Reverse Engineering of Polynomial Models of Gene Regulatory Networks

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Reverse Engineering of Polynomial Models of Gene
Regulatory Networks

• Lecture 4 Jan-Feb 04
• Dr. Eduardo Mendoza
• Physics Department
• Mathematics Department Center for
  NanoScience
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de

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Topics

• Reverse Engineering of systems
• Reverse Engineering Algorithm for Polynomial
  Models
• Application to the segment polarity GRN of
  Drosophila melanogaster
• Concluding remarks

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Reference

• Laubenbacher, R., Stiegler, B
• A Computational Algebra Approach to the Reverse
  Engineering of Gene Regulatory Networks (to
  appear in Journal of Theor. Biology)

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1. Reverse Engineering of Systems

• Systems identification in Engineering goal is to
  construct a system with prescribed dynamical
  properties
• In Systems Biology, one is interested in
  identifying as closely as possible a unique
  biological system that has been observed
  experimentally
• In both cases sparsity of available measurements
  will leave the system underdetermined

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Reverse Engineering of Networks

• Modeling framework for a network with n (known)
  nodes
• time-discrete
• finite set X of possible values for each node
• choose X to be a finite field k (with k ps, p
  prime)
• Given are
• set of state transitions (usually in form of
  time series)

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Reverse Engineering (contd)

• The problem is to find a function f kn ? kn
  (finite dynamical system) such that

• Each time series is then called a trace of f
• f is typically not unique (unless all state
  transitions of the system are specified)
• ? Choice between a large set of solutions
  necessary

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Polynomial models

• Write f (f1 ,f2 ,...,fn). Then each
  coordinate function fi can be expressed
  (uniquely) as an element of kX1 ,X2 ,...,Xn,
  i.e.
• fi Sa ai,a Xa and aj k
• i.e. Every finite dynamical system over k is
  simply a vector of polynomial functions with
  coefficients in k.

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Criterion for model selection

• Choose each fi minimal in the sense that

• Note this criterion is different from the
  usual sparsest network criterion (e.g. Yeung et
  al, 2002 or the REVEAL algorithm by Liang et al
  ) a network such that each node takes inputs
  from as few variables as possible

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Benefits of polynomial approach
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2. Polynomial Reverse Engineering Algorithm

• It suffices to consider the case of one time
  series.

Step1
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Step 1 (contd)
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Step 2 Compute vanishing ideal
Gröbner basis !
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Step 3 find reduction f of f0
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Complexity of the algorithm
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3. Application to Drosophila SPN

• Use the Albert-Othmer model to test if reverse
  algorithm works
• i.e. Given time series data, can the reverse
  algorithm uncover the A-O model?

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Polynomial equations (1)
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Equations with additional variables
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Laubenbacher-Stiegler Model
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Summary of test results
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Example of predicted relationship
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Data Discretization (1)
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Data Discretization (2)
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Gepasi Simulation (1)
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Gepasi Simulation (2)
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Perspectives for MS Theses

• Interesting to test the LS algorithm for other
  biological networks
• use DVD as tool
• investigate choice of p gt 2
• begin with existing Boolean or GLN networks
• best in conjunction with an experimentalist
• Open question ways to use results of coding
  theory?

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Thanks for your attention !

• Questions?