs (i,i+3)
System
Econformation
Ehb
Ehb
Ehb
(first turn, i—i+3)
(infinite chain)
(cooperativity)
Polyalanine
5.9
-3.5
-8.6
-5.1
Polyglycine
7.2
-4.1
-9.9
-5.8
Hydrogen bond strength as calculated in a cluster approach
4
-5.9 kcal/mol polyalanine a-helix
-5.9 kcal/mol polyglycine a-helix
1
The back bone significantly affects the strength of neighboring hb’s
Without back bone the hb energy increases ~ 50 %
J.Ireta, J. Neugebauer, M. Scheffler, A. Rojo, M. Galván J. Phys. Chem B, 107, 1432 (2003)
Occurrence of the (Z,q) Values in Crystals of Proteins
q
Z
It is possible to
estimate the (Z, q)
parameters for a
residue in a realistic
structure of a
protein
right
handed
helices
left
handed
helices
Extended
conformations
0.5
1.0
1.5
2.0
2.5
Z (Å)
3.0
3.5
4.0
60
90
120
150
180
210
240
q (degrees)
270
300
The values for (Z, q) cluster along the minimum energy pathway of the potential
energy surface of an infinitely long polypeptide
Occurrence of the (Z,q) Values in Crystals of Proteins
right-handed
conformations
% of residues
3
left-handed conformations
2.5
a-helix
2
60% of the residues are in a
right handed conformation
1.5
1
310-helix
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Z (Å)
18% of the residues in right-handed conformations adopt a 310-helical
structure
The majority of residues in left-handed conformations are in an extended
structure (they may be forming b-sheets)
Origin of the Left-handed Twist in the Extended
Conformations
Stability
(kcal/mol)
2
Alanine in a
fully extended
structure with
the amide
group planar
N
C
H
C
O
C
Nitrogen
pyramidalization
1.5
1
Alanine in a fully
extended structure
0.5
0
Glycine in a fully
extended structure
-0.5
140
150
160
right-handed
170
180
q (degrees)
190
200
210
left-handed
220
Phonon Dispersion Spectrum of Polyalanine
Dotted lines
unscaled
frequencies
(factor 1.02)
Amide A band
(N-H stretching)
Solid lines
scaled
frequencies
Amide 1 band
(C=O stretching)
Amide 2 band
(C-N stretching
N-H bending)
L. Ismer
The a-helix is a true minimum (zero imaginary frequencies)
Specific Heat of the Polyalanine  -helix
Theoretical results compared with experimental values
*
*
[1]
experiment
[1]
[2]
force field [2]
[3]
force field [3]
DFT-PBE [4]
experiment [3]
DFT results [6]
force field results [4]
force field results [5]
[1] M. Daurel et al., Biopol. 14, 801 (1975)
[2] B. Fanconi et al., Biopol. 10, 1277 (1971)
[1] [3]
M.V.K.
Daurel
etetal.,
801 (1975)
Datye
al. Biopol.
JCP 84, 14,
12 (1986)
[2] B. Fanconi et al., Biopol. 10, 1277 (1971)
[3] V.K. Datye et al. JCP 84, 12 (1986)
DFT accurately describes the heat capacity (at low temperatures)
Possible reasons for remaining differences: van der Waals, anharmonicity
L. Ismer, J. Ireta, S.Boeck and J. Neugebauer, PRE 71, 031911 (2005)
Alanine
∆F (kcal/mol)
Free Energy of the
Helical Conformations
(room temp.)
∆Evib(0 K)
(unfolding
temp.)
∆Etot
Temperature (K)
∆F (kcal/mol)
Glycine
The a-helix is the lowest-energy
structure even at high
temperature
L. Ismer
Temperature (K)
Force Fields
Class I Force-Fields:
V (r ) 
 k b  b    kq q  q 
2
b
2
0
0
bonds
angles
Two-body
interaction
 qi q j Aij Cij 
  k cosn     1   
 12  6 
rij
rij
rij 
torsions
nonbond 

pairs
Three-body
interaction
q
2
1
b
4
3
Three-body
interaction
Two-body
interaction
rij

5
1
Two-body
interaction
Four-body
interaction
Force constants adapted
to match normal-modes
frequencies for a number
of peptide fragments
4
Four-body
interaction
2 Charges
3
Lennard-Jones
Parameters
Obtained from ab-initio
calculations, usually
HF/6-31G*
Fitting to reproduce densities
and heats of vaporization in
liquid simulations
Fitting to reproduce ab-initio (HF or MP2)
potential energy surfaces
DFT vs Force Fields
DFT-PBE
AMBER
p
a
310
CHARMM27
both force fields
predict the a-helix to
be the most stable
conformation
only AMBER reproduces
all the helical minima
M. John
p
a
M. John
310
p
a
Ending Effects
PolyGly
PolyAla
-3
Ehb (kcal/mol)
-4
Electrostatic potential
-5.4 kcal/mol, N=7
-5
-
+
-6
-7
-8
Ehb , 
-9
~ 1 kcal/mol
Ehb, 
-10
2
4
6
8
10
12
14
16
18
20
Helix axis
Number of peptide units
PolyAla
PolyGly
cooperativity
3
 Ehb (kcal/mol )
0
third turn
-2
second turn
-3
First turn
-4
-5
2
4
6 8 10 12 14 16 18 20
Number of peptide units
8
5
2
-1
9
6
4
7
10
1
Helix axis
After the second turn the hydrogen bond
strength increases smoothly
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
M. John
Conclusions
Infinitely long chains of polypeptides are realistic models to study
the secondary structure of proteins in combination with electronic
structure methods.
These models allow to properly include the cooperative effect of
hydrogen bonding, which is crucial to describe the folded
conformations.
Moreover fine details of the structure of proteins like the lefthandeness of extended conformations are explained by these models
Acknowledgements
Franziska Grzegorzewski:
Calculations of left-handed helices
Lars Ismer:
Phonons
Marcus John:
Forcefields
Matthias Scheffler
Marcelo Galván
Arturo Rojo
Jörg Neugebauer
Ramachandran Plot
BLYP/TZVP Ramachandran Plot1
of the Alanine dipeptide
Dihedral Angles
f
y
C5 (FES)
(-150.0, 158.8)
1.77 kcal/mol
R
C7eq (27)
(-83.8, 75.1)
Ground State
a
There is no minimum associated with the
a-helix conformation
Hydrogen bonds are missing
Allowed regions where repulsion
among atoms is negligible
1. R. Vargas et al J. Phys. Chem. A 106, 3213 (2002)
a-helix: The Success Of a Theoretical Prediction
Antecedents:
X-ray diffraction spectra of fibrous proteins
(a-keratin, b-keratin found e.g. in hair)
Pauling-Corey Model (1950):
a helical conformation
where planar peptides
are connected by
hydrogen bonds
D. A. Eisenber, “The discovery of the a-helix and bsheet, the principal structural features of proteins”,
Proc. Natl. Acad. Sci. USA 100, 11207 (2003)
The Peptide Bond
The resonant model, theoretical model proposed by L. Pauling
R1
Single
bond
state
R1
Single
bond
Ca
H
C
Ca
C
N
Ca
O
+
N
O
-
double
bond
H
Double
bond
state
(zwitterion)
Ca
R2
R2
The peptide bond has a partial double
bond character
Rn-1
C
H
C
O
Peptide group characteristics
N
C
Planar
Rn
Rigid
Peptide group
Protein Structure
(20 different aminoacids)
Primary structure
(amino acid sequence)
The biological function of proteins crucially
depends on their structural conformation
secondary structure
(b-sheet)
The Importance of Cooperativity

hb
NEconformation  E
elastic energy
N  A  0
A = 4 for p-helix
A = 3 for a-helix
A = 2 for 310-helix
stabilization energy
E (kcal/mol)
40
Chains containing at least
10 peptide units are stable
in a-helical conformation
p
20
a
310
Short alanine helices prefer
a 310 conformation
0
-20
N peptide units
-40
0
5
10
15
20
25
M. John