Kin Sum Leung

Jean-Marie Galano

Yu Fung Yau

Camille Oger

Thierry Durand

Chung-Yung Lee

Walnut-Enriched Diet Elevated α-Linolenic Acid, Phytoprostanes, and Phytofurans in Rat Liver and Heart Tissues and Modulated Antiinflammatory Lipid Mediators in the Liver

Keywords: walnut, α-linolenic acid, phytoprostanes, phytofurans, neuroprostanes

α-Linolenic acid (ALA) and its non-enzymatic oxidized products, namely, phytoprostanes and phytofurans, are found in some nuts. The uptake and deposition of these compounds are not well-defined. Walnut has high ALA and a considerable amount of phytoprostanes and phytofurans compared to other common nuts. When fed to rodents, ALA and eicosapentaenoic acid levels increased in the liver and heart tissues compared to the control diet. Furthermore, phytoprostanes and phytofurans were elevated 3fold in both tissues after a walnut diet, indicating that they are not only contributed from the diet but also generated through in vivo autoxidation of ALA found in the walnuts. It was further noted that a walnut diet reduced 5-F 2t -isoprostanes and 12hydroxyeicosatetraenoic acid and induced 4-F 4t -neuroprostane and significant amounts of anti-inflammatory hydroxydocosahexaenoic acid in the liver only. Altogether, high ALA in a walnut diet elevated phytoprostanes and phytofurans in the liver and heart tissues and showed the regulation of anti-inflammatory lipid mediators in the liver only.

INTRODUCTION

Regular consumption of nuts alleviates cardiovascular risk factors, such as dyslipidemia, type 2 diabetes, and metabolic syndrome, and the risk of coronary heart disease (CHD). [START_REF] Sabate | Nut consumption and blood lipid levels: A pooled analysis of 25 intervention trials[END_REF][START_REF] Kendall | metabolic syndrome and diabetes[END_REF][START_REF] Luo | Nut consumption and risk of type 2 diabetes, cardiovascular disease, and all-cause mortality: A systematic review and meta-analysis[END_REF] Nuts are rich in unsaturated fatty acids, protein, fibers, plant sterols, minerals, and antioxidants, including vitamins and phenolic compounds. The macronutrient profiles in most nuts are similar. However, the ratios of monounsaturated and polyunsaturated fatty acids, types and amounts of phenolic compounds, vitamins, and minerals are different in each nut; therefore, consumption of different nuts contributes to different health benefits. [START_REF] Ros | Health benefits of nut consumption[END_REF] Walnuts are rich in fatty acids and contain a profound number of phytochemicals, including phenolic acids, flavonoids, lignin, tannins, etc. [START_REF] Vinson | Nuts, especially walnuts, have both antioxidant quantity and efficacy and exhibit significant potential health benefits[END_REF] Among all tree nuts, walnuts consist of the lowest ratio of saturated fatty acids (SFAs) and the highest ratio of omega-3 (n-3) and omega-6 (n-6) polyunsaturated fatty acids (PUFAs) to total fatty acids. [START_REF] Maguire | Fatty acid profile, tocopherol, squalene and phytosterol content of walnuts, almonds, peanuts, hazelnuts and the macadamia nut[END_REF] It is believed that walnut consumption is important for improving liver and heart health. Barrera et al. reported that continuous walnut intake for 3 months can reduce liver damage markers, including alanine transaminase (ALT), aspartate transaminase (AST) and γ-glutamyl transferase (GGT), in patients with non-alcoholic fatty liver disease, [START_REF] Barrera | The role of diet and nutritional intervention for the management of patients with NAFLD[END_REF] while Han et al. reported that low intake of nuts in the diet is associated with a significantly higher risk of non-alcoholic fatty liver disease. [START_REF] Han | Associations between intakes of individual nutrients or whole food groups and non-alcoholic fatty liver disease among Korean adults[END_REF] Multiple studies revealed that regular consumption of walnut (30-100 g per day) reduced the risk of cardiovascular diseases [START_REF] Manninen | Lipid alterations and decline in the incidence of coronary heart disease in the Helsinki Heart Study[END_REF][START_REF] Feldman | The scientific evidence for a beneficial health relationship between walnuts and coronary heart disease[END_REF] as well as decreased aortic endothelin 1 and other cardiovascular disease risk factors in hamsters. [START_REF] Davis | Walnuts reduce aortic ET-1 mRNA levels in hamsters fed a high-fat, atherogenic diet[END_REF] However, studies relating to walnut and its health benefits to the liver and heart are not conclusive.

Phytoprostanes (PhytoPs) are predominantly formed through non-enzymatic oxidation of α-linolenic acid (ALA). Insofar, researchers have successfully characterized different kinds of PhytoPs in plant foods, including gulupa shells, [START_REF] Medina | Quantification of phytoprostanesbioactive oxylipinsand phenolic compounds of Passif lora edulis Sims shell using UHPLC-QqQ-MS/MS and LC-IT-DAD-MS/MS[END_REF] leaves, [START_REF] Yonny | Thermal Stress in Melon Plants: Phytoprostanes and Phytofurans as Oxidative Stress Biomarkers and the Effect of Antioxidant Supplementation[END_REF] some common nuts, [START_REF] Carrasco-Del Amor | Impact of processing conditions on the phytoprostanes profile of three types of nut kernels[END_REF] and oil samples, [START_REF] Collado-Gonzalez | New UHPLC-QqQ-MS/MS method for quantitative and qualitative determination of free phytoprostanes in foodstuffs of commercial olive and sunflower oils[END_REF] which were further shown to be reliable in vivo oxidative stress biomarkers for plants. Recently, Minghetti et al. reported PhytoPs to be bioactive in mammalian cells. The group showed that B 1 -PhytoP enhanced metabolic activity and prevented human neuroblastoma (SH-SY5Y) cell death when induced with oxidative stress using H 2 O 2 . They also observed that 16-B 1 -PhytoP protected SH-SY5Y cells against oxidative injury by activating PPAR-γ, a ligand-dependent transcription factor that regulates several antioxidant enzymes. [START_REF] Minghetti | Nonenzymatic oxygenated metabolites of α-linolenic acid B1-and L1-phytoprostanes protect immature neurons from oxidant injury and promote differentiation of oligodendrocyte progenitors through PPAR-γ activation[END_REF] Recently, it was also discovered that glycerol-bound PhytoPs can contribute to allergic inflammation. [START_REF] Gonzaĺez Roldań | Lipid Mediators From Timothy Grass Pollen Contribute to the Effector Phase of Allergy and Prime Dendritic Cells for Glycolipid Presentation[END_REF] During the non-enzymatic oxidation process, when exposed to high oxygen tension, the release of phytofurans (PhytoFs) becomes predominant. We recently had access to PhytoFs thanks to our total synthesis work. [START_REF] Cuyamendous | Synthesis and discovery of phytofurans: Metabolites of α-linolenic acid peroxidation[END_REF][START_REF] Cuyamendous | The novelty of phytofurans, isofurans, dihomo-isofurans and neurofurans: Discovery, synthesis and potential application[END_REF] Regardless, the biological role of PhytoFs is not well-understood. To date, PhytoPs and PhytoFs were characterized in some common nut and seed samples [START_REF] Cuyamendous | Synthesis and discovery of phytofurans: Metabolites of α-linolenic acid peroxidation[END_REF] and leaves of a melon plant. [START_REF] Yonny | Thermal Stress in Melon Plants: Phytoprostanes and Phytofurans as Oxidative Stress Biomarkers and the Effect of Antioxidant Supplementation[END_REF] There is a lack of information in the development of PhytoPs and PhytoFs in vivo when food rich in ALA is consumed. Furthermore, it is unknown if ALA autoxidizes in vivo to produce PhytoPs and PhytoFs for a functional role or absorbed from food resources. In this study, we measured ALA, PhytoPs, and PhytoFs in some common nuts (almond, peanut, pine nut, and walnut) found in the Asian diet. Among the nuts, walnut had a rich content of ALA and the highest amount of total PhytoPs (9-D 1t -PhytoP, 9-F 1t -PhytoP, 16-F 1t -PhytoP, 9-L 1 -PhytoP, and 16-B 1 -PhytoP) and PhytoF [16(RS)-13-epi-ST-Δ 14 -9-PhytoF]. To further understand the uptake and absorption, a walnut diet was fed to Sprague Dawley (SD) rats and then profiled for n-6 and n-3 PUFAs and related lipid mediators in the liver and heart tissues using targeted lipidomics. Here, we report for the first time that ALA in walnut increased n-3 PUFA levels as well as its PhytoP and PhytoF metabolites in the liver and heart tissues. Our observation suggests that ALA from walnut contributed to the development of PhytoPs and PhytoFs in vivo and was not influenced by the content in walnut. Furthermore, rich ALA elevated anti-inflammatory lipid mediators and reduced inflammatory lipid mediators, suggesting its health benefits in the human diet.

MATERIALS AND METHODS

Materials. All solvents used in the experiments were highperformance liquid chromatography (HPLC)-grade. PUFA and oxidized lipid product standards and heavy labeled internal standards were purchased from Cayman Chemical Co. (Ann Arbor, MI, U.S.A.) or synthesized by the Institute des Biomolećules Max Mousseron (IBMM, Montpellier, France).

Animal Experiment. Eight-week-old male Sprague Dawley rats (n = 16) were obtained from Charles River Laboratories International, Wilmington, MA, U.S.A., and bred in the Animal House, School of Biological Sciences, The University of Hong Kong (HKU). The rats were randomly assigned into two groups, control and walnut, with 8 rats in each group. The control group was fed with Open Standard Diet D11112221 (Research Diets, New Brunswick, NJ, U.S.A.), and the walnut group were fed with a walnut-enriched diet (9.5% ground walnut, w/w). All diets were isocaloric (18% kcal of protein, 65% kcal of carbohydrate, and 16% kcal of fat) made by Research Diets, and the composition details are shown in Table S1 of the Supporting Information.

The use of live animals and the protocols of the animal studies were approved by the Department of Health of the Hong Kong Special Administrative Region of the People's Republic of China (HKSAR) Government and the committee of the Use of Live Animal in Teaching and Research of HKU (CULATR 3733-15). The rats were separated into groups of two in a solid-bottom polycarbonate cage with autoclaved hardwood bedding, with free access to deionized, reverse-osmosis-treated water and sterile rodent diets ad libitum. The animals experienced 12 h light and dark cycles in a room set to 20-25 °C with 35-70% relative humidity. The body weight, feed intake, and signs of abnormality were monitored throughout the study. The animal experiments commenced after 1 week of acclimatization and were complied with the local institutional ethical guidelines. After 28 days of feeding, the animals were sacrificed via cervical dislocation. Liver and heart tissues were immediately harvested, washed with phosphate-buffered saline (PBS) containing 0.005% (w/v) butylated hydroxytoluene (BHT) and 0.005% (w/v) indomethacin, and stored at -80 °C.

Lipid Extraction in Nuts and Animal Tissues. Four different kinds of common nuts, including almond (organic, U.S.A.), peanut (organic, China), pine nut (organic, Italy), and walnut (organic, shelled, Italy), were purchased from a local food market. These nuts were raw and organic, without any heat treatments or the addition of seasonings and food additives. The samples were finely ground by an electronic grinder (Kenwood AT320A), and PUFAs and oxidized lipid products were extracted by Folch extraction and purified by solid-phase extraction (SPE) according to previous studies, with modifications. [START_REF] Folch | A simple method for the isolation and purification of total lipides from animal tissues[END_REF][START_REF] Leung | Profiling of Omega-Polyunsaturated Fatty Acids and Their Oxidized Products in Salmon after Different Cooking Methods[END_REF] In brief, 0.05 g of ground nuts or animal tissues (liver and heart) were homogenized in 5 mL of cold Folch solution (chloroform/methanol, 2:1, v/v) with 0.01% BHT (w/v) at 24 000 rpm in 20 s bursts on ice using a polytron benchtop homogenizer (T25, Ultra-Turrax, IKA, Germany). After the homogenization, phase separation was induced by 2 mL of 0.9% aqueous sodium chloride. It was then centrifuged at 4 °C and 3000g for 10 min, and the lower organic phase was collected in a glass vial. The extracts were dried completely under a stream of nitrogen, and hydrolysis was performed using 1 mL of 1 M methanolic potassium hydroxide with 0.01% BHT, 1 mL of PBS (pH 7.4), and 100 μL of internal standard mix (Table S2 of the Supporting Information) in the dark overnight. The alkaline hydrolysis was terminated by adding 200 μL of 5 N hydrochloric acid, 0.5 mL of methanol, 2.7 mL of 40 mM aqueous formic acid, and 4 mL of 20 mM aqueous formic acid. The hydrolyzed samples were then purified by SPE using 60 mg MAX columns (Oasis, Waters, Milford, MA, U.S.A.). The purified samples were dried under a stream of nitrogen gas and resuspended afterward with 100 μL of acetonitrile. The final extracts were aliquoted into autosampler vials with an insert and were immediately analyzed using liquid chromatography-tandem mass spectrometry (LC-MS/MS).

Quantification of PUFAs and Oxidized Lipid Products by LC-MS/MS. PUFAs and oxidized lipid products were analyzed according to previous studies, with modifications. [START_REF] Leung | Profiling of Omega-Polyunsaturated Fatty Acids and Their Oxidized Products in Salmon after Different Cooking Methods[END_REF][START_REF] Dupuy | Simultaneous quantitative profiling of 20 isoprostanoids from omega-3 and omega-6 polyunsaturated fatty acids by LC-MS/ MS in various biological samples[END_REF][START_REF] Lee | Prenatal exposure to the contaminant perfluorooctane sulfonate elevates lipid peroxidation during mouse fetal development but not in the pregnant dam[END_REF] Briefly, LC-MS/MS analysis was performed using a 1290 Infinity LC system (Agilent Technologies, Santa Clara, CA, U.S.A.) with a Kinetex C18 column (150 × 21 mm, 2.6 μm, Phenomenex, Torrance, CA, U.S.A.). Milli-Q water/formic acid (99.9:0.1, v/v) and acetonitrile/formic acid (99.9:0.1, v/v) were used as solvents A and B, respectively, for the mobile phases in the LC system. The flow rate was 0.3 μL/min, and a stepwise gradient elution was set as follows: first, 20% B was kept for the first 2 min, then a gradual elevation of B to 98% was achieved from 2 to 10 min, and 98% B was maintained for 5 min. Finally, the mobile phase was set back to 20% B from 15 to 15.1 min and kept until 20 min to prepare for the next run. The autosampler was operated at 4 °C, and the injection volume of each sample was 10 μL.

The LC system coupled to a QTrap 3200 mass spectrometer (Sciex Applied Biosystems, Waltham, MA, U.S.A.) with electrospray ionization (ESI) was used. The ESI was run in negative atmospheric pressure chemical ionization (APCI) mode, and the analytes were detected by MS/MS using multiple reaction monitoring (MRM). The source temperature was set at 325 °C, the spray voltage was set to -3000 V and nitrogen gas was used as curtain gas. The concentration of the sample analytes was determined by associating the peak area of the analyte with the peak area of the corresponding internal standard. The deuterated form of the analytes where available was used as internal standards for quantitation. The complete set of m/z for each analyte is described in Table S2 of the Supporting Information. The m/z values are derived from the standards available commercially and synthesized by IBMM, and the others are from the LIPID MAPS database and literature reports. For analytes without deuterated standards, the peak area of the deuterated standards having the closest chemical structure were used as the internal standard for the quantitation of those analytes.

Reverse Transcription and Quantitative Real-Time Polymerase Chain Reaction (qPCR). Total RNA was extracted and purified from the rat heart tissue using TRIzol reagent (Invitrogen, Carlsbad, CA, U.S.A.), and the cDNA was generated from RNA using a high-capacity RNA-to-cDNA kit (Applied Biosystems, Foster City, CA, U.S.A.). The extraction procedures are according to the manual of the manufacturer. A real-time PCR reaction mixture was prepared by adding 1 μL of sample cDNA (∼40 ng/μL), 1 μL of forward and reverse primer mix (10 μM, Invitrogen, CA, U.S.A.), 3 μL of diethylpyrocarbonate (DEPC) water, and 5 μL of 2× SYBR green PCR master mix (Applied Biosystems, Foster City, CA, U.S.A.) to a 96-well plate. The reaction was run by a Step-One Plus real-time thermal cycler (Applied Biosystems, Foster City, CA, U.S.A.), and the PCR conditions were as follows: first, a hot start at 95 °C for 15 min, then denaturation with 40 cycles of 95 °C for 15 s, and finally annealing at 60 °C for 1 min. All of the samples were run in triplicate, and the experiments were repeated thrice for data confirmation. The results obtained were calculated using the ΔΔC q method and recorded as fold change. The primers used in this study (Table S3 of the Supporting Information) were in-house-designed and validated using 2% agarose gel electrophoresis. A single band represents that the primers have a high specificity, while two or more bands observed in the gel indicated that the primers need to be redesigned.

Statistical Analysis. All data were analyzed by GraphPad Prism (version 7 for PC, GraphPad Software, Inc., La Jolla, CA, U.S.A.) and reported as the mean ± standard deviation (SD). Data were analyzed by an unpaired, non-parametric t test (Mann-Whitney test, two tailed). A p value of <0.05 was denoted as statistically significant.

RESULTS

ALA, PhytoP, and PhytoF Contents of the Nut Samples. The concentrations of ALA and its oxidized lipid products PhytoPs and PhytoFs were quantified in four different common nuts purchased in local food markets (Figures 1 and2). Among the samples, walnut had the highest amount of ALA as well as total PhytoPs (Figure 1). Not all types of the PhytoPs were detectable in the nut samples (Figure 2). Only 9-F 1t -PhytoP, 9-L 1 -PhytoP, and 16(RS)-13-epi-ST-Δ 14 -9-PhytoF were measurable in the nut samples, where the highest concentration was 9-F 1t -PhytoP. Notably, 16-F 1t -PhytoP was only found in walnut. Altogether, walnut was the only sample that contained all types of PhytoPs and one PhytoF and the highest concentration of PhytoPs among the four nut samples. Therefore, it was selected for further investigation on the uptake in the liver and heart tissues.

Walnut Diet Did Not Affect Rat Weight Gain and Food Intake. The physical characteristics of the rats during the feeding period showed that the flavor and smell of the walnut enriched in the diet did not affect the appetite of the rats because there was no significant difference in chow consumption for both groups. In addition, since the diets were formulated to be isocaloric, the body weight gain of the rats between control and walnut groups also did not show a significant difference (Figure S1 of the Supporting Information). The average daily walnut diet consumption was 29.48 g; thus, the rats consumed 2.80 g of walnut per day containing approximately 0.27 g of ALA. For the control diet, soybean oil was the main fat contributor in the diet. The average daily control diet consumption was 26.99 g thus, 1.78 g of soybean oil per day was taken, which contained approximately 0.07 g of ALA. Therefore, the walnut diet contained almost 4 times higher ALA than the control diet.

n-3 PUFAs and Their Oxidized PUFAs Were Upregulated, while Those of n-6 PUFAs Were Downregulated by the Walnut Diet in the Liver Tissue. PUFAs and their oxidized products were measured in the rat liver to evaluate if the walnut-enriched diet altered the lipid metabolism. It was found that the walnut diet reduced n-6 PUFA levels, including arachidonic acid (AA) and adrenic acid (AdA), while elevated n-3 PUFA levels, including ALA, eicosapentaenoic acid (EPA), and n-3 docosapentaenoic acid (n-3 DPA), compared to the control. However, the docosahexaenoic acid (DHA) level was not modified by the walnut diet (Figure 3).

Similar to PUFAs, the walnut diet significantly downregulated n-6 PUFA-related enzymatic oxidized products, in particular 12-HETE, while it significantly upregulated n-3 PUFA-related enzymatic oxidized products, including 7-, 11-, and 17-HDHA, compared to the control. Levels of 5-, 8-, and 15-HETE and 4-and 14-HDHA were not modified by the walnut diet. Moreover, the total lipoxygenase (LOX)-mediated hydroxyeicosatetraenoic acid (HETE) level was significantly lower than total LOX-mediated hydroxydocosahexaenoic acids (HDHAs) after walnut diet consumption compared to the control (Table 1).

For non-enzymatic oxidized products of PUFAs, only the 5-F 2t -IsoP level derived from AA in the liver was significantly lowered by the walnut diet. There was no change for 5-F 3t -IsoP and 15-F 3t -IsoP levels derived from EPA in the liver by the walnut diet compared to the control (Table 2). Those from DHA, including 4-, 10-, and 13-F 4t -NeuroPs and 16-HDHA, and all PhytoPs (9-D 1t -PhytoP, 9-F 1t -PhytoP, 16-F 1t -PhytoP, 9-L 1 -PhytoP, and 16-B 1 -PhytoP) and PhytoF [16(RS)-13-epi-ST-Δ 14 -9-PhytoF] derived from ALA were significantly elevated by the walnut diet compared to the control (Table 2). Levels of 10-F 4t -NeuroP and 8-HDHA were not modified by the walnut diet.

Walnut Diet Mainly Altered PUFAs, PhytoPs, and PhytoFs in the Heart Tissue. The walnut diet significantly upregulated ALA, EPA, and DPA n-3 levels but not DHA, while it downregulated AA and AdA in the heart tissue compared to the control (Figure 4). For enzymatic oxidized products, only the 15-HETE level was found to be reduced and the 4-HDHA level was found to be elevated by the walnut diet compared to the control. Other enzymatic oxidized products (5-, 8-, and 12-HETE and 7-, 11-, 14-, and 17-HDHA) did not change in the heart tissue after the walnut diet compared to the control (Table 3).

For non-enzymatic oxidized products, all PhytoPs (9-D 1t -PhytoP, 9-F 1t -PhytoP, 16-F 1t -PhytoP, 9-L 1 -PhytoP, and 16-B 1 -PhytoP) and PhytoF [16(RS)-13-epi-ST-Δ 14 -9-PhytoF] levels, which are derived from ALA, showed significant elevation after the walnut diet (Table 4). Levels of nonenzymatic oxidized products derived from AA (5-F 2t -IsoP and 15-F 2t -IsoP), EPA (5-F 3t -IsoP and 15-F 3t -IsoP), and DHA (4-F 4t -and 10-F 4t -NeuroP and 8-, 10-, 13-, and 16-HDHA) did not have any remarkable changes by the walnut diet compared to the control (Table 4).

No Significant Changes Were Observed in Gene Expression Related to Inflammation and Antioxidant Enzyme after Walnut Supplementation. The measurement of the oxidized PUFA products indicates that anti-inflammatory (HDHA) and pro-inflammatory (HETE) lipid mediators were up-and downregulated, respectively, by the walnut diet. Further evaluation of gene expressions related to inflammation, including interleukin (IL)-6, IL-1β, and tumor necrosis factor α (TNF-α), and gene expression related to antioxidant enzyme catalase were elucidated using RT-PCR (Figure S3 of the Supporting Information). Surprisingly, none of the gene expressions was affected by the walnut diet compared to the control.

DISCUSSION

Long-chain n-3 PUFAs, including ALA, EPA, and DHA, are important to our health as they are essential fatty acids and also, protect against chronic diseases, such as CVD, neurodegenerative disease, and inflammatory-related diseases. [START_REF] Rajaram | Health benefits of plant-derived α-linolenic acid[END_REF][START_REF] Tapiero | Polyunsaturated fatty acids (PUFA) and eicosanoids in human health and pathologies[END_REF][START_REF] Calder | n-3 polyunsaturated fatty acids, inflammation, and inflammatory diseases[END_REF][START_REF] Thomas | Omega-3 Fatty Acids in Early Prevention of Inflammatory Neurodegenerative Disease: A Focus on Alzheimer's Disease[END_REF] Consumption of dietary n-3 PUFAs is necessary since ALA cannot be synthesized in the human body due to the absence of Δ 12 and Δ 15 desaturase enzymes. [START_REF] Burdge | Conversion of α-linolenic acid to longer-chain polyunsaturated fatty acids in human adults[END_REF] ALA can be obtained from plant-based food, such as plant oils, nuts, and seeds. However, it is known that the skipped diene bonds in the structure of ALA readily oxidize non-enzymatically to release PhytoPs and PhytoFs. [START_REF] Cuyamendous | Synthesis and discovery of phytofurans: Metabolites of α-linolenic acid peroxidation[END_REF][START_REF] Jahn | A cautionary note on the correct structure assignment of phytoprostanes and the emergence of a new prostane ring system[END_REF] Nonetheless, it is unclear if PhytoPs and PhytoFs found in vivo are contributed by the diet or attributed by the autoxidation of ALA. Presently, a handful of literature could be found related to PhytoPs, and the majority of the studies focused on oxidative stress and higher plants. [START_REF] Medina | Quantification of phytoprostanesbioactive oxylipinsand phenolic compounds of Passif lora edulis Sims shell using UHPLC-QqQ-MS/MS and LC-IT-DAD-MS/MS[END_REF][START_REF] Carrasco-Del Amor | Impact of processing conditions on the phytoprostanes profile of three types of nut kernels[END_REF][START_REF] Collado-Gonzalez | New UHPLC-QqQ-MS/MS method for quantitative and qualitative determination of free phytoprostanes in foodstuffs of commercial olive and sunflower oils[END_REF] Reports on PhytoFs are even less because they were only recently synthesized and discovered by Cuyamendous et al. [START_REF] Yonny | Thermal Stress in Melon Plants: Phytoprostanes and Phytofurans as Oxidative Stress Biomarkers and the Effect of Antioxidant Supplementation[END_REF][START_REF] Cuyamendous | Synthesis and discovery of phytofurans: Metabolites of α-linolenic acid peroxidation[END_REF][START_REF] Cuyamendous | The novelty of phytofurans, isofurans, dihomo-isofurans and neurofurans: Discovery, synthesis and potential application[END_REF] In this study, the levels of ALA, PhytoPs, and one PhytoF were measured in edible nuts, including almond, peanut, pine nut, and walnut. A high level of ALA (>10 μg/g) was found in walnut; a medium level of ALA (1-10 μg/g) was found in pine nut, and a low level of ALA (<0.1 μg/g) was found in almond and peanut. In and also all types of PhytoPs, whereas for PhytoF, all nuts contain similar levels. This is probably because the formation of PhytoFs requires high oxygen tension, which normally will not exist in common packaging or storage conditions of these nuts. [START_REF] Cuyamendous | The novelty of phytofurans, isofurans, dihomo-isofurans and neurofurans: Discovery, synthesis and potential application[END_REF] More so, it is likely that these nuts were not under oxidative stress during growth.

To elucidate the etiology of PhytoPs and PhytoF, walnut was chosen for the diet in the in vivo study because it had the highest ALA, PhytoP, and PhytoF levels among the nut samples tested. Both the liver and heart tissues were found to have high levels of n-3 PUFAs, except DHA, and lower levels of n-6 PUFAs compared to the control. The changes in the PUFA profile in both tissues were probably due to concentrated ALA in walnut. When absorbed, it is likely that ALA and linoleic acid (LA) competed in the metabolism for the same set of desaturating and elongating enzymes for the formation of EPA and DHA and AA, respectively. Studies showed that altering the ratio of ALA/LA in the diet can regulate the levels of n-3 and n-6 PUFAs in animal tissues. [START_REF] Blank | Optimizing DHA levels in piglets by lowering the linoleic acid to αlinolenic acid ratio[END_REF][START_REF] Tu | Omega-3 long chain fatty acid synthesis is regulated more by substrate levels than gene expression[END_REF] Consequently, as shown in our study, increasing ALA in the diet can upregulate the levels of ALA, n-3 DPA, and EPA in the tissues. Despite this, the walnut diet did not upregulate DHA in the tissues. In fact, several studies had same findings. Gibson et al. even reported that the DHA levels in plasma negatively correlated with the level of ALA in the diet. [START_REF] Gibson | Docosahexaenoic acid synthesis from α-linolenic acid is inhibited by diets high in polyunsaturated fatty acids. Prostaglandins, Leukotrienes Essent[END_REF] It is suggested that ALA can inhibit the formation of DHA by competing with the same enzyme for the conversion, i.e., Δ 6 desaturase, [START_REF] Voss | The metabolism of 7,10,13,16,19-docosapentaenoic acid to 4,7,10,13,16,19-docosahexaenoic acid in rat liver is independent of a 4-desaturase[END_REF] which is the reason why a high level of ALA in the diet can suppress DHA formation and accumulation in the tissues.

Increasing the levels of ALA in the diet also modified the oxidized PUFA products in the tissues. From our findings, it indicated that LOX-mediated HETEs generated from AA were reduced, while LOX-mediated HDHA generated from DHA increased. The reduction in pro-inflammatory LOX-mediated oxidized products from AA and the elevation in antiinflammatory LOX-mediated oxidized products from DHA may be due to the downregulation of AA in the liver after consuming a walnut diet.

Both Jump et al. and Endo et al. suggested that in vivo elevation of n-3 PUFAs, in particular EPA and DHA, through the diet will increase the assimilation of n-3 PUFAs into the sn2 position of membrane phospholipids. [START_REF] Jump | Omega-3 fatty acid supplementation and cardiovascular disease[END_REF][START_REF] Endo | Cardioprotective mechanism of omega-3 polyunsaturated fatty acids[END_REF] Because cellular phospholipase A2 will release the PUFAs from membrane phospholipids at the sn2 position for the substrate of cyclooxygenase (COX), LOX, and cytochrome P450 (CYP450) enzymes, the elevation of n-3 PUFAs at the sn2 position of the membrane phospholipid will upregulate the generation of anti-inflammatory oxidized products from n-3 PUFAs and also downregulate the generation of proinflammatory eicosanoids from n-6 PUFAs. Several cell studies also reported that EPA was able to compete with AA to generate less pro-aggregatory thromboxane TXA3 instead of TXA2 via COX and generate less pro-inflammatory eicosanoid LTB5 instead of LTB4 via LOX. [START_REF] Takamura | Long-Term Administration of Eicosapentaenoic Acid Improves Post-Myocardial Infarction Cardiac Remodeling in Mice by Regulating Macrophage Polarization[END_REF][START_REF] Lockette | Vascular reactivity and high dietary eicosapentaenoic acid[END_REF][START_REF] Terano | Biosynthesis and biological activity of leukotriene B5[END_REF] In addition, during the enzymatic oxidation process of AA, EPA, and DHA, they rely on the same set of enzymes, including LOX, COX, and CYP450. As a result, the reduction of AA by the walnut diet potentially increased the competition of EPA and DHA for these enzymes; therefore, it is plausible that the oxidation products of EPA and DHA are increased, even though DHA was not upregulated in the tissues.

Nonetheless, more types of eicosanoids, including prostaglandins, leukotrienes, and thromboxanes, from both n-3 and n-6 PUFAs need to be determined in the liver to have a full understanding on changes of enzymatic PUFA oxidation after increasing levels of ALA, PhytoPs, and PhytoFs from diets.

Non-enzymatic oxidized products from AA, mainly the 5-F 2t -IsoP level, were also reduced in the liver by the walnut diet and probably due to the reduction of AA and the antioxidative properties of phytochemicals in walnut. Walnuts contain high levels of numerous phytochemicals. Several literature reported that tannins, flavonoids, and phenolic acids in walnuts contribute to the antioxidative properties of walnut, which can help reduce oxidation of unsaturated fats. [START_REF] Carvalho | Human cancer cell antiproliferative and antioxidant activities of Juglans regia L. Food[END_REF][START_REF] Bolling | Tree nut phytochemicals: Composition, antioxidant capacity, bioactivity, impact factors. A systematic review of almonds, Brazils, cashews, hazelnuts, macadamias, pecans, pine nuts, pistachios and walnuts[END_REF] Non-enzymatic oxidized products from DHA, including 4-F 4t -NeuroP, 10-HDHA, 13-HDHA, and 16-HDHA, were augmented after the walnut diet, even though the diet did not elevate the DHA concentration in the liver. Milne et al. suggested that production of F 4 -NeuroPs from DHA is dominant compared to F 2 -IsoPs from AA in tissues that contain a higher concentration of DHA. [START_REF] Milne | Elevated oxidation of docosahexaenoic acid, 22:6 (n-3), in brain regions of rats undergoing ethanol withdrawal[END_REF] It is plausible that the reduction of AA as a result of the walnut diet increased the ratio of DHA/AA in the liver tissue, which subsequently increased the generation of 4-F 4t -NeuroP. The elevation of 4-F 4t -NeuroP potentially modulated HDHA in the liver, which was noted by Lee et al., who showed that infusion of 4-F 4t -NeuroP in rats upregulated HDHA in the liver and brain. [START_REF] Lee | Nonenzymatic oxygenated metabolite of docosahexaenoic acid, 4(RS)-4-F4t -neuroprostane, acts as a bioactive lipid molecule in neuronal cells[END_REF] Surprisingly, 4-F 4t -NeuroP and HDHA levels were unaffected by the walnut diet in the heart tissue. The disparity between the liver and heart tissues is unclear. Regardless, it was found by Roy et al. that 4-F 4t -NeuroP displayed antiarrhythmic property and protected the heart from ischemia-reperfusion injury. [START_REF] Roy | Nonenzymatic lipid mediators, neuroprostanes, exert the antiarrhythmic properties of docosahexaenoic acid[END_REF] On the other hand, the walnut diet successfully elevated n-3 PUFAs and reduced n-6 PUFAs in the heart tissue, suggesting that n-3 PUFAs, namely, ALA, EPA, and DPA n-3 , generated in the liver were transported to the heart tissue. Furthermore, 5-F 2t -IsoP but not 15-F 2t -IsoP was significantly reduced by the walnut diet in the liver; the latter is a standard biomarker to assess in vivo oxidative stress status. [START_REF] Morrow | The isoprostanes: Unique bioactive products of lipid peroxidation[END_REF] Additionally, both 5-F 2t -IsoP and 15-F 2t -IsoP were not modified by the walnut diet in the heart. To corroborate this observation, the gene expression of antioxidant enzyme catalase was evaluated in the heart tissue and was evidently shown not to be affected by the walnut diet. We further analyzed inflammation-related gene expressions, including IL-1β, IL-6, and TNF-α, in the heart tissue, and as like our findings for HDHA levels, they were not altered. Altogether, the walnut diet did not improve the antioxidant capacity nor anti-inflammation properties in the heart.

The most obvious findings with the walnut diet are the marked elevation of PhytoPs and PhtyoFs in both liver and heart tissues. The total PhytoP amount consumed from the walnut diet by the rats per week was approximately 42 ng on average, while the total amount of the PhytoPs in the liver tissues of the walnut diet group was 122 ng higher than the control group on average (data not shown); this is 3-fold higher than the weekly PhytoP consumption from the diet. It could be postulated that the elevation of PhytoPs in tissues was attributed by not only PhytoPs and PhytoFs in the walnut but also the ALA increment in the walnut diet.

This study suggested that regular consumption of walnut, containing high ALA, increases the n-3/n-6 PUFA ratio in the liver and heart tissues and induces non-enzymatic autoxidation of ALA to release PhytoPs. Furthermore, ALA induced anti-inflammatory (HDHA) and reduced pro-inflammatory (HETE) lipid mediators; whether this modification is related to PhytoPs or related to other antioxidant components, such as phytochemicals, of the walnuts requires further elucidation.

ASSOCIATED CONTENT

* sı Supporting Information

Figure 1 .

 1 Figure 1. Concentrations of ALA and total PhytoPs measured in common nuts. Data are presented as the mean ± SD (n = 4). The concentration is shown as micrograms per gram of nut samples for (A) ALA and nanograms per gram of nut samples for (B) total PhytoPs in nut samples.

Figure 2 .

 2 Figure 2. Concentrations of PhytoPs and PhytoF measured in common nuts. Data are presented as the mean ± SD (n = 4). The concentration is shown as nanograms per gram of nut samples.

Figure 3 .

 3 Figure 3. Levels of PUFAs measured in the liver tissue. Graphs represent (A) arachidonic acid, (B) adrenic acid, (C) α-linolenic acid, (D) eicosapentaenoic acid, (E) n-3 docosapentaenoic acid, and (F) docosahexaenoic acid measured in the rat liver tissue. The concentration is shown as micrograms per gram of tissue samples. Data are presented as the mean ± SD (n = 8). ( * * ) p < 0.01 compared to the control.

24 4 .

 4 83 ± 1.29 d a Data are presented as the mean ± SD (n = 8). HDHA, hydroxydocosahexaenoic acid; IsoP, isoprostane; and NeuroP, neuroprostane. The concentration is shown as nanograms per gram of tissue samples. b p < 0.01 compared to the control. c p < 0.05 compared to the control. d p < 0.001 compared to the control.

Figure 4 .

 4 Figure 4. Levels of PUFAs measured in the heart tissue. Graphs represent (A) arachidonic acid, (B) adrenic acid, (C) α-linolenic acid, (D) eicosapentaenoic acid, (E) n-3-type docosapentaenoic acid, and (F) docosahexaenoic acid measured in the rat heart tissue. Data are presented as the mean ± SD (n = 8). The concentration is shown as micrograms per gram of tissue samples. ( * * ) p < 0.01 compared to the control.

Table 1 .

 1 Levels of Enzymatic Oxidized PUFA Products Measured in the Liver Tissue a

		control	walnut
	Enzymatic Oxidized Products from AA
	5-HETE	0.11 ± 0.06	0.07 ± 0.01
	8-HETE	0.96 ± 0.17	0.79 ± 0.18
	12-HETE	2.17 ± 0.18	1.64 ± 0.27 b
	15-HETE	0.09 ± 0.02	0.08 ± 0.02
	total LOX-mediated HETEs	3.36 ± 0.21	2.57 ± 0.46 c
	Enzymatic Oxidized Products from DHA
	4-HDHA	3.25 ± 1.51	3.21 ± 0.73
	7-HDHA	0.04 ± 0.02	0.11 ± 0.04 b
	11-HDHA	17.63 ± 8.62	39.69 ± 17.84 c
	14-HDHA	6.89 ± 1.95	15.78 ± 10.57
	17-HDHA	45.20 ± 22.68	105.07 ± 29.42 c
	total LOX-mediated HDHAs	73.02 ± 20.21	163.87 ± 51.07 b
	a Data are presented as the mean ± SD (n = 8). HETE,
	hydroxyeicosatetraenoic acid; HDHA, hydroxydocosahexaenoic acid;

LOX, lipoxygenase; total LOX-mediated HETEs, sum of 5-, 8-, 12-, and 15-HETE; and total LOX-mediated HDHAs, sum of 4-, 7-, 11-, 14-, and 17-HDHA. The concentration is shown as nanograms per gram of tissue samples. b p < 0.01 compared to the control. c p < 0.05 compared to the control.

Table 2 .

 2 Levels of Non-enzymatic Oxidized PUFA Products Measured in the Liver Tissue a

		control	walnut
	Non-enzymatic Oxidized Products from AA
	5-F 2t -IsoP	3.88 ± 1.49	0.86 ± 0.37 b
	15-F 2t -IsoP	12.85 ± 1.88	11.56 ± 1.30
	Non-enzymatic Oxidized Products from ALA
	9-D 1t -PhytoP	13.17 ± 6.91	57.26 ± 18.10 b
	9-F 1t -PhytoP	8.14 ± 3.32	31.28 ± 2.80 b
	16-F 1t -PhytoP	3.70 ± 1.97	21.55 ± 5.37 b
	9-L 1 -PhytoP	4.54 ± 2.89	17.25 ± 2.65 b
	16-B 1 -PhytoP	1.56 ± 1.71	26.05 ± 9.36 b
	16(RS)-13-ST-Δ 14 -9-PhytoF	0.72 ± 0.19	7.40 ± 0.35 b
	Non-enzymatic Oxidized Products from EPA
	5-F 3t -IsoP	3.43 ± 0.87	2.67 ± 1.69
	15-F 3t -IsoP	1.51 ± 1.11	1.82 ± 1.15
	Non-enzymatic Oxidized Products from DHA
	4-F 4t -NeuroP	5.49 ± 0.92	10.07 ± 3.84 c
	10-F 4t -NeuroP	8.88 ± 1.11	11.98 ± 2.50
	8-HDHA	6.12 ± 2.44	6.24 ± 1.73
	10-HDHA	0.25 ± 0.09	0.54 ± 0.22 c
	13-HDHA	0.46 ± 0.13	1.13 ± 0.32 b
	16-HDHA	0.84 ± 0.

Table 3 .

 3 Levels of Enzymatic Oxidized PUFA Products Measured in the Heart Tissue a

		control	walnut
	Enzymatic Oxidized Products from AA
	5-HETE	0.48 ± 0.12	0.36 ± 0.07
	8-HETE	1.09 ± 0.25	0.77 ± 0.09
	12-HETE	4.87 ± 1.27	6.10 ± 1.87
	15-HETE	0.53 ± 0.08	0.25 ± 0.10 b
	total LOX-mediated HETEs	6.97 ± 1.52	7.47 ± 1.93
	Enzymatic Oxidized Products from DHA
	4-HDHA	0.08 ± 0.05	0.20 ± 0.07 b
	7-HDHA	0.36 ± 0.14	0.58 ± 0.12
	11-HDHA	1.38 ± 0.89	1.82 ± 0.05
	14-HDHA	0.45 ± 0.07	0.60 ± 0.27
	17-HDHA	6.39 ± 0.79	8.29 ± 3.47
	total LOX-mediated HDHAs	8.66 ± 1.52	11.49 ± 3.77
	a Data are presented as the mean ± SD (n = 8). HETE,
	hydroxyeicosatetraenoic acid; HDHA, hydroxydocosahexaenoic acid;

LOX, lipoxygenase; total LOX-mediated HETEs, sum of 5-, 8-, 12-, and 15-HETE; and total LOX-mediated HDHAs, sum of 4-, 7-, 11-, 14-, and 17-HDHA. The concentration is shown as nanograms per gram of tissue samples. b p < 0.01 compared to the control.

Table 4 .

 4 Levels of Non-enzymatic Oxidized PUFA Products Measured in the Heart Tissue a

		control	walnut
	Non-enzymatic Oxidized Products from AA
	5-F 2t -IsoP	12.61 ± 0.99	11.20 ± 6.63
	15-F 2t -IsoP	144.99 ± 24.29	110.67 ± 28.78
	Non-enzymatic Oxidized Products from ALA
	9-D 1t -PhytoP	1.69 ± 0.86	3.62 ± 1.03 b
	9-F 1t -PhytoP	0.30 ± 0.11	1.74 ± 0.56 b
	16-F 1t -PhytoP	0.77 ± 0.39	11.91 ± 1.20 b
	9-L 1 -PhytoP	0.53 ± 0.22	2.11 ± 0.93 b
	16-B 1 -PhytoP	1.14 ± 0.56	18.21 ± 5.04 b
	16(RS)-13-ST-Δ 14 -9-PhytoF	0.74 ± 0.42	3.57 ± 0.94 b
	Non-enzymatic Oxidized Products from EPA
	5-F 3t -IsoP	5.94 ± 2.00	8.03 ± 0.98
	15-F 3t -IsoP	0.65 ± 0.34	0.72 ± 0.37
	Non-enzymatic Oxidized Products from DHA
	4-F 4t -NeuroP	10.15 ± 2.56	12.00 ± 3.50
	10-F 4t -NeuroP	13.83 ± 5.18	11.63 ± 3.86
	8-HDHA	0.83 ± 0.13	1.13 ± 0.28
	10-HDHA	0.08 ± 0.04	0.09 ± 0.02
	13-HDHA	0.14 ± 0.08	0.24 ± 0.03
	16-HDHA	2.71 ± 0.53	3.65 ± 0.93

a Data are presented as the mean ± SD (n = 8). HDHA, hydroxydocosahexaenoic acid; IsoP, isoprostane; and NeuroP, neuroprostane. The concentration is shown as nanograms per gram of tissue samples. b p < 0.01 compared to the control.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jafc.0c06690.

Composition of control and walnut diets (Table S1), MRM transition measured by LC-MS/MS of PUFAs and their oxidized product standards commercially available (Cayman Chemical, Ann Arbor, MI, U.S.A.) and synthesized by Durand's group (Table S2), primers used for real-time PCR (Table S3), typical LC-MS/MS chromatogram of (A) ALA and (B) 16(RS)-13-epi-ST-Δ 14 -9-phytofuran and sample mass spectra with the structure of (C) ALA and (D) 16(RS)-13-epi-ST-Δ 14 -9phytofuran investigated in the study (Figure S1), physical characteristics of rats over 4 weeks of feeding (Figure S2), and gene expressions of IL-1β, IL-6, TNFα, and catalase measured in the heart tissue (Figure S3) (PDF)

AUTHOR INFORMATION Corresponding Author

Jetty Chung-Yung Lee


Emmanuel Bernuau

Wilfrid Perruquetti

Emmanuel Moulay

Retraction obstruction to time-varying stabilization

Keywords: Retraction obstruction, Lyapunov stability, stabilization

come

Introduction

Topological obstruction to stabilization is a long standing problem of control theory detailed in the introduction of [START_REF] Moulay | Conley index condition for asymptotic stability[END_REF]. There are two main topological obstructions to continuous stabilization. First, the Brockett obstruction is a local obstruction related to the structure of the controlled systems involving the nature of feedback controls [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF]. Then, the retraction obstruction is a global obstruction related to the structure of the underlying state space: if the state space of the system has the structure of a vector bundle over a compact manifold, no continuous static feedback can globally stabilize an equilibrium. This result has been proved and its consequences studied in detail in [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF].

The Brockett condition, which is necessary in the case of continuous time-invariant feedback controls, does not remain necessary for driftless controllable systems with time-varying feedback. The existence of such feedbacks has been proved in [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF], while [START_REF] Pomet | Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift[END_REF] gave an explicit de-Email addresses: emmanuel.bernuau@ec-lille.fr (Emmanuel Bernuau), wilfrid.perruquetti@inria.fr (Wilfrid Perruquetti), emmanuel.moulay@univ-poitiers.fr (Emmanuel Moulay).

sign under an additional condition on the Control Lie Algebra (see [START_REF] Pomet | Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift[END_REF]Assumption 1]). A natural question is, hence, to wonder whether continuous time-varying feedback controls could also avoid the retraction obstruction as suggested in the introduction of [START_REF] Nakamura | Minimum projection method for nonsmooth control lyapunov function design on general manifolds[END_REF].

In [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF]Remark 1], the authors mention that their result also handle the case of dynamic feedback. Indeed a dynamic feedback is usually seen as a dynamic extension where the augmented state is stable. Estimation of parameters or observer-based control are in this scope. In that case, the result of [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF] is applicable directly, with the method exposed in their remark. Nevertheless, a timevarying static feedback is also a dynamic extension using a timer τ = 1. However in this situation, there is no convergence to a single equilibrium point, but to a submanifold; the result of [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF] is therefore not applicable in this context, following [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF]Remark 1].

The aim of this paper is to prove that, in the second case, the obstruction still remains: a time-varying feedback control which is globally asymptotically uniformly stabilizing a system defined on a fiber bundle with a compact manifold as its base space does not exist.

Retraction obstruction

By a manifold we mean a smooth, positive dimensional, connected manifold without boundary. The definition of a fiber bundle is given for instance in [START_REF] Abraham | Foundations of Mechanics[END_REF].

Our purpose is to link the topological property of contractibility of a manifold to the existence of a globally asymptotically stable equilibrium. Let us introduce the definitions we will be using and some useful properties.

Definition 1 Let E be a topological space and x 0 ∈ E. A retraction of E on x 0 is a continuous mapping h : [0, 1] × E → E such that for all x ∈ E, h(0, x) = x and h(1, x) = x 0 . A topological space E is said to be contractible if there exists a retraction of E.

Proposition 2 [8, Section 2.4] No compact manifold is contractible. Consider a controlled system defined by ẏ = f 0 (y, u) y ∈ N , u ∈ U, (1)
with N a manifold and U a set of admissible controls, and where f 0 is a continuous vector field.

We wonder about the global stabilizability of the system via a time-varying feedback u(y, t). To do that, we will set τ = 1 and look at the partial asymptotic stability of the closed-loop system:

ẏ = f 0 (y, u(y, τ )) τ = 1 . (2)
Hence, let us introduce the definitions of partial stability we will be using. Those definitions are adapted from [START_REF] Haddad | Nonlinear dynamical systems and control -A Lyapunov based approach[END_REF].

Definition 3 Let M = N × T be a product manifold. Consider f = (f 1 , f 2
) a forward complete continuous vector field on M with the property of unicity of solutions in forward time. We denote by Φ the semiflow of f and p 1 the canonical projection on N .

(1) We say that y ∞ ∈ N is a partial equilibrium if for all τ ∈ T , we have f 1 (y ∞ , τ ) = 0. (2) A partial equilibrium y ∞ ∈ N is said to be partially stable uniformly in τ if for all U ⊂ N neighborhood of y ∞ there exists V ⊂ N a neighborhood of y ∞ such that for all y ∈ V and for all τ ∈ T , p 1 •Φ(t, (y, τ )) ∈ U for all t ≥ 0. (3) A partial equilibrium y ∞ ∈ N is said to be partially globally asymptotically stable uniformly in τ if it is partially stable uniformly in τ and if for all (y, τ ) ∈ M we have p 1 • Φ(t, (y, τ )) → y ∞ when t → +∞.

Remark 4

The last item of definition 3 is slightly different from the more standard ones. Indeed, to prove our result, we only need the stability to be uniform with respect to τ . The uniformity with respect to τ of the convergence, which is usually required, is not necessary here.

The following definition, inspired by [START_REF] Khalil | Nonlinear Systems[END_REF]Chapter 12] about time-varying stabilizability, is here given in the partial stability context.

Definition 5

The system (1) is said to be globally asymptotically uniformly stabilizable by means of a continuous generalized time-varying feedback if there exist a point y ∞ ∈ N , a manifold T , a continuous mapping f 2 : N × T → TT with f 2 (y, τ ) ∈ T τ T for all y ∈ N and all τ ∈ T and a continuous control law u(y, τ ) such that y ∞ is a partial equilibrium of the closed loop system ( ẏ, τ ) = (f 0 (y, u(y, τ )), f 2 (y, τ )) and is partially globally asymptotically stable uniformly in τ .

Remark 6 Let us note that, taking T = R and f 2 (y, τ ) = 1, this definition boils down to the definition of global asymptotic stabilization by means of a continuous time-varying feedback. In the generalized time-varying setting, the variable τ can be stable or not, scalar or vector, bounded or not.

In [3, Theorem 1], the authors use Proposition 2 to prove that if a manifold N admits a structure of fiber bundle over a compact manifold, then no continuous vector field over N can have an unique globally asymptotically stable equilibrium. Hence, they conclude that the system (1) cannot be globally asymptotically stabilized by means of a state feedback. Let us prove now that [3, Theorem 1] can be extended in the following way to the time-varying setting.

Theorem 7 Assume that N is a manifold with a structure of fiber bundle over a compact manifold Q. If there exists a continuous section of the bundle, then the system (1) is not globally asymptotically uniformly stabilizable by means of a continuous generalized time-varying feedback in such a way that the augmented vector field has the forward unicity of solutions property.

Proof. Ad absurdum, assume that there exists a continuous dynamic feedback which globally asymptotically uniformly stabilizes the system (1) in such a way that the augmented vector field has the forward unicity of solutions. Let us denote by τ the added variable, and τ ∈ T .

We have τ = f 2 (y, τ ), and the system (1) can be rewritten in an extended form, with f 1 (y, τ ) = f 0 (y, u(y, τ )):

ẏ τ = f 1 (y, τ ) f 2 (y, τ ) . ( 3
)
The equation (3) defines a continuous vector field f on the manifold M = N × T , with the forward unicity of solutions property. Moreover, there exists a partially globally asymptotically stable equilibrium uniformly in τ denoted by y ∞ ∈ N . Let us denote π 0 : N → Q the fiber bundle projection. Set p 1 : M → N the first canonical projection and set π = π 0 • p 1 . We denote q ∞ = π 0 (y ∞ ). Similarly, fix τ 0 in T and set σ(q) = (σ 0 (q), τ 0 ) where σ 0 : Q → N is a continuous section of π 0 . Clearly, σ is a continuous section of π. We also note that the manifold M trivially inherits a structure of fiber bundle over Q with projection π.

The vector field f is continuous and has the forward unicity of solutions property. Therefore, it admits a semiflow Φ [START_REF] Bhatia | Local semi-dynamical systems[END_REF]. Let us denote:

h : [0, 1] × Q → Q (λ, q) →    π • Φ ln 1 1-λ , σ(q) if λ = 1 q ∞ if λ = 1
Since we clearly have h(0, q) = q and h(1, q) = q ∞ , let us prove the continuity of h. This mapping is obviously continuous on [0, 1) × Q.

Let us show the continuity at (1, q) for q ∈ Q. Let (λ n , q n ) ∈ [0, 1) × Q be a sequence of points converging to (1, q). We set

t n = ln 1 1 -λ n , x n = σ(q n ), x = σ(q).
We have t n → +∞ and, by continuity of the section, x n → x. Let U ⊂ Q be a neighborhood of q ∞ and U 0 = π -1 0 (U ) ⊂ N the corresponding neighborhood of y ∞ . By partial stability uniformly in τ , there exists V 0 ⊂ N a neighborhood of y ∞ such that for all y ∈ V 0 , all τ ∈ T and all t ≥ 0, we have p 1 • Φ(t, (y, τ )) ⊂ U 0 . On the other hand, the partial attractivity of y ∞ means that p 1 • Φ(t, x) → y ∞ when t → ∞. Thus, there exists T > 0 such that p 1 • Φ(T, x) ∈ V 0 . By continuity, there exists N 1 > 0 such that for all n > N 1 we have p 1 • Φ(T, x n ) ∈ V 0 . Therefore, for all t ≥ T , we have p 1 • Φ(t, x n ) ∈ U 0 . But t n → +∞, so there exists N 2 > 0 such that for all n > N 2 , t n > T . Thus, for all n > N = max(N 1 , N 2 ), we have

p 1 • Φ(t n , x n ) ∈ U 0 . Hence, for all n > N , h(λ n , q n ) = π 0 •p 1 •Φ(t n , x n ) ∈ U .
Since U is an arbitrary neighborhood of q ∞ , the mapping h is continuous. However, the mapping h defines a retraction of the compact manifold Q on q ∞ , which leads to the expected contradiction thanks to Proposition 2.

Example 8 Let us consider the following system, defined on the circle:

θ = u, θ ∈ S 1 . (4)
By using the angular velocity ω = θ we can rewrite the system as θ = ω ω = u .

(5)

Here the state space is the tangent space of the circle, denoted by TS 1 . The tangent space of a manifold has always a structure of vector bundle over that manifold, and therefore TS 1 has a structure of vector bundle over the compact manifold S 1 with projection π 0 given by:

π 0 (θ, ω) = θ.
Moreover, being a vector bundle, the tangent bundle admits σ 0 , the zero section, as a continuous section. One may wonder if it is possible to design a continuous feedback control u(θ, ω, t) globally stabilizing a state (θ 0 , 0) such that the closed-loop system has uniqueness of solution in forward time.

Since S 1 is compact, from Theorem 7 the system (5) cannot be globally asymptotically uniformly stabilized.

Finally, taking into account Theorem 7 and [3, Theorem 1], we have the following result: consider the system (1) defined on a manifold with a structure of fiber bundle over a compact manifold. If there exists a continuous section of the bundle, then no continuous dynamic feedback can globally asymptotically uniformly stabilize the system in such a way that the closed loop system has the forward unicity of solutions.

Conclusion

This paper extends [3, Theorem 1] to the case of timevarying feedback control. We prove that under mild assumptions, no continuous time-varying feedback control can avoid the retraction obstruction; that is, no continuous time-varying feedback control can globally asymptotically uniformly stabilize an equilibrium on a state space which has a structure of fiber bundle over a compact manifold.

This topological obstruction on compact manifolds prevents us from having continuous globally asymptotically uniformly stabilizing feedback (neither static nor time-varying nor dynamic). Moreover it is proved in [START_REF] Mayhew | On the topological structure of attraction basins for differential inclusions[END_REF] that the obstruction still remains for discontinuous autonomous vector fields or differential inclusions. Finally the topological obstruction appears to be a strong constraint on stabilization, and few possibilities remain. First, since the uniformity property of the partial stability is indeed being used in our proof, the possibility of non-uniform global stabilizability still remains open. Second, hybrid feedbacks can be considered as suggested in [START_REF] Mayhew | On the topological structure of attraction basins for differential inclusions[END_REF]. Finally, other notions of solutions for discontinuous systems exist; some of them might not inherit the same obstruction to global stabilization.

Acknowledgement

We would also like to thank the reviewers for their help in the improvement of the paper.


E Moulay
email: emmanuel.moulay@univ-poitiers.fr

Morse theory and Lyapunov stability on manifolds

Keywords: Morse theory, Lyapunov stability, Lyapunov functions

come

Introduction

The main goal of this paper is to see the implications of Morse theory for the problem of Lyapunov stability on manifolds. Indeed, stability and stabilization for nonlinear systems evolving on manifolds are a new research field (see for instance [START_REF] Malisoff | Global stabilization for systems evolving on manifolds[END_REF]). In differential topology, Morse theory developed by John Milnor in the twentieth century gives a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. It has also been used and mentioned in some papers dedicated to Lyapunov stability in control theory (see [START_REF] Malisoff | Global stabilization for systems evolving on manifolds[END_REF][START_REF] Rifford | The stabilization problem: AGAS and SRS feedbacks in "Optimal Control, Stabilization and Nonsmooth Analysis[END_REF][START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF][START_REF] Wilson | The structure of the level surfaces of a lyapunov function[END_REF]). Nevertheless, a paper that links Morse theory and Lyapunov stability in control theory has never been addressed in the literature.

The paper is organized as follows. Some preliminaries are outlined in Section 2. In Section 3, we analyze the Morse Lyapunov functions. In Section 4, we consider the first Morse theorem known as the deformation lemma and its application for dynamical systems with asymptotically stable sets. In Section 5, we recall the Brown-Stallings lemma which leads to a necessary condition of asymptotic stability and the second Morse theorem dedicated to dynamical systems with a single critical point. We show why this second Morse theorem is not really accurate in control theory due to a more precise result proved by Sontag in [START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF]. We provide the third Morse Theorem with the Reeb Theorem and their applications for stability of dynamical systems with multi critical points in Section 6. Finally, Morse inequalities lead to a necessary condition for the existence of a Morse Lyapunov function.

Notations and definitions

In the paper, M denotes a manifold of finite dimension n. Consider a continuous vector field f on M with the property that for every x ∈ M, there exists a unique right maximally defined integral curve of f starting at x, and, furthermore, every right maximally defined integral curve of f is defined on [0, +∞). In this case, the integral curves of f are jointly continuous functions of time and initial condition and thus define a continuous semi-flow

ϕ : [0, +∞) × M → M satisfying ϕ (0, x) = x ϕ (t 1 , ϕ (t 2 , x)) = ϕ (t 1 + t 2 , x)
for all t 1 , t 2 ∈ [0, +∞) and x ∈ M (see for instance [START_REF] Bhatia | Local semi-dynamical systems[END_REF]Theorem 3.4]). The manifold M, called the state space, and the semi flow ϕ, called the evolution function, lead to the notion of dynamical system denoted in short by (M, ϕ) (see for instance [START_REF] Bhatia | Local semi-dynamical systems[END_REF] and [START_REF] Bhatia | Stability Theory of Dynamical Systems[END_REF] for a general definition of topological dynamical systems).

T p M denotes the tangent space to M at p. In a general way, we study the stability of invariant sets.

Definition 1. Let I ⊂ M, I is an invariant set if for all x ∈ I and t ∈ R ≥0 , ϕ (t, x) ∈ I. If I is a point, I is called an equilibrium point of the dynamical system (M, ϕ).

Let us recall the notions of stability given in [START_REF] Bhatia | Stability Theory of Dynamical Systems[END_REF].

Definition 2. I ⊂ M is stable if every open neighborhood U 1 ⊂ M of I, there exist an open neighborhood U 2 ⊂ Mof I such that ϕ (t, U 2 ) ⊆ U 1 for all t ≥ 0, where ϕ (t, U) = {ϕ (t, x) : x ∈ U} .
An invariant set I is asymptotically stable if:

• I is stable,

• I is attractive: i.e. for all x ∈ I there exist an open neighborhood N ⊂ M of I such that for all x ∈ N , ϕ (t, x) → I as t + ∞.

The domain of attraction is denoted by

A = {x ∈ M : ϕ (t, x) → I as t → +∞} .
Besides, I is globally asymptotically stable if N = M.

The lie derivative of V : M → R along f : M → M is defined by

L f V : R n → R, L f V (x) = dV p (f (p)) .
Definition 3. Suppose that K is an invariant set of the dynamical system (M, ϕ). A continuous function

V : A → R ≥0 is a Lyapunov function if 1. V (x) > 0 for all x ∈ A \ K, 2. V (x) = 0 for all x ∈ K, 3. V is proper, that is, V -1 (B) is compact for every compact subset B of R ≥0 , 4. V is strictly decreasing along orbits of ϕ, that is, V (ϕ (t, x)) < V (x) for all t > 0 and x ∈ A \ K. If V is differentiable, Condition 4 is replaced by L f V (x) < 0 for all x ∈ A\K.
We recall some definitions of Morse theory given in [START_REF] Milnor | Morse Theory[END_REF].

Definition 4. Let V : M → R be a smooth function. A critical point of V is a point p ∈ M at which the differential dV p : T p M → T V (p) R has rank zero, i.e. if in any local coordinate system (x 1 , . . . , x n ) around p one has ∂V ∂x 1 (p) , . . . , ∂V ∂x n (p) = 0.
For each critical point p of V , there exists a bilinear symmetric form H p (V ) on T p M, called the Hessian of p (see [START_REF] Milnor | Morse Theory[END_REF]Chapter 2]). Definition 5. A critical point p is a non-degenerate critical point if the Hessian H p (V ) is a non-degenerate bilinear form, i.e. if in any local coordinate system (x 1 , . . . , x n ) around p, the Hessian matrix

∂ 2 V ∂xi∂xj (p) 1≤i,j≤n is non- degenerate.
The dimension of the subspace of T p M on which H p (V ) is negative definite is called the Morse index of V at p and is denoted by ind (V, p).

A C 2 function V : M → R is a Morse function if all its critical points are non-degenerate.

The level sets of a function V : M → R are

M a = V -1 ((-∞, a]), M a,b = V -1 ([a, b]).
Let us recall some topological definitions given in [START_REF] Bredon | Topology and Geometry[END_REF][START_REF] Hatcher | Algebraic Topology[END_REF].

Definition 6. A topological space is a n-cell if it is homeomorph to R n .
A space X is contractible if it is homotopy equivalent to the one-point space.

A subspace A of X is called a deformation retract of X if there exists a continuous function

h : [0, 1] × X → X such that for all x ∈ X, a ∈ A h (0, x) = x, h (1, x) ∈ A, h (1, a) = a.
The k-th Betti number of M denoted by b k is the the rank of the k-th homology group H k (M). The Euler Characteristic of M is defined by

χ (M) = k i=1 (-1) k b k .
For a precise definition of the homology group and a survey about homology theory, the reader may refer to [9, Chapter 2].

Morse Lyapunov functions

Let us prove a result dedicated to isolated critical points of a smooth Lyapunov functions.

Lemma 7. Suppose that x e is an equilibrium point of the dynamical system (M, ϕ). If V : M → R is a differentiable Lyapunov function for the system then x e is the only critical point of V .

Proof. Suppose that V contains another critical point x c in the domain of attraction. Due to the definition of a Lyapunov function, we have L f V (x c ) = 0. But it contradicts the fact that for a Lyapunov function with a single equilibrium point x = x e , L f V (x) < 0 for all x = x e .

Let us recall the Morse Lemma given for instance in [7, Lemma 2.2].

Theorem 8 (Morse Lemma). Let p ∈ M be a non-degenerate critical point of a smooth function V : M → R. There exists a local coordinate system (x 1 , . . . , x n ) in a neighborhood N ⊂ M of p with x i (p) = 0 for all 1 ≤ i ≤ n and such that for x ∈ N ,

V (x) = V (p) -x 2 1 -. . . -x 2 i + x 2 i+1 + . . . + x 2 n
where i = ind (V, p).

We may now characterize the Morse Lyapunov functions.

Corollary 9. Let p ∈ M be an equilibrium point of (M, ϕ) and V : M → R ≥0 a Morse Lyapunov function. There exists a local coordinate system (x 1 , . . . , x n ) around p such that V is locally the canonical quadratic Lyapunov function

V (x) = x 2 1 + . . . + x 2 n with ind (V, p) = 0.
Proof. As p ∈ M is an equilibrium point and due to Lemma 7, p ∈ M is the only critical point of V . Moreover, V is supposed to be a Morse function and thus by using Theorem 8,

V (x) = V (p) -x 2 1 -. . . -x 2 i + x 2 i+1 + . . . + x 2 n .
As a Lyapunov function is positive definite, we have the result.

Level sets of a Lyapunov function

The following theorem is known as the deformation lemma and is the first Morse theorem. It can be founded in [7, Theorem 3.1].

Theorem 12 (Deformation Lemma). Let

V : M → R be a smooth function and a,b ∈ V (M) such that a < b. Assume that M a,b is compact and does not contain critical point of V , then M a is diffeomorphic to M b . Moreover, M a is a deformation retract of M b .
By using Theorem 12, we deduce a result dedicated to dynamical systems with asymptotically stable sets.

Corollary 13. Let M be a smooth Riemannian manifold. If M contains a closed invariant asymptotically stable set, then for all a,b ∈ V (M), M a is diffeomorphic to M b and M a is a deformation retract of M b where V is a smooth Lyapunov function.

Proof. If M is a smooth Riemannian manifold and K ⊂ M a closed invariant asymptotically stable set, there exists a smooth Lyapunov function V : M → R ≥0 for the dynamical system given in [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF]Theorem 3.2] and [START_REF] Nadzieja | Construction of a smooth lyapunov function for an asymptotically stable set[END_REF]. As V is proper, M a,b is compact. Due to Lemma 7, M a,b does not contain critical point of V and the result follows from Theorem 12.

A similar result already exists when

M = R n (see [4, Theorem 1.2].

Systems with a single critical point

In this section, we recall some results dedicated to systems with a single critical point. Let us recall the Brown-Stallings lemma which can be founded in [10, Lemma 3].

Theorem 14 (Brown-Stallings Lemma). Let M be a paracompact manifold such that every compact subset is contained in an open set diffeomorphic to Euclidean space. Then M itself is diffeomorphic to Euclidean space. Theorem 17 (Morse Theorem). Let V : M → R be a Morse function, p a critical point such that ind (V, p) = i and c = V (p). If there exists > 0 such that M c-,c+ is compact and does not contain other critical point p, then M c-∪ e i is a deformation retract of M c+ where e i is an i-cell (in particular, M c+ has the homotopy type of M c-with an i-cell attached).

In the case of a single critical point, Theorem 17 implies that the level sets of a Morse Lyapunov function are homotopic. But in the case of an asymptotically stable point of a dynamical system, and thus the existence of a Lyapunov function only, there exists a more precise result which asserts that the domain of attraction is contractible. This result has been given by Sontag in [3, Theorem 21].

Theorem 18 (Sontag Theorem). Let us consider the dynamical system (M, ϕ) with an equilibrium point x e ∈ M. Suppose that x e is asymptotically stable. Then the domain of attraction of x e , which is given by

A = x ∈ M : lim t→+∞ ϕ (t, x) = x e is contractible.
Theorem 18 remains true for all systems even if M is a general topological space contrary to Corollary 15 and Theorem 17. It is proved by using the flow as a deformation retract. However, this result cannot be extended to multi equilibria contrary to Morse theory.

Systems with multi critical points

The following third Morse theorem given in [START_REF] Milnor | Morse Theory[END_REF]Theorem 3.5] is a generalization of Theorem 17 to multi critical points. In this case, the manifold has the shape of a CW-complex.

Theorem 19 (Morse Theorem). If V : M → R is a Morse function such that M a is compact for each a ∈ R then M has the homotopy type of a CWcomplex with one i-cell for each critical point of index i.

For more details on CW-complex, the reader may refer to [START_REF] Hatcher | Algebraic Topology[END_REF].

Corollary 20. Suppose that the dynamical system (M, ϕ) has several equilibria (x 1 , . . . , x k ). If there exists a Morse Lyapunov function V : M → R ≥0 then {x 1 , . . . , x k } is a retract of the domain of attraction.

Proof. Due to Lemma 7 and Theorem 19, M has the homotopy type of 1≤i≤k {e i }. Moreover, as V is a Lyapunov function, we deduce with Corollary 9 that M has the homotopy type of {x 1 , . . . , x k }.

As an application of Theorem 19, there is the theorem of Reeb which can be found in [10, Theorem 1'].

Theorem 21 (Reeb Theorem). Suppose that M is compact without boundary, if V : M → R is a smooth function with only two critical points, then M is homeomorphic to the n-sphere S n .

Remark 22. The reader may notice that the critical points are allowed to be degenerate.

Let us apply the Theorem of Reeb to Lyapunov stability of a dynamical system having two equilibria.

Corollary 23. Let M be a compact manifold without boundary and (x 1 , x 2 ) the only two equilibria of the dynamical systems (M, ϕ). If I = {x 1 , x 2 } is asymptotically stable for the dynamical system (M, ϕ), then M is homeomorphic to the n-sphere S n .

Proof. There exists a smooth Lyapunov function V : M → R ≥0 for the dynamical system given in [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF]Theorem 3.2] and [START_REF] Nadzieja | Construction of a smooth lyapunov function for an asymptotically stable set[END_REF] with only two critical points due to Proposition 7. The Theorem 21 of Reeb leads to the result.

Example 24. As an example, consider the global set stabilization of the spacecraft attitude [START_REF] Li | Global set stabilisation of the spacecraft attitude using finite-time control technique[END_REF]. In this case, there exists an unwinding phenomenon due to the two equilibria and Theorem 18. But, Corollary 23 is more precise in the case of two equilibria. The system of the spacecraft attitude with two equilibria cannot be locally stabilized when the state space is not homeomorphic to S n .

Theorem 25 (Morse Inequalities). Let m k be the number of critical points of a Morse function V with index k, then we have

b k ≤ m k ∀k, j i=0 (-1) j-i b i ≤ j i=0
(-1) j-i m i ∀j,

χ (M) = k (-1) k b k = k (-1) k m k .
Let us give a necessary condition for the existence of a Morse Lyapunov function based on the Euler characteristic which is a topological invariant. Remark 27. This result implies that if χ (M) = k then there is no Morse Lyapunov function for the dynamical system. If M is a compact oriented differentiable manifold, the Euler characteristic can be computed by using the Poincaré-Hopf Theorem given for instance in [START_REF] Barden | An Introduction to Differential Manifolds[END_REF]Theorem 7.6.5]

Remark 10 .

 10 The Morse Lyapunov function plays the role of the canonical quadratic Lyapunov function for systems defined on manifolds. Let us give an example of Morse functions given in [10, Theorem 6.6]. Example 11. Let M be manifold in Euclidean space. For almost all p ∈ R n , the quadratic function L p : M → R defined by L p (x) = x -p 2 is a Morse function.

  The Brown-Stallings lemma leads to the following necessary condition of asymptotic stability which can be founded in[START_REF] Wilson | The structure of the level surfaces of a lyapunov function[END_REF] Theorem 2.2].Corollary 15. Let M be a paracompact manifold. The domain of attraction of an asymptotically stable equilibrium point is diffeomorphic to an Euclidean space.Remark 16. Corollary 15 has been mention in[START_REF] Sontag | Mathematical Control Theory: Deterministic Finite Dimensional Systems[END_REF] Page 22]. If M ⊂ R n , then the domain of attraction is diffeomorphic to R n .Let us recall the second Morse theorem given in[START_REF] Milnor | Morse Theory[END_REF] Theorem 3.2].

Corollary 26 .

 26 Let us consider the dynamical system (M, ϕ) with several equilibria (x 1 , . . . , x k ). If there exists a Morse Lyapunov functionV : M → R ≥0 then χ (M) = k ≥ b 0 .Proof. If there exists a Morse Lyapunov function V , due to Lemma 7 and Corollary 9, (x 1 , . . . , x k ) are the only critical points with indices 0. Then, by using Theorem 25, it leads to χ (M) = m 0 = k. The first inequality of Theorem 25 ensures that b 0 ≤ m 0 = k.

Acknowledgment

The author wishes to thank Pr. Mihai Tibar, from the Department of Mathematics at the University of Lille, for introducing him to Morse theory.