A unified statistical framework for sequence comparison and structure comparison

1
A unified statistical frameworkfor sequence
comparison and structure comparison

• Michael Levitt Mark Gerstein

2
Statistics Introduction

• Statistics is the discipline which deals with
  inference in the presence of variation
• Given a score, how significant is it?
• Ho , HA , Critical Region, P-value
• Extreme Value Distribution-maximum over all
  sequence scores is distributed as Extreme Value
  Distribution
• Reason why extreme value distribution is useful
• maximize score over all possible random
  alignments

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Introduction

• Given sequence and structural scores, develop
  hypothesis testing framework
• Ho Two proteins compared are unrelated
• Distribution of scores of unrelated proteins
  determined empirically using PDB data at 40
  sequence identity
• No assumption of background distribution

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Sequence Comparison Framework

• Sequence score determined by SSEARCH and BLOSUM
  50 substitution matrix
• Sseq (sequence score), n and m (lengths of two
  sequences compared) in p.d.f.
• Compared all possible pairs to determine
  empirically the p.d.f.

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P.D.F. for Sequence Score
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Cross Section of p.d.f for constant ln(nm)
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Density Distribution for constant ln(nm)

• Density distribution follows extreme value
  distribution exp(-Z exp(-Z)) pcseq(Z)
• Z(Sseq - µseq)/oseq
• µseq a ln(nm) b model average a and b
  fitted to the observed density by least squares
• oseq a

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Comparison to BLAST and FASTA statistics

• Critical region to determine p-value for model
  Pseq(zgtZ)
• Comparison of model p-values with BLAST p-value
  found BLAST p-value higher than model
• FASTA statistic better coverage, more error than
  model

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Structure Comparison Algorithm
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Structure Comparison Framework

• The score obtained from the structure comparison
  algorithm is Sstr
• P.d.f. for Sstr used N (number of residues
  matched) and Sstr (pairs which scored high were
  removed)
• Kept N fixed and fitted extreme value
  distribution to density using all N

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Comparison with RMS

• RMS deviation in alpha-carbon after least squares
  fit is traditional method
• RMS score used to determine p.d.f. with ln(RMS
  score) and N
• Comparison of RMS with Sstr found RMS worse than
  S in coverage and accuracy

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Comparison with RMS (cont.)

• Three reasons
• Sstr depends most strongly on best-fitting atoms
  RMS depends most on worst-fitting atoms
• Sstr penalizes gaps RMS does not
• Sstr is analogous to Sseq in the sense that both
  use dynamic-programming

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Comparison of Structure and Sequence Comparison
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Concluding Remarks

• Significance of sequence structure score can be
  calculated from any structural alignment program
• This method of statistical significance is
  between FASTA and BLAST methods

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Efficient Detection of Three-Dimensional
Structural Motifs in Biological Macromolecules By
Computer Vision Techniques

• Ruth Nussinov Haim J. Wolfson

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Introduction

• One of the earlier papers addressing structure
  comparison
• Based on computer vision techniques ( geometric
  hashing paradigm)
• No a priori predefined motif assumed
• Advantage Can be parallelized

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Problem

• Given 3D coordinates of atoms of two molecules,
  find a rigid transformation (rotation and
  translation allowed) so that a large number of
  atoms of one molecule match the atoms of the
  other molecule
• Closely related to 3D rigid object recognition

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Geometric Hashing ParadigmRepresentation of
Geometric Constraints

• Proteins represented as points using coordinate
  frames (minimal representation of coordinate
  frames)
• Pick three noncolinear points to define a plane
  (RS) and construct orthogonal 3D coordinate
  system based on RS

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Representation of Geometric Constraints (cont.)

• Define orthonormal vectors w.r.t. RS so that any
  point can be represented as a linear combination
  of the orthonormal vectors
• To remove dependence on particular RS (may
  preclude recognition if at least one of the RS
  points does not match with input substructure),
  represent the m points in all basis triplets
  (I.e. all orthonormal vectors) with all possible
  RS

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Algorithm for Representation of Geometric
Constraints

• For each RS
• Compute orthonormal 3D basis associated with each
  RS
• Compute coordinates of all other points in
  coordinate frame defined by 3D basis
• For each point define address of hash table with
  labels and measurements
• Use each address to enter hash table with pair
  (model, RS)

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Determining Hash Table Entries with Model M1 and
Points 4 and 1 as Basis
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Locations of Hash Table Entries for Model M1
after all bases, RS
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Geometric Hashing Matching

• Given observed object
• 1. Choose an RS and compute 3D basis associated
  with RS
• 2. Compute the coordinates of the other observed
  object points in 3D basis
• 3. For each point, enter hash table at address
  defined by labels and measurements and label and
  coordinate of new point

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Geometric Hashing Matching (cont.)

• For step 3 Tally a vote for model and RS for
  each entry found at address can histogram all
  hash table entries which received one or more
  votes
• 4. If no pair scores high (determine by
  threshold), then go to 1, and begin with
  different RS of the observed object

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Geometric Hashing Matching (cont.)

• 5. Consider all the models from step 4 and find
  rigid motion that gives best least squares match
• 6. Transform the model point set according to the
  transformation of step 5 and check consistency of
  all biological information (I.e. match labeling)

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Modifications to Algorithm

• Could modify voting scheme, modify representation
  of coordinate axes to 2D coordinate axes (reduces
  worst case running time analysis), could apply
  representation of atoms to alpha-carbons only (no
  labeling allowed), could group atoms together
  into a single unit and analyze structures using
  these atom groups

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Algorithm Performance

• Experimented with bacterial proteins, bovine
  pancreas protein, calcium binding protein, bovine
  liver protein, and protein from hen egg
• All experiments were favorable to excellent
  results in terms of fit

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Conclusion

• Algorithm needs O(N x m4) for hash table (can be
  big for large N, m)
• Running time for algorithm can also be long
• Can be parallelized (ie. representation stage
  independent of matching stage)
• Sequence order independent (ie. Insensitive to
  gaps, insertions, deletions)