Malgorzata Kloc
email: mkloc@houstonmethodist.org

Ahmed Uosef

Jarek Wosik

Jacek Kubiak

Rafik M Ghobrial

Virus interactions with the actin cytoskeleton-what we know and do not know about SARS-CoV-2

Keywords: actin, viruses, SARS-Cov-2, COVID-19, endocytosis, Golgi, ER, cell junctions, syncytium, directional transmission

Actin cytoskeleton and actin-dependent molecular and cellular events are responsible for the organization of the eukaryotic cells and their functions. All viruses, including SARS-COV-2, depend on host cell organelles and molecular components for invasion and propagation. Thus, it is not surprising that they also interact, at many levels, with actin cytoskeleton of the host. There are many studies on how different viruses reconfigure and manipulate actin cytoskeleton of the host during successive steps of their life cycle. However, we know relatively little about the SARS-CoV-2 virus interactions with actin cytoskeleton. Here we describe the subversion of actin cytoskeleton strategies frequently used by different viruses during entry, assembly, and egress from the host cell. We emphasize what is known and unknown about SARS-CoV-2 in these subjects. This review should encourage investigation of SARS-Cov-2 functions in the uncharted territories and eventually help developing novel antiviral therapies or mitigate the severity of the COVID-19 disease.

Introduction

In all eukaryotic cells, the actin cytoskeleton is indispensable for the anchoring and organization of the subcellular compartments and organelles. It regulates most cell functions, including cellcell adhesion, cell movement, and cell division [START_REF] Dominguez | Actin structure and function[END_REF][START_REF] Kloc | RhoA Pathway and Actin Regulation of the Golgi/Centriole Complex[END_REF][START_REF] Pollard | Actin, a central player in cell shape and movement[END_REF]. The cortical actin participates in all events related to the expression and presentation of membrane/cell surface molecules, the formation and movement of endocytic/phagocytic vesicles, virus entry, exocytosis, and virus egress.

Additionally, the actin in the cell nucleus regulates DNA repair, chromatin remodeling and condensation status, and gene transcription [START_REF] Kelpsch | Nuclear Actin: From Discovery to Function[END_REF][START_REF] Kloc | New Insights into Cellular Functions of Nuclear Actin[END_REF]. Two main forms of actin in the eukaryotic cells are the monomeric globular G-actin and polymerized filamentous F-actin. Actin polymerization and G/F actin dynamics are regulated by the small GTPase RhoA and actin interacting proteins [START_REF] Kloc | RhoA Pathway and Actin Regulation of the Golgi/Centriole Complex[END_REF][START_REF] Molinie | The Arp2/3 Regulatory System and Its Deregulation in Cancer[END_REF][START_REF] Moreau | A complex of N-WASP and WIP integrates signalling cascades that lead to actin polymerization[END_REF][START_REF] Weaver | Integration of signals to the Arp2/3 complex[END_REF][START_REF] Romero | Actin polymerization downstream of integrins: signaling pathways and mechanotransduction[END_REF][START_REF] Watanabe | Inside view of cell locomotion through single-molecule: fast F-/G-actin cycle and G-actin regulation of polymer restoration[END_REF][START_REF] Borovac | Regulation of actin dynamics during structural plasticity of dendritic spines: Signaling messengers and actin-binding proteins[END_REF][START_REF] Lee | Molecular dynamics study of interactions between polymorphic actin filaments and gelsolin segment-1[END_REF][START_REF] Schafer | Dynamin2 and cortactin regulate actin assembly and filament organization[END_REF][START_REF] Edwards | Fascins, a family of actin bundling proteins[END_REF]. Because of the pleiotropy of actin function, it is not surprising that the life cycle of viruses, from the point of cell entry to the egress of the progeny, is intimately connected to the actin cytoskeleton of the host cell, and that viruses hijack various actin functions and modify or redirect them to enhance self-propagation. All studied viruses reconfigure and manipulate actin cytoskeleton of host to optimize the production of the progeny.

It is known that RNA viruses can promote actin polymerization and reorganization [START_REF] Swaine | CDC42 use in viral cell entry processes by RNA viruses[END_REF]. For example, the transmissible gastroenteritis virus (TGEV) and porcine hemagglutinating encephalomyelitis (PHEV) coronaviruses use Rac1/Cdc42 small GTPases of the RhoA family to induce actin polymerization [START_REF] Swaine | CDC42 use in viral cell entry processes by RNA viruses[END_REF][START_REF] Hu | The epidermal growth factor receptor regulates cofilin activity and promotes transmissible gastroenteritis virus entry into intestinal epithelial cells[END_REF][START_REF] Lv | Porcine hemagglutinating encephalomyelitis virus activation of the integrin _5_1-FAKcofilin pathway causes cytoskeletal rearrangement to promote its invasion of N2a cells[END_REF]. Additionally, the virus manipulates the cell adhesion complexes and modifies intercellular contacts, allowing a directional cell-to-cell spreading. The different strategies used by different viruses to manipulate actin cytoskeleton indicate that these strategies evolved independently [START_REF] Taylor | Subversion of the actin cytoskeleton during viral infection[END_REF][START_REF] Wen | Cytoskeleton-a crucial key in host cell for coronavirus infection[END_REF]. Most of the studied actin-virus interactions describe viruses other than relatively novel SARS-CoV-2. Thus, here we describe the subversion of actin cytoskeleton strategies used by various viruses during entry, assembly, and egress from the host cells and point out what (if anything) we know about SARS-CoV-2 on those subjects. This review should encourage the exploration of SARS-Cov-2 behavior and functions in these uncharted areas and eventually help to develop novel anti-viral therapeutic or prevention strategies.

SARS-CoV-2 structure

The SARS-CoV-2 is a spherical structure of ~ 100nm diameter. In the center of the virus, there is a tightly packed positive-sense, single-strand, 29,811 nucleotide-long RNA, which is the socalled "translation-ready," i.e., after the entry into the host cell translates (using the synthetic machinery of the host) a set of viral proteins necessary for the RNA replication and formation of new copies of the virus [START_REF] Bar-On | SARS-CoV-2 (COVID-19) by the numbers[END_REF][START_REF] Haque | A comprehensive review about SARS-CoV-2[END_REF][START_REF] Kim | The Architecture of SARS-CoV-2 Transcriptome[END_REF][START_REF] Schmidt | The SARS-CoV-2 RNA-protein interactome in infected human cells[END_REF][START_REF] Wang | SARS-CoV-2: Structure, Biology, and Structure-Based Therapeutics Development[END_REF]. The RNA has 10-14 open reading frames (ORFs) and codes for 24-27 proteins [START_REF] Bar-On | SARS-CoV-2 (COVID-19) by the numbers[END_REF][START_REF] Finkel | The coding capacity of SARS-CoV-2[END_REF][START_REF] Katsnelson | What do we know about the novel coronavirus's 29 proteins?[END_REF]. The viral transcription and replication also rely on intricate longrange RNA-RNA interactions with the host RNA [START_REF] Ziv | The Short-and Long-Range RNA-RNA Interactome of SARS-CoV-2[END_REF]. Viral RNA is surrounded by the protective coat, called the capsid, built of 1000 copies of RNA-binding nucleocapsid (N) protein (Fig. 1). The N protein is highly phosphorylated, has a molecular weight of 46.5 kDa, and through the binding to the genomic packaging signal facilitates the packaging of the viral genome within the capsid [START_REF] Molenkamp | Identification of a specific interaction between the coronavirus mouse hepatitis virus A59 nucleocapsid protein and packaging signal[END_REF][START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF]. The capsid is surrounded by the membrane envelope, which derives from the membrane of the host cells but has inserted viral proteins: ~1000 copies of the membrane (M) protein, ~20 copies of the envelope (E) protein, and ~100 copies of the spike (S) protein (Fig. 1). The M protein has a molecular weight of 25-30 kDa, and it forms a dimer that shapes the viral membrane and promotes its binding to the capsid [START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF][START_REF] Neuman | A structural analysis of M protein in coronavirus assembly and morphology[END_REF]. The E protein has a molecular weight of 8-12 kDa and an ion channel activity. It is crucial for virus assembly and its release from the host cell [START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF]. The S proteins form the viral spikes. Each spike contains three copies of S protein and has a length of ~ 10nm [START_REF] Bar-On | SARS-CoV-2 (COVID-19) by the numbers[END_REF][START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF]. The S protein has a molecular weight of 150 kDa, is N-glycosylated, and belongs to the class I fusion family of proteins, which facilitate binding to the receptors and allow membrane fusion [START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF][START_REF] Kloc | Exaptation of Retroviral Syncytin for Development of Syncytialized Placenta, Its Limited Homology to the SARS-CoV-2 Spike Protein and Arguments against Disturbing Narrative in the Context of COVID-19 Vaccination[END_REF]. The S protein has S1 and S2 subunits. The S1 has the receptor-binding domain, and the S2 mediates membrane fusion [START_REF] Huang | Structural and functional properties of SARS-CoV-2 spike protein: potential antivirus drug development for COVID-19[END_REF].

The S protein-containing spikes have a high affinity to the angiotensin-converting enzyme 2 (ACE2) receptors on the host cells and facilitate entry of the virus [START_REF] Kim | SARS-CoV-2 Evolutionary Adaptation toward Host Entry and Recognition of Receptor O-Acetyl Sialylation in Virus-Host Interaction[END_REF][START_REF] Seyran | The structural basis of accelerated host cell entry by SARS-CoV-2 †[END_REF][START_REF] Zhang | Entry, egress and vertical transmission of SARS-CoV-2[END_REF]. The binding of the virus to the host cells involves two steps, first is the docking of the virus to the ACE2, and second is the activation of the spike protein by the proteolytic cleavage at two sites by the hostencoded proteases [START_REF] Braga | Drugs that inhibit TMEM16 proteins block SARS-CoV-2 spike-induced syncytia[END_REF][START_REF] Hoffmann | Priming time: how cellular proteases arm coronavirus spike proteins[END_REF]. The proteolytic cleavage is also a two-step process, the first cleavage occurs at the boundary of the S1and S2 subunits, and the second exposes the S2 subunit. The first cleavage can occur before or after the binding to the receptor. The second cleavage occurs before or after endocytosis, through the activity of the serine protease TMPRSS2 or endosomal proteases, respectively. The second cleavage primes the S2 subunit for the fusion of the virus with the cell membrane. In the SARS-Cov-2 and MERS-CoV, the boundary between S1 and S2 subunits has several amino acids, which can also be cleaved by another serine protease, the furin. This stretch of amino acids is absent in the SARS-CoV [START_REF] Huang | Structural and functional properties of SARS-CoV-2 spike protein: potential antivirus drug development for COVID-19[END_REF][START_REF] Braga | Drugs that inhibit TMEM16 proteins block SARS-CoV-2 spike-induced syncytia[END_REF][START_REF] Hoffmann | Priming time: how cellular proteases arm coronavirus spike proteins[END_REF]. Additionally, the S protein has the sialic acid (Sia)-binding domain, which interacts with the sialic acid-containing molecules on the surface of the host cell and facilitates the virus entry [START_REF] Seyran | The structural basis of accelerated host cell entry by SARS-CoV-2 †[END_REF]. It seems that this dual feature of S protein is one of the reasons for the high transmissibility of the SARS-CoV-2 [START_REF] Kim | SARS-CoV-2 Evolutionary Adaptation toward Host Entry and Recognition of Receptor O-Acetyl Sialylation in Virus-Host Interaction[END_REF][START_REF] Sun | The Role of Cell Surface Sialic Acids for SARS-CoV-2 Infection[END_REF]. Recent studies indicate that the recruitment of the SARS-CoV-2 to the cell surface is also facilitated by the heparan sulfate (HS) proteoglycan surface molecules. The HS interact with the Spike protein, increasing virus concentration and promoting engagement with the ACE2 receptors [START_REF] Zhang | Heparan sulfate assists SARS-CoV-2 in cell entry and can be targeted by approved drugs in vitro[END_REF]. Interestingly, the HCoV-OC43 and HCoV-HKU1coronaviruses, which like SARS-CoV-2, belong to the beta-CoVs family [START_REF] Liu | Human Coronavirus-229E, -OC43, -NL63, and -HKU1 (Coronaviridae)[END_REF], have an additional gene that expresses the hemagglutinin-esterase (HE) dimer protein [START_REF] Rabaan | SARS-CoV-2, SARS-CoV, and MERS-COV: A comparative overview[END_REF]. In these viruses, the S protein-containing spikes are interspersed with the shorter ~8nm stubby projections composed of HE protein. The HE protein has a dual function [START_REF] Soltani | The hemagglutinin-esterase gene in human coronaviruses SARS-CoV-2, HKU1 and OC43[END_REF], it has a receptor-binding lectin domain that is specific for Oacetylated sialic acid (O-Ac-Sia) and promotes attachment of the virus to the sialic acidcontaining molecules on the surface of a host cell, and a receptor destroying sialate-O-acetyl esterase that promotes virus release [START_REF] Kim | SARS-CoV-2 Evolutionary Adaptation toward Host Entry and Recognition of Receptor O-Acetyl Sialylation in Virus-Host Interaction[END_REF]. It seems that the S and HE proteins are functionally interdependent and co-evolved to optimize the balance between virus attachment and release [START_REF] Lang | Coronavirus hemagglutinin-esterase and spike proteins coevolve for functional balance and optimal virion avidity[END_REF]. The binding to the receptors and the virus entry into the cell (assessed for SARS-Cov-1) lasts ~10 min [START_REF] Bar-On | SARS-CoV-2 (COVID-19) by the numbers[END_REF]. The process of making new viruses within the cell lasts around 10 hours and ends with the release of ~ 1000 newly formed viruses [START_REF] Bar-On | SARS-CoV-2 (COVID-19) by the numbers[END_REF]. Below, we summarize what happens inside the infected cell, from the moment of virus attachment to the receptor till the release of the progeny, and how the actin cytoskeleton of the host is involved in these processes.

Actin-driven "surfing" of the virus

Some viruses, such as murine leukemia virus, human immunodeficiency virus, vesicular stomatitis virus, and human papilloma virus slide (surf) on the surface of host cell protrusions (filopodia), which contain actin filaments, toward the cell membrane territory with the highest endocytic activity (43) (Fig 2A). Although the virus can enter cells at the membrane located outside of filopodia, the filopodia-dependent entry, depending on the inherent ability of filopodia-region to bind and engulf various molecules, including viruses, is much more efficient and enhances virus entry and infection of the host [START_REF] Lehmann | Actin-and myosin-driven movement of viruses along filopodia precedes their entry into cells[END_REF]. Confocal time-lapse microscopy of murine leukemia virus (MLV) infecting human embryonic kidney HEK 293 cells showed that the virus, after attaching to the tip of filopodia, moved downward toward the cell's body [START_REF] Lehmann | Actin-and myosin-driven movement of viruses along filopodia precedes their entry into cells[END_REF].

Mechanism initiating the surfing is mainly unknown but certainly involves contact of the virus with its receptor(s) [START_REF] Lehmann | Actin-and myosin-driven movement of viruses along filopodia precedes their entry into cells[END_REF]. The surfing itself is powered by the contraction (retrograde F-actin flow) of filopodium's actin filaments by the myosin II motors located within the cortical actin at the base of the filopodium. This creates the dragging force that moves virus/receptor complexes to the base of the filopodium. Cytochalasin D, which blocks the barbed ends of actin filaments, inhibits surfing [START_REF] Taylor | Subversion of the actin cytoskeleton during viral infection[END_REF][START_REF] Lehmann | Actin-and myosin-driven movement of viruses along filopodia precedes their entry into cells[END_REF]. Although there is no description of SARS-CoV-2 surfing, the highresolution scanning microscopy showed that SARS-CoV-2 adheres to the surface of filopodia-like projections of the African green monkey kidney Vero cells, which would suggest a possibility of surfing [START_REF] Caldas | Ultrastructural analysis of SARS-CoV-2 interactions with the host cell via high resolution scanning electron microscopy[END_REF]. Recently, Seyran et al. [START_REF] Seyran | The structural basis of accelerated host cell entry by SARS-CoV-2 †[END_REF] described that prior the binding to the ACE2 receptor, the SARS-CoV-2 S protein interacts (through the flat sialic acid-binding domain at the N-terminal domain of the S1 subunit) with the sialic acid layer of the epithelium, allowing the virus to move on the epithelium surface to find and interact with the ACE2 receptor. It remains unknown if such movement of the virus has anything to do with, described above surfing perpetuated by the retrograde F-actin flow.

3.Virus entry into the cell

The coronaviruses can enter cells using various routes such as phagocytosis, micropinocytosis, or membrane fusion [START_REF] Wen | Cytoskeleton-a crucial key in host cell for coronavirus infection[END_REF]. However, recent studies showed that the preferable mode of entry of the SARS-CoV-2 virus is through the ACE2 receptor and clathrin-mediated endocytosis [START_REF] Bayati | SARS-CoV-2 infects cells after viral entry via clathrin-mediated endocytosis[END_REF], which depends on a proper arrangement of cortical actin (Fig. 3). Owczarek et al. [START_REF] Owczarek | Early events during human coronavirus OC43 entry to the cell[END_REF] showed that both stabilization or depolymerization of actin interferes with endocytosis and virus entry of the common coronavirus OC43. During virus entry, the virus/receptor complex recruits clathrin, which coats the invaginating membrane pit. The initiation of membrane invagination is controlled by the multiprotein complex containing the membrane-associated protein intersectin -1 that directs interactions between the endocytic vesicles and actin cytoskeleton, the epidermal growth factor receptor substrate 15 (Eps15) that is a scaffolding adaptor protein, and the membrane-sculpting F-BAR domain-containing Fer/Cip4 homology domain-only proteins 1 and 2 (FCHo1/2), [START_REF] Owczarek | Early events during human coronavirus OC43 entry to the cell[END_REF]. The FCHo1/2 recruits interscetin-1 and Eps15 and induces membrane invagination [START_REF] Owczarek | Early events during human coronavirus OC43 entry to the cell[END_REF][START_REF] Henne | FCHo proteins are nucleators of clathrin-mediated endocytosis[END_REF]. The Eps15 (48) recruits the adaptor protein 2 (AP2; [START_REF] Mcmahon | Molecular mechanism and physiological functions of clathrinmediated endocytosis[END_REF], which recruits clathrin to form a lattice coating the membrane pit. The "neck" of the pit attracts amphiphysin protein [START_REF] Peter | BAR domains as sensors of membrane curvature: the amphiphysin BAR structure[END_REF], which in turn recruits the dynamin, which via binding and hydrolyzing GTP severs the "neck" of the pit from the cell membrane [START_REF] Hill | The role of dynamin and its binding partners in coated pit invagination and scission[END_REF]. The scission results in the formation of the clathrin-coated endosomal vesicle that contains the receptor and the virus (Fig. 3A); [START_REF] Owczarek | Early events during human coronavirus OC43 entry to the cell[END_REF][START_REF] Kaksonen | Mechanisms of clathrin-mediated endocytosis[END_REF].

4.The endosomal phase of the virus

The next step after the formation of the endosome containing SARS-CoV-2/receptor complex is the acidification of the endosome through the proton pump-dependent release of Ca 2+ from the endosome via intracellular messenger nicotinic acid adenine dinucleotide phosphate (NAADP) sensitive two-pore channel (TPC2) [START_REF] Petersen | Endocytic uptake of SARS-CoV-2: the critical roles of pH, Ca2+, and NAADP[END_REF]. The acidification of the endosome occurs during the maturation of the early endosome, to the late endosome. In the late endosome, the complex of S protein and ACE2 receptor is cleaved by cathepsin-L, which promotes the fusion of viral and endosome membrane and the release of the viral RNA to the host-cell cytoplasm (Fig. 3A); [START_REF] Blaess | COVID-19/SARS-CoV-2 Infection: Lysosomes and Lysosomotropism Implicate New Treatment Strategies and Personal Risks[END_REF].

Studies showed that endosomal acidification inhibitors, such as chloroquine or bafilomycin A1 drastically reduce RNA release and replication of SARS-CoV-2 [START_REF] Shang | Inhibitors of endosomal acidification suppress SARS-CoV-2 replication and relieve viral pneumonia in hACE2 transgenic mice[END_REF]. The spatial arrangement of the endosomes and their trafficking within the cell, and the recycling of the endosomal membranes back to the cell surface, are supported by the actin filaments and microtubules and are powered by various motor proteins [START_REF] Granger | The role of the cytoskeleton and molecular motors in endosomal dynamics[END_REF].

5.Virus replication and translation

Released viral RNA (often called the genomic RNA, gRNA) undergoes replication, transcription, and translation to produce viral proteins and progeny RNAs [START_REF] Kim | The Architecture of SARS-CoV-2 Transcriptome[END_REF][START_REF] V'kovski | Coronavirus biology and replication: implications for SARS-CoV-2[END_REF]; (Fig. 3B). First, the two large open reading frames, ORF1a and ORF1b of viral RNA, are translated into the polyproteins pp1a and pp1ab, which after co-translational and post-translational modification are converted into the non-structural proteins (nsps), which participate in viral replication and transcription. [START_REF] Kim | The Architecture of SARS-CoV-2 Transcriptome[END_REF][START_REF] V'kovski | Coronavirus biology and replication: implications for SARS-CoV-2[END_REF][START_REF] Alexandersen | SARS-CoV-2 genomic and subgenomic RNAs in diagnostic samples are not an indicator of active replication[END_REF]. In the next step, viral gRNA generates (with the help of the nsps) the negative-sense RNA intermediates, which are the templates for synthesizing gRNAs of the next generation and subgenomic RNAs (sgRNAs). The sgRNAs will be translated into the viral structural proteins (N, E, M, and S), while the gRNAs will be assembled with the structural proteins to form the new progeny of the virus (Fig. 3B). A recent analysis of the SARS0-CoV-2 transcriptome showed that the viral RNA also has several ORFs of unknown functions [START_REF] Kim | The Architecture of SARS-CoV-2 Transcriptome[END_REF].

In all positive-strand (+) RNA viruses, the replication of viral RNA and its transcription occurs within the virus-induced replication organelle, situated in the vicinity of the host cell nucleus (Fig. 3B). This replication compartment, which was recently found in the cells infected by the SARS-CoV (59), creates a protective niche for viral replication and transcription. It derives from the ER or other membranous structures of the host cell and consists of double-membrane vesicles (DMVs), double-membrane cups, called the spherules (DMSs), convoluted membranes (CMs), or/and tubules (57, 60, 61; 62). We do not know if actin participates in the establishment and maintenance of the replication organelle. However, the actin cytoskeleton involvement in the formation and structure of all membranous organelles in eukaryotic cells (2) would imply so. It is known that in the eukaryotic cells, actin regulates various transcription factors and that the binding to actin is necessary for the activity of all RNA polymerases [START_REF] Kloc | New Insights into Cellular Functions of Nuclear Actin[END_REF]. This implies that also the transcription of viral RNA should be actin-dependent. This assumption is supported by the older studies showing that actin cytoskeleton is involved in the replication and transcription of human parainfluenza virus type 3 [START_REF] De | Involvement of actin microfilaments in the transcription/replication of human parainfluenza virus type 3: possible role of actin in other viruses[END_REF]. Moreover, the mutational analysis of respiratory syncytial virus (RSV) indicates that actin and profilin are necessary for viral transcription and that the divalent-cation-binding domain of actin is necessary for the binding and activation of viral RNA synthesis [START_REF] Burke | Profilin is required for optimal actindependent transcription of respiratory syncytial virus genome RNA[END_REF][START_REF] Harpen | Mutational analysis reveals a noncontractile but interactive role of actin and profilin in viral RNA-dependent RNA synthesis[END_REF]. Additionally, studies indicate that also the viruses replicating in the nucleus interact with nuclear actin for replication, assembly, and export of mRNA/ capsids [START_REF] Cibulka | Nuclear actin and lamins in viral infections[END_REF]. The fact that the interactions with actin are necessary for many viral functions suggests that the SARS-CoV-2 RNA transcription might also be actin-dependent despite that, to our knowledge, there is no direct information available on the subject.

Virus assembly and egress

The newly synthesized viral proteins and RNAs are transported to the ER/Golgi intermediate compartment (ERGIC, also referred to as the vesicular-tubular cluster (VTC; 67) where the SARS-CoV-2 RNA coated with N proteins are enveloped with the host-cell membrane containing viral M, E, and S proteins (Fig. 3C). From the ERGIC, the virus enters the trans-Golgi network (TGN)/Golgi complex for posttranslational modification of the proteins [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF]. Some time ago, a peculiar structure called "the juxtanuclear actin ring" was described in the coronavirusinfected cells (Fig. 3B). The ring, built of actin filament layers surrounding the cell nucleus and the adjacent ER and Golgi compartments facilitates viral genome replication and protein synthesis [START_REF] Wen | Cytoskeleton-a crucial key in host cell for coronavirus infection[END_REF][START_REF] Zhao | Transmissible gastroenteritis virus and porcine epidemic diarrhoea virus infection induces dramatic changes in the tight junctions and microfilaments of polarized IPEC-J2 cells[END_REF]. The sequestration of the molecules needed for virus replication, transcription, and translation by the actin ring may concentrate them in the vicinity of ER and Golgi for further processing. Another function of perinuclear actin ring was described in human adenovirus infection. In this case, the adenovirus early region 4 ORF4 protein (E4orf4) induces, via myosin II motor/ Src kinase/ Cdc42/N-Wasp/ RhoA kinase, and Rac1 pathways, actin polymerization around the nucleus. Subsequently, the actin ring, by recruiting endosomes, disrupting vesicular trafficking, and membrane dynamics, leads to the activation of an endo-lysosomal-based cell death pathway [START_REF] Robert | Adenovirus E4orf4 hijacks rho GTPase-dependent actin dynamics to kill cells: a role for endosome-associated actin assembly[END_REF].

The egress of the newly assembled viruses from the cell occurs via exocytosis [START_REF] Blaess | COVID-19/SARS-CoV-2 Infection: Lysosomes and Lysosomotropism Implicate New Treatment Strategies and Personal Risks[END_REF][START_REF] Cohen | Identification of a Golgi complex-targeting signal in the cytoplasmic tail of the severe acute respiratory syndrome coronavirus envelope protein[END_REF][START_REF] Mcbride | The cytoplasmic tail of the severe acute respiratory syndrome coronavirus spike protein contains a novel endoplasmic reticulum retrieval signal that binds COPI and promotes interaction with membrane protein[END_REF][START_REF] Perrier | The C-terminal domain of the MERS coronavirus M protein contains a trans-Golgi network localization signal[END_REF], (Fig. 3C) Recent studies showed that SARS-CoV-2 and other β-Coronaviruses use a unique exit strategy. Instead of using the conventional secretory pathway used by other viruses, they use the deacidified lysosomes to egress from the cell [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF]. The exact mechanism of deacidification of the lysosomes, which deactivates the lytic enzymes is still unknown. Ghosh et al [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF] suggest that the deacidification of lysosomes is either a direct consequence of too much cargo (viruses) and/or a dysfunctional proton pump, or that some of the viral proteins have a viroporin function that by the formation of hydrophilic pores modifies membrane permeability and Ca 2+ homeostasis leading to lysosome deacidification. Also, it is still unknown how the viruses translocate from the TGN/Golgi to the lysosomes. Ghosh et al. [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF] propose two possible routes.

One route is the transfer from the Golgi to the multivesicular bodies (MVB), which mature into the lysosomes via the activity of the Rab7 GTPase pathway (Fig. 3C). The fact that the inhibition of Rab7 reduces both the lysosome formation and the egress of the virus support this option [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF].

Another, more complicated option is the retrograde transport of the virus from the Golgi back to the ER/ERGIC and then to the lysosomes. The modification of the second option is the microphagy, when the fragments of ER containing the virus are directly engulfed by the lysosomes [START_REF] Ghosh | β-Coronaviruses Use Lysosomes for Egress Instead of the Biosynthetic Secretory Pathway[END_REF]. Studies on the avian infectious bronchitis coronavirus showed that the budding of the virus from the cell membrane and the egress are actin filament-dependent and involve the interaction between actin and the amino acids A159 and K160 of the viral membrane M protein [START_REF] Wen | Cytoskeleton-a crucial key in host cell for coronavirus infection[END_REF][START_REF] Wang | Interaction of the coronavirus infectious bronchitis virus membrane protein with beta-actin and its implication in virion assembly and budding[END_REF]. The scanning electron and atomic force microscopy showed that just before its egress the SARS-CoV induces the formation of pseudopodia and thickening of the cortical actin, which by bending the membrane expels the virus progeny from the host cell (75).

Actin comets

Many viruses, such as Vaccinia, Ebola, Marburg, and insect baculoviruses, use the actin "tails" or the so-called "actin comets" to be propelled within the cytoplasm to the cell periphery and/or to the outside of the infected cell (Fig. 2B). Viral proteins induce actin nucleation via the activity of the Arp2/3 complex. This is followed by the interaction with the myosin and kinesin motors and actin polymerization/depolymerization cycles, which propel the virus. Because the cometassociated viruses were noticed not only inside the cytoplasm but also in the vicinity of the cellto-cell junctions, it has been proposed that the propelling force of the actin comet overcomes the actin reinforced cellular junctions allowing delivery of the virus to the adjacent cell [START_REF] Strauss | A virus joins the movement. Intracellular pathogens[END_REF][START_REF] Rietdorf | Kinesin-dependent movement on microtubules precedes actinbased motility of vaccinia virus[END_REF][START_REF] Labudová | Cell-to-cell transport in viral families: faster than usual[END_REF]. So far, there is no information if the coronaviruses and SARS-CoV-2 use the actin comets for the movement inside and out of the cell.

Actin and a direct cell-to-cell transmission of the virus

Cell-to-cell direct transmission that is faster and much more efficient than a cell-free spread, and allows circumventing immune response, especially antibody-mediated immunity of the host, is efficiently used by many viruses, including the SARS-CoV-2 (70, 78, 79, 80). Below we describe several possible routes for a direct transmission of the virus between cells. All these routes of transmission involve the actin cytoskeleton of the host cells.

a. Tunneling nanotubes (TNTs)

Tunneling nanotubes (TNTs) are long, thin (50-1500 nm in diameter), F-actin-based open-ended channels used by the neighboring cells to exchange large cargo, such as protein complexes, lipids, nucleic acids, vesicles, pathogens, and organelles (Golgi, ER, lysosomes, endosomes, and mitochondria) at the distance of hundreds of micrometers [START_REF] Labudová | Cell-to-cell transport in viral families: faster than usual[END_REF][START_REF] Mothes | Virus cell-to-cell transmission[END_REF][START_REF] Dupont | Tunneling Nanotubes: Intimate Communication between Myeloid Cells[END_REF][START_REF] Kloc | Exogenous Molecule and Organelle Delivery in Oogenesis[END_REF][START_REF] Rustom | Nanotubular highways for intercellular organelle transport[END_REF][START_REF] Mccoy-Simandle | Exosomes and nanotubes: control of immune cell communication[END_REF][START_REF] Tiwari | Role of Tunneling Nanotubes in Viral Infection, Neurodegenerative Disease, and Cancer[END_REF], (Fig. 4) The ticker TNTs, which are used for the transport of very large cargo contain besides the F-actin also the microtubules. The traveling speed of the cargo through the TNTs is very fast, reaching ~ 7 nm/sec [START_REF] Tiwari | Role of Tunneling Nanotubes in Viral Infection, Neurodegenerative Disease, and Cancer[END_REF]. Viruses can also "surf", docking by the heparan sulfate (HS) receptors, on the surface of TNTs for long distances (Fig. 4). The same receptors can also induce the fusion of the virus with the TNT membrane and induce endocytic internalization of the virus into the TNT lumen ( 85), (Fig. 4). Actin polymerization required for the formation of TNTs is regulated by the Rho GTPase signaling. including Rac1/Cdc42/ Pak1 pathways [START_REF] Zhang | Rho GTPases and the emerging role of tunneling nanotubes in physiology and disease[END_REF]. Many viruses such as HIV-1, influenza, and coronaviruses induce the formation of TNTs to spread between cells [START_REF] Tiwari | Role of Tunneling Nanotubes in Viral Infection, Neurodegenerative Disease, and Cancer[END_REF][START_REF] Zhang | Rho GTPases and the emerging role of tunneling nanotubes in physiology and disease[END_REF].

Interestingly, the TNTs formed by Mycobacterium tuberculosis (known to enhance HIV-1 pathology) are used by the co-infecting HIV for the cell-to-cell transfer [START_REF] Zhang | Rho GTPases and the emerging role of tunneling nanotubes in physiology and disease[END_REF]. We do not know if the SARS-Cov-2 uses TNTs to spread between cells. However, the observation that murine hepatitis (MHV) coronavirus and SARS-CoV-1 promote the formation of filopodia and membrane ruffling using Rac1/Cdc42/Pak1pathways, and the ability of SARS-CoV-2 to bind to the surface HS, suggest that they can also induce the formation of TNTs. Certainly, this should be studied further in the context of SARS-CoV-2 as it may offer novel therapeutic targets for preventing or fighting COVID-19.

b. Virological synapse

Many viruses, including human T cell leukemia virus type 1(HTLV-1), human immunodeficiency virus type 1 (HIV-l), and SIV retroviruses spread from cell to cell through the virally induced structure called the virological synapse (VS) or the infectious synapse, which shares similarities with the neurological, immunological, and plant synapses [START_REF] Garcia | Virological Synapse for Cell-Cell Spread of Viruses[END_REF][START_REF] Martin | Virological synapsemediated spread of human immunodeficiency virus type 1 between T cells is sensitive to entry inhibition[END_REF][START_REF] Piguet | Dangerous liaisons at the virological synapse[END_REF][START_REF] Sewald | In vivo imaging of virological synapses[END_REF]. During the synapse formation, the pre-synaptic and post-synaptic cells adhere to each other, but their membranes do not fuse. The virological synapse (VS) directly transmits viruses from the infected (pre-synaptic) to the uninfected (post-synaptic) cell [START_REF] Garcia | Virological Synapse for Cell-Cell Spread of Viruses[END_REF]. Also, the HIV-1, after being captured and internalized by the dendritic cells, is transferred through the immunological synapse to the interacting T cells. However, studies indicate that not only internalized virus is being transmitted, but also the virus attached to the surface of the dendritic cells. This type of viral transmission is called trans-infection [START_REF] Cavrois | In vitro derived dendritic cells transinfect CD4 T cells primarily with surface-bound HIV-1 virions[END_REF].

The formation of any synapse, including the VS, involves the polarization of the F-actin filaments and the Rho GTPase signaling [START_REF] Dustin | The immunological synapse and the actin cytoskeleton: molecular hardware for T cell signaling[END_REF][START_REF] Felts | 3D visualization of HIV transfer at the virological synapse between dendritic cells and T cells[END_REF][START_REF] Haller | Nef does not inhibit F-actin remodelling and HIV-1 cell-cell transmission at the T lymphocyte virological synapse[END_REF][START_REF] Kloc | The newly found functions of MTOC in immunological response[END_REF]. The actin network at the synapse plays not only the structural function but is a scaffold for the signaling molecules and the clustering of the virus [START_REF] Dustin | The immunological synapse and the actin cytoskeleton: molecular hardware for T cell signaling[END_REF][START_REF] Felts | 3D visualization of HIV transfer at the virological synapse between dendritic cells and T cells[END_REF]. It seems that at least in the case of HIV-1, the synapse contains a local, F-actin-depleted zone that facilitates virus entry by removing the physical cortical actin barrier [START_REF] Vasiliver-Shamis | Human immunodeficiency virus type 1 envelope gp120-induced partial T-cell receptor signaling creates an F-actindepleted zone in the virological synapse[END_REF]. Besides the retroviruses, the coronaviruses also use the VS for a direct cell-to-cell transmission. Yang et al. [START_REF] Yang | pH-dependent entry of severe acute respiratory syndrome coronavirus is mediated by the spike glycoprotein and enhanced by dendritic cell transfer through DC-SIGN[END_REF] described the VS transmission of SARS-coronavirus (SARS-CoV) between dendritic cells (DC) and target cells. Recently, more and more studies suggest that the ability of the SARS-CoV2 to invade the neural system involves the transmission of the virus through the neurological synapses (98-100).

c. Virus interaction with the cellular junctions

A direct transfer of viruses between cells can also occur through intercellular junctions. Many has a PDZ-binding motif (PBM). The binding of viral PBM to the PALS1 (protein associated with Lin seven 1) present in human cell junctions relocates PALS1 from the junctions to the ERGIC compartment, where the virus is assembled. Envelope protein also binds and dislocates other PDZ-containing junctional proteins, including the adherens junction protein syntenin and the tight junction ZO-113 (Zonula Occludens-113) protein, which regulates the actin-relevant RhoA/ROCK pathway (114). Disruption of the junctional proteins in the inter-epithelial junctions of the lungs causes vascular leakage, damage of alveoli, and induces acute respiratory distress syndrome (ARDS) in COVID-19 patients (112, 115).

Virally induced syncytia

The pathological examination of the lungs of the patients with severe cases of COVID-19 infection shows the presence of SARS-CoV-2-infected, multinucleated syncytia of pneumocytes (116, 117). Additionally, these patients are often severely lymphopenic, i. e. they have a deficit of lymphocytes, which decreases their ability to fight the infection (118, 119). It is known that many viruses, including poxviruses, paramyxoviruses, herpes viruses, and retroviruses, induce the fusion of the infected cells into the multinuclear syncytia, which eventually die (120). Recent studies showed that SARS-CoV-2 infection causes the formation of the giant syncytia and the destruction of the tight junctions in the lung airway epithelium (113). There are two main types of syncytia: 1. homotypic syncytia that contain the same type of cells, and 2. heterotypic syncytia that result from the fusion of different types of cells. Recently, Zhang al. (119) described another type of syncytia in the lungs of COCID-19 patients. These are the syncytia of the lung cells containing the internalized lymphocytes (Fig. 5). The cell fusion leading to the formation of the syncytia requires the rupture of the membranes of the fusing partners followed by the membrane rejoining. The whole process is facilitated by the fusogenic proteins and involves the restructuring of the supportive platform of the actin cytoskeleton underlying the cell membranes.

The Spike protein of SARS-Cov-2 is a fusogen with homology to the mammalian syncytin-1, which is involved in the formation of the syncytial layers of the placenta [START_REF] Kloc | Exaptation of Retroviral Syncytin for Development of Syncytialized Placenta, Its Limited Homology to the SARS-CoV-2 Spike Protein and Arguments against Disturbing Narrative in the Context of COVID-19 Vaccination[END_REF]. The model and the details of the fusion process are described in ref. [START_REF] Kloc | Exaptation of Retroviral Syncytin for Development of Syncytialized Placenta, Its Limited Homology to the SARS-CoV-2 Spike Protein and Arguments against Disturbing Narrative in the Context of COVID-19 Vaccination[END_REF] and 119. The virally induced syncytia not only enhance virus spreading but may also serve as a tool for the elimination of the immune cells, which fight the virus. Studies of COVID-19 patients showed that SARS-CoV-2 induces both homotypic syncytia of the pneumocytes, and the heterotypic cell structures, which contain fused lung cells with the engulfed lymphocytes inside [START_REF] Braga | Drugs that inhibit TMEM16 proteins block SARS-CoV-2 spike-induced syncytia[END_REF]119). These studies propose that the SARS-CoV-2 infected cells express the viral S (spike) protein on their surface. The interaction of the spike protein with the ACE2 receptor of the neighboring cell activates the calcium-activated lipid scramblase TMEM16F (anoctamin 6) that exposes the phosphatidylserine on the cell surface and triggers the formation of syncytium, which, subsequently, engulfs the immune cells (116,122). These studies suggest that syncytium-mediated lymphocyte internalization and elimination may be one of the causes of lymphopenia in COVID-19 patients (119-121). Recent studies indicate the antihelminthic drug niclosamide, suppress the activity of TMEM16F and inhibits the spike-induced syncytia formation in SARS-CoV-2 (123).

Conclusions

The information presented here indicates that the interactions of the viruses, including SARS-CoV-2, with the actin cytoskeleton and the actin-related cell functions and signaling pathways, are of crucial importance for virus functions, infectivity, and pathogenicity. It is also clear that the knowledge on the SARS-CoV-2 interactions with the actin and other components of the cytoskeleton is still very limited and requires further studies. A better understanding of these interactions, processes, and the participating viral and cellular components can help design novel antiviral therapies and/or mitigate the severity of COVID-19 symptoms and damaging, long-term effects. The conventional approaches to fight viral infections include vaccinations, antiviral drugs, and preventing the overactivation of the immune response. The latter is especially important in COVID-19 disease, in which the overproduction of inflammatory factors causes cytokine storm in the patients' lungs (124). The recent analysis of gene expression profile in the SARS-CoV/SARS-CoV-2 infection using the Through Gene Ontology (GO) and MetaCore showed that the upregulation of actin/cytoskeleton remodeling via the RhoA pathway is crucial for the infection (125). Thus, we propose that the interference with the actin-dependent function of virus and/or host immune cells using RhoA pathway inhibitors, such as the multiple sclerosis drugs Fingolimod and Syponimod, might be a novel therapeutical approach in fighting viral diseases (126). The binding of the S protein of the SARS-CoV-2 to the heparan sulfate (HS) proteoglycan molecule (which acts as a co-receptor for the ACE2 binding) present on the cell surface, induces the formation of a tunneling nanotube (TNT) between the infected cell and the neighboring noninfected cell. The virus is transferred within the lumen of the TNT. The TNT contains actin filaments. The Rho GTPase signaling including Rac1/Cdc42/ Pak1 pathways regulates actin polymerization, which is required for the formation of TNT. The virus can also, after binding to the HS, surf on the surface of TNT. The virus on the surface of TNT can also internalize into the TNT interior. Although there is no data that the SARS-CoV-2 spreads through the TNTs, the fact that its S protein binds to the HS, and that a closely related SARS-CoV-1 can induce various types of cytoplasmic extensions, suggests so. interaction initiates a cascade of events, which eventually lead to the apposition of neighboring cell membranes, membrane scission, and cell fusion. The interaction of the spike protein with the ACE2 receptor of the neighboring cell activates the calcium-activated lipid scramblase TMEM16F (anoctamin 6) that exposes phosphatidylserine on the cell surface and triggers the formation of the syncytium. The SARS-CoV-2 induces not only the formation of the pneumocyte syncytium but also the formation of heterotypic syncytia when the pneumocytederived syncytia engulf the lymphocytes. The internalized lymphocytes undergo apoptosis and die. SARS-C0V-2 uses this method to eliminate from the lungs the virus-fighting lymphocytes.

  viruses evolved various strategies to alter different types of cell junctions to allow viral penetration. There are three main categories of cell junctions. The tight (occluding) junctions support the exchange of small hydrophilic molecules. The anchoring junctions (adherens junctions, desmosomes, focal adhesions, and hemidesmosomes) attach cells to each other or the extracellular matrix. Adherens junctions and focal adhesions are connected to the bundles of actin filaments, while desmosomes and hemidesmosomes are connected to the intermediate filaments. The gap (communicating) junctions transfer small (below 1000Da) proteins, amino acids, nucleotides, small metabolites, ions, regulatory molecules, and secondary messengers (82, 101, 102). Studies showed that hepatitis C virus and retroviruses may enter through the tight junctions, human papilloma virus (HPV) through the adherens junctions, and HIV may modify gap junctions for the entry (103). Retroviruses modify the distribution of tight junction proteins occludin and claudin-1 and destroy tight junctions via RhoA/ROCK / MLC signaling pathways (103-105). Hepatitis C virus also affects occludin and claudin-1 and these proteins may also promote endocytosis of the virus (103, 106, 107). The HPV enters through the adherens junctions. It modifies the distribution of adherens junction proteins such as β-catenin, disrupting its association with E-cadherin and actin cytoskeleton (103, 108). The HIV hijacks the gap junctions to spread apoptosis-inducing factors to the uninfected cells and destroys, via mitogenactivated protein kinase (MAPK) pathway, the adherens and tight junctions (103, 109). Studies showed that the intermediate filament protein vimentin connected to the adherence junctions is essential for the viruses' life cycle, including entry, transport, replication, assembly, and egress (110). The intermediate filaments, actin, and microtubules are also crucial for the entry and propagation of dengue virus, Zika virus, and West Nile virus (111). Recent studies showed that the alterations of cell junctions induced by SARS-CoV-2 are highly damaging for the patients (112,113). Humans have over 100 proteins containing the PDZ (post-synaptic density protein) domains, some of which are the components of the cell junctions. The envelope (E) protein of SARS-CoV-2 and SARS-CoV-1 (called the SARS-CoV before the emergence of SARS-CoV-2)

Figure 1 .

 1 Figure 1. SARS-CoV-2 structureThe SARS-CoV-2 consists of the nucleocapsid and the envelope. The nucleocapsid contains ss(+)RNA surrounded by the RNA-binding nucleocapsid (N) proteins. The nucleocapsid is surrounded by the lipid bilayer membrane, which derives from the cell membrane of the host cell but has inserted viral proteins: membrane (M) proteins (M), envelope (E) proteins, and the spike (S) proteins.

Figure 2 .

 2 Figure 2. Hypothetical SARS-CoV-2 surfing and actin comets. A) Filopodia contain the bundles of actin filaments. The virus bound to its receptor surfs on the surface of the cell membrane, from the tip to the bottom of the filopodia, and to the membrane region with the highest endocytic activity for a subsequent internalization. The motor proteins located at the bottom of the filopodium cause actin filament contraction and retrograde movement, which propels the receptor-bound virus. B) Viruses can also induce actin polymerization and the formation of the actin filament comets (tails), which propel the virus within the cell toward the cell surface or expel it from the cell. Viral proteins induce actin nucleation via the activity of the Arp2/3 complex, and the myosin and kinesin motor-dependent actin polymerization/depolymerization cycles propel the virus. Although these types of movement are used by many different viruses, so far, we do not know if SARS-CoV-2 may surf or use the actin comets for the movement.

Figure 3 .

 3 Figure 3. SARS-CoV-2 life cycle A)Diagram summarizes virus endocytosis. The preferable mode of SARS-CoV-2 entry to the host cell is the clathrin-mediated endocytosis. The binding of the virus to the ACE2 receptor induces, through the action of membrane sculpting complex (FCHo1/2, interscetin-1 and Eps15), the rearrangement of the cortical actin, membrane bending, and recruitment of clathrin to the surface of the membrane pit. Subsequently, the "neck" of the pit recruits the dynamin, which cuts off the "neck" of the pit from the cell membrane. The resulting clathrin-coated early endosome contains the virus and the receptor. The early endosome matures through the acidification into the late endosome. In the late endosome, the virus is cleaved off the receptor by cathepsin-L, which also promotes viral and endosomal membrane fusion. The receptor recycles back to the cell membrane, and the viral genomic RNA (gRNA) is released into the cytoplasm.The movement of endocytic vesicles and their spatial distribution within cell cytoplasm are actin/motor protein dependent. B) Summary of the post-endocytic phase of the virus life cycle.Released viral RNA (gRNA) is translated into the non-structural proteins (nsps), which will be

Figure 4 .

 4 Figure 4. Hypothetical TNT-dependent transfer of the SARS-CoV-2. The binding of the S protein of the SARS-CoV-2 to the heparan sulfate (HS) proteoglycan

Figure 5 .

 5 Figure 5. SARS-CoV-2 induced syncytia After the cell infection and release of the viral RNA, the infected cell synthesizes viral Spike (S) protein. The S protein expressed on the surface of the infected cell interacts with the ACE2 receptor of the uninfected cell. Because the viral S protein has fusogenic properties, this

Acknowledgments

Some of the drawings used to make figures were from the Servier Medical ART: SMART, smart.servier.com.

Conflict of interests

None of the authors has a conflict of interests


Mattia Melosso

Luca Bizzocchi

Houda Gazzeh

Francesca Tonolo

Jean-Claude Guillemin

Silvia Alessandrini

Victor M Rivilla

Luca Dore

Vincenzo Barone

Cristina Puzzarini

Víctor M Rivilla

Gas-phase identification of (Z)-1,2-ethenediol, a key prebiotic intermediate in the formose reaction

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

The origin of life from simple raw materials, i.e. abiogenesis, is still a hot, highly debated topic involving many research fields. While a general and robust solution to this problem is still sought, it is nowadays widely accepted that abiogenesis was not a smooth process, but more likely a dynamic scenario involving a number of successive steps and occurring across different geological eras and environments. 1 Indeed, none of the main accredited theories for the origin of life, i.e. primordial soup, hydrothermal vents, and extraterrestrial origin of biological building blocks, is able to fully explain how bio-relevant molecules could have formed and survived in hostile conditions, although each of them provides unique pieces to complete the overall puzzle. In this context, the extraterrestrial formation (and the subsequent delivery on Earth) of the main building blocks of life, such as amino acids and sugars, is increasingly recognized as an important pathway. 2 One of the most important routes towards the synthesis of prebiotic species starting from simple organic material is the formose reaction, which is thought to be responsible for the formation of simple carbohydrates, such as ribose, on early Earth environments. 3 The formose reaction uses formaldehyde (H2CO, 1) as starting material and proceeds through aldol, reverse-aldol, and isomerization reactions, eventually forming glycolaldehyde (HOCH2CHO, 2), glyceraldehyde, and a number of four, five, and six-carbon atoms sugars. [4][5][6] This process, firstly discovered by Butlerov in 1861 6 , leads to the formation of fundamental biological units found in RNA and cofactors (e.g., ATP and NADH). However, the formose reaction exhibits some limits in aqueous solution, 1 thereby reducing its possible role for the sugar formation on early Earth conditions. Conversely, it is conceivable that formose-like reactions might have played an important role in the gas-phase and dust-grain surface chemistry occurring in the interstellar medium (ISM). 7 The discovery of an enantiomeric excess of L-amino acids in the Murchison meteorite 8 and then in many carbonaceous chondrites further supports this hypothesis, as L-amino acids are found to act as catalyst in the formose reaction, thus promoting the formation of sugars belonging to the D-series. 9 These findings seem to suggest that an enantiomeric imbalance has possibly emerged elsewhere and could have then been transferred to Earth by meteoritic and cometary bombardment. Therefore, it is not unrealistic to think that prebiotic molecules formed in the ISM could have played a role in setting the stage of life on early Earth. This hypothesis is also supported by the detection of interstellar glycolaldehyde 10 and many other complex, bio-relevant species in molecular clouds, star-forming regions, and Solar-type protostars. [11][12][13][14][15] Such observations demonstrate the existence of formation mechanisms of organic molecules compatible with the harsh conditions of the ISM (T = 10-100 K, number densities ≈ 10 4 cm -3 ), whose detailed comprehension still requires a huge effort and, in particular, the Please do not adjust margins identification of key intermediate species linking small organic species to bio-relevant molecules. Hydroxymethylene (HCOH, 3) is a prototypical short-lived species involved in sugar formation in non-aqueous conditions that escaped detection for a long time. 16 It is thought to be responsible for an efficient initiation of formose-like reactions, 17 especially at low temperatures such as those found in the ISM. In a similar manner, 1,2-ethenediol (HO-CH=CH-OH, 4) has been suggested being deeply involved in the process of sugar formation in the ISM, e.g. by producing glyceraldehyde through its reaction with formaldehyde. 18 Diol 4, the higher energy and highly reactive enol of 2, has been recently identified in an experiment simulating methanolbearing interstellar ice-analogues exposed to ionizing radiation. 18 Similarly to other prebiotic species, [19][20][21] 4 can be formed on the icy mantles of interstellar dust grains before being released into the gas phase through thermal desorption or energetic processes, such as strong UV irradiation. This suggests that 4 is likely present in the interstellar medium. However, its detection is currently not possible, because its rotational spectrum is unknown to date. Indeed, it should be noted that the 90% of our astrochemical inventory has been discovered through the observation of rotational transitions 22 . Compared to other techniques, e.g., electronic or vibrational spectroscopy, which are more suitable for the identification of functional groups, rotational spectroscopy allows for distinguishing among different isotopologues, structural isomers, and even conformers. In this context, it is noteworthy that both 1 and 2, as well as two additional C2H4O2 isomers (acetic acid and methyl formate), have been identified in space thanks to the observation of their spectroscopic signatures 10,23,24 , as it has been the case for halogenated, 25 aromatic, 26,27 or chiral molecules. 28 Here, in order to enable a step forward in the understanding of sugar formation in the ISM, we report the gas-phase synthesis of the Z isomer of 4 via flash vacuum pyrolysis (FVP) of bis-exo-5-norbornene-2,3-diol (C7H10O2, 5, see Scheme 1) and its spectroscopic characterization.

The synthesis is based on the general retro-Diels-Alder reaction proposed by Turecek 29 . The Z isomer of 4 (hereafter Z-4) has been characterized via measurements of its spectrum using a frequency-modulation submillimeter-spectrometer and its comparison with simulations based on high-level quantumchemical calculations (see ESI for details).

The spectrum, recorded between 80-125 GHz and 240-375 GHz during the FVP of 5 at 750°C, revealed absorptions due to the by-product cyclopentadiene (c-C5H6, 6) and the more stable tautomeric form 2. 30 However, the most intense features detected in the spectrum (Fig. 1) do not belong to any of these or other known molecules, while they match very well the spectral patterns predicted for the Z-(syn,anti) form of 4. ‡ With the help of theoretical predictions (based on coupledcluster theory § ) and using the SPCAT 31 subroutine of the CALPGM suite, the signals around 257 GHz could be unambiguously assigned to the strongest µa-allowed transitions of Z-4 (µa = 1.96 D), shifted with respect to predictions by only 0.1%. This first assignment has been used to refine the spectral predictions, in order to reduce the discrepancy between the computed and observed spectra and to facilitate the interpretation of the subsequent measurements. In addition, we found that a mild heating of 5 decreases the intensity of the interfering lines produced by 2. Under the best conditions, the Z-4 to 2 abundance ratio is about 6:1. The identification of Z-4 was further confirmed by the presence of a tiny splitting in most of the absorption features. These splittings are a clear evidence of a tunneling motion occurring within the molecular frame, as expected in view of the similarity of Z-4 with ethylene glycol (HO-(CH2)2-OH, 7). 32 In particular, the -OH moieties undergo a concerted large amplitude tunneling motion between two equivalent positions of Z-(syn,anti)-4, thus splitting each rotational energy level into two. As a consequence, the transitions allowed by the dipole moment component oriented along the exchange axis must occur between the two inversion states and show a 10:6 intensity ratio obeying to spin-statistics (Fig. 2, see also Sec. 2 in the ESI). On the other hand, moderately intense b-type transitions (μb = 0.62 D) take place within the inversion states. In agreement with the results of Kleimeier et al. 18 , our computed equilibrium geometry of Z-4 indicates a non-planar structure (the C=C-O-H dihedral angle being about 25°), which should result in a moderate dipole moment component along the c-axis of the molecule (μc = 0.97 D). However, no μc-allowed transitions could be observed in the spectrum. While this fact Scheme 1 -(Z)-1,2-ethenediol (Z-4) and cyclopentadiene (6) generated in the flash vacuum pyrolysis of bis-exo-5-norbornene-1,2-diol (5). Other molecules of interest are reported for the sake of clarity: formaldehyde (1), glycolaldehyde (2), hydroxymethylene (3), and ethylene glycol (7). Please do not adjust margins may appear puzzling at first, the experimental evidence is explained by the very low barrier to planarity (≈ 0.2 kJ mol -1 at the B2PLYP/aug-cc-pVTZ level of theory; see ESI for details), ruled by the motion of a single H atom. Hence, in the ground vibrational state the molecule is effectively planar, as confirmed by its inertia defect, whose value is close to zero and slightly negative (see Table 1), a clear sign of near planarity together with the low frequency out of plane motion 33 . From the analysis of the spectrum, more than 1100 transitions have been assigned to Z-4 and analysed with an effective Hamiltonian that accounts for centrifugal distortion as well as tunneling effects. The optimized spectroscopic constants are in excellent agreement with the corresponding computed values (Table 1). This, together with the observed tunneling splittings, provides strong evidence for the correct identification of Z-4 and for its structure and internal dynamics. The small energy difference between the two inversion substates is a clear hint of a high-barrier separating the two identical conformation of Z-(syn,anti)-4. In the aGg' form of 7, this energy difference is four orders of magnitude larger, 34 because the single C-C bond lowers the barrier height considerably. The accurate spectroscopic characterization of Z-4 now permits its search in the ISM. The observation of 4 would shed light on how formose-like reactions can proceed under interstellar conditions, whether following the mechanism proposed by Ricardo et al. 35 or a different pathway, 5 and can provide additional insights on sugar formation in extraterrestrial environments. Being the highly energetic tautomeric form of 2, the diol 4 can promote chemical reactions when no external energetic sources are available, as it is the case at very low temperatures. This opens up reaction channels that otherwise would be closed and expands the horizon of sugars formation in the ISM. With both 1 and 2 already detected in different astronomical environments, 10,23,36,37 the presence of 4 in the interstellar medium appears more than plausible. The interest in 4 is, however, not limited to prebiotic and interstellar chemistry; enols are also key intermediates of tropospheric acids via ketoenol photo-tautomerization mechanisms 38 and of hydrocarbon oxidation processes, 39 thus widening the number of applications for which our gas-phase spectroscopic identification is invaluable. From another point of view, this work demonstrates the feasibility of producing 4 directly in the gas phase through the retro-Diels-Alder reaction of 5, thereby allowing its spectroscopic characterization. Although the coupling of FVP with high-resolution spectroscopic techniques has been widely employed in the past for the study of enols, 40, 41 carbenes 42,43 , and other unstable species, [44][45][46] this is the first time that an enediol is observed directly in the gas phase by rotational spectroscopy. The analysis of the spectrum, supported by quantum-chemical calculations, has revealed a vibrationallyaveraged planar structure of Z-(syn,anti)-4, which undergoes a large amplitude tunneling motion that exchanges the position of the two hydroxyl groups. The experimental findings are in excellent agreement with results issuing from high-level quantum-chemical calculations. In view of the important information that can be retrieved from the spectroscopic characterization in an isolated environment, and the concrete possibility to search for the title molecule in the ISM, we are currently extending the approach presented here to a number of highly reactive species not yet observed in the gas phase.

Fig. 1 -

 1 Fig.1-Portion of the millimeter-wave spectrum around 257 GHz. The experimental spectrum (black trace) is compared with the spectral simulation of glycolaldehyde (2, blue trace), cyclopentadiene (6, green trace), and the computational guess obtained for 1,2-ethenediol (Z-4, orange trace). The intensities of the simulated spectra have been adjusted in order to match roughly the experimental spectrum.

  Université de Monastir, Avenue Taher Hadded B.P 56, Monastir 5000, Tunisie. f. Centro de Astrobiología (CSIC-INTA), Ctra. de Ajalvir Km. 4, Torrejón de Ardoz, 28850 Madrid, Spain.

	a. Dipartimento di Chimica "Giacomo Ciamician", Università di Bologna, Via F. Selmi
	2, 40126 Bologna, Italy. E-mail: mattia.melosso2@unibo.it;
	cristina.puzzarini@unibo.it
	b. Scuola Superiore Meridionale, Università di Napoli Federico II, Largo San
	Marcellino 10, 80138 Naples, Italy.
	c. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy.
	d. Univ Rennes, Ecole Nationale Supérieure de Chimie de Rennes, CNRS, ISCR -
	UMR6226, F-35000 Rennes, France.

e. g. INAF -Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Florence, Italy. † Electronic Supplementary Information (ESI) available: See DOI: 10.1039/x0xx00000x

Table 1

 1 Comparison of experimental and quantum-chemical spectroscopic parameters of (Z)-1,2-ethenediol. Numbers in parentheses represent the error in the unit of the last quoted digit. When the error is not reported, the parameter is kept fixed to the corresponding computed value. [b] See ESI for details about the level of theory employed.

	Constant	Unit	Value [a]	Theory [b]	Constant	Unit	Value [a]
	A	MHz	19507.2(2)	19597.9	E*	MHz	0.361(3)
	B	MHz	6312.1(2)	6298.6	E*J	kHz	0.12(1)
	C	MHz	4772.4533(2)	4766.8	E*K	kHz	0.75(5)
	DJ	kHz	6.9479(4)	7.21	E*JJ	Hz	0.10(1)
	DJK	kHz	-36.977(6)	-39.15	E*JK	Hz	0.7(1)
	DK	kHz	103.10(1)	107.88	E*KK	Hz	1.6(2)
	d1	kHz	-2.2929(2)	-2.35	Fab	MHz	-218(6)
	d2	kHz	-0.15145(4)	-0.15	Δ	u Å 2	-0.078
	HJ	mHz	3.1(1)	-0.26
	HJK	mHz	112.7(5)	171.
	HKJ	mHz	-1378.(5)	-1660.
	HK	mHz	3490.	3490.
	h1	mHz	2.9(1)	1.74
	h2	mHz	1.080	1.080
	h3	mHz	0.243	0.243
	LKKJ	mHz	0.074(6)
	[a]

Acknowledgements

This study was supported by Bologna University (RFO funds) and the Italian Space Agency (ASI; 'Life in Space' project, N. 2019-3-U.0). The SMART@SNS Laboratory (http://smart.sns.it) is acknowledged for providing high-performance computing facilities. J.-C.G. thanks the Centre National d'Etudes Spatiales (CNES) and the "Programme National Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP cofunded by CEA and CNES for a grant. V.M.R. acknowledges support from the Comunidad de Madrid through the Atracción de Talento Investigador Modalidad 1 (Doctores con experiencia) Grant (COOL: Cosmic Origins Of Life; 2019-T1/TIC-15379).

Notes and references ‡ Diol 4 can exist in two different stereoisomeric forms, namely E and Z, with the latter being more stable by about 20 kJ mol -1 . 47 The Z isomer possesses two configurations, the preferred (syn, anti) form and the higher energetic (anti, anti) conformer.


F Aimo

S Krämer

M Klanjšek

M Horvatić

C Berthier

H Kikuchi

Spin Configuration in the 1/3 Magnetization Plateau of Azurite Determined by NMR

Keywords: numbers: 75.10.Pq, 75.25.+z, 67.80.dk, 76.60.-k

High magnetic field 63,65 Cu NMR spectra were used to determine the local spin polarization in the 1/3 magnetization plateau of azurite, Cu3(CO3)2(OH)2, which is a model system for the distorted diamond antiferromagnetic spin-1/2 chain. The spin part of the hyperfine field of the Cu2 (dimer) sites is found to be field independent, negative and strongly anisotropic, corresponding to ≈10 % of fully polarized spin in a d-orbital. This is close to the expected configuration of the "quantum" plateau, where a singlet state is stabilized on the dimer. However, the observed non-zero spin polarization points to some triplet admixture, induced by strong asymmetry of the diamond bonds J1 and J3.

The natural mineral azurite, Cu 3 (CO 3 ) 2 (OH) 2 , has been recently recognized [1] as a model system for the frustrated antiferromagnetic Heisenberg spin-1/2 chain of "distorted diamond" geometry defined in Fig. 1. Its most prominent feature is a large plateau in the magnetization curve at 1/3 of the saturation magnetization, which extends from 11 to 30 T when the applied magnetic field (H 0 ) is perpendicular to the chains. Such a "1/3 plateau" is usually associated with a classical collinear up-up-down (uud) type of spin arrangement, or rather to a quantum state which has this classical analogue. For example, a uud state is predicted for spins on a two-dimensional triangular lattice and observed in the Cs 2 CuBr 4 compound [2]. The 1/3 plateau in azurite is proposed to be of fundamentally different, "00u" type, where the dominant J 2 coupling ensures that the two "dimer" spins on the Cu2 sites (see Fig. 1) are in a singlet state, while the third "monomer" (Cu1) spin is completely polarized by the field. As this state is based on the presence of a singlet, it is of pure quantum nature without a classical analogue. Azurite is a good candidate to be the first system exhibiting such a 1/3 plateau state, but a direct experimental evidence is still missing. The point is that both types of plateaus are predicted for a diamond chain, the 00u type driven by dominant J 2 coupling and an analogue of the uud state in presence of dominant J 1 and J 3 [3,4]. The azurite is close to the phase boundary between them, and there is a controversy on the J values proposed from the magnetization, specific heat and neutron scattering experiments [1,5,6,7]. The two different plateau types are distinguished by very different local spin polarizations, which can in principle be directly accessed by performing nuclear magnetic resonance (NMR) on the on-site copper 63,65 Cu nuclei. In this letter we present such NMR data which show that in the 1/3 plateau the dimer spins are nearly in the singlet configuration and thus confirm the 00u type of plateau. We find a small non-zero spin polarization of these sites, estimated to approximately 10 % of full polarization, which points to an important asymmetry of J 1 and J 3 couplings. The observed polarization is incompatible with a uud type of plateau, in which the dimer spins are strongly polarized.

In general, the copper NMR spectrum of a single crystal consists of 6 NMR lines per each non-equivalent Cu site, corresponding to three transitions between energy levels of a spin I = 3/2 nucleus for each of the two 63 Cu and 65 Cu isotopes. In the crystallographic structure of azurite, shown in Fig. 1, we recognize two different copper sites in two equivalent chains of different orientation with respect to the arbitrary direction of the applied magnetic field H 0 (α). We also note that the two Cu2 sites of each dimer are expected to be undistinguishable by NMR, which is ensured by the inversion symmetry with respect to the center of each dimer. We thus expect that 63,65 Cu NMR spectrum has 6 × 2 × 2 = 24 NMR lines. Single crystal spectra presented in Fig. 2 contain only 12 lines, meaning that we observe only one of the two Cu sites. We recall that these spectra are taken in the plateau phase, so that they are not affected by the Néel ordering appearing at 1.9 K at field values below the plateau phase [8]. A standard way for the identification of the observed site is to compare the symmetry of the local electric field gradient (EFG) tensor determined by NMR to what is expected from the local symmetry of the four nearest neighboring (NN) oxygen atoms (see Fig. 1). This rather technical procedure, explained in detail in the following paragraph, unambiguously demonstrates that only Cu2, i.e. the dimer site is observed by NMR. Each copper isotope (I = 3/2) generates a triplet of NMR lines whose average frequency reflects the Zeeman coupling to the total "effective" magnetic field H eff , while the line splitting is induced by the "quadrupolar" coupling to the local EFG. The corresponding nuclear spin Hamiltonian,

Cu2 Cu1 O Z H J 1 J 3 J 2 b c* c a H( )
H = γI•H eff + hν Q [3I 2 Z -I(I + 1)+ η(I 2 + + I 2 -)/2]/6
, is uniquely defined by 5 parameters: the EFG tensor described by the quadrupolar coupling ν Q and its asymmetry parameter η, and H eff and its direction (ϑ EFG , ϕ EFG ) with respect to the principal axes (X,Y,Z) of the EFG tensor [START_REF] Abragam | The Principles of Nuclear Magnetism[END_REF]. Knowing the gyromagnetic ratios for the two isotopes, 63 γ and 65 γ, as well as the ratio of their quadrupolar couplings, 63 ν Q / 65 ν Q = 1.0805, these 5 parameters can be fit to provide the observed 6 NMR frequencies (for each chain). The ν Q and η parameters do not depend on the orientation of magnetic field and, in particular, they are common to spectra from two chains shown in Fig. 1. In this experiment we performed in-situ rotation of the crystal around the axis that was close to the crystal a-axis, and have taken several complete spectra at different rotation angles α (see Fig. 1), as shown in Fig. 2. The smallest NMR line widths and thus the most precise fits are obtained when ϑ EFG ∼ = 90 • , which for chain 1 corresponds to α ∼ = 45 • . This particular orientation has therefore been used to determine 63 ν Q = 36.5 MHz and η = 0.085, and the obtained EFG values have been successfully used to produce all the other fits -this time by fitting only three parameters (H eff , ϑ EFG , ϕ EFG ) for each set of 6 NMR frequencies (see Fig. 2). The experimentally obtained ϑ EFG (α) dependence could then be compared to a simple approximate estimate for this quantity, based on the crystallographic structure. We know that for an ideal tetragonal coordination a pure d X 2 -Y 2 orbital pointing towards 4 NN oxygens generates axially symmetric EFG with the strongest principal axis along the Z direction. Therefore, the best estimate for the Z axis is the normal to the plane approximately defined by 4 NN oxygens (see Fig. 1). The direction of H eff is approximated by the direction of the applied field, supposing that the rotation axis is precisely the a-axis of the crystal. These estimates of Z and H eff directions define the angle ϑ Theory , and its rotation dependence ϑ Theory (α) is plotted in Fig. 3 for both Cu sites and both chains, in comparison with the experimental ϑ EFG (α) values. Neglecting small offset due to various approximations, from these data one clearly identifies that the observed NMR signal corresponds to dimer Cu2 sites and is incompatible with the Cu1 sites. The principal information obtained from the fits is the spin part of the hyperfine field,

H spin = H eff -(1 + K orb ) H 0 = A gµ B S ,
induced by the local spin polarization S through the hyperfine coupling tensor A. The orbital (Van Vleck) shift tensor K orb is here a minor correction, because typical A values for a copper spin are as large as A Z ≈ -20 T/µ B , with large anisotropy A Z /A ⊥ ≈ 10. Knowing that the EFG and the hyperfine shift tensors are dominantly determined by the same wave function, we expect that the principal axes of both tensors are approximately the same, so that the H spin vs. ϑ EFG dependence provides a complete information on the local spin polarization. In Fig. 3 we have plotted the rotational dependence of the experimental NMR line shift (H eff -H 0 ) and its extrapolation by a sinusoidal fit, together with an estimate of the orbital shift (K Z orb ≈ 1.3 %, K ⊥ orb ≈ 0.3 %). From these data we estimate the spininduced hyperfine field to be H spin ≈ -2.0 T, with an anisotropy H spin /H ⊥ spin ≈ 11 (where and ⊥ refer to the principal axes of this tensor). This corresponds to about 10 % spin polarization of a typical Cu 2+ d X 2 -Y 2 orbital. The error in these values is estimated to be ≈ 20 %, dominantly from the extrapolation of the angular dependence to H spin [10]. In particular, in Fig. 3 we clearly see that the maximum of the experimental line shift is shifted by 18 • from the expected ϑ EFG = 90 • value. This means that the principal axes of the EFG and the hyperfine tensors are not really parallel, pointing to a departure from the simplified picture of pure d X 2 -Y 2 orbital. Indeed, the electronic density observed by x-ray diffraction suggest significant admixture of other orbitals [START_REF] Belokoneva | [END_REF].

In a true magnetization plateau the magnetization should not vary with the magnetic field. In order to test this most prominent feature of the plateau in azurite, we performed very high field measurements of the copper NMR spectra, in the field range 17-28 T and at 1.4 K, for H 0 applied close to the c-axis. Because the effects of the quadrupolar coupling and the hyperfine shift are en-tangled in the NMR spectra, the shift can only be determined by the complete NMR fits as explained in the third paragraph. The line positions and fits shown in Fig. 4 indeed confirm that H spin is to a high precision field independent in the plateau. For a field variation from 19 to 26 T (i.e. 37 %) the change in the measured |H spin | is found to be (1 ± 1) %, where the precision is limited by our estimate of the orbital shift tensor. This information might be important to constrain the possible effects of Dzyaloshinski-Moria (DM) interaction terms, which may induce some weak field dependence of the spin polarizations. The DM interaction on the dimer bond has been invoked to explain strong anisotropy of the width of the plateau [1]. However, the presence of an inversion center at the center of the dimer precludes such a term, and only DM interaction on J 1 and J 3 exchange paths are possible. Whether or not they can explain the observed anisotropy has not yet been studied theoretically.

Despite considerable efforts, we could not observe the NMR signal from the monomer Cu1 spin. We have tried to find it at very low temperature in order to minimize the spin fluctuations. In this way the longitudinal (T -1 1 ) relaxation rate was reduced, however, the same does not necessarily apply to the transverse (T -1

2 ) relaxation [START_REF] Berthier | Well known example is the relaxation in the superconducting state of high Tc superconductors[END_REF]. To provide the correct total polarization of the 1/3 plateau, the spin polarization at the Cu1 site has to be ≈ 80 %, that is ≈8 times more than at the dimer sites. We can then roughly estimate that the corresponding T -1 2 ratio is of the order of 8 2 = 64. As the dimer T 2 values are in the 10-100 µs range (depending on the orientation), this factor is enough to reduce the Cu1 T 2 below the 687 MHz/T. Lines are linear fits to the observed field dependence, and symbols fits to this linear interpolation at 19, 22.5 and 26 T, which confirm that the spin polarization of the dimer Cu2 site is magnetic field independent. Color/symbol code is the same as in Fig. 2.

experimental dead time for the observation of an NMR signal. This provides a reasonable explanation why Cu1 spin could not be observed, but also an important hint on the system: the longitudinal spin fluctuations (effective in T 2 relaxation) are probably not gapped. Finally, we remark that here we have only considered the standard on-site hyperfine coupling to the copper spin, and not the transferred hyperfine coupling which could in principle couple the observed nuclear spin at the dimer site to the neighboring (strongly polarized) monomer spin. This latter mechanism typically relies on almost negligible admixture (∼1 %) of the on-site s-wave orbital in the extended Wannier wave function belonging to the neighboring electronic spin. While the on-site spin polarization induced in this way is negligible (∼1 %), very high hyperfine coupling of an s-wave orbital (∼200 T/µ B ) can in principle provide significant transferred hyperfine field. However, this field is positive and isotropic, in obvious contradiction to what is observed in azurite. Negative and strongly anisotropic H spin necessarily implies significant on-site spin polarization of the Cu orbital.

To provide a simple discussion for the observed polarization of dimer spins, we note that the approximate wave functions proposed for the 1/3 plateau of the diamond chain [3] can be generalized to represent an arbitrary mixture of the three single-spin-flip states, Ψ(∓, β) = cos β [ |↓↑↑ ∓ |↑↓↑ ]/ √ 2sin β |↑↑↓ , which by construction has correct total and local spin polarization of µ B and µ B ×(sin 2 β, sin 2 β, cos 2 βsin 2 β), respectively. In particular, there is equal spin polarization on the two dimer spins. In this notation the two reference plateau states [3] are 00u = Ψ(-, 0) and uud = Ψ(+, arccos(1/ √ 3)), where the sign difference corresponds to the different symmetry. (Note that at least two unit cells should be taken into account to properly represent all symmetries of the system.) Using Ψ(∓, β) as a trial function to minimize the energy by optimizing β, one can easily see that the pure singlet 00u (β = 0) state is obtained only for the symmetric diamond couplings J 1 = J 3 , while deviation from this case necessarily leads to some admixture of the triplet, meaning some non-zero polarization of the dimer spins (β = 0). Observed polarization on the dimer site means that this admixture is significant, sin β ≈ √ 0.1 ≈ 0.3 [START_REF]This corresponds closely to the state Ψ(-, arcsin(1/3)) which was identified[END_REF], and thus should correspond to an important asymmetry of couplings. A correct estimate of the corresponding J coupling values should rely on numerical solutions of the spin Hamiltonian, relating the observed spin polarization to the corresponding constraint on the J couplings, say the J 3 /J 1 vs. J 2 /J 1 dependence [START_REF]One compatible set of values is[END_REF]. Two available points predicting the correct local spin polarization, J 3 /J 1 = -0.5 [START_REF]This corresponds closely to the state Ψ(-, arcsin(1/3)) which was identified[END_REF] and +0.4 [START_REF]One compatible set of values is[END_REF], show that from NMR we cannot directly conclude if one of the couplings is ferromagnetic or not. However, if NMR data is combined with other constrains, as the width of the plateau from the magnetization data [1] and the energy of excitations from the neutron scat-tering data [6], one should clearly define the couplings, or indicate whether the diamond chain model is too simple to describe azurite. Here we recall a possible influence of inter chain couplings.

In conclusion, by copper NMR in the 1/3 magnetization plateau of azurite we have determined the local spin polarization of the dimer spins to be ≈ 0.1 µ B . This provides the first direct evidence for a "quantum" type of a 1/3 plateau having no classical analogue, which consists of dimers in a singlet state and fully polarized monomers. The deviation from ideal zero polarization of the dimer implies important asymmetry of the diamond couplings, J 1 = J 3 , and provides a strong constraint for the determination of their values.

FIG. 1 :

 1 FIG. 1: (color online) Diamond chains formed by the exchange interactions, Jij Si •Sj , between S = 1/2 spins of Cu 2+ ions in the crystal structure of azurite. There are two equivalent but differently oriented chains, related by the ac-plane of mirror symmetry. Chains contain "monomer" spins on the Cu1 sites coupled by J1 and J3 interactions to each spin of the dimer formed by the two Cu2 sites, mutually coupled by J2. There is an inversion symmetry on each Cu1 site and at the center of each dimer. For each Cu site 4 nearest neighboring oxygen atoms (connected by thin lines) define approximately the plane of the local symmetry of the wave functions and of the corresponding EFG tensor. Dotted line vectors define the rotation angle α, and the angle ϑ between the magnetic field and the Z principal axis of the EFG tensor. C and H atoms are not shown.

  FIG. 2: (color online) Rotation dependence of copper NMR spectra of azurite at T = 1.5 K. Magnetic field of 15.0 T is applied perpendicular to the rotation axis that was close to the crystal a-axis. (a) NMR spectrum taken at α = 85 • and the corresponding fit (vertical lines), as explained in the text. (b) Angular dependence of the observed line positions (crosses, with lines to guide the eye) and the corresponding fits (open symbols: circles for the central transitions and squares and diamonds for the satellites). Color code is given in the figure: dark (light) grey lines for the chain 1 (2) and solid (dashed) lines for the 63(65) Cu isotope. Vertical dotted line denotes H0 || ac-plane orientation, where both chains are identical.

FIG. 3 :

 3 FIG.3:(color online) Left scale (black lines and symbols): rotation dependence of the NMR shift obtained from the highest frequency 65 Cu NMR line (open circles), the fit to these data (solid line) and an estimate of the orbital shift (dashed line). These results provide an estimate of the field H spin ≈ -2.0 T and its anisotropy H spin /H ⊥ spin ≈ 11, which corresponds to 10 % spin polarization in a d-orbital. Right scale (color / grey-scale lines and symbols): ϑEF G deduced by fitting the line positions shown in Fig.2(b) (solid squares and diamonds), following closely the predictions for the dimer Cu2 site (solid lines) and not the Cu1 site (short-dashed lines), as explained in the text. Experimental ϑEF G(α) dependence (diamonds) is used to define the upper horizontal scale.

FIG. 4 :

 4 FIG.4:(color online) Magnetic field dependence of the63,65 Cu NMR line positions in azurite at 1.4 K, for the field orientation close to the c-axis. The spectra were taken at constant frequency by sweeping the field, and results plotted in the reduced frequency scale to eliminate the dominant field dependence γ H0, with γ = ( 63 γ + 65 γ)/2 = 11.687 MHz/T. Lines are linear fits to the observed field dependence, and symbols fits to this linear interpolation at 19, 22.5 and 26 T, which confirm that the spin polarization of the dimer Cu2 site is magnetic field independent. Color/symbol code is the same as in Fig.2.

We acknowledge fruitful discussions with B. Grenier, A. Honecker, O. Janson, F. Mila, H. Ohta, T. Sakai and T. Tonegawa. Part of this work has been supported by the French ANR project NEMSICOM and by European Commission through the Transnational Access Specific Support Action (Contract No. RITA-CT-2003-505474) and the EuroMagNET network (Contract No. RII3-CT-2004-506239).


Yayu Huang

Davi Savietto

David Labatut

Sébastien Pujol

Julien Breda

Sylvie Combes

Thierry Gidenne

Laurence Fortun-Lamothe

Location of burrows in housing: effects on their occupation rate and health of growing rabbits

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