use of quaternions in biomolecular structure analysis
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
• 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.
2/
||arccos1)( 1
ii dqdq
is
Quaternion-Based Straightness
• 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:
Quaternion-Based Straightness
2/
|2/|1)( 2
is
2/
||arccos1)( 1
ii dqdq
is
• So our definition of straightness is just a simple quaternion measure:
Quaternion-Based Straightness
||
1)( 2is
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 Straightness
Quaternion-Based Straightness
Quaternion-Based 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 = 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.]
Quaternion-Based Straightness
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.]
Quaternion-Based Straightness
Anomalies – very high straightness for “unstructured” groups
Twenty Common Amino Acids
Amino acid Total average C-straightness
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
Twenty Common Amino Acids
Amino acid Total average C-straightness
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
Twenty Common Amino Acids
Amino acid Total average C-straightness
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
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
http://Jmol.sourceforge.net