Identifying Structural Motifs in Proteins

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Identifying Structural Motifs in Proteins

• Rohit Singh
• Joint work with Mitul Saha

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The Big Picture small motifs
Active Sites are preserved across proteins
with similar functions
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The Big Picture large motifs
Even bigger motifs are often conserved.
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Oh, BTW

• There are two different issues here
• Find the best match for the motif in the protein
• Extensively studied in vision/graphics
• Is the match significant ?
• For small motifs a good match is more likely
• What is probability of a match against a random
  protein being this good ? (cf. BLAST)

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Whats in it for a CS guy ?

• The problem of matching two point-sets has many
  applications
• Most current algorithms geared towards points
  that are indistinguishable (e.g. points on a
  mesh)
• There are few rigorous results on the
  significance of matches

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So what have we done ?

• Towards a more rigorous approach for scoring the
  quality of a match (between motif and protein)
• Provide a method that is capable of finding the
  optimum match based on these criteria

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Problem Description

• Given a motif and a protein, for each point in
  the motif, find a corresponding point in the
  protein.
• Given these correspondences, find the best
  transformation (rotation and translation only) of
  the motif that aligns it to the protein.
• Optimize over all possible correspondences

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Oh, BTW

• Given two sets of k points, easy to find the
  optimal rotation and translation that minimizes
  the least sum-of-squared error (also RMSD).
• Boils down to finding the largest eigenvalue of a
  4x4 matrix.

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Previous Work

• Brute Force approach match edges of same length.
• Geometric Hashing

Pennec Ayache, Bioinformatics, 1998
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What is missing ?

• Ad hoc Try to minimize a quantity that is only
  indirectly related to the least square error or
  RMSD.
• Hard to evaluate the quality of partial matches
• Brute Force methods infeasible for larger motifs
• Geometric Hashing requires significant
  preprocessing

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Estimating the error

• Model the alignment problem as a regression
  problem

Y model set (protein) T data set (motif) g
transformation (rottrans)

• Which error criterion to use ?
• Least Mean Squared Error (also RMSD)
• LSE is not good when you have outliers.
• what to do ?

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Robust error estimation

• LSE larger error terms have disproportionate
  influence.
• Use a function to reduce the effect of larger
  error terms (M-estimators)

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Its an optimization problem!

• Consider the case of full matching
• Domain set of all possible correspondences
  between points on the motif and points on the
  protein
• Range given a particular set of corresponding
  points, the minimum error in aligning those point
  sets.
• Goal find the global minimum of this function!

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Looking for global minimum

• Our approach
• Prune the search space to a small and plausible
  sub-space
• Find (most) of the local minima in this sub-space
  quickly
• Choose the minimum over these local minima

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Finding local minima is easyICP

• Iterative Closest Point (Besl-McKay)

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ICP contd

• ICP is guaranteed to converge to a local minimum
• But depends a lot on initial seeding
• Convergence is quick 4-5 iterations
• ICP movie

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Pruning the search space

• Every point in motif/protein has some features
• Amino acid type, element type, sec. structure,
  hydrophobic/polar, substitutable
• Assume a point with feature X can only match
  another point with feature X (or Y,Z,W)
• Assume some features are more frequent than
  others

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Our Approach

• Find the feature that is least frequent in
  protein.
• For each occurrence of the feature
• Seed ICP appropriately. Find local minimum.
• Look around a few more times
• Return the best answer you have

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Observations

• Will always find a perfect match, if it exists.
• Moreover, will find such a match quickly.
• The error is directly interpretable in RMSD terms

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Does it work ?
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contd
Trypsin active site against Trypsin like proteins
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contd
Trypsin active site against kinases
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What about partial matching ?

• Basic idea is the same pruningICP
• Replace least squared error estimates by
  M-estimator based errors.
• Problem How to find the optimal
  rotation/translation that minimizes this new
  variety of error criterion?
• Answer weighted LSE ?
• Is there a better way ?

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RANSAC
Choice of the parameters has statistical
justification
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Plain Vanilla (Least Squares)
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M-estimator weighted LSE
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M-estimator RANSAC
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contd

• Data for distorted trypsin active site against
  ten different trypsins

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Future Work

• Test on larger motifs secondary structure
  elements
• Choice of better features
• A theoretical guarantee about the quality of
  results
• Explore different criteria for partial matching

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Thanks!