Lara

François Nicolle

REFLECTIONS ON DIGITAL IDENTITY IN THE ONTOLOGICAL CYBERSPACE Ziyed GUELMAMI

Keywords: Cyberspace, self-disclosure, digital identity

This essay introduces an ontological discussion of cyberspace. Challenging the objective, subjective and relational perspectives of spaces and place, the Heideggerian precognitive approach contends that "lived spaces" are constituted by an individual's purpose, objectives and aspirations. An existential analysis is carried out to show how a different kind of spatialization is achieved by human beings in cyberspace ("cyber-spatiality") and how it affects the way they disclose themselves online. Heidegger's ground-breaking perspective proves itself useful to provide a deeper understanding of the way consumers experience online space.

INTRODUCTION

Today, people dwell regularly in virtual worlds for many purposes : shopping, entertainment, social bonding, information searching or dating for instance. The nature of the space where our conscience is thrown in does not seem relevant to the way we experience reality. Navigation on the "cyberspace" has become commonplace in Western societies and changed the way we interact with the world and the way we construct our identity. The traditional perspectives of space, namely the objective, subjective and relational approaches of space, seem inadequate to explore the peculiar spatiality characterizing cyberspace, that is, how people phenomenologically relate to their environment through technology and virtual interfaces. Philosophical debates on the nature of space have been durably affected by the Cartesian perspective opposing the subject (conceptualized as a self-contained inner entity) to the object (the world "out there"). Consequently, space is often understood as an objective and immuable "container" or as a subjective representation produced by the human mind. The lack of conceptual tools to understand the elusive nature of cyberspace is especially problematic in a time of frenetic technological development, where new spaces of human interaction flourish online. Thus, an important issue emerges concerning the spatial nature of these online spaces and their role in transforming identities and communities.

DISCUSSING THE NEBULOUS NATURE OF CYBERSPACE

Although digital consumption has been the focus of many marketing papers, very few addressed the problem of the nature of cyberspace. There is no consensual definition of this concept neither in consumer research nor in computer sciences despite the wide use of the word. The few existing definitions are often descriptive and narrative and consequently lack scientific precision [START_REF] Cicognani | On the Linguistic Nature of Cyberspace and Virtual Communities[END_REF]. Therefore, the word is becoming more and more meaningless [START_REF] Strate | The varieties of cyberspace: Problems in definition and delimitation[END_REF]. Cyberspace is in fact often seen as a network of interconnected electronic devices (e.g. [START_REF] Hemetsberger | Creative Cyborgs: How Consumers Use the Internet For Self-Realization[END_REF]. The emphasis is sometimes put on the informational dimension of cyberspace, as in Cicognani's definition, cyberspace as an "electronic fluxus of information" (1998, p.19). The social dimension of cyberspace has also been early identified by researchers and became an important point of focus for many researches (e.g. [START_REF] Venkatesh | Cyberspace: The Next Marketing Frontier (?) Questions and Issues[END_REF][START_REF] Kalay | The role of place in cyberspace[END_REF][START_REF] Zhang | Reconceptualizing Cyberspace:" Real" Places in Digital Space[END_REF]. One important particularity of cyberspace is that it allows a subject-decentered, fragmented and non-geographically bounded interaction [START_REF] Venkatesh | Cyberspace: The Next Marketing Frontier (?) Questions and Issues[END_REF], making it the most fertile ground for building and sustaining a postmodern communitas. The current development of social media suggests that social bonding is becoming the most crucial feature of cyberspace, rather than information exchange. These researches seem to not fully appreciate the postmodern nature of cyberspace, which has penetrated every space of human existence, becoming "realer than real" [START_REF] Venkatesh | Cyberspace: The Next Marketing Frontier (?) Questions and Issues[END_REF]. Some notable works discussed the possibility for cyberspace to being compared to "real space". But conclusions somehow differ. For example, [START_REF] Zhang | Reconceptualizing Cyberspace:" Real" Places in Digital Space[END_REF] argue that even if the comparison between cyberspace and space/place is legitimate, « cyberspace is not a 'space' but a 'place'. The 'space' in cyberspace is no more than a metaphor for a 'place' constituted of 'place-like' units » (p. 99). Like some others researchers, they conclude that cyberspace must be understood as a spatial metaphor and is not space per se. On the contrary, [START_REF] Coyne | Designing information technology in the postmodern age[END_REF] precisely describes how cyberspace may be understood as a world, a space or places. Moreover, he contends that cyberspace shares many common features with physical space in that it is a container, it is measurable space and enabling and constraining human interaction (1995, p.155). This point of view really questions the intuitive distinction that many researchers make between space and cyberspace. This type of space, which is the product of past events, covers every object in relation with human existence and defines possibilities of action as much as interdictions. According to him, these levels are deeply interconnected, relying on one another. Just like "real space", cyberspace can be described according to the interconnection of these dimensions. Cyberspace is largely depending on its physical constituents : an expansion of the number of connected devices, visual interfaces and servers automatically means an expansion of cyberspace. Like urban space, cyberspace is built according to logical considerations on human needs and thus depends on mental space. Finally, as a formidable medium for social interaction, cyberspace can be understood in terms of social space. Cyberspace is the sum of numerous online places defined by their social utility and by their own implicit and explicit social rules : social media for self expression and communication, online multiplayer games for entertainment, community forums for socialization, educating oneself, etc.

From different theoretical perspectives, cyberspace is equivalent to "real", offline space. Nevertheless, as Venkatesh, Meamber and Firat's work suggests, the latter allows biological life and the former does not. The latter is directly and naturally accessible to our senses and the former is technology-mediated. As essential as these specificities may be, we should not put aside the fact that cyberspace allows and significantly expands human existence. This idea is closely related to Don Ihde's postphenomenological perspective of technology (1993). According to Ihde, technology allows human beings to access to qualities and dimensions of reality that would be otherwise out of reach. Ihde asserts that human beings develop relationships to reality in a precognitive fashion, before the theoretical constitution of the subject and of the object. Therefore, human beings and their world are always "mutually constitutive in their fundamental interrelation" (1993, p.3). Technology must not be regarded as a mere mediator, between the subject and the world, but rather as a constitutive element in the subject's relationship to reality. According to postphenomenology, virtual worlds must be apprehended as epistemological and ontological instruments [START_REF] Gualeni | Augmented Ontologies or How to Philosophize with a Digital Hammer[END_REF]. Following this approach, in the following, cyberspace will be considered from an ontological standpoint and defined as an Internet-mediated space of human existence.

Heidegger's theory of space is useful to apprehend cyberspace as "lived space" [START_REF] Schatzki | Martin Heidegger: theorist of space[END_REF]. According to Heidegger, the main constitutive feature of human existence is Being-in-the-world, that is to say, Dasein's working out its self-understanding throughout existence. Therefore, there are many ways of Being-in-the-world. Heidegger also claimed that the primary way Dasein is Being-in is practical rather than cognitive. That means that people are primarily "busy" using "equipment" (Zeug, for instance a computer, files, softwares, etc.) rather than acquiring knowledge and theorizing about things. Indeed, Heidegger explains that the process of knowing the world is based on a practical way of Being-in that consists in using things for defined purposes. This approach tackles the problem of space in a very novel way : space is constituted by things that allow people to fulfill their goals. From this perspective, cyberspace appears as a legitimate and "real" space. In the next section, we call "cyber-spatiality", the peculiar way for humans to apprehend cyberspace as a space of existence.

IMPLICATIONS OF BEING IN THE CYBERSPACE ON THE CONSUMERS' DIGITAL IDENTITY CONSTRUCTION

Cyberspace springs from technology that is created by humans for humans. Consequently, the technology and interface that support cyberspace are designed in an intelligible way for us : once the computer is turned on, the interface is supposed to make sense for the user. Cyberspace is underlain by an incalculable number of softwares that mimic features of human existence [START_REF] Thrift | The Automatic Production of Space[END_REF], however its fundamentally Internetmediated nature implies that Dasein does not relate to cyber-objects and cyber-sociality in the same way than in the material world. According to Heidegger, Dasein is characterized by "de-severance" (Ent-fernung), which means that Dasein has a natural tendency to bring things close, to make them familiar and ready to use (no matter how far these things are situated). Consequently, de-severance tends to create familiarity with distant things such as virtual objects (forums, broadcasted events, online games, etc.) and people (online acquaintances, geographically distant friends). Moreover, as Heidegger suggests, media have the power to expand one's world-around by providing an increasing number of familiar and ready to use objects. Human consciousness and knowledge of the world are especially enhanced by cyberspace as it allows people to grasp circumspectively and bring closer an infinity of scattered objects related to their concernful dealings. Cyberspace is structured to provide this readiness and actionability of knowledge in the more efficient way possible. That is why cyberspace can be characterized as a "teleotopia". Teleotopic spaces are phenomenologically constructed and structured according to specific projects and purposes.

A large literature on online behavior shows that people tend to conduct themselves somehow differently in cyberspace than in "real" world [START_REF] Christopherson | The positive and negative implications of anonymity in Internet social interactions:"On the Internet, Nobody Knows[END_REF]. This difference may be explained by the peculiar spatialization people experience in cyber social spaces. Obviously, the non-physical presence and the relative anonymity characterizing online browsing -and therefore, cyber-spatiality -play an important part in the specificity of cyber-communication. These two factors influence the degree to which one frees himself from social norms and roles [START_REF] Postmes | Behavior online: does anonymous computer communication reduce gender inequality[END_REF]. Cyber-spatiality is ubiquity : one can be present in different cyberplaces at the same time, but necessarily in an ephemeral and ghost-like way. Freedom and ubiquity in cyberspace allow people to easily express a multiplicity of identities online [START_REF] Iteanu | L'identité numérique en question[END_REF]. People undertake an identity play in cyber social places implying the superposition or the interchangeability of these multiple identities [START_REF] Pierre | Génétique de l'identité numérique[END_REF]. Digital identities are constructed as representations as much for other people than for oneself [START_REF] Castells | Who are you on the internet ? Digital identity : an exhibition concerning technosocial uses of the net[END_REF][START_REF] Pierre | Génétique de l'identité numérique[END_REF]. The construction of digital identities involves an introspection, a cogito, that questions one's true identity. For instance, the creation of a personal profile on a dating app leads to thinking about oneself and selecting scattered elements among many facets of one's identity. The de-severance characterizing Dasein allows a person to bind to different online communities in order to fit in. This phenomenon is quite interesting when compared to that of online activism for differentiating oneself (Donath, 2007). One's world-around is constantly shifting particularly because of the endlessness of information streams and the instability of online social relations. As much as in the offline world, people need to carry out different identity performances according to the various contexts they have to deal with. The freedom and ubiquity characterizing cyber-spatiality allow them to evolve easily and simultaneously in otherwise inaccessible social spaces. Consequently, people construct their identities online in a very peculiar way by binding with others from afar, by fitting in social groups they are not familiar with and by performing their identities as they see fit.

In the last section, the implications of these theoretical considerations for consumer research are discussed.

CONCLUSION : IMPLICATIONS FOR CONSUMER RESEARCH

First, this research provides epistemological contributions. By shedding light on the ontological dimension of cyberspace, we argued that considerations on the "real" or "virtual" nature of cyberspace are not relevant anymore. For instance, browsing should not be regarded as an activity (what most researches implicitly postulate) but as a technologically mediated way of Being-in. Consequently, connecting to Facebook before even getting out of bed (as related by [START_REF] Llamas | Living in a Digital World[END_REF] is not more surprising than gently telling your wife or husband "good morning" as you wake up. Being-in-cyberspace is not an activity, that it why cyberspace imbues every single part of our existence. It is just a way to be. This cyber-ontology also has theoretical implications about the identity of the digital consumer. The analysis of cyberspace according to Heidegger's philosophy pretty much agrees with Kozinets' (2013) judicious point : the digital consumer that we have been looking for with much determination is not some kind of unfathomable entity, it is just us. Nevertheless, this point seems to underestimate ontological value and influence of cyberspace. This paper argues that, in cyberspace, Dasein is characterized by cyber-spatiality. This means that the way we relate to our environment -the world-around -is necessarily altered by cyberspace. The cyberspatialized consumer is an ontologically augmented person [START_REF] Gualeni | Augmented Ontologies or How to Philosophize with a Digital Hammer[END_REF] because he has an almost unlimited and immediate access to "equipment". But as Kozinets (2013) suggests it, we must not engage ourselves in some kind of post-humanist poetics and praising blindly the advent of a technologically transcended "cyborg". As we argued earlier, cyberspace is a plan of human existence, and therefore a place of mistrust as much as a place for sharing. Many people see their lives enhanced by cyberspace but many get also scammed and cyberbullied in cyberspace [START_REF] Moore | Anonymity and roles associated with aggressive posts in an online forum[END_REF]. Cyber-spatiality makes us much more aware of the world, but it can be for the worst.

In conclusion, this essay aimed at showing that cyberspace can and should be regarded as a real space, a lived space where identity projects are carried out in a very peculiar way. We should embrace cyberspace as the important part of human existence that it is and leave behind the outdated question of its reality. By focusing on its growth as a legitimate plan of human existence and its very real influence on people's identities, we can hope for tremendous developments in our understanding of this fascinating phenomenon.


Nathan P De Jong

Michael C Rudolph

Matthew Jackman

Rachel R Sharp

Ken Jones

Julie Houck

Zhaoxing Pan

Jane Reusch

Paul Maclean

Daniel H Bessesen

Audrey Bergouignan
email: audrey.bergouignan@iphc.cnrs.fr

Short-Term Adaptations in Skeletal Muscle Mitochondrial Oxidative Capacity and Metabolic Pathways to Breaking Up Sedentary Behaviors in Overweight or Obese Adults

Keywords: Sedentary behavior, Mitochondria, Skeletal muscle molecular adaptations, mRNA sequencing, active breaks, obesity

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INTRODUCTION

Prolonged sedentary behaviors (SB), defined as "any waking behavior characterized by an energy expenditure ≤ 1.5 metabolic equivalents while in a sitting, reclining, or lying posture" [START_REF] Tremblay | Sedentary Behavior Research Network (SBRN) -Terminology Consensus Project process and outcome[END_REF], are detrimental for health. It has been negatively associated with common risk factors for cardiometabolic disease such as waist circumference, body mass index (BMI, kg/m 2 ), 2 hr postprandial glycemia, and fasting plasma glucose, triglycerides (TG) and C-reactive protein. Conversely, frequently breaking up SB with short bouts of physical activity (PA) confers metabolic health benefits independent of total time spent in moderate to vigorous physical activity (MVPA) and total daily sitting time [START_REF] Healy | Breaks in sedentary time: beneficial associations with metabolic risk[END_REF][START_REF] Healy | Sedentary time and cardio-metabolic biomarkers in US adults: NHANES 2003-06[END_REF]. It is now well established that, compared to sedentary control conditions, acute (≤ 12 hr) and short-term (≤ 4 days) exposures to active breaks reduce postprandial glucose and insulin concentrations compared to a sitting control in adults who are lean, with obesity and type 2 diabetes [START_REF] Blankenship | Effects of subtracting sitting versus adding exercise on glycemic control and variability in sedentary office workers. Applied physiology, nutrition, and metabolism = Physiologie appliquee[END_REF][START_REF] Dempsey | Interrupting prolonged sitting in type 2 diabetes: nocturnal persistence of improved glycaemic control[END_REF][START_REF] Dempsey | Benefits for Type 2 Diabetes of Interrupting Prolonged Sitting With Brief Bouts of Light Walking or Simple Resistance Activities[END_REF][START_REF] Dunstan | Breaking Up Prolonged Sitting Reduces Postprandial Glucose and Insulin Responses[END_REF][START_REF] Holmstrup | Multiple short bouts of exercise over 12-h period reduce glucose excursions more than an energy-matched single bout of exercise[END_REF][START_REF] Homer | Regular activity breaks combined with physical activity improve postprandial plasma triglyceride, nonesterified fatty acid, and insulin responses in healthy, normal weight adults: A randomized crossover trial[END_REF][START_REF] Larsen | Breaking up of prolonged sitting over three days sustains, but does not enhance, lowering of postprandial plasma glucose and insulin in overweight and obese adults[END_REF][START_REF] Peddie | Breaking prolonged sitting reduces postprandial glycemia in healthy, normal-weight adults: a randomized crossover trial[END_REF][START_REF] Van Dijk | Effect of moderateintensity exercise versus activities of daily living on 24-hour blood glucose homeostasis in male patients with type 2 diabetes[END_REF].

To investigate the independent effects of the active breaks, the metabolic response of short-frequent PA bouts performed throughout the day have been compared to those elicited by a time-matched single continuous bout [START_REF] Holmstrup | Multiple short bouts of exercise over 12-h period reduce glucose excursions more than an energy-matched single bout of exercise[END_REF][START_REF] Homer | Regular activity breaks combined with physical activity improve postprandial plasma triglyceride, nonesterified fatty acid, and insulin responses in healthy, normal weight adults: A randomized crossover trial[END_REF][START_REF] Peddie | Breaking prolonged sitting reduces postprandial glycemia in healthy, normal-weight adults: a randomized crossover trial[END_REF]. Those studies have highlighted that at same total active time and/or energy expenditure, differential metabolic effects are elicited by these different PA modalities. Short-frequent bouts of PA decreases postprandial plasma glucose and insulin concentration, while a singlecontinuous bout of PA reduces postprandial TG levels, both compared to prolonged sedentary periods [START_REF] Plaisance | Postprandial triglyceride responses to aerobic exercise and extended-release niacin[END_REF]. In a randomized cross-over trial, we further showed that breaking up SB with short-frequent bouts of moderate-intensity walking increases 24 hr carbohydrate (CHO) oxidation, while a single-continuous bout of moderate-intensity walking increases 24 hr fatty acid (FA) oxidation in physically inactive adults with overweight or obesity [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF]. The mechanisms underlying these differential metabolic effects are however poorly known [START_REF] Dunstan | Television viewing time and mortality: the Australian Diabetes, Obesity and Lifestyle Study (AusDiab)[END_REF].

So far, only two human clinical trials have investigated some of the underlying mechanisms at the skeletal muscle; skeletal muscle being a critical site for glucose disposal and nutrient metabolism both at rest and during PA [START_REF] Egan | Exercise metabolism and the molecular regulation of skeletal muscle adaptation[END_REF]. Acute light-and moderate-intensity walking breaks (2-min bouts every 20 min for 5 hr) have been associated with differentially expressed genes involved in carbohydrate (CHO) and lipid metabolism (pyruvate dehydrogenase kinase 4 [PDK4] and nicotinamide Nmethyltransferase [NNMT], respectively) but also in cellular development, growth, and proliferation (myogenic factor 6 [MYF6], epithelial membrane protein 1 [EMP1], and dynein, light chain, LC8-type 1, transcript variant 1 [DYNLL1]) in adults with overweight or obesity [START_REF] Latouche | Effects of breaking up prolonged sitting on skeletal muscle gene expression[END_REF]. Performing these frequent active breaks for 3 days was shown to activate both non-oxidative and oxidative glucose uptake via insulinindependent and dependent pathways [START_REF] Bergouignan | Frequent interruptions of sedentary time modulates contraction-and insulin-stimulated glucose uptake pathways in muscle: Ancillary analysis from randomized clinical trials[END_REF], which may optimize insulin action and both glucose oxidation and storage [START_REF] Skovbro | High-fat feeding inhibits exercise-induced increase in mitochondrial respiratory flux in skeletal muscle[END_REF]. It remains unclear whether these molecular adaptations align with or are different from those triggered by a time-matched singlecontinuous bout of PA, and whether they can explain the differential metabolic responses observed at the whole-body level.

In this ancillary study, we tested the hypothesis that 4-days of breaking up SB with short-frequent bouts of PA would enhance mitochondrial respiration in the presence of CHO-associated substrates and skeletal muscle would increase expression profiles favoring CHO metabolism. Because prolonged PA is known to be associated with increases in whole-body fat oxidation rates especially in post-absorptive conditions (15), we further hypothesized that 4-days following a daily single-continuous bout of PA matched for total active time would exhibit greater lipid-linked mitochondrial respiration and expression of genes associated with the regulation of lipid metabolism.

Our findings provide muscle-specific evidence that molecular and functional adaptations depend on the terms of PA, i.e. frequency, duration and/or volume, and could contribute to whole-body energetics to improve metabolic health outcomes.

METHODS

Experimental Protocol

The detailed description of the study design and main outcomes have been published previously [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF]. Following the screening visit, three interventions were completed by each participant in a randomized balanced order. Each intervention period lasted 4-days. The first 3-days in free-living conditions and the 4 th day was in a wholeroom calorimeter. All interventions were separated by a 28-day wash out period. Study procedures were performed during the same phase of the menstrual cycle in all women.

All the visits were conducted at the Clinical and Translational Research Center of University of Colorado Hospital. This study was approved by the Colorado Multiple Institutional Review Board and listed at ClinicalTrials.gov: NCT02258438.

Participants

Study volunteers were recruited from October 2014 to October 2016. Inclusion criteria included: age 19 -45 yr, BMI 27 -33 kg/m 2 , weight stable for ≥ 3 months, and being sedentary defined as sitting ≥ 6 h/day. Women were premenopausal but could use oral contraceptives. Exclusion criteria included clinically diagnosed diabetes, consumption of glucose-and/or lipid-lowering medication, dyslipidemia, smoking, or being physically active (> self-reported 150 min/wk MVPA).

Screening Visit

Written informed consent was obtained at the screening visit. Following verification of eligibility criteria, participants completed a treadmill walk test to determine a selfperceived moderate-intensity walking pace (mph). Participants were asked to walk at a similar moderate-intensity pace in free-living conditions and completed the same treadmill walking pace in the whole-room calorimeter.

Study Interventions

Participants were randomized to one of three possible trial-condition sequences using balanced blocks, separately prepared for male and female participants. The study statistician (Z.P.) prepared the computer generated randomization assignments and placed them in sealed envelopes which were opened the day before the first intervention to reveal the order of study interventions.

Sedentary behaviors (SED): During the 3-day free-living period, participants were asked to refrain from structured exercise and maintain habitual levels of daily PA and SB. On day 4, participants remained sedentary in the whole-room calorimeter.

Sedentary + single-continuous bout of walking (ONE): During the 3-day free-living period, participants were instructed to perform 45-min of moderate-intensity walking once per day and to maintain habitual levels of daily PA and SB the remainder of the day. On day 4, participants performed one 45-min bout of moderate intensity walking on a treadmill at 10:00 AM but otherwise remained sedentary in the whole-room calorimeter.

Sedentary + microbouts of walking (MICRO): During the 3-day free-living period, participants were instructed to perform a 5-min bout of moderate intensity walking each hour for 9 consecutive hours and to maintain habitual levels of daily PA and SB the remainder of the day. On day 4, participants performed a 5-min bout of moderateintensity treadmill walking every hour for 9 consecutive hours from 10:00AM to 6:00PM and remained sedentary the rest of the day.

Compliance was verified with objectively measured time spent sitting and lying, stepping and daily steps with ActivPAL™ PA monitors (PAL Technologies Ltd., Glasgow, Scotland) and PA intensity was confirmed with ActiGraph GT3X tri-axial accelerometer (ActiGraph, Pensacola, FL, USA). Data on time spent sedentary and daily patterns of PA have been previously published [START_REF] Jong | Breaking up Sedentary Time in Overweight/Obese Adults on Work Days and Non-Work Days: Results from a Feasibility Study[END_REF].

Standardized Diet

Each participant consumed a 3-day standardized diet leading up to the whole-room calorimeter study visit. The macronutrient content of the control diet was 30% fat, 55% CHO, and 15% protein of total energy intake. On day 4, the macronutrient composition was 20% fat, 65% CHO and 15% protein. Daily energy needs were calculated based on resting metabolic rate and fat free mass measured by dual energy X-ray absorptiometry (DXA, Hologic Delphi-W, Bedford, MA), as previously reported [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF]. By design, participants received the same amount of food across the three interventions and as a result were in similar state of negative energy balance in the two PA trials (8).

Skeletal Muscle Biopsy

Following an overnight fast, a vastus lateralis skeletal muscle biopsy was collected under local anesthesia (2% wt/vol lidocaine HCl) using the Bergstrom biopsy needle technique on the morning of day 5. From the total biopsy sample, ~10-20 mg was placed immediately in ice cold BIOPS solution (10mM Ca-EGTA buffer, 0.1M free calcium, 20mM imidazole, 20mM taurine, 50mM potassium 2-[N-morpholino]-ethanesulfonic acid, 0.5mM dithiothreitol, 6.56mM MgCl2, 5.77mM ATP, and 15mM phosphocreatine [PCr], pH 7.1) and ~40-60 mg was snap frozen in liquid nitrogen and stored at -80 o C for future analysis.

Preparation of Permeabilized Muscle Fiber Bundles

Muscle fiber bundles were gently teased apart in a petri dish containing ice cold BIOPS solution with fine-nosed forceps and a dissecting microscope. Approximately 2-5 mg fiber bundles were then permeabilized in saponin (5 mg/ml) for 30 minutes at 4°C and then washed once for 10 minutes at 4°C with Mir05 respiration medium (0.5mM EGTA, 3mM MgCl2•6H2O, 60mM K-lactobionate, 20mM taurine, 10mM KH2PO4, 20mM HEPES, 110mM sucrose, and 1g/L BSA, pH 7.1), all on an orbital shaker [START_REF] Pesta | High-resolution respirometry: OXPHOS protocols for human cells and permeabilized fibers from small biopsies of human muscle[END_REF]. The permeabilized muscle fiber bundles were blotted, weighed, and placed in the highresolution respirometry (HRR) chambers of an Oxygraph 2K (Oroboros Inc., Innsbruck, Austria) for analysis.

Skeletal Muscle Mitochondrial Respiration Protocols

HRR of permeabilized muscle fibers offers an integrative ex vivo measure of the dynamics of coupled metabolic pathways [START_REF] Gnaiger | Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis[END_REF]. Quantification of oxygen consumption in permeabilized fiber bundles was conducted at 37°C in the oxygen concentration above 200 nmol O2/ml over a ~2 hr period. Substrate-uncoupler-inhibitor titration (SUIT) protocols for CHO and FA respiration was performed following stabilization of the O2 trace. The addition of Cytochrome C was used to confirm the integrity of the outer mitochondrial membrane and the quality control threshold was set at ≤ 10% increase in respiration. Steady state O2 flux for each respiratory state was determined and normalized to dry fiber bundle weight using Datlab 5 software (Oroboros Inc.).

In the CHO SUIT, LEAK state without adenylates respiration (i.e., substrate driven non-phosphorylating inner membrane proton leak) was determined by the addition of saturating concentrations of pyruvate (2 M) and malate (0.4 M). State 3 coupled respiration supported by electron flux through complex I was measured with the addition of ADP (0.5 M). Oxidative phosphorylation (OXPHOS) was achieved by adding saturating concentrations glutamate (2 M) and succinate (1 M). The addition of oligomycin (5 mM) inhibited ATP synthase and titrations of the uncoupler FCCP (carbonyl cyanide-p-trifluoromethoxyphenylhydrazone 0.1 mM) were performed to assess maximal electron transport system (ETS c1+c2) respiration. Lastly, complex I inhibitor rotenone (1 mM) was then added to measure the rate of respiration through complex II alone (ETS c2). Finishing, malonic acid (2 M) and antimycin A (5 mM) were added to inhibit complex II and III, respectively. With total electron transport chain respiration inhibition, residual oxygen consumption was then quantified to be a correction factor for all mitochondrial respiratory rates.

In the FA SUIT, LEAK state without adenylates respiration was measured by the addition of octanoylcarnitine (0.1 M) and malate (0.4 M). This was followed by ADP (0.5 mM) for State 3 coupled respiration and then pyruvate (2 M), glutamate (2 M) and succinate (1 M) for OXPHOS. The remainder of the FA SUIT protocol was the same as the CHO SUIT.

For both substrate specific SUIT protocols, the respiratory control ratio (RCR) was calculated (State 3/LEAK) to determine the degree of coupled respiration and the flux control ratio (FCR, each respiratory state normalized to maximal respiratory rate, i.e. ETS c1+c2) to determine oxygen flux in different respiratory control states.

RNA sequencing

Samples were sequenced as previously described [START_REF] Hohos | High-fat diet-induced dysregulation of ovarian gene expression is restored with chronic omega-3 fatty acid supplementation[END_REF][START_REF] Presby | Regular exercise potentiates energetically expensive hepatic de novo lipogenesis during early weight regain[END_REF]. Briefly, frozen muscle samples were mechanically homogenized (~40-60 mg) in 1mL of TRIzol, and RNA was isolated according to manufacturer's instructions (Quick-DNA/RNA MiniPrep Plus Kit catalog# ZD7003, Research Products International, Mt. Prospect, IL, USA). RNA samples were submitted to the University of Colorado Denver's Genomics and Microarray Core where quality was assessed with the Agilent 2100 Bioanalyzer both before and after library preparation, which used Nugen's Universal Plus mRNA-seq kit according to manufacturer's protocols (NuGEN, Redwood City, CA). Sequencing was performed with the NovaSEQ6000 Ilumina sequencing platform as 150 nucleotide paired end reads, sequenced at 80 million paired reads per sample. Samples were run in two batches. Generated reads were trimmed, mapped to the human genome (GRCh38) using gSNAP, fragments per kilobase of transcript per million mapped reads (FPKM) were calculated using Cufflinks, and R for discovery of differential gene expression [START_REF] Trapnell | Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation[END_REF][START_REF] Wu | Fast and SNP-tolerant detection of complex variants and splicing in short reads[END_REF]. Genes were removed if their FPKM expression was less than 5 in either group of the statistical comparisons or if the difference in means was > 5. Raw mRNA sequencing data has been archived under the GEO accession number GSE186715.

Pathway Analysis

Canonical pathway and pathway comparison was conducted as previously described using IPA (Ingenuity Pathway Analysis, Qiagen) [START_REF] Carli | Single Cell RNA Sequencing of Human Milk-Derived Cells Reveals Sub-Populations of Mammary Epithelial Cells with Molecular Signatures of Progenitor and Mature States: a Novel, Non-invasive Framework for Investigating Human Lactation Physiology[END_REF]. Briefly, pathway analysis was performed using statistically significant differentially expressed genes with p-values of <0.05. Comparison analyses of the differentially expressed gene (DEG) lists between PA modes (ONE v SED, MICRO v SED, and MICRO v ONE) were used to calculate biological pathways and functions that were in common among DEG lists. The output lists were ordered by pathway enrichment significance (Benjamini-Hochberg false discovery rate [FDR]) and pathways were colored by predicted activation Z-score (red = positive, blue = negative, white = N/A).

Data and Statistical Analysis

The appropriateness of normality assumption for ANOVA and linear mixed effects model analysis was verified using Shapiro-Wilk test and visual examination of residual plots. One-way analysis of variance (ANOVA) was used to test for differences in participant characteristics between three subgroups: participants from the parent study (N=20) [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF], participants included for Oroboros experiments to measure mitochondrial respiration in permeabilized muscle fibers (N=19) and participants for RNAseq analysis for molecular analysis (N=8). All initial respiratory states (LEAK, State 3, OXPHOS, ETS c1+c2 and ETS c2) that qualified for inclusion in data analysis (≤ 10% change in respiration after the addition of cytochrome C) were assessed with the ROUT method to identify and remove outliers from the data set [START_REF] Motulsky | Detecting outliers when fitting data with nonlinear regression -a new method based on robust nonlinear regression and the false discovery rate[END_REF]. Linear mixed models tested for between-intervention differences in permeabilized muscle fiber respiratory states (LEAK, State 3, OXPHOS, ETS c1+c2 and ETS c2), and ratios (RCR and FCR) with intervention as repeated effect, sequence, period, and intervention as fixed effects and participants as random effects with variance components as covariance structure. A P ≤ 0.1 was considered significant for model interactions. The least significant difference post-hoc test was used to examine between-intervention differences, a P ≤ 0.05 was considered significant. Differentially expressed genes were analyzed with unpaired ttest between (SED vs. MICRO, SED vs. ONE, and ONE vs. MICRO) followed by FDR which was considered significant ≤ 0.05 by the Benjamini and Hochberg multiple testing correction. Spearman correlational analysis stratified by intervention was performed on mitochondrial oxygen consumption dependent variables with previously published whole-body metabolic outcomes as independent variables [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF]. Linear mixed effect model and between-group comparison of correlations were fit to the dependent variables for the three interventions simultaneously while accounting for the correlation of repeated observations from the same participants. P ≤ 0.1 was considered significant for model interactions and P ≤ 0.05 was considered significant for between-group differences in correlation. Data are provided as mean ± SD, unless specified otherwise.

RESULTS

Participant Characteristics

As previously reported [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF], twenty participants (n=10 males and 10 females) completed all study related procedures, mean age was 32.4 ± 6.3 yr and BMI 30.6 ± 2.9 kg/m 2 . A subsample of participants had fasting muscle biopsies harvested for measurement of mitochondrial oxygen consumption in permeabilized muscle fibers (N=19, 9 males and 10 females) and a subsample for RNA sequencing and pathway analysis (N=8, 4 males and 4 females). No differences in the participant characteristics were noted between the two subsamples and the sample from the parent clinical trial (Table 1).

Mitochondrial capacity for fat oxidation is enhanced by a single continuous bout of PA, but not by multiple short bouts of PA

In the FA SUIT, no between-intervention differences in mitochondrial respiratory states were observed (Figure 1A), but RCR (i.e., degree of mitochondrial coupling) differed across interventions (intervention effect: P=0.008). In the presence of FAassociated substrates (Oct + M + ADP; Figure 1E), the RCR was higher after completing 4-days of ONE (4.8 ± 2.5) compared to both SED (2.3 ± 1.0, P=0.002) and MICRO (3.1 ± 1.1, P=0.024). The higher coupling in the presence of FA-associated substrates was likely driven by a decrease in LEAK respiration. LEAK FCR (i.e., contribution of mitochondrial respiratory state to overall flux; Figure 1C) differed across the three interventions (intervention effect: P = 0.020); it was lower in ONE (0.035 ± 0.008) compared to SED (0.075 ± 0.010, P=0.006), but not to MICRO (0.052 ± 0.005, P=0.21). In the CHO SUIT, there were no between-intervention differences observed for mitochondrial respiratory states (Figure 1B), FCR (Figure 1D), and RCR (Figure 1F).

Correlations between skeletal muscle mitochondrial oxidative capacity and whole-body metabolic outcomes

Mitochondrial coupling (i.e., RCR) in the presence of lipid-associated substrates was positively associated with fasting insulin concentrations in both PA interventions (slope ONE = 0.65, P = 0.0008 and slope MICRO = 0.24, P = 0.02; Table 2). The slopes of these associations were significantly different from SED (slope ONE vs. SED: P = 0.015 and slope MICRO vs SED: P = 0.05). Furthermore, ONE and MICRO differed (P = 0.03), the association was stronger in ONE than in MICRO. Similarly, mitochondrial coupling in the presence of lipid-associated substrates was positively associated with fasting insulin/glucose, an index of insulin sensitivity, in both PA interventions (slope ONE = 58.11, P = 0.001 and MICRO = 23.24, P = 0.012). The slope for ONE was significantly different from SED (slope ONE vs. SED: P = 0.002) and the slope for MICRO tended to differ from the one observed in SED (slope MICRO vs SED: P = 0.07). Sleeping CHO oxidation (i.e., proxy of fasting CHO oxidation) was negatively correlated with CHOsupported LEAK respiration (slope = -36.06, P = 0.03) but not with OXPHOS and ETS c1+c2 (P for interaction: OXPHOS P = 0.401 and ETS c1+c2 P = 0.194). Sleeping fat oxidation (i.e. proxy of fasting fat oxidation) was not correlated with fat-supported LEAK, OXPHOS, and ETS c1+c2 (P for interaction P = 0.919, P = 0.357, and P = 0.115, respectively).

Differential gene expression leads to similar pathway enrichment for MICRO and ONE with more biological pathways enriched in ONE

Differentially expressed genes were calculated (FDR ≤ 0.05) relative to the SED condition, and 1,126 DEG were identified in the ONE and 678 DEG were identified in the MICRO (Figure 2A). About 14% of DEG were shared between ONE and MICRO (Figure 2B). Ingenuity Pathway Analysis software was used to evaluate biological pathways and functions (Figure 3), providing a global view of the gene expression differences for muscle adaptations to PA mode. Significantly different gene lists were analyzed using the Comparison Analysis tool in IPA, and the pathway enrichment scores (sum of -log10(FDR)) were calculated as an indication of the proportion of DEGs that populate a given pathway. The fold changes of individual DEGs within a pathway were used to calculate whether a given pathway was predicted to be activated (red), inhibited (blue), or no prediction (white) [START_REF] Carli | Single Cell RNA Sequencing of Human Milk-Derived Cells Reveals Sub-Populations of Mammary Epithelial Cells with Molecular Signatures of Progenitor and Mature States: a Novel, Non-invasive Framework for Investigating Human Lactation Physiology[END_REF]. This pathway analysis revealed similar DEG enrichment and predicted activation states for ONE and MICRO compared to SED.

The Oxidative Phosphorylation pathway was predicted to be activated (SED vs ONE: P < 0.0001; SED vs MICRO: P < 0.0001) and the Sirtuin Signaling pathway was predicted to be inhibited (SED vs ONE: P < 0.0001; SED vs MICRO: P < 0.0001). Critical cellular energy and metabolic sensor pathways were significantly enriched following ONE (P < 0.01, parenthetical values) but not MICRO (P > 0.05) when compared to SED. For instance, the mTOR, ERK/MAPK, AMPK, Nrf2-mediated oxidative stress response, insulin receptor, calcium signaling, necroptosis, and HIF1α pathways were enriched following ONE but not MICRO. Comparing both PA interventions (MICRO vs ONE) several pathways were enriched including Sirtuin Signaling, Oxidative Phosphorylation, mTOR, ERK/MAPK, AMPK, and NRF2-mediated oxidative stress response pathways (P < 0.01 for all, parenthetical values). Oxidative Phosphorylation pathway was predicted to be less activated following MICRO (MICRO vs ONE activation z-score: -3.357, P < 0.001). Additionally, while these pathways were not enriched, calcium signaling and necroptosis signaling pathways were upregulated for MICRO (P < 0.05). All pathways expressed by the interventions are listed in Supplementary Tables 4 and5.

When comparing the significantly different genes modulated in the Oxidative

Phosphorylation pathways for ONE or MICRO (relative to SED), more genes were similarly expressed in common between ONE and MICRO (N = 19/45) while some were unique only to MICRO (N=10/45) or ONE (N=16/45) (Figure 4A). Several genes regulated in common between both PA trials, included downregulation of the mitochondria trafficking inesin-binding protein 1 (TRAK1) and apoptosis activator Bcl2associated agonist of cell death (BCL2), mitochondrial ATP synthase subunit epsilon 5E (ATP5E), cytochrome c oxidase genes (COX6, -7A2 and -7B), and NADH dehydrogenase subcomplex subunits (NDUFA2, -4, -8, 11 and -13, NDUFB5, -9, NDUFS3, and NDUFV2), the mitochondrial fission 1 protein (FIS1), and the mitochondrial peroxiredoxin-5 (PDX5).

The Sirtuin Signaling pathway, critical for sensing cellular energy status and metabolism (4), was also regulated following both ONE and MICRO. Significantly different genes that were regulated in common between ONE and MICRO included forkhead box protein O4 (FOXO4, decreased), mitochondrial import inner membrane translocase subunits (TIMM8, -23 and -50, and PAM16), and the mitochondrial cholesterol transport protein (TSPO) (Figure 4B). Significantly regulated genes in the ONE included 5'-AMP-activated protein kinase catalytic subunit alpha-1 (PRKAA1, decreased), superoxide dismutase 1 and 2 (increased SOD1 and decreased SOD2), lipogenic transcription factor sterol regulatory element-binding protein 1 (SREBF1, decreased), and inflammatory response signal transducer and activator of transcription 3 (STAT3, decreased). Furthermore, ONE upregulated unique mitochondrial import inner membrane translocase subunits (TIMM13 and -17A), as well as mitochondrial import receptor subunits (TOMM6, -7, and 22), and several MAP kinase signal transduction genes (MAPK1, -3, and -12). In contrast to ONE, MICRO had unique regulation of mitochondrial isocitrate dehydrogenase 2 (IDH2, decreased), skeletal muscle autophagy regulators forkhead box protein O3 (FOXO3, decreased) and ubiquitin-like-conjugating enzyme (ATG3, increased), microtubule components (TUBA1A, TUBA1B, and TUB8), as well as complex V ATP synthase subunit gamma (ATP5C1). All significant DEGs organized by mode of PA are presented in supplemental data (Supplemental Tables 1,2, 3).

DISCUSSION

In this randomized cross-over study, we compared the functional and molecular adaptations in skeletal muscle biopsies to short-frequent bouts of PA spread throughout the day to a time-matched single-continuous bout of PA, both compared to sedentary control conditions, in adults with overweight or obesity. We identified changes in the regulation of gene expression and mitochondrial functional capacity that are in common to both PA interventions and others that are unique to either ONE or MICRO. This indicates that biological pathways in skeletal muscle differentially respond to the terms of PA, i.e. volume, frequency and/or duration of the bouts.

Similar enrichment and activation of two key biological pathways known to regulate substrate oxidation, Oxidative Phosphorylation pathway (including electron transport chain components) and Sirtuin Signaling pathway, were observed after ONE and MICRO suggesting that adaptations within these pathways depend on total daily active time rather than bouts frequency and/or duration. These changes were observed 14 hr after the last microbout of PA and after 21 hr after the long continuous bout of PA, indicating a sustained response beyond the acute period. A potential regulator of muscle adaptation to acute forms of PA is FOXO4, that was commonly downregulated following both ONE and MICRO. FOXO4 and family members are transcription factors known to carry out the inhibitory effects of insulin and IGF-1 (30) potentially supporting improved insulin sensitivity in skeletal muscle following PA. This is in line with the improved insulin sensitivity noted at the whole-body level after both ONE and MICRO [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF].

Contrary to our hypothesis, we observed no functional adaptations in mitochondrial oxidative capacity in the presence of CHO-associated substrates. This suggests that over the short-term short-frequent bouts of PA can trigger changes in the regulation of gene expression in pathways associated with the regulation of metabolism but these changes do not translate into higher mitochondrial function. This further suggests that mechanisms others than adaptations in mitochondrial oxidative capacity may contribute to the increase in 24 hr CHO oxidation we observed previously in MICRO [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF]. Because short-frequent bouts of brisk walking were completed throughout the day, participants were essentially in the postprandial state. The greater oxidative rates of CHO may therefore be the simple result of CHO availability to provide energy for body movements. Of note, substrate driven LEAK respiratory state in the presence of CHOassociated substrate was negatively correlated with sleeping CHO oxidation (an index of fasting CHO oxidation) and no associations were observed for OXPHOS and ETS mitochondrial respiratory states. A potential early mechanism for adaptations in mitochondrial function may be the decrease in substrate driven LEAK respiration through a more robust electron transport chain in which more protons are used for ATP production. The mitochondrial adaptations to MICRO may therefore take time. Longerterm studies will be needed to test if short-frequent bouts of PA can ultimately enhance overall skeletal muscle mitochondrial function.

We found that ONE, but not MICRO, resulted in a higher RCR when permeabilized muscle fibers were provided lipid-associated substrates, suggesting an improved coupling efficiency. The observed higher RCR in the presence of lipid-associated substrates may be due to a decrease in LEAK state FCR which indicates a reduced contribution of the LEAK respiratory state to overall electron flux and improved coupling efficiency because less protons are lost to non-coupled respiration. Besides one study testing the effect of an acute 60-min bout of PA in 21 healthy untrained male participants [START_REF] Skovbro | High-fat feeding inhibits exercise-induced increase in mitochondrial respiratory flux in skeletal muscle[END_REF], the evidence on acute and short-term adaptions in mitochondrial oxidative capacity are limited. Therefore, it is interesting to note that adaptations can be detected after 4-days of PA. In addition, this improved mitochondrial oxidative capacity for lipids along with the molecular adaptations favoring substrate oxidation may contribute to the higher 24 hr FA oxidation we observed previously in response to the single-continuous bout intervention [START_REF] Jong | Effect of frequent interruptions of sedentary time on nutrient metabolism in sedentary overweight male and female adults[END_REF].

The daily single-continuous bout intervention also had more biological pathways that were enriched with DEGs than the multiple short PA bouts intervention (Figure 3). These include the AMPK signaling pathway which is sensitive to changes in ATP turnover, calcium signaling pathway which is sensitive to oscillations in calcium concentrations from skeletal muscle contractile function, and HIF1α signaling pathway which is sensitive to the intracellular partial pressure of oxygen [START_REF] Egan | Exercise metabolism and the molecular regulation of skeletal muscle adaptation[END_REF]. Additionally, pathways associated with skeletal muscle nutrient uptake, growth, differentiation, survival, and the oxidative stress response were enriched after the daily singlecontinuous bout of PA intervention, including Insulin Receptor signaling, mTOR, ERK/MAPK signaling and NRF2-mediated Oxidative Stress Response. This constellation of enriched pathways after the single-continuous PA bout intervention is known to be associated with PGC-1α [START_REF] Long | Exercise-induced mitogen-activated protein kinase signalling in skeletal muscle[END_REF][START_REF] Cunningham | mTOR controls mitochondrial oxidative function through a YY1-PGC-1alpha transcriptional complex[END_REF][START_REF] Musci | Exercise-Induced Mitohormesis for the Maintenance of Skeletal Muscle and Healthspan Extension[END_REF][START_REF] Schieke | The mammalian target of rapamycin (mTOR) pathway regulates mitochondrial oxygen consumption and oxidative capacity[END_REF], a master regulator of mitochondrial biogenesis [START_REF] Halling | PGC-1α-mediated regulation of mitochondrial function and physiological implications. Applied physiology, nutrition, and metabolism = Physiologie appliquee[END_REF], which may explain why we observed higher mitochondrial coupling. Of note, these results are confounded by an energy deficit as energy expended during PA was not replaced and future studies will need to confirm if similar results are observed in stable energy balance.

The enrichment of several PA associated transcriptional response pathways may be due to the sustained nature of the single-continuous PA bout intervention more than total active time. The longer bout length may create a greater challenge to skeletal muscle homeostasis therefore eliciting a greater adaptive stress response. This might be particularly true for the Oxidative Phosphorylation pathway because when comparing both PA interventions together this pathway was predicted to be less expressed in shortfrequent bouts intervention. However, the Calcium Signaling pathway was predicted to be positively expressed following the exposure to frequent active breaks suggesting that the frequency rather than length of the PA bouts may impact this pathway. Sarcolemma depolarization during muscle contraction increases cytosolic Ca++ thereby activating the calcium signaling pathway. The frequent nature of the PA bouts may trigger waves of cytosolic Ca++ release that leads to increased expression of the Calcium Signaling pathway which has been shown to be associated with mitochondrial biogenesis [START_REF] Ojuka | Role of calcium and AMP kinase in the regulation of mitochondrial biogenesis and GLUT4 levels in muscle[END_REF].

A strength of the study was the randomized cross-over design of the interventions because each participant served as their own control and all females were studied in the same phase of their menstrual cycle. We were able to compare the respective effects of interrupting sedentary time from increased energy expenditure and/or PA by including a time-matched control. Compliance with each intervention period was verified with PA monitors. The results of this ancillary study should also be considered along with several limitations. Muscle biopsies were harvested at two different time intervals after the last bout of PA (21 hr post the single-continuous PA bout and 14 hr post the last bout of the short-frequent PA bouts intervention). The current experiments quantified mitochondrial respiratory capacity and not the actual respiration directly following completion of each 4-day PA trial. Additionally, the sample size for RNA sequencing was small and RNA sequencing results were not confirmed with RT-quantitative PCR due to limitations of muscle tissue availability.

CONCLUSION

Breaking up SB with short-frequent bouts of PA spread throughout the day over the short-term (4-days) is a sufficient stimulus to promote changes in the regulation of skeletal muscle gene expression in pathways associated with substrate oxidation. These changes were similar to those elicited by a time-matched daily single-continuous bout of PA, suggesting that total daily active time and/or energy expenditure is a primary trigger for those pathways. However, these molecular changes translated into an improved capacity for mitochondrial fat oxidation and were associated with higher expression in muscle-contraction transcription signaling pathways only when PA was performed as a single-continuous bout. In contrast, the Calcium Signaling pathway was only activated when PA was completed as multiple short-frequent bouts. Therefore, this study showed for the first time that biological pathways in skeletal muscle are differentially regulated by the terms of PA, i.e. total active time, bout length and bout frequency. Future studies will need to determine whether these differential responses are sustained over time and translate into differential functional and whole-body changes.

Tables

ONE vs SED

MICRO vs SED

MICRO vs ONE

FIGURES Figure 1 LEAKFigure 1 :

 11 FIGURESFigure1

Figure 2 :

 2 FIGURE 2

FIGURE 3

 3 FIGURE 3

Figure 3 :

 3 Figure 3: Comparison of metabolic and exercise-responsive pathways. Pathway enrichment was calculated -log10(FDR) for each comparison (value indicated in parentheses). For pathway ranking, total FDR score was calculated as the sum across the row for each FDR score. Pathway activation state colored by z-score (white = N/A, red = activated, blue = inhibited). A portion (n=8 -4m/4f) of the skeletal muscle samples were analyzed by RNA sequencing to identify differentially expressed genes and pathway analysis was performed with Ingenuity Pathway Analysis (IPA, Qiagen Inc.). Skeletal muscle biopsies were collected in the fasting state from adults with overweight-to-obesity adults (n = 8, 4m/4f) after completing three 4-day study interventions: SED: sedentary control, ONE: single-continuous bout of 45-min/day of moderate-intensity physical activity, MICRO: hourly 5-min bouts of moderate intensity physical activity performed for 9 consecutive hours per day. * Significant zscore: P < 0.05. # Significant pathway enrichment: P < 0.05.

FIGURE 4 Figure 4 .

 44 FIGURE 4

Table 1 : Participant characteristics

 1

	Muscle mitochondrial	Muscle molecular
	respiration	adaptations
	(Oroboros)	(RNA seq)

Table 2 : Relationship between mitochondrial oxidative capacity and whole-body metabolic measures

 2 Spearman correlational analysis between fasting insulin concentration and mitochondrial coupling in the presence of lipidassociated substrates stratified by intervention. A linear mixed effect model was fit to the dependent variable for each trial simultaneously while accounting for the correlation of repeated observations from the same participant. Under the linear mixed effect model, slopes of the dependent variable vs independent variable were estimated and between-group comparison of the slopes were conducted. The between-group difference in slope assesses the difference between the two groups with respect to the correlation. SED -sedentary control, ONE -single-continuous bout of 45-min/day of moderate-intensity physical activity, MICRO -hourly 5-min bouts of moderate intensity physical activity performed for 9 consecutive hours per day. g/min: gram per minute substrate oxidation. Significant model interaction: * P ≤ 0.05, ** P ≤ 0.01. Significant betweenintervention: # P ≤ 0.05, ## P ≤ 0.01.

	Dependent	Independent Variable	Intervention	Slope	P-value	P	for
	Variable			Estim		interacti
				ate		on
	RCR fat	Fasting Insulin	SED	0.08	0.96	0.003**
			ONE	0.65	0.008 ##
			MICRO	0.24	0.02 #
			SED -ONE	-0.64	0.015 #
			SED -BREAK	-0.24	0.05 #
			ONE -BREAK	0.40	0.03 #

Signaling Pathway Oxidative Phosphorylation mTOR Signaling ERK/MAPK Signaling AMPK Signaling NRF2-mediated Oxidative Stress Response Insulin Receptor Signaling Calcium Signaling Necroptosis Signaling Pathway HIF1α Signaling Fatty Acid β-oxidation I PI3K/AKT Signaling Glycolysis I Z-Score (11.02 # ) (11.43 # ) (6.78 # )

	*					-2 0 2 4
	*	(8.81 # )	*	(11.01 # )	*	(3.86 # )
	*	(12.21 # )				(10.26 # )
		(4.53 # )				(2.07 # )
		(2.41 # )				(3.56 # )
		(2.64 # )				(2.12 # )
		(3.16 # )
		(2.91 # )			*
	*	(2.38 # )	*		*
		(2.36 # )

*

CONFLICTS OF INTEREST:

There is nothing to disclose for any of the authors.


Melani Barrios
email: melani@fceia.unr.edu.ar

Gabriela Reyero

Mabel Tidball

Necessary conditions to a fractional variational problem

Keywords: Fractional Derivatives and Integrals, Fractional Ordinary Differential Equations, Variational Problems. Mathematics Subject Classification 26A33, 34A08, 58E25

In order to solve fractional variational problems, there exist two theorems of necessary conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that involves only Caputo derivatives. In this article, we make a comparison solving a particular fractional variational problem with both methods to obtain some conclusions about which method gives the optimal solution.

Introduction

The fractional variational calculus is a recent field, started in 1997, where classical variational problems are considered, but in the presence of fractional derivatives or integrals [START_REF] Agrawal | Formulation of euler-lagrange equations for fractional variational problems[END_REF][START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives[END_REF][START_REF] Malinowska | Introduction to the fractional calculus of variations[END_REF].

In the last years numerous works have been developed tending to extend the theory of the variational calculus in order to be able to be applied to problems of fractional variational calculus. This is fundamentally due, on the one hand, to an important development of the fractional calculus both from the mathematical point of view and its applications in other areas (electricity, magnetism, mechanics, dynamics of fluids, medicine, etc, [START_REF] Almeida | A fractional calculus of variations for multiple integrals with application to vibrating string[END_REF][START_REF] Bastos | Calculus of variations involving caputo-fabrizio fractional differentiation[END_REF][START_REF] Goos | On the initialboundary-value problem for the time-fractional diffusion equation on the real positive semiaxis[END_REF][START_REF] Goos | Mathematical analysis of a cauchy problem for the time-fractional diffusion-wave equation with α ∈ (0, 2)[END_REF][START_REF] Hilfer | Applications of fractional calculus in physics[END_REF][START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF]), which has led to great growth in its study in recent decades. On the other hand, the fractional differential equations establish models far superior to those that use differential equations with integer derivatives because they incorporate into the model issues of memory [START_REF] Ferrari | Study of a fractional-order model for hiv infection of cd4+ t-cells with treatment[END_REF] or later effects that are neglected in the models with classical derivative.

There are several definitions of fractional derivatives [START_REF] Diethelm | The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type[END_REF][START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF]. The most commonly used are the Riemann-Liouville fractional derivative and the Caputo fractional derivative. It is important to remark that while the Riemann-Liouville fractional derivatives [START_REF] Oldham | The fractional calculus theory and applications of differentiation and integration to arbitrary order[END_REF] are historically the most studied approach to fractional calculus, the Caputo [START_REF] Caputo | Linear models of dissipation whose q is almost frequency independent-ii[END_REF][START_REF] Caputo | Linear models of dissipation in anelastic solids[END_REF] approach to fractional derivatives is the most popular among physicists and scientists, because the differential equations defined in terms of Caputo derivatives require regular initial and boundary conditions. Furthermore, differential equations with Riemann-Liouville derivatives require nonstandard fractional initial and boundary conditions that lead, in general, to singular solutions, thus limiting their application in physics and science [START_REF] Herrmann | Fractional calculus: an introduction for physicists[END_REF][START_REF] Hilfer | Applications of fractional calculus in physics[END_REF].

In order to solve fractional variational problems, there exist two theorems of optimality conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives [START_REF] Agrawal | Generalized euler-lagrange equations and transversality conditions for fvps in terms of the caputo derivative[END_REF][START_REF] Almeida | Isoperimetric problems of the calculus of variations with fractional derivatives[END_REF][START_REF] Almeida | A fractional calculus of variations for multiple integrals with application to vibrating string[END_REF][START_REF] Almeida | Fractional euler-lagrange differential equations via caputo derivatives[END_REF][START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives[END_REF][START_REF] Baleanu | On exact solutions of a class of fractional eulerlagrange equations[END_REF][START_REF] Bastos | Necessary optimality conditions for fractional difference problems of the calculus of variations[END_REF][START_REF] Malinowska | Advanced methods in the fractional calculus of variations[END_REF][START_REF] Malinowska | Generalized natural boundary conditions for fractional variational problems in terms of the caputo derivative[END_REF][START_REF] Malinowska | Introduction to the fractional calculus of variations[END_REF][START_REF] Odzijewicz | Fractional variational calculus with classical and combined caputo derivatives[END_REF], and other Euler-Lagrange equation that involves only Caputo derivatives [START_REF] Antunes | Analysis of a class of boundary value problems depending on left and right caputo fractional derivatives[END_REF][START_REF] Barrios | An euler-lagrange equation only depending on derivatives of caputo for fractional variational problems with classical derivatives[END_REF][START_REF] Barrios | Estabilidad de una generalización de ecuaciones logísticas fraccionarias[END_REF][START_REF] Blaszczyk | Numerical solution of euler-lagrange equation with caputo derivatives[END_REF][START_REF] Kamocki | Variational methods for a fractional dirichlet problem involving jumarie's derivative[END_REF][START_REF] Lazo | The dubois-reymond fundamental lemma of the fractional calculus of variations and an euler-lagrange equation involving only derivatives of caputo[END_REF].

In the present work we will make a comparison between the solutions of the two Euler-Lagrange equations.

The paper is organized as follows: some basic definitions of fractional derivatives and fractional variational problems are shown in section two. Section three presents a particular fractional variational problem, solutions of it using both methods and the comparison between them. We end this paper with our conclusions.

Mathematical tools

Introduction to fractional calculus

In this section, we present some definitions and properties of the Caputo and Riemann-Liouville fractional calculus. For more details on the subject and applications, we refer the reader to [START_REF] Diethelm | The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type[END_REF][START_REF] Oldham | The fractional calculus theory and applications of differentiation and integration to arbitrary order[END_REF][START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF].

Definition 1. The Mittag Leffler function with parameters α, β, is defined by

E α,β (z) = ∞ k=0 z k Γ(αk + β) (1)
for all z ∈ C.

Definition 2. The Gamma function, Γ : (0, ∞) → R, is defined by

Γ(x) = ∞ 0 s x-1 e -s ds. (2)
Definition 3. The Riemann-Liouville fractional integral operator of order

α ∈ R + 0 is defined in L 1 [a, b] by a I α x [f ](x) = 1 Γ(α) x a (x -s) α-1 f (s) ds. ( 3
) Definition 4. If f ∈ L 1 [a, b],
the left and right Riemann-Liouville fractional derivatives of order α ∈ R + 0 are defined, respectively, by

RL a D α x [f ](x) = 1 Γ(n -α) d n dx n x a (x -s) n-1-α f (s)ds and RL x D α b [f ](x) = (-1) n Γ(n -α) d n dx n b x (s -x) n-1-α f (s)ds, with n = α . Definition 5. If d n f dx n ∈ L 1 [a, b]
, the left and right Caputo fractional derivatives of order α ∈ R + 0 are defined, respectively, by

C a D α x [f ](x) = 1 Γ(n -α) x a (x -s) n-1-α d n ds n f (s)ds and C x D α b [f ](x) = (-1) n Γ(n -α) b x (s -x) n-1-α d n ds n f (s)ds, with n = α .
Now some different properties of the Riemann-Liouville and Caputo derivatives will be seen.

Remark 1. (Relation between the Riemann-Liouville and the Caputo fractional derivatives) Considering 0 < α < 1 and assuming that f is such that RL

a D α x [f ], RL x D α b [f ], C a D α x [f ] and C x D α b [f ] exist, then C a D α x [f ](x) = RL a D α x [f ](x) - f (a) 1 -α (x -a) -α and C x D α b [f ](x) = RL x D α b [f ](x) - f (b) 1 -α (b -x) -α . If f (a) = 0 then C a D α x [f ](x) = RL a D α x [f ](x) and if f (b) = 0 then C x D α b [f ](x) = RL x D α b [f ](x). Remark 2.
An important difference between Riemann-Liouville derivatives and Caputo derivatives is that, being K an arbitrary constant,

C a D α x K = 0 , C x D α b K = 0, however RL a D α x K = K Γ(1 -α) (x -a) -α , RL x D α b K = K Γ(1 -α) (b -x) -α , RL a D α x (x -a) α-1 = 0, RL x D α b (b -x) α-1 = 0.
In this sense, the Caputo fractional derivatives are similar to the classical derivatives.

Theorem 1. (Integration by parts. See [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF])

Let 0 < α < 1. Let f ∈ C 1 ([a, b]) and g ∈ L 1 ([a, b]). Then, b a g(x) C a D α x f (x) dx = b a f (x) RL x D α b g(x) dx + x I 1-α b g(x)f (x) b a and b a g(x) C x D α b f (x) dx = b a f (x) RL a D α x g(x) dx -a I 1-α x g(x)f (x) b a . Moreover, if f (a) = f (b) = 0, we have that b a g(x) C a D α x f (x) dx = b a f (x) RL x D α b g(x) dx and b a g(x) C x D α b f (x) dx = b a f (x) RL a D α x g(x) dx.

Fractional variational problems

Consider the following problem of the fractional calculus of variations which consists in finding a function y ∈ α a E that optimizes (minimizes or maximizes) the functional

J(y) = b a L(x, y, C a D α x y) dx (4) with a Lagrangian L ∈ C 1 ([a, b] × R 2 ) and α a E = {y : [a, b] → R : y ∈ C 1 ([a, b]), C a D α x y ∈ C([a, b])},
subject to the boundary conditions: y(a) = y a , y(b) = y b . Now Euler-Lagrange equations for this problem will be stated. In the first one appears both Caputo and Riemann-Liouville derivatives (Theorem 2), meanwhile the second one only depends on Caputo derivatives (Theorem 4).

The proof of the following theorem is in [START_REF] Malinowska | Introduction to the fractional calculus of variations[END_REF].

Theorem 2. If y is a local optimizer to the above problem, then y satisfies the next Euler-Lagrange equation:

∂L ∂y + RL x D α b ∂L ∂ C a D α x y = 0. ( 5
)
Remark 3. Equation ( 5) is said to involve Caputo and Riemann-Liouville derivatives. This is a consequence of the Lagrange method to optimize functionals: the application of integration by parts (Theorem 1) for Caputo derivatives in the Gateaux derivative of the functional relates Caputo with Riemann-Liouville derivatives.

Remark 4. Equation ( 5) is only a necessary condition to existence of the solution. We are now interested in finding sufficient conditions. Typically, some conditions of convexity over the Lagrangian are needed. Definition 6. We say that f (x, y, u) is convex in S ⊆ R 3 if f y and f u exist and are continuous, and the condition

f (x, y + y 1 , u + u 1 ) -f (x, y, u) ≥ f y (x, y, u)y 1 + f u (x, y, u)u 1 ,
holds for every (x, y, u), (x, y + y 1 , u + u 1 ) ∈ S.

The following theorem is valid only for the solution of the Euler-Lagrange equation involving Riemann-Liouville and Caputo derivatives [START_REF] Almeida | Fractional euler-lagrange differential equations via caputo derivatives[END_REF]. Its proof can be seen at [START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives[END_REF].

Theorem 3. Suppose that the function L(x, y, u) is convex in [a, b]×R 2 .
Then each solution y of the fractional Euler-Lagrange equation ( 5) minimizes (4), when restricted to the boundary conditions y(a) = y a and y(b) = y b .

Following [START_REF] Lazo | The dubois-reymond fundamental lemma of the fractional calculus of variations and an euler-lagrange equation involving only derivatives of caputo[END_REF], in the below theorem, we will see an Euler-Lagrange fractional differential equation only depending on Caputo derivatives.

+ C x D α b ∂L ∂ C a D α x y = 0. ( 6
)
Remark 5. We can see that the equation ( 6) depends only on the Caputo derivatives. It is worth noting the importance that

L ∈ C 2 ([a, b] × R 2 )
, without this the result would not be valid. As we remarked before, the advantage of this new formulation is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives.

From now on, when we work with the Euler-Lagrange equation that uses derivatives of Caputo and Riemann-Liouville (5), we will abbreviate it with C-RL and when we use the Euler-Lagrange equation that uses only derivatives of Caputo [START_REF] Almeida | Necessary and sufficient conditions for the fractional calculus of variations with caputo derivatives[END_REF], we will abbreviate it with C-C. Remark 6. Unlike the equation ( 5), at the moment, there are not sufficient conditions for the equation ( 6) which only involves Caputo derivatives. Now we present an example that we are going to solve using these two different methods, in order to make comparisons.

Example

The scope of this section is to present two different candidates to be a solution for a particular problem that arise from solving the two Euler-Lagrange equations presented in the previous section.

First, we are going to solve the classical case, where only appears an integer derivative, and then we are going to deal with the fractional case.

Classical case

The classical problem consist in finding a function y ∈ a E that optimizes (minimizes or maximizes) the functional

J(y) = 1 0 (y (x)) 2 -24 y(x) dx, y(0) = 0 , y(1) = 0,
where

a E = {y : [a, b] → R : y ∈ C 1 ([a, b])}.
To solve this (refer to [START_REF] Van Brunt | The Calculus of Variations[END_REF]), we consider the Lagrangian L(x, y, y ) = (y ) 2 -24 y.

Its Euler-Lagrange equation is

∂L ∂y - ∂ ∂x ∂L ∂y = 0, that is, y (x) = -12.
Solving this equation and taking into account that y(0) = y(1) = 0, we obtain the solution

y(x) = -6x 2 + 6x. ( 8
)

Fractional case

The fractional problem consist in finding a function y ∈ α a E that optimizes (minimizes or maximizes) the functional

J(y) = 1 0 C 0 D α x [y] (x) 2 -24 y(x) dx, y(0) = 0 , y(1) = 0, where α a E = {y : [a, b] → R : y ∈ C 1 ([a, b]), C a D α x y ∈ C([a, b])}.
To solve this we consider the Lagrangian

L(x, y, C 0 D α x [y]) = C 0 D α x [y] 2 -24 y. (9)
Like we said before, we are going to solve it using two methods, one with the C-RL Euler-Lagrange ( 5) and the other one with the C-C Euler-Lagrange equation (6).

Resolution by C-RL equation

Applying the equation ( 5), we obtain

∂L ∂y + RL x D α 1 ∂L ∂ C 0 D α x [y] = 0 -24 + RL x D α 1 2 C 0 D α x [y] = 0 RL x D α 1 C 0 D α x [y] = 12. RL x D α 1 (1 -x) β = Γ(1 + β) Γ(1 + β -α) (1 -x) β-α
and the property RL x D α 1 (1 -x) α-1 = 0, which we have seen on remark 2, considering β = α we can conclude

C 0 D α x [y] (x) = 12 Γ(1 + α) (1 -x) α + c 1 (1 -x) α-1 (10)
where c 1 ∈ R.

Taking into account the following equalities

(-1) n n-1 j=0 (α -j) = Γ(n -α) Γ(-α) , (-1) n n-1 j=0 (α -1 -j) = Γ(n -α + 1) Γ(-α + 1) , we can write (1 -x) α = ∞ n=0 (-1) n n-1 j=0 (α -j) n! x n = ∞ n=0 Γ(n -α) Γ(-α)
x n n! ,

(1 -x) α-1 = ∞ n=0 (-1) n n-1 j=0 (α -1 -j) n! x n = ∞ n=0 Γ(n -α + 1) Γ(-α + 1)
x n n! , replacing this in [START_REF] Barrios | An euler-lagrange equation only depending on derivatives of caputo for fractional variational problems with classical derivatives[END_REF],

C 0 D α x [y] (x) = 12 Γ(1 + α) ∞ n=0 Γ(n -α) Γ(-α) x n n! + c 1 ∞ n=0 Γ(n -α + 1) Γ(-α + 1)
x n n! .

Considering C 0 D α x x β = Γ(1+β) Γ(1+β-α)
x β-α and the linearity of the Caputo derivative, we obtain

y(x) = 12 Γ(1 + α) 2 x α ∞ n=0 Γ(n + 1)Γ(n -α)Γ(1 + α) Γ(1)Γ(-α)Γ(1 + n + α) x n n! + + c 1 Γ(1 + α) x α ∞ n=0 Γ(n + 1)Γ(n -α + 1)Γ(1 + α) Γ(1)Γ(1 -α)Γ(1 + n + α) x n n! + c 2 ,
where c 2 ∈ R.

Using the definition of the Hypergeometric function of parameters a, b, c [START_REF] Erdlyi | Higher transcendental functions[END_REF]:

2 F 1 (a, b, c, x) = ∞ n=0 Γ(a + n)Γ(b + n)Γ(c) Γ(a)Γ(b)Γ(c + n) x n n! , (11)
we can rewrite the solution as

y(x) = 12 Γ(1 + α) 2 x α 2 F 1 (1, -α, 1 + α, x)+ + c 1 Γ(1 + α) x α 2 F 1 (1, 1 -α, 1 + α, x) + c 2 .
Taking into account that y(0) = y(1) = 0, we obtain

y RL (x) = 12 Γ(1 + α) 2 x α 2 F 1 (1, -α, 1 + α, x)- - 6 Γ(1 + α) 2 x α 2 F 1 (1, 1 -α, 1 + α, 1) 2 F 1 (1, 1 -α, 1 + α, x). ( 12
)
Remark 7. This solution is valid only for α > 0.5 since otherwise the solution tends to infinity and does not satisfy the terminal condition.

Finally, since L(x, y, u) = u 2 -24y is a convex function, indeed L(x, y + y 1 , u + u 1 )-L(x, y, u) = (u + u 1 ) 2 -24(y -y 1 ) -u 2 + 24y = = u 2 + 2uu 1 + u 2 1 -24y -24y 1 -u 2 + 24y = = u 2 1 + 2uu 1 -24y 1 ≥ -24y 1 + 2uu 1 = ∂ 2 L(x, y, u, v)y 1 + ∂ 3 L(x, y, u, v)u 1 ,
it is verified for every (x, y, u), (x, y + y 1 , u + u 1 ) ∈ [0, 1] × R 2 , applying the theorem 3, y RL minimizes the problem for 0.5 < α ≤ 1.

Remark 8. We can notice that Theorem 3 only works for functions y that satisfy the C-RL Euler-Lagrange equation, but furthermore they must satisfy the boundary conditions. In the case of not satisfying the boundary conditions (as in the case of 0 < α < 0.5), the theorem does not work. Remark 9. We can observed that the solution [START_REF] Bastos | Calculus of variations involving caputo-fabrizio fractional differentiation[END_REF] tends to [START_REF] Baleanu | Fractional calculus: models and numerical methods[END_REF] when α tends to 1. This means that when α = 1, we recover the solution of the classical problem.

Resolution by C-C equation

As the Lagrangian (9) L ∈ C 2 ([0, 1] × R 2 ), we can apply the Theorem 4. Then using the equation ( 6), we obtain ∂L ∂y

+ C x D α 1 ∂L ∂ C 0 D α x [y] = 0 -24 + C x D α 1 2 C 0 D α x [y] = 0 C x D α 1 C 0 D α x [y] = 12.
By definition,

C x D α 1 (1 -x) β = Γ(1 + β) Γ(1 + β -α) (1 -x) β-α
and the property in remark 2 that, unlike the Riemann-Liouville derivative,

C x D α 1 [d 1 ] = 0, for every d 1 ∈ R, considering β = α we can conclude C 0 D α x [y] (x) = 12 Γ(1 + α) (1 -x) α + d 1 . (13)
Note that in this step this equation is different from [START_REF] Barrios | An euler-lagrange equation only depending on derivatives of caputo for fractional variational problems with classical derivatives[END_REF], and that is why we are going to obtain two different solutions. Now we can write

12 Γ(1 + α) (1 -x) α = 12 Γ(1 + α) ∞ n=0 (-1) n n-1 j=0 (α -j) n! x n .
Taking into account the following equality

(-1) n n-1 j=0 (α -j) = Γ(n -α) Γ(-α) , we obtain 12 Γ(1 + α) (1 -x) α = 12 Γ(1 + α) ∞ n=0 Γ(n -α) Γ(-α) x n n! .
Replacing this in [START_REF] Bastos | Necessary optimality conditions for fractional difference problems of the calculus of variations[END_REF],

C 0 D α x [y] (x) = 12 Γ(1 + α) ∞ n=0 Γ(n -α) Γ(-α) x n n! + d 1 . Considering C 0 D α x x β = Γ(1+β) Γ(1+β-α)
x β-α and the linearity of the Caputo derivative, we obtain

y(x) = 12 Γ(1 + α) ∞ n=0 Γ(n -α) Γ(-α) Γ(1 + n) Γ(1 + n + α) x n+α n! + d 1 x α + d 2 ,
where d 2 ∈ R.

Using the definition (11) of the Hypergeometric function of parameters a, b, c, we can rewrite the solution as

y(x) = 12 Γ(1 + α) 2 x α 2 F 1 (1, -α, 1 + α, x) + d 1 x α + d 2 .
Taking into account that y(0) = y(1) = 0, we obtain

y C (x) = 12 Γ(1 + α) 2 x α 2 F 1 (1, -α, 1 + α, x) - 6 Γ(1 + α) 2 x α . ( 14
)
Remark 10. Unlike y RL in ( 12), y C is valid for every 0 < α ≤ 1. However, we can not ensure that it is a minimum of the problem because there are no sufficient conditions theorem for the C-C Euler-Lagrange equation.

Remark 11. We can observe that the solution (14) tends to (8) when α tends to 1. This means that both solutions of each Euler-Lagrange equations, y RL and y C tend to the solution of the classical Euler-Lagrange equation when α = 1.

Comparison between methods

In this section we are going to show some graphics in order to compare the solutions obtained from the different methods. Figure 1 presents the convergence of both solutions y RL [START_REF] Bastos | Calculus of variations involving caputo-fabrizio fractional differentiation[END_REF] in the left and y C (14) in the right, when we take limit as α approaches one. We can see that both converge to the classical solution y (8). Remark 13. In this figure we can see how the difference between the shapes of both solutions becomes more remarkable when α approaches 0.5, where the solutions y RL diverge as we saw in Remark 7. Table 1 presents the values obtained in each case. To calculate the integrand we approximate the Caputo fractional derivatives of both y C and y RL . For this we use a method of L 1 type that can be seen in [START_REF] Baleanu | Fractional calculus: models and numerical methods[END_REF][START_REF] Li | Numerical methods for fractional calculus[END_REF]. This method consists of making a regular partition of the interval [0, 1] as 0 = x 0 ≤ x 1 ≤ ... ≤ x m = 1, of size h > 0 sufficiently small, and then approximating the Caputo derivative as follows:

C 0 D α x [y] (x m ) = m-1 k=0 b m-k-1 (y(x k+1 ) -y(x k )),
where

b k = h -α Γ(2 -α) (k + 1) 1-α -k 1-α .
Then, to calculate the integrals, we use the Riemann sums approximation. 1 we can see that as we get closer to α = 0.5, the difference between the values is very large, being the minimum the solution of the C-RL equation, while for values 0 < α ≤ 0.5 obviously we only have the solution of the C-C equation, since it is the only one that verifies the border conditions.

Conclusions

In this article, two theorems of necessary conditions to solve fractional variational problems were studied: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives (C-RL), and other Euler-Lagrange equation that involves only Caputo derivatives (C-C). A particular example was presented in order to make a comparison between both conditions.

We were able to get several conclusions. The first thing is that we were able to verify that for 0.5 < α ≤ 1, the minimum was obtained from the solution of the C-RL Euler-Lagrange equation, as suggested by the Theorem 3. Now, for 0 < α ≤ 0.5, the C-RL Euler-Lagrange equation did not provide us with a solution, while C-C equation did. However, we wonder, can we ensure that the solution of the C-C equation (y C ) is the optimal solution for the problem at least for these values of α? The answer to this question is NO. Although in [START_REF] Lazo | The dubois-reymond fundamental lemma of the fractional calculus of variations and an euler-lagrange equation involving only derivatives of caputo[END_REF] it was shown that the solution y C is a critical solution of the problem, and we also saw that its shape is graphically more similar to the shape of the classical solution, we cannot ensure that y C is an optimal solution or even for the cases in which 0 < α ≤ 0.5. If there was a Theorem of Sufficient Conditions for the C-C equations, our example would not verify it, because if it did, this Theorem would be in contradiction with the Theorem of Sufficient Conditions for the C-RL equations (Theorem 3). This means that the convexity conditions over the Lagrangian does not reach to obtain sufficient conditions for the C-C equations. Then, if there were other conditions and such a Theorem existed, in order not to contradict Theorem 3, it should depends on the value of α.

In other hand, we can observe that when α = 1, y C was the classical solution and it was the minimum of the problem, but when α decreased, when 0.5 < α < 1, these solutions were not minimums, because the y RL solutions were. There is a discontinuity in the y C solution when α goes to 1. Then, why y C would be the minimum solutions when 0 < α < 0.5? If they were minimums, it would exist another discontinuity of these solutions with respect to α.

In conclusion, while working with C-C equations make the work easier when it comes to calculations, many times we have to be careful with the implementation of this method since C-RL Euler-Lagrange equations are the ones that truly provide us with the optimal solution.

Theorem 4 .

 4 Let y be an optimizer of (4) with L ∈ C 2 ([a, b] × R 2 ) subject to boundary conditions y(a) = y a , y(b) = y b , then y satisfies the fractional Euler-Lagrange differential equation ∂L ∂y

Figure 1 :

 1 Figure 1: Convergence of y RL and y C solutions

Figure 2

 2 Figure 2 presents a comparison between both solutions y RL (12) and y C (14), for different values of α.

Figure 3

 3 Figure 3 presents the solution y C (14) for α = 0.4.

Figure 2 :Figure 3 : 4 Remark 14 .

 23414 Figure 2: Comparison of the C-RL and C-C solutions

Table 1 :

 1 Values obtained with C-RL and C-CRemark 15. In Table

	α	C-RL	C-C
	1	-12.1752	-12.1752
	0.95	-16.4431	-14.3133
	0.9	-17.3685	-16.7006
	0.8	-36.6555	-22.2567
	0.7	-60.2608	-28.9016
	0.55	-127.9983	-40.9804
	0.4 the solution does not exist -55.5863

Acknowledgments

This work was partially supported by Universidad Nacional de Rosario through the projects ING568 "Problemas de Control Óptimo Fraccionario". The first author was also supported by CONICET through a PhD fellowship.