State estimation for gene networks with intrinsic and extrinsic noise: A case study on E.coli arabinose uptake dynamics 

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Ordinary Dierential Equation (ODE) Models

Very likely, the most used formalism for modelling gene regulatory networks is that of ordinary dierential equations (ODEs). These models can generally be studied and analysed using tools developed for nonlinear systems, in order to investigate dynamics, bifurcation behaviour, system stability [65, 108]. Example of biological models involving the ODE formalism can be found in [65, 95, 105].
More specically, when ODEs are used to model gene expression, the cellular concentration of proteins, mRNAs and other molecules are represented by non-negative continuous time variables. Notably, to model the typical transcription-translation process, the ODEs formalism makes use of two equations for any given gene i: one (3.2) to model the dynamics of the transcribed mRNA concentration and the other (3.3) for the concentration of the corresponding translated protein. Hence, given a GRN of n genes, and let mi, pi be the concentrations of mRNA and protein for the associated gene i, respectively, we have [139]: dmi dt = Fi(fR i (p1); fR i (p2); :::; fR i (pn)) 􀀀 imi.

Quasi-steady-state assumption of mRNA concentration

Very often in the literature, when modelling gene expression, it is assumed that the main gene expression regulation is at the transcriptional level. This hypothesis stems from the fact that|in some GRN|the mRNA dynamics is much faster than protein dynamics, i.e. mRNA concentration reaches its equilibrium faster than that of protein. This is oftentimes due to the fact that i i , that is mRNA degrades much faster than protein (typical mRNA half-lives are 2 􀀀 6 minutes, while those of proteins are on the order of hours [14]).

Piecewise Linear (PL) models

Piecewise linear (PL) models also consist on systems of dierential equations, but dierently from classical ODE models, their vector elds have (nitely many) points of discontinuity. This is because the PL system state space is divided into regions (domains) in which the vector eld may assume dierent expressions. However, these expressions must be ane or linear in each variable.
PL systems are a class of qualitative models, which can be used to facilitate the analysis of large classical ODE GRN models (see Section 3.2). In fact, intuitively, PL models can be derived from ODE models (3.2)-(3.3) with Hill functions (3.4)-(3.4) by letting the Hill coecient tends to innity. In this case the Hill functions h+, h􀀀 turn into step functions s+; s􀀀, respectively: lim n!1 h+(x; ; n) = s+(x; ) = 8< : 1 if x > 0 if x < .

Dynamical study of PL systems

The dynamics of PL systems can be studied in the n-dimensional state-space = 1 2 ::: n, where each i is dened by i = fxi 2 R0j0 xi maxig for some maximum concentration value maxi. A protein encoded by a gene will be involved in dierent interactions at dierent concentration thresholds, so for each variable xi, we assume there are pi ordered thresholds 1 i ; :::; pi i (we also dene 0 i = 0 and pi+1 i = maxi). The (n 􀀀 1)-dimensional hyper-planes dened by these thresholds partition into hyper-rectangular regions we call domains. Specically, a domain D is dened to be a set D = D1 ::: Dn, where Di is one of the following: Di = xi 2 ij0 xi < 1 i.

Solutions and Stability in Switching Domains

To provide the existence and the possibility for solutions to be continued on all domains, we have to dene the right-hand side of system (3.13) at the points of discontinuity of the function f. To this end, we use a construction originally proposed by Filippov [73] and then applied to PL systems ([90]). The method consists of extending the system (3.13) as a dierential inclusion, x_ 2 H(x);

Table of contents :

1 Introduction 
1.1 Motivations
1.2 Contributions
2 Notes on Molecular Cell Biology 
2.1 The Cell
2.1.1 Prokaryotes and Eukaryotes
2.1.2 E. coli as model organism
2.2 Gene expression: from DNA to Protein
2.2.1 Transcription: from gene to RNA
2.2.2 Translation: from RNA to protein
2.3 Regulation of Gene expression
2.3.1 Transcriptional control
2.3.2 Post-transcriptional control
2.4 Measurement Techniques
2.4.1 mRNA quantication
2.4.2 Protein quantication
2.4.3 Measurement limitations
3 Modelling Genetic Regulatory Network Systems 
3.1 Boolean Models
3.1.1 Synchronous and Asynchronous networks
3.1.2 Graph theoretical representation
3.1.3 Example: Boolean bistable switch
3.2 Ordinary Dierential Equation (ODE) Models
3.2.1 Quasi-steady-state assumption of mRNA concentration
3.2.2 Example: ODE bistable switch
3.3 Piecewise Linear (PL) models
3.3.1 Dynamical study of PL systems
3.3.2 Solutions and Stability in Regular Domains
3.3.3 Solutions and Stability in Switching Domains
3.3.4 Example: PL bistable switch
3.4 Stochastic Models
3.4.1 The Chemical Master Equation (CME) Stochastic simulation algorithm (SSA)
3.4.2 The chemical Langevin equation (CLE)
3.4.3 Example: CME and CLE bistable switch
3.5 Final comments
3.5.1 Deterministic Vs stochastic models
3.5.2 Quantitative Vs qualitative models
4 A Simple Model to Control Growth Rate of Synthetic E. coli during the Exponential Phase: Model Analysis and Parameter Estimation 
4.1 Introduction
4.2 The Open-loop Model
4.2.1 Growth rate
4.2.2 cAMP-CRP activation
4.2.3 CRP synthesis
4.2.4 CGEM synthesis
4.2.5 Proteins removal
4.3 Qualitative Analysis of the Open-loop Model
4.3.1 Open-loop model in glucose growth
4.4 Growth rate expression for exponential phase
4.5 In silico Identiability Analysis of Growth Rate
4.5.1 Problem Statement
4.5.2 Generation of Simulated Data Sets
4.5.3 Model Parametrization and Global Optimization
4.5.4 In Silico Practical Identiability Analysis
4.6 Conclusions
5 Controlling bacterial growth: in silico feedback law design to re-wire the genetic network 
5.1 Introduction
5.2 Piecewise linear models with dilution
5.2.1 Solutions in Regular Domains
5.2.2 Solutions in Switching Domains
5.2.3 Equilibria and Stability in Regular Domains
5.2.4 Equilibria and Stability in Switching Domains
5.3 Introduction to the control problem
5.4 Open-loop model
5.4.1 Growth rate
5.4.2 cAMP-CRP activation
5.4.3 CRP synthesis
5.4.4 RNAP synthesis
5.4.5 CRP and RNAP removal
5.5 Qualitative analysis of the open-loop system
5.5.1 Open-loop system in glucose growth
5.5.2 Open-loop system under an alternative carbon source
5.6 Closed-loop model
5.7 Qualitative analysis of the closed-loop system
5.7.1 Closed-loop system in glucose growth
5.7.2 Closed-loop system in maltose growth
5.8 Inverse Diauxie
5.9 Conclusions
6 Switched piecewise quadratic models of biological networks: application to control of bacterial growth 
6.1 Introduction
6.2 Piecewise Linear systems overview
6.3 The growth rate model
6.4 The Switched Piecewise Quadratic (SPQ) system
6.5 The PQ subsystem: dynamical study
6.5.1 Solutions and Stability in Regular Domains
6.5.2 Solutions and Stability in Threshold Domains
6.6 Stability Analysis of the SPQ system
6.7 Open loop control of the RNAP-ribosomes system
6.7.1 SPQ model of the open-loop control system
6.8 Conclusion
7 Attractor computation using interconnected Boolean networks: testing growth models in E. Coli 
7.1 Introduction
7.2 Methodology
7.2.1 From discrete to Boolean models
7.2.2 Dynamics of Boolean models
7.2.3 Interconnection of Boolean models
Transition graphs and semi-attractors
The asymptotic graph
7.2.4 Attractors of an interconnection
7.3 Application: a model for E. Coli growth mechanism
7.3.1 E. Coli nutritional stress response module
7.3.2 The cellular growth module
7.3.3 System interconnection
7.4 Results
7.4.1 General properties
7.4.2 Growth Rate limited by ribosomes or RNA polymerase
7.4.3 Growth Rate limited by bulk proteins
7.4.4 Model discrimination
7.4.5 Dynamical behaviour
7.5 Conclusions
8 A coarse-grained dynamical model of E. coli gene expression machinery at varying growth rates 
8.1 Introduction
8.2 E. coli GEM network: biological description
8.2.1 Ribosomes synthesis and function
8.2.2 RNAP synthesis and function
8.2.3 Proteins synthesis and function
8.3 Mathematical background
8.3.1 Transcription
8.3.2 Translation Translation of nascent mRNA Translation of completed mRNA Comments on ribosome engaged in translation
8.3.3 Final conclusions
8.4 E. coli GEM dynamical model
8.4.1 rnn gene expression model
8.4.2 rpoBC gene expression model
8.4.3 bulk gene expression model
8.4.4 Complete dynamical model of E. coli GEM
8.5 Model calibration
8.5.1 Experimental data
8.5.2 Parameters taken from literature
8.5.3 Calculated growth-rate-dependent parameters
Average DNA per cell:
Individual gene copy number per cell: Promoter concentration of rnn operon Promoter concentration of rpoBC genes Promoter concentration of bulk genes Promoter concentration of non-specic binding sites
8.5.4 Estimated parameters
8.6 Free RNAP and Free ribosomes
8.7 Model reduction
8.8 Conclusions
9 State estimation for gene networks with intrinsic and extrinsic noise: A case study on E.coli arabinose uptake dynamics 
9.1 Introduction
9.2 Stochastic modelling of genetic networks
9.3 Case study: E.coli arabinose uptake dynamics
9.4 Gene network state estimation
9.4.1 The Square-Root Unscented Kalman Filter
Measurement update
SRUKF Initialization
9.5 State estimation: Simulation results for the E.coli arabinose uptake system
9.5.1 Comparison of SRUKF and PF
9.5.2 Performance of the SRUKF in presence of extrinsic noise
9.6 Conclusions
10 Conclusions and Perspectives 
10.1 Qualitative models
10.2 Qualitative control strategies
10.3 Quantitative models
10.4 Parameter estimation
10.5 Stochastic models and state estimation
10.6 Perspectives
10.6.1 Qualitative control: application to real data
10.6.2 Identiability, sensitivity analysis and validation of GEM model .
10.6.3 Combining qualitative and quantitative formalisms for control purposes
10.6.4 Further investigation of dynamical growth rate models
10.6.5 Filtering applications of GRN models
A List of Publications 

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