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 MCStrajectories 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 cin 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 Protein’s Structure 3n coordinates of Cα 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 Cα atoms by its centroid 3n coordinates 3m coordinates

10. 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 • 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 9 structurally diverse proteins of size 38 -76 residues:

12. dRMS m cRMS 4 8 12 16 20 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.

16. 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

17. 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

18. 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

19. 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

20. 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?

21. 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!

22. 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

23. 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

24. Random Chains c5 c7 • The dimensions are uncorrelated • Average behavior can be approximated by normal variables: c2 c6 c8 cn-1 c0 c4 c1 c3

25. 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 [ 1 ] [ 5 ] [ 9 7 2 6 5 1 4 6 ] [ 5 1 -2 -11 -2 2 1]

26. 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

27. Discarding lowest levels of detail coeeficients m-averaging  (m = 2v) 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

28. Random Chains and Proteins