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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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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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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S0
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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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Extension to PBNs

• For each xi there are predictor sets Fi
  fj(i)j1,,l(i)
• Realization of PBN f1,, fN
• cj(i) - probability of predictor fj(i) predicting
  xi
• Independence not necessary, i.e.

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Network Selection Probability

• Matrix K - Realizations of PBN
• Calculate Transition Probabilities between
  different GAPs (Gene Activity Profiles).

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Example
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K
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Markovian Behavior
Pr (011) -gt(100) P0 (1-1-1)(1-1-0)(1-
1-0) P1 (1-1-1)(1-1-0)(1-0-0)
P2 (1-1-1)(1-0-0)(1-0-0) P3
(1-1-1)(1-0-0)(1-0-0) P2 P3
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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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Inference of PBNs

• Select best set of predictors using their
  Coefficient of Determination (COD)
• ei - error of best estimate of xi in the absence
  of any conditional variables.
• e(Xi, fk(i)(Xk(i))) error on using conditional
  estimator fk(i)(Xk(i)) for calculating Xi

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Dynamics of PBNs
Draw the PBN Homogeneous Markov chain with
Stationary Distribution
Steady-state (limiting) distribution
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Relationship to Bayesian Networks

• Bayesian Network predicts gene xi given its
  parents Pa(xi).
• To predict the binary expression of gene xi

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Influence of Genes

• Major vs Minor Impact
• In terms of Function
• In terms of target gene Overall influence

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Example

• Influence of x2 on x1, requires f1(1) and f2(1)
• Assume Uniform
• distribution

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Sensitivities of Genes

• Sensitivity of a function at vector x
• Average sensitivity of function (with respect to
  Distribution D)
• If given influence matrix collective effect

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Questions?