nature-inspired smart info systems westra: piecewise linear dynamic modeling and identification of...

Westra: Piecewise Linear Dynamic Modeling and Identification of Gene-Protein Interaction Networks 1

Nature-inspired Smart Info Systems

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

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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



Question: Can gene regulatory networks be reconstructed from

time series of observations of (partial) genome wide and protein

concentrations?

1. Problem formulation



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



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.



2.1 Modeling the molecular dynamics and reaction kinetics as Stochastic Differential Equations




2.2 Gene-Protein Interaction Networks as Piecewise Linear Models




2.3 Problems concerning the identifiability of PieceWise Linear models




3. The Implications of Stochastic fluctuations and Deterministic Chaos




3.1 Stochastic fluctuations




3.2 Noise-induced control in single-cell gene expression




Influence of stochastic fluctuations on the evolution of the expression of two coupled genes.

.



3.3 Deterministic Chaos




4. Identification of Interactions between Genes and Proteins




4.2 The identification of PIECEWISE linear networks by L1-minimization




Gene-Protein Interaction Networks asPiecewise Linear Models

The general case is complex and approximate

Strongly dependent on unknown microscopic details

Relevant parameters are unidentified and thus unknown



2. Modeling of PWL Systems as subspace models

Global dynamics:

Local attractors (uniform, cycles, strange)

Basins of Attraction

Each BoA is a

subsystem Σi

“checkpoints”State space

Σ1

Σ2Σ3

Σ4

Σ5

Σ6



Modeling of PWL Systems as subspace models

State vector moves through state space

driven by local dynamics (attractor, repeller) and inputs

in each subsystem Σ1

the dynamics is governed by the local equilibria.

approximation of subsystem as linear statespace model:

State space



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 problemseverely 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 ofthe ensemble of genes. This can be avoided by persistently exciting inputs.



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.



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



The identification of PIECEWISE linear networks by L1-minimization

K linear time-invariant subsystems {Σ1, Σ2, .., ΣK}Continuous/Discrete time




Weights wkj indicate membership of observation #k

to subsystem Σj :



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.



(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



1. unknown switching times,regular sampling intervals,poor data, known state derivatives

This is similar to simple linear case



This can thus be written as:



with:



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.



Linear L1-criterion:



With linear L1-criterion E1 the problem can be formulated as LP-problem:

LP1: compute H1,H2 from simple linear case

LP2: A and B, using E1-criterion and extra constraints for W, H1,H2,

LP3: compute optimal weights W, using E1-criterion with constraints for W, H1,H2, A and B



2. unknown switching times,regular sampling intervals,poor data, unknown state derivatives

Use same philosophy as mentioned before



Subspace dynamics and linear L1-criterion :



System parameters and empirical data :



Quadratic Programming problem QP :

Problem: not well-posed: i.e.: Jacobian becomes zero and ill-conditioned near optimum



Therefore split in TWO Linear Programming problems:



In case of sparse interactions replace LP1 with:



Performance of robust Identification approach

Artificially produced data reconstructed with this approach

Compare reconstructed and original data



The influence of increasing intrinsic noise on the identifiability.



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.



a: Number of errors versus M, b: Computation time versus M

N = 50000, k = 10, m = 0.



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 σA

N = 10000, k = 10, m = 5, M = 150, measuring noise B = 0.



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



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



Centroids after clustering 23 clusters



Gene ontology



Gene ontologyCluster 1GO 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 2GO 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 22GO 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 23GO 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.



Reonstructed network



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’



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



Dynamics:

– implicit system dynamics:

– linear statespace model makes gene interaction explicit:



Dynamics:

– estimate gene-gene interaction matrix A from empirical data:



reactions



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



Real network structure: implicit

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agent



Real network structure: explicit

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slave slaveslave

mastermaster



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Reconstructed network structure: low noise

master master

slave slave slave



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Reconstructed network structure: moderate noise

slave slaveslave

mastermaster



Reconstructed network structure: high noise (an example)

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slave masterslave

slavemaster



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’



The deterministic Tyson-Novak model.



The stochastic Tyson-Novak model.



Example: stochastic Tyson-Novak model



5. Epilogue: Lessons from Nature




Discussion …

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nature-inspired smart info systems westra: piecewise linear dynamic modeling and identification of...

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