1. Joel Ireta Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin Infinite Polypeptides: An Approach to Study the Secondary Structure of Proteins

2. Multiscale Modeling of Proteins Atomistic model Reduced models The structure results from a subtle interplay between covalent bonds and non- covalent 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 coarsening

3. 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 Noncovalent Interactions 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

4. Techniques accounting for the electronic correlation are needed for an accurate description of the hydrogen bonds H Hydrogen Bond Nature O N O H N Projection of the electrostatic potential on a charge density isosurface. System: alanine peptide dimers forming a hydrogen bond Attractive part : electrostatic induction and charge transfer ? Repulsion part: electronic exchange interaction r hb  BD H A

5. Density Functional Theory Energy is a functional of the electronic density,  (r) kinetic energy of non-interacting electrons Electron-nucleus interaction nuclei-nuclei interaction exchange-correlation energy (non-classical electron-electron interaction) LDA GGA Pseudopotential approximation classical electron-electron interaction only valence electrons are treated explicitly core electrons are included by using a pseudopotential

6. DFT Accuracy and the Hydrogen Bond Directionality r hb  BD H A C O N H H H 2 formamide H C C C H H H H H H NO 2 N-Methyl Acetamide C C C N O H H H H H H H 2 N-N dimethyl formamide With increasing deviation from a linear arrangement of the hydrogen bonds, the accuracy of the DFT-PBE decreases. J.Ireta, J. Neugebauer, M. Scheffler J. Phys. Chem A, 108, 5692 (2004)

7. + Hydrogen Bonds are Cooperative + + E (kcal/mol) Formamide units An infinite network of hbs strengthens each individual bond by more than a factor of two Formamide chain

8. Ending Effects hbs in the chain r hb (Å) CN H O Electrostatic Potential n=2 n=3 n=4 n=5 n=6 n=7

9. Solvent Helix Stability  -helix Capping R 1 R 2 Capping Hydrogen bonds Helix dipole q- q+ + - Open questions: Is the helix conformation intrinsically stable? Is there a free energy minimum corresponding to  an isolate  helical conformation? Are the hydrogen bonds strong enough to stabilize the helical conformation? Why do different amino acids have different propensity to form helices ? C N C O R1R1 R2R2 Glycine Does not form helices C C N C O R1R1 R2R2 Alanine the highest propensity to form helices Side group Side group

10. 1 2 3 4 5 Helix-Coil Transition Random coil Denaturation ( unfolding ) helix Temperature Solvent Pressure The formation of a helix can be divided in two steps: 1. helix nucleation: 2. helix propagation: 1 4 2 3 Experimental observations: Helix formation may not be a two-state process

11. Reference system: Fully extended structure (FES) Model One-dimensional crystal Stability  z r Unit cell Unit cell Stability hb per peptide unit Peptide unit

12. Potential Energy Surface at 0 K  E (kcal/mol) left handed helices right handed helices Extended conformations folded conformations -3 0 3 6  (degrees) Z (Å) folding unfolding right handed left handed

13. Stability (kcal/mol) Z (Å)  -helix  -helix    helix    conformation fully extended structure right handed left handed Minimum Energy Pathway (i, i + 1) (i, i + 2) (i, i + 3) (i, i + 4)

14. H N C O R R    Equilibrium structure of polyalanine in  -helix conformation Good agreement between calculated and experimental parameters! NO <HOC < NHO hb Pitch  -Helix Geometry

15. Trajectory z (Å)  (deg)  -helix  -helix    helix    helix Fold Unfold Structural transitions occur in approximately two steps: 1)mainly a change in the length 2) delayed adjustment of the twist glycine alanine

16.  -helix without hb E hb = Hydrogen bond energy Hydrogen Bond Strength Stability Fully extended structure (FES)  -helix = Energy per peptide unit infinite chain finite chain N=3 (  -helices ) N=2 ( 3 10 -helices )

17. Hydrogen Bond Strength in Infinite Helices with Different ( L,  ) Parameters Z  Number of hbs per PU E hb (kcal/mol) 1.1780.01-10.4 1.3283.12-3.9 1.5098.21-8.6 1.71102.92-3.3 1.95120.01-7.7 3 10     (3 10 ) (ts 1 ) (ts 2 ) ts 1 ts 2 Z (Å)  E (kcal/mol) N H C O N C O H N H Transition state (ts) hb Bifurcated hbs Ground state hb PU i PU i+n PU i PU i+n PU i+n-1 The helix with the strongest hbs is not the lowest energy structure J. Ireta, J. Neugebauer, M. Scheffler, A. Rojo, M. Galvan J. Am. Chem. Soc. in press

18. Hydrogen Bond Cooperativity in  -Helix (kcal/mol)  -helix hbs (i,i+3) -5.9 kcal/mol polyalanine  -helix -5.9 kcal/mol polyglycine  -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 increases ~ 50 % J.Ireta, J. Neugebauer, M. Scheffler, A. Rojo, M. Galv á n J. Phys. Chem B, 107, 1432 (2003)

19. Z  Occurrence of the (Z,  ) Values in Crystals of Proteins It is possible to estimate the (Z,  ) parameters for a residue in a realistic structure of a protein Z (Å)  (degrees) Extended conformations left handed helices right handed helices The values for (Z,  ) cluster along the minimum energy pathway of the potential energy surface of an infinitely long polypeptide

20. Z (Å) % of residues right-handed conformations left-handed conformations  -helix 3 10 -helix 60% of the residues are in a right handed conformation The majority of residues in left-handed conformations are in an extended structure (they may be forming  -sheets) 18% of the residues in right-handed conformations adopt a 3 10 -helical structure Occurrence of the (Z,  ) Values in Crystals of Proteins

21. H N C C C O Origin of the Left-handed Twist in the Extended Conformations  (degrees) Glycine in a fully extended structure Alanine in a fully extended structure Alanine in a fully extended structure with the amide group planar left-handed right-handed Nitrogen pyramidalization Stability (kcal/mol)

22. Phonon Dispersion Spectrum of Polyalanine Dotted lines unscaled frequencies (factor 1.02) Solid lines scaled frequencies Amide A band (N-H stretching) Amide 1 band (C=O stretching) Amide 2 band (C-N stretching N-H bending) L. Ismer The  -helix is a true minimum (zero imaginary frequencies)

23. Specific Heat of the Polyalanine  -helix [1] [2] [3] [1] M. Daurel et al., Biopol. 14, 801 (1975) [2] B. Fanconi et al., Biopol. 10, 1277 (1971) [3] V.K. Datye et al. JCP 84, 12 (1986) Theoretical results compared with experimental values experiment [3] DFT results [6] force field results [4] force field results [5] * experiment [1] force field [2] force field [3] * [1] M. Daurel et al., Biopol. 14, 801 (1975) [2] B. Fanconi et al., Biopol. 10, 1277 (1971) [3] V.K. Datye et al. JCP 84, 12 (1986) Possible reasons for remaining differences: van der Waals, anharmonicity DFT-PBE [4] L. Ismer, J. Ireta, S.Boeck and J. Neugebauer, PRE 71, 031911 (2005) DFT accurately describes the heat capacity (at low temperatures)

24. ∆ F (kcal/mol) (room temp.)(unfolding temp.) ∆E vib (0 K) ∆E tot Temperature (K) ∆ F (kcal/mol) Alanine Glycine Free Energy of the Helical Conformations The  -helix is the lowest-energy structure even at high temperature L. Ismer

25. Force Fields Class I Force-Fields: Two-body interaction Three-body interaction Four-body interaction Two-body interaction 1 23 4 5 b   r ij Two-body interaction Three-body interaction Force constants adapted to match normal-modes frequencies for a number of peptide fragments Charges Obtained from ab-initio calculations, usually HF/6-31G* Lennard-Jones Parameters Fitting to reproduce densities and heats of vaporization in liquid simulations Four-body interaction 1 2 3 4 Fitting to reproduce ab-initio (HF or MP2) potential energy surfaces

26. M. John CHARMM27 AMBER DFT-PBE   3 10 DFT vs Force Fields both force fields predict the  -helix to be the most stable conformation only AMBER reproduces all the helical minima

27.   3 10   M. John

28. 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 cooperativity PolyGly PolyAla First turn second turn third turn + - Electrostatic potential Helix axis Ending Effects PolyGly PolyAla -5.4 kcal/mol, N=7 E hb,  ~ 1 kcal/mol

29. M. John

30. 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 left- handeness 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

31. R   Dihedral Angles Ramachandran Plot Allowed regions where repulsion among atoms is negligible 1. R. Vargas et al J. Phys. Chem. A 106, 3213 (2002)  C7 eq (2 7 ) (-83.8, 75.1) Ground State C5 (FES) (-150.0, 158.8) 1.77 kcal/mol There is no minimum associated with the  -helix conformation Hydrogen bonds are missing BLYP/TZVP Ramachandran Plot 1 of the Alanine dipeptide

32.  -helix: The Success Of a Theoretical Prediction Antecedents: X-ray diffraction spectra of fibrous proteins (  -keratin,  -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  -helix and  - sheet, the principal structural features of proteins”, Proc. Natl. Acad. Sci. USA 100, 11207 (2003)

33. The Peptide Bond C H C O N C RnRn R n-1 The peptide bond has a partial double bond character Peptide group characteristics Planar Rigid Peptide group The resonant model, theoretical model proposed by L. Pauling R1R1 R1R1 C N O H CC CC C N O H CC CC R2R2 R2R2 - + Single bond double bond Single bond state Double bond state (zwitterion)

34. secondary structure (  -sheet) Protein Structure Primary structure (amino acid sequence) The biological function of proteins crucially depends on their structural conformation (20 different aminoacids)

35. The Importance of Cooperativity stabilization energy elastic energy A = 4 for  -helix A = 3 for  -helix A = 2 for 3 10 -helix Chains containing at least 10 peptide units are stable in  -helical conformation Short alanine helices prefer a 3 10 conformation E (kcal/mol) N peptide units   3 10

36. M. John