1. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks1 Ronald L. Westra, Department of Mathematics Lars Eijssen, Joyce Corvers, Department of Genetics Maastricht University On the identifiability of piecewise linear gene-protein networks relative to noise and chaos G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ1Σ1 Σ2Σ2

2. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks2 1. Background and problem formulation 2. Modeling and identification of gene/proteins interactions 3. The implications of stochastic fluctuations and deterministic chaos 5. Example 1: Application on artificial reaction model 5. Example 2: Application on Tyson-Novak model for fission yeast 5. Example 3: Application on fission yeast expression data 6. Conclusions Items in this Presentation

3. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks3 Question: Can gene regulatory networks be reconstructed from time series of observations of (partial) genome wide and protein concentrations? 1. Problem formulation

4. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks4 Relation between mathematical model and phys-chem-biol reality Macroscopic complexity from simple microscopic interactions Approximate modeling as partitioned in subsystems with local dynamics Modeling of subsystems as piecewise linear systems (PWL) PWL-Identification algorithms: network reconstruction from (partial) expression and RNA/protein data Experimental conditions of poor data: lots of gene but little data The role of stochasticity and chaos on the identifiability Problems in modeling and identification

5. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks5 2. Modeling the Interactions between Genes and Proteins Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

6. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks6 2.1 Modeling the molecular dynamics and reaction kinetics as Stochastic Differential Equations Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

7. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks7 2.2 Gene-Protein Interaction Networks as Piecewise Linear Models Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

8. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks8 2.3 Problems concerning the identifiability of PieceWise Linear models Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

9. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks9 3. The Implications of Stochastic fluctuations and Deterministic Chaos Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

10. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks10 3.1 Stochastic fluctuations Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

11. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks11 3.2 Noise-induced control in single-cell gene expression Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

12. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks12 Influence of stochastic fluctuations on the evolution of the expression of two coupled genes..

13. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks13 3.3 Deterministic Chaos Prerequisite for the successful reconstruction of gene- protein networks is the way in which the dynamics of their interactions is modeled.

14. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks14 4. Identification of Interactions between Genes and Proteins Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

15. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks15 4.2 The identification of PIECEWISE linear networks by L 1 -minimization Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

16. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks16 Gene-Protein Interaction Networks as Piecewise Linear Models The general case is complex and approximate Strongly dependent on unknown microscopic details Relevant parameters are unidentified and thus unknown

17. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks17 2. Modeling of PWL Systems as subspace models Global dynamics: Local attractors (uniform, cycles, strange) Basins of Attraction Each BoA is a subsystem Σ i “checkpoints” State space Σ1Σ1 Σ2Σ2 Σ3Σ3 Σ4Σ4 Σ5Σ5 Σ6Σ6

18. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks18 Modeling of PWL Systems as subspace models State vector moves through state space driven by local dynamics (attractor, repeller) and inputs in each subsystem Σ 1 the dynamics is governed by the local equilibria. approximation of subsystem as linear statespace model: State space

19. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks19 Problems concerning the identifiability of Piecewise Linear models 1. Due to the huge costs and efforts involved in the experiments, only a limited number of time points are available in the data. Together with the high dimensionality of the system, this makes the problem severely under-determined. 2. In the time series many genes exhibit strong correlation in their time-evolution, which is not per se indicative for a strong coupling between these genes but rather induced by the over-all dynamics of the ensemble of genes. This can be avoided by persistently exciting inputs.

20. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks20 Problems concerning the identifiability of Piecewise Linear models 3. Not all genes are observed in the experiment, and certainly most of the RNAs and proteins are not considered. therefore, there are many hidden states. 4. Effects of stochastic fluctuations on genes with low transcription factors are severe and will obscure their true dependencies.

21. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks21 Such are the problems relating to the identifiability of piecewise linear systems: Are conditions for modeling rate equations met? High stochasticity and chaos Are piecewise linear approximations a valid metaphor? Problems with stochastic modeling

22. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks22 The identification of PIECEWISE linear networks by L 1 -minimization K linear time-invariant subsystems {Σ 1, Σ 2,.., Σ K } Continuous/Discrete time

23. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks23 4.2 The identification of PIECEWISE linear networks by L 1 -minimization Weights w kj indicate membership of observation #k to subsystem Σ j :

24. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks24 Rich and Poor data poor data : not sufficient empirical data is available to reliably estimate all system parameters, i.e. the resulting identification problem is under- determined.

25. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks25 (un)known switching times, regular sampling intervals, rich / poor data, Identification of PWL models with known switching times and regular sampling intervals from rich data Identification of PWL models with known switching times and regular sampling intervals from poor data

26. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks26 1. unknown switching times, regular sampling intervals, poor data, known state derivatives This is similar to simple linear case

27. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks27 This can thus be written as:

28. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks28 with:

29. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks29 with: The approach is as follows: (i)initialize A, B, and W, (ii)perform the iteration: 1. Compute H1 and H2, using the simple linear system approach 2. Using fixed W, compute A and B, 3. Using fixed A and B, compute W until: (iii) criterion E has converged sufficiently – or a maximum number of iterations.

30. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks30 Linear L1-criterion:

31. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks31 With linear L 1 -criterion E 1 the problem can be formulated as LP-problem: LP1: compute H 1,H 2 from simple linear case LP2: A and B, using E 1 -criterion and extra constraints for W, H 1,H 2, LP3: compute optimal weights W, using E 1 -criterion with constraints for W, H 1,H 2, A and B

32. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks32 2. unknown switching times, regular sampling intervals, poor data, unknown state derivatives Use same philosophy as mentioned before

33. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks33 Subspace dynamics and linear L1-criterion :

34. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks34 System parameters and empirical data :

35. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks35 Quadratic Programming problem QP : Problem: not well-posed: i.e.: Jacobian becomes zero and ill-conditioned near optimum

36. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks36 Therefore split in TWO Linear Programming problems:

37. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks37 In case of sparse interactions replace LP1 with:

38. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks38 Performance of robust Identification approach Artificially produced data reconstructed with this approach Compare reconstructed and original data

39. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks39 The influence of increasing intrinsic noise on the identifiability.

40. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks40 a: CPU-time Tc as a function of the problem size N, b: Number of errors as a function of the number of nonzero entries k, M = 150, m = 5, N = 50000.

41. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks41 a: Number of errors versus M, b: Computation time versus M N = 50000, k = 10, m = 0.

42. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks42 a: Minimal number of measurements Mmin required to compute A free of error versus the problem size N, b: Number of errors as a function of the intrinsic noise level σ A N = 10000, k = 10, m = 5, M = 150, measuring noise B = 0.

43. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks43 Example 1 : how to apply this method on current data sets Spellman et al. data for cell-cycle of fission yeast : Components: 6179 genes measured for 18-24 irregular time instants Processing: fuzzy C-means, gene annotation with Go term finder and Fatigo, net recontruction with identification algorithm

44. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks44 Spellman et al. data for cell-cycle of fission yeast : Processing: Selection of most up/down-regulated genes: 3107 from 6179 Clustering: fuzzy C-means: best outcome 23 clusters Gene annotation with Go term finder (4th level) and Fatigo, both for biological process and cellular component Net recontruction with identification algorithm on 23 clusters

45. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks45 Centroids after clustering 23 clusters

46. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks46 Gene ontology

47. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks47 Gene ontology Cluster 1 GO Term Finder: The genes are involved in spindle pole during the cell cycle, with relations to microtubuli and chromosomal structure. FatiGO: The main cellular component is the chromosome. Cluster 2 GO Term Finder: The genes are involved in proliferation and replications, especially bud neck and polarized growth. FatiGO: The results found by the GO Term Finder are confirmed. ……………. Cluster 22 GO Term Finder: Only a few annotations are found and there are many unknown genes. The genes are involved in respiration and reproduction. The main cellular components are the actin/cortical skeleton and the mitochondrial inner membrane. FatiGO: No further clear annotations are found. Cluster 23 GO Term Finder: The genes are involved in RNA processing. The main cellular components are the nucleus, the RNA polymerase complex and the ribonucleoprotein complex. FatiGO: The main cellular component is the ribonucleoprotein complex.

48. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks48 Reonstructed network

49. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks49 Example 2: artificial data of hierarchic/sparse network Artificial reaction network with: Components: 2 master genes with high transcription rates 3 slave genes with low transcription rates 4 agents (= RNA or proteins). Processes: stimulation, inhibition, transcription, and reactions between ‘agents’

50. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks50 Dynamics : – large hierarchic and sparse network – implicit relation between genes with expression x through agents (= proteins, RNA) with concentration a – system near equilibrium and small perturbations – inputs: persistent excitation u

51. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks51 Dynamics : – implicit system dynamics: – linear statespace model makes gene interaction explicit:

52. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks52 Dynamics : – estimate gene-gene interaction matrix A from empirical data:

53. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks53 reactions

54. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks54 reactions

55. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks55 reactions

56. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks56 Matlab-simulation y(1) = - 0.03*x(1) + 0.2*(1-x(1))*a(2)^2 - 0.2*x(1)*a(3) ; y(2) = - 0.05*x(2) + 0.3*(1-x(2))*a(1) - 0.1*x(2)*a(4) ; y(3) = - 0.02*x(3) + 0.1*(1-x(3))*a(2) - 0.1*x(3)*a(1) ; y(4) = - 0.01*x(4) + 0.2*(1-x(4))*a(1)*a(2) - 0.2*x(4)*a(3)^2; y(5) = - 0.02*x(5) + 0.3*(1-x(5))*a(3) - 0.1*x(5)*a(1); y(6) = - 0.02*a(1) + 0.4*x(1) - 0.2*a(1)*a(2) - 0.1*a(1)*a(3)^3; y(7) = - 0.01*a(2) + 0.15*x(2) - 0.2*a(1)*a(2); y(8) = - 0.01*a(3) + 0.2*a(1)*a(2) - 0.1*a(1)*a(3)^3; y(9) = - 0.05*a(4) + 0.9*a(1)*a(3); rate equations

57. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks57 Real network structure: implicit 21 a 3 4 5 d b c p g gene agent

58. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks58 Real network structure: explicit 21 3 45 slave master

59. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks59

60. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks60

61. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks61

62. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks62 21 3 45 Reconstructed network structure: low noise master slave

63. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks63 21 3 45 Reconstructed network structure: moderate noise slave master

64. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks64 Reconstructed network structure: high noise (an example) 21 3 45 slavemasterslave master

65. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks65 Example 3: data of Tyson-Novak math. model for cell cycle Tyson-Novak model for cell-cycle of fission yeast : Components: 9 agents (= RNA or proteins). Processes: stimulation, inhibition, transcription, and reactions between ‘agents’

66. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks66 The deterministic Tyson-Novak model.

67. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks67 The stochastic Tyson-Novak model.

68. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks68 Example: stochastic Tyson-Novak model

69. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks69 Example: stochastic Tyson-Novak model

70. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks70 4.2 The identification of PIECEWISE linear networks by L 1 -minimization Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

71. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks71 5. Epilogue: Lessons from Nature Prerequisite for the successful reconstruction of gene-protein networks is the way in which the dynamics of their interactions is modeled.

72. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks72 Discussion …  G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ1Σ1 Σ2Σ2