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Protein Structural Prediction
Performance of Structure Prediction
Methods
TRILOGY: Sequence–Structure Patterns
•
•
Identify short sequence–structure patterns 3 amino acids
Find statistically significant ones (hypergeometric distribution)

Correct for multiple trials
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These patterns may have structural or functional importance
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2.
Pseq:
Pstr:
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Start with short patterns of 3 amino acids
R1xa-bR2xc-dR3
3 C – C distances, & 3 C – C vectors
{V, I, L, M}, {F, Y, W}, {D, E}, {K, R, H}, {N, Q}, {S, T}, {A, G, S}
•
Extend to longer patterns
Bradley et al. PNAS 99:8500-8505, 2002
TRILOGY
TRILOGY: Extension
Glue together two 3-aa patterns that overlap in 2 amino acids
P-score = i:Mpat,…,min(Mseq, Mstr) C(Mseq, i) C(T – Mseq, Mstr – i) C(T, Mstr)-1
NAD/RAD binding motif
found in several folds
-- unit found in three
proteins with the TIMbarrel fold
Helix-hairpin-helix
DNA-binding motif
TRILOGY: Longer Patterns
Type-II  turn between
unpaired  strands
Four Cysteines forming 4
S-S disulfide bonds
A fold with repeated
aligned -sheets
Three strands of an antiparallel -sheet
A -hairpin connected
with a crossover to
a third -strand
Small Libraries of Structural
Fragments for Representing Protein
Structures
Fragment Libraries For Structure Modeling
known
structures
fragment
library
…
protein
sequence
predicted
structure
Small Libraries of Protein Fragments
Kolodny, Koehl, Guibas, Levitt, JMB 2002
Goal:
Small “alphabet” of protein structural fragments that can be used to represent
any structure
1.
2.
3.
Generate fragments from known proteins
Cluster fragments to identify common structural motifs
Test library accuracy on proteins not in the initial set
f
Small Libraries of Protein Fragments
Dataset: 200 unique protein domains with most reliable & distinct structures from SCOP

•
36,397 residues
Divide each protein domain into consecutive fragments beginning at random initial
position
Library: Four sets of backbone fragments

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4, 5, 6, and 7-residue long fragments
Cluster the resulting small structures into k clusters using cRMS, and applying k-means
clustering with simulated annealing


f
Cluster with k-means
Iteratively break & join clusters with simulated annealing to optimize total variance Σ(x – μ)2
Evaluating the Quality of a Library
•
Test set of 145 highly reliable protein
structures (Park & Levitt)
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Protein structures broken into set of
overlapping fragments of length f
•
Find for each protein fragment the most
similar fragment in the library (cRMS)
Local Fit: Average cRMS value over all
fragments in all proteins in the test set
Global Fit: Find “best” composition of
structure out of overlapping fragments
 Complexity is O(|Library|N)
 Greedy approach extends the C best
structures so far from pos’n 1 to N
Results
C=
Protein Side-Chain Packing
• Problem: given the backbone
coordinates of a protein, predict the
coordinates of the side-chain atoms
• Method: decompose a protein
structure into very small blocks
Slide credits: Jimbo Xu
Protein Structure Prediction
• Stage 1: Backbone
Prediction
 Ab initio folding
 Homology modeling
 Protein threading
• Stage 2: Loop
Modeling
• Stage 3: Side-Chain
Packing
• Stage 4: Structure
Refinement
The picture is adapted from http://www.cs.ucdavis.edu/~koehl/ProModel/fillgap.html
Slide credits: Jimbo Xu
Side-Chain Packing
0.3
0.2
0.3
0.7
0.1
0.4
0.1
0.1
0.6
clash
Each residue has many possible side-chain positions
Each possible position is called a rotamer
Need to avoid atomic clashes
Slide credits: Jimbo Xu
Energy Function
Assume rotamer A(i) is assigned to
residue i. The side-chain packing
quality is measured by
 S (i, A(i))   P(i, j, A(i), A( j))
clash penalty
10
i
clash penalty
0.82 1
occurring preference
The higher the occurring probability,
the smaller the value
d a ,b
ra  rb
d a ,b : distance between two atoms
ra , rb :atom radii
Minimize the energy function to obtain the best side-chain packing.
Slide credits: Jimbo Xu
Related Work
• NP-hard [Akutsu, 1997; Pierce et al., 2002] and NP-complete to
achieve an approximation ratio O(N) [Chazelle et al, 2004]
• Dead-End Elimination: eliminate rotamers one-by-one
• SCWRL: biconnected decomposition of a protein structure [Dunbrack
et al., 2003]
 One of the most popular side-chain packing programs
• Linear integer programming [Althaus et al, 2000; Eriksson et al, 2001;
Kingsford et al, 2004]
• Semidefinite programming [Chazelle et al, 2004]
Slide credits: Jimbo Xu
Algorithm Overview
• Model the potential atomic clash relationship
using a residue interaction graph
• Decompose a residue interaction graph into
many small subgraphs
• Do side-chain packing to each subgraph almost
independently
Slide credits: Jimbo Xu
Residue Interaction Graph
h
b
s
m
a
e
l
Vertices:
Each residue is a vertex
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Edges:
Two residues interact if there
is a potential clash between
their rotamer atoms
f
d
c
•
k
i
j
Residue Interaction Graph
Slide credits: Jimbo Xu
Key Observations
• A residue interaction graph is a geometric neighborhood graph
 Each rotamer is bound to its backbone position by a constant distance
 No interaction edge between two residues if distance > D
• D: constant depending on rotamer diameter
• A residue interaction graph is sparse!
Slide credits: Jimbo Xu
Tree Decomposition
[Robertson & Seymour, 1986]
• Definition. A tree decomposition (T, X) of a graph G = (V, E):
 T=(I, F) is a tree with node set I and edge set F
 X is a set of subsets of V, the components; Union of elts. in X = V
 1-to-1 mapping between I and X
 For any edge (v,w) in E, there is at least one X(i) in X s.t. v, w are in X(i)
 In tree T, if node j is on the path from i to k, then X(i) ∩ X(k)  X(j)
• Tree width is defined to be the maximal component size minus 1
Slide credits: Jimbo Xu
Tree Decomposition
[Robertson & Seymour, 1986]
Greedy: minimum degree heuristic
b
f
d
c
h
m
a
c
e
l
1.
2.
3.
4.
5.
k
i
j
f
d
abd
g
h
g
m
a
e
l
i
j
k
Choose the vertex with minimal degree
The chosen vertex and its neighbors form a component
Add one edge to any two neighbors of the chosen vertex
Remove the chosen vertex
Repeat the above steps until the graph is empty
Slide credits: Jimbo Xu
Tree Decomposition (Cont’d)
h
b
f
d
c
g
m
a
e
l
Tree Decomposition
k
abd
i
acd
clk
cdem
fg
h
defm
eij
j
Tree width:
size of maximal component – 1
Slide credits: Jimbo Xu
Side-Chain Packing Algorithm
Xir
Xr
Xq
Top-to-Bottom:
Extract the optimal assignment
Xi
Xp
Xji
Xli
Xj
Bottom-to-Top:
Calculate the minimal energy function
Xl
Time complexity:
Exponential in tree width, linear in
graph size
A tree decomposition rooted at Xr
Score of component Xi
F ( X i , A( X ir )) 
min F ( X
A( X i  X r )
Score of subtree rooted at Xi
j
, A( X ji ))  F ( X l , A( X li ))  Score( X i , A( X i ))
Score of subtree rooted at Xl
Score of subtree rooted at Xj
Slide credits: Jimbo Xu
Empirical Component Size Distribution
Tested on the 180 proteins used by SCWRL 3.0.
Components with size ≤ 2 ignored.
Slide credits: Jimbo Xu
Result
) << 
Theoretical time complexity: O( N
 is the average number rotamers for each residue.
N 2 / 3 log N
N
CPU time (seconds)
protein
size
SCWRL
SCATD
speedup
1gai
472
266
3
88
1a8i
812
184
9
20
1b0p
2462
300
21
14
1bu7
910
56
8
7
1xwl
580
27
5
5
Five times faster on
average, tested on 180
proteins used by
SCWRL
Same prediction
accuracy as SCWRL
3.0
Slide credits: Jimbo Xu
Accuracy
1
0.95
0.9
0.85
0.8
0.75
SCATD
SCWRL
0.7
0.65
0.6
0.55
0.5
ASN ASP CYS
HIS
ILE
SER TYR VAL
A prediction is judged correct if its deviation from
the experimental value is within 40 degree.
Slide credits: Jimbo Xu