Vigdis Aas
email: vigdis.aas@hf.hio.no

Nina P Hessvik
email: n.p.hessvik@farmasi.uio.no

Marianne Wettergreen

Andreas W Hvam- Men

Stefan Hallén
email: stefan.hallen@astrazeneca.com

G Hege Thoresen
email: hege.thoresen@farmasi.uio.no

Arild C Rustan
email: arild.rustan@farmasi.uio.no

Andreas W Hvammen

Chronic hyperglycemia reduces substrate oxidation and impairs metabolic switching of human myotubes

Keywords: Myotubes, Skeletal Muscle, Energy metabolism, Mitochondria

Skeletal muscle of insulin resistant individuals is characterized by lower fasting lipid oxidation and reduced ability to switch between lipid and glucose oxidation. The purpose of the present study was to examine if chronic hyperglycemia would impair metabolic switching of myotubes. Human myotubes were treated with or without chronic hyperglycemia (20 mmol/l glucose for 4 days), and metabolism of [ 14 C]oleic acid (OA) and [ 14 C]glucose was studied. Myotubes exposed to chronic hyperglycemia showed a significantly reduced OA uptake and oxidation to CO 2 , whereas acid-soluble metabolites were increased compared to normoglycemic cells (5.5 mmol/l glucose). Glucose suppressibility, the ability of acute glucose (5 mmol/l) to suppress lipid oxidation, was 50 % in normoglycemic cells and reduced to 21 % by hyperglycemia. Adaptability, the capacity to increase lipid oxidation with increasing fatty acid availability, was not affected by hyperglycemia. Glucose uptake and oxidation were reduced by about 40 % after hyperglycemia, and oxidation of glucose in presence of mitochondrial uncouplers showed that net and maximal oxidative capacities were significantly reduced. Hyperglycemia also abolished insulin-stimulated glucose uptake. Moreover, ATP concentration was reduced by 25 % after hyperglycemia. However, none of the measured mitochondrial genes were downregulated nor was mitochondrial DNA content.

Microarray and real-time RT-PCR showed that no genes were significantly regulated by chronic hyperglycemia. Addition of chronic lactate reduced both glucose and OA oxidation to the same extent as hyperglycemia. In conclusions, chronic hyperglycemia reduced substrate oxidation in skeletal muscle cells and impaired metabolic switching. The effect is most likely due to an induced mitochondrial dysfunction.

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INTRODUCTION

Metabolic flexibility is the capacity of the body to switch between lipid oxidation in the fasting state to carbohydrate oxidation in the fed state and vice versa, i.e. the capacity to adjust fuel consumption according to availability [START_REF] Kelley | Fuel selection in human skeletal muscle in insulin resistance: a reexamination[END_REF]. Insulin resistance and type 2 diabetes (T2D) have been associated with impaired ability to switch between lipid and glucose oxidation [START_REF] Kelley | Skeletal muscle fatty acid metabolism in association with insulin resistance, obesity, and weight loss[END_REF][START_REF] Corpeleijn | Impaired skeletal muscle substrate oxidation in glucose-intolerant men improves after weight loss[END_REF], and it has been postulated that this impaired metabolic switching is an intrinsic property of skeletal muscle [START_REF] Ukropcova | Dynamic changes in fat oxidation in human primary myocytes mirror metabolic characteristics of the donor[END_REF]. Ukropcova et al showed that the metabolic phenotype of the donor was maintained in myotube cultures after removal of in vivo neuroendocrine factors [START_REF] Ukropcova | Dynamic changes in fat oxidation in human primary myocytes mirror metabolic characteristics of the donor[END_REF]. However, metabolic flexibility in vivo has been shown to improve by weight loss [START_REF] Corpeleijn | Impaired skeletal muscle substrate oxidation in glucose-intolerant men improves after weight loss[END_REF], indicating that lifestyle factors can affect metabolic flexibility. We recently showed that treatment with n-3 fatty acids improved metabolic switching of myotubes [START_REF] Hessvik | Metabolic switching of human myotubes is improved by n-3 fatty acids[END_REF].

Whether exposure to chronic hyperglycemia affects metabolic flexibility of skeletal muscle is at present unknown.

There are numerous reports on the effect of environmental and dietary factors on development of intramuscular lipid accumulation and insulin resistance. Skeletal muscle of obese and insulin resistant subjects is characterized by increased intramyocellular lipids (IMCL) and reduced mitochondrial oxidative capacity [START_REF] Petersen | Impaired mitochondrial activity in the insulin-resistant offspring of patients with type 2 diabetes[END_REF][START_REF] He | Skeletal muscle lipid content and oxidative enzyme activity in relation to muscle fiber type in type 2 diabetes and obesity[END_REF]. We have previously shown that myotubes from type 2 diabetic subjects have reduced fatty acid oxidation after culturing in a non-diabetic environment, implying a genetic defect [START_REF] Gaster | Reduced lipid oxidation in skeletal muscle from type 2 diabetic subjects may be of genetic origin: evidence from cultured myotubes[END_REF]. Experiments in rats confirmed that there is an inherited connection between oxidative capacity and a diabetic phenotype, as rats bred to obtain low aerobic capacity were insulin resistant and displayed a diabetic metabolic profile [START_REF] Wisloff | Cardiovascular risk factors emerge after artificial selection for low aerobic capacity[END_REF]. These animals also expressed reduced level of several proteins involved in mitochondrial biogenesis, suggesting a genetic association between mitochondrial dysfunction and insulin resistance. Several studies have shown reduced mitochondrial content and/or mitochondrial dysfunction in skeletal muscle of T2D subjects [START_REF] Kelley | Dysfunction of mitochondria in human skeletal muscle in type 2 diabetes[END_REF][START_REF] Mogensen | Mitochondrial respiration is decreased in skeletal muscle of patients with type 2 diabetes[END_REF][START_REF] Lowell | Mitochondrial dysfunction and type 2 diabetes[END_REF]. There is substantial support for the hypothesis that reduced oxidative capacity might be the fundamental cause of IMCL accumulation and insulin resistance [START_REF] Mogensen | Mitochondrial respiration is decreased in skeletal muscle of patients with type 2 diabetes[END_REF][START_REF] Befroy | Impaired mitochondrial substrate oxidation in muscle of insulin-resistant offspring of type 2 diabetic patients[END_REF]. However, the possibility that mitochondrial dysfunction rather is the consequence of insulin resistance can not be excluded.

In rat, chronic glucose infusion induced insulin resistance in skeletal muscle concurrent with increased triacylglycerol (TAG), malonyl-CoA and long-chain fatty acyl-CoA (LCA-CoA) levels in muscle [START_REF] Laybutt | Muscle lipid accumulation and protein kinase C activation in the insulin-resistant chronically glucose-infused rat[END_REF]. Lipid intermediates, such as fatty acyl-CoA, ceramides and diacylglycerol, are linked to muscle insulin resistance both in vivo and in several in vitro models [START_REF] Timmers | Muscular diacylglycerol metabolism and insulin resistance[END_REF]. Glucose oversupply has also been associated with a range of

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ACCEPTED MANUSCRIPT 5 metabolic changes, such as increased lactate production [START_REF] Shaham | Metabolic profiling of the human response to a glucose challenge reveals distinct axes of insulin sensitivity[END_REF][START_REF] Laybutt | Specific adaptations in muscle and adipose tissue in response to chronic systemic glucose oversupply in rats[END_REF], and it has been shown that diabetic and obese patients have increased plasma lactate levels [START_REF] Chen | Plasma lactate concentration in obesity and type 2 diabetes[END_REF][START_REF] Lovejoy | Lactate generation following glucose ingestion: relation to obesity, carbohydrate tolerance and insulin sensitivity[END_REF]. The role of lactate is not known, but it is most likely more complex than being a simple waste product of metabolism (reviewed in [START_REF] Sola-Penna | Metabolic regulation by lactate[END_REF][START_REF] Philp | Lactate--a signal coordinating cell and systemic function[END_REF]). It has been hypothesized that lactate is a signaling molecule directing metabolic activity. In muscle, lactate inhibits 6-phosphofructo-1-kinase and consequently decreases glucose consumption [START_REF] Zancan | Fructose-2,6bisphosphate counteracts guanidinium chloride-, thermal-, and ATP-induced dissociation of skeletal muscle key glycolytic enzyme 6-phosphofructo-1-kinase: A structural mechanism for PFK allosteric regulation[END_REF].

Chronic hyperglycemia (HG) has been shown to contribute to insulin resistance in skeletal muscle [START_REF] Zierath | The effect of hyperglycaemia on glucose disposal and insulin signal transduction in skeletal muscle[END_REF]. The mechanism, however, by which HG induces insulin resistance is not clear. We have previously shown that treatment of human myotubes with chronic HG reduced acute glucose uptake and glycogen synthesis. This reduction accompanied accumulation of TAG in the cells, an increased de novo lipogenesis and increased acyl-CoA:1.2-diacylglycerol acyltransferase (DGAT) activity, whereas total cell content of glycogen was unchanged [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF] .

The effect of HG was maximal after 4 days and reversible, at least when it came to glycogen synthesis [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF]. Exposure of 3T3-L1 adipocytes to HG has been found to induce insulin resistance and loss of mitochondrial membrane potential [START_REF] Gao | Mitochondrial dysfunction is induced by high levels of glucose and free fatty acids in 3T3-L1 adipocytes[END_REF]. Moreover, mitochondria became smaller and more compact, whereas mitochondrial DNA was unaffected by HG [START_REF] Gao | Mitochondrial dysfunction is induced by high levels of glucose and free fatty acids in 3T3-L1 adipocytes[END_REF],

indicating that HG can induce mitochondrial dysfunction. Therefore, we hypothesized that exposure to chronic HG induces some kind of mitochondrial dysfunction leading to impaired metabolic switching of myotubes.

A C C E P T E

Substrate oxidation assay

The muscle cells were cultured on 96-well CellBIND ® microplates as described above.

Growth medium was completely removed before addition of substrates. Substrate, [1-14 C]glucose (37 kBq/ml), [6-14 C]glucose (37 kBq/ml), [ 14 C(U)]glucose (37 kBq/ml) or [1-

14 C]oleic acid (37 kBq/ml), was given in DPBS with 10 mmol/l HEPES and 1 mmol/l Lcarnitine (only added with oleic acid). Oleic acid was bound to BSA at a ratio of 2.5/1. A 96well UNIFILTER ® microplate was mounted on top of the CellBIND ® plate as described before [START_REF] Wensaas | Cell-based multiwell assays for the detection of substrate accumulation and oxidation[END_REF], and the cells were incubated at 37°C for 4 h. The CO 2 trapped in the filter was counted by liquid scintillation in a MicroBeta™ Trilux scintillation counter (PerkinElmer).

The remaining cell-associated radioactivity was also assessed by liquid scintillation, and the sum of CO 2 and cell-associated radioactivity was considered as total substrate uptake.

Oxidation of [1-14 C]oleic acid to acid-soluble metabolites (ASM) was measured by acidic precipitation of the incubation media. CO 2 production from [1-14 C]oleic acid was measured in the presence or absence of various compounds known to modify mitochondrial function;

dinitrophenol (DNP 100 µmol/l) and carbonylcyanide-4-trifluoromethoxyphenylhydrazone (FCCP 0.6 mmol/l) (both mitochondrial uncouplers), and oligomycin (ATP synthase inhibitor, 1 µg/ml). From these experiments we calculated baseline (routine) oxidation (R) and uncoupled oxidation with FCCP (E), as well as baseline oxidation in the presence of [START_REF] Hutter | Senescenceassociated changes in respiration and oxidative phosphorylation in primary human fibroblasts[END_REF]. The net routine flux control ratio was calculated as netR=(R-L)/E.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT oligomycin (L)
Protein content in each well was determined [START_REF] Bradford | A rapid and sensitive method for quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF], and the data are presented as CO 2 /mg protein, cell-associated substrate/mg protein or ASM/mg protein.

Lipid distribution

Myotubes were incubated with [1-14 C]oleic acid (18.5 kBq/ml, 0.2 mmol/l) for 4 h before they were harvested into ice-cold water, centrifuged (1000 g, 5 min), resuspended in distilled water and sonicated. Cell-associated lipids were extracted with chloroform:methanol and separated by thin layer chromatography as described by Folch et al. [START_REF] Folch | A simple method for the isolation and purification of total lipids from animal tissues[END_REF].

RNA isolation and analysis of gene expression by real-time RT-PCR

Human skeletal muscle cells were washed, trypsinized and pelleted before total RNA was isolated using Agilent Total RNA isolation kit (Agilent Technologies, Santa Clara, CA, USA)

according to the supplier's protocol. Total RNA (0.1 µg/µl) was reversely transcribed with oligo primers using a PerkinElmer Thermal Cycler 9600 (25 C for 10 min, 37 C for 1 h, 99 C for 5 min) and a TaqMan reverse-transcription reagents kit (Applied Biosystems, Warrington, UK). Real-time PCR was performed using an ABI PRISM 7000 Detection System (Applied Biosystems, Warrington, UK). DNA expression was determined by SYBR ® Green, and primers were designed using Primer Express ® (Applied Biosystems, Warrington, UK) (sequences in Table 1). Each target were quantified in triplicates and carried out in a 25 µl reaction volume according to the supplier's protocol. All assays were run for 40 cycles (95 C for 12 s followed by 60

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C for 60 s). The transcription levels were normalized to the reference genes GAPDH and 36B4.

Microarray

Total RNA was prepared from primary myotubes from three donors using Agilent Total RNA isolation kit according to the supplier's protocol (Agilent Technologies). RNA was used individually and RNA integrity was checked on chip analysis (Agilent 2100 bioanalyzer, Agilent Technologies, Santa Clara, CA, USA) according to the manufacturer's instructions.

RNA was judged as suitable for array hybridization only if samples exhibited intact bands corresponding to the 18S and 28S ribosomal RNA subunits, and displayed no chromosomal peaks or RNA degradation products (RNA Integrity Number > 9.0). cRNA synthesis was performed using Illumina TotalPrep RNA Amplification (San Diego, CA, USA) according to after NG treatment (control), followed by identification of differentially expressed probe sets using Intensity-based moderated t-statistics [START_REF] Sartor | Intensity-based hierarchical Bayes method improves testing for differentially expressed genes in microarray experiments[END_REF]. P-values were corrected for multiple testing by using Benjamini and Hochberg's false discovery rate (FDR) method [START_REF] Benjamini | Controlling the False Discovery Rate: a Practical and Powerful Approach to Multiple Testing[END_REF]. Probe sets that satisfied the criterion of FDR < 10 % (q-value < 0.1) and fold-change >1.2 or < -1.2, were considered to be significantly regulated. GEO accession number GSE19620.

Immunoblotting

Aliquots with 20 g cell protein (total cell lysate prepared in Laemmli buffer) were separated by SDS-PAGE (10 % resolving gel) and transferred to polyvinylidene difluoride membranes.

The membranes were immunoblotted with AMPK (1:1000), phospho-AMPK (1:1000), ACC Cell medium lactate concentration was measured with a radiometer ABL 715 blood gas analyzer (Diamond Diagnostics, Holliston, MA, USA). Medium from 6 well plates were harvested after 48 h and analyzed without dilution.

Protein synthesis

The cells were grown in CytoStar-T scintillating microplates and incubated with [ 14 C(U)]leucine (37 kBq/ml, 0.8 mmol/l) on day 8 after initiated differentiation. Incorporation of [ 14 C]leucine was followed for 10 h with scintillation proximity assay (SPA) technology.

Uptake and accumulation of leucine in cells treated with hyperglycemia (20 mmol/l glucose for 4 days) were compared to normoglycemic control cells (5.5 mmol/l glucose).

Statistics

All data are presented as mean ± SEM. Statistical comparison between different treatments was performed by Student's t-test and linear mixed models (LMM) analysis (SPSS ver. 16.0).

The parameter of interest was entered as the dependent variable and pretreatment (HG and NG) and acute treatments were entered as fixed variables. Differences were considered statistically significant at p<0.05. All experiments were performed with at least triplicate observations, and replicate experiments are performed on cells from different donors and given as n.

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RESULTS

Effect of acute glucose on oleic acid oxidation

Simultaneous accumulation and oxidation of oleic acid (OA) was determined in human skeletal muscle cells (myotubes) as previously described [START_REF] Wensaas | Cell-based multiwell assays for the detection of substrate accumulation and oxidation[END_REF]. The suppressive effect of acute addition (4 h during CO 2 trapping) of glucose on [1-14 C]OA oxidation was evident at 0.5 mmol/l glucose (Fig. 1), and at 5 mmol/l glucose OA oxidation was suppressed by 53 %. OA oxidation was not significantly suppressed by acute addition of 5 mmol/l deoxyglucose (DOG), 1.43 (± 0.83) nmol/mg protein of CO 2 was trapped in absence and 1.07 (± 0.55) nmol/mg protein in presence of 5 mmol/l DOG (p=0.31, n=3), indicating that the suppressive effect of glucose on OA oxidation is due to some downstream metabolites in the glycolytic pathway.

Effect of chronic hyperglycemia on oleic acid metabolism

To examine whether chronic hyperglycemia affects the suppressive effect of glucose on OA metabolism, myotubes were pretreated with hyperglycemia (HG, 20 mmol/l glucose) for 4 days and OA (0-300 µmol/l) metabolism was measured in the absence and presence of acute glucose (5 mmol/l) (Fig. 2). As shown in Fig. 1, acute glucose (5 mmol/l) suppressed OAoxidation to CO 2 by 50 % in normoglycemic cells (Fig. 2A, p<0.0005 overall effect), whereas this suppressive effect of acute glucose was not significant in hyperglycemic cells (Fig. 2B).

Chronic exposure to HG reduced complete oxidation of OA to CO 2 by 47 % compared to exposure to NG (Fig. 2A and B, p<0.0005, overall effect), indicating that oxidation was already suppressed after chronic HG exposure. Incomplete OA oxidation to acid-soluble metabolites (ASM) was however increased by 22 (± 8) % in HG cells compared to NG cells (p=0.02, data not shown). ASM consist mainly of tricarboxylic acid cycle metabolites and reflect incomplete oxidation of fatty acids. Acute glucose did not suppress ASM production, neither in NG nor HG cells (data not shown).

Cellular uptake of OA in NG cells, assessed as the sum of cell-associated and CO 2trapped radioactivity, increased with increasing concentrations of OA (20-300 µmol/l) (p<0.0005) and in presence of glucose (p<0.0005) (Fig. 2C). HG cells on the other hand showed no significantly increased OA uptake with increasing OA concentrations (p=0.12) or in presence of glucose (p=0.56) (Fig. 2D), showing a reduced sensitivity towards acute glucose after chronic HG exposure. The overall cellular uptake of OA was also significantly reduced by 32 % in HG cells (p=0.004). However, myotubes exposed to HG showed 21 %

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increased incorporation of OA into triacylglycerol compared to NG cells (p=0.02, data not shown), in accordance with previous findings [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF]. The other lipid classes (free fatty acid, diacylglycerol, phospholipids, and cholesterol ester) were not significantly affected by HG pretreatment.

The fraction of OA that was oxidized (CO 2 /cellular uptake) was independent of OA concentration, 4.7 (± 0.5) % in NG cells and 4.4 (± 0.5) % in HG cells in absence of glucose.

Glucose reduced this fraction significantly in NG cells to 1.5 (± 0.2) % (p<0.0005), whereas there was no significant reduction (2.6 (± 1.0) % (p=0.17)) in HG cells. The observed effects of HG on metabolism, was not due to hypertonicity as previously shown [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF].

Glucose suppressibility and adaptability in myotubes treated with chronic hyperglycemia

To investigate whether HG affected metabolic switching, the parameters suppressibility and adaptability were calculated according to previous definitions [START_REF] Hessvik | Metabolic switching of human myotubes is improved by n-3 fatty acids[END_REF]. Glucose suppressibility, the ability of acutely added glucose to suppress lipid oxidation, was markedly reduced to 21 % in HG cells compared to 50 % in NG cells (Fig. 3A) (p=0.03). Adaptability, defined as the capacity to increase lipid oxidation upon exposure to increasing fatty acid concentrations, was present in both NG and HG cells (p=0.04 overall effect). The adaptability seemed to be reduced in cells pretreated with HG, although not significantly (Fig. 3B).

Effect of oleic acid on glucose metabolism in normoglycemic and hyperglycemic cells

The effect of hyperglycemia on glucose metabolism was also examined. Glucose uptake, assessed as the sum of cell-associated and CO 2 -trapped radioactivity from [6-14 C]glucose, was reduced by exposure to hyperglycemia for 4 days (Fig. 4A), as reported before using a standard deoxyglucose uptake assay [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF]. The overall glucose uptake in HG cells was 61 % of the amount measured in NG cells (p=0.03).

To evaluate the effect of fatty acids on glucose metabolism, oleic acid (OA) was added to the cell culture media during acute glucose uptake (4 h) in concentrations up to 300 µmol/l.

Glucose uptake seemed to increase with increasing OA concentrations, although not significantly (p=0.06), and there was no difference between NG and HG cells (Fig. 4A).

Oxidation of glucose to CO 2 was reduced by HG to the same degree as glucose uptake, to 62 % of control cells (p=0.03, Fig. 4B). There was no significant effect of acute oleic acid addition on glucose oxidation. The fraction of glucose that was oxidized (CO 2 /cellular uptake) was constant and not dependent on chronic hyperglycemia or concentration of oleic acid (data

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13 not shown). The effect of hyperglycemia on glucose metabolism was confirmed with two other isotopes of glucose, [1-14 C]glucose (data not shown) and [ 14 C(U)]glucose (Fig. 5B).

Effect of chronic hyperglycemia on mitochondrial capacity

To determine the mitochondrial capacity of myotubes pretreated with NG or HG, complete oleic acid oxidation and glucose oxidation to CO 2 was measured in presence of the mitochondrial uncoupler 2.4-dinitrophenol (DNP). DNP (100 µmol/l) increased OA oxidation 1.5-fold in both NG and HG cells in absence of glucose (Fig. 5A). In accordance with the results in Fig. 2 The maximal capacity to oxidize [ 14 C(U)]glucose was strongly compromised in HG cells (Fig. 5B) to only 47 % of the capacity seen in NG cells (p<0.001). In presence of DNP, the suppressive effect of OA on glucose oxidation (OA suppressibility) was also evident. OA suppressibility was significantly lower in HG compared to NG cells (23 % vs. 42 % in HG and NG cells, respectively, Fig. 5C).

Gene regulation by chronic hyperglycemia

To further elucidate the molecular mechanism behind the effects of hyperglycemia on glucose and lipid metabolism, we examined by real-time RT-PCR and microarray analysis, gene expression in NG and HG cells. Some of the essential genes involved in carbohydrate (ChREBP, PDK-4) and fatty acid metabolism (FAS, SCD-1, ACC) and fatty acid uptake (CD36) were examined by real-time RT-PCR (Fig. 6). HG did not affect the gene expression level of any of the examined genes. Neither were genes associated with mitochondrial function (CPT1, CYC1, UCP-2 and UCP-3) changed (Fig. 6). The gene 36B4 was used as reference gene, and similar results were obtained when gene expression were related to the reference gene glyceraldehyd-3-phosphate dehydrogenase (GAPDH) (data not shown).

Microarray analysis of gene expression surprisingly showed that no genes were significantly

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regulated by 4 days of chronic HG (data not shown, Illumina arrays with > 48 000 probes, GSE19620). However, it must be emphasized that cells from only 3 subjects were included in these experiments, and that might be too few to detect small changes.

Mitochondrial DNA, ATP and lactate concentration

Since mitochondrial capacity seemed to be reduced after chronic HG compared to NG, we wanted to examine the mitochondrial content, by measuring mitochondrial DNA, of the myotubes. Mitochondrial DNA relative to nuclear DNA was not altered by HG (Fig. 7A).

However, cells exposed to HG contained 25 % less ATP than NG cells (Fig. 7B, p=0.03).

To examine whether the HG effect could be explained by an increased production of lactate, myotubes were pretreated with sodium lactate (1 or 10 mmol/l) for 1-4 days before glucose oxidation was measured (Fig. 8A). The effect of HG on glucose oxidation was mimicked by lactate, at 10 mmol/l lactate for 4 days glucose oxidation was reduced by 44 % (p=0.03). The effect of lactate on glucose uptake was similar to the effects on oxidation (Fig. 8B). Preincubation with sodium lactate (10 mmol/l) also reduced OA oxidation to the same extent as HG (data not shown). Addition of sodium lactate to the cell media did not change pH. For comparison the effect of lactic acid (0.01, 0.1 or 0.2 mmol/l) was examined on glucose oxidation and uptake, and a similar effect as of HG and lactate treatments was observed (Fig. 8C andD). In this case, pH of the culture medium was reduced to 6.8. The calculated concentration of non-dissociated lactic acid in the media was similar after addition of lactate and lactic acid. To exclude an effect of acidification, HCl was added in an amount that gave the same reduction in pH as lactic acid, but HCl had no effect on glucose oxidation (data not shown). Insulin-stimulated glucose uptake was abolished by HG (Fig. 9). Lactate seemed to reduce the insulin responses as well, although not significantly (p=0.14). However, the lactate concentration in the cell media was only slightly increased in cells exposed to HG for 4 days, 4.0 (± 0.6) mmol/l versus 3.7 (± 0.5) mmol/l in media from NG cells (n=5, p=0.08). The average pH value in the cell media was similar, 7.2 (± 0.2) in both HG and NG.

This implied a specific effect of lactate independent of pH.

Impaired glucose oxidation, reduced ATP concentrations and increased lactate production could be explained by a dysfunctional pyruvate dehydrogenase (PDH) complex.

To test whether HG impaired PDH activity, glucose oxidation was measured after incubation with a new PDHK inhibitor, AZD7545 (10-100 nM for 2-4 days). The HG-induced reduction of glucose oxidation was not abolished by the PDHK inhibitor. Glucose oxidation was reduced by 22 % after chronic HG to 13.8 (± 0.9) nmol CO 2 /mg protein in cells without

Activation of AMPK

AMPK is known to be an important energy sensor in cells. The involvement of AMPK in HG-treated myotubes was therefore of interest. HG had no effect on either AMPK or ACC phosphorylation compared to NG control cells (data not shown). However, pretreatment with deoxyglucose (DOG) increased phosphorylation of both AMPK and ACC to the same extent in NG and HG cells, showing responses to acute energy depletion.

Effect of chronic hyperglycemia on protein synthesis

To explore the effect of HG on cellular viability and energy requirement, protein synthesis was assessed as [ 14 C(U)]leucine uptake and accumulation, measured by a scintillation proximity assay (SPA). Protein synthesis was about 2-fold increased after chronic HG (p=0.02 overall effect), indicating that hyperglycemia did not cause a reduced energy requirement (Fig. 10). Leucine uptake and accumulation was stimulated by insulin (100 nmol/l) in both NG and HG cells. After 10 h insulin-stimulated [ 14 C(U)]leucine accumulation was increased by 35.6 (± 7.4) nmol/mg protein in NG cells and by 24.7 (± 10.2) nmol/mg protein in HG cells, showing an impaired relative insulin response on protein synthesis in HGtreated cells (p=0.04, data not shown).

DISCUSSION

In the present study we showed that chronic exposure of differentiated human skeletal muscle cells (myotubes) to hyperglycemia (HG) significantly reduced both glucose and oleic acid oxidation and reduced the suppressive effect of acute glucose on oleic acid oxidation. Glucose oxidation in presence of a mitochondrial uncoupler (DNP) was also markedly reduced, and the same was cellular ATP concentration. Mitochondrial content, as measured by mitochondrial DNA, was unchanged and so was the expression level of several mitochondrial genes. Moreover, HG did not affect the expression level of any genes measured by real-time RT-PCR or microarray. Addition of lactate to control cells mimicked the effect of HG on inducing insulin resistance and reducing substrate oxidation. In addition, lactate concentration in cell media was slightly increased after hyperglycemia. These results imply that chronic hyperglycemia can induce impaired metabolic switching and reduce substrate oxidation in healthy myotubes without changing gene expression.

Metabolic switching of myotubes, characterized by the parameters suppressibility and adaptability, has been suggested to be of intrinsic origin and to correlate with in vivo characteristics [START_REF] Ukropcova | Dynamic changes in fat oxidation in human primary myocytes mirror metabolic characteristics of the donor[END_REF]. However, metabolic switching can also be affected by extrinsic factors, as we recently showed that n-3 fatty acids improve metabolic switching of myotubes [START_REF] Hessvik | Metabolic switching of human myotubes is improved by n-3 fatty acids[END_REF]. In the present study, glucose suppressibility (the suppressive effect of acute glucose on oleic acid oxidation) was significantly reduced by chronic HG. Adaptability (the capacity to increase fatty acid oxidation with increasing fatty acid availability) seemed to be reduced by HG exposure, although not significantly. This induced impairment of metabolic switching was not observed in the opposite situation; oleic acid suppressibility of glucose oxidation at baseline conditions. However, in presence of DNP, at maximal mitochondrial electron transport system (ETS) capacity, a suppressive effect of oleic acid on glucose oxidation was evident after both NG and chronic HG pretreatment of the cells, although significantly lower after HG. A possible explanation could be that glycolytic metabolism is preferred in these cells, and that a substrate competition was evident only at maximum flux through the ETS caused by DNP.

The acute glucose suppression of lipid oxidation has similarities with the switch from fatty acid to carbohydrate oxidation in the postprandial phase and at high intensity exercise.

The mechanism behind this substrate switch is not known, but the suppressive effect on CO 2 but not on ASM indicates an effect downstream of fatty acid -oxidation. The mechanism is probably due to a mitochondrial competition between carbohydrates and fatty acids. This could be mediated by malonyl-CoA inhibiting CPT1 and thereby entry and oxidation of fatty acids in mitochondria [START_REF] Sidossis | Glucose and insulin-induced inhibition of fatty acid oxidation: the glucose-fatty acid cycle reversed[END_REF][START_REF] Hue | The Randle cycle revisited: a new head for an old hat[END_REF]. Another explanation focus on the ETS, and is based on that catabolism of carbohydrates produces redox equivalents (NADH) feeding electrons into complex I, whereas catabolism of fatty acids produces redox equivalents feeding electrons into both complex I and II. Substrate competition at ETS level could possibly be explained by differences in relative activity of the complexes and in feedback control on PDH and βoxidation by redox state [START_REF] Sahlin | Turning down lipid oxidation during heavy exercise--what is the mechanism?[END_REF]. Our results on substrate oxidation in presence of DNP showed that maximal oxidation capacity of glucose, but not OA, was compromised by HG, suggesting a defect at the complex I level or upstream. Measurements of mitochondrial function using OA as substrate further revealed that 21 % less of ETS capacity was activated for ATP production after HG treatment.

The present results indicate that chronic exposure of myotubes to high glucose concentration impairs mitochondrial function. Complete oxidation of glucose and oleic acid to carbondioxide was reduced. Incomplete oxidation of oleic acid, i.e. production of acid-soluble metabolites (ASM), was increased, as was oleic acid distribution into intracellular triacylglycerol (TAG), in accordance with previous findings [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF]. The increased storage of oleic acid as TAG could partly explain why HG suppresses oleic acid oxidation. Decreased complete oxidation in combination with increased incomplete oxidation of fatty acids as observed after chronic HG exposure has also been reported to occur in myotubes from obese type 2 diabetic individuals [START_REF] Wensaas | Fatty acid incubation of myotubes from humans with type 2 diabetes leads to enhanced release of betaoxidation products because of impaired fatty acid oxidation: effects of tetradecylthioacetic acid and eicosapentaenoic acid[END_REF]. Moreover, an imbalance between β-oxidation and TCA cycle or electron transport chain activity has been suggested to lead to increased incomplete fatty acid oxidation and accumulation of lipid-derived metabolites in the mitochondria, further resulting in mitochondrial stress and development of skeletal muscle insulin resistance [START_REF] Muoio | Obesity-related derangements in metabolic regulation[END_REF][START_REF] Muoio | Mechanisms of disease: molecular and metabolic mechanisms of insulin resistance and beta-cell failure in type 2 diabetes[END_REF][START_REF] Koves | Mitochondrial overload and incomplete fatty acid oxidation contribute to skeletal muscle insulin resistance[END_REF]. Thus, the observed impaired substrate oxidation might be due to an affected step somewhere in the TCA cycle or electron transport chain machinery, leading to accumulation of lipid-derived metabolites in the mitochondria and possibly impaired metabolic switching.

Glycolytic products such as lactate and pyruvate might play a regulatory role in the mechanism behind metabolic switching. The effect of HG on glucose uptake and oxidation was mimicked by extracellular addition of lactate. The activity of CPT-1 is strongly pHdependent, and could possibly be inhibited by lactate [START_REF] Starritt | Sensitivity of CPT I to malonyl-CoA in trained and untrained human skeletal muscle[END_REF]. However, in the present study addition of lactate also inhibited glucose oxidation and this effect was rather specific since acidification by HCl did not decrease glucose oxidation. In addition, lactate did not change the pH of the cell culture medium. A specific effect of lactate is also supported by the lactic acid experiments. Recently, lactate was found to be the ligand for the orphan G-protein-

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

coupled receptor GPR81, and it was shown that lactate regulated adipocyte lipolysis through this receptor [START_REF] Liu | Lactate inhibits lipolysis in fat cells through activation of an orphan G-protein-coupled receptor, GPR81[END_REF]. In muscle lactate is known to decrease glucose consumption and glycolytic flux [START_REF] Costa Leite | Lactate favours the dissociation of skeletal muscle 6-phosphofructo-1-kinase tetramers downregulating the enzyme and muscle glycolysis[END_REF]. Although lactate concentration was only slightly increased in cell media of myotubes exposed to chronic HG, it suggests that glucose oversupply pushes the glycolytic pathway. Intramyocellular lactate concentration was not measured, but could be increased to a greater extent. Possibly, recycling of lactate back into glucose could also explain the reduced ATP level, and there exist evidence that this futile cycle is active in myotubes [START_REF] Jin | Evidence for reverse flux through pyruvate kinase in skeletal muscle[END_REF]. An increased lactate production can be caused by increased lactate dehydrogenase activity, reduced activity of pyruvate dehydrogenase (PDH) or lack of acetyl CoA. PDH activity is controlled by PDH kinase, and if HG increased PDHK activity and thereby reduced PDH activity, an inhibitor of PDHK might then relieve the HG effect. The PDHK inhibitor, AZD7545, however showed no effect on the reduced glucose oxidation caused by HG.

Changed PDH activity by any other mechanism can however not be excluded by these experiments. Whether the effect of HG is mediated by lactate is uncertain, and should be more thoroughly studied. Any effect of lactate or chronic HG on lipolysis in muscle is not known and should also be examined. The present finding of increased storage of oleic acid as triacylglycerol (TAG), together with previous observations of increased TAG content in myotubes treated with chronic hyperglycemia [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF] could imply that lipolysis might be affected. In theory, glucose oversupply could also possibly increase cell glycogen content and thereby favor utilization of glucose derived from stored glycogen and suppress lipid oxidation. However, based on previous results we have no reasons to believe that glycogen content is increased after HG in absence of insulin [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF].

The reduced mitochondrial capacity could not be explained by altered gene expression or by a decreased number of mitochondria. In fact, the protein synthesis experiments showed that the cells in energy surplus were more actively accumulating leucine, indicating high metabolic activity. This is in accordance with a previous study showing increased protein synthesis in hyperglycemic diabetic patients [START_REF] Gougeon | Effects of oral hypoglycemic agents and diet on protein metabolism in type 2 diabetes[END_REF]. Resistance towards insulin-stimulated protein anabolism has also been reported in type 2 diabetics [START_REF] Pereira | Insulin resistance of protein metabolism in type 2 diabetes[END_REF]. Insulin-stimulated protein synthesis was reduced in cells pretreated with HG confirming insulin resistance, which was also observed on glucose uptake. Why the muscle cells then turn down energy production (ATP) and increases energy storage is not known. Of course, other energy demanding processes than protein synthesis could be affected by HG, such as calcium recycling or other membrane-associated pumping activities.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

19 Surprisingly, no genes were found to be significantly regulated by chronic HG. These results suggest that the metabolic effects of HG might be mediated by altered protein activities caused by some form of post-translational modification, e.g. phosphorylations, glycations or acetylations. Previously, we have reported that neither phosphorylation of PKB [START_REF] Aas | Chronic hyperglycaemia promotes lipogenesis and triacylglycerol accumulation in human skeletal muscle cells[END_REF] nor ERK [START_REF] Aas | Lipid metabolism in human skeletal muscle cells: effects of palmitate and chronic hyperglycaemia[END_REF] is affected by chronic HG. Glucose infusion to rats has been shown to induce increased PKC activation [START_REF] Houdali | Prolonged glucose infusion into conscious rats inhibits early steps in insulin signalling and induces translocation of GLUT4 and protein kinase C in skeletal muscle[END_REF] and to reduce phosphorylation and activation of AMPK, an important sensor of energy status and regulator of energy metabolism in cells [START_REF] Kraegen | Increased malonyl-CoA and diacylglycerol content and reduced AMPK activity accompany insulin resistance induced by glucose infusion in muscle and liver of rats[END_REF]. In the present study, chronic HG did not change phosphorylation of AMPK or its substrate ACC.

These data suggest that although energy uptake and oxidation is reduced, the cells are in an energy neutral state. Protein expression and activity of diacylglycerol kinase has been shown to be decreased in type 2 diabetic subjects and normalized upon correction of hyperglycemia [START_REF] Chibalin | Downregulation of diacylglycerol kinase delta contributes to hyperglycemia-induced insulin resistance[END_REF]. This kinase is thought to be involved in the metabolic switch between lipid and glucose metabolism and might play a role in the metabolic inflexibility seen in type 2 diabetes. Other enzymatic changes and protein modifications by HG such as glycosylations or glycations can not be excluded and are currently under investigation.

In conclusion, chronic hyperglycemia impairs substrate oxidation in skeletal muscle cells and reduces the metabolic switching of the cells. This shows that hyperglycemia can induce a diabetic-resembling phenotype in former healthy cells. The effect is most likely due to an induced mitochondrial dysfunction or post-translational modifications.

A C C E P T E

ACCEPTED MANUSCRIPT

Research highlights of the manuscript "Chronic hyperglycemia reduces substrate oxidation and impairs metabolic switching of human myotubes"; -Chronic hyperglycemia (HG, 20 mmol/l glucose for 4 days) reduced both glucose oxidation and oleic acid oxidation by 39 % and 47 %, respectively. -Glucose suppressibility of oleic acid oxidation was significantly reduced after HG, from 50 % in control cells to 21 % after HG. -Incomplete oxidation of oleic acid was increased by 22 % after HG.

-The maximal mitochondrial capacity was reduced after HG without any changes in mitochondrial content or mitochondrial gene expression. Assessed by micro array, no genes at all were significantly regulated by HG after 4 days. -The electron transport system capacity activated for ATP production was significantly lowered by HG, and the myotube content of ATP was also reduced. -These effects of HG were not followed by activation of AMPK.

-However, the effect of HG could be mimicked by addition of lactate.

-The HG treated cells were not in a state of low energy requirement, since protein synthesis in fact was increased after HG.

In conclusion, chronic hyperglycemia reduces substrate oxidation and impairs metabolic switching. The effect seems to be mediated by some mitochondrial defect without changes in mitochondrial gene expression or mitochondrial content.

  , penicillin/streptomycin (10,000 IE/10 mg/ml), HEPES, amphotericin B, ), oligomycin and extracellular matrix (ECM) gel were from Sigma (St.Louis, MO, USA). Dulbecco's Modified Eagle's Medium low glucose (DMEM), DMEM-Glutamax, foetal calf serum (FCS), Dulbecco's Phosphate Buffered Saline (DPBS) and trypsin/EDTA (0.05 %) were from Gibco/Invitrogen (Grand Island, NY, USA). Ultroser G was from Pall Corporation (St-Germain-en-Laye Cedex France) and lactic acid was from Apotekproduksjon AS (Oslo, Norway). D-[6-14 C]glucose (56 mCi/mmol -2.1 GBq/mmol), D-[1-14 C]glucose (54 mCi/mmol -2.0 GBq/mmol), D-[ 14 C(U)]glucose (10 mCi/mmol -0.37 GBq/mmol), [1-14 C]oleic acid (53 mCi/mmol -2.0 GBq/mmol) and L-[ 14 C(U)]leucine (318 mCi/mmol -11.766 GBq/mmol) were provided by either by American Radiolabeled Chemicals Inc. (St.Louis, MO, USA) or by NEN Radiochemicals, PerkinElmer (Boston, MA, USA). Insulin Actrapid A C C E P T E D M A N U S C R I P T was from Novo Nordisk (Bagsvaerd, Denmark). The antibodies against AMPK (#2532), phosphorylated AMPK (Thr172, #2531), ACC (#3662) and phosphorylated ACC (Ser79, #3661) were from Cell Signaling Technology (Beverly, MA, US). The pyruvate dehydrogenase kinase (PDHK) inhibitor AZD7545 was a kind gift from CVGI Research Area, AstraZeneca (Alderley Park, UK). Corning ® CellBIND ® microplates were from Corning B.V. Life Sciences (Schipol-Rijk, The Netherlands), 96-well UNIFILTER ® microplate from Whatman (Middlesex, UK), and 96-well Wallac Isoplates, TopSeal for 96-well micoplates and the scintillation liquid Optiphase Supermix was from PerkinElmer (Waltham, MA, USA). The ATPlite 1 step kit was from PerkinElmer Life and Analytical Sciences (Shelton, CT, USA

  s protocol. Hybridization, washing, and scanning of Illumina Human-6 Express BeadChips version 3 arrays (> 48 000 probes) were according to standard Illumina protocols (San Diego, CA, USA). Data extraction and quality control was performed using BeadStudio version 3.1.3.0 (Illumina) and the Gene Expression module 3.2.7. Arrays were normalized using quantile normalization, and expression estimates were calculated by GC robust multiarray average background adjustment. Fold change (FC) was calculated as log2transformed expression level after HG treatment divided by log2-transformed expression level

2. 7 .

 7 Mitochondrial DNA copy measurement DNA was isolated from whole cell lysates by PureGene DNA isolation kit (Qiagene, Germantown, MD, USA). Quantification of mitochondrial DNA (mtDNA) relative to nuclear DNA (nDNA) was done by real-time qPCR with TaqMan probes using a Stratagene 3000mxp. Primer and probe sequences were taken from He et al [32], Bogacka et al [33] and Bai et al [34], and confirmed with Primer3. Each target were quantified in triplicates and carried out in a 25 l reaction volume (12.5 µl 2 x qPCR mix (dNTP, HotGoldStar DNA polymerase, MgCl 2 (5 mmol/l), ROX), 0.75 µl fwd primer (300 nmol/l), 0.75 µl rew primer (300 nmol/l), 0.5 µl probe (200 nmol/l), 5 µl template (10 ng), 5.5 µl H 2 O). All assays were run for 95ºC in 10 min followed by 45 cycles of 95 C for 15 s and 60 C for 60 s. The threshold cycle number (Ct) was used for calculating the relative copy number (Rc) by expressing differences in Ct of the mtDNA and nDNA qPCR: Rc = 2 ∆Ct [35-36].

( 1 : 10 ( 2 . 9 .

 11029 1000) and phospho-ACC (1:1000) antibodies (Cell Signaling Technology, Beverly, MA, US) over night. Immunoreactive bands were visualized with enhanced chemiluminescence A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT ECL) (Amersham Biosciences, Buckinghamshire, UK), and quantified with Gel-Pro Analyzer (version 2.0) software. Intracellular ATP and lactate concentrations Intracellular concentration of ATP in NG and HG cells was measured with ATPlite 1step kit from PerkinElmer Life and Analytical Sciences (Shelton, CT, USA). Luminescence was detected with Victor 3 Wallac 1420 multilabel counter (PerkinElmer). ATP concentration was quantified according to a standard curve and given as A C C E P T E D M A N U S C R I P T M ATP/mg cell protein. DNP (100 µmol/l) which was added as a control, reduced ATP concentration by 30 %.

15 PDKH

 15 inhibition and by 23 % to 10.6 (± 1.8) nmol CO 2 /mg protein in myotubes pretreated for 4 days with a combination of HG and 100 nM AZD7545 (n=3). The activity of the inhibitor was confirmed by an acute stimulatory effect on glucose oxidation. AZD7545 (100 nM for 4 h) increased glucose oxidation by 45 % in NG cells and by 35 % in HG cells.

Figure 1 .

 1 Figure legends

Figure 2 . 14 C

 214 Figure 2. Oleic acid metabolism in normoglycemic (NG) and hyperglycemic (HG) cells in absence and presence of acute glucose (5 mmol/l). Myotubes were pretreated with 5.5 mmol/l glucose (NG) or 20 mmol/l glucose (HG) for 4 days before they were incubated with [1-14 C]oleic acid (OA) (37 kBq/ml, 20-300 µmol/l) for 4 h in absence or presence of acute glucose (5 mmol/l). CO 2 (A, B) was trapped in a filter and counted by liquid scintillation as described in Figure1. Uptake of OA was assessed as the sum of oxidized OA (trapped CO 2 ) and remaining cell-associated radioactivity (C, D). Data are given as mean ± SEM (n=3-5).

Figure 3 .

 3 Figure 3. Suppressibility (A) and adaptability (B) of oleic acid (OA) oxidation in NG and HG cells. Suppressibility was calculated as [1-(OA oxidation (100 µmol/l) in presence of 5mmol/l glucose/OA oxidation at 0 mmol/l glucose)]*100 %. Adaptability was calculated as fold increase in OA oxidation from 20 µmol/l OA, here 0 mmol/l and 5 mmol/l glucose are merged. These data were calculated from the results in Figure2. * p<0.05 vs. NG.

Figure 4 .

 4 Figure 4. Glucose metabolism at increasing oleic acid concentrations in normoglycemic (NG)and hyperglycemic (HG) cells. Myotubes were pretreated with 5.5 mmol/l glucose (NG) or 20 mmol/l glucose (HG) for 4 days before incubation with [6-14 C]glucose (37 kBq/ml, 5 mmol/l) for 4 h in presence of 0-300 µmol/l oleic acid during CO 2 trapping. Glucose uptake assessed as the sum of cell-associated glucose and trapped CO 2 (A) and CO 2 production from glucose (B) were assessed as described in Experimetal. Data are given as mean ± SEM (n=3).

Figure 5 .

 5 Figure 5. Oxidation of oleic acid and glucose in presence of the mitochondrial uncoupler 2.4dinitrophenol (DNP). Myotubes were pretreated with 5.5 mmol/l glucose (NG) or 20 mmol/l glucose (HG) for 4 days before they were incubated with radiolabeled substrates in the absence or presence of DNP (100 µmol/l) for 4 h. (A) [1-14 C]oleic acid (OA) (37 kBq/ml, 100 µmol/l) in absence or presence of acute glucose (5 mmol/l) and (B) [ 14 C(U)]glucose (37 kBq/ml) in absence or presence of acute OA (100 µmol/l). (C) Oleic acid suppressibility of

Figure 6 .

 6 Figure 6. Gene expression in myotubes after pretreatment with hyperglycemia (HG) for 4 days relative to normoglycemic (NG) cells. Data are given as mean ± SEM (n=4). Expression of the fatty acid transporter CD36, fatty acid synthase (FAS), stearoyl-CoA desaturase (SCD-1), acetyl CoA carboxylase (ACC), carbohydrate responsive element binding protein (ChREBP), cytochrome C (CYC1), carnitine palmitoyltransferase (CPT-1A, -1B), pyruvate dehydrogenase kinase (PDK-4) and uncoupling proteins (UCP-2, -3) were studied relative to the reference gene 36B4 by real-time PCR as described in Experimental.

Figure 7 .

 7 Figure 7. Mitochondrial DNA (A) and ATP concentration (B) in human myotubes. The myotubes were grown in normoglycemic medium (NG) or in medium containing 20 mmol/l glucose (HG) for 4 days before mitochondrial DNA (mtDNA) relative to nuclear DNA (nDNA) was measured by real-time PCR, and ATP concentration was measured with an ATPlite 1step kit as described in Experimental. Data are given as mean (± SEM) (n=5-7). * p<0.05 vs. NG.

Figure 8 .

 8 Figure 8. Effect of sodium lactate and lactic acid on glucose oxidation and uptake. Myotubes were preincubated with sodium lactate (1 and 10 mmol/l) (A, B) or lactic acid (10-200 µmol/l) (C, D) for 4 days, before CO 2 production from [ 14 C(U)]glucose (37 kBq/ml, 5 mmol/l) was measured for 4 h. Glucose uptake was assessed as the sum of cell-associated glucose and trapped CO 2 . Data are given as mean (± SEM) (n=4). * p<0.05 vs. NG.

Figure 9 . 29 Figure 10 . 30 Figure 1 A 39 Figure 10 A

 929103013910 Figure 9. Insulin-stimulated glucose uptake in myotubes. Myotubes were preincubated with normoglycemia (5.5 mmol/l glucose, NG), hyperglycemia (20 mmol/l glucose, HG), sodium lactate (10 mmol/l) or lactic acid (0.01 mmol/l) for 4 days, before insulin-stimulated (100 nmol/l) glucose uptake was measured. CO 2 production from [ 14 C(U)]glucose (37 kBq/ml, 5 mmol/l) was measured for 4 h, and glucose uptake was assessed as the sum of cell-associated glucose and trapped CO 2 . Insulin-stimulated glucose uptake is given as % of basal (glucose uptake in absence of insulin). Data are given as mean (± SEM) (n=4). * p<0.05 vs. NG.

ACCEPTED MANUSCRIPT 7 2.2. Human skeletal muscle cell cultures

	Satellite cells were isolated from the M. obliquus internus abdominis of 8 healthy donors, age
	39.9 (± 2.9) years, body mass index 23.5 (± 1.4) kg/m 2 , fasting glucose 5.3 (± 0.2) mmol/l, insulin, plasma lipids and blood pressure within normal range and no family history of diabetes. The biopsies were obtained with informed consent and approval by the National Committee for Research Ethics, Oslo, Norway. The cells were cultured in DMEM-Glutamax (5.5 mmol/l glucose) with 2 % FCS, 2 % Ultroser G, penicillin/streptomycin (P/S) and amphotericin B until 70-80 % confluence. Myoblast differentiation to myotubes was then induced by changing medium to DMEM-Glutamax (5.5 mmol/l glucose) with 2 % FCS, 34 pmol/l insulin, P/S and amphotericin B. Experiments were performed after 8 days of differentiation, and pre-incubation with hyperglycemia (20 mmol/l glucose) was started after M A N U S C R I P T 4 days.
	A C C E P T E D
	CYC1, CPT-1A, CPT-1B, UCP-2 and UCP-3 and the housekeeping genes GAPDH and 36B4
	were provided by Invitrogen (Carlsbad, CA, USA). SYBR green and TaqMan reverse
	transcription kit reagents were obtained from Applied Biosystems (Warrington, UK). Agilent
	Total RNA isolation kit was purchased from Agilent Technologies (Santa Clara, CA, USA).

). Bio-Rad Protein Assay Dye Reagent was from Bio-Rad Laboratories (NY, USA). Thin layer chromatography (TLC) plates (Silica gel) were from Merck (Darmstadt, Germany). The primers for CD36, FAS, SCD-1, ACC, ChREBP, PDK-4, Primers and probes for ND1 and B2M and qPCR MasterMix Plus Low ROX were from Eurogentec (Seraing, Belgium). Puregene DNA isolation kit was from Gentra Systems, Qiagen (Germantown, MD, USA).

  , glucose suppressed OA oxidation by about 50 % in NG cells, and 20 % in HG cells. There was no statistically significant difference between NG and HG in maximal fatty acid oxidation in presence of DNP. With DNP, acute glucose suppressed OA oxidation by 39 % in NG cells and by 24 % in HG cells. Additional experiments on OA oxidation were performed to further investigate mitochondrial energy production. By use of compounds

modifying substrate oxidation and calculating net routine flux control ratio (netR) (see Experimental) we observed that 52 % of electron transport system (ETS) capacity was activated for ATP production in NG cells, whereas this was decreased to 41 % after HG treatment (p=0.04, n=3).

ACKNOWLEDGEMENTS

We thank Mari-Ann Baltzersen and Christine Østraat Buen for excellent technical assistance, and Farshid Ezligini and Kjersti Roen Eriksen at Rikshospitalet University Hospital for doing the lactate measurements.

FUNDING

The present work was granted by AstraZeneca, The Norwegian Diabetes Foundation, Freia Chocolade Fabriks Medical Foundation, and The Anders Jahre's Foundation.


David Déharbe

Pascal Fontaine

Bruno Woltzenlogel

David Deharbe

P❛s❝❛ • ✳❋♦♥t❛✐♥❡❅•♦r✐❛✳❢r Bruno

Woltzenlogel Paleo

Quantifier Inference Rules for SMT proofs *

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In the typical architecture of an SMT-solver, the core automated reasoner is a propositional SAT-solver, and quantifiers are handled by independent modules [START_REF] Ge | Solving quantified verification conditions using satisfiability modulo theories[END_REF]. In veriT [START_REF] Bouton | veriT: an open, trustable and efficient SMT-solver[END_REF], essentially universal quantifiers are handled by an instantiation module, which heuristically chooses terms to instantiate such quantified variables. The instantiation module is called on-demand as rarely as possible (to reduce the number of generated instances) and only on essentially universally quantified subformulas. Essentially existential quantifiers, on the other hand, are handled by a skolemization module that is called only in a pre-processing phase and replaces all the essentially existentially quantified variables by skolem terms.

Currently, these modules are not proof-producing: if the input problem contains quantifiers that require skolemization, the proof produced by veriT will take as starting point the skolemized formula. If the instantiation module is called, generated instances will be used to deduce unsatisfiability, and the proof produced by veriT will contain holes. This paper discusses advantages and disadvantages of possible inference rules to handle quantifiers in the proof format of the SMT-solver veriT. We believe veriT's instantiation module behaves mostly like those in other solvers that handle quantifiers, e.g. CVC3 [START_REF] Barrett | CVC3[END_REF] or Z3 [START_REF] Mendonça De Moura | Z3: An efficient smt solver[END_REF]. We thus believe that the following discussion is relevant in the larger context of SMT solving for quantified formulas.

We aim at developing inference rules for skolemization and instantiation that take into account the following requirements:

• Proof size: the proofs produced by the skolemization and by the instantiation modules should be as short as possible, relative to the size of the formula that needs to be skolemized or instantiated.

• Faithfulness to the inner workings of the quantification modules: the proposed inference rules should reflect what actually happens inside the solver, so that they can also be used for precisely tracing executions; from a tool engineering perspective, this is important for debugging, profiling and maintainability.

• Ease of programming: our solver has a generic framework for proof production; it is desirable that the new inference rules comply with this framework.

• Compatibility with the proof format: veriT's input format follows the SMT-LIB standard; its output format also obeys the proposed proof format [START_REF] Besson | A flexible proof format for SMT: a proposal[END_REF]. Hence the inference rules should be expressible in this format.

• Fine-grainedness and simplicity: the inference rules should describe instantiation and skolemization in steps that are as small and simple as possible.

• Generality: inference rules having a broader range of applicability should be preferred to overspecialized inference rules that can be used only for specific purposes.

• Elegance: the proposed inference rules should fit into the style of rules already existing in the solver.

• Complexity of proof-checking: it should be possible for an external proof checker to efficiently check instances of the proposed inference rules.

• User-friendliness: the proposed inference rules should be suitable for the users of the solver and their applications.

In this paper we discuss some alternative inference rules, focusing on the more objective and more easily measurable criteria mentioned above (e.g. proof size, fine-grainedness and complexity of proofchecking). And we leave the more subjective criteria for future work.

The Proof Format

veriT's proof format follows a proposed format [START_REF] Besson | A flexible proof format for SMT: a proposal[END_REF] in the philosophy of the SMT-LIB standard [START_REF] Barrett | The SMT-LIB standard : Version 2.0[END_REF]. Its grammar is partially shown below. Clauses are sets of arbitrary formulas (not only literals), and inference rules have an arbitrary number of clauses as premises and a single clause as conclusion. Optionally, an inference rule may also take terms and attributes as arguments.

gen_clause :

:= clause_id | ✭ rule_id (✿❝•❛✉s❡s ✭ gen_clause * ✮ | ✿❛••✲❝•❛✉s❡s)? (✿t❡r♠s ✭ term * ✮)? attribute * (✿❝♦♥❝•✉s✐♦♥ clause )?✮ | ✭s✉❜♣r♦♦❢ proofstep * (✿❝♦♥❝•✉s✐♦♥ clause )? ✮
This document describes inference rules abstractly, using a proof-theoretical notation that is independent from any concrete proof format. The translation from this notation to the proof format is easy. An inference with the form

Γ 1 . . . Γ n Γ rule_id(term * ; attribute * ) becomes ✭rule_id (✿❝•❛✉s❡s ✭η(Γ 1 ) . . . η(Γ n )✮)? (✿t❡r♠s ✭ term * ✮)? attribute * (✿❝♦♥❝•✉s✐♦♥ Γ)?✮
where η(Γ i ) is either the clause id of Γ i or an inference rule instance that derives Γ i or a subproof that derives Γ i .

3 Rules for Instantiation of Essentially Universal Quantifiers

Essentially universal quantifiers are universal quantifiers that occur with positive polarity or existential quantifiers that occur with negative polarity. Essentially universally quantified variables can be instantiated by any term of the suitable sort. When a satisfying assignment that does not generate any theory conflict contains an essentially universally quantified formula, the instantiation module generates and returns singleton clauses whose only formulas are instances of an instantiation axiom schema. The instantiation terms are usually chosen by a heuristic based on E-matching [START_REF] Detlefs | Simplify: a theorem prover for program checking[END_REF][START_REF] Moskal | E-matching for fun and profit[END_REF]. This heuristic selects ground terms that appear in the literals composing the satisfying assignment, sometimes based on annotations called triggers (i.e. sets of term patterns). If a quantified formula is of the form ∀x

1 . . . . .∀x n .A[x 1 , . . . , x n ] or ¬∃x 1 . . . . .∃x n .A[x 1 , . . . , x n ] -with A[_]
being a formula not starting with an essentially universal quantifier -the instantiation module instantiates not only the first universally quantified variable x 1 , but all the variables x i at once. This instantiation heuristic (based on E-matching) is incomplete, and a simple clause set that is unsatisfiable but irrefutable due to this incompleteness is simple: {∀x.P(x) ; ∀y.¬P(y)}.

To find a refutation, the solver shall instantiate x and y to the same arbitrary term t and resolve the two unit clauses with each other. Taking these remarks into account, the most straightforward inference rules for the instantiation module would be:

∀ x.F( x) → F( a) forall_inst_axiom F( a) → ∃ x.F( x) exists_inst_axiom
where x denotes a sequence of variables x 1 , . . . , x n and a a sequence of terms a 1 , . . . , a n of suitable sort.

An obvious and easily implementable idea to improve these rules is to combine them with the clause form transformation rule for implication, as shown below. This reduces the size of proofs, since it eliminates the need to always apply the implication rules after the instantiation axioms.

¬∀ x.F( x), F( a) forall_inst_cnf_axiom ¬F( a), ∃ x.F( x) exists_inst_cnf_axiom
In first-order resolution proofs, it is usual to follow a convention that considers variables to be implicitly universally quantified. Universal quantifiers then simply do not (need to) appear in the proof. One might wonder if it would be desirable to adopt a similar convention in the presented proof format and rules. The answer is negative: because the SMT-LIB standard does not enforce any naming convention to distinguish identifiers for constants and for free variables, a proof checker would not be able to (easily) tell whether a given identifier (e.g. x) is a constant or a variable. With explicit rules for omiting the quantifiers, a more sophisticated proof-checker could keep track of which identifiers are variables. However, this would imply an undesirable loss of simplicity of the proof format and of the proof checker.

Rules for Skolemization of Essentially Existential Quantifiers

Essentially existential quantifiers are existential quantifiers that occur with positive polarity or universal quantifiers that occur with negative polarity. veriT eliminates them by skolemization during a preprocessing phase. The simplest solution would be to disregard this kind of pre-processing in the proof production. However, this would go against the style of veriT's proof format, since veriT does produce proofs for other pre-processing tasks such as clause form transformation. Another simple solution consists of having a single macro inference rule that skolemizes all essentially existential quantifiers:

F sk(F) skolemize_all
where sk(F) is any skolemization of F.

The rule skolemize_all is simple to implement in veriT and simple to check by an independent proof checker. The proof checker just needs to traverse F and sk(F) once, checking that each essentially existential quantifier is eliminated and the quantified variables it binds are replaced by skolem terms headed by skolem symbols that do not occur anywhere else in the proof and whose arguments are (a subset, depending on the skolemization algorithm used, of) variables bound by essentially universal quantifiers having scope over the eliminated quantifier. This rule is also convenient from the point of view of size, since it is clearly linear in the size |sk(F)| of sk(F). |sk(F)|, however, is in the worst case

Θ(|F| 2 ), if F is a tree-formula. To see that |sk(F)| is O(|F| 2
), just note that the number of essentially existential quantifiers in |F| is O(|F|) and each of these quantifiers is replaced by a skolem-term of size O(|F|). To see that in the worst case |sk(F)| is Ω(|F| 2 ), just consider the following example sequence:

F n = ∀x 1 . . . ∀x n ∃y 1 . . . ∃y n .P(x 1 , . . . , x n , y 1 , . . . , y n )
If F is a dag-formula, skolemization may require it to be transformed to an exponentially bigger treeformula first. In this case, the worst-case size of |sk(F)| is Θ(2 |F| )). An example where this happens is available in [START_REF] Fontaine | Techniques for Verification of Concurrent Systems with Invariants[END_REF].

However, the rule skolemize_all has the disadvantage of being very coarse-grained, since it skolemizes the whole formula at once. In trying to develop more fine-grained inference rules, it would be desirable to have something analogous to the rules forall_inst_axiom and exists_inst_axiom. This could be attempted with rules such as the following:

∃x.F(x) → F( f new (x 1 , . . . , x n )) exists_skolem_axiom F( f new (x 1 , . . . , x n )) → ∀x.F(x) forall_skolem_axiom ¬∃x.F(x), F( f new (x 1 , . . . , x n )) exists_skolem_cnf_axiom ¬F( f new (x 1 , . . . , x n )), ∀x.F(x) forall_skolem_cnf_axiom
where x 1 , . . . , x n are the free variables occurring in F(x), f new is a fresh new skolem symbol, not occurring anywhere else in the proof. The rules above are validity-preserving only if we just consider models in which f new (x 1 , . . . , x n ) has a fixed interpretation as the witness of the essentially existentially quantified variable it replaces (if such a witness exists). Otherwise, these rules are merely satisfiability-preserving.

In the case of instantiation of essentially universal quantifiers, when a quantified formula g (e.g. ∀x.F(x)) needs to be instantiated, g is one of the formulas in a clause c (e.g. Γ, ∀x.F(x)). So, after stating an instantiation axiom clause c ′ (e.g. ¬∀x.F(x), F(a)), we can do the actual instantiation of g in c simply by resolving c with c ′ . This replaces g by its instance in c. However, in the case of skolemization, a quantified formula g (e.g. ∃x.F(x)) may often occur not shallowly as a direct formula of a clause c but more deeply as a subformula of a formula in c (e.g. Γ, ∀y.∃x.F(x)). Therefore, replacing g by its instance in c cannot be done simply by resolution with a skolemization axiom. To overcome this problem, a deep version of resolution is proposed, so that one of the resolved formulas can occur arbitrarily deep inside another formula. This rule may be used in the more general case of the replacement of a deep occurrence of a subformula by another:

Γ, ¬F 1 , F 2 ∆, F + (F 1 ) Γ, ∆, F + (F 2 ) deep_resolution+ Γ, ¬F 1 , F 2 ∆, F -(F 2 ) Γ, ∆, F -(F 1 ) deep_resolution-
where the signs + andindicate the polarity of the annotated subformula.

Note that deep_resolution+ and deep_resolution-are analogous to deep applications of modus ponens and modus tollens, but fit better in the style of veriT's proof format, which is based on resolution.

Γ, F 1 → F 2 ∆, F + [F 1 ] Γ, ∆, F + [F 2 ] deep_modus_ponens Γ, F 1 → F 2 ∆, F -[F 2 ] Γ, ∆, F -[F 1 ] deep_modus_tollens Γ, F 1 ↔ F 2 ∆, F[F 1 ] Γ, ∆, F[F 2 ] deep_replacement
This approach with skolemization axioms and deep resolution has many problems, though. Firstly, there is a significant increase in the size of proofs: if m quantifiers need to be skolemized and for the sake of fine-grainedness a deep replacement is performed separately for each of the quantifiers, then there will be Θ(m) inferences, whose conclusions are of size O(|sk(F)|). Consequently, there is also a significant increase in the proof-checking time. Anti-prenexing the quantifiers as much as possible could reduce this problem in the average case.

Secondly and perhaps more seriously, proof-checking the skolemization axiom rules and the deep resolution rules depends on being capable of distinguishing identifiers of free variables and constants, and the SMT-LIB standard does not enforce any distinction. To understand this issue, consider the following formulas:

F 1 := ∀x.∃y.P(x, c, y) F 2 := ∀c.∀x.∃y.P(x, c, y)

and consider the following exists_skolem_axiom:

∃y P(x, c, y) → P(x, c, f new (x))
Proof-checking this axiom depends on being able to tell whether c is a variable or a constant, for if it were a variable, then c should also have been listed as an argument of f new . Moreover, both F 1 and F 2 might occur in a proof, and then c occurs both as a constant identifier and as a variable identifier. In such cases, the proof-checker should be able to accept this skolem axiom, it should be able to accept a deep resolution (or modus ponens) with F 1 (concluding ∀x.P(x, c, f new (x))), but it should also be able to reject an incorrect deep resolution (or modus ponens) with F 2 (unsoundly concluding ∀x.P(x, c, f new (x)) instead of ∀x.∀c.P(x, c, f new (x, c))). This simple example shows that the combination of skolem axioms, deep replacement rules and no distinction between identifiers for constants and for variables leads to unsoundness. This could be fixed by requiring non-local side conditions in the deep replacement rules, so that a deep replacement is only allowed if any skolem function symbol occuring in the replacing formula has as arguments all the identifiers that occur free in the replacing formula and that become bound after the replacement. Although this is technically feasible, it is questionable whether the increase in the complexity of proof-checking is a reasonable price to pay for the fine-grainedness and elegance provided by the deep replacement rules and the skolem axioms. This means that the proof checker would not be able the check the correctness of a skolemization axiom inference locally; it would not be able to verify whether the list of arguments of the skolem function, which ought to contain all the free variables (but not the constants) of F(x), is correct. The proof checker would only be able to tell whether the arguments of the skolem function are variables or constants when the deep resolutions are performed. Proof-checking would not be just local anymore, since the proof checker would need to keep track of some global correctness conditions. One way to keep proof-checking local, while still keeping fine-grainedness would be to combine the skolemization axiom rule and the deep resolution rule into a single unary inference rule, as follows:

Γ, G + [∃x.F(x)] Γ, G + [F( f new (x 1 , . . . , x n ))] exists_skolem Γ, G -[∀x.F(x)] Γ, G -[F( f new (x 1 , . . . , x n ))] forall_skolem
where x 1 , . . . , x n are the free variables of F(x) that are bound in G(Qx.F(x)).

Another approach would be to give up using skolem terms and use Hilbert's epsilon terms instead. A problem with this approach is that the size of the transformed formula epsilon(F) for a first-order formula F is in the worst-case Ω(2 |F| ) for tree-like F. This lower bound can be easily proved by considering the sequence of linearly growing formulas F n := ∃x 1 . . . ∃x n .P(x 1 , . . . , x n ) and checking that the sequence epsilon(F n ) grows indeed in Ω(2 n ). For dag-like F, another exponential blow up is possible, since the formula may need to be transformed to a tree, and hence the size is Ω( 2

2 |F| ). ∃ x.F( x) → F(ε x.F( x)) epsilon_axiom_1 F(ε x.¬F( x)) → ∀ x.F( x) epsilon_axiom_2
Yet another alternative worth considering would be to stop doing skolemization as a pre-processing step altogether, and do it only on demand, when an essentially existentially quantified formula occurs shallowly as a direct formula of a clause. In this case, no deep replacement would be necessary, and skolem terms would be always just skolem constants. Equivalently, strong quantifier rules or axioms that instantiate the essentially existentially quantified variables by eigen-variables could be used.

Conclusions

In this paper we have presented a few alternative inference rules for handling quantifiers in the proof format of veriT. We showed that each alternative has advantages but also disadvantages with respect to the requirements mentioned in the introduction. Therefore, we found none of the alternatives completely satisfactory.

Since it seems to be difficult to find the right balance to satisfy most if not all the requirements simultaneously, we have implemented some of these alternative rules in veriT selecting those that seemed to fit better within the existing proof style of that tool. The quantifier instantiation module produces instances of the rules forall_inst_axiom and exists_inst_axiom to justify lemmas that are added to the Boolean satisfiability solver. The clauses they introduce are then combined with existing rules for CNF transformation and resolution. Skolemization is applied to the input formula only on essentially existentially quantified variables occuring at the outermost level, and thus only produce skolem constants. Also, since the formulas generated by the quantifier instantiation module might reveal essentially existential quantifiers at the outermost level, skolemization is also applied to instances. In both cases, exists_skolem_cnf_axiom and forall_skolem_cnf_axiom are used in the proof, and the resulting clauses are further combined using deep_resolution rules.

This work was supported by the ANR DeCert and the SMT-SAVeS Projects


Francis Ribaud

Stéphane Vento

A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

Keywords: KdV-like equations, Cauchy problem AMS Classification: 35Q53, 35B65, 35Q60

In this note we study the generalized 2D Zakharov-Kuznetsov equations ∂ t u + ∆∂ x u + u k ∂ x u = 0 for k ≥ 2. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces

Introduction

In this short note, we are interested with the Cauchy problem associated to the generalized Zakharov-Kuznetsov (gZK) equations u t + ∆u x + u k u x = 0, (1.1) in two-dimensional space and for k = 2, 3, 4, .... These equations are natural multi-dimensional generalizations of the well-known generalized Kortewegde Vries equations and have been derived in [START_REF] Zakharov | On three dimensional solitons[END_REF] when k = 1 to model the propagation of nonlinear ionic-sonic waves in a magnetized plasma. We give sharp results concerning the well-posedness issue in standard Sobolev spaces H s (R 2 ) for suitable s ∈ R. This work follows and use similar technics as in the paper [START_REF] Ribaud | Well-posedness results for the 3D Zakharov-Kuznetsov equation[END_REF] where we proved that the 3D associated problem for k = 1 is locally well-posed in

H 1 + (R 3 ).
Remark that the Sobolev spaces Ḣs (R 2 ) are invariant by the natural rescaling of the equation as soon as s = s

k := 1 -2/k. Thus a natural question is whether (gZK) is well-posed in H s (R 2 ) for s > s k . 1 Theorem 1.1. For any u 0 ∈ H s (R 2 ) with    s > 1/4 if k = 2, s > 5/12 if k = 3, s > 1 -2/k if k ≥ 4,
there exist T > 0, a Banach space X s T and a unique solution u of the Cauchy problem associated to (1.1) with u(0

) = u 0 such that u ∈ X s T ∩ C b ([0, T ], H s (R 2 )). Moreover, the flow-map u 0 → u is Lipschitz on every bounded set of H s (R 2 ).
This theorem improves the recent works of Farah, Linares and Pastor in [START_REF] Farah | A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results[END_REF]- [START_REF] Linares | Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation[END_REF]- [START_REF] Linares | Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation[END_REF] where local well-posedness was obtained in

H s (R 2 ) for s > 3/4 if 2 ≤ k ≤ 8 and s > s k if k > 8.
In view of the ill-posedness result obtained in [START_REF] Linares | Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation[END_REF], Theorem 1.1 is optimal (up to the end point) for k ≥ 4 whereas in the particular cases k = 2 and 3, we still have a gap (respectively 1/4 and 1/12) compared with the scaling index. Concerning the end point s = s k , local well-posedness could perhaps be reached by following the strategy developed in [START_REF] Vento | Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space[END_REF], but with a flow-map only continuous.

In a standard way our proof is based on a fixed point scheme applied to the Duhamel formulation of (1.1):

u(t) = U (t)u 0 - 1 k + 1 t 0 U (t -t ′ )∂ x (u k+1 )(t ′ )dt ′ . (1.2)
where U (t) = e -t∆∂x denotes the propagator associated with the linear part of (1.1). Following the works of Kenig, Ponce and Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] on the KdV equation, we use in a crucial way some sharp dispersive estimates for free solutions. More precisely, these estimates are the well known Kato smoothing effect

∇U (t)ϕ L ∞ x L 2 yT ϕ L 2 , (1.3)
which allows to gain one derivative in each spatial direction, and the maximal in time estimate

U (t)ϕ L 4 x L ∞ yT ϕ H s , s > 3/4, (1.4)
proved in [START_REF] Linares | Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation[END_REF]. On the other hand, similarly to the generalized KdV equations, the previous bound is no more sufficient to deal with low non-linearities (k = 2, 3) and we need the following L 2 x based maximal in time estimate (see [START_REF] Faminskii | The Cauchy problem for the Zakharov-Kuznetsov equation[END_REF])

U (t)ϕ L 2 x L ∞ yT ϕ H s , s > 3/4. (1.5)
2 Proof of the main result 2.1 The case k ≥ 4

As mentioned in the introduction, we want to take advantage of the L 4

x L ∞ yT linear estimate (1.4). This motivates the choice of our resolution space:

X s T = {u ∈ C b ([0, T ], H s (R 2 )) : u X s T < ∞}, where u X s T = u L ∞ T H s xy + ∇ s+1 u L ∞ x L 2 yT + ∇ s-3/4 + u L 4 x L ∞ yT .
Combining estimates (1.3)-(1.4) as well as the straightforward bound

U (t)ϕ L ∞ T L 2 ϕ L 2 , (2.1)
we get

U (t)ϕ X s T ϕ H s . (2.2)
Note that the bound for the second term can be handled by using a low-high frequencies decomposition and next Bernstein inequality and estimate (1.3).

Having the linear part under control, it remains to deal with the integral term. It is not too hard to adapt the proofs of Propositions 3.5-3.6-3.7 in [START_REF] Ribaud | Well-posedness results for the 3D Zakharov-Kuznetsov equation[END_REF] to deduce

t 0 U (t -t ′ )∂ x u k+1 (t ′ )dt ′ X s T ∇ s-1 ∂ x u k+1 L 1 x L 2 yT . (2.3)
The multi-dimensional version of Theorem A.13 in [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] applies and leads to

∇ s-1 ∂ x u k+1 L 1 x L 2 yT ∇ s u k+1 L 1 x L 2 yT ∇ s u L 7 + x L 14/3 - yT u k L 7k/6 - x L 7k/2 + yT . (2.4)
We claim that the first product in the right hand side of (2.4) is controlled by the X s T norm of u. Indeed, an interpolation argument shows that

∇ α u L p x L q yT u X s T (2.5)
as soon as there exists θ ∈ [0, 1] such that

1 p = 1 -θ 4 , 1 q = θ 2 , α = s + 7θ -3 4 - .
Taking α = s, i.e. θ = 3/7 + , it follows that

∇ s u L 7 + x L 14/3 - yT u X s T .
If we choose now θ = 4/7k in (2.5), we infer

∇ (s-3/4+1/k) -u L ( 1 4 -1 7k ) -1 x L 7k/2 yT u X s T .
In order to get the desired contraction factor, we will interpolate this inequality with the bound

∇ (s-1) + u L ∞ - xyT T 0 + ∇ s u L ∞ T L 2 xy T 0 + u X s T . (2.6)
This leads to

∇ (s-3/4+1/k) -u L (( 1 4 -1 7k ) -1 ) + x L 7k/2 + yT T 0 + u X s T . (2.7)
By virtue of the Sobolev inequalities, we get

u L 7k/6 - x L 7k/2 + yT ∇ (1/4-1/k) + u L (( 1 4 -1 7k ) -1 ) + x L 7k/2 + yT ∇ (s-3/4+1/k) -u L (( 1 4 -1 7k ) -1 ) + x L 7k/2 + yT T 0 + u X s T (2.8) for s -3/4 + 1/k > 1/4 -1/k, that is s > s k . Gathering together (2.2)- (2.3)-(2.4)-(2.7
) and (2.8) we infer that

F (u) X s T u 0 H s + T 0 + u k+1 X s T ,
where F (u) denote the right hand side of (1.2). The well-posedness result follows then from standard arguments.

The case k = 2

The proof in this case follows the same lines as in the case k ≥ 4, but with the L 4

x norm replaced with a L 2 x maximal in time norm. So let us endow the X s T space with the norm

u X s T = u L ∞ T H s xy + ∇ s+1 u L ∞ x L 2 yT + ∇ s-3/4 + u L 2 x L ∞ yT ,
for any s > 1/4. Using now (1.5), we easily see that

U (t)u 0 X s T u 0 H s , (2.9)
and

t 0 U (t -t ′ )∂ x u 3 (t ′ )dt ′ X s T ∇ s-1 ∂ x u 3 L 1 x L 2 yT .
(2.10)

Again, the fractional Leibniz rule yields the bound

∇ s-1 ∂ x u 3 L 1 x L 2 yT ∇ s u L 7/2 + x L 14/3 - yT u 2 L 14/5 - x L 7 + yT .
By interpolation, we get

∇ α u L p x L q yT u X s T (2.11)
for α, p and q satisfying

1 p = 1 -θ 2 , 1 q = θ 2 , α = s + 7θ - 3 4
for 0 ≤ θ ≤ 1. On one hand, we deduce that

∇ s-1 ∂ x u 3 L 1 x L 2 yT u X s T ,
were we took θ = 3/7 + in (2.11). On the other hand, for θ = 2/7 -, we infer

∇ (s-1/4) -u L 14/5 - x L 7 + yT u X s T ,
which interpolated with (2.6) gives

∇ (s-1/4) -u L 14/5 - x L 7 + yT T 0 + u X s T .
This yields the desired result for s > 1/4.

The case k = 3

Now we consider the intermediate case k = 3. To prove our result, we define the resolution space as the intersection of the two previous spaces, i.e. equipped with the norm

u X s T = u L ∞ T H s xy + ∇ s+1 u L ∞ x L 2 yT + ∇ s-3/4 + u (L 2 x ∩L 4 x )L ∞ yT .
Actually we don't require the full range L 2 x ∩ L 4 x for the maximal in time norm, but only ∇ s-3/4 + u L 3

x L ∞ yT u X s T .

(2.12)

According to (2.2)-(2.3)-(2.9)-(2.10) we have

F (u) X s T u 0 H s + ∇ s u 4 L 1 x L 2 yT .
Using again the Leibniz rule for fractional derivatives, we infer .

∇ s u 4 L 1 x L 2
From an interpolation argument with (2.12), we easily check that both these norms are acceptable as soon as s > 5/12.


Frédéric Berthommier
email: frederic.berthommier@gipsa-lab.grenoble-inp.fr

Jean-Luc Schwartz
email: jean-luc.schwartz@gipsa-lab.grenoble-inp.fr

Audiovisual streaming in voicing perception: new evidence for a low-level interaction between audio and visual modalities

Keywords: auditory streaming, voicing perception, multisensory fusion, AV speech perception, scene analysis

published or not. The documents may come

Audiovisual streaming in voicing perception: new evidence for a low-level interaction between audio and visual modalities

Frédéric Berthommier, Jean-Luc

Introduction

The voicing cue has been used for showing an early interaction between audition and vision [START_REF] Schwartz | Seeing to hear better: Evidence for early audiovisual interactions in speech identification[END_REF] since this is a non visible feature. When added to crowd noise (irregular, unpredictable, composed of many overlapped speeches), the effect of vision of the lips was to favor the detection of the pre-voicing component (PVC) and then the grouping of the feature with the following consonant, then perceived as voiced. This effect due to the vision was attributed to the anticipatory movement of the lips, allowing focusing on the target PVC and the identification of the voiced consonant, instead of the unvoiced counterpart (e.g. /d/ vs. /t/). Moreover, the interfering noise carries little confusion with the target speech and no redundancy at all with visible movements of lips. The task was to discriminate speech and crowd noise similar to non speech. In another experiment using two concurrent speeches, it has been shown by [START_REF] Driver | Enhancement of selective listening by illusory mislocation of speech sounds due to lipreading[END_REF] that the intelligibility of the background speech can be increased by the vision of the foreground speaker. The confusion between foreground and background is also greatly decreased by the contribution of visual information and the segmentation of the two concurrent speeches is improved. This effect is rather uncontrolled and this is likely a consequence of phonetic integration. Recent experiments (Devergie et al., 2011) show an enhancement of the auditory streaming of vowel sequences by visible lip movement. The new paradigm we propose (Fig. 1) is based on a careful control of the confusion/intersect between foreground and background, as well as of the speech content, in order to analyze how this intersection is processed under the influence of vision thanks to the exclusive allocation rule [START_REF] Bregman | Auditory scene analysis[END_REF]: the target PVC will be assigned to the background stream or to the phoneme to form a /b/. New evidence in favor of low-level functional interactions between audition and vision are provided in the following.

Experiment procedure 2.1. Material

First, a continuum between /p/ and /b/ is realized by varying the amplitude of a unique pre-voicing segment obtained from a /b/. This target PVC is spliced with a /p/ at four different log-scaled amplitude levels [0.125 0.25 0.5 1] relative to the original level and the first element is the original /p/ (Fig. 2). Since this is log-scaled, the five indexes, from 1 to 5, are taken as quantitative support of the PVC amplitude (PVCA). Amplitude is set at 0 for PVCA=1 and at 1 for PVCA=5. Firstly, an audio-visual sequence of 8 /pa/ and 8 /ba/ items, randomized to avoid any visual bias, was recorded by a male speaker at a regular rhythm. The total duration is 49.2s and the time of occurrence of the 16 plosives was measured at the time of release. The audio content was generated by insertion of the same /pa/ sample at each of these 16 mounting points, so as to preserve the audiovisual synchrony. Then, the target PVC was added at the same points with a randomly selected level within the five amplitudes defined before. The syllable rate was about 0.3 Hz, allowing large pauses in which 34 isolated PVCs with constant amplitude at 0.125 (PVCA=2) have been inserted regularly in the "stream" condition so that the PVC rate including the target PVC was about 1Hz (Fig. 3). Using this principle, we prepared a 2*2 factorial design, Clear/Stream and Audio/AV, by concatenation of 20 sequences of 16 syllables, allowing a total of 320 stimuli. In the "Clear" condition there is no supplementary stream of PVCs. In the "Audio" condition, the generated audio is mounted with a static image of lips at intermediate aperture. In the "AV" condition this is mounted with the original video track, including lip-opening gestures. The 5 levels are randomly distributed across the "Clear" and the "Stream" condition separately so that the 2 distributions are paired between A and AV in order to avoid a bias. The 4 conditions are also randomized across the 20 sequences.

Experiment and data processing

The 16*4*5=320 target stimuli are presented with headphones in a soundproof room, in a same long movie having 16.4 mn duration, using the Presentation software with a 2 forced-choice response paradigm. The task was to press online a left arrow key for /p/ identification and a right arrow key for /b/, for each occurrence. Eleven subjects (5 males and 6 females) performed the experiment. They were not previously informed about the existence of the background stream so that they didn't pay too much attention to it because its amplitude is weak. The response time was also recorded and used for identification of missing responses as well as for further analysis. The response delay was calculated according to the mounting points described before. When 2 responses were given during the interval separating two consecutive mounting points, or when no response was given, the response was considered as missing. Only a few responses (15 altogether for all subjects) were missing.

Results and discussion

Analysis of identification responses

The average response rate for /p/ according to the PVC amplitude is displayed on Fig. 4. The /p/ response rate decreases rapidly when the amplitude of the target PVC increases for the "Clear" condition, for all conditions, but with shifts of the global curves between conditions. The shift of "Stream-Audio" (S-A) relative to "Clear-Audio" (C-A) is about one point of PVCA. Basically, this means that the target PVC needs to have an amplitude twice larger than the amplitude of the PVC competitors set at (PVCA=2) to "escape" from the stream and lead to a /b/ response at the same rate. Decrease is much slower with a sustained /p/ response rate in the S-AV condition, indicating a strong interaction with the visual input. It has been checked that visual /pa/ and /ba/ have the same response rates. The statistical data analysis is performed using a persubject probit fit for each of the four conditions, with PVCA as a controlled variable. For each subject, abscissa of intersect between the probit fit and the 50% response point (P50%) is measured for each of the 4 conditions. A three ways ANOVA is applied on these data with subject included as a factor to remove acrosssubject variability. The effect of the "Stream" factor is highly significant (p<0.001) and the interaction between A/AV and stream factor is also significant (p=0.011). The post-hoc analysis with error bars in Fig. 5 shows that S-A is significantly different from Clear conditions as well as from S-AV. The two clear conditions are not significantly different. This indicates that streaming has an effect in the audio modality, greatly reinforced in the audio-visual modality.

To explain the decrease of the /b/ response rate in the "Stream" condition, an adaptation mechanism could be invoked. With a high rate of voicing cue, the units responsive for /b/ could be saturated, leading to an increase in the rate of /p/ responses. Adaptation could occur either at a pre-phonetic level with units specific for the voicing features or at a phonetic level by adaptation of the units responsive for the voiced consonants, including the /b/. Adaptation at a phonetic level is discarded because there is no reason to observe a supplementary adaptation of the /b/ unit in the AV condition. On the other hand, the streaming of sounds according to their regularity is a well-known phenomenon (van Noorden, 1975, [START_REF] Bregman | Auditory scene analysis[END_REF]) and the target PVC could be embedded in the stream of the other PVCs because it is well interleaved (see Fig. 3). In this case, this is a general mechanism available for all sounds and the level of streaming could be the primary auditory cortex or lower auditory areas [START_REF] Micheyl | The role of auditory cortex in the formation of auditory streams[END_REF]. The effect of vision is either due to an increase of the embedding of the target PVC in the primary levels and by its exclusion at the higher phonetic levels. The presence of the target PVC in the speech stream is not apparent in the visual stream, which could favor its rejection because it is already integrated in the audio background in the primary levels. Whatever the underlying mechanism, this is a demonstration of interplay between a low-level auditory mechanism and multimodal integration for feature assignment.

Analysis of response time

The average response time over all subjects are plotted according to the PVCA in Figure 6. We observe that the curves have their minimum at PVCA=1 and 5 and a maximum in between. For each condition, the position of the maximum appears to be related to the P50% point present in Figure 4. In other words, the maximum of response time is related to the maximum of uncertainty. To assess this relationship, abscissa of the maximum (Max RT) has been evaluated subject by subject for each condition after a fine spline interpolation. This allows evaluating continuous values of PVCA after a local processing of the peaks. The plotting of the 4*11 maximal RT positions vs. the P50% points shows a strong linear relationship (Fig. 7). The correlation is 0.88 after removal of a unique spurious point (circled in Fig. 7). This high correlation shows that the response time is mainly explained by the degree of uncertainty and not by the involvement of supplementary delays, e.g. due to audiovisual integration. There is no difference between C-A and C-AV, and the difference between S-A and S-AV is related to the shift of the identification curve, not due to audiovisual integration.

Conclusion

We have found a strong interaction between the vision of the lips movements and the auditory streaming of the pre-voicing component. When a stream of PVCs is presented in the background, the target PVC is significantly embedded in this stream in the audio-only condition, and a greater proportion of /p/ is perceived, due to its exclusive allocation to the stream. The vision of the lips does not favor the speech specific grouping of the target PVC together with the simultaneous /p/ to form a /b/ but on the contrary aids to discard it from the foreground speech and to favor its embedding in the audio background stream (Fig. 8). Remarkably, the sequential grouping of the PVCs is favored against the simultaneous grouping of "target PVC" and /p/. This is because the /p/ only is visible and redundant (both sound and video carry information about the /p/). Then, the grouping of non-visible vs. visible components is improved, rather than the grouping of speech vs. background components, which is the alternative solution. This could be related to direct influences coming from multimodal regions on primary areas as well as the result of higher-level mechanisms. The target PVC might be simply gated in speech specific/phonetic levels under the influence of vision and then rejected to primary levels, without any per se interaction with primary auditory levels.

Figure 1 :

 1 Figure 1: Aim of the new paradigm.

Figure 2 :

 2 Figure 2: The /p/-/b/ continuum with PVC amplitude.

Figure 3 :

 3 Figure 3: Sample.wav "Stream" sequence with PVCA noted.

Figure 4 :

 4 Figure 4: Average results of identification responses.

Figure 5 :

 5 Figure 5: Interaction between the 2 factors.

Figure 6 :

 6 Figure 6: Response time vs. PVCA.

Figure 7 :

 7 Figure 7: Correlation between Max RT and P50%

Figure 8 :

 8 Figure 8: Effect of vision of the lips movement.

Acknowledgements

This work was supported by the French National Research Agency (ANR) through funding of the MULTISTAP project (MULTISTability and binding in Audition and sPeech: ANR-08-BLAN-0167 MULTISTAP).