Joel Ireta Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin

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

2. Multiscale Modeling of Proteins coarsening Reduced models Atomistic model 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

3. 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

4. Hydrogen Bond Nature H rhb q A D B O N H H 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 O N Techniques accounting for the electronic correlation are needed for an accurate descriptionof the hydrogen bonds

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

6. H rhb q A D B H O N H H H H C C C H H H H N O C H H H C H H N H C N-Methyl Acetamide C H 2 H O 2 N-N dimethyl formamide 2 formamide DFT Accuracy and the Hydrogen Bond Directionality 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 + Formamide chain + + E (kcal/mol) An infinite network of hbs strengthens each individual bond by more than a factor of two Formamide units

8. Ending Effects H rhb (Å) O N C n=2 n=3 n=4 n=5 n=6 n=7 hbs in the chain Electrostatic Potential

9. Helix Stability Open questions: Is the helix conformation intrinsically stable? Is there a free energy minimum corresponding toan isolatehelical conformation? Are the hydrogen bonds strong enough to stabilize the helical conformation? Why do different amino acids have different propensity to form helices ? Solvent Capping R1 q- + Hydrogen bonds Helix dipole a-helix - q+ R2 Capping R1 R1 N C N C C C C R2 O R2 O Side group Glycine Alanine Side group Does not form helices the highest propensity to form helices

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

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

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

13. Minimum Energy Pathway 27-conformation 310-helix (i, i + 1) (i, i + 2) p-helix Stability (kcal/mol) (i, i + 4) a-helix (i, i + 3) left handed fully extended structure right handed Z (Å)

14. O R y w C <HOC N f H R NO hb Pitch <NHO a-Helix Geometry Equilibrium structure of polyalanine in a-helix conformation Good agreement between calculated and experimental parameters!

15. Trajectory q (deg) 27-helix p-helix 310-helix a-helix alanine glycine Fold Unfold z (Å) • Structural transitions occur in approximately two steps: • mainly a change in the length • 2) delayed adjustment of the twist

16. a-helix without hb Fully extended structure (FES) Ehb = Hydrogen bond energy Stability = Energy per peptide unit N=3 ( a-helices ) N=2 ( 310-helices ) finite chain infinite chain Hydrogen Bond Strength a-helix

17. Hydrogen Bond Strength in Infinite Helices with Different ( L, q ) Parameters C C O H N Ground state hb PU i+n PU i (p) (ts1) Transition state (ts) (a) H N PU i+n-1 (ts2) hb (310) Bifurcated hbs DE (kcal/mol) O hb PU i+n PU i H N ts1 ts2 The helix with the strongest hbs is not the lowest energy structure a p 310 Z (Å) J. Ireta, J. Neugebauer, M. Scheffler, A. Rojo, M. Galvan J. Am. Chem. Soc. in press

18. 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 Hydrogen Bond Cooperativity in a-Helix (kcal/mol) a-helix hbs (i,i+3) 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)

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

20. Occurrence of the (Z,q) Values in Crystals of Proteins right-handed conformations % of residues left-handed conformations a-helix 60% of the residues are in a right handed conformation 310-helix 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)

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

22. 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)

23. Specific Heat of the Polyalanine α-helix Theoretical results compared with experimental values [1] experiment [1] force field [2] force field [3] * [2] [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) [3] V.K. Datye et al. JCP 84, 12 (1986) [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) 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)

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

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

26. DFT-PBE AMBER p 310 a CHARMM27 DFT vs Force Fields both force fields predict the a-helix to be the most stable conformation only AMBER reproduces all the helical minima M. John

27. p 310 p a a M. John

28. PolyGly PolyAla Electrostatic potential -5.4 kcal/mol, N=7 - + Ehb ,  ~ 1 kcal/mol Ehb,  Helix axis cooperativity 9 PolyGly PolyAla 6 3 8 5 10 2 7 4 third turn 1 second turn Helix axis After the second turn the hydrogen bond strength increases smoothly First turn 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 Ending Effects

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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. Ramachandran Plot Dihedral Angles y f R BLYP/TZVP Ramachandran Plot1 of the Alanine dipeptide C5 (FES) (-150.0, 158.8) 1.77 kcal/mol 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)

32. 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 b-sheet, the principal structural features of proteins”, Proc. Natl. Acad. Sci. USA 100, 11207 (2003)

33. The Peptide Bond Single bond Ca Ca H H Single bond state + Double bond state (zwitterion) C N C N double bond Ca O Ca O - R2 R2 Rn-1 The peptide bond has a partial double bond character Peptide group characteristics Planar Rigid C H N C O C Rn Peptide group The resonant model, theoretical model proposed by L. Pauling R1 R1

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

35. The Importance of Cooperativity A = 4 for p-helix A = 3 for a-helix A = 2 for 310-helix elastic energy stabilization energy E (kcal/mol) Chains containing at least 10 peptide units are stable in a-helical conformation p a 310 Short alanine helices prefer a 310 conformation N peptide units

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