Approximation of Protein Structure for Fast Similarity Measures

1
Approximation of Protein Structure for Fast
Similarity Measures

• Fabian Schwarzer
• Itay Lotan
• Stanford University

2
Comparing Protein Structures
Same protein
vs.
Analysis of MDS and MCS trajectories
Graph-based methods
Structure prediction applications

• Evaluating decoy sets
• Clustering predictions (Shortle et al,
  Biophysics 98)

Stochastic Roadmap Simulation (Apaydin et al,
RECOMB 02)
http//folding.stanford.edu
3
k Nearest-Neighbors Problem
Given a set S of conformations of a protein and a
query conformation c, find the k conformations
in S most similar to c.
Can be done in N
size of S L time to compare two conformations
4
k Nearest-Neighbors Problem
What if needed for all c in S ?
- too much time

• Can be improved by
• Reducing L
• A more efficient algorithm

5
Our Solution

• Reduce structure description

Approximate but fast similarity measures
Reduce description further
Efficient nearest-neighbor algorithms can be used
6
Description of a Proteins Structure
3n coordinates of Ca atoms (n Number of
residues)
7
Similarity Measures - cRMS

• The RMS of the distances between corresponding
  atoms after the two conformations are optimally
  aligned

Computed in O(n) time
8
Similarity Measures - dRMS

• The Euclidean distance between the
  intra-molecular distances matrices of the two
  conformations

Computed in O(n2) time
9
m-Averaged Approximation

• Cut chain into m pieces
• Replace each sequence of n/m Ca atoms by its
  centroid

3n coordinates
3m coordinates
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Why m-Averaging?

• Averaging reduces description of random chains
  with small error
• Demonstrated through Haar wavelet analysis
• Protein backbones behave on average like random
  chains
• Chain topology
• Limited compactness

11
Evaluation Test Sets
9 structurally diverse proteins of size 38 -76
residues

• Decoy sets conformations from the Park-Levitt
  set (Park Levitt, JMB 96), N 10,000
• Random sets conformations generated by the
  program FOLDTRAJ (Feldman Hogue, Proteins 00),
  N 5000

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Decoy Sets Correlation
0.37 0.73
0.40 0.86
0.84 0.98
0.70 0.94
0.98 0.99
0.92 0.96
0.98 0.99
0.92 0.98
0.98 0.99
0.93 0.97
Higher Correlation for random sets!
13
Speed-up for Decoy Sets

• Between 5X and 8X for cRMS (m 8)
• Between 9X and 36X for dRMS (m 12)
• with very small error

For random sets the speed-up for dRMS was between
25X and 64X (m 8)
14
Efficient Nearest-Neighbor Algorithms

• There are efficient nearest-neighbor algorithms,
  but they are not compatible with similarity
  measures
• cRMS is not a Euclidean metric
• dRMS uses a space of dimensionality
  n(n-1)/2

15
Further Dimensionality Reduction of dRMS

• kd-trees require dimension ? 20
• m-averaging with dRMS is not enough
• Reduce further using SVD

SVD A tool for principal component analysis.
Computes directions of greatest variance.
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Reduction Using SVD

• Stack m-averaged distances matrices as vectors
• Compute the SVD of entire set
• Project onto most important singular vectors

dRMS is thus reduced to ?20 dimensions
Without m-averaging SVD can be too costly
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Testing the Method

• Use decoy sets (N 10,000)
• m-averaging with (m 16)
• Project onto 20 largest PCs (more than 95 of
  variance)
• Each conformation represented by 20 numbers

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Results

• For k 10, 25, 100
• Decoy sets 80 correct furthest
  NN off by 10 - 20 (0.7Å 1.5Å)
• 1CTF, with N 100,000 ? similar results
• Random sets ? 90 correct with smaller error (5
  - 10)

When precision is important use as pre-filter
with larger k than needed
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Running Time

• N 100,000
• k 100, for each conformation

Brute-force
84 hours Brute-force m-averaging
4.8 hours Brute-force m-averaging SVD 41
minutes Kd-tree m-averaging SVD 19
minutes
kd-trees will have more impact for larger sets
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Structural Classification
Computing the similarity between structures of
two different proteins is more involved
2MM1
1IRD
vs.
The correspondence problem Which parts of the
two structures should be compared?
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STRUCTAL (Gerstein Levitt 98)

• Compute optimal correspondence using dynamic
  programming
• Optimally align the corresponding parts in space
  to minimize cRMS
• Repeat until convergence

O(n1n2) time
Result depends on initial correspondence!
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STRUCTAL m-averaging

• Compute similarity for structures of same SCOP
  super-family with and without m-averaging

correlation
speed-up
n/m
3
0.60 0.66
7
0.44 0.58
19
5
8
0.35 0.57
46
NN results were disappointing
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Conclusion

• Fast computation of similarity measures
• Trade-off between speed and precision
• Exploits chain topology and limited compactness
  of proteins
• Allows use of efficient nearest-neighbor
  algorithms
• Can be used as pre-filter when precision is
  important

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Random Chains
c5
c7
c2
c6
c8
cn-1
c0
c4
c1
c3

• The dimensions are uncorrelated
• Average behavior can be approximated by normal
  variables

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1-D Haar Wavelet Transform

• Recursive averaging and differencing of the values

Detail Coefficients
Level
Averages
9 7 2 6 5 1 4 6
3
2
8 4 3 5
1 -2 2 -1
1
6 4
-2 -1
0
5
1
9 7 2 6 5 1 4 6
5 1 -2 -1 1 -2 2 1
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Haar Wavelets and Compression
When discarding detail coefficients the
approximation error is the root of the sum of the
squares of the discarded coefficients

• Compress by discarding smallest coefficients

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Transform of Random Chains
For random chains the pdf of the detail
coefficients is Coefficients expected to be
ordered!
Discard coefficients starting at lowest level
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Random Chains and Proteins