Probabilistic clustering of sequences: Inferring new bacterial regulons by comparative genomics

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Probabilistic clustering of sequences Inferring
newbacterial regulons by comparative genomics

• Erik van Nimwegen , Mihaela Zavolan, Nikolaus
  Rajewsky , and Eric D. Siggia
• In PNAS 2002

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Contents

• Synopsis
• Introduction
• Model
• Classifiability VS Clusterability
• Implementation
• Results
• Conclusions

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Synopsis

• Why Cluster Regulons?
• Genome-wide comparisons of Enteric Bacteria Yield
  Conserved Putative Regulatory Sites
• Large Datasets
• How to Assign Sites to Clusters?
• Assumption Regulatory Sites can be represented
  as samples from Weight Matrices(WM)
• Monte Carlo Sampling of a Probability
  Distribution to Partition and Align DNA Sequences
  into Clusters
• Determine Number of Clusters and Assign Some
  Significance Metric to Them
• When is Clustering Feasible?
• Set of All Putative Sites for a Single Genome
  Infeasible
• Solution Use Sites from All Genomes

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Introduction - Background

• Why do we need to sequence non-coding regions of
  the genome?
• They help in information about gene regulation
• Smaller than normal coding regions and difficult
  to identify
• Examples
• In E.Coli
• 60-80 sites of known binding and regulatory sites
• 300, actual number according to protein sequence
  homology
• What is a Weight Matrix(WM)?
• Describes Protein Binding Sites in Bacterial
  Genomes
• Gives Probability of Finding Base Alpha at
  position I of Binding Site

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Existing Computational Strategies

• I Identify sets of similar sequences in the
  regulatory regions of functionally related groups
  of genes.
• II Identify repetitive patterns within an entire
  genome
• Some disadvantages
• Those using WM cannot process genome scale data
  representing thousands of Transcription Factors
• Other schemes not suitable for processing sites
  inferred from interspecies comparison

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Some Proposed Improvements

• Partition Entire Set of Sites at Once
• Infer Number of Cluster Internally
• Assign Partitions to All Subsequences of Clusters
• Theoretical Limit on Clusterability of Sets of
  Regulatory Sites Derived
• What is Clusterability?
• A set of Sites is said to be Clusterable if
• It is possible to infer which sites are from the
  same WM

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Model

• Scores in Motif Finding Techniques
• Calculate Information Score of Estimated WM
• Where ba is the background frequency of base a,
  and the wia are the WM probabilities estimated
  from the sequences in the alignment
• Task Cluster together binding sites of a number
  of unknown TFs

2 Ways of Grouping
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Model (Contd.)

• Tasks
• Assign Probabilities to All Partitions
• The Probability of a partition is the product of
  the probabilities, for each cluster
• Step 1 Consider first the conditional
  probability P(Sw) that a set of n length l
  sequences S was drawn from a given WM w
• Step 2 Calculate probability P(S) that all
  sequences in S came from some w

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Model (Contd.)

• Step 3 Define for any partition C of a data set
  of sequences D into clusters Sc the likelihood
  P(DC) that all sequences in each Sc were drawn
  from a single WM
• Step 4 Calculate Posterior Probability

Where p(C) is the prior distribution over
partitions
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Classifiability

• Classifiability
• What is Correct Regulation of Gene Expression?
• TFs should bind preferentially to their own sites
• P(sw) gt P(sw) Called as Classification Task
• Given WMs and a set of Sequences Sampled From
  Them
• Assign a Sequence to the WM that maximizes P(sw)
• WM that maximizes P(sw) is the WM from which
  sequence s was sampled

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Clusterability

• Suppose there are nG sequences
• Obtained by Sampling n times in G different
  sequences
• Calculate Probability m of n samples co-cluster.
  How?
• Sum P(CD) over all partitions of C in which m
  samples of w occur together
• If for this set G, for more than half of WMs
• m gtn/2, then this set is clusterable

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Analytical and Numerical Evaluations of
Classifiability and Clusterability

• Given I, the fraction of the space of 4l
  sequences filled by the binding sites for this WM
  is e-2I
• I is measure of specificity of WM
• Results show
• exp(2I) directly proportional to (1/G) for
  classification
• exp(2I) directly proportional to (1/G2) for
  clustering a set of n 3 binding sites
• Clustering is impossible for sites near the
  classification threshold

Clusterability
N3
N5
N10
N15
Classifiability
I Vs G
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Implementation

• Monte Carlo Random Walk to Sample Distribution
• Choose a mini-WM and consider assigning it to a
  randomly chosen cluster
• Increase P(CD)
• Generates Random Clusters

Move from Partition C to C
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Impact on Clusters

• Generates Dynamic Clusters
• What happens when a pair of mini-Wms are moved
  together?
• Clusters may evaporate
• New clusters may form
• Task
• To identify significant clusters
• By finding sets of mini-WMs that are grouped
  together persistently during the Monte Carlo
  sampling
• Membership Stability
• Ideal Case Finding Stable Core Members
• Reality Constantly Drifting Clusters with
  Uncorrelated Memberships

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Identifying Candidate Clusters

• Searching Maximum Likelihood Partition
• Use Simulated Annealing to Maximize P(CD)
• P(DC) raised to Power Beta(Beta3 Ideally)
• Testing Significance of ML Clusters
• By Sampling P(CD)
• Measures
• Probability p(k) that k members co-cluster
• Mean Size of Cluster
• Minimum Length of Interval
• Clusters with Significant

ML Partitions Obtained by Annealing
The number of co-clustering members of an ML
cluster is the maximum number of mini-WMs from
the ML cluster that co-occur in a single cluster
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Modifications for Large Data Sets

• Several Monte Carlo Random Walks
• Measure that Each Pair of Mini-WMs co-clusters
• Construct Graph
• Nodes correspond to Mini-WMS
• Edges between mini-WMs i and j exist if and only
  if their co-clustering probability
  (1/2)
• Candidate Clusters ? Connected Components of
  Graph
• Calculate Probabilistic Cluster Membership
• Calculate p(k)
• Cluster significance is derived from p(k)

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Data Sets Used

• Uses Different Prokaryotic Genomes
• Short Sequences 15 to 25 Bases
• Use Genomes of E. coli,Actinobacillus
  actinomycetemcomitans, Haemophilus influenzae,
  Pseudomonas aeruginosa, Shewanella putrefaciens,
  Salmonella typhimurium, Thiobacillus
  ferrooxidans, V. cholerae, Y. pestis,Klebsiella
  pneumoniae

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Operations

• Extend
• Expand Alignments to Length 32
• Pad Bases From Genomes
• Replace Sequences of Closely Related Species by
  their Consensus
• Processing on E.Coli
• Align all known E. coli sites for each TF into
  its own mini-WM
• Add 56 Mini-WMs to this Set
• Create Additional Test Set Consisting of
• 397 Known Binding Sites and E. coli sequences of
  the top 2,000 unannotated mini-WMs

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Results

• Test Set of 397 Binding Sites
• Sample P(CD) and Measure How Well the Sites
  Cluster
• Significant Clusters Obtained for 24 of 53 TFs
• Twenty two TFs have three or fewer sites in the
  test set, and with the exception of trpR their
  sites did not cluster significantly
• Comparison
• Inferred Results(Annealing) VS Results From Site
  Annotation
• Good Agreement Between Both Results
• Likelihood for Partition Obtained By Annealing is
  Higher than that by Site Annotation
• Improvements of Proposed Work
• Recovers almost half of all regulons for which
  binding sites are known and the large majority of
  regulons for which there are more than three
  sites known

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Results for Large Datasets

• Annealed state and significance statistics not
  converged fully within running times
• 1010 steps, taking a week on a workstation per
  run
• Solution
• Converge Using Pair Statistics
• Example
• Around 365 clusters on average
• Connectivity graph gave 274 components containing
  1,139 out of 2,056 mini-WMs
• 1 Space Filled By Top 45 Clusters
• Top 80 Clusters Fill 10 of Space
• Next 115 Clusters Fill 39 of Space

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Conclusions and Discussion

• New Inference Procedure for Probabilistically
  Partitioning a Set of DNA Sequences into Clusters
• Assumption All WMs are of Fixed Length
• Predictions from applying the algorithm to data
  sets of putative regulatory sites extracted from
  enteric bacteria
• 100 new regulons in E. coli, containing 500
  binding sites, and 50 binding sites for known TFs
• In addition to TF binding sites RNA stems
  controlling translation and termination motifs
  are present in Predicted Regulons