Protein Structure Space

Patrice Koehl

Computer Science and Genome Center

http://www.cs.ucdavis.edu/~koehl/

From Sequence to Function

KKAVINGEQIRSISDLHQTLKKWELALPEYYGENLDALWDCLTGVEYPLVLEWRQFEQSKQLTENGAESVLQVFREAKAEGCDITI

Sequence

Structure

Function

ligand

Protein Structure Space1CTF 1TIM 1K3R

1A1O 1NIK 1AON

68 AA 247 AA 268 AA

384 AA 4504 AA 8337 AA

Outline

•Protein Shape DescriptorsDifferential Geometry Tools

•Classifying Proteins

The Shapes of Protein Structures

•Protein Structure SpaceDimension?

•Complexity of Protein Structures

Are Proteins 3D, or 1D objects?

Outline

•Protein Shape DescriptorsDifferential Geometry Tools

•Classifying Proteins

The Shapes of Protein Structures

•Protein Structure SpaceDimension?

•Complexity of Protein Structures

Are Proteins 3D, or 1D objects?

Classification of Protein Structure: CATH

C

A

T

AlphaMixed AlphaBeta

Beta

Sandwich

Tim BarrelOther Barrel

Super RollBarrel

Protein Structure Space

Test set2,930 proteins out of 23,000 proteins in PDBNo sequence similarity (Fasta E-value < e-4)

Reference structural similarity defined from CATH769 folds

104,000 pairs of similar structures out of 4,600,000 pairs

Performance measure: ROC curve(Receiver Operating Characteristic)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0...

...0...

...0

1

1

N

N

d

d

D XXG T= X

Distance Matrix Metric Matrix Points in Space

Projecting Protein Structure Space

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0...1

...01

110

D

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0...0

...01

010

D

k

k

k

k

Class

Fold

Projecting Protein Structure Space

Protein Structure Similarity

Root mean square distance: cRMS:

( ) ∑=

−−=N

iii TRba

NBAcRMS

1

21,

N: number of equivalent atoms between A and BR, T: rigid transformation that minimizes cRMS.

Eigenvalues of the Metric Matrix:

Protein Structure Classes

Measure of Structure Similarity:cRMS after Optimal Superposition(Structal)

A Picture of the Protein Structure Space

Proteins

Proteins

α and Proteins

A Picture of the Protein Structure Space

1a81G2

1sfcK0

1repC21bdo00

2bi6H0

Proteins

Proteins

α and Proteins

Outline

•Protein Shape DescriptorsDifferential Geometry Tools

•Classifying Proteins

The Shapes of Protein Structures

•Protein Structure SpaceDimension?

•Complexity of Protein Structures

Are Proteins 3D, or 1D objects?

0 10 20 30 40 50 60 70 80 90 100

Rate of true negatives (%)

100

90

80

70

60

50

30

20

Rat

e of

true

pos

itive

s (%

)

40

10

Random measureArea = 0.5

“Perfect” measureArea = 1.0

Protein Fold SpaceROC Analysis

(Receiver Operating Characteristic)

Protein Fold SpaceROC Analysis

(Receiver Operating Characteristic)

True positives

pairs of proteins that belong to the same T class of CATH

True negatives

pairs of proteins that belong to the same C class, but not thesame T class.

Rate of true negatives (%)

Rat

e of

true

pos

itive

s (%

)

Protein Fold Space

Fasta: 0.54

CATH Class : 0.51

CATH Fold20 : 0.98 Fold20: first 20coordinates derivedfrom the CATHfold matrix

CATH class: first 3coordinates derivedfrom the CATHclass matrix

Rate of true negatives (%)

Rat

e of

true

pos

itive

s (%

)

Protein Fold Space

Fasta: 0.54Fasta: 0.54

Structal: 0.88

Protein Structure Features

()zyxzyxˆsin2),(),,( dR =

{ }),,(min)(),(

zyxxzy

R=ρ

y

x

z

R(x,y,z)

{ })(min xx

ρ=Δ

Global radius of curvature:

Thickness:

(Gonzalez & Maddocks, PNAS, 1999, 96:4769)

Thickness of a protein structure

= 2.60 Ǻ

Curvature Feature Vector

p

zyxpp dCdCdCzyxR

U/1

),,(

1⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫∫∫

[ ]543215 UUUUUC =

Performance of the Curvature Feature Vector

Curvature vector performs better than fasta.

Needs more features to match Structal.

Rate of true negatives (%)

Rat

e of

true

pos

itive

s (%

)

Fasta: 0.54

Structal: 0.88

C5: 0.65

Protein Structure Features: Writhing

∫∫=2 2121 ),(

4

1dtdtttWr ω

π

Fain and Røgen, PNAS, 100: 119 (2003)

( )3

21

121121

)()(

)('),()(),('det),(

tt

tttttt

γγ

γγγγω

−

−=

Writhing Number

+ -

Sign of Crossing

(t1)

(t2)

[ ]|||| 12121110 WrWrWrWrW =

Writhe Feature Vector for Each Protein

1

Protein Structure Features: Writhing

W10 Writhe performs better than C5 Curvature

Rat

e of

true

pos

itive

s (%

)

Rate of true negatives (%)

Fasta: 0.54

C5: 0.65

Structal: 0.88

W10: 0.77

Outline

•Protein Shape DescriptorsDifferential Geometry Tools

•Classifying Proteins

The Shapes of Protein Structures

•Protein Structure SpaceDimension?

•Complexity of Protein Structures

Are Proteins 3D, or 1D objects?

Clustering Protein Fragments to Extract a Small Set of Representatives (a Library)

data clustereddata

library

(Simulated annealing K means)

Generating an approximate structure

Fragment library

A B C D

Generating an approximate structure

Fragment library

A B C D

Fragment library

Generating an approximate structure

A B C D

Fragment library

Generating an approximate structure

A B C D

Fragment library

Generating an approximate structure

A B C D

Structural Sequence:

AC

Fitting Protein Structures

better

100 fragments of length 50.91 Ǻ cRMS

50 fragments of length 72.78 Ǻ cRMS

Complexity(states/residue)

Ave

rage

cR

MS

dis

tan

ce

Longer fragments give better fit at same complexity

Fragment Size:7 residues6 residues5 residues4 residues

(Kolodny, Koehl, Guibas, Levitt, J. Mol. Biol.,323, 297 2002)

3

1

−= LNC N: number of fragmentsL: size of each fragment

Choosing the “right” library

Size L N such that Complexity=20

7 160000

6 8000

5 400

4 20

cRMS model-experimental structure cRMS model-experimental structure

0.2 0.6 1.0 0.2 0.6 1.0

# of

str

uct

ure

sA Structural Alphabet for Protein Backbone

Pro

tein

siz

e

Fragment size: 4Number of fragment: 20

Structural Alphabet: Application to Structure Comparison

cRMS = 1Å

Collaborators• Michael Levitt

(Computational Biology)

Stanford University

• Marc Delarue

(Biophysics)

Institut Pasteur, Paris

• Rachel Kolodny

(Computer Science)

Columbia University

• Herbert Edelsbrunner

(Math/Computer Science)

Duke University

• Peter Roegen

(Math)

DTU, Denmark

• Joel Hass

(Math)

UC Davis

Thank You