Explore Biological Pathways from Noisy Array Data by Directed Acyclic Boolean Networks

1
Explore Biological Pathways from Noisy Array Data
by Directed Acyclic Boolean Networks

• Lei M. Li
• University of Southern California
• Email lilei_at_hto.usc.edu
• Henry Horng-Shing Lu
• National Chiao Tung University
• Email hslu_at_stat.nctu.edu.tw
• Journal of Computational Biology (2005)

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Biochips DNA Microarrays

• Experiment Designs
• Image Processing
• Normalization
• Gene Selection
• Clustering
• Classification
• Pathway/Network Analysis

http//microarray.vai.org/Quality_control/images/H
A68_156_TestHyb.jpg
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Cell Cycle
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Pathway Reconstruction

• Explore pathways of biological elements from
  microarray or other array data?
• Structure of pathway sub-structure
• Beyond similarity scores
• Computational implementation
• Statistical significance
• Control of false positives and false negatives

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Boolean Networks

• Kauffman (1969, 1974, 1977, 1979), Kauffman and
  Glass (1973)
• Gene Networks (Liang et al., 1998 Akutsu et al.,
  2000a-b, 2003 Shmulevich et al., 2002a-c,
  2003a-b, 2004 Kim et al., 2002 Datta et al.,
  2003, 2004 Hashimoto et al., 2004, Li and Lu,
  2005)
• Random Boolean Networks (http//www.activewebs.ch/
  schwarzer/rbn/index.htm)
• Probabilistic Boolean Networks(http//www2.mdande
  rson.org/app/ilya/PBN/PBN.htm)
• SPAN s-p-scores Associated with a Networks (Li
  and Lu, 2005)

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An Example of Boolean Networks
1
2
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From Boolean to Probabilistic Boolean Networks
as models of Genetic regulatory Networks, Ilya
et al., Proceeding of IEEE, 2002.
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Dynamics of Boolean Networks
Time Step t 1 1 0 1 0
011 001 010 011 011
From Boolean to Probabilistic Boolean Networks
as models of Genetic regulatory Networks, Ilya
et al., Proceeding of IEEE, 2002.
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Attractors and BasinsProliferation, Apoptosis,
Differentiation
From Boolean to Probabilistic Boolean Networks
as models of Genetic regulatory Networks, Ilya
et al., Proceeding of IEEE, 2002.
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Dynamical Patterns
http//www.geocities.com/jaap_bax/boolean.html
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Java Demo

• An Introduction to Complex Systems Torsten Reil,
  Department of Zoology, University of Oxford
  (http//users.ox.ac.uk/quee0818/compl
  exity/complexity.html)

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Challenging Issues

• Random Boolean Networks (http//www.activewebs.ch/
  schwarzer/rbn/index.htm)
• Probabilistic Boolean Networks(http//www2.mdande
  rson.org/app/ilya/PBN/PBN.htm)
• Reconstruction of Boolean Networks
• SPAN s-p-scores Associated with a Networks (Li
  and Lu, 2005)

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Directed Acyclic Boolean (DAB) Networks

• Objects binary elements and their Boolean duals
  A and its dual ,on and off status of a gene
• Relations
• Prerequisite
• Similarity AB
• Negative-similarity
• Not trivial in the presence of measurement errors

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Graph Representation

• Representation directed graph
• Ground set two vertices for one element, A and
  it dual
• Directed relation , A is prerequisite
  for B
• Undirected relation A-B, A is similar to B
• Redundancy covering pairs
• Acyclic and no conflict never

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An Example
Diagram
The table of state values
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How Many DAB Networks?

• Example
• A DAB corresponds to a subset of on-off states
• How many feasible DAB networks?
• Super-exponential

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Sampling from DAB Networks

• Sample with replacement from the table of states
• Exhaustive sample
• Arrange them in a binary array
• How to identify pairwise relations?

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Six Patterns
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Sampling and Design Issues

• How will this count strategy work if we sample
  only a fraction of the state space?
• Assumption no measurement error
• No false negative pairwise relations
• False positive pairwise relations can happen
• Selection bias a big problem!
• Design random if not exhaustive

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Miclassification Errors

• An exhaustive sample from the state space with
  replacement-
• Data-a binary array-
• An example of binary array data

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Problem and Strategy

• Data array with measurement error
• Goal reconstruct the DAB networks
• Strategy
• Consider each pair of elements
• find the most likely relation
• assign a significance score to this relation
• s-p-score
• Put together pairwise relations by ranking
• s-p-score

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Pairwsie Relation in 2 by 2 Tables
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Misclassified Data
Counts with error
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Probability Splitting
Observations with errors
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Diagonal Models and s-scores

• EM Algorithm
• Expectation-redistribute the count in each cell
• Asymptotics classical results

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Triangular Models and p-scores

• Computation EM algorithm
• The full model is saturated with parameters
• Will the likelihood method work? How and Why?

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Model Selection and s-p-scores

• Model selection
• The smaller the score, the more support to the
  null hypothesis
• Between the two diagonal models, select the
  smaller s-score
• Among the four triangular models, select the
  smallest p-score
• s-p-score the score of the model that minimizes
  BIC
• SPAN s-p-scores Associated with a Networks

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Statistical Inferences

• Statistical tests type I and II errors.
• Good controls of false negatives and positives.
• S-p-scores play the role of test statistics
• We expect those relations with smaller
  s-p-scores and real, and those with larger scores
  are false.
• Irregularity

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Control of False Negatives

• We expect the p-score is a good estimate of p
  under the null hypothesis
• Accuracy of estimates
• The asymptotic of MLE works under the null
  hypothesis
• Fisher information matrix

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Asymptotic Variances

• Irregularity one singularity point
• Fix eliminate house-keeping and silent genes

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Control of False Positives

• Why not likelihood ratio test?
• Steins lemma the chance of type II error does
  not go to zero

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Reconstruction of DAB Networks

• The s-p-score does not make too much sense if we
  have two elements without a prior knowledge of p.
• They are more meaningful if we have many
  elements.
• Rank the s-p scores in the ascending order.
• Watch list of pairwise relations.
• How to determine the threshold? Know biology.

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Simulation
simulate a data set of 76 samples with flipping
probability p0.05
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Reconstruction from Simulation
Estimated DAB networks
True DAB networks
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Yeast MARK pathway (Robert et al., 2000,
Science)
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Conclusion and Discussion

• DAB directed acyclic Boolean networks
• SPAN s-p-scores an exploratory tool with the
  control of false positives and negatives
• Future studies
• Discretization
• Relations involving more than two elements
• Time course data
• Dynamics and evolution
• Integrations with other methods
• Incorporation of related data and evidences

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Acknowledgements

• Professor Wing H. Wong, Michael Waterman, and
  Simon Tavaré.
• Institute of Pure and Applied Mathematics, UCLA.
• CEGS grant from NIH in USA for Lei M. Li.
• Grants from National Science Council in Taiwan
  for Henry H.-S. Lu.