Maëlle Cartel

Pierre-Luc Mouchel

Mathilde Gotanègre

Laure David

Sarah Bertoli

Véronique Mansat-De Mas

Arnaud Besson

Jean-Emmanuel Sarry

Stéphane Manenti

Christine Didier
email: christine.didier@inserm.fr

Acute myeloid leukemia Inhibition of ubiquitin-specific protease 7 sensitizes acute myeloid leukemia to chemotherapy

Resistance of acute myeloid leukemia (AML) to therapeutic agents is frequent. Consequently, the mechanisms leading to this resistance must be understood and addressed. In this paper, we demonstrate that inhibition of deubiquitinylase USP7 significantly reduces cell proliferation in vitro and in vivo, blocks DNA replication progression and increases cell death in AML. Transcriptomic dataset analyses reveal that a USP7 gene signature is highly enriched in cells from AML patients at relapse, as well as in residual blasts from patient-derived xenograft (PDX) models treated with clinically relevant doses of cytarabine, which indicates a relationship between USP7 expression and resistance to therapy. Accordingly, singlecell analysis of AML patient samples at relapse versus at diagnosis showed that a gene signature of the pre-existing subpopulation responsible for relapse is enriched in transcriptomes of patients with a high USP7 level. Furthermore, we found that USP7 interacts and modulates CHK1 protein levels and functions in AML. Finally, we demonstrated that USP7 inhibition acts in synergy with cytarabine to kill AML cell lines and primary cells of patients with high USP7 levels. Altogether, these data demonstrate that USP7 is both a marker of resistance to chemotherapy and a potential therapeutic target in overcoming resistance to treatment.

Introduction

Acute myeloid leukemia (AML) originates from the transformation and clonal expansion of undifferentiated hematopoietic progenitors, characterized by altered growth, differentiation, and proliferation capacities, which result in failure of bone marrow hematopoietic functions. Although a majority of AML patients initially respond to standard induction therapy, a protocol that combines cytarabine (AraC) with an anthracycline, relapses are common and overall survival remains very poor [START_REF] Döhner | Acute myeloid leukemia[END_REF]. Whole-genome sequencing analyses have highlighted the molecular heterogeneity of AML and allow for risk-based stratification [START_REF] Papaemmanuil | Genomic classification and prognosis in acute myeloid leukemia[END_REF]. For some mutations resulting in oncogenic signaling, specific inhibitors have been developed and are included in clinical strategies [START_REF] Yang | Use of FLT3 inhibitors in acute myeloid leukemia remission induction or salvage therapy: systematic review and meta-analysis[END_REF][START_REF] Dinardo | Durable remissions with ivosidenib in IDH1mutated relapsed or refractory AML[END_REF]. Unfortunately, in the majority of cases these strategies show only transient and limited efficacy due to a variety of resistance mechanisms that occur in minor subpopulations of resistant leukemic cells which can initiate relapse [START_REF] Zhou | Resistance to FLT3 inhibitors in acute myeloid leukemia: molecular mechanisms and resensitizing strategies[END_REF]. Consequently, identification of these resistance mechanisms and new potential drug targets is urgently needed in AML.

Ubiquitylation is an essential post-translational modification involved in protein stability, localization, interactions, and activity. This process is closely regulated, antagonized by cysteine protease deubiquitinating enzymes (or DUBs), and is involved in multiple signaling [START_REF] Harrigan | Deubiquitylating enzymes and drug discovery: emerging opportunities[END_REF]. Ubiquitin-specific protease 7 (USP7, also known as herpesvirus-associated ubiquitinspecific protease, or HAUSP) belongs to the largest subfamily of DUB and is one of the most studied, disease-associated DUBs [START_REF] Basu | Cellular homeostasis or tumorigenesis: USP7 playing the double agent[END_REF]. It has been noted that USP7 participates in the regulation of apoptosis and senescence by modulating the p53 pathway, either by directly deubiquitinating p53 or by stabilizing MDM2, an E3-ubiquitin ligase that ubiquitinates and targets p53 for proteasomal degradation [START_REF] Li | A dynamic role of HAUSP in the p53-Mdm2 pathway[END_REF]. Independently of its action on p53, USP7 modulates various pathways in homeostasis or during oncogenesis by targeting a large panel of substrates involved in cell cycle, DNA damage response, immune response, signal transduction, neuronal differentiation, and epigenetic regulation [START_REF] Basu | Cellular homeostasis or tumorigenesis: USP7 playing the double agent[END_REF][START_REF] Bhattacharya | Emerging insights into HAUSP (USP7) in physiology, cancer and other diseases[END_REF]. Other notable targets of USP7 include the tumor suppressor PTEN [START_REF] Song | The deubiquitinylation and localization of PTEN are regulated by a HAUSP-PML network[END_REF] and the transcription factor FOXO4 [START_REF] Van Der Horst | FOXO4 transcriptional activity is regulated by monoubiquitination and USP7/HAUSP[END_REF]. Through deubiquitination, USP7 triggers their nuclear exclusion and inactivation. Interestingly, in chronic myeloid leukemia, BCR-ABL phosphorylates USP7 and enhances its deubiquitinating activity on PTEN [START_REF] Morotti | BCR-ABL disrupts PTEN nuclear-cytoplasmic shuttling through phosphorylation-dependent activation of HAUSP[END_REF]. Therefore, it is evident that USP7 is at the center of a complex network, and recent efforts have focused on the discovery and development of small molecule inhibitors of this protein [START_REF] D'arcy | Deubiquitinase inhibition as a cancer therapeutic strategy[END_REF][START_REF] Kategaya | USP7 small-molecule inhibitors interfere with ubiquitin binding[END_REF][START_REF] Turnbull | Molecular basis of USP7 inhibition by selective small-molecule inhibitors[END_REF][START_REF] Lamberto | Structure-guided development of a potent and selective non-covalent active-site inhibitor of USP7[END_REF][START_REF] Pozhidaeva | USP7-specific inhibitors target and modify the enzyme's active site via distinct chemical mechanisms[END_REF][START_REF] Desroses | The next step forward in ubiquitin-specific protease 7 selective inhibition[END_REF][START_REF] Gavory | Discovery and characterization of highly potent and selective allosteric USP7 inhibitors[END_REF]. These inhibitors were found to enhance apoptosis in chronic lymphocytic leukemia [START_REF] Carra | Therapeutic Inhibition of USP7-PTEN network in chronic lymphocytic leukemia: a strategy to overcoming TP53 mutated/ deleted clones[END_REF] and multiple myeloma [START_REF] Chauhan | A small molecule inhibitor of ubiquitin-specific protease-7 induces apoptosis in multiple myeloma cells and overcomes bortezomib resistance[END_REF] and to reduce neuroblastoma growth in vivo [START_REF] Fan | USP7 inhibitor P22077 inhibits neuroblastoma growth via inducing p53-mediated apoptosis[END_REF]. In this context, we hypothesize that a possible dysregulation of USP7 may contribute to resistance in AML.

In this study, we first showed that pharmacological or RNA interference-mediated inhibition of USP7 significantly reduced AML cell proliferation in vitro and in vivo, impaired replication fork progression and increased AML cell death. Furthermore, our results demonstrate that USP7 expression is a predictive marker of resistance to chemotherapy in AML. We also report that USP7 is a major regulator of CHK1 protein in AML through its deubiquitinating activity, which leads to CHK1 stabilization. Finally, we found that USP7 inhibition potentiates cytarabine (AraC) treatment by increasing leukemic cell death, and independently of p53 status. Altogether, these results suggest that targeting USP7 may represent a novel therapeutic strategy in AML.

Materials and methods

Proximity ligation assay (PLA)

For PLA, cells were washed twice with 1× PBS, fixed with 4% PFA and then permeabilized. AML cells were seeded on coverslips pretreated with Cell Tack (ref 354240, lot 7033571, Corning). Using the Duolink kit by Sigma, the following steps are described according to Adam et al. [START_REF] Adam | A PIM-CHK1 signaling pathway regulates PLK1 phosphorylation and function during mitosis[END_REF]. The PLA signal was detected with a Zeiss confocal microscope (LSM 780). A series of Z-stack confocal microscopy images was taken, and quantification of the images was carried out using ImageJ software.

For immunofluorescence, the staining protocol used is described by David et al. [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF].

Clonogenic assay

Clonogenic assays were performed on different AML primary samples, as previously described [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF].

Tumor xenografts in NOD SCID gamma (NSG) mice NOD/LtSz-SCID/ IL-2Rγ chain null (NSG) mice were bred at the UMS006 in Toulouse (France) using breeders obtained from Charles River. All animal experiment protocols were approved by the Institutional Animal Care and Use Ethics Committee of the UMS006 and the Midi-Pyrénées Region (approval 2017071314596526). Mice (6-9 weeks old) were sublethally treated with busulfan (30 mg/kg) 24 h before injection of leukemic cells. Leukemic OCI-AML3 cell line was thawed at room temperature, washed twice in PBS, and suspended in Hanks balanced salt solution at a final concentration of 2 million cells per 200 μl of Hanks balanced salt solution per mouse for tail vein injection. Twenty-two days after injection, the mice were randomly split into two groups: one group was treated with 100 µl corn oil with 10% DMSO as a vehicle (n = 7), and one group was treated with 30 mg/kg of P22077 diluted in 100 µl corn oil with 10% DMSO (n = 8). Mice were treated by daily IP injections of P22077 or the vehicle for 5 days. Daily monitoring of mice for symptoms of disease (ruffled coat, hunched back, weakness and reduced mobility) determined the time to euthanize animals with signs of distress. NSG mice were euthanized in accordance with European ethics committee protocols. Overall mouse survival was established under these conditions.

For patient-derived xenograft (PDX), NSG mice (6-9 weeks old) were sublethally treated with busulfan (30 mg/kg) 24 h before injection of human leukemic cells. Mice were engrafted by an injection of 3•10 6 primary AML cells in the tail vein. Twenty-two weeks after injection, peripheral blood was obtained to determine the fraction of human blasts using flow cytometry. Mice were then randomly split into two groups: one group was treated with 100 µl PBS containing 10-15% DMSO as a vehicle (n = 6), and one group was treated with 15 mg/kg of P22077 diluted in 100 µl PBS with DMSO (n = 7). Mice were treated 3 days per week for 2 weeks and daily for 1 week by IP injections.

NSG mice were euthanized in accordance with European ethics committee protocols. Bone marrow (mixed from tibias and femurs) and spleen were dissected in a sterile environment and flushed in Hanks balanced salt solution with 1% FBS. Blasts from bone marrow and spleen were labeled with FITC-conjugated anti-hCD3, PE-conjugated anti-hCD33, PerCP-Cy5.5-conjugated anti-mCD45.1, and APC-H7-conjugated anti-hCD45 (all antibodies from Becton Dickinson, BD, except for FITC-conjugated anti-hCD3 from Ozyme Biolegend) to determine the fraction of human blasts (hCD45+mCD45.1-hCD33+ cells) using flow cytometry. Analyses were performed on a Life Science Research II (LSR II) flow cytometer with DIVA software (BD). The number of AML cells in the total tumor burden (in bone marrow and spleen) was determined by using CountBright beads (Invitrogen) and applying the described protocol.

RNA sequence analysis

Single-cell transcriptomics

A blood cell sample was collected from one patient at diagnosis and at relapse following chemotherapeutic induction. AML cells were purified by Ficoll centrifugation and AML blast cells were then sorted based on the expression of CD45+, CD33+ and annexinV-. Five hundred cells were used per condition for 10x Genomics single cell assaying as recommended by 10x Genomics. Sequencing was performed with an Illumina High seq 3000. The raw data were clustered with the 10x Genomics cell ranger pipeline.

Transcriptomic signatures and data mining

Two USP7 signatures were generated from transcriptomes of AML patients with high versus low expression of USP7 from two independent databases (TCGA Network [START_REF] Network | Genomic and epigenomic landscapes of adult de novo acute myeloid leukemia[END_REF], and Verhaak database GSE6891 [START_REF] Verhaak | Prediction of molecular subtypes in acute myeloid leukemia based on gene expression profiling[END_REF]). Gene Set Enrichment Analysis (GSEA) using USP7 signature was performed from the relapse and diagnosis transcriptomes according to Hackl et al. [START_REF] Hackl | A gene expression profile associated with relapse of cytogenetically normal acute myeloid leukemia is enriched for leukemia stem cell genes[END_REF], the high and low responder in mice according to Farge T et al. [START_REF] Farge | Chemotherapy-resistant human acute myeloid leukemia cells are not enriched for leukemic stem cells but require oxidative metabolism[END_REF], and the LSC and bulk transcriptomes according to Eppert et al. [START_REF] Eppert | Stem cell gene expression programs influence clinical outcome in human leukemia[END_REF]. The analysis was performed using the Broad Institute GSEA3.0 software.

DNA fiber spreading

DNA fiber spreading was performed on leukemic cell line, as previously described [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF].

Statistical analysis

At least three independent experiments were carried out to generate each dataset and statistical analyses were performed as described in the figures, using the Prism software package (GraphPad Software). Results are expressed as mean ± SEM. Differences were considered significant for the following p values: *P < 0.05; **P < 0.01; ***P < 0.001; ****P < 0.0001.

Results

Targeting USP7 in leukemic cell lines

We first investigated the impact of USP7 silencing or pharmacological inhibition on the proliferation and viability of leukemic cell lines. USP7 silencing in HL-60 cells (Supplementary Fig. 1A) significantly reduced cell proliferation (Fig. 1a) by increasing apoptotic cell death (Fig. 1b) and the proportion of cells in the G1 phase (Fig. 1c).

In order to assess the importance of p53 status in this response, USP7 pharmacological inhibition (P22077) was then tested in both HL-60 (p53-) and OCI-AML3 (p53+) cell lines (Fig. 1d-f andg-i, respectively and Supplementary Fig. 1B). This confirmed the results obtained by RNA interference, and indicated that the p53 pathway is in all probability not involved in these USP7 inhibition effects on leukemic cells. As a control of USP7 pharmacological inhibition efficiency, we demonstrated that polyubiquitination of the serine-threonine kinase CHK1, one of the welldescribed USP7 substrates [START_REF] De Vega | USP7 controls Chk1 protein stability by direct deubiquitination[END_REF][START_REF] Zhang | ATM-mediated stabilization of ZEB1 promotes DNA damage response and radioresistance through CHK1[END_REF], was increased, and that in turn, CHK1 protein level was decreased in both leukemic cell lines (Supplementary Fig. 1C-E).

The effect of USP7 inhibition in vivo was then studied in an AML xenograft model. Briefly, 2 million OCI-AML3 cells were injected into the tail vein of immunodeficient adult NSG mice after pre-conditioning with a sub-lethal treatment of busulfan one day prior to injection in order to establish AML disease. USP7 inhibitor treatment by intraperitoneal injection over a 5-day span significantly improved overall mouse survival (***P = 0.0001) compared with vehicle-treated mice (Fig. 1j). Altogether, these results show that USP7 inhibition significantly reduces leukemic cells proliferation and impairs their viability in vitro and in vivo.

USP7 inhibition impacts primary AML cell viability without affecting normal cells

We then performed similar experiments on primary AML cells and normal peripheral blood mononuclear cells (PBMC). As shown in Fig. 2a, USP7 inhibition with P22077 significantly reduced cell viability in three different primary AML samples, while PBMCs were barely affected by the treatment. It should be noted that primary AML and PBMCs classically do not proliferate under these culture conditions. Altogether, these results suggest that USP7 controls leukemic cell viability without affecting normal hematopoietic cell survival.

Subsequently, to study the effect of USP7 inhibition in vivo, we used the PDX model. Briefly, 3 million unsorted primary AML cells were injected into adult NSG mice (6-9 weeks) after pre-conditioning with a sub-lethal treatment of busulfan 1 day prior to injection. Engraftment efficiency was monitored in peripheral blood by flow cytometry analysis of human CD45 + and CD33 + positive cells at 22 weeks after xenotransplantation (Fig. 2b and Supplementary Fig. 2A) before starting treatment and for cohort randomization. Mice showing at least 20 AML human blasts per microliter of peripheral blood were assigned to experimental groups in order to obtain balanced average engraftment levels in each cohort at the initiation of therapy (Fig. 2c). PDX mice were treated by intraperitoneal injection with 15 mg/kg P22077, three times per week for 2 weeks and daily for 1 week (Fig. 2b). After 3 weeks of treatment, a significant reduction in total tumor burden in bone marrow and spleen was observed in USP7 inhibitortreated versus untreated mice (Fig. 2d). In addition, a significant reduction in spleen size was monitored in USP7 inhibitor-treated mice compared with untreated mice (Fig. 2e,f). It is important to note that P22077 was used at a well-tolerated dose, which was monitored by mouse body weight and the viability of normal hematopoietic cells in murine bone marrow (Fig. 2g,h). Otherwise, the USP7 inhibitor does not affect normal hematopoietic murine cells in the spleen, as determined by the number of CD45.1positive cells (Supplementary Fig. 2B).

USP7 abundance, transcriptomic gene signature and AML patient outcome

We then determined USP7 protein levels in a cohort of 57 primary AML samples by western blot analysis. Ten of these samples were not usable (degraded or did not have enough actin signal, annotated as not included (NI) in Fig. 4a). Therefore, we performed the analysis on 47 patient samples. As shown in Fig. 3a, USP7 protein level was highly variable in AML samples. Interestingly, a similar analysis was performed in a panel of six AML cell lines (Supplementary Fig. 3A), and the same heterogeneous expression was shown.

Therefore, we divided the patient data into two groups according to their median USP7 expression: low or high protein expressers (Table 1). There were no significant differences between patients with low and high USP7 protein expression in terms of age, sex, cytogenetics, or NPM1 status (Table 1). We also found that USP7 expression seems to correlate with FLT3-ITD mutation. Furthermore, we also observed that samples with high USP7 expression presented an increase in white blood cells, which is consistent with a frequent clinical observation in AML patients harboring mutations in the FLT3 tyrosine-kinase receptor. Consequently, we investigated the impact of FLT3-ITD on USP7 abundance in two human FLT3-ITD-positive cell lines, MOLM-14 and MV4-11. Pharmacological inhibition of FLT3 activity in these leukemic cells with AC220 (quizartinib), a specific FLT3 inhibitor approved by the US Food and Drug Administration for the treatment of patients with relapse/refractory FLT3-ITD AML, by 2, 6 or 24 h incubation was not followed by a significant decrease in USP7 level (Supplementary Fig. 3B).

Given our results and that USP7 has been implicated in transcription regulation [START_REF] Desroses | The next step forward in ubiquitin-specific protease 7 selective inhibition[END_REF][START_REF] Chauhan | A small molecule inhibitor of ubiquitin-specific protease-7 induces apoptosis in multiple myeloma cells and overcomes bortezomib resistance[END_REF][START_REF] An | USP7 inhibitor P5091 inhibits wnt signaling and colorectal tumor growth[END_REF], we then questioned whether high USP7 abundance AML cells display a specific USP7-related transcriptomic signature. To test this hypothesis, USP7 abundance (transcript and protein) was quantified in AML patient samples. We found that USP7 transcript abundance and USP7 protein levels were highly correlated (R = 0.8073, **P = 0.0085) (Fig. 3b). We then defined a USP7-specific gene signature (41 genes upregulated, Supplementary Table 1) from The Cancer Genome Atlas transcriptomic database [START_REF] Network | Genomic and epigenomic landscapes of adult de novo acute myeloid leukemia[END_REF], and found that this signature is highly enriched in the transcriptome of AML patient samples at relapse (HACKL cohort, GEO: GSE66525 [START_REF] Hackl | A gene expression profile associated with relapse of cytogenetically normal acute myeloid leukemia is enriched for leukemia stem cell genes[END_REF]) (Fig. 3c). We also observed a similar enrichment in transcriptomes of cytarabine-resistant human AML cells purified from PDX models treated with clinically relevant doses of cytarabine (GEO: GSE97631 [START_REF] Farge | Chemotherapy-resistant human acute myeloid leukemia cells are not enriched for leukemic stem cells but require oxidative metabolism[END_REF]) (Fig. 3d). These data highlight a correlation between high USP7 transcriptomic signature, AML cell resistance to chemotherapeutic drugs and relapse. Therefore, USP7 signature could represent a new predictive marker of chemoresistance and relapse in AML.

To further characterize the heterogeneity of primary AML samples, we performed single-cell RNA sequencing of AML cells collected either at diagnosis or at relapse from an AML patient (IM10) (Fig. 3e) treated with a combination of anthracycline and cytarabine. Based on the difference in their gene expression, we identified two distinct transcriptional clusters of cells at diagnosis and relapse (Fig. 3f), and observed that at relapse, the decrease was striking in cluster 2 compared with cluster 1 (Fig. 3f, right panel), which implies that the signature of cluster 1 corresponds to the gene signature of the chemoresistant cells in this patient. Interrogation of two public transcriptomic datasets established on diagnosis of AML patients (TCGA Network [START_REF] Network | Genomic and epigenomic landscapes of adult de novo acute myeloid leukemia[END_REF], and Verhaak database GSE6891 [START_REF] Verhaak | Prediction of molecular subtypes in acute myeloid leukemia based on gene expression profiling[END_REF]) revealed that this chemoresistant gene signature was enriched in AML samples with high USP7 abundance (Fig. 3g,h). USP7 protein abundance was high in primary cells from these patients (Supplementary Fig. 3C), which is consistent with the data described above. These data strongly suggest that high USP7 levels are associated with chemoresistance in AML cells, and with the initiation of relapse following standard chemotherapy treatment. Overall, these data reveal that this high USP7 signature is a predictive marker of resistance and relapse in AML cells.

USP7 and CHK1 protein expression are correlated in AML

As previously described, CHEK1 expression is an independent prognostic marker in AML and the high protein abundance correlates with resistance to cytarabine in primary AML cells [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF]. In order to determine whether USP7 participates in chemoresistance of leukemic cells by modulation of CHK1 abundance/stability, we first investigated their protein levels in our cohort of 57 primary AML samples. Both CHK1 and USP7 protein expression levels were heterogeneous in AML cells (Fig. 4a,b). Based on the median expression of CHK1 protein, we then separated the cohort into two groups of high CHK1-(Fig. 4a,c, exemplified by sample # 29) and low CHK1-(Fig. 4a and Supplementary Fig. 3D, exemplified by samples #28 and #30) expressing samples. A significant correlation between CHK1 and USP7 protein levels was found in high CHK1 (Fig. 4c) but not in the low CHK1 samples (Supplementary Fig. 3D). Overall, these data strongly suggest a functional link between USP7 and CHK1 in leukemic cells that express high CHK1 levels.

USP7 interacts with and stabilizes CHK1 protein through K48 deubiquitination in AML cells

To determine whether USP7 can regulate CHK1 in leukemic cells, we first immunoprecipitated CHK1 from HL-60 and HEL leukemic cell lines (Fig. 5a, b, respectively). In both cell lines USP7 co-immunoprecipitated with CHK1. This demonstrates the presence of both proteins in the same complex.

This interaction was further confirmed by PLA used to visualize the proximity between two proteins. Through this approach, we confirmed that CHK1 and USP7 are colocalized in HL-60 cells, as shown by the presence of PLA dots in cells (Fig. 5c). The absence of staining in the presence of a single PLA antibody was confirmed as recommended (Supplementary Fig. 4a). Moreover, CHK1 or USP7 silencing with siRNA significantly reduced the number of PLA dots, which demonstrates the specificity of this signal (Fig. 5c).

Since we demonstrated that USP7 and CHK1 interact in leukemic cells, we then determined whether USP7 can affect CHK1 ubiquitination in AML cells. First, we performed experiments by overexpressing HA-tagged ubiquitin in HL-60 cells and using siRNA against CHK1 or USP7 by treatment with the proteasome inhibitor MG132 to stabilize ubiquitinated forms of the protein. Endogenous CHK1 was then immunoprecipitated and immunoblotted with CHK1 and HA-antibodies. The HA-ubiquitin signal present in immunoprecipitated CHK1 was higher when USP7 was silenced (Fig. 5d, and Supplementary Fig. 4B). To further support this observation, endogenous CHK1 was immunoprecipitated from HL-60 cells treated with P22077 in the presence of the proteasome inhibitor MG132. These fractions were immunoblotted and probed with an anti-ubiquitin antibody. The results indicate that CHK1 was more abundantly ubiquitinated following USP7 inhibition compared with the control, which suggests that USP7 is a negative regulator of CHK1 ubiquitination (Supplementary Fig. 1C). Collectively, these results indicate that USP7 is an important mediator of CHK1 deubiquitination in leukemic cells.

We then investigated whether USP7 can regulate CHK1 protein level or its subcellular localization in leukemic cells.

Fig. 2 USP7 inhibition impacts primary leukemic cell proliferation and viability in vitro and in vivo, without impacting normal cells. a Peripheral blood mononuclear cells (PBMC, n = 3) and AML primary samples (n = 3) were treated with 10 μM P22077 for 24 h, and cell viability was assessed by flow cytometry with annexinV/cell viability staining, using a MACSQuant VYB flow cytometer and data were analyzed with FlowJo software. b Schematic diagram of the treatment regimen and schedule used to treat NSG-based PDX models with P22077. Peripheral blood engraftment was assessed after 22 weeks and mice were assigned to experimental groups. Six mice were treated with the vehicle (PBS + DMSO) or 15 mg/kg/day P22077 via intraperitoneal route. After 3 weeks of treatment, mice were sacrificed in order to characterize viable residual AML cells. c Engraftment efficiency was measured in peripheral blood by flow cytometry analysis at 22 weeks after xenotransplantation. Mice with at least 20 human AML blasts/µl of blood were assigned to experimental groups to obtain balanced average engraftment levels in each cohort at initiation of therapy. Six mice were assigned to the vehicle group and seven mice were assigned to the P22077 treatment group. For statistical analyses, a t-test was used, ns not significant. d The total number of viable human AML cells expressing CD45+ and CD33+ was analyzed and quantified using flow cytometry in the bone marrow and spleen of P22077-treated AMLxenografted mice compared with vehicle-treated AML-xenografted mice (total cell tumor burden in millions). For statistical analysis, a nonparametric t-test was used. *P ≤ 0.05. e, f Spleen size was monitored in USP7 inhibitor condition (P22077, n = 7) compared with untreated mice (Vehicle, n = 6). For statistical analysis, a nonparametric t-test was used. **P ≤ 0.01 g The body weight of the mice was monitored during the experiment. The mean of each group is presented and male and female mice are differentiated. h Murine viable cells were measured by flow cytometry analysis (CD45.1+ ANXV-cells) to determine the toxicity of P22077 in bone marrow. For statistical analyses, a t-test was used, ns not significant.

Having performed USP7 pharmacological inhibition in HL-60 (p53-) and OCI-AML3 (p53+) (Supplementary Fig. 1D,E), we went further and silenced USP7 expression by RNA interference in HL-60 and HEL (p53+) cell lines, and observed a significant decrease in CHK1 protein levels (Supplementary Fig. 4C and4D, respectively), although only half of the USP7 proteins were downregulated with these two siRNA sequences in HL-60. Using immunofluorescence, we Fig. 3 USP7 gene expression and transcriptomic gene set. a USP7 protein levels were determined by immunoblotting in 57 primary AML samples. Actin was used as the loading control and extracts from KG1a leukemic cell line served as a between-gel control. Total USP7 protein level quantification is reported. b Linear regression analysis for the correlation between USP7 mRNA level and USP7 protein level, analyzed in nine AML primary samples. c GSEA of the high USP7 gene signature in transcriptomes of patients with AML at relapse (red) compared with at diagnosis (blue). d GSEA of the high USP7 gene signature in transcriptome of patients with AML that are low (red) compared with high (blue) responders in PDX models. e Schematic diagram of primary AML samples from one patient (IM10) at diagnosis and at relapse following chemotherapeutic induction. AML blast cells were sorted based on the CD45+, CD33+ and ANXV-expression and then processed for single-cell 3′ RNA sequencing analysis. f SNE plot of scRNAseq data with cells colored according to K-means cluster assignment of the cells from IM10 at diagnosis, and at relapse. g Gene set enrichment analysis (GSEA) of cluster 1 signature generated using the cells at diagnosis and at relapse was performed from the transcriptomes of the TCGA database. h The same analysis as in g was performed from the transcriptome of the Verhack database. also confirmed that ~50% of the CHK1 protein signal disappears after 24 h of USP7 inhibition, without any changes in CHK1 subcellular localization (Supplementary Fig. 4E). Together, these experiments show that genetic or pharmacological inhibition of USP7 decreases CHK1 protein levels in three different leukemic cell lines, independently of p53 status, which demonstrates the role of USP7 deubiquitinase activity on CHK1 stability.

To complete our investigation, experiments were performed to test whether the half-life of CHK1 is regulated by USP7 in leukemic cells. To this end, HL-60 cells were treated with P22077 and the half-life of CHK1 was measured in the presence of 50 μg/ml cycloheximide (CHX) to inhibit protein biogenesis. As shown in Fig. 5e, USP7 inhibition significantly shortened the half-life of CHK1 protein after 6 h of P22077 treatment, without affecting CHEK1 mRNA expression (Supplementary Fig. 4F).

Finally, to further understand which poly-ubiquitin chain on CHK1 is removed by USP7, we immunoprecipitated an equal amount of K48-linked poly-ubiquitin or K63 linked poly-ubiquitin proteins with specific antibodies in HL-60 cells. The results indicated that K48-linked ubiquitin species conjugated on CHK1 represent the major form of ubiquitin moieties (Fig. 5f). Moreover, inhibition of USP7 by P22077 resulted in a significant increase in the K48 poly-ubiquitylation of CHK1 in HL-60 cells (Fig. 5g). Overall, our data demonstrate that USP7 stabilizes CHK1 by preventing proteasomal degradation by removing K48linked poly-ubiquitin chains in leukemic cells.

On the whole, our results predict that targeting USP7 in leukemic cells, which exhibit a strong dependence on CHK1 level for proliferation [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF], can impair their survival.

Targeting USP7 overcomes cytarabine resistance in AML

To explore and confirm whether USP7 participates in the chemoresistance of leukemic cells, we examined the capacity to sensitize HL-60 and OCI-AML3 cells to cytarabine (AraC) treatment through USP7 inhibition. As shown in Fig. 6a,b, P22077 synergistically (Supplementary Fig. 4G) enhanced cell death induced by cytarabine in these resistant cell lines in vitro, independently of p53 status.

To further support our hypothesis, we quantified the capacity of primary AML cells to form colonies in a methylcellulose-based semi-solid medium, when exposed to clinically relevant concentrations of cytarabine (10 nM) combined with the P22077 inhibitor (Fig. 6c,d). Based on the median expression of USP7 protein, we separated this cohort into two groups of high USP7 and low USP7 expressing samples. Leukemic cells with high USP7 levels (Fig. 6c) tended to be more resistant to cytarabine compared with cells that expressed lower levels of USP7 (Fig. 6d), which is consistent with our transcriptomics analyses Statistical analysis was established by the unpaired t-test for age and white blood cell count and then a Fisher's exact tests were applied to determine the P value.

(Fig. 3). It is striking to note that USP7 inhibition potentiated the effects of cytarabine in cells that expressed high levels of USP7, leading to reduced colony formation, while it did not modify the sensitivity to cytarabine of cells that expressed low levels of USP7. Our results indicate that P22077 significantly enhances the cytotoxic effect of cytarabine on leukemic cells by reducing their colony-forming potential. In order to determine whether USP7 participates in the chemoresistance of leukemic cells by modulation of CHK1 stability, we then monitored the CHK1/USP7 association by PLA in AML cells treated with cytarabine. A significant increase in the proximity between the two proteins was visualized in cytarabinetreated HL-60 cells (Fig. 6e,f). The specificity of the interaction was confirmed by using a USP7 inhibitor (P22077) that induced a significant decrease in CHK1 protein level (Supplementary Fig. 4E), dramatically reducing the number of PLA dots per cell (Fig. 6e,f). Similar results were obtained with the combination of cytarabine and P22077. Altogether, these data suggest that USP7 plays a role in promoting CHK1 stability in response to DNA-damaging treatment.

Fig. 4 USP7 and CHK1 protein expression correlate in primary AML samples. a CHK1 and USP7 protein levels were determined by immunoblot and actin was used as a loading control in 46 primary AML samples. KG1a cell line extract was used as an internal control between gels (CTL). Samples were considered "high CHK1 abundance" if the average protein abundance value was higher than the median. b Linear regression analysis for the correlation between CHK1 and USP7 protein levels in primary AML samples. c Same analysis as in b with 21 primary AML samples with high CHK1 abundance.

Consequently, we further examined whether a USP7/ CHK1 axis is involved in AML cell chemoresistance by measuring cell death in response to CHK1 inhibitor (SCH900776) treatment alone or in combination with cytarabine and P22077 (Fig. 6g). As described earlier (Fig. 6a,b), P22077 significantly increased cell death after cytarabine treatment. Interestingly, the addition of SCH900776 in P22077 and cytarabine-treated cells did not increase cell death, which suggests that P22077induced cell death is mediated by the USP7/CHK1 axis. Despite the large spectrum of USP7 substrates, we showed that the cytotoxic effect caused by USP7 inhibition is mediated, at least in part, by a decrease in CHK1 expression. Overall, these results indicate that P22077 could be a relevant anti-leukemic drug in overcoming chemoresistance.

Finally, to further understand the molecular mechanisms underlying these effects, we explored the contribution of this USP7/CHK1 axis in the DNA replication fork progression in AML cells. We used the DNA fiber spreading assay for this purpose, in keeping with previous reports that clearly documented the role of USP7 or CHK1 in the DNA replication process [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF][START_REF] Petermann | CHK1 requirement for high global rates of replication fork progression during normal vertebrate S phase[END_REF][START_REF] Lecona | USP7 is a SUMO deubiquitinase essential for DNA replication[END_REF]. HL-60 cells were cultured in the presence of a USP7 inhibitor (P22077), a CHK1 inhibitor (SCH900776), or a combination of both for 16 h, then the thymidine analogs CldU and IdU were sequentially pulsed for an equal amount of time (20 min each) resulting in the labeling of nascent DNA tracks in vivo. With this method, we monitored replication fork progression as indicated by the presence of nascent DNA at the level of individual replicating DNA molecules. A reduction in nascent DNA track length is indicative of stalled DNA replication. USP7 or CHK1 inhibition led to a significant and similar reduction in the rate of ongoing replication forks in HL-60 cells (Fig. 6h). Interestingly, concomitant inhibition of CHK1 and USP7 did not further increase the replication fork blockage, which supports the hypothesis that USP7 and CHK1 are in the same signaling pathway of DNA replication progression in HL-60 cells.

Discussion

Recent studies have reported that the USP7 level is elevated in several types of tumors, and that USP7 overexpression is often predictive of a poor prognosis [START_REF] Bhattacharya | Emerging insights into HAUSP (USP7) in physiology, cancer and other diseases[END_REF][START_REF] An | USP7 inhibitor P5091 inhibits wnt signaling and colorectal tumor growth[END_REF][START_REF] Zhao | USP7 overexpression predicts poor prognosis in lung squamous cell carcinoma and large cell carcinoma[END_REF][START_REF] Ma | Ubiquitin-specific protease 7 expression is a prognostic factor in epithelial ovarian cancer and correlates with lymph node metastasis[END_REF]. Several studies consequently suggested USP7 as an attractive pharmaceutical target for various cancers, and pharmacological inhibitors of this enzyme have recently been developed [START_REF] Bhattacharya | Emerging insights into HAUSP (USP7) in physiology, cancer and other diseases[END_REF][START_REF] D'arcy | Deubiquitinase inhibition as a cancer therapeutic strategy[END_REF][START_REF] Kategaya | USP7 small-molecule inhibitors interfere with ubiquitin binding[END_REF][START_REF] Turnbull | Molecular basis of USP7 inhibition by selective small-molecule inhibitors[END_REF][START_REF] Lamberto | Structure-guided development of a potent and selective non-covalent active-site inhibitor of USP7[END_REF][START_REF] Pozhidaeva | USP7-specific inhibitors target and modify the enzyme's active site via distinct chemical mechanisms[END_REF][START_REF] Desroses | The next step forward in ubiquitin-specific protease 7 selective inhibition[END_REF][START_REF] Gavory | Discovery and characterization of highly potent and selective allosteric USP7 inhibitors[END_REF]. In this study, we showed that USP7 plays a key role in AML chemoresistance. Furthermore, we observed heterogeneous USP7 levels in AML cell lines and primary samples and established that patients with a high USP7 signature were more prone to chemoresistance and relapse. Experiments using single-cell analysis support these findings by showing that the resistant gene signature is enriched in AML samples that exhibit high USP7 levels, which reveals that this chemoresistant cell population preexists at diagnosis and is enriched following first line treatment. Consequently, these data suggest that USP7 could be a new prognostic marker and a potential therapeutic target for AML. Our finding that USP7 inhibition in vitro and in vivo significantly suppresses leukemic cell growth, alone and in combination with chemotherapy, independently of p53 status supports this notion. Although p53 can be stabilized by USP7 [START_REF] Li | A dynamic role of HAUSP in the p53-Mdm2 pathway[END_REF], it appears that in AML, which does not frequently present p53 mutations, cells do not rely on p53 stabilization. Despite the large spectrum of USP7 substrates, we showed that the cytotoxic effect caused by USP7 inhibition is mediated, at least in part, by a decrease in CHK1 function due to similar anti-leukemic effects with the CHK1 inhibitor SCH900776. Our data clearly demonstrate that the USP7/CHK1 axis plays an important part in DNA replication progression and leukemic proliferation and survival. Based on our recent study in which we showed that high CHK1 protein levels contribute to cellular resistance to cytarabine, in part by facilitating fork progression or stalled fork restart [START_REF] David | CHK1 as a therapeutic target to bypass chemoresistance in AML[END_REF], and based on the present results, we now suggest that high CHK1 levels, stabilized by USP7, allow cells to survive by enabling them to adapt to DNA replication stress induced by cytarabine treatment. Therefore, we believe that dysregulation of a USP7/CHK1 axis may be a new "Achilles heel" in AML Fig. 5 USP7 interacts and deubiquitinates CHK1 in leukemic cells. a Whole-cell lysates from HL-60 cells were immunoprecipitated with anti-CHK1 antibody or an irrelevant Immunoglobulin (IgG) and then immunoblotted with antibodies against the indicated proteins. b A similar experiment to a was performed with anti-CHK1 antibody from siRNA control (CTL) or siRNA against CHK1 transfected HEL cells for 24 h. c Proximity ligation assays were performed using the CHK1 and USP7 antibodies on HL-60 cells transfected with the indicated siRNA for 48 h. DNA was counterstained with DAPI. Z-stack images were acquired by confocal microscopy. Scale bar, 1 µm. The quantification of foci per cell is presented. Quantification of the images was performed with ZEN and ImageJ software. For co-staining of CHK1 and USP7: mean = 3.75 ± 0.22 foci/cell from 1180 nuclei analyzed under siRNA control condition (n = 4), mean = 1.85 ± 0.12 foci/cell from 1126 nuclei analyzed under siRNA CHK1 condition (n = 4), and mean = 2.69 ± 0.21 foci/cell from 650 nuclei analyzed under siRNA USP7 condition (n = 3). P values were determined using the Mann-Whitney test, with ****P ≤ 0.0001. d HL-60 cells were transfected with HA-Ub. After 72 h, a new transfection round was performed with siRNA control (CTL) or siRNA against CHK1 or USP7 for another 48 h. Finally, 6 h before harvesting, cells were treated with 10 µg/ml MG132. Cell extracts were immunoprecipitated with anti-CHK1 antibody followed by western blot analysis with the indicated antibodies. e HL-60 were treated with 50 μg/ml cycloheximide (CHX) in the presence or absence of 10 μM P22077. Lysates were prepared at the indicated times. Representative western blot is presented and quantification from three independent experiments is shown graphically below. **P ≤ 0.01. f HL-60 cellular extracts were prepared for immunoprecipitation (IP) with the indicated antibodies followed by western blotting with anti-CHK1 or anti-ubiquitin antibody. g HL-60 cells were treated with or without 10 μM P22077 for 24 h. Six hours before being harvested, cells were treated with 10 µg/ml MG132 and processed for immunoprecipitation (IP) with an irrelevant immunoglobulin (IgG) or an anti-ubiquitin (Lys48-specific) antibody, followed by immunoblotting analysis using anti-CHK1 antibody. cells. In fact, USP7 inhibition has a profound impact on proliferation and viability in leukemic cells that express high USP7 levels compared with cells that express low USP7 levels, or in PBMC. This characteristic could provide a useful therapeutic window through which this class of inhibitors could be proposed for clinical trials.

Although we documented the heterogeneity of USP7 abundance in primary AML samples, the mechanisms by which USP7 protein and mRNA expressions are controlled in leukemic cells remain to be defined. It will be important to investigate the mechanisms that control USP7 expression in the context of AML heterogeneity. In-depth analyses of the clinical and biological characteristics of AML patients revealed that USP7 expression significantly correlates with FLT3-ITD mutation. However, in our studies, we did not observe direct regulation of USP7 expression by the FLT3 tyrosine-kinase receptor, but this specific aspect will require further investigations in the future. Nevertheless, we established that USP7 transcript and protein abundance are highly correlated in AML samples, which suggests that the majority of USP7 regulation might occur at the transcriptional level. Previous reports have documented the transcriptional regulation of USP7, for instance in T-cell acute lymphoblastic leukemia (T-ALL) in which the transcription factor NOTCH1 induces USP7 gene expression, and where USP7 controls the stability of NOTCH1 and the JMJD3 histone demethylase through a positive regulatory loop [START_REF] Shan | USP7 deubiquitinates and stabilizes NOTCH1 in T-cell acute lymphoblastic leukemia[END_REF][START_REF] Jin | USP7 cooperates with NOTCH1 to drive the oncogenic transcriptional program in T cell leukemia[END_REF]. It is noteworthy that the NOTCH-signaling pathway has been reported to play oncogenic and tumor suppressor functions in the hematopoietic system, although the biological and clinical relevance remains unclear in AML [START_REF] Lobry | Notch signaling: switching an oncogene to a tumor suppressor[END_REF]. A potential regulatory and functional link between USP7 and NOTCH1 is supported by our transcriptomic analyses considering that Delta-like non-canonical Notch ligand 1 (DLK1) was upregulated in cells enriched in USP7 gene signature. DLK1 encodes the NOTCH activator Delta1 which is involved in primary and model AML cell growth [START_REF] Wang | NOTCH activation promotes glycosyltransferase expression in human myeloid leukemia cells[END_REF]. Furthermore, previous analyses show a significantly shorter survival of patients with high NOTCH1 or Delta1 expression, suggesting that activation of the NOTCH pathway may be associated with a poor prognosis in AML [START_REF] Xu | Activation of notch signal pathway is associated with a poorer prognosis in acute myeloid leukemia[END_REF][START_REF] Kamga | Notch signaling drives bone marrow stromal cellmediated chemoresistance in acute myeloid leukemia[END_REF]. All these studies, combined with our data, suggest the existence of a NOTCH/USP7/CHK1 axis involved in AML chemoresistance, which should be investigated in future studies.

Finally, the large spectrum of USP7 substrates and its implication in fundamental cellular and developmental processes define USP7 as a highly important molecule. Depending on the targeted substrates and the cellular context, USP7 might be either a tumor-suppressive or an oncogenic protein. Consequently, it will be important to understand how the acute leukemic cells highjack USP7 functions.

In summary, our study provides evidence that USP7 is an important component in chemoresistant AML cells. Moreover, a combination of a USP7 inhibitor with cytarabine induces synergistic anti-leukemic activity. Overall, our study opens the possibility that targeting USP7 is a promising therapeutic strategy in treating AML and supports the inclusion of USP7 inhibitors in clinical trials aimed at overcoming chemoresistance in AML.

Fig. 6 USP7 inhibition potentiates cytarabine treatment in AML cells. a HL-60 cells were treated with P22077 (10 µM), cytarabine (AraC, 300 nM) or a combination of both drugs, for 24 h. Cell death was assessed by flow cytometry with annexinV/cell viability staining using a MACSQuant VYB flow cytometer and FlowJo software. Statistical analysis was performed by unpaired t-test (n = 4). *P ≤ 0.05 and **P ≤ 0.01. b OCI-AML3 cells were treated as indicated for 24 h. Cell death was assessed as in a. Statistics were established by unpaired t-test (n = 3). *P ≤ 0.05, **P ≤ 0.01 and ns not significant. c, d Analyses of the clonogenic properties of high USP7 abundance (n = 6 primary samples, left panel) and low USP7 abundance (n = 5 primary samples, right panel) in AML blast cells with continuous exposure to 10 nM cytarabine (AraC) alone or in combination with 5 µM USP7 inhibitor (P22077). Colony formation was assessed after 7 days and represented as the ratio of the number of clones between untreated and treated conditions. For statistical analyses, an unpaired Wilcoxon test was used. *P ≤ 0.05, and ns not significant. e Proximity ligation assays were performed using anti-CHK1 and anti-USP7 antibodies in HL-60 cells treated with 10 µM of P22077 and/or 300 nM of cytarabine for 16 h. DNA was counterstained with DAPI. Z-stack images were acquired by confocal microscopy. f Quantification of PLA foci from three independent experiments was performed with ZEN and ImageJ software. For co-staining of CHK1 and USP7: mean = 1.07 ± 0.07 foci/cell from 677 nuclei analyzed for the untreated condition (NT); mean = 3.31 ± 0.21 foci/cell from 743 nuclei analyzed for cytarabine treatment; mean=0.68 ± 0.04 foci/cell from 604 nuclei analyzed for P22077 treatment and mean = 1.24 ± 0.10 foci/cell from 407 nuclei analyzed for combined treatment. P-values were determined using the unpaired t-test, ****P < 0.0001. g HL-60 cells were treated with the indicated treatment for 24 h. Cell death was assessed by flow cytometry with annexinV/cell viability staining using a MACSQuant VYB flow cytometer and FlowJo software. Statistical analysis was established by unpaired t-test (n = 4). *P ≤ 0.05 and **P ≤ 0.01. In parallel, cellular extracts were collected and expression of the indicated proteins was assessed by western blotting (see under the graph). A representative western blot is presented and ratios for CHK1 protein level normalized to actin loading control are indicated below. h DNA fiber assay analysis showing iododeoxyuridine (IdU) track length (μm). HL-60 cells were treated with 10 µM of P22077 or 1 μM SCH900776 or the combined treatment for 16 h and were then labeled with CldU and IdU for 20 min. Mean (±SD) track lengths. Data are based on two independent experiments and a total of 260 (NT; mean = 14.34 ± 2.50), 299 (P22077; mean = 8.81 ± 2.87), 278 (SCH900776; mean = 8.99 ± 2.30) and 215 (combined treatment; mean = 9.16 ± 2.54) fibers were scored. Statistical analysis was performed using the unpaired t-test with the Welch's correction, ****P < 0.00001.

Fig. 1 USP7

 1 Fig. 1 USP7 Inhibition impacts leukemic cell proliferation and viability in vitro and in vivo. a HL-60 cells were transfected with the control (CTL) or USP7 siRNA for 48 h, cell quantity was then assessed by Trypan Blue staining. For statistical analyses, a nonparametric t-test was used. *P ≤ 0.05 and **P ≤ 0.01. n = 5. b In similar experiments to the data in a, cell death was assessed by flow cytometry with annexinV/cell viability staining using a MACSQuant VYB flow cytometer and raw data were analyzed with FlowJo software. Data represent the mean ± SEM of four experiments. Statistical significance was determined by student's unpaired t test. c In similar experiments as described in a, cell cycle distribution was determined using propidium iodide (PI) staining and analyzed by flow cytometry using a MACSQuant VYB flow cytometer and completed by analyses with FlowJo software. Representative cell cycle distribution profiles from three independent experiments are shown. d HL-60 cells were treated as indicated and 48 h after cell quantity was assessed by Trypan Blue staining. Statistical analyses were performed as in a (n = 5). e In similar experiments to the data in d, cell death was assessed by flow cytometry using annexinV/cell viability staining. Statistical significance was determined as in b (n = 3). f In similar experiments to the data in d, cell cycle distribution was determined using propidium iodide (PI) staining. Representative cell cycle distribution profiles from three independent experiments are shown. g OCI-AML3 cells were treated as indicated and 48 h after the number of cells was assessed by Trypan Blue staining. Statistical analyses were performed as in a (n = 6). h In similar experiments to the data in g, cell death was assessed by flow cytometry using annexinV/cell viability staining. Statistical significance was determined as in b (n = 5). i In similar experiments to the data in g, cell cycle distribution was determined using propidium iodide (PI) staining. Representative cell cycle distribution profiles from three independent experiments are shown. j Kaplan-Meier curves of mice survival were established for NSG mice engrafted with OCI-AML3, and treated with P22077 (30 mg/kg/day) or the vehicle (DMSO + corn oil) for 5 days. For statistical analysis, a Mantel-Cox test was used. ***P ≤ 0.001.

Table 1

 1 Biological properties of primary AML samples. A summary of the main biological properties of primary AML samples used in this study.

		USP7 expression
	Total	Low	High	P value
		(<median value)	(>median value)
		n	n

Acknowledgements We would like to sincerely thank Thomas Farge for his technical support during in vivo experiments and Alison Boutet for technical assistance. We thank Laetitia Ligat and Dr Marie Tosolini at the CRCT microscopy facility and Manon Farcé at the CRCT cytometry facility for their helpful discussions of the analyses and the development of macro used in the ImageJ software. We gratefully acknowledge Dr Ze'ev Ronai for the critical review of the manuscript, and sincerely thank all members of the "Cell Cycle and Cancer" team for their critical review during the entire study. The authors sincerely thank Chloé Laplagne for the technical PBMC preparation and Dr Marie Jeanne Pillaire for insightful discussions on DNA fiber spreading experiments. Maëlle Cartel is a recipient of a fellowship from the Ligue Nationale contre le Cancer. We are grateful to our healthcare professionals for their boundless investment during the COVID-19 crisis.

Author contributions MC, PLM, MG, LD, JES, SM, and CD conceived and designed the study; MC, PLM, MG, LD, and CD performed experiments; SB and VMDM. contributed clinical samples; MC, AB, JES, SM, and CD wrote the manuscript; AB, JES, SM, and CD insured administrative, technical and material support; SM and CD supervised the project.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Ricardo A Salido-Ruiz
email: ricardo.salido@cucei.udg.mx

Radu Ranta
email: radu.ranta@univ-lorraine.fr

Gundars Korats

Steven Le Cam

Laurent Koessler

Valerie Louis-Dorr

A unified weighted minimum norm solution for the reference inverse problem in EEG

Keywords: EEG, Reference potential, Inverse problems

. In this work we do not aim to further compare these techniques but to enlighten the fact that both solutions can be derived from a general inverse problem formalism for reference estimation. In particular, we show that the AR is the minimum norm solution, while REST is a weighted minimum norm including some approximate propagation model. AR is thus a particular case of REST, which itself uses a particular formulation of the source estimation inverse problem.

Introduction

The EEG signals measure the potential difference between an electrode, placed somewhere on the head surface, and a reference electrode, placed somewhere else on the body. Ideally, the reference electrode should be placed in the most electrically inactive position, far from the region of interest. In practice, the reference is contaminated with unknown local and/or propagated electrical activity. Still, most of the studies need reference free potentials for better results [START_REF] Geselowitz | The zero of potential[END_REF][START_REF] Dien | Issues in the application of the average reference: review, critiques and recommendations[END_REF][START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF][START_REF] Kayser | In search of the Rosetta Stone for scalp EEG: Converging on reference-free techniques[END_REF][START_REF] Nunez | EEG coherency. I: Statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales[END_REF]. For example, cortical connections are estimated using coherence and phase delays among EEG channels, and the reference potential might strongly affect these methods1 [START_REF] Nunez | EEG coherency. I: Statistics, reference electrode, volume conduction, Laplacians, cortical imaging, and interpretation at multiple scales[END_REF][START_REF] Essl | EEG coherence and reference signals: experimental results and mathematical explanations[END_REF][START_REF] Nunez | EEG coherency II: experimental comparisons of multiple measures[END_REF][START_REF] Hu | Lecture Notes in Computer Science: Advances in Neural Networks[END_REF][START_REF] Hu | On the recording reference contribution to EEG correlation, phase synchorony, and coherence[END_REF]. A previous study showed that even Blind Source Separation techniques might be affected by the reference problem [START_REF] Salido-Ruiz | EEG montage analysis in the blind source separation framework[END_REF].

This paper focuses on EEG recorded using a scalp reference. In this case, the reference electrode captures the same kind of activity as any other electrode, that is a mixture of brain sources. In this context, the most popular solution for cancelling the reference potential is to subtract the spatial average of the recorded signals from all measurements. This solution, known as average reference montage (AR) was challenged by the Reference Standardization Technique (REST) [START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF]. Several authors compared the two solutions (or other re-referencing techniques) [START_REF] Zhai | A study on the reference electrode standardization technique for a realistic head model[END_REF][START_REF] Yao | Evaluation of different cortical source localization methods using simulated and experimental EEG data[END_REF][START_REF] Nunez | REST: A Good Idea but Not the Gold Standard[END_REF][START_REF] Kayser | In search of the Rosetta Stone for scalp EEG: Converging on reference-free techniques[END_REF][START_REF] Salido-Ruiz | Problmes inverses contraints en EEG : applications aux potentiels absolus et l'influence du signal de rfrence dans l'analyse de l'EEG[END_REF][START_REF] Liu | Estimating a neutral reference for electroencephalographic recordings: The importance of using a high-density montage and a realistic head model[END_REF][START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF], the overall conclusion being that REST is deemed to yield better results, although this is not always the case, especially when the head model is prone to inaccuracies. The goal of this paper is to theoretically show that both AR and REST belong to the same family of inverse problems widely encountered in EEG analysis, i.e., weighted minimum norms. If the same idea was recently put forward by [START_REF] Hu | Unified Bayesian estimator of EEG reference at infinity: rREST (Regularized Reference Electrode Standardization Technique)[END_REF][START_REF] Hu | The statistics of eegunipolar references: Derivations and properties[END_REF], our paper proposes an alternative simple algebraic proof (another algebraic proof can be found also in [START_REF] Salido-Ruiz | Problmes inverses contraints en EEG : applications aux potentiels absolus et l'influence du signal de rfrence dans l'analyse de l'EEG[END_REF], in French). To avoid redundancies with the previously cited papers, we do not propose an extensive simulation protocol in order to asses if or when one of the solutions should be used, but we present an evaluation of the difference between them depending on the used head model (necessary for REST).

EEG measurement model

Consider the classical EEG linear model given by:

x = As (1)
where x ∈ R m×1 is a vector with unknown real potentials under each electrode with respect to infinity, from here defined as absolute potentials for the m electrodes, A ∈ R m×p is the mixing lead-field matrix and s ∈ R p×1 is the source vector. Note that the reference potential is included among the m; for convenience, it will be assumed in the rest of the paper that this potential is the m-th one. In EEG, the matrix A represents the head model, obtained either by analytical approximations (e.g. spherical head models) or numerically, for realistic head models obtained after MRI segmentation of head structures (brain, CSF, skull, scalp). The sources vector s contains the amplitudes of the dipoles that model the neural sources [START_REF] Baillet | Electromagnetic brain mapping[END_REF]. For these realistic models, possible source emplacements depend on the brain volume discretization but, regardless of the used head model, the number of sources p is far bigger than the number of electrodes m (p m). Equation (1) introduces the so-called instantaneous mixture model, with the matrix A time-invariant and of rank m. For real signals recorded in time, i.e., having n 1 time samples, the vectors x (s) become matrices m × n (p × n), and we consider during all the paper that x is full-row rank m.

The actually measured potentials are given by the common reference montage (CR), with potentials x CR modeled by subtracting the potential of the chosen reference electrode from the other electrodes. This can be seen as a matrix transform of the absolute potentials from (1):

x CR = T CR x = T CR As = A CR s (2)
with T CR the m -1 rank matrix:

T CR = I m-1 -1 m-1 (3)
where I m-1 is the (m -1) × (m -1) identity matrix and 1 m-1 ∈ R m-1×1 is a vector of 1's. As mentioned above and with no loss of generality, we assume the reference electrode potential as the last absolute potential in EEG linear model [START_REF] Baillet | Brainstorm: a matlab toolbox for the processing of meg and eeg signals[END_REF]. Note that, unlike x in (1, the dimension of x CR is m -1 (number of available signals).

The most employed solutions propose to cancel the reference by transforming the CR into other EEG montages (average reference -AR, bipolar -BM, Laplacian -LM) [START_REF] Hu | Automatic identification and removal of scalp reference signal for intracranial EEGs based on Independent Component Analysis[END_REF][START_REF] Zhai | A study on the reference electrode standardization technique for a realistic head model[END_REF][START_REF] Geselowitz | The zero of potential[END_REF][START_REF] Libenson | Practical approach to electroencephalography[END_REF][START_REF] Hjorth | An on-line transformation of EEG scalp potentials into orthogonal source derivations[END_REF]. A more elaborated solution, based on head modeling, is the reference standardization (REST) [START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF]. Among these, only AR and REST propose absolute potential estimations, the BM and the LM being local estimators (along with the reference potential, they also cancel propagated activities originating in far situated regions). We will only focus here on the former.

Average Reference (AR)

The classical rationale behind average reference montages (AR) is that, under the hypothesis of equidistributed electrodes covering the whole head, the sum of all electrode potentials on the scalp should be zero (see [START_REF] Bertrand | A theoretical justification of the average reference in topographic evoked potential studies[END_REF] for a proof for the spherical head model).

In practice, as the only available signals are x CR , AR signals are obtained by calculating the sum of the m -1 x CR signals, dividing it by the total number of electrodes m and subtracting it from each measurement [START_REF] Offner | The EEG as potential mapping: The value of the average monopolar reference[END_REF][START_REF] Osselton | Acquisition of EEG data by bipolar, unipolar and average reference methods: a theoretical comparison[END_REF][START_REF] Dien | Issues in the application of the average reference: review, critiques and recommendations[END_REF]. More formally, the AR is obtained by averaging over all electrodes (reference electrode included with a null potential (i.e., its potential with respect to itself):

x AR = I m - 1 m 1 m 1 T m x CR 0 = T AR x CR (4)
with

T AR = I m-1 -1 m 1 m-1 1 T m-1 1 m 1 T m (5)
As we can see, x AR has a dimension of m and rank of m -1, i.e., it preserves the rank of x CR (one less than x). In practice, the average signal is included as the last row (x m,AR )2 .

Reference Electrode Standardization Technique (REST)

The REST method [START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF][START_REF] Zhai | A study on the reference electrode standardization technique for a realistic head model[END_REF] allows to estimate absolute potentials at any point on the scalp by solving a forward problem with "equivalent sources" s constrained to a dipolar layer completely surrounding the actual sources, i.e., the brain. Several versions can be theoretically proposed, depending on the configuration chosen for the dipolar layer (e.g., a sphere circumscribing the whole brain, or the numerically approximated cortical surface). The chosen dipolar layer will yield a specific forward model between the dipoles situated on this layer and the actual electrodes placed on the head surface. Let this model be Ã. The equivalent dipolar sources on this layer are estimated from scalp EEG recordings (equivalently in CR or in AR) by using a simple inverse problem formalism. We give below only the CR based approach:

ŝ = Ã+ CR x CR (6)
with + designating the classical Moore-Penrose pseudo-inverse. Of course, ÃCR depends on a mixing model (see Eq. ( 3)), here assumed à . A forward model using the assumed à and the estimated equivalent sources ŝ yield the REST estimations of the EEG absolute potentials:

x REST = Ãŝ = Ã Ã+ CR x CR = T REST x CR (7)
It is important to recall that different REST solutions can be obtained by choosing different dipolar layer configurations and thus different à models.

Unified Inverse Problem Modeling

As seen above, two main solutions exist for estimating the absolute potentials: average montage and REST. Three relatively recent comparative studies [START_REF] Kayser | In search of the Rosetta Stone for scalp EEG: Converging on reference-free techniques[END_REF][START_REF] Nunez | REST: A Good Idea but Not the Gold Standard[END_REF][START_REF] Liu | Estimating a neutral reference for electroencephalographic recordings: The importance of using a high-density montage and a realistic head model[END_REF] conclude that both solutions present valid theoretical arguments and that both are acceptable. Our aim is to deepen the analysis, to show the connections between the two techniques and their integration in a larger inverse problem framework.

Consider the case of a EEG recording with a cephalic reference as given in (Eq. 2). The estimation of absolute potentials x from the measured x CR and matrix transformation T CR , is an ill-posed inverse problem somehow similar to the classical EEG source estimation with a known mixing model T CR .

Thus, the absolute potentials inverse problem writes as follows:

x = Tx CR = TT CR x (8)
where the unknown matrix T ∈ R m×m-1 is a generalized inverse of the common reference transformation matrix T CR :

T = W -1 T T CR T CR W -1 T T CR -1 (9)
with W a weighting matrix allowing infinite solutions. In the next paragraphs we describe several particular Weighted Minimum Norm (WMN) solutions.

WMN Optimal Solution: Oracle1

The underdetermined system (8) has an infinite number of solutions. In theory, the best inverse transform T can be estimated by minimizing the leastsquares error between absolute and measured potentials. This optimal transform T O1 can be obtained by multiplying absolute potentials x by the Moore-Penrose pseudo inverse of measured potentials x CR :

T O1 = xx T CR x CR x T CR -1 = AΣ s A T T T CR T CR AΣ s A T T T CR -1 (10)
where Σ x is the absolute potentials covariance matrix, A the head model and Σ s the source covariance matrix (for completeness, the proof in given in the appendix. This optimal solution is equivalent to [START_REF] Hallez | Review on solving the forward problem in EEG source analysis[END_REF] for

W -1 = Σ x = AΣ s A T .
Of course, it remains theoretical, because neither the absolute potentials covariance matrix nor, equivalently, the propagation coefficients between the actual sources s and the sensors (depending on the sources positions and orientations and on the head model) and the source covariance matrix, are known in practice.

WMN Sub-Optimal Solutions: Oracle2 and REST

If, in practice, Σ s is not known, rather accurate head models can be obtained for A for all possible source locations inside the brain volume.

Ignoring the source covariance matrix but supposing that we have some a priori knowledge on their positions and orientations, as well as a good head model (i.e., making Σ s = I p but assuming a known A), one obtains:

T O2 = AA T T T CR T CR AA T T T CR -1 = A [T CR A] + = AA + CR (11)
A step further towards more realistic situations is to ignore the source positions and orientations, but to still assume a complete head model, i.e., a mixing matrix A C corresponding to a complete lead-field matrix for all possible source configurations (of course, in practice it will be computed for a grid of points inside the brain volume). The equations are exactly the same as [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF], with a complete mixing matrix A C replacing A (which is an incomplete lead-field matrix, in the sense that it accounts only for the a priori known source positions and orientations). Finally, as pointed out in [START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF], distributing equivalent sources on a layer surrounding the actual brain sources, theoretically yields another mixing model A R used in the REST solution.

If Oracle1 and Oracle2 cannot be used in practice (because they are based on unknown information on the actual sources), it is noteworthy that REST-like solutions (either based on a complete volume model or an equivalent layer) could be in principle used, because they only need a propagation model between some chosen sources and the surface electrodes, and these models can be estimated with more or less accuracy from imaging techniques and physical considerations (see e.g. [START_REF] Hallez | Review on solving the forward problem in EEG source analysis[END_REF] for a review of the forward problem in EEG)).

Finally, it is easy to see that all these weighted minimum norm solutions share the same equation [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF], which enlightens the fact that any full-rank matrix à can be used to construct a generalized inverse of T CR . Indeed,

T CR Ã ÃT T T CR T CR Ã ÃT T T CR -1 = I m-1
regardless of the accuracy of the model Ã. In this sense, a completely false or random model à will yield false estimates of the absolute potentials x, but they will still verify x CR = T CR x.

Minimum Norm Solution, MN

Within this context, the most evident solution of equation ( 9), without any a priori information about the mixture or the sources (i.e., neither on A nor on the source covariance), is the Minimum Norm Solution (MN) obtained when W = I m . Then, an estimation of absolute potentials is given by:

x = T T CR T CR T T CR -1 x CR = T + CR x CR (12)
with T + CR the Moore-Penrose pseudo-inverse of the common reference transformation matrix T CR .

Proposition. The minimum norm solution to the inverse reference problem is the AR solution from (5):

T + CR = T AR Proof. By Sherman-Morrison formula and using the definition of the Moore-6 Penrose pseudo-inverse and the expression of T CR (3), one can write:

T + CR = T T CR (T CR T T CR ) -1 = I m-1 -1 T m-1 I m-1 -1 m-1 I m-1 -1 T m-1 -1 = I m-1 -1 T m-1 I m-1 + 1 m-1 1 T m-1 -1 = I m-1 -1 T m-1 I m-1 -1 m-1 1 T m-1 1 + 1 T m-1 1 m-1 = I m-1 -1 T m-1 I m-1 -1 m-1 1 m 1 T m-1 = T AR (13)
In the first row, we use block-matrix multiplication, while in the second we use the Sherman-Morrison formula (or more generally the Woodbury identity), see for example [START_REF] Petersen | The matrix cookbook[END_REF]). Readers interested by a second algebraic proof are directed to [START_REF] Salido-Ruiz | Problmes inverses contraints en EEG : applications aux potentiels absolus et l'influence du signal de rfrence dans l'analyse de l'EEG[END_REF]. Note that the AR montage obtained by pseudo-inverting T CR can be seen as another particular case of REST, obtained for a dipolar layer having the head geometry and situated infinitely close below the electrodes (closely mimicking an à = I m ).

Results

The aim of this section is to briefly present numerical results, both on simulated and real data, supporting the previous analysis.

Simulation

The simulations presented here illustrate the fact that the accuracy of different estimations of the absolute potentials, obtained using the previously described approaches. These estimated potentials are compared with simulated ground truth potentials, computed as follows: a three shell mesh model (Colin 27) was extracted from Brainstorm [START_REF] Baillet | Brainstorm: a matlab toolbox for the processing of meg and eeg signals[END_REF] toolbox in order to have a realistic geometry. The cortical layer mesh (inner shell) had 642 nodes with 13mm mean distance between neighbouring points, while the scalp mesh (outer shell) had 1922 nodes.

A regularly spaced grid was constructed inside the inner shell (brain), with a 13mm distance between neighbouring points, which yielded 1774 points. Several source configurations were tested, namely using p = {10, 100, 500} dipolar brain sources randomly chosen among the inner shell grid points. The orientations were random, and the time courses were simulated as spatially and temporally correlated Gaussians of length N s = m 8 + 1 × f s seconds (rule derived from [START_REF] Korats | Applying ICA in EEG: Choice of the window length and of the decorrelation method[END_REF]), f s being sampling frequency and m the number of scalp electrodes. Three sensor setups were tested using m = {64, 128, 256} scalp electrodes placed on the head surface (outer shell) according to the 10-10 system, one of them being the reference electrode. Sensor coordinates were also extracted from the Brainstorm toolbox and snapped to the mesh vertices by nearest-neighbour rule. The absolute EEG scalp potentials were simulated by projecting the sources of interest to the sensors using the BEM model implemented in the Helsinki toolbox [START_REF] Stenroos | A matlab library for solving quasi-static volume conduction problems using the boundary element method[END_REF], with conductivity ratios of 40:1 between the brain and the skull and 1:1 between brain and scalp. Simulated absolute potentials x were transformed into measured potentials x CR by (2). One hundred simulations were performed for every configuration (position, orientations and time courses of the dipoles).

In order to test the influence of the model accuracy on the results, we tested the different weighted inverse solutions, from the fully specified Oracle [START_REF] Hjorth | An on-line transformation of EEG scalp potentials into orthogonal source derivations[END_REF] to the model-free average reference (4), passing through REST and volume REST v [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF]. For REST v , the lead-field A C was computed for the complete grid of 1774 volume points inside the brain shell and all three orientations (m×5322). For REST, the lead-field A was computed for the cortical layer of 642 points and for dipoles orthogonal to the scalp (m × 642). For completeness, we also implemented a REST-like solution, in which we used a random A mixing model (m × 642, generated as a spatially correlated uniform random variables), with no physical significance. We re-emphasize here that all models (including the random one) perfectly explain the CR measurements. The evaluation criterion was the relative error introduced in [START_REF] Yao | A method to standardize a reference of scalp EEG recordings to a point at infinity[END_REF]:

RE = X -X / X ( 14
)
where X contains the absolute potentials, X are the estimates and • is the Frobenius norm.

The simumation results confirm the hypothesis that the accuracy of the estimated zero-referenced potentials is proportional to the amount of additional correct information injected in the solution. Completely wrong models (random A) yield the worst solutions, far below the average reference AR (they are not figured here in order to ease the comparisons among the other methods). In principle, one should use as complete models as possible (Figs 2). If we put aside the Oracles, very accurate but impossible to use in practice, the complete full-volume REST v model generally outperforms the cortical REST, at least for the same spatial density of the lead-field matrix (not shown here, a denser cortical layer REST also improves the estimations)3 .

Real signals

According to our previous argumentation, REST and AR solutions should converge to the same solution when the dipolar layer used for REST approaches the head surface and thus the sensors. We have evaluated their relationship using real data, without any previous knowledge on the sources. The EEG signals (resting state) were recorded at the Nancy University Hospital (CHU Nancy) during routine clinical evaluation of an epileptic patient (XXXXXXXXX). Seventysix electrodes were placed according to the 10-10 system, the reference electrode being chosen, for clinical reasons at FPz. The head (scalp, outer skull, inner skull) were computed from the MRI of the patient using the BrainStorm pipeline (1922 nodes and 3840 faces for each surface). Several REST solutions were computed for the absolute potentials, for dipolar layers placed at different depths with respect to the head surface, but outside the brain (inner skull) mesh (see figure 2. The geometry of the layer was the same as the one of the scalp, in order to be able to keep a constant distance between the sensors and the layer, except in the lower part of the brain, where we considered a flat surface 10 mm outside the inner skull. The number of dipoles on the layer was constant and the same as the number of points on the scalp mesh (1922). We used the same criterion RE for evaluating the difference between REST and AR montages, REST being considered the ground truth (X in ( 14)) and AR the estimate X. The results are given in table 1, for different depths of the dipolar layer. As it can be seen, the closer the dipolar layer to the surface, the smaller the "distance" (the relative difference, in Forbenius norm) between the AR and REST estimates (for numerical reasons, dipolar layers closer to the surface than 3mm yield worse results, in the sense that the REST solution is less accurate than the simple average montage AR).

Figure 2: Partial head mesh for the studied epileptic patient. To ease the visualization, only the scalp and the inner skull are represented, along with the dipolar layer used for REST (here at 8mm depth below the scalp surface)

Discussion

This work presents a unified inverse problem framework for the reference problem in surface EEG, which is an algebraic alternative to the approach recetly proposed by [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF][START_REF] Hu | The statistics of eegunipolar references: Derivations and properties[END_REF]. The main difference is that our approach is purely deterministic and based only on linear algebra (i.e., matrix pseudo-inverse estimation, the minimum norm solution and weighted minimum norm in the least-squares (LS) sense [START_REF] Demmel | Applied numerical linear algebra[END_REF][START_REF] Trefethen | Numerical linear algebra[END_REF][START_REF] Noble | Applied linear algebra[END_REF] while Hu et al. approaches are semi [START_REF] Hu | The statistics of eegunipolar references: Derivations and properties[END_REF] or completely non deterministic [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF]. In other words, our approach belongs to the family of LS estimators and does not imply any distributional assumptions as in frequentists statistics (maximum likelihood estimates, ML) of Bayesian inference (maximum a posteriori, MAP) [START_REF] Myung | Tutorial on maximum likelihood estimation[END_REF]. To the best of our knowledge, this algebraic approach was not proposed elsewhere, even if the same unified framework paradigm for the reference problem was discussed in [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF][START_REF] Hu | The statistics of eegunipolar references: Derivations and properties[END_REF].

The LS solution is estimated when an inverse problem has no exact solution and it can be extended to the general linear inverse problem as presented in [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF]. The solution to general linear inverse problems from the point of view of LS yields to the Moore-Penrose pseudo-inverse. The absolute potentials estimation by pseudo-inversion is an underdetermined inverse problem, as the number of absolute potentials to be estimated is greater than the number of available signals (m with respect to m -1 in the notations used in this paper). Consequently, the problem has an infinite number of solutions, parametrized by the a priori information that can be injected. The different parametrizations lead to different so-called Weighted MN solutions, having different accuracies depending on the quality of the injected a priori information, as we have shown in the previous section.

WXWXWXWXWXWWX do we need all that follows, till the Conclusion ? WXWXWXWX

In the other hand ML solutions of the linear inverse problem assumes data with multivariate Gaussian distribution, unknown model parameters and if we know the data covariance matrix the model parameters can be estimated [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF]. Here, is the main difference ML solution seeks to estimate the model parameters in order to find the inverse problem solution while WMN seek to estimate the inverse problem solution from a known model. ML obtains the optimum values for the model parameters by maximizing the probability that the observed data are in fact observed [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF]. This means that the ML solution of the model parameters is the weighted least squares (WLS) solution with the weighting matrix as the inverse of the covariance matrix of the observed data. Now, in the case of the underdetermined problem the LS inverse does not exist. This, is because the distribution of the observed data has no well-defined maximum with respect to variations of the model parameters. Thus, in order to solve this underdetermined problem ML needs to add a priori information that allows the distribution to have a well-defined peak. The way of doing this is explained in [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF], in resume the ML problem for an exact theory translates into finding the maximum of the joint distribution of the a priori distribution of data and the a priori probability distribution of model parameters. Note that if the a priori probability distribution for the model parameters is much more certain than that of the observed data, then the estimate of model parameters (the maximum likelihood point) tends to be close to the a priori model parameters. On the other hand, if the data are far more certain than the model parameters, then the estimates of the model parameters primarily reflect information contained in the data [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF]. In this context the ML solution strongly depends of the probability distribution of the model parameters.

MAP solution comes from Bayesian inference statistics and it works on a posterior distribution not only the likelihood (i.e., the probability of the data given the model parameters assuming true population of model parameters). The posterior distribution in change, is the probability of the model parameters given the data.

The algebraic proof that we propose allows to uncover a family of WMN solutions, with in principle different performances from the methods for inverse problem solutions based on inference statistics such as ML and MAP solutions. Our presented solution enlightens the fact that a good model yields the best solution, a false model yields a wrong solution and the model-free AR is somewhere in the middle, as REST (which outperforms AR if the model is right). As we have discussed here, ML can be used to estimate a model that can be used in the WMN solution in order to obtain solutions comparable with REST or even better. MAP solutions can be used in the same way.

A second major difference with other similar works such as [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF][START_REF] Hu | The statistics of eegunipolar references: Derivations and properties[END_REF] is how we tackle the inverse problem of the reference. First, we start estimating the generalized inverse of the common reference transformation matrix (i.e., the matrix which is implicitly multiplying the unknown zero-reference potentials), formally is the pre-multiplication of the reference transformation matrix with the inaccessible zero-reference EEG potentials. The difference between our reference transformation matrix T CR is that is a non-square matrix. In the work presented by Hu in [START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF] they present their equivalent matrix H or Tr with the difference that is a square matrix pre-multiplying the and is a square matrix (i.e., I N e -1f t , where f is a vector of the same length as vector 1, vector of ones). Our solution was obtained in one step because it is well known that the generalized inverse has a known form, thus the inverse of our transformation matrix is presented in eq.( 9) with W a weighting matrix allowing infinite solutions. If W is the identity we obtain the minimum norm solution in the least squares sense which yields to the solution given by the average reference. This approach also allows to replace W for any other matrix which describes the variance of the signals to be estimated, that we need to know a priori and can be obtained from the head model. WXWXWXWXWXWXWXXWXWXWX

Conclusions

As mentioned, our aim is not to present new simulation results, for extensive simulation results and comparisons (including the effects of model errors, geometry or noise errors), the reader is referred to [START_REF] Zhai | A study on the reference electrode standardization technique for a realistic head model[END_REF][START_REF] Yao | Evaluation of different cortical source localization methods using simulated and experimental EEG data[END_REF][START_REF] Nunez | REST: A Good Idea but Not the Gold Standard[END_REF][START_REF] Kayser | In search of the Rosetta Stone for scalp EEG: Converging on reference-free techniques[END_REF][START_REF] Liu | Estimating a neutral reference for electroencephalographic recordings: The importance of using a high-density montage and a realistic head model[END_REF][START_REF] Chella | Impact of the reference choice on scalp eeg connectivity estimation[END_REF]. The main contribution of the work presented in this paper is the reformulation of different reference estimation methods. Indeed, all methods, from the best possible model based Oracle1 to the model-free average reference, can be seen as ill-posed inverse problems aiming to estimate absolute potentials from the measurements. In particular, we have shown that the AR is the minimum-norm solution to this problem, while the other methods are weighted norms, more or less accurate depending on the amount and on the precision of the injected a priori information.

Figure 1 :

 1 Figure 1: Relative errors RE. The bars represent mean errors over 100 simulations, the standard deviations being given by the vertical whiskers on the top of each bar. Only the 128 sensors setup is shown here, but the results are similar for the other tested montages Depth 10mm 8mm 6mm 4mm RE 0.136 0.101 0.082 0.078

1 = 1 = 1 = 1 =

 1111 Demonstration of TO T O = xx T CR x CR x T CR xx T T T CR T CR xx T T T CR -R x T T CR T CR R x T T CR -As(As) T T T CR T CR As(As) T T T CR -AR s A T T T CR T CR AR s A T T T CR -1

Table 1 :

 1 Relative error between the AR and REST estimations, depending on the depth of the dipolar layer

Although some of these studies were focused on intracranial EEG, the effect of the reference potential is the same. Note though that for iEEG, the reference estimation techniques are different, as they assume that the reference is sufficiently far from the measuring electrodes to be considered independent/uncorrelated[START_REF] Hu | Automatic identification and removal of scalp reference signal for intracranial EEGs based on Independent Component Analysis[END_REF]

, 21, 31].

Note that in[START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF], the formulation is done differently (i.e., , they use a T R of dimension N e × N e, m = N e in our case), but they are equivalent (i.e., , one of the rows of T R has only zeros). Our rationale was to start,as in classical inverse problems, from the actual measurements x CR and not from the absolute potentials. Still, the two formulations yield the same conclusions.

Note that Oracle solutions cannot function if the number of sources is below the number of sensors because of the matrix inversion in[START_REF] Hjorth | An on-line transformation of EEG scalp potentials into orthogonal source derivations[END_REF] and[START_REF] Hu | How do reference montage and electrodes setup affect the measured scalp eeg potentials?[END_REF].

Acknowledgements

The authors would like to thank for the funding to the National Council of Science and Technology of Mexico (CONACyT) and the Latvian National Research Programme "The next generation of information and communication technologies" (NexIT).