Protein Planes

Bob Fraser

Protein Folding 882 Project

November, 2006

Overview

• Motivation

• Points to examine

• Preliminary results

• Further work

Cα trace problem

• Given: only approximate positions of the Cα atoms of a protein

• Aim: Construct the entire backbone of the protein– This is an open problem!

Cα trace problem

• Why do it?

• Some PDB files contain only Cα atoms.

• More importantly, many predictive approaches are incremental, and begin by producing the Cα trace.

Cα trace problem

• Possible solutions:– De novo, CHARMM fields (Correra 90)– Fragment matching (Levitt 92)– Maximize hydrogen bonding (Scherega et al.

93)

• If we know the dimensions corresponding to the peptide plane, why can’t we just fit these to the Cα?

Cα trace problem

• Known as idealized covalent geometry– Used by Engh & Huber (91) for X-ray

crystallography refinement– Supplemented by including dihedral

information (Payne 93, Blundell 03)

• All methods achieve <1Å rmsd, ~0.5Å rmsd is good.

• Perhaps including more information about the plane could further improve results.

The peptide plane

NOTE! 1 plane ~=1 residue

The peptide plane

• We know that these dimensions are very regular…

but how regular?

Can I depend on these dimensions for predictions where accuracy is important and errors may be cumulative?

The idea

• Cα form the connectors at the corners of each plane.

• Given a regular length of the plane, we could fit all the bond lengths and angles to be ideal.

The task

• Survey the structures in the PDB, and determine how close the known structures adhere to these values.

• There is a question regarding what this is actually measuring, but this is our standard knowledge base.– Could think of it as verifying the PDB

structures.– When we find irregularities, try to find why

What to measure?

• Length of plane (Cα – Cα distance)

• Bond lengths

• Bond angles

• Dihedral angles (Φ,Ψ,ω)

• Angle between helix axis and plane

• Beta axis and plane angle?

Length of plane (Cα – Cα distance)

• The so-called bond distance when given a Cα trace.

• If all bond angles and lengths are fixed, this distance should also be constant.

• Let’s check this distance in the PDB, and determine the average, standard deviation, maximum and minimum values found.

Bond lengths

• These values should be considered relatively fixed.

• Variance in bonds is more likely attributable to measurement error

• Let’s measure anyhow to see what we find

• Measure for :

N-Cα , Cα-C, C-O, C-N

Bond angles

• These values should be considered relatively fixed, as with lengths

• Measure for :

• N-Cα-C

• Cα-C-O

• Cα-C-N

• O-C-N

• C-N-Cα

Aside: Forces equation

Dihedral Angles

• This is where it gets more interesting!

• For the peptide plane, we are particularly interested in omega (ω).

• For planes, expect only values close to 180° for trans configuration, and 0° for cis configuration.

• Let’s check the PDB files to verify this

Dihedral Angle

• Given: 4 points

• To find: Determine how close the points are to co-planarity

• Use 3 points to define a plane, the 4th forms a vector with one of the first 3.

• cis and trans configurations are co-planar, we’ll see how many others we find.

Omega

• Measure absolute difference from cis/trans• We’ll call anything within 15° as in those

classes.

C

O

Cα

N

Cα

N-C

A = N-C X C-Cα A

ω

N-C

B = Cα-N X N-CB

Phi and Psi

• These angles can vary widely, as shown in the Ramachandran plot.

• It could be interesting to do this survey again however if time permits however…

Angle between helix axis and plane

• It is assumed that the planar regions for amino acids in a helix are parallel to the axis of the helix.

• Let’s put this to the test!

• How do we measure the axis of helix?– It is a subjective measure– We’ll use the method of Walther et al. (96), it

provides a local helix axis

Walther axis calculation

Plane-axis angle

• Now we have a peptide plane and the helix axis, so we can once again find the angle between them easily.

• This same method could be applied to beta strands fairly easily.

• We should expect that some pattern should arise since beta strands are have regular patterns, particularly when in beta sheets.

Data Analysis

• Use the entire PDB database as a source, and a subset of 159 proteins for the preliminary study.

• Several choices for the parsing method– My previous code– BALL– Matlab’s pdbread() function

Preliminary Results

• Plane Length

• Quite a bit of variance…• Why?

cis/trans

• The length of the plane is quite different between the two!

• So, we can treat them as two cases:

cis trans

Bond lengths

N-Cα Cα-C C-O C-N

Average 1.4553 1.5214 1.2313 1.3278

Std.Dev. 0.0171 0.0184 0.0125 0.0132

Instances 38325 38325 38325 37848

Minimum 1.3453 1.3975 1.1633 1.1683

Maximum 2.7398 3.0729 2.1397 1.8167

Textbook 1.45 1.52 1.23 1.33

Bond angles

N-Cα-C Cα-C-O Cα-C-N O-C-N C-N-Cα

Average 111.31 120.8 116.33 122.79 121.42

Std.Dev. 3.374 1.4482 1.7595 1.437 2.1339

Instances 38325 38325 37848 37848 37848

Minimum 56.87 61.127 99.355 98.651 80.913

Maximum 171.09 130.04 140.04 136.46 142.59

Textbook 111 121 116 123 122

Omega

cis trans Other

Average 1.8748 178.17 155.74

Std.Dev. 3.234 1.8835 23.205

Instances 61 37698 89

Notes: - [Other]>[cis] - all cis have Proline residues, though 1322 Prolines total

The most offensive residue!!!

Effects of deleting this residue

• Averages not affected, only min/max and standard deviation, ie.:

• Cα-C-O angle: σ 1.44->1.14, min 61.1->94.6

• N-Cα-C angle: min 56.9->83.6

• C-O bond: max 2.14->1.34• Cα-C bond: max 3.07->2.75• Plane length: max 5.02->4.43• It was in the trans class for ω, however.

Preliminary Results

• Data is very consistent in PDB, and support the theoretical values.

• Using such values should be acceptable.

• Tough to predict cis configuration though!– Pro is always involved (so far), but other

residue is mixed (19 Leu, 7 Tyr, etc.). – The difference is in the length of the plane.

Future Work

• Incorporate secondary structure information for determining axis/plane angles and phi/psi angles.

• Run analysis on full PDB

• Develop algorithm for using secondary structure to solve trace problem.

• Test on randomized Cα traces to determine whether it is effective.

Thanks!

Selected References– M.A. DePristo, P.I.W. de Bakker, R.P. Shetty, and T.L. Blundell.

Discrete restraint-based protein modeling and the C -trace problem. Protein Science, 12:2032-2046, 2003.

– A. Liwo, M.R. Pincus, R.J. Wawak, S. Rackovsky, and H.A. Scheraga. Calculation of protein backbone geometry from alpha-carbon coordinates based on peptide-group dipole alignment.

Protein Sci., 2(10):1697-1714, 1993. – G.A. Petsko and D. Ringe. Protein Structure and Function. New

Science Press Ltd, London, 2004.– D. Walther, F. Eisenhaber, and P. Argos. Principles of helix-helix

packing in proteins: the helical lattice superimposition model. J.Mol.Biol., 255: 536-553, 1996.