Research article

A mathematical model for the robust blood glucose tracking

  • In this paper, we study the problem of the robust blood glucose tracking. Tracking here means that the error between a state variable of a system under control and its desired time-varying reference converges to zero over time. Robustness here means that a controller designed for a system can tolerate a small variation of the system parameters. Since the parameters in the blood glucose regulation system differ in people, such a robust controller is useful in the insulin pump technology: an insulin pump equipped with such a robust controller could be used in a group of people. Thus, in our study, parameter uncertainties are introduced into a mathematical model of the blood glucose regulation system. Using an actual blood glucose level as feedback and an exogenous glucose input and a desired glucose reference as feedforward, we design a robust feedback and feedforward controller, which drives the blood glucose to track the desired time-varying glucose reference for any small uncertainties. Numerical simulations with published experimental blood glucose data are conducted to further confirm our theoretical results.

    Citation: Weijiu Liu. A mathematical model for the robust blood glucose tracking[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 759-781. doi: 10.3934/mbe.2019036

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  • In this paper, we study the problem of the robust blood glucose tracking. Tracking here means that the error between a state variable of a system under control and its desired time-varying reference converges to zero over time. Robustness here means that a controller designed for a system can tolerate a small variation of the system parameters. Since the parameters in the blood glucose regulation system differ in people, such a robust controller is useful in the insulin pump technology: an insulin pump equipped with such a robust controller could be used in a group of people. Thus, in our study, parameter uncertainties are introduced into a mathematical model of the blood glucose regulation system. Using an actual blood glucose level as feedback and an exogenous glucose input and a desired glucose reference as feedforward, we design a robust feedback and feedforward controller, which drives the blood glucose to track the desired time-varying glucose reference for any small uncertainties. Numerical simulations with published experimental blood glucose data are conducted to further confirm our theoretical results.


    1. Introduction

    A wide range of cellular functionsincluding cell proliferation, cell survival and cell fate commitment has been shown to be regulated by the activity of Notch signalling in nearly all multicellular organisms [1,2,3,4]. Accordingly with these pleiotropic effects, there is extensiveevidence linking the deregulation of Notch pathway with multiple diseasesand cancer [4,5,6,7,8]. Despite the greatprogress in elucidating the mechanisms and molecularfeatures of this signalling cascade during the last decades, the explanation for the diversityof cellular responsestriggered by the pathway is not fully understood.

    Since the first allele of Notch gene was identified in Drosophila melanogaster by T H Morganin 1917 [9], this genetically powerful model organism has provided solid explanations for much of what we know about Notch-dependent intercellular signalling [6]. In addition, the evolutionary conservation between flies and humans has ensured the functional significance of these findings for obtaining a better understanding of human development and disease. Numerous studies have highlighted the power of using the development of the Drosophila imaginal discs as an in vivo system model for analyzing how cooperative interactions between the Notch pathwayand other signals contribute to regulate cell proliferation in epithelia. Moreover, these structures have been employed to define the molecular mechanisms underlying tumor development and metastasis when Notch signalling is perturbed [10]. As occurs inmammals, Drosophila Notch activation can promote tumor growth by causing abnormal cell proliferation, not only autonomously, but also triggering proliferation in the normal surrounding cells. This effect is pleiotropic and context-dependent. Cancer is primarily a disease caused by loss of normal cellular growth regulation and this is often caused by the loss of cell proliferation control. To elucidate how alteration in the activity of Notch leads to the loss of proliferative control is fundamental to define the role that Notch plays in the intricatemachinery that regulates the cell cycle and cell proliferation during normal development. In this review we will focus on recent advances and knowledge on the function of Notch signalling in regulating cell proliferation during normal development of the imaginal discs. The mechanisms of normal Notch signalling regulating cell proliferation may provide useful insights to understand how alteration in this pathway can induce tumorogenesis.

    2. Description of the Notch signalling pathway

    In Drosophila the activation of the Notch signalling pathwaydepends on the interaction between the uniqueNotch receptor identified in this insect(in mammals there are 4 Notch orthologs, termed NOTCH1-4)with its ligands Delta (Dl) (Delta-like [Dll1], Dll3 and Dll4 in mammals) and Serrate (Ser) (Jagged, Jag 1 and Jag2 in mammals). The Notch receptor is a single-pass transmembrane molecule synthesized as a precursorform that is cleaved by Furin-like convertase (S1 cleavage) to generate a non-covalently linked heterodimer [11]. This heterodimer is composed of two subunits. One of these subunits consists of the major portion of the extracellular domain (NEC) fragment, and the other is the C-terminal region that contains the remainderof the NEC domain, the transmembrane domain and the intracellular domain(NICD) [3,4]. The activation of the receptor initiatesa proteolytic process of the receptorand the subsequent release of the Notch intracellular domain (NICD) [4,6,8,12,13]. The cleavageof the receptor is performed by different proteasesin a sequential manner, firstly by members of the ADAM metalloprotease Kuzbanian (Kuz)/TACE family, and later by the presenilin (PS)-dependent gamma-secretase complex (S3 cleavage) to release a soluble intercellular Notch fragment (NICD). This domain then translocates to the nucleusto participate in concert with the transcription factor CSL (CBF1-Suppressor of Hairless (Su(H)), and Mastermind (Mam), in the transcriptional activation of Notch target genes [2,3,41].

    The Notch (NICD) is subject to a variety of post-translational modifications, including phosphorylation, ubiquitylation, hydroxylation and acetylation. These modifications can modulate the relative strength of receptor-ligand interaction, as well as the outcomeof the signal transduction, and therefore the final cellular response induced by the activation of the pathway [4,6,13,14]. Another process that modulates the activity of Notch signalling is the endocytic trafficking [15,16,17,18]. Ligand endocytosis contributes to generate the physical forces necessary to dissociate and activate the receptor, and activated receptors enter endosomes to signal. Moreover, endosomal trafficking is involved in down-regulating Notch receptors that have not been activated. Disruption of the function of genes encoding for endocytic componentscause different tumors [15,16,17,19]. According to whether the gene affected is required early or late in the endocyticpathway the tumor phenotype differ. Thus, mutations in genes that act at the level of early endosome fusion, such as Rab5, the syntaxin, Avalanche (Avl) and Rabenosyn-5, cause autonomous tumor growth of wholly mutant tissue [20,21], but clonesof mutant cells for these genes fail to overgrow. In contrast, loss of later endosomal trafficking components, such as ESCRTI (TSG101)and ESCRTII (Vps25), exhibit a non-autonomous tumor phenotype whereby mutant clonesinduce proliferation in adjacent wild-type tissue [22,23,24,25,26] reviewed [23,25]. The difference between these two types of endocytic tumorsseems to be dueto their differential effects on Notch signalling [18,20,22,23,25,26,27,28]. Thus, whereaslack of Notch internalization in early endocytic mutants does not appear to promote Notch signalling [18,20,21] the disruption of ESCRTI and ESCRTII results in ectopic Notch activation and the induction of the expression of the secreted JAK/STAT ligand Unpaired (Upd) [18,22,23,25,26,27], which acts to promote growth in adjacentwild-type tissue (see below). The influence of the endocytictrafficking on Notch signalling has been coveredin several reviews [15,16,17,19].

    3. Notch signalling coordinates the growth of the eye discs

    The adult eye, antennaand part of head cuticle of Drosophila originate from a commonstructure known as eye-antennal discs. The discs develop from approximately 70 ectodermic cells that invaginate from the dorsal pouch of the embryo forming a flat epithelial sac-like structure [29,30]. The few cells that constitute the primordium of the eye disc start to proliferate during the larval stages, reaching an approximate number of 97000 cells at end of the pupal stage [31]. During larval development, the eye-antennal disc is divided into different linage units known as compartments [32,33]. The boundaries between different compartments act as a signalling centersnecessary for the growth and patterning of the discs [34,35]. One of the borders that play a fundamental role during eye development is the Dorsal-Ventral (D/V) boundary.

    In the early stage of eye disc development, before the definition of the D/V border, the function of Notch signalling is necessary to separate the presumptive eye territory from the antenna field and promote proliferation. The elimination of Notch function during this stage truncates completely the development of the eye field and, simultaneously duplicates the presumptive region of the antenna [36,37]. This phenotype can be at least partially rescued by over-expressing the cell cycle regulator Cyclin-E (CycE) [37]. Interestingly, this role of Notch signalling participating in the early specification of different territories is not exclusivefor the eye field, and a similar mode of action has been reported to take place at early stages of wingdevelopment (see below and Figure 1).

    Figure 1. Notchinduced growth and territorial specification. Illustration of the subdivision of the wing and eye-antenna imaginal discs into distinct territories. The wing disc is subdivided in wing and body wall primordia by the antagonistic signalling pathwaysWg (blue) and EGFR (red). The eye-antenna imaginal disc is subdivided into antenna and eye primordia by the activity of EGFR(red) and Eya (green), respectively. Notch signalling induced cell proliferation promotes the subdivision of thewing and eye-antenna imaginal discs in distinct territories.

    Different signalling events occur at the D/V boundaryto control and coordinate the growth of the eye discs during larval development. A key proliferative signal is established around a maximal concentration of Notch activity along this boundary, this region is known as ‘organizer’ [35,38]. Although the Notchreceptor is expressedin almost all the cells of the eye-antennal disc, its activityis limited to the D/V boundary due to the function of genes whose expression is restricted either to the dorsal or ventral compartments of the eye disc [38,39,40,41]. The sequential and localized expression of genes such as panier (pnr), wingless (wg) and iroquois (iro-C) determine dorsal fate of the eye primordium[38,42,43]. Concomitantly, the activity of the iro-C into the dorsal region represses the expression of fringe (fng), thus restricting its expression to the ventral half of the eye primordium [38,39,40,42] (Figure 2). Opposing fng-positive and fng-negative cells establishthe molecular mechanism that functionally defines the D/V boundary. The fng geneencodes for a glycosyltransferase that enzymatically modifies Notch modulating the ability of this receptorto interact with its ligands Dl and Ser [44,45,46]. Fng inhibits the abilityof Ser to signal through Notch and at the same time potentiates Notch-Dl interactions, therebygenerating the local activation of Notch along the D/V boundary. This localized activation of Notch promotes global growth throughout the eye discs from this early stage of eye development and onwards [38,47]. Any disruption affectingthe D/V boundary specification impedes eye formation, while ectopic generation of D/V boundaries hold the potential to create de novo organizers that frequentlycause eye duplications or enlargements of the eye field [38,39,40,41].

    Figure 2. Role of Notch in autonomous and non-autonomous control of cell proliferation in the eye disc. Schematic representation of an early second (left), and third (right) larval instar eye discs. The subdivision of the eye discs in Dorsal (yellow), and Ventral (orange) compartments depends on the restricted expression of different genes that lead to the activation of Notch signalling at the D/V boundary. (A) Model suggesting that the function of the Jak/Stat pathway is required down-stream of the Notch signalling. Accordingly to this model, Upd is activated at the organizer by Notch signalling to promote global eye growth. (B) Alternative model suggesting that Jak/Stat activation is necessary for defining the organizer, and therefore it functions up-stream of Notch signalling.

    3. Non-autonomous effects of Notch signalling in control of cell proliferation during eye development

    Although Notch activity is restricted to the D/V boundary at the early phases of larvaldevelopment, the influenceof Notch signallingon proliferation spans throughout the eye disc. This implies the existence of a relay signal that expands the growth promoting effect of Notch to the entire eye disc. It has been proposed that the Pax protein encoded by the gene eye gone (eyg) is a major effector of Notch signalling at the D/V boundary[48,49]. In second instar eye discs eyg transcription is induced by Notch signalling in a slicewithin the eye primordium straddling the D/V boundary[48,49]. Over-expressing eyg cancompensate loss of Notchactivity, whereas loss of eyg blocked the function of Notch signalling. Accordingly, it has been proposed that eyg is required downstream of Notch signalling for eye growth(Figure 2) [48,49]. Since eyg encodes for a transcription factor, other protein/s must operate downstream of eyg to relay the D/V organizer proliferative signal throughoutthe eye disc [49]. One of the proteins that might be mediating this functionis Unpaired (Upd), the ligand of the JAK/STAT pathway [50]. Considering these data, a model hasbeen proposed that suggests that Notch signalling and eyg induce the expression of upd at the organizer, then this secreted protein would diffuse throughout the eye disc acting over long distance to promotecell proliferation [48,51,52] (Figure 2A). The fact that in an eyg nullmutant background the ectopicactivation of Notch signalling can still induceproliferation in the eye discs, suggestthat Notch could also control proliferation by an eyg-independent mechanism. Moreover, as Notch signalling can induce upd, but not eyg, in the posterior eye discs margin, the activation of upd can also be through an eyg-independent mechanism [48]. The functional link from Notch signalling to eyg andupd would explain how the localized activation of Notch could regulate global growth of the eye disc (Figure 2A). However, other authors have questioned this model, as they have found that the growth defects caused by the elimination of Notch signallingcannot be rescued by the activation of the upd /JAK/STAT (Figure 2B) [53]. These authors proposethat the activation of JAK/STATpathway by Upd precedes the formation of the organizer. Thus, in early eye development, upd would be antagonizing Wg and Hedgehog(Hh) signalling in the ventral domain and therefore repressing iro-C in this domain. Consistently with this model, the over-expression of upd induces overgrowth associated with iro-C inhibition and the appearance of new D/V boundaries [53]. Thus, in an initial phase JAK/STAT pathway would be necessary to define the position of the organizer, and therefore it would be functioning upstream of Notch signalling. Once the D/V boundary is defined Notch signalling would be locallyactivated along this border to establish the organizer, and therefore to induce global eye growth [53]. Interestingly, these authorshave reported that oneof the factors that is, at least partially, executing this global growth response induced by Notch signalling is the Golgi transmembrane type II glycoprotein Four-jointed (Fj) (Figure 2B)[54]. The ectopic expression of fj partially rescues the proliferative defects caused by reduced Notch activity [53]. The expression of fj is activated by Notch and JAK/STAT pathway at the D/V boundary in agraded manner (Figure 2B). This expressionis complementary to the graded expressionof the tumor suppressorgene dachsous (ds). This gene encodes for an atypical cadherin that binds to, and activates the protocadherin Fat, that in turn initiatesthe Hippo pathway signal. The function of this pathwayrestrains disc growth byinhibiting cell proliferation [55,56]. Fj kinase activityphosphorylates the cadherindomains of Fat and Ds during their transit through the Golgi, decreasing the affinity of Ds for Ft, and therefore reducing the activity of Hippo pathway [56]. Thus, Fj might be functioning as a regulatory hub, integrating Notch and Hippo pathway during the growthof the eye discs [53].

    During eye disc development Hh and Notchsignalling display an antagonistic interaction that plays an important role in the definitionof the final size of the eye. In early second larval stage hh is expressed in marginal cells of the eye disc, and its product is secreted in an oppositegradient to that of the growth-promoting factors produced in the organizer[47,57] (Figure 2B). The functionof this signalling pathway is crucial for retinal patterning and also to regulate eye growth [47]. In a recent genetic screening, Da Ros and co-workers[57] have found that the over-expression of the conserved microRNAmir-7 enhances the overgrowth eye phenotype caused by the ectopic expression of Dl. These authors proposed that the cooperative action between mir-7 and Notch pathway reduces the activity of Hh signalling pathway by down-regulating two functionally redundant Hh receptors, Interference hedgehog (Ihog) and Brother of Ihog (Boi) [58,59,60,61]. ihog is directly silenced by miR-7, whereasboi is repressed by Notch signalling during eye growth. In this context Hh signalling would be necessary to reduce cell proliferation (Figure 2B). These results contrast with the function that it has been proposed for Hh signalling in promoting cell proliferation during eye discs development [62]. However, recent reports suggestthat when Hh pathwayis activated in clones of cells, the mutant cellsare eliminated by apoptosis, whereas the surrounding wild-type cells over-proliferate [63]. Interestingly, mutations in members of the Hh pathway have been associated with human cancers [64]. This suggests that in addition to the well-known oncogenic role of Hh signalling, this pathway might also have a tumor-suppressor function. Accordingly, the down- regulation of components of Hh signalling enhancesthe phenotype caused by the ectopic activation of Notch, resultingin severe disc overgrowth. Moreover, the tumor-like overgrowths induced by the ectopic expression of Dl andmir-7 are suppressed when Hh signal is increased [57]. These results reveal a cooperative antagonistic interaction between these two pathwaysin the control of cell proliferation. Considering that these signalling pathways, and their regulation, are highly conserved between flies andhumans, it is likely that this mechanismmight play a role in the development of some human cancers.

    All these models coincide with the idea that different signalling pathways mediate the Notch-induced global eye growth (Figure 2). These signalspromote cell proliferation throughout the eye discs in response to the local activation of Notch at the D/V boundary. Thus, the mechanisms underlying Notch-induced tumorigenesis might be, at least partially, explained by the interaction between Notch signalling and these signalling pathways(Hh, JAK/STAT and the Hippo pathway).

    4. Cell autonomous effect of Notch regulating cell proliferation during eye disc development

    4.1. Second mitotic wave

    During eye disc development the onset of retinal differentiation is associated with the formation of anindentation in the most posteriorregion of the eye disc, known as morphogenetic furrow (MF). Since progressively anterior cells undergo this constriction, the furrow takes on a wave-likeappearance moving from posterior region ofthe eye discs to the anterior [65,66,67]. The furrow sweeps across the disc, leaving in itswake the developing clusters of cells that will become the individual unitsof the compound eye, known as ommatidia. Moreover, furrow initiationpromotes cell cycle synchronization, as cells anteriorto the furrow proliferate randomly, but as soon as they enter the MF they arrest in G1 [68,69,70]. After the passage of the furrow, most of the cellsthat are not part of the cluster of cells that have initiated the process of differentiation undergo a single round of cell division, known as the second mitoticwave (SMW). After this, most cells will not divide again [66,71]. This additional round of cell division supplies unspecified precursorcells to each ommatidium [66]. Because the cells only undergo one round of division at this stage, and because of the progressive mode of development, all stages of the cell cycle can be observedeasily in a single eye disc (Figure 3).

    Figure 3. Second Mitotic wave regulation. Third instar eye imaginal discs, showing the different phases of the cell cycle. BrdU incorporation (green) marks S phase, CycB staining (red) indicates G2 phase and staining with antibodies against PH3 (red) marks mitosis. Photoreceptors are staining with Elav (blue). In the morphogenetic furrow cells are arrested in G1, and are therefore devoid of mitotic cells expressing PH3; posterior to the furrow, the SMW is marked by a narrow band of mitotic cells. Model of SMW regulation by different signals: Delta/Notch signalling, controls the onset of S-phase in the SMW by regulating the activity of dE2F, and the expression of CycA. Su(H) (asterisk) independently of Notch blocks the activity of Dap for promoting CycE function. The later G2/M checkpoint in the SMW is overcome by EGFR signalling that triggers the expression of Stg. Model modified from [72].

    Different studies report that there is a specificNotch requirement for cells to enter S-phase in the SMW [52,72,73,74]. These studies have shown that clones of cells containing loss of function allelesof Notch, Dl or Su(H) do not enter S-phaseand consequently fail toincorporate Bromodeoxyuridine (5-bromo-2'-deoxyuridine, BrdU) or express Cyclin-B (CycB) in the region posterior to the MF [72,73]. Consistently, the ectopic activation of Notch signalling or the over-expression of an activated form of Su(H) in cells posterior to the MF expands the zone of BrdU incorporation in the SMW, and increases the number of mitotic cells posterior to the MF [52,72,73]. This cell-autonomous effect ofNotch is independent of the function of this signalling pathway inducing the proliferative signals associated with theorganizer [52]. The function of Notch signalling promoting S phase entry in the SMW depends on at least two processes(Figure 3). Firstly, the de-repression of the inhibition of dE2F1 by Retinoblastoma factor(Rbf). Secondly, the activation of the expression of Cyclin-A (CycA) by Notch signalling [72]. However, the co-over-expression of CycA and dE2F1 is not sufficient to rescue S-phase in Notch mutant cells, implyingthe existence of, at least, one other targetgene by which Notch triggers the onset of S-phase [72]. An obviouscandidate for being a Notch effector in this transition is CycE, which is a primary trigger of S-phase [75]. However, it has been suggestedthat Notch signalling might be controlling this transition independently of this cyclin, since high levels of CycE are observed in Notch mutant cells [72,73]. CycE functionsthrough regulating the activityof its partner Cdk2. Thus, thelevel of expression of CycE does not always correlate with the activity of the complex formed by CycE/Cdk2kinase, since the function or expression of Cdk2 can be modified independently of CycE. In fact Cdk2 mutant cells accumulate high levels of CycE but fail to enter S phase in the SMW [74]. In addition, CycE/Cdk2 kinase activity can be inhibited by the p21/p27family of Cdk inhibitor, Dacapo(Dap). The over-expression of dap blocks the S phase transition, though mutant cells accumulate high levels of CycE protein [76,77]. It has been reported that Su(H) mutant cells are arrested in G1 in the SMW, in part becausedap is up-regulated. Thus, G1/S progression in Su(H) mutant cells can be induced by expressing CycE together with Cdk2 or eliminating dap in Su(H) mutant cells [74]. The up-regulation of dap in Su(H) mutant cells is dependent upon the basic helix-loop-helix (bHLH) proneural protein Daughterless (Da) [74]. Surprisingly, in clones of Notch mutant cells the levels of Dap are reduced [72,73], suggesting that Dap accumulation is not contributing to the G1 arrestin these mutantcells. These observations suggest that the mechanisms involvedin the retention of Notch and Su(H) mutant cellsin G1 in the SMW might be different (Figure 3) [74]. This possible mechanismshould be further analyzed, since it might be a novelregulatory process, throughwhich Su(H) could regulatecell proliferation.

    The movementof the MF depends on the secretedproteins Hh and Decapentaplegic (Dpp). Hh is secreted from differentiating cells behind the MF and induces the expression of Dpp, which then also participates in furrow propagation [78,79,80,81,82]. Posterior-to-anteriorspread of both proteins (Hh and Dpp) is also required to arrest cells inG1 ahead of the MF, and to activate the expression of Dl in an uniform band of cells [62,68,72,73,82]. As cells emerge from the MF, Dl activates to the Notch receptorto promote G1/S progression in all cells, although only uncommitted cells enter S-phase in the SMW, since Epidermal growth factor receptor (EGFR) activity blocks the cell cycle progression in G1 inthe cells that are part of the precluster [72,73,83].

    4.2. Hh and Notch signalling in the control of cell proliferation in the SMW

    Hh signalling plays a fundamental role in initiatingand coordinating the onset of S-phase in the SMW. This signalling pathway activates several branches to promote G1/S transition duringthis coordinated division. On one hand, it promotes transcription of CycE and CycD [84], principal regulators of the S-phase onset, and on the other hand, together with Dpp, leads to the expression of Dl in the furrow (Figure 3) [72,73]. In addition to this function, Hh signalling is also requiredto regulate the expression of the basic helix-loop-helix (bHLH) proteinAtonal (Ato) [80,85,86]. This transcription factor is essential for the differentiation of the photoreceptors [87]. Interestingly, Bossuyt and co-workers [88] have identified an anti-oncogenic function for ato during eye development. These authors report that the loss of ato strongly enhancesthe formation and progression of tumors generated when Notch is activatedin "sensitized" flies, whereas, ectopic expression of ato strongly inhibitstumor formation in these flies [88]. This suggests that tumor formationmight require loss of the capacity of the cells to inducecell fate commitment and differentiation.

    ato is first expressedin a band of cells in front of the MF in response to Hh signalling. In the MF, this stripe of cells is refined into clusters of cells expressing high levels of ato (intermediate groups). Finally, from these groups the expression of ato is restricted to one single cell in each cluster, which will become the R8 photoreceptor [47]. The functionof Notch is necessary to restrict the expression of ato toonly one cell in each group. The Hh pathway is required to repress ato expression between the nascent proneural clusters [86]. Thus, both signalling pathwayscoordinately define the final pattern of expression of ato [47]. Interestingly, this repressive functionis mediated by the bHLH protein Da (Figure 4). This factor is dynamically regulated in the furrow by several mechanisms, including Hh, and Notch signalling pathways. In the region posterior to the MF, Notchsignalling up-regulates the expression of da in some cells to repress the expression of ato, to define the R8 pattern(Figure 4) [89]. Interestingly, the regulation of the expression of the cyclin inhibitor dap in the SMW depends on Da [74,90]. All these data suggest a tight link between the role of Notch signalling regulating proliferation in the SMW and its function controlling differentiation.

    Figure 4. Schematic representation of the genetic interactions between Notch/atonal/da in the R8 equivalence group. The function of Notch signalling regulating cell proliferation in the equivalence group is tightly linked with its function controlling the differentiation of the photoreceptors. In the cells of the equivalence group that do not differentiate as R8s, Notch signalling is involved in the down-regulation of Atonal at the same time as it promotes G1/S progression.

    5. Autonomous requirements of Notch signalling in the regulation of cell proliferation in the ventral region of the eye discs

    The cell autonomous effect of Notch signalling regulating cell proliferation seems to be restricted to cells posteriorto the MF, as cell proliferation ahead of the furrow is not detectably impairedby loss of Notch. The factthat clones of Notch and Dl mutant cells can be generated implies that they are not required for the earlier, unpatterned proliferation in the region anterior to the MF [52,72]. However, different reports have shown that the growth of the ventralregion of the eye discs depends on the function of ventral genes like Lobe (L) and its downstream target Ser. The depletion of the functionof these genes in the whole eyedisc result in selectivegrowth defects in the ventral half of the eye [38,91]. In addition, other reports suggestthat ventral Notch clones show disrupted upd transcription. These results argue that there is Notch activation not only at the D/V-midline but also in ventraltissue, and that Notch activity in ventral tissue is necessary for upd expression [52]. Surprisingly, Ser loss-of-function clones do not exhibit any phenotypes in the eye [41,92,93]. It has been suggested that Ser might function as a diffusible factor, or there may be other functionally redundantSer-like genes. The functionof Ser and Notch in this contextis not clear, further analysis is necessary to clarify the role of these factors in the control of cell proliferation in the ventralregion of the eye disc.

    6. Notch pathway activation in the wing disc

    The dynamic patternof Notch pathwayactivation, as observedby the transcriptional response of its targets like E(spl)mβ or the synthetic Notch Responding Element (NRE) [94,95,96] reflects the differentNotch requirements during wing development. As previously describedfor the eye disc, as development progresses the wing discs is subdivided into different compartments. In early second larval stage the discs is subdivided into two compartments, Dorsal and Ventral. This division depends on the function of the gene apterous (ap), which is expressed specifically in dorsal cells [97,98,99,100,101]. Some aspects of the definition of the D/V border are conservedin the developing eye and the wing, although the precise activating mechanisms differ. One important difference between both discs is that in the eye discs, comparedto the wing disc, the D/V axis is inverted. Thus, the restricted expression of ap in the dorsal region activates the dorsal-specific expression of Ser and fng. As in the eye discs, Fng inhibits Notch activation by Ser, but in this case in the dorsal compartment, restricting Ser to signal to ventral cells [102,103,104,105,106,107]. Contrary to the negative function of Fng in Notch-Ser interaction, Fng binds to Notch and promote Notch-Dl interaction, therefore restricting Notch activation at the D/V border [44,106,107,108,109,110]. As previously described in the eye discs, the D/V boundary in the wing discs also functions as an organizing “center” influencing global growth and patterning of the wing [97,101]. The growth response induced by Notch at the D/V boundary of the wing disc is, at least partially, mediatedthrough the expression of target genes such as wg, Cut and vestigial (vg) [104,108,109,111,112,113,114]. Wg promotes wing development at least in part, by feeding an auto-regulatory loop of vg, which is required for wing specification and growth [111,115]. Thereforein the absence of Notch activityat the D/V boundary, the wing does not grow due to a failure of wing cell fate specification [96,109,111,112,114,116,117,118]. In the wing disc, it has been proposed that the effects of Notch signalling on cell proliferation is not just a consequence of the activation of these targetgenes [96,117,119,120,121] (see below). In accordance to this more generalrole of Notch controlling wing growth, mutantclones for Notch do not proliferate normally in the wing, even when they are far away from the D/V border, suggesting that Notch is required for the proliferation of the whole wing primordia [122].

    The effectsof Notch in control of the cell cycle and proliferation are context-dependent as it occurs in the eye discs. For example atthe D/V boundary Notch has a negative impact on cell proliferation defining the so-called Zone of Non-Proliferation Cells (ZNCs), while in other regionsNotch can influencegrowth autonomously and non-autonomously (see below). ThereforeNotch pathway can have a very different outcome on cell proliferation depending on the time and spatial locationof its activation, and its synergy with other signalling pathways.

    7. Early Notch induced cell proliferation promotes territorial subdivision of wing disc

    An important developmental decision that occurs during wing discs development, apart from the A/P and D/V compartmentalization, is the specification of the wing pouchand the body wall fates [123,124,125]. One of the earliest roles of Notch during wing disc development is to separate the mutually antagonistic effects of two signallingpathways, EGFR and Wg that are required for body wall and wing specification, respectively [96,123,125]. In the early wing disc primordium, Vein the ligand of the EGFR pathway, is expressed in dorsal body wall cells, and restricts the expression of Wg to the distal domain of the disc, delimiting the region of the disc that is specified as wing cell fate (Figure 1) [126,125,127]. Growth induced by Notch activity pull apart the mutually repressive effects of the EGFR and Wg pathways leading to wing cell fate specification [96]. Interestingly, this effect of Notch in early wing disc growth, and wing development can be rescueby the expression of the cell cycle regulator CycE (G1/S)or the universal G2/M regulator string (stg, the cdc25 homologue) in conditions of reduced Notch function [96]. ThereforeNotch seems to control wing disc growth modulating the expression of these cell cycle regulators. This early role of Notch precedes the function of this pathway inducingthe D/V organizer centre. It is remarkable the similarities that exist between this processand that observed during the definition of eye primordium(see above) (Figure 1). This suggests that likely the mechanisms involved in the regulation of both processesare conserved between both discs.

    8. Autonomous control of cell proliferation by Notch and the zone of non-proliferation cells

    Proliferation of the wing disc proceedsrandomly at a rapid rate that slowly decays atthe end of third instar stage. Just before pupal formation, the cells of the discs are arrested in the G2 phase of the cell cycle. However, a band of cells atthe D/V boundary are arrested in G1 around 30h before the rest of the cells of the discs stop proliferate. This effect is as a consequence of aregulatory program involvedin sensory organ formation, and defines the ZNCs [128,129]. The pattern of cell proliferation at these stages correlates with the pattern of activity of the mutuallyexclusive Notch and Wg signalling pathways. The activation of Notch at the D/V boundary activates wg that signals to non-boundary cells, while boundary cells are refractory to Wg-signalling through Cut repression [130,131]. At the D/V boundary Notch activity blocks cells in G1 by repressing the G1/S regulators bantam microRNA and dmyc, both factors are necessary to activate the dE2F1 transcription factor. In non-boundary cells, which are actively proliferating, the response to Notch is reduced by the activityof Wg alleviating the block in G1 (Figure 5) [131,130]. Interestingly, cells that are unable to respond to Notch and Wg signalling are still able to proliferate [131]. It has been suggested that in anterior cells, Wg induces a G2arrest in Dorsal and Ventralcells close to the D/V boundary by inducing the expression of the bHLH proneural transcription factors achaete (ac) and scute (sc) leading to the repression of stg [129]. However, in at recent report [132] it has been proposedthat the capacity of Sc to repress stg expression in the ZNC could be mediatedby Da. Da functions as a transcriptional repressor of stg, thereby regulating G2/M transition. The over-expression of sc in the wing discs induces high levels of Da that in turn block the expression of stg [132]. These data suggest that, as occurs during the SMW, the function of Notch signalling regulating cell proliferation in the wing margin is tightlylinked with its function controlling the differentiation of the sensory organs that appear in this region. The bHLH family of transcription factors plays an important role mediating the functionof Notch signalling in the regulation of these processes [133]. This family of proteins coordinate cell cycle control, cell lineage commitment and cell differentiation [134].

    Figure 5. Role of Notch in autonomous and non-autonomous control of cell proliferation in the wing imaginal disc. Schematic representation of a wing imaginal disc with the activity of Notch pathway in purple at the D/V boundary, Wg (blue) and Vg (green) in the pouch and Jak/Stat activation pathway in orange. At the zone of non-proliferation cells, Notch blocks cell proliferation autonomously though the repression of the bantam microRNA and dmyc. At the pouch, Notch’s ability to induce the expression of dmyc and CycE is repressed by Vg in a feed-forward repression mechanisms. In the proximal pouch and the hinge, Notch cooperates with the Jak/Stat pathway to promote cell proliferation.

    Interestingly, it has been reported that most of Notch functionsduring wing disc development seem to be mediated by this family of transcription factors. Thus, the bHLH proteins encodingby E(spl)-C and deadpan (dpn) are requiredat the wing margin for wg andCut expression while repressdmyc and therefore G1/S transtition. Moreover, these factorsalso mediate the function of Notch signallingregulating cell proliferation in the rest of the wing pouch, and in the hinge region [135,136].

    9. Notch autonomous and non-autonomous control of cell proliferation

    The analysis of the effectscaused by the ectopic activation of Notch duringwing disc proliferation reflects the context-dependent nature of Notch signalling. Thus, clones of cells expressing, either the intracellular domain of Notch (NICD), that function as a constitutively activatedform of the receptor, or the ligandDl, stimulate cell proliferation [119,121,120]. In the distal region of the wing pouch, these clones induce non-autonomous over-growth, as result of the inductionof ectopic D/V boundaries. However the size of these clones is smaller than controlclones. This later effect is likely caused by the function of Notch in the specification of the ZNC [119,120,121,129,131,135]. Cloneslocated in the proximal region of the wingand in the wing hinge, induce large over-growths, both autonomously and non-autonomously. The proliferative effects of theseclones are not caused by the generation of ectopic D/V boundaries [119,121].

    The non-autonomous effects on cell proliferation producedby Notch signalling from the D/V boundary, or when this pathway is ectopically expressed in the hinge, likely depend on a secreted relay factor/s. The obvious candidates to be mediating these non-autonomous effectsare the Notch target genes wg andvg, which are expressed at the D/V boundary [102,104,105,108,109,137]. However, different results suggest that only partof the proliferative effects inducedby Notch can be attributed to wg expression. Thus, wg is not ectopically expressed in all proximal clones, and in the absence of Wg function, Notch activation can induce proliferation in some proximalclones and in the hinge [121]. In addition, the ectopic expression of wg is not sufficient to reproduce the effects caused when Notch signalling is activated [117,119,121,138]. Wg’s role in controlling cell proliferation is controversial and has been extensively discussed elsewhere [117,119,121,138,139,140].

    All the complex effects on wing disc growth mediated by Notch are indicative of the context dependence that this pathway has regulating cell proliferation. To solve this problem Djiane and co-workers [136] have used a genome-wide approach to characterize the repertoire of genes directly activatedby Notch that could be mediating Notch’s effects on tissue growth. Among the genes identified in this analysis they have found: cell proliferation regulators (CycE and stg), cell signalling molecules (wg and upd), and wing growth regulators (sd and vg). RNAi knockdownof most of the genes identified in this study suppressed to some extent the growth defects caused by Notch pathway activation. The authorsdefined 4 different classesof target genes. The first was a largely cell-autonomous response genes (e. g. wg). The second group comprised genes that respond to Notch activation both autonomously and non-autonomously (e. g. sd and fj). Interestingly fj has been also identified as a Notch target gene during eye development (see above). The third group includes genes that present a complex response to Notch signalling activation. Thus, these genes are not autonomously expressed in all the cells where Notch is active in the wing pouch, whereas are both autonomously and non-autonomously up-regulated in the hinge and pleural regions (e. g. CycE). Finally, group four is constituted by genes that are up-regulated in broad domainsat the hinge and pleural regionstips, but are not responsive to Notch in the wing pouch (e. g. dmyc). These data implies the existence of mechanisms to limitthe capacity of Notchsignalling to up-regulate the expression of the target genes of group 3 and 4 in centralwing pouch. One of the factorsthat is mediating this effect is the TAE transcriptional factor Scalloped (Sd) [141]. This gene is activated by Notch, and it has been classified in group 2 [136]. Sd forms a complex with the nuclear protein Vg [141,142]. This complex is requiredfor wing specification and growth [118]. Sd is also one of the binding partners of Yorkie, the transcriptional effector of the Hippo pathway that regulates tissue growth and cell survival(reviewed [143]). Therefore, changes in the balance between the amountsof Sd/Vg/Yki proteinsmight influence the target genes activated, and the proliferative response of wing cells. In this way, Notch pathway activation or down-regulation has a profoundimpact in the expression of expanded (ex), one of the best-characterized Yki targets [144,145]. Notch activationrepresses ex expression just in the wing pouch while Notch down-regulation de-represses it in the D/V marginwhere it is not expressed [144]. According to this feed-forward repression model, Notch dependent activation of vg, inducesthe formation of the Sd/Vg repressive complex in the pouch preventing the activation of Yorkie targets such as ex and the cell cycle regulators dmyc and CycE or the apoptosis regulator diap1 (Figure 5) [136].

    Similarly, the proteins of the E(spl)-C, well known targets of Notch, are also involved in a negative feed-forward onto other Notch targets like dmyc, restricting the capacity of Notchsignalling to activategene expression to particular regions of wing pouch [136]. However, other Notch targets like ex or diap1 are not regulated by the E(spl)-C, suggesting that differenttargets depend on different negative feed-forward mechanisms [136,144].

    The ectopic activation of Notch in the hinge and pleuralregions promotes non-autonomous cell proliferation and activation of dmyc, CycE or diap1 [119,120,121,131,135]. In this region the activation of Notch induces high levels of Jak/Stat pathway activity [136,146]. A feed-forward mechanism between the Notch and Jak/Stat pathways has been proposed to coordinate the non-autonomous growth at the periphery of the pouch. This processdepends on the secreted Jak/Statligands Upd1, Upd2 and Upd3 that would function as a secreted relay factors. Interestingly, these Jak/Stat ligands are also regulated by Notch. Therefore, a feed-forward mechanism between these two pathways has been proposed to explain the non-autonomous growth and expression of CycE, dmyc and diap1 (Figure 5) [136]. However, a functional link between these pathwaysis still missing.

    10. Notch growth regulation in the leg disc

    Notchsignaling establishes boundaries that separate each leg segment, as it does in the D/V boundary of the wing and in the eye discs. In the leg disc, the Notch pathway is activated in 9 concentric rings that prefigure the leg joints along the Proximo-Distal (PD) axis (Figure 6) [95,147,148]. The expression of the Notch ligands, Dl and Ser, is also segmentally controlled by the combination of the leg " gap" genes and PD tarsal genes, in proximal cells to the end of each segment [147,149]. The Notchpathway is activated in a row of cells adjacent and distal to Dl and Ser expression domains at the presumptive joints. Apart for being completely requiredfor the formation of all joints, Notch also controls the growth of the leg in a non-autonomous manner. Mutant cells for components of the Notch pathway that span two segments are usually associated with joint and growth defects [95,147,148]. Interestingly, the leg size reduction involved mutant and non-mutant tissue. Moreover, clones that locally activate the pathway induced leg overgrowths composed of mutant and wild type tissue, suggesting that as in the wing and eye discs, Notch control leg growth in an autonomous and non-autonomous manner [95,147,148]. It has been proposed that Notch pathway activation at the joint induces an unidentified signal that acts non-autonomously to promote cell survival and growth [150]. However it is still unknownif in the leg disc, Notch controls cell proliferation, cell shape changes or orientedcell division that could be the cause of the growth defects observed in Notch mutant legs.

    Figure 6. Leg growth control by Notch: Drosophila leg with the 10 segments separated by the joints. The cartoon represents a joint where the proximal expression of the Notch ligand (Dl) activate the Notch pathway (N*) in the joint and controls the growth of the adjacent segments.



    11. Notch and its relationship with epigenetic modifiers

    The combination of different ex vivo andin vivo RNAiscreenings [151], as well as genome-wide [136] approaches have identifieddistinct epigenetic factors that are functionally related to Notch signalling. Interestingly the alteration of the functionof some of these factors synergize with the gain of Notch signalling to foment hyperplastic and neoplastic growth. One example is the BTB/POZ containing transcriptional repressors longitudinals lacking (lola) and pipsqueak (psq). It has beenshown that these proteins cooperate with Notch in the formationof tumors and invasion after Dl over-expression in the wing and eye imaginal disc [152]. Both lola and psq function as epigenetic silencers that together with co-expression with Dl shutdown the expression of Rbf, a tumor suppressor gene that negatively regulates dE2F1, promoting metastatic tumors [152]. The importance of epigeneticregulation in the control of the function of Notch signalling, and its relation with tumor formation has been recently reviewed [10].

    12. Effects of loss of apico-basal polarity in Notch signaling

    In Drosophila, Scribble (Scrib), Discs large (Dlg) and Lethal giant larvae (Lgl), cooperatively establish, and maintainapico-basal cell polarity. Loss-of-function mutations in these genes resultin neoplastic overgrowth, along with a loss of apico-basal cell polarity and differentiation. Though scrib mutant clones shown ectopic expression of CycE, and excessive cell proliferation, they do not become overgrown because they are eliminated by Jun N-terminal kinase(JNK)-dependent apoptosis [153]. However, the ectopic activation of Notch in these clonesinduces unrestrained tissue overgrowth [153,154,155,156,157,158]. The mechanistic details about how Notch promotes this effect is unknown, however it has been proposedthat the activation of Notch in scrib mutant cells blocks differentiation and apoptosis, and also promotes JNK-mediated tumor overgrowth and invasion. aPKC signalling is also requiredto promote tumor overgrowthwhen Notch is ectopically expressed in scrib mutant clones, through either increased cell proliferation or cellsurvival [158].

    It is not clear what the relationship is between cell polarity and tissue growth. Different publications have suggested that at least the apico-basal polarity determinants Lgl/aPKC and Crb can establishapico-basal cell polarity, and control tissue growth, independently [159,160,161,162]. The effectson proliferation and survival of these factors are via the Salvador/Warts/Hippo (SWH) tumor suppressor pathway [160,161,162]. Therefore, Notch signalling mightbe cooperating with the Hippo pathway to promote tumor growth in mutant cellsrather than loss of apico-basal cell polarity. In addition, it has been recently published that Lgl regulates endocytosis to restrict vesicleacidification and prevent ectopic ligand-dependent Notch signaling [28]. Thus, the disruption of apico-basal cell polarity can affect different processes, and compromise the activity of multiple signalling pathways, that eventually can cooperate with Notch signalling to promote tumor formation.

    13. Notch and tumor formation

    Although it is well defined that Notch 1 is an oncogene in human T lymphoblastic leukemias/lymphomas (T-ALL), there is little evidenceto support a causative role for Notch in the initiation of tumorigenesis in human solid cancers [5,6,7,8]. Considering the clear relationship between proliferative events and Notch activation, it has been proposed that the hyper-proliferative statesinduced by Notch, frequently in synergy with other signals, can eventually lead to bona fide oncogenicevents, even though Notch activation does not lead to cancer per se [5]. Therefore, the identification of the mechanisms by which Notch signalling control cell proliferation during normal development can help us to better understandhow alterations in the activityof Notch signalling may lead to oncogenic events in differentpathological conditions. The analysis of the function of Notch signalling during the development of Drosophila has been fundamental for the identification of such mechanisms. Interestingly the development of some tumors in humans seems to be initiatedby mechanisms similar to those by which Notch signalling induces normal growth. For example, the functionof Notch regulating the activityof different cell cycle regulatorsand cyclins, seems to be involved in the initiation of some cancers. Thus, increased levels of the Notch ligand JAG1, which commonly occur in breast cancers, promotes cell cycle progression by inducing CycD1 through Notchsignalling [163]. Moreover, in mice homozygous-null for Ccnd3 (CycD), the ectopic expression of Notch failed to induce T-ALL, suggesting a fundamental role for D type cyclins in Notch-mediated transformation [8,164]. Notch signalling also drives cell cycle entry by transcriptionally activating myc, that is a direct target of Notch and contributes to cell cycle progression in T-ALL [8,165,166], as well as in Notch-induced mouse mammary tumors [167]. Moreover, Notch can promotecell cycle progression repressing the transcription of CDKIs p27 and p57 through HES1 in different cell types [8]. In T-ALL, Notch directs the transcription of the E3 ubiquitin ligase S phase kinase-associated protein 2 (SKP2), which leads to decreased p27 protein levelsand increased cell proliferation [168].

    Potential involvement of Notch signalling in other human cancers is frequently associated with the cooperative action between Notch and other signalling pathways (Wg, Jak/Stat). As we have describedin this review, these pathwaysare functionally related to Notch signalling in the control of cell proliferation during normal development [6,8].

    14. Conclusions

    There are many precedents to demonstrate the utility of the development of the imaginal discs of Drosophila as a model to reveal the mechanistic details of how Notch signalling controls cell proliferation. The definition of these mechanisms is fundamental to establish how perturbations in the activity of Notch signalling can lead to an excess of cell proliferation, the primary caused of cancer. The studies in Drosophila have shown that Notch can regulate the activity and/or function of different factors that control the cell cycle. Interestingly the function of some of these cell cycle regulators have been found that to be altered in human cancer associated with disruption of Notch signalling. In addition the function of Notch in synergy with other signalling pathways can regulate different factors involved in the control of cell proliferation.

    An important aspect of the function of Notch regulating cell proliferation is its context-dependent nature. Different studies in Drosophila have revealed that this effect is at least partially caused by different feed-forward mechanism between Notch signalling and other genes and signalling pathways. Therefore the final outcome of the activity of Notch signalling in the regulation of cell proliferation depends on the interaction between this pathway and different genes, and signals.

    Acknowledgments

    We thank L A Baena-Lopez for their helpfulcomments and constructive criticisms. This work was supported by grants from MINECO BFU2011-23224 to AB and MINECOBFU2012-34353 to CE.

    Conflict of Interest

    The authors declare no conflict of interest for this work.



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