Satish Sen

Hélène Raingard

Stéphanie Dumont

Andries Kalsbeek

Patrick Vuillez

Etienne Challet
email: challet@inci-cnrs.unistra.fr

Ultradian feeding in mice not only affects the peripheral clock in the liver, but also the master clock in the brain

Keywords: circadian rhythm, feeding, 6-meal schedule, clock gene, suprachiasmatic nucleus

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Restricted feeding during the resting period causes pronounced shifts in a number of peripheral clocks, but not the central clock in the suprachiasmatic nucleus (SCN). By contrast, daily caloric restriction impacts also the light-entrained SCN clock, as indicated by shifted oscillations of clock (PER1) and clock-controlled (vasopressin) proteins. To determine if these SCN changes are due to the metabolic or timing cues of the restricted feeding, mice were challenged with an ultradian 6-meals schedule (1 food access every 4 h) to abolish the daily periodicity of feeding. Mice fed with ultradian feeding that lost <10% body mass (i.e. isocaloric) displayed 1.5-h phase-advance of body temperature rhythm, but remained mostly nocturnal, together with up-regulated vasopressin and down-regulated PER1 and PER2 levels in the SCN. Hepatic expression of clock genes (Per2, Rev-erbα, and Clock) and Fgf21 was, respectively, phase-advanced and up-regulated by ultradian feeding. Mice fed with ultradian feeding that lost >10% body mass (i.e. hypocaloric) became more diurnal, hypothermic in late night, and displayed larger (3.5 h) advance of body temperature rhythm, more reduced PER1 expression in the SCN, and further modified gene expression in the liver (e.g. larger phase-advance of Per2 and up-regulated levels of Pgc-1α). While glucose rhythmicity was lost under ultradian feeding, the phase of daily rhythms in liver glycogen and plasma corticosterone (albeit increased in amplitude) remained unchanged. In conclusion, the additional impact of hypocaloric conditions on the SCN are mainly due to the metabolic and not the timing effects of restricted daytime feeding.

Introduction

Biological rhythms are under the control of circadian oscillators, including a master circadian clock located in the suprachiasmatic nucleus (SCN) of the anterior hypothalamus and peripheral oscillators present in almost every cell of the body [START_REF] Bray | The role of cell-specific circadian clocks in metabolism and disease[END_REF]. The underlying molecular mechanism of the clock is based on transcriptional and translational feedback loops consisting of positive and negative elements [START_REF] Reppert | Molecular analysis of mammalian circadian rhythms[END_REF]. When heterodimerized, the positive limb elements BMAL1 and CLOCK activate transcription of the negative elements (Period (Per)1, 2, 3 and Cryptochrome (Cry)1, 2) that in turn inhibit BMAL1/CLOCK transactivation. In parallel, other clock genes such as Rev-erbα, β and Ror α, β, whose transcription is also activated by BMAL1/CLOCK, modulate Bmal1 and Clock transcription [START_REF] Cho | Regulation of circadian behaviour and metabolism by REV-ERB-alpha and REV-ERB-beta[END_REF][START_REF] Crumbley | Direct regulation of CLOCK expression by REV-ERB[END_REF][START_REF] Preitner | The orphan nuclear receptor REV-ERBalpha controls circadian transcription within the positive limb of the mammalian circadian oscillator[END_REF].

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Light perceived by the retina is the most potent synchronizer of the circadian rhythm produced by the molecular clock mechanism within the SCN. The molecular clockwork regulates the rhythmic transcription of clock-controlled genes, such as the 55 gene coding for neuropeptide Arginine Vasopressin (Avp) [START_REF] Jin | A molecular mechanism regulating rhythmic output from the suprachiasmatic circadian clock[END_REF]. The output of the SCN controls the timing of peripheral clocks via nervous, hormonal and behavioral cues [START_REF] Froy | Circadian rhythms, aging, and life span in mammals[END_REF]. Food access restricted to the usual resting period can 60 phase-shift circadian oscillations in a number of peripheral organs and brain regions outside the SCN, while the SCN master clock remains synchronized to the light-dark cycle [START_REF] Damiola | Restricted 1080 feeding uncouples circadian oscillators in peripheral tissues from the central pacemaker in the suprachiasmatic nucleus[END_REF][START_REF] Feillet | Forebrain oscillators ticking with different clock hands[END_REF][START_REF] Stokkan | Entrainment of the circadian clock in the liver by feeding[END_REF]. However, when daytime restricted feeding is combined with caloric restriction, the master clock is affected, as assessed by phase-advances in daily rhythms of body temperature, activity rhythm, and pineal melatonin, as well as by altered photic resetting [START_REF] Challet | Interactions between light, mealtime and 1060 calorie restriction to control daily timing in mammals[END_REF]. Moreover, daily caloric restriction leads to phase-shifts in daily oscillations of clock (PER1) and clock-controlled (AVP) proteins in the SCN [START_REF] Mendoza | Circadian and photic 1180 regulation of clock and clock-controlled proteins in the suprachiasmatic nuclei of calorie-restricted mice[END_REF]. To avoid the synchronizing effects of daily restricted feeding, a protocol has been developed using a feeding regimen of six 10-min food accesses equally distributed over 24 h (i.e. one 10-min meal every 4 h) [START_REF] Kalsbeek | Circadian control of insulin secretion is independent of the temporal distribution of feeding[END_REF]. In nocturnal rats under light-dark conditions, this ultradian 6-meals-a-day feeding schedule does not modify the phase of locomotor activity rhythm, but if food access to the 6-meals is shortened to cause body mass loss, rats become partially active during daytime due to a phaseadvance of the rest/activity rhythm [START_REF] Mendoza | Daily meal timing is not necessary for resetting the main circadian 1170 clock by calorie restriction[END_REF].

One recent rat study showed that peripheral clock gene rhythms are still present during ultradian 6-meals-a-day feeding, despite changes in amplitude and phase (Su et al., 2016b). Another study suggested that in mice, the peripheral clocks remain unaffected by meal timing when each meal is given equally spaced either 2, 3, 4 or 6 times per day. However, if meal frequency is unevenly distributed, i.e. with unequal intervals between the meals, then the phase of peripheral clock genes changes, especially in the kidney. Moreover, that study also showed that ultradian 6-meal feeding coupled to caloric restriction was able to produce phase-advances of peripheral clocks inversely proportional to the degree of energy intake [START_REF] Kuroda | Meal frequency patterns determine the phase of mouse peripheral circadian clocks[END_REF].

In the present study, we aimed at investigating further whether it is the daily timing of feeding and fasting or metabolic cues associated with caloric restriction that affects the central and peripheral clocks. For that purpose, we challenged mice with a 6-meals-a-day feeding schedule (combined with isocaloric or hypocaloric conditions) and studied their behavioural and physiological changes, as well as expression of clock and clock-controlled genes in the master clock and liver.

Materials and methods

Animals and housing

Seventy-six 5-week old male C57BL/6J mice (Janvier labs, Le Genest-Saint-Isle, France) were used for this study. The animals were housed in individual cages equipped with a wheel, at an ambient temperature of 23 ± 2°C under 12:12 h light-dark conditions (lights on at 7:00 AM (defining Zeitgeber Time (ZT) 0) and off at 19:00 PM (=ZT12)). In a group of 46 animals, access to food was automatically controlled by electronic timers for six cages at a time. Thirty animals served as controls and had ad libitum 125 access to food.

All experiments were performed in accordance with the U.S. National Institutes of Health Guide for the Care and Use of Laboratory Animals (1996)

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, the French National Law (implementing 130 the European Communities Council Directive 86/ 609/EEC) and approved in advance by the Regional Ethical Committee of Strasbourg for Animal Experimentation (AL/50/57/02/13) and in compliance with the ethical standards of the jour-135 nal [START_REF] Portaluppi | Ethics and methods for biological rhythm research on animals and human beings[END_REF].

Surgery

Mice were anesthetized with isoflurane (Vetflurane, Virbac 3% powered by 0.2 l/min O2) to implant a transponder (Minimitter, Vitalview, Sunriver, OR, 140 USA) in the abdominal cavity to record body temperature and general cage activity. The abdomen was shaved and sprayed with antiseptic (DermaSpray, Bayer) before an incision in the skin and muscle (8-10 mm) was made. Once the transponder was inserted into the abdominal cavity, the muscle layer was stitched with surgical sutures (Filapeau, 3.0) and anti-inflammatory medication was provided in drinking water (Metacam, 0.2 mg/ml, 0.1 ml) for 2 days.

Experimental procedure

After surgery, the animals were placed in experimental cages for 2 weeks with drinking water and food ad libitum. After this, mice were habituated to an ultradian schedule of six meals each day with one food 155 access every 4 h (ZT2, ZT6, ZT10, ZT14, ZT18 and ZT22). Access to food during restricted feeding was set automatically by the Food Planning system based on a food basket allowing and preventing food access in the low and upper position, respectively 160 (Intellibio, Seichamps, France). Lifting and fall of the food basket being associated with a brief motor noise, these auditory cues may have signaled food availability to the mice. The smaller mesh size of the trough compared to the size of the food pellets pre-165 vented any food hoarding in the cage. The duration of food access was reduced every 4 days gradually from 6 × 1 h, via 6 × 30 min, 6 × 20 min to 6 × 15 min. This protocol was based on previous studies in rats [START_REF] Kalsbeek | Circadian control of insulin secretion is independent of the temporal distribution of feeding[END_REF]Mendoza et al., 170 2008). The fact that mice were fed every day at the same times has probably improved their ability to adjust to ultradian feeding, as opposed to irregular meal times [START_REF] Valle | Detrimental effects of irregular meals on rats' ability to adjust to meal feeding[END_REF]. Food intake during daytime and nighttime was measured twice (at the steps 175 of 6 × 1 h and 6 × 15 min) to evaluate the day-night pattern of food intake. The body mass was measured weekly. At the end of two weeks of feeding according to the 6 × 15 min protocol, two groups were categorized according to individual adaptation to 180 the paradigm, eventually leading to body mass loss.

A cut-off at 10% body mass loss allowed to distinguish an isocaloric group including animals with less than 10% of body mass loss (mean: 5.4 ± 0.5%; n total = 24; n = 4 per ZT) and a hypocaloric group in which 185 animals lost 10% or more (up to 25%) body mass (mean: 15.5 ± 1.1%; n total = 22; n = 3-4 per ZT). Animals of the control group were kept with food and water ad libitum (n = 5 per ZT).

Immunohistochemistry

190 At the end of the experiment, animals were sacrificed with an overdose of pentobarbital. Mice fed with ultradian 6-meals schedule were sampled every 4 h between food accesses (i.e. ZT0, ZT4, ZT8, ZT12, ZT16 and ZT20) to limit direct effects of feeding 195 while avoiding prolonged fasting. Control mice fed ad libitum were sacrificed at the same times. Blood was sampled by intracardiac puncture, liver was sampled in the right lobe, and the heart was perfused with 50 mL of 0.9% saline followed by 50 mL of 4% Mm00520708_m1, Pparα: Mm01208835 m1 and Pgc-1α: Mm00440939_m1) were used for all the genes with 1µl of cDNA in the reaction mixture of 20 μl. Each reaction PCR was done in duplicate. A dilution curve was prepared of pooled cDNA samples using log10 standards to calculate the amplification efficiency for each primer set (values were between 1.85-1.99). Data were normalized to Tbp (Mm00446971_m1) and analysed the comparative cycle threshold (Ct) method RQ = 2-ΔΔCt. ΔΔCt = ΔCt sample -ΔCt reference [START_REF] Pfaffl | A new mathematical model for relative quantification in real-time RT-PCR[END_REF] with efficiency corrections. Transcript levels were calculated relative to the mean of ZT 0 samples.

Plasma metabolic parameters

Plasma samples were obtained after centrifugation of fresh blood collected with 4% EDTA (10 µL for 1 mL of blood) and centrifuged for 10 min (5000 rpm at 4°C). Plasma glucose was evaluated with GOD-PAP kit (Biolabo, Maizy, France). The ACS-ACOD method (NEFA-HR2; Wako, Osaka, Japan) was used for assaying plasma non-esterified fatty acids (NEFA). Plasma concentrations of corticosterone were determined by a Rat/Mouse Corticosterone EIA kit (AC-14F1, IDS EURL, Paris, FRANCE). The limit of sensitivity of the assay was 0.55 ng/mL.

Hepatic glycogen assay

Samples of fresh liver were flash-frozen in liquid nitrogen. Hepatic glycogen was quantified according to the method developed by Murat and Serfaty [START_REF] Murat | Simple enzymatic determination of polysaccharide (glycogen) content of animal tissues[END_REF].

Statistical analysis

Data are presented as mean ± standard error of the mean (SEM). Statistical analysis was performed by SigmaPlot (version 12, SPSS Inc, Chicago, IL, USA). Significance was defined at p < 0.05.

Two-way analysis of variance (ANOVA) with or without repeated measures (RM) were performed to assess the effect of Feeding conditions (food ad libitum, ultradian iso-or ultradian hypo-caloric feeding) and the effect of Time (either baseline versus experimental condition, or time of day), and the Interaction between these factors. Oneway analysis of variance (ANOVA) was performed to assess the effect of Time in the separate feeding groups. When appropriate, post-hoc analysis was performed using Fisher LSD method. For assessing daily rhythmicity, we used a cosinor analysis to determine mean level, amplitude and acrophase of the considered parameter with SigmaPlot software (Jandel Scientific,Chicago, IL). Data were fitted to the following regression: [y = A + B•cos(2π(x -C)/ 24)], where A is the mean level, B the amplitude and C the acrophase of the rhythm.

Results

Food intake

When mice were fed with 6 meals of 1 h each, they ate most of their food during their active period (night time) and less during their resting period (p < 0.001). Post-hoc analysis showed a significant effect of Time (night vs. day; Fisher LSD test, p < 0.05) (Figure 1A). 340 isocaloric mice, both during day and night (Group: isocaloric vs. hypocaloric, p < 0.001; Time: night vs. day, p = 0.056; Interaction: p = 0.88) (Figure 1B). Activity levels being similar in both groups (see below), the reduced food intake and larger loss of 345 body mass in the hypocaloric group seem to be due to a poorer adaptation of these mice to six 10-min meals schedule per day (e.g. less efficient feeding strategy or capacity) compared to isocaloric mice.

Behavioral and physiological outputs of the 350 central clock

Wheel-running activity Two-way ANOVA-RM revealed that during the 6 meals-a-day feeding schedule, daytime activity was more enhanced in hypocaloric animals than in isocaloric mice as compared to ad libitum baseline conditions (Feeding: baseline with food ad libitum vs. 6-meal schedule, p < 0.001; effect of Group: isocaloric vs. hypocaloric, p < 0.001 and Interaction, p < 0.001; Figures 2 and3A). The proportion of nocturnal activity was lower in the 6 meals-a-day as compared to ad libitum feeding conditions, and the decrease during 6-meals feeding was more marked in the hypocaloric group compared to the isocaloric group (Feeding effect: p < 0.001; Group effect: p < 0.001 and Interaction: p < 0.001; Figure 3B). The total wheelrunning activity was increased during ultradian 15min 6-meal feeding as compared to ad libitum baseline conditions (Feeding: baseline with food ad libitum vs. 6-meal schedule, p < 0.001), while there was no difference between the isocaloric and hypocaloric groups (Group: p = 0.27; Figure 3C). Body temperature Two-way ANOVA-RM performed on mean body temperature showed a significant effect of Feeding (p < 0.001), a significant effect of Group (p < 0.001) and a significant Interaction (p < 0.001). Post-hoc analysis (Fisher LSD test, p < 0.05) showed that in mice fed with the ultradian 6-meal schedule mean body temperature decreased with a deeper drop in the hypocaloric group (-1.2°C) than in the isocaloric group (-0.2°C), compared to baseline values (Figures 4 and5A).

The amplitude of the body temperature rhythm was increased by the ultradian 6-meal schedule, both in hypo-or isocaloric conditions (Feeding: p < 0.001; Group: p = 0.81; Interaction, p = 0.019) (Figure 5B).

The acrophase of the body temperature rhythm showed a phase-advance during the ultradian 6-meals schedule, with earlier values in the hypocaloric group (i.e. +3.5 h compared to baseline peak) compared to the isocaloric group (i.e. +1.5 h compared to baseline; Feeding: p < 0.001; Group: p < 0.006; and Interaction, p < 0.001) (Figure 5C).

Daily expression of clock and clock-controlled proteins in the SCN

PER1

The expression level of the clock protein PER1 was assessed in the SCN of mice fed ad libitum and in animals fed according the ultradian 6-meal schedule (hypocaloric and isocaloric groups; Figure 6). There were significant effects of Time of day and Feeding, and a significant Interaction between the two parameters (p < 0.001 for the three factors, Table 1). The significant Feeding effect was caused by a reduction in mean levels of PER1 in the isocaloric and hypocaloric groups, as compared to ad libitum fed controls. One-way ANOVA analysis showed a significant effect of Time in mice fed ad libitum and in the animals fed with ultradian 6-meal schedule (Table 1). Cosinor analysis confirmed the daily rhythmicity of PER1 levels in the SCN of all three groups (peak around ZT14) (Figure 7A, Table 2), as well as the down-regulated expression in mice fed with the ultradian 6-meals schedule, with the most reduced levels in the hypocaloric group (Table 2). In addition, the ultradian 6-meals schedule reduced the amplitude of PER1 expression rhythm in the SCN. Together these results clearly show that the 6 meal feeding conditions affect the master clock.

PER2

Expression of PER2, another core clock protein, was also investigated in the SCN of control mice and those fed according the ultradian 6-meals schedule (Figure 6). Two-way ANOVA revealed significant effects of Time of day and Feeding (p < 0.001 for the two factors Table 1). The One-way ANOVA analysis showed a significant effect of Time in mice fed ad libitum and the animals fed with the ultradian 6-meal schedule (Table 1). Mean levels of PER2 were reduced in both groups of animals fed with the 6-meal schedule, as compared to control mice fed ad libitum (Table 2). Cosinor analysis showed a significant rhythmic expression of PER2 in all three experimental groups, with a peak expression in early 440 night (around ZT13-14) (Figure 7B, Table 2).

Vasopressin (AVP)

Expression of AVP, a clock-controlled protein, was examined in the SCN of ad libitum fed mice, and animals fed according to the ultradian 6-meal 445 schedule (Figure 6). Two-way ANOVA showed a significant effect of Feeding condition (p < 0.001) and a significant Interaction (p = 0.005; Table 1). One-way ANOVA analysis showed a significant effect of Time in mice fed ad libitum, but not in 450 animals fed with the ultradian 6-meal schedule (Table 1). The mean levels of AVP expression in the SCN were up-regulated by 30% throughout the 24 h cycle in both the isocaloric and hypocaloric group, compared to the control group fed ad libitum (Table 2). Cosinor analysis detected a significant rhythm in the ad libitum fed group, with an acrophase at dusk, around ZT11, but not in the isocaloric and hypocaloric groups (Figure 7C, Table 2). The lack of rhythmicity in the isocaloric 6-meals schedule groups indicates that the ultradian feeding schedule may have a major impact on SCN functioning.

Rhythms of plasma metabolites

Plasma glucose

While the main effect of Time of day was not significant for plasma glucose (Table 1), significant differences among groups did occur (Feeding (p < 0.001) and Interaction (p < 0.01)). The oneway ANOVA analysis showed a significant effect of Time in mice fed ad libitum, but no effect of Time in the animals fed with ultradian 6-meal schedule (Table 1). Cosinor analysis detected a significant rhythm in plasma glucose only in ad libitum group (Table 3). As compared to ad libitum control values, the mean levels of plasma glucose were unchanged and reduced in isocaloric and hypocaloric groups, respectively (Figure 8A, Table 3). These results indicate that the ultradian 6-meal schedule causes arrhythmicity of the plasma glucose rhythm in both the isocaloric and hypocaloric group and an additional hypoglycemia in the hypocaloric group.

Non-esterified fatty acids (NEFA) Two-way ANOVA showed a significant effect of Time of Day (p = 0.034) and Feeding condition (p = 0.020), but no Interaction (Table 1). One-way ANOVA analysis showed a significant effect of Time only in the isocaloric group, but not in the animals fed ad libitum (Table 1). Cosinor analysis showed no significant 490 daily rhythms for plasma NEFA levels (Figure 8B, Table 3).

Plasma corticosterone

The daily corticosterone rhythm was significantly affected by the different feeding conditions Feeding (p < 0.001) and Interaction (p < 0.003; Table 1). The one-way ANOVA analysis showed a significant effect of Time in all 3 feeding groups (Table 1), as did the cosinor regression (Table 3). Remarkably, the ultradian 6-meal schedule did not modify its basal levels at 500 dawn. By contrast, ultradian feeding caused an increase of the daily amplitude of corticosterone variations, especially in the hypocaloric group (Figure 8C, Table 3).

Liver glycogen 505 Two-way ANOVA showed significant effects of Time (p < 0.001), but not of Feeding or Interaction (Table 1). One-way ANOVA analysis showed a significant effect of Time in fed ad libitum control, isocaloric and hypocaloric groups 510 (Table 1). Cosinor regression showed a significant rhythm in all 3 groups (Table 3; Figure 8D). Of note, in contrast to plasma glucose, neither the phase, nor the levels of liver glycogen were affected by the ultradian 6-meal feeding. Table 1. p values for the effects of time, feeding and interaction (two-way ANOVA) and effect of time for each feeding condition (one-way ANOVA).

Per2

The one-way ANOVA analysis showed a significant effect of Time in all 3 feeding groups 520 (Table 1) and so did the cosinor regressions (Table 4). However, the significant effects of Feeding (p < 0.001) and Interaction (p < 0.001); Table 1) indicate that these rhythms differed between groups. Indeed, Per2 expression was 525 down-regulated and markedly phase-advanced by the ultradian 6-meal feeding in both hypocaloric (7 h) and isocaloric groups (4 h), as compared to control mice fed ad libitum (Figure 9A, Table 4).

Clock 530 Expression of the circadian gene Clock in the liver was still rhythmic in mice fed according to the ultradian 6-meal schedule (Time of day, p < 0.001; Tables 1 and4). The significant effects of Feeding (p < 0.001) and Interaction (p < 0.001) indicated a 535 down-regulated Clock expression only in isocaloric 6-meal fed mice (Figure 9B, Tables 1 and4). Moreover, hepatic expression of Clock was phaseadvanced, respectively, by 3.5 and 1.2 h in the isocaloric and hypocaloric groups, as compared to 540 control mice fed ad libitum (Figure 9B, Table 4)

Rev-erbα

The hepatic expression of Rev-erbα (also known as Nr1d1) showed rhythmic expression for all feeding conditions (Time of day, p < 0.001; Feeding (p = 0.03) 545 and Interaction (p < 0.001); Tables 1 and4). Rev-erbα expression was phase-advanced, respectively, by ~3.5 and 4.7 h in the hypocaloric and isocaloric groups, as compared to control mice fed ad libitum (Figure 9C, Table 4).

550 Sirt1 Two-way ANOVA showed a significant effect of Time for Sirt1 expression in the liver (p < 0.001; Table 1), but no effect of Feeding or Interaction.

One-way ANOVA analysis showed a significant 555 effect of Time only in the hypocaloric group of mice but not in the mice fed ad libitum or the isocaloric group (Table 1). Cosinor regression showed a significant rhythm in ad libitum and hypocaloric groups (Table 5). Sirt1 expression was not significantly phase-shifted in mice fed with the ultradian 6-meal schedule, as compared to control mice fed ad libitum (Figure 9D, Table 5).

Pparα

The expression of the metabolic gene peroxisomeproliferative-activated-receptor alpha (Pparα) in the liver was rhythmic only in the isocaloric group and the ad libitum fed animals as indicated by the one-way ANOVA (Table 1). Moreover there was a significant effect of Feeding (p < 0.002) and

Interaction (p < 0.007) (Table 1). Cosinor regression indicated significant rhythms in the ad libitum and isocaloric groups (Table 5). Pparα expression was down-regulated in isocaloric 6-meal fed group (Figure 9E, Table 5).

Pgc-1α

The hepatic expression of Pgc-1α showed significant effects of Time (p = 0.007; Table 1) and Feeding (p < 0.001). The effect of Feeding was caused by the up-regulated Pgc-1α expression during ultradian 6-meal feeding, especially in the hypocaloric group, as compared to control mice fed ad libitum (Figure 9F, Table 1). One-way ANOVA analysis showed a significant effect of Time only in the mice fed ad libitum, but not in the animals fed with ultradian 6-meal schedule (Table 1) and significant cosinor regressions in all the groups (Table 5).

Fgf21

The significant effect of Feeding (p < 0.001) and

Interaction (p = 0.017) was caused by Fgf21 Fitted lines show significant cosine regressions (see methods). ~effect of time of day (p < 0.05), # effect of feeding (p < 0.05) and x interaction between feeding and time of day (p < 0.05).

expression being up-regulated by the ultradian 6-meal schedule (Figure 9G, Table 1). Fgf21 in liver showed rhythmic expression in ad libitum and hypocaloric group (Table 5). One-way 595 ANOVA analysis showed a significant effect of Time only in hypocaloric group of mice fed under ultradian 6 meal schedule (Table 1).

Discussion

Challenging mice with a 6-meals-a-day feeding 600 schedule (i.e. one food access every 4 h throughout the 24-h light-dark cycle) markedly disrupts the circadian organization in the SCN and the liver, thus demonstrating that the daily rhythm of feeding and fasting profoundly affects the functioning 605 of the peripheral and master clock system.

Therefore, the 6-meals-a-day feeding schedule, i.e. the absence of a clear daily rhythm of feeding and fasting, results in a partial desynchronization (uncoupling) within the multi-oscillatory network, 610 with stronger changes occurring when ultradian feeding was combined with caloric restriction. In addition, caloric restriction results in a strong phase-advance of the nocturnal pattern of wheelrunning activity, thereby leading to a more diurnal 615 phenotype in locomotor activity. Hence we concluded that ultradian periodicity of the meals and hypocaloric condition cause significant changes in the SCN and peripheral clocks, and that metabolic cues associated with caloric restriction modulate 620 the temporal niche of activity.

Physiological and behavioural changes

Mice fed according to the ultradian feeding schedule were initially given food access for 1 h every 4 h (6 × 1 h per 24 h). During this stage of the 6 meals-a-day feeding protocol, animals show a normal pattern of food intake characterized by high nocturnal intake, but when each food access was shortened to 15 min (6 × 15 min per 24 h) the mice eat equal amounts of food during day and night. Similar adaptations to such unusual feeding schedules have been initially demonstrated in rats fed with an ultradian 6-meals schedule of 6 × 10 min each [START_REF] Kalsbeek | Circadian control of insulin secretion is independent of the temporal distribution of feeding[END_REF]. More recently, another team applied this protocol in mice [START_REF] Kuroda | Meal frequency patterns determine the phase of mouse peripheral circadian clocks[END_REF]. In the present study, the fact that about half of the mice on the ultradian feeding schedule lost body mass led us to subdivide them into two groups, isocaloric and hypocaloric mice (with a cut-off for 10% of body mass loss). The similar levels of wheel-running activity between the two groups indicate that the differences observed in the hypocaloric group are likely due to reduced energy intake and related metabolic changes.

While the isocaloric group of mice maintained essentially a nocturnal pattern of wheel-running activity under the ultradian feeding schedule, the behavioral pattern is very different in the hypocaloric group. These mice became partially diurnal, being inactive during late night and shifting their activity onsets toward daytime. These data suggest that the SCN is affected by hypocaloric feeding. A daily hypocaloric single meal has previously been shown to induce phase-advances of the rest-activity rhythm in both rats and mice, independent of the time of hypocaloric feeding [START_REF] Challet | Phaseadvanced daily rhythms of melatonin, body temperature, and locomotor activity in food-restricted rats fed during 1065 daytime[END_REF][START_REF] Challet | Entrainment in calorie-restricted mice: Conflicting zeitgebers and freerunning conditions[END_REF]. Also, energetic constraints associated with the so-called work for food paradigm lead to shifts of the nocturnal activity pattern of mice to diurnality [START_REF] Hut | Working for food shifts nocturnal mouse activity into the day[END_REF][START_REF] Van Der Vinne | Cold and hunger induce diurnality in a nocturnal mammal[END_REF]. When diurnal grass rats (Arvicanthis ansorgei) are fed with daily hypocaloric feeding at night, they become partially nocturnal [START_REF] Mendoza | Setting the main circadian clock of a diurnal mammal by hypocaloric feeding[END_REF]. In addition to the dramatic reorganization of rest-activity rhythm in the hypocaloric mice, it is worth mentioning that these mice also display some anticipatory activity prior to food access at ZT6 (Figure 2B) that is not seen in the isocaloric group (Figure 2A). A previous work studied behavioral changes of calorie restricted mice fed with a daily amount equal to 60% of ad libitum conditions using automatic feeders that delivered food pellets every 4 h. Compared to control mice (i.e. fed ad libitum in addition to the food pellets supplied by the feeders), calorie restricted mice displayed increased daytime activity, as in our hypocaloric group, and anticipation to the meals, especially those occurring during daytime [START_REF] Luby | Food anticipa-1160 tory activity behavior of mice across a wide range of circadian and non-circadian intervals[END_REF].

A drop in the mean body temperature was detected when mice were fed according to the ultradian feeding rhythm. Such drops in body temperature are also seen when rats and mice are fasted [START_REF] Nagashima | Effects of fasting on thermoregulatory processes and the daily oscillations in rats[END_REF][START_REF] Tokizawa | Estimation of the core temperature control during ambient temperature changes and the influence of circadian rhythm and metabolic conditions in mice[END_REF] and other conditions with energetic challenges (e.g. [START_REF] Hut | Working for food shifts nocturnal mouse activity into the day[END_REF], likely helping these animals to minimize their energy expenditure. Previously a drop in mean body temperature and a phase-advance was also observed in rats challenged with ultradian 6-meal schedules combined with caloric restriction [START_REF] Mendoza | Daily meal timing is not necessary for resetting the main circadian 1170 clock by calorie restriction[END_REF]. In the current study the acrophase of the body temperature rhythm was shifted toward daytime during the 6-meals-a-day feeding schedule, with a larger phase-advance in the hypocaloric fed mice compared to the isocaloric group. The sensitivity of the SCN clock to temperature cues has been a subject of recent debate because some studies found that the SCN may be shifted or not by temperature cycles [START_REF] Buhr | Temperature as a universal resetting cue for mammalian circadian oscilla-1050 tors[END_REF][START_REF] Herzog | Circadian entrainment to temperature, but not light, in the isolated suprachiasmatic nucleus[END_REF]. The fact that the rest-activity rhythm was only phase-advanced in the hypocaloric group may be due to the deeper hypothermia in the calorie-restricted mice, i.e. the SCN and/or its downstream structures (e.g. secondary brain clocks) 705 would be sensitive to the shifting effects of this deeper hypothermia.

Glucose metabolism

In contrast to the daily profiles of plasma NEFA, the daily rhythm of plasma glucose was highly altered by 710 the 6-meal feeding schedule, with differential effects depending on whether the mice belonged to the hypocaloric or isocaloric group. In mice with ad libitum access to food, plasma glucose levels rise during daytime, as previously observed in other stu-715 dies (e.g. [START_REF] Ahren | Diurnal variation in circulating leptin is dependent on gender, food intake and circulating insulin in mice[END_REF][START_REF] Grosbellet | Leptin modulates the daily rhythmicity of blood glucose[END_REF]. While rats fed with the 6-meal schedule maintain their daily rhythm in plasma glucose at the same phase as rats fed ad libitum (La [START_REF] Fleur | A suprachiasmatic nucleus generated rhythm in basal glucose 1150 concentrations[END_REF], in mice fed with 6-meal schedule daily plasma glucose levels became 720 arrhythmic. In the hypocaloric group, in addition a relative hypoglycaemia occurred, most likely due to the caloric restriction [START_REF] Mahoney | Caloric restriction in C57BL/6J mice mimics therapeutic fasting in humans[END_REF].

Liver glycogen is crucial for maintaining blood glucose homeostasis. However, contrary to plasma 725 glucose levels, the daily profiles in hepatic glycogen content remained unaltered, whatever the feeding condition of the mice. These findings suggest that daily rhythmicity of glycogen synthesis in these animals is dependent on SCN outputs (other than AVP) 730 and/or light signals, most likely conveyed by autonomic inputs [START_REF] Cailotto | Daily rhythms in metabolic liver enzymes and plasma glucose require a balance in the autonomic output to the liver[END_REF]. Daytime restricted feeding in rats, however, results not only in increased gluconeogenesis, but also in a 12-h shift in the daily variations of hepatic glycogen content 735 [START_REF] Perez-Mendoza | Daytime restricted feeding modifies the daily variations of liver gluconeogenesis: adaptations in biochemical and endocrine regulators[END_REF], thus indicating that daily restricted feeding can impact timing of glycogen synthesis. In the present study, the unaltered daily rhythm of glycogen content in the liver as opposed to the altered rhythms of plasma glucose suggest that 740 changes in glycemia during ultradian feeding are mainly due to an altered timing in glucose utilization (e.g. by the skeletal muscles and/or the brain).

Plasma corticosterone

Contrary to the restricted-feeding paradigm, the ultra-745 dian 6-meal feeding schedule does not induce any pre-feeding rise of corticosterone, but increases the amplitude of daily rhythm of plasma corticosterone in rats [START_REF] Fleur | A suprachiasmatic nucleus generated rhythm in basal glucose 1150 concentrations[END_REF]. In the present study performed in mice, the daily rhythm of plasma corticos-750 terone also showed a profound increase in amplitude during ultradian feeding conditions, but no shift of its acrophase. As for liver glycogen content, these findings suggest that daily rhythmicity of corticosterone concentrations during 6-meal schedule is controlled 755 by SCN outputs (that would be different in that situation from the classical involvement of SCN AVP; [START_REF] Kalsbeek | Circadian rhythms in the hypothalamo-pituitary-adrenal (HPA) axis[END_REF] and/or light signals probably conveyed by autonomic inputs [START_REF] Husse | The light-dark cycle controls peripheral rhythmicity in mice with a genetically ablated suprachiasmatic nucleus clock[END_REF][START_REF] Ishida | Light activates the adrenal gland: timing of gene expression and glucocorticoid release[END_REF]. A disappearance of the circadian 760 trough of circulating glucocorticoids is a hallmark of chronic stress [START_REF] Dallman | Bottomed out: metabolic significance of the circadian trough in glucocorticoid concentrations[END_REF]. The low morning levels of plasma corticosterone during 6-meals schedule (both the iso-and hypo-caloric groups) indicate that the basal activity of the hypothalamic-765 pituitary-adrenal (HPA) axis is unaltered. By contrast, mice challenged with work for food display a large increase of corticosterone levels throughout daytime (van der [START_REF] Van Der Vinne | Cold and hunger induce diurnality in a nocturnal mammal[END_REF], suggesting hyperactivity of the HPA axis. Thus, 6-meals schedule and work for 770 food paradigms differentially affect the HPA axis. The body mass loss may explain the increased evening peak of corticosterone during the ultradian feeding schedule, as mildly food-restricted rats fed at dusk also show an increased amplitude of the plasma corticos-775 terone rhythm that seems to result from an increased sensitivity of ACTH to CRH [START_REF] Garcia-Belenguer | Facilitation and feedback in the hypothalamo-pituitaryadrenal axis during food restriction in rats[END_REF].

Clock and metabolic gene expression in the liver

In food-restricted animals fed during the resting 780 phase, meal feeding is well known to phase-shift peripheral clocks, especially those in the liver [START_REF] Damiola | Restricted 1080 feeding uncouples circadian oscillators in peripheral tissues from the central pacemaker in the suprachiasmatic nucleus[END_REF][START_REF] Stokkan | Entrainment of the circadian clock in the liver by feeding[END_REF]. By contrast, in response to ultradian feeding, the liver clock is not affected as a whole (Kuroda 785 et al., 2012;Su et al., 2016a). Instead, as confirmed in the present study, distinct transcriptional changes are detected according to the clock gene considered (i.e. Clock, Rev-erbα, and Per2).

First, the phase of oscillations of the circadian 790 gene Clock was only slightly phase-advanced (+1-3 h) by the 6-meal feeding schedule. As CLOCK controls rhythmic synthesis of hepatic glycogen via transcriptional activation of Glycogen synthase 2 [START_REF] Doi | CLOCK regulates circadian rhythms of hepatic glycogen synthesis through transcrip-1085 tional activation of Gys2[END_REF], it is worth noting that hepatic variations in glycogen content and Clock expression do not keep exactly the same phase-relationship according to feeding conditions, suggesting that rhythmic expression of Glycogen synthase 2 is also controlled by transcriptional factors other than CLOCK.

Second, diurnal expression of Rev-erbα in the liver was more phase-advanced (+3-5 h) by the 6meals schedule, independent of the metabolic (isoor hypocaloric) status of the mice. Rhythmic transcription of Clock being controlled by REV-ERBα [START_REF] Crumbley | Direct regulation of CLOCK expression by REV-ERB[END_REF], the smaller phaseadvance of liver Clock oscillations than those of Rev-erbα raise the possibility of post-translational modifications delaying transcriptional activity of REV-ERBα. A daily hypocaloric meal also induces a phase-advanced hepatic expression of Rev-erbα [START_REF] Feillet | Lack of food anticipation in Per2 mutant mice[END_REF], suggesting that this common effect with ultradian feeding is unrelated to daily feeding signals, but may result from other cues (e.g. shifted metabolic or thermic cues; see below). Furthermore, an apparent ~4-h phaseadvance of this gene has also been reported in voles fed according to an ultradian rhythm (one meal every 2.5 h; van der [START_REF] Van Der Veen | Impact of behavior on central and peripheral circadian clocks in the common vole Microtus arvalis, a mammal with ultradian rhythms[END_REF]. Previous reports showed that liver expression of Rev-erbα is downregulated by refeeding after overnight fasting [START_REF] Oike | Feeding cues and injected nutrients induce acute expression of multiple clock genes in the mouse liver[END_REF][START_REF] Tahara | Refeeding after fasting elicits insulin-dependent regulation of Per2 and Rev-erbalpha with shifts in the liver clock[END_REF]. This modulatory effect does not readily explain the observed pattern of Rev-erbα expression under 6meal schedule, probably due to the short period of fasting (~4 h) between each food access). Together, these results indicate daily and ultradian feeding cues modulate the diurnal expression of Rev-erbα in the liver.

Third, the 6-meals schedule (isocaloric group) lead to prominent phase-advances (+4 h) of Per2 expression in the liver, thus confirming previous data in PER2::LUC mice challenged with ultradian 6-meal feeding [START_REF] Kuroda | Meal frequency patterns determine the phase of mouse peripheral circadian clocks[END_REF]. An earlier rise of liver Per2 mRNA levels has also been reported in rats under 6-meal schedule [START_REF] Cailotto | The suprachiasmatic nucleus controls the daily variation of plasma glucose via the autonomic output to the liver: are the clock genes involved?[END_REF] as well as in mice challenged with work for food [START_REF] Van Der Vinne | Cold and hunger induce diurnality in a nocturnal mammal[END_REF]. Oxyntomodulin released by the gut after food intake up-regulates Per2 expression in the liver [START_REF] Landgraf | Oxyntomodulin regulates resetting of the liver circadian clock by food[END_REF]. This modulatory effect could partly explain the dampened amplitude of hepatic expression of Per2 during 6-meal schedule, though it did not abolish the daily rhythmicity (as would be expected if oxyntomodulin was the sole factor involved because food intake would trigger Per2 transcription every 4 h, leading to constitutive levels throughout 24 h). The larger shift in the hypocaloric group (+7 h) is also in accordance with previous findings in PER2::LUC mice challenged with ultradian 6-meal feeding combined to caloric restriction [START_REF] Kuroda | Meal frequency patterns determine the phase of mouse peripheral circadian clocks[END_REF]. These results show that hepatic oscillations of Per2 are regulated not only by meal-timing cues, but also by metabolic cues associated with a reduction in the total energy intake. Among these cues, thermic signals due to phase-advanced body temperature (and deeper hypothermia in the case of ultradian hypocaloric feeding) may directly modulate liver expression of Per2 [START_REF] Kornmann | System-driven and oscillator-dependent circadian transcription in mice with a conditionally active liver clock[END_REF] which may, in turn, phase-advance the whole liver clock. Several metabolic genes in the liver are regulated by the molecular clock, such as Sirt1, Pparα, Pgc-1α and Fgf21. In turn, SIRT1 causes deacetylation and degradation of PER2 and modulates transcription of clock genes (such as Bmal1, Per2 and Cry1) due to NAD + dependent protein deacetylase activity [START_REF] Asher | SIRT1 regulates circadian clock gene expression through PER2 deacetylation[END_REF]. SIRT1 also regulates Pparα expression [START_REF] Purushotham | Hepatocytespecific deletion of SIRT1 alters fatty acid metabolism and results in hepatic steatosis and inflammation[END_REF] and interacts with Pgc-1α in the liver [START_REF] Rodgers | Nutrient control of glucose homeostasis through a complex of PGC-1alpha and SIRT1[END_REF]. The hepatic expression of these metabolic genes was affected by ultradian feeding. However, the precise changes (mean, phase and/or amplitude) markedly differ according to the genes, suggesting a combination of direct effects of 6-meal schedule conditions and modulation of phase-control by the liver clock (that was phase-advanced by ultradian feeding).

Regarding Pgc-1α transcription, it was increased during 6-meals schedule in the isocaloric group, and even more in the hypocaloric group, confirming previous data in mice challenged with another hypocaloric condition without daily feeding cues (i.e. alternate days of fasting for 3 months; [START_REF] Ranhotra | Long-term caloric restriction up-regulates PPAR gamma co-activator 1 alpha (PGC-1alpha) expression in mice[END_REF]. Levels of Sirt1 mRNA levels tended to be increased in the hypocaloric group during ultradian feeding, an effect possibly linked to negative energy balance as up-regulated expression of Sirt1 is clearly observed in 24-h fasted mice [START_REF] Hayashida | Fasting promotes the expression of SIRT1, an NAD+ -dependent protein deacetylase, via activation of PPARalpha in mice[END_REF]. In addition, the phase of diurnal expression of Sirt1 and Pgc-1α in the mouse liver was not affected by ultradian feeding, suggesting some uncoupling with the liver clockwork. Hepatic expression of Pparα was globally down-regulated by the isoca-895 loric 6-meals schedule in contrast to the fastinginduced up-regulation of this gene [START_REF] Hayashida | Fasting promotes the expression of SIRT1, an NAD+ -dependent protein deacetylase, via activation of PPARalpha in mice[END_REF], indicating that the duration of fasting is a critical parameter for transactivation of this gene. 900 Fgf21 expression is highly induced in the liver of mice fasted over 24 h [START_REF] Lundasen | PPARalpha is a key regulator of hepatic FGF21[END_REF]. Here, Fgf21 expression was also increased during the 6-meals schedule, without a positive correlation with Pparα, one of its transcriptional regu-905 lators [START_REF] Lundasen | PPARalpha is a key regulator of hepatic FGF21[END_REF][START_REF] Oishi | Circadian expression of FGF21 is induced by PPARalpha activation in the mouse 1195 liver[END_REF]. Because FGF21 can be released in the bloodstream and act on the central clock [START_REF] Bookout | FGF21 regulates metabolism and circadian behavior by acting on 1045 the nervous system[END_REF], we assayed plasma FGF21 in mice challenged with the ultradian 6-meal schedule.

910

In spite of higher hepatic expression, plasma levels of FGF21 were not increased during ultradian feeding (data not shown). This suggests that the ~4-h duration of fasting between 2 meals during ultradian feeding was too short to allow 915 FGF21 synthesis and/or release. In any case, these findings seem to rule out a critical role of FGF21 as a hepatokine that would convey temporal cues associated with ultradian feeding to the brain.

920

Clock and clock-controlled proteins in the SCN

In contrast to the high sensitivity of peripheral clocks to the synchronizing effects of meal time, the SCN clock remains synchronized to the light-925 dark cycle in food-restricted rodents fed during daytime [START_REF] Damiola | Restricted 1080 feeding uncouples circadian oscillators in peripheral tissues from the central pacemaker in the suprachiasmatic nucleus[END_REF][START_REF] Stokkan | Entrainment of the circadian clock in the liver by feeding[END_REF], leading to the current view that the SCN is insensitive to feeding cues. The present study challenges this concept by highlighting molecular 930 changes in the SCN of mice fed with an ultradian feeding schedule. These changes, including upregulated daily expression of the clock-controlled protein AVP and dampened oscillations of the two clock proteins PER1 and PER2, reveal that the 935 daily alternation of feeding and fasting also participates in the normal functioning of the SCN. For AVP and PER2, the reducing effects of ultradian feeding on the amplitude of their oscillations were comparable between the isocaloric and hypocaloric groups, while the magnitude of the downregulation of PER1 was more pronounced in the hypocaloric group. Of note, ultradian feeding combined with hypocaloric conditions does not produce the same temporal alterations in the SCN as those observed after daily caloric restriction (i.e. phaseadvances in daily oscillations of PER1 and AVP, [START_REF] Mendoza | Circadian and photic 1180 regulation of clock and clock-controlled proteins in the suprachiasmatic nuclei of calorie-restricted mice[END_REF]or non-significant increased oscillations of PER2, van der Vinne et al., 2014), suggesting that caloric restriction is able to shift the SCN clock only when it is combined with a timed daily feeding cue. The effects of ultradian feeding under hypocaloric conditions, however, led to phase-advances of rest/activity rhythm comparable to those induced by daily caloric restriction [START_REF] Mendoza | Feeding cues alter clock gene oscillations and photic responses in the suprachiasmatic nuclei of mice exposed to a light/dark cycle[END_REF] or by work for food [START_REF] Van Der Vinne | Cold and hunger induce diurnality in a nocturnal mammal[END_REF]. This finding confirms that the behavioral phase-advances do not rely on the diurnal timing of a daily feeding cue (absent under ultradian feeding; [START_REF] Mendoza | Daily meal timing is not necessary for resetting the main circadian 1170 clock by calorie restriction[END_REF] and suggests that the neural changes leading to a more diurnal temporal niche of activity under food shortage occur in SCN pathways downstream to the clockwork (e.g. transforming growth factor α or prokineticin 2 signaling) and/or its downstream structures (i.e. secondary brain clocks).

Common voles often display ultradian rhythms of rest-activity and feeding-fasting. Furthermore, the number of SCN cells containing AVP has been negatively correlated with robustness of circadian rhythmicity in this species [START_REF] Gerkema | Expression of circadian rhythmicity correlates with the number of arginine-vasopressin-immunoreactive cells in the suprachiasmatic nucleus of common voles, Microtus arvalis[END_REF]. Our data actually suggest that in mice, ultradian feeding may trigger up-regulated levels of AVP in the SCN.

Previous studies have already shown that daytime restricted feeding in rats modifies the daily pattern of AVP release from the SCN (i.e. delayed onset and earlier offset; Kalsbeek et al., 1998). Furthermore, daily caloric restriction also resulted in a phase-advance of the AVP rhythm in the SCN of mice fed at midday [START_REF] Mendoza | Circadian and photic 1180 regulation of clock and clock-controlled proteins in the suprachiasmatic nuclei of calorie-restricted mice[END_REF]. When vasopressin signalling is altered, interneuronal coupling and the daily amplitude of transcriptional activity in the SCN is compromised (e.g. [START_REF] Li | Vasopressin receptor 1155 V1a regulates circadian rhythms of locomotor activity and expression of clock-controlled genes in the suprachiasmatic nuclei[END_REF][START_REF] Yamaguchi | Mice genetically deficient in vasopressin V1a and V1b receptors are resistant to jet lag[END_REF]. The weaker oscillations of AVP during ultradian feeding may thus feedback within the SCN to the molecular clockwork and participate in dampened SCN oscillations, as observed here for a reduced amplitude of PER1 and PER2 expression.

AVP released from the SCN terminals modulates the daily timing of glucocorticoid secretion by the adrenal glands, via inhibitory effects in the dorsomedial hypothalamic nuclei and the subparaventricular hypothalamic region [START_REF] Kalsbeek | A diurnal rhythm of stimulatory input to the hypothalamopituitary-adrenal system as revealed by timed intrahypothalamic administration of the vasopressin V1 antagonist[END_REF]. In mice fed according to the ultradian feeding schedule, the daily rhythm of plasma corticosterone is increased in amplitude and not shifted, while AVP protein expression is markedly increased. This apparent discrepancy raises the intriguing possibility that AVP in the SCN is actually synthesized, but not released as usual, in response to the 6-meal feeding schedule. In fact, a similar conclusion was drawn by the group of Gerkema because they showed that the increased 1005 levels of AVP in the SCN of arrhythmic voles are due to a decreased synaptic release of AVP [START_REF] Jansen | Being circadian or not: vasopressin release in cultured SCN mirrors behavior in adult voles[END_REF].

To sum up, this study highlights that when the daily feeding rhythm is abolished by introducing an 1010 ultradian periodicity of food intake, SCN function and daily rhythmicity of clock and metabolic genes in the liver are modified, glucose arrhythmicity occurs and amplitude of corticosterone rhythm increases. The additional impact of hypocaloric conditions under ultradian feeding highlights specific changes (not dependent on daily feeding cues or overnight fasting), including phase-advances of rest/activity and body temperature rhythms, more marked reduction of PER1 in the SCN and modified gene expression in the liver (e.g. larger shift of Per2 and up-regulated levels of Pgc-1α). Thus, not only the timing of feeding and fasting, but also metabolic cues associated with hypocaloric conditions may affect the central and peripheral clocks.

la Recherche Scientifique and University of Strasbourg (P.V.

1035 and E.C.) and University of Amsterdam (A.K.).

Figure 1 .Figure 2 .

 12 Figure 1. Food intake in mice fed with ultradian 6-meal schedules. (A) Day and night food intake during 6 × 1 h ultradian schedules in isocaloric and hypocaloric groups. (B) Day and night food intake during 6 × 15 min ultradian schedules for isocaloric and hypocaloric groups. Means (± SEM) lacking common letters are significantly different (Post-hoc test after ANOVA; p < 0.05).

Figure 3 .

 3 Figure 3. Changes in the levels of wheel-running activity between baseline conditions with with food ad libitum and 6 × 15 min ultradian schedules for isocaloric and hypocaloric groups. (A) Daytime wheel-running activity, expressed in % of the total activity. (B) Nocturnal wheelrunning activity, expressed in % of the total activity. (C) Total wheel-running activity (per 24 h), expressed in numbers of wheel revolutions. Means (± SEM) lacking common letters are significantly different (Post-hoc test after 2-way ANOVA-RM; p < 0.05).

Figure 4 .

 4 Figure 4. Daily body temperature in two mice submitted to ultradian 6-meal schedules, belonging either to the isocaloric (panel A) or hypocaloric groups (panel B; same mice as in Figure 2). The horizontal lines show the successive shortening in duration of food access and the vertical lines show the imposed timing of the 6 meals every 4 h, starting at ZT2 (Zeitgeber 2, i.e. 2 h after lights on). The white area on the 4th day of 6 × 15 min feeding schedule is due to a failure in data acquisition (~12 h). Note the increase in daytime thermogenesis, and decrease in nocturnal thermogenesis of the mouse of the hypocaloric group (panel B). The mean daily profiles of body temperature rhythm are during food ad libitum (panel C) and 6 × 15 min ultradian schedules (panel D) for isocaloric (black squares) and hypocaloric groups (grey triangles). During 6-meals schedule (panel D) rectangle boxes indicate the time of food access. Note the diet-induced thermogenesis after each meal in panel D. Rectangle boxes are also drawn in panel C for visual comparison.

Figure 5 .

 5 Figure 5. Changes in body temperature between baseline conditions with food ad libitum and 6 × 15 min ultradian schedules for isocaloric and hypocaloric groups, including mean body temperature (A), amplitude rhythm (B), and acrophase (C). Means (± SEM) lacking common letters are significantly different (Post-hoc test after 2-way ANOVA-RM; p < 0.05).

Figure 6 .

 6 Figure 6. Photomicrographs of PER1, PER2 and AVP immunostaining in the SCN of ad libitum fed, Isocaloric and Hypocaloric groups. Frames indicate the time points with largest differences among groups. Scale bar, 200 μm. ZT, Zeitgeber time (ZT0 and ZT12 defining lights on and off, respectively).

Figure 7 .

 7 Figure 7. Daily profiles of expression of clock and clock-controlled proteins in the SCN of fed ad libitum (white circle), ultradian isocaloric (black square) and hypocaloric (dark grey triangle) groups. (A) PER1 expression; (B) PER2 expression; (C) AVP expression. Fitted lines show significant cosine regressions (see methods). ~effect of time of day (p < 0.05), # effect of feeding (p < 0.05) and × interaction between feeding and time of day (p < 0.05).

Figure 8 .

 8 Figure 8. Daily profiles of plasma glucose, non-esterified fatty acids, corticosterone, and hepatic glycogen in fed ad libitum (white circle), ultradian isocaloric (black square) and hypocaloric (dark grey triangle) groups. (A) Plasma glucose, (B) Plasma non-esterified fatty acids (NEFA) (C) Plasma corticosterone (D) Hepatic glycogen. Fitted lines show significant cosine regressions (see methods). ~effect of time of day (p < 0.05), # effect of feeding (p < 0.05) and x interaction between feeding and time of day (p < 0.05).

Figure 9 .

 9 Figure 9. Daily profiles of hepatic expression of clock and metabolic genes in fed ad libitum (white circle), ultradian isocaloric (black square) and hypocaloric (dark grey triangle) groups. (A) Per2, (B) Clock, (C) Rev-erbα, (D) Sirt1, (E) Pparα, (F) Pgc-1α and (G) Fgf21.Fitted lines show significant cosine regressions (see methods). ~effect of time of day (p < 0.05), # effect of feeding (p < 0.05) and x interaction between feeding and time of day (p < 0.05).

Table 2 .

 2 Parameters of cosinor regressions of daily expression of clock and clock-controlled proteins in the SCN.

			SCN PER1 (n cells)		SCN PER2 (n cells)			SCN AVP (n cells)
			Mean	SEM	p	Mean	SEM	p	Mean	SEM	p
	Ad libitum (n=30)	a	191.06*	9.17	<0.001	114.92*	3.81	<0.001	42.43 #	1.92*	= 0.040
		b	111.14*	13.03		85.96	5.40		7.32	2.70
		c	13.43	0.44		13.44	0.23		10.60	1.42
	Isocaloric (n=24)	a	117.79 ^11.08	<0.001	83.79	8.53	<0.001	55.37	1.45	= 0.799
		b	47.27	15.67		70.08	11.75		-	-
		c	13.69	1.26		14.22	0.67		-	-
	Hypocaloric (n=22)	a	84.93	8.04	<0.001	68.00	7.61	<0.001	53.56	1.57	= 0.135
		b	39.18	11.58		68.48	10.99		-	-
		c	15.14	1.08		14.34	0.58		-	-

Table 2

 2

shows the three parameters of cosinor regressions, including a) the mean level, b) the amplitude, and c) the acrophase of the rhythm (see Methods for details). For the acrophase, the reference time is Zeitgeber 0 (i.e., lights on).*Ad libitum group is significantly different from Isocaloric and Hypocaloric groups (p < 0.05). ^Isocaloric group is significantly different from Hypocaloric group (p < 0.05). # Ad libitum group is significantly different from Hypocaloric group (p < 0.05). P values in the right columns indicate the significance of the cosinor fit. the liver

Table 3 .

 3 Parameters of cosinor regressions of daily variations of plasma glucose, non-esterified fatty acids, liver glycogen, and plasma corticosterone. Table 3 shows the three parameters of cosinor regressions, including a) the mean level, b) the amplitude, and c) the acrophase of the rhythm (see Methods for details). For the acrophase, the reference time is Zeitgeber 0 (i.e., lights on).*Ad libitum group is significantly different from Isocaloric and Hypocaloric groups (p < 0.05). ^Isocaloric group is significantly different from Hypocaloric group (p < 0.05). # Ad libitum group is significantly different from Hypocaloric group (p < 0.05). £ Hypocaloric group is different from Isocaloric and Ad libitum fed groups (p < 0.05). § Ad libitum group is different from Isocaloric group (p < 0.05). € Isocaloric is different from Ad libitum. NEFA, non-esterified fatty acids and glucose mmol/L, Millimoles per Litre. Liver glycogen mg/100 mg, Milligram of glycogen per 100 milligrams of liver. Plasma corticosterone ng/ml, Nanogram per Milliliter. p values on the right columns indicate significance of the cosinor fit. Non-significant parameters are not shown (-).

			Plasma Glucose (mmol/L) Plasma NEFA (mmol/L) Liver Glycogen (mg/100mg) Plasma Corticosterone (ng/mL)
			Mean	SEM	p	Mean SEM	p	Mean	SEM	p	Mean	SEM	p
	Ad libitum (n = 30)	a 12.02 # 0.49 = 0.022 0.13 0.013 = 0.601 3.95	0.26	<0.001	114.49*	8.53	<0.001
		b	2.05	0.69		-	-		3.22 §	0.38		70.93	11.88
	Isocaloric (n = 24)	c a 11.59 ^0.42 = 0.124 0.18 € 0.01 = 0.706 3.32 3.23 1.29 -1.28 -	0.44 0.24	= 0.003	11.55 96.71	0.66 18.015	<0.001
		b	-	-		-	-		1.84	0.36		127.98	25.48
		c	-	-		-	-		0.80	0.69		10.09	0.76
	Hypocaloric (n = 22) a	8.82	0.59 = 0.131 0.14 0.01 = 0.587 4.00	0.34	<0.001	203.03	19.90	<0.001
		b	-	-	-	-	-		2.18	0.48		201.20 £	27.87
		c	-	-	-	-	-		2.93	0.85		11.37	0.54

Table 4 .

 4 Parameters of cosinor regressions of daily expression of clock genes in the liver.

				Liver Per2 (a.u.)			Liver Clock (a.u.)			Liver Clock (a.u.)
			Mean	SEM	p	Mean	SEM	p	Mean	SEM	p
	Ad libitum (n = 30)	a	2.18*	0.11	<0.001	0.76 §	0.02	<0.001	9.51	1.24	<0.001
		b	2.09*	0.16		0.30	0.03		13.40	1.81
		c	15.75*	0.28		22.66*	0.45		7.10*	0.49
	Isocaloric (n = 24)	a	1.41	0.14	<0.001	0.55 ^0.02	<0.001	6.53	0.99	<0.001
		b	1.06	0.20		0.21	0.03		8.37	1.40
		c	11.81 ^0.71		19.19 ^0.59		3.56	0.64
	Hypocaloric(n = 22)	a	1.70	0.17	<0.001	0.75	0.03	= 0.004	9.47	1.35	<0.001
		b	1.08	0.24		0.21	0.05		9.60	1.95
		c	8.71	1.88		21.42	0.99		2.37	0.74

Table 4 shows the three parameters of cosinor regressions, including a) the mean level, b) the amplitude, and c) the acrophase of the rhythm (see Methods for details). For the acrophase, the reference time is Zeitgeber 0 (i.e., lights on).*Ad libitum group is significantly different from Isocaloric and Hypocaloric groups (p < 0.05). ^Isocaloric group is significantly different from Hypocaloric group (p < 0.05). § Ad libitum group is different from Isocaloric group (p < 0.05). a.u., arbitrary units. P values on the right columns indicate significance of the cosinor analysis.

Table 5 .

 5 Parameters of cosinor regressions of daily expression of metabolic genes in the liver. Table 5 shows the three parameters of cosinor regressions, including a) the mean level, b) the amplitude, and c) the acrophase of the rhythm (see Methods for details). For the acrophase, the reference time is Zeitgeber 0 (i.e., lights on).*Ad libitum group is significantly different from Isocaloric and Hypocaloric groups (p < 0.05). ^Isocaloric group is significantly different from Hypocaloric group (p < 0.05). # Ad libitum group is significantly different from Hypocaloric group (p < 0.05). § Ad libitum group is different from Isocaloric group (p < 0.05). £ Hypocaloric group is different from Isocaloric and Ad libitum fed groups (p < 0.05). € Isocaloric is different from Ad libitum. ¥ Hypocaloric is different from Ad libitum. a.u., arbitrary units. p values on the right columns indicate significance of the cosinor analysis. Non-significant parameters are not shown (-).

			Liver Sirt1 (a.u.)	Liver Pparα (a.u.)	Liver Pgc-1α (a.u.)	Liver Fgf21 (a.u.)
			Mean	SEM	p	Mean	SEM	p	Mean	SEM	p	Mean	SEM	p
	Ad libitum (n = 30)	a	1.27	0.05	= 0.050	1.24 §	0.05	<0.001	2.08*	0.11	= 0.002	1.61	0.23	= 0.009
		b	0.18	0.07		0.37	0.07		0.61	0.16		1.13 #	0.33
		c	12.23	1.55		12.09 §	0.76		10.88	1.01		6.93 #	0.65
	Isocaloric (n = 24)	a	1.33	0.04	= 0.066	0.88	0.04	= 0.015	3.27 ^0.26	= 0.042	6.14 €	0.74	= 0.326
		b	-	-		0.20	0.06		0.98	0.36		-	-
		c	-	-		7.43	1.20		11.23	1.4		-	-
	Hypocaloric (n = 22)	a	1.45	0.06	= 0.003	1.15	0.12	= 0.364	5.25	0.75	= 0.033	5.57 ¥	0.71	= 0.003
		b	0.35	0.09		-	-		2.97 £	1.04		3.99	1.01
		c	14.41	0.94		-	-		9.09	1.39		23.44	0.95

Mice on the 15-min 6-meals-a-day feeding schedule, both the hypocaloric and isocaloric group, ate equal amounts of food during day and night. However, the amount of food ingested by the hypocaloric group was significantly lower (~20%) than that by the

Acknowledgements

We thank Sylviane Gourmelen and Dr. Dominique Sage-Ciocca for their expert assistance with animal care and actimetry, respectively. We are also indebted to Pr. Paul Pévet for continuous support.

Declaration of interest

The authors report no conflict of interest. This work was supported by a doctoral fellowship from "Neurotime" Erasmus Mundus program (S.S.), and grants from Centre National de


E A Hernandez-Vargas

C Parra-Rojas

S Olaru

Switching Logistic Maps to Design Cycling Approaches Against Antimicrobial Resistance

Antimicrobial resistance is a major threat to global health, in particular, new SARS-CoV-2 variants during the COVID-19 pandemic. Scheduling cycling therapies by targeting phenotypic states associated with specific mutations can help us to eradicate pathogenic variants. In this paper, we introduce a logistic switching model to abstract mutation networks of collateral resistance. We found conditions for which the unstable zero-equilibrium of the logistic maps can be stabilized through a switching signal. That is, persistent populations can be eradicated through tailored switching regimes.

Starting from an optimal-control formulation, the switching policies show their potential in the stabilization of the zeroequilibrium for dynamics governed by logistic maps. Simulation results show the applicability of Parrondo's Paradox to design cycling therapies against drug resistance.

I. INTRODUCTION

Antimicrobials such as antibiotics and antivirals are powerful antidotes to fight against infections. However, the misuse and overuse of drugs have led to drug resistance, which can be roughly defined as the ability of a microorganism to replicate in the presence of a drug. For instance, during the COVID-19 pandemic, there have been new variants such as the B.1.1.7 in the UK and B. 1.351 in South Africa which are more resistant to neutralization by convalescent plasma and vaccine sera [START_REF] Wang | Antibody Resistance of SARS-CoV-2 Variants B.1.351 and B.1.1.7[END_REF].

Modeling of infectious diseases has been developed at different scales and pursuing different goals [START_REF] Hernandez-Vargas | A new view of multiscale stochastic impulsive systems for modeling and control of epidemics[END_REF]. Between-hosts models have helped to propose new vaccination strategies or support public health strategies [START_REF] Weitz | Modeling shield immunity to reduce COVID-19 epidemic spread[END_REF], [START_REF] Tetteh | Network Models to Evaluate Vaccine Strategies towards Herd Immunity in COVID-19[END_REF]. On the other hand, for within-host infection, mathematical modeling has been used to capture the dynamics of different infectious diseases inside the host and understand the interaction of the pathogen and the immune system, as well as the scheduling of therapies [START_REF] Rivadeneira | Impulsive Zone Model Predictive Control ( iZMPC ) for Therapeutic Treatments : application to HIV dynamics[END_REF], [START_REF] Hernandez-Vargas | Modeling and Control of Infectious Diseases: with MATLAB and R[END_REF]. Most mathematical models to represent microbial dynamics are shown to be based on variations of the classical Verhulst logistic growth equation. For instance, the logistic model has served as a key mathematical tool in representing the growth of tumors [START_REF] Adam | A Survey of Models for Tumor-Immune System Dynamics, ser. Modeling and Simulation in Science[END_REF] and microbes [START_REF] Juška | Growth of microbial populations: Mathematical modeling, laboratory exercises, and modelbased data analysis[END_REF].

The logistic model indicates that a stable population has a saturation level, known as the carrying capacity which forms a numerical upper bound on the growth size [START_REF] Tsoularis | Analysis of logistic growth models[END_REF]. While theoretical approaches to mitigate drug resistance have been mainly developed at between-host level [START_REF] Blanquart | Evolutionary epidemiology models to predict the dynamics of antibiotic resistance[END_REF], too little has been directed at investigating within-host strategies against antimicrobial resistance [START_REF] Hernandez-Vargas | Modeling and Control of Infectious Diseases: with MATLAB and R[END_REF].

Here, for a given mutation network, we introduce a logistic switching map to capture the drug resistance dynamics of bacteria. This model is instrumental to design control strategies to minimize the ability of a bacterial sub-population to survive a drug concentration, known as persistence. However, designing switching strategies in dynamical systems is not trivial and leads to a context similar to the so-called Parrondo's Paradox [START_REF] Maier | Switching induced oscillations in the logistic map[END_REF], in which two losing games are combined in a determined order as to obtain a winning game [START_REF] Harmer | Losing strategies can win by Parrondo's paradox[END_REF]. Previous numerical simulations of switching logistic maps [START_REF] Maier | Switching induced oscillations in the logistic map[END_REF] have shown that switching decisions may follow either the "undesirable + undesirable = desirable" or the "chaotic+chaotic = order" dynamics.

In the next sections, the mathematical abstraction of antimicrobial resistance dynamics is formulated in the form of switched systems. Subsequently, evolutionary and control algorithms are used to find sub-optimal switching policies. Simulation results show the applicability of Parrondo's Paradox to design cycling therapies against drug resistance.

II. LOGISTIC SWITCHING MAPS

Throughout, R denotes the field of reals, R n stands for the vector space of all n-tuples of real numbers. R n×n is the space of n × n matrices with real entries. N denotes the set of natural numbers, while the restriction to positive natural numbers bounded by q will be used under the compact notation N q = {n ∈ N : 1 ≤ n ≤ q}. For x in R n , x i denotes the i th component of x, and the notation x 0 means that

x i ≥ 0 for 1 ≤ i ≤ n. R n + = {x ∈ R n : x 0} denotes the non-negative orthant in R n . The transpose is represented as A .
Using a switching signal σ(t), one can formally deal with a selection mechanism among q ∈ N possible subsystems. These subsystems are represented mathematically by a family of functions f i (•) : R n -→ R n , i ∈ N q which characterize the right hand side of ordinary differential equation. Based on these elements, one can define a switched non-linear autonomous system as follows:

ẋ(t) = f σ(t) (x(t)), (1)
where x(t) ∈ X is the system state at time t, and the state space X is a closed subset of R n . Given the initial condition x(0) = x 0 at time t 0 = 0, the solution of (1) is denoted x(t, x 0 , σ(t)).

Problem 1: In this work, σ(•) is considered as the only manipulable control signal for the system (1). The primary goal would be to design, whenever this is possible, a switching path to make the origin of the system (1) stable. However, for practical application of the policy described in the next sections and biological reasons, we relax the problem to the design of a sub-optimal switching path σ(•) that yields a minimal cost functional J in a fix horizon of time (t f ).

A. Modeling Antimicrobial Resistance as Logistic Maps

To abstract the dynamics of the different genotypes during drug therapy, the following switched logistic model is described:

ẋi (t) = ρ i,σ(t) x i (t) 1 - x i (t) K -δx i (t)+µ η j =i m i,j x j (t).
(2) The population of different η pathogenic variants are represented by the state variable x i . ρ i,σ(t) is the proliferation rate of the variant i under the treatment σ(t) ∈ {1, 2, 3, ..., N } which can be changed at any time t. N is the total number of possible drug therapies that can be administered. The maximum carrying capacity is K. δ is the clearance of the variant x i .

The mutation rate is represented by µ. The genetic connections between genotypes are represented by m i,j ∈ {0, 1}, that is, m i,j = 1 if and only if it is possible for genotype j to mutate into genotype i. An illustrative example to represent the mutation network for n variant and m therapies is presented in Figure 1. However, more complex mutation trees can be considered.

III. STABILITY ANALYSIS WITHOUT SWITCHING OF THERAPIES

Let us consider the system (2) and focus on the case of a single therapy. Without loss of generality, we simplify the notation by taking ρ i,σ(t) → ρ i , and normalize the bacterial populations by the carrying capacity of the system,

x i /K → x i , so that ẋi (t) = ρ i x i (t) (1 -x i (t)) -δx i (t) + µ η j =i m i,j x j (t). (3)
The strains that mutate act as a source to one or more strains, whereas each strain can have at most one source, with the case of zero sources corresponding to the wildtype. In other words, we do not take into account the case when two or more different strains can yield exactly the same genotype upon mutation, since this is unlikely in a real-life scenario. At the same time, and for the same reason, all backwards mutations are forbidden, so that if m j,i = 0 then necessarily m i,j = 0. More generally, there is no way of coming back to strain i starting from strain i; that is, once a mutation from strain i to strain j occurs, there exists no possible sequence of L mutations x j1 → x j2 → . . . → x jL → x i for any length L. The equilibrium points of the system with n strains, which we denote by (x 1 , . . . , xn ), can be found recursively by solving for ẋi = 0 in Eq. ( 3), and have the form

xi =            0, max λ i ρ i , 0 , si = 0 λ i + λ 2 i + 4µρ i si 2ρ i , si > 0 (4)
Here, we have denoted by si the equilibria of the strain that acts as a source of x i ; this means that, if x i results from the mutation of a certain x k , then si = xk . At the same time,

λ i = ρ i -δ -µc i (5)
where c i is the number of "offsprings" of strain i. Theorem 1: λ i > 0 ensures the existence of the equilibrium point with xi > 0 and xk > 0 for all x k that can be reached starting from x i , otherwise xk = 0 for λ i < 0. This point is locally stable if, additionally, λ k < 0 for all other k.

Proof: The Jacobian of the system evaluated at a given equilibrium point can be decomposed as J = D + µM 1 , where M 1 contains the off-diagonal elements of M , and thus only encodes the mutations between strains. The entries of D are given by

d ii = ρ i -δ -µc i -2ρ i xi
The properties derived above for the constraints on the mutation patterns imply that M 1 can be thought of as the adjacency matrix of a directed acyclic graph (DAG) [START_REF] Mckay | Acyclic digraphs and eigenvalues of (0, 1)-matrices[END_REF]; therefore, there exists a topological ordering of the strains that renders M 1 strictly upper triangular [START_REF] Nicholson | Matrices with permanent equal to one[END_REF]. In other words, we can relabel the strains so that for every mutation from x i to x j , we have i < j. The Jacobian becomes upper triangular, and its eigenvalues will be given by the entries of D. At the origin, the i-th eigenvalue will correspond to λ i from Eq. ( 5). On the other hand, the eigenvalues associated with non-zero coordinates are given by

∆ i = -λ 2 i + 4µρ i si
and they are always negative. As a consequence, whenever a strain x i is able to persist by itself -which translates into λ i > 0 -then all of its offsprings will also persist, as seen from Eq. ( 4), even if they would be eradicated in isolation. If all other strains in the system are unable to persist in isolation, the eigenvalues associated to the equilibrium point where xi > 0, with xk > 0 if k corresponds to a offspring of i (that is, sk = xi ) and xk = 0 otherwise, will be given by

-λ i < 0 ∆ k < 0 sk = xi λ k < 0 sk = xi
and the equilibrium is thus asymptotically stable.

Theorem 2: If we explicitly separate the zero and nonzero equilibrium coordinates-so that the i-th element of an equilibrium point is either 0 or xi -then the set defined by

Ω = {x ∈ R n |0 ≤ x i ≤ xi ∀i}
is positively invariant with respect to the system described by Eq. (3). Equivalently, for any given initial condition x 0 ∈ Ω, the solution to (3) satisfies x(t) ∈ Ω ∀t > 0.

Proof: Let us recall that

ẋ = f (x) = (f 1 (x), . . . , f n (x)),
and consider the boundaries of Ω, given by

B (1) i = {x|x i = 0, 0 < x j < xj ∀j = i} B (2) i = {x|x i = xi , 0 < x j < xj ∀j = i}
The invariance condition [START_REF] Blanchini | Set invariance in control[END_REF] translates into f • n ≥ 0 along the boundaries above, where n is a normal vector pointing towards Ω. We note that the normal vectors that correspond to each boundary are given by n(1)

i = êi for B (1) i and n(2) i = -ê i for B (2)
i , where êi is the i-th unitary vector, (ê i ) j = δ ij , with δ ij the Kronecker delta.

Choosing an initial condition x 0 along B

i , with

x j = p j ∈ [0, xj ] ∀j = i we have f (x 0 ) • n(1) i = f i (x 0 ) = µ η j =i p j m i,j ≥ 0 ∀i
Similarly, for the case of B

i , we choose an initial condition x 0 with x i = xi and x j = p j ∈ [0, xj ] ∀j = i; this yields

f (x 0 ) • n(2) i = -f i (x 0 ) = -ρ i xi (1 -xi ) + δ xi + µc i x i + µ η j =i m i,j p j
Now, we use the fact that (x 1 , . . . , xn ) is an equilibrium point of the system, so that

ρ i xi (1 -xi ) -δ xi -µc i xi + µ η j =i m i,j xj = 0
and

-ρ i xi (1 -xi ) + δ xi + µc i xi = µ η j =i m i,j xj Hence, f (x 0 ) • n(2) i = µ η j =i m i,j (x j -p j ) ≥ 0 ∀i since 0 ≤ p j ≤ xj ∀j = i.
Therefore, Ω is positively invariant with respect to the system (3).

Corollary 1: Given the system (3) with µ = 0, the equilibrium at the origin (x k = 0) is globally asymptotically stable when ρ i < δ ∀i ∈ {1, . . . , n}. Proof: We have seen above that the origin is a locally asymptotically stable equilibrium point of (3) when ρ i < δ + µc i , ∀i simultaneously. In order to establish the global stability of the origin in Ω, let us choose a Lyapunov function

V (x) = η i x i
We note that V ≥ 0, ∀x ∈ Ω, and V = 0 iff x = 0. Now,

V (x) = η i ẋi = η i ρ i x i (1 -x i ) -δx i = η i (ρ i -δ)x i - η i ρ i x 2 i (6)
From the expression above, ρ i < δ ∀i ensures that V ≤ 0 ∀x ∈ Ω, and V = 0 iff x = 0. Therefore, the origin is globally asymptotically stable under this condition.

Remark 1: The condition above arises trivially from considering all strains as independent of one another, in absence of mutations, and it is more restrictive than the condition found previously, since mutations have a stabilizing effect on the origin. However, Eq. ( 6) can be employed as a basis to establish a switching policy in order to ensure the eradication of the pathogen in the case with mutations under a set of different available therapies.

IV. SWITCHING THERAPIES Based on the general model ( 2), the problem formulation can then be broadly described as the design of a switching policy for the gradual eradication of pathogenic strains. To formally state the problem we need the following definitions:

Definition 1: A switching policy is the cycling choice between therapies σ with a period T σ , which can be represented by a piecewise constant function σ(t) : R → {1, 2, . . . , N }. σ(t) remains constant for all t ∈ [kT σ , (k + 1)T σ ).

Definition 2: A periodic cycling policy is the periodic switching between therapies i with regular intervals T i . For example, for the case of two therapies with periods T 1 and T 2 , we have σ(T 1 ) = 1 and σ(T 2 ) = 2, then the periodic cycling policy sequence is

σ = {σ = 1 T1 , σ = 2 T2 , • • • , σ = 1 T1 , σ = 2 T2 }. (7)
Next, we describe the design of switching policies of the system (2) in a discrete-time version. To this end, we will divide the analysis into two parts. The first is assuming the variants do not mutate that is µ = 0, this is still an interesting biological scenario as the main goal is to eradicate different variants of a pathogenic micro-organism. The second part will consider mutation, as µ = 0 for the case of not backward mutations, as from the evolution point of view, it is very difficult that this type of mutation occurs.

A. Switching Therapies Assuming No Mutations (µ = 0)

Assuming that the variants do not present mutations, that is µ = 0, we can re-parameterize the model (2) for a fixed period of treatment (σ(t) = σ) as follows:

ẋi (t) = b i,σ x i (t) 1 - x i (t) a i,σ (8)
where b i,σ = ρ i,σ -δ and a i,σ = Kb i,σ /ρ i,σ . Equation ( 8) has the conventional logistic form which can be solved analytically or by the nearly exact method [START_REF] Kwessi | Nearly exact discretization of single species population models[END_REF], thus the recursive form of ( 2) is as follows:

z i (k + 1) = a i,σ z i (k) z i (k) + (a i,σ -z i (k))e -bi,σTσ (9)
where k is the discrete-time step, and T σ is the period where the therapy σ is provided. Theorem 3: Let the dynamics in (2) without mutation (µ = 0) and a periodic cycle (T σ ) for each of the therapies (σ = 1, ..., N ). The different variants will be eradicated if and only if the following condition is satisfied

N σ=1 ρ i,σ T σ T < δ ( 10
)
where T = N σ=1 T σ . Proof: Considering the initial condition z 0 = z(0) = 0 and the first two cycles of therapies σ = 1, 2 with their respective period T σ=1,2 , by construction, we have the Jacobian matrix J σ=1,2 at the origin as follows:

J σ=1,2 = e bi,1T1+bi,2T2 (11)
In a similar vein, we can generalize the Jacobian J σ=1,..,N in the origin as follows,

J σ=1,...,N = e N σ=1 bi,σTσ (12)
as b i,σ = ρ i,σ -δ and and we are interested in the stability of the origin, we check the exponential term to be less than 1, then

N σ=1 (ρ i,σ -δ)T σ ≤ 0 (13) N σ=1 ρ i,σ T σ ≤ δT (14) N σ=1 ρ i,σ T σ T < δ ( 15
)
Remark 2: The condition (10) can additionally serve to design the duration of the therapies (T σ ) to eradicate all different variants during the course of an infection if the following inequality system has a solution

                               N σ=1 ρ 1,σ T σ < δT N σ=1 ρ 2,σ T σ < δT . . . N σ=1 ρ n,σ T σ < δT.
This can be formulated as a feasibility problem in a linear programming framework. While this could provide a sufficient condition for a possible success of a cycling therapy, the unfeasible solution does not necessarily imply the failure of a switching trajectory to stabilize the origin.

Corollary 2: For a cycle of the therapies with equal time duration, the stability condition [START_REF] Blanquart | Evolutionary epidemiology models to predict the dynamics of antibiotic resistance[END_REF] can be further simplified to N σ=1 ρ i,σ N < δ Proof: Following the Theorem 3, we have that the period of each therapy is equal, that is T 1 = T 2 = ... = T N . Then, N σ=1 T σ = N T σ . Therefore condition [START_REF] Blanquart | Evolutionary epidemiology models to predict the dynamics of antibiotic resistance[END_REF] becomes independent of T σ .

Remark 3: The corollary 2 points out that in order to eradicate a variant i with cycling therapies, the average expansion of the variant i through the cycling should be less than its contraction.

B. Switching Therapies Assuming Mutations (µ = 0)

For the case with mutation (µ = 0), a condition similar to [START_REF] Blanquart | Evolutionary epidemiology models to predict the dynamics of antibiotic resistance[END_REF] is difficult to prove. Model Predictive Control (MPC) appears to be suitable for a sub-optimal application to the biomedical application, due to its robustness to disturbances, model uncertainties and the capability of handling constraints [START_REF] Hernandez-Vargas | Modeling and Control of Infectious Diseases: with MATLAB and R[END_REF], [START_REF] Hernandez-Vargas | Switching Strategies to Mitigate HIV Mutation[END_REF], [START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF], [START_REF] Zurakowski | A model predictive control based scheduling method for HIV therapy[END_REF]. Thus, the selection of therapies based on the MPC will be employed on similar reasoning as in our previous work based on switched linear systems [START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF]. In summary, if the total population size is small enough during a finite time of treatment, then there is a significant probability that the population becomes zero. Therefore, we consider the cost

J := c x(t f ) ( 16
)
where c is the column vector with all ones. The final time, t f , is the time point at which the cost function will be evaluated. This cost should be minimized under the action of the switching rule. The decision time to switch therapy is T c . Thus, the control algorithm predicts the future dynamic behavior of the system over a prediction horizon T p and computes an open-loop optimal control sequence over the prediction horizon. Then the first decision of the open-loop optimal control is applied. The problem is written as follows: Problem 2: The internal variables of the controller will be denoted as (x, σ), with constraints

x(k) ∈ Ω ⊆ R n and σ(•) ∈ U ⊆ N n q . Find min σ J(x(k), σ(•); T c , T p ), with J(x(k), σ(•); T p , T c ) := cx(t + T p )
subject to:

ẋ(τ ) = A σ(τ ) x(τ ) x(τ ) = x(k) σ(τ ) ∈ U, ∀τ ∈ [k, k + T c ] σ(τ ) = σ(τ + T c ), ∀τ ∈ [k + T c , k + T p ] x(τ ) ∈ Ω, ∀τ ∈ [k, k + T p ]
This MPC problem involves a nonlinear and complex optimization problem, the global optimization algorithm known as differential evolution [START_REF] Storn | Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces[END_REF] is applied here for the implementation of the algorithm without further tuning of the solver.

V. NUMERICAL SIMULATIONS

In a similar way to [START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF], simple illustrative examples are considered based on 4 genetic variants and two different therapies. Thus, η = 4, and 2 drug therapies, N = 2. The strain i = 1 is considered the Wild Type (WT) which is susceptible to most of the drugs. The Genotype 1 (G 1 ) is resistant to therapy 1 but it is susceptible to therapy 2. The Genotype 2 (G 2 ) is resistant to therapy 2, but it is susceptible to therapy 1. The Highly Resistant Genotype (R G ) is a genotype with a low replication rate, but it is resistant to all drug therapies. Pathogen clearance rate is fixed, δ = 0.24 day -1 , which corresponds to a half life of slightly less than 3 days [START_REF] Perelson | Mathematical Analysis of HIV-1 Dynamics in Vivo[END_REF]. Typical mutation rates are of the order of µ = 10 -4 . The following mutation matrix is considered: Three different scenarios are proposed as in the Table I. The first scenario, the most ideal case, describes a stabilizable switched system with a full symmetry between G 1 and G 2 , in the sense that first therapy inhibits G 2 with the same intensity that therapy 2 inhibits G 1 . In practice, a small difference in relative replication ability is expected. The second scenario shows a system that can not be stabilized with symmetry between G 1 and G 2 (similar to Case 1) but there is a resistant genotype that can not be stabilized under switching. The third scenario is a switched system that can be stabilized under switching and it has full asymmetry for replication rates in G 1 and G 2 . In general, the finite set of possible To compute optimal switching trajectories, we consider a "brute force" approach, which calculates all possible combinations of the therapies (N c = t f /T c ) and then finding the minimum for the cost function [START_REF] Kwessi | Nearly exact discretization of single species population models[END_REF]. However, this approach is computational unfeasible for checking N c > 15. Thus, in order to check larger treatment combinations we consider the differential evolution algorithm [START_REF] Storn | Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces[END_REF].

M u =     0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0     (17)
Assuming a prediction horizon of 100 days (T p ), we consider an MPC strategy as discussed in Problem 2. Table II illustrates treatment scenarios of 200 days (t f ) with decision time to switch therapy of 20 days (T c ), that is N c = 10, 2 10 combinations. Based on the column of "brute force" results in the Table II, we can conclude that the DE algorithm can find optimal trajectories. While we can not guarantee that the DE algorithm will find the optimal solution for any example, it might find solutions very close to the optimal. In a similar fashion, the MPC was able to find optimal switching trajectories in cases 1 and 2 but not in the case 3. However, MPC finds a solution that is very close to the optimal. ) is rapidly eradicated. The genotypes 2 (x 2 ) and 3 (x 3 ) are gradually eradicated by switching between therapies. It can be observed that the switching rule does not have a clear pattern. The resistant genotype (x 4 ) is also eradicated due to the appropriate switching between the therapies. Total pathogen load is represented in the cost function J.

The advantage of the MPC would be the computational time, thus we could explore short decision times. Thus, in Fig. 2 we illustrate the performance of MPC for 500 days (t f ) with decision time to switch therapy of 5 days (T c ), that is N c = 100, 2 100 combinations. This scenario would not be computationally feasible for a "brute force" approach. The upper panel in Fig. 2 shows how the MPC strategy can decrease the pathogen load of the different genotypes. The lower panel in Fig. 2 presents the switching trajectory, which does not have any intuitive pattern, thus highlighting the potential of MPC and other control strategies.

VI. CONCLUSIONS

This paper introduced a switching logistic model with the potential to be the basis for scheduling antimicrobial therapies to mitigate resistance. Considering the case of absence of switching between treatments when backward mutations are forbidden, we derived conditions of global stability presented in Theorem 1. Assuming variants without mutations during treatments, conditions for which unstable zero-equilibrium of the logistic maps can be stabilized through a periodic switching signal are derived in Theorem 3 and Corollary 2. For the case with mutations, switching strategies based on receding horizon optimization such as the MPC were formulated in Problem 2. Numerical results highlighted that the MPC strategy will perform very close to optimal control policies while computational resources are largely decreased.

Fig. 1 .

 1 Fig. 1. Illustrative mutation tree for n variants and m therapies. WT represents the wild type. Different variants would mutate as effect of a given therapy which is indicated with color squares. The appearance of a highly resistant variant (black circle HR) is hypothesized as o consequence of a sequence of mutations, meaning that it is resistant to all available therapies.

Fig. 2 .

 2 Fig.2. MPC results for simulation scenario 3. This considers a simulation time of 500 days with a decision time of every 5 days. The wildtype (x 1 ) is rapidly eradicated. The genotypes 2 (x 2 ) and 3 (x 3 ) are gradually eradicated by switching between therapies. It can be observed that the switching rule does not have a clear pattern. The resistant genotype (x 4 ) is also eradicated due to the appropriate switching between the therapies. Total pathogen load is represented in the cost function J.

TABLE II TOTAL

 II PATHOGEN LOAD AT THE END OF TREATMENT OF 200 DAYS WITHA POSSIBILITY TO SWITCH EVERY 20 DAYS.

		Optimal	Optimal
	Scenario	(Brute Force)	(DE Algorithm)	MPC
	1	2.3615×10 5	2.3615×10 5	2.3615×10 5
	2	1.8851×10 3	1.8851×10 3	1.8851×10 3
	3	0.1040	0.1040	0.1141
	control values causes problems for many control techniques.
	Nevertheless in the cause of MPC, having a finite set of
	options may be an advantage in making the optimization
	easier to solve.

*This work was funded by the Universidad Nacional Autonoma de Mexico (UNAM) -PAPIIT with the number IA102521.