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

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

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.

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.

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.

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.

Andy Hanson, Indiana University

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

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.

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

Reference Frames The frame has both translational and rotational aspects.

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

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

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

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

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

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

Local Helical Axes

Quaternion Difference Map

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

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.

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

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

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

select *; color straightness

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

Quaternion-Based Straightness

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

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

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

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

Helix residuesSheet residuesUnstructured residues Total average C-straightness , σ = , σ = , σ = Total average P-straightness , σ = , σ = , σ = [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.] Quaternion-Based Straightness For the entire PDB database, straightness correlates well with DSSP-calculated secondary structure.

PDB IDC- straightness P- straightness Description 2HI Aberrant bonds between carbonyl oxygen and peptide nitrogen atoms 1NH Aberrant bonds between carbonyl oxygen atoms 1KIL Helix designation missing 3FX Problem with helix connection designations 3HEZ0.8444Not calculable Disconnected helix fragments [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.] Quaternion-Based Straightness Anomalies – very high straightness for “unstructured” groups

Twenty Common Amino Acids Amino acidTotal average C-straightness Amino acidTotal average C-straightness ILE0.7325CYS LEU0.7257TYR VAL0.7215LYS ALA0.7192THR MET0.7149HIS GLU0.7000SER GLN0.6967ASP TRP0.6860ASN ARG0.6839PRO PHE0.6802GLY0.5315

Twenty Common Amino Acids Amino acidTotal average C-straightness Amino acidTotal average C-straightness ILE0.7325CYS LEU0.7257TYR VAL0.7215LYS ALA0.7192THR MET0.7149HIS GLU0.7000SER GLN0.6967ASP TRP0.6860ASN ARG0.6839PRO PHE0.6802GLY0.5315

Twenty Common Amino Acids Amino acidTotal average C-straightness Amino acidTotal average C-straightness ILE0.7325CYS LEU0.7257TYR VAL0.7215LYS ALA0.7192THR MET0.7149HIS GLU0.7000SER GLN0.6967ASP TRP0.6860ASN ARG0.6839PRO PHE0.6802GLY0.5315

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.

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.

Visualization Can Drive Research Future directions: –Natural extension to nucleic acids

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

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

Acknowledgments Andrew Hanson, Indiana University Howard Hughes Medical Institute Jmol user community