Riko Koyama

Tiphaine Mannic
email: tiphaine.mannic@gmail.com

Jumpei Ito
email: jumpei.ito@kcl.ac.uk

Laurence Amar
email: laurence.amar@aphp.fr

Maria-Christina Zennaro
email: maria-christina.zennaro@inserm.fr

Michel Florian Rossier
email: michel.rossier@hopitalvs.ch

Andrés Maturana
email: maturana@agr.nagoya-u.ac.jp

Andrés Daniel Maturana

MicroRNA-204 Is Necessary for Aldosterone-Stimulated T-Type Calcium Channel Expression in Cardiomyocytes

Keywords: microRNA, T-channels, aldosterone, cardiomyocytes

Introduction

Mineralocorticoids, in particular aldosterone, are associated with the development of cardiac hypertrophy [START_REF] Duprez | Influence of arterial blood pressure and aldosterone on left ventricular hypertrophy in moderate essential hypertension[END_REF] and heart failure [START_REF] Swedberg | Hormones regulating cardiovascular function in patients with severe congestive heart failure and their relation to mortality. CONSENSUS Trial Study Group[END_REF]. Clinical trials have demonstrated that mineralocorticoid receptor (MR) antagonists, such as spironolactone, improve the cure rate of patients with heart failure [START_REF] Pitt | The effect of spironolactone on morbidity and mortality in patients with severe heart failure. Randomized Aldactone Evaluation Study Investigators[END_REF]. Patients with primary aldosteronism have a thicker left ventricular wall and a higher left ventricular mass index compared to patients with essential hypertension despite both groups having similar blood pressure [START_REF] Rossi | Changes in left ventricular anatomy and function in hypertension and primary aldosteronism[END_REF]. In addition to studies on patients, the link between hyperaldosteronism and heart failure progression has also been documented in experimental animals [START_REF] Bathgate-Siryk | Negative impact of β-arrestin-1 on post-myocardial infarction heart failure via cardiac and adrenal-dependent neurohormonal mechanisms[END_REF][START_REF] Lymperopoulos | Adrenal beta-arrestin 1 inhibition in vivo attenuates post-myocardial infarction progression to heart failure and adverse remodeling via reduction of circulating aldosterone levels[END_REF]. Aldosterone might therefore have a direct effect on cardiovascular tissue in addition to its hypertensive effect. Indeed, MR activation in cardiac myocytes results in the upregulation of numerous ion channels including voltage-gated Na + channels [START_REF] Boixel | Aldosterone increases voltage-gated sodium current in ventricular myocytes[END_REF], voltage-gated K + channels [START_REF] Perrier | Mineralocorticoid receptor antagonism prevents the electrical remodeling that precedes cellular hypertrophy after myocardial infarction[END_REF], voltage-gated Ca 2+ channels [START_REF] Benitah | Aldosterone upregulates Ca 2+ current in adult rat cardiomyocytes[END_REF][START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF][START_REF] Maturana | Role of the T-Type Calcium Channel Ca(V)3.2 in the Chronotropic Action of Corticosteroids in Isolated Rat Ventricular Myocytes[END_REF], and hyperpolarization-activated cyclic nucleotide-gated channels HCN4 [START_REF] Song | Aldosterone modulates I(f) current through gene expression in cultured neonatal rat ventricular myocytes[END_REF], which promotes hypertrophy development and heart failure [START_REF] Perrier | Mineralocorticoid receptor antagonism prevents the electrical remodeling that precedes cellular hypertrophy after myocardial infarction[END_REF][START_REF] Rossier | Contribution of Mineralocorticoid and Glucocorticoid Receptors to the Chronotropic and Hypertrophic Actions of Aldosterone in Neonatal Rat Ventricular Myocytes[END_REF][START_REF] Tomaselli | Electrophysiological remodeling in hypertrophy and heart failure[END_REF]. In particular, it has been shown that cardiac T-type voltage-gated Ca 2+ channel expression and activity are upregulated by aldosterone in neonatal rat cardiomyocytes [START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF][START_REF] Maturana | Role of the T-Type Calcium Channel Ca(V)3.2 in the Chronotropic Action of Corticosteroids in Isolated Rat Ventricular Myocytes[END_REF][START_REF] Ferron | T-type Ca2+ signalling regulates aldosterone-induced CREB activation and cell death through PP2A activation in neonatal cardiomyocytes[END_REF]. In the adult heart, T-type Ca 2+ channels drive the pacemaker depolarization in the sinoatrial node [START_REF] Hagiwara | Contribution of two types of calcium currents to the pacemaker potentials of rabbit sino-atrial node cells[END_REF][START_REF] Vassort | Role of T-type Ca2+ channels in the heart[END_REF][START_REF] Ono | Pathophysiological significance of T-type Ca2+ channels: Properties and functional roles of T-type Ca2+ channels in cardiac pacemaking[END_REF]. T-type Ca 2+ channel re-expression in the adult ventricle has been associated with the development of cardiac hypertrophy and heart failure [START_REF] Chiang | The Ca(v)3.2 T-Type Ca2+ Channel Is Required for Pressure Overload-Induced Cardiac Hypertrophy in Mice[END_REF][START_REF] Horiba | T-type Ca2+ channel blockers prevent cardiac cell hypertrophy through an inhibition of calcineurin-NFAT3 activation as well as L-type Ca2+ channel blockers[END_REF].

The regulatory mechanisms of T-type Ca 2+ channel expression by aldosterone require several glucocorticoid response elements (GREs) on the promoter of CACNA1G (Calcium Voltage-Gated Channel Subunit Alpha1 G), the gene encoding for CaV3.1, one of the two T-type Ca 2+ channels expressed in the heart. Mutations of GRE1 abolish CACNA1G transcriptional activation induced by aldosterone stimulation [START_REF] Benmohamed | Regulation of T-type Cav3.1 channels expression by synthetic glucocorticoid dexamethasone in neonatal cardiac myocytes[END_REF]. However, the regulatory mechanisms of the CACNA1H (Calcium Voltage-Gated Channel Subunit Alpha1 H) gene which encodes CaV3.2, the other cardiac T-type Ca 2+ channel, by aldosterone remain elusive. MicroRNAs (miRNAs) are single-stranded RNA molecules composed of about 22 non-coding nucleotides that regulate the translation of their target mRNAs [START_REF] Zardo | Transcriptional targeting by microRNA-Polycomb complexes A novel route in cell fate determination[END_REF]. The contribution of microRNA to cardiovascular development and diseases has been widely explored [START_REF] Hata | Functions of microRNAs in cardiovascular biology and disease[END_REF]. Conditional deletion of Dicer, which is an essential component in the miRNA biogenesis, causes sudden death, accompanied by cardiac remodeling in postnatal murine cardiomyocytes [START_REF] Martins | Conditional Dicer gene deletion in the postnatal myocardium provokes spontaneous cardiac remodeling[END_REF]. Targeted Dicer deletion in adult mice hearts provokes hypertrophy, cardiac dysfunction, and fetal gene reactivation [START_REF] Martins | Conditional Dicer gene deletion in the postnatal myocardium provokes spontaneous cardiac remodeling[END_REF]. Moreover, the expression of several miRNAs is altered in hypertrophied mice hearts [START_REF] Sayed | MicroRNAs play an essential role in the development of cardiac hypertrophy[END_REF].

The role of particular miRNAs in cardiac hypertrophy has also been investigated. Overexpression of miR-208a in the heart is sufficient to induce cardiac hypertrophic growth in mice, due to the inhibition of two hypertrophic negative regulators, thyroid hormone-associated protein 1 and myostatin [START_REF] Callis | MicroRNA-208a is a regulator of cardiac hypertrophy and conduction in mice[END_REF]. The expression of miR-23a is upregulated in hypertrophied cardiomyocytes [START_REF] Lin | miR-23a functions downstream of NFATc3 to regulate cardiac hypertrophy[END_REF]. MiR-23a is regulated by NFATc3 (Nuclear Factor of Activated T Cells 3) which mediates the calcineurin hypertrophic signal and targets the anti-hypertrophic protein muscle-specific ring finger protein 1 (MuRF1) [START_REF] Lin | miR-23a functions downstream of NFATc3 to regulate cardiac hypertrophy[END_REF].

Some studies have demonstrated the interplay between ion channels and miRNAs. For example, the expression of the type II inositol 1,4,5-trisphosphate receptor (IP3RII), an intracellular calcium channel located in the ER, is regulated by miR-133a, resulting in an altered calcium signaling in hypertrophied cardiomyocytes [START_REF] Drawnel | Mutual antagonism between IP3RII and miRNA-133a regulates calcium signals and cardiac hypertrophy[END_REF]. The inhibition of miR-133a in cardiomyocytes increases IP3RII levels, hypertrophic marker gene levels, and cell surface area [START_REF] Drawnel | Mutual antagonism between IP3RII and miRNA-133a regulates calcium signals and cardiac hypertrophy[END_REF]. Defining the interplay of miRNAs and ion channels might provide important insights into the regulatory mechanisms of cardiac hypertrophy.

Given the apparent contribution of miRNAs in cardiovascular diseases, we hypothesized that specific miRNAs could mediate some effects of aldosterone in cardiomyocytes, in particular, the expression of CaV3.1 and CaV3.2 T-type calcium channels.

In an initial study, we found that aldosterone stimulates the expression of several miRNAs, with miR-204 being the most upregulated one. The overexpression of miR-204 led to a marked increase in the spontaneous beating frequencies of isolated neonatal rat cardiomyocytes, mimicking our previously reported effect of aldosterone on the rate of spontaneous contractions. Concomitantly, miR-204, by itself, stimulated an increase in T-type calcium currents and mRNA expression of both CaV3.1 and CaV3.2. In addition, the protein expression of a known repressor of CaV3.2 [START_REF] Kuwahara | NRSF regulates the fetal cardiac gene program and maintains normal cardiac structure and function[END_REF], REST-NRSF (RE1-silencing transcription factor also known as neuron-restrictive silence factor), was found to be reduced upon overexpression of miR-204. The repression of miR-204 expression led to the suppression of an aldosterone-stimulated increase in T-type calcium currents and CaV3.1 and CaV3.2 mRNA expression. Taken together, these results uncover a mechanical pathway of CaV3.2 T-type calcium channel stimulated expression by aldosterone involving the inhibition of REST-NRSF by miR-204 upregulation in rat cardiomyocytes.

Results

Aldosterone Stimulates miR-204 Expression

Aldosterone has been shown to upregulate T-type Ca 2+ channel expression and activity in cardiomyocytes, as well as the cell spontaneous beating frequency [START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF][START_REF] Maturana | Role of the T-Type Calcium Channel Ca(V)3.2 in the Chronotropic Action of Corticosteroids in Isolated Rat Ventricular Myocytes[END_REF]. We hypothesized that some specific miRNAs might play a role in the aldosterone-stimulated increase of T-type Ca 2+ channel expression in ventricular cardiomyocytes. A microarray screening, allowing the simultaneous measurement of the expression levels of 2 × 381 miRNAs, was performed in naïve or aldosterone-stimulated (1 µM for 3 or 24 h) neonatal rat cardiomyocytes. A high, supra-physiological concentration of aldosterone was used throughout the present study in order to activate both MR and GR receptors [START_REF] Rossier | Contribution of Mineralocorticoid and Glucocorticoid Receptors to the Chronotropic and Hypertrophic Actions of Aldosterone in Neonatal Rat Ventricular Myocytes[END_REF]. The results showed that, under these conditions, the most upregulated miRNA by 3 h aldosterone was miR-204 (Figure 1A). In contrast to other miRNAs, miR-204 remained markedly elevated even after 24 h of stimulation with the hormone. We then precisely evaluated the miR-204 expression levels in aldosterone-treated cardiomyocytes by quantitative PCR. Aldosterone (1 µM for 24 h) significantly raised miR-204 expression levels in cardiomyocytes (Figure 1B). We then verified that the effect of aldosterone on miR-204 is specifically mediated by the activation of the mineralocorticoid receptor (MR). Spironolactone (10 µM for 24 h), a strong MR antagonist, prevented miR-204 upregulation by aldosterone in cardiomyocytes (Figure 1B). miR-204, by itself, stimulated an increase in T-type calcium currents and mRNA expression of both CaV3.1 and CaV3.2. In addition, the protein expression of a known repressor of CaV3.2 [START_REF] Kuwahara | NRSF regulates the fetal cardiac gene program and maintains normal cardiac structure and function[END_REF], REST-NRSF (RE1-silencing transcription factor also known as neuron-restrictive silence factor), was found to be reduced upon overexpression of miR-204. The repression of miR-204 expression led to the suppression of an aldosterone-stimulated increase in T-type calcium currents and CaV3.1 and CaV3.2 mRNA expression. Taken together, these results uncover a mechanical pathway of CaV3.2 T-type calcium channel stimulated expression by aldosterone involving the inhibition of REST-NRSF by miR-204 upregulation in rat cardiomyocytes.

Results

Aldosterone Stimulates miR-204 Expression

Aldosterone has been shown to upregulate T-type Ca 2+ channel expression and activity in cardiomyocytes, as well as the cell spontaneous beating frequency [START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF][START_REF] Maturana | Role of the T-Type Calcium Channel Ca(V)3.2 in the Chronotropic Action of Corticosteroids in Isolated Rat Ventricular Myocytes[END_REF]. We hypothesized that some specific miRNAs might play a role in the aldosterone-stimulated increase of T-type Ca 2+ channel expression in ventricular cardiomyocytes. A microarray screening, allowing the simultaneous measurement of the expression levels of 2 × 381 miRNAs, was performed in naïve or aldosteronestimulated (1 µM for 3 or 24 h) neonatal rat cardiomyocytes. A high, supra-physiological concentration of aldosterone was used throughout the present study in order to activate both MR and GR receptors [START_REF] Rossier | Contribution of Mineralocorticoid and Glucocorticoid Receptors to the Chronotropic and Hypertrophic Actions of Aldosterone in Neonatal Rat Ventricular Myocytes[END_REF]. The results showed that, under these conditions, the most upregulated miRNA by 3 h aldosterone was miR-204 (Figure 1A). In contrast to other miRNAs, miR-204 remained markedly elevated even after 24 h of stimulation with the hormone. We then precisely evaluated the miR-204 expression levels in aldosterone-treated cardiomyocytes by quantitative PCR. Aldosterone (1 µM for 24 h) significantly raised miR-204 expression levels in cardiomyocytes (Figure 1B). We then verified that the effect of aldosterone on miR-204 is specifically mediated by the activation of the mineralocorticoid receptor (MR). Spironolactone (10 µM for 24 h), a strong MR antagonist, prevented miR-204 upregulation by aldosterone in cardiomyocytes (Figure 1B).

These results, therefore, suggest that aldosterone stimulates an increase in miR204 expression in cardiomyocytes via MR activation. Total RNA was then collected, and miRNA content was analyzed by qPCR on a Taqman low-density array (TLDA) covering more than 300 murine miRNA targets. The response to aldosterone of each miRNA was expressed as fold increase, calculated versus the control amount determined in unstimulated cells. The graph shows the mean values from two experiments performed from two independent cell preparations (the arrow indicates results differing by more than 35% from one experiment to the other). The miRNAs displaying higher signal increases in response to 3 h of exposure to aldosterone are shown. (B) The graph shows the relative expression of miR-204 determined in control rat neonatal ventricular cardiomyocytes, and in cells treated for 24 h with aldosterone alone (1 µM), aldosterone and spironolactone (10 µM), or spironolactone alone. The miR-204 expression levels were determined by real-time quantitative PCR using specific primers and normalized to U6 SnRNA. The bars and error bars indicate the means + SEMs (n = 3). ** means p < 0.01 vs. control, and * p < 0.05. These results, therefore, suggest that aldosterone stimulates an increase in miR204 expression in cardiomyocytes via MR activation.

Chronotropic Action of miR-204 in Isolated Cardiomyocytes

We previously showed that aldosterone stimulation of isolated cardiomyocytes results in a marked increase of their spontaneous beating frequencies [START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF]. We thus tested whether miR-204 overexpression can mimic the effect of aldosterone after 24 h of transfection (Figure 2). As previously reported, stimulation of 1 µM aldosterone for 24 h significantly increases the spontaneous beating rate of cardiomyocytes as compared to the control condition. Neither the expression of GFP alone, nor GFP with a control miRNA, affected the beating rate. However, miR-204 overexpression significantly increased the beating rate of cardiomyocytes (Figure 2B). In these experiments, the presence of a plasmid coding for GFP allowed us to select only transfected cells; indeed, only green fluorescent cells were considered for measuring the beating frequency (Figure 2A). These results suggest that miR-204 plays a role in the chronotropic effect of aldosterone in cardiomyocytes.

Chronotropic Action of miR-204 in Isolated Cardiomyocytes

We previously showed that aldosterone stimulation of isolated cardiomyocytes results in a marked increase of their spontaneous beating frequencies [START_REF] Lalevee | Aldosterone increases T-type calcium channel expression and in vitro beating frequency in neonatal rat cardiomyocytes[END_REF]. We thus tested whether miR-204 overexpression can mimic the effect of aldosterone after 24 h of transfection (Figure 2). As previously reported, stimulation of 1 µM aldosterone for 24 h significantly increases the spontaneous beating rate of cardiomyocytes as compared to the control condition. Neither the expression of GFP alone, nor GFP with a control miRNA, affected the beating rate. However, miR-204 overexpression significantly increased the beating rate of cardiomyocytes (Figure 2B). In these experiments, the presence of a plasmid coding for GFP allowed us to select only transfected cells; indeed, only green fluorescent cells were considered for measuring the beating frequency (Figure 2A). These results suggest that miR-204 plays a role in the chronotropic effect of aldosterone in cardiomyocytes. coding for GFP either alone or in the presence of miR-204 or a control microRNA (miR-ctl). Images show representative cell fluorescence observed 24 h after cell transfection under various experimental conditions. (B) The cardiomyocyte beating frequency was determined by counting individual cell contractions under the microscope 24 h after cell transfection and/or incubation under several experimental conditions (indicated below the graph by the presence of a + sign). When cells were co-transfected with the GFP plasmid, only the beating frequency of fluorescent cells was determined. Data are means ± SEMs calculated from two independent cell preparations in which myocyte beating frequency was determined in 8 to 16 different fields for each condition. The statistical significance of the difference versus respective controls (* p < 0.05, ** p < 0.01, and *** p < 0.001) was assessed by unpaired Student's t-tests.

T-Type Ca 2+ Channel Expression in miR-204-Overexpressing Cardiomyocytes

We next tested whether miR-204 overexpression can mimic the effect of aldosterone on T-type Ca 2+ channel expression and activity. Overexpression of miR-204 was achieved by adenoviral transduction (Figure S1). The overexpression of miR-204 for 24 h resulted in a significant increase in CaV3.2 mRNA levels (Figure 3). The expression of the pore subunit of the T-type, CaV3.1, and L-type, CaV1.2, voltage-gated calcium channels were also affected by miR-204, but the effect in these cases remained below statistical significance. While the responses to aldosterone and miR-204 were quite similar on CaV1.2 and CaV3.1 expression, the hormone was much more efficient at inducing an elevation of CaV3.2, suggesting that aldosterone recruits additional mechanisms for increasing the expression of the latter channel.

T-Type Ca 2+ Channel Expression in miR-204-Overexpressing Cardiomyocytes

We next tested whether miR-204 overexpression can mimic the effect of aldosterone on T-type Ca 2+ channel expression and activity. Overexpression of miR-204 was achieved by adenoviral transduction (Figure S1). The overexpression of miR-204 for 24 h resulted in a significant increase in CaV3.2 mRNA levels (Figure 3). The expression of the pore subunit of the T-type, CaV3.1, and L-type, CaV1.2, voltage-gated calcium channels were also affected by miR-204, but the effect in these cases remained below statistical significance. While the responses to aldosterone and miR-204 were quite similar on CaV1.2 and CaV3.1 expression, the hormone was much more efficient at inducing an elevation of CaV3.2, suggesting that aldosterone recruits additional mechanisms for increasing the expression of the latter channel. Next, T-type Ca 2+ current densities were determined by analyzing the difference between total Ca 2+ currents measured from a holding potential at -90 mV and L-type specific Ca 2+ currents evoked from a holding potential at -40 mV in the whole cell configuration of the patch clamp technique. Aldosterone treatment (1 µM for 24 h) significantly and markedly increased both T-type and L-type Ca 2+ currents between -20 mV and +10 mV (Figure 4). The overexpression of miRNA-204 partially mimicked the effect of aldosterone, while cells infected with the luciferase gene displayed currents similar to those measured in control, naïve cells. Taken together, these results suggest that miR-204 increases the expression of the L-type channel subunit CaV1.2, as well as both of the T-type channel subunits, CaV3.1 and CaV3.2, downstream of MR activation in neonatal rat cardiomyocytes. They also suggest that Ca 2+ channel regulation in response to aldosterone requires a not yet identified miR-204 target protein that is expected to downregulate Ca 2+ channel expression.

MicroRNA-204 Is Necessary for the Aldosterone-Induced Ca 2+ Channel Upregulation.

In order to confirm that miR-204 expression is necessary to allow aldosterone to stimulate L-type and T-type Ca 2+ channel upregulation, we used an antisense RNA to specifically target miR-204 (named antagomir-204). Neonatal rat cardiomyocytes were transfected with antagomir-204 to block miR-204 expression induced by the hormone, while a control antisense RNA was used in control cells. After 24 h of transfection, cardiomyocytes were stimulated with aldosterone (1 µM) for an additional period of 24 h. The mRNA levels of L-type and both T-type Ca 2+ channel pore subunits, CaV1.2, CaV3.1, and, CaV3.2, were then measured by qPCR (Figure 5). Aldosterone (1 µM for 24 h) induced a significant increase in CaV3.2 mRNA levels in cells pretreated (37.5 nM for 24 h) with the control antisense RNA (Ctrl Ihn + Aldo) but this response was almost completely prevented in Taken together, these results suggest that miR-204 increases the expression of the L-type channel subunit CaV1.2, as well as both of the T-type channel subunits, CaV3.1 and CaV3.2, downstream of MR activation in neonatal rat cardiomyocytes. They also suggest that Ca 2+ channel regulation in response to aldosterone requires a not yet identified miR-204 target protein that is expected to downregulate Ca 2+ channel expression.

MicroRNA-204 Is Necessary for the Aldosterone-Induced Ca 2+ Channel Upregulation

In order to confirm that miR-204 expression is necessary to allow aldosterone to stimulate L-type and T-type Ca 2+ channel upregulation, we used an antisense RNA to specifically target miR-204 (named antagomir-204). Neonatal rat cardiomyocytes were transfected with antagomir-204 to block miR-204 expression induced by the hormone, while a control antisense RNA was used in control cells. After 24 h of transfection, cardiomyocytes were stimulated with aldosterone (1 µM) for an additional period of 24 h. The mRNA levels of L-type and both T-type Ca 2+ channel pore subunits, CaV1.2, CaV3.1, and, CaV3.2, were then measured by qPCR (Figure 5). Aldosterone (1 µM for 24 h) induced a significant increase in CaV3.2 mRNA levels in cells pretreated (37.5 nM for 24 h) with the control antisense RNA (Ctrl Ihn + Aldo) but this response was almost completely prevented in antagomiR-204-treated (37.5 nM for 24 h) cardiomyocytes (Inh + Aldo). The mRNA levels of CaV1.2 and CaV3.1 were slightly increased by aldosterone in control cells, without reaching statistical significance, and this increase was prevented when cardiomyocytes were treated with antagomir-204. Next, we tested whether the increase in aldosterone-stimulated T-type Ca 2+ currents was also affected by miR-204 inhibition (Figure 6). Aldosterone (1 µM for 24 h) induced a significant increase in L-type and T-type Ca 2+ current densities in cardiomyocytes pretreated with a control antisense RNA oligonucleotide (37.5 nM for 24 h). In contrast, in antagomir-204-pretreated (37.5 nM for 24 h) cardiomyocytes, aldosterone-induced stimulation was prevented on both L-type and T-type Ca 2+ currents (Figure 6). Cell pretreatment with antagomiR-204 or the control antisense RNA, in the absence of aldosterone, had no effect on the basal L-type and T-type current densities (Figure S2). Next, we tested whether the increase in aldosterone-stimulated T-type Ca 2+ currents was also affected by miR-204 inhibition (Figure 6). Aldosterone (1 µM for 24 h) induced a significant increase in L-type and T-type Ca 2+ current densities in cardiomyocytes pretreated with a control antisense RNA oligonucleotide (37.5 nM for 24 h). In contrast, in antagomir-204-pretreated (37.5 nM for 24 h) cardiomyocytes, aldosterone-induced stimulation was prevented on both L-type and T-type Ca 2+ currents (Figure 6). Cell pretreatment with antagomiR-204 or the control antisense RNA, in the absence of aldosterone, had no effect on the basal L-type and T-type current densities (Figure S2). These results thus demonstrate that miR-204 is required for the aldosterone-stimulated expression of the T-type Ca 2+ channels pore subunit CaV3.2.

MicroRNA-204 Mimics the Aldosterone-Induced Downregulation of NRSF

Because miRNAs promote the downregulation of their specific targets [START_REF] Hata | Functions of microRNAs in cardiovascular biology and disease[END_REF], we hypothesized that in order to promote increased T-channel expression upon aldosterone stimulation, miRNA204 must downregulate the expression of one or several negative transcriptional regulators of T-channels that are already expressed in resting cells and therefore prevent channel expression. REST-NRSF is a known transcriptional repressor that blocks the transcription and expression of the CaV3.2 T-type channel in cardiac cells [START_REF] Kuwahara | NRSF regulates the fetal cardiac gene program and maintains normal cardiac structure and function[END_REF]. We therefore tested whether REST-NRSF protein expression was affected These results thus demonstrate that miR-204 is required for the aldosterone-stimulated expression of the T-type Ca 2+ channels pore subunit CaV3.2.

MicroRNA-204 Mimics the Aldosterone-Induced Downregulation of NRSF

Because miRNAs promote the downregulation of their specific targets [START_REF] Hata | Functions of microRNAs in cardiovascular biology and disease[END_REF], we hypothesized that in order to promote increased T-channel expression upon aldosterone stimulation, miRNA204 must downregulate the expression of one or several negative transcriptional regulators of T-channels that are already expressed in resting cells and therefore prevent channel expression. REST-NRSF is a known transcriptional repressor that blocks the transcription and expression of the CaV3.2 T-type channel in cardiac cells [START_REF] Kuwahara | NRSF regulates the fetal cardiac gene program and maintains normal cardiac structure and function[END_REF]. We therefore tested whether REST-NRSF protein expression was affected by miR-204 (Figure 7A). Aldosterone (1 µM) stimulation resulted in a marked reduction of REST-NRSF expression. Similarly, cardiomyocytes overexpressing miR-204 also showed a reduced expression of the REST-NRSF protein. Interestingly, neither aldosterone or miR-204 affected the messenger RNA coding for REST-NRSF, a signature of protein translation regulation by miRNA (Figure 7B).

by miR-204 (Figure 7A). Aldosterone (1 µM) stimulation resulted in a marked reduction of REST-NRSF expression. Similarly, cardiomyocytes overexpressing miR-204 also showed a reduced expression of the REST-NRSF protein. Interestingly, neither aldosterone or miR-204 affected the messenger RNA coding for REST-NRSF, a signature of protein translation regulation by miRNA (Figure 7B).

These results thus strongly suggest that miR-204 targets REST-NRSF and reduces its expression, an effect resulting in the release of the chronic transcriptional repression of CaV3.2 by this protein.

Materials and Methods

Experimental Animals

The care and handling of rats for the present study were in conformity with the Guide for the Care and Use of Laboratory Animals published by the NIH (publication no. 85- [START_REF] Hata | Functions of microRNAs in cardiovascular biology and disease[END_REF]1996) and approved by the Animal Care and Use Committees of Nagoya University (Elucidating the mechnisms of gene expression in cardiac diseases, 2018031374, 30 March 2018).

Cell Culture, Stimulation, and Transfection

The hearts from sacrificed Wistar/ST three-day-old neonatal rats were removed after dissection. To isolate cardiomyocytes, cardiac muscle tissue was digested in 10 mL phosphate-buffered saline (PBS) containing 0.12% type 2 collagenase at 37 °C and pelleted by centrifugation at 950 rpm for 5 min. The cell pellet was then suspended in Dulbecco's modified Eagle's medium with 10% fetal bovine serum, 1% penicillin-streptomycin, and 100 nM 5-bromodeoxyuridine (BrDU) and washed five times. Cells were then plated on 10 cm dishes for 1 h to separate cardiomyocytes from fibroblasts. Finally, purified cardiomyocytes were cultured in Dulbecco's modified Eagle's medium with 10% fetal bovine serum, 1% penicillin-streptomycin, and 100 nM BrDU in 5% CO2 at 37 °C for 24 h. The culture media was changed to Dulbecco's modified Eagle's medium containing 1% penicillinstreptomycin, 100 nM BrDU, 1 µg/mL insulin, and 5 µg/mL transferrin.

After 24 h, cells were stimulated with 1 µM aldosterone and/or 10 µM spironolactone, infected with miR-204 adenovirus and short hairpins targeting luciferase (shLuc) adenovirus, or transfected with miR-204 inhibitor, luciferase siRNA, and 40 ng of plasmids containing GFP (Green Fluorescent Protein) using Lipofectamine 3000. Adenovirus for human miR-204 was purchased from ViGene (Rockville, MD, USA). Recombinant human adenoviruses-5 encoding shLuc were constructed, These results thus strongly suggest that miR-204 targets REST-NRSF and reduces its expression, an effect resulting in the release of the chronic transcriptional repression of CaV3.2 by this protein.

Materials and Methods

Experimental Animals

The care and handling of rats for the present study were in conformity with the Guide for the Care and Use of Laboratory Animals published by the NIH (publication no. 85- [START_REF] Hata | Functions of microRNAs in cardiovascular biology and disease[END_REF]1996) and approved by the Animal Care and Use Committees of Nagoya University (Elucidating the mechnisms of gene expression in cardiac diseases, 2018031374, 30 March 2018).

Cell Culture, Stimulation, and Transfection

The hearts from sacrificed Wistar/ST three-day-old neonatal rats were removed after dissection. To isolate cardiomyocytes, cardiac muscle tissue was digested in 10 mL phosphate-buffered saline (PBS) containing 0.12% type 2 collagenase at 37 • C and pelleted by centrifugation at 950 rpm for 5 min. The cell pellet was then suspended in Dulbecco's modified Eagle's medium with 10% fetal bovine serum, 1% penicillin-streptomycin, and 100 nM 5-bromodeoxyuridine (BrDU) and washed five times. Cells were then plated on 10 cm dishes for 1 h to separate cardiomyocytes from fibroblasts. Finally, purified cardiomyocytes were cultured in Dulbecco's modified Eagle's medium with 10% fetal bovine serum, 1% penicillin-streptomycin, and 100 nM BrDU in 5% CO 2 at 37 • C for 24 h. The culture media was changed to Dulbecco's modified Eagle's medium containing 1% penicillin-streptomycin, 100 nM BrDU, 1 µg/mL insulin, and 5 µg/mL transferrin.

After 24 h, cells were stimulated with 1 µM aldosterone and/or 10 µM spironolactone, infected with miR-204 adenovirus and short hairpins targeting luciferase (shLuc) adenovirus, or transfected with miR-204 inhibitor, luciferase siRNA, and 40 ng of plasmids containing GFP (Green Fluorescent Protein) using Lipofectamine 3000. Adenovirus for human miR-204 was purchased from ViGene (Rockville, MD, USA). Recombinant human adenoviruses-5 encoding shLuc were constructed, amplified, and purified as previously described [START_REF] Maturana | Role of the T-Type Calcium Channel Ca(V)3.2 in the Chronotropic Action of Corticosteroids in Isolated Rat Ventricular Myocytes[END_REF]. MiScript miRNA inhibitor was obtained from QIAGEN (Hilden, Germany). Luciferase siRNA (sense strand, 5 -CUUACGCUGAGUACUUCGATT-3 ; antisense strand, 5 -UCGAAGUACUCAGCGUAAGTT-3 ; where lower cases indicate DNA) was purchased from Gene Design Inc. (Osaka, Japan).

MicroRNA Preparation and Screening by Microarray

Total RNA was prepared using Trizol (Invitrogen, Carlsbad, CA, USA), and miRNA were isolated using an RNeasy MinElute CLEAN UP kit (Qiagen) or a Mirvana kit (Thermo Fisher Waltham, MA, USA). Global miRNA expression was then determined using TaqMan Low Density Arrays (TLDA) (Applied Biosystems, Foster, CA, USA), according to the manufacturer's protocols on an AB7900 platform. Total RNA from cardiomyocytes was extracted using TRIreagent (Sigma-Aldrich, St. Louis, MO, USA) according to the manufacturer's indications. Total RNA was reverse transcribed using ReverTra Ace qPCR RT Kit (Toyobo, Osaka, Japan). Real-time quantitative PCR was performed with the synthesized cDNA (50 ng of RNA contents), 250 nM primers, and THUNDERBIRD SYBR qPCR Mix (Toyobo, Osaka, Japan) at a final volume of 25 µL. Stratagene Mx3000P (Agilent Technologies, Santa Clara, CA, USA) was used for the PCR reaction. The mRNA levels were normalized to those coding for glyceraldehyde 3-phosphate dehydrogenase (GAPDH). The primer sequences are indicated in Table 1. MicroRNAs from cardiomyocytes and isolated mice hearts were extracted using mirVana miRNA Isolation Kit (Ambion, Thermo Fisher, Waltham, MA, USA) according to the manufacturer's indications. cDNAs were synthesized with 10 ng of total miRNAs using the Taqman miRNA reverse transcription kit (Applied Biosystems, Foster City, CA, USA). Real-time quantitative PCR was performed with synthesized cDNA (0.89 ng of total RNA contents), Taqman small RNA assay and Taqman universal PCR Master MixII (Applied Biosystems, Foster City, CA, USA) in a final volume of 20 µL. A 7300 real-time PCR system (Applied Biosystems, Foster City, CA, USA) was used for the PCR reaction. The miR204 levels were normalized to U6 SnRNA.

Protein Isolation and Western Blot

Whole cell proteins were isolated from neonatal cardiomyocytes using an ice-cold RIPA buffer (50 mM Tris-HCL, 150 mM NaCl, 5 mM EDTA, 1% Triton, 0.5% Na-deoxicolate) supplemented with complete protease inhibitor from (Roche, Basel, Switzerland). Protein concentrations were determined by BCA assay (Thermo Fisher Scientific, Walmath, MA, USA). Twenty micrograms of protein was loaded into a 7.5% SDS-polyacrylamide gel and separated by electrophoresis. Proteins were then transferred on polyvinylidene fluoride (PVDF) membrane. The PVDF membrane was immunoblotted with a rabbit polyclonal antibody targeting RESF-NRSF (1/1000) (Cell Signaling, Danvers, MA, USA) and a monoclonal antibody targeting GAPDH linked with a horseradish peroxidase (HRP) (Cell Signaling, Danvers, MA, USA) for detection. To detect NRSF, an anti-rabbit IgG coupled with HRP (GE Healthcare, Chicago, IL, USA) was used. Chemiluminescence was detected using an ECL detection kit (Millipore Corp., Burlington, MA, USA), and images were captured with a cooled-CCD camera of a LAS-1000 plus luminescent image analyzer (Fujifilm, Tokyo, Japan).

Patch Clamp

Cardiomyocytes plated on poly-L-lysine coated glass bottom dishes stimulated by aldosterone (1 µM), in the presence or absence of spironolactone (10 µM) and infected with adenovirus for miR-204 or shLuc, or transfected with miR-204 inhibitor, luciferase siRNA, and GFP plasmids.

Patch clamp micropipettes from borosilicate glass capillaries with 0.86 mm diameters (Harvard apparatus, Holliston, MA, USA) were fabricated using a puller PC-10 (Narishige, Tokyo, Japan). The snap of the micropipette tip was fire polished using a microforge MF-900 (Narishige, Tokyo, Japan). Cells were placed in a bath solution containing (in mM) 125 N-methyl-D-glucamine, 5 4-aminopyridine, 20 tetraethylammonium chloride (TEA Cl), 2 CaCl 2 , 10 D -glucose, and 10 HEPES (pH 7.4 with TEAOH). A dish was placed on an inverted microscope Olympus IX71 (Olympus, Tokyo, Japan). The micropipette was filled with a pipette solution containing (in mM) 130 CsCl, 10 EGTA, 25 HEPES, 3 Mg-ATP, and 0.4 Li-GTP (pH 7.2 adjusted with CsOH).

After formation of the whole-cell patch clamp configuration, Ca 2+ currents were measured in the voltage-clamp mode using an Axopatch 200B amplifier (Molecular Devices, St. Jose, CA, USA). The cell was held at a membrane potential of -80 mV and depolarized by 10 mV voltage steps up to +80 mV to measure T-and L-type Ca 2+ currents. Then, the cell was held at a membrane potential of -40 mV and depolarized by 10 mV voltage steps from -60 mV to 70 mV to measure isolated L-type currents (after T channel inactivation). The leak was subtracted automatically by Clamp 10 software (Molecular Devices, St. Jose, CA, USA).

Data Analysis

Statistical analysis was performed by one-way ANOVA followed with the Turkey-Kramer's test or by unpaired Student's t-tests. Data are presented as means ± SEMs (* means p < 0.05, ** p < 0.01, and *** p < 0.001 compared to control samples).

Discussion

We found here that miR-204 is the main microRNA upregulated by aldosterone in cardiomyocytes. The increase of miR-204 expression occurs downstream of MR activation. In turn, our results suggest that miR-204 contributes to the increased expression levels of T-type Ca 2+ channels and, therefore, to the spontaneous cardiomyocyte beating frequency. In the heart, multiple microRNAs regulate the expression of ion channels that contribute to the development of cardiac pathologies, such as arrhythmias and atrial fibrillation. MicroRNAs can directly target ion channel transcripts and downregulate their expression. The HCN4 channel is a target of miR-423-5p in the heart sinus node [START_REF] Souza | Targeting miR-423-5p Reverses Exercise Training-Induced HCN4 Channel Remodeling and Sinus Bradycardia[END_REF]. Both miR-328 and miR-21 were found to downregulate CANA1C and CACNB2 genes, which encoding, respectively, for the pore and for an auxiliary subunit of the L-type voltage-gated Ca 2+ channel [START_REF] Barana | Chronic atrial fibrillation increases microRNA-21 in human atrial myocytes decreasing L-type calcium current[END_REF][START_REF] Lu | MicroRNA-328 contributes to adverse electrical remodeling in atrial fibrillation[END_REF]. In contrast, miR-219 promotes increased expression of SCN5A, encoding for the pore subunit of the voltage-gated sodium channel, probably through indirect targeting of the SCN5A transcripts [START_REF] Daimi | Regulation of SCN5A by microRNAs: MiR-219 modulates SCN5A transcript expression and the effects of flecainide intoxication in mice[END_REF]. Our results show that miR-204 promotes the increased expression of both CACNA1G and CACNA1H mRNA, and, in this way, upregulates T-type Ca 2+ currents. A similar effect has been observed for CACNA1C and L-type currents, although to a lesser extent. Therefore, miR-204 apparently plays a role downstream of MR activation for the re-expression of T-type channels. We hypothesized that miR-204 does not directly target T-channel encoding genes, but probably, some putative protein that negatively regulates channel expression. Recent studies have shown that miR-204 regulates the expression of transcription factors, such as forkhead box C1 (FOXC1) and myeloid ecotropic viral integration site 1 homolog 2 (MEIS2), which are both transcription factors involved in the eye development [START_REF] Paylakhi | FOXC1 in human trabecular meshwork cells is involved in regulatory pathway that includes miR-204, MEIS2, and ITG β1[END_REF]. In pancreatic beta cells, miR-204 targets the insulin transcription factor MAFA [START_REF] Xu | Thioredoxin-interacting protein regulates insulin transcription through microRNA-204[END_REF]. The increased expression of T-type calcium channel in response to miR-204 suggests that the miRNA targets some intermediate negative transcriptional regulator. Indeed, our results show that aldosterone and miR-204 overexpression inhibit the expression of REST-NRSF protein, a transcriptional repressor known to target CaV3.2 expression in cardiomyocytes [START_REF] Kuwahara | NRSF regulates the fetal cardiac gene program and maintains normal cardiac structure and function[END_REF]. This protein thus appears to be a relevant candidate for linking miR-204 increase and T-channel upregulation. However, the possibility that other transcriptional regulators, also playing a role in T-channel upregulation, might be targeted by miR-204 remains to be tested.

In a recent study performed in vivo, aldosterone was found to stimulate the expression of multiple miRNAs, miR-21 being the most upregulated in the left ventricle of the heart [START_REF] Ball | Role and Regulation of MicroRNAs in Aldosterone-Mediated Cardiac Injury and Dysfunction in Male Rats[END_REF]. This difference from our present study might be due to the duration of aldosterone stimulation and the presence of other cell types in the left ventricle of the heart in vivo (fibroblasts and macrophages). Indeed, in the present study, the stimulation was acute, for 3 to 24 h, whereas the stimulation performed in vivo in the study of Ball et al. was chronic, with treatments occurring over two weeks. Aldosterone might stimulate a variety of microRNAs at different times that contribute to the dynamic cardiomyocyte response.

MicroRNAs also play a role in the development of the heart. In particular, miR-204 has been shown to promote the proliferation and differentiation of human-derived cardiomyocyte progenitor cells [START_REF] Xiao | MicroRNA-204 is required for differentiation of human-derived cardiomyocyte progenitor cells[END_REF]. In vitro, miR-204 also stimulates the proliferation of rat neonatal and adult cardiomyocytes. In addition, the overexpression of miR-204, specifically in the heart ventricles of mice, has been shown to lead to heart hypertrophy [START_REF] Liang | miRNA-204 drives cardiomyocyte proliferation via targeting Jarid2[END_REF].

Our results suggest that miR-204 is upregulated downstream of MR upon aldosterone stimulation of cardiomyocytes (Figure 1B). The effect of aldosterone was prevented by spironolactone, an MR antagonist, suggesting that miR-204 upregulation is induced via MR activation in aldosterone-stimulated cardiomyocytes. We cannot exclude the involvement of additional receptors, in particular, the glucocorticoid receptor. Further studies are required to confirm the exclusivity of MR in the activation of miR-204. It would be also worthwhile to determine whether the miR-204 promoter, located on TRPM3 (Transient Pore Receptor Member 3) gene [START_REF] Hall | TRPM3 and miR-204 Establish a Regulatory Circuit that Controls Oncogenic Autophagy in Clear Cell Renal Cell Carcinoma[END_REF], has specific steroid hormone-response elements, which MR binds to for the regulation of gene transcription [START_REF] Bienvenu | Mineralocorticoid receptors and the heart, multiple cell types and multiple mechanisms: A focus on the cardiomyocyte[END_REF].

In conclusion, our study strongly suggests that miR-204 expression in ventricular cardiomyocytes is necessary for aldosterone-induced T-channel upregulation, leading to an increase in low voltage-activated Ca 2+ currents and therefore, to the acceleration of ventricular cell spontaneous beatings. Because this chronotropic action of aldosterone is recognized as a mechanism that possibly leads to ventricular electrical dysfunction, targeting miR-204 could provide new insights into the prevention of cardiac arrhythmia and heart failure.

Figure 1 .

 1 Figure 1. MicroRNA regulation by aldosterone in rat neonate ventricular cardiomyocytes. (A) Isolated ventricular cardiomyocytes were incubated for 3 h or 24 h with or without aldosterone (1000 nmol/L). Total RNA was then collected, and miRNA content was analyzed by qPCR on a Taqman low-density

Figure 1 .

 1 Figure 1. MicroRNA regulation by aldosterone in rat neonate ventricular cardiomyocytes. (A) Isolated ventricular cardiomyocytes were incubated for 3 h or 24 h with or without aldosterone (1000 nmol/L).

Figure 2 .

 2 Figure 2. MiR-204 mimics the chronotropic action of aldosterone in cardiomyocytes. (A) Neonate rat ventricular cardiomyocytes in culture were electroporated and immediately transfected with a plasmid coding for GFP either alone or in the presence of miR-204 or a control microRNA (miR-ctl). Images show representative cell fluorescence observed 24 h after cell transfection under various experimental conditions. (B) The cardiomyocyte beating frequency was determined by counting individual cell contractions under the microscope 24 h after cell transfection and/or incubation under several experimental conditions (indicated below the graph by the presence of a + sign). When cells

Figure 2 .

 2 Figure 2. MiR-204 mimics the chronotropic action of aldosterone in cardiomyocytes. (A) Neonate rat ventricular cardiomyocytes in culture were electroporated and immediately transfected with a plasmid

Figure 3 .

 3 Figure 3. Overexpression of miR-204 increases the expression of calcium channel alpha1 subunits. The mRNA levels for L-type and T-type Ca 2+ channel pore subunits CaV1.2, CaV3.1, and CaV3.2 was determined in control cells and in aldosterone (aldo, 1 µM for 24 h)-treated, miR-204 adenovirusinfected (AdmiR-204), or shLuc adenovirus-infected cardiomyocytes (NC). Bars and error bars indicate the means + SEMs (n = 3). * means p < 0.05 vs. control. Next, T-type Ca 2+ current densities were determined by analyzing the difference between total Ca 2+ currents measured from a holding potential at -90 mV and L-type specific Ca 2+ currents evoked from a holding potential at -40 mV in the whole cell configuration of the patch clamp technique. Aldosterone treatment (1 µM for 24 h) significantly and markedly increased both T-type and L-type

Figure 3 .

 3 Figure 3. Overexpression of miR-204 increases the expression of calcium channel alpha1 subunits. The mRNA levels for L-type and T-type Ca 2+ channel pore subunits CaV1.2, CaV3.1, and CaV3.2 was determined in control cells and in aldosterone (aldo, 1 µM for 24 h)-treated, miR-204 adenovirus-infected (AdmiR-204), or shLuc adenovirus-infected cardiomyocytes (NC). Bars and error bars indicate the means + SEMs (n = 3). * means p < 0.05 vs. control.

  Int. J. Mol. Sci. 2018, 19, x FOR PEER REVIEW 6 of 14

Figure 4 .

 4 Figure 4. T-type and L-type Ca 2+ currents are increased upon miR-204 overexpression. The graphs show the current density-voltage relationship obtained from whole cell patch clamp experiments in control (Ctrl), aldosterone (1 µM for 24 h)-treated (Aldo), miR-204 adenovirus-infected (Admir-204), and luc-adenovirus-infected cardiomyocytes (Ad-luc) obtained from holding potential at -90 mV (upper panel), reflecting the total (L-type and T-type) calcium current, or at -40 mV (middle panel), reflecting L-type current. The difference plots, reflecting T-type Ca 2+ currents, are shown in the lower panel. Traces are representative examples of current recordings at -20 mV depolarization steps. Scale bars are 100 pA (vertical) and 25 ms (horizontal). Data are means ± SEMs (n = 11-16). The bar graph represents the means + SEMs of the T current density recorded at -10 mV in control (1), aldosterone (1 µM for 24 h)-treated (2), miR-204 adenovirus-infected (3), and luc-adenovirus-infected (4) cardiomyocytes (* p < 0.05).

Figure 4 .

 4 Figure 4. T-type and L-type Ca 2+ currents are increased upon miR-204 overexpression. The graphs show the current density-voltage relationship obtained from whole cell patch clamp experiments in control (Ctrl), aldosterone (1 µM for 24 h)-treated (Aldo), miR-204 adenovirus-infected (Admir-204), and luc-adenovirus-infected cardiomyocytes (Ad-luc) obtained from holding potential at -90 mV (upper panel), reflecting the total (L-type and T-type) calcium current, or at -40 mV (middle panel), reflecting L-type current. The difference plots, reflecting T-type Ca 2+ currents, are shown in the lower panel. Traces are representative examples of current recordings at -20 mV depolarization steps. Scale bars are 100 pA (vertical) and 25 ms (horizontal). Data are means ± SEMs (n = 11-16). The bar graph represents the means + SEMs of the T current density recorded at -10 mV in control (1), aldosterone (1 µM for 24 h)-treated (2), miR-204 adenovirus-infected (3), and luc-adenovirus-infected (4) cardiomyocytes (* p < 0.05).

  Int. J. Mol. Sci. 2018, 19, x FOR PEER REVIEW 7 of 14 antagomiR-204-treated (37.5 nM for 24 h) cardiomyocytes (Inh + Aldo). The mRNA levels of CaV1.2 and CaV3.1 were slightly increased by aldosterone in control cells, without reaching statistical significance, and this increase was prevented when cardiomyocytes were treated with antagomir-204.

Figure 5 .

 5 Figure 5. Inhibition of miR-204 expression prevents aldosterone-induced increase of calcium channel α1 subunits. Graphs show mRNA levels for L-type and T-type Ca 2+ channel pore subunits-α1C, α1G, and α1H-in control cells (Ctrl), as well as in antagomiR-204 inhibitor (37.5 nM for 24 h)-transfected (Inh), antgomiR-204 inhibitor (37.5 nM for 24 h)-transfected + aldosterone (1 µM for 24 h)-stimulated (Inh+Aldo), luciferase siRNA (37.5 nM for 24 h)-transfected (Ctrl Inh), and luciferase siRNA (37.5 nM for 24 h)-transfected + aldosterone (1 µM for 24 h)-stimulated (Ctrl Inh + Aldo) cardiomyocytes. The bars and error bars indicate the means ± SEMs (n = 3). ** means p < 0.01 vs. control.

Figure 5 .

 5 Figure 5. Inhibition of miR-204 expression prevents aldosterone-induced increase of calcium channel α1 subunits. Graphs show mRNA levels for L-type and T-type Ca 2+ channel pore subunits-α1C, α1G, and α1H-in control cells (Ctrl), as well as in antagomiR-204 inhibitor (37.5 nM for 24 h)-transfected (Inh), antgomiR-204 inhibitor (37.5 nM for 24 h)-transfected + aldosterone (1 µM for 24 h)-stimulated (Inh+Aldo), luciferase siRNA (37.5 nM for 24 h)-transfected (Ctrl Inh), and luciferase siRNA (37.5 nM for 24 h)-transfected + aldosterone (1 µM for 24 h)-stimulated (Ctrl Inh + Aldo) cardiomyocytes. The bars and error bars indicate the means ± SEMs (n = 3). ** means p < 0.01 vs. control.

Figure 6 .

 6 Figure 6. Inhibition of miR-204 expression prevents an aldosterone-induced increase of Ca 2+ currents. Calcium currents were recorded in the whole cell configuration of the patch clamp technique in cardiomyocytes transfected (37.5 nM for 24 h) either with the antagomir targeting miR-204 (miR204Inh + Aldo) or with luciferase siRNA (Ctrl Inh + Aldo) and stimulated for 24 h with aldosterone (1 µM). Control cells (Ctrl) were naive (non-transfected) and resting (unstimulated) cardiomyocytes. The graphs show the current density-voltage relationship recorded from a holding potential at -90 mV (upper panel), reflecting the total (L-type and T-type) calcium current, or at -40 mV (middle panel), reflecting the L-type current. The difference plots that specifically reflect T-type Ca 2+ currents are shown in the lower panel. The traces shown on the left are representative examples of recordings at -10 mV depolarization steps. Data are means + SEMs from 15 (Ctrl) or 12 (Ctrl Inh or miR204Ihn) independent cells. The bar graph represents the means and SEMs of T current density recorded at -10 mV in control cells (1), control inhibitor plus aldosterone (2), and antagomir targeting miR-204 plus aldosterone (3) (* p < 0.05).

Figure 6 .

 6 Figure 6. Inhibition of miR-204 expression prevents an aldosterone-induced increase of Ca 2+ currents. Calcium currents were recorded in the whole cell configuration of the patch clamp technique in cardiomyocytes transfected (37.5 nM for 24 h) either with the antagomir targeting miR-204 (miR204Inh + Aldo) or with luciferase siRNA (Ctrl Inh + Aldo) and stimulated for 24 h with aldosterone (1 µM). Control cells (Ctrl) were naive (non-transfected) and resting (unstimulated) cardiomyocytes. The graphs show the current density-voltage relationship recorded from a holding potential at -90 mV (upper panel), reflecting the total (L-type and T-type) calcium current, or at -40 mV (middle panel), reflecting the L-type current. The difference plots that specifically reflect T-type Ca 2+ currents are shown in the lower panel. The traces shown on the left are representative examples of recordings at -10 mV depolarization steps. Data are means + SEMs from 15 (Ctrl) or 12 (Ctrl Inh or miR204Ihn) independent cells. The bar graph represents the means and SEMs of T current density recorded at -10 mV in control cells (1), control inhibitor plus aldosterone (2), and antagomir targeting miR-204 plus aldosterone (3) (* p < 0.05).

Figure 7 .

 7 Figure 7. Aldosterone stimulation and miR-204 overexpression reduce REST-NRSF protein expression. (A) The pictures represent immunoblotting which shows the expression of REST-NRSF (upper panel) and GAPDH (lower panel) detected with specific antibodies. Cardiomyocytes were incubated for 24 h with either 1 µM aldosterone (Aldo) or ethanol (1 µL), or infected with miR-204 adenovirus (AdmiR204), or shLuc adenovirus (AdLuc) for the same period. The picture is representative of three experiments performed from three different cell preparations. (B) The bar graph shows the mRNA expression level of the transcriptional repressor NRSF determined by RT-qPCR in control, aldosterone (1 µM), miR204 adenovirus infected or shLuc adenovirus infected cardiomyocytes. The bars and error bars indicate the means + SEMs (n = 3).

Figure 7 .

 7 Figure 7. Aldosterone stimulation and miR-204 overexpression reduce REST-NRSF protein expression. (A) The pictures represent immunoblotting which shows the expression of REST-NRSF (upper panel) and GAPDH (lower panel) detected with specific antibodies. Cardiomyocytes were incubated for 24 h with either 1 µM aldosterone (Aldo) or ethanol (1 µL), or infected with miR-204 adenovirus (AdmiR204), or shLuc adenovirus (AdLuc) for the same period. The picture is representative of three experiments performed from three different cell preparations. (B) The bar graph shows the mRNA expression level of the transcriptional repressor NRSF determined by RT-qPCR in control, aldosterone (1 µM), miR204 adenovirus infected or shLuc adenovirus infected cardiomyocytes. The bars and error bars indicate the means + SEMs (n = 3).

3. 4 .

 4 RNA Isolation and Real-Time Quantitative Reverse Transcription (RT)-Polymerase Chain Reaction (PCR)

  the sequences of the primer pairs used for the real-time quantitative PCR experiments in the present study. Fw: forward primer Rev: reverse primer.

Supplementary Materials:

  It can be found in the following link: http://www.mdpi.com/1422-0067/19/10/ 2941/s1. Author Contributions: All authors contributed to the work of the present study. R.K., T.M., I.J., L.A., M.C.Z., M.F.R., and A.D.M. performed experiments, analyzed and interpreted data. R.K., A.D.M., and M.F.R. drafted the manuscript. M.F.R., M.C.Z., and A.D.M. reviewed and edited the manuscript. M.F.R. and A.D.M. conceptualized the study.

  The graph shows the relative expression of miR-204 determined in control rat neonatal ventricular cardiomyocytes, and in cells treated for 24 h with aldosterone alone (1 µM), aldosterone and spironolactone (10 µM), or spironolactone alone. The miR-204 expression levels were determined by real-time quantitative PCR using specific primers and normalized to U6 SnRNA. The bars and error bars indicate the means + SEMs (n = 3). ** means p < 0.01 vs. control, and * p < 0.05.

	Int. J. Mol. Sci. 2018, 19, x FOR PEER REVIEW	4 of 14
	array (TLDA) covering more than 300 murine miRNA targets. The response to aldosterone of each
	miRNA was expressed as fold increase, calculated versus the control amount determined in
	unstimulated cells. The graph shows the mean values from two experiments performed from two
	independent cell preparations (the arrow indicates results differing by more than 35% from one
	experiment to the other). The miRNAs displaying higher signal increases in response to 3 h of
	exposure to aldosterone are shown. (B)

Table 1 .

 1 Primers Used in Real-Time Quantitative PCR.

Int. J. Mol. Sci. 2018, 19, x FOR PEER REVIEW

Funding: M.F.R. is recipient of a grant from the Fondation Endocrinologie (Geneva, Switzerland).

Conflicts of Interest:

The authors declare no conflict of interest.


Lledos Benjamin
email: benjamin.lledos@math.univ-toulouse.fr

A UNIQUENESS RESULT FOR A TRANSLATION INVARIANT PROBLEM IN THE CALCULUS OF VARIATIONS

We present a uniqueness result of uniformly continuous solutions for a general minimization problem in the Calculus of Variations. We minimize the functional I λ (u) := Ω φ(∇u)+λu with φ a convex but not necessarily strictly convex function, Ω an open set of R N with N ∈ N * and λ ∈ R. The proof is based on the two following main points : the functional I λ is invariant under translations and we assume that the function φ is not affine on any non-empty open set. This provides a shorter proof and/or an extension for some already known uniqueness results for functionals of the type I λ that are presented in the article.

Mathematics Subject Classification: 35A02, 49N99.

1. Introduction 1.1. Presentation of the problem. We study a uniqueness result for a non-strictly convex problem in the calculus of variations in dimension N ≥ 1. We consider functionals of the following form: Here, ψ : R N → R is a continuous function in W 1,1 loc (R N ) and E(Ω, ψ) is a subset of W 1,1 ψ (Ω), namely the set of functions in W 1,1 (Ω) such that the extension outside Ω by ψ : R N → R is in W 1,1 loc (R N ). We work with the set E(Ω, ψ) to cover many situations: in the following E(Ω, ψ) can stand for W 1,1 ψ (Ω), W 1,p ψ (Ω), W 1,p ψ (Ω) ∩ C 0 (Ω), C 0,1 ψ (Ω) or {v ∈ W 1,1 ψ (Ω), ∇v(x) ∈ K for a.e. x ∈ Ω} with K a convex set.

We assume that I λ is well defined on E(Ω, ψ) with values in R ∪ {+∞} and that there exists v ∈ E(Ω, ψ) such that I λ (v) < +∞.

We point out that φ is not necessarily strictly convex. Hence, no general result guarantees uniqueness. Another remarkable feature of the main result Theorem 1.2 is that φ(z) can be equal to +∞ for some z ∈ R N . Hence, this framework covers also the case when we put a pointwise constraint on the gradient. In the situations studied in [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] and [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF], P λ is the convexification of a non-convex problem that arises in shape optimization. In these cases (see also [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF]) uniqueness for P λ can be useful to prove existence or non-existence of a minimizer for related non-convex problems.

1.2. Main result. In order to present the main result of this paper we introduce the following sets: Definition 1.1. We say that a convex set F ∈ R N is an affine part of φ if ∀z, ξ ∈ F we have:

φ(z) < +∞, φ(ξ) < +∞ and φ z + ξ 2 = φ(z) + φ(ξ) 2 .
The dimension of F is the dimension of the affine space generated by F .

We make the two following assumptions on the set E(Ω, ψ):

(A1) The set E(Ω, ψ) is invariant under translation: if v ∈ E(Ω, ψ) then v(•) = v(• + τ ) + α ∈ E Ω τ , ψ(• + τ ) + α with Ω τ = Ω -τ . (A2) If u ∈ E(Ω, ψ) and v ∈ E(V, u) with V a non empty open set in Ω, then the extension of v by u is in E(Ω, ψ).
We make one assumption on φ: (A3) The dimensions of the affine parts of φ are at most N -1.

We now present the main theorem of this article: Theorem 1.2. Under those three assumptions, the problem P λ admits at most one uniformly continuous minimizer.

The proof of Theorem 1.2 which is relatively short and simple, unifies results that are described in Section 2. For some of them it is even an extension to higher dimensions or to a more general set Ω.

The first two assumptions (A1) and (A2) are relatively standard. For instance, they are satisfied when

E(Ω, ψ) is W 1,1 ψ (Ω), W 1,p ψ (Ω), W 1,p ψ (Ω) ∩ C 0 (Ω), C 0,1 (Ω) or {v ∈ W 1,1 ψ (Ω), ∇v(x) ∈ K for a.e. x ∈ K} with K a convex set.
However, the last assumption (A3) on the dimensions of the affine parts of φ is necessary for the statement to be true. It is easy to find counterexamples when this condition is not satisfied. For instance if we consider the problem when N = 1, Ω = (0, 1), ψ(0) = 0, ψ(1) = 1, λ = 0 and φ(•) = | • |, every increasing function that satisfies the boundary condition is a solution. Another example of non uniqueness when the last assumption is not satisfied can be found in [START_REF] Cellina | Comparison results and estimates on the gradient without strict convexity[END_REF]Example 6]

with φ(•) = (| • | -1) 2 + .
In this article, we prove a uniqueness result but we do not provide any result about the existence or about the regularity of minimizers. In the cases where the existence is provided by the direct methods and the uniform continuity is standard, we obtain the following corollary of Theorem 1.2: Corollary 1.3. We assume that the problem P λ admits a minimizer and every minimizer is uniformly continuous. Under the three assumptions (A1), (A2) and (A3), the problem P λ has a unique minimizer.

That is the case for instance when Ω is smooth, ψ smooth and φ satisfies a p-growth condition, some examples are given in Section 2. In some problems like in [START_REF] Bousquet | Boundary continuity of solutions to a basic problem in the calculus of variations[END_REF], if a uniformly continuous minimizer exists then all the minimizers are uniformly continuous. In this case we have: Corollary 1.4. We assume that if the problem P λ has a uniformly continuous minimizer then every minimizer is uniformly continuous. Under the three assumptions (A1), (A2) and (A3) if a minimizer of P λ is uniformly continuous then it is the only minimizer.

The main idea of the proof comes from [START_REF] De Silva | Minimizers of convex functionals arising in random surfaces[END_REF]Section 4]. In this paper, De Silva and Savin consider the minimization in dimension two of

I(u) = Ω φ(∇u)dx on E(Ω, ψ) = C 0,1
ψ (Ω) with φ a convex function with domain K := {z ∈ R 2 , φ(z) < +∞} a closed polygon. In [START_REF] De Silva | Minimizers of convex functionals arising in random surfaces[END_REF]Proposition 4.5], the authors use the fact the problem is invariant under translation to compare the gradient of a minimizer at a point with the gradient of another minimizer at another point.

1.3. Plan of the paper. In the next section, we present some cases where we can apply Theorem 1.2. In Section 3, we present some classical results for functionals of the type (1.1). The last section is devoted to the proof of Theorem 1.2.

Applications

In this section we apply Theorem 1.2 to obtain uniqueness results for some functionals arising in the literature and we conclude by a subsection on a comparison principle.

2.1. Radial functions. In this subsection, we assume that φ(z) = g(||z||) with g : R → R a convex function such that g(0) < g(t) for all t ̸ = 0 and || • || a strictly convex norm.

Proposition 2.1. For such a φ, the assumption (A3) is satisfied.

Proof. Since g is strictly increasing and || • || is strictly convex, the lower level sets of φ are strictly convex. We assume by contradiction that there exists z ∈ R N and ϵ > 0 such that B ϵ (z) ⊂ F where B ϵ (z) := {y ∈ R N such that ||y -z|| < ϵ} and F is an affine part of φ. Without loss of generality, we can assume that z ̸ = 0. Hence, there exists

z ′ ∈ B ϵ (z)\{z} such that ||z ′ || = ||z||. Since z, z ′ ∈ F , φ is affine on [z, z ′ ]. Since φ(z) = φ(z ′ ), the segment [z, z ′ ]
is contained in the boundary of a lower level set of φ. That is a contradiction. Thus, the interior of F is empty. Hence, the dimension of F is at most N -1. □

In the rest of the subsection, we detail some examples where the norm considered is the Euclidean norm | • |. If the set E(Ω, ψ) is a subset of uniformly continuous functions that are in W 1,1 (Ω) then every minimizer is uniformly continuous. In this case, Theorem 1.2 becomes: P λ admits at most one minimizer. For instance that is the case in [10, Theorem 3] where the set considered is C 0,1 ψ (Ω). A minimization among Lipschitz continuous functions also appears in [9, Theorem 1.1] but only for the dimension two. Hence, Theorem 1.2 allows to generalize this result to higher dimensions.

This theorem can also be used to prove the uniqueness part for the least gradient problem considered in [START_REF] Sternberg | Existence, uniqueness, and regularity for functions of least gradient[END_REF]:

To minimize Ω |∇u|dx on {u ∈ BV (Ω) ∩ C 0 (Ω), u = ψ on ∂Ω}.
In fact, we can not directly apply Theorem 1.2 because E(Ω, ψ) ⊈ W 1,1 (Ω) but [12, Theorem 5.9] provides that if Ω and ψ are smooth, then the minimizers are globally Lipschitz continuous on Ω.

In the articles [START_REF] Lussardi | A uniqueness result for a class of non-strictly convex variational problems[END_REF], [START_REF] Marcellini | A relation between existence of minima for nonconvex integrals and uniqueness for non strictly convex integrals of the calculus of variations[END_REF] and [START_REF] Sternberg | Existence, uniqueness, and regularity for functions of least gradient[END_REF] the question of existence is much harder. Now, we look at problems where the direct methods guarantee the existence of a minimizer.

If φ satisfies a p-growth condition: there exist C 1 , α > 0 and p > 1 such that (2.1)

C 1 |z| p -α ≤ φ(z)
for every z ∈ R N then by [4, Theorem 3.30], P λ admits at least one minimizer on W 1,p ψ (Ω). In [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF], we have:

(2.2) φ(z) = |z| if |z| ≤ 1, 1 2 (|z| 2 + 1) if |z| > 1,
and in [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF], up to some rescaling we have :

φ(y) =      1 2 |z| 2 if |z| ≤ 1, |z| -1 2 if 1 < |z| < 2, 1 4 (|z| 2 + 2) if 2 ≤ |z|.
In both cases, φ is a radial convex function with 0 as strict minimum that satisfies the assumption (A3) and a p-growth condition. Hence, these two problems, or any problem of this type, admit at least a minimizer. It remains to show that every minimizer is continuous.

By 2. An interesting fact here is that we do not assume that the boundary of Ω is connected in contrast with [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] and [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF]. In this way, Theorem 1.2 extends the main results of [START_REF] Bouchitté | On a degenerate problem in the calculus of variations[END_REF] and [START_REF] Kawohl | Analysis and numerical studies of a problem of shape design[END_REF] with a shorter proof.

2.2.

When Ω is convex and φ is superlinear. In this subsection we use some results from [START_REF] Bousquet | Boundary continuity of solutions to a basic problem in the calculus of variations[END_REF]. We assume that Ω is convex, λ = 0, ψ is continuous and φ is superlinear:

lim |z|→+∞ φ(z) |z| = +∞.
Under those assumptions, [2, Theorem 1.4] ensures that P λ has at least one continuous solution on Ω when E(Ω, ψ) = W 1,1 ψ (Ω). Hence if φ satisfies the assumption (A3) then Theorem 1.2 provides that P λ has a unique continuous minimizer.

Moreover, if we assume that the affine parts of φ have a uniformly bounded diameter and that there exists a uniformly continuous minimizer of P λ then by [2, Proposition 1.6], see also Lemma 3.7, every minimizer is uniformly continuous on Ω. In this case if φ satisfies the assumption (A3) then by Corollary 1.4, P λ has a unique minimizer and this minimizer is uniformly continuous on Ω.

2.3.

Functions defined on a bounded convex set. In this subsection we assume that K is a bounded convex set of R N , φ(z) = φ(z) + χ K (z) with φ : R N → R a convex function that satisfies the assumption (A3) on K and

χ K (z) = 0 if z ∈ K, +∞ if z / ∈ K.
In this case, for every minimizer u and for a.e. x ∈ Ω, ∇u(x) ∈ K. Hence, every minimizer is globally Lipschitz continuous on Ω. Thus, Theorem 1.2 becomes: P λ admits at most one minimizer on E(Ω, ψ).

The main source of inspiration for this paper comes from [START_REF] De Silva | Minimizers of convex functionals arising in random surfaces[END_REF]Section 4]. In this article, as mentioned in the introduction, φ is defined on a closed convex polygon in R 2 and φ ≡ +∞ outside. The function is strictly convex inside the polygon. Hence, φ satisfies the assumption (A3). Thus, if the minimizer exists, it is the only one. We can generalize it to higher dimensions with a polytope instead of a polygon. This type of formulation φ(z) = φ(z) + χ K (z) can appear when we introduce a constraint on the gradients of the minimizers. For instance in [START_REF] Sternberg | The Constrained Least Gradient Problem in Rn[END_REF], the authors consider the least gradient problem with φ(z) = |z| for all z ∈ R N but the minimization is done on {u ∈ C 0,1 (Ω), |∇u| ≤ 1 a.e.}. In this case, we can set K = B 1 (0) and apply Theorem 1.2 to obtain uniqueness if a minimizer exists.

Comparison principle.

If we assume that the minimizers of P λ are uniformly continuous then we have the following comparison principle: Proposition 2.3. Let ψ 1 and ψ 2 be two boundary conditions on ∂Ω. If assumptions (A1), (A2), (A3) are satisfied and ψ 1 ≤ ψ 2 on ∂Ω then u 1 ≤ u 2 where u 1 is the uniformly continuous solution of P λ on E(Ω, ψ 1 ) and u 2 is the uniformly continuous solution of P λ on E(Ω, ψ 2 ).

Proof. By assumptions on ψ 1 and ψ 2 , Lemma 3.3 and Lemma 3.4 below imply that min{u 1 , u 2 } is a solution of P λ on E(Ω, ψ 1 ). Hence, by Theorem 1.2 we have min{u 1 , u 2 } = u 1 and thus u 1 ≤ u 2 . □

Preliminary results

In this section, we recall some classical and simple results that we use in the proof of Theorem 1.2. The first lemma shows that the problem is invariant under translations as in [7, Section 1.1]. Lemma 3.1. If u is a minimizer of P λ on E(Ω, ψ) then for every τ ∈ R N and α ∈ R, u(• + τ ) + α is a minimizer on E(Ω τ , ψ + α) where Ω τ := Ω -τ .

Proof. We consider v ∈ E(Ω τ , ψ + α). By assumption (A1), v(x) := v(x -τ ) -α is in E(Ω, ψ) and thus Since v can be any function in E(Ω τ , ψ + α) we have the desired result. □ Remark 3.2. This lemma uses the invariance under translations of this type of functionals with λ constant. Hence, it seems difficult to generalize Theorem 1.2 to more general problems. When λ is not constant, we can use different ideas to prove uniqueness for specific functionals. For instance, in an upcoming paper, we prove uniqueness for φ as in (2.2) with λ having small oscillations by comparing the level sets of two minimizers.

(3.1) Ω φ(∇ v(x)) + λ v(x)dx ≥ Ω φ(∇u(x)) + λu(x)dx. But Ωτ φ(∇v(y)) + λv(y)dy = Ω φ(∇ v(x)) + λ( v(x) + α)dx = Ω φ(∇ v(x)) + λ v(x)dx + αλ|Ω|.

By the inverse change of variables applied to u and (3.1) we obtain

The following lemma establishes that a minimizer on Ω is still a minimizer on the subsets of Ω.

Lemma 3.3. Let u be a minimizer of P λ on E(Ω, ψ). For every non empty open set V ⊂ Ω, u| V is a minimizer on V among all the functions in E(V, u).

Proof. We assume by contradiction that this is not the case. There exists v ∈ E(V, u) such that

V φ(∇v(x)) + λv(x)dx < V φ(∇u(x)) + λu(x)dx.
If we define w(x) = u(x) if x / ∈ V and w(x) = v(x) if x ∈ V then by assumption (A2), w ∈ E(Ω, ψ) and I λ (w) < I λ (u). This contradicts the fact that u is a minimizer. □

In the next lemma, we prove that the extension of a minimizer by a compatible minimizer is still a minimizer. Lemma 3.4. Let u be a minimizer of P λ on E(Ω, ψ) and

V ⊂ Ω a non empty open set. If v is a minimizer of P λ on E(V, u) then w = v on V, u on Ω\V,
is a minimizer of P λ on E(Ω, ψ).

Proof.

By assumption (A2) on E, w ∈ E(Ω, ψ). We compare I λ (u) and I λ (w): By Lemma 3.3, V φ(∇u) + λu = V φ(∇v) + λv. Hence, I λ (u) = I λ (w). Thus, w is a minimizer of P λ on E(Ω, ψ). □

I λ (u) = Ω φ(∇u) + λu = V φ (
We introduce the notion of point of approximate continuity that we use in the proof of Theorem 1.2. Definition 3.5. Let f : Ω → R m be a measurable function, we say that x 1 is a point of approximate continuity if for every η > 0

(3.2) lim r→0 |{x ∈ B r (x 1 ), |f (x) -f (x 1 )| ≥ η}| |B r (x 1 )| = 0.
We recall a classical result that can be found in [6, Section 1.7, Theorem 3].

Proposition 3.6. Let f : Ω → R m be a measurable function. Hence, a.e. x ∈ Ω is a point of approximate continuity of f .

In the last result of this section, we show that given three minimizers, φ is affine on the simplex generated by their gradients at almost every point: Lemma 3.7. Let u, v and w be three minimizers of P λ on E(Ω, ψ). Then φ is affine on the simplex with vertices [∇u(x), ∇v(x), ∇w(x)] for a.e. x ∈ Ω.

Proof. Since u is a solution of P λ ,

I λ (u) ≤ I λ u + v + w 3 .
By the convexity of φ,

I λ u + v + w 3 = Ω φ ∇u + ∇v + ∇w 3 + λ u + v + w 3 dx ≤ 1 3 Ω (φ(∇u) + λu)dx + 1 3 Ω (φ(∇v) + λv)dx + 1 3 Ω (φ(∇w) + λw)dx = 1 3 I λ (u) + 1 3 I λ (v) + 1 3 I λ (w).
Since v and w are other solutions,

I λ (u) = 1 3 I λ (u) + 1 3 I λ (v) + 1 3 I λ (w).
This implies that

Ω φ ∇u + ∇v + ∇w 3 dx = Ω 1 3 (φ(∇u) + φ(∇v) + φ(∇w))dx.
Hence for a.e. x ∈ Ω,

φ ∇u(x) + ∇v(x) + ∇w(x) 3 = 1 3 φ(∇u(x)) + φ(∇v(x)) + φ(∇w(x)) .
Hence, for a.e. x ∈ Ω, φ is affine on the simplex with vertices [∇u(x), ∇v(x), ∇w(x)]. □ Remark 3.8. The fact that φ is affine on the simplex with vertices [∇u(x), ∇v(x), ∇w(x)] is equivalent to ∇u(x), ∇v(x), ∇w(x) ∈ F where F is an affine part of φ.

Remark 3.9. The number of minimizers considered in Lemma 3.7 can be changed. For instance, we can prove it with n ∈ N minimizers. In this case we obtain that φ is affine on the convex hull of the gradients of the solutions and they are still in the same affine part F of φ.

Proof of Theorem 1.2

The proof of Theorem 1.2 is divided into three steps. The first one is very similar to the proof of [START_REF] De Silva | Minimizers of convex functionals arising in random surfaces[END_REF]Proposition 4.5].

Step 1 Let u 1 , u 2 be two uniformly continuous minimizers of P λ on E(Ω, ψ). We call ω a common modulus of continuity of u 1 and u 2 namely: lim

w→0 + ω(x) = 0, ω is increasing and ω(a+b) ≤ ω(a)+ω(b)
for every a, b ∈ R + . We assume by contradiction that there exist x 0 ∈ Ω and ϵ > 0 such that u 1 (x 0 ) -u 2 (x 0 ) = ϵ.

We consider ρ > 0 such that ω(ρ) < ϵ 6 . For every y ∈ Ω, u 1 (y) -u 2 (y) ≤ 2ω(dist(y, ∂Ω)). Hence,

ω(2ρ) ≤ ϵ 3 < ϵ 2 = u 1 (x 0 )-u 2 (x 0 ) 2
≤ ω(dist(x 0 , ∂Ω)). Thus, B 2ρ (x 0 ) ⋐ Ω. For every τ ∈ R N such that |τ | < ρ and for every x ∈ B ρ (x 0 ) we have u 1 (x) ≥ u 1 (x 0 ) -ω(ρ) and u 2 (x + τ ) ≤ u 2 (x 0 ) + 2ω(ρ). Hence, we have that (4.1)

B ρ (x 0 ) ⊂ {x ∈ Ω ∩ Ω τ , u 2 (x + τ ) + ϵ 2 < u 1 (x)}
and for the same reasons:

(4.2) B ρ (x 0 ) ⊂ {x ∈ Ω ∩ Ω τ , u 1 (x + τ ) - ϵ 2 > u 2 (x)}
with Ω τ = Ω -τ . For every x ∈ Ω ∩ Ω τ , we have

|u 1 (x)-u 2 (x+τ )| ≤ ω(ρ)+min{|u 1 (x)-u 2 (x)|, |u 1 (x+τ )-u 2 (x+τ )|} ≤ ω(ρ)+2ω dist(x, ∂(Ω∩Ω τ )) .
Since ω(ρ) < ϵ 6 we have:

(4.3) {x ∈ Ω ∩ Ω τ , u 2 (x + τ ) + ϵ 2 < u 1 (x)} ⋐ Ω ∩ Ω τ
and for the same reasons:

(4.4) {x ∈ Ω ∩ Ω τ , u 1 (x + τ ) - ϵ 2 > u 2 (x)} ⋐ Ω ∩ Ω τ .
We extend the function v τ (x) := min{u 1 (x), u 2 (x + τ ) + ϵ 2 } defined on Ω ∩ Ω τ by u 1 to Ω. By assumptions (A1), (A2) and the inclusion (4.3), this extension belongs to E(Ω, ψ). Thanks to the inclusion (4.4), if we extend w τ (x) := max{u 1 (x + τ ) -ϵ 2 , u 2 (x)} by u 2 , we also have that w τ ∈ E(Ω, ψ). By Lemma 3.

∈ Ω ∩ Ω τ , u 2 (x + τ ) + ϵ 2 < u 1 (x)}, u 1 and u 2 (• + τ ) + ϵ
2 are solutions of the same problem. By Lemma 3.4, v τ is a minimizer of P λ on E(Ω, ψ). With the same arguments we can prove that w τ is also a minimizer of P λ on E(Ω, ψ).

Step 2 In this crucial step of the proof we show that there exists an (N -1)-dimensional set containing the essential ranges of the gradients of any minimizer. We consider x 1 ∈ B ρ 2 (x 0 ) and x ′ 1 ∈ B ρ 2 (x 0 ) two points of approximate continuity of (∇u 1 , φ(∇u 1 )) and x 2 ∈ B ρ 2 (x 0 ) a point of approximate continuity of (∇u 2 , φ(∇u 2 )). By Proposition 3.6 and the fact that ∇u 1 , ∇u 2 and φ are measurable, that is the case for a.e. (x 1 , x ′ 1 , x 2 ) ∈ B ρ 2 (x 0 ) 3 . We fix τ := x 2 -x 1 and τ ′ := x ′ 1 -x 1 . Then by definition of v τ and w τ ′ , x 1 is a point of approximate continuity of (∇v τ , φ(∇v τ )) and (∇w τ ′ , φ(∇w τ ′ )).

We define f : x → ∇u 1 (x), ∇w τ ′ (x), ∇v τ (x), φ(∇u 1 (x)), φ(∇w τ ′ (x)), φ(∇v τ (x)) . Since x 1 is a point of approximate continuity of f , we have for every n ∈ N * :

lim r→0 |{x ∈ B r (x 1 ), |f (x) -f (x 1 )| > 1 n }| |B r (x 1 )| = 0.
Thus, for every n ∈ N * , there exists x ∈ B rn (x 1 ) such that |f (x) -f (x 1 )| ≤ 1 n . By Lemma 3.7 we can further assume that (4.5)

φ ∇u 1 (x) + ∇w τ ′ (x) + ∇v τ (x) 3 = φ(∇u 1 (x)) + φ(∇w τ ′ (x)) + φ(∇v τ (x)) 3 < +∞.
We introduce K := {z ∈ R N , φ(z) < +∞} the convex domain of φ and we distinguish two cases:

a) If the triplet (x 1 , x ′ 1 , x 2 ) is such that ∇u 1 (x 1 )+∇w τ ′ (x 1 )+∇vτ (x 1 ) 3
∈ Int(K), then by continuity of φ on Int(K) we have

φ ∇u 1 (x 1 ) + ∇w τ ′ (x 1 ) + ∇v τ (x 1 ) 3 = φ(∇u 1 (x 1 )) + φ(∇w τ ′ (x 1 )) + φ(∇v τ (x 1 )) 3 .
By the definition of w τ ′ , v τ and (4.1)-(4.2), this is equivalent to ∇u 1 (x 1 ), ∇u 1 (x ′ 1 ) and ∇u 2 (x 2 ) being in the same affine part of φ. Now, we prove that φ is affine on the convex hull of F := {∇u 1 (x) with x ∈ X 1 and ∇u 2 (x) with x ∈ X 2 } with X i the set of points of approximate continuity of (∇u i , φ(∇u i )). If we consider y ∈ B ρ 2 (x 0 ) a point of approximate continuity of (∇u i , φ(∇u i )) and y ′ ∈ B ρ 2 (x 0 ) a point of approximate continuity of (∇u j , φ(∇u j )) with i, j ∈ {1, 2} we have that (4.6) ∇u 1 (x 1 ) + ∇u 1 (x ′ 1 ) + ∇u 2 (x 2 ) + ∇u i (y) + ∇u j (y ′ ) 5 ∈ Int(K).

By using the functions v y-x 1 if i = 2 and w y-x 1 if i = 1, and similarly for j, we obtain the analogue of (4.5) for these five minimizers, see Remark 3.9. Hence, we can argue as above and rely on (4.6) to get that the fives points ∇u 1 (x 1 ), ∇u 1 (x ′ 1 ), ∇u 2 (x 2 ), ∇u i (y) and ∇u j (y ′ ) belong to the same affine part of φ. Thus, φ is affine on the segment [∇u i (y), ∇u j (y ′ )]. Hence, φ is affine on the convex hull of F . Thus by assumption (A3), there exists F an (N -1)-dimensional affine space such that for a.e. x ∈ B ρ 2 (x 0 ), ∇u 1 (x) and ∇u 2 (x) belong to F. b) If there is no such triplet then for every couple (x 1 , x ′ 1 ) of points of approximate continuity, we have that ∇u(x 1 ), ∇u(x ′ 1 ) belong to the same face of ∂K, namely the intersection of an affine hyperplane with ∂K. Otherwise the convex combination would be in Int(K) contradicting the fact that we are not in case a). Hence, there exists an affine hyperplane F such that for a.e. x ∈ B ρ 2 (x 0 ), ∇u 1 (x) is in F. The fact that we are not in case a) gives that for a.e. x ∈ B ρ 2 (x 0 ), ∇u 2 (x) is in F too.

Step 3 Since the dimension of F is at most N -1 there exists a normal vector η to F. The scalar product ⟨∇u i (x), η⟩ = C is constant in B ρ 2 (x 0 ) and is independent of i = 1, 2. Thus, u 1 -u 2 = ϵ > 0 on (x 0 + Rη) ∩ B ρ 2 (x 0 ). Hence, the compact set E ϵ := (u 1 -u 2 ) -1 (ϵ) ⋐ Ω has no extreme points in Ω.

We consider x 0 ∈ E ϵ such that | x 0 | = max

(1. 1 )

 1 I λ : u → Ω φ(∇u(x)) + λu(x)dx with λ ∈ R, Ω a bounded open set of R N and φ : R N → R ∪ {+∞} a measurable convex function. We consider the problem: P λ : To minimize I λ (v) on E(Ω, ψ).

  Ωτ φ(∇v(y)) + λv(y)dy ≥ Ωτ φ ∇(u(y + τ ) + α) + λ(u(y + τ ) + α)dy.

  ∇u) + λu + Ω\V φ(∇u) + λu and I λ (w) = Ω φ(∇w) + λw = V φ(∇v) + λv + Ω\V φ(∇u) + λu.

  1 and Lemma 3.3, on the open set {x

x∈Eϵ|x|.

  Since E ϵ has no extreme points, there exist y 1 and y 2 in E ϵ such that x 0 ∈ (y 1 , y 2 ). By the triangle inequality, either|y 1 | > | x 0 | or |y 2 | > | x 0 |.That is a contradiction. Hence, we have u 1 = u 2 on Ω.

A UNIQUENESS RESULT FOR A TRANSLATION INVARIANT PROBLEM IN THE CALCULUS OF VARIATIONS.