Use of quaternions in biomolecular structure analysis

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

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






• 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.

2/

||arccos1)( 1

ii dqdq

is


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

2/

||arccos1)( 1

ii dqdq

is

• In fact, in quaternion algebra, the distance between two quaternions can be expressed in terms of the quaternion second derivative:


2/

|2/|1)( 2

is

2/

||arccos1)( 1

ii dqdq

is

• So our definition of straightness is just a simple quaternion measure:


||

1)( 2is


• select *; color straightness


• select not helix and not sheet and straightness > 0.85; color 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 = 0.9997.]

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

||

1)( 2is

],[112 )( iiii

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

Helix residues Sheet residues Unstructured residues

Total average C-straightness

0.8526, σ = 0.2234

0.7697, σ = 0.2210

0.3874, σ = 0.4310

Total average P-straightness

0.8660, σ = 0.1742

0.7326, σ = 0.2181

0.3564, σ = 0.4136

[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.]


For the entire PDB database, straightness correlates well with DSSP-calculated secondary structure.

PDB ID C-straightness

P-straightness

Description

2HI5 0.9528 0.9210 Aberrant bonds between carbonyl oxygen and peptide nitrogen atoms

1NH4 0.9517 0.9440 Aberrant bonds between carbonyl oxygen atoms

1KIL 0.9142 0.9102 Helix designation missing

3FX0 0.9037 0.8086 Problem with helix connection designations

3HEZ 0.8444 Not 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.]


Anomalies – very high straightness for “unstructured” groups

Twenty Common Amino Acids

Amino acid Total average C-straightness


ILE 0.7325 CYS 0.6779

LEU 0.7257 TYR 0.6727

VAL 0.7215 LYS 0.6695

ALA 0.7192 THR 0.6500

MET 0.7149 HIS 0.6492

GLU 0.7000 SER 0.6321

GLN 0.6967 ASP 0.6270

TRP 0.6860 ASN 0.6161

ARG 0.6839 PRO 0.5444

PHE 0.6802 GLY 0.5315




ILE 0.7325 CYS 0.6779

LEU 0.7257 TYR 0.6727

VAL 0.7215 LYS 0.6695

ALA 0.7192 THR 0.6500

MET 0.7149 HIS 0.6492

GLU 0.7000 SER 0.6321

GLN 0.6967 ASP 0.6270

TRP 0.6860 ASN 0.6161

ARG 0.6839 PRO 0.5444

PHE 0.6802 GLY 0.5315




ILE 0.7325 CYS 0.6779

LEU 0.7257 TYR 0.6727

VAL 0.7215 LYS 0.6695

ALA 0.7192 THR 0.6500

MET 0.7149 HIS 0.6492

GLU 0.7000 SER 0.6321

GLN 0.6967 ASP 0.6270

TRP 0.6860 ASN 0.6161

ARG 0.6839 PRO 0.5444

PHE 0.6802 GLY 0.5315


• The bottom line:



• The bottom line:


– Quaternion-based straightness offers a simple quantitative measure of biomolecular structure.


• Future directions:

– Natural extension to nucleic acids



– Natural extension to nucleic acids– Define “motifs” based on quaternions



– 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

[email protected]

http://Jmol.sourceforge.net

mailto:[email protected]

http://jmol.sourceforge.net/

Documents

Use of quaternions in biomolecular structure analysis