Claire Marquilly

Ana Boulanger

Germain U Busto

Laure Pasquer

Robert P Zinzen

Bassem A Hassan

Lee G Fradkin

Thomas Preat

Jean-Maurice Dura

Respective contribution of Appl to mushroom body axon growth and long-term memory in Drosophila

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

Introduction

Alzheimer's disease (AD) is associated with extracellular accumulation of amyloid fibrils derived from the Amyloid Precursor Protein (APP). APPs have therefore been intensely investigated, however their physiological function in the brain remains unclear and controversial [START_REF] Soldano | Beyond pathology: APP, brain development and Alzheimer's disease[END_REF][START_REF] Nicolas | Amyloid precursor protein and neural development[END_REF][START_REF] Selkoe | The amyloid hypothesis of Alzheimer's disease at 25 years[END_REF]. Although there are three paralogues in mammals (APP, APLP1 and APLP2), Drosophila encodes a single APP homologue, called Appl, that is expressed in all neurons throughout development. It has been shown that the type I transmembrane protein coding gene Appl is a conserved neuronal modulator of a Wnt planar cell polarity (Wnt/PCP) pathway, a regulator of cellular orientation within the plane of an epithelium [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF]. It has been proposed that Appl is part of the membrane complex formed by the core PCP receptors Fz (frizzled) and Vang (Van Gogh) [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF][START_REF] Liu | The amyloid precursor protein is a conserved Wnt receptor[END_REF].

The mushroom bodies (MBs) are two bilaterally symmetric structures in the central brain that are required for learning and memory [START_REF] Lin | The making of the Drosophila mushroom body[END_REF]. Each MB is comprised of 2000 neurons that arise from 4 neuroblasts. Three types of neurons appear sequentially during development: the embryonic/early larval γ, the larval α'β', and the late larval/pupal αβ type. Each αβ neuron projects an axon that branches to send an α branch dorsally, which contributes to the formation of the α lobe, and a β branch medially, which contributes to the formation of the β lobe [START_REF] Lee | Development of the Drosophila mushroom bodies: sequential generation of three distinct types of neurons from a neuroblast[END_REF]. So far, Appl d is the only reported null Appl allele and it results from a synthetic genomic deletion removing the Appl locus [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF]. The Appl d -bearing chromosome was selected, after γ irradiation, as a translocation of a partial duplication of the X chromosome on the Y chromosome to a X chromosome terminal deficiency (S1 Fig) . Appl d flies are viable, fertile and display no gross structural defects in the brain [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF]. However, the Appl signaling pathway is required for proper axon outgrowth in the MBs since Appl d flies display modestly-penetrant axonal defects in αβ neurons. In particular, Appl is required cell-autonomously for β-branch axon outgrowth [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF][START_REF] Cassar | Analysis of Amyloid Precursor Protein Function in Drosophila melanogaster[END_REF]. Using in vivo inducible RNA interference strategies, it was shown that Appl knockdown in the MBs results in loss of memory following the association of odorants with electric shocks [START_REF] Goguel | Drosophila amyloid precursor protein-like is required for long-term memory[END_REF]. Nevertheless, the memory defect of Appl null flies cannot be assessed for aversive memory because Appl d flies, unlike wild-type flies, show an abnormal response to electric shock [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF].

The ventral nervous system defective (vnd) gene, which is immediately proximal to Appl, encodes a Nk2-class homeodomain transcription factor, that acts in a context-dependent manner as an activator or repressor and is essential for the development of the nervous system. The vnd gene encodes two proteins: Vnd-A and Vnd-B whose mRNAs arise from two different promoters. These two proteins differ in their aminoterminal domains and are identical in the remainder of their sequences. While Vnd-A is a transcription repressor from promoters containing Nk-2 binding sites, Vnd-B directly activates transcription [START_REF] Stepchenko | The alternative protein isoform NK2B, encoded by the vnd/NK-2 proneural gene, directly activates transcription and is expressed following the start of cells differentiation[END_REF]. Flies bearing a null allele of vnd in hemi-or homozygous condition are embryonic lethal [START_REF] Jimenez | vnd, a gene required for early neurogenesis of Drosophila, encodes a homeodomain protein[END_REF].

Interestingly, we found that the Appl d chromosome also genetically affects vnd function. To genetically dissect Appl and vnd functions, we generated CRISPR alleles that precisely deletes the Appl gene without affecting vnd function (Appl C2.1 ) and that precisely deletes each one of the two vnd transcripts without affecting Appl (vnd C∆A and vnd C∆B ). We showed here that vnd-A, but not vnd-B, is also required for MB β-branch axon outgrowth. Unexpectedly, vnd is expressed in neurons close to, but not within, the MB during development and is required non-cellautonomously, likely by promoting the production of one (or several) secreted factor(s) involved in the MB axon outgrowth. To analyze the exact developmental and memory defects due to the specific loss of the Appl function only, we used Appl C2.1 and first confirmed the partial requirement of Appl in MB β-branch axon outgrowth. Interestingly, Appl C2.1 flies, in contrast to Appl d flies, showed a normal response to electric shock but a complete loss of long term memory independently from potential defects during brain development.

3

Generating new Appl CRISPR alleles

The complex chromosome structure associated with Appl d complicates its subsequent genetic manipulation (S1 Fig) . Therefore, we intended to produce new null Appl alleles by removing the entire Appl transcriptional unit via CRISPR/Cas9-mediated deletion [START_REF] Doudna | Genome editing. The new frontier of genome engineering with CRISPR-Cas9[END_REF]. We recovered Appl C1.4 and Appl C2.1 , two CRISPR Appl null alleles (Fig 1 andS2 Fig), following an adaptation of a published protocol in order to produce circa 50kb deletions [START_REF] Port | Optimized CRISPR/Cas tools for efficient germline and somatic genome engineering in Drosophila[END_REF]. Appl d MBs display a modestly-penetrant cell-autonomous axon growth defect of the β-branches which manifests as an absence of the β lobe [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF][START_REF] Liu | The amyloid precursor protein is a conserved Wnt receptor[END_REF]. Surprisingly, we found a strong difference in the penetrance of the absence of the β lobe phenotype in the two Appl CRISPR alleles. Although 9% of Appl C2. 1 MBs lacked the β lobe , slightly lower than the 14.5% described for Appl d [START_REF] Marquilly | Htt is a repressor of Abl activity required for APP-induced axonal growth[END_REF], Appl C1.4 MBs displayed a much higher penetrance (66%) of this phenotype (Fig 1). We determined the precise deletions present in the two alleles via sequencing and found that while the Appl C2.1 deletion precisely removed the Appl transcriptional unit, the Appl C1. 4 allele also removed a part of the vnd transcriptional unit (Fig 1). We then similarly mapped the Appl d deletion and found that, unlike Appl C2.1 , it removes most of the intergenic region between Appl and vnd which may influence vnd function (Fig 1). We then assessed, by genetic complementation tests, if the different Appl null alleles possibly affected vnd function. We used vnd A a molecularly-characterized lethal allele which impacts both vnd-B and vnd-A transcripts (Fig 2) [START_REF] Haelterman | Large-scale identification of chemically induced mutations in Drosophila melanogaster[END_REF]. Although Appl C2.1 /vnd A MBs displayed no anatomical MB phenotypes, Appl d /vnd A and Appl C1.4 /vnd A MBs displayed 14% and 69% of β lobe absence, respectively (Fig 2). Flies heterozygous for any of the four mutations displayed no MB phenotypes: vnd A /+ (n = 72), Appl C1. 4 /+ (n = 60), Appl C2.1 /+ (n = 42) and Appl d /+ previously described [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF]. Therefore, Appl d and Appl C1.4 but not Appl C2.1 affect vnd functions in MB development. Taken together, these data strongly suggest that vnd, the gene just proximal to Appl, is also involved in the MB β-branch axon growth and likely to a much greater extent than Appl itself.

Generation of new vnd CRISPR alleles deleted either for the vnd-B or for the vnd-A function

There are two vnd transcripts, vnd-A and vnd-B that produce two different Vnd proteins, respectively Vnd-A and Vnd-B (Fig 2). It was proposed that, while Vnd-A has a main role during embryogenesis, Vnd-B is associated with metamorphosis [START_REF] Stepchenko | The alternative protein isoform NK2B, encoded by the vnd/NK-2 proneural gene, directly activates transcription and is expressed following the start of cells differentiation[END_REF]. In order to determine which Vnd isoform is required for MB β-branch axon growth, we produced the new vnd alleles vnd C∆A 6 ) which affect both isoforms. Also, Appl C1. 4 /vnd C∆A female MBs displayed a strong phenotype of β lobe absence with a penetrance (64%) similar to those of Appl C1.4 /vnd A female MBs (Fig 4). Genetically, Appl C1.4 /vnd C∆A ; Dp-Appl + /+ females have two wild-type doses of Appl + and the β lobe absence phenotype (69%) was not distinguishable from that observed in Appl C1.4 /vnd C∆A ; +/+ females (64%) and therefore be entirely due to vnd lack-of-function. Supporting this hypothesis, we observed a complete rescue when an extra dose of vnd + is supplied in Appl C1. 4 /vnd C∆A ; Dpvnd + /+ females (Fig 4). Appl C1.4 /Y males have no Appl + function (null allele) and also display reduced vnd + function. Moreover, Appl C1.4 /Y male MB phenotype is moderately rescued by the presence of an extra dose of Appl + and strongly rescued by an extra dose of vnd + (Fig 4). Taken together, these results strongly indicate that the vnd-A transcript but not the vnd-B transcript is specifically required for MB β-branch axon growth.

Vnd is expressed around the MBs and is not required within the MBs

In order to know where vnd is expressed, we employed a vnd-T2A-GAL4 line where GAL4 is under the control of endogenous vnd regulatory sequences via CRISPR gene targeting so that GAL4 and vnd are translated from the same mRNA transcript [START_REF] Lee | Conservation and divergence of related neuronal lineages in the Drosophila central brain[END_REF][START_REF] Chen | An enhanced gene targeting toolkit for Drosophila: Golic+[END_REF]. UAS-GFP labelling revealed a pattern in the ventral ganglion of the third larval instar central nervous system (L3 CNS) similar to that described with immunostaining with antibodies against Vnd [START_REF] Stepchenko | The alternative protein isoform NK2B, encoded by the vnd/NK-2 proneural gene, directly activates transcription and is expressed following the start of cells differentiation[END_REF] thus validating the use of the vnd-T2A-GAL4 line as a bona fide vnd reporter (S4 Fig) . GFP was detected, in the developing brain, close to the MBs visualized by anti-Fas2 staining, from L3 to 24 h after puparium formation (APF) (Fig 5A,D'). Staining reveals two different morphological structures: one organized in a honeycomb pattern and one containing filamentous structures with neurite appearance. To determine the identity of the honeycomb structures, we performed DAPI staining at the L3 stage. We observed a significant DAPI staining in these structures indicating that they likely correspond to cell nuclei (Fig 5EE'). Moreover, these cell nuclei are Vnd positive and Repo negative (Fig 5F ,G') indicating that these cells correspond to neurons, rather than glia. The neurites corresponding to these cell nuclei are very close to the developing MB medial lobe from which the adult β lobe develops (Fig 5). Noticeably, we could not observe any GFP labeling within the MBs themselves. This unexpected expression pattern suggests a non-cell-autonomous role for Vnd in the MB β-branch axon growth. We tested this hypothesis by MARCM mosaic analysis which allows the generation of homozygous vnd loss-of-function MB clones in an otherwise heterozygous genetic background and overcome organismal lethality [START_REF] Lee | Development of the Drosophila mushroom bodies: sequential generation of three distinct types of neurons from a neuroblast[END_REF][START_REF] Lee | Mosaic analysis with a repressible cell marker for studies of gene function in neuronal morphogenesis[END_REF]. Mitotic recombination was induced in late-stage embryos/early first instar larvae and the clones were analyzed at the adult stage. We obtained 20 MB vnd A/A clones that include the αβ neurons. All the 20 vnd mutant clones displayed a β-branch axon growth that looked identical to that in wild-type clones (S5 Fig). Moreover, we obtained similar results, namely a normal β-branch axon growth, with two additional different null alleles, vnd ∆38 (8 clones) and vnd 6 (4 clones) (S5 Fig) . Taken together, these results demonstrate that, although vnd function is strongly required for MB βbranch axon growth, vnd is neither required nor expressed in the MBs themselves. Therefore, it is most likely that Vnd regulates MB axon growth by a non-cell-autonomous mechanism.

Appl-null flies have no long-term memory strictly due to the absence of Appl in adult MBs

Although it is clear that Appl function is required for aversive olfactory memory, it has not yet been possible to assess the memory of Appl-null flies , because Appl d flies exhibit impaired electric shock avoidance [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF][START_REF] Bourdet | The full-length form of the Drosophila amyloid precursor protein is involved in memory formation[END_REF]; In such flies a defect in the aversive memory assay could result from an impairment in electric shock perception during conditioning and therefore not represent true memory defects. We showed here that Appl d also affects vnd function (Fig 2 ) and hence wondered if this impairment in electric shock avoidance was really due to the lack of the Appl function itself. In order to test this hypothesis, we evaluated the long-term memory (LTM) performance of Appl C2.1 /Y flies which are null for Appl but wild-type for vnd. The electric shock avoidance and olfactory acuity appears wild-type in Appl C2.1 flies (Fig 6). Interestingly, Appl C2.1 flies showed a complete loss of LTM, and this memory defect was fully rescued by the restricted expression of an UAS-Appl transgene in adult MBs ( Fig 6). No rescue was detected in the non-induced 18°C controls (S6 Fig) . These results conclusively demonstrate that the specific and complete loss of Appl affects long-term memory independently from any potential defects during brain development.

Discussion

It has been proposed that Appl is part of the membrane complex formed by the core PCP proteins Fz and Vang [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF]. Appl d was, previous to this report, the only Appl null allele described. Appl d mutant flies are completely lacking Appl function and exhibit 14.5 % MB β-lobe loss [START_REF] Marquilly | Htt is a repressor of Abl activity required for APP-induced axonal growth[END_REF] in accordance with what has been reported [START_REF] Soldano | The Drosophila Homologue of the Amyloid Precursor Protein Is a Conserved Modulator of Wnt PCP Signaling[END_REF][START_REF] Liu | The amyloid precursor protein is a conserved Wnt receptor[END_REF]. This modestly-penetrant axonal phenotype due to the lack of Appl could be due to partially redundant function provided by the other transmembrane receptors of the complex. In particular, flies homozygous for the loss of function allele Vang stbm-6 exhibit 50% β-lobe loss [START_REF] Liu | The amyloid precursor protein is a conserved Wnt receptor[END_REF]. As the Appl d allele involves complex chromosomic rearrangements [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF], we decided to leverage CRISPR in order to produce a definitive deletion of around 50 kb to specifically eliminate the Appl transcriptional unit. We recovered two Appl complete deletions: Appl C1.4 and Appl C2.1 removing around 52 kb and 48 kb and exhibiting 66 % and 9 % β-lobe loss respectively. DNA sequences analysis revealed that, while the Appl C2.1 deletion removes only Appl sequences, the Appl C1. 4 deletion additionally removes a part of the proximally located transcriptional unit corresponding to the vnd gene (Fig 1). This indicates that in addition to Appl, vnd is likely involved in the β-branch axon outgrowth. Moreover, as can be expected from the extent of the deletions and based on genetic complementation tests, Appl C1. 4 deletion, but not Appl C2.1 deletion, affects vnd function. Most importantly, the Appl d deletion, which removes most of the Appl coding sequence and also the intergenic region between Appl and vnd, also affects vnd function (Fig 1 and2). The role of vnd in β-branch axon outgrowth is complicated by the fact that the vnd viable mutant alleles (Appl d and Appl C1.4 deletions), are associated with the lack of Appl function. Nevertheless, employing a small (less than 100 kb) genomic Appl + duplication, we found that flies lacking only vnd function exhibit 69% β-lobe loss that is completely rescued by a small genomic vnd + duplication (Fig 4). This demonstrates the requirement of vnd function for MB β-branch axon outgrowth. To better understand the roles of both Vnd isoforms in the MB β-branch axon outgrowth, we produced two mutations, vnd C∆B and vnd C∆A , which remove the vnd-B and the vnd-A transcript respectively (Fig 3

and S3 Fig). Surprisingly, although

Vnd-A has a significant role in MB axon growth, apparently Vnd-B plays little to no role in this process. Noticeably, MB phenotype of Appl C1. 4 /vnd C∆A flies is similar in penetrance to the one of Appl C1.4 /vnd A , indicating that the lack of vnd-A function seems to have the same effect than the complete lack of both vnd functions.

It may have been anticipated that Vnd, a transcription factor, was required within the MBs and was probably being part of the Appl signaling pathway in order to act on axon growth. Consequently, Vnd should have been expressed within the MBs and the vnd function was expected to be cell-autonomous. However, endogenous vnd transcription monitored by GAL4 expression that is translated from the same mRNA transcript and validated by an anti-Vnd antibody show no sign of expression within the MBs. Rather, vnd is transcribed within a neuronal brain structure in close proximity to the developing MB medial lobe from which the adult β lobe develops (Fig 5). Also, using MARCM mosaic analysis, we show that β axons extended from MB clones null for vnd function exhibit wild-type growth patterns (S5 Fig) demonstrating a non-cell-autonomous requirement for Vnd. Therefore, we favor the hypothesis where vnd expressed in a MB surrounding brain structure is generating, directly or indirectly, one or several secreted factors that will act on the axon growth of the developing β medial lobe.

Prior to this study, it was not possible to evaluate the effects of a complete lack of Appl in aversive olfactory memory due to the failure of Appl d mutants to show normal shock reactivity [START_REF] Luo | Human amyloid precursor protein ameliorates behavioral deficit of flies deleted for Appl gene[END_REF][START_REF] Bourdet | The full-length form of the Drosophila amyloid precursor protein is involved in memory formation[END_REF]. It was therefore hypothesized that it was the lack of Appl per se which was the cause of this impairment in electric shock avoidance. We reevaluated this hypothesis with the Appl C2.1 allele which removes only the Appl function. Flies bearing only a deletion of Appl react normally to electric shock and olfactory cues similarly to control flies but show a total lack of 24h memory after a long-term conditioning (Fig 6). Therefore, it is not the lack of Appl function in the Appl d mutants which is the origin of the impairment in electric shock avoidance. Importantly, the memory defect can be completely rescued by the expression of Appl + solely in the postdevelopment adult MBs. This demonstrates that a complete lack of Appl results in a specific memory defect independently of any role that Appl might have in the developing brain. Consequently, the modest mutant MB axonal growth morphological phenotype displayed by the Appl C2.1 allele likely has no measurable impact on long-term memory. This is in accordance with the requirement of the vertical lobes but not the median lobes in aversive long-term memory [START_REF] Pascual | Localization of long-term memory within the Drosophila mushroom body[END_REF].

The accumulation of amyloid-β (Aβ) is a hallmark feature of AD [START_REF] Palop | Amyloid-beta-induced neuronal dysfunction in Alzheimer's disease: from synapses toward neural networks[END_REF][START_REF] Musiek | Three dimensions of the amyloid hypothesis: time, space and 'wingmen[END_REF]. Nevertheless, the production of Aβ peptides is at the expense of the full-length APP protein levels, whose roles in AD are, as yet, unclear. The Drosophila memory defect due to the complete lack of Appl may therefore be relevant to the memory defects associated with human AD. Therefore, our observation that the memory defect resulting from the complete lack of Appl could be completely rescued by the expression of Appl + in a specific adult brain region, thus bypassing any potential role of APPL in development could be of significance to our understanding of the molecular basis of AD.

Materials and methods

Drosophila stocks

All crosses were performed on standard culture medium at 25 ° C. Except where otherwise stated, all alleles and transgenes have been described previously (http://flystocks.bio.indiana.edu/). The following alleles were used: Appl C1.4 , Appl C2.1 , vnd C∆A and vnd C∆B were generated in this study. Apply + (BL #56073), Appl d (BL #43632), Vang stbm-6 (BL #6918) and vnd A (BL #57139), vnd ∆38 [24](Chu 1998), vnd 6 [12](Jimenez 1995). The following transgenes were used: UAS-mCD8::GFP (BL #5137), UAS-Appl (BL #38403 ) and the duplications resulting from genomic transgenes Dp-Appl + (BL #32288) and Dp-vnd + (BL #30219) both inserted into the same attP site at 65B2. We used three GAL4 lines: vnd-T2A-GAL4 which reveals vnd transcription, c739-GAL4 (BL #7362) expressed in adult αβ MB neurons [START_REF] Aso | The mushroom body of adult Drosophila characterized by GAL4 drivers[END_REF] and VT30559 expressed in adult MB neurons [START_REF] Placais | Upregulated energy metabolism in the Drosophila mushroom body is the trigger for long-term memory[END_REF]. Recombinant chromosomes were obtained by standard genetic procedures and were molecularly verified.

CRISPR-Cas9 strategy

All guide RNA sequences (single guide RNA (sgRNA)) were selected using the algorithm targetfinder.flycrispr.neuro.brown.edu/ containing 20 nucleotides each (PAM excluded) and are predicted to have zero off-targets. We selected one pair of sgRNA for the Appl gene and two pairs of sgRNA for the vnd gene. For Appl, the pair is targeting the all transcriptional unit (S2 Fig) . For vnd, each pair is targeting either the A specific region of vnd or the B specific region of vnd (S3 Fig). We used the following oligonucleotide pairs: CRISPR-1 Appl fwd and CRISPR-1 Appl rev to target the all Appl region, CRISPR-1 vnd A fwd and CRISPR-1 vnd A rev to target the A region of vnd, CRISPR-1 vnd B fwd and CRISPR-1 vnd B rev to target the B region of vnd (see the corresponding oligonucleotide sequences in S2 and S3 Fig. ). We introduced two sgRNA sequences into pCFD4 [START_REF] Port | Optimized CRISPR/Cas tools for efficient germline and somatic genome engineering in Drosophila[END_REF], a gift from Simon Bullock (Addgene plasmid # 49411) by Gibson Assembly (New England Biolabs) following the detailed protocol at crisprflydesign.org. For PCR amplification, we used the protocol described on that website. Construct injection was performed by Bestgene (Chino Hills, CA) and all the transgenes were inserted into the same attP site (VK00027 at 89E11). Being aware that the DNA excision of nearly 50 kb might be rare, we have decided to use a positive screen before validation by genomic PCR. We have used a stock where the Appl gene is marked by a y + construct. The deletion of the Appl gene should therefore be associated to y -phenotype. Transgenic males expressing the Appl sgRNAs and bearing an isogenized X chromosome (y Apply + Act-Cas9 w*) were crossed to FM7c/ph 0 w females. 180 crosses were set up and only two crosses gave rise to y Appl deletion? Act-Cas9 w*/FM7c females indicating 1% of CRISPR efficacy. From each of these two crosses, a single y Appl deletion Act-Cas9 w* /FM7c female was crossed with FM7c males to make a stock (Appl C1. 4 and Appl C2. 1 where Cas9 was removed). The deletion were then validated by genomic PCR using two couples of primers: Appl C1.4 fwd (GAGCCAGATACACAAGCACA) / Appl C1.4 rev (GGCTTTGTTTACTTCCTGGC) and Appl C2.1 fwd (TCCTACTACGTTCCACAATC) / Appl C2.1 rev (TAATGCCCAACATATCCAAC). The precise endpoints of the deletion were determined by sequencing (Genewiz, France). Transgenic males expressing the different vnd sgRNAs were crossed to y nos-Cas9 w* females bearing an isogenized X chromosome. 100 crosses were set up for each sgRNA pair, with up to 5 males (because of poor viability) containing both the sgRNAs and nos-Cas9, and 5 FM7c/ph 0 w females. From each selected cross, a single y vnd deletion? nos-Cas9 w* /FM7c female was crossed with FM7c males to make a stock which was validated for the presence of an indel by genomic PCR with primers flanking the anticipated deletion and the precise endpoints of the deletion were determined by sequencing (Genewiz, France) using vnd-specific primers: vnd C∆A fwd (ccaacaaagccgagagtctct) / vnd C∆A rev (cgggaatttctaagccagggt) and vnd C∆B fwd (cgatttggggcgttgtgagta) / vnd C∆B rev (gttgggctttaatccgggagt). A CRISPR efficacy of at least 5% was obtained in both cases. We kept a single stock from each set of CRISPR experiment: vnd C∆A and vnd C∆B where Cas9 was removed.

Brain dissection, immunostaining and MARCM mosaic analysis

Adult fly heads and thoraxes were fixed for 1 h in 3.7% formaldehyde in phosphate-buffered saline (PBS) and brains were dissected in PBS. For larval and pupal brains, brains were first dissected in PBS and then fixed for 15 min in 3.7% formaldehyde in PBS. They were then treated for immunostaining as described [START_REF] Lee | Mosaic analysis with a repressible cell marker for studies of gene function in neuronal morphogenesis[END_REF][START_REF] Boulanger | ftz-f1 and Hr39 opposing roles on EcR expression during Drosophila mushroom body neuron remodeling[END_REF]. Antibodies, obtained from the Developmental Studies Hybridoma Bank, were used at the following dilutions: Mouse monoclonal anti-Fas2 (1D4) 1:10 , mouse monoclonal anti-Repo (8D1.2) 1:10 and guinea pig polyclonal anti-vnd (1/1000). Goat secondary antibodies conjugated to Cy3 against mouse or guinea pig IgG (Jackson Immunoresearch laboratory) were used at 1:300 for detection. DAPI (Sigma) was used after secondary antibody washes. Tissues were incubated 10 minutes at room temperature in a 1/1000 solution from a stock containing 1 mg/ml of DAPI solution, then washed. To generate clones in the MB, we used the MARCM technique [START_REF] Lee | Mosaic analysis with a repressible cell marker for studies of gene function in neuronal morphogenesis[END_REF]. First instar larvae were heat-shocked at 37°C for 1 h. Adult brains were fixed for 15 min in 3,7% formaldehyde in PBS before dissection and GFP visualization.

Production of the anti-Vnd antiserum

Full length vnd-RA was amplified via PCR from cDNA and cloned into a pRSET bacterial expression vector (ThermoFisher) in frame with an N-terminal 6x-His tag, amplified in DH5alpha and transformed into BL21 competent cells for protein production. His-tagged VndA was purified over a Nickel-NTA agarose resin (Qiagen) according to manufacturer's recommendation. Purified protein was injected into guinea pigs by Squarix GmbH (Marl, Germany, www.squarix.de), selected by assuring that pre-bleeds did not produce signal in Western blots of fly extracts. Successive bleeds were screened for distinct signal in Western blots using embryo extracts and by immunohistochemistry, where signal was expected post 2.5hrs AED.

Behavior experiments

For behavior experiments, flies were raised on standard medium at 18°C and 60% humidity in a 12-h light/dark cycle. We used the TARGET system [START_REF] Mcguire | Spatiotemporal rescue of memory dysfunction in Drosophila[END_REF] to inducibly express UAS-Appl + construct exclusively in adult flies and not during development. 0-3 day old flies were transferred to fresh bottles containing standard medium and kept at 30.5°C for 2 days before conditioning. For noninduced experiments, flies were transferred to fresh bottles containing standard medium and kept at 18°C before conditioning. All the behavior experiments, including the sample sizes, were conducted similarly to other studies [START_REF] De Tredern | Glial glucose fuels the neuronal pentose phosphate pathway for long-term memory[END_REF]. Groups of 20-50 flies were subjected to olfactory conditioning protocol consisting of five associative cycles spaced by 15min inter-trial intervals (5x spaced training). Conditioning was performed at 25°C using previously described barrel-type machines that allow the parallel training of up to 6 groups [START_REF] Pascual | Localization of long-term memory within the Drosophila mushroom body[END_REF]. Throughout the conditioning protocol, each barrel was plugged into a constant air flow at 2 L/min. For a single cycle of associative training, flies were first exposed to an odorant (the CS+) for 1 min while 12 pulses of 5s long 60 V electric shocks were delivered; flies were then exposed 45 s later to a second odorant without shocks (the CS-) for 1 min. The odorants 3-octanol (Fluka 74878, Sigma-Aldrich) and 4methylcyclohexanol (Fluka 66360, Sigma-Aldrich), diluted in paraffin oil to a final concentration of 2.79•10-1 g/L, were alternately used as conditioned stimuli. During unpaired conditionings, the odor and shock stimuli were delivered separately in time, with shocks occurring 3 min before the first odorant. Flies were kept at 18°C on standard medium between conditioning and the memory test. The memory test was performed at 25°C in a T-maze apparatus [START_REF] Tully | Classical conditioning and retention in normal and mutant Drosophila melanogaster[END_REF] [START_REF] Tully | Classical conditioning and retention in normal and mutant Drosophila melanogaster[END_REF], 24 h after spaced training. Each arm of the T-maze was connected to a bottle containing 3-octanol and 4methylcyclohexanol, diluted in paraffin oil to a final concentration identical to the one used for conditioning. Flies were given 1 min to choose between either arm of the T-maze. A performance score was calculated as the number of flies avoiding the conditioned odor minus the number of flies preferring the conditioned odor, divided by the total number of flies. A single performance index value is the average of two scores obtained from two groups of genotypically identical flies conditioned in two reciprocal experiments, using either odorant (3-octanol or 4methylcyclohexanol) as the CS + . The indicated 'n' is the number of independent performance index values for each genotype. The shock response tests were performed at 25°C by placing flies in two connected compartments; electric shocks were provided in only one of the compartments. Flies were given 1 min to move freely in these compartments, after which they were trapped, collected, and counted. The compartment where the electric shocks were delivered was alternated between two consecutive groups. Shock avoidance was calculated as for the memory test. Because the delivery of electric shocks can modify olfactory acuity, our olfactory avoidance tests were performed at 25°C on flies that had first been presented another odor paired with electric shocks. Innate odor avoidance was measured in a T-maze similar to those used for memory tests, in which one arm of the T-maze was connected to a bottle with odor diluted in paraffin oil and the other arm was connected to a bottle with paraffin oil only. Naive flies were given the choice between the two arms during 1 min. The odor-interlaced side was alternated for successively tested groups. Odor concentrations used in this assay were the same as for the memory assays. At these concentrations, both odorants are innately repulsive.

Microscopy and image processing

Images were acquired at room temperature using a Zeiss LSM 780 ) equipped with a 40x PLAN apochromatic 1.3 oil-immersion differential interference contrast objective lens and a Leica SP8 laser scanning confocal microscopes (MRI Platform, Institute of Human Genetics, Montpellier, France. The immersion medium used was Immersol 518F. The acquisition software used was Zen 2011 (black edition) for the Zeiss and LasX for the Leica. Contrast and relative intensities of the green (GFP), of the red (Cy3) and of the DAPI channels were processed with ImageJ (Fiji) software. Settings were optimized for detection without saturating the signal.

Statistics

Comparison between two groups expressing a qualitative variable was analyzed for statistical significance using the Chi 2 test. (BiostaTGV: http://biostatgv.sentiweb.fr/?module=tests). For behavior experiments, all data are presented as mean ± SEM. 2 groups of about 30 flies were reciprocally conditioned, using respectively octanol or methylcyclohexanol as the CS + . The memory score was calculated from the performance of two groups as described above, which represents a single experimental replicate. Comparisons of the data series between two conditions were achieved by a two-tailed unpaired t-test. Comparisons between more than two distinct groups were made using a one-way ANOVA test, followed by Tukey pairwise comparisons between the experimental groups and their controls. ANOVA results are presented as the value of the Fisher distribution F(x,y) obtained from the data, where x is the number of degrees of freedom between groups and y is the total number of degrees of freedom for the distribution. In the figures, asterisks illustrate the significance level of the t-test, or of the least significant pairwise comparison following an ANOVA. Values of p < 0.05 were considered to be significant. Statistical significance was defined as: ns = not statistically different, **p<0.01 and****p<0.0001.

Figure legends

Fig 1 .Fig 2 .

 12 Fig 1. Generating new Appl CRISPR alleles. (A) Schematic representation of Appl (blue) and vnd (green) genes and transcripts. White boxes represent 5'UTR and 3'UTR, blue and green boxes represent Appl and vnd coding sequences

Figure 2 Appl

 2 Figure 2

Figure 4 CD

 4 Figure 4

  and vnd C∆B by CRISPR/Cas9 genomic engineering that eliminate either the vnd-A or the vnd-B isoform, respectively (Fig 3 and S3 Fig). Males bearing a vnd C∆B mutant allele (vnd C∆B /Y), as well as vnd C∆B / Appl C1.4 females, are viable and have essentially wild-type MBs (Fig 3). Males bearing a vnd C∆A mutant allele (vnd C∆A /Y) are embryonic lethal similarly to other vnd null alleles (vnd A , vnd ∆38 and vnd

  1 /Y flies. For octanol, data presented as mean ± SEM. ttest n=12, t22 = 1.12, P = 0.26, ns: not significant. For methylcyclohexanol, data presented as mean ± SEM. t-test n=12, t22 = 1.51, P = 0.15, ns: not significant. (B) Shock reactivity was normal in Appl C2.1 /Y flies. Data presented as mean ± SEM. t-test. n=12, t22 = 1.47, P = 0.16, ns: not significant. (C) Flies only null for Appl have no measurable LTM which is fully rescued by the restricted expression of Appl in adult MBs. Data presented as mean ± SEM. n = 22-23, F(4, 108) = 16,41, P < 0.0001. Significance level of t-test or least significant pairwise comparison following ANOVA: ****p < 0.0001, ns: not significant. which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint this version posted September 3, 2023. ; https://doi.org/10.1101/2023.09.01.555727 doi: bioRxiv preprint

					Figure 1
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	C	WT	0%	D	Appl C1.4 / Y	66%	E	Appl C2.1 / Y	9%

(

  4 / vnd C∆B vnd C∆B / Y which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint this version posted September 3, 2023. ; https://doi.org/10.1101/2023.09.01.555727 doi: bioRxiv preprint

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  Appl C2.1 / Y ; UAS-Appl / + Appl C2.1 / Y ; tubP-Gal80 ts / UAS-Appl ; VT30559 / +

								Figure 6
	A					B			C
			Olfactory Acuity		Shock Avoidance		Long Term Memory
		1.0	n.s.		n.s.		1.0	n.s.		0.4
	Avoidance Index	0.2 0.4 0.6 0.8				Avoidance Index	0.2 0.4 0.6 0.8		24h memory score	0.1 0.2 0.3	****
		0.0	12	12	12	12	0.0	12	12	0.0	****
			Octanol	Méthylcyclohexanol				23	22	23	23	22
				+/ Y		Appl C2.1 / Y	+/ Y ; tubP-Gal80 ts / + ; VT30559 / +

Acknowledgments

We thank Aymeric Chartier (IGH, Montpellier) for fruitful discussions in order to obtain Appl CRISPR alleles and Natasha Danda and Nuran Demirbas for technical assistance. We thank the Bloomington Drosophila Stock Center (Indiana University), the BioCampus RAM-Drosophila facility and the BioCampus imaging facility MRI (Montpellier, France). We acknowledge BestGene and Genewiz for transgene service and DNA sequencing respectively. The 1D4 anti-Fasciclin II hybridoma and the 8D12 anti-Repo monoclonal antibody developed by Corey Goodman were obtained from the Developmental Studies Hybridoma Bank, created by the NICHD of the NIH and maintained at the University of Iowa, Department of Biology, Iowa City, IA 52242. We thank Chris Doe for the vnd ∆38 and vnd 6 alleles and Tzumin Lee for the vnd-T2A-GAL4 line. Funding: C.M. was supported by a PhD grant from the Ministère de l'Enseignement Supérieur et de la Recherche. C.M. and G.U.B. were supported by the Fondation pour la Recherche Médicale (FRM) respectively for the 4 th PhD year and for a 3 year post-doctoral fellowship. Work in the laboratory of J.-M.D. was supported by the Centre National de la Recherche Scientifique (CNRS), the Association pour la Recherche sur le Cancer (grants PJA 20151203422), the FRM (Programme "EQUIPES FRM2016" project DEQ20160334870) and the ANR 2021 ORIO. Work in the laboratory of T.P. was supported by the Fonds ESPCI and the Fondation Langlois.


Pradere

M M Groz

A Sommier

E Abisset-Chavanne

J C Batsale

C Pradere

Flash Method and Bayesian inference for measurement of thermophysical fields

Flash method and Bayesian inference for measurement of thermophysical fields

Introduction

Thermal diffusivity is one of the main parameters in the thermal characterization of materials. The method proposed by Parker [START_REF] Parker | Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity[END_REF] is one of the oldest thermal methods used for the characterization of diffusion by rear face flash. The hypotheses considered in Parker's method are that the excitation flux is uniform over the entire surface and is of short duration (Dirac-type excitation in time). The material is also considered adiabatic. Under these conditions, the maximum sensitivity to thermal dispersion is reached at the half-rise time t 1/2 of the normalized thermogram. Parker's method is still widely used because of its implementation simplicity and its large application domain because it can be used for insulators as well as for conductors. However, the assumptions that it requires are not always realistic; in particular, adiabatic boundary conditions are rarely verified. Indeed, the conductive heat losses at material surfaces are rarely negligible. Since this method was presented, several methods were subsequently proposed to take into account the thermal losses for diffusivity estimation. Among them, the method proposed by Balageas [START_REF] Balageas | Flash thermal diffusivity measurements using a novel temperature-time history analysis[END_REF] was to focus the study near the time origin, where the influence of heat loss is low. Other methods proposed by Degiovanni et al. use particular points of the thermogram during the rising time or the temporal moments between two points to estimate the diffusivity [3, 4, 5]. Another example is the method of partial times with convective losses proposed by Degiovanni [6] or the approach of asymptotic expansions proposed by Mourand [7]. Most of these methods use particular points or areas of the thermograms to estimate the diffusivity. Other works proposed estimating the thermal diffusivity with whole thermograms using least-squares methods by minimizing the residual composed of the norm of the difference between the measured data and the theoretical model [8,9,10,[START_REF] Sheindlin | Advances in the use of laser-flash techniques for thermal diffusivity measurement[END_REF]. Most recently, a method was developed [12,13] to calculate the thermal diffusivity as well as a global estimation of the heat losses. These parameters are estimated through the Biot numbers and calculated from the ratio between the integrals of the complete temperature trends of the two faces. One of the difficulties of all these methods is their sensitivity to noise level. During the last two decades, Bayesian methodologies were developed [START_REF] Fudym | Bayesian approach for thermal diffusivity mapping from infrared images with spatially random heat pulse heating[END_REF][START_REF] Berger | Bayesian inference for estimating thermal properties of a historic building wall[END_REF], which have the advantage of being minimally sensitive to noise. In these works, the Markov chain Monte Carlo method (MCMC) is used. In this paper, a method based on Bayesian inference is proposed to locally and conjointly estimate the two parameters on each point of the material with whole thermograms without using the MCMC methods but with the generation of a base that represents the physical model. This method is compared to the classical minimizations method using least squares, and the performances are compared, particularly on the estimation precision and the time computation of the algorithm. At the end of this work, a methodology is also proposed to quantify the average value of the thermal conductivity of a material.

2 Methodology for the simultaneous estimation of the heat loss and the diffusivity

Experimental setup

Originally, the experimental setup proposed by Parker was made with a thermocouple. Now, with the advance of the technology, a classical rear face flash system is associated with an IR camera. The setup used in this paper is presented in figure 1. The flash lamp comes from Uniblitz and has an energy E of 1600 J. The lamp is synchronized with the IR camera by using an analogical TTL link to perform a pretrigger mode before the flash. The IR camera is an MCT longwave (λ = 9-11µm) FLIR SC7600 with a matrix sensor of 320×256 pixels and a pitch of 30×30 µm 2 . With the lenses used and the sample distance, the resulting spatial resolution was 280×280 µm 2 per pixel.

Direct model

In the rear flash experiment, the sample is subjected to a heat pulse of energy Q 0 (J•m -2 ) on its front face z = 0. The convective losses are modeled by the heat transfer coefficients h 0 and h e (W•m -2 •K -1 ) for the front and rear faces, respectively, as illustrated by figure 2.

Direct model

The governing equations [16] are then given by:

                   ∂T (z, t) ∂t = a ∂ 2 T (z, t) ∂z 2 -λ ∂T (z, t) ∂z z=0 = Q 0 δ(t) -h 0 T (0, t) -λ ∂T (z, t) ∂z z=e = h e T (e, t) (1)
where a is the diffusivity coefficient (m

2 •s -1 ), λ is the thermal conductivity (W•m -1 • K -1
) and e the thickness of the material (m).

In the Laplace domain, equation 1 takes the following form:

                   p θ(z, p) = a d 2 θ(z, p) dz 2 -λ dθ(z, p) dz z=0 = Q 0 -h 0 θ(0, p) -λ dθ(z, p) dz z=e = h e θ(e, p) (2)
As presented in equation 3, we can introduce the τ (s) parameter and the dimensionless Biot numbers for the rear face Bi 0 and the front face Bi e :

τ = e 2 a ; Bi 0 = h 0 e λ ; Bi e = h e e λ , (3)
Equation 2 can easily be solved in terms of the rear face Laplace temperature:

θ(e, p) = e λ 2Q 0 √ pτ e -√ pτ Bi 0 + √ pτ Bi e + √ pτ - √ pτ -Bi 0 √ pτ -Bi e e -2 √ pτ (4)
In this paper, the heat losses are assumed to be identical on the front and rear faces; therefore, Bi 0 = Bi e = Bi. As λ = aρC p , where ρ (kg•m -3 ) and C p (J•K -1 •kg -1 ) are the density and specific heat of the material, respectively, equation 4 can be simplified and written as:

θ(e, p) = 2Q 0 eρC p τ √ pτ e -√ pτ Bi + √ pτ 2 - √ pτ -Bi 2 e -2 √ pτ (5)
A numerical inverse Laplace transform can then be performed on equation 5 by using the Stehfest algorithm [START_REF] Stehfest | Algorithm 368: Numerical inversion of laplace transforms [d5[END_REF]16] to calculate the temperature in the temporal domain.

In the flash method, the quantity of energy Q 0 is hard to quantify because it is not directly related to the light energy quantity that is radiated by the flash lamp. Actually, the energy received by the sample is also a function of the optical properties as the absorption of the sample. To remove this parameter, the temperature is normalized. The thermogram is then normalized by the maximum value of the temperature, which leads to a temperature variation between 0 and 1. Thanks to this normalization, the temperature is only dependent on the Biot number and the parameter τ .

Sensitivity study with respect to the parameters

In this paper, the objective is the estimation of both the diffusivity a and the heat losses h on a sample. However, the parameters τ and Bi have the advantage of being more universal. Indeed, a fixed value of τ can correspond to different couples (a, e) of diffusivities and thicknesses. Therefore, the estimation of the parameter τ enables the estimation of the diffusivity a if the thickness e is known, or inversely, the estimation of the thickness e if the diffusivity a is known. Likewise, a fixed Biot number can correspond to a large number of couples (h, e, λ), and the estimation of each of these parameters can be made. In particular, if the thickness e is known, a characteristic length corresponding to a convecto-conductive ratio can be estimated. Thus, throughout the rest of the paper, the study will be conducted with the parameters τ and Bi.

A sensitivity study with respect to the parameters τ and Bi is therefore performed in this part to highlight the influence of each of these parameters on the temperature. Several thermal responses are computed with the help of equation 5 by varying these parameters. For a fixed Biot number, the higher the value of τ (i.e. the smaller the diffusivity a for a fixed thickness), the later the maximum temperature is reached. Predictably, as concluded by Parker [START_REF] Parker | Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity[END_REF], the maximum sensitivity with respect to the τ parameter, given by the maximum of the partial derivatives illustrated in figure 3.c, is reached at the half rise time t 1/2 of each normalized thermogram (illustrated in figure 3.a).

Conversely, for a fixed parameter τ , the higher the Biot number (i.e. the higher the heat losses), the sooner the maximum temperature is reached. The maximum sensitivity with respect to the Biot number is reached after the rise time of the temperature. Predictably, for small values of Bi, the sensitivity of the thermal response with respect to this parameter is low, but the higher the values of Bi are, the less the influence of this parameter is negligible. To emphasize this point, the estimation of the τ parameter with the Parker method is performed for different values of Biot numbers between 0 and 2 and for τ = 100. The method proposed by Parker [START_REF] Parker | Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity[END_REF] consists of determining the thermal diffusivity a of the material with the help of equation 6:

a = 0.139 e 2 t 1/2 , (6)
where a (m•s -2 ) is the diffusivity coefficient and e (m) is the depth of the material. This equation can be written with the τ parameter as:

τ = t 1/2 0.139 . ( 7
)
The Parker estimation of the τ parameters and the absolute errors performed as percentages are illustrated in figure 4 for the different Biot numbers. τ is underestimated when Bi is a nonnull value (i.e. in presence of heat losses). Moreover, the higher Bi is, the higher the error performed on the estimation of τ . For Bi = 2, the error is almost 40%, which is nonnegligible. This result is explained by the fact that the maximum temperature is reached sooner with increasing values of Bi, and therefore, the value t 1/2 is underestimated. The estimation of the Biot number is thus required to obtain a good estimation of the τ parameter.

Estimation methodology with Bayesian inference

The method generally used for the estimation of both of the parameters is an inversion. The Levenberg-Marquardt (LM) algorithm [18] is usually computed for this problem because of its ease of implementation. However, for a large number of measures, the inversion algorithm can be time consuming. In this paper, a methodology based on Bayesian inference is proposed as an alternative and compared to LM inversion.

The problem of sample characteristics estimation can be defined in each pixel of the image as follows: knowing the measured temperature evolution T m (t) at z = e (front face), determine the Biot number Bi and the parameter τ , i.e., the couple β = (τ, Bi).

The methodology proposed in this paper uses Bayesian inference, a probabilistic method based on the Bayes relation [19]:

π posterior (β|T m (t)) = π(T m (t)|β) • π prior (β) π(T m (t)) (8)
• π posterior (β|T m (t)) is the posterior probability density, i.e. the conditional probability of β, given the measured temperature T m (t).

• π prior (β) is the prior density, which is the a priori information about β prior to the measurements.

• π(T m (t)|β) is the likelihood function, which expresses the likelihood of different temperature measurement outcomes T m (t) with β given.

• π(T m (t)) is the model evidence or the marginal probability density of the measurements, which plays the role of a normalizing constant.

Bayesian inference is then a stochastic approach that results in a probability density function providing a probability weight to each possible variable β. In this paper, the evidence is not considered. Indeed, even if its computation does not represent a tricky issue due to the analytical nature of the direct problem, the knowledge of the more probable β in each pixel, independent of the normalization, is sufficient to reconstruct the sample diffusivity and loss coefficient fields.

For the prior, which allows us to account for the a priori knowledge of β, only a limitation of the parametric space (τ, Bi) in the variation range of the parameters is considered to be as general as possible. The prior is taken as a uniform probability of the variation range of each parameter [τ min , τ max ] and [Bi min , Bi max ] with a null probability outside. Finally, in this case, the posterior probability density is directly linked to the likelihood function [20]:

π posterior (β|T m (t)) ∝ π(T m (t)|β), β ∈ [τ min , τ max ] × [Bi min , Bi max ] (9)
The likelihood function π(T m (t)|β) is defined as follows:

π(T m (t)|β) ∝ exp - 1 2Γ 2 ||T m (t) -T r (t, β)|| 2 2 , (10)
where T r (t, β) is the front face temperature of reference computed using the direct model defined in section 2.2 for β ∈ [τ min , τ max ] × [Bi min , Bi max ], and T m (t) is the measured front face temperature at each pixel. Γ is the standart deviation and is set at 1 in this paper.

In this case, as the forward problem is analytical and few parameters are considered, the likelihood function is built by directly sampling the discretized parametric space (τ, Bi). This allows us to not have to rely on Markov chain Monte Carlo methods, which are commonly used when the large computation time of the forward problem prevents the exploration of the entire parameter space [20, [START_REF] Carlin | Bayesian model choice via markov chain monte carlo methods[END_REF][START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]23].

3 Presentation and study of the method

Application on numerical examples

In this section, the thermograms are numerically computed for different (τ, Bi) to evaluate the performance of the proposed Bayesian methodology. To study the Bayesian method, three thermal responses are studied with different couples of parameters (τ , Bi):

• T 1 , with τ = 100 s and Bi = 0.1.

• T 2 , with τ = 100 s and Bi = 0.5.

• T 3 , with τ = 50 s and Bi = 1.

Generation of the numerical reference base

For the Bayesian method, a numerical base that contains the temperature of reference T r (t, β) is used in equation 10. According to the needs, this base can be calculated beforehand in the form of an abacus or can be generated at the same time as the estimation. The only essential point of this base is that the time interval and the time discretization of thermograms in the base must be the same as those of the measured thermograms T m (t). Conversely, regarding the choice of the parameters β, there are no real constraints for the base, except that it is recommended to have a sufficiently broad spectrum so that it can approach the real solution.

For the generation of the base, the idea is to use the Parker method to focus the Bayesian estimation around a first estimation τ 0 instead of searching in the whole range of existing parameters τ . For the example illustrated by figure 5, as it is supposed we have a better a priori knowledge for the diffusivity than for the heat loss thanks to the Parker method, the base is generated with 100 values of τ linearly distributed on the interval [20, 200] and 100 values of Bi linearly distributed on the interval [0, 2] to have 10000 different thermograms with a unique couple (τ, Bi). To understand the physical correspondence, for a sample of thickness e = 1 cm, τ varying from 20 to 200 corresponds to the diffusivity a varying on the interval [0.5, 5] × 10 -6 m 2 •s -1 . Bi varies from 0 to 2 for a sample of conductivity

λ = 1 W•m -1 •K -1
(a PVC, for example), corresponds to the heat loss h (W•m -2 •K -1 ) varying on the interval [0, 200].

The complete calculation of such a base takes 5.1 s on a 2.3 GHz quad-core i7 laptop with 16 GHz of RAM.

Implementation of the estimation algorithm

The Bayesian method, as depicted in part 2.4, is then performed for each of the three thermograms and for each noise level, illustrated by figure 5 with the help of the generated base.

For the three thermograms, the likelihoods are then computed from equation 10. The posterior probabilities are obtained for each couple (τ, Bi) of the base. The obtained repartition of posterior probabilities for the thermogram T 3 is presented in figure 6.a for the case with SNR = 200 (low noise level) and in figure 6.b for SNR = 20 (high noise level). The reconstructed thermograms with the estimated parameters fit the measured thermograms with a very good accuracy, for any noise level, for the Bayesian method. However, for the high noise level, the minimization can lead to a bad estimation, as seen for the thermogram T 1 . To compare the errors made by the two methods, a parametric study for large numbers of τ and Bi was performed for different noise levels.

Parametric study and noise influence

In this part, the two methods are performed for broad values of couples (τ , Bi) and for different noise levels. As in the previous part, the "real thermograms" are computed numerically, and white noise is added to simulate a real measure. Then, the estimation of the couple (τ , Bi) is made with the two methods (minimization and Bayesian). For the minimization, the initial conditions are given by Parker for τ , and Bi takes n equal to 1. This methodology is performed for each couple (τ , Bi), and the absolute error is calculated between the estimated and real parameters. The fields of errors are illustrated in figure 8 for each parameter and for the two methods for an SNR = 20.

For the minimization, for SNR = 20, there is a large area located for τ ∈ [60, 120] and for Bi ∈ [0.1, 0.6] where the error is high, both for the τ and the Bi parameter. As shown in figure 8.a., the error is approximately 35% for the τ parameter, and in figure 8.c, it can be higher than 100% for the Bi parameter. For the Bayesian method, this area is not specific, and the errors are much lower: less than 5% for τ and less than 20%, as illustrated in figure 8 For the minimization method, this "area of error" is dependent on the initial condition of the minimization. The location of this area is indeed decaled for different initial conditions for Bi. Likewise, the intensity of the error is dependent on the SNR. To illustrate these results, the evolution of the error as a function of the SNR is presented in figure 9 for point 1, which is located in figure 8. This point corresponds to the couple (τ , Bi) = (70, 0.3). For both methods, the error made on the τ parameter is represented with a continuous line, whereas the error made on Bi is represented with a dashed line. For the minimization method, the error decreases when the SNR increases.

13/21 For the Bayesian method, the errors are almost constant and almost independent of the noise level. For the Bayesian method, it is also important to note that the error depends on the chosen interval between two parameters of the generated base because it gives precision to the estimation. It can be noted that it is always possible to generate a second refined numerical base around the estimated couple (τ, Bi) to obtain a better estimation if needed, in the same manner as presented in [24].

Another point to discuss is the time computation of the algorithm. Once the base is generated, the Bayesian algorithm to perform the estimations on the three thermograms takes 0.07 s, whereas the LM algorithm takes 0.18 s. In this case, the difference is negligible, but for larger values of thermograms, the difference will be important, as will be presented in the next part.

Thus, the Bayesian method enables the estimation (τ, Bi) parameters with very good precision. This method is easy to implement and minimally sensitive to noise.

Experimental case on high insulated media

In this part, the results obtained on an experimental case are presented. The studied sample is composed of two juxtaposed materials of the same thickness e = 3 mm but different diffusivities: a PVC and a foam, as illustrated by figure 10. The theoretical values of diffusivity for the PVC is around 2 × 10 -7 m 2 •s -1 whereas it is 1.5 × 10 -7 m 2 •s -1 for the Foam. The objective is to determine the parameters τ and Bi on each pixel of the surface. In this example, a foam and a PVC were chosen because the thermal conductivity for an insulator (foam) tends toward zero, and therefore, the Biot number increases. The ratio between the two Biot numbers will enable the estimation of the thermal conductivity of the foam, as will be presented at the end of this part. The experimental setup used for this example is presented in part 2.1. The study was focused on the sample on a rectangle of dimensions 8 × 7 cm 2 , with a resolution of 280 µm 2 per pixel. The camera frequency rate was fixed to 10 Hz, and the final acquisition time was fixed at 60 s.

On each pixel, a first estimation τ 0 is made with the Parker method, and the obtained results are illustrated in figure 11. For both materials, the estimations are similar, τ ≈ 40 s at each pixel. The distinctive line that can be seen in figure 11 at y = 0, where the τ estimation seems lower, is an artefact because the junction between the two materials (foam + PVC) was not perfect.

Then, to perform the Bayesian methodology, the implementation on a base, as depicted in part 2.4, is performed. It can be observed that the results on the estimations for the couple (τ, Bi) are very similar for the two different methods of estimation. The depth of each material is 3 mm, then the estima-tion of the thermal diffusivity can be made on each point with equation 3. The average value of the diffusivity is 1.99 × 10 -7 m 2 •s -1 for the PVC and for the foam it is 1.47 × 10 -7 m 2 •s -1 which are consistent with the theoretical values.

To evaluate the accuracy of the estimation, the focus is on points P P V C and P F oam of the sample, the first one on the PVC and the second on the foam, represented in figure 12.a. The thermograms measured by the IR camera are illustrated in figure 13 as well as their fit with the estimated (τ, Bi) for both the minimization and the Bayesian methodology. The estimation of the τ parameter with the Parker method and the reconstructed parameter is also presented. The Parker estimation leads to the same value of τ for the two pixels (PVC and Foam). The two resulting thermograms are then superposed and do not fit the measured thermograms, even during the rising time of temperature, as shown in figure 13. These results are consistent with the conclusion drawn by part 2.3. Figure 13 shows that the estimation of the couple (τ, Bi) is better with the Bayesian method than with the LM minimization because the reconstructed thermograms with the estimated parameters fit the measured thermograms with better accuracy.

Moreover, the high difference between the two methods is the computation time: the complete estimation of the couple (τ, Bi) on each pixel of the sample was taking at 2 h 35 min for the LM algorithm, whereas it was only 20 min with the Bayesian methodology, with similar precision. This difference is explained by the type of algorithm. Minimization methods such as the LM algorithm require the inversion of matrices, which is of high complexity, whereas Bayesian methodology is based on the subtraction of thermograms, which is of minimal complexity.

Relative estimation of the thermal conductivity

The Biot number is dependent on the heat loss h, the thickness e and the thermal conductivity λ of the studied material. In the previous experiment, the thickness e was the same for the PVC and the foam. The two materials were placed in the same environment; therefore, the heat loss coefficient h can be assumed to be identical. Then, the ratio between the Biot numbers leads to the ratio between the thermal conductivity of the two materials, as presented by equation 11:

Bi F oam Bi P V C = λ P V C λ F oam (11)
Thus, the estimation of the couples (τ, Bi) for two juxtaposed materials enables a ratio between the thermal conductivities of these two materials. If the conductivity of one of the materials is known, the second conductivity can then be estimated.

The ratio between the Biot numbers and the mean Biot number of the PVC is determined on the results obtained in figure 12.d and illustrated in figure 14. The mean value of Bi P V C is 0.11 in this example. The mean Biot number ratio obtained for the foam is 26.9, which means that the thermal conductivity of the PVC is 26.9 times greater than the thermal conductivity of the foam. This result is consistent with the materials used; the thermal conductivity of a PVC is approximately 10 -1 W.m -1 . K -1 , whereas the foam used is approximately 2-3×10 -2 W.m -1 .K -1 . This setup can then be used as a first estimation method for the thermal conductivity of any material in addition to the thermal diffusivity. This methodology can be compared to the hot disk transient plane source (TPS) methods [25,26,27] that allow the estimation of the thermal conductivity, the thermal diffusivity and the specific heat capacity in a single measurement. Conductivity and diffusivity are tested directly, and specific heat is calculated from the former two.

Conclusion

A method based on Bayesian inference was proposed to conjointly estimate the following two fields of thermophysical parameters: a thermal characteristic time τ directly linked to the thermal diffusivity and the thickness and the Biot number, directly linked to the heat loss and the thermal conductivity. This method is robust to noise and leads to very good estimations of the parameters, with results comparable to a classical minimization method. This method is, however, very fast and less time consuming. A setup and methodology are also presented to estimate the average value of the thermal conductivity of an unknown material.

Fig. 1 :

 1 Fig. 1: Schema of the experimental setup.

Fig. 2 :

 2 Fig. 2: Schema of the material.

  Figure 3.a presents the normalized thermograms for Bi = 0.2 and for different τ between 10 and 100, whereas Figure 3.b illustrates the normalized thermograms for τ = 100 and for Bi varying between 0 and 2. The study of the influence of each of the two parameters (τ, Bi) is performed with the help of the partial derivatives of the thermal responses with respect to these two parameters. Figure 3.c illustrates the partial derivatives ∂T /∂τ of each of the thermograms of Figure 3.a, whereas Figure 3.d illustrates the partial derivatives ∂T /∂Bi of each of the thermograms of Figure 3.b.

Fig. 3 :

 3 Fig. 3: Sensitivity study with respect to the parameters τ and Bi. Normalized temperatures for (a) Bi = 0.2 constant and different values of τ and (b) τ = 100 constant and different values of Biot number. Partial derivatives of the thermograms with respect to (c) the parameter τ for a fixed Biot number and (d) the Biot number for a fixed τ .

Fig. 4 :

 4 Fig. 4: Estimation of the τ parameter (diffusivities) with the Parker method for different Biot numbers (heat losses) and the corresponding absolute errors (in percentage).

Fig. 5 :

 5 Fig. 5: Normalized thermograms for the three different couples (τ, Bi) presented: (a) SNR = 200 and (b) SNR = 20.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Posterior probabilities for the thermogram T 3 for each value of the β couple: Representation in image (a) for SNR = 200 and (b) for SNR = 20. Representation in 3D (c) for SNR = 200 and (d) for SNR = 20.

Fig. 8 :

 8 Fig. 8: Fields of error estimations on the parameters τ and Bi for SNR = 10. Error on the estimation of τ with a) the minimization and b) the Bayesian method. Error on the estimation of Bi with c) the minimization and d) the Bayesian method.

Fig. 9 :

 9 Fig. 9: Errors in the estimation of τ and Bi as a function of the SNR (logarithmic scale) for point 1 presented in figure 8.

Fig. 10 :

 10 Fig. 10: Photography of the studied material.

Fig. 11 :

 11 Fig. 11: Estimation of the parameters τ on each pixel of the sample with the Parker method: a) with a scale between 20 and 100 and b) with a scale centered at approximately 40.

Fig. 12 :

 12 Fig. 12: Estimation of the parameters τ on each pixel of the experimental material by a) minimization with the LM algorithm and b) Bayesian method. Estimation of the adimensional Biot number Bi on each pixel of the sample with c) minimization and d) Bayesian method.

Fig. 13 :

 13 Fig. 13: Measured thermograms at pixels P P V C and P F oam of figure 12.a and their fit with the Bayesian method and with LM minimization.

Fig. 14 :

 14 Fig. 14: Ratio between the Biot numbers of the Foam and the mean Biot numbers of the PVC leading to the ratio values of the thermal conductivity between the two materials.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author declarations

Conflict of interest

The authors have no conflicts to disclose.