Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks1 Ronald L. Westra, Ralf L. M. Peeters, Department of Mathematics Maastricht University Robust Identification of Piecewise Linear Gene-Protein Interaction Networks NISIS conference, Albufeira, October 4, 2005

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear 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 fission yeast expression data 5. Example 2: Application on artificial reaction model 5. Example 3: Application on Tyson-Novak model for fission yeast 6. Conclusions Items in this Presentation

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks3 Questions: * Can we identify (= reconstruct) gene regulatory networks from time series of observations of (partial) genome wide and protein concentrations? * What is the influence of intrinsic noise and deterministic chaos on the identifiability of such networks? 1. Problem formulation

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks8 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks9 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks10 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks11 PWL Systems as flexible networks For different biological processes the subsystem defines a different network structures G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ2Σ2 G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 Σ1Σ1

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks12 3. Identification of Interactions between Genes and Proteins

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks13 The identifiability of Piecewise Linear models from Microarray Timeseries Sequence of genome- wide expression profiles at consequent instants become more realistic with decreasing costs …

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks14 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks15 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks16 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks17 The identification of PIECEWISE linear networks by L 1 -minimization K linear time-invariant subsystems {Σ 1, Σ 2,.., Σ K } Continuous/Discrete time

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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks19 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks20 (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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks21 1. unknown switching times, regular sampling intervals, poor data, known state derivatives This is similar to simple linear case

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks22 This can thus be written as:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks23 with:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks24 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks25 Linear L1-criterion:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks26 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks27 2. unknown switching times, regular sampling intervals, poor data, unknown state derivatives Use same philosophy as mentioned before

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks28 Subspace dynamics and linear L1-criterion :

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks29 System parameters and empirical data :

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks30 Quadratic Programming problem QP : Problem: not well-posed: i.e.: Jacobian becomes zero and ill-conditioned near optimum

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks31 Therefore split in TWO Linear Programming problems:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks32 In case of sparse interactions replace LP1 with:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks33 Performance of the robust Identification approach Artificially produced data reconstructed with this approach Compare reconstructed and original data Here some results …

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks34 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 =

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks35 a: Number of errors versus M, b: Computation time versus M N = 50000, k = 10, m = 0.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks36 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks37 The influence of increasing intrinsic noise on the identifiability.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks38 4. The Implications of Stochastic fluctuations and Deterministic Chaos 4.1 Stochastic fluctuations : * some experimental and numerical results

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks39 the evolution of the expression of two coupled genes. The genes, with expressions x 1 and x 2, are coupled as: with zero-mean Gaussian stochastic noise. Influence of increasing intrinsic noise level. The time steps dt relate to the strength of the noise. Stochastic fluctuations

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks40 Influence of stochastic fluctuations on the evolution of the expression of two coupled genes.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks41 Noise-induced control in single-cell gene expression 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks42 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks43 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks44 4. The Implications of Stochastic fluctuations and Deterministic Chaos 4.1 Stochastic fluctuations : * some experimental and numerical results 4.2 Deterministic chaos: * some remarks

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks45 Gene-Protein system with coupling strength λ Consider a gene with expression x(t) that is coupled to an stimulating protein with density a(t). With f a sigmoid function. Now consider the limit gene states x(t) as function of λ

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks46 Feigenbaum bifurcation as λ increases

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks47 Complex deterministic chaotic behaviour

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks48 Deterministic Chaos in a Gene-Protein system This chaotic behaviour is beneficial for identification as it provides many independent data points to explore the dynamics in the basin of attraction. In this way, chaos acts as the persistently exciting inputs in the linear-convergent case.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks49 Applications

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks50 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 irregular time instants Processing: fuzzy C-means, gene annotation with Go term finder and Fatigo, net recontruction with identification algorithm

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks51 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks52 Centroids after clustering 23 clusters

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks53 Gene ontology

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks54 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.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks55 Reonstructed network

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks56 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’

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks57 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

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks58 Dynamics : – implicit system dynamics: – linear statespace model makes gene interaction explicit:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks59 Dynamics : – estimate gene-gene interaction matrix A from empirical data:

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks60 reactions

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks61 reactions

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks62 reactions

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks63 Matlab-simulation y(1) = *x(1) + 0.2*(1-x(1))*a(2)^ *x(1)*a(3) ; y(2) = *x(2) + 0.3*(1-x(2))*a(1) - 0.1*x(2)*a(4) ; y(3) = *x(3) + 0.1*(1-x(3))*a(2) - 0.1*x(3)*a(1) ; y(4) = *x(4) + 0.2*(1-x(4))*a(1)*a(2) - 0.2*x(4)*a(3)^2; y(5) = *x(5) + 0.3*(1-x(5))*a(3) - 0.1*x(5)*a(1); y(6) = *a(1) + 0.4*x(1) - 0.2*a(1)*a(2) - 0.1*a(1)*a(3)^3; y(7) = *a(2) *x(2) - 0.2*a(1)*a(2); y(8) = *a(3) + 0.2*a(1)*a(2) - 0.1*a(1)*a(3)^3; y(9) = *a(4) + 0.9*a(1)*a(3); rate equations

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks64 Real network structure: implicit 21 a d b c p g gene agent

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks65 Real network structure: explicit slave master

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks66

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks67

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks68

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks Reconstructed network structure: low noise master slave

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks Reconstructed network structure: moderate noise slave master

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks71 Reconstructed network structure: high noise (an example) slavemasterslave master

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks72 Example 3: influence of noise Artificial model for infection with deterministic chaos and no intrinsic noise : Components: 2 agents (= RNA or proteins). Processes: stimulation, inhibition, transcription, and reactions between ‘agents’

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks73 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’

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks74 The deterministic Tyson-Novak model.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks75 The stochastic Tyson-Novak model.

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks76 Example: stochastic Tyson-Novak model

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks77 Example: stochastic Tyson-Novak model

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks78 5. Conclusions The piecewise linear system is an attractive metaphor for modeling dynamic gene-protein interactions Robust identification can efficiently reconstruct network structures of PWL systems for ‘poor’ data Stochastic fluctuations mostly affect slave genes with low transcription rates Strongest links (e.g. master genes) are most resistant to increasing noise

Nature-inspired Smart Info Systems Westra: Robust Identification of Piecewise Linear Gene-Protein Interaction Networks79 Discussion …  G2G2G2G2 G1G1G1G1 P2P2P2P2 P1P1P1P1 P3P3P3P3 G3G3G3G3 G4G4G4G4 G1G1G1G1 P5P5P5P5 P4P4P4P4 P3P3P3P3 G3G3G3G3 G6G6G6G6 Σ1Σ1 Σ2Σ2