Piecewise constant control strategy with uncertain measurements 

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Biological context

Gene regulatory networks

A gene is a functional unit composed of bases of DNA (Deoxyribonucleic Acid) that encodes any type of information determining the phenotype of a living organism (see figure 2.1.1). This information is transmitted to the organism through a process called “gene expression”, mainly composed of two steps: transcription and translation [6].
The transcription step describes the process by which the sequence of DNA composing a gene is copied into mRNA (messenger Ribonucleic Acid) by a specific enzyme called RNA polymerase that binds to the starting point of the DNA branch, called promoter, and slides along the gene up to a terminal point called the untranslated region, in order to scan and produce a copy of the information. The mRNA is then decoded by a macromolecule called Ribosome that synthesizes a long chain of amino acids forming a protein: this step is the translation (see figure 2.1.1). In a naive way, it will be considered that one gene codes for one protein through this gene expression machinery. In fact, the reality may be much more complex: a single gene often codes for different proteins depending on the splicing process during which some parts of mRNA are removed before translation. However, this phenomenon goes far beyond the scope of this manuscript. More information about gene expression can be found in [6].
Proteins are essential for living organisms as they perform most cellular functions. These macro-molecules are indeed involved in transport and storage (such as Hemoglobin that carries oxygen throughout the whole organism), in structures (such as Actin that forms microfilaments for cell shape), in body defense (such as Antibodies that bind and neutralize specific pathogens), in chem-ical reactions (such as enzymes that bind to substrates to convert them into different molecules of interest) and in communication (such as hormones that transmit signals between organs), to name but a few. This manuscript will focus on a specific group of proteins involved in gene expression regulation, called transcription factors (TF).
Transcription factors are proteins able to control the rate of transcription of a specific gene in order to ensure that it is expressed at the right place, at the right time and in the right amount within the organism. They are often divided in two groups: the activators that boost the transcription of one gene, and repressors that instead decrease the transcription of one gene. Usually, the transcription factor induces the regulation by binding to a specific region of the gene under control, called binding site or promoter (see figure 2.1.2). Once bound, an activator is able to increase the likelihood of transcription of the gene by taping itself to RNA polymerase with strong protein-protein interactions and thus helping its recruitement. Conversely, the repressors either prevent physically the RNA polymerase to bind to the promoter of the gene and transcribe it into mRNA (called DNA-binding repression), or prevent the translation of mRNA into protein by binding to the mRNA directly (called RNA-binding repression) (see [6] for more details).
A transcription factor is not always specific to a unique gene. It may even be sometimes both activator or inhibitor, depending on the cellular context and on the other transcription factors or proteins that may bind to the same binding site, called associated co-factors [6]. It follows that a unique gene is most of the time controlled by multiple simultaneous transcription factors leading to a combinatorial regulation of transcription, and determining whether the gene is up or down-regulated is a hard task.
Usually, all these interactions between genes, mRNA and proteins at the level of a reaction, of a cell, of an organ or of a whole organism, form a large network called “gene regulatory network”. These networks are composed of nodes and edges, where nodes represent proteins or genes and the edges summarize regulatory relationships between these nodes and may either be direct or indirect, activations or repressions [9] (see figure 2.1.3). Studying gene regulatory networks allow to explain and understand how the genotype shapes physiological and phenotypic observations. However, when considering that the number of human genes has been estimated between 20000 and 25000, and the number of transcription factors is approximately 2600, it is easy to imagine how large and how complex these regulatory networks may become. To circumvent this problem, it has been observed that these complex networks display abundant recurrent and repetitive motifs composed of a small number of genes highly interconnected, called building blocks (see figure 2.1.3). Indeed, several types of motifs were found to happen more often in gene regulatory networks than in randomly generated networks that share the same topological properties [90]. These different recurrent patterns often play central roles in biological functions, and this manuscript will focus on one particular essential motif called “feedback loops”.

Genetic feedback loops

Genetic feedback loops have the topology of a ring: the expression of each gene in the network is regulated by one previous gene and regulates the expression of a following gene, so that the genes in the network are coupled successively and form a directed cycle (see figure 2.2.1). Two main groups of feedback loops exist depending on the number of repressions within the network. If the number of repressions is even, the loop is called “positive feedback loop”. In this case, one gene indirectly activates its own expression via the activations and the even number of repressions within the cycle. Conversely, if the number of repressions is odd, the loop is called “negative feedback loop” and in this case a gene indirectly represses its own expression [114] (see figure 2.2.1).
Negative feedback loops have been shown to be essential for two different mechanisms: homeosta-sis and biological oscillations. Homeostasis is a vital function that allows to maintain relatively constant the biological internal operating conditions despite environmental or molecular fluctua-tions. A lot of common and popular features are under the control of homeostatic mechanisms in the human body, such as the body temperature, blood pressure, blood sugar level [106], or ATP (Adenosine triphosphate) that plays a central role in providing energy for many processes in living cells [68]… Negative loops are also responsible for the emergence of self-sustained oscillations and endogenous biological clocks that coordinate periodically different biological functions [40]. From the macro-scale to the micro-scale, oscillatory behaviors can be observed everywhere. For example, the circadian clock allows organisms to anticipate and adapt to environmental changes by generating 24-hours self-sustained oscillations coupled to day-night cycles in a wide variety of genes, molecules and internal parameters such as the sleep-wake cycle or the body temperature [7, 104]. Similarly, the cell cycle is composed of different phases during which a single cell goes through the duplication of its genetic material in order to divide and seems synchronized with the circadian clock with a period of 24 hours [6, 46]. As a last example, the female menstrual cycle is also a complicated oscillator that controls the reproductive system through different hormones that vary with a period of approximately 28 days [106]. All these phenomena can emerge through an underlying genetic regulation composed of negative loops. Interestingly, some systems highlight both the homeostatic and oscillatory behaviour, depending on internal conditions. This is the case for the regulation of a protein called p53. This protein has been discovered in 1979 and it is now clear that it plays an essential role in living organisms: p53 is indeed involved in tumor suppression, apoptosis, DNA repair and acts as a transcription factor that regulates a huge amount of genetic pathways (see for example [73] for a complete review). In healthy organisms and unstressed conditions, this protein is kept at low levels thanks to tight homeostatic control mechanisms [21]. In various stress conditions however, such as the presence of malignant cells or in case of DNA damage, it has been observed that the concentration of p53 starts to oscillate [78, 85, 23]. These sustained oscillations have been interpreted as essential for DNA repair or tumor suppression [122]. These two main dynamical be-haviors have been partly explained through a negative regulation of p53 by another protein called Mdm2.
While negative feedback loops tend to reduce the effect of a small disturbance, positive feedback loops have an inverse effect and exacerbate perturbations. For this reason, they have been shown to be responsible for multi-stability, leading to differentiation processes or cell decision making
[114]. Cell differentiation allows undifferentiated cells, called stem cells, to differentiate into any specialized cells with specific functions and is a life-long process, from development stages to repair phases (see the left sketch in figure 2.3.1). This multi-stability property also allows drastically different fates and decisions from one cell to another, even in the case of daughter cells in similar genetic and molecular environments [6]. A famous and essential example of cell decision making is apoptosis, a process during which an old or damaged cell dies. It has been interestingly observed that sister cells do not always respond similarly to various apoptosis signals: some of them indeed go through the apoptotic phases and die while others do not react. This difference in decision is partly explained by an auto-activation of a family of enzymes called Caspases that become activated when cell death is programmed [6].

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Importance of understanding and controlling these loops

As highlighted in the previous sections, gene regulatory feedback loops govern a large amount of vital biological functions. For this reason, a better understanding of their underlying mechanisms as well as new strategies for the control of their dynamics would lead to an improvement regarding insight into diseases and pharmacological treatments [110].
Disrupted negative feedback loops have now been identified in many diseases. For example, an homeostasis disruption, called a dyshomeostasis, may result in the emergence of undesirable sus-tained oscillations with sometimes dramatic consequences for the organism. For instance, due to its important role in apoptosis, the protein p53 introduced in the previous section has been shown to be tightly regulated in healthy organisms in order to prevent extreme expression levels. Indeed, inappropriate activity of p53 with too high or too low concentrations can lead to various diseases, such as neurodegenerative disorders characterized by a neuronal loss like Alzheimer [113], or early embryonic lethality [21]. These types of disruptions are not restricted to the genetic scale: at the neuron scale for example, periodic firing patterns with under- and over-stimulation are known to be involved in various kinds of cerebral damage [65, 51]. Due to the underlying role of genetic net-works, it seems likely that homeostatic behaviors at any scale may often hide a genetic homeostatic control. For example, in [53], it has been shown that the coenzyme PLP (Pyridoxal phosphate: a catalyst derived from a vitamin) is tightly regulated through a genetic cascade in order to guarantee homeostasis of neurotransmitters. In case of extreme PLP activities, the consequent dyshomeostasis of neurotransmitters (such as serotonin or dopamine) has been shown to provoke epileptic attacks.
The reverse scenario, namely the disruption of a biological oscillator, seems also involved in many disorders. For example, it has been observed that many diseases such as cancers [76] or neurodegen-erative disorders [91] are susceptible to cause a disruption of the circadian clock. Alternatively, the synthetic generation of circadian rhythms in disrupted organisms has been proved to be effective for the slowdown of disease progression [76]. For these reasons, the circadian clock is now considered as a promising tool for therapeutic progress, and especially for cancer treatments. All these examples support the high interest of finding new strategies for the control of biological negative feedback loops.
Similarly, the control of biological positive feedback loops opens up vast biotechnological appli-cations and opportunities. For example, it has been shown recently that cell differentiation is a reversible process [24]: scientists have indeed been able to turn a differentiated cell back into an undifferentiated stem cell. However, this phenomenon called “dedifferentiation” is not yet well understood (see the right sketch in figure 2.3.1). Being able to control and understand dedifferenti-ation processes may allow to produce and store a large amount of stem cells. These non-specialized cells are helpful in many domains. For example, cancers are often induced by an abnormal cell growth and division that invade and spread widely throughout the organism, and the control and understanding of differentiation processes may reveal how and why such diseases start and develop. Furthermore, these cells are more and more used in cancer treatments as they are able to replace any damaged cell. Indeed, chemotherapy or radiotherapy that are responsible for the destruction of cancer cells do not make the distinction with healthy cells. In order to restore a normal amount of healthy cells, patients with typical diseases such as Leukemia for example, can receive stem cell transplants. Finally, the control of positive feedback loops can also be useful for programmed cell differentiation in order to force a cell to differentiate into any particular cell type. This may be interesting in the context of tissue regeneration for example, where a great amount of specialized cells is needed.

Biological design and control tools

Synthetic biology has made significant progress over the past few years and has offered multiple tools for the design and the control of genetic motifs [103].
The idea of synthetic biology emerged at the end of the 20th century, but the first synthetic circuits were really engineered at the beginning of the 21st century. This recent multidisciplinary field derived from genetics, biophysics but also computer and control engineering, in order to create, re-design, control and program fully artificial biological systems. Most of the time, some DNA building blocks or sequences are stitched and assembled artificially together thanks to different enzymes that are able to both cleave and facilitate the joining of DNA strands, and these new modules are then inserted in living organisms for different purposes. Thanks to synthetic modification of DNA, a lot of transcriptional and translational parameters may be tuned: for example, by changing the promoter of a gene, its transcription rate may be modified, whereas its degradation rate may be impacted by a modification of its stop codon.
A famous and extensively used application of synthetic biology concerns gene expression measure-ment. Indeed, understanding and studying a genetic motif often induces the investigation of genetic expression changes. In 2008, the three chemists Osamu Shimomura, Roger Tsien and Martin Chalfie received the Nobel Prize for their study on bioluminescent jellyfish. He discovered a protein called GFP (green fluorescent protein) that exhibits green fluorescence after exposure to blue light. Nowa-days, this protein or other similar fluorescent proteins such as YFP, RFP, CFP (for respectively Yellow, Red and Cyan fluorescent protein) are widely used as reporter genes for the measurement of gene expression. One way to do this in practice is to fuse the coding sequence of the reporter gene downstream or upstream of the coding sequence of the gene of interest. In this way, both genes are expressed to the same extent (see figure 2.4.1): a chimeric protein is produced and recapitulates the expression level of the gene of interest. The expression of the reporter gene can be quantified thanks to fluorescence microscopy. As the light is proportional to the abundance of protein expression, the intensity of fluorescence leads to a partial estimation of the temporal and spatial gene expression level within the cell. In this way, gene expression can be evaluated by quantifying levels of the gene product, which usually consists in the corresponding coding proteins.

Table of contents :

1 Motivations and organization of the manuscript 
2 Biological context 
2.1 Gene regulatory networks
2.2 Genetic feedback loops
2.3 Importance of understanding and controlling these loops
2.4 Biological design and control tools
2.5 Conclusion
3 Mathematical modeling of genetic feedback loops 
3.1 The deterministic and ordinary differential equation framework
3.2 Non-linear ordinary differential systems
3.2.1 Modeling transcription and translation
3.2.2 ODE model for genetic feedback loops in dimension N
3.2.3 Steady states
3.2.4 Local Stability
3.2.5 Monotone dynamical systems
3.2.6 Monotone cyclic feedback systems
3.2.7 Numerical illustrations
3.3 Piecewise affine differential systems
3.4 Boolean systems
3.5 Conclusion
4 Classical control strategy 
4.1 Introduction
4.2 The controlled model
4.3 A new methodology for global results
4.4 Schwarzian derivatives
4.5 Conditions on
4.6 Numerical illustrations
4.7 Conclusion
5 Saturated control strategy 
5.1 Introduction
5.2 The controlled model
5.3 Global asymptotic stability
5.4 Conclusion
6 Piecewise constant control strategy 
6.1 Introduction
6.2 The controlled model
6.3 Global convergence
6.3.1 Global convergence for the negative loop
6.3.2 Global convergence for the positive loop
6.4 Global stability
6.4.1 Global stability for the negative loop
6.4.2 Global stability for the positive loop
6.5 The PWC control inside Hill functions: an illustration
6.5.1 The controlled Toggle Switch model
6.5.2 Global results
6.6 A trade-off between speed of convergence and strength of inputs
6.7 Conclusion
7 Piecewise constant control strategy with uncertain measurements 
7.1 Introduction
7.2 The controlled model
7.3 Global convergence
7.3.1 Global convergence for the negative loop
7.3.2 Global convergence for the positive loop
7.4 A synthetic example: the Repressilator
7.5 A biological example: the p53-Mdm2 negative loop
7.6 The PWC control with uncertainties inside Hill functions: an illustration
7.6.1 The controlled Toggle Switch model
7.6.2 Global results
7.7 Conclusion
8 Design of synthetic modifications 
8.1 Introduction
8.2 The controlled model
8.3 Global asymptotic stability
8.4 Global convergence towards an undifferentiated region for the positive loop
8.5 Conclusion
9 A new problematic: the emergence of oscillations
9.1 Introduction
9.2 A biological motivation: the circadian clock
9.2.1 A reduced circadian clock model
9.3 A synthetic modification of the loop
9.4 A PWC control strategy
9.5 Application to the circadian clock
9.6 Conclusion
10 Conclusions and perspectives 

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