Martin Villalba
email: martin.villalba@inserm.fr

Nerea Allende

Nerea Allende-Vega
email: nerea.allende-vega@inserm.fr

Metabolic stress controls mutant p53 R248Q stability in acute myeloid leukemia cells

Keywords: metabolic stress, mutant p53, dichloroacetate (DCA), 17-AAG, autophagy, acute myeloid leukemia (AML)

Metabolic stress controls mutant p53 R248Q stability in acute myeloid leukemia cells

Introduction

Wild-type p53 (wt p53) was refereed once as "the guardian of the genome" for its important role as a tumor suppressor gene [START_REF] Lane | T antigen is bound to a host protein in SV40transformed cells[END_REF] . Today p53 is not only known as a tumor suppressor but also a master regulator of many cellular processes such as cell cycle, apoptosis, DNA repair, inflammation and metabolism [START_REF] Gudkov | Prospective therapeutic applications of p53 inhibitors[END_REF] . The p53 gene is the most frequent target for mutation in human cancer, including hematological malignancies [START_REF] Brosh | When mutants gain new powers: news from the mutant p53 field[END_REF] .

The frequency of p53 mutations in acute myeloid leukemia (AML) is approximately 10%. However, in AML with complex karyotype, the rate of p53 mutations and/or deletions is almost 70% [START_REF] Kadia | TP53 mutations in newly diagnosed acute myeloid leukemia: Clinicomolecular characteristics, response to therapy, and outcomes[END_REF] . Furthermore, p53 mutations are associated with poor prognosis and decreased survival in AML.

Mutations are found in all coding exons of the p53 gene, but most of them are located in the DNA-binding domain, with the most common in codons 175, 245, 248, 273 and 282. These are the "hot spot" residues, which are very frequently mutated in all types of cancer [START_REF] Cho | Crystal structure of a p53 tumor suppressor-DNA complex: understanding tumorigenic mutations[END_REF] . These mutations do not always correlate with loss of function of p53 and can actively promote tumor growth by gain-of-function (GOF) mechanism [START_REF] Kim | Mutant p53 partners in crime[END_REF][START_REF] Muller | p53 and its mutants in tumor cell migration and invasion[END_REF][START_REF] Oren | Mutant p53 gain-of-function in cancer[END_REF] . The important role of GOF by mutant p53 (mt p53) is further supported by the finding that patients carrying missense mutation and expressing mt p53 in the germline have a significantly earlier cancer onset than patients with mutations in TP53 that result in loss of p53 protein [START_REF] Bougeard | Molecular basis of the Li-Fraumeni syndrome: an update from the French LFS families[END_REF][START_REF] Zerdoumi | Drastic effect of germline TP53 missense mutations in Li-Fraumeni patients[END_REF] . Moreover, mt p53 accumulation is critical for p53 oncogenic GOF that actively contributes to cancer development and progression [START_REF] Suh | Multiple stress signals activate mutant p53 in vivo[END_REF] .

R248 is mutated into three amino acids R248Q, R248W and R248L [START_REF] Petitjean | Impact of mutant p53 functional properties on TP53 mutation patterns and tumor phenotype: lessons from recent developments in the IARC TP53 database[END_REF] . Interestingly, p53-R248Q, but not p53-R248W, confers invasive ability when overexpressed in p53null cells [START_REF] Yoshikawa | Mutant p53 R248Q but not R248W enhances in vitro invasiveness of human lung cancer NCI-H1299 cells[END_REF] . Thus, not only the position of the mutation but also the nature of the substitution may influence the activity of the resulting mt p53 protein. In fact, mutant R248Q induces more aggressive tumors in mice compare with other hotspot mutants [START_REF] Hanel | Two hot spot mutant p53 mouse models display differential gain of function in tumorigenesis[END_REF][START_REF] Mello | Not all p53 gain-of-function mutants are created equal[END_REF][START_REF] Ng | R248Q mutation--Beyond p53-DNA binding[END_REF] . R248Q has a greater tendency to aggregate and can seed the aggregation of wt p53. In breast cancer samples, R248Q aggregates into prion-like amyloid oligomers sequestrating and inactivating wt p53 [START_REF] Bom | Mutant p53 aggregates into prion-like amyloid oligomers and fibrils: implications for cancer[END_REF] . Codon 248 of the p53 protein is most frequently mutated in pancreatic tumors (based on cBioPortal), in lymphomas [START_REF] Peller | TP53 in hematological cancer: low incidence of mutations with significant clinical relevance[END_REF] , myelodysplastic syndromes (MSD) and AML [START_REF] Lindsley | Prognostic Mutations in Myelodysplastic Syndrome after Stem-Cell Transplantation[END_REF][START_REF] Rucker | TP53 alterations in acute myeloid leukemia with complex karyotype correlate with specific copy number alterations, monosomal karyotype, and dismal outcome[END_REF] . In summary, it is essential to further study mechanisms reducing the function of this p53 mutant, but with a minimal effect on wt p53. Wt p53 stability is mainly control by the proteasome-ubiquitin pathway, however it is still unclear which pathway degrades mt p53. In response to different stresses, both wt and mt p53 accumulate in cells. While wt p53 returns to basal level following recovery from stress, mt p53 remains stable [START_REF] Vijayakumaran | Regulation of Mutant p53 Protein Expression[END_REF] . Certain mt p53 proteins accumulate to high levels in tumor cells [START_REF] Bartek | Genetic and immunochemical analysis of mutant p53 in human breast cancer cell lines[END_REF] due to its interaction with the chaperones Hsp70 and Hsp90. Hsp90 inactivates the E3 ligases MDM2 and CHIP, impairing proteasomal degradation of mt p53 [START_REF] Li | Functional inactivation of endogenous MDM2 and CHIP by HSP90 causes aberrant stabilization of mutant p53 in human cancer cells[END_REF] . mt p53 degradation also occurs by different types of autophagy: macroautophagy and Chaperone-Mediated Autophagy (CMA) [START_REF] Choudhury | Dissecting the pathways that destabilize mutant p53: the proteasome or autophagy?[END_REF] . Macroautophagy, induced by glucose restriction or by proteasomal inhibition, promotes mt p53 degradation [START_REF] Rodriguez | Dietary downregulation of mutant p53 levels via glucose restriction: mechanisms and implications for tumor therapy[END_REF] . When nutritional deprivation inhibits macroautophagy, CMA is activated and induces mt p53 degradation [START_REF] Vakifahmetoglu-Norberg | Chaperone-mediated autophagy degrades mutant p53[END_REF] . For further complexity, mt p53 can inhibit autophagy [START_REF] Morselli | Mutant p53 protein localized in the cytoplasm inhibits autophagy[END_REF][START_REF] Cordani | Mutant p53 proteins counteract autophagic mechanism sensitizing cancer cells to mTOR inhibition[END_REF] .

One approach to target mt p53 is to reduce mt p53 levels with little effect on wt p53 using compounds that promote degradation of mt p53 such as the Hsp90 inhibitor 17-AAG [START_REF] Li | Functional inactivation of endogenous MDM2 and CHIP by HSP90 causes aberrant stabilization of mutant p53 in human cancer cells[END_REF][START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . 17-AAG is a geldanamycin analogue, currently in clinical trials as anticancer drug that triggers the activation of a heat shock response, promotes proteasome degradation and induces the autophagic pathway [START_REF] Zhang | Targeting multiple signal transduction pathways through inhibition of Hsp90[END_REF][START_REF] Riedel | 17-AAG induces cytoplasmic alpha-synuclein aggregate clearance by induction of autophagy[END_REF] . [START_REF] Neckers | Heat-shock protein 90 inhibitors as novel cancer chemotherapeutic agents[END_REF] .

In this study, we uncover different mechanisms that promote mutant p53-R248Q depletion in different cellular contexts. In tumors growing in normal, no stress, conditions, 17-AAG eliminates R248Q through macroautophagy. However, in tumors with macroautophagy inhibition and high stability of mt p53, 17-AAG still was able to induce mt p53 degradation through CMA. Also we showed that metabolic stress caused the pyruvate dehydrogenase kinase-1 (PDK1) inhibitor dichloroacetate (DCA) promotes higher accumulation and stabilization of R248Q protein by increasing its interaction with the Hsp90 chaperone machinery. Furthermore, accumulation of R248Q prevents macroautophagy by inhibiting the expression of several macroautophagy genes. Our data demonstrate that there is a negative feedback loop between macroautophagy and mt p53. Under DCA-induced metabolic stress, when macroautophagy is largely reduced, 17-AAG induces mt p53 degradation through CMA.

Results

Different effect of DCA and 17-AAG on R248Q stability

DCA causes metabolic stress by inhibiting PDK1 in AML cells [START_REF] Khan | Human Leukemic Cells performing Oxidative Phosphorylation (OXPHOS) Generate an Antioxidant Response Independently of Reactive Oxygen species (ROS) Production[END_REF][START_REF] Khan | Mitochondrial Complex I activity signals antioxidant response through ERK5[END_REF] . This forces cells to decrease glycolysis and increase oxidative phosphorylation (OXPHOS) [START_REF] Khan | Human Leukemic Cells performing Oxidative Phosphorylation (OXPHOS) Generate an Antioxidant Response Independently of Reactive Oxygen species (ROS) Production[END_REF][START_REF] Khan | Mitochondrial Complex I activity signals antioxidant response through ERK5[END_REF][START_REF] Khan | The PDK1 Inhibitor Dichloroacetate Controls Cholesterol Homeostasis Through the ERK5/MEF2 Pathway[END_REF] . DCA induces wt p53 transcriptional activity via AMPK and its efficacy to cause cell cycle arrest depends on p53 status [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . Besides, we observed that both wt p53 and mt p53 protein levels accumulated after DCA treatment including in the NB4 cell line, which carries p53 R248Q [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . We extensively investigated the functional activity of wt p53 after DCA treatment and found that p53 induced cell cycle arrest in G0/G1 phase, although failed to induce programmed cell death (PCD). Cell cycle arrest involved p53 transcriptional activity because we observed upregulation of MDM2 and p21 mRNAs [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] (Supplementary Fig. 1a). We also described in this work that DCA-induced metabolic stress depended in wt p53 and involved mRNA expression of its metabolic targets GLS2, SCO2 and AMBKβ [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . Moreover, DCA increased ROS expression and disturbed oxygen consumption [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] .

We confirmed here that DCA induced accumulation of mt p53 protein without affecting p53 mRNA (Fig. 1a and Supplementary Fig. 1b). Hsp90 inhibition by 17-AAG promoted R248Q degradation (Fig. 1b). Surprisingly, co-treatment with DCA and 17-AAG was more effective than 17-AAG alone (Fig. 1c). This effect was only observed at protein level. R248Q mRNA was not affected by DCA + 17-AAG co-treatment (Supplementary Fig. 1c). This shows that DCA effects mainly rely in protein stability as previously proposed [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] .

To understand how these two drugs, with individually opposite effects on mt p53 protein levels, synergy to eliminate R248Q, we study the stability of R248Q in presence of DCA and 17-AAG alone and in combination. First, we determined whether the augmentation in R248Q protein correspond with an increase of stability following DCA treatment. To measure the half-life of mtp53, NB4 cells were treated for 24h with 10 mM DCA. Next, we added at different times the inhibitor of protein synthesis cycloheximide and analyzed the levels of p53 by western blotting. As shown in Fig. 1d, the stability of R248Q was higher after DCA treatment suggesting that DCA interferes with mt p53 degradation. Then, we examined the stability of R248Q after treatment of DCA and/or 17-AAG. The half-life of R248Q decreased with 17-AAG (Fig. 1e). DCA+17-AAG co-treatment further reduced R248Q level. These data indicate that decrease of R248Q protein levels is due to decrease in stability upon DCA+17-AAG treatment and not to inhibition of mtp53 mRNA expression (Supplementary fig. 1c).

Hsp90 controls R248Q stability

There are two main routes of protein degradation in eukaryotes, the ubiquitinproteasome and the autophagy-lysosome pathways. The first predominantly regulates the stability of wt p53. Little is known about the degradation of mutant p53. In general, mt p53 proteins show increased stability compared to the wt protein due to their interaction with the Hsp90 chaperone complex [START_REF] Li | Functional inactivation of endogenous MDM2 and CHIP by HSP90 causes aberrant stabilization of mutant p53 in human cancer cells[END_REF] . We compared the endogenous levels of p53 protein from two AML cell lines with different p53 status, OCI-AML3 cells (wt p53) and NB4 cells (R248Q). NB4 cell line showed higher expression of p53 (Fig. 2a).

The proteasome inhibitor MG132 induced accumulation of wt p53 but not of p53-R248Q (Fig. 2b). The magnitude of the effect of MG132 treatment indicates that the proteasome is the major route for wt p53 degradation, but not for R248Q. This result suggests that different degradation pathways control the stability of wt p53 and R248Q.

Next, we examined the role of the lysosome pathway by using the lysosome inhibitor chloroquine (CQ) and wortmannin, an inhibitor of the early stage of macroautophagy.

Both inhibitors caused accumulation of R248Q protein levels (Fig. 2c). The increase of the macroautophagy markers, LC3-I and LC3-II detected after treatment of wortmannin and CQ respectively, indicates that both compounds were functional. MG132 did not affect LC3-I and LC3-II or mt p53 levels. Hence R248Q degradation is mainly dependent on the lysosomal, but not on the proteasomal pathway.

The molecular chaperone Hsp90 maintains the conformation, stability and activity of several oncogenic proteins including specific mt p53 proteins [START_REF] Whitesell | HSP90 and the chaperoning of cancer[END_REF] . As 17-AAG promotes degradation and decreases stability of R248Q (Fig. 1b), we studied the role of Hsp90.

We monitored the p53-Hsp90 interaction by co-immunoprecipitation experiments after treatment with DCA and/or 17-AAG (Fig. 2d). This co-immunopecipitation experiment was performed for 8 hours to avoid the complete degradation of p53. NB4 cell extracts were used for immunoprecipitation with the anti-p53 antibody (DO-1) and the eluates blotted against Hsp90. R248Q interacted with the chaperone Hsp90 (Fig. 2d,lane1). As expected, inhibition of Hsp90 with 17-AAG reduced the interaction of R248Q with Hsp90 (Fig. 2d, lane 2) while in contrast DCA increased the R248Q-Hsp90 binding (Fig. 2d, lane 3). DCA+17-AAG co-treatment decreased Hsp90-R248Q interaction compared to DCA alone (Fig. 2d, lane 4). These results indicate that Hsp90 is an important regulator of R248Q stability.

17-AAG induces macroautophagy in AML cell lines

17-AAG is currently in clinical trials as an anticancer drug that specifically inhibits Hsp90 functions [START_REF] Neckers | Heat-shock protein 90 inhibitors as novel cancer chemotherapeutic agents[END_REF] , but up-regulates other HSPs. 17-AAG promotes mt p53 proteasomal degradation [START_REF] Li | Functional inactivation of endogenous MDM2 and CHIP by HSP90 causes aberrant stabilization of mutant p53 in human cancer cells[END_REF] and autophagy to remove aggregates [START_REF] Riedel | 17-AAG induces cytoplasmic alpha-synuclein aggregate clearance by induction of autophagy[END_REF] . To investigate whether 17-AAG enhances proteasomal or lysosomal degradation in wt p53 and mt p53 AML cells, these were exposed to 17-AAG at different doses for 24h and then treated with the proteasomal inhibitor MG132 or the macroautophagy inhibitor bafilomycin A1 (Baf A1) for 4 hours before analyzing samples by western blotting. Baf A1 is an inhibitor of the late phase of autophagy that prevents the fusion between autophagosomes and lysosomes. In NB4 cells, inhibition of the proteasome with MG132 failed to accumulate mt p53, instead resulted in a reduction (Fig. 3a). In contrast, inhibition of macroautophagy by Baf A1 blocked mt p53 degradation mediated by 17-AAG. In OCI-AML-3 wt p53 was accumulated only in the presence of MG132.

Next, we examined whether the macroautophagy pathway mediates R248Q degradation by 17-AAG. We treated NB4 cells with 17-AAG for 24h and then with 50 nM wortmannin and 50 uM CQ for 4h. In contrast to DCA, both wortmannin and CQ were able to rescue degradation caused by 17-AAG (Fig. 3b). This suggests that 17-AAG causes the elimination of R248Q through induction of macroautophagy.

We next determined the effect of 17-AAG and DCA on macroautophagy flux in AML cells with different p53 status. We monitored autophagy flux by the addition of CQ (Fig. 3c). Protein extracts were analyzed by western blotting for LC3-II, a well establish macroautophagy marker. In OCI-AML-3, the autophagy flux was maintained after treatment of any of these drugs. We observed an increase in LC3-II levels after treatment with 17-AAG in NB4 cells (R248Q). Interestingly, a reduction of autophagy flux was observed after DCA treatment (Fig. 3c).

Macroautophagy is a complex sequence of biological events leading to formation, maturation, and fusion of autophagosomes with lysosomes to allow the degradation and recycling of cellular components [START_REF] Viry | Autophagy: an adaptive metabolic response to stress shaping the antitumor immunity[END_REF] . All these events are controlled by many autophagyrelated genes (ATGs), which are mostly transcriptionally induced by different stimuli such as nutritional deprivation, infections or metabolic and oncogenic stress [START_REF] Galluzzi | Autophagy in malignant transformation and cancer progression[END_REF] . To determine the mechanism for the increase in autophagy flux upon 17-AAG treatment in NB4 cells, we monitored mRNA levels of some essential autophagy genes such as ATG5, BECN1 and two different isoforms of LC3 (Fig. 3d). BECN1 and LC3 mRNAs were significantly increased indicating that 17-AAG may up-regulate macroautophagy by increasing the transcription of autophagy genes. Taken together all these results suggest that the degradation of R248Q by 17-AAG is dependent on macroautophagy and not on proteasomal mechanisms. Thus, the stability of wt p53 and mt p53 is controlled by different proteolytic mechanisms.

DCA inhibits macroautophagy in the presence of R248Q

Depending on the stress and wt p53 location, i.e. nuclear or cytoplasmic, macroautophagy can be either stimulated or inhibited by this tumor suppressor [START_REF] Tasdemir | Regulation of autophagy by cytoplasmic p53[END_REF] .

Mutant p53, which is mainly cytoplasmic, causes macroautophagy inhibition by repressing the expression of some critical autophagy genes [START_REF] Vakifahmetoglu-Norberg | Chaperone-mediated autophagy degrades mutant p53[END_REF][START_REF] Morselli | Mutant p53 protein localized in the cytoplasm inhibits autophagy[END_REF][START_REF] Cordani | Mutant p53 proteins counteract autophagic mechanism sensitizing cancer cells to mTOR inhibition[END_REF][START_REF] Morselli | p53 inhibits autophagy by interacting with the human ortholog of yeast Atg17, RB1CC1/FIP200[END_REF] . We studied the autophagy flux after DCA treatment in cell lines with different status of p53. Autophagy flux was again monitored by detection of LC3-II under CQ treatment. In wt (OCI-AML3) and null (HL60) p53 cell lines, DCA did not basically affect this flux. However, in the NB4 cell line carrying the R248Q p53 mutant, we observed a marked decrease in LC3-II levels (Fig. 4a).

Our results show that the effect of DCA on macroautophagy depends on the p53 status and illustrate that the presence of R248Q may impair macroautophagy. These data also suggest that there is a negative feed back loop between macroautophagy and R248Q.

Macroautophagy controls mt p53 protein levels and at the same time mt p53 has an inhibitory effect on macroautophagy.

Next, we monitored the effect of the co-treatment of DCA with 17-AAG on macroautophagy by evaluating the autophagy flux in cells with different p53 status of (Fig. 4b) .We did not observe any significant change in autophagy in OCI-AML-3 (wt p53) and HL-60 cells (p53 null) after DCA and/or 17-AAG treatments. Surprisingly, in NB4 cells the combination DCA and 17-AAG further decrease the autophagy flux compared to DCA alone and it was co-related with an enhanced reduction of R248Q protein. Hence, decreasing mt p53 does not rescue the inhibition induced by metabolic stress on macroautophagy flux. Moreover, CQ partially blocked R248Q degradation induced by 17-AAG and DCA+17-AAG indicating that the lysosomal pathway could be responsible for R248Q removal after these treatments. Therefore, there should be an alternative pathway to degrade mt p53 in conditions where macroautophagy is inhibited.

Chaperone-mediated autophagy degrades R248Q under metabolic stress

CMA can modulate the degradation of multiple Hsp90 client proteins [START_REF] Han | Molecular Chaperone HSP90 Is Necessary to Prevent Cellular Senescence via Lysosomal Degradation of p14ARF[END_REF] . p53-R248Q significantly decreased with the DCA+17-AAG combination under conditions where macroautophagy was impaired (Figs. 1 and4b), suggesting a possible role for CMA. In agreement with this hypothesis, 17-AAG increased Hsc70 protein levels even in the presence of DCA in mt and wt p53 cell lines (Fig. 5a). This finding is important because Hsc70 is the only known chaperone to mediate substrate targeting for CMA [START_REF] Chiang | A role for a 70-kilodalton heat shock protein in lysosomal degradation of intracellular proteins[END_REF] . To investigate whether CMA is responsible for R248Q degradation, we tested the interaction of R248Q with Hsc70 (Fig. 5b). Both 17-AAG and DCA increased the interaction between R248Q and Hsc70, and this interaction was preserved with the cotreatment DCA+17-AAG. 17-AAG decreased the interaction of Hsc70 and Hsp90. Our results indicate that inhibiting Hsp90 function has no effect on the binding of R248Q with Hsc70 and neither Hsc70 activity. Importantly, Hsp90 inhibition may be required to induce CMA by 17-AAG. This data also suggests that different pathways are engaged by Hsp90 and Hsc70 to control the stability of R248Q (Fig. 5b).

We further investigated which type of degradation (proteasomal, macroautophagy or CMA) was induced in the presence of DCA and 17-AAG. For this purpose, NB4 cells were treated with DCA and/or 17-AAG with the addition of MG132, 3-Methyladenine (3-MA), wortmannin and CQ (Fig 5c). We used hence three inhibitors to target autophagy at different steps. 3-MA and wortmannin inhibits autophagosome formation and autophagic sequestration, respectively. CQ blocks lysosomal degradation. Only the presence of CQ prevented the degradation of R248Q, indicating that under these conditions, R248Q degradation is mainly lysosomal and probably through CMA.

Finally, to determine whether the presence Hsc70 is essential for R248Q degradation, we knockdown Hsc70 using two specific siRNAs (Fig. 5d). Downregulation of Hsc70 blocked the degradation of R248Q by 17-AAG and DCA+17-AAG, indicating that Hsc70 plays a vital role in the stability of R248Q.

Our results propose that CMA could modulate the levels of R248Q under metabolic stress conditions. 17-AAG controls two lysosomal degradation pathways: in normal unstressed conditions, 17-AAG induces macroautophagy and under metabolic stress conditions promotes CMA (graphical abstract).

The inhibitory effect of DCA on autophagy depends on p53 status

The presence of mt p53 can have an adverse impact on macroautophagy impairing autolysosome formation [START_REF] Morselli | Mutant p53 protein localized in the cytoplasm inhibits autophagy[END_REF] . Mutants of p53, including R248Q, can counteract autophagy on various phases of the process [START_REF] Cordani | Mutant p53 proteins counteract autophagic mechanism sensitizing cancer cells to mTOR inhibition[END_REF] . To further investigate whether DCA enhances macroautophagy inhibition via mt p53, we monitored the expression of some essential autophagy genes such as ATG5, ATG12, BECN1 and two different isoforms for LC3 (Fig. 6). A decrease of mRNA expression of all these genes was only observed in the NB4 cell line expressing R248Q. No decrease on gene expression was detected in OCI-AML-3 cells (wt p53) or HL60 cell (p53 null). ATG12 mRNA increased about 2-3 folds in the HL60 cell (p53 null). Hence, DCA inhibits expression of autophagy gene in the presence of R248Q.

Mutant R248Q has an inhibitory effect on macroautophagy

To further investigate whether DCA inhibits macroautophagy via mt p53, we silenced or overexpressed R248Q in the previously portrayed AML cell lines (Fig. 7). First, we analyzed in NB4 cells the effect of DCA on the mRNA levels of different autophagy genes after reduction of R248Q by siRNA (Fig. 7a). R248Q down-regulation efficiently enhanced the expression of the autophagy genes indicating that the presence of R248Q inhibited the macroautophagy pathway. The knockdown of R248Q partially blocked the inhibitory effect of DCA. In particular, the inhibition of BCEN1 mRNA by DCA was entirely prevented after reduction of R248Q. Beclin-1, the protein encoded by BCEN1 gene, is a crucial component of nucleation and maturation of macroautophagy pathway, one of the early steps of macroautophagy. ATG5/ATG12 and LC3 mediate the elongation of the phagophore [START_REF] Kang | The Beclin 1 network regulates autophagy and apoptosis[END_REF] . Based on this data, R248Q could have an inhibitory effect on the formation of the autophagosome at the early steps of the macroautophagy pathway. In addition, DCA did not increase mt p53 mRNA in NB4 cells treated with control siRNA (Fig. 7a). This shows that DCA effects mainly rely in protein stability as previously proposed [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] .

Hence, we next checked the autophagy flux after R248Q down-regulation (Fig. 7b). As we previously observed, DCA reduced autophagy flux in control siRNA transfected cells. Interestingly, the flux was restored after p53 knockdown. Furthermore, ectopic expression of wt p53 and R248Q in p53-null HL60 cells caused a reduction of the autophagy flux in untreated cells (Fig. 7c). In the presence of DCA, only R248Q further decreased the autophagy flux. All these results demonstrate the repressive role of mutant p53-R248Q in macroautophagy and suggest that the DCA-induced inhibition of macroautophagy could be due to the increase in mutant p53-R248Q levels.

Co-treatment of DCA and 17-AAG caused inhibition of macroautophagy in p53 dependent manner

To investigate the effect of wt p53 or R248Q on macroautophagy after treatment with DCA and/or 17-AAG, we carried out experiments to overexpress and knockdown both genes. The efficiency of the knockdown of R248Q was confirmed by qPCR and western blot analysis in NB4 cells (Fig. 8a). Whereas in cells transfected with siRNA control DCA+17-AAG decreased the autophagy flux (Fig. 8b, top panel), reduction of R248Q by siRNA re-established the autophagy flux (Fig. 8b, bottom panel). This data demonstrate a critical role of R248Q in inhibiting autophagy. To further explore the role of R248Q in autophagy, HL60 cells were transfected with plasmids encoding wt p53 or R248Q. Autophagy flux analysis showed that when R248Q was overexpressed, autophagy flux decreased after DCA+17-AAG co-treatment (Fig. 8c, bottom panel).

However, overexpression of wt p53 had little or no effect on autophagy flux (Fig. 8c, top panel). These data are consistent with the idea that this mt p53 has an inhibitory effect on the autophagy flux stimulated by metabolism stress.

Discussion

Cells growing in low resources or under metabolic stress use autophagy to survive. We show here that metabolic stress regulates different forms of autophagy depending on p53 status. DCA induces metabolic stress and p53 accumulation. In cells expressing wt p53 or lacking p53 expression, DCA does not affect the autophagy flux. In cells expressing the p53 mutation R248Q, metabolic stress represses macroautophagy and promotes p53-R248Q accumulation. In a feed back loop, R248Q represses further macroautophagy inducing even higher accumulation of R248Q protein levels. We show that 17-AAG short-circuits this loop by inhibiting Hsp90, which releases R248Q bound to Hsc70. In this condition, Hsc70 induces massive R248Q degradation by CMA.

Interestingly, wt p53 is not affected by this treatment because is not mainly stabilized by Hsp90. Therefore; the DCA+17-AAG is effective to eliminate mt p53.

It is now established that mt p53 acquires oncogenic functions to drive cell migration, invasion, and metastasis and p53 mutation does not represent the equivalent of p53 loss [START_REF] Muller | p53 and its mutants in tumor cell migration and invasion[END_REF] . Furthermore, not all p53 mutants have the same GOF activities. Different p53 mutations impart unique activities to stimulate the development of various tumor types.

Therefore, it is essential to study each p53 mutant independently to devise different target and therapeutic strategies. It has been reported that mt p53 could be targeted for proteasomal or autophagy degradation [START_REF] Choudhury | Dissecting the pathways that destabilize mutant p53: the proteasome or autophagy?[END_REF][START_REF] Rodriguez | Dietary downregulation of mutant p53 levels via glucose restriction: mechanisms and implications for tumor therapy[END_REF] , however remain unclear how mutant R248Q is degraded. In the present study, we investigated different strategies to promote the elimination of R248Q in different cellular conditions. We specifically focus on p53-R248Q mutant because is one of the most frequently found in a wide range of human cancers, especially in AML. It would be interesting to investigate if other p53 mutants are degraded by the same mechanisms than p53-R248Q. Although, it is possible that different degradation mechanisms will be activated depending on the physiological conditions, e.x. metabolic stress, and depending on the mutation. Specific mt p53 proteins accumulate to high levels in tumor cells due to defects in their degradation.

Additionally, multiple stress signals can induce their stabilization and promote its GOF. This favors the development of more aggressive tumors [START_REF] Suh | Multiple stress signals activate mutant p53 in vivo[END_REF] . mt p53 stabilization could be due to interaction with Hsp70 and Hsp90 chaperones that protect mt p53 from degradation [START_REF] Li | Functional inactivation of endogenous MDM2 and CHIP by HSP90 causes aberrant stabilization of mutant p53 in human cancer cells[END_REF] .

We found that autophagy and not proteasomal degradation is the primary pathway responsible for the effective elimination of R248Q. The use of DCA and 17-AAG allowed us to determine the alternative types of the autophagy pathway engaged to degrade R248Q under different conditions. DCA stabilizes R248Q by enhancing its binding with the Hsp90 chaperone and inhibiting the macroautophagy pathway. In contrast, inhibition of Hsp90 function by 17-AAG induces macroautophagy and promotes R248Q degradation. When macroautophagy is repressed under confluent conditions, CMA degrades mt p53 [START_REF] Vakifahmetoglu-Norberg | Chaperone-mediated autophagy degrades mutant p53[END_REF] CMA contributes to energetic cellular balance and it is activated by metabolic stress [START_REF] Tasset | Role of chaperone-mediated autophagy in metabolism[END_REF] . We show here that DCA-induced PDK1 inhibition constrains macroautophagy and by inducing metabolic stress activates CMA. This explains why DCA+17-AAG causes higher R248Q destabilization than 17-AAG alone.

The Hsc70 cytosolic chaperone is essential in mediating CMA [START_REF] Bandyopadhyay | The chaperonemediated autophagy receptor organizes in dynamic protein complexes at the lysosomal membrane[END_REF] . It is also part of Hsp90 complex, and it can be found associated with different mt p53 proteins [START_REF] Vakifahmetoglu-Norberg | Chaperone-mediated autophagy degrades mutant p53[END_REF][START_REF] Akakura | A role for Hsc70 in regulating nucleocytoplasmic transport of a temperature-sensitive p53 (p53Val-135)[END_REF] . We propose that DCA pre-treatment increases the interaction of R248Q with the chaperone complex, including Hsc70, leading to its stabilization and inhibition of macroautophagy. The subsequent addition of 17-AAG caused the release of Hsp90 from the complex but without affecting the interaction with Hsc70. This promotes CMAmediated R248Q degradation. Similar results were found in a study using an oxazoline analog of apratoxin A (oz-apraA) [START_REF] Shen | Cyclodepsipeptide toxin promotes the degradation of Hsp90 client proteins through chaperone-mediated autophagy[END_REF] . By inhibiting Hsp90 function, oz-apraA increases the interaction of Hsp90 clients to Hsc70/Hsp70 chaperones and promotes their degradation by CMA.

The crosstalk between different forms of autophagy pathways has been reported by various studies and the Hsc70 chaperone has been proposed as a candidate for acting as a cross-talking molecule between macroautophagy and CMA [START_REF] Shen | Cyclodepsipeptide toxin promotes the degradation of Hsp90 client proteins through chaperone-mediated autophagy[END_REF] . Cells with impaired CMA function were able to promote protein degradation through up-regulation of macroautophagy [START_REF] Massey | Consequences of the selective blockage of chaperone-mediated autophagy[END_REF] and vice versa, inhibition of macroautophagy also contributes to further induction of CMA [START_REF] Bandyopadhyay | The chaperonemediated autophagy receptor organizes in dynamic protein complexes at the lysosomal membrane[END_REF] . Macroautophagy and CMA communicate with each other and Hsc70 may be a fundamental element for this crosstalk. [START_REF] Wu | Crosstalk Between Macroautophagy and Chaperone-Mediated Autophagy: Implications for the Treatment of Neurological Diseases[END_REF] In summary, we provide different strategies to eliminate mutant R248Q p53 using 17-AAG. Our data show that 17-AAG induces macroautophagy to eliminate R248Q under normal growth conditions. Energy stress stimulates the accumulation of R248Q with molecular chaperones and enhances its inhibitory effect on macroautophagy. Under this stress condition, 17-AAG still can remove R248Q protein through CMA pathway.

Understanding the mechanism of mt p53 degradation may help in the development of new therapeutic approaches, and it will be useful for the treatment of patients carrying p53 mutations.

Materials and Methods

Cell cultures

AML (OCI-AML-3, NB4 and HL60) cell lines were cultured as previously described [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . OCI-AML-3 cells harbor wt p53, NB4 carries mt p53 (R248Q) and HL60 is a p53 -/-cell line. Cells were grown at 37°C and 5% CO 2 in a humidified atmosphere.

Reagents, siRNAs, plasmids and transfection

DCA was purchased from Santa Cruz. 17-AAG was from Selleck. MG132 was purchased from Calbiochem. Cycloheximide, wortmannin, bafilomycin A, chloroquine and 3MA were purchased from Sigma-Aldrich. Wt and mutant p53 constructs were a gift from Dr. Shannon C Kenney. Transfection of wt p53 and R248Q was carried out using Lipofectamine RNAiMAX (Invitrogen) in Opti-MEM (Invitrogen), according to the manufacturer's instructions. p53 siRNA was a gift from Dr Xirodimas and it was ON-TARGETplus SMARTpools (mixture of 4siRNA) from Dharmacon. Two siRNA duplexes were used to knockdown Hsc70 as previously described [START_REF] Boschelli | A cell-based screen for inhibitors of protein folding and degradation[END_REF] . siRNAs were synthesized by Eurofins MWG Operon. Cells (2 x 10 6 in 100 μl) were transfected with 100 nM Hsc70 siRNA, p53 siRNA or control siRNA by electroporation using Amaxa SF Cell line 4D-Nucleofector kit (Lonza Bioscience). Cells were harvested 24 to 72 h post-transfection.

Cell proliferation, viability

Cell viability and cell numbers were determined using the Muse® Cell Analyzer (Millipore) as previously described [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] .

Western blot analysis

Primary antibodies against HSP90 (C45G5) and β-actin were purchased from Cell Signaling Technology. LC3B antibody was from GenTex. HSC70 (13D3) was purchased from Abcam. The anti-p53 antibody (DO-1) was a gift from Dr Xirodimas.

Cell extracts were lysed in 2x SDS sample buffer. Proteins were resolved by SDS-PAGE and transferred to nitrocellulose or PVDF membranes using the Trans-Blot® Turbo™ Transfer System (Bio-Rad). Peroxidase-coupled anti-mouse and anti-rabbit secondary antibodies were used at a dilution of 1:10.000 (Sigma). Bound antibodies were detected by enhanced chemiluminescence (Millipore). To obtain the image for the western blotting the Molecular Imager Gel Doc XRS system (Biorad) was used, this system provides a reliable and sensitive imaging of chemilunescence western blots. For the quantification and analysis, we used the Image Lab Software (Biorad), a powerful and specific software for acquisition, analysis and quantification of blot images.

Measure of p53 half-life

To measure the half-life of mt p53, NB4 cell lines were treated with 10 mM DCA and/or 1µM 17-AAG. After 24 hours, the protein synthesis inhibitor cycloheximide (20 µg/ml), was added at different time points. Cells extracts were prepared and the remaining mt p53 protein levels determined by Western blotting. For the quantification and analysis of the protein levels, we used the Image Lab Software (Biorad).

Immunoprecipitation

Cells were lysed in NP-40 buffer: 0.5 % NP-40, 50 mM HEPES pH 7.5, 100 mM NaCl, 5 mM EDTA, 1 mM DTT and complete protease inhibitor mixture (Roche). Lysates were used for co-immunoprecipitation using 1μg of antibodies: anti-p53 (DO-1) or anti-HSC70 as indicated. Dynabeads protein G (Thermo Fisher) was used following 20 manufacturer's instructions.

RT-PCR and DNA sequencing

Total RNA was extracted using NucleoSpin RNA isolation columns (Macherey-Nagel), reverse transcription was carried out using random primers. Quantitative PCR was performed as described previously [START_REF] Allende-Vega | MdmX is a substrate for the deubiquitinating enzyme USP2a[END_REF] with SsoADV SYBR Green qPCR SuperMix (Biorad) and a CFX Connect TM Real-Time qPCR machine (Biorad). p53 and actin primers were previously described [START_REF] Vega | The presence of wild type p53 in hematological cancers improves the efficacy of combinational therapy targeting metabolism[END_REF] . Beclin 1, ATG5, LC3a and LC3b 52 and ATG12 28 primers were previously described. All samples were normalized to β-actin mRNA levels.

Quantification of autophagy flux

Flux was monitored by the addition of CQ (50 µM) for 4 h at the end of the incubation periods. Flux was determined by subtracting control sample from CQ treated samples, thus reflecting the amount of LC3-II that accumulated in the 4 h following CQ addition.

Statistical analysis

Statistical analysis was performed using the Student's t test: *p<0.05; **p<0.01; ***p<0.001. Values were expressed as the mean ± the standard error of the mean (SEM).

Author Contribution

N. A-V. conducted and designed the experiments; N.A-V and M.V. wrote the paper.

Figure titles and legends

Figure 1 .

 1 Figure 1. DCA and 17-AAG have different effect on p53-R248Q stability. (a) NB4 (mutant p53-R248Q) cells were incubated with the indicated concentrations of DCA for 24 h. p53 protein levels were analyzed by western blot analysis. (b) Decrease of p53 protein levels in NB4 cells after treatment with different concentrations of 17-AAG for 24 h. (c) NB4 cells were treated with DCA (10 mM) and/or 17-AAG (1 µM) for 24 h before protein analysis by western blotting. (d) NB4 cells were treated with 10 mM DCA and after 24 h incubated with 20 μg/mL cycloheximide (CHX) and harvested at the indicated times. At the indicated times after the addition of cycloheximide, the level of p53 was analyzed by western blotting. (e) NB4 cells were treated for 24 with 10 mM DCA and/or 17-AAG (1 µM) and after 24 h treated with CHX as in (d). Graphs represent the quantification of the western blots.

Figure 2 .

 2 Figure 2. Different mechanisms control wild type and mutant p53 stabilities. (a) Comparison of the endogenous p53 protein levels from two AML cell lines with different p53 status, OCI-AML3 cells (wt p53) and NB4 cells (R248Q). (b) The proteasome inhibitor MG132 induces accumulation of wt p53 but not of the R248Q mutant protein. NB4 cells were incubated with MG132 (10 µM) for 6 h before analyzing p53 levels. (c) Autophagy inhibition induces accumulation of p53-R248Q. NB4 cells were incubated for 4 h with the autophagy inhibitors chloroquine (50 µM) and wortmannin (50 mM) and MG132 before analyzing p53 levels. (d) Coimmunoprecipitation of p53 and Hsp90 after treatment with DCA and/or 17-AAG. NB4 cell extracts from cells treated for 8 h with 1µM 17-AAG and 10mM DCA were used for immunoprecipitation with the anti-p53 antibody (DO-1) and analyzed by western blotting with anti-Hsp90 or anti-p53 antibodies as indicated.

Figure 3 . 17 -

 317 Figure 3. 17-AAG induces macroautophagy in NB4 cells. (a) OCI-AML-3 (wtp53) and NB4 (R248Q) cells were treated with different doses of 17-AAG for 24h and then treated with the proteasomal inhibitor MG132 (10 µM) or the macroautophagy inhibitor bafilomycin A1 (Baf A1; 100 nM) for 4 hours before analyzing samples by western blotting. (b) NB4 cells were incubated with or without 17-AAG for 24h and then with two autophagy inhibitors: 50 nM wortmannin and 50 uM chloroquine for 4h. (c) Increase in LC3-II levels after treatment with 17-AAG in NB4 cells. Chloroquine was added for 4 h after treatment for 24 h with 17-AAG (1 µM) or DCA (10 mM). Protein extracts were analyzed by western blotting for the depicted proteins. (d) RT-qPCR analysis of autophagy genes (atg5, becn1 and LC3) in NB4 cells treated with different concentration of 17-AAG for 24 h; *p<0.05; **p<0.01; ***p<0.001.

Figure 4 .

 4 Figure 4. The effect of DCA on macroautophagy depends on the p53 status. (a) Autophagy flux after DCA treatment in cell lines with different status of p53. AML cell lines were cultured in the presence of different concentrations of DCA for 24 h followed by 4h of CQ (50 μM). Autophagy flux was monitored by detection of LC3-II under CQ treatment. (b) Effect of 17-AAG alone or in combination with DCA and different status of p53 on autophagy flux. AML cells were treated with 10 mM DCA, and after one hour, 17-AAG was added (1 μM) for 24 h. Autophagy flux was monitored by the addition of 50 μM CQ for 4h before collecting the cells for western blot analysis. Graphs represent the quantification of the western blots after giving a standard value of 1 to the control.

Figure 5 .

 5 Figure 5. CMA degrades R248Q under metabolic stress. (a) Cells were treated with DCA (10 mM) and/or 17-AAG (1 µM) for 24 h before protein analysis by western blotting. (b) R248Q interacts with Hsc70 chaperone. NB4 cells were treated with DCA (10 mM) and/or 17-AAG (1 µM) for 8 h, harvested and proteins in the extracts were immunoprecipitated with anti-HSC70 antibody (13D3). Immunoprecipitates were analyzed by Western blotting with anti-Hsc70, anti-p53 and anti-HSsp90 as indicated. (c) Only inhibition of autophagy degradation by CQ prevented the elimination of R248Q. NB4 cells were treated with DCA and/or 17-AAG as in (a) and with MG132 (10 µM), 3-MA (10 mM), wortmannin (50 nM) and CQ (50 µM) for 12 h. D) Hsc70 is essential for R248Q degradation. NB4 cells were transfected with control siRNA and two siRNA against Hsc70. 72 h later they were incubated with 10 mM DCA and/or 17-AAG (1 µM) for 24 h and protein expression was analyzed by western blotting.

Figure 6 .

 6 Figure 6. DCA inhibits expression of macroautophagy genes via mutant p53. Different AML cell lines were treated for 24 h with different concentrations of DCA. Expression of autophagy genes (BECN1, ATG5, ATG12, LC3) was analyzed by RT-qPCR.

Figure 7 .

 7 Figure 7. R248Q mediates inhibition of macroautophagy induced by.DCA. (a) p53 knockdown partially attenuates the inhibitory effect of DCA on the autophagy pathway. NB4 cells were transfected with control siRNA or siRNA targeting p53 and forty hours later were incubated with 10 mM DCA for a further 24 h. The effect on autophagy gene expression was analyzed by RT-qPCR. (b) Autophagy flux is restored after R248Q down-regulation. NB4 cells were transfected with control siRNA or p53 siRNA. 72 h later they were incubated with 10 mM DCA for 24 h. Autophagy flux was monitored by

Figure 8 .

 8 Figure 8. The combination DCA+17-AAG causes inhibition of macroautophagy in the presence of R248Q. (a) NB4 cells were transfected with control siRNA and p53 siRNA. Knockdown efficiency was validated by qPCR and western blot. (b) DCA+17-AAG co-treatment causes inhibition of autophagy flux in presence of R248Q. After 72 h transfection with the indicated siRNAs, NB4 cells were treated with 10 mM DCA and/or 1 µM 17-AAG. Autophagy flux was monitored by detection of LC3-II under CQ treatment. (c) R248Q inhibits autophagy flux stimulated by metabolism stress. HL60 cells were transfected with plasmids encoding wt p53 or R248Q and after 24 h, DCA and/or 17-AAG were added for 24 h. CQ was used to monitor autophagy flux. Graphs represent the quantification of the western blots.

Acknowledgments

This work was supported by the PRT-K program 2018 (MV; 2018-021) and Canceropole GSO Emergence (MV; 2018/2019).

Additional Information

Competing Interests

The authors declare no competing interests.


K Beauvois

H Godfrin

E Krotscheck +#

R E Zillich

Transport and Phonon Damping in 4 He

)) compared experimental results for S(k, ω) from high-precision neutron scattering experiments and theoretical results using the "dynamic many-body theory" (DMBT), showing excellent agreement over the whole experimentally accessible pressure regime. This paper focuses on the specific aspect of the propagation of low-energy phonons. We report calculations of the phonon meanfree path and phonon life time in liquid 4 He as a function of wave length and pressure. Historically, the question was of interest for experiments of quantum evaporation. More recently, there is interest in the potential use of 4 He as a detector for low-energy dark matter (K. Schulz and Kathryn M. Zurek, Phys. Rev. Lett. 117, 121302 (2016)). While the mean free path of long wave length phonons is large, phonons of intermediate energy can have a short mean free path of the order of µm. Comparison of different levels of theory indicate that reliable predictions of the phonon mean free path can be made only by using the most advanced many-body method available, namely, DMBT.

Introduction

It has been known for a long time [1]- [4] that low-energy phonons in liquid 4 He display an anomalous dispersion relation which allows these phonons to decay. Precise neutron scattering measurements of the phonon dispersion in liquid 4 He [5] provide the phonon dispersion relation for all experimentally accessible densites with unprecedented accuracy. They confirm the finding of earlier work that the phonon dispersion relation is anomalous up to densities of about ρ = 0.0245 Å-3 . These data agree very well with our recent theoretical results for the dynamics of 4 He based on time-dependent multiparticle correlations [6]; our methods should therefore also be capable of quantitative microscopic predictions for the phonon lifetime. We apply our many-body theory of inelastic scattering, previously applied to scattering off 4 He droplets [7] and the surface of 4 He [8]. In the following sections, we briefly review both the theoretical methods used to calculate the relevant ground state properties (Section 2) and the dynamic features (Section 3) of the 4 He liquid, and the data analysis to obtain accurate values of the phonon dispersion coefficient.

The investigations are, among others, of interest because multiple scattering in superfluid 4 He has been proposed as a detection mechanism for low-mass dark matter [9]- [START_REF] Romani | Probing sub-GeV dark matter with superfluid helium at HeRALD[END_REF], among other proposed detectors [START_REF] Tiffenberg | [END_REF]. Such a detector design requires accurate knowledge of the propagation of low-energy phonons within the medium. In particular, when the phonon dispersion relation is anomalous, phonons are damped and have a finite mean free path.

2 Ground State Structure of 4 He Microscopic calculation of properties of many-body systems begin with an accurate calculation of ground-state properties. For 4 He it is adequate to begin with a non-relativistic Hamiltonian

H = - h2 2m ∑ i ∇ 2 i + ∑ i< j v(|r i -r j |) (1)
where the pair-wise interaction v(r) is taken from Aziz et al. [14].

The most efficient evaluation of ground state properties is done by the variational Jastrow-Feenberg ansatz for the ground state:

Ψ 0 (r 1 , . . ., r N ) = exp 1 2 ∑ i< j u 2 (r i , r j ) + ∑ i< j<k u 3 (r i , r j , r k ) . (2)
The correlation functions u i (r 1 , . . . , r i ) are obtained by minimizing the ground state energy E 0 δ δ u i (r 1 , . . ., r i )

Ψ 0 H Ψ 0 Ψ 0 Ψ 0 = 0 . (3)
The method is known as Jastrow-Feenberg-Euler-Lagrange (JF-EL) method. The key quantity obtained from such a ground state calculation and used for the calculation of the dynamics is the static structure function S(k). We show in Fig. 1 a comparison of different calculations and experiments.

Many-Body Dynamics

Dynamics is treated at the same level as the ground state: We write the dynamic wave function as containing a small, time-dependent component

|Ψ (t) = e -iE 0 t/h e 1 2 δU(t) |Ψ 0 [ Ψ 0 |e 1 2 δU † (t) e 1 2 δU(t) |Ψ 0 ] 1/2 , ( 4
)
where Ψ 0 is the ground state, and δU(t) is an excitation operator that is written, for the case of bosons, in exactly the same manner as the ground state wave function: PIMC [START_REF] Boronat | Microscopic Approaches to Quantum Liquids in Confined Geometries[END_REF] HNC-EL [6] Svensson et al. [17] Robkoff and Hallock [18] Cowley and Woods [START_REF] Cowley | [END_REF] Gibbs [19] Fig. 1 (color online) Static structure function S(k) for 4 He at equilibrium density. We compare JF-EL, Monte Carlo [START_REF] Boronat | Microscopic Approaches to Quantum Liquids in Confined Geometries[END_REF], and experiments [START_REF] Cowley | [END_REF]17,18,19]. The figure is from Ref. 6.

δU(t) = ∑ i δ u 1 (r i ;t) + ∑ i< j δ u 2 (r i , r j ;t) + . . . (5
The amplitudes δ u i (r 1 , . . . r i ;t) are determined by the time-dependent generalization of the Ritz' variational principle:

δ δ u i (r 1 , . . .r i ;t) dt Ψ (t)|H -ih∂ t |Ψ (t) = 0 . (6)
Assuming that δU(t) is a small perturbation of the ground state correlations, we can linearize the equations of motion for δ u i (r i , . . .;t), leading to the density-density response function χ(k, ω) from which we obtain the dynamic structure function S(k, ω) = ℑm χ(k, ω):

χ(q, ω) = S(q) hω -ε F (q) -Σ (q, hω) + S(q) -hω -ε F (q) -Σ (q, -hω) , (7)
where ε F (k) = h2 k 2 /2mS(k) is the Feynman excitation spectrum, and the self-energy is given by an integral equation

Σ (q, hω) = 1 2 d 3 k 1 d 3 k 2 (2π) 3 ρ δ (q -k 1 -k 2 ) Ṽ3 (q; k 1 , k 2 ) 2 hω -ε F (k 1 ) -Σ (k 1 , hω -ε F (k 2 )) -ε F (k 2 ) -Σ (k 2 , hω -ε F (k 1 )
) .

(8) Ṽ3 (k; p, q) is the three-phonon vertex

Ṽ3 (q; k 1 , k 2 ) = h2 2m S(k 1 )S(k 2 ) S(q) q • k 1 X(k 1 ) + q • k 2 X(k 2 ) -q 2 X3 (q, k 1 , k 2 ) , (9)
where X(k) = 1 -1/S(k). X3 (q, k 1 , k 2 ) is the fully irreducible three-phonon coupling matrix element. In the simplest approximation, X3 (q, k 1 , k 2 ) is replaced by the three-body correlation ũ3 (q, k 1 , k 2 ); this approximation ensures that long-wavelength properties of the excitation spectrum are preserved [20]. Improved calculations [21] sum a 3-point integral equation to ensure that exact properties of X3 (q, k 1 , k 2 ) as q → 0 and of the Fourier transform X 3 (r 1 , r 2 , r 3 ) for |r 1r 2 | → 0 and |r 1r 3 | → 0 are satisfied [21]. We include these corrections routinely, they have a visible effect only for wave vectors between the maxon and the roton. The CBF-Brillouin-Wigner (CBF-BW) approximation [22] is obtained by omitting the self-energy corrections in the energy denominator of Eq. ( 8).

The implementation of the method outlined only briefly here has led to an unprecedented agreement between theoretical predictions [6] and experimental results [23] which are still being explored [5]. ) Fig. 2 A comparison of theoretical and experimental data for the dynamic structure function S(k,ω) from experiments [23,5] and two versions of our theory: The middle figure is based on the solution of the integral equation ( 8), the lowest figure shows the CBF-BW approximation; the only difference to previous work [22] are more accurate input functions S(k) and X 3 (r 1 ,r 2 ,r 3 ).

In Fig. 2, we compare the experimental dynamic structure function S(k, ω) (top panel) [23] with results from DMBT calculations (middle panel) and CBF-BW calculation [22] (bottom panel). In the CBF-BW calculation we have, of course, used the best available values for S(k) in the three-phonon vertex (9) and included the irreducible part X3 (q; k 1 , k 2 ) as described in Ref. 21. Thus, the numerical values are different from those of Ref. 22 but the physics described is basically the same.

The most significant difference between the CBF-BW approximation and the full solution of the integral equation ( 8) is the void regime above the phonon-maxon-roton dispersion relation. This is caused by the fact that, in the CBF-BW approximation, excitations decay into Feynman phonons ε F (k) and not into the physical phonons. Furthermore, CBF-BW overestimates the excitation energies in the maxon region, while DMBT agrees with the experiment.

Phonon dispersion

The theoretical phonon dispersion relation is obtained by solving the implicit equation

ε 0 (k) = ε F (k) + Σ (k, ε 0 (k)) . ( 10
)
At long wave lengths, the phonon dispersion relation is

ε 0 (k) ≈ hck(1 -γk 2 ) ( 11
)
where c is the speed of sound and γ is the phonon dispersion coefficient. If γ < 0, we speak of anomalous dispersion, long-wavelength phonons are damped.

To obtain the dispersion coefficient γ we have fitted both experimental and theoretical data by a polynomial of the form

ε 0 (k) = hck 1 -γk 2 + α 3 k 3 + α 4 k 4 , (12)
from which the phonon dispersion coefficient was extracted. The polynomial form turned out to be more flexible than the Padé approximation [1,2]

ε 0 (k) = hck 1 -γk 2 1 -k 2 /Q 2 a 1 + k 2 /Q 2 b . (13)
In particular, the Padé approximation [START_REF] Tiffenberg | [END_REF] does not contain the term proportional to k 3 which can be calculated analytically from the asymptotic form of the microscopic two-body interaction. Assuming the typical asymptotic form V (r) = C 6 r -6 , one arrives at [24,25]

α 3 = π 2 24 ρ mc 2 C 6 . (14)
We should note that fitting procedure should not be understood as a rigorous low-k expansion in the sense of a Taylor expansion of the dispersion relation ε 0 (k) around k = 0, but rather as a fit to the data in the theoretically and experimentally relevant regime. The fit works well for k < 0.6 Å-1 .

To explain the fact that the above fitting procedure should not be considered to be a rigorous Taylor expansion, we must review a little more of the theoretical background. The Feynman spectrum ε F (k) is derived from a Bogoliubov formula Ṽp-h (k) (meV)

ε F (k) = t 2 (k) + 2t(k) Ṽp-h (k) (15)
ε F (k) (meV) k (Å -1 )
Ṽp-h (k) moment expansion (17) Feynman spectrum (15) Spectrum with moment expansion Fig. 3 The figure shows the full particle-hole interaction Ṽp-h (k) (solid red line, left scale) and the lowmomentum expansion (17) (solid blue line) at a density ρ = 0.0215 Å-3 . Also shown are the Feynman dispersion relations obtained from the Bogoliubov relation ( 15) for these two cases (red and blue dashed lines, right scale).

where t(k) = h2 k 2 /2m is the kinetic energy, and

Ṽp-h (k) ≡ ρ d 3 rV p-h (r)e ik•r (16)
is the "particle-hole" interaction or, in the language of Aldrich and Pines [26] the "pseudopotential". For what follows we only need the property that, for large distances, V p-h (r) falls off like the bare interaction, i.e. V p-h (r)

∼ C 6 r -6 for r → ∞. Then, Ṽp-h (k) has the expansion Ṽp-h (k) = Ṽp-h (0+) +V 2 k 2 + π 2 12 ρC 6 k 3 ( 17
)
where Ṽp-h (0+) = ρ d 3 rV (r) = mc 2 is related to the speed of sound, and V 2 is the second moment V 2 = -ρ 6 d 3 rV p-h (r)r 2 . Higher moments do not exist due to the van der Waals tail.

Fig. 4 shows the full Ṽp-h (k) the way it is used in the Bogoliubov formula [START_REF] Boronat | Microscopic Approaches to Quantum Liquids in Confined Geometries[END_REF] and the expansion (17) as calculated from the potential moments. Evidently, the agreement is very good only in the regime 0 ≤ k ≤ 0.1 Å-1 which is inaccessible to neutron scattering measurements. The figure also shows the Feynman spectrum ε F (k) as obtained from the full Ṽp-h (k) and from the moment expansion.

Both the DMBT and the CBF-BW results for γ agree quite well with the experimental data, whereas the Feynman approximation predicts anomalous dispersion at all densities.

The calculated phase velocities c = ε 0 (k)/hk are given in Fig. 4 for densities covering the full range between the saturated vapor pressure and solidification. Also shown are experimental data [23,5] at four pressures corresponding to the same the same density range. Clearly the agreement is excellent. Only a very small shift in density is observed between the theoretical calculations and the experimental results, already discussed in previous publications [23,5].

The polynomial expression [START_REF] Romani | Probing sub-GeV dark matter with superfluid helium at HeRALD[END_REF] provides an excellent fit of the theoretical curves in the small wave vector range. In practice, fits of the experimental curves were done in the range 0.18 < k < 0.6 Å-1 , the lower bound being determined by the neutron detectors smallest angle. The highest bound was determined as the maximum wave vector where stable fits could be obtained using the form [START_REF] Romani | Probing sub-GeV dark matter with superfluid helium at HeRALD[END_REF]. This was also verified for the theoretical curves; for the latter, the fit range could be extended to k = 0, without affecting significantly the results of the fits, as seen in Fig. 5.

The experimental results have been analyzed using the polynomial expression [START_REF] Romani | Probing sub-GeV dark matter with superfluid helium at HeRALD[END_REF] to determine the dispersion coefficient γ for several 4 He densities. The speed of sound c determined by the fit agrees well with the well know values measured by ultrasonic or thermodynamic techniques [27,28], as can be seen in the extrapolations to k = 0 in Fig. 5. The agreement is not perfect, however, and this affects the value of the next term in the expansion, the dispersion coefficient γ. We thus show in Fig. 5 the curves for γ determined using either the fitted values of c, or the ultrasonic ones.

For completeness, we have also extracted other data from both the calculations and the experiments. Table 1 gives the calculated coefficients hc, the speed of sound c, the Grüneisen-Constant u and the dispersion coefficient γ. Static quantities like c and u can also be obtained from Monte Carlo simulations, Evidently, the agreement between the DMBT results and the experiment is excellent at all densities. The dispersion coefficient γ turns positive above ρ ≈ 0.0245 Å-3 , meaning that long-wavelength phonons can propagate freely only at high pressures. The Feynman approximation predicts, on the other hand, a negative dispersion coefficient at all densites.

ρ ( Å-3 ) hc (meV Å) c (m/sec) u γ F γ c u γ JF-

Phonon Mean Free Path

If γ < 0, a phonon of energy/momentum (hω, k) can decay into two phonons of lower energy and longer wave length. As a consequence, the self-energy becomes complex on the phonon dispersion relation. The onset of the imaginary part of an excitation of energy and momentum (hω, k) is at the critical energy hω crit = 2ε 0 (k/2), where ε 0 (k) is the dispersion relation calculated with DMBT, see eq. (10). At a corresponding critical wave number k crit determined by solving 2ε 0 (k/2) = ε 0 (k) for k, a phonon can decay into two phonons, paral- lel to the original phonon but with half the wave number. At this point the imaginary part of the self-energy is [29,6]

Σ (k, hω) = - | Ṽ3 (k; -k/2, -k/2)| 2 k 16πρε ′ 0 (k/2) 2ε 0 (k/2) -hω ε ′′ 0 (k/2) . ( 18
)
Phonons with lower wave numbers, k < k crit , have more decay channels because the wave vectors of the produced phonons need not be parallel to the original phonon; this angular spread is discussed further below. Phonons with wave numbers k > k crit do not decay into pairs of phonons and have infinite life-time in the DMBT approximation. However, higher-order processes, ı.e. decay into three and more phonons, lead to a long, but finite life-time also for k > k crit .

The life time of phonons with k < k crit can be readily calculated from the imaginary part of the self-energy,

τ(k) = h ℑm[Σ (k, hε 0 (k), k)] -1 (19)
The mean free path of a phonon, important for the suggested application of superfluid 4 He for dark matter detection mentioned above, can be obtained from

d(k) = v g (k) τ(k) (20)
where v g = dω(k) dk is the group velocity. Evidently, two things are needed for a reliable theoretical prediction of the phonon life time τ(k) and of the mean free path d(k). (1) An accurate dispersion coefficient is required for the low-momentum kinematics,

k = q + p (21) ε 0 (k) = ε 0 (q) + ε 0 (p) (22)
which in particular determines the range of momenta where phonon damping occurs. (2)

The three-phonon vertex V 3 is required for an accurate damping strength. The comparison of the dynamic structure function between experiment and the DMBT result in Fig. 2 shows that DMBT is accurate enough that eqns.( 19) and ( 20) indeed provide a reliable theoretical prediction of phonon life time and mean free path. Figs. 6 show our results for the phonon mean free path d(k) in both CBF-BW approximation (left panel) and in DMBT (right panel) for several helium densities ρ. The density range is smaller in the latter case, because the dispersion relation is not anomalous anymore for ρ 0.0245 Å-3 , see the dispersion coefficient γ shown in Fig. 5. In the CBF-BW calculation phonons decay by producing Feynman-phonons, which overestimates the anomalous dispersion and which have a negative γ for all densities considered here (see Fig. 5). Therefore CBF-BW yields a finite mean free path even for ρ = 0.0260 Å-3 . In fact d(k) is almost independent of ρ and thus of the pressure in the CBF-BW approximation, while the much improved DMBT result shows that the range of momenta where phonons can decay strongly depends on ρ, hence on the pressure. While CBF-BW predicts a minimal decay length of about 0.1µm for all densities, the DMBT results show that the minimal decay length is 1 -10µm around the equilibrium density, and increases significantly with density and pressure, because γ crosses zero and becomes positive at high density, Fig. 5, until the phonons do not decay anymore for densities higher than 0.0240 Å-3 ,

k (Å -1 ) ρ=0.0215Å -3 ρ=0.0220Å -3 ρ=0.0225Å -3 ρ=0.0230Å -3 ρ=0.0235Å -3 ρ=0.0240Å -3 ρ=0.0245Å -3 ρ=0.0250Å -3 ρ=0.0255Å -3 ρ=0.0260Å -3 0 0.2 0.4 0.6 0.8 1 k (Å -1 ) ρ=0.0215Å -3 ρ=0.0220Å -3 ρ=0.0225Å -3 ρ=0.0230Å -3 ρ=0.0235Å -3 ρ=0.0240Å -3
Fig. 6 The phonon mean free path d(k) as function of the wave number for different pressures. The left panel shows the result in CBF-BW approximation and the right one shows the improved result obtained with the DMBT method which shows that the phonon mean free path depends strongly on the pressure.

While the DMBT calculation predicts a much smaller range of wave numbers at which phonons are dampened, the values for mean free path d(k) at different densities are almost identical for a given k, if there is damping at all. In fact, the d(k) values predicted by CBF-BW and DMBT are essentially the same. The reason is that the lifetime and thus the mean free path are mostly determined by the vertex V 3 , which is the same for CBF-BW and DMBT. The strength of damping does depend on the first derivative of the dispersion which, however, for the small k range relevant here is very similar in all approximations (Feynman, CBF-BW, and DMBT) -only the higher order derivatives, essential for the correct decay kinematics, are sensitive to the approximation.

Inelastic currents

Anomalous dispersion is a prerequisite for phonon decay, but the deviation of the dispersion relation from a linear dispersion is, in the wave number regime under consideration here, hardly visible, see Fig. 4. Therefore, a phonon decays into a pair of more or less collinear phonons. The angular distribution of these decay phonons can be obtained from the transport current, which is the second order expectation value of the current operator in the fluctuations δU(r 1 , . . . , r N ,t)

j (2) (r,t) ≡ 1 4 Ψ 0 δU * ĵ(r) δU Ψ 0 = hN 8mi d 3 r 2 . . . d 3 r N Ψ 2 0 [δU * (t)∇ 1 δU(t) -c.c.] (23)
The derivation of workable formulas is somewhat tedious, some essential steps will be presented in the Appendix. A detailed derivation for the case of impurity scattering off the 4 He surface may be found in Ref. 8, and for the case of impurity and 4 He scattering in Ref. 7.

The decay of a phonon with wave vector k produces two phonons according to the kinematics ( 21) and (22). We are interested in a measure for the probability that a decay product is ejected in direction e (e is a unit vector with Euler angles θ and ϕ). For this Fig. 7 The angular dependence of the phonons produced by the decay of an initial phonon with wave number k. θ is the angle of the decay phonons with respect to the direction of the initial phonon. The density is ρ = 0.0215 Å-3 .

purpose we calculate the rate of inelastic phonon current in direction e, for which we obtain

d dt j (2) (e) = 1 mS(k) d 3 q (2π) 3 ρ q δ (e - q q ) ℑm σ (k, q, ω) (24)
The δ (eq/q) obviously selects only phonons with wave vector q parallel to the direction of interest. Here σ (k, q, ω) is

σ (k, q, ω) = 1 2 Ṽ (k, k -q, q) 2 hω -ε 0 (|k -q|) -ε 0 (q) + iη (25)
The integration over q yields the self energy

d 3 q
(2π) 3 ρ σ (q; k, ω) = Σ (k, ω), see Eq. (8). For the calculation of σ (k, q, ω), we have replaced the non-local self-energy appearing in Eq. ( 8) by the dispersion relation (10), this is legitimate in the low momentum regime under consideration.

Fig. 6 shows, at the density ρ = 0.0215 Å-3 , the DMBT prediction for the rate with which a phonon with wave number k decays into phonons with direction θ with respect to k (obviously, the rate does not depend on ϕ). The figure summarizes both the kinematics and the strength of phonon damping by decay into lower momentum phonons. In agreement with the decay length in the right panel of Fig. 6, the decay rate is highest for phonons with momentum k close to 0.5 Å-1 . Fig. 6 shows that these higher energy phonons decay into lower energy phonons which lie in a narrow cone (small θ ). Phonons with k = 0.2 -0.3 Å-1 decay at a smaller rate (have a larger decay length, see Fig. 6), but generate phonons in a larger cone up to angles of θ = 20 • .

Conclusion

The dispersion and damping of low momemtum phonons in superfluid helium-4 is of practical interest for cosmological applications in weakly interacting particle detectors to test hypotheses for dark matter. We presented a comparison of the low momentum dispersion in 4 He between experimental data and the results from dynamic many-body theory (DMBT) and find excellent agreement. The dispersion obtained with DMBT is a significant improvement over the CBF prediction in the full momentum and energy range of interest. This allows us to make quantitative predictions for the decay of phonons due to the anomalous dispersion without phenomenological or experimental input parameters. The phonon mean free path is similar to CBF-BW results, but the CBF-BW approximation fails to predict the pressure dependence, leading to decay lengths as short as 0.1µm. This shortcoming is remedied by the DMBT which shows that the minimal decay length is strongly pressure dependent.

Transport current

We present in this appendix the essential steps of the derivation of the transport currents; the details of the derivation are rather similar to the ones for the impurity currents derived in Refs. 8 and 7. We derive the inelastic current at the level of pair fluctuations, i.e. we truncate the expansion (5) at the two-body level. To derive the integral equation ( 8) one needs to include fluctuations to all orders [6]; our result will be plausible enough such that we can avoid these complications.

As in the derivation of the linearized equations of motion for the correlation operator ( 5), leading to the density-density response function χ(k,ω), Eq. ( 7), it is convenient to apply an weak harmonic perturbation (e.g. induced by a neutron beam)

U ext (r;t) = U ext (r)e -iωt +U * ext (r)e iωt e ηt (26)
As usual, the perturbation is switched on adiabatically with an infinitesimal rate η > 0. After linearization, the positive and negative frequency terms leads to a corresponding response of the correlation operator δU(t) split into (complex) positive and negative frequency terms; similarly for the response of one-body, two-body densities, δ ρ n .

Inserting the correlation operator ( 5) at the two-body level in the transport currents ( 23) gives (we omit the time dependence for clarity)

j (2) (r) = h 4mi δ ρ * 1 (r 1 )∇ 1 δ u 1 (r 1 ) + d 3 r 2 δ ρ * 2 (r 1 ,r 2 )∇ 1 δ u 2 (r 1 ,r 2 ) . (27)
where we have used that we are only interested in the time-averaged currents, since only those are observable by a detector. While looking very simple, the expression (27) for the current contains both the fluctuations of the correlation and of the densities. In accordance with the derivation of the self energy Σ (k, hω), summarized in the text, we express δ u 1 and δ ρ 2 in terms of δ ρ 1 and δ u 2 . It turns out advantageous to introduce δ v 1 (r;t)

such that δ ρ 1 (r 1 ) = ρ [S * δ v 1 ] (r 1 ) . (28)
where we define the convolution product with the static structure function

[S * f ] (r 1 ) = f (r 1 ) + ρ d 3 r 2 h(r 1 ,r 2 ) f (r 2 ) . (29)
In linear order, we can then write

δ v 1 (r) = δ u 1 (r) + δ w 1 (r) δ w 1 (r) = ρ d 3 r 2 h(r 1 ,r 2 )δ u 2 (r 1 ,r 2 ) + ρ 2 2 d 3 r 2 d 3 r 3 Y (r 1 ;r 2 ,r 3 )δ u 2 (r 2 ,r 3 ) , (30)
where the Y (r 1 ;r 2 ,r 3 ) is the set of those contributions to the three-body distribution function that is non-nodal in the point r 1 ; details can be found in Ref. [6].

In all further manipulations, we use the "convolution approximation" which is the simplest approximation for high-order distribution functions that maintains the exact long-wavelength properties. In convolution approximation, Y (r 1 ;r 2 ,r 3 ) is simply

Y (r 1 ;r 2 ,r 3 ) = h(r 1 ,r 2 )h(r 1 ,r 3 ) . (31)
Using Eqs. ( 30) and (31) and expressing δ ρ 2 (r 1 ,r 2 ) in terms of δ ρ 1 (r) and the fluctuation of the pair distribution function δ g(r 1 ,r 2 ), δ ρ 2 (r 1 ,r 2 ) = ρ [δ ρ 1 (r 1 ) + ρ 1 (r 2 ] g(r 1 ,r 2 ) + ρ 2 g(r 1 ,r 2 ) we obtain j (2)

Finally we eliminate δ g(r 1 ,r 2 ) using the convolution approximation:

δ g(r 1 ,r 2 ) = d 3 r 3 h(r 1 ,r 3 )h(r 2 ,r 3 )δ ρ 1 (r 3 ) + [S * δ u 2 * S] (r 1 ,r 2 ) .

(33) which yields the length expression for the current The first term (34) is the current in Feynman approximation which is the elastic channel. The second and third terms, (35) and (36), are correlations between the density fluctuation δ ρ 1 and the pair correlation fluctuation δ u 2 . We will show below that only the fourth term (37) contains the decay rate of an elementary excitation (in the above derivation given in Feynman approximation). We now need explicit expressions for the one-and two-body fluctuations. These come from solving the equations of motion, formally given by Eq. ( 6), in detail given in Refs. 8, 6. We can assume that the perturbation U ext (r) is a plane wave with wave vector q which induces a plane wave density fluctuation, propagating with phase velocity given by the dispersion relation, ω 0 (q) = ε 0 (q)/h: δ ρ(r;t) = λ ρe i(q•r-ω 0 (q)t) e ηt (38)

j (2) (r 1 ) = h 4mi δ ρ * 1 (r 1 )∇ 1 δ v 1 (r 1 ) (34)
+ hρ 4mi d 3 r 2 ∇ 1 [δ u 2 * S] (r 1 ,r 2 )h(r 1 ,r 2 )δ ρ * 1 (r 2 ) (35)
+ hρ 4mi δ ρ * 1 (r 1 ) d 3 r 2 [S * δ u 2 ](r 1 ,r 2 )∇ 1 h(r 1 ,r 2 ) (36)
where λ ≪ 1 is an arbitrary strength factor. The two-body correlation fluctuations are then coupled to the density fluctuation according to the equations of motion. Their spatial Fourier transform is given by ũ2 (k 1 ,k 2 ) = λ (2π) 3 δ (q + k 1 + k 2 ) ε 0 (q) -ε 0 (k 1 ) -ε 0 (k 1 ) + ihη Ṽ (q;k 1 ,k 2 ) S(q)S(k 1 )S(k 2 ) e -iω 0 (q)t)+ηt (39)

We finally use Eq. (38) for δ ρ 1 (r) and Eq. (39) for δ u 2 (r 1 ,r 2 ) in the current j (2) (r) above. The fourth term (37) which becomes λ 2 h 4m

d 3 k 1 d 3 k 2 (2π) 3 ρ δ (q + k 1 + k 2 ) k 1 1 S(q) Ṽ (q;k 1 ,k 2 ) 2 (ε 0 (q) -ε 0 (k 1 ) -ε 0 (k 2 )) 2 + h2 η 2 e 2ηt

)

Fig. 4

 4 Fig.4Dispersion relation for several4 He densities, comparing experiment (markers for pressures P=0, 5, 10, 24 bar, corresponding to densities 0.0218,0.0230,0.0239 and 0.0258 Å-3 ), and DMBT (lines, for several densities from 0.0210 to 0.0260 Å-3 as indicated in the legend). Thick lines indicate theoretical curves for densities close to the experimental values given above. The black dots at k = 0 indicate the sound velocities determined by ultrasonic techniques[27].

Fig. 5

 5 Fig.5Dispersion coefficient γ for several4 He densities, comparing neutron scattering experiments (blue and black dots and lines correspond to different data analyses described in the text), ultrasonic measurements[3] (green triangles) and DMBT results (red lines and solid and open squares). Similar results are found for the different fit ranges 0.0 -0.6 Å-1 (dash-dotted red line) and 0.2 -0.6 Å-1 (solid red line). We also show the dispersion coefficient coming from the Feynman spectrum (15) (light blue line).

1 (r 1 )ρ 2 2 ∇ 1 d

 1121 ∇ 1 δ v 1 (r 1 ) -ρ d 3 r 2 δ u 2 (r 1 ,r 2 )∇ 1 h(r 1 ,r 2 ) -3 r 2 d 3 r 3 h(r 1 ,r 2 )h(r 1 ,r 3 )δ u 2 (r 2 ,r 3 ) +ρ d 3 r 2 h(r 1 ,r 2 )∇ 1 δ u 2 (r 1 ,r 2 )δ ρ * 1 (r 2 ) +ρ 2 d 3 r 2 δ g * (r 1 ,r 2 )∇ 1 δ u 2 (r 1 ,r 2 ) .

+ h 4mi d 3 r 2 [S * δ u * 2 *

 22 S] (r 1 ,r 2 )∇ 1 δ u 2 (r 1 ,r 2 ) (37)

Table 1 The

 1

			EL			DMBT		expt./DMC
	0.0210	1.466	222.7	2.738	-2.221	-1.804	211.5	3.080	-1.85
	0.0215	1.561	237.2	2.628	-1.946	-1.495	227.0	2.944	-1.56
	0.0220	1.657	251.7	2.534	-1.710	-1.223	242.6	2.832	-1.29
	0.0225	1.752	266.2	2.453	-1.510	-0.906	258.2	2.738	-1.04
	0.0230	1.848	280.7	2.384	-1.333	-0.760	274.0	2.657	-0.81
	0.0235	1.944	295.3	2.323	-1.182	-0.541	289.9	2.588	-0.60
	0.0240	2.040	309.9	2.269	-1.046	-0.330	305.9	2.529
	0.0245	2.136	324.6	2.222	-0.925	-0.127	322.1	2.476
	0.0250	2.234	339.4	2.179	-0.815	0.087	338.5	2.430
	0.0255	2.331	354.2	2.141	-0.713	0.311	355.0	2.388
	0.0260	2.429	369.1	2.107	-0.619	0.571	371.7	2.532

table shows in columns 2-6 the calculated values hc, the speed of sound c, the Grüneisen constant u and the phonon dispersion coefficient γ F in Feynman approximation, and as obtained by fitting the dispersion relation by the form

[START_REF] Romani | Probing sub-GeV dark matter with superfluid helium at HeRALD[END_REF]

. Col. 7-9 give the corresponding quantities obtained, as far as possible, from experiments or from Monte Carlo simulations. The experimental results for the dispersion coefficient γ have been obtained by interpolating the data of Ref. 3 at the densities in col. 1 by a quadratic polynomial.

In the limit of adiabatically switching on the perturbation, η → 0, we need to calculate the rate with which the inelastic current is generated, i.e. the time derivative. Using

we obtain the rate of the total inelastic current d dt j (2) apart from the arbitrary strength factor λ 2 . Selecting only the inelastic phonon current (the decay products) in a specific direction e relative to the incoming phonon with wave vector q, we obtain the rate Eq. (24). Note that the two mixed term contributions to j (2) (r), Eqs. ( 35) and (36), do not have a finite contribution to the rate in the adiabatic limit.


Régis Portalez -Altimesh -

Florent Duguet
email: altimesh-florent.duguet@altimesh.com

Using CLANG/LLVM Vectorization to Generate Mixed Precision Source Code

CUDA Source -with half2 intrinsics

At Supercomputing 2015, NVIDIA announced Jetson TX1, a mobile supercomputer, offering up to 1 TFLOPs of compute power for a power envelope typical of embedded devices. Targeting image processing and deep learning, this platform is the first available to natively expose mixed precision instructions. However, the new mixed precision unit requires that operations on 16-bit precision floating points are done in pairs. Hence, approaching peak performance level requires usage of the half2 type which pairs two values in a single register.

In this work, we present an approach that makes use of existing vectorization tool developed for CPU code optimization to further generate CUDA source code that uses half2 intrinsic functions, hence enabling mixed precision hardware usage with little effort. Using this approach, we are able to generate efficient CUDA code from a single scalar version of the code. This source to source code translation may be used in many application fields for different numeric types. Moreover, this approach shows very nice boundary effects such as better memory access pattern and instruction level parallelism. ALIGNED LOADS / STORES -CLANG/LLVM identifies alignment form attributes such as __attribute__((align_value(x))), and transforms into appropriate vector load operations. Gets transformed to a 128 bits load which provides best memory latency and bandwidth performances.

BOOLEAN OPERATIONS -In the code, invF function gets inlined. It contains a if which is transformed and a select llvm instruction is issued. Select makes conditional assignment based on a mask. This instruction is transformed in a mul and fma using the mask value as 0 o 1.

VECTOR ARITHMETIC OPERATIONS -CLANG/LLVM generates arithmetic code using vector of floats, initialy aimed CPU vector units. These instructions get reused and transformed into instructions using half2 vector intrinsics available with mixed precision.

The Jetson TX1 and the upcoming Pascal are capable of mixed precision. This feature aimed at deep learning and image processing offers a 16-bit floating-point arithmetic unit with improved performances. The way this unit is exposed [START_REF]Mixed Precision[END_REF] is similar to SSE instructions, that is each operation is performed on a pair of values. SSE instructions, and more generally SIMD instructions have been around for more than a decade.

Compilers and optimizers are familiar with these instruction sets and generate code that makes use of those. The LLVM compiler [START_REF]LLVM Compiler infrastructure[END_REF] is an example of choice: indeed, its intermediate representation performs the vectorization stage operating on small vector registers [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF]. Then, these vectors are translated into small vector instructions by the backend, depending on the target platform -ARM with NEON, Intel x86 with SSE or AVX, IBM Power with VSX.

Building on the analogy between mixed precision instructions and SIMD instructions, we make use of the vectorization unit of CLANG operating on C++ together with LLVM to generate intermediate language with small vectors. However, a few modifications have to be made: 16-bits floating-point precision (a.k.a. half) is not supported on most SIMD units, hence, the input source code is written using 32-bit precision (float). In the transformation from LLVM intermediate to CUDA, we convert float32 into float16 automatically.

The SIMD unit used by LLVM is aligned with known architecture sizes which are at least 128 bits (SSE, NEON, VSX). However, mixed precision is 32 bits with two halves. The transformation is not a one-to-one: a kind of emulation library has to be written. Some operations assumed to exist in the SIMD instruction set of a GPU do not exist in mixed precision, such as select [START_REF]LLVM select[END_REF] (x86 : vblend, AltiVec/VSX: vec_sel). These operations have to be written in the emulation library.

As illustrated here, 128-bit aligned memory access provide best bandwidth. Hence, we use the 8-entries wide small vector registers from LLVM, as 8 halves have a 128 -bit memory footprint.

In order to benefit from the 128-bit aligned loads, even using 16-bit floating-point precision, we configure the LLVM optimizer to use 8entry wide SIMD instructions and registers. Then, we write a small vector emulation library which operates on a custom type half8. We then need to write the appropriate operators on these entries.

We illustrate here the details of the select operation, most others being quite straightforward.

  __restrict red, float* ALIGN32 __restrict green, float*ALIGN32 __restrict blue, const float* ALIGN32 __restrict L, const float* ALIGN32 __restrict a, const float* ALIGN32 __restrict b, const int width, const int height, const int stride) { const float inv116 = 1.0f / 116.0f; const float inv200 = 1.0f / 200.0f; const float inv500 = 1.0f / 500.0f; for (int y = threadIdx.y + blockDim.y * blockIdx.y ; y < height ; y += blockDim.y * gridDim.y) { for (int x = threadIdx.x + blockDim.x * blockIdx.x ; x < width/8 ; x += blockDim.x * gridDim.x) { int index = x + y * stride; #pragma clang loop vectorize(enable) interleave(enable) for(int u = 0; u < 8; ++u) { float lL = L[8*index + u] ; float la = a[8*index + u] ; float lb = b[8*index + u] ; float fy = (lL + 16.0f) * inv116 ; float fx = fy + (la * inv500 ) ; float fz = fy -(lb * inv200) ; float lX = 5.6506753f * invf(fx) ; float lY = 5.6506753f * invf(fy) ; float lZ = 5.6506753f * invf(fz) ; float lR = 0.4184657f * lX + (-0.15866f) * lY + (-0.0828349f) * lZ ; float lG = (-0.091168f) * lX + 0.252431f * lY + 0.015707521f * lZ ; float lB = 0.0009208f * lX + (-0.0025498f) * lY + 0.178598f * lZ ; red[8*index + u] = (lR) ; green[8*index + u] = (lG) ; blue[8*index + u] = (lB) ; ...................................................................> .L_4: /*0170*/ XMAD R0, R18.reuse, c[0x0] [0x164], R23; /*0178*/ XMAD.MRG R3, R18.reuse, c[0x0] [0x164].H1, RZ; /*0188*/ XMAD.PSL.CBCC R19, R18.H1, R3.H1, R0; /*0190*/ ISCADD R12, R19.reuse, c[0x0][0x150], 0x5; /*0198*/ LDG.128 R12, [R12]; /*01a8*/ ISCADD R8, R19, c[0x0][0x154], 0x5; /*01b0*/ { ISCADD R4, R19, c[0x0][0x158], 0x5; /*01b8*/ LDG.128 R8, [R8]; } /*01c8*/ { HADD2_32I R17, RZ.H0_H0, 0.2069091796875, 0.2069091796875; /*01d0*/ LDG.128 R4, [R4]; } /*01d8*/ LLVM source <...................................................> vector.body: ; preds = %for.body.6.us %9 = getelementptr inbounds float, float* %L, i64 %8, !dbg !125 %10 = bitcast float* %9 to <8 x float>*, !dbg !125 %wide.load = load <8 x float>, <8 x float>* %10, align 32, !dbg !125, !tbaa !126 %11 = getelementptr inbounds float, float* %a, i64 %8, !dbg !128 %12 = bitcast float* %11 to <8 x float>*, !dbg !128 %wide.load137 = load <8 x float>, <8 x float>* %12, align 32, !dbg !128, !tbaa !126 %13 = getelementptr inbounds float, float* %b, i64 %8, !dbg !129 %14 = bitcast float* %13 to <8 x float>*, !dbg !129 %wide.load138 = load <8 x float>, <8 x float>* %14, align 32, !dbg !129, !tbaa !126 %15 = fadd <8 x float> %wide.load, <float 1.600000e+01, float 1.600000e+01, float 1.600000e+01, float 1.600000e+01, float 1.600000e+01, float 1.600000e+01, float 1.600000e+01, float 1.600000e+01>, !dbg !130 %16 = fmul <8 x float> %15, <float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000, float 0x3F81A7B960000000>, !dbg !131 <...................................................> %26 = select <8 x i1> %21, <8 x float> %23, <8 x float> %25, !dbg !144 %27 = fmul <8 x float> %26, <float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000, float 0x40169A4AA0000000>, !dbg !145 %28 = fcmp ogt <8 x float> %16, <float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000, float 0x3FCA7B9600000000>, !dbg !146

Aligned 128 bits loads improve memory bandwidth on all architectures here, on Maxwell (GeForce GTX 980) template<> hyb_device hyb_inline half8 select<half8, halfbool8>( const halfbool8& m, const half8& l, const half8& r) { half8 res ; res.vh2[0] = __hsub2 (l.vh2[0], r.vh2[0]) ; res.vh2 [START_REF]Mixed Precision[END_REF] = __hsub2 (l.vh2 [START_REF]Mixed Precision[END_REF], r.vh2 [START_REF]Mixed Precision[END_REF]) ; res.vh2 [START_REF]LLVM Compiler infrastructure[END_REF] = __hsub2 (l.vh2 [START_REF]LLVM Compiler infrastructure[END_REF], r.vh2 [START_REF]LLVM Compiler infrastructure[END_REF]) ; res.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF] = __hsub2 (l.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF], r.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF]) ; res.vh2[0] = __hfma2 (m.vh2[0], res.vh2[0], r.vh2[0]) ; res.vh2 [START_REF]Mixed Precision[END_REF] = __hfma2 (m.vh2 [START_REF]Mixed Precision[END_REF], res.vh2 [START_REF]Mixed Precision[END_REF], r.vh2 [START_REF]Mixed Precision[END_REF]) ; res.vh2 [START_REF]LLVM Compiler infrastructure[END_REF] = __hfma2 (m.vh2 [START_REF]LLVM Compiler infrastructure[END_REF], res.vh2 [START_REF]LLVM Compiler infrastructure[END_REF], r.vh2 [START_REF]LLVM Compiler infrastructure[END_REF]) ; res.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF] = __hfma2 (m.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF], res.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF], r.vh2 [START_REF]Vector LLVA: A Virtual Vector Instruction Set for Media Processing[END_REF] When operating on images, some algorithms operate on different color spaces. We illustrate here the conversion between CIE-L*a*b* [START_REF]Color Space from CIE[END_REF] to RGB color-space, via the CIE-XYZ color space. We apply this work to this color transformation. Note that the transformation calls a function that contains an if clause, which results in a select in LLVM intermediate language.

Other image processing algorithms such as filters could be applied, however, the benefit of this approach is limited to the vectorization capability of the LLVM optimizer.

REFERENCES DISCUSSION

This approach reuses major contributions to the compilers and optimizer research fields. Indeed, the data alignment and automatic vectorization including transformation of conditions into select instructions are very hard to identify and implement. The implementation of small vector library and float to half transformation is pretty straightforward and does not show major technical lock.

While this work is motivated by the mixed-precision intrinsics library, and its SIMD nature, it can also be used without the final float to half transformation. Indeed, on Kepler for instance, more execution units than scheduling units are present, which requires the scheduler to issue two fp32 operations in a single clock cycle to benefit from the full compute power. As a result, using a small vector library with several floats is a good way to offer the scheduler opportunities for ILP. Another significant improvement is the use of aligned 128-bit loads. Using the same input code compiled with CUDA, on a Maxwell architecture leads to bandwidth improvement (as illustrated above). When using 32-bit loads, the global read buffer gets saturated while reaching only 46% of memory bandwidth. A higher bandwidth usage is immediately obtained, without code change, automatically transforming 32-bit loads into 128-bit loads.

(In this example, the loop is unrolled and both kernels are run with the same configuration and require the same number of registers).

These performance improvements may come at some supplemental cost: operating on four or eight values at a time may lead to more register pressure. A heuristic needs to be found to identify whether the benefit of using 128-bit loads is not lost with register spilling. Further improving the float to half transformation tool including a vector-size transformation might actually help in this operation. However, this would require significant code analysis as loop iterations count would be different.