Gene Regulatory Networks - the Boolean Approach

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Gene Regulatory Networks - the Boolean Approach

• Andrey Zhdanov
• Based on the papers by Tatsuya Akutsu et al
• and others

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Gene Regulatory Networks - the Boolean Approach

• Gene Expressions Revisited

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Gene Expressions Revisited

• One of the major subjects of study in cell
• biology is the behaviour of proteins the
• workhorses of a cell.

Myoglobin molecule
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Gene Expressions Revisited

• We are interested in analysing protein
• expression levels amounts of different
• proteins synthesized by the cell.

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Gene Expressions Revisited

• The blueprints for all possible proteins that
• can be synthesized by a cell genes are
• stored in the cell's nucleus.
• Only small fraction of all possible proteins is
• synthesized in each cell.

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Gene Expressions Revisited

• Proteins are synthesized from genes by the
• process of transcription and translation.

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Gene Expressions Revisited

• We estimate protein expression levels
• indirectly by measuring gene expression
• levels (amounts of mRNA produced for a
• certain gene) with DNA chips.

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Gene Expressions Revisited

• This approach makes a number of
• assumptions
• Genes exist and are easily identifiable
• Each protein is encoded by a single gene
• Protein expression (amount of protein produced)
  is determined by the corresponding gene
  expression (amount of mRNA produced)
• These assumptions do not always hold (but
• we use them anyway -)

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Gene Regulatory Networks - the Boolean Approach

• Gene Regulatory Networks

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Gene Regulatory Networks

• We want to use protein (or gene) expression
• measurements to understand the mechanisms
• regulating proteins' production.
• Note that there is certain circularity to our
  logic
• since we made certain assumptions about
• these very same mechanisms in order to
• measure protein expressions.

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Gene Regulatory Networks

• In the talks by Shahar and Leon we have
• seen the regulatory network approach to
• modelling the protein expression mechanisms.
• In his talk Oded has introduced tools for time
• series analysis that can be applied to our
• problem.

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Gene Regulatory Networks

• We are looking for a formal model of the
• protein expression control mechanism that
• can serve as a framework for a rigorous
• treatment of the problem.
• To that end we assume that production rate of
• a certain protein at any given time is regulated
• only by the amount of other proteins within the
• cell at that time.

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Gene Regulatory Networks

• Example

Protein B
Protein D
inhibits
excites
Protein A
excites
Protein C
Expression level
Protein A
Protein B
Protein C
Protein D
time
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Gene Regulatory Networks

• Treating the gene expressions as real-valued
• functions of continuous time variable leads to
• the system of differential equations as the
• model for the gene regulatory network.

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Gene Regulatory Networks - the Boolean Approach

• Boolean Regulatory Networks

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Boolean Regulatory Networks

• To facilitate the treatment of the problem we
• further simplify our model to the Boolean
• Regulatory Network. We assume
• Discrete time and synchronous update model
• Genes expression level is binary

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Boolean Regulatory Networks

• More formally, a boolean network
• consists of a set of nodes representing genes
• and a list of boolean functions
• where is computes boolean
  function
• of nodes and assigns the output to

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Boolean Regulatory Networks

• The state of the network at time t is defined by
• assignment of 0s and 1s to the node variables.
• The state of each node at time t1 is
• calculated from the states of the nodes
• at time t according to

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Boolean Regulatory Networks

• Boolean regulatory network can be visualized
• by the means of wiring diagram

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Boolean Regulatory Networks

• Since the networks state at t1 is completely
• determined by its state at t, we can treat the
• gene expressions time series as an unordered
• set of input / output pairs.
• We say that the network is consistent with a
• set of input/output pairs if for each pair
  setting
• the network to the input state at time t causes
• it to reach the output state at t1.

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Boolean Regulatory Networks

• We can now start formulating some of the
• fundamental problems for our model.
• CONSISTENCY Given the number of nodes
• and set of input/output pairs, decide whether
• there is a boolean network consistent with the
• pairs.

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Boolean Regulatory Networks

• COUNTING Given the number of nodes
• and set of input/output pairs, count the number
• of boolean networks consistent with the
• pairs.

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Boolean Regulatory Networks

• ENUMERATION Given the number of nodes
• and set of input/output pairs, output all the
• boolean networks consistent with the pairs.

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Boolean Regulatory Networks

• IDENTIFICATION Given the number of nodes
• and set of input/output pairs, decide whether
• there is a unique boolean network consistent
• with the pairs and output one if exists.

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Boolean Regulatory Networks

• The four problems presented above are
• closely related. We address them in the
• straightforward manner by constructing all
• possible boolean networks and checking them
• on all the input/output pairs.
• To make this task computationally feasible we
• need yet another assumption we assume
• that the networks indegree is bounded by
• some constant K.

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Boolean Regulatory Networks

• Some of the results
• The complexity of the brute-force algorithm for
• the CONSISTENCY problem is
• Where is the number of nodes (genes) and
• is the number of input/output pairs.
• The results for the other problems are similar.

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Boolean Regulatory Networks

• Another theoretical result concerns the
• number of input/output pairs required to
• uniquely identify a boolean network.
• Again, to facilitate calculations, we make an
• unrealistic assumption we assume that the
• input/output pairs are randomly drawn from a
• uniform distribution.

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Boolean Regulatory Networks

• Theorem If
  input/output
• expressions are drawn from a uniform
• distribution, the probability that there are more
• than one boolean network consistent with
• them is at most

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Boolean Regulatory Networks

• Conclusions
• Boolean gene expression networks represent
• a relatively simple model of the gene
• expression control mechanisms of the cell.
• However, despite many (often unrealistic)
• simplifying assumptions, this model has not
• yielded any interesting theoretical results yet,
• which indicates the intristic difficulty of
• modeling gene expression mechanisms.