Use of quaternions in biomolecular structure analysis

1. Use of quaternions in biomolecular structure analysis Robert M. Hanson, Daniel Kohler, and Steven Braun Department of Chemistry, St. Olaf College Northfield, MN 55057 August 19, 2009 238th ACS National Meeting Washington, DC

2. Protein Secondary Structure • My research interest is in describing, visualizing, and quantifying protein and nucleic acid secondary structure, particularly in relation to substrate binding.

3. Protein Secondary Structure • As the current principal developer and project manager of the Jmol molecular visualization project, I get requests periodically for new visualization ideas.

4. The Jmol Molecular Visualization Project • As the current principal developer and project manager of the Jmol molecular visualization project, I get requests periodically for new visualization ideas.

5. The Jmol Molecular Visualization Project • As the current principal developer and project manager of the Jmol molecular visualization project, I get requests periodically for new visualization ideas.

6. The Jmol Molecular Visualization Project • As the current principal developer and project manager of the Jmol molecular visualization project, I get requests periodically for new visualization ideas.

7. Andy Hanson, Indiana University

8. Outline • Reference Frames • Quaternions • Local Helical Axes • Quaternion-Based “Straightness”

9. Visualization Can Drive Research • The main point: • Sometimes a good visualization can lead to interesting findings in basic research that otherwise simply would not be considered.

10. Reference Frames • The basic idea is that each amino acid residue can be assigned a “frame” that describes its position and orientation in space.

11. Reference Frames • The frame has both translational and rotational aspects.

12. Quaternion Frames • A quaternion is a set of four numbers. • Unit quaternions can describe rotations.

13. Quaternion Frames • The choice of frame is (seemingly) arbitrary. “P” “C” “N”

14. Local Helical Axes • The quaternion difference describes how one gets from one frame to the next. This is the local helical axis.

15. Local Helical Axes • The quaternion difference describes how one gets from one frame to the next. This is the local helical axis.

16. Local Helical Axes • Strings of local helical axes identify actual “helices.”

17. Local Helical Axes • Sheet strands are also technically helical as well.

18. Local Helical Axes

19. Quaternion Difference Map

20. Straightness • The quaternion differences can be used to unambiguously define how “straight” a helix is.

21. Quaternion-Based Straightness • The dot product of two vectors expresses how well they are aligned. This suggests a definition of “straightness” based on quaternion dot products.

22. Quaternion-Based Straightness • The “arccos” business here just allows us to turn the dot product into a distance measure – on the four-dimensional hypersphere!

23. Quaternion-Based Straightness • In fact, in quaternion algebra, the distance between two quaternions can be expressed in terms of the quaternion second derivative:

24. Quaternion-Based Straightness • So our definition of straightness is just a simple quaternion measure:

25. Quaternion-Based Straightness • select *; color straightness

26. Quaternion-Based Straightness • select not helix and not sheet and straightness > 0.85; color straightness

27. Quaternion-Based Straightness

28. Quaternion-Based Straightness

29. Quaternion-Based Straightness

30. Quaternion-Based Straightness

31. Quaternion-Based Straightness

33. Quaternion-Based P Straightness • We have found several interesting aspects of straightness. Among them are two relationships to well-known “Ramachandran angles.” For P-straightness: where

34. [Figure 5. Correlation of quaternion- and Ramachandran-based P-straightness for protein 2CQO. R² = 0.9997.]

35. Quaternion-Based C Straightness • We have found several interesting aspects of straightness. Among them are two relationships to well-known “Ramachandran angles.” For C-straightness: and

36. [Figure 7. Correlation between quaternion- and Ramachandran-based C-straightness for protein 2CQO. R² ≈ 1.]

37. Quaternion-Based Straightness For the entire PDB database, straightness correlates well with DSSP-calculated secondary structure. [Table 1. Summarizes overall average C-straightness and P-straightness measures for all within(helix), within(sheet), and (protein and not helix and not sheet) residues in the Protein Data Bank.]

38. Quaternion-Based Straightness Anomalies – very high straightness for “unstructured” groups [Table 2. Some structures where overall average straightness is high but labels in the PDB file result in the misappropriation of secondary structure. In this way, straightness can check for errors in PDB files.]

39. Twenty Common Amino Acids

40. Twenty Common Amino Acids

41. Twenty Common Amino Acids

42. Visualization Can Drive Research • The bottom line: • Sometimes a good visualization can lead to interesting findings in basic research that otherwise simply would not be considered.

43. Visualization Can Drive Research • The bottom line: • Sometimes a good visualization can lead to interesting findings in basic research that otherwise simply would not be considered. • Quaternion-based straightness offers a simple quantitative measure of biomolecular structure.

44. Visualization Can Drive Research • Future directions: • Natural extension to nucleic acids

45. Visualization Can Drive Research • Future directions: • Natural extension to nucleic acids • Define “motifs” based on quaternions

46. Visualization Can Drive Research • Future directions: • Natural extension to nucleic acids • Define “motifs” based on quaternions • Extension to molecular dynamics calculations and ligand binding

47. Acknowledgments • Andrew Hanson, Indiana University • Howard Hughes Medical Institute • Jmol user community hansonr@stolaf.edu http://Jmol.sourceforge.net