Clémence Gruget

Jeff Coleman

Shyam Oscar D Bello

Shyam S Krishnakumar

Eric Perez

James E Rothman

Frédéric Pincet
email: pincet@lps.ens.fr

Stephen H Donaldson
email: steve.donaldson@phys.ens.fr

Frederic Pincet

Rearrangements under confinement lead to increased binding energy of Synaptotagmin-1 with anionic membranes in Mg 2+ and Ca 2+

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Introduction

Upon arrival of an action potential at the neuronal synapse, calcium ions (Ca 2+ ) enter the cytosol of the neuron, triggering the soluble N-ethylmaleimide-sensitive factor activating protein receptor (SNARE) proteins to fully zipper, leading to fusion of pre-docked vesicles containing neurotransmitters [START_REF] Chapman | Synaptotagmin: A Ca2+ sensor that triggers exocytosis?[END_REF][START_REF] Südhof | Membrane fusion: Grappling with SNARE and SM proteins[END_REF]. Synaptotagmin-1 (Syt1), a synaptic vesicle associated protein, has been identified as the principal Ca 2+ sensor that activates SNAREs following Ca 2+ influx [START_REF] Chapman | Synaptotagmin: A Ca2+ sensor that triggers exocytosis?[END_REF][START_REF] Brose | Synaptotagmin: a calcium sensor on the synaptic vesicle surface[END_REF][START_REF] Martens | How Synaptotagmin Promotes Membrane Fusion[END_REF]. Syt1 has two Ca 2+ binding C2 domains (C2A and C2B), connected by a 9-residue flexible linker domain, and a 61-residue linker domain between the transmembrane domain and the tandem C2 domains.

Syt1 is known to interact with the plasma membrane both before and after Ca 2+ binding. A polylysine stretch in C2B has been shown to interact with anionic lipids in the absence of Ca 2+ [START_REF] Bai | PIP2 increases the speed of response of synaptotagmin and steers its membrane-penetration activity toward the plasma membrane[END_REF][START_REF] Van Bogaart G Den | Phosphatidylinositol 4,5bisphosphate increases Ca2+ affinity of synaptotagmin-1 by 40-fold[END_REF][START_REF] Kuo | The Calcium-Dependent and Calcium-Independent Membrane Binding of Synaptotagmin 1: Two Modes of C2B Binding[END_REF][START_REF] Araç | Close membrane-membrane proximity induced by Ca2+-dependent multivalent binding of synaptotagmin-1 to phospholipids[END_REF]. Upon Ca 2+ influx, three Ca 2+ bind in C2A and two Ca 2+ bind in C2B. Ca 2+ binding to the anionic pocket effectively neutralizes electrostatic repulsion between the binding site and the target membrane [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF], which allows non-polar residues nearby the Ca 2+ binding sites (also referred to as hydrophobic loops) to insert into the membrane [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF][START_REF] Kuo | Phosphatidylinositol 4,5-bisphosphate alters synaptotagmin 1 membrane docking and drives opposing bilayers closer together[END_REF][START_REF] Hui | Ca2+ Triggered Simultaneous Membrane Penetration of the Tandem C2-Domains of Synaptotagmin I[END_REF][START_REF] Herrick | Position of synaptotagmin I at the membrane interface: Cooperative interactions of tandem C2 domains[END_REF], serving as the power stroke to activate fusion.

The precise biochemical and biophysical mechanisms for the Syt1 Ca 2+ -trigger remain unclear. While mutating the C2A Ca 2+ binding site causes a significant decrease to evoked neurotransmitter release in vivo, an analogous mutation in C2B effectively abolishes evoked release [START_REF] Mackler | The C(2)B Ca(2+)-binding motif of synaptotagmin is required for synaptic transmission in vivo[END_REF][START_REF] Paddock | Membrane Penetration by Synaptotagmin Is Required for Coupling Calcium Binding to Vesicle Fusion In Vivo[END_REF]. Several studies illustrate that Syt1 can oligomerize and modulate bending of the target membrane, and that C2B drives these processes [START_REF] Martens | How Synaptotagmin Promotes Membrane Fusion[END_REF][START_REF] Hui | Synaptotagmin-Mediated Bending of the Target Membrane Is a Critical Step in Ca 2 + -Regulated Fusion[END_REF][START_REF] Wang | Calcium sensitive ring-like oligomers formed by synaptotagmin[END_REF]. Syt1 is therefore thought to function at least partially by performing bending work on the target membrane, providing a highly curved membrane to ease the high energy barrier associated with membrane fusion. Syt1 may also act as a clamp in the absence of Ca 2+ , by forming oligomerized structures which keep the SNAREs away from the vesicle-membrane contact zone [START_REF] Wang | Calcium sensitive ring-like oligomers formed by synaptotagmin[END_REF][START_REF] Zanetti | Ring-like oligomers of Synaptotagmins and related C2 domain proteins[END_REF][START_REF] Wang | Circular oligomerization is an intrinsic property of synaptotagmin[END_REF][START_REF] Rothman | Hypothesis: Buttressed Rings Assemble, Clamp, and Release SNAREpins for Synaptic Transmission[END_REF].

The kinetics and assembly of soluble C2AB (i.e., Syt1 without the transmembrane domain and linker domain, containing only the soluble C2A and C2B domains) with anionic membranes have been assessed using stopped-flow fluorescence resonance energy transfer (FRET) and microscale thermophoresis (MST) measurements, revealing strong binding between soluble C2AB and membranes containing phosphatidylserine (PS) and phosphatidylinosotinol (PIP2) in the absence of Ca 2+ , attributed to the polybasic patch binding to PIP2/PS [START_REF] Van Bogaart G Den | Phosphatidylinositol 4,5bisphosphate increases Ca2+ affinity of synaptotagmin-1 by 40-fold[END_REF][START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF]. The presence of PIP2 leads to an increase in the Ca 2+ affinity of Syt1, possibly due to conformational changes induced by PIP2 binding [START_REF] Van Bogaart G Den | Phosphatidylinositol 4,5bisphosphate increases Ca2+ affinity of synaptotagmin-1 by 40-fold[END_REF]. Ca 2+ binding by Syt1 also leads to a large decrease in the off-rate of Syt1 from membranes [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF], presumably due to insertion of the non-polar residues in the C2AB Ca 2+ binding loops. A recent paper measured energetics of single-molecule C2AB interactions with optical tweezers in the presence of Ca 2+ [START_REF] Ma | Single-molecule force spectroscopy of protein-membrane interactions[END_REF], but reported a lack of binding in the absence of Ca 2+ . A previous single-molecule AFM study attached Syt1 to an AFM tip and performed adhesion force measurements at PC/PS membranes [START_REF] Takahashi | Interaction of synaptotagmin with lipid bilayers, analyzed by single-molecule force spectroscopy[END_REF], but no binding energetics or distance-dependent Syt1-membrane interactions were reported. Additionally, while Mg 2+ is known to bind in the Syt1 Ca 2+binding sites in the absence of Ca 2+ [START_REF] Zhou | Architecture of the synaptotagmin-SNARE machinery for neuronal exocytosis[END_REF] and has subtle effects on Syt1-membrane interactions [START_REF] Schiavo | Calcium-dependent switching of the specificity of phosphoinositide binding to synaptotagmin[END_REF][START_REF] Lee | Dynamic Ca2+ Dependent Stimulation of Vesicle Fusion by Membrane-Anchored Synaptotagmin 1[END_REF], no studies have directly compared the binding energies in Mg 2+ and Ca 2+ . Therefore, there remains a lack of direct probes of the long-range and short-range behaviors and energetics of divalent ion-dependent Syt1 binding at anionic membranes, especially with Syt1 embedded in a membrane, as occurs under physiological conditions. We directly measured the interaction energy between a lipid membrane decorated with cytoplasmic Syt1 and an anionic membrane composed of PC/PS/PIP2 using a surface forces apparatus (SFA), as shown schematically in Figure 1. To isolate the Syt1-target membrane interaction, which has been shown to be more productive for fusion [START_REF] Bai | PIP2 increases the speed of response of synaptotagmin and steers its membrane-penetration activity toward the plasma membrane[END_REF][START_REF] Lee | Dynamic Ca2+ Dependent Stimulation of Vesicle Fusion by Membrane-Anchored Synaptotagmin 1[END_REF][START_REF] Stein | Synaptotagmin activates membrane fusion through a Ca2+-dependent trans interaction with phospholipids[END_REF][START_REF] Park | Controlling synaptotagmin activity by electrostatic screening[END_REF], we have included only PC and Syt1 in the protein-containing membrane. By using EGTA, Mg 2+ , or Ca 2+ in the buffer, the effects of divalent ions are elucidated during Syt1 confinement, docking, and unbinding at an anionic membrane surface, including binding energetics, kinetics, and molecular rearrangements.

Materials and Methods

Protein expression and purification

The DNA construct used in this study was generated by cloning the cytoplasmic domain (residues 83 to 421) of rat synaptotagmin-1 into pGEX6p-1 (GE Healthcare, Marlborough, MA) using restriction sites XhoI and NotI. A 12x histidine residue tag was added upstream (N-terminal of the protein) using BamHI and XhoI. Two residues, C277A and E269C, were mutated using a QuikChange mutagenesis kit (Agilent Technologies, Santa Clara, CA) to allow for subsequent fluorescent labelling. The construct was transformed and grown in Escherichia coli BL21(DE3) to an OD 600 ~0.8 and the expression was induced with 0.5 mM isopropyl b-D-1thiogalactopyranoside (IPTG). The cells were harvested after 4 hr at 37˚C and suspended in lysis buffer (25 mM HEPES, pH 7.4, 500 mM KCl, 1 mM MgCl2, 1 mM CaCl2, 15mM Imidazole, 0.4 mM TCEP, 10% glycerol, 1% Triton X-100, protease inhibitors). The sample was lysed using a cell disrupter, and the lysate was supplemented with 0.1% polyethylimine before centrifugation (35,000 rpm for 30 min). The supernatant was loaded onto Ni-NTA (Qiagen, Valencia, CA) beads (3-4 hours or overnight at 4˚C) with 10 ul of Benzonase (2000 units). The beads were washed with 20mL of lysis buffer with 0.1% Triton X-100, then re-suspended in 5ml of lysis buffer supplemented with 10 μg/mL of DNAse I, 10 μg/mL of RNAseA and 10μl of Benzonase, and incubated at room temperature for 1 hour. Subsequently, the beads were rinsed quickly with 10 mL of high salt buffer (25 mM HEPES, pH 7.4, 1M KCl, 1 mM MgCl2, 1 mM CaCl2, 15mM Imidazole, 0.4 mM TCEP, 10% glycerol) to remove nucleotide contamination, and washed several times with 25 mM HEPES, pH 7.4, 500 mM KCl, 50mM Imidazole, 1 mM MgCl2, 1 mM CaCl2, 0.4 mM TCEP, 10% glycerol.

The protein was eluted off the nickel beads in 25 mM HEPES, pH 7.4, 400 mM KCl, 500 mM Imidazole, 0.5 mM CaCl2, 0.4 mM TCEP, 10% glycerol. The GST tag was cleaved overnight at 4°C using Prescission protease, and then removed with a 1 hour room temperature incubation in Glutathione-Sepharose (Thermo Fisher Scientific, Grant Island, NY). The protein was then run on a size exclusion chromatography column (Superdex 75 16/60 High load) equilibrated with 25 mM HEPES, pH 7.4, 150 mM KCl, 0.4 mM TCEP and further purified by anionic exchange (Mono-S) chromatography. All chromatography was carried out with AKTA (GE Healthcare, Marlborough, MA). The protein concentration was determined with a Bradford assay using BSA as a standard.

The 260 nm/280 nm ratios were measured to check nucleotide contamination. The protein was flash frozen and stored at -80˚C with 20% glycerol.

Surface forces measurements

The force-distance measurements were done with a home-built SFA similar to the original Israelachvili design [START_REF] Israelachvili | Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0-100 nm[END_REF]. Briefly, back-silvered mica surfaces were glued on cylindrical glass disks (R ~ 2 cm) with UV-cured glue (NOA81, Norland Optics), then a monolayer of 1,2-dimyristoylsn-glycero-3-phosphoethanolamine (DMPE) was deposited on both surfaces at an area/molecule of 0.4 nm 2 using a home-built Langmuir-Blodgett trough [START_REF] Perez | A simple, Cheap, Clean, Reliable, Linear, Sensitive, Low-Drift Transducer for Surface Pressure[END_REF]. DMPE binds strongly to mica, creating a stable inner monolayer on both surfaces. Next, on one surface we deposited an outer layer of 95% 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) and 5% 1,2-dioleoyl-sn-

glycero-3-[(N-(5-amino-1carboxypentyl)iminodiacetic acid)succinyl] nickel (DGS-NTA-Ni) with
an area/molecule of 0.4 nm 2 , and on the other surface an outer layer of 80% POPC, 15% 1,2dioleoyl-sn-glycero-3-phospho-L-serine (DOPS), and 5% L-α-phosphatidylinositol-4,5bisphosphate (PIP2) was deposited at 0.5 nm 2 . The POPC/DOPS/PIP2 membrane was kept immersed in 25 mM HEPES, 150 mM KCl, with 0.5 mM of EGTA, and in certain cases, 0.5 mM of free Ca 2+ or 1.0 mM of free Mg 2+ buffer (calculated using Maxchelator, maxchelator.stanford.edu).

The 95% POPC, 5% DGS-NTA-Ni membrane was immersed in a small vial of the same buffer (~3 mL volume) into which ~5 uL of ~2 mg/mL 12xHis-Syt1 was injected and mixed well via pipet. After 1 hr of protein immersion, the small vial was transferred twice into clean buffer solution (~200 mL volume) to remove unbound protein. Finally both surfaces were carefully transferred under buffer into the SFA chamber. One surface was mounted on a spring with the other on a stiff mount in a crossed-cylinder geometry. The distance was measured via multiple beam interferometry and the force by spring deflection. For each condition we measured at least 3 independent experimental setups, with at least 8 independent contact locations and at least 2 different protein batches to demonstrate reproducibility. Error bars represent standard errors over the independent contact locations.

Results

Effect of anionic lipids on Syt1 interactions with lipid membranes

We measured the forces during approach and separation of a Syt1-decorated membrane and an anionic membrane. In the Syt1-membrane, accounting for 5% Ni-NTA and 2-3 histidines binding per Ni-NTA gives a maximum surface density of ~2.5x10 16 molecules of Syt1 per m 2 corresponding to ~40 nm 2 /Syt1 molecule or ~1.3 Syt1 copies per every 100 lipids in the outer leaflet. As such, the experiment simulates a synaptic vesicle approaching the anionic plasma membrane, albeit in the absence of SNAREs and other regulatory proteins, in order to isolate the pure Syt1-membrane interactions.

An SFA measurement consists of approaching and separating the surfaces while measuring the distance interferometrically (distance resolution ~1 Å) and measuring the corresponding forces with a cantilever spring (force resolution ~100 nN). In the standard procedure, we apply a 1 hr contact time (tc = 1 hr) between the end of the approach and the beginning of the separation, with shorter tc in specific cases. The surfaces are approached and separated quasi-statically, such that the distance and force are measured simultaneously every ~10 s. The average speed of approach/separation outside of the interaction zone (i.e. D > 50 nm) is ~1 nm/sec, which captures the quasi-equilibrium interaction force profile between Syt1 and the opposing membrane. The surfaces were initially immersed in 150 mM KCl, 25 mM HEPES buffer, and 0.5 mM EGTA. To first examine the effects of anionic lipids on the binding of Syt1 to membranes, we adjusted the composition of the opposing "target" membrane. We measured the force-distance interactions of Syt1 approaching an anionic (PS/PIP2) membrane surface, and compared these interactions with Syt1 approaching a neutral (100% POPC) membrane surface (Fig. 2). By convention, repulsive forces have a positive sign while attractive forces have a negative sign. Figure 2. Interaction force vs. distance measurement (approach and separation) by SFA for Syt1 membrane with a 100% POPC membrane (blue circles), and a Syt1-coated membrane with an 80% POPC, 15% DOPS, and 5% PIP2 membrane (red squares). Filled symbols are during approach while empty symbols are during separation. In the Syt1-membrane cartoons (upper right), the Ca 2+ binding loops of C2A and C2B are indicated by the yellow sites, and the polybasic patch on C2B is indicated by the red site.

During the approach of Syt1, a significant force was first measured at D ~ 25-30 nm, which roughly corresponds to the fully stretched Syt1 [START_REF] Lin | Control of membrane gaps by synaptotagmin-Ca2+ measured with a novel membrane distance ruler[END_REF]. Electrostatic interactions at this range are vanishingly small (Debye length ~0.8 nm), so this interaction is attributed to a steric interaction between the Syt1 chains and the anionic membrane surface. As the surfaces are pushed further together (a primitive mimic of the SNARE/Munc13 complex which brings the vesicle and plasma membrane together from large distances), an exponential repulsion is measured. These observations indicate that in spite of the structured C2A and C2B domains, the 61-residue linker between the transmembrane domain and C2A combined with the 9-residue linker between C2A and C2B are unstructured when not under confinement, and the isolated protein behaves similar to a random coil. We therefore apply the polymer mushroom model, which has also been applied to SNARE proteins [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF],

𝐹 𝑅 ⁄ 72𝜋Γ 𝑘 𝑇𝑒 √ / , ( 1
)
where F is the measured force, R ~ 2 cm is the radius of the surfaces, Γ is the surface density of Syt1, kB is Boltzmann's constant, T = 298 K is the temperature, D is the distance between the membranes (see Fig. 1), and Rg is the radius of gyration of Syt1. By fitting this equation to the measured F/R vs. D curves, we measure Γ and Rg.

The approach curves (filled points, Fig. 2) are nearly equivalent for both the PC membranes and the anionic membranes, revealing that the long-range interactions between Syt1 and membranes do not depend on the membrane charge. These repulsive forces follow the mushroom model closely, and the measurement reveals a surface coverage Γ ~1x10 16 molecules/m 2 and radius of gyration Rg ~6±0.5 nm for both conditions (Fig. 2, black curve). However, upon separating the two surfaces, drastic differences are observed between the PC lipids and anionic lipid membranes.

For the PC membrane, a small hysteresis is measured (compared to the approach curve), but no adhesion is observed. This suggests that the Syt1 molecules are in a slightly compressed mode upon separation, but they do not bind specifically to the PC lipids. Conversely, for the anionic membrane, a strong adhesion force Fad is measured, as the force at which the spring experiences an instability and a so-called jump-out of contact is observed (indicated by the "jump-out" arrow in Fig. 2). By applying Derjaguin's Approximation, which is valid for R >> D and if the interactions decrease at least as 1/D 2 (and when the surfaces are not deformed or flattened), we find the surface energy per unit area, W, as

𝑊 . ( 2
)
The normalized adhesion force is measured as Fad/R = -1.6 mN/m, corresponding to an adhesion energy Wad = -0.25 mJ/m 2 . Then, using this adhesion energy, one can find the energy per molecule of Syt1, by

𝐸 𝑊/𝛤 . (3)
As such, we measure the energy per molecule as a function of distance between the membranes, allowing for distance-dependent probing of the energetics of confinement and binding of Syt1. In the case of Syt1 binding to the anionic lipid membrane in 0.5 mM EGTA, Wad = -0.25 mJ/m 2 and Γ = 1.3x10 16 molecules/m 2 give a binding energy ESyt1~5 kBT. These values were taken from the force-distance measurement presented in Fig. 2. Calculating the average and standard error over a representative sample of 8 independent measurements gives ESyt1 = 5.8±0.9 kBT in EGTA. Note that the sign of the binding energy is reported as positive by convention as the magnitude of the adhesion (i.e., negative) force and energy.

Effects of divalent cations Mg 2+ and Ca 2+ on Syt1 interactions with lipid membranes

Next, we measured energy vs. distance curves separately in 0.5 mM EGTA, 0.5 mM free Mg 2+ , and 0.5 mM free Ca 2+ , to examine the effects of Mg 2+ and Ca 2+ on the interaction between Syt1 and the anionic membrane (Fig. 3). As Syt1 approaches the membrane, an exponential repulsive force profile was observed again (Fig. 3A,B), which closely follows the mushroom model with Rg = 6±0.5 nm in all 3 cases (Fig. 3B), indicating that the divalent ions do not have a significant effect on the extended structure before Syt1 contacts the membrane. The surfaces are driven together until F/R ~5 mN/m, where the distance remaining between the bilayers is D = 5.2±0.3 nm (i.e., the confined Syt1 thickness), a similar value to thicknesses previously measured for Syt1

bridging between liposomes [START_REF] Araç | Close membrane-membrane proximity induced by Ca2+-dependent multivalent binding of synaptotagmin-1 to phospholipids[END_REF][START_REF] Kuo | Phosphatidylinositol 4,5-bisphosphate alters synaptotagmin 1 membrane docking and drives opposing bilayers closer together[END_REF][START_REF] Lin | Control of membrane gaps by synaptotagmin-Ca2+ measured with a novel membrane distance ruler[END_REF][START_REF] Seven | Prevalent mechanism of membrane bridging by synaptotagmin-1[END_REF]. As observed by the repulsive interactions (Fig. 3A,B), confining Syt1 to this level has an energetic cost of ~15-20 kBT. This energy might be overcome by binding to SNAREs or Munc13 in vivo, or if Syt1 localizes away from the center of the contact zone of the highly curved vesicle, then this barrier could be significantly lower. The binding energies at short contact times, tc = 0 min, were also measured, as shown in Figure 3C. These results show the same trend as the binding energies for tc = 60 min, although the trend is less pronounced. The experimental subset performed in 0.5 mM free Ca 2+ and 1 mM free Mg 2+ gave ESyt1 = 7.9 ± 0.9 kBT for tc = 0 min, again virtually the same as the value measured in Ca 2+ alone (ESyt1 = 7.2 ± 0.6 kBT). The binding kinetics, which plateau between tc = 30-60 min, are shown in supporting information (Section S1). Syt1 triggers Ca 2+ dependent fusion remarkably quickly in vivo, often in less than a millisecond [START_REF] Südhof | Membrane fusion: Grappling with SNARE and SM proteins[END_REF]. The slow equilibration in the SFA measurement, over the course of ~1 hr, is clearly non-biological. This timescale for equilibration was also observed for SNARE proteins [START_REF] Li | Energetics and dynamics of SNAREpin folding across lipid bilayers[END_REF] and is likely due to the surface geometry, which provides confinement to nm-level distances over hundreds of μm 2 compared to hundreds of nm 2 in the synaptic vesicle. However, the slow kinetics are in fact advantageous in the present measurements because it enables observation of slow molecular rearrangements. This confinement effect may be enhanced by the Syt1 concentration which is around 10x larger than the average Syt1 content of a synaptic vesicle [START_REF] Takamori | Molecular Anatomy of a Trafficking Organelle[END_REF]. Nonetheless, if all Syt1 are bound to the plasma membrane, e.g.

as in the recently proposed ring-shaped oligomers model [START_REF] Rothman | Hypothesis: Buttressed Rings Assemble, Clamp, and Release SNAREpins for Synaptic Transmission[END_REF], their local concentration is increased up to the same order of magnitude as the concentration in the current SFA experiments. In any case, using 10x smaller Syt1 concentration in the present measurements would decrease the magnitudes of Γ and Wad accordingly, making accurate measurement of ESyt1 impossible.

By measuring the distance at the final point of approach and first point of separation at the same applied force, we obtain the distance shift during the contact time, ∆D = Dseparation,initial-Dapproach,final, as shown in detail in the supporting information (Section S2). A negative value for ∆D indicates an inward distance shift (i.e., the surfaces become closer together). For tc = 0 min, ∆D is zero within experimental error. However, for tc = 1 hr, ∆D increases in magnitude from -0.7 nm in EGTA, to -1.2 nm in Mg 2+ , and to -1.8 nm in Ca 2+ , as shown in Figure 3D. Therefore, the binding energies were correlated with ∆D, indicating that relatively slow molecular rearrangements during confinement lead to the increased adhesion, discussed in more detail below.

Discussion

Most previous measurements of Syt1-membrane binding have utilized only the soluble C2AB to measure association with a membrane. Importantly, we include the entire cytoplasmic domain of Syt1 and attach it directly to a membrane, providing a more precise mimic of Syt1 in synaptic vesicles approaching the plasma membrane. Under physiological conditions, the Syt1 interaction with cis PS lipids is known to be screened by ATP, such that only the trans interaction is productive [START_REF] Park | Controlling synaptotagmin activity by electrostatic screening[END_REF]. Therefore, by including only PC in the cis membrane, we measure the Syt1trans interaction under the physiologically-relevant topology.

Previous solution phase measurements indicate that Ca 2+ -independent binding of C2AB to anionic membranes occurs through the C2B polylysine patch, while the C2A plays a small or negligible role in lipid binding [START_REF] Bai | PIP2 increases the speed of response of synaptotagmin and steers its membrane-penetration activity toward the plasma membrane[END_REF][START_REF] Kuo | The Calcium-Dependent and Calcium-Independent Membrane Binding of Synaptotagmin 1: Two Modes of C2B Binding[END_REF][START_REF] Schiavo | Calcium-dependent switching of the specificity of phosphoinositide binding to synaptotagmin[END_REF]. Therefore, the adhesion measured here between Syt1 and the anionic membrane in EGTA, ESyt1 = 5.8 ± 0.9 kBT, likely arises primarily from the binding of the polylysine patch of C2B with the anionic PS and PIP2 lipid headgroups. Several groups have measured association or dissociation constants via isothermal calorimetry or MST, which can be used to calculate a binding free energy. With similar lipid compositions the dissociation constant measurements of Syt1 with membranes are reported to be in the range of ~5-10 kBT in EGTA [START_REF] Van Bogaart G Den | Phosphatidylinositol 4,5bisphosphate increases Ca2+ affinity of synaptotagmin-1 by 40-fold[END_REF][START_REF] Kuo | The Calcium-Dependent and Calcium-Independent Membrane Binding of Synaptotagmin 1: Two Modes of C2B Binding[END_REF].

A recent single molecule study reported no binding between C2AB and an anionic membrane in the absence of Ca 2+ , possibly because the Ca 2+ -independent binding requires participation of multiple molecules [START_REF] Ma | Single-molecule force spectroscopy of protein-membrane interactions[END_REF]. The value reported here (5.8 ± 0.9 kBT) is in the lower range of values reported thus far, which is perhaps expected. By including the entire cytosolic domain and anchoring it to a membrane, we have reduced degrees of freedom compared to the solution phase measurements. Similarly, the extreme confinement due to the close apposition of both membranes reduces the degrees of freedom for Syt1 even further and provides a model closer to the crowded and confined situation in vivo.

A simple screened Coulomb interaction to model the polybasic patch as a cation of valence +4 and the PIP2 as an anion of valence -3 allows for a simplified view of this interaction. As shown in the supporting information (Section S3), this ion-ion interaction is fully attractive but becomes significant compared to the thermal energy only for separations ~1 nm. The contact energy (i.e., adhesion or binding energy) is ~8 kBT. A combined experimental-theoretical study of polylysine binding at anionic membrane surfaces found that each lysine provided about 1.7 kBT to the total binding energy [START_REF] Ben-Tal | Binding of small basic peptides to membranes containing acidic lipids: theoretical models and experimental results[END_REF], such that a 4-lysine stretch as found in C2B should bind with ~7 kBT. These simplified models are in rough agreement with the experimentally measured value, indicating that the C2B-membrane interaction is almost purely accounted for as a charge-charge interaction between the polybasic patch and the anionic lipids.

The increased binding of Syt1 in the presence of Ca 2+ is also expected from previous work [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF][START_REF] Ma | Single-molecule force spectroscopy of protein-membrane interactions[END_REF], which indicates that Syt1 reorients to insert the hydrophobic residues of the Ca 2+ loop into the membrane. A recent optical tweezers study reported a binding energy of ~12 kBT in the presence of 0.5 mM free Ca 2+ for C2AB binding to an anionic membrane [START_REF] Ma | Single-molecule force spectroscopy of protein-membrane interactions[END_REF]. A wide range of dissociation constants, and therefore binding energetics, have been measured by traditional biological assays for soluble C2AB with anionic membranes, with a maximum value of ~25 kBT [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF]. The value measured here in 0.5 mM Ca 2+ , ESyt1 = 17.6 ± 1.4, is again smaller than the traditional assays, although is potentially more representative physiologically because Syt1 is anchored to the membrane.

The increase in the binding energy upon addition of Mg 2+ (compared to in EGTA) has not been previously reported and is potentially due to a weaker interaction of the Ca 2+ binding pocket in the presence of Mg 2+ that results in partial insertion of the hydrophobic loops. The membrane binding energy in Mg 2+ is between the binding energies measured in Ca 2+ and EGTA. One possible explanation for the weaker overall interaction in Mg 2+ compared to Ca 2+ is that partial hydrophobic loop insertion occurs in concert with binding of the polybasic patch. The other possibility is that the polybasic patch is no longer bound in the presence of Mg 2+ and the binding energy originates entirely from the partial loop insertion. These possibilities are presented schematically in Figure 4 and discussed further directly below. We cannot distinguish between these in the current measurements and more detailed structural measurements could help elucidate the mechanism.

Since we obtained similar binding energies for experiments in Ca 2+ and both Ca 2+ and Mg 2+

simultaneously, it appears that the Mg 2+ occupies the Ca 2+ sites in the absence of Ca 2+ , but Ca 2+ outcompetes Mg 2+ when they are both present, such that Syt1 reaches the full Ca 2+ -dependent binding energy.

Due to the geometry in the SFA, the first Syt1-membrane interaction is through the C2B polybasic patch which orients the C2B and provides a similar initial binding conformation in all 3 conditions, as shown in Figure 4. The measurements of ∆D indicate that Syt1 molecules slowly rearrange to find their final conformation. Even for tc=0 min, the surfaces remain at ~5 nm level confinement for ~5 minutes, allowing some small fraction of molecules to change conformation and leading to the observed adhesion increase for the tc=0 min case over the 3 conditions. In EGTA, the small increase in binding energy over tc=1 hr combined with the small value for ∆D implicate non-specific rearrangements. In Ca 2+ , the C2B hydrophobic loops prefer an upright orientation during loop insertion, allowing C2A to rotate and align parallel with the C2B during the equilibration to insert its hydrophobic loops in the membrane, leading to the measured large increase in binding and more dramatic rearrangements of ∆D = -1.8 nm in Ca 2+ . With Mg 2+ , the C2B again contacts the membrane first via the C2B polybasic patch, with the additional possibility of partial loop insertion. Rotation during the contact time results in a similar side-by-side configuration of C2AB, but due to only partial hydrophobic loop insertion, the distance shift is limited to ∆D = -1.2 nm and the binding energy increase is modest. With a physiological concentration of ~1 mM Mg 2+ , the measurements in Mg 2+ potentially reveal mechanisms of Syt1 action. Several previous studies have suggested that the presence of Mg 2+ is important for the Ca 2+ sensitivity of Syt1 [START_REF] Zhou | Architecture of the synaptotagmin-SNARE machinery for neuronal exocytosis[END_REF][START_REF] Schiavo | Calcium-dependent switching of the specificity of phosphoinositide binding to synaptotagmin[END_REF][START_REF] Lee | Dynamic Ca2+ Dependent Stimulation of Vesicle Fusion by Membrane-Anchored Synaptotagmin 1[END_REF]. Rearrangements of potential physiological importance measured here include the transition from polybasic patch binding, to partial loop insertion in Mg 2+ , to full loop insertion in Ca 2+ (Fig. 4). Additionally, if C2A is held adjacent to the cis membrane by SNAREs or other conformational factors, the rearrangement from C2B polybasic binding/partial loop insertion in Mg 2+ to full loop insertion in Ca 2+ may be physiologically relevant. These rearrangements are not necessarily sequential and might be multimodal such that during loop insertion, the polybasic patch interaction persists to some degree [START_REF] Pérez-Lara | PtdInsP2 and PtdSer cooperate to trap synaptotagmin-1 to the plasma membrane in the presence of calcium[END_REF]. The physiological role of these Mg 2+ effects merit further exploration in a functional context.

While these measurements are consistent with the ring model [START_REF] Wang | Calcium sensitive ring-like oligomers formed by synaptotagmin[END_REF][START_REF] Wang | Circular oligomerization is an intrinsic property of synaptotagmin[END_REF][START_REF] Rothman | Hypothesis: Buttressed Rings Assemble, Clamp, and Release SNAREpins for Synaptic Transmission[END_REF], no direct evidence of Syt1 rings or oligomers was found. It is difficult to envision how oligomerization would impact the measured results in SFA. While the slow rearrangements might reflect oligomerization, they also simply might be an effect of the large scale confinement over many μm 2 . Similarly, the measured Rg is ~2-3x larger than the expected Rg for Syt1 [START_REF] Guan | A synaptotagmin suppressor screen indicates SNARE binding controls the timing and Ca2+cooperativity of vesicle fusion[END_REF][START_REF] Liu | Linker mutations reveal the complexity of synaptotagmin 1 action during synaptic transmission[END_REF], which could be a subtle signature of oligomerization, but also might simply result from differences between measuring Rg for the full cytoplasmic domain between 2 surfaces (as done here) vs. measuring Rg of C2AB in solution. The transition from the initial contact state in Mg 2+ to the equilibrated loop insertion in Ca 2+ is consistent with the ring model.

We directly measured Syt1-membrane binding energies and interaction mechanisms, along with confinement and molecular rearrangement details of Syt1-membrane interactions. Future measurements focusing on Syt1 mutants and more realistic lipid compositions will help to precisely assess the roles of different Syt1 binding sites. While the current results suggest that the hydrophobic loops play a role in the presence of Mg 2+ , additional structural and biochemical work is required to elucidate the precise nature of this interaction. The inclusion of SNAREs and observation of Syt1 loop-insertion, in correlation with measurements of distance-dependent binding energetics, could help elucidate the precise mechanistic details of Syt1 action in fast Ca 2+triggered synaptic transmission.

Figure 1 .

 1 Figure 1. A schematic of the SFA experiment. Syt1 coated membrane (top) interacts with an anionic membrane (bottom), consisting of 80mol% POPC, 15mol% DOPS, and 5mol% PIP2. The Ca 2+ binding loops of C2A and C2B are indicated by the yellow sites, and the polybasic patch on C2B is indicated by the red site.

Figure 3 .

 3 Figure 3. Force-distance measurements by SFA in the presence of EGTA, Mg 2+ , and Ca 2+ for (A) approach and separation on normal axes and (B) approach forces only on a semilog plot. (C) Summary of binding interactions at short and long contact times, and (D) summary of the distance shift during the 1 hr contact time for each condition.

Figure

  Figure3A,B shows selected force runs; every individual measurement results in a measurement of Γ, Rg, Wad, and therefore ESyt1. Typically, Γ ranges from about ~5x1015 to ~2x1016 molecules/m 2 , while Rg is in the range 5-8 nm. When the surfaces are separated from each other, divalent ions lead to significant increases to the binding energy (Fig.3A,C). Addition of Ca 2+ is known to cause insertion of the hydrophobic residues near the Ca 2+ binding sites of C2AB, and the binding energy increases to ESyt1 = 17.6 ± 1.4 kBT. Notably, the binding increases in Mg 2+ , from ESyt1 = 5.8 ± 0.9 kBT in EGTA to ESyt1 = 10.0 ± 1.1 kBT in Mg 2+ . This increase in the binding energy implicates interaction of the hydrophobic loops when Syt1 coordinates Mg 2+ , possibly due to a weaker hydrophobic loop interaction compared to Ca 2+ (see discussion for details). A subset of experiments were performed in buffer containing 0.5 mM free Ca 2+ and 1 mM free Mg 2+ over 2

Figure 4 .

 4 Figure 4. Schematic mechanism for the Syt1-membrane rearrangements measured in this study.The Syt1 is anchored to the top membrane. Important residues of Syt1 are labeled as follows: red, C2B polybasic; yellow, C2B Ca 2+ binding; green, C2A Ca 2+ binding; pink, C2A and C2B hydrophobic loops. Membrane contacts are denoted by red circles (polybasic patch), blue circles (partial hydrophobic loop insertion), blue ovals (full hydrophobic loop insertion). Black arrows denote the slow measured rearrangements (over the course of 1 hr), while hypothesized fast (~msec) potential physiological transitions are denoted by the heavy red arrows (labeled "fast"). See text for detailed mechanistic explanation.

ACKNOWLEDGEMENTS

This work was supported by Agence Nationale de la Recherche (ANR) ANR-14-1CHN-0022-01 to JER. CG was supported by the Fondation pour la Recherche Médicale (FRM) FDT20170437276. SHD was supported by LabEX ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.


Sunghun Pothier
email: hugues.pothier@cea.fr

A Levy Park

J Yeyati

P Nygård

Krogstrup

L Tosi

C Metzger

M F Goffman

C Urbina

H Pothier

Sunghun Park

A Levy Yeyati

J Nygård

P Krogstrup

M Y Kupriyanov

E Ilichev

Spin-orbit splitting of Andreev states revealed by microwave spectroscopy

Spin-orbit splitting of Andreev states revealed by microwave spectroscopy

Introduction. The Josephson supercurrent that flows through a weak link between two superconductors is a direct and generic manifestation of the coherence of the many-body superconducting state. The link can be a thin insulating barrier, a small piece of normal metal, a constriction or any other type of coherent conductor, but regardless of its specific nature the supercurrent is a periodic function of the phase difference δ between the electrodes [1]. However, the exact function is determined by the geometry and material properties of the weak link. A unifying microscopic description of the effect has been achieved in terms of the spectrum of discrete quasiparticle states that form at the weak link: the Andreev bound states (ABS) [2][START_REF] Beenakker | Josephson supercurrent through a superconducting quantum point contact shorter than the coherence length[END_REF][START_REF] Furusaki | Dc Josephson effect and Andreev reflection[END_REF][START_REF] Bagwell | Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity[END_REF]. The electrodynamics of an arbitrary Josephson weak link in a circuit is not only governed by the phase difference but depends also on the occupation of these states. Spectroscopy experiments on carbon nanotubes [START_REF] Pillet | Revealing the electronic structure of a carbon nanotube carrying a supercurrent[END_REF], atomic contacts [START_REF] Bretheau | Exciting Andreev pairs in a superconducting atomic contact[END_REF][START_REF] Bretheau | Supercurrent spectroscopy of Andreev states[END_REF][START_REF] Janvier | Coherent manipulation of Andreev states in superconducting atomic contacts[END_REF] and semiconducting nanowires [START_REF] Lee | Spin-resolved Andreev levels and parity crossings in hybrid superconductorsemiconductor nanostructures[END_REF][START_REF] Van Woerkom | Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions[END_REF][START_REF] Hays | Direct microwave measurement of Andreevbound-state dynamics in a proximitized semiconducting nanowire[END_REF] have clearly revealed these fermionic states, each of which can be occupied at most by two quasiparticles. The role of spin in these excitations is a topical issue in the rapidly growing fields of hybrid superconducting devices [START_REF] Michelsen | Manipulation with Andreev states in spin active mesoscopic Josephson junctions[END_REF][START_REF] Franceschi | Hybrid superconductor -quantum dot devices[END_REF][START_REF] Linder | Superconducting spintronics[END_REF] and of topological superconductivity [START_REF] Prada | Measuring Majorana nonlocality and spin structure with a quantum dot[END_REF][START_REF] Zazunov | Josephson effect in multiterminal topological junctions[END_REF][START_REF] Deng | Nonlocality of Majorana modes in hybrid nanowires[END_REF][START_REF] Hart | Controlled finite momentum pairing and spatially varying order parameter in proximitized HgTe quantum wells[END_REF]. It has been predicted that for finite-length weak links the combination of a phase difference, which breaks time-reversal symmetry, and of spinorbit coupling, which breaks spin-rotation symmetry, is enough to lift the spin degeneracy, giving therefore rise to spin-dependent Josephson supercurrents even in the absence of an external magnetic field [START_REF] Chtchelkatchev | Andreev quantum dots for spin manipulation[END_REF][START_REF] Padurariu | Theoretical proposal for superconducting spin qubits[END_REF][START_REF] Béri | Splitting of Andreev levels in a Josephson junction by spinorbit cbezuglioupling[END_REF][START_REF] Cayao | SNS junctions in nanowires with spin-orbit coupling: Role of confinement and helicity on the subgap spectrum[END_REF]. Here we report the first observation of transitions between zerofield spin-split ABS.

ABS and spin-orbit interaction. Andreev bound states are formed from the coherent Andreev reflections that quasiparticles undergo at both ends of a weak link. Quasiparticles acquire a phase at each of these Andreev reflections and while propagating along the weak link of length L. Therefore, the ABS energies depends on δ, on the transmission probabilities for electrons through the weak link and on the ratio λ = L/ξ where ξ is the superconducting coherence length. Assuming ballistic propagation, ξ = v F /∆ is given in terms of the velocity v F of quasiparticles at the Fermi level within the weak link and of the energy gap ∆ of the superconducting electrodes. In a short junction, defined by L ξ, each conduction channel of the weak link, with transmission probability τ, gives rise to a single spin-degenerate Andreev level at energy E A = ∆ 1 -τ sin 2 (δ/2) [START_REF] Beenakker | Josephson supercurrent through a superconducting quantum point contact shorter than the coherence length[END_REF][START_REF] Furusaki | Dc Josephson effect and Andreev reflection[END_REF][START_REF] Bagwell | Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity[END_REF]. This simple limit has been probed in experiments on aluminum superconducting atomic contacts, using three different methods: Josephson spectroscopy [START_REF] Bretheau | Exciting Andreev pairs in a superconducting atomic contact[END_REF], switching current spectroscopy [START_REF] Bretheau | Supercurrent spectroscopy of Andreev states[END_REF] and microwave spectroscopy in a circuit-QED setup [START_REF] Janvier | Coherent manipulation of Andreev states in superconducting atomic contacts[END_REF]. The spectrum of Andreev states in a weak link with a sizable spin-orbit coupling has already been probed in two experiments on InAs nanowires [START_REF] Van Woerkom | Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions[END_REF][START_REF] Hays | Direct microwave measurement of Andreevbound-state dynamics in a proximitized semiconducting nanowire[END_REF]. Both experiments were performed in the limit L ξ. In Ref. [START_REF] Hays | Direct microwave measurement of Andreevbound-state dynamics in a proximitized semiconducting nanowire[END_REF], the zero-field spectrum was probed using a circuit-QED setup and no effect of spin-orbit interaction was reported. In Ref. [START_REF] Van Woerkom | Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions[END_REF], where spectra at finite field were obtained by Josephson spectroscopy, spin-orbit interaction enters in the interpretation of the spectra when the Zeeman energy is comparable to the superconducting gap [START_REF] Van Heck | Zeeman and spin-orbit effects in the Andreev spectra of nanowire junctions[END_REF]. In the following, we consider a finite-length weak link with Rashba spin-orbit interaction (Fig. 1(a)), and show that spin-split Andreev states require at least two transverse subbands. We first discuss the case of a purely one-dimensional weak link. As shown by the green lines in Fig. 1(b) spin-orbit interaction splits the dispersion relation (assumed to be parabolic) according to the electron spin direction [START_REF] Yu | Properties of a 2D electron gas with lifted spectral degeneracy[END_REF]. Andreev reflections (AR) at the superconductors couples electrons (full circles) with holes (open circles) of opposite spins and velocities. When the transmission probability across the wire is perfect (τ = 1), Andreev bound states arise when the total accumulated phase along closed paths that involve two AR and the propagation of an electron and a hole in opposite directions (Fig. 1(c)) is a multiple of 2π [2]. Figure 1(d) shows, in the excitation representation, the energy of the resulting ABS as a function of δ. ABS built with right-(left-) moving electrons are shown with thin solid (dashed) lines in Fig. 1(c&d). Note that the existence of two ABS at some phases is just a finite length effect [START_REF] Bagwell | Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity[END_REF] (here L/ξ = 0.8) and that ABS remain spindegenerate as the spatial phase acquired by the electron and the Andreev-reflected hole is the same for both spin directions. Backscattering in the weak link (τ = 1), due either to impurities or to the spatial variation of the elec-trostatic potential along the wire, couples electrons (as well as holes) of the same spin travelling in opposite directions, leading to avoided crossings at the points indicated by the open blue circles in Fig. 1(d). One obtains in this case two distinct Andreev states (thick solid lines), which remain spin-degenerate. This is no longer the case in the presence of a second transverse subband, even if just the lowest one is actually occupied [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF][START_REF] Reynoso | Spin-orbit-induced chirality of Andreev states in Josephson junctions[END_REF][START_REF] Yokoyama | Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires[END_REF][START_REF] Murani | Andreev spectrum with high spin-orbit interactions: revealing spin splitting and topologically protected crossings[END_REF]. Figure 1(f) shows how spin-orbit coupling hybridizes the spin-split dispersion relations of the two subbands (around the crossing points of 1↑ with 2↓ and of 1↓ with 2↑) [START_REF] Moroz | Effect of the spinorbit interaction on the band structure and conductance of quasi-one-dimensional systems[END_REF][START_REF] Governale | Spin accumulation in quantum wires with strong Rashba spin-orbit coupling[END_REF]. The new dispersion relations become non-parabolic and are characterized by different energy-dependent spin textures [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF][START_REF] Reynoso | Spin-orbit-induced chirality of Andreev states in Josephson junctions[END_REF][START_REF] Yokoyama | Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires[END_REF][START_REF] Murani | Andreev spectrum with high spin-orbit interactions: revealing spin splitting and topologically protected crossings[END_REF][START_REF] Moroz | Effect of the spinorbit interaction on the band structure and conductance of quasi-one-dimensional systems[END_REF][START_REF] Governale | Spin accumulation in quantum wires with strong Rashba spin-orbit coupling[END_REF]. We focus on a situation in which only the two lowest ones (m 1 and m 2 in the figure) are occupied. Importantly, their associated Fermi velocities are different. When τ = 1, this leads, as illustrated by Fig. 1(e,g), to two families of ABS represented by black and red thin lines, built from states with different spin textures. As before, backscattering leads to avoided crossings at the points indicated by the blue open circles in Fig. 1(e). The resulting ABS group in manifolds of spin-split states represented by the thick black lines. In the absence of a magnetic field, the states remain degenerate at δ = 0 and π. Figure 2 shows parity-conserving transitions that can be induced by absorption of a microwave photon, at a given phase. Red arrows (Fig. 2(a)) correspond to pair transitions in which the system is initially in the ground state and a pair of quasiparticles is created, either in one manifold or in different ones. Green arrows (Fig. 2(b)) correspond to single-particle transitions where a trapped quasiparticle [START_REF] Zgirski | Evidence for long-lived quasiparticles trapped in superconducting point contacts[END_REF] already occupying an Andreev state is excited to another one [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]33] which can be in the same or in another ABS manifold. The corresponding transition energies in the absorption spectrum for both the pair and single-particle cases are shown in Fig. 2(c), as a function of the phase difference δ. Pair transitions that create two quasiparticles in the same energy manifold do not carry information on the spin structure. On the contrary, pair and single-particle transitions involving different energy manifolds produce peculiar bundles of four distinct lines all crossing at δ = 0 and δ = π. They are a direct signature of the spin splitting of ABS. Finally, single-particle transitions within a manifold give rise to bundles of 2 lines. As discussed below, some of these transitions are accessible in our experiment.

↑ (d) 1-𝜏 𝑘 𝑥 AR AR (a) E (b) (c) L 𝑥 𝑦 𝛿/2 -∆ ∆ AR AR µ ↑ ↑ 𝑥 0 AR AR m 1 m 2 1 2 1 2 E ↓ ↑ ↑ ↓ (f) (g) 𝑦 𝛿/2 -𝛿/2 -∆ ∆ AR AR µ ↓ ↓ ↓ ↓ 1-𝜏 (
Figure 3 shows a spectrum measured on an InAs nanowire weak link between aluminum electrodes. The plot shows at which frequency f 1 microwave photons are absorbed, as a function of the phase difference δ across the weak link (see description of the experiment below). This is a very rich spectrum, but here we point two salient features highlighted with color lines on the right half side of the figure. The red line corresponds to a pair transition, with extrema at δ = 0 and δ = π. The frequency f 1 (δ = 0) = 26.5 GHz is much smaller than twice the gap of aluminum 2∆/h = 88 GHz, as expected for a junction longer than the coherence length. To the best of our knowledge, this is the first observation of a discrete Andreev spectrum in the long-junction limit. The observation of the bundle of lines (in green) with crossings at δ = 0 and δ = π that clearly corresponds to single-particle transitions shown in Fig. 2(c) is the central result of this work.

Experimental setup. The measurements are obtained using the circuit-QED setup shown in Fig. 4(d), and performed at ∼ 40 mK in a pulse-tube dilution refrigerator. The superconducting weak link was obtained by etching away, over a 370-nm-long section, the 25-nm-thick aluminum shell that fully covers a 140-nm-diameter InAs nanowire [START_REF] Krogstrup | Epitaxy of semiconductorsuperconductor nanowires[END_REF][START_REF] Chang | Hard gap in epitaxial semiconductor-superconductor nanowires[END_REF][START_REF] Goffman | Conduction channels of an InAs-Al nanowire Josephson weak link[END_REF] (see Figs. 4(a) and (b)). A side-gate allows tuning the charge carrier density and the electrostatic potential in the nanowire and therefore the Andreev spectra [START_REF] Van Woerkom | Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions[END_REF]. The weak link is part of an aluminum loop of area S ∼ 10 3 µm 2 , which has a connection to ground to define a reference for the gate voltage (see Fig. 4(c)). The phase δ across the weak link is imposed by a small magnetic field B z (< 5µT) perpendicular to the sample plane: δ = B z S/ϕ 0 , with ϕ 0 = /2e the reduced flux quantum. Two additional coils are used to apply a magnetic field in the plane of the sample. The loop is inductively coupled to the shorted end of a λ/4 microwave resonator made out of Nb, with resonance frequency f 0 ≈ 3.26 GHz and internal quality factor Q int ≈ 3 × 10 5 . A continuous signal at frequency f 0 is sent through a coplanar transmission line coupled to the resonator (coupling quality factor Q c ≈ 1.7 × 10 5 ), and the two quadratures I and Q of the transmitted signal are measured using homodyne detection (see Fig. 4(d)). Andreev excitations in the weak link are induced by a microwave signal of frequency f 1 applied on the side gate. The corresponding microwave source is chopped at 3.3 kHz, and the response in I and Q is detected using two lock-ins, with an integration time of 0.1 s. This response is expressed in terms of the corresponding frequency shift f -f 0 in the resonator (see Appendix A3). The fact that single-particle transitions are observed (see Fig. 3) means that during part of the measurement time Andreev states are occupied by a single quasiparticle. This is in agreement with previous experiments, in which the fluctuation rates for the occupation of Andreev states by out-of-equilibrium quasiparticles were found to be in the 10 ms -1 range [START_REF] Janvier | Coherent manipulation of Andreev states in superconducting atomic contacts[END_REF][START_REF] Hays | Direct microwave measurement of Andreevbound-state dynamics in a proximitized semiconducting nanowire[END_REF][START_REF] Zgirski | Evidence for long-lived quasiparticles trapped in superconducting point contacts[END_REF]. Note that in contrast to an excitation that couples to the phase across the contact through the resonator [START_REF] Janvier | Coherent manipulation of Andreev states in superconducting atomic contacts[END_REF][START_REF] Van Heck | Zeeman and spin-orbit effects in the Andreev spectra of nanowire junctions[END_REF][START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF], exciting through the gate allows to drive transitions away from δ = π and at frequencies very far detuned from that of the resonator. Spectroscopy at zero magnetic field. Figure 5(a) presents another spectrum taken at zero magnetic field (apart from the tiny perpendicular field B z < 5 µT required for the phase biasing of the weak link), at V g = 0.5 V. In comparison with the spectrum in Fig. 3, pair transitions are hardly visible in Fig. 5. Bundles of lines corresponding to single-particle transitions have crossings at 7.1, 14.0 and 22.4 GHz at δ = 0 and 9, 21.5 and 26.0 GHz at δ = π. Here, as in Fig. 3 (see Appendix A2), replicas of transition lines shifted by f 0 are also visible (bundle of lines near f 1 = 11 GHz and around δ = 0). They correspond to transitions involving the absorption of a photon from the resonator. Remarkably, the sign of the response appears correlated with the curvature of the transition lines. This suggests that the signal is mainly associated with a change in the effective inductance of the nanowire weak link. Additional work is needed to confirm this interpretation. We focus on the bundle of lines between 13 and 23 GHz for which the effect of a magnetic field B was also explored. Green lines in Fig. 5

τ cos ((λ 1 -λ 2 ) ∓ δ) + (1 -τ ) cos((λ 1 + λ 2 ) x r ) = cos(2 arccos -(λ 1 + λ 2 ) ) (1)
where x r = 2x 0 /L. It should be noticed that Eq. (1) for λ 1 = λ 2 reduces to the known result for a single quantum channel without spin-orbit [START_REF] Bagwell | Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity[END_REF][START_REF] Samuelsson | Nonequilibrium Josephson effect in mesoscopic ballistic multiterminal SNS junctions[END_REF]. The fit in Fig. 5 for the gap of Al). These values can be related to microscopic parameters, in particular to the intensity α of the Rashba spin-orbit interaction entering in the Hamiltonian of the system as H R = -α(k x σ y -k y σ x ) (with σ x,y Pauli matrices acting in the spin) [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]. Assuming a parabolic transverse confinement potential, an effective wire diameter of W = 140 nm, and an effective junction length of L = 370 nm, the values of λ 1,2 are obtained for µ = 422 µeV (measured from the bottom of the band) and α = 38 meV nm, a value consistent with previous estimations [START_REF] Fasth | Direct Measurement of the Spin-Orbit Interaction in a Two-Electron InAs Nanowire Quantum Dot[END_REF][START_REF] Scherübl | Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires[END_REF]. However, we stress that this estimation is model-dependent: very similar fits of the data can be obtained using a double-barrier model (with scattering barriers located at the left (x = -L/2) and right (x = L/2) edges of the wire) with λ 1 = 1.1 and λ 2 = 1.9, leading to α = 32 meV nm. For both models, we get only two manifolds of Andreev levels in the spectrum, and only these four single-particle transitions are expected in this frequency window (transitions within a manifold are all below 3.5 GHz). The other observed bundles of transitions are attributed to other conduction channels: although we considered till now only one occupied transverse subband, the same effect of spin-dependent velocities is found if several subbands cross the Fermi level.

A more elaborate model, together with a realistic modeling of the bands of the nanowire, is required to treat this situation and obtain a quantitative fit of the whole spectra.

Spin character of ABS. The splitting of the ABS and the associated transitions in the absence of a Zeeman field reveal the difference in the Fermi velocities v 1 and v 2 , arising from the spin-orbit coupling in the multi-channel wire. To further confirm that this is indeed a spin effect we probe the ABS spectra under a finite magnetic field, and in particular as a function of the orientation of the field with respect to the nanowire axis [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]. Figure 5(c) shows the spectrum in presence of an in-plane magnetic field with amplitudes B = 0, 2.6 and 4.4 mT applied at an angle of -45 • with respect to the wire axis. The symmetry around δ = 0 and δ = π is lost. This is accounted for by an extension of the single-barrier model at finite magnetic field (green lines) and assuming an anisotropic g-factor: g ⊥ = 12 and g / / = 8 (see below and Appendix A1).

The specific effects of a parallel and of a perpendicular magnetic field on the ABS are shown in Fig. 6. When the field is perpendicular to the wire (B ⊥ x), the ABS spectrum becomes asymmetric (this is related to the physics of ϕ 0 junctions [START_REF] Reynoso | Spin-orbit-induced chirality of Andreev states in Josephson junctions[END_REF]), as observed in Fig. 6(b,d). The field is directly acting in the quantization direction of the spinsplit transverse subbands (gray parabolas in Fig. 1(f)) from which the ABS are constructed, leading to Zeeman shifts of the energies. When the field is along the wire axis B//x, and thus perpendicular to the spin quantization direction, it mixes the spin textures and lifts partly the degeneracies at δ = 0 and δ = π (see Fig. 7 in Appendix). The spectrum of ABS is then modified, but remains symmetric [40] around δ = 0 and π, see Fig. 6(a,c). Keeping the same parameters as in Fig. 5, the value of the g-factor was taken as a fit parameter for all the data with per- pendicular field, and for all the data with parallel field, leading to two distinct values of the g-factor: g ⊥ = 12 and g / / = 8 (see Appendix). Green lines show the resulting best fits. Concluding remarks. The results reported here show that the quasiparticle spin can be a relevant degree of freedom in Josephson weak links, even in the absence of a magnetic field. This work leaves several open questions. Would a more realistic modeling of the nanowire [START_REF] Degtyarev | Features of electron gas in InAs nanowires imposed by interplay between nanowire geometry, doping and surface states[END_REF][START_REF] Zuo | Supercurrent Interference in Few-Mode Nanowire Josephson Junctions[END_REF][START_REF] Antipov | Effects of Gate-Induced Electric Fields on Semiconductor Majorana Nanowires[END_REF][START_REF] Winkler | A unified numerical approach to semiconductor-superconductor heterostructures[END_REF] allow for a precise determination of spin-orbit interaction from the measured spectra? We need to understand, along the lines of [START_REF] Gorelik | Microwave-Induced Somersault Effect in Flow of Josephson Current through a Quantum Constriction[END_REF] for example, the coupling between the microwave photons and the ABS when the excitation is induced through an electric field modulation, as done here, instead of a phase modulation [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]33,[START_REF] Despósito | Controlled dephasing of Andreev states in superconducting quantum point contacts[END_REF]. In particular, what are the selection rules? Are transitions between ABS belonging to the same manifold allowed? Can one observe pair transitions leading to states with quasiparticles in different manifolds? What determines the signal amplitude? Independently of the answer to these questions, the observation of spin-resolved transitions between ABS constitutes a first step towards the manipulation of the spin of a single superconducting quasiparticle [START_REF] Chtchelkatchev | Andreev quantum dots for spin manipulation[END_REF][START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]. Would the spin coherence time of a localized quasiparticle be different from that of a propagating one [47]? Finally, we think that the experimental strategy used here could allow probing a topological phase with Majorana bound states at larger magnetic fields [33].

Technical support from P. Sénat is gratefully acknowledged. We thank M. Devoret, M. Hays and K. Serniak for sharing their results on a similar experiment and for discussions. We thank A. Reynoso for providing us codes related to his work [START_REF] Reynoso | Spin-orbit-induced chirality of Andreev states in Josephson junctions[END_REF] and for useful discussions. We also acknowledge discussions with C ¸. Girit, H. Bouchiat, A. Murani and our colleagues from the Quantronics group. We thank P. The nanowire is described by the Hamiltonian H 3D consisting of a kinetic energy, a confining harmonic potential in y and z-directions with a confinement width W (effective diameter of the nanowire) and Rashba spin-orbit coupling with intensity α,

H 3D = p 2 x + p 2 y + p 2 z 2m * + 2 (y 2 + z 2 ) 2m * (W/2) 4 + α(-k x σ y + k y σ x ), (2)
where m * is the effective mass and σ x,y are the Pauli matrices for spin. We consider two spin-full transverse subbands denoted by nσ, with n = 1, 2 and σ =↑, ↓, arising from the confining potential in the transverse direction (gray parabolas in Fig. 1(f)) under the effect of the Rashba spin-orbit coupling. The energy dispersion relations of the resulting lowest subbands (green lines labelled m 1 and m 2 in Fig. 1(f)) are [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]

E s (k x ) = 2 k 2 x 2m * + E ⊥ 1 + E ⊥ 2 2 - E ⊥ 1 -E ⊥ 2 2 -sαk x 2 + η 2 , (3)
where s = -1 corresponds to m 1 and s = +1 to m 2 , and E ⊥ n = 4 2 n/(m * W 2 ). η = √ 2α/W is the strength of the subband mixing due to the Rashba spin-orbit coupling. In accordance to the estimated nanowire diameter we take W ∼ 140 nm, which leads to E ⊥ 2 -E ⊥ 1 ∼ 0.68 meV for the subband separation. Particle backscattering within the nanowire is accounted for by either a single delta-like potential barrier located at some arbitrary position x 0 or by potential barriers localized at both ends (x = ±L/2).

The linearized Bogoliubov-de Gennes equation around the chemical potential µ is

H 0 + H b ∆(x)e iδ(x) ∆(x)e -iδ(x) -H 0 -H b Ψ(x) = E A Ψ(x) (4)
with the basis Ψ(x) = (ψ e +,R (x),

ψ e +,L (x), ψ e -,R (x), ψ e -,L (x), ψ h +,R (x), ψ h +,L (x), ψ h -,R (x), ψ h -,L (x))
, where R(L) refers to the right-moving (left-moving) electron (e) or hole (h) in the bands m 1 (-), m 2 (+). Here H 0 is the Hamiltonian for electrons in the nanowire

H 0 =     -i v 1 ∂ x -v 1 k F 1 0 0 0 0 i v 2 ∂ x -v 2 k F 2 0 0 0 0 -i v 2 ∂ x -v 2 k F 2 0 0 0 0 i v 1 ∂ x -v 1 k F 1     , (5)
where v j=1,2 are the Fermi velocities given by

v j = k F j m * + (-1) j α E ⊥ 1 /2 -(-1) j αk F j E ⊥ 1 /2 -(-1) j αk F j 2 + η 2 , (6)
and k F j are the Fermi wave vectors satisfying E s (k F j ) = µ. We note that if there is no subband mixing, i.e. η = 0 (gray parabolas in Fig. 1(f)), Eqs. ( 3) and ( 6)

show that k F 1 -k F 2 = 2m * α/ 2 and v 1 -v 2 = (k F 1 -k F 2 ) /m * -2α/ = 0,
indicating clearly that the Fermi velocities are the same. The potential scattering term H b is modeled as

H b = U b (x)     1 cos[(θ 1 -θ 2 )/2] 0 0 cos[(θ 1 -θ 2 )/2] 1 0 0 0 0 1 cos[(θ 1 -θ 2 )/2] 0 0 cos[(θ 1 -θ 2 )/2] 1     , (7)
where

U b (x) = U 0 δ(x -x 0 ) for a single barrier at x = x 0 U L δ(x + L/2) + U R δ(x -L/2) for barriers at x = -L/2 and x = L/2, (8)
and θ j=1,2 = arccos[(-1) j ( k Fj /m * -v j )/α] characterize the mixing with the higher subbands, i.e. cos(θ j /2) and sin(θ j /2) determine the weight of the states on the hybridized subbands and therefore their spin texture. The superconducting order parameter ∆(x)e iδ(x) in Eq. ( 4) is given by ∆e -iδ/2 at x < -L/2, ∆e iδ/2 at x > L/2, and zero otherwise, where δ is the superconducting phase difference. Ballistic regime. In the absence of particle backscattering, the phase accumulated in the Andreev reflection processes at x = -L/2 and x = L/2, as illustrated in Fig. 1(c), leads to the following transcendental equation for the energy = E A /∆ of the ABS as a function of δ :

sin( λ 1 -sδ/2 -arccos ) sin( λ 2 + sδ/2 -arccos ) = 0, (9)
where λ j=1,2 = L∆/( v j ). For 1, there are two sets of solutions given by

↑ (δ) = 1 1+λ1 δ 2 + l + 1 2 π ↓ (δ) = 1 1+λ1 -δ 2 + l + 1 2 π , (δ) = 1 1+λ2 δ 2 + l + 1 2 π (δ) = 1 1+λ2 -δ 2 + l + 1 2 π
, with integers l and l . The ballistic ABS are represented by the thin lines (black and red) in Fig. 1(e). Single barrier model. In this case, the effect of the barrier can be taken into account as an additional boundary condition at x = x 0 ,

Ψ(x 0 + 0 + ) =     M 12 0 0 0 0 M 21 0 0 0 0 M 12 0 0 0 0 M 21     Ψ(x 0 -0 + ), (10)
where 0 + is a positive infinitesimal and M ij is the 2 × 2 matrix given by

M ij = 1 t     tt -rr v j v i r e iϕ - v i v j re -iϕ 1     (11)
with ϕ = ((k

F 1 + k F 2 ) + (λ 1 + λ 2 ) /L)x 0 .
The reflection and transmission coefficients are determined by

te -iua = t e iua = cos d + iu s sin d d -1 , re -iϕ = r e iϕ = -i √ u 1 u 2 sin d d cos θ 1 -θ 2 2 √ tt , d = 1 2 u 2 1 + u 2 2 -2u 1 u 2 cos(θ 1 -θ 2 ), (12)
where

v 0 = v 1 v 2 /U 0 , u j = v j /v 0 , u s = (u 1 + u 2 )/2
, and u a = (u 1 -u 2 )/2. From the continuity conditions at x = ±L/2 and Eq. ( 10) we find the transcendental equation ( 1) where τ = |t| 2 . As already noticed in the main text, Eq. (1) leads to split ABS when v 1 = v 2 , except for δ = 0, π where the ABS degeneracy is protected by time-reversal symmetry. Double barrier model. In this case, there are two boundary conditions similar as Eq. ( 10) at the NS interfaces, which results in the transcendental equation sin(˜ 1 -arccos ) sin(˜ 2 -arccos ) =(2 -τ L -τ R ) sin(˜ 1 ) sin(˜ 2 )

-(1 -τ L )(1 -τ R ) sin(˜ 1 + arccos ) sin(˜ 2 + arccos ) -2 (1 -τ L )(1 -τ R ) cos(ϕ tot )(1 -2 ), (13)
where ˜ j = λ j +(-1) j sδ/2, τ L,R are the transmission probabilities at x = ∓L/2, θ ν are the scattering phases acquired at the interfaces (ν ≡ L, R):

θ ν = arg cos d ν + i sin d ν d ν v s v ν , (14)
where d ν and v ν are defined as d in Eq. ( 12) replacing U 0 by U ν . Finally, we note

ϕ tot = (k F 1 + k F 2 )L -(θ L + θ R )
the total accumulated phase. For the estimations quoted in the main text we have assumed two identical barriers, i.e. τ L = τ R = τ . Magnetic field effect. Information on the ABS spin structure can be inferred from their behavior in the presence of a finite magnetic field. This behavior depends strongly on the orientation of the field with respect to the nanowire axis [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]. We consider a magnetic field lying in the xy-plane. The y-component B y (parallel to the spin states of the transverse subbands without RSO) shifts the energy of the subbands depending on the spin states and modifies the Fermi wave vectors as illustrated in Fig. 7(c). They thus satisfy

E s (k F ) = 2 k 2 F 2m * + E ⊥ 1 + E ⊥ 2 2 - E ⊥ 1 -E ⊥ 2 2 -s αk F - gµ B 2 B y 2 + η 2 = µ. (15)
On the other hand, the x-component B x mixes opposite-spin states thus opening a gap at the crossings points as illustrated in Fig. 7(a). We include this effect perturbatively [START_REF] Park | Andreev spin qubits in multichannel Rashba nanowires[END_REF]. For both, B//x and B ⊥ x cases, the resulting ABS and the corresponding transition lines are shown in the middle and bottom rows of Fig. 7.

Fitting strategy The transcendental equations (Eqs. (1) and ( 13)) for the single and double barrier models contain dimensionless parameters with which we fit the experimental data at zero magnetic field:

(a) (b) (c) B // x ↓ 1 E ↓ 1 𝑘 𝑥 µ 2 ↓ 2 ↓ B = 0 E 𝑘 𝑥 ↓ 1 ↓ 1 2 ↓ 2 ↓ µ B x µ E 𝑘 𝑥 ↓ 1 ↓ 1 2 ↓ 2 ↓ FIG. 7.
Effect of an in-plane magnetic field on the band structure (top row), the Andreev levels (bottom row, left) and the excitation spectrum (bottom row, right). (b) reference graphs at zero field; (a) field applied along the wire axis; (c) field applied perpendicularly to the wire axis. The field effect on the band structure is exaggerated for clarity. The model parameters for the Andreev levels and the excitation spectrum are the same as in Fig. 5 and B = 10 mT.

• λ 1 , λ 2 , τ , and x r for the single barrier model,

• λ 1 , λ 2 , τ , and ϕ tot for the double barrier model.

We then deduce the physical parameters, α, µ (measured from the bottom of the lowest band), L, and U 0 (or U L/R ) using Eqs. (3), ( 6) and ( 12), and assuming that the nanowire diameter is fixed at W = 140 nm. We further set m * = 0.023 m e where m e is the bare electron mass. For the experimental data in Fig. 5, the single barrier model gives λ 1 = 1.3, λ 2 = 2.3, τ = 0.295, and x r = 0.52, resulting in the microscopic parameters α = 53 meV nm, µ = 255 µeV, U 0 = 92 meV nm, L = 332 nm. Using the double barrier model, we get λ 1 = 1.1, λ 2 = 1.9, τ = 0.52, ϕ tot = 0.93 (Mod 2π), α = 36 meV nm, µ = 427 µeV, U L = U R = 130 meV nm, L = 314 nm. Another possibility is to fix the length of the junction L to the length of the uncovered section of the InAs nanowire, 370 nm, which leads to α = 38 meV nm and µ = 422 µeV for the single barrier model (α = 32 meV nm and µ = 580 µeV for the double barrier model). However, in the single barrier model one cannot find values of U 0 leading to the corresponding τ . This is due to the fact that in our simplified model for the scattering matrix, processes involving the higher subbands are neglected, thus limiting its validity to small values of U 0 . In order to fit the finite magnetic field data, in addition to the parameters determined at zero magnetic field, one needs the g-factors in the parallel and perpendicular directions, g / / and g ⊥ . We used all the data taken with field in the parallel and in the perpendicular directions, and calculated the correlation function between the images of the measured spectra (taking the absolute value of the response f -f 0 ) and theory, using various values of g / / and g ⊥ . Figure 8 shows the dependence of the correlation functions with g / / and g ⊥ . Best agreement is found for g / / = 8 and g ⊥ = 12, which are within the range of values reported in the literature [START_REF] Björk | Tunable effective g-factor in InAs nanowire quantum dots[END_REF][START_REF] Deacon | Electrically tuned g tensor in an InAs selfassembled quantum dot[END_REF][START_REF] Samuel D'hollosy | g-factor anisotropy in nanowire-based InAs quantum dots[END_REF][START_REF] Vaitiekenas | Effective g Factor of Subgap States in Hybrid Nanowires[END_REF]. Note that the determination of g / / is less accurate, and that, overall, g / / = 4 gives a similar correlation as g / / = 8, but agreement is worse at largest values of B / / where the effect is the strongest.

A2. Fit of the data at Vg = -0.89 V Many features of the data taken at V g = -0.89 V (Fig. 3) can be accounted for by the single-barrier model. This is shown in Fig. 9, where we compare the data with the results of theory using λ 1 = 2.81, λ 2 = 4.7, τ = 0.25 and x r = 0.17. The Andreev spectrum obtained with this set of parameters (Fig. 9(c)) presents 3 manifolds of spinsplit states, leading to 3 bundles of 4 lines associated to single-particle transitions between manifolds (green lines in Fig. 9(b)). They are in good agreement with transition lines at least partly visible in the data. In addition, the pair transition corresponding to two quasiparticles excited in the lowest manifold gives rise to an even transition which falls in the frequency range of the data, and roughly corresponds to a transition visible in the data. Assuming a fixed length L = 370 nm and using the model of Eq. ( 2) we deduce the microscopic parameters α = 43.7 meV nm and µ = 102 µeV (measured from the bottom of the band). However, these values should be taken with care since the linearization of the dispersion relation is not valid for energies close to ∆ when µ ∆.

A3. Measurement calibration

The measurement is performed by chopping with a square-wave the excitation signal applied on the gate and recording with lock-in detectors the corresponding modulation of the response of the circuit on the two quadratures I and Q. We interpret these modulations as arising from shifts of the resonator frequency. To calibrate this effect, we measured how the DC values of I and Q change for small variations of the measurement frequency f 0 around 3.26 GHz. Figure 10 shows two examples of the gate voltage-dependence of the spectrum at phase δ = π, with reference spectra as a function of phase. In both spectra, single-particle transitions appear white at δ = π, whereas pair transitions appear black. When V g is changed, both types of lines move up and down, but do not change color. Both types of transitions are observed in the frequency window 2-20 GHz at almost all values of V g . A remarkable feature is that black and white lines move "out of phase", which can be understood from the effect of V g on the transmission τ : when τ decreases, the distance between the two lowest manifolds decreases at δ = π, so that the transition energy for single-particle transitions decreases; at the same time, the energy of the lowest manifold increases, and so does the transition energy for pair transitions.

FIG. 1 .

 1 FIG. 1. Effect of the Rashba spin-orbit coupling (RSO) on Andreev levels. (a) Weak link of length L between superconductors with phase difference δ. Blue star symbolizes a scatterer at position x0. (b) Dispersion relation for a purely one-dimensional weak link in the presence of RSO (green solid lines, labels ↑↓ indicate spin in y-direction). Density of states of superconducting electrodes is sketched at both ends of the wire. (c) Andreev reflections (AR) at the superconductors couples electrons (full circles) with holes (open circles) of opposite spins and velocities, leading to the formation of ABS. Blue arrows indicate reflections due to a scatterer. (d) Energy of ABS (excitation representation). Thin lines in (c) and (d): transmission τ = 1, ABS formed from right-moving electrons and left-moving holes (solid) or the opposite (dashed). Backscattering (τ = 1) leads to opening of gaps at the crossings highlighted with blue circles in (d). Resulting spin-degenerate Andreev levels are shown with thick solid lines. (e-g) Effect of RSO in the presence of two transverse subbands, only the lowest one being occupied. (f) Dispersion relation (subband spacing and superconducting gap are in a ratio that roughly corresponds to our experiments). Grey solid lines labelled 1↑↓ and 2↑↓ are dispersion relations for uncoupled subbands. RSO couples states of different subbands and opposite spins, leading to hybridized bands (green solid lines) with energy-dependent spin textures. Fermi level µ is such that only the lowest energy bands m1 and m2 are occupied. AR couples for example a fast electron from m2 to a fast hole (in black), and a slow electron from m1 to a slow hole (in red). (g) Construction of ABS: black and red loops are characterized by different absolute velocities. Spins pointing in different directions symbolize spin textures of the bands. Thin red and black lines, solid and dashed in (e,g): ABS at τ = 1, associated with different spin textures. Thick black lines in (e): ABS when crossings highlighted with blue circles are avoided due to backscattering.

FIG. 2 .

 2 FIG. 2. Possible parity-conserving transitions in a weak link with spin-split ABS (levels positions correspond to the phase indicated with an arrow in Fig. 1(e)). Blue line corresponds to the ground state. (a) Pair transitions. A pair of quasiparticles are created from the ground state, either both in the same manifold (solid arrows) or not (dashed arrows). (b) Single-particle transitions. A quasiparticle already present in one ABS (solid dot) is excited to another ABS, either in the same (dotted arrows) or in another (solid arrows) manifold. (c) Corresponding transition energies, as a function of the phase difference δ across the weak link. (Transitions involving quasiparticles in the continuum are not represented).

FIG. 3 .

 3 FIG. 3. Microwave excitation spectrum measured at a gate voltage Vg = -0.89 V. The grey scale represents the frequency change f -f0 of a resonator coupled to the weak link when a microwave excitation at frequency f1 is applied, as a function of the phase difference δ across the weak link. In the right half of the figure, some transition lines are highlighted. Red line corresponds to a pair transition, green lines are single-particle transitions.

FIG. 4 .

 4 FIG. 4. Experimental setup. (a) False-color scanning electron microscope image of the InAs-Al core-shell nanowire. The Al shell (grey) was removed over 370 nm to form the weak link between the superconducting electrodes. A close-by side electrode (Au, yellow) is used to gate the InAs exposed region (green). (b),(c) The nanowire is connected to Al leads that form a loop. This loop is located close to the shorted end of a coplanar wave guide (CPW) resonator. (d) The CPW resonator is probed by sending through a bus line a continuous microwave tone at its resonant frequency f0 = 3.26 GHz and demodulating the transmitted signal, yielding quadratures I and Q. Microwaves inducing Andreev transitions are applied through the side gate (frequency f1) using a bias tee, the DC port being used to apply a DC voltage Vg.

  (b) are fits of the data at B = 0 with a simple model that accounts for two bands with different Fermi velocities v 1 and v 2 , and the presence of a single scatterer in the wire (see Appendix A1). The model parameters are λ j=1,2 = L∆/( v j ) and the position x 0 ∈ [-L/2, L/2] of the scatterer of transmission τ . ABS are found at energies E = ∆, with solution of the transcendental equation (see Appendix A1):

  (b) corresponds to λ 1 = 1.3, λ 2 = 2.3, τ = 0.295 and x r = 0.525 (we take ∆ = 182 µeV= h × 44 GHz

FIG. 5 .

 5 FIG. 5. Excitation spectra at Vg = 0.5 V. (a) Large scale spectrum at zero magnetic field. (b) Zoom on the same data, with fits (see text). (c) Dependence of the spectrum with the amplitude B of an in-plane magnetic field applied at an angle of -45 • with respect to the nanowire axis. Green lines are fits (see text).

FIG. 6 .

 6 FIG.6. Effect of an in-plane magnetic field on the ABS excitation spectrum around δ = 0. The Andreev states correspond to the same gate voltage as in Fig.5. Field is applied parallel (a,c) or perpendicular (b,d) to the wire. Green lines are the result from the theory, using g ⊥ = 12 and g / / = 8 (see text).

  Orfila and S. Delprat for nanofabrication support. This work has been supported ANR contract JETS, by the Renatech network, by the Spanish MINECO through Grant No. FIS2014-55486-P, FIS2017-84860-R and through the "María de Maeztu" Programme for Units of Excellence in R&D (MDM-2014-0377). Center for Quantum Devices is supported by the Danish National Research Foundation. L. Tosi was supported by the Marie Sk lodowska-Curie individual fellowship grant 705467. P. Krogstrup acknowledges support from Microsoft Quantum and the European Research Council (ERC) under the grant agreement No.716655 (HEMs-DAM) Appendix A1. Details on the theoretical model and the fitting parameters

FIG. 8 .

 8 FIG.8. Correlation (normalized to maximum value) between finite magnetic field data and theory as a function of g factor, for field direction parallel (blue) or perpendicular (red) to the nanowire.

FIG. 9 .

 9 FIG. 9. (a) Data at Vg = -0.89 V, with yellow arrows pointing to transition lines that are replicas of lines appearing exactly 3.26 GHz above. (b) Same data superimposed with predictions of the single barrier model, using parameters corresponding to the spectrum of ABS shown in (c). Single-particle transitions (green lines) between the three manifolds, labelled 1,2,3 in (c) are visible in the data. Red line in (b) is the pair transition leading to two quasiparticles in manifold 1.

2 πFIG. 10 .

 210 FIG. 10. Spectrum as a function of gate voltage at δ = π (right of (a), left of (b)), for two intervals of gate voltage, and taken during different cool-downs. (a) left panel and (b) right panel: phase dependence at the gate voltages corresponding to Vg = -0.24 V (a) (resp. Vg = -1.21 V (b)), which correspond to the leftmost (resp. rightmost) gate voltages of the panels showing the Vg dependence.