Nathalie Bouquier1

Sophie Sakkaki1

Fabrice Raynaud1

Anne-Laure Hemonnot- Girard1

Vincent Seube1

Vincent Compan1

Federica Bertaso1

Julie Perroy1

Enora Moutin1

The Shank3Venus=Venus knock in mouse enables isoform-specific functional studies of Shank3a

Keywords: Shank3, knock in, ASD model, isoform-specific tagging, Shank31C, Shank3a

Background: Shank3 is a scaffolding protein essential for the organization and function of the glutamatergic postsynapse. Monogenic mutations in SHANK3 gene are among the leading genetic causes of Autism Spectrum Disorders (ASD). The multiplicity of Shank3 isoforms seems to generate as much functional diversity and yet, there are no tools to study endogenous Shank3 proteins in an isoform-specific manner.

Methods:

In this study, we created a novel transgenic mouse line, the Shank3Venus=Venus knock in mouse, which allows to monitor the endogenous expression of the major Shank3 isoform in the brain, the full-length Shank3a isoform.

Results:

We show that the endogenous Venus-Shank3a protein is localized in spines and is mainly expressed in the striatum, hippocampus and cortex of the developing and adult brain. We show that Shank3Venus=C and Shank3Venus=Venus mice have no behavioral deficiency. We further crossed Shank3Venus=Venus mice with Shank31C=1C mice, a model of ASD, to track the Venus-tagged wild-type copy of Shank3a in physiological (Shank3Venus=C) and pathological (Shank3Venus=1C) conditions. We report a developmental delay in brain expression of the Venus-Shank3a isoform in Shank3Venus=1C mice, compared to Shank3Venus=C control mice.

Conclusion:

Altogether, our results show that the Shank3Venus=Venus mouse line is a powerful tool to study endogenous Shank3a expression, in physiological conditions and in ASD.

Introduction

Autism Spectrum Disorders (ASDs) affect about 1% of the human population. Patients with ASD have impairments in social communication and interaction and present repetitive and stereotyped behaviors (American Psychiatric Association, 2013). A significant proportion of ASD is due to genetic mutations, which has given hope for treatment [START_REF] Kleijer | Neurobiology of autism gene products: Towards pathogenesis[END_REF]. The discovery of monogenic forms of ASD offers a unique opportunity to explore the molecular and cellular mechanisms underlying ASD. The striking convergence of "ASD genes" onto glutamatergic signaling [START_REF] Connor | Synaptic basis of social dysfunction: A focus on postsynaptic proteins linking group-I mGluRs with AMPARs and NMDARs[END_REF][START_REF] Lee | Review excitation / inhibition imbalance in animal models of autism spectrum disorders[END_REF][START_REF] Brown | Clustering the autisms using glutamate synapse protein interaction networks from cortical and hippocampal tissue of seven mouse models[END_REF] supports the synaptic hypothesis as a leading cause of autism [START_REF] Bourgeron | From the genetic architecture to synaptic plasticity in autism spectrum disorder[END_REF], with the prominent role played by postsynaptic proteins linking group I mGlu receptors with AMPA and NMDA receptors [START_REF] Connor | Synaptic basis of social dysfunction: A focus on postsynaptic proteins linking group-I mGluRs with AMPARs and NMDARs[END_REF]. Heterozygous mutations in the SHANK3 gene are among the leading monogenic causes of ASD and can lead to Phelan McDermid syndrome (PMS) [START_REF] Durand | Mutations in the gene encoding the synaptic scaffolding protein SHANK3 are associated with autism spectrum disorders[END_REF][START_REF] Jiang | Modeling autism by SHANK Gene mutations in mice[END_REF][START_REF] Sala | Shank synaptic scaffold proteins: Keys to understanding the pathogenesis of autism and other synaptic disorders[END_REF][START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF]. PM is characterized by a significant delay in expressive speech, intellectual disability, hypotonia, dysmorphic facial features, increased tolerance to pain, epilepsy, and autism-like behavior [START_REF] Costales | Phelan -McDermid syndrome and SHANK3: Implications for treatment[END_REF].

The SHANK3 gene is located on mouse chromosome 15E3 and human chromosome 22q13.3. There are at least 6 different Shank3 protein isoforms, Shank3a to Shank3f (Figure 1A), the full-length isoform (Shank3a) being the most abundant form in the brain [START_REF] Wang | Transcriptional and functional complexity of Shank3 provides a molecular framework to understand the phenotypic heterogeneity of SHANK3 causing autism and Shank3 mutant mice[END_REF]. Shank3a is also the longest isoform and includes a Shank/ProSAP N-terminal domain (SPN), 6 ankyrin repeats (ANK), a Src Homology 3 domain (SH3), a PDZ (PSD-95/Dlg1/ZO-1) domain, a proline-rich region (Pro-rich), and a sterile alpha motif (SAM) domain at the Cterminus [START_REF] Naisbitt | Shank, a Novel family of postsynaptic density proteins that binds to the NMDA Receptor / PSD-95 / GKAP complex and cortactin[END_REF][START_REF] Tu | Coupling of mGluR/Homer and PSD-95 complexes by the Shank family of postsynaptic density proteins[END_REF]. These protein-protein interaction domains multiply the possibility of interactions of Shank3a protein with dozens of synaptic proteins, making it a key component of the glutamatergic post-synaptic architecture. Unfortunately, due to the lack of specific antibodies or genetic tools, there is currently no possibility to study this Shank3a isoform exclusively.

In an attempt to study the physiological role of endogenous Shank3a, we generated a Shank3Venus=Venus knock in (KI) mouse. This KI strategy is adding a tag on the protein of interest while bypassing the need to overexpress the coding gene, thus providing information on the endogenous protein at physiological expression levels in an intact cellular context. In addition, only the major isoform, Shank3a, is tagged in this mouse line, allowing to focus on Shank3a expression and function amongst all other isoforms. Finally, this new mouse line can be crossed with Shank3 mutant mice to study the repercussion of such mutations on Venus-Shank3a wild type protein in heterozygous mouse models of ASD. This strategy, focusing on the physiological role of endogenous Shank3, complements the previous attempts to understand how alterations of the Shank3 protein can lead to the development of ASD [START_REF] Yoo | Shank mutant mice as an animal model of autism[END_REF][START_REF] Sala | Shank synaptic scaffold proteins: Keys to understanding the pathogenesis of autism and other synaptic disorders[END_REF][START_REF] Varghese | Autism spectrum disorder: Neuropathology and animal models[END_REF].

Materials and methods

Generation of Shank3Venus=Venus KI mice at the Mouse Clinical Institute and genotyping

The targeting vector was constructed as follows. A 3.6 kb 30 homology arm fragment encompassing exons 3-7 was amplified by PCR and subcloned in an Mouse Clinical Institute (MCI) proprietary vector. This vector bares a floxed Neomycin resistance cassette associated with a Cre auto-excision transgene that allows the excision of the whole cassette in the chimera's male germ line. A 1.5 kb fragment corresponding to the fusion of two PCRs amplicons (the Venus tag sequence and a 740-bps genomic fragment encompassing exon 2) was cloned in a second step using the endogenous SrfI site in 50 and finally, a 3.6 kb (GC rich as covering the 50 UTR of the gene) fragment corresponding to the 50 homology arms was amplified by PCR and subcloned in step 2 plasmid to generate the final targeting construct. The linearized construct was electroporated in C57BL/6N mouse embryonic stem (ES) cells. After selection, targeted clones were identified by PCR using external primers and further confirmed by Southern blot with a Neo probe (50 and 30 digests) as well as a 30 external probe. Three positive ES clones were injected into BALB/cN blastocysts. One gave germ line transmission. Resulting male chimeras were bred with wildtype C57BL/6N females. Germline transmission of the knock in allele with a direct excision of the floxed selection cassette was obtained. The allele nomenclature (according to Mouse Genome Informatics) is Shank3tm1.Venus/Ics. 2019112715218160 v4 and APAFIS#23476-2020010613546503 v4). The Shank31C=1C mice (Jackson Laboratory, Bar Harbor, ME, USA, stock #018398) have a deletion of the Shank3 30 terminal part, starting just before exon 21. The resulting protein is truncated before the Homer-binding domain. Homozygous Shank3Venus=Venus mice were bred with homozygous Shank31C=1C to obtain Shank3Venus=1C mice (Figure 5A). Heterozygous Shank3Venus=C mice were crossed together to obtain Shank3Venus=Venus mice, Shank3Venus=C mice and Shank3C=C mice. Mice were grouped after weaning with respect to their sex.

Western blot analysis

Mice of both sexes were anesthetized with isoflurane. Brain structures were solubilized in a 10% SDS solution supplemented with protease inhibitors (Roche Diagnostics, Germany).

Samples were sonicated and centrifuged. Concentration of solubilized proteins was quantified using BCA assay (Sigma-Aldrich, USA). All samples were then set at the same volume and concentration for the rest of the protocol and 30ug of proteins were loaded for each condition. Proteins were eluted in Laemmli sample buffer, resolved on a tris-acetate 3-8% gradient gel (BioRad, USA) enabling the detection of all expected bands at different molecular weights into the same gel (from Venus-Shank3a to GAPDH). Proteins were transferred onto a nitrocellulose membrane, which was cut in two pieces upper and lower than 75 kDa and then incubated with a mouse anti-GFP antibody and anti-GAPDH antibody for the upper and lower part, respectively. After this first immunostaining, we applied 0.5% azide-containing solution to the upper part of the membrane to quench the chemiluminescence and incubated the rabbit anti-Shank3 antibody for a second revelation of this upper part of the gel. Immunoblot detection was performed using the following primary antibodies: GFP (mouse monoclonal from Clontech, USA, reference Cat.#632381, dilution 1:500), GAPDH (rabbit polyclonal from Santa Cruz reference, dilution 1:25,000), Shank3 (rabbit polyclonal from Santa Cruz, USA, reference SC-30193, dilution 1:500). A Mann-Whitney test was used for experiments comparing 2 conditions and a non-parametric Kruskal-Wallis test with uncorrected Dunn's post-test for those comparing more than 2 conditions.

Hippocampal primary cell culture

Cultures were prepared from postnatal Shank3Venus=Venus mice as previously described [START_REF] Moutin | Procedures for culturing and genetically manipulating murine hippocampal postnatal neurons[END_REF]. Briefly, hippocampi were mechanically and enzymatically dissociated with papain (Sigma-Aldrich, USA) and hippocampal cells were seeded in Neurobasal-A medium (Gibco, USA) supplemented with B-27, Glutamax, L-glutamine, antibiotics and Fetal Bovine Serum (all media from Gibco, ThermoFisher Scientific, USA). After 2 days in culture, cytosine b-D-arabinofuranoside hydrochloride (Sigma-Aldrich, USA) was added to curb glia proliferation. The day after, 75% of the medium was replaced by BrainPhys medium (Stemcell Technologies, Canada) supplemented with B-27, Glutamax and antibiotics.

Immunocytochemistry

Shank3Venus=Venus hippocampal primary cultures were fixed at Day In Vitro 15 with 4% paraformaldehyde for 10 min and then permeabilized and blocked with a 3% BSA, 0.1% Triton X-100, PBS solution (blocking buffer) for 1 h at room temperature. Cultures were then incubated overnight at 4°C with primary antibodies against GFP (TP401, Biolabs, USA) to stain Venus, and against MAP2 (M4403, Sigma-Aldrich, USA) diluted in blocking buffer to final concentrations of 1:1,000. After washes, cells were incubated with secondary antibodies for 2 h at room temperature, washed, incubated with Hoechst 33258 (B2883, Sigma-Aldrich, USA) diluted in water for 5 min at room temperature, mounted on slides and observed under an Axio-Imager Z1 microscope equipped with appropriate epifluorescence and filters (Carl Zeiss, Germany).

Behavioral experiments

In order to characterize the psychomotor development of Shank3Venus=Venus mice, weight, flipping, cliff avoidance and walking tests were performed each day from P0 to P15 on both male and female pups. Self-grooming, Three chambers sociability test, Open field and Marble burying were performed on male and female mice aged 12-14 weeks.

Weight:

The mice were weighed each day on a precision balance.

Flipping: Mice were placed on their backs. Those remaining on their backs for at least one minute were scored 0. Those that flipped onto their stomachs within 30 s to 1 min, 15-30 s, or 5-15 s were scored 1, 2, 3, respectively. Mice that flipped over in less than 5 s received a score of 4.

Cliff avoidance: Mice were placed on the edge of a Plexiglas platform over a 20 cm cliff with their nose and front legs over the edge. Mice that moved away from the edge within 30 s received the maximal score of 1, while mice that did not move got the score of 0.

Walking: Mice were observed for 1 min. Mice that remained immobile or moved in circles received a score of 0. Mice that walked but inconsistently and asymmetrically received a score of 1. Mice that walked symmetrically but slowly got a score of 2 while those that walked correctly and quickly received a maximum score of 3.

Open Field consisted of a square arena (50 cm 50 cm). Mice were placed in the center of the arena and left to explore freely for 10 min. Total distance traveled, speed and time spend in the center zone (defined as a 25 cm side square) were measured during 10 min by video tracking (Ethovision, Noldus, Netherlands).

Self-grooming:

Mice were placed in a 20 cm Plexiglas square arena and video monitored. Time spend self-grooming was measured visually during 10 min.

Marble test: Mice were placed individually in a clean cage with 5 cm thick bedding together with 15 marbles evenly distributed on top of the bedding. After 30 min in the cage, mice were removed and we evaluated the level of marble burying according to the following score: 0 for a totally buried marble; 1 for a half-buried marble and 2 for a totally visible marble. We also evaluated the level of interaction with the marbles counting marbles that were untouched during the test.

Three chambers sociability test: Mice were tested individually in a three chambers test apparatus (rectangular 40 60plexiglass, Panlab, Spain). The apparatus was cleaned with alcohol 30% between trials. Each of the two side chambers contained a cage (9 cm in diameter) in which social (stranger mouse) and non-social (object) stimuli could be confined. Testing consisted of three trials: trial 1 (habituation), the test mouse was placed in the middle chamber and then could freely explore the test box for 10 min with the cages empty. After which they returned to their home cage for 1 h before proceeding with the next trial. During trial 2 (social preference), an unknown mouse (stranger 1: social stimulus) was introduced in one of the cages in one side chamber, whereas an unknown object (object: non-social stimulus) was enclosed in the other cage in the opposite side chamber. Location of stranger 1 in the left or right-side chamber was balanced across subjects. The test mice were allowed to explore during 10 min. After 1 h in their home cage, mice were exposed to trial 3. During trial 3 (social novelty), the object present in trial 2 was replaced by a novel unfamiliar mouse (stranger 2: novel social stimulus). Again, the test mouse was allowed to explore during 10min. Behavior was video monitored during all trials (Ethovision, Noldus, Netherlands) to analyze general activity (distance traveled, speed). Social preference index was calculated as the ratio of time spent in the side chamber containing the social stimulus by the time spent in both side chamber during trial 2. Social novelty recognition index was calculated as the ratio of time spent in the side chamber containing the novel social stimulus by the time spent in both side chamber during trial 2.

Statistical analysis:

For pups, we analyzed the evolution of mice proportion over time using Generalized Linear Model (GLM) (Ordinal Logit model, SPSS R IBM R ) with score as dependent variable and genotype and days as covariates. We also used GLM to compare the evolution of weight over time with weight as dependent variable and genotype and days as covariate (Linear Scale Response model, SPSS R IBM R ).

For the adults' tests, differences between the three genotypes groups were analyzed using One-way ANOVA or Kruskal-Wallis accordingly to their distribution (Graph Pad Prism 9.3.0).

Results

Generation of a Shank3Venus=Venus KI mouse line allowing the exclusive detection of Shank3a isoform

In mice, the SHANK3 gene is composed of 22 exons (Figure 1A). Multiple intragenic promoters, at least six (see [START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF] for review), lead to six major protein isoforms: Shank3a to Shank3f. We have generated the Shank3Venus=Venus KI mouse line by inserting the DNA coding for Venus fluorescent tag at the beginning of exon1 of SHANK3 gene, just after the ATG start codon (Figure 1B). This strategy enables to tag exclusively the endogenous Shank3 variants that are produced by the first promoter: variants of the Shank3a isoform only, Shank3b to f being generated by promotors located downstream Venus DNA coding sequence. To check for the effective expression of this tagged version of Shank3, we performed Western blots on brain samples of adult Shank3Venus=Venus mice with two different primary antibodies. The first one was raised against Venus and the second one against the C-terminal part of Shank3 [START_REF] Verpelli | Importance of Shank3 protein in regulating metabotropic glutamate receptor 5 (mGluR5) expression and signaling at synapses[END_REF] (which recognizes Shank3a, c/d e, and f isoforms). Consistently, Western blot analysis of adult Shank3Venus=Venus mouse brains showed only one double band (Figure 1C), with an apparent molecular weight corresponding to the expected size of full-length Venus-Shank3a (theoretical Venus-Shank3a molecular weight: 212 kDa). As expected, the shorter isoforms lacking at least the N-terminal exon 1 and exon2 (Shank 3b to f) could not be detected using an anti-Venus antibody. These shorter bands are present when using the anti-Shank3 antibody labeling all Shank3 isoforms conserving the C-terminal part, excepted for Shank3f which molecular weight is around 10 kDa [START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF] and cannot be detected in our experimental conditions (Figure 1C). Furthermore, as already reported for Shank3c, Shank3d, and Shank3e, additional transcripts may be produced by alternative splicing of exons (exon 11, exon 12, exon 18, exon 21, and exon 22). For example, for Shank3c, alternative splicing gives rise to Shank3c1, Shank3c3, and Shank3c4 [START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF]. We noticed that the Venus-Shank3 band was indeed a double band, which may be the result of alternative splicing. To complete the characterization of endogenous Venus-Shank3 expression, immunocytochemistry on hippocampal neuronal cultures from Shank3Venus=Venus mice shows that labeled endogenous Venus-Shank3 proteins are expressed in a punctate manner along dendrites (Figure 1D), as expected from the endogenous expression of Shank3 in dendritic spines [START_REF] Tu | Coupling of mGluR/Homer and PSD-95 complexes by the Shank family of postsynaptic density proteins[END_REF].

Shank3Venus=Venus KI mice exhibit typical development and behavior

We evaluated Shank3Venus=C and Shank3Venus=Venus mice metabolic and sensorimotor development by tracking mice weight, sensorimotor reflexes (flipping and cliff avoidance tests), and walking abilities (Figure 2). We used mice from postnatal day 0 (P0) to 15 (P15) to detect possible early deficits. We found no significant difference in weight from birth until P15 between Shank3Venus=C and Shank3C=C littermates. Similarly, flipping, cliff avoidance, and walking scores were not significantly affected in Shank3Venus=C mice compared to WT mice. Because Shank3a mutations are associated with ASD like phenotypes, to rule-out any behavioral effect of the fusion of the Venus tag on Shank3a, we further characterized 12-14 weeks adult mice behaviors in tests known to reveal autistic-like phenotypes at this age. We evaluated mice social interests using the three chambers test. We found no difference between Shank3C=C, Shank3Venus=C, and Shank3Venus=Venus mice in social preference nor in social novelty, highlighting the absence of atypical social interest in these KI mice (Figure 3A). We assessed stereotyped behaviors and anxiety measuring selfgrooming, locomotion in the open field and mice propensity to interact with marbles. The time spent grooming was not significantly different between Shank3C=C, Shank3Venus=C, and Shank3Venus=Venus mice (Figure 3B), suggesting the absence of stereotyped behavior and no atypical anxiety level in Venustagged Shank3a mice. The time spent in the center zone of an open field (Figure 3C) was not significantly different between genotypes either, suggesting the absence of atypical anxiety. We however noticed a slight but statistically significant decrease in burying score of Shank3Venus=C heterozygous mice compared to Shank3C=C in the marble test (Figure 3D) meaning that this genotype tends to burry more marbles. Yet, the propensity to interact with the marbles (number of untouched marbles) remained the same. As Marble burying test can be used as a proxy to measure anxiety, the outcome of this test reflects complex behavior combining anxiety together with novelty avoidance, stereotypical movements and general locomotion. That is why we performed supplemental behavioral tests for this genotype: Elevated Plus Maze test confirmed that anxiety-like behaviors of Shank3Venus=C adult mice were typical (Supplementary Figure 1A). Performance in the Cyclotron (Supplementary Figure 1B) and Rotarod (Supplementary Figure 1C) showed that motor activity, motor coordination and endurance were typical as well. Aforementioned absence of hyper self-grooming ruled out stereotypical movements' abnormalities. Altogether, these behavioral experiments show that Shank3Venus=C and Shank3Venus=Venus KI mice behave as WT mice. In future studies, Shank3Venus=Venus KI mice can then be crossed with Shank3 mutant mouse models of ASD to understand the consequences of a Shank3 ASD mutation on the molecular dynamics of Venus-Shank3a at synapses in heterozygous mice.

Shank3Venus=Venus KI mice exhibit typical expression of Shank3 isoforms in the brain

To compare the expression of the different Shank3 isoforms in Shank3Venus=Venus (Figure 4A) and wild type mice (Figure 4B), Western blotting analysis was conducted on adult brain samples using Shank3 antibody. The Shank3a isoform is highly expressed in the striatum, hippocampus and cortex (Figures 4A',B') while the c/d isoforms predominate in the cerebellum (Figures 4A",B"). Shank3e is present but weakly expressed in all brain areas studied. Relative to the other variants, it is most highly expressed in the striatum. We did not find major differences in Shank3 variants expression profile between Shank3Venus=Venus mice and Shank3C=C littermates. The absence of signal using the anti-Venus antibody on Western blot from Shank3C=C mouse brain extracts (Figure 4B), which does not express Venus-tagged proteins, confirmed the specificity of endogenous Venus-Shank3a isoform detection by Venus tagging.

Overall, these experiments using Shank3Venus=Venus mice confirm a predominant expression of Shank3a in the striatum, hippocampus and cortex.

Crossing Shank3Venus=Venus KI mice with the Shank31C=1C mouse model of ASD reveals altered expression of the wild type Shank3a isoform in ASD

The Shank31C mouse line displays a mutation found in humans and is commonly used to investigate autisticlike phenotypes [START_REF] Kouser | Loss of predominant Shank3 isoforms results in hippocampusdependent impairments in behavior and synaptic transmission[END_REF][START_REF] Duffney | Autism-like deficits in Shank3-deficient mice are rescued by targeting actin regulators[END_REF][START_REF] Moutin | Restoring glutamate receptosome dynamics at synapses rescues autismlike deficits in Shank3deficient mice[END_REF]. We crossed Shank3Venus=Venus mice with Shank31C=1C or Shank3C=C mouse strains to compare heterozygous mice expressing either Venus-Shank3 and untagged wild type Shank3 (Shank3Venus=C) or Venus-Shank3 and untagged Shank31C (Shank3Venus=1C), Figure 5A. This allowed us to study Venus-Shank3a isoform expression in the context of ASD. In Shank3Venus=C mice, Venus-Shank3a was poorly expressed until postnatal day 7 in all studied brain regions (Figure 5B). At postnatal day 14, it considerably increased and reached a plateau at one month. These Venus-Shank3a expression kinetics were modified in Shank3Venus=1C mice. Indeed, the expression of Venus-Shank3a appeared to be delayed in the striatum and hippocampus, until one month. To better characterize this developmental gap in Venus-Shank3a expression, we directly compared Venus-Shank3a expression between the two genotypes at the critical period of synaptogenesis in juvenile (P7-14) mice (Figure 5C). Venus-Shank3a expression in the juvenile group (P7-14) was normalized by Shank3a expression in the adult group (1-3M), in both Shank3Venus=C and Shank3Venus=1C genotypes. We confirmed a significant decrease of Venus-Shank3a expression in juvenile mice when co-expressed with Shank13C, in the striatal and hippocampal regions. This alteration was absent when analyzing the full brain and only a tendency in the cortex. Altogether, these experiments show that the Shank3Venus=Venus KI mouse line can be crossed with ASD Shank3-mutant mice to track the expression of Venus-Shank3a isoform.

The first results obtained with Shank3Venus=Venus KI mouse line revealed a delay in Venus-Shank3a expression during the postnatal critical period of development (P14) in the Shank31C mouse model of ASD.

Discussion

Shank3 is a key scaffolding protein that organizes glutamatergic postsynapse architecture and function [START_REF] Boeckers | ProSAP/Shank proteins -A family of higher order organizing molecules of the postsynaptic density with an emerging role in human neurological disease[END_REF][START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF]. Many Shank3 isoforms are expressed in the brain, each one has its own unique specificity of brain area, cell type or even subcellular expression and each interacts with a specific plethora of proteins, strongly suggesting differential functions for each isoform [START_REF] Wang | Transcriptional and functional complexity of Shank3 provides a molecular framework to understand the phenotypic heterogeneity of SHANK3 causing autism and Shank3 mutant mice[END_REF][START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF]. The longest and major Shank3 isoform has never been selectively studied due to the lack of specific antibodies or genetic tools enabling to isolate its functions from that of other isoforms. Here, we developed a transgenic Shank3Venus=Venus KI mouse to specifically study this Shank3a isoform, in its intact cellular context, at endogenous expression levels. Moreover, as a proof of concept, we show that this mouse line can be crossed with Shank3 mutant mice to study the Venus-Shank3a wild type protein in heterozygous mice models of ASD.

FIGURE Although studies with overexpressed proteins have been essential so far to highlight different Shank isoforms specific functions [START_REF] Han | SHANK3 overexpression causes manic-like behaviour with unique pharmacogenetic properties[END_REF][START_REF] Eltokhi | Imbalanced post-and extrasynaptic SHANK2A functions during development affect social behavior in SHANK2-mediated neuropsychiatric disorders[END_REF], genome editing is a powerful technique for studying gene expression and functions in an intact cellular environment. The main advantage of this strategy precisely resides in its non-invasive nature, which circumvents protein overexpression frequently resulting from the use of mammaliancell gene delivery vectors, such as viruses. Hence, tagging the endogenous protein minimizes experimental bias which may arise from aberrant protein expression or from deficient proteinprotein interactions when the stoichiometry of functional complexes is altered. In agreement with these considerations, we found that the endogenous Venus-Shank3a protein in the KI mice displays similar developmental expression profile to the untagged endogenous Shank3a [START_REF] Wang | Transcriptional and functional complexity of Shank3 provides a molecular framework to understand the phenotypic heterogeneity of SHANK3 causing autism and Shank3 mutant mice[END_REF][START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF], it is correctly targeted to dendritic spines, the overall metabolic and sensorimotor development of the Shank3Venus=Venus KI mice is typical and adult Shank3Venus=Venus KI mice do not display autistic like features. On the other hand, respecting endogenous expression levels can complicate the performance and analysis of experiments that rely on the fluorescence of a small number of Venus-tagged proteins at the single cell level. We by-passed the problem of weak fluorescence by amplifying the Venus fluorescence using fluorescent antibodies raised against Venus. Overall, we showed that the Shank3Venus=Venus KI mouse line provides a potential tool to elucidate the localization of Shank3a endogenous proteins in neurons by Venus tagging without altering Shank3 isoforms expression and preserving typical development and behavior of the mice. The morphology of Shank3Venus=Venus hippocampal neurons in culture is spiny, with a punctiform distribution of Venus-Shank3 at synapses. These mice will represent a powerful tool, by circumventing the lack of isoform specific antibodies, to investigate molecular dysfunctions at the synaptic level in various shankopathies (by crossing this line with mice carrying Shank3 mutations).

In a first set of experiments, Western blot analysis of the transgene expression revealed that only the Shank3a isoform was tagged with Venus. This observation is coherent with the genome editing chosen strategy, which was to add the coding sequence of Venus in frame with Shank3 exon1. Shank3 exon1 is absent from Shank3 b to f isoforms, each of them being produced by gene transcription starting at specific promoters located downstream of exon2. Hence as expected, only the isoform driven by the first promoter (upstream exon1) could display the Venus tag, i.e., the Venus-Shank3a isoform. However, downstream from the first transcription start site, SHANK3 gene has alternative options of splicing, which could have resulted in the generation of a wide array of mRNA transcripts and protein isoforms. Alternative splicing of exon (exon 11, exon 12, exon 18, exon 21, and exon 22) have indeed already been shown to trigger the variety of additional transcripts for Shank3c, 3d, and 3e [START_REF] Jiang | Modeling autism by SHANK Gene mutations in mice[END_REF]. Our biochemical data revealed a unique band for Venus-Shank3 (Figure 1C), the Shank3a full-length, which appears, however, as a doublet. Hence, even if no alternative Shank3a isoform has been reported for so far [START_REF] Monteiro | SHANK proteins: Roles at the synapse and in autism spectrum disorder[END_REF], this doublet could indeed arise from alternative splicing. Alternatively, post-translational modifications, such as Shank3a phosphorylation [START_REF] Wu | A bidirectional switch in the shank3 phosphorylation state biases synapses toward up or down scaling[END_REF], would similarly induce a slight shift in molecular weight. Future experiments may test these hypotheses, which are not mutually exclusive.

Consistent with previous reports [START_REF] Lim | Characterization of the Shank family of synaptic proteins. Multiple genes, alternative splicing, and differential expression in brain and development[END_REF][START_REF] Wang | Transcriptional and functional complexity of Shank3 provides a molecular framework to understand the phenotypic heterogeneity of SHANK3 causing autism and Shank3 mutant mice[END_REF][START_REF] Lee | Shank3-mutant mice lacking exon 9 show altered excitation/inhibition balance, enhanced rearing, and spatial memory deficit[END_REF][START_REF] Mei | Adult restoration of Shank3 expression rescues selective autistic-like phenotypes[END_REF], we found that the different Shank3 protein isoforms are differently expressed according to developmental stages and in a brain region specific manner, suggesting the existence of isoform specific functions. Systematic analysis of isoform-specific binding partners and isoform-specific brain expression patterns is an important step for future research. In that attempt, we used the herein generated Shank3Venus=Venus KI mouse line in which endogenous Shank3a full-length isoform is specifically tagged to follow its neurodevelopmental expression. Our data show that Venus-Shank3a is mainly expressed in the cortex, striatum and hippocampus at early developmental stage when synaptogenesis occurs (P14). This specific brain area expression persists in the adult brain with an additional delayed expression in the adult cortex. We found no expression of Venus-Shank3a in the cerebellum, or a marginal one.

Given the potential importance of the SHANK3 gene in ASD [START_REF] Betancur | SHANK3 haploinsufficiency: A " common " but underdiagnosed highly penetrant monogenic cause of autism spectrum disorders[END_REF][START_REF] Carbonetto | A blueprint for research on shankopathies: A view from research on autism spectrum disorder[END_REF][START_REF] Leblond | Meta-analysis of SHANK mutations in autism spectrum disorders : A gradient of severity in cognitive impairments[END_REF][START_REF] Sala | Shank synaptic scaffold proteins: Keys to understanding the pathogenesis of autism and other synaptic disorders[END_REF], it is crucial to understand more about its physiological role at the synapse and how it is disrupted by mutations. In humans, SHANK3 gene (22q13.3) deletion triggers neurodevelopmental disorders like Phelan-McDermid syndrome (PMS), characterized by autistic like behaviors, hypotonia and delayed or absent speech [START_REF] Bonaglia | Disruption of the ProSAP2 gene in a t(12;22)(q24.1;q13.3) is associated with the 22q13.3 deletion syndrome[END_REF][START_REF] Phelan | 22q13 deletion syndrome[END_REF][START_REF] Phelan | Deletion 22q13.3 syndrome[END_REF][START_REF] Phelan | The 22q13.3 deletion syndrome (Phelan-McDermid syndrome)[END_REF]. Shank3 protein truncation is thought to cause the core neurodevelopmental and behavioral deficits that are observed in patients. By definition, in this heterozygous condition only one of the two Shank3 alleles displays the mutation. This means that the core PMS symptoms could either be triggered by Shank3 haploinsufficiency [START_REF] Phelan | The 22q13.3 deletion syndrome (Phelan-McDermid syndrome)[END_REF], and/or by a dominant negative effect of the Shank3 mutant form (resulting from aberrant trafficking to the dendritic spines or lack of interaction with functional partners). The latter hypothesis is more frequently investigated [START_REF] Yoo | Shank mutant mice as an animal model of autism[END_REF][START_REF] Sala | Shank synaptic scaffold proteins: Keys to understanding the pathogenesis of autism and other synaptic disorders[END_REF][START_REF] Varghese | Autism spectrum disorder: Neuropathology and animal models[END_REF]. Our strategy, focusing on the expression and role of the nonmutated endogenous Venus-Shank3a when co-expressed with the truncated untagged form, Shank31C, complements the previous attempts to understand how alteration of the Shank3 protein can lead to the development of ASD. Here we showed that Venus-Shank3a expression is delayed during brain development when co-expressed with Shank31C. In particular, Venus-Shank3a expression is deficient around P7-14, a period during which Shank3 is essential because of its role in the synaptogenesis and neuronal wiring of the developing brain [START_REF] Sala | Regulation of dendritic spine morphology and synaptic function by Shank and Homer[END_REF][START_REF] Sala | Shank synaptic scaffold proteins: Keys to understanding the pathogenesis of autism and other synaptic disorders[END_REF][START_REF] Roussignol | Shank expression is sufficient to induce functional dendritic spine synapses in aspiny neurons[END_REF][START_REF] Durand | SHANK3 mutations identified in autism lead to modification of dendritic spine morphology via an actin-dependent mechanism[END_REF][START_REF] Macgillavry | Shankcortactin interactions control actin dynamics to maintain flexibility of neuronal spines and synapses[END_REF][START_REF] Sarowar | Actin-dependent alterations of dendritic spine morphology in shankopathies[END_REF][START_REF] Pagani | Deletion of autism risk gene Shank3 disrupts prefrontal connectivity functional[END_REF]. We observed this delayed expression in the cortex, striatum and hippocampus. Defect in neuronal connectivity within the limbic system could explain, at least in part, the cognitive deficits associated with ASD. Hence, ASD may also depend on the ability of the wild type Shank3 allele to be typically expressed and engaged into functional complexes at synapses. Similarly, the difference in impairment severity might be explained not only by the expression pattern of Shank3 mutated form but also by the extent to which the spared Shank3 protein can (or cannot) compensate for its loss.

To conclude, the Shank3Venus=Venus KI mouse line enables to track the Shank3a isoform in physiological and ASD conditions. This mouse line will be further useful to screen for differential Venus-Shank3a protein interactors in physiological and pathological conditions. Understanding Shank3 proteins function from an isoform specific perspective may also help to explain how different SHANK3 gene mutations may result in distinct phenotypic consequences in different transgenic mice [START_REF] Monteiro | Shank3 mutations and HCN channelopathy: One size does not fit all[END_REF]. Importantly, isoform-specific studies will help understanding the clinical conditions of patients with SHANK3 mutations, for whom genotype-phenotype stratification will help the design of specific pharmacological agents.

FIGURE 1

 1 FIGURE 1Generation of the Shank3Venus=Venus knock in mouse line and detection of Venus Shank3a isoform expression. (A) Schematic representation of the mouse Shank3 gene structure and protein domains of the different protein isoforms. Sites of intragenic promoters and alternative splicing exons are represented by arrows. SPN for Shank3/ProSAP N-terminal domain (1-75 aa); ANK for Ankyrin repeat domain (148-345 aa); SH3 for src Homology-3 domain (470-529 aa); PDZ for PSD-95/DLG/ZO-1 domain (570-664 aa); PRO for proline-rich domain (831-1417 aa); SAM (1667-1730 aa). (B) Schematic representation of the Venus-Shank3 protein domains and steps of Shank3Venus=Venus KI mouse line creation. Please note that the Venus coding sequence was inserted at the beginning of exon 1 (in frame with and just after the ATG), enabling the exclusive Venus-tagging of the Shank3 variants that are generated by the first promoter, Shank3a isoforms. (C) Total brain lysates from 3 different Shank3Venus=Venus and Shank3C=C adult mice were immunoblotted for Venus (left) and Shank3 (right) expressions. Expected molecular weight for untagged Shank3a, 3c/d and 3e isoforms are indicated on the right side of the blot (a, c/d, and e). (D) Immunocytochemistry images from Day In Vitro 15 Shank3Venus=Venus hippocampal primary cultures stained with antibodies against the dendritic marker MAP2 (red), Venus (green), and the nuclear marker Hoechst (blue).

FIGURE 2

 2 FIGURE 2 Psychomotor ability test battery in Shank3Venus=Venus, Shank3Venus=C and Shank3C=C KI pups. All behavioral experiments were performed on at least 18 mice per group. (A) Weight. Pups were weighed from birth to P15. Data are mean +/-SEM. (B) Walking. Mice

FIGURE 3

 3 FIGURE 3 Behavioral test battery in Shank3Venus=Venus, Shank3Venus=C, and Shank3C=C KI adult mice. (A) Three chambers sociability test. Social preference index: time spent in social stimulus side chamber divided by time spent in both chambers. Social novelty index: time spent in the novel social stimulus side chamber divided by time spent in both chambers. Oneway ANOVA, p-values = 0.71 and 0.51 for social preference and social novelty indexes respectively. (B) Self-grooming. Time spent by the mice grooming during a 10 min period. Kruskal-Wallis test, p-value = 0.48. (C) Open Field. Total distance traveled by the mice during the 10 min test. One-way ANOVA, p-values = 0.13. Time spent in the center zone. One-way ANOVA, p-values = 0.95; latency to the first entry into the center zone. Kruskal Wallis, p-value = 0.38. (D) Marble burying. Number of marbles untouched. Kruskal-Wallis, p-value = 0.11. Score of marble burying. One-Way ANOVA, p-value = 0.004. Bars are mean +/-SEM. from 19 Shank3C=C, 21 Shank3Venus=C, and 9 Shank3Venus=Venus except for Open field's time spent in the center zone and latency to the center zone where data are from 17 Shank3C=C, 19 Shank3Venus=C, and 9 Shank3Venus=Venus.

FIGURE 4

 4 FIGURE 4 Shank3 protein isoforms expression in different brain areas. (A, Top A') Total lysates of striatum (Str), hippocampus (Hipp), cerebellum (Cb), cortex (Cx), and full brain from 3 different Shank3Venus=Venus adult mice immunoblotted for Venus, Shank3 and GAPDH. (Bottom A') Summary graph of the Shank3a/GAPDH ratios (quantified using Shank3 antibody). (A") Graphs present the Shank3a, Shank3c/d and Shank3e over GAPDH ratios in the full brain or the different brain areas (quantified with Shank3 antibody). (B) Same as panel (A) but in 3 Shank3C=C adult mice. For panels (A,B) data are mean _ SEM, *indicates p-value < 0.05, **p < 0.01, non-parametric Kruskal-Wallis test with uncorrected Dunn's post-test.

  All animal procedures were conducted in accordance with the European Communities Council Directive, supervised by the IGF institute's local Animal Welfare Unit (A34-172-41) and approved by the French Ministry of Research (agreement numbers: APAFIS#23357-

	The	primers	used	for	genotyping	have	the	following	sequence:
	GGTACGGCGAGATCGCAAAGG and CTCTCTCC GCCGGGAACAG. The size of the
	PCR products is 862 bp for the KI allele and 133 bp for the WT allele
	Animal handling

Acknowledgments

The mouse mutant line was established at the Mouse Clinical Institute (Institut Clinique de la Souris, MCI/ICS) in the Genetic Engineering and Model Validation. We thank the iExplore animal facility (IGF, Montpellier). We thank Muriel Asari for illustrations (IGF, Montpellier).

Funding

This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (JP, grant agreement No. 646788).

Data availability statement

The original contributions presented in this study are included in the article/Supplementary material, further inquiries can be directed to the corresponding authors.

Ethics statement

The animal study was reviewed and approved by the French Ministry of Research (APAFIS#23357-2019112715218160 v4 APAFIS#23476-2020010613546503 v4). Written informed consent was obtained from the owners for the participation of their animals in this study.

Author contributions

EM and JP conceived research and wrote the manuscript. JP and FR designed the genome editing strategy. NB, SS, FR, FB, JP, and EM designed research experiments. NB, SS, and EM performed mice genotyping. VC and EM performed immunocytochemistry. A-LH-G performed tissue dissections. SS, EM, and FB performed behavioral experiments. VS, NB, and EM performed Western blots. JP supervised the project. All authors contributed to the preparation of the manuscript and approved it.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Luis S Froufe-Pérez

William Guerin

Rémi Carminati

Robin Kaiser

Threshold of a Random Laser with Cold Atoms

Keywords: numbers: 42, 25, Dd,42, 55, Zz

We address the problem of achieving an optical random laser with a cloud of cold atoms, in which gain and scattering are provided by the same atoms. The lasing threshold can be defined using the on-resonance optical thickness b0 as a single critical parameter. We predict the threshold quantitatively, as well as power and frequency of the emitted light, using two different light transport models and the atomic polarizability of a strongly-pumped two-level atom. We find a critical b0 on the order of 300, which is within reach of state-of-the-art cold-atom experiments. Interestingly, we find that random lasing can already occur in a regime of relatively low scattering.

Random lasing occurs when the optical feedback due to multiple scattering in a gain medium is strong enough so that gain in the sample volume overcomes losses through the surface. Since its theoretical prediction by Letokhov [1], great efforts have been made to experimentally demonstrate this effect in different kinds of systems [2,3,4,5,6], as well as to understand the fundamentals of random lasing [7,8]. The broad interest of this topic is driven by potential applications (see [9] and references therein) and by its connections to the subject of Anderson localization [10]. State-of-the-art random lasers [9] are usually based on condensed matter systems, and feedback is provided by a disordered scattering medium, while gain is provided by an active material lying in the host medium or inside the scatterers. In general, scattering and gain are related to different physical entities.

Another system that can be considered for achieving random lasing is a cold atomic vapor, using magnetooptical traps, where radiation trapping [11] as well as lasing [12,13] have been demonstrated. One advantage is the ability to easily model the microscopic response of the system components, which can be extremely valuable to fully understand the physics of random lasers. However, in such system, the ability to combine gain and multiple scattering at the same time is not obvious, as both should be provided by the same atoms. The purpose of this Letter is to address this issue quantitatively. Note that even though new interesting features appear when coherent feedback is involved [14], we will consider only incoherent (intensity) feedback. Following Letokhov's theory, we consider a homogenous, disordered and active medium of size L. The random lasing threshold is governed by two characteristic lengths: the elastic scattering mean free path ℓ sc [15,16] and the linear gain length ℓ g (ℓ g < 0 in the cases of absorption or inelastic scattering). In the diffusive regime, defined as L ≫ ℓ sc , the lasing threshold is reached when the unfolded path length, L 2 /ℓ sc , becomes larger than the gain length. More precisely, the threshold is given by [1,17] L eff > βπ ℓ sc ℓ g /3, where β is a numerical factor that depends on the geometry of the sample (β = 1 for a slab, β = 2 for a sphere), and L eff = ηL is the effective length of the sample, taking into account the extrapolation length [15]. Another important length scale is the extinction length, as measured by the forward transmission of a beam through the sample, T = e -L/ℓex . The extinction length is given by ℓ -1 ex = ℓ -1 sc -ℓ -1 g . Let us consider now a homogeneous atomic vapor, constituted by atoms of polarizability α(ω) at density ρ, submitted to a homogeneous pump field. The extinction length and the scattering mean free path are related to their corresponding cross-section σ by ℓ -1 ex,sc = ρ σ ex,sc , with σ ex (ω) = k 0 ×Im[α(ω)] and σ sc (ω) = k 4 0 /6π×|α(ω)| 2 [18]. As we consider only quasi-resonant light, we use only the wave vector k 0 = ω 0 /c with ω 0 the atomic frequency. We also define a dimensionless atomic polarizability α such that α = α × 6π/k 3 0 (we omit the dependence on ω in the following). As it is an intrinsic parameter of the cloud, it is convenient to use the on-resonance optical thickness b 0 = ρσ 0 L, where σ 0 = 6π/k 2 0 is the resonant scattering cross-section (without pump laser). Using these quantities, the threshold condition writes

ηb 0 > βπ 3|α| 2 (|α| 2 -Im(α)) . (1)
Moreover, we have L/ℓ sc = b 0 |α| 2 and η = 1 + 2ξ/ [L/ℓ sc + 2(β -1)ξ] with ξ ≃ 0.71 for L > ℓ sc [START_REF] Case | Linear transport theory[END_REF][START_REF] Drozdowicz | [END_REF]. Note that deeply in the diffusive regime (L ≫ ℓ sc ), η ∼ 1. Eq. ( 1) is the first result of this Letter. It shows, in the diffusive regime, the existence of a threshold of random lasing as soon as the medium exhibits gain, i.e., |α| 2 -Im(α) > 0. This threshold is given by a critical onresonance optical thickness, expressed as a function of the atomic polarizability only. Interestingly, the condition Im(α) < 0, corresponding to single-pass amplification (T > 1), is not a necessary condition.

The previous result is general and does not depend on a particular pumping mechanism or atomic model. Let us now specify a gain model that will allow numerical evaluations of the lasing threshold and of the features of the emitted light. We shall use the simplest case of strongly-pumped two-level atoms, for which the normalized atomic polarizability at frequency ω can be written analytically (assuming a weak "probe" intensity) [21],

α(δ, ∆, Ω) = - 1 2 1 + 4∆ 2 1 + 4∆ 2 + 2Ω 2 × (δ + i)(δ -∆ + i/2) -Ω 2 δ/(2∆ -i) (δ + i)(δ -∆ + i/2)(δ + ∆ + i/2) -Ω 2 (δ + i/2)
.

(

) 2
In this expression, ∆ = (ω p -ω 0 )/Γ is the normalized detuning between the pump frequency ω p and the atomic transition ω 0 of linewidth Γ, δ = (ω-ω p )/Γ is the normalized detuning between the considered "probe" frequency and the pump, and Ω is the Rabi frequency, normalized by Γ, associated with the pump-atom interaction. For a strong enough pumping power, this atomic polarizability allows for single-pass gain, when Im(α)¡0. This gain mechanism is referred as "Mollow gain" [13,21] and corresponds to a three-photon transition (population inversion in the dressed-state basis).

For each couple of pumping parameters {∆, Ω}, the use of the polarizability (2) into the threshold condition (1) allows the calculation of the critical on-resonance optical thickness b 0 as a function of δ. Then, the minimum of b 0 and the corresponding δ determine the optical thickness b 0cr that the cloud must overcome to allow lasing, and the frequency δ RL of the random laser at threshold. The result is presented in Fig. 1 for a spherical geometry (β = 2). The result for b 0cr is independent of the sign of ∆ and we only show the region ∆ > 0. The minimum optical thickness that allows lasing is found to be b 0cr ≈ 200 and is obtained for a large range of parameters, approximately along the line Ω ≈ 3∆. The optimum laser-pump detuning is near the gain line of the transmission spectrum, i.e., δ RL ∼ sign(∆) √ ∆ 2 + Ω 2 (a small shift compared to the maximum gain condition is due to the additional constraint of combined gain and scattering).

The obtained critical optical thickness is achievable with current technology [22], showing that random lasing is possible in a system of cold atoms with Mollow gain. As this result has been obtained using the diffusion approximation, the condition L/ℓ sc = b 0 |α| 2 ≫ 1 must be satisfied. This is not the case in the full range of random lasing parameters that we have found. For example, with ∆ ≈ 1 and Ω ≈ 3, the critical optical thickness is almost minimum, b 0cr = 213, but L/ℓ sc ≈ 0.44. In this case, the threshold defined by Eq. ( 1) is at best unjustified, at worst wrong. In order to identify in Fig. 1 the region in which the approach should be valid, we have hatched the area corresponding to L/ℓ sc < 3. Note that random lasing is still expected in this region, but for a larger on-resonance optical thickness, that would allow the fulfillment of the diffusive condition. The minimum optical thickness in the region of parameters compatible a priori with the diffusion approximation is 347, and is located in the vicinity of {∆ = 1, Ω = 1.2}.

This first evaluation demonstrates the need for a more refined transport model. In the following, we use the approach introduced in Ref. [23], that is based on the radiative transfer equation (RTE). The RTE is a Boltzmanntype transport equation [START_REF] Chandrasekhar | Radiative Transfer[END_REF], that has a larger range of validity with respect to the ratio L/ℓ sc than the diffusion equation [START_REF] Elaloufi | [END_REF].

Letokhov's diffusive theory [1,17] and the RTE-based theory [23] of random lasing both rely on a modal expansion of the solution of the transport equation. In order to compare the predictions of both models, we focus on the slab geometry (β = 1) since the modal expansion of the RTE is well known in this case [START_REF] Case | Linear transport theory[END_REF] (to our knowledge, no simple expansion is available for a sphere in the RTE approach). The modal approach consists in looking for solutions of the form Ψ s (z, u, t) = φ κ,s (u) exp(iκz) exp(st), where Ψ(z, u, t) is the specific intensity (z is the distance from the slab surface and u denotes a propagation direction). For a given real κ, s(κ) and φ κ,s form a set of eigenvalues and eigenfunctions of the RTE. If one denotes by s 0 (κ) the eigenvalue corresponding to the mode with the longest lifetime in the passive system, a laser instability appears when s 0 (κ) > 0 in the presence of gain. The lasing threshold is defined by the condition s 0 (κ) = 0. For isotropic scattering, this eigenvalue has an analytical expression valid for κ ℓ sc < π/2 [START_REF] Case | Linear transport theory[END_REF]23]:

s 0 (κ)/c = ℓ -1 g -ℓ -1 sc -κ/ tan(κ ℓ sc ) (3)
where c is the energy velocity. For a slab of width L, the dominant mode corresponds to κ = π/L eff = π/(L+2ξℓ sc ).

In practice, this determination of κ is meaningful as long as ξ = 0.71 can be taken as a constant (independent on L), which is the case for L > ℓ sc . This condition sets the limit of accuracy of the modal RTE approach. The diffusive result is recovered from the RTE approach in the limit κℓ sc ≪ 1 [START_REF] Elaloufi | [END_REF]. A first order expansion of Eq. (3) yields s The comparison between the RTE and diffusive approaches deserves two comments. Firstly, the gain contribution to s 0 (κ) (first term in Eq. ( 3)) is the same in both models. Secondly, the scattering contribution (second term in Eq. ( 3)) is larger in the RTE model by a factor of at most 1.13 (when L ∼ ℓ sc ). Thus, the correction introduced by the RTE model, compared to the diffusion approximation, is relatively small, as it corresponds to an increase of η b 0cr of at most a few percents. This means that the diffusive model gives accurate results down to L ℓ sc , and that in cold-atom systems, random lasing can occur even in a regime of low scattering.

In Fig. 2, we compare the minimum optical thickness obtained with both models for the slab geometry and with the diffusive model for the sphere geometry. To put forward the domain of validity in each case, we plot the results as a function of L/ℓ sc . As expected, in the range L > ℓ sc the threshold predicted by the RTE for the slab geometry is only slightly larger than the one given by the diffusion approximation, so that the two curves can hardly be distinguished. For the sphere geometry, we dashed the part corresponding to the domain where the diffusive model is a priori not reliable, i.e., L/ℓ sc < 3. Nevertheless, by generalizing the conclusion obtained with the slab geometry, we reasonably expect the threshold to be located between 250 and 300.

Let us now turn to a first characterization of such a random laser. An important quantity to be investigated is the emitted power as a function of the pumping power. In the stationary regime (continuous pumping) we numerically solve the optical Bloch equations for a strongly-pumped two-level atom (without using the weak "probe" approximation that leads to Eq. ( 2)) to obtain the polarizability at the lasing frequency, including the gain saturation induced by the random laser intensity. Above threshold, the laser intensity in the medium I (in) RL ∝ |Ω RL | 2 is determined by the condition s 0 (κ, |Ω RL | 2 ) = 0 (s 0 would be positive without gain saturation). The obtained intensity is analogous to the intra-cavity intensity of a standard laser, and thus does not correspond to the emitted power P (out) RL . At equilibrium, gain compensates losses, and P (out) RL is equal to the generated power, related to the gain cross-section σ g , i.e., P

(out) RL ∝ σ g |Ω RL | 2 with σ g = σ 0 |α| 2 -Im(α) .
In order to know if the laser signal can be extracted from the background fluorescence, it is particularly relevant to compare the emitted laser power with the pumpinduced fluorescence P Fluo ∝ σ 0 |Ω| 2 /(1 + 4∆ 2 + 2|Ω| 2 ). From this, we compute the ratio

P (out) RL P Fluo = |Ω RL | 2 |Ω| 2 |α| 2 -Im(α) 1 + 4∆ 2 + 2|Ω| 2 .
(4) We plot the result in Fig. 3 as a function of |Ω| 2 , for a pump detuning ∆ = 1. To obtain Eq. ( 4), we assume that both pump and laser intensities are homogeneously distributed across the whole system. We also consider only the optimum random laser frequency, thus neglecting the spectral width of the random laser or any interaction between different random laser frequencies. Hence we neglect several effects as mode competition [26] and inelastic scattering of the laser light. Nevertheless we think that the order or magnitude of the ratio laserto-fluorescence powers can be realistic for actual experiments, at least as long as only one mode of the laser is active [26]. For the chosen set of parameters, this ratio is more than 5% and hence laser emission should be measurable. Its distinction from the pump-induced fluorescence can be made by looking at the spectrum of the emitted light. Another interesting prediction of this model is that the laser emission frequency shifts as the pump intensity is increased [Fig. 3]. This corresponds to the shift of the maximum gain of the Mollow polarizability.

|Ω| 2 δ RL 1 1
In summary, we have established the possibility of achieving random lasing with cold atoms. The random laser threshold is described by a single critical parameter, the on-resonance optical thickness b 0 . In the particular case of a gain mechanism based on a strongly-pumped two-level atom (Mollow gain), our model predicts a critical b 0 ∼ 300. Such an optical thickness is achievable in current cold-atoms experiments, e.g. by using crossed dipole traps [22]. We have also determined the basic features of the emitted light above threshold, showing that the random laser emission should be measurable.

Another interesting result is that, due to the large gain, lasing can be obtained with a low feedback (low amount of scattering, i.e., L ∼ ℓ sc ). This regime is similar to that encountered in certain semiconductor lasers with a very poor cavity, and is different from the working regime of random lasers realized to date. This new regime could be numerically investigated by RTE-based simulations [START_REF] Elaloufi | [END_REF].

Finally, let us stress that the model developed here has several limitations, so that the numbers should be considered as first-order estimates. Firstly, we have considered monochromatic pumping, thus neglecting inelastic scattering from the pump. The inelastically-scattered photons may have a non-negligible influence on the atomic response, as shown in [27]. Secondly, the RTE model needs to be extended to a sphere geometry, and to a medium with inhomogeneous density and/or pumping. This would require a full numerical solution of coupled RTEs for the pump and probe beams [23]. We also outline that the case of the Mollow gain was chosen for the sake of simplicity, whereas other gain mechanisms might be more adapted for the search of random lasing, such as Raman gain or parametric gain [13]. Each gain mechanism has its advantages and drawbacks, but the degrees of freedom they offer, together with the first estimates presented here, make us confident that the experimental realization of a cold-atom random laser is possible with current technology.

The authors thank S. Skipetrov, J. J. Sáenz, R. Pierrat and F. Michaud for useful discussions. L.S.F. acknowledges the financial support of Spanish ministry of science and innovation through its Juan de la Cierva program. This work is supported by ANR-06-BLAN-0096.

FIG. 1 :

 1 FIG. 1: Threshold of random lasing based on Mollow gain [21], calculated for each pair of pumping parameters ∆ (detuning) and Ω (Rabi frequency) with Eqs. (1,2). Only the ∆ > 0 part is represented. (a) Critical optical thickness b0cr to allow lasing. (b) Detuning δRL of the random laser from the pump frequency. The black area corresponds to a forbidden region (no gain). The hatched part corresponds to parameters for which the diffusion approximation is a priori not reliable.

  /c = ℓ -1 g -κ 2 ℓ sc /3, where the superscript (DA) stands for diffusion approximation. The condition s (DA) 0 (κ = π/L eff ) = 0 leads to Letokhov's threshold, with β = 1.

FIG. 2 :

 2 FIG. 2: Critical optical thickness for different geometries and transport models. The lower, red curve corresponds to the slab geometry (width L), with RTE model (continuous line) and diffusive model (open circle). The upper, blue curve corresponds to the sphere geometry (diameter L), with the diffusive model. The part with L/ℓsc < 3 is dashed, as the model may not be reliable.

FIG. 3 :

 3 FIG. 3: Continuous line: Emitted random laser power normalized to the pump fluorescence power, as a function of the pump intensity. Dashed line: Normalized laser detuning δRL. The random medium is a spherical cloud of two level atoms with an on-resonance optical-thickness b0 = 650.

* Electronic address: Robin.Kaiser@inln.cnrs.fr


André De Palma
email: andre.de-palma@cyu.fr

Karim Kilani

Best, worst, and Best&worst choice probabilities for logit and reverse logit models

Keywords: Best-worst scaling experiments, Logit model, Random utility models, Reverse logit model JEL classification. C25, C35

This paper builds upon the work of Professor Marley, who, since the beginning of his long research career, has proposed rigorous axiomatics in the area of probabilistic choice models. Our study concentrates on models that can be applied to best and worst choice scaling experiments. We focus on those among these models that are based on strong assumptions about the underlying ranking of the alternatives with which the individual is assumed to be endowed when making the choice. Taking advantage of an inclusion-exclusion identity that we showed a few years ago, we propose a variety of best-worst choice probability models that could be implemented in software packages that are flourishing in this field.

Introduction

Professor Tony Marley, to whom we pay a well-deserved tribute in this special issue of this journal, is undoubtedly the pioneer of probabilistic choice models in which we are interested in the best choice, the worst choice, and/or both the best and the worst choice. His theoretical contributions on the subject can be traced back to [START_REF] Anthony | Some probabilistic models of simple choice and ranking[END_REF] which can be considered as the seminal paper on this topic. This first theoretical work has gained in maturity due to Marley's collaboration with Jordan Louviere who had seen very early and in an independent way, the interest of best-worst scaling experiments. Their joint work Marley and [START_REF] Anthony | Some probabilistic models of best, worst, and best-worst choices[END_REF] published in the Journal of Mathematical Psychology set new milestones in this area.

Since then, the number of works on Best-worst scaling experiments has not stopped growing. Goggle Scholar currently lists more than six hundred published papers on the subject in areas as diverse as Health, Marketing, Transportation, etc. In 2015, their joint book [START_REF] Louviere | Best-Worst Scaling: Theory, Methods and Applications[END_REF] published with Terry Flynn as a third author, makes a major synthesis of theoretical and empirical contributions of the moment.

Meanwhile, a number of packages of R or Python have emerged that allow users to implement and estimate such models. The most remarkable and comprehensive is probably the Apollo package [see [START_REF] Hess | Apollo: A flexible, powerful and customisable freeware package for choice model estimation and application[END_REF]. The majority of the packages implement the MaxDiff model that constructs the probability of the simultaneous best and worst choices as the product of a logit best choice probability in the whole choice set, multiplied by a probability, of the same logit type but with negative sign scales, which represents the worst choice probability in the set to the exclusion of the best choice. We propose here other models almost as simple, but with different theoretical foundations.

In this article, we rely in particular on the research of Marley and Louviere [2005]. Our objective is to revisit the foundations of these models, with reference to the theory of choice developed by Luce [1959]. Our objective is to better position these models, by contributing to the enrichment of the existing literature on the subject, in particular with respect to the classification and rationalization of these models.

In this paper, we focus only on models where the individual is supposed to have a strict underlying ranking of the alternatives, but represented in a probabilistic way by the modeler. This is the approach we took in de Palma et al. [2017], even if we had a slightly stronger assumption than here of a Random Utility Model [RUM, see [START_REF] Anderson | Discrete choice theory of product differentiation[END_REF], McFadden, 1981[START_REF] Mcfadden | Economic choices[END_REF], which is not really necessary.

We provide here a new general and rather simple proof of an inclusionexclusion type identity which seems to us fundamental and which would benefit from being better known. This would open up many new perspectives for models designed to analyze best and worst choice experiments. Indeed, we have shown that the probabilities of the best and worst choices are totally linked by this form of inclusion-exclusion identity. Any worst choice probability can be obtained from the best choice probabilities by an alternating identity, and vice versa.

The content of the rest of the paper, which will highlight this modeling, is described in the paragraphs that immediately follow.

In Section 2, we state the underlying rationality assumption required in the models we study. We then define the best and worst choice probabilities in this framework. We then formulate and prove the identity that links these two types of choice probabilities.

In Section 3, we recall the Luce's choice axiom [START_REF] Duncan | Individual Choice Behavior: A Theoretical Analysis[END_REF] In section 4, we address the question of the best and worst choices being considered simultaneously. We refer to their probabilities as the best&worst choice probabilities. We show that when the MNL is generated through a RUM, even if the utilities are correlated as in the model we call Gumbel-Strauss model, we obtain best&worst choice probabilities that are a product of a logit probability by an alternating sum of logit probabilities. From this, a variety of best and worst probabilities emerge, enriching the models already known for estimating experiments where the respondents simultaneously reveal their best and worst choices. Individuals' preferences for the alternatives are assumed random and are represented in the following way: An individual selected at random from a population strictly corresponds to ranked set of alternatives. This signifies that the individual orders a number of options according to those they prefer the most, to those least preferred, and that two alternatives are never equally desired. It is also feasible that an individual who has to reveal their ranking of alternatives may modify the order in a seemingly random fashion from one experiment to another. This last point is often adopted in mathematical psychology, while the first definition is favored in econometrics.

More formally, a strict linear order, which is similar to a permutation or a ranking of the alternatives of T , is denoted using the binary relation symbol ≻.

We must read the notation for events in the following manner:

1 x ≻ y ⇐⇒ x is better than y ⇐⇒ y is worse than x, (1)
where x and y are two different alternatives belonging to T . The random nature of preferences is represented by a probability distribution P (•) on the universe Ω consisting of all the strict linear orders (or rankings for short) on the choice set T . Recall that the number of rankings of the alternatives of T is factorial

|T |.
The very broad hypotheses we make in this paper are specified below:

Assumption 1. Individual preferences are represented by a probability distribution P (•) on the set of strict linear orders over the total set of alternatives T .

Best and worst choice probabilities

A more general notation than that we described for two alternatives in (1), instead puts into practice the use of a subset of alternatives. Let x be any alternative of T , and Y be any nonempty subset of alternatives within T , which

does not contain x, i.e. x / ∈ Y .
The following event

(x ≻ Y ) ≡ y∈Y (x ≻ y), (2)
must be understood as the subset of rankings where the alternative x is strictly preferred over all the alternatives contained within Y . The probabilities of these events are referred to as the best choice probabilities.

In a symmetrical manner, we can define the following event

(Y ≻ x) ≡ y∈Y (y ≻ x), (3)
where x this time represents the worst of all the alternatives contained within Y . We refer to the probabilities of such events as the worst choice probabilities.

The identity relating the best and the worst choice probabilities

We show that the events in which x is considered as better or worse than other alternatives is closely related according to particular identities. First of all, let us define the best and the worst choice probabilities in the master choice set T as being respectively the following positive numbers

b x ≡ P (x ≻ T \ {x}); w x ≡ P (T \ {x} ≻ x), x ∈ T. (4)
Note that they satisfy the following normalization identity

x∈T b x = x∈T w x = 1. ( 5
)
In this article we shall use the following fairly compact notations

b Y ≡ y∈Y b y ; w Y ≡ y∈Y w y , ∅ ⊆ Y ⊆ T. (6)
By definition, we have b ∅ = w ∅ = 0 and b T = w T = 1.

First note that if only two alternatives x and y are involved, the event or the subset of rankings such that (y ≻ x) is the complement of (x ≻ y). By the complement rule in probability, we obtain the following identity for the binary case

P (y ≻ x) = 1 -P (x ≻ y). (7)
Conversely, we can also interpret this simple relation as the probability that y is the best choice rewritten as a function of the probability that y is the worst choice.

Then, for three different alternatives x,y,z ∈ T , as long as the complement of the intersection is the same as the union of the complement, the subset of rankings such that ({y,z} ≻ x) is the complement of (x ≻ y) ∪ (x ≻ z).

Applying both the complement and addition rules in probability, we get the following identity for the trinary case

P ({y,z} ≻ x) = 1 -P (x ≻ y) -P (x ≻ z) + P (x ≻ {y,z}). (8)
The left side is the worst choice probability associated to the alternative x, while the right side contains nothing but best choice probabilities for this same alternative x. We can also rearrange and rewrite this using simple algebra and binary case identity (7). The previous identity becomes

P (x ≻ {y,z}) = 1 -P (y ≻ x) -P (z ≻ x) + P ({y,z} ≻ x).
The generalization of such binary and trinary identities is possible as demonstrated in the next theorem. We will artificially and for notational convenience extend the notations ( 2) and (3) to the case where Y is the empty set by setting

P (x ≻ ∅) = P (∅ ≻ x) ≡ 1.
We have:

Theorem 1. Under Assumption 1, the worst choice probabilities are related to the best choice probabilities by the following identity:

P (Y ≻ x) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | P (x ≻ Y ′ ). (9)
Conversely, the best choice probabilities are related to the worst choice probabilities by:

P (x ≻ Y ) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | P (Y ′ ≻ x). ( 10
)
Proof. First note that I(Y ≻ x) = y∈Y I(y ≻ x),where I(•) is the event indicator function. The above displayed expression can be rewritten as follows

I(Y ≻ x) = y∈Y (1 -I(x ≻ y)).
Expansion of the product of the above expression yields

I(Y ≻ x) = 1 + Y ′ ⊆Y (-1) |Y ′ | y∈Y ′ I(x ≻ y),
which can be rewritten as follows

I(Y ≻ x) = 1 + Y ′ ⊆Y (-1) |Y ′ | I(x ≻ Y ′ ).
Since by convention we set: (x ≻ ∅) ≡ Ω so that I(x ≻ ∅) = 1, the above displayed equation, can be rewritten in a slightly more compact form by adding the empty set to the sum of its RHS

I(Y ≻ x) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(x ≻ Y ′ ).
Applying the expectation operator to both sides of that equation, and by the linearity of that operator, the indicator function I(•) can be replaced by the probability function P (•), obtaining (9). The proof of (10) goes along the same lines as above by systematically switching the positions of x, Y and Y ′ with respect to the symbol ≻.

Models à la Luce

We introduce here some axiomatic models in the same vein as Luce [1959]. We start by the standard Luce model of which the hypotheses shed light on the best choice probabilities, and we recall the classic form of best choice probabilities induced by this axiom. Thanks to our identity outlined in Theorem 1, the form of worst choice probabilities is subsequently obtained. Next, we propose an axiomatic symmetry, that of Luce, but one which supports the worst choice probabilities called the Reverse Luce Axiom. We derive this according to the same principle as for the standard model, both for the worst choice probabilities and best choice probabilities introduced by our identity. Finally, in our last subsection, we demonstrate a model that verifies that both axioms are perfectly symmetrical and offer little interest from a practical point of view.

Luce's axiom

We start by recalling one of the fundamental results of axiomatic random preferences, that is to say, Luce's axiom.

Axiom 1 (Luce's axiom). The best choice probabilities satisfy, for any x,y ∈ T, x ̸ = y, and any Z ⊂ T \ {x,y}, the following property:

P (x ≻ {y} ∪ Z) P (y ≻ {x} ∪ Z) = b x b y . (11)
This axiom suggests that ratio between the probability that x is the best choice and probability that y is the best choice, does not depend on the presence of other alternatives. In other words, the alternatives of Z, are described as irrelevant alternatives. This ratio must, among other things, correspond with the ratio of choice probabilities when all the alternatives of the master set T are implicated in the choice. We sometimes refer to this axiom as Independence of Irrelevant Alternatives (IIA) property.

One of the fundamental results published in Luce [1959] and which is the basic of the MNL, is reiterated and demonstrated once again in the following proposition:

Theorem 2 (Luce, 1959). The best choice probabilities satisfy Luce's choice axiom (Axiom 1) iff the best choice probabilities have the MNL form given by:

P (x ≻ Y ) = b x b x + b Y . ( 12
)
Proof. By the law of total probability, we have the following normalizing identity

P (x ≻ Y ) + y∈Y P (y ≻ {x} ∪ Y \ {y}) = 1.
Then, by Axiom 1, we have

P (y ≻ {x} ∪ Y \ {y}) P (x ≻ Y ) = P (y ≻ {x} ∪ (Y \ {y}) P (x ≻ {y} ∪ (Y \ {y}) = b y b x , implying P (y ≻ {x} ∪ Y \ {y})) = b y b x P (x ≻ Y ).
Therefore, the above normalizing identity leads to

P (x ≻ Y ) = 1 1 + b y /b x ,
which can be rewritten as in (12).

We can go on to use the previously outlined theorems so as to deduce the form of worst choice probabilities. We adopt the usual convention that an empty sum is zero, i.e. y∈∅ b y = 0.

Proposition 1. Luce's axiom (Axiom 1) holds iff the worst choice probabilities have the following form:

P (Y ≻ x) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | b x b x + b Y ′ . ( 13
)
Proof. Applying Identity (9) of Theorem 1 with the best , we get the worst choice probabilities formula provided in (13).

The obtained result is a necessary and sufficient condition as long as Theorem 1 gives us a relation which links the worst choice probabilities to the best choice probabilities whose form is given by Theorem 2, which itself gives necessary and sufficient conditions.

Reverse Luce's axiom

A natural extension of the previous MNL involves transferring the IIA property from the best choice probabilities to the worst choice probabilities. We call this the Reverse Luce choice axiom that we will now outline.

Axiom 2 (Reverse Luce's choice axiom). The worst choice probabilities satisfy, for any different alternatives x,y ∈ T , and any Z ⊂ T \ {x,y}, the following property:

P ({x} ∪ Z ≻ y) P ({y} ∪ Z ≻ x) = w x w y . ( 14
)
It appears obvious that this axiom implies that this time the worst choice probabilities take the MNL form.

Theorem 3. The reverse Luce's choice axiom (Axiom 2) holds iff the worst choice probabilities have the RMNL form given by:

P (Y ≻ x) = w x w x + w Y . ( 15
)
Proof. The proof goes along the same lines as the proof of Theorem 2. In a straightforward manner, we can systematically change the positions of the alternatives x and y, besides the subset Y in comparison to the preference ≻.

In the same way as outlined previously, we can combine our fundamental theorem, Theorem 1, with Theorem 3, which expresses the form of the worst choice probabilities, in order to obtain the form of the best choice probabilities.

Corollary 1. Luce's axiom (Axiom 2) holds iff the best choice probabilities have the following form:

P (x ≻ Y ) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | w x w x + w Y ′ .
Recall that this formulation of best choice probabilities was obtained by [START_REF] Anderson | Reverse discrete choice models[END_REF] who conceived it by inverting the signs of the random utility variables in an Additive Random Utility Model (ARUM). Our use here is rather in the spirit of fundamental theory with respect to more general Luce models and their ilk.

An impossibility result

It seems reasonable within the context of this paper to ask ourselves what model is produced when we simultaneously apply the two Axioms 1 and 2? We obtain the following:

Theorem 4 (Impossibility result). The probabilistic preferences satisfy simultaneously Axioms 1 and 2 iff:

P (x ≻ Y ) = P (Y ≻ x) = 1 1 + |Y | . ( 16
)
Proof. If the conditions for Axioms 1 and 2 are met, the probabilities for binary choices is written as:

P (x ≻ y) = b x b x + b y = w y w x + w y ,
this entails the equality of the following ratios w y /w x = b x /b y . Taking into account the constraints of normalization given by ( 5), we arrive at

w x = 1/b x y∈T 1/b y . ( 17
)
We can now remind ourselves of the identity given by Eq. ( 8), which is applicable when we have three different alternatives x,y,z ∈ T . Therefore bringing us to:

1 - b x b x + b y - b x b x + b z + b x b x + b y + b z - 1/b x 1/b x + 1/b y + 1/b z = 0,
which can be rewritten as follows

1 - b x b x + b y - b x b x + b z + b x b x + b y + b z - b y b z b y b z + b x b z + b x b y = 0.
Regrouping the first three terms as well as the fourth and fifth terms, we obtain

b y b z -b 2 x (b x + b y )(b x + b z ) + b 2 x b z + b 2 x b y -b 2 y b z -b y b 2 z (b x + b y + b z )(b y b z + b x b z + b x b y ) = 0,
which can be further factorized as follows

(b y b z -b 2 x ) 1 (b x + b y )(b x + b z ) - b y + b z (b x + b y + b z )(b y b z + b x b z + b x b y ) = 0.
After some algebraic manipulations, the second term into parentheses can be factorized, obtaining 17), we have the same form w x = 1/n. Finally, using the Eqs. ( 12) and ( 13), we obtain Eq. ( 16).

(b y b z -b 2 x ) b x b y b z (b x + b y )(b x + b z )(b x + b y + b z )(b y b z + b x b z + b x b y ) = 0.
So this result gives us a strong impossibility result, describing how it is not possible to construct a model which satisfies Luce's two axioms at the same time -the original and the reverse -other than in a purely symmetric model, limiting this construction. We will discover in the next section that it is possible to obtain all the best and worse probabilities for each axiom thanks to a general identity that connects these probabilities.

Best&worst choice probabilities

We are now interested in the simultaneous choice of the best and worst alternatives. More formally, if x and z are two different alternatives of T , and Y is a non-empty subset of T that does not contain either of these two alternatives, we are interested in computing the probability that x is considered the best alternative while z is the worst alternative when the choice set contains x,Y, and z. The probabilities of these events will be called best&worst choice probabilities and are defined by

P (x ≻ Y ≻ z) ≡ P ((x ≻ Y ) ∩ (Y ≻ z)). (18)
The aim here is not to reconstruct an exhaustive theory of identities that can exist in all choice subsets of T as we have done for the best choice and the worst choice probabilities separately. Extensive discussions are made in Marley and [START_REF] Anthony | Some probabilistic models of best, worst, and best-worst choices[END_REF], in de Palma et al. [2017] and Delle Site et al. [2019].

It is also worth noting in the same vein of results and as a complement the interesting result of [START_REF] Colonius | A representation theorem for finite best-worst random utility models[END_REF] which gives necessary and sufficient conditions for a system of best&worst choice probabilities, i.e. all probabilities of the type P (x ≻ Y ≻ z), so that they are rationalizable. The author thus obtains an extension of the result of [START_REF] Falmagne | A representation theorem for finite random scale systems[END_REF] who showed that the non-negativity of the polynomials exhibited in Block and Marschak which are derived from the best choice pobabilities is a necessary and sufficient condition for rationalizability [see also the short proof of [START_REF] Fiorini | A short proof of a theorem of falmagne[END_REF] Our goal here is rather to identify forms of best&worst choice probabilities that remain compatible with the Luce axiom for best choice probabilities (Axiom 1), or equivalently, models compatible with the reverse Luce axiom for worst choice probabilities (Axiom 2). These models could be interesting to test for future empirical applications.

Lemma 1. Under Assumption 1, the best&worst choice probabilities are related to joint best choice probabilities by the following identity:

P (x ≻ Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | P ((x ≻ (Y \ Y ′ ) ∪ {z}) ∩ (z ≻ Y ′ )). ( 19
)
Similarly, the best&worst are related to joint worst choice probabilities by:

P (x ≻ Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | P ((Y ′ ≻ x) ∩ {x} ∪ ((Y \ Y ′ ) ≻ z)). (20)
Proof. First, we note that by transitivity of the preference relation ≻, we have

(x ≻ Y ≻ z) = (x ≻ Y ∪ {z}) ∩ (Y ≻ z),
which implies, by using the indicator function I(•), the following equality

I(x ≻ Y ≻ z) = I(x ≻ Y ∪ {z}) × I(Y ≻ z).
Using the fact that

I(Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(z ≻ Y ′ ),
we get then

I(x ≻ Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(x ≻ Y ∪ {z}) × I(z ≻ Y ′ ).
For any y ∈ Y ′ , if z ≻ y and x ≻ z, we have necessarily x ≻ y which allows us to remove this redundancy by writing

I(x ≻ Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(x ≻ (Y \ Y ′ ) ∪ {z}) × I(z ≻ Y ′ ).
Applying the mathematical expectation to both members of this equality and by linearity of expectation, we obtain Eq. ( 19).

Alternately, we can also write

I(x ≻ Y ≻ z) = I(x ≻ Y ) ∩ ({x} ∪ Y ≻ z).
Then using the identity

I(x ≻ Y ) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(Y ′ ≻ x),
and by similar arguments as above, we get

I(x ≻ Y ≻ z) = ∅⊆Y ′ ⊆Y (-1) |Y ′ | I(Y ′ ≻ x) × I({x} ∪ (Y \ Y ′ ) ≻ z).
Applying the mathematical expectation then gives us Eq. ( 20).

Thus, the best&worst choice probabilities in the formulation (19) appear as an alternating sum of joint best choice probabilities where z is preferred to a subset of Y while x is preferred to all the other alternatives, including z.

In the formulation (20), it is rather an alternating sum of joint worst choice probabilities where x is worse than any subset of Y while z is worse than all the remaining alternatives, including x.

At this stage, Lemma 1 is not fully operational with such broad assumptions about preferences, even if we restrict ourselves to models where Axiom 1 (resp.

Axiom 2) is satisfied. de Palma et al. [2017] have successfully used this type of identity to compute the best&worst choice probabilities in the generalized exteme value model.

The Gumbel-Strauss model

In this section we will compute the various choice probabilities seen above in a class of RUMs, i.e. in models where each alternative x is assigned a specific random utility U x , where the best choice probabilities verify the axiom 1. We are interested in a particular RUM where the multivariate distribution of the utility vector is described by its CDF given by

P x∈T (U x ≤ u x ) = exp   - x∈T b x e -αux 1/α   , (21)
where α ≥ 1. Thus, the margins are Gumbel distributions, and α allows to introduce correlations between utilities. We notice that the limiting case α = 1 corresponds to the independence of the utilities.

We will refer to this model as the Gumbel-Strauss model insofar as the distribution of random utilities is a generalized Gumbel distribution, and that the authorship of the use of this distribution in the framework of RUMs is attributed to [START_REF] Strauss | Some results on random utility models[END_REF].

We will again demonstrate that the best choice probabilities do indeed take the form MNL. The proof will be inspiring for upcoming results.

Lemma 2. The best choice probabilities corresponding to the Gumbel-Strauss model, where the utilities have a CDF given by ( 21), take the MNL form given by ( 12).

Proof. By definition, we have

P (x ≻ Y ) ≡ P   y∈Y (U y ≤ U x )   = P (U Y ≤ U x ),
where U Y ≡ max y∈Y U y . The CDF of the couple composed of the variables U x and U Y and given by F

2 (u x ,u Y ) ≡ P (U x ≤ u x ,U Y ≤ u Y ), satisfies F 2 (u x ,u Y ) = exp   -b x e -αux + b Y e -αu Y 1/α   .
We then have to calculate the following double integral

P (x ≻ Y ) = +∞ -∞ ux -∞ ∂ 2 F 2 (u x ,u Y ) ∂u x ∂u Y du Y du x .
A first integration of the inner integral yields the following simplification

P (x ≻ Y ) = +∞ -∞ ∂F 2 (u x ,u x ) ∂u x du x . As ∂F 2 (u x ,u x ) ∂u x = b x b x + b Y Ae -ux exp -Ae -ux .
where

θ Z =                P (U x ≥ U T \{x,z} , Z = ∅; P (U x ≥ max(U X ,U z ); U z ≥ U Z ), ∅ ⊊ Z ⊊ T \ {x,z}; P (U x ≥ U z ≥ U T \{x,z} ) Z = T \ {x,z},
where X ≡ T \ {x,z} \ Z. We have clearly

θ ∅ = b x b z b z + b ∅
since it corresponds to the probability that x is the best choice in T . We also have θ T \{x,z} is the probability that x is the best choice and z is the second-best choice in T . It satisfies

θ T \{x,z} = P (U z ≥ U T \{x,z} )-P (U z ≥ U T \{z} ) = b z 1 -b x -b z = b x b z b z + b T \{x,z}
.

More generally, we shall prove that we always have θ Z = b x b z /(b z + b Z ). For ∅ ⊊ Z ⊊ T \ {x,z}, we can then use the multivariate cumulative distribution function of the quadruplet (U x ,U X ,U z ,U Z ) which is given by

F 4 (u x ,u X ,u z ,u Z ) = exp -b x e -αux + b X e -αu X + b z e -αuz + b Z e -αu Z 1/α .
We need to compute the quadruple integral given below

θ Z = +∞ -∞ ux -∞ ux -∞ uz -∞ ∂ 4 F 4 (u x ,u X ,u z ,u Z ) ∂u x ∂u X ∂u z ∂u Z du Z du z du X du x .
By simplifying two inner integrals, we obtain a double integral

θ Z = +∞ -∞ ux -∞ ∂ 2 F 4 (u x ,u x ,u z ,u z ) ∂u x ∂u z du z du x .
By differentiating F 4 twice, we obtain

∂ 2 F 4 (u x ,u x ,u z ,u z ) ∂u x ∂u z = b x b z e -αux e -αuz (A + α -1)A 1-2α e -A ,
where A is a function of the couple (u x ,u z ) defined by

A ≡ [(b x + b X )e -αux + (b z + b Z )e -αuz ] 1/α ,
By performing a first change of variable at the level of the inner integral, and noticing that we have b

x + b X + b z + b Z = b T = 1, we get θ Z = C × b x b z b z + b Z where C is a constant equal to C ≡ ∞ -∞ ∞ e -u (A + α -1)A -α e -A dAe -αu du.
We have thus proven that

P (x ≻ T \ {x,z} ≻ z) = C × b x ∅⊆Z⊆T \{x,z} (-1) |Z| b z b z + b Z .
It remains to show to finally prove our result that the constant C is indeed 1.

Noting that the alternating sum is nothing but a worst choice probability, we have P (x ≻ T \ {x,z} ≻ z) = C × b x P (T \ {x,z} ≻ z).

By summing over all of the z ∈ T \ {x} both members of the previous equation, we find

b x = C × b x ,
which indeed implies that C = 1.

As a conclusion to this section, we note that the best&worst probabilities for the Gumbel-Strauss model take the following form P (x ≻ T \ {x,z} ≻ z) = P (x ≻ T \ {x}) × P (T \ {x,z} ≻ z).

This multiplicative form specific to the logit model generated by independent Gumbel distributions, which corresponds to α = 1 for the distribution given by ( 21), had been previously pointed out [Marley and Louviere, 2005, Proposition 9]. Here we generalize this interesting result in two ways: first, we determine the complete analytical form of the best&worst choice probabilities, and second, we show that this result remains valid even if the utilities are correlated in some way.

Alternatively, one can, without any further difficulty, consider the reverse model of the one given by Equation ( 22) which would have best&worst choice probabilities given by

P (x ≻ T \ {x,z} ≻ z) = w z ∅⊆X⊆T \{x,z} (-1) |X| w x w x + w X . ( 23
)

Concluding remarks

In this paper we conducted a review of models that involve the best, the worst, and the best&worst choices in a choice set. We have focused on models that assume an underlying rationality, i.e. these probabilities are induced by a random order of preference over the alternatives.

It can be interesting in empirical applications to use models that have an explicit expression of these various probabilities. This is why we have concentrated our efforts on models of the MNL type where all the probabilities of interest to us are explicit and where a worst choice probability has an alternating form of best choice probabilities.

The best and worst probabilities considered simultaneously are the products of logit and alternating logit probabilities. We have also proposed reversed models where the worst probabilities have an MNL expression while the best probabilities have the alternating form. In future research, it would be interesting to implement these models in packages such as Apollo and to confront them with the current models already implemented, such as the MaxDiff model. This would open up interesting and new research paths.

  based on the best choice probabilities -in this case, the induced probabilities of the standard Luce model are often called the multinomial logit model (MNL). We state another axiom of the same type, this time based on worst choice probabilities. The corresponding probabilities of this alternative construct are what we will refer to as the reverse Luce model or Reverse Multinomial Logit Model (RMNL). The above identity allows us to explicitly determine the worst choice probabilities in an MNL and the best choice probabilities in an RMNL. Next, we show that a model bringing together the two axioms of the standard Luce model and the inverse Luce model produces a fully symmetric model. This is of little use from a practical point of view and represents a somewhat impossibility result.

Finally

  , in Section 5, we summarize our results and propose theoretical and empirical research avenues to further extend the scope of best and worst scaling experiments. 2 The best and worst choice probabilities identity The choice setting The total master choice set, denoted by T , is assumed to be finite and containing at least three alternatives (n ≡ |T | ≥ 3, where | • | denotes set cardinality).

  Therefore, we have necessary that: b 2 x = b y b z . This type of equality is also valid for any permutation of the alternatives x,y,z, for instance providing us with b 2 y = b x b z . Using the ratios of last two equality relationships, we can identify: b x /b y = 1. In view of the normalization (5) we can deduct b x = 1/n. Next, using Eq. (

A more rigorous approach for the notation of x ≻ y would be {≻∈ Ω : x ≻ y}. To preserve notation, we use the former expression.

where A ≡ (b x + b Y ) 1/α > 0, then using the fact that +∞ -∞ Ae -ux exp -Ae -ux du x = 1, we obtain

An immediate consequence of the previous result is the following one which gives the form of the worst choice probabilities for this model:

Proposition 2. The worst choice probabilities corresponding to the Gumbel-Strauss model, take the alternate form given by (13).

Proof. Since the best choice probabilities are of the MNL type, it is sufficient to use the Proposition 1 which provides the alternating form of the worst choice probabilities.

We are now fully prepared for the computation of the simultaneous best and worst choice probabilities for the Gumbel-Strauss model. We have derived the following result:

Proposition 3. The best&worst choice probabilities corresponding to the Gumbel-Strauss model, take the alternate form given by

Proof. We will use for our proof the Identity (19) of Lemma 1. It allows us to write the probabilities of the best and worst choice as follows