Reduction and stability analysis of a transcription-translation model of RNA polymerase 

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Visualization of process activities

For models as complex as the mammalian circadian clock model, it is advantageous to qualitatively visualize process activities or inactivities during the system dynamics. Chapter 6. Principal process analysis and its robustness to parameter changes 65 PPA allows to visually summarize this information in one figure with the help of graphical tools. They are described below. Boolean Process Map: shows the time-dependent activity of processes, ordered by variables, during the whole system dynamics [t0, T]. Active, resp. inactive, processes are depicted by a white, resp. black, bar.
Dynamical Process Map: displays the activity of processes and their interactions with variables. In this map, we distinguish three types of process activity to take account of system equations sharing common processes. Variables (represented by boxes) are connected by processes (arrows), which can be either inactive (shown in black), active for all the variables involved (red) or, active for some variables involved and inactive for the others (yellow). 3-D Process Map: depicts qualitatively for each process, the time-dependent evolution of its intensity. Process activities are averaged per hour, leading to the discretization of time. Vertical bars represent process weights for each hour. Their color code represents the intensity of process weights relatively to the other weights.
Example: Figure 6.2 shows the boolean process map for variable x14 with its specific switching times (Panel A), the dynamical process map for the time intervals between t11 4,4 and t11 4,5 (Panel B), and the 3-D process map with the evolution of seven processes during time, discretized for each hour (Panel C). The nuclear import of protein BMAL1 is the strongest process.

First model reduction

The first step of PPA identifies always inactive processes and remove them from the original system. The threshold value δ must be chosen in the range [0,1], preferentially at a low value to avoid neglecting important processes. Otherwise the dynamics of the new system would change significantly. The objective is to obtain g(xr), the function approximating f(x) and including less processes.
We introduce the ODE system (6.6), which approximates system (6.2): x˙ r = g (xr, pr) (6.6) where xr = (xr 1, xr 2, …, xr n) ǫ Rn is the vector of component concentrations, x0 = (x01, x02, …, x0n) ǫ Rn the vector of their initial values, and pr ǫ Rc, where c ≤ b is the vector of parameters. The model reduction approach relies basically on the following theorem: if the vector fields of two systems are close (f(x) ≈ g(x)), then the solutions of the original and approximated systems are close during some time interval under the assumptions on the Lipschitz conditions listed in [57, p. 96, Th. 3.4].

Global sensitivity analysis

The simplification method described above is performed for a fixed set of parameters and initial conditions. In Chapter 7 we will study the robustness of PPA with respect to variations of the initial conditions of the system. Here we focus instead on the effect of varying parameter values on the quality of the method.
To that aim, we performed a global sensitivity analysis on the global relative errors (6.8) between the original and reduced models. In a first analysis, we considered the errors defined as previously, for the six model outputs (eMP ,eMC ,eMB,ePTot ,eCTot ,eBTot); then, in a more detailed analysis, we computed the global relative error for each variable (ei, i = 1, …, 16). The method we used is based on a factorial design on the uncertain parameters [35, Ch.3, pp. 69-209], analysis of variance (ANOVA) and principal component analysis (PCA) [69].
First, we explored the parameter space using factorial design. We varied Nf = 51 parameters of the model [74] (see Section 6.3). We chose Nl = 2 levels for each parameter pf (or factor): p− f = 0.8 pf and p+ f = 1.2 pf . A full factorial design, defined as all possible combinations of the parameter levels, would be necessary to estimate the main effects and interactions of all parameters. Such a full design corresponds to N Nf l = 251 parameter combinations and would necessitate the same number of model simulations to compute the corresponding outputs, which are far too many. Thus we implemented a fractional factorial design [27], which is a subset (fraction) of the full design, chosen in order to estimate all main effects αf and two-way interactions βfk of the following linear statistical model linking the error eh to the parameters pf : eh = μ + X fαf + X f X k6=f βfk + ǫh.

Principal process analysis and first reduction

We applied PPA to identify major processes along the system dynamics. We decomposed each ordinary differential equation in processes, as shown in Equation (6.4) for BMAL1. Each process has a biological interpretation and corresponds to a regulatory process or a biochemical reaction. We then calculated the relative weight of each process using Equation (6.5) and set a low threshold δ = 0.1 (see Section 6.2.3 for the choice of this value). We collected the switching times (values given in Appendix D.2) and then built a boolean process map to visualize the activity/inactivity of each process (see Figure 6.4). We obtained a first reduction of the model by neglecting 24 out of 76 processes, which were always inactive (32% of all processes). They correspond to mRNA and protein basal degradations; cytosolic dephosphorylations of CRY, BMAL1, and PER-CRY; PERCRY- CLOCK-BMAL1 dissociation in the nucleus; and BMAL1 dephosphorylation in the nucleus. The list of neglected processes is shown in Appendix D.3.

Principal process analysis and model reduction based on initial conditions in a rectangle

To increase the robustness of the method, the initial condition is chosen in some region, then we  ompute if the activity/inactivity of the process fi,j – and consequently the reduced system g(xr) – changes.
We divide the variable space into rectangles, and then apply the technique in each domain: for simplicity, we consider in this paper a system with two variables (x1, x2) and a logarithmic subdivision, corresponding to order of magnitude from the modeling point of view. The grid is shown in Figure 7.1.
Every point θm,n = (θm 1 , θn 2 ) corresponds the value (10m, 10n): for example the point θ2,0 = (102, 100) = (100, 1). We call Bm,n the rectangle delimited by the four vertices θm,n, θm+1,n, θm+1,n+1 and θm,n+1 (shown in Figure 7.2): inside of it, every process fi,j(x, p) is limited horizontally fi,j(θm,n 1 , p) < fi,j(x, p) < fi,j(θm+1,n 1 , p) and vertically fi,j(θm,n 2 , p) < fi,j(x, p) < fij(θm,n+1 2 , p). To compute a global bound for the weights in the rectangle, we need the following assumption for the processes. Below, all the functions are supposed to be locally Lipschitz; by “fixed sign”, we mean that the functions are either non-negative, or non-positive, or zero.

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Table of contents :

1 Introduction 
1.1 Motivations
1.2 Approach
1.3 Organization of the manuscript and contributions
2 Introduction (en fran¸cais) 
3 Notes on molecular cell biology 
3.1 Escherichia coli
3.2 Growth of E. coli
3.3 Gene expression
3.3.1 Transcription
3.3.2 Translation
3.3.3 mRNA degradation
3.4 Regulation of gene expression in E. coli
4 Modeling genetic regulatory network systems 
4.1 Ordinary differential equation models
4.1.1 Modeling transcription-translation
4.1.2 Quasi-steady-state assumption of mRNA concentration
4.2 Analysis of a genetic bistable switch
4.2.1 Phase plane analysis
4.2.2 Jacobian matrix
4.2.3 Piece-wise affine linear system
4.3 Parameter sensitivity analysis
4.3.1 Local sensitivity analysis
4.3.2 Global sensitivity analysis
4.4 Parameter fitting
5 Reduction and stability analysis of a transcription-translation model of RNA polymerase 
5.1 Introduction
5.2 The coupled transcription-translation model of RNA polymerase
5.2.1 Description of the model
5.2.2 Full equation
5.3 Time-scale reduction (fast-slow behavior)
5.3.1 Parameter values for the coupled transcription-translation models of RNA polymerase
5.3.2 Separation of the full system into “fast” and “slow” variables
5.4 Verification of the applicability of the Tikhonov’s theorem for the fast subsystems
5.5 Application of the Tikhonov’s theorem
5.6 Dynamical study of the reduced system
5.6.1 Simulations of the full and the reduced system
5.6.2 Equilibria of the reduced system
5.6.3 Stability of equilibria
5.7 Applications to other models
5.8 Comparison with a classical model of RNA polymerase
5.9 System with a variable growth rate
5.10 Conclusion
6 Principal process analysis and its robustness to parameter changes 
6.1 Introduction
6.2 Methodology
6.2.1 Principal process analysis (PPA)
6.2.2 Visualization of process activities
6.2.3 First model reduction
6.2.4 Creation of chains of sub-models
6.2.5 Global sensitivity analysis
6.3 Model description
6.4 Principal process analysis and first reduction
6.5 Creation of sub-models
6.6 Parameter influence
6.7 Conclusion
7 Principal process analysis and reduction of biological models with different orders of magnitude
7.1 Introduction
7.2 Methodology
7.2.1 Principal process analysis and model reduction
7.2.2 Principal process analysis and model reduction based on initial conditions in a rectangle
7.2.3 Possible transitions between domains
7.3 The gene expression model
7.4 Model reduction from an initial condition
7.5 Model reduction in a rectangle
7.6 Conclusion
8 Principal process analysis applied to a model of endocrine toxicity induced by Fluopyram 
8.1 Introduction
8.2 Methodology
8.2.1 Absolute principal process analysis
8.2.2 Visualization of the process activity
8.3 Hierarchical graph
8.4 Model
8.4.1 Blood compartment
8.4.2 Liver compartment
8.4.3 Brain compartment
8.4.4 Thyroid compartment
8.4.5 The data
8.4.6 Different experiments in silico
8.5 Absolute principal process analysis on the experiment 1A
8.6 Absolute principal process analysis on the experiment 2B
8.7 Conclusion and future steps
9 Model and control of the gene expression machinery in E. coli 
9.1 Introduction
9.2 The model
9.2.1 Growth rate
9.3 The effect of IPTG on E. coli growth
9.4 Model analysis with three-level PPA
9.4.1 Methodology
9.4.2 Different applications Nutrient stress condition IPTG stress condition
9.5 Conclusion
10 Single-cell model calibration of growth control experiments in E. coli 
10.1 Introduction
10.2 Model
10.3 Methodology
10.3.1 Data
10.3.2 Extraction of cellular profiles
10.3.3 Calculation of average cell profiles
10.3.4 Calibration of the model
10.4 Results
10.4.1 Cellular profiles
10.4.2 Calibration of the average cell model
10.4.3 Calibration of the single-cell models
10.4.4 Comparison
10.5 Conclusion
11 Conclusion and perspectives 
11.1 Classical tools for the analysis and reduction of biological models
11.2 New tools for the analysis and reduction of biological systems
11.3 Design and analysis of the gene expression machinery in E. coli
11.4 Single-cell and average cell calibration of the gene expression machinery control in E. coli
12 Conclusion et perspectives (en fran¸cais) 
12.1 Outils classiques pour l’analyse et la r´eduction des mod`eles biologiques
12.2 Nouveaux outils pour l’analyse et la r´eduction des syst`emes biologiques .
12.3 Mod´elisation et analyse du m´ecanisme d’expression des g`enes dans E. coli
12.4 Calibration d’un mod`ele de contrˆole de la machinerie d’expression des g`enes dans E. coli en utilisant les profils de la cellule individuelle et la moyenne
A List of publications 
B First application of principal process analysis on biological models 
B.1 Methodology
B.1.1 Principal process analysis (PPA)
B.1.2 Visualization of process activities
B.1.3 First model reduction
B.1.4 Creation of chains of sub-models
B.2 Model for circadian rhythms in Drosophila
B.2.1 Description
B.2.2 Model reduction
B.2.3 Qualitative tool: heat process map
B.2.4 Creation of sub-models based on time windows
B.3 Model for the influence of RKIP on the ERK signaling pathway
B.3.1 Description
B.3.2 Model reduction
B.3.3 Qualitative tool: 3-D process map
B.4 Conclusions
B.5 Supplementary materials
C Supplementary materials of Chapter 5 
C.1 Monotone systems
C.2 Tikhonov’s theorem
D Supplementary materials of Chapter 6 
D.1 Full mammalian model
D.2 Switching times
D.3 Neglected procceses
D.3.1 First reduced model
D.3.2 Second reduced model: sub-models
D.4 Dynamical process maps
E Supplementary materials of Chapter 8 
E.1 Full dynamics of the experiments 1A, 2A, 2B
F Supplementary materials of Chapter 9 
F.1 Parameters and initial values of the GEM model
G Supplementary materials of Chapter 10 
G.1 Model parameters for each c calibration


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