Introduction to Probabilistic Boolean Networks

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Introduction to Probabilistic Boolean Networks
Ina Sen May 28 , 2008

• 1. From Boolean to Probabilistic Boolean Networks
  as Models of Gene Regulatory Networks
• 2. Probabilistic Boolean Networks a rule based
  uncertainty model for gene regulatory network

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Model Considerations

• To what extent does the model represent reality?
• Is the right type of data being used to infer
  the model?
• What does one hope to learn from the model?

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Boolean Network Terms

• Maximum Connectivity
• Gene Activity Profile
• Attractors / Basins of Attraction
• Structural Stability
• Canalyzing Function
• Ordered regime vs chaotic regime
• Complex regime

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Boolean Network Dynamics

• G(V,F) containing n genes x1, x2,, xn and
  initial joint probability distribution D(x), x in
  0,1n
• Joint probability distribution after one step of
  network
• Thus, Dt1 Y(Dt)
• where Y 0,12n -gt 0,12n

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Representations

• If Dt1,Dt be represented as 1 x 2n vectors
• Let A be defined as 2n x 2n matrix
• function C gives the integer binary vector.
• Matrix A has exactly 1 non-zero entry in each row

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Probabilistic Boolean Networks

• BNs assume deterministic nature of predictive
  function, may not be true given
• Biological uncertainty
• Experimental noise
• Interacting latent variables
• Resolve overfitting

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Mathematical Definition

• Given genes V x1, x2,, xn, for each xi in V
  there is a set of boolean functions Fi

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Extension to PBNs

• Realization of PBN
• Probability of choosing a particular predictor
• Not necessary that Boolean functions composing
  the network to be independent.

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Network Selection Probability

• Define matrix K
• Calculate Transition Probabilities between
  different GAPs (Gene Activity Profiles).

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Markovian Behavior
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Probabilities

• For any GAP x, there exists some GAP x such that
  
• Thus,
• for any i 1,,2n.
• A becomes a Markov matrix and PBN a
  homogeneous Markov Process, i.e. having
  transition probabilities invariant with time.

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PBN Example
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Next Time

• Inference of PBNs
• Dynamics of PBNs
• Relationship to Bayesian Networks
• Influence Sensitivities of Genes in PBNs
• Example as discussed in the paper.

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Questions / Suggestions?