The Statistical Significance of Maxgap Clusters

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The Statistical Significance of Max-gap Clusters

• Rose Hoberman
• David Sankoff
• Dannie Durand

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Glycolysis Pathway
Glycolysis Clusters
Clostridium acetobutylicum
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Gene Clustering for Functional Inference in
Bacterial Genomes

• The Use of Gene Clusters to Infer Functional
  Coupling, Overbeek et al., PNAS 96 2896-2901,
  1999.

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original genome
large scale duplication or speciation event
rearrangement, mutation
Gene content and order are preserved
Similarity in gene content Neither content nor
order is strictly preserved
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Evolution of gene order conservation in
prokaryotes Tamames, Genome Biology 2, 2001
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Evolution of gene order conservation in
prokaryotes Tamames, Genome Biology 2, 2001
Gene insertion/loss
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Evolution of gene order conservation in
prokaryotes Tamames, Genome Biology 2, 2001
Gene insertion/loss
Local rearrangement
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Two Possible Questions

• Given a set of genes that we believe are
  functionally related, determine if they cluster
  together spatially more than we would expect by
  chance
• Identify all significantly conserved gene
  clusters as a starting point for making
  functional inferences

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Two Possible Questions

• Given a set of genes that we believe are
  functionally related, determine if they cluster
  together spatially more than we would expect by
  chance
• Identify all significantly conserved gene
  clusters as a starting point for making
  functional inferences

Reference set scenario
Whole genome comparison
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Reference Set Scenario
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Reference Set Scenario

• Model of a genome
• G 1, , n an ordered set of n unique genes
• assume genes do not overlap
• chromosome breaks ignored

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Reference Set Scenario

• Model of a genome
• G 1, , n an ordered set of n unique genes
• assume genes do not overlap
• chromosome breaks ignored
• Reference gene scenario
• m genes of interest (in red) are pre-specified
• want to find clusters of (a subset of) these
  genes

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Whole Genome Scenario
Given two genomes G 1, , n and H 1, ,
n
Find all significant clusters of at least k
homologs in close proximity in both genomes?
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Outline

• What formalisms do we need to address these
  questions?
• Definitions formulate a cluster definition
• Algorithms identifying clusters in real data
• Statistics assess the significance of one or
  more clusters
• Reference set scenario
• Whole genome comparison
• Conclusion

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Why develop a formal statistical model?

• Understand trends and verify that they match our
  expectations
• Choose parameters effectively
• Statistical tests for data analysis

Typically researchers use randomization tests to
estimate statistical significance
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Cluster Definitions

• An intuitive notion of a cluster is a group of
  genes
• occurring in close proximity
• neither gene content nor order is strictly
  conserved
• Algorithms and statistics require a formal
  definition.
• What properties are desirable?
• Do existing definitions have these properties?

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size 3 genes

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3

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length 6

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3
• length number of genes between first and last
  red genes
• Example cluster length 6

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length 6

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3
• length number of genes between first and last
  red genes
• Example cluster length 6

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density 6/11

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3
• length number of genes between first and last
  red genes
• Example cluster length 6
• density proportion of red genes (size/length)
• Example density 0.5

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density 6/11

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3
• length number of genes between first and last
  red genes
• Example cluster length 6
• density proportion of red genes (size/length)
• Example density 0.5

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gap 4 genes

• Possible Cluster Parameters
• size number of red genes in the cluster
• Example cluster size 3
• length number of genes between first and last
  red genes
• Example cluster length 6
• density proportion of red genes (size/length)
• compactness maximum gap between adjacent red
  genes

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Max-Gap Cluster
gap?? g

• Commonly used in analysis of genomic data
• Desirable properties
• Ensures minimum local density
• Extensible doesnt artificially limit cluster
  length
• Disjoint clusters will not overlap

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Outline

• Formalisms
• Reference set scenario
• Whole genome comparison
• Conclusion

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Formalisms

• Definitions formulate a cluster definition
• Algorithms identify clusters in real data
• Statistics assess the significance of a cluster

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A Statistical Model

• Given
• a genome G 1, , n unique genes
• a set of m reference genes
• a maximum-gap size g
• Null hypothesis
• Random gene order
• Alternate hypotheses
• Evolutionary history
• Functional selection

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Statistics of Max-Gap Gene Clusters

• We provide
• analytical and dynamic programming solutions
• to determine cluster significance exactly
• for the reference set scenario
• Hoberman, Sankoff and Durand. In Proceedings of
  the RECOMB Satellite Workshop on Comparative
  Genomics'', J. Lagergren, ed., Lecture Notes in
  Bioinformatics, Springer Verlag, in press.
• Hoberman, Sankoff, Durand. Submitted to RECOMB
  2005.

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Test Statistic Complete Clusters

• The probability of observing all m reference
  genes in a max-gap cluster in G

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Test Statistic Incomplete Clusters

• The probability of observing at least h of the m
  reference genes in a max-gap cluster in G

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Cluster significance
n 1000, m50
n 500, h m/2

• n number genes in each genome
• m number of genes shared between the two
  genomes
• g maximum allowed gap size
• h size of cluster (e.g. number of red genes)

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Significant Parameter Values (a 0.0001)
n 500
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Significant Parameter Values (a 0.0001)
n 500
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Outline

• Formalisms
• Reference set scenario
• Whole genome comparison
• Conclusion

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Formalisms

• Definitions formulate a cluster definition
• Algorithms identify clusters in real data
• Statistics assess the significance of one or
  more clusters

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Whole genome comparison
g?? 10
g?? 10

• Find all sets of genes that form max-gap clusters
  in both genomes.

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Properties of Max-Gap Clusters for Whole Genome
Comparison

• Clusters are locally dense in both genomes
• Clusters are still guaranteed to be disjoint.
• The definition is symmetric with respect to genome

Most existing cluster algorithms are not
symmetric!
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Algorithms Finding Max-Gap Clusters

• If g 2
• There is no valid max-gap cluster of size two or
  three
• There is a valid max-gap cluster of size four

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Algorithms Finding Max-Gap Clusters

• A consequence of this is that a greedy iterative
  approach will not find all max-gap clusters
• Specifically, larger clusters that dont contain
  smaller ones will not be found

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Algorithms Finding Max-Gap Clusters

• There is an efficient divide-and-conquer
  algorithm to find all max-gap clusters (Bergeron
  et al, 2002)
• Since algorithms are generally not stated
  formally in application papers, we dont know
  whether people are actually getting what they
  think theyre getting

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Formalisms

• Definitions formulate a cluster definition
• Algorithms identify clusters in real data
• Statistics assess the significance of one or
  more clusters

Work in Progress
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Statistics Whole genome comparison
g?? 10
g?? 10

• What is the probability that at least k genes
  form a max-gap cluster in both genomes?

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Statistics Whole genome comparison
g?? 10
g?? 10

• What is the probability that at least k genes
  form a max-gap cluster in both genomes?
• Assuming identical gene content, the probability
  of finding a max-gap cluster of size at least k
  is always one!

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An Example

Example g 1
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An Example
Example g 1
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An Example
Example g 1
A cluster of size k does not necessarily contain
a cluster of size k-1
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An Example
Example g 1
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An Example
Example g 1

• When gene content is identical, there will always
  be a cluster of size n

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An Example
Example g 1

• When gene content is identical, there will always
  be a cluster of size n
• Therefore, for all k, there will always be a
  cluster of size at least k

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An Example
Example g 1

• When gene content is identical, there will always
  be a cluster of size n
• Therefore, for all k, there will always be a
  cluster of size at least k
• Therefore, the probability of finding a cluster
  of size at least k is always one!

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Relaxing the Assumption of Identical Gene Content

• Assume only m of the n genes in each genome are
  shared
• If the longest run of non-shared genes is less
  than g then we are still guaranteed to find a
  complete cluster

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• More generally
• Simulations of randomly ordered genomes show that
  large clusters may be very likely to occur merely
  by chance

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Unexpected Statistical Trends

• There can be a significant probability of finding
  a cluster that includes all homologous gene pairs
• The significance of a cluster of size k can be
  less than that of a cluster of size k-1
• Probabilities are not monotonic
• Large clusters may not be significant

n 1000, m 250, g20
Probability of a cluster of size 250 50
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Outline

• Formalisms
• Reference set scenario
• Whole genome comparison
• Conclusion

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Clusters Are Used in Many Other Applications
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Max-Gap Clusters are Especially Common
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• Formal statistical models allow us to
• understand trends and verify that they match our
  expectations,
• choose parameters effectively
• conduct statistical tests for data analysis
• Formal statistical models require
• a formal cluster definition
• a search procedure to find clusters
• These issues are more complicated than they might
  seem!

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Summary

• Results statistical tests of significance for
  max-gap clusters
• Reference set scenario
• Genome comparison (work in progress)
• We need to
• explicitly consider the cluster properties we
  would like our definitions to satisfy
• rigorously evaluate whether our definition meets
  these requirements
• carefully prove that our search procedures match
  our stated definitions

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• Thank You