Ronald L' Westra

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Systems Theoretical Modelling of Genetic
Pathways

• Ronald L. Westra
• Systems Theory Group
• Department Mathematics
• Maastricht University

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Dynamic Genetic Pathways

• What is a Genetic Pathway
• Static versus dynamic view on Genetic Pathway
• Modelling Genetic Pathway
•  Some approaches to modelling dynamic pathways
• Framework for System Theoretic Approach

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Genetic Pathway

• Network of those genes that are connected by
  causal relations in their expressions
• Input few microarray data of gene expression as
  function of some experiments
• Implicit assumption of convergent behaviour

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Static Genetic Pathways

• Start with isolated microarray data
• Reconstruct causal relations with conditional
  probability models, e.g.
• Bayesian belief networks
• Bootstrap methods

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Reconstruction of Genetic Pathway
Microarray experiments
Gene expression data
Exp1 Exp2 Exp3 Exp4 Gen1 1.8 -0.2 1.7 -0.4 Gen2 -
1.1 -1.9 1.0 1.6 Gen3 0.4 1.3 -1.3 -1.8 Gen4 -0.1
-0.2 -0.4 -0.6 Gen5 1.6 0.5 1.7 1.3 Gen6 1.0 1.2 1
.7 -2.0
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Reconstructed Genetic Pathway
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Problems with Genetic Pathways

• Equilibrium
• oscillation (N-cycle)
• punctuated (intermitted stasis)
• Non-equilibrium
• pulse reaction
• growth
• Chaotic
• carcinogenesis

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Problems with Genetic Pathways

• Emergent and complex behaviour
• Cooperative behaviour due to multigene
  interaction
• upward complexity

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Problems with Genetic Pathways

• In vivo the expression of a specific gene varies
  over time
• Moreover a gene can be expressed because
• being expressed is its default value
• it is part of another active pathway

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What exactly represents the measured value of
gene expression

• Gene expression measured at
• certain moment sampling
• during some time averaging

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Conclusion Genetic Pathways are Dynamic
Systems
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What happens in a Genetic pathway ?

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Dynamic Genetic Pathways

• Example of dynamic GP
• in non-equilibrium is
• during growth
• Model is the regulation of the Endo16 gene in sea
  urchin (Strongylocentrotis purpuratus)
• Eric H. Davidson,
• C-H Yu,
• Caltech (http//www.its.caltech.edu)

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Dynamic Genetic Pathways

• Place of Endo16 in Sea urchin genome map

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Dynamic Genetic Pathways

• Sea urchin -Endo16 related gene and gene products
  expressions

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Dynamic Genetic Pathways

• Circuit diagram for Endo16 transcription

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Dynamic Genetic Pathways

• Decision rules for Endo16 Dynamics

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Modelling Dynamic Genetic Pathways

• The if-then modelling by Davidson and Yuh is
  efficient but phenomenological
• Can we provide deep models for dynamic gene
  networks?

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Modelling Dynamic Genetic Pathways

• Upinder Bhalla (National Center Biological
  Sciences, India) and Ravi Iyengar (Mount Sinai
  school of Medicine, New York) neural
  networks
• Kurt Kohn (National Cancer Institute, Bethesda,
  USA) electrical circuits

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Modelling Dynamic Genetic Pathways

• Upinder Bhalla (National Center Biological
  Sciences, India) and Ravi Iyengar (Mount Sinai
  school of Medicine, New York) neural
  networks

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Modelling Dynamic Genetic Pathways
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Bahalla and Iyengar used 15 genetic circuits in
their model
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Bahalla and Iyengar used GENESIS neural network
simulator to model 15 genetic circuits

• Validation of model on a testset
• open neural network simulation
• filled real data

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Bhalla, Iyengar Approach

• conclusion
• Modelling DGP as neural network provides good
  fit, but
• model can be sub-optimal
• each new case must be trained separately
• black-box, no deep model

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Modelling Dynamic Genetic Pathways

• Kurt Kohn (National Cancer Institute, Bethesda,
  USA) electrical circuits

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• Gene networks as electric logic circuits

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• Gene networks as electric logic circuits

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• Gene control clock

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• Gene control clock

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Modelling Dynamic Genetic Pathways

• Systems Theory compartimental time- delayed
  dynamical system

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Modelling Dynamic Genetic Pathways
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Modelling Dynamic Genetic Pathways

• xk Present expression of gene k .
• It results from past expressions of potentially
  all other genes with certain transfer function
  G and parameters q (time delay, coupling
  strength, threshold)

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Modelling Dynamic Genetic Pathways

• Several problems
• e.g. how to model the environment?
• Transfer-function parameter set
• Input (in case of external agent, eg toxic)

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Modelling Dynamic Genetic Pathways

• Basic model

xk expression of gene k uk external inputs
(eg toxic agents) yk observable output (eg
proteine) nk noise q parameter set(time
delay, coupling strength, threshold)
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Modelling Dynamic Genetic Pathways

• GP as directed and weighted graph

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Use of System Model

• Validate known Genetic pathway
• Calculate relevant constants as
  gene-gene-coupling parameters, relative
  thresholds, effective time-delays
• Qualitatively explain observed complex
  behaviour from the model
• Reconstruct genetic pathways from individual
  dynamic gene-expressions

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Bilinear Systems Approach

approach 1 Linear autoregression ARX approach
2 subspace identification N4SID
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Experimental Data

• model for a dynamic genetic pathway induction
  of multiple gene expression changes in the human
  hepatoma HepG2 cell line by the established human
  carcinogen benzo(a)pyrene.
• 2 series measurement each 5 minutes during 120
  minutes
• Ma costs 50600 euro 30.000 euro

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Preliminary Results

• Cross validation of specific gene expression

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Conclusions

• Genetic Pathways are dynamic systems
• In vivo micro array measurement can be obscured
  by dynamic behaviour of gene expression
• Modelling of DGP with NN results in black box
• Modelling of DGP with electrical circuits is
  successful but only in forward direction
• Modelling with Systems Identification approach
  allows for forward and backward modelling
• Modelling with Systems Identification approach
  allows for reconstruction of GP from dynamical
  data
• Disadvantage many measurements necessary,
  sensitive to hidden parameters and missing values

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• Ronald L. Westra
• Systems Theory Group
• Department Mathematics
• Maastricht University
• Westra_at_math.unimaas.nl