Lydia Mata-Cantero

María Jesús Chaparro
email: mjcmaria.j.chaparro@gsk.com

Gonzalo Colmenarejo
email: colmenarejo.gonzalo@gmail.com

Concepción Cid

Álvaro Cortés
email: alvarocortesc@gmail.com

Manuel S Rodriguez
email: manuel.rodriguez@itav.fr

Julio Martín
email: julio.j.martin@gsk.com

Javier Gamo

Maria G Gomez-Lorenzo

F Javier Gamo

Maria G Gomez- Lorenzo

X Protein

Identification of small molecules disrupting the ubiquitin proteasome system in malaria Running tittle: Inhibition of P. falciparum Ubiquitin Proteasome System

Keywords: UPS, ubiquitin proteasome system, DUB, deubiquitinase, TUBEs, tandem ubiquitin binding entities, UBA, ubiquitin-associated domains, iRBCs, infected red blood cells, HTS, high throughput screening, deviation, Ub, ubiquitin, TCAMs, Tres Cantos antimalarial Set, PFI, Property Forecast Index, IFI, Inhibition Frequency Index, ACTs, artemisinin combination therapies, NA, non-adjusted Plasmodium falciparum, proteasome, UPS, inhibitor, screening, ubiquitylation

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The most severe form of malaria is caused by the protozoan parasite Plasmodium falciparum. This disease is one of the deadliest infectious diseases in the world, causing more than 400,000 deaths per year [START_REF] Who | World Malaria Report[END_REF]. Current first-line treatment is based on artemisinin combination therapies (ACTs). However, the malaria parasite has developed resistance against all widely used antimalarials, and ACTs are not an exception. Increasing drug resistance has led to an urgent need for developing new therapies effective not only on acute infection with high parasite loads in blood, but also on different stages of the parasite to block transmission. Thus, identification of drugs targeting different mechanisms in different stages of the parasite is crucial for malaria eradication [START_REF] Gamo | Antimalarial drug resistance: new treatments options for Plasmodium[END_REF].

In recent years, P. falciparum ubiquitin proteasome system (UPS) has been considered a promising target for drug development [START_REF] Ng | Protein Degradation Systems as Antimalarial Therapeutic Targets[END_REF][START_REF] Aminake | The proteasome of malaria parasites: A multistage drug target for chemotherapeutic intervention?[END_REF]. Ubiquitylation is a post-translational change in which ubiquitin (Ub) is covalently bound to the target protein. This process requires the sequential action of three groups of enzymes, E1 ubiquitin-activating enzymes, E2 ubiquitin-conjugating enzymes, and E3 ligases. E3 ligases are the main responsible for substrate specificity leading to the transfer of ubiquitin from the E2 ligase to the final substrate. This ubiquitylation process can also be reverted by the action of deubiquitylation enzymes (DUBs). The cooperative interplay between the ubiquitylation and deubiquitylation processes determines the prevalence of certain ubiquitylated proteins in the cell and, thus, the activation/deactivation of specific functions [START_REF] Komander | The emerging complexity of protein ubiquitination[END_REF].

Eukaryotic 26S proteasome consists of a 20S core, responsible for catalytic activity, and two regulatory 19S particles located at both sides that control entry of ubiquitylated proteins into the proteolytic core. 20S comprises four heptameric rings, of which the outer rings are formed by alpha subunits and the inner rings by catalytic beta subunits, β1 (caspase-like), β2 (trypsin-like) and β5 (chymotrypsin-like) [START_REF] Bedford | Assembly, structure, and function of the 26S proteasome[END_REF].

UPS controls key processes in eukaryotic cells including apoptosis, cell cycle and DNA repair. Thus, UPS deregulation has been related to multiple diseases such as cancer, neurological disorders, and pathogen infections [START_REF] Schmidt | Regulation of proteasome activity in health and disease[END_REF]. In 2003, the proteasome inhibitor bortezomib was the first drug within the UPS approved by the US Food and Drug Administration for the treatment of multiple myeloma. Several UPS inhibitors have subsequently been launched or tested in clinical trials [START_REF] Xolalpa | Targeting the ubiquitin proteasome system: beyond proteasome inhibition[END_REF][START_REF] Mata-Cantero | The ubiquitin proteasome system (UPS) as a cancer drug target: emerging mechanisms and therapeutics[END_REF].

In addition to its proven tractability as drug target, different studies have revealed the essentiality of P. falciparum UPS in its survival and transmission [START_REF] Tschan | Broad-spectrum antimalarial activity of peptido sulfonyl fluorides, a new class of proteasome inhibitors[END_REF][START_REF] Li | Validation of the proteasome as a therapeutic target in Plasmodium using an epoxyketone inhibitor with parasite-specific toxicity[END_REF]. Irreversible inhibition of parasite growth on all parasite stages has been seen using different types of human proteasome inhibitors, some of them within the nanomolar range [START_REF] Ng | Protein Degradation Systems as Antimalarial Therapeutic Targets[END_REF]. Moreover, they also show a synergistic behavior in resistant parasites when combined with artemisinin [START_REF] Ng | Protein Degradation Systems as Antimalarial Therapeutic Targets[END_REF][START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF]. While P. falciparum proteasome exhibits a high degree of homology with its human counterpart, some divergences have been found in the β2 active site by cryo-electron microscopy [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF][START_REF] Li | Assessing subunit dependency of the Plasmodium proteasome using small molecule inhibitors and active site probes[END_REF]. These divergences are being exploited to develop inhibitors with selectivity for the parasite proteasome to avoid side effects [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF][START_REF] Li | Assessing subunit dependency of the Plasmodium proteasome using small molecule inhibitors and active site probes[END_REF][START_REF] Xie | Target Validation and Identification of Novel Boronate Inhibitors of the Plasmodium falciparum Proteasome[END_REF].

On the other hand, most studies of P. falciparum UPS are focused on the P. falciparum proteasome remaining the UPS upstream components understudied. Although all the main UPS components have been identified in the parasite genome in silico, only some of them have been tested experimentally [START_REF] Mata-Cantero | New insights into host-parasite ubiquitin proteome dynamics in P. falciparum infected red blood cells using a TUBEs-MS approach[END_REF][START_REF] Artavanis-Tsakonas | Characterization and structural studies of the Plasmodium falciparum ubiquitin and Nedd8 hydrolase UCHL3[END_REF]. While E1 and E2 paralogs are highly conserved, E3 ligases and DUBs are highly divergent, thus offering the possibility to develop selective drugs.

The objective of this study was to identify inhibitors with potential to interfere with any of the steps involved in the parasite UPS. For that, two previously reported cellular assays that quantitatively measure changes in ubiquitylated protein levels [START_REF] Mata-Cantero | Development of two novel high-throughput assays to quantify ubiquitylated proteins in cell lysates: application to screening of new anti-malarials[END_REF] were used. As a key advantage, these methods do not require extensive or previous knowledge of the targets they are hitting. A high throughput screening (HTS) campaign was run with the Tres Cantos AntiMalarial Set (TCAMS), a GSK compound collection of ca 13K compounds that inhibits P. falciparum in vitro growth [START_REF] Gamo | Thousands of chemical starting points for antimalarial lead identification[END_REF]. As a result, we were able to identify new chemical entities that inhibit the parasite proteasome, validating the approach used. Moreover, compounds acting through other components of the parasite UPS were also identified. All these hits can be followed up to develop new antimalarial drugs with novel modes of action.

Results and discussion

High throughput screening at single shot with AlphaLISA assay

Two cellular assays previously developed were used to run an HTS campaign [START_REF] Mata-Cantero | Development of two novel high-throughput assays to quantify ubiquitylated proteins in cell lysates: application to screening of new anti-malarials[END_REF].

Progression cascade of the whole HTS can be found in the graphical abstract. In both assays, the signal measured is proportional to the amount of ubiquitylated proteins amount present in P. falciparum cells, allowing the identification of UPS inhibitors that are able to change the total levels of ubiquitylated proteins in cells.

The homogeneous AlphaLISA assay was selected to perform the primary screening campaign, as it can be run in 1536-well plates and requires no washing steps. AlphaLISA is a cellular assay where compounds are incubated with infected red blood cells (iRBCs) for 1 hour. This time of incubation was chosen to rule out that the changes measured in the total pool of ubiquitylated protein after treatment are the result of the effect of the compound on other pathways instead of on the UPS.

As the assay is cell-based the selected compound collection consisted of the TCAM set [START_REF] Gamo | Thousands of chemical starting points for antimalarial lead identification[END_REF] and other compounds from the GlaxoSmithKline HTS screening collection which inhibited P. falciparum in vitro growth in previous whole cell phenotypic screenings. A total of 14,747 compounds were tested. The aim was to identify compounds that inhibit parasite growth through the UPS, as the mechanisms of action of most compounds in this collection set remain unknown.

Assay was performed in duplicate at a final assay concentration of 10 µM. Two controls per plate were used: Control 1, samples in absence of compounds (basal levels of ubiquitylated proteins in the cell) and Control 2, samples in presence of proteasome inhibitor MG132 at a concentration that produces complete proteasome inhibition [START_REF] Kreidenweiss | Comprehensive study of proteasome inhibitors against Plasmodium falciparum laboratory strains and field isolates from Gabon[END_REF]. The AlphaLISA signal obtained for each compound was normalized as percent stimulation compared to Control 1, corresponding to 100% stimulation. Control 2 was not used to normalize the percent inhibition of each compound because the levels of ubiquitylated protein accumulation could vary depending on the target affected within the UPS. MG132 increased ubiquitylated protein levels in the cells approximately 1.8 times as compared to Control 1 (Figure 1A) (180% stimulation) and was used to control the assay performance for each plate, that was in the range of previously reported data [START_REF] Mata-Cantero | Development of two novel high-throughput assays to quantify ubiquitylated proteins in cell lysates: application to screening of new anti-malarials[END_REF].

Patterns in plate responses were corrected using an in-house developed algorithm [START_REF] Coma | Statistics and decision making in high-throughput screening[END_REF]. The percentage of stimulation found for the compounds followed a normal distribution in both replicates (Figure 1B). Most compounds and Control 1 samples showed a percentage of stimulation centered around 100% (inactive compounds). The right tail in pink correspond to the actives hits of the assay. These compounds have a percentage of stimulation above the robust cut-off in at least one of the replicates. Robust cut-off was estimated through a robust algorithm [START_REF] Leach | Selecting Diverse Sets of Compounds: Hierarchical Clustering[END_REF] that considers the mean value of Control 1 plus three times its standard deviation. The resulting cut-offs for both replicates were very similar (132% and 133% for replicates 1 and 2 respectively). Figure 1C shows the correlation of the percentage stimulation between both replicates. In addition, a Pareto ranking was calculated with the two responses, to estimate a round cut-off for this bivariate distribution. Compounds labeled in yellow are the ones rescued in this way, giving a total hit rate of 6.8% of all compounds tested. shows the correlation of the pEC50s obtained for each replicate. Compounds displaying a maximum percent stimulation higher than 131% were considered active compounds. Actives in two replicates (n=2) are represented in black, those active in one replicate and inactive in the other one in pink (n=1), and actives in one replicate but whose curves could not be fitted (nonadjusted NA) in the other replicate appear in grey. The black line is the perfect correlation (y=x), while dotted lines represent ± 1 log difference of perfect correlation.

Active n=2 Active n=1 NA

A total of 996 compounds were considered actives in at least one of the replicates. They were progressed to dose response experiments to determine their potency. The quality of the assay was assessed with different proteasome inhibitor being in the rage of previously reported data (17) (Table 1). The minimum percentage of stimulation considered out of the noise of the assay was 131%, because this corresponded to the statistical cut-off in both replicates.

Compounds displaying curves with a maximal asymptote above this percent stimulation in at least one replicate were considered active compounds. Thus, 544 compounds met these criteria, giving a confirmation rate of 54.6%.

Figure 1D depicts the correlation between pEC50s obtained per replicate. 438 compounds were positive in both replicates, showing a pEC50 correlation within the range of pEC50replicateX-pEC50replicateZ<1, represented by grey dotted lines. The other 106 compounds were active in one of the replicates, while in the other replicate they were inactive (pEC50<4, pink) or could not be fitted with the analysis software due to variability (grey, non-adjusted, NA).

Hit confirmation with DELFIA assay

Hits from the AlphaLISA assay were further tested in the orthogonal DELFIA assay [START_REF] Mata-Cantero | Development of two novel high-throughput assays to quantify ubiquitylated proteins in cell lysates: application to screening of new anti-malarials[END_REF]. This heterogeneous assay was performed to confirm the effect of the hits on the UPS and to rule out non-specific interference in the AlphaLISA assay as washes are performed between each addition step.

Hits from the AlphaLISA assay (544 compounds) were tested at dose response per duplicate (Figure 2). Assay behavior was similar to that of AlphaLISA in terms of signal to background and cut-off. The results of the DELFIA assay showed a greater variability in replicates due to the extensive washing steps but, at the same time, the stringent conditions allowed for robust validation of the inhibitors found in the primary screening with the AlphaLISA assay.

Presence of the proteasome inhibitor MG132 led to increase ubiquitylated protein levels up to 185%, (Figure 2A). Compounds with a maximum asymptote higher than 131% were active (robust cut-off). Figure 2B shows the correlation of the pEC50s obtained between DELFIA replicates. Only 106 hits were active in both replicates. Curves from many compounds could not be fitted in one of the replicates due to assay variability or because they were close to the response background (labeled as NA in Figure 2B). * Some inhibition observed (~50% activity) Table 1. Results for tool compound proteasome inhibitors in AMC assay for human THP1 and P. falciparum samples.

Figure 2C shows the correlation of the pEC50s found in the AlphaLISA and DELFIA assays.

311 compounds were confirmed in the DELFIA assay in one or both replicates, revealing a confirmation rate of 57.1% with the AlphaLISA assay. Compounds active in more than two replicates in total (more than two active replicates in either AlphaLISA or DELFIA), and with a pEC50 higher than 4.5 (black) showed the best correlation, as they were the most reliable hits.

Comparison of UPS and whole cell assays (Figure 2D) indicate that compounds are less potent in UPS assays, probably due to differences in the incubation times with the compounds, 1 hour vs. 48 hours respectively.

The results of both assays were used to select compounds for further studies. Thus, the 311 compounds considered hits were tested in proteolytic activity assays of the proteasome which clearly differentiated two groups: compounds whose target is the proteasome (26S) and compounds that probably affect other UPS components upstream the proteasome (see graphical abstract).

Proteasome inhibitors

Proteasome activity assays based on fluorescence readout were performed as described [START_REF] Kisselev | Monitoring activity and inhibition of 26S proteasomes with fluorogenic peptide substrates[END_REF]. Compounds showing autofluorescence at 460 nm were also tested in the Proteasome-Glo assay whose readout is luminescence. Proteasome-enriched fractions from P. falciparum extracts were used as proteasome source as previously described [START_REF] Kisselev | Monitoring activity and inhibition of 26S proteasomes with fluorogenic peptide substrates[END_REF]. All compounds were tested at single shot at 50 µM in the chymotrypsin-like, caspase-like, and trypsin-like proteasome activity assays. Positive compounds were then subjected to dose response assays to assess their potency, in duplicate. Sixteen compounds were active in at least one of the activities tested and with an inhibition rate higher than 50%. Proteasome activity assays were also performed on human proteasome-enriched fractions from THP1 cells to assess selectivity. Proteasome inhibitors MG132 and bortezomib were included as controls in all experiments (Table 1). The identification of proteasome inhibitors was expected and validated the screening rational.

Most of the proteasome inhibitors found were not selective for the P. falciparum proteasome, which hampers further interest from a drug discovery point of view in malaria. However, results

showed that 6 compounds belonging to the same cluster with a cyanamide group were selective for the P. falciparum proteasome. Three of them belong to the TCAM set: TCMDC-133787, TCMDC-133806, and TCMDC-138496. All compounds inhibited the trypsin-like and caspase-like proteasome activities in P. falciparum, but not the chymotrypsin-like activity, while they had no effect on any human proteasome activities (Table 2). These compounds showed no cytotoxicity in the HepG2 assay. The Property Forecast Index (PFI) was calculated as a predictor of developability of compounds based on their physicochemical properties [START_REF] Young | Getting physical in drug discovery II: the impact of chromatographic hydrophobicity measurements and aromaticity[END_REF]. Compounds with a PFI<8 are considered that met the "drug-like" criteria. PFI of cyanamide cluster is less than 8 having a MW < 500, cLogP < 5, and a number of aromatic rings < 4.

A selection of analogs of these compounds were tested in the three proteasome activities of the parasite to better understand their structure-activity relationship (SAR) (data not shown).

Compounds without the cyanamide group were inactive in the proteasome activity assays, which strongly suggests that this group is an essential component of the pharmacophore. SAR results also underlined the importance of other parts of the molecule, such as the sulfone group. The least potent compounds in the proteasome, such as GSK´459, do not have this group (Table 2), although in this case the decrease in potency is not reflected in whole cell activity. Activity improved when the position of the heteroatoms or the chiral center were changed, as in compound TCMDC-138496.

The subunit specificity of the proteasome inhibitors identified was assessed using the human active-site probe Me4BodipyFL-Ahx3Leu3VS (Figure 3A). This probe covalently reacts with all three β catalytic subunits in human proteasomes [START_REF] Berkers | Probing the specificity and activity profiles of the proteasome inhibitors bortezomib and delanzomib[END_REF], but only with the trypsin-like (β2) and chymotrypsin-like (β5) in P. falciparum [START_REF] Li | Validation of the proteasome as a therapeutic target in Plasmodium using an epoxyketone inhibitor with parasite-specific toxicity[END_REF]. The β subunits not bound to the inhibitors are labeled with the probe and then visualized by fluorescence. Treatment with our compounds caused the disappearance of the band corresponding to trypsin-like activity in P. falciparum but not in human fractions (Figure 3A). The human proteasome inhibitor bortezomib led to disappearance of all bands in human fractions, but only of the chymotrypsin-like band in P. falciparum, in agreement with the results reported in activity assays (Table 1). All bands in the human and parasite proteasomes were lost after treatment with the non-specific inhibitor MG132 (Figure 3A and Table 1). These results confirmed the selectivity of our compounds against trypsin-like activity (β2) in P.

falciparum, previously shown in the proteasome activity assays. Caspase-like activity could not be confirmed, as it is not labeled by the probe in the P. falciparum proteasome [START_REF] Li | Validation of the proteasome as a therapeutic target in Plasmodium using an epoxyketone inhibitor with parasite-specific toxicity[END_REF][START_REF] Li | Assessing subunit dependency of the Plasmodium proteasome using small molecule inhibitors and active site probes[END_REF].

Table 2. Summary of data for cyanamide proteasome inhibitors

To get some insight into the selectivity profile and mechanism of action of these compounds, molecular modeling studies were performed exploiting the recently published structural information about P. falciparum proteasome [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF] and internal crystallographic complexes with other cyanamides and different serine and cysteine proteases [START_REF] Laine | Discovery of novel cyanamide-based inhibitors of cathepsin C[END_REF]. According to this information, these compounds could be acting as covalent ligands with the nitrile warhead. However, their selectivity towards β1 and β2 activities appears to point to a specific recognition mechanism.

Based on this hypothesis, we modeled complexes of GSK'737 and TCMDC-133787 with the β1, β2 and β5 units of the human and P. falciparum proteasomes using a covalent docking workflow followed by manual refinement and energy minimization. In our proposed binding mode (Figure 3B), the compounds are attached to the catalytic threonine residue after a reaction with the cyano group of the ligand, that is then interacting with the terminal amine group of the protein, similarly to other available complexes for cysteine proteases [START_REF] Laine | Discovery of novel cyanamide-based inhibitors of cathepsin C[END_REF], in which the nitrogen atom adopts a role as a hydrogen bond acceptor. In our model, the P1 cavity would be partially occupied by the pyrrolidine ring as a proline analogue. This might explain the negligible activity of these compounds towards β5 for both species, due to a possible requirement for a small hydrophobic residue (e.g. leucine) in that position for this catalytic unit [START_REF] Rawlings | MEROPS: the database of proteolytic enzymes, their substrates and inhibitors[END_REF]. The different linkers (sulfonamide, methyl sulfone or carbamate) would establish hydrogen bond interactions with the backbone atoms of residues Ala-49 and Thr-21 (P. falciparum β2 unit numbering scheme) in a similar manner to other known inhibitors such as WLW-vs derivatives (PDB ID 5FMG) [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF] or bortezomib (PDB ID 5LF3) [START_REF] Schrader | The inhibition mechanism of human 20S proteasomes enables nextgeneration inhibitor design[END_REF]. The geometry imposed by the threonine-warhead complex would explain the different hydrogen bond patterns that the linkers could establish and, possibly, the potency drop for GSK'459. Finally, all compounds have a bulky, hydrophobic group (i.e. naphthalene derivatives, aryl halides, and 1,3 benzodioxol) that resembles, in different degrees, a tryptophan residue and that would be located in the P3 sub pocket. Recently, it has been discovered that the P3 sub pocket in P. falciparum β catalytic units, especially β2, is significantly larger than that in their human homologues (Figure 3C), opening new opportunities to achieve selective inhibitors by introducing bulky groups into that position (e.g. tryptophan in WLL-vs inhibitor [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF][START_REF] Li | The cryo-EM structure of the Plasmodium falciparum 20S proteasome and its use in the fight against malaria[END_REF]).

According to our proposed binding mode, the presence of these groups in the reported compounds could be an example of such strategy and one of the reasons for their selectivity against the P. falciparum proteasome To determine the effect on parasite growth of this family of trypsin-like and caspase-like proteasome inhibitors, iRBCs synchronized in rings or mature trophozoites/schizonts were treated for 24 and 48 hours with the compounds TCMDC-133787 and GSK´459. Bortezomib, chloroquine and atovaquone were used as controls (Figure 4). Bortezomib is a human proteasome inhibitor that inhibits the chymotrypsin-like activity. Treatment of parasites with bortezomib resulted in parasite growth arrest in a similar way to chloroquine either in rings or mature forms. However, treatment with our trypsin-like and caspase-like proteasome inhibitors, TCMDC-133787 and GSK´459 did not prevent progression to the next parasite stage (Figure 4). In humans, in vitro and in vivo protein degradation is significantly reduced only when either the trypsin-like or caspase-like sites are inhibited together with the chymotrypsin-like sites [START_REF] Kisselev | Importance of the different proteolytic sites of the proteasome and the efficacy of inhibitors varies with the protein substrate[END_REF]. The same may occur in the parasite. It has been reported that the most significant parasite killing results from coinhibition of the chymotrypsin-like activity with the trypsin-like or caspase-like proteasome activities [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF][START_REF] Li | Assessing subunit dependency of the Plasmodium proteasome using small molecule inhibitors and active site probes[END_REF]. Co-inhibition allows for attenuating parasite growth at any point in the intraerythrocytic cycle. Chymotrypsin-like activity is essential for fast inhibition of growth being implicated in parasite schizogony, while inhibition of trypsin-like activity alone does not significantly affect parasite growth as it has been confirmed with our cyanamides. Parasitemia remained constant during the 48 hours tested, while in the control culture parasitemia increased more than 6-fold. They seemed to have no effect on trophozoites or early schizonts, but prevented increase in parasitemia, maybe because they affected merozoite egress, the invasion process, or early ring stages. The fact that not all parasite stages are equally sensitive to inhibition of the different proteolytic activities of the 20S core may suggest that each proteasome activity might be regulating a specific pool of ubiquitylated proteins [START_REF] Kisselev | Importance of the different proteolytic sites of the proteasome and the efficacy of inhibitors varies with the protein substrate[END_REF].

On the other hand, some proteasome inhibitors can also react with other serine or cysteine proteases because the proteasome has a hydrolytic mechanism similar to those proteases [START_REF] Kisselev | Proteasome inhibitors: from research tools to drug candidates[END_REF].

In fact, although cyanamides were selective for P. falciparum proteasome, it has previously been reported that they are inhibitors of different kinds of cathepsins, which are cysteine proteases. A cyanamide-based inhibitor analogue to TCMDC-133787 was reported by our colleagues at GSK as a novel human cathepsin C inhibitor for the treatment of neutrophil-dominated inflammatory diseases such as chronic obstructive pulmonary disease and cystic fibrosis [START_REF] Laine | Discovery of novel cyanamide-based inhibitors of cathepsin C[END_REF]. Selectivity for different cathepsins (cathepsins C, B, K, L, and S), all cysteine proteases, was achieved by changing different parts of the molecules. In the case of cyanamides identified here, the selectivity window in terms of cytotoxicity was higher than 100-fold in most cases. Further studies of the action of these compounds on different human cathepsins should be done to assess a good toxicological profile and to ensure the non-toxic effects in vivo.

As regards activity against other cysteine proteases of the parasite, it is known that P.

falciparum has different families of cysteine proteases such as falcipains, which have been considered potential drug targets because they are involved in the hemoglobin degradation process. Treatment of cultured P. falciparum parasites with broadly active cysteine protease inhibitors such as leupeptin or E-64 causes the food vacuole to swell and fill with dark-staining material [START_REF] Prasad | Blocking Plasmodium falciparum development via dual inhibition of hemoglobin degradation and the ubiquitin proteasome system by MG132[END_REF] due to accumulation of large amounts of undegraded hemoglobin, with maximum activity achieved in mature trophozoites and schizonts. Absence of a food vacuole defect in Figure 4 suggests that inhibition of cysteine proteases in this digestive organelle is not causing the inhibition of parasite growth.

All these results suggest that compounds belonging to the cyanamide cluster are good candidates to be administered in combination with other antimalarials such as artemisinin.

Selective inhibitors acting upon the parasite trypsin-like activity have previously been shown to enhance killing by artemisinin of both sensitive and resistant parasites without host toxicity [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF].

In contrast to previously reported selective proteasome inhibitors, cyanamides are small molecules that are not peptide mimetics and have good PhysChem properties.

Non-proteasome inhibitors

In addition to proteasome inhibitors, our strategy has allowed the identification of compounds that increased ubiquitylated protein levels through other mechanisms that could be directly associated with parasite growth inhibition. This confirms published results showing that accumulation of ubiquitylated proteins can lead to parasite death [START_REF] Bridgford | Artemisinin kills malaria parasites by damaging proteins and inhibiting the proteasome[END_REF].

295 compounds were inactive in all proteasome activity assays, forming the group of nonproteasome inhibitors (graphical abstract). Among them, 169 hits were active in the AlphaLISA and DELFIA assays in at least three replicates and with a pEC50 higher than 4.5 (Figure 2C, black dots). This restrictive selection criteria was applied to choose the best compounds of this group. In a first stage, the structures were clustered using a complete linkage hierarchical algorithm [START_REF] Leach | Selecting Diverse Sets of Compounds: Hierarchical Clustering[END_REF], which resulted in 64 chemical clusters, 36 of them singletons, with a wide variety of chemotypes. Further inspection revealed that most of them, irrespective of the cluster, contained a phenol group (such as phenol benzylamines), and/or were amino alcohols, including quinine and amino-phenyl-ethanol derivatives. Six were the most populated (82 compounds in total) clusters, with more than 3 members per cluster (Figure 5). The numbers of compounds within each cluster ranged from 38 in cluster 1 as the most populated and 4 in cluster 6 as the less populated. Clusters 1, 3 and 5 had with amino alcohols (pink dots). Most compounds in cluster 1 had a hydroxy group in benzyl position. The substitution of the amine could be aliphatic chains or aromatic rings. Phenol Aminoalcohol Aminoalcohol + phenol Clusters 3 and 5 shared a quinine scaffold. Moreover, cluster 5 had a long chain and was highly substituted in the right-hand side. The mechanism of action of quinine remains unclear, but the drug has been attributed an inhibitory role of hemozoin biocrystallization like chloroquine.

However, some mutations have been found in the HECT E3 ligase PfUT (33) in quinine-resistant parasites [START_REF] Sanchez | A HECT ubiquitin-protein ligase as a novel candidate gene for altered quinine and quinidine responses in Plasmodium falciparum[END_REF], connecting the resistance with the UPS. The lower correlation seen with the whole cell assay here could point to an additional mechanism involving the UPS system of the parasite.

Clusters 2 and 4 contained compounds with the phenol group (Figure 5, grey dots). In addition, cluster 2 included structures similar to described protease inhibitors with benzamides substituted and an additional hydroxy group in para position, validating our approach. The interplay of phenols with the UPS has previously been reported in human cells. Natural phenolic compounds regulate the UPS during oxidative stress [START_REF] Chang | Regulation of ubiquitin and 26S proteasome mediated by phenolic compounds during oxidative stress[END_REF] and induce growth arrest or apoptosis probably by regulating the UPS. For example, compounds such as methyl gallate, gallic acid, and tannic acid increase ubiquitylated protein levels in treated human cells. They have an effect on the E1 conjugating enzyme activity and the 26S proteasome through their OH group, and also interact with other proteins such as protein S5a and DUBs USP47 and USP15 [START_REF] Chang | Phenolic compounds stage an interplay between the ubiquitin-proteasome system and ubiquitin signal autophagic degradation for the ubiquitin-based cancer chemoprevention[END_REF]. The correlation of the pIC50s obtained in the UPS and in the whole cell assays may suggest that phenols present in the compounds could be affecting one or more components of the UPS.

Regarding the physicochemical profile, more than half the compounds showed a good PFI index and had no historical annotations of liabilities found in GSK internal databases (redox, promiscuity, reactivity, etc.), especially compounds belonging to clusters 1 (34/38), 3 (10/10), and 4 (7/9). All compounds had less than 4 aromatic rings and most of them had a MW < 500 and a cLogP < 5. Regarding cytotoxicity, data in HepG2 showed that 77 of the 82 compounds were noncytotoxic.

Figure 5 shows the correlation between AlphaLISA and P. falciparum 3D7A whole-cell assays pEC50s. The difference in pEC50 values was less than one for all of them, exhibiting almost the same potency in both assays despite the fact that treatment with compounds lasted only one hour. Phenols belonging to clusters 2 and 4 were near to perfect correlation, strongly suggesting that parasite growth and UPS inhibitions are related. Amino alcohols present in clusters 3 and 5 also showed a good correlation but, in all cases, compounds were more potent in the whole cell assay, possibly indicating that there are other targets involved in their mode of action or that more time is needed for compounds to exert their action. Some dose response curves are presented in Figure 5 as an example of the results found for each cluster.

Little is known about components of the UPS in P. falciparum. Future identification of the proteins targeted by these non-proteasome inhibitor compounds could reveal novel, validated, tractable and selective targets which could be used in new drug discovery programs. Further studies, like chemogenomic or proteomic methods (e.g. global shift thermal profiling) among others, are required to identify which targets are being hit by the inhibitors found.

Conclusion

Results from the screening campaign shown here revealed compounds hitting different targets within the UPS pathway. They represent excellent starting points for drug discovery programs, particularly compounds belonging to the proteasome inhibitors, as the target they are hitting has been identified. Moreover, non-proteasome inhibitors described could be used as tools to identify new tractable antimalarial targets.

Materials and methods

P. falciparum cultures

P. falciparum 3D7A strain from the Malaria Research and Reference Reagent Resource Center (MR4) was used to perform all assays reported. Parasites were grown in T150 flasks with fresh red blood cells (RBCs) at hematocrit 1% (volume percentage of RBCs) as described in [START_REF] Prasad | Blocking Plasmodium falciparum development via dual inhibition of hemoglobin degradation and the ubiquitin proteasome system by MG132[END_REF].

Preparation of assay plates

Compounds were dissolved in 100% DMSO. Final DMSO concentration in the assays was were left empty. For single shot assays, final compound concentration in the assay was 10 µM in AlphaLISA and 50 µM in the proteasome assays. For dose response determinations in any of the assays described, the highest concentration was 100 μM, and 11 points diluted 1:3 in DMSO were tested in all the assays performed by duplicate.

Purification of iRBCs with mature trophozoites/schizonts

Purified, highly synchronized P. falciparum-infected red blood cells (iRBCs) with mature forms were required to perform the HTS campaign. The previously described protocol was used for this purpose [START_REF] Mata-Cantero | Magnetic isolation of Plasmodium falciparum schizonts iRBCs to generate a high parasitaemia and synchronized in vitro culture[END_REF]. As control of inhibition, part of the culture was treated with 1.5 µM of MG132 dissolved in DMSO. Culture was dispensed into AlphaLISA and DELFIA plates as described in each section. All this experimental work was carried out under biosafety level 3 procedures.

TUBE-AlphaLISA and DELFIA assays

A Multidrop Combi liquid handler (Thermo Electron Corporation) was used for each reagent addition to assay plates. The protocol was followed as described by Mata-Cantero et al [START_REF] Mata-Cantero | Development of two novel high-throughput assays to quantify ubiquitylated proteins in cell lysates: application to screening of new anti-malarials[END_REF].

Data analysis

Data were normalized to percentage of stimulation using the following equation:

% 𝑆𝑡𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛(𝑥) = 𝑅𝑥 |𝑅 𝐶𝑡𝑟𝑙1 | * 100
Rx is the assay response measured for compound X, while RCtrl1 is calculated as the average of control 1 (columns with 1% DMSO) in the same plate of compound X. Data and assay performance statistics, such as signal to background ratio (S/B), robust standard deviation (SD), and coefficient of variation (CV= (SD/average)*100) were calculated using ActivityBase XE (IDBS) templates. Robust cut-offs at robust mean + 3*SD of Control 1 (estimated through a robust algorithm) [START_REF]Robust Statistics-How Not to Reject Outliers[END_REF] were used to mark the hits. Before that, patterns present in the plates were detected and removed using an in-house developed algorithm [START_REF] Coma | Statistics and decision making in high-throughput screening[END_REF]. pEC50 values (pEC50 = -log10(EC50)) were obtained with ActivityBase XE non-linear regression bundle. Data visualization and further analysis were performed using TIBCO Spotfire 3.2 software and Microsoft Excel.

Selection criteria and data mining

For single shot experiments, hits had a response above the statistical cut-off (robust mean + 3*SD response of Control 1) of each replicate. In AlphaLISA and DELFIA dose response assays, compounds with a maximum asymptote higher than 131% were considered active compounds (actives). This is based on the response and standard deviation obtained in the experiments calculated as stated above. Compounds with pEC50 values lower than 4 (> 100 µM)

were considered inactive. GSK proprietary compound screening databases were queried for historical data of the hits obtained. For proteasome activity assays described in a following section, compounds showing a maximum percentage of inhibition higher than 50% in at least one proteasome activity were considered actives.

Proteasome-enriched fractions

The protocol described by Kisselev et al [START_REF] Kisselev | Monitoring activity and inhibition of 26S proteasomes with fluorogenic peptide substrates[END_REF] was adapted to obtain proteasome-enriched fractions present in purified P. falciparum cells and THP1 crude extracts. Briefly, P. falciparum cultures were synchronized with 5% sorbitol (w/v) and scaled up for five days. Culture was harvested by centrifugation at 700 g for 5 minutes, and parasites were isolated by breaking RBCs with 0.01% saponin (w/v) as previously described. At this point, parasite and THP1 cell were equally processed. Cells were resuspended in sucrose buffer (50 mM Tris buffer pH 7.5, 5 mM MgCl2, 1 mM EDTA, 50 mM NaCl, 250 mM sucrose, 2 mM ATP, 1 mM DTT) and lysed by nitrogen cavitation for 15 minutes (110 bar of 1500 psi). Lysates were centrifuged at 20,000 g for 30 minutes at 4ºC, and supernatants containing soluble protein were subjected to an ultracentrifugation step at 300,000 g for 2 hours at 4ºC. The resulting proteasome-containing pellets were resuspended in sucrose buffer and left on ice for 30 minutes to complete solubilization. Samples were centrifuged for 10 minutes at 20,000 g to remove insoluble material.

Proteins contained in supernatant, the proteasome-enriched fraction, were measured using the Bradford method.

Proteasome activity assays

Proteasome activities were assayed with fluorogenic or luminescence peptide substrates using previously described methods [START_REF] Kisselev | Monitoring activity and inhibition of 26S proteasomes with fluorogenic peptide substrates[END_REF][START_REF] Rivett | Multicatalytic endopeptidase complex: proteasome[END_REF] or the Proteasome-Glo assays (Promega) respectively. In fluorescence assays, proteasome-enriched fractions from P. falciparum cells were dissolved in sucrose buffer at 0.1 mg/mL to measure the three proteasome activities. THP1 proteasome-enriched fractions were adjusted to 0.1 mg/mL for the chymotrypsin-like activity and to 0.2 mg/mL for the caspase-like and trypsin-like activities. 4 µL of proteasome-enriched fractions at the required concentration were added to compounds and incubated for 1 hour (384-well, low volume plates). Then, 4 µL of peptides 100 µM Suc-LLVY-AMC (Sigma), 200 µM Z-Leu-Leu-Glu-AMC (Sigma) or 200 µM Ac-RLR-AMC (Boston) in assay buffer (50 mM Tris buffer pH 7.5, 5 mM MgCl2, 1 mM EDTA, 50 mM NaCl, 1 mM DTT, 2 mM ATP, BSA 0.1 mg/ml) were added to the plates to measure the chymotrypsin-like, caspase-like or trypsin-like proteasome activities respectively. Proteasome inhibitors epoxomicin and MG132 at 10 µM were included as controls.

In the Proteasome-Glo assay, proteasome-enriched fractions were assayed in HEPES buffer (10 mM HEPES, 1 mM DTT, 2 mM ATP, BSA 0.1 mg/ml) following manufacturer protocols.

Proteasome concentration for P. falciparum was kept at 0.1 mg/mL, while THP1 proteasome was used at 0.05 mg/mL for chymotrypsin-like and trypsin-like activities, and at 0.1 mg/mL for caspase-like activity. Volumes and order of addition were the same as in the fluorescence assay.

Fluorescence (λex/em = 380/460 nm) or luminescence were measured in the Envision plate reader (Perkin Elmer). For dose response experiments, the first compound concentration tested was 100 µM diluting the compounds 1:3 in DSMO (final DMSO concentration, 1%).

Proteasome active-site probes

For all proteasome labeling experiments, the active-site probe Me4BodipyFL-Ahx3Leu3VS (UbiQ) was used at a final concentration of 500 nM in sucrose buffer. THP1 (20 μg) or P. falciparum (10 μg) proteasome-enriched fractions were incubated with compounds at 100 μM, or DMSO as control, for 1 hour in sucrose buffer. The active-site probe was added to proteasome enriched fractions previously treated with DMSO or compounds and incubated for 2 hours at RT.

Samples were then denatured by addition of Laemmli sample buffer, boiled for 10 minutes and run on a 15% SDS-PAGE gel. Bands were visualized using the fluorescein channel in a Chemidoc reader (Biorad) (λex/em = 480/530 nm).

Whole cell inhibition assay

The parasite growth inhibition assay was performed using the standard 3 H-hypoxanthine incorporation method. This assay relies on parasite incorporation of labeled hypoxanthine that is proportional to P. falciparum growth [START_REF] De Cozar | Development of a Novel High-Density [3H]Hypoxanthine Scintillation Proximity Assay To Assess Plasmodium falciparum Growth[END_REF].

Parasite inhibition phenotype

P. falciparum cultures were synchronized in schizonts by performing 70% Percoll gradient centrifugation. Next day, culture in rings at 2% parasitemia and 2% hematocrit was treated with compounds at 10 times their IC50s in the whole cell assay, 0.01 µM atovaquone, 0.25 µM chloroquine, 1 µM bortezomib (Selleckchem), 5 µM TCMDC-133787, and 15 µM GSK'459, at a final DMSO concentration of 0.5%. After 24 hours, parasitemia was quantified and photographs were taken of Giemsa-stained thin blood smears. The same experiment was carried out treating the parasites in mature trophozoites/schizonts.

HepG2 cytotoxicity assay

The acute cytotoxic effect of compounds administered to growing human liver-derived HepG2 cells (ATCC HB-8065) was determined using the luminescence assay previously described [START_REF] Crouch | the use of ATP bioluminescence as a measure of cell-proliferation and cytotoxicity[END_REF].

Molecular modeling

The structure of the three catalytic β1, β2, and β5 subunits of the P. falciparum proteasome were obtained from the cryo-EM derived structure (PDB ID 5FMG) [START_REF] Li | Structure-and function-based design of Plasmodiumselective proteasome inhibitors[END_REF][START_REF] Li | The cryo-EM structure of the Plasmodium falciparum 20S proteasome and its use in the fight against malaria[END_REF], while human subunits were obtained from the apo X-ray crystal (4R3O) [START_REF] Harshbarger | Crystal structure of the human 20S proteasome in complex with carfilzomib[END_REF]. The Schrodinger 2017-3 suite was used to pre-process protein structure, model the missing side chains, and assign optimal protonation states. 3D models for the compounds were generated with Ligprep, followed by a conformational analysis to generate low-energy states for input in the docking protocol. The covalent docking module in Maestro was used to generate and score different poses with a covalent bond between the cyano group and the oxygen in the Thr-1 side chain (nucleophilic addition to a triple bond).

The resulting poses for the 6 systems (β1, β2, and β5 for human and P. falciparum) were manually analyzed and corrected based on information from internal crystal complexes with other cysteine

Figure 1 .

 1 Figure 1. AlphaLISA screening campaign. Compounds were tested in the AlphaLISA assay in duplicate at 10 µM. Two controls per plate were included and normalized as percent stimulation. Control 1 corresponded to the signal obtained in absence of inhibition (basal levels of ubiquitylated proteins corresponding to 100%), while Control 2 was the signal measured in wells treated with the proteasome inhibitor MG132 (1.5 µM). A. Bars represent the average percent stimulation for each control in the replicates. Average signal to background obtained per plate (S/B=Average % stimulation control 2 wells / Average % stimulation control 1 wells) is given in bold for each replicate. B. Number of wells for a given percent stimulation. All compounds tested (14,747) plus control 1 wells are represented. Black color corresponds to control 1 wells, while wells treated with compounds appear in grey. Compounds with percent stimulation above the statistical cut-off for each replicate being considered active hits are depicted in pink. C. Correlation of the percent stimulation obtained for each replicate. Inactive compounds are colored grey, and active hits in at least one replicate are labeled in pink as in B. Yellow compounds were rescued after calculating a Pareto round cut-off. The line indicates the perfect correlation (y=x). D. Active hits were then assayed at dose response starting at 100 µM (1:3 dilutions) per duplicate. Graph

Figure 2 .

 2 Figure 2. Hit confirmation in DELFIA assay. Hits from the AlphaLISA campaign were tested in the DELFIA assay in duplicate in dose response (first point 100 µM). Two controls were included per plate and normalized as percent stimulation. Control 1 corresponds to the signal obtained in the absence of inhibition (basal levels of ubiquitylated proteins corresponding to 100%), while Control 2 is the signal measured in wells treated with the proteasome inhibitor MG132 (2 µM). A. Bars represent the average percent stimulation per replicate. Average of signal to background obtained per plate (S/B=average % stimulation Control 2 wells/average % stimulation Control 1 wells) is indicated in bold for each replicate. B. Correlation of the pEC50s recorded for each replicate. Compounds displaying a maximum percent stimulation higher than 131% were actives. In black, actives in two replicates (n=2); in pink, actives in only one replicate (n=1); and in grey, compounds active in one replicate but whose curves could not be fitted in the other replicate (non-adjusted NA). C. Correlation of the average pEC50s recorded in the AlphaLISA and DELFIA assays per compound. Compounds labeled in black were active in more than three replicates (n=2 in AlphaLISA and/or DELFIA) and had a pEC50>4.5 in all replicates. The black line indicates the perfect correlation (y=x), while dotted lines represent ± 1 log difference of perfect correlation. D. Correlation of the average pEC50s recorded in the AlphaLISA and whole cell assays per compound. Labels are the same than in C.

  Figure 2

Figure 3 .

 3 Figure 3. Validation of proteasome inhibitors binding. A. Proteasome active-site probes. The proteasome active-site probe Me4BodipyFL-Ahx3Leu3VS was used reacting with the trypsinlike (β2) and chymotrypsin-like (β5) proteasome activities in P. falciparum and with the three β catalytic subunits in THP1 in the absence of compounds. Proteasome-enriched fractions from P. falciparum and THP1 cells were treated with compounds for 1 hour at 100 µM. The probe was then added labelling the proteasome active sites not previously bound to the compounds. B. Modelling of GSK'737 binding to P. falciparum and human proteasomes. Proposed binding mode of compound GSK'737 to P. falciparum β2 unit (trypsin-like activity). The compound is predicted to bind covalently to the catalytic threonine residue, and to establish hydrogen bonds with the backbone of residues Thr-21 and Ala-49. C. Comparison of the binding modes in human and P. falciparum P1 and P3 sub pockets are highlighted.

Figure 4 .

 4 Figure 4. Effect of proteasome inhibitors identified on the intraerythrocytic parasite stage. Synchronized iRBCs with rings or mature trophozoites/schizonts were treated for 24 and 48 hours with compounds. Photographs of Giemsa-stained thin blood smears were taken to observe the phenotype and the effect on parasite in vitro growth. Chloroquine and atovaquone were included as controls of fast-and slow-acting compounds respectively with modes of action unrelated to UPS inhibition.

Figure 5 .

 5 Figure 5. Non-proteasome inhibitors. The six more populated clusters with their scaffolds that share the compounds belonging to each cluster are presented. Graphs show the correlation between the average pEC50 found in the UPS AlphaLISA assay and the pIC50 that inhibits parasite in vitro growth (whole cell). The black line is the perfect correlation (y=x), the pink and orange dotted lines represent the ±0.5 log and ±1 log difference of perfect correlation respectively. Compounds in grey have a phenol group, those in pink an amino alcohol group, and those in orange both kinds of groups in their structure. Dose response curves for one of the compounds belonging to each cluster are shown as examples. Percent stimulations recorded in the AlphaLISA (black) and DELFIA (pink) assays for a representative of the cluster are plotted versus compound concentration.

1%.

  AlphaLISA assay plates (Greiner 1536-well white plates) were prepared with compounds by adding 20 nL of each compound per well with an Echo dispenser, except for columns 11 and 12, which were filled with 20 nL of DMSO, and columns 35 and 36, which were left empty. 200 nL of each compound were added to 384-well flat bottom black plates for DELFIA assays, while 40 nL were dispensed to 384-well low volume black plates for proteasome assays. Columns 6 and 18

Acknowledgements

Human RBCs were supplied by the Blood Transfusion Center of CAM (Madrid, Spain) and Banc de Sang i Teixits (Barcelona, Spain). Human biological samples were sourced ethically, and their research use complied with the terms of the informed consent under an IRB/EC approved protocol. We also thank Jim Brown for critically review of the manuscript. GSK has funded this work.


Johannes Berger
email: johannes.berger@uni-hamburg.de

Azat M Gainutdinov
email: azat.gainutdinov@lmpt.univ-tours.fr

Ingo Runkel
email: ingo.runkel@uni-hamburg.de

MODIFIED TRACES FOR QUASI-HOPF ALGEBRAS

published or not. The documents may come

Modified traces were introduced in [START_REF] Geer | Virelizier Traces on ideals in pivotal categories[END_REF][START_REF] Geer | Ambidextrous objects and trace functions for nonsemisimple categories[END_REF]. They are a generalisation of the categorical trace in a pivotal linear category. The latter is defined on the whole category and expressed in terms of duality morphisms and the pivotal structure. The former are only defined on a suitable tensor ideal of the pivotal category, but they have better non-degeneracy properties than the categorical trace.

The example of interest to us is that of a pivotal and unimodular finite tensor category C, and its tensor ideal Proj(C) of projective objects. If C is not semisimple, the categorical trace vanishes identically on Proj(C). For the modified trace, the opposite happens: there exists a non-zero modified trace on Proj(C) that induces a non-degenerate pairing on Hom-spaces [START_REF] Costantino | Some remarks on the unrolled quantum group of sl(2)[END_REF][START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF][START_REF] Geer | M-traces in (non-unimodular) pivotal categories[END_REF]. This observation has important applications to link invariants [GPT, BBGe], to threedimensional topological field theories [DGP], as well as to vertex operator algebras with non-semisimple representation theory in the context of a conjectural Verlinde formula [START_REF] Gainutdinov | The non-semisimple Verlinde formula and pseudo-trace functions[END_REF][START_REF] Creutzig | Logarithmic conformal field theory, log-modular tensor categories and modular forms[END_REF][START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF].

While there are explicit constructions of modified traces [START_REF] Geer | Ambidextrous objects and trace functions for nonsemisimple categories[END_REF], these can be tedious to deploy in examples. In [BBGa] a simple description of modified traces was found in the case that C = H-mod is the category of finite-dimensional representations of a finite-dimensional pivotal unimodular Hopf algebra H: there is a one-to-one correspondence between left/right modified traces and left/right cointegrals of the Hopf algebra. In this paper we generalise this result to quasi-Hopf algebras. Since the definition of cointegrals for quasi-Hopf algebras is more complicated than that for Hopf algebras [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF][START_REF] Bulacu | Integrals for (dual) quasi-Hopf algebras[END_REF][START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF], the treatment is more technical than that in [BBGa], but the method of proof is the same. Other generalisations of [BBGa] have been given in [Ha, FOG].

Let us state our main result in more detail for right cointegrals and right modified traces. The results for the left variant are analogous and can be found in the main text. Our conventions for quasi-Hopf algebras are given in Section 3, so we will be brief here. Let k be a field and let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over k with pivot g ∈ H. For λ ∈ H * let λ be defined by λ(h) = λ(gh) for all h ∈ H. Then λ is a right cointegral if and only if

λ(h) 1 = λ ⊗ g q R ∆(h) p R (1.1)
for all h ∈ H (Corollary 3.8). Here, q R = Ψ 1 ⊗S -1 (αΨ 3 )Ψ 2 , p R = Φ 1 ⊗Φ 2 βS(Φ 3 ), Φ is the coassociator of H, Ψ its inverse, and α, β ∈ H are the evaluation and coevaluation element.

A right modified trace on Proj(C) is a collection of linear maps {t P : End C (P ) → k} P ∈Proj(C) (1.2) satisfying cyclicity and compatibility with the categorical trace (or rather with the right partial categorical trace, see Section 2). The family t • is uniquely determined by its value on a projective generator. A right modified trace provides a pairing Let now C = H-mod, the pivotal finite tensor category of finite-dimensional left H-modules. We can take the left regular module H as projective generator. A non-degenerate right modified trace on Proj(C) exists if and only if H is unimodular (Theorem 4.5 (1)). In this case we prove (Theorem 4.5 (2)):

C(M, P ) × C(P, M ) → k, (f, g) → t P (f • g),
Theorem 1.1. Let H be a finite-dimensional pivotal unimodular quasi-Hopf algebra over a field k. There is a one-to-one correspondence between right modified traces t • on Proj(H-mod) and right cointegrals λ ∈ H * via

t H (f ) = λ f (1)
for all f ∈ End H (H) .

In particular, such traces exist and are unique up to scalar multiples. Every nonzero right modified trace t • on Proj(H-mod) is non-degenerate.

As an illustration we consider the family Q(N, β) of symplectic fermion quasi-Hopf algebras introduced in [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]. Here N ∈ Z >0 and β ∈ C satisfies β 4 = (-1) N . Q(N, β) is a non-semisimple factorisable ribbon quasi-Hopf algebra, and so Q(N, β)-mod is a (non-semisimple) modular tensor category. Conjecturally, for β = e -iπN/4 this modular tensor category is equivalent to the category of representations of the even part of the vertex operator super algebra of N pairs of symplectic fermions [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Conj. 6.8]. In Section 5 we give explicit formulas for the cointegrals on Q(N, β) (left and right cointegrals coincide in this case), as well as for the modified trace on projective modules (again the left and right variant coincide). This paper is organised as follows. In Sections 2 and 3 we review the necessary background on modified traces and on cointegrals in quasi-Hopf algebras, respectively. In Section 4 we prove Theorem 1.1, and in Section 5 we illustrate the result in the example of the symplectic fermion quasi-Hopf algebras.

While we were writing the present paper, the paper [SS] by Shibata and Shimizu appeared, which also contains a proof of Theorem 1.1.

We denote the pivotal structure by δ V : V → V ∨∨ and will choose right duals and left duals to be identical as objects. We write

← - ev V : V ∨ ⊗ V → 1 , ← -- coev V : 1 → V ⊗ V ∨ -→ ev V : V ⊗ V ∨ → 1 , --→ coev V : 1 → V ∨ ⊗ V , (2.1)
for the left (←) and right (→) evaluation and coevaluation maps. They are related by

-→ ev V = ← - ev V ∨ • (δ V ⊗ id V ∨ ), etc. Given a morphism f : A ⊗ C → B ⊗ C in C,
its right partial trace over C is defined to be (we omit writing '⊗', say 1 instead of id, and write ∼ -→ for the coherence isomorphisms)

tr r C (f ) = A ∼ -→ A1 1 ← -- coev C -----→ A(CC ∨ ) ∼ -→ (AC)C ∨ f 1 -→ (BC)C ∨ ∼ -→ B(CC ∨ ) 1 -→ ev C ----→ B1 ∼ -→ B . (2.2)
Analogously, the left partial trace over C of a morphism g :

C ⊗ A → C ⊗ B is tr l C (g) = A ∼ -→ 1A --→ coev C 1 -----→ (C ∨ C)A ∼ -→ C ∨ (CA) 1 g -→ C ∨ (CB) ∼ -→ (C ∨ C)B ← - ev C 1 ----→ 1B ∼ -→ B . (2.3)
The original definition of modified traces from [START_REF] Geer | Virelizier Traces on ideals in pivotal categories[END_REF][START_REF] Geer | Ambidextrous objects and trace functions for nonsemisimple categories[END_REF] is for general tensor ideals, but here we will restrict our attention to the tensor ideal Proj(C) ⊂ C , (2.4) the full subcategory consisting of all projective objects in C. We note that unless C is semisimple, the categorical trace vanishes identically on Proj(C), see e.g. [START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF]Rem. 4.6 In case k is algebraically closed and C is unimodular (i.e. the socle and top of the projective cover of the tensor unit 1 are both the tensor unit), non-zero left and right modified traces as above exist and are unique up to scalars [START_REF] Geer | M-traces in (non-unimodular) pivotal categories[END_REF]Sec. 5.3]. Furthermore, these left/right modified traces are non-degenerate. This significantly generalises earlier existence and uniqueness results, see e.g. [START_REF] Geer | Ambidextrous objects and trace functions for nonsemisimple categories[END_REF][START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF][START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF].

We will focus on C being H-mod, the category of finite-dimensional modules over a pivotal unimodular quasi-Hopf algebra. In this situation we obtain existence, uniqueness and non-degeneracy of non-zero left/right modified traces without requiring k to be algebraically closed (Theorem 4.5 below), and by using methods different from those in [START_REF] Geer | M-traces in (non-unimodular) pivotal categories[END_REF].

Cointegrals for quasi-Hopf algebras

In this section we recall some definitions and properties related to quasi-Hopf algebras that we shall need. We start by giving our conventions for quasi-Hopf algebras, and then proceed to define integrals, cointegrals and symmetrised cointegrals for quasi-Hopf algebras (which have to be pivotal and unimodular in the latter case).

Throughout this section, H denotes a finite-dimensional quasi-Hopf algebra over k.

Quasi-Hopf algebras. The antipode of H is denoted by S and the coassociator and its inverse by Φ and Ψ = Φ -1 . The evaluation and coevaluation element are α, β ∈ H, and without loss of generality we assume

ε(α) = 1 = ε(β) .
(3.1)

Using sumless Sweedler notation for both (iterated) coproducts and elements in tensor powers of H, we write for example

∆(h) = h (1) ⊗ h (2) , (∆ ⊗ id)(∆(h)) = h (1,1) ⊗ h (1,2) ⊗ h (2) , Φ = Φ 1 ⊗ Φ 2 ⊗ Φ 3 ∈ H ⊗3 (3.2)
Following the conventions from [START_REF] Farsad | SL(2, Z)-action for ribbon quasi-Hopf algebras[END_REF]Sec. 6] for the axioms, the comultiplication ∆ :

H → H ⊗ H satisfies (∆ ⊗ id) (∆(h)) • Φ = Φ • (id ⊗∆) (∆(h)) , (3.3) or in index notation h (1,1) Φ 1 ⊗ h (1,2) Φ 2 ⊗ h (2) Φ 3 = Φ 1 h (1) ⊗ Φ 2 h (2,1) ⊗ Φ 3 h (2,2) (3.4)
for all h ∈ H. Note that this is the opposite of the convention that, for example, [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF][START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF] are using.

The category C := H-mod of finite-dimensional H-modules is a monoidal category with associator

Φ U,V,W : U ⊗ (V ⊗ W ) → (U ⊗ V ) ⊗ W u ⊗ v ⊗ w → Φ 1 u ⊗ Φ 2 v ⊗ Φ 3 w .
(3.5)

If x ∈ H ⊗ H, then we will frequently write

x 21 := τ (x) = x 2 ⊗ x 1 ∈ H ⊗ H , (3.6)
where τ is the tensor flip of vector spaces. This notation is extended to higher tensor powers in the obvious way. Following [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF], for h, a ∈ H, f ∈ H * we write

(h f )(a) = f (ah) , f h = h (1) f (h (2) ) , (f h)(a) = f (ha) , h f = f (h (1) )h (2) . (3.7)
One can also consider the opposite and the coopposite quasi-Hopf algebras H op (with opposite multiplication) and H cop (with opposite comultiplication). These algebras become quasi-Hopf algebras after modifying the defining data according to

S op = S cop = S -1 , Φ op = Ψ, α op = S -1 (β), β op = S -1 (α), Φ cop = Ψ 321 = Ψ 3 ⊗ Ψ 2 ⊗ Ψ 1 , α cop = S -1 (α) and β cop = S -1 (β).
Dual modules. For V ∈ C the left dual V ∨ is the dual vector space V * , with action

h.v * , w = v * , S(h)w for all h ∈ H, v * ∈ V * , w ∈ V, (3.8)
and the (left) rigid structure on H-mod is then defined as follows. The left evaluation are ← -

ev V : V ∨ ⊗ V → 1 , ← - ev V (v * ⊗ v) = v * , αv , (3.9)
respectively. Using a basis {v i } of V with corresponding dual basis {v i } we can write the left coevaluation as

← -- coev V : 1 → V ⊗ V ∨ , ← -- coev V (1) = i βv i ⊗ v i (3.10)
One can define right duals analogously in terms of S -1 , but we will not do this here as below we will work with pivotal quasi-Hopf algebras, where we will use the pivotal structure to define right duals.

We extend the hook notation from (3.7) to dual vector spaces, so that for example the action on the left dual of the H-module V could then be written as

v * S(h), since h.v * | w = v * | S(h)w = v * S(h) | w (3.11) Φ Ψ β α H H H H = p R q R H H H H = id H⊗H Figure 1.
The equalities in (3.14) follow from the zig-zag identities for duals in H-mod, where both sides are tensored with the identity. We show this above for first equality in (3.14), where we have written out id H times the zig-zag identity for H in H-mod as a string diagram in Vect.

for all h ∈ H, w ∈ V , v * ∈ V * .
The regular left and right action by an element h ∈ H is denoted by l h and r h , respectively, so that for all a ∈ H l h (a) = ha and r h (a) = ah .

(3.12)

We will make use of the special elements q R , p R , q L , p L ∈ H ⊗ H, which have already appeared in [Dr], and have been further used in [START_REF] Hausser | Doubles of quasi-quantum groups[END_REF] and subsequent papers concerning quasi-Hopf algebras. They are defined as (3.13) and satisfy the identities

q R := Ψ 1 ⊗ S -1 (αΨ 3 )Ψ 2 , p R := Φ 1 ⊗ Φ 2 βS(Φ 3 ) , q L := S(Φ 1 )αΦ 2 ⊗ Φ 3 , p L := Ψ 2 S -1 (Ψ 1 β) ⊗ Ψ 3 ,
∆(q R 1 )p R [1 ⊗ S(q R 2 )] = 1 ⊗ 1 , [1 ⊗ S -1 (p R 2 )]q R ∆(p R 1 ) = 1 ⊗ 1 , ∆(q L 2 )p L [S -1 (q L 1 ) ⊗ 1] = 1 ⊗ 1 , [S(p L 1 ) ⊗ 1]q L ∆(p L 2 ) = 1 ⊗ 1 (3.14)
and

(1 ⊗ S -1 (a (2) ))q R ∆(a (1) ) = (a ⊗ 1)q R , ∆(a (1) )p R (1 ⊗ S(a (2) )) = p R (a ⊗ 1) . (3.15)
These identities are most easily understood using the standard graphical calculus2 for rigid monoidal categories, see for example in Figure 1. More about this may be found in [START_REF] Hausser | Doubles of quasi-quantum groups[END_REF]Sec. 2].

Pivotal quasi-Hopf algebras. To state the definition of a pivotal element, we need to recall the Drinfeld twist, which is an invertible element

f ∈ H ⊗ H such that f ∆(S(a))f -1 = (S ⊗ S) (∆ cop (a)) , (3.16)
for all a ∈ H. It corresponds to the canonical natural isomorphism

γ W,V : W ∨ ⊗ V ∨ ∼ -→ (V ⊗ W ) ∨ (3.17) of H-modules, via (γ W,V (ϕ ⊗ ψ)) (v ⊗ w) = (ψ ⊗ ϕ)(f 1 v ⊗ f 2 w) , (3.18) for all v ∈ V, w ∈ W, ϕ ∈ W * , ψ ∈ V * .
For more details see e.g. [START_REF] Farsad | SL(2, Z)-action for ribbon quasi-Hopf algebras[END_REF]Sec. 6.2]. We will not need the explicit description of f here, but let us remark that in our conventions we have

(ε ⊗ id)(f ) = 1 = (id ⊗ε)(f ) . (3.19)
We can now state (see [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Def. 4.1] for the Hopf case and [START_REF] Bulacu | On sovereign, balanced and ribbon quasi-Hopf algebras[END_REF]Def. 3.2] for the quasi-Hopf case, where the name "sovereign" is used instead).

Definition 3.1. H is called pivotal if there is an invertible element g ∈ H, called the pivot, satisfying ∆(g) = f -1 • (S ⊗ S)(f 21 ) • (g ⊗ g) , (3.20)
and such that S 2 (h) = ghg -1 , for all h ∈ H.

The pivot satisfies [START_REF] Bulacu | On sovereign, balanced and ribbon quasi-Hopf algebras[END_REF]Prop. 3.12] ε(g) = 1 and S(g) = g -1 .

(3.21)

We remark that the second property stems from the more general fact that in every pivotal category we have the identity δ -1 V ∨ = (δ V ) ∨ , see e.g. [START_REF] Selinger | A survey of graphical languages for monoidal categories[END_REF]Lem. 4.11]. Remark 3.2.

(1) A pivot is not necessarily unique. For example, if z is central, invertible and satisfies ∆(z) = z ⊗ z, then zg is also a pivot. We will indicate our choice of pivot by saying that (H, g) is a pivotal quasi-Hopf algebra. (2) If (H, g) is pivotal, then indeed H-mod is pivotal. The pivotal structure is the monoidal natural isomorphism with components

δ V : V → V ∨∨ , δ V = δ Vect V • l g , V ∈ H-mod , (3.22)
where δ Vect is the canonical pivotal structure of underlying vector spaces. In fact, the set of pivotal structures on H-mod is in bijective correspondence with the set of pivots for H. For a proof see e.g. [START_REF] Bulacu | Involutory quasi-Hopf algebras[END_REF]Prop. 3.2], and note that our pivot is their inverse pivot.

Using the pivotal structure we define right duals in the standard way. Namely, as an object the right dual ∨ V of an object V is just the left dual V ∨ . The right evaluation and coevaluation are then defined as -→ ev

V := ← - ev V ∨ • (δ V ⊗ id V ∨ ) , --→ coev V := id V ∨ ⊗δ -1 V • ← -- coev V ∨ . (3.23)
Explicitly, for example,

-→ ev V (v ⊗ w * ) = gv , w * S(α) = S(α)gv , w * = gS -1 (α)v , w * , (3.24)
where in the last step we used S(h)g = gS -1 (h) for h ∈ H.

Integrals and cointegrals.

A left integral 3 for H is an element Λ of H, such that hΛ = ε(h)Λ for all h ∈ H. One similarly defines a right integral for H to be a left integral for H op . The spaces of left and right integrals are always one-dimensional for a finite-dimensional quasi-Hopf algebra, see for example [START_REF] Bulacu | Integrals for (dual) quasi-Hopf algebras[END_REF]Sec. 2].

The difference between left and right integrals is measured by the modulus of H. This is the unique algebra morphism γ : H → k such that for any left (resp. right) integral Λ (resp. Λ ) we have Λh = γ(h)Λ (resp. hΛ = γ -1 (h)Λ ). Left and right integrals coincide if and only if the modulus of H is given by the counit ε, and in that case we say that H is unimodular.

Remark 3.3. The category H-mod is unimodular if and only if H is unimodular, see [ENO, Rem.

3.2].

There is also the 'dual' notion of cointegrals for quasi-Hopf algebras, proposed in [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF] and further studied in [START_REF] Bulacu | Integrals for (dual) quasi-Hopf algebras[END_REF][START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF]. To state their definition we need the elements U, V ∈ H ⊗ H, given by

U := f -1 (S ⊗ S)(q R 21 ) , V := (S -1 ⊗ S -1 ) f 21 p R 21 . (3.25)
We also set

u = (γ ⊗ S 2 )(V). (3.26) Remark 3.4. It is easy to see that (q R ) cop = q L 21 , (p R ) cop = p L 21 , (q L ) cop = q R 21 , and (p L ) cop = p R 21 . (3.27) Likewise one finds f cop = (S -1 ⊗ S -1 )(f ), cf. [BC2, Sec. 3],
and therefore

U cop := (S -1 ⊗ S -1 )(q L f -1 ) , V cop := (S ⊗ S) p L f 21 , u cop = (γ ⊗ S -2 )(V cop ) (3.28)
If H is pivotal, then g cop = g -1 .

3 In [BBGa] this is called a left cointegral, and what we call cointegral here is called integral there (all in the case of Hopf algebras). We follow the conventions in e.g. [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF][START_REF] Bulacu | Integrals for (dual) quasi-Hopf algebras[END_REF][START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF], from where we take the definition of cointegrals for quasi-Hopf algebras.

Paraphrasing [START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF]Eq. (3.6) and Def. 3.4] we now define: Definition 3.5. Let γ be the modulus of H.

• A left cointegral for H is an element λ l ∈ H * satisfying

(id ⊗λ l )(V∆(h)U) = γ(Φ 1 )λ l (hS(Φ 2 ))Φ 3 (3.29)
for all h ∈ H. • A right cointegral for H is a left cointegral for H cop . In full detail this means that λ r ∈ H * is a right cointegral for H if and only if

(id ⊗λ r ) V cop ∆ cop (h)U cop = γ(Ψ 3 )λ r (hS -1 (Ψ 2 ))Ψ 1 , (3.30)
for all h ∈ H.

Cointegrals for quasi-Hopf algebras have a categorical interpretation, see [SS, BGR]. Cointegrals satisfy a number of properties, and for convenience we collect the ones we will need in the following proposition.

Proposition 3.6. Let γ be the modulus of H.

(1) Left (resp. right) cointegrals exist and are unique up to scalar.

(2) Non-zero left (resp. right) cointegrals are non-degenerate forms on H.

(3) Let λ l be a left cointegral. Then, for all a, b ∈ H (3.32)

λ l (S -1 (a)b) = λ l (bS(a γ)) (3.31) (4)
(5) Let λ r be a right cointegral. Then

λ = (λ r u) • S (3.33)
is a left cointegral. (6) Let λ l be a left cointegral. Then

λ = (λ l u cop ) • S -1 (3.34) is a right cointegral.
Proof. The proofs of (1)-( 4) can be found in [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF]Sec. 4,5]. The last two points are [START_REF] Bulacu | On integrals and cointegrals for quasi-Hopf algebras[END_REF]Prop. 4.3].

The preceding proposition allows us to give the following equivalent characterisation of cointegrals. 4,5 Lemma 3.7. Suppose that H is pivotal, Let λ ∈ H * and set We thank the anonymous referee for explaining to us the improved result in Lemma 3.7, which in the originally submitted manuscript was only formulated in the unimodular case.

5 The shifted left cointegral λ l also appears in [START_REF] Shibata | Categorical aspects of cointegrals on quasi-Hopf algebras[END_REF]Sec. 6.4] in relation to γ-twisted module traces.

(1) λ is a left cointegral if and only if

(id ⊗ λ l ) q L ∆(h)p L = γ(Ψ 3 ) λ l (Ψ 2 h) • gS -1 (Ψ 1 ) (3.36)
(2) λ is a right cointegral if and only if

( λ r ⊗ id) q R ∆(h)p R = γ(Φ 1 ) λ r (Φ 2 h) • g -1 S(Φ 3 ) (3.37)
Proof. We will prove the second part, the first statement is completely analogous. Let λ r ∈ H * be a right cointegral. By Proposition 3.6 (5), we thus have

λ r = λ r ug = (λ l • S -1 ) g = (g -1 λ l ) • S -1 . (3.38)
for a left cointegral λ l . Using this equality and evaluating the left cointegral equation (3.29) on S -1 (h)g -1 for h ∈ H gives

(id ⊗λ l )(V∆(S -1 (h)g -1 )U) = γ(Φ 1 ) λ r (Φ 2 h)Φ 3 . (3.39)
We have

∆(g -1 )U = (g -1 ⊗ g -1 )(S ⊗ S)(q R 21 f -1 21 ) = (S -1 ⊗ S -1 )(q R 21 f -1 21 )(g -1 ⊗ g -1 ). (3.40) Using V = (S -1 ⊗ S -1 )(f 21 p R 21 ) and ∆(S -1 (h)) = (S -1 ⊗ S -1 )(f 21 ∆ cop (h)f -1 21 ) (3.41)
we immediately simplify (3.39) to

γ(Φ 1 ) λ r (Φ 2 h)Φ 3 = (id ⊗λ l ) (S -1 ⊗ S -1 ) q R 21 ∆ cop (h)p R 21 • (g -1 ⊗ g -1 ) = ((r g -1 • S -1 ) ⊗ λ r ) q R 21 ∆ cop (h)p R 21 = ((S -1 • l g -1 ) ⊗ λ r ) q R 21 ∆ cop (h)p R 21 (3.42)
Then, applying S on both sides and multiplying with g -1 on the left gives (3.45)

( λ r ⊗ id) q R ∆(h)p R = γ(Φ 1 ) λ r (Φ 2 h) • g -1 S(Φ 3 ), ( 3
The second identity can be seen using points (3) and ( 5) of the same proposition.

Symmetrised cointegrals. For the rest of this section we assume:

(H, g) is pivotal and unimodular .

Note that in this case the elements u and u cop from Proposition 3.6 are both equal to 1. The following immediate corollary to Proposition 3.7 will be useful when comparing to modified traces.

Corollary 3.8. Let λ ∈ H * . Then

(1) λ is a left cointegral if and only if

λ l (h) 1 = g -1 ⊗ λ l (q L ∆(h)p L ) (3.46) for all h ∈ H. (2) λ is a right cointegral if and only if λ r (h) 1 = λ r ⊗ g (q R ∆(h)p R ) (3.47)
for all h ∈ H.

Following [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Sec. 4], we call λ l and λ r the symmetrised left and right cointegral, respectively. The adjective "symmetrised" is justified by the following corollary, which follows from Proposition 3.6 (2), equation (3.45), and the fact that here we assume H to be unimodular. Corollary 3.9 ( [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Prop. 4.4]). The non-zero symmetrised left (resp. right) cointegrals are non-degenerate symmetric linear forms on H. Remark 3.10. Let H be a finite-dimensional Hopf algebra. Following [Ra], for any grouplike element g ∈ H, one can define the left ideal L g ⊆ H * of left gcointegrals (called left g-integrals in [Ra]) as

L g = {ϕ ∈ H * | (id ⊗ϕ)(∆(h)) = ϕ(h)g ∀h ∈ H} (3.48)
These ideals are all one-dimensional [START_REF] Radford | Finiteness conditions for a Hopf algebra with a nonzero integral[END_REF]Prop. 3]. Indeed, note that L 1 is the space of left cointegrals. Then the linear isomorphism

L g ϕ → (ϕ h) ∈ L h -1 g for all grouplike g, h, (3.49)
shows L g ∼ = L 1 . Similarly one may define the space R g of right g-cointegrals.

Thus, if (H, g) is a pivotal Hopf algebra, Lemma 3.7 reduces to the statement (λ l g -1 ) ∈ L g and (λ r g) ∈ R g -1 .

(3.50)

For unimodular H, a symmetrized left cointegral is therefore a left g-cointegral, and a symmetrized right cointegral is a right g -1 -cointegral.

Modified traces for quasi-Hopf algebras

Throughout this section H will be a finite-dimensional quasi-Hopf algebra over k.

Tensoring with the regular representation. Let V ∈ H-mod. We denote by ε V the vector space V with trivial H-module structure, i.e. hv = ε(h)v for h ∈ H, v ∈ V . Recall the definition of the elements p R , etc., from (3.13). We need the following generalisation of [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Thm. 5.1] to quasi-Hopf algebras (see also [START_REF] Schauenburg | Hopf modules and the double of a Quasi-Hopf Algebra[END_REF]Sec. 2.3]).

Proposition 4.1.

(1) The map

φ r : H ⊗ ε V → H ⊗ V, h ⊗ v → ∆(h)p R • (1 ⊗ v) = h (1) p R 1 ⊗ h (2) p R 2 v is an isomorphism of H-modules, with inverse ψ r : H ⊗ V → H ⊗ ε V, h ⊗ v → (id ⊗S)(q R ∆(h)) • (1 ⊗ v) .
(2) The map

φ l : ε V ⊗ H → V ⊗ H, v ⊗ h → ∆(h)p L • (v ⊗ 1)
is an isomorphism of H-modules, with inverse

ψ l : V ⊗ H → ε V ⊗ H, v ⊗ h → (S -1 ⊗ id)(q L ∆(h)) • (v ⊗ 1) .
Proof. We only prove the first part, the second part is completely analogous. It is obvious that φ r is an intertwiner, so we only need to show that ψ r is a two-sided inverse.

Recall the second identity in (3.15), which can be graphically represented as

p R = p R . (4.1)
Using pictures we compute the composition φ r • ψ r :

ψ r φ r = q R p R = q R p R (4.1) = q R p R (3.14) = (4.2)
Similarly one shows ψ r • φ r = id. Since φ r is an intertwiner and bijective, ψ r necessarily is an intertwiner as well and we are done.

As a consequence of the previous considerations we have the following lemma.

Lemma 4.2. Let W be an H-module. The map

Ξ : H op ⊗ k End k (W ) → End H (H ⊗ W ) , a ⊗ m → φ r • (r a ⊗ m) • ψ r (4.3)
is an algebra isomorphism.

Proof. Since for any algebra A we have the algebra isomorphism

A op ∼ = End A (A), a → r a , (4.4)
together with isomorphism property from Proposition 4.1 we see that the prescription Ξ : (a ⊗ m) → φ r • (r a ⊗ m) • ψ r is bijective, and thus the isomorphism is established. It remains to be shown that the isomorphism is one of algebras.

The multiplication for the endomorphism algebras is just composition, and for

H op ⊗ k End k (W )
we simply take the one induced by the tensor product of kalgebras. Then the calculation

Ξ(a ⊗ m) • Ξ(b ⊗ n) = φ r • (r a ⊗ m) • ψ r • φ r (r b ⊗ n) • ψ r = φ r • ((r a • r b ) ⊗ (m • n)) • ψ r = Ξ((a ⊗ m) • (b ⊗ n)) (4.5)
shows that Ξ indeed preserves the algebra structure.

A similar result holds for End H (W ⊗ H).

The main theorem. We will need the following extension result for symmetric linear forms: Let A be a finite-dimensional unital k-algebra. By a family of trace maps {t P : End A (P ) → k} P ∈A-pmod (as opposed to left/right modified traces) we mean a family as in Definition 2.1 (i), which, however, only satisfies condition 1 (cyclicity) and not conditions 2 or 2' (which do not make sense in A-mod). We have [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Prop. 2.4], see also [START_REF] Gainutdinov | Projective objects and the modified trace in factorisable finite tensor categories[END_REF]Prop. 5.8]):

Proposition 4.3. Let A be a finite-dimensional unital k-algebra. Then a symmetric linear form t on A extends uniquely to a family of trace maps {t P : End A (P ) → k} P ∈A-pmod , given by

t P (f ) = n i=1 t((b i • f • a i )(1)), f ∈ End A (P ), (4.6)
where n depends on P , and a i :

A → P , b i : P → A satisfy id P = n i=1 a i • b i . (4.7)
In particular

t A (r x ) = t(x), x ∈ A . (4.8)
The next lemma is an instance of the Reduction Lemma [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Lem. 3.2] when one takes C = H-mod and H as projective generator.

Lemma 4.4. Let H be pivotal with pivot g. A symmetric linear function t on H extends to a right modified trace on H-pmod if and only if for all f ∈ End H (H ⊗H)

t H⊗H (f ) = t H (tr r H (f )) (4.9)
holds, where t P is as in Proposition 4.3, for P ∈ H-pmod. Similarly, t extends to a left modified trace on Proj( H M) if and only if

t H⊗H (f ) = t H tr l H (f ) (4.10) holds for all f ∈ End H (H ⊗ H).
We denote the subspace of symmetric forms t ∈ H * which extend to a right/left modified trace on H-pmod by Sym r/l tr .

(4.11)

Given t ∈ Sym r/l tr , the corresponding modified trace t • takes the value End H (H) → k , f → t H (f ) = t(f (1)) (4.12)
on the left regular module H.

We can now state the main theorem of our paper. Parts 2 and 3 generalise [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Thm. 1] to the setting of quasi-Hopf algebras. A stronger version of Part 1 was shown for Hopf algebras in [START_REF] Orozco | Module traces and Hopf group-coalgebras[END_REF]Cor. 6.1].

Theorem 4.5. Let (H, g) be a finite-dimensional pivotal quasi-Hopf algebra over k. We have:

(1) A non-degenerate left (right) modified trace on H-pmod exists if and only if H is unimodular. Suppose now that H is in addition unimodular. Then:

(2) Sym r/l tr is equal to the space of symmetrised right/left cointegrals. In particular, dim(Sym r/l tr ) = 1.

(3) A non-zero element of Sym r/l tr extends to a non-degenerate right/left modified trace on H-pmod.

Proof. (1) If t • is a non-degenerate left or right modified trace, then H h → t H (r h ) ∈ k is a non-degenerate symmetric linear form on H. Unimodularity follows from [START_REF] Hausser | Integral theory for quasi-Hopf algebras[END_REF]Prop. 5.6]. The converse direction amounts to parts 2 and 3.

(2) Suppose now that H is unimodular, and let t • be a family of trace maps on H-pmod (not necessarily left/right modified traces). Let t ∈ H * be the symmetric form on H which corresponds to t • via Proposition 4.3. We will now compute both sides of (4.9) in Lemma 4.4 separately and then use that lemma to prove the statement.

Let W ∈ H-mod and f ∈ End H (H ⊗ W ). where a ⊗ m is a simple tensor in H op ⊗ End k (W ). For simplicity and without loss of generality we will assume that f actually corresponds to the simple tensor a ⊗ m. By cyclicity of t • we get

t g f q R p R = t g q R φ r a m ψ r p R = t g q R a m p R
t H⊗W (f ) = t H⊗W (φ r • (r a ⊗ m) • ψ r ) = t H⊗ ε W (r a ⊗ m) = tr k (m) t(a) , (4.14)
where tr k (m) is the trace of the linear operator m. This can be seen by choosing any basis of W and considering a decomposition of H ⊗ ε W into (dim W ) copies of H. t H (tr r W (f )): Here we use that t H (tr r W (f )) = t (tr r W (f )(1)) and then rewrite the resulting expression as in Figure 2. Altogether, this gives (4.15) where ρ : H ⊗ k W → W is the action of H on W . Since (4.14) and (4.15) hold in particular for W = H, the left regular module, and for all a, m, we can rephrase condition (4.9) in Lemma 4.4 as follows: the symmetric linear form t on H extends to a right modified trace on H-pmod if and only if t(a)1 = (t ⊗ g) q R ∆(a)p R .

t H (tr r W (f )) = t(q R 1 a (1) p R 1 )tr k ρ(gq R 2 a (2) p R 2 ⊗ -) • m ,
(4.16)

But this is just the defining equation (3.47) for a symmetrised right cointegral.

The left version of the proof is completely analogous and uses (3.46). (3) By Proposition 3.9 the symmetrised right/left cointegrals are non-degenerate. It is shown in [START_REF] Beliakova | Modified trace is a symmetrised integral[END_REF]Thm. 2.6] that this implies that the corresponding right/left modified traces are non-degenerate in the sense of Definition 2.1 (ii).

Example: symplectic fermion quasi-Hopf algebra

In this section we will use Theorem 4.5 to compute the modified trace for the socalled symplectic fermion quasi-Hopf algebras defined in [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]. One reason that these quasi-Hopf algebras are of interest is their relation to a fundamental example of logarithmic two-dimensional conformal field theories, namely the symplectic fermion conformal field theory, see [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF] for more details and references.

Quasi-Hopf structure. The family of symplectic fermion ribbon quasi-Hopf algebras Q = Q(N, β), where N is a non-zero natural number and β ∈ C satisfies β 4 = (-1) N , is defined as follows [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Sec. 3]. As a C-algebra, Q is a unital associative algebra generated by

{ K, f i | 1 ≤ i ≤ N, = ± } .
(5.1)

With the elements

e 0 = 1 2 (1 + K 2 ) , e 1 = 1 2 (1 -K 2 ) (5.2)
we can write the defining relations for Q as

{f ± i , K} = 0 , {f + i , f - j } = δ i,j e 1 , {f ± i , f ± j } = 0 , K 4 = 1 , (5.3)
where {-, -} is the anticommutator. Then e 0 , e 1 are central orthogonal idempotents with e 0 + e 1 = 1. The dimension of Q is 2 2N +2 . It is enough to specify the quasi-Hopf algebra structure on generators. The coproduct is

∆(K) = K ⊗ K -(1 + (-1) N ) e 1 K ⊗ e 1 K , ∆(f ± i ) = f ± i ⊗ 1 + ω ± ⊗ f ± i , (5.4)
where ω ± = (e 0 ± ie 1 )K. The counit is

ε(K) = 1 , ε(f ± i ) = 0 .
(5.5)

We introduce

β ± = e 0 + β 2 (±iK) N e 1 (5.6)
to define the coassociator and its inverse as

Φ ±1 = 1 ⊗ 1 ⊗ 1 + e 1 ⊗ e 1 ⊗ e 0 (K N -1) + e 1 (β ± -1) .
(5.7)

Finally, the antipode S and the evaluation and coevaluation elements α and β are given by

S(K) = K (-1) N = (e 0 + (-1) N e 1 )K , α = 1 , S(f ± i ) = f ± i (e 0 ± (-1) N ie 1 )K , β = β + .
(5.8)

For convenience we also state the inverse antipode on generators:

S -1 (K) = K (-1) N , S -1 (f ± i ) = ω ± f ± i .
(5.9)

Note that S(β ± ) = S -1 (β ± ) = β ∓ , and

β + β -= 1. The pivot of Q is 6 g = (e 0 + (-i) N +1 e 1 K N )K .
(5.10) ¿From [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Eq. (3.35)] we know that the Drinfeld twist and its inverse are given by

f ±1 = e 0 ⊗ 1 + e 1 ⊗ e 0 K N + e 1 β ∓ ⊗ e 1 .
(5.11)

One furthermore computes

q R = 1 ⊗ 1 + e 1 ⊗ e 1 (β -1) , p R = 1 ⊗ 1 + e 0 ⊗ e 1 (β -1) , q L = 1 ⊗ 1 + e 1 ⊗ e 0 (K N -1) + e 1 (β -1) , p L = β -⊗ 1 + e 1 β -⊗ e 0 (K N -1) + e 1 (β --1) .
(5.12)

For later use, we fix a basis of Q. The basis elements are The quasi-Hopf algebra Q can be equipped with an R-matrix and a ribbon element, turning it into a ribbon quasi-Hopf algebra. In [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Prop. 3.2] it was shown that it is in fact a factorisable ribbon quasi-Hopf algebra. Factorisability implies unimodularity [START_REF] Bulacu | Factorizable quasi-Hopf algebras[END_REF]Sec. 6], giving another argument showing that Q is unimodular. A ribbon category is in particular pivotal. The pivot in (5.10) was obtained as g = v -1 u, where v is the ribbon element and u is the Drinfeld element.

B a, b,i :=   | a| j=1 f + a j     | b| k=1 f - b k   K i , ( 5
Modified trace. We will see that the spaces of left and right modified traces coincide for Q. To compute the modified trace explicitly, we first find the (also coinciding) left and right symmetrised cointegrals via Corollary 3.8. Then we employ Theorem 4.5 and the relation (4.12) to obtain the value of the modified trace on the projective generator Q. Proof. We will verify that λ r satisfies both conditions in Corollary 3.8. To this end, we first note that the coproduct takes the following form on elements of the above basis: ∆(B l, a,i ) = B l, a,i ⊗ K i + ω (5.18) where in each tensor factor in "(lower terms)" the number of f + 's is strictly less than | a|, or the number of f -'s is strictly less than | b|, or both. Therefore, both sides of the two conditions in Corollary 3.8 vanish identically unless one chooses h = B N , N ,i , i ∈ {0, 1, 2, 3}. In these four cases a straightforward computation shows that the conditions in Corollary 3.8 hold.

Note that because β 4 = (-1) N , for odd N only one of the two summands in (5.17) is present, the other coefficient is zero. For N even, both summands are present.

Since the symmetrised cointegral is two-sided, so is the corresponding modified trace. By (4.12) the explicit value of the modified trace on f ∈ End Q (Q) is t Q (f ) = λ(f (1)) ,

(5.19) with λ as in (5.17).

The modified trace has also been computed by a different method in [GR2, Sec. 9], namely by using the existence of a simple projective object in Q-mod.

There, the modified trace is given on the four indecomposable projectives. To relate the two computations, first note that the central idempotents of Q are e 0 , e ± 1 = 1 2 e 1 (1 (5.20) see [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Sec. 3.6]. The decomposition of the right regular module Q is

± β -1 v) = 1 2 e 1 1 ∓ iK N k=1 (1 -2f + k f - k ) ,
Q = P 0+ ⊕ P 0-⊕ X ⊕2 N 1+ ⊕ X ⊕2 N 1- , (5.21)
where P 0± are the projective covers of the two one-dimensional simple modules of Q and X 1± are projective simple objects of dimension 2 N [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Sec. 3.7].

The projections to P 0± are given by right-multiplication with the (non-central) idempotents e ± 0 = 1 2 (1 ± K)e 0 . The central idempotents e ± 1 project to the direct sums X ⊕2 N 1± . Set

x ± = N j=1 f + j f - j e ± 0 , y ± = e ± 1 .
(5.22)

Note that x ± and y ± are central in Q [START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF]Sec. 3.6]. It is straightforward to compute the modified trace of r x ± , r y ± ∈ End Q (Q): (5.23) where r h denotes the right multiplication with h ∈ Q, cf. (3.12). This agrees with [GR2, Sec. 9] up to a normalisation factor of 1 2 (-1) 1 2 N (N +1) . Since g is of order two, the left and right cointegrals also agree. One can compute the cointegral for Q by shifting the symmetrised cointegral from Proposition 5.1 by g. Similar to the symmetrised cointegral, it is non-vanishing only on the top components, and with (5.25)

t Q (r x ± ) = ± 1 2 (-1) 1 2 N (N -1) β 2 , t Q (r y ± ) = ± 1 2 (-1) 1 2 N (N -1) (-2) N ,
a ± = β 2 ± δ N,

  Let λ r be a right cointegral. Then, for all a, b ∈ H λ r (S(a)b) = λ r bS -1 (γ a) .

  = λ r (S(ab)g) = λ r (S(a)gS -1 (γ b)) = λ r (S(a)S(γ b)g) = λ l ((γ b)a).

Figure 2 .

 2 Figure2. Calculating the left hand side of (4.15). The string diagrams are all in Vect. To arrive at the initial string diagram, recall the expression (3.24) for the right evaluation map. The first step is just substitution of f from (4.13). In the second step, we use that ψ r (p R 1 ⊗ p R 2 w) = 1 ⊗ w for all w ∈ W (which in turn is immediate from ψ r • φ r = id) and substitute the definition of φ r .

  .13) where i ∈ Z 4 , and a, b are strictly ordered multi-indices of lengths 0 ≤ | a|, | b| ≤ N .By "strictly ordered" we mean that for a = (a 1 , a 2 , . . . , a | a| ) we have 1 ≤ a 1 < . . . < a | a| ≤ N , and similarly for b. The element corresponding to B a, b,i in the dual basis is denoted by left and a right integral in Q[START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF] Sec. 3.5]. In particular, Q is unimodular.

  Proposition 5.1. The linear formλ r = (β 2 + i) • B N , N ,1 * + (β 2 -i) • B N , N ,3 * (5.17)is simultaneously a left and a right symmetrised cointegral for Q.

-

  K i ⊗ B l, a,i + (lower terms) × e 0 ⊗ e 0 + e 0 ⊗ e 1 + e 1 ⊗ e 0 + (-1) (N +1)i e 1 ⊗ e 1 ,

  even , b ± = ±iδ N,odd(5.24) it can be expressed asλ = a + B N , N ,0 * + b + B N , N ,1 * + a -B N , N ,2 * + b -B N , N ,3 * .

  for all P ∈ Proj(C) and M ∈ C. If all these pairings are non-degenerate, t • is called non-degenerate.

  ].

	2 . (Left Partial Trace Property) If P ∈ Proj(C) and V ∈ C, then
	for all f ∈ End C (V ⊗ P )
	t V ⊗P (f ) = t P tr l V (f ) .	(2.8)
	If such a family t • satisfies both the left and the right partial trace property,
	then it is simply called a modified trace.
	ii) A right (resp. left) modified trace t • is non-degenerate if the pairings
	Hom C (M, P ) × Hom C (P, M ) → k,	(f, g) → t P (f • g) ,	(2.9)
	are non-degenerate for all M ∈ C, P ∈ Proj(C).
	Definition 2.1.
	i) A right (left) modified trace on Proj(C) is a family of linear functions
	{t P : End C (P ) → k} P ∈Proj(C)	(2.5)
	satisfying two conditions, cyclicity and right (left) partial trace property, given
	as follows.
	1. (Cyclicity) If P, P ∈ Proj(C), then for all f : P → P , g : P → P
	we have
	t P (g • f ) = t P (f • g) .	(2.6)
	2. (Right Partial Trace Property) If P ∈ Proj(C) and V ∈ C,
	then for all f ∈ End C (P ⊗ V )
	t P ⊗V (f ) = t P (tr r V (f )) .	(2.7)

Our string diagrams are read from bottom to top. The further conventions we use for the graphical notation are detailed e.g. in[START_REF] Farsad | SL(2, Z)-action for ribbon quasi-Hopf algebras[END_REF] Sec. 2.1, 2.2, 6.1].

The symbol g has a slightly different meaning in[START_REF] Farsad | The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, Z)-action on its centre[END_REF], and so the expression for g stated there differs from the one given here.

Acknowledgements: We thank Ehud Meir and Tobias Ohrmann for useful discussions. We are grateful to the anonymous referee for helpful suggestions and for providing us with an improved version of Lemma 3.7. AMG thanks the CNRS and ANR project JCJC ANR-18-CE40-0001 for support. JB is supported by the Research Training Group 1670 of the DFG.

Throughout this paper we fix a field k.

Modified traces

In this section we review the definition of modified traces from [GKP1,GPV,GKP2] and recall some of their properties. Throughout this section let C be a pivotal finite tensor category over k.