Marlène Lac

Geneviève Tavernier

Ph.D Cedric Moro
email: cedric.moro@inserm.fr

Does housing temperature influence glucose regulation and muscle-fat crosstalk in mice? 1 2

Keywords: Thermoneutrality, brown adipose tissue, glucose metabolism, myokines, mouse 27 model 28 29 Highlights 30 Thermoneutrality, brown adipose tissue, glucose metabolism, myokines, mouse 27

The robustness of scientific results is partly based on their reproducibility. Working with animal models, particularly in the field of metabolism, requires to avoid any source of stress to rule out a maximum of bias. Housing at room temperature is sufficient to induce thermal stress activating key thermogenic organs such as brown adipose tissue (BAT) and skeletal muscle.

BAT covers most of the non-shivering thermogenesis in mice and burns a variety of fuels such as glucose and lipids. A high prevalence of BAT is associated with a strong protection against type 2 diabetes risk in humans, implying that BAT plays a key role in glucose homeostasis.

However, thermal stress is poorly and inconsistently considered in experimental research. This thermal stress can significantly impede interpretation of phenotypes by favoring compensatory signaling pathways. Indeed, various studies revealed that thermoneutrality is essential to study metabolism in mice in order to reach a suitable level of "humanization". In this review, we briefly discuss if and how ambient temperature influence blood glucose homeostasis through BAT and muscle-fat crosstalk.

Dear Claude, Thank you for the review of our manuscript BIOCHI-D-22-00551R1 entitled "Does housing temperature influence glucose regulation and muscle-fat crosstalk in mice?". We specifically addressed Reviewer 2 remaining comments and comprehensively revised the manuscript. We thank again the reviewers for their insightful and constructive comments and hope that the paper will be now acceptable for publication in the journal.

Sincerely, Cedric Moro, Ph.D. On behalf of all co-authors Reviewer #2: The manuscript has been substantially revised, however, some problems are still present:

We sincerely thank the reviewer for the positive and constructive assessment of the manuscript. 1) On line 57 of the tracked version, you mention that main heat producing organs are " cardiac and skeletal muscles, brain, liver, and adipose tissues". White adipose tissue is definitely not an important producer of heat, modify the text so that it is more accurate.

We agree. Sorry for this omission. The sentence has been changed (line 56).

2) On line 63-64 of the tracked version you use the argument that after the stimulation by norepinephrine BAT is the main source of heat, while the muscle temperature decreases. This is not surprising given the fact that NE is a specific activator of BAT, but you cannot draw from the result of this experiment the conclusion that shivering is not significant contributor to the cold-induced heat production. This is correct. We modified the sentence as follow:

When mice are exposed to cold (4-6°C), shivering slightly increase (~30%) as recorded by electromyography [5]. However, the contribution of shivering to cold-induced energy expenditure and heat production was minor in comparison to cold-induced non shivering thermogenesis mostly produced by brown fat. Thus, when directly recording tissue temperature with specific probes, norepinephrine increases markedly heat production in BAT (+1°C) while temperature in skeletal muscle did not change significantly [5].

3) On line 204 of the tracked version you use BMR for conditions of measurement of metabolic rate under temperatures below termoneutrality. By definition, BMR is measured at thermoneutrality. Yes, we agree and have revised the sentence accordingly for "metabolic rate" (line 190). 4) On line 217 of the tracked version you mention the fact that "... baseline energy expenditure at room temperature was significantly higher in IL-6-/-mice than in wild-type animals...". This fact does not correspond to the sentence on line 221 of the tracked version ("... oxygen consumption at room temperature was identical in wild-type and IL-6-/-mice..." and also does not correspond to the data given in this paper. Sorry for the confusion. "… baseline energy expenditure" was meant for "RER". The sentence has been changed accordingly (lines 203-205).

Introduction

The living conditions on the surface of the earth are different all around the globe. Indeed, 37 animals and especially mammals have to cope with drastic temperature changes during the year 38 or sometimes during a day [1]. The maintenance of their body temperature at a constant value 39 is essential for metabolic and vital functions. This maintenance of body temperature is energy 40 consuming; at "low" ambient temperatures, the body must provide enough substrates, such as 41 glucose and fatty acids to produce heat. In contrast, when the ambient temperature is "high", 42 the body must increase its energy expenditure in order to dissipate the heat. An ambient "third" 43 temperature exists, the thermoneutrality, over which metabolic rate is minimum and constant, 44 with a balance at equilibrium between heat loss and heat production [2]. However, most induced energy expenditure and heat production was minor in comparison to cold-induced non 60 shivering thermogenesis mostly produced by brown fat. Thus, when directly recording tissue 61 temperature with specific probes, norepinephrine increases markedly heat production in BAT 62 (+1°C) while temperature in skeletal muscle did not change significantly [5]. Brown adipose 63 tissue (BAT) ensures most of the non-shivering thermogenesis. BAT is present in rodents and 64 in infants, but also in adult humans [6]. Indeed, the numerous BAT mitochondria contain 65 uncoupling protein 1 (UCP1) [7]. This protein can be inhibited by purine-based nucleotides, or 66 activated by fatty acids [8]. When UCP1 is activated by fatty acids, the proton gradient 67 produced by the mitochondrial respiratory chain is dissipated, allowing heat production. The 68 muscle is also responsible for thermogenesis through high frequency muscular contraction, also 69 called shivering thermogenesis [9]. Its major role in maintaining body temperature during 70 thermal stress is explained by its large mass in the body reaching up to 35% of total body weight.

71

White adipose tissue (WAT) also plays a role in non-shivering thermogenesis. It has the ability 72 to convert partially into a heat producing tissue, by increasing mitochondria and thermogenic 73 processes. This phenomenon, called browning or beigeing, is characterized by an increase in 74 the expression of UCP1. It occurs under conditions of cold exposure and activation of the 75 sympathetic nervous system [10] or the atrial natriuretic peptide pathway [11]. Others 76 thermogenic mechanisms can occur independently of UCP1 in WAT involving creatine [12] 77 and calcium [5] futile cycles. The energy requirement essential for the survival of living 78 organisms defines the basal metabolic rate (BMR). To measure this parameter, it is essential 79 that people are in fasted conditions, lying down, without clothing protection and awake. Above 80 all, it is necessary to be at an ambient temperature that does not require heat production or 81 dissipation [13].The ratio of daily energy expenditure to BMR is ~2-fold higher in mice housed 82 at standard animal care facility (between 21°C and 23°C) (Figure 1). The use of rodent models in research does not allow reproduction of the same metabolic 84 responses as in humans. Minimizing the involvement of thermogenesis in metabolism is 85 therefore essential to "humanize" the animals and thus facilitate the extrapolation of results [3].

86

In mice, thermoneutrality varies from 28°C to 32°C, that is the reason why the term of 87 thermoneutral zone is often used [14]. In fact, in mice, during the light phases or rest phases, 88 the temperature zone where energy expenditure was minimal extends over 4°C. During the dark 89 phases or active phases, it converges towards a point. This point separates temperatures that 90 lead to an increase in body temperature and temperatures that conversely lead to a body 91 temperature loss. The decrease in body temperature is counteracted by an increase in energy 92 expenditure [14]. To rule out the thermogenic role of BAT, experiments must be carried out at 93 the temperature that corresponds to a zero thermoregulatory balance. However, although 94 housing animals at 21°C is recognized as a significant thermal stress, it has also been shown 95 that thermoneutrality must be regulated in a very specific way and depends on many factors, 96 such as the number of animals by cage, the resources available (enrichments and nesting 97 materials). It is also necessary to take into account the active and resting phases, and to adapt 98 the temperature to these phases. According to Speakman and colleagues, the ideal housing 99 temperature would be 25°C in mice arguing that humans usually live 3-5°C below 100 thermoneutrality [2]. This implies that the ratio of daily energy expenditure to BMR is about 101 1.8 in mice at 25°C, mimicking the ratio found in free-living humans.

102

To define the thermoneutral zone, one must evaluate the temperature at which the blood flow 103 from the center of the body to the skin provides as much heat as the flow from the skin to the 104 core of the body. Thermoneutrality is therefore not strictly identical to thermal comfort 105 temperature which is influenced by behaviors in humans [15]. In mice, thermoneutrality is used 106 and typically ranges between 29 and 32°C in metabolic research. Differences in 107 thermoneutrality temperatures between mice and humans are mainly explained by the large 108 difference in body size [14]. Housing mice at thermoneutrality can help unveil metabolic 109 phenotype since they spend less energy for maintaining their body temperature constant [16].

110

This lower energy expenditure makes them more sensitive to the development of obesity [17] 111 and less sensitive to fasting-induced weight loss [18]

Influence of thermoneutrality on blood glucose control 125

Housing temperature is also of major importance in the study of metabolism, because many 126 organs, nervous and hormonal systems are sensitive to temperature variations. It is well known 127 that mice fed a high-fat diet (HFD) develop glucose intolerance within a few days regardless of 128 housing temperature. However, several factors such as blood glucose, food intake and energy 129 expenditure are significantly decreased when housed at thermoneutrality. Indeed, we observed 130 that C57BL/6J mice had reduced fasting blood glucose levels when housed at 30°C compared 131 to 21°C. Overall their glucose excursion during an oral glucose tolerance test was significantly 132 lower, although their responsiveness (AUC OGTT) to the glucose load was similar (Figure 2).

133

This observation is in agreement with previous studies [19,22]. Some parameters were studied 134 and described in wild-type C57BL/6J male mice, fed either a standard diet or a HFD and during 135 an acute exposure (6h) to 15, 20, 25 or 30°C [22]. As expected, main differences were shown 136 between mice fed a HFD vs standard diet, i.e. fasting blood glucose and insulin were increased 137 by the HFD and only fasting insulin was increased in obese mice housed at 30°C compared to 138 other temperatures. Furthermore, glucose excursion during an OGTT was increased in HFD-139 fed mice housed at 15°C, 20°C or 25°C when compared to mice housed at 30°C. However, 140 housing temperature did not seem to influence insulin tolerance [22]. Another study with acute 141 response to changes in housing conditions combined with HFD was carried out. In addition to 142 classical in vivo tests, body composition was analysed. The increase in percent fat mass seen at 143 thermoneutrality was linked to an increase in subcutaneous and perigonadal fat pads while BAT 144 mass was reduced [23]. This effect was accompanied by a reduced BAT expression of 145 thermogenic genes such as Ucp1, Pparγ (peroxisome proliferator-activated receptor-gamma) 146 and Pdrm16 (PR domain containing 16). Interestingly, in vivo glucose uptake measured using 147 radiolabelled 2-deoxyglucose was mostly reduced in BAT at room temperature (22°C) after 1-148 day of HFD, while it was mostly reduced in soleus muscle at thermoneutrality. Altogether, these 149 studies suggest that thermoneutrality significantly influences blood glucose regulation while 150 increasing the susceptibility of mice to develop obesity and T2D in response to HFD.

151

As previously discussed, UCP1 plays a major role in heat production by uncoupling the 152 respiratory electron chain to ATP production. It was therefore expected that Ucp1 -/-mice would 153 become obese. First analyses failed to validate this assertion [24]. More than 10 years later, 154 Ucp1 -/-mice were housed at thermoneutrality (30°C) and the results were remarkable, Ucp1 -/- 155 mice became obese even when fed a standard diet [16]. This also indicates that robust demonstrated that loss of UCP1 function does not promote diet-induced obesity in mice [25].

159

In UCP1 deficient mice, administration of CL316,243, a specific beta3 agonist, did not change 160 glucose uptake while O2 consumption was decreased compared to wild-type mice. Glucose 161 uptake in BAT thus appears to be uncorrelated with heat production via UCP1 [26]. steatosis, independently of changes in food intake [28]. In addition, white adipose tissue-173 specific inactivation of D2 in euthyroid mice lead to excess body weight and fat mass gain on 174 a HFD at room temperature due to impaired BAT thermogenesis and reduced systemic lipid 175 oxidation [29]. This "obese phenotype" is lost when mice are housed at thermoneutrality, thus 176 unraveling the role of thermogenesis in this process. In contrast, skeletal muscle (SKM) D2 were studied during stress and cold exposure. At room temperature, the respiratory exchange 203 ratio (RER) was significantly higher in IL-6 -/-compared to wild-type mice, reflecting a fat 204 utilization rate significantly lower in IL-6 -/-mice. At thermoneutrality, the RER was similar in 205 wild-type and IL-6 -/-mice. There was a trend for decreased baseline oxygen consumption in 206 IL-6 -/-mice at thermoneutrality, whereas oxygen consumption at room temperature was 207 identical in wild-type and IL-6 -/-mice [31]. In summary, irrespective of the housing 208 temperature, IL6-deficient mice exhibit reduced cold-and stress-induced EE, indicating that 209 endogenous IL6 is a key regulator of this process. In addition, Knudsen et al. demonstrated 210 using IL-6 -/-mice that IL-6 is required for a full induction of UCP1 protein expression in 211 response to cold exposure and influences the UCP1 protein content iWAT of both untrained 212 and exercise trained animals [32].

213

FGF21 has been identified as a myokine correlating with insulin resistance in humans [33].

214

FGF21 deficiency in mice was shown to result in impaired ability to adapt to long-term cold 215 exposure with diminished browning of WAT [34]. Inversely, systemic administration of FGF21 216 in obese mice resulted in reduced adiposity, improved glycemic control, as well as increased 217 energy expenditure. Altogether, these observations suggest that FGF21 may be a key factor The living conditions on the surface of the earth are different all around the globe. Indeed, 37 animals and especially mammals have to cope with drastic temperature changes during the year 38 or sometimes during a day [1]. The maintenance of their body temperature at a constant value 39 is essential for metabolic and vital functions. This maintenance of body temperature is energy 40 consuming; at "low" ambient temperatures, the body must provide enough substrates, such as 41 glucose and fatty acids to produce heat. In contrast, when the ambient temperature is "high", 42 the body must increase its energy expenditure in order to dissipate the heat. An ambient "third" 43 temperature exists, the thermoneutrality, over which metabolic rate is minimum and constant, 44 with a balance at equilibrium between heat loss and heat production [2]. However, most

The use of rodent models in research does not allow reproduction of the same metabolic 84 responses as in humans. Minimizing the involvement of thermogenesis in metabolism is 85 therefore essential to "humanize" the animals and thus facilitate the extrapolation of results [3].

86

In mice, thermoneutrality varies from 28°C to 32°C, that is the reason why the term of 87 thermoneutral zone is often used [14]. In fact, in mice, during the light phases or rest phases, 88 the temperature zone where energy expenditure was minimal extends over 4°C. During the dark 89 phases or active phases, it converges towards a point. This point separates temperatures that 90 lead to an increase in body temperature and temperatures that conversely lead to a body 91 temperature loss. The decrease in body temperature is counteracted by an increase in energy 92 expenditure [14]. To rule out the thermogenic role of BAT, experiments must be carried out at 93 the temperature that corresponds to a zero thermoregulatory balance. However, although 94 housing animals at 21°C is recognized as a significant thermal stress, it has also been shown 95 that thermoneutrality must be regulated in a very specific way and depends on many factors, 96 such as the number of animals by cage, the resources available (enrichments and nesting 97 materials). It is also necessary to take into account the active and resting phases, and to adapt 98 the temperature to these phases. According to Speakman and colleagues, the ideal housing 99 temperature would be 25°C in mice arguing that humans usually live 3-5°C below 100 thermoneutrality [2]. This implies that the ratio of daily energy expenditure to BMR is about 101 1.8 in mice at 25°C, mimicking the ratio found in free-living humans.

102

To define the thermoneutral zone, one must evaluate the temperature at which the blood flow 103 from the center of the body to the skin provides as much heat as the flow from the skin to the 104 core of the body. Thermoneutrality is therefore not strictly identical to thermal comfort 105 temperature which is influenced by behaviors in humans [15]. In mice, thermoneutrality is used 106 and typically ranges between 29 and 32°C in metabolic research. Differences in 107 thermoneutrality temperatures between mice and humans are mainly explained by the large 108 difference in body size [14]. Housing mice at thermoneutrality can help unveil metabolic 109 phenotype since they spend less energy for maintaining their body temperature constant [16].

110

This lower energy expenditure makes them more sensitive to the development of obesity [17]

Influence of thermoneutrality on blood glucose control 125

Housing temperature is also of major importance in the study of metabolism, because many 126 organs, nervous and hormonal systems are sensitive to temperature variations. It is well known 127 that mice fed a high-fat diet (HFD) develop glucose intolerance within a few days regardless of 128 housing temperature. However, several factors such as blood glucose, food intake and energy 129 expenditure are significantly decreased when housed at thermoneutrality. Indeed, we observed 130 that C57BL/6J mice had reduced fasting blood glucose levels when housed at 30°C compared 131 to 21°C. Overall their glucose excursion during an oral glucose tolerance test was significantly 132 lower, although their responsiveness (AUC OGTT) to the glucose load was similar (Figure 2).

133

This observation is in agreement with previous studies [19,22]. Some parameters were studied 134 and described in wild-type C57BL/6J male mice, fed either a standard diet or a HFD and during 135 an acute exposure (6h) to 15, 20, 25 or 30°C [22]. As expected, main differences were shown 136 between mice fed a HFD vs standard diet, i.e. fasting blood glucose and insulin were increased 137 by the HFD and only fasting insulin was increased in obese mice housed at 30°C compared to 138 other temperatures. Furthermore, glucose excursion during an OGTT was increased in HFD-139 fed mice housed at 15°C, 20°C or 25°C when compared to mice housed at 30°C. However, 140 housing temperature did not seem to influence insulin tolerance [22]. Another study with acute 141 response to changes in housing conditions combined with HFD was carried out. In addition to 142 classical in vivo tests, body composition was analysed. The increase in percent fat mass seen at 143 thermoneutrality was linked to an increase in subcutaneous and perigonadal fat pads while BAT 144 mass was reduced [23]. This effect was accompanied by a reduced BAT expression of 145 thermogenic genes such as Ucp1, Pparγ (peroxisome proliferator-activated receptor-gamma) 146 and Pdrm16 (PR domain containing 16). Interestingly, in vivo glucose uptake measured using 147 radiolabelled 2-deoxyglucose was mostly reduced in BAT at room temperature (22°C) after 1-148 day of HFD, while it was mostly reduced in soleus muscle at thermoneutrality. Altogether, these 149 studies suggest that thermoneutrality significantly influences blood glucose regulation while 150 increasing the susceptibility of mice to develop obesity and T2D in response to HFD.

151

As previously discussed, UCP1 plays a major role in heat production by uncoupling the 152 respiratory electron chain to ATP production. It was therefore expected that Ucp1 -/-mice would 153 become obese. First analyses failed to validate this assertion [24]. More than 10 years later, 154 Ucp1 -/-mice were housed at thermoneutrality (30°C) and the results were remarkable, Ucp1 -/- 155 mice became obese even when fed a standard diet [16]. This also indicates that robust demonstrated that loss of UCP1 function does not promote diet-induced obesity in mice [25].

159

In UCP1 deficient mice, administration of CL316,243, a specific beta3 agonist, did not change 160 glucose uptake while O2 consumption was decreased compared to wild-type mice. Glucose 161 uptake in BAT thus appears to be uncorrelated with heat production via UCP1 [26]. steatosis, independently of changes in food intake [28]. In addition, white adipose tissue-173 specific inactivation of D2 in euthyroid mice lead to excess body weight and fat mass gain on 174 a HFD at room temperature due to impaired BAT thermogenesis and reduced systemic lipid 175 oxidation [29]. This "obese phenotype" is lost when mice are housed at thermoneutrality, thus 176 unraveling the role of thermogenesis in this process. In contrast, skeletal muscle (SKM) D2 Human skeletal muscle is known to secrete various myokines [30]. Among those, several such were studied during stress and cold exposure. At room temperature, the respiratory exchange 203 ratio (RER) was significantly higher in IL-6 -/-compared to wild-type mice, reflecting a fat 204 utilization rate significantly lower in IL-6 -/-mice. At thermoneutrality, the RER was similar in 205 wild-type and IL-6 -/-mice. There was a trend for decreased baseline oxygen consumption in 206 IL-6 -/-mice at thermoneutrality, whereas oxygen consumption at room temperature was 207 identical in wild-type and IL-6 -/-mice [31]. In summary, irrespective of the housing 208 temperature, IL6-deficient mice exhibit reduced cold-and stress-induced EE, indicating that 209 endogenous IL6 is a key regulator of this process. In addition, Knudsen et al. demonstrated 210 using IL-6 -/-mice that IL-6 is required for a full induction of UCP1 protein expression in 211 response to cold exposure and influences the UCP1 protein content iWAT of both untrained 212 and exercise trained animals [32].

213

FGF21 has been identified as a myokine correlating with insulin resistance in humans [33].

214

FGF21 deficiency in mice was shown to result in impaired ability to adapt to long-term cold 215 exposure with diminished browning of WAT [34]. Inversely, systemic administration of FGF21 216 in obese mice resulted in reduced adiposity, improved glycemic control, as well as increased 217 energy expenditure. Altogether, these observations suggest that FGF21 may be a key factor Housing at thermoneutrality, in addition to revealing phenotypes, could also help potentiating 265 certain therapeutic approaches. For instance, a recent study compared numerous PPAR 266 activators for their ability to induce WAT browning in vivo in lean and obese mice at room 267 temperature and at thermoneutrality. Among them, tesaglitazar, a dual PPARα/γ activator, 268 induced the highest increase of Ucp1 mRNA levels in WAT at both housing conditions [START_REF] Kroon | 402 PPARgamma and PPARalpha synergize to induce robust browning of white fat in vivo[END_REF]. In

156

  compensating mechanisms occur when housing mice below thermoneutrality. More recently, 157 Dieckmann et al. produced a new transgenic mouse line of UCP1 knockout mice and 158

162

  Cold-induced energy expenditure is controlled by several neuro-hormonal mechanisms 163 including the sympathetic nervous system (SNS), the natriuretic peptides and thyroid hormones. 164 The sympathetic nervous system stimulates the cAMP-inducible type 2 deiodinase (D2) that 165 amplifies thyroid hormone signaling in BAT by locally converting the prohormone T4 to the 166 active form of thyroid hormone T3. Mice with targeted inactivation of D2 (D2KO mice) exhibit 167 impaired BAT thermogenesis and hypothermia during cold exposure [27]. However, D2KO 168 mice housed at room temperature do not develop obesity when compared to WT animals both 169 when fed a chow or a HFD The residual activation of BAT caused by thermal stress at room 170 temperature likely mask the lack of D2. Interestingly, after 60 days of ad libitum HFD feeding 171 at thermoneutrality, D2KO mice exhibited developed obesity, glucose intolerance and hepatic 172

4 .

 4 177does not seem to play a significant metabolic role given that SKM-D2KO animals exhibit no 178 particular metabolic phenotype.179In summary, several studies highlight that housing temperature is a key factor in the study of 180 "humanized" mice. For a more accurate transposition of findings related to glucose regulation 181 and T2D risk from mice to humans, experiments in mice should be performed in at 182 thermoneutrality. Our data are in agreement with previously published work indicating that 183 glucose tolerance is strongly influenced by housing temperature. Although mice housed at 184 thermoneutrality have reduced fasting blood glucose levels, they are more susceptible to diet-185 induced obesity and T2D. 186 187 Influence of thermoneutrality on muscle-fat crosstalk 188 We previously discussed that housing mice at ambient temperature below thermoneutrality lead 189 to a significant increase in metabolic rate since mice must defend their core temperature by 190 activating both shivering and non-shivering thermogenesis. Thus, shivering-related muscle 191 contraction may serve to augment BAT activity and WAT browning through circulating 192 secreted signals, i.e. referred as myokines. Shivering does not seem to occur at room 193 temperature and can be recorded by electromyography when temperature fall below 7°C [4]. 194 Human skeletal muscle is known to secrete various myokines [30]. Among those, several such 195 as Interleukin-6 (IL-6), Fibroblast growth factor-21 (FGF21), Irisin, Meteorin-like, and β-196 Aminoisobutyric Acid (BAIBA) have been suggested to target WAT and increase non-197 shivering thermogenesis. 198 IL-6 deficient mice display increased body weight and body fat mass without significant 199 changes in food intake. Thus it is unknown how endogenous IL-6 suppresses body fat, but there 200 is growing evidence that central IL-6 increases sympathetic outflow. To elucidate the possible 201 interactions between endogenous IL-6 and the SNS, the responses of wild type and IL-6 -/-mice 202

257Figures Legend 405 Figure 1 . 408 Figure 2 . 29 Highlights 30  31  32  33 

 405140822930313233 Figures Legend

111

  and less sensitive to fasting-induced weight loss[18].McKie et al. demonstrated that housing 112 temperature influences exercise-induced increase in mitochondrial biogenesis and thermogenic 113 gene expression in inguinal WAT while others tissues such as BAT and skeletal muscle remain 114 unaffected [19]. Remarkably mice housed at thermoneutrality run greater distance on a wheel 115 compared to mice housed at room temperature. Conventional housing temperatures are 116 considered protective against obesity [20]. In a recent study, authors have shown morphological 117 differences both in BAT and muscle according to the housing temperature of mice. At 118 thermoneutrality, mitochondria in both tissues were loosely packed with irregular cristae, while 119 adaptation to cold initiated ultrastructural remodeling of mitochondria including acquisition of 120 more elaborate cristae structure [21]. In summary, housing temperature is critical to unveil 121 metabolic phenotypes in mice since the sustained increase in metabolic rate observed at room 122 temperature favor compensatory mechanisms.123 124

156

  compensating mechanisms occur when housing mice below thermoneutrality. More recently, 157 Dieckmann et al. produced a new transgenic mouse line of UCP1 knockout mice and 158

162

  Cold-induced energy expenditure is controlled by several neuro-hormonal mechanisms 163 including the sympathetic nervous system (SNS), the natriuretic peptides and thyroid hormones. 164 The sympathetic nervous system stimulates the cAMP-inducible type 2 deiodinase (D2) that 165 amplifies thyroid hormone signaling in BAT by locally converting the prohormone T4 to the 166 active form of thyroid hormone T3. Mice with targeted inactivation of D2 (D2KO mice) exhibit 167 impaired BAT thermogenesis and hypothermia during cold exposure [27]. However, D2KO 168 mice housed at room temperature do not develop obesity when compared to WT animals both 169 when fed a chow or a HFD The residual activation of BAT caused by thermal stress at room 170 temperature likely mask the lack of D2. Interestingly, after 60 days of ad libitum HFD feeding 171 at thermoneutrality, D2KO mice exhibited developed obesity, glucose intolerance and hepatic 172

177 4 .

 4 does not seem to play a significant metabolic role given that SKM-D2KO animals exhibit no 178 particular metabolic phenotype.179In summary, several studies highlight that housing temperature is a key factor in the study of 180 "humanized" mice. For a more accurate transposition of findings related to glucose regulation 181 and T2D risk from mice to humans, experiments in mice should be performed in at 182 thermoneutrality. Our data are in agreement with previously published work indicating that 183 glucose tolerance is strongly influenced by housing temperature. Although mice housed at 184 thermoneutrality have reduced fasting blood glucose levels, they are more susceptible to diet-185 induced obesity and T2D. 186 187 Influence of thermoneutrality on muscle-fat crosstalk 188 We previously discussed that housing mice at ambient temperature below thermoneutrality lead 189 to a significant increase in metabolic rate since mice must defend their core temperature by 190 activating both shivering and non-shivering thermogenesis. Thus, shivering-related muscle 191 contraction may serve to augment BAT activity and WAT browning through circulating 192 secreted signals, i.e. referred as myokines. Shivering does not seem to occur at room 193 temperature and can be recorded by electromyography when temperature fall below 7°C [4].

  194

195 as Interleukin- 6 (

 6 IL-6), Fibroblast growth factor-21 (FGF21), Irisin, Meteorin-like, and β-196 Aminoisobutyric Acid (BAIBA) have been suggested to target WAT and increase non-197 shivering thermogenesis. 198 IL-6 deficient mice display increased body weight and body fat mass without significant 199 changes in food intake. Thus it is unknown how endogenous IL-6 suppresses body fat, but there 200 is growing evidence that central IL-6 increases sympathetic outflow. To elucidate the possible 201 interactions between endogenous IL-6 and the SNS, the responses of wild type and IL-6 -/-mice 202

  218linking UCP1 expression to improved glucose metabolism. As discussed earlier, UCP1 KO 219 mice were resistant to diet-induced obesity below thermoneutrality and displayed elevated 220 levels of serum FGF21. Generation of UCP1/FGF21 double knockout (DKO) mice to 221 investigate the role of FGF21 signaling during HFD-induced body weight gain was undertaken 222 [35]. Maintaining these mice during the twelve weeks of HFD intervention at thermoneutrality 223 led to identical body weight and body fat gain in all genotypes (WT, FGF21 KO, UCP1 KO, 224 and DKO) coinciding with the lack of FGF21 effects in BAT and iWAT. Collectively, this 225 study characterizes the effects of endogenous FGF21 that acts as master regulator to protect 226 from diet-induced obesity in the absence of UCP1 [36]. 227 Irisin derives from the cleavage of Fibronectin type III domain-containing protein 5 (FNDC5) 228 produced by skeletal muscle during physical exercise [37]. This myokine induces myogenesis 229 and WAT browning together with a concomitant increase in energy expenditure. Although a 230 controversy exists on the physiological role of irisin in humans, administration of irisin in mice 231 can positively affect glucose homeostasis, lipid profile and other metabolic parameters related 232 to obesity. Celi and colleagues [38] suggested that irisin secretion could have evolved from 233 shivering-related muscle contraction during cold exposure, serving to augment brown fat 234 thermogenesis in concert with FGF21 also in humans. 235 Meteorin-like (Metrnl) is induced after exercise and cold exposure in skeletal muscle and 236 adipose tissue. Elevated circulating Metrnl causes an increase in whole-body energy 237 expenditure associated with WAT browning and improved glucose tolerance in obese/diabetic 238 mice [39]. Metrnl expression was measured in BAT, WAT depots of mice chronically housed 239 at 30°C or after an acute (6h, 24h) and chronic challenge (2 wk) at 4°C. Acute cold exposure 240 significantly elevated Metrnl gene expression in BAT and WAT. Adenovirus-mediated 241 overexpression of Metrnl significantly improved glucose tolerance when compared to control 242 mice injected with Lac Z. Collectively, these data illustrate that increases in circulating Metrnl 243 causes an increase in energy expenditure and an improvement in glucose homeostasis in 244 obese/diabetic mice [39].

  245β-amino-isobutyric acid (BAIBA) is a non-protein β-amino acid, a metabolite of thymine, 246 released by skeletal myocytes[40]. BAIBA can also be generated by catabolism of the 247 branched-chain amino acid valine. BAIBA enhances WAT browning via PPARα, and may 248 contribute to exercise-induced protection from metabolic diseases. It has also been found that 249 BAIBA attenuates insulin resistance, inhibits inflammation and induces fatty acid oxidation in 250 skeletal muscle via the AMPK (5'-AMP-activated protein kinase)-PPARβ pathway [40]. 251 In summary, WAT browning through circulating secreted signals from shivering-related muscle 252 contraction significantly contribute to non-shivering thermogenesis, energy balance and blood 253 glucose homeostasis. This muscle-adipose crosstalk may represent a thermogenic, cold-254 activated endocrine axis, that is exploitable for obesity and T2D therapeutics development.

257

  Rodents are essential models in preclinical research. They have provided important knowledge 258 on disease mechanisms. However, they have been most often used at room temperature, a 259 temperature known to cause thermal stress. Housing temperature is a critical factor to take into 260 account when studying metabolism as it can significantly mask or unmask phenotypes. For 261 instance, thermoneutrality revealed a frank metabolic phenotype of Ucp1 -/-mice. The 262 compensatory mechanisms occurring at room temperature at least in part involve a muscle-fat 263 crosstalk orchestrated by secreted signals through shivering-related muscle contraction.

  264

Figure 1 . 408 Figure 2 .

 14082 Figures Legend

  Figure 1Figure

  . McKie et al. demonstrated that housing

	112
	113	temperature influences exercise-induced increase in mitochondrial biogenesis and thermogenic
	114	gene expression in inguinal WAT while others tissues such as BAT and skeletal muscle remain
	115	unaffected [19]. Remarkably mice housed at thermoneutrality run greater distance on a wheel
	116	compared to mice housed at room temperature. Conventional housing temperatures are
	117	considered protective against obesity [20]. In a recent study, authors have shown morphological
	118	differences both in BAT and muscle according to the housing temperature of mice. At
	119	thermoneutrality, mitochondria in both tissues were loosely packed with irregular cristae, while
	120	adaptation to cold initiated ultrastructural remodeling of mitochondria including acquisition of
	121	more elaborate cristae structure [21]. In summary, housing temperature is critical to unveil
	122	metabolic phenotypes in mice since the sustained increase in metabolic rate observed at room
	123	temperature favor compensatory mechanisms.
	124

Acknowledgements

275 Some of the work that is presented here was supported by grants from the National Research Acknowledgements 275 Some of the work that is presented here was supported by grants from the National Research

Notions of thermogenesis and thermoneutrality

52

The body temperature is maintained at a constant value within the organism by means of a 53 balance that allows the production and dissipation of heat. Heat is virtually produced by every 54 organ having a baseline metabolism. Main heat producing organs involve cardiac and skeletal 55 muscles, brain, liver and brown and beige fat. In presence of thermal stress, both shivering and 56 non-shivering thermogenesis is activated. Shivering does not seem to occur significantly at 57 room temperature [4]. When mice are exposed to cold (4-6°C), shivering slightly increase 58 (~30%) as recorded by electromyography [5]. However, the contribution of shivering to cold-59 induced energy expenditure and heat production was minor in comparison to cold-induced non 60 shivering thermogenesis mostly produced by brown fat. Thus, when directly recording tissue 61 temperature with specific probes, norepinephrine increases markedly heat production in BAT 62 (+1°C) while temperature in skeletal muscle did not change significantly [5]. Brown adipose 63 tissue (BAT) ensures most of the non-shivering thermogenesis. BAT is present in rodents and 64 in infants, but also in adult humans [6]. Indeed, the numerous BAT mitochondria contain 65 uncoupling protein 1 (UCP1) [7]. This protein can be inhibited by purine-based nucleotides, or 66 activated by fatty acids [8]. When UCP1 is activated by fatty acids, the proton gradient 67 produced by the mitochondrial respiratory chain is dissipated, allowing heat production. The 68 muscle is also responsible for thermogenesis through high frequency muscular contraction, also 69 called shivering thermogenesis [9]. Its major role in maintaining body temperature during 70 thermal stress is explained by its large mass in the body reaching up to 35% of total body weight.

71

White adipose tissue (WAT) also plays a role in non-shivering thermogenesis. It has the ability 72 to convert partially into a heat producing tissue, by increasing mitochondria and thermogenic 73 processes. This phenomenon, called browning or beigeing, is characterized by an increase in 74 the expression of UCP1. It occurs under conditions of cold exposure and activation of the 75 sympathetic nervous system [10] or the atrial natriuretic peptide pathway [11]. Others 76 thermogenic mechanisms can occur independently of UCP1 in WAT involving creatine [12] 77 and calcium [5] futile cycles. The energy requirement essential for the survival of living 78 organisms defines the basal metabolic rate (BMR). To measure this parameter, it is essential 79 that people are in fasted conditions, lying down, without clothing protection and awake. Above 80 all, it is necessary to be at an ambient temperature that does not require heat production or 81 dissipation [13].The ratio of daily energy expenditure to BMR is ~2-fold higher in mice housed 82 at standard animal care facility (between 21°C and 23°C) (Figure 1).


Jean B Lasserre
email: lasserre@laas.fr

ON REPRESENTATIONS OF THE FEASIBLE SET IN CONVEX OPTIMIZATION

Keywords: 1991 Mathematics Subject Classification, 90C25 65K05, Key words and phrases, Convex programming; KKT conditions

We consider the convex optimization problem minx{f (x) : g j (x) ≤ 0, j = 1, . . . , m} where f is convex, the feasible set K is convex and Slater's condition holds, but the functions g j 's are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the g j 's are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation.

Introduction

Given differentiable functions f, g j : R n → R, j = 1, . . . , m, consider the following convex optimization problem:

(1.1)

f * := inf x { f (x) : x ∈ K }
where f is convex and the feasible set K ⊂ R n is convex and represented in the form:

(1.2) K = { x ∈ R n : g j (x) ≤ 0, j = 1, . . . , m }.

Convex optimization usually refers to minimizing a convex function over a convex set without precising its representation (see e.g. Ben-Tal and Nemirovsky [1, Definition 5.1.1] or Bertsekas et al. [START_REF] Bertsekas | Convex Analysis and Optimization[END_REF]Chapter 2]), and it is well-known that convexity of the function f and of the set K imply that every local minimum is a global minimum. An elementary proof only uses the geometry of K, not its representation by the defining functions g j ; see e.g. The convex set K may be represented by different choices of the (not necessarily convex) defining functions g j , j = 1, . . . , m. For instance, the set

K := {x ∈ R 2 : 1 -x 1 x 2 ≤ 0; x ≥ 0} is convex but the function x → 1 -x 1 x 2 is not convex on R 2 + .
Of course, depending on the choice of the defining functions (g j ), several properties may or may not hold. In particular, the celebrated Karush-Kuhn-Tucker (KKT) optimality conditions depend on the representation of

K. Recall that x ∈ K is a KKT point if (1.3) ∇f (x) + m j=1
λ j ∇g j (x) = 0 and λ j g j (x) = 0, j = 1, . . . , m, for some nonnegative vector λ ∈ R m . (More precisely (x, λ) is a KKT point.)

Convex programming refers to the situation where f is convex and the defining functions g j of K are also convex. See for instance Ben-Tal and Nemirovsky [1, p. 335 [6, p. 217-218], and Hiriart-Urruty [START_REF] Hiriart-Urruty | Optimisation et Analyse Convexe[END_REF].

A crucial feature of convex programming is that when Slater's condition holds 1 , the KKT optimality conditions (1.3) are necessary and sufficient, which shows that a representation of the convex set K with convex functions (g j ) has some very attractive features.

The purpose of this note is to show that in fact, when K is convex and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation. Indeed, we show that if K is convex and Slater's condition holds then the KKT optimality conditions (1.3) are also necessary and sufficient for all representations of K that satisfy a mild nondegeneracy condition, no matter if the g j 's are convex. So this attractive feature is not specific to representations of K with convex functions.

That a KKT point is a local (hence global) minimizer follows easily from the convexity of K. More delicate is the fact that any local (hence global) minimizer is a KKT point. Various constraint qualifications are usually required to hold at a minimizer, and when the g j 's are convex the simple Slater's condition is enough. Here we show that Slater's condition is also sufficient for all representations of K that satisfy a mild additional nondegeneracy assumption on the boundary of K. Moreover under Slater's condition this mild nondegeneracy assumption is automatically satisfied if the g j 's are convex.

Main result

Let K ⊂ R n be as in (1.2). We first start with the following non degeneracy assumption: Assumption 2.1 (nondegeneracy). For every j = 1, . . . , m, (2.1) ∇g j (x) = 0, whenever x ∈ K and g j (x) = 0.

Observe that under Slater's condition, (2.1) is automatically satisfied if g j is convex. Indeed if g j (x) = 0 and ∇g j (x) = 0 then by convexity 0 is the global minimum of g j on R n . Hence there is no x 0 ∈ K with g j (x 0 ) < 0. We next state the following characterization of convexity. Lemma 2.2. With K ⊂ R n as in (1.2), let Assumption 2.1 and Slater's condition both hold for K. Then K is convex if and only if for every j = 1, . . . , m:

(2.2) ∇g j (x), y -x ≤ 0, ∀ x, y ∈ K with g j (x) = 0.
Proof. Only if part. Assume that K is convex and ∇g j (x), yx > 0 for some j ∈ {1, . . . , m} and some x, y ∈ K with g j (x) = 0. Then g j (x + t(yx)) > 0 for all sufficiently small t, in contradiction with x + t(yx) ∈ K for all 0 ≤ t ≤ 1 (by convexity of K). If part. By (2.2), at every point x on the boundary of K, there exists a supporting hyperplane for K. As K is closed with nonempty interior, by [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF][Th. 1.3.3] the set K is convex 2 .

1 Slater's condition holds for K if for some x 0 ∈ K, g j (x 0 ) < 0 for every j = 1, . . . , m. 2 The author wishes to thank Prof. L. Tuncel for providing him with the reference [START_REF] Schneider | Convex Bodies: The Brunn-Minkowski Theory[END_REF].

Theorem 2.3. Consider the nonlinear programming problem (1.1) and let Assumption 2.1 and Slater's condition both hold. If f is convex then every minimizer is a KKT point and conversely, every KKT point is a minimizer.

Proof. Let x * ∈ K be a minimizer (hence a global minimizer) with f * = f (x * ). We first prove that x * is a KKT point. The Fritz-John optimality conditions state that λ 0 ∇f (x * ) + m j=1 λ j ∇g j (x * ) = 0; λ j g j (x * ) = 0, j = 1, . . . , m, for some non trivial nonnegative vector 0 = λ ∈ R m+1 . See e.g. Hiriart-Urruty [START_REF] Hiriart-Urruty | Optimisation et Analyse Convexe[END_REF]Th. page 77] or Polyak [START_REF] Polyak | Introduction to Optimization[END_REF]Theor. 1,p. 271]. We next prove that λ 0 = 0. Suppose that λ 0 = 0 and let J := {j ∈ {1, . . . , m} : λ j > 0}. As λ = 0 and λ 0 = 0, the set J is nonempty. Next, as g j (x 0 ) < 0 for every j = 1, . . . , m, there is some ρ > 0 such that B(x 0 , ρ) := {z ∈ R n : zx 0 < ρ} ⊂ K and g j (z) < 0 for all z ∈ B(x 0 , ρ) and all j ∈ J. Therefore we obtain j∈J λ j ∇g j (x * ), zx * = 0 ∀ z ∈ B(x 0 , ρ), which, by Lemma 2.2, implies that ∇g j (x * ), zx * = 0 for every j ∈ J and every z ∈ B(x 0 , ρ). But this clearly implies that ∇g j (x * ) = 0 for every j ∈ J, in contradiction with Assumption 2.1. Hence λ 0 > 0 and we may and will set λ 0 = 1, so that the KKT conditions hold at x * .

Conversely, let x ∈ K be an arbitrary KKT point, i.e., x ∈ K satisfies

∇f (x) + m j=1 λ j ∇g j (x) = 0; λ j g j (x) = 0, j = 1, . . . , m,
for some nonnegative vector λ ∈ R m . Suppose that there exists y ∈ K with f (y) < f (x). Then we obtain the contradiction:

0 > f (y) -f (x) ≥ ∇f (x), y -x [by convexity of f ] = - m j=1 λ j ∇g j (x), y -x ≥ 0
where the last inequality follows from λ ≥ 0 and Lemma 2.2. Hence x is a minimizer.

Hence if K is convex and both Assumption 2.1 and Slater's condition hold, there is a one-to-one correspondence between KKT points and minimizers. That is, the KKT optimality conditions are necessary and sufficient for all representations of K that satisfy Slater's condition and Assumption 2.1.

However there is an important additional property when all the defining functions g j are convex. Dual methods of the type sup

λ∈R m +    inf x f (x) + m j=1 λ j g j (x)    , are well defined because x → f (x) + m j=1 λ j g j (x) is a convex function. In par- ticular, the Lagrangian x → L f (x) := f (x) -f * + m j=1 λ j g j (x), defined from an arbitrary KKT point (x * , λ) ∈ K × R m
+ , is convex and nonnegative on R n , with x * being a global minimizer. If the g j 's are not convex this is not true in general.

Example 1. Let n = 2 and consider the problem

P : f * = min { f (x) : a -x 1 x 2 ≤ 0; Ax ≤ b; x ≥ 0 }, where a > 0, A ∈ R m×n , b ∈ R m ,
and f is convex and differentiable. The set

K := {x ∈ R 2 : a -x 1 x 2 ≤ 0; Ax ≤ b; x ≥ 0 }
is convex and it is straightforward to check that Assumption 2.1 holds. Therefore, by Theorem 2.3, if Slater's condition holds, every KKT point is a global minimizer. However, the Lagrangian

x → f (x) -f * + ψ(a -x 1 x 2 ) + λ, Ax -b -µ, x ,
with nonnegative (ψ, λ, µ) ∈ R × R m × R n , may not be convex whenever ψ = 0 (for instance if f is linear). On the other hand, notice that K has the equivalent convex representation

K := x ∈ R 2 : x 1 √ a √ a x 2 0; Ax ≤ b ,
where for a real symmetric matrix B, the notation B 0 stands for B is positive semidefinite.

A topic of further investigation is concerned with computational efficiency. Can efficient algorithms be devised for some class of convex problems (1.1) where the defining functions g j of K are not necessarily convex?