Georgi Muskhelishvili
email: g.muskhelishvili@icloud.com

Raphaël Forquet
email: raphael.forquet@insa-lyon.fr

Sylvie Reverchon
email: sylvie.reverchon-pescheux@insa-lyon.fr

Sam Meyer
email: sam.meyer@insa-lyon.fr

William Nasser
email: william.nasser@insa-lyon.fr

Coherent Domains of Transcription Coordinate Gene Expression During Bacterial Growth and Adaptation

Keywords: bacteria, chromosomal origin-to-terminus axis, DNA thermodynamic stability, DNA supercoiling, nucleoid-associated proteins, transcription, chromosomal domains, genetic regulation

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Introduction

Coordination of the bacterial chromosome structure and function is mediated by changes in genomic DNA supercoiling associated with DNA replication and its cessation and corresponding alterations of transcription. The mechanism coordinating genomic transcription is not fully understood, but recent findings made it increasingly clear that the genetic regulation of bacterial growth cannot be satisfactorily explained on the basis of the classical operon model. First of all, genomic transcription turned out to be both pervasive and fuzzy, so that the operon model does not hold the promise to explain all the alterations of genetic activity during bacterial growth and adaptation, as thought before [START_REF] Reverchon | Dickeya ecology, environment sensing and regulation of virulence programme[END_REF][START_REF] Wade | Pervasive transcription: Illuminating the dark matter of bacterial transcriptomes[END_REF][START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF][START_REF] Junier | Conserved Units of Co-Expression in Bacterial Genomes: An Evolutionary Insight into Transcriptional Regulation[END_REF][START_REF] Junier | Universal and idiosyncratic characteristic lengths in bacterial genomes[END_REF][START_REF] Yus | Determination of the Gene Regulatory Network of a Genome-Reduced Bacterium Highlights Alternative Regulation Independent of Transcription Factors[END_REF]. Secondly, positional effects on reporter gene expression were observed in Escherichia coli [START_REF] Bryant | Chromosome position effects on gene expression in Escherichia coli K-12[END_REF][START_REF] Brambilla | Gene regulation by H-NS as a function of growth conditions depends on chromosomal position in Escherichia coli[END_REF][START_REF] Scholz | High-Resolution Mapping of the Escherichia coli Chromosome Reveals Positions of High and Low Transcription[END_REF], consistent with the notion that genetic expression not only is regulated in digital (discontinuous) mode-that is, by classical transcription factor (TF)-target gene (TG) interactions-but also is subject to the analog (continuous) mode of control, mediated by cell physiology-dependent changes of chromosomal DNA supercoiling [START_REF] Junier | Conserved Units of Co-Expression in Bacterial Genomes: An Evolutionary Insight into Transcriptional Regulation[END_REF][START_REF] Hatfield | DNA topology-mediated control of global gene expression in Escherichia coli[END_REF][START_REF] Blot | Homeostatic regulation of supercoiling sensitivity coordinates transcription of the bacterial genome[END_REF][START_REF] Marr | Dissecting the logical types of network control in gene expression profiles[END_REF][START_REF] Sonnenschein | Ranges of control in the transcriptional regulation of Escherichia coli[END_REF][START_REF] Sonnenschein | Analog regulation of metabolic demand[END_REF][START_REF] Berthoumieux | Shared control of gene expression in bacteria by transcription factors and global physiology of the cell[END_REF]. Furthermore, recent studies discovered a local impact of transcription-coupled supercoil diffusion (TCSD) on transcription initiation from nearby promoters [START_REF] Meyer | Torsion-mediated interaction between adjacent genes[END_REF][START_REF] Sobetzko | Transcription-coupled DNA supercoiling dictates the chromosomal arrangement of bacterial genes[END_REF][START_REF] El Houdaigui | Bacterial genome architecture shapes global transcriptional regulation by DNA supercoiling[END_REF], whereas high-throughput studies discovered coherent domains of transcription spanning chromosomal regions from several kilobases to several hundred kilobases [START_REF] Jeong | Spatial patterns of transcriptional activity in the chromosome of Escherichia coli[END_REF][START_REF] Peter | Genomic transcriptional response to loss of chromosomal supercoiling in Escherichia coli[END_REF][START_REF] Berger | Coordination of genomic structure and transcription by the main bacterial nucleoid-associated protein HU[END_REF][START_REF] Ferrandiz | The genome of Streptococcus pneumoniae is organized in topology-reacting gene clusters[END_REF][START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF][START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF], thus calling for completely new models of transcriptional regulation [START_REF] Marr | Dissecting the logical types of network control in gene expression profiles[END_REF][START_REF] Meyer | Chromosomal organization of transcription: In a nutshell[END_REF][START_REF] Reverchon | Rethinking the bacterial genetic regulation[END_REF]. Among the factors so far implicated in the formation of chromosomal expression domains are strong transcription units [START_REF] Booker | DNA topology of highly transcribed operons in Salmonella enterica serovar Typhimurium[END_REF][START_REF] Le | High-resolution mapping of the spatial organization of a bacterial chromosome[END_REF], regulatory interactions [START_REF] Fritsche | A model for Escherichia coli chromosome packaging supports transcription factor-induced DNA domain formation[END_REF], binding effects of nucleoid-associated proteins (NAPs) and other DNA structuring proteins including macrodomain-specific DNA binding proteins [START_REF] Dame | Chromosomal macrodomains and associated proteins: Implications for DNA organization and replication in gram negative bacteria[END_REF][START_REF] Scolari | Gene clusters reflecting macrodomain structure respond to nucleoid perturbations[END_REF][START_REF] Lioy | Multiscale Structuring of the E. coli Chromosome by Nucleoid-Associated and Condensin Proteins[END_REF], DNA topoisomerases [START_REF] Meyer | Chromosomal organization of transcription: In a nutshell[END_REF], and last but not least, the peculiar sequence organization of chromosomal regions [START_REF] Ferrandiz | The genome of Streptococcus pneumoniae is organized in topology-reacting gene clusters[END_REF][START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF][START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF].

Whatever the nature of the factors determining the subdivision of the chromosome into separate expression domains, recent studies made it obvious that the coordination of genomic function involves the temporal emergence of coherent domains of transcription (CODOs, formerly denoted as DCTs [START_REF] Meyer | Chromosomal organization of transcription: In a nutshell[END_REF]) comprising large (up to megabase-sized) chromosomal regions and resembling the so-called pathogenicity islands with peculiar sequence organization, that can form "archipelagos" in the genome [START_REF] Hacker | Pathogenicity islands and the evolution of microbes[END_REF][START_REF] Bouyioukos | From multiple pathogenicity islands to a unique organized pathogenicity archipelago[END_REF]. Understanding the mechanism organizing the CODOs in the genome requires an integrative (holistic) study revealing the links between the spatial patterns of gene transcription, the structural dynamics of the corresponding chromosomal regions, and the global sequence organization of genome.

Model of Regulation of Gene Expression during the E. coli Growth Cycle-A Brief Overview

In E. coli, the role of chromosomal sequence organization in the regulation of the genetic function is quite conspicuous. Initiation of replication at the chromosomal origin of replication (OriC) requires high negative superhelicity both in vitro and in vivo and is assisted by several DNA architectural proteins including highly abundant NAPs, such as integration host factor (IHF), factor for inversion stimulation (FIS), and histone-like protein first isolated from E. coli strain U13 (HU) [START_REF] Crooke | coli minichromosome replication: Regulation of initiation at oriC[END_REF][START_REF] Gille | The FIS protein binds and bends the origin of chromosomal DNA replication, oriC, of Escherichia coli[END_REF][START_REF] Margulies | The FIS protein fails to block the binding of DnaA protein to oriC, the Escherichia coli chromosomal origin[END_REF][START_REF] Ryan | IHF and HU stimulate assembly of pre-replication complexes at Escherichia coli oriC by two different mechanisms[END_REF][START_REF] Ryan | Escherichia coli prereplication complex assembly is regulated by dynamic interplay among Fis, IHF and DnaA[END_REF][START_REF] Chodavarapu | Escherichia coli Dps interacts with DnaA protein to impede initiation: A model of adaptive mutation[END_REF][START_REF] Magnan | Regulation of DNA Replication Initiation by Chromosome Structure[END_REF][START_REF] Wolanski | oriC-encoded instructions for the initiation of bacterial chromosome replication[END_REF]. Importantly, in γ-proteobacteria in general and in E. coli in particular, the sequences around OriC demonstrate a higher thermodynamic stability compared to those around the chromosomal terminus (Ter) of replication (Figure 1). In E. coli, the G/C-rich OriC end is activated early on commitment of cells to exponential growth [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF]. This early activation of the OriC end is associated with the activation of DNA gyrase, which introduces negative supercoils in the DNA [START_REF] Van Workum | DNA supercoiling depends on the phosphorylation potential in Escherichia coli[END_REF] and increases the expression of supercoiling-dependent strong transcription units (transcriptons) encoding ribosomal RNA (rrn) and protein genes, as well as of the RpoZ (ω) subunit of RNA polymerase (RNAP), which stabilizes the vegetative (σ 70 ) holoenzyme assembly [START_REF] Mukherjee | GroEL is involved in activation of Escherichia coli RNA polymerase devoid of the omega subunit in vivo[END_REF][START_REF] Minakhin | Bacterial RNA polymerase subunit omega and eukaryotic RNA polymerase subunit RPB6 are sequence, structural, and functional homologs and promote RNA polymerase assembly[END_REF][START_REF] Oostra | Enhancement of ribosomal ribonucleic acid synthesis by deoxyribonucleic acid gyrase activity in Escherichia coli[END_REF] and the activator of rrn transcription FIS [START_REF] Oostra | Enhancement of ribosomal ribonucleic acid synthesis by deoxyribonucleic acid gyrase activity in Escherichia coli[END_REF][START_REF] Bosch | FIS-dependent trans-activation of tRNA and rRNA operons of Escherichia coli[END_REF][START_REF] Ross | coli Fis protein activates ribosomal RNA transcription in vitro and in vivo[END_REF][START_REF] Ohlsen | Interrelated effects of DNA supercoiling, ppGpp, and low salt on melting within the Escherichia coli ribosomal RNA rrnB P1 promoter[END_REF][START_REF] Sander | Mechanisms of upstream activation of the rrnD promoter P1 of Escherichia coli[END_REF][START_REF] Appleman | Activation of Escherichia coli rRNA transcription by FIS during a growth cycle[END_REF][START_REF] Schneider | The expression of the Escherichia coli fis gene is strongly dependent on the superhelical density of DNA[END_REF][START_REF] Zhi | Fis stabilizes the interaction between RNA polymerase and the ribosomal promoter rrnB P1, leading to transcriptional activation[END_REF][START_REF] Zhang | Effects of Fis on ribosome synthesis and activity and on rRNA promoter activities in Escherichia coli[END_REF][START_REF] Choi | Effects of FIS protein on rnpB transcription in Escherichia coli[END_REF], among other genes involved in boosting growth. Activation of these strong transcriptons requires high negative superhelicity, supplied on the one hand by DNA gyrase, demonstrating a decreasing gradient of binding sites from OriC to Ter [START_REF] Jeong | Spatial patterns of transcriptional activity in the chromosome of Escherichia coli[END_REF][START_REF] Berger | Coordination of genomic structure and transcription by the main bacterial nucleoid-associated protein HU[END_REF] and on the other hand, by increased production of negative supercoils in the wake of both the translocating replisomes and numerous RNAP molecules engaged in the transcription of stable RNA operons, all of which are oriented towards the Ter [START_REF] Liu | Supercoiling of the DNA template during transcription[END_REF][START_REF] Yu | Replication-induced supercoiling: A neglected DNA transaction regulator?[END_REF][START_REF] Muskhelishvili | DNA structure and bacterial nucleoid-associated proteins[END_REF]. Notably, these powerful DNA translocases also increase the positive superhelicity ahead of them, which can be counterbalanced by increased gyrase activity observed downstream of the highly transcribed operons [START_REF] Yang | DNA gyrase binds to the family of prokaryotic repetitive extragenic palindromic sequences[END_REF][START_REF] Sutormin | Single-nucleotide-resolution mapping of DNA gyrase cleavage sites across the Escherichia coli genome[END_REF]. Although direct experimental evidence is still lacking, all these effects are thought to lead to higher negative superhelicity in the chromosomal OriC end compared to the Ter end [START_REF] Sobetzko | Gene order and chromosome dynamics coordinate spatiotemporal gene expression during the bacterial growth cycle[END_REF], thus facilitating the utilization of thermodynamically stable (G/C-richer) DNA sequences for transcription. Indeed, in cells lacking the major NAP HU, the chromosomal rrn domain, encompassing OriC together with the most distal rrn operons on both chromosomal arms, is activated aberrantly on the expense of genes requiring high negative superhelicity for transcription, suggesting that HU is involved in buffering the accumulation of excessive negative superhelicity in the OriC end, which otherwise, could impair the coordinated transcription of the genome [START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF][START_REF] Berger | Coordination of genomic structure and transcription by the main bacterial nucleoid-associated protein HU[END_REF][START_REF] Ferrandiz | HU of Streptococcus pneumoniae Is Essential for the Preservation of DNA Supercoiling[END_REF]. The NAP FIS, abundant during the commitment of cells to fast exponential growth, exerts a more local coordinating effect by stabilizing DNA loops at upstream activating sequences (UAS) of stable RNA operons and buffering them at deviations of superhelical density [START_REF] Rochman | Promoter protection by a transcription factor acting as a local topological homeostat[END_REF][START_REF] Muskhelishvili | Transcription factor as a topological homeostat[END_REF].

On cessation of replication, when the processes increasing the supercoil dynamics in the G/C-richer chromosomal Ori end subside, the less G/C-rich Ter end gains a competitive advantage in terms of energy requirements for duplex melting associated with the initiation of transcription. This switch in the relative activity of chromosomal ends is coordinated with a change in RNAP holoenzyme composition and promoter recognition specificity [START_REF] Ishihama | Functional modulation of Escherichia coli RNA polymerase[END_REF][START_REF] Typas | The molecular basis of selective promoter activation by the sigmaS subunit of RNA polymerase[END_REF][START_REF] Typas | Differential ability of sigma(s) and sigma70 of Escherichia coli to utilize promoters containing half or full UP-element sites[END_REF], apparently assisted by abundant NAPs, demonstrating a high frequency of repressor sites in the Ori end, such as heat-stable nucleoid-structuring protein (H-NS) and leucine-responsive protein (Lrp) [START_REF] Sobetzko | Gene order and chromosome dynamics coordinate spatiotemporal gene expression during the bacterial growth cycle[END_REF], as well as increased expression of DNA-binding protein from starving cells (Dps), the most abundant stationary-phase NAP involved in condensing the DNA [START_REF] Chodavarapu | Escherichia coli Dps interacts with DnaA protein to impede initiation: A model of adaptive mutation[END_REF][START_REF] Ohniwa | Dynamic state of DNA topology is essential for genome condensation in bacteria[END_REF]. H-NS and Lrp also synergize to shut down stable RNA promoters on depletion of nutrients [START_REF] Pul | LRP and H-NS-cooperative partners for transcription regulation at Escherichia coli rRNA promoters[END_REF]. Furthermore, at this stage, the negative superhelicity of the Ter end is increased [START_REF] Lal | Genome scale patterns of supercoiling in a bacterial chromosome[END_REF]. Thus, ultimately, the successive transcriptional activation of chromosomal poles during the E. coli growth cycle appears to be governed by coupling the static gradient of DNA thermodynamic stability with the dynamic gradient of negative superhelicity along the chromosomal OriC-Ter axis, which acts as a coordinate system [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF][START_REF] Sobetzko | Gene order and chromosome dynamics coordinate spatiotemporal gene expression during the bacterial growth cycle[END_REF].

Importantly, the chromosomal OriC and Ter ends also differ in terms of genetic function. The former is enriched in anabolic genes and the latter in catabolic genes [START_REF] Nigatu | Relationship between digital information and thermodynamic stability in bacterial genomes[END_REF]. More compellingly, the temporal expression of genes is tightly coupled to their spatial arrangement in the chromosome-a feature that is conserved in both the γand the α-proteobacteria [START_REF] Sobetzko | Gene order and chromosome dynamics coordinate spatiotemporal gene expression during the bacterial growth cycle[END_REF][START_REF] Brinza | Genomic analysis of the regulatory elements and links with intrinsic DNA structural properties in the shrunken genome of Buchnera[END_REF]. Thus, in E. coli, the spatial order of genes along the chromosomal OriC-Ter axis and the temporal expression of genetic function is coupled with a strategic organization of sequences with distinct thermodynamic stability around the opposite chromosomal poles. This superimposition of the gradients of DNA thermodynamic stability and supercoiling on a conserved chromosomal gene order with anabolic and catabolic functions, allocated respectively to the OriC and the Ter ends, provides a bona fide device for the spatiotemporal coordination of genomic expression during the E. coli growth cycle [START_REF] Muskhelishvili | Order from the order: How a spatiotemporal genetic program is encoded in a 2D genetic map of the bacterial chromosome[END_REF]. The dynamic patterns of CODOs emerging during the E. coli growth cycle and featuring distinct sequence organization are consistent with this notion [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF]. However, the genomic sequence organization in the commensal bacterium E. coli is likely streamlined for fast growth. The question is whether a similar organization can be revealed in related bacteria with more sophisticated lifestyles, such as, e.g., bacterial pathogens.

One excellent model to explore this question is the plant pathogenic bacterium Dickeya dadantii, which during plant infection, has to cope with various types of severe environmental stress [START_REF] Jiang | Global transcriptional response of Dickeya dadantii to environmental stimuli relevant to the plant infection[END_REF][START_REF] Reverchon | Virulence Program of a Bacterial Plant Pathogen: The Dickeya Model[END_REF].

Model of Regulation of Gene Expression during the Pathogenic Growth of D. dadantii

The sequence organization of D. dadantii genome markedly differs from that of E. coli. In particular, it features four (two in each chromosomal arm) thermodynamically distinct DNA regions. These regions appear to be highly conserved, although they have suffered extensive rearrangements during the evolution of Dickeya species (Figure 2). [START_REF] Duprey | The phytopathogenic nature of Dickeya aquatica 174/2 and the dynamic early evolution of Dickeya pathogenicity[END_REF]). (B) Distribution of DNA average negative melting energy in circular Dickeya genomes. The average melting energy was calculated using the parameters of SantaLucia [START_REF] Santalucia | A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics[END_REF] with a scanning window size of 500 kb sliding by 4 kb as in Figure 1 and Reference [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF]. Color-coded in blue and red are, respectively, the high and low average negative melting energy (same color scale as in Figure 1). The G/C-rich SD1 and SD2 regions and the A/T-rich LD1 and LD2 regions are indicated in Dickeya dadantii. Ter: terminus of replication.

Conservation of these regions, despite the fact that some of the Dickeya genus representatives experienced a substantial change in their habitat, suggests that they are important for the adaptation and maintenance of pathogenicity. Indeed, recent studies in the D. dadantii model demonstrated that the transcriptional response of this pathogen to various kinds of stress produces coherent spatially confined gene expression patterns dubbed the "stress response" domains [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF]. These latter comprise DNA sequences of different thermodynamic stability and thus closely resemble the CODOs of E. coli. Since these CODOs distinctly respond to alterations of DNA supercoiling, they also closely resemble the topological domains observed in the human pathogen Streptococcus pneumoniae [START_REF] Ferrandiz | The genome of Streptococcus pneumoniae is organized in topology-reacting gene clusters[END_REF].

The detection methodology of CODOs is based on exploring the correspondence between the genomic distributions of DNA analog parameters (such as gene expression density, average negative melting/stacking energy, supercoiling response of the expressed genes, preferential orientation of the transcriptons, relative frequency of binding sites for NAPs) and the distribution of digital parameters (unique genes and genetic functions) [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF]. Ultimately, this methodology intends to correlate genomic sequence organization with genetic function. In D. dadantii, the distribution of each of these parameters was measured in the transcriptomes obtained from cells grown under distinct stress conditions mimicking the environment during pathogenic growth; subsequently, the correlations between their spatial distribution patterns were mapped in the genome to determine common, statistically significant boundaries. Analyses of transcriptomes obtained under 32 different growth conditions identified 11 different CODOs, emerging in specific constellations in response to the applied stress [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF][START_REF] Reverchon | Rethinking the bacterial genetic regulation[END_REF]. In the OriC end of the chromosome, the boundaries of some of these "stress response" CODOs closely coincide with the positions of stable RNA operons [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF], in keeping with findings in E. coli [START_REF] Booker | DNA topology of highly transcribed operons in Salmonella enterica serovar Typhimurium[END_REF]. However, the crucial question is whether it is possible to identify signatures in the basic design of genomic sequence organization that explain the dynamic behavior of the CODOs.

In E. coli, the measurements of the distribution of DNA analog (i.e., structure-related) and digital (i.e., genetic code-related) parameters of the genomic sequence revealed a strict anti-correlation between thermodynamic stability and codon variability in the OriC and Ter ends of the genome. In the G/C-rich OriC end, the randomness of codon distribution is low, whereas in the relatively A/T-rich Ter region, where thermodynamic stability is low, codon randomness is high [START_REF] Nigatu | Relationship between digital information and thermodynamic stability in bacterial genomes[END_REF]. This is because due to the degeneracy of the genetic code, G/C-rich codons are used, preferentially, in the G/C-rich region, whereas in A/T-rich regions, all the codons are used equally. Application of this approach to D. dadantii revealed the same regularity: the G/C-rich SD1 and SD2 regions demonstrated low sequence randomness, whereas the A/T-rich LD1 and LD2 regions demonstrated high sequence randomness (Figure 3). Furthermore, the genomic distributions of these two characteristics of DNA sequences revealed extended domains featuring an opposite skew of thermodynamic stability and codon randomness. Strikingly, these static genomic patterns largely coincided with the dynamic borders of the CODOs derived from the correlated changes of analog parameters in both the time-resolved transcriptomes of E. coli and the stress response transcriptomes of D. dadantii (Figure 3). Given the difference in the used approaches, it is highly unlikely that this coincidence is accidental. It rather suggests that not only the genetic code, but also the dynamic behavior of the chromosome is inscribed in the primary sequence organization of the genome. Importantly, while most of the CODOs comprise sequences of distinct thermodynamic stability, they also harbor specific adaptation and virulence determinants dedicated to coping with various kinds of environmental stress [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF][START_REF] Meyer | Chromosomal organization of transcription: In a nutshell[END_REF]. A similar organization of function in isolated gene clusters and chromosomal topological domains was observed in Salmonella [START_REF] Webber | Clinically relevant mutant DNA gyrase alters supercoiling, changes the transcriptome, and confers multidrug resistance[END_REF], S. pneumoniae [START_REF] Ferrandiz | HU of Streptococcus pneumoniae Is Essential for the Preservation of DNA Supercoiling[END_REF], and Streptomyces [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF][START_REF] Szafran | Transcriptional Response of Streptomyces coelicolor to Rapid Chromosome Relaxation or Long-Term Supercoiling Imbalance[END_REF]. Taken together, these findings support the notion that the CODOs represent genuine sub-chromosomal structural-functional entities involved in coordinated responses of the genome to environmental change. 1, Figure 2, and Reference [START_REF] Nigatu | Relationship between digital information and thermodynamic stability in bacterial genomes[END_REF]). Individual CODOs (d1 to d11) are indicated above the graphs. In E. coli, the CODOs d7 and d8 comprise the Ter region, and OriC is located in the middle of d1. In D. dadantii, the four thermodynamically distinct regions LD1, SD1, LD2, and SD2 are comprised in the CODOs d3, d5, d7, and d9 respectively. OriC is at the border between d11 and d1, whereas Ter is located in the middle of d6. The borders of the CODOs (vertical lines superimposed on the GC content/sequence randomness curves) were defined on the basis of correlation between various analog DNA information types retrieved from the transcriptomes of E. coli [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF] and D. dadantii [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF].

Testing the Dynamical Behavior of CODOs by Reporter Constructs.

While we still lack a clear understanding of the mechanisms leading to their emergence, most of the CODOs feature genes that are characterized by a distinct average negative melting energy, prefer a particular supercoiling regimen, and/or coherently respond to particular NAPs. For example, the CODO d7 of D. dadantii comprising the LD2 region contains DNA of low thermodynamic stability (see Figure 2). Under conditions of osmotic stress, the differentially expressed genes in this CODO are mainly downregulated (Figure 4; blue square in the upper row, marked "dens" for gene expression density), whereas the majority of the still expressed genes are encoded by sequences of high average negative melting energy (red square in the upper row, marked "melt") and activated by DNA relaxation (red square in the upper row, marked "rel"). However, on application of oxidative stress, the genes in the same CODO d7 are mainly upregulated, whereby the expressed genes are predominantly encoded by sequences of low average negative melting energy and activated by high negative superhelicity, FIS and H-NS (red columns in the lower row). corresponding to four different growth media. The parameters of expressed genes are color-coded (red for high, and blue for low values): dens: gene expression density, i.e., high gene density indicates a high proportion of genes upregulated, whereas low gene density indicates a high proportion of genes downregulated; melt: average negative melting energy of expressed sequences (note that unlike in Figures 1 and2, red indicates high negative melting energy, and blue indicates low negative melting energy); hyp: proportion of genes activated by high negative supercoiling; rel: proportion of genes activated by low negative supercoiling; FIS: proportion of factor for inversion stimulation (FIS)-activated genes; H-NS: proportion of heat-stable nucleoid-structuring protein (H-NS)-activated (de-repressed) genes. Adapted from Reference [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF].

In other words, we see that depending on the kind of challenge, the same chromosomal region produces distinct specific responses. Since the organization of CODOs reveals a combined input of several factors pertinent to the modulation of DNA supercoil dynamics, it is likely that their formation reflects changing structural dynamics of the corresponding chromosomal regions induced by environmental stress. In principle, such dynamic alterations can be examined using reporter gene constructs capable to sense regional changes of chromosomal supercoil dynamics. More specifically, the question is whether a reporter gene inserted into a particular CODO will reflect its activity (defined as the average response of its genes to a particular environmental signal). Since the emergence of a CODO is correlated with a directionally coherent alteration (either activation or repression) of gene expression, a faithful reporter construct should comply with the behavior of the CODO it is targeted to. For example, peculiar influence of the encompassing chromosomal regions on the inserted reporter cassettes has been reported in S. pneumoniae, where the topoisomerase genes implicated in the physiological control of supercoiling were found to be harbored by topological domains distinctly responding to DNA relaxation [START_REF] Ferrandiz | Role of global and local topology in the regulation of gene expression in Streptococcus pneumoniae[END_REF].

In D. dadantii, the correlation between domain behavior and reporter activity was examined using the supercoiling-sensitive fisP-YFP construct, which was previously shown to faithfully reproduce the regional differences in supercoil dynamics induced by deletion of HU in E. coli [START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF]. D. dadantii CODOs d5 and d7 emerge synchronously and harbor adaptation/virulence determinants, the expression of which significantly correlates with that of the CODOs [START_REF] Meyer | Chromosomal organization of transcription: In a nutshell[END_REF]. These two CODOs respond similarly to oxidative stress but respond in opposite ways to osmotic stress. Accordingly, it was observed that the fisP-YFP reporter inserted in d5 and d7 demonstrates a behavior reflecting that of the targeted CODOs, i.e., activation under oxidative stress in both CODOs and activation in d5 vs repression in d7 under osmotic stress (Figure 5A). Notably, in both of these CODOs, the activity of the fisP-YFP reporter declined after the relaxation of DNA induced by novobiocin treatment, as expected. However, when this construct was inserted into the CODO d8, which is "asynchronous" in a sense that it does not appear in any of the examined growth conditions in combination with either d5 or d7, the reporter responded poorly to both oxidative and osmotic stresses, whereas DNA relaxation by novobiocin increased instead of decreasing the reporter activity. At the same time, the response of the fisP-YFP reporter to acidic stress was uniform and did not depend on the behavior of the targeted CODO. This versatile response of the fisP-YFP reporter markedly differed from that of the dpsP-YFP construct, in which the reporter is under the control of the promoter of the dps gene encoding Dps, the major stationary-phase NAP. In E. coli, this latter construct demonstrated relative independence from the chromosomal context [START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF] and was also much less discriminative between the domains in D. dadantii (Figure 5B).

However, why does the fisP-YFP construct inserted in CODOs d5 and d7, as well as these CODOs themselves, respond to osmotic stress in opposite ways? Osmotic shock stimulates K + uptake [START_REF] Sutherland | Osmotic regulation of transcription: Induction of the proU betaine transport gene is dependent on accumulation of intracellular potassium[END_REF], compensated by an increase in internal glutamate concentration [START_REF] Dinnbier | Transient accumulation of potassium glutamate and its replacement by trehalose during adaptation of growing cells of Escherichia coli K-12 to elevated sodium chloride concentrations[END_REF]. This change in the internal milieu could affect protein occupancy in CODOs d7 and d5, comprising sequences of, respectively, low (LD2) and high (SD2) thermodynamic stability (see Figure 2). Indeed, protein occupancy depends on both DNA composition and sequence [START_REF] Olson | DNA sequence-dependent deformability deduced from protein-DNA crystal complexes[END_REF][START_REF] Brogaard | A map of nucleosome positions in yeast at base-pair resolution[END_REF]. Osmotic shock, besides redirecting RNAP binding in the genome [START_REF] Sun | Density of sigma70 promoter-like sites in the intergenic regions dictates the redistribution of RNA polymerase during osmotic stress in Escherichia coli[END_REF], could also affect the binding of NAPs, which depends on both sequence composition and DNA topology [START_REF] Schneider | FIS modulates growth phase-dependent topological transitions of DNA in Escherichia coli[END_REF][START_REF] Liu | Physical properties of DNA may direct the binding of nucleoid-associated proteins along the E. coli genome[END_REF][START_REF] Ouafa | The nucleoid-associated proteins H-NS and FIS modulate the DNA supercoiling response of the pel genes, the major virulence factors in the plant pathogen bacterium Dickeya dadantii[END_REF][START_REF] Duprey | Temporal control of Dickeya dadantii main virulence gene expression by growth phase-dependent alteration of regulatory nucleoprotein complexes[END_REF][START_REF] Japaridze | Hyperplectonemes: A Higher Order Compact and Dynamic DNA Self-Organization[END_REF]. For example, the binding of the global repressor H-NS [START_REF] Dorman | the genome sentinel[END_REF] is modulated by ionic composition, sequence organization, and DNA topology [START_REF] Ouafa | The nucleoid-associated proteins H-NS and FIS modulate the DNA supercoiling response of the pel genes, the major virulence factors in the plant pathogen bacterium Dickeya dadantii[END_REF][START_REF] Japaridze | Spatial organization of DNA sequences directs the assembly of bacterial chromatin by a nucleoid-associated protein[END_REF][START_REF] Liu | A divalent switch drives H-NS/DNA-binding conformations between stiffening and bridging modes[END_REF][START_REF] Van Der Valk | Mechanism of environmentally driven conformational changes that modulate H-NS DNA-bridging activity[END_REF][START_REF] Joyeux | Role of Salt Valency in the Switch of H-NS Proteins between DNA-Bridging and DNA-Stiffening Modes[END_REF][START_REF] Will | Modulation of H-NS transcriptional silencing by magnesium[END_REF][START_REF] Rafiei | Growth Phase Dependent Chromosome Condensation and H-NS Protein Redistribution in Escherichia coli Under Osmotic Stress[END_REF]. H-NS preferentially binds A/T-rich sequences and represses transcription by bridging DNA duplexes and forming inter-wound filaments [START_REF] Lang | High-affinity DNA binding sites for H-NS provide a molecular basis for selective silencing within proteobacterial genomes[END_REF][START_REF] Maurer | A systematic in vitro study of nucleoprotein complexes formed by bacterial nucleoid-associated proteins revealing novel types of DNA organization[END_REF]. This bias of H-NS binding could explain the opposite effects of the osmotic shock on the G/C-rich CODO d5 and the A/T-rich CODO d7 [START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF], as well as the "faithful" response of the fisP-YFP reporter inserted into these CODOs-a conjecture that can be tested, for example, by examining the reporter behavior in the hns mutant strain.

It is noteworthy that, while it is generally accepted that alterations of transcript patterns are adaptive, there are reports about the absence of correlation between the transcript pattern and the expressed function. For example, in a study comparing the expression and mutant fitness of biosynthetic genes in several bacteria including the metal-reducing bacterium Shewanella oneidensis, E. coli, the ethanol-producing bacterium Zymomonas mobilis, and the sulfate-reducing bacterium Desulfovibrio alaskensis it was reported that there is little correlation between the importance of genes for optimal growth or fitness and their upregulation [100]. Also, a recent study of the plant pathogen Pseudomonas syringae reported the absence of correlation between mutant fitness in planta and either the magnitude of expression or the degree of induction for most genes in planta compared to in vitro conditions [101]. However, fitness is measured cumulatively over several growth cycles, whereas expression is measured during a single growth/infection cycle, making direct comparisons difficult. In principle, the lack of strict coherence between the impact of gene mutations on fitness and the expression of the genes concerned could be explained by the existence of several genes or loci acting in parallel on the same process. The highly reproducible expression patterns of the CODOs observed under conditions of applied environmental stress or modulation of DNA topology by topoisomerase inhibitors and/or poisons suggest that functionally linked genes including virulence genes [START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF][START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF][START_REF] Ferrandiz | Role of global and local topology in the regulation of gene expression in Streptococcus pneumoniae[END_REF] are co-transcribed within a variable chromosomal context, which is beyond the detection capacity of the approaches examining the comparative fitness of individual mutants. (A) Expression of the supercoiling-dependent fisP-YFP reporter construct reflects the behavior of the synchronously emerging CODOs d5 and d7 in response to oxidative and osmotic stress. The response of the same reporter in the "asynchronously" emerging CODO d8, with the exception of the response to acidic stress, is entirely different. (B) The response of the dpsP-YFP construct to perturbations is less discriminative between the domains. The domain behavior is indicated, with the red arrow corresponding to upregulation, the blue arrow to downregulation, and the green line to no change in gene expression density. The fisP-yfp and dps-yfp constructs were obtained by PCR amplification from the pUCter fis-yfp and pUCter dps-yfp plasmids [START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF], respectively, and contain the E. coli fis operon promoter sequence (comprising 300 bp upstream from the transcription start site at +1 and the downstream dusB ORF) and the E. coli dps promoter sequence (comprising 500 bp upstream from the transcription start site at +1) cloned upstream of the yfp gene, followed by an rrnB terminator and a cat gene conferring chloramphenicol (Cm) resistance. The PCR fragments containing the fisP-yfp-Cm R and dpsP-yfp-Cm R cassettes were cloned in pGEMT plasmids, electroporated in D. dadantii cells, and inserted in d5 (between the convergent ousA and ORF46989), d7 (between the convergent fliT and fliE genes), and d8 (between the tandem pelA and pelE genes) by marker-exchange recombination. The recombinants were selected in low-phosphate medium in the presence of chloramphenicol, under conditions in which pGEMT derivatives are highly unstable. Expression of the fisP-yfp or dpsP-yfp fusions in different chromosomal domains was measured by a spectrofluorimeter (TECAN Infinite F200) with emission/excitation wavelengths of 548/505nm, respectively.

Conclusions

The sequence organization of the bacterial genomes reflects the necessity of facile accessibility of genomic regions encoding particular functions to regulatory factors during bacterial growth and adaptation to changing environmental conditions. We have argued that both in the commensal E. coli and in the plant pathogen Dickeya, this coordination of structure and function is achieved by the strategic organization of large chromosomal domains of gene expression containing sequences with distinct thermodynamic stability and supercoiling response, termed CODOs. The Dickeya CODOs and similar regions in other pathogens harbor various adaptation/virulence determinants that are co-regulated with and expressed within the CODOs, emerging in responses to environmentally induced changes of DNA supercoiling. Since global DNA supercoiling changes as a function of metabolic state, while the latter depends on growth conditions, it is likely that specific constellations of CODOs emerging in response to environmental change aid the facile adaptation of pathogenic bacteria to stress. We believe that further studies of the coupling between the physicochemical properties of DNA, genomic expression dynamics, and genetic function will pave the way towards understanding how the coordinated transcriptional response to environmental change is encoded in the primary sequence organization of bacterial genomes.

Figure 1 .

 1 Figure 1. Distribution of DNA average negative melting energy in the circular genomes of γ-proteobacteria and Escherichia coli. Color-coded in blue and red represent, respectively, the high and low average negative melting energies of the sequences. The outer ring indicates, by color, the chromosomal macrodomains: Ori (green), Right (red), Left (dark blue), Ter (light blue), LNS (Left Non-Structured region) and RNS (Right Non-Structured region) in black. OriC: origin of replication. Scanning window size is 500 kb sliding by 4 kb. Adapted from Reference [23].

Figure 2 .

 2 Figure 2. (A) Phylogenetic tree of the various Dickeya species (adapted from Reference[START_REF] Duprey | The phytopathogenic nature of Dickeya aquatica 174/2 and the dynamic early evolution of Dickeya pathogenicity[END_REF]). (B) Distribution of DNA average negative melting energy in circular Dickeya genomes. The average melting energy was calculated using the parameters of SantaLucia[START_REF] Santalucia | A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics[END_REF] with a scanning window size of 500 kb sliding by 4 kb as in Figure1and Reference[START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF]. Color-coded in blue and red are, respectively, the high and low average negative melting energy (same color scale as in Figure1). The G/C-rich SD1 and SD2 regions and the A/T-rich LD1 and LD2 regions are indicated in Dickeya dadantii. Ter: terminus of replication.

Figure 3 .

 3 Figure 3. Mapping of correlations between G/C content and distributions of codons (sequence randomness) in the E. coli (left panel A) and D. dadantii (right panel B) genomes. The distributions were calculated using 500 kb windows sliding by 4 kb (as in Figure 1, Figure 2, and Reference [71]).Individual CODOs (d1 to d11) are indicated above the graphs. In E. coli, the CODOs d7 and d8 comprise the Ter region, and OriC is located in the middle of d1. In D. dadantii, the four thermodynamically distinct regions LD1, SD1, LD2, and SD2 are comprised in the CODOs d3, d5, d7, and d9 respectively. OriC is at the border between d11 and d1, whereas Ter is located in the middle of d6. The borders of the CODOs (vertical lines superimposed on the GC content/sequence randomness curves) were defined on the basis of correlation between various analog DNA information types retrieved from the transcriptomes of E. coli[START_REF] Sobetzko | DNA thermodynamic stability and supercoil dynamics determine the gene expression program during the bacterial growth cycle[END_REF] and D. dadantii[START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF].

Figure 4 .

 4 Figure 4. Response of the CODO d7 (560 kb) of D. dadantii to osmotic (upper row) and oxidative (lower row) stress. The domain is represented as a square consisting of four rectangles (numbered 1 to 4)corresponding to four different growth media. The parameters of expressed genes are color-coded (red for high, and blue for low values): dens: gene expression density, i.e., high gene density indicates a high proportion of genes upregulated, whereas low gene density indicates a high proportion of genes downregulated; melt: average negative melting energy of expressed sequences (note that unlike in Figures1 and 2, red indicates high negative melting energy, and blue indicates low negative melting energy); hyp: proportion of genes activated by high negative supercoiling; rel: proportion of genes activated by low negative supercoiling; FIS: proportion of factor for inversion stimulation (FIS)-activated genes; H-NS: proportion of heat-stable nucleoid-structuring protein (H-NS)-activated (de-repressed) genes. Adapted from Reference[START_REF] Jiang | Chromosomal "stress-response" domains govern the spatiotemporal expression of the bacterial virulence program[END_REF].

Figure 5 .

 5 Figure 5. Response of reporter gene constructs to perturbations of chromosomal supercoiling.(A) Expression of the supercoiling-dependent fisP-YFP reporter construct reflects the behavior of the synchronously emerging CODOs d5 and d7 in response to oxidative and osmotic stress. The response of the same reporter in the "asynchronously" emerging CODO d8, with the exception of the response to acidic stress, is entirely different. (B) The response of the dpsP-YFP construct to perturbations is less discriminative between the domains. The domain behavior is indicated, with the red arrow corresponding to upregulation, the blue arrow to downregulation, and the green line to no change in gene expression density. The fisP-yfp and dps-yfp constructs were obtained by PCR amplification from the pUCter fis-yfp and pUCter dps-yfp plasmids[START_REF] Berger | Genes on a Wire: The Nucleoid-Associated Protein HU Insulates Transcription Units in Escherichia coli[END_REF], respectively, and contain the E. coli fis operon promoter sequence (comprising 300 bp upstream from the transcription start site at +1 and the downstream dusB ORF) and the E. coli dps promoter sequence (comprising 500 bp upstream from the transcription start site at +1) cloned upstream of the yfp gene, followed by an rrnB terminator and a cat gene conferring chloramphenicol (Cm) resistance. The PCR fragments containing the fisP-yfp-Cm R and dpsP-yfp-Cm R cassettes were cloned in pGEMT plasmids, electroporated in D. dadantii cells, and inserted in d5 (between the convergent ousA and ORF46989), d7 (between the convergent fliT and fliE genes), and d8 (between the tandem pelA and pelE genes) by marker-exchange recombination. The recombinants were selected in low-phosphate medium in the presence of chloramphenicol, under conditions in which pGEMT derivatives are highly unstable. Expression of the fisP-yfp or dpsP-yfp fusions in different chromosomal domains was measured by a spectrofluorimeter (TECAN Infinite F200) with emission/excitation wavelengths of 548/505nm, respectively.

Microorganisms 2019, 7, 694; doi:10.3390/microorganisms7120694 www.mdpi.com/journal/microorganisms

Microorganisms 2019, 7, 694

Funding: This work was in part supported by the INSA invited researcher grant to G.M. The article processing charge was funded by INSAVALOR.

Conflicts of Interest:

The authors declare no conflict of interest.


Dilshad Surroop

Pascal Combes

Philippe Martin

Pierre Rouchon

Third-order virtual measurements with signal injection

come

I. INTRODUCTION

Signal injection is a control technique that has become widely used for the "sensorless" control of electrical motors at low velocity since its introduction by [START_REF] Jansen | Transducerless position and velocity estimation in induction and salient AC machines[END_REF], [START_REF] Corley | Rotor position and velocity estimation for a salient-pole permanent magnet synchronous machine at standstill and high speeds[END_REF] ("sensorless" meaning only the currents are measured, but neither the rotor position nor its velocity). Its consists in superimposing a fastvarying signal on the control, which creates some small ripple in the measured currents; this ripple contains information about the rotor position that can be used to suitably control the motor. At first sight, the method might seem peculiar to electrical motors, with a usually somewhat heuristic analysis. The essence of the method was then conceptualized in [START_REF] Combes | Adding virtual measurements by signal injection[END_REF] as the creation of new "virtual measurements" which can be extracted from the actual measured output, hence providing means to overcome observability degeneracies; the main ingredient of the analysis is second-order averaging, following the ideas introduced in [START_REF] Jebai | Sensorless position estimation and control of permanent-magnet synchronous motors using a saturation model[END_REF]. Developments following these ideas, with an application to magnetic levitation systems, can be found in [START_REF] Yi | Relaxing the conditions for parameter estimation-based observers of nonlinear systems via signal injection[END_REF], [START_REF] Yi | An adaptive observer for sensorless control of the levitated ball using signal injection[END_REF].

The purpose of this paper is to extend [START_REF] Combes | Adding virtual measurements by signal injection[END_REF] by showing that more virtual measurements can be produced with a finer analysis of the ripple thanks to third-order averaging. To keep the computations as simple as possible and focus on the important ideas, we restrict to Single-Input Single-Output systems with a linear dynamics; nonlinear Multiple-Input Multiple-Output systems could nevertheless be addressed along the same lines. More precisely, consider the system

ẋ = Ax + Bu (1a) y = h(x), (1b)
where (x, y, u) belongs to a compact subset of R n × R × R, and A, B are constant matrices; the measured output y = h(x) is assumed smooth enough (e.g. at least C 3 ). We show that by superimposing to the control u a periodic signal with small period ε, we can make available the so-called virtual measurements

Y 1 = H 1 (x) := εh (x)B (2a) Y 2 = H 2 (x) := ε 2 2 (h (x)B) B (2b)
Y 3 = H 3 (x) := ε 2 h (x)AB, (2c)
which can be used in addition to Y 0 = H 0 (x) := h(x) to control (1a). The contribution with respect to [START_REF] Combes | Adding virtual measurements by signal injection[END_REF] is threefold:

• an analysis of the output ripple by third-order averaging, with a simpler derivation (section II) • a much more elaborated procedure to extract the virtual measurements from the output ripple (section III) • a numerical simulation demonstrating the method is indeed effective, even though the output ripple may be very small and buried into noise (section IV).

II. SIGNAL INJECTION AND THIRD-ORDER AVERAGING

Assume we have designed a suitable control law

u = α η, Y, t η = a η, Y, t ,
where η ∈ R q and Y is the vector (Y 0 , Y 1 , Y 2 , Y 3 ). In other words, the closed-loop system

ẋ = Ax + Bα η, H(x), t (3a) η = a η, H(x), t (3b)
has the desired exponentially stable behaviour, where H := (H 0 , H 1 , H 2 , H 3 ). We have changed the notation of the state to (x, η), so as to distinguish between the solutions of (3) and of (5) below. Now consider the modified control law

u = α η, H(x), t + s 0 t ε (4a) η = a η, H(x), t (4b)
H(x) = H x -εBs 1 t ε -ε 2 ABs 2 t ε + O ∞ (ε 3 ), ( 4c
)
where s 0 is a 1-periodic function with zero mean, s 1 is the primitive of s 0 with zero mean, and s 2 the primitive of s 1 with zero mean (notice s 1 and s 2 are also 1-periodic); and O ∞ denotes the uniform "big O" symbol of analysis, namely f (z, ε) = O ∞ (ε) if there exists K > 0 independant of z and ε such that f (z, ε) ≤ Kε. The closed-loop system then reads

ẋ = Ax + Bα η, H(x), t + Bs 0 t ε (5a) η = a η, H(x), t (5b)
H(x) = H x -εBs 1 t ε -ε 2 ABs 2 t ε + O ∞ (ε 3 ). (5c)
Theorem 1: Let x(t), η(t) and x(t), η(t) be respectively the solutions of ( 5) and (3), with initial condition x(0), η(0) and x(0), η(0) = x(0) -εBs 1 (0)ε 2 ABs 2 (0), η(0) . Then for all t ≥ 0,

x(t) = x(t) + εBs 1 t ε + ε 2 ABs 2 t ε + O ∞ (ε 3 ) (6a) η(t) = η(t) + O ∞ (ε 3 ) (6b) y(t) = H 0 x(t) + H 1 x(t) s 1 t ε + H 2 x(t) s 2 1 t ε + H 3 x(t) s 2 t ε + O ∞ (ε 3 ).
(6c) Proof: The proof is an application of higher-order averaging for differential equations [7, section 2.9], with slow time dependance [7, section 3.3]: consider the two equations

dX dσ (σ) = εf 1 (X, εσ, σ) + O ∞ (ε 4 ) d X dσ (σ) = εf 1 ( X, εσ, σ) + ε 3 k 3 ( X, εσ, σ) + O ∞ (ε 4 ),
where f 1 , k 3 are T -periodic with respect to their third variable and k 3 has zero-mean with respect to its third variable; according to [7, theorem 2.9.2], the solutions of these two equations starting from the same initial condition are related by

X(σ) = X(σ) + O ∞ (ε 3 ) (7)
on the timescale 1/ε. It is possible to extend this relation to an infinite timescale, provided that the averaged system has an exponentially stable equilibrium, and that the initial condition is in a compact subset of the region of attraction of this equilibrium, along the lines of [1, lemma 2].

In our case, we first rewrite (5) in the fast timescale

σ := t/ε dx dσ = ε Ax + Bα η, H(x), εσ + Bs 0 (σ) (8a) dη dσ = εa η, H(x), εσ . (8b)
We introduce the new coordinates ( x, η) such that

x η = x η -ε Bs 1 (σ) 0 -ε 2 ABs 2 (σ) 0 . (9)
Notice (5c) simply reads

H(x) = H( x) + O ∞ (ε 3 ).
Differentiating with respect to σ and expanding to third order yields the system in the new coordinates

˙ x = ε A x + Bα η, H( x), εσ (10a) + ε 3 A 2 Bs 2 (σ) + O ∞ (ε 4 ) ˙ η = εa η, H( x), εσ + O ∞ (ε 4 ). (10b)
By [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF], we then get

x = x + O ∞ (ε 3 ) η = η + O ∞ (ε 3 ).
Replacing ( x, η) by (x, η) using transformation (9) in the two previous equations, we have the desired result

x = x + εBs 1 (σ) + ε 2 ABs 2 (σ) + O ∞ (ε 3 ) η = η + O ∞ (ε 3 ).
Plugging this expression for x in y = h(x) and Taylor expanding to second order yields (6c). The practical use of the theorem is the following. Assume we can compute a third-order estimate Y of Y from the knowledge of only the actual measurement y. Using the relation between x and x, we have

Y = H(x) + O ∞ (ε 3 ) = H(x) + O ∞ (ε 3 )
The control law with signal injection

u = α η, Y , t + s 0 t ε (11a) η = a η, Y , t (11b)
is then practically implementable and will control (1) as desired in average.

III. DEMODULATION OF THE VIRTUAL MEASUREMENTS

In this section, we show how to estimate the virtual measurements from the ripple in the actual measurement. Consider the composite signal

y(t) = Y 0 (t) + Y 1 (t)s 1 t ε + Y 2 (t)s 2 1 t ε + Y 3 (t)s 2 t ε + O ∞ (ε 3 ),
where Y 0 , . . . , Y 3 are in C 3 and Y

(3) 0 , . . . , Y

(3) 3 are bounded. Theorem 2 below states that Y 0 , . . . , Y 3 can be estimated at third order thanks to periodic low-pass filters.

A. Decomposition on an orthogonal basis

We first rewrite y as a decomposition on the periodic orthogonal signals (1, s 1 , s 2 , S 1 ),

y(t) = Y 0 (t) + Y 1 (t)s 1 t ε + Y 2 (t)S 1 t ε + Y 3 (t)s 2 t ε + O ∞ (ε 3 ). (12) The signals 1, s 1 , s 2 are orthogonal (for the scalar product f, g = 1 0 f (τ )g(τ )dτ ); indeed, 1, s 1 = 1, s 2 = 0 since s 1 ,s 2 have zero mean, and s 1 , s 2 = 1 2 1 0 (s 2 1 ) (σ)dσ = 0. S 1 is then obtained from s 2 1 by Gram-Schmidt orthogonal- ization, S 1 (t) := s 2 1 (t) -s 2 1 - s 2 1 ,s1 s 2 1 s 1 (t) - s 2 1 ,s2 s 2 2 s 2 (t).
As a consequence, the "coordinates" Y i are

Y 0 (t) = Y 0 (t) + s 2 1 Y 2 (t) Y 1 (t) = Y 1 (t) + s 2 1 ,s1 s 2 1 Y 2 (t) Y 2 (t) = Y 2 (t) Y 3 (t) = Y 3 (t) + s 2 1 ,s2 s 2 2 Y 2 (t).
B. Extraction of the Y i using iterated moving averages

We now turn to the design of the demodulating filters, which are based on iterated moving averages of the form

M 1 ϕ (t) := 1 ε t t-ε ϕ(σ)dσ M k ϕ (t) := 1 ε t t-ε M k-1 ϕ (σ)dσ, k > 1.
We first recall a basic result on finite differences. Lemma 1: Let ϕ be C 3 with ϕ (3) bounded. Then the p thorder backward difference

∆ p ϕ (t) := p i=0 p i (-1) i ϕ(t -iε) satisfies ∆ p ϕ (3-p) ∞ ε p ϕ (3)
∞ , p = 0, . . . , 3;

ϕ ψ means there exists K > 0 such that ϕ ≤ Kψ.

Proof: For simplicity, we just prove as an example the case p = 2. By the Taylor-Lagrange formula, there exists

t 1 ∈ [t -ε, t] and t 2 ∈ [t -2ε, t] such that, ϕ (t -ε) = ϕ (t) -εϕ (t) + ε 2 2 ϕ (3) (t 1 ) ϕ (t -2ε) = ϕ (t) -2εϕ (t) + 2ε 2 ϕ (3) (t 2 )
Therefore,

∆ p ϕ (t) = -ε 2 ϕ (3) (t 1 ) + 2ε 2 ϕ (3) (t 2 ); hence the desired inequality ∆ p ϕ ∞ ≤ 3ε 2 ϕ (3) ∞ .
This lemma is used to prove the following result, which has an important role on the filter design. Lemma 2: Let ϕ be in C 3 with ϕ (3) bounded, and ζ 0 be a 1-periodic function with zero mean. Then,

M 3 ϕ ζ0 ∞ ε 3 ϕ (3) ∞ ζ 3 ∞ ,
with ζ j+1 the zero-mean primitive of ζ j , and ζ(t) := ζ( t ε ). Proof: Integrating by parts three times and using the periodicity of ζ j gives

M ϕ ζ0 (t) = ∆ 1 ϕ (t)ζ 1 t ε -ε∆ 1 ϕ (t)ζ 2 t ε + ε 2 ∆ 1 ϕ (t)ζ 3 t ε -ε 2 t t-ε ϕ (3) (σ)ζ 3 σ ε dσ. ( 13
)
We now dominate the last two terms: clearly,

ε 2 t t-ε ϕ (3) (σ)ζ 3 σ ε ≤ ε 3 ϕ (3) ∞ ζ 3 ∞ ;
lemma 1 applied to the third term yields

ε 2 ∆ 1 ϕ (t)ζ 3 t ε ε 3 ϕ (3) ∞ ζ 3 ∞ .
Notice that if f ∞ ≤ K then M p f ∞ ≤ K; therefore applying M 1 and M 2 to these two terms yields the same bounds.

Consider next the second term of (13); its moving average is

-εM ∆ 1 ϕ ζ2 (t) = -ε∆ 2 ϕ ζ 3 t ε + ε t t-ε ∆ 1 ϕ (σ)ζ 3 σ ε dσ.
Using again lemma 1, the two terms of the right-hand side are similarly dominated by ϕ (3) ∞ ζ 3 ∞ , and so are their moving averages.

Finally, consider the first term of (13). Its moving average is

M ∆ 1 ϕ ζ1 (t) = ∆ 2 ϕ (t)ζ 2 t ε -ε∆ 2 ϕ (t)ζ 3 t ε + ε t t-ε ∆ 1 ϕ (σ)ζ 3 σ ε dσ.
Likewise, by lemma 1, the last two terms of the right-hand side are also dominated by ϕ (3) ∞ ζ 3 ∞ , and so are their moving average; as for the first term, integrating by parts its moving average yields

M ∆ 2 ϕ ζ2 (t) = ∆ 3 ϕ (t)ζ 3 t ε - t t-ε ∆ 2 ϕ (σ)ζ 3 σ ε dσ.
Using lemma 1 to the right-hand side ends the proof. We then recover the Y i from linear combination of shifted M 3 y thanks to the following lemma. Lemma 3: Let ϕ be in C 3 with ϕ (3) bounded. Define

P ϕ (t) := 17 4 M 3 ϕ (t) -5M 3 ϕ (t -ε) + 7 4 M 3 ϕ (t -2ε),
Then the operator ϕ → P ϕ is the identity up to third order,

P ϕ (t) = ϕ(t) + O ∞ (ε 3
). Proof: Again, consider first a single moving average of ϕ. Doing a change of variable in the integral and computing the Taylor expansion of ϕ gives

1 ε t t-ε ϕ (σ)dσ = 1 ε ε 0 ϕ(t -σ)dσ = 1 ε ε 0 ϕ(t) -σϕ (t) + σ 2 2 ϕ (t) dσ + O ∞ (ε 3 ) = ϕ(t) - ε 2 ϕ (t) + ε 2 6 ϕ (t) + O ∞ (ε 3 ).
We iterate the previous calculations to get the expressions of M 2 ϕ and M 3 ϕ ,

M 2 ϕ (t) = M ϕ (t) - ε 2 M ϕ (t) + ε 2 6 M ϕ (t) + O ∞ (ε 3 ) + ϕ(t) -εϕ (t) + 7 12 ϕ (t) + O ∞ (ε 3 ) M 3 ϕ (t) = ϕ(t) - 3 2 εϕ (t) + 15 12 ε 2 ϕ (t) + O ∞ (ε 3 ).
Finally, we compute the shifted triple moving average for k = 0, 1, 2. This yields Let α = 1 0 0 D -1 = 17/4 -5 7/4 . This yields the desired conclusion

   M 3 ϕ (t) M 3 ϕ (t -ε) M 3 ϕ (t -2ε)    =    1 -3 2 15 12 1 -5 2 39 12 1 -7 2 75 12    D    ϕ(t) εϕ (t) ε 2 ϕ (t)    + O ∞ (ε 3 )
P ϕ (t) = αDϕ(t) = ϕ(t) + O ∞ (ε 3 ).
Combining the three lemmas, we estimate the Y i . Theorem 2: Considering the operator P defined in lemma 3, we have an estimate for each of the Y i , namely Proof: Let first determine the estimate for Y 2 . Computing P y Š1 , we get

Y 2 (t) := 1 S 2 1 P y Š1 (t) (14a) = Y 2 (t) + O ∞ (ε 3 ) Y 3 (t) := 1 s 2 2 P yš2 (t) - s 2 1 ,s2 s 2 2 Y 2 (t) (14b) = Y 3 (t) + O ∞ (ε 3 ) Y 1 (t) := 1 s 2 1 P yš1 (t) - s 2 1 ,s1 s 2 1 Y 2 (t) (14c) = Y 1 (t) + O ∞ (ε 3 ) Y 0 (t) := P y (t) -s 2 1 Y 2 (t) (14d) = Y 0 (t) + O ∞ (ε 3 ).
P y Š1 (t) = P Y0 Š1 (t) + P Y1 š1 Š1 (t) + P Y2( Š2 1 -S 2 1 ) (t) + P Y2S 2 1 (t) + P Y3 š2 Š1 (t) + O ∞ (ε 3 ).
Since P is a linear combination of shifted M 3 and S 1 , s 1 S 1 , s 2 S 1 and S 2 1 -S 2 1 have zero mean, we have by lemma 2

P y Š1 (t) = P Y2S 2 1 (t) + O ∞ (ε 3 ).
Consequently, using lemma 3, we have the following estimate for Y 2 (which is equal to Y 2 )

Y 2 (t) = 1 S 2 1 P y Š1 (t) = Y 2 (t) + O ∞ (ε 3 ).
Let compute Y 3 . The same calculations for P yš2 provide

P yš2 (t) = s 2 2 Y 3 (t) = s 2 2 Y 3 (t) + s 2 1 ,s2 s 2 2 Y 2 (t) , that is, by definition of Y 3 (14b) Y 3 (t) = Y 3 (t) + O ∞ (ε 3 ).
Following the previous lines, we get the estimates Y 0 , Y 1 .

IV. A WORKED EXAMPLE

We illustrate the interest of the method on the system

ẋ1 = x 2 (15a) ẋ2 = x 3 (15b) ẋ3 = u + d (15c) y = x 1 x 2 + x 3 3 3 , ( 15d
)
where d is an unknown disturbance. We would like x 1 to track the reference x ref

1
while rejecting the disturbance, with a time response of about a few time units. We want to operate around steady state, i.e. near states of the form (x ss 1 , 0, 0) and inputs such that u ss + d ss = 0. Notice the system is not first-order observable around steady state because of the

Y 1 = ε x 2 x 1 x 2 3   0 0 1   = εx 2 3 Y 2 = ε 2 2 0 0 1   0 1 0 1 0 0 0 0 2x 3     0 0 1   = ε 2 x 3 Y 3 = ε 2 x 2 x 1 x 2 3   0 1 0   = ε 2 x 1
hence first-order observability is restored thanks to Y 2 and Y 3 , without even considering y and Y 1 . Notice thirdorder averaging is paramount, since the virtual measurement Y 1 stemming from second-order averaging is still degenerate. With Y 3 = ε 2 x 1 and Y 2 = ε 2 x 3 , the system is completely linear and can therefore be easily controlled. We design a

= M η + N x ref 1 (t) + L Y 3 /ε 2 Y 2 /ε 2 .
Following section II, this yields the implementable control law for (15a)-(15c), which is a particular case of (11),

u = -Kη + s 0 ( t ε ) η = M η + N x ref 1 (t) + L Y 3 /ε 2 Y 2 /ε 2 ,
where Y 2 and Y 3 are obtained from the actual measurement (15d) by the demodulation procedure of section III. The injected signal s 0 is a square wave of amplitude 1 and frequency 100, which ensures the oscillation is fast with respect to the time constants of the closed-loop system; n := 2 is used in the demodulating filter.

The test scenario is the following: at t = 0 the system starts at rest at the origin, with the reference x ref slope 10 -3 , meaning the system must slowly move while nearly first-order unobservable; finally, at t = 180, x ref 1 is a filtered step so as to quickly return to 0.

Without noise on the actual output y, the performance is excellent: the reference is tracked and the disturbance is rejected (Fig. 1). In fact the system behaves nearly as in the ideal situation where it is controlled directly from x 1 , x 3 without signal injection (Fig. 2); the ripple is visible on x 3 because it is directly affected by the input, but is much smaller on x 1 , x 2 , as anticipated by the averaging analysis. A zoomed view for 37.3 ≤ t ≤ 37.5 gives a better insight of the signals: With a measured output y m corrupted by noise, one thus might fear the ripple is much too small to be useful; this is not the case since the demodulation process is essentially a low-pass filter. Indeed, the system performs as desired (Fig. 7 and8), even though the output ripple is buried into noise (Fig. 6). The measurement noise used in the simulation is a band-limited white noise with noise power 1 × 10 -18 and sample time 1 × 10 -5 .

V. CONCLUSION

We have shown that it is possible to produce "virtual outputs" thanks to the injection of fast-varying periodic probing signal. These virtual measurements can be used to simplify the design of a feeedback law, in particular when the system observability degenerates. The analysis relies on third-order averaging, and significantly extends the approach of [START_REF] Combes | Adding virtual measurements by signal injection[END_REF]. For simplicity, we have restricted to Single-Input Single-Output systems with linear dynamics; in a future work we will generalize the approach to nonlinear Multi-Input Multi-Output systems.

Fig. 1 .

 1 Fig. 1. x 1 and estimate Ŷ3 /ε 2 (top); x 2 (middle); x 3 and estimate Ŷ2 /ε 2 (bottom).

Fig. 2 .

 2 Fig. 2. Difference between x (signal injection) and x i (ideal control).

Fig. 3 .

 3 Fig. 3. Control (top) and measured output (bottom)

Fig. 4 .

 4 Fig. 4. Zoom on Fig. 2.

Fig. 5 .

 5 Fig. 5. Zoom on Fig. 3.

Fig. 6 .Y 3

 63 Fig. 6. Control (top) and measured output (bottom) with noise

1

  set to 0; at t = 20, a step disturbance d of magnitude -0.25 is applied; for 40 ≤ t ≤ 140, x ref 1 is a slow (filtered) ramp with

Fig. 7 .

 7 Fig. 7. x 1 and estimate Ŷ3 /ε 2 (top); x 2 (middle); x 3 and estimate Ŷ2 /ε 2 (bottom).

  Fig. 4 illustrates (6a); Fig. 5(top) shows the square wave in the control signal; Fig. 5(bottom) shows the ripple in the actual output is really tiny near steady state, with a strange form caused by the nonlinearity of this output.

Fig. 8 .

 8 Fig. 8. Difference between x (signal injection) and x i (ideal control).