Multiple Structure Alignment by Optimal RMSD Implies that the Average Structure is a Consensus

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Multiple Structure Alignment by Optimal RMSD
Implies that the Average Structure is a Consensus

• Xueyi Wang and Jack Snoeyink
• Department of Computer Science
• UNC - Chapel Hill

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Outline

• Multiple structure alignment
• Average structure and wRMSD
• Minimizing wRMSD
• Results

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Protein Structure is Important

• Structures are better conserved than sequences.
• Similar structures may have similar functions.
• Protein structures increase exponentially.

Protein number
Year
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Uses of Structure Alignment

• Pairwise structure alignment
• -- Search for homologous protein
• -- Protein structure classification
• Multiple structure alignment
• -- Identify structural conserved region
• -- Provide clues for building evolutionary tree
• -- Determine consensus structure for protein
  families

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Multiple Structure Alignment

• Structure abstracted as points or vectors
• Optimization methods
• -- Some methods do pairwise structure alignment
  first, then use heuristics to integrate multiple
  structures.
• -- Some methods align all structures at once.
• Target functions usually some form of RMSD
  extended from RMSD for pairwise alignment

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Outline

• Multiple structure alignment
• Average structure and wRMSD
• Minimizing wRMSD
• Results

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Average Structure and wRMSD

• Definitions of average structure and RMSD.
• Four properties of the average structure --
• Average is the best consensus!
• wRMSD.

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Definitions

• n structures (proteins) S1, S2, , Sn
• m points (atoms) in each structure with
  correspondence, e.g. Si has pi1, pi2, , pim.
• Average structure has points
• RMSD ,
• or SD

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Property 1

• For any aligned position k, the SD from all
  structures to any point qk equals the SD to the
  average point plus the SD from to qk

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Property 2

• The SD for all pairs equals SD to the average
  structure

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Property 3

• For all n structures, the SD to the average
  structure is less than the SD to any other
  structure.

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Property 4

• The best consensus of true structure is the one
  with the minimum SD to the average structure .

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wRMSD

• Add Position weight wk ? 0 on each aligned
  position k.
• Weighted RMSD
• All the properties still hold.

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Outline

• Multiple structure alignment
• Average structure and wRMSD
• Minimizing wRMSD
• Results

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Minimizing wRMSD

• Translate and Rotate structures in 3D space.
• For structure Si, define Ri as a 3?3 rotation
  matrix and Ti as a 3?1 translation vector.
• Target function

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Algorithm
1. Translate all the structures to the origin. 2.
Compute the average structure and the
wRMSD. 3. Align each structure Si to by
Horns method. 4. Compute new average structure
and wRMSDnew. 5. If wRMSD wRMSDnew lt ?,
the algorithm terminates otherwise, set
and wRMSD wRMSDnew, go to step 3.
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Algorithm

• wRMSD decreases in each iteration.
• Time complexity O(n m) for each iteration, where
  n is the number of structures and m is the number
  of points in each structure.
• is maximized when the algorithm ends.
• Implemented using MATLAB.

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Outline

• Multiple structure alignment
• Average structure and wRMSD
• Minimizing wRMSD
• Results

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Performance Test

• Test set 23 protein families from HOMSTRAD, w/
  10 structures and total aligned length 100.
• All 5,000 tests for each family converge to the
  same local minima, where each begin with random
  rotations.

• Maximum iterations 6.
• Maximum running time 40 milliseconds.

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Consensus Structure
grs (pyridine nucleotide-disulphide
oxidoreductases class)
all 11 aligned proteins
consensus structure
Structure with minimum RMSD
Structure with maximum RMSD
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Determine Conserved Core

• Multiple structure alignment by optimal RMSD has
  outlier effects.
• Solution -- Another iterative algorithm based on
  the previous algorithm
• -- Align all the structures by previous
  algorithm.
• -- Find the aligned position that the average
  distance to the average point are larger than a
  threshold, and set the weights to zero.

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Conserved Core
grs (pyridine nucleotide-disulphide
oxidoreductases class)
proteasome A-type and B-type
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Conclusions and Future Work

• Conclusions
• -- Average structure is the consensus.
• -- Algorithm for minimizing wRMSD.
• -- Algorithm for finding conserved region.
• Future work
• -- Minimizing RMSD for gapped multiple structure
  alignment (submitted).

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Acknowledgements

• Jun (Luke) Huan.
• UNC Bioinformatics and Computational Biology
  training fellowship.
• NIH grant GM-074127.

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The End