1. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks1 Ronald L. Westra Department of Mathematics Maastricht University Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks 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 the interactions between genes and proteins 3. The implications of stochastic fluctuations and deterministic chaos 4. Identification of interactions between genes and proteins 5. Lessons from nature 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 Biochemical reactions and kinematic rate equations: this is a microscopic reality: (in)elastic collisions, electrostatic forces, “binding” this is a statistic average. true only under some conditions

7. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks7 Conditions for modeling reactions as rate equations The law of large numbers. In inhomogeneous mixtures or in slow reactions as in gene-, RNA-, and protein-interactions this will not (always) be true. Hence; the problem is stochastic. The Maxwell velocity distribution should apply, otherwise details of the velocity distribution will enter. This condition is not met for macromolecules in a cytoplasm. The distribution of the internal degrees of freedom of the constituents, like rotational and vibrational energies, must have the same ’temperature’ as the Maxwell velocity distribution, otherwise it will influence the rate of the collisions that result in a chemical reaction. This condition is not met by gene/RNA/protein interactions.

8. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks8 2.2 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

9. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks9 1. Qualitative Piecewise Linear Models Following Hidde de Jong et al. (2002, 2004) b il sum of step-functions s +,–

10. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks10 Qualitative Piecewise Linear Models – H. de Jong Example with kind permission from: Hidde de Jong (J.Comp.Biol.2002)

11. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks11 Qualitative Piecewise Linear Models – H. de Jong Example with kind permission from: Hidde de Jong (J.Comp.Biol.2002)

12. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks12 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

13. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks13 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

14. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks14 2.3 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.

15. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks15 2.3 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.

16. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks16 3. The Implications of Stochastic fluctuations and Deterministic Chaos Experimental and theoretical studies now indicate that stochasticity plays a vital role in the self- organized control of biological processes.

17. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks17 Influence of stochastic fluctuations on the evolution of the expression of two coupled genes. Stochasticity appears detrimental to the process

18. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks18 3.1 Stochastic fluctuations i. In experimental work on E. coli, Elowitz and Swain found that low intracellular copy numbers of molecules can limit the precision of gene regulation. They found that: genotypic identical cells exhibit substantial phenotypic variation this variation arises from stochasticity in gene expression this variation is essential for many biological processes prime factors in stochasticity are: transcription rate, regulatory dynamics, and genetic factors.

19. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks19 ii. In stochastic simulations on Drosophila and Neurospora, Goldbeter and Gonze found that robust circadian oscillations can emerge at the cellular level, even when only a few tens of mRNA and protein molecules are involved. This shows how autoregulation processes at the cellular level allow the emergence of a coherent biological rhythm out of molecular noise.

20. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks20 3.1 Stochastic fluctuations iii. Steuer found that the addition of noise to a deterministic simulation model of the cell-cycle in fission yeast (Tyson-Novak model) could explain several experimental findings, such as the existence of quantized cycle times in double-mutant wee1−cdc25 cells. Moreover, he found that his stochastic model led to the emergence of noise induced oscillations.

21. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks21 The deterministic and stochastic Tyson-Novak model.

22. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks22 Stochastic fluctuations Nature is robust to noise and exploits noise in controlling quasi-cyclic processes by hierarchic and sparse organization: masters and slaves Different roles in different sub-systems Therefore flexible network structure

23. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks23 Flexible network structure with chancing master-slaves G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ1Σ1 Σ2Σ2

24. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks24 Example

25. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks25 Example

26. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks26 Example

27. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks27 Example 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);

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

29. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks29

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

31. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks31 4. Identification of Interactions between Genes and Proteins Based on the (sub) space models of piecewise linear systems.

32. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks32 4.1 The identification of SIMPLE linear networks by partial L 1 -minimization First we study the case of K = 1, i.e. one subsystem We assume a hierarchic, non-symmetric, and sparse system with linear state space dynamics

33. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks33 General subspace dynamics The evolution of the n-dimensional state space vector x (gene expressions) depend on p-dim inputs u, parameters θ and white noise ξ.

34. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks34 Linearized form of subsystem Σ l First order linear approximation of system separates state vector x and inputs u.

35. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks35 Data from experiments & observations Empirical data : Hankel-matrices

36. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks36 Relation between Hankel matrices The relation between the state derivatives, states and inputs is also applicable to the data matrices.

37. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks37 Row-by-row relation Set of N decoupled linear systems of size Mx(N+m) α is sparse, β not necessarily

38. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks38 Reformulation: A: Hankel matrices X and U, x: rows of A and B, b: row of state derivatives

39. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks39 Solution to partial sparse case Primal problem

40. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks40 Partial sparse case – dual approach Dual problem

41. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks41 Performance of dual partial approach Artificially produced data reconstructed with this approach Compare reconstructed and original data

42. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks42 The influence of increasing intrinsic noise on the identifiability.

43. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks43 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.

44. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks44 a: Number of errors versus M, b: Computation time versus M N = 50000, k = 10, m = 0.

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

46. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks46 Second we study the general case of K > 1, i.e. multiple subsystem Again: we assume hierarchic, non-symmetric, and sparse systems with linear state space dynamics 4.2 The identification of PIECEWISE linear networks by L 1 -minimization

47. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks47 The identification of PIECEWISE linear networks by L 1 -minimization K linear time-invariant subsystems {Σ 1, Σ 2,.., Σ K } Continuous/Discrete time

48. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks48 4.2 The identification of PIECEWISE linear networks by L 1 -minimization Weights w kj indicate membership of observation #k to subsystem Σ j :

49. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks49 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.

50. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks50 (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

51. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks51 1. unknown switching times, regular sampling intervals, poor data, known state derivatives This is similar to simple linear case

52. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks52 This can thus be written as:

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

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

55. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks55 Quadratic L2-criterion versus Linear L1-criterion:

56. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks56 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

57. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks57 2. unknown switching times, regular sampling intervals, poor data, unknown state derivatives Use same philosophy as mentioned before

58. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks58 Subspace dynamics and linear L1-criterion :

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

60. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks60 Quadratic Programming problem QP : Problem: not well-posed: i.e.: Jacobian becomes zero and ill-conditioned near optimum

61. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks61 Therefore split in TWO Linear Programming problems:

62. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks62 In case of sparse interactions replace LP1 with:

63. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks63 3. unknown switching times, irregular sampling intervals, poor data, unknown state derivatives Now, the problem has become more complex …

64. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks64 Continuous subspace dynamics and quadratic L2-criterion :

65. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks65 Iteration in space of system parameters Step 1:

66. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks66 Step 2

67. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks67 Step 3

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 5. Epilogue: Lessons from Nature Is it realistic to expect that gene-protein-RNA network structures can ever be reconstructed from gene expression data and RNA and protein concentrations?

71. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks71 i.The role of stochasticity in the self-organized control of biological processes In the course of evolution, Nature has developed a considerable robustness towards stochastic fluctuations and chaos on the molecular level. Indeed, Nature even has succeeded to utilize this, potentially negative, condition to its own benefit. Experimental and theoretical studies now indicate that stochasticity plays a vital role in the self-organized control of biological processes. Stochastic fluctuations at the molecular level induce coherent oscillations at the cellular level. The autoregulation processes of the gene network thus allow for the emergence of biological rhythm out of noise.

72. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks72 ii. The Natural way of organization: Hierarchic row sparsity and collective behavior Organization of gene regulatory networks by hierarchy and row-sparsity master genes versus slave genes : master genes control the global dynamics collective behavior with a limited degree of freedom : this collective behavior falsely suggests that the system is not sparse at all, the composition of the gene clusters differs for different biological processes different biological processes = partitioned state space with local stable attractors = piecewise linear systems.

73. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks73 iii. Modeling and reconstructing Nature: the final questions how sharp is the real division between ’masters’ and ’slaves’? is Biological control characterized by a flexible network structures? is the partitioned state space a good metaphor for modeling Nature? are piecewise linear systems a good approximation of this metaphor? can robust identification techniques reveal the underlying network topology, despite all the intrinsic and extrinsic noise, chaos, and effects due to collective behavior?

74. Nature-inspired Smart Info Systems Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks74 Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks  G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ1Σ1 Σ2Σ2