L Bubier

R J Lu

E Williams

Wim E E Chesler

Crusio

Anna Delprato

Marie-Paule Algéo

Brice Bonheur

Jason A Bubier

Lu Lu

Robert W Williams

Elissa J Chesler

Wim E Crusio

QTL and Systems Genetics Analysis of Mouse Grooming and Behavioral Responses to Novelty in an Open Field

Keywords: Exploratory behavior, BXD recombinant inbred mice, quantitative trait loci, complex traits, systems genetics, Dab1, genotype x sex interaction, epistatic interaction, repetitive behavior, mouse chromosome 4

QTL and systems genetics analysis of mouse grooming and behavioral responses to novelty in an open field

4 at ~104 Mb that modulates grooming duration in both males and females (LRS values of ~18, explaining 25% and 14% of the variance, respectively) and a suggestive QTL modulating locomotion that maps to the same locus. Bioinformatic analysis indicates Disabled 1 (Dab1, a key protein in the reelin signaling pathway) as a particularly strong candidate gene modulating these behaviors. We also found two highly suggestive QTLs for a sex by strain interaction for grooming duration on chromosomes 13 and 17. In addition, we identified a pairwise epistatic interaction between loci on chromosomes 12 at 36-37 Mb and 14 at 34-36 Mb that influences rearing frequency in males.

Introduction

Analyzing behaviors of rodents in a novel open field is a classic test traditionally used to assess emotionality [START_REF] Hall | Emotional behavior in the rat. I. Defecation and urination as measures of individual differences in emotionality[END_REF], Hall, 1936a, Hall, 1936b). The behavior displayed is considered to be the result of two conflicting drives: curiosity and fear [START_REF] Barnett | Activity, exploration, curiosity and fear: An ethological study[END_REF], but is now more often interpreted in terms of exploratory behavior, anxiety, and locomotor activity. Exploratory behavior is elicited by novel stimuli and consists of behavioral acts and postures that permit an animal to collect information about new aspects of the environment [START_REF] Crusio | The genetic architecture of behavioural responses to novelty in mice[END_REF]. Some repetitive behaviors, such as grooming or drug-induced stereotypy, can compete with these basic exploratory behaviors. Thus, behavior in an open field has become an important aspect of the development of models of several human disorders including autism spectrum (ASD) and substance-use disorders, underscoring the importance of identifying gene variants and neural networks regulating these behaviors.

Behavior in an open field was among the first behavioral phenotypes to be analyzed by quantitative genetic methods. Classical Mendelian crosses as well as diallel crosses invariably indicated significant genetic variation among inbred strains [START_REF] Broadhurst | Psychogenetics of emotionality in the rat[END_REF][START_REF] Crusio | Genetic dissection of mouse exploratory behavior[END_REF][START_REF] Crusio | The genetics of exploratory behavior[END_REF][START_REF] Crusio | The genetic architecture of behavioural responses to novelty in mice[END_REF][START_REF] Henderson | Predicting relationships between psychological constructs and genetic characters: an analysis of changing genetic influences on activity in mice[END_REF][START_REF] Willis-Owen | The genetic basis of emotional behaviour in mice[END_REF]. The genetic architecture of the different behaviors consisted of additive-genetic effects as well as ambidirectional dominance, a pattern consistent with the action of multiple polymorphic genes [START_REF] Crusio | The genetics of exploratory behavior[END_REF]. Factor analyses of genetic correlations reveals the presence of three underlying processes: exploration, anxiety, and self-maintaining behavior (grooming frequency and duration; [START_REF] Crusio | Genetic dissection of mouse exploratory behavior[END_REF].

When molecular-genetic techniques became available that made it possible to determine the approximate locus of some of these polygenes (now termed Quantitative Trait Loci -QTLs), open-field behavior again was among the first behavioral phenotypes to be so investigated [START_REF] Flint | A simple genetic basis for a complex psychological trait in laboratory mice[END_REF][START_REF] Gershenfeld | Mapping quantitative trait loci for open-field behavior in mice[END_REF][START_REF] Mathis | Genetic analysis of anxiety-related behaviors and responses to benzodiazepine-related drugs in AXB and BXA recombinant inbred mouse strains[END_REF][START_REF] Mott | Simultaneous detection and fine mapping of quantitative trait loci in mice using heterogeneous stocks[END_REF][START_REF] Talbot | High-resolution mapping of quantitative trait loci in outbred mice[END_REF]. Several loci have been mapped, although few genes have actually been identified [START_REF] Logan | High-precision genetic mapping of behavioral traits in the diversity outbred mouse population[END_REF][START_REF] Williams | An evolutionarily conserved role for the aryl hydrocarbon receptor in the regulation of movement[END_REF][START_REF] Yalcin | Genetic dissection of a behavioral quantitative trait locus shows that Rgs2 modulates anxiety in mice[END_REF]. Most of these studies were carried out using different heterogeneous populations, necessitating the genotyping at many marker loci of every individual mouse studied, whereas the other studies could use only the relatively moderate numbers of recombinant inbred (RI) strains that were then available. RI strains have the advantage that all of them have already been densely genotyped [START_REF] Shifman | A high-resolution single nucleotide polymorphism genetic map of the mouse genome[END_REF][START_REF] Williams | The genetic structure of recombinant inbred mice: high-resolution consensus maps for complex trait analysis[END_REF]. In addition, because RI strains are fully inbred, they can be easily shared among a research community. Phenotypical data obtained by different groups can be fruitfully combined, opening the possibility to test replicability, GXE effects, and to exploit powerful bioinformatic resources to study trait covariance, pleiotropy, and genetic modulation. For example, extensive mRNA expression datasets are available for the BXD RIs for different brain regions, and these can be jointly analyzed along with behavioral, neural, or physiological phenotypes [START_REF] Chesler | Complex trait analysis of gene expression uncovers polygenic and pleiotropic networks that modulate nervous system function[END_REF][START_REF] Mulligan | GeneNetwork: A Toolbox for Systems Genetics[END_REF]. Until relatively recently, however, the number of RI strains was too small to allow mapping with sufficient power and precision. This changed about a decade ago when an extended set of RI strains became available, derived from the C57BL/6J and DBA/2J strains (BXD RIs; [START_REF] Peirce | A new set of BXD recombinant inbred lines from advanced intercross populations in mice[END_REF]. A large number of widely-studied behaviors have been assessed in this expanded set of RI strains [START_REF] Philip | High-throughput behavioral phenotyping in the expanded panel of BXD recombinant inbred strains[END_REF], revealing precise QTLs, though many specific aspects of behavior have yet to be examined. We therefore decided to use this resource for a large-scale systems-genetics study comprising 53 BXD strains aimed at localizing QTLs for behaviors displayed in a novel open field.

Materials and Methods

Subjects

Breeding pairs of 53 BXD strains were acquired from the University of Tennessee Health Center (Memphis, TN, USA) and the Center for Neurogenomics and Cognitive Research (Free University, Amsterdam, Netherlands). Breeding pairs of the parental strains (C57BL/6J and DBA/2J) were obtained from Charles River (L'Arbresle, France). Our aim was to test 10 males and 10 females of each strain in the open field. Because of logistical problems (for example, some strains not breeding well), we did not achieve this goal for 16l strains. Exact information about sample sizes per sex and strain is presented in supplementary worksheet 1. Briefly, at an age of 11 +/-1 weeks, we observed the behavior of 914 mice: 451 females from 52 BXD strains and 463 males from 53 BXD strains, as well as 10 males and 10 females for each one of the parental strains. All animals used were housed and bred in the SPF mouse facility of the University of Bordeaux (Pessac) in a climate-controlled breeding room (temperature: 21+/-1 °C, humidity: 55+/-10%, 12 hour light-dark cycle with lights on at 7 am). Food (Safe, type 113, sterilized) and water (softened, sterilized) were available ad libitum. Animals were housed 2-4 in same sex/ genotype groups in clear plastic cages (162×406×176 mm, Tecniplast) filled with poplar wood shavings (Souralit).

Open field

Procedure-The open field was a rectangular cage (109*49*49 cm) with a clear Plexiglas front panel that was placed in a brightly-lit room (~300 lx). Thirty minutes prior to the test mice were taken to the experimental room. All tests took place between 8:30 am and 5:30 pm. Animals were placed in the center and, starting 5 sec later, observed directly and continuously for 20 min. Overall locomotor activity and time spent in the thigmotaxis zone (surface within 5 cm of the walls) were recorded automatically using the Ethovision program (version 3, over the course of the project successively upgraded to version 8; Noldus, Wageningen, The Netherlands). Rearing (standing upright on the hind legs, while forepaws are free); leaning (standing upright on the hind legs, one or two forepaws against the wall), grooming frequency (number of grooming episodes), grooming duration, and jumping, were scored manually in Ethovision by using a computer keyboard with several keys coding for the different behaviors. Grooming included all self-cleaning activities such as tail and ventrum licking, facial wiping, etc. Defecation was quantified by counting the number of fecal boli deposited during a session. The floor was wiped with water in between sessions, but not rinsed.

On any test day, animals of a given sex were tested as one batch first, followed by the batch of the other sex. The order of testing for males and females was alternated between test days.

Statistical analysis

All statistical analyses were performed with SAS 9.3 (Sas Institute Inc., 1987). Data are reported as means +/-SEM. To determine strain and sex effects, we used two-way ANOVAs with strain and sex as between-subjects factors. As a measure of the strength of strain differences, we estimated the intraclass correlation as follows. The between-strain and within-strain variance components were derived from the expected mean squares from oneway ANOVAs that were run separately for each sex using the SAS procedure GLM. The estimate of the intraclass correlation then equals r i = V between strains /(V between strains + V within strains ). The percentage of the variance between means explained by a certain QTL was calculated as the square of the correlation between the strain means and the genetic marker with the highest LRS score (note that to some extent this will be an overestimation, due to the 'Beavis effect'; see [START_REF] Xu | Theoretical basis of the Beavis effect[END_REF].

Bioinformatics-All of the genetic analyses were done in GeneNetwork (www.genenetwork.org; Parker et al., 2017) which is an open access bioinformatics resource for systems genetics that exists as both a repository for genetic, genomic, and phenotypic data together with a suite of statistical programs for data analysis that includes mapping and evaluating QTLs, examining phenotype/genotype correlations, and building interaction networks [START_REF] Mulligan | GeneNetwork: A Toolbox for Systems Genetics[END_REF]. Genes located within significant QTL intervals were screened with the QTLminer component of GeneNetwork [START_REF] Alberts | QTLminer: identifying genes regulating quantitative traits[END_REF], the Mouse Genome Informatics/Strains, SNPs and Polymorphisms database (www.informatics.jax.org/strains_SNPs.shtml), and the NCBI dbSNP database (www.ncbi.nlm.nih.gov/snp). Functional associations and gene ontology for candidate genes were further assessed using Gene (www.ncbi.nlm.nih.gov/gene), DAVID (version 6.8; david.abcc.ncifcrf.gov), WebGestalt (bioinfo.vanderbilt.edu/webgestalt/), and literature mining using PubMed (www.ncbi.nlm.nih.gov/pubmed). In order to prioritize positional candidates within the QTL intervals, the GeneWeaver database (www.geneweaver.org) was queried to find gene sets from previous studies from other labs associated with grooming and rearing. The gene sets were further analyzed using tools available at the GeneWeaver database to identify genes that are highly represented among these gene sets. The results of the GeneWeaver analysis were exported to the STRING database of known and predicted protein-protein interactions (string-db.org/) to identify the relationship of positional candidates to previously reported grooming related genes.

QTL mapping-The QTL mapping module of GeneNetwork was used to identify QTLs for the open-field data. This module enables interval mapping, composite interval mapping, and a pairwise scan option to identify epistatic effects. QTL significance was assessed using the likelihood ratio statistic (LRS) obtained after 5000 permutations and 2000 bootstrap tests. QTLs were deemed significant if P genome-wide <0.05 and suggestive if P genome-wide <0.63, which yields, on average, one false positive per genome scan. Male and female data were analyzed separately. In cases where interactions between strain and sex were present (i.e., sex differences vary over strains), we mapped the QTLs responsible for these interactions by analyzing the sex differences (measured as the difference between the male and female strain means). In all cases, outliers were winsorized.

The presence of cis-eQTLs for the genes within the QTL intervals was determined by correlating phenotype data with brain tissue-specific gene expression levels. For this, we used publicly-available data obtained from GeneNetwork, using the following datasets:

Results

Raw data

All data have been deposited in the GeneNetwork database and are publicly available. GeneNetwork trait IDs (male ID followed by female ID): Locomotion: 17528, 17538; Distance covered near the wall as a percentage of total distance covered: 17529, 17539, Percentage time spent near the wall: 17530, 17540; Leaning against the wall: 17531, 17541; Rearing: 17532, 17542; Grooming, bout frequency: 17533, 17543;Grooming, duration: 17534, 17544;Jumping: 17536, 17546;and Defecation: 17537, 17547. Strain means for the different variables are presented in Figure 1 and Supplementary Figure 5.

Strain and sex differences

For all variables measured, strain differences were significant. Differences between sexes were observed for percent distance in the thigmotaxis zone, rearing, grooming frequency, grooming duration, and defecation. Significant sex*strain interactions were observed for locomotion, rearing, grooming duration, and defecation (locomotion: strain: F

Mapping of significant QTLs

The clear strain differences indicate the presence of significant genetic variation, offering the opportunity to attempt to localize QTLs for these phenotypes. A significant QTL peak was obtained for grooming duration on Chr 4 in both males and females (LRS=17.9 and LRS=18.1, respectively; Figure 2A andB) accounting for 25% of the variance in males and 14% in females (Figure 1). Although B6 animals show smaller grooming durations than D2 animals, strains that inherited the B haplotype at this locus had increased grooming time. The QTL region, including peaks and shoulders, for males spanned 97-112 Mb and the highest LRS value occurred at 104 Mb, whereas for females the interval on Chr 4 ranged from 101-125 Mb and peaked at ~103 Mb. Composite interval mapping controlling for this QTL in males and females did not reveal any secondary QTLs.

QTL Region Analysis

GeneNetwork was queried for the set of all traits with LRS > 10 occurring within the same Chr 4 interval. From this search, the trait that mapped most closely to the region of our QTL interval was locomotor activity (horizontal distance travelled) that had also been measured in an open field [START_REF] Philip | High-throughput behavioral phenotyping in the expanded panel of BXD recombinant inbred strains[END_REF]. The QTL interval for locomotion in that study mapped within the interval that we observed here for grooming duration (Figure 2C) and the two traits correlated inversely (Figure 2D).

The QTL interval was reviewed to define potential candidate genes based on currently known functional associations with phenotypes related to grooming and locomotor activity (Supplementary worksheet 2). Within the 97-112 Mb region of Chr 4 there are 146 genes and predicted genes (GeneNetwork database queried September 28, 2016 and MGI database queried September 29, 2016). Of the genes for which functional information is currently available, only Dab1 is associated with both locomotor and grooming phenotypes. Elavl4 and Lepr are connected with locomotor behavior. Lrp8, the reelin receptor, also occurs within the QTL interval but it is not known to be associated with either phenotype.

Dab1 is a key component of the reelin signaling pathway, involved in neuronal migration and positioning during development. In the QTL interval, Dab1 occurs around the QTL peak at ~103.839/104.166 Mb (SNP: rs32341666 single nucleotide variant G/T, with B6 having a G and D2 a T).

Genes within the QTL interval were also examined for the presence of cis-eQTLs in brainrelated expression datasets specific for the BXDs (Supplementary worksheet 3). If such cis-eQTLs exist for any of the genes implicated in the previous section, this would increase the likelihood that these genes modulate the trait, since cis regulation indicates a difference in gene expression levels. For Dab1, there were no cis-eQTLs in either amygdala, cerebellum, hippocampus, hypothalamus, pre-frontal cortex, striatum, or whole brain. There were however Dab1 cis-eQTLs in two datasets of exon level gene expression in the hippocampus and striatum. There were 4 other genes within the Chr. 4 interval associated with behavior (Elavl4, Sgip1, Insl5, Lepr), 2 of which (Elavl4, Sgip1) also showed cis-eQTLs in the hippocampus and striatum exon-level datasets.

Grooming positional candidate evaluation

To further evaluate the genes within the QTL interval for an association with grooming, fifteen gene sets from previous experiments related to grooming behavior were retrieved from the GeneWeaver database: (GS181097: Mouse GO:2000821 regulation of grooming behavior, GS185500: Mouse GO:0007625 grooming behavior, GS197887: Human GO: 2000821 regulation of grooming behavior, GS202324: Human GO:0007625 grooming behavior, GS238017: Human [MeSH] Grooming, : D006120 GS224143: Rat Anxiety related response QTL 19 [Anxrr19 Published QTL Chr 10], GS224144: Rat Anxiety related response QTL 18 [Anxrr18 Published QTL Chr 2], GS224524: Rat Anxiety related response QTL 20 [Anxrr20 Published QTL Chr 18], GS163449: Mouse MP:0001441 increased grooming behavior, GS166329: Mouse MP:0009327 abnormal maternal grooming, GS166392: Mouse MP:0001440 abnormal grooming behavior, GS168966: Mouse MP: 0001442 decreased grooming behavior, GS168968: Mouse MP:0001443 poor grooming). Two gene sets were omitted from further analysis (GS136298: Maternal Performance and GS236065: [MeSH] Hair Preparations) due to lack of relevance.

In the first analysis, the union of these gene sets was obtained and intersected with the complete list of positional candidates to determine whether any of the positional candidates had already been associated with grooming. No elements were found in this intersection. Therefore we tested the putative relationship with Dab1 to grooming-related genes as follows: Among the 13 gene sets above, highly represented genes were found using the "Gene Set Graph" tool in GeneWeaver (Figure 3A). The 9 most highly connected genes were Dlg4, Hoxb8, Shank3, Nrxn1, Hprt, Ppt1, Ddo, Nmur2, Oxt, of which Dlg4, Hoxb8, Shank3, Hprt, and Ppt1 are associated with both grooming and locomotor activity. The set of 9 genes was entered into EMBL STRING along with the candidate Dab1. The network was expanded by one step and a limit of 5 interactors to search for the existence of a short path between the candidate and known (experimentally-derived) grooming genes. Dlg4, the most frequently annotated gene in GeneWeaver grooming data is one step from Dab1 through the Grin1, Grin2a, and Grin2b NMDA subunit genes (Figure 3B). Therefore it is a plausible functional candidate for grooming behavior based on molecular proximity to known genes. The relationship is through known interactions among homologues of Dab1 and NMDA subunits in other species, and co-occurrence in PubMed abstracts.

Mapping of suggestive QTLs

For females, suggestive QTLs were found for locomotion (Chr 4 and Chr 9), percent thigmotaxis time (Chr 4 and Chr 15), percent thigmotaxis distance (Chr 3, Chr 4, Chr 5, Chr 8, Chr 11, and Chr 13), grooming frequency (Chr 4, Chr 18), grooming duration (Chr 18), leaning (Chr 5), and defecation (Chr 4) (Supplementary Figure 6). For males suggestive QTLs were found for locomotion (Chr 4), percent thigmotaxis time (Chr 4), percent thigmotaxis distance (Chr 3), grooming frequency (Chr 3 and Chr 18), leaning (Chr 4 and Chr 13), defecation (Chr 9) and rearing (Chr 15) (Supplementary Figure 6).

Suggestive QTL peaks common between males and females occurred for locomotion (Chr 4), grooming frequency (Chr 18), percent thigmotaxis time (Chr 4), and percent thigmotaxis distance (Chr 3; Supplementary Figure 6).

Mapping of QTLs for sex by strain interactions

For grooming duration, defecation, locomotion, and rearing we found significant sex by strain interactions, warranting a search for the underlying genes. QTL mapping of these data revealed highly suggestive QTLs for grooming duration on Chr 13 (LRS=15.54, 30-46 Mb) and Chr 17 (LRS=15.54, 72-78 Mb) and some barely suggestive QTLs for the other traits (defecation, locomotion, and rearing; Supplementary Figure 8). The genes within the Chr 13 and Chr 17 intervals were examined for association with sex related characteristics. The Chr 13 interval contained 102 genes. Two of these, Bmp6 (Bone morphogenetic protein 6) and Mak (male germ cell association kinase) are linked with male genitalia development and male germ cell processes, respectively. The Chr 17 interval was narrow and only contained 20 genes, one of them, Srd5a2 (steroid alpha-reductase 2) is associated with sexual differentiation and male and female genitalia development. Bmp6, Mak, and Srd5a2 each differ at 2 or 3 SNP polymorphisms between the parental strains, C57BL/6J and DBA/2J.

Multivariate analyses

A factor analysis of open-field traits was performed on strain means for the BXD and parental lines (Table 1) in order to clarify the multivariate structure of the data. For both the male and female data sets two factors had an eigenvalue >1, which were subjected to an orthoblique Harris-Kaiser rotation (Table 1). Results for males and females were very similar. Factor 1 shows positive loadings of both thigmotaxis-related variables and may represent anxiety. Factor 2 had sizeable loadings for grooming (duration and frequency) and defecation, as well as for jumping, leaning, and locomotion, but with an opposite sign. This factor may represent exploration. No strong loadings were found for rearing in females, but it loaded on Factor 1 in males, with a sign opposite to thigmotactic tendencies.

QTL mapping of combined open-field traits

The factor scores were used to detect possible additional QTLs (Supplementary Figure 7). No significant QTLs were obtained for any of the male or female factors. For females, suggestive QTLs occurred on Chr 3 and Chr 4 for Factor 1 and Chr 4 for Factor 2. For males, a suggestive QTL was obtained on Chr 4 for Factor 2. The Factor 2 QTLs occurred around the same Mb location observed for the significant QTLs produced for grooming duration (and locomotion) on Chr 4.

Epistatic QTLs

Epistatic effects are non-additive interactions between multiple QTL loci where the combined effect is either greater or smaller than the summed effects of each gene alone [START_REF] Mather | Biometrical Genetics The Study of Continuous Variation[END_REF]. A significant pairwise-epistatic interaction was found for rearing in males between loci on Chr 12 and Chr 14 (LRS 1=1.557, LRS 2=0.878, LRS interact =33.754, LRS full=35.908, p<0.05, Figure 4).

A QTL miner search using GeneNetwork at the 36. 628917 -37.595106 Mb location (MGI: 35.925620-36.689444 Mb) on Chr 12 rendered 9 or 10 genes within the interval: Agr3: anterior gradient 3, Agr2: anterior gradient 2, Tspan13: tetraspanin 13, Bzw2: basic leucine zipper and W2 domains 2, Ankmy2: ankyrin repeat and MYND (myeloid, Nervy, and DEAF-1)-type Zn 2+ finger domain containing 2, Lrrc72: leucine rich repeat containing 72, Sostdc1: sclerostin domain containing 1 Ispd: isoprenoid synthase domain containing, and two unclassified genes, D630036H23Rik and 1700101O05Rik. Additional support for the Chr 12 locus influencing rearing was found in a query of the GeneWeaver database where a gene list and interval associated with a Chr 12 QTL for rearing overlaps with the interval identified in this study (Gene Set-GS136584/PMID:14694905).

For the Chr 14 interval, a QTL miner search using GeneNetwork at the 34.026418 -34.609256 Mb location (MGI: 33214026-33640754) resulted in 5 or 6 genes: Arhgap22: Rho GTPase activating protein 22, Mapk8: mitogen-activated protein kinase 8, Ptpn20: protein tyrosine phosphatase, non-receptor type 20, Frmpd2: FERM and PDZ domain containing 2, as well as two unclassified genes, 6030458A19Rik and A930006J02Rik. In an attempt to find a possible functional association among genes from both chromosomal regions, a PubMed literature search for co-occurrence and ontology resulted in 2 genes with known neurological associations, Mapk8 and Bzw2. Another potential link could be Arhgap22 and Agr2 which are implicated in cell growth and cell proliferation. Only a weak epistatic interaction was observed for rearing in females (LRS1= 0.037, LRS2 =4.242, LRS interact = 25.024, LRS full = 29.557, ns).

Discussion

In this study we aimed to identify genetic components of open-field behaviors using the extended BXD RI strains. We found one significant QTL: for grooming duration on Chr 4 for both males and females. Inspection of data from other laboratories available in the GeneNetwork database showed that a significant QTL was previously found in this interval for locomotion-related traits in an open-field. Of the candidate genes in this region, only Dab1 has been associated with both grooming and locomotion. Dab1 scm (scrambler) is a spontaneous mutant that produces a truncated form of the Dab1 protein due to aberrant splicing. Scrambler mice have neuroanatomical abnormalities including degeneration of the cerebellum, hippocampus, and neocortex, with concomitant behavioral problems including abnormal motor coordination, balance, and gait [START_REF] Sweet | Scrambler, a new neurological mutation of the mouse with abnormalities of neuronal migration[END_REF]. Similar problems are not found in conditional Dab1 KO mice, which have normal Dab1 expression during development, although Dab1 deficiency in adulthood induces hyperactivity and decreased anxiety [START_REF] Imai | Dorsal Forebrain-Specific Deficiency of Reelin-Dab1 Signal Causes Behavioral Abnormalities Related to Psychiatric Disorders[END_REF]. In addition, it has been observed that scrambler mice have reduced grooming durations compared to wild-type controls [START_REF] Strazielle | Abnormal grooming activity in Dab1(scm) (scrambler) mutant mice[END_REF]. The Dab1 gene undergoes complex splicing which generates many functional isoforms with different combinations of signaling domains [START_REF] Gao | Reelin-Disabled-1 signaling in neuronal migration: splicing takes the stage[END_REF]. The Dab1 variant 2 relevant to this study, results from an alternative splicing event that excludes exon 7 which is predicted to delete 13 amino acids containing a consensus tyrosine phosphorylation site.

Based on the foregoing, we propose Dab1 as a likely gene candidate for the Chr 4 QTL. The significance of this finding is further supported by GWAS in human that have implicated DAB1 in ASD [START_REF] Shen | Association and gene-gene interactions study of reelin signaling pathway related genes with autism in the Han Chinese population[END_REF], as well as with plasma amyloid beta concentrations, one of the pathological hallmarks of Alzheimer disease [START_REF] Chouraki | A genome-wide association meta-analysis of plasma Ab peptides concentrations in the elderly[END_REF]. A possible pathway for the involvement of Dab1 in grooming is the association of Dab1 with NMDA receptors [START_REF] Chen | Reelin modulates NMDA receptor activity in cortical neurons[END_REF] through its connection with Grin subunits (cf. Figure 3b). In turn, NMDA receptors have been implicated in the regulation of grooming and other repetitive behaviors [START_REF] Jaramillo | Altered Striatal Synaptic Function and Abnormal Behaviour in Shank3 Exon4-9 Deletion Mouse Model of Autism[END_REF][START_REF] Kim | Neuronal mechanisms and circuits underlying repetitive behaviors in mouse models of autism spectrum disorder[END_REF].

The implication of Dab1 and the reelin pathway in this QTL for grooming duration is intriguing because this pathway is known to be associated with human autism. Grooming has been the subject of detailed genetic analyses due to its association with neuropsychiatric disorders such as Alzheimer's disease, Parkinson's disease, Huntington's disease, Obsessive Compulsive Disorder, Attention Deficit Disorder, and ASD [START_REF] Kalueff | Neurobiology of rodent self-grooming and its value for translational neuroscience[END_REF][START_REF] Roth | Potential translational targets revealed by linking mouse grooming behavioral phenotypes to gene expression using public databases[END_REF]. Self-grooming represents more than 15% of the waking activity in mice and after sleep is their main activity. Mice also groom in stressful situations such as novelty-based anxiety tests (e.g., elevated plus maze and open field), holeboard-decision making, etc. (reviewed in [START_REF] Kalueff | Neurobiology of rodent self-grooming and its value for translational neuroscience[END_REF][START_REF] Roth | Potential translational targets revealed by linking mouse grooming behavioral phenotypes to gene expression using public databases[END_REF]. Our findings provide evidence for a construct validity of grooming as an ASD-related behavior in the mouse and further highlight the utility of the BXD RI population, systems genetic analysis, and our candidate gene prioritization and network augmentation approach as a means of studying this disorder.

Further findings from this study remain to be resolved. We also found two highly suggestive QTLs for the sex by strain interaction for grooming duration on Chr 13 and 17. Several strong candidates within these QTL intervals are Bmp6, Mak, and Srd5a2, associated with sexual differentiation, male germ cell processes, and male/female genitalia development.

There was a significant epistatic interaction between loci on Chr 12 and 14 for rearing behavior. Although these loci contain only few genes, no strong candidates could be identified. Two possible interacting gene pairs are Mapk8/Bzw2 and Arhgap22/Agr2. The Ahr gene in the Chr 12 interval has been reported as modulating both horizontal and vertical movement [START_REF] Williams | An evolutionarily conserved role for the aryl hydrocarbon receptor in the regulation of movement[END_REF]. However, the effect found in that study was a main effect of the QTL whereas the effect that we report here is only significant in interaction with the Chr 14 locus and specific for rearing. No effect of the Chr 12 locus was found on locomotor activity in our study.

Only suggestive QTLs were obtained for several of the other traits, despite the large number of strains used in this study. Most likely, this indicates a polygenic inheritance with each gene having only a very small contribution to the trait, confirming the results of previous research [START_REF] Crusio | Genetic dissection of mouse exploratory behavior[END_REF][START_REF] Crusio | The genetics of exploratory behavior[END_REF][START_REF] Crusio | The genetic architecture of behavioural responses to novelty in mice[END_REF].

In summary, the present study demonstrates the power of the BXD RIS approach for QTL mapping and analysis of behavioral traits that share both face and biomolecular construct validity with human behavioral disorders. The wealth of information available on these strains, ranging from a plethora of behavioral, neuronal, and physiological phenotypes to extensive gene-expression data from different brain regions, allows a powerful system genetics approach [START_REF] Schughart | Systems Genetics. Methods and Protocols[END_REF] to identify the genetic and molecular

  amygdala [INIA Amygdala cohort Affy MoGene 1.0 ST Mar 2011 RMA], cerebellum [SJUT Cerebellum mRNA M430 (March 2005) RMA] hippocampus [Hippocampus Consortium M430v2 (June 2006) RMA], hypothalamus [INIA Hypothalamus Affy MoGene 1.0 ST (Nov10) RMA], pre-frontal cortex [VCU BXD PFC SAL M430 2.0 (Dec 2006) RMA], striatum [HBP Rosen Striatum M430V2 (April 2005) RMA], whole brain [INIA Brain mRNA 430 (June 2006) RMA, hippocampus exon [UMUTaffy Hippocampus exon (Feb 2009) RMA], striatum exon [HQF Striatum Affy Mouse Exon 1.0ST (Dec 2009) RMA]. All these datasets were based on pooled samples of males and females (except the one for the pre-frontal cortex which used males only).

Figure 1 .

 1 Figure 1. BXD strain differences for grooming duration in the open field Variation in grooming duration time in the BXD mice and parental strains. C57BL/6J orange bars and DBA/2J green bars; Males (A) and females (B). Bar graphs represent the means (+/-SEM). Intra-class correlations and overall means are shown. The y-axis represents grooming duration time in seconds with BXD strains rank ordered from low to high. For enlarged versions indicating strain names, see Figures S1 A and B.

Figure 2 .

 2 Figure 2. Whole genome scans of trait data for grooming duration and locomotion A-C: Whole genome scans for grooming duration (A: males, B: females) and locomotion (C: females). The x-axis represents chromosome number and megabase position (tick marks represent 25 Mb increments, each chromosome starting at 0) and the y-axis represents the likelihood ratio statistic (LRS) of linkage. Blue lines represent LRS across the genome. The pink and gray horizontal lines are approximate threshold values which are used to assess whether a peak is significant (P<0.05) or suggestive (P<0.63) respectively. D: Spearman rank correlation plot for grooming duration and locomotion. For enlarged versions, see Figures S2 A-D.

Figure 3 .

 3 Figure 3. Relationship of Dab1 with grooming-related genes GeneSet graph intersection (A). Analysis of thirteen gene sets associated with grooming behavior revealed 9 highly represented genes (Dlg4, Hoxb8, Shank3, Nrxn1, Hprt, Ppt1, Ddo, Nmur2, Oxt). STRING molecular interaction network (B) illustrating the shortest path of the highly represented genes associated with grooming behavior and Dab1. Known interactions indicated by pink edges are experimentally confirmed, and those indicated by blue edges are from curated databases. Relationships indicated by green edges are from text mining, and black edges from gene co-expression analysis. For enlarged versions, see Figures S3 A and B.

Figure 4 .

 4 Figure 4. Two-dimensional plot of LRS scores for pairs of loci involved in epistatic interaction for rearing in males The lower right half of the plot gives a summary of LRS values of the full model, representing cumulative effects of both loci and their possible interaction. The upper left side of the plot indicates the LRS values for the presence of epistatic interactions. Both sets of x and y axes are labeled with chromosome number. Zoomed thumbnail indicates Chr12/ Chr14 interaction colored in red. For an enlarged version, see Figure S4.

  P<0.001; jumping: strain: F 53,804 =3.47, P<0.001; sex: F 1,804 =0.59, ns; strain * sex: F 53,804 =0.55, ns; defecation: strain: F 53,804 =8.06, P<0.001; sex: F 1,804 =36.04, P<0.001; strain * sex: F 53,804 =1.72, P<0.01).

	53,804 =17.60,
	P<0.001; sex: F 1,804 =0.31, ns; strain * sex: F 53,804 =2.13, P<0.001; percent distance in the
	thigmotaxis zone: strain: F 53,804 =11.76, P<0.001; sex: F 1,804 =4.92, P<0.05; strain * sex:
	F 53,804 =1.10, ns; percent time in the thigmotaxis zone: strain: F 53,804 =9.94, P<0.001; sex:
	F 1,804 =2.97, ns; strain * sex: F 53,804 =1.24, ns; leaning: strain: F 53,804 =8.35, P<0.001; sex:
	F 1,804 =0.04, ns; strain * sex: F 53,804 =0.96, ns; rearing: strain: F 53,804 =8.94, P<0.001; sex:
	F 1,804 =25.68, P<0.001; strain * sex: F 53,804 =1.46, P<0.05; grooming frequency: strain:
	F 53,804 =4.46, P<0.001; sex: F 1,804 =11.55, P<0.001; strain * sex: F 53,804 =1.22, ns; grooming
	duration: strain: F 53,804 =8.55, P<0.001; sex: F 1,804 =20.08, P<0.001; strain * sex:
	F 53,804 =2.88,

Genes Brain Behav. Author manuscript; available in PMC 2018 November 01.

Acknowledgments

This study was supported by the following grants: NIMH R01 MH072920 to WEC, NIAAA U01 AA016662 and U01 AA013499 to RWW, NIAAA R01 AA018776 to EJC, and NIAAA R01 AA021951 to LL. Drs. Sabine Spijker and August B. Smit (Free University of Amsterdam) from the Neuro-BSIK Mouse Phenomics Consortium (BSIK03053) generously provided several BXD strains (for the origin of these strains, see Loos et al., 2014). We thank Raphael Pineau and Laetitia Medan for expert animal care and Alexis Cornuez for help with behavioral testing.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.

Author Manuscript

Delprato et al.

Page 18

Table 1 Factor analysis of open field-related variables


Xavier Cousin

Keywords: Chromatic number, stability number, graph theory Nombre chromatique, nombre de stabilité, théorie de graphes

Graph Cartesian sum: pivot between chromatic number and stability number Summary: We describe still another equivalence between stability number and chromatic number. We transform the concerned graph using a Cartesian sum and based on an algorithm to find the corresponding number.

Introduction

We first remind some notions and naming convention. It allows introducing the "Cartesian sum" of graphs.

Then from any graph, knowing a research algorithm of the chromatic number, we deduce an algorithm to find the number of stability of this graph by building a particular graph based on Cartesian sum of the initial graph with claws. We show the opposite: from a known algorithm of the number of stability research, we deduce an algorithm to find the chromatic number of any graph.

In literature, we find the way from stability number to chromatic number [1] but not necessarily the other way from chromatic number to stability number using the same mechanism. Here, we remind and clarify the both ways.

Reminders and conventions

• G = (X, A) is the graph which has X as set of vertices and A as set of edges.

• |Y| is the cardinal of a set Y.

• We note d G (x) the degree of x vertex on graph G.

• A vertex clique is a set of vertices linked together in pairs by edges. We note  the maximum number of a clique.

• We name  the stability number; it is the maximum set of independent vertices (no edge between these vertices two by two). It's the opposite or the complement of the clique.

• N is the vertices number of G (or |X|) and M the edges number (or |A|).

• χ is the chromatic number of G corresponding to the minimum number of colours to strongly colour the vertices. In the books of Claude Berge [1] [2], this number is noted  (G).

• K q represents the order clique q.

• G is noted as the complement of G.

• We note  the minimum number of cliques that partition all the vertices.

• G1 + G2 denotes the Cartesian sum of two graphs (cf. [1] [2]).

Reminder: Cartesian sum of graph. A Cartesian sum G e of a graph G with another graph R is a graph consisting of the union of k duplications of the graph G, where k is the cardinal of X R and edges of R connecting the vertices of the different duplicates (Gi). We can see graphically a duplication Gi in a horizontal axis and a duplication Rj in the vertical plane.

Here we will mainly use the Cartesian sum G + K q .

Example : Let a graph G with X = {pa, pb, ca, cb} and A= {(pa, ca), (pa, cb), (pb, ca)}, G e = G + K 3 = {(pa1, pa2), (pa1, pa3), (pa2, pa3), (pb1, pb2), (pb1, pb3), (pb2, pb3), (ca1, ca2), (ca1, ca3), (ca2, ca3), (cb1, cb2), (cb1, cb3), (cb2, cb3)}).

Xavier Cousin 2/4

It is the Cartesian sum, G + K 3 which gives graphically:

pa pb ca cb G pa1 pb1 cb1 ca1 G 1 pa2 pb2 cb2 ca2 G 2 pa3 pb3 cb3 ca3 G 3 edges of K 3 linked to cb G e

Resulting graph

The edges set {(cb1, cb2), (cb1, cb3), (cb2, cb3)} will be named K 3 (cb) linked to the vertex cb of G.

Remark: for the chromatic number research χ, we will associate the colour i to the i th duplication.

Equivalence between stability number and chromatic number

We show how to use a supposedly known algorithm to find the chromatic number in another to compute the number of stability. We also explain the opposite, that is to say how to obtain the chromatic number knowing a method to find the number of stability.

a) From stability number to chromatic number

Theorem 4 : - to χ (Reminder [1]) For a simple graph G = (X, A), to search χ in a graph G = (X, A) is to look for the stability number  iteratively in Cartesian sums G e of graphs G e = (G + Kq) on q going from an upper bound (max degree for example) to 1.

In summary, we work iteratively q from upper bound to 1 by searching  (G+Kq). We find χG=i when i is such that  (G+K i-1 )≠|X| and  (G+K i )=|X|.

Proof: This theorem is is an augmented version of Berge's theorem and corollary ( [1]).

We want to find χ in a simple graph and we know a bound (at worst the maximum degree).

We take a graph G e = (G+Kq) with q initialised with an upper bound, q duplications of G and a set of edges such that all duplicated vertices xi (from 1 to q) coming from the same basis vertex x of G then linked two by two (cliques).

To find a maximum stable set equal to in G e indicates if there is or not a q-colouring (colouring to q colours) in G. If  = |X| then there is a q-colouring. The reader will verify for himself that it is impossible to have a  >|X| due to the cliques between vertices of G e coming from the same base vertex. Of all the duplicate vertices from the same base vertex, only one will belong to the stable set. We can thus confuse the doubled vertex xi of G e of the maximum stable set with the reference vertex, x of G. The vertices of this maximal stable set of G e belonging to the same i th duplication of G, will give the i th colour to the vertices of corresponding reference.

Example: Let's take the example of the figure of the reminder. The maximum degree is equal to 2, the upper bound of χ is 2. We thus have the graph resulting from the following figure with the search for a stable maximum set (here {cb1, ca1, pa2 , pb2}) and the induced colouring (ca and cb of the same colour  and pa, pb of another colour ):

pa1 pb1 cb1 ca1 G1 pa2 pb2 cb2 ca2 G2 G e

Search of χ

We go through the construction process of G e with q = upper bound down to q = 1. We know that there exists qstaining with q = upper bound, so  initial = |X|. As soon as current  courant <|X| of G then q-colouring is no longer possible. So the chromatic number χ of G is equal to q + 1 (q is the current value).

b) From chromatic number to stability number

Theorem 5 : -χ to  To search the stability number  in a simple graph G = (X, A) amounts to looking for the number χ iteratively in Cartesian Cartesian graphs, , q going from |X| to 1.

We work iteratively on q of |X| at 1 while searching  (

) and when for one i, χ( ) ≠ |X| andχ (

) = |X| we have  (G)=i.
Proof:

The principle is identical to the previous one. We build a graph G e with the complementary of G named and a clique Kq : G e = + K q . By iterating on q and looking for the  (G e ), we find  ( ).  If q >  ( ), (G e )=|X| because all cliques of have a cardinal lower than the one of clique Kq.  Si q ≤  ( ), (G e )<|X|. Thus we can find the number  ( ) with an algorithm to find  .

In our case, we have an algorithm to find χ, so we reason on the complementary of G e , the χ( ) corresponding to  (G e ).

We obtain the  ( ) because the basis graph is the complementary of G and we have  (G) =  ( ).

Synthesis

Here is a summary of the main results with G=(X,A) and G+K q =  where G i =(X i , A i ) i th copy of G and K q (x) copy of K q linked to a vertex x of X :   to χ ;  i, 1  i  upper_bound (max degree),  (G+K i-1 )≠|X| and  (G+K i )=|X|, and χG=i.

Conclusion

We have presented here how from the chromatic number of any graph find its stability number and vice versa.

These equivalences go through an iterative search on particular graphs consisting of the Cartesian sum of graph and a clique.

In another paper, we will show how to transform the chromatic number research into chromatic index research.



  χ to ;  i, 1  i  |X|, χ ( ) ≠ |X| and χ( ) = |X|, and  (G)=i.


Independent Fausto Galetto
email: fausto.galetto@gmail.com

Collatz-Syracuse Conjecture, proof by Reliability Integral Theory

Keywords: Quality Methods, Numerical Methods, Hailstone Conjecture, SPQR

In a previous paper we showed a proof (probabilistic) by merging the infinite number of states into three SuperStates: the merged process was still a Markov process easily solvable. Now we provide a different probabilistic proof of the Conjecture via the Reliability Integral Theory and the SPQR Principle. We devise a "ideal machine" (Gedanken Experiment) which, as quickly as one wants, makes transitions between the SuperStates and finally ends into the "Collatz Cycle" where it stays forever.

Introduction

The author during 2019 presented several proofs of the conjecture; all but one are probabilistic: the last one [START_REF] Galetto | Collatz Conjecture, new proof by merging into three SuperStates[END_REF] merged the infinite states of the Markov Process (related to the conjecture) into three Superstates. In this paper we deal further with the three Superstates and provide a simple formula giving the probability of entering the "Collatz Cycle". In a previous paper [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF] we provided a probabilistic proof of the Conjecture; later, after we saw the interesting paper [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF]; both the papers tried to prove the Hailstone Conjecture using Markov processes. In another paper [START_REF] Galetto | Syracuse_Collatz Conjecture: Comparison of two Markov approaches towards the proof[END_REF], we compared the two probabilistic methods using the Reliability Integral Theory [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] and the SPQR Principle [START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF]. Any probabilistic method makes "probable" the proof, but it is not really a mathematical proof. To overcome such a drawback later we showed a non-probabilistic proof using Flow Graphs and the SPQR Principle [START_REF] Galetto | Proof of Syracuse_Collatz Conjecture by Flow Graphs[END_REF].

The Collatz problem (also called the 3x+1 mapping, hailstone problem, Syracuse problem, ...), posed by L. Collatz in 1937, states that the system of the two difference equations, involving natural numbers, [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF] given the initial condition y 0 (any integer positive number) arrives after some (n is a number not known in advance) "continued" iterations to the value y n =1. It is considered a very difficult problem to be solved. As done in previous papers, we name "state of the system" the integer positive number generated by (1); so we see that the problem is transformed into the following:

given any initial state y 0 the system makes a certain number n of transitions (n is a number not known in advance) and finally it ends into the state y n =1. We can associate to any state of the system y k an edge e k+1 =(y k , y k+1 ) traversed at time k+1 the index of the row of a matrix P and to state y k+1 the index of the column of the same matrix P; then we can describe the graph by the matrix P with entries 1 related to the arrow of the transition y k y k+1 for any edge e k+1 =(y k , y k+1 ). Then for any state of the system y k there is an infinite dimensional row vector u(k), with all entries u i (k)=0, but one entry u y (k)=1, related to the edge e k+1 =(y k , y k+1 ): it is a unit vector of vector space. The vector u(k) refers to the k-th iteration of a mapping T. The mapping T is provided by an infinite-dimensional matrix P=[a ij ], named transition matrix (with infinite rows and columns); rows and columns are indexed by the natural numbers (states of the system) 1, 2, 3, 4, ..., n, n+1, ...; every a ij entry is 0, except , ........ [START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF] where the indexes i and j are given by ( 1), for the arrows e k+1 =(y k , y k+1 ). The P matrix has a 3 by 3 submatrix (with rows and columns indexed by the numbers 1, 2, 4) such that  when the system is in the state 1, the next transition is to state 4: 1  4  when the system is in the state 2, the next transition is to state 1: 2  1  when the system is in the state 4, the next transition is to state 2: 4  2 All this means that when the system enters one of those 3 states [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF][START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] it never leaves out of them, the system (or the process) circulates in the set 1, 2, 4 forever. It is a "periodic process". The set 1, 2, 4 is the Collatz Cycle.

In the paper [START_REF] Galetto | Collatz Conjecture, new proof by merging into three SuperStates[END_REF] we could arrange the (infinite) matrix P with a SuperState SS 0 made of the 3 states [START_REF] Carletti | Quantifying the degree of average contraction of Collatz orbits[END_REF][START_REF] Galetto | Proof of the Syracuse_Collatz Conjecture[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] (the Collatz Cycle), a SuperState SS 1 made of the infinite EVEN states [6, 8, 10, …], a SuperState SS 2 made of the infinite ODD states [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF][START_REF] Galetto | Proof of Syracuse_Collatz Conjecture by Flow Graphs[END_REF], …] The matrix P was be partitioned into 6 submatrices, written simply as where P 00 , P 11 and P 22 are square matrices.

The submatrix P 00 is orthogonal: its inverse is its transpose = . It is important to notice that P 3 , the 3 rd power of the matrix P, is such that the submatrix is the identity matrix; when the system reaches the set 1, 2, 4 of the states it remains there forever. It follows that = = . The matrices P 00 , P 11 and P 22 are square matrices, while the others are rectangular.

The transition graph of the merged Markov Process

The process is bound to enter the SuperState SS 0 =1, 2, 4 because the rectangular submatrix in the middle left corner has only one 1 entry [the other entries are all 0] and the rectangular submatrix in lower left corner has only 0 entries. The "periodic process" circulating in the set 1, 2, 4 is ruled by the submatrix P 00 . The graph of the transitions is given in figure 2.

In the figure 3 we show the flow graph of the 3 SuperStates SS 0 , SS 1 and SS 2 (of the merged process) and the transitions between them; notice that there are three arrows from SS 1 , one back to SS 2 , one forward to SS 0 and one re-entering into SS 1 (which accounts for the internal transitions within SS 1 ). The merged process is ruled by a matrix P merged as the following where the transition probabilities are shown (we shall see later how to find the probabilities p 10 and p 11 ). IF p 10 >0, the matrix P merged provides the "steady state probability vector" =[1, 0, 0] solution of the relationship =P merged , which states that the process stays forever in the SuperState SS 0 after entering it. After entering SS 0 the probability of being in the states [making the SuperState SS 0 ] 1, 2, 4 (Collatz cycle) is given by the "steady state probability vector" *=[1/3, 1/3, 1/3] solution of the relationship *=*P 00 . Now we consider our Gedanken Experiment [the "ideal machine" which makes transitions between the SuperStates (fig, 3), as quickly as one wants, and finally ends into SS 0 the "Collatz Cycle" where it stays forever].

The transitions are ruled by transition rates  1 (from SS 2 to SS 1 ),  (from SS 1 to SS 2 ),  0 (from

SS0 SS1 SS2

SS 1 to SS 0 ); the transition rates are not probabilities! Let's see how our "ideal machine" works: when it is SS 2 it goes next into SS 1 after a random time ruled by the density transition probability  1 exp(- 1 t) while when it is SS 1 arises a competition between the random time T 0 , ruled by the density transition probability  0 exp(- 0 t) [to go to SS 0 ] and the random time T 2 , ruled by the density transition probability exp(-t) [to go to SS 2 ]; if T 0 <T 2 , then the machine goes to SS 0 (on the contrary it goes back to SS 2 ). According the Reliability Integral Theory [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF] we define the functions R 2 (t) and R 1 (t) as the probabilities that the machine does not enter SS 0 (the Collatz Cycle) at time t, GIVEN that it started in SS 2 or SS 1 , respectively, at time 0 (when the machine begins its operation).

The two integral equations providing the probabilities R 2 (t) and R 1 (t) are

The solution R 2 (t) of the RIT integral equations [START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF][START_REF] Galetto | Reliability and Maintenance, Scientific Methods, Practical Approach[END_REF]

is ........ (3)
where s 1 and s 2 are the absolute values of the negative roots of the characteristic equation

.... (4)

Since -s 1 and -s 2 are negative (whichever are the positive parameters  0 ,  1 and ) R 2 (t)0 as t: the Collatz Cycle (the superstate SS 0 ) is eventually entered as time pass by.

The transition rates for the merged Markov Process

Now we try to define the positive parameters  0 ,  1 and  of the Markov process in relation with the Collatz problem.

The rates refer to the number of transitions between the superstates, per unit time: it is the speed of our ideal machine. Since the speed is at our will we can fix  0 as we want; let's fix  1 =1/ns, 1 transition per nanosecond from SS 2 to SS 1 which mean a mean time 1/ 1 =1 ns. This consequence of this choice is that the mean time 1/( 0 +) for any transition out of the susperstate SS 1 (from SS 1 to SS 2 and from SS 1 to SS 0 ) and plus the mean time between transitions within the susperstate SS 1 must be 1 ns. To be consistent with the matrix P merged the "instantaneous" transition probability 0. The transition rate  0 is related to the transition probability p 10 of the P merged matrix: p 10 = 0 /( 0 +). We derive  0 =p 10 /(1-p 10 ). In the paper [START_REF] Galetto | Collatz Conjecture, new proof by merging into three SuperStates[END_REF] we set p 10 =1/[2(81m)], where m=3y 0 +1 with the initial condition y 0 (any integer positive odd number).

The figure 4 show two cases. The Mean Time To SS 0 (from SS 2 ) are respectively 151108 ns (with y 0 ) and 451108 ns (with y' 0 ) and (y 0 <y' 0 ).

Conclusion

Having applied to an "ideal machine" (Gedanken Experiment) the SPQR («Semper Paratus ad Qualitatem et Rationem») Principle and Reliability Integral Theory, the author thinks that his new probabilistic method is able to provide the proof of the Syracuse_Collatz Conjecture [he did already probabilistically in previous papers]. [START_REF] Galetto | The SPQR («Semper Paratus ad Qualitatem et Rationem[END_REF][START_REF] Galetto | Syracuse_Collatz Conjecture: Comparison of two Markov approaches towards the proof[END_REF][START_REF] Galetto | Proof of Syracuse_Collatz Conjecture by Flow Graphs[END_REF] The "ideal machine" can be as fast as one wants, to go from the initial state y 0 to the superstate SS 0 : it all depends on the transition rate  1 and on the mean time 1/ 1 .

Figure 2 .Figure 3 .

 23 Figure 2. The graph of the transitions within and between the SuperStates SS 0 , SS 1 and SS 2 (only few of the total transitions are shown)

Figure 4 .

 4 Figure 4. The "Reliabilities" from the SuperState SS 2 to SS 0 , for two initial values y 0 and y' 0 (y 0 <y' 0 )

  5 from SS 1 to SS 2 , we must put =0.5 transitions/ns, that is, in mean, it takes 2 transitions to go from SS 1 to SS 2 , because, in mean, 1 transition is spent within the susperstate SS 1 [transition from one state of SS 1 to another state of SS 1 ].


Independent Fausto Galetto
email: fausto.galetto@gmail.com

The Tetralogy of Thermodynamics Thermodynamics made SIMPLE

Following the historical development of thermodynamics ideas we will present firstly the 0 th Principle, secondly the 2 nd Principle, thirdly the 1 st Principle and finally the 3 rd Principle. Then we show that entropy is proportional to a numerical function F(X), where X is the ratio of energies and that the fundamental Boltzmann's constant is the atom of Entropy. We think to have given a "good" insight on Entropy. We condense all ideas in the Thermodynamics Tetrahedron.

Introduction

We try here to provide the basics of Thermodynamics and make them simple so that every "logical" person can understand.

In order to understand deeply the argument, the reader can (but it is not necessary to) refer to the documents "The Mathematics of Thermodynamics, Addendum to Mathematics of Thermodynamics, Linhart ideas on Entropy versus "classical Entropy". Proof of Linhart nonsense, Horstmann Thermodynamics versus Mathematics_FIRST part, Horstmann Errors on Thermodynamics." [see references] The basic ideas on which Thermodynamics is founded are set into 4 Principles (now fixed), from the 0 th to the 3 rd one; each principle defines a fundamental fact of experience.

Here we present immediately all, with their characteristic variables; their explanation will follow later.

The four Principles of Thermodynamics The zero-th Principle: there exists a physical (measurable) variable, named Temperature (T), defining the thermal equilibrium the Entropy at zero absolute temperature is the same for all substances,

S(T=0)=constant

The important variables to be defined are: Temperature, Absolute Temperature, Energy, Entropy

We must add to the previous variables another one: the HEAT.

These Principles were defined after Carnot work on Heat Engines; actually Carnot used the 2 nd and the 0 th ones without giving them any name. The 1 st Principle came later than the 2 nd ; some books do not mention the 0 th one, assuming that it is not necessary. The 3 rd Principle is the last one. This author prefers to use the wording Principles instead of Laws (found in many documents).

Let's start our journey with some known facts that we experience.

Fist known fact

Everyone, either in summer or in winter, feels differently: in summer we feel hot, while in winter we feel cold. We attribute these feelings to a physical quantity named temperature (we use the symbol t, for the time being; later we use T, the absolute temperature).

We are able to distinguish between hot objects and cold objects; we say that hot objects have a higher temperature t H than the temperature t C of the cold objects: t H > t C .

To deal scientifically with any physical quantity we must be able to measure it.

Second known fact

Regarding the temperature, we experience the fact that if we mix a cold quantity of matter (think to a cube of ice…) with a "hot" quantity of water (think to a glass of water…), after some time, we feel the (mixed) matter + water at the same intermediate temperature t, and we say that they are in thermal equilibrium. Thermal equilibrium is an equivalence class: IF the object A is in thermal equilibrium with an object C AND another object B is in thermal equilibrium with the same object C, THEN the objects A and B are in thermal equilibrium with each other. This is named the Zeroth Principle of thermodynamics. This principle allows us to devise an object C [or more objects C] able to "measure" the temperature: it is named "thermometer". A common thermometer in everyday use consists of a mass of liquid-usually mercury or alcohol-that expands into a glass capillary tube when is put into contact with "hot" objects.

Third known fact

If we take in our hand the right end of a little spoon and we put the other end on a flame, almost immediately we feel our hand "burning"… Fourth known fact If we have fever we feel hotter than when we are healthy… All these facts (and many others!) show that there exists a physical variable able to "measure" the hotness and the coldness of any object; IF we put in contact any "hot" with any "cold" objects, after a certain time, the two objects reach the same temperature: they are in thermal equilibrium (they have the same level of hotness and coldness). We must logically admit that "something (even if we do not really know what it is) is flowing from the hot object (at higher temperature t H ) to the cold object (at lower temperature t C )": the "something" flowing has a special name heat. [we know that heat is energy, BUT we pretend not to know, for the time being!) Heating is the way to transfer heat. At the beginning of Thermodynamics, near 1824, S. N. Carnot used the name caloric (for that "something"); you will see later why we say this. Caloric was related to the "puissance of feu" (power of fire), according to Carnot. Fire was the main source of "energy" at that time… The discovery of fire was one of the first great humans innovations, made in prehistory [the period between the first stone tools (about 3.3 million years ago) and the invention of writing systems].

Fire allowed humans to do many activities as producing light and heat, cooking food, … and metal "fabrication", … Up to today many devices use "heat" to make their intended work; they do not use fire anymore; they use energy generated by heat in combustion chambers (e.g. turbines, engines, …)

In our journey we follow history: we deal firstly with the 2 nd Principle before the 1 st Principle!!!

The 2 nd Principle of Thermodynamics

The Newcomen Steam Engine was one of the most interesting pieces of technology developed during the 1700's. About 65 years after this engine invention, J. Watt made an improvement which improved the efficiency of the machine. Several technical improvement have been taken since then, in order to improve the efficiency of heat engines. Anyway before S. L. Carnot, in 1824, nobody gave any important "theoretical" contribution to the performance of heat engines. Carnot did not consider the "mechanical" details of the engines and gave due attention to the "significant ("theoretical") aspects" of heat engines; they were: A. Caloric (Heat) is supplied in, at High temperature T H [Q H is the quantity of Caloric (Heat) delivered to the engine] B. The engine transforms caloric into (mechanical) work C. Caloric (Heat) is delivered out, at Low temperature T C (<T H ) [Q C is the quantity of Caloric (Heat) delivered out of the engine] No other idea was considered important for Carnot. Notice that Carnot assumed that it is impossible to "extract work" from heat at a single temperature. In other words, the process of making work go into heat can take place at a given temperature, one cannot reverse it to get the work back again. Although, at that time, everybody thought that, according to the caloric theory, the quantities of Caloric Q H and Q C would have to be the same, Carnot did not say that Q H was equal to Q C because he did not believe it; that, in spite of the analogy between the motive power 1) of a heat engine and 2) of a water wheel, where two levels of height of a waterfall are needed to get the wheel move [and the quantity of water discharged by the wheel at the bottom level is the same as originally entered at the top level]. He invented an ideal engine, named Carnot Engine, working according the Carnot Cycle, made of two isotherm and two adiabatic lines, as in figure 1 Adiabatic transformation pv k =const Figure 1. Carnot Cycle on the left [an isotherm line is such that the temperature is constant, an adiabatic line is such that there is not heat transfer]

Notice that Carnot could not say how much quantity of heat was Q C , because he did not know the first Principle. Following his ideas Carnot could prove that, for his ideal engine, We can get it deriving from (1) the ratio Q H /Q C =T H /T C ; let a Carnot Engine working between the boiling water temperature and melting ice temperature: IF Q H =1000 calorie (we use this because we follow the "caloric" ideas! 1 calorie=4.186 J) then Q C =732 (approximately) and hence the ratio is T H /T C =1000/732; now, by setting T H -T C =100, we get T H =373 (approximately) and T C =273 (approximately). The true value is T C =273.15 K (Kelvin degrees). Writing (1) as (1 bis) and we set Q/T=S, named entropy (concept and name unknown to Carnot, devised by Clausius later) we see that (1 ter) that is, in a reversible Carnot Cycle is zero and hence the entropy variation S B -S A =S, for any reversible transformation depend only by the points A (initial) and B (final) of the transformation (see figure 1) and not on the way we go from A to B. This very important point means that Entropy is a State Function [it depends only on the state A (initial) and on the state B (final) of the system making the transformation]. Notice the wording of Clausius: from En-ERG-y he set En-TROP-y. At his time energy was measured in "erg"! TROP is related to the Greek word "trope" meaning transformation. For an infinitesimal reversible transformation we set *

where the symbol d*Q means that the quantity of heat d*Q is not an exact differential (it depends on the transformation considered), while dS is an exact differential. Hence the integral, for a cyclic reversible process is

∮ = ∮ * = 0 (3)
For cyclic NONreversible processes we have

∮ * > 0 ( 3 bis)
The second Principle of Thermodynamics is then the following For ANY cyclic processe we have

∮ * ≥ 0 (4)
ℎ Notice that this was not the Carnot Principle because he did not know the concept of entropy: he defined his Principle as "heat cannot be taken in at a certain temperature and converted into work with no other change in the system or the surroundings".

There are many ways to define the 2 nd Principle; you can find them in [START_REF] Galetto | The Mathematics of Thermodynamics[END_REF][START_REF] Galetto | Addendum to Mathematics of Thermodynamics[END_REF].

We presented fist the 2 nd Principle; now we go to the first one.

The 1 st Principle of Thermodynamics

In order to understand the argument, it would be useful to refer to the document "The Mathematics of Thermodynamics", and to revise the known facts given above and there about the temperature, the thermal equilibrium, the transformation of energy (mechanical, kinetic, gravitational, electromagnetic, nuclear, …, heat) and the state variables. Experiments of B. Thompson and J. P. Joule showed that one can generate as much heat as wanted by friction. Joule's experiments proved, with an experimental error <5%, the "linear" relationship between mechanical work and heat (mechanical equivalent of heat, 1 calorie=4.186 joule). R. Mayer and later H. Helmoltz (1847) thougth that there had to be a principle ruling the equivalence of the energies involved: the 1 st Principle of Thermodynamics was born.

All the researchers (but very few) agree that the 1 st Principle of Thermodynamics states the Conservation of Energy (which was unknown to Carnot). In Thermodynamics it is written as dU = d*W + d*Q

(5) [in differential form it is (for an infinitesimal transformation)] where  dU is the exact differential of the state function U, named Internal Energy  d*W is a differential form, NOT exact, of the Work (W) done on the system  d*Q is a differential form, NOT exact, of the quantity of Heat (Q) supplied to the system So it is now clear that what Carnot named "caloric" was actually Energy, "thermal energy" if we like to call it. All the variables in [START_REF] Galetto | Horstmann Thermodynamics versus Mathematics_FIRST part[END_REF] are measured in J (Joule).

In Physics, there are many conservation "laws": the energy conservation, the momenta conservation, the angular momenta conservation, the charge conservation, the electron lepton number conservation, the baryon number conservation, the muon lepton conservation, the tau lepton number conservation, the charm conservation, the strangeness conservation.

Being the Internal Energy a state function [depending only on the endpoints of a transformation] it is true that for a cyclic process, reversible or irreversible, we have

∮ = 0 (6)
This 1 st principle stating that a change in internal energy in a system can occur as a result of energy transfer by heat, by work, or by both, is essentially presented in one form and it is very important, but it makes no distinction between processes that occur spontaneously and those that do not. The figure 2 The situation depicted in figure 2 shows that there is a hierarchy between the various forms of energy: heat has a lower hierarchy than work; heat is a degraded form of energy, because < 50% of heat can be transformed into work. This strong limitation gave origin to the 2 nd Principle of Thermodynamics (we saw first in section 2).

It is now clear that a Carnot Engine, operating between two temperatures (absolute …) T 1 >T 2 , with quantities of heat (energy!) Q 1 (supplied to the engine) and Q 2 (removed from the engine), can make work -W=Q 1 -Q 2 (see figure 1; it is the area, with negative sign, enclosed by the Carnot Cycle).

Unfortunately, there are people saying the following NONSENSE (people NOT cited here …; notice the word "Laws" and not Principles): … the First and the Second Laws of Thermodynamics are thus not independent from each other, but they are connected to each other with a kind of dialectic … kind of relationship… … and then connect this with the ideas … … formulated by Isaac Newton -as his Third Law: "For every action, there is an equal and opposite reaction".

Figure 2 show the "great asymmetry" of the two INDEPENDENT Principles: that's why BOTH must be taken into account in any actual process!!! Today we know that heat is kinetic energy and temperature T as well is related to kinetic energy. The following figure shows clearly all the types of exchanges happening at the boundary of a system with its surrounding. We end this section by citing the various energies (besides the ones cited in figure 3) present in any body: the thermal energy (kinetic energy) related to the body temperature, the chemical energy related to the molecules, nuclear energy related to the atoms nuclei and body mass.

The 3 rd Principle of Thermodynamics

It is the most recent. M. Plank stated it this way (first form): Entropy of all pure, perfectly crystalline homogeneous materials is 0 at T=0. And what about non_pure, non_perfectly crystalline homogeneous materials? Then we can say that, at any temperature greater than zero, the entropy of every substance will be greater than zero.

Another way (second form) is

It is impossible to reduce the temperature of a material body to the absolute zero of temperature in a finite number of operations.

In spite of that, we know that specific heat varies with temperature. If, however, temperature intervals are not too great, the temperature variation can be ignored and c can be treated as a constant. However near T=0 we know that the specific heat c V decreases with the temperature. We argue then a (very probable) physical fact that the specific heat C V 0 as T0, so that the following integral is * *

(S 0 is the integration constant). From Quantum Mechanics it is possible to deduce that S 0 could be >0 at T=0 [although the second form]. We do not pursue further the matter.

Entropy insight

We now provide a very interesting interpretation of the physical significance of Entropy.

We did not entered into the Statistical Mechanics, but we shall use one of its findings. We know that the kinetic energy for an atom of mass m 0 is k B T, for any degree of freedom

The ratio within (…) is an adimensional variable dx.

Hence by integration we have * ( )

where X a numerical variable and F(X) a suitable increasing function of X, with F(0)/k B =S 0 . Therefore we see that the fundamental Boltzmann's constant is the atom of Entropy.

Remembering the Boltzmann formula (found near 1900) S = k B ln()

we have F(X)=ln(). A final result is that entropy can be seen as the unavailable energy per unit temperature:

¢¢ ¢¢

Conclusion

Following the historical development of thermodynamics ideas we presented firstly the 0 th Principle, secondly the 2 nd Principle, thirdly the 1 st Principle and finally the 3 rd Principle. Then we showed that entropy is proportional to a numerical function F(X), where X is the ratio of energies.

We showed that the fundamental Boltzmann's constant is the atom of Entropy.

We entitled the paper "The Tetralogy of Thermodynamics, Thermodynamics made SIMPLE" because we gave the fundamental ideas form which the whole Thermodynamic Science was developed.

  combines the 1 st and the 2 nd Principles and shows what can happen in practice: asymmetry between Work and Heat…

Figure 2 .

 2 Figure 2. asymmetry between Work and Heat…

Figure 3 .

 3 Figure 3. types of exchanges happening at the boundary of a system with its surrounding

Figure 4 .

 4 Figure 4. The Thermodynamic Tetrahedron

  THEN the same engine, operating in the reversed cycle, from the work -W provides the quantity of heat -Q H (at temperature T H ) while absorbing -Q C (at temperature T C ). Absolute temperature needs a scale of measure.

	so that the efficiency was
	(2)

NOTICE that, in order formulae

[START_REF] Galetto | The Mathematics of Thermodynamics[END_REF]

and (2) hold, T C and T H MUST be absolute temperatures. Absolute temperature, what's that? Moreover, (1) is related only to the reversible Carnot Cycle! A reversible Engine is one that provides the same quantities of heat and work with reversed signs [any + become -, and vice versa] (see figure1 with reversed arrows): IF from Q H and Q C the engine provides the work W,

  . [k B is the Boltzmann's constant; k B =R/N A =1.3806504(24)*x10 -23 J/K, N A =Avogadro Number =6.02214179(30)*x10 23 particles/mol, Gas constant R=8.314472(15)* J/(mol K); the numbers (ab)* parentheses for the values represent the uncertainties of the last two digits]We know as well that the quantity of heat Q is kinetic energy. We can rewrite (3) as *