People

1
Title

• People

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Outline

• Motivation
• Theoretical foundations
• Biological extensions
• Implementation
• Validation techniques
• Results from yeast

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Motivation

• Post-genomics, want to understand organisms
  protein-protein interaction network
• Model network as a probabilistic graph
• Edge weights represent interaction probabilities
• Interested in protein signaling cascades
• Show up as simple paths in the graph
• Want to find biologically interesting paths
  efficiently
• Score paths, with high scores reflecting
  importance
• Extended graph algorithms provide speed

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Theoretical Foundation

• Finding long, simple paths is NP-Complete
• Reduce from TSP
• Requirement for paths to be simple is what drives
  hardness
• Color-Coding is a randomized, dynamic-programming
  based algorithm for finding paths of fixed length
• Developed by Alon et all

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Color-Coding

• Need to eliminate restriction for paths to be
  simple
• Instead, randomly color graph and require paths
  be colorful (containing exactly one vertex of
  each color)
• Number of colors fixed length of paths
• A colorful path is always simple
• The problem of finding colorful paths can be
  solved with dynamic programming

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Color-Coding DP

• Key point a colorful path of length k contains a
  colorful path of length k-1.
• Store path information at each node for each
  subset of k colors
• Only 2k color subsets, rather than O(nk) node
  subsets
• Runtime is O(2kkm) ltlt O(knk) brute force
• Space is O(2kn) ltlt O(knk) brute force

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PICTURE
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Monte Carlo Details

• A colorful path is simple, but a simple path may
  not be colorful under a given coloring
• Solution run multiple independent trials
• After one trial,

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Adding Biology

• Color-Coding gives an algorithmic basis, now
  introduce biologically motivated extensions
• Can set the start or end of path by type
• Eg screening by Gene Ontology categories
• Can force the inclusion of a protein on the path
  by giving it a unique color
• Using counters, can specify path must contain
  between x and y proteins of a given type
• Computational cost multiplicative in y per counter

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Adding Biology - Segmented Paths

• Pathways may be ordered
• Signaling pathways going from the membrane, to
  nuclear proteins and finally transcription
  factors
• Assign each protein an integer label based on
  biological information, build path out of ordered
  sequences of labeled proteins
• Now only need to constrain color collisions among
  proteins with the same label
• If path length is equally split among labels,
  probability of correct coloring rises
• Modifications allow for inability to assign
  proteins to unique labels

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Adding Biology - More Structures

• Modifications to the Color-Coding recurrence
  allow for the discovery beyond simple paths
• Example Two-terminal series-parallel graphs
• Capture parallel signaling pathways
• PICTURE

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Generating Edge Weights

• So far, have glossed over how weights
  (probabilities) on the protein graph are assigned
• Here, use our previous work, generate logistic
  function of three variables (for a pair of
  proteins)
• Number of times interaction between them was
  experimental observed
• Pearson correlation coefficient of expressions
  (for corresponding genes)
• Their small world clustering coefficient
• Using training data from MIPS as gold standard
  for training relative weighting
• Taking log of weights makes path score additive

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Application

• Tested our simple path implementation with the
  yeast interaction network
• 4,500 vertices, 14,500 edges
• Based on interaction data from Database of
  Interacting Proteins (Feb 2004)
• Varied path length, much faster than brute force
  at longer end of spectrum (14x for paths of
  length 9)
• Focus on paths from membrane proteins to
  transcription factors

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Validation Techniques

• Three methods of validation
• Two statistical
• Functional enrichment p-value based on how many
  proteins in the path are similar (by GO category)
• Weight p-value compares weights of paths to those
  found when the protein graph undergoes random
  degree-preserving shuffling
• Lastly, search for expected pathways
• MAP-Kinase, ubiquitin-ligation

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BIO ABOUT MAPK AND ETC?
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Statistical Results

• 100 best length 8 paths _at_ 99.9 success
• 100 normal/2000 random paths used for weight
  p-value

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MAPK Recovery Results
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Conclusion

• Presented efficient, color-coding based
  algorithms for finding simple paths
• Added biological extensions, other structures
• Applied our algorithms to yeast
• Shown 60 of discovered pathways were
  significantly enriched
• Recovered known MAP-Kinase pathways

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(Future Work?)