1. Protein Rigidity and Flexibility: Applications to Folding A.J. Rader University of Pittsburgh Center for Computational Biology & Bioinformatics

2. How can one characterize the flexibility of proteins? Molecular dynamics Comparing different conformations Identifying domains within proteins Jacobs, et al. PROT 44 (2001) Gerstein and Krebs Nucl. Acids. Res. 26 (1998) http://molmovdb.mbb.yale.edu/molmovdb/

3. What is Flexibility? Flexibility, as presented here, is determined directly from the mechanical properties of the structure studied. The number of floppy modes, F, measures the number of degrees of freedom and quantifies the flexibility of the network (protein). Floppy modes are zero frequency modes corresponding to deformations which cost no energy. Maxwell counting is a mean field approximation for the fraction of floppy modes as a function of, the mean coordination, of the network.

4. Inspiration from Network Glasses Is the universal rigidity phase transition seen in network glasses at = 2.40 also seen in proteins? (suggests mean coordination is a relevant parameter) Is protein folding/unfolding related to a rigidity phase transition in proteins?

5. 2D Maxwell constraint counting 4 atoms, 4 bonds F=2*4 - 4 - 3 = 1 Floppy 4 atoms, 6 bonds F=2*4 - 6 - 3 = -1 Rigid & Stressed 4 atoms, 5 bonds F=2*4 - 5 - 3 = 0 Isostatically Rigid F = 2N – N bonds - 3

6. Laman’s Theorem In 2D: rigidity is uniquely found by building the network up one bond at a time and checking for redundant bonds in all subgraphs of the network. A redundant bond is found whenever b > 2n-3 in a subgraph. Laman, J. Engrg. Math. 4:331 (1970). Led to the pebble game algorithm. Jacobs & Thorpe, Phys. Rev. Lett. 75:4051 (1996). 3D generalization ( b > 3n-6) is only valid for the graphs of the type that also contain bond-bending constraints (next-nearest-neighbor constraints). These bond-bending constraints correspond to fixing the chemical bond angle. Tay & Whiteley, Struct. Topo. 9:31 (1985); Jacobs, J. Phys. A. 31:6653 (1998).

7. 3D Maxwell constraint counting (on bond-bending networks) 5 Atoms6 Atoms7 Atoms 5*3 -5 -5 -6 = -1 Rigid 6*3 -6 -6 -6 = 0 Isostatic 7*3 -7 -7 -6 = 1 Flexible F = 3N – N bonds – N angles -6

8. Applying Maxwell counting to Proteins  FIRST* *Floppy Inclusions & Rigid Substructure Topography Jacobs, et al. PROT 44 (2001) http://firstweb.pa.msu.edu/

9. Model Fix physically inspired constraints and count number of floppy modes and mean coordination. D H A B  Covalent bond Chemical bond angle Hydrogen bond Hydrophobic contact Dihedral angles left free to rotate  degrees of freedom (floppy modes)

10. Hydrogen Bonds The energy is assigned according to C N H A d r   N H O C Adds 3 constraints per H-bond

11. R ≤ 0.25 Å R CC Adds 2 constraints per tether Hydrophobic Contact Hydrophobic Tether Hydrophobic Tethers

12. The Protein Model Always present: covalent bonds (with bond- bending constraints) peptide bonds locked hydrophobic tethers Variable interactions: inclusion of a hydrogen bond depends on the temperature

13. Hespenheide, et al. JMGM (2002) Rigid Cluster Decomposition

14. Unfolding with FIRST Hydrogen bonds are removed one at a time based upon energy to simulate thermal denaturation. Removal of hydrogen bonds mimics unfolding pathways. (non-covalent interactions in the native state contain information about folding ) Native State Hydrophobic Collapse Formation of hydrogen bonds and salt bridges Denatured Folding Unfolding Breaking of hydrogen bonds and salt bridges

15. Rigidity Lost Rader, et al. PNAS 99 (2002)

16. Hydrogen Bond Dilution

17. Fraction of floppy modes

18. Differentiate… once, twice

19. Protein Dataset

20. Folding Results from FIRST Native state hydrogen bonds and hydrophobic interactions encode information about folding. Results for the rigidity phase transition in glass networks give a critical mean coordination of c = 2.385. A similar plot of flexibility versus mean coordination exists for a set of 26 diverse protein structures with the folding transition at T = 2.405 ± 0.015. Higher mean coordination implies rigid and folded, lower implies flexible and unfolded. The jump in df/d suggests a first-order phase transition within proteins.

21. Application to Rhodopsin Question of interest: what is the conformational change induced by retinal isomerization? Proposed activation mechanism: opening of helical bundle toward cytoplasmic ends of the helices, predominantly involving helices III,VI, & VII.

23. 2.3942.3902.376 N hb 11810131

24. Rhodopsin Folding Core

25. Rhodopsin Summary Greater flexibility observed at the cytoplasmic ends of all transmembrane (TM) helices. Folding Core is formed by parts of TM helices III,IV,V; 2 nd  -sheet; and some extracellular loops. residues 9,10,22-27,102-116,166-171,175-180,185- 188,203-207,211* Flexible portions consistent with proposed activation mechanism: movement of cytoplasmic ends of helices III,VI, & VII.

26. Acknowledgements Proteins and Rigidity Claire Vieille Harini Krishnamurthy Ming Lei Maria Zavodszky Mykyta Chubynsky Funding NSF, DOE, and NIH MSU Center for Biological Modeling. CCBB, School of Medicine University of Pittsburgh MSU and FIRST development Michael F. Thorpe Leslie A. Kuhn Don Jacobs Brandon Hespenheide CCBB and Rhodopsin Ivet Bahar Judith Klein-Seetharaman Basak Isin