Modeling regulatory networks in cells using Bayesian networks

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Modeling regulatory networks in cells using
Bayesian networks
Golan Yona
Department of Computer Science Cornell University
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Outline

• Regulatory networks
• Expression data
• Bayesian Networks
• What
• Why
• How
• Learning networks from expression data
• Using Bayesian networks to analyze expression
  data (Friedman et al)

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Regulatory networks

• KEGG Regulatory
• Pathways

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Metabolic pathways

• KEGG Metabolic
• Pathways

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Expression Arrays

• Measure the expression levels of thousands of
  genes in a cell under specific conditions (e.g.
  cell cycle) simultaneously
• Each cell has the same genomic data but different
  subsets of proteins are being expressed in
  different cells and at the same cell under
  different conditions.
• Protein level is controlled by controlling
• transcription initiation
• mRNA transcription
• mRNA transport
• splicing
• post-translational modifications
• degradation of mRNA and proteins.
• Microarray measure the level of mRNA, thus
  providing an indirect evidence for the control of
  protein levels

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Micro Spotting pin
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• Some are over-expressed (red), some
  under-expressed (green) measured with respect to
  a control group of genes (fixed genes)
• Different pathways are activated under different
  conditions

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Goals

• Recover protein interactions and sub-networks
  that correspond to regulatory networks in the
  cell.
• Basic assumption some genes are dependent on
  others while others exhibit independence or
  conditional independence
• The means Bayesian networks. Capable of modeling
  the statistical dependencies between different
  variables (genes)
• Different from clustering analysis ..
• Applicable when the dependency between genes is
  local
• Problems data is noisy, partial, sometimes
  misleading (translation, activation), not enough
  to ensure statistically significant models, time
  scale

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Bayesian Networks

• A compromise between the assumption of complete
  dependency and complete conditional independence
  (Naïve Bayes)
• Less constraining yet still tractable
• We know something about the statistical
  dependencies between features but not necessarily
  about the type of the underlying distributions

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Example
Oil pressure In engine
Oil temp.
Fan speed
Engine temp.
Air pressure In tire
Coolant temp.
Smoke
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Bayesian Networks

• Also called belief nets
• A graph description of dependencies and
  independencies between variables. Each node
  corresponds to a variable (gene).
• The graph is directed and acyclic
• The variables are discrete
• A variable can take on a value from a set of
  values a1,a2, e.g. on/off
• The probability of a specific value P(ai) and ?i
  P(ai) 1
• A link joining node A to node C is directional
  and represents the set of conditional
  probabilities P(cj/ai) causality (the
  probability that C is on when A is off)
• The network is described in terms of its nodes
  and edges and the conditional probability
  distributions associated with each node.

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Network structure

• For every node A
• The parents of A is the set of immediate
  predecessors of A
• The children of A is the set of immediate
  successors of A
• B is a descendant of A if there is a directed
  path from A to B
• Conditional probability
• Network assertions
• The value of a variable depends on its parents
• A variable is conditionally independent of it
  non-descendants given its parents

Parents
Eng cold, Eng cold, Eng hot, Eng hot Fan
fast Fan slow Fan fast Fan slow High 0.1 0.4 0.6 0
.9 Low 0.9 0.6 0.4 0.1
Coolant temp.
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Calculating the probability of an assignment

• The network describes the joint probability
  distribution of all variables (some conditionally
  independent and some are not)
• Depends on the structure!
• The probability of a specific assignment of
  values y1,y2,yn for the variables Y1,Y2,,Yn
• This is the likelihood of the data given the
  model.
• All you need to know is..

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Learning Bayesian network from data

• Given the data set with specific assignments for
  variables (on/off for each gene), how can we find
  the most probable network structure that explains
  the data (the best match to the data)?
• How to quantify a match?
• Note that there are two aspects of the network
  that we need to learn
• Structure (nodes, edges)
• Conditional probability distributions
• Common strategy assign a score to each network G

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• Common strategy assign a score to each network G
• Pick the network that maximizes the score

Likelihood prior
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Learning

• The likelihood of the data given the model is
  estimated by averaging over all possible
  assignments of parameters (conditional
  probabilities) to G
• Summation over all possible assignments for
  conditional probabilities. The major contribution
  is from the set estimated from the data
• Given a specific structure, for every node we
  lookup its parents and calculate the empirical
  conditional probability distribution

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Model selection

• The second term (log prior) is a measure for the
  complexity of the model (through uncertainty)
• Occam razor entia non sunt multiplicanda
  praeter necessitatem(thou shall not multiply
  entities)
• MDL principle
• In the papers discussed here it is being ignored

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In search for the best network

• In theory test different structures, calculate
  the probability of assignment to variables for
  each network structure, and output the network
  that maximizes the likelihood of the data given
  the network.
• Impossible in practice the number of possible
  networks over n genes is
• For the yeast genome with 6000 genes this is gt
  105,000,000

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Possible solution

• Apply a heuristic local greedy search Start
  with a random network and locally improve it, by
  testing perturbations over the original
  structure.
• Test one edge at a time, by adding, removing or
  reversing the edge, and testing its affect on the
  score. If the score improves - accept

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How to learn from expression data

• Two types of features learned from multiple
  networks
• First - a gene Y is in the Markov blanket of X
  (two genes are involved in the same biological
  process. No other gene mediates the dependence)
• Problem of unobserved variables that can
  intermediate the interaction
• Second type a gene X is ancestor of Y (based on
  all networks that are learned)

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Application to the Yeast Cell cycle data

• Expression level measurements for 6177 genes
  along different time points in six cell cycles
  altogether 76 measurements for each gene
• Only 800 genes vary during cell cycle and 250
  cluster into 8 fairly distinct classes.
• Networks are learned for the 800 genes
• Confidence values based on the set of networks
  learned from different bootstrap sets

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Typical sub-network
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Biological significance

• Order relations there are a few dominant genes
  that appear before many others, e.g. genes that
  are involved in cell cycle control and initiation.

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• Most are nuclear proteins, but also cytoplasm
  membrane proteins (budding and sporulation)
• Some DNA repair proteins (prerequisite for
  transcription)
• RSR1 initiator of signal trunsduction cascades
  in the cell

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Biological significance

• Markov connection functionally related

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• Most pairs have similar functions (verified
  sometimes through transitivity)
• Some are physically adjacent on the chromosome
• Some relations cannot be detected directly from
  expression data
• Detect conditional independence group of genes
  that are expressed similarly, but one is a parent
  of all others and there are no connections
  between the others the parent is a control gene
  (e.g. CLN2 early cell cycle control gene, that
  controls RNR3, SVS1, SRO4 and RAD41 that are
  functionally unrelated).

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Conclusions

• A powerful tool, but
• not enough data
• Computational problems
• Learning algorithms
• Authors decompose networks into basic elements
  again
• Many possible extensions