Approximation of Protein Structure for Fast Similarity

Measures

Itay LotanFabian Schwarzer

Comparing Protein Structures

vs.Same protein:

Analysis of MDS

and MCS

trajectories

http://folding.stanford.edu

Structure prediction applications• Evaluating decoy sets

• Clustering predictions (Shortle et al, Biophysics ’98)

Graph-based Methods

Stochastic Roadmap Simulation (Apaydin et al, RECOMB ’02)

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

(log )O N k L

k Nearest-Neighbors Problem

What if needed for all c in S ?

2 (log )O N k L - too much time

Can be improved by:

1. Reducing L

2. A more efficient algorithm

Our Solution

Reduce structure description

Approximate but fast similarity measures

Efficient nearest-neighbor algorithms can be used

Reduce description further

Description of a Protein’s Structure

3n coordinates of Cα atoms (n – Number of residues)

m-Averaged Approximation Cut chain into pieces of length m Replace each sequence of m Cα

atoms by its centroid

3n coordinates

3n/m coordinates

Similarity Measures - cRMS

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

2

21

1( , ) min

n

T i ii

cRMS P Q p Tqn

Computed in O(n) time

Similarity Measures - dRMS

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

2

2 1

2( , )

( 1)

n iP Qij ij

i j

dRMS P Q d dn n

Computed in O(n2) time

1. Decoy sets: conformations from the Park-Levitt set (Park et al, JMB ’97), N =10,000

2. Random sets: conformations generated by the program FOLDTRAJ (Feldman & Hogue, Proteins ’00), N = 5000

Evaluation: Test Sets

8 structurally diverse proteins of size 54 -76 residues:

Decoy Sets Correlation

m cRMS dRMS

4

6

9

12

3 0.99

0.98 – 0.990.92 – 0.990.81 – 0.980.54 – 0.92

0.96 – 0.980.94 – 0.970.78 – 0.930.65 – 0.960.52 – 0.69Higher Correlation for random

sets!

Speed-up for Decoy Sets

9x for cRMS (m = 9) 36x for dRMS (m = 6)with very small error

For random sets the speed-up for dRMS goes up to 81x (m = 9)

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

Further Dimensionality Reduction of dRMS

kd-trees require dimension 20m-averaging with dRMS is not enough

Reduce further using SVD

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

Reduction Using SVD

1. Stack m-averaged distance matrices as vectors

2. Compute the SVD of entire set3. Project onto most important

singular vectors

dRMS is thus reduced to 20 dimensionsWithout m-averaging SVD can be too costly

Testing the Method

Use decoy sets (N = 10,000) and random sets (N = 5,000)

m-averaging with (m = 4) Project onto 16 PCs for decoys, 12

PCs for random sets Find k = 10, 25, 100 NNs for 250

conformations in each set

Results Decoy sets:

~77% correct Furthest approximate NN off by 10% - 15% ~4k approximate NNs contain all true k NNs

Random sets: 71%, 76%, 84% correct respectively Furthest approximate NN off by 5% - 10% ~3k approximate NNs contain all true k NNs

More Results: N = 100,000

1CTF decoys: ~70% correct Furthest approximate NN off by ~20% ~6k approximate NNs contain all true k NNs

1CTF random: 46%, 48%, 60% correct respectively Furthest approximate NN off by ~16% ~7k approximate NNs contain all true k NNs

Running Time

N = 100,000, m=4, PC = 16

Find k = 100 for each conformation

Brute-force: ~84 hoursBrute-force + m-averaging: ~4.8 hoursBrute-force + m-averaging + SVD: 41

minutesKd-tree + m-averaging + SVD: 19 minutes

kd-trees will have more impact for larger sets

Structural ClassificationComputing the similarity between structures of two different proteins is more involved:

The correspondence problem:

Which parts of the two structures should be compared?

1IRD 2MM1

vs.

STRUCTAL (Subbiah et al, ’93)

1. Compute optimal correspondence using dynamic programming

2. Optimally align the corresponding parts in space to minimize cRMS

3. Repeat until convergence

O(n1n2) time

STRUCTAL + m-averaging 256 protein domains (180 – 420 res) 3691 good matches (Sandelin’s PROTOFARM) 6375 random pairs Compute SAS scores (cRMS/length*100)

m correlation3

4

5

0.81

0.77

0.70

speed-up~9x

~16x~25x

OK (P < 0.005) BAD (P > 0.005)

SAS score

Num

ber

of

pair

s

MisclassificationP ≤ 0.005 P ≤ 0.001

Total FP FN Total FP FN

m=1410(4%)

176 225457

(4.5%)

215 242

m=3802(8%)

274 528660

(6.5%)

262 398

m=4993

(10%)248 745

870(8.5%

)282 588

m=51602(16%)

186 14161129(11%)

148 981

Random Chains

c0c1

c2

c3c4

c5

cn-1c6

c7

c8

The dimensions are uncorrelated Average behavior can be

approximated by normal variables:

1 (0,1)i i N l c c

1-D Haar Wavelet Transform

Recursive averaging and differencing of the values

Level AveragesDetail Coefficients

[ 9 7 2 6 5 1 4 6 ]

[ 8 4 3 5 ]

[ 6 4 ]

[ 5 ]

[ 1 -2 2 -1 ]

[ -2 -1 ]

[ 1 ]

3

2

1

0

[ 9 7 2 6 5 1 4 6 ]

[ 5 1 -2 -1 1 -2 2 1 ]

pdf of the detail coefficients is:

Coefficients expected to be ordered!

Discard coefficients starting at lowest level

Transform of Random Chains

m-averaging

Discarding lowest levels of detail coeeficients

logm

( ) 0, 4j jd N O

Random Chains and Proteins

Protein backbones behave on average like random chains Chain topology Limited compactness

Conclusion

Fast computation of similarity measures

Trade-off between speed and precision Exploits chain topology of proteins and

limited compactness Allows use of efficient nearest-neighbor

algorithms Can be used as filter when precision is

important