1. Systems Theoretical Modelling of Genetic Pathways Ronald L. Westra Systems Theory Group Department Mathematics Maastricht University

2. 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

3. 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

4. Static Genetic Pathways Start with isolated microarray data Reconstruct causal relations with conditional probability models, e.g.: Bayesian belief networks Bootstrap methods

5. Reconstruction of Genetic Pathway Exp1Exp2Exp3Exp4 Gen11.8-0.21.7-0.4 Gen2-1.1-1.91.01.6 Gen30.41.3-1.3-1.8 Gen4-0.1-0.2-0.4-0.6 Gen51.60.51.71.3 Gen61.01.21.7-2.0 Microarray experiments Gene expression data

6. Reconstructed Genetic Pathway G2 G1 G4 G5 G6 G3

7. Problems with Genetic Pathways Equilibrium: * oscillation (N-cycle) * punctuated (intermitted stasis) Non-equilibrium: * pulse reaction * growth Chaotic: * carcinogenesis

8. Problems with Genetic Pathways Emergent and complex behaviour Cooperative behaviour due to multigene interaction upward complexity

9. 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

10. Gene expression measured at: * certain moment: sampling * during some time: averaging What exactly represents the measured value of gene expression

11. Conclusion: Genetic Pathways are Dynamic Systems

12. Pr2 j 2 : strength of proteine current Pr2 Pr1 j 1 : strength of proteine current Pr1 cause at t = 0: G1 x 1 : degree of expression of G1G2 G1 Biochemical Environment Pr1 Pr2 result at t = T: G2 x 2 : degree of expression of G2 What happens in a Genetic pathway ?

13. 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)

14. Dynamic Genetic Pathways Place of Endo16 in Sea urchin genome map

15. Dynamic Genetic Pathways Sea urchin - Endo16 related gene and gene products expressions

16. Dynamic Genetic Pathways Circuit diagram for Endo16 transcription

17. Dynamic Genetic Pathways Decision rules for Endo16 Dynamics

18. 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?

19. 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 Modelling Dynamic Genetic Pathways

20. Upinder Bhalla (National Center Biological Sciences, India) and Ravi Iyengar (Mount Sinai school of Medicine, New York) : neural networks Modelling Dynamic Genetic Pathways

21. Modelling Dynamic Genetic Pathways

22. Bahalla and Iyengar used 15 genetic circuits in their model

23. 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

24. 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

25. Kurt Kohn (National Cancer Institute, Bethesda, USA): electrical circuits Modelling Dynamic Genetic Pathways

26. Gene networks as electric logic circuits

28. Gene control clock

30. Systems Theory: compartimental time- delayed dynamical system Modelling Dynamic Genetic Pathways

31. Modelling Dynamic Genetic Pathways Pr2 j 2 : strength of proteine current Pr2 Pr1 j 1 : strength of proteine current Pr1 cause at t = 0: G1 x 1 : degree of expression of G1 G2G2G2G2 G1G1G1G1 Biochemical Environment Pr1 Pr2 result at t = T: G2 x 2 : degree of expression of G2

32. Modelling Dynamic Genetic Pathways x k = Present expression of gene k. It results from past expressions of – potentially all – other genes with certain transfer function G and parameters  (time delay, coupling strength, threshold)

33. Modelling Dynamic Genetic Pathways Several problems e.g. : how to model the environment? 1.Transfer-function – parameter set 2.Input (in case of external agent, eg toxic)

34. Modelling Dynamic Genetic Pathways Basic model: x k = expression of gene k u k = external inputs (eg toxic agents) y k = observable output (eg proteine) n k = noise  = parameter set(time delay, coupling strength, threshold)

35. Modelling Dynamic Genetic Pathways GP as directed and weighted graph G2 G1 G4 G5 G6 G3 +0.23 +1.03 +0.48 -0.71 +0.75 +0.36 -0.84 +0.19 +0.27 -0.18

36. Use of System Model 1.Validate “known” Genetic pathway 2.Calculate relevant constants as gene-gene-coupling parameters, relative thresholds, effective time- delays 3.Qualitatively “explain” observed complex behaviour from the model 4.Reconstruct genetic pathways from individual dynamic gene- expressions

37. approach 1 : Linear autoregression ARX approach 2 : subspace identification N4SID Bilinear Systems Approach

38. 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 ~50*600 euro : 30.000 euro

39. Preliminary Results Cross validation of specific gene expression

40. 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

41. Ronald L. Westra Systems Theory Group Department Mathematics Maastricht University Westra@math.unimaas.nl