Tengfei Zhu

Geneviève Corraze

Elisabeth Plagnes-Juan

Jérôme Montfort

Julien Bobe

Edwige Quillet

Mathilde Dupont-Nivet

Sandrine Skiba-Cassy
email: sandrine.skiba@inra.fr

Jérome Montfort

MicroRNAs related to cholesterol metabolism affected by vegetable diet in rainbow trout (Oncorhynchus mykiss) from control and selected lines

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Increasing demand for world fisheries production has resulted in a great expansion of aquaculture over the last decades, accounting for about half of the world global seafood production (FAO, 2016). As capture fisheries are limited and showed no increase since the mid-1980, the increasing demand for fish in the future will be inevitably supplied by aquaculture. However, the production of fishmeal and fish oil coming from wild fishery stocks and constituting important traditional ingredients in aquafeed is also going to keep stable or even decrease [START_REF] Jackson | Fishmeal and fish oil and its role in sustainable aquaculture[END_REF]. Therefore, the utilization of plant ingredients to replace fishmeal and fish oil has been considered as the best solution to support the sustainable development of aquaculture in recent years [START_REF] Tacon | Aquaculture feeds: addressing the long-term sustainability of the sector[END_REF] and evident progress has been made regarding the substitution of fishmeal and fish oil by plant raw materials in salmonids and marine fish [START_REF] Benedito-Palos | High levels of vegetable oils in plant protein-rich diets fed to gilthead sea bream (Sparus aurata L.): growth performance, muscle fatty acid profiles and histological alterations of target tissues[END_REF][START_REF] Torstensen | Novel production of Atlantic salmon (Salmo salar) protein based on combined replacement of fish meal and fish oil with plant meal and vegetable oil blends[END_REF].

However, studies have indicated that total replacement of fishmeal and fish oil by the plant-based diets still reduced growth performances and flesh quality [START_REF] Geay | Effects of the total replacement of fish-based diet with plant-based diet on the hepatic transcriptome of two European sea bass (Dicentrarchus labrax) half-sibfamilies showing different growth rates with the plant-based diet[END_REF][START_REF] Panserat | Differential gene expression after total replacement of dietary fish meal and fish oil by plant products in rainbow trout (Oncorhynchus mykiss) liver[END_REF]. In order to further improve the use of plant food in carnivorous fish like salmonids, research efforts have been more recently focused on the selective breeding. Indeed, the selective breeding could be the alternative solution to enhance fish adaptation to the plant-based diets and has been well confirmed at a research level. For example, it has been demonstrated that survival rate and mean body weight were improved in the first generation of rainbow trout selected for the ability to adapt to a totally plant-based diet (Le [START_REF] Boucher | Selection for adaptation to dietary shifts: towards sustainable breeding of carnivorous fish[END_REF]. Moreover, in Atlantic salmon, the deposition and/or retention of dietary n-3 long-chain polyunsaturated fatty acids (LC-PUFA) in flesh, such as eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA), is a highly heritable trait [START_REF] Leaver | Heritability and mechanisms of n-3 long chain polyunsaturated fatty acid deposition in the flesh of Atlantic salmon[END_REF] and Nofima has also carried out profitable work in this respect [START_REF] Horn | Breeding for More Omega-3 in Salmon[END_REF], prompting further interest in exploring genotype-nutrient interactions.

The replacement of fishmeal and fish oil by plant ingredients caused great variations in nutrient supply, including the reduction of cholesterol and n-3 LC-PUFA. Cholesterol is derived from the animal organisms, so it is absent in plant ingredients [START_REF] Tocher | The role of phospholipids in nutrition and metabolism of teleost fish[END_REF]. In animals, cholesterol represents an important component of the membrane to modulate the fluidity and permeability and it is also the precursor of many biologically active compounds, including bile acids, steroid hormones and vitamin D [START_REF] Goedeke | Regulation of cholesterol homeostasis[END_REF].

To maintain cholesterol homeostasis, the expression of genes related to cholesterol metabolism is tightly controlled at both transcriptional and post-transcriptional level. The sterol regulatory element-binding proteins (SREBPs) and the liver X receptors (LXRs) are known to be well involved in the transcriptional regulation of cholesterol metabolism by targeting genes involved in the cholesterol synthesis (hmgcr, 3-hydroxy-3-methylglutaryl-CoA, reductase) and efflux (abca1, ATP-binding-cassette transporter-A1, abcg5 and abcg8, ATP-binding-cassette transporter G5 and G8 and cyp7a1, cytochrome P450 cholesterol 7a-hydroxylase), respectively [START_REF] Sakakura | Sterol regulatory element-binding proteins induce an entire pathway of cholesterol synthesis[END_REF][START_REF] Zhao | Liver X receptor in cholesterol metabolism[END_REF]. At the post-transcriptional level, a class of small non-protein coding RNAs of 19-24 nucleotides, designated as microRNA (miRNA), has been found to regulate gene expression either by mRNA decay or translational repression [START_REF] Iwakawa | The functions of microRNAs: mRNA decay and translational repression[END_REF]. Among the wide numbers of miRNAs, miR-1 [START_REF] Zhong | MicroRNA-1 and microRNA-206 suppress LXRα-induced lipogenesis in hepatocytes[END_REF], miR-33a [START_REF] Gerin | Expression of miR-33 from an SREBP2 intron inhibits cholesterol export and fatty acid oxidation[END_REF][START_REF] Najafi-Shoushtari | MicroRNA-33 and the SREBP host genes cooperate to control cholesterol homeostasis[END_REF], miR-122 [START_REF] Esau | miR-122 regulation of lipid metabolism revealed by in vivo antisense targeting[END_REF]), miR-128 (Adlakha et al., 2013) and miR-223 [START_REF] Vickers | MicroRNA-223 coordinates cholesterol homeostasis[END_REF][START_REF] Wang | MicroRNAs 185, 96, and 223 repress selective high-density lipoprotein cholesterol uptake through posttranscriptional inhibition[END_REF] were reported to be involved in the regulation of cholesterol and lipid metabolism in mammals.

The ability to grow on the plant-based diet has been proved to be genetically variable in fish (Le [START_REF] Boucher | Plant-based diet in rainbow trout (Oncorhynchus mykiss Walbaum): are there genotype-diet interactions for main production traits when fish are fed marine vs. plant-based diets from the first meal?[END_REF][START_REF] Quinton | Genetic relationships of body composition and feed utilization traits in European whitefish (L.) and implications for selective breeding in fishmeal-and soybean meal-based diet environments[END_REF]. Although the growth potential has been effectively enhanced by selective breeding, the molecular mechanisms underlying this selection still remain unknown. In the present study, we used two lines of rainbow trout, the selected line which had better adaptation to the plant-based diet and the non-selected control line in order to investigate the potential role of miRNA, such as miR-1, miR-33a, miR-122, miR-128 and miR-223, in the adaptation to the plant-based diet. Therefore, the hepatic expression of these miRNAs and the genes related to cholesterol metabolism were analyzed. Furthermore, as miR-33a, miR-128 and miR-223 were found to be affected by the plant-based diet or selective breeding, we expanded our study to some genes that were identified in silico as potential direct targets of these miRNAs based on the complementarity between the seed sequence of the miRNAs and the 3′ untranslated region (UTR) region of the rainbow trout transcripts.

Materials and methods

Ethics statement

Experimentation was conducted in the INRA experimental facilities (Peima facilities, Sizun, France and UMR Numéa, St-Pée-sur-Nivelle, France) authorized for animal experimentation by the French veterinary service which is the competent authority (B 29-277-02 and A 64-495-1). The experiments were in strict accordance with EU legal frameworks related to the protection of animals used for scientific research (Directive 2010/63/EU) and according to the National Guidelines for Animal Care of the French Ministry of Research (decree n°2013-118, February 1st, 2013). The scientists in charge of the experimentation received training and personal authorization (N°B64 10,003 and A29 102). In agreement with ethical committees "Comité d'Ethique Aquitaine Poissons Oiseaux" (C2EA-73) and "Comité d'Ethique Finistérien en Expérimentation Animale" (C2EA-74), the experiment reported here does not need approval by a specific ethical committee since it implies only classical rearing practices with all diets used in the experiment formulated to cover the nutritional requirements of rainbow trout (National Research Council, 2011). Fish were monitored daily during the experiment. If any clinical symptoms (i.e. morphological abnormality, restlessness or uncoordinated movements) were observed, fish were sedated by immersion in 2% benzocaine solution and then euthanized by immersion in a 6% benzocaine solution (anesthetic overdose) during 3 min.

Fish, diet and experimental design

The experiment was conducted on two groups of rainbow trout: one line (S) which has been selected for better ability to grow and survive with a totally plant-based diet for three generations [START_REF] Callet | Successful selection of rainbow trout (Oncorhynchus mykiss) on their ability to grow with a diet completely devoid of fishmeal and fish oil, and correlated changes in nutritional traits[END_REF][START_REF] Boucher | Selection for adaptation to dietary shifts: towards sustainable breeding of carnivorous fish[END_REF]) and a control line (C) maintained with high effective size and without artificial selection. 3600 eggs from S line and 3600 eggs from C line were obtained by within line mating (18 dams and 31 sires for each line). At 19 days post fertilization (dpf), the eyed eggs were randomly distributed into 12 tanks (0.25 m 3 ) with 600 eyed eggs in each tank. The fish were reared in the 12 flow-through tanks (three tanks per treatment) in INRA experimental facilities (PEIMA, Sizun, France) at 11 °C and under artificial photoperiod condition (from 8 am to 8 pm). From the first-feeding stage, S and C fish were fed ad libitum with either a plant-based diet totally devoid of marine products (V diet) or a diet containing fishmeal and fish oil (M diet) during six months. Diets were manufactured in our experimental facilities of Donzacq (France) using a twin-screw extruder (Clextral, France). The M diet contained fishmeal and fish oil as main protein and lipid source, respectively. The V diet was formulated with a blend of vegetable oils (palm, rapeseed and linseed oils) and a blend of plant protein sources (corn and wheat gluten, soybean meal, soy protein concentrate, light white lupin, dehulled pea and extruded whole wheat). Diets were formulated to fulfill the requirements of rainbow trout according to NRC recommendations (National Research Council, 2011). Synthetic L-ly- sine, L-arginine, dicalcium-phosphate and soy-lecithin were added to the V diet to correct the deficiency in essential amino acids, phosphorous and phospholipid supply. Diet formulations and compositions are described in Table 1.

Sampling procedure

At the end of the feeding trial, fish were anesthetized with benzocaine (30 mg/L) and killed by a sharp blow to the head. Six fish were sampled from each tank at 8 h and 16 h after the last meal, respectively. Blood was removed from the caudal vein into EDTA syringes and centrifuged (1000 g,1 0 -15 min); then recovered plasma was immediately frozen in liquid nitrogen and kept at -80 °C for the measurement of cholesterol and triglycerides. Liver was dissected and weighed for hepato-somatic index determination and then immediately frozen in liquid nitrogen and kept at -80 °C. About 100 mg was cut from the whole liver for RNA extraction and then subsequent gene and miRNA expression analyses.

Diet and whole body composition analysis

Proximate analysis of the experimental diets was determined according to Association of Official Analytical Chemists (AOAC, 2000)a s follows: dry matter was analyzed by drying the samples to constant weight at 105 °C for 24 h. Crude protein was determined using the Kjeldahl method after acid digestion and estimated by multiplying nitrogen by 6.25. Crude lipid was quantified by petroleum diethyl ether extraction using Soxhlet method. Gross energy content was determined in an adiabatic bomb calorimeter (IKA). Ash was examined by combustion in a muffle furnace at 550 °C for 16 h. Dietary sterols were assayed on a total lipid extract by means of the Liebermann-Burchard method (Stadtman, 1957).

Analysis of plasma metabolites

Plasma metabolite concentration was measured using commercial kits, triglycerides (PAP 150, bioMérieux, France) and cholesterol (CHOD-PAP, Sobioda, France), adapted to microplate format, according to the recommendations of the manufacturer.

In silico identification of miRNA target gene

Putative miRNA targets were predicted using the miRanda algorithm (www.ncbi.nlm.nih.gov/pubmed/15502875)a p p l i e dt os t u d i e dm a t u r e miRNA sequences and rainbow trout 3'UTR regions. Rainbow trout 3'UTR regions were predicted using a set of 78,415 contigs originating from de novo assembled RNA-seq data publicly available in the PhyloFish online database (www.ncbi.nlm.nih.gov/pubmed/27189481). In order to focus on the 3'UTR of the contigs, zebrafish reference proteins obtained from ensembl (www.ensembl.org) were blasted onto the rainbow trout PhyloFish contigs. The region dowstream of the alignment region on each contig was subsequently kept for further analysis using miRanda with default parameters.

Gene expression analysis

Quantitative RT-PCR gene expression analyses were performed on liver samples (n = 6) collected 8 h and 16 h after the last meal. Genes analyzed in the present study were: HMG-CoA reductase (hmgcr), HMG-CoA synthase (hmgcs), lanosterol 14α-demethylase (cyp51) and 7-dehydrocholesterol reductase (dhcr7) for cholesterol biosynthesis, ATPbinding cassette transporter G5 (abcg5) and ATP-binding cassette transporter G8 (abcg8) for cholesterol excretion, cholesterol 7α-hydroxylase (cyp7a1) for bile acid synthesis, ATP-binding cassette transporter A1 (abca1) for plasma cholesterol efflux, sterol regulatory elementbinding protein 2 (srebp-2) and liver X receptor α (lxrα) as transcriptional factors, acyl-CoA cholesterol acyltransferase 1 (acat1) and acyl-CoA cholesterol acyltransferase 2 (acat2) for cholesterol esterification, fatty acid synthase (fas) and the transcription factor sterol regulatory element-binding protein 1c (srebp-1c) for lipogenesis and glucokinase (gck) for glycolysis although known as the first step of lipogenesis. Analysis was also expanded to genes identified as potential direct targets of miRNAs, including the genes targeted by miR-33a (retinol dehydrogenase 14a (rdh14a), caspase 6 apoptosis-related cysteine peptidase like 2 (casp6l2), leucine carboxyl methyltransferase 2 (lcmt2)), the genes targeted by miR-122 (cytochrome P450 family 27 subfamily A polypeptide 7 (cyp27a7), methyltransferase like 5 (mettl5), histone deacetylase 3 (hdac3), hypoxia-inducible factor 1 α subunit a (hif1aa)), and the genes targeted by miR-128 (phosphodiesterase 4B cAMP-specifica( pde4ba) and RAR-related orphan receptor A paralog a (roraa)). Total RNA was extracted as previously described [START_REF] Mennigen | Ontogenetic expression of metabolic genes and microRNAs in rainbow trout alevins during the transition from the endogenous to the exogenous feeding period[END_REF] using the Trizol reagent (Invitrogen, Carlsbad, CA, USA) according to the manufacturer's instructions and was quantified by spectrophotometry (absorbance at 260 nm). The integrity of the samples was assessed using agarose gel electrophoresis. 1 μg of total RNA was used for cDNA synthesis. The SuperScript III RNaseH-reverse transcriptase kit (Invitrogen) with oligo dT random primers (Promega, Charbonniéres, France) was used to synthesize cDNA (n = 6 for cholesterol metabolism genes in 6 h). The primer sequences used for qRT-PCR analyses are listed in the Table 4 andTable 5. Quantitative RT-PCR assays were performed on the Roche LightCycler 480 II system (Roche Diagnostics, Neuilly sur Seine, France). The assays were carried out using a reaction mix of 6 μL per sample containing 2 μL of 76 times diluted cDNA, 0.24 μL of each primer (10 μM), 3 μL of LightCycler 480 SYBR® Green I Master mix and 0.52 μL DNAse/RNAse free water (5 Prime GmbH, Hamburg, Germany). The PCR protocol was initiated at 95 °C for 10 min for initial denaturation of the cDNA and hot-start Taqpolymerase activation, followed by 45 cycles of a three-step amplification program (15 s at 95 °C, 10s at melting temperature Tm (60-65 °C), 15 s at 72 °C), according to the primer set used. Melting curves were systematically monitored (5 s at 95 °C, 1 min at 65 °C, temperature gradient at 0.11 °C/s from 65 to 97 °C) at the end of the last amplification cycle to confirm the specificity of the amplification reaction. Each PCR assay included replicate samples (duplicate of reverse transcription and PCR amplification) and negative controls (RT-and cDNA-free samples, respectively). As the expression of 18 s was not significantly different among treatments at 8 h (data not shown) and the expression of ef1α was not significantly different among treatments at 16 h (data not shown), the 18S ribosomal RNA (18 s) and elongation factor 1α (ef1α) was used for the normalization of the genes at 8 h and 16 h, respectively. Relative quantification of target gene expression was determined using the E-Method from the LightCycler 480 software (version SW 1.5; Roche Diagnostics). In all cases, PCR efficiency (E) measured by the slope of a standard curve with serial dilutions of cDNA ranged between 1.8 and 2.

miRNA expression analysis

Several miRNAs including miR-1-3p, miR-33a-5p, miR-122-5p, miR-128-3p and miR-223-3p were detected by the TaqMan® Advanced miRNA Assays (A25576, Applied Biosystems, ThermoFisher) in liver collected 8 h and 16 h after the last meal. The spike miR-39 (Caenorhabditis. elegans) was used as an exogenous control for normalization as previously described [START_REF] Tay | Identification of reference miRNAs in plasma useful for the study of oestrogen-responsive miRNAs associated with acquired Protein S deficiency in pregnancy[END_REF] and has been shown to be present at relatively constant levels among the four treatments. Total RNA in liver was obtained in the same way as that for gene expression analysis. 80 ng RNA were used for the poly(A) tailing, ligation and reverse transcription reactions to synthesize the cDNA of b Mineral premix (g or mg/kg diet): calcium carbonate (40% Ca), 2.15 g; magnesium oxide (60% Mg), 1.24 g; ferric citrate, 0.2 g; potassium iodide (75% I), 0.4 mg; zinc sulfate (36% Zn), 0.4 g; copper sulfate (25% Cu), 0.3 g; manganese sulfate (33% Mn), 0.3 g; dibasic calcium phosphate (20% Ca, 18% P), 5 g; cobalt sulfate, 2 mg; sodium selenite (30% Se), 3 mg; KCl, 0.9 g; and NaCl, 0.4 g (UPAE, INRA).

c Vitamin premix (IU or mg/kg diet): DL-α-tocopherol acetate, 60 IU; sodium menadione bisulphate, 5 mg; retinyl acetate, 15,000 IU; DL-cholecalciferol, 3000 IU; thiamin, 15 mg; riboflavin, 30 mg; pyridoxine, 15 mg; B12, 0.05 mg; nicotinic acid, 175 mg; folic acid, 500 mg; inositol, 1000 mg; biotin,2.5 mg; calcium panthotenate,50 mg; and choline chloride, 2000 mg (UPAE, INRA).

all miRNAs followed by a miR-Amp reaction for cDNA pre-amplification according to the manufacturer's instruction. PCR was performed in a reaction mix of 6 μL containing 2 μL cDNA (200 times diluted for liver cDNA and 50 times diluted for plasma cDNA), 2.67 μL 2× Fast Advanced Master mix (Applied Biosystems, Foster city, California, USA), 0.27 μL TaqMan® Advanced miRNA Assay (20×) (Applied Biosystems, Foster city, California, USA) and 1.06 μL DNAse/RNAse free water (5 Prime GmbH, Hamburg, Germany). The sequences of the probes were listed in the Table 3. The PCR protocol was initiated at 95 °C for 20s for initial denaturation of the cDNA and the enzyme activation, followed by 50 cycles of a 2 steps amplification program (3 s at 95 °C for denaturation, 30s at 60 °C for annealing). Each PCR assay included replicates for each sample (duplicates of reverse transcription and PCR amplification) and negative controls (reverse transcriptase free and RNA free samples). Relative quantification of the target miRNA was determined using the E-Method from the LightCycler 480 software (version SW 1.5; Roche Diagnostics). PCR efficiency measured by the slope of a standard curve with serial dilutions of miRNA cDNA ranged between 1.8 and 2.

Statistical analysis

Results are expressed as means ± SD. Statistical analyses were carried out using two-way ANOVA (line, diet effects and line*diet interaction), followed by a Tukey test for post hoc analysis. Normality was assessed using the Shapiro-Wilk test, while homogeneity of variance was determined using Levene's test. For all statistical analysis, the level of significance was set at P < .05. Statistical analysis was performed using R software [START_REF] Fox | Rcmdr: R Commander[END_REF].

Results

Body weight, hepatosomatic index, cholesterol and triglyceride in plasma

After 6 months of feeding, the final body weight of rainbow trout fed the M diet was significantly higher than that fed the V diet. As expected, the fish from the S line reached a higher body weight than those in the C group regardless the dietary treatment. However, as already shown by Le [START_REF] Boucher | Selection for adaptation to dietary shifts: towards sustainable breeding of carnivorous fish[END_REF] and [START_REF] Callet | Successful selection of rainbow trout (Oncorhynchus mykiss) on their ability to grow with a diet completely devoid of fishmeal and fish oil, and correlated changes in nutritional traits[END_REF], the selection response was significantly higher when fish were fed the V diet than those fed the M diet. This means that when we selected fish fed 100% V diet for better survival and growth, two kinds of response occurred: 1) general selection for growth which will be expressed whatever the diet, 2) specific selection linked to the V diet. The hepatosomatic index was significantly higher in trout fed the M diet and was not affected by the lines (Table 2).

The cholesterol level in plasma was higher in trout fed the M diet at both 8 h and 16 h after the last meal. It was significantly affected by the line at 16 h after the last meal, with a significantly lower concentration of cholesterol in the plasma in the S line regardless of the diet. The level of triglycerides in plasma was higher in trout fed the M diet than those fed the V diet but only at 16 h after the meal. No significant line effect was found for plasma triglycerides (Table 2).

Hepatic expression of miRNAs

Expression of miR-1-3p, miR-33a-5p, miR-122-5p, miR-128-3p and miR-223-3p was assessed in liver 8 h and 16 h after the last meal (Fig. 1). The expression of miR-1 and miR-223 was not affected by diet nor by line. Interestingly, miR-33a was expressed significantly lower in the liver of fish fed the M diet at both 8 h and 16 h. The expression of miR-122 and miR-128 significantly increased in the S line compared to the C line at 8 h after the meal. miR-122 was also found to be less expressed in fish fed the V diet but only at 16 h.

Hepatic expression of genes related to cholesterol and lipid metabolism

We analyzed the hepatic expression of genes related to cholesterol and lipid metabolism or potentially regulated by miR-1-3p, miR-33a-5p, miR-122-5p, miR-128-3p and miR-223-3p (Fig. 2). With a high concordance among genes involved in cholesterol synthesis, the expression of hmgcr, hmgcs, cyp51 and dhcr7 significantly increased in the liver of fish fed the V diet at both 8 h and 16 h after the last meal. However, none of them were affected by the line. Correspondingly, the expression of their regulator srebp-2 showed the same significant variation with higher expression in fish fed the V diet regardless of the lines at both 8 h and 16 h. The genes involved in cholesterol efflux were shown to be affected either by diet or line. The V diet significantly increased the expression of abcg5 at 8 h, cyp7a1 and abca1 at both 8 h and 16 h after the meal. On the contrary, selective breeding for better adaptation to the plant-based diet leaded to decreased expression of abcg8 at 16 h after the meal. The only gene not affected by the treatments was lxrα, which was not affected neither by diet nor line. Regarding the genes involved in cholesterol esterification, a significant interaction between diets and lines was found for acat1 which was higher expressed in the fish from the selected line fed the M diet. A significantly higher expression of acat2 was observed in both lines fed the V diet at 8 h after the meal. Whatever the line, the expression of fas Table 2 Morphometric parameters in rainbow trout from selected (S) line and control (C) line fed the marine (M) diet and vegetable (V) diet (C-M represents for the rainbow trout from C line fed the M diet, etc.); and srebp-1c was higher in fish fed the V diet at both 8 h and 16 h after the meal, however, the expression of gck was much lower in fish fed the V diet at 8 h and 16 h. The expression of gck was also affected by the line, with much higher expression in the S line at 8 h. In summary, the expression of most genes related to cholesterol and lipid metabolism was affected by the V diet rather than the lines, showing a higher expression in both lines fed the V diet.

C-M C-V S-M S-

Expression of target genes by corresponding miRNAs in liver

Based on sequence complementarity between the seed sequences of miR-33a-5p, miR-122-5p and miR-128-3p and 3'UTR region of rainbow trout mRNAs, a list of potential direct targets of these previous cited miRNAs was established (Supplementary data S1-3). We analyzed the expression of rdh14a, casp6l2 and lcmt2 as potential direct targets of miR-33a (Fig. 3A). Among the three genes targeted by miR-33a, only the expression of casp6l2 was detected to be affected by diet and line. The expression of casp6l2 was significantly lower in fish fed the V diet at both 8 h and 16 h after the meal and significantly higher in S line only at 8 h after the meal. However, no significant difference was found in the expression of rdh14a and lcmt2. Regarding the genes targeted by miR-122 (cyp27a7, mettl5, hdac3 and hif1aa)(Fig. 3B), we observed that feeding trout with the V diet resulted in decreased expression of cyp27a7 at 8 h after the meal. A significant interaction between diet and line effects was found for mettl5 at 16 h after the meal, showing a higher expression in the S line than the C line when fish were fed the M diet but a lower expression in the S line when fish were fed the V diet. On the other hand, the expression of hdac3 and hif1aa was not affected by diets or lines at both time points. We finally analyzed pde4ba and roraa as putative direct targets of miR-128 (Fig. 3C). Both genes expressed at much lower level in the fish fed the V diet at 8 h after the meal. This lowering effect of the V diet was maintained 16 h after the meal for pde4ba. Finally, results showed that the expression of pde4ba at 16 h after the meal was also affected by the line, with a lower expression in the S line.

Correlations between miRNAs and their putative target genes

Pearson correlation analysis showed that miR-33a negatively correlated with casp6l2 and miR-122 positively correlated with cyp27a7 significantly 8 h after the meal, however, other pairs of miRNAs and their target genes did not show any significant correlations (Table 6). Pearson correlation coefficient larger than zero is positive correlation, while Pearson correlation coefficient less than zero is negative correlation.

⁎ P < .05 is statistically significant with one star.

Discussion

Selective breeding in fish for better adaptation to plant-based diet represents a promising future in aquaculture when increasingly more fishmeal and fish oil are replaced by vegetable ingredients in aquafeeds. Recent selective breeding program developed in rainbow trout has proven that individual variability in the ability to survive and grow on the plant-based diet can be effectively used to promote fish production (Le [START_REF] Boucher | Plant-based diet in rainbow trout (Oncorhynchus mykiss Walbaum): are there genotype-diet interactions for main production traits when fish are fed marine vs. plant-based diets from the first meal?[END_REF]. In order to decipher the molecular mechanisms underlying this adaptation to the plant-based diet totally devoid of marine ingredients, we examined the expression of five miRNAs potentially involved in the regulation of lipid and cholesterol metabolism in the liver of control and selected rainbow. We also analyzed the hepatic expression of putative indirect and direct target genes of these miRNAs.

Our study demonstrated that the hepatic expression of miRNAs could be affected either in fish fed the 100% plant-based diet or in line selected for better adaptation to the plant-based diet by selective breeding. The vegetable diet increased the hepatic expression of miR-33a, while selective breeding for better survival and growth on the totally plant-based diet enhanced the expression of miR-122 and miR-128. However, the miR-1 (reported to inhibit lxrα expression [START_REF] Zhong | MicroRNA-1 and microRNA-206 suppress LXRα-induced lipogenesis in hepatocytes[END_REF], which may result in cellular cholesterol increase via reducing cholesterol efflux and conversion to bile acids) and miR-223 (known to reduce cellular cholesterol and promote cholesterol efflux by directly repressing hmgcs and indirectly increasing abca1 and cyp7a1, respectively [START_REF] Vickers | MicroRNA-223 coordinates cholesterol homeostasis[END_REF]), were not modified by the diet nor the selective breeding.

One particularity of the miR-33a in human is that its gene is located in the intron of srebp-2, the gene encoding SREBP-2 protein [START_REF] Marquart | miR-33 links SREBP-2 induction to repression of sterol transporters[END_REF]. This particularity seems to be conserved across animal species, since miR-33a is also positioned in an intron of the srebp-2 gene in rainbow trout. The concomitance and co-expression of srebp-2 and miR-33a could be the reason why the expression of both miR-33a and srebp-2 were found to be concurrently higher in trout fed the plantbased diet. In mammals, miR-33a could repress the cholesterol efflux by targeting abcg5 and abcg8 (involved in direct cholesterol discharge into the bile), cyp7a1 (key enzyme in bile acid synthesis) and abca1 (contributing to hepatic cholesterol efflux to plasma) [START_REF] Bommer | Regulation of lipid homeostasis by the bifunctional SREBF2-miR33a locus[END_REF][START_REF] Gerin | Expression of miR-33 from an SREBP2 intron inhibits cholesterol export and fatty acid oxidation[END_REF][START_REF] Li | Regulation of cholesterol and bile acid homeostasis by the cholesterol 7α-hydroxylase/steroid response elementbinding protein 2/microRNA-33a axis in mice[END_REF]. The increasing expression of miR-33a might indicate the repression of cholesterol export and bile acid synthesis [START_REF] Gerin | Expression of miR-33 from an SREBP2 intron inhibits cholesterol export and fatty acid oxidation[END_REF][START_REF] Li | Regulation of cholesterol and bile acid homeostasis by the cholesterol 7α-hydroxylase/steroid response elementbinding protein 2/microRNA-33a axis in mice[END_REF] when trout were fed the plant-based diet. However, the higher expression of cholesterol efflux genes (abca1, cyp7a1 and abcg5) found in trout fed the V diet in the present study goes against this hypothesis. This discrepancy may be attributed to the variety of factors acting at transcriptional or post-transcriptional levels that orchestrate the regulation of gene expression together. For example, in mammals, it was reported that srebp-2 positively regulated the expression of abca1 [START_REF] Wong | SREBP-2 positively regulates transcription of the cholesterol efflux gene, ABCA1, by generating oxysterol ligands for LXR[END_REF] and the unusually long 3'UTR of abca1 also harbors many binding sites of other miRNAs [START_REF] Ramirez | MicroRNA-758 regulates cholesterol efflux through posttranscriptional repression of ATP-binding cassette transporter A1[END_REF] that probably also contribute to the regulation of abca1. Therefore, in the present study, the higher expression of abca1 in trout fed the V diet may be due to the higher expression of srebp-2 and/or the involvement of other miRNAs counteracting the effect of the miR-33a.

miR-122, a liver specific miRNA, has been suggested to play roles in cholesterol and lipid metabolism [START_REF] Esau | miR-122 regulation of lipid metabolism revealed by in vivo antisense targeting[END_REF][START_REF] Krützfeldt | Silencing of microRNAs in vivo with 'antagomirs[END_REF]. In contrast to most other miRNAs inhibiting gene expression, miR-122 works to promote the expression of genes involved in cholesterol and fatty acid synthesis through a mechanism still unknown (Aryal et al., 2017;[START_REF] Iliopoulos | MicroRNA-370 controls the expression of microRNA-122 and Cpt1alpha and affects lipid metabolism[END_REF][START_REF] Krützfeldt | Silencing of microRNAs in vivo with 'antagomirs[END_REF]. In the present study, miR-122 expression was shown to be affected by the selective breeding with enhanced expression in the selected line compared to the control line at 8 h after the meal. However, the increase in miR-122 in the selected line was not associated with a concomitant enhanced expression of genes involved in cholesterol biosynthesis (hmgcr, hmgcs, cyp51 and dhcr7) or lipogenesis (fas and srebp-1c). Similarly, the promoting effect of the vegetable diet on the expression of the foregoing genes was not accompanied by an enhanced expression of miR-122. Therefore, the present study confirms our previous results that in vivo inhibition of miR-122 by antagomir in trout did not affect the expression of genes involved in cholesterol synthesis [START_REF] Mennigen | Metabolic consequences of microRNA-122 inhibition in rainbow trout, Oncorhynchus mykiss[END_REF].

Similar to miR-122, the expression of miR-128 increased in the selected line regardless of the diets. This increase was only observed 8 h after the meal, confirming our previous investigations that miRNAs could be under postprandial nutritional control in fish [START_REF] Mennigen | Postprandial regulation of hepatic microRNAs predicted to target the insulin pathway in rainbow trout[END_REF]. miR-128 is considered as an actor to regulate cholesterol homeostasis by limiting cholesterol efflux (Adlakha et al., 2013). Cholesterol efflux is a mechanism implemented by cells to eliminate excess cholesterol and prevent cellular cholesterol accumulation. This efflux is mediated by the ABCA1, of which expression has been shown to be inhibited at both mRNA and protein level by miR-128 through direct binding to the 3'UTR region of abca1 mRNA (Adlakha et al., 2013). The positive effect of the selective breeding on the expression of miR-128 seems to be consistent with the lower plasma cholesterol level measured in the selected line at 16 h after the meal. However, the higher expression of miR-128 but lower plasma cholesterol in selected line was inconsistent with the result of abca1 expression at mRNA level, which was not affected by selective breeding. Therefore, further investigations are still needed to determine if abca1 expression could be affected by the miR-128 at protein level rather than mRNA level.

In the present study, the expressions of miR-33a, miR-122 and miR-128 in liver were all affected either by diets or lines, while miR-1 and miR-223 were not affected by either line or diet. Therefore, we further utilized the 3'UTR of these three miRNAs to identify their putative Fig. 2. Expression of genes involved in cholesterol synthesis (A), cholesterol efflux (B), cholesterol regulation (C), cholesterol esterification (D), and fatty acid synthesis (E) at 8 h and 16 h in the liver of rainbow trout from selected (S) line and control (C) line fed the marine (M) diet and (V) vegetable diet. The selected line was selected for better survival and growth performance with a 100% plant-based diet during three generations. The control line was the common trout without any artificial selection. Expression values were normalized by 18 s at 8 h and ef1α at 16 h. Values are means (n = 6), with their standard deviations represented by vertical bars. (Results of two-way analysis of variance are indicated with the P value). One star/circle represents a P value < .05 for diets/lines, respectively; two stars/circles represent a P value < .01 for diets/lines, respectively. direct target genes and then analyzed the expression of the suggested genes to see if they could be regulated by the corresponding miRNAs.

In contrary to the increased expression of miR-33a in trout fed the V diet, its putative direct target casp6l2 showed significantly lower expression in trout fed the V diet regardless of time point, indicating the expression of casp6l2 could be post-transcriptionally inhibited by miR-33a. The negative correlation between casp6l2 and miR-33a is also consistent with this speculation. CASP6L2 belongs to the caspase family of cysteine proteases associated with immune responses and apoptosis. It is known that the activation of caspases proceeds to the production of pro-inflammatory cytokines and apoptosis [START_REF] Siegel | Caspases at the crossroads of immune-cell life and death[END_REF]. As the immune function is affected in salmonids when marine ingredients are replaced by plant ingredients in aquafeeds [START_REF] Martin | Nutrigenomics and immune function in fish: new insights from omics technologies[END_REF], the regulation of casp6l2 by diet composition is not surprising.

Among the miR-122 putative target genes analyzed in the present study, only cyp27a7 was found to be affected, with a lower expression in fish fed the V diet. In the present study, the expression of miR-122 was subjected to an up-regulation linked to selective breeding 8 h after the last meal and a down-regulation by the plant-based diet 16 h after the last meal regardless of the fish line. As no significant effect of selective breeding was recorded for cyp27a7 expression, its direct regulation by miR-122 remains questionable. Correspondingly, correlation analysis did not show the negative correlation between miR-122 and cyp27a7, either. The physiological role of CYP27A7 still remains to be determined. However, like the sterol 27-hydroxylase (CYP27A), a member of the same cytochrome family, CYP27A7 may be involved in the cholesterol and bile acid metabolism as well [START_REF] Dubrac | Role of CYP27A in cholesterol and bile acid metabolism[END_REF]. In this respect, its lower expression in fish fed the V diet is in agreement with the impact of vegetable diet on cholesterol and bile acid metabolism.

Though the expression of miR-128 was up-regulated in the selected line 8 h after the last meal, the expression of pde4ba and roraa, two potential direct targets of miR-128, was not affected by selective breeding. This inconsistency may suggest that miR-128 is not involved in the regulation of these genes at mRNA level. The absence of correlation between miR-128 and pde4ba and roraa also supports this speculation. On the other hand, the expression of these genes did, however, decrease in the trout fed the V diet. PDE4B belongs to a family of four genes (from PDE4A to D) that act to hydrolyze cAMP and switch off cAMP signaling cascades [START_REF] Houslay | PDE4 cAMP phosphodiesterases: modular enzymes that orchestrate signalling cross-talk, desensitization and compartmentalization[END_REF]. A recent study indicated that an increased hepatic expression of pde4b could alter cAMP signaling, impair fatty acid oxidation and predispose to liver steatosis (Avila et al., 2016). A current phenotypic trait observed in trout fed the totally plant-based diet showed that the decrease in hepatosomatic index was related to the reduction of lipid content in the liver [START_REF] Lazzarotto | Three-year breeding cycle of rainbow trout (Oncorhynchus mykiss) fed a plant-based diet, totally free of marine resources: Consequences for reproduction, fatty acid composition and progeny survival[END_REF]. The reduced expression of pde4ba observed in trout fed the plant-based diet is thus in agreement with the reduced accumulation of fat in the liver, which may be consistent with the role assigned to PDE4B in mammals. RORAA is a member of the retinoid-related orphan receptor (ROR) subfamily. These receptors play critical roles in development, immunity, circadian rhythm, and cellular metabolism [START_REF] Jetten | Retinoid-related orphan receptors (RORs): critical roles in development, immunity, circadian rhythm, and cellular metabolism[END_REF]. It is reported that impaired expression of rorα in natural sg/sg mutant mice leads to reduced plasma cholesterol level by limiting cholesterol efflux. These mice also exhibited reduced expression of srebp-1c as well as of its target gene, fatty acid synthase (fas) [START_REF] Lau | The orphan nuclear receptor, RORα, regulates gene expression that controls lipid metabolism: staggerer (sg/sg) mice are resistant to diet-induced obesity[END_REF]. In the present study, because the lower expression of roraa but higher expression of abca1, abcg5, srebp-1c and fas in fish fed the plant-based diet was inconsistent with the results in mice study, further investigations are needed to better understand the role of these retinoid-related orphan receptors in trout metabolism.

Among the genes that were chosen as putative targets of miR-33a,

-122 and -128, some of them, including rdh14, lcmt2, mttl5, hdac3 and hif1aa, were neither affected by fish lines nor diets. It has been established that miRNAs regulate gene expression via two main mechanisms depending on the degree of complementarity between miRNA and its target gene. When complementarity is high, mRNA is cleaved and degraded through the RNA-induced silencing complex (RISC).

Conversely, low complementarity resulted in translational inhibition of the mRNA target and reduction of the abundance of the encoded protein [START_REF] Zeng | MicroRNAs and small interfering RNAs can inhibit mRNA expression by similar mechanisms[END_REF]. Thus it is necessary to conduct additional studies at protein level to clearly evaluate the involvement of miR-33a, -122 or -128 in the regulation of these genes.

In conclusion, the present study demonstrated that hepatic expression of miRNAs may be affected by selective breeding of rainbow trout for better adaptation to plant-based diets as well as by total dietary replacement of fishmeal and fish oil by plant ingredients. The miR-33a was found to be greatly induced by plant-based diet whereas miR-122 and miR-128 were affected by selective breeding. This study opens new perspective to evaluate the role of miRNAs in fish nutrition and selective breeding. However, the present study is restricted to the investigation of gene expression at mRNA level instead of protein due to the unavailability of antibodies for Western blot in fish. Even though some relevant regulations were observed between miRNAs and their potential target genes, additional investigations necessitating the development of antibodies are essential to further establish the role of miRNAs in rainbow trout metabolism. Finally, the present study also highlighted some new genes affected by plant-based diet, thus linking this change in dietary composition to, among others, immune response, apoptosis and cAMP signaling pathways. The identification of these new molecular targets in the replacement of marine ingredients by plant products in rainbow trout may aid in opening new fields of investigation to better understand the mechanisms of adaptation to plantbased diet in carnivorous fish.

Fig. 1 .

 1 Fig. 1. miRNAs expression involved in cholesterol metabolism at 8 h and 16 h including miR-1, miR-33a, miR-122, miR-128 and miR-223 in the liver of rainbow trout from selected (S) line and control (C) line fed the marine (M) diet and (V) vegetable diet. The selected line was selected for better survival and growth performance with a 100% plant-based diet during three generations. The control line was the common trout without any artificial selection. Expression values were normalized by spike miR-39. Values are means (n = 6), with their standard deviations represented by vertical bars. (Results of two-way analysis of variance are indicated with the P value). One star/circle represents a P value < .05 for diets/lines, respectively; two stars/circles represent a P value < .01 for diets/lines, respectively.

Fig. 2 .

 2 Fig. 2. (continued)

Fig. 3 .

 3 Fig. 3. Expression of possible target genes by miR-33a (A), miR-122 (B) and miR-128 (C) identified in silico at 8 h and 16 h in the liver of rainbow trout from selected (S) line and control (C) line fed the marine (M) diet and (V) vegetable diet. The selected line was selected for better survival and growth performance with a 100% plant-based diet during three generations. The control line was the common trout without any artificial selection. Expression values were normalized by 18 s at 8 h and ef1α at 16 h. Values are means (n = 6), with their standard deviations represented by vertical bars. (Results of two-way analysis of variance are indicated with the P value). One star/circle represents a P value < .05 for diets/lines, respectively; two stars/circles represent a P value < .01 for diets/lines, respectively.

Table 1

 1 Ingredients and proximate composition of the diets.

	Ingredients (%)	M	V
	Fish meal	58.42	0.00
	Corn gluten	0.00	16.03
	Wheat gluten	0.00	17.72
	Soybean meal	0.00	10.13
	Soy protein concentrate	0.00	15.19
	Light white lupin	0.00	6.75
	Dehulled pea	0.00	4.30
	Extruded whole wheat,	25.26	3.38
	Fish oil	14.09	0.00
	Vegetable oils a	0.00	15.61
	Mineral premix b	1.12	1.18
	Vitamin premix c	1.12	1.18
	Soy lecithin	0.00	2.11
	L-lysine	0.00	1.52
	L-methionine	0.00	0.34
	CaHPO4.2H2O	0.00	3.21
	Attractant Mix	0.00	1.35
	Analytical composition
	Dry Matter (DM, %)	90.13	94.41
	Crude protein (% DM)	46.98	50.01
	Lipid (% DM)	18.80	19.49
	Sterols (% DM)	0.77	0.35
	Energy (kJ/g DM)	23.24	23.84
	Ash (% DM)	8.17	6.25

a Vegetable oils: palm oil (30%), rapeseed oil (55%), linseed oil (15%).

Table 3

 3 Sequences of the probes in TaqMan ® Advanced miRNA Assays.

	Assay	Assay ID	Probe sequences
	miR-1-3p	477820_mir	TGGAATGTAAAGAAGTATGTAT
	miR-33a-5p	478347_mir	GTGCATTGTAGTTGCATTGCA
	miR-39-3p	478293_mir	TCACCGGGTGTAAATCAGCTTG
	miR-122-5p	477855_mir	TGGAGTGTGACAATGGTGTTTG
	miR-128-3p	477892_mir	TCACAGTGAACCGGTCTCTTT
	miR-223-3p	477983_mir	TGTCAGTTTGTCAAATACCCCA
	miR-451a	478107_mir	AAACCGTTACCATTACTGAGTT

Table 4

 4 Primer sequences of cholesterol metabolic and lipogenic genes. Primer sequences of miRNA target genes.

	Gene	Forward primer	Reverse primer	Genoscope a or Genbank accession numbers
				Paralogue 1	Paralogue 2
	abca1	CAGGAAAGACGAGCACCTT	TCTGCCACCTCACACACTTC	GSONMG00078741001	GSONMG00074045001
	abcg5	CACCGACATGGAGACAGAAA	GACAGATGGAAGGGGATGAA	GSONMG00075025001	/
	abcg8	GATACCAGGGTTCCAGAGCA	CCAGAAACAGAGGGACCAGA	GSONMG00075024001	/
	acat1	GGCAAGCCTGATGTGGTAGT	ACCGTGCCATTCTCCTTCTG	GSONMG00033938001	/
	acat2	TGCTTGTTGTCCCTGGGTTT	GTGTGGCTGTGACGTGTTTC	GSONMG00054446001	GSONMG00054447001
	cyp51	CCCGTTGTCAGCTTTACCA	GCATTGAGATCTTCGTTCTTGC	GSONMG00031182001	GSONMG00044416001
	cyp7a1	ACGTCCGAGTGGCTAAAGAG	GGTCAAAGTGGAGCATCTGG	AB675933.1 GSONMG00066448001	AB675934.1 GSONMG00037174001
	dhcr7	GTAACCCACCAGACCCAAGA	CCTCTCCTATGCAGCCAAC	GSONMG00025402001	GSONMG00039624001
	fas	TGATCTGAAGGCCCGTGTCA	GGGTGACGTTGCCGTGGTAT	GSONMG00062364001
	gck	GCACGGCTGAGATGCTCTTTG	GCCTTGAACCCTTTGGTCCAG	GSONMG00033781001	GSONMG00012878001
	hmgcr	GACCATTTGGGAGCTTGTGT	GAACGGTGAATGTGCTGTGT	GSONMG00016350001	/
	hmgcs	AGTGGCAAAGAGAGGGTGTG	TTCTGGTTGGAGACGAGGAG	GSONMG00010243001	/
	lxrα	TGCAGCAGCCGTATGTGGA	GCGGCGGGAGCTTCTTGTC	GSONMG00014026001	GSONMG00064070001
	srebp-1c	CATGCGCAGGTTGTTTCTT	GATGTGTTCGTGTGGGACTG	XM_021624594.1
	srebp-2	TAGGCCCCAAAGGGATAAAG	TCAGACACGACGAGCACAA	GSONMG00039651001	GSONMG00061885001
	18 s	CGGAGGTTCGAAGACGATCA	TCGCTAGTTGGCATCGTTTAT
	ef1α	TCCTCTTGGTCGTTTCGCTG	ACCCGAGGGACATCCTGTG	AF498320.1	/

a https://www.genoscope.cns.fr/trout

Table 5

Table 6

 6 Correlations between miRNAs and their potential target genes.

	miRNAs	Potential target gene	Pearson value
			8 h	16 h
	miR-33a	rdh14a	-0.31	0.3051
		casp6l2	-0.47⁎	-0.3813
		lcmt2	-0.06	-0.0878
	miR-122	cyp27a7	0.42⁎	0.0442
		mettl5	0.21	0.3137
		hdac3	-0.13	0.1321
		hif1aa	0.08	0.259
	miR-128	pde4ba	-0.25	0.3258
		roraa	-0.09	0.9701

Acknowledgements

The authors thank F. Sandres and F. Terrier for manufacturing the experimental diets (INRA, Donzaq, France), L. Labbé and T. Kerneis for fish rearing and A. Herman for technical assistance in the laboratory.

Financial support

This work was carried out within the framework of the AGREENFISH project (project ANR-13-ADAP-001) funded by the French National Research Agency (ANR). T. Zhu gratefully acknowledges the financial assistance provided by the China Scholarship Council (CSC, File No. 201406330071) for his doctoral fellowship.

Conflict of interest

None.

Authorship SS, GC: formulated research questions and designed the study. EQ, MDN: provided the lines. JM, JB: provided information on direct target gene of miRNA. TZ, EPJ: performed the analysis. TZ, SS, GC: analyzed the data. TZ, SS, GC wrote the paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.aquaculture.2018.08.058.

Adlakha, Y.K., Khanna, S., Singh, R., Singh, V.P., Agrawal, A., Saini, N., 2013. Proapoptotic miRNA-128-2 modulates ABCA1, ABCG1 and RXRα expression and cholesterol homeostasis. Cell Death Dis. 4, e780. AOAC, 2000. Official Methods of Analysis of AOAC International. Gaithersburg, MD USA. Aryal, B., Singh, A.K., Rotllan, N., Price, N., Fernández-Hernando, C., 2017. MicroRNAs andlipid metabolism. Curr. Opin. Lipidol. 28, 273-280. Avila, D.V., Barker, D.F., Zhang, J., McClain, C.J., Barve, S., Gobejishvili, L., 2016. Dysregulation of hepatic cAMP levels via altered Pde4b expression plays a critical role in alcohol-induced steatosis. J. Pathol. 240, 96-107.


Sergio Ramírez

Miguel Romero

Camilo Rocha

Frank Valencia

Real-time Rewriting Logic Semantics for Spatial Concurrent Constraint Programming

Process calculi provide a language in which the structure of terms represents the structure of processes together with an operational semantics to represent computational steps. This paper uses rewriting logic for specifying and analyzing a process calculus for concurrent constraint programming (ccp), combining spatial and real-time behavior. In these systems, agents can run processes in different computational spaces (e.g., containers) while subject to real-time requirements (e.g., upper bounds in the execution time of a given operation), which can be specified with both discrete and dense linear time. The real-time rewriting logic semantics is fully executable in Maude with the help of rewriting modulo SMT: partial information (i.e., constraints) in the specification is represented by quantifier-free formulas on the shared variables of the system that are under the control of SMT decision procedures. The approach is used to symbolically analyze existential real-time reachability properties of process calculi in the presence of spatial hierarchies for sharing information and knowledge.

Introduction

Concurrent constraint programming (ccp) [START_REF] Saraswat | Semantic foundations of concurrent constraint programming[END_REF] is a well-established process model for concurrency based upon the shared-variables communication model. Its basic intuitions arise mostly from logic; in fact, ccp processes can be interpreted both as concurrent computational entities and logic specifications (e.g., process composition can be seen as parallel execution and as conjunction). In ccp, agents can interact by posting (or telling) partial information in a medium such as a centralized store. Partial information is represented by constraints (e.g., x > 42) on the shared variables of the system. The other way in which agents can interact is by querying (or asking) about partial information entailed by the store. This provides the synchronization mechanism of the model: asking agents are suspended until there is enough information in the store to answer their query. As other mature models of concurrency, ccp has been extended to capture aspects such as mobility [START_REF] Chiarugi | Stochastic modelling of non markovian dynamics in biochemical reactions[END_REF][START_REF] Gilbert | Concurrent constraint programming with process mobility[END_REF][START_REF] Guzmán | Belief, knowledge, lies and other utterances in an algebra for space and extrusion[END_REF], stochastic behavior [START_REF] Gupta | Stochastic processes as concurrent constraint programs[END_REF], and -most prominentlytemporal behavior [START_REF] De Boer | A timed concurrent constraint language[END_REF][START_REF] Gupta | Computing with continuous change[END_REF][START_REF] Nielsen | Temporal concurrent constraint programming: Denotation, logic and applications[END_REF][START_REF] Saraswat | Foundations of timed concurrent constraint programming[END_REF][START_REF] Saraswat | Default timed concurrent constraint programming[END_REF] for timed and reactive computations, where processes can be constrained also by unit delays and time-out conditions.

However, due to their centralised notion of store, all the previously-mentioned extensions are unsuitable for today's systems where information and processes can be spatially distributed among certain groups of agents. Examples of these systems include agents posting and querying information in the presence of spatial hierarchies for sharing information and knowledge, such as friend circles and shared albums in social networks, or shared folders in cloud storage. Recently, the authors of [START_REF] Knight | Spatial and Epistemic Modalities in Constraint-Based Process Calculi[END_REF] enhanced and generalized the theory of ccp for systems with spatial distribution of information in the novel spatial concurrent constraint programming (sccp), where computational hierarchical spaces can be assigned to belong to agents. In sccp, each space may have ccp processes and other (sub) spaces, processes can post and query information in their given space (i.e., locally), and may as well move from one space to another.

As an example, consider the tree-like structures depicted in Figure 1. They correspond to hierarchical computational spaces of, e.g., virtual containerization (i.e., virtual machines inside other virtual machines). Each one of these spaces is endowed with an agent identifier (either root or a natural number) and a local store (i.e., a constraint), and the processes can be executed and spawned concurrently inside any space, with the potential to traverse the structure, querying and posting information locally, and even creating new spaces. The sccp calculus enables the formal modeling of such scenarios and of transitions that can lead from an initial system state (e.g., Figure 1a) to a final state (e.g., Figure 1b) by means of an operational semantics [START_REF] Knight | Spatial and Epistemic Modalities in Constraint-Based Process Calculi[END_REF]. This paper presents the real-time sccp calculus (rtsccp), a generalization of sccp with timing constraints. The intended models of rtsccp are spatially-distributed multiagent reactive systems that may have different computing capabilities (e.g., virtual containers with heterogeneous bandwidth and main memory configurations) and be subject to real-time requirements (e.g., upper bounds in the execution time of a given operation). The formal semantics of rtsccp is given in the form of a real-time rewriting logic semantics that is executable in the Maude system [START_REF] Clavel | All about Maude -A High-Performance Logical Framework: How to Specify, Program, and Verify Systems in Rewriting Logic[END_REF]. As such, the real-time rewriting logic specification can be subject to automatic reachability analysis and LTL modelchecking, and thus enables the formal analysis of timing behavior for agents distributed in hierarchical spaces, such as fault-tolerance and consistency.

(root) W = 9 (0) X ≥ 11
In the rtsccp real-time rewriting logic semantics, flat configurations of object-like terms encode the hierarchical structure of spaces, and equational and rewrite rules both axiomatize the concurrent computational steps of processes. Time attributes are associ-ated to process-store interaction, as well as to process mobility in the space structure, by means of maps from agents to non-negative real quantities; these choices can be interpreted to denote, as previously mentioned, upper bounds in the execution time of the given operations. The underlying constraint system of sccp is materialized with the help of the rewriting modulo SMT [START_REF] Rocha | Rewriting modulo SMT and open system analysis[END_REF] approach, with constraints being quantifier-free formulas over Boolean and integer shared variables, and information entailment queried as semantic inference and automatically delivered by the SMT-based decision procedures. The main contribution of this work can also be seen as yet another interesting use of rewriting logic as a semantic framework: the support in rewriting logic for realtime systems [START_REF] Ölveczky | Specification of real-time and hybrid systems in rewriting logic[END_REF] and open systems [START_REF] Rocha | Rewriting modulo SMT and open system analysis[END_REF], make of the rtsccp real-time rewriting logic semantics a symbolic and fully executable specification in Maude, that is both sound and complete (relative to initial semantics) for reachability analysis in spatial constraint systems with discrete and dense linear timing constraints.

Outline. Section 2 summarizes some preliminaries on constraint systems and rewriting logic. The real-time rewriting logic semantics for rtsccp is presented in Section 3. The use of the semantics for symbolic reachability analysis is illustrated with some examples in Section 4. Additional related work and concluding remarks are included in Section 5. The rewriting logic specification for rtsccp and the examples are available at http://escher.javerianacali.edu.co/rtsccpe/index.html

Preliminaries

This section briefly explains the basics of constraint and spatial constraint systems with extrusion. It also presents a summary of order-sorted rewriting logic [START_REF] Meseguer | Conditional rewriting logic as a unified model of concurrency[END_REF], a semantic framework that unifies a wide range of models of concurrency.

A constraint system (cs) C is a complete algebraic lattice C = (Con, ), where the elements in Con are called constraints and the order entailment of information: d c (or, c d) asserts that the constraint c contains at least as much information as the constraint d. The symbols , true, and false denote the least upper bound (lub) operation, the bottom, and the top element of C, respectively. An n-agent spatial constraint system (n-scs) [START_REF] Knight | Spatial and Epistemic Modalities in Constraint-Based Process Calculi[END_REF] C is a cs (Con, ) equipped with n self-maps [•] 1 , . . . , [•] n over its set of constraints Con satisfying, for each function [•]

i : Con → Con: [true] i = true and [c d] i = [c] i [d] i , for each c, d ∈ Con.
An n-agent spatial constraint system with extrusion (n-scse) [START_REF] Guzmán | Belief, knowledge, lies and other utterances in an algebra for space and extrusion[END_REF] is an n-scs C equipped with n self-maps ↑ 1 , . . . , ↑ n over Con,

written (C, ↑ 1 , . . . , ↑ n ), such that each ↑ i is the right inverse of [•] i .
A rewrite theory is a tuple R = (Σ, E B, R) with: (i) (Σ, E B) an order-sorted equational theory with signature Σ, E a set of equations over T Σ , and B a set of structural axioms -disjoint from the set of equations E -over T Σ for which there is a finitary matching algorithm (e.g., associativity, commutativity, and identity, or combinations of them); and (ii) R a finite set of (possibly conditional) rewrite rules over T Σ (X). Intuitively, R specifies a concurrent system whose states are elements of the set T Σ/E B of Σ-terms modulo E B and whose concurrent transitions are axiomatized by the rules R according to the inference rules of rewriting logic [START_REF] Bruni | Semantic foundations for generalized rewrite theories[END_REF]. In particular, for t, u ∈ T Σ representing states of the concurrent system described by R, a transition from t to u is captured by a formula of the form t -→ R u; the symbol -→ R denotes the binary rewrite relation induced by R over T Σ/E B and T R = (T Σ/E B , → R ) denotes the initial reachability model of R.

The rewriting logic semantics of a language L is a rewrite theory R L = (Σ L , E L B L , R L ) where → R L provides a step-by-step formal description of L's observable runto-completion mechanisms. The conceptual distinction between equations and rules in R L has important consequences that are captured by rewriting logic's abstraction dial [START_REF] Meseguer | The rewriting logic semantics project: A progress report[END_REF]. Setting the level of abstraction in which all the interleaving behavior of evaluations in L is observable, corresponds to the special case in which the dial is turned down to its minimum position by having E L B L = ∅. The abstraction dial can also be turned up to its maximal position as the special case in which R L = ∅, thus obtaining an equational semantics of L without observable transitions. The rewriting logic semantics presented in this paper is faithful in the sense that such an abstraction dial is set at a position that exactly captures the interleaving behavior of the concurrency model.

The real-time rewrite theory presented in this work is time-robust, namely: (i) in any given state, time can advance either any amount up to a specific instant in time or not at all; and (ii) instantaneous rules (i.e., those that are not tick rules and are supposed to take zero time) can only be applied when the system has advanced the maximal possible amount of time before any timed action can become enabled. Under these two assumptions and by using the maximal time sampling strategy, unbounded and timebounded search and model checking are sound and complete with respect to timed fair paths [START_REF] Ölveczky | Abstraction and Completeness for Real-Time Maude[END_REF]. They exclude paths with an infinite sequence of tick steps where, at each step, time could have advanced to time r (the duration of the first step in a path) or beyond, but with a total path duration less than r. Also are excluded those 'unfair' paths containing an infinite and consecutive sequence of 0-time ticks over a state on which an instantaneous rule can be applied. Note that a time-robust system may have Zeno paths, where the sum of the durations of an infinite number of tick steps is bounded. By restricting the computations to time-bounded prefixes only a finite set of states can be reached from an initial state, so that the target real-time specification does not exhibit any Zeno behavior and temporal properties can be model checked.

Satisfiability Modulo Theories (SMT) studies methods for checking satisfiability of first-order formulas in specific models. In this work, the representation of the constraint system is based on SMT solving technology. Given an many-sorted equational theory E 0 = (Σ 0 , E 0 ) and a set of variables X 0 ⊆ X over the sorts in Σ 0 , the formulas under consideration are in the set QF Σ 0 (X 0 ) of quantifier-free Σ 0 -formulas: each formula being a Boolean combination of Σ 0 -equation with variables in X 0 (i.e., atoms). The terms in T Σ 0 /E 0 (X) are called built-ins and represent the portion of the specification that will be handled by the SMT solver (i.e., they are semantic data types). In this setting, an SMT instance is a formula φ ∈ QF Σ 0 (X 0 ) and the initial algebra T E + 0 , where

E + 0 is a decidable extension of E 0 such that φ is satisfiable in T E + 0 iff there exists σ : X 0 -→ T Σ 0 such that T E 0 | = φσ.
Maude [START_REF] Clavel | All about Maude -A High-Performance Logical Framework: How to Specify, Program, and Verify Systems in Rewriting Logic[END_REF] is a language and tool supporting the formal specification, execution, and analysis of concurrent systems specified as rewrite theories, including those with realtime semantics (see [START_REF] Ölveczky | Abstraction and Completeness for Real-Time Maude[END_REF]) and those with built-ins as proposed in the rewriting modulo SMT approach (see [START_REF] Rocha | Rewriting modulo SMT and open system analysis[END_REF]).

This section introduces the rtsccp real-time rewriting logic semantics in the form of a real-time rewrite theory R, detailing some aspects of to its syntax and transitions.

Figure 2 depicts the module structure of R, where a triple-line arrow ( ) represents module importation by protecting and a single-line arrow (→) module importation by inclusion. The difference between these two importing modes is that the former allows surjectivity (junk) and injectivity (confusion) [START_REF] Clavel | All about Maude -A High-Performance Logical Framework: How to Specify, Program, and Verify Systems in Rewriting Logic[END_REF].

System States

The tree-like structure of the hierarchical spaces is represented as a flat configuration of object-like terms encoding the state of execution of the agents. The hierarchical relationships among spaces are specified by common prefixes as part of an agent's name.

In an observable state, each agent's space is represented by a set of object-like terms: some encoding the state of execution of all its processes and exactly one object representing its local store. The object-based system is represented using Maude's predefined module CONFIGURATION imported in SCCP-STATE. The object and class identifiers are:

subsorts Nat Aid < Oid . ops agent process Simulation : -> Cid . op {_} : Configuration -> Sys .

The system states are represented by the topsort Sys with argument a configuration of objects containing the setup of each one of the agents in the system. A Configuration is a multiset of objects with set union denoted by juxtaposition and identity none. There are two types of object identifiers: agent identifiers (Aid) for identifying agents and their hierarchical structure, and natural numbers (Nat) for some additional identification used internally in R. There are three types of class identifiers, namely, for agents, processes, and a simulation object. A simulation object specifies the attributes required for the real-time simulation of the system, such as the global time and the scheduler.

Each agent has one attribute, namely, its a store, and each process has two attributes: an universal identifier (used internally for execution purposes) and the command (i.e., process) it is executing:

op store :_ : Boolean -> Attribute [ctor] . op Uid :_ : Nat -> Attribute [ctor] . op command :_ : SCCPCmd -> Attribute [ctor] .
The syntax of commands is presented in Section 3.2. Section 3.3 explains how quantifier free formulas are used to represent constraints (as Boolean) and the entailment relation is encoded with the help of SMT-based decisions procedures.

Finally, the attributes of the simulation object include the global time (attribute gtime); the priority queue of the system commands to be processed, ordered by timeto-execution (attribute pqueue); the collection of pending commands, i.e., ask commands that are waiting for its guarding constraint to become activate (attribute pend); the counter for assigning the next internal identifier when spawning a new process (attribute nextID); a flag that is on whenever a tick rule needs to be applied (attribute flg); and a collection of maps containing the time it takes to process certain commands relative to the space where they are executed (attributes MAsk, MTell, MSp, and MExt). The sort Time, as it is often the case in Real-time Maude [START_REF] Ölveczky | Specification of real-time and hybrid systems in rewriting logic[END_REF], can be used to represent either discrete or dense linear time, while Ttime is the name of the Maude view that is used to instantiate parameterized sorts with time: As mentioned before, the qualified identifiers of agents are used to encode the hierarchical structure of spaces (sort Aid). The root of any tree is denoted by constant root and any other qualified name corresponds to a dot-separated list natural numbers (sort Nat), organized from left to right:

op root : -> Aid . op _._ : Nat Aid -> Aid .
Example 1. In this syntax, the container system depicted in Figure 1a can be specified as follows:

Commands

The following EBNF-like notation defines the process-like syntax of commands:

P ::= 0 | tell(c) | ask(c) → P | P P | [P] i | ↑ i (P)
where c is a constraint and i an agent identifier. The tell(c) command posts the constraint c to the local store (once a constraint is added, it cannot be removed from the store so that the store grows monotonically). The command ask(c) → P queries if c can be deduced from the information in the local store; if so, the agent behaves like P, otherwise, it remains blocked until more information is added to the store. A basic ccp process like-language usually adds parallel composition (P Q) combining processes P and Q concurrently. The command [P] i indicates that command P must be executed inside the agent i's space: any information that P produces is available to other commands that execute within the same space. The command ↑ i (P) denotes that P is to be run outside the space of agent i and the information posted by P is going to be stored in the parent of agent i. The SCCP-SYNTAX module includes the syntax of commands in rtsccp:

Time Scaffolding

The real-time behavior in R associates timing behavior to those commands that interact with stores (i.e., tell and ask commands) and to commands that involve mobility among the space structure of the system (i.e., [_] _ and ↑_(_)). More precisely, tell and ask commands can take time when posting and querying, respectively, from a store. Moving the execution of a command inside an agent and extruding from a space can also take up time. Such times are given by the time maps MTell (for tell), MAsk (for ask), MSp (for [_] _ ), and MExt (for ↑_(_)), and can be consulted using the getTimeCmd function. For example, MTell(i) denotes the time it takes to execute a tell command inside the agent's i space. The run-to-completion time of commands is simulated with the help of a leftist heap that keeps track of all the active commands that are waiting for the global timer to advance. One motivation to use leftist heaps is that insertion, removal, and querying are defined without the need of structural axioms, which may result in performance gains during execution. A leftist heap [START_REF] Okasaki | Purely Functional Data Structures[END_REF] is a heap-ordered binary tree that satisfies the leftist property: the rank (i.e., the length of its rightmost path to a leaf) of any left child is at least as large as the rank of its right sibling. Each entry in the heap is a pair (i, t) where i is an unique identifier of a process and t the time it needs to start executing. At the beginning, all the processes belong to the heap and they are ordered with respect to their execution time. A process is executed when its execution time is the minimum time of all the processes that are pending to complete their transitions. The leftist heap is implemented as a parameterized container in the functional module LEFTIST-HEAP{X :: STRICT-TOTAL-ORDER}, with admissible parameters only being strict total orders: sort Heap{X} NeHeap{X} . subsort NeHeap{X} < Heap{X} . op empty : -> Heap{X} [ctor] . op T(_,_,_,_) : Nat X$Elt Heap{X} Heap{X} -> NeHeap{X} [ctor] . op isEmpty : Heap{X} -> Bool . eq isEmpty(empty) = true . eq isEmpty(T(Ra,E,L,R)) = false . op rank : Heap{X} -> Nat . eq rank(empty) = 0 . eq rank(T(Ra,E,L,R)) = Ra . op makeT : X$Elt Heap{X} Heap{X} -> NeHeap{X} . eq makeT(E,L,R)

= if rank(L) >= rank(R) then T(rank(R) + 1,E,L,R) else T(rank(L) + 1,E,R,L) fi .
Heaps are constructed from the constant empty and the T(_,_,_,) function symbol: the first argument is the rank of the tree (sort Nat), the second one the label (sort X$Elt), and the third and fourth ones the left and right children (sort Heap{X}), respectively. In the semantics of rtsccp, the sort X$Elt is instantiated with the sort of pairs of the form (I, T ), where I is an internal process identifier and T is the run-completion time of such a process. Auxiliary operations include isEmpty, rank, and makeT, which are used to verify whether a heap is empty, compute the rank of a given heap, and create a heap out of two heaps, respectively. Other key operations on leftist heaps are the merging of two heaps (function merge), inserting an element in a heap (function insert), removing an element from a heap (function deleteMin), and finding the element at the top of a non-empty heap (function findMin).

op merge : Heap{X} Heap{X} -> Heap{X} . eq merge(empty, L) = L . eq merge(L, empty) = L . eq merge(T(Ra,E,L,R),T(Ra',E',L',R'))

= if (E < E' or E == E') then makeT(E,L,merge(R,T(Ra',E',L',R'))) else makeT(E',L',merge(T(Ra,E,L,R),R')) fi . op insert : X$Elt Heap{X} -> NeHeap{X} . eq insert(E,L) = merge(T(1,E,empty,empty),L) . op deleteMin : NeHeap{X} -> Heap{X} . eq deleteMin(T(Ra,E,L,R)) = merge(L,R) . op findMin : NeHeap{X} -> X$Elt . eq findMin(T(Ra,E,L,R)) = E .

The Constraint System

In this rewriting logic semantics, the sort Boolean (available in the current version of Maude from the INTEGER module) defines the data type used to represent rtsccp's constraints. Terms of sort Boolean are quantifier-free formulas built from variables ranging over the Booleans and integers, and the usual function symbols. The current version of Maude is integrated with the CVC4 [START_REF] Barrett | CVC4[END_REF] and Yices2 [START_REF] Dutertre | Yices 2[END_REF] SMT solvers, which can be queried via the meta-level. In this semantics, queries to the SMT solvers are encapsulated by functions check-sat and check-unsat: op check-sat : Boolean -> Bool . eq check-sat(B) = metaCheck(['INTEGER], upTerm(B)) . op check-unsat : Boolean -> Bool . eq check-unsat(B) = not(check-sat(B)) .

The function invocation check-sat(B) returns true only if B is satisfiable. Otherwise, it returns false if it is unsatisfiable or undefined if the SMT solver cannot decide. Note that function invocation check-unsat(B) returns true only if B is unsatisfiable. Therefore, the rewriting logic semantics of rtsccp instantiates the constraint system C = (Con, ) by having quantifier-free formulas, modulo the semantic equivalence in T E + 0 (i.e., the model implemented in the SMT solver extending the initial model T E 0 ), as the constraints Con and semantic validity relative to T E + 0 as the entailment relation . More precisely, if Γ is a finite set of terms of sort Boolean and φ is term of sort Boolean, the following equivalence holds: Γ φ iff check-unsat γ∈Γ γ ∧ ¬φ . In order to make a direct relation between the entailment relation and the Maude syntax, the operator entails is defined as follows:

op entails : Boolean Boolean -> Bool . eq entails(C1:Boolean, C2:Boolean) = check-unsat(C1:Boolean and not(C2:Boolean)) .

System Transitions

The tick rule models time elapse in the system [START_REF] Ölveczky | Specification of real-time and hybrid systems in rewriting logic[END_REF]: where the the auxiliary operation delta reduce T0 units the execution time of every command in the heap P: op delta : Heap{Tuple} Time -> Heap{Tuple} . eq delta(empty,T') = empty . eq delta(T(N,((I,T)),P,P0),T') = T(N,((I,T monus T')),delta(P,T'),delta(P0,T')) .

When the [tick] rule is fired, the global time is incremented in T0 units, where T0 is the minimum time present in the priority queue P, which is modified by removing the process with the minimum execution time. It also adds the pending commands to the priority queue. The pending commands are ask commands that, although they have been activated already for execution, have not been able to execute because their guard has not been met by the state of the corresponding local stores. The tick rule puts all these pending process back in the main queue, so that their guards can be checked again and be executed or put back in the pending queue. Figure 3 depicts the possible transitions an ask command can take between being in the priority queue, in the pending queue, and finally executing. The rules [ask] and [delay] are introduced below.

pqueue pending The invisible transitions of the semantics are specified with the help of equational rules. Namely, one for removing a 0 command from a configuration and another one to join the contents of two stores of the same space (i.e., two stores with the same Aid). The latter type of transition is important because when a new process is spawned in a agent's space, a store with the empty constraint (i.e., true) is created for that space. If such a space existed before, then the idea is that the newly created store is subsumed by the existing one. Note that neither of the invisible transitions takes time, i.e., they are really instantaneous, and they axiomatize structural properties of commands.

[tick] [delay] exec [ask]
The goal of this section is to explain how the rewriting logic semantics R of rtsccp and rewriting modulo SMT can be used as an automatic mechanism for solving existential reachability goals in the initial model T R . This approach can be useful for symbolically proving or disproving real-time safety properties of T R . The approach presented in this section mainly relies on Maude's search command, but it can be easily extended to be useful in the more general setting of Maude's LTL Model Checker.

The examples presented in this section use the following time functions for the processes: For example, querying the store at the root agent takes 1/20 time units. Fault tolerance is the property that ensures a system to continue operating properly in the event of the failure; consistency means that a local failure does not propagate to the entire system. In R, this means that if a store becomes inconsistent, it is not the case that such an inconsistency spreads to the entire system. Of course, inconsistencies can appear in other stores due to some unrelated reasons.

MAsk : root |-> 1/
Searching an inconsistent store can be easily implemented with the help of R and Maude's search command. The answer of this command in the positive would mean that from some initial state, there is a state in which a store becomes inconsistent at some point of execution within a given time interval. Taking advantage of R and the rewriting modulo SMT approach, also is possible to know when a store is inconsistent. As an example, consider the container system in Figure 1a Note that a store is inconsistent if it is unsatisfiable, thereby checking whether a store is inconsistent is accomplished with the function check-unsat. The aforementioned command searches for an inconsistent store during the first 1.5 units of time of the system's execution. This command does not find an inconsistent store between the first 1.5 units of time in any of the 56 reachable states. However, it is possible to make a store inconsistent by adding inconsistent information, for example by adding the process tell(X <= 10) in 0. The output for the search command is: There are 238 reachable states (from the initial state) and 74 of them have an inconsistent store between the first 1.5 units of time. The first inconsistency appears in 0.5 time units, and the last one in 1.3 times units. Note that, the system continues evolving even though there is an inconsistency. It is possible to verify that there are states with consistent and inconsistent stores at the same time by slightly modifying the above search command.

Knowledge inference refers to acquiring new knowledge from existing facts. In the setting of R, this means that from a given initial state an agent, at some point, has gained enough information to infer new facts. A positive answer to such a query, means that from some initial state, at some moment during execution, there is at least one agent that has gained enough information to infer the given facts. As an example, consider the container system in Figure 1a It checks if there is a state, reachable from the given initial state, in which some store logically implies Y < 15 in the time interval (0, 2). This command does not find a container with enough information in such time interval in any of the 56 reachable states. However, if the time interval in the command is changed to (2, 3) the query finds two solutions: In addition to the related work included in Section 1, it is important to mention other related research in the area of timing semantics for concurrent constraint programming. An extension of concurrent constraint programming in [START_REF] Saraswat | Foundations of timed concurrent constraint programming[END_REF] presents a timed asynchronous computation model and propose an implementation using loop-free deterministic finite automata, a declarative framework for reactive systems where time is represented as discrete time units. More recently, J. A. Pérez and C. Rueda [START_REF] Pérez | Non-determinism and Probabilities in Timed Concurrent Constraint Programming[END_REF] propose an operational semantics based on probabilistic automaton, extending the work in [START_REF] Saraswat | Foundations of timed concurrent constraint programming[END_REF], with probabilistic and non-deterministic choices for processes. The inclusion of stochastic information for processes proposed by J. Aranda et al. in [START_REF] Aranda | Stochastic Behavior and Explicit Discrete Time in Concurrent Constraint Programming[END_REF] associates to each computation a random variable determining its time duration: given a set of competing actions, the fastest action is executed, that is, the one with the shortest duration. Finally, G. Sarria and C. Rueda [START_REF] Sarria | Real-time concurrent constraint programming[END_REF] present a real-time extension of ccp with application to music interaction.

In the realm of rewriting logic, Degano et al. [START_REF] Degano | A causal semantics for CCS via rewriting logic[END_REF] provide a rewriting logic semantics for Milner's CCS with interleaving behavior. Additionally, a set of axioms is defined for a logical characterization of the concurrency of CCS processes. In [START_REF] Verdejo | Two Case Studies of Semantics Execution in Maude: CCS and LOTOS[END_REF], the authors use rewriting logic to represent the semantics of CCS and a modal logic for describing local capabilities of CCS processes. In particular, they study how to make executable the SOS semantics of CSS and present a fully executable specification of the semantics. More recently, M. Romero and C. Rocha [START_REF] Romero | Reachability analysis for spatial concurrent constraint systems with extrusion[END_REF] have proposed a symbolic rewriting logic semantics of the spatial modality of ccp with extrusion.

This paper has presented a real-time rewriting logic semantics for spatial concurrent constraint programming (rtsccp) that is fully executable in the Maude system. The intended models of rtsccp are spatially-distributed multi-agent reactive systems that may have different computing capabilities and be subject to real-time requirements. In this setting, time attributes are associated to process-store interaction, as well as to process mobility in the space structure, by means of maps from agents to non-negative real quantities. Details about the underlying constraint system have been given as materialized with the help of rewriting modulo SMT. Furthermore, examples of reachability analysis performed on this semantics have been given to illustrate certain aspects of the timing behavior of agents distributed across hierarchical spaces, such as fault-tolerance and consistency.

Future work can span in two directions. One interesting direction to follow is to pursue challenging case studies in which, with the help of the real-time rewriting logic semantics for rtsccp presented in this work, other key aspects of spatially distributed concurrent processes such as privacy can be analyzed. The other direction, is to pursue a more general and fully symbolic rewriting logic semantics for rtsccp where time information can also be modeled as shared variables under the control of the SMT decision procedures. In such a setting, interesting properties of real-time systems such as missed deadlines and deadlocks could be fully analyzed, e.g., for infinitely many initial states in a system.

5 (

 5 a) Initial state of the system.

1 (

 1 b) Final state of the system.

Fig. 1 :

 1 Fig. 1: A containerization example.

Fig. 2 :

 2 Fig. 2: Module hierarchy of rtsccp

  op gtime :_ : Time -> Attribute [ctor] . op pqueue :_ : Heap{Tuple} -> Attribute [ctor] . op pend :_ : Heap{Tuple} -> Attribute [ctor] . op nextID :_ : Nat -> Attribute [ctor] . op flg :_ : Bool -> Attribute [ctor] . op MAsk :_ : Map{Aid, Ttime} -> Attribute [ctor] . op MTell :_ : Map{Aid, Ttime} -> Attribute [ctor] . op MSP :_ : Map{Aid, Ttime} -> Attribute [ctor] . op MExt :_ : Map{Aid, Ttime} -> Attribute [ctor] .

<

  root : agent | store : (W:Integer === (9).Integer) > < 0 . root : agent | store : (X:Integer >= 11) > < 0 . 1 . root : agent | store : (Y:Integer > 5) > < 1 . root : agent | store : true > < 2 . root : agent | store : true >

  . op ask_->_ : Boolean SCCPCmd -> SCCPCmd . op _||_ : SCCPCmd SCCPCmd -> SCCPCmd [assoc comm gather (e E)

  op fTime : Map{Aid, Ttime} Aid -> Time . eq fTime(M:Map{Aid, Ttime}, L1) = if $hasMapping(M, L1) then M[L1] else 0 fi . op getTimeCmd : Attribute Attribute Attribute Attribute SCCPCmd Aid -> Time . eq getTimeCmd(MTell: MT, MAsk: MA, MIn: MI, MOut: MO, tell(B1), L1) = fTime(MT, L1) . eq getTimeCmd(MTell: MT, MAsk: MA, MIn: MI, MOut: MO, C1 in I1, L1) = fTime(MI, L1) . eq getTimeCmd(MTell: MT, MAsk: MA, MIn: MI, MOut: MO, C1 out I1, L1) = fTime(MO, L1) . eq getTimeCmd(MTell: MT, MAsk: MA, MIn: MI, MOut: MO, C1, L1) = 0 [owise] .

  crl [tick] :{ X < I : Simulation | pqueue : P, gtime : T, flg : true, pend : P0, Atts > } => { X < I : Simulation | pqueue : merge(delta(deleteMin(P),T0),P0), gtime : (T plus T0), flg : false, pend : empty, Atts > } if T0 := p2(findMin(P)) .

Fig. 3 :

 3 Fig. 3: Possible transitions for ask commands.

  20, (0 . root) |-> 1/10, (0 . 1 . root) |-> 3/20, (1 . root) |-> 1/10, (2 . root) |-> 1/10 MTell : root |-> 1/10, (0 . root) |-> 3/20, (0 . 1 . root) |-> 1/5, (1 . root) |-> 3/20, (2 . root) |-> 3/20 MSP : root |-> 1/2, (0 . root) |-> 7/10, (0 . 1 . root) |-> 4/5, (1 . root) |-> 13/20, (2 . root) |-> 3/5 MExt : root |-> 1/2, (0 . root) |-> 13/20, (0 . 1 . root) |-> 1, (1 . root) |-> 1/2, (2 . root) |-> 3/5

  and the following search command with a the time interval [0, 1.5): search in SCCP : { < root : agent | store : (W:Integer === (9).Integer) > < 0 . root : agent | store : (X:Integer >= 11) > < 0 . 1 . root : agent | store : (Y:Integer > 5) > < 1 . root: agent | store: true > < 2 . root: agent | store: true > < root : process | UID : 1, command : ((((ask X:Integer > 2 -> (tell(Y:Integer < 10) in 0 in 1 out 0)) in 0) || ((tell(Z:Integer =/== (10).Integer) || (tell(T:Integer === 1) in 3)) in 2)) || (tell(X:Integer <= 10) in 0)) > < 1 : Simulation | gtime : 0,pqueue : T(1,((1,0)),empty,empty), pend : empty,nextID : 19, flg : false, ... (time maps)} =>* { < 1 : Simulation | gtime : T:Time, Atts:AttributeSet > < A:Aid : agent | store : B0:Boolean > C:Configuration } such that check-unsat(B0:Boolean) and T < 3/2 .

Solution 1 (

 1 state 159) states: 160 rewrites: 16666 in 876ms cpu (875ms real) (19025 rewrites/second) C:Configuration --> < root:agent | store:(W:Integer === (9).Integer) > < 0 . 1 . root : agent | store : (Y:Integer > 5) > < 1 . root:agent | store:true > < 2 . root:agent | store:true > < 2 . root:process | UID: 28, command: tell(Z:Integer =/== (10).Integer) > < 2 . root:process | UID: 27, command: (tell(T:Integer === (1).Integer) in 3) > < 0 . root:process | UID : 24, command: (tell(Y:Integer < 10) in 0 in 1 out 0) > A:Aid --> 0 . root B0 --> X:Integer >= (11).Integer and X:Integer <= (10).Integer Atts --> pqueue : T(2,(25,1/20),T(1,(24,4/5),empty,empty),T(1,(28,1/10), T(1,(27,3/5),empty,empty),empty)),pend : empty,nextID : 29, flg : true, ... (time maps) T --> 1/2

  and the following search command: search in SCCP : { < root : agent | store : (W:Integer === (9).Integer) > < 0 . root : agent | store : (X:Integer >= 11) > < 0 . 1 . root : agent | store : (Y:Integer > 5) > < 1 . root : agent | store : true > < 2 . root : agent | store : true > < root : process | UID : 1, command : (((ask X:Integer > 2 -> (tell(Y:Integer < 10) in 0 in 1 out 0)) in 0) || ((tell(Z:Integer =/== (10).Integer) || (tell(T:Integer === 1) in 3)) in 2)) > < 1 : Simulation | gtime : 0,pqueue : T(1,((1,0)),empty,empty), pend : empty, nextID : 19, flg : false, ... (time maps) > } =>* { < 1 : Simulation | gtime : T:Time, Atts:AttributeSet > < A:Aid : agent | store : B0:Boolean > C:Configuration } such that entails(B0:Boolean, Y:Integer < 15) and T:Time > 0 and T:Time < 2 .

Solution 1 ( 2 Solution 2 (

 122 state 54) states: 55 rewrites: 7466 in 360ms cpu (358ms real) (20738 rewrites/second) C:Configuration --> < root:agent | store:(W:Integer === (9).Integer) > < 0 . root : agent | store : (X:Integer >= 11) > < 1 . root : agent | store : true > < 2 . root : agent | store : (Z:Integer =/== (10).Integer) > < 3 . 2 . root : agent | store : (T:Integer === (1).Integer) > A:Aid --> 0 . 1 . root B0:Boolean --> Y:Integer > (5).Integer and Y:Integer < (10).Integer Atts:AttributeSet --> pqueue : T(1,(29,1/10),empty,empty),pend : empty, nextID : 30,flg : true, ... (time maps) T:Time --> 5/state 55) states: 56 rewrites: 7601 in 368ms cpu (366ms real) (20654 rewrites/second) C:Configuration --> < root:agent | store:(W:Integer === (9).Integer) > < 0 . root : agent | store : (X:Integer >= 11) > < 1 . root : agent | store : true > < 2 . root : agent | store : (Z:Integer =/== (10).Integer) > < 3 . 2 . root : agent | store : (T:Integer === (1).Integer) > A:Aid --> 0 . 1 . root B0:Boolean --> Y:Integer > (5).Integer and Y:Integer < (10).Integer Atts:AttributeSet --> pqueue : empty,pend : empty,nextID : 30, flg : false, ... (time maps) T:Time --> 13/5 ... The webpage at http://escher.javerianacali.edu.co/rtsccpe/index.html contains more details about this example and other examples about reachability analysis with R, including knowledge inference and equivalence of knowledge.

Acknowledgments. The authors would like to thank the anonymous referees for their helpful comments. The first author was partially supported by Colciencias via the project

eq { < L0: process | command: 0, Atts > X } = { none X } . eq < L0: agent | store: B0 > < L0: agent | store: B1 > = < L0: agent | store: (B0 and B1) > .

The following six rules capture the concurrent observable behavior in R: The [tell] rule implements the execution semantics of a tell command by posting the given constraint in the local store and by removing such a command from the configuration. The [parallel] rule implements the semantics for parallel composition of process by spawning the two process in the current space. Rule [space] creates a new agent's space denoted by N0.L0, with an empty store (i.e., true), beside the execution of program C0 within the new agent's space. The [extrusion] rule executes C0 in the parent's space L0. The [ask] rule executes a command C1 when the guard B1 in ask B1 -> C1 holds: that is, when B1 is a semantic consequence of the contents B0 of the local store. Note that the semantic consequence relation of the constraint system is queried by asking the SMT solver. The [delay] rule represents the negative answer for the ask rule: whenever B0 does not entail B1 or B1 is not a semantic consequence of the contents B0 of the store. The [delay] rule moves an ask command into the pending heap, where it will remain until the tick rule executes again.