Jürgen Sonnemann
email: juergen.sonnemann@med.uni-jena.de

Stephanie Greßmann

Sabine Becker

Susan Wittig

Mareike Schmudde

James F Beck

The histone deacetylase inhibitor vorinostat induces calreticulin exposure in childhood brain tumour cells in vitro

Keywords: Bortezomib, Calreticulin, Childhood brain tumour, Histone deacetylase inhibitor, TRAIL, Vorinostat

Purpose It has recently been recognised that anticancer chemotherapy can elicit an immunogenic form of apoptosis characterised by the exposure of calreticulin (CRT) on the surface of dying tumour cells, entailing an immune response that contributes to the therapeutic outcome. CRT exposure has been found to be induced by anthracyclins and oxaliplatin, but not by other proapoptotic antineoplastic agents including etoposide, camptothecin and cisplatin. In this study, we examined the histone deacetylase inhibitor vorinostat for its capability to stimulate CRT exposure in tumour cells. Methods Childhood tumour cells, i.e. the brain tumour cell lines PFSK and DAOY and the Ewing's sarcoma cell line CADO-ES-1, were treated with vorinostat, and CRT exposure was determined by Xow cytometric analysis of CRT immunoXuorescence. Combination eVects of vorinostat/TRAIL and vorinostat/bortezomib were also assessed. Results Vorinostat treatment induced CRT exposure in PFSK and DAOY cells, but not in caspase-8-deWcient CADO-ES-1 cells. CRT exposure could be prevented by the pan-caspase inhibitor z-VAD-fmk and by brefeldin A, an inhibitor of Golgi-mediated transport.

Conclusion

Vorinostat has the capacity to elicit CRT exposure, suggesting its usefulness as immunogenic antitumour agent.

Introduction

Histone deacetylase inhibitors (HDACi), such as vorinostat (suberoylanilide hydroxamic acid, SAHA), are an emerging class of promising new anticancer agents. They induce apoptosis, growth arrest and diVerentiation in tumour cells while leaving healthy cells relatively unscathed [START_REF] Lane | Histone deacetylase inhibitors in cancer therapy[END_REF]. Their clinical potential is underscored by the fact that vorinostat has been approved by the US Food and Drug Administration for treatment of cutaneous T-cell lymphoma [START_REF] Richon | Development of vorinostat: current applications and future perspectives for cancer therapy[END_REF].

Engagement of antitumour immunity is another promising option to improve the management of malignant disease [START_REF] Zitvogel | The anticancer immune response: indispensable for therapeutic success?[END_REF]. Although antineoplastic chemotherapy has long been regarded to induce an immunologically silent or even immunosuppressive response, recent evidence suggests that chemotherapeutic drugs can in fact have immunostimulatory properties. For example, the proteasome inhibitor bortezomib promotes the exposure of heat shock protein 90 on the surface of dying myeloma cells, which leads to the maturation of dendritic cells allowing the latter to activate T cells [START_REF] Spisek | Bortezomib enhances dendritic cell (DC)-mediated induction of immunity to human myeloma via exposure of cell surface heat shock protein 90 on dying tumor cells: therapeutic implications[END_REF].

123

Calreticulin (CRT) is another protein whose exposure on the plasma membrane of apoptotic cells can result in the activation of immune cells [START_REF] Gardai | Cell-surface calreticulin initiates clearance of viable or apoptotic cells through trans-activation of LRP on the phagocyte[END_REF][START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF]. CRT usually resides in the endoplasmic reticulum (ER); chemotherapy, however, may cause apoptotic cancer cells to translocate CRT to their surface. The presence of CRT on the surface of tumour cells then directs their uptake by dendritic cells that induce an antitumour immune response [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF]. But among the cytotoxic agents tested, only few-the anthracyclins and oxaliplatin-induce CRT translocation and in turn immunogenic cell death, whereas others-e.g. mitomycin C, etoposide, camptothecin and cisplatin-do not [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF][START_REF] Tesniere | Immunogenic death of colon cancer cells treated with oxaliplatin[END_REF].

HDACi can potentially also stimulate an anticancer immune response. For instance, they have been shown to induce tumour cell-surface expression of NKG2D ligands and boost tumour cell recognition and lysis by NK cells [START_REF] Armeanu | Natural killer cellmediated lysis of hepatoma cells via speciWc induction of NKG2D ligands by the histone deacetylase inhibitor sodium valproate[END_REF][START_REF] Skov | Cancer cells become susceptible to natural killer cell killing after exposure to histone deacetylase inhibitors due to glycogen synthase kinase-3-dependent expression of MHC class I-related chain A and B[END_REF][START_REF] Schmudde | Histone deacetylase inhibitors sensitize tumour cells for cytotoxic eVects of natural killer cells[END_REF], to upregulate the expression of MHC class I and class II proteins [START_REF] Magner | Activation of MHC class I, II, and CD40 gene expression by histone deacetylase inhibitors[END_REF] as well as of costimulatory molecules [START_REF] Maeda | Up-regulation of costimulatory/adhesion molecules by histone deacetylase inhibitors in acute myeloid leukemia cells[END_REF], and to enhance antigen processing and presentation [START_REF] Setiadi | Epigenetic enhancement of antigen processing and presentation promotes immune recognition of tumors[END_REF]. However, it has not yet been studied whether HDACi cause CRT translocation and, this way, an immunogenic activity.

To explore this issue, we investigated whether vorinostat could induce CRT exposure in childhood brain tumour cells. Brain tumours are the most common solid malignancies of childhood and a leading cause of cancer-related death in children. Despite advances in therapy of childhood brain tumours in general over the last few decades, the prognosis remains dismal for a subset of patients, particularly those with disseminated disease [START_REF] Packer | Childhood brain tumors: accomplishments and ongoing challenges[END_REF]. In addition, current therapies are still frequently associated with severe long-term sequelae. Thus, the development of treatment strategies that may improve the chance of survival as well as lessen side eVects is of paramount importance. The utilisation of the immune system to target brain tumours may be a valuable option [START_REF] Mitchell | Immunotherapy of malignant brain tumors[END_REF].

We examined a possible CRT-exposing activity of vorinostat in two paediatric brain tumour cell lines: the supratentorial primitive neuroectodermal tumour (sPNET) cell line PFSK and the medulloblastoma cell line DAOY. We have recently shown that vorinostat induces apoptosis in these cell lines [START_REF] Kumar | Histone deacetylase inhibitors induce cell death in supratentorial primitive neuroectodermal tumor cells[END_REF][START_REF] Sonnemann | Histone deacetylase inhibitors induce cell death and enhance the susceptibility to ionizing radiation, etoposide, and TRAIL in medulloblastoma cells[END_REF]. In the present report, we show that vorinostat treatment also elicits CRT exposure.

Materials and methods

Reagents

Vorinostat, mitomycin C, brefeldin A and z-VAD-fmk were purchased from Alexis (Grünberg, Germany). Bortezomib was purchased from LC Laboratories (Woburn, MA, USA). TRAIL was purchased from Peprotech (Hamburg, Germany).

Cell lines DAOY and PFSK cells were a gift from Dr. M. Grotzer (Zurich, Switzerland) and maintained in Improved MEM Zinc Option (Invitrogen, Karlsruhe, Germany) or RPMI 1640 (PAA, Cölbe, Germany), respectively. CADO-ES-1 cells were obtained from the DSMZ (Braunschweig, Germany) and maintained in RPMI 1640. Media were supplemented with 10% foetal calf serum, 2 mM L-glutamine, 100 units/ml penicillin G sodium and 100 g/ml streptomycin sulphate (PAA). CADO-ES-1 cells were cultivated in collagen-coated (5 g/cm 2 ; Roche, Mannheim, Germany) tissue culture Xasks. Cells were cultivated at 37°C in a humidiWed 5% CO 2 incubator and routinely passaged when 90% con-Xuent. Cell viability was determined by the trypan blue exclusion test. Cells were regularly inspected to be free of mycoplasma with the PCR mycoplasma detection kit from Applichem (Darmstadt, Germany).

Flow cytometric detection of cell-surface CRT

The cells were plated at 1 £ 10 5 cells in 12-well plates and treated with vorinostat for 24 h. Brefeldin A or z-VAD-fmk was applied 1 h before treatment; in the combination experiments, vorinostat was applied 4 h before TRAIL or bortezomib. Cells were harvested with 0.2% EDTA, washed twice with PBS, and incubated with the mouse anti-CRT monoclonal antibody FMC 75 (2 g in 100 l PBS/2.5% BSA; Abcam, Cambridge, UK) for 45 min at 4°C in the dark. Cells were washed twice and stained with a FITCconjugated goat anti-mouse IgG antibody (20 g/ml in 2.5% BSA; Dianova, Hamburg, Germany) for 45 min at 4°C in the dark. After washing, 50,000 cells were analysed using a BD (Heidelberg, Germany) FACSCanto II; cells were gated on viable cells. A mouse IgG (20 g/ml in 2.5% BSA; eBioscience, San Diego, CA, USA) was used as isotype-matched control. Frequency histograms were generated using Flowjo software (Ashland, OR, USA). Data are shown as fold induction after calculation of the mean Xuorescence intensities of anti-CRT-stained cells divided by isotype-matched IgG-stained cells.

Immunoblot detection of total CRT

After a 24-h incubation with vorinostat, cells were lysed on ice for 15 min in 40 mM Tris-HCl (pH 7.4), 150 mM NaCl, 1% Triton X-100, 0.5% sodium deoxycholate and 0.1% SDS supplemented with a protease inhibitor cocktail (Roche) followed by brief soniWcation. Protein concentration was assayed using bicinchoninic acid (Pierce, Rockford, IL, USA) according to the manufacturer's instructions. Concentration of 30 g of total cellular protein per lane was separated by standard SDS-PAGE on 10% gel and 123 electrophoretically transferred to nitrocellulose membrane (Whatman, Dassel, Germany). After blocking in PBS containing 5% dry milk and 0.05% Tween 20, CRT was immunodetected using FMC 75 antibody (dilution 1:1,000). Equal loading of protein was veriWed by detection of GAPDH using mouse anti-GAPDH monoclonal antibody (dilution 1:10,000; Biodesign International, Saco, ME, USA). Peroxidase-conjugated goat anti-mouse IgG (dilution 1:25,000; Dianova) was used as secondary antibody followed by enhanced chemiluminescence detection (Amersham, Freiburg, Germany) of speciWc signals.

Statistical analysis

Statistical signiWcance of diVerences between experimental groups was determined using the paired two-tailed Student's t test.

Results

CRT exposure on PFSK sPNET cells

To test whether vorinostat elicits CRT exposure in paediatric brain tumour cells, we initially determined CRT surface expression on PFSK cells. Figure 1a, b demonstrates that vorinostat induced CRT exposure in a dose-dependent manner, as monitored by Xow cytometric analysis of viable cells for CRT immunoXuorescence staining. To exclude a passive leakage of CRT to the plasma membrane, cells were pretreated with brefeldin A, an inhibitor of protein transport from the ER to the Golgi apparatus: Inhibition of ER-Golgi traYc abolished vorinostat-elicited CRT exposure. Because chemotherapy-mediated CRT translocation has been shown to involve caspases [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF], PFSK cells were pretreated with the pan-caspase inhibitor z-VAD-fmk: Caspase inhibition prevented the CRT-exposing activity of vorinostat.

CRT exposure on DAOY medulloblastoma cells

Next, we studied the eVect of vorinostat on CRT exposure in DAOY cells. As shown in Fig. 1c, vorinostat treatment also provoked a concentration-dependent CRT surface expression in these cells. As a negative control, DAOY cells were also treated with mitomycin C, which has been reported to exert no CRT-exposing eVect [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF]. Mitomycin C did not increase CRT exposure in DAOY cells. In addition, we examined whether CRT exposure in vorinostat-treated cells was accompanied by a general rise of CRT abundance. Immunoblot detection of CRT in total cell lysates showed that vorinostat treatment did not result in an increase in total CRT (Fig. 1d).

CRT exposure on CADO-ES-1 Ewing's sarcoma cells Caspase-8 has been shown to be necessary for anthracyclinand oxaliplatin-induced CRT exposure [START_REF] Panaretakis | Mechanisms of pre-apoptotic calreticulin exposure in immunogenic cell death[END_REF]. To shed light on the relevance of caspase-8 for HDACi-induced CRT exposure, we employed the Ewing's sarcoma cell line CADO-ES-1, a cell line deWcient in caspase-8 expression [START_REF] Fulda | Sensitization for death receptor-or drug-induced apoptosis by re-expression of caspase-8 through demethylation or gene transfer[END_REF], but apoptotically responsive to HDACi treatment [START_REF] Sonnemann | Histone deacetylase inhibitors induce cell death and enhance the apoptosis-inducing activity of TRAIL in Ewing's sarcoma cells[END_REF]. Figure 1e illustrates that vorinostat had no eVect on CRT surface expression on CADO-ES-1 cells.

CRT exposure after vorinostat/TRAIL or vorinostat/bortezomib combination treatment A large number of preclinical studies have demonstrated synergistic proapoptotic activity of vorinostat with other antineoplastic agents, such as the death receptor ligand TRAIL or the proteasome inhibitor bortezomib [START_REF] Richon | Development of vorinostat: current applications and future perspectives for cancer therapy[END_REF]. We thus evaluated whether vorinostat could also synergise with TRAIL or bortezomib in triggering CRT exposure. In consistence with the observation that TRAIL elicits CRT exposure in Hela cells [START_REF] Panaretakis | Mechanisms of pre-apoptotic calreticulin exposure in immunogenic cell death[END_REF], we found that TRAIL evoked CRT surface expression on DAOY cells (Fig. 2a). Yet the vorinostat/TRAIL combination treatment produced a merely additive eVect. However, when vorinostat was combined with bortezomib, a trend to overadditive induction of CRT exposure was observed (Fig. 2b).

Discussion

To the best of our knowledge, we herein report for the Wrst time that treatment with an HDACi leads to CRT surface exposition on tumour cells. Given the limited number of chemotherapeutics with proven CRT-exposing activitythe anthracyclins and oxaliplatin [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF][START_REF] Tesniere | Immunogenic death of colon cancer cells treated with oxaliplatin[END_REF]-this entails a substantial expansion of antineoplastic agents endowed with the potential to elicit a CRT-dependent antitumour immune response.

In our study, we have shown that vorinostat induced CRT exposure in a concentration-dependent fashion. Several lines of evidence presented indicate that vorinostat promotes CRT exposure by the same mechanism as the known CRT-exposing compounds. First, we could not detect a rise of total CRT expression, in concordance with the Wnding that exposure of CRT induced by anthracyclins does not stem from a general increase in CRT abundance [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF]. Second, CRT translocation has been demonstrated to occur via the ER-Golgi pathway [START_REF] Panaretakis | Mechanisms of pre-apoptotic calreticulin exposure in immunogenic cell death[END_REF]. Our results show that the vorinostat-mediated CRT exposure could be prevented by brefeldin A, suggesting that CRT translocation induced by vorinostat also depends on anterograde ER-Golgi traYc. Third, in line with the observation that chemotherapy-elicited

123

CRT surface expression depends on caspase activity [START_REF] Obeid | Calreticulin exposure dictates the immunogenicity of cancer cell death[END_REF], vorinostat-induced CRT exposure could be abolished by z-VAD-fmk. In addition, vorinostat failed to promote CRT exposure in caspase-8-deWcient CADO-ES-1 cellswhich are otherwise susceptible to HDACi-treatment [START_REF] Sonnemann | Histone deacetylase inhibitors induce cell death and enhance the apoptosis-inducing activity of TRAIL in Ewing's sarcoma cells[END_REF]-consistent with the Wnding that caspase-8 is necessary for CRT exposure, but not apoptosis, induced by anthracyclins and oxaliplatin [START_REF] Panaretakis | Mechanisms of pre-apoptotic calreticulin exposure in immunogenic cell death[END_REF].

A multitude of preclinical studies have documented that HDACi have great potential to augment the anticancer eYciency of other therapeutic regimens, such as chemoand radiotherapy [START_REF] Lane | Histone deacetylase inhibitors in cancer therapy[END_REF][START_REF] Richon | Development of vorinostat: current applications and future perspectives for cancer therapy[END_REF]. With respect to the use of TRAIL for the treatment of childhood tumours, we have previously shown that vorinostat and TRAIL synergise to elicit cell death in medulloblastoma, Ewing's sarcoma and hepatoblastoma [START_REF] Sonnemann | Histone deacetylase inhibitors induce cell death and enhance the susceptibility to ionizing radiation, etoposide, and TRAIL in medulloblastoma cells[END_REF][START_REF] Sonnemann | Histone deacetylase inhibitors induce cell death and enhance the apoptosis-inducing activity of TRAIL in Ewing's sarcoma cells[END_REF][START_REF] Dzieran | DiVerential responsiveness of human hepatoma cells versus normal hepatocytes to TRAIL in combination with either histone deacetylase inhibitors or conventional cytostatics[END_REF]. TRAIL seemed furthermore to be particularly interesting since it has been demonstrated to stimulate CRT exposure in carcinoma cells [START_REF] Panaretakis | Mechanisms of pre-apoptotic calreticulin exposure in immunogenic cell death[END_REF]. In the attempt to Wnd out whether vorinostat and TRAIL synergised in inducing CRT exposure, however, we detected a merely additive eVect. In contrast, the combination of vorinostat with bortezomib-the latter has been reported to elicit immunogenic cell death by inducing exposure of heat shock protein 90 [START_REF] Spisek | Bortezomib enhances dendritic cell (DC)-mediated induction of immunity to human myeloma via exposure of cell surface heat shock protein 90 on dying tumor cells: therapeutic implications[END_REF]-produced a trend to enhanced induction of CRT surface expression. These Wndings show that the combination of two CRT-exposing compounds does not necessarily result in the potentiation of CRT exposure, but they also show that it can in principle be augmented by a suitable combination treatment.

Again to the best of our knowledge, we here have also shown for the Wrst time that childhood brain tumour cells can respond to antineoplastic agents by exposing CRT. Therefore, our results suggest that the concept of immunochemotherapy [START_REF] Zitvogel | The anticancer immune response: indispensable for therapeutic success?[END_REF] may be a therapeutic option in these diYcult to treat tumours. Chemotherapy of brain tumours is in general complicated in that the blood-brain barrier obstructs the penetration of most drugs into the brain [START_REF] Muldoon | Chemotherapy delivery issues in central nervous system malignancy: a reality check[END_REF]. However, this shortcoming does supposedly not apply to vorinostat, for it has been shown to cross the blood-brain barrier in mice [START_REF] Hockly | Suberoylanilide hydroxamic acid, a histone deacetylase inhibitor, ameliorates motor deWcits in a mouse model of Huntington's disease[END_REF]. To sum up, our study on the one hand extends the notion that HDACi have immunomodulatory activity that may contribute to mediating their antitumour eVects [START_REF] Armeanu | Natural killer cellmediated lysis of hepatoma cells via speciWc induction of NKG2D ligands by the histone deacetylase inhibitor sodium valproate[END_REF][START_REF] Skov | Cancer cells become susceptible to natural killer cell killing after exposure to histone deacetylase inhibitors due to glycogen synthase kinase-3-dependent expression of MHC class I-related chain A and B[END_REF][START_REF] Schmudde | Histone deacetylase inhibitors sensitize tumour cells for cytotoxic eVects of natural killer cells[END_REF][START_REF] Magner | Activation of MHC class I, II, and CD40 gene expression by histone deacetylase inhibitors[END_REF][START_REF] Maeda | Up-regulation of costimulatory/adhesion molecules by histone deacetylase inhibitors in acute myeloid leukemia cells[END_REF][START_REF] Setiadi | Epigenetic enhancement of antigen processing and presentation promotes immune recognition of tumors[END_REF] and on the other supports their potential usefulness in paediatric oncology [START_REF] Fruhwald | The epigenetics of cancer in children[END_REF].

Fig. 1

 1 Fig. 1 Induction of CRT exposure by vorinostat. Cells were treated with vorinostat for 24 h. a, b, c, e, CRT cell-surface expression was determined by Xow cytometric analysis of viable cells. a, Cells were stained with CRT antibody (solid line) or isotype control (Wlled histogram). b, c, e, CRT mean Xuorescence is depicted as fold induction after calculation of the mean Xuorescence intensities of anti-CRT-stained cells divided by isotype-matched IgG-stained cells; mean § SEM. of each 3 experiments are shown (*treatment vs. control; # brefeldin A and z-VAD-fmk, respectively, vs. 5 M vorinostat). d, total CRT expression was determined by immunoblot analysis of cell lysates

Fig. 2

 2 Fig. 2 Induction of CRT exposure by vorinostat in combination with TRAIL or bortezomib. 4 h after administration of vorinostat, DAOY cells were treated with TRAIL (a) or bortezomib (b) for 24 h. CRT cell-surface expression was determined by Xow cytometric analysis of viable cells. CRT mean Xuorescence is depicted as fold induction after calculation of the mean Xuorescence intensities of anti-CRT-stained cells divided by isotype-matched IgGstained cells; mean § SEM. of each 3 experiments are shown (*treatment vs. control; # combination treatment vs. vorinostat; § combination treatment vs. TRAIL or bortezomib, respectively)

Acknowledgments

We gratefully acknowledge the input and discussions of Barbara Bröker (Institute for Immunology, University of Greifswald). This work was supported by a grant from the "Wilhelm Sander-Stiftung, Neustadt/Donau". S.G. received a fellowship from the "IZKF des Universitätsklinikum Jena"; M.S. received a fellowship from the "Alfried Krupp Wissenschaftskolleg Greifswald der Alfried Krupp von Bohlen und Halbach-Stiftung".


Jerzy Warminski
email: j.warminski@pollub.pl

Nonlinear Normal Modes of a Self-Excited System Driven by Parametric and External Excitations

Keywords: nonlinear vibrations, nonlinear normal modes, parametric resonances, self-excitation, model reduction NODY9679_source.doc

Vibrations of a nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.

Introduction

Self-excited systems are one of the most interesting and common in mechanical engineering theory and practice [START_REF] Nayfeh | Nonlinear Oscillations[END_REF]. The response of this type of systems is represented by periodic limit cycles which can be stable, then the self-excitation is called soft, or unstable, called hard or catastrophic.

Another, well known and intensively studied are parametric vibrations which are generated by periodically changing in time parameters. Dynamics of a parametrically excited system is observed by, so called, parametric resonances where vibration amplitudes increase reaching large values [START_REF] Cartmell | An Introduction to Linear, Parametric and Nonlinear Vibrations[END_REF].

Interesting dynamical phenomena can be observed when both of mentioned above vibrations meet together at the same time. Then interactions between two different types of vibration take place. Such a situation may arise in many mechanical engineering structures.

As an example we can mention dynamics of a helicopter's rotor blade, while during the rotation due to varying blade incidence angle, the stiffness of the blade is changing periodically. Under flutter conditions caused by air flow the interaction of both parametric and self-excitation takes place. Moreover, due to nonlinear coupling various vibration modes are coupled as well.

Tondl [START_REF] Tondl | On the interaction between self-excited and parametric vibrations[END_REF] and co-authors [START_REF] Tondl | Cancelling of elf-Excited Vibrations by Means of Parametric Excitation[END_REF], [START_REF] Tondl | The effect of parametric excitation on a self-excited three-mass system[END_REF] considerably contributed in the analysis of interactions of parametric and self-excited vibrations. They studied dynamics of self-excited systems of van der Pol's type driven by Mathieu parametric excitation. It has been found there that in wide intervals of excitation frequency the response of the system is quasi-periodic, and observed by quasi-periodic limit cycles on Poincaré maps. However, near the parametric resonances, self-excited frequency is quenched and the response of the system is periodic, represented by singular points on Poincaré maps. Those regions are sometimes called "frequency synchronisation regions", due to the fact of the synchronisation of self-and parametric excitation frequencies. Extension of the works concerning parametric and self excited systems on bifurcation analysis and also chaotic or hyperchaotic dynamics can be found e.g. in [START_REF] Abadi | Nonlinear Dynamics of Self-Excitation in Autoparametric Systems[END_REF]- [START_REF] Verhurlst | Quenchening of self-excited vibrations[END_REF].

The behaviour of the parametrically and self-excited system can completely change if the system is additionally forced by external force. In many practical examples it can happen that frequency of the external force is equal to half of the parametric excitation frequency. As an example we can mention a rotating shaft with a rectangular cross-section and unbalanced mass (Fig. 1). During the rotation the stiffness of the system changes twice (parametric excitation) while the external inertia force once per revolution. Fig. 1 Example of an externally and parametrically excited system with 1:2 frequencies ratio. Such a situation may appear also in other examples e.g. in sagged cable dynamics [START_REF] Benedettini | Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions[END_REF]- [START_REF] Luongo | Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables[END_REF] or gear systems [START_REF] Warminski | Regular and Chaotic Vibrations of Parametrically and Self-Excited Systems with Ideal and Non-Ideal Energy Sources[END_REF], vibrations of bars, beams and plates [START_REF] Nayfeh | Nonlinear Interactions, Analytical, Computational and Experimental Methods[END_REF] as well as in microelectromechanical systems (MEMS) [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF]. If the considered structure is additionally damped by a nonlinear force e.g. nonlinear friction or is placed in a fluid flow, which yield in flutter vibration, then we can find a system where also self-excited terms may appear. Differential equation of motion of one degree of freedom system with self-, parametric and external excitation can be written in form ( ) ( )(

) 3 1 cos 2 cos x f x x x t q t γ μ ω ω + + + - = (1)
In the considered model the external force excites the system exactly inside the principal parametric resonance, since in this region the response of the system is subharmonic with respect to the parametric excitation frequency. The dynamics of such a system was investigated in papers [START_REF] Szabelski | The parametric self excited non-linear system vibrations analysis with the inertial excitation[END_REF], [START_REF] Szabelski | The self-excited system vibrations with the parametric and external excitations[END_REF]. The nonlinear damping was there assumed as the Rayleigh's nonlinear function ( )

3 f x x x α β = -+
. The results indicated very important quantitative changes in the principal parametric resonance regions. The resonance curves received by the multiple scale of time method for the system without external force ( 0 q = ) and driven by external force ( 0 q > ) are presented in Fig. 2 , external force amplitude 0 q = (Fig. 2a) or 0.05 q = (Fig. 2b). The response of the self-excited system driven parametrically is mainly quasi-periodic apart from, so called, frequency synchronisation regions where self-excited vibrations are quenched by parametric excitation and the single harmonic oscillations are observed. This phenomenon is visible near the parametric resonance regions. Fig. 2(a). Nevertheless, if the self-excited system is driven parametrically and externally then its response changes quantitatively and qualitatively. We see in Fig. 2(b) that inside this region the resonance curve possesses an internal loop (red colour). Near the frequency about 1.07 five possible solutions are possible. The stability analysis shows that only two upper among five are stable. Of course to get the stable solution it is necessary to put initial conditions inside their basins of attractions. Possible existence of five equilibria for one DOF parametric system with negative damping has been also presented in [START_REF] Rand | Parametric Resonance of Hopf Bifurcation[END_REF]. Similar phenomena has been observed in [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF] for one DOF MEMS resonator modelled by a van der Pol-Duffing system driven by parametric and external excitation. The bifurcation analysis revealed regions of the five solutions appearance.

The problem of existence of the additional solutions is more complicated for models with many degrees of freedom. For many DOF systems additional interactions between vibrating modes may appear (apart from that resulted from various vibrations types).

The results for two DOF model [START_REF] Szabelski | Vibrations of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation[END_REF] confirm existence of the internal loop inside the main parametric region. However, to get this result a classical linear normal modes approach has been applied there. It has been shown that there is an essential difference between numerical and analytical results. This linear decoupling, which is usually used for weakly nonlinear system, was not sufficient for the nonlinear system with parametric and selfexcitation. Therefore, for a proper decoupling of the system another technique is required.

Application of nonlinear normal modes (NNMs) [START_REF] Rosenberg | Normal Modes of Non-linear dual-mode systems[END_REF], [START_REF] Vakakis | Non-Linear Normal Modes (NNMs) and Their Applications in Vibration Theory: an Overview[END_REF] is the promising approach which allows for separation of the structure for single DOF oscillators. Such constructed oscillators should include all information related with the nonlinear system's dynamics.

Various techniques of NNM have been presented and discussed on serious of conferences directly devoted to this topic, the first conference organised by C.H.Lamarque [START_REF] Warminski | Non-Linear Vibration Modes of Coupled Parametrically and Self-Excited Oscillators[END_REF], the second by A.Vakais [START_REF] Warminski | Analysis of Different Vibrations Types by Application of Nonlinear Normal Modes[END_REF] and the third by G.Rega [START_REF] Warminski | Nonlinear Normal Modes of Coupled Self-Excited Oscillators Under Parametric Resonance condition[END_REF]. Nonlinear Normal Modes of a strongly nonlinear autonomous system, conservative or damped, are formulated in [START_REF] Shaw | Normal Modes for Non-Linear Vibratory Systems[END_REF]. This formulation is based on the centre-manifold approach and it allows for model decoupling by taking into account, both, the displacement and velocity of a chosen set of coordinates, which are so called master coordinates while rest are slave coordinates constituted by modal surfaces. This methodology can be applied for nonlinear but autonomous systems. The centre manifold reduction has been successfully applied for coupled self-excited systems [START_REF] Warminski | Nonlinear normal modes of coupled self-excited oscillators in regular and chaotic vibration regimes[END_REF].

Complexity of NNM formulation increases if the system is non-autonomous. Then time has to be introduced as an additional coordinate. To get the modal equation it is necessary to apply a special numerical Galerkin-based technique taking into account a time coordinate [START_REF] Jiang | Nonlinear normal modes for vibratory systems under harmonic excitation[END_REF]. This method allows for finding NNMs for every instant of time.

Another proposal of NNM formulation, defined for the nonlinear systems with weak damping, around the resonance region is presented in [START_REF] Szemplinska-Stupnicka | One-modal Analysis of Resonant Vibrations of Nonlinear Systems[END_REF], [START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF]. The resonant normal modes were found as a function of amplitude for the stationary response instead of functions of displacement and velocity. This approach allows for the system reduction around of resonance zones. The response is represented by a single oscillator with modal mass and frequency, constrained by an amplitude dependent relationship. Similar formulation has been presented

in [START_REF] Bellizzi | A new formulation for existence and calculation of nonlinear normal modes[END_REF] with application to undamped nonlinear and autonomous mechanical systems. The resonant periodic response has been defined by a single degree of freedom system with period of motion depending on the amplitude. The approach presented in [START_REF] Szemplinska-Stupnicka | One-modal Analysis of Resonant Vibrations of Nonlinear Systems[END_REF]- [START_REF] Bellizzi | A new formulation for existence and calculation of nonlinear normal modes[END_REF], seems be also attractive for the nonlinear the self-and parametrically or/and externally excited system. The study of self-and parametrically excited system without external force influence has been published in [START_REF] Warminski | Nonlinear Vibration Modes of Parametrically Coupled Self-Excited Oscillators[END_REF].

This paper presents resonant nonlinear normal modes based on the approach formulated in [START_REF] Warminski | Nonlinear Vibration Modes of Parametrically Coupled Self-Excited Oscillators[END_REF]. The normal modes are constructed on the same basis of nonlinear natural oscillations.

However, in the present case the analysis is extended to the system with two degrees of freedom which includes self-, parametric and external excitations. Such a system may exhibit a multi-solutions resonance curve comparing to those presented in [START_REF] Warminski | Nonlinear Vibration Modes of Parametrically Coupled Self-Excited Oscillators[END_REF] where only single classical resonances took place. This study is carried out for the purpose of proper reduction of the multi-degree of freedom model into set of separate single oscillators near the principal parametric resonance region. The separated nonlinear modes should exhibit appearance of the internal loop inside the parametric resonance, similar to whose described for that of one DOF.

Model of the system

Let's consider two coupled nonlinear oscillators (I) and (II) composed of mass, nonlinear spring of Duffing type and nonlinear damping which represents self-excitation (Fig. 3). The oscillators are coupled parametrically by a spring with periodically changing stiffness.

Besides, the first oscillator is driven by a harmonic force.

( )( ) ( ) ( ) ( ) 1 2 3 1 1 1 1 1 1 1 12 1 2 3 2 2 2 2 2 2 12 2 1 2 cos 2 cos cos 2 0 d d m X X X X t X X q t f f m X X X X t X X δ γ δ μ ω δ γ δ μ ω + + + - - = Ω + + + + - - - = (2)
Nonlinear damping is represented by Rayleigh's function

3 1 1 1 1 1 d f x x α β = - + and 3 2 2 2 2 2 d f x x α β = - + .
The coupling stiffness is linear of Mathieu type with periodically varying term ( )( )

12 12 1 2 cos 2 f t X X δ μ ω = - - .
In mechanical engineering systems this kind of periodically varied stiffness may appear e.g. in gear boxes due to changing of mesh stiffness during operation.

The main feature of the considered model is that nonlinear damping produces selfexcitation represented by limit cycles on the phase plane and the parametric excitation of frequency 2ω yields in instability regions so called parametric resonances. Near the principal parametric resonance the response of the system is subharmonic (1:2) with corresponding frequency ω . Such resonances appear when excitation frequency is close to one of the natural frequencies

01 ω ω ≈ or 02 ω ω ≈ .
Because we want the system to be driven by external harmonic force exactly inside the principal parametric resonance, therefore frequency Ω is taken as half of parametric excitation, ω Ω = . Physical justification of such a model is presented in the introduction.

Nonlinear Normal Modes Formulation

The nonlinear normal modes are formulated on the basis of the observation of the physical system response [START_REF] Szabelski | Vibrations of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation[END_REF]. The interaction between parametric and self-excited vibration results mainly in quasi-periodic motion composed by influence of both vibration types. On the Poincaré map this kind of motion is represented by quasi-periodic limit cycles. It means that the self-excitation dominates in those regions exhibiting its main features. However, approaching the parametric resonances, near their neighbourhood, after the second kind of Hopf bifurcation (Nejmark-Sacker bifurcation), the self-excitation is quenched, and the system response is periodic.

To formulate the nonlinear normal modes we will concentrate only in the neighbourhood of the main parametric resonance region. Taking into account that the selfexcitation (nonlinear damping) is quenched we can assume that the vibration modes are close to the mode of nonlinear conservative system. Therefore, at first we will formulate nonlinear normal modes of a nonlinear conservative system and then we will extend them to the selfexcited model driven by parametric and external excitations. The equations of motion of the conservative nonlinear system takes form ( ) ( )

3 1 1 1 1 1 1 12 1 2 3 2 2 2 2 2 2 12 1 2 0 0 m X X X X X m X X X X X δ γ δ δ γ δ + + + - = + + - - = (3)
In contrast to the pure linear system we define the vibrations modes as functions of amplitude.

One of the mode coefficients can be taken arbitrary and the second is expressed as a function of amplitude. Therefore, the coefficient 1 u is taken as a constant and for convenience is equal to one ( 1 1 u = ) in further analysis, while the second is a function of amplitude ( )

2 2
u u a = .

The solution of the set (3) is sought by the harmonics balance method (HBM). Only the first harmonic of the response is taken into account therefore the solution is assumed in form

1 0 2 2 0 cos ( )cos X t a X au a t ω ω = = (4)
where 0 ω is a natural vibration frequency of the nonlinear system. According to definitions introduced in papers related to NNMs the coordinate 1 X is called "master coordinate" while 2 X is so called "slave" coordinate (see e.g. [START_REF] Shaw | Normal Modes for Non-Linear Vibratory Systems[END_REF], [START_REF] Vakakis | Non-Linear Normal Modes (NNMs) and Their Applications in Vibration Theory: an Overview[END_REF]).

Substituting the solutions (4) into (3), putting and expanding trigonometric functions we get

- + + + - + = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ - + + + - + = ⎠ (5)
Taking into account the first harmonic approximation, we neglect the terms with 0 cos3 t ω .

We may note that this reduction does not reject nonlinear terms in total, only in the second order approximation. After such simplification we get ( ) ( )

2 2 1 0 1 12 1 2 12 2 2 3 1 2 2 2 0 2 1 2 2 2 3 0 4 3 0 4 m a u u m a u a ω δ δ γ δ δ ω δ δ γ - + + - = - + + + + - = + (6)
where coefficient ( )

2 2
u u a ≡ is a function of amplitude. Eliminating natural frequency 0 ω from above equations we get

( ) ( )( ) ( )
2 3 2 2 1 1 2 2 2 1 2 2 12 2 2 1 1 2 3 1 0 4 a m u m u u m m u u m m γ γ δ δ δ + - + - - + = (7)
Equation ( 7) is a modal equation which allows for determination the mode coefficients of the nonlinear conservative system. From [START_REF] Bakri | Parametric excitation in non-linear dynamics[END_REF] we can find two roots for 2 u , ( )

21
u a and ( )

22
u a , for the first and the second mode respectively. However, because it is easier to determine inverse relation we find ( )

2 a u which takes form ( )( ) ( ) ( ) 2 1 2 2 1 2 2 2 1 1 2 2 2 2 1 1 2 2 1 2 3 u m mu u m m a u m m u δ δ δ γ γ - + - - = - (8)
Two real solutions for a represent the nonlinear curves of ( )

2
a u for the first and the second nonlinear normal modes, respectively. We have to bear in mind that natural frequency of nonlinear system 0 ω is also amplitude dependent. Eliminating in (6) vibration mode 2 u , we can find

6 4 2 3 0 2 0 1 0 0 0 ω ω ω Γ +Γ +Γ +Γ = (9)
where The solutions which tend in a limit to frequencies of the linear model, Having the modal coefficient ( )

3 3 1 2 3 3 12 3 4 m a γ δ Γ = - , ( ) 3
2 j
u a , the modal solutions take form The formulated above modes of the nonlinear conservative system will be applied to the system (2) for determination of the resonance curves.

Vibrations of a driven parametrically and externally coupled self-excited system

To transform the system from generalized into normal coordinates we make an assumption that inside the principal parametric resonance the vibration modes are close to the nonlinear normal modes formulated in previous chapter for the nonlinear conservative system.

Substituting [START_REF] Verhurlst | A note on higher order averaging[END_REF] into a set of the original equations [START_REF] Cartmell | An Introduction to Linear, Parametric and Nonlinear Vibrations[END_REF] and next multiplying the first equation by 1 j u (which in this case is equal to one) and the second by 2 j u (j=1,2), then adding them, we get

( ) ( ) ( ) 2 2 2 4 3 1 2 2 1 2 2 2 1 2 1 2 2
(1 ) , ,

j j j j j j j j j j m m u Y u u Y u Y F Y Y t δ δ δ γ γ ⎡ ⎤ + + + + - + + = ⎣ ⎦ (11)
where

( ) ( ) ( ) 2 2 4 3 2 1 2 2 1 2 2 1 cos 2 cos j j j j j j j F u Y t u Y u Y q t μ ω α α β β ω = - + + - + + , j=1,2.
Taking into account that for the conservative system ( ) , , 0

j j j
F Y Y t = and then substituting the periodic solution [START_REF] Verhurlst | Quenchening of self-excited vibrations[END_REF] we get relationship which represents natural frequency of the nonlinear system

0 cos j j Y a t ω = into (
( ) ( ) 2 2 2 2 4 0 1 1 2 2 2 2 1 2 2 2 1 2 2 1 3 1 4 j j j j j u u a u m m u ω δ δ δ γ γ ⎤ ⎡ = + - + + + ⎥ ⎢ ⎣ + ⎦ (12)
This equation corresponds to the frequency found from [START_REF] Abadi | Nonlinear Dynamics of Self-Excitation in Autoparametric Systems[END_REF], therefore the equation of motion may be written in nonlinear normal coordinates as

( ) ( ) 2 0 ( ) ( ) , , j j j j j j j j M a Y M a Y F Y Y t a ω + = (13)
where ( )

2 1 2 2 j j M m a m u + =
is the modal mass of the system for the first and the second vibration mode for 1, 2 j = , respectively. Function ( ) , ,

j j j
F Y Y t can also be expressed as

3 cos 2 cos j j j j j j j q j F C Y t C Y C Y C t μ α β ω ω = + - + (14)
where ( )

2 2 1 j j C u µ µ = - , 2 1 2 2 j j C u α α α = + , 4 1 2 2 j j C u β β β = + , qj C q = ( 15
)
are modal coefficients representing parametric, self-and external excitation. The essential assumption made to transform the equations (3) into ( 13) is that the system's response is periodic up to the first balanced harmonic. The modes are called resonant nonlinear normal modes because they can be used only for regions of a single frequency response. For a selfexited system driven by parametric and external excitations they are used for the frequency locking zones ([ 3]- [START_REF] Tondl | The effect of parametric excitation on a self-excited three-mass system[END_REF], [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF]) characterized by periodic solutions. The formulated modes in approximation decouple the system also if amplitude and phase of the solution are slow functions of time.

To get resonance solution it is necessary to solve a single equation [START_REF] Srinil | Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables[END_REF] together with constrain equation (7).

Analytical Solutions of Self, Parametrically and Externally Driven System

Analytical solutions of Eq.( 13) due to its nonlinear nature can be determined by application of approximation methods. The Multiple Scale of time method [START_REF] Nayfeh | Nonlinear Oscillations[END_REF] is used to find the approximate solution. Introducing a formal small parameter ε, equation [START_REF] Srinil | Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables[END_REF] is written in the form ( )

( ) 2 0 ( ) ( ) , , j j j j j j j j a M a Y M a Y F Y Y t ω ε + = for j=1,2 (16)
Function [START_REF] Luongo | Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables[END_REF] is expressed by

j j F F ε =
, and parameters: μ εμ = ,

1 1 α εα = , 1 1 β εβ = , 2 2 α εα = , 2 2
β εβ = and consequently

j j C C µ µ ε = , j j C C α α ε = , j j C C β β ε = , qj qj C C ε = .
According to the formulation presented in Chapter 4 it is assumed that the response of the system is periodic. However, assuming that amplitude and phase of the solution of (16) are slow functions of time i.e. if during one period of the fast scale there is a small change of amplitude (the slow scale) the presented approach can be used also in the neighbourhood of the resonance (frequency locking) zones. Solution of Eq.( 16) is sought in form of a series of the small parameter ε

0 0 1 1 0 1 ( , ) ( , ) ( , ) j j j Y t Y T T Y T T ε ε = + +… (17)
where ( )

0 0 1 , j Y T T , ( )
1 0 1
, j Y T T are zero and first order functions of time. Time is also expressed by the series of the small parameter,

0 1 t T T ε = + +… (18)
where 0 T and 1 T are respectively fast and slow scale of time. Such time definitions results in the following formulae for the first and the second time derivatives

2 2 0 1 2 0 1 2 d ... d D D D t T T T ε ε ε ε ∂ ∂ ∂ = + + = + + ∂ ∂ ∂ (19) 2 2 2 2 0 0 1 0 2 1 2 d 2 (2 ) ... d D D D D D D t μ ε = + + + + (20)
where

m m n n D T ∂ = ∂
means m order partial derivative with respect to n scale of time.

As it has been assumed earlier we sought the solutions around the principal parametric resonance, where the self-excitation is quenched by dominating parametric vibrations, therefore, bearing in mind that parametric excitation frequency is equal to 2ω and external excitation ω , we can write

2 2 0 j j ω ω εσ = + (21)
where j σ is frequency detuning parameter around the first and the second natural frequency

0 j ω , j=1,2.
Substituting solution [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF] and taking into account the derivatives definitions [START_REF] Szabelski | The self-excited system vibrations with the parametric and external excitations[END_REF], [START_REF] Rand | Parametric Resonance of Hopf Bifurcation[END_REF], after grouping terms with respect to ε order we get a set of differential equations in successive perturbation orders 0 ε -order

2 2 0 0 0 0 j j D Y Y ω + = (22) 1 ε -order 2 2 0 1 1 3 0 0 1 0 0 1 0 0 1 0 0 2 ( ) cos 2 cos / j j j j j j j j jj q j j j D Y Y Y D D Y C D D Y C D D Y C Y t C t M α β μ ω σ ω ω + = ⎡ ⎤ - + - + + ⎣ ⎦ (23)
Solution of Eq.( 22) is written in complex form

0 0 1 1 1 0 1 1 0 ( , ) ( ) exp( ) ( ) exp( ) j j j Y T T A T i T A T i T ω ω = + - (24)
where

1 i = -is an imaginary unit, 1 j A , 1 j
A is complex amplitude and it's conjugate. Solution [START_REF] Warminski | Non-Linear Vibration Modes of Coupled Parametrically and Self-Excited Oscillators[END_REF] is substituted into [START_REF] Vakakis | Non-Linear Normal Modes (NNMs) and Their Applications in Vibration Theory: an Overview[END_REF] and then, after grouping terms in proper exponential functions, we get

0 0 1 3 2 2 3 3 0 1 1 1 1 3 2 1 1 1 1 1 1 2 1 2 3 2 1 2 j j i T j j j j j j j q j i T j j j j j j j j j A C D Y Y e i A C M C C e i DA A i A C i A c M A A C c μ ω β μ ω α β ω ω ω σ ω ω ⎛ ⎞ + = - - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ ⎛ ⎞ + - - + - + + ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ (25)
where cc means complex conjugate functions to those written in the equation. To eliminate secular terms, the components near 0 ( ) exp i T ω must vanish, thus

1 3 2 1 1 1 1 1 1 1 2 3 0 2 2 j j q j j j j j j j j j j C C i D A A i A C i A C M A A μ α β ω σ ω ω ⎛ ⎞ - - + - + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (26)
Expressing complex amplitude j A by the polar form

1 j i j j A a e φ = (27)
and then separating real and imaginary parts of (26) we get modulation equations for amplitude a j and phase φ j

3 3 1 3 1 1 sin 2 sin 2 8 4 2 1 1 1 cos 2 cos 2 4 2 j j j j j j j j q j j j j j j j j j j q j M a C a C a C a C M a M a C a C α β μ μ ω ω ω φ φ ω φ σ φ φ = - - - = - - - (28)
In a steady state 0 j a = , 0 j φ = . Therefore, from equations [START_REF] Warminski | Nonlinear normal modes of coupled self-excited oscillators in regular and chaotic vibration regimes[END_REF] we can find the resonance curve around the principal parametric resonances

2 2 2 2 2 2 2 4 2 0 qj j j j qj qj j j j j C a C C C a C µ µ λ - -Λ + + -Λ = (29)
where

2 3 1 3 4 j j j j j a C a C α β λ ω ω ⎛ ⎞ = - ⎜ ⎟ ⎝ ⎠ , ( ) 2 2 2 0 j j j M λ ω ω = - , and
( ) ( ) 2 2 2 2 2 2 2 1 2 2 4 2 j q j j j j j j j j q j j C a C a C C C µ µ µ λ λ λ Λ = - + +
To get vibration amplitudes in the resonance regions it is necessary to solve equation [START_REF] Jiang | Nonlinear normal modes for vibratory systems under harmonic excitation[END_REF] together with nonlinear modal forms of the considered system formulated by Eq.( 7).

Introducing the particular solution of (25) into [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF] yields

( ) ( ) ( ) 2 cos cos 3 sin 3 3 16 2 j j j j j j j a a Y a t C t C t M μ β ω φ ε ω φ ω ω φ ⎡ ⎤ = + + - + + + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (30)
where the amplitude a and the phase φ are determined from modulation equations [START_REF] Warminski | Nonlinear normal modes of coupled self-excited oscillators in regular and chaotic vibration regimes[END_REF] or for a steady state from algebraic equation (29).

Numerical Calculations

Numerical calculations are carried out for the nonlinear conservative model data

1 2 1 2 1 2 1 2 1, 2, 1, 1, 0.3, 0.1, 0.1 m
m δ δ δ γ γ = = = = = = = (31)
and for self-excitation parameters

1 1 2 2
, 0.01 0.05, 0.01, 0.05

β α α β = = = = (32)
while amplitude of parametric and external excitation are

0.2 µ = , 0 0.2 q ≤ ≤ (33)
To evaluate results obtained by the proposed method the numerical calculations are also performed for linear normal modes. The physical coordinates X are transformed to normal coordinates Y by linear transformation

0 X = u Y (34)
where 0 u is a linear modal matrix

110 120 0 210 220 u u u u ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ u ( 35
)
For the assumed data [START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF] coefficients of modal vectors takes values Taking into account the derived Eq. (7), which constitutes relationship of the mode coefficient The vertical dashed lines arising from these points represent linear modal lines which are amplitude independent. To solve the equation [START_REF] Nayfeh | Nonlinear Interactions, Analytical, Computational and Experimental Methods[END_REF] apart from modal coefficients also natural frequencies of the nonlinear model have to be determined from Eq.( 12). Fig. 5. Vibration natural frequencies versus amplitude; parameters given by [START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF] Solid lines in Fig. 5 presents course of frequency for free vibration of a nonlinear system with respect to amplitude, dashed line points out amplitude independent linear natural frequencies.

To construct the response based on nonlinear normal modes and related with it nonlinear resonant curves we have to solve Eq.( 29) together with modal relationships [START_REF] Bakri | Parametric excitation in non-linear dynamics[END_REF] and nonlinear frequencies [START_REF] Benedettini | Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions[END_REF]. Instead of Eqs, [START_REF] Bakri | Parametric excitation in non-linear dynamics[END_REF], [START_REF] Benedettini | Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions[END_REF] the set of algebraic equations (6) may be taken for numerical convenience.

At first, correctness of the nonlinear normal modes is demonstrated for a nonlinear conservative system without any excitations or damping. In Fig. 6 we see free vibrations of coupled nonlinear oscillators in physical (generalised) and normal coordinates determined by linear transformation (34). Initial conditions have been selected to activate only the first normal coordinate. Therefore, for simulations, initial conditions have values:

( ) 1 X 0 10.0 = , unstable. It is worth to mention that apart from steady state response, the sixth quasi-periodic motion also takes place. This solution is not presented in the figure because it has not been determined by the proposed analytical approach. The quasi-periodic motion requires a special treatment. The curves presented in Fig. 8(a) have been found by the nonlinear normal mode transformation. Therefore, each amplitude has been determined together with modal and frequency dependencies presented in Fig. 8(b) and (c) respectively. For amplitude equal to zero the modal coefficient takes values equal to the linear modes, however for 0 a > the modes is a nonlinear function (Fig. 8b).

The equivalent resonance curve got by linear normal modes approach is shown in Fig. 9 for comparison. We see that solutions differ essentially. The internal loop is located on the left hand side of the main resonance branch and is much larger than that got by nonlinear normal mode formulation.

Conclusions

The paper presents a technique for resonant nonlinear normal modes formulation with application to a self-and parametrically excited system. It is shown that NNMs defined on the basis of free vibrations of a nonlinear system give results which are much closer to direct numerical simulation, comparing with classical LNMs. A width of synchronisation regions, near the principal parametric resonance fits very well to the regions found by numerical simulations, presented in the bifurcation diagram. Resonance curves, obtained by nonlinear and linear normal modes are essentially different. Correctness of the applied technique is confirmed by time histories analysis for various set of parameters. An effort of numerical solving of a set of nonlinear algebraic equations is a cost which is paid for the accuracy increase. Out of the synchronisation regions, due to strong influence of self-excitation, another method for NNMs formulation is required. The method should take into account also influence of velocity on the vibration modes.

Figures Caption

Fig. 1 Example of an externally and parametrically excited system with 1:2 frequencies ratio.

  (a) and (b), respectively.

Fig. 2

 2 Fig.2 Resonance curve of a self-excited system driven parametrically (a) and driven parametrically and externally (b); 0.01 α = , 0.05 β = , 0.1 γ = , 0.1 µ = , 0.05 q =

Fig. 3

 3 Fig. 3 Model of coupled self-excited oscillators driven by parametric and external excitations

Equation ( 9 )

 9 has three real roots for 2 0 ω , however only two of them have physical meaning.

=

  will be taken for the further consideration. Thus, we found the formulae for the natural frequencies

  frequencies of the linear system 01 0.766091 ω = , 02 1.16752 ω = , where index 0 underlines the linear transformation.

Fig. 4 .

 4 Fig.4. Vibration modes versus amplitude; parameters given by[START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF]

2 u

 2 versus vibration amplitude, we can draw the nonlinear modal curves as functions of amplitude. The solid line on the right hand side of the Fig.4 presents the first vibration mode, while on the left hand side the second mode is plotted. Of course, for amplitude equal zero 0 a = value of the vibration mode corresponds to linear mode coefficients 210 u or 220 u (36).

Fig. 10

 10 Fig.10 Bifurcation diagram of the physical coordinate X 1 versus excitation frequency;

  To validate the received results the original Eqs.(2) have been solved directly by numerical methods. The bifurcation diagram of the physical coordinate 1 X versus excitation frequency is presented in Fig.10. The phenomenon of internal loop appearance is clearly visible near the principal parametric resonance around the second natural frequency for the main branch of the resonance curve as well as internal loop fit very well to the resonance regions obtained by nonlinear normal modes approach.

Fig. 2 Fig. 3 Fig. 4 .Fig. 5 .Fig. 6 .

 23456 Fig.2 Resonance curve of a self-excited system driven parametrically (a) and driven parametrically and externally (b); 0.01 α = , 0.05 β = , 0.1 γ = , 0.1 µ = , 0.05 q =

Acknowledgements

Financial support of Structural Funds in the Operational Programme -Innovative Economy (IE OP) financed from the European Regional Development Fund Project No POIG.0101.02-00-015/08 is gratefully acknowledged.

Y 0 0 = . We note that assumed values are high in the aim to expose the nonlinear effects (Fig. 6a,b). For a linear system only the first mode is activated while the second is equal to zero. Application of the linear normal modes to the nonlinear model is acceptable only for relatively small amplitudes.

Fig. 6. Free vibrations in physical coordinates X 1 , X 2 and linear normal coordinates Y 1 , Y 2 ; initial conditions ( )

Y 0 0 = ; parameters given by [START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF].

For the considered example the linear transformation leads to large differences. ). However, after a few seconds the coordinate increases and goes to very high values (Fig. 6d). The linear transformation leads definitely to wrong results. Therefore, for large displacements the nonlinear transformation is required.

The same analysis has been repeated but applying nonlinear transformation [START_REF] Verhurlst | A note on higher order averaging[END_REF]. It means that modal coefficient 21 u has been calculated taking into account vibration amplitude.

Initial condition for physical coordinates take values ( ) , ( )

Y 0 0 = , parameters given by [START_REF] Szemplinska-Stupnicka | Normal Co-ordinates in Analysis of Main Resonances of the Non-linear Systems with Many Degrees of Freedom[END_REF].

Time histories of X 1 and X 2 generated for initial conditions ( ) 1 X 0 10.0 = , ( ) Presented above formulation is used for resonant curves determination of the self-, parametrically and externally excited system near the main parametric resonances. The solutions are received from (13) by using analytical solutions [START_REF] Jiang | Nonlinear normal modes for vibratory systems under harmonic excitation[END_REF] assuming that in the resonance the vibrations modes are close to the mentioned above NNMs found for the nonlinear conservative system. Solving together [START_REF] Jiang | Nonlinear normal modes for vibratory systems under harmonic excitation[END_REF] and Eq.( 7) we get the resonance curve presented in Fig. 8(a). , 0.01 0.05, 0.01, 0.05

This curve has been plotted for the principal parametric resonance region near the second natural frequency. The resonance is chosen because the nonlinear effects are very well demonstrated there. In the Fig. 8 we can see behaviour of the system similar to that of one degree of freedom model presented in the introduction and papers [START_REF] Pandey | Frequency Locking in a Forced Mathieu-van der Pol-Duffing System[END_REF], [START_REF] Szabelski | The self-excited system vibrations with the parametric and external excitations[END_REF]. External force acting together with parametric and self-excitation changes classical resonance curve of the considered model. We see that in the main parametric resonance additional internal loop arises. In the frequency region, near 1.25 ω = , five solutions are possible. However, numerical check up shows that only two upper solutions are stable, while three lower are ( )


Elodie Mutel

Aya Abdul-Wahed

Nirilanto Ramamonjisoa

Anne Stefanutti

Isabelle Houberdon

Sophie Cavassila

Frank Pilleul

Olivier Beuf

Amandine Gautier-Stein

Armelle Penhoat

Gilles Mithieux

Dr Fabienne Rajas
email: fabienne.rajas@univ-lyon1.fr

Targeted deletion of liver glucose-6 phosphatase mimics glycogen storage disease type 1a including development of multiple adenomas

Keywords: glycogen storage disease, glucose-6 phosphatase, endogenous glucose production, hepatic steatosis, hepatic adenomas

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Targeted deletion of liver glucose-6 phosphatase mimics glycogen storage disease type 1a including development of multiple adenomas.

Introduction

Glycogen storage disease type I (GSD1) is a metabolic disorder characterized by severe hypoglycaemia associated with hepatic, renal and intestinal disorders [START_REF] Chou | Mutations in the glucose-6-phosphatase-alpha (G6PC) gene that cause type Ia glycogen storage disease[END_REF]. This inborn disease is due to a deficiency in glucose-6 phosphatase (G6Pase) activity. G6Pase is an essential enzyme of endogenous glucose production. Patients affected with GSD1 are therefore unable to maintain blood glucose concentration outside the time of meals and suffer from severe hypoglycaemic episodes if they are not managed by continuous feeding [START_REF] Chou | Type I glycogen storage diseases: disorders of the glucose-6-phosphatase complex[END_REF][START_REF] Moses | Historical highlights and unsolved problems in glycogen storage disease type 1[END_REF][START_REF] Ozen | Glycogen storage diseases: new perspectives[END_REF]. GSD type 1a and type 1b respectively correspond to inherited mutations in the genes that encode the catalytic subunit (G6PC) and the transporter unit (G6PT) of the G6Pase system [START_REF] Chou | Mutations in the glucose-6-phosphatase-alpha (G6PC) gene that cause type Ia glycogen storage disease[END_REF][START_REF] Chou | Type I glycogen storage diseases: disorders of the glucose-6-phosphatase complex[END_REF][START_REF] Bruni | Enzymatic characterization of four new mutations in the glucose-6 phosphatase (G6PC) gene which cause glycogen storage disease type 1a[END_REF][START_REF] Bozon | Mutation analysis in 24 French patients with glycogen storage disease type 1a[END_REF]. The G6pc gene is expressed in the liver, kidney, and intestine, and these are the only organs that release glucose into the blood circulation [START_REF] Croset | Rat small intestine is an insulin-sensitive gluconeogenic organ[END_REF][START_REF] Mithieux | Induction of control genes in intestinal gluconeogenesis is sequential during fasting and maximal in diabetes[END_REF][START_REF] Mithieux | A novel role for glucose 6-phosphatase in the small intestine in the control of glucose homeostasis[END_REF][START_REF] Rajas | The glucose-6 phosphatase gene is expressed in human and rat small intestine: regulation of expression in fasted and diabetic rats[END_REF].

G6Pase deficiency leads to accumulation of glucose-6 phosphate (G6P), glycogen, and triglycerides in the liver and kidneys of patients. This results in marked hepatomegaly and nephromegaly. In the liver, the long term complications of G6Pase deficiency include focal nodular hyperplasia and more often hepatocellular adenomas (HCA) with risk of malignant transformation [START_REF] Cassiman | An adult male patient with multiple adenomas and a hepatocellular carcinoma: Mild Glycogen Storage Disease type Ia[END_REF][START_REF] Franco | Hepatocellular carcinoma in glycogen storage disease type Ia: a case series[END_REF]. These tumours are often bleeding, multiple and non-encapsulated.

The current treatments to prevent hypoglycaemia are based on nocturnal nutrition [START_REF] Heller | Nutritional therapy for glycogen storage diseases[END_REF].

When HCA or liver failure occur, the management strategies are resection of the tumour and/or liver transplantation [START_REF] Labrune | Hepatocellular adenomas in glycogen storage disease type I and III: a series of 43 patients and review of the literature[END_REF][START_REF] Matern | Liver transplantation for glycogen storage disease types I, III, and IV[END_REF].

To gain further insights into the long-term mechanisms of the disease and to evaluate potential treatment strategies, an animal model with sufficient life expectancy is required.

Identification of the G6pc gene in 1993 has enabled a G6pc-knockout mouse model to be developed [START_REF] Lei | Glucose-6-phosphatase dependent substrate transport in the glycogen storage disease type-1a mouse[END_REF][START_REF] Shelly | Isolation of the gene for murine glucose-6-phosphatase, the enzyme deficient in glycogen storage disease type 1A[END_REF], as well as the evaluation of gene therapy approaches to correct G6pc deficiency ( [START_REF] Ghosh | Long-term correction of murine glycogen storage disease type Ia by recombinant adenoassociated virus-1-mediated gene transfer[END_REF][START_REF] Yiu | Complete Normalization of Hepatic G6PC Deficiency in Murine Glycogen Storage Disease Type Ia Using Gene Therapy[END_REF], see [START_REF] Chou | Gene therapy for type I glycogen storage diseases[END_REF] for review). However, long-term complications cannot be evaluated because even with continuous glucose therapy, the total G6pc knock-out mice rarely live to be over 3 months of age [START_REF] Lei | Glucose-6-phosphatase dependent substrate transport in the glycogen storage disease type-1a mouse[END_REF]. To investigate the long-term biology and pathogenesis of GSD-1a, we generated liver-specific G6pc -/-mice by an inducible CRE (CRE ERT2 ) strategy. We give a detailed characterization of this novel GSD1a mouse model over 18 months following inactivation of the G6pc gene in the liver.

Generation of B6.g6pc lox mice

The targeting vector used for the homologous recombination strategy is shown in Figure 1A and contained a 4.5 kb 5' homologous g6pc arm, a 3 kb 3' homologous g6pc arm and a neomycin resistance (Neo R ) cassette. Two loxP sites flank G6pc exon 3 and two Frt sites flank Neo R . We electroporated 129S2/SvPas mouse embryonic stem (ES) cells with the linearized construct. After selection, targeted clones were identified by PCR with external primers, with confirmation by Southern blotting with 5' and 3' external probes. Two positive ES clones were injected into C57BL/6J blastocysts. The male chimeras were mated with transgenic females expressing Flp recombinase to delete the Neo R cassette. Then, the B6;g6pc lox/w mice were backcrossed with the C57Bl/6J strain for 10 generations (Charles Rivers Laboratories).

Generation of liver-specific G6pc-null mice

The B6.g6pc lox/w mice were crossed with transgenic mice expressing the inducible CRE ERT2 recombinase under the control of the serum albumin promoter to confer its liverspecific expression (B6. SA creERT2/w ) [START_REF] Schuler | Efficient temporally controlled targeted somatic mutagenesis in hepatocytes of the mouse[END_REF]. B6. SA creERT2/w mice expressed, only in the liver, a recombinant Cre recombinase fused to a mutated ligand-binding domain of the estrogen receptor resulting in a tamoxifen-dependent Cre-recombinase [START_REF] Feil | Ligand-activated site-specific recombination in mice[END_REF][START_REF] Nagy | Cre recombinase: the universal reagent for genome tailoring[END_REF]. To induce the excision of G6pc exon 3, male adult (6 to 8 weeks old) B6.g6pc lox/lox .SA creERT2/w and B6.g6pc lox/w .SA creERT2/w mice were injected intraperitoneally once daily with 100 µl of tamoxifen (10 mg/ml, Sigma-Aldrich) on five consecutive days, to obtain L-g6pc -/-and L- Mouse genotypes were determined from genomic DNA by PCR, with specific primers (Table 1, see Figure 1). Genomic DNA was extracted from tissues in PCR Direct Lysis buffer (Euromedex).

Magnetic resonance imaging

As previously described [START_REF] Baboi | Magnetic resonance imaging follow-up of liver growth of neuroendocrine tumors in an experimental mouse model[END_REF][START_REF] Beuf | Multimodal imaging for the detection and characterization of liver lesions in a mouse model of neuroendocrine tumor[END_REF], we performed MRI scans on mice anesthetized by isoflurane inhalation. This was done using a horizontal Biospec-7T Bruker system (Bruker, Ettlingen, Germany) with a cylindrical volumetric coil of 32 mm internal diameter (Rapid Biomedical, Würzburg, Germany). The body temperature of the mice was maintained at 37°C by warm water circulation. A pressure sensor was placed on the abdomen to monitor the respiratory cycle and to synchronize image acquisition with respiratory motion using a

Trigger Unit HR V2.0 (Rapid Biomedical, Würzburg, Germany). Scout scans were performed to identify regions of interest and the graphical prescriptions for the following scan. Then, T2-weighted contrast images were acquired and analyzed in the axial plane based on a fat suppressed (FS) rapid acquisition with relaxation enhanced (RARE) sequence with the following parameters: echo time (TE) 40.4 ms, field of view (FOV) 30 x 30 mm 2 , matrix 256 x 192, 36 or 48 slices, slice thickness 0.5 mm, RARE factor =8.

As previously described [START_REF] Baboi | Magnetic resonance imaging follow-up of liver growth of neuroendocrine tumors in an experimental mouse model[END_REF], the sequence was synchronized with respiration using balanced acquisitions over several respiratory periods with an effective repetition time (TR) of about 6 sec. Reading of MR images to detect adenomas was performed by a trained radiologist who had over fifteen years of experience.

Metabolic studies

Blood was withdrawn from the tail vein and collected into EDTA or sodium fluorure/oxalate (anticoagulant). Plasma glucose, triglycerides, total cholesterol, uric acid and lactate concentrations were determined with Biomérieux colorimetric kits. Glucose-6 phosphate and glycogen determinations were carried out on frozen tissue homogenates as previously described [START_REF] Guignot | Mechanisms by which insulin, associated or not with glucose, may inhibit hepatic glucose production in the rat[END_REF][START_REF] Mithieux | Intrahepatic mechanisms underlying the effect of metformin in decreasing basal glucose production in rats fed a high-fat diet[END_REF].

Gene expression analysis

Mice were killed by cervical dislocation 6 h after food removal. The frozen tissues were sampled, and G6Pase and PEPCK activities determined as previously described [START_REF] Rajas | The glucose-6 phosphatase gene is expressed in human and rat small intestine: regulation of expression in fasted and diabetic rats[END_REF][START_REF] Daniele | Phosphatidylinositol 3-kinase translocates onto liver endoplasmic reticulum and may account for the inhibition of glucose-6-phosphatase during refeeding[END_REF][START_REF] Mithieux | Liver microsomal glucose-6phosphatase is competitively inhibited by the lipid products of phosphatidylinositol 3kinase[END_REF][START_REF] Rajas | Induction of PEPCK gene expression in insulinopenia in rat small intestine[END_REF]. Immunoblotting was carried out using purified anti-rat G6PC (use at 1:1,500) or anti-PEPCK (use at 1:5,000; Santa Cruz Biotechnology) antibodies [START_REF] Rajas | Immunocytochemical localization of glucose 6-phosphatase and cytosolic phosphoenolpyruvate carboxykinase in gluconeogenic tissues reveals unsuspected metabolic zonation[END_REF]. Total RNAs were isolated from tissues with TRIzol reagent (Invitrogen). Reverse transcription and real-time PCR were performed as previously described [START_REF] Pillot | Protein feeding promotes redistribution of endogenous glucose production to the kidney and potentiates its suppression by insulin[END_REF], using sequence-specific primers (Table 1). The mouse ribosomal protein mL19 transcript (Rpl19) was used as a reference.

Histological analysis

Tissues were fixed with 10% formalin and embedded in paraffin. Cross-sections (5 µm thick) were cut and stained with hematoxylin and eosin (H&E), periodic Acid Schiff (PAS)

and picrosirius red staining. The slides were examined on a Coolscope microscope (Nikon, Tokyo, Japan).

Statistical analyses

Results are reported as means ± SEM. The various groups were compared by Mann-Whitney tests. Differences were considered statistically significant if P<0.05.

RESULTS

Generation of a time-dependent and liver-specific deficiency of G6Pase in mice

We disrupted G6Pase activity in the liver, by time dependent (conditional) and tissuespecific invalidation of the G6pc gene, based on a CRE-lox strategy. As endogenous glucose production, especially from the liver, is critical during the neonatal period (because of the low content of glucose in milk) [START_REF] Chatelain | Development and regulation of glucose-6-phosphatase gene expression in rat liver, intestine, and kidney: in vivo and in vitro studies in cultured fetal hepatocytes[END_REF][START_REF] Girard | Adaptations of glucose and fatty acid metabolism during perinatal period and suckling-weaning transition[END_REF], we have induced gene deletion in adult mice.

Male B6.g6pc lox/w .SA creERT2/w and B6.g6pc lox/lox .SA creERT2/w mice (see genotyping in Figure 1B) were therefore treated with tamoxifen at 7-8 weeks of age. This resulted in the specific excision of exon 3 from both G6pc alleles, in the liver (Figure 1C). The liver-specific deletion of exon 3 in G6pc was confirmed at the mRNA level by RT-PCR on mRNA from the liver, kidney, and intestine of L-G6pc -/-and L-G6pc +/-mice. Truncated G6pc mRNA was only amplified in the liver from L-G6pc -/-and L-G6pc +/-mice (Figure 1D). Hepatic G6Pase activity was undetectable after one month of gene deletion (0.31±0.13 U/g of protein in L-G6pc -/-versus 68.0±2.1 U/g of protein in control mice) (Figure 2A). Liver G6Pase activity remained undetectable in L-G6pc-/-mice (0.30±0.08 U/g of protein in L-g6pc -/-) after 18 months, confirming the persistence of the G6pc inactivation. Western blot analyses confirmed the absence of the G6PC protein from the livers of L-G6pc -/-mice (Figure 2A). In heterozygous mice (L-G6pc +/-), G6Pase activity was only half of that in control mice (39.2±2.9U/g of protein in L-g6pc +/-) and was correlated with the amount of G6PC protein detected on western blots (Figure 2A). Transcription of truncated G6pc mRNA appeared to be up regulated, with three times more G6pc mRNA in the liver of L-G6pc -/-mice than in controls (Figure 2B). This implies that there is a compensatory increase in gene expression in response to the absence of G6PC protein. The expression of cytosolic phosphoenolpyruvate carboxykinase (PEPCK-c, Pck1) was unaltered (Figures 2C and2D).

the transporter subunit of G6Pase remained unchanged (Figure 2E). Expression of the ubiquitous (non gluconeogenic) glucose-6-phosphatase-related protein (G6PC3 or UGRP) was also unaffected (Figure 2F).

Metabolic effects of hepatic G6pc deficiency in mice

The male and female L-G6pc -/-mice showed similar survival rates to C57Bl/6J control (wt)

and L-G6pc +/-heterozygous mice (data not shown). However, in contrast to total G6pc -/- mice, which are reported to have growth retardation [START_REF] Lei | Glucose-6-phosphatase dependent substrate transport in the glycogen storage disease type-1a mouse[END_REF], the loss of G6Pase specifically in the liver during adulthood had only marginal effect, if any, on weight gain (Figure 3A).

Therefore, L-G6pc -/-mice are fully viable and do not require glucose therapy to survive [START_REF] Lei | Glucose-6-phosphatase dependent substrate transport in the glycogen storage disease type-1a mouse[END_REF]. After some hours of fasting, the blood glucose concentration in L-G6pc -/-mice could be lowered to about 3 mM (data not shown). This is consistent with the crucial role of hepatic glycogenolysis in endogenous glucose production during the early post-absorptive state. However, in the fed state, mean blood glucose concentration in L-G6pc -/-mice was similar to L-G6pc -/+ and control mice (Figure 3B). The blood glucose concentrations of L-G6pc -/+ mice were similar to those of control mice (around 150 mg/dL), even after 6h of fasting (data not shown). This is consistent with the absence of blood glucose problems and any pathological phenotype in heterozygous human patients [START_REF] Moses | Historical highlights and unsolved problems in glycogen storage disease type 1[END_REF].

Biochemical analyses revealed highly disturbed plasma parameters just after deletion of G6pc in the liver. After ten days, L-G6pc -/-mice had plasma triglyceride and uric acid concentrations that were three times as high as those in control mice as well as plasma cholesterol concentrations that were twice as high (Figures 3C-E). L-G6pc -/-mice also had a plasma lactic acid concentration that had increased by 50% compared with control mice (Figure 3F). These concentrations remained high one month after gene deletion, but they then normalized with time; all metabolic plasma parameters except for cholesterol were normalized after 18 months (Figures 3C-F).

LG6pc-/-mice develop hepatomegaly and steatosis

As in GSD-1a patients and total G6pc knock-out mice, loss of hepatic G6PC led to accumulation of hepatic glucose-6 phosphate (G6P), glycogen, and triglycerides, which resulted in hepatomegaly and steatosis (Figure 4). The livers of L-G6pc -/-mice were enlarged and pale (Figure 4A) and accounted for about 8% of total body mass, versus only 4% in control mice (Figure 4C). The deposition of glycogen increased between 10 days and one month after deletion, to reach about 50 mg of glycogen/g in L-G6pc -/-liver at 18 months (Figures 4E). Accumulation of triglycerides in L-G6pc -/-livers also increased with time (Figure 4F). Histological observations of L-G6Ppc -/-livers confirmed marked steatosis, with large hepatocytes containing large lipid vacuoles, mainly in the periportal area (Figure 5F-G). As a result, the portal space was small and distorted (Figure 5G). Marked glycogen accumulation was also observed by periodic acid-Schiff staining (Figure 5H). Picrosirius red staining of the L-G6pc -/-livers did not reveal any areas of fibrosis (Figure 5I). In contrast with total G6pc knock-out [START_REF] Lei | Glucose-6-phosphatase dependent substrate transport in the glycogen storage disease type-1a mouse[END_REF], the kidneys and intestine in L-G6pc -/-mice did not accumulate any glycogen. After 18 months, the weight (0.16±0.01g for L-G6pc -/-versus 0.15±0.1g for controls) and histological features of L-G6pc -/-kidneys were also completely normal (Figure 5J).

Late appearance of hepatocellular adenomas in L-G6pc -/-mice

Because of the short lifespan of total G6pc -/-mice in the absence of glucose or gene therapy treatment, there was no possibility of detecting HCA. However, in the present study, using MRI, we were able to evaluate the development of nodules in the liver of L-G6pc -/-mice. MRI was performed every three months, from 3 to 18 months after gene deletion. In less than 15% (3 out of 26) of L-G6PC -/-livers, small lesions could be detected as early as 9 months (data not shown). In about 30% (8 out of 26) of L-G6PC -/-mice, small nodules in the liver of about 1 mm diameter were first detected after one year (Figure 6A).

At 15-16 months, 40-50% of L-G6pc -/-mice had liver nodules between 1 and 3 mm diameter. At 18 months, all (n=17) L-G6pc -/-mice had developed liver nodules between 1 and 10 mm diameter (Figures 6 B andC). There was at least one, and up to 5 large nodules (>4-5 mm) in each liver at this time (Figure 6 6H). More than 20 biopsies of GSD1a livers (n=17 mice) were analyzed by immunohistochemistry (H&S staining). No portal spaces were seen within the nodules.

In most cases (80%), the hepatocytes within the nodule were large and swollen, with pale cytoplasms (Figures 6 G and 6I). Areas of necrosis and inflammation were seen within the nodules (Figure 6I). The tumour shown in Figure 6G proved to be an adenoma. In a few cases (about 20%), the hepatocytes within the nodules were small, suggesting the development of a "precarcinoma" lesion (Figures 6H and6J). Areas of necrosis and numerous foci of inflammation were also present within these nodules (data not shown). In no case, pericellular fibrosis was observed around hepatic nodules. Moreover, no atypia was noted in any of the nodules. No mice developed hepatocellular carcinoma in the course of this study.

DISCUSSION

The maintenance of blood glucose levels within a narrow range (about 1g/L) is a critical physiological function. Outside of the time of meals and during fasting, endogenous glucose production is wholly responsible for maintaining plasma glucose levels. Glucose can be produced only in the liver, kidney, and intestine, because these are the only organs known to express G6Pase, the key enzyme for endogenous glucose production (for review see [START_REF] Mithieux | A novel role for glucose 6-phosphatase in the small intestine in the control of glucose homeostasis[END_REF]). Among these organs, the liver has a special role because it is able to rapidly mobilize glucose stored under the form of glycogen. Although the liver is the main glucoseproducing organ under post-absorptive conditions, we hypothesized that the inactivation of G6pc in the liver only would not be lethal, because of compensatory glucose production by the kidneys and intestine. This type of compensation has already been reported during the anhepatic phase of liver transplantation in humans [START_REF] Battezzati | Nonhepatic glucose production in humans[END_REF], and has been suggested in mice with inactivation of Pck1 in the liver [START_REF] She | Mechanisms by which liver-specific PEPCK knockout mice preserve euglycemia during starvation[END_REF]. In line with our hypothesis, L-G6pc -/-mice are normoglycemic in the fed state. This strongly suggests that glucose release by the kidney and intestine is sufficient to exquisitely regulate blood glucose concentration in L-G6pc -/- mice when unrestrained food is available. Furthermore, L-G6pc -/-mice had a normal survival rate, which allowed us, for the first time, to evaluate the development of GSD1a pathology in the livers of the mice, up to 18 months of age.

As expected, the liver of L-G6pc -/-mice is rapidly enlarged, due to the accumulation of glucose-6 phosphate, glycogen, and triglycerides. In parallel, there is a rapid increase in plasma triglycerides, cholesterol, lactic, and uric acids, as also observed in human GSD1a patients and in total G6pc knock-out mice [START_REF] Moses | Historical highlights and unsolved problems in glycogen storage disease type 1[END_REF][START_REF] Ozen | Glycogen storage diseases: new perspectives[END_REF]. Together with the absence of kidney and gut deficiency and phenotype in L-G6pc -/-mice, this strongly suggests that the liver deficiency may be fully responsible for the deregulation of plasma parameters in GSD1a.

Whereas some liver parameters tended to worsen with time (e.g. liver TG content), others (blood parameters) progressively decreased towards control values (e.g. uricemia and lactatemia). This suggests that there are some compensatory mechanisms taking place over time to correct the primary defaults initiated by the liver deficiency. Interestingly, a comparable improvement of blood biochemical profile was also observed after some months in the total G6pc knock-out mice surviving thanks to pre-weaning management with glucose injection and post-weaning management with Nutragel and glucose added to the drinking water [START_REF] Salganik | A detailed characterization of the adult mouse model of glycogen storage disease Ia[END_REF]. However, as in LG6pc -/-mice, liver histology and glycogen accumulation did not improve with aging [START_REF] Salganik | A detailed characterization of the adult mouse model of glycogen storage disease Ia[END_REF]. Therefore, it is probable that a satisfying control of blood glucose could explain this amelioration (dependent either on exogenous supply or on compensation by extrahepatic endogenous sources).This is also in line with the observation that GSD1a pathology is better tolerated with patient aging, and that some of the metabolic plasma parameters have been reported to be at least partially corrected with time [START_REF] Moses | Historical highlights and unsolved problems in glycogen storage disease type 1[END_REF].

To detect the putative appearance of hepatic adenomas as soon as possible while preserving the living animal, we used MRI, a powerful imaging technique, which allowed us to identify hepatic nodules about 1 mm in diameter. The first nodules were detected at 9 months of age, and after 18 months all livers in L-G6pc -/-mice showed at least one largesized lesion. As in human GSD1a, the adenomas were multiple, non-encapsulated, and in about 20% of cases, showed features of pre-carcinomas. It should be noted that adenomas were first detected at 9 months, a time when several plasma parameters had been normalized. However, this late detection cannot exclude that the development of adenomas may have started much earlier but they were not large enough to be detected by MRI. However, it is important to note that the development of adenoma appeared rather lately, whilst liver steatosis tended to worsen (see Figure 4). In the same time, several plasma parameters were markedly corrected, suggesting that the development of HCA was dependent on the alteration of liver metabolism and not on the deregulation of the plasma parameters. Moreover, intrahepatic metabolic disturbances induced by G6Pase deficiency had the capacity to initiate adenoma formation in all of the livers studied.

Furthermore, even if hepatocellular carcinoma was not observed, a substantial proportion of the adenomas could be prone to develop into carcinomas.

In conclusion, we report the characterization of a novel animal model of GSD1a, which is exclusively targeted to the liver. We demonstrate that L-G6Ppc -/-mice have all the hepatic symptoms of the human pathology, including hepatomegaly, steatosis, glycogen overload.

L-G6pc -/-mice also have the expected dysregulations of plasma metabolites related to liver function such as triglycerides, cholesterol, lactic and uric acids. Our animal GSD1a model is perfectly viable without treatment and is normoglycaemic when food is available.

Because of this, for the first time we are able to report that the development of adenomas is bound to happen with time in GSD1a livers. In addition, a significant proportion of these adenomas exhibit the features of pre-carcinomas. Because of its viability, this new animal model will be extremely useful for a better understanding of the pathological mechanisms of GSD1a. It will also be of value in assessing the effects of food quality on delaying the deleterious consequences of the disease (e.g. adenomas). Our model should be able to help refine the different gene therapy approaches aimed at correcting G6Pase expression in the liver. Finally, in the near future, the CRE-lox approach that was used to generate the liver-deficient model, will allow us to assess the consequences of kidney or intestine pathology. This will be based on the use of a CRE recombinase expressed under the control of adapted gut or kidney gene promoters to specifically target the deficiency to either organ. between the G6pc w locus and the targeting vector in ES cells, Neo R was removed to obtain the G6pc lox allele through FLP recombination. Exon 3 of G6pc was specifically excised by CRE recombination (G6pc del ). Specific primers (Table 1) for genotyping were identified (p1, Histological analysis of two different tumours from L-G6pc -/-liver stained with H&S (G-J).

Panel I corresponds to a higher magnification of the adenoma showed on panel G and panel J corresponds to the tumour of panel H. Arrows in panels H and J point the edge of the tumour, revealing the absence of fibrous encapsulation.

Table 1

Oligodeoxyribonucleotide primer sequences for PCR (top) and qPCR (bottom). S = sense and AS= antisense sequences.

  E-F). Panels 6 A and B show the development of one nodule on MRI from 12 months (6A) through to 18 months (6B) following gene deletion. No obvious necrosis or significant haemorrhage was seen. The nodules were well demarcated from the surrounding liver, without a fibrous capsule (Figures 6G and

Figure 1 .

 1 Figure 1. Targeting strategy used to generate liver-specific g6pc null mice (L-g6pc -/-). (A) Scheme of the wild-type G6pc allele (G6pc w ), located on chromosome 11, and the

  p2 and p3). (B) PCR genotype analysis on the basis of tail DNA, using the p2 and p3 primers, which yielded products of 385 and 486 bp for the wild-type and floxed G6pc alleles, respectively. Specific primers were used to amplify a 320 bp Cre fragment. The expected sizes are shown on the left of the panel. (C) PCR genotype analysis on the basis of tail and liver DNA, using p1 and p3 primers. Fragments of 1189, 1029 and 595 bp correspond to the G6pc lox , G6pc w and G6pc del alleles, respectively. (D) Liver-specific excision of G6pc exon 3. RT-PCR analysis on liver, kidney or intestine mRNA, from L-g6pc - /-, L-g6pc +/-and L-g6pc +/+ (C57Bl/6J) mice, using G6pc exon 2 and exon 5 primers. A 761 bp DNA fragment is obtained from the wt allele and the deletion of exon 3 yields a 655 bp DNA fragment. Comparable data were obtained at 10 days or 18 months after tamoxifen treatment.

Figure 2 .

 2 Figure 2. Glucose-6-phosphatase is absent in the liver of liver-specific G6pc-null mice (L-G6pc -/-). (A and C) G6Pase and PEPCK-c activities were assayed in the liver of

Figure 3 .

 3 Figure 3. Body weight and plasmatic parameters of L-G6pc -/-mice. (A) Growth rates of L-G6pc -/-(black symbols) and wt (white symbols) male and female mice. Body weight

Figure 4 .

 4 Figure 4. Liver-specific G6pc null mice develop liver hepatomegaly and steatosis. (A) Livers of L-G6pc -/-and wt mice. (B) Liver weight, (C) Liver weight compared to the

Figure 5 .

 5 Figure 5. Histology and glycogen storage analyses of the liver and the kidneys of

Figure 6 .

 6 Figure 6. Development of liver nodules in L-G6pc -/-mice after one year of gene

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Figure 6

 6 Figure 5

ACKNOWLEDGMENTS

We would like to thank the Mouse Clinical Institute (Strasbourg, France) for invaluable help in generating the B6.g6pc lox mice. This work received financial support from the "Association Française contre les Myopathies" (Paris, France) and the "GIS-Institut des Maladies Rares" (Paris, France). We also thank Prof. Pierre Chambon and Dr. Daniel Metzger (Mouse Clinical Institute, Strasbourg, France) for generously providing B6. SA CreERT2 transgenic mice, Nicolas Gadot and Prof. Jean-Yves Scoazec (ANIPATH, Faculty of Medecine Laennec, Lyons, France) for carrying out the histology study, Angèle Chamousset and Jean-Michel Vicat for animal care (ALECS, Faculty of Medecine Laennec, IFR62 Lyon-Est, Lyons), Jean-Baptiste Langlois for MRI acquisitions (Animage Platform, Cermep, Lyons) and the members of the CECIL Platform (Faculty of Medecine Laennec, IFR62 Lyon-Est, Lyons). This work was supported by research grants from the "Agence Nationale de la Recherche" (ANR-07-MRAR-011-01) and the "Association Francophone des Glycogénoses".